hyperon-nuclear interactions from su(3) chiral effective field … · 2020. 3. 9. · and hyperon...
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REVIEWpublished: xx January 2020
doi: 10.3389/fphy.2020.00012
Edited by:
Ruprecht Machleidt,
University of Idaho, United States
Reviewed by:
Theodoros Gaitanos,
Aristotle University of Thessaloniki,
Greece
Francesca Gulminelli,
University of Caen Normandy, France
*Correspondence:
Stefan Petschauer
Specialty section:
This article was submitted to
Nuclear Physics,
a section of the journal
Frontiers in Physics
Received: 13 October 2019
Accepted: 13 January 2020
Published: xx January 2020
Citation:
Petschauer S, Haidenbauer J,
Kaiser N, Meißner U-G and Weise W
(2020) Hyperon-Nuclear Interactions
From SU(3) Chiral Effective Field
Theory. Front. Phys. 8:12.
doi: 10.3389/fphy.2020.00012
Hyperon-Nuclear Interactions FromSU(3) Chiral Effective Field TheoryStefan Petschauer 1,2*, Johann Haidenbauer 3, Norbert Kaiser 2, Ulf-G. Meißner 4,3,5 and
Wolfram Weise 2
1 TNG Technology Consulting GmbH, Unterföhring, Germany, 2 Physik Department, Technische Universität München,
Garching, Germany, 3 Institute for Advanced Simulation and Jülich Center for Hadron Physics, Institut für Kernphysik,
Forschungszentrum Jülich, Jülich, Germany, 4 Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for
Theoretical Physics, Universität Bonn, Bonn, Germany, 5 Tbilisi State University, Tbilisi, Georgia
The interaction between hyperons and nucleons has a wide range of applications in
strangeness nuclear physics and is a topic of continuing great interest. These interactions
are not only important for hyperon-nucleon scattering but also essential as basic input
to studies of hyperon-nuclear few- and many-body systems including hypernuclei and
neutron star matter. We review the systematic derivation and construction of such
baryonic forces from the symmetries of quantum chromodynamics 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.
Keywords: chiral Lagrangian, effective field theory, hyperon-nucleon interaction, flavor SU(3) symmetry,
strangeness nuclear physics
1. INTRODUCTION
Strangeness nuclear physics is an important topic of ongoing research, addressing for examplescattering of baryons including strangeness, properties of hypernuclei, or strangeness in infinitenuclear matter and in neutron star matter. The theoretical foundation for such investigations areinteraction potentials between nucleons and strange baryons such as the ! hyperon.
Nuclear many-body systems are (mainly) governed by the strong interaction, described at thefundamental level by quantum chromodynamics (QCD). The elementary degrees of freedom ofQCD are quarks and gluons. However, in the low-energy regime of QCD quarks and gluons areconfined into colorless hadrons. This is the region where (hyper-)nuclear systems are formed. Inthis region QCD cannot be solved in a perturbative way. Lattice QCD is approaching this problemvia large-scale numerical simulations: the (Euclidean) space-time is discretized and QCD is solvedon a finite grid [1–4]. Since the seminal work of Weinberg [5, 6] chiral effective field theory (χEFT)has become a powerful tool for calculating systematically the strong interaction dynamics for low-energy hadronic processes [7–9]. Chiral EFT employs the same symmetries and symmetry breakingpatterns at low-energies as QCD, but it uses the proper degrees of freedom, namely hadrons insteadof quarks and gluons. In combination with an appropriate expansion in small external momenta,the results can be improved systematically, by going to higher order in the power counting, andat the same time theoretical errors can be estimated. Furthermore, two- and three-baryon forcescan be constructed in a consistent fashion. The unresolved short-distance dynamics is encoded inχEFT in contact terms, with a priori unknown low-energy constants (LECs).
The NN interaction is empirically known to very high precision. Corresponding two-nucleonpotentials have been derived to high accuracy in phenomenological approaches [10–12]. Nowadaysthe systematic theory to construct nuclear forces is χEFT [13, 14]. (Note however that there are stilldebates about the Weinberg power counting schemes and how it is employed in practice [15–17]).
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In contrast, the YN interaction is presently not known in suchdetail. The scarce experimental data (about 35 data points forlow-energy total cross sections) do not allow for a uniquedetermination of the hyperon-nucleon interaction. The limitedaccuracy of the YN scattering data does not permit a uniquephase shift analysis. However, at experimental facilities such as J-PARC in Japan or later at FAIR in Germany, a significant amountof beam time will be devoted to strangeness nuclear physics.Various phenomenological approaches have been employedto describe the YN interaction, in particular boson-exchangemodels [18–23] or quark models [24–26]. However, giventhe poor experimental data base, these interactions differconsiderably from each other. Obviously there is a need for amore systematic investigation based on the underlying theoryof the strong interaction, QCD. Some aspects of YN scatteringand hyperon mass shifts in nuclear matter using EFT methodshave been covered in Savage and Wise [27] and Korpa et al.[28]. The YN interaction has been investigated at leadingorder (LO) in SU(3) χEFT [29–31] by extending the verysuccessful χEFT framework for the nucleonic sector [13, 14]to the strangeness sector. This work has been extended tonext-to-leading order (NLO) in Petschauer and Kaiser [32],Haidenbauer et al. [33, 34] where an excellent description ofthe strangeness −1 sector has been achieved, comparable tomost advanced phenomenological hyperon-nucleon interactionmodels. An extension to systems with more strangeness hasbeen done in Haidenbauer et al. [35, 36] and Haidenbauer andMeißner [37]. Systems including decuplet baryons have beeninvestigated in Haidenbauer et al. [38] at leading order in non-relativistic χEFT. Recently calculations within leading ordercovariant χEFT have been performed for YN interactions inthe strangeness sector [39–43] with comparable results (see also[44]). It is worth to briefly discuss the differences between thecovariant and the heavy-baryon approach. In the latter, due to theexpansion in the inverse of the baryon masses, some terms arerelegated to higher orders. Also, it can happen that the analyticstructure is distorted in the strict heavy-baryon limit. This caneasily be remedied by including the kinetic energy term in thebaryon propagator [45]. In what follows, we will present resultsbased on the heavy-baryon approach.
Numerous advanced few- and many-body techniques havebeen developed to employ such phenomenological or chiralinteractions, in order to calculate the properties of nuclearsystems with and without strangeness. For example, systemswith three or four particles can be reliably treated by Faddeev-Yakubovsky theory [46–49], somewhat heavier (hyper)nucleiwith approaches like the no-core-shell model [50–55]. In thenucleonic sector many-body approaches such as QuantumMonte Carlo calculations [56–58], or nuclear lattice simulations[59–61] have been successfully applied and can be extendedto the strangeness sector. Furthermore, nuclear matter iswell described by many-body perturbation theory with chirallow-momentum interactions [62–64]. Concerning ! and #
hyperons in nuclear matter, specific long-range processesrelated to two-pion exchange between hyperons and nucleonsin the nuclear medium have been studied in Kaiser andWeise [65] and Kaiser [66]. Conventional Brueckner theory
[67–69] at first order in the hole-line expansion, the so-calledBruecker-Hartree-Fock approximation, has been widely appliedto calculations of hypernuclear matter [20, 24, 70, 71] employingphenomenological two-body potentials. This approach is alsoused in investigations of neutron star matter [72–74]. Recently,corresponding calculations of the properties of hyperons innuclear matter have been also performed with chiral YNinteraction potentials [37, 75, 76].
Employing the high precision NN interactions describedabove, even “simple” nuclear systems such as triton cannotbe described satisfactorily with two-body interactions alone.The introduction of three-nucleon forces (3NF) substantiallyimproves this situation [77–80] and also in the context of infinitenuclear matter 3NF are essential to achieve saturation of nuclearmatter. These 3NF are introduced either phenomenologically,such as the families of Tuscon-Melbourne [81, 82], Brazilian[83], or Urbana-Illinois [84, 85] 3NF, or constructed accordingto the basic principles of χEFT [78, 86–94]. Within an EFTapproach, 3NF arise naturally and consistently together withtwo-nucleon forces. Chiral three-nucleon forces are importantin order to get saturation of nuclear matter from chirallow-momentum two-body interactions treated in many-bodyperturbation theory [63]. In the strangeness sectors the situationis similar: Three-baryon forces (3BF), especially the !NNinteraction, seem to be important for a satisfactorily descriptionof hypernuclei and hypernuclear matter [58, 95–103]. Especiallyin the context of neutron stars, 3BF are frequently discussed.The observation of two-solar-mass neutron stars [104, 105] setsstrong constraints on the stiffness of the equation-of-state (EoS)of dense baryonic matter [106–110]. The analysis of recentlyobserved gravitational wave signals from a two merging neutronstars [111, 112] provides further conditions, by constraining thetidal deformability of neutron star matter.
A naive introduction of!-hyperons as an additional baryonicdegree of freedomwould soften the EoS such that it is not possibleto stabilize a two-solar-mass neutron star against gravitationalcollapse [113]. To solve this so-called hyperon puzzle, several ad-hoc mechanisms have so far been invoked, e.g., through vectormeson exchange [114, 115], multi-Pomeron exchange [116] ora suitably adjusted repulsive !NN three-body interaction [117–119]. Clearly, a more systematic approach to the three-baryoninteraction within χEFT is needed, to estimate whether the 3BFcan provide the necessary repulsion and thus keep the equation-of-state sufficiently stiff. A first step in this direction was done inPetschauer et al. [120], where the leading 3BFs have been derivedwithin SU(3) χEFT. The corresponding low-energy constantshave been estimated by decuplet saturation in Petschauer et al.[121]. The effect of these estimated 3BF has been investigated inPetschauer et al. [121] and Kohno [122].
In this review article we present, on a basic level, theemergence of nuclear interactions in the strangeness sectorfrom the perspective of (heavy-baryon) chiral effective fieldtheory. After a brief introduction to SU(3) χEFT in section2, we present how the interaction between hyperons andnucleons is derived at NLO from these basic principles for two-baryon interactions (section 3) and for three-baryon interactions(section 4). In section 5, applications of these potentials are briefly
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reviewed for YN scattering, infinite nuclear matter, hypernuclei,and neutron star matter.
2. SU(3) CHIRAL EFFECTIVE FIELDTHEORY
An effective field theory (EFT) is a low-energy approximation toa more fundamental theory. Physical quantities can be calculatedin terms of a low-energy expansion in powers of small energiesand momenta over some characteristic large scale. The basic ideaof an EFT is to include the relevant degrees of freedom explicitly,while heavier (frozen) degrees of freedom are integrated out.An effective Lagrangian is obtained by constructing the mostgeneral Lagrangian including the active degrees of freedom, thatis consistent with the symmetries of the underlying fundamentaltheory [6]. At a given order in the expansion, the theory ischaracterized by a finite number of coupling constants, calledlow-energy constants (LECs). The LECs encode the unresolvedshort-distance dynamics and furthermore allow for an order-by-order renormalization of the theory. These constants are a prioriunknown, but once determined from one experiment or fromthe underlying theory, predictions for physical observables canbe made. However, due to the low-energy expansion and thetruncation of degrees of freedom, an EFT has only a limited rangeof validity.
The underlying theory of chiral effective field theory isquantum chromodynamics. QCD is characterized by twoimportant properties. For high energies the (running) couplingstrength of QCDbecomes weak, hence a perturbative approach inthe high-energy regime of QCD is possible. This famous featureis called asymptotic freedom of QCD and originates from thenon-Abelian structure of QCD. However, at low energies andmomenta the coupling strength of QCD is of order one, and aperturbative approach is no longer possible. This is the regionof non-perturbative QCD, in which we are interested in. Severalstrategies to approach this regime have been developed, suchas lattice simulations, Dyson-Schwinger equations, QCD sumrules or chiral perturbation theory. The second important featureof QCD is the so-called color confinement: isolated quarks andgluons are not observed in nature, but only color-singlet objects.These color-neutral particles, the hadrons, are the active degreesof freedom in χEFT.
But already before QCD was established, the ideas ofan effective field theory were used in the context of thestrong interaction. In the 60’s the Ward identities related tospontaneously broken chiral symmetry were explored by usingcurrent algebra methods (e.g., [123]). The group-theoreticalfoundations for constructing phenomenological Lagrangians inthe presence of spontaneous symmetry breaking have beendeveloped byWeinberg [5], Coleman et al. [124], and Callan et al.[125]. With Weinberg’s seminal paper [6] it became clear howto systematically construct an EFT and generate loop correctionsto tree level results. This method was improved later by Gasserand Leutwyler [7, 126]. A systematic introduction of nucleons asdegrees of freedom was done by Gasser et al. [8]. They showedthat a fully relativistic treatment of nucleons is problematic, as
the nucleon mass does not vanish in the chiral limit and thusadds an extra scale. A solution for this problem was proposedby Jenkins and Manohar [127] by considering baryons as heavystatic sources. This approach was further developed using asystematic path-integral framework in Bernard et al. [128]. Thenucleon-nucleon interaction and related topics were consideredby Weinberg [86]. Nowadays χEFT is used as a powerful toolfor calculating systematically the strong interaction dynamics ofhadronic processes, such as the accurate description of nuclearforces [13, 14].
In this section, we give a short introduction to the underlyingsymmetries of QCD and their breaking pattern. The basicconcepts of χEFT are explained, especially the explicit degreesof freedom and the connection to the symmetries of QCD.We state in more detail how the chiral Lagrangian can beconstructed from basic principles. However, it is beyond thescope of this work to give a detailed introduction to χEFT andQCD. Rather we will introduce only the concepts necessary forthe derivation of hyperon-nuclear forces. We follow [9, 13, 14,129–131] and refer the reader for more details to these references(and references therein).
2.1. Low-Energy QuantumChromodynamicsLet us start the discussion with the QCD Lagrangian
LQCD =∑
f=u,d,s,c,b,t
qf(
i /D−mf
)
qf −1
4Gµν,aG
µνa , (1)
with the six quark flavors f and the gluonic field-strengthtensor Gµν,a(x). The gauge covariant derivative is defined by
Dµ = 1∂µ − igAaµ
λa2 , where Aa
µ(x) are the gluon fields andλa the Gell-Mann matrices. The QCD Lagrangian is symmetricunder the local color gauge symmetry, under global Lorentztransformations, and the discrete symmetries parity, chargeconjugation, and time reversal. In the following we will introducethe so-called chiral symmetry, an approximate global continuoussymmetry of the QCD Lagrangian. The chiral symmetry isessential for chiral effective field theory. In view of the applicationto low energies, we divide the quarks into three light quarks u, d, sand three heavy quarks c, b, t, since the quark masses fulfill ahierarchical ordering:
mu,md,ms " 1 GeV ≤ mc,mb,mt . (2)
At energies andmomenta well below 1GeV, the heavy quarks canbe treated effectively as static. Therefore, the light quarks are theonly active degrees of freedom of QCD for the low-energy regionwe are interested in. In the following we approximate the QCDLagrangian by using only the three light quarks. Compared tocharacteristic hadronic scales, such as the nucleon mass (MN ≈939 MeV), the light quark masses are small. Therefore, a goodstarting point for our discussion of low-energy QCD are masslessquarks mu = md = ms = 0, which is referred to as the chirallimit. The QCD Lagrangian becomes in the chiral limit
L0QCD =
∑
f=u,d,s
qf i /Dqf −1
4Gµν,aG
µνa . (3)
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Now each quark field qf (x) is decomposed into itschiral components
qf ,L = PL qf , qf ,R = PR qf . (4)
using the left- and right-handed projection operators
PL = 1
2(1− γ5) , PR = 1
2(1+ γ5) , (5)
with the chirality matrix γ5. These projectors are called left- andright-handed since in the chiral limit they project the free quark
fields on helicity eigenstates, h qL,R = ± qL,R, with h = %σ · %p /∣∣%p∣∣.
For massless free fermions helicity is equal to chirality.Collecting the three quark-flavor fields q = (qu, qd, qs) and
equivalently for the left and right handed components, we canexpress the QCD Lagrangian in the chiral limit as
L0QCD = qRi /DqR + qLi /DqL −
1
4Gµν,aG
µνa . (6)
Obviously the right- and left-handed components of the masslessquarks are separated. The Lagrangian is invariant under aglobal transformation
qL → L qL , qR → R qR , (7)
with independent unitary 3 × 3 matrices L and R acting inflavor space. This means that L 0
QCD possesses (at the classical,unquantized level) a global U(3)L×U(3)R symmetry, isomorphicto a global SU(3)L×U(1)L×SU(3)R×U(1)R symmetry.U(1)L×U(1)R are often rewritten into a vector and an axial-vector partU(1)V × U(1)A, named after the transformation behavior of thecorresponding conserved currents under parity transformation.The flavor-singlet vector current originates from rotations ofthe left- and right-handed quark fields with the same phase(“V = L + R ”) and the corresponding conserved charge isthe baryon number. After quantization, the conservation of theflavor-singlet axial vector current, with transformations of left-an right-handed quark fields with opposite phase (“A = L−R ”),gets broken due to the so-called Adler-Bell-Jackiw anomaly [132,133]. The symmetry group SU(3)L × SU(3)R refers to the chiralsymmetry. Similarly the conserved currents can be rewritten intoflavor-octet vector and flavor-octet axial-vector currents, wherethe vector currents correspond to the diagonal subgroup SU(3)Vof SU(3)L × SU(3)R with L = R.
