Scattering of decuplet baryons in chiral effective field theory
J. Haidenbauer S. Petschauer N. Kaiser Ulf-G. Meißner W. Weise


















  ./figures/ ./graphs/ 
 






















  






t>c<Institute for Advanced Simulation and Jülich Center for Hadron Physics,
Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany

Physik Department, Technische Universität München, D-85747 Garching, Germany

Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center
for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
A formalism for treating the scattering of decuplet baryons in chiral 
effective field theory is developed. The minimal Lagrangian and potentials in 
leading-order SU(3) chiral effective field theory for the
interactions of octet baryons (B) and decuplet baryons (D) for the 
transitions BB→ BB, BB↔ DB, DB→ DB, BB↔ DD,
DB↔ DD, and DD→ DD are provided. 
As an application of the formalism we compare with results from
lattice QCD simulations for ΩΩ and NΩ scattering. 
Implications of our results pertinent to  the quest for dibaryons are discussed. 
12.39.Fe13.75.Ev12.38.Gc14.20.Pt
Scattering of decuplet baryons in chiral effective field theory 


    
J. Haidenbauer1
S. Petschauer2
N. Kaiser2
Ulf-G. Meißner 3,1
W. Weise2
    December 30, 2023
===========================================================================



§ INTRODUCTION
An important and basically timeless topic in hadron physics is the possible existence 
of dibaryons <cit.>. 
Historically, the so-called dibaryons have primarily been conceived as tightly bound 
six-quark objects. However, the terminology is also frequently adopted for 
shallow (or not so shallow)
bound states of two baryons, as in practice it might be difficult to distinguish 
between these six-quark and two-baryon configurations. 
We note that methods to disentangle compact multi-quark states from loosely 
bound molecular systems have recently been reviewed in ref. <cit.>.

A famous dibaryon proposed to be of the first category is the
H-particle, predicted by Jaffe within the bag model as a compact |uuddss⟩
state <cit.>. The H-dibaryon has the quantum numbers of the ΛΛ 
system in an S-wave state, namely strangeness S=-2, isospin I=0, and J^P=0^+. 
After decades of failed efforts to establish its existence experimentally,
it regained popularity a few years ago due to lattice simulations that provided
evidence for its presence <cit.>, 
though only for quark masses that are larger than the physical ones.

Definitely of the second category is the deuteron, a bound state in the coupled ^3S_1-^3D_1 
partial waves of the neutron-proton (np) system. Also of that category is a possible
(unstable) quasibound state in the ^3S_1-^3D_1 partial waves of the Σ N channel 
with isospin I=1/2 which appears as a pronounced cusp-like enhancement in the 
Λ p invariant-mass spectrum of reactions like 
K^-d →π^-Λ p and pp→ K^+Λ p very close to the Σ N threshold,
see the review <cit.>.  

In refs. <cit.> it has been shown that chiral 
effective field theory (EFT), an approach initially suggested for 
deriving and describing the forces between 
nucleons <cit.>, 
can be straightforwardly extended to the interaction of baryons with strange­ness. 

In this framework the symmetries of QCD together with the appropriate low-energy 
degrees of freedom are exploited to construct the baryon-baryon interactions. Moreover,
there is an underlying power counting that allows one to improve the results
systematically by going progressively to higher orders in a perturbative expansion.

This approach is well prepared to shed light on the H-dibaryon, should it indeed exist, 
and on other possible dibaryons in the strange­ness S=-3 and -4 sectors, as
demonstrated in refs. <cit.>.
In particular, one can study implications of the imposed (approximate)
SU(3) symmetry and further explore the dependence of the properties of such dibaryons 
on the pion[The Gell-Mann-Oakes-Renner relation states that the squared pion 
mass is proportional to the average light quark mass. Therefore, the notions
“quark mass dependence” and “pion mass dependence”are used synonymously.]
and baryon masses. The latter aspect is important since, as mentioned, the pertinent lattice QCD 
(LQCD) calculations were not performed for physical quark masses. 

In the present paper we extend the investigations of 
refs. <cit.>
to decuplet baryons. 
The main goal is to provide the basic elements for treating the scattering of
octet baryons with decuplet baryons (BD) and of two decuplet baryons (DD) 
within SU(3) chiral EFT. Here we restrict ourselves, as a first step, to leading order (LO) 
in the Weinberg counting <cit.>.  
In this case the interaction potential
is given by single pseudoscalar-meson (π, η, K) exchange plus
four-baryon contact terms without derivatives <cit.>. The latter
represent the short-ranged part of the baryon-baryon force and involve
low-energy constants (LECs),  parameters to be fixed by fits to data.

The resulting potential is then used in a regularized Lippmann-Schwin­ger (LS) 
equation to generate possible bound states and to evaluate two-body
scattering amplitudes. 

As in refs. <cit.>
we assume the (approximate) validity of SU(3) flavor symmetry for the coupling strengths involved[SU(3) 
breaking effects are nonetheless incorporated using physical baryon and meson masses.]. 
This assumption allows one to establish relations between the forces in 
channels with different isospin and strange­ness, and we recapitulate the
implications of SU(3) symmetry for the interactions in the BD and DD systems.
In addition we investigate the consequences of SU(3) symmetry for transitions from 
these systems to states composed of two octet baryons (BB). 

Note that a work in similar spirit but aiming at the charm and beauty sector,
where instead heavy quark spin symmetry plays a fundamental role, has been 
presented recently <cit.>. 
 
A primary motivation for our study comes from recent LQCD calculations
for  and N scattering
<cit.>. 
In some of those computations signals for possible dibaryons were found 
<cit.>.
Thus, as a first application of our formalism we analyze the results for 
 and N scattering presented in those works. We also discuss 
implications on possible other dibaryon states, notably in the , 
, and  systems.  

Apart from indications for dibaryons in the  and N
systems from LQCD calculations, 
there are further longstanding claims for dibaryons in those channels from 
studies based on the constituent quark model, see 
refs. <cit.> for more recent efforts.
Possible bound states in Ξ and Ξ^*<cit.>, 
and <cit.> have been discussed, too. 

And finally, there has been a noticeable revival of dibaryons in the  sector,
in the context of measurements of the reactions pn→ d π^0π^0 and 
pn→ d π^+π^- by the WASA-at-COSY collaboration in Jülich  
<cit.>. A resonance in the ^3D_3-^3G_3NN partial 
wave required to describe the data <cit.>, termed 
d^*(2380) dibaryon, 
could be a reflection of an S-wave quasibound state in the  system 
<cit.>. 

In the analysis of the LQCD simulations we follow closely the strategy 
of our previous works <cit.>:
(i) the LECs, i.e. the only free parameters in the potential, are determined by 
a fit to LQCD results (phase shifts, scattering lengths) employing the inherent 
baryon and meson masses of the lattice simulation;
(ii) Results at the physical point are obtained via a calculation in which
the pertinent physical masses of the mesons are substituted in the evaluation of 
the potential and those of the baryons in the baryon-baryon propagators appearing
in the LS equation. 
Such an “extrapolation”can be expected to work only qualitatively, given the still 
unphysically large hadron masses in the LQCD simulations for , 
N, and . Nonetheless  recent (and still preliminary) lattice
results for the H-dibaryon, and for  and NΞ phase shifts, respectively, 
close to the physical point <cit.> reveal that the 
prediction/extrapolation in refs. <cit.>, 
performed in the way described above, may be fairly realistic. In this context let
us mention that also for other baryon-baryon channels LQCD simulations have been 
performed for almost physical quark masses <cit.> 
and preliminary results for phase shifts have become available [Notably, 
the method employed in these works is yet under discussion in the LQCD community.]
for some channels. 

This paper is structured as follows:

A basic outline of our formalism is given in sec. <ref>. 
Specifically, we establish the underlying Lagrangians for constructing the
interactions in the BB, BD, and DD sectors at leading order in chiral EFT. 
Furthermore, the essentials of the imposed SU(3) flavor symmetry are described. 
Finally, explicit expressions for the potentials resulting from meson-exchange and
from the four-baryon contact terms are given. 

In sec. <ref>, as exemplary applications of our formalism,
results for selected DD and BD reactions are presented, where we focus
on such reaction channels for which predictions from lattice QCD simulations 
for phase shifts and/or scattering lengths are available: 
 scattering in the ^1S_0 partial wave 
and N in the ^5S_2 partial wave. Furthermore, we explore the 
 system with total angular momentum J=3 and isospin I=0, 
and the possible emergence of a bound state that could be associated 
with the d^*(2380) dibaryon.

A brief summary and an outlook is presented in sec. <ref>. 

Technical details of our calculation are summarized in three appendices.







§ FORMALISM


Our calculation is based on a non-relativistic approach in leading-order SU(3) chiral EFT.
Possible (isospin violating) π^0-η or  Σ^0-Λ  mixing is neglected.
We use the conventional building blocks as laid down, e.g., in 
refs. <cit.>, with the pseudoscalar meson 
octet represented by the 3×3 matrix in flavor space:

    ϕ=
     [  π^0 + η/√(3)       √(2)π^+      √(2) K^+;       √(2)π^- -π^0 + η/√(3)      √(2) K^0;      √(2) K^-      √(2)K̅^0      -2η/√(3) ] .

The octet baryons are represented by the3×3 matrix

    B=
     [  Σ^0/√(2) + Λ/√(6)                Σ^+                  p;                Σ^- -Σ^0/√(2) + Λ/√(6)                  n;                Ξ^-                Ξ^0           -2Λ/√(6) ]  .
The decuplet baryons are represented by the totally symmetric three-index tensor T:

    T_111   =Δ^++ ,  T_112=1√(3)Δ^+ ,  T_122=1√(3)Δ^0 ,  T_222=Δ^- ,
    
    T_113   =1√(3)Σ^*+ ,  T_123=1√(6)Σ^*0 ,  T_223=1√(3)Σ^*- ,
    
    T_133   =1√(3)Ξ^*0 ,  T_233=1√(3)Ξ^*- ,
    
    T_333   =Ω^-  .
The construction of the chiral Lagrangian requires a complete set of spin operators, covering all combinations of spin 1/2 and spin 3/2 states.
These have been established employing the Wigner-Eckart theorem.
One obtains for spin 1/2 the identity matrix  and σ^i, i.e. the usual Pauli matrices, corresponding to scalar and vector operators.
For the spin 1/2 to spin 3/2 transition operators, the resulting 4×2 matrices connect the two-component spinors of octet baryons with the four-component spinors of decuplet baryons <cit.>. 
The corresponding vector and rank-2 tensor operators are S^i† and S^ij†, respectively.The reverse transition, 3/2 to 1/2, is described by the Hermitian-conju­gate spin transition operators S^i and S^ij.
Finally, the 4×4 
matrices , Σ^i, Σ^ij, Σ^ijk, corresponding in this order to scalar, vector, rank-2 and rank-3 tensor operators, are introduced for the description of the spin 3/2 sector.

