A relativistic model for the non-mesonic weak decay of the {{}_{\Lambda}^{12}C} hypernucleus

# A relativistic model for the non-mesonic weak decay of the 12ΛC hypernucleus

Francesco Conti Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia and
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy
Andrea Meucci Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia and
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy
Carlotta Giusti Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia and
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy
Franco Davide Pacati Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia and
Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, I-27100 Pavia, Italy
July 16, 2019
###### Abstract

A fully relativistic finite nucleus wave-function approach to the non-mesonic weak decay of the hypernucleus is presented. The model is based on the calculation of the amplitudes of the tree-level Feynman diagrams for the process and includes one-pion exchange and one-kaon exchange diagrams. The pseudo-scalar and pseudo-vector choices for the vertex structure are compared. Final-state interactions between each one of the outgoing nucleons and the residual nucleus are accounted for by a complex phenomenological optical potential. Initial and final short-range correlations are included by means of phenomenological correlation functions. Numerical results are presented and discussed for the total non-mesonic decay width , the ratio, the intrinsic asymmetry parameter, and the kinetic energy and angular spectra.

###### pacs:
21.80.+a Hypernuclei; 24.10.Jv Relativistic Models

## I Introduction

The birth of hypernuclear physics dates back to 1952 HypFirst () when the first hypernuclear fragment originated from the collision of a high-energy cosmic proton and a nucleus of the photographic emulsion exposed to cosmic rays was observed through its weak decays, revealing the presence of an unstable particle: this was interpreted as due to the formation of a nucleus in which a neutron is replaced by the hyperon, i.e., the lightest strange baryon. A hypernucleus is a bound system of neutrons, protons, and one or more hyperons. Only the lightest hyperon, the , is stable with respect to esoenergetic strong and electromagnetic processes in nuclear systems. Therefore, the most stable hypernuclei are those made up of nucleons and a particle. We denote with a hypernucleus with protons, neutrons, and a (-hypernucleus).

Hypernuclei represent a unique laboratory for the study of strong and weak interactions of hyperons and nucleons through the investigation of hypernuclear structure and decay. The particle, which does not have to obey the Pauli principle, is an ideal low-energy probe of the nuclear environment which allows a deepening of classical nuclear physics subjects, such as the role of nuclear shell models and the dynamical origin of the nuclear spin-orbit interaction. Hypernuclear physics also establishes a bridge between nuclear and hadronic physics, since many related issues can in principle unravel the role played by quarks and gluons partonic degrees of freedom inside nuclei. In this direction, the study of hybrid theories combining meson-exchange mechanisms with direct quark interactions have the potentiality to teach us something on the confinement phenomenon, an issue still far from being satisfactorily understood.

In -hypernuclei the can decay via either a mesonic or a non-mesonic strangeness-changing weak interaction process. In the nuclear medium the mesonic decay, , which is the same decay of a free , is strongly suppressed, but in the lightest hypernuclei, by the effect of the Pauli principle on the produced nucleon, whose momentum ( 100 MeV/) is well below the Fermi momentum. In the non-mesonic weak decay (NMWD) the pion produced in the weak transition is virtual and gets absorbed by neighbor nucleons. Then, two or three nucleons with high momenta ( 400 MeV/) are emitted. We can distinguish between one and two-nucleon induced decays, according to whether the interacts with a single nucleon, either a proton, (decay width ), or a neutron, (), or with a pair of correlated nucleons, (). Mesons heavier than the pion can also mediate these transitions. The NMWD process is only possible in the nuclear environment and represents the dominant decay channel in hypernuclei beyond the -shell. The total weak decay rate is given by the sum of the mesonic () and non-mesonic () contributions:

 Γtot=Γm+Γnm, (1)

with

 Γnm=Γ1+Γ2,Γ1=Γp+Γn. (2)

The fundamental interest in the NMWD mode is that it provides a unique tool to study the weak strangeness changing () baryon-baryon interaction , in particular its parity conserving part, that is much more difficult to study with the weak transition, that is overwhelmed by the parity-conserving strong interaction. Since no stable hyperon beams are available, the weak process can be investigated only with bound strange systems. The study of the inverse process would however be useful.

Although the relevance of the NMWD channel was recognized since the early days of hypernuclear physics, only in recent years the field has experienced great advances due to the conception and realization of innovative experiments and to the development of elaborated theoretical models stab1 (); rev1 (); rev2 (); rev2a (); rev3 (); rev4 (); rev5 ().

For many years the main open problem in the decay of hypernuclei has been the puzzle, i.e., the disagreement between theoretical predictions and experimental results of the ratio between the neutron- and proton-induced decay widths: for all the considered hypernuclei the experimental ratio, in the range , was strongly underestimated (by about one order of magnitude) by the theoretical results. The ratio directly depends on the isospin structure of the weak process driving the hypernuclear decay. The analysis of the ratio is a complicated task, due to difficulties in the experimental extractions, which require the detection of the decay products, especially neutrons, and to the presence of additional competing effects, such as final-state interactions (FSI) of the outgoing nucleons and two-nucleon induced decays, which could in part mask and modify the original information.