After the introduction of small non-vanishing quark masses,the quark mass term of the QCD Lagrangian Equation (1) can beexpressed as
LM = −qMq = −(
qRMqL + qLMqR)
, (8)
with the diagonal quark mass matrix M = diag(mu,md,ms).Left- and right-handed quark fields are mixed in LM andthe chiral symmetry is explicitly broken. The baryon numberis still conserved, but the flavor-octet vector and axial-vectorcurrents are no longer conserved. The axial-vector current is notconserved for any small quark masses. However, the flavor-octet
vector current remains conserved, if the quark masses are equal,mu = md = ms, referred to as the (flavor) SU(3) limit.
Another crucial aspect of QCD is the so-called spontaneouschiral symmetry breaking. The chiral symmetry of the Lagrangianis not a symmetry of the ground state of the system, theQCD vacuum. The structure of the hadron spectrum allowsto conclude that the chiral symmetry SU(3)L × SU(3)R isspontaneously broken to its vectorial subgroup SU(3)V, the so-called Nambu-Goldstone realization of the chiral symmetry. Thespontaneous breaking of chiral symmetry can be characterizedby a non-vanishing chiral quark condensate 〈qq〉 *= 0,i.e., the vacuum involves strong correlations of scalar quark-antiquark pairs.
The eight Goldstone bosons corresponding to thespontaneous symmetry breaking of the chiral symmetry areidentified with the eight lightest hadrons, the pseudoscalarmesons (π±,π0,K±,K0, K0, η). They are pseudoscalar particles,due to the parity transformation behavior of the flavor-octetaxial-vector currents. The explicit chiral symmetry breaking dueto non-vanishing quark masses leads to non-zero masses of thepseudoscalar mesons. However, there is a substantial mass gap,between the masses of the pseudoscalar mesons and the lightesthadrons of the remaining hadronic spectrum. For non-vanishingbut equal quark masses, SU(3)V remains a symmetry of theground state. In this context SU(3)V is often called the flavorgroup SU(3), which provides the basis for the classification oflow-lying hadrons in multiplets. In the following we will considerthe so-called isospin symmetric limit, with mu = md *= ms. Theremaining symmetry is the SU(2) isospin symmetry. An essentialfeature of low-energy QCD is, that the pseudoscalar mesonsinteract weakly at low energies. This is a direct consequenceof their Goldstone-boson nature. This feature allows for theconstruction of a low-energy effective field theory enabling asystematic expansion in small momenta and quark masses.
Let us introduce onemore tool for the systematic developmentof χEFT called the external-field method. The chiral symmetrygives rise to so-called chiral Ward identities: relations betweenthe divergence of Green functions that include a symmetrycurrent (vector or axial-vector currents) to linear combinationsof Green functions. Even if the symmetry is explicitly broken,Ward identities related to the symmetry breaking term exist. Thechiral Ward identities do not rely on perturbation theory, but arealso valid in the non-perturbative region of QCD. The external-field method is an elegant way to formally combine all chiralWard identities in terms of invariance properties of a generatingfunctional. Following the procedure of Gasser and Leutwyler[7, 126] we introduce (color neutral) external fields, s(x), p(x),vµ(x), aµ(x), of the form of Hermitian 3 × 3 matrices thatcouple to scalar, pseudoscalar, vector, and axial-vector currentsof quarks:
L = L0QCD + Lext
= L0QCD + qγ µ(vµ + γ5aµ)q− q(s− iγ5p)q . (9)
All chiral Ward identities are encoded in the correspondinggenerating functional, if the global chiral symmetry SU(3)L ×SU(3)R of L 0
QCD is promoted to a local gauge symmetry of
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TABLE 1 | Transformation properties of the external fields under parity and charge
conjugation.
vµ aµ s p
P Pµν v
ν - Pµνa
ν s -p
C -vµ+ aµ+ s+ p+
For P a change of the spatial arguments (t, %x ) → (t,−%x ) is implied and we defined the
matrix Pµν = diag(+1,−1,−1,−1).
L [134]. Since L 0QCD is only invariant under the global
chiral symmetry, the external fields have to fulfill a suitabletransformation behavior:
vµ + aµ → R(vµ + aµ)R† + iR∂µR
† ,
vµ − aµ → L(vµ − aµ)L† + i L∂µL
† ,
s+ i p → R(
s+ i p)
L† ,
s− i p → L(
s− i p)
R† , (10)
where L(x) and R(x) are (independent) space-time-dependentelements of SU(3)L and SU(3)R.
Furthermore, we still require the full Lagrangian L to beinvariant under P, C, and T. As the transformation properties ofthe quarks are well-known, the transformation behavior of theexternal fields can be determined and is displayed in Table 1.Time reversal symmetry is not considered explicitly, since it isautomatically fulfilled due to the CPT theorem.
Another central aspect of the external-field method is theaddition of terms to the three-flavor QCD Lagrangian in thechiral limit, L 0
QCD. Non-vanishing current quark masses andtherefore the explicit breaking of chiral symmetry can be includedby setting the scalar field equal to the quark mass matrix, s(x) =M = diag (mu,md,ms). Similarly electroweak interactions can beintroduced through appropriate external vector and axial vectorfields. This feature is important, to systematically include explicitchiral symmetry breaking or couplings to electroweak gaugefields into the chiral effective Lagrangian.
2.2. Explicit Degrees of FreedomIn the low-energy regime of QCD, hadrons are the observablestates. The active degrees of freedom of χEFT are identified asthe pseudoscalar Goldstone-boson octet. The soft scale of thelow-energy expansion is given by the small external momentaand the small masses of the pseudo-Goldstone bosons, while thelarge scale is a typical hadronic scale of about 1 GeV. The effectiveLagrangian has to fulfill the same symmetry properties as QCD:invariance under Lorentz and parity transformations, chargeconjugation and time reversal symmetry. Especially the chiralsymmetry and its spontaneous symmetry breaking has to beincorporated. Using the external-field method, the same externalfields v, a, s, p as in Equation (9), with the same transformationbehavior, are included in the effective Lagrangian.
As the QCD vacuum is approximately invariant under theflavor symmetry group SU(3), one expects the hadrons toorganize themselves in multiplets of irreducible representationsof SU(3). The pseudoscalar mesons form an octet (cf. Figure 1).
FIGURE 1 | Pseudoscalar meson octet (JP = 0−), baryon octet (JP = 1/2+),
and baryon decuplet (JP = 3/2+).
The members of the octet are characterized by the strangenessquantum number S and the third component I3 of the isospin.The symbol η stands for the octet component (η8). As anapproximation we identify η8 with the physical η, ignoringpossible mixing with the singlet state η1. For the lowest-lyingbaryons one finds an octet and a decuplet (see also Figure 1).In the following we summarize how these explicit degrees offreedom are included in the chiral Lagrangian in the standardnon-linear realization of chiral symmetry [124, 125].
The chiral symmetry group SU(3)L×SU(3)R is spontaneouslybroken to its diagonal subgroup SU(3)V. Therefore, theGoldstone-boson octet should transform under SU(3)L × SU(3)Rsuch that an irreducible 8-representation results for SU(3)V. Aconvenient choice to describe the pseudoscalar mesons underthese conditions is a unitary 3 × 3 matrix U(x) in flavor space,which fulfills
U†U = 1 , detU = 1 . (11)
The transformation behavior under chiral symmetry reads
U → RUL† , (12)
where L(x), R(x) are elements of SU(3)L,R. An explicitparametrization of U(x) in terms of the pseudoscalar mesons isgiven by
U(x) = exp[
iφ(x)/f0]
, (13)
with the traceless Hermitian matrix
φ(x) =8∑
a=1
φa(x)λa
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=
π0 + 1√3η
√2π+ √
2K+√2π− −π0 + 1√
3η
√2K0
√2K− √
2K0 − 2√3η
. (14)
The constant f0 is the decay constant of the pseudoscalarGoldstone bosons in the chiral limit. For a transformation ofthe subgroup SU(3)V with L = R = V , the meson matrix Utransforms as
U → VUV† , (15)
i.e., the mesons φa(x) transform in the adjoint (irreducible)8-representation of SU(3). The parity transformation behavior
of the pseudoscalar mesons is φa(t, %x )P→ −φa(t,−%x ) or,
equivalently, U(t, %x ) P→ U†(t,−%x ). Under charge conjugationthe particle fields are mapped to antiparticle fields, leading to
UC→ U+.The octet baryons are described by Dirac spinor fields and
represented in a traceless 3× 3 matrix B(x) in flavor space,
B =8∑
a=1
Baλa√2
=
1√2#0 + 1√
6! #+ p
#− − 1√2#0 + 1√
6! n
,− ,0 − 2√6!
. (16)
We use the convenient [135] non-linear realization of chiralsymmetry for the baryons, which lifts the well-knownflavor transformations to the chiral symmetry group. Thematrix B(x) transforms under the chiral symmetry groupSU(3)L × SU(3)R as
B → KBK† , (17)
with the SU(3)-valued compensator field
K (L,R,U) =√
LU†R†R√U . (18)
Note that K (L,R,U) also depends on the meson matrix U. Thesquare root of the meson matrix,
u =√U , (19)
transforms as u →√RUL† = RuK† = KuL†.
For transformations under the subgroup SU(3)V the baryonstransform as an octet, i.e., the adjoint representation of SU(3):
B → VBV† . (20)
The octet-baryon fields transform under parity and charge
conjugation as Ba (t, %x )P→ γ0Ba (t,−%x ) and Bα,a
C→ Cαβ Bβ ,a
with the Dirac-spinor indices α,β , and with C = iγ2γ0.
A natural choice to represent the decuplet baryons is a totallysymmetric three-index tensor T. It transforms under the chiralsymmetry SU(3)L × SU(3)R as
Tabc → KadKbeKcf Tdef , (21)
with the compensator field K(L,R,U) of Equation (18). Foran SU(3)V transformation the decuplet fields transform as anirreducible representation of SU(3):
Tabc → VadVbeVcf Tdef . (22)
The physical fields are assigned to the following components ofthe totally antisymmetric tensor:
T111 = /++ ,T112 = 1√3/+ ,T122 = 1√
3/0 ,T222 = /− ,
T113 = 1√3#∗+ , T123 = 1√
6#∗0 , T223 = 1√
3#∗− ,
T133 = 1√3,∗0 , T233 = 1√
3,∗− ,
T333 = 0− . (23)
Since decuplet baryons are spin-3/2 particles, each component isexpressed through Rarita-Schwinger fields. Within the scope ofthis article, decuplet baryons are only used for estimating LECsvia decuplet resonance saturation. In that case it is sufficient totreat them in their non-relativistic form, where no complicationswith the Rarita-Schwinger formalism arise.
Now the representation of the explicit degrees of freedom andtheir transformation behavior are established. Together with theexternal fields the construction of the chiral effective Lagrangianis straightforward.
2.3. Construction of the Chiral LagrangianThe chiral Lagrangian can be ordered according to the number ofbaryon fields:
Leff = Lφ + LB + LBB + LBBB + . . . , (24)
where Lφ denotes the purely mesonic part of the Lagrangian.Each part is organized in the number of small momenta (i.e.,derivatives) or small meson masses, e.g.,
Lφ = L(2)φ + L
(4)φ + L
(6)φ + . . . . (25)
Lφ has been constructed to O(q6) in Fearing and Scherer [136]and Bijnens et al. [137]. The chiral Lagrangian for the baryon-number-one sector has been investigated in various works.The chiral effective pion-nucleon Lagrangian of order O(q4)has been constructed in Fettes et al. [138]. The three-flavorLorentz invariant chiral meson-baryon Lagrangians LB at orderO(q2) and O(q3) have been first formulated in Krause [139]and were later completed in Oller et al. [140] and Frink andMeißner [141]. Concerning the nucleon-nucleon contact terms,the relativistically invariant contact Lagrangian at orderO(q2) fortwo flavors (without any external fields) has been constructed inGirlanda et al. [142]. The baryon-baryon interaction Lagrangian
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LBB has been considered up to NLO in Savage and Wise [27],Polinder et al. [29], and Petschauer and Kaiser [32]. Furthermorethe leading three-baryon contact interaction Lagrangian LBBB
has been derived in Petschauer et al. [120].We follow closely Petschauer and Kaiser [32] to summarize
the basic procedure for constructing systematically the three-flavor chiral effective Lagrangian [124, 125] with the inclusionof external fields [7, 126]. The effective chiral Lagrangian hasto fulfill all discrete and continuous symmetries of the stronginteraction. Therefore, it has to be invariant under parity (P),charge conjugation (C), Hermitian conjugation (H), and theproper, orthochronous Lorentz transformations. Time reversalsymmetry is then automatically fulfilled via the CPT theorem.Especially local chiral symmetry has to be fulfilled. A commonway to construct the chiral Lagrangian is to define so-calledbuilding blocks, from which the effective Lagrangian can bedetermined as an invariant polynomial. Considering the chiraltransformation properties, a convenient choice for the buildingblocks is
uµ = i[
u†(
∂µ − i rµ)
u− u(
∂µ − i lµ)
u†]
,
χ± = u†χu† ± uχ†u ,
f±µν = uf Lµνu† ± u†f Rµνu , (26)
with the combination
χ = 2B0(
s+ i p)
, (27)
containing the new parameter B0 and the external scalar andpseudoscalar fields. One defines external field strength tensors by
f Rµν = ∂µrν − ∂νrµ − i[
rµ, rν]
,
f Lµν = ∂µlν − ∂ν lµ − i[
lµ, lν]
, (28)
where the fields
rµ = vµ + aµ , lµ = vµ − aµ , (29)
describe right handed and left handed external vector fields. Inthe absence of flavor singlet couplings one can assume 〈aµ〉 =〈vµ〉 = 0, where 〈. . . 〉 denotes the flavor trace. Therefore, thefields uµ and f±µν in Equation (26) are all traceless.
Using the transformation behavior of the pseudoscalar mesonsand octet baryons in Equations (12) and (17), and thetransformation properties of the external fields in Equation (10),one can determine the transformation behavior of the buildingblocks. All building blocks A, and therefore all products ofthese, transform according to the adjoint (octet) representationof SU(3), i.e., A → KAK†. Note that traces of products of suchbuilding blocks are invariant under local chiral symmetry, sinceK†K = 1. The chiral covariant derivative of such a building blockA is given by
DµA = ∂µA+[
1µ,A]
, (30)
with the chiral connection
1µ = 1
2
[
u†(
∂µ − i rµ)
u+ u(
∂µ − i lµ)
u†]
. (31)
The covariant derivative transforms homogeneously under thechiral group as DµA → K
(
DµA)
K†. The chiral covariantderivative of the baryon field B is given by Equation (30) as well.
A Lorentz-covariant power counting scheme has beenintroduced by Krause [139]. Due to the large baryon mass M0
in the chiral limit, a time-derivative acting on a baryon field Bcannot be counted as small. Only baryon three-momenta aresmall on typical chiral scales. This leads to the following countingrules for baryon fields and their covariant derivatives,
B , B , DµB ∼ O(
q0)
,(
i /D−M0)
B ∼ O(
q)
. (32)
The chiral dimension of the chiral building blocks and baryonbilinears B1B are given in Table 2. A covariant derivative actingon a building block (but not on B) raises the chiral dimensionby one.
A building block A transforms under parity, chargeconjugation and Hermitian conjugation as
AP = (−1)pA , AC = (−1)cA+ , A† = (−1)hA , (33)
with the exponents (modulo two) p, c, h ∈ {0, 1} given inTable 2A, and + denotes the transpose of a (flavor) matrix.A sign change of the spatial argument, (t, %x) → (t,−%x), isimplied in the fields in case of parity transformation P. Lorentzindices transform with the matrix Pµ
ν = diag(+1,−1,−1,−1)under parity transformation, e.g., (uµ)P = (−1)pPµ
νuν . Thetransformation behavior of commutators and anticommutatorsof two building blocks A1, A2 is the same as for building blockand should therefore be used instead of simple products, e.g.,
[A1,A2]C± = (−1)c1+c2 (A+
1 A+2 ± A+
2 A+1 )
= ±(−1)c1+c2 [A1,A2]+± . (34)
The behavior under Hermitian conjugation is the same.The basis elements of the Dirac algebra forming the baryon
bilinears transform as
γ01γ0 = (−1)p11 , C−11C = (−1)c11+ ,
γ01†γ0 = (−1)h11 , (35)
where the exponents p1 , c1 , h1 ∈ {0, 1} can be found inTable 2B.As before, Lorentz indices of baryon bilinears transform with thematrix Pµ
ν under parity.Due to the identity
[
Dµ,Dν
]
A = 1
4
[[
uµ, uν
]
,A]
− i
2
[
f+µν ,A]
(36)
it is sufficient to use only totally symmetrized products ofcovariant derivatives, Dαβγ ...A, for any building block A (orbaryon field B). Moreover, because of the relation
Dνuµ − Dµuν = f−µν , (37)
only the symmetrized covariant derivative acting on uµ need tobe taken into account,
hµν = Dµuν + Dνuµ . (38)
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TABLE 2 | Behavior under parity, charge conjugation, and Hermitian conjugation
as well as the chiral dimensions of chiral building blocks and baryon bilinears
B1B [140].
