ΣΣ → YN transitions in the decay of 6 ΛΛ He

Interuniversity Master’s Degree in Nuclear Physics Master Thesis ΣΣ →Y N transitions in the decay of 6 ΛΛHe Ariadna Aixut L´opez Superviors: Assumpta Parre˜no Garc´ıa (UB) Laura Tol´os Rigueiro (ITP, FIAS, ICE) Universitat de Barcelona Quantum Physics and Astrophysics Department December 2019 2 ”If a theory is complicated, its wrong” Richard Feynman 3 Acknowledgments Aquest treball ha representat per a mi un abans i un despr´es en la meva manera de veure la f´ısica. I ´es que la recerca no es tanca mai definitivament, sin´o que sempre hi haur`a una nova manera d’enfocar una recerca i nous c`alculs per a desenvolupar o afinar. M’agradaria agra¨ır a les dues meves tutores, l’Assumpta i la Laura, tot l’esfor¸c i dedicaci´o que han volcat en aquest treball. Sense elles, no hag´ues estat possible. M’agradaria agra¨ır a l’Assumpta, en especial, tots els moments en que ha estat m´es que una tutora i m’ha recolzat, cuidat i esperonat quan m´es ho necessitava, des que ens vam con`eixer quan jo nom´es era al tercer any del grau. Gr`acies a les dues de tot cor. ´ Es obligatori agrair tamb´e al Jordi Maneu tota la feina feta per al meu treball, tant davant d’un teclat com d’un full en blanc. Sense menys import`ancia, gr`acies a l’Anna per recolzar-me i ajudar-me en aquest treball, i en tot el cam´ı que hem fet juntes fins aqu´ı. Com sempre, no pot faltar una menci´o als meus pares, l’Ernest i la Rosa, sense els quals no hagu´es arribat mai on s´oc. 4 Contents I. Introduction 5 II. Decay rate 9 III. One Meson Exchange Model 12 A. Exchange Diagrams 13 B. Vertices and Final Potential Expression 14 C. Coupling Constants 19 D. Isospin dependence 23 E. Spin dependence 28 IV. Discussion of results 31 Appendices 34 A. r-space integrals 34 B. Isospin matrix elements 37 C. Example of calculation of coupling constants 41 1. Strong example: the Σ−π+→Σ0vertex 41 2. Weak PV example: the Σ+π−→nvertex 42 3. Weak PC example: the Σ+π−→nvertex 43 4. Total coupling constants contribution. 43 5 I. Introduction A hypernucleus is a bound system of nucleons and at least one hyperon, where hyperons are baryons containing strange quarks. The presence of the strange quark makes hyperons spaecially attractive to study since these systems are unstable with respect to the weak force, leading to the appearance of weak decays which do not conserve neither parity, isospin nor flavour. Traditionally, the study of the decay of single-Λ hypernuclei has captured most of the attention [1][2][3] , since they are the lightest strange bound systems and consequently the easiest to produce. Hypernuclei are usually produced with hadronic reactions in excited states. To reach the ground state, these nuclei typically experiment a deexcitation process which often includes the emission of particles or electromagnetic radiation. Once in the ground state, these nuclei decay through weak interaction processes, as the ones studied in the present work. A Λ hyperon in free space decays weakly, with a lifetime of τΛ= (2.631 ±0.020) ×10−10 s, leading to nucleons and pions (mesonic decay mode) in the final state with an approximate ratio Γ(Λ →pπ−)/Γ(Λ →nπ0)∼2 , which is compatible with a change in isospin along the decay process of ∆I= 1/2. The observed suppression of ∆I= 3/2 transitions over ∆I= 1/2 ones has motivated a number of theoretical investigations, as that of Ref. [4], but a fundamental understanding of its origin is still missing. When the hyperon is bound in the nuclear medium this decay mode is Pauli blocked, since the emerging nucleon tries to access momentum states that are already occupied by the surrounding nucleons, and non-mesonic decay modes induced by the presence of these nucleons (ΛN→NN) become dominant. Recently, the attention has been focused on doubly strange systems which offer a richer scenario that includes additional decay channels. In the present work we study the decay of 6 ΛΛHe. This hypernucleus was first observed in the KEK-E373 experiment [5], which aroused great interest on theoretical and experimental studies involving the ΛΛ interaction. Since then, other ΛΛ - hypernuclei have been observed, allowing for a precise extraction of the ΛΛ binding energy, as well as many other nuclear properties. More recently, the E07 experiment at J-PARC [6, 7] aims to produce ΛΛ nuclei throughout the capture of Ξ hyperons, with the subsequent strong conversion into ΛΛ pairs. It is estimated that the statistical sample will be an order of magnitude greater than the one in