"CMS Tracker Alignment"

A magyar CMS-csoport szemináriuma

Vesztergombi György (RMKI)

2011 okt. 17-én, hétfõn, 13.30

1. Millipede 2. Z-mass problem3. Parity4. Overlap5. Rigid body6. Recursive global alignment

Outlines

Brainstorming

MILLIPEDE

Data driven

20 000 tracker modules: x,y,z and

Single track fit ~ few 1 000 000 samples

General 2 with multi-million unknown parameter

Data sets:

cosmicsmini-bias isolated muonsZ->

WEAK modes????

The CMS (Compact Muon Solenoid) by construction posseses axial symmetry which is preserved in case of small pT Z decays, because the muons are emitted practically back-to-back.

The weak mode problem in alignment is connected to the existence of this axial symmetry, which is assumed to be solved by using the Z two-body massconstraint, because the TWIST

= cdist * z (1)

produces mass-shift in the two-body decay reconstruction. This shift can be included in the 2 fit as additional term weighted with the reconstructed width of theresonance.

One should not forget, however, the existence of additional symmetries such as P, parity and C, charge-conjugation. In case of the application of the distortion formule (1) one gets CP= 1, because the sign of the muon and the z-coordinate is changing simultaneously, giving (-1)*(-1) = 1. The result of this distortion is that the change ofthree-momentum of both decay particles will have the same sign, producing the mass-shift (see Fig. 1).

x

y ~

ideal

distorted

TWIST

pdistB = pideal -dppdist

A = pideal -dp

MASS corrected = pdist A + pdist

B = pideal –dp +pideal –dp = 2 * pideal - 2*dp = Mideal - 2*dp

x

z

A

B

There is, however, a CP=-1 solution, too. One can also use the following distortion formule

= a*|z| (2)

where the absolute value of z-coordinate is applied. Fig. 2 illustrates the 2 versionof these distortions.

In case of distortion (2) the momentum change of the muons has different sign withpractically equal in size which leaves the invariant mass unchanged in the first approximation (see Fig. 3), which has the DRAMATIC consequence that it will not influence the 2 value!!!! One finds „anti-twist” preference, because there is no gradient pointing toward decreasing the size of the twist.

+

+

+

-

TWIST

ANTI-TWIST

-

+

+

+

x

z

x

y ~

distorted

ANTI-TWIST

ideal

pdist B = pideal -dppdist A = pideal

+dp

MASS corrected = pdist A + pdist

B = pideal +dp +pideal –dp = 2 * pideal = Mideal

In practice the alignment fit is done with 2 sets of data, where the Z sample always represents only a minority. In a given -region the concrete gradient is determined by the majority of tracks, if it is directing along the CP=-1 solution, then the fit will not be influenced by Z mass constraint. This gives the explanation for the results of Joerg Behr:

„- alignment procedure does not fully correct for the twist

- impact of TwoBody DecayTrajectory is very small „

Talk on CMS Tracker Alignment Workshop, Hamburg May 30, 2011, page 6.

One can conclude:

- if the Cosmics data prefer at given region the twist correction, then Z tracks will add a bit more push toward that direction

- if the Cosmics data prefer at a given region „anti-twist” correction, then Z tracks will not increase 2, thus they will not influence the fitting procedure

- the overall size the correction will depend on the Cosmics, if it is completely independent -region-by--region, then one expects that in half of the regions will go the fit in the right direction producing a half corrected result.

Due to the axial symmetry one should regard Z-decays only in the (x,z) plane

The magnetic field will deviate the tracks in y-direction.

The twist and anti-twist distortions in will also go to y-direction.

For illustration Z decay is shown at = 1 ( = 45o) on Fig.4.

(Of course, at zero rapidity there is no effect of the twists.)

At high momenta in the (x,y) projection the relevant part of the circle can be approximated by a parabola:

circle: x2 + (y-R)2 = R2 parabola: y = a*x2 , where a= 1/ (2R)

In 3-dim space the helix can be parametrized around z=0 as

x = k*z and y = a*(k*z)2 =b*z2

In first approximation one gets also: yideal = =b*z2 .