(A) Chiral building blocks
p c h O
uµ 1 0 0 O(
q1)
f+µν 0 1 0 O(
q2)
f−µν 1 0 0 O(
q2)
χ+ 0 0 0 O(
q2)
χ− 1 0 1 O(
q2)
(B) Baryon bilinears B!B
1 p c h O
1 0 0 0 O(
q0)
γ5 1 0 1 O(
q1)
γµ 0 1 0 O(
q0)
γ5γµ 1 0 0 O(
q0)
σµν 0 1 0 O(
q0)
Finally, the chiral effective Lagrangian can be constructedby taking traces (and products of traces) of differentpolynomials in the building blocks, so that they areinvariant under chiral symmetry, Lorentz transformations,C and P.
2.3.1. Leading-Order Meson LagrangianAs a first example, we show the leading-order purelymesonic Lagrangian. From the general constructionprinciples discussed above, one obtains for the leading-ordereffective Lagrangian
L(2)φ =
f 204〈uµu
µ + χ+〉 . (39)
Note that there is no contribution of order O(q0). This isconsistent with the vanishing interaction of the Goldstone bosonsin the chiral limit at zero momenta.
Before we continue with the meson-baryon interactionLagrangian, let us elaborate on the leading chiral Lagrangian inthe purely mesonic sector without external fields, but with non-vanishing quark masses in the isospin limit: vµ(x) = aµ(x) =p(x) = 0 and s(x) = M = diag (m,m,ms). Inserting thedefinitions of the building blocks, Equation (39) becomes withthese restrictions:
L(2)φ =
f 204〈∂µU∂µU†〉+ 1
2B0f
20 〈MU† + UM〉 . (40)
The physical decay constants fπ *= fK *= fη differ from thedecay constant of the pseudoscalar Goldstone bosons in the chirallimit f0 in terms of order (m,ms): fφ = f0 {1+O (m,ms)}. Theconstant B0 is related to the chiral quark condensate. Already
from this leading-order Lagrangian famous relations such as the(reformulated) Gell-Mann–Oakes–Renner relations
m2π = 2mB0 +O(m2
q) ,
m2K = (m+ms)B0 +O(m2
q) ,
m2η = 2
3(m+ 2ms)B0 +O(m2
q) , (41)
or the Gell-Mann–Okubo mass formula, 4m2K = 3m2
η +m2π , can
be derived systematically.
2.3.2. Leading-Order Meson-Baryon InteractionLagrangian
The leading-order meson-baryon interaction Lagrangian L(1)B is
of orderO(q) and reads1
L(1)B =〈B
(
i /D−MB)
B〉+ D
2〈Bγ
µγ5{uµ,B}〉
+ F
2〈Bγ
µγ5[
uµ,B]
〉 . (42)
The constant MB is the mass of the baryon octet in the chirallimit. The two new constants D and F are called axial-vectorcoupling constants. Their values can be obtained from semi-leptonic hyperon decays and are roughly D ≈ 0.8 and F ≈0.5 [143]. The sum of the two constants is related to the axial-vector coupling constant of nucleons, gA = D + F = 1.27,obtained from neutron beta decay. At lowest order the pion-nucleon coupling constant gπN is connected to the axial-vectorcoupling constant by the Goldberger-Treiman relation, gπNfπ =gAMN . The covariant derivative in Equation (42) includes thefield 1µ, which leads to a vertex between two octet baryonsand two mesons, whereas the terms containing uµ lead to avertex between two octet baryons and one meson. Different
octet-baryon masses appear first in L(2)B due to explicit chiral
symmetry breaking and renormalization and lead to correctionslinear in the quark masses:
Mi = MB +O(m,ms) . (43)
2.4. Weinberg Power Counting SchemeAs stated before, an effective field theory has an infinite numberof terms in the effective Lagrangian and for a fixed process aninfinite number of diagrams contribute. Therefore, it is crucialto have a power counting scheme, to assign the importance of aterm. Then, to a certain order in the power counting, only a finitenumber of terms contribute and the observables can be calculatedto a given accuracy.
First, let us discuss the power counting scheme of χEFT in thepure meson sector, i.e., only the pseudoscalar Goldstone bosonsare explicit degrees of freedom. The chiral dimension ν of aFeynman diagram represents the order in the low-momentumexpansion, (q/!χ )ν . The symbol q is generic for a small externalmeson momentum or a small meson mass. The scale of chiral
1Note that an overall plus sign in front of the constants D and F is chosen,consistent with the conventions in SU(2) χEFT [13].
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FIGURE 2 | Graphical representation of the Lippmann-Schwinger equation.
FIGURE 3 | Example of a planar box diagram. It contains an reducible part
equivalent to the iteration of two one-meson exchange diagrams, as
generated by the Lippmann-Schwinger equation. Additionally it contains a
genuine irreducible contribution that is part of the effective potential.
symmetry breaking !χ is often estimated as 4π fπ ≈ 1 GeV oras the mass of the lowest-lying resonance, Mρ ≈ 770 MeV. Asimple dimensional analysis leads to the following expression forthe chiral dimension of a connected Feynman diagram [6]:
ν = 2+ 2L+∑
i
vi/i , /i = di − 2 . (44)
The number of Goldstone boson loops is denoted by L and vi isthe number of vertices with vertex dimension /i. The symbol distands for the number of derivatives or meson mass insertions atthe vertex, i.e., the vertex originates from a term of the Lagrangianof the orderO(qdi ).
With the introduction of baryons in the chiral effectiveLagrangian, the power counting is more complicated. The largebaryon mass comes as an extra scale and destroys the one-to-one correspondence between the loop and the small momentumexpansion. Jenkins and Manohar used methods from heavy-quark effective field theory to solve this problem [127]. Basicallythey considered baryons as heavy, static sources. This leads to adescription of the baryons in the extreme non-relativistic limitwith an expansion in powers of the inverse baryon mass, calledheavy-baryon chiral perturbation theory.
Furthermore, in the two-baryon sector, additional featuresarise. Reducible Feynman diagrams are enhanced due to thepresence of small kinetic energy denominators resulting frompurely baryonic intermediate states. These graphs hint at the non-perturbative aspects in few-body problems, such as the existenceof shallow bound states, and must be summed up to all orders. Assuggested by Weinberg [86, 87], the baryons can be treated non-relativistically and the power counting scheme can be applied toan effective potential V , that contains only irreducible Feynmandiagrams. Terms with the inverse baryon mass M−1
B may be
FIGURE 4 | Examples for reducible (left) and irreducible (right) three-baryon
interactions for !NN. The thick dashed line cuts the reducible diagram in two
two-body interaction parts.
counted as
q
MB∝( q
!χ
)2. (45)
The resulting effective potential is the input for quantummechanical few-body calculations. In case of the baryon-baryoninteraction the effective potential is inserted into the Lippmann-Schwinger equation and solved for bound and scattering states.This is graphically shown in Figures 2, 3. The T-matrix isobtained from the infinite series of ladder diagrams withthe effective potential V . In this way the omitted reduciblediagrams 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 reducible diagrams such ason the left-hand side of Figure 4, are generated automatically andare not part of the effective potential. One should distinguish suchiterated two-body interactions, from irreducible three-baryonforces, as shown on the right-hand side of Figure 4.
After these considerations, a consistent power countingscheme for the effective potential V is possible. The soft scale qin the low-momentum expansion (q/!χ )ν denotes now smallexternal meson four-momenta, small external baryon three-momenta or the small meson masses. Naive dimensional analysisleads to the generalization of Equation (44):
ν = 2− B+ 2L+∑
i
vi/i , /i = di +1
2bi − 2 , (46)
where B is the number of external baryons and bi is the number ofinternal baryon lines at the considered vertex. However, Equation(46) has an unwanted dependence on the baryon number, due tothe normalization of baryon states. Such an effect can be avoidedby assigning the chiral dimension to the transition operatorinstead of thematrix elements. This leads to the addition of 3B−6to the formula for the chiral dimension, which leaves the B = 2case unaltered, and one obtains (see for example [9, 13, 14, 130])
ν = −4+ 2B+ 2L+∑
i
vi/i , /i = di +1
2bi − 2 . (47)
Following this scheme one arrives at the hierarchy of baryonicforces shown in Figure 5. The leading-order (ν = 0) potential
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FIGURE 5 | Hierarchy of baryonic forces. Solid lines are baryons, dashed lines are pseudoscalar mesons. Solid dots, filled circles, and squares denote vertices with
/i = 0, 1, and 2, respectively.
is given by one-meson-exchange diagrams and non-derivativefour-baryon contact terms. At next-to-leading order (ν = 2)higher order contact terms and two-meson-exchange diagramswith intermediate octet baryons contribute. Finally, at next-to-next-to-leading order (ν = 3) the three-baryon forces start tocontribute. Diagrams that lead to mass and coupling constantrenormalization are not shown.
3. BARYON-BARYON INTERACTIONPOTENTIALS
This section is devoted to the baryon-baryon interactionpotentials up to next-to-leading order, constructed from thediagrams shown in Figure 5. Contributions arise from contactinteraction, one- and two-Goldstone-boson exchange. Theconstructed potentials serve not only as input for the descriptionof baryon-baryon scattering, but are also basis for few- andmany-body calculations. We give also a brief introductionto common meson-exchange models and the difference tointeraction potentials from χEFT.
3.1. Baryon-Baryon Contact TermsThe chiral Lagrangian necessary for the contact vertices shownin Figure 6 can be constructed straightforwardly according tothe principles outlined in section 2. For pure baryon-baryonscattering processes, no pseudoscalar mesons are involved in thecontact vertices and almost all external fields can be dropped.Covariant derivatives Dµ reduce to ordinary derivatives ∂µ. Theonly surviving external field is χ+, which is responsible for theinclusion of quark masses into the chiral Lagrangian:
FIGURE 6 | Leading-order and next-to-leading-order baryon-baryon contact
vertices.
χ+2
= χ = 2B0
mu 0 00 md 00 0 ms
≈
m2π 0 00 m2
π 00 0 2m2
K −m2π
, (48)
where in the last step the Gell-Mann–Oakes–Renner relations,Equation (41), have been used. In flavor space the possible termsare of the schematic form
〈BBBB〉 , 〈BBBB〉 , 〈BB〉〈BB〉 , 〈BB〉〈BB〉 , (49)
and terms where the field χ is inserted such as
〈BχBBB〉 , 〈BBχ BB〉 , 〈BχB〉〈BB〉 , . . . (50)
where in both cases appropriate structures in Dirac space have tobe inserted. For the case of the non-relativistic power countingit would also be sufficient, to insert the corresponding structures
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in spin-momentum space. The terms involving χ lead to explicitSU(3) symmetry breaking at NLO linear in the quark masses. Aset of linearly independent Lagrangian terms up toO(q2) for purebaryon-baryon interaction in non-relativistic power countingcan be found in Petschauer and Kaiser [32].
After a non-relativistic expansion up toO(q2) the four-baryoncontact Lagrangian leads to potentials in spin and momentumspace. A convenient operator basis is given by [29]:
P1 = 1 ,
P2 = %σ1 · %σ2 ,
P3 = (%σ1 · %q )(%σ2 · %q )−1
3(%σ1 · %σ2)%q 2 ,
P4 =i
2(%σ1 + %σ2) · %n ,
P5 = (%σ1 · %n )(%σ2 · %n ) ,
P6 =i
2(%σ1 − %σ2) · %n ,
P7 = (%σ1 · %k )(%σ2 · %q )+ (%σ1 · %q )(%σ2 · %k ) ,
P8 = (%σ1 · %k )(%σ2 · %q )− (%σ1 · %q )(%σ2 · %k ) , (51)
with %σ1,2 the Pauli spin matrices and with the vectors
%k = 1
2(%pf + %pi) , %q = %pf − %pi , %n = %pi × %pf . (52)
The momenta %pf and %pi are the initial and final state momentain the center-of-mass frame. In order to obtain the minimalset of Lagrangian terms in the non-relativistic power countingof Petschauer and Kaiser [32], the potentials have beendecomposed into partial waves. The formulas for the partialwave projection of a general interaction V =
∑8j=1 VjPj
can be found in the appendix of Polinder et al. [29]. Foreach partial wave one produces a non-square matrix whichconnects the Lagrangian constants with the different baryon-baryon channels. Lagrangian terms are considered as redundantif their omission does not lower the rank of this matrix. Forthe determination of the potential not only direct contributionshave to be considered, but also additional structures fromexchanged final state baryons, where the negative spin-exchangeoperator −P(σ ) = − 1
2 (1+ %σ1 · %σ2) is applied. In the end6 momentum-independent terms at LO contribute, and aretherefore only visible in 1S0 and 3S1 partial waves. At NLO 22terms contribute that contain only baryon fields and derivatives,and are therefore SU(3) symmetric. The other 12 terms atNLO include the diagonal matrix χ and produce explicit SU(3)symmetry breaking.
In Table 3, the non-vanishing transitions projected ontopartial waves in the isospin basis are shown (cf. [29–32]).The pertinent constants are redefined according to the relevantirreducible SU(3) representations. This comes about in thefollowing way. Baryons form a flavor octet and the tensor productof two baryons decomposes into irreducible representationsas follows:
8⊗ 8 = 27s ⊕ 10a ⊕ 10∗a ⊕ 8s ⊕ 8a ⊕ 1s , (53)
where the irreducible representations 27s, 8s, 1s are symmetricand 10a, 10∗a , 8a are antisymmetric with respect to theexchange of both baryons. Due to the generalized Pauliprinciple, the symmetric flavor representations 27s, 8s, 1s haveto combine with the space-spin antisymmetric partial waves1S0, 3P0, 3P1, 3P2, . . . (L + S even). The antisymmetricflavor representations 10a, 10∗a , 8a combine with the space-spin symmetric partial waves 3S1, 1P1, 3D1 ↔ 3S1, . . .
(L + S odd). Transitions can only occur between equalirreducible representations. Hence, transitions between space-spin antisymmetric partial waves up to O(q2) involve the 15constants c27,8s,11S0
, c27,8s,11S0, c27,8s,13P0
, c27,8s,13P1, and c27,8s,13P2
, whereas
transitions between space-spin symmetric partial waves involve
the 12 constants c8a,10,10∗
3S1, c8a,10,10
∗3S1
, c8a,10,10∗
1P1, and c8a,10,10
∗3D1-3S1
. The
constants with a tilde denote leading-order constants, whereasthe ones without tilde are at NLO. The spin singlet-triplettransitions 1P1 ↔ 3P1 is perfectly allowed by SU(3) symmetrysince it is related to transitions between the irreduciblerepresentations 8a and 8s. Such a transition originated from theantisymmetric spin-orbit operator P6 and its Fierz-transformedcounterpart P8 and the single corresponding low-energy constantis denoted by c8as. In case of the NN interaction such transitionsare forbidden by isospin symmetry. The constants c27,8s,11S0
and
c8a,10,10∗
3S1fulfill the same SU(3) relations as the constants c27,8s,11S0
and c8a,10,10∗
3S1in Table 3. SU(3) breaking terms linear in the
quark masses appears only in the S-waves, 1S0, 3S1, and areproportional m2
K − m2π . The corresponding 12 constants are
c1,...,12χ . The SU(3) symmetry relations in Table 3 can alsobe derived by group theoretical considerations [29, 144–146].Clearly, for the SU(3)-breaking part this is not possible and thesecontributions have to be derived from the chiral Lagrangian.
In order to obtain the complete partial-wave projectedpotentials, some entries in Table 3 have to be multiplied withadditional momentum factors. The leading order constants c ijreceive no further factor. For the next-to-leading-order constants(without tilde and without χ) the contributions to the partialwaves 1S0, 3S1 have to be multiplied with a factor p2i + p2f .
The contribution to the partial waves 1S0, 3S1 from constants
cjχ has to be multiplied with (m2
K − m2π ). The partial waves
3P0, 3P1, 3P2, 1P1, 1P1 ↔ 3P1 get multiplied with the factor pipf .