experiment KEK-E373, allowing for a more accurate extraction of the strong and weak baryon-baryon (BB) interaction observables in the strange sector. Due to the strong interaction acting on the baryons, the ΛNand ΛΛ pairs in the initial hypernucleus can mix with other baryon-baryon (BB) channels with either one-unit or two-units of strangeness, ΣNand ΞNand ΣΣ respectively. The different components of the wave functions are contained in the solution of the G-matrix equations. These initial two-body particle states will in turn decay via weak processes leading to BB states which leave the system with enough momenta to be treated with a T-matrix formalism, where no medium corrections are considered. 6 In Ref. [8] the decay of 6 ΛΛHe induced by ΛN→NN and ΛΛ →ΛN(ΣN) transitions was studied. In this work, the decay rates to the different final states were calculated to be: ΓNN = 0.96ΓΛ, ΓΛN= 3.58 ×10−2ΓΛand ΓΣN= 3.0×10−3ΓΛ, where ΓΛ= 3.8×109s−1corresponds to the decay rate of a Λ in free space. The total contribution added up to a final value for the rate of Γ=0.96ΓΛ, dominated by the exchange of πand the Kmesons in the weak model. As it can be seen from Fig. 1 (taken from Ref. [9]), the dominant component of the ΛΛ wave function is the diagonal ΛΛ −ΛΛ one, with the ΛΛ −ΞNbeing the quantitatively second in importance in the relevant region. This fact motivated the work of Refs. [9, 10], with the inclusion of the ΞN→Y N decay channel in the formalism, which modified the results presented in [8]. While the ΛNcontribution experimented a decrease, giving a value of ΓΛN= 1.31 ×10−2ΓΛ, the ΣNcontribution increased its value to ΓΣN= 2.67 ×10−2ΓΛ, adding to a total contribution of Γ=3.98 ×10−2ΓΛ. Nevertheless, a complete study of the decay process must include all the components of the wave function. The remaining contribution is the one coming from the ΛΛ−ΣΣ channel, for which one expects an even smaller contribution. In any case, it is necessary to give a realistic estimation of its value as it can give us additional information on the baryn-baryon interaction as compared to the other channels. -0.1 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 Ψ (fm-3/2) r (fm) Harmonic oscillator From GΛΛ - ΛΛ From GΛΛ - ΞN From GΛΛ - ΣΣ Figure 1: Comparison of the harmonic oscillator ΛΛ radial wave function in 6 ΛΛHe with the different components of this same wave function obtained from a G-matrix calculation, and corresponding to the 1S0channel for a value of the relative momentum of 100 MeV. Figure taken from Ref.[9] In the present work we study the (ΛΛ−ΣΣ) →Y N decay mode in 6 ΛΛHe . To describe the weak transition we use a one-meson-exchange model which includes the contribution of the pseudoscalar πand ηmesons. The evaluation of the corresponding Feynman amplitude requires the knowledge 7 of a set of strong and weak baryon-baryon-meson coupling constants. Only the ones involving decays into pions with enough phase-space can be directly extracted from experiments. To obtain the rest of couplings one must use effective lagrangians, based on the use of SU(3) flavor symmetry, formulated for the strong and weak interaction among hadrons. For an easier understanding of the diagrams we are using, the baryons and mesons relevant for this work are shown in Tables I and II [11]. Due to the lack of time, Kmesons are not considered in the present work and their inclusion is postponed. The structure of this master thesis is the following. In Chapter II the expression for the decay rate is presented. In Chapter III the One Meson Exchange model used in our calculations of the two-body weak transition is explained, including the representation of the exchange diagrams and corresponding strong and weak vertices. Once the weak transition potential is derived, the coupling constants are computed using effective lagrangians and the pole model. Last two sections of Chaper III are dedicated to obtain the matrix elements for the isospin and the spin operators. Finally, in Chapter IV we discuss the results obtained and present some conclusions. We have added some appendices at the end to illustrate some specific computations, as well as the integral for the radial contributions. Particle Q Quark composition I I3S Mass (MeV) Mean Life τ(s) Non-Leptonic Decay Modes Λ 0 uds 0 0 -1 1115.683 ±0.006 (2.632 ±0.020) ·10−10 pπ−/nπ0 Σ00 uds 1 0 -1 1192.642 ±0.024 (7.4±0.07) ·10−20 - Σ+1 uus 1 1 -1 1189.37 ±0.07 (0.8018 ±0.0026) ·10−10 pπ0/nπ+ Σ−-1 dds 1 -1 -1 1197.449 ±0.030 (1.479 ±0.011) ·10−10 nπ− n0 udd 1 2−1 