# Track Seeding¶

To reduce the time needed to reconstruct particle tracks, a track seed (henceforth: seed) is created which serves as initial direction for the track reconstruction algorithm (henceforth: the tracking). The tracking then tries to find all measurements belonging to a single particle in this direction in order to reconstruct the track. This means, if no seed exists for a particle, this particle will not be reconstructed. On the other hand, finding too many seeds which either correspond to a particle for which another seed already exists or which does not correspond to a particle at all increases the time needed for tracking.

A good seeding algorithm therefore has the following properties:

1. It finds at least one seed for each particle that should be found

2. It doesn’t find many seeds which do NOT correspond to particles

3. It doesn’t find many seeds per particle

The most typical way to create seeds is to combine measurements. In a homogeneous magnetic field, 3 measurements perfectly describe the helical path of a charged particle. One such triplet of measurements would then constitute a seed and define in close bounds where the tracking needs to look for additional measurements to create a track spanning the whole detector. The difficulty is in choosing the correct measurements, as a helix can be fitted through any 3 measurements in a collision event with potentially tens of thousands of measurements. Therefore, a number of constraints or “cuts” are defined to reduce the number of candidates. Cuts may define where particles originate or the range of energy of particles to be found or otherwise restrict the combination of measurements for seed creation.

## Acts Implementation¶

The seeding implementation in Core/include/Acts/Seeding/ is based on the ATLAS track seeding. It was rewritten with a focus on parallelism and maintainability and as detector agnostic as possible, only assuming a (near) homogeneous magnetic field with particles originating from the central detector region. Cuts are configurable and can be plugged in as algorithm which is called by the seeding. The seeding works on measurements or “SpacePoints” (SP), which need to provide x,y,z coordinates with the z axis being along the magnetic field, and x and y. The interaction region must be close to $$x=y=0$$, such that the interaction region has a smaller detector radius $$r = \sqrt(x^2+y^2)$$ than the measurements closest to the interaction region, see also Fig. 3.

Fig. 3 Example detector with x,y,z coordinate system. While x and y coordinates must be 0 close to the interaction region, z can be arbitrary. The magnetic field must be along the z-axis such that charged particles are deflected in phi around the z-axis.

Fig. 4 The same detector as above but in x/y coordinate system. Helices show up as circles in this projection, as shown by the purple particle path. Three measurements define a circle, which is used to calculate radius (corresponding to the particle energy), curvature direction in $$\phi$$ corresponding to the particle charge, and impact parameters.

Fig. 5 The same detector as above but mapped onto an r/z coordinate system. This projection is used to calculate the pseudorapidity $$\eta$$ of a seed (in the code calculated in $$cot \theta$$ for speed), e.g. to test if two combinations (SP bottom, SP middle) and (SP middle, SP top) have similar pseudorapidity and are therefore compatible with the same particle track.

Three iterators over SP need to be passed to the public createSeedsForGroup function in the Seedfinder class. The seedfinder will then attempt to create seeds, with each seed containing exactly one SP returned by each of the three iterators.

• SPs from the first iterator are always used as measurement of a seed with the smallest detector radius r,

• SPs from the second iterator are only used as measurement of a seed with r between the r of the first and the third iterator

• SPs from the third iterator are always used as measurement with the largest r in a seed.

Warning

Note that the seeding algorithm breaks down for particles with a particle track whose helix diameter is smaller than the detector radius until which seeds are to be created. This is due to ordering assumptions of SP locations as well as due to approximations which become inaccurate for lower energy particles.

The createSeedsForGroup function then iterates over middle SP, and within this loop separately iterates once over bottom SP and once over top SP. Within each of the nested loops, bottom SP - middle SP respectively middle SP - top SP are tested for compatibility by applying cuts that can be tested with two SP only (pseudorapidity, origin along z-axis, distance in r between SP).