The entries for 3S1 → 3D1 and 3D1 → 3S1 have to be multipliedwith p2i and p
2f , respectively. For example, one obtains for theNN
interaction in the 1S0 partial wave:
〈NN, 1S0|V|NN, 1S0〉
= c271S0 + c271S0 (p2i + p2f )+
1
2c1χ (m
2K −m2
π ) , (54)
or for the ,N → ## interaction with total isospin I = 0 in the1P1 → 3P1 partial wave:
〈##, 3P1|V|,N, 1P1〉 = 2√3c8aspipf . (55)
When restricting to the NN channel the well-known twoleading and seven next-to-leading order low-energy constants
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U(3)relationsofpure
baryo
n-baryo
nco
ntact
term
sfornon-van
ishingpartialw
aves
upto
O(q
2)innon-relativistic
power
countin
gforch
annelsdes
cribed
bystrangen
essSan
dtotaliso
spinI[32].
SI
Transition
j∈
{1S0,3P0,3P1,3P2}
j∈
{3S1,1P1,3S1
↔3D
1}
1P1
→3P1
3P1
→1P1
1S0
χ3S1
χ
00
NN→
NN
0c10∗
j0
00
c7 χ 2
1NN→
NN
c27j
00
0c1 χ 2
0
-11 2
!N→
!N
1 10(9c27j
+c8sj)
1 2(c
10∗
j+c8aj)
−c8as
−c8as
c2 χ
c8 χ
1 2!N→
#N
−3 10(c
27j
−c8sj)
1 2(c
10∗
j−c8aj)
−3c8as
c8as
−c3 χ
−c9 χ
1 2#N→
#N
1 10(c
27j
+9c8sj)
1 2(c
10∗
j+c8aj)
3c8as
3c8as
c4 χ
c10
χ
3 2#N→
#N
c27j
c10j
00
c1 χ 4
−c7 χ 4
-20
!!
→!
!1 40(5c1 j+
27c27j
+8c8sj)
00
0c5 χ 2
0
0!
!→
,N
1 20(5c1 j−
9c27j
+4c8sj)
00
2c8as
3c1 χ 4−
3c2 χ−c3 χ+
3c5 χ 4
0
0!
!→
##
−√3
40(5c1 j+
3c27j
−8c8sj)
00
00
0
0,N→
,N
1 10(5c1 j+
3c27j
+2c8sj)
c8aj
2c8as
2c8as
2c1 χ 3−
3c2 χ+
c4 χ 3+
9c5 χ 8
c11
χ
0,N→
##
√3
20(−
5c1 j+c27j
+4c8sj)
02√3c8as
0−
c1 χ
4√3+
√3c3 χ+
c4 χ √3
0
0#
#→
##
1 40(15c1 j+c27j
+24c8sj)
00
00
0
1,N→
,N
1 5(2c27j
+3c8sj)
1 3(c
10j
+c10∗
j+c8aj)
−2c8as
−2c8as
c6 χ
c12
χ
1,N→
##
01
3√2(c
10j
+c10∗
j−
2c8aj)
02√2c8as
0√2c10
χ−
c7 χ
2√2−
√2c9 χ
1,N→
#!
√6 5(c
27j
−c8sj)
1 √6(c
10j
−c10∗
j)
2√
2 3c8as
0−
1 3
√
2 3c1 χ+√
3 2c2 χ−
c4 χ
3√6−√
2 3c6 χ
c10
χ √6+√
2 3c12
χ+
c7 χ
2√6−√
3 2c8 χ+√
2 3c9 χ
1#
!→
#!
1 5(3c27j
+2c8sj)
1 2(c
10j
+c10∗
j)
00
−c1 χ 9+
4c3 χ 3+
4c4 χ 9+
2c6 χ 3
4c10
χ 3+
2c12
χ 3−
c7 χ 3−
4c9 χ 3
1#
!→
##
01
2√3(c
10j
−c10∗
j)
04 √3c8as
00
1#
#→
##
01 6(c
10j
+c10∗
j+
4c8aj)
00
00
2#
#→
##
c27j
00
00
0
-31 2
,!
→,
!1 10(9c27j
+c8sj)
1 2(c
10j
+c8aj)
−c8as
−c8as
−55c1 χ
72
+2c2 χ+
7c3 χ 6+
c4 χ 18+
3c5 χ
32
+c6 χ 12
11c10
χ
12
+3c11
χ 4+
25c12
χ
12
+5c7 χ
24
−7c8 χ 4−
c9 χ 6
1 2,
!→
,#
−3 10(c
27j
−c8sj)
1 2(c
10j
−c8aj)
−3c8as
c8as
11c1 χ
24
−3c2 χ 2−
c3 χ 2−
c4 χ 3+
9c5 χ
32
+c6 χ 4
9c10
χ 4−
3c11
χ 4+
5c12
χ 4−
c7 χ 8−
3c8 χ 4−
c9 χ 2
1 2,
#→
,#
1 10(c
27j
+9c8sj)
1 2(c
10j
+c8aj)
3c8as
3c8as
11c1 χ
24
−3c2 χ+
5c3 χ 2+
c4 χ 6+
27c5 χ
32
+3c6 χ 4
5c10
χ 4+
3c11
χ 4+
3c12
χ 4−
c7 χ 8−
3c8 χ 4−
3c9 χ 2
3 2,
#→
,#
c27j
c10∗
j0
0−
2c1 χ 3+
3c2 χ 2+c3 χ+
c4 χ 6
3c10
χ 2−c7 χ+
3c8 χ 2−
3c9 χ
-40
,,
→,
,0
c10j
00
05c10
χ+
4c12
χ−
3c8 χ−
2c9 χ
1,
,→
,,
c27j
00
0−
4c1 χ 3+
3c2 χ+
2c3 χ+
c4 χ 3
0
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of Epelbaum et al. [147] are recovered, which contribute to thepartial waves 1S0, 3S1, 1P1, 3P0, 3P1, 3P2, 3S1 ↔ 3D1.
Note, that the SU(3) relations in Table 3 are general relationsthat have to be fulfilled by the baryon-baryon potential inthe SU(3) limit, i.e., mπ = mK = mη. This feature canbe used as a check for the inclusion of the loop diagrams.Another feature is, that the SU(3) relations contain only afew constants in each partial wave. For example, in the 1S0partial wave only the constants c271S0
, c8s1S0, c11S0
are present. If
these constants are fixed in some of the baryons channels,predictions for other channels can be made. This has, forinstance, been used in Haidenbauer et al. [35], where theexistence of ##, #, and ,, bound states has been studiedwithin SU(3) χEFT.
3.2. One- and Two-Meson-ExchangeContributionsIn the last section, we have addressed the short-range partof the baryon-baryon interaction via contact terms. Let usnow analyze the long- and mid-range part of the interaction,generated by one- and two-meson-exchange as determinedin Haidenbauer et al. [33]. The contributing diagrams up toNLO are shown in Figure 5, which displays the hierarchy ofbaryonic forces.
The vertices, necessary for the construction of thesediagrams stem from the leading-order meson-baryon interaction
Lagrangian L(1)B in Equation (42). The vertex between two
baryons and one meson emerges from the part
D
2〈Bγ µγ5{uµ,B}〉+
F
2〈Bγ µγ5
[
uµ,B]
〉
= − 1
2f0
8∑
i,j,k=1
NBiBjφk (Biγµγ5Bj)(∂µφk)+O(φ3) , (56)
where we have used uµ = − 1f0
∂µφ+O(φ3) and have rewritten the
pertinent part of the Lagrangian in terms of the physical mesonand baryon fields
φi ∈{
π0,π+,π−,K+,K−,K0, K0, η}
,
Bi ∈{
n, p,#0,#+,#−,!,,0,,−} . (57)
The factors NBiBjφk are linear combinations of the axial vectorcoupling constantsD and F with certain SU(3) coefficients. Thesefactors vary for different combinations of the involved baryonsand mesons and can be obtained easily by multiplying out thebaryon and meson flavor matrices. In a similar way, we obtainthe (Weinberg-Tomozawa) vertex between two baryons and two
mesons from the covariant derivative in L(1)B , leading to
〈Biγ µ[
1µ,B]
〉
= i
8f 20
8∑
i,j,k,l=1
NBiφkBjφl (BiγµBj)(φk∂µφl)+O(φ4) , (58)
where 1µ = 18f 20
[φ, ∂µφ]+O(φ4) was used.
The calculation of the baryon-baryon potentials is donein the center-of-mass frame and in the isospin limit mu =md. To obtain the contribution of the Feynman diagrams tothe non-relativistic potential, we perform an expansion in theinverse baryon mass 1/MB. If loops are involved, the integrandis expanded before integrating over the loop momenta. Thisproduces results that are equivalent to the usual heavy-baryonformalism. In the case of the two-meson-exchange diagrams atone-loop level, ultraviolet divergences are treated by dimensionalregularization, which introduces a scale λ. In dimensionalregularization divergences are isolated as terms proportional to
R = 2
d − 4+ γE − 1− ln (4π) , (59)
with d *= 4 the space-time dimension and the Euler-Mascheroniconstant γE ≈ 0.5772. These terms can be absorbed by thecontact terms.
According to Equations (56) and (58) the vertices have thesame 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 amasterformula, where the proper masses of the exchanged mesonshave to be inserted, and which has to be multiplied with anappropriate SU(3) factor N. In the following we will presentthe analytic formulas for the one- and two-meson-exchangediagrams, introduced in Haidenbauer et al. [33]. The pertinentSU(3) factors will be displayed next to the considered Feynmandiagram (cf. Figure 7). The results will be given in terms of acentral potential (VC), a spin-spin potential (%σ1 · %σ2 VS) and atensor-type potential (%σ1 · %q %σ2 · %q VT). The momentum transfer isq =
∣∣%pf − %pi
∣∣, with %pi and %pf the initial and final state momenta
in the center-of-mass frame.Note that the presented results apply only to direct diagrams.
This is for example the case for the leading-order one-eta
exchange in the !n interaction, i.e., for !(%pi)n(−%pi)η−→
!(%pf )n(−%pf ). An example of a crossed diagram is the one-kaon
exchange in the process !(%pi)n(−%pi)K−→ n(−%pf )!(%pf ), where the
nucleon and the hyperon in the final state are interchanged andstrangeness is exchanged. In such cases, %pf is replaced by−%pf andthe momentum transfer in the potentials is q =
∣∣%pf + %pi
∣∣. Due
to the exchange of fermions in the final states a minus sign arises,and additionally the spin-exchange operator P(σ ) = 1
2 (1+ %σ1 · %σ2)has to be applied. The remaining structure of the potentials staysthe same (see also the discussion in section 4).
The leading-order contribution comes from the one-mesonexchange diagram in Figure 7A. It contributes only to the tensor-type potential:
VomeT (q) = − N
4f 20
1
q2 +m2 − iε. (60)
The symbol M in the SU(3) coefficient N denotes the charge-conjugated meson of mesonM in particle basis (e.g., π+ ↔ π−).
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FIGURE 7 | One- and two-meson-exchange contributions and corresponding SU(3) factors. (A) One-meson exchange, (B) planar box, (C) crossed box, (D) left
triangle, (E) right triangle, and (F) football diagram.
At next-to-leading order the two-meson exchange diagramsstart to contribute. The planar box in Figure 7B contains anirreducible part and a reducible part coming from the iteration ofthe one-meson exchange to second order. Inserting the potentialinto the Lippmann-Schwinger equation generates the reduciblepart; it is therefore not part of the potential (see also section 2.4).The irreducible part is obtained from the residues at the polesof the meson propagators, disregarding the (far distant) poles ofthe baryon propagators. With the masses of the two exchangedmesons set tom1 andm2, the irreducible potentials can be writtenin closed analytical form,
Vplanar boxirr, C (q) = N
3072π2f 40
{
5
3q2
+(
m21 −m2
2
)2
q2+ 16
(
m21 +m2
2
)
+[
23q2 + 45(
m21 +m2
2
)](
R+ 2 ln
√m1m2
λ
)
+m2
1 −m22
q4
[
12q4 +(
m21 −m2
2
)2
− 9q2(
m21 +m2
2
)]
lnm1
m2
+ 2
w2(
q)
[
23q4 −(
m21 −m2
2
)4
q4+ 56
(
m21 +m2
2
)
q2
+ 8m2
1 +m22
q2(
m21 −m2
2
)2
+ 2(
21m41 + 22m2
1m22 + 21m4
2
)]
L(
q)
}
, (61)
Vplanar boxirr, T
(
q)
= − 1
q2Vplanar boxirr, S (q)
= − N
128π2f 40
[
L(
q)
− 1
2−
m21 −m2
2
2q2ln
m1
m2
+ R
2+ ln
√m1m2
λ
]
(62)
where we have defined the functions
w(
q)
= 1
q
√(
q2 + (m1 +m2)2) (
q2 + (m1 −m2)2)
,
L(
q)
=w(
q)
2qln
[
qw(
q)
+ q2]2 −
(
m21 −m2
2
)2
4m1m2q2. (63)
The relation between the spin-spin and tensor-type potentialfollows from the identity (%σ1 × %q ) · (%σ2 × %q ) = q2 %σ1 · %σ2 − (%σ1 ·%q ) (%σ2 · %q ).
One should note that all potentials shown above are finitealso in the limit q → 0. Terms proportional to 1/q2 or1/q4 are canceled by opposite terms in the functions L(q) andw(q) in the limit of small q. For numerical calculations it isadvantageous to perform an expansion of the potentials in apower series for small q in order to implement directly thiscancellation. For equal meson masses the expressions for thepotentials reduce to the results in Kaiser et al. [148]. This isthe case for the NN interaction of Epelbaum et al. [147, 149,
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150] and Entem and Machleidt [151] based on χEFT, whereonly contributions from two-pion exchange need to be takeninto account.
In actual applications of these potentials such as inHaidenbauer et al. [33], only the non-polynomial part ofEquations (61) and (62) is taken into account, i.e., thepieces proportional to L(q) and to 1/q2 and 1/q4. Thepolynomial part is equivalent to the LO and NLO contactterms and, therefore, does not need to be considered. Thecontributions proportional to the divergence R are likewiseomitted. Their effect is absorbed by the contact terms ora renormalization of the coupling constants, see, e.g., thecorresponding discussion in Appendix A of Epelbaum et al. [149]for the NN case.
These statements above apply also to the other contributionsto the potential described below.
The crossed box diagrams in Figure 7C contribute to thecentral, spin-spin, and tensor-type potentials. The similarstructure with some differences in the kinematics of the planarand crossed box diagram leads to relations between them.Obviously, the crossed box has no iterated part. The potentialsof the crossed box are equal to the potentials of the irreduciblepart of the planar box, up to a sign in the central potential:
Vcrossed boxC (q) = −V
planar boxC, irr (q) ,
Vcrossed boxT (q) = − 1
q2Vcrossed boxS (q) = V
planar boxT, irr (q) . (64)
The two triangle diagrams, Figures 7D,E, constitute potentials,that are of equal form with different SU(3) factors N. Thecorresponding central potential reads
VtriangleC (q) = − N
3072π2f 40
{
− 2(
m21 +m2
2
)
+(
m21 −m2
2
)2
q2− 13
3q2
+[
8(
m21 +m2
2
)
−2(
m21 −m2
2
)2
q2+ 10q2
]
L(
q)
+m2
1 −m22
q4
[(
m21 −m2
2
)2 − 3(
m21 +m2
2
)
q2]
lnm1
m2
+[
9(
m21 +m2
2
)
+ 5q2](
R+ 2 ln
√m1m2
λ
)}
. (65)
The football diagrams in Figure 7F also contributes only to thecentral potential. One finds
V footballC (q) = N
3072π2f 40
{
− 2(
m21 +m2
2
)
−(
m21 −m2
2
)2
2q2− 5
6q2 + w2 (q
)
L(
q)
+ 1
2
[
3(
m21 +m2
2
)
+ q2](
R+ 2 ln
√m1m2
λ
)
−m2
1 −m22
2q4
[(
m21 −m2
2
)2 + 3(
m21 +m2
2
)
q2]
lnm1
m2
}
. (66)
3.3. Meson-Exchange ModelsEarlier investigations of the baryon-baryon interactions has beendone within phenomenological meson-exchange potentials suchas the Jülich [18, 19, 21], Nijmegen [20, 22, 23], or Ehime[152, 153] potentials. As we use two of them for comparison, wegive a brief introduction to these type of models.
