Z mass constraint with twists in twist

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

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

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

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

Piecewise Alignment by OVERLAPPING tracks

Simplified ONE-dimensional case

z1 z2 z3

+/- dz +/-dz +/-dz

Overlappingmodules

No magnetic field: (m,b) track fitted from the other planes: x = m *z + b

xj

xj+1

Assumption : distortion is occuring only in x-direction:

j-th module is shifted by „sj”

xjmeas + sj = xfitted = m (z1-dz) + b

xj+1meas + sj+1 = xfitted = m (z1+dz) + b

By subtraction one gets a recursive formule :

sj+1 = sj + xjmeas – xj+1

meas (+/-) 2 * m * dz

The sign depends on the parity of „j”

For many tracks one can get with small error the expected value: <sj+1 - sj >

ALL the PLANE becomes a RIGID body with single „s1” free parameter

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

Local relative shift calculation is in first order independent from „xfitted” calculation, because

| zi – zj | >> dz thus „m” is not a sensitive parameter

In case of 3-dim the actual „dz” will depend on the shift parameter,therefore one should solve the equation explicitely.

Track selection:

- Multiple-scattering is not critical if „dz” is small -> low momentum allowed

- Hadron tracks as good as muons

- Question of cluster size dependence, crossing angle studies

- Include vertex for RIGID body alignments