Study of the $ {{\boldsymbol J}/\boldsymbol\psi\bf\to \boldsymbol\Lambda\bar{\boldsymbol\Sigma}^{\bf 0}\boldsymbol\eta}$ reaction

The BESIII collaboration has recently reported results on the $ J/\psi \to \Lambda \bar \Sigma^0 \eta +\mathrm{c.c} $ reaction [1], in which a clear signal for the $ \Lambda(1670) $ excitation is observed in the $ \eta \Lambda $ mass distribution, accounting for practically all the strength of this quantity. The reaction violates isospin, providing an opportunity to investigate the origin of isospin violation, a relevant issue in the strong interaction, which conserves this quantity.

The $ \Lambda(1670) $ lies very close to the $ \eta \Lambda $ threshold, making it a candidate molecular state. However, although proximity to a threshold favors molecular components [2–4], it is not a sufficient condition for a state to have a molecular nature [5–7].

For a long time, the $ \Lambda(1670) $ has been identified as a dynamically generated state, together with the two $ \Lambda(1405) $ states, arising from the interaction of the coupled channels $ \bar{K} N $, $ \pi \Sigma $, $ \pi \Lambda $, $ \eta \Sigma $, $ \eta \Lambda $, and $ K \Xi $ [8, 9]. The $ \Lambda(1670) $ was found in [9] to correspond to a state that couples most strongly to $ K \Xi $, followed by $ \eta \Lambda $ and then $ \bar{K} N $, while coupling very weakly to $ \pi \Sigma $.

The nature of a resonance as a dynamically generated state from coupled-channel interactions allows for larger isospin violation than expected for elementary particles in reactions involving these states. This is indeed the case for the small width of the $ D^*_{s0}(2317) $ resonance, due to the isospin-forbidden decay into $ D_s \pi^0 $. In fact, the $ D^*_{s0}(2317) $ width, assuming the $ D^*_{s0}(2317) $ to be an ordinary state, is much smaller than that obtained when it is considered a molecular state, mostly due to the $ KD $ interaction [10−17]. To understand the reason, one must return to early works [18], which show that an important source of isospin violation arises from loops containing kaons, due to the different masses of the charged and neutral kaons. Kaonic loops have been identified as the source of isospin violation in many reactions [19−27]. In the case of the $ \Lambda(1670) $ state, there are loops from $ K^- p $ and $ \bar{K}^0 n $, as well as loops from $ K^+ \Xi^- $ and $ K^0 \Xi^0 $, which also have different masses. It is therefore interesting to examine how these channels influence isospin violation in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay. This is the purpose of the present work, where we show that the influence of the $ K \Xi $ channels is much more important than that of the $ \bar{K} N $ channels.

The $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction can be considered a complement to two other reactions, $ \Lambda^+_c \to \pi^+ K^- p $ [28] and $ K^- p \to \eta \Lambda $ [29], which were studied in [30], from which the $ \eta \Lambda $ scattering length and effective range were extracted. We use here the same input as in the study of the $ \Lambda^+_c \to \pi^+ K^- p $ reaction, and the agreement obtained with the data here corroborates the findings and conclusions obtained in [30], including the precise values of $ a,r_0 $ extracted for $ \eta \Lambda $ scattering close to threshold in [30].

The starting point is the strong decay of $ J/\psi $ into $ \bar{\Sigma}^0 B P $, where B is a baryon from the $ 1/2^+ $ octet and P is a pseudoscalar meson from the $ 0^- $ octet. The $ \bar{\Sigma}^0 B P $ system initially satisfies isospin symmetry, which implies that $ BP $ has isospin $ I=1 $. The $ BP $ interaction then ultimately leads to $ \eta \Lambda $ with isospin violation, which occurs through loops that do not cancel because of the different masses of the $ BP $ states belonging to the same isospin multiplet.

To determine which $ BP $ states are produced in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction, we consider that $ J/\psi $ is an SU(3) singlet in the $ u,d,s $ sector. Then, writing the matrices for the $ q\bar{q} $ states of the octet in terms of mesons and those for the $ qqq $ states in terms of the baryons of the octet, we have

There are several ways to construct SU(3) singlet structures with these matrices by taking traces of products, such as $ \langle \bar{B} B P \rangle $, $ \langle \bar{B} P B \rangle $, $ \langle \bar{B} B\rangle \langle P \rangle $, $ \langle \bar{B} P\rangle \langle B \rangle $, $ \langle B P\rangle \langle \bar{B} \rangle $, $ \langle \bar{B} \rangle \langle B \rangle \langle P \rangle $. However, using elements of heavy quark symmetry, one finds that the largest contribution comes from the structures containing fewer traces [31, 32]; therefore, we retain only these terms. In Ref. [31], it is shown that one should expect a reduction factor of 1/$ N_c $, where $ N_c $ is the number of colors, in an amplitude that contains an extra trace. In the present case, in addition to the two terms that we have, $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $, we could have the $ \langle \bar{B} B\rangle \langle P \rangle $ structure, because P, unlike B and $ \bar{B} $, has a nonzero trace. One may wonder why such a structure is not present in the work of [33], where the most general structure has two terms, which are equivalent to those used here. The reason is that in [33], one takes the P matrix corresponding to the octet of pseudoscalars, for which P also has zero trace. Here, we consider the $ \eta-\eta^\prime $ mixing of [34], and the trace of P is nonzero. Nevertheless, because of the large-$ N_c $ argument and the small $ \eta-\eta^\prime $ mixing, we consider that the $ \langle \bar{B} B\rangle \langle P \rangle $ term can be neglected, and we return to the two structures used in [33]. Thus, we consider the $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $ structures in our study. We find

However, since it was found in [9] that the $ \pi\Sigma $ states couple very weakly to the $ \Lambda(1670) $, we neglect these channels and find

With the isospin phase convention used in our approach, $ (K^+,K^0),(\bar{K}^0,-K^-) $, the $ p K^{-} - n \bar{K}^0 $ combination in Eq. (3) has $ I=1 $, as required, so that, when combined with the $ \bar{\Sigma}^0 $, it can yield the $ I=0 $ of the $ J/\psi $. Both $ \Sigma^0 \eta $ and $ \Lambda \pi^0 $ have $ I=1 $ and therefore cannot undergo a strong-interaction transition to $ \eta\Lambda $. However, $ K^{-} p $ and $ n \bar{K}^0 $ can lead to $ I=0 $, since the loops of $ K^{-} p $ and $ n \bar{K}^0 $ involved in the final-state interaction do not cancel. For practical purposes, we can simply take the object produced in the first step of the $ J/\psi $ decay

We now consider the $ \langle \bar{B} P B \rangle $ structure. We obtain

and, neglecting the $ \pi \Sigma $ channels as before, as well as the $ \pi^0 \Lambda $ and $ \eta \Sigma^0 $ channels, which cannot transition to an $ I=0 $ state, we obtain

Once again, considering the isospin doublet $ (\Xi^0,-\Xi^-) $ in analogy to $ (\bar{K}^0,-K^-) $, we see that the $ K \Xi $ combination in Eq. (6) has $ I=1 $, as it should. However, in the final-state interaction producing $ \eta\Lambda $, the $ K^+ \Xi^- $ and $ K^0 \Xi^0 $ loops do not cancel. Thus, we obtain an isospin-violating mechanism that can ultimately produce $ \eta\Lambda $, with the shape of the $ \Lambda(1670) $.

When considering isospin-violation effects, one cannot neglect the fact that electromagnetic interactions also violate isospin and can contribute to the processes studied. No study of the present reaction has addressed such effects, but related work has been performed for $ J/\psi $ decay into baryon-antibaryon pairs [35]. In that study, a general operator with seven free parameters is used to fit the data, and the operator explicitly contains an electromagnetic term. Of relevance to us is the result in Table 5 of that paper, which shows that the strong interaction alone gives rates about two orders of magnitude larger than the electromagnetic terms alone and, by itself, provides a fair reproduction of the data. The relative effect of electromagnetic contributions could be larger in isospin-violating processes, but this possibility is not discussed there. Insight can be gained from studies in which isospin violation from loops and electromagnetic effects are considered together, as in the analysis of $ f_0(980) $-$ a_0(980) $ mixing in [36]. That paper concludes that kaon mass differences in the loops generate the dominant mixing effect, whereas direct electromagnetic contributions are subleading. One should not forget that the mass differences between states with different charges in the same isospin multiplet are partly due to electromagnetic interactions [37] (see also related work on this issue in [38]). However, the relevant question here is whether considering loops with different particle masses is sufficient to determine isospin breaking. According to [36], this is the case in their problem, and we shall assume that it is also the case here.

The two structures $ \langle \bar{B} P B \rangle $ and $ \langle \bar{B} B P \rangle $ are different, as shown in Eqs. (4), (6), and we assign them different weights, $ A, B $, respectively. Thus, for practical purposes, we have

The transition amplitude for $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ is then given by the diagrams shown in Fig. 1,

Figure 1.  Diagrams involving final-state interactions of $ PB $ pairs leading to the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay.

The interaction of the coupled channels leading to the formation of the $ \Lambda(1670) $ occurs in S-wave; hence, we keep the pair $ BP $ in S-wave. However, $ \bar{\Sigma}^0 \eta \Lambda $ has the combination $ J^P, \frac{1}{2}^- 0^- \frac{1}{2}^+ $, which has positive parity; therefore, a P-wave is required to match the parity of the $ J/\psi $. This is achieved by contracting the $ \epsilon^\mu $ polarization vector of the $ J/\psi $ with the momentum of $ \bar{\Sigma}^0 $. Hence, our transition amplitude is

with

Evaluating the sum and average over polarizations of the F factor, we obtain

with $ \mathit{\boldsymbol{p}}^2_{\Sigma^0} $ evaluated in the $ J/\psi $ rest frame.

The starting point is the strong decay of $ J/\psi $ into $ \bar{\Sigma}^0 B P $, where B is a baryon from the $ 1/2^+ $ octet and P is a pseudoscalar meson from the $ 0^- $ octet. The $ \bar{\Sigma}^0 B P $ system initially satisfies isospin symmetry, which implies that $ BP $ has isospin $ I=1 $. The $ BP $ interaction then ultimately leads to $ \eta \Lambda $ with isospin violation, which occurs through loops that do not cancel because of the different masses of the $ BP $ states belonging to the same isospin multiplet.

