High-density kaonic-proton matter (KPM) composed of Lambda* equiv K-p multiplets and its astrophysical connections

We propose and examine a new high-density composite of $\Lambda^* \equiv K^-p = (s \bar{u}) \otimes (uud)$, which may be called {\it Kaonic Proton Matter (KPM)}, or simply, {\it $\Lambda^* $-Matter}, where substantial shrinkage of baryonic bound systems originating from the strong attraction of the $(\bar{K}N)^{I=0}$ interaction takes place, providing a ground-state neutral baryonic system with a huge energy gap. The mass of an ensemble of $(K^-p)_m$, where $m$, the number of the $K^-p$ pair, is larger than $m \approx 10$, is predicted to drop down below its corresponding neutron ensemble, $(n)_m$, since the attractive interaction is further increased by the Heitler-London type molecular covalency, as well as by chiral symmetry restoration of the QCD vacuum. Since the seed clusters ($K^-p$, $K^-pp$ and $K^-K^-pp$) are short-lived, the formation of such a stabilized relic ensemble, $(K^-p)_m$, may be conceived during the Big-Bang Quark Gluon Plasma (QGP) period in the early universe before the hadronization and $quark$-$anti$-$quark$ annihilation proceed. At the final stage of baryogenesis a substantial amount of primordial ($\bar{u},\bar{d}$)'s are transferred and captured into {\it KPM}, where the anti-quarks find places to survive forever. The expected {\it KPM} state may be {\it cold, dense and neutral $\bar{q} q$-hybrid ({\it Quark Gluon Bound (QGB)}) states, $[s(\bar{u} \otimes u) ud]_m$,} to which the relic of the disappearing anti-quarks plays an essential role as hidden components. The {\it KPM} may also be produced during the formation and decay of neutron stars.


Introduction
In the present paper we propose and examine a new high-density neutral matter, anti-Kaonic Proton Matter (KPM), composed of hitherto known units of which may be called KPM, or simply, Λ * -Matter (Λ * -M). Its free unit, Λ * , first predicted by Dalitz and Tuan [1], has been identified to be a known resonance state of Λ(1405) with a mass of M = 1405 MeV/c 2 [2]. Its spectacular nature was not fully realized before. The present investigation arises from our recent theoretical finding of high-density anti-Kaonic (K) few-body Nuclear Clusters (KNC) [3][4][5][6][7][8][9][10], where nuclear systems with a density of ρ ≈ 3ρ 0 (ρ 0 being the normal nuclear density, 0.17 fm −3 ) are spontaneously formed, driven by the strong (KN ) I=0 attraction without the aid of gravity. We start our discussion from empirical information concerning the most important building blocks: K − p (= Λ * ), K − pp and K − K − pp.
iii) In K − pp ∼ Λ * p and K − K − pp ∼ Λ * Λ * a molecular analogy stands even for the systems of nuclear interactions [7], and the Heitler-London type covalent bonding effect [17] plays an important role as wide-ranging multiple bonding forces [9].
iv) The spontaneous nuclear shrinkage causes an enhancement of theKN interaction by Chiral Symmetry Restoration (CSR) that iterates further production of higher nuclear densities, and thus of larger kaonic binding energies and decreased masses of the KPM ensemble. v) Thus, the joint effect of the multiple bonding of Λ * and the CSR may cause a large energy gap, where the ground state of the Λ * multiplet may become well below that of the corresponding neutron ensemble: The double kaonic cluster, K − K − pp, initially predicted by [6], shows a well developed deeply bound structure of two Λ * 's, whereas they persist to keep the identification as Λ * (= K − p). Here, we comment on the interaction of the two Λ * (= K − p)'s. The original migrating exchange force of Heitler and London [17] was considered between two fermionic electrons in H + -H 0 and H 0 -H 0 molecules. In the present case, on the contrary, the migrating particles are bosonic K − mesons, the wave function of which is where the two protons sit on sites a and b which are separated by a distance of D. Then, the exchange interaction is obtained as as given by a g-matrix in [3]. If Kis assumed to be a fermion: Fermion covalent bonds cancel each other.
Boson covalent bonds are always added. If Kis assumed to be a fermion: Fermion covalent bonds cancel each other.
Boson covalent bonds are always added.
If Kis assumed to be a fermion: Fermion covalent bonds cancel each other. This ∆U (D), shown in Fig. 1, is a bonding potential due to doubly migrating K − 's, and is about twice as strong as the one from single K − migration in K − pp, discussed in [7]. On the other hand, if we artificially assume the migrating particles to be spinless fermions, the two terms of Eq. (.3) should be subtracted, and would yield a much weaker bonding. It is noted that the bonding from multi-K − migrations is always additively constructed due to the bosonic nature of K − . In this way, the K − 's bring about a much stronger binding effect. We have obtained the effective potential between the two Λ * (= K − p)'s by folding the bonding potential, ∆U , the K − K − repulsive potential, V KK , and a realistic N N potential having a repulsive core, with the internal K − p distribution of Λ * .
We applied the effective Λ * -Λ * interaction thus obtained to calculate the binding energies of multiple (Λ * ) m systems that approximate multiple (K − p) m states. Here, we take into account the possible combinations of Λ * -Λ * bonding, as the number of bonding increases with the multiplicity being 2, 6, 12, 20, 30, .., for m = 2, 3, 4, 5, 6, .., respectively. The results obtained by using a variational method (ATMS [20] employed in [7]) are shown in Fig. 2, which indicate that the energy level per each Λ * of a multiple (Λ * ) m state drops down, and finally exceeds the threshold level of free Λ emission, when the Λ * multiplicity becomes larger than some critical number. The number is estimated to be 10, if the effect of CSR (discussed in the next section) that will enhance the assumed basic KN interaction is taken into account. Such a multiplet as (Λ * ) m>10 could be stable against any strong-interaction decay. Figure 3 shows a stable ensemble of Λ * 's together with Heitler-London type covalent bonding of bosonic K − , that produces super-strong nuclear interaction [7]. A mean-field model for multiKfs in nuclei is employed in [18], but lacks just this multi-bonding mechanism of the super-strong nuclear attraction, which gives a drastic non-linear decrease of M [(K − p) m ] as m increases. It should be mentioned that the K − in a nucleus cannot keep to hold its independent-particle motion in mean field by yielding a marked (Λ * = K − p) cluster correlation. In fact, the K − in a nucleus does not satisfy the ghealingh condition for independent-particle motion discussed by Gomes et al. [19].
In order to see the effect of CSR on the size of the basic Λ * Λ * system, distributions of the Λ * -Λ * distance obtained from Faddeev-Yakubovsky calculation for K − K − pp [10] is also shown in Fig. 3.

