December 13,1993 CDF/TOP/ANA/2378 revised version "KINEMATICAL ANALYSIS OF b-TAGGED EVENTS WITH NON-GAUSSIAN ERRORS" (a la CDF1993-modified Phys.Rev.D47, 967 (1993).) S. Behrends/ Brandeis K. Sliwa / Tufts The b-tagged events were re-fitted with non-gaussian errors, according to "Response_Functions" which represent the standard approach taken by the QCD Group for summarizing the effects of jet energy degradation and resolution. A Response Function (RF) is defined as the probability distibution for finding a JETS Et value, at a given true jet Et; true jet Et is defined, as usual for the QCD Group, as the sum of the Et's of all final state particles from the fragmented parton that point within the clustering cone. The clustering cone size is 0.7. In addition, the RF at a given true Et is weighted by the JETCLU clustering efficiency (approx. 90/% at 10 GeV true Et). The Response_Functions can be used in conjunction with a quick MC, to simulate detector response to jets, or can be used to define RMS measurment errors -- not to be confused with the systematic errors assigned to jet Et -- on JETS Et. More details, and examples of Response Functions, can be found in CDF-1650 and references therein. The analysis of the b-tagged events was performed in two runs, "Gauss" and RESPONSE_FUNCTION". For each run the grid in the parameter space spanned by the jet errors +- 3 sigma (jet energies and errors according to Brian's list) was kept fixed in size, but the probability for every point in the grid was assigned either using a standard gaussian or Steve's "Response_Function". The difference between the mass values obtained with gaussian and non-gaussian error parametrizations are small, in general well within errors on the mass values obtained from the fits. The values of the likelihood for different combinations in a given events change as well, in some events leading to a different combination becoming the one with the best likelihood (for example in event 40758_44414). The combinations "chosen" by b-tagging are marked *. The values of the three dynamical factors used in my method are listed as well (Xb is the factor due to the structure functions; lep is the factor due to the angular distribution of a lepton from t decay; and Pt is the factor due to the transverse momentum of the ttbar system - a new parametrization (fit to HERWIG MC data) is used here.) The FORTRAN code of the RESPONSE_FUNCTION routines are enclosed after the table in which the results are compared. Also enclosed in the new parametrization of the transverse momentum distribution of the t-tbar pair. XMT is the transverse momentum of the t-tbar pair divided by the top quark mass, and it scales with top quark mass. assgn. Gauss Response_Function l