On the validity of the ∆I = 1 / 2 Isospin Rule and the Block–Dalitz Model in Hypernuclear Non–mesonic Weak Decay

On the validity of the ∆I =1/2 Isospin Rule and the Block–Dalitz Model in Hypernuclear Non–mesonic Weak Decay ` Oscar Hern´andez Mart´ın Master’s Thesis advisors: Assumpta Parre˜no, ` Angels Ramos Departament de F´ısica Qu`antica i Astrof´ısica and Institut de Ci`encies del Cosmos, Universitat de Barcelona, E–08028 Barcelona, Spain November 29, 2020 Abstract A study of the non–mesonic weak decay of light hypernuclei is performed on the basis of the Block–Dalitz approach which allows one to extract the elementary ΛN→nN decay amplitudes and rates without detailed knowledge of the microscopic interaction mechanism. The present work analyses the validity of this approach using decay rates calculated within a one–meson–exchange transition potential supplemented by a two–pion–exchange mechanism. It is found that although the Block–Dalitz model is a reliable approximation for predicting the decay rates of s–shell hypernuclei, it fails to reproduce the 5 ΛHe asymmetry parameter. Possible ∆I= 3/2 isospin contributions to the non–mesonic decay rates are also investigated within the factorization approximation. Unfortunately, the experimental data are still of limited precision and therefore, it is not possible to extract the degree of violation of the ∆I= 1/2 isospin rule. The present study introduces improvements on preceding theoretical models and, together with forthcoming J–PARC data on the non–mesonic decay of four– body hypernuclei, it will play an important role in establishing the detailed spin–isospin dependence of the ΛN→nN process as well as in the design of theoretical models. Acknowledgment M’agradaria agrair a les meves tutores, l’Assumpta i l’` Angels, la seva amable ajuda i guia en la realitzaci´o d’aquest treball. No hauria estat possible sense elles. Tamb´e m’agradaria agrair a les meves amigues i amics tota l’ajuda i estima que m’han donat en tota la meva formaci´o acad`emica. Per ´ultim, per`o no menys important, agrair el suport incondicional de la meva fam´ılia. Gr`acies de tot cor. 1 Contents 1 Introduction 4 2 Hypernuclear weak decay 8 2.1 Non–mesonic decay rate . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Two–body Spin-Isospin states . . . . . . . . . . . . . . . . 11 2.1.2 ∆I= 1/2rule......................... 14 2.1.3 Initial ΛNwavefunction .................. 15 2.1.4 Final NN wavefunction................... 16 2.2 Weak Transition Potential . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Decay rate asymmetry . . . . . . . . . . . . . . . . . . . . . . . . 21 3 Block–Dalitz model for s–shell hypernuclei 24 3.1 Pure ∆I= 1/2 and ∆I= 3/2 transitions . . . . . . . . . . . . . 31 4 Results 32 4.1 On the reliability of the Block–Dalitz model . . . . . . . . . . . 32 4.2 Four–body hypernuclei and test of the ∆I= 1/2 ......... 37 5 Conclusions 43 Appendices 48 Appendix A The Block–Dalitz non–mesonic decay rates 48 A.1 3 ΛHrate ................................ 48 A.2 4 ΛHrate ................................ 51 A.3 4 ΛHerate................................ 53 A.4 5 ΛHerate................................ 53 Appendix B Spin matrix elements 56 B.1 Spin–Spin Transition . . . . . . . . . . . . . . . . . . . . . . . . . 56 B.2 TensorTransition........................... 56 B.3 PV Transition: Pseudoscalar Mesons . . . . . . . . . . . . . . . . 56 B.4 PV Transition: Vector Mesons . . . . . . . . . . . . . . . . . . . 56 Appendix C The asymmetry parameter 57 C.1 Aterms ................................ 58 C.1.1 aa∗combination ....................... 58 C.1.2 ab∗combination ....................... 59 C.1.3 ac∗combination ....................... 59 C.1.4 ad∗combination ....................... 59 C.1.5 ae∗combination ....................... 60 C.1.6 af∗combination........................ 61 C.2 Bterms ................................ 61 C.2.1 ba∗combination ....................... 61 C.2.2 bb∗combination ....................... 62 2 C.2.3 bc∗combination ....................... 62 C.2.4 bd∗combination ....................... 63 C.2.5 be∗combination ....................... 64 C.2.6 bf∗combination ....................... 65 C.3 Cterms ................................ 65 C.3.1 ca∗combination ....................... 65 C.3.2 cb∗combination ....................... 66 C.3.3 cc∗combination ....................... 66 C.3.4 cd∗combination ....................... 66 C.3.5 ce∗combination ....................... 67 C.3.6 cf∗combination........................ 67 C.4 Dterms ................................ 68 C.4.1 da∗combination ....................... 68 C.4.2 db∗combination ....................... 68 C.4.3 dc∗combination ....................... 69 C.4.4 dd∗combination ....................... 70 C.4.5 de∗combination ....................... 70 C.4.6 df∗combination ....................... 71 C.5 Eterms ................................ 72 C.5.1 ea∗combination ....................... 72 C.5.2 eb∗combination ....................... 72 C.5.3 ec∗combination ....................... 73 C.5.4 ed∗combination ....................... 74 C.5.5 ee∗combination ....................... 75 C.5.6 ef∗combination........................ 76 C.6 Fterms ................................ 77 C.6.1 fa∗combination........................ 77 C.6.2 fb∗combination ....................... 77 C.6.3 fc∗combination........................ 78 C.6.4 fd∗combination ....................... 78 C.6.5 fe∗combination........................ 79 C.6.6 ff∗combination........................ 79 3 1 Introduction The strangeness quantum number, S, is conserved in processes mediated by the strong and electromagnetic interactions, but not in those involving the weak force. Baryons which contain this quantum number are called hyperons. The Λ baryon is the lightest hyperon, composed by u, d and s quarks and plays an important role in a variety of processes involving strangeness. For instance, hyperons may appear in the inner core of neutron stars and they are important players in their evolution and balance, which is given by the weak interactions. The mass of the Λ particle is mΛ= 1115.68 MeV and it has zero charge and isospin. With a mean free path and a lifetime of the order of 10 cm and 10−10 seconds, respectively, hyperons decay through weak processes (except the Σ0 which decay via electromagnetic processes) which do not conserve parity, isospin or strangeness quantum number. A hypernucleus is a bound system which contains nucleons (N) and one or more strange baryons (Y). An important point to describe the hypernuclear structure is the knowledge of the Y N and Y Y interactions in the same way that the NN interaction serves to know the nuclear structure. Although we could obtain more direct information of the Y N and Y Y interaction from cross section measurements, these are very difficult due to the short hyperon mean free path. For this reason, quantitative information of these interactions comes mainly from the study of production and decay of hypernuclei. The existence of these systems gives a new vision to the traditional nuclear physics, as for example the possibility of new selection rules and states with new symmetries. In fact, the study of strange nuclei have opens the door to studies of other exotic systems, as the ones in the charm sector. In addition, a precise knowledge of the baryon–baryon interactions in the strange sector would allow to determine the amount of flavor SU(3) symmetry breaking due to the different values of the quark masses in the light u, d and s sector. Hypernuclear physics was born in 1952 when Marian Danysz and Jerzy Pniewski discovered the first hypernucleus. They observed a delayed disintegration of a heavy fragment when they were working with emulsion chamber experiments [1]. These observations based on emulsion and bubble chambers motivated new experiments of hypernuclear production. With the advent of particle accelerators, beams of particles (usually kaons and pions) collide with a nucleon producing hyperons among the fragments of these collisions, which are then captured by a nucleus. Different reaction mechanisms are used: strangeness exchange reactions such as n(π−, K−)Λ, and associated strangeness production processes such as the hadronic n(π+, K+)Λ reaction or the electroproduction n(e, e0K+)Λ reaction. In these processes, hypernuclei can be created in some excited state but they reach their ground state via mechanisms mediated by the strong interaction, the electromagnetic one or by particle emission [2]. The Λ decay in free space proceeds mainly via the mesonic mode, Λ →Nπ, in which a meson (pion, in this case) and a nucleon are detected in the final 4 state. This decay mode is larger than the semi–leptonic and weak radiative ones by a factor of 103, so the Λ hyperon decays via two different mesonic channels: Λ →p+π−(63.9%) and Λ →n+π0(35.8%) [3]. The experimental ratio for the free decay rates, Γfree π−/Γfree π0≈1.78, is very close to 2 and strongly suggests the dominance of ∆I= 1/2 isospin transitions over the ∆I= 3/2 ones, which is commonly known as the ∆I= 1/2 rule for weak processes. Although this rule is well established experimentally and it turns to be valid for different processes that involve strangeness (such as the decay of the kaon meson [4]), its dynamical origin is not yet understood. Neither is it if the ∆I= 3/2 suppression is a universal feature of all weak decay processes. In case of finding a significant relevance of ∆I= 3/2 terms in the strangeness physics sector it would be the first indication of violation of the ∆I= 1/2 selection rule. Such experimental evidence has not been found yet. When the Λ particle is embedded in a nuclear medium, the mesonic decay mode becomes Pauli blocked because the final nucleon momentum (pN≈100 MeV/c) is smaller than the Fermi one (kF≈270 MeV/c). As we shall see in the following, although this mode is blocked in infinite nuclear matter, it can