20939.565413 ± 0.000006 880.2±1.0 - p1 uud 1 2 1 20938.272081 ± 0.000006 >2·1029 (years) - Table I: Physical properties of the (JP)=(1 2 +) baryons used in the present work. 8 Particle Q Quark composition I I3S Mass (MeV) Mean Life τ(s) π00 (u¯u−d¯ d)/√2 1 0 0 134.9770 ±0.0005 (8.52 ±0.18) ·10−17 π+1u¯ d1 1 0 139.57061 ±0.00024 (2.6033 ±0.0005) ·10−8 π−-1 d¯u1 -1 0 139.57061 ±0.00024 (2.6033 ±0.0005) ·10−8 η0c1(u¯u−d¯ d) + c2(s¯s) 0 0 0 547.862 ±0.017 - Table II: Physical properties of the pseudoscalar (JP) = (0−) mesons used in the present work. 9 II. Decay rate The aim of this work is to obtain the decay rate expression of ΛΛ-hypernuclei when the ΛΛ−ΣΣ strong coupled wave function is considered. The nonmesonic weak decay mechanism is studied within the one boson exchange model. If we suppose that the initial hypernucleus is at rest, we can express our decay rate as [12]: Γ = Zd3k1 (2π)3Zd3k2 (2π)3X m1m2X nuc.res 1 2J+ 1 X MJ (2π)δ(E)|M|2,(1) where Mis the nuclear transition amplitude M=<N Y 4Z|ˆ O|6 ΛΛHe >with ˆ Othe two-body operator responsible of the transition ΛΛ →Y N,k1and k2are the two final particles momenta, J is the quantum number for the initial hypernucleus, MJits projection and the δ(E) stands for the energy conservation. The first sum is over the spin projections of the two final particles whereas the second is over the quantum numbers of 4He. The two initial integrals Rd3k1 (2π)3Rd3k2 (2π)3(2π)δ(E) are to be done over the two final particles, it is, over all final momenta possibilities. To compute the weak transition amplitude, we need to decouple the ΛΛ pair from the initial hypernuclear wave function. This is relatively simple, since the two Λ0sare weakly coupled to the residual 4He core. It is known experimentally that the ΛΛ system has a quantum number L= 0, which contributes to the total wave function with a symmetric factor. Because these two particles are identical fermions, its compulsory for their total wave function to be antisymmetric. Then, an antisymmetric factor for the spin is needed, and therefore they must be coupled to S= 0. |Ψ(6 ΛΛHe) >=|6 ΛΛHe >J=S=L=T=0=|ΛΛ >J=S=L=T=0 ⊗|4He >J=S=L=T=0 (2) Our final wave function contains the two baryons coming out from the two-body weak process, and a residual 4-particle state which has to coincide with the initial 4He core, acting as spectator in the weak reaction. <Ψf|=< N|⊗ < Y |⊗ <4Z|= =X SMS <1 2m1 1 2m2|SMS>< SMS|X TMT <1 2t1T2t2|TMT>< TMT|<~ kN,~ kY|<4Z|= =X SMSX TMT <1 2m1 1 2m2|SMS>< 1 2t1T2t2|TMT>< SMS|< T MT|<~ kN,~ kY|<4Z|.(3) ∆I= 1/2 variations in the weak transition can be incoporated through the coupling of the |1/2−1/2>isospurion to one of the Λ’s in the initial ΛΛ pair, that fixes the isospin state of the 16 The potential is the sum of these two terms including G2 WgSfactor and the isospin operator, which will be given at the end of this section: Vm(~q) = −G2 WgS1 ~q 2+m2 m A 2M(~σ2·~q) + B~σ1·~q 2¯ M 1 ~q 2+m2 m 1 2E(~σ2·~q)ˆ O(T) = =−G2 WgS 1 ~q 2+m2 m 1 2M(~σ2·~q)A+B~σ1·~q 2¯ Mˆ O(T).(18) In order to calculate the transition amplitude we have to perform a Fourier transformation. The calculation of the PV contribution is given by: V0(~r) = 1 (2π)3Zei~q~r ~q 2+m2(~σ ·~q)d3q=σj∇j i(2π)3Zei~q~r ~q 2+m2d3q=−1 4iπ(~σ2·ˆr)e−mr r(m+1 r),(19) which can be solved using the Cauchy’s residue theorem. The PC potential in r-space is similarly calculated: V00(~r) = 1 (2π)3Zei~q~r ~q 2+m2(~σ1·~q)(~σ2·~q)d3q. (20) Here we use the relation (~σ1·~q)(~σ2·~q) = {σ1qσ2q−1 3σ1σ2q2}+1 3σ1σ2q2=S12 +1 3σ1σ2q2to split the PC potential in a central and a tensor contribution. Note that, as explained in section [III E], the tensor operator will not contribute because the ΣΣ system is in the singlet state. Again, using the Cauchy’s residue theorem, we obtain the PC potential V00(~r) = σ1σ2 3(2π)3 2π ir Z∞ −∞ eiqr ~q 2+m2q3dq =−σ1σ2 6π e−mr rm2.(21) The final potential then becomes V(~r) = −G2 WgS−A 2M(1 4iπ(~σ2·ˆr)e−mr r(m+1 r)) + B 2¯ M2M(−σ1σ2 6π e−mr rm2)ˆ O(T) = =−G2 WgS 1 2M4π e−mr r−A(1 i(~σ2·ˆr)(m+1 r)) + B ¯ M(−σ1σ2 3m2)ˆ O(T).(22) Both PC and PV contributions can be splited in three terms: one containing the spin operator, another for the isospin one, and a final term including all radial dependences. For our case: V(~r) = h~σ1~σ2ˆ VSS(r) + ~σ2·ˆrVP V (r)iˆ O(T),(23) VSS =G2 WgS B 2¯ M2M 1 3e−mr 2πr m2 VPV (r) = G2 WgS A 2Mme−mr 4iπr 1 + m r,(24) where ˆ O(T) is the isospin operator. 