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Wavelet Analysis of E852 Experimental Data
V. L. Korotkikh  and L. I. Sarycheva 
 Scobeltsyn Institute of Nuclear Physics, Moscow State University, Moscou 119899, Russia
Abstract. A calculation of background in the hp 0 mass spectrum by the wavelet analysis is
presented. The advantages and shortcomings of wavelet method are discussed.
INTRODUCTION
A wavelet analysis [1, 2] became a necessary mathematical tool in many investigations
of the nonstationary (in time) or in homogeneous (in space) signals (see [3] and many
references in it). The wavelets can distinct the local characteristics of a signal by chang­
ing a scale (dilations) and by translations over the whole region in which it is studied.
Due to the completeness of the system, they also allow for the niverse transformation to
be done. So, there is a possibility to separate a signal (as some local ingomogeneity) and
a smooth background.
A large final state statistics of E852 experiment [4, 5, 6] in the p p interactions at
18 GeV/c allows to use the wavelet analysis both for one­dimensional and for two­
dimensional distributions. For one­dimensional mass distribution it can help to estimate
the background under a resonant peak of signal. It is important because the partial wave
analysis (PWA) fits the sum of waves and background to the data. If we calculate the
mass dependence of incorrectly then it can be smeared over some waves and give false
effects. It is specially important in a case of the rare exotic effects which are comparable
with the background in a mass dependence of hp 0 system [? ] produced in the p p
interaction as an example. One should stress that the wavelet analysis does not pretend
ot explain the underlying dynamics and physics nature.
BASIC FORMULAE
The discrete wavelets were used. The scaling (j) and oscillation (y) functions are
defined as
j(x) =
p
2
2M 1
Å
k=0
h k j(2x k); y(x) =
p
2
2M 1
Å
k=0
g k j(2x k); (1)
where the coefficients h k and g k are related by
g k = ( 1) k h 2M k 1 : (2)

The integer M defines the number of coefficients of finite support. The dilated and
translated versions of scaling and oscillation function are
j j;k (x) = 2 j=2 j(2 j x k); y j;k (x) = 2 j=2 y(2 j x k): (3)
The interger j sets j­th resolution level and determines the size of cells over x­
dependence. The index k is a translation parameter.
Any function f (x) can be decomposed by the wavelet functions
f (x) =
2 jn 1
Å
k=0
s j;k j j;k (x) +
j max
Å
j= j n
2 j 1
Å
k=0
d j;k y j;k (x); (4)
where j n and j max are a minimum and maximum resolution level to be considered. The
coefficients s j;k and d j;k carry information about the content of function f (x) at various
scales.
We use a fast wavelet algorithm for direct and reverse transformation. The calculation
of s j;k and d j;k is fulfiled by
s j 0 +1;k = Å m
hm s j 0 ;2k+m ; d j 0 +1;k = Å m
gm s j 0 ;2k+m ; (5)
where j 0 = j max j is a reverse resolution index. For start of calculation at j 0 = 0 we
take
s 0;k = f (x k ); k = 1;2; : : : ; k max (6)
at maximum resolution level, k max = 2 j max (at minimum size of cell). All results below
are obtained with the orthohonal D 8 Daubechies [1] wavelets (M = 4) and j max = 5. The
coefficient values h k (k = 0;1; : : :; 7) are taken from the work [3].
WAVELET ANALYSIS OF BACKGROUND
Let's begin to study the background in a simple example:
f (x) = f BW (x) + f BG (x); (7)
where f BW (x) is a Breit­Wigner function (our signal) and f BG (x) is a polinomial of
second order. We take the parameters of function which are close to real experimental
data (sii below). Then we do the wavelet analysis (5) and the reverse transformation
(synthesis). The coefficients d j;k give a contribution of oscillations (fluctuations) at
various seales. The synthesis with all d j;k gives exact values of function f (x). Let's
use the cut of d j;k :
d 0
j;k = d 0;0 ; if j = 0; k = 0; (8)
After calculation of new set of d 0
j;k by the fast algotithm of synthesis with this cut we get
a smooth function WL(x).
Let's do the wavelet analysis of functions f BW (x) and get d (BW )
0;0 . Then we define a
background BCGR(x), calculated by the wavelet analysis, as
BCGR(X) = aWL(x)+b; (9)