These sets of matrices for the different types of transitions form a basis.
More details on the construction of the spin operators and their explicit expressions 
can be found in Appendix <ref>.

For the two-body interactions considered in the following, tensor products of two of 
these spin operators are involved.




 §.§ Lagrangians generating meson exchange


In this subsection we consider the meson-baryon Lagrangians for the construction of the 
leading-order meson exchange interactions of octet and decuplet baryons.

The SU(3) invariant Lagrangian for octet baryons coupled to the pseudoscalar meson octet is given by
<cit.>
    ℒ_BBϕ   = -D/2f_0 tr( B̅σ⃗·{∇⃗ϕ,B}) - F/2f_0 tr(B̅σ⃗·[∇⃗ϕ,B])   ,

where f_0 is the weak pseudoscalar-meson decay constant f_0 ≈ f_π.
The coupling constants F and D satisfy F + D = g_A,
with g_A the axial-vector strength measured in neutron β-decay[Empirical 
values of these constants are f_π≃ 92.2 MeV and g_A = 1.272± 0.002<cit.>.].
Introducing standard isospin operators, the Lagrangian can be written in its well known
form <cit.>:

    ℒ_BBϕ   =   -f_NNπN̅N· 
    +if_ΣΣπ×·
          -f_ΛΣπ[Λ̅+Λ]·-f_ΞΞπΞ̅Ξ·
          -f_Λ NK[N̅Λ K+Λ̅ N K^†]
    
          
    -f_ΞΛ K[Ξ̅Λ K_c+Λ̅Ξ K_c^†]
    
          
    -f_Σ NK[· K^†N+N̅ K·]
    
          
    -f_ΞΣ K[· K_c^†Ξ+Ξ̅ K_c·]
    
          
    -f_NNη_8N̅ Nη
          
    -f_ΛΛη_8Λ̅Λη-f_ΣΣη_8·η
           
    -f_ΞΞη_8Ξ̅Ξη .


Here, we have introduced the isospin doublets

    N=([ p; n ]) ,  Ξ=([ Ξ^0; Ξ^- ]) ,  
    K=([ K^+; K^0 ]) ,   K_c=([   ^0; -K^- ]) .

The signs have been chosen according to the conventions of ref. <cit.>.
With regard to isovectors the assignment ^+ = -|1,1⟩, etc., is used so 
that the inner product of  (or ) defined 
in spherical components reads

    ·=∑_m (-1)^mΣ_mΣ_-m=
    Σ^+Σ^-+Σ^0Σ^0+Σ^-Σ^+ .
Spin- and momentum operators (σ·∇) have been omitted in eq. (<ref>) to 
simplify the presentation. That structure is the same for all B_iB_jϕ_k vertices. 

The coupling constants introduced in eq. (<ref>) are given by 

    [         f_NNπ  =               f,       f_NNη_8  =  1/√(3)(4α -1)f,;         f_Λ NK =  -1/√(3)(1+2α)f,         f_ΞΞπ  =        -(1-2α)f,;       f_ΞΞη_8  = -1/√(3)(1+2α )f,         f_ΞΛ K =   1/√(3)(4α-1)f,;         f_ΛΣπ  =    2/√(3)(1-α)f,       f_ΣΣη_8  =   2/√(3)(1-α )f,;         f_Σ NK =         (1-2α)f,         f_ΣΣπ  =            2α f,;       f_ΛΛη_8  =  -2/√(3)(1-α )f,         f_ΞΣ K =              -f. ]

with f≡ g_A/(2f_0) and α≡ F/(F+D). 

In the present work we use the same values as in our previous LO study of the YN system <cit.>,
namely f_0=93 MeV, g_A = 1.26, α=0.4. 

The Lagrangian involving an octet and a decuplet baryon is written in the form 
<cit.>
    ℒ_DBϕ   = C/f_0∑_a,b,c,d,e=1^3 ϵ_abc( T̅_adeS⃗^ †·(∇⃗ϕ_db)B_ec
                       +
    B̅_ceS⃗·(∇⃗ϕ_bd)T_ade)   .

The coupling constant C can be fixed from the decay Δ→π N. 
One findsC=3g_A/4<cit.>. The vector spin operator 
 consists of 4× 2 matrices representing the transition
from spin 1/2 to 3/2. Their explicit forms are given in Appendix <ref>. 

Omitting again the spin-momentum dependent part of the Lagrangian, eq. (<ref>),
one can cast the remainder into a simple form using isospin operators

    ℒ_DBϕ   =   -f_NΔπΔ̅N· 
    +if_Σπ×·
          -f_ΛπΛ·-f_ΞπΞ·
          
    -f_N K· K^†N
    -f_ΣΔ KΔ̅ K·
          
    -f_Λ KΛ K_c
    -f_Σ K K_c·
          
    -f_Ξ K· K_c^†Ξ
    -f_Ξ KΞ K^†
          
    -f_Ση_8·η
    -f_Ξη_8Ξη
          
    +   h.c. ,

where the Δ isobar is represented by an isospin quartet, i.e. 
and  and  by isospin states analogous to those in eqs. (<ref>)  
and (<ref>).
The SU(3) relations for the coupling constants of octet to decuplet 
baryons are expressed in terms of the coupling constant f_BDand read

    f_NΔπ = 
    f_Ξ K = 
    -f_ΣΔ K   =    f_BD, 
    
    f_Σπ =
    f_N K =
    f_Σ K =
    f_Ξ K   =    -1/√(6) f_BD, 
    
    f_Λπ =
    -f_Λ K =
    -f_Ση_8 =
    f_Ξη_8   =   1/√(2) f_BD, 
    
    f_Ξπ   =   1/√(6) f_BD .

The relation between the coupling constants in eqs. (<ref>)
and (<ref>) is f_BD = √(2) C/f_0 and results from the different normalization
of the isospin transition operator , cf. eq. (<ref>) in 
Appendix <ref>, and the tensor representation for the decuplet baryons
in eq. (<ref>). 

The Nπ coupling constant h_A, commonly used in studies of π N scattering
within chiral perturbation theory <cit.>, coincides with f_BD. 
Accordingly, the choice of C mentioned above is in line with the large-N_c value 
h_A = 3g_A/(2√(2)).

The Lagrangian involving two decuplet baryons and a meson expressed in the conventional building 
blocks is given by

    ℒ_DDϕ   = H/f_0∑_a,b,c,d=1^3 (T̅_abcΣ⃗T_abd) · (∇⃗ϕ_cd)   ,

with a new coupling constant H and the spin-3/2
operator Σ (see appendix A).  Although the same
symbol is used for the isotriplet of Σ^+,0,- hyperons, the
respective meaning of  Σ should be clear from the context.  

Casting the Lagrangian eq. (<ref>) into a simple form utilizing isospin operators leads to 

    ℒ_DDϕ   =   -f_π· 
    +if_π×·
          -f_π·
          
    -f_ K[· K^†
      + K·]
    
          
    -f_ K[· K_c^†+ K_c·]
    
          
    -f_ K[ K^† +  K]
    
          
    -f_η_8η
    -f_η_8·η
           
    -f_η_8η  
    -f_η_8η ,

omitting once again the spin-momentum part. 
Here the SU(3) relations for the coupling constants are:

    f_π = f_π   =    f_DD, 
    
    -f_ K = f_ K   =   √(6) f_DD, 
    
    f_π = f_ K   =    2f_DD, 
    
    f_η_8 = -f_η_8 =-f_η_8/2    =   √(3) f_DD, 
    
    f_η_8   =    0  ,
where the normalization of the decuplet-baryon
tensor and the definition of the isospin 3/2 operator as given in
Appendix <ref> imply f_DD = H / (3 f_0). We note that a remarkable correlation between the 
leading πΔ N and πΔΔ couplings beyond the large-N_C expansion has
recently been found <cit.>.

The general form of the LO meson-exchange potential is analogous to that for BB systems 
<cit.>, namely 

    V_P   =   -(_1· q)(_2· q)/ q^2+m^2_P ,

where m_P is the mass of the exchanged pseudoscalar meson P. 
The symbol 𝒪_i stands generically for the spin operators
at the two vertices. 
For spin 1/2 → 1/2 it is given by ,
for 3/2 → 3/2 by ,
for 1/2 → 3/2 by , and 
for 3/2 → 1/2 by .

The transferred momentum q is defined in terms of the final and initial 
center-of-mass (c.m.) momenta of the baryons, p' and p, as 
q= p'- p. 
As already mentioned in the Introduction, we use either the physical masses of the exchanged 
pseudoscalar mesons or masses corresponding to the lattice QCD simulations. 
Thus, the explicit SU(3) breaking reflected in the mass splitting between 
the pseudoscalar mesons is taken into account. 

Following our previous works <cit.> 
the η meson is identified with the octet state η_8 and its 
physical mass is used. A possible coupling of two octet baryons to a
singlet meson (η_0) which would introduce a further coupling constant 
<cit.>, is ignored in the present context. As far as the vertex of
two decuplet baryons and a meson is concerned, since 
according to the decomposition 
10⊗10 = 64⊕27⊕8⊕1<cit.>,
one can also couple the two decuplet baryons to a singlet and then with a singlet
meson to obtain an SU(3) invariant. Again this would introduce a further coupling 
constant which we likewise ignore.

The actual LO meson-exchange potential is obtained by multiplying the 
spin-momentum part of the potential, eq. (<ref>) with the  
coupling constants as given in eqs. (<ref>), (<ref>), or (<ref>)
and with the pertinent isospin factors, i.e. 

