Ratio between two $\Lambda$ and $\bar{\Lambda}$ production mechanisms in $p$ scattering

We consider $\Lambda$ and $\bar{\Lambda}$ production in a wide range of proton scattering experiments. The produced $\Lambda$ and $\bar{\Lambda}$ may or may not contain a diquark remnant of the beam proton. The ratio of these two production mechanisms is found to be a simple universal function $r = [ \kappa/(y_p - y) ]^i$ of the rapidity difference $y_p - y$ of the beam proton and the produced $\Lambda$ or $\bar{\Lambda}$, valid over four orders of magnitude, from $r \approx 0.01$ to $r \approx 100$, with $\kappa = 2.86 \pm 0.03 \pm 0.07$, and $i = 4.39 \pm 0.06 \pm 0.15$.

We consider Λ andΛ production in a wide range of proton scattering experiments. The produced Λ andΛ may or may not contain a diquark remnant of the beam proton. The ratio of these two production mechanisms is found to be a simple universal function r = [κ/(yp − y)] i of the rapidity difference yp − y of the beam proton and the produced Λ orΛ, valid over four orders of magnitude, from r ≈ 0.01 to r ≈ 100, with κ = 2.86 ± 0.03 ± 0.07, and i = 4.39 ± 0.06 ± 0.15.
TheΛ/Λ production ratio measured in a wide range of proton scattering experiments pZ → Λ(Λ)X has been found to be a universal function f (y p − y) of "rapidity loss" y p − y, where y p and y are, respectively, the rapidities of the beam proton and the produced Λ orΛ [1,2]. The function f (y p − y) is observed to be independent, or depends only weakly, on the total center of mass energy √ s of the two colliding hadrons in the range 0.024 to 7 TeV, on the target Z = p,p, Be or Pb, on the transverse momentum p T of the Λ orΛ, or on sample composition [2]. We consider the picture in which an s quark produced in the scattering may coalesce with a ud diquark remnant of the beam proton and produce a Λ [2][3][4][5][6].
Let n Λ (y) and nΛ(y) be the distributions of Λ's andΛ's as a function of rapidity y in the center of mass frame of the two colliding hadrons. Rapidity is defined so that the p beam has positive rapidity y p . We write these distributions as follows: n Λ (y) = n Λα1 (y) + n Λα2 (y) + n Λβ (y), nΛ(y) = nΛ α1 (y) + nΛ α2 (y) + nΛ β (y), where n Λα1 (y) is the distribution of Λ's containing a diquark remnant of beam 1, n Λα2 (y) is the distribution of Λ's containing a diquark remnant of beam 2, and n Λβ (y) is the distribution of Λ's containing no beam remnant, and similarly forΛ's. Production mechanism β has no memory of the beams and hence n Λβ (y) = nΛ β (y) ≡ n β (y) = n β (−y). The distribution n β (y) is approximately independent of y within the "rapidity plateau" −y max < y < y max as shown schematically in Fig. 1. See also Fig. 2 of [2], and Fig. 50.4 of [7]. We assume that beam 1 is a proton beam, so nΛ α1 = 0. If beam 2 is a p beam, nΛ α2 = 0. If beam 2 is ap beam, n Λα2 = 0. We now consider events with y > y min so that nΛ α2 and n Λα2 can be neglected. TheΛ/Λ production ratio can then be written as: where the ratio, r, of the yields of the two production mechanisms α and β is The purpose of this note is to point out that the ratio r can be fit over four orders of magnitude, from r ≈ 0.01 to r ≈ 100, with a simple universal function with only two parameters κ and i: (4) Figure 2 presents the ratios r = 1/f − 1 measured in a wide range of proton scattering experiments. The data points with y < 0.75 of the DØ pp experiment were omitted because for them we can not neglect nΛ α2 (y). The data point of the STAR pp experiment has y = 0, so n Λα1 = n Λα2 . Therefore we have divided 1/f − 1 by 2.
The parameters κ and i have a simple interpretation and can be read off the log (y p − y) vs log r graph in Fig. 2: κ ≈ 2.8 is the rapidity loss at which r = 1, and −i ≈ −4.4 is the slope of the straight line in Fig. 2. For y p − y < κ production mechanism α dominates. For y p − y > κ production mechanism β dominates.
The fit to all of the data in Fig. 2 obtains κ = 2.79 ± 0.03 and i = 4.54 ± 0.08 with χ 2 = 637 for 121 degrees of freedom. The large χ 2 is due to tension between the data points of different experiments as can be seen in Fig.  2. Omitting the R-603 and R-607 measurements, which have some data points off the rapidity plateau, obtains κ = 2.86 ± 0.03 and i = 4.39 ± 0.06 with χ 2 = 342 for 102 degrees of freedom. This is the fit shown in Fig.  2. Fitting only the E8 Pb data points where production mechanism α dominates obtains κ = 2.93 ± 0.15 and i = 4.06 ± 0.30 with χ 2 = 10.6 for 13 degrees of freedom. Fitting all data with y p − y > 2.8, where production mechanism β dominates, obtains κ = 2.94 ± 0.10 and i = 4.23 ± 0.25 with χ 2 = 37 for 32 degrees of freedom. In conclusion, we see no significant departure from Eq. (4) at either end of the data range. Our final estimate from several fits is κ = 2.86 ± 0.03 ± 0.07, i = 4.39 ± 0.06 ± 0.15, where the first uncertainty is statistical from the fit, and the second uncertainty is systematic and accounts for different data selections for the fits.
From √ s = 0.024 to 7 TeV the cross-sections σ(pp) and σ(pp), and the width of the rapidity plateau 2y max , increase by approximately a factor 2 [7], so n β (y) is approximately independent of √ s and y on the rapidity plateau. We conclude that the probability density that a p scatters and becomes a Λ with rapidity y is proportional to [κ/(y p − y)] i . This result should also be valid for Λ c , Λ b , Σ + , etc.