17 To obtain the isospin matrix elements we enforce the ∆T=1 2rule through the coupling of isospurion to the initial hyperon in the weak vertex |˜ Y >≡ |1 2−1 2>⊗|Y > . (25) Given that the Σ is an isospin triplet, the coupling with the isospurion gives us two spurion states, one with T= 1/2 and the other with T= 3/2: |˜ Σ−>=|Σ−>⊗ 0 1!=|1 2−1 2>⊗|1−1>=|3 2−3 2> |˜ Σ0>=|Σ0>⊗ 0 1!=|1 2−1 2>⊗|10 >=r2 3|3 2−1 2>+1 √3|1 2−1 2> |˜ Σ+>=|Σ+>⊗ 0 1!=|1 2−1 2>⊗|11 >=1 √3|3 2 1 2>+r1 3|1 2 1 2> . (26) Since we are only considering the exchange of non-strange mesons (π, η), the ΣΣ →Y N transition will always have the weak vertex located in the ΣNbaryonic line. This vertex can be expressed as MΣ→Nm =h¯ ψN(AΣ1 2+BΣ1 2γ5) + ¯ ψN(AΣ3 2+BΣ3 2γ5)iˆ O(T)·~ Φm˜ Σ,(27) where the first term represents the decay from the isospin 1/2 of the Σ spurion, and the second one involves Tinstead of τ, that is the decay from the isospin 3/2 component of the Σ spurion. Only when a πis the meson being interchanged this two cases will appear in contraposition with the ηcase, where the isospin can only come from a 1/2→1/2 transition. This is explained more accurately in section [III D]. Putting together all the contributions the final potential is given by VΣ(~q) = −GFm2 m gS 2M" AΣ1 2+BΣ1 2 2¯ M~σ1~q!+ AΣ3 2+BΣ3 2 2¯ M~σ1~q!#ˆ O(T)~σ2~q ~q 2+m2 m .(28) Where GFm2 m=G2 W. Now we apply the same relation as before ((~σ1·~q)(~σ2·~q) = {σ1qσ2q− 1 3σ1σ2q2}+1 3σ1σ2q2=S12 +1 3σ1σ2q2), and do a Fourier transform to obtain the potential decomposition as before. The structure is the same, but taking into account that we have two different options for each 18 isospin operator. V1/2 SS (r) = G2 F B1/2 2¯ M2M 1 3e−mr 2πr m2 V1/2 PV (r) = G2 FgA1/2 2Mme−mr 4iπr 1 + m r V3/2 SS (r) = G2 F B3/2 2¯ M2M 1 3e−mr 2πr m2 V3/2 PV (r) = G2 FgA3/2 2Mme−mr 4iπr 1 + m r.(29) 19 C. Coupling Constants At this point, we must derive the values of the weak and strong baryon-baryon-meson (BBφ) constants that appear in the expression of our potential. To do that, we will use the appropriate effective Lagrangians in each sector. For the strong vertices the Lagrangian at lowest order is given by the expression: Lstrong = Tr[ ¯ B(iγµ∇µ−MB)B] + D 2Tr[ ¯ Bγµγ5{uµ, B}] + F 2Tr[ ¯ Bγµγ5[uµ, B]],(30) with Band ¯ Bthe baryon and anti-baryon matrices: B=    1 √2Σ0+1 √6Λ Σ+p Σ−−1 √2Σ0+1 √6Λn Ξ−Ξ0−2 √6Λ     (31) ¯ B=    1 √2¯ Σ0+1 √6¯ Λ¯ Σ−¯ Ξ− ¯ Σ+−1 √2¯ Σ0+1 √6¯ Λ¯ Ξ0 ¯p¯n−2 √6¯ Λ     .(32) The operators appearing in Eq. (30) are defined as: ∇µ=∂µ+ [Γµ, B] uµ=i(u†∂µu−u∂µu†) Γµ=i(u†∂µu+u∂µu†).(33) The umatrix contains the meson fields, φ, through the expression : u=eiφ/√2f≈1 + i √2fφ(x) u†=e−iφ/√2f≈1−i √2fφ(x)†.(34) which has been expanded to first order to account for the exchange of a single meson. The explicit expressions for the meson matrix is: φ=    1 √2π0+1 √6η π+K+ π−−1 √2π0+1 √6η K0 K−¯ K0−2 √6η     .(35) 20 One can see that at lowest order, the contribution of the Γµand uµterms are: Γµ=i(u†∂µu+u∂µu†) = 0 (36) and uµ=i(u†∂µu−u∂µu†) = −√2 f∂µφ, (37) allowing us to rewirte the first term in the Lagrangian of Eq.(30) as: ¯ B(iγµ∇µ−MB)B=i¯ Bγµ∇µB−¯ BMBB= =i¯ Bγµ∂µB+i¯ Bγµ[Γµ, b]B−¯ BMBB=i¯ Bγµ∂µB−¯ BMBB, (38) contributing only to the kinetic energy and the mass term of the baryons in diagonal channels. The BBφ coupling constants will be given by the second and third terms in Eq.(30), where the constants Dand Fare the octet baryon to meson couplings, with values D= 0.85 MeV, F= 0.52 MeV [13][14]. To exemplify the procedure we have added in Appendix (C 1) the explicit calculation of the strong Σ−π+→Σ0vertex. Our results for the strong coupling constants required to perform the present calculation, which includes only the exchange of the πand ηmesons, are presented in Table III. Note that, for completeness, we have added the ones corresponding to the kaon, although its contribution has not been evaluated in the present work. For the weak vertices, one must perform an equivalent calculation but using the appropriate weak lagrangian. It is important to note that the effective lagrangian can only give us the parityviolating coupling constants [15] because it does not contain the CP transformation properties needed to compute PC constants. LPV =GFm2 π√2fπhDTr[ ¯ B{ξ†hξ, B}] + hFTr[ ¯ B[ξ†hξ, B]],(39) where GFm2 π= 2.21 ×10−7,fπ= 93 (MeV) is the pion decay constant, hD=−1.69 ×10−7and hF= 3.26×10−7constants taken from [16], his a matrix which effectively accounts for the change in strangeness, h=    000 001 000     ,(40) and the meson fields are included in the ξmatrix in analogy with the umatrix appearing in the strong Lagrangian, giving at leading order: ξ†hξ =h+ [h, φ]i √2f.