Conventional meson-exchange models of the YN interactionare usually also based on the assumption of SU(3) flavorsymmetry for the occurring coupling constants, and in somecases even on the SU(6) symmetry of the quark model [18, 19]. Inthe derivation of the meson-exchange contributions one followsessentially the same procedure as outlined in section 3.2 for thecase of pseudoscalar mesons. Besides the lowest pseudoscalar-meson multiplet also the exchanges of vector mesons (ρ, ω, K∗),of scalar mesons (σ (f0(500)),...), or even of axial-vector mesons(a1(1270),...) [22, 23] are included. The spin-space structure ofthe corresponding Lagrangians that enter into Equation (42)and subsequently into Equation (56) differ and, accordingly,the final expressions for the corresponding contributions to theYN interaction potentials differ too. Details can be found inHolzenkamp et al. [18] and Rijken et al. [20, 22]. We want toemphasize that even for pseudoscalar mesons the final result forthe interaction potentials differs, in general, from the expressiongiven in Equation (60). Contrary to the chiral EFT approach,recoil, and relativistic corrections are often kept in meson-exchange models because no power counting rules are applied.Moreover, in case of the Jülich potential pseudoscalar couplingis assumed for the meson-baryon interaction Lagrangian forthe pseudoscalar mesons instead of the pseudovector coupling(Equation 42) dictated by chiral symmetry. Note that in someYN potentials of the Jülich group [18, 19] contributions fromtwo-meson exchanges are included. The ESC08 and ESC16potentials [22, 23] include likewise contributions from two-meson exchange, in particular, so-called meson-pair diagramsanalog to the ones shown in Figures 7D–F.
The major conceptual difference between the various meson-exchange models consists in the treatment of the scalar-mesonsector. This simply reflects the fact that, unlike for pseudoscalarand vector mesons, so far there is no general agreement aboutwhat are the actual members of the lowest lying scalar-mesonSU(3) multiplet. Therefore, besides the question of the massesof the exchange particles it also remains unclear whether andhow the relations for the coupling constants should be specified.As a consequence, different prescriptions for describing thescalar sector, whose contributions play a crucial role in anybaryon-baryon interaction at intermediate ranges, were adoptedby the various authors who published meson-exchange modelsof the YN interaction. For example, the Nijmegen group viewsthis interaction as being generated by genuine scalar-mesonexchange. In their models NSC97 [20] and ESC08 (ESC16) [22,23] a scalar SU(3) nonet is exchanged—namely, two isospin-0mesons [an ε(760) and the f0(980)] an isospin-1 meson (a0(980))
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FIGURE 8 | Leading three-baryon interactions: contact term, one-meson
exchange, and two-meson exchange. Filled circles and solid dots denote
vertices with /i = 1 and /i = 0, respectively.
and an isospin-1/2 strange meson κ with a mass of 1,000 MeV.In the initial YN models of the Jülich group [18, 19] a σ (witha mass of ≈ 550 MeV) is included which is viewed as arisingfrom correlated ππ exchange. In practice, however, the couplingstrength of this fictitious σ to the baryons is treated as a freeparameter and fitted to the data. In the latest meson-exchangeYN potential presented by the Jülich group [21] a microscopicmodel of correlated ππ and KK exchange [154] is utilized to fixthe contributions in the scalar-isoscalar (σ ) and vector-isovector(ρ) channels.
Let us mention for completeness that meson-exchange modelsare typically equipped with phenomenological form factors inorder to cut off the potential for large momenta (short distances).For example, in case of the YN models of the Jülich group theinteraction is supplemented with form factors for each meson-baryon-baryon vertex (cf. [18, 19] for details). Those form factorsare meant to take into account the extended hadron structureand are parameterized in the conventional monopole or dipoleform. In case of the Nijmegen potentials a Gaussian form factoris used. In addition there is some additional sophisticated short-range phenomenology that controls the interaction at shortdistances [22, 23].
4. THREE-BARYON INTERACTIONPOTENTIALS
Three-nucleon forces are an essential ingredient for a properdescription of nuclei and nuclear matter with low-momentumtwo-body interactions. Similarly, three-baryon forces, especiallythe !NN interaction, are expected to play an importantrole in nuclear systems with strangeness. Their introductionin calculations of light hypernuclei seems to be required.Furthermore, the introduction of 3BF is traded as a possiblesolution to the hyperon puzzle (see section 1). However, so faronly phenomenological 3BF have been employed. In this sectionwe present the leading irreducible three-baryon interactionsfrom SU(3) chiral effective field theory as derived in Petschaueret al. [120]. We show the minimal effective Lagrangian requiredfor the pertinent vertices. Furthermore the estimation of thecorresponding LECs through decuplet saturation and an effectivedensity-dependent two-baryon potential will be covered [121].
According to the power counting in Equation (47) the 3BFarise formally at NNLO in the chiral expansion, as can beseen from the hierarchy of baryonic forces in Figure 5. Three
types of diagrams contribute: three-baryon contact terms, one-meson and two-meson exchange diagrams (cf. Figure 8). Notethat a two-meson exchange diagram, such as in Figure 8, witha (leading order) Weinberg-Tomozawa vertex in the middle,would formally be a NLO contribution. However, as in thenucleonic sector, this contribution is kinematically suppresseddue to the fact that the involved meson energies are differencesof baryon kinetic energies. Anyway, parts of these NNLOcontributions get promoted to NLO by the introduction ofintermediate decuplet baryons, so that it becomes appropriateto use these three-body interactions together with the NLO two-body interaction of section 3. As already stated, the irreduciblecontributions to the chiral potential are presented. In contrastto typical phenomenological calculations, diagrams as on theleft side of Figure 4 do not lead to a genuine three-bodypotential, but are an iteration of the two-baryon potential.Such diagrams will be incorporated automatically when solving,e.g., the Faddeev (or Yakubovsky) equations within a coupled-channel approach. The three-body potentials derived from SU(3)χEFT are expected to shed light on the effect of 3BFs inhypernuclear systems. Especially in calculations about lighthypernuclei these potentials can be implemented within reliablefew-body techniques [48, 49, 51, 52].
4.1. Contact InteractionIn the following we consider the leading three-baryon contactinteraction. Following the discussion in section 2.3 thecorresponding Lagrangian can be constructed. The inclusion ofexternal fields is not necessary, as we are interested in the purelybaryonic contact term. One ends up with the following possiblestructures in flavor space [120]
〈BBBBBB〉 , 〈BBBBBB〉 , 〈BBBBBB〉 ,〈BBBBBB〉 , 〈BBBB〉〈BB〉 , 〈BBBB〉〈BB〉 ,〈BBBB〉〈BB〉 , 〈BBB〉〈BBB〉 , 〈BBB〉〈BBB〉 ,〈BB〉〈BB〉〈BB〉 , 〈BB〉〈BB〉〈BB〉 , (67)
with possible Dirac structures
1⊗ 1⊗ 1 , 1⊗ γ5γµ ⊗ γ5γµ , γ5γ
µ ⊗ 1⊗ γ5γµ ,
γ5γµ ⊗ γ5γµ ⊗ 1 , γ5γµ ⊗ i σµν ⊗ γ5γν , (68)
leading to the following operators in the three-body spin space
1 , %σ1 · %σ2 , %σ1 · %σ3 , %σ2 · %σ3 , i %σ1 · (%σ2 × %σ3) . (69)
All combinations of these possibilities leads to a (largelyovercomplete) set of terms for the leading covariant Lagrangian.Note that in Petschauer et al. [120] the starting pointis a covariant Lagrangian, but the minimal non-relativisticLagrangian is the goal. Hence, only Dirac structures leading toindependent (non-relativistic) spin operators are relevant.
Let us consider the process B1B2B3 → B4B5B6,where the Bi are baryons in the particle basis, Bi ∈{n, p,!,#+,#0,#−,,0,,−}. The contact potential V hasto be derived within a threefold spin space for this process.
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The operators in spin-space 1 is defined to act between thetwo-component Pauli spinors of B1 and B4. In the same way,spin-space 2 belongs to B2 and B5, and spin-space 3 to B3and B6. For a fixed spin configuration the potential can becalculated from
χ(1)B4
†χ(2)B5
†χ(3)B6
†V χ
(1)B1
χ(2)B2
χ(3)B3
, (70)
where the superscript of a spinor denotes the spin space andthe subscript denotes the baryon to which the spinor belongs.The potential is obtained as V = −〈B4B5B6| L |B1B2B3〉,where the contact Lagrangian L has to be inserted, and the36 Wick contractions need to be performed. The number36 corresponds to the 3! × 3! possibilities to arrange thethree initial and three final baryons into Dirac bilinears. Oneobtains six direct terms, where the baryon bilinears combinethe baryon pairs 1–4, 2–5, and 3–6, as shown in Equation(70). For the other 30 Wick contractions, the resulting potentialis not fitting to the form of Equation (70), because thewrong baryon pairs are connected in a separate spin space.Hence, an appropriate exchange of the spin wave functionsin the final state has to be performed. This is achieved bymultiplying the potential with the well-known spin-exchange
operators P(σ )ij = 12 (1 + %σi · %σj). Furthermore, additional minus
signs arise from the interchange of anti-commuting baryonfields. The full potential is then obtained by adding up all 36contributions to the potential. One obtains a potential that fulfillsautomatically the generalized Pauli principle and that is fullyanti-symmetrized.
In order to obtain a minimal set of Lagrangian termsof the final potential matrix have been eliminated until therank of the final potential matrix (consisting of multipleLagrangian terms and the spin structures in Equation69) matches the number of terms in the Lagrangian. Theminimal non-relativistic six-baryon contact Lagrangianis [120]
L = −C1〈BaBbBcBaBbBc〉+C2〈BaBbBaBcBbBc〉−C3〈BaBbBaBbBcBc〉+C4〈BaBaBbBbBcBc〉−C5〈BaBbBaBb〉 〈BcBc〉
−C6
(
〈BaBbBcBa(σ iB)b(σiB)c〉
+ 〈BcBbBa(σ iB)c(σiB)bBa〉
)
+C7
(
〈BaBbBaBc(σ iB)b(σiB)c〉
+ 〈BcBb(σ iB)cBa(σiB)bBa〉
)
−C8
(
〈BaBbBa(σ iB)bBc(σiB)c〉
+ 〈BbBa(σ iB)bBaBc(σiB)c〉
)
+C9〈BaBaBb(σ iB)bBc(σiB)c〉
−C10
(
〈BaBbBa(σ iB)b〉 〈Bc(σ iB)c〉
TABLE 4 | Irreducible representations for three-baryon states with strangeness S
and isospin I in partial waves |2S+1LJ〉, with the total spin S = 12 ,
32 , the angular
momentum L = 0, and the total angular momentum J = 12 ,
32 [120].
States (S,I) 2S1/2 4S3/2
NNN (0, 12 ) 35
!NN,#NN (−1,0) 10, 35 10a
!NN,#NN (−1,1) 27, 35 27a
#NN (−1,2) 35
!!N,#!N,##N,,NN (−2, 12 ) 8,10,27,35 8a, 10a, 27a
#!N,##N,,NN (−2, 32 ) 10,27, 35, 35 10a, 27a
##N (−2, 52 ) 35
!!!,##!,###,,!N,,#N (−3,0) 8,27 1a,8a,27a
#!!,##!,###,,!N,,#N (−3,1) 8, 10, 10,27,35, 35 8a,10a,10a, 27a
##!,###,,#N (−3,2) 27,35, 35 27a
,!!,,#!,,##,,,N (−4, 12 ) 8,10,27,35 8a, 10a, 27a
,#!,,##,,,N (−4, 32 ) 10,27, 35, 35 10a, 27a
,## (−4, 52 ) 35
,,!,,,# (−5,0) 10, 35 10a
,,!,,,# (−5,1) 27, 35 27a
,,# (−5,2) 35
,,, (−6, 12 ) 35
+ 〈BbBa(σ iB)bBa〉 〈Bc(σ iB)c〉)
−C11〈BaBbBc(σ iB)aBb(σiB)c〉
+C12〈BaBb(σ iB)aBcBb(σiB)c〉
−C13〈BaBb(σ iB)a(σiB)bBcBc〉
−C14〈BaBb(σ iB)a(σiB)b〉 〈BcBc〉
− i εijkC15〈BaBbBc(σ iB)a(σjB)b(σ
kB)c〉+ i εijkC16〈BaBb(σ iB)aBc(σ
jB)b(σkB)c〉
− i εijkC17〈BaBb(σ iB)a(σjB)bBc(σ
kB)c〉+ i εijkC18〈Ba(σ iB)aBb(σ
jB)bBc(σkB)c〉 , (71)
with vector indices i, j, k and two-component spinor indicesa, b, c. In total 18 low-energy constants C1 . . .C18 are present.The low-energy constant E of the six-nucleon contactterm (cf. [78]) can be expressed through these LECs byE = 2(C4 − C9).
As in the two-body sector, group theoretical considerationscan deliver valuable constrains on the resulting potentials. Inflavor space the three octet baryons form the 512-dimensionaltensor product 8 ⊗ 8 ⊗ 8, which decomposes into the followingirreducible SU(3) representations
8⊗ 8⊗ 8 =64⊕ (35⊕ 35)2 ⊕ 276 ⊕ (10⊕ 10)4 ⊕ 88 ⊕ 12 , (72)
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where the multiplicity of an irreducible representations isdenoted by subscripts. In spin space one obtain for the productof three doublets
2⊗ 2⊗ 2 = 22 ⊕ 4 . (73)
Transitions are only allowed between irreducible representationsof the same type. Analogous to Dover and Feshbach [145] forthe two-baryon sector, the contributions of different irreduciblerepresentations to three-baryon multiplets in Table 4 can beestablished. At leading order only transitions between S-wavesare possible, since the potentials are momentum-independent.Due to the Pauli principle the totally symmetric spin-quartet 4must combine with the totally antisymmetric part of 8⊗ 8⊗ 8 inflavor space,
Alt3(8) = 56a = 27a + 10a + 10a + 8a + 1a . (74)
It follows, that these totally antisymmetric irreduciblerepresentations are present only in states with total spin3/2. The totally symmetric part of 8⊗ 8⊗ 8 leads to
Sym3(8) = 120s = 64s + 27s + 10s + 10s + 8s + 1s . (75)
However, the totally symmetric flavor part has no totallyantisymmetric counterpart in spin space, hence theserepresentations do not contribute to the potential. In Table 4,these restrictions obtained by the generalized Pauli principlehave already be incorporated. The potentials of Petschauer et al.[120] (decomposed in isospin basis and partial waves) fulfill therestrictions of Table 4. For example the combination of LECsrelated to the representation 35 is present in theNNN interactionas well as in the ,,# (−5, 2) interaction.
4.2. One-Meson Exchange ComponentThe meson-baryon couplings in the one-meson exchangediagram of Figure 8 emerges from the leading-order chiral
Lagrangian L(1)B (see Equation 56). The other vertex involves
four baryon fields and one pseudoscalar-meson field. InPetschauer et al. [120], an overcomplete set of terms for thecorresponding Lagrangian has been constructed. In order toobtain the complete minimal Lagrangian from the overcompleteset of terms, the matrix elements of the process B1B2 → B3B4φ1
has been considered in Petschauer et al. [120]. The correspondingspin operators in the potential are
%σ1 · %q , %σ2 · %q , i (%σ1 × %σ2) · %q , (76)
where %q denotes the momentum of the emitted meson.Redundant term are removed until the rank of the potentialmatrix formed by all transitions and spin operators matchesthe number of terms in the Lagrangian. One ends up with theminimal non-relativistic chiral Lagrangian
L = D1/f0〈Ba(∇ iφ)BaBb(σiB)b〉
+ D2/f0(
〈BaBa(∇ iφ)Bb(σiB)b〉
+ 〈BaBaBb(σ iB)b(∇ iφ)〉)
FIGURE 9 | Generic meson-exchange diagrams. The wiggly line symbolized
the four-baryon contact vertex, to illustrate the baryon bilinears. (A) Generic
one-meson exchange diagram. (B) Generic two-meson exchange diagram.
+ D3/f0〈Bb(∇ iφ)(σ iB)bBaBa〉
− D4/f0(
〈Ba(∇ iφ)BbBa(σiB)b〉
+ 〈BbBa(σ iB)b(∇ iφ)Ba〉)
− D5/f0(
〈BaBb(∇ iφ)Ba(σiB)b〉
+ 〈BbBa(∇ iφ)(σ iB)bBa〉)
− D6/f0(
〈Bb(∇ iφ)Ba(σiB)bBa〉
+ 〈BaBbBa(∇ iφ)(σ iB)b〉)
− D7/f0(
〈BaBbBa(σ iB)b(∇ iφ)〉
+ 〈BbBa(σ iB)bBa(∇ iφ)〉)
+ D8/f0〈Ba(∇ iφ)Ba〉〈Bb(σ iB)b〉+ D9/f0〈BaBa(∇ iφ)〉〈Bb(σ iB)b〉+ D10/f0〈Bb(∇ iφ)(σ iB)b〉〈BaBa〉+ i εijkD11/f0〈Ba(σ iB)a(∇kφ)Bb(σ
jB)b〉
− i εijkD12/f0(
〈Ba(∇kφ)Bb(σiB)a(σ
jB)b〉
− 〈BbBa(σ jB)b(∇kφ)(σ iB)a〉)
− i εijkD13/f0〈BaBb(∇kφ)(σ iB)a(σjB)b〉
− i εijkD14/f0〈BaBb(σ iB)a(σjB)b(∇kφ)〉 , (77)
with two-component spinor indices a and b and 3-vector indicesi, j, and k. For all possible strangeness sectors S = −4 . . . 0 oneobtains in total 14 low-energy constants D1 . . .D14. The low-energy constant of the corresponding vertex in the nucleonicsector D is related to the LECs above by D = 4(D1 − D3 +D8 − D10).2
To obtain the 3BF one-meson-exchange diagram, the genericone-meson-exchange diagram in Figure 9A can be investigated.
