Absolute photon energy calibration using π 0 s

Transcription

1 Absolute photon energy calibration using π s A.D. Bukin BINP, Novosibirsk June 1, 25 BaBar Note #571 BaBar Note #582 analytically First idea of deconvolution algorithm Idea to use special fitting functions which can be convoluted BaBar Note #588 This work. Attempt to estimate the systematic limits for deconvolution algorithm

2 Choice of fitting curve for asymmetric peaks Good fitting curve should have the following properties: Allow asymmetry of peak over wide range from Gaussian distribution to almost triangle Continuity and smoothness in the peak region in order to provide low bias in peak position determination Provide reasonable χ 2 for most distributions over π mass and photon energy It would be convenient to have peak position among the fit parameters of the curve A.D. Bukin EMC calib. meeting June 1, 25 2

3 Convolution of Gaussian with exponential distribution Convolution of Gaussian G(x; x g, σ g = (x xg 2 1 2σ e g 2 2πσg and highly asymmetric distribution 1 λ F exp (x; λ = exp ( x λ, x λ, x, λ >, can be written analytically F g1λ (η; x p, σ g, λ = + = 1 2 λ exp [ σ 2 g 2λ 2 F exp (ω; λg(η ω; x g, σ g dω = ] [ ( (η xg λ 1 erf + 2σg λ + η x g λ ] σ g 2 λ A.D. Bukin EMC calib. meeting June 1, 25 3

4 Convolution of Gaussian with exponential distribution F g2λ (η η λ = 5, 1,, 1, 5 x g = ; σ g = 1; A.D. Bukin EMC calib. meeting June 1, 25 4

5 Fitting photon energy distribution For high statistics the most adequate the following fitting function was admitted: F 3g1λ (η; A, ψ 1, ψ 2, x p, σ g1, σ g2, σ g3, λ 1, λ 2, λ 3 = = A {ψ 1 F g1λ (η; x p, σ g1, λ (1 ψ 1 ψ 2 F g1λ (η; x p, σ g2, λ (1 ψ 1 (1 ψ 2 F g1λ (η; x p, σ g3, λ 3 } where 9 parameters ψ 1, ψ 2, x p, σ g1, σ g2, σ g3, λ 1, λ 2, λ 3 are form parameters, and A is a normalization factor. A.D. Bukin EMC calib. meeting June 1, 25 5

6 Fitting γγ inv. mass distribution ( z = ln m γγ m π 1 2 ln E γ1 E (true γ1 + ln E γ2 E (true γ2 = 1 2 (η 1 + η 2. For symmetric decays we have the convolution of identical unknown photon energy distributions. Supposing that they are of the type F 3g1λ, we can find the convolution analytically and get fitting function for z-distribution: F ms (z; A, ψ 1, ψ 2, x p, σ g1, σ g2, σ g3, λ 1, λ 2, λ 3 where 9 parameters ψ 1, ψ 2, x p, σ g1, σ g2, σ g3, λ 1, λ 2, λ 3 are actually the parameters of unknown energy distribution, and A is a normalization factor. A.D. Bukin EMC calib. meeting June 1, 25 6

7 Event samples Number of photons nphotons Entries e Mean RMS Underflow Overflow Integral 7.725e e+7 Number of photons nphotons Entries e Mean RMS Underflow Overflow Integral 7.725e e Number of photons Number of photons Monte Carlo: generic B B events Data: Run-1 (all events A.D. Bukin EMC calib. meeting June 1, 25 7

8 Photon energy distribution PhotE, GeV h2 Entries e e+7 Mean RMS PhotE, GeV 7 1 h2 Entries e e+7 Mean RMS Underflow Overflow 2.565e e+4 Integral 1.238e e Underflow Overflow 2.565e e+4 Integral 1.238e e E γ, GeV E γ, GeV N γ 1, red dashed line for Monte Carlo A.D. Bukin EMC calib. meeting June 1, 25 8

9 Angular distributions of the photons with Eγ (48, 52 MeV h2g Phi vs Theta h2g Phi vs Theta Entries Mean x 1.13 Mean y.4868 Mean x Mean y.743 RMS x RMS y RMS x.5942 Entries Integral 1.35e RMS y Integral 7.963e P3h i P3h i e2.4 2 T h ta Monte Carlo A.D. Bukin EMC calib. meeting e2.4 2 T h ta Data June 1, 25 9

10 E γ (8, 12 MeV, N γ 1, MC, true angles 3 1 /m π mixed events 3 1 /m π,true angles 18 lnm2mtrubg Entries e+7 1 lnm2mtruang Entries e+7 16 Mean Mean RMS.3496 Underflow 1.773e+6 8 RMS.3447 Underflow 1.31e Overflow 5.28e+4 Integral 1.866e+7 6 Overflow 9.888e+4 Integral 1.22e /m π mixed events / m π, true angles χ 2 /n D = 588.4/369, x p = A.D. Bukin EMC calib. meeting June 1, 25 1

