Cross Section

The production cross section for $V_0$ particles in the acceptance of the HERA-B detector can be expressed as follows:

$$\sigma_{V_0}=\frac{1}{Br(V_0) L} \iint \frac{N(p_t^2,x_F)}{\epsilon(p_t^2,x_F)} \, \mathrm{d} p_t^2\, \mathrm{d} x_F$$ (6.19)

where $N(p_t^2,x_F)$ is the number of observed $V_0$s in bins of $p_t^2$ and $x_F$ and $\epsilon(p_t^2,x_F)$ is the total acceptance which includes geometrical acceptance of the detector and reconstruction efficiency (see Section 6.4) for the considered kinematical range. The branching ratios $Br(V_0)$ are taken from Ref. [#!Pdg!#] , and $L$ is the luminosity which has been calculated for each wire separately, as it is described in 6.5.

The inclusive differential cross section per nucleon $\mathrm {d} \sigma _{pA} / \mathrm {d}x_F$ for various targets and target materials are shown in Table 6.4 for the $x_F$ range [-0.12,0]. The inclusive differential cross section per nucleon $\mathrm {d} \sigma _{pA} / \mathrm {d}x_F$ for various targets and target materials are shown in Table 6.5 for the positive $x_F$ range.

Table 6.4: The inclusive differential cross sections for the $-0.12 \le x_F < 0$ range for the production of $K^0_S, \, \,\Lambda$ and $\bar \Lambda$ particles. The total cross sections obtained by extrapolation to the $-1. \le x_F \le 1$ range are listed.

Particle	Target	$\mathrm {d} \sigma _{pA} / \mathrm {d}x_F$ (mb)	$\sigma_{pA}$ (mb)
	C	337.1 $\pm$ 17.6 $\pm$ 60.	135.2 $\pm$ 7.1 $\pm$ 20.5
$K^0_S$	Ti	1709.2 $\pm$ 95.3 $\pm$ 227.8	693.7 $\pm$ 38.7 $\pm$ 92.5
	W	4521.2 $\pm$ 241.3 $\pm$ 617.4	1835. $\pm$ 98. $\pm$ 250.8
	C	85.6 $\pm$ 4.4 $\pm$ 15.1	61.2 $\pm$ 3.2 $\pm$ 11.
$\Lambda$	Ti	428.3 $\pm$ 22.9 $\pm$ 67.1	306.2 $\pm$ 16.4 $\pm$ 47.9
	W	1078.2 $\pm$ 55.9 $\pm$ 194.1	767.9 $\pm$ 39.9 $\pm$ 138.2
	C	42.1 $\pm$ 2.3 $\pm$ 6.5	14.8 $\pm$ 0.8 $\pm$ 2.7
$\bar \Lambda$	Ti	185.5 $\pm$ 11.9 $\pm$ 35.3	73.9 $\pm$ 4.2 $\pm$ 14.9
	W	550.7 $\pm$ 29.7 $\pm$ 85.4	193.4 $\pm$ 10.4 $\pm$ 36.0

Table 6.5: The inclusive differential production cross section $\mathrm {d} \sigma _{pA} / \mathrm {d}x_F$ in $mb$ for $K^0_S, \, \,\Lambda$ and $\bar \Lambda$ particles measured on three different targets for positive $x_F$ range.

$\triangle\, x_F$	C	Ti	W
$K^0_S$
0. - 0.015	665.3 $\pm$ 7.5 $\pm$ 133.1	3056.5 $\pm$ 41.8 $\pm$ 611.3	7850.4 $\pm$ 71.7 $\pm$ 1570.1
0.015 - 0.03	648.3 $\pm$ 14.5 $\pm$ 129.7	2878.1 $\pm$ 78.9 $\pm$ 575.6	7439.9 $\pm$ 137.2 $\pm$ 1487.8
0.03 - 0.045	665.9 $\pm$ 33.1 $\pm$ 133.2	2773.3 $\pm$ 177.6 $\pm$ 554.6	6757.9 $\pm$ 289.9 $\pm$ 1351.6
0.045 - 0.06	626.8 $\pm$ 65.2 $\pm$ 125.4	3387.6 $\pm$ 554.7 $\pm$ 677.5	8482.1 $\pm$ 861.5 $\pm$ 1696.4
$\Lambda$
0. - 0.015	171.2 $\pm$ 8.3 $\pm$ 34.2	1149.3 $\pm$ 78.1 $\pm$ 229.8	2595.1 $\pm$ 102.4 $\pm$ 519.2
0.015 - 0.03	236.2 $\pm$ 19.6 $\pm$ 47.2	1217.6 $\pm$ 131.8 $\pm$ 243.5	2947.2 $\pm$ 188.3 $\pm$ 589.4
0.03 - 0.045	269.5 $\pm$ 44.5 $\pm$ 53.9	1327.1 $\pm$ 257.6 $\pm$ 265.4	4412.2 $\pm$ 588.2 $\pm$ 882.4
$\bar \Lambda$
0. - 0.015	108.1 $\pm$ 6.4 $\pm$ 21.6	533.3 $\pm$ 41.1 $\pm$ 106.6	1345.9 $\pm$ 67.1 $\pm$ 269.2
0.015 - 0.03	111.5 $\pm$ 11.4 $\pm$ 22.3	646.1 $\pm$ 87.3 $\pm$ 129.2	1352.9 $\pm$ 112.7 $\pm$ 270.6
0.03 - 0.045	105.5 $\pm$ 17.9 $\pm$ 21.1	917.5 $\pm$ 277.4 $\pm$ 183.5	2263.7 $\pm$ 420.2 $\pm$ 452.7

