5 A L I G N M E N T O F T H E P I L E - U PS E N S O R S

As described in the previous chapter, the Pile-Up (PU) system has been successfully com-missioned and various studies have been performed to optimise the system readout and toalign its output signals in time. We present here a study of the sensors’ spatial alignment.

If the PU sensors are displaced, strips of adjacent sensors might no longer lie in circles. Forthe VELO sensors, a displacement of the order of tens of micrometers in the hit (r, z) positionis seen to affect the vertexing performance [96]. Although the resolution requirement on thevertexing at the L0 trigger level is less stringent as for offline reconstruction, a large PUmisalignment can degrade the performance of the vertex algorithm. Therefore, the Pile-Upsensors are aligned to optimise the performance of this algorithm.

In this section, the method used to evaluate the PU sensors’ misalignment relative tothe VELO sensors is described. The study is based on a χ2 minimisation of the residualsbetween PU hits and reconstructed tracks that intersect the Pile-Up sensor planes. It makesuse of the Pile-Up analog information and assumes a pre-aligned VELO detector.

The obtained misalignment parameters have been implemented and used in the LHCbConditions DataBase (see Sec. 3.6) to allow corrected vertexing. The achieved precision ispresented, while the resulting improvement on the vertexing performance is presented inCh. 6.

5.1 The residualA residual is the signed distance of a single measurement (hit) to a track, either defined asdistance of closest approach in 3D-space or distance measured in the plane of the detectoritself. In the following, we will deal with plane residuals r and introduce a unit vector n,which lies in the plane of the detector and is orthogonal to the measurement strips. If mrepresents the measurement (for instance, the strip position in r) and ~xt is the calculatedintersection point between the track and the detector plane (the “track point”), the residualcan be written as

r = m− n ·~xt . (56)

We refer to biased residuals if the hit (measurement) is used in the track fit, and unbiasedresiduals if the track is fitted without using the hit.

5.2 Misalignment effect on the residualsIn the case of a perfectly aligned detector, the difference between the track point and thehit measurement is only due to the detector resolution, and the average residual on all mea-surements tends to zero. In a realistic case instead, the average residual might not be zeroand depends on the detector misalignment: by studying this dependency, the misalignmentcorrections can be determined.

The transformation between an ideally aligned detector (ideal) and a misaligned one (mis)is equivalent to the transformation that goes from a local reference frame (loc), fixed to a

61

62 Alignment of the Pile-Up sensors

detector element, to the global frame (glo), defined in the LHCb experiment. If ~xloc is a pointin the detector local frame (a sensor frame for the Pile-Up case) and ~xglo is a point in theglobal LHCb frame, we can introduce a transformation matrix Mglo

loc to go from one to theother:

~xglo = Mgloloc~x

loc , (57)

which is equivalent to~xmis = Mmis

ideal~xideal . (58)

The degrees of freedom of a detector element in LHCb are the translations along the Carte-sian axes (δx, δy, δz) and the rotations with respect to the same axes. The transformationmatrix can thus be split in a rotation matrix R and a translation matrix (vector) T of thecoordinates from the local to the global frame [97]:

~xglo = R~xloc + T . (59)

The rotation matrix follows the convention

R = Rx(α) · Ry(β) · Rz(γ) , (60)

where (α, β, γ) are the Euler rotation angles around the (x, y, z)-axes respectively. Thedependency of the residuals on the transformation between aligned and misaligned framesfor the VELO and Pile-Up case is illustrated in the next section.

5.3 A method for the PU-to-VELO alignmentFor the VELO detectors, the alignment method proceeds in three steps: the evaluation ofthe relative misalignments of the two sensors within one module, the evaluation of themisalignments within each detector half and the evaluation of the relative misalignments ofthe two halves.

For the Pile-Up detector we will focus on the evaluation of the sensor-to-sensor (PU-sensor to VELO-sensor) misalignment. The alignment study is performed using the spatialinformation from Pile-Up hits and VELO tracks extrapolated to the PU sensor planes.