occur in finite nuclei under different circumstances, being less important as the mass number increases [7]. The same medium which is responsible for the suppression of the mesonic mode is also responsible for the appearance of a new decay mode, the weak non–mesonic decay which becomes dominant in hypernuclei with A≥5. Hence, except for the very lightest ones, Λ–hypernuclei decay mainly through the non–mesonic decay, in which mesons are not detected in the final state and involves decay channels induced by one (ΛN→nN) and two (ΛNN →nNN) nucleons, the two nucleon–induced channel representing the 10 −15% of the total non–mesonic weak decay rate [5]. Relevant quantities for comparison between theory and experiment are the total decay width, which can be expressed in terms of the mesonic and non–mesonic decay widths, the ratio between the neutron–induced (Λn→nn) and proton– induced (Λp→np) decays, Γn/Γp, and the asymmetry in the angular distribution of the nucleons from the weak decay of hypernuclei. This asymmetry arises from the interference between the parity–conserving (PC) and parity–violating (PV) amplitudes on the weak decay mechanism, as we will see in Section 3. The total decay rate of a Λ–hypernucleus is then: ΓT= ΓM+ ΓNM ,(1) where ΓM= Γπ−+ Γπ0,ΓNM = Γp+ Γn+ Γ2,(2) where Γ2stands for the two–nucleon induced channel (Λnn →nnn, Λnp →nnp, Λpp →npp). Although it is true that the total width is well reproduced by different models, one of the problems that hypernuclear studies had to face for many years was to 5 reconcile experimental and theoretical values for the Γn/Γpand for the asymmetry parameter [2,6–8]. There were many efforts devoted to provide a theoretical explanation of the large experimental values obtained for this ratio. These experimental results were quite scarce and not precise, so they presented large uncertainties due to, in part, the difficulty of detecting the products of non– mesonic decay, especially the neutrons. Essentially, two kinds of approaches are available for predicting the decay rates and the asymmetry parameters of Λ– hypernuclei. The first one consists in a finite nucleus formalism where by using a weak coupling scheme and a shell model approach, the many–body transition amplitude is expressed in terms of the two–body amplitudes calculated using a meson–exchange potential. Other approaches include a many–body technique in which the calculation is performed in infinite nuclear matter and then it is extended to finite nuclei through the local density approximation. [6,9]. Given the impossibility of generating stable hyperon beams, hypernuclear decay offers us the best scenario to study the weak ΛN→nN interaction. Moreover, using the change of strangeness as a signature, this mechanism allows us to extract more information than the strangeness 0 counterpart, the weak NN interaction, since one can explore not only the the parity–violating part of baryon–baryon interaction but also the parity–conserving one. The non–mesonic decay can be understood in terms of the free–space mechanism, where the virtual pion emitted at the weak vertex, ΛNπ, is absorbed by a nucleon in the system. Although this one–pion-exchange process (OPE) generates good estimations for the total non–mesonic decay rate, the model fails to reproduce the Γn/Γpratio. The reason lies in the tensor dominance of the pion-exchange mechanism, which favors the p–induced process leading to np pairs in the final state. Due to the large momentum of the final nucleon ( k≈400MeV/c), it is advisable to include not only long–range contributions (those generated by the pion) but also short–range contributions generated by heavier mesons. The OPE model was generalized to a one–meson–exchange model (OME) including the mesons of the pseudoscalar (η,π,K) and the vector octets (ρ,wand K∗) [10]. Although the OME model describes the non– mesonic rates and the ratio Γn/Γpsatisfactorily, it predicts a too large and negative asymmetry parameter. Fortunately, recently these problems were resolved [8]. The degree of development reached in the last years permitted to achieve a reasonable agreement between theory [11–23] and experiment [24–32]. In particular, within a finite nucleus framework which adopted a OME model supplemented by a chirally motivated (correlated plus uncorrelated) two–pion exchange models (TPE), it was possible to reproduce the asymmetry and the decay rates of 5 ΛHe and 5 ΛC [11]. As we will see in the following, the correlated (2π/σ) and uncorrelated (2π) two–pion exchanges drastically modify the asymmetry parameter while exerting a moderate effect on the decay rates. However, although the OME+TPE model was able to reproduce the measured observables of non–mesonic decay, the discrepancies between various models fall within the large error bars of the experimental data, meaning that the weak decay of hy- 6 pernuclei is not yet properly understood. Other models have been developed to address some of the observed discrepancies between theory and experiments. Among them, hybrid models combining quark and meson degrees of freedom [33] or calculations based on effective field theories [34]. In the present work, we will study the weak non–mesonic decay based on the OME and OME+TPE models for different hypernuclei using the finite nucleus approach. The Λ particle can interact with the nucleons of the s,por higher shells, involving a relative angular momentum of the ΛNpair L6= 0. In our case, we will consider s–shell hypernuclei since it has the merit of considerably reducing the number of possible states for the initial state (and thus the final state) by considering the ΛNpair in a relative angular momentum of L= 0. Possible violations of the ∆I= 1/2 rule are discussed using the phenomenological Block–Dalitz model, whose limits of applicability have not been discussed to date. Although this approach starts from a contact interaction which does not consider the long–range effects of the exchange model, we will see that it will be extremely good in reproducing the decay rates of light hypernuclei in terms of a few elementary spin–isospin rates for the Λn→nn and Λp→np processes without detailed knowledge of the microscopic interaction mechanism. This work is organized as follows. In Section 2 we briefly detail the main characteristics of the mesonic and the non–mesonic weak decay, as well as the finite nucleus formalism used for the calculation of the decay rates and asymmetries. The weak transition potential is based on one–meson–exchange and one–meson– exchange plus two–pion–exchange. In Section 3 the approach of Block–Dalitz to the non–mesonic weak decay of s–shell hypernuclei is discussed. Numerical results obtained with the finite nucleus calculation for the decay rates and the asymmetries of four– and five–body hypernuclei are presented in Section 4, together with a discussion on the reliability of the Block–Dalitz model. Moreover, possible violations of the selection isospin rule ∆I= 1/2 are investigated by adopting the factorization approximation of Ref. [35]. Finally, some concluding remarks are given in Section 5. 7 2 Hypernuclear weak decay In order to discuss the weak decay modes of hypernuclei, it is convenient to introduce the Fermi gas model. This model defines the properties of an infinite system of non–interacting identical fermions which obey the Pauli exclusion principle. In the isospin picture, nuclear matter is made up of nucleons (proton and neutron are two states of the same particle, the nucleon) with mono–particular states described by plane wave functions. The energy and the single–particle nucleon wave function read: Φ~ k=1 √Vei~ k~rχs msχt mt, E =~2k2 2mN ,(3) where the two χ’s denote the spin and isospin states. According to the Pauli principle, each nucleon in this system occupies a different mono–particular state with one of the two possible values of the third spin component, ms±1/2 and one of the two possible values of isospin, mt±1/2. The maximum value of |~ k|corresponding to the occupied states in the ground state is the Fermi momentum, kF. Therefore, the density reads: ρ(r) = X k<kF|Φ~ k|2=2 3π2k3 F.(4) Using the normal nuclear matter density, ρ(r)≈0.16 fm−3, one obtains kF= 270 MeV/c . The Λ hyperon production takes place by collision reactions between different kind of mesons (pions and kaons) and a nucleus by means of processes such as the n(K−, π−)Λ and n(π+, K+)Λ reactions. The creation of Λ–hypernuclei requires that the Λ particle created in these reactions remains in the nuclear system. As we already mentioned, the free Λ decays mainly via two mesonic weak decay modes (Λ →π−p, π0n). Using energy–momentum conservation, it is possible to evaluate the Q–value and the final momentum: mΛ≈p~p2+m2 π+q~p2+m2 N,(5) Q≈mΛ−mπ−mN≈35 MeV ,(6) p≈100 MeV/c.(7) Inside the hypernucleus, the hyperon and the nucleon feel the action of attractive mean fields UNand UΛ, which come from the NN and ΛNinteractions, respectively. The binding energies of the recoil nucleon (BN≃ −8 MeV) and of the Λ (BΛ≥ −27 MeV) affect the above expressions [36]. Qbound =Q+BΛ−BN is smaller than the free Q–value and hence tends to reduce p. Since the Fermi momentum is kF≈270 MeV/c, the mesonic weak decay process becomes Pauli blocked due to the fact that final momentum (~p) is not large 8 Denoting by Γfree π0the mesonic Λ →n+π0decay width and by Γfree π−the mesonic Λ→p+π−ones, the ratio of rates for ∆I= 1/2 yields: Γfree π− Γfree π0≈| hπ−p|ˆ T1/2,91/2|Λi |2 | hπ0n|ˆ T1/2,91/2|Λi |2=p2/3 p1/3 2 = 2 ,(27) while for ∆I= 3/2 processes one finds: Γfree π− Γfree π0≈| hπ−p|ˆ T3/2,91/2|Λi |2 | hπ0n|ˆ T3/2,91/2|Λi |2=p1/3 p2/3 2 =1 2,(28) where the ˆ T1/2and ˆ T3/2operators change 1/2 and 3/2 units of isospin, respectively. Experimentally the ratio of the relevant widths is Γfree π−/Γfree π0≈1.78 [2]. As we can see above, only in the first case seems the experimental prediction to be recovered, although there is room for some amount of ∆I= 3/2 contributions. 