sign in sign up
<<
Home , Baryon, Hyperon, Quantum chromodynamics, Strong interaction

-nuclear from SU(3) chiral effective field theory

Stefan Petschauer,1, 2, ∗ Johann Haidenbauer,3 Norbert Kaiser,2 Ulf-G. Meißner,4, 3, 5 and Wolfram Weise2 1TNG Technology Consulting GmbH, D-85774 Unterföhring, Germany 2Physik Department, Technische Universität München, D-85747 Garching, Germany 3Institute for Advanced Simulation and Jülich Center for , Institut für Kernphysik, Forschungszentrum Jülich, D-52425 Jülich, Germany 4Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for , Universität Bonn, D-53115 Bonn, Germany 5Tbilisi State University, 0186 Tbilisi, Georgia (Dated: March 17, 2020) The between and has a wide range of applications in and is a topic of continuing great interest. These interactions are not only important for hyperon- scattering but also essential as basic input to studies of hyperon-nuclear few- and many-body systems including hypernuclei and . We review the systematic derivation and construction of such baryonic from the symmetries of chromodynam- ics within non-relativistic SU(3) chiral effective field theory. Several applications of the resulting potentials are presented for topics of current interest in strangeness nuclear physics.

PACS numbers: 12.39.Fe 13.75.Ev 14.20.Jn 21.30.-x 21.65.+f Keywords: chiral Lagrangian, effective field theory, hyperon-nucleon interaction, flavor SU(3) , strangeness nuclear physics

CONTENTS VI. Conclusions 32

I. Introduction1 Acknowledgments 33

II. SU(3) chiral effective field theory3 References 33 A. Low-energy quantum chromodynamics3 B. Explicit degrees of freedom5 C. Construction of the chiral Lagrangian7 I. INTRODUCTION 1. Leading-order Lagrangian8 2. Leading-order meson- interaction Strangeness nuclear physics is an important topic of Lagrangian9 ongoing research, addressing for example scattering of D. Weinberg power counting scheme9 including strangeness, properties of hypernuclei or strangeness in infinite and in neutron III. Baryon-baryon interaction potentials 10 star matter. The theoretical foundation for such investi- A. Baryon-baryon contact terms 10 gations are interaction potentials between nucleons and B. One- and two-meson-exchange contributions 12 strange baryons such as the Λ hyperon. C. Meson-exchange models 16 Nuclear many-body systems are (mainly) governed by the , described at the fundamental level IV. Three-baryon interaction potentials 17 by (QCD). The elementary A. Contact interaction 17 degrees of freedom of QCD are and . How- B. One-meson exchange component 18 ever, in the low-energy regime of QCD quarks and glu- C. Two-meson exchange component 20 ons are confined into colorless . This is the re- arXiv:2002.00424v2 [nucl-th] 14 Mar 2020 D. ΛNN three-baryon potentials 21 gion where (hyper-)nuclear systems are formed. In this E. Three-baryon through decuplet region QCD cannot be solved in a perturbative way. Lat- saturation 21 tice QCD is approaching this problem via large-scale nu- F. Effective in-medium two-baryon interaction 24 merical simulations: the (Euclidean) space-time is dis- cretized and QCD is solved on a finite grid [1–4]. Since V. Applications 25 the seminal work of Weinberg [5,6] chiral effective field A. Hyperon-nucleon and hyperon-hyperon theory (χEFT) has become a powerful tool for calculating scattering 25 systematically the strong interaction dynamics for low- B. Hyperons in nuclear matter 28 energy hadronic processes [7–9]. Chiral EFT employs C. Hypernuclei and hyperons in neutron 31 the same symmetries and patterns at low-energies as QCD, but it uses the proper degrees of freedom, namely hadrons instead of quarks and gluons. In combination with an appropriate expansion in small ∗ [email protected] external momenta, the results can be improved system- 2 atically, by going to higher order in the power counting, or chiral interactions, in order to calculate the proper- and at the same time theoretical errors can be estimated. ties of nuclear systems with and without strangeness. Furthermore, two- and three-baryon forces can be con- For example, systems with three or four can structed in a consistent fashion. The unresolved short- be reliably treated by Faddeev-Yakubovsky theory [46– distance dynamics is encoded in χEFT in contact terms, 49], somewhat heavier (hyper)nuclei with approaches like with a priori unknown low-energy constants (LECs). the no-core-shell model [50–55]. In the nucleonic sector The NN interaction is empirically known to very high many-body approaches such as Quantum Monte Carlo precision. Corresponding two-nucleon potentials have calculations [56–58], or nuclear lattice simulations [59– been derived to high accuracy in phenomenological ap- 61] have been successfully applied and can be extended proaches [10–12]. Nowadays the systematic theory to to the strangeness sector. Furthermore, nuclear mat- construct nuclear forces is χEFT [13, 14]. (Note however ter is well described by many-body perturbation theory that there are still debates about the Weinberg power with chiral low-momentum interactions [62–64]. Con- counting schemes and how it is employed in practice [15– cerning Λ and Σ hyperons in nuclear matter, specific 17].) In contrast, the YN interaction is presently not long-range processes related to two- exchange be- known in such detail. The scarce experimental data tween hyperons and nucleons in the nuclear medium have (about 35 data points for low-energy total cross sections) been studied in Refs. [65, 66]. Conventional Brueckner do not allow for a unique determination of the hyperon- theory [67–69] at first order in the hole-line expansion, nucleon interaction. The limited accuracy of the YN the so-called Bruecker-Hartree-Fock approximation, has scattering data does not permit a unique phase shift anal- been widely applied to calculations of hypernuclear mat- ysis. However, at experimental facilities such as J-PARC ter [20, 24, 70, 71] employing phenomenological two-body in Japan or later at FAIR in Germany, a significant potentials. This approach is also used in investigations amount of beam time will be devoted to strangeness nu- of matter [72–74]. Recently, corresponding clear physics. Various phenomenological approaches have calculations of the properties of hyperons in nuclear mat- been employed to describe the YN interaction, in partic- ter have been also performed with chiral YN interaction ular -exchange models [18–23] or models [24– potentials [37, 75, 76]. 26]. However, given the poor experimental data base, Employing the high precision NN interactions de- these interactions differ considerably from each other. scribed above, even "simple" nuclear systems such as Obviously there is a need for a more systematic inves- triton cannot be described satisfactorily with two-body tigation based on the underlying theory of the strong interactions alone. The introduction of three-nucleon interaction, QCD. Some aspects of YN scattering and forces (3NF) substantially improves this situation [77–80] hyperon mass shifts in nuclear matter using EFT meth- and also in the context of infinite nuclear matter 3NF are ods have been covered in Refs. [27, 28]. The YN in- essential to achieve saturation of nuclear matter. These teraction has been investigated at leading order (LO) 3NF are introduced either phenomenologically, such as in SU(3) χEFT [29–31] by extending the very success- the families of Tuscon-Melbourne [81, 82], Brazilian [83] ful χEFT framework for the nucleonic sector [13, 14] to or Urbana-Illinois [84, 85] 3NF, or constructed accord- the strangeness sector. This work has been extended to ing to the basic principles of χEFT [78, 86–94]. Within next-to-leading order (NLO) in Refs. [32–34] where an ex- an EFT approach, 3NF arise naturally and consistently cellent description of the strangeness −1 sector has been together with two-nucleon forces. Chiral three-nucleon achieved, comparable to most advanced phenomenolog- forces are important in order to get saturation of nu- ical hyperon-nucleon interaction models. An extension clear matter from chiral low-momentum two-body inter- to systems with more strangeness has been done in actions treated in many-body perturbation theory [63]. Refs. [35–37]. Systems including decuplet baryons have In the strangeness sectors the situation is similar: Three- been investigated in Ref. [38] at leading order in non- baryon forces (3BF), especially the ΛNN interaction, relativistic χEFT. Recently calculations within leading seem to be important for a satisfactorily description of order covariant χEFT have been performed for YN in- hypernuclei and hypernuclear matter [58, 95–103]. Espe- teractions in the strangeness sector [39–43] with compa- cially in the context of neutron stars, 3BF are frequently rable results, see also Ref. [44]. It is worth to briefly dis- discussed. The observation of two-solar-mass neutron cuss the differences between the covariant and the heavy- stars [104, 105] sets strong constraints on the stiffness baryon approach. In the latter, due to the expansion in of the equation-of-state (EoS) of dense baryonic matter the inverse of the baryon masses, some terms are rel- [106–110]. The analysis of recently observed gravitational egated to higher orders. Also, it can happen that the wave signals from a two merging neutron stars [111, 112] analytic structure is distorted in the strict heavy-baryon provides further conditions, by constraining the tidal de- limit. This can easily be remedied by including the ki- formability of neutron star matter. netic energy term in the baryon [45]. In what A naive introduction of Λ-hyperons as an additional follows, we will present results based on the heavy-baryon baryonic degree of freedom would soften the EoS such approach. that it is not possible to stabilize a two-solar-mass neu- Numerous advanced few- and many-body techniques tron star against gravitational collapse [113]. To solve have been developed to employ such phenomenological this so-called hyperon puzzle, several ad-hoc mechanisms 3 have so far been invoked, e.g., through turbative approach is no longer possible. This is the re- exchange [114, 115], multi- exchange [116] or gion of non-perturbative QCD, in which we are interested a suitably adjusted repulsive ΛNN three-body interac- in. Several strategies to approach this regime have been tion [117–119]. Clearly, a more systematic approach developed, such as lattice simulations, Dyson-Schwinger to the three-baryon interaction within χEFT is needed, equations, QCD sum rules or chiral perturbation theory. to estimate whether the 3BF can provide the neces- The second important feature of QCD is the so-called sary repulsion and thus keep the equation-of-state suf- color confinement: isolated quarks and gluons are not ficiently stiff. A first step in this direction was done observed in nature, but only color-singlet objects. These in Ref. [120], where the leading 3BFs have been derived color-neutral particles, the hadrons, are the active de- within SU(3) χEFT. The corresponding low-energy con- grees of freedom in χEFT. stants have been estimated by decuplet saturation in But already before QCD was established, the ideas Ref. [121]. The effect of these estimated 3BF has been of an effective field theory were used in the context of investigated in Refs. [121, 122]. the strong interaction. In the sixties the Ward iden- In this review article we present, on a basic level, the tities related to spontaneously broken chiral symmetry emergence of nuclear interactions in the strangeness sec- were explored by using methods, e.g., tor from the perspective of (heavy-baryon) chiral effective by Adler and Dashen [123]. The -theoretical foun- field theory. After a brief introduction to SU(3) χEFT dations for constructing phenomenological Lagrangians in Sec.II, we present how the interaction between hy- in the presence of spontaneous symmetry breaking have perons and nucleons is derived at NLO from these ba- been developed by Weinberg [5] and Callan, Coleman, sic principles for two-baryon interactions (Sec.III) and Wess and Zumino [124, 125]. With Weinberg’s seminal for three-baryon interactions (Sec.IV). In Sec.V appli- paper [6] it became clear how to systematically construct cations of these potentials are briefly reviewed for YN an EFT and generate loop corrections to tree level re- scattering, infinite nuclear matter, hypernuclei and neu- sults. This method was improved later by Gasser and tron star matter. Leutwyler [7, 126]. A systematic introduction of nucle- ons as degrees of freedom was done by Gasser, Sainio and Svarc [8]. They showed that a fully relativistic treatment II. SU(3) CHIRAL EFFECTIVE THEORY of nucleons is problematic, as the nucleon mass does not vanish in the chiral limit and thus adds an extra scale. An effective field theory (EFT) is a low-energy approx- A solution for this problem was proposed by Jenkins and imation to a more fundamental theory. Physical quan- Manohar [127] by considering baryons as heavy static tities can be calculated in terms of a low-energy expan- sources. This approach was further developed using a sion in powers of small energies and momenta over some systematic path-integral framework in Ref. [128]. The characteristic large scale. The basic idea of an EFT is to nucleon-nucleon interaction and related topics were con- include the relevant degrees of freedom explicitly, while sidered by Weinberg in Ref. [86]. Nowadays χEFT is heavier (frozen) degrees of freedom are integrated out. used as a powerful tool for calculating systematically the An effective Lagrangian is obtained by constructing the strong interaction dynamics of hadronic processes, such most general Lagrangian including the active degrees of as the accurate description of nuclear forces [13, 14]. freedom, that is consistent with the symmetries of the un- In this section, we give a short introduction to the un- derlying fundamental theory [6]. At a given order in the derlying symmetries of QCD and their breaking pattern. expansion, the theory is characterized by a finite num- The basic concepts of χEFT are explained, especially the ber of constants, called low-energy constants explicit degrees of freedom and the connection to the (LECs). The LECs encode the unresolved short-distance symmetries of QCD. We state in more detail how the chi- dynamics and furthermore allow for an order-by-order ral Lagrangian can be constructed from basic principles. of the theory. These constants are a pri- However, it is beyond the scope of this work to give a de- ori unknown, but once determined from one experiment tailed introduction to χEFT and QCD. Rather we will or from the underlying theory, predictions for physical introduce only the concepts necessary for the derivation observables can be made. However, due to the low-energy of hyperon-nuclear forces. We follow Refs. [9, 13, 14, 129– expansion and the truncation of degrees of freedom, an 131] and refer the reader for more details to these refer- EFT has only a limited range of validity. ences (and references therein). The underlying theory of chiral effective field theory is quantum chromodynamics. QCD is characterized by two important properties. For high energies the (running) A. Low-energy quantum chromodynamics coupling strength of QCD becomes weak, hence a per- turbative approach in the high-energy regime of QCD is possible. This famous feature is called asymptotic free- Let us start the discussion with the QCD Lagrangian dom of QCD and originates from the non-Abelian struc- X  1 µν ture of QCD. However, at low energies and momenta LQCD = q¯f iD/ − mf qf − Gµν,aG , (1) 4 a the coupling strength of QCD is of order one, and a per- f=u,d,s,c,b,t 4 with the six quark flavors f and the gluonic field-strength with independent unitary 3 × 3 matrices L and acting 0 tensor Gµν,a(x). The gauge covariant derivative is de- in flavor space. This means that LQCD possesses (at the 1 a λa a fined by Dµ = ∂µ − igAµ 2 , where Aµ(x) are the classical, unquantized level) a global U(3)L ×U(3)R sym- fields and λa the Gell-Mann matrices. The QCD La- metry, isomorphic to a global SU(3)L ×U(1)L ×SU(3)R × grangian is symmetric under the local color gauge sym- U(1)R symmetry. U(1)L × U(1)R are often rewritten into metry, under global Lorentz transformations, and the dis- a vector and an axial-vector part U(1)V × U(1)A, named crete symmetries , conjugation and time re- after the transformation behavior of the corresponding versal. In the following we will introduce the so-called conserved currents under parity transformation. The chiral symmetry, an approximate global continuous sym- flavor-singlet vector current originates from rotations of metry of the QCD Lagrangian. The chiral symmetry is the left- and right-handed quark fields with the same essential for chiral effective field theory. In view of the phase (“V = L + R ”) and the corresponding conserved application to low energies, we divide the quarks into charge is the . After , the con- three quarks u, d, s and three heavy quarks c, b, t, servation of the flavor-singlet axial vector current, with since the quark masses fulfill a hierarchical ordering: transformations of left- an right-handed quark fields with opposite phase (“A = L − R ”), gets broken due to the mu, md, ms  1 GeV ≤ mc, mb, mt . (2) so-called Adler-Bell-Jackiw [132, 133]. The sym- metry group SU(3)L × SU(3)R refers to the chiral sym- At energies and momenta well below 1 GeV, the heavy metry. Similarly the conserved currents can be rewritten quarks can be treated effectively as static. Therefore, into flavor-octet vector and flavor-octet axial-vector cur- the light quarks are the only active degrees of freedom rents, where the vector currents correspond to the diag- of QCD for the low-energy region we are interested in. onal subgroup SU(3)V of SU(3)L × SU(3)R with L = R. In the following we approximate the QCD Lagrangian After the introduction of small non-vanishing quark by using only the three light quarks. Compared to masses, the quark mass term of the QCD Lagrangian (1) characteristic hadronic scales, such as the nucleon mass can be expressed as (MN ≈ 939 MeV), the light quark masses are small. Therefore, a good starting point for our discussion of low- LM = −qMq¯ = − (¯qRMqL +q ¯LMqR) , (8) energy QCD are massless quarks mu = md = ms = 0, which is referred to as the chiral limit. The QCD La- with the diagonal quark mass M = grangian becomes in the chiral limit diag(mu, md, ms). Left- and right-handed quark fields are mixed in LM and the chiral symmetry is 0 X 1 µν explicitly broken. The baryon number is still conserved, L = q¯f iDq/ f − Gµν,aG . (3) QCD 4 a but the flavor-octet vector and axial-vector currents f=u,d,s are no longer conserved. The axial-vector current is Now each quark field qf (x) is decomposed into its chiral not conserved for any small quark masses. However, components the flavor-octet vector current remains conserved, if the quark masses are equal, mu = md = ms, referred to as qf,L = PL qf , qf,R = PR qf . (4) the (flavor) SU(3) limit. Another crucial aspect of QCD is the so-called spon- using the left- and right-handed projection operators taneous . The chiral symme- 1 1 try of the Lagrangian is not a symmetry of the ground PL = (1 − γ5) ,PR = (1 + γ5) , (5) state of the system, the QCD . The structure 2 2 of the hadron spectrum allows to conclude that the chi- with the matrix γ5. These projectors are ral symmetry SU(3)L × SU(3)R is spontaneously broken called left- and right-handed since in the chiral limit to its vectorial subgroup SU(3)V, the so-called Nambu- they project the free quark fields on helicity eigenstates, Goldstone realization of the chiral symmetry. The spon- hˆ qL,R = ± qL,R, with hˆ = ~σ · ~p/ |~p |. For massless free taneous breaking of chiral symmetry can be characterized helicity is equal to chirality. by a non-vanishing chiral quark condensate hqq¯ i= 6 0, i.e., Collecting the three quark-flavor fields q = (qu, qd, qs) the vacuum involves strong correlations of scalar quark- and equivalently for the left and right handed compo- antiquark pairs. nents, we can express the QCD Lagrangian in the chiral The eight Goldstone corresponding to the spon- limit as taneous symmetry breaking of the chiral symmetry are identified with the eight lightest hadrons, the pseu- 0 1 µν ± 0 ± 0 ¯ 0 LQCD =q ¯RiDq/ R +q ¯LiDq/ L − Gµν,aGa . (6) doscalar (π , π ,K ,K , K , η). They are pseu- 4 doscalar particles, due to the parity transformation be- Obviously the right- and left-handed components of the havior of the flavor-octet axial-vector currents. The ex- massless quarks are separated. The Lagrangian is invari- plicit chiral symmetry breaking due to non-vanishing ant under a global transformation quark masses to non-zero masses of the pseu- doscalar mesons. However, there is a substantial mass qL → L qL , qR → R qR , (7) gap, between the masses of the pseudoscalar mesons and 5 the lightest hadrons of the remaining hadronic spectrum. vµ aµ s p For non-vanishing but equal quark masses, SU(3)V re- µ ν µ ν PP ν v −P ν a s −p mains a symmetry of the . In this context C −vµ> aµ> s> p> SU(3)V is often called the flavor group SU(3), which pro- vides the basis for the classification of low-lying hadrons Table I: Transformation properties of the external fields in . In the following we will consider the so- under parity and charge conjugation. For P a change of called symmetric limit, with m = m =6 m . u d s the spatial arguments (t, ~x ) → (t, −~x ) is implied and The remaining symmetry is the SU(2) isospin symmetry. we defined the matrix P µ = diag(+1, −1, −1, −1). An essential feature of low-energy QCD is, that the pseu- ν doscalar mesons interact weakly at low energies. This is a direct consequence of their Goldstone-boson nature. This feature allows for the construction of a low-energy effec- Another central aspect of the external-field method tive field theory enabling a systematic expansion in small is the addition of terms to the three-flavor QCD La- 0 momenta and quark masses. grangian in the chiral limit, LQCD. Non-vanishing cur- Let us introduce one more tool for the systematic de- rent quark masses and therefore the explicit breaking of velopment of χEFT called the external-field method. The chiral symmetry can be included by setting the scalar chiral symmetry gives rise to so-called chiral Ward iden- field equal to the quark mass matrix, s(x) = M = tities: relations between the divergence of Green func- diag (mu, md, ms). Similarly electroweak interactions can tions that include a symmetry current (vector or axial- be introduced through appropriate external vector and vector currents) to linear combinations of Green func- axial vector fields. This feature is important, to system- tions. Even if the symmetry is explicitly broken, Ward atically include explicit chiral symmetry breaking or cou- identities related to the symmetry breaking term exist. plings to electroweak gauge fields into the chiral effective The chiral Ward identities do not rely on perturbation Lagrangian. theory, but are also valid in the non-perturbative region of QCD. The external-field method is an elegant way to formally combine all chiral Ward identities in terms of in- B. Explicit degrees of freedom variance properties of a generating functional. Following the procedure of Gasser and Leutwyler [7, 126] we in- In the low-energy regime of QCD, hadrons are the ob- troduce (color neutral) external fields, s(x), p(x), vµ(x), servable states. The active degrees of freedom of χEFT aµ(x), of the form of Hermitian 3×3 matrices that couple are identified as the pseudoscalar Goldstone-boson octet. to scalar, pseudoscalar, vector and axial-vector currents The soft scale of the low-energy expansion is given by of quarks: the small external momenta and the small masses of the pseudo-Goldstone bosons, while the large scale is = 0 + L LQCD Lext a typical hadronic scale of about 1 GeV. The effective 0 µ = LQCD +qγ ¯ (vµ + γ5aµ)q − q¯(s − iγ5p)q . (9) Lagrangian has to fulfill the same symmetry properties as QCD: invariance under Lorentz and parity transfor- All chiral Ward identities are encoded in the correspond- mations, charge conjugation and time reversal symme- ing generating functional, if the global chiral symmetry try. Especially the chiral symmetry and its spontaneous 0 SU(3)L × SU(3)R of LQCD is promoted to a local gauge symmetry breaking has to be incorporated. Using the 0 symmetry of L [134]. Since LQCD is only invariant un- external-field method, the same external fields v, a, s, p der the global chiral symmetry, the external fields have as in Eq. (9), with the same transformation behavior, to fulfill a suitable transformation behavior: are included in the effective Lagrangian. As the QCD vacuum is approximately invariant un- † † vµ + aµ → R(vµ + aµ)R + i R∂µR , der the flavor symmetry group SU(3), one expects the † † hadrons to organize themselves in multiplets of irre- vµ − aµ → L(vµ − aµ)L + i L∂µL , ducible representations of . The pseudoscalar † SU(3) s + i p → R (s + i p) L , mesons form an octet, cf. Fig.1. The members of the s − i p → L (s − i p) R† , (10) octet are characterized by the strangeness quantum num- ber S and the third component I3 of the isospin. The where L(x) and R(x) are (independent) space-time- symbol η stands for the octet component (η8). As an dependent elements of SU(3)L and SU(3)R. approximation we identify η8 with the physical η, ignor- Furthermore, we still require the full Lagrangian L to ing possible mixing with the η1. For the be invariant under P , C and T . As the transformation lowest-lying baryons one finds an octet and a decuplet, properties of the quarks are well-known, the transforma- see also Fig.1. In the following we summarize how these tion behavior of the external fields can be determined and explicit degrees of freedom are included in the chiral La- is displayed in Tab.I. Time reversal symmetry is not con- grangian in the standard non-linear realization of chiral sidered explicitly, since it is automatically fulfilled due to symmetry [124, 125]. the CPT theorem. The chiral symmetry group SU(3)L × SU(3)R is spon- 6

