1. Ososkov G. Wavelet analysis CBM Collaboration meeting Wavelet application for handling invariant mass spectra Gennady Ososkov LIT JINR, Dubna Semeon Lebedev GSI, Darmstadt and LIT JINR, Dubna 12th CBM Collaboration meeting Oct. 13-18 2008, JINR Dubna

2. Ososkov G. Wavelet analysis CBM Collaboration meeting Why do we need wavelets for handling invariant mass spectra? -we need them when S/B ratio is << 1 Smoothing after background subtraction without losing any essential information resonance indicating even in presence of background evaluating peak parameters

3. Ososkov G. Wavelet analysis CBM Collaboration meeting Estimating peak parameters in G 2 wavelet domain How it works: after background subtracting we have a noisy spectrum It is transformed by G 2 into wavelet domain, where we look for the wavelet surface maximum (b max a max) and then fit this surface by the analytical formula for W G2 (a,b;x 0,σ)g starting fit from x 0 =b max and Eventually, we should find the maximum of this fitted surface and use its coordinates as estimations of peak parameters From them we can obtain halfwidth, amplitude and even the integral peak has bell-shape form

4. Ososkov G. Wavelet analysis CBM Collaboration meeting Real problems 1. CBM spectra Λc invariant mass spectrum (by courtesy of Iou.Vassilev) and its G 2 spectrum more and more detailed Wavelet method results: A=15.0 σ =0.0116 mean=2.2840 I w =0.435 PDG m=2.285 I gauss =0.365 (19% less)

5. Ososkov G. Wavelet analysis CBM Collaboration meeting Real problems 2. CBM spectra Low-mass dileptons (muon channel) ω. Gauss fit of reco signal M=0.7785 σ =0.0125 A=1.8166 I g =0.0569 ω. Wavelets M=0.7700 σ =0.0143 A=1.8430 I w =0.0598 - ω– wavelet spectrum ω.ω. ω-meson φ-meson Even φ- and mesons have been visible in the wavelet space, so we could extract their parameters. Thanks to Anna Kiseleva

6. Ososkov G. Wavelet analysis CBM Collaboration meeting Real problems 3. Resonance structure Carbon p, D K.Abraamyan, V.Toneev et al., arXiv:0806/0806.2790 Resonance structure study in γγ invariant mass spectra in dC-Interactions A resonance structure in the invariant mass spectrum of two photons at Mγγ = 360 MeV was observed in the reaction dC → γ +γ+X at momentum P d =2.75 GeV/c per nucleon. Schematic view of PHOTON-2 setup. S1, S2 are scintillation counters. Dubna Nuclotron, Photon-2 setup π 0 decay from dC into γγ has bulky background. Trerefore models with channels η,ή, NN-bremsstrahlung, ω, Δ → Nγ, Σ →Λγ etc producing this background has been simulated, but the only generator which took into account the R resonance with small opening angle gave good correspondence with experimental data.

7. Ososkov G. Wavelet analysis CBM Collaboration meeting Gaussian wavelet application for resonance structure study Invariant mass distributions of γγ pairs satisfying the above criteria without (upper panel) and with (bottom panel) the background subtraction. Selection criteria to background subtraction: the number of photons in an event, Nγ =2; the energies of photons, Eγ ≥ 100 MeV; the summed energy ≤ 1.5 GeV

8. Ososkov G. Wavelet analysis CBM Collaboration meeting The invariant mass distribution of γγ pairs and the biparametric distribution of the GW of the 8-th order for dC interactions. The distribution is obtained with an additional condition for photon energies Eγ1/Eγ2 > 0.8 and binning in 2MeV. Therefore, the presented results of the continuous wavelet analysis with vanishing momenta confirm the finding of a peak at Mγγ ∼ (2 − 3)m π in the γγ invariant mass distribution obtained within the standard method with the subtraction of the background from mixing events. rresonant structure wasn’t observed in pC-interactions at 5.5 Gev/c