2.1.3 Initial ΛNwave function Although the total hypernuclear system has been decomposed in terms of angular momentum coupling between the ΛNpair and the (A92) nuclear core, it is still necessary to know the initial and final two–body wave functions. As for the Λ and Nwave functions in the initial state, we use the solutions of a harmonic oscillator potential, with appropriate values of the oscillator parameters bΛand bNthat are adjusted to reproduce the binding energy of the Λ–particle in the considered hypernucleus and the form factor for the (A–1)–particle core, respectively. For 5 ΛHe we use bΛ= 1.85 fm and bN= 1.39 fm, while bΛ= 2.04 fm and bN= 1.54 fm are taken for 4 ΛHe and 4 ΛH. Working in the center of mass frame system, the initial Λ and Nwave functions can be written in terms of relative (rel) and center of mass (CM) harmonic oscillator wave functions describing the ΛNsystem as: ΦΛ 100(~r1 b)ΦN 100(~r2 b) = Φrel 100(~r √2b)ΦCM 100 (~ R √2b),(29) where b=bΛ+bN 2is the average oscillator parameter. Consequently, the two– body transition in Eq. (24) can be factorized in terms of other transition amplitudes, which depend on relative (Nr,Lr) and CM (NR,LR) principal and orbital angular momentum quantum numbers of the ΛNand NN systems: tΛN→NN =X NrLrNRLR X(NrLrNRLRlΛlN)tNrLrNRLR ΛN→NN ,(30) where X(NrLrNRLR, lΛlN) are called the Moshinky brackets. For lΛ=lN= 0 the value is X(101000) = 1 [39]. 15 Using this approach, the initial nucleon and Λ are assumed to be independent but there is a correlation between them due to the action of the strong interaction. To take this initial interactions into account, the harmonic oscillator wave function is replaced by a correlated ΛNwave function which is obtained by multiplying the uncorrelated Λ and Nwave functions by a function that is adjusted to microscopic G–matrix calculations in nuclear matter. For the baryon–baryon strong interactions we take the Nijmegen soft–core model, version NSC97f [40], which has been used with success in hypernuclear structure and decay calculations. We follow the Ref. [10], where the correlation function is parametrized as: fΛN(r) = (1 −exp−r2/a2)n+br2exp−r2/c2,(31) with a= 0.5 fm, b= 0.25 fm, c= 1.28 fm and n= 2. 2.1.4 Final NN wave function The final wave function of the two outgoing free nucleons can be written as h~r1~r2|~ k1~ k1s1ms1s2ms2t1mt1t2mt2i=ei~ k1~r1ei~ k2~r2χs1 ms1χs2 ms2χt1 mt1χt2 mt2(32) where the four χ’s are the spin and isospin states. We use relative and CM coordinates in such a way that the final state is rewritten as follows: h~ R~r |~ P~ kSMSTMTi=ei~ P~ Rei~ k~rχS MSχT T3,(33) where χS MSand χT T3are the aforementioned total spin and isospin wave function, respectively. In order to incorporate the antisymmetrization of the two–nucleon wave function, it is necessary to include the exchange of their coordinates and quantum numbers. This involves exchanging ~ k→9 ~ kand adding the factor (−1)S+T. Therefore, Eq. (33) is rewritten as: h~ R~r |~ P~ kSMSTMTi=1 √2ei~ P~ Rei~ k~r −(−1)S+Te−i~ k~rχS MSχT MT.(34) Moreover, it is also possible to incorporate interaction effects between the two outgoing nucleons by substituting the plane wave function by a distorted wave function, which is the solution of the Lippmann–Schwinger equation with an appropriate NN potential. The Lippmann–Schwinger equation is |Φ(±)i=|φi+1 E−H0±iV|Φ(±)i,(35) where |φirepresents a solution of the free Hamiltonian H0,Eis the energy of the two final nucleon state represented by |Φ(±)i. The plus (minus) signs denotes states at infinite time before (after) the interaction. Multiplying Eq. (35) by the potential Vand defining V|Φ(±)i ≡ T±|φi, one obtains the T–matrix equation. T±=V+V G± 0T±,(36) 16 Table 2: Possible 2S+1LJchannels in the weak decay (WD) of s–shell hypernuclei incorporating the strong final state interaction (FSI) ΛN(Lr) WD NN (L0) FSI NN (L) 1S0−→ 1S0−→ 1S0 3S1−→ 3S1−→ 3S1,3D1 3S1−→ 3D1−→ 3D1,3S1 1S0−→ 3P0−→ 3P0 3S1−→ 1P1−→ 1P1 3S1−→ 3P1−→ 3P1 where G± 0= (E−H0±i)91is the free–space Green’s function. Once the T–matrix equation has been solved and using the previous equations and definitions, the correlated NN wave function (Φ) is obtained. We will denote as Φ~ k in the relative and center of mass coordinates. Therefore, the free wave function ei~ k~r, which is a solution of Schr¨odinger equation, is replaced by the correlated wave function of the two nucleons Φ~ k. In Table 2 we present all the possible final states starting from ΛNinitial states with L= 0 and considering the final state interaction after the action of the weak transition potential. The strong interaction conserves, among other properties, parity (−1)Land total angular momentum (∆J= 0). Specifically, the tensor component (∆L= 2) of the strong final state interaction produces a mixing between 3S1and 3D1NN final state whereas the central part (∆L= 0) does not produce an overlap between different channels. Based on these considerations, the matrix elements of Eq. (30) are: tNrLrNRLR ΛN→NN =1 √2Zd3RZd3re−i~ P·~ RΦ∗ ~ k(~r)χ†S MSχ†T T3V(~r)ΦCM NRLR ~ R b/√2! ×Φrel NrLr~r √2bχS0 MS0χT0 T30 = (2π)3/2ΦCM NRLR~ Pb √2trel , (37) with ΦCM NRLR~ Pb √2is the Fourier transform of the ΛNcenter of mass oscillator wave function and trel =1 √2Zd3rΦ∗ ~ k(~r)χ†S MSχ†T T3V(~r)Φrel NrLr~r √2bχS0 MS0χT0 T30.(38) 17 2.2 Weak Transition Potential The weak transition potential is built from the exchange of virtual mesons belonging to the ground state pseudoscalar (π,ρ,K) and vector octets (K∗,ω, η) [10], to which we add the uncorrelated (2π) and correlated (2π/σ) two–pion– exchange [11]. One of the merits of this weak potential was that for the first time it was possible to reproduce the experimental data of both the asymmetry parameters and the decay rates for 5 ΛHe and 12 ΛC. In this work we show the final expressions of the transition potentials. The one-meson exchange contributions are obtained from the corresponding Feynman amplitudes, displayed in Figure 1, conveniently Fourier transformed to coordinate space. In order to account for the finite size of the hadrons involved in the transition, form factors are included at the weak and strong vertices of these amplitudes. More details are given in Ref. [10]. We note that, while the baryon–baryon strong coupling constants are taken from the model NSC97f, in the present work we employ monopole–type form factors at the meson–baryon– baryon vertices, typical of the Juelich BB interaction models [41] with similar cut–off values, instead of the exponential–type form–factor of the Nijmegen models adopted in [11]. This new procedure gives rise to similar results as those obtained in Ref. [11] and has the advantage of permitting a more straightforward comparison with the parameters employed by other models in the literature. Within the OME model, the decomposition of the weak interaction potential for the ΛN→nN transition is given in the following way: V(~r) = X i Vi(~r) = X iX α V(i) α(r)ˆ Oαˆ I(i) α,(39) where the index iruns over all exchanged mesons and the index αover the different spin–space operational structures. The spin operators ˆ Oαare: ˆ Oα=           ˆ 1 central spin-independent (C) ~σ1·~σ2central spin-dependent (SS) S12(~r) = 3(~σ1·~r)(~σ2·~r)−~σ1·~σ2tensor (T) ~σ2·~r PV for pseudoescalar mesons (~σ1×~σ2)·~r PV for vector mesons and the isospin operators ˆ I(i) αare: ˆ I(i) α=   ˆ 1 isoescalar mesons (η, ω) ~τ1·~τ2isovector mesons (π, ρ) linear combination of ˆ 1 and ~τ1·~τ2isodoublet mesons (K, K∗) 18 Figure 1: Feynman diagrams corresponding to the weak ΛN→NN transition amplitudes mediated by the exchange of the non–strange (π, ρ, η, ω) and the strange (K, K∗) mesons. The circle and the square stand for a weak and a strong vertex, respectively. The different pieces of V(i) αcan be written in terms of Yukawa or Yukawa–like functions multiplied by certain factors, K(i) α, which contain weak and strong coupling constants: V(i) C(r) = K(i) C exp−µir 4πr ≡K(i) CVC(r, µi),(40) V(i) SS (r) = K(i) SS 1 3µ2 i exp−µir 4πr −δ(r)≡K(i) SSVSS(r, µi),(41) V(i) T(r) = K(i) T 1 3µ2 iexp−µir1 + 3 µir+3 (µir)2≡K(i) TVT(r, µi),(42) V(i) P V (r) = K(i) P V µiexp−µir1 + 1 µir≡KP V VP V (r, µi),(43) where µidenotes the mass of the different mesons. The expressions for K(i) αare given in [10]. 19 Figure 2: Uncorrelated [(a) and (b)] and correlated [(c)] two–pion–exchange Feynman diagrams corresponding to the weak ΛN→NN transition amplitudes. The circle and the square stand for a weak and a strong vertex, respectively. This OME potential is supplemented by chirally motivated contributions of uncorrelated (2π) and correlated (2π/σ) two–pion–exchange mechanism. The relevant diagrams are shown in Fig. 2. The two–pion–exchange (TPE) interaction takes place only in the scalar–isoscalar channel. This is due to the fact that the ρmeson is already represented in the OME mechanism and, therefore, the vector–isovector channel (2π/ρ) is not implemented here. Also, the intermediate NN states are not considered in these diagrams to avoid double counting when including the NN strong correlations. Furthermore, we note that the full TPE potential is of pure parity–conserving nature. Once the expression for the potential is known, what remains to be done in Eq. (38) is to perform a partial–wave expansion of the final wave function, so that the relative ΛN→nN amplitude trel in Eq. (38) can be written as follows: trel =1 √2X iX αX LL0J 4πi−L0hLMLSMS|JMJiYLML(ˆ kr) ×hLrMLrS0MS0|JMJih(L0S)JMJ|ˆ Oα|(LrS0)JMJi ×hTT3|ˆ Ii α|T0T30iZr2drΦ∗J LL0(kr, r)Vi α(r) ×Φrel NrLr(r/√2b), (44) where the function Φ∗J LL0(kr, r) is the scattering solution of the two final nucleons moving under the influence of the strong interaction. The explicit expressions for the expectation value of the spin–space piece of ˆ Oαcan be found in Appendix B. 20 2.3 Decay rate asymmetry Λ–hypernuclei can be created with a certain degree of polarization depending on the kinematic conditions