Three Baryon Interaction Generated by Determinant Interaction of Quarks

We discuss the three-baryon interaction generated by the determinant interaction of quarks, known as the Kobayashi-Maskawa-'t Hooft (KMT) interaction. The expectation value of the KMT interaction operator is calculated in fully-antisymmetrized quark-cluster model wave functions for one-, two- and three-octet baryon states. The three-baryon potential from the KMT interaction is found to be repulsive for $NN\Lambda$ and $N\Lambda\Lambda$ systems, while it is zero for the $NNN$ system. The strength and range of the three-baryon potential are found to be comparable to those for the $NNN$ three-body potential obtained in lattice QCD simulations. The contribution to the $\Lambda$ single particle potential in nuclear matter is found to be 0.28 MeV and 0.73 MeV in neutron matter and symmetric nuclear matter at normal nuclear density, respectively. These repulsive forces are not enough to solve the hyperon puzzle, but may be measured in high-precision hyperisotope experiments.


Introduction
The discovery of two-solar-mass neutron stars [1,2] has cast doubt on the equation of state (EOS) based on conventional nuclear physics. EOSs with nucleons and leptons for neutron star matter can support neutron stars whose masses are M ∼ 2M , where M is the solar mass. With hyperons (Y ), however, the EOS generally becomes much softer and many of proposed EOSs cannot support 2M neutron stars [3][4][5][6][7]. This problem, referred to as the hyperon puzzle, has been attracting much attention.
Several mechanisms have been proposed so far to solve the hyperon puzzle. One of the ideas is to assume that the crossover transition [8,9] from nuclear matter to quark matter takes place at relatively low density, 2 − 3ρ 0 , with ρ 0 0.16 fm −3 being the normal nuclear matter density, instead of the often assumed first order deconfinement phase transition at high density. In this case, the transition may occur at lower density than the onset of hyperon mixing via the weak interaction, and then prevents the matter from softening owing to emergence of the hyperons. The crossover nature of the transition also circumvents the softening from the first order phase transition. This crossover picture can be examined by making use of high-energy heavy-ion collision experiments. Indeed, the direct flow collapse in Au+Au collisions at √ s N N = 11.5 GeV [10] seems to suggest the sudden softening of † Report No: YITP-  c The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
EOS [11], which could be caused by the first order phase transition. Further studies including the isospin asymmetry dependence of the phase transition order are necessary to examine the crossover transition in dense neutron-rich matter. Another idea is to modify the hyperonnucleon (Y N ) interaction. Within the flavor SU(3) coupling scheme but off the flavor-spin SU(6) couplings, one can obtain a stiff hyperonic matter EOS [12] with parameters including largess contents in nucleons, which however contradicts the recent lattice QCD calculations; see Ref. [13] and references therein. The third idea is to introduce the three-baryon (3B) interaction involving hyperons If the three-nucleon (3N ) force is repulsive and there exists repulsive 3B forces involving hyperons, the EOS can support the 2M neutron stars by suppressing the hyperon mixing at high densities. The role of the 3N force has been extensively discussed in nuclear physics. Ab initio calculations with two-nucleon interactions cannot reproduce the binding energies of threenucleon systems (t and 3 He) and the nuclear matter saturation point, and it has been noticed that three-nucleon (3N ) force is necessary to reproduce these fundamental aspects of nucleon many-body systems [14][15][16][17][18][19][20][21]. The 3N force contains two-pion P-wave exchange term with an intermediate ∆ excitation (Fujita-Miyazawa force) [14] and shorter-range phenomenological term [15]. One of the physical pictures of the short-range 3N force is the multi-pomeron exchange [16]. Recently, 3N force has been derived systematically in the chiral effective field theory, where energy-independent 3N forces appear at the next-to-next-to-leading order [17,18]. Microscopic calculations including these 3N forces show that the 3N force is relevant to the nuclear matter saturation and the nuclear matter EOS at high densities [15,16,[19][20][21], and also to deuteron-proton scattering observables [22]. Moreover, there haven been remarkable progress in the lattice QCD simulation for the 3N force based on the Nambu-Bethe-Salpeter wave function, i.e. HAL QCD method [23]. At present, such lattice QCD simulations have been performed for larger π masses than the physical π mass and then their data still have large error-bars. Cooperations between lattice QCD simulations, experiments and phenomenological approaches including present study should play an important role to understand nature of the 3N force and also other 3B forces in future.