wwj M /dM L M /dM L Xb lep Pt --------------------------------------------------------------------------- 40758_44414 4 123 128.0 0.50e-10 127.6 .10e-8 3.5 0.12 0.93 * 4 231 173.5/8.2 0.78e-2 173.0/6.8 .12e-1 1.1 0.34 1.0 4 132 178.0 0.10e-1 177.6 .20e-1 1.0 0.35 1.1 3 124 167.5 0.72e-12 167.4 .32e-10 4.8 0.07 0.59 3 241 181.5 0.59e-7 176.6 .72e-7 3.2 0.19 0.12 3 142 184.5 0.20e-6 180.0 .31e-6 3.5 0.19 0.22 2 134 169.7 0.26e-4 167.6 .28e-4 1.7 0.32 0.20 2 341 174.7 0.56e-4 173.6 .74e-4 1.4 0.36 0.14 --------------------------------------------------------------------------- 43096_47223 * 1 342 167.8/7.5 0.30e-4 166.0/6.4 .26e-4 0.4 0.39 0.89 --------------------------------------------------------------------------- 43351_266423* 3 124 146.9/6.3 0.39e-2 147.9/7.2 .72e-2 1.3 0.33 0.56 --------------------------------------------------------------------------- 43421_65648 4 123 127.7 0.66e-3 127.8 .21e-2 20. 0.37 1.1 4 231 132.9 0.15e-5 127.6 .72e-2 20. 0.32 1.2 3 124 114.3 0.55e-4 114.4 .20e-3 21. 0.15 0.41 3 142 144.9 0.25 145.4 .41 9.2 0.38 1.0 2 341 128.7 0.17e-1 128.6 .26e-1 13. 0.43 0.87 1 234 108.8 0.75e-4 113.0 .14e-2 18. 0.16 1.1 * 1 342 142.0 0.38e-1 144.0 .59e-1 6.3 0.40 0.99 --------------------------------------------------------------------------- 45070_262116 4 123 103.8 0.73e-3 103.5 .15e-2 33. 0.38 0.55 * 4 231 143.0 0.11 142.2 .30 8.4 0.44 0.95 3 124 105.0 0.49e-3 105.3 .10e-2 28. 0.45 0.56 3 142 136.1 0.38e-2 139.2 .89e-2 9.5 0.44 0.68 --------------------------------------------------------------------------- 45610_139604 3 142 177.9 0.11e-1 180.1 .62e-2 3.8 0.43 0.78 2 134 125.5 0.19e-9 125.6 .12e-8 12. 0.04 0.24 * 2 341 187.3 0.69e-15 189.7 .51e-6 2.3 0.43 0.82 2 143 155.6 0.70e-1 156.8 .22 4.8 0.27 0.94 --------------------------------------------------------------------------- 45705_54765 4 123 156.0 0.54e-2 156.2 .10e-1 1.5 0.41 0.75 4 231 139.8 0.26e-4 141.0 .52e-3 2.5 0.34 0.63 4 132 141.7 0.11e-2 141.9 .15e-2 1.7 0.35 0.64 3 124 154.3 0.14e-2 151.6 .21e-2 2.2 0.41 0.59 3 241 132.1 0.10e-6 121.5 .52e-7 6.0 0.21 0.21 2 134 162.7 0.13e-4 157.9 .70e-5 4.1 0.32 0.28 * 1 234 169.0 0.19e-3 170.2 .37e-2 2.5 0.40 0.95 --------------------------------------------------------------------------- 45753_79414 4 123 139.0 0.14e-4 140.0 .61e-4 7.1 0.41 0.29 * 4 231 135.8 0.55e-5 136.1 .17e-4 8.9 0.39 0.21 3 142 185.2 0.38e-32 186.2 .11e-3 4.9 0.28 1.5 2 341 no solution 170.3 .67e-15 0.6 0.42 0.71 2 143 138.6 0.60e-33 139.3 .17e-4 5.3 0.34 0.86 --------------------------------------------------------------------------- 45879_123158 4 123 132.9 0.79e-13 132.7 .41e-7 15. 