S S i.e., the mesons φ (x) transform in the adjoint (irre- 6 K0 K+ 6 n p a +1 0 ducible) 8-representation of SU(3). The parity transfor-  T  T P  q 0 q T  q 0 q T mation behavior of the pseudoscalar mesons is φa(t, ~x ) → − π + − Σ + P 0 π  T π −1 Σ  T Σ −φ (t, −~x ) or, equivalently, U(t, ~x ) → U †(t, −~x ). Un- T η  T Λ  a qT  qqc qT  qqc der charge conjugation the fields are mapped to C −1 T  −2 T  fields, leading to U → U >. − ¯ 0 − 0 K K - Ξ Ξ - The octet baryons are described by Dirac spinor fields q 1 1q I3 q 1 1q I3 −1 − 2 0 2 1 −1 − 2 0 2 1 and represented in a traceless 3 × 3 matrix B(x) in flavor space, S 6∆− ∆0 ∆+ ∆++ 0 8 T  X Baλa qq q q B = √ T Σ∗− Σ∗0 Σ∗+ 2 −1 T  a=1  √1 0 √1 +  qT  qq 2 Σ + 6 ΛΣ p ∗− ∗0 −2 T Ξ Ξ =  Σ− − √1 Σ0 + √1 Λ n  . (16) T   2 6  q q Ξ− Ξ0 − √2 Λ T  − −3 TΩ 6 - − 3 −1 − 1 0q 1 1 3 I3 We use the convenient [135] non-linear realization of chi- 2 2 2 2 ral symmetry for the baryons, which lifts the well-known Figure 1: octet (J P = 0−), baryon flavor transformations to the chiral symmetry group. The octet (J P = 1/2+) and baryon decuplet (J P = 3/2+). matrix B(x) transforms under the chiral symmetry group SU(3)L × SU(3)R as

† taneously broken to its diagonal subgroup SU(3)V. B → KBK , (17) Therefore the Goldstone-boson octet should transform under SU(3)L × SU(3)R such that an irreducible 8- with the SU(3)-valued compensator field representation results for SU(3) . A convenient choice V √ √ to describe the pseudoscalar mesons under these condi- K (L, R, U) = LU †R†R U. (18) tions is a unitary 3×3 matrix U(x) in flavor space, which fulfills Note that K (L, R, U) also depends on the meson matrix † U. The square root of the meson matrix, U U = 1 , det U = 1 . (11) √ The transformation behavior under chiral symmetry u = U, (19) reads √ transforms as u → RUL† = RuK† = KuL†. For U → RUL† , (12) transformations under the subgroup SU(3)V the baryons transform as an octet, i.e., the adjoint representation of where L(x), R(x) are elements of SU(3)L,R. An explicit parametrization of U(x) in terms of the pseudoscalar SU(3): mesons is given by B → VBV † . (20) U(x) = exp [i φ(x)/f0] , (13) The octet-baryon fields transform under parity and with the traceless Hermitian matrix P 0 charge conjugation as Ba (t, ~x ) → γ Ba (t, −~x ) and 8 C X Bα,a → CαβB¯β,a with the Dirac-spinor indices α, β and φ(x) = φa(x)λa with C = iγ2γ0. a=1 √ √ A natural choice to represent the decuplet baryons is π0 + √1 η 2π+ 2K+ √ 3 √ a totally symmetric three-index tensor T . It transforms − 0 1 0 =  2π −π + √ η 2K  . (14) under the chiral symmetry SU(3)L × SU(3)R as  √ √ 3  − ¯ 0 − √2 2K 2K 3 η Tabc → KadKbeKcf Tdef , (21)

The constant f0 is the decay constant of the pseudoscalar Goldstone bosons in the chiral limit. For a transforma- with the compensator field K(L, R, U) of Eq. (18). For an transformation the decuplet fields transform tion of the subgroup SU(3)V with L = R = V , the meson SU(3)V matrix U transforms as as an irreducible representation of SU(3):

† U → VUV , (15) Tabc → VadVbeVcf Tdef . (22) 7

The physical fields are assigned to the following compo- fulfilled via the CPT theorem. Especially local chiral nents of the totally antisymmetric tensor: symmetry has to be fulfilled. A common way to con- struct the chiral Lagrangian is to define so-called build- 111 ++ 112 √1 + 122 √1 0 222 − T = ∆ ,T = 3 ∆ ,T = 3 ∆ ,T = ∆ , ing blocks, from which the effective Lagrangian can be T 113 = √1 Σ∗+ ,T 123 = √1 Σ∗0 ,T 223 = √1 Σ∗− , determined as an invariant polynomial. Considering the 3 6 3 chiral transformation properties, a convenient choice for 133 √1 ∗0 233 √1 ∗− T = 3 Ξ ,T = 3 Ξ , the building blocks is 333 − T = Ω . (23)  † † uµ = i u (∂µ − i rµ) u − u (∂µ − i lµ) u , † † † Since decuplet baryons are -3/2 particles, each com- χ± = u χu ± uχ u , ponent is expressed through Rarita-Schwinger fields. f ± = uf L u† ± u†f R u , (26) Within the scope of this article, decuplet baryons are µν µν µν only used for estimating LECs via decuplet resonance with the combination saturation. In that case it is sufficient to treat them in their non-relativistic form, where no complications with χ = 2B0 (s + i p) , (27) the Rarita-Schwinger formalism arise. Now the representation of the explicit degrees of free- containing the new parameter B0 and the external scalar dom and their transformation behavior are established. and pseudoscalar fields. One defines external field Together with the external fields the construction of the strength tensors by chiral effective Lagrangian is straightforward. R fµν = ∂µrν − ∂ν rµ − i [rµ, rν ] , L fµν = ∂µlν − ∂ν lµ − i [lµ, lν ] , (28) C. Construction of the chiral Lagrangian where the fields The chiral Lagrangian can be ordered according to the number of baryon fields: rµ = vµ + aµ , lµ = vµ − aµ , (29)

Leff = Lφ + LB + LBB + LBBB + ..., (24) describe right handed and left handed external vector fields. In the absence of flavor singlet couplings one can where Lφ denotes the purely mesonic part of the La- assume haµi = hvµi = 0, where h... i denotes the flavor ± grangian. Each part is organized in the number of small trace. Therefore, the fields uµ and fµν in Eq. (26) are all momenta (i.e., derivatives) or small meson masses, e.g., traceless. Using the transformation behavior of the pseudoscalar (2) (4) (6) Lφ = Lφ + Lφ + Lφ + .... (25) mesons and octet baryons in Eq. (12) and Eq. (17), and the transformation properties of the external fields in 6 Lφ has been constructed to O(q ) in Refs. [136, 137]. Eq. (10), one can determine the transformation behavior The chiral Lagrangian for the baryon-number-one sector of the building blocks. All building blocks A, and there- has been investigated in various works. The chiral ef- fore all products of these, transform according to the ad- 4 fective pion-nucleon Lagrangian of order O(q ) has been joint (octet) representation of SU(3), i.e., A → KAK†. constructed in Ref. [138]. The three-flavor Lorentz in- Note that traces of products of such building blocks are variant chiral meson-baryon Lagrangians LB at order invariant under local chiral symmetry, since K†K = 1. 2 3 O(q ) and O(q ) have been first formulated in Ref. [139] The chiral covariant derivative of such a building block and were later completed in Refs. [140, 141]. Concern- A is given by ing the nucleon-nucleon contact terms, the relativistically 2 invariant contact Lagrangian at order O(q ) for two fla- DµA = ∂µA + [Γµ,A] , (30) vors (without any external fields) has been constructed in Ref. [142]. The baryon-baryon interaction Lagrangian with the chiral connection LBB has been considered up to NLO in Refs. [27, 29, 32]. 1  † † Furthermore the leading three-baryon contact interaction Γµ = u (∂µ − i rµ) u + u (∂µ − i lµ) u . (31) 2 Lagrangian LBBB has been derived in Ref. [120]. We follow closely Ref. [32] to summarize the basic pro- The covariant derivative transforms homogeneously un- † cedure for constructing systematically the three-flavor der the chiral group as DµA → K (DµA) K . The chi- chiral effective Lagrangian [124, 125] with the inclusion ral covariant derivative of the baryon field B is given by of external fields [7, 126]. The effective chiral Lagrangian Eq. (30) as well. has to fulfill all discrete and continuous symmetries of the A Lorentz-covariant power counting scheme has been strong interaction. Therefore it has to be invariant under introduced by Krause in Ref. [139]. Due to the large parity (P ), charge conjugation (C), Hermitian conjuga- baryon mass M0 in the chiral limit, a time-derivative act- tion (H) and the proper, orthochronous Lorentz transfor- ing on a baryon field B cannot be counted as small. Only mations. Time reversal symmetry is then automatically baryon three-momenta are small on typical chiral scales. 8

p c h O where the exponents pΓ, cΓ, hΓ ∈ {0, 1} can be found in 1 Tab. II(b). As before, Lorentz indices of baryon bilinears uµ 1 0 0 O q µ + 2 transform with the matrix P ν under parity. fµν 0 1 0 O q − 2 Due to the identity fµν 1 0 0 O q 2 χ+ 0 0 0 O q 1 i  +  χ 1 0 1 O q2 [Dµ,Dν ] A = [[uµ, uν ] ,A] − f ,A (36) − 4 2 µν (a) Chiral building blocks it is sufficient to use only totally symmetrized products of covariant derivatives, Dαβγ...A, for any building block Γ p c h O A (or baryon field B). Moreover, because of the relation 0 1 0 0 0 O q D u − D u = f − , (37) 1 ν µ µ ν µν γ5 1 0 1 O q 0 γµ 0 1 0 O q only the symmetrized covariant derivative acting on uµ 0 γ5γµ 1 0 0 O q need to be taken into account, 0 σµν 0 1 0 O q hµν = Dµuν + Dν uµ . (38) (b) Baryon bilinears B¯ΓB Finally, the chiral effective Lagrangian can be con- Table II: Behavior under parity, charge conjugation and structed by taking traces (and products of traces) of dif- Hermitian conjugation as well as the chiral dimensions ferent polynomials in the building blocks, so that they of chiral building blocks and baryon bilinears B¯ΓB are invariant under chiral symmetry, Lorentz transfor- [140]. mations, C and P .

This leads to the following counting rules for baryon fields 1. Leading-order meson Lagrangian and their covariant derivatives, As a first example, we show the leading-order purely 0  B, B,D¯ µB ∼ O q , iD/ − M0 B ∼ O (q) . mesonic Lagrangian. From the general construction prin- (32) ciples discussed above, one obtains for the leading-order The chiral dimension of the chiral building blocks and effective Lagrangian baryon bilinears B¯ΓB are given in Tab.II. A covariant 2 derivative acting on a building block (but not on B) raises (2) f0 µ L = huµu + χ+i . (39) the chiral dimension by one. φ 4 A building block A transforms under parity, charge Note that there is no contribution of order O(q0). This is conjugation and Hermitian conjugation as consistent with the vanishing interaction of the Goldstone AP = (−1)pA,AC = (−1)cA> ,A† = (−1)hA, bosons in the chiral limit at zero momenta. (33) Before we continue with the meson-baryon interaction with the exponents (modulo two) p, c, h ∈ {0, 1} given Lagrangian, let us elaborate on the leading chiral La- in Tab. II(a), and > denotes the transpose of a (fla- grangian in the purely mesonic sector without external fields, but with non-vanishing quark masses in the isospin vor) matrix. A sign change of the spatial argument, µ µ (t, ~x) → (t, −~x), is implied in the fields in case of parity limit: v (x) = a (x) = p(x) = 0 and s(x) = M = transformation P . Lorentz indices transform with the diag (m, m, ms). Inserting the definitions of the building µ blocks, Eq. (39) becomes with these restrictions: matrix P ν = diag(+1, −1, −1, −1) under parity trans- µ P p µ ν formation, e.g., (u ) = (−1) P ν u . The transforma- 2 (2) f0 µ † 1 2 † tion behavior of commutators and anticommutators of L = h∂µU∂ U i + B0f hMU + UMi . (40) φ 4 2 0 two building blocks A1,A2 is the same as for build- ing block and should therefore be used instead of simple The physical decay constants fπ =6 fK =6 fη differ from products, e.g., the decay constant of the pseudoscalar Goldstone bosons in the chiral limit f0 in terms of order (m, ms): fφ = C c1+c2 > > > > [A1,A2]± = (−1) (A1 A2 ± A2 A1 ) f0 {1 + O (m, ms)}. The constant B0 is related to the c1+c2 > chiral quark condensate. Already from this leading-order = ±(−1) [A1,A2] . (34) ± Lagrangian famous relations such as the (reformulated) The behavior under Hermitian conjugation is the same. Gell-Mann–Oakes–Renner relations The basis elements of the Dirac algebra forming the 2 2 baryon bilinears transform as mπ = 2mB0 + O(mq) , 2 2 m = (m + ms) B0 + O(m ) , pΓ −1 cΓ > K q γ0Γγ0 = (−1) Γ ,C ΓC = (−1) Γ , 2 2 2 † hΓ m = (m + 2ms) B0 + O(m ) , (41) γ0Γ γ0 = (−1) Γ , (35) η 3 q 9

2 2 or the Gell-Mann–Okubo mass formula, 4mK = 3mη + m2 , can be derived systematically. V π V T = V + + V + ··· V 2. Leading-order meson-baryon interaction Lagrangian V

The leading-order meson-baryon interaction La- (1) 1 Figure 2: Graphical representation of the grangian LB is of order O(q) and reads Lippmann-Schwinger equation.

(1)  D µ L =hB¯ iD/ − MB Bi + hBγ¯ γ5{uµ,B}i B 2 Λ N F µ η + hBγ¯ γ5 [uµ,B]i . (42) 2

The constant MB is the mass of the baryon octet in Λ N the chiral limit. The two new constants D and F are η called axial-vector coupling constants. Their values can Λ N be obtained from semi-leptonic hyperon decays and are Figure 3: Example of a planar box diagram. It contains roughly D ≈ 0.8 and F ≈ 0.5 [143]. The sum of the an reducible part equivalent to the iteration of two two constants is related to the axial-vector coupling con- one-meson exchange diagrams, as generated by the stant of nucleons, g = D + F = 1.27, obtained from A Lippmann-Schwinger equation. Additionally it contains neutron beta decay. At lowest order the pion-nucleon a genuine irreducible contribution that is part of the g is connected to the axial-vector πN effective potential. coupling constant by the Goldberger-Treiman relation, gπN fπ = gAMN . The covariant derivative in Eq. (42) in- cludes the field Γµ, which leads to a vertex between two octet baryons and two mesons, whereas the terms con- of chiral symmetry breaking Λχ is often estimated as 4πfπ ≈ 1 GeV or as the mass of the lowest-lying res- taining uµ to a vertex between two octet baryons and one meson. Different octet-baryon masses appear onance, Mρ ≈ 770 MeV. A simple dimensional analysis leads to the following expression for the chiral dimension first in (2) due to explicit chiral symmetry breaking LB of a connected [6]: and renormalization and lead to corrections linear in the quark masses: X ν = 2 + 2L + vi∆i , ∆i = di − 2 . (44) i Mi = MB + O(m, ms) . (43) The number of loops is denoted by L and vi is the number of vertices with vertex dimension D. Weinberg power counting scheme ∆i. The symbol di stands for the number of derivatives or meson mass insertions at the vertex, i.e., the vertex As stated before, an effective field theory has an infinite originates from a term of the Lagrangian of the order number of terms in the effective Lagrangian and for a O(qdi ). fixed process an infinite number of diagrams contribute. With the introduction of baryons in the chiral effec- Therefore, it is crucial to have a power counting scheme, tive Lagrangian, the power counting is more compli- to assign the importance of a term. Then, to a certain cated. The large baryon mass comes as an extra scale order in the power counting, only a finite number of terms and destroys the one-to-one correspondence between the contribute and the observables can be calculated to a loop and the small momentum expansion. Jenkins and given accuracy. Manohar used methods from heavy-quark effective field First, let us discuss the power counting scheme of theory to solve this problem [127]. Basically they con- χEFT in the pure meson sector, i.e., only the pseu- sidered baryons as heavy, static sources. This leads to a doscalar Goldstone bosons are explicit degrees of free- description of the baryons in the extreme non-relativistic dom. The chiral dimension ν of a Feynman diagram limit with an expansion in powers of the inverse baryon represents the order in the low-momentum expansion, mass, called heavy-baryon chiral perturbation theory. ν (q/Λχ) . The symbol q is generic for a small external Furthermore, in the two-baryon sector, additional fea- meson momentum or a small meson mass. The scale tures arise. Reducible Feynman diagrams are enhanced due to the presence of small kinetic energy denomina- tors resulting from purely baryonic intermediate states. These graphs hint at the non-perturbative aspects in few- 1 Note that an overall plus sign in front of the constants D and F body problems, such as the existence of shallow bound is chosen, consistent with the conventions in SU(2) χEFT [13]. states, and must be summed up to all orders. As sug- 10

N Λ N N Λ N N Λ N B = 2 case unaltered, and one obtains (see for example Refs. [9, 13, 14, 130]) π π

∗ π π X 1 Σ ν = −4+2B +2L+ vi∆i , ∆i = di + bi −2 . (47) Σ 2 i π π Following this scheme one arrives at the hierarchy of N Λ N N Λ N N Λ N baryonic forces shown in Fig.5. The leading-order Figure 4: Examples for reducible (left) and irreducible (ν = 0) potential is given by one-meson-exchange di- (right) three-baryon interactions for ΛNN. The thick agrams and non-derivative four-baryon contact terms. dashed line cuts the reducible diagram in two two-body At next-to-leading order (ν = 2) higher order contact interaction parts. terms and two-meson-exchange diagrams with interme- diate octet baryons contribute. Finally, at next-to-next- to-leading order (ν = 3) the three-baryon forces start gested by Weinberg [86, 87], the baryons can be treated to contribute. Diagrams that lead to mass and coupling non-relativistically and the power counting scheme can constant renormalization are not shown. be applied to an effective potential V , that contains only irreducible Feynman diagrams. Terms with the inverse −1 baryon mass MB may be counted as III. BARYON-BARYON INTERACTION POTENTIALS q  q 2 ∝ . (45) MB Λχ This section is devoted to the baryon-baryon interac- tion potentials up to next-to-leading order, constructed The resulting effective potential is the input for quan- from the diagrams shown in Fig.5. Contributions arise tum mechanical few-body calculations. In case of the from contact interaction, one- and two-Goldstone-boson baryon-baryon interaction the effective potential is in- exchange. The constructed potentials serve not only as serted into the Lippmann-Schwinger equation and solved input for the description of baryon-baryon scattering, but for bound and scattering states. This is graphically are also basis for few- and many-body calculations. We shown in Fig.2 and Fig.3. The T -matrix is obtained give also a brief introduction to common meson-exchange from the infinite series of ladder diagrams with the ef- models and the difference to interaction potentials from fective potential V . In this way the omitted reducible χEFT. diagrams are regained. In the many-body sector, e.g., Faddeev (or Yakubovsky) equations are typically solved within a coupled-channel approach. In a similar way re- A. Baryon-baryon contact terms ducible diagrams such as on the left-hand side of Fig.4, are generated automatically and are not part of the effec- tive potential. One should distinguish such iterated two- The chiral Lagrangian necessary for the contact ver- body interactions, from irreducible three-baryon forces, tices shown in Fig.6 can be constructed straightfor- as shown on the right-hand side of Fig.4. wardly according to the principles outlined in Sec.II. For After these considerations, a consistent power count- pure baryon-baryon scattering processes, no pseudoscalar ing scheme for the effective potential V is possible. The mesons are involved in the contact vertices and almost all ν external fields can be dropped. Covariant derivatives Dµ soft scale q in the low-momentum expansion (q/Λχ) de- notes now small external meson four-momenta, small ex- reduce to ordinary derivatives ∂µ. The only surviving ternal baryon three-momenta or the small meson masses. external field is χ+, which is responsible for the inclusion Naive dimensional analysis leads to the generalization of of quark masses into the chiral Lagrangian: Eq. (44):   mu 0 0 χ+ X 1 = χ = 2B0  0 md 0  ν = 2 − B + 2L + vi∆i , ∆i = di + bi − 2 , (46) 2 2 0 0 ms i  2  mπ 0 0 2 where B is the number of external baryons and bi is the ≈  0 mπ 0  , (48) 2 2 number of internal baryon lines at the considered ver- 0 0 2mK − mπ tex. However, Eq. (46) has an unwanted dependence on the baryon number, due to the normalization of baryon where in the last step the Gell-Mann–Oakes–Renner re- states. Such an effect can be avoided by assigning the lations, Eq. (41), have been used. In flavor space the chiral dimension to the transition operator instead of the possible terms are of the schematic form matrix elements. This leads to the addition of 3B − 6 to the formula for the chiral dimension, which leaves the hBB¯ BB¯ i , hB¯BBB¯ i , hBB¯ ihBB¯ i , hB¯B¯ihBBi , (49) 11

two-baryon force three-baryon force

LO

NLO

NNLO ···

Figure 5: Hierarchy of baryonic forces. lines are baryons, dashed lines are pseudoscalar mesons. Solid dots, filled circles and squares denote vertices with ∆i = 0, 1 and 2, respectively.