Equation 56 relies on the measured hit position at the strip m (measurement). Usingadditional information from the track, we can build a spacepoint ~xm on the plane of thePU-sensor, associated to m. In fact, the measurement itself only “carries” information on ther-coordinate of the hit. To build a space point associated to the measurement, a φ coordinateis obtained from the associated VELO track. The spacepoint is expressed in cylindricalcoordinates as ~xm = (m, φtrack(z), z), where m, z are measured from the Pile-Up hit andφtrack(z) is taken from the track extrapolated to the plane of the sensor. In this way

m = n ·~xm , (61)

and Eq. 56 becomesr = n ·~xm − n ·~xt = n · (~xm −~xt) . (62)

Note that the value of the residual is defined as the signed distance between the measure-ment (strip) and the track point. The sign depends on the chosen convention and in the fol-lowing we assume a positive residual if the r-coordinate of ~xm is larger than the r-coordinateof ~xt.

5.3 A method for the PU-to-VELO alignment 63

Equation 62 can be expressed in global coordinates, which are coordinates in the mis-aligned frame:

rglo = nglo · (~xglom −~xglo

t ) , (63)

Alternatively in local coordinates, which can be associated to the coordinates in the ideallyaligned frame, we have

rloc = nloc · (~xlocm −~xloc

t ) , (64)

In this frame the average residual on all measurements tends to zero, that is⟨nloc · (~xloc

m −~xloct )⟩=⟨

nloc · (~xidealm −~xideal

t )⟩= 0 , (65)

where ~xidealm and ~xideal

t are the points of the hit and track interception with the sensor whichwould be measured without misalignments.

In the case that the local-to-global transformation represents the actual misalignment, rgloand rloc are identical. Therefore, we can determine the needed misalignment corrections bycalculating ∆r = rglo − rloc as a function of the misalignment parameters and minimising itto be zero:

∆r = n · (~xglom −~xloc

m )− n · (~xglot −~xloc

t ) . (66)

In this passage we use nglo ≈ nloc = n, which is true on average for all residuals, in theapproximation of small rotations along the axes (in the Cartesian product, this leads to asecond order correction to r).

The contribution (~xglom −~xloc

m ) to the residual depends on the matrix transformation frommisaligned to aligned/ideal frame. This is the combination of the three rotations along theaxes

Rx(α) =

1 0 00 cosα −sinα0 sinα cosα

Ry(β) =

cosβ 0 sinβ0 1 0

−sinβ 0 cosβ

Rz(γ) =

cosγ −sinγ 0sinγ cosγ 0

0 0 1

, (67)

and the contribution for the translations (Eq. 68)

Tidealmis = δTideal

mis =

δxδyδz

. (68)

Working in the approximation of infinitesimal rotations:

Ridealmis ≈ 1 + δRideal

mis

δRidealmis =

0 −δγ δβδγ 0 −δα−δβ δα 0

. (69)

To express ~xloc in the local sensor frame, it is convenient to use cylindrical (r, φ) coordinates:

~xloc =

rcosφrsinφ

0

(70)

64 Alignment of the Pile-Up sensors

and hence the contribution to the residual becomes:

~xglom −~xloc

m = (1 + δRgloloc )~x

locm + δTglo

loc −~xlocm =

δγ · rsinφ−δγ · rcosφ

δβ · rcosφ− δα · rsinφ

− δx

δyδz

. (71)

At this point we introduce the track slopes tx and ty along the x and y axis respectively, asthey are reconstructed by the VELO. They are defined as

tx =

(dxdz

), ty =

(dydz

). (72)

The contribution (~xglot − ~xloc

t ) to Eq. 66 can then be evaluated using the vector of the trackslopes as

~xglot −~xloc

t ≈

tx · δzty · δz1 · δz

, (73)

where δz = (~xglot −~xloc

t )|z. The z component of ~xlocm is by definition equal to that of ~xloc

t :

~xlocm |z= ~xloc

t |z (74)

and the z component of ~xglom is required to be equal to that of ~xglo

t too, in order to align thesensor: ⟨

~xglom |z

⟩=⟨~xglo

t |z⟩

. (75)

Therefore we write:∆z = (~xglo

m −~xlocm )|z (76)

and from Eq. 71:∆z = δβ · rcosφ− δα · rsinφ− δz . (77)

Defining the quantity (tx ex · n + ty ey · n) ≡ σ, we obtain for Eq. 66:

∆r ≈[δγ(rsinφ)ex · n− δγ(rcosφ)ey · n− δxex · n− δyey · n

](78)

+ [δα(rsinφ)σ− δβ(rcosφ)σ + δz · σ] , (79)

where the term in the first brackets is the contribution from (~xglom −~xloc

m ) and the term in thesecond brackets is the contribution from (~xglo

t −~xloct ).

