Final State Interactions in the Near-Threshold Production of Kaons from Proton-Proton Collisions?
A. Sibirtsev? and W. Cassing Institut f??r Theoretische Physik, Universit??t Giessen u a D-35392 Giessen, Germany
arXiv:nucl-th/9802025v2 12 Feb 1998
Abstract We analyse the pp ?ú p??K cross section recently measured at COSY arguing that the enhancement of the production cross section at energies close to the reaction threshold should be due to the ??p ?nal state interaction. We ?nd that the experimental ??p elastic scattering data as well as the predictions from the J??lich-Bonn model are in reasonable agreement with the new u + -meson production. We propose to study directly the ?nal state results on K interaction by measurements of the cross section as a function of the hyperon momentum in the ??p cm system.
PACS: 25.40.-h; 25.40.Ep; 25.40.Ve Keywords: Nucleon induced reactions; inelastic proton scattering; kaon production
Supported by Forschungszentrum J?? lich u On leave from the Institute of Theoretical and Experimental Physics, 117259 Moscow, Russia.
Recently, the COSY-11  and COSY-TOF Collaborations  measured the pp ?ú p??K + cross section at energies close to the reaction threshold. The experimental K + yield found is twice as large as predicted in Ref.  and by F??ldt and a Wilkin  for excess energies ? < 7 MeV whereas both models reproduce the data for ? > 50 MeV. Here we will argue that this enhancement is due to ??p Final State Interactions (FSI). In Ref.  the total cross section for the pp ?ú p??K + reaction has been calculated within the One-Boson-Exchange (OBE) model including both the pion and kaon exchanges in line with the analysis performed at high energies . In the latter work the FSI between the ??-hyperon and the proton has been neglected for reasons of simplicity but also due to the large uncertainty in the low energy ??p scattering cross section [6, 7]. The free parameters of the model, i.e. the coupling constants and the cut-o? parameters of the form factors for the NN?Ð and NY K vertices, were ?tted to the experimental data taken at high energies . On the other hand, F??ldt and Wilkin  calculated the energy dependence of a + the pp ?ú p??K cross section by using only the one pion exchange in line with the calculations from Ref. . Moreover, it was assumed that the production amplitude is constant and is related to the pp ?ú pp?Ç reaction. The FSI between the ??-hyperon and the proton then was incorporated via the e?ective range approximation. With only
a single free parameter ?xed by the ?Ç production data, the calculations from Ref.  reasonably reproduced the pp ?ú p??K + cross section at ? = 2 MeV  available at that time. Note, however, that the FSI correction factor at this energy is about ? 14 and therefore the production amplitude due to the pion exchange itself is very small. On the other hand, Tsushima et al.  calculated the pp ?ú p??K + production amplitude in a microscopic model which illustrates that near threshold the pion exchange contribution to the reaction is small and underestimates the most recent data from COSY [1, 2] by about a factor ? 10. Our present work is to clarify these partly con?icting results. We recall that the total cross section for the reaction pp ?ú p??K + is obtained by integrating the di?erential cross section d2 ?Ò/dtds1 over the available phase space, ?Ò= dtds1 d2 ?Ò 1 = 9 3 2 dtds1 2?Ð q s qK dtds1 ?Ì |M(t, s1 )|2 . s1 (1)