So far 3B forces involving hyperons have not been known well. The currently available hypernuclear data are not enough to determine the 3B force involving hyperons precisely at the level of the 3N force, then we have to rely on some models or assumptions. In Ref. [16], the repulsive 3B forces are assumed to work universally for Y N N , Y Y N , Y Y Y as well as for N N N in the multi-pomeron exchange mechanism. By fixing the multi-pomeron exchange strength by the nuclear elastic scattering, the nuclear matter saturation and hypernuclear separation energies are reproduced in the G-matrix calculations, and the 2M neutron stars are found to be supported. In Ref. [24], the authors adopt a phenomenological hyperonnucleon potential, which contains the ΛN forces consistent with the Λp scattering data and strongly repulsive Wigner type ΛN N force [25]. The Λ separation energies are well reproduced in the auxiliary field diffusion Monte-Carlo calculations, and the EOS fitted to the Monte-Carlo results can support the 2M neutron stars when the ΛN N forces are included [26]. In this treatment, the Λ separation energies in heavy Λ hypernuclei are calculated to be more than 60 MeV with the two-body (2B) forces, while they are reduced to be less than 30 MeV with the 3B forces. The effects of the 3B potential are very large, and it is natural to deduce that the four-and more baryon potential would be as important as the 2/20 3B potential. In Ref. [27], internal structure modification of baryons is taken into account in the framework of the chiral quark meson coupling model. The modification leads to the σ dependence of the baryon-σ coupling, or the baryon-σ-σ multi-body coupling, where σ is the isoscalar scalar meson field. Because of the multi-body coupling and the Fock term contribution, hyperons are suppressed at high densities, and 2M neutron stars can be supported within the observation errors. While these approaches are successful in explaining the existence of 2M neutron stars with EOS including hyperons, it is preferable to derive the 3B interaction from a view point of the quark dynamics.
One of the possible origins of the 3B interaction is the determinant interaction of quarks, referred to as the Kobayashi-Maskawa-'t Hooft (KMT) interaction [28][29][30][31], where det denotes the determinant with respects to the flavor indices, i and j. With threeflavors of quarks (N f = 3), the KMT interaction replaces three quarks having different flavors with three quarks, then it can generate the 3B interaction; The 3B system having at least one u, d and s quarks can be connected by the KMT interaction as shown schematically in

Fig. 2 Other interaction terms from the KMT interaction
The KMT interaction was introduced in order to account for the axial U (1) (U(1) A ) anomaly [28,29], and it was found to be generated by instantons [30,31]. The strength of the KMT interaction g D can be fixed by reproducing basic quantities of mesons. For 3/20 example, g D was determined to be g D Λ 5 = −9.29 [32] or g D Λ 5 = −12.36 [33] in the Nambu-Jona-Lasinio (NJL) model [34,35] by fitting the π, K and η masses and the pion decay constant with other model parameters. Since the U(1) A anomaly pushes up the η mass (m η ), the value of g D is essentially fixed by m η [32].
From the above considerations, the 3B interaction from the KMT interaction has favorable features to resolve the hyperon puzzle. It acts only on systems including strange quarks, and it is probably repulsive because of the negative value of g D . In the meanfield treatment for the NJL model, the KMT interaction pushes up the constituent quark mass [32], where m u,d,s represents the current quark mass, g s is the fourfermion scalar coupling, and qq < 0 is the quark condensate. The KMT interaction is found to act repulsively also in the ΛΛ interaction [36]. While the color-magnetic interaction is strongly attractive in the flavor singlet six-quark state, the H dibaryon [37], the KMT interaction acts repulsively and the mass of H may be pushed up above the ΛΛ threshold.
In this article, we discuss the 3B interaction generated by the KMT interaction in the quark cluster model. We adopt here the non-relativistic treatment for the wave functions of quarks and the KMT interaction. We consider the non-relativistic wave functions of quarks in baryons, and the spatial wave function is assumed to be (0s) 3 ; all quarks are assumed to be in the s-wave state of the harmonic oscillator potential and to have the same spatial extension. In the non-relativistic treatment of the KMT interaction, we ignore the γ 5 part in Φ ij , then the KMT interaction becomes the zero-range three-body interaction among quarks (six-fermi interaction), which modifies the flavor but does not modify the spin and color of quarks.
This paper is organized as follows. In Sec. 2, we evaluate the norm of the one-, two-, and three-baryon wave functions in the quark cluster model. In Sec. 3, we quantify the 3B potential generated by the KMT interaction based on the three-baryon wave function and discuss its implication to the hyperon puzzle. Section 4 is devoted to the summary.