0.23 0.92 4 231 169.0 0.19e-17 160.1 .12e-6 7.0 0.40 0.40 2 134 158.4 0.10e-7 157.8 .18e-6 4.4 0.02 5.5 * 2 341 165.3 0.23e-4 164.3 .34e-4 2.8 0.11 0.57 1 342 179.9 0.34e-2 181.0 .48e-2 1.6 0.41 1.0 1 243 182.8 0.25e-2 181.6 .56e-2 0.9 0.40 1.1 --------------------------------------------------------------------------- 45880_31838 4 231 112.6 0.34e-1 112.2 .26e-6 21. 0.49 0.81 4 132 158.0 0.61e-8 159.1 .14e-7 5.1 0.37 0.10 3 124 146.0 0.15e-1 146.3 .24e-1 4.3 0.42 0.83 * 3 241 124.2 0.62e-3 124.2 .66e-3 5.9 0.47 0.74 3 142 162.6 0.27e-3 163.1 .22e-2 4.4 0.37 1.0 2 134 131.7 0.10e-5 132.9 .48e-5 9.2 0.42 0.08 2 341 144.8 0.43e-3 147.1 .12e-2 6.8 0.44 0.35 2 143 165.2 0.11e-3 163.8 .81e-3 5.3 0.43 0.73 --------------------------------------------------------------------------- Steve Behrend's RESPONSE_FUNCTION.FOR --------------------------------------------------------------------------- Real Function RESPONSE_FUNCTION(ETRUE,EMEAS) Implicit None Real ROOT2PI, EDEGRADE, SLP1, SLP2, SIG Real XNORM_LOG, PROB_LOG Real ETRUE, EMEAS REAL SIGMA, DEGRADE, SLOPE, FZERO External SIGMA, DEGRADE, DFREQ, SLOPE, FZERO Double Precision DFREQ, CERN, ARG Data ROOT2PI /2.506628/ C EDEGRADE = DEGRADE(ETRUE) C C>>>Find upward (SLP1) and downward (SPL2) slopes: SLP1 = SLOPE(ETRUE,1) SLP2 = SLOPE(ETRUE,2) SIG = SIGMA(ETRUE) C C>>>Evaluate first smeared exponential (upwards): IF (SLP1 .EQ. 0) GOTO 200 XNORM_LOG =(0.5*(SIG/SLP1)**2.)-.5-(EMEAS-EDEGRADE+SLP2/2)/SLP1 ARG = (EDEGRADE - EMEAS - (SLP1 + SLP2)/2)/SIG + (SIG/SLP1) CERN = DFREQ(ARG) If (SLP1 .Gt. 0.0) CERN = 1.0D+00 - CERN If (CERN .Le. 1.0D-37) Then 200 SIG = SQRT(SIG**2 + SLP1**2) C ARG = (EDEGRADE - EMEAS - SLP1 - SLP2)/SIG ARG = (EDEGRADE - EMEAS)/SIG ARG = - (ARG**2.)/2. RESPONSE_FUNCTION = 1.0D+00/(SIG*ROOT2PI) * DEXP(ARG) Else PROB_LOG = XNORM_LOG + DLOG(CERN) RESPONSE_FUNCTION = (1./ABS(SLP1)) * EXP(PROB_LOG) EndIf C C>>>Evaluate second smeared exponential (downwards): IF (SLP2 .EQ. 0) GOTO 201 XNORM_LOG =(0.5*(SIG/SLP2)**2.)-.5-(EMEAS-EDEGRADE+SLP1/2)/SLP2 ARG = (EDEGRADE - EMEAS - (SLP1 + SLP2)/2)/SIG + (SIG/SLP2) CERN = DFREQ(ARG) If (SLP2 .Gt. 0.0) CERN = 1.0D+00 - CERN If (CERN .Le. 1.0D-37) Then 201 SIG = SQRT(SIG**2 + SLP2**2) C ARG = (EDEGRADE - EMEAS - SLP1 - SLP2)/SIG ARG = (EDEGRADE - EMEAS)/SIG ARG = - (ARG**2.)/2. RESPONSE_FUNCTION = (RESPONSE_FUNCTION & + 1.0D+00/(SIG*ROOT2PI) * DEXP(ARG))/2 Else PROB_LOG = XNORM_LOG + DLOG(CERN) RESPONSE_FUNCTION = (RESPONSE_FUNCTION & + (1./ABS(SLP2)) * EXP(PROB_LOG))/2 EndIf RESPONSE_FUNCTION = ABS(1 - FZERO(ETRUE)) * RESPONSE_FUNCTION Return End C C / / / / / / / / / / / / / / / / / / / / C Real Function DEGRADE(ETRUE) Implicit None Real ECM, ETRUE Real A, B, C, D, E, F, a2, OFFSET PARAMETER (A = 0.471) PARAMETER (B = 0.788) PARAMETER (C = 9.831E-04) PARAMETER (D = -3.573E-06) PARAMETER (E = 4.103E-09) PARAMETER (A2 = -0.63) c