9. Ososkov G. Wavelet analysis CBM Collaboration meeting Continuous or discrete wavelets? Continuous wavelets are remarkably resistant to noise (robust), but because of their non-orthogonality one obtains non-admissible signal distortions after inverse transform. Besides, real signals to be analysed by computer are always discrete. So orthogonal discrete wavelets look preferable. The discrete wavelet transform (DWT) was built by Mallat as multi-resolution analysis. It consists in representing a given data as a signal decomposition into basis functions φ and ψ, which must be compact. Various types of discrete wavelets One of Daubechie’s wavelets Coiflet – most symmetric

10. Ososkov G. Wavelet analysis CBM Collaboration meeting DWT Peak Finder Denoising by DWT shrinking Wavelet shrinkage means, certain wavelet coefficients are reduced to zero, so after the inverse transform one can eliminate from the spectrum part of high and low frequencies. Our innovation is the adaptive shrinkage, i.e. λ k = 3σ k where k is decomposition level (k=scale 1,...,scale n ), σ k is RMS of W ψ for this level (recall: sample size is 2n) The discrete wavelets are a good tool for background eliminating and peak detecting. However the main problem of wavelet applications was the absence of corresponding C++ software in any of available frameworks. So we had to build it ourselves and Simeon accomplished that successfully. An example of Daub2 spectrum

11. Ososkov G. Wavelet analysis CBM Collaboration meeting CBM spectra. Result 3. Low-mass dileptons (muon channel) Its fragment after adaptive shrinking with specially chosen multipliers M in λ k = Mσ k for each level k of DWT Original inv. mass spectrum Result of adaptive shrinking with λ k = 3σ k inevitable Gibbs edge effect no backround! ω-meson φ-meson

12. Ososkov G. Wavelet analysis CBM Collaboration meeting 1.Wavelet approach looks quite applicable for handling CBM invariant mass spectra with small S/B ratio 2.Discrete wavelets could be considered as a new tool for detecting the peak existence. 3.Algorithms and programs have been developed for estimating resonance peak parameters on the basis of gaussian continuous wavelets 4.Algorithms and programs have been developed for resonance peak detecting on the basis of discrete wavelets 5.First attempts of the wavelet applications to CBM open charm and meson data are very promising Summary and outlook

13. Ososkov G. Wavelet analysis CBM Collaboration meeting What to do Commit, eventually, wavelet software into the SVN repository Tuning of running software in close contacts with physicists interested in peak finding business Furnish this software by algorithms for automatic peak search and their parameters estimation

14. Ososkov G. Wavelet analysis CBM Collaboration meeting Thank you for attention!

15. Ososkov G. Wavelet analysis CBM Collaboration meeting Recall to wavelet introduction One-dimensional wavelet transform (WT) of the signal f(x) has 2D form where the function  is the wavelet, b is a displacement (shift), and a is a scale. Condition C ψ < ∞ guarantees the existence of  and the wavelet inverse transform. Due to freedom in  choice, many different wavelets were invented. The family of continuous wavelets with vanishing momenta is presented here by Gaussian wavelets, which are generated by derivatives of Gaussian function Two of them, we use, are and Most known wavelet G 2 is named “the Mexican hat”

16. Ososkov G. Wavelet analysis CBM Collaboration meeting Recall to wavelet introduction (cont) Applicatios for extracting special features of mixed and contaminated signal G 2 wavelet spectrum of this signal Filtering results. Noise is removed and high frequency part perfectly localized An example of the signal with a localized high frequency part and considerable contamination then wavelet filtering is applied Filtering works in the wavelet domain by thresholding of scales, to be eliminated or extracted, and then by making the inverse transform

17. Ososkov G. Wavelet analysis CBM Collaboration meeting Peak parameter estimating by gaussian wavelets When a signal is bell-shaped one, it can be approximated by a gaussian Thus, we can work directly in the wavelet domain instead of time/space domain and use this analytical formula for W G2 (a,b;x 0,σ)g surface in order to fit it to the surface, obtained for a real invariant mass spectrum. The most remarkable point is: since the fitting parameters x 0 and σ, can be estimated directly in the G 2 domain, we do not need the inverse transform! Then it can be derived analytically that its wavelet transformation looks as the corresponding wavelet. For instance, for G 2 (x) one has Considering W G2 as a function of the dilation b we obtain its maximum and then solving the equation we obtain.