employed in the production reactions. This means that the hypernucleus has a total angular momentum (JI) in a preferred direction in space, known as polarization axis. JIcomes from the addition between the spin of the Λ–particle and the nuclear core angular momentum, JC. The two dominant decay mechanisms are the proton– and neutron–induced decay, where we have added a vector index in the Λ to indicate that the hypernucleus is polarized: ~ Λp→np ~ Λn→nn Protons from the proton–stimulated decay are emitted asymmetrically with respect the polarization axis of Λ–hypernuclei. We denote as θthe angle between the direction of the outgoing protons and the polarization axis. This asymmetric emission means that the number of protons emerging in a certain direction (θ) is different from the number of protons in the opposite direction (180◦+θ). As we will see later, this asymmetry comes from the interference between the PC and PV amplitudes. Therefore, the study of polarized hypernuclei provides us with extra information on the non–mesonic weak decay, such as the relative phases between PC and PV amplitudes, which puts additional restrictions to the theoretical models. In principle, there is also an asymmetry with respect to the neutrons emitted in the proton–stimulated decay, but its measurement is hard to reach. In addition to this, it should be pointed out that there is no neutron– asymmetry from neutron–stimulated decay because one can not distinguish the two final neutrons. From now on we will focus on the asymmetry generated in the accessible ~ Λp→ np process. It is obtained from the intensity of protons emitted along a direction forming an angle θwith respect to the hypernuclear polarization axis [37]: I(θ, JI)≡Tr[Mρ(JI)M†](θ) = X F,M,M0hF;θ|M|I;JI, MIi ×hI;JI, MI|ρ(JI)|I;JI, M0ihI;JI, M0|M†|F;θi, (45) where Mis the operator describing the elementary transition, |I;JI, MIiis the initial hypernuclear state, MIdenoting the third component of the total hypernuclear spin JIand |F;θiis the final state, which is composed by the residual nucleus and the two outgoing nucleons, and ρ(JI) is the density matrix of the polarized hypernucleus. For pure vector polarization perpendicular to the plane of the n(K+, π+)Λ reaction producing the hypernucleus, the density matrix is given in Ref. [7] as: ρ(J) = 1 2J+ 1[1 + Py(JI)Sy 3 JI+ 1],(46) 21 where Pyis the hypernuclear polarization, which depends on the kinematics and dynamics of the hypernuclear production reaction, and Syis the J–spin operator along the polarization axis. From Eq. (46) one can obtain the proton distribution in the following form: I(θ, JI) = I0(JI)[1 + A(θ, JI)] = I0(JI)[1 + Py(JI)Ap(θ, JI)] ,(47) where I0is the isotopic intensity for an unpolarized hypernucleus and Apis the hypernuclear asymmetry parameter which read: I0(JI) = Tr(MM†) 2JI+ 1 ≡Γp,(48) A(θ, JI) = Py(JI)3 JI+ 1 Tr(MSyM†)(θ) Tr(MM†).(49) It is clear that, by construction, we can write down two properties of I(θ, JI) and A(θ, JI). Zπ 0 dθI(θ, JI) = I0(JI).(50) Zπ 0 dθA(θ, JI)=0.(51) From the last equation, the function A(θ, JI) can be expressed as a series of odd powers of cos θ. If we truncate to first order in the expansion, we have: A(θ, JH) = Ccos θ . (52) It is clear that if the hypernucleus is generated without polarization, there is no asymmetry in the protons emitted in proton-stimulated decay. This means that A(θ, JH) = 0. For this reason, and as shown in Eq. (47), the constant C can be expressed as a product of the hypernucleus polarization, Pytimes a remainder quantity, called by exclusion as the asymmetry parameter, Ap. C≡PyAp.(53) Using these considerations, the Eq. (47) is rewritten as follows: I(θ, JI) = I0(JI)[1 + A(θ, JI)] = I0(JI)[1 + Py(JI)Ap(JI) cos(θ)] .(54) The shell model weak–coupling scheme allows rewriting the asymmetry in terms of the polarization of the hyperon spin, pΛand the so–called intrinsic asymmetry parameter, aΛ: A=PyApcos(θ) = pΛaΛcos(θ).(55) This is as if we considered that the process responsible for the appearance of the asymmetry is the elementary process ~ Λp→np taking place in the nuclear medium. In other words, the polarization and the asymmetry parameter of the entire hypernucleus are referred to as those of the elemental decay process. 22 This representation implies that the asymmetry parameter should be practically independent of the decaying hypernucleus. Several calculations [2, 6, 10, 11, 15, 17, 37, 42] of the asymmetry parameter of different hypernuclei show only moderate dependence on the hypernuclear structure. Although this property is also supported by experimental data [28], one has to note that there are still large error bars. Using the weak–coupling scheme, the relation between the hypernuclear and the Λ polarization is: pΛ=−J J+1 Pyif J=JC−1 2 Pyif J=JC+1 2 (56) From Eq. (55) it then follows: aΛ=−J+1 JApif J=JC−1 2 Apif J=JC+1 2 (57) where aΛcan be reinterpreted as an intrinsic value of the process as mentioned above. Note that, for example, in case of 5 Λ~ He, J= 1/2 and JC=0, the intrinsic asymmetry parameter is aΛ=Ap(5 Λ~ He). Note also that aΛ=Ap= 0 if J= 0. In relation to this last result, the four–body hypernuclei have J= 0 and, therefore, the asymmetry parameter vanish in this approach. In Appendix C one can find the expression of the asymmetry parameter for s–shell hypernuclei in terms of the elementary amplitudes appearing in Table 1. From all these considerations we can rewrite Eq. (54) as: I(θ, JI) = I0(JI) [1 + pΛaΛcos θ].(58) Experimentally, the asymmetry is determined by measuring I(0◦) and I(180◦) as: aΛ=1 pΛ I(0◦, JI)−I(180◦, JI) I(0◦, JI) + I(180◦, JI).(59) 23 3 Block–Dalitz model for s–shell hypernuclei An approach was proposed by Block and Dalitz [43] which allows one to determine the neutron– and proton–induced decays in terms of a few rates for the elementary process ΛN→nN in s–shell hypernuclei. As we will see in this chapter, the relationship between this elementary rates is affected by the isospin change involved in the non–mesonic weak decay. In principle, both ∆I= 1/2 and ∆I= 1/3 transitions are possible, but most of the models impose the phenomenological ∆I= 1/2 rule observed in hadronic weak processes. In previous works these rates were determined phenomenologically by fitting the available data on s–shell hypernuclei for total and partial decay widths [44–49]. The usual outcome was the large error bars prevent from drawing any definitive conclusions. For this reason, more precise measurements of the decay rates are necessary, especially for four–body hypernuclei, in order to obtain reliable conclusions on the identification of possible violation of the ∆I= 1/2 selection rule. Fortunately, an upcoming experiment at J-PARC will measure these decay rates more precisely [50]. The Block–Dalitz approach obtains the decay rates employing simple arguments. In terms of classical physics, the interaction probability of a particle which crosses an infinite homogeneous system is dP =ds/λ, where ds is the thickness, λ= 1/(σρ) is the mean free path of this particle, σis the cross section and ρis the particle density of the homogeneous system. In this way, one can evaluate the width for the non–mesonic weak decay by: ΓNM =dPΛN→nN dt =ds λdt =σρds dt =σρv , (60) where vis the speed of Λ particle in the rest frame of the system. Within a semiclassical approximation for a finite nuclei one can evaluate the width by performing an average over spin and isospin states. In addition, if one assumes the zero range approximation of the non–mesonic process, the decay rate is proportional to the overlap between the Λ and nuclear densities [2]. ΓNM =hσviZd(~r)ρ(~r)|ψΛ(~r)|2=hσviρA,(61) being ρAthe nuclear density normalized to the mass number A=Z+Nat the position of the Λ particle and ψΛ(~r) is the Λ wave function normalized to unity. Consequently, in the Block and Dalitz approach, the non–mesonic width ΓNM = Γn+ Γpof the hypernucleus A+1 ΛZturns out to be factorized into a density–dependent factor (ρA) and a term incorporating the dynamics of the non–mesonic decay [43]. Γ(A+1 ΛZ) = R(A+1 ΛZ)ρA=NRn(A+1 ΛZ) + ZRp(A+1 ΛZ) AρA,(62) where Rdenotes a rate (per unit nucleon density at the Λ position) average over spin and isospin and is given in the second equality in Eq. (62) in terms of 24 3.1 Pure ∆I= 1/2and ∆I= 3/2transitions We have already mentioned the characteristics in case of assuming that the ΛN→nN transition occurs with a ∆I= 1/2 and ∆I= 3/2 pure isospin change. For pure ∆I= 1/2, Eqs. (84) and (85) taking δ0= 0 and δ1= 0 give: Rn1 Rp1 =2 1 + 2≤Rn0 Rp0 = 2 .(95) For pure ∆I= 3/2 one has (δ0→ ∞ and δ1→ ∞): Rn1 Rp1 =Rn0 Rp0 =1 2.(96) We note that, although the ratio Rn0/Rp0= 2 is implied by the ∆I= 1/2, Rn0/Rp0also equals 2 for a particular mixture of ∆I= 1/2 and ∆I= 3/2 transitions (using δ0=−4 in Eq. (84)). Similarly occurs for pure ∆I= 3/2 transitions, with the ratio Rn0/Rp0taking a value of 1/2 for a specific combination of ∆I= 1/2 and ∆I= 3/2 transitions. Therefore, a result with Rn0/Rp0= 2 does not necessarily imply the validity of the ∆I= 1/2 rule, although a result different from 2 would certainly imply the violability of this selection rule. Two equalities among the hypernuclear decay rates can be obtained from Eq. (95) and Eqs. (64)–(67) for pure ∆I= 1/2 transitions: Rn0 Rp0≡Γn(4 ΛHe) Γp(4 ΛH) = 2 ,(97) which can be equivalently rewritten as: Rn0 Rp0≡ Γn Γp (4 ΛH)Γn Γp (4 ΛHe) Γn Γp (5 ΛHe) = 2 .(98) 31 4 Results 4.1 On the reliability of the Block–Dalitz model In this section we show how a finite calculation of four– and five–body hypernuclei allows one to study the reliability of the Block–Dalitz approach. First, we evaluate the amplitudes for the proton–induced non–mesonic decay of 5 ΛHe with the shell model calculation. The results are given in Table 3 and, as we previously commented, we have employed two weak transition potentials: the one–meson–exchange (OME) and one–meson–exchange plus two–pion–exchange (OME+TPE) models that were built imposing the ∆I= 1/2 transitions rule. The non–vanishing neutron–induced amplitudes, equally as in Eq. (70), verify: An=√2Ap, Bn=√2Bp, Fn=√2Fp.