COMMON/RESPONSE/ECM DATA ECM/1800./ c OFFSET = 0 IF (ABS(ECM-546.) .LT. 10.) OFFSET = A2 If (ETRUE .Ge. 14.5) Then !was 10 8/29/91 DEGRADE = A + B*(ETRUE) + & C*(ETRUE**2) + D*(ETRUE**3) + E*(ETRUE**4) + OFFSET Else DEGRADE = 2.033 + 0.597*ETRUE + 0.00669*(ETRUE**2) + OFFSET EndIf c Return End C C / / / / / / / / / / / / / / / / / / / / C Real Function SLOPE(ETRUE,ISLOPE) COMMON/RESPONSE/ECM DATA ECM/1800./ C SLOPE = 0 IF (ISLOPE .EQ. 1) THEN SLOPE = 2.164 + 0.01576*ETRUE + (-1.611E-05)*ETRUE**2 IF (ABS(ECM-546.) .LT. 10.) THEN !ECM = 546 SLOPE = SLOPE - 0.48*1.2 !Effect of 546 UE vs. 1800 END IF ELSE IF (ISLOPE .EQ. 2) THEN SLOPE = 0.605 - 0.05046*ETRUE - ( 6.378E-05)*ETRUE**2 IF (SLOPE .GT. -0.2) SLOPE = -0.200 !Low Et has small downward tail END IF RETURN END C C / / / / / / / / / / / / / / / / / / / / C Real Function SIGMA(ETRUE) Implicit None Real ETRUE, EDUM C EDUM = ETRUE If (ETRUE .Lt. 4.) EDUM = 4. SIGMA = 0.320*SQRT(EDUM) + (0.0264)*EDUM + 1.039 - (3.934/EDUM) C Return End C C / / / / / / / / / / / / / / / / / / / / C Real Function FZERO(ETRUE) Implicit None REAL ETRUE INTEGER NEFF DATA NEFF/14/ REAL ET_EFF(14),EFF_JETS(14), EFF_MAX DATA ET_EFF/ & 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, & 12.5, 15.5, 20.5, 26.5/ DATA EFF_JETS/ & 0.079,0.222,0.403,0.548,0.670,0.747,0.796,0.850,0.879,0.895, & 0.934,0.953,0.974,0.981/ LOGICAL LFIRST DATA LFIRST/.TRUE./ REAL XINTERP2 External XINTERP2 C IF (LFIRST) THEN LFIRST = .FALSE. EFF_MAX = XINTERP2(NEFF,ET_EFF,EFF_JETS,ET_EFF(NEFF)) !Max JETS eff END IF FZERO = XINTERP2(NEFF,ET_EFF,EFF_JETS,ETRUE)/EFF_MAX FZERO = 1 - FZERO IF (FZERO .LT. 0) FZERO = 0. C RETURN END C C / / / / / / / / / / / / / / / / / / / / C FUNCTION XINTERP2(NELEMENTS,XAXIS,YAXIS,X) REAL XINTERP2, XAXIS(*),YAXIS(*) IF (X .LT. XAXIS(1)) THEN XINTERP2 = YAXIS(1) RETURN END IF DO IBIN = 1, NELEMENTS-1 IF((X .GE. XAXIS(IBIN)) .AND. (X .LT. XAXIS(IBIN+1))) THEN XINTERP2 = YAXIS(IBIN) & + ((X-XAXIS(IBIN))/(XAXIS(IBIN+1)-XAXIS(IBIN)) ) & * (YAXIS(IBIN+1) - YAXIS(IBIN)) RETURN ENDIF END DO XINTERP2 = YAXIS(NELEMENTS) RETURN END --------------------------------------------------------------------------- simple Gaussian probability routine used in "Gauss" studies. --------------------------------------------------------------------------- subroutine Gauss(en,ej,sigma,pgauss) c Normalized Gaussian function real en,ej,sigma,pi,pgauss data pi /3.141593/ c pgauss=(exp(-(((en-ej)/sigma)**2)/2))/(sigma*sqrt(2.0*pi)) return end --------------------------------------------------------------------------- xmt=pt_ttb/top_m c ISAJET ttbar pt distribution (Kuni Kondo's fit) c prob_ttb=128.95*xmt*exp(6.7*xmt**2-13.39*xmt) c K Sliwa's fit to HERWIG (and keep xmt<0.5) prob_ttb=611.*xmt*exp(21.*xmt**2-28.*xmt)