1 2 P3 = (~σ1 · ~q )(~σ2 · ~q ) − (~σ1 · ~σ2)~q , 3 i P4 = (~σ1 + ~σ2) · ~n, 2 P5 = (~σ1 · ~n )(~σ2 · ~n ) , Figure 6: Leading-order and next-to-leading-order i P6 = (~σ1 − ~σ2) · ~n, baryon-baryon contact vertices. 2

P7 = (~σ1 · ~k )(~σ2 · ~q ) + (~σ1 · ~q )(~σ2 · ~k ) , ~ ~ and terms where the field χ is inserted such as P8 = (~σ1 · k )(~σ2 · ~q ) − (~σ1 · ~q )(~σ2 · k ) , (51) with the Pauli spin matrices and with the vectors hBχB¯ BB¯ i , hBBχ¯ BB¯ i , hBχB¯ ihBB¯ i ,... (50) ~σ1,2 1 where in both cases appropriate structures in Dirac space ~k = (~pf + ~pi) , ~q = ~pf − ~pi , ~n = ~pi × ~pf . (52) have to be inserted. For the case of the non-relativistic 2 power counting it would also be sufficient, to insert the The momenta ~pf and ~pi are the initial and final state corresponding structures in spin-momentum space. The momenta in the center-of-mass frame. In order to ob- terms involving χ lead to explicit SU(3) symmetry break- tain the minimal set of Lagrangian terms in the non- ing at NLO linear in the quark masses. A set of lin- relativistic power counting of Ref. [32], the potentials 2 early independent Lagrangian terms up to O(q ) for have been decomposed into partial waves. The formu- pure baryon-baryon interaction in non-relativistic power las for the partial wave projection of a general interaction counting can be found in Ref. [32]. P8 V = VjPj can be found in the appendix of Ref. [29]. 2 j=1 After a non-relativistic expansion up to O(q ) the four- For each partial wave one produces a non-square matrix baryon contact Lagrangian leads to potentials in spin and which connects the Lagrangian constants with the differ- momentum space. A convenient operator basis is given ent baryon-baryon channels. Lagrangian terms are con- by [29]: sidered as redundant if their omission does not lower the rank of this matrix. For the determination of the po- 1 P1 = , tential not only direct contributions have to be consid- P2 = ~σ1 · ~σ2 , ered, but also additional structures from exchanged final 12 state baryons, where the negative spin-exchange oper- next-to-leading-order constants (without tilde and with- (σ) 1 1 1 3 ator −P = − 2 ( + ~σ1 · ~σ2) is applied. In the end 6 out χ) the contributions to the partial waves S0, S1 2 2 momentum-independent terms at LO contribute, and are have to be multiplied with a factor pi + pf . The contri- therefore only visible in 1 and 3 partial waves. At 1 3 j S0 S1 bution to the partial waves S0, S1 from constants cχ NLO 22 terms contribute that contain only baryon fields 2 2 has to be multiplied with (mK − mπ). The partial waves and derivatives, and are therefore SU(3) symmetric. The 3 3 3 1 1 3 P0, P1, P2, P1, P1 ↔ P1 get multiplied with the other 12 terms terms at NLO include the diagonal matrix 3 3 3 3 factor pipf . The entries for S1 → D1 and D1 → S1 χ and produce explicit SU(3) symmetry breaking. 2 2 have to be multiplied with pi and pf , respectively. For In Tab.III the non-vanishing transitions projected 1 example, one obtains for the NN interaction in the S0 onto partial waves in the isospin basis are shown, cf. partial wave: Refs. [29–32]. The pertinent constants are redefined ac- cording to the relevant irreducible SU(3) representations. 1 1 hNN, S0|Vˆ |NN, S0i This comes about in the following way. Baryons form a 27 27 2 2 1 1 2 2 flavor octet and the tensor product of two baryons de- 1 1 − (54) =c ˜ S0 + c S0 (pi + pf ) + cχ(mK mπ) , composes into irreducible representations as follows: 2 or for the → interaction with total isospin ∗ ΞN ΣΣ I = 0 8 ⊗ 8 27 ⊕ 10 ⊕ 10 ⊕ 8 ⊕ 8 ⊕ 1 (53) 1 3 = s a a s a s , in the P1 → P1 partial wave: √ where the irreducible representations 27s, 8s, 1s are h 3 | ˆ | 1 i 8as (55) ∗ ΣΣ, P1 V ΞN, P1 = 2 3c pipf . symmetric and 10a, 10a, 8a are antisymmetric with respect to the exchange of both baryons. Due to the When restricting to the NN channel the well-known two generalized Pauli principle, the symmetric flavor repre- leading and seven next-to-leading order low-energy con- sentations 27s, 8s, 1s have to combine with the space- stants of Ref. [147] are recovered, which contribute to the 1 3 3 3 1 3 1 3 3 3 3 3 spin antisymmetric partial waves S0, P0, P1, P2,... partial waves S0, S1, P1, P0, P1, P2, S1 ↔ D1. (L + S even). The antisymmetric flavor representations Note, that the SU(3) relations in Tab.III are general ∗ 10a, 10a, 8a combine with the space-spin symmetric relations that have to be fulfilled by the baryon-baryon 3 1 3 3 partial waves S1, P1, D1 ↔ S1,... (L + S odd). potential in the SU(3) limit, i.e., mπ = mK = mη. This Transitions can only occur between equal irreducible rep- feature can be used as a check for the inclusion of the resentations. Hence, transitions between space-spin an- loop diagrams. Another feature is, that the SU(3) rela- tisymmetric partial waves up to O(q2) involve the 15 tions contain only a few constants in each partial wave. 27,8s,1 27,8s,1 27,8s,1 27,8s,1 27,8s,1 1 constants c˜1S , c1S , c3P , c3P and c3P , For example, in the S0 partial wave only the constants 0 0 0 1 2 27 8s 1 whereas transitions between space-spin symmetric par- c˜1S , c˜1S , c˜1S are present. If these constants are fixed in 8a,10,10∗ 8a,10,10∗ 0 0 0 tial waves involve the 12 constants c˜3 , c3 , some of the baryons channels, predictions for other chan- S1 S1 8a,10,10∗ 8a,10,10∗ nels can be made. This has, for instance, been used in c1 and c3 3 . The constants with a tilde de- P1 D1- S1 Ref. [35], where the existence of , and bound note leading-order constants, whereas the ones without ΣΣ ΣΞ ΞΞ states has been studied within SU(3) EFT. tilde are at NLO. The spin singlet-triplet transitions χ 1 3 P1 ↔ P1 is perfectly allowed by SU(3) symmetry since it is related to transitions between the irreducible repre- B. One- and two-meson-exchange contributions sentations 8a and 8s. Such a transition originated from the antisymmetric spin-orbit operator P6 and its Fierz- In the last section, we have addressed the short-range transformed counterpart P8 and the single correspond- ing low-energy constant is denoted by c8as. In case of part of the baryon-baryon interaction via contact terms. the NN interaction such transitions are forbidden by Let us now analyze the long- and mid-range part of the 27,8s,1 8a,10,10∗ interaction, generated by one- and two-meson-exchange isospin symmetry. The constants c˜1 and c˜3 S0 S1 27,8s,1 as determined in Ref. [33]. The contributing diagrams up fulfill the same SU(3) relations as the constants c1 S0 to NLO are shown in Fig.5, which displays the hierarchy 8a,10,10∗ and c3 in Tab.III. SU(3) breaking terms linear in of baryonic forces. S1 1 3 The vertices, necessary for the construction of these di- the quark masses appears only in the S-waves, S0, S1, 2 2 agrams stem from the leading-order meson-baryon inter- and are proportional mK − mπ. The corresponding 12 1,...,12 (1) constants are cχ . The SU(3) symmetry relations in action Lagrangian LB in Eq. (42). The vertex between Tab.III can also be derived by group theoretical consid- two baryons and one meson emerges from the part erations [29, 144–146]. Clearly, for the SU(3)-breaking D µ F µ part this is not possible and these contributions have to hBγ¯ γ5{uµ,B}i + hBγ¯ γ5 [uµ,B]i be derived from the chiral Lagrangian. 2 2 8 In order to obtain the complete partial-wave projected 1 X − ¯ µ O 3 potentials, some entries in Tab.III have to be multi- = NBiBj φk (Biγ γ5Bj)(∂µφk) + (φ ) , 2f0 plied with additional momentum factors. The leading i,j,k=1 i (56) order constants c˜j receive no further factor. For the 13 9 χ 9 χ 6 c 9 χ c 9 χ 2 2 c c 2 3 3 − − q − 8 χ 4 c 8 χ 9 χ 7 + 8 χ 4 9 χ c 9 χ 4 c c 3 9 χ c 3 c 3 8 χ 2 4 c − c 3 2 − 3 2 − − 7 χ − − √ c 24 7 χ 8 χ 8 5 7 χ q 8 7 χ c 3 c 8 χ c − c 2 c 3 3 χ + − 2 − 9 χ 7 χ − 4 − 7 χ c 8 χ 7 χ − c 6 1 √ 2 10 χ 12 χ 11 χ 0 0 0 0 0 c c c + 12 χ 7 χ 2 c c c 12 χ S √ − c c 4 12 χ − 12 χ c 12 4 12 χ 2 3 3 7 χ c c 5 c 25 3 − c 2 + + − 10 χ + + + + 4 c 12 χ 2 11 χ c 10 χ 4 11 χ 11 χ 2 10 χ c 4 4 10 χ c 3 c c 2 3 c 3 √ c 3 3 3 4 q − + + + 10 χ 4 10 χ 10 χ c 4 6 c c 9 12 10 χ 5 √ c 11 6 χ 6 χ 4 12 c c 6 χ 6 χ 4 3 c c 2 3 + in non-relativistic power counting for + + 5 χ q ) c 6 χ 5 χ 4 χ 32 3 5 χ 4 χ 3 c c 3 6 5 χ c 5 χ 2 c c 32 3 − 8 32 2 c 4 c 4 χ 9 q 9 27 3 c √ 6 + ( + + 4 χ + + √ c + + 3 χ + 4 χ + O 3 3 χ 18 c c 4 χ 4 χ c 3 9 c 3 χ 4 χ 4 χ 3 χ 3 6 c 4 c c c − c χ + + [ 32 ]. 3 χ 3 + 2 2 χ − 4 χ 6 χ 2 χ 5 χ 1 χ 1 χ c − 0 2 4 2 + + + 0 0 0 3 χ 2 χ c c c c c c c √ 2 χ 6 I c 2 c S 3 χ − 2 χ 3 2 2 2 χ c 7 3 χ 3 χ 3 c 1 c 3 2 c c c + 4 3 5 3 q + + 3 − + 3 1 χ − − + √ 2 χ + + c 1 χ 2 χ 4 1 χ c 3 c 2 c 3 1 χ c 2 χ 1 χ 1 χ 1 χ 2 9 3 4 c 3 4 c c c c − 3 2 3 2 3 − + 2 − − − − 1 χ q 1 χ c c 1 3 72 1 χ 24 c 55 11 24 − 11 − 1 and total isospin P as 1 as as 8 as as 8 as as as as 8 S c as as 8 8 8 8 8 8 c c 0 0 8 8 2 c c → c c c c 2 3 c c 4 3 2 3 1 − − √ √ − P 3 1 P as as 3 8 as as as 8 as as as as as c 8 8 8 c 8 8 8 8 8 c c c 0 00 0 0 00 0 0 0 5 2 3 3 c c → c c c 3 2 3 2 1 − − √ q − − − P 1 } 1 ) 0 2 a ) 0 0 0 0 D ) 8 j 3 a a c 8 j ) 0 8 j ) 2 2 c ) ) ) 3 ∗ ) 0 0 c ∗ ) ) ) 3 ↔ a a a ∗ a a a 10 j − 8 j 8 j 8 j 1 10 j + 8 j 8 j 8 j c 10 j + 4 c c c c ∗ c c c S c ∗ ∗ 3 ∗ ∗ − a 10 j + − + − , + − + 10 j 10 j 10 j 8 j 0 0 0 0 0 0 0 0 0 0 0 c + 10 j 10 j 10 j 1 c ∗ ∗ ∗ c c c c 10 j c c 10 j P 10 j 10 j 10 j c + 10 j 10 j 10 j 10 j c + 1 c c c ( c + c c c ( ( ( ( , ( 3 ( ( ( 10 j 6 1 1 1 2 2 1 2 10 j 1 1 1 2 1 2 2 2 1 1 c √ 10 j √ c S ( 2 c ( channels described by strangeness 3 ( 2 1 3 1 6 1 √ 3 ∈ { j ) 0 0 0 0 0 } ) 0 2 ) 0) 0 0 0 0 0 0 0 s )) 0 0 2 2 s s s 8 j s s 8 j 8 j 8 j P c 8 j 8 j c c c 3 8 c c ) ) ) ) ) ) ) ) , ) s s s s s s s s 1 s 8 j 8 j − 8 j 8 j 8 j 8 j 8 j 8 j + 4 8 j + 8 c c P + 4 + 2 c c c c c c c + 24 3 27 j 27 j 27 j , − − c + + 27 j 27 j − c c 0 27 j + 3 + 2 c c + 9 + 9 27 j 27 j 27 j 27 j 27 j 0 c P 27 j 27 j 9 c c c 27 j 27 j + 27 j 3 c c + 3 c c 27 j 27 j 27 j 27 j c + ( ( , 1 j − + 3 c c c c ( 1 j + 27 0 c 1 j (9 ( (9 ( 3 3 c 6 1 j 1 j 10 10 5 c S (2 (3 5 1 j c c 1 1 1 1 √ 1 c 10 10 10 10 1 5 5 1 (5 − − − (5 (5 3 ( (5 (15 40 3 √ 1 1 ∈ { 20 10 1 1 20 40 40 √ j − N N c N N N N N ΣΛ ΣΣ 0 ΣΣ ΣΣ ΣΣ Ξ ΣΣ 0 NN ΛΛ Ξ ΞΞ 0 Λ Σ Σ ΞΛ ΞΣ ΞΣ NN c ΞΞ ΣΛ ΣΣ 0 ΞΣ Σ ΣΣ Ξ → → → → → → → → → → → → → → → → → → → → → → → → → N N N N N N N N N transition ΞΛ ΞΛ ΞΣ ΞΣ Λ Λ Σ Σ NN NN 3 1 1 2 2 1 2 2 2 1 2 1 2 3 2 1 0 ΣΣ 2 ΣΣ 1 Ξ 0 ΛΛ 1 Ξ 1 ΣΛ 0 Ξ 1 ΣΣ 1 0 ΛΛ 0 Ξ 1 Ξ 1 ΞΞ 1 ΣΛ Table III: SU(3) relations of pure baryon-baryon contact terms for non-vanishing partial waves up to 1 2 0 ΛΛ 3 4 0 ΞΞ 0 0 SI − − − − 14

1 3 where we have used uµ = − ∂µφ + O(φ ) and have Note that the presented results apply only to di- f0 rewritten the pertinent part of the Lagrangian in terms rect diagrams. This is for example the case for the of the physical meson and baryon fields leading-order one-eta exchange in the Λn interaction, η i.e., for Λ(~p )n(−~p ) −→ Λ(~p )n(−~p ). An example of  0 + − + − 0 ¯ 0 i i f f φi ∈ π , π , π ,K ,K ,K , K , η , a crossed diagram is the one- exchange in the pro-  0 + − 0 − K Bi ∈ n, p, Σ , Σ , Σ , Λ, Ξ , Ξ . (57) cess Λ(~pi)n(−~pi) −→ n(−~pf )Λ(~pf ), where the nucleon and the hyperon in the final state are interchanged and The factors NB B φ are linear combinations of the axial i j k strangeness is exchanged. In such cases, ~pf is replaced vector coupling constants D and F with certain SU(3) by −~pf and the momentum transfer in the potentials coefficients. These factors vary for different combinations is q = |~pf + ~pi |. Due to the exchange of fermions in of the involved baryons and mesons and can be obtained the final states a minus sign arises, and additionally the easily by multiplying out the baryon and meson flavor (σ) 1 1 spin-exchange operator P = 2 ( + ~σ1 · ~σ2) has to be matrices. In a similar way, we obtain the (Weinberg- applied. The remaining structure of the potentials stays Tomozawa) vertex between two baryons and two mesons the same (see also the discussion in Sec.IV). (1) from the covariant derivative in LB , leading to The leading-order contribution comes from the one- ¯ µ hBiγ [Γµ,B]i meson exchange diagram in Fig. 7(a). It contributes only 8 to the tensor-type potential: i X µ 4 = NB φ B φ (B¯iγ Bj)(φk∂µφl) + O(φ ) , 8f 2 i k j l 0 i,j,k,l=1 (58) ome N 1 1 4 VT (q) = − 2 2 2 . (60) where Γµ = 2 [φ, ∂µφ] + O(φ ) was used. 4f q + m − i 8f0 0 The calculation of the baryon-baryon potentials is done in the center-of-mass frame and in the isospin limit mu = md. To obtain the contribution of the Feynman diagrams to the non-relativistic potential, we perform an The symbol M¯ in the SU(3) coefficient N denotes the charge-conjugated meson of meson in particle basis expansion in the inverse baryon mass 1/MB. If loops M + − are involved, the integrand is expanded before integrat- (e.g., π ↔ π ). ing over the loop momenta. This produces results that are equivalent to the usual heavy-baryon formalism. In At next-to-leading order the two-meson exchange dia- the case of the two-meson-exchange diagrams at one-loop grams start to contribute. The planar box in Fig. 7(b) level, ultraviolet divergences are treated by dimensional contains an irreducible part and a reducible part com- regularization, which introduces a scale λ. In dimensional ing from the iteration of the one-meson exchange to sec- regularization divergences are isolated as terms propor- ond order. Inserting the potential into the Lippmann- tional to Schwinger equation generates the reducible part; it is therefore not part of the potential, see also Subsec.IID. 2 The irreducible part is obtained from the residues at the R = + γE − 1 − ln (4π) , (59) d − 4 poles of the meson , disregarding the (far dis- tant) poles of the baryon propagators. With the masses with d =6 4 the space-time dimension and the Euler- of the two exchanged mesons set to m1 and m2, the ir- Mascheroni constant γE ≈ 0.5772. These terms can be absorbed by the contact terms. reducible potentials can be written in closed analytical According to Eqs. (56) and (58) the vertices have the form, same form for different combinations of baryons and mesons, just their prefactors change. Therefore, the one- and two-pseudoscalar-meson exchange potentials can be ( given by a master formula, where the proper masses of planar box N 5 2 Virr, C (q) = 2 4 q the exchanged mesons have to be inserted, and which has 3072π f0 3 to be multiplied with an appropriate SU(3) factor N. In 2 2 − 2 the following we will present the analytic formulas for m1 m2 2 2 + + 16 m1 + m2 the one- and two-meson-exchange diagrams, introduced q2  √  in Ref. [33]. The pertinent SU(3) factors will be displayed m1m2 + 23q2 + 45 m2 + m2 R + 2 ln next to the considered Feynman diagram, cf. Fig.7. The 1 2 λ results will be given in terms of a central potential (VC), a 2 2  m1 − m2 4 2 22 spin-spin potential (~σ1·~σ2 VS) and a tensor-type potential + 12q + m − m q4 1 2 (~σ1 ·~q~σ2 ·~qVT ). The momentum transfer is q = |~pf − ~pi |,  with ~pi and ~pf the initial and final state momenta in the 2 2 2 m1 − 9q m1 + m2 ln center-of-mass frame. m2 15

B3 B4 B3 B4 B3 B4

M2 M2

Bil Bir Bil Bir M M1 M1

B1 B2 B1 B2 B1 B2 N = N ¯ N N = N ¯ N BilB1M1 B3BilM2 BilB1M1 B3BilM2 N = NB3B1M¯ NB4B2M ¯ ¯ ×NBir B2M1 NB4Bir M2 ×NBir B2M2 NB4Bir M1 (a) One-meson exchange (b) Planar box (c) Crossed box

B3 B4 B3 B4 B3 B4

M2 M2 M2

Bi Bi

M1 M1 M1

B1 B2 B1 B2 B1 B2 ¯ ¯ ¯ ¯ N = NB3M2B1M1 NBiB2M1 NB4BiM2 N = NBiB1M1 NB3BiM2 NB4M2B2M1 N = NB3M¯ 2B1M1 NB4M¯ 1B2M2 (d) Left triangle (e) Right triangle (f) Football diagram

Figure 7: One- and two-meson-exchange contributions and corresponding SU(3) factors.