2This LEC D has not to be confused with the axial-vector coupling constant D inEquation (56).
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It involves the baryons i, j, k in the initial state, the baryons l,m, nin the final state and an exchanged meson φ. The contact vertexon the right is pictorially separated into two parts to indicate thatbaryon j–m and k–n are in the same bilinear. The spin spacescorresponding to the baryon bilinears are denoted by A,B,C.
On obtains a generic potential of the form
V = 1
2f 20
%σA · %qli%q 2li +m2
φ
(
N1 %σC · %qli + N2i (%σB × %σC) · %qli)
, (78)
with the momentum transfer %qli = %pl − %pi carried bythe exchanged meson. The constants N1 and N2 are linearcombinations of low-energy constants.
The complete one-meson exchange three-baryon potential forthe process B1B2B3 → B4B5B6 is finally obtained by summingup the 36 permutations of initial-state and final-state baryonsfor a fixed meson and by summing over all mesons φ ∈{
π0,π+,π−,K+,K−,K0, K0, η}
. Additional minus signs arisefrom interchanging fermions and some diagrams need to bemultiplied by spin exchange operators in order to be consistentwith the form set up in Equation (70). As defined before, thebaryons B1, B2, and B3 belong to the spin-spaces 1, 2, and 3,respectively.
4.3. Two-Meson Exchange ComponentThe two-meson exchange diagram of Figure 8 includes the vertexarising from the Lagrangian in Equation (56). Furthermore thewell-known O(q2) meson-baryon Lagrangian [139] is necessary.For the two-meson exchange diagram of Figure 8 we need inaddition to the Lagrangian in Equation (56) the well-knownO(q2) meson-baryon Lagrangian [139]. The relevant terms are[140]
L = bD〈B{χ+,B}〉+ bF〈B[χ+,B]〉+ b0〈BB〉 〈χ+〉+ b1〈B[uµ, [uµ,B]]〉+ b2〈B{uµ, {uµ,B}}〉+ b3〈B{uµ, [uµ,B]}〉+ b4〈BB〉 〈uµuµ〉+ id1〈B{[uµ, uν], σµνB}〉+ id2〈B[[uµ, uν], σµνB]〉+ id3〈Buµ〉〈uνσµνB〉 , (79)
with uµ = − 1f0
∂µφ + O(φ3) and χ+ = 2χ − 14f 20
{φ, {φ,χ}}+O(φ4), where
χ =
m2π 0 00 m2
π 00 0 2m2
K −m2π
. (80)
The terms proportional to bD, bF , b0 break explicitly SU(3) flavorsymmetry, because of different meson masses mK *= mπ . TheLECs of Equation (79) are related to the conventional LECs ofthe nucleonic sector by [155, 156]
c1 =1
2(2b0 + bD + bF) ,
c3 = b1 + b2 + b3 + 2b4 ,
c4 = 4(d1 + d2) . (81)
To obtain the potential of the two-meson exchange diagram ofFigure 8, the generic diagram of Figure 9B can be considered.It includes the baryons i, j, k in the initial state, the baryonsl,m, n in the final state, and two exchanged mesons φ1 andφ2. The spin spaces corresponding to the baryon bilinears aredenoted by A,B,C and they are aligned with the three initialbaryons. The momentum transfers carried by the virtual mesonsare %qli = %pl − %pi and %qnk = %pn − %pk. One obtains the generictransition amplitude
V = − 1
4f 40
%σA · %qli %σC · %qnk(%q 2
li +m2φ1)(%q 2
nk +m2φ2)
×(
N′1 + N′
2 %qli · %qnk + N′3 i (%qli × %qnk) · %σB
)
, (82)
with N′i linear combinations of the low-energy constants of the
three involved vertices. The complete three-body potential for atransition B1B2B3 → B4B5B6 can be calculated by summing upthe contributions of all 18 distinguable Feynman diagrams andby summing over all possible exchanged mesons. If the baryonlines are not in the configuration 1–4, 2–5, and 3–6 additional(negative) spin-exchange operators have to be included.
4.4. #NN Three-Baryon PotentialsIn order to give a concrete example the explicit expressionfor the !NN three-body potentials in spin-, isospin-, andmomentum-space are presented for the contact interaction andone- and two-pion exchange contributions [120]. The potentialsare calculated in the particle basis and afterwards rewritten intoisospin operators.
The !NN contact interaction is described by thefollowing potential
V!NNct = C′
1 (1− %σ2 · %σ3)(3+ %τ2 · %τ3)+ C′
2 %σ1 · (%σ2 + %σ3) (1− %τ2 · %τ3)+ C′
3 (3+ %σ2 · %σ3)(1− %τ2 · %τ3) , (83)
where the primed constants are linear combinations of C1 . . .C18
of Equation (71). The symbols %σ and %τ denote the usual Paulimatrices in spin and isospin space. The constant C′
1 appears onlyin the transition with total isospin I = 1. The constants C′
2 andC′3 contribute for total isospin I = 0.For the !NN one-pion exchange three-body potentials,
various diagrams are absent due to the vanishing !!π-vertex,which is forbidden by isospin symmetry. One obtains thefollowing potential
V!NNOPE =− gA
2f 20
×( %σ2 · %q52%q 252 +m2
π
%τ2 · %τ3[
(D′1 %σ1 + D′
2 %σ3) · %q52]
+ %σ3 · %q63%q 263 +m2
π
%τ2 · %τ3[
(D′1 %σ1 + D′
2 %σ2) · %q63]
+ P(σ )23 P(τ )23 P(σ )13
%σ2 · %q62%q 262 +m2
π
%τ2 · %τ3
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×[
−D′1 + D′
2
2(%σ1 + %σ3) · %q62
+D′1 − D′
2
2i (%σ3 × %σ1) · %q62
]
+ P(σ )23 P(τ )23 P(σ )12
%σ3 · %q53%q 253 +m2
π
%τ2 · %τ3
×[
−D′1 + D′
2
2(%σ1 + %σ2) · %q53
−D′1 − D′
2
2i (%σ1 × %σ2) · %q53
])
, (84)
with only two constants D′1 and D′
2, which are linearcombinations of the constants D1 . . .D14. Exchange operators in
spin space P(σ )ij = 12 (1 + %σi · %σj) and in isospin space P(τ )ij =
12 (1+ %τi · %τj) have been introduced.
The !NN three-body interaction generated by two-pionexchange is given by
V!NNTPE =
g2A3f 40
%σ3 · %q63 %σ2 · %q52(%q 2
63 +m2π )(%q 2
52 +m2π )
%τ2 · %τ3
×(
− (3b0 + bD)m2π + (2b2 + 3b4) %q63 · %q52
)
− P(σ )23 P(τ )23
g2A3f 40
%σ3 · %q53 %σ2 · %q62(%q 2
53 +m2π )(%q 2
62 +m2π )
%τ2 · %τ3
×(
− (3b0 + bD)m2π + (2b2 + 3b4) %q53 · %q62
)
. (85)
Due to the vanishing of the!!π vertex, only those two diagramscontribute, where the (final and initial)! hyperon are attached tothe central baryon line.
4.5. Three-Baryon Force Through DecupletSaturationLow-energy two- and three-body interactions derived fromSU(2) χEFT are used consistently in combination with eachother in nuclear few- and many-body calculations. The apriori unknown low-energy constants are fitted, for example,to NN scattering data and 3N observables such as 3-bodybinding energies [78]. Some of these LECs are, however, largecompared to their order of magnitude as expected from thehierarchy of nuclear forces in Figure 5. This feature has itsphysical origin in strong couplings of the πN-system to thelow-lying /(1232)-resonance. It is therefore, natural to includethe /(1232)-isobar as an explicit degree of freedom in thechiral Lagrangian (cf. [157–159]). The small mass differencebetween nucleons and deltas (293 MeV) introduces a small scale,which can be included consistently in the chiral power countingscheme and the hierarchy of nuclear forces. The dominantparts of the three-nucleon interaction mediated by two-pionexchange at NNLO are then promoted to NLO through thedelta contributions. The appearance of the inverse mass splittingexplains the large numerical values of the corresponding LECs[13, 160].
In SU(3) χEFT the situation is similar. In systems withstrangeness S = −1 like !NN, resonances such as the
spin-3/2 #∗(1385)-resonance play a similar role as the / inthe NNN system, as depicted in Figure 4 on the right side.The small decuplet-octet mass splitting (in the chiral limit),/ : = M10 − M8, is counted together with external momentaand meson masses as O(q) and thus parts of the NNLOthree-baryon interaction are promoted to NLO by the explicitinclusion of the baryon decuplet, as illustrated in Figure 10.It is therefore likewise compelling to treat the three-baryoninteraction together with the NLO hyperon-nucleon interactionof section 3. Note that in the nucleonic sector, only thetwo-pion exchange diagram with an intermediate /-isobar isallowed. Other diagrams are forbidden due to the Pauli principle,as we will show later. For three flavors more particles areinvolved and, in general, also the other diagrams (contact andone-meson exchange) with intermediate decuplet baryons inFigure 10 appear.
The large number of unknown LECs presented in the previoussubsections is related to the multitude of three-baryonmultiplets,with strangeness ranging from 0 to −6. For selected processesonly a small subset of these constants contributes as has beenexemplified for the !NN three-body interaction. In this sectionwe present the estimation of these LECs by resonance saturationas done in Petschauer et al. [121].
The leading-order non-relativistic interaction Lagrangianbetween octet and decuplet baryons (see, e.g., [161]) is
L = C
f0
3∑
a,b,c,d,e=1
εabc
(
Tade%S † ·(
%∇φdb
)
Bec
+ Bce%S ·(
%∇φbd
)
Tade
)
, (86)
where the decuplet baryons are represented by the totallysymmetric three-index tensor T (cf. Equation 23). At this orderonly a single LEC C appears. Typically the (large-Nc) valueC = 3
4 gA ≈ 1 is used, as it leads to a decay width 1(/ →πN) = 110.6 MeV that is in good agreement with the empiricalvalue of 1(/ → πN) = (115 ± 5) MeV [158]. The spin 1
2
to 32 transition operators %S connect the two-component spinors
of octet baryons with the four-component spinors of decupletbaryons (see e.g., [162]). In their explicit form they are given as2× 4 transition matrices
S1 =(
− 1√2
0 1√6
0
0 − 1√6
0 1√2
)
,
S2 =(
− i√2
0 − i√6
0
0 − i√6
0 − i√2
)
,
S3 =
0√
23 0 0
0 0√
23 0
. (87)
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FIGURE 10 | Hierarchy of three-baryon forces with explicit introduction of the baryon decuplet (represented by double lines).
FIGURE 11 | Saturation via decuplet resonances. (A) Saturation of the six-baryon contact interaction. (B) Saturation of the BB → BBφ vertex. (C) Saturation of the
NLO baryon-meson vertex.
These operators fulfill the relation SiSj† = 13 (2δij − iεijkσk).
A non-relativistic B∗BBB Lagrangian with a minimal set ofterms is given by [121]:
L = H1
3∑
a,b,c,d,e,f=1
εabc[(
Tade%S†Bdb)
·(
Bfc %σBef)
+(
Bbd%S Tade
)
·(
Bfe %σBcf) ]
+H2
3∑
a,b,c,d,e,f=1
εabc[(
Tade%S†Bfb)
·(
Bdc %σBef)
+(
Bbf %S Tade
)
·(
Bfe %σBcd) ]
, (88)
with the LECsH1 andH2. Again one can employ group theory tojustify the number of two constants for a transition BB → B∗B. Inflavor space the two initial octet baryons form the tensor product8⊗8, and in spin space they form the product 2⊗2. These tensorproducts can be decomposed into irreducible representations:
8⊗ 8 = 27⊕ 8s ⊕ 1︸ ︷︷ ︸
symmetric
⊕ 10⊕ 10∗ ⊕ 8a︸ ︷︷ ︸
antisymmetric
,
2⊗ 2 = 1a ⊕ 3s . (89)
In the final state, having a decuplet and an octet baryon, thesituation is similar:
10⊗ 8 = 35⊕ 27⊕ 10⊕ 8 ,
4⊗ 2 = 3⊕ 5 . (90)
As seen in the previous sections, at leading order only S-wavetransitions occur, as no momenta are involved. Transitions areonly allowed between the same types of irreducible (flavor andspin) representations. Therefore, in spin space the representation3 has to be chosen. Because of the Pauli principle in theinitial state, the symmetric 3 in spin space combines with theantisymmetric representations 10, 10∗, 8a in flavor space. Butonly 10 and 8a have a counterpart in the final state flavor space.This number of two allowed transitions matches the numberof two LECs in the minimal Lagrangian. Another interestingobservation can be made from Equations (89) and (90). For NNstates only the representations 27 and 10∗ can contribute, as canbe seen, e.g., in Table 3. But these representations combine eitherwith the wrong spin, or have no counterpart in the final state.
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Therefore, NN → /N transitions in S-waves are not allowedbecause of the Pauli principle.
Having the above two interaction types at hand, one canestimate the low-energy constants of the leading three-baryoninteraction by decuplet saturation using the diagrams shown inFigure 10. At this order, where no loops are involved, one justneeds to evaluate the diagrams with an intermediate decupletbaryon and the diagrams without decuplet baryons and comparethem with each other.
In order to estimate the LECs of the six-baryon contactLagrangian of Equation (71), one can consider the processB1B2B3 → B4B5B6 as depicted in Figure 11A. The left sideof Figure 11A has already been introduced in the previoussubsection and can be obtained by performing all 36 Wickcontractions. For the diagrams on right side of Figure 11Athe procedure is similar. After summing over all intermediatedecuplet baryons B∗, the full three-body potential of all possiblecombinations of baryons on the left side of Figure 11A can becompared with the ones on the right side. In the end the 18 LECsof the six-baryon contact Lagrangian C1, . . . ,C18 of Equation(71) can be expressed as linear combinations of the combinationsH21 , H
22 and H1H2 and are proportional to the inverse average
decuplet-octet baryon mass splitting 1// [121].Since we are at the leading order only tree-level diagrams
are involved and we can estimate the LECs of the one-meson-exchange part of the three-baryon forces already on the levelof the vertices, as depicted in Figure 11B. We consider thetransition matrix elements of the process B1B2 → B3B4φ andstart with the left side of Figure 11B. After doing all possibleWick contractions, summing over all intermediate decupletbaryons, and comparing the left side of Figure 11B with the righthand side for all combinations of baryons and mesons, the LECscan be estimated. The LECs of the minimal non-relativistic chiralLagrangian for the four-baryon vertex including one meson ofEquation (77) D1, . . . ,D14 are then proportional to C// and tolinear combinations of H1 and H2 [121].
The last class of diagrams is the three-body interaction withtwo-meson exchange. As done for the one-meson exchange, theunknown LECs can be saturated directly on the level of thevertex and one can consider the process B1φ1 → B2φ2 asshown in Figure 11C. A direct comparison of the transitionmatrix elements for all combinations of baryons and mesonsafter summing over all intermediate decuplet baryons B∗ leadsto the following contributions to the LECs of the meson-baryonLagrangian in Equation (79):
bD = 0 , bF = 0 , b0 = 0 ,
b1 =7C2
36/, b2 =
C2
4/, b3 = − C2
3/, b4 = − C2
2/,
d1 =C2
12/, d2 =
C2
36/, d3 = − C2
6/, (91)
These findings are consistent with the /(1232) contribution tothe LECs c1, c3, c4 (see Equation 81) in the nucleonic sector[157, 160]:
c1 = 0 , c3 = −2c4 = −g2A2/
. (92)
FIGURE 12 | Effective two-baryon interaction from genuine three-baryon
forces. Contributions arise from two-pion exchange (1), (2a), (2b), (3), one-pion
exchange (4), (5a), (5b), and the contact interaction (6).
Employing the LECs obtained via decuplet saturation, theconstants of the !NN interaction (contact interaction, one-pionand two-pion exchange) of section 4.4 can be evaluated:
C′1 = C′
3 =(H1 + 3H2)2
72/, C′
2 = 0 ,
D′1 = 0 , D′
2 =2C(H1 + 3H2)
9/,
3b0 + bD = 0 , 2b2 + 3b4 = − C2
/. (93)
Obviously, the only unknown constant here is the combinationH′ = H1 + 3H2. It is also interesting to see, that the (positive)sign of the constantsC′
i for the contact interaction is already fixed,independently of the values of the two LECs H1 and H2.
