bGesellschaft für Schwerionenforschung (GSI), 64220 Darmstadt, Germany
of the scattering angle and the total center-of-mass energy up to 4.2 GeV/c2.
A simple parametrization is obtained, suitable for fast sampling in MC
simulations.Phase-shift analyses of pp elastic scattering cross sections are available
(e.g. Refs. [2,3]), yielding elaborate
parametrizations that are generally in good agreement with the data. Nonetheless,
the complexity of such analyses makes them cumbersome to use for fast MC
simulations. Within this context, a few-parameter global function
of the center-of-mass (cm) scattering angle 
and the total cm energy 
is most convenient [4].
=90±86.5
deg that become less prominent with increasing invariant mass, as a relatively
flat valley in-between acquires more features. The peaks are due
to excluding the Coulomb potential at forward (backward) angles, in order
to avoid diverging cross sections. This is of no consequence for HADES
simulations since very forward angles are inaccessible. The SAID spectra
are in excellent agreement with the data for finite angles up to about
3 GeV incident proton laboratory kinetic energies.
The two peaks are fitted with the symmetric Gaussian-like function
 |
(1) |
requiring three independent coefficients, 
0-2.
The intermediate region was fitted with a sixth-order even polynomial
whose rank was optimized to minimize the reduced 
2
function:
 |
(2) |
To prevent Eq. (2) from interfering with the fitting
of the peaks by Eq. (1), an "envelope" function was
devised as a weight for the former:
 |
(3) |
This is effectively a step function, equal to unity over the ``flat'' region of the spectrum, and decaying rapidly near the onset of the peaks. The cutoff parameter of 78 deg, once determined by trial-and-error, was fixed.
Last, adding the quadratic function
 |
(4) |
was found to improve the fitting by smoothing out the transition between the peaks and the intermediate region.
The full function used for fitting the SAID
cm angular-distribution spectra (Fig. 1) is:
 |
(5) |
.
The two peaks rise far more steeply for the lowest few total cm energies
as compared to the rest, necessitating fitting independently in two regions
of mpp, namely [1.9,2.1] and (2.1,4.2] GeV/c2.
All the coefficients were fitted as polynomials fni
 |
(6) |
with rank ni varying for each coefficient i=0-7,
with the exception of 0
in the lower-mass region for which
 |
(7) |
was used instead. In the lower-mass region 0,3
were fitted up to 2.3 and 2.2 GeV/c2 respectively, to supply
enough data points for the free parameters, although above 2.1 GeV/c2
the large-mass parametrization is preferred (Fig. 2).
The full parametrization
(Table 1), obtained by substituting the invariant-mass
dependent coefficients 0-7(mpp)
into Eq. (5), is in good agreement with the SAID
spectra (dotted curves in Fig. 1).
Noting that
is an ansatz for 
,
the total cross section
 |
(8) |
is also parametrized (Table 1), above 2.1 GeV/c2
by fitting with f
>=f5
Eq.
(6), and below, with
 |
(9) |
is fixed by the beam and target kinematics, the scattering angle
is sampled from
(Table 1), and the final spectrum is normalized by 
(Eq. (8)) to yield differential cross sections (mb).
This algorithm has been coded into the MC package Pluto++
[4] , which uses the Rejection Method (RM) for random
sampling (see e.g. [7]). The latter establishes a criterion
for sampling ,
with the aid of two random numbers from a flat distribution, and a test
function g()
greater than
and with an indefinite integral that is invertible in .