    V^ℬ_1ℬ_2→ℬ_1'ℬ_2'_OBE   =   ∑_π,η,K 
    f_ℬ_1ℬ_1'Pf_ℬ_2ℬ_2'P ℐ_P^ℬ_1ℬ_2→ℬ_1'ℬ_2'  V_P ,

where ℬ stands here for octet or decuplet baryons. 

A complete list of isospin factors for the DD case is provided in 
sec. <ref> where DD scattering results 
are presented for various channels and partial waves. 
Here, we refrain from listing those for BDscattering and for the BB→ BD, BB→ DD, 
and BD→ DD transitions. The ones for diagrams 
involving π and/or η exchange can be straightforwardly calculated by using 
eq. (27) of ref. <cit.>. Furthermore, many coefficients can be also inferred from the tables in 
refs. <cit.> since for transitions of baryons with the
same isospins in the initial- and final states, the isospin coefficients are also the
same. Note, however that symmetrization factors (√(2) and/or a (-1)^I-I_1-I_2 from
recoupling) may differ and that has to be taken into account appropriately. 

Some isospin coefficients for BB→ BD can be found in refs. <cit.>. 

An explicit representation of the potentials in the particle basis can be found in Appendix <ref>. This form of the potential has been used, in particular, to obtain the SU(3) relations in 
Appendix <ref>, starting from the contact term Lagrangians.






 §.§ Partial wave decomposition of the meson-exchange contribution
 

The partial wave projection of the meson-exchange contribution to the potential is performed using 
the helicity basis, see refs. <cit.> or Appendix B of <cit.>. The partial-wave projected potential is given by 


    ⟨λ_1'λ_2'| V^J(p',p) |λ_1λ_2  ⟩
          = 2π∫_-1^+1 d(cosθ) d^J_λ,λ'(θ ) 
    ⟨λ_1'λ_2'| V(p⃗',p⃗) |λ_1λ_2  ⟩ ,

with λ=λ_1-λ_2, λ'=λ_1'-λ_2'. 
Here |λ_i⟩ and |λ_i'⟩ represent the helicity states of 
the incoming and outgoing baryons. The total angular momentum is denoted by J. 
The quantities d^J_λ,λ'(θ ) are the reduced rotation
matrices. We calculate those numerically based on the definition of the d-functions 
in terms of Jacobi polynomials <cit.>. Analytic expressions for the d-functions needed for J up to 2 
can be found, e.g. in ref. <cit.>. An incomplete set of the 
d^J_λ,λ''s with λ = 3is listed in ref. <cit.>. 

The amplitudes in the standard JMLS basis are then constructed via a
unitary transformation, also described in Appendix B of <cit.>: 


    ⟨ JML'S'| V^J(p',p) |JMLS  ⟩ =  ∑_λ_1λ_2,λ_1'λ_2'
          ⟨ JML'S'| JMλ_1'λ_2' ⟩⟨λ_1'λ_2'| V^J(p',p) |λ_1λ_2  ⟩⟨ JMλ_1λ_2 |JMLS  ⟩,


with 

    ⟨ JMLS|JMλ_1λ_2⟩ = √(2L+1/2J+1)⟨ L0Sλ|Jλ⟩⟨ s_1λ_1 s_2 -λ_2|Sλ⟩,
 
where λ = λ_1 -λ_2
and ⟨ s_1λ_1 s_2λ_2 | Sλ⟩ denotes a
 Clebsch-Gordan coefficient.
 
The wave function of the decuplet baryons is constructed from 
the standard Rarita-Schwinger spinor  

    ψ^μ (p⃗, Λ) = ∑_λ_1,λ_2⟨ 1λ_1 1/2λ_2 | 3/2Λ⟩ϵ^μ (p⃗, λ_1) u (p⃗, λ_2)  .


In the non-relativistic (static) approach that we follow here, the 
polarization vectors ϵ^μ(p⃗, λ_1) reduce to <cit.>
    ϵ^i (p⃗,± 1) = 1/√(2)(
    [ ∓ 1;  -i;   0;     ]),  ϵ^i (p⃗,0) = 
    (
    [ 0; 0; 1;   ])
 
and u (p⃗, λ_2)reduces to a two-component spinor χ(λ_2). 

Then the matrix element of the operator ·q⃗ 
representing the spin structure of the DDϕ vertex required for the
DD→ DD, ND→ ND and ND→ DD transition potentials 
is given in the helicity basis by

    ⟨Λ' |  ·q⃗ | Λ⟩   =    3∑_λ_1',λ_2',λ_1,λ_2⟨ 1λ_1'1/2λ_2' | 3/2Λ'⟩⟨ 1λ_1 1/2λ_2 | 3/2Λ⟩
       ×   ϵ^* (p⃗ ', λ_1') ϵ (p⃗, λ_1)  ⟨λ_2' |  σ⃗·q⃗ | λ_2⟩ .


The helicity matrix elements for ·q⃗ can be found, 
e.g., in ref. <cit.>. 

Analytic expressions of helicity amplitudes for the transitions BB→ BD and BB→ DD 
that involve the operators ·q⃗ and/or
·q⃗
can easily be deduced from those in refs. <cit.>.
Note that an alternative method to perform
the partial-wave projections, which exploits rotational invariance  by
averaging over the total angular momentum projection M, has been
formulated in ref. <cit.>.



 §.§ Lagrangians involving contact terms


The construction of the minimal Lagrangian collecting the contact terms at leading order for the various combinations of octet 
and decuplet baryons is achieved by writing down all chirally invariant flavor structures combined with all possible spin 
structures.
This minimal set is obtained by eliminating redundant terms until the rank of the matrix formed by all transitions matches the number of terms in the Lagrangian.
Redundant terms are deleted in such a way that one obtains a maximal number of contact terms with different spin structures and a minimal number of terms with different flavor structures
[
In the case of DB→ DB we eliminated the “exchange” spin structures S⃗⊗S⃗^† 
and S^αβ⊗ S^αβ† by including a larger number of flavor structures.
].

In the following we present the minimal leading-order Lagrangians in the non-relativistic limit and rewritten in terms of particle fields, using conventional matrix notation.
For the low-energy constants c^f_ij of the contact terms, the subscript ij denotes the number of decuplet baryons 
in the initial and final state.
Note that Lagrangian terms for ℒ_BBBB can be found in <cit.> 
and the terms for ℒ_DBBB have been constructed in <cit.>.
We have:


    ℒ_BBBB =       ^1∑_a,b,c,d=1^3 (B̅_ab B_bc)(B̅_cd B_da) 
       + ^2∑_a,b,c,d=1^3 (B̅_abσ⃗B_bc)·(B̅_cdσ⃗B_da) 
       + ^3∑_a,b,c,d=1^3 (B̅_ab B_cd)(B̅_bc B_da) [0] 
       + ^4∑_a,b,c,d=1^3 (B̅_abσ⃗B_cd)·(B̅_bcσ⃗B_da) 
       + ^5∑_a,b,c,d=1^3 (B̅_ab B_ba)(B̅_cd B_dc) 
       + ^6∑_a,b,c,d=1^3 (B̅_abσ⃗B_ba)·(B̅_cdσ⃗B_dc)
      .

    ℒ_DBBB =       ^1∑_a,b,c,d,e,f=1^3 ϵ_abc[
    (T̅_adeS⃗^† B_db)·(B̅_fcσ⃗B_ef)+ 
                           (B̅_bdS⃗  T_ade)·(B̅_feσ⃗B_cf)
    ]
       +^2
    ∑_a,b,c,d,e,f=1^3 ϵ_abc[
    (T̅_adeS⃗^† B_fb)·(B̅_dcσ⃗B_ef)+ 
                           (B̅_bfS⃗  T_ade)·(B̅_feσ⃗B_cd)
    ]  .

    ℒ_DBDB =       ^1∑_a,b,c,d,e=1^3 (T̅_abc T_abc)(B̅_de B_ed)
       + ^2∑_a,b,c,d,e=1^3 (T̅_abcΣ⃗  T_abc)·(B̅_deσ⃗  B_ed) 
       +^3∑_a,b,c,d,e=1^3 (T̅_abc T_abd)(B̅_ce B_ed)
       + ^4∑_a,b,c,d,e=1^3 (T̅_abcΣ⃗  T_abd)·(B̅_ceσ⃗  B_ed) [0] 
       +^5∑_a,b,c,d,e=1^3 (T̅_abc T_abd)(B̅_ed B_ce)
       + ^6∑_a,b,c,d,e=1^3 (T̅_abcΣ⃗  T_abd)·(B̅_edσ⃗  B_ce) 
       +^7∑_a,b,c,d,e=1^3 (T̅_abc T_ade)(B̅_bd B_ce)
       + ^8∑_a,b,c,d,e=1^3 (T̅_abcΣ⃗  T_ade)·(B̅_bdσ⃗  B_ce)
     .

    ℒ_DDBB = 
       =
    ^1∑_a,b,c,d,e,f,g,h=1^3 ϵ_abcϵ_def[
    (T̅_adgS⃗^† B_gc)·(T̅_behS⃗^† B_hf) 
                       +
    (B̅_cgS⃗  T_adg)·(B̅_fhS⃗  T_beh)
    ]
       +^2
    ∑_a,b,c,d,e,f,g,h=1^3 ϵ_abcϵ_def[
    (T̅_adgS^αβ† B_gc)(T̅_behS^αβ† B_hf) 
                   +
    (B̅_cgS^αβ T_adg)(B̅_fhS^αβ T_beh)
    ]  .

    ℒ_DDDB =       ^1∑_a,b,c,d,e,f,g=1^3 ϵ_abc[
    (T̅_adeΣ⃗T_def)·(T̅_bfgS⃗^† B_gc) 
                       +
    (T̅_defΣ⃗T_ade)·(B̅_cgS⃗  T_bfg)
    ]
       +^2∑_a,b,c,d,e,f,g=1^3 ϵ_abc[
    (T̅_adeΣ^αβ T_def)(T̅_bfgS^αβ† B_gc) 
                       +
    (T̅_defΣ^αβ T_ade)(B̅_cgS^αβ T_bfg)
    ]  .