In order to be comparable with results from other experiments the obtained cross sections have to be extrapolated to the full $x_F$ range [-1.,1.]. The extrapolation was based on the measurements in the negative $x_F$ region by using the parameterization $\mathrm{d} \sigma_{pA} / \mathrm{d}x_F \varpropto (1-x_F)^n$ [#!brod!#]. The parameter $n$ is constant and values for different strange particles are taken from the measurements of inclusive strange-particle production done by other experiments [#!Adam!#]. The resulting production cross sections $\sigma_{pA}$ for $V_0$ are listed in Table 6.4.

The differential cross sections $\mathrm {d} \sigma _{pA} / \mathrm {d}p_t^2$ are listed in Table 6.7. The results are plotted in Fig. 6.18, fits are done by the function

$$\frac{\mathrm{d} \sigma_{pA}}{\mathrm{d}p_t^2}=\sigma B exp(-B p_t^2),$$ (6.20)

where B is a parameter independent from $x_F$ and $p_t^2$ [#!zavm!#]. Measured parameter B for different target materials are listed in Table 6.6. The measured $p_t^2$ spectra is well described by the Eq. 6.20.

Table 6.6: The values of the parameter B obtained by fitting the differential cross section.

$B (GeV/c)^{-2}$
	C	Ti	W
$K^0_S$	3.3 $\pm$ 0.3	3.1 $\pm$ 0.3	3.1 $\pm$ 0.3
$\Lambda$	2.1 $\pm$ 0.2	2. $\pm$ 0.3	2.0 $\pm$ 0.2
$\bar \Lambda$	2.3 $\pm$ 0.2	2.1 $\pm$ 0.2	2.1 $\pm$ 0.2

The differential production cross sections $\mathrm {d} \sigma _{pA} / \mathrm {d}p_t^2 \mathrm {d} x_F$ are listed in Table 6.8-6.10. Only statistical errors are quoted here.

Table 6.7: The inclusive differential production cross section $\mathrm {d} \sigma _{pA} / \mathrm {d}p_t^2$ in $mb/(GeV/c)^2$ for $K^0_S, \, \,\Lambda$ and $\bar \Lambda$ particles measured on three different targets. The $p_t^2$ bins ( $\triangle\, p_t^2$) are in $(GeV/c)^2$.