To determine which $ BP $ states are produced in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction, we consider that $ J/\psi $ is an SU(3) singlet in the $ u,d,s $ sector. Then, writing the matrices for the $ q\bar{q} $ states of the octet in terms of mesons and those for the $ qqq $ states in terms of the baryons of the octet, we have

There are several ways to construct SU(3) singlet structures with these matrices by taking traces of products, such as $ \langle \bar{B} B P \rangle $, $ \langle \bar{B} P B \rangle $, $ \langle \bar{B} B\rangle \langle P \rangle $, $ \langle \bar{B} P\rangle \langle B \rangle $, $ \langle B P\rangle \langle \bar{B} \rangle $, $ \langle \bar{B} \rangle \langle B \rangle \langle P \rangle $. However, using elements of heavy quark symmetry, one finds that the largest contribution comes from the structures containing fewer traces [31, 32]; therefore, we retain only these terms. In Ref. [31], it is shown that one should expect a reduction factor of 1/$ N_c $, where $ N_c $ is the number of colors, in an amplitude that contains an extra trace. In the present case, in addition to the two terms that we have, $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $, we could have the $ \langle \bar{B} B\rangle \langle P \rangle $ structure, because P, unlike B and $ \bar{B} $, has a nonzero trace. One may wonder why such a structure is not present in the work of [33], where the most general structure has two terms, which are equivalent to those used here. The reason is that in [33], one takes the P matrix corresponding to the octet of pseudoscalars, for which P also has zero trace. Here, we consider the $ \eta-\eta^\prime $ mixing of [34], and the trace of P is nonzero. Nevertheless, because of the large-$ N_c $ argument and the small $ \eta-\eta^\prime $ mixing, we consider that the $ \langle \bar{B} B\rangle \langle P \rangle $ term can be neglected, and we return to the two structures used in [33]. Thus, we consider the $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $ structures in our study. We find

However, since it was found in [9] that the $ \pi\Sigma $ states couple very weakly to the $ \Lambda(1670) $, we neglect these channels and find

With the isospin phase convention used in our approach, $ (K^+,K^0),(\bar{K}^0,-K^-) $, the $ p K^{-} - n \bar{K}^0 $ combination in Eq. (3) has $ I=1 $, as required, so that, when combined with the $ \bar{\Sigma}^0 $, it can yield the $ I=0 $ of the $ J/\psi $. Both $ \Sigma^0 \eta $ and $ \Lambda \pi^0 $ have $ I=1 $ and therefore cannot undergo a strong-interaction transition to $ \eta\Lambda $. However, $ K^{-} p $ and $ n \bar{K}^0 $ can lead to $ I=0 $, since the loops of $ K^{-} p $ and $ n \bar{K}^0 $ involved in the final-state interaction do not cancel. For practical purposes, we can simply take the object produced in the first step of the $ J/\psi $ decay

We now consider the $ \langle \bar{B} P B \rangle $ structure. We obtain

and, neglecting the $ \pi \Sigma $ channels as before, as well as the $ \pi^0 \Lambda $ and $ \eta \Sigma^0 $ channels, which cannot transition to an $ I=0 $ state, we obtain

Once again, considering the isospin doublet $ (\Xi^0,-\Xi^-) $ in analogy to $ (\bar{K}^0,-K^-) $, we see that the $ K \Xi $ combination in Eq. (6) has $ I=1 $, as it should. However, in the final-state interaction producing $ \eta\Lambda $, the $ K^+ \Xi^- $ and $ K^0 \Xi^0 $ loops do not cancel. Thus, we obtain an isospin-violating mechanism that can ultimately produce $ \eta\Lambda $, with the shape of the $ \Lambda(1670) $.

When considering isospin-violation effects, one cannot neglect the fact that electromagnetic interactions also violate isospin and can contribute to the processes studied. No study of the present reaction has addressed such effects, but related work has been performed for $ J/\psi $ decay into baryon-antibaryon pairs [35]. In that study, a general operator with seven free parameters is used to fit the data, and the operator explicitly contains an electromagnetic term. Of relevance to us is the result in Table 5 of that paper, which shows that the strong interaction alone gives rates about two orders of magnitude larger than the electromagnetic terms alone and, by itself, provides a fair reproduction of the data. The relative effect of electromagnetic contributions could be larger in isospin-violating processes, but this possibility is not discussed there. Insight can be gained from studies in which isospin violation from loops and electromagnetic effects are considered together, as in the analysis of $ f_0(980) $-$ a_0(980) $ mixing in [36]. That paper concludes that kaon mass differences in the loops generate the dominant mixing effect, whereas direct electromagnetic contributions are subleading. One should not forget that the mass differences between states with different charges in the same isospin multiplet are partly due to electromagnetic interactions [37] (see also related work on this issue in [38]). However, the relevant question here is whether considering loops with different particle masses is sufficient to determine isospin breaking. According to [36], this is the case in their problem, and we shall assume that it is also the case here.

The two structures $ \langle \bar{B} P B \rangle $ and $ \langle \bar{B} B P \rangle $ are different, as shown in Eqs. (4), (6), and we assign them different weights, $ A, B $, respectively. Thus, for practical purposes, we have

The transition amplitude for $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ is then given by the diagrams shown in Fig. 1,

Figure 1.  Diagrams involving final-state interactions of $ PB $ pairs leading to the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay.

The interaction of the coupled channels leading to the formation of the $ \Lambda(1670) $ occurs in S-wave; hence, we keep the pair $ BP $ in S-wave. However, $ \bar{\Sigma}^0 \eta \Lambda $ has the combination $ J^P, \frac{1}{2}^- 0^- \frac{1}{2}^+ $, which has positive parity; therefore, a P-wave is required to match the parity of the $ J/\psi $. This is achieved by contracting the $ \epsilon^\mu $ polarization vector of the $ J/\psi $ with the momentum of $ \bar{\Sigma}^0 $. Hence, our transition amplitude is

with

Evaluating the sum and average over polarizations of the F factor, we obtain

with $ \mathit{\boldsymbol{p}}^2_{\Sigma^0} $ evaluated in the $ J/\psi $ rest frame.

The starting point is the strong decay of $ J/\psi $ into $ \bar{\Sigma}^0 B P $, where B is a baryon from the $ 1/2^+ $ octet and P is a pseudoscalar meson from the $ 0^- $ octet. The $ \bar{\Sigma}^0 B P $ system initially satisfies isospin symmetry, which implies that $ BP $ has isospin $ I=1 $. The $ BP $ interaction then ultimately leads to $ \eta \Lambda $ with isospin violation, which occurs through loops that do not cancel because of the different masses of the $ BP $ states belonging to the same isospin multiplet.

To determine which $ BP $ states are produced in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction, we consider that $ J/\psi $ is an SU(3) singlet in the $ u,d,s $ sector. Then, writing the matrices for the $ q\bar{q} $ states of the octet in terms of mesons and those for the $ qqq $ states in terms of the baryons of the octet, we have

There are several ways to construct SU(3) singlet structures with these matrices by taking traces of products, such as $ \langle \bar{B} B P \rangle $, $ \langle \bar{B} P B \rangle $, $ \langle \bar{B} B\rangle \langle P \rangle $, $ \langle \bar{B} P\rangle \langle B \rangle $, $ \langle B P\rangle \langle \bar{B} \rangle $, $ \langle \bar{B} \rangle \langle B \rangle \langle P \rangle $. However, using elements of heavy quark symmetry, one finds that the largest contribution comes from the structures containing fewer traces [31, 32]; therefore, we retain only these terms. In Ref. [31], it is shown that one should expect a reduction factor of 1/$ N_c $, where $ N_c $ is the number of colors, in an amplitude that contains an extra trace. In the present case, in addition to the two terms that we have, $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $, we could have the $ \langle \bar{B} B\rangle \langle P \rangle $ structure, because P, unlike B and $ \bar{B} $, has a nonzero trace. One may wonder why such a structure is not present in the work of [33], where the most general structure has two terms, which are equivalent to those used here. The reason is that in [33], one takes the P matrix corresponding to the octet of pseudoscalars, for which P also has zero trace. Here, we consider the $ \eta-\eta^\prime $ mixing of [34], and the trace of P is nonzero. Nevertheless, because of the large-$ N_c $ argument and the small $ \eta-\eta^\prime $ mixing, we consider that the $ \langle \bar{B} B\rangle \langle P \rangle $ term can be neglected, and we return to the two structures used in [33]. Thus, we consider the $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $ structures in our study. We find

However, since it was found in [9] that the $ \pi\Sigma $ states couple very weakly to the $ \Lambda(1670) $, we neglect these channels and find

With the isospin phase convention used in our approach, $ (K^+,K^0),(\bar{K}^0,-K^-) $, the $ p K^{-} - n \bar{K}^0 $ combination in Eq. (3) has $ I=1 $, as required, so that, when combined with the $ \bar{\Sigma}^0 $, it can yield the $ I=0 $ of the $ J/\psi $. Both $ \Sigma^0 \eta $ and $ \Lambda \pi^0 $ have $ I=1 $ and therefore cannot undergo a strong-interaction transition to $ \eta\Lambda $. However, $ K^{-} p $ and $ n \bar{K}^0 $ can lead to $ I=0 $, since the loops of $ K^{-} p $ and $ n \bar{K}^0 $ involved in the final-state interaction do not cancel. For practical purposes, we can simply take the object produced in the first step of the $ J/\psi $ decay

We now consider the $ \langle \bar{B} P B \rangle $ structure. We obtain

and, neglecting the $ \pi \Sigma $ channels as before, as well as the $ \pi^0 \Lambda $ and $ \eta \Sigma^0 $ channels, which cannot transition to an $ I=0 $ state, we obtain

Once again, considering the isospin doublet $ (\Xi^0,-\Xi^-) $ in analogy to $ (\bar{K}^0,-K^-) $, we see that the $ K \Xi $ combination in Eq. (6) has $ I=1 $, as it should. However, in the final-state interaction producing $ \eta\Lambda $, the $ K^+ \Xi^- $ and $ K^0 \Xi^0 $ loops do not cancel. Thus, we obtain an isospin-violating mechanism that can ultimately produce $ \eta\Lambda $, with the shape of the $ \Lambda(1670) $.

When considering isospin-violation effects, one cannot neglect the fact that electromagnetic interactions also violate isospin and can contribute to the processes studied. No study of the present reaction has addressed such effects, but related work has been performed for $ J/\psi $ decay into baryon-antibaryon pairs [35]. In that study, a general operator with seven free parameters is used to fit the data, and the operator explicitly contains an electromagnetic term. Of relevance to us is the result in Table 5 of that paper, which shows that the strong interaction alone gives rates about two orders of magnitude larger than the electromagnetic terms alone and, by itself, provides a fair reproduction of the data. The relative effect of electromagnetic contributions could be larger in isospin-violating processes, but this possibility is not discussed there. Insight can be gained from studies in which isospin violation from loops and electromagnetic effects are considered together, as in the analysis of $ f_0(980) $-$ a_0(980) $ mixing in [36]. That paper concludes that kaon mass differences in the loops generate the dominant mixing effect, whereas direct electromagnetic contributions are subleading. One should not forget that the mass differences between states with different charges in the same isospin multiplet are partly due to electromagnetic interactions [37] (see also related work on this issue in [38]). However, the relevant question here is whether considering loops with different particle masses is sufficient to determine isospin breaking. According to [36], this is the case in their problem, and we shall assume that it is also the case here.