(99) Within the shell–model approach, the partial non–mesonic decay rates of 5 ΛHe read: Γn(5 ΛHe) = |An|2+|Bn|2+|Fn|2= 2(|Ap|2+|Bp|2+|Fp|2),(100) Γp(5 ΛHe) = |Ap|2+|Bp|2+|Cp|2+|Dp|2+|Ep|2+|Fp|2. Before proceeding, it is necessary to discuss the difference between the uppercase amplitudes shown here and the lowercase ones, which were introduced in the previous chapter. The uppercase amplitudes include the effect of the realistic NN final state interaction (FSI) and details of the hypernuclear structure. For this reason, uppercase amplitudes are denoted as hypernuclear amplitudes. As we commented in Section 2.1.4, FSI introduce a mixing between the 3S1and 3D1 NN final state after the action of the weak transition because other transitions mediated by the strong interaction are possible. This mixing only affects the parity–conserving amplitudes Cpand Dpbecause the tensor component (∆L= 2) of the NN strong interaction couples relative angular states having the same parity and total angular momentum. These amplitudes are modified as follows: Cp=Cp(3S1→3S1→3S1) + Cp(3S1→3D1→3S1), Dp=Dp(3S1→3S1→3D1) + Dp(3S1→3D1→3D1), where the first arrow in each amplitude denotes the weak transition Λp→np and the second one denotes the strong transition np →np. It is expected that, by introducing a TPE central potential such as the one of Ref. [11], their effects would only be seen on the Apand Cpamplitudes. However, as we can see in Table 3, the TPE mechanism also modifies the tensor transition amplitude Dp due to the mixing mentioned above. In order to obtain a relationship between the elementary amplitudes of the Block–Dalitz model (apto fp) and the hypernuclear amplitudes (Apto Fp), it 32 Table 3: Proton–induced non–mesonic weak decay amplitudes for 5 ΛHe obtained with the one–meson–exchange (OME) and one–meson–exchange plus two–pion– exchange (OME+TPE) finite nucleus approach. OME OME+TPE Ap:1S0→1S0−0.1057 +0.0823 Bp:1S0→3P0+0.0056 +0.0056 Cp:3S1→3S1−0.1818 +0.0739 Dp:3S1→3D1−0.1483 −0.2772 Ep:3S1→1P1+0.4593 +0.4593 Fp:3S1→3P1+0.2045 +0.2045 is convenient to express Eq. (67) as follows: Γn(5 ΛHe) = 2(|ap|2+|bp|2+|fp|2)ρ4 8,(101) Γp(5 ΛHe) = (|ap|2+|bp|2+|cp|2+|dp|2+|ep|2+|fp|2)ρ4 8. From Eq. (100) and (101) one immediately obtains the elementary Block–Dalitz amplitudes from the calculated hypernuclear ones: ap=Apr8 ρ4 , bp=Bpr8 ρ4 , cp=Cpr8 3ρ4 ,(102) dp=Dpr8 3ρ4 , ep=Epr8 3ρ4 , fp=Fpr8 3ρ4 . Therefore, by construction, with the Block–Dalitz amplitudes of Eq. (102) one reproduces the 5 ΛHe calculated decay rates, as can seen in Table 5 when comparing the Block–Dalitz results with those of the OME (lines OME and OME–BD) and the OME+TPE (lines OME+TPE and OME+TPE–BD) models. The decay amplitudes of Table 3 together with the Block–Dalitz approach allows one to extend the prediction to the other s–shell hypernuclei. The procedure is the following: by comparing Eqs. (100) with Eq. (67) one first obtains the RNJ rates and, subsequently, these rates are employed in Eqs. (64)–(66) to obtain the decay widths for 3 ΛH, 4 ΛH and 4 ΛHe. The results for the RNJ rates obtained with the OME and OME+TPE models are reported in Table 4, where the value ρ4= 0.045 fm−3, evaluated in Ref. [47] through a quark model based Λ wave–function and a Gaussian nuclear density that reproduced the experimental mean square radius of 4He, has been taken. The isospin–dependence of the ΛN→nN process is summarized by the ratios Rn0/Rp0and Rn1/Rn1. The first one equals 2 for both models due to the 33 Table 4: Elementary ΛN→nN Block–Dalitz decay rates RNJ (in units of fm−3) obtained with the one–meson–exchange (OME) and one–meson–exchange plus two–pion–exchange (OME+TPE) weak potential models enforcing the ∆I= 1/2 isospin rule. Model Rn0Rp0Rn1Rp1 Rn0 Rp0 Rn1 Rp1 Rn1 Rn0 Rp1 Rp0 OME 3.98 1.99 4.96 18.24 2.00 0.27 1.24 9.16 OME+TPE 2.42 1.21 4.96 19.86 2.00 0.25 2.05 16.41 enforced ∆I= 1/2 rule. The second one gives, via Eq. (95), ||= 2.53 for the OME while ||= 2.65 for the OME+TPE potential. These results imply that, by inspecting the definition of the parameter in Eq. (83), the ΛN→nN reduced amplitudes into isospin 0 and 1 final states are of the same order of magnitude, that is, the ∆I= 1/2 transition gives the same probability of leaving the final state in a isosinglet–state or in a isotriplet–state. The spin–dependence of the ΛN→nN process is instead summarized by the ratios Rn1/Rn0and Rp1/Rp0. We observe that the TPE terms contribute in the same amount in both ratios, almost doubling their value compared to the OME result. This is due to the fact that the TPE terms decrease the central amplitude Apand, therefore, reduce the value of Rn0and Rp0with the same factor. On the other hand, the calculation predicts the dominance of the spin–triplet channels , especially for the proton–induced transitions and the OME+TPE model. As we previously mentioned, the RNJ ’s values adjusted for the five–body hypernuclei 5 ΛHe allows one to obtain the Block–Dalitz decay rates to three– and four–body hypernuclei. In this work we limit to 4 ΛH and 4 ΛHe since for hypertriton (3 ΛH) there is still no data available on non–mesonic decay rates, which complicates the discussion of the results. Moreover, a realistic theoretical approach implies knowing the exact solution of a three–body problem. Let us point out that the lifetime of the 3 ΛH has been a controversial observable for many years, since several experiments have claimed it to be as much as 30% shorter [53–55] than the lifetime of the free Λ, which is the value that one expects due to loosely bound Λ in the hypertriton. Although some theoretical works assigned a possible explanation to the effect of pion final–state interactions [56], recent experiments [57] have obtained a value within 1σof the free Λ lifetime, being also compatible with the result obtained in a recent theoretical study that employs state–of–the–art nuclear and hypernuclear hamiltonians, considers ΣNN admixtures in the wave–function and accounts for pion distortion [58]. The non–mesonic decay channels of the hypertriton have been theoretically established to represent only about 1.5% of the total decay rate [59] and would not help in explaining an hypothetical smaller lifetime than that of the free Λ. The results for four–body hypernuclei have been obtained from Eqs. (65)–(66) 34 and are summarized in Table 5 (lines OME–BD and OME+TPE–BD) together with the results of the finite nucleus calculations (lines OME and OME+TPE). The value employed for the density ρ3(= 0.0259 fm−3) has been fixed in order to reproduce the OME+TPE finite nucleus prediction for ΓNM(4 ΛHe). From Table 5, we observe that the OME and OME+TPE models predict similar values for both the Γnand Γprates of 5 ΛHe, while this does not hold for Γnin 4 ΛHe and Γpin 4 ΛH. This is due to the fact that the OME+TPE model reduces the spin–singlet rates Rn0and Rp0in relation to those of the OME model, while taking approximately equal values for Rn1and Rp1in both models. The lack of spin–triplet Rn1contribution in Γn(4 ΛHe) and Rp1in Γn(4 ΛH) but present in Γp(4 ΛHe) and Γp(4 ΛH), respectively, aggravates the effect. On the contrary, this does not take place in 5 ΛHe because the spin–triplet contributions dominate in the rates Γn(5 ΛHe) and Γp(5 ΛHe). The decay rates obtained with the Block–Dalitz model for four–body hypernuclei are very similar to the ones calculated within the finite nucleus calculation. Despite the fact that no assumptions were made on the dynamics of the ΛN→nN interaction mechanism and in view of the results obtained in Table 5, we conclude that the Block–Dalitz model can be used to infer directly from the experimental hypernuclear rates the spin–isospin dependence of the non–mesonic weak decay. We think that this result is especially important in view of the forthcoming E22 J–PARC experiment [50]. In contrast to what is obtained for the decay rates, Table 5 shows that the Block– Dalitz model is unable to reproduce the asymmetry parameter obtained with the finite nucleus calculation for 5 ΛHe. We have seen that the two approaches turn out to predict different contributions for all the interference terms building up the asymmetry. Note that the asymmetry parameter aΛin the Block–Dalitz model is given by Eq. (68) which is strictly valid only for the ~ Λp→np free space process and the effects of the FSI and the hypernuclear structure have not been included. In view of the results in Table 5, we may conclude that by looking at the decay rates there is a reasonable agreement between the finite nucleus OME and OME+TPE results and the available data. However, the OME badly fails in reproducing the experimental asymmetry parameter for 5 ΛHe. We note that the negative value of the asymmetry in the OME model comes mainly from the interference between the Apand Epamplitudes and the sum of interferences of the Fpamplitude with the Cpand Dpones. On the other hand, the TPE mechanism transforms the negative interferences ApEp,BpCpand FpCpinto positive contributions that cancel the negative interferences FpDpand BpDp. In consequence, the large and negative aΛin the OME model becomes a positive value in the OME+TPE one. The OME+TPE model corrects the OME defect to reproduce aΛbut it seems to slightly imbalance the decay rates. Specifically, the OME+TPE model predicts 35 Table 5: The s–shell hypernuclei non–mesonic weak decay rates (in units of the free Λ decay width, ΓΛ= 3.8×109s−1) and intrinsic asymmetry parameters predicted by the finite nucleus calculation and the Block–Dalitz model (BD) for the one–meson–exchange (OME) and one–meson–exchange plus two–pion– exchange (OME+TPE) weak potential models. Results are shown for pure ∆I= 1/2 decays and by including ∆I= 3/2 contributions evaluated with the factorization approximation of