2 24 2  m − m the potentials in a power series for small q in order to + 23q4 − 1 2 + 56 m2 + m2 q2 w2 (q) q4 1 2 implement directly this cancellation. For equal meson 2 2 masses the expressions for the potentials reduce to the m1 + m2 2 22 results in Refs. [148]. This is the case for the NN inter- + 8 2 m1 − m2 q action of Refs. [147, 149–151] based on χEFT, where only  ) contributions from two-pion exchange need to be taken 4 2 2 4 + 2 21m1 + 22m1m2 + 21m2 L (q) , into account. (61) 1 In actual applications of these potentials such as in V planar box (q) = − V planar box(q) irr, T q2 irr, S Ref. [33], only the non-polynomial part of Eqs. (61) and " (62) is taken into account, i.e., the pieces proportional 2 2 2 4 N 1 m1 − m2 m1 to L(q) and to 1/q and 1/q . The polynomial part is = − 2 4 L (q) − − 2 ln 128π f0 2 2q m2 equivalent to the LO and NLO contact terms and, there- √ # fore, does not need to be considered. The contributions R m1m2 + + ln (62) proportional to the divergence R are likewise omitted. 2 λ Their effect is absorbed by the contact terms or a renor- malization of the coupling constants, see, e.g., the corre- where we have defined the functions sponding discussion in Appendix A of Ref. [149] for the r    NN case. 1 2 2 2 2 w (q) = q + (m1 + m2) q + (m1 − m2) , q  22 2 22 w (q) qw (q) + q − m1 − m2 These statements above apply also to the other contri- L (q) = ln 2 . (63) butions to the potential described below. 2q 4m1m2q The relation between the spin-spin and tensor-type po- tential follows from the identity (~σ1 × ~q ) · (~σ2 × ~q ) = The crossed box diagrams in Fig. 7(c) contribute to 2 q ~σ1 · ~σ2 − (~σ1 · ~q )(~σ2 · ~q ). the central, spin-spin, and tensor-type potentials. The One should note that all potentials shown above are similar structure with some differences in the kinematics finite also in the limit q → 0. Terms proportional to 1/q2 of the planar and crossed box diagram leads to relations or 1/q4 are canceled by opposite terms in the functions between them. Obviously, the crossed box has no iterated L(q) and w(q) in the limit of small q. For numerical cal- part. The potentials of the crossed box are equal to the culations it is advantageous to perform an expansion of potentials of the irreducible part of the planar box, up to 16 a sign in the central potential: also the exchanges of vector mesons (ρ, ω, K∗), of scalar mesons (σ (f0(500)), ...), or even of axial-vector crossed box planar box VC (q) = −VC, irr (q) , mesons (a1(1270), ...) [22, 23] are included. The spin- 1 space structure of the corresponding Lagrangians that V crossed box(q) = − V crossed box(q) = V planar box(q) . T q2 S T, irr enter into Eq. (42) and subsequently into Eq. (56) dif- (64) fer and, accordingly, the final expressions for the corre- sponding contributions to the YN interaction potentials The two triangle diagrams, Figs. 7(d) and 7(e), con- differ too. Details can be found in Refs. [18, 20, 22]. We stitute potentials, that are of equal form with different want to emphasize that even for pseudoscalar mesons the SU(3) factors N. The corresponding central potential final result for the interaction potentials differs, in gen- reads eral, from the expression given in Eq. (60). Contrary to ( the chiral EFT approach, recoil and relativistic correc- triangle N 2 2 tions are often kept in meson-exchange models because VC (q) = − 2 4 − 2 m1 + m2 3072π f0 no power counting rules are applied. Moreover, in case of the Jülich potential pseudoscalar coupling is assumed for 2 22 m − m 13 + 1 2 − q2 the meson-baryon interaction Lagrangian for the pseu- q2 3 doscalar mesons instead of the pseudovector coupling " 2 # (42) dictated by chiral symmetry. Note that in some 2 m2 − m2 2 2 1 2 2 potentials of the Jülich group [18, 19] contributions + 8 m1 + m2 − 2 + 10q L (q) YN q from two-meson exchanges are included. The ESC08 and 2 2 m − m h 2 i m ESC16 potentials [22, 23] include likewise contributions + 1 2 m2 − m2 − 3 m2 + m2 q2 ln 1 q4 1 2 1 2 m from two-meson exchange, in particular, so-called meson- 2 pair diagrams analog to the ones shown in Figs. 7(d),  √  )  2 2 2 m1m2 7(e), and 7(f). + 9 m1 + m2 + 5q R + 2 ln . (65) λ The major conceptual difference between the various meson-exchange models consists in the treatment of the The football diagrams in Fig. 7(f) also contributes only scalar-meson sector. This simply reflects the fact that, to the central potential. One finds unlike for pseudoscalar and vector mesons, so far there ( is no general agreement about what are the actual mem- football N 2 2 bers of the lowest lying scalar-meson SU(3) multiplet. VC (q) = 2 4 − 2 m1 + m2 3072π f0 Therefore, besides the question of the masses of the ex- 2 22 change particles it also remains unclear whether and how m − m 5 − 1 2 − q2 + w2 (q) L (q) the relations for the coupling constants should be spec- 2q2 6 ified. As a consequence, different prescriptions for de-  √  1 m1m2 scribing the scalar sector, whose contributions play a + 3 m2 + m2 + q2 R + 2 ln 2 1 2 λ crucial role in any baryon-baryon interaction at interme- diate ranges, were adopted by the various authors who 2 2 ) m1 − m2 h 2 22 2 2 2i m1 published meson-exchange models of the YN interaction. − 4 m1 − m2 + 3 m1 + m2 q ln . 2q m2 For example, the Nijmegen group views this interaction (66) as being generated by genuine scalar-meson exchange. In their models NSC97 [20] and ESC08 (ESC16) [22, 23] a scalar SU(3) nonet is exchanged - namely, two isospin- C. Meson-exchange models 0 mesons (an (760) and the f0(980)) an isospin-1 me- son (a0(980)) and an isospin-1/2 strange meson κ with Earlier investigations of the baryon-baryon interac- a mass of 1000 MeV. In the initial YN models of the tions has been done within phenomenological meson- Jülich group [18, 19] a σ (with a mass of ≈ 550 MeV) exchange potentials such as the Jülich [18, 19, 21], Ni- is included which is viewed as arising from correlated ππ jmegen [20, 22, 23], or Ehime [152, 153] potentials. As exchange. In practice, however, the coupling strength of we use two of them for comparison, we give a brief intro- this fictitious σ to the baryons is treated as a free param- duction to these type of models. eter and fitted to the data. In the latest meson-exchange Conventional meson-exchange models of the YN in- YN potential presented by the Jülich group [21] a micro- ¯ teraction are usually also based on the assumption of scopic model of correlated ππ and KK exchange [154] is SU(3) flavor symmetry for the occurring coupling con- utilized to fix the contributions in the scalar-isoscalar (σ) stants, and in some cases even on the SU(6) symmetry of and vector-isovector (ρ) channels. the [18, 19]. In the derivation of the meson- Let us mention for completeness that meson-exchange exchange contributions one follows essentially the same models are typically equipped with phenomenological procedure as outlined in Sect.IIIB for the case of pseu- form factors in order to cut off the potential for large doscalar mesons. Besides the lowest pseudoscalar-meson momenta (short distances). For example, in case of the 17

propriate to use these three-body interactions together with the NLO two-body interaction of Sec.III. As al- ready stated, the irreducible contributions to the chi- ral potential are presented. In contrast to typical phe- nomenological calculations, diagrams as on the left side of Fig.4 do not lead to a genuine three-body potential, but are an iteration of the two-baryon potential. Such Figure 8: Leading three-baryon interactions: contact diagrams will be incorporated automatically when solv- term, one-meson exchange and two-meson exchange. ing, e.g., the Faddeev (or Yakubovsky) equations within Filled circles and solid dots denote vertices with ∆i = 1 a coupled-channel approach. The three-body potentials and ∆ = 0, respectively. i derived from SU(3) χEFT are expected to shed light on the effect of 3BFs in hypernuclear systems. Especially in calculations about light hypernuclei these potentials YN models of the Jülich group the interaction is supple- can be implemented within reliable few-body techniques mented with form factors for each meson-baryon-baryon [48, 49, 51, 52]. vertex, cf. [18, 19] for details. Those form factors are meant to take into account the extended hadron struc- ture and are parametrized in the conventional monopole A. Contact interaction or dipole form. In case of the Nijmegen potentials a Gaussian form factor is used. In addition there is some In the following we consider the leading three-baryon additional sophisticated short-range phenomenology that contact interaction. Following the discussion in Sub- controls the interaction at short distances [22, 23]. sec.IIC the corresponding Lagrangian can be con- structed. The inclusion of external fields is not neces- sary, as we are interested in the purely baryonic contact IV. THREE-BARYON INTERACTION term. One ends up with the following possible structures POTENTIALS in flavor space [120] hB¯B¯BBBB¯ i , hB¯BB¯ BBB¯ i , hB¯BBB¯ BB¯ i , Three-nucleon forces are an essential ingredient for hBB¯ BB¯ BB¯ i , hB¯BBB¯ ihBB¯ i , hBB¯ BB¯ ihBB¯ i , a proper description of nuclei and nuclear matter with low-momentum two-body interactions. Similarly, three- hB¯B¯BB¯ ihBBi , hB¯B¯B¯ihBBBi , hB¯BB¯ ihBBB¯ i , baryon forces, especially the ΛNN interaction, are ex- hBB¯ ihBB¯ ihBB¯ i , hB¯B¯ihBB¯ ihBBi , (67) pected to play an important role in nuclear systems with strangeness. Their introduction in calculations of light with possible Dirac structures hypernuclei seems to be required. Furthermore, the in- µ µ 1 ⊗ 1 ⊗ 1 , 1 ⊗ γ5γ ⊗ γ5γµ , γ5γ ⊗ 1 ⊗ γ5γµ , troduction of 3BF is traded as a possible solution to the µ 1 µν hyperon puzzle (see Sec.I). However, so far only phe- γ5γ ⊗ γ5γµ ⊗ , γ5γµ ⊗ i σ ⊗ γ5γν , (68) nomenological 3BF have been employed. In this section leading to the following operators in the three-body spin we present the leading irreducible three-baryon interac- space tions from SU(3) chiral effective field theory as derived 1 in Ref. [120]. We show the minimal effective Lagrangian , ~σ1 · ~σ2 , ~σ1 · ~σ3 , ~σ2 · ~σ3 , i ~σ1 · (~σ2 × ~σ3) . (69) required for the pertinent vertices. Furthermore the esti- All combinations of these possibilities leads to a (largely mation of the corresponding LECs through decuplet sat- overcomplete) set of terms for the leading covariant La- uration and an effective density-dependent two-baryon grangian. Note that in Ref. [120] the starting point is a potential will be covered [121]. covariant Lagrangian, but the minimal non-relativistic According to the power counting in Eq. (47) the 3BF Lagrangian is the goal. Hence, only Dirac structures arise formally at NNLO in the chiral expansion, as can leading to independent (non-relativistic) spin operators be seen from the hierarchy of baryonic forces in Fig.5. are relevant. Three types of diagrams contribute: three-baryon con- Let us consider the process B1B2B3 → B4B5B6, tact terms, one-meson and two-meson exchange dia- where the Bi are baryons in the particle basis, Bi ∈ grams, cf. Fig.8. Note that a two-meson exchange dia- {n, p, Λ, Σ+, Σ0, Σ−, Ξ0, Ξ−}. The contact potential V gram, such as in Fig.8, with a (leading order) Weinberg- has to be derived within a threefold spin space for this Tomozawa vertex in the middle, would formally be a process. The operators in spin-space 1 is defined to act NLO contribution. However, as in the nucleonic sector, between the two-component Pauli spinors of B1 and B4. this contribution is kinematically suppressed due to the In the same way, spin-space 2 belongs to B2 and B5, and fact that the involved meson energies are differences of spin-space 3 to B3 and B6. For a fixed spin configuration baryon kinetic energies. Anyway, parts of these NNLO the potential can be calculated from contributions get promoted to NLO by the introduction † † † of intermediate decuplet baryons, so that it becomes ap- (1) (2) (3) (1) (2) (3) (70) χB4 χB5 χB6 V χB1 χB2 χB3 , 18

ijk i j k where the superscript of a spinor denotes the spin space − i  C15hB¯aB¯bB¯c(σ B)a(σ B)b(σ B)ci and the subscript denotes the baryon to which the + i ijkC hB¯ B¯ (σiB) B¯ (σjB) (σkB) i spinor belongs. The potential is obtained as V = 16 a b a c b c ijk ¯ ¯ i j ¯ k −hB4B5B6| L |B1B2B3i, where the contact Lagrangian − i  C17hBaBb(σ B)a(σ B)bBc(σ B)ci has to be inserted, and the 36 Wick contractions need ijk i j k L + i  C18hB¯a(σ B)aB¯b(σ B)bB¯c(σ B)ci , (71) to be performed. The number 36 corresponds to the 3! × 3! possibilities to arrange the three initial and three with vector indices i, j, k and two-component spinor in- final baryons into Dirac bilinears. One obtains six direct dices a, b, c. In total 18 low-energy constants C1 ...C18 terms, where the baryon bilinears combine the baryon are present. The low-energy constant E of the six- pairs 1–4, 2–5 and 3–6, as shown in Eq. (70). For the nucleon contact term (cf. Ref. [78]) can be expressed other 30 Wick contractions, the resulting potential is not through these LECs by E = 2(C4 − C9). fitting to the form of Eq. (70), because the wrong baryon As in the two-body sector, group theoretical consider- pairs are connected in a separate spin space. Hence, an ations can deliver valueable constrains on the resulting appropriate exchange of the spin wave functions in the potentials. In flavor space the three octet baryons form final state has to be performed. This is achieved by mul- the 512-dimensional tensor product 8 ⊗ 8 ⊗ 8, which de- tiplying the potential with the well-known spin-exchange composes into the following irreducible SU(3) represen- (σ) 1 1 tations operators Pij = 2 ( + ~σi · ~σj). Furthermore additional minus signs arise from the interchange of anticommut- 8 ⊗ 8 ⊗ 8 = ing baryon fields. The full potential is then obtained by adding up all 36 contributions to the potential. One 64 ⊕ (35 ⊕ 35)2 ⊕ 276 ⊕ (10 ⊕ 10)4 ⊕ 88 ⊕ 12 , obtains a potential that fulfills automatically the gener- (72) alized Pauli principle and that is fully antisymmetrized. where the multiplicity of an irreducible representations is In order to obtain a minimal set of Lagrangian terms of denoted by subscripts. In spin space one obtain for the the final potential matrix have been eliminated until the product of three doublets rank of the final potential matrix (consisting of multiple Lagrangian terms and the spin structures in Eq. (69)) 2 ⊗ 2 ⊗ 2 = 22 ⊕ 4 . (73) matches the number of terms in the Lagrangian. The Transitions are only allowed between irreducible repre- minimal non-relativistic six-baryon contact Lagrangian sentations of the same type. Analogous to Ref. [145] for is [120] the two-baryon sector, the contributions of different ir- reducible representations to three-baryon multiplets in L = − C1hB¯aB¯bB¯cBaBbBci Tab.IV can be established. At leading order only transi- ¯ ¯ ¯ + C2hBaBbBaBcBbBci tions between S-waves are possible, since the potentials − C3hB¯aB¯bBaBbB¯cBci are momentum-independent. Due to the Pauli principle the totally symmetric spin-quartet 4 must combine with + C hB¯ B B¯ B B¯ B i 4 a a b b c c the totally antisymmetric part of 8⊗8⊗8 in flavor space, − C5hB¯aB¯bBaBbi hB¯cBci  Alt3(8) = 56a = 27a + 10a + 10a + 8a + 1a . (74) − C hB¯ B¯ B¯ B (σiB) (σiB) i 6 a b c a b c It follows, that these totally antisymmetric irreducible  + hB¯ B¯ B¯ (σiB) (σiB) B i representations are present only in states with total spin c b a c b a 3/2. The totally symmetric part of 8 ⊗ 8 ⊗ 8 leads to  i i + C7 hB¯aB¯bBaB¯c(σ B)b(σ B)ci Sym3(8) = 120s = 64s +27s +10s +10s +8s +1s . (75) i i  + hB¯cB¯b(σ B)cB¯a(σ B)bBai However, the totally symmetric flavor part has no totally antisymmetric counterpart in spin space, hence these  i i − C8 hB¯aB¯bBa(σ B)bB¯c(σ B)ci representations do not contribute to the potential. In Tab.IV these restrictions obtained by the generalized i i  + hB¯bB¯a(σ B)bBaB¯c(σ B)ci Pauli principle have already be incorporated. The po- tentials of Ref. [120] (decomposed in isospin basis and ¯ ¯ i ¯ i + C9hBaBaBb(σ B)bBc(σ B)ci partial waves) fulfill the restrictions of Tab.IV. For ex-  i i ample the combination of LECs related to the represen- − C10 hB¯aB¯bBa(σ B)bi hB¯c(σ B)ci tation 35 is present in the NNN interaction as well as i i  + hB¯bB¯a(σ B)bBai hB¯c(σ B)ci in the ΞΞΣ (−5, 2) interaction. i i − C11hB¯aB¯bB¯c(σ B)aBb(σ B)ci i i B. One-meson exchange component + C12hB¯aB¯b(σ B)aB¯cBb(σ B)ci ¯ ¯ i i ¯ − C13hBaBb(σ B)a(σ B)bBcBci The meson-baryon couplings in the one-meson ex- i i − C14hB¯aB¯b(σ B)a(σ B)bi hB¯cBci change diagram of Fig.8 emerges from the leading-order 19

2 4 states (S, I) S1/2 S3/2 1 NNN (0, 2 ) 35

ΛNN, ΣNN (−1, 0) 10, 35 10a ΛNN, ΣNN (−1, 1) 27, 35 27a ΣNN (−1, 2) 35 1 ΛΛN, ΣΛN, ΣΣN, ΞNN (−2, 2 ) 8, 10, 27, 35 8a, 10a, 27a 3 ΣΛN, ΣΣN, ΞNN (−2, 2 ) 10, 27, 35, 35 10a, 27a 5 ΣΣN (−2, 2 ) 35

ΛΛΛ, ΣΣΛ, ΣΣΣ, ΞΛN, ΞΣN (−3, 0) 8, 27 1a, 8a, 27a ΣΛΛ, ΣΣΛ, ΣΣΣ, ΞΛN, ΞΣN (−3, 1) 8, 10, 10, 27, 35, 35 8a, 10a, 10a, 27a ΣΣΛ, ΣΣΣ, ΞΣN (−3, 2) 27, 35, 35 27a 1 ΞΛΛ, ΞΣΛ, ΞΣΣ, ΞΞN (−4, 2 ) 8, 10, 27, 35 8a, 10a, 27a 3 ΞΣΛ, ΞΣΣ, ΞΞN (−4, 2 ) 10, 27, 35, 35 10a, 27a 5 ΞΣΣ (−4, 2 ) 35

ΞΞΛ, ΞΞΣ (−5, 0) 10, 35 10a ΞΞΛ, ΞΞΣ (−5, 1) 27, 35 27a ΞΞΣ (−5, 2) 35 1 ΞΞΞ (−6, 2 ) 35

Table IV: Irreducible representations for three-baryon states with strangeness S and isospin I in partial waves 2S+1 1 3 1 3 | LJ i, with the total spin S = 2 , 2 , the L = 0 and the total angular momentum J = 2 , 2 [120].