In the first theoretical calculations the one-pion-exchange (OPE) nonrelativistic picture was adopted as a natural starting point in the description of the process, mainly on the basis of its success in predicting the basic features of the strong interaction. The first OPE models were able to reproduce the non-mesonic decay width but predicted too small ratios L1 (); L3 (); L4 (); L4b (); L2 (); L4c (). It thus seemed that the theoretical approaches tend to underestimate and overestimate . A solution of the puzzle then requires devising dynamical effects able to increase the -induced channel and decrease the -induced one.

In the following years the theoretical framework was improved including the exchange of all the pseudo-scalar and vector mesons, in the form of a full one-meson-exchange (OME) model, or properly simulating additional effects, above all initial short-range correlations (SRC) and FSI, by means of direct quark mechanisms and many-body techniques L1 (); L3 (); L4b (); L2 (); L8 (); L5 (); L6 (); L7 (); DI3 (); DI3b (); DI3c (); hqm (); Dq1 (); Dq2 (). In particular, the inclusion of exchanges seems essential to improve the agreement between theory and experiments. Only a few of these calculations have been able to predict a sizeable increase of the ratio L2 (); L4c (); L8 (); Dq2 (), but no fundamental progress has been achieved concerning the deep dynamical origin of the puzzle.

The situation has been considerably clarified during the very last years, thanks to considerable progress in both experimental techniques exp1 (); exp2 (); exp3 (); exp4 (); exp5 (); exp6 (); exp7 () and theoretical treatments L2 (); L4c (); L8 (); Dq2 (); th1 (); th2 (); th3 (); th4 (); jun (); Asth1 (); th5 (); last (); Bauer:2010tk (). From the experimental point of view, the new generation of KEK experiments has been able to measure the fundamental observables for the and hypernuclei with much more precision as compared with the “old” data, also providing the first results of simultaneous one-proton and one-neutron energy spectra, which can be directly compared with model calculations. Very recently, it has also been possible to obtain for the first time coincidence measurements of the nucleon pairs emitted in the non-mesonic decay, with valuable information on the corresponding angular and energy correlations. These new data further refine our experimental knowledge of the hypernuclear decay rates, also allowing a cleaner and more reliable extraction of the ratio. From the theoretical point of view, crucial steps towards the solution of the puzzle have been carried out, mainly through a non-trivial reanalysis of the pure experimental results by means of a proper consideration of FSI and rescattering mechanisms, inside the nuclear medium, for the outgoing nucleons, as well as of the two-nucleon induced channel. This strict interplay between theory and experiments is at the basis of the present belief that the puzzle has been solved. In particular, this is due to the study of nucleon coincidence observables, recently measured at KEK exp5 (); exp6 (), whose weak-decay-model independent analysis carried out in th1 (); th2 () yields values of around for the and hypernuclei, in satisfactory agreement with the most recent theoretical evaluations L2 (); L4c (); L8 (); Dq2 (). New, more precise results are expected from forthcoming experiments at DANE daphne () and J-PARC jparc ().

Another intriguing issue is represented by the asymmetry of the angular emission of non-mesonic decay protons from polarized hypernuclei. The large momentum transfer involved in the reaction can be exploited to produce final hypernuclear states characterized by a relevant amount of spin-polarization, preferentially aligned along the axis normal to the reaction plane pol1 (); pol2 (). The hypernuclear polarization mainly descends from a non-negligible spin-flip term in the elementary scattering process, which in turn interferes with the spin-non-flip contribution asym (): in free space, and for GeV and , the final hyperon polarization is about 75%.

Polarization observables represent a natural playground to test the present knowledge of the NMWD reaction mechanism, being strictly related to the spin-parity structure of the elementary interaction. Indeed, by focusing on the -induced channel, experiments with polarized hypernuclei revealed the existence of an asymmetry in the angular distributions of the emitted protons with respect to the hypernuclear polarization direction. Such an asymmetry originates from an interference effect between parity-violating and parity-conserving amplitudes for the elementary process, and can thus complement the experimental information on the and partial decay rates, which are instead mainly determined by the parity-conserving contributions. As for the ratio, FSI could play a crucial role in determining the measured value of this observable.