    ℒ_DDDD =     + ^1∑_a,b,c,d,f=1^3 (T̅_abc T_abc)(T̅_def T_def) 
       + ^2∑_a,b,c,d,f=1^3 (T̅_abcΣ⃗T_abc)·(T̅_defΣ⃗T_def) 
        + ^3∑_a,b,c,d,f=1^3 (T̅_abcΣ^αβ T_abc)(T̅_defΣ^αβ T_def) 
       + ^4∑_a,b,c,d,f=1^3 (T̅_abcΣ^αβγ T_abc)(T̅_defΣ^αβγ T_def) [0] 
        + ^5∑_a,b,c,d,f=1^3 (T̅_abc T_abd)(T̅_def T_cef) 
       + ^6∑_a,b,c,d,f=1^3 (T̅_abcΣ⃗T_abd)·(T̅_defΣ⃗T_cef) 
        + ^7∑_a,b,c,d,f=1^3 (T̅_abcΣ^αβ T_abd)(T̅_defΣ^αβ T_cef) 
       + ^8∑_a,b,c,d,f=1^3 (T̅_abcΣ^αβγ T_abd)(T̅_defΣ^αβγ T_cef)
     .


 §.§ Group theoretical considerations


Group theory can be used to deduce the number of terms in the chiral contact Lagrangian.
First of all we need to express the two-baryon states in terms of irreducible representations of 
SU(3) (flavor) and SU(2) (spin).
For octet-octet baryon states BB the decompositions in flavor and spin space are, respectively:

    8⊗8 =
    27⊕8_𝐬⊕1⊕10⊕10⊕8_𝐚 ,2⊗2 = 1⊕3 ,

where in flavor space the representations 27,8_𝐬,1 are symmetric and 
the representations 10,10,8_𝐚 are antisymmetric. 
In spin space 1 is antisymmetric and 3 is symmetric.

For decuplet-octet baryon states DB one obtains

    10⊗8 = 35⊕27⊕10⊕8 ,4⊗2 = 3⊕5 .


Decuplet-decuplet baryon states DD take the form

    10⊗10 = 35⊕10⊕28⊕27 ,4⊗4 =1⊕5⊕3⊕7 .


Combining this information with the Pauli principle 
(for 8⊗8 and 10⊗10) 
and the fact that, at leading order, only transitions between the same flavor and spin irreducible representations can occur, we can write down the possible combinations of flavor and spin representations (flavor,spin) in which the transition can occur:

    BB→ BB    (27,1),(8_s,1),(1,1),(10,3),(10^*,3),(8_a,3) 
    
    BB→ DB    (10,3),(8,3) 
    
    BB→ DD    (27,1),(10,3) 
    
    DB→ DB    (35,3),(27,3),(10,3),(8,3),(35,5),(27,5),
        (10,5),(8,5) 
    
    DB→ DD    (35,3),(27,5) 
    
    DD→ DD    (35,3),(35,7),(10,3),(10,7),(28,1),(28,5), 
        (27,1),(27,5)

The resulting number of different (flavor,spin) representations 
is 6 for BB→ BB, 8 for DB→ DB, 8 for DD→ DD, 
and 2 each for the transitions  BB→ DB,  BB→ DD, and  DB→ DD. 
This fits well to the number of low-energy constants of the minimal Lagrangians 
in sec. <ref>. It serves as a non-trivial check of our calculation.

The SU(3) relations for the various two-body S-wave contact 
interactions among octet and decuplet baryons are summarized in Appendix C.
Furthermore, the relations between the LECs of the Lagrangians and the irreducible representations are given.
The SU(3) relations for the decuplet-decuplet interaction can be found already in Table <ref> as they are of prime interest for applications discussed in the following sections. 
All those relations are relevant for checking the consistent construction of the LO Lagrangian 
and potentials.
By an approach analogous to the one in ref. <cit.>, we can establish which 
irreducible representations contribute to a specific strangeness-isospin channel.
These relations have to conform with corresponding results that follow directly from the isoscalar factors of the 
8⊗8, 10⊗8, and 10⊗10
representations given in ref. <cit.>[Note in particular the symmetry relations 
between 8⊗10 and 10⊗8 as given in 
Table I of ref. <cit.>.]. 
They are also important for actual calculations as they specify
how different strangeness and isospin channels are connected with each other 
via SU(3) symmetry. This aspect will be further exploited in sec. <ref>.  



 §.§ Partial wave decomposition of contact spin structures


In the following we present the explicit forms of the contact potentials V_cont
involving the most general two-body spin operators for the non-vanishing transitions 
between partial waves ^2S+1L_J at leading order.
Following the method described in ref. <cit.> one finds (with constants a_i):

 0.2cm 
BB→ BB:  V_cont = a_1 + a_2σ⃗_1·σ⃗_2
    V_^1S_0 = ⟨^1S_0|V_cont|^1S_0⟩   = a_1 - 3a_2 
    
    V_^3S_1 = ⟨^3S_1|V_cont|^3S_1⟩   = a_1 + a_2
BB← DB:  V_cont = a_1 S⃗^†_1 ·σ⃗_2 
        BB→ DB:  V_cont = a_1 S⃗_1·σ⃗_2
    V_^3S_1 = ⟨^3S_1|V_cont|^3S_1⟩   = -2√(2/3)a_1
BB← DD:  V_cont = a_1S⃗^†_1·S⃗^†_2 + a_2S^ij†_1S^ij†_2 
        BB→ DD:  V_cont = a_1S⃗_1·S⃗_2 + a_2S^ij_1S^ij_2
    V_^1S_0 = ⟨^1S_0|V_cont|^1S_0⟩   = -√(2) a_1 + 5√(2)/3a_2 
    
    V_^3S_1 = ⟨^3S_1|V_cont|^3S_1⟩   = -√(10)/3 ( a_1 + a_2 )
DB→ DB:  V_cont = a_1 + a_2Σ⃗_1·σ⃗_2
    V_^3S_1 =⟨^3S_1|V_cont|^3S_1⟩   = a_1 - 5 a_2 
    
    V_^5S_2 =⟨^5S_2|V_cont|^5S_2⟩   = a_1 + 3 a_2
DB→ BD:  V_cont = a_1S⃗_1·S⃗^†_2 + a_2S^ij_1S^ij†_2
    V_^3S_1= ⟨^3S_1|V_cont|^3S_1⟩   = 1/3a_1 - 5/3 a_2 
    
    V_^5S_2= ⟨^5S_2|V_cont|^5S_2⟩   = a_1 + 1/3 a_2
DB← DD:  V_cont= a_1Σ⃗_1·S⃗^†_2 + a_2Σ^ij_1S^ij†_2
         DB→ DD:  V_cont=a_1Σ⃗_1·S⃗_2 + a_2Σ^ij_1S^ij_2
    V_^3S_1=⟨^3S_1|V_cont|^3S_1⟩   = 2√(5/3) a_1 + √(10) a_2 
    
    V_^5S_2=⟨^5S_2|V_cont|^5S_2⟩   = 2√(3) a_1 - √(2) a_2
DD→ DD:  V_cont = a_1 + a_2Σ⃗_1·Σ⃗_2 + 
a_3Σ^ij_1Σ^ij_2 + a_4Σ^ijk_1Σ^ijk_2
    V_^1S_0=⟨^1S_0|V_cont|^1S_0⟩   = a_1 - 15a_2 + 15/2a_3 -350/3a_4
    
    V_^3S_1=⟨^3S_1|V_cont|^3S_1⟩   = a_1 - 11a_2 + 3/2a_3 +70a_4
    
    V_^5S_2=⟨^5S_2|V_cont|^5S_2⟩   = a_1 - 3a_2 - 9/2a_3 -70/3a_4
    
    V_^7S_3=⟨^7S_3|V_cont|^7S_3⟩   = a_1 + 9a_2 + 3/2a_3 +10/3a_4

 

The actual potential for a specific channel and partial wave is a combination of low-energy constants 
according to Table <ref> and Tables <ref> - <ref> in Appendix <ref>.





§ APPLICATION TO LATTICE QCD RESULTS


In this section we exemplify how the formalism developed above 
can be used to analyze results from lattice QCD computations. 

It should be understood that this study has primarily illustrative character.
Lattice results for interactions involving decuplet baryons are so far restricted to unphysically large pion masses. It is obvious that LO chiral EFT can give only a qualitative (but nonetheless instructive) picture. More elaborate treatments, at next-to-leading order and beyond, will be feasible as lattice simulations proceed (close) to physical masses and become increasingly accurate.

The reaction amplitude for a specific potential 
V=V_OBE+V_cont is obtained by solving a corresponding LS 
equation in partial-wave projected form <cit.>,

    T^L” L'(p”,p';E)=V^L”L'(p”,p')+
    
          ∑_L∫_0^∞dpp^2/(2π)^3  V^L”L(p”,p)
    2μ/k^2-p^2+iηT^LL'(p,p';E) ,

where the tensor coupling between different orbital angular momenta 
L”, L' is taken into account. 
μ is the reduced mass, i.e. μ = M_1 M_2/(M_1+ M_2), with M_1, M_2 being the 
masses of the baryons in the intermediate state. The on-shell momentum in the intermediate 
state, k, and the kinetic energy E in the center-of-mass frame  
are given by the relation √(s) = E + M_1 + M_2 = √(M^2_1+k^2)+√(M^2_2+k^2). 
The scattering amplitude (<ref>) generally includes couplings between different ℬℬ
channels <cit.>. 
In the applications described in this section we consider systems without such
coupled channels, hence corresponding indices and summations in eq. (<ref>) have been omitted.

For the simulation of the lattice QCD results the respective baryon masses as given
in the corresponding publications are employed. These will be given in each case considered as we proceed. 

In the calculations at the physical point we use the following (isospin averaged) 
baryon masses: M_N=938.92 MeV, M_=1232 MeV, M_=1385 MeV, M_=1530 MeV, 
and M_=1672.45 MeV. The LS equation is solved in the isospin basis and the 
Coulomb interaction is ignored (as in the lattice QCD calculations).
 