(41) As before, an explicit example of the calculation is included in Appendix C 2, and the values of the relevant coupling constants for the weak sector are presented in Table V. 21 Channel D 2Tr[ ¯ Bγµγ5{uµ, B}] + F 2Tr[ ¯ Bγµγ5[uµ, B]] Numerical values Σ−π+→Σ0+Fγµγ5¯ Σ0∂µπ+Σ−1 f13.36 Σ+π−→Σ0-Fγµγ5¯ Σ0∂µπ−Σ+1 f- 13.32 Σ0π0→Σ00 0 Σ0η→Σ0Dγµγ5¯ Σ0∂µηΣ01 √3f12.59 Σ0π−→Σ−Fγµγ5¯ Σ−∂µπ−Σ01 f13.36 Σ0π0→Λ Dγµγ5¯ Λ∂µπ0Σ01 √3f12.18 Σ−π+→Λ +Dγµγ5¯ Λ∂µπ+Σ−1 √3f12.21 Σ−π0→Σ−-Fγµγ5¯ Σ−∂µπ0Σ−1 f-13.39 Σ+π−→Λ Dγµγ5¯ Λ∂µπ−Σ−1 √3f12.16 Σ0K0→n-D¯nγµγ5∂µK0Σ01 2f+ F¯nγµγ5∂µK0Σ01 2f-3.78 Σ0K+→pD¯pγµγ5∂µK+Σ01 2f-F¯pγµγ5∂µK+K01 2f3.78 Σ+K0→pD¯pγµγ5∂µK0Σ+1 √2f-F¯pγµγ5∂µK0K+1 √2f5.34 Σ−K+→nD¯nγµγ5∂µK+Σ−1 √2f-F¯nγµγ5∂µK+Σ−1 √2f5.36 Σ−η→Σ−D¯ Σ−γµγ5∂µηΣ−1 √3f12.64 Table III: Values of the strong baryon-baryon-meson coupling constants involving pseudoscalar mesons. The numerical values are presented in terms of the corresponding baryons and meson. The weak PC coupling constants can be obtained with the use of the pole model, as explained in Ref. [15]. To illustrate the method, we give the expression for the coupling entering the diagram depicted in Fig.(5), where the weak transition has been moved from the vertex to the corresponding baryon line. The first diagram on the right hand side of the figure represents the same weak process but with the weak transition shifted after the strong interaction takes place, while in the second diagram it is shifted onto the baryonic line right before the strong coupling occurs. The final expression for this particular vertex is given by: gΣ+Λπ− 1 mn−mΛ AΛn+gΣ+Σ0π− 1 mn−m0 Σ AΣ0n+gpnπ− 1 mΣ+−mp AΣp.(42) In the equation above, the gconstants stand for the strong BBM couplings at each of the strong vertices, the denominator contains the mass difference between the baryons appearing in the baryon line where the weak vertex occurs, and the Atake into account these weak transitions. For this specific case, since the pion can connect two Σ baryons or a Σ and a Λ baryon through 22 Channel D 2Tr[ ¯ Bγµγ5{uµ, B}] + F 2Tr[ ¯ Bγµγ5[uµ, B]] Numerical value nπ0→n ∂µπ0(D+F)1 2fn¯n13.84 nπ+→p ∂µπ+(D+F)n¯p1 √2f19.56 Σ0π+→Σ+−∂µπ+Σ0F¯ Σ+1 f-13.32 Σ0K−→Ξ−∂µK−Σ0(D+F)¯ Ξ−1 2f18.52 nK−→Σ−∂µK−(D−F)n¯ Σ−1 √2f5.36 Σ0¯ K0→Ξ0Σ0∂µ¯ K0¯ Ξ0(D+F)1 2f18.47 n¯ K0→Σ0n∂µ¯ K0¯ Σ0(−D+F)1 2f-3.78 Σ+π0→Σ+FΣ+∂µπ0¯ Σ+1 f13.30 pπ0→p p¯p∂µπ0(D+F)1 2f13.82 pπ−→n p∂µπ−¯n(D+F)1 √2f19.56 Σ−¯ K0→Ξ−Σ−∂µ¯ K0¯ Ξ−(D+F)1 √2f26.24 Σ+K−→Ξ0Σ+∂µK−¯ Ξ−(D+F)1 √2f26.09 pK−→Λ−p∂µK−¯ Λ(D+ 3F)√3 6f-15.37 pK−→Σ0p¯ Σ0∂µK−(D−F)1 2f5.35 nη →n−n∂µη¯n(D−3F)√3 6f4.14 Table IV: Strong coupling constants entering the expression of the PC weak vertices through the pole model. The numerical values are presented in terms of the corresponding baryons and meson. the strong interaction, we get an additional term, depicted as the third. 23 Channel hDTr[ ¯ B{ξ†hξ, B}] + hFTr[ ¯ B[ξ†hξ, B]] Numerical values Σ0π0→n−hD−hF 2fΣ0(2f+iπ0)¯nfmGFm2 m4.95×10−7 Σ0π+→p−hD−hF √2fΣ0iπ+¯pfmGFm2 m3.50×10−7 Σ0η→n(hD−hF) 2fΣ0(−2f+√3iη)¯nfmGFm2 m-4.29×10−7 Σ0¯ K0→Λ−hD √3fΣ0¯ K0i¯ ΛfmGFm2 m9.75×10−8 Σ0K−→Σ−√2 fΣ0iK−¯ Σ−fmGFm2 m4.61×10−7 Σ0¯ K0→Σ0hD fΣ0¯ K0¯ Σ0ifmGFm2 m-1.69×10−7 Σ+η→phD−hF √2f(2f−√iη)¯pΣ+fmGFm2 m6.06×10−7 Σ+π0→phD−hF √2f(2f+iπ0)¯pΣ+fmGFm2 m-3.5×10−7 Σ−π+→n−hD−hF fi¯nΣ−π+fmGFm2 m4.95×10−7 Σ+π−→n0 0 Σ−¯ K0→Σ−hD+hF f¯ K0i¯ Σ−Σ−fmGFm2 m1.57×10−7 Σ0K−→Λq2 3hDi¯ ΛK−Σ+fmGFm2 m-1.38×10−7 Σ+K−→Σ0−√2 fi¯ Σ0K−Σ+fmGFm2 m-4.61×10−7 Table V: Values of the weak PV baryon-baryon-meson coupling constants involving pseudoscalar mesons. Note that, for completeness, we have added the ones corresponding to the kaon, although its contribution has not been evaluated in the present work. The numerical value is always the term containing the meson. The results are given in terms of GFm2 mand the corresponding baryons and meson. D. Isospin dependence In analogy with the formalism presented in Ref. [9], we write the general structure of the isospin operator which enters Eq. (8) as: g1<1 2tNf , T tY|ˆ O1|1 2t, T2t2>< 1 2t|1t1,1 2−1 2>+ +g2<1 2tNf , T tY|ˆ O2|3 2t, T2t2>< 3 2t|1t1,1 2−1 2>+ +g3<1 2tNf , T tY|ˆ O3|1 2t, T2t2>< 1 2t|1t1,1 2−1 2> , (43) where the Clebsh-Gordan coefficients on the right in each line represent the coupling of he isospurion (|1 2−1 2i) to the Σ entering the weak vertex (|1t1i), to give an isospurion state (˜ Σ) with isospin projection t=t1−1 2. We also recall here that |T2t2istands for the isospin state corresponding to 24 Σ+n π+ (a) = Σ+Σ0 n π− (b) + Σ+p n π− (c) + Σ+Λ n π− (d) Figure 5: Ilustrative example of using a pole model. Channel