aa r X i v : . [ nu c l - e x ] F e b EPJ manuscript No. (will be inserted by the editor)
The quest for light multineutron systems
F. Miguel Marqu´es and Jaume Carbonell LPC Caen, Normandie Universit´e, ENSICAEN, Universit´e de Caen, CNRS/IN2P3, 14050 Caen, France Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, FranceReceived: date / Revised version: date
Abstract.
The long history of the research concerning the possible existence of bound or resonant states inlight multineutron systems, essentially n and n, is reviewed. Both the experimental and the theoreticalpoints of view have been considered, with the aim of showing a clear picture of all the different detectionand calculation techniques that have been used, with particular emphasis in the issues that have beenfound. Finally, some aspects of the present and future research in this field are discussed. PACS.
Trineutron, Tetraneutron, Multineutron, Few-Body Systems, Few-Nucleon Problem, ab initio
February 23, 2021
Contents
Understanding the structure of nuclei and how their prop-erties emerge from the underlying forces between nucleonsis a major goal of modern nuclear physics. Light nucleihave long played a fundamental role in this respect, andthose exhibiting very asymmetric neutron-to-proton ratioshave proven to be particularly sensitive to details of thetwo- and few-body forces used in nuclear models. There-fore, the question about the existence of multineutrons, the most extreme combinations one can find, raises manyexperimental and theoretical challenges. The discovery ofsuch neutral systems as bound or resonant states wouldhave far-reaching implications for many facets of nuclearphysics, from the nature of the force itself up to the wayit builds nuclei, as we will see in Sec. 3, and also for themodeling of neutron stars (see for example Ref. [1]).The quest for neutral nuclei may be traced back to theearly 1960s [2]. However, the last two decades have wit-nessed a renewed and enhanced interest in studying lightneutron systems. This is essentially due to two experimen-tal results claiming the possible observation of bound orlow-lying resonant tetraneutron states [3,4], which moti-vated new experiments but mostly new theoretical studies,leading to progress in the computation of the exact solu-tions of the few-nucleon system in the continuum (see forexample Refs. [5,6,7,8,9,10]).The main difficulty in the study of multineutron statesis that there are no particle-stable substructures, unlikestandard charged nuclei. If the dineutron were bound, onecould more comfortably inquire about the existence of n(as n + n) or n (as n+ n), both experimentally and the-oretically. Despite some recent speculations [11] this seemshowever totally excluded, making the progress in this fieldparticularly non trivial. This intrinsic difficulty explainsthe lack of strong experimental evidence concerning theexistence of such states, and leads on the theoretical sideto contradictory results for the very same systems, evenwhen described using the same nn interaction.As the title says, we have restricted this review pa-per to small clusters of neutrons, basically n, n and n,the ones that can be realistically accessed in experimentsand that can be theoretically tackled with exact ab initio methods. We have deliberately omitted all the interestingstudies concerning nuclear matter and neutron droplets,involving a large number of particles, for they require a F. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems very different approach from the experimental as well asthe theoretical points of view. The interested reader canfind helpful reviews in Refs. [12,13,14,15,16].The paper is divided in two main sections, correspond-ing to the review of the experimental (Sec. 2) and theo-retical (Sec. 3) works. On the experimental side, we havetried to class the long series of very different experiments,across a wide range of techniques, into a few categoriesfor clarity, and in doing so we have restricted ourselves toopenly accessible works. In the theoretical part we havefocused our efforts mainly on the results obtained duringthe last twenty years, which also coincide with the mainexperimental signals. We will close the paper with someconclusions and perspectives in the field.
Detecting neutral particles represents an experimental chal-lenge in general. Charged particles crossing a detector ma-terial interact with the atomic electron clouds, leading todetection efficiencies of 100%. Neutral particles, however,must interact with a nucleus, a much less probable pro-cess, leading to typical detection efficiencies (for similarlysized detectors) of few per cent. Moreover, the detectionof x neutrons becomes exponentially harder, since the effi-ciency will decrease roughly as ε xn ≈ ( ε n ) x , as shown inFig. 1. To make things worse, the latter is just an upperlimit due to cross-talk effects [17] (a neutron may inter-act several times), that require the application of rejectionalgorithms with subsequent losses of total efficiency.Even if one obviates the detection problem, the factthat neutrons are difficult to guide and unstable them-selves does not allow us to build a system of several neu-trons out of its components. Therefore, all the experimen-tal approaches must face the challenge of overcoming thosetwo issues: how to build multineutrons and then how todetect them. In reviewing all the existing works since the1960s we have followed the chronology when possible, butat the same time we have tried to organize them in sub-sections according to the probe they used.Paradoxically, even if the aim of these experiments isthe ‘observation’ of systems of several neutrons, almostall of them have in common the absence of any neutrondetection, due to the aforementioned very low ε xn efficien-cies. We have identified basically two main categories ofexperiments according to their principle: a ○ −−−−−−→ b ○ ⇓⇑ c ○ −−−−−−→ ( A n) missing mass a ○ −−−−−−→ b ○ ⇓⇑ ( A n) −−−−−−→ ( X ) two step In both of them one must detect in the final state onlyone charged particle, b ○ , either directly or indirectly.On the left panel, representing the missing-mass cat-egory, the multineutron is sought to be formed in a two-body collision a ○ + c ○ leading to a two-body final state b ○ + A n. Only in that case, the detection of b ○ can sign the
110 1 2 3 4 5 6 x (detected neutrons) ( ε ) x [ % ] Fig. 1.
The neutron efficiency to the power of the number ofdetected neutrons, as a function of the latter. From bottom totop, the lines/symbols correspond to neutron efficiencies ε n of 10, 20, 30, 40 and 50%, respectively. The vertical scale goesfrom 1 h to 60%. In practice, due to cross-talk effects the multi-neutron ( x >
1) efficiency will be even lower, ε xn < ( ε n ) x . population of states in the missing multineutron systemthrough the unique constraints of two-body kinematics.The main advantages (+) and issues ( − ) of the missing-mass technique are:+ It only requires the detection of one charged particle.+ The multineutron mass number A is well defined.+ Both bound and resonant A n states can be probed. − The cross-sections that bring all the protons into b ○ without breaking it are generally (extremely) low. − Any beam or target contaminant different from a ○ or c ○ would lead to a missing partner(s) of b ○ that is nota multineutron.On the right panel above, representing the two-stepcategory, a bound multineutron is supposed to be pro-duced in a previous reaction, and then one seeks to signits subsequent interaction with a nucleus a ○ that inducesa transformation into b ○ . The advantages and issues ofthe two-step category are:+ It only requires the detection of one charged particle. − Only bound A n states can lead to this second reaction. − There is no sensitivity to the multineutron energy. − Only a lower limit of the mass number A can be in-ferred from the transformation into b ○ . − A contaminant different from a ○ could lead to b ○ with-out requiring a multineutron. − The generally uncontrolled previous reaction may pro-duce a huge background of many particles, that couldbe eventually responsible for the production of b ○ .Concerning the mass number of the multineutron can-didate systems, although n is the simplest one, the pair-ing effects observed along the neutron dripline suggest . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 3 that , , n could more likely exhibit bound or resonantstates. However, the availability of beams and targets,and the significance of the cross-sections involved, havemostly limited the search to , n. This is particularly truein the missing-mass technique, since the charged partnersdefine the multineutron mass number and A > n separation energy ( S n ) of bound nuclei,since a higher tetraneutron binding would make them un-bound. For most of the quest this value was S n ( He) =3 . S n ( B) = 1 . α + n would bethe first particle threshold in He. As the breakup of Heis dominated by the He channel [19], the tetraneutron, ifbound, should be so by less than 1 MeV.Before starting the detailed review of the experiments,note that we will only address the results that have beenpublished in accessible international journals. Moreover,we will not cover experiments that tried to probe the ex-istence of multineutrons indirectly through the search ofanalog states in neighboring nuclei, like , H and , He.
One of the cleanest probes is the double charge exchange(DCX) reaction ( π − , π + ), which falls within the missing-mass category. The incoming negative pion becomes pos-itive, changing two protons in the target into neutrons.By measuring the momenta of the pions on a , He tar-get, one can measure the missing mass of the , n system.Obviously, since no other helium isotopes can be used astargets, no other multineutrons are accessible.To our knowledge, this axis started in 1965 when Gilly et al. searched for the He( π − , π + ) n reaction [20]. Theyfound no evidence of the tetraneutron, that would mani-fest itself through peaks in the π + spectrum due to two-body kinematics. The spectrum could be explained by in-troducing a final-state interaction (FSI) between two ofthe neutrons, as can be seen in Fig. 2 (red curve).In 1970 Sperinde et al. used a He target and searchedfor the He( π − , π + ) n reaction [21]. No evidence for abound trineutron was found, but an enhancement at low3 n energies was observed. The same enhancement was ob-served in 1974 [22], but it was finally explained by the FSIbetween two of the neutrons or between all of them in atwo-step DCX [23].In 1976 Bistirlich et al. used a variant of this tech-nique by studying the single charge exchange (SCX) onhydrogen, H( π − , γ )3 n [24]. After the subtraction of thedifferent contaminants present in the tritium cell, no ev-idence for a trineutron was found in the γ spectrum. In T( π - ) [ MeV ] d σ / d Ω d E [ µ b / s r / M e V ] -10 0 10 E(4n) [ MeV ] T( π + ) = 176 MeV [ (2n)+n+n ][ (2n)+(2n) ][ n+n+n+n ] Fig. 2.
Experimental results for the reaction He( π − , π + )4 n at θ = 0 ◦ . The curves represent the phase space for the finalstates n + n + n + n , (2 n ) + n + n , and (2 n ) + (2 n ), normalizedat 232 MeV and including the experimental conditions. Thesmall axis represents the energy of the neutrons around the 4 n threshold. Adapted from Ref. [20]. et al. undertook the same study with bet-ter statistics and resolution [25], and confirmed that notrineutron was needed to describe the observed spectrum,only the 2 n FSI was required.In 1984 Ungar et al. revisited the tetraneutron withthe He( π − , π + ) n reaction [26]. As a test, they also mea-sured the C( π − , π + ) Be reaction, in which the forma-tion of states in Be appeared clearly as a peak in the π + spectrum (Fig. 3, triangles). In the 4 n channel, however,no apparent peak was observed (Fig. 3, circles). A fewevents laid in the expected region of bound n, althoughat a rate similar to the events beyond that region, whichare kinematically forbidden. The latter background wasdue to imperfect rejection of the events in which the π + decayed inside the spectrometer. In search of a possibleresonant behavior, the spectrum below the 4 n thresholdwas found to exhibit a broad enhancement with respect tofour-neutron phase space, but it was consistent with theFSI of two neutron pairs [26].In 1986, Stetz et al. extended the search to the trineu-tron with the reactions , He( π − , π + ) , n [27], at severalenergies and angles, but again no evidence of trineutronor tetraneutron states was found.In 1989 Gorringe et al. repeated Ungar’s experimentat lower energy, to reduce the continuum DCX contribu-tion, but at a higher angle, corresponding to a higher mo-mentum transfer to the neutrons [28]. After backgroundsubtraction, 6 counts remained in the possible bound nwindow, but those were consistent with the estimated con-tinuum contribution. Only an upper limit of the produc-tion cross-section could be deduced.In 1997, a systematic study of the He( π − , π + )3 n re-action by Yuly et al. [29] at 120–240 MeV did not show F. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems P( π + ) [ MeV/c ] d σ / d Ω d P [ nb / s r / ( M e V / c ) ] He( π - , π + ) C( π - , π + ) Be ( ÷ T( π - ) = 165 MeV Fig. 3.