The two structures $ \langle \bar{B} P B \rangle $ and $ \langle \bar{B} B P \rangle $ are different, as shown in Eqs. (4), (6), and we assign them different weights, $ A, B $, respectively. Thus, for practical purposes, we have

The transition amplitude for $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ is then given by the diagrams shown in Fig. 1,

Figure 1.  Diagrams involving final-state interactions of $ PB $ pairs leading to the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay.

The interaction of the coupled channels leading to the formation of the $ \Lambda(1670) $ occurs in S-wave; hence, we keep the pair $ BP $ in S-wave. However, $ \bar{\Sigma}^0 \eta \Lambda $ has the combination $ J^P, \frac{1}{2}^- 0^- \frac{1}{2}^+ $, which has positive parity; therefore, a P-wave is required to match the parity of the $ J/\psi $. This is achieved by contracting the $ \epsilon^\mu $ polarization vector of the $ J/\psi $ with the momentum of $ \bar{\Sigma}^0 $. Hence, our transition amplitude is

with

Evaluating the sum and average over polarizations of the F factor, we obtain

with $ \mathit{\boldsymbol{p}}^2_{\Sigma^0} $ evaluated in the $ J/\psi $ rest frame.

The starting point is the strong decay of $ J/\psi $ into $ \bar{\Sigma}^0 B P $, where B is a baryon from the $ 1/2^+ $ octet and P is a pseudoscalar meson from the $ 0^- $ octet. The $ \bar{\Sigma}^0 B P $ system initially satisfies isospin symmetry, which implies that $ BP $ has isospin $ I=1 $. The $ BP $ interaction then ultimately leads to $ \eta \Lambda $ with isospin violation, which occurs through loops that do not cancel because of the different masses of the $ BP $ states belonging to the same isospin multiplet.

To determine which $ BP $ states are produced in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction, we consider that $ J/\psi $ is an SU(3) singlet in the $ u,d,s $ sector. Then, writing the matrices for the $ q\bar{q} $ states of the octet in terms of mesons and those for the $ qqq $ states in terms of the baryons of the octet, we have

There are several ways to construct SU(3) singlet structures with these matrices by taking traces of products, such as $ \langle \bar{B} B P \rangle $, $ \langle \bar{B} P B \rangle $, $ \langle \bar{B} B\rangle \langle P \rangle $, $ \langle \bar{B} P\rangle \langle B \rangle $, $ \langle B P\rangle \langle \bar{B} \rangle $, $ \langle \bar{B} \rangle \langle B \rangle \langle P \rangle $. However, using elements of heavy quark symmetry, one finds that the largest contribution comes from the structures containing fewer traces [31, 32]; therefore, we retain only these terms. In Ref. [31], it is shown that one should expect a reduction factor of 1/$ N_c $, where $ N_c $ is the number of colors, in an amplitude that contains an extra trace. In the present case, in addition to the two terms that we have, $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $, we could have the $ \langle \bar{B} B\rangle \langle P \rangle $ structure, because P, unlike B and $ \bar{B} $, has a nonzero trace. One may wonder why such a structure is not present in the work of [33], where the most general structure has two terms, which are equivalent to those used here. The reason is that in [33], one takes the P matrix corresponding to the octet of pseudoscalars, for which P also has zero trace. Here, we consider the $ \eta-\eta^\prime $ mixing of [34], and the trace of P is nonzero. Nevertheless, because of the large-$ N_c $ argument and the small $ \eta-\eta^\prime $ mixing, we consider that the $ \langle \bar{B} B\rangle \langle P \rangle $ term can be neglected, and we return to the two structures used in [33]. Thus, we consider the $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $ structures in our study. We find

However, since it was found in [9] that the $ \pi\Sigma $ states couple very weakly to the $ \Lambda(1670) $, we neglect these channels and find

With the isospin phase convention used in our approach, $ (K^+,K^0),(\bar{K}^0,-K^-) $, the $ p K^{-} - n \bar{K}^0 $ combination in Eq. (3) has $ I=1 $, as required, so that, when combined with the $ \bar{\Sigma}^0 $, it can yield the $ I=0 $ of the $ J/\psi $. Both $ \Sigma^0 \eta $ and $ \Lambda \pi^0 $ have $ I=1 $ and therefore cannot undergo a strong-interaction transition to $ \eta\Lambda $. However, $ K^{-} p $ and $ n \bar{K}^0 $ can lead to $ I=0 $, since the loops of $ K^{-} p $ and $ n \bar{K}^0 $ involved in the final-state interaction do not cancel. For practical purposes, we can simply take the object produced in the first step of the $ J/\psi $ decay

We now consider the $ \langle \bar{B} P B \rangle $ structure. We obtain

and, neglecting the $ \pi \Sigma $ channels as before, as well as the $ \pi^0 \Lambda $ and $ \eta \Sigma^0 $ channels, which cannot transition to an $ I=0 $ state, we obtain

Once again, considering the isospin doublet $ (\Xi^0,-\Xi^-) $ in analogy to $ (\bar{K}^0,-K^-) $, we see that the $ K \Xi $ combination in Eq. (6) has $ I=1 $, as it should. However, in the final-state interaction producing $ \eta\Lambda $, the $ K^+ \Xi^- $ and $ K^0 \Xi^0 $ loops do not cancel. Thus, we obtain an isospin-violating mechanism that can ultimately produce $ \eta\Lambda $, with the shape of the $ \Lambda(1670) $.

When considering isospin-violation effects, one cannot neglect the fact that electromagnetic interactions also violate isospin and can contribute to the processes studied. No study of the present reaction has addressed such effects, but related work has been performed for $ J/\psi $ decay into baryon-antibaryon pairs [35]. In that study, a general operator with seven free parameters is used to fit the data, and the operator explicitly contains an electromagnetic term. Of relevance to us is the result in Table 5 of that paper, which shows that the strong interaction alone gives rates about two orders of magnitude larger than the electromagnetic terms alone and, by itself, provides a fair reproduction of the data. The relative effect of electromagnetic contributions could be larger in isospin-violating processes, but this possibility is not discussed there. Insight can be gained from studies in which isospin violation from loops and electromagnetic effects are considered together, as in the analysis of $ f_0(980) $-$ a_0(980) $ mixing in [36]. That paper concludes that kaon mass differences in the loops generate the dominant mixing effect, whereas direct electromagnetic contributions are subleading. One should not forget that the mass differences between states with different charges in the same isospin multiplet are partly due to electromagnetic interactions [37] (see also related work on this issue in [38]). However, the relevant question here is whether considering loops with different particle masses is sufficient to determine isospin breaking. According to [36], this is the case in their problem, and we shall assume that it is also the case here.

The two structures $ \langle \bar{B} P B \rangle $ and $ \langle \bar{B} B P \rangle $ are different, as shown in Eqs. (4), (6), and we assign them different weights, $ A, B $, respectively. Thus, for practical purposes, we have

The transition amplitude for $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ is then given by the diagrams shown in Fig. 1,

Figure 1.  Diagrams involving final-state interactions of $ PB $ pairs leading to the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay.

The interaction of the coupled channels leading to the formation of the $ \Lambda(1670) $ occurs in S-wave; hence, we keep the pair $ BP $ in S-wave. However, $ \bar{\Sigma}^0 \eta \Lambda $ has the combination $ J^P, \frac{1}{2}^- 0^- \frac{1}{2}^+ $, which has positive parity; therefore, a P-wave is required to match the parity of the $ J/\psi $. This is achieved by contracting the $ \epsilon^\mu $ polarization vector of the $ J/\psi $ with the momentum of $ \bar{\Sigma}^0 $. Hence, our transition amplitude is

with

Evaluating the sum and average over polarizations of the F factor, we obtain

with $ \mathit{\boldsymbol{p}}^2_{\Sigma^0} $ evaluated in the $ J/\psi $ rest frame.

The starting point is the strong decay of $ J/\psi $ into $ \bar{\Sigma}^0 B P $, where B is a baryon from the $ 1/2^+ $ octet and P is a pseudoscalar meson from the $ 0^- $ octet. The $ \bar{\Sigma}^0 B P $ system initially satisfies isospin symmetry, which implies that $ BP $ has isospin $ I=1 $. The $ BP $ interaction then ultimately leads to $ \eta \Lambda $ with isospin violation, which occurs through loops that do not cancel because of the different masses of the $ BP $ states belonging to the same isospin multiplet.

To determine which $ BP $ states are produced in the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ reaction, we consider that $ J/\psi $ is an SU(3) singlet in the $ u,d,s $ sector. Then, writing the matrices for the $ q\bar{q} $ states of the octet in terms of mesons and those for the $ qqq $ states in terms of the baryons of the octet, we have

There are several ways to construct SU(3) singlet structures with these matrices by taking traces of products, such as $ \langle \bar{B} B P \rangle $, $ \langle \bar{B} P B \rangle $, $ \langle \bar{B} B\rangle \langle P \rangle $, $ \langle \bar{B} P\rangle \langle B \rangle $, $ \langle B P\rangle \langle \bar{B} \rangle $, $ \langle \bar{B} \rangle \langle B \rangle \langle P \rangle $. However, using elements of heavy quark symmetry, one finds that the largest contribution comes from the structures containing fewer traces [31, 32]; therefore, we retain only these terms. In Ref. [31], it is shown that one should expect a reduction factor of 1/$ N_c $, where $ N_c $ is the number of colors, in an amplitude that contains an extra trace. In the present case, in addition to the two terms that we have, $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $, we could have the $ \langle \bar{B} B\rangle \langle P \rangle $ structure, because P, unlike B and $ \bar{B} $, has a nonzero trace. One may wonder why such a structure is not present in the work of [33], where the most general structure has two terms, which are equivalent to those used here. The reason is that in [33], one takes the P matrix corresponding to the octet of pseudoscalars, for which P also has zero trace. Here, we consider the $ \eta-\eta^\prime $ mixing of [34], and the trace of P is nonzero. Nevertheless, because of the large-$ N_c $ argument and the small $ \eta-\eta^\prime $ mixing, we consider that the $ \langle \bar{B} B\rangle \langle P \rangle $ term can be neglected, and we return to the two structures used in [33]. Thus, we consider the $ \langle \bar{B} B P \rangle $ and $ \langle \bar{B} P B \rangle $ structures in our study. We find

However, since it was found in [9] that the $ \pi\Sigma $ states couple very weakly to the $ \Lambda(1670) $, we neglect these channels and find

With the isospin phase convention used in our approach, $ (K^+,K^0),(\bar{K}^0,-K^-) $, the $ p K^{-} - n \bar{K}^0 $ combination in Eq. (3) has $ I=1 $, as required, so that, when combined with the $ \bar{\Sigma}^0 $, it can yield the $ I=0 $ of the $ J/\psi $. Both $ \Sigma^0 \eta $ and $ \Lambda \pi^0 $ have $ I=1 $ and therefore cannot undergo a strong-interaction transition to $ \eta\Lambda $. However, $ K^{-} p $ and $ n \bar{K}^0 $ can lead to $ I=0 $, since the loops of $ K^{-} p $ and $ n \bar{K}^0 $ involved in the final-state interaction do not cancel. For practical purposes, we can simply take the object produced in the first step of the $ J/\psi $ decay

We now consider the $ \langle \bar{B} P B \rangle $ structure. We obtain

and, neglecting the $ \pi \Sigma $ channels as before, as well as the $ \pi^0 \Lambda $ and $ \eta \Sigma^0 $ channels, which cannot transition to an $ I=0 $ state, we obtain

Once again, considering the isospin doublet $ (\Xi^0,-\Xi^-) $ in analogy to $ (\bar{K}^0,-K^-) $, we see that the $ K \Xi $ combination in Eq. (6) has $ I=1 $, as it should. However, in the final-state interaction producing $ \eta\Lambda $, the $ K^+ \Xi^- $ and $ K^0 \Xi^0 $ loops do not cancel. Thus, we obtain an isospin-violating mechanism that can ultimately produce $ \eta\Lambda $, with the shape of the $ \Lambda(1670) $.