hDTr[ ¯ B{ξ†hξ, B}] + hFTr[ ¯ B[ξ†hξ, B]] Numerical value Σ0π0→n gnnπ01 mΣ0−mnAΣ0n+gΣ0Σ0π01 mn−mΣ0AΣ0n2.25×10−5 Σ0π+→p gnpπ+1 mΣ0−mnAΣ0n+gΣ0Σ+π+1 mp−mΣ+AΣ+p9.80×10−8 Σ0η→n gnnη 1 mΣ0−mnAΣ0n+gΣ0Σ0η1 mn−mΣ0AΣ0n-1.44×10−6 Σ0¯ K0→ΛgΣ0Ξ0¯ K01 mΛ−mΞ0AΞ0Λ+gnΛ¯ Σ0−mn 1 mΣ0−mnAΣ0n1.21×10−6 Σ0K−→Σ−gΣ0Ξ−K− 1 mΣ−−mΞ− AΞ−Σ−+gnΣ−K− 1 mΣ0−mnAΣ0n4.18×10−6 Σ0¯ K0→Σ0gΣ0Ξ0¯ K01 mΣ0−mΞ0AΞ0Σ0+gnΣ0¯ K01 mΣ0−mnAΣ0n1.7×10−6 Σ+η→p gΣ+Σ+η1 mp−mΣ+AΣ+p+gppη 1 mΣ+−mpAΣ+p2.57×10−6 Σ+π0→pgΣ+Σ+π01 mp−mΣ+AΣ+p+gppπ01 m+ Σ−mpAΣ+p-1.26×10−7 Σ−π+→n gΣ−Λπ+1 mn−mΛAΛn+gΣ−Σ0π+1 mn−mΣ0AΣ0n-7.90×10−7 Σ+π−→n gpnπ− 1 mΣ+−mpAΣ+pgΣ+Λπ− 1 mn−mΛAΛn+gΣ+Σ0π− 1 mn−mΣ0AΣ0n-9.99×10−7 Σ−¯ K0→Σ−gΣ−Ξ−¯ K01 mΣ−−mΞ− AΞ−Σ−6.70×10−6 Σ0K−→ΛgΣ+Ξ0K− 1 mΛ−mΞ0AΞ0Λ+gpΛK− 1 mΣ+−mpAΣ+p1.14×10−5 Σ+K−→Σ0gΣ+Ξ0K− 1 mΣ0−mΞ0AΞ0Σ0+gpΣ0K− 1 mΣ+−mpAΣ+p2.40×10−6 Table VI: Values of the weak PC baryon-baryon-meson coupling constants involving pseudoscalar mesons. Note that, for completeness, we have added the ones corresponding to the kaon, although its contribution has not been evaluated in the present work. the Σ entering the strong vertex, while h1 2tNf |and hT tY|represent the final nucleon and hyperon 25 isospin states respectively. The operational structures in Eq. (43) are: ˆ O1=~τ1·~ T2,ˆ O2=~ T1·~ T2,ˆ O3=ˆ I1·ˆ I2.(44) but not all of them contribute to all final states. When our final state is given by a nΛ pair we will only have contributions from the two first operators, due to the fact that the Λ has T= 0 and thus, it is not possible to connect the corresponding states with an identity operator. This is illustrated in Figure 6, where only the two possibilities that are allowed are drawn. All three operators contribute when the final state is NΣ, as shown in Figure 7. ˜ Σ1(T= 1/2) ~τ1N(T= 1/2) Σ2(T= 1) Λ(T= 0) ~ T2 m (a) ˜ Σ1(T= 3/2) ~ T1N(T= 1/2) Σ2(T= 1) Λ(T= 0) ~ T2 m (b) Figure 6: Diagrams showing the different operators that contribute to a final nΛ state. ˜ Σ1(T= 1/2) ~ T1N(T= 1/2) Σ2(T= 1) Σ(T= 1) ~ T2 m (a) ˜ Σ1(T= 1/2) ˆ I1N(T= 1/2) Σ2(T= 1) Σ(T= 1) ˆ I2 m (b) ˜ Σ1(T= 3/2) ~ T1N(T= 1/2) Σ2(T= 1) Σ(T= 1) ~ T2 m (c) Figure 7: Diagrams showing the different isospin operators that contribute to a final NΣ state. The matrix elements for these operators can be easily obtained. Explicit calculations are included in section B, and here only the final results are given. The connection between 3 2and 1 2 isospin states is given by: T1 3 2 1 2 = 0 0 −1 √30 0 0 0 −1!T0 3 2 1 2 =  0q2 30 0 0 0 q2 30 T−1 3 2 1 2 = −1 0 0 0 01 √30 0!,(45) while ~τ matrices connect 1 2states: τ1 1 2 1 2 = 0√2 0 0 !τ−1 1 2 1 2 = 0 0 √2 0!τ0 1 2 1 2 = 1 0 0−1!.(46) 32 Γ(ΛΛ −ΛΛ) Γ(ΛΛ −ΞN) Γ(ΛΛ −ΣΣ) π3.96 3.35 0.02 η0.02 6.42 ×10−46.25 ×10−3 π+η2.37 1.66 0.01 Table XII: PV/PC contributions ratio for the three wave function components and for the exchange of π,ηand both. In this section the results from all the previous computations are presented. This is an exploratory study of the contribution of the πand ηmesons to the decay rate produced by the ΛΛ −ΣΣ component. All of the results are computed in a previously on Refs. [10, 12]. As it can be seen on the results, the transition rate for Σ0nand Σ−pdiffers only by a factor 2. This is due to the fact that they have the same final state structure except for the Clebsch-Gordan coefficient for the spin. For Σ0nthe Clebsch-Gordan is 1 √3, while for Σ−pis q2 3. Since we are dealing with a transition amplitude, all this values are squared, leading to this factor 2. Regarding the contribution of each pseudoscalar to the final decay rate, the ηexchange is negligible as compared to the pion exchange. For the (ΛΛ −ΛΛ) value it represents a 10,71% of the total contribution and similar values are obtained for (ΛΛ −ΞN) which is a 7,3%, and also for the total contribution which leads to a 6,75%. For the case (ΛΛ −ΣΣ) the exchange of η contributes in a 54,9% and cannot be neglected. Also the ηmeson exchange has a remarkable role in the Λnfinal state in all cases, specially in the (ΛΛ −ΛΛ) channel. For the pions and also for the contribution of both, the ΣNfinal state has a larger decay rate, as it can be seen in Table (X). If we compute the PV/PC ratio, as displayed in table (ref), it is easy to see the leading contribution for each meson. In all channels except (ΛΛ −ΣΣ), the largest contribution to PV comes from the πexchange up to a 5 factor, while for the ηmeson exchange, the contribution to PC can be up to three orders of magnitude bigger than PV. When the two exchanges are taken into account, the ratio balances again in favour