The integral in the LS equation (<ref>) is divergent for the chiral potentials
specified above. A regularization needs to introduced. We utilize here the same prescription 
as in our YN studies, where the potentials in the LS equation are cut off in momentum space by multiplication with a regulator function, 
f(p',p) = exp[-(p'^4+p^4)/Λ^4], 
so that the high-momentum components of the baryon and pseudoscalar meson fields are removed. 
In the present study we employ the cut-off scales500 and 700 MeV, in line with the 
range considered in our LO study of the N and N interactions <cit.>. 
The variation of the results with the cutoff reflect uncertainties that will be indicated by bands. 
Clearly, a better error analysis based e.g. on the approach advocated in ref. <cit.>
should be done once more lattice data and/or higher order calculations are available.

At LO the only dependence of the potential V on the pion mass or on other meson masses comes from 
the meson propagators in eq. (<ref>). The SU(3) breaking manifested in 
the masses of the octet and/or decuplet baryons does not affect the potential itself 
at this order <cit.>.  
However, those masses enter the LS equation (<ref>) and, therefore, influence
the actual result for the reaction amplitude T.

It is important to take this effect into account, as argued in 
refs. <cit.>. After all, the binding energy of
a possible bound state results from a delicate interplay between the potential energy 
(that depends on the pion mass) and the kinetic energy in the baryon-baryon Green's
function (that is affected by the baryon masses). 
 
At next-to-leading order the contact terms as well as the coupling 
constants depend on the quark masses or, equivalently, on 
the pion mass. For details, we refer to 
refs. <cit.>, 
where the quark mass dependence of the NN interaction has been investigated;
see also ref. <cit.>. However, as mentioned, for the present limited applications it is sufficient to stay at the LO level of single pseudoscalar meson exchange plus contact term with a much restricted set of low-energy constants.ForNN 
scattering <cit.>
the pion-exchange contribution plays an essential role.
In contrast, for some of the systems considered here such as or N, the 
only contributor to pseudoscalar-meson exchange is the η meson. 
Variations of the pion mass (or the SU(3) breaking due to the 
small pion mass as compared to the K and η masses) are expected 
to be less important for such systems. 

In the present work we follow closely the strategy utilized in our study/analysis 
of lattice QCD simulations for the H-di­baryon <cit.>
and other possible bound states of two-body systems involving octet baryons in 
the sectors with strangeness S=-2, -3 and -4<cit.>.  
We aim at a reproduction of the phase shifts and/or scattering lengths from the
LQCD calculation within LO chiral EFT. Thereby we employ the masses of the 
pseudoscalar mesons and
the baryons corresponding to the lattice simulation in our calculation and 
fix the low-energy constant associated with the strength of the contact term
by a fit to the lattice data. 

Results at the physical point are then obtained replacing the masses
of the mesons in the potential by their physical values and correspondingly
those of the baryons in the LS equation. 



 §.§ ΔΔπ coupling constant


An essential ingredient of the calculation is the ΔΔπ coupling constant.
Its value, together with the imposed SU(3) symmetry relations, fixes the 
strengths of all contributions from pseudo­scalar-meson exchange to the DD interaction. 
Unfortunately, unlike the NNπ and NΔπ coupling constants, the value 
for f_ΔΔπ is not constrained by experimental information. 

A wide range of values for the ΔΔπ coupling constant can be found 
in the literature <cit.>. 
In a variety of calculations the quark model value<cit.>
is used which amounts to f_π = f_NNπ/5<cit.>
for the normalization of the spin- and isospin operators as defined in Appendix <ref>. 

Large N_c arguments lead to g_1 = 9g_A/5<cit.>, 
where g_1 is the π coupling constant commonly used in chiral 
perturbation theory. 
Taking into account the different normalization of the
spin- and isospin operators, this corresponds likewise to f_π = f_NNπ/5. 
The result in <cit.> using QCD sum rules is g_1 = 0.885± 0.15
based on eqs. (4) and (10) of that work, i.e. roughly half of the large N_c 
prediction. Recent lattice QCD calculations indicate values around g_1 ≈ 0.6 
(g_1 ≃ g_A/2) <cit.> for m_π≈ 300 MeV.  
In a new study of pion-nucleon scattering within chiral perturbation theory
up to third order that includes the Δ resonance explicitly <cit.> 
a value g_1 =1.21± 0.46 ± 0.39 was deduced. The large uncertainty 
indicates the difficulty to pin down the π coupling constant reliably 
from such a calculation.  See, however, ref. <cit.>.



 §.§ Partial waves and isospin factors


An overview of partial waves up to total angular momentum 
J=3 is provided in Table <ref>.
In general, denoting the states as |JMLS⟩, 
L being the orbital angular momentum and S the total spin,
the BD system (spin 1/2 ⊗ 3/2) can be in the following two sets 
of four states

    |JMJ1⟩,  |JMJ2⟩,  |JM(J± 2)2⟩; 
          |JM(J± 1)1⟩,  |JM(J± 1)2⟩,

which differ by parity and, therefore, do not couple. 
For DD (3/2 ⊗ 3/2) there are four such sets, 

    |JMJ0⟩,  |JMJ2⟩,  |JM(J± 2)2⟩; 
          |JMJ1⟩,  |JMJ3⟩,  |JM(J± 2)3⟩; 
          |JM(J± 1)1⟩,  |JM(J ± 3)3⟩,  |JM(J ± 1)3⟩; 
          |JM(J ± 1)2⟩,

where again there is no coupling between states with different parity. 
The other sets decouple too, as long as the interaction is given only
by pseudoscalar-meson exchange, eq. (<ref>). 



We focus here on (coupled) partial waves that involve 
S-wave states, i.e., where contributions from contact terms 
at leading order arise. The DD system can be in the following S-wave 
states: ^1S_0, ^3S_1, ^5S_2, ^7S_3. In channels with identical
particles the Pauli principle reduces the number of possible states.
For example, in the  system only the S waves ^1S_0 and ^5S_2 
are allowed. In case of ΔΔ the spin-space odd states (^1S_0, ^5S_2) 
can have total isospin I=1 and 3 while the spin-space even states 
(^3S_1, ^7S_3) can have total isospin I=0 and 2. 

There is no restriction from the Pauli principle for the BD system 
so that one has the S-wave states ^3S_1 and ^5S_2  
in all channels. 



In Table <ref> the isospin factors that enter into the evaluation
of the meson-exchange contribution to the DD potential are summarized. 
For strangeness S=-2 and -4 there are channels with non-identical 
and with identical particles which require special treatment. 

Specifically, a proper symmetrization is required which can be
achieved by introducing the flavor-exchange operator P_f, cf. 
the procedure applied in the analogous BB case with S=-2
(, Ξ N, etc.) described in ref. <cit.>. 



 §.§  and SU(3) related channels
The Paul principle implies that the only allowed S-wave states for  are
^1S_0 and ^5S_2. Moreover, the associated potentials depend only on a single 
LEC for each partial wave, namely the one corresponding to the SU(3) irreducible 
representation  28, see Table <ref>.

There are lattice QCD simulations for  scattering by two groups. 
The earlier one by Buchoff et al. <cit.> corresponds to a pion 
mass of m_π≈ 390 MeV and an  mass close to the physical value.  
It suggests a fairly weak and repulsive interaction in the ^1S_0 state
with a scattering length of a = (0.16± 0.22) fm in the ^1S_0 partial wave. 
The results of the HAL QCD collaboration, published soon after, turned
out to be qualitatively different <cit.>. 
That calculation suggests a strongly attractive interaction for the ^1S_0
channel, with phase shifts comparable to those of the ^1S_0 partial wave
in neutron-proton scattering. Indeed, taking into account the large
statistical errors, even an  bound state might be supported.
The lattice set-up in ref. <cit.> corresponds to the masses:
m_π = 701 MeV, m_K = 789 MeV, M_ = 1966 MeV, quite far from the physical point.

Results of our analysis of the HAL QCD results are presented in Fig. <ref> 
as a function of the kinetic energy in the center-of-mass frame. These are achieved 
by appropriately adjusting the LEC C^28, ^1S_0_22 (denoted simply by C^28
in the following discussion) to the lattice data for each cutoff. The variation of the 
results with the cutoffs Λ=500-700 MeV is indicated by uncertainty bands.

Fig. <ref>(a) shows a fit to the central HAL QCD prediction 
(hatched/blue band) while 
(b) is based on a fit to the upper limit given for the ^1S_0 phase shift. 
In this case an  bound state with binding energy around 1 MeV is produced.  
Since the η mass is not given in ref. <cit.>, we assume
that m_η≈ m_K. 

In principle, the GMO mass formula could have been used to fix m_η based on the 
values of m_K and m_π. However, we refrain from doing so in this illustrative study. 

Anyway, it should be said that variations of m_ηby20 or 30 MeV have very little influence on the actual
results and can be accommodated by a slight readjustment of C^28.   

The extrapolation to the physical point is indicated by dark (red)  
bands in
Fig. <ref>. Obviously, the interaction becomes less attractive 
and for neither of the two cases considered there is a bound state.

It should be noted that preliminary results for  corresponding to
a pion mass close to the physical point (m_π≈ 145 MeV) reported 
recently by the HAL QCD collaboration <cit.> 
suggest that there could be indeed a bound state, with a binding energy 
roughly comparable to that of the deuteron.  However, the corresponding 
mass of the  baryon is not specified so that it remains unclear how 
close the latter is to its physical value. 

In any case, it will be interesting to see the final result. 


The results in Fig. <ref> were obtained with a π coupling
constant corresponding to g_1 =1.5. In the course of fitting to the lattice 
predictions we varied g_1 and it turned out that values around 1.3-1.5 
allowed for the best reproduction of the energy dependence suggested by
the lattice calculation. For larger values, specifically for values
close to the large N_c prediction, g_1 =9 g_A/5 ≈ 2.3, we
observed a very strong energy dependence of the ^1S_0 phase shift
close to threshold that is not in line with the lattice data. 
It is caused by a dramatic increase in the coupling between the 
^1S_0 and ^5D_0 partial waves. The larger coupling constant
increases the strength of the tensor force due to η exchange 
so that the mixing angle becomes larger than the ^1S_0 phase shift
itself already at small energies. We consider such a scenario as not
realistic. But it should be noted at the same time that the conclusions of ref. <cit.>
are based on an effective purely central  potential, used for calculating the ^1S_0 phase shift. 
Accordingly, the role of the coupling to the ^5D_0 channel in the evaluation of the lattice
data remains unclear. 