$\triangle\, p_t^2$	C	Ti	W
$K^0_S$
0. - 0.2	327.1 $\pm$ 16.6 $\pm$ 55.6	1549.5 $\pm$ 79.3 $\pm$ 263.4	4042.8 $\pm$ 204.4 $\pm$ 687.2
0.2 - 0.4	121.2 $\pm$ 6.2 $\pm$ 20.6	580.7 $\pm$ 30.2 $\pm$ 98.7	1542.1 $\pm$ 78.6 $\pm$ 262.1
0.4 - 0.6	56. $\pm$ 2.9 $\pm$ 9.5	281.2 $\pm$ 15.2 $\pm$ 47.8	743.8 $\pm$ 38.5 $\pm$ 126.4
0.6 - 0.8	29.5 $\pm$ 1.6 $\pm$ 5.	146.5 $\pm$ 8.3 $\pm$ 24.9	415.1 $\pm$ 22. $\pm$ 70.5
0.8 - 1.0	18.2 $\pm$ 1.1 $\pm$ 3.1	93.3 $\pm$ 5.8 $\pm$ 15.8	253.3 $\pm$ 14. $\pm$ 43.01
1.0 - 1.2	11.1 $\pm$ 0.7 $\pm$ 1.8	67.7 $\pm$ 4.9 $\pm$ 11.5	169.7 $\pm$ 9.9 $\pm$ 28.8
$\Lambda$
0. - 0.2	81.2 $\pm$ 4.5 $\pm$ 13.7	494.9 $\pm$ 28.1 $\pm$ 84.1	919.7 $\pm$ 48.7 $\pm$ 156.3
0.2 - 0.4	50.6 $\pm$ 2.8 $\pm$ 8.6	296.7 $\pm$ 17.1 $\pm$ 50.4	577.9 $\pm$ 30.7 $\pm$ 98.2
0.4 - 0.6	32.7 $\pm$ 1.9 $\pm$ 5.5	179.1 $\pm$ 11.2 $\pm$ 30.4	430.8 $\pm$ 23.9 $\pm$ 73.2
0.6 - 0.8	20.7 $\pm$ 1.4 $\pm$ 3.5	114.7 $\pm$ 8.1 $\pm$ 19.5	269.1 $\pm$ 15.9 $\pm$ 45.7
0.8 - 1.0	14.8 $\pm$ 1.1 $\pm$ 2.5	90.1 $\pm$ 8. $\pm$ 15.3	221.7 $\pm$ 14.6 $\pm$ 37.7
1.0 - 1.2	12.3 $\pm$ 1.2 $\pm$ 2.1	77.1 $\pm$ 8.8 $\pm$ 13.1	142.6 $\pm$ 10.7 $\pm$ 24.2
$\bar \Lambda$
0. - 0.2	25.1 $\pm$ 1.5 $\pm$ 4.2	118.2 $\pm$ 7.2 $\pm$ 20.1	317.2 $\pm$ 17.6 $\pm$ 53.9
0.2 - 0.4	16.1 $\pm$ 0.9 $\pm$ 2.7	79.6 $\pm$ 4.9 $\pm$ 13.5	200.2 $\pm$ 11.1 $\pm$ 34.
0.4 - 0.6	9.7 $\pm$ 0.6 $\pm$ 1.6	51.3 $\pm$ 3.5 $\pm$ 8.7	126.6 $\pm$ 7.4 $\pm$ 21.5
0.6 - 0.8	6.1 $\pm$ 0.4 $\pm$ 1.	27.5 $\pm$ 2.2 $\pm$ 4.6	87.6 $\pm$ 5.6 $\pm$ 14.8
0.8 - 1.0	3.5 $\pm$ 0.3 $\pm$ 0.5	19.9 $\pm$ 2.1 $\pm$ 3.3	56.7 $\pm$ 4.1 $\pm$ 9.6
1.0 - 1.2	2.6 $\pm$ 0.2 $\pm$ 0.4	19.1 $\pm$ 2.6 $\pm$ 3.2	36.8 $\pm$ 3.1 $\pm$ 6.2

Figure 6.18: The differential production cross sections $\mathrm {d} \sigma _{pA} / \mathrm {d}p_t^2$ for $V_0$ for three target materials (Carbon, Titanium and Tungsten).

The ratios of production cross sections from the measurements listed above for mid-rapidity for Carbon wire are

$$\frac{\sigma (K^0_S)}{\sigma (\Lambda)}=5.9 \pm 0.3 , \, \, \frac{\sigma (\bar \Lambda)}{\sigma ( \Lambda)}=0.68 \pm 0.07.$$

Fig. 6.19 shows the ratios for all three different target materials used.

Figure: The ratio of $\sigma (\bar \Lambda) / \sigma (\Lambda)$ determined at mid-rapidity for the used targets.

The dependences of production cross sections on the atomic number of the target material are shown in Fig. 6.20 and fitted by the $\sigma_{pA} \varpropto A^{\alpha}$. Production cross sections and $\alpha$ obtained from the fit are listed in Table 6.11.

Figure 6.20: The $V_0$ total production cross section as a function of atomic mass A of the target material. The solid lines show fits by the $\sigma _{pA} \varpropto A_{\alpha }$ function.

Table 6.11: Production cross section per nucleon for $V_0$ and results of the dependences of production cross sections on the atomic number.

	$K^0_S$	$\Lambda$	$\bar \Lambda$
$\sigma_{pN}$ (mb)	13.2 $\pm$ 0.91 $\pm$ 2.3	6.5 $\pm$ 0.57 $\pm$ 0.94	1.6 $\pm$ 0.17 $\pm$ 0.28
$\alpha$	0.961 $\pm$ 0.026	0.927 $\pm$ 0.021	0.93378 $\pm$ 0.026

The production cross sections per nucleon as a function of the atomic number of the target material are shown in Fig. 6.21

Figure 6.21: The $V_0$ total production cross section per nucleon as a function of atomic mass A of the target material.