Experimental results for the reactions He( π − , π + )4 n at θ = 0 ◦ (circles) and C( π − , π + ) Be at θ = 8 ◦ (triangles,divided by 2). The curves are fits to guide the eye, with aWoods-Saxon distribution only (red) plus an additional Gaus-sian function (blue). The peak in the C channel correspondsto the formation of the Be ground and two first excited states,and the range in yellow in the He channel to the region ex-pected for a bound tetraneutron. Adapted from Ref. [26]. evidence for a trineutron, and indicated that the DCX pro-cess proceeds as a sequential SCX one. In 1999, Gr¨ater et al. studied the same reaction at lower energies, 65–120 MeV, and found no evidence either [30].For the sake of completeness, let us mention two spe-cial, non-conventional uses of the negative-pion probe. Ear-lier, in 1979, Chultem et al. had proposed an original useof a π − beam in a two-step process [31]. They hoped toproduce a bound tetraneutron in the Pb( π − , π + ) n re-action, by DCX on an α cluster inside lead, and then mea-sure the tetraneutron absorption by another lead nucleus,leading to Pb. However, they found no α particles fromits decay chain into Bi and then
Po. Finally, in 1991Gornov et al. studied the 3 n missing mass in the reactions Be( π − , t He) and Be( π − , d He) [32]. The missing-massspectrum was tentatively described using a very broadtrineutron resonance at 3 MeV, but the very limited res-olution did not allow to draw firm conclusions.In summary, after more than 30 years of DCX exper-iments with pion beams, only upper limits following thenon-observation of multineutrons have been set, and thetechnique is not being presently used. However, the DCXtechnique has been recently revisited by Kisamori et al. using exotic nuclei [4], as we will see in Sec. 2.5.
The DCX pion reactions are a very clean and powerfulprobe, although the cross-sections involved are extremelylow. From the start of the multineutron program a sec-ond avenue was taken, exploiting the two-step principledescribed previously. In the first step, the multineutron issupposed to be produced in a high-flux reaction, not neces-sarily well-characterized, like spallation or induced fission.Assuming that a bound multineutron was thus produced,together with many other particles, we may use it in a sec-ondary two-body reaction that transforms a given samplein a unique way. Therefore, we need to demonstrate thatthe sample was transformed, but above all that it couldnot be transformed in any other alternative way.Already in 1963, Schiffer et al. irradiated samples ofnitrogen and aluminum in a nuclear reactor, searching forthe reactions N( n , n ) N and Al( n , t ) Mg [33]. Ifbound tetraneutrons had been produced in the fission ofthe uranium fuel, they might have observed the β -delayedneutron decay of N and the γ -rays from Mg and itsdaughter Al. However, none of them were significantlyobserved above the background.In 1965 Cierjacks et al. carried out a similar experi-ment, with samples of nitrogen, oxygen, magnesium andother heavier elements surrounding a uranium target bom-barded with deuterons [34]. For the three lighter samples,they searched for bound tetraneutrons emitted in uraniumfission through the reactions N( n , n ) N, O( n , t ) Nand Mg( n , n ) Mg. However, they were not able toobserve the corresponding β -delayed neutrons or γ -raysabove the background.In 1968 Fujikawa et al. searched for bound trineutronsin the reaction n + Li, that would transform a bariumsample through the secondary reaction
Ba( n , n ) Ba[35]. Barium was in the form of a high-purity BaCO pow-der, and the potential production of Ba would havebeen identified through the γ -rays of its daughter, La.After chemical separation of lanthanum from the powder,no γ -rays were observed, and an upper limit of the cross-section was provided.In 1977 the multineutron search seemed to come toa happy end at CERN. D´etraz obtained the first posi-tive result after exposing a natural zinc sample to a tung-sten block irradiated with a proton beam of 24 GeV [36](Fig. 4). Block and sample were separated by an aluminumscreen, supposed to be thick enough to stop any chargedparticle that could have been produced, and the reactionthat was sought was , , , , Zn( A n , xn ) Zn, i.e. thetransformation of a natural zinc isotope into Zn throughthe interaction with a neutral cluster. The relatively longhalf-life of Zn (46.5 h) and its daughter Ga (14.1 h),and the several γ -rays emitted by the latter, made it pos-sible to remove the sample from the high-activity area andperform a clean measurement of those γ -rays.After several-day exposure, the 100 g of natural zincwere sent by air freight to Orsay, where a Ge(Li) detec-tor revealed the unambiguous production of Zn throughfive prominent γ -rays of its daughter Ga. This observa-tion required up to four chemical separations of gallium, . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 5 ➤ ➤ ➤ ➤ ➤ ➤ ➤ ➤ CERN Orsay p (24 GeV) Al screen W t a r g e t (64 − Zn sample Zn ?( → Ga * → γ )Ge(Li) ? Fig. 4.
Schematic view of D´etraz’s experiment [36]. In the col-lision of protons and tungsten a neutral cluster was supposedto be produced, go through a screen, and induce ( A n , xn ) re-actions on natural zinc. The new Zn isotope was detectedthrough its decay into Ga ∗ with a germanium detector. due to the high activity induced in the sample by otherbeam-induced reactions. The production of Zn couldhave been the result of any (
A >
1) bound multineutron,but since the previous searches for A = 2–4 had failed,D´etraz concluded that most likely bound hexa- or octa-neutrons ( A = 6 ,
8) had been produced [36].Only a few months later, Turkevich et al. tried to con-firm D´etraz’s exciting results using a similar principle [37].As the first step, they irradiated uranium with 700 MeVprotons, and exposed a lead sample behind an aluminumscreen, again supposed to stop any charged particle. Forthe second step, they used the one that was described inthe previous section for Chultem [31], the search for the
Pb( A n , xn ) Pb reaction through the α particles fromits decay into Bi and
Po. No trace of “polyneutrons”was found, shedding doubt on D´etraz’s results.In 1980, de Boer et al. followed suit. Since Turkevichconcluded that their results would have not been sensitiveto multineutrons with
A < n, and not , n as he had suggested [36]. Therefore, de Boer irradi-ated a block of tellurium with He ions and searched forthe interaction of bound tetraneutrons in the same blockthrough the
Te( n , n ) Te reaction [38]. No γ -raysfrom the decay of Te were observed, and de Boer con-cluded, in agreement with D´etraz, that the latter had infact underestimated the transmission of tritons throughhis aluminum shield and that the reaction observed hadbeen (nat)
Zn( t, p ) Zn, without any multineutron [38].This tree-year period of excitement ended up with somefrustration, but the multineutron quest went on. However,following these results, the two-step or activation probewas abandoned for more than thirty years in favor of thecleaner missing-mass experiments.Anyhow, in 2012 Novatsky et al. revisited the tech-nique by inducing uranium fission with 62 MeV α parti-cles, and then searched for the interaction of bound multi-neutrons within strontium and aluminum samples shieldedby thin Kapton and beryllium layers [39,40]. The authorsclaimed they had observed the reaction Sr( A n , xn ) Sr, through γ -rays from the daughter Y [39], and the reac-tion Al( A n , p xn ) Mg, through γ -rays from the decayinto Al and Si [40]. They concluded that, due to thelack of evidence for a bound tetraneutron, their resultsshould “certainly” correspond to heavier clusters ( A > A third research axis was also opened in the early years.Like the pion DCX, it relied on the cleaner missing-massprinciple. However, instead of changing two protons intotwo neutrons, which limited the search to , He targets,the multineutron system was probed in a transfer reactionbetween two stable nuclei, increasing the possible combi-nations. But as in the DCX reaction, the cross-sectionswere expected to be very low, since one needed to: trans-fer all the protons away from one nucleus; sometimes bringsome neutrons back in the opposite sense; and in any eventlead exclusively to a two-body final state.In 1965 Ajdaˇci´c et al. started to use this probe witha simple ( n, p ) transfer reaction on the triton [41]. Witha 14 MeV neutron beam they detected protons from the H( n, p )3 n reaction. Some events were observed at a miss-ing mass of about 1 MeV below the 3 n threshold, the firstcandidates of a (quite) bound trineutron. However, know-ing that the very first experiments had failed to find abound tetraneutron, more likely to exist due to pairing,they concluded that their result was “highly improbable”.One year later, Thornton et al. repeated the same exper-iment with 21 MeV neutrons and better resolution [42],and found no evidence for a bound trineutron.In 1968 Ohlsen et al. used a triton beam and searchedalready for a more complex transfer reaction, H( t, He)3 n [43]. The missing mass reconstructed from He lead to adeviation from four-body phase space, only at forward an-gles, that could be consistent with a low-energy trineutronresonance. They were not able to exclude, however, an ef-fect from the reaction mechanism itself.In 1974 Cerny et al. started to use heavier nuclei andsearched for the tri- and tetraneutron in the reactions Li( Li , C)3 n and Li( Li , C)4 n [44]. Concerning thetrineutron, the intense C channel led to a very highstatistics spectrum. Unlike the Li( Li , B) H channel,that exhibited several structures, the 3 n missing-mass spec-trum could be well described by four-body phase space,plus some small peaks from known target contaminants(that lead to C partners different from 3 n ). F. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems E( C) [ MeV ] C o un t s / k e V * * * Li gs13 B gs E( Li) = 79.6 MeV
Fig. 5.
