1. Z mass constraint with twists in twist

2. The CMS (Compact Muon Solenoid) by construction posseses axial symmetry which is preserved in case of small p T 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 mass constraint, because the TWIST  = c dist * 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 the resonance. 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 of three-momentum of both decay particles will have the same sign, producing the mass-shift (see Fig. 1).

3. x y ~  ideal distorted TWIST p dist B = p ideal -dpp dist A = p ideal -dp MASS corrected = p dist A + p dist B = p ideal –dp +p ideal –dp = 2 * p ideal - 2*dp = M ideal - 2*dp x z A B

4. 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 version of these distortions. In case of distortion (2) the momentum change of the muons has different sign with practically 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.

5. +  -  TWIST ANTI-TWIST - + + +

6. x z x y ~  distorted ANTI-TWIST ideal p dist B = p ideal -dp p dist A = p ideal +dp MASS corrected = p dist A + p dist B = p ideal +dp +p ideal –dp = 2 * p ideal = M ideal

7. 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.

8. 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 (  = 45 o ) 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: x 2 + (y-R) 2 = R 2 parabola: y = a*x 2, 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*z 2 In first approximation one gets also: y ideal =  =b*z 2.

9. After distortion: y dist =  =b*z 2 + c dist *z = b*( z+ c dist /(2b)) 2 - (c dist /2) 2 /b Formally this corresponds to the same parabola which is shifted in z-direction by  z = c dist /(2b) and by  y = (c dist /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 curvature assuming the parabolic form using 2 points: 1. point: origo (x o,y o ) = (0,0) 2. point: measured middle point (x c, y c ) = (k*z c, b*z c 2 + c dist *z c ) y c = b* z c 2 + c dist *z c = b fit * z c 2 which gives b fit = b + c dist /z c.

10. In this approximation the momentum after the distortion will be: p dist = p ideal * b/b fit = p ideal * b / ( b + c dist /Z c ) If one takes into account the charge q of the particle and the sign of z-coordinate: p dist = p ideal * b / ( b + SIGN * c dist /|Z c |) where SIGN twist = sign( charge) * sign (Z c ) SIGN anti-twist = sign( charge) corresponding to the CP symmetry of the twist applied.

11. p dist = p ideal * b/b fit ideal distorted fitted z y ~  IDEAL TRACK y ideal =  =b*z 2. DISTORTED TRACK: y dist =  =b*z 2 + c dist *z = b*( z+ c dist /(2b)) 2 - (c dist /2) 2 /b FITTED TRACK: y c = b* z c 2 + c dist *z c = b fit * z c 2 ( -c dist /(2b), -(c dist /2) 2 /b ) zczc Shifted parabola vertex

12. European Union law: Subsidiarity is an organizing principle that matters ought to be handled by the smallest, lowest or least centralized competent authority....

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

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

18. Piecewise Alignment by OVERLAPPING tracks Simplified ONE-dimensional case z 1 z 2 z 3 +/- dz+/-dz+/-dz Overlapping modules No magnetic field: (m,b) track fitted from the other planes: x = m *z + b xjxj x j+1

19. Assumption : distortion is occuring only in x-direction: j-th module is shifted by „s j ” x j meas + s j = x fitted = m (z 1 -dz) + b x j+1 meas + s j+1 = x fitted = m (z 1 +dz) + b By subtraction one gets a recursive formule : s j+1 = s j + x j meas – x j+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: ALL the PLANE becomes a RIGID body with single „s 1 ” free parameter

20. 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

21. Local relative shift calculation is in first order independent from „x fitted ” calculation, because | z i – z j | >> 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