4.6. Effective In-medium Two-BaryonInteractionIn this subsection we summarize how the effect of three-body force in the presence of a (hyper)nuclear medium canbe incorporated in an effective baryon-baryon potential. InHolt et al. [163], the density-dependent corrections to theNN interaction have been calculated from the leading chiralthree-nucleon forces. This work has been extended to thestrangeness sector in Petschauer et al. [121]. In order to obtainan effective baryon-baryon interaction from the irreducible 3BFsin Figure 8, two baryon lines have been closed, which representsdiagrammatically the sum over occupied states within the Fermisea. Such a “medium insertion” is symbolized by short doublelines on a baryon propagator. All types of diagrams arising thisway are shown in Figure 12.
In Petschauer et al. [121], the calculation is restricted tothe contact term and to the contributions from one- and two-pion exchange processes, as they are expected to be dominant.When computing the diagrams of Figure 8 the medium insertion
corresponds to a factor −2πδ(k0)θ(kf − |%k|). Furthermore, anadditional minus sign comes from a closed fermion loop. The
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effective two-body interaction can also be calculated from theexpressions for the three-baryon potentials of Petschauer et al.[120] via the relation
V12 =∑
B
trσ3
∫
|%k|≤kBf
d3k
(2π)3V123 , (94)
where trσ3 denotes the spin trace over the third particle and wherea summation over all baryon species B in the Fermi sea (withFermi momentum kBf ) is done.
As an example of such an effective interaction, we displaythe effective !N interaction in nuclear matter (with ρp *=ρn) derived in Petschauer et al. [121]. It is determined fromtwo-pion-exchange, one-pion-exchange and contact!NN three-body forces. Only the expressions for the !n potential are shownas the !p potential can be obtained just by interchanging theFermi momenta k
pf with knf (or the densities ρp with ρn) in the
expressions for !n. In the following formulas the sum over thecontributions from the protons and neutrons in the Fermi seais already employed. Furthermore, the values of the LECs arealready estimated via decuplet saturation (see section 4.5). Thetopologies (1), (2a), and (2b) vanish here because of the non-existence of an isospin-symmetric !!π vertex. One obtains thedensity-dependent !n potential in a nuclear medium
Vmed,ππ!n =
C2g2A12π2f 40 /
{
1
4
[8
3(knf
3 + 2kpf
3)− 4(q2 + 2m2)10(p)− 2q211(p)
+ (q2 + 2m2)2G0(p, q)]
+ i
2(%q× %p ) · %σ2
(
210(p)+ 211(p)
− (q2 + 2m2)(G0(p, q)+ 2G1(p, q)))}
, (95)
Vmed,π!n = gACH′
54π2f 20 /
(
2(knf3 + 2k
pf
3)− 3m210(p)
)
, (96)
Vmed,ct!n = H′2
18/(ρn + 2ρp) . (97)
The different topologies related to two-pion exchange [(1), (2a),(2b), (3)] and one-pion exchange [(4), (5a), (5b)] have alreadybeen combined in Vmed,ππ and Vmed,π , respectively. The densityand momentum dependent loop functions 1i(p) and Gi(p, q) canbe found in Petschauer et al. [121]. The only spin-dependentterm is the one proportional to %σ2 = 1
2 (%σ1 + %σ2) − 12 (%σ1 − %σ2)
and therefore one recognizes a symmetric and an antisymmetricspin-orbit potential of equal but opposite strength.
5. APPLICATIONS
5.1. Hyperon-Nucleon andHyperon-Hyperon ScatteringWith the hyperon-nucleon potentials outlined in section 3hyperon-nucleon scattering processes can be investigated. Thevery successful approach to the nucleon-nucleon interactionof Epelbaum et al. [147, 149, 150] within SU(2) χEFT, hasbeen extended to the leading-order baryon-baryon interaction inPolinder et al. [29, 30] and Haidenbauer and Meißner [31] bythe Bonn-Jülich group. In Haidenbauer et al. [33, 35, 36], thisapproach has been extended to next-to-leading order in SU(3)chiral effective field theory. As mentioned in section 2.4, thechiral power counting is applied to the potential, where only two-particle irreducible diagrams contribute. These potentials arethen inserted into a regularized Lippmann-Schwinger equationto obtain the reaction amplitude (or T-matrix). In contrast tothe NN interaction, the Lippmann-Schwinger equation for theYN interaction involves not only coupled partial waves, but alsocoupled two-baryon channels. The coupled-channel Lippmann-Schwinger equation in the particle basis reads after partial-wavedecomposition (see also Figure 2)
Tρ′′ρ′ ,Jν′′ν′ (k′′, k′;
√s) = Vρ′′ρ′,J
ν′′ν′ (k′′, k′)
+∑
ρ,ν
∫ ∞
0
dk k2
(2π)3Vρ′′ρ ,J
ν′′ν (k′′, k)
× 2µν
k2ν − k2 + iεTρρ′,J
νν′ (k, k′;√s) , (98)
where J denotes the conserved total angular momentum. Thecoupled two-particle channels (!p,#+n,#0p,. . . ) are labeledby ν, and the partial waves (1S0, 3P0, . . .) by ρ. Furthermore,µν is the reduced baryon mass in channel ν. In Haidenbaueret al. [33], a non-relativistic scattering equation has been chosento ensure that the potential can also be applied consistentlyto Faddeev and Faddeev-Yakubovsky calculations in the few-body sector, and to (hyper-) nuclear matter calculations withinthe conventional Brueckner-Hartree-Fock formalism (see section5.2). Nevertheless, the relativistic relation between the on-shellmomentum kν and the center-of-mass energy has been used,√s =
√
M2B1,ν
+ k2ν +√
M2B2,ν
+ k2ν , in order to get the two-
particle thresholds at their correct positions. The physical baryonmasses have been used in the Lippmann-Schwinger equation,which introduces some additional SU(3) symmetry breaking.Relativistic kinematics has also been used to relate the laboratorymomentum plab of the hyperon to the center-of-mass energy√s. The Coulomb interaction has been implemented by the
use of the Vincent-Phatak method [147, 164]. Similar to thenucleonic sector at NLO [147], a regulator function of the formfR(!) = exp[−(k′4 + k4)/!4] is employed to cut off the high-energy components of the potential. For higher orders in thechiral power counting, higher powers than 4 in the exponent offR have to be used. This ensures that the regulator introducesonly contributions, that are beyond the given order. The cutoff! is varied in the range (500 . . . 700) MeV, i.e., comparableto what was used for the NN interaction in Epelbaum et al.
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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
! = (500 . . .650) MeV [33], while the green (light) band are results to LO for ! = (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].
[147]. The resulting bands represent the cutoff dependence, afterreadjusting the contact parameters, and thus could be viewed as alower bound on the theoretical uncertainty. Recently, improvedschemes to estimate the theoretical uncertainty were proposedand applied to the NN interaction [165–168]. Some illustrativeresults for YN scattering based on the method by Epelbaumet al. [166, 168] have been included in Haidenbauer et al. [34].However, such schemes require higher orders than NLO in thechiral power counting if one wants to address questions like theconvergence of the expansion.
The partial-wave contributions of the meson-exchangediagrams are obtained by employing the partial-wavedecomposition formulas of Polinder et al. [29]. For furtherremarks on the employed approximations and the fitting strategywe refer the reader to Haidenbauer et al. [33]. As can be seenin Table 3, one gets for the YN contact terms five independentLO constants, acting in the S-waves, eight additional constantsat NLO in the S-waves, and nine NLO constant acting in theP-waves. The contact terms represent the unresolved short-distance dynamics, and the corresponding low-energy constantsare fitted to the “standard” set of 36 YN empirical data points[169–174]. The hypertriton (3!H) binding energy has beenchosen as a further input. It determines the relative strengthof the spin-singlet and spin-triplet S-wave contributions of the!p interaction. Due to the sparse and inaccurate experimentaldata, the obtained fit of the low-energy constants is not unique.For instance, the YN data can be described equally well witha repulsive or an attractive interaction in the 3S1 partial waveof the #N interaction with isospin I = 3/2. However, recentcalculations from lattice QCD [3, 4] suggest a repulsive 3S1phase shift in the #N I = 3/2 channel, hence the repulsivesolution has been adopted. Furthermore, this is consistentwith empirical information from #−-production on nuclei,
which point to a repulsive #-nucleus potential (see alsosection 5.2).
In the following we present some of the results of Haidenbaueret al. [33]. For comparison, results of the Jülich ’04 [21] andthe Nijmegen [20] meson-exchange models are also shown inthe figures. In Figure 13, the total cross sections as functions ofplab for various YN interactions are presented. The experimentaldata is well reproduced at NLO. Especially the results in the!p channel are in line with the data points (also at higherenergies) and the energy dependence in the #+p channel issignificantly improved at NLO. It is also interesting to note thatthe NLO results are now closer to the phenomenological Jülich’04model than at LO. One expects the theoretical uncertainties tobecome smaller, when going to higher order in the chiral powercounting. This is reflected in the fact, that the bands at NLO areconsiderably smaller than at LO. These bands represent only thecutoff dependence and therefore constitute a lower bound on thetheoretical error.
In Table 5, the scattering lengths and effective rangeparameters for the !p and #+p interactions in the 1S0 and3S1 partial waves are given. Result for LO [29] and NLOχEFT [33], for the Jülich ’04 model [21] and for the NijmegenNSC97f potential [20] are shown. The NLO !p scatteringlengths are larger than for the LO calculation, and closer tothe values obtained by the meson-exchange models. The triplet#+p scattering length is positive in the LO as well as theNLO calculation, which indicates a repulsive interaction in thischannel. Also given in Table 5 is the hypertriton binding energy,calculated with the corresponding chiral potentials. As statedbefore, the hypertriton binding energy was part of the fittingprocedure and values close to the experimental value could beachieved. The predictions for the 3
!H binding energy are basedon the Faddeev equations in momentum space, as described in
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TABLE 5 | 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].
NLO LO Jül ’04 NSC97f
! [MeV] 450 500 550 600 650 700 600
a!ps −2.90 −2.91 −2.91 −2.91 −2.90 −2.90 −1.91 −2.56 −2.60
r!ps 2.64 2.86 2.84 2.78 2.65 2.56 1.40 2.74 3.05
a!pt −1.70 −1.61 −1.52 −1.54 −1.51 −1.48 −1.23 −1.67 −1.72
r!pt 3.44 3.05 2.83 2.72 2.64 2.62 2.13 2.93 3.32
a#+ps −3.58 −3.59 −3.60 −3.56 −3.46 −3.49 −2.32 −3.60 −4.35
r#+ps 3.49 3.59 3.56 3.54 3.53 3.45 3.60 3.24 3.16
a#+pt 0.48 0.49 0.49 0.49 0.48 0.49 0.65 0.31 −0.25
r#+pt −4.98 −5.18 −5.03 −5.08 −5.41 −5.18 −2.78 −12.2 −28.9
(3!H) EB −2.39 −2.33 −2.30 −2.30 −2.30 −2.32 −2.34 −2.27 −2.30
The binding energies for the hypertriton are calculated using the Idaho-N3LO NN potential
[151]. The experimental value for the 3!H binding energy is −2.354(50) MeV.
Nogga [49, 175]. Note that genuine (irreducible) three-baryoninteractions were not included in this calculation. However, in theemployed coupled-channel formalism, effects like the important!-# conversion process are naturally included. It is important todistinguish such iterated two-body interactions, from irreduciblethree-baryon forces, as exemplified in Figure 4.
Predictions for S- and P-wave phase shifts δ as a functionof plab for !p and #+p scattering are shown in Figure 14. The1S0 !p phase shift from the NLO χEFT calculation is closerto the phenomenological Jülich ’04 model than the LO result.It points to moderate attraction at low momenta and strongrepulsion at higher momenta. At NLO the phase shift has astronger downward bending at higher momenta compared to LOor the Jülich ’04model. As stated before, more repulsion at higherenergies is a welcome feature in view of neutron star matter with!-hyperons as additional baryonic degree of freedom. The 3S1!p phase shift, part of the S-matrix for the coupled 3S1-3D1
system, changes qualitatively from LO to NLO. The 3S1 phaseshift of the NLO interaction passes through 90◦ slightly belowthe #N threshold, which indicates the presence of an unstablebound state in the #N system. For the LO interaction and theJülich ’04 model no passing through 90◦ occurs and a cusp ispredicted, that is caused by an inelastic virtual state in the #Nsystem. These effects are also reflected by a strong increase ofthe !p cross section close to the #N threshold (see Figure 13).The 3S1 #N phase shift for the NLO interaction is moderatelyrepulsive and comparable to the LO phase shift.
Recently an alternative NLO χEFT potential for YN scatteringhas been presented [34]. In that work a different strategy forfixing the low-energy constants that determine the strengthof the contact interactions is adopted. The objective of thatexploration was to reduce the number of LECs that need tobe fixed in a fit to the !N and #N data by inferring some ofthem from the NN sector via the underlying SU(3) symmetry (cf.section 3.1). Indeed, correlations between the LO and NLO LECsof the S-waves, i.e., between the c’s and c’s, had been observed
already in the initial YN study [33] and indicated that a uniquedetermination of them by considering the existing !N and #Ndata alone is not possible. It may be not unexpected in view ofthose correlations, that the variant considered in Haidenbaueret al. [34] yields practically equivalent results for !N and#N scattering observables. However, it differs considerable inthe strength of the !N → #N transition potential andthat becomes manifest in applications to few- and many-bodysystems [34, 176].
There is very little empirical information about baryon-baryon systems with S = −2, i.e., about the interaction in the!!, ##, !#, and ,N channels. Actually, all one can find inthe literature [36] are a few values and upper bounds for the,−p elastic and inelastic cross sections [177, 178]. In additionthere are constraints on the strength of the !! interactionfrom the separation energy of the 6
!!He hypernucleus [179].Furthermore estimates for the !! 1S0 scattering length existfrom analyses of the!! invariant mass measured in the reaction12C(K−,K+!!X) [180] and of !! correlations measured inrelativistic heavy-ion collisions [181].
Despite the rather poor experimental situation, it turned outthat SU(3)-symmetry breaking contact terms that arise at NLO(see section 3.1), need to be taken into account when going fromstrangeness S = −1 to S = −2 in order to achieve agreementwith the available measurements and upper bounds for the !!
and ,N cross sections [36]. This concerns, in particular, the LECc1χ that appears in the 1S0 partial wave (cf. Equation 54). Actually,
its value can be fixed by considering the pp and #+p systems,as shown in Haidenbauer et al. [35], and then employed in the!! system.
Selected results for the strangeness S = −2 sector arepresented in Figure 15. Further results and a detailed descriptionof the interactions can be found in Polinder et al. [30],Haidenbauer et al. [35, 36], and Haidenbauer and Meißner[37]. Interestingly, the results based on the LO interaction fromPolinder et al. [30] (green/gray bands) are consistent with allempirical constraints. The cross sections at LO are basicallygenuine predictions that follow from SU(3) symmetry utilizingLECs fixed from a fit to the !N and #N data on the LO level.The !! 1S0 scattering length predicted by the NLO interactionis a!! = −0.70 · · · −0.62 fm [36]. These values are wellwithin the range found in the aforementioned analyses whichare a!! = (−1.2 ± 0.6) fm [180] and −1.92 < a!! < −0.50fm [181], respectively. The values for the ,0p and ,0n S-wavescattering length are likewise small and typically in the order of±0.3 ∼ ±0.6 fm [36] and indicate that the ,N interaction has tobe relatively weak in order to be in accordance with the availableempirical constraints. Indeed, the present results obtained inchiral EFT up to NLO imply that the published values and upperbounds for the ,−p elastic and inelastic cross sections [177, 178]practically rule out a somewhat stronger attractive ,N force.
Also for ,N scattering an alternative NLO χEFTpotential has been presented recently [37]. Here the aimis to explore the possibility to establish a ,N interactionthat is still in line with all the experimental constraints for!! and ,N scattering, but at the same time is somewhatmore attractive. Recent experimental evidence for the
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FIGURE 14 | Various S- and P-wave phase shifts δ as a function of plab for the !p and #+p interaction [33]. Same description of curves as in Figure 13.
FIGURE 15 | ,−p induced cross sections. The bands represent results at NLO (red/black) [36] and LO (green/gray) [30]. Experiments are from Ahn et al. [178], Kim
et al. [unpublished data], and Aoki et al. [177]. Upper limits are indicated by arrows.
existence of ,-hypernuclei [182] suggests that the in-medium interaction of the ,-hyperon should be moderatelyattractive [183].