If both compatible bottom and top SP have been found, test each bottom SP, middle SP, top SP triplet combination in a triple nested loop. A major part of this is the calculation of the helix circle. In order to perform calculations only once, the circle calculation is spread out over the three loops.:

for (auto spM : middleSPs) {

// contains parameters required to calculate circle with linear equation
// ...for bottom-middle
std::vector<LinCircle> linCircleBottom;
// ...for middle-top
std::vector<LinCircle> linCircleTop;
transformCoordinates(compatBottomSP, *spM, true, linCircleBottom);
transformCoordinates(compatTopSP, *spM, false, linCircleTop);

// create vectors here to avoid reallocation in each loop
std::vector<const InternalSpacePoint<external_spacepoint_t>*> topSpVec;
std::vector<float> curvatures;
std::vector<float> impactParameters;

std::vector<std::pair<
float, std::unique_ptr<const InternalSeed<external_spacepoint_t>>>>
seedsPerSpM;
size_t numBotSP = compatBottomSP.size();
size_t numTopSP = compatTopSP.size();

for (size_t b = 0; b < numBotSP; b++) {
auto lb = linCircleBottom[b];
float Zob = lb.Zo;
float cotThetaB = lb.cotTheta;
float Vb = lb.V;
float Ub = lb.U;
float ErB = lb.Er;
float iDeltaRB = lb.iDeltaR;

// 1+(cot^2(theta)) = 1/sin^2(theta)
float iSinTheta2 = (1. + cotThetaB * cotThetaB);
// calculate max scattering for min momentum at the seed's theta angle
// scaling scatteringAngle^2 by sin^2(theta) to convert pT^2 to p^2
// accurate would be taking 1/atan(thetaBottom)-1/atan(thetaTop) <
// scattering
// but to avoid trig functions we approximate cot by scaling by
// 1/sin^4(theta)
// resolving with pT to p scaling --> only divide by sin^2(theta)
// max approximation error for allowed scattering angles of 0.04 rad at
// eta=infinity: ~8.5%
float scatteringInRegion2 = m_config.maxScatteringAngle2 * iSinTheta2;
// multiply the squared sigma onto the squared scattering
scatteringInRegion2 *=
m_config.sigmaScattering * m_config.sigmaScattering;

// clear all vectors used in each inner for loop
topSpVec.clear();
curvatures.clear();
impactParameters.clear();
for (size_t t = 0; t < numTopSP; t++) {
auto lt = linCircleTop[t];

// protects against division by 0
float dU = lt.U - Ub;
if (dU == 0.) {
continue;
}
// A and B are evaluated as a function of the circumference parameters
// x_0 and y_0
float A = (lt.V - Vb) / dU;
float S2 = 1. + A * A;
float B = Vb - A * Ub;
float B2 = B * B;
// sqrt(S2)/B = 2 * helixradius
if (S2 < B2 * m_config.minHelixDiameter2) {
continue;
}
// 1/helixradius: (B/sqrt(S2))/2 (we leave everything squared)
float iHelixDiameter2 = B2 / S2;


To calculate the helix circle in the x-y plane, the x,y coordinates are transformed into a U/V plane in order to calculate the circle with a linear instead of a quadratic equation for speed. From the helix circle, particle energy and impact parameters can be estimated.

The scattering calculation is also spread over the nested loops to avoid redoing calculations. First, the maximum allowed scattering at the configured minimum transverse momentum (pT) cut is calculated and scaled by the pseudorapidity of the bottomSP-middleSP dupletto get the minimum momentum of the duplet. This duplet’s pseudorapidity is used for later calculation of the scattering for the triplet as well.:

// 1+(cot^2(theta)) = 1/sin^2(theta)
float iSinTheta2 = (1. + cotThetaB * cotThetaB);
// calculate max scattering for min momentum at the seed's theta angle
// scaling scatteringAngle^2 by sin^2(theta) to convert pT^2 to p^2
// accurate would be taking 1/atan(thetaBottom)-1/atan(thetaTop) <
// scattering
// but to avoid trig functions we approximate cot by scaling by
// 1/sin^4(theta)
// resolving with pT to p scaling --> only divide by sin^2(theta)
// max approximation error for allowed scattering angles of 0.04 rad at
// eta=infinity: ~8.5%
float scatteringInRegion2 = m_config.maxScatteringAngle2 * iSinTheta2;
// multiply the squared sigma onto the squared scattering
scatteringInRegion2 *=
m_config.sigmaScattering * m_config.sigmaScattering;


The following code block calculates if the triplet forms a nearly straight line in the r/z plane (see Fig. 5) as the particle path in the r/z plane is unaffected by the magnetic field 1. This is split in two (may be revised in the future); the first test occurs before the calculation of the helix circle. Therefore, the deviation from a straight line is compared to the maximum allowed scattering at minimum pT scaled by the forward angle (as calculated above). Both the check against min pT as the check against the calculated pT (discussed further below) take the correlated measurement uncertainty into account.:

// add errors of spB-spM and spM-spT pairs and add the correlation term
// for errors on spM
float error2 = lt.Er + ErB +
2 * (cotThetaB * lt.cotTheta * varianceRM + varianceZM) *
iDeltaRB * lt.iDeltaR;

float deltaCotTheta = cotThetaB - lt.cotTheta;
float deltaCotTheta2 = deltaCotTheta * deltaCotTheta;
float error;
float dCotThetaMinusError2;
// if the error is larger than the difference in theta, no need to
// compare with scattering
if (deltaCotTheta2 - error2 > 0) {
deltaCotTheta = std::abs(deltaCotTheta);
// if deltaTheta larger than the scattering for the lower pT cut, skip
error = std::sqrt(error2);
dCotThetaMinusError2 =
deltaCotTheta2 + error2 - 2 * deltaCotTheta * error;
// avoid taking root of scatteringInRegion
// if left side of ">" is positive, both sides of unequality can be
// squared
// (scattering is always positive)

if (dCotThetaMinusError2 > scatteringInRegion2) {
continue;
}
}


Now the check for scattering with calculated particle momentum. Momentum is calculated from the pT and the pseudorapidity. This must be $$\geq$$ the lower pT cut, and therefore allows $$\leq$$ scattering compared to the previous check, as the scattering decreases linearly with particle energy:

// calculate scattering for p(T) calculated from seed curvature
float pT2scatter = 4 * iHelixDiameter2 * m_config.pT2perRadius;
// TODO: include upper pT limit for scatter calc
// convert p(T) to p scaling by sin^2(theta) AND scale by 1/sin^4(theta)
float p2scatter = pT2scatter * iSinTheta2;
// if deltaTheta larger than allowed scattering for calculated pT, skip
if ((deltaCotTheta2 - error2 > 0) &&
(dCotThetaMinusError2 >
p2scatter * m_config.sigmaScattering * m_config.sigmaScattering)) {
continue;
}


The last cut applied in this function is on the so-called impact parameters, which is the distance of the perigee of a track from the interaction region in mm of detector radius. It is calculated and cut on before storing all top SP compatible with both the current middle SP and current bottom SP.:

// A and B allow calculation of impact params in U/V plane with linear
// function
// (in contrast to having to solve a quadratic function in x/y plane)
float Im = std::abs((A - B * rM) * rM);

if (Im <= m_config.impactMax) {
topSpVec.push_back(compatTopSP[t]);
// inverse diameter is signed depending if the curvature is
// positive/negative in phi
curvatures.push_back(B / std::sqrt(S2));
impactParameters.push_back(Im);
}


The bottom SP and middle SP as well as the collection of top SP is passed to SeedFilter::filterSeeds_2SpFixed, whose collected output for the current middle SP with all compatible bottom SP and top SP is then passed to SeedFilter::filterSeeds_1SpFixed.

## SeedFilter::filterSeeds_2SpFixed¶

This function assigns a weight (which should correspond to the likelihood that a seed is good) to all seeds and calls the detector specific cuts to apply a hard cut or modify the weight. The weight is a “soft cut”, in that it is only used to discard tracks if many seeds are created for the same middle SP in SeedFilter::filterSeeds_1SpFixed

The weight is influenced by:

1. The impact parameter (the higher the distance the worse)

2. The number of seeds which may belong to the same particle track

3. Optional detector specific cuts.

The impact parameter is multiplied by the configured factor and subtracted from the weight, as seeds with higher impact parameters are assumed to be less likely to stem from a particle than another seed using the same middle SP with smaller impact parameters.

The number of seeds only differing in top SP which have similar helix radius and the same sign (i.e. the same charge) is used to increase the weight, as it means that more than three measurements that may be from the same particle have been found. The measurements must have a minimum distance in detector radius, such that measurements from the same layer cannot be counted towards the increased weight. The number of found compatible seeds is multiplied by a configured factor and added to the weight.

The optional detector specific cuts can use the weight from 1. and 2. and the three SP to apply a hard cut or change the weight of a seed.

## SeedFilter::filterSeeds_1SpFixed¶

This function allows the detector specific cuts to filter on the basis of all seeds with a common middle SP and limits the number of seeds per middle SP to the configured limit. It sorts the seeds by weight and, to achieve a well-defined ordering in the rare case weights are equal, sorts them by location. The ordering by location is only done to make sure reimplementations (such as the GPU code) are comparable and return the bitwise exactly same result.

## Footnotes¶

1

approximately, this is one of the reasons the algorithm breaks down for low energy particles.