The weak strangeness production reaction $pn \to p\Lambda$ in a one-boson-exchange model

The weak production of Lambdas in nucleon-nucleon scattering is studied in a meson-exchange framework. The weak transition operator for the $NN \to N \Lambda$ reaction is identical to a previously developed weak strangeness-changing transition potential $\Lambda N \to NN$ that describes the nonmesonic decay of hypernuclei. The initial $NN$ and final $YN$ state interaction has been included by using realistic baryon-baryon forces that describe the available elastic scattering data. The total and differential cross sections as well as the parity-violating asymmetry are studied for the reaction $pn \to p\Lambda$. These observables are found to be sensitive to the choice of the strong interaction potential and the structure of the weak transition potential.


I. INTRODUCTION
Over the last several decades, the Standard Model of weak interactions has been thoroughly tested by a vast amount of data for leptonic and semi-leptonic decays and reactions.
Hadronic weak interactions are in general more difficult to study experimentally since they are usually obscured by the presence of the much larger strong interaction. This requires employing processes in which the strong force cannot participate due to overriding symmetry principles. In the case of the weak nucleon-nucleon interaction it was realized more than 40 years ago [1] that the current-current form of the weak interaction dictates the presence of a weak transition between nucleons which would lead to parity impurities in nuclear states which are of first order in the weak coupling. Using these parity nonconserving observables in many experiments on nuclear gamma and alpha transitions, polarized NN scattering, as well as the recent first measurement of the nuclear anapole moment [2], much has been learned about the weak nucleon-nucleon interaction [3].
The situation is very different for the flavor-changing baryon-baryon interaction. Soon after the discovery of hypernuclei it was recognized that the Λ inside the nuclear medium decays not through its Pauli-blocked mesonic decay channel, but predominantly through the ∆S =1 nonmesonic transition ΛN → NN, thus opening a door to the study of the weak strangeness-changing hyperon-nucleon force. However, experimental progress in this field of weak hypernuclear decays has been slow until recently due to the difficult multi-coincidence, low count-rate nature of these measurements. In recent years the situation has improved significantly due to a series of new experiments at BNL and KEK. On the theoretical side it became clear that in order to unambiguously extract the weak ΛN → NN transition potential significant effort must be spent to account for the nuclear structure effects as accurately as possible. When this task was recently completed [4] it became apparent that major discrepancies between theory and experiment cannot be due to the underlying nuclear structure but have to arise from the nature of the weak transition potential itself.
Even with the nuclear structure input under control it has become desirable to measure the process NN → NΛ directly since the hypernuclear decay can only probe the reaction at one well-defined kinematic setting. Though considered impossible for many years since this process is reduced in cross section by around 12 orders of magnitude from the standard elastic NN scattering, recent progress in experimental and accelerator technology may have brought measuring this process within reach [5]. It is therefore timely to provide predictions of various observables based on the transition potential used in the nonmesonic hypernuclear decay.
In Sec. II of this paper, we present the expressions for the matrix elements and the cross section. Sec. III briefly describes the transition operator derived in Ref. [4]. Our results are discussed in Sec. IV and summarized in Sec. V.

II. DIFFERENTIAL CROSS-SECTION
The differential cross section per unit solid angle in the center-of-mass system for the reaction pn → pΛ as depicted in Fig. 1 is given by the expression with √ s = E 1 + E 2 = E 3 + E 4 the total available energy in the center-of-mass system and p I and p F the relative momenta of the particles in the initial and final states respectively.
In a plane wave Born approximation (PWBA) the weak transition matrix elements read where the overline stands for the antisymmetric combination of the two particle states {1} and {2} and V w is the nonrelativistic weak transition potential.
Accounting for the locality of the weak potential, the direct term of the previous matrix elements can be written in the distorted wave Born approximation (DWBA) as where V w ( r ) contains the radial, angular and isospin dependence of the weak transition potential and Ψ N N ) stands for the distorted ΛN (NN) wave function. In Sec. III it is shown how the weak potential can be decomposed as where the index i sums over mesons and α over the different spin channels. The radial part of the potential is denoted by V (i) α (r), the piece containing the angular and spin dependence by V (i) α (r) andÎ (i) α denotes the appropriate isospin operator for each meson. Using the partial wave decomposition for the distorted waves, working in the coupled basis formalism, LS(J) (see appendix), and assumingp I parallel to the z-axis, the modulus squared of the weak matrix elements for the pn → pΛ reaction finally reads The distorted radial wave functions in the above equation are generated from a T-matrix which is constructed using the nucleon-nucleon (NN) and hyperon-nucleon (Y N) strong potentials. We make use of the Nijmegen 93 [6] and Bonn B [7] NN potentials and the Nijmegen soft-core [8] and Jülich [9] Y N potentials. Comparison between the results obtained using the different interaction models is made in Sec. IV.