The asymmetry puzzle concerns the strong disagreement between theoretical predictions and experimental extractions of the so-called intrinsic asymmetry parameter . The first asymmetry measurements pol1 (); pol2 () with limited statistics gave large uncertainties and even inconsistent results. The very recent and more accurate data from KEK-E508 rev2a (); Asexp2 (); Asexp3 () favour small values of , compatible with a vanishing value. Moreover, the observed asymmetry parameters are negative for and positive (and smaller, in absolute value) for . Theoretical models generally predict negative and larger values of . FSI effects do not improve the agreement with data GarbarinoAsymmetry (). The inclusion, within the usual framework of nonrelativistic OME models, of the exchange of correlated and uncorrelated pion pairs GarbarinoTPE () greatly improves the situation. Indeed, it only slightly modifies the non-mesonic decay rates and the ratio, but the modification in the strength and sign of some relevant decay amplitudes is crucial and yields asymmetry parameters which lie well within the experimental observations. In particular, a small and positive value is now predicted for in . This important achievement justifies the claim that also the asymmetry puzzle has finally found a solution.

Recent experimental and theoretical studies have led to a deeper understanding of some fundamental aspects of the NMWD of -hypernuclei. From a theoretical point of view, the standard approach towards these topics has been strictly nonrelativistic, with both nuclear matter and finite nuclei calculations converging towards similar conclusions: nonrelativistic full one-meson-exchange plus two-pion-exchange models, based on the polarization-propagator method (PPM) PPM2 (); PPM3 () or on the wave-function method (WFM) L4 (); L2 (); L4c (), seem able to reproduce all the relevant observables for the and light-medium hypernuclei. A crucial contribution to this achievement is however due to a non-trivial theoretical analysis of KEK most recent coincidence data, based on the proper consideration and simulation of nuclear FSI and two-nucleon induced decays th2 (); GarbarinoTPE (); ICC (); ICCcorr (). In those nonrelativistic models many theoretical ingredients are included with unavoidable approximations. Initial-state interactions and strong SRC are treated in a phenomenological way, though based on microscopic models calculations. The inclusion of the full pseudo-scalar and vector mesons spectra, in particular the strange and mesons, in the context of complete OME nonrelativistic calculations, somewhat clashes with the still poor knowledge of the weak --meson coupling constants for mesons heavier than the pion. Their evaluation requires model calculations which unavoidably introduce a certain degree of uncertainty in the corresponding conclusions about the ratio and other observables. In order to reproduce few observables, i.e., the decay rates , , and , and the asymmetry parameter for and , these models need to include many dynamical effects, such as the exchange of all the possible mesons, plus two-pion exchange, plus phenomenological mesons, plus the corresponding interferences, and, moreover, strong nuclear medium effects in the form of non-trivial FSI. Although the inclusion of many theoretical ingredients can be considered as a natural and desirable refinement of the simple OPE models, all the improvements do not seem to provide a significantly deeper insight into the decay dynamics. Despite all the theoretical efforts, the solution of the and parameter puzzles seems due to effects, such as distortion, scattering and absorption of the primary nucleons by the surrounding nuclear medium, rather than to the weak-strong interactions driving the elementary or process. The re-analysis of the recent KEK experimental data exp1 (); exp5 (); exp6 (); E307corr () and the corresponding extraction of are indeed completely independent of the weak-decay-mechanisms th2 (); th5 (), but depend strongly on the model adopted to describe FSI and on somewhat arbitrary assumptions, e.g. on the ratio between two-nucleon and one-nucleon induced non-mesonic decay rates.

The presently available experimental information on hypernuclear decay is still limited and affected by uncertainties of both experimental and theoretical nature. Moreover, single-nucleon spectra seem to point at a possible systematic protons underestimation th5 (). The new generation of experiments planned in various laboratories worldwide is expected to produce more precise data on the already studied observables as well as new valuable information in the form of differential energy and angular decay particles spectra.

In spite of the recent important achievements, the NMWD of hypernuclei deserves further experimental and theoretical investigation. From the theoretical point of view, the role of relativity is almost unexplored. But for a few calculations in relPS (); rel1 (); rel2 (); Cmes2 () no fully relativistic model has been exploited to draw definite conclusions about the role of relativity in the description of the weak decay dynamics.

In this paper we present a fully relativistic model for the NMWD of PHD (). The adopted framework consists of a finite nucleus WFM approach based on Dirac phenomenology. As a first step the model includes only OPE and one-kaon-exchange (OKE) diagrams, and is limited to one-nucleon induced decay. We are aware that the neglected contributions could play an important role in the decays. Our aim is to explain all the at least qualitative features of the hypernuclear NMWD with a conceptually simple model, in terms of a few physical mechanisms and free parameters. We stress that, dealing with a fully relativistic treatment of the weak dynamics based on the calculation of Feynman diagrams within a covariant formalism, it is quite difficult to directly compare such an approach and its results to standard nonrelativistic OME calculations. We will thus rather focus on the internal coherence and on the theoretical motivations of the model.

The model is presented in Sec. 2. Numerical results for the total non-mesonic decay width , the ratio, the intrinsic asymmetry parameter, as well as for kinetic energy and angular spectra are presented and discussed in Sec. 3. The sensitivity to the choice of the main theoretical ingredients is investigated. The theoretical predictions of the model are compared with the most recent experimental results. Some conclusions are drawn in Sec. 4.