Finally, we express the unit vector n in local coordinates. It lies on the sensor plane, isorthogonal to the tangent of the R-strip at the hit position ~xloc

m and points radially outwards,according to the residual sign convention. If (ex, ey, ez) are the unit vectors parallel to theCartesian axes, we have1

n = cosφex + sinφey . (80)

Note that φ is the azimuthal coordinate of xloct , which is already used to build the spacepoint

~xm associated to the measured hit position at the r-strip.The obtained relation is:

σ = txcosφ + tysinφ∆r ≈ −δxcosφ− δysinφ + δα(rsinφ)σ− δβ(rcosφ)σ + δz · σ . (81)

1 n · ez = 0, according to the approximation nglo ≈ nloc = n.

5.4 Implementation of the PU sensors alignment 65

In the following we present an alignment study of the Pile-Up sensors to determine theδx, δy, δz offsets and the δα, δβ rotation corrections by minimising ∆r.

5.4 Implementation of the PU sensors alignmentThe alignment of the Pile-Up sensors relative to the VELO-sensors is performed using thetrack–based procedure explained in the previous section. The algorithm makes use of thesoftware reconstruction, is based on the Pile-Up analog information and on the track posi-tion. The resulting residuals are always unbiased, since the LHCb tracking procedure doesnot make use of PU hits.

All variables have been calculated in the local sensor frame: the global to local trans-formations for each PU sensor are defined in this section. The selection applied to obtainthe input sample and the fitting procedure are presented below, together with the resultsobtained using 2010 data.

5.4.1 Global to local transformations for PU sensorsIn the reconstruction procedure all variables are commonly evaluated in the global frame,but the geometry parameters stored in the LHCb Conditions DataBase have to be expressedin the local frame. Therefore we need to define explicitly the transformations to go from theglobal frame to each Pile-Up sensor frame. As seen in Eq. (59), such transformations are acombination of rotations and translations.

For the PU sensors, labelled according to their sensor ID, the axes rotations are definedaccording to the LHCb convention:

Rlocglo,128 =

+1 0 00 +1 00 0 +1

Rlocglo,129 =

−1 0 00 −1 00 0 +1

Rloc

glo,130 =

+1 0 00 −1 00 0 −1

Rlocglo,131 =

−1 0 00 +1 00 0 −1

. (82)

It is useful to calculate how the slopes of a track change from the global to a certain localframe: if ~x1 and ~x2 are points on the same track (direction), the distance between them willbe

∆~xloc = Rlocglo∆~xglo ⇒

∆xloc

∆yloc

∆zloc

= Rlocglo

∆xglo

∆yglo

∆zglo

. (83)

From the slope definition (72) and by a unit change ∆zglo = 1: txty1

loc

· ∆zloc = Rlocglo

txty1

glo

. (84)

This transformation is used to evaluate the correct expression of σ (see Eq. 81) in the localPU sensor frame.

66 Alignment of the Pile-Up sensors

Figure 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Image from the LHCb event display, showing an event with tracks reconstructed through the VELO modules and thematching hits on PU sensors (at the very left).

5.4.2 Algorithm implementationTo evaluate the Pile-Up system misalignment, we implemented the following algorithmin the LHCb software framework2, starting from a set of standard tracks reconstructed inLHCb3:

1. In all events, only tracks containing a node –that is a track hit used in the track fittingprocedure– on VELO sensor 0 for the A side or sensor 1 for the C side are selected,see Fig. 37. In this way, only reconstructed tracks pointing in the backward direction,which are more likely to be extrapolated to the PU acceptance, are used.

2. A selection is applied to clean the track sample from low quality tracks; see nextsection for a detailed description of the selection cuts.

3. Each track is extrapolated to the nominal z-position4 of a certain Pile-Up sensor, wherethe track extrapolation point is expressed in the coordinates of the local sensor frame.