(1)  i i chiral Lagrangian LB , see Eq. (56). The other vertex − D6/f0 hB¯b(∇ φ)B¯a(σ B)bBai involves four baryon fields and one pseudoscalar-meson i i  field. In Ref. [120] an overcomplete set of terms for the + hB¯aB¯bBa(∇ φ)(σ B)bi corresponding Lagrangian has been constructed. In or-  i i der to obtain the complete minimal Lagrangian from the − D7/f0 hB¯aB¯bBa(σ B)b(∇ φ)i overcomplete set of terms, the matrix elements of the pro-  ¯ ¯ i i cess B1B2 → B3B4φ1 has been considered in Ref. [120]. + hBbBa(σ B)bBa(∇ φ)i The corresponding spin operators in the potential are i i + D8/f0hB¯a(∇ φ)BaihB¯b(σ B)bi ¯ i ¯ i ~σ1 · ~q, ~σ2 · ~q, i (~σ1 × ~σ2) · ~q, (76) + D9/f0hBaBa(∇ φ)ihBb(σ B)bi i i + D10/f0hB¯b(∇ φ)(σ B)bihB¯aBai where ~q denotes the momentum of the emitted meson. ijk h ¯ i ∇k ¯ j i Redundant term are removed until the rank of the po- + i  D11/f0 Ba(σ B)a( φ)Bb(σ B)b ijk  k i j tential matrix formed by all transitions and spin oper- − i  D12/f0 hB¯a(∇ φ)B¯b(σ B)a(σ B)bi ators matches the number of terms in the Lagrangian. j k i  One ends up with the minimal non-relativistic chiral La- − hB¯bB¯a(σ B)b(∇ φ)(σ B)ai grangian ijk k i j − i  D13/f0hB¯aB¯b(∇ φ)(σ B)a(σ B)bi ¯ i ¯ i ijk i j k L = D1/f0hBa(∇ φ)BaBb(σ B)bi − i  D14/f0hB¯aB¯b(σ B)a(σ B)b(∇ φ)i , (77)  i i + D2/f0 hB¯aBa(∇ φ)B¯b(σ B)bi with two-component spinor indices a and b and 3-vector indices i, j and k. For all possible strangeness sectors i i  + hB¯aBaB¯b(σ B)b(∇ φ)i S = −4 ... 0 one obtains in total 14 low-energy constants D ...D . The low-energy constant of the correspond- h ¯ ∇i i ¯ i 1 14 + D3/f0 Bb( φ)(σ B)bBaBa ing vertex in the nucleonic sector D is related to the LECs  i i above by − − .2 − D4/f0 hB¯a(∇ φ)B¯bBa(σ B)bi D = 4(D1 D3 + D8 D10) To obtain the 3BF one-meson-exchange diagram, the i i  + hB¯bB¯a(σ B)b(∇ φ)Bai generic one-meson-exchange diagram in Fig. 9(a) can be

 i i − D5/f0 hB¯aB¯b(∇ φ)Ba(σ B)bi 2 This LEC D has not to be confused with the axial-vector coupling i i  + hB¯bB¯a(∇ φ)(σ B)bBai constant D in Eq. (56). 20 l m n l m n

φ φ1 φ2

i j k i j k ABC ABC (a) Generic one-meson exchange diagram (b) Generic two-meson exchange diagram

Figure 9: Generic meson-exchange diagrams. The wiggly line symbolized the four-baryon contact vertex, to illustrate the baryon bilinears. investigated. It involves the baryons i, j, k in the ini- grangian [139]. The relevant terms are [140] tial state, the baryons l, m, n in the final state and an ¯ ¯ ¯ exchanged meson φ. The contact vertex on the right L = bDhB{χ+,B}i + bF hB[χ+,B]i + b0hBBi hχ+i µ µ is pictorially separated into two parts to indicate that + b1hB¯[u , [uµ,B]]i + b2hB¯{u , {uµ,B}}i baryon j–m and k–n are in the same bilinear. The spin + b hB¯{uµ, [u ,B]}i + b hBB¯ i huµu i spaces corresponding to the baryon bilinears are denoted 3 µ 4 µ ¯ µ ν by A, B, C. + id1hB{[u , u ], σµν B}i + id hB¯[[uµ, uν ], σ B]i On obtains a generic potential of the form 2 µν µ ν + id3hBu¯ ihu σµν Bi , (79)

1 3 1 with uµ = − ∂µφ+O(φ ) and χ+ = 2χ− 2 {φ, {φ, χ}} 1 ~σA · ~qli   f0 4f0 · × · (78) 4 V = 2 2 2 N1~σC ~qli +N2i (~σB ~σC ) ~qli , + O(φ ), where 2f0 ~qli + mφ  2  mπ 0 0 2 χ =  0 mπ 0  . (80) with the momentum transfer ~qli = ~pl − ~pi carried by the 2 2 0 0 2mK − mπ exchanged meson. The constants N1 and N2 are linear combinations of low-energy constants. The terms proportional to bD, bF , b0 break explicitly SU(3) flavor symmetry, because of different meson masses The complete one-meson exchange three-baryon po- mK =6 mπ. The LECs of Eq. (79) are related to the con- tential for the process B1B2B3 → B4B5B6 is finally ob- ventional LECs of the nucleonic sector by [155, 156] tained by summing up the 36 permutations of initial-state and final-state baryons for a fixed meson and by summing 1  0 + − + − 0 0 c1 = (2b0 + bD + bF ) , over all mesons φ ∈ π , π , π ,K ,K ,K , K¯ , η . 2 Additional minus signs arise from interchanging fermions c3 = b1 + b2 + b3 + 2b4 , and some diagrams need to be multiplied by spin ex- c = 4(d + d ) . (81) change operators in order to be consistent with the form 4 1 2 set up in Eq. (70). As defined before, the baryons B1, B2 To obtain the potential of the two-meson exchange di- and B3 belong to the spin-spaces 1, 2 and 3, respectively. agram of Fig.8, the generic diagram of Fig. 9(b) can be considered. It includes the baryons i, j, k in the ini- tial state, the baryons l, m, n in the final state, and two exchanged mesons φ1 and φ2. The spin spaces corre- sponding to the baryon bilinears are denoted by A, B, C and they are aligned with the three initial baryons. The C. Two-meson exchange component momentum transfers carried by the virtual mesons are ~qli = ~pl − ~pi and ~qnk = ~pn − ~pk. One obtains the generic transition amplitude The two-meson exchange diagram of Fig.8 includes 1 ~σA · ~qli ~σC · ~qnk the vertex arising from the Lagrangian in Eq. (56). Fur- V = − 4 2 2 2 2 2 4f (~q + m )(~q + m ) thermore the the well-known O(q ) meson-baryon La- 0 li φ1 nk φ2 grangian [139] is necessary. For the two-meson exchange  0 0 0  × N + N ~qli · ~qnk + N i (~qli × ~qnk) · ~σB , diagram of Fig.8 we need in addition to the Lagrangian 1 2 3 in Eq. (56) the well-known O(q2) meson-baryon La- (82) 21

0 (σ) 1 1 with Ni linear combinations of the low-energy constants erators in spin space Pij = 2 ( + ~σi · ~σj) and in isospin of the three involved vertices. The complete three-body (τ) space P = 1 (1 + ~τ · ~τ ) have been introduced. potential for a transition → can be ij 2 i j B1B2B3 B4B5B6 The ΛNN three-body interaction generated by two- calculated by summing up the contributions of all 18 dis- pion exchange is given by tinguable Feynman diagrams and by summing over all possible exchanged mesons. If the baryon lines are not in 2 ΛNN gA ~σ3 · ~q63 ~σ2 · ~q52 V = ~τ2 · ~τ3 the configuration 1–4, 2–5 and 3–6 additional (negative) TPE 3f 4 (~q 2 + m2 )(~q 2 + m2 ) spin-exchange operators have to be included. 0 63 π 52 π  2  × − (3b0 + bD)mπ + (2b2 + 3b4) ~q63 · ~q52 2 D. ΛNN three-baryon potentials (σ) (τ) gA ~σ3 · ~q53 ~σ2 · ~q62 − P23 P23 4 2 2 2 2 ~τ2 · ~τ3 3f0 (~q53 + mπ)(~q62 + mπ) In order to give a concrete example the explicit expres-  2  × − (3b0 + bD)mπ + (2b2 + 3b4) ~q53 · ~q62 . sion for the ΛNN three-body potentials in spin-, isospin- and momentum-space are presented for the contact in- (85) teraction and one- and two-pion exchange contributions Due to the vanishing of the ΛΛπ vertex, only those two [120]. The potentials are calculated in the particle basis diagrams contribute, where the (final and initial) Λ hy- and afterwards rewritten into isospin operators. peron are attached to the central baryon line. The ΛNN contact interaction is described by the fol- lowing potential ΛNN 0 1 E. Three-baryon force through decuplet saturation Vct = C1 ( − ~σ2 · ~σ3)(3 + ~τ2 · ~τ3) + C0 ~σ · (~σ + ~σ )(1 − ~τ · ~τ ) 2 1 2 3 2 3 Low-energy two- and three-body interactions derived 0 1 + C3 (3 + ~σ2 · ~σ3)( − ~τ2 · ~τ3) , (83) from SU(2) χEFT are used consistently in combination where the primed constants are linear combinations of with each other in nuclear few- and many-body calcu- lations. The a priori unknown low-energy constants are C1 ...C18 of Eq. (71). The symbols ~σ and ~τ denote the usual Pauli matrices in spin and isospin space. The con- fitted, for example, to NN scattering data and 3N ob- 0 servables such as 3-body binding energies [78]. Some of stant C1 appears only in the transition with total isospin 0 0 these LECs are, however, large compared to their or- I = 1. The constants C2 and C3 contribute for total isospin I = 0. der of magnitude as expected from the hierarchy of nu- For the ΛNN one-pion exchange three-body poten- clear forces in Fig.5. This feature has its physical origin tials, various diagrams are absent due to the vanish- in strong couplings of the πN-system to the low-lying ing ΛΛπ-vertex, which is forbidden by isospin symmetry. ∆(1232)-resonance. It is therefore natural to include the One obtains the following potential ∆(1232)-isobar as an explicit degree of freedom in the chiral Lagrangian (cf. Refs. [157–159]). The small mass ΛNN gA difference between nucleons and deltas (293 MeV) intro- VOPE = − 2 2f0 duces a small scale, which can be included consistently  ~σ · ~q h i in the chiral power counting scheme and the hierarchy of × 2 52 · 0 0 · 2 2 ~τ2 ~τ3 (D1~σ1 + D2~σ3) ~q52 nuclear forces. The dominant parts of the three-nucleon ~q52 + mπ interaction mediated by two-pion exchange at NNLO are ~σ · ~q h i 3 63 0 0 then promoted to NLO through the delta contributions. + 2 2 ~τ2 · ~τ3 (D1~σ1 + D2~σ2) · ~q63 ~q63 + mπ The appearance of the inverse mass splitting explains (σ) (τ) (σ) ~σ2 · ~q62 the large numerical values of the corresponding LECs + P23 P23 P13 2 2 ~τ2 · ~τ3 ~q62 + mπ [13, 160]. 0 0 In SU(3) χEFT the situation is similar. In systems h D1 + D2 × − (~σ1 + ~σ3) · ~q62 with strangeness S = −1 like ΛNN, resonances such as 2 ∗ 0 0 the spin-3/2 Σ (1385)-resonance play a similar role as the D1 − D2 i + i (~σ3 × ~σ1) · ~q62 ∆ in the NNN system, as depicted in Fig.4 on the right 2 side. The small decuplet-octet mass splitting (in the chi- (σ) (τ) (σ) ~σ3 · ~q53 ral limit), ∆ := M10−M8, is counted together with exter- + P23 P23 P12 2 2 ~τ2 · ~τ3 ~q53 + mπ nal momenta and meson masses as O(q) and thus parts 0 0 h D1 + D2 of the NNLO three-baryon interaction are promoted to × − (~σ1 + ~σ2) · ~q53 2 NLO by the explicit inclusion of the baryon decuplet, as 0 0  illustrated in Fig. 10. It is therefore likewise compelling D1 − D2 i − i (~σ1 × ~σ2) · ~q53 , (84) 2 to treat the three-baryon interaction together with the NLO hyperon-nucleon interaction of Sec.III. Note that 0 0 with only two constants D1 and D2, which are linear in the nucleonic sector, only the two-pion exchange dia- combinations of the constants D1 ...D14. Exchange op- gram with an intermediate ∆-isobar is allowed. Other di- 22 agrams are forbidden due to the Pauli principle, as we will set of terms is given by [121]: show later. For three flavors more particles are involved and, in general, also the other diagrams (contact and one- 3   X  ¯ ~† ¯  meson exchange) with intermediate decuplet baryons in L = abc TadeS Bdb · Bfc~σBef a,b,c, Fig. 10 appear. d,e,f=1     The large number of unknown LECs presented in the + B¯bdST~ ade · B¯fe~σBcf previous subsections is related to the multitude of three- 3 baryon multiplets, with strangeness ranging from 0 to X  †  + H   T¯ S~ B · B¯ ~σB  −6. For selected processes only a small subset of these 2 abc ade fb dc ef a,b,c, constants contributes as has been exemplified for the d,e,f=1 ΛNN three-body interaction. In this section we present   ¯ ~ ¯   the estimation of these LECs by resonance saturation as + Bbf STade · Bfe~σBcd , done in Ref. [121]. (88)

The leading-order non-relativistic interaction La- with the LECs H1 and H2. Again one can employ group grangian between octet and decuplet baryons (see, e.g., theory to justify the number of two constants for a tran- Ref. [161]) is sition BB → B∗B. In flavor space the two initial octet baryons form the tensor product 8 ⊗ 8, and in spin space they form the product 2 ⊗ 2. These tensor products can be decomposed into irreducible representations: 3   C X ¯ ~ † ~ ∗ L = abc TadeS · ∇φdb Bec 8 ⊗ 8 = 27 ⊕ 8s ⊕ 1 ⊕ 10 ⊕ 10 ⊕ 8a , f0 a,b,c,d,e=1 | {z } | {z } symmetric antisymmetric !   2 ⊗ 2 = 1a ⊕ 3s . (89) + B¯ceS~ · ∇~ φbd Tade , (86) In the final state, having a decuplet and an octet baryon, the situation is similar: where the decuplet baryons are represented by the to- 10 ⊗ 8 = 35 ⊕ 27 ⊕ 10 ⊕ 8 , tally symmetric three-index tensor T , cf. Eq. (23). At 4 ⊗ 2 = 3 ⊕ 5 . (90) this order only a single LEC C appears. Typically the As seen in the previous sections, at leading order only (large-N ) value C = 3 g ≈ 1 is used, as it leads to a c 4 A -wave transitions occur, as no momenta are involved. decay width Γ(∆ → πN) = 110.6 MeV that is in good S Transitions are only allowed between the same types of agreement with the empirical value of Γ(∆ → πN) = irreducible (flavor and spin) representations. Therefore, (115±5) MeV [158]. The spin 1 to 3 transition operators 2 2 in spin space the representation 3 has to be chosen. Be- ~ S connect the two-component spinors of octet baryons cause of the Pauli principle in the initial state, the sym- with the four-component spinors of decuplet baryons (see metric 3 in spin space combines with the antisymmet- e.g., [162]). In their explicit form they are given as 2 × 4 ric representations 10, 10∗, 8 in flavor space. But only transition matrices a 10 and 8a have a counterpart in the final state flavor space. This number of two allowed transitions matches the number of two LECs in the minimal Lagrangian. An- ! − √1 √1 other interesting observation can be made from Eqs. (89) 2 0 6 0 S1 = 1 1 , and (90). For NN states only the representations 27 0 − √ 0 √ 6 2 and 10∗ can contribute, as can be seen, e.g., in Tab.III. ! − √i − √i But these representations combine either with the wrong 2 0 6 0 S2 = , spin, or have no counterpart in the final state. There- 0 − √i 0 − √i 6 2 fore, NN → ∆N transitions in S-waves are not allowed  q 2  because of the Pauli principle. 0 3 0 0 S3 = . (87) Having the above two interaction types at hand, one  q 2  0 0 3 0 can estimate the low-energy constants of the leading three-baryon interaction by decuplet saturation using the diagrams shown in Fig. 10. At this order, where no loops are involved, one just needs to evaluate the diagrams with † 1 an intermediate decuplet baryon and the diagrams with- These operators fulfill the relation SiSj = 3 (2δij − iijkσk). out decuplet baryons and compare them with each other. In order to estimate the LECs of the six-baryon con- A non-relativistic B∗BBB Lagrangian with a minimal tact Lagrangian of Eq. (71), one can consider the process 23

three-baryon force decuplet-less EFT decuplet-contribution

LO

NLO

NNLO ···

Figure 10: Hierarchy of three-baryon forces with explicit introduction of the baryon decuplet (represented by double lines).

= = + = +

(a) Saturation of the (b) Saturation of the BB → BBφ (c) Saturation of the NLO baryon-meson vertex six-baryon contact interaction vertex

Figure 11: Saturation via decuplet resonances.

B1B2B3 → B4B5B6 as depicted in Fig. 11(a). The left tions, summing over all intermediate decuplet baryons, side of Fig. 11(a) has already been introduced in the pre- and comparing the left side of Fig. 11(b) with the right vious subsection and can be obtained by performing all hand side for all combinations of baryons and mesons, the 36 Wick contractions. For the diagrams on right side of LECs can be estimated. The LECs of the minimal non- Fig. 11(a) the procedure is similar. After summing over relativistic chiral Lagrangian for the four-baryon vertex ∗ all intermediate decuplet baryons B , the full three-body including one meson of Eq. (77) D1,...,D14 are then potential of all possible combinations of baryons on the proportional to C/∆ and to linear combinations of H1 left side of Fig. 11(a) can be compared with the ones on and H2 [121]. the right side. In the end the 18 LECs of the six-baryon contact Lagrangian C1,...,C18 of Eq. (71) can be ex- 2 pressed as linear combinations of the combinations H1 , 2 H2 and H1H2 and are proportional to the inverse average The last class of diagrams is the three-body interaction decuplet-octet baryon mass splitting 1/∆ [121]. with two-meson exchange. As done for the one-meson ex- Since we are at the leading order only tree-level di- change, the unknown LECs can be saturated directly on agrams are involved and we can estimate the LECs of the level of the vertex and one can consider the process the one-meson-exchange part of the three-baryon forces B1φ1 → B2φ2 as shown in Fig. 11(c). A direct compari- already on the level of the vertices, as depicted in son of the transition matrix elements for all combinations Fig. 11(b). We consider the transition matrix elements of baryons and mesons after summing over all interme- ∗ of the process B1B2 → B3B4φ and start with the left diate decuplet baryons B leads to the following contri- side of Fig. 11(b). After doing all possible Wick contrac- butions to the LECs of the meson-baryon Lagrangian in 24

corrections to the NN interaction have been calculated from the leading chiral three-nucleon forces. This work has been extended to the strangeness sector in Ref. [121]. In order to obtain an effective baryon-baryon interaction from the irreducible 3BFs in Fig.8, two baryon lines (1) (2a) (2b) (3) have been closed, which represents diagrammatically the sum over occupied states within the Fermi sea. Such a “medium insertion” is symbolized by short double lines on a baryon propagator. All types of diagrams arising this way are shown in Fig. 12. In Ref. [121] the calculation is restricted to the contact (4) (5a) (5b) (6) term and to the contributions from one- and two-pion ex- change processes, as they are expected to be dominant. Figure 12: Effective two-baryon interaction from When computing the diagrams of Fig.8 the medium in- genuine three-baryon forces. Contributions arise from sertion corresponds to a factor −2πδ(k0)θ(kf − |~k|). Fur- two-pion exchange (1), (2a), (2b), (3), one-pion thermore, an additional minus sign comes from a closed exchange (4), (5a), (5b) and the contact interaction (6). loop. The effective two-body interaction can also be calculated from the expressions for the three-baryon potentials of Ref. [120] via the relation

Eq. (79): X Z d3k V12 = trσ3 3 V123 , (94) ~ B (2π) bD = 0 , bF = 0 , b0 = 0 , B |k|≤kf 7C2 C2 C2 C2 b1 = , b2 = , b3 = − , b4 = − , where trσ3 denotes the spin trace over the third particle 36∆ 4∆ 3∆ 2∆ and where a summation over all baryon species B in the 2 2 2 B C C C Fermi sea (with Fermi momentum kf ) is done. d1 = , d2 = , d3 = − , (91) 12∆ 36∆ 6∆ As an example of such an effective interaction, we dis- play the the effective ΛN interaction in nuclear matter These findings are consistent with the ∆(1232) contribu- (with ρ =6 ρ ) derived in Ref. [121]. It is determined tion to the LECs c , c , c (see Eq. (81)) in the nucleonic p n 1 3 4 from two-pion-exchange, one-pion-exchange and contact sector [157, 160]: ΛNN three-body forces. Only the expressions for the Λn 2 potential are shown as the Λp potential can be obtained gA p c1 = 0 , c3 = −2c4 = − . (92) just by interchanging the Fermi momenta with n (or 2∆ kf kf the densities ρp with ρn) in the expressions for Λn. In the Employing the LECs obtained via decuplet saturation, following formulas the sum over the contributions from the constants of the ΛNN interaction (contact interac- the and in the Fermi sea is already em- tion, one-pion and two-pion exchange) of Subsec.IVD ployed. Furthermore, the values of the LECs are already can be evaluated: estimated via decuplet saturation, see Subsec.IVE. The 2 topologies (1), (2a) and (2b) vanish here because of the 0 0 (H1 + 3H2) 0 C1 = C3 = ,C2 = 0 , non-existence of an isospin-symmetric vertex. One 72∆ ΛΛπ obtains the density-dependent potential in a nuclear 2C(H + 3H ) Λn D0 = 0 ,D0 = 1 2 , medium 1 2 9∆ 2 2 2 ( C med,ππ C g 3b0 + bD = 0 , 2b2 + 3b4 = − . (93) V = A ∆ Λn 2 4 12π f0 ∆ Obviously, the only unknown constant here is the combi- 1h8 n3 p3 2 2 2 0 (k + 2k ) − 4(q + 2m )Γ˜0(p) − 2q Γ˜1(p) nation H = H1 + 3H2. It is also interesting to see, that 4 3 f f the (positive) sign of the constants C0 for the contact in- i i 2 2 2 ˜ teraction is already fixed, independently of the values of + (q + 2m ) G0(p, q) the two LECs H1 and H2. i  + (~q × ~p ) · ~σ2 2Γ˜0(p) + 2Γ˜1(p) 2 ) F. Effective in-medium two-baryon interaction 2 2  − (q + 2m )(G˜0(p, q) + 2G˜1(p, q)) , (95)

In this subsection we summarize how the effect of three-body force in the presence of a (hyper)nuclear 0 medium can be incorporated in an effective baryon- med,π gACH n3 p3 2 ˜  VΛn = 2 2 2(kf + 2kf ) − 3m Γ0(p) , (96) baryon potential. In Ref. [163] the density-dependent 54π f0 ∆ 25