## Ii Model

In this Section we present a fully relativistic finite nucleus wave-function approach to study the NMWD of the hypernucleus. Our model is based on a fully relativistic evaluation of the elementary amplitude for the process, which, at least in the impulse approximation, is the fundamental interaction responsible for the NMWD. Covariant, complex amplitudes are calculated in terms of proper Feynman diagrams. The tree-level diagram involves a weak and a strong current, connected by the exchange of a single virtual meson. Integrations over the spatial positions of the two vertices as well as over the transferred 3-momentum are performed. As a first approximation, only OPE and OKE diagrams are considered. Possible two-nucleon induced contributions are neglected, even if they could play an important role in the hypernuclear decay phenomenology.

Interested readers can find further details about the present model in the PhD thesis of Ref. PHD (), where an extensive analysis of the adopted formalism as well as of the involved theoretical ingredients is provided.

In the calculation of the hypernuclear decay rate the Feynman amplitude must then be properly included into a many-body treatment for nuclear structure. The amplitude is therefore only a part of the complete calculation, but it is the basic ingredient of the model and involves all the relevant information on the dynamical mechanisms driving the decay process.

Short range correlations are also included in the model, coherently with what commonly done in most nonrelativistic calculations, since the relatively high nucleon energies involved in the hypernuclear NMWD can in principle probe quite small baryon-baryon distances, where strong interactions may be active and play an important role. Following a phenomenological approach, we have chosen to include initial SRC effects by means of a multiplicative local and energy-independent function, whose general form relPS () provides an excellent parametrization of a realistic correlation function obtained from a G-matrix nonrelativistic calculation Gmatrix (); Halderson (). The problem of ensuring a correct implementation of such a nonrelativistic SRC function within a relativistic, covariant formalism has been addressed in Ref. SRC () and shown to be tightly connected with the choice of the interaction vertices. For full generality, we also choose to account for possible strong short range interactions acting on the two final emitted nucleons, again adopting a simple phenomenological average correlation function L2 () which provides a good description of nucleon pairs in finalSRC () as calculated with the Reid soft-core interaction finalSRC2 (); such final-state SRC could in principle play an important role, and they complement the final-state interactions between each of one of the two emitted nucleons and the residual nulceus, that is accounted for in our model by a relativistic complex optical potential.

### ii.1 Coupling ambiguities

In order to devise a relativistic treatment of the elementary process, great care must be devoted to the choice of the Dirac-Lorentz structure for the strong and weak parity-conserving vertices. The pseudo-scalar () prescription, that consists in a Dirac structure, and the pseudo-vector () one, that contains a axial-vector structure, are in principle equivalent, at least for positive energy on-shell states, because they descend from equivalent Lagrangians. However, ambiguities arise when one tries to take into account SRC in terms of a multiplicative local and energy-independent function . Such ambiguities are not of dynamical origin and should not be mistaken as relativistic effects: they are simply bound to the phenomenological way of including (initial and final) short range correlations, by matching a nonrelativistic correlation function within a relativistic Feynman diagram approach. The crucial observation, in this regard, is that it is possible to give theoretical reasons SRC () to prefer the coupling in its modified version where the -derivative operates on the propagator , over the coupling and also over the standard one, where the -derivative acts on the matrix element. On the one hand, the choice permits to recover, in the nonrelativistic limit, the standard OPE potential, multiplied by , which is commonly used as the starting point in nonrelativistic calculations, whereas the and couplings yield a simple Yukawa function in the same limit: this allows, at least in principle, a comparison between relativistic and nonrelativistic results. On the other hand, a microscopic model of (initial) SRC effects, adopting standard vertices and introducing an additional -exchange mechanism simultaneous to the OPE dominant one, produces a result analogous to what can be derived in a phenomenological tree-level approach contemplating the inclusion of a SRC function, provided in this case the modified derivative coupling, rather than the one, is employed. The main feature is the development of an explicit dependence of the interaction matrix elements on the exchanged three-momentum (through the momentum involved in the corresponding loop integrals, in the microscopic model, or the derivative effect of the coupling, in the tree-level phenomenological approach). When dealing with a more complex model for nuclear structure, we do not generally use positive energy on-shell states. Still the general message keeps its validity, though the details of the explicit calculations may be different. In order to correctly treat SRC, nuclear currents showing a dependence on the 3-momentum transfer are needed, which in the simple model above correspond to matrix elements between external spinors and intermediate spinors carrying the momentum of the intermediate state excited by the heavy meson. This can be achieved using the coupling acting on the pion field, while the use of the or standard couplings, as done for instance in Ref. relPS (), would generate nuclear currents independent of , corresponding, within the considered simple SRC model, to matrix elements between spinors all carrying the external momenta.