Energy spectrum of C from the Li( Li , C)4 n reac-tion at θ = 7 . ◦ . Known contaminant reactions are indicatedeither explicitly or with an asterisk. The red curve correspondsto five-body phase space, and the range in yellow to the regionexpected for a bound tetraneutron. Adapted from Ref. [44]. In the tetraneutron channel, however, the low C pro-duction led to a poor separation from the tail of the muchstronger C distribution. The resulting 4 n missing-massspectrum could be described by five-body phase space plusthe known contaminants, as can be seen in Fig. 5. Al-though some events are visible in the possible region forbound n, the signal was not significant with respect tothe background level. They concluded that the purity wasstill much worse than in the DCX experiments, but thatby using different beam-target combinations the way hadbeen opened to heavier multineutrons.In 1988 Belozyorov et al. improved on the main issuesof Cerny’s work, the target purity and fragment identifica-tion, and searched for the trineutron in the Li( B , O)3 n reaction and for the tetraneutron in the Li( B , O)4 n , Li( Be , N)4 n and Be( Be , O)4 n reactions [45]. In allthe reactions on Li, the missing-mass spectra above the3 n and 4 n thresholds were well described by the corre-sponding four- and five-body phase space, showing no ev-idence for multineutron resonances. Although the two re-actions leading to O showed a few events below the 4 n threshold, they were consistent with the background dueto pulse pileup or to beryllium target impurities [45].In 1995 Bohlen et al. used a C beam in order toprobe very neutron-rich missing masses, among them thetrineutron in the reaction H( C , N)3 n on a CD tar-get [46]. The missing mass was fully described below thethreshold by the carbon contribution and above it by thedecay of a broad N resonance. Finally, in 2005 Alek-sandrov et al. repeated Cerny’s experiment, with similarbeam energy and target, and obtained the same negativeresults for both the tri- and tetraneutron [47].This technique appeared as a good compromise be-tween pion DCX and activation. It could access heavymultineutrons from a variety of beam-target combinations, and the potential signals were supposed to be unambigu-ous. However, the absence in practice of clear signals ledtowards a need for higher purities, and the technique hasbeen put aside for the last fifteen years.
The experiments performed in the XX century used mainlystable beams and targets. The beams could thus be veryintense, but building a neutral system from balanced com-binations of protons and neutrons required reactions withvery low cross-sections. Moreover, the potential multineu-tron signal often shared parts of the spectra with back-ground from contaminant species, and due to the lowcross-sections used the background contributions becametoo important for a signal to be clearly established.In 2002 Marqu´es et al. proposed at the GANIL facilitya new technique that could solve those issues [3]. With theadvent of radioactive secondary beams, the possible pre-formation of multineutrons inside very neutron-rich nucleiwas considered, similar to the preformation of α particlesin the process of α decay. Within this scenario, the untilthen complex formation step of multineutrons could bereduced to the breakup of one of those nuclei, with an in-crease in cross-section of several orders of magnitude (mb,compared to the nb or pb of the previous probes) due tothe weak binding of these clusters.The radioactive beam was Be, in which the Be+4 n threshold is at only 5 MeV. Following the breakup on acarbon target at 35 MeV/N, the detection of the Befragment provided a clean signature of the channel. Forthe detection of a potentially liberated tetraneutron clus-ter, the principle was similar to that used by Chadwick inthe discovery of the neutron [48]: deduce the mass of theneutral particle from the recoil induced by elastic scatter-ing on charged particles. The recoil energy E p of a protonin the organic scintillator detector is related to the en-ergy per nucleon E n of the incoming A n cluster, obtainedfrom its time of flight: ( E p /E n ) A / ( A + 1) . Sincethe dineutron is unbound, and the trineutron should alsobe due to pairing, the measurement of proton recoils overthe incoming neutron energy of 1.5–2.5 could only be at-tributed to a bound tetraneutron.The method was also applied to data from Li and B beams, but only in the case of Be some events wereobserved with characteristics consistent with the produc-tion and detection of a multineutron cluster. The 6 eventsare shown in Fig. 6 in yellow, with proton recoils 1.4–2.2times higher than those expected for individual neutrons,and appear all in coincidence with the detection of a Befragment (note the absence of events in the most abundant Be channel). Special care was taken to estimate the ef-fects of pileup, i.e. the detection of several neutrons inthe same detector, and it was found that it could at mostaccount for some 10% of the observed signal. The conclu-sion for the most probable scenario was the formation ofa bound tetraneutron in coincidence with Be [3].As we will see in Sec. 3, this result triggered severaltheoretical calculations that could not explain the possible . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 7
HHeLi Be E p / E n P I D [ a r b . un i t s ] (E p /E n ) max = f(A) Fig. 6.
Scatter plot of the particle identification parameterPID vs the proton recoil in the neutron detector (normalizedto the neutron energy) for the reaction ( Be , X + n ). Thedotted lines show the region centered on the Be peak andwith E p /E n > .
4, and the 6 events in yellow are candidatesto the formation of a bound tetraneutron. The scale on theupper axis shows the maximum proton recoil as a function ofthe multineutron mass number. Adapted from Ref. [3]. binding of the tetraneutron. Moreover, another work ques-tioned the probe itself [49], arguing that a weakly boundtetraneutron would rather undergo breakup than elasticscattering. Marqu´es et al. addressed both issues [50], find-ing that the signal observed could be generated also by alow-energy tetraneutron resonance ( E . et al. in 1964 as a potential probe of boundmultineutrons, but no abnormal signals were observed atthat time in the bombardment of a Ca target with Cand He beams [51]. On the other hand, another experi-ment in 1971 by Koral et al. had already tried to detectbound trineutrons from abnormal induced recoils [52]. Fol-lowing the bombardment of a lithium target with neutronssimilar to the one in Ref. [35], they searched abnormal re-coils of He ions by comparing the time of flight and thepulse height signals in a helium scintillator, the same prin-ciple used at GANIL, but no evidence was found.The decrease in beam intensities at GANIL and theaging of the neutron detector used did not allow an un-ambiguous confirmation of this result. Other experimentsat GANIL using the missing-mass technique were not ableto find positive signals either. After some theoretical worksin the early 2000s, the field became quiet. E [ MeV ] C o un t s / M e V background × Fig. 7.
Missing-mass spectrum of the He( He , Be)4 n reac-tion. The solid (red) curve represents the sum of the directdecay of correlated 2 n pairs plus the estimated background.The dashed (blue) curve represents only the latter, multipliedby a factor of 10 in order to make it visible. The 4 events atthreshold are highlighted in yellow. Adapted from Ref. [4]. In 2016, Kisamori et al. proposed a new probe at RIKEN: He( He , Be) n, a DCX reaction using exotic nuclei [4].Sending a very intense He beam at 186 MeV/N onto aliquid He target, the exit channel was selected throughthe detection in an spectrometer of the two α particlesfrom the decay in flight of Be. The large Q value of the( He , Be) reaction almost compensated the binding en-ergy of the α particle and allowed for the formation ofa 4 n system with small momentum transfer. The authorsexpected in this way to enhance the odds of a weakly inter-acting tetraneutron in the final state, that would remainin the target area (and would not be directly detected).The 4 n missing-mass spectrum (Fig. 7) showed 4 eventsvery close to threshold. The relative energy and angle be-tween the two α particles were consistent with the forma-tion of the Be ground state. The background was esti-mated from the probability of having two He beam par-ticles from the same bunch breaking up, due to the highbeam intensity, and leading to the detection of two in-dependent α particles at small angle that could mimic a Be decay. It was found to be uniformly spread over thewhole range of missing mass (blue curve in Fig. 7), withan integrated value of about 2 events [4].The red curve in Fig. 7 corresponds to a calculationof the direct decay of the 4 n final state within a wavepacket similar to the initial He, including the interac-tion between neutrons and between neutron pairs [4]. Itincludes also the estimated background described above,not visible at the scale of the figure. This curve, withoutthe hypothesis of a tetraneutron resonance state, clearlycannot explain the events observed around the 4 n thresh-old. Note that there was only 1 event in the kinematically F. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems forbidden region (at about −
20 MeV in Fig. 7), an addi-tional indication of the low background.The 4 events in the region 0 < E n < E ( n) = 0 . ± . Γ < . As mentioned when introducing the experimental tech-niques, paradoxically almost none of the many experi-ments already reviewed has tried to detect the neutrons.With the recent improvements in beam intensities andneutron detection efficiencies in the world leading facil-ities, trying to detect the neutrons from the decay of amultineutron system seems a logical next step, which inaddition would give access to eventual neutron correla-tions in their decay: a ○ −−−−−−→ X ⇓⇑ c ○ −−−−−−→ ( A n) −→ A n ○ neutron detection For the sake of completeness, we note that in 2016Bystritsky et al. claimed the first direct detection of multi-neutrons with A = 6 , Hecounters, with a total estimated efficiency of ε n ∼ U sample, and searched for the neutronsthat would stem from the multineutron cluster decay ofuranium. Although they based their claim in the detec-tion of few events with 5 neutron hits, they also con-cluded that the confirmation would require an increase ofthe statistics, and a reduction of the cosmic and naturalbackgrounds, by at least one order of magnitude [53].The Radioactive Isotope Beam Factory (RIBF) of theRIKEN Nishina Center provides nowadays the highest in-tensities of light neutron-rich beams, together with highneutron-detection efficiencies. Since the 2016 result, twoexperiments have been undertaken at RIKEN aiming atthe detection, for the first time, of all the neutrons emittedin a multineutron ( x >
2) decay. They benefited both fromthe combined capabilities of the NEBULA [54] and Neu-LAND demonstrator [55] neutron detector arrays. Withan average neutron efficiency of ε n ∼ ε n ∼ O, one of the grails of nuclear structure physics, using year of publication nu m b e r o f e xp e r i m e n t s G A N I L R I KE NC E RN A n n Fig. 8.