After distortion:

ydist = =b*z2 + cdist *z = b*( z+ cdist /(2b))2- (cdist /2)2/b

Formally this corresponds to the same parabola which is shifted in z-direction by

z = cdist /(2b) and by y = (cdist /2)2/b in y-direction.

Thus if one uses the measured coordinates for the fit, the curvature at the bottom of the parabola will be the same.

BUT!! There is a physics constraint: the trajectory should pass by the beam spot. In this case one can get a good estimate for the curvatureassuming the parabolic form using 2 points:

1. point: origo (xo,yo) = (0,0)

2. point: measured middle point (xc, yc) = (k*zc, b*zc2 + cdist *zc)

yc = b* zc2 + cdist *zc = bfit * zc

2 which gives bfit = b + cdist /zc .

In this approximation the momentum after the distortion will be:

pdist = pideal * b/bfit = pideal * b / ( b + cdist/Zc)

If one takes into account the charge q of the particle and the sign of z-coordinate:

pdist = pideal * b / ( b + SIGN * cdist/|Zc|)

where

SIGNtwist = sign( charge) * sign (Zc)

SIGNanti-twist = sign( charge)

corresponding to the CP symmetry of the twist applied.

pdist = pideal * b/bfit

ideal

distorted

fitted

z

y ~

IDEAL TRACK yideal = =b*z2 .

DISTORTED TRACK:

ydist = =b*z2 + cdist *z = b*( z+ cdist /(2b))2- (cdist /2)2/b

FITTED TRACK: yc = b* zc2 + cdist *zc = bfit * zc

2

( -cdist /(2b), -(cdist /2)2/b )zc

Shifted parabola

vertex

Single track alignment ensures relatively independent fitting prcedures different ( ) regions, therefore one can have separate twist or anti-twist results as fuction of. In Fig. 5 one can see 2 regions. Fig. 6 shows a possible general case.

aa

aa

bb

bb

TWIST

ANTI-TWIST

Different ( ) regions can have different twist , anti-twist regions

European Union law:

Subsidiarity is an organizing principle that matters ought to be handled by

the smallest, lowest or least centralized competent authority. ...

Points from all tracks in Layer #1 Only points from tracks in overlap region

FPix # of tracks: 54368FPix # of selected tracks: 2795, 5.14089 %

Local method for correction

Let us assume that we have an approximately good alignment and want to check is there any twist in the system.

One can use a completely -symmetric source: gammas from the primary vertex

Let us assume that the coordinate system is centered on the beam spot:

proton beams are along z-axis, (xbeam,ybeam)= (0,0)

Let us take 2 overlapping modules A and B in zrange ( zmin,zmax)

The aim is to check the relative difference of A and B

Physics tool gamma-conversion in module A, using the e+/e- pairs.

The converted pair topologically can be reconstructed, as the distance between modules A and B is minimal, therefore one can see only a single track, pointing exactly toward the vertex point,THUS the difference directly gives A - B = corections of B relative to A. By recursive steps one can calibrate the whole circle.

e+

e-

A

B

C

B - A =

C - B =BC

Vertex

Module: A

m

MA

RA

s

r

m

r

mr= -

s = m * tg ()

Measured coordinate: s

Recursive fitting

Simplified 2-dim model with gammas:

Module „A”

Module „B”

mA

mB

s1

s2

s3

The recursive procedure for fitting relative position of module „j” with respect to module „j+1” can be generalized for 3-dimensional case.

In Barrel one obtains wheels with one free ladder module

In Forward one obtains disks with one free blade module

One should repeat the procedure with the new alignment parameters to get consistent fitted parmeters.

Overall fit can be performed on larger objects with drastically reduced number of parameters with much less freedom for twists and other criminalities.

Ideal tool for CROSS-CHECK of existing fitting procedures