Ref. [35]. Comparison with the most recent data is also given 5 ΛHe Model ΓnΓpΓNM = Γn+ ΓpΓn/ΓpaΛ OME 0.106 0.319 0.425 0.332 -0.516 OME–BD 0.106 0.319 0.425 0.332 -0.605 OME ∆I= 3/2 0.086–0.145 0.308–0.367 0.425–0.453 0.236–0.468 -0.761– -0.162 OME+TPE 0.097 0.342 0.439 0.284 +0.070 OME+TPE–BD 0.097 0.342 0.439 0.284 +0.111 OME+TPE ∆I= 3/2 0.084–0.129 0.335–0.383 0.439–0.468 0.220–0.381 -0.249–+0.346 KEK–E462 [25,26] 0.424±0.024 0.45±0.11±0.03 KEK–E462 [28] 0.07±0.08+0.08 −0.00 4 ΛHe Model ΓnΓpΓNM = Γn+ ΓpΓn/Γp OME 0.032 0.248 0.280 0.128 OME–BD 0.034 0.245 0.279 0.140 OME ∆I= 3/2 0.004–0.086 0.241–0.282 0.276–0.328 0.014–0.359 OME+TPE 0.016 0.267 0.283 0.062 OME+TPE–BD 0.021 0.262 0.283 0.080 OME+TPE ∆I= 3/2 0.001–0.059 0.263–0.294 0.281–0.324 0.003–0.220 BNL07 [29] ≤0.035 0.180 ±0.028 0.177 ±0.029 ≤0.19 BNL98 [60] 0.04 ±0.02 0.16 ±0.02 0.20 ±0.03 0.25 ±0.13 KEK98 [61] 0.01+0.04 −0.01 0.16 ±0.02 0.17 ±0.05 0.06+0.28 −0.06 4 ΛH Model ΓnΓpΓNM = Γn+ ΓpΓn/Γp OME 0.080 0.016 0.095 4.992 OME–BD 0.081 0.017 0.099 4.734 OME ∆I= 3/2 0.066–0.107 2E-4–0.084 0.093–0.150 0.785–411.292 OME+TPE 0.072 0.008 0.080 8.764 OME+TPE–BD 0.075 0.010 0.086 7.264 OME+TPE ∆I= 3/2 0.064–0.093 0.001–0.064 0.079–0.128 1.008–117.359 BNL98 [60] 0.17 ±0.11 a relative large value of Γp(5 ΛHe), thus increasing the value of ΓNM(5 ΛHe) and reducing Γn/Γp(5 ΛHe) away from the central value of the experimental results. However, one has to note that in several cases the available data are of limited 36 precision, lacking or even missing. For this reason and despite the scarcity experimental data, we may conclude that the OME+TPE model obtains results approximately in accordance with the experimental data. Future experimental data on the decay rates and Γn/Γpratio, especially in four–body hypernuclei will be of utmost importance in order to infer the best potential model for the ΛN→nN weak decay. Furthermore, an improvement of experimental data could represent an important advance on the discussion of the reliability of the Block–Dalitz model, since as we previously commented, Eqs. (91)–(94) turn out to be useful for testing it. 4.2 Four–body hypernuclei and test of the ∆I= 1/2 The results discussed so far have been obtained enforcing the ∆I= 1/2 isospin rule for all the meson–baryon weak couplings involved in the meson–exchange weak transition potential (OME and OME+TPE). As we previously mentioned, the transition ΛN→nN is assumed to proceed via the exchange of virtual pseudoescalar and vector mesons. The transition involves a weak (HW BB0M) and a strong vertex (HS BB0M). Now, we study possible contributions of the ∆I= 3/2 terms to the non–mesonic weak decay. The BB0Mweak couplings for the ∆I= 3/2 transitions are negligible for pions and this result is generalized to the entire pseudoscalar octet. Therefore, the weak vertices that will be modified to include the violation of the ∆I= 1/2 rule are ΛNρ, ΛNK∗and ΛNω. Unfortunately, these couplings are not known experimentally. They will be obtained from the factorization approximation, in which each BB0Mvertex is factorized into the product of baryon–to–baryon and vacuum–to–meson matrix elements of quarks currents. The ∆I= 3/2 ΛNω coupling vanishes in the factorization approximation [62] . In order to take into account the limitations of this approximation, we allow for a variation of up to a factor ±3 (sM) in the ∆I= 3/2ρand K∗couplings. Since the relative sign between the ∆I= 1/2 and ∆I= 3/2 amplitudes is not predicted within this approach we allow for both possibilities. For details on the factorization approximation we refer to Refs. [35,62]. The results for the decay rates and the asymmetry parameter of four– and five– body hypernuclei are also given in Table 5 for the OME and OME+TPE models including ∆I= 3/2 contributions. The range of variability of the ∆I= 3/2 contributions depends on these scaling factors sρand sK∗∈[−3,+3]. From Table 5 we see that the effect of ∆I= 3/2 contributions is moderate in all the decay rates except for Γn(4 ΛHe), Γp(4 ΛH) and the asymmetry parameter. A comparison between our results and the experimental data can inform us about possible violations of the ∆I= 1/2 rule. Unfortunately, as we mentioned in Section 4.1, the available data are scarce or still present large experimental error bars and therefore, it is not appropriate to draw conclusions yet. We hope that the future E22 experiment at J–PARC [50] can obtain improved data, especially for 4 ΛHe and 4 ΛH. 37 Table 6: Predictions for the isospin ratios Rn0/Rp0and corresponding value of δ0≡AJ=0 3/2(If= 1)/AJ=0 1/2(If= 1) obtained with the OME+TPE model introducing different ∆I= 3/2 contributions. The value of δ0is obtained from Eq. (84). sρsK∗Rn0/Rp0δ0 −3−3 0.18 0.88 −3 0 3.89 −0.21 −3 3 9.68 −0.44 0−3 0.07 1.18 0 0 2 0 0 3 7.29 −0.38 3−3 0.01 1.56 3 0 0.83 0.31 3 3 3.74 −0.20 We have seen in Section 3.1 that the available data of Γn(4 ΛHe) and Γn(4 ΛH) are especially useful for testing the ∆I= 1/2 isospin selection rule. As we see in Eq. (97), the ratio Rn0/Rp0can be obtained by the OME and OME+TPE finite nucleus calculation as Γn(4 ΛHe)/Γp(4 ΛH) and can be expressed, via Eq. (84), as a function of the ratio δ0of Eq. (81) between the ∆I= 3/2 and ∆I= 1/2 ΛN→nN reduced amplitudes for isospin 1 final states. In Table 6 we show the numerical values of Rn0/Rp0and δ0obtained for the different contributions of the sρand sK∗scaling factors. Pure ∆I= 1/2 transitions corresponds to δ0= 0, Rn0/Rp0= 2 and sK∗=sρ= 0. As we mentioned, Rn0/Rp0= 2 does not imply that the ∆I= 1/2 rule is valid but deviations from this value are clearly signal of the violation of this selection rule. By a simple inspection of Figure 3 and Table 6 we conclude that the Rn0/Rp0is very sensitive to the ∆I= 3/2 contributions. We have seen that the ratio Rn1/Rp1is not sensitive to the ∆I= 3/2 transitions. This is due to the fact that the only transition with total spin J= 1 that leaves the final state with If= 1 (note that ˆ T3/2acting on an I= 1/2 initial state cannot leave the final state at If= 0) is the one corresponding to the parity– violating (PV) amplitude F,1S1→3P1. The PV operator for vector mesons is given in Section 2.2 and has the following form: ˆ OPV = (~σ1×~σ2)·~r. With the quantum numbers of the initial (Lr,S0) and final state (L,S) we see that the vector mesons (ρand K∗) do not contribute to this amplitude because the 9j symbol appearing in this matrix element is equal to 0, so (See Appendix B.4): h(LS)JMJ|ˆ O|(LrS0)JMJ)i= 0 .(103) This is the reason why the ratio Rn1/Rp1is insensitive to 1S1→3P1∆I= 3/2 contributions. Based on these conclusions, we will study the degree of violation of the ∆I= 1/2 rule in terms of the parameter δ0, which measures the strength 38 of ∆I= 3/2 contributions. Figure 3: The ratio Rn0/Rp0is given versus δ0=AJ=0 3/2(If= 1)/AJ=0 1/2(If= 1). The following figures show the dependence of the the asymmetry parameter, the total non–mesonic decay rate and the Γn/Γpratio of 5 ΛHe on the amount of ∆I= 1/2 violation, measured by the parameter δ0, for the OME and OME+TPE models. The experimental value together with the margin of error (faded grey box) are also shown. In Figure 4 it can be seen that the asymmetry parameter is very sensitive to the meson–exchange model. As we already commented, the OME+TPE model transforms the large and negative value of the OME model into a positive value, more aligned with the experimental result. The ∆I= 3/2 maintain the positive sign of the asymmetry parameter as long as δ0≥ −0.17. Moreover, the asymmetry parameter becomes relatively sensitive to δ0and changes from −0.3 to 0.2 in OME+TPE and from −0.3 to −0.6 in OME for the considered δ0range. As for the total rate ΓNM and Γn/Γpratio, we observe in Figure 5 and Figure 6, 39 Figure 4: The asymmetry parameter aΛof 5 ΛHe obtained with OME and OME+TPE model is given versus δ0=AJ=0 3/2(If= 1)/AJ=0 1/2(If= 1). Faded grey box represents the margin of error of the experimental central value represented by the continuous line (exp). respectively, that they are relatively insensitive to the model, while Γn/Γpis moderately sensitive to δ0. To finalize, we conclude that more precise data of the hypernuclear decay observables are needed in order to learn about the details of the non–mesonic weak decay. In this respect, the asymmetry parameter stands out for being strongly sensitive to the model, and the present measured value has already discarded some of the existing models in the literature. In order to learn about the ∆I= 1/2 isospin rule, it is necessary to obtain Rn0/Rp0with good precision because a deviation of the value 2 would indicate the presence of ∆I= 3/2 terms. Thanks to the Block–Dalitz model validated in the present work, we can identify the former ratio to the ratio Γn(4 ΛHe)/Γp(4 ΛH), built up from the rates of light hypernuclei. 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First, to study the hypernucleus we will use the weak coupling scheme. As we have already mentioned, the hypernuclear state is the result of the coupling between the Λ and the nuclear core states: |A ΛZi=|Λi⊗|Corei.(104) To study the non–mesonic weak decay, it will be necessary to decouple one of the nucleons of the core, leaving a residual (A−1) nuclear system. Being both, Λ and N, in the lower energy state of the ground state hypernucleus, only two initial ΛNstates are possible, the singlet 1S0and the triplet 3S1states. A.1 3 ΛHrate The hypertriton 3 ΛH is a bound state of a proton, a neutron and a Λ with total spin JH= 1/2 and MH=±1/2. The initial state is defined by |3 ΛHi ≡| 1 2MHi=|Λi⊗ | npi.