### ii.2 Pseudo-scalar couplings

As a first example, we employ a coupling for the strong vertex and for the parity-conserving part of the weak interaction. The fundamental process can then be decomposed into a weak vertex, governed by the weak Hamiltonian

 (3)

and a strong vertex, driven by the Hamiltonian

 H(s)NNπ=igNNπ¯Ψ(s)Nγ5% \boldmathτ⋅ϕπΨ(b)N. (4)

The Dirac spinors and are the wave functions of the bound hyperon and nucleon inside the hypernucleus, is the Dirac spinor representing the scattering wave function of each one of the two final nucleons, is the vector formed by the three Pauli matrices, and is the isovector pion field. The Fermi weak constant and the pion mass give . The empirical constants and are adjusted to the free decay and determine the strengths of the parity-violating and parity-conserving non-mesonic weak rates, respectively. Finally, is the strong coupling. The initial and final nucleon fields, and , are defined in space-spin-isospin space and they are described by a space-spin part times a two component isospinor. In addition, the field is represented as a pure state to enforce the empirical selection rule.

The relativistic Feynman amplitude for the two-body matrix element describing the transition, driven by the exchange of a virtual pion, can be written as

 T(PS)fi,π = iGFm2πgNNπ∫d4x∫d4yfiniΛN(|\boldmathx−\boldmathy|) (5) × [¯Ψ(s)\boldmathk1,ms1,mt1(x)(Aπ+Bπγ5)τa1Ψ(b)αΛ,μΛ,−1/2(x)] × δabΔπ(x−y)ffinNN(|% \boldmathx−\boldmathy|) × [¯Ψ(s)\boldmathk2,ms2,mt2(y)γ5τb2Ψ(b)αN,μN,mtN(y)],

where and are the bound and nucleon wave functions, with quantum numbers and total spin (isospin) projections (, ), and , with , are the scattering wave functions for the two final nucleons emitted in the hypernuclear NMWD, with asymptotic momenta and spin (isospin) projections (). In both the initial and the final baryon wave functions it is possible to factor out the isospin 2-spinors as well as the energy-dependent exponentials: , where is the total energy of the considered baryon. The factor represents a short-range two-body correlation function acting on the initial and baryons, and similarly describes possible short range interactions between the two final nucleons emerging from the interaction vertex. is the Fourier transform of the product of the pion propagator with the vertex form factors (supposed to be equal for the strong and weak vertices), i.e.,

 Δπ(x−y)=∫d4q(2π)4eiq⋅(x−y)q2−m2π+iεF2π(q2) . (6)

After performing time integrations in Eq. (5) and taking advantage of the part of the integral in Eq. (6), we get for the relativistic amplitude the expression

 T(PS)fi,π = iGFm2πgNNπI∫d3%\boldmath$x$∫d3\boldmathyfiniΛN(|%\boldmath$x$−\boldmathy|) (7) × [¯ψ(s)\boldmathk1,ms1(%\boldmath$x$)(Aπ+Bπγ5)ψ(b)αΛ,μΛ(\boldmathx)] × Δπ(|\boldmathx−\boldmathy|)ffinNN(|\boldmathx−\boldmathy|)[¯ψ(s)\boldmathk2,ms2(\boldmathy)γ5ψ(b)αN,μN(\boldmathy)] × (2π)δ(E1+E2−EΛ−EN) ,

where

 Δπ(|\boldmathx−\boldmathy|)≡∫d3\boldmathq(2π)3e−i\boldmathq⋅(%\boldmath$x$−\boldmathy)(q0)2−\boldmathq2−m2π+iεF2π((q0)2−% \boldmathq2)∣∣∣q0=˜q0, (8)

with and is an isospin factor that depends on the considered decay channel (either - or - induced), i.e.,

 I ≡ [(χmt11/2)†% \boldmathτ1χ−1/21/2]⋅[(χmt21/2)†\boldmathτ2χmtN1/2] . (9)

It is easy to check that is different from zero only for the charge-conserving processes and . For the calculation of the integral over the 3-momentum transfer in Eq. (7) we choose a monopolar form factor, i.e.,

 Fπ(q2)≡Λ2π−m2πΛ2π−q2, (10)

where MeV is the pion mass and GeV is the cut-off parameter relPS (). The initial correlation function adopted in our calculations is relPS ()

 finiΛN(r)=(1−e−r2/a2)n+br2e−r2/c2, (11)

with , while the final correlation function is chosen as L2 ()

 ffinNN(r)=1−j0(qcr), (12)

where is the first spherical Bessel function, and fm. The two correlation functions of Eqs. (11) and (12) are plotted in Fig. 1. As a consequance of the approximations adopted in the present calculation, we could then, for practical purposes, treat initial and final SRC as a whole, in terms of an overall correlation functions defined as

 f(|\boldmathx−\boldmathy|)=finiΛN(|% \boldmathx−\boldmathy|)ffinNN(|\boldmathx−% \boldmathy|). (13)

