1 Introduction

The relation between the yield of neutral kaons and the charged ones is commonly expected to be in the form (see Refs. [1,2,3]):

$$\begin{aligned} K^{0}_{s}= (K^++ K^-)/2 \end{aligned}$$
(1)

where \(K^{0}_{s}\), \(K^+\), \(K^-\) are the mean multiplicities of the corresponding kaon species. This formula comes from the isospin symmetry. However it is not valid for isospin asymmetric initial states.

In Sect. 2, we show a compilation of experimental data on neutral and charged kaon production in \(p+p\) collisions for different \(\sqrt{s}\) range.

In Sect. 3, we present a model based on the parton structure (valence and sea quarks) of different kaons and initial hadrons, the so-called Quark Parton Model (QPM). Such model was used about 30 years ago in the design phase of the CNGS neutrino beam and kaons beams for NA31 experiment [4].

In this paper, we show that this model works much better for \(p+p\) interaction in a large energy range than Eq. (1).

2 Relation between charged and neutral kaon production

The world data compilation of the average number of produced \(K^{0}_{s}\) per event in \(p+p\) collisions is shown in Fig. 1 with red squares, whereas the blue ones correspond to the number of \(K^{0}_{s}\) extracted from \(K^+\) and \(K^-\) production using Eq. (1).

Blue squares are systematically located above the red squares. This shows that Eq. (1) is not adequate for \(p+p\) collisions.

The relation between charged and neutral kaons can be derived using the so called Smushkevich rule: “For all particle involved in isospin-conserving relation all members of isospin multiplets are produced in equal numbers if and only if the initial population is uniform” [7,8,9,10]. It means that it is made up of equal numbers of the members of any multiplets involved. This is obviously not the case for \(p+p\) collisions.

The kaons form two isospin doublets: \(K^+-K^0\)and \(K^--\bar{K^0}\). Therefore e.g. in \({\bar{p}}\)+p collisions, which are obviously uniform we can expect the same number of \(K^+\) and \(K^0\) as well as the same number of \(K^-\) and \(\bar{K^0}\):

$$\begin{aligned} K^+= & {} K^0\end{aligned}$$
(2)
$$\begin{aligned} K^-= & {} \bar{K^0}\end{aligned}$$
(3)

therefore,

$$\begin{aligned} K^++ K^-= K^0+ \bar{K^0}\end{aligned}$$
(4)

which leads to Eq. (1). This formula is not valid for the nonuniform isospin initial state \(p+p\) with two members of NN doublet with \(I_{3}=+1/2\), therefore the observation of a discrepancy between Eq. (1) and \(p+p\) data is not surprising.

In this case, when the initial population is nonuniform, a model based on the quark structure of the participating particles was proposed.

3 Derivation of Quark Parton Model formula

The quark structure of K mesons is as follows:

$$\begin{aligned} K^+= & {} u{\bar{s}} \end{aligned}$$
(5)
$$\begin{aligned} K^-= & {} {\bar{u}}s \end{aligned}$$
(6)
$$\begin{aligned} K^0= & {} d{\bar{s}} \end{aligned}$$
(7)
$$\begin{aligned} \bar{K^0}= & {} {\bar{d}}s \end{aligned}$$
(8)
$$\begin{aligned} K^{0}_{s}= & {} (K^0+ \bar{K^0})/2 = ( d{\bar{s}} + {\bar{d}}s)/2 \end{aligned}$$
(9)

In nucleon–nucleon interaction, u and d valence quarks exist in the initial state. For example, \(K^+\) can be produced with either valence or sea quarks while \(K^-\) with the sea quarks only:

$$\begin{aligned} K^+= & {} u_v\bar{s_s} + u_s\bar{s_s} \end{aligned}$$
(10)
$$\begin{aligned} K^-= & {} \bar{u_s}s_s \end{aligned}$$
(11)
$$\begin{aligned} K^0= & {} d_v\bar{s_s} + d_s\bar{s_s} \end{aligned}$$
(12)
$$\begin{aligned} \bar{K^0}= & {} \bar{d_s}s_s \end{aligned}$$
(13)
$$\begin{aligned} K^{0}_{s}= & {} (d_v\bar{s_s} + d_s\bar{s_s} + \bar{d_s}s_s)/2 \end{aligned}$$
(14)

To find a relation between charged and neutral kaons yields we assume the conservation of isospin symmetry. Therefore, there is no difference between up and down quarks (\(u_v = d_v\)). We assume as well that in the first approximation the production of sea quark pairs \(u{\bar{u}}\) and \(d{\bar{d}}\) is equally probable. It is consistent with flavor independence of processes such as \(g\rightarrow q{\bar{q}}\).