(105) As we have already commented, antisymmetry of the final NN system imposes some restrictions to the relevant quantum numbers. The np pair is in total orbital momentum L= 0 and therefore, J=S. Due to the fact that the Λ is an isosinglet state (tΛ= 0) the total isospin of the initial hypernucleus is the same as the nuclear core. Experimentally, there is no evidence of Λnn and Λpp bound states, which would imply that the ΛNN system could form an isotriplet (I= 1) state. The np pair, and consequently the hypertriton, are in isospin singlet states (I= 0). Therefore, only the S= 1 np state is allowed, leading to a total spin and projection of J= 1 and M=±1,0. Taking JH= 1/2 and jΛ= 1/2 we can write: |1 2MHi=X mΛ,Mh1 2mΛ1M|1 2MHi | 1 2mΛiΛ|1Minp .(106) Without loss of generality, one can impose MH= +1 2and therefore, MH=1 2=mΛ+M⇒mΛ=1 2−M |1 2 1 2i=X Mh1 2 1 2 9M1M|1 2 1 2i | 1 2 1 2 9MiΛ|1Minp .(107) Now, we decouple the np pair in the neutron and the proton states. |1Minp =X mnX mp =h1 2mn 1 2mp|1 2 1 2i | 1 2mni | 1 2mpi, 48 so, the initial hypernuclear state is rewritten as |1 2 1 2i=X M,mph1 2 1 2 9M1M|1 2 1 2ih1 2M9mp 1 2mp|1Mi |1 2 1 2 9MiΛ|1 2M9mpin|1 2mpip. (108) To describe the Λn→nn and Λp→np transitions, one needs to couple the Λ with a neutron nand with a proton p, respectively. We will denote the coupled ΛNstate as |SiMii: Λn:|1 2 1 2 9MiΛ|1 2M9mpin=X Si,Mih1 2 1 2−M1 2M−mp|SiMii | SiMiiΛn, (109) where Mi= 1/2−M+M−mp= 1/2−mp⇒mp= 1/2−Mi. Λp:|1 2 1 2 9MiΛ|1 2mpip=X Si,Mih1 2 1 2 9M1 2mp|SiMii | SiMiiΛp, (110) where Mi= 1/2−M+mp⇒mp=M+Mi−1/2 Rewriting the sum in Eq. (108) one obtains: |1 2 1 2i=X SiX MiX Mh1 2 1 2 9M1M|1 2 1 2ih1 2M+Mi911 2 1 2 9Mi|1Mi h1 2 1 2 9M1 2M+Mi−1 2|SiMii | SiMiiΛn|1 2 1 2 9Miip, (111) for the Λn→nn transition and |1 2 1 2i=X SiX MiX Mh1 2 1 2 9M1M|1 2 1 2ih1 2 1 2 9Mi 1 2M+Mi9 1 2|1Mi h1 2 1 2 9M1 2M+Mi9 1 2|SiMii | SiMiiΛp|1 2 1 2 9Miin, (112) for the Λp→np one. We focus on the Λn→nn transition, but the Λp→np case proceeds analogously. If Si= 0 ⇒Mi= 0 and M=±1,0, the contribution to Eq. (111) reads: 49 X Mh1 2 1 2 9M1M|1 2 1 2ih1 2M9 1 2 1 2 1 2|1Mi ×h1 2 1 2 9M1 2M9 1 2|0 0i | 0 0iΛn|1 2 1 2ip =h1 2−1 21 1 |1 2 1 2ih1 2 1 2 1 2 1 2|1 1ih1 2−1 2 1 2 1 2|0 0i | 0 0iΛn|1 2 1 2ip +h1 2 1 21 0 |1 2 1 2ih1 2−1 2 1 2 1 2|1 0ih1 2 1 2 1 2−1 2|0 0i | 0 0iΛn|1 2 1 2ip +h1 2 3 21−1|1 2 1 2ih1 2−3 2 1 2 1 2|1−1i ×h1 2−1 2 1 2−3 2|0 0i | 0 0iΛn|1 2 1 2ip = −r2 3·1·−1 √2+1 √3·1 √2·1 √2+ 0!|0 0iΛn|1 2 1 2ip =√3 2|0 0iΛn|1 2 1 2ip. (113) Then, the probability is calculated as follows: P=h1 2 1 2|1 2 1 2i 2 Si=0 =3 4.(114) If Si= 1 ⇒Mi=±1,0 and M=±1,0, the contribution to Eq. (111) reads: X MiX Mh1 2 1 2 9M1M|1 2 1 2ih1 2M+Mi9 1 2 1 2 1 2 9Mi|1 0i h1 2 1 2 9M1 2M+Mi9 1 2|1Mi|1Mii|1 2 1 2 9Mii. (115) From the previous equation, we can extract some restrictions: •The state |1 2 1 2−Miiimplies that the Mi=−1 spin projection is forbidden. •The CG coefficient h1 2 1 2−M1M|1 2 1 2iimplies that the M=−1 spin projection is forbidden. •M+Mi−1 2=±1 2⇒M+Mi= 1 or M+Mi= 0 . To sum up, if Mi= 1 ⇒M= 0 and if Mi= 0 ⇒M= 0,1. Therefore, 50 h1 2 1 21 0 |1 2 1 2ih1 2 1 2 1 2−1 2|1 0ih1 2 1 2 1 2 1 2|1 1i | 1 1i | 1 2−1 2i +h1 2−1 21 1 |1 2 1 2ih1 2 1 2 1 2 1 2|1 1ih1 2−1 2 1 2 1 2|1 0i | 1 0i | 1 2 1 2i +h1 2 1 21 0 |1 2 1 2ih1 2−1 2 1 2 1 2|1 0ih1 2 1 2 1 2−1 2|1 0i | 1 0i | 1 2 1 2i =1 √3·1 √2·1|1 1i | 1 2−1 2i + −r2 3·1·1 √2+1 √3·1 √2·1 √2!|1 0i | 1 2 1 2i =1 √6|1 1i|1 2−1 2i− 1 2√3|1 0i | 1 2 1 2i. (116) The probability is: P=h1 2 1 2|1 2 1 2i 2 Si=1 =1 √62 +−1 2√32 =1 4.(117) As a result, the n–induced and p–induced rates read: Rn=3 4Rn0+1 4Rn1. Rp=3 4Rp0+1 4Rp1.(118) Finally, if we go back to Eq. (62), the total width for 3 ΛH reads: ΓNM(3 ΛH) = NRn(3 ΛH) + ZRp(3 ΛH) 2ρ2= (3Rn0+Rn1+ 3Rp0+Rp1)ρ2 8. A.2 4 ΛH rate The 4 ΛH is a bound state of a proton, two neutrons and a Λ with total spin JH= 0 and MH= 0. Assuming a weak coupling scheme, the initial hypernuclear state is defined as |4 ΛHi ≡| 0 0i=|Λi⊗ | pnni.(119) The nuclear core composed by three baryons (nnp) can only be coupled to Jc= 1/2 due to the total angular momentum conservation (0 = 1/2Λ⊗1/2nnp). In consequence, the initial hypernuclear state can be treated as |0 0i=|1 2mΛiΛ|1 2MJinnp ,(120) with 0 = mΛ+MJ. To define the n–induced and the p–induced transitions, it is necessary to study the nuclear core. Assuming the pairing hypothesis of nuclear physics, the nuclear core can be treated as the coupling between nn and pstates. |nnpi=|nni⊗ | pi(121) 51 As we already commented, the total antisymmetric wave function of the nn pair requires J= 0. In accordance with these considerations, the initial state described in Eq. (120) can be rewritten as |0 0i=|1 2mΛiΛ|1 2mpip|0 0inn ,(122) where mΛ+mp= 0. In the first place, we study the p–induced transition. Starting from Eq. (122), the Λp→np transition is described with the coupling between Λ and pstates. We will denote the Λpstate with |SiMii. The angular momentum conservation requires that the Λppair is in a spin–singlet state (Si= 0) in order to satisfy JH= 0. As a consequence, the probability of this configuration is P(Si= 0) = 1 .(123) In the second place, we study the n–induced process. Starting from the initial state defined in Eq. (122), the procedure will be as follows. To describe the Λn→nn transition it is necessary to uncouple the nn pair and subsequently, to define the |SiMiistate coupling the Λ with one n: |0 0inn =X mnm0 n h1 2mn 1 2m0 n|0 0i | 1 2mnin|1 2m0 nin =X mnh1 2mn 1 2 9mn|0 0i | 1 2mnin|1 2−mnin, (124) where 0 = mn+m0 n⇒m0 n=−mn. Now, we couple the Λnpair in a |SiMiistate. |1 2mΛiΛ|1 2mnin=X SiMih1 2mΛ 1 2mn|SiMii | SiMiiΛn.(125) As a result, the 4 ΛH hypernuclear state can be written as: |0 0i=X mnX SiMih1 2mn 1 2−mn|0 0ih1 2mΛ 1 2mn|SiMii |1 2mpip|1 2−mnin|SiMiiΛn, (126) where Mi=mΛ+mn. If Si= 0 ⇒Mi= 0 and mΛ=−mn. Using mΛ= 1/2 without loss of generality, the Si= 0 contribution from the above equation reads: h1 2−1 2 1 2 1 2|0 0ih1 2 1 2 1 2−1 2|0 0i | 1 2−1 2ip|1 2 1 2in|0 0iΛn =−1 √2·1 √2|1 2mpip|1 2 1 2in|00iΛn =−1 2|1 2−1 2ip|1 2 1 2in|0 0iΛn. (127) 52 The probability of this configuration is: P(Si= 0) = |h0 0 |0 0i|2 Si=0 =1 4,(128) and for completeness, P(Si= 1) = 1 −P(Si= 0) = 3 4.(129) As a result, the n–induced and p–induced rates read: Rn=1 4Rn0+3 4Rn1. Rp=Rp0.(130) Finally, if we go back to Eq. (62), the total width for 4 ΛH reads: ΓNM(4 ΛH) = NRn(4 ΛH) + ZRp(4 ΛH) 3ρ3= (Rn0+ 3Rn1+ 2Rp0)ρ3 6.(131) A.3 4 ΛHe rate This case is completely analogous to the 4 ΛH, but exchanging the role of the neutrons by that of the protons. Therefore, the n–induced and p–induced rates read: Rn=Rn0. Rp=1 4Rp0+3 4Rp1.(132) Finally, if we go back to Eq. (62), the total width for 4 ΛHe is Γ(4 ΛHe) = NRn(4 ΛHe) + ZRp(4 ΛHe) 3ρ3= (Rp0+ 3Rp1+ 2Rn0)ρ3 6.(133) A.4 5 ΛHe rate The 5 ΛHe is a bound state composed by a Λ, two protons and two neutrons. The total initial angular momentum is JH= 1/2 and we assume MH= 1/2 without loss of generality. So, the initial state is defined as |5 ΛHei ≡| 1 2 1 2i=|Λi⊗ | nnppi.(134) As it is well known, the nuclear core is the 4He state and has J= 0. Assuming the pairing hypothesis of nuclear physics, we will describe this system from the coupling between nn and pp pairs. |nnppi ≡| 4Hei ≡ |nni⊗|ppi(135) 53 As we have already commented, the nn and pp pairs are in L= 0 and I= 1 states, therefore S=J= 0. By using these considerations, the total hypernuclear state reads: |1 2 1 2i=|1 2mΛiΛ|0 0inn |0 0ipp ,(136) with MH=1 2=mΛ. As in the previous cases, the procedure will be as follows. First of all, it will be advisable to decouple the nn and pp pairs in two single nucleon states and then, it will be necessary to couple the Λ with nand the Λ with pstates to study the n–induced and the p–induced processes, respectively. |0 0inn =X mnh1 2mn 1 2−mn|0 0i | 1 2mnin|1 2−mnin,(137) |0 0ipp =X mph1 2mp 1 2−mp|0 0i | 1 2mpip|1 2−mpip.(138) We analyse the n–induced process, but the p–induced one proceeds analogously. The initial state is defined as: |1 2 1 2i=|Λi⊗ | 00inn⊗ | 00ipp .(139) After decoupling |00inn, as shown in Eq. (137), we couple Λ with nto the |SiMiistate: |1 2mΛiΛ|1 2mpin=X SiMih1 2mΛ 1 2mn|SiMii | SiMiiΛn,(140) where Mi=mΛ+mn. Therefore, the total initial state read: |1 2 1 2i=X mnX SiMih1 2mn 1 2−mn|0 0ih1 2mΛ 1 2mn|SiMii |1 2−mnin|SiMiiΛn|0 0ipp (141) If Si= 0 ⇒Mi= 0 and mΛ=−mnwith mΛ= 1/2. Then, the Si= 0 contribution reads: h1 2−1 2 1 2 1 2|0 0ih1 2 1 2 1 2−1 2|0 0i | 1 2 1 2in|0 0iΛn|0 0ipp =−1 2|1 2 1 2in|0 0iΛn|0 0ipp (142) The probability of this configuration is: P(Si= 0) = h1 2 1 2|1 2 1 2i 2 =1 4,(143) 54 and for completeness , P(Si= 1) = 1 −P(Si= 0) = 3 4.(144) An analogous procedure is followed for the p–induced process. As a result, the n–induced and p–induced rates read: Rn=1 4Rn0+3 4Rn1. Rp=1 4Rp0+3 4Rp1.(145) Finally, if we go back to Eq. (62), the total width for 5 ΛHe read: Γ(5 ΛHe) = NRn(5 ΛHe) + ZRp(5 ΛHe) 4ρ4= (Rn0+ 3Rn1+Rp0+ 3Rp1)ρ4 8.