of PV. For the (ΛΛ −ΣΣ) channel, the exchange of a π, an ηas well as the combination of both exchanges contributes more significantly to the PC term. As we can see in Table (X), the role played by ΛΛ −ΣΣ component of the wave function is not very significant at the level of the total rate, being one order of magnitude below to the corresponding ΓΛΛ and ΓΞN. Regarding the rates of the (ΛΛ −ΛΛ) →Λn, the sum of both contributions is constructive when leading to final Λnstate while is slightly destructive when dealing with (ΛΛ −ΣN) states. On the contrary, the ΛΛ −ΞNcomponent of the of the wave function shows a constructive interference for all final states. The behaviour we obtained for the ΛΛ −ΣΣ contribution is similar to the one shown for the ΛΛ −ΛΛ one. 33 It is difficult to reconcile this numbers with what we see from Fig. (1) since we would expect much more similar values for ΓΣΣ and ΓΞN, and much more difference between these rates and ΓΛΛ. Our calculations of the decay rates have been based on the exchange of πand η. In order to have a complete picture of the contribution of the ΣΣ −Y N transitions to the decay of 6 ΛΛHe, we need to incorporate the exchange of not only πand η, but also Kand vector mesons, such as ρ,ω and K∗. This is out of the scope of the present thesis and will be addressed in a future work. 34 Appendices A. r-space integrals In this section we compute PSMS|<~ k, SMS|ˆ O|ψREL, S0= 0 MS0= 0 >|2=|< ψ1S0 Y N (~ k,~r)|ˆ O|ψ1S0 ΣΣ (~r)>|2appearing in Eq. (8). We calculate separately the terms for S= 1 and for S= 0, taking in account the results from the previous section. To illustrate the calculation, we take for the initial wavefunctions harmonic oscillator wavefunctions, while for the outgoing ones take plane waves. Nevertheless, the final result has been computed using the correlated wave functions obtained solving the appropriate G-matrix and T-matrix equations. ψ1S0 ΣΣ (~r) = s4 √πb3e−r2/2b2Y00(ˆr) ψ1S0 Y N (~ k,~r) = ei~ k·~r =X LML 4πiLjL(kr, r)Y∗ LML(ˆ kr)YLML(ˆr),(A1) with ba parameter of the plane wave functions. For S= 0, the spin-dependent term, we have: < ψ1S0 Y N (~ k,~r)|ˆ O|ψ1S0 ΣΣ (~r)>= =1 (2π)3/2Zd3rψREL fVSS(~r)ψREL 0< S = 0 MS= 0|~σ1·~σ2|S0= 0 MS0= 0 >= =1 (2π)3/2Zd3rψREL fVSS(~r)ψREL 0(−3). (A2) Where we know from Eq. (24): VSS =G2 WgS B 2¯ M2M 1 3e−mr 2πr m2=V0 S e−mr 2πr (A3) < ψ1S0 Y N (~ k,~r)|ˆ O|ψ1S0 ΣΣ (~r)>= =1 (2π)3/2Zdrr2Zdˆr X LML 4πiLjL(kr)Y∗ LML(ˆ k)YLML(ˆr)  ×G2 WgS B 2¯ M2M 1 3e−mr 2πr m2 2 π1/4b3/2e−r2/2b2Y00(ˆr)(−3).(A4) Integrating the terms depending on ˆr: ZdˆrYLML(ˆr)Y00(ˆr) = 1 √4πδL0δML0,(A5) 35 So our integral becomes: < ψ1S0 Y N (~ k,~r)|ˆ O|ψ S 0 ΣΣ(~r)>= =−1 (4π)5/4Y∗ 00(ˆ k)G2 WgS B ¯ MM m2 b3/2Zdrr2j0(kr)e−mr re−r2/2b2= =−1 (4π)5/4Y∗ 00(ˆ k)G2 WgS B ¯ MM m2 b3/2Zdrr2sin(kr) kr e−mr re−r2/2b2= =−1 (4π)5/4Y∗ 00(ˆ k)G2 WgS B ¯ MM m2 b3/2Zdr sin(kr) ke−mre−r2/2b2.(A6) For S= 1 we take again from Eq. (24): VPV (r) = G2 WgS A 2Mme−mr 4iπr 1 + m r(A7) < ψ3P0 Y N (~ k,~r)|ˆ O|ψ1S0 ΣΣ (~r)>= =1 (2π)3/2Zd3rψREL f∗VPV (~r)ψREL 0<1MS|~σ2·~r|S0= 0 MS0= 0 >= =1 (2π)3/2Zdrr2Zdˆr X LML 4πiLjL(kr)Y∗ LML(ˆ k)YLML(ˆr) G2 WgS Am 2M e−mr 4iπr 1 + m r ×2 π1/4b3/2e−r2/2b2Y00(ˆr)<1MS|~σ2·~r|S0= 0 MS0= 0 >= =1 (2π)3/2Zdrr2Zdˆr X LML 4πiLjL(kr)Y∗ LML(ˆ k)YLML(ˆr)  ×G2 WgS Am 2M e−mr 4iπr 1 + m r 2 π1/4b3/2e−r2/2b2Y00(ˆr) ×"X m10 <1MS|1 2m10 1 2(MS−m10)>< 1 2m10 1 2−m10|S0= 0 MS0= 0 > ×(−1)−MS√3<1 2(−m10)1MS|1 2(MS−m10)>r4π 3Y1MS(ˆr)#.(A8) Performing the Rdˆrintegral as we did before: ZdˆrYLML(ˆr)Y00(ˆr)Y1MS(ˆr) = 1 √4πZdˆrYLML(ˆr)Y1MS(ˆr) = 1 √4πδL1δMLMS,(A9) we can simplify our expression: 36 < ψ3P0 Y N (~ k,~r)|ˆ O|ψ1S0 ΣΣ (~r)>= =1 (2π)3/2Y∗ LML(ˆ k)G2 WgS Am M (−1)MS π1/4b3/2 × X m10 <1MS|1 2m10 1 2(MS−m10)>< 1 2m10 1 2−m10|S0= 0 MS0= 0 > ×<1 2(−m10)1MS|1 2(MS−m10)>Zdrr2j1(kr)e−mr re−r2/2b21 + m r= =1 (2π)3/2Y∗ LML(ˆ k)G2 WgS Am M (−1)MS π1/4b3/2 X m10 <1MS|1 2m10 1 2(MS−m10)> ×<1 2m10 1 2−m10|S0= 0 MS0= 0 >< 1 2(−m10)1MS|1 2(MS−m10)> ×Zdr sin(kr) k2r−cos(kr)e−mre−r2/2b21 + m r.(A10) 37 B. Isospin matrix elements In this section we work with the isospin part of our decay rate expression: <1 2−1 2|ˆ O(T)|1 2−1 2>= =<1 2−1 2|T10t10T20t20 >< T10t10T20t20|ˆ O(T)|T1t1T2t2>< T1t1T2t2|1 2−1 2> . (B1) For that, we need to know the reduced matrix elements of the operators themselves. In this section, we are using the Wigner-Eckart theorem (Eq. 55): < jm|Tk q|j0m0>=(−1)2k< j0m0kq|jm >< j|Tk|j0> √2j+ 1 ,(B2) to get rid of mdependence. To calculate < j|Tk|j0>we are going to use two properties. The first one will be the relation between cartesian and spherical components: <3 2mA|~τAN |1 2mN>=<1 2mN1−1|3 2mA>1 √2    1 i 0     + +<1 2mN10|3 2mA>    0 0 1     +<1 2mN11|3 2mA>−1 √2    1 −i 0     , (B3) so: <3 2mA|τx|1 2mN>=1 √2(<1 2mN1−1|3 2mA>−<1 2mN11|3 2mA>) <3 2mA|τz|1 2mN>=<1 2mN10|3 2mA> <3 2mA|τy|1 2mN>=i √2(<1 2mN1−1|3 2mA>+<1 2mN11|3 2mA>).