Stability and stiffness of KP M
Here, we consider the basic stability of KP M . The longevity of KP M depends on its stiffness against the addition of external foreign substances and the subtraction of internal components. The total mass of the Λ * multiplet in the preceding section, M [(Λ * ) m ] per baryon, is well approximated as with ∆U av. = −135 MeV for m = 4 ∼ 8. Then, the Λ * separation energy, S m (Λ * ), for the (Λ * ) m → free Λ * + (Λ * ) m−1 process is given by It is noted that S m is 2-times larger than BE m (binding energy per Λ * ), that is the mass difference between a free Λ * and a bound Λ * in (Λ * ) m , due to a rearrangement of the (Λ * ) m−1 cluster. The S 6 (Λ * ) is estimated to be 675 MeV, which is almost 2-orders of magnitude larger than the nucleon separation energy of about 8 MeV from usual nuclear systems. S m (Λ * ) becomes larger with m > 6. As for the weak decay of (Λ * ) m , the Q-value of the (Λ * ) m → n + (Λ * ) m−1 non-leptonic process is given by  [20]. The corresponding nuclear densities and neutron Fermi levels are also shown, indicating that the Λ * in the (Λ * )6 cannot decay to a neutron in neutron matter at 3.2 times the normal density ρ0. due to negative Q-values, though the mass of Λ * in (Λ * ) 6 is still heavier than the neutron mass, as shown in Fig. 2. Only (5 ∼ 6) × {Λ * → n} take place through simultaneous weak decays, which are profoundly suppressed by the decay multiplicity.

