4.4 INTERRELACIÓN DE LOS INDICADORES DE QUIEBRA Y DE
4.4.1 COMPORTAMIENTO DE LOS INDICADORES DE QUIEBRA Y DE
The problem of track fitting is to determine the helix parameters corresponding to a charged particle track that has left hits in the detector. The central concept of theHTTfast track fitting procedure is that this relationship is approximately linear over a sufficiently small region in the space of helix parameters. In practice this is done by defining regions where each of the particular linear maps are valid (sectors) and by obtaining the linear coefficients (constants) for each map to be used in the fitting procedure. The constant banks can then be loaded into the local memory on HTT
Letxi denote the vector of hit coordinates associated with a given track, wherei∈ {1, . . . , N}
withN the total number of coordinates. The individual elements of the vector may correspond to local hit measurements in unique layers of the tracking detectors or to measurements in different dimensions by the same detector layer (i.e. for the silicon pixel detector). Let qi denote the five
true helix parameters of the track, taken here as (pT, η, φ, d0, z0). The goal is to express the helix parameters as a linear function of the hits
qi≈qˆi≡cijxj+ki (5.2)
so that the helix parametersq0
i of a similar track with hit parametersx0i can be well-approximated
by applying the same fit constantscij andkias for the original track. In theHTTsystem, the same
constants are used for all sets of hits which are recorded in the same set of unique silicon sensors, usually distributed across multiple physical detector layers.
For each sector, fit constants can be obtained from a large sample of simulated muons, where both the hit information and truth track parameters are known. Henceforth consider a single sector, and letcij andkidenote the constants to be determined. The constants can be found by minimizing
the square difference between the trueqi and approximated ˆqitrack parameters
h(qi−qˆi)2i=h(qi−(cijxj+ki))2i (5.3)
over all tracks. Differentiating this expression with respect tokl gives
0 = ∂ ∂klh (qi−qˆi)2i (5.4) = ∂ ∂kl k2−2cijhxiikj−2kihqii (5.5) = 2(kl−clihxii − hqli) (5.6) (5.7) yielding the relation that
ki=cijhxji+hqii (5.8)
Now, differentiating with respect toclm gives
0 = ∂ ∂clmh (qi−qˆi)2i (5.9) = ∂ ∂clm [cijckjhxixki −2kicijhxji −2cijhxiqji] (5.10) = 2(clihxixmi −klhxmi − hxlqmi) (5.11) (5.12)
5. Hardware Tracking for the Trigger 66
Combining this with Equation5.8 and definingVij=hxixji − hxiihxjithis gives
0 = clihxixmi −(clihxii+hqli)hxmi − hxlqmi (5.13) = cliVim− hqlihxmi − hxlqmi (5.14) = cliVimVmj−1−(hqlihxmi+hxlqmi)Vmj−1 (5.15) (5.16) and thus cij = (hqiihxki+hxiqki)Vkj−1. (5.17)
The quality of a fit is excellent while xi is near the average value hxii, with the error being
proportional to their square difference due to the limitation of the linear approximation. A useful quality metric is the χ2, which further accounts for correlations among the coordinates and is computed as
χ2= (xi− hxii)Vij−1(xj− hxji) (5.18)
In order to fit tracks and compute a χ2 value for each one, one must store the constantsc
ij (5N
numbers), ki (5 numbers), and Vij−1 (N(N + 1)/2 numbers, as V is symmetric). The number of
parameters stored for V may be reduced by accounting for correlations among hits due to the constraint that they originate from a physical track. Thus, the eigenvalue decomposition of the (symmetric) matrixV−1 =UTΛU is useful, with U a matrix of unit-normalized eigenvectors (let
the componentsuij ofU be theicomponent of vector j) and Λ a diagonal matrix of eigenvalues.
Five eigenvalues will be nearly zero due to the track constraint, which can be dropped from theχ2 computation. Thus, definingAij =uij andBi =uijhxji, withi= 1, . . . , N −5 indexing only the
eigenvector components associated with the non-singular eigenvalues, theχ2 can be approximated as
χ2≈(Aijxj−Bi)2 (5.19)
This requires storingN(N−5)+(N−5) = (N+1)(N−5) numbers. However, this is only fewer than theN(N+ 1)/2 numbers before eigen-decomposition forN <10. The real benefit of this approach is in reducing the number of sums and multiplications the FPGA must compute. Multiplying an
m×n-dimensional matrix n×p-dimensional matrix requires mnp multiplications and m(n−1)p
sums. Thus the originalχ2requiresN2+N multiplications andN2+N
−1 adds (N×N byN×1 then 1×N byN×1) while the approximate calculation requires (N−5)N+(N−5) = (N+1)(N−5) multiplies and (N−5)N+ (N−5) + (N−6) = (N+ 2)(N−5)−1 adds, both improvements for all
N. In the case of tracks contained entirely in the barrel region of the ITk detectors (|η|.1), hits in five pixel and four double-sided strip layers may be used, so that the hit vector is 18-dimensional. This requires 5·18 + 5 = 95 constants to be stored for track fitting (using 5N = 90 multiplies and 5(N−1) + 5 = 5N= 90 adds per fit). In addition, 247 constants are needed for theχ2calculation, each of which uses 247 multiplies and 259 adds (versus 152 constants, 380 multiplies, and 379 adds without eigen-decomposition).
In the first stage, the hits to be fit are determined by the matched patterns. If hits are present in allN layers, the corresponding constant set is retrieved based on the pattern identifier so that the fit proceeds with the full coordinate set. The trackχ2 is also computed, with tracks only retained if a loose requirement is satisfied.
Fits with< N layers must be conducted when either hits are not present in each of theN layers (missing coordinates) or if the χ2 for the N-layer track is poor (a recovery fit). This requires a modification to the N-layer fit procedure either by: using a separate, lower-dimensional constant set, or; substitutingdummy hit coordinates into the fully-dimensional constant set. Both strategies have been investigated in the context of the HTT system, with each incurring costs in different parts of the system (memory versus fit resources). In the latter configuration, hit coordinates are guessed which minimize theχ2of the corresponding track fit. At this point, the resulting first-state fit tracks may be passed back to theEF or extrapolated into the remaining layers of the detector for an expanded second-stage fit, making use of the additional coordinates.
An overlap removal (historically denoted as the ‘Ghostbuster’ or hit warrior (HW) algorithm) may also be applied (at several possible stages) to reduce the number of tracks which share common coordinates. The algorithms removes tracks which share hits in common, deciding which to retain based on theχ2 of each fit and the number of hits associated to each track. Often first-stage tracks are required to have at least three unique hits.