### ii.3 Pseudo-vector couplings

When we use derivative couplings, the equivalent of Eq. (5) is

 T(PV′)fi,π = iGFm2πgNNπ∫d4x∫d4y (14) × [¯Ψ(s)\boldmathk1,ms1,mt1(x)(Aπ−iBπ2¯Mγ5γμ∂xμ)τa1Ψ(b)αΛ,μΛ,−1/2(x)] × δabΔπ(x−y)f(|\boldmathx−\boldmathy|) × [¯Ψ(s)\boldmathk2,ms2,mt2(y)(−i2MNγ5γν∂yν)τb2Ψ(b)αN,μN,mtN(y)],

where , with GeV, and now is defined as in Eq. (13). In Eq. (14) the space-time derivatives act just on the pion propagator (given in Eq. (8)) and not on the SRC function , which is considered as a phenomenological ingredient entering Eq. (14) in a factorized form. Thus, the involved derivatives translate into multiplicative terms under the integral over . After the time integrations in Eq. (14), we obtain the final expression of the relativistic amplitude

 T(PV′)fi,π = iGFm2πgNNπI∫d3%\boldmath$x$∫d3\boldmathy (15) × [∫d3\boldmathq(2π)3e−i\boldmathq⋅(\boldmathx−\boldmathy)q2−m2π+iεF2π(q2)f(|% \boldmathx−\boldmathy|) ×[¯ψ(s)\boldmathk1,ms1(\boldmathx)(Aπ+Bπ2¯Mγ5/q)ψ(b)αΛ,μΛ(\boldmathx)] ×[¯ψ(s)\boldmathk2,ms2(\boldmathy)(−12MNγ5/q)ψ(b)αN,μN(\boldmathy)] ×(2π)δ(E1+E2−EΛ−EN)] ,

where , , and are defined in Eqs. (9), (10), and (13), respectively. Eq. (15) must be evaluated at .

The crucial difference between Eq. (7) and Eq. (15) is that in Eq. (15), obtained adopting derivative couplings at the vertices, the matrix elements between the initial bound and the final scattering states explicitly depend on the 3-momentum transfer , while using couplings the matrix elements in Eq. (7) are independent of .

Though representing a computational complication, the -dependence of the matrix elements is a desirable feature in connection with the problem of correctly including short range correlations in a fully relativistic formalism as the one developed here. The use of vertices, which produces baryonic matrix elements only depending on the external variables, is not coherent with a simple but significant model of the physical mechanism behind SRC, based on the simultaneous exchange of a pion plus one or more heavy mesons. The correlated Feynman amplitudes involve box (or more complex) diagrams and one expects that the interaction matrix elements explicitly depend on the momentum involved in the corresponding loop integrals. This in turn represents a strong motivation to consider couplings as the most appropriate ones for a fully relativistic approach to the NMWD of -hypernuclei, since they prove able to mimic such a physical effect.

### ii.4 Initial- and final-state wave functions

The main theoretical ingredients entering the relativistic amplitudes of Eqs. (7) and (15) are the vertices operators and the initial and final baryon wave functions. Since we adopt a covariant description for the strong and weak interaction operators, the involved wave functions are required to be 4-spinors. Their explicit expressions are obtained within the framework of Dirac phenomenology in presence of scalar and vector relativistic potentials. In the calculations presented in this work the bound nucleon states are taken as self-consistent Dirac-Hartree solutions derived within a relativistic mean field approach, employing a relativistic Lagrangian containing , , and mesons contributions boundwf (); Serot (); adfx (); lala (); sha (). Slight modifications also permit to adapt such an approach to the determination of the initial wave function and binding energy. The explicit form of the bound-state wave functions reads

 ψμnκ(r)=(gnκ(|r|)Yμκ(Ω)ifnκ(|r|)Yμ−κ(Ω)), (16)

where the 2-components spin-orbital is written as

 Yμκ(Ω)≡∑μl,μs=±1/2(lμl1/2μs∣∣jμ)Yμll(Ω)χμs1/2, (17)

with

 j=|κ|−12and{l=κifκ>0l=−κ−1ifκ<0; (18)

is the radial quantum number and determines both the total and the orbital angular momentum quantum numbers. The normalization of the radial wave functions is given by

 (19)

The outgoing nucleons wave functions are calculated by means of the relativistic energy-dependent complex optical potentials of Ref. chc (), which fits proton elastic-scattering data on several nuclei in an energy range up to 1040 MeV. In the explicit construction of the ejectile states, the direct Pauli reduction method is followed. It is well known that a Dirac 4-spinor, commonly represented in terms of its two Pauli 2-spinor components