When considering isospin-violation effects, one cannot neglect the fact that electromagnetic interactions also violate isospin and can contribute to the processes studied. No study of the present reaction has addressed such effects, but related work has been performed for $ J/\psi $ decay into baryon-antibaryon pairs [35]. In that study, a general operator with seven free parameters is used to fit the data, and the operator explicitly contains an electromagnetic term. Of relevance to us is the result in Table 5 of that paper, which shows that the strong interaction alone gives rates about two orders of magnitude larger than the electromagnetic terms alone and, by itself, provides a fair reproduction of the data. The relative effect of electromagnetic contributions could be larger in isospin-violating processes, but this possibility is not discussed there. Insight can be gained from studies in which isospin violation from loops and electromagnetic effects are considered together, as in the analysis of $ f_0(980) $-$ a_0(980) $ mixing in [36]. That paper concludes that kaon mass differences in the loops generate the dominant mixing effect, whereas direct electromagnetic contributions are subleading. One should not forget that the mass differences between states with different charges in the same isospin multiplet are partly due to electromagnetic interactions [37] (see also related work on this issue in [38]). However, the relevant question here is whether considering loops with different particle masses is sufficient to determine isospin breaking. According to [36], this is the case in their problem, and we shall assume that it is also the case here.

The two structures $ \langle \bar{B} P B \rangle $ and $ \langle \bar{B} B P \rangle $ are different, as shown in Eqs. (4), (6), and we assign them different weights, $ A, B $, respectively. Thus, for practical purposes, we have

The transition amplitude for $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ is then given by the diagrams shown in Fig. 1,

Figure 1.  Diagrams involving final-state interactions of $ PB $ pairs leading to the $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $ decay.

The interaction of the coupled channels leading to the formation of the $ \Lambda(1670) $ occurs in S-wave; hence, we keep the pair $ BP $ in S-wave. However, $ \bar{\Sigma}^0 \eta \Lambda $ has the combination $ J^P, \frac{1}{2}^- 0^- \frac{1}{2}^+ $, which has positive parity; therefore, a P-wave is required to match the parity of the $ J/\psi $. This is achieved by contracting the $ \epsilon^\mu $ polarization vector of the $ J/\psi $ with the momentum of $ \bar{\Sigma}^0 $. Hence, our transition amplitude is

with

Evaluating the sum and average over polarizations of the F factor, we obtain

with $ \mathit{\boldsymbol{p}}^2_{\Sigma^0} $ evaluated in the $ J/\psi $ rest frame.

In Ref. [1], the fit to the data included a small contribution from the $ \Lambda(1810) \, (1/2^+) $, which we also include here. This contribution appears mainly at higher masses in the $ \eta \Lambda $ spectrum and does not distort the shape of the $ \Lambda(1670) $. The production of $ \Lambda(1810)(1/2^+) $ in $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) $ does not violate parity and can proceed via an S-wave. By contrast, the decay $ \Lambda(1810)(1/2^+) \rightarrow \eta\Lambda $ requires a P-wave and proceeds through an operator $ \vec{\sigma} \vec{p}_\eta $, as in the $ NN\pi $ vertex. The mechanism for $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) \rightarrow \bar{\Sigma}^0 \eta \Lambda $ is depicted in Fig. 2.

Figure 2.  Mechanism of $ J/\psi \rightarrow \bar{\Sigma}^0 \eta \Lambda $ via $ \Lambda(1810)(1/2^+) $ excitation.

Since this mechanism does not interfere with the former one, we sum the contributions incoherently. The contribution from Fig. 2 can be written in terms of a $ t' $ amplitude such that

with

In Ref. [1], the fit to the data included a small contribution from the $ \Lambda(1810) \, (1/2^+) $, which we also include here. This contribution appears mainly at higher masses in the $ \eta \Lambda $ spectrum and does not distort the shape of the $ \Lambda(1670) $. The production of $ \Lambda(1810)(1/2^+) $ in $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) $ does not violate parity and can proceed via an S-wave. By contrast, the decay $ \Lambda(1810)(1/2^+) \rightarrow \eta\Lambda $ requires a P-wave and proceeds through an operator $ \vec{\sigma} \vec{p}_\eta $, as in the $ NN\pi $ vertex. The mechanism for $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) \rightarrow \bar{\Sigma}^0 \eta \Lambda $ is depicted in Fig. 2.

Figure 2.  Mechanism of $ J/\psi \rightarrow \bar{\Sigma}^0 \eta \Lambda $ via $ \Lambda(1810)(1/2^+) $ excitation.

Since this mechanism does not interfere with the former one, we sum the contributions incoherently. The contribution from Fig. 2 can be written in terms of a $ t' $ amplitude such that

with

In Ref. [1], the fit to the data included a small contribution from the $ \Lambda(1810) \, (1/2^+) $, which we also include here. This contribution appears mainly at higher masses in the $ \eta \Lambda $ spectrum and does not distort the shape of the $ \Lambda(1670) $. The production of $ \Lambda(1810)(1/2^+) $ in $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) $ does not violate parity and can proceed via an S-wave. By contrast, the decay $ \Lambda(1810)(1/2^+) \rightarrow \eta\Lambda $ requires a P-wave and proceeds through an operator $ \vec{\sigma} \vec{p}_\eta $, as in the $ NN\pi $ vertex. The mechanism for $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) \rightarrow \bar{\Sigma}^0 \eta \Lambda $ is depicted in Fig. 2.

Figure 2.  Mechanism of $ J/\psi \rightarrow \bar{\Sigma}^0 \eta \Lambda $ via $ \Lambda(1810)(1/2^+) $ excitation.

Since this mechanism does not interfere with the former one, we sum the contributions incoherently. The contribution from Fig. 2 can be written in terms of a $ t' $ amplitude such that

with

In Ref. [1], the fit to the data included a small contribution from the $ \Lambda(1810) \, (1/2^+) $, which we also include here. This contribution appears mainly at higher masses in the $ \eta \Lambda $ spectrum and does not distort the shape of the $ \Lambda(1670) $. The production of $ \Lambda(1810)(1/2^+) $ in $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) $ does not violate parity and can proceed via an S-wave. By contrast, the decay $ \Lambda(1810)(1/2^+) \rightarrow \eta\Lambda $ requires a P-wave and proceeds through an operator $ \vec{\sigma} \vec{p}_\eta $, as in the $ NN\pi $ vertex. The mechanism for $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) \rightarrow \bar{\Sigma}^0 \eta \Lambda $ is depicted in Fig. 2.

Figure 2.  Mechanism of $ J/\psi \rightarrow \bar{\Sigma}^0 \eta \Lambda $ via $ \Lambda(1810)(1/2^+) $ excitation.

Since this mechanism does not interfere with the former one, we sum the contributions incoherently. The contribution from Fig. 2 can be written in terms of a $ t' $ amplitude such that

with

In Ref. [1], the fit to the data included a small contribution from the $ \Lambda(1810) \, (1/2^+) $, which we also include here. This contribution appears mainly at higher masses in the $ \eta \Lambda $ spectrum and does not distort the shape of the $ \Lambda(1670) $. The production of $ \Lambda(1810)(1/2^+) $ in $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) $ does not violate parity and can proceed via an S-wave. By contrast, the decay $ \Lambda(1810)(1/2^+) \rightarrow \eta\Lambda $ requires a P-wave and proceeds through an operator $ \vec{\sigma} \vec{p}_\eta $, as in the $ NN\pi $ vertex. The mechanism for $ J/\psi \rightarrow \bar{\Sigma}^0 \Lambda(1810)(1/2^+) \rightarrow \bar{\Sigma}^0 \eta \Lambda $ is depicted in Fig. 2.

Figure 2.  Mechanism of $ J/\psi \rightarrow \bar{\Sigma}^0 \eta \Lambda $ via $ \Lambda(1810)(1/2^+) $ excitation.

Since this mechanism does not interfere with the former one, we sum the contributions incoherently. The contribution from Fig. 2 can be written in terms of a $ t' $ amplitude such that

with

We use the PDG formula [39], adapted to the normalization convention of Mandl and Shaw [40]

where we assign the labels $ \bar{\Sigma}^0 (3) $, $ \eta (1) $, and $ \Lambda (2) $. In Eq. (14), we have

where $ |t'|^2 $ is given by Eq. (12).

We use the PDG formula [39], adapted to the normalization convention of Mandl and Shaw [40]

where we assign the labels $ \bar{\Sigma}^0 (3) $, $ \eta (1) $, and $ \Lambda (2) $. In Eq. (14), we have

where $ |t'|^2 $ is given by Eq. (12).

We use the PDG formula [39], adapted to the normalization convention of Mandl and Shaw [40]

where we assign the labels $ \bar{\Sigma}^0 (3) $, $ \eta (1) $, and $ \Lambda (2) $. In Eq. (14), we have

where $ |t'|^2 $ is given by Eq. (12).

We use the PDG formula [39], adapted to the normalization convention of Mandl and Shaw [40]

where we assign the labels $ \bar{\Sigma}^0 (3) $, $ \eta (1) $, and $ \Lambda (2) $. In Eq. (14), we have

where $ |t'|^2 $ is given by Eq. (12).

We use the PDG formula [39], adapted to the normalization convention of Mandl and Shaw [40]

where we assign the labels $ \bar{\Sigma}^0 (3) $, $ \eta (1) $, and $ \Lambda (2) $. In Eq. (14), we have

where $ |t'|^2 $ is given by Eq. (12).