One baryon state
We consider here the flavor-spin SU(6) wave function of octet baryons. For example, the color, flavor, spin wave function for n ↑ , n ↓ and Λ ↑ are given as where ε abc is the Levi-Civita tensor with respect to color indices. The first 1/ √ 3! factor is the color symmetry factor, which is common to all baryons. The normalization factor is determined to normalize the fully antisymmetrized wave function, The antisymmetrizer A is defined as 4/20 Table   1 Octet baryon wave functions with the spin-up state. The fully antisymmetrized wave function is given as where P denotes permutation, q i is the quark wave function, and sgn(P ) is the sign (or signature) of the permutation P . The flavor and spin wave functions for octet baryons are summarized in Table 1 for completeness. The norm of the one baryon state is obtained as where 3) are the flavor-spin coefficient and component of the one-baryon wave function, respectively. The antisymmetrizer A acts on the color, flavor, spin, and spatial coordinate wave functions, while the symmetrizer in the flavor-spin-spatial (fss) coordinates, S fss , does not exchange color indices. In the last line, the matrix element is obtained in the flavor, spin, and spatial coordinate wave functions, F P (φ i , φ j ) = φ i | P φ j fss . For example, one of the elements looks like provided that the spatial wave function is common to all quarks and normalized.

Two baryon states
Two-baryon states are defined as the product of two-baryon wave functions fully antisymmetrized in color, flavor, spin and spatial coordinates. We show the case of n ↑ n ↓ , as an example, 5/20 The norm of the above two-baryon state is obtained as The color and flavor-spin-spatial factors are shown by C P and F P . The exchange of quarks among baryons gives rise to a sign factor, which cannot be absorbed in the Levi-Civita tensor ε, where 1 a (2 d ) shows one of the quarks in the first (second) baryon having the color index a (d). The color factor is obtained as the contraction of ε. For the case of the second term in Eq. (11), the first (second) three colors in the bra state need to be abd (cef ) to have a finite matrix element, and we obtain The color factor in the two-baryon case is found to be C P = 1, −1/3, 1/3, −1 for zero, one, two and three quark exchanges between baryons, respectively. The flavor-spin-spatial factor for the permutation P is provided that two baryons are located at the same spatial point.

Three baryon states
Three-baryon states are defined in a way similar to the two-baryon states. For the n ↑ n ↓ Λ ↑ state, the three-baryon wave function reads The norm of three-baryon states are obtained as where and C P and F P are the color and flavor-spin-spatial factors, respectively.
Calculation of the color factor for the three-baryon states is straightforward but lengthy. We show one of the non-trivial examples, In this case the color factor is C P = −1/9. Exchanges of quarks among three baryons can be categorized into the following types. We schematically show the corresponding diagrams for these types in Fig. 3.
We find that the color factor is ±1 when there is no quark exchange among baryons (a1, a2, a3). When one baryon keeps its three quarks (b1-b4), the color factor is ±1/3. In the case where all three baryons are involved in quark exchanges, the color factor is ±1/9 in most of the cases (c1-c3, d1-d3). The single exception is (e), C P = 0, where the ket state is made of three baryons, each of which contains one quark from each baryon in the bra state. We summarize the color factor in Table 2, and the norm in some of two-and three-baryon systems in Table 3.
The norm of the three-baryon states can be more elegantly evaluated by using the reduction formula of the antisymmetrizer for the three octet-baryon states [38,39], where P ij denotes the exchange operator of the i-th and j-th quark, P ijk denotes the cyclic exchange operator of the ijk-th quarks, and P is the baryon exchange operator. Equation (17) takes care of 1680 permutations, which act on (3!) 3 = 216 terms in the product of three antisymmetrized baryon wave functions and result in 1680 × 216 = 9! permutations. In our categorization the first, second, third and fourth terms in the first line of Eq. (17) correspond to (a1-a3), (b1-b4), (d1-d3), and (c1-c3), respectively, and the second line corresponds to type (e).

KMT operator and its expectation value
We now evaluate the KMT matrix elements in one-, two-and three-baryon systems. We concentrate on the (0s) 6 and (0s) 9 configurations of quarks for two-and three-baryon systems, respectively, which correspond to the cases where two-or three-baryons are located at the same spatial point.