4. The Pile-Up cluster closest in r to the track extrapolation point is selected. Since thereis no φ information on the cluster position, the cluster is required to belong to thesame φ-sector as the track extrapolation point, evaluated in the local frame.

5. The residual is measured in the local frame as the distance in r of the extrapolated hitto the chosen cluster (r-of-cluster minus r-of-track point). The r-coordinate positionof the cluster is taken as the r of the “more significant” strip in the cluster, in terms ofcollected charge5. This distance represents the ∆r residual of Eq. 81, to be minimised.

2 In the package Velo/VeloRecMonitors of the Vetra application, see Sec. 3.6 .3 The used track container is Rec/Track/Best.4 The z coordinate used is the value stored in the database.5 rOfStrip obtained from the central strip of the cluster, weighted with the interstripFraction ex-

tracted by use of the Fractional Position Tool.

5.4 Implementation of the PU sensors alignment 67

6. Only residuals smaller than 600 µm in absolute value are used for the subsequentfitting procedure; this cut is chosen in order to reduce the number of random combi-nations of tracks and clusters, which may be source of systematics.

5.4.3 Track selectionIn Fig. 38 the distributions of track extrapolation points on the (x, y) plane for the four PUsensors are shown, before the alignment procedure. All tracks containing a node on VELOsensors 0 or 1 are used. The colour distribution shows, as expected, that the track occupancyis higher at lower r values and lower for higher |z|-position sensors. The PU sensors placedfurther away from the VELO –Fig. 38 (a) and (b)– have a lower track occupancy because ofa smaller overlap in acceptance of VELO and PU.

The track selection aims at cleaning the track sample from low momentum tracks, ghosttracks and badly reconstructed ones. In addition, we want to reduce the contribution tothe residual estimate coming from combinatorics, i.e. random matches between good VELOtracks and Pile-Up hits. Both selections are described in the following sections.

Selection on track qualityAfter requiring a node on VELO sensors 0 or 1, the track sample contains only tracks of typeVelo6, as these belong to particles travelling in the negative z-direction. Velo tracks are 3Dtracks obtained from VELO hits in both r and φ sensors.

To investigate the track quality of the sample, we first look at the track χ2 probability (seeFig. 39). The distribution shows a non-flat shape, peaking around zero. This can be relatedto badly reconstructed tracks, either due to unaccounted multiple scattering or to wronghits assigned to a track. These effects can be reduced in both cases by setting7 a lower cuton the χ2 probability, prob(χ2)>0.01, eliminating the peak in the first bin of Fig. 39 (a).

In Fig. 39 (b) the distribution of track χ2/nDoF for PU sensor 130 is shown after applyingthe quality cut prob(χ2)>0.01. An additional selection requiring χ2/nDoF<2 is then made,to further enhance the quality of the track sample. Another safe cut is implemented inorder to clean the sample from ghost tracks, that are “non-existing” tracks reconstructedfrom random combinations of hits. In this case we require a minimum number of clusters(measurements) to be used by the tracking algorithm (n>6). A maximum threshold on thenumber of clusters is furthermore fixed at n<21, as a general track quality selection to avoidbadly reconstructed tracks in very high occupancy events. Figure 40 shows the distributionof the number of VELO clusters n assigned to each track in the sample for PU sensor 130;similar distributions are obtained for the other sensors. The number of clusters is most likelyeven because each VELO module is composed of a pair of (R, φ) sensors.

Selection on fiducial volumeFurther cuts on the track sample are needed to reduce combinatorics that may occur whena track is extrapolated very close to the border of a sensor or to the boundary between twosectors. For this reason, it is useful to cut on the (x, r)-coordinates of the track extrapolationpoint, to exclude points close to the sensor border, and on the φ-coordinates, to excludepoints close to the boundaries between contiguous sectors of the same sensor8. The distri-

6 See Sec. 3.3 for a description of the tracking algorithms and classification in LHCb.7 Equivalently, one could place a higher threshold on χ2/nDoF.8 This leads to a cut of about ±1 degree around the boundaries at φ=0, ±π/4, ±π/2.