As shown in Eq. (10), the amplitudes $ t_{\bar{K}N,\eta\Lambda} $ and $ t_{K\Xi,\eta\Lambda} $ are required. They are calculated using the factorized Bethe–Salpeter equation for the coupled channels $ K^{-} p $, $ \bar{K}^{0} n $, $ \pi^0 \Lambda $, $ \pi^0 \Sigma^0 $, $ \eta \Lambda $, $ \eta \Sigma^0 $, $ \pi^+ \Sigma^- $, $ \pi^- \Sigma^+ $, $ K^+ \Xi^- $, and $ K^0 \Xi^0 $.

with G denoting the diagonal meson-baryon loop function, regularised with a cutoff, and $ V_{ij} $ the transition potential between the channels, as evaluated in [8]. The $ f_i $ constants and $ q_{\rm max} $ for the channels have been taken from [30], where they were fitted to the precise data for $ \Lambda^+_c \to \pi^+ K^- p $ [28]. We use them here to maintain consistency with the data from this reaction, in which the $ \Lambda(1670) $ resonance was clearly observed in the $ K^- p $ spectrum.

As shown in Eq. (10), the amplitudes $ t_{\bar{K}N,\eta\Lambda} $ and $ t_{K\Xi,\eta\Lambda} $ are required. They are calculated using the factorized Bethe–Salpeter equation for the coupled channels $ K^{-} p $, $ \bar{K}^{0} n $, $ \pi^0 \Lambda $, $ \pi^0 \Sigma^0 $, $ \eta \Lambda $, $ \eta \Sigma^0 $, $ \pi^+ \Sigma^- $, $ \pi^- \Sigma^+ $, $ K^+ \Xi^- $, and $ K^0 \Xi^0 $.

with G denoting the diagonal meson-baryon loop function, regularised with a cutoff, and $ V_{ij} $ the transition potential between the channels, as evaluated in [8]. The $ f_i $ constants and $ q_{\rm max} $ for the channels have been taken from [30], where they were fitted to the precise data for $ \Lambda^+_c \to \pi^+ K^- p $ [28]. We use them here to maintain consistency with the data from this reaction, in which the $ \Lambda(1670) $ resonance was clearly observed in the $ K^- p $ spectrum.

As shown in Eq. (10), the amplitudes $ t_{\bar{K}N,\eta\Lambda} $ and $ t_{K\Xi,\eta\Lambda} $ are required. They are calculated using the factorized Bethe–Salpeter equation for the coupled channels $ K^{-} p $, $ \bar{K}^{0} n $, $ \pi^0 \Lambda $, $ \pi^0 \Sigma^0 $, $ \eta \Lambda $, $ \eta \Sigma^0 $, $ \pi^+ \Sigma^- $, $ \pi^- \Sigma^+ $, $ K^+ \Xi^- $, and $ K^0 \Xi^0 $.

with G denoting the diagonal meson-baryon loop function, regularised with a cutoff, and $ V_{ij} $ the transition potential between the channels, as evaluated in [8]. The $ f_i $ constants and $ q_{\rm max} $ for the channels have been taken from [30], where they were fitted to the precise data for $ \Lambda^+_c \to \pi^+ K^- p $ [28]. We use them here to maintain consistency with the data from this reaction, in which the $ \Lambda(1670) $ resonance was clearly observed in the $ K^- p $ spectrum.

As shown in Eq. (10), the amplitudes $ t_{\bar{K}N,\eta\Lambda} $ and $ t_{K\Xi,\eta\Lambda} $ are required. They are calculated using the factorized Bethe–Salpeter equation for the coupled channels $ K^{-} p $, $ \bar{K}^{0} n $, $ \pi^0 \Lambda $, $ \pi^0 \Sigma^0 $, $ \eta \Lambda $, $ \eta \Sigma^0 $, $ \pi^+ \Sigma^- $, $ \pi^- \Sigma^+ $, $ K^+ \Xi^- $, and $ K^0 \Xi^0 $.

with G denoting the diagonal meson-baryon loop function, regularised with a cutoff, and $ V_{ij} $ the transition potential between the channels, as evaluated in [8]. The $ f_i $ constants and $ q_{\rm max} $ for the channels have been taken from [30], where they were fitted to the precise data for $ \Lambda^+_c \to \pi^+ K^- p $ [28]. We use them here to maintain consistency with the data from this reaction, in which the $ \Lambda(1670) $ resonance was clearly observed in the $ K^- p $ spectrum.

As shown in Eq. (10), the amplitudes $ t_{\bar{K}N,\eta\Lambda} $ and $ t_{K\Xi,\eta\Lambda} $ are required. They are calculated using the factorized Bethe–Salpeter equation for the coupled channels $ K^{-} p $, $ \bar{K}^{0} n $, $ \pi^0 \Lambda $, $ \pi^0 \Sigma^0 $, $ \eta \Lambda $, $ \eta \Sigma^0 $, $ \pi^+ \Sigma^- $, $ \pi^- \Sigma^+ $, $ K^+ \Xi^- $, and $ K^0 \Xi^0 $.

with G denoting the diagonal meson-baryon loop function, regularised with a cutoff, and $ V_{ij} $ the transition potential between the channels, as evaluated in [8]. The $ f_i $ constants and $ q_{\rm max} $ for the channels have been taken from [30], where they were fitted to the precise data for $ \Lambda^+_c \to \pi^+ K^- p $ [28]. We use them here to maintain consistency with the data from this reaction, in which the $ \Lambda(1670) $ resonance was clearly observed in the $ K^- p $ spectrum.

We first examine the relevance of the A and B parameters of Eq. (10) to the mass distributions. This is shown in Fig. 3, where the contribution of the B term, involving the $ K \Xi $ amplitudes, has a weight an order of magnitude larger than that of the A term. Thus, the isospin violation is mainly associated with the $ K \Xi $ amplitudes rather than those of $ \bar{K}N $. The second conclusion is that the shapes of the distributions obtained with the A term alone or the B term alone are practically identical. We also find that linear combinations of the A and B terms give the same distributions, up to a global normalization. This is not surprising, since in coupled channels the pole of the resonance comes from $ {\rm det} [1-VG]=0 $, and the $ t_{ij} $ matrices of Eq. (16) go as $ A_{ij}/{\rm det} [1-VG]=0 $, with $ A_{ij} $ some matrix. Hence, all amplitudes share $ ({\rm det} [1-VG])^{-1} $, and close to the pole the amplitudes are similar up to a normalization. The same conclusion is reached if the amplitudes are parameterized in terms of the couplings as $T_{ij}\simeq g_i g_j/(M_{\rm inv} -M_R+ {\rm i} \Gamma/2)$.

Figure 3.  (color online) The three distinct mass distributions for which A or B is equal to zero ($ C=0 $).

The next step is to compare with experiment. We perform a best fit to the data by setting $ A=B $, after establishing that the B term dominates and that the shape is the same for any linear combination of the A and B terms. ¹ Essentially, we determine the relative magnitude of the C term of Eq. (12) and a global normalization. The results are shown in Fig. 4. The fit to the data is good, with the same quality as the fit performed in the experiment [1], and the contribution of the $ \Lambda(1810)(1/2^+) $ state is also very similar to that in the experimental fit. The novelty of our work consists in showing the dynamical origin of the $ \Lambda(1670) $ excitation in this isospin-forbidden reaction, namely the mass differences between the charged and neutral $ \bar{K} N $ and $ K \Xi $ modes.

Figure 4.  (color online) Results for the decay $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $.

We would like to make some comments concerning the imperfect agreement with the data, particularly the $ \eta \bar{\Sigma} $ mass distribution in Fig. 4. One might consider a contribution from some excited Σ states, but the PDG shows that the excited Σ states known so far in the energy region from 1750 MeV to 1900 MeV do not decay into $ \eta \Sigma $, and the $ \Sigma(1900) $ has only a branching fraction of 1 % to $ \eta \Sigma $. Therefore, other sources should be examined to improve the present distribution. However, given that these results arise from strong cancellations of terms, the agreement should be considered sufficiently good. In particular, it is good enough for our purposes, for which the main conclusion is tied to the large signal for the $ \Lambda(1670) $ that the reaction shows in the $ \eta \Lambda $ mass distribution. Since the isospin-violating contributions come from loops in rescattering in our approach, the appearance of the $ \Lambda(1670) $ is tied to this rescattering, indicating that it is generated by the pseudoscalar-baryon interaction and providing strong support for the nature of this resonance as dynamically generated, rather than as a standard three-quark state.

It should be noted that the amplitudes obtained with the chiral unitary approach [8, 30] are reliable only for energies not exceeding $ 1.78 \mathrm{\; GeV} $ in the $ \eta\Lambda $ invariant mass. Thus, as is commonly done [41], we gradually suppress the amplitudes above a certain energy. We take

with $ M_{\rm cut}=1.78 \mathrm{\; GeV} $, $ \alpha=5.4 \mathrm{\; GeV}^{-1} $, as in [41]. Note that this simply reduces the strength of the $ \eta\Lambda $ mass distribution in a region where its contribution is very small and dominated by the $ \Lambda(1810)(1/2^+) $.

To gain insight into the cancellations occurring in the isospin-forbidden amplitude, we plot in Fig. 5 the relevant $ t_{ij} $ matrix elements $ t_{K^- p,\eta \Lambda} $, $ t_{\bar{K}^0 n,\eta \Lambda} $, $ t_{K^+ \Xi^-,\eta \Lambda} $, and $ t_{{K}^0 \Xi^0,\eta \Lambda} $ appearing in Eq. (10). The roles of $ Re\, t_{\bar{K}N,\eta \Lambda} $, $ Im\, t_{\bar{K}N,\eta \Lambda} $ and $ Re\, t_{K \Xi,\eta \Lambda} $, $ Im\, t_{K \Xi,\eta \Lambda} $ are exchanged, a feature often encountered in coupled channels for inelastic amplitudes. The important observation is that the magnitudes of $ Gt $ are remarkably similar, apart from an overall sign and normalization. Thus, combinations of A and B with opposite signs in Eq. (10) lead to larger decay widths than when A and B have the same sign. In any case, the figures explain why the terms in Eq. (10) with the $ t_{K \Xi,\eta \Lambda} $ amplitudes dominate. Indeed, $ |Im\, t_{K \Xi,\eta \Lambda}| $ is about an order of magnitude larger than $ |Im\, t_{\bar{K}N,\eta \Lambda}| $. We also observe in Fig. 5 that there are small differences between the charged and neutral channels in the t matrix. It is also useful to plot the results for $ Gt $ entering the evaluation of Eq. (10). We show the results in Fig. 6. Re $ G t $ and Im $ G t $ are now similar in shape, up to an overall sign. This is a consequence of the fact that Im $ G_{\bar{K}N} $ dominates $ G_{\bar{K}N} $, and hence in $ G_{\bar{K}N} t_{\bar{K}N,\eta \Lambda} $ the roles of the real and imaginary parts are now reversed with respect to $ t_{\bar{K}N,\eta \Lambda} $. The differences observed in $ Gt $ for the different channels are responsible for the isospin violation in the reaction studied.

Figure 5.  (color online) Differences between the charged and neutral channels for the $ t_i $ terms.

Figure 6.  (color online) Differences between the charged and neutral channels in the $ G_i t_i $ terms.