(B4) With this, we can express it in spherical components: τ1=−1 √2(τx+iτy) =<1 2mN11|3 2mA>=<3 2mA|τ1|1 2mN> τ0=τz=<1 2mN10|3 2mA>=<3 2mA|τ0|1 2mN> τ−1=1 √2(τx−iτy)<1 2mN1−1|3 2mA>=<3 2mA|τ−1|1 2mN> . (B5) The second property comes from the Wigner-Eckart theorem mentioned previously: <3 2mA|τ0|1 2mN>=(−1)2·0<1 2mN10|3 2mA>< 3 2||τ|1 2> 2.(B6) 38 Combining these two expressions we get: <1 2mN10|3 2mA>=(−1)2·0<1 2mN10|3 2mA>< 3 2||τ|1 2> 2(B7) <3 2||τ|1 2>= 2.(B8) Due to the fact that the reduced matrix elements do not depend on the projection values for the isospin, it can be obtained with all other projections, leading to the same exact result. For the reduced matrix elements involving integer numbers we used [10]: <00|Tk|1m >= (−1)k<001 −k|1m >= (−1)kδmk (B9a) <1m|Tk|00 >=δmk (B9b) <1m0|Tk|1m >=√2<1m0|1m1k > . (B9c) We are now capable of obtaining the matrix for each operator. We are going to compute T, an operator that allows us to connect T= 3/2 states and T= 1/2 states, and τthat allows us to connect two T= 1/2 states. <1 2 1 2|T1|3 2−1 2>=<3 2−1 211|1 2 1 2>(−1)2·11 √2<1 2|T1|3 2>= =1 √6(−1)2·11 √2<1 2|T1|3 2>=1 √6·1·1 √2(−2) = −1 √3 <1 2−1 2|T1|3 2−3 2>=<3 2−3 211|1 2−1 2>(−1)2·11 √2<1 2|T1|3 2>= =1 √2(−1)2·11 √2<1 2|T1|3 2>=1 √2·1·1 √2(−2) = −1 (B10) T1= 0 0 −1√3 0 0 0 0 −1!(B11) <1 2 1 2|T0|3 2 1 2>=<3 2 1 210|1 2 1 2>(−1)2·11 √2<1 2|T0|3 2>= =−1 √3(−1)2·11 √2<1 2|T0|3 2>=−1 √3·1·1 √2(−2) = r2 3 <1 2−1 2|T0|3 2−1 2>=<3 2−1 210|1 2−1 2>(−1)2·11 √2<1 2|T0|3 2>= =1 √2(−1)2·11 √2<1 2|T0|3 2>=1 √2·1·1 √2(−2) = r2 3(B12) 39 T0=  0q2 30 0 0 0 q2 30 (B13) <1 2 1 2|T−1|3 2 3 2>=<3 2 3 21−1|1 2 1 2>(−1)2·11 √2<1 2|T−1|3 2>= =−1 √2(−1)2·11 √2<1 2|T−1|3 2>=−1 √2·1·1 √2(−2) = −1 <1 2−1 2|T−1|3 2 1 2>=<3 2 1 21−1|1 2−1 2>(−1)2·11 √2<1 2|T−1|3 2>= =1 √6(−1)2·11 √2<1 2|T−1|3 2>=−1 √3·1·1 √2(−2) = −1 √3(B14) T−1= −1 0 0 0 01 √30 0!.(B15) We can do the same for the transition 1 2→1 2with a τoperator: <1 2−1 2|τ1|1 2 1 2>=<1 2 1 211|1 2−1 2>(−1)2·11 √2<1 2|τ1|1 2>= =−r2 3(−1)2·11 √2<1 2|τ1|3 2>=−r2 3·1·1 √2(√6) = −√2 (B16) τ1= 0√2 0 0 !(B17) <1 2 1 2|τ−1|1 2−1 2>=<1 2−1 211|1 2 1 2>(−1)2·11 √2<1 2|τ−1|1 2>= =r2 3(−1)2·11 √2<1 2|τ−1|3 2>=r2 3·1·1 √2(√6) = √2 (B18) τ−1= 0 0 √2 0!(B19) <1 2−1 2|τ0|1 2−1 2>=<1 2−1 211|1 2−1 2>(−1)2·11 √2<1 2|τ0|1 2>= =1 √3(−1)2·11 √2<1 2|τ0|3 2>=1 √3·1·1 √2(√6) = 1 <1 2 1 2|τ0|1 2 1 2>=<1 2 1 211|1 2 1 2>(−1)2·11 √2<1 2|τ0|1 2>= =−1 √3(−1)2·11 √2<1 2|τ0|3 2>=−1 √3·1·1 √2(√6) = −1 (B20) 40 τ0= 1 0 0−1!.(B21) The last step is to obtain each value for the different diagrams we get at the very beginning of this work. The general expression can be obtained combining all the terms obtained before: <1 2−1 2|ˆ O(T)|1 2−1 2>= =<1 2−1 2|T10t10T20t20 >< T10t10T20t20|ˆ O(T)|T1t1T2t2>< T1t1T2t2|1 2−1 2>= =<1 2−1 2|T10t10T20t20 >< T10t10T20t20|A1·A2|T1t1T2t2>< T1t1T2t2|1 2−1 2>= =<1 2−1 2|T10t10T20t20 >< T1t1T2t2|1 2−1 2>X µ (−1)µ ×< T10t10|Aµ 1|T1t1>< T20t20|A−µ 2|T2t2>, (B22) where here the operator Acan be both τor Tdepending on the isospin value of initial and final states. 41 C. Example of calculation of coupling constants 1. Strong example: the Σ−π+→Σ0vertex As an example, the computations for the Σ−π+→Σ0vertex strong coupling constant are explicitly written down. As we previously said, only the two last terms of the Lagrangian will contribute to the final value: D 2Tr[ ¯ Bγµγ5{uµ, B}] + F 2Tr[ ¯ Bγµγ5[uµ, B]],(C1) with the particular values for B,¯ Band φ: B=    0 0 0 Σ−0 0 0 0 0     ¯ B=    1 √2¯ Σ00 0 0−1 √2¯ Σ00 0 0 0     φ=    0π+0 0 0 0 0 0 0     .(C2) The first step is to compute both the commutator and anticommutator for each term we need. Using uµ=i(u†∂µu−u∂µu†) = −√2 f∂µφwe get: {uµ, B}=−√2 f∂µ(π+)Σ−       010 000 000        000 100 000     +    000 100 000        010 000 000        = =−√2 f∂µ(π+)Σ−       100 010 000        (C3) [uµ, B] = −√2 f∂µ(π+)Σ−       010 000 000        000 100 000    −    000 100 000        010 000 000        = =−√2 f∂µ(π+)Σ−       100 0−1 0 000        .(C4) Both traces may now be evaluated, taking in account that γ5and γµare in Dirac space, which allows us to take this terms out of the trace: D 2Tr[ ¯ Bγµγ5{uµ, B}] = D 2γµγ5Tr[ ¯ B{uµ, B}] = =D 2γµγ5 −√2 f∂µ(π+)Σ−(1 √2 ¯ Σ0)!Tr     100 0−1 0 000     = 0 (C5)