68 Alignment of the Pile-Up sensors

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 38 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of track extrapolation points on the (x, y) plane for the four PU sensors, fortracks containing a node on VELO sensor 0 or 1, but before applying any further trackselection. The colour scale represents the number of entries per bin of 250 µm x 250 µm. Onthe first plot, the orientation of the (x, y) axes in the local sensor frame is also shown. The φcoordinate in this frame varies from −π/2 to +π/2.

5.4 Implementation of the PU sensors alignment 69

(a) (b)

Figure 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .The distribution of track χ2 probability (a), for tracks extrapolated to PU sensor 130, show that the track sampleis contaminated with bad quality tracks. Similar distributions are obtained for the other PU sensors. In (b) thedistribution of track χ2/nDoF for PU sensor 130 is shown after applying a quality cut of P(χ2)>0.01

Table 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Track selection cuts.

prob(χ2) prob(χ2) > 0.01χ2/nDoF χ2/nDoF < 2number of clusters n < 21 and n eventrack r 18 mm ≤ r ≤ 41 mmtrack x x≥5 mm

track φ(| φ |≥ π

180 and | φ |≤ (π4 −

π180 ))

or(| φ |≤ (π

4 + π180 ) and | φ |≤ (π

2 −π

180 ))

butions of these variables are shown in Fig. 41 for sensor 128. Please note that from here onthe sensor local coordinate system is used, unless stated otherwise.

A summary of the chosen cuts is listed in Tab. 4. After the track selection is applied, thedistribution of track extrapolation points on the (x, y) PU sensor plane is shown in Fig. 42

for all four PU sensors.

5.4.4 Residual distribution and fitting implementationTo evaluate the residuals with the procedure previously described, a sample of 50000 eventsacquired in 2010 has been processed: the obtained residuals follow the distribution shownin Fig. 43, where they are plotted on the (x, y) plane of each sensor. It is clear from these

70 Alignment of the Pile-Up sensors

Figure 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of the number of clusters assigned to each track (in black), for tracks used in the analysis of PU sensor130; note that the number of clusters is most likely even because each VELO module is composed of a pair of (R, φ)sensors. The distribution obtained after the cut on the number of clusters is shown in grey.

plots that especially sensor 128 is misaligned and all the other sensors require non-negligiblecorrections, too.

As suggested by Eq. 81, we have a handle on the misalignments by studying the residualsversus the track φ-coordinate. To first order, this distribution shows a cosine-sine depen-dence9 on φtrack and gives direct information on the relative x-y translations; the strongerthe (δx, δy) misalignment, the more evident is the angular dependence. The distributionsobtained before applying any misalignment correction are shown in Fig. 44. A clear mis-alignment effect due to translations in x or y is seen in sensor 128, while this effect is lessprominent for the other sensors. The misalignments of sensors 130 and 131, visible in Fig.43, might be dominated by other degrees of freedom.

The residual is expressed as a linear combination of functions of the track parameters(φ, r, slopes tx and ty) and the residual distributions are fitted with a multi-dimensional fit,following the relation listed in Eq. 81. The fit is performed using the ROOT TLinearFitter

tool [138]. In this way, all degrees of freedom of the sensor misalignment, δx, δy, δz, δα, δβ,can be determined at the same time and correlations are taken into account by the fit proce-dure.

The error assigned to the residual σres is chosen to be equal to the hit resolution. TheVELO hit resolution for a cluster depends both on the strip pitch and the projected angle, i.e.the angle at which the particle crosses the sensor [98]. For the Pile-Up, it is approximatedwith the strip pitch dependence only, i.e. with the binary resolution (pitch/

√12). Therefore

the hit resolution varies with the r-coordinate according to the known linear dependence ofthe pitch size [68]:

σres[µm] =

[40.0 +

(101.6− 40.0) · (r[µm]− 8190)4195− 8190

]· 1√

12. (85)

9 It shows a dependence on a linear combination of sines and cosines.

5.4 Implementation of the PU sensors alignment 71

(a) track r-coordinate (b) track φ-coordinate

(c) track x-coordinate

Figure 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of the track r, φ and x coordinate for PU sensor 128, in the local frame; similar distributions are obtainedfor the other sensors. The distribution in φ shows a dip around 0 radians, due to a lower track reconstruction efficiencyon the plane y = 0.