Results based on the lattice calculation in ref. <cit.> are 
indicated by the gray (green) band in Fig. <ref>, where we
fixed the LEC by a fit to the scattering length a = 0.16 fm.
Since the relevant masses in that study are already fairly close to the 
physical point (M_ = 1632 MeV <cit.>, m_η = 587 MeV <cit.>) 
we show only the evaluation for physical masses. 
The obvious discrepancy between the 
predictions in refs. <cit.> and <cit.> cannot be 
resolved by the present study. 



Once the LEC C^28 is fixed by a fit to the  channel,
SU(3) symmetry can be exploited to obtain corresponding ^1S_0 results/predictions 
for  (I=3),  (I=5/2), and  (I=1/2). 
The interaction in all those cases is determined by the same LEC,
cf. Table <ref>, together with contributions from meson exchange.
And in all those cases there is no coupling to BB or BD channels. 

The corresponding phase shifts are presented in Figs. <ref>-<ref>.
They illustrate the amount of SU(3) breaking that arises in our calculation from the
mass differences between π, η and K, and between the decuplet baryons. 
Clearly, given that the  and  resonances are fairly broad, the  and  
phase shifts are interesting just for purely academic reasons. This is different for the  
because its width is only around 10 MeV, i.e. comparable to the one of the ω meson, 
so that one can view it practically as a stable particle.  

While in case of  scattering only the η-meson can contribute, 
η as well as K exchange is possible for , see Table <ref>.
However, given that m_K≈ m_η there is not much difference in the actual 
potentials. Since the reduced mass μ of the latter system is smaller we 
still expect that the attraction should be slightly weaker, cf. Fig. <ref>. 
Concerning  and  there is a sizable contribution from pion exchange
so that the corresponding potentials are more strongly influenced by the SU(3) breaking
due to the meson masses. At the same time the reduced masses for these systems
are noticeably smaller. Both effects together lead to interactions that are
appreciably less attractive, as can be seen from the phase shifts
in Figs. <ref> and <ref>.
The predictions based on the lattice result of ref. <cit.> 
(cf. green/gray bands) suggest a basically repulsive interaction for all
considered systems. Only in the  and  channels and for energies
close to the threshold does one observe slightly positive values for the ^1S_0 phase shift 
signalling a weakly attractive tail of the potential. 

In this context, let us mention that quark-model studies <cit.>, but also 
Faddeev-type calculations <cit.>, often suggest the existence of a  dibaryon 
with I=3. Recently, an experimental search for such a dibaryon state has been performed by the 
WASA-at-COSY collaboration in the reaction pp→ ppπ^+π^+π^-π^- <cit.>. 
However, no clear-cut evidence for such a state was found.




 §.§  with J=3 and I=0


Dibaryon candidates for the  state with J=3, I=0 are often 
mentioned together with the ones with J=0, I=3 discussed above. 

In particular, model studies point out a close connection between the  
states with I(J^P)=0(3^+) and its I(J^P)=3(0^+) mirror state   
<cit.>.

Indeed, at least with regard to the S-wave interactions the product of the 
expectation values for the spin and isospin operators is identical for the two 
channels so that the corresponding potential due to pion-exchange is the same. 
However, from Table <ref> one can see that the states actually belong 
to different SU(3) irreducible representations, so that the interactions
involve different LECs. 

Furthermore, for the state with J=3 there are many more coupled
partial waves, see Table <ref>. And finally, the  state with 
J=3, I=0 couples to the NN system. 

Nonetheless, the  state with J=3, I=0 is of special interest 
because, as discussed in the Introduction, the d^*(2380) dibaryon candidate 
seen by the WASA-at-COSY collaboration <cit.> could be a quasibound state
produced by this system. Indeed, the analysis of data on quasifree polarized 
np scattering presented in that paper suggests the d^*(2380) dibaryon to be 
a resonance in the ^3D_3-^3G_3NN partial wave. This state 
can couple to the  system with the partial waves 
^7S_3-^3D_3-^7D_3-^3G_3-^7G_3-^7I_3, see Table <ref>.

The position of the resonance pole given by the WASA-at-COSY collaboration is 
2380± 10 - i (40± 5) MeV. Since the nominal  threshold is at 2464 MeV
this would imply a binding energy in the order of 84± 10 MeV. 
Of course this naive binding energy assignment ignores the width associated with the Δ→π N decay channels.
For an overview of the rapidly growing literature on the d^*(2380) dibaryons
see for example ref. <cit.>.
 
Addressing the experimental results directly is beyond the scope of the
present study. Fortunately, there are also lattice QCD calculations for 
the  system in question by the HAL QCD collaboration to which we can 
connect. It should be said, however, that so far only preliminary results are 
available that have been presented at conferences <cit.>.
Accordingly, we emphasize that our considerations here have likewise
preliminary character.

The  lattice QCD calculation of the ^7S_3 partial wave with I=0
corresponds to a pion mass of m_π = 1015 MeV <cit.>. 
Unfortunately, the pertinent value for M_ is not given. Supposedly the 
lattice set-up with m_π = 1015 MeV corresponds to an SU(3) symmetric calculation 
where the octet baryon mass is M_B=2030 MeV <cit.>, and so we assume 
that using this mass is a meaningful option. 

The results of the HAL QCD collaboration for the ^7S_3 partial wave is shown in 
Fig. <ref> together with our fits. The LEC is fixed in such a way that 
the results at low energies are well reproduced. The hypothesis of a  bound 
state <cit.> is taken over and imposed in the fitting procedure. 
Again the cutoff values Λ=500 and 700 MeV have been adopted. 

Given that the lattice calculation is for a rather large pion mass, the phase shifts
are basically determined by the contact term in ^7S_3 alone. The binding 
energy implied by the fit is around 50-100 MeV. 

When physical masses are used the results for the two cutoffs show different trends, see 
Fig. <ref>. In one case, Λ=500 MeV (dash-dotted 
line), there is very little change and the binding energy is still around 75 MeV. 
This happens to be close to the d^*(2380) but this coincidence should of course not be overinterpreted. 
In the other case (solid line), 
the attraction is strongly reduced and only a fairly shallow bound state 
with a binding energy of about2.2 MeV survives. 

Lowering the  mass to its physical value weakens the attraction because
the reduced mass becomes smaller and along with it the contribution of the loop
integral in the LS equation (<ref>). 
At the same time, reducing the pion mass increases the tensor coupling to the
other partial waves, in particular, to the two D waves. Thereby, the 
attraction is increased so that there is a compensating effect. 
Obviously, the details of this compensation depends strongly on the cutoff 
so that no clear trend emerges. One has to keep in mind that the ^7S_3 state 
can couple to many other partial waves, see Table <ref>, which 
increases the sensitivity to the employed cutoff mass. 

A further complication is certainly the circumstance that the so far available lattice 
QCD calculation is for rather large masses. An extrapolation over a large mass region 
and, moreover, based on an LO calculation cannot be very reliable. Nevertheless, even for masses closer to the
physical point it will be a challenge to perform an accurate extrapolation
in view of the complicated angular-momentum structure of that state. 

Anyway, based on the present results one might still conclude that the existence 
of a quasibound  state with J=3, I=0 is at least not totally implausible. 
We should add that, given the very preliminary character of the  LQCD calculation,
we ignored here the coupling to NN (in the ^3D_3-^3G_3 partial wave, 
cf. Table <ref>) and the fact that the Δ has a sizable width, i.e. 
there is a coupling to the NNπ and NNππ continuum <cit.>. 
Both should have a significant influence on the location of the state and 
certainly on its width. 




 §.§ N with J=2

Finally, let us consider the N interaction in the ^5S_2
partial wave. Also for this case results from a lattice QCD
calculation by the HAL QCD collaboration are available <cit.>.
The lattice set-up corresponds to the masses:
M_ = 2105 MeV, M_N = 1806 MeV m_π = 875 MeV, m_K = 916 MeV 
(again we assume that m_η≈ m_K). 

The N system is interesting because it involves only stable particles
(stable against hadronic decay) so that, in principle, even scattering
experiments are feasible. However, unlike  discussed above, 
N can couple to various other BB and BD channels.
Specifically, the N^5S_2 system (with threshold at √(s)=2611 MeV) 
can couple to ΛΞ (^1D_2, threshold at 2434 MeV), 
ΣΞ (^1D_2, threshold at 2511 MeV) 
and ΛΞπ (^3P_2s, threshold at 2574 MeV). 
In addition, and more unfavorable for a concrete calculation, it couples
to ΛΞ^* (threshold at 2647 MeV), ΣΞ^* (threshold at 2725 MeV), 
and ΞΣ^* (threshold at 2703 MeV) in all partial waves, 
see Table <ref>. 
Thus all four LECs that contribute to 
BD scattering (cf. Table <ref>) are needed. There is no way to 
determine the individual values of those four LECs from the N results in 
ref. <cit.>. 

Indeed, the only “selective” LEC in BD scattering is C^35 which, once 
fixed, could be used to relate the interactions for N (I=2),  (I=5/2), 
and Ξ (I=1/2). 
Unfortunately, lattice results are not available for any of
these channels.

We focus in the following on the only BD lattice results available
and ignore all couplings of N to other channels. Then only one specific
combination of LECs enter, see Table <ref>, which can be fixed by 
a fit to the lattice predictions. Note that the 
interaction in the N system is particularly simple because, apart from the
contact interaction, only η-meson exchange can contribute. For other BD
reactions there are contributions from direct diagrams, involving BBϕ 
and DDϕ vertices, and from exchange diagrams that involve BDϕ and 
DBϕ vertices. 
 
Fits to the lattice calculation are presented in Fig. <ref>.
Again the standard cutoffsΛ=500 MeV and 700 MeV are
adopted. The lattice results of ref. <cit.> support the
existence of a bound state in the N system. The binding energy 
estimated from our fits is in the range9-11 MeV. When extrapolating
to the physical point we observe the same difficulty as already in the
 case above. There is no clear trend because the results depend
significantly on the cutoff. In one case (Λ=500 MeV, dash-dotted line) 
the bound state would survive, with a binding energy of around 2 MeV, while 
for the other cutoff (solid line) it disappears. Thus conclusions - even 
qualitative ones - are difficult to draw. 
This is even more the case if we recall the various other channels that
can couple to N and that are open at the physical point. In view of these
additional channels, it is to be expected that the dynamics in the N system
is highly complex. It will therefore
be challenging to establish the existence of a bound
state in this system from a lattice calculation. Actually, the 
situation is even more involved than the one for the H-dibaryon, where there is 
a delicate interplay between the interactions in the , Ξ N,
and  channels 
<cit.>. 