11 E γ (8, 12 MeV, N γ 1, MC, true angles (check the solution lneg2egtru Entries e+7 Mean RMS.1217 Underflow 3.194e+6 Overflow 1.994e+5 Integral 8.119e / E true lneg2egtru Entries e+7 Mean RMS.1217 Underflow 3.194e+6 Overflow 1.994e+5 Integral 8.119e / E true χ 2 /n D = /199 x p = x p = χ 2 /n D = /19 x p =.1385 ±.54 A.D. Bukin EMC calib. meeting June 1, 25 11

12 E γ (8, 12 MeV, N γ 1, MC, true angles /m π mixed events 12 lnm2mtrubg Entries Mean RMS.3494 Underflow 1.156e+5 8 Overflow 3354 Integral 1.216e /m π,true angles 7 lnm2mtruang 6 Entries Mean RMS.3399 Underflow 6.82e+4 4 Overflow 7474 Integral 5.712e /m π mixed events / m π, true angles χ 2 /n D = 338.9/369, x p =.1537 ±.11 Distribution vs E γ γ : ɛ p = 1.13, FWHM =.117 A.D. Bukin EMC calib. meeting June 1, 25 12

13 E γ (8, 12 MeV, N γ 1, MC, true angles (check the solution 25 lneg2egtru Entries Mean RMS.1214 Underflow 4.879e+5 15 Overflow 2.554e+4 Integral 9.617e lneg2egtru Entries Mean RMS.1214 Underflow 4.879e+5 15 Overflow 2.554e+4 Integral 9.617e / E true / E true χ 2 /n D = 56916/199 χ 2 /n D = 34.8/19 x p =.1537 ±.11 x p =.128 ±.16 x p =.26 ±.11 A.D. Bukin EMC calib. meeting June 1, 25 13

14 E γ (8, 12 MeV, N γ 1, MC, measured angles /m π mixed events /m π 12 lnm2mtrubg Entries Mean RMS.3494 Underflow 1.156e+5 8 Overflow 3354 Integral 1.216e lnm2mtru Entries Mean RMS.3387 Underflow 5.68e+4 Overflow 152 Integral 5.812e /m π mixed events / m π χ 2 /n D = 37.3/369, x p = Distribution vs E γ γ : ɛ p = 1.11, FWHM =.127 A.D. Bukin EMC calib. meeting June 1, 25 14

15 E γ (8, 12 MeV, N γ 1, MC, measured angles (check the solution 25 lneg2egtru Entries lneg2egtru Entries Mean RMS Mean RMS.1214 Underflow 4.879e+5 Underflow 4.879e+5 15 Overflow 2.554e+4 Integral 9.617e+5 15 Overflow 2.554e+4 Integral 9.617e / E true / E true χ 2 /n D = 2329/199 χ 2 /n D = 146/19 x p = x p = x p =.123 ±.15 A.D. Bukin EMC calib. meeting June 1, 25 15

16 E γ (8, 12 MeV, N γ 1, Data /m π mixed events lnm2mtrubg Entries Mean RMS.3492 Underflow 2.68e+4 Overflow 593 Integral 2.116e /m π mixed events /m π lnm2mtru Entries Mean RMS.3423 Underflow 2.457e+4 Overflow 477 Integral 1.972e / m π χ 2 /n D = 385.9/369, x p = Distribution vs E γ γ : ɛ p = 1.3, FWHM =.155 A.D. Bukin EMC calib. meeting June 1, 25 16

17 E γ (12, 16 MeV, N γ 1, MC, true angles /m π mixed events 6 lnm2mtrubg 5 Entries Mean RMS.3649 Underflow 3.79e+4 Overflow 1.868e+5 3 Integral 4.49e+5 2 /m π,true angles 25 lnm2mtruang Entries Mean RMS.352 Underflow 1.652e+4 15 Overflow 8.395e+4 Integral 2.159e /m π mixed events / m π, true angles χ 2 /n D = 324.6/369, x p =.1399 ±.36 Distribution vs E γ γ : ɛ p = 1.13, FWHM =.95 A.D. Bukin EMC calib. meeting June 1, 25 17

18 E γ (12, 16 MeV, N γ 1, MC, true angles, Eg in (12,16 (check the solution, Eg in (12, lneg2egtru Entries Mean RMS.1229 Underflow 2.711e+5 Overflow 2.152e+4 Integral 7.795e / E true lneg2egtru Entries Mean RMS.1229 Underflow 2.711e+5 Overflow 2.152e+4 Integral 7.795e / E true χ 2 /n D = 1151/199 χ 2 /n D = 271.5/19 x p =.1399 ±.36 x p =.138 ±.14 x p =.9 ±.4 A.D. Bukin EMC calib. meeting June 1, 25 18

19 E γ (12, 16 MeV, N γ 1, MC, measured angles /m π mixed events 6 lnm2mtrubg 5 Entries Mean.4673 RMS Underflow 3.79e+4 Overflow 1.868e+5 3 Integral 4.49e+5 /m π 25 lnm2mtru Entries Mean.32 2 RMS.3539 Underflow 1.575e+4 15 Overflow 8.217e+4 Integral 2.184e /m π mixed events / m π χ 2 /n D = 311.6/369, x p =.1335 ±.14 Distribution vs E γ γ : ɛ p = 1.11, FWHM =.13 A.D. Bukin EMC calib. meeting June 1, 25 19