We first examine the relevance of the A and B parameters of Eq. (10) to the mass distributions. This is shown in Fig. 3, where the contribution of the B term, involving the $ K \Xi $ amplitudes, has a weight an order of magnitude larger than that of the A term. Thus, the isospin violation is mainly associated with the $ K \Xi $ amplitudes rather than those of $ \bar{K}N $. The second conclusion is that the shapes of the distributions obtained with the A term alone or the B term alone are practically identical. We also find that linear combinations of the A and B terms give the same distributions, up to a global normalization. This is not surprising, since in coupled channels the pole of the resonance comes from $ {\rm det} [1-VG]=0 $, and the $ t_{ij} $ matrices of Eq. (16) go as $ A_{ij}/{\rm det} [1-VG]=0 $, with $ A_{ij} $ some matrix. Hence, all amplitudes share $ ({\rm det} [1-VG])^{-1} $, and close to the pole the amplitudes are similar up to a normalization. The same conclusion is reached if the amplitudes are parameterized in terms of the couplings as $T_{ij}\simeq g_i g_j/(M_{\rm inv} -M_R+ {\rm i} \Gamma/2)$.

Figure 3.  (color online) The three distinct mass distributions for which A or B is equal to zero ($ C=0 $).

The next step is to compare with experiment. We perform a best fit to the data by setting $ A=B $, after establishing that the B term dominates and that the shape is the same for any linear combination of the A and B terms. ¹ Essentially, we determine the relative magnitude of the C term of Eq. (12) and a global normalization. The results are shown in Fig. 4. The fit to the data is good, with the same quality as the fit performed in the experiment [1], and the contribution of the $ \Lambda(1810)(1/2^+) $ state is also very similar to that in the experimental fit. The novelty of our work consists in showing the dynamical origin of the $ \Lambda(1670) $ excitation in this isospin-forbidden reaction, namely the mass differences between the charged and neutral $ \bar{K} N $ and $ K \Xi $ modes.

Figure 4.  (color online) Results for the decay $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $.

We would like to make some comments concerning the imperfect agreement with the data, particularly the $ \eta \bar{\Sigma} $ mass distribution in Fig. 4. One might consider a contribution from some excited Σ states, but the PDG shows that the excited Σ states known so far in the energy region from 1750 MeV to 1900 MeV do not decay into $ \eta \Sigma $, and the $ \Sigma(1900) $ has only a branching fraction of 1 % to $ \eta \Sigma $. Therefore, other sources should be examined to improve the present distribution. However, given that these results arise from strong cancellations of terms, the agreement should be considered sufficiently good. In particular, it is good enough for our purposes, for which the main conclusion is tied to the large signal for the $ \Lambda(1670) $ that the reaction shows in the $ \eta \Lambda $ mass distribution. Since the isospin-violating contributions come from loops in rescattering in our approach, the appearance of the $ \Lambda(1670) $ is tied to this rescattering, indicating that it is generated by the pseudoscalar-baryon interaction and providing strong support for the nature of this resonance as dynamically generated, rather than as a standard three-quark state.

It should be noted that the amplitudes obtained with the chiral unitary approach [8, 30] are reliable only for energies not exceeding $ 1.78 \mathrm{\; GeV} $ in the $ \eta\Lambda $ invariant mass. Thus, as is commonly done [41], we gradually suppress the amplitudes above a certain energy. We take

with $ M_{\rm cut}=1.78 \mathrm{\; GeV} $, $ \alpha=5.4 \mathrm{\; GeV}^{-1} $, as in [41]. Note that this simply reduces the strength of the $ \eta\Lambda $ mass distribution in a region where its contribution is very small and dominated by the $ \Lambda(1810)(1/2^+) $.

To gain insight into the cancellations occurring in the isospin-forbidden amplitude, we plot in Fig. 5 the relevant $ t_{ij} $ matrix elements $ t_{K^- p,\eta \Lambda} $, $ t_{\bar{K}^0 n,\eta \Lambda} $, $ t_{K^+ \Xi^-,\eta \Lambda} $, and $ t_{{K}^0 \Xi^0,\eta \Lambda} $ appearing in Eq. (10). The roles of $ Re\, t_{\bar{K}N,\eta \Lambda} $, $ Im\, t_{\bar{K}N,\eta \Lambda} $ and $ Re\, t_{K \Xi,\eta \Lambda} $, $ Im\, t_{K \Xi,\eta \Lambda} $ are exchanged, a feature often encountered in coupled channels for inelastic amplitudes. The important observation is that the magnitudes of $ Gt $ are remarkably similar, apart from an overall sign and normalization. Thus, combinations of A and B with opposite signs in Eq. (10) lead to larger decay widths than when A and B have the same sign. In any case, the figures explain why the terms in Eq. (10) with the $ t_{K \Xi,\eta \Lambda} $ amplitudes dominate. Indeed, $ |Im\, t_{K \Xi,\eta \Lambda}| $ is about an order of magnitude larger than $ |Im\, t_{\bar{K}N,\eta \Lambda}| $. We also observe in Fig. 5 that there are small differences between the charged and neutral channels in the t matrix. It is also useful to plot the results for $ Gt $ entering the evaluation of Eq. (10). We show the results in Fig. 6. Re $ G t $ and Im $ G t $ are now similar in shape, up to an overall sign. This is a consequence of the fact that Im $ G_{\bar{K}N} $ dominates $ G_{\bar{K}N} $, and hence in $ G_{\bar{K}N} t_{\bar{K}N,\eta \Lambda} $ the roles of the real and imaginary parts are now reversed with respect to $ t_{\bar{K}N,\eta \Lambda} $. The differences observed in $ Gt $ for the different channels are responsible for the isospin violation in the reaction studied.

Figure 5.  (color online) Differences between the charged and neutral channels for the $ t_i $ terms.

Figure 6.  (color online) Differences between the charged and neutral channels in the $ G_i t_i $ terms.

We first examine the relevance of the A and B parameters of Eq. (10) to the mass distributions. This is shown in Fig. 3, where the contribution of the B term, involving the $ K \Xi $ amplitudes, has a weight an order of magnitude larger than that of the A term. Thus, the isospin violation is mainly associated with the $ K \Xi $ amplitudes rather than those of $ \bar{K}N $. The second conclusion is that the shapes of the distributions obtained with the A term alone or the B term alone are practically identical. We also find that linear combinations of the A and B terms give the same distributions, up to a global normalization. This is not surprising, since in coupled channels the pole of the resonance comes from $ {\rm det} [1-VG]=0 $, and the $ t_{ij} $ matrices of Eq. (16) go as $ A_{ij}/{\rm det} [1-VG]=0 $, with $ A_{ij} $ some matrix. Hence, all amplitudes share $ ({\rm det} [1-VG])^{-1} $, and close to the pole the amplitudes are similar up to a normalization. The same conclusion is reached if the amplitudes are parameterized in terms of the couplings as $T_{ij}\simeq g_i g_j/(M_{\rm inv} -M_R+ {\rm i} \Gamma/2)$.

Figure 3.  (color online) The three distinct mass distributions for which A or B is equal to zero ($ C=0 $).

The next step is to compare with experiment. We perform a best fit to the data by setting $ A=B $, after establishing that the B term dominates and that the shape is the same for any linear combination of the A and B terms. ¹ Essentially, we determine the relative magnitude of the C term of Eq. (12) and a global normalization. The results are shown in Fig. 4. The fit to the data is good, with the same quality as the fit performed in the experiment [1], and the contribution of the $ \Lambda(1810)(1/2^+) $ state is also very similar to that in the experimental fit. The novelty of our work consists in showing the dynamical origin of the $ \Lambda(1670) $ excitation in this isospin-forbidden reaction, namely the mass differences between the charged and neutral $ \bar{K} N $ and $ K \Xi $ modes.

Figure 4.  (color online) Results for the decay $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $.

We would like to make some comments concerning the imperfect agreement with the data, particularly the $ \eta \bar{\Sigma} $ mass distribution in Fig. 4. One might consider a contribution from some excited Σ states, but the PDG shows that the excited Σ states known so far in the energy region from 1750 MeV to 1900 MeV do not decay into $ \eta \Sigma $, and the $ \Sigma(1900) $ has only a branching fraction of 1 % to $ \eta \Sigma $. Therefore, other sources should be examined to improve the present distribution. However, given that these results arise from strong cancellations of terms, the agreement should be considered sufficiently good. In particular, it is good enough for our purposes, for which the main conclusion is tied to the large signal for the $ \Lambda(1670) $ that the reaction shows in the $ \eta \Lambda $ mass distribution. Since the isospin-violating contributions come from loops in rescattering in our approach, the appearance of the $ \Lambda(1670) $ is tied to this rescattering, indicating that it is generated by the pseudoscalar-baryon interaction and providing strong support for the nature of this resonance as dynamically generated, rather than as a standard three-quark state.

It should be noted that the amplitudes obtained with the chiral unitary approach [8, 30] are reliable only for energies not exceeding $ 1.78 \mathrm{\; GeV} $ in the $ \eta\Lambda $ invariant mass. Thus, as is commonly done [41], we gradually suppress the amplitudes above a certain energy. We take

with $ M_{\rm cut}=1.78 \mathrm{\; GeV} $, $ \alpha=5.4 \mathrm{\; GeV}^{-1} $, as in [41]. Note that this simply reduces the strength of the $ \eta\Lambda $ mass distribution in a region where its contribution is very small and dominated by the $ \Lambda(1810)(1/2^+) $.

To gain insight into the cancellations occurring in the isospin-forbidden amplitude, we plot in Fig. 5 the relevant $ t_{ij} $ matrix elements $ t_{K^- p,\eta \Lambda} $, $ t_{\bar{K}^0 n,\eta \Lambda} $, $ t_{K^+ \Xi^-,\eta \Lambda} $, and $ t_{{K}^0 \Xi^0,\eta \Lambda} $ appearing in Eq. (10). The roles of $ Re\, t_{\bar{K}N,\eta \Lambda} $, $ Im\, t_{\bar{K}N,\eta \Lambda} $ and $ Re\, t_{K \Xi,\eta \Lambda} $, $ Im\, t_{K \Xi,\eta \Lambda} $ are exchanged, a feature often encountered in coupled channels for inelastic amplitudes. The important observation is that the magnitudes of $ Gt $ are remarkably similar, apart from an overall sign and normalization. Thus, combinations of A and B with opposite signs in Eq. (10) lead to larger decay widths than when A and B have the same sign. In any case, the figures explain why the terms in Eq. (10) with the $ t_{K \Xi,\eta \Lambda} $ amplitudes dominate. Indeed, $ |Im\, t_{K \Xi,\eta \Lambda}| $ is about an order of magnitude larger than $ |Im\, t_{\bar{K}N,\eta \Lambda}| $. We also observe in Fig. 5 that there are small differences between the charged and neutral channels in the t matrix. It is also useful to plot the results for $ Gt $ entering the evaluation of Eq. (10). We show the results in Fig. 6. Re $ G t $ and Im $ G t $ are now similar in shape, up to an overall sign. This is a consequence of the fact that Im $ G_{\bar{K}N} $ dominates $ G_{\bar{K}N} $, and hence in $ G_{\bar{K}N} t_{\bar{K}N,\eta \Lambda} $ the roles of the real and imaginary parts are now reversed with respect to $ t_{\bar{K}N,\eta \Lambda} $. The differences observed in $ Gt $ for the different channels are responsible for the isospin violation in the reaction studied.