Chiral symmetry restoration forKN
The recent experimental data on K − pp from DISTO [11] and J-PARC E27 [12] gave a binding energy of about 100 MeV, which is a factor of 2 larger than the original prediction [4] based on the empirical Λ(1405) mass. A recent Faddeev-Yakubovsky calculation [10] shows that the observed binding energy corresponds to an effectivē KN interaction which is about 17% more attractive than that assumed in the original prediction. Here, we consider the origin of this enhanced interaction in terms of the chiral-symmetry restoration (CSR) effect [21][22][23][24][25][26][27].
In general, when CSR takes place in dense nuclear medium, the quark condensate decreases toward zero, and the (KN ) I=0 interaction is expected to increase in magnitude. A naive qualitative estimate was made in [10] by employing a model of Brown, Kubodera and Rho (BKR) [27]. Figure 4 shows the estimated quark condensate (straight line) and the enhancement factors FK N I=0 as functions of the nuclear density ρ(r), where ω is a "QCD-vacuum clearing factor". In the case ofKN I=0 , a drastic situation takes place [10]; FK N increases and amplifies the binding energy and shrinks the nucleus furthermore, leaving less and less room for the QCD vacuum with further increasing the ω and FK N factor nonlinearly. An enhancement factor of 1.5 corresponding to the density ρ/ρ 0 ≈ 2 produces an enormous multiplication of the binding energy of K − K − pp [10]. Although the above estimate is very rough, the CSR effect in combination with the Heitler-London type enhancement is expected to bring the KPM mass of moderate multiplicity (m ∼ 10) well below the nucleon mass,.
How can KPM be formed? As the KPM seed clusters, K − p, K − pp and K − K − pp, are short-lived with Γ ∼ 100 MeV, they cannot survive during the cascading collisions toward heavier clusters. Exceptional cases might take place during the initial phase of the early universe, where quarks (u, d, s,ū,d,s) and gluons are produced in quark gluon plasma (QGP) at extreme high temperatures and densities, but probably before the hadronization stage, as illustrated in Fig. 5. Since the KPM seeds, particularly, K − p ≡ Λ * and K − pp ≡ Λ * p, are distinctly deeply bound with binding energies of around 50 ∼ 100 MeV, whereas other quarks and hadrons are relatively shallowly bound, we expect that during the course of decreasing temperatures (kT ≈ 100 MeV, and in expansion), the seeds are likely to become deep quasi-stable self-trapping centers, and recombined with other seeds that have just been born nearby. The star-like red objects illustrated in Fig. 5 (b) represent such just-born fresh composites of Λ * multiplets with m ∼ 10. They undergo further combinations to become a large-scale more stable KP M . This process is in competition with the branching ratio of Λ * formation {sū + uub → s(ūu)ud} to the normalqq annihilation background,q + q ↔ g ′ s in the early universe. Certainly, such competition occurs in the QCD level, and we need more knowledge on its answer.
It is to be noted that the basic unit of KP M , {(sū)(uud)}, involves aū-u pair, which is essential in producing this deeply bound system. This system possesses oneū quark per unit that has been transferred from the primordial QGP phase. Figure 5 (d), (e), and (f) shows symbolically (d) a disappearing ANTI-MATTER sector that involves originally unboundq beforeqq annihilation, and (f) a dominating MATTER sector of relative baryon density around 2 × 10 −8 , resulting afterqq annihilation and baryogenesis. During the anti-quark disappearing stage relic and stable composites of {(sū)(uud)} are formed, and constitute a (e) HYBRID sector. In other words, a substantial fraction of anti-particles may remain being hidden relic in the KPM phase as an unknown astronomical object. Formation of Quark-Gluon Bound (QGB) states Annihilating, but still surviving, anti-quarks contribute to forming seeds for KPM: Λ * ≡ [s(ūu)ud]. This particle-anti-particle hybrid state has very strong attractive interactions with surrounding similar species; thus, multiple Λ * states are composited and their mutual fusions take place in a short time and on a large scale, as if it occurred in a sudden phase transition.
The above Λ * 's that are defined as K − p in the language of hadrons may be born directly from constituents of QGP from the beginning, but eventually become cooled so as to be changed into the new phase: Quark Gluon Bound (QGB) states. While being cooled furthermore, its QGB phase may remain unchanged. Whether KP M could form a macroscopic object or not, the possibility of KP M fragments as low-temperature QGB states should be an extremely interesting problem, as no such quark-gluon bound states at low temperatures have been experienced so far either empirically or theoretically.

Production of KPM from supernova explosions
Finally, we consider possible population of KPM in connection with neutron stars [n] m , which is somewhat similar to kaon condensation as discussed by Kaplan and Nelson [30] and Brown et. al. [31]. The neutron stars (NS) once produced may proceed to KPM in gentle multiple decay processes that occur slowly: [n] NS → [K − p] KP M + (ν +ν) ′ s. (.9) On the other hand, precursors of supernova explosion may undergo explosive processes toward not only to neutron-star (NS) formation but also to KPM formation: This latter process has never been considered nor observed. It may be an interesting process, as we may anticipate some astronomical observational signals.

Concluding remarks
Very recently, new experiments have been carried out to search for hadron production in extremely high-energy Pb + Pb collisions at LHC-ALICE [32,33], where the most important precursor K − K − pp toward KPM (see Fig. 2) can be investigated. Such a precursor can also be produced in the reactions (p + p → Λ * + Λ * + K + + K + ) at lab energies of around 7 GeV [28,29]. One can also study the (Λ * ) m multiplets with moderate multiplicity, m, in heavy-ion reactions, which are expected to exist as metastable fragments with various lifetimes. They might include important QGB fragments.