Here s is the squared invariant mass of the colliding protons, q is the proton momentum in the center-of-mass while t is the squared 4-momentum transfered from the initial proton to the ?nal hyperon or proton in case of the kaon or pion exchange, respectively. Moreover, s1 is the squared invariant mass of the Kp or K?? system, respectively, while qK is the kaon momentum in the corresponding center-of-mass system. In Eq. (1) |M| is the amplitude of the reaction which is an analytical function of t and s1 . Let us start with the experimental pp ?ú p??K + cross section and extract the reaction amplitude averaged over t and s1 by taking |M|2 out of the integral in Eq. (1). The average reaction amplitude then can be determined by comparing ?Ò from Eq. (1) with the corresponding experimental cross section from Refs. [1, 2, 9]. The results for the average matrix element |M| are shown in Fig. 1 as function of 2
?Ì the excess energy ? = s ? mp ? m?? ? mK , with mp , m?? and mK being the mass of the proton, ??-hyperon and kaon, respectively. Obviously, the matrix element |M| as evaluated from the data is not constant and decreases substantially with the excess energy ?. Following the Watson-Migdal approximation the total reaction amplitude can be factorized in terms of the production |Mprod | and FSI amplitude |AF SI |. Since at high energies the FSI is negligible, the latter amplitude should converge to 1 for ? ?ú ?Þ. Within the OBE model the squared production amplitude, for instance for the pion exchange, is given as [3, 5, 8]1
2 |Mprod (t, s1 )|2 = gN N ?Ð
t (t ? ?Ì2 )2
??2 ? ?Ì2 ?Ð ??2 ? t ?Ð
|A?Ð0 p?ú??K + (s1 )|2 ,
where gN N ?Ð is the coupling constant and ?Ì is the pion mass. The form factor of the NN?Ð vertex is given in brackets with ???Ð denoting the cut-o? parameter. In Eq. (2) |A| is the amplitude for the reaction ?Ð 0 p ?ú ??K + , which can be calculated microscopically within the resonance model2 or evaluated from the experimental data as q?Ð |A?Ð0 p?ú??K + (s1 )|2 = 16?Ð s1 (3) ?Ò?Ð0 p?ú??K + (s1 ), qK where q?Ð is the pion momentum in the ??K center-of-mass system and ?Ò?Ð0 p?ú??K + is the physical cross section. ?Ì Since m?? + mK ?Ü s1 ?Ü m?? + mK + ?, thus close to the pp ?ú p??K + reaction threshold the amplitude |A?Ð0 p?ú??K + | is almost constant. Moreover, the 4-momentum transfer squared t is a slowly varying function of energy at low ? as shown in Fig. 2a). Therefore it is a quite reasonable approximation to assume that the production amplitude |Mprod | is almost constant near the threshold. Similar arguments can be set up for the kaon exchange amplitude. We thus conclude that the deviation of the reaction amplitude |M| (shown in Fig. 1) from a constant value is due to the FSI. A similar conclusion is obtained by analysing the pp ?ú pp?Ç reaction cross section  in the same way as illustrated in Fig. 3. Moreover, the experimental data indicate that for kaon production the FSI correction is substantially smaller than for ?Ç-meson production. We note that in principle one should account for the FSI between all ?nal particles produced in the reaction pp ?ú p??K + ; here we assume that the ??p interaction is much stronger than the K + p and K + ?? interaction, respectively. Following the original idea of Chew and Low  the amplitude |AF SI | is related to the on mass-shell elastic scattering amplitude |Ael | for the reaction ??p ?ú ??p, which can be calculated from the corresponding physical cross section in analogy to Eq. (3). The strength of the FSI depends upon the ??-hyperon momentum in the ??p center-of-mass system and for ?xed excess energy the relative momentum q?? extends from zero to its maximal value as shown in Fig. 2b). Thus to calculate the total cross section for the reaction pp ?ú p??K + at ? < 100 MeV [1, 2] one needs to know the ??p scattering amplitude for 0 ?Ý q?? ?Ü 400 MeV/c.
One should add the amplitude corresponding to the exchange graph. Since the resonance properties are ?tted to the experimental data, both the resonance model and Eq. (3) should give the same results.