The solid histogram represents the number of experi-ments reviewed that were searching for multineutrons (36 total,mainly for tri- and tetraneutrons) as a function of the year ofpublication of the results. In yellow are those searching specif-ically for the tetraneutron. The stars represent the three posi-tive signals reported, the empty one that was refuted [36] andthe two solid ones that have not been contested yet [3,4]. Thepale Gaussians guide the eye through the recurring pattern. the reaction H( F , O)4 n [56]. The moderate intensityof the very exotic F beam lead to some 100 complete4 n events. Although the analysis of the results is still inprogress, it seems that the decay of the ground state of O would proceed through the narrow O ground state,i.e. it would be a sequential 2 n -2 n decay. As such, no in-formation should be derived concerning multineutrons.The second experiment used a similar setup and tech-nique, the proton removal from a beam and the detectionof a fragment plus four neutrons, but this time in search ofthe H ground-state energy and its potential tetraneutrondecay with the reaction H( He , H)4 n [57]. With respectto the previous experiment, the much higher intensity ofthe He beam will lead to several orders of magnitudeincrease in statistics, providing tens of thousands of 4 n events, while the absence of low-lying , , H resonancesshould allow for the observation of a direct 4 n decay, andthe eventual correlations within. In this section we have reviewed experiments taking placeover a sixty-year period, all searching for evidence of multi-neutron existence. Their chronology (Fig. 8) exhibits sev-eral trends. Even if the techniques have been diverse andtheir sensitivity has increased with time, we can see a re-curring pattern of ‘bunches’ during the first forty years,with experiments accumulating from mid to end of eachdecade. Some experiments at mid-decade triggered others,and then the overall negative results lead to a stop in theprogram, until someone else restarted it a few years later.Towards the end of the century the number of experiments . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 9 in each bunch decreased, showing signs of exhaustion dueto the lack of positive signals.The number of experiments in the present century hasbeen much lower, although two positive signals of a boundor low-lying resonant tetraneutron were obtained. As saidin the introduction those signals have renewed the inter-est in the field, both experimentally and, and as we willsee in the next section, theoretically. Therefore, for thenext extension of Fig. 8 we expect an upcoming significant‘bunch’ of results. Taking into account the increasing ac-curacy and sensitivity of these new experiments, the nextfew years will possibly see the end of this quest whateverthe outcome, at least with respect to , n.Moreover, besides the aforementioned experiments aim-ing at the detection of four neutrons already carried out[56,57], complementary missing-mass experiments are alsobeing programmed at RIKEN, without neutron detectionbut with increased sensitivity. The tetraneutron has al-ready been revisited using the same reaction as in 2016, He( He , Be) n, with several improvements in the exper-imental conditions [58]. It has also been probed in theknockout of an α particle off He at backward angles inquasi-free conditions, H( He , pα ) n [59], avoiding the FSIof the 4 n with the other particles in the reaction. There arealso new plans to probe the trineutron, with the reaction H( t, He) n [60].Among those missing-mass experiments, some are al-ready searching for the next heavier system, the hexaneu-tron. One has already been carried out, knocking out two α particles from Be in the reactions H( , Be , pαα ) , n[61], and a second one is planned in a near future, knockingout an α particle and a proton from Li in the reactionH( Li , ppα ) n [62]. Depending on their results, future ex-periments could be planned in order to study neutron cor-relations in the decay of hexaneutron states.However, the hexaneutron seems to represent a massfrontier difficult to cross in the laboratory. In the nextsection we will see how theoretical calculations have beendealing with the lightest multineutrons, and how theycould help us go beyond A = 6 in order to understandpossible binding energy trends in these systems. The theoretical interest in bound or resonant multineutronstates is as old as nuclear theory, and its development goesin parallel with the technical possibility to obtain accuratequantum mechanical solutions of few interacting particlesusing reliable interactions. This is an essential ingredientwhen dealing with systems very close to, or above, theirdissociation threshold. The progress in this domain hasbeen slow. The first rigorous formulation of the three-bodyproblem in Quantum Mechanics dates from 1960 [63], andthe first realistic solution took still several years [64]. As ithas been the case in the experiments, the main efforts ofthe theory have been concentrated on , n, and to a lesserextent on the next heavier systems, , n.Even if we can trace the origin of both theoretical andexperimental programs back to the 1960s, it seems clear that the theoretical community has significantly increasedits interest in 3 n and 4 n systems following each of the twoexperimental signals described in the previous section. Wewill thus review the many theoretical works from that per-spective. We will describe the activity prior to GANIL re-sults on the breakup of Be into Be+4 n [3], then reviewthe different approaches that were triggered by this result,and finally discuss the many calculations that followed thesecond signal in the DCX reaction He( He , Be)4 n [4].But before moving to the specific works, let us start withsome general considerations about the theoretical treat-ment of the few-neutron problem. The theoretical study of A n systems has faced two hard-wearing difficulties. The first one is finding a reliable methodto solve exactly the A -body problem ( A = 3 , ... ) forloosely-bound states as well as for resonant states, whichcould be embedded in the continuum and very far from thephysical region. By “exactly” we mean a rigorous mathe-matical formulation of the quantum mechanical problemand a uniformly convergent numerical scheme which couldlead to the required accuracy. This implies an ab initio solution, i.e. only in terms of the nn interaction. We an-ticipate that for such diluted systems the three-nucleonforces (3NF) should play a negligible role.Since such systems, if they manifest at all in nature,would be a subtle balance between the attractive nn inter-action and the Pauli ‘repulsion’ and live in the continuum,any uncontrolled approximation, in the antisymmetry orin the boundary conditions, could be fatal and lead to illu-sionary conclusions. This constraint delayed the first seri-ous attempt to study the 3 n system to the end of the 60s,by Mitra et al. [65], and this was still based on an ad-hocmethod they had developed for solving the Schrodingerequation with a rank-1 separable interaction, which fur-thermore turned to be not very realistic.The second difficulty is related to the fact that, un-til the contrary is proven, A n has no bound subsystems.Therefore, the usual scattering theory with two-clusterasymptotic channels cannot be applied to obtain the scat-tering amplitude and eventually the location of the cor-responding poles. The only pertinent quantity, the am-plitude for the 3 → → quite widely adopted definition of resonance as a rapidincrease of a phase-shift crossing π/ A n resonance is determining the location of the rele-vant S -matrix pole, or any equivalent quantity like the A -neutron Green function, in the complex-energy mani-fold (or complex-momentum plane). This allows to definethe intrinsic parameters of the resonance in the usual way: E = E R − i Γ k = + i | k | ) anda virtual ( k = − i | k | ) state appear as singularities inthe imaginary k -axis, in the upper (bound) and lower (vir-tual) half planes. They are mapped onto the two Riemannsheets E I and E II of the E -surface, in the right panel, re-spectively at the points E ∈ E I and E ∈ E II . Note thatthe energies of the virtual states are negative, as are forbound states, despite being in the continuum. The upperhalf k -plane is mapped onto the first (“physical”) Riemannsheet E I , and the lower one onto the second (“unphysi-cal”) Riemann sheet E II , which are glued to each otherby the positive real axis E > k -plane, k = k R − ik I with k I >
0, and are mapped onto E inthe unphysical sheet E II . They appear always by pairs( k , k ′ = − k ∗ ) symmetric with respect to the imaginary k -axis. Note that Re( E ) > k ) < Re( k ). Thesituation is more complicated in presence of open channelswith open thresholds as well as dynamical cuts [66], butwhat is described above is enough for our purposes. Theinterested reader can find a more detailed description inthe standard books of scattering theory [67,68,69,70,71].Whether or not this A n singularity could manifest inany observable, and in which way, must be considered as adifferent (and difficult) problem, especially taking into ac-count that such states can be produced in complex nuclearreactions that could mask, generally in an unknown way,the intrinsic parameters ( E R , Γ ) of an eventual resonance. The research for an eventual bound state is relatively easy.If a direct bound state calculation gives a negative eigen-state of the Hamiltonian, the system is bound. On the con-trary, we conclude that there is no bound state... within Γ k E E E k k E=kE=k k−plane E−surface E I E II γ Γ γ Fig. 9.
Complex momentum plane and energy manifold. the accuracy of the method, which provides then a lowerlimit of the binding energy. The method for computingbound states must be accurate enough to deal with looselybound systems, i.e. quite extended objects in configurationspace. We would like to point out from now that no seriouscalculation has ever predicted a 3 n or 4 n bound state.However, it is worth noting that the aforementionedpioneering calculation of Mitra et al. [65] was quite op-timistic. They realized that a pure S-wave nn interac-tion generates a strong repulsion in the 3 n system dueto the Pauli principle, but concluded that “a moderatelyattractive P force ... is enough to produce a bound nstate”, and added that the “range and strength of theforce seem to be in good accord with the P phase-shiftdata of Bryan and Scott”, one of the most realistic po-tentials available at that time. Their conclusion, certainlydue to the simplicity of their dynamical approach and in-teractions, would be denied very soon.In the absence of bound states, the optimal choice is tofind the location of the nearest singularity in the contin-uum, i.e. the parameters ( E R , Γ ) of a virtual or resonantstate with all the caveats mentioned. This is however adifficult task when working in configuration space, mainlydue to the exponentially increasing boundary conditions.The ‘regalian’ way to avoid the latter is the Complex Scal-ing Method (CSM) [72,73,74]. In the two-body case it con-sists of an analytic continuation of the Schr¨odinger equa-tion by means of a complex rotation of the coordinates( r → r e iθ ), which transforms the exponentially increasingboundary conditions of a resonance into square integrableones, provided the “rotation angle” θ is larger than halfthe argument θ r of the complex resonance energy (1):0 < θ − θ r < π θ r = Γ E R Under these conditions, illustrated in Fig. 10, the res-onant state can be computed by means of the standardbound-state problem techniques. This approach, supple-mented with many variants, has been successfully extendedto A = 3 , , . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 11 r Re(E)Im(E) θ r E Fig. 10.
The principle of the Complex Scaling Method. tion angle, in turn limited by the analytical cuts of thetwo-body interaction, and this restricts the practical ap-plication of CSM to relatively narrow states [77].To circumvent this problem, one possibility is to intro-duce an additional binding to the physical system understudy until it gets artificially bound. One can then removeadiabatically the ‘artifact’ and follow the evolution of thesystem in the continuum until it reaches the physical state.A commonly used artifact consists of a “scaling factor” s in front of the nn interaction, which can eventually dependon each partial wave α ≡ { L, S, J } : V αs α ( r ) = s α V αnn ( r ) (2)and increase s until the multineutron system binds. Notethat only moderate s values are required to bind the dineu-tron with all realistic and semi-realistic V nn models, asshown in Fig. 11. For instance, the critical value to get B nn > s = 1 .
080 using the Argonne AV18 poten-tial [80] and s = 1 .