Figure 5.  (color online) Differences between the charged and neutral channels for the $ t_i $ terms.

Figure 6.  (color online) Differences between the charged and neutral channels in the $ G_i t_i $ terms.

We first examine the relevance of the A and B parameters of Eq. (10) to the mass distributions. This is shown in Fig. 3, where the contribution of the B term, involving the $ K \Xi $ amplitudes, has a weight an order of magnitude larger than that of the A term. Thus, the isospin violation is mainly associated with the $ K \Xi $ amplitudes rather than those of $ \bar{K}N $. The second conclusion is that the shapes of the distributions obtained with the A term alone or the B term alone are practically identical. We also find that linear combinations of the A and B terms give the same distributions, up to a global normalization. This is not surprising, since in coupled channels the pole of the resonance comes from $ {\rm det} [1-VG]=0 $, and the $ t_{ij} $ matrices of Eq. (16) go as $ A_{ij}/{\rm det} [1-VG]=0 $, with $ A_{ij} $ some matrix. Hence, all amplitudes share $ ({\rm det} [1-VG])^{-1} $, and close to the pole the amplitudes are similar up to a normalization. The same conclusion is reached if the amplitudes are parameterized in terms of the couplings as $T_{ij}\simeq g_i g_j/(M_{\rm inv} -M_R+ {\rm i} \Gamma/2)$.

Figure 3.  (color online) The three distinct mass distributions for which A or B is equal to zero ($ C=0 $).

The next step is to compare with experiment. We perform a best fit to the data by setting $ A=B $, after establishing that the B term dominates and that the shape is the same for any linear combination of the A and B terms. ¹ Essentially, we determine the relative magnitude of the C term of Eq. (12) and a global normalization. The results are shown in Fig. 4. The fit to the data is good, with the same quality as the fit performed in the experiment [1], and the contribution of the $ \Lambda(1810)(1/2^+) $ state is also very similar to that in the experimental fit. The novelty of our work consists in showing the dynamical origin of the $ \Lambda(1670) $ excitation in this isospin-forbidden reaction, namely the mass differences between the charged and neutral $ \bar{K} N $ and $ K \Xi $ modes.

Figure 4.  (color online) Results for the decay $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $.

We would like to make some comments concerning the imperfect agreement with the data, particularly the $ \eta \bar{\Sigma} $ mass distribution in Fig. 4. One might consider a contribution from some excited Σ states, but the PDG shows that the excited Σ states known so far in the energy region from 1750 MeV to 1900 MeV do not decay into $ \eta \Sigma $, and the $ \Sigma(1900) $ has only a branching fraction of 1 % to $ \eta \Sigma $. Therefore, other sources should be examined to improve the present distribution. However, given that these results arise from strong cancellations of terms, the agreement should be considered sufficiently good. In particular, it is good enough for our purposes, for which the main conclusion is tied to the large signal for the $ \Lambda(1670) $ that the reaction shows in the $ \eta \Lambda $ mass distribution. Since the isospin-violating contributions come from loops in rescattering in our approach, the appearance of the $ \Lambda(1670) $ is tied to this rescattering, indicating that it is generated by the pseudoscalar-baryon interaction and providing strong support for the nature of this resonance as dynamically generated, rather than as a standard three-quark state.

It should be noted that the amplitudes obtained with the chiral unitary approach [8, 30] are reliable only for energies not exceeding $ 1.78 \mathrm{\; GeV} $ in the $ \eta\Lambda $ invariant mass. Thus, as is commonly done [41], we gradually suppress the amplitudes above a certain energy. We take

with $ M_{\rm cut}=1.78 \mathrm{\; GeV} $, $ \alpha=5.4 \mathrm{\; GeV}^{-1} $, as in [41]. Note that this simply reduces the strength of the $ \eta\Lambda $ mass distribution in a region where its contribution is very small and dominated by the $ \Lambda(1810)(1/2^+) $.

To gain insight into the cancellations occurring in the isospin-forbidden amplitude, we plot in Fig. 5 the relevant $ t_{ij} $ matrix elements $ t_{K^- p,\eta \Lambda} $, $ t_{\bar{K}^0 n,\eta \Lambda} $, $ t_{K^+ \Xi^-,\eta \Lambda} $, and $ t_{{K}^0 \Xi^0,\eta \Lambda} $ appearing in Eq. (10). The roles of $ Re\, t_{\bar{K}N,\eta \Lambda} $, $ Im\, t_{\bar{K}N,\eta \Lambda} $ and $ Re\, t_{K \Xi,\eta \Lambda} $, $ Im\, t_{K \Xi,\eta \Lambda} $ are exchanged, a feature often encountered in coupled channels for inelastic amplitudes. The important observation is that the magnitudes of $ Gt $ are remarkably similar, apart from an overall sign and normalization. Thus, combinations of A and B with opposite signs in Eq. (10) lead to larger decay widths than when A and B have the same sign. In any case, the figures explain why the terms in Eq. (10) with the $ t_{K \Xi,\eta \Lambda} $ amplitudes dominate. Indeed, $ |Im\, t_{K \Xi,\eta \Lambda}| $ is about an order of magnitude larger than $ |Im\, t_{\bar{K}N,\eta \Lambda}| $. We also observe in Fig. 5 that there are small differences between the charged and neutral channels in the t matrix. It is also useful to plot the results for $ Gt $ entering the evaluation of Eq. (10). We show the results in Fig. 6. Re $ G t $ and Im $ G t $ are now similar in shape, up to an overall sign. This is a consequence of the fact that Im $ G_{\bar{K}N} $ dominates $ G_{\bar{K}N} $, and hence in $ G_{\bar{K}N} t_{\bar{K}N,\eta \Lambda} $ the roles of the real and imaginary parts are now reversed with respect to $ t_{\bar{K}N,\eta \Lambda} $. The differences observed in $ Gt $ for the different channels are responsible for the isospin violation in the reaction studied.

Figure 5.  (color online) Differences between the charged and neutral channels for the $ t_i $ terms.

Figure 6.  (color online) Differences between the charged and neutral channels in the $ G_i t_i $ terms.

We first examine the relevance of the A and B parameters of Eq. (10) to the mass distributions. This is shown in Fig. 3, where the contribution of the B term, involving the $ K \Xi $ amplitudes, has a weight an order of magnitude larger than that of the A term. Thus, the isospin violation is mainly associated with the $ K \Xi $ amplitudes rather than those of $ \bar{K}N $. The second conclusion is that the shapes of the distributions obtained with the A term alone or the B term alone are practically identical. We also find that linear combinations of the A and B terms give the same distributions, up to a global normalization. This is not surprising, since in coupled channels the pole of the resonance comes from $ {\rm det} [1-VG]=0 $, and the $ t_{ij} $ matrices of Eq. (16) go as $ A_{ij}/{\rm det} [1-VG]=0 $, with $ A_{ij} $ some matrix. Hence, all amplitudes share $ ({\rm det} [1-VG])^{-1} $, and close to the pole the amplitudes are similar up to a normalization. The same conclusion is reached if the amplitudes are parameterized in terms of the couplings as $T_{ij}\simeq g_i g_j/(M_{\rm inv} -M_R+ {\rm i} \Gamma/2)$.

Figure 3.  (color online) The three distinct mass distributions for which A or B is equal to zero ($ C=0 $).

The next step is to compare with experiment. We perform a best fit to the data by setting $ A=B $, after establishing that the B term dominates and that the shape is the same for any linear combination of the A and B terms. ¹ Essentially, we determine the relative magnitude of the C term of Eq. (12) and a global normalization. The results are shown in Fig. 4. The fit to the data is good, with the same quality as the fit performed in the experiment [1], and the contribution of the $ \Lambda(1810)(1/2^+) $ state is also very similar to that in the experimental fit. The novelty of our work consists in showing the dynamical origin of the $ \Lambda(1670) $ excitation in this isospin-forbidden reaction, namely the mass differences between the charged and neutral $ \bar{K} N $ and $ K \Xi $ modes.

Figure 4.  (color online) Results for the decay $ J/\psi\to \Lambda\bar{\Sigma}^0\eta $.

We would like to make some comments concerning the imperfect agreement with the data, particularly the $ \eta \bar{\Sigma} $ mass distribution in Fig. 4. One might consider a contribution from some excited Σ states, but the PDG shows that the excited Σ states known so far in the energy region from 1750 MeV to 1900 MeV do not decay into $ \eta \Sigma $, and the $ \Sigma(1900) $ has only a branching fraction of 1 % to $ \eta \Sigma $. Therefore, other sources should be examined to improve the present distribution. However, given that these results arise from strong cancellations of terms, the agreement should be considered sufficiently good. In particular, it is good enough for our purposes, for which the main conclusion is tied to the large signal for the $ \Lambda(1670) $ that the reaction shows in the $ \eta \Lambda $ mass distribution. Since the isospin-violating contributions come from loops in rescattering in our approach, the appearance of the $ \Lambda(1670) $ is tied to this rescattering, indicating that it is generated by the pseudoscalar-baryon interaction and providing strong support for the nature of this resonance as dynamically generated, rather than as a standard three-quark state.

It should be noted that the amplitudes obtained with the chiral unitary approach [8, 30] are reliable only for energies not exceeding $ 1.78 \mathrm{\; GeV} $ in the $ \eta\Lambda $ invariant mass. Thus, as is commonly done [41], we gradually suppress the amplitudes above a certain energy. We take

with $ M_{\rm cut}=1.78 \mathrm{\; GeV} $, $ \alpha=5.4 \mathrm{\; GeV}^{-1} $, as in [41]. Note that this simply reduces the strength of the $ \eta\Lambda $ mass distribution in a region where its contribution is very small and dominated by the $ \Lambda(1810)(1/2^+) $.