5.2. Hyperons in Nuclear MatterExperimental investigations of nuclear many-body systemsincluding strange baryons, for instance, the spectroscopy of
hypernuclei, provide important constraints on the underlyinghyperon-nucleon interaction. The analysis of data for single !-hypernuclei over a wide range in mass number leads to the result,that the attractive ! single-particle potential is about half asdeep (≈ −28 MeV) as the one for nucleons [184, 185]. At thesame time the !-nuclear spin-orbit interaction is found to beexceptionally weak [186, 187]. Recently, the repulsive nature of
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the #-nuclear potential has been experimentally established in#−-formation reactions on heavy nuclei [188]. Baryon-baryonpotentials derived within χEFT as presented in section 3 areconsistent with these observations [75, 76]. In this sectionwe summarize results of hyperons in infinite homogeneousnuclear matter of Petschauer et al. [76] obtained by employingthe interaction potentials from χEFT as microscopic input.The many-body problem is solved within first-order Bruecknertheory. A detailed introduction can be found, e.g., in Day [69],Baldo [189], and Fetter and Walecka [190].
Brueckner theory is founded on the so-called Goldstoneexpansion, a linked-cluster perturbation series for the groundstate energy of a fermionic many-body system. Let us considera system of A identical fermions, described by the Hamiltonian
H = T + V , (99)
where T is the kinetic part and V corresponds to the two-bodyinteraction. The goal is to calculate the ground state energy ofthis interacting A-body system. It is advantageous to introduceda so-called auxiliary potential, or single-particle potential,U. TheHamiltonian is then split into two parts
H = (T + U)+ (V − U) = H0 +H1 , (100)
the unperturbed part H0 and the perturbed part H1. One expectsthe perturbed part to be small, if the single particle potentialdescribes well the averaged effect of the medium on the particle.In fact, the proper introduction of the auxiliary potential is crucialfor the convergence of Brueckner theory.
Conventional nucleon-nucleon potentials exhibit a strongshort-range repulsion that leads to very large matrix elements.Hence, the Goldstone expansion in the form described above willnot converge for such hard-core potentials. One way to approachthis problem is the introduction of the so-called Bruecknerreaction matrix, or G-matrix. The idea behind it is illustratedin Figure 16A. Instead of only using the simple interaction,an infinite number of diagrams with increasing number ofinteractions is summed up. This defines theG-matrix interaction,which is, in contrast to the bare potential, weak and of reasonablerange. In a mathematical way, the reaction matrix is defined bythe Bethe-Goldstone equation:
G(ω) = V + VQ
ω −H0 + iεG(ω) , (101)
with the so-called starting energy ω. The Pauli operator Qensures, that the intermediate states are from outside theFermi sea. As shown in Figure 16A, this equation represents aresummation of the ladder diagrams to all orders. The arisingG-matrix interaction is an effective interaction of two particlesin the presence of the medium. The medium effects come insolely through the Pauli operator and the energy denominatorvia the single-particle potentials. If we set the single-particlepotentials to zero and omit the Pauli operator (Q = 1), werecover the usual Lippmann-Schwinger equation for two-bodyscattering in vacuum (see also Figure 2). This medium effecton the intermediate states is denoted by horizontal double
lines in Figure 16A. An appropriate expansion using the G-matrix interaction instead of the bare potential is the so-calledBrueckner-Bethe-Goldstone expansion, or hole-line expansion.
Finally, the form of the auxiliary potential U needs to bechosen. This choice is important for the convergence of the hole-line expansion. Bethe et al. [191] showed for nuclear matterthat important higher-order diagrams cancel each other if theauxiliary potential is taken as
Um = Re∑
n≤A
〈mn|G(ω = ωo.s.)|mn〉A , (102)
where the Brueckner reactionmatrix is evaluated on-shell, i.e., thestarting energy is equal to the energy of the two particles m, n inthe initial state:
ωo.s. = E1(k1)+ E2(k2) ,
EBi (ki) = Mi +k2i2Mi
+ ReUi(ki) . (103)
Pictorially Equation (102) means, that the single-particlepotential can be obtained by taking the on-shell G-matrixinteraction and by closing one of the baryon lines, as illustrated inFigure 16B. Note that this implies a non-trivial self-consistencyproblem. On the one hand, U is calculated from the G-matrixelements via Equation (102), and on the other hand the startingenergy of the G-matrix elements depends on U through thesingle-particle energies Ei in Equation (103).
At the (leading) level of two hole-lines, called Brueckner-Hartree-Fock approximation (BHF), the total energy is given by
E =∑
n≤A
〈n|T|n〉+ 1
2
∑
m,n≤A
〈mn|G|mn〉A
=∑
n≤A
〈n|T|n〉+ 1
2
∑
n≤A
〈n|U|n〉 , (104)
i.e., the ground-state energy E can be calculated directly after thesingle-particle potential has been determined.
The definition ofU in Equation (102) applies only to occupiedstates within the Fermi sea. For intermediate-state energiesabove the Fermi sea, typically two choices for the single-particlepotential are employed. In the so-called gap choice, the single-particle potential is given by Equation (102) for k ≤ kF andset to zero for k > kF , implying a “gap” (discontinuity) inthe single-particle potential. Then only the free particle energies(M + %p 2/2M) of the intermediate states appear in the energydenominator of the Bethe-Goldstone equation (Equation 101)since the Pauli-blocking operator is zero for momenta below theFermi momentum. In the so-called continuous choice Equation(102) is used for the whole momentum range, hence the single-particle potentials enter also into the energy denominator. InSong et al. [192], the equation of state in symmetric nuclearmatter has been considered. It has been shown, that theresult including three hole-lines is almost independent of thechoice of the auxiliary potential. Furthermore, the two-holeline result with the continuous choice comes out closer to
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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.
the three hole-line result, than the two-hole line calculationwith the gap choice. Another advantage of the continuouschoice for intermediate spectra is that it allows for a reliabledetermination of the single-particle potentials including theirimaginary parts [70]. The results presented here employ thecontinuous choice.
In the following we present some results of Petschaueret al. [76] for the in-medium properties of hyperons, basedon the YN interaction derived from SU(3) χEFT at NLO.The same potential V as in the Lippmann-Schwinger equation(Equation 98) for free scattering is used. However, as inHaidenbauer and Meißner [75] the contact term c8as forthe antisymmetric spin-orbit force in the YN interaction,allowing spin singlet-triplet transitions, has been fitted to theweak !-nuclear spin-orbit interaction [193, 194]. Additionally,for the ease of comparison, the G-matrix results obtainedwith two phenomenological YN potentials, namely of theJülich ’04 [21] and the Nijmegen NSC97f [20] meson-exchangemodels, are given. Note that, like the EFT potentials, thesephenomenological YN interactions produce a bound hypertriton[49]. For more details about derivation and the commonlyemployed approximations, we refer the reader to Reuber et al.[19], Rijken et al. [20], Kohno et al. [24], Schulze et al. [70],Vidaña et al. [71].
Let us start with the properties of hyperons in symmetricnuclear matter. Figure 17 shows the momentum dependenceof the real parts of the ! single-particle potential. The valuesfor the depth of the ! single-particle potential U!(k =0) at saturation density, kF = 1.35 fm−1, at NLO arebetween 27.0 and 28.3 MeV. In the Brueckner-Hartree-Fockapproximation the binding energy of a hyperon in infinitenuclear matter is given by BY (∞) = −UY (k = 0). Theresults of the LO and NLO calculation are consistent with theempirical value of about U!(0) ≈ −28 MeV [184, 185]. Thephenomenological models (Jülich ’04, Nijmegen NSC97f) leadto more attractive values of U!(0) = (−35 . . . − 50) MeV,where the main difference is due to the contribution in the3S1 partial wave. In contrast to LO, at NLO the ! single-particle potential at NLO turns to repulsion at fairly lowmomenta around k ≈ 2 fm−1, which is also the case for theNSC97f potential.
An important quantity of the interaction of hyperons withheavy nuclei is the strength of the !-nuclear spin-orbit coupling.
It is experimentally well established [186, 187] that the !-nucleus spin-orbit force is very small. For the YN interaction ofHaidenbauer and Meißner [75] it was indeed possible to tunethe strength of the antisymmetric spin-orbit contact interaction(via the constant c8as), generating a spin singlet-triplet mixing(1P1 ↔ 3P1), in a way to achieve such a small nuclear spin-orbitpotential.
Results for # hyperons in isospin-symmetric nuclear matterat saturation density are also displayed in Figure 17. Analysesof data on (π−,K+) spectra related to #− formation in heavynuclei lead to the observation, that the #-nuclear potential insymmetric nuclear matter is moderately repulsive [188]. The LOas well as the NLO results are consistent with this observation.Meson-exchangemodels often fail to produce such a repulsive#-nuclear potential. The imaginary part of the #-nuclear potentialat saturation density is consistent with the empirical value of−16 MeV as extracted from #−-atom data [195]. The imaginarypotential is mainly induced by the #N to !N conversion innuclear matter. The bands representing the cutoff dependence ofthe chiral potentials, become smaller when going to higher orderin the chiral expansion.
In Figure 18, the density dependence of the depth of thenuclear mean-field of ! or # hyperons at rest (k = 0). In orderto see the influence of the composition nuclear matter on thesingle-particle potentials, results for isospin-symmetric nuclearmatter, asymmetric nuclear matter with ρp = 0.25ρ and pureneutron matter are shown. The single-particle potential of the! hyperon is almost independent of the composition of thenuclear medium, because of its isosinglet nature. Furthermore,it is attractive over the whole considered range of density 0.5 ≤ρ/ρ0 ≤ 1.5. In symmetric nuclear matter, the three # hyperonsbehave almost identical (up to small differences from the masssplittings). When introducing isospin asymmetry in the nuclearmedium a splitting of the single-particle potentials occurs dueto the strong isospin dependence of the #N interaction. Thesplittings among the #+, #0, and #− potentials have a non-linear dependence on the isospin asymmetry which goes beyondthe usual (linear) parametrization in terms of an isovector Lanepotential [196].
Recently the in-medium properties of the , have beeninvestigated for ,N potentials from χEFT [37]. For amore extensive discussion and further applications seeKohno [197].
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FIGURE 17 | Momentum dependence of the real part of the single-particle potential of a ! hyperon and of the real and imaginary parts of the single-particle potential
of a # hyperon in isospin-symmetric nuclear matter at saturation density, kF = 1.35 fm−1 [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 ! = 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.
5.3. Hypernuclei and Hyperons in NeutronStarsThe density-dependent single-particle potentials of hyperonsinteracting with nucleons in nuclear and neutron matter findtheir applications in several areas of high current interest:the physics of hypernuclei and the role played by hyperonsin dense baryonic matter as it is realized in the core ofneutron stars.
From hypernuclear spectroscopy, the deduced attractivestrength of the phenomenological !-nuclear Woods-Saxonpotential is U0 8 −30 MeV at the nuclear center [183].This provides an important constraint for U!(k = 0) atρ = ρ0. The non-existence of bound #-hypernuclei, onthe other hand, is consistent with the repulsive nature of the#-nuclear potential as shown in Figure 18. In this context
effects of YNN three-body forces are a key issue. Whiletheir contributions at normal nuclear densities characteristic ofhypernuclei are significant but modest, they play an increasinglyimportant role when extrapolating to high baryon densities inneutron stars.
First calculations of hyperon-nuclear potentials based onchiral SU(3) EFT and using Brueckner theory have beenreported in Haidenbauer et al. [176] and Kohno [122].Further investigations of (finite) ! hypernuclei utilizingthe EFT interactions can be found in Haidenbauer andVidana [198], based on the formalism described in Vidaña[199]. For even lighter hypernuclei, the interactions are alsocurrently studied [200, 201]. Examples of three- and four-body results can be found in Haidenbauer et al. [34, 202] andNogga [49, 175].
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FIGURE 19 | The ! single-particle potential U!(p! = 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 ! = 450−500 MeV. The dotted (blue) band includes the density-dependent !N-interaction derived from the !NN
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 Yamamoto et al. [204].
Here we give a brief survey of !-nuclear interactions forhypernuclei and extrapolations to high densities relevant toneutron stars, with special focus on the role of the (a prioriunknown) contact terms of the !NN three-body force. Detailscan be found in Haidenbauer et al. [176]. Further extended workincluding explicit 3-body coupled channels (!NN ↔ #NN) inthe Brueckner-Bethe-Goldstone equation is proceeding [203].
Results for the density dependence of the ! single-particlepotential are presented in Figure 19 for symmetric nuclearmatter (Figure 19A) and for neutron matter (Figure 19B).Predictions from the chiral SU(3) EFT interactions (bands) areshown in comparison with those for meson-exchange YN modelsconstructed by the Jülich [21] (dashed line) and Nijmegen [20](dash-dotted line) groups. One observes an onset of repulsiveeffects around the saturation density of nuclear matter, i.e., ρ =ρ0. The repulsion increases strongly as the density increases.Already around ρ ≈ 2ρ0, U!(0, ρ) turns over to net repulsion.
Let us discuss possible implications for neutron stars. It shouldbe clear that it is mandatory to include the !N–#N couplingin the pertinent calculations. This represents a challenging tasksince standard microscopic calculations without this couplingare already quite complex. However, without the !N–#Ncoupling, which has such a strong influence on the in-mediumproperties of hyperons, it will be difficult if not impossible to drawreliable conclusions.
The majority of YN-interactions employed so far inmicroscopic calculations of neutron stars have properties similarto those of the Jülich ’04 model. In such calculations, hyperonsstart appearing in the core of neutron stars typically atrelatively low densities around (2 − 3)ρ0 [113, 119]. Thiscauses the so-called hyperon puzzle: a strong softening of theequation-of-state, such that the maximum neutron star mass falls
far below the constraint provided by the existence of severalneutron stars with masses around 2M9. Assume now that naturefavors a scenario with a weak diagonal !N-interaction and astrong !N–#N coupling as predicted by SU(3) chiral EFT.The present study demonstrates that, in this case, the ! single-particle potential U!(k = 0, ρ) based on chiral EFT two-bodyinteractions is already repulsive at densities ρ ∼ (2 − 3)ρ0. Theone of the #-hyperon is likewise repulsive [35]. We thus expectthat the appearance of hyperons in neutron stars will be shiftedto much higher densities. In addition there is a repulsive density-dependent effective !N-interaction that arises within the samechiral EFT framework from the leading chiral YNN three-baryonforces. This enhances the aforementioned repulsive effect further.It makes the appearance of !-hyperons in neutron star matterenergetically unfavorable. In summary, all these aspects takentogether may well point to a solution of the hyperon puzzle inneutron stars without resorting to exotic mechanisms.
6. CONCLUSIONS
In this review we have presented the basics to derive the forcesbetween octet baryons (N,!,#,,) at next-to-leading order inSU(3) chiral effective field theory. The connection of SU(3) χEFTto quantum chromodynamics via the chiral symmetry and itssymmetry breaking patterns, and the change of the degrees offreedom has been shown. The construction principles of thechiral effective Lagrangian and the external-field method havebeen presented and the Weinberg power-counting scheme hasbeen introduced.
Within SU(3) χEFT the baryon-baryon interaction potentialshave been considered at NLO. The effective baryon-baryon
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potentials include contributions from pure four-baryon contactterms, one-meson-exchange diagrams, and two-meson-exchangediagrams at one-loop level. The leading three-baryon forces,which formally start to contribute at NNLO, consist of athree-baryon contact interaction, a one-meson exchange and atwo-meson exchange component. We have presented explicitlypotentials for the !NN interaction in the spin and isospinbasis. The emerging low-energy constants can be estimatedvia decuplet saturation, which leads to a promotion of someparts of the three-baryon forces to NLO. The expressions ofthe corresponding effective two-body potential in the nuclearmedium has been presented.
In the second part of this review we have presented selectedapplications of these potentials. An excellent description of theavailable YN data has been achieved with χEFT, comparableto the most advanced phenomenological models. Furthermore,in studies of the properties of hyperons in isospin symmetricand asymmetric infinite nuclear matter, the chiral baryon-baryon potentials at NLO are consistent with the empiricalknowledge about hyperon-nuclear single-particle potentials. Theexceptionally weak !-nuclear spin-orbit force is found to berelated to the contact term responsible for an antisymmetric
spin-orbit interaction. Concerning hypernuclei and neutron starspromising results have been obtained and could point to asolution of the hyperon puzzle in neutron stars.
In summary, χEFT is an appropriate tool for constructingthe interaction among baryons in a systematic way. It sets theframework for many promising applications in strangeness-nuclear physics.
AUTHOR CONTRIBUTIONS
All authors listed have made a substantial, direct and intellectualcontribution to the work, and approved it for publication.
FUNDING
This work was supported in part by the DeutscheForschungsgemeinschaft (DFG) through the funds provided tothe Sino-German Collaborative Research Center Symmetriesand the Emergence of Structure in QCD (Grant No. TRR110),by the CAS President’s International Fellowship Initiative (PIFI)(Grant No. 2018DM0034), and by the VolkswagenStiftung(Grant No. 93562).
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Conflict of Interest: SP is employed by the company TNG Technology ConsultingGmbH.
The remaining authors declare that the research was conducted in the absence ofany commercial or financial relationships that could be construed as a potentialconflict of interest.
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