 ψk,ms(r)=(ϕk,ms(r)χk,ms(r)), (20)

can be written in terms of its positive energy component as

 ψk,ms(r)=⎛⎜ ⎜⎝ϕk,ms(r)[(σ⋅k)MN+E+S(|r|)−V(|r|)]ϕk,ms(r)⎞⎟ ⎟⎠, (21)

where and are the scalar and vector potentials for the final nucleon with energy . The upper component can be related to a Schrödinger-like wave function by the Darwin factor , i.e.,

 ϕ(r)≡√D(|r|)˜ϕ(r), (22)

with

 D(|r|)≡MN+E+S(|r|)−V(|r|)MN+E. (23)

The two-component wave function is solution of a Schrödinger equation containing equivalent central and spin-orbit potentials, which are functions of the energy-dependent relativistic scalar and vector potentials and . Its general form is given by

 ˜ϕk,ms(r) = √MN+E2E∑lmljμ4πil[ulj(|r|)Yμlj;|k|(Ωr)] (24) ×(lml1/2ms∣∣jμ)Yml∗l(Ωk).

### ii.5 Decay rates

In the complete calculation of the total and partial decay rates, as well as of polarization observables, the dynamical information on the elementary process, given by the amplitudes in Eq. (7) or Eq. (15), are included in a many-body calculation for nuclear structure. The weak non-mesonic total decay rate is defined as relPS (); NR1 ()

 Γnm = ∫d3\boldmathk1(2π)3∫d3\boldmathk2(2π)3(2π)δ(MH−ER−E1−E2) (25)

The energy-conserving delta function connects the sum of the asymptotic energies of the two outgoing nucleons, coming from the underlying microscopic process, with the difference between the initial hypernucleus mass and the total energy of the residual -particle system after the decay. A sum over is also usually understood. Integration over the phase spaces of the two final nucleons is needed, since the decay rate is a fully inclusive observable. Moreover, the sums in Eq. (25) encode an average over the initial hypernucleus spin projections , where is the hypernucleus total spin, a sum over all the spin and isospin quantum numbers of the residual -system, , as well as a sum over the spin and isospin projections of the two outgoing nucleons and , respectively. If we choose a reference frame in which, for instance, the -axis is aligned along the momentum , and exploiting the energy-conservation in the delta function, the six-dimensional integral in Eq. 25 can be reduced to a two-dimensional integral, one over the energy of one of the two final nucleons and the other one over the relative angle between the momenta of the two nucleons (due to azimuthal symmetry), which can be performed numerically.

The expression for the NMWD rate can be decomposed into a sum over - and -induced decay processes without any interference effects, i.e.,

 Γnm=∑mtNΓnm[mtN]=Γ(p)nm+Γ(n)nm, (26)

where is defined as in Eq. (25) and is evaluated with a fixed value of the initial-nucleon isospin projection, for -induced and for -induced channels. Actually, in each term of the quantum numbers are fixed, so that would involve products of the kind , where, in principle, also interference effects, , are allowed. However, the non diagonal products with are necessarily zero, since if one of the two amplitudes is non-zero the other one must vanish as a consequence of the charge-conservation isospin factor of Eq. (9) (same final state but different initial states, or ). Therefore, only the diagonal terms, , contribute and without interferences the coherent sum over becomes an incoherent one.

The nuclear transition amplitude, from the initial hypernuclear state to the final state of an residual nucleus and the two outgoing nucleons, is defined as

 Mfi=⟨f|^MΛN→NN|i⟩ (27)

and can be represented in terms of the elementary two-body relativistic Feynman amplitude, , which contains all the relevant information about the weak-strong dynamics driving the global decay process. The final -particle state must be further specified and decomposed into products of antisymmetric two-nucleon and residual -nucleon wave functions. An explicit decomposition for the initial hypernuclear wave function can be developed following the approach introduced in Ref. relPS (), which is based on a weak-coupling scheme, i.e., the isoscalar hyperon is assumed to be in the ground state and it only couples to the ground-state wave function of the -nucleon core. As discussed in Ref. relPS (), this weak-coupling approximation has been able to yield quite good results in hypernuclear shell-model calculations weakcoupl ().

The final expression for is

 ∣∣Mfi∣∣2[mtN] = (TRMTR1/2mtN∣∣THMTH)2 (28) × ∑jNA⟨JcTH{∣∣JRTR,jNmtN⟩2 × ×(JcMc1/2mΛ∣∣JHMJH)2∣∣TAfi,π∣∣2],

where for and for , are the spin-isospin quantum numbers for the initial hypernucleus, are the initial-nucleon total spin and its third component, are the same quantum numbers for the -nucleon core, and, finally, is the initial total spin projection. In Eq. (28) are the real coefficients of fractional parentage (c.f.p.), which allow the decomposition of the initial -nucleon core wave functions in terms of states involving a single nucleon coupled to a residual -nucleon state. The factor is produced by the combination of initial- and final-state antisymmetrization factors with the number of pairs contributing to the total decay rate. Eq. (28) neglects possible quantum interference effects between different values of (and ), namely we are ruling out interferences between different shells ( and ) for the initial nucleon. Thus the calculation does not require the c.f.p., but only the spectroscopic factors , that can be taken, e.g. from Ref. relPS ().