FIGURE 1. Calculation of background by help of wavelet analysis. a) Setted distribution: sum of Breit­
Wigner and background (solid line), only background (dotted line). b) Background from wave analysis
(solid line), setted background (dotted line).
where a and b are free parameters which are found by fit sum ( f BW (x) +BCGR(x)) to
the function (7). The fig. 1b demonstrates the copmarison the real background f BG (x)
and the calculated BCGR(x). The correspondence is not exact but satisfactory. The value
of parameter a = 0:3 is close to the ratio d (BG)
0;0 =d 0;0 = d (BG)
0;0 =(d
(BG)
0;0 + d (BW )
0;0 ), which is
equal to 0.24. So, the sense of parameter a is the part of wavelet oscillations in the
background at minimum resolution level.
Now consider the real experimental data. We have about 20000 events in the reaction
p p ! hp 0 n, h ! p + p p 0 at 18 GeV/c (E852 experiment) [6]. Our aim is the estima­
tion of background in the mass spectrum of p 0 p 0 p + p ­meson system. One of the way
is a method of "side bands". Two bands near a peak of h­meson at m = 0:549 GeV in
mass spectrum of p 0 p + p ­system are selected (fig. 2). The sum of their widths is equal
to the central width of h. Then two distributions over mass of four mesons (p 0 p 0 p + p )

FIGURE 2. Mass distribution of p 0 p + p ­system in the reaction p p ! p + p p 0 p 0 n at 18 GeV/c.
Central band is a region of h­meson. Left and right bands are used for event selection which corresponds
to the background in mass spectrum of hp 0 ­system, when h decays as h ! p 0 p + p . Mass m(p 0 p + p ) is
given at GeV/c 2 .
are built (fig. 3). One is the distribution with events, in which mass of p + p p 0 mesons is
in the eta­mass region (central band in fig. 2). Other distribution, called the background,
corresponds to the events in which mass m(p 0 p + p ) is in the side bands regions in fig.
2.
Let's estimate the background in mass spectrum of (p 0 p 0 p + p ) system by the wavelet
analysis in such way as in the example above. We use the cut (8) of d j;k and calculate
WL(x) for the experimental data in fig. 3. Then we fit the sum of background (9) and two
Breit­Wigner functions to the data and find the parameters a and b (a = 0:21, b = 135).
The curve of background, calculated by wavelet analysis , is similar to the back­
ground from side band method, but it is more smooth (fig. 3). The wavelet analysis of
background allows to remove a peak mear mass m = 1:32 GeV/c which can be false
events (perhaps, from decay a 2 ! p 0 p + p )in the estimation of background by side

FIGURE 3. Comparison of two background estimations. Line is the background from wavelet analysis.
Histogram is the background from side bands method. Crosses are the data of E852 experiment on hp 0 ­
system.
bands method. But we have to emphasize that the wavelet analysis doesn't give an abso­
lute value of background.
CONCLUSIONS
A simple one­dimensional experimental distribution was considered. There are no sharp
peaks or many peaks with various widths (strong oscillations). The region of function
argument is short. The analysis with discrete wavelets restores rather good the smooth
background (Fig.3). There is no necessity to have any additional information besides the
distribution in itself. Of course, we can describe the same background by the polynomial
as in our artificial example (fig.1).
But we demonstrate that the wavelet analysis gives a good result. It will be certainly a

value analysis in more complex cases with various peak widths and in the large function
argument. We suppose that the wavelet analysis will be useful when the background
cannot be describe by polynomial.
Let's do some remarks after our experience of wavelet application. It is interesting to
continue the present study using other wavelets and more refined cuts of the coefficients
d j;k . There is a problem how the wavelet analysis can describe the interference effects
between Breit­Wigner signal and background. It is useful also to do a combined analysis
of the wavelet decomposition and the fit the resonance parameters of signal. There is also
a problem of taking into account the experimental errors in the frame of wavelet analysis.
ACKNOWLEDGMENTS
Authors thank I. M. Dremin, O. V. Ivanov and V. A. Nechitailo for very helpful discus­
sion of wavelet method.
REFERENCES
1. Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1991.
2. Meyer, Y., Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993.
3. Dremin, I. M., Ivanov, O. V., and Nechitailo, V. A., Usp. Fiz. Nauk, 171, 465 (2001).
4. Teige, S., Phys. Rev., p. 012001 (1998).
5. Chung, S. U., Phys. Rev., p. 092001 (1999).
6. Korotkikh, V. L., and Sarycheva, L. I., Nucl. Phys., p. 413 (2000).