The efficiency of the RM effectively reflects in the ratio of the area under the distribution over that of the test function. In the limit of complete overlap the minimum of two flat random number calls suffices. Most commonly a straight line over the distribution function is used as the test function, a valid but generally inefficient choice. To improve the efficiency, a test function comprising of four line segments, defined by five points (Table 2), is employed (Fig. 3). A typical MC simulation of elastic pp scattering with the parametrization presented here and Pluto++ is shown in Fig. 4.
of elastic pp scattering differential cross sections has been obtained,
in terms of the cm scattering angle and the total cm energy, which is valid
over the entire range of available data for incident proton momenta up
to ~7.2 GeV/c. This provides an alternative to the complex phase-shift
analyses, suitable for use with fast MC simulation codes. The SAID spectra
are for most angles reproduced to within 10-20%, comparable to experimental
uncertainties, with larger uncertainties near minima. Simulations of elastic
pp scattering are of interest in conjunction with spectrometer studies
and detector calibration, due to the well-known systematics of this process.
[1] P. Salabura for the HADES collaboration, Acta Phys. Polon. B27 (1996) 421.
[2] V.G.J. Stoks, R.A.M. Klomp, M.C.M. Renmeester, and J.J. de Swart, Phys. Rev. C 48 (1993) 792. http://nn-online.sci.kun.nl/NN/index.html
[3] A. Arndt and R.L. Workman, Few Body Syst. Suppl. 7 (1994) 64.
[4] M.A. Kagarlis, in preparation.
[5] Rene Brun and Fons Rademakers, Proceedings AIHENP'96 Workshop, Lausanne, Sep. 1996; Nucl. Inst. & Meth. in Phys. Res. A389 (1997) 81.
[6] F.James, CERN Program Library Long Writeup D506, CERN, 1998.
[7] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C : The Art of Scientific Computing, Cambridge University Press (2nd edition), Cambridge, January 1993.
[8] K.A. Jenkins, L.E. Price, R. Klem, R.J. Miller, P. Schreiner, M.L. Marshak, E.A. Peterson, and K. Ruddick, Phys. Rev. D 21 (1980) 2445.
[9] I. Ambats, D.S. Ayres, R. Dichord, A.F. Greene, S.L. Kramer, A. Lesnik, D.R. Rust, C.E.W. Ward, A.B. Wicklund, and D.D. Yovanovitch, Phys. Rev. D 9 (1974) 1179.
[10] T. Fujii, G.B. Chadwick, G.B. Collins, P.J. Duke, N.C. Hien, M.A.R. Kemp, and F. Turkot, Phys. Rev. 128 (1962) 1836.
[11] W.M. Preston, R. Wilson, and J.C. Street, Phys. Rev. 188 (1960) 579.
[12] Y.I. Azimov et al., Sov. Phys. JETP
15
(1962) 299.
tabulated below yields cm differential cross sections (mb/sr) of elastic
proton-proton scattering in terms of the cm scattering angle
(deg) and the total cm energy mpp (GeV/c2)
up to 4.2 GeV/c2. The coefficients 0-7
of Eq. (5) are fitted by the n+1 and 4-parameter
functions fn(mpp) and f'0(mpp),
Eqs. (6,7) respectively. The total
cross section (mpp)
(mb) is used for the absolute normalization of the spectra, and it is fitted
with fn(mpp) for n=5, and f<
of Eq. (9), for the two invariant-mass regions.