02 √ q q med,ct H 2 2 2 2 has been used, s = MB + kν + MB + kν , in VΛn = (ρn + 2ρp) . (97) 1,ν 2,ν 18∆ order to get the two-particle thresholds at their correct The different topologies related to two-pion exchange positions. The physical baryon masses have been used ((1), (2a), (2b), (3)) and one-pion exchange ((4), (5a), in the Lippmann-Schwinger equation, which introduces (5b)) have already been combined in V med,ππ and some additional SU(3) symmetry breaking. Relativistic V med,π, respectively. The density and momentum de- kinematics has also been used to relate the laboratory ˜ ˜ pendent loop functions Γi(p) and Gi(p, q) can be found momentum√ plab of the hyperon to the center-of-mass en- in Ref. [121]. The only spin-dependent term is the one ergy s. The Coulomb interaction has been implemented 1 1 proportional to ~σ2 = 2 (~σ1 + ~σ2) − 2 (~σ1 − ~σ2) and there- by the use of the Vincent-Phatak method [147, 164]. fore one recognizes a symmetric and an antisymmetric Similar to the nucleonic sector at NLO [147], a regula- 04 4 4 spin-orbit potential of equal but opposite strength. tor function of the form fR(Λ) = exp[−(k + k )/Λ ] is employed to cut off the high-energy components of the potential. For higher orders in the chiral power count- V. APPLICATIONS ing, higher powers than 4 in the exponent of fR have to be used. This ensures that the regulator introduces A. Hyperon-nucleon and hyperon-hyperon only contributions, that are beyond the given order. The scattering cutoff Λ is varied in the range (500 ... 700) MeV, i.e., comparable to what was used for the NN interaction in With the hyperon-nucleon potentials outlined in Ref. [147]. The resulting bands represent the cutoff de- Sec.III hyperon-nucleon scattering processes can be in- pendence, after readjusting the contact parameters, and vestigated. The very successful approach to the nucleon- thus could be viewed as a lower bound on the theoreti- nucleon interaction of Refs. [147, 149, 150] within SU(2) cal uncertainty. Recently, improved schemes to estimate χEFT, has been extended to the leading-order baryon- the theoretical uncertainty were proposed and applied to baryon interaction in Refs. [29–31] by the Bonn-Jülich the NN interaction [165–168]. Some illustrative results group. In Refs. [33, 35, 36] this approach has been ex- for YN scattering based on the method by Epelbaum et tended to next-to-leading order in SU(3) chiral effective al. [166, 168] have been included in Ref. [34]. However, field theory. As mentioned in Subsec.IID the chiral such schemes require higher orders than NLO in the chi- power counting is applied to the potential, where only ral power counting if one wants to address questions like two-particle irreducible diagrams contribute. These po- the convergence of the expansion. tentials are then inserted into a regularized Lippmann- The partial-wave contributions of the meson-exchange Schwinger equation to obtain the reaction amplitude diagrams are obtained by employing the partial-wave de- (or T -matrix). In contrast to the NN interaction, the composition formulas of Ref. [29]. For further remarks on Lippmann-Schwinger equation for the YN interaction in- the employed approximations and the fitting strategy we volves not only coupled partial waves, but also coupled refer the reader to Ref. [33]. As can be seen in Tab.III, two-baryon channels. The coupled-channel Lippmann- one gets for the YN contact terms five independent LO Schwinger equation in the particle basis reads after constants, acting in the S-waves, eight additional con- partial-wave decomposition (see also Fig.2) stants at NLO in the S-waves, and nine NLO constant acting in the P -waves. The contact terms represent the 00 0 √ 00 0 ρ ρ ,J 00 0 ρ ρ ,J 00 0 unresolved short-distance dynamics, and the correspond- Tν00ν0 (k , k ; s) = Vν00ν0 (k , k ) ∞ 2 ing low-energy constants are fitted to the “standard” set Z dk k 00 X ρ ρ ,J 00 of 36 empirical data points [169–174]. The hyper- + 3 Vν00ν (k , k) YN 0 (2π) 3 ρ,ν triton (ΛH) has been chosen as a further 0 input. It determines the relative strength of the spin- 2µν ρρ ,J 0 √ × 2 2 Tνν0 (k, k ; s) , singlet and spin-triplet S-wave contributions of the Λp kν − k + i (98) interaction. Due to the sparse and inaccurate experi- mental data, the obtained fit of the low-energy constants where J denotes the conserved total angular momentum. is not unique. For instance, the YN data can be de- The coupled two-particle channels (Λp, Σ+n, Σ0p,...) scribed equally well with a repulsive or an attractive in- 1 3 3 are labeled by ν, and the partial waves ( S0, P0,...) teraction in the S1 partial wave of the ΣN interaction by ρ. Furthermore, µν is the reduced baryon mass in with isospin I = 3/2. However, recent calculations from 3 channel ν. In Ref. [33] a non-relativistic scattering equa- lattice QCD [3,4] suggest a repulsive S1 phase shift in tion has been chosen to ensure that the potential can the ΣNI = 3/2 channel, hence the repulsive solution has also be applied consistently to Faddeev and Faddeev- been adopted. Furthermore, this is consistent with em- Yakubovsky calculations in the few-body sector, and pirical information from Σ−-production on nuclei, which to (hyper-) nuclear matter calculations within the con- point to a repulsive Σ-nucleus potential (see also Sub- ventional Brueckner-Hartree-Fock formalism (see Sub- sec.VB). sec.VB). Nevertheless, the relativistic relation between In the following we present some of the results of the on-shell momentum kν and the center-of-mass energy Ref. [33]. For comparison, results of the Jülich ’04 [21] 26

+ + Λp -> Λp Λp -> Λp Σ p -> Σ p

300 70 250 + 0 Σ n -> <- Σ p Sechi-Zorn et al. Eisele et al. 60 Kadyk et al. Kadyk et al. Alexander et al. Hauptman et al. 200

50 200

150 40 (mb) (mb) (mb)

σ σ 30 σ 100

100 20

50

10

0 0 0 100 200 300 400 500 600 700 800 900 500 600 700 800 100 120 140 160 180 plab (MeV/c) plab (MeV/c) plab (MeV/c)

Figure 13: Total cross section σ as a function of plab. The red (dark) band shows the chiral EFT results to NLO for variations of the cutoff in the range Λ = (500 ... 650) MeV [33], while the green (light) band are results to LO for Λ = (550 ... 700) MeV [29]. The dashed curves are the result of the Jülich ’04 meson-exchange potential [21], the dash-dotted curves of the Nijmegen NSC97f potential [20]. and the Nijmegen [20] meson-exchange models are also space, as described in Refs. [49, 175]. Note that genuine shown in the figures. In Fig. 13 the total cross sections (irreducible) three-baryon interactions were not included as functions of plab for various YN interactions are pre- in this calculation. However, in the employed coupled- sented. The experimental data is well reproduced at channel formalism, effects like the important Λ-Σ con- NLO. Especially the results in the Λp channel are in line version process are naturally included. It is important with the data points (also at higher energies) and the to distinguish such iterated two-body interactions, from energy dependence in the Σ+p channel is significantly irreducible three-baryon forces, as exemplified in Fig.4. improved at NLO. It is also interesting to note that Predictions for S- and P -wave phase shifts δ as a func- the NLO results are now closer to the phenomenological + tion of plab for Λp and Σ p scattering are shown in Jülich ’04 model than at LO. One expects the theoreti- 1 Fig. 14. The S0 Λp phase shift from the NLO χEFT cal uncertainties to become smaller, when going to higher calculation is closer to the phenomenological Jülich ’04 order in the chiral power counting. This is reflected in model than the LO result. It points to moderate attrac- the fact, that the bands at NLO are considerably smaller tion at low momenta and strong repulsion at higher mo- than at LO. These bands represent only the cutoff de- menta. At NLO the phase shift has a stronger downward pendence and therefore constitute a lower bound on the bending at higher momenta compared to LO or the Jülich theoretical error. ’04 model. As stated before, more repulsion at higher en- In Tab.V the scattering lengths and effective range ergies is a welcome feature in view of neutron star matter + 1 with Λ-hyperons as additional baryonic degree of free- parameters for the Λp and Σ p interactions in the S0 3 3 dom. The S1 Λp phase shift, part of the S-matrix for and S1 partial waves are given. Result for LO [29] and 3 3 the coupled S1- D1 system, changes qualitatively from NLO χEFT [33], for the Jülich ’04 model [21] and for the 3 LO to NLO. The S1 phase shift of the NLO interac- Nijmegen NSC97f potential [20] are shown. The NLO Λp ◦ scattering lengths are larger than for the LO calculation, tion passes through 90 slightly below the ΣN threshold, and closer to the values obtained by the meson-exchange which indicates the presence of an unstable + in the ΣN system. For the LO interaction and the Jülich models. The triplet Σ p scattering length is positive ◦ in the LO as well as the NLO calculation, which indi- ’04 model no passing through 90 occurs and a cusp is cates a repulsive interaction in this channel. Also given predicted, that is caused by an inelastic virtual state in in Tab.V is the binding energy, calculated the ΣN system. These effects are also reflected by a strong increase of the Λp cross section close to the ΣN with the corresponding chiral potentials. As stated be- 3 fore, the hypertriton binding energy was part of the fit- threshold, see Fig. 13. The S1 ΣN phase shift for the ting procedure and values close to the experimental value NLO interaction is moderately repulsive and comparable 3 to the LO phase shift. could be achieved. The predictions for the ΛH binding energy are based on the Faddeev equations in momentum Recently an alternative NLO χEFT potential for YN 27

NLO LO Jül ’04 NSC97f Λ [MeV] 450 500 550 600 650 700 600 Λp as −2.90 −2.91 −2.91 −2.91 −2.90 −2.90 −1.91 −2.56 −2.60 Λp rs 2.64 2.86 2.84 2.78 2.65 2.56 1.40 2.74 3.05 Λp at −1.70 −1.61 −1.52 −1.54 −1.51 −1.48 −1.23 −1.67 −1.72 Λp rt 3.44 3.05 2.83 2.72 2.64 2.62 2.13 2.93 3.32 Σ+p as −3.58 −3.59 −3.60 −3.56 −3.46 −3.49 −2.32 −3.60 −4.35 Σ+p rs 3.49 3.59 3.56 3.54 3.53 3.45 3.60 3.24 3.16 Σ+p at 0.48 0.49 0.49 0.49 0.48 0.49 0.65 0.31 −0.25 Σ+p rt −4.98 −5.18 −5.03 −5.08 −5.41 −5.18 −2.78 −12.2 −28.9 3 (ΛH) EB −2.39 −2.33 −2.30 −2.30 −2.30 −2.32 −2.34 −2.27 −2.30

Table V: The YN singlet (s) and triplet (t) scattering length a and effective range r (in fm) and the hypertriton binding energy EB (in MeV) [33]. The binding energies for the hypertriton are calculated using the Idaho-N3LO 3 NN potential [151]. The experimental value for the ΛH binding energy is -2.354(50) MeV.

1 Λp S Λ 3 Σ 3 0 p S1 N S1 (I=3/2) 180 40 30

150 20 30 10 120 20 0 90 -10 10

(degrees) (degrees) 60 (degrees) -20 δ δ δ

0 -30 30 -40 -10 0 -50

-20 -30 -60 0 200 400 600 800 0 200 400 600 800 0 100 200 300 400 500 600 plab (MeV/c) plab (MeV/c) plab (MeV/c)

+ Figure 14: Various S- and P -wave phase shifts δ as a function of plab for the Λp and Σ p interaction [33]. Same description of curves as in Fig. 13. scattering has been presented [34]. In that work a dif- transition potential and that becomes manifest in appli- ferent strategy for fixing the low-energy constants that cations to few- and many-body systems [34, 176]. determine the strength of the contact interactions is There is very little empirical information about adopted. The objective of that exploration was to reduce baryon-baryon systems with S = −2, i.e., about the in- the number of LECs that need to be fixed in a fit to the teraction in the ΛΛ, ΣΣ, ΛΣ, and ΞN channels. Ac- ΛN and ΣN data by inferring some of them from the NN tually, all one can find in the literature [36] are a few sector via the underlying SU(3) symmetry, cf. Sect.IIIA. values and upper bounds for the Ξ−p elastic and inelas- Indeed, correlations between the LO and NLO LECs of tic cross sections [177, 179]. In addition there are con- the S-waves, i.e., between the c˜’s and c’s, had been ob- straints on the strength of the ΛΛ interaction from the 6 served already in the initial YN study [33] and indicated separation energy of the ΛΛHe [180]. Fur- 1 that a unique determination of them by considering the thermore estimates for the ΛΛ S0 scattering length exist existing ΛN and ΣN data alone is not possible. It may from analyses of the ΛΛ invariant mass measured in the be not unexpected in view of those correlations, that the reaction 12C(K−,K+ΛΛX) [181] and of ΛΛ correlations variant considered in [34] yields practically equivalent re- measured in relativistic heavy-ion collisions [182]. sults for ΛN and ΣN scattering observables. However, Despite the rather poor experimental situation, it it differs considerable in the strength of the ΛN → ΣN turned out that SU(3)-symmetry breaking contact terms 28

70 100 − − − Ξ p -> ΛΛ 90 Ξ p -> Ξ p 60 80 Ahn (2006) Ahn (2006) 50 70 Kim (2015) 60 40 50 (mb) (mb)

σ 30 σ 40

20 30

---> --->

20

---> ---> 10 10

0 0 0 200 400 600 800 0 200 400 600 800 plab (MeV/c) plab (MeV/c)

100 100 − 0 − 0 0 90 Ξ p -> Ξ n 90 Ξ p -> ΛΛ,Ξ n,Σ Λ 80 80 Ahn (2006) Aoki (1998) 70 70

60 60

50 50 (mb) (mb) σ σ 40 40

30 30

20 20

10 10

0 0 0 200 400 600 800 0 200 400 600 800 plab (MeV/c) plab (MeV/c)

Figure 15: Ξ−p induced cross sections. The bands represent results at NLO (red/black) [36] and LO (green/grey) [30] . Experiments are from Ahn et al. [177], Kim et al. [178], and Aoki et al. [179]. Upper limits are indicated by arrows. that arise at NLO, see Sect.IIIA, need to be taken into relatively weak in order to be in accordance with the account when going from strangeness S = −1 to S = −2 available empirical constraints. Indeed, the present re- in order to achieve agreement with the available mea- sults obtained in chiral EFT up to NLO imply that the surements and upper bounds for the ΛΛ and ΞN cross published values and upper bounds for the Ξ−p elastic 1 sections [36]. This concerns, in particular, the LEC cχ and inelastic cross sections [177, 179] practically rule out 1 that appears in the S0 partial wave, cf. Eq. (54). Ac- a somewhat stronger attractive ΞN force. tually, its value can be fixed by considering the pp and Also for ΞN scattering an alternative NLO χEFT po- Σ+p systems, as shown in Ref. [35], and then employed tential has been presented recently [37]. Here the aim is in the ΛΛ system. to explore the possibility to establish a ΞN interaction that is still in line with all the experimental constraints Selected results for the strangeness S = −2 sector are presented in Fig. 15. Further results and a detailed de- for ΛΛ and ΞN scattering, but at the same time is some- scription of the interactions can be found in Refs. [30, 35– what more attractive. Recent experimental evidence for 37]. Interestingly, the results based on the LO interaction the existence of Ξ-hypernuclei [183] suggests that the in- from Ref. [30] (green/grey bands) are consistent with all medium interaction of the Ξ-hyperon should be moder- empirical constraints. The cross sections at LO are basi- ately attractive [184]. cally genuine predictions that follow from SU(3) symme- try utilizing LECs fixed from a fit to the ΛN and ΣN data 1 B. Hyperons in nuclear matter on the LO level. The ΛΛ S0 scattering length predicted by the NLO interaction is aΛΛ = −0.70 · · · −0.62 fm [36]. These values are well within the range found in the afore- Experimental investigations of nuclear many-body sys- mentioned analyses which are aΛΛ = (−1.2±0.6) fm [181] tems including strange baryons, for instance, the spec- and −1.92 < aΛΛ < −0.50 fm [182], respectively. The troscopy of hypernuclei, provide important constraints on values for the Ξ0p and Ξ0n S-wave scattering length are the underlying hyperon-nucleon interaction. The analy- likewise small and typically in the order of ±0.3 ∼ ±0.6 sis of data for single Λ-hypernuclei over a wide range fm [36] and indicate that the ΞN interaction has to be in leads to the result, that the attrac- 29 tive Λ single-particle potential is about half as deep The arising G-matrix interaction is an effective interac- (≈ −28 MeV) as the one for nucleons [185, 186]. At the tion of two particles in the presence of the medium. The same time the Λ-nuclear spin-orbit interaction is found to medium effects come in solely through the Pauli oper- be exceptionally weak [187, 188]. Recently, the repulsive ator and the energy denominator via the single-particle nature of the Σ-nuclear potential has been experimentally potentials. If we set the single-particle potentials to zero established in Σ−-formation reactions on heavy nuclei and omit the Pauli operator (Q = 1), we recover the [189]. Baryon-baryon potentials derived within χEFT as usual Lippmann-Schwinger equation for two-body scat- presented in Sec.III are consistent with these observa- tering in vacuum, see also Fig.2. This medium effect on tions [75, 76]. In this section we summarize results of hy- the intermediate states is denoted by horizontal double perons in infinite homogeneous nuclear matter of Ref. [76] lines in Fig. 16(a). An appropriate expansion using the obtained by employing the interaction potentials from G-matrix interaction instead of the bare potential is the χEFT as microscopic input. The many-body problem so-called Brueckner-Bethe-Goldstone expansion, or hole- is solved within first-order Brueckner theory. A detailed line expansion. introduction can be found, e.g., in Refs. [69, 190, 191]. Finally, the form of the auxiliary potential U needs to Brueckner theory is founded on the so-called Gold- be chosen. This choice is important for the convergence stone expansion, a linked-cluster perturbation series for of the hole-line expansion. Bethe, Brandow and Petschek the ground state energy of a fermionic many-body sys- [192] showed for nuclear matter that important higher- tem. Let us consider a system of A identical fermions, order diagrams cancel each other if the auxiliary potential described by the Hamiltonian is taken as X H = T + V, (99) Um = Re hmn|G(ω = ωo.s.)|mniA , (102) n≤A where T is the kinetic part and V corresponds to the where the Brueckner reaction matrix is evaluated on- two-body interaction. The goal is to calculate the ground shell, i.e., the starting energy is equal to the energy of state energy of this interacting -body system. It is ad- A the two particles m, n in the initial state: vantageous to introduced a so-called auxiliary potential, or single-particle potential, U. The Hamiltonian is then ωo.s. = E1(k1) + E2(k2) , split into two parts 2 ki EBi (ki) = Mi + + Re Ui(ki) . (103) 2Mi H = (T + U) + (V − U) = H0 + H1 , (100) Pictorially Eq. (102) means, that the single-particle po- the unperturbed part H0 and the perturbed part H1. tential can be obtained by taking the on-shell G-matrix One expects the perturbed part to be small, if the single interaction and by closing one of the baryon lines, as illus- particle potential describes well the averaged effect of the trated in Fig. 16(b). Note that this implies a non-trivial medium on the particle. In fact, the proper introduction self-consistency problem. On the one hand, U is calcu- of the auxiliary potential is crucial for the convergence of lated from the G-matrix elements via Eq. (102), and on Brueckner theory. the other hand the starting energy of the G-matrix ele- Conventional nucleon-nucleon potentials exhibit a ments depends on U through the single-particle energies strong short-range repulsion that leads to very large ma- Ei in Eq. (103). trix elements. Hence, the Goldstone expansion in the At the (leading) level of two hole-lines, called form described above will not converge for such hard- Brueckner-Hartree-Fock approximation (BHF), the total core potentials. One way to approach this problem is energy is given by the introduction of the so-called Brueckner reaction ma- trix, or G-matrix. The idea behind it is illustrated in X 1 X E = hn|T |ni + hmn|G|mniA Fig. 16(a). Instead of only using the simple interaction, 2 n≤A m,n≤A an infinite number of diagrams with increasing number X 1 X of interactions is summed up. This defines the G-matrix = hn|T |ni + hn|U|ni , (104) 2 interaction, which is, in contrast to the bare potential, n≤A n≤A weak and of reasonable range. In a mathematical way, the reaction matrix is defined by the Bethe-Goldstone i.e., the ground-state energy E can be calculated directly equation: after the single-particle potential has been determined. The definition of U in Eq. (102) applies only to oc- Q cupied states within the Fermi sea. For intermediate- G(ω) = V + V G(ω) , (101) state energies above the Fermi sea, typically two choices ω − H0 + i for the single-particle potential are employed. In the so- with the so-called starting energy ω. The Pauli operator called gap choice, the single-particle potential is given by Q ensures, that the intermediate states are from outside Eq. (102) for k ≤ kF and set to zero for k > kF , imply- the Fermi sea. As shown in Fig. 16(a) this equation repre- ing a “gap” (discontinuity) in the single-particle poten- sents a resummation of the ladder diagrams to all orders. tial. Then only the free particle energies (M + ~p 2/2M) 30

V V G(ω) = V + + V + ··· U = G(ωo.s.) k < kF V V

(a) Bethe-Goldstone equation (b) Single particle potential

Figure 16: Graphical representation of the determination of the single-particle potential from the G-matrix interaction (b) and of the Bethe-Goldstone equation (a). The symbol ωo.s denotes the on-shell starting energy.