72 Alignment of the Pile-Up sensors

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 42 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of track extrapolation points on the (x, y) plane for the four PU sensors, after applying the track selection.The colour scale represents the number of entries per bin of 250 µm x 250 µm. The histograms in (b) and (d), obtainedfrom the same track sample, show a lower occupancy in a specific area in φ; this is due to a dead link on VELOφ-sensor 67.

5.4 Implementation of the PU sensors alignment 73

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Residuals (in mm) as function of position for each PU sensors (x, y) plane, before applying any misalignment correc-tion. Each bin in (x, y) is the average residual collected for that bin.

74 Alignment of the Pile-Up sensors

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Residual distributions against the track φ-coordinate for the four PU sensors, before applying any misalignmentcorrection and after implementing the track selection. To first order, the distribution shows a cosine-sine dependenceon φtrack . A clear misalignment effect due to translations in x or y is seen in sensor 128.

5.4 Implementation of the PU sensors alignment 75

5.4.5 ResultsAfter an initial fit to the distributions, the procedure is iterated four times, at each stepapplying the new misalignment corrections as incremental corrections.

The iteration of the fitting procedure aims at improving the misalignment corrections andproves the convergence of the method. In Fig. 45 the corrections (δx, δy, δz) obtained for fourconsecutive iterations are shown indeed to converge to zero and confirm the effectiveness ofthe algorithm.

The results for all the obtained misalignments are listed in Tab. 5 for the translations andTab. 6 for the rotations. Each value is obtained as the sum of the corrections extracted ateach iteration of the fit. The statistical errors depend on the error on the residuals; they areobtained from the covariance matrix of the multidimensional fit and are propagated to thefinal sum of corrections. As was expected from Fig. 44, a large offset in y is found for sensor128.

Table 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .PU sensors’ translation misalignments.

Sensor δx ± σδx (mm) δy ± σδy (mm) δz ± σδz (mm)128 +0.016± 0.001 −0.235± 0.001 −0.341± 0.012129 +0.028± 0.001 −0.026± 0.001 −0.269± 0.013130 +0.046± 0.001 −0.012± 0.001 +0.562± 0.007131 +0.083± 0.001 +0.013± 0.001 −0.352± 0.007

Table 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .PU sensors’ rotation misalignments.

Sensor δα ± σδα(rad) δβ ± σδβ

(rad)

128 −0.0007± 0.0004 −0.0092± 0.0005129 +0.0037± 0.0005 −0.0051± 0.0005130 +0.0003± 0.002 −0.0141± 0.0003131 +0.0109± 0.0003 +0.0211± 0.0003

Figures 46 and 47 show the mean of the residuals evaluated after implementing all themisalignment corrections in the database; the residuals are plotted against (x, y) and φrespectively. As expected, applying the corrected alignment information results in a morehomogeneous distribution and in a movement of the mean value towards zero.

Figure 48 shows a comparison between the residual distributions before and after apply-ing the misalignment corrections; again, the distributions get narrower and better centeredaround zero. By fitting the distribution with a Gaussian in the range −0.2<z<0.2 mm, weobtain an r.m.s. between 70 and 90 µm. This is larger than the maximum binary resolution(102/

√12 ≈ 30 µm), an indication of some remaining effect.

The non-Gaussian shape of the distribution is due to the use of residuals that correspondto different strip pitches and projected angles [98].

76 Alignment of the Pile-Up sensors

(a) δx misalignment correction (b) δy misalignment correction

(c) δz misalignment correction (d) δα misalignment correction

(e) δβ misalignment correction

Figure 45 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Misalignment corrections and corresponding errors obtained for the three translations and two rotations for PU sensor128. Each point corresponds to a different algorithm iteration: at every step the corrections are applied incrementally.Similar distributions are obtained for the other PU sensors.

5.4 Implementation of the PU sensors alignment 77

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 46 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Map of the residuals (indicated on the colour scale, in mm) on the PU sensors (x, y) plane, after applying all themisalignment corrections. The residual spread is now visibly reduced and the mean value moves towards zero.

78 Alignment of the Pile-Up sensors

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of the residuals against the track φ-coordinate for the four PU sensors, after applying all the misalignmentcorrections. The plots no longer show a cosine-sine dependence and demonstrate the success of the procedure incorrecting for x and y translations.