| Region I. 1.9 < mpp < 2.1 GeV/c2 (28 parameters) | |||||||||||
| (mpp) |
fn | c0 | c1 | c2 | c3 | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 |
f'0 | 90.16161 | -42.54657 | 250.6459 | -82.64069 | ||||||
| 1 |
f1 | 81.46621 | 2.442514 | ||||||||
| 2 |
f1 | -1.613531 | 1.424601 | ||||||||
| 3 |
f3 | 222.9465 | -125.6755 | -23.42175 | 15.85506 | ||||||
| 4
×102 |
f2 | 1.675475 | -1.718991 | 0.4285591 | |||||||
| 5×104 |
f2 | -2.464428 | 2.413957 | -0.5908636 | |||||||
| 6×108 |
f2 | 3.266005 | -3.199807 | 0.7833496 | |||||||
| 7×10 |
f2 | 3.624606 | -3.551761 | 0.8713306 | |||||||
| ×10-1 |
f< |
8.62782 | -4.02031 | -1584.07 | 2923.57 | ||||||
| Region II. 2.1 < mpp < 4.2 GeV/c2 (55 parameters) | |||||||||||
| (mpp) |
fn | c0 | c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 |
| 0 |
f2 | 208.6132 | -85.75014 | 9.606524 | |||||||
| 1×102 |
f3 | 9014.328 | -249.0616 | 26.35260 | 3.374141 | ||||||
| 2×10 |
f1 | 1.403383 | 4.544666 | ||||||||
| 3×102 |
f9 | 816000.1 | -1924544. | 1962241. | -1126850. | 397754.5 | -88173.33 | 11948.51 | -896.5965 | 27.03873 | .1756284 |
| 4×105 |
f9 | -535576.2 | 1179326. | -1056764. | 478358.4 | -100020.3 | -1995.474 | 5989.377 | -1358.871 | 135.3221 | -5.257673 |
| 5×107 |
f7 | 6069.216 | -15344.59 | 15584.70 | -8343.774 | 2567.415 | -458.0696 | 44.27438 | -1.805952 | ||
| 6×1011 |
f5 | 4072.224 | -5742.969 | 3032.050 | -734.3411 | 78.63649 | -2.664602 | ||||
| 7×104 |
f5 | 470.0798 | -1108.074 | 924.0620 | -362.0080 | 70.10400 | -5.363294 | ||||
| ×10-2 |
f5 | -264.737 | 402.894 | -146.488 | -12.4763 | 14.8093 | -1.90930 | ||||
Table 2: The
five reference points ,
below define the test function g()
of Fig. (3). The prefactor in the second column is
to be multiplied by
(Table 1) evaluated at the angle of the first column,
with
fixed by the beam and target kinematics.
|
(deg) |
×(mb/sr) |
| 0.0 | 1.1 |
| 3.7 | 1.1 |
| 10.5 | 1.2 |
| 36.0 | 1.3 |
| 90.0 | 1.4 |
)
of Eq. (5), and the spectra reproduced from the parametrization
of Table 1 are shown as solid (blue)
and dotted (red) curves respectively.
Fig 2: The coefficients 0-7
are derived from fitting the differential cross section spectra to Eq.
(5) (open blue circles).
The (red) curves result from fitting these
coefficients to the invariant-mass dependent functions of Table 1.
The index of each coefficient, as well as the number of required parameters
for the two itotal cm-energy regions, are indicated in each panel. The
units are consistent with differential cross sections in mb/sr, for angles
in deg and
in GeV/c2.
Fig 3: The distribution function ,
with mpp=2.994 GeV/c2 corresponding to beam
protons of Tlab=2.9 GeV, is shown (dotted red
line) with the test function g()
(solid green curve) defined by the five points
of Table 2 as discussed in the text. Due to the symmetry
about =90
deg only half of the spectrum is shown. Although the sampling efficiency
depends on
and mpp, the ratio of the areas under the two curves
is indicative of the efficiency averaged over angles, for the case in hand
67.9%. In practive, this means that 67.9% of the time the first attempt
at sampling is successful.
Fig 4: A MC simulation of Nevt=50k
elastic pp scattering events is shown (red
histogram), for mpp=2.994 GeV/c2 corresponding
to incident protons of Tlab=2.9 GeV, with the parametrization
of Table 1 implemented in the code Pluto++
[4]. The calculation shown requires 11.2 CPU seconds
on a 200 MHz Linux PC. The plotted data are a compilation from Refs. [8-12]
for Tlab=2.8 - 3.0 GeV. The binning is chosen for optimum
matching in the display of the simulation and the data, with the number
of bins (135) roughly equal to that of the available data points (145).
The simulation is scaled by the factor
(see Eqs. (8-9) and the related discussion).