20 E γ (12, 16 MeV, N γ 1, MC, measured angles (check the solution, Eg in (12, lneg2egtru Entries Mean RMS.1229 Underflow 2.711e+5 Overflow 2.152e+4 Integral 7.795e / E true, Eg in (12, lneg2egtru Entries Mean RMS.1229 Underflow 2.711e+5 Overflow 2.152e+4 Integral 7.795e / E true χ 2 /n D = 37567/199 χ 2 /n D = 3652/19 x p =.133 ±.14 x p =.1248 ±.13 x p =.8 ±.14 A.D. Bukin EMC calib. meeting June 1, 25 2

21 E γ (12, 16 MeV, N γ 1, Data /m π mixed events /m π 12 lnm2mtrubg Entries lnm2mtru Entries Mean.4778 RMS.3641 Underflow 6787 Overflow 4.35e+4 Integral 9.638e Mean.1221 RMS.3627 Underflow 1.4e+4 Overflow 3.679e+4 Integral 9.66e /m π mixed events / m π χ 2 /n D = 391.4/369, x p = Distribution vs E γ γ : ɛ p = 1.11, FWHM =.145 A.D. Bukin EMC calib. meeting June 1, 25 21

22 E γ (2, 24 MeV, N γ 1, MC, true angles /m π mixed events /m π,true angles lnm2mtrubg Entries Mean.5343 RMS.3656 Underflow 4584 Overflow 1.525e+5 Integral 6.929e+4 9 lnm2mtruang 8 Entries Mean RMS Underflow Overflow 6.652e+4 Integral 3.994e /m π mixed events / m π, true angles χ 2 /n D = 377.4/369, x p =.92 ±.13 Distribution vs E γ γ : ɛ p = 1.8, FWHM =.84 A.D. Bukin EMC calib. meeting June 1, 25 22

23 E γ (2, 24 MeV, N γ 1, MC, true angles (check the solution, Eg in (2,24, Eg in (2,24 18 lneg2egtru 18 lneg2egtru 16 Entries Entries Mean RMS.155 Underflow 1.255e Mean RMS.155 Underflow 1.255e+5 1 Overflow 1.321e+4 1 Overflow 1.321e+4 8 Integral 4.969e+5 8 Integral 4.969e / E true / E true χ 2 /n D = 7391/199 χ 2 /n D = 594.5/19 x p =.92 ±.13 x p =.1292 ±.14 x p =.37 ±.13 A.D. Bukin EMC calib. meeting June 1, 25 23

24 E γ (2, 24 MeV, N γ 1, MC, measured angles /m π mixed events 1 lnm2mtrubg Entries Mean.5343 RMS.3656 Underflow 4584 Overflow 1.525e+5 Integral 6.929e /m π mixed events /m π lnm2mtru Entries Mean.2379 RMS.3315 Underflow 2475 Overflow 6.621e+4 Integral 4.5e / m π χ 2 /n D = 361.3/369, x p =.7 ±.14 Distribution vs E γ γ : ɛ p = 1.5, FWHM =.12 A.D. Bukin EMC calib. meeting June 1, 25 24

25 E γ (2, 24 MeV, N γ 1, MC, measured angles, Eg in (2,24 (check the solution, Eg in (2, lneg2egtru Entries Mean RMS lneg2egtru Entries Mean RMS Underflow 1.255e+5 Overflow 1.321e+4 Integral 4.969e Underflow 1.255e+5 Overflow 1.321e+4 Integral 4.969e / E true / E true χ 2 /n D = 44814/199 χ 2 /n D = 769/19 x p =.7 ±.14 x p =.1259 ±.13 x p =.56 ±.14 A.D. Bukin EMC calib. meeting June 1, 25 25

26 E γ (2, 24 MeV, N γ 1, Data /m π mixed events /m π lnm2mtrubg Entries Mean.5792 RMS.3649 Underflow 2342 Overflow 8.82e+4 Integral 3.657e lnm2mtru Entries 155 Mean RMS.3872 Underflow 1.56e+4 Overflow 9.521e+4 Integral 4.419e /m π mixed events / m π χ 2 /n D = 4.7/369, x p = Distribution vs E γ γ : ɛ p = 1.15, FWHM =.93 A.D. Bukin EMC calib. meeting June 1, 25 26

27 Conclusion A function F 3g1λ (x; A, ψ 1, ψ 2, x p, σ g1 σ g2, λ proved to be a good asymmetric fitting function for the photon energy distribution. For several energy bins deconvolution algorithm of absolute energy calibration was tested. This procedure for Monte Carlo events provides the estimates of systematic errors due to ignorance of angular fluctuations. Problems: Inconsistency of combinatorial background and estimate of background formed by photons of neighbouring events. High level of correlation of the parameters of fitting function makes the minimization of χ 2 a difficult task for MINUIT. A.D. Bukin EMC calib. meeting June 1, 25 27