In the non-relativistic treatment of the KMT interaction (2), we ignore terms which involve γ 5 and replace γ 0 with a unit matrix in Φ ij , 8/20  Then we obtain the matrix element of the KMT interaction for the three-quark states having a common color configuration as, 9/20 where the operatorT i,j replaces the flavor j with the flavor i, and ϕ α (x) (ϕ α (x)) shows the spatial part of the α-th quark wave function in the bra (ket) state. We find the quantum mechanical KMT interaction operator as, The operatorT i,j α now acts on the α-th quark. For more generic product wave functions with the same color configurations, φ = α q α and φ = α q α , the KMT matrix element is found to be where σ α , f α and ϕ α (σ α , f α and ϕ α ) are the spin, flavor and spatial wave functions for the α-th quark in φ (φ ), respectively. The KMT matrix elements in baryon systems are obtained as a sum of those for the product wave functions, where c I , f I and ϕ I (c J , f J and ϕ J ) are the flavor-spin coefficients, flavor configurations, and spatial wave functions of the I-th (J-th) component of the baryon wave function in the bra (ket) state, respectively, and C P is the color factor. In Table 3, we show the norm and the diagonal matrix elements of the KMT operator, in baryons sitting at x = 0. Spatial wave functions are assumed to be the same for all quarks, then the norm becomes zero when the two baryons have the same flavors and spins.  For one-baryon states, the diagonal matrix elements of the KMT operator are found to be zero.
We have summed up the contributions of all the quark permutations, 3! = 6, 6! = 720 and 9! = 362880 for one-, two-, and three-baryon(s), respectively. We note that F KMT αβγ is an integer, C P is a multiple of 1/9 and c * I c J is a multiple of (1/12) n−nΛ × (1/2) nΛ for the diagonal matrix elements in n baryon systems with n Λ being the number of Λ. Thus N A and T A are rational numbers. The genuine two-and three-baryon part of the matrix elements are evaluated by subtracting the contribution in subsystems. For example, for n ↑ n ↓ Λ ↑ threebaryon systems, we subtract the N Λ contributions, where T = T A /N A . We find that the expectation values of the KMT operator take positive values of 3 − 20 for 2B and 3B systems including hyperons. Thus the KMT interaction is confirmed to generate a 3B repulsive potential when we have hyperons in 3B systems.
The expectation value of the KMT operator, T 2B and T 3B , is found to be spindependent: It is larger for larger spin states, for instance, T 2B ((N Λ) S=1 ) > T 2B ((N Λ) S=0 ) and T 3B ((N N Λ) S=3/2 ) > T 3B ((N N Λ) S=1/2 ). Since the KMT operator does not change the quark spin in the nonrelativistic treatment, more quark pairs have the same spin in the bra and ket and larger matrix element T A will appear in larger spin states. We note that the matrix element in the spin quartet state, T 3B ((pnΛ) S=3/2 ), is much larger than others.

Baryon potentials from the KMT interaction
We evaluate the 3B potential from the KMT interaction (KMT-3B potential) as the expectation value of the 3B part of the KMT interaction operator (21) in three-baryons located 11/20 at R 1 , R 2 and R 3 , 1 where V A /N A | 3B denotes the 3B potential part of the expectation value. The spatial part of the intrinsic baryon wave function is assumed to be (0s) 3 ; all quarks are assumed to be in the s-wave state of the harmonic oscillator potential and to have the same spatial extension, where R is the position of the baryon and the size parameter ν is related with the size of the quark wave functions b as ν = 1/(2b 2 ). The strength of the KMT matrix element for three baryons located at the same position (R 1,2,3 = 0) reads By using the parameters, g D Λ 5 = −9.29 and Λ = 631.4 MeV [32], b = 0.6 fm [40] or b = 0.5562 fm [41], we find If we take g D Λ 5 = −12.36 and Λ = 602.3 MeV [33], V 0 becomes about 1.68 times larger than the values (33) because V 0 is linearly proportional to g D . At finite R 1,2,3 , the spatial matrix element of the KMT operator is given as where R ij = R i − R j and R i (i = a ∼ f ) is one of R 1 , R 2 and R 3 . When the antisymmetrization effects on the spatial wave function are ignored, other spatial matrix elements ( ϕ µ | ϕ µ in Eq. (25)) cancels with those from the norm. 2 This prescription provides correct results when the baryons are separated enough or three-baryons are located at the same spatial point. The KMT-3B potential is then given as 1 Strictly speaking, this is the expectation value of the potential and not the potential itself. As long as the extension of the baryon wave function is large enough compared with the intrinsic extension of the quark wave function, however, the present treatment gives a good estimate of the potential.