5.4 Implementation of the PU sensors alignment 79

After alignmentEntries 67042Mean 0.00022RMS 0.12

residual (mm)-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.005

0.01

0.015

0.02

0.025

After alignmentEntries 67042Mean 0.00022RMS 0.12

Before alignmentEntries 66944Mean -0.0089RMS 0.18

(a) sensor 128

Before alignmentEntries 59330Mean -0.0065RMS 0.12

residual (mm)-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.005

0.01

0.015

0.02

0.025

Before alignmentEntries 59330Mean -0.0065RMS 0.12

After alignmentEntries 59312Mean 3.5e-05RMS 0.12

(b) sensor 129

After alignmentEntries 82566Mean 0.00052RMS 0.09

residual (mm)-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04 After alignmentEntries 82566Mean 0.00052RMS 0.09

Before alignmentEntries 82818Mean 0.078RMS 0.1

(c) sensor 130

After alignmentEntries 74081Mean -0.00034RMS 0.093

residual (mm)-0.6 -0.4 -0.2 0 0.2 0.4 0.60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04After alignment

Entries 74081Mean -0.00034RMS 0.093

Before alignmentEntries 74486Mean -0.042RMS 0.096

(d) sensor 131

Figure 48 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of the residuals for the four PU sensors, before (black) and after (blue) applying the misalignment correc-tions. The residual spread is reduced and the mean value is centered on zero for all sensors. The non-Gaussian shapeof the distributions is due to the use of residuals that correspond to different strip pitches and projected angles [98].

80 Alignment of the Pile-Up sensors

(a) sensor 128 (b) sensor 129

(c) sensor 130 (d) sensor 131

Figure 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Alignment precision for the four PU sensors for x and y translations, obtained from the projection of the residualsdistribution against φtrack . From the r.m.s. of the histograms, we derive that the precision achieved is between 2 and 5

µm.

5.4.6 Remaining misalignments and stabilityFrom the shape of the residual distribution versus a certain degree of freedom we can assessthe quality of the alignment achieved for that DOF. For instance, the distribution versus φ,obtained from the profile of the histograms in Fig. 47, gives information on the remainingmisalignment in x and y.

We project the mean of this profile distribution on the residual axis to investigate theremaining spread of the residuals. Figure 49 shows the projections of the residual meansbinned in φ (with 31 bins) and can be considered as a good estimate of the alignmentprecision for x and y translations. From the r.m.s. of the histograms we conclude that theprecision achieved for the alignment in x and y of the PU sensors is between 2 and 5 µm.

Finally, a check on the system alignment stability is performed to spot potential move-ments of the system over time. The algorithm is applied on several datasets distributed oversix months of data acquisition: Fig. 50 shows the results obtained for PU sensor 129 interms of misalignment corrections in (δx, δy, δz). These corrections are all consistent withina few µm except one point in z (corresponding to a sample acquired in a period that reg-

5.5 Conclusions 81

(a) ∆x misalignment (b) ∆y misalignment

(c) ∆z misalignment

Figure 50 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Distribution of the misalignment corrections for the three translations over six months of data acquisition (PU sensor129); the run number of the dataset used to extract the correction is shown on the x-axis.

istered anomalous relative temperature variations); the system alignment can be thereforeconsidered stable over time.

5.5 ConclusionsIn order to improve the performance of the PU vertexing algorithm we studied the spacealignment of the detector. The PU sensors have been aligned using Velo tracks, by imple-menting a (PU)sensor-to-(VELO)sensor relative alignment algorithm. This compares trackextrapolations of an already aligned VELO detector to hits in the Pile-Up system.

We determined the misalignment corrections and corresponding errors for the three trans-lations along the Cartesian axes (δx, δy, δz) and for the rotations (δα, δβ) with respect to thex and y-axes. The corrections obtained are mainly of the order of tens of µm for the transla-tions along x and y, while they are one order of magnitude higher for the translation along

82 Alignment of the Pile-Up sensors

z. Applying the corrections will considerably improve the z-position estimate of the PUvertices, as shown in the following chapter (see Sec. 6.4).