087 using Nijmegen Reid93 [81]. Forthe phenomenological CD-MT13 [82,83], it is only slightlylarger, s = 1 . V of the system a term of the form: V = A X i Fig. 11. Scaling factors (2) needed to bind 2 n in the S statefor some nn interactions. AV18 and Reid93 give s & . The first one is the introduction of spurious processesthat can modify the evolution of the system. For example,if one artificially binds n by introducing a scaling factor s in the nn interaction, the dineutron will be quickly boundby several tens of MeV. For s ≈ 3, a typical value re-quired to bind n, B nn ≈ 100 MeV. Therefore, a multineu-tron state built in such a way will contain one or severalsubsystems with very peculiar properties (e.g. extremelycompact), that will certainly affect its evolution into thecontinuum. What one believes to be a bound n state willbe in fact a system strongly decaying into n + n.If the method used to solve the three-body problemdoes not allow to properly account for open scatteringchannels, like when using basis of square integrable func-tions, this fact will not be detected in the numerical calcu-lations but will influence the properties and the evolutionof a state bound by brute force in an uncontrolled way. Inany event, the evolution of the multineutron system whendecreasing s will be totally dominated by this process andone may attribute to n resonances what corresponds toan unphysical decay n → n + n. The same will happenwhen studying the tetraneutron with the correspondingspurious channels n → n + n and n → n+ n.The appearance of these spurious thresholds can beavoided by adding an A -neutron force involving only thetotal number of neutrons of the studied system. For exam-ple, for n we can introduce a 3 n force that we can choosefor simplicity of hyper-radial form: W n ( ρ ) = − W f ( ρ ) ρ = x + y (3)where f ( ρ ) is an asymptotically decreasing function of ρ (like e − ρ or Yukawa) and ( x, y ) are the intrinsic Jacobicoordinates. This artifact does not modify any subsystemthreshold and the evolution W → V T ( r ) = V e ( r − R ) /a (4) E2nE3n + E + E E3n0E2nE4n + Fig. 12. Possible decay channels of 3 n and 4 n in a trap. a common practice in the research on neutron droplets[14,15,16]. For small multineutrons, the evolution of theirbinding energies as a function of the potential strength V has been used to draw conclusions about eventual 3 n and 4 n resonances. However, this approach presents someinstabilities, as we illustrate for the simplest case of 3 n in the upper panel of Fig. 12. If B n > B n for a givenset ( V , R, a ) of the confining potential (4), the n willdecay into a dimer, staying in the well, plus an outgoingneutron with kinetic energy proportional to B n − B n . Ifthe methods used to compute the bound state account forthe full quantum dynamics of the system, the supposedconfined multineutron state must decay. This fact maystrongly impact the extrapolation pattern. Since the pres-ence of thresholds is associated with algebraic branchingpoints, the open decay channels will generate sharp struc-tures in the extrapolation function and the results of anyextrapolation procedure become unreliable. The bindingenergy of multineutrons in a trap is in fact only fictitious,as are the properties extrapolated from their evolution asa function of the trap parameters. The same will happenfor a n state, with respect to the open thresholds n + nand 2 n + n (Fig. 12, lower panel).The second mechanism that may induce parasitic phe-nomena is related to the evolution from a bound state toa state in the continuum. It is not a mere extrapolationfrom negative to positive values in an energy axis. As al-ready noted, the quantum mechanical energy continuumis not an axis, even not a plane, but a Riemann manifoldwith dynamical cuts and delicate analytic properties.The evolution of a bound state into the continuum canbe properly described by the so-called Analytic Continu-ation on the Coupling Constant Method (ACCCM) [71].It consists in adding to the Hamiltonian H , for which weaim to determine the parameters of a resonance, a per-turbation λW . We can then make use of the analytic de-pendence of the spectrum of the ‘perturbed’ Hamiltonian H λ = H + λW on the coupling constant λ . If k is thewave number of a bound state with negative energy E of H λ ( E = ~ k / µ ), its dependence on λ is given by the λλ λ k k L R Re(k) Im(k) kk 22 3 c Bound states λ>λ c λ k=0 ResonanceAntiresonance λ<λ c k Fig. 13. Schematic view of the ACCC method. An ensembleof bound state positions ( λ n , k n ) is computed for λ > λ c , acritical branching point where k = 0, and the trajectory of the S -matrix resonance pole for λ < λ c is determined. Pad´e-like analytic expansion: k ( λ ) = i P N ( z ) Q M ( z ) = i c + c z + c z + . . . c N z N d z + d z + . . . d M z M (5)where P N and Q M are polynomials on the complex vari-able z = √ λ − λ c . The critical value λ c corresponds to azero-energy bound state that moves into the continuumand to an algebraic branching point of the complex func-tion (5).ACCCM is schematically illustrated in Fig. 13 and pro-ceeds as follows. In a first step, a series of bound statevalues ( λ n , k n ) are computed, with λ > λ c and k n purelyimaginary. They are inserted in the formal expansion (5)to determine the N + M unknown coefficients ( c i , d i ),which are real. The k ( λ ) dependence being fixed, one candecrease λ → H in the complex plane. Note that for λ < λ c , z is a com-plex number, as will be k and the corresponding E . Thismethod has been successfully used to compute the posi-tion of the nearest 3 n and 4 n resonances [84,85,86], aswell as of the , H isotopes [87]. It requires an accuratedetermination of the bound state ( λ n , k n ) parameters anda careful stability study as a function of the degrees N and M of the Pad´e expansion (5).After this overview of the general issues related to thecalculation of few-neutron systems, in the following we willreview the most relevant theoretical works that have beenpublished prior, in between and after the two experimentalsignals observed at GANIL in 2002 and RIKEN in 2016. Before 2002 we find in the literature several works devotedto 3 n and 4 n systems which, although far from being unin-teresting, can be considered as preliminary or incompletefrom the physical point of view, either due to the quantummechanical treatment of the three-body problem and/or . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 13 to a very simplistic nn interaction. As we noted in theintroduction, the first result to our knowledge is from Mi-tra et al. back to 1966 [65], using a technique for solvingthe three-body Schr¨odinger equation with rank-1 separa-ble S- and P-wave nn interactions. Their optimistic con-clusion about a bound n was immediately contested byOkamoto et al. [88] and Barbi [89], showing with varia-tional approaches that based on their knowledge of theNN interaction the trineutron was unbound by at least10 MeV, and that the depth of the required nn potentialwell to bound it should be unphysically large.The first ‘exact’ 3 n calculation solving the Faddeevequations in momentum space, although using a rank-1separable Yamaguchi force, is due to Gl¨ockle in 1978 [90].It is also the first time that the trajectory of the complex-energy n resonance as a function of a scaling factor s waspresented, together with a careful discussion of the ana-lytic continuation properties of the scattering amplitude.The critical value for binding 3 n was found to be s = 4 . V nn , the critical S-wave scaling fac-tor to bind 3 n was slightly lowered to s = 3 . 7, but stillremained unacceptably large. They also determined thatwhen using a realistic nn interaction the most favorablecandidate was J π = 3 / − , dominated by a PF force butwith a scaling factor of s ≈ 4. The other candidates were J π = 1 / − , / + , / − , by order and in contradictionwith Mitra [65], who had found the positive-parity statethe most favorable one. Despite all these refinements theyreached, however, the very same conclusion: no chancefor a 3 n bound state and a “low energy resonance is ex-tremely unlikely”. They ended with a non-prophetical re-mark: “The [required modifications in V nn ] are not sosmall that one likes to push forward a very expensive ex-periment to measure a three-neutron resonance”. Wouldit be enough to discourage any further experimental ortheoretical research? Certainly not!We are aware of other works that appeared during thefollowing decade [92,93,94,95,96], but we are not going tocomment on them with further detail for they representstill a preliminary stage, due to the interactions and/orthe techniques used to solve the three-body problem.At the end of this period, two works devoted to 3 n and 4 n resonances were published, which deserve specialattention for they were also based on exact solutions ofthe three- and four-body problem.In 1997 Sofianos et al. [97], within the framework ofthe hyper-spherical formalism and CSM, computed theposition of the 3 n and 4 n resonances in the complex-momentum plane, identified with the zeros of the Jostfunction or the S -matrix poles. The nn interaction was -40-30-20-100 -10 -5 0 5 Re(E) [ MeV ] I m ( E ) [ M e V ] n(3/2 - ) Fig. 14. S -matrix pole trajectory of 3 n in J = 3 / − with arank-2 separable potential. Adapted from Ref. [102]. limited to two different central S-wave potentials, one purelyattractive Yukawa and the semi-realistic MT13 [82]. Withthe MT13 potential, the lowest-lying 3 n resonance was E = − . − . i MeV with J π = 1 / − (degeneratedwith 3 / − within the considered potentials), and it waslocated in the third quadrant of the second Riemann sheet(Fig. 9) . The search for a 4 n resonance with the hyper-spherical method gave for the nearest state J π = 0 + and E = − . − . i MeV, i.e. also located in the thirdquadrant of the second Riemann sheet.Although limited by using the “minimal approxima-tion” in the hyper-spherical expansion and a semi-realisticNN potential, the conclusion from this work was clear.First, any 3 n and 4 n possible bound state is very far, sinceone requires an unphysically large enhancement factor inthe NN interaction (typically s ≈ n and 4 n with the MT13 potential), well beyond anypossible uncertainties of the models used. Second, the res-onant states, although well identified, are located in theunphysical region with no possibility to manifest in anscattering experiment.In 1999 Witala and Gl¨ockle [98,99] published a solu-tion of the Faddeev equations in configuration space forthe 3 n system. The nn interaction was taken from thephenomenological GPdT model [100] and from the real-istic Reid93 Nijmegen potential [81], which includes the S , P and PF channels, all attractive. They intro-duced a different scaling factor s α in each partial waveof the nn potential, except in S to keep the dineutronunbound. They first forced the 3 n system into a boundstate and decreased s α progressively following the res-onant trajectory in the complex-momentum plane. The J π = 1 / − , / − , / + resonance positions were obtainedby using the so-called Smooth Exterior Complex ScalingMethod [101].Due to numerical instabilities they could not reach thephysical case with all s = 1. Their conclusion was however Note that there is a misprint in Table 1 of Ref. [97], thereal part of the resonance energy appears with opposite sign.4 F. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems similar to the previous works: the 3 n resonance energieswill have such a large imaginary part that one should notexpect to see any trace of them in experiments. In 2002, few months after GANIL publication of the nsignal [3], there was another paper from Gl¨ockle et al. concerning the 3 n system [102]. They used rank-2 separa-ble nn forces ( S , P , P , D ) and solved the Faddeevequations, which were analytically continued into the un-physical energy sheet below the positive real-energy axis.The trajectories of the 3 n S -matrix poles in the J = 1 / ± and 3 / ± states were traced as a function of the scalingfactors of the nn forces (Fig. 14), but only in the P , P and D channels in order to keep 2 n unbound. The finalpositions of the S -matrix poles, once the artificial factorsremoved, were found to be in all cases far from the pos-itive real-energy axis, which provided a strong indicationfor the nonexistence of nearby 3 n resonances.This was the last of a series of papers which were pio-neers on this topic [90,91,98]. The overall conclusion wasthat 3 n resonances could not exist near the physical re-gion. The clarity of these works and the convincing argu-ments put forward should have been sufficient to discour-age once and for all any claim for a 3 n resonance. As wewill see, this is unfortunately far from being the case.Immediately after GANIL publication of the eventualobservation of a tetraneutron, three consecutive paperswere published [103,104,105] as a reaction to this surpris-ing result. They agreed in the impossibility of a boundtetraneutron, based on the fact that the modifications ofthe NN (and eventually NNN) interaction required to ob-tain a n bound state were so large that the description ofall the surrounding nuclear chart ( He, H, He...) wouldbe destroyed, and not by a small quantity but by severaltens of MeV. If the conclusion was unanimous, the meth-ods used were however quite different.In 2003 Timofeyuk [103] used Hyperespherical Har-monics (HH) formalism and a quite simplistic NN inter-action (Reichstein and Tang [106]) to compute the lowestdiagonalized hyper-radial potential V KK ′ ( ρ ) in the modelspace K max = 16, and all of them where found to bemonotonously repulsive for the 2 n , 4 n , 6 n and 8 n systems.The possibility of a narrow resonance was also excludedin any of these systems. In a previous preprint [107] thesame author had proposed alternative transfer reactions toinvestigate multineutron states and evaluated some crosssection using the DWBA approximation.Later in 2003 Bertulani et al. [104] considered a to-tally different approach. They computed the interactionbetween two 2 n dimers artificially bound in a simplisticNN model (Volkov [108]), hoping to find a possible at-traction among them, as one could expect between twobosonic-like ( S ) systems. Whatever they tried, they sys-tematically obtained a dimer-dimer repulsion. They con-cluded about the impossibility to explain any 4 n boundor resonant state within this model, “although more com-plex variational approaches still can be explored”, as well -12-10-8-6-4-2024 -5 -4 -3 -2 -1 0 V [ MeV ] 〈 H 〉 [ M e V ] AV18 + IL2 + external well R = R = R = Fig. 15. Energies of 4 n in a Woods-Saxon trap (4) as a func-tion of the potential strength V . The results obtained withseveral values of R are used to extrapolate linearly into thecontinuum. Adapted from Ref. [105]. as any possible T = 2 4 n forces. Despite the simplicityof the model and the crude approximations used in ob-taining the dimer-dimer potential, their conclusion turnsout to be quite general as it will be discussed later in theframework of EFT (Effective Field Theory) for fermionsin the unitary limit.A few months earlier Pieper [105] had applied GreenFunction Monte-Carlo (GFMC) methods to study the pos-sible existence of a n bound state. He used the NN Ar-gonne V18 interaction supplemented with the Illinois IL23NF [109], which had been very successful in reproduc-ing the binding energies of A = 2–10 nuclei. The conclu-sion was the same than in previous works: any attempt toforce a n bound state would have “devastating effects” inthe description of the nuclear chart. However, Pieper sur-prisingly claimed a possible n resonance at about only2 MeV. These two conclusions seemed somehow contra-dictory: the 4 n Hamiltonian could hardly accommodateat the same time the absolute impossibility of a boundstate and a near-threshold resonance.The possible resonant state was determined by com-puting the energy of n bound states in a Woods-Saxontrap (4) with fixed a [105]. Using different values of R ,the potential strength V was progressively decreased un-til the 4 n became unbound, and the last energy valueswere extrapolated towards the absence of trap at V = 0(Fig. 15). All radii led to similar extrapolated values of E ∼ et al. published a work devotedto 4 n and H in the framework of the HH method [110]. . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 15 -10-8-6-4-2 2 4 6 E Re ( MeV ) PadØ (4,4) PadØ (3,3) PadØ (2,2) E I m g ( M e V ) PadØ (1,1) J =0 - Fig. 16. Selected resonance trajectories for the 3 n (upper)and 4 n (lower) states as a function of the strength W of theartifact. Results were obtained by combining ACCC and CS(solid points) methods. Adapted from Refs. [84,85]. An interesting critical review of 4 n results can be foundin Sec. 4 of that paper. They explained why no indicationof 4 n FSI was found in the pion DCX reactions, and con-cluded that any resonant state, if existing at all, must bevery broad. Furthermore, they computed with a reactionmodel how the position of the broad 4 n peak can be shifteddepending on the reaction used to observe it. They pointedout, with illustrative examples from standard QuantumMechanics textbooks, that sharp resonance-like structuresin the continuum can be formed even if only repulsive po-tentials are present in the system.In 2005 Lazauskas et al. , motivated by the paradoxi-cal results of Pieper, undertook a series of works to locatethe nearest 3 n [84] and 4 n [85] resonances in the complex-energy plane using a realistic nn Reid93 interaction withCSM and ACCCM, adapted to properly deal with thecontinuum. The technique used for solving the three- andfour-body problem was the Faddeev-Yakubovsky (FY) equa-tions in configuration space, which were already tested insimilar systems like He, p + H, n + He, H+ H and n + H[111]. The only approximations made were the number ofgrid points and the partial-wave expansion in both in-teractions, and the FY amplitudes which were carefullychecked. Several J π states were examined, being the near-est ones 3 / − , / − , / + for 3 n and 0 + for 4 n . The trajectories in the complex plane were plotted asa function of the strength of an artificial three- and four-body force of the type (3), starting from a bound state( W = W c ) and moving to the continuum by decreasingthe strength until the physical value at W = 0. All ofthem were located very far from the physical region wellbefore W was removed, lying in the third quadrant of thesecond sheet, i.e. Im(E) < < 0, often knownas “sub-threshold resonances” (Fig. 9). We have displayedin the upper panel of Fig. 16 the J π =3 / − , 1 / − , 1 / + trajectories of 3 n obtained with the CSM and ACCCM.The lower panel shows the 0 − trajectory of 4n using AC-CCM, where the convergence of the method as a functionof the degree ( N, M ) in (5) can be observed.The conclusion was that there could be no observable3 n or 4 n resonance, a fortiori bound state, with realisticnuclear forces, in close agreement with the previous 3 n work [102] despite the different interaction and methodsused. It was pointed out that Pieper’s conclusions werecertainly due to the extrapolation procedure used.For the sake of completeness, we note that in 2008Anagnostatos [112] claimed that n and n could be particle-stable. Within the framework of the so-called IsomorphicShell Model, mixing semi-classical and classical mechan-ics, he concluded that, even if the results depended on thestrength of V nn , at least low-lying resonant states could bereasonably expected, with n being more favored than n.These conclusions remain however very doubtful, for thevery peculiar approach used to solve the few-body quan-tum mechanical problem in a classical way. After RIKEN publication of a new tetraneutron signal [4],the theoretical interest on this field was renewed and evenenhanced. We remind that this new result was in fact alsocompatible with a bound state, while GANIL result wasfound to be also compatible with a low-lying resonance[50], and they were therefore in agreement.In the same 2016, Hiyama et al. [86] published a jointeffort between the authors of Refs. [84,85] and theoristsfrom RIKEN, working independently and using differentapproaches for solving the few-nucleon problem. The workfrom 2005 [85] let no hope for the RIKEN signal to be at-tributable to a 4 n resonance, and it was natural to clarifynot only the tension with the experimental results but alsowith Pieper’s conclusion.The philosophy, as well as the computational methods,were different. From the physical point of view, the strat-egy was to keep unchanged realistic V nn forces and the wellcontrolled V nnn forces in the T = 1 / V nnn in the T = 3 / T = 3 / T = 1 / 2. The advan-tage of this approach was to preserve the good description t h r e s ho l d –36 –32 –28 –24–20–16 Re(E) (MeV) I m ( E ) ( M e V ) –3–5 a) resonance n J =0 + π Fig. 17. n resonance trajectory for the J π = 0 + state. Circlescorrespond to AV8 ′ potential and triangles to INOY04 [122].The 3NF strength W (3 / 2) was changed from − 37 to − 16 MeVin steps of 1 MeV for AV8 ′ , and from − 36 to − 24 MeV insteps of 2 MeV for INOY04. The RIKEN resonance region isindicated by the arrow at the top. by the modern nuclear Hamiltonians of all the systems not(or little) sensitive to this 3NF component.From the technical point of view, the 4 n solutions inconfiguration space were obtained in two different ways:by using the variational Gaussian Expansion Method [113,114,115,116,117,118] and by solving FY equation in con-figuration space [84,85]. The complex energy solutionswhere computed using both the CS and ACCC methods.The 3NF, for both T channels, were a sum of two hyper-radial Gaussians (one attractive and one repulsive): V Nijk = / X T =1 / X n =1 W n ( T ) e − ( r ij + r jk + r ki ) /b n P ijk ( T ) (6)where P ijk ( T ) is a projection operator for the total three-nucleon T state. The parameters ( W n , b n ) for the T = 1 / He( e, e ′ ) He ∗ elec-tromagnetic transition form factors [119], in conjunctionwith the AV8 ′ version of the Argonne NN potential [120].For the T = 3 / T = 1 / W (3 / W (3 / 2) at which 4 n becomesbound was determined for J π = 0 + , 2 + , 1 + , 0 − , 1 − ... (byorder of appearance). By decreasing this value, all of themmoved into the resonance region, i.e. into the second Rie-mann sheet of the energy surface with E = E R − iE I . Itwas found however that the strength required to repro-duce RIKEN results for the most favorable case (0 + ) was W (3 / ∈ [ − , − 30] MeV, which is totally unphysical.To have an idea, we can compare it with the correspond-ing W (1 / ≈ T = 3 / W (1 / 2) is responsible for about 5 MeVof extra binding in He, the additional energy required forthe 4 n system to become a resonant state is of the orderof 100 MeV. Furthermore, it was found that starting from W (3 / ≈ − 18 MeV the well-known unbound states Li, He ∗ and H become bound.The trajectory of the n(0 + ) resonant state in thecomplex-energy plane as a function of W (3 / 2) is shownin Fig. 17. Already at half the strength required to re-produce RIKEN result, the real part of the resonance is E R ≈ Γ ≈ 10 MeV. But even thesevalues are unphysically large, and they completely missthe description of the well-known n + H cross-section [86].The 3 n was also studied in detail, and the conclusions werethat the corresponding resonant states were located evenfurther from the physical region than the 4 n ones.In 2017 the same authors completed the previous work[123,124] by computing in their numerical simulations thestrength function used by RIKEN experimental group, fol-lowing their suggestion. While the works of 2005 [84,85]had located the pole position of the nearest n and nresonances generated by realistic interactions, the ques-tion asked in 2017 was whether or not one could accountfor a 4 n near-threshold resonance by modifying the mostunknown part of the nuclear Hamiltonian: the 3NF for T = 3 / 2. Despite the different motivations and methods,all the conclusions were in line with the pre-GANIL eraworks: any near-threshold observable resonance in the n and n system (and a fortiori a bound state) must bedefinitely excluded. Concerning the RIKEN result, it waspointed out that there exist quantum mechanical mecha-nisms, other than S -matrix poles, that could generate alow-energy enhancement in a scattering cross-section.All in all, it seemed clear at this stage that Pieper’sprediction of a few MeV tetraneutron resonance should beconsidered as an isolated case in the theoretical landscape,maybe a consequence of the trap artifact or of his cavalierextrapolation to the continuum, but in any event devoidof any physical ground. However, if this was clear for somepeople, it was not clear for everybody.Indeed, in the same 2016 Shirokov et al. [125] founda n resonance with parameters E R = 0 . 