(146) 55 Appendix B Spin matrix elements For completeness, in this section we show the spin matrix coefficients appearing in the ΛN→nN weak transition potential [10]. We denote the initial pair of ΛNwith spin and angular momentum S0and Lrand a final NN pair with spin and angular momentum Sand L0. The total spin angular momentum and its third projection are denoted by Jand MJ, respectively. B.1 Spin–Spin Transition h(L0S)JMJ|ˆ Oα|(LrS0)JMJi= (2S(S+ 1) −3)δLrL0δS0S B.2 Tensor Transition h(L0S)JMJ|ˆ Oα|(LrS0)JMJi=SJ LrL0δLrL0δS0S where SJ LrL0are given in the following table. SJ LrL0L0=J+ 1 L0=J L0=J−1 Lr=J+ 1 −2(J+2) 2J+1 06√J(J+1) 2J+1 Lr=J0 0 0 Lr=J−16√J(J+1) 2J+1 0−2(J−1) 2J+1 B.3 PV Transition: Pseudoscalar Mesons h(L0S)JMJ|ˆ Oα|(LrS0)JMJi= (−1)J+1−L0√6p2S0+ 1 ×√2S+ 1p2Lr+ 1h10Lr0|L00i1/2 1/2S0 S1 1/2L0Lr1 S0S J B.4 PV Transition: Vector Mesons h(L0S)JMJ|ˆ Oα|(LrS0)JMJi= i(−1)J−L0+S)6√6p2S0+ 1 ×√2S+ 1p2Lr+ 1h10Lr0|L00iL0Lr1 S0S J  1 1 1 1/2 1/2S 1/2 1/2S0  56 Tr[~p2·~σ2ˆ Tˆ T†] = bc∗ 32 Tr[p2l(−3I1σ2lσ2j−3σ1iσ1jσ2lσ2i+σ1jσ1kσ2lσ2k −σ1iσ1jσ1kσ2lσ2iσ2k+σ1iσ1kσ2lσ2iσ2jσ2k)qj] =bc∗ 32 [p2l(−12δlj −12δlj + 4δlj + 4ijklik + 4δik(δliδjk −δlj δik +δlkδij))qj] =bc∗ 32 [p2l(−20δlj −4·2δlj + 4(δlj −3δlj +δlj)qj] =bc∗ 32 [p2l(−32δlj)qj] = −bc∗~p2·~q (189) C.2.4 bd∗combination ˆ Tˆ T†=bd∗ 16√2(1 −~σ1·~σ2)( ~σ1−~σ2)·q(3 ~σ1·p ~σ2·q−~σ1·~σ2) =bd∗ 16√2(1 −σ1iσ2i)(σ1j−σ2j)qj(3σ1kqkσ2mqm−σ1nσ2n) =bd∗ 16√2[(3σ1jσ1kσ2m−3σ1kσ2jσ2m−3σ1iσ1jσ1kσ2iσ2m + 3σ1iσ1kσ2iσ2jσ2m)qjqkqm+ (σ1nσ2jσ2n−σ1jσ1nσ2n +σ1iσ1jσ1nσ2iσ2n−σ1iσ1nσ2iσ2jσ2n)qj] (190) Tr[ˆ Tˆ T†] = bd∗ 16√2Tr[(−3σ1iσ1jσ1kσ2iσ2m+ 3σ1iσ1kσ2iσ2jσ2m)qjqkqm + (σ1iσ1jσ1nσ2iσ2n−σ1iσ1nσ2iσ2jσ2n)qj] = 0 (191) Tr[~p2·~σ2ˆ Tˆ T†] = bd∗ 16√2Tr[p2l[(3σ1jσ1kσ2lσ2m−3σ1iσ1jσ1kσ2lσ2iσ2m + 3σ1iσ1kσ2lσ2iσ2jσ2m)qjqkqm+ (−σ1jσ1nσ2lσ2n +σ1iσ1jσ1nσ2lσ2iσ2n−σ1iσ1nσ2lσ2iσ2jσ2n)qj]] =bd∗ 16√2(p2l[(12δjkδlm + 12ijklim + 12δik(δliδjm −δlj δim +δlmδij))qjqkqm + (−4δlj −4ijnlin −4δin(δliδjn −δljδin +δlnδij))qj]] =bd∗ 16√2[p2l[12δjkδlm −12(δljδkm −δjmδkl) + 12(δlkδjm −δljδkm +δlmδkj))qjqkqm+ 8δljqj]] =bd∗ 16√2[32~p2·~q] = 2 √2bd∗~p2·~q =√2bd∗~p2·~q (192) 63 C.2.5 be∗combination ˆ Tˆ T†=−√3 64 be∗(1 −~σ1·~σ2)( ~σ1−~σ2)·q(~σ1−~σ2)·q(3 + ~σ1·~σ2) =−√3 64 be∗(1 −σ1iσ2i)(σ1j−σ2j)(3σ1k−3σ2k+σ1kσ1mσ2m −σ1mσ2kσ2m)qjqk=−√3 64 be∗(1 −σ1iσ2i)(3σ1jσ1k−3σ1jσ2k +σ1jσ1kσ1mσ2m−σ1jσ1mσ2kσ2m−3σ1kσ2j+ 3σ2jσ2k −σ1kσ1mσ2jσ2m+σ1mσ2jσ2kσ2m)qjqk (193) Tr[ˆ Tˆ T†] = −√3 64 be∗Tr[(3σ1jσ1kI2−σ1jσ1mσ2kσ2m+ 3I1σ2jσ2k −σ1kσ1mσ2jσ2m+ 3σ1iσ1jσ2iσ2k−σ1iσ1jσ1kσ1mσ2iσ2m +σ1iσ1jσ1mσ2iσ2kσ2m+ 3σ1iσ1kσ2iσ2j +σ1iσ1kσ1mσ2iσ2jσ2m−σ1iσ1mσ2iσ2jσ2kσ2m)qjqk] =−√3 64 be∗[(12δjk −4δjk + 12δjk −4δjk + 12δjk −12δjk −8δjk + 12δjk −8δjk −12δjk)qjqk] = 0 (194) Tr[~p2·~σ2ˆ Tˆ T†] = −√3 64 be∗Tr[p2l(σ2l−σ1iσ2lσ2i)(3σ1jσ1k−3σ1jσ2k +σ1jσ1kσ1mσ2m−σ1jσ1mσ2kσ2m−3σ1kσ2j+ 3σ2jσ2k −σ1kσ1mσ2lσ2kσ2m+σ1mσ2jσ2kσ2m)qjqk] =−√3 64 be∗Tr[p2l(σ1jσ1kσ1mσ2lσ2m −σ1jσ1mσ2lσ2kσ2m+ 3σ2lσ2jσ2k−σ1kσ1mσ12lσ2jσ2m −3σ1iσ1jσ1kσ2lσ2i+ 3σ1iσ1jσ1kσ2lσ2i −σ1iσ1jσ1kσ1mσ2lσ2iσ2m+σ1iσ1jσ1mσ2lσ2iσ2kσ2m + 3σ1iσ1kσ2lσ2iσ2j+σ1iσ1kσ1mσ2lσ2iσ2jσ2m −σ1iσ1mσ2lσ2iσ2jσ2kσ2m)qjqk] =−√3 64 be∗[ip2l(4jkl −4lkj + 12ljk −4ljk −12ljk + 12ljk −4ljk + 4lkj + 4ljk + 4kjl + 12lkj + 4lkj + 4jkl −4jkl −4lkj −4lkj)qjqk]=0 (195) 64 C.2.6 bf∗combination ˆ Tˆ T†=√6 32 bf∗(1 −~σ1·~σ2)( ~σ1−~σ2)·q(~σ1+~σ2)·q =√6 32 bf∗(1 −σ1iσ2i)(σ1j−σ2j)qj(σ1k+σ2k)qk =√6 32 bf∗(σ1j−σ2j−σ1iσ1jσ2i+σ1iσ2iσ2j)(σ1k+σ2k)qjqk =√6 32 bf∗(σ1jσ1k+σ1jσ2k−σ2jσ1k−σ2jσ2k−σ1iσ1jσ1kσ2i −σ1iσ1jσ2iσ2k+σ1iσ1kσ2iσ2j+σ1iσ2iσ2jσ2k)qjqk (196) Tr[ˆ Tˆ T†] = √6 32 bf∗T r[(σ1jσ1kI2−I1σ2jσ2k−σ1iσ1jσ2iσ2k +σ1iσ1kσ2iσ2j)qjqk] = √6 32 bf∗[(2δjk2−2·2δjk −2δij2δik + 2δik2δij)qjqk]=0 (197) Tr[~p2·~σ2ˆ Tˆ T†] = √6 32 bf∗T r[P2l(−I1σ2lσ2jσ2k−σ1iσ1jσ1kσ2lσ2i −σ1iσ1jσ2lσ2iσ2k+σ1iσ1kσ2lσ2iσ2j)qjqk] =√6 32 bf∗[p2l(−2·2iljk −2iijk2δli −2δij2ilik + 2δik2ilij)qlqk] =√6 32 bf∗[p2l(−16iljk)qjpk] = −√6 2bf∗~p2·(~q ×~q)=0 (198) C.3 C terms C.3.1 ca∗combination ˆ Tˆ T†=ca∗ 16 (3 + ~σ1·~σ2)(1 −~σ1·~σ2) TT ∗=ca∗ 16 (3 −2~σ1·~σ2−3+2~σ1·~σ2)=0 (199) Tr[ˆ Tˆ T†] = 0 (200) Tr[~p2·~σ2ˆ Tˆ T†] = 0 (201) 65 C.3.2 cb∗combination ˆ Tˆ T†=cb∗ 32 (3 + ~σ1·~σ2)( ~σ1−~σ2)·q(1 −~σ1·~σ2) =cb∗ 32 (3 + σ1iσ2i)(σ1j−σ2j−σ1jσ1kσ2k+σ1kσ2jσ2k)qj =cb∗ 32 (3σ1j−3σ2j−3σ1jσ1kσ2k+ 3σ1kσ2jσ2k +σ1iσ1jσ2i−σ1iσ2iσ2j−σ1iσ1jσ1kσ2iσ2k+σ1iσ1kσ2iσ2jσ2k)qj (202) Tr[ˆ Tˆ T†] = cb∗ 32 Tr[(−σ1iσ1jσ1kσ2iσ2k+σ1iσ1kσ2iσ2jσ2k)qj] = 0 (203) Tr[~p2·~σ2ˆ Tˆ T†] = cb∗ 32 Tr[p2l(−3I1σ2lσ2j−3σ1jσ1kσ2lσ2k+σ1iσ1jσ2lσ2i −σ1iσ1jσ1kσ2lσ2iσ2k+σ1iσ1kσ2lσ2iσ2jσ2k)qj] =cb∗ 32 [p2k(−3·2·2δlj −3·2δjk2δlk + 2δij2δli −2iijk2ilik + 2δik2(δliδjk −δljδik +δlkδij ))qj] =cb∗ 32 [p2l(−12δlj −12δlj + 4δlj −4·2δlj + 4(δlj −3δlj +δlj)qj)] = −cb∗(~p2·~q) (204) C.3.3 cc∗combination ˆ Tˆ T†=|c|2 16 (3 + ~σ1·~σ2)(3 + ~σ1·~σ2) =|c|2 16 (9 + 6 ~σ1·~σ2+ ( ~σ1·~σ2)2) =|c|2 16 (12 + 4( ~σ1·~σ2)) = |c|2 4(3 + ~σ1·~σ2) (205) Tr[ˆ Tˆ T†] = |c|2 4Tr[3 + ~σ1·~σ2] = |c|2 4Tr[3I1I2+σ1iσ2i]=3|c|2(206) Tr[~p2·~σ2ˆ Tˆ T†] = |c|2 4Tr[p2(3I1σ2k+σ1iσ2kσ2i)] = 0 (207) C.3.4 cd∗combination ˆ Tˆ T†=cd∗ 8√2(3 + ~σ1·~σ2)(3 ~σ1·q ~σ2·q−~σ1·~σ2) =cd∗ 8√2(3 + σ1iσ2i)(3σ1jqjσ2kqk−σ1mσ2m) =cd∗ 8√2[(9σ1jσ2k+ 3σ1iσ1jσ2iσ2k)qjqk−3σ1mσ2m−σ1iσ1mσ2iσ2m] (208) 66 Tr[ˆ Tˆ T†] = cd∗ 8√2Tr[3σ1iσ1jσ2iσ2kqjqk−σ1iσ1mσ2iσ2m] =cd∗ 8√2[12δjkqjqk−4δimδim] = 0 (209) Tr[~p2·~σ2ˆ Tˆ T†] = cd∗ 8√2Tr[p2l(3σ1iσ1jσ2lσ2iσ2kqjqk−σ1iσ1mσ2lσ2iσ2m)] =cd∗ 8√2[p2l(12iδijljkqjqk−4iδimlim)] =cd∗ 8√212[~p2·(~q ×~q)] = 0 (210) C.3.5 ce∗combination ˆ Tˆ T†=−√3 32 ce∗(3 + ~σ1·~σ2)( ~σ1−~σ2)·q(3 + ~σ1·~σ2) =−√3 32 ce∗(3 + σ1iσ2i)(σ1j−σ2j)qj(3 + σ1kσ2k) =−√3 32 ce∗(3 + σ1iσ2i)(3σ1j−3σ2j+σ1jσ1kσ2k−σ1kσ2jσ2k)qj =−√3 32 ce∗(9σ1j−9σ2j+ 3σ1jσ1kσ2k−3σ1kσ2jσ2k + 3σ1iσ1jσ2i−3σ1iσ2iσ2j+σ1iσ1jσ1kσ2iσ2k−σ1iσ1kσ2iσ2jσ2k)qj (211) Tr[ˆ Tˆ T†] = −√3 32 ce∗Tr[(σ1iσ1jσ1kσ2iσ2k−σ1iσ1kσ2iσ2jσ2k)qj] = 0 (212) Tr[~p2·~σ2ˆ Tˆ T†] = −√3 32 ce∗Tr[p2l(−9I1σ2lσ2j+ 3σ1jσ1kσ2lσ2k+ 3σ1iσ1jσ2lσ2i +σ1iσ1jσ1kσ2lσ2iσ2k−σ1iσ1kσ2lσ2iσ2jσ2k)qj] =−√3 32 ce∗[p2l(−9·2·2δlj + 3 ·2δjk2δlk + 3 ·2δij2δli + 2iijk2ilik −2δik2(δliδjk −δljδik +δlkδij )qj] =−√3 32 ce∗[p2l(−36δlj + 12δlj + 12δlk + 8δlj −4(δlj −3δlj +δlj))qj]=0 (213) C.3.6 cf∗combination ˆ Tˆ T†=√6 16 cf∗(3 + ~σ1·~σ2)( ~σ1+~σ2)·q =√6 16 cf∗(3 + σ1iσ2i)(σ1j+σ2j)qj =√6 16 cf∗(3σ1j+ 3σ2j+σ1iσ1jσ2i+σ1iσ2iσ2j)qj (214) 67 Tr[ˆ Tˆ T†] = 0 (215) Tr[~p2·~σ2ˆ Tˆ T†] = √6 16 cf∗T r[p2k(3I1σ2lσ2j+σ1iσ1jσ2lσ2i)qj] =√6 16 cf∗[p2k(3 ·2·2δkj + 2δij2δki)qj] =√6 16 cf∗[p2k(16δkj)qj] = √6cf∗(~p2·~q) (216) C.4 D terms C.4.1 da∗combination ˆ Tˆ T†=da∗ 8√2(3 ~σ1·q ~σ2·q−~σ1·~σ2)(1 −~σ1·~σ2) =da∗ 8√2(3σ1iqiσ2jqj−σ1kσ2k)(1 −σ1mσ2m) =da∗ 8√2[(3σ1iσ2j−3σ1iσ1mσ2jσ2m)qiqj−σ1kσ2k +σ1kσ1mσ2kσ2m] (217) Tr[ˆ Tˆ T†] = ad∗ 8√2Tr[−3σ1iσ1mσ2jσ2mqiqj+σ1kσ1mσ2kσ2m] =ad∗ 8√2[−3·2δim2δjmqiqj+ 2δkm2δkm] =ad∗ 8√2[−12δijqiqj+ 4δkmδkm] = 0 (218) Tr[~p2·~σ2ˆ Tˆ T†] = ad∗ 8√2Tr[P2l(−3σ1iσ1mσ2lσ2jσ2mqiqj+σ1kσ1mσ2lσ2kσ2m)] =ad∗ 8√2[p2l(−3·2δim2iljmqiqj+ 2δkm2iikm)] =ad∗ 8√2[p2l(−12iljiqiqj)] = −ad∗ 8√212[~p2·(~q ×~q)] = 0 (219) C.4.2 db∗combination ˆ Tˆ T†=db∗ 16√2(3 ~σ1·q ~σ2·q−~σ1·~σ2)( ~σ1−~σ2)·~q(1 −~σ1·~σ2) =db∗ 16√2(3σ1iσ2jqiqj−σ1kσ2k)(σ1m−σ2m)(1 −σ1nσ2n)qm =db∗ 16√2[(3σ1iσ1mσ2j−3σ1iσ2jσ2m−3σ1iσ1mσ1nσ2jσ2n + 3σ1iσ1nσ2jσ2mσ2n)qiqjqm+ (−σ1kσ1mσ2k+σ1kσ2kσ2m +σ1kσ1mσ1nσ2kσ2n−σ1kσ1nσ2kσ2mσ2n)qm] (220) 68 Tr[ˆ Tˆ T†] = db∗ 16√2Tr[(−3σ1iσ1mσ1nσ2jσ2n+ 3σ1iσ1nσ2jσ2mσ2n)qiqjqm + (σ1kσ1mσ1nσ2kσ2n−σ1kσ1nσ2kσ2mσ2n)qm] =db∗ 16√2[(−12iimnδjn + 12ijmnδin)qiqjqm + (4ikmnδkn −4ikmnδkn)qm]=0 (221) Tr[~p2·~σ2ˆ Tˆ T†] = db∗ 16√2Tr[p2l[(3σ1iσ1mσ2lσ2j−3σ1iσ1mσ1nσ2lσ2jσ2n + 3σ1iσ1nσ2lσ2jσ2mσ2n)qiqjqm+ (−σ1kσ1mσ2lσ2k +σ1kσ1mσ1nσ2lσ2kσ2n−σ1kσ1nσ2lσ2kσ2mσ2n)qm]] =db∗ 16√2[p2l[(12δimδlj + 12(δilδmj −δijδml) + 12δin(δljδmn −δlmδjn +δlnδjm))qiqjqm] + (−4δlm + 8δlm + 4δlm)qm =db∗ 16√2[(24q2+ 8)~p2·~q] = √2db∗(~p2·~q) (222) C.4.3 dc∗combination ˆ Tˆ T†=dc∗ 8√2(3 ~σ1·q ~σ2·q−~σ1·~σ2)(3 + ~σ1·~σ2) =dc∗ 8√2(3σ1iqiσ2jqj−σ1kσ2k)(3 + σ1mσ2m) =dc∗ 8√2[(9σ1iσ2j+ 3σ1iσ1mσ2jσ2m)qiqj−3σ1kσ2k−σ1kσ1mσ2kσ2m] (223) Tr[ˆ Tˆ T†] = dc∗ 8√2Tr[3σ1iσ1mσ2jσ2mqiqj−σ1kσ1mσ2kσ2m] =dc∗ 8√2[12δijqiqj−4δkmδkm] = 0 (224) Tr[~p2·~σ2ˆ Tˆ T†] = dc∗ 8√2Tr[p2l(3σ1iσ1mσ2lσ2jσ2mqiqj−σ1kσ1mσ2lσ2kσ2m)] =dc∗ 8√2[p2l(12iδimljm −4iδkmlkm)] = 0 (225) 69 C.4.4 dd∗combination ˆ Tˆ T†=|d|2 8(S12(p))2=|d|2 8(4S2−2S12) =|d|2 8(4(3 2+~σ1·~σ2 2)−2(3 ~σ1·p ~σ2·p−~σ1·~σ2) =|d|2 8(6 + 4 ~σ1·~σ2−6~σ1·p ~σ2·p) (226) Tr[ˆ Tˆ T†] = |d|2 8Tr(6) = 3 4|d|2Tr(I)=3|d|2(227) Tr[~p2·~σ2ˆ Tˆ T†] = 0 (228) C.4.5 de∗combination ˆ Tˆ T†=−√3 √2 de∗ 16 (3 ~σ1·q ~σ2·q−~σ1·~σ2)( ~σ1−~σ2)·q(3 + ~σ1·~σ2) =−√3 √2 de∗ 16 (3σ1iqiσ2jqj−σ1kσ2k) ×(3σ1m−3σ2m+σ1mσ1nσ2n−σ1nσ2mσ2n)qm =−√3 √2 de∗ 16 [(9σ1iσ1mσ2j−9σ1iσ2jσ2m+ 3σ1iσ1mσ1nσ2jσ2n −3σ1iσ1nσ2jσ2mσ2n)qiqjqm+ (3σ1kσ2kσ2m−3σ1kσ1mσ2k −σ1kσ1mσ1nσ2kσ2n+σ1kσ1nσ2kσ2mσ2n)qm] (229) Tr[ˆ Tˆ T†] = −√3 √2 de∗ 16 Tr[(3σ1iσ1mσ1nσ2jσ2n−3σ1iσ1nσ2jσ2mσ2n)qiqjqm + (−σ1kσ1mσ1nσ2kσ2n+σ1kσ1nσ2kσ2mσ2n)qm] =−√3 √2 de∗ 16 [(12iimnδjn −12ijmnδin)qiqjqm + (−4iδknkmn + 4iδknkmn)qm]=0 (230) 70 Tr[~p2·~σ2ˆ Tˆ T†] = −√3 √2 de∗ 16 Tr[p2l[(9σ1iσ1mσ2lσ2j+ 3σ1iσ1mσ1nσ2lσ2jσ2n −3σ1iσ1nσ2lσ2jσ2mσ2n)qiqjqm+ (−3σ1kσ1mσ2lσ2k −σ1kσ1mσ1nσ2lσ2kσ2n+σ1kσ1nσ2lσ2kσ2mσ2n)qm]] =−√3 √2 de∗ 16 [p2l[(36δimδlj −12(δilδmj −δijδml) −12δin(δljδmn −δlmδjn +δlnδjm)qiqjqm] + (−12δlm −8δlm −4δlm)qm] =−√3 √2 de∗ 16 [24q2~p2·~q −24~p2·~q] = 0 (231) C.4.6 df∗combination ˆ Tˆ T†=√3 8df∗(3 ~σ1·q ~σ2·q−~σ1·~σ2)( ~σ1+~σ2)·q =√3 8df∗(3σ1ipiσ2jpj−σ1kσ2k)(σ1m+σ2m)pm =√3 8df∗[(3σ1iσ1mσ2j+ 3σ1iσ2jσ2m)pipjpm −(σ1kσ1mσ2k+σ1kσ2kσ2m)pm] (232) Tr[ˆ Tˆ T†] = 0 (233) Tr[~p2·~σ2ˆ Tˆ T†] = √3 8df∗Tr[P2l(3σ1iσ1mσ2lσ2jpipjpm−σ1kσ1mσ2lσ2kpm)] =√3 8df∗[p2l(12δimδljpipjpm−4δkmδlmpm)] =√3 8df∗[12q2~p2·~q −4~p2·~q] =√3df∗(~p2·~q) (234) 71 C.5 E terms C.5.1 ea∗combination ˆ Tˆ T†=−√3 32 ea∗(3 + ~σ1·~σ2)( ~σ1−~σ2)q(1 −~σ1·~σ2) =−√3 32 ea∗(3 + σ1iσ2i)(σ1j−σ2j)qj(1 −σ1kσ2k) =−√3 32 ea∗(3σ1j−3σ2j+σ1iσ1jσ2i−σ1iσ2iσ2j)(1 −σ1kσ2k)qj =−√3 32 ea∗(3σ1j−3σ2j+σ1iσ1jσ2i−σ1iσ2iσ2j −3σ1jσ1kσ2k+ 3σ1kσ2jσ2k−σ1iσ1jσ1kσ2iσ2k+σ1iσ1kσ2iσ2jσ2k)qj (235) Tr[ˆ Tˆ T†] = −√3 32 ea∗Tr[(−σ1iσ1jσ1kσ2iσ2k+σ1iσ1kσ2iσ2jσ2k)qj]=0 (236) Tr[~p2·~σ2ˆ Tˆ T†] = −√3 32 ea∗Tr[p2l(−3I1σ2lσ2j+σ1iσ1jσ2lσ2i−3σ1jσ1kσ2lσ2k −σ1iσ1jσ1kσ2lσ2iσ2k+σ1iσ1kσ2lσ2iσ2jσ2k)qj] =−√3 32 ea∗[p2l(−3·2·2δlj + 2δij2δli −3·2δjk2δlk −2iijk2ilik + 2δik2(δliδjk −δlkδik +δlkδij ))qj] =−√3 32 ea∗[p2l(−12δlj + 4δlj −12δlj −4·2δlj + 4(δlj −3δlj +δlj))pj] = −√3 32 ea∗[p2l(−32δlj)qj] =√3ea∗(~p2·~q) (237) C.5.2 eb∗combination ˆ Tˆ T†=√3 64 eb∗(3 + ~σ1·~σ2)( ~σ1−~σ2)·q(~σ1−~σ2)·q(1 −~σ1·~σ2) =√3 64 eb∗(3 + σ1iσ2i)(σ1j−σ2j)(σ1k−σ2k−σ1kσ1mσ2m +σ1mσ2kσ2m)qjqk=√3 64 eb∗(3 + σ1iσ2i)(σ1jσ1k−σ1jσ2k −σ1jσ1kσ1mσ2m+σ1jσ1mσ2kσ2m−σ1kσ2j +σ2jσ2k+σ1kσ1mσ2jσ2m−σ1mσ2jσ2kσ2m)qjqk (238) 72 C.6.5 fe∗combination ˆ Tˆ T†=−√18 32 fe∗(~σ1+~σ2)·q(~σ1−~σ2)·q(3 + ~σ1·~σ2) =−√18 32 fe∗(σ1i+σ2i)qi(σ1j−σ2j)qj(3 + σ1kσ2k) =−√18 32 fe∗(σ1i+σ2i)(3σ1j−3σ2j+σ1jσ1kσ2k−σ1kσ2jσ2k)qiqj =−√18 32 fe∗(3σ1iσ1j−3σ1iσ2j+σ1iσ1jσ1kσ2k−σ1iσ1kσ2jσ2k + 3σ1jσ2i−3σ2iσ2j+σ1jσ1kσ2iσ2k−σ1kσ2iσ2jσ2k)qiqj (265) Tr[ˆ Tˆ T†] = −√18 32 fe∗T r[(3σ1iσ1jI2−σ1iσ1kσ2jσ2k −3I1σ2iσ2j+σ1jσ1kσ2iσ2k)qiqj]=0 (266) Tr[~p2·~σ2ˆ Tˆ T†] = −√18 32 fe∗T r[p2l(σ1iσ1jσ1kσ2lσ2k−σ1iσ1kσ2lσ2jσ2k −3I1σ2lσ2iσ2j+σ1jσ1kσ2lσ2iσ2k)qiqj] =−√18 32 fe∗[p2l(2iijk2δlk −2δik2iljk −3·2·2ilij + 2δjk2ilij)qiqj] =−√18 8fe∗[p2l(iijl −ilji −3ilij +ilij)qiqj]=0 (267) C.6.6 ff∗combination ˆ Tˆ T†=6 16|f|2(~σ1+~σ2)·q(~σ1+~σ2)·q =6 16|f|2(σ1i+σ2i)qi(σ1j+σ2j)qj =6 16|f|2(σ1iσ1j+σ2iσ2j+σ1iσ2j+σ2iσ1j)pipj (268) Tr[ˆ Tˆ T†] = 6 16|f|2Tr[(σ1iσ1jI2+I1σ2iσ2j)] =6 16|f|2[(2 ·2δij + 2 ·2δij)qiqj] = 3|f|2 (269) Tr[~p2·~σ2ˆ Tˆ T†] = 6 16|f|2Tr[p2l(I1σ2lσ2iσ2j)qiqj] =6 16|f|2[p2l(2 ·2ilijqiqj)] =6 4|f|2[~p2·(~q ×~q)] = 0 (270) 79