2 For a more serious discussion, we need to evaluate the spatial factor R KMT αβγ in Eq. (25) including the overlaps of other quarks, as well as those in the norm. In addition, the KMT matrix element can be finite also in those configurations having T = 0 at zero distance. For example, the n ↑ n ↑ Λ ↑ configuration has a zero norm and a zero KMT matrix element due to the Pauli blocking when three baryons are located at the same point, but will have a finite norm and a finite KMT matrix element at finite distances. These are beyond the scope of this paper, and we concentrate on the potentials in N N Λ and N ΛΛ channels having finite KMT matrix elements at zero distance. In Fig. 4, we show the KMT-3B potential V KMT 3B for the N N Λ system in (nn) S=0 Λ and (npΛ) S=3/2 channels with a linear configuration in which three baryons are located at R 1 = (−r, 0, 0), R 2 = (0, 0, 0) and R 3 = (r, 0, 0). We adopt V 0 values evaluated in Eq. (33). The KMT-N N Λ potential at r = 0 has the height of 5.8 (9.2) MeV in (nn) S=0 Λ channel and 24.2 (38.2) MeV in (npΛ) S=3/2 channel for b = 0.6(0.5562) fm, respectively, and the potential range is determined by the baryon size.
We compare the KMT-N N Λ potential with the 3N potential obtained by using the lattice QCD simulation (LQCD-3N potential) [23]. It is interesting to find that the KMT-N N Λ potential and the LQCD-3N potential have the same order of strengths and ranges, while the system is different and they do not need to agree. One of the differences is the short range behavior; the KMT-N N Λ potential has the peak at r = 0, whereas the LQCD-3N potential is suppressed at r = 0.

KMT-3B potential energy in nuclear matter
We shall now evaluate the KMT-3B potential energy per baryon in nuclear matter. We assume that the density of each baryons is constant, then we get the energy per baryon W KMT 3B from the KMT-3B potential as where ρ B is the baryon density and Ω denotes the spatial volume. It should be noted that W 0 is independent of the baryon size b. The KMT-3B potential energy per baryon in nΛ matter is given as where Y n and Y Λ represent the density fractions of n and Λ, Y n = ρ n /ρ B and Y Λ = ρ Λ /ρ B , respectively, and T 3B (N N Λ) and T 3B (N ΛΛ) are the spin-averaged KMT matrix elements, where (S, T ) = (0, 1) pair is taken for N N Λ. Note that Y n + Y Λ = 1 in nΛ matter. For npΛ matter, we find where T 3B (npΛ) is the spin-averaged KMT matrix element in the npΛ system, Note that Y n + Y p + Y Λ = 1 in the npΛ matter.
In the left panel of Fig. 5, we show the KMT-3B potential energy per baryon in nΛ and npΛ matter. Each shaded area corresponds to 0.1 ≤ Y Λ ≤ 0.3 and neutron and proton fractions are taken to be Y n = 1 − Y Λ and Y n = Y p = (1 − Y Λ )/2 for nΛ and npΛ matter, respectively. The KMT-3B potential energy per baryon amounts to be W KMT 3B = 0.027 and 0.064 MeV at (ρ B , Y Λ ) = (ρ 0 , 0.1) in nΛ and npΛ matter, respectively. At higher density and larger Λ fraction, the KMT-3B potential is found to reduce the nuclear symmetry energy slightly; It may be instructive to compare the above potential energy with that in quark matter having the same uds quark fractions, It should be noted that only the residual interaction contribution is considered here, and other contributions coming from the condensates are not counted. In the right panel of Fig. 5, we compare the coefficients of the potential energy per baryon, W KMT /[W 0 (ρ B /ρ 0 ) 2 ] in nΛ, npΛ and quark matter. At small strange quark fraction (small Y Λ ), we note that the quark matter estimate agrees with the baryonic matter estimate in npΛ matter, while baryonic matter estimate results in weaker KMT-3B repulsion in nΛ matter. Finally, we discuss the KMT-3B potential contribution to the Λ single particle energy, In nΛ and pnΛ matter, we find In Fig. 6, we show the Λ single particle potential in nΛ and pnΛ matter. The neutron and proton fractions in npΛ matter is again taken to be Y n = Y p = (1 − Y Λ )/2. The average pnΛ matrix element is larger than that of nnΛ, since we have more combinations of uds in pnΛ matter. As a result, Λ feels more repulsion in symmetric matter than in pure neutron matter. The KMT-3B potential contribution to the Λ single particle energy is found to be U KMT Λ(nΛ) 0.28 MeV and U KMT Λ(pnΛ) 0.73 MeV at ρ B = ρ 0 in neutron matter and symmetric nuclear matter (nΛ and pnΛ matter at Y Λ = 0), respectively. These contributions are not large, but would be measurable in high-resolution experiments on the Λ separation energy of hyperisotopes such as 40 Λ K and 48 Λ K [42]. 15/20

Summary
We have evaluated the expectation value of the determinant interaction of quarks, the Kobayashi-Maskawa [28,29] and 't Hooft [30,31] (KMT) interaction, in three-baryon systems, and have discussed the three-baryon potential from the KMT interaction. The KMT vertex gives rise to such three-quark interaction that all the u, d and s quarks need to participate. Then the KMT interaction is expected to generate potentials among three baryons which include at least one hyperon. The KMT interaction is responsible to the U (1) A anomaly, and its strength is determined from the η mass [32,33]. The negative sign of the strength parameter in the Lagrangian generates a repulsive three-quark interaction.