III. THE WEAK POTENTIAL
In Ref. [4] a one-boson-exchange model is developed to describe the ΛN → NN transition, where the pseudoscalar π, η,K and vector ρ, ω and K * mesons mediate the interaction.
We use this model in order to describe the present inverse reaction. In this study we have refrained from considering the transition pn → NΣ → pΛ, since it was found to be an order of magnitude smaller than that for the direct Λ production [10].
The nonrelativistic reduction of the Feynman amplitude, corresponding to the diagram depicted in Fig. 1, leads to the nonrelativistic weak potential in momentum space, which for the exchange of pseudoscalar mesons takes the form where G F m 2 π = 2.21 × 10 −7 is the Fermi coupling constant, q is the momentum carried by the meson (M) directed towards the strong vertex, g = g NNM the strong coupling constant for the NNM vertex, µ the meson mass, M N the nucleon mass and M the average between the nucleon and Λ masses. The operatorsÂ andB contain, in addition to the weak coupling constants, the particular isospin structure corresponding to the exchanged meson.
In the case of vector meson exchange the weak potential takes the form with F 1 = g V NNM , F 2 = g T NNM the strong coupling constants andα,β andε the weak coupling constants that also contain the appropriate isospin operator of the particular meson.
Performing a Fourier transform of Eqs. (6) and (7), and using the relation ( (7), one obtains the weak transition potential in coordinate space, which can be cast into the form where the index i runs over the different mesons exchanged (i = 1, . . . , 6 represents π, η,K,ρ, ω,K * ) and α over the different spin operators denoted by C (central spin independent), SS (central spin dependent), T (tensor) and P V (parity violating). In the above expression, n i = 1(0) refers to the pseudoscalar (vector) mesons. In the case of isovector mesons (π, ρ) the isospin factor is τ 1 τ 2 and for isoscalar mesons (η,ω) this factor is just1 for all spin structure pieces of the potential. In the case of isodoublet mesons (K, K * ) there are contributions proportional to1 and to τ 1 τ 2 that depend on the coupling constants and, therefore, on the spin structure piece of the potential denoted by α. For K-exchange we haveÎ (3) and for K * -exchangê The different pieces V (i) α (r), with α = C, SS, T, P V , are given by where µ i denotes the mass of the different mesons. The expressions for K (i) α , which contain factors and coupling constants, are given in Table I. The explicit values of the coupling constants can be found in Table III of Ref. [4].
is used at each vertex, where the value of the cut-off, Λ i , depends on the meson. These values are the same ones as those of the strong Jülich Y N interaction [9], since the Nijmegen model distinguishes form factors only in terms of the transition channel. The resulting expressions for the regularized potentials, which were given already in Ref. [4], will be repeated here in order to correct for some misprints. The effect of form factors is included by making the following replacements in Eqs. (11) to (14)