of the intermediate states appear in the energy denom- −UY (k = 0). The results of the LO and NLO calcu- inator of the Bethe-Goldstone equation (101) since the lation are consistent with the empirical value of about Pauli-blocking operator is zero for momenta below the UΛ(0) ≈ −28 MeV [185, 186]. The phenomenological Fermi momentum. In the so-called continuous choice models (Jülich ’04, Nijmegen NSC97f) lead to more at- Eq. (102) is used for the whole momentum range, hence tractive values of UΛ(0) = (−35 ... − 50) MeV, where 3 the single-particle potentials enter also into the energy the main difference is due to the contribution in the S1 denominator. In Ref. [193] the in sym- partial wave. In contrast to LO, at NLO the Λ single- metric nuclear matter has been considered. It has been particle potential at NLO turns to repulsion at fairly low shown, that the result including three hole-lines is al- momenta around k ≈ 2 fm−1, which is also the case for most independent of the choice of the auxiliary potential. the NSC97f potential. Furthermore the two-hole line result with the continuous An important quantity of the interaction of hyper- choice comes out closer to the three hole-line result, than ons with heavy nuclei is the strength of the Λ-nuclear the two-hole line calculation with the gap choice. An- spin-orbit coupling. It is experimentally well established other advantage of the continuous choice for intermedi- [187, 188] that the Λ-nucleus spin-orbit force is very ate spectra is that it allows for a reliable determination small. For the YN interaction of Ref. [75] it was indeed of the single-particle potentials including their imaginary possible to tune the strength of the antisymmetric spin- parts [70]. The results presented here employ the contin- orbit contact interaction (via the constant c8as), gener- uous choice. 1 3 ating a spin singlet-triplet mixing ( P1 ↔ P1), in a way In the following we present some results of Ref. [76] to achieve such a small nuclear spin-orbit potential. for the in-medium properties of hyperons, based on the Results for Σ hyperons in isospin-symmetric nuclear YN interaction derived from SU(3) χEFT at NLO. The matter at saturation density are also displayed in Fig. 17. same potential V as in the Lippmann-Schwinger equation Analyses of data on (π−,K+) spectra related to Σ− for- (98) for free scattering is used. However, as in Ref. [75] mation in heavy nuclei lead to the observation, that the the contact term c8as for the antisymmetric spin-orbit Σ-nuclear potential in symmetric nuclear matter is mod- force in the YN interaction, allowing spin singlet-triplet erately repulsive [189]. The LO as well as the NLO results transitions, has been fitted to the weak Λ-nuclear spin- are consistent with this observation. Meson-exchange orbit interaction [194, 195]. Additionally, for the ease of models often fail to produce such a repulsive Σ-nuclear comparison, the G-matrix results obtained with two phe- potential. The imaginary part of the Σ-nuclear poten- nomenological YN potentials, namely of the Jülich ’04 tial at saturation density is consistent with the empirical [21] and the Nijmegen NSC97f [20] meson-exchange mod- value of −16 MeV as extracted from Σ−- data [196]. els, are given. Note that, like the EFT potentials, these The imaginary potential is mainly induced by the ΣN to phenomenological YN interactions produce a bound hy- ΛN conversion in nuclear matter. The bands represent- pertriton [49]. For more details about derivation and the ing the cutoff dependence of the chiral potentials, become commonly employed approximations, we refer the reader smaller when going to higher order in the chiral expan- to Refs. [19, 20, 24, 70, 71]. sion. Let us start with the properties of hyperons in symmet- In Fig. 18 the density dependence of the depth of the ric nuclear matter. Fig. 17 shows the momentum depen- nuclear mean-field of Λ or Σ hyperons at rest (k = 0). dence of the real parts of the Λ single-particle potential. In order to see the influence of the composition nu- The values for the depth of the Λ single-particle poten- clear matter on the single-particle potentials, results for −1 tial UΛ(k = 0) at saturation density, kF = 1.35 fm , at isospin-symmetric nuclear matter, asymmetric nuclear NLO are between 27.0 and 28.3 MeV. In the Brueckner- matter with ρp = 0.25ρ and pure neutron matter are Hartree-Fock approximation the binding energy of a hy- shown. The single-particle potential of the Λ hyperon peron in infinite nuclear matter is given by BY (∞) = is almost independent of the composition of the nuclear 31

Figure 17: Momentum dependence of the real part of the single-particle potential of a Λ hyperon and of the real and imaginary parts of the single-particle potential of a Σ hyperon in isospin-symmetric nuclear matter at saturation −1 density, kF = 1.35 fm [76]. The red band, green band, blue dashed curve and red dash-dotted curve are for χEFT NLO, χEFT LO, the Jülich ’04 model and the NSC97f model, respectively.

Figure 18: Density dependence of the hyperon single-particle potentials at k = 0 with different compositions of the nuclear matter, calculated in χEFT at NLO with a cutoff Λ = 600 MeV [76]. The green solid, red dashed and blue dash-dotted curves are for ρp = 0.5ρ, ρp = 0.25ρ and ρp = 0, respectively. medium, because of its isosinglet nature. Furthermore, investigated for ΞN potentials from χEFT [37]. For a it is attractive over the whole considered range of den- more extensive discussion and further applications see sity 0.5 ≤ ρ/ρ0 ≤ 1.5. In symmetric nuclear matter, the Ref. [198]. three Σ hyperons behave almost identical (up to small differences from the mass splittings). When introduc- ing isospin in the nuclear medium a splitting of the single-particle potentials occurs due to the strong C. Hypernuclei and hyperons in neutron stars isospin dependence of the ΣN interaction. The splittings among the Σ+, Σ0 and Σ− potentials have a non-linear The density-dependent single-particle potentials of hy- dependence on the isospin asymmetry which goes beyond perons interacting with nucleons in nuclear and neutron the usual (linear) parametrization in terms of an isovec- matter find their applications in several areas of high tor Lane potential [197]. current interest: the physics of hypernuclei and the role played by hyperons in dense baryonic matter as it is re- Recently the in-medium properties of the Ξ have been alized in the core of neutron stars. 32

From hypernuclear spectroscopy, the deduced attrac- constraint provided by the existence of several neutron tive strength of the phenomenological Λ-nuclear Woods- stars with masses around 2M . Assume now that nature Saxon potential is U0 ' −30 MeV at the nuclear cen- favors a scenario with a weak diagonal ΛN-interaction ter [184]. This provides an important constraint for and a strong ΛN–ΣN coupling as predicted by SU(3) UΛ(k = 0) at ρ = ρ0. The non-existence of bound Σ- chiral EFT. The present study demonstrates that, in this hypernuclei, on the other hand, is consistent with the case, the Λ single-particle potential UΛ(k = 0, ρ) based repulsive nature of the Σ-nuclear potential as shown in on chiral EFT two-body interactions is already repulsive Fig. 18. In this context effects of YNN three-body forces at densities ρ ∼ (2 − 3)ρ0. The one of the Σ-hyperon is are a key issue. While their contributions at normal nu- likewise repulsive [35]. We thus expect that the appear- clear densities characteristic of hypernuclei are signifi- ance of hyperons in neutron stars will be shifted to much cant but modest, they play an increasingly important higher densities. In addition there is a repulsive density- role when extrapolating to high baryon densities in neu- dependent effective ΛN-interaction that arises within the tron stars. same chiral EFT framework from the leading chiral YNN First calculations of hyperon-nuclear potentials based three-baryon forces. This enhances the aforementioned on chiral SU(3) EFT and using Brueckner theory have repulsive effect further. It makes the appearance of Λ- been reported in Ref. [122, 176]. Further investigations of hyperons in neutron star matter energetically unfavor- (finite) Λ hypernuclei utilizing the EFT interactions can able. In summary, all these aspects taken together may be found in [199], based on the formalism described in well point to a solution of the hyperon puzzle in neutron Ref. [200]. For even lighter hypernuclei, the interactions stars without resorting to exotic mechanisms. are also currently studied [201, 202]. Examples of three- and four-body results can be found in Refs. [34, 49, 175, 203]. VI. CONCLUSIONS Here we give a brief survey of Λ-nuclear interactions for hypernuclei and extrapolations to high densities rel- In this review we have presented the basics to derive vant to neutron stars, with special focus on the role of the forces between octet baryons (N, Λ, Σ, Ξ) at next-to- the (a priori unknown) contact terms of the ΛNN three- leading order in SU(3) chiral effective field theory. The body force. Details can be found in Ref. [176]. Further connection of SU(3) χEFT to quantum chromodynamics extended work including explicit 3-body coupled chan- via the chiral symmetry and its symmetry breaking pat- nels (ΛNN ↔ ΣNN) in the Brueckner-Bethe-Goldstone terns, and the change of the degrees of freedom has been equation is proceeding [204]. shown. The construction principles of the chiral effec- Results for the density dependence of the Λ single- tive Lagrangian and the external-field method have been particle potential are presented in Fig. 19 for symmet- presented and the Weinberg power-counting scheme has ric nuclear matter (a) and for neutron matter (b). Pre- been introduced. dictions from the chiral SU(3) EFT interactions (bands) Within SU(3) χEFT the baryon-baryon interaction are shown in comparison with those for meson-exchange potentials have been considered at NLO. The effec- YN models constructed by the Jülich [21] (dashed line) tive baryon-baryon potentials include contributions from and Nijmegen [20] (dash-dotted line) groups. One ob- pure four-baryon contact terms, one-meson-exchange di- serves an onset of repulsive effects around the saturation agrams, and two-meson-exchange diagrams at one-loop density of nuclear matter, i.e., ρ = ρ0. The repulsion in- level. The leading three-baryon forces, which formally creases strongly as the density increases. Already around start to contribute at NNLO, consist of a three-baryon ρ ≈ 2ρ0, UΛ(0, ρ) turns over to net repulsion. contact interaction, a one-meson exchange and a two- Let us discuss possible implications for neutron stars. meson exchange component. We have presented explic- It should be clear that it is mandatory to include the ΛN– itly potentials for the ΛNN interaction in the spin and ΣN coupling in the pertinent calculations. This repre- isospin basis. The emerging low-energy constants can be sents a challenging task since standard microscopic calcu- estimated via decuplet saturation, which leads to a pro- lations without this coupling are already quite complex. motion of some parts of the three-baryon forces to NLO. However, without the ΛN–ΣN coupling, which has such The expressions of the corresponding effective two-body a strong influence on the in-medium properties of hyper- potential in the nuclear medium has been presented. ons, it will be difficult if not impossible to draw reliable In the second part of this review we have presented se- conclusions. lected applications of these potentials. An excellent de- The majority of YN-interactions employed so far in scription of the available YN data has been achieved with microscopic calculations of neutron stars have properties χEFT, comparable to the most advanced phenomenolog- similar to those of the Jülich ’04 model. In such cal- ical models. Furthermore, in studies of the properties of culations, hyperons start appearing in the core of neu- hyperons in isospin symmetric and asymmetric infinite tron stars typically at relatively low densities around nuclear matter, the chiral baryon-baryon potentials at (2 − 3)ρ0 [113, 119]. This causes the so-called hyperon NLO are consistent with the empirical knowledge about puzzle: a strong softening of the equation-of-state, such hyperon-nuclear single-particle potentials. The excep- that the maximum neutron star mass falls far below the tionally weak Λ-nuclear spin-orbit force is found to be re- 33

20 20

0 0 (MeV) (MeV) Λ Λ U U -20 -20

-40 -40

0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 ρ / ρ ρ / ρ 0 0

Figure 19: The Λ single-particle potential UΛ(pΛ = 0, ρ) as a function of ρ/ρ0 in symmetric nuclear matter (a) and in neutron matter (b). The solid (red) band shows the chiral EFT results at NLO for cutoff variations Λ = 450-500 MeV. The dotted (blue) band includes the density-dependent ΛN-interaction derived from the ΛNN three-body force. The dashed curve is the result of the Jülich ’04 meson-exchange model [21], the dash-dotted curve that of the Nijmegen NSC97f potential [20], taken from Ref. [205]. lated to the contact term responsible for an antisymmet- ACKNOWLEDGMENTS ric spin-orbit interaction. Concerning hypernuclei and neutron stars promising results have been obtained and This work is supported in part by the Deutsche could point to a solution of the hyperon puzzle in neutron Forschungsgemeinschaft (DFG) through the funds pro- stars. vided to the Sino-German Collaborative Research Cen- ter “Symmetries and the Emergence of Structure in In summary, χEFT is an appropriate tool for con- QCD" (Grant No. TRR110), by the CAS Presi- structing the interaction among baryons in a systematic dent’s International Fellowship Initiative (PIFI) (Grant way. It sets the framework for many promising applica- No. 2018DM0034), and by the VolkswagenStiftung tions in strangeness-nuclear physics. (Grant No. 93562).

[1] S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, T. Inoue, N. Ishii, [5] S. Weinberg, 166, 1568 (1968). K. Murano, H. Nemura, and K. Sasaki, Progress of [6] S. Weinberg, Physica 96A, 327 (1979). Theoretical and 2012, 01A105 [7] J. Gasser and H. Leutwyler, Annals of Physics 158, 142 (2012), arXiv:1206.5088. (1984). [2] S. R. Beane, W. Detmold, K. Orginos, and M. J. Sav- [8] J. Gasser, M. E. Sainio, and A. Svarc, Nuclear Physics age, Progress in Particle and Nuclear Physics 66, 1 B 307, 779 (1988). (2011), arXiv:1004.2935. [9] V. Bernard, N. Kaiser, and U.-G. Meißner, Interna- [3] S. R. Beane, E. Chang, S. D. Cohen, W. Detmold, H.- tional Journal of E 04, 193 (1995), W. Lin, T. C. Luu, K. Orginos, A. Parreño, M. J. Sav- arXiv:hep-ph/9501384 [hep-ph]. age, and A. Walker-Loud, 109, [10] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, 172001 (2012), arXiv:1204.3606. and J. J. de Swart, Physical Review C 49, 2950 (1994), [4] H. Nemura et al., Proceedings, 35th International arXiv:9406039 [nucl-th]. Symposium on Lattice Field Theory (Lattice 2017): [11] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys- Granada, Spain, June 18-24, 2017, EPJ Web Conf. 175, ical Review C 51, 38 (1995), arXiv:9408016 [nucl-th]. 05030 (2018), arXiv:1711.07003 [hep-lat]. [12] R. Machleidt, Physical Review C 63, 024001 (2001), 34

arXiv:0006014 [nucl-th]. Review D 94, 014029 (2016), arXiv:1603.07802. [13] E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, [40] X.-L. Ren, K.-W. Li, L.-S. Geng, B. Long, P. Ring, Reviews of Modern Physics 81, 1773 (2009), and J. Meng, Chinese Physics C 42, 014103 (2018), arXiv:0811.1338. arXiv:1611.08475. [14] R. Machleidt and D. R. Entem, Physics Reports 503, 1 [41] K.-W. Li, X.-L. Ren, L.-S. Geng, and B.-W. Long, Chi- (2011), arXiv:1105.2919. nese Physics C 42, 014105 (2018), arXiv:1612.08482. [15] D. B. Kaplan, M. J. Savage, and M. B. Wise, Phys.Lett. [42] J. Song, K.-W. Li, and L.-S. Geng, Physical Review C B424, 390 (1998), arXiv:9801034 [nucl-th]. 97, 065201 (2018), arXiv:1802.04433. [16] A. Nogga, R. Timmermans, and U. van Kolck, [43] K.-W. Li, T. Hyodo, and L.-S. Geng, Phys. Rev. C98, Phys.Rev. C72, 054006 (2005), arXiv:0506005 [nucl-th]. 065203 (2018), arXiv:1809.03199 [nucl-th]. [17] E. Epelbaum and U.-G. Meißner, Few Body Syst. 54, [44] X. L. Ren, E. Epelbaum, and J. Gegelia, (2019), 2175 (2013), arXiv:nucl-th/0609037 [nucl-th]. arXiv:1911.05616 [nucl-th]. [18] B. Holzenkamp, K. Holinde, and J. Speth, Nuclear [45] V. Bernard, N. Kaiser, U.-G. Meißner, and A. Schmidt, Physics A 500, 485 (1989). Z. Phys. A348, 317 (1994), arXiv:hep-ph/9311354 [hep- [19] A. G. Reuber, K. Holinde, and J. Speth, Nuclear ph]. Physics A 570, 543 (1994). [46] K. Miyagawa and W. Glöckle, Phys. Rev. C48, 2576 [20] T. A. Rijken, V. G. J. Stoks, and Y. Yamamoto, Phys- (1993). ical Review C 59, 21 (1999), arXiv:9807082 [nucl-th]. [47] K. Miyagawa, H. Kamada, W. Glöckle, and V. G. J. [21] J. Haidenbauer and U.-G. Meißner, Physical Review C Stoks, Phys. Rev. C51, 2905 (1995). 72, 044005 (2005), arXiv:0506019 [nucl-th]. [48] A. Nogga, H. Kamada, and W. Glöckle, Physical review [22] T. A. Rijken, M. M. Nagels, and Y. Yamamoto, letters 88, 172501 (2002), arXiv:0112060 [nucl-th]. Progress of Theoretical Physics Supplement 185, 14 [49] A. Nogga, Proceedings, 22nd European Conference on (2010), arXiv:9401004 [nucl-th]. Few-Body Problems in Physics (EFB22): Cracow, [23] M. M. Nagels, T. A. Rijken, and Y. Yamamoto, Phys. Poland, September 9-13, 2013, Few Body Syst. 55, 757 Rev. C99, 044003 (2019), arXiv:1501.06636 [nucl-th]. (2014). [24] M. Kohno, Y. Fujiwara, T. Fujita, C. Nakamoto, [50] S. K. Bogner, R. J. Furnstahl, and A. Schwenk, Prog. and Y. Suzuki, Nuclear Physics A 674, 229 (2000), Part. Nucl. Phys. 65, 94 (2010), arXiv:0912.3688 [nucl- arXiv:9912059 [nucl-th]. th]. [25] Y. Fujiwara, Y. Suzuki, and C. Nakamoto, Progress [51] R. Wirth, D. Gazda, P. Navratil, A. Calci, J. Langham- in Particle and Nuclear Physics 58, 439 (2007), mer, and R. Roth, Physical Review Letters 113, 192502 arXiv:0607013 [nucl-th]. (2014), arXiv:1403.3067. [26] H. Garcilazo, T. Fernandez-Carames, and A. Valcarce, [52] R. Wirth and R. Roth, Phys. Rev. Lett. 117, 182501 Phys. Rev. C75, 034002 (2007), arXiv:hep-ph/0701275 (2016), arXiv:1605.08677 [nucl-th]. [hep-ph]. [53] R. Wirth, D. Gazda, P. Navritil, and R. Roth, Phys. [27] M. J. Savage and M. B. Wise, Physical Review D 53, Rev. C 97, 064315 (2018), arXiv:1712.05694 [nucl-th]. 349 (1996), arXiv:9507288 [hep-ph]. [54] D. Gazda and A. Gal, Phys. Rev. Lett. 116, 122501 [28] C. Korpa, A. Dieperink, and R. Timmermans, (2016), arXiv:1512.01049 [nucl-th]. Phys.Rev. C65, 015208 (2002), arXiv:0109072 [nucl-th]. [55] D. Gazda and A. Gal, Nucl. Phys. A954, 161 (2016), [29] H. Polinder, J. Haidenbauer, and U.-G. Meißner, Nu- arXiv:1604.03434 [nucl-th]. clear Physics A 779, 244 (2006), arXiv:nucl-th/0605050 [56] S. Gandolfi, F. Pederiva, S. Fantoni, and K. E. [nucl-th]. Schmidt, Physical Review Letters 98, 102503 (2007), [30] H. Polinder, J. Haidenbauer, and U.-G. Meißner, arXiv:0607022 [nucl-th]. Physics Letters B 653, 29 (2007), arXiv:0705.3753. [57] S. Gandolfi, A. Y. Illarionov, K. E. Schmidt, F. Ped- [31] J. Haidenbauer and U.-G. Meißner, Physics Letters B eriva, and S. Fantoni, Physical Review C 79, 054005 684, 275 (2010), arXiv:0907.1395. (2009), arXiv:0903.2610. [32] S. Petschauer and N. Kaiser, Nuclear Physics A 916, 1 [58] D. Lonardoni, F. Pederiva, and S. Gandolfi, Physical (2013), arXiv:1305.3427. Review C 89, 014314 (2014), arXiv:1312.3844. [33] J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. [59] B. Borasoy, E. Epelbaum, H. Krebs, D. Lee, and U.- Meißner, A. Nogga, and W. Weise, Nuclear Physics G. Meißner, The European Physical Journal A 31, 105 A 915, 24 (2013), arXiv:1304.5339. (2007), arXiv:0611087 [nucl-th]. [34] J. Haidenbauer, U.-G. Meißner, and A. Nogga, ac- [60] E. Epelbaum, H. Krebs, D. Lee, and U.-G. cepted for publication in Eur. Phys. J. A (2020), Meißner, Physical Review Letters 106, 192501 (2011), arXiv:1906.11681 [nucl-th]. arXiv:1101.2547. [35] J. Haidenbauer, U.-G. Meißner, and S. Petschauer, [61] T. A. Lähde and U.-G. Meißner, Lect. Notes Phys. 957, The European Physical Journal A 51, 17 (2015), 1 (2019). arXiv:1412.2991. [62] J. W. Holt, N. Kaiser, and W. Weise, Progress in Parti- [36] J. Haidenbauer, U.-G. Meißner, and S. Petschauer, Nu- cle and Nuclear Physics 73, 35 (2013), arXiv:1304.6350. clear Physics A 954, 273 (2016), arXiv:1511.05859. [63] L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, L. E. [37] J. Haidenbauer and U. G. Meißner, The European Phys- Marcucci, and F. Sammarruca, Physical Review C 89, ical Journal A 55, 23 (2019), arXiv:1810.04883. 044321 (2014), arXiv:1402.0965. [38] J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. [64] F. Sammarruca, L. Coraggio, J. W. Holt, N. Itaco, Meißner, and W. Weise, The European Physical Jour- R. Machleidt, and L. E. Marcucci, Physical Review nal C 77, 760 (2017), arXiv:1708.08071. C 91, 054311 (2015), arXiv:1411.0136. [39] K.-W. Li, X.-L. Ren, L.-S. Geng, and B. Long, Physical [65] N. Kaiser and W. Weise, Physical Review C 71, 015203 35