To gain insight into the cancellations occurring in the isospin-forbidden amplitude, we plot in Fig. 5 the relevant $ t_{ij} $ matrix elements $ t_{K^- p,\eta \Lambda} $, $ t_{\bar{K}^0 n,\eta \Lambda} $, $ t_{K^+ \Xi^-,\eta \Lambda} $, and $ t_{{K}^0 \Xi^0,\eta \Lambda} $ appearing in Eq. (10). The roles of $ Re\, t_{\bar{K}N,\eta \Lambda} $, $ Im\, t_{\bar{K}N,\eta \Lambda} $ and $ Re\, t_{K \Xi,\eta \Lambda} $, $ Im\, t_{K \Xi,\eta \Lambda} $ are exchanged, a feature often encountered in coupled channels for inelastic amplitudes. The important observation is that the magnitudes of $ Gt $ are remarkably similar, apart from an overall sign and normalization. Thus, combinations of A and B with opposite signs in Eq. (10) lead to larger decay widths than when A and B have the same sign. In any case, the figures explain why the terms in Eq. (10) with the $ t_{K \Xi,\eta \Lambda} $ amplitudes dominate. Indeed, $ |Im\, t_{K \Xi,\eta \Lambda}| $ is about an order of magnitude larger than $ |Im\, t_{\bar{K}N,\eta \Lambda}| $. We also observe in Fig. 5 that there are small differences between the charged and neutral channels in the t matrix. It is also useful to plot the results for $ Gt $ entering the evaluation of Eq. (10). We show the results in Fig. 6. Re $ G t $ and Im $ G t $ are now similar in shape, up to an overall sign. This is a consequence of the fact that Im $ G_{\bar{K}N} $ dominates $ G_{\bar{K}N} $, and hence in $ G_{\bar{K}N} t_{\bar{K}N,\eta \Lambda} $ the roles of the real and imaginary parts are now reversed with respect to $ t_{\bar{K}N,\eta \Lambda} $. The differences observed in $ Gt $ for the different channels are responsible for the isospin violation in the reaction studied.

Figure 5.  (color online) Differences between the charged and neutral channels for the $ t_i $ terms.

Figure 6.  (color online) Differences between the charged and neutral channels in the $ G_i t_i $ terms.

We have theoretically studied the $ J/\psi \to \Lambda\bar \Sigma^0 \eta $ reaction, which was recently measured by the BESIII collaboration. In this reaction, the $ \Lambda(1670) $ resonance is excited and accounts for nearly all the strength of the mass distributions. Taking advantage of the fact that the $ \Lambda(1670) $ resonance is one of the states generated by coupled-channel interactions within the chiral unitary approach, we have studied the reaction from this perspective and examined the dynamical origin of the isospin violation. As in many other reactions, this violation is attributed to the effect of loops involving kaons, primarily the $ K \Xi $ loops, due to the incomplete cancellation of terms in isospin-allowed amplitudes arising from the different masses of charged and neutral particles belonging to the same isospin multiplets. To evaluate the amplitudes, we first consider the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, both belonging to SU(3) octets. For this purpose, we treat the $ J/\psi $ as an SU(3) singlet and identify the dominant structures of an antibaryon, a baryon, and a pseudoscalar that couple to an SU(3) singlet. Using the couplings obtained in the study of the meson-baryon chiral unitary approach, the isospin violation is ultimately linked to the effect of the $ K^- p, \bar K^0 n $ and $ K^+ \Xi^-,K^0 \Xi^0 $ channels. The present study shows that the $ K^+ \Xi^-,K^0 \Xi^0 $ channels are mainly responsible for the strength of the mass distributions.

Up to a global normalization, which we cannot determine using only the SU(3) symmetry argument applied to the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, the mass distributions that we obtain are in good agreement with the experimental mass distributions. This agreement is particularly good when the contribution of the $ \Lambda(1810) (1/2^+) $ is included, which we fitted to the experimental data and which is very similar to that obtained in the experimental fit. In summary, we have obtained an appealing and plausible explanation for this interesting isospin-violating reaction, based on previous findings on meson-baryon interactions provided by the chiral unitary approach.

We have theoretically studied the $ J/\psi \to \Lambda\bar \Sigma^0 \eta $ reaction, which was recently measured by the BESIII collaboration. In this reaction, the $ \Lambda(1670) $ resonance is excited and accounts for nearly all the strength of the mass distributions. Taking advantage of the fact that the $ \Lambda(1670) $ resonance is one of the states generated by coupled-channel interactions within the chiral unitary approach, we have studied the reaction from this perspective and examined the dynamical origin of the isospin violation. As in many other reactions, this violation is attributed to the effect of loops involving kaons, primarily the $ K \Xi $ loops, due to the incomplete cancellation of terms in isospin-allowed amplitudes arising from the different masses of charged and neutral particles belonging to the same isospin multiplets. To evaluate the amplitudes, we first consider the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, both belonging to SU(3) octets. For this purpose, we treat the $ J/\psi $ as an SU(3) singlet and identify the dominant structures of an antibaryon, a baryon, and a pseudoscalar that couple to an SU(3) singlet. Using the couplings obtained in the study of the meson-baryon chiral unitary approach, the isospin violation is ultimately linked to the effect of the $ K^- p, \bar K^0 n $ and $ K^+ \Xi^-,K^0 \Xi^0 $ channels. The present study shows that the $ K^+ \Xi^-,K^0 \Xi^0 $ channels are mainly responsible for the strength of the mass distributions.

Up to a global normalization, which we cannot determine using only the SU(3) symmetry argument applied to the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, the mass distributions that we obtain are in good agreement with the experimental mass distributions. This agreement is particularly good when the contribution of the $ \Lambda(1810) (1/2^+) $ is included, which we fitted to the experimental data and which is very similar to that obtained in the experimental fit. In summary, we have obtained an appealing and plausible explanation for this interesting isospin-violating reaction, based on previous findings on meson-baryon interactions provided by the chiral unitary approach.

We have theoretically studied the $ J/\psi \to \Lambda\bar \Sigma^0 \eta $ reaction, which was recently measured by the BESIII collaboration. In this reaction, the $ \Lambda(1670) $ resonance is excited and accounts for nearly all the strength of the mass distributions. Taking advantage of the fact that the $ \Lambda(1670) $ resonance is one of the states generated by coupled-channel interactions within the chiral unitary approach, we have studied the reaction from this perspective and examined the dynamical origin of the isospin violation. As in many other reactions, this violation is attributed to the effect of loops involving kaons, primarily the $ K \Xi $ loops, due to the incomplete cancellation of terms in isospin-allowed amplitudes arising from the different masses of charged and neutral particles belonging to the same isospin multiplets. To evaluate the amplitudes, we first consider the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, both belonging to SU(3) octets. For this purpose, we treat the $ J/\psi $ as an SU(3) singlet and identify the dominant structures of an antibaryon, a baryon, and a pseudoscalar that couple to an SU(3) singlet. Using the couplings obtained in the study of the meson-baryon chiral unitary approach, the isospin violation is ultimately linked to the effect of the $ K^- p, \bar K^0 n $ and $ K^+ \Xi^-,K^0 \Xi^0 $ channels. The present study shows that the $ K^+ \Xi^-,K^0 \Xi^0 $ channels are mainly responsible for the strength of the mass distributions.

Up to a global normalization, which we cannot determine using only the SU(3) symmetry argument applied to the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, the mass distributions that we obtain are in good agreement with the experimental mass distributions. This agreement is particularly good when the contribution of the $ \Lambda(1810) (1/2^+) $ is included, which we fitted to the experimental data and which is very similar to that obtained in the experimental fit. In summary, we have obtained an appealing and plausible explanation for this interesting isospin-violating reaction, based on previous findings on meson-baryon interactions provided by the chiral unitary approach.

We have theoretically studied the $ J/\psi \to \Lambda\bar \Sigma^0 \eta $ reaction, which was recently measured by the BESIII collaboration. In this reaction, the $ \Lambda(1670) $ resonance is excited and accounts for nearly all the strength of the mass distributions. Taking advantage of the fact that the $ \Lambda(1670) $ resonance is one of the states generated by coupled-channel interactions within the chiral unitary approach, we have studied the reaction from this perspective and examined the dynamical origin of the isospin violation. As in many other reactions, this violation is attributed to the effect of loops involving kaons, primarily the $ K \Xi $ loops, due to the incomplete cancellation of terms in isospin-allowed amplitudes arising from the different masses of charged and neutral particles belonging to the same isospin multiplets. To evaluate the amplitudes, we first consider the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, both belonging to SU(3) octets. For this purpose, we treat the $ J/\psi $ as an SU(3) singlet and identify the dominant structures of an antibaryon, a baryon, and a pseudoscalar that couple to an SU(3) singlet. Using the couplings obtained in the study of the meson-baryon chiral unitary approach, the isospin violation is ultimately linked to the effect of the $ K^- p, \bar K^0 n $ and $ K^+ \Xi^-,K^0 \Xi^0 $ channels. The present study shows that the $ K^+ \Xi^-,K^0 \Xi^0 $ channels are mainly responsible for the strength of the mass distributions.

Up to a global normalization, which we cannot determine using only the SU(3) symmetry argument applied to the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, the mass distributions that we obtain are in good agreement with the experimental mass distributions. This agreement is particularly good when the contribution of the $ \Lambda(1810) (1/2^+) $ is included, which we fitted to the experimental data and which is very similar to that obtained in the experimental fit. In summary, we have obtained an appealing and plausible explanation for this interesting isospin-violating reaction, based on previous findings on meson-baryon interactions provided by the chiral unitary approach.

We have theoretically studied the $ J/\psi \to \Lambda\bar \Sigma^0 \eta $ reaction, which was recently measured by the BESIII collaboration. In this reaction, the $ \Lambda(1670) $ resonance is excited and accounts for nearly all the strength of the mass distributions. Taking advantage of the fact that the $ \Lambda(1670) $ resonance is one of the states generated by coupled-channel interactions within the chiral unitary approach, we have studied the reaction from this perspective and examined the dynamical origin of the isospin violation. As in many other reactions, this violation is attributed to the effect of loops involving kaons, primarily the $ K \Xi $ loops, due to the incomplete cancellation of terms in isospin-allowed amplitudes arising from the different masses of charged and neutral particles belonging to the same isospin multiplets. To evaluate the amplitudes, we first consider the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, both belonging to SU(3) octets. For this purpose, we treat the $ J/\psi $ as an SU(3) singlet and identify the dominant structures of an antibaryon, a baryon, and a pseudoscalar that couple to an SU(3) singlet. Using the couplings obtained in the study of the meson-baryon chiral unitary approach, the isospin violation is ultimately linked to the effect of the $ K^- p, \bar K^0 n $ and $ K^+ \Xi^-,K^0 \Xi^0 $ channels. The present study shows that the $ K^+ \Xi^-,K^0 \Xi^0 $ channels are mainly responsible for the strength of the mass distributions.

Up to a global normalization, which we cannot determine using only the SU(3) symmetry argument applied to the $ J/\psi $ decay into $ \bar \Sigma^0 $ and a pair of pseudoscalar-baryon states, the mass distributions that we obtain are in good agreement with the experimental mass distributions. This agreement is particularly good when the contribution of the $ \Lambda(1810) (1/2^+) $ is included, which we fitted to the experimental data and which is very similar to that obtained in the experimental fit. In summary, we have obtained an appealing and plausible explanation for this interesting isospin-violating reaction, based on previous findings on meson-baryon interactions provided by the chiral unitary approach.