IV. RESULTS
The total cross section for the pn → pΛ reaction including only one-pion-exchange in the weak transition potential is shown in Fig. 2 Fig. 3. Fig. 3a shows the cross sections calculated using the Nijmegen potentials for getting the NN and ΛN distorted waves and Fig. 3b shows those using the Bonn B NN and Jülich A ΛN potentials. In Fig. 3a we see that adding the ρ meson to one-pion-exchange considerably reduces the cross section in the kinematic region studied here. Including the remaining mesons does not change the cross section much, which ends up being about a factor 2 smaller than that for one-pion only. This effect appears to be surprisingly different from the moderate reduction of 15% found for the decay rate of hypernuclei when the effect of adding heavier mesons to the one-pion-exchange mechanism was studied [4]. However, we As can be seen in Fig. 3b, Fig. 3b), the Jülich A model shows a clear enhancement close to the pΣ threshold.
Adding the ρ meson furthermore yields a strong reduction of the cross section. On the other hand, the addition of the remaining mesons produces a substantial enhancement which gives rise to a final cross section not very different from the pion-only result. We note here that the pion-only cross section shown in Fig. 3 is close to the results obtained in Ref. [10] with one-pion-exchange, but we do not find the tremendous enhancement of a factor of 3 in the cross section when the ρ meson is included. This is most likely due to the different models for the weak ρNΛ vertices used in the ρ-exchange mechanism. Their weak coupling constants are larger than the ones used here by a factor of more than 2 for the parity conserving ones (α ρ and β ρ ) and a factor 3.5 for the parity violating one (ε ρ ).
Let us now explore the origin of the peak in the cross section around the Σ threshold.
Nuclear matter calculations already showed some time ago [11][12][13]  A measure of the amount of parity violation in the weak Λ production is given by the asymmetry A defined as where σ + (σ − ) is the cross section for positive (negative) helicity of the incoming proton. In It is clear that these strong Y N potentials are not sufficiently constrained by the small amount of total cross section data on Y N scattering. Hence the different Y N models which produce the total cross sections for Y N scattering equally well, give rise to very different predictions when applied to other reactions that are sensitive to the Y N interaction, like hypernuclear structure calculations, studies of nuclear matter with strangeness, or the weak transition pn → pΛ studied here. More data on Y N scattering, especially on differential cross sections and polarization observables, is highly desirable in order to constraint the Y N interactions sufficiently well so that the pn → pΛ reaction can be used to learn about the weak four fermion interaction. In particular, the two-step pn → NΣ → pΛ transition, not considered in the present work, could be studied after having a better knowledge of the Y N interaction.

V. SUMMARY
We have studied the weak strangeness production reaction pn → pΛ in a distorted wave Born approximation formalism using the one-meson-exchange model for the weak transition consisting of six mesons viz. the π, ρ, η, ω,K and K * . The distorted wave functions are written in terms of partial wave expansions and are generated using the different available potentials for nucleon-nucleon and hyperon-nucleon interactions.
The total cross sections are sensitive to the model ingredients of the weak transition operator. Including the ρ meson decreases the pion-only cross sections by a factor of 2 or more. The effect of the remaining mesons depends on the strong potentials employed to distort the pΛ states, giving rise to cross sections that can be either very close or a factor 2 smaller than the pion-only results.
The kinematical region explored by the free pn → pΛ reaction is much larger than that by the inverse reaction, the nonmesonic decay ΛN → NN taking place inside hypernuclei.
The heavier mesons contribute very differently and are more important in the free reaction compared to the nonmesonic decay due to the different behavior of the ΛN wave function inside a hypernucleus compared to that in free space.
The total cross section computed with the Nijmegen ΛN wave functions show a step-like behavior around 1140 MeV/c beam momentum, where the strong ΛN → ΣN transition opens up. The Jülich ΛN results show a dramatic peak in this region. However, the peak in our results is not as pronounced as the one found in earlier work [10]. The major contribution to these cross sections comes from the ΛN partial waves in the 3 S 1 − 3 D 1 coupled channels.
We find the pn → pΛ reaction to be very sensitive to the type of model used for the strong hyperon-nucleon interaction. Hence, more data on Y N scattering for observables other than the total cross sections are needed to constrain the Y N interaction models and use the pn → Λp reaction to extract the weak four fermion interaction.
The cross sections for the pn → pΛ reaction are of the order of 10 −12 mb and are at the borderline of feasibility for the existing experimental facilities.

VII. APPENDIX
The Lippmann-Schwinger equation allows us to obtain the scattered wave function for a pair of particles moving under the influence of the strong interaction. The scattered states will then be given by for the incoming NN states and by for the outgoing ΛN states. The previous equations are written in terms of the T-matrix which is obtained from and fulfills | Φ NN and | Φ ΛN being the corresponding unperturbed states for the NN and ΛN systems, respectively. Note that Eq. (24) should have been written as Projecting Eqns. (20) and (21) into r-space and performing a partial-wave decomposition, the distorted waves can be written as where the quantities J JM LS (r) are the generalized spherical harmonics. The radial wave functions in the above equations are generated numerically using the T-matrix constructed from the appropriate potentials and are given as follows