Repulsive potential among three-baryons including hyperons may help to solve the hyperon puzzle of two-solar-mass neutron stars [1,2]. The expectation value of the KMT operator (Eq. (27)) is obtained for the two-and threebaryon (2B and 3B) states consisting of nucleons and Λ baryons, where baryons are assumed to be located at the same spatial point. We have adopted nonrelativistic (0s) 3 wave functions for octet baryons, where a common spatial wave function is assumed for all quarks. 2B and 3B states are given as the product of baryon wave functions antisymmetrized under the quark exchanges. The KMT operator is found to take positive expectation values of 3 − 20 for 2B and 3B systems consisting of N and at least one Λ. Thus the KMT interaction is confirmed to generate 3B repulsive potential in N N Λ and N ΛΛ systems. The expectation value of the KMT operator, T 2B and T 3B , is found to be larger for the larger spin state, T 2B ((N Λ) S=1 ) > T 2B ((N Λ) S=0 ) and T 3B ((N N Λ) S=3/2 ) > T 3B ((N N Λ) S=1/2 ). Since the KMT operator does not change the quark spin in the nonrelativistic treatment, there can be more quark pairs having the same spin in the bra and ket of larger spin states and the expectation value of the KMT operator tends to be larger. We also note that the matrix element in the spin quartet state, T 3B ((pnΛ) S=3/2 , is much larger than others.
We have evaluated the 3B potential from the KMT interaction (KMT-3B potential). The strength of the 3B potential is calculated by using the harmonic oscillator wave function of quarks adopted in the quark cluster model analyses of 2B potentials [40,41] and the strength of the KMT interaction obtained from the η mass analyses [32]. The obtained N N Λ potential from the KMT interaction has the height of 5.8 − 9.2 MeV in (nn) S=0 Λ channel and 24.2 − 38.2 MeV in (npΛ) S=3/2 channel at zero distance, and the potential range is determined by the baryon size. The strength and the range of the 3B potential in the linear configuration are found to be similar to those from the 3N potential obtained in the lattice QCD simulation [23] except the short range region.
The KMT-3B potential energy per baryon in nuclear matter W KMT 3B and the KMT-3B potential contribution to the Λ single particle potential U KMT Λ are also estimated. The KMT-3B contribution to the equation of state, W KMT 3B , is found to be small; W KMT 3B 0.05 MeV and 0.1 MeV, for nΛ and npΛ matter at (ρ B , Y Λ ) = (ρ 0 , 0.1), respectively. The KMT-3B contribution to the Λ single particle potential, U KMT Λ , is not large but would be visible; U KMT Λ 0.56 MeV and 1.2 MeV in nΛ and npΛ matter at ρ B = ρ 0 , respectively. The difference is small, but may be detectable in high-precision experiment on the separation energies of hyperisotopes such as 40 Λ K and 48 Λ K [42]. At ρ B = 3ρ 0 , U KMT Λ amounts to 2.5 MeV and 6.6 MeV in nΛ and npΛ matter, respectively. In order to solve the hyperon puzzle of massive 16/20 neutron stars, we need to find the mechanism to generate additional Λ single particle potential ∆U Λ ∼ 100 MeV at around ρ B = 3ρ 0 , and the KMT-3B potential explains 2 − 7% of the required repulsion. Other three-quark interactions such as the confinement potential [43] can also contribute to the 3B potential.