§ SUMMARY AND OUTLOOK


In this paper we have derived the general form of the baryon-baryon interaction 
involving octet and decuplet baryons in a chiral effective field theory approach 
based on the Weinberg power counting. The present work is an extension of earlier
studies on the nucleon-nucleon <cit.>, 
hyperon-nucleon <cit.>
and hyperon-hyperon <cit.> systems
within the same framework. Specific attention 
is paid to the connections between the interactions in the various baryon-baryon
channels that follow from the underlying (approximate) SU(3) flavor symmetry.
The leading-order potential presented in this paper consists of two components: 
i) long-ranged one-pseudo­scalar-meson exchanges (π, K, η)
with coupling constants at the various baryon-baryon-meson vertices
related via SU(3) symmetry; ii) short-ranged four-baryon contact terms without derivatives. 
For the latter the most general, minimal and SU(3) invariant Lagrangian has been derived.  

The number of independent contact terms at LO amounts to: six for the
scattering of two octet baryons (as already established in<cit.>),
eight for the scattering of two decuplet baryons, eight for 
the scattering of an octet baryon on a decuplet baryon, and  
two (each) for the transitions between these systems. 

The low-energy constants associated with those contact terms need to
be determined from scattering data. Given the lack of empirical information
on the scattering of decuplet baryons we have illustrated how
lattice QCD simulations <cit.> 
can be used to constrain or even fix some of the LECs. Admittedly, since the presently available 
lattice QCD calculations for scattering of decuplet baryons still involve large pion masses 
(in general m_π≈ 700 - 1000 MeV) and large baryon masses,
the considered extrapolations to the physical point have to be taken with a grain of salt.
Nonetheless, it is clear from the presented applications that chiral EFT 
provides a useful tool to analyze lattice results once calculations 
corresponding to masses closer to the physical point will become available.

Finally, since the  can decay only through weak interactions, in principle, actual experiments 
where  baryons are scattered on nucleons are feasible, analogous to 
those performed for the N and N systems. 
Indeed, there are plans for studying the N interaction experimentally 
at J-PARC <cit.>, where it is intended to produce
the  in the reaction K^-p → K^+ K^0. 
There is also a proposal to establish a secondary K^0_L beam at JLab 
that can then be used for producing s  <cit.>.
 baryons can also be produced in the reaction p̅p→, a case for
the P̅ANDA project at the FAIR facility 
in Darmstadt <cit.>. 

Furthermore, heavy-ion collisions could allow one to access information on the
N<cit.> but also the  interaction. 

Another and completely different field of application for our formalism might
be in studies of pion production in NN collisions. Here, an explicit inclusion 
of the direct N and  interactions could be a sensible next step for 
refining the treatment of the reaction NN → NNπ within 
chiral perturbation theory <cit.>.  
Furthermore, a better knowledge of the Δ N and ΔΔ
interaction could shed light on the so-called Δ puzzle in neutron
star matter, see e.g. ref. <cit.>.
 
 Acknowledgments:
We acknowledge stimulating discussions with Tom Luu. 
This work is supported in part by the DFG and the NSFC through
funds provided to the Sino-German CRC 110 “Symmetries and
the Emergence of Structure in QCD”. The work of UGM was also
supported by the Chinese Academy of Sciences (CAS) President's
International Fellowship Initiative (PIFI) (Grant No. 2017VMA025).





§ SPIN MATRICES


In this appendix we summarize the spin transition operators involving spin 1/2 and spin 3/2 states.
In order to construct these matrices (cf. ref. <cit.>), we use the Wigner-Eckart 
theorem, which relates matrix elements of a spherical tensor operator to a reduced matrix element 
and Clebsch-Gordon coefficients. For writing a Cartesian tensor (up to rank 3) in terms of spherical 
tensors, we express the usual scalar , vector x^i, irreducible tensor of rank two 
x^ix^j - r⃗^ 2δ^ij/3 and irreducible tensor of rank 
three 5x^ix^jx^k - (x^iδ^jk+x^jδ^ik+x^kδ^ij)r⃗^ 2 into spherical 
harmonics of the same rank.

Following this approach, we obtain for the vector spin matrices (see also refs. <cit.>)
[Note that in Wiringa et al. <cit.>S is defined as 4× 2
matrix, which is equivalent to S^† in the present paper.]:

    σ_1 =[ 0 1; 1 0 ] ,   σ_2 =[  0 -i;  i  0 ] ,   σ_3 =[  1  0;  0 -1 ] ,

    S_1 = [ -1/√(2)       0  1/√(6)       0;       0 -1/√(6)       0  1/√(2) ],S_2 = [ -i/√(2)       0 -i/√(6)       0;       0 -i/√(6)       0 -i/√(2) ],
       S_3 = [      0 √(2/3)      0      0;      0      0 √(2/3)      0 ],

    Σ_1 = [    0 √(3)    0    0; √(3)    0    2    0;    0    2    0 √(3);    0    0 √(3)    0 ],Σ_2 = [       0 -i √(3)       0       0;  i √(3)       0    -2 i       0;       0     2 i       0 -i √(3);       0       0  i √(3)       0 ],
       Σ_3 = [  3  0  0  0;  0  1  0  0;  0  0 -1  0;  0  0  0 -3 ].

The analogous isospin matrices corresponding to the spin matrices σ⃗, S⃗ 
and Σ⃗ are denoted by τ⃗, T⃗ and θ⃗.

The spin matrices with rank two and three can be expressed through vector and scalar spin matrices:

    S^ij =    -1/√(6)( σ^i S^j + σ^j S^i) ,S^ij† = -1/√(6)( S^i†σ^j + S^j†σ^i ) , 
    Σ^ij =   1/8( Σ^i Σ^j + Σ^j Σ^i - 10δ^ij)
    = δ^ij - 3/2( S^i†S^j + S^j†S^i ) , 
    Σ^ijk =   1/36√(3)( 5( Σ^i Σ^j Σ^k + Σ^k Σ^i Σ^j + Σ^j Σ^k Σ^i +Σ^i Σ^k Σ^j 
       + Σ^j Σ^i Σ^k + Σ^k Σ^j Σ^i)
    -82( Σ^i δ^jk + Σ^j δ^ik + Σ^k δ^ij ) ) .


§ TWO-BODY POTENTIALS IN PARTICLE REPRESENTATION


In this section, we show the two-body interaction potentials in particle basis for the transitions
BB→ BB, BB↔ DB, DB→ DB, BB↔ DD,
DB↔ DD, and DD→ DD.

First, we write the Lagrangians presented in sections <ref> and <ref> in terms of their particle fields, as defined in equations (<ref>), (<ref>) and (<ref>).
The physical fields comprise the sets:

    ϕ_i    ∈{π^0,π^+,π^-,K^+,K^-,K^0,K̅^0,η} ,
    
     B_i    ∈{n,p,Σ^0,Σ^+,Σ^-,Λ,Ξ^0,Ξ^-} ,
    
     B^*_i    ∈{Δ^-,Δ^0,Δ^+,Δ^++,Σ^*0,Σ^*+,Σ^*-,Ξ^*0,Ξ^*-,Ω^-} .


For the octet-baryon meson Lagrangian one obtains

    ℒ_BBϕ = -1/2f_0∑_i,j,k N_B_iB_jϕ_k (B̅_i σ⃗B_j) · (∇⃗ϕ_k)  ,

where the sum over i,j,k runs over all particles fields, specified in eq. (<ref>).
Furthermore, we have introduced the SU(3) factors N. 
These factors can be easily obtained by multiplying the flavor matrices and taking the traces.
For example N_ΛΛη can be calculated as (see eq. (<ref>))

    N_ΛΛη = D  tr(Y^⊤_Λ{ Y_η,Y_Λ})
    + F  tr( Y^⊤_Λ[Y_η,Y_Λ])  = -2D √(3) ,

with the flavor matrices

    Y_Λ :=
     [  1/√(6)       0       0;       0  1/√(6)       0;       0       0 -2/√(6) ] ,  
    Y_η :=
     [  1√(3)      0      0;      0  1√(3)      0;      0      0 -2√(3) ] .

We will use the representation in terms of SU(3) factors N in the following, as it allows for a simple 
presentation of the two-baryon potentials.

For the corresponding Lagrangian terms involving decuplet-baryons one obtains in the same way

    ℒ_DBϕ   = C/f_0∑_i,j,k N_B_i^*B_jϕ_k[
    (B̅_i^* S⃗^ † B_j) ·(∇⃗ϕ_k) 
                       +
    (B̅_j S⃗ B_i^*) ·(∇⃗ϕ_k^†)
    ]  ,

    ℒ_DDϕ = H/f_0∑_i,j,k N_B^*_iB^*_jϕ_k (B̅^*_i Σ⃗B^*_j) · (∇⃗ϕ_k)  .


The minimal set of contact Lagrangian terms of sec. <ref> can also be rewritten in terms of particle fields, with SU(3) factors N:

    ℒ_BBBB =      ∑_i,j,k,l N^f_B_iB_jB_kB_l(B̅_iB_j)(B̅_kB_l) 
       +∑_i,j,k,l N^f_B_iB_jB_kB_l(B̅_iσ⃗B_j)·(B̅_kσ⃗B_l)  .

    ℒ_DBBB = ∑_i,j,k,l N^f_B_i^*B_jB_kB_l[
       (B̅_i^*S⃗^† B_j)·(B̅_kσ⃗B_l) 
       +(B̅_jS⃗  B_i^*)·(B̅_lσ⃗B_k)
    ]  .

    ℒ_DBDB = +    ∑_i,j,k,l N^f_B^*_iB^*_jB_kB_l(B̅^*_i B^*_j)(B̅_k B_l) 
    
    +   ∑_i,j,k,l N^f_B^*_iB^*_jB_kB_l(B̅^*_iΣ⃗B^*_j)·(B̅_kσ⃗B_l)
     .