Fig. 1
figure 1

Multiplicity per event of neutral kaons \(K^{0}_{s}\) and charged kaons \(\frac{K^++K^-}{2}\) in inelastic \(p+p\) interactions as a function of collision energy in the center of mass reference frame. Data points taken from [2, 5, 6]

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$$\begin{aligned} u_s = d_s = \bar{u_s} = \bar{d_s} = \alpha \end{aligned}$$
(15)

but for \(g\rightarrow s{\bar{s}}\) the available phase space is smaller because of significantly larger mass:

$$\begin{aligned} s_s = \bar{s_s} = \gamma \end{aligned}$$
(16)

we obtain:

$$\begin{aligned} K^{0}_{s}= & {} (d_v \gamma + \alpha \gamma + \alpha \gamma )/2 \end{aligned}$$
(17)
$$\begin{aligned} K^+= & {} u_v \gamma + \alpha \gamma \end{aligned}$$
(18)
$$\begin{aligned} K^-= & {} \alpha \gamma \end{aligned}$$
(19)

We still should multiply \(d_v\) and \(u_v\) by the number of corresponding valence quarks for the proton–proton, proton–neutron or neutron-neutron. After some simple arithmetic:

$$\begin{aligned} p\text {+}p: K^{0}_{s}&= \frac{1}{4}(K^++ 3K^-) \end{aligned}$$
(20)
$$\begin{aligned} n\text {+}n: K^{0}_{s}&= K^+\end{aligned}$$
(21)
$$\begin{aligned} p\text {+}n: K^{0}_{s}&= \frac{1}{2}(K^++ K^-) \end{aligned}$$
(22)

We would like to compare the validity of the two models represented by Eqs. (1) and (20) using a quantity, \(R_{K}\), corresponding to the normalized (w.r. to \(K^{0}_{s}\) yield) difference between the number of expected \(K^{0}_{s}\) from charged kaons yields and the number of directly measured \(K^{0}_{s}\).

From Eq. (1), we obtain:

$$\begin{aligned} R_{K} = \frac{K^++ K^-- 2K^{0}_{s}}{2K^{0}_{s}} \end{aligned}$$

while using Eq. (20):

$$\begin{aligned} R_{K} = \frac{K^++ 3K^-- 4K^{0}_{s}}{4K^{0}_{s}}. \end{aligned}$$

We calculate both ratios for different values of \(\sqrt{s}\). The results are shown in Fig. 2.

Only such data points were shown for which the \(K^+\), \(K^-\) and \(K^{0}_{s}\) multiplicities were measured at the same or close energy. One can see that the QPM formula leads to much better agreement with the line drawn at zero. Points are shown with the associated total uncertainties (statistical and systematical uncertainties) with the exception of the second point at 4.9 GeV where only statistical uncertainty was provided.

One can observe a convergence of the expectations for both models for higher energies. Another issue is the comparison with experimental data – it is problematic to draw any statistically sound conclusion due to large uncertainties for this energy range.

Fig. 2
figure 2

Difference, normalized by \(K^{0}_{s}\), between a relation based on multiplicities of charged kaons given by the two models discussed in this paper (Eqs. 1 and 20) and \(K^{0}_{s}\) yields as a function of the center of mass energy of colliding protons

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Additionally, there is indication that QPM (Eq. (20)) is describing \(p+p\) data well (and better than Eq. (1)) in different rapidity bins as shown in Fig. 3. The values of \(R_{K}\) related to kaon multiplicities averaged on the whole rapidity region corresponding to \(\sqrt{s}=17.27\) GeV plotted in Fig. 2 show that QPM performs better than the simple average formula (\(R_{K}\) is closer to zero). In Fig. 3, one can observe that this effect is especially pronounced for the mid-rapidity region.

Fig. 3
figure 3

Rapidity distribution dn/dy of \(K^{0}_{s}\) mesons in inelastic \(p+p\) interactions at 158 GeV/c. Solid red-coloured circles correspond to the NA61/SHINE results from Ref. [5] (systematic uncertainties not shown on the plot), black squares represent results from Brick et al. at FNAL [11], blue full diamonds show results obtained from Eq. (20) using charged kaon yields recently measured by NA61/SHINE at the same beam momentum [6] while empty blue diamonds correspond to Eq. (1). This is a modified version of the figure taken from Ref. [5]

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4 Summary

We have noticed that the widespread relation between the average number of produced charged and neutral kaons holds only for a colliding system that is an uniform population w.r. to the isospin (i.e. that consists of an equal number of all members of a given isospin multiplet). This is not the case for \(p+p\) collisions therefore one should not expect that Eq. (1) will describe the data accurately.

We have shown that a better agreement with world data for a wide energy range is obtained if one uses a relation derived from simple considerations about the quark structure of kaons and nucleons Eq. (20).