(2005), arXiv:nucl-th/0410062 [nucl-th]. [95] R. Bhaduri, B. Loiseau, and Y. Nogami, Nuclear [66] N. Kaiser, Physical Review C 71, 068201 (2005), Physics B 3, 380 (1967). arXiv:0601101 [nucl-th]. [96] R. Bhaduri, B. Loiseau, and Y. Nogami, Annals of [67] K. Brueckner, C. A. Levinson, and H. Mahmoud, Phys- Physics 44, 57 (1967). ical Review 95, 217 (1954). [97] A. Gal, J. Soper, and R. Dalitz, Annals of Physics 63, [68] K. Brueckner and C. A. Levinson, Physical Review 97, 53 (1971). 1344 (1955). [98] A. Gal, J. Soper, and R. Dalitz, Annals of Physics 72, [69] B. D. Day, Reviews of Modern Physics 39, 719 (1967). 445 (1972). [70] H.-J. Schulze, M. Baldo, U. Lombardo, J. Cugnon, and [99] A. Gal, J. Soper, and R. Dalitz, Annals of Physics 113, A. Lejeune, Physical Review C 57, 704 (1998). 79 (1978). [71] I. Vidaña, A. Polls, A. Ramos, M. Hjorth-Jensen, and [100] A. Bodmer and Q. Usmani, Nuclear Physics A 477, 621 V. G. J. Stoks, Physical Review C 61, 025802 (2000), (1988). arXiv:9909019 [nucl-th]. [101] A. A. Usmani, Physical Review C 52, 1773 (1995), [72] M. Baldo, G. F. Burgio, and H.-J. Schulze, Physical arXiv:9503023 [nucl-th]. Review C 61, 055801 (2000), arXiv:9912066 [nucl-th]. [102] D. Lonardoni, S. Gandolfi, and F. Pederiva, Physical [73] H.-J. Schulze, A. Polls, A. Ramos, and I. Vidaña, Phys- Review C 87, 041303 (2013), arXiv:1301.7472. ical Review C 73, 058801 (2006). [103] D. Logoteta, I. Vidana, and I. Bombaci, Eur. Phys. J. [74] H.-J. Schulze and T. A. Rijken, Physical Review C 84, A55, 207 (2019), arXiv:1906.11722 [nucl-th]. 035801 (2011). [104] P. B. Demorest, T. T. Pennucci, S. M. Ransom, M. S. E. [75] J. Haidenbauer and U.-G. Meißner, Nuclear Physics A Roberts, and J. W. T. Hessels, Nature 467, 1081 936, 29 (2015), arXiv:1411.3114. (2010), arXiv:1010.5788. [76] S. Petschauer, J. Haidenbauer, N. Kaiser, U.-G. [105] J. Antoniadis et al., Science 340, 1233232 (2013), Meißner, and W. Weise, The European Physical Jour- arXiv:1304.6875. nal A 52, 15 (2016), arXiv:1507.08808. [106] K. Hebeler, J. M. Lattimer, C. J. Pethick, and [77] S. C. Pieper and R. B. Wiringa, Annual Review of Nu- A. Schwenk, Physical Review Letters 105, 161102 clear and Particle Science 51, 53 (2001), arXiv:0103005 (2010), arXiv:1007.1746. [nucl-th]. [107] T. Hell and W. Weise, Physical Review C 90, 045801 [78] E. Epelbaum, A. Nogga, W. Glöckle, H. Kamada, U.-G. (2014), arXiv:1402.4098v2. Meißner, and H. Witala, Physical Review C 66, 064001 [108] A. W. Steiner, J. M. Lattimer, and E. F. Brown, (2002), arXiv:0208023 [nucl-th]. arXiv:1510.07515 (2015), 10.1140/epja/i2016-16018-1. [79] N. Kalantar-Nayestanaki, E. Epelbaum, J. G. Mess- [109] I. Vidaña, Proceedings, 15th International Conference chendorp, and A. Nogga, Rept. Prog. Phys. 75, 016301 on Strangeness in Quark Matter (SQM 2015): Dubna, (2012), arXiv:1108.1227 [nucl-th]. Moscow region, Russia, July 6-11, 2015, J. Phys. Conf. [80] H.-W. Hammer, A. Nogga, and A. Schwenk, Rev. Mod. Ser. 668, 012031 (2016), arXiv:1509.03587 [nucl-th]. Phys. 85, 197 (2013), arXiv:1210.4273 [nucl-th]. [110] I. Vidaña, Proc. Roy. Soc. Lond. A474, 0145 (2018), [81] B. H. J. McKellar and R. Rajaraman, Physical Review arXiv:1803.00504 [nucl-th]. Letters 21, 450 (1968). [111] B. P. Abbott et al. (LIGO Scientific, Virgo), Phys. Rev. [82] S. A. Coon, M. D. Scadron, and B. R. Barrett, Nuclear Lett. 119, 161101 (2017), arXiv:1710.05832 [gr-qc]. Physics A 242, 467 (1975). [112] B. P. Abbott et al. (LIGO Scientific, Virgo, Fermi [83] H. T. Coelho, T. K. Das, and M. R. Robilotta, Physical GBM, INTEGRAL, IceCube, AstroSat Cadmium Review C 28, 1812 (1983). Zinc Telluride Imager Team, IPN, Insight-Hxmt, [84] B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. ANTARES, Swift, AGILE Team, 1M2H Team, Dark Pieper, and R. B. Wiringa, Physical Review C 56, 1720 Energy Camera GW-EM, DES, DLT40, GRAWITA, (1997). Fermi-LAT, ATCA, ASKAP, Las Cumbres Observa- [85] S. C. Pieper, V. R. Pandharipande, R. B. Wiringa, tory Group, OzGrav, DWF (Deeper Wider Faster and J. Carlson, Physical Review C 64, 014001 (2001), Program), AST3, CAASTRO, VINROUGE, MAS- arXiv:0102004 [nucl-th]. TER, J-GEM, GROWTH, JAGWAR, CaltechNRAO, [86] S. Weinberg, Physics Letters B 251, 288 (1990). TTU-NRAO, NuSTAR, Pan-STARRS, MAXI Team, [87] S. Weinberg, Nuclear Physics B 363, 3 (1991). TZAC Consortium, KU, Nordic Optical Telescope, [88] S. Weinberg, Physics Letters B 295, 114 (1992), ePESSTO, GROND, Texas Tech University, SALT arXiv:9209257 [hep-ph]. Group, TOROS, BOOTES, MWA, CALET, IKI-GW [89] U. van Kolck, Physical Review C 49, 2932 (1994). Follow-up, H.E.S.S., LOFAR, LWA, HAWC, Pierre [90] S. Ishikawa and M. R. Robilotta, Physical Review C 76, Auger, ALMA, Euro VLBI Team, Pi of Sky, Chan- 014006 (2007), arXiv:0704.0711. dra Team at McGill University, DFN, ATLAS Tele- [91] V. Bernard, E. Epelbaum, H. Krebs, and U.- scopes, High Time Resolution Survey, RIMAS, G. Meißner, Physical Review C 77, 064004 (2008), RATIR, SKA South Africa/MeerKAT), Astrophys. J. arXiv:0712.1967. 848, L12 (2017), arXiv:1710.05833 [astro-ph.HE]. [92] V. Bernard, E. Epelbaum, H. Krebs, and U.- [113] H. Djapo, B.-J. Schaefer, and J. Wambach, Physical G. Meißner, Physical Review C 84, 054001 (2011), Review C 81, 035803 (2010), arXiv:0811.2939. arXiv:1108.3816. [114] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, [93] H. Krebs, A. Gasparyan, and E. Epelbaum, Physical Nuclear Physics A 881, 62 (2012), arXiv:1111.6049. Review C 85, 054006 (2012), arXiv:1203.0067. [115] S. Weissenborn, D. Chatterjee, and J. Schaffner-Bielich, [94] H. Krebs, A. Gasparyan, and E. Epelbaum, Physical Physical Review C 85, 065802 (2012), arXiv:1112.0234. Review C 87, 054007 (2013), arXiv:1302.2872. [116] Y. Yamamoto, T. Furumoto, N. Yasutake, and 36

T. A. Rijken, Physical Review C 90, 045805 (2014), [144] S. Iwao, Il 34, 1167 (1964). arXiv:arXiv:1406.4332v1. [145] C. B. Dover and H. Feshbach, Annals of Physics 198, [117] T. Takatsuka, S. Nishizaki, and R. Tamagaki, Progress 321 (1990). of Theoretical Physics Supplement 174, 80 (2008). [146] C. B. Dover and H. Feshbach, Annals of Physics 217, [118] I. Vidaña, D. Logoteta, C. Providência, A. Polls, 51 (1992). and I. Bombaci, Europhys. Lett. 94, 11002 (2011), [147] E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nu- arXiv:1006.5660. clear Physics A 747, 362 (2005), arXiv:nucl-th/0405048 [119] D. Lonardoni, A. Lovato, S. Gandolfi, and F. Ped- [nucl-th]. eriva, Physical Review Letters 114, 092301 (2015), [148] N. Kaiser, R. Brockmann, and W. Weise, Nuclear arXiv:1407.4448. Physics A 625, 758 (1997), arXiv:nucl-th/9706045 [120] S. Petschauer, N. Kaiser, J. Haidenbauer, U.-G. [nucl-th]. Meißner, and W. Weise, Physical Review C 93, 014001 [149] E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nuclear (2016), arXiv:1511.02095. Physics A 637, 107 (1998), arXiv:9801064 [nucl-th]. [121] S. Petschauer, J. Haidenbauer, N. Kaiser, U.-G. [150] E. Epelbaum, W. Glöckle, and U.-G. Meißner, Nuclear Meißner, and W. Weise, Nuclear Physics A 957, 347 Physics A 671, 295 (2000), arXiv:9910064 [nucl-th]. (2017), arXiv:1607.04307. [151] D. R. Entem and R. Machleidt, Physical Review C 68, [122] M. Kohno, Physical Review C 97, 035206 (2018), 041001 (2003), arXiv:0304018 [nucl-th]. arXiv:1802.05388. [152] K. Tominaga, T. Ueda, M. Yamaguchi, N. Kijima, [123] S. L. Adler and R. F. Dashen, Current algebras and D. Okamoto, K. Miyagawa, and T. Yamada, Nucl. applications to (Benjamin, New York, Phys. A642, 483 (1998). 1968). [153] K. Tominaga and T. Ueda, Nucl. Phys. A693, 731 [124] S. Coleman, J. Wess, and B. Zumino, Physical Review (2001). 177, 2239 (1969). [154] A. Reuber, K. Holinde, H.-C. Kim, and J. Speth, Nucl. [125] C. G. Callan, S. Coleman, J. Wess, and B. Zumino, Phys. A608, 243 (1996), arXiv:nucl-th/9511011 [nucl- Physical Review 177, 2247 (1969). th]. [126] J. Gasser and H. Leutwyler, Nuclear Physics B 250, 465 [155] M. Frink and U.-G. Meißner, Journal of High Energy (1985). Physics 0407, 028 (2004), arXiv:0404018 [hep-lat]. [127] E. Jenkins and A. V. Manohar, Physics Letters B 255, [156] M. Mai, P. Bruns, B. Kubis, and U.-G. Meißner, Phys- 558 (1991). ical Review D 80, 094006 (2009), arXiv:0905.2810. [128] V. Bernard, N. Kaiser, J. Kambor, and U.-G. Meißner, [157] V. Bernard, N. Kaiser, and U.-G. Meißner, Nuclear Nucl. Phys. B388, 315 (1992). Physics A 615, 483 (1997), arXiv:9611253 [hep-ph]. [129] A. W. Thomas and W. Weise, The Structure of the Nu- [158] N. Kaiser, S. Gerstendörfer, and W. Weise, Nucl.Phys. cleon (Wiley-VCH Verlag, Berlin, 2001). A637, 395 (1998), arXiv:nucl-th/9802071 [nucl-th]. [130] S. Scherer and M. Schindler, Lecture Notes in Physics [159] H. Krebs, E. Epelbaum, and U.-G. Meißner, The Euro- 830, 1 (2012). pean Physical Journal A 32, 127 (2007), arXiv:0703087 [131] S. Petschauer, Baryonic forces and hyperons in nuclear [nucl-th]. matter from SU(3) chiral effective field theory, Ph.d. [160] E. Epelbaum, H. Krebs, and U.-G. Meißner, Nuclear thesis, Technische Universität München, München, Ger- Physics A 806, 65 (2008), arXiv:0712.1969. many (2015). [161] K. Sasaki, E. Oset, and M. Vacas, Physical Review C [132] S. L. Adler, Physical Review 177, 2426 (1969). 74, 064002 (2006), arXiv:nucl-th/0607068 [nucl-th]. [133] J. S. Bell and R. Jackiw, Il Nuovo Cimento A 60, 47 [162] T. E. O. Ericson and W. Weise, and Nuclei (1969). (Clarendon Press, Oxford, 1988). [134] H. Leutwyler, Annals of Physics 235, 165 (1994), [163] J. W. Holt, N. Kaiser, and W. Weise, Physical Review arXiv:hep-ph/9311274 [hep-ph]. C 81, 024002 (2010), arXiv:0910.1249 [nucl-th]. [135] H. Georgi, Weak Interactions and Modern Particle The- [164] C. M. Vincent and S. Phatak, Physical Review C 10, ory (Benjamin / Cummings, Reading, MA, 1984). 391 (1974). [136] H. W. Fearing and S. Scherer, Physical Review D 53, [165] R. J. Furnstahl, D. R. Phillips, and S. Wesolowski,J. 315 (1996), arXiv:hep-ph/9408346 [hep-ph]. Phys. G 42, 034028 (2015), arXiv:1407.0657. [137] J. Bijnens, G. Colangelo, and G. Ecker, Journal of High [166] E. Epelbaum, H. Krebs, and U.-G. Meißner, The Euro- Energy Physics 1999, 020 (1999), arXiv:9902437 [hep- pean Physical Journal A 51, 53 (2015), arXiv:1412.0142. ph]. [167] R. J. Furnstahl, N. Klco, D. R. Phillips, and [138] N. Fettes, U.-G. Meißner, M. Mojzis, and S. Steininger, S. Wesolowski, Physical Review C 92, 024005 (2015), Annals of Physics 283, 273 (2000), arXiv:0001308 [hep- arXiv:1506.01343. ph]. [168] S. Binder et al. (LENPIC), Phys. Rev. C 93, 044002 [139] A. Krause, Helv. Phys. Acta 63, 3 (1990). (2016), arXiv:1505.07218 [nucl-th]. [140] J. A. Oller, M. Verbeni, and J. Prades, Journal [169] B. Sechi-Zorn, B. Kehoe, J. Twitty, and R. A. Burn- of High Energy Physics 2006, 079 (2006), arXiv:hep- stein, Physical Review 175, 1735 (1968). ph/0608204 [hep-ph]. [170] G. Alexander, U. Karshon, A. Shapira, G. Yekutieli, [141] M. Frink and U.-G. Meißner, The European Physical R. Engelmann, H. Filthuth, and W. Lughofer, Physical Journal A 29, 255 (2006), arXiv:0609256 [hep-ph]. Review 173, 1452 (1968). [142] L. Girlanda, S. Pastore, R. Schiavilla, and M. Viviani, [171] R. Engelmann, H. Filthuth, V. Hepp, and E. Kluge, Physical Review C 81, 034005 (2010), arXiv:1001.3676. Physics Letters 21, 587 (1966). [143] B. Borasoy, Physical Review D 59, 054021 (1999), [172] F. Eisele, H. Filthuth, W. Föhlisch, V. Hepp, and arXiv:9811411 [hep-ph]. G. Zech, Physics Letters B 37, 204 (1971). 37

[173] V. Hepp and H. Schleich, Z. Phys. 214, 71 (1968). [189] E. Friedman and A. Gal, Physics Reports 452, 89 [174] D. Stephen,, Ph.D. thesis, University of Massachusetts (2007), arXiv:0705.3965. (1970). [190] M. Baldo, Nuclear methods and the nuclear equation of [175] A. Nogga, Nuclear Physics A 914, 140 (2013). state, International Review of Nuclear Physics, Vol. 8 [176] J. Haidenbauer, U. G. Meißner, N. Kaiser, and (World Scientific Publishing Co. Pte. Ltd., 1999). W. Weise, The European Physical Journal A 53, 121 [191] A. L. Fetter and J. D. Walecka, of (2017), arXiv:1612.03758. Many-Particle Systems, Dover Books on Physics (Dover [177] J. K. Ahn et al., Phys. Lett. B633, 214 (2006), Publications, New York, 2003). arXiv:nucl-ex/0502010 [nucl-ex]. [192] H. A. Bethe, B. H. Brandow, and A. G. Petschek, Phys- [178] S. J. Kim et al., Unpublished. ical Review 129, 225 (1963). [179] S. Aoki et al., Nucl. Phys. A644, 365 (1998). [193] H. Q. Song, M. Baldo, G. Giansiracusa, and U. Lom- [180] H. Takahashi et al., Phys. Rev. Lett. 87, 212502 (2001). bardo, Physical Review Letters 81, 1584 (1998). [181] A. M. Gasparyan, J. Haidenbauer, and C. Hanhart, [194] A. Gal, Progress of Theoretical Physics Supplement Phys. Rev. C85, 015204 (2012), arXiv:1111.0513 [nucl- 186, 270 (2010), arXiv:1008.3510 [nucl-th]. th]. [195] E. Botta, T. Bressani, and G. Garbarino, The European [182] A. Ohnishi, K. Morita, K. Miyahara, and T. Hyodo, Physical Journal A 48, 41 (2012), arXiv:1203.5707. Nucl. Phys. A954, 294 (2016), arXiv:1603.05761 [nucl- [196] C. B. Dover, D. J. Millener, and A. Gal, Physics Re- th]. ports 184, 1 (1989). [183] K. Nakazawa et al., PTEP 2015, 033D02 (2015). [197] J. Dabrowski, Physical Review C 60, 025205 (1999). [184] A. Gal, E. V. Hungerford, and D. J. Millener, Rev. [198] M. Kohno, Phys. Rev. C 100, 024313 (2019), Mod. Phys. 88, 035004 (2016), arXiv:1605.00557 [nucl- arXiv:1908.01934 [nucl-th]. th]. [199] J. Haidenbauer and I. Vidana, (2019), [185] D. J. Millener, C. B. Dover, and A. Gal, Physical Re- doi.org/10.1140/epja/s10050-020-00055-6, view C 38, 2700 (1988). arXiv:1910.02695 [nucl-th]. [186] Y. Yamamoto, H. Band¯o, and J. Zofka, Progress of [200] I. Vidaña, Nucl. Phys. A958, 48 (2017), Theoretical Physics 80, 757 (1988). arXiv:1603.05635 [nucl-th]. [187] S. Ajimura, H. Hayakawa, T. Kishimoto, H. Kohri, [201] A. Nogga, Proceedings, 13th International Conference K. Matsuoka, S. Minami, T. Mori, K. Morikubo, E. Saji, on Hypernuclear and Physics (HYP A. Sakaguchi, Y. Shimizu, M. Sumihama, R. E. Chrien, 2018): Portsmouth Virginia, USA, June 24-29, 2018, M. May, P. H. Pile, A. Rusek, R. Sutter, P. M. Eugenio, AIP Conf. Proc. 2130, 030004 (2019). G. B. Franklin, P. Khaustov, K. Paschke, B. P. Quinn, [202] H. Le, J. Haidenbauer, U.-G. Meißner, and A. Nogga, R. A. Schumacher, J. Franz, T. Fukuda, H. Noumi, Phys. Lett. B801, 135189 (2020), arXiv:1909.02882 H. Outa, L. Gan, L. Tang, L. Yuan, H. Tamura, [nucl-th]. J. Nakano, T. Tamagawa, K. Tanida, and R. I. Sawafta, [203] J. Haidenbauer, U.-G. Meißner, A. Nogga, and Physical Review Letters 86, 4255 (2001). H. Polinder, Lect.Notes Phys. Lecture Notes in Physics, [188] H. Akikawa, S. Ajimura, R. E. Chrien, P. M. Eu- 724, 113 (2007), arXiv:0702015 [nucl-th]. genio, G. B. Franklin, J. Franz, L. Gang, K. Imai, [204] D. Gerstung, N. Kaiser, and W. Weise, (2020), P. Khaustov, M. May, P. H. Pile, B. P. Quinn, A. Rusek, arXiv:2001.10563 [nucl-th]. J. Sasao, R. I. Sawafta, H. Schmitt, H. Tamura, L. Tang, [205] Y. Yamamoto, S. Nishizaki, and T. Takatsuka, Prog. K. Tanida, L. Yuan, S. H. Zhou, L. H. Zhu, and X. F. Theor. Phys. 103, 981 (2000). Zhu, Physical Review Letters 88, 082501 (2002).