    ℒ_DDBB = +    ∑_i,j,k,l N^1_B^*_iB_jB^*_kB_l[
    (B̅^*_iS⃗^† B_j)·(B̅^*_kS⃗^† B_l) 
                     +
    (B̅_jS⃗ B^*_i)·(B̅_lS⃗ B^*_k)
    ] 
    
    +     ∑_i,j,k,l N^2_B^*_iB_jB^*_kB_l[
    (B̅^*_iS^αβ† B_j)(B̅^*_kS^αβ† B_l) 
                     +
    (B̅_jS^αβ B^*_i)(B̅_lS^αβ B^*_k)
    ]  .

    ℒ_DDDB = +    ∑_i,j,k,l N^1_B^*_iB^*_jB^*_kB_l[
    (B̅^*_iΣ⃗B^*_j)·(B̅^*_kS⃗^† B_l)
                     +
    (B̅^*_jΣ⃗B^*_i)·(B̅_lS⃗ B^*_k)
    ] 
    
    +    ∑_i,j,k,l N^2_B^*_iB^*_jB^*_kB_l[
    (B̅^*_iΣ^αβ B^*_j)(B̅^*_k S^αβ† B_l)
                     +
    (B̅^*_jΣ^αβ B^*_i)(B̅_l S^αβ B^*_k)
    ]  .

    ℒ_DDDD = +    ∑_i,j,k,l N^f_B^*_iB^*_jB^*_kB^*_l(B̅^*_iB^*_j)(B̅^*_kB^*_l) 
    
    +   ∑_i,j,k,l N^f_B^*_iB^*_jB^*_kB^*_l(B̅^*_iΣ⃗B^*_j)·(B̅^*_kΣ⃗B^*_l) 
    
    +   ∑_i,j,k,l N^f_B^*_iB^*_jB^*_kB^*_l(B̅^*_i Σ^αβ B^*_j)(B̅^*_kΣ^αβ B^*_l) 
    
    +   ∑_i,j,k,l N^f_B^*_iB^*_jB^*_kB^*_l(B̅^*_iΣ^αβγ B^*_j)(B̅^*_kΣ^αβγ B^*_l)
     .




Let us now come to the calculation of the two-body interaction potentials
based on the Lagrangian in particle basis as shown above.
The calculation is done in the center-of-mass frame with momentum assignments A(p⃗ )B(-p⃗ )→ C(p⃗^ ')D(-p⃗^ ').
We use the common definitions q⃗ = p⃗^ ' - p⃗ and k⃗ = p⃗^ ' + p⃗, where q⃗ appears in the direct one-meson exchange and k⃗ appears in the exchanged one-meson exchange[
We choose the conventions [B_i,B^*_j]_+ = B_iB^*_j + B^*_jB_i = 0 (and [B_i,B_j]_+ = [B^*_i,B^*_j]_+ = 0), i.e., all (octet and decuplet) baryons anti-commute.].

For all considered transitions we show in the first line, how the potential is calculated from Feynman diagrams.
In the Feynman diagrams themselves, particles that are vertically above each other are defined to be in the same baryon bilinear.
Octet baryons are denoted by single lines, decuplet baryons by double lines, and mesons by dashed lines.

Below the Feynman diagrams, we provide the full potentials in particle basis, for a general assignment of baryons B_i, decuplet baryons B_i^* and mesons ϕ.
For a clearer presentation, we introduce prefactors X_i, which are linear combinations of SU(3) coefficients N and low-energy constants.
The factor X_1 is always related to the direct one-meson exchange, X_2 is related to the exchanged one-meson exchange and the remaining X_i concern contact interaction with various spin-structures.
The X_i for the considered interactions are summarized in tables <ref> and <ref>.

For the construction of the potentials we also need various spin exchange operators, defined by 
P^(σ)|χ_1,χ_2⟩ = |χ_2,χ_1⟩,
where the first position in |.,.⟩ is spin space 1 (or baryon bilinear 1) and the second position in |.,.⟩ is in spin space 2 (or baryon bilinear 2).
For two spin-1/2 states the well-known spin exchange operator is given by

    = ^† = 1/2 (  + σ⃗_1·σ⃗_2 )  .

For two spin-3/2 states, the spin exchange operator can be expressed through

    = ^† =
    1/4 +
    1/20Σ^i_1Σ^i_2 +
    1/6Σ_1^ijΣ_2^ij +
    3/200Σ_1^ijkΣ_2^ijk .

The spin exchange operator exchanging a spin-1/2 state and a spin-3/2 state is given by[
The spin exchange operator fulfills |1/2,3/2⟩=|3/2,1/2⟩ and |3/2,1/2⟩=|1/2,3/2⟩.
]
    = 3/4( S⃗_1^ †·S⃗_2 + S_1^† ij S_2^ij)  .




Let us now list all considered transitions and their potentials, starting with the well-known potential 
involving only octet baryons, and with an increasing number of involved decuplet baryons.

  * B_1B_2→ B_3B_4 (with bilinears in spin space B_1-B_3 and B_2-B_4):

    V =   B_1B_2B_3B_4BBBBome - ·B_1B_2B_4B_3BBBBome +
    B_1B_2B_3B_4BBBBct - ·B_1B_2B_4B_3BBBBct
    
    =    
    -X_1/4f^2_0σ⃗_1·q⃗ σ⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2/4f^2_0σ⃗_1·k⃗ σ⃗_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3  + X_4 σ⃗_1·σ⃗_2  ,

  * B_1B_2→ B_3^*B_4: 

    V =   B_1B_2B_3^*B_4BBDBome - ·B_1B_2B_4B_3^*BBBDome +
    B_1B_2B_3^*B_4BBDBct - ·B_1B_2B_4B_3^*BBBDct
    
    =   C X_1/2f^2_0S⃗^ †_1·q⃗ σ⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    -C X_2/2f^2_0σ⃗_1·k⃗ S⃗^ †_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3 S⃗^ †_1 ·σ⃗_2  ,

  * B_1^*B_2→ B_3B_4: 

    V =   B^*_1B_2B_3B_4DBBBome - ·B^*_1B_2B_4B_3DBBBome +
    B^*_1B_2B_3B_4DBBBct - ·B^*_1B_2B_4B_3DBBBct
    
    =   C X_1/2f^2_0S⃗_1·q⃗ σ⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    -C X_2/2f^2_0S⃗_1·k⃗ σ⃗_2·k⃗/k⃗^ 2+m^2_ϕ
    + X_3 S⃗_1 ·σ⃗_2  ,

  * B_1^*B_2→ B^*_3B_4: 

    V    = 
    B^*_1B_2B_3^*B_4DBDBome - ·B^*_1B_2B_4B_3^*DBBDome +
    B^*_1B_2B_3^*B_4DBDBct - ·B^*_1B_2B_4B_3^*DBBDct
    
    =   X_1 H/2f^2_0Σ⃗_1·q⃗ σ⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2 C^2/f^2_0S⃗_1·k⃗ S⃗^ †_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3  + X_4 Σ⃗_1·σ⃗_2  ,

  * B_1B_2→ B^*_3B^*_4: 

    V =   B_1B_2B_3^*B^*_4BBDDome - ·B_1B_2B^*_4B_3^*BBDDome +
    B_1B_2B_3^*B^*_4BBDDct - ·B_1B_2B^*_4B_3^*BBDDct
    
    =   
    -X_1 C^2/f^2_0S⃗^ †_1·q⃗ S⃗^ †_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2 C^2/f^2_0S⃗^ †_1·k⃗ S⃗^ †_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3 S⃗^†_1 ·S⃗^†_2 + X_4 S^αβ†_1 S^αβ†_2  ,

  * B^*_1B^*_2→ B_3B_4: 

    V =   B^*_1B^*_2B_3B_4DDBBome - ·B^*_1B^*_2B_4B_3DDBBome +
    B^*_1B^*_2B_3B_4DDBBct - ·B^*_1B^*_2B_4B_3DDBBct
    
    =    
    -X_1C^2/f^2_0S⃗_1·q⃗ S⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2C^2/f^2_0S⃗_1·k⃗ S⃗_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3 S⃗_1 ·S⃗_2 + X_4 S^αβ_1 S^αβ_2  ,

  * B^*_1B_2→ B^*_3B^*_4: 

    V =   B^*_1B_2B_3^*B^*_4DBDDome - ·B^*_1B_2B^*_4B_3^*DBDDome +
    B^*_1B_2B_3^*B^*_4DBDDct - ·B^*_1B_2B^*_4B_3^*DBDDct
    
    =    
    -X_1C H/f^2_0Σ⃗_1·q⃗ S⃗^ †_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2C H/f^2_0Σ⃗_1·k⃗ S⃗^ †_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3 Σ⃗_1·S⃗_2^† + X_4 Σ_1^αβS_2^αβ† ,

  * B^*_1B^*_2→ B^*_3B_4: 

    V =   B^*_1B^*_2B_3^*B_4DDDBome - ·B^*_1B^*_2B_4B_3^*DDBDome +
    B^*_1B^*_2B_3^*B_4DDDBct - ·B^*_1B^*_2B_4B_3^*DDBDct
    
    =    
    -X_1C H/f^2_0Σ⃗_1·q⃗ S⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2C H/f^2_0S⃗_1·k⃗ Σ⃗_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3 Σ⃗_1·S⃗_2 + X_4 Σ_1^αβS_2^αβ ,

  * B^*_1B^*_2→ B^*_3B^*_4: 

    V =   B^*_1B^*_2B_3^*B^*_4DDDDome - ·B^*_1B^*_2B^*_4B_3^*DDDDome +
    B^*_1B^*_2B_3^*B^*_4DDDDct - ·B^*_1B^*_2B^*_4B_3^*DDDDct
    
    =   
    -X_1H^2/f^2_0Σ⃗_1·q⃗ Σ⃗_2·q⃗/q⃗^ 2+m^2_ϕ
    +X_2H^2/f^2_0Σ⃗_1·k⃗ Σ⃗_2·k⃗/k⃗^ 2+m^2_ϕ
       + X_3  + X_4 Σ⃗_1·Σ⃗_2
    + X_5 Σ_1^αβΣ_2^αβ + X_6 Σ_1^αβγΣ_2^αβγ ,
f>c<F>p8.cm<