### ii.6 Antisymmetrization and isospin factors

A crucial role in determining the ratio is played by the isospin content of the model, namely the factors defined in Eq. (9) in terms of the SU(2) isospin operators (generally represented by the Pauli matrices) and of the corresponding isospin 2-spinors for the initial and as well as for the two final nucleons.

Taking advantage of the isospin selection rule, from the isospin point of view, the behaves like a neutron state. We can then explicitly represent the isospin spinors for the , and baryons as

 χp≡χmtN=1/21/2=(10),χn≡χmtN=−1/21/2=(01),χΛ=χn. (29)

With these definitions, the isospin factors can be evaluated for all the possible combinations of the , , and isospin projection quantum numbers. They are non-zero only for those processes in which charge is conserved, namely and . We obtain

 I[mtN=1/2,mt1=−1/2,mt2=1/2]≡I(d)Λp→np=−1, (30) I[mtN=1/2,mt1=1/2,mt2=−1/2]≡I(e)Λp→np=2, (31) I[mtN=−1/2,mt1=−1/2,mt2=−1/2]≡I(d)Λn→nn=I(e)Λn→nn=1. (32)

All the others possibilities imply charge violation and give zero. The and apices refer to the direct and exchange diagrams of the relativistic Feynman amplitudes for the elementary processes.

The use of the isospin formalism means that we are treating the neutron and the proton as two indistinguishable particles; therefore the final state is composed of two identical particles and this requires the antisymmetrization of the amplitude. The antisymmetrization acts on the two final nucleons, exchanging their spin-isospin quantum numbers, , and their momenta within the matrix elements defining the complex amplitude. We can thus define

 TAfi,π≡T(d)fi,π−T(e)fi,π, (33)

where is the Feynman amplitude for the direct diagram, given by Eqs. (5) or (12), while represents the Feynman amplitude for the exchange diagram, obtained from the same Eqs. (5) or (12), but with the interchanges , and . In addition, the antisymmetrization involves different factors for the direct and exchange diagrams: and for a final pair, and for a final pair. Taking advantage of the factorization , the antisymmetrized Feynman amplitudes can be written as

 Λp→np:TAfi,π = I(d)Λp→np˜T(d)fi,π−I(e)Λp→np˜T(e)fi,π (34) = (−)[˜T(d)fi,π+2˜T(e)fi,π], Λn→nn:TAfi,π = I(d)Λn→nn˜T(d)fi,π−I(e)Λn→nn˜T(e)fi,π (35) = [˜T(d)fi,π−˜T(e)fi,π].

We stress that many complex amplitudes with different quantum numbers contribute to the calculation of the nuclear transition amplitude. It is therefore difficult to make simple estimates of the final result.

### ii.7 Asymmetries in polarized hypernuclei decay

The main effect that is obtained with polarized hypernuclei is given by the angular asymmetry in the distribution of the emitted protons with respect to the direction of the hypernuclear polarization. It can be shown relPS () that the non-mesonic partial decay rate for the proton-induced process can be written as

 Γ(p)nm=12JH+1∑MJHσ(JH,MJH)≡I0(JH), (36)

where is the intensity of protons emitted along the quantization axis for a spin projection of the hypernuclear total spin . In terms of the isotropic intensity for an unpolarized hypernucleus, , the intensity of protons emitted in the non-mesonic decay of a polarized hypernucleus (through the elementary process) along a direction forming an angle with the polarization axis is defined by

 I(Θ,JH)=I0(JH)[1+Py(JH)Ay(JH)cosΘ], (37)

where is the hypernuclear polarization and the hypernuclear asymmetry parameter, both depending on the specific hypernucleus under consideration. The asymmetry is a property of the non-mesonic decay and it only depends on the dynamical mechanism driving the weak decay. In contrast, also depends on the kinematical and dynamical features of the associated production reaction. The explicit expression for reads

 Ay(JH)≡3JH+1∑MJHσ(JH,MJH)MJH∑MJHσ(JH,MJH), (38)

in terms of the quantities defined in Eq. (36).

Within the framework of the shell-model weak coupling scheme, supposing that the hyperon sits in the orbital and interacts (weakly) only with the nuclear core ground-state, angular momentum algebra can be employed to relate the polarization of the spin inside the hypernucleus to the hypernuclear polarization

 pΛ(JH)=⎧⎪ ⎪⎨⎪ ⎪⎩−JHJH+1Py(JH),ifJH=Jc−12,Py(JH),ifJH=Jc+12, (39)

where denotes the total spin of the