Fig. 4 shows the amplitude |Ael | extracted from the experimental data  plotted as a function of q?? . Actually, there are no data below 50 MeV/c and the experimental results have large error bars. The amplitude |Ael |, furthermore, can be calculated in the hyperon-nucleon interaction model developed by Holzenkamp, Reuber, Holinde and Speth . Within the Bonn-J?? lich approach the ??p elastic scattering cross
u section is given in the e?ective-range formalism as ?Ò??p?ú??p =
?Ð 3?Ð + 2 , 2 2 2 + (?1/as + 0.5rs q?? ) q?? + (?1/at + 0.5rt q?? )2
where a is the scattering length and r is the e?ective range, while the indices s and t stand for singlet and triplet ??p states. The scattering amplitude |Ael | was calculated ? with a and r parameters from Ref.  (model A) and is shown in Fig. 4 by the dashed line. Actually the e?ective range approximation is valid at low energies and can not be extended to q?? > 200 MeV/c. Following both the low energy prediction from the Bonn-J?? lich model and the experimental data we can ?t the ??p scattering amplitude u as ?Á?Â (5) |Ael | = C 1 + 2 q?? + ?Á2 /4 with ?Á = 170 MeV, ?Â = 130 MeV and C = 87. The result is shown by the fat solid line in Fig. 4. The FSI amplitude now is proportional to the ??p elastic scattering amplitude, but normalized such that |AF SI | ?ú 1 at large excess energies ?. The solid line in Fig. 1 shows our result calculated with the |AF SI | averaged over the phase space distribution for q?? and the production amplitude |Mprod | = 1.05 fm. The dashed line in Fig. 1 illustrates the result obtained with the prescription for FSI from Ref. [4, 14] as 2 4?Ât2 2?Âs + , (6) |AF SI |2 = 2 2 (?Ás + ?Ás + 2?Ì?)2 (?Át + ?Át + 2?Ì?)2 where ?Ì is the reduced mass in the ??p system and the parameters ?Á and ?Â were evaluated from the scattering length and the e?ective range for the singlet and spin-triplet ??p interaction . Fig. 1 illustrates a good agreement between the experimental data from COSY [1, 2] and our FSI approach and proves the strong in?uence of the ??p interaction in the ?nal state. Actually, the FSI e?ect can be observed directly by measuring the pp ?ú p??K + cross section as a function of the momentum q?? . Fig. 5 shows this di?erential cross section calculated3 by the phase space alone (dashed lines) and with FSI correction (solid lines) for the excess energies ? relevant for COSY-11 and COSY-TOF experiments. The enhancement at low q?? is due to the FSI and is more pronounced for the range of TOF energies. We conclude, that the pp ?ú p??K + cross section at energies close to the reaction threshold is strongly e?ected by the hyperon-nucleon ?nal state interaction. However, this e?ect is less pronounced as in the pp ?ú pp?Ç reaction close to threshold since the ??N interaction is weaker than the pp interaction at low relative momenta.
The results are normalized to the experimental total cross section.
Further experimental studies are necessary for the direct measurement of the FSI as a function of the relative momentum q?? as well as more upgrade calculations on the Y p interaction, which are
under study in J?? lich . u
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Figure 1: The average amplitude for the pp ?ú p??K + reaction as a function of the excess energy. The symbols show the results extracted from the experimental data of Refs. [1, 2, 9]. The lines are our calculations with FSI as described in the text.
Figure 2: The range of the 4-momentum transfer squared (a) and maximal hyperon momentum in the ??p cm system (b) as function of the excess energy ?.
Figure 3: The average amplitude for the pp ?ú pp?Ç reaction. The symbols show the results extracted from the experimental data . The dashed line shows the production amplitude while the solid line is the total amplitude corrected by FSI (6) with parameters in line with the pp interaction.
Figure 4: The ??p elastic scattering amplitude as a function of the hyperon momentum q?? in the center-of-mass system. The dots show the experimental data ; the fat solid line is our ?t while the dashed line shows the e?ective range approximation with parameters from the J?? lich-Bonn model . u
Figure 5: The pp ?ú p??K + cross section as a function of the hyperon momentum in the ??p cm system calculated for two values of the excess energy ?. The solid lines show our results with FSI correction while the dashed lines are the pure phase space distributions.