84 MeV and Γ = 1 . 38 MeV, in perfect agreement with RIKEN results.They used their non-local NN interaction JISP16 [126]and attempted several methods to describe the n statepredicted by Pieper [105]. The JISP16 V NN is expandedin terms of HO radial basis R nl as: V LL ′ SJ ( r, r ′ ) = n ( L ) X n =0 n ′ ( L ) X n ′ =0 R nL ( r ) C LL ′ SJnn ′ R nL ′ ( r ′ )with n ( L ) = n ′ ( L ) = 4. It is thus a rank-4 separableinteraction, whose C nn ′ coefficients and b parameter ofthe HO were adjusted to successfully reproduce the low-energy properties of NN and some light nuclei. It can beconsidered as a high-precision potential. . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 17 First they tried to solve the 4 n problem with the No-Core Shell Model (NCSM) by scaling the interaction andtracking the lowest state as a function of the scaling factor s . They performed extrapolations to the unbound regimesimilarly to Pieper [105] and found results in quantita-tive agreement, i.e. a resonance at E R ∼ E R ∼ . Γ ∼ . n in the continuum. In particular, in order tocalculate the 4 → S -matrix resonant parameters, theyused a Single-State Harmonic Oscillator Representationof Scattering Equations (SS-HORSE), an extension of theNCSM Hamiltonian in an HO basis by an infinite kineticenergy matrix. For this extension they used the so-called“democratic decay approximation”, which describes the4 n continuum in terms of an hyper-spherical harmonics(HH) basis, limited in practice to its “minimal approxi-mation”, i.e. with hyper-spherical momentum K = 2.The SS-HORSE approach provides the 4 → S ( E ), and the corresponding phase shifts δ ( E ), at the positive eigen-energies of the NCSM Hamil-tonian, from which they extrapolated the position of the S -matrix poles following some analytic expressions. Theseresults depend critically in the parametrization of S ( E )in terms of four adjusted parameters. The values so ob-tained for the resonance parameters of the S -matrix polewere considered “surprisingly small”. Their final result,almost exactly the same than RIKEN one, was obtainedby further adding a false pole plus an additional fittingparameter, and was presented as a “prediction”.The findings of Shirokov, in line with Pieper’s, were instrong contradiction with all the previous ones obtainedwith direct FY and Gaussian methods, and brought evenmore confusion to an already quite confuse situation.In 2017 Gandolfi et al. [127] used Pieper’s methods(Woods-Saxon trap, QMC techniques and extrapolationto the continuum) and reached a similar result: a 4 n near-threshold resonance at E R = 2 . n resonance was “lower in en-ergy than the tetraneutron one”, at E R = 1 . -6-4-20246810 1 1.2 1.4 1.6 1.8 2 s E [ M e V ] DMRG (n.o.)NCGSM-3p3h (n.o.)NCGSM-2p2h (n.o.)NCGSM-2p2h(exp.) n(0 + ) N3LO λ = -1 Fig. 18. Evolution of the energy and width (shaded area) ofthe 4 n system with the scaling of the N3LO interaction from 2to 1. The circles represent the NCGSM results with two neu-trons in the continuum, which is used to generate the NCGSMresults based on natural orbitals with two (triangles) and three(squares) neutrons in the continuum. The DMRG results with-out truncations are represented by stars. The RIKEN experi-mental energy is indicated by the solid circle, and the yellowarea shows the uncertainty. Adapted from Ref. [128]. One interesting conclusion was that, due to the ex-treme diluteness of the system, the roles played by the3NF and the details of the NN one were very small. Asa check of their trap results, they repeated the calcula-tion by introducing an overall scaling factor s on the nn interaction. Surprisingly, they found that a value of only s = 1 . n , in sharp contradictionwith all the previous works where an s value 2–3 timesbigger had been found. Pieper had also concluded thatchanges more dramatic than 30% were needed. Even so,the extrapolated result at the physical value s = 1 was E = 2 . . 0) MeV, consistent with their trap results.Also in 2017, Fossez et al. [128] used two ab initio techniques and several NN interactions (chiral EFT andJISP16) to study the existence of a 4 n resonance, which“would deeply impact our understanding of nuclear mat-ter”. The first technique was the NCGSM (as in Ref. [125]),which is a NCSM extension into the complex-energy planewith couplings to the continuum by means of the Berggrenbasis, that incorporates some selected resonances. The sec-ond technique was the Density Matrix RenormalizationGroup (DMRG), an alternative way to solve the nuclearmany-body problem in the continuum. Both methods weresupplemented by the use of natural orbitals and a newidentification technique for broad resonances.The nn interaction was scaled by a global factor s inorder to bind the 4 n system, which was then decreased un-til it reached the physical state s = 1. The results dependstrongly on the particular technique (NCGSM or DMRG)and the different approximations made (see Fig. 18). For example, at s = 2 the 4 n system is bound in all the approx-imations, but its binding energy ranges from ≈ E R ≈ Γ ≈ s ≈ . 5. They observed a clear trend in growing width asthe 4 n continuum was more accurately described. Theyeven noted that “the opening of new decay channels andthe presence of continuum states in the configuration mix-ing above the threshold is expected to make the width ex-plode when s → 1, especially in the DMRG results whereall decay channels are open”.In view of this dispersion it is very difficult for thereader to draw a conclusion. In any event, the authorsconfirmed the existence of an S -matrix pole of the 4 n sys-tem associated with J π = 0 + . They concluded that whilethe energy position of a 4 n resonance might be compatiblewith the RIKEN value when s → 1, including more thantwo particles in the continuum suggested that the widthwould be larger than Γ ≈ s and theanomalous transition from an artificially bound state intothe continuum that it generates. The appearance of n and n open channels can substantially modify the propertiesof the computed resonance when the system goes into thecontinuum. One may find intriguing that in all the con-sidered schemes the critical scaling factor determining the4 n threshold lies in the range s ∼ . s = 1 . n [129] and 4 n [130] statesin the framework of AGS equations in momentum space[131]. This formalism, equivalent to FY equations, pro-vides rigorous solutions for the three- and four-body prob-lem and describes very accurately all the correspondingnuclear systems around the breakup thresholds [132,133,134]. An important advantage of the AGS approach withrespect to previous calculations is its ability to computethe resonance position and its effect on the scattering am-plitudes that could lead to observables in a scattering pro-cess, in particular the 3 n → n and 4 n → n transitionamplitudes.The trineutron [129] was studied using several realis-tic nn interactions and with a scaling factor s α only inthe non-zero angular momentum partial waves, in orderto avoid a bound n in S that would open a threshold inthe 3 n system and complicate the analysis. For sufficientlylarge values of s α , a n bound state was found for several J π and at critical values within s ≈ nn interaction model. When decreasing s α , a resonancebehavior was observed in all the transition amplitudes, as | 〈 P | T / - | S 〉 | E (MeV) Fig. 19. n transition amplitude for different values of thescaling factor in PF partial wave, taken from Ref. [129]. -2000-10000 〈 k y | T | k z , 〉 ( f m ) Re Im NLOSRGSRG( S ) -2000-100001000 0 5 10 15 〈 k z | T | k z , 〉 ( f m ) E (MeV) ReIm 01 0 5 | 〈 k z |T |k z, 〉 | k z Fig. 20. Physical 4 n transition amplitude as a function of theenergy, taken from Ref. [130]. shown in Fig. 19 for the PF case. Clearly, the resonantbehavior for s = 6 . s and com-pletely disappears for the physical interaction strength.Note that even with s = 4 the amplitude is totally flat.The corresponding pole trajectories were also extracted,and for the Reid93 interaction they matched the resultsfrom FY calculations [84]. Deltuva concluded that thereare no physically observable 3 n states consistent with thepresently accepted interaction models, and that the poletrajectories were similar for all the realistic nn potentialsand in close agreement with Refs. [84,86].The tetraneutron [130] was studied using the same rig-orous methodology. The 4 n transition operators T βα in the0 + state were calculated using a physical nn potential, i.e. s α = 1. As in the 3 n case it showed no indication of res-onances, but for sufficiently large s α a resonant behaviorwas clearly seen in all matrix elements. The tetraneutronbecame bound at s = 5 . 29, thus lower in energy than the . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 19 trineutron, bound only at s > 6. The results were found tobe almost independent of the particular nn potential, as inprevious works. Interestingly, it was found that even in theabsence of any 4 n resonance, some 4 → n subsystem in the final state,like the He( He , Be) one measured at RIKEN.The conclusions of these two papers were in close agree-ment with the exact FY methods [84,85,86,97,98], butin sharp contradiction with Refs. [105,125,128], and evenwith Ref. [127] that had used the same EFT potential.In 2019 Deltuva and Lazauskas shed some light on thisstriking difference [135,136]. The first paper was a criticalComment [135] to the conclusions of Ref. [127]. In partic-ular, they analyzed quantitatively some serious shortcom-ings related to confining neutrons in a Woods-Saxon trap(4) and to the misleading extrapolation procedure withouttaking into account the analytical structure of the thresh-olds. Two problems were pointed out:1. For some trap parameters [127], e.g. R = 6 fm and a = 0 . 65 fm, n starts being bound at V ∼ . 09 MeV.Thus, when n is supposed to become bound in thetrap, n is already bound by several MeV (see Ap-pendix for details) and n should in fact decay into n(s) and maybe even n, provided the theoretical dy-namical method was able to account for open scatter-ing channels. In any case, the 4 n states of Refs. [105,127] fulfilling E n < E n were not really bound states,but some discretized continuum states that do notevolve into a resonance. The extrapolation of their en-ergies into the continuum is misleading .2. Even assuming that the 4 n state in the trap was reallybound, the evolution into a continuum state involvesbranching at each threshold with discontinuity in thesecond derivative of energy with respect to a strengthparameter. Polynomial extrapolations neglect this dis-continuity and are thus conceptually incorrect. Thenon-trivial evolution of the simple 2 n case was con-sidered in detail, and is reproduced in our Appendix.Gandolfi et al. argued [137] that they had not claimedthe existence of 3 n or 4 n resonances, nor seen evidence ofopen decay channels in the neutron trap, and that theirextrapolation had worked in some selected examples.In the second paper [136] Deltuva and Lazauskas stud-ied the consequences of a bound n, obtained by enhanc-ing V nn with a scaling factor, in the evolution towardsthe continuum of an artificially bound n state. The re-sults were obtained by combining the FY [84,85] and AGS[130] formalisms. Two different cases were considered:1. The same scaling factor s was used in all partial waves.They found that a 0 + bound n indeed emerges for adeep enough n, with s ≈ . s , thisartificially bound n, although having negative energy, It is indeed very disappointing that this trivial kinematicalfact could have been ignored not only by the authors but alsopassed through the referee procedure of top level journals. is not longer bound but decays into two n, in con-tradiction with Refs. [127,128] that found a bound nfor s ≈ . 5. Furthermore, by continuing to decrease s ,they found that the 4 n system will never evolve into aresonance but into a virtual state, similar to the NN S case. The binding is then dominated by the S-waveand it is clear that there is no way to build a 4 n reso-nance with a global scaling factor.2. The scaling factor for the S interaction was fixed to s = 1 . B nn = 0 . 316 MeV) and the main extrabinding was provided by scaling P- and higher angu-lar momentum partial waves. This breaking of S-wavedominance is essential to produce P-wave resonancesin the final state. Such a resonance is indeed formedwith s P ≈ . 9, but when slowly removing the scalingfactor the resonant character disappeared well beforereaching the physical case s = 1.Their main result, the fact that one scaling factor cannotgenerate a 4 n resonance, is enlightening and very relevantin view of the many studies that use this artifact.Later in 2019 Li et al. [138] contradicted the two pre-vious works. They used ab initio NCGSM and concludedthat n and n are both observable resonant states. The nresonance parameters were ( E, Γ ) n = (2 . , . 38) MeV.They even claimed that the n was lower in energy andnarrower, ( E, Γ ) n = (1 . , . 91) MeV, and strongly en-couraged experimentalists to search for the trineutron atlow energy. These calculations, that pretend to be “nearlyexact”, are however based in the same trap methods ofRefs. [105,127], and we have already discussed the kind ofredhibitory problems from which these methods suffer.The multineutron field is still very active, as shown bythe two additional works devoted to this topic that havebeen already published in 2020, and with which we willclose this theoretical review.In the first one, Higgins et al. [139] studied the 3 n and4 n systems within the adiabatic hyper-spherical frame-work. The corresponding adiabatic potential-energy curvewas analyzed and found to be repulsive, in agreement withTimofeyuk [103]. They concluded that there is no sign ofa low-energy resonance for any of these systems. However,they observed in both of them some low-energy enhance-ment of the Wigner-Smith “time delay”, which for a sin-gle adiabatic potential is defined as Q ( E ) ≡ d [ δ ( E )] /dE ,that could provide a hint to understand the GANIL andRIKEN near-threshold enhancements.Later in 2020 Ishikawa [140], using FY and realistic NNpotential (AV18), found no evidence of a 3 n resonance.Moreover, he examined different methods used to extrap-olate the 3 n energy from an artificially bound state to thecontinuum, and found that the enhancement of nn P-wavealone or the use of 3NF do not generate spurious 3 n reso-nances. However, using external trapping potentials leadsto positive 3 n energy results, which without a further care-ful analysis may be considered as a resonance while theyare in fact a general defect of the trapping method. Inthis way, Ishikawa explained the contradictory results ofRefs. [127,138] concerning 3 n states. At first glance, the striking disagreement among differ-ent theoretical works presented in this section is obvious,shocking, and could be even devastating for the credibilityof the field. From a general point of view one can distin-guish two main families of results, whose conclusions areorthogonal to each other.In the first group, the existence of near-threshold 3 n and 4 n resonant states is claimed. In the case of the tetra-neutron, those works seem thus in agreement with thetwo experimental signals reported at GANIL and RIKEN.However, for the second group this is a scientific nonsensesince an explicit calculation of these states finds them veryfar from the physical region, in the third quadrant of theunphysical energy sheet. According to them, the recentexperimental results, if confirmed, in no way can be at-tributed to a 3 n or 4 n resonant state and must have an-other origin, still to be clarified.If these two points of view can hardly be more oppo-site, there is at least a general consensus in that the originof such contradictory conclusions must not be found in thedifferent interactions used. Indeed, for these low-energyphysical states with a large spatial extension, any interac-tion providing acceptable low-energy parameters leads tosimilar qualitative results. Furthermore, the role of three-neutron forces is negligible. The differences among themmust rather be found in the methods used to solve thefew-nucleon problem and/or in the way they access thefew-neutron continuum.All results obtained using FY or the AGS equationsagree with each other about the non-existence of any near-threshold n and n. These approaches represent nowa-days the most rigorous framework for solving the few-body problem, and include in a natural way all the contin-uum states, without additional approximations or ansatzs.Their agreement is not only qualitative but also quanti-tative, in particular when computing the critical values ofthe scaling factors or the trajectories of the resonance po-sition in the complex-energy surface. Moreover, this agree-ment is extended to the variational Gaussian Expansionmethods, when the resonant states are tackled with CSMor ACCCM, and to HH, where they unanimously find astrong repulsion in the adiabatic potential curves disal-lowing any bound or resonant state.Furthermore, we would like to mention the EFT de-velopments in multifermion systems close to the unitarylimit, as it is the case of multineutron states, where theyfound a universal dimer-dimer and fermion-dimer strongrepulsion [141,142,143], in contrast with any resonant stateand in line with the above mentioned series of works.This impressive bulk of coherent results is in frank con-tradiction with the two GFMC results, as well as with theNCSM-SS-HORSE and the two NCGSM works, the latterthree being based on many-body techniques.GFMC [105,127] is accurate for computing bound statesand has been able to deal with a larger number of parti-cles than FY and AGS, but it is clearly less adapted toscattering problems, in particular the computation of res-onances. Moreover, we have shown that their conclusions are strongly weakened due to two important additionalissues: the use of traps and the method employed to ex-trapolate the (supposed) bound state into the continuum.NCSM-SS-HORSE [125] is based on several many-bodytechniques that were not checked in similar few-nucleonscattering problems. Moreover, the 4 n continuum is treatedin a very indirect way, with the “democratic decay” hy-pothesis for extracting the 4 n → n amplitude (neglectingthe FSI between virtual dineutrons [136]) and a non-trivialextrapolation procedure to extract the complex parame-ters of an eventual resonance. Finally, they needed thefurther addition of a “false pole” in order to be in perfectagreement with the previously published RIKEN result.The more comprehensive NCGSM-DMRG work [128]exhibits a manifest dependence in the approximations usedto solve the four-body problem, and their global scalingfactor could reach the physical case only for the less accu-rate of them. However, they observe a clear trend of an in-creasing width of the state, which according to the authors“exploded when s → n in a given reaction cross-section, either breakupor DCX. We have presented in this section convincing ar-guments against the attribution of this enhancement toa resonance, at least in the standard way a resonance isunderstood in nuclear, hadronic and particle physics.However, as it has been emphasized many times andby many authors [85,86,110,124,130,139], there are otherquantum mechanical mechanisms that generate resonance-like structures, even for repulsive interactions. Deltuva’sresults of Fig. 20 show it explicitly in the 4 n → n non-resonant transition amplitude. A much simpler case is il-lustrated in Fig. 21, corresponding to the P-wave scatter-ing of two identical particles ( ~ /m = 1) on a repulsivepotential well V ( r ) = + V × Θ ( R − r ), where Θ is theHeaviside step function. By changing the parameter R ,one can find a rich variety of resonance-like structures,with the corresponding E -dependent phase shift crossing π/ . Ignoring the underlyingdynamical content, these structures can be attributed to aresonant state, very close to binding. We believe that thiskind of behavior is also manifest in the time-delay analysisof Ref. [139].Whether or not the experimental search of this quan-tum mechanical enhancements is pertinent enough to jus- The same potential was considered in [110] to illustrate anS-wave resonance-like behavior.. Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 21 E σ R=5R=6R=7R=8R=10 V =1 Fig. 21. Resonant P-wave cross-section in a repulsive potentialof the form V ( r ) = + V when r < R , for different values of R . tify great investments can be submitted to debate. How-ever, associating these enhancements to physical resonantstates would be misleading. Waiting for the next theoret-ical paper claiming near-threshold 3 n and 4 n resonances,it seems clear that after all these decades of efforts anddiscussion a general consensus based on rigorous solutionsof the problem is emerging to conclude... what was alreadyclear from the first Gl¨ockle’s work in 1978. “Sit finis libris, non finis quaerendi” (this may be the endof the book, but not the end of the quest), was an elegantway to close an open question in the scholastic times.Despite all the controversies and debates, this quest isdefinitely a fascinating story. Think of the distant 1960s,with experimentalists trying to do ‘magic’ with pions inorder to create neutron matter in the laboratory; othersgoing to nuclear reactors, looking for their grail in such aradioactive environment; others manipulating ion beams,hoping to add an remove nucleons against all odds. Inparallel, theoreticians starting the first three-body cal-culations, despite lacking at that time the sophisticatedtools and inputs that we have seen were needed. Sixtyyears later the quest is still fascinating... but still open. Iffrom the theoretical point of view the question of a nar-row 3 n and 4 n resonance seems closed, there is an urgentneed for a definitive experimental conclusion concerningthe GANIL and RIKEN events.Nuclear theory, and in particular the description oflight nuclei from the underlying forces between nucleons,has seen an enormous progress working hand in hand withexperiments. The main theoretical models have been bornfrom the known properties of stable nuclei, and then theyhave been refined and updated with the increasing knowl-edge about more and more exotic isotopes. In the field ofneutral nuclei, however, the theory advances almost blind.The only two experimental signals [3,4] are still weak anddo not provide any firm and precise observable that couldbenchmark the different methods and techniques. In this respect, it should be a priority to confront thecontroversies surrounding the tri- and tetraneutron calcu-lations with the help of high-statistics, unambiguous ex-perimental results. New techniques or experiments thatwould provide still another weak and/or ambiguous signaldo not seem the best way to unlock progress in the field. Itis our hope that the new experiments already (or soon tobe) undertaken at RIKEN will bring the very much neededreliable reference points for the theoretical conclusions tobe adapted. If the positive signals were refuted, then thecalculations predicting resonances should be reevaluated.If they were confirmed though, then the detailed charac-teristics of the signals (such as energies, widths, or angularcorrelations) should be explained.Once the 3 n and 4 n systems will be clarified, bothfrom theory and experiment, it will be easier to move onthe heavier systems, in particular 6 n and 8 n . Even if thelightest multineutron states were not observable, a perti-nent question remains: when increasing the mass number,can at some point a multineutron manifest as a boundor resonant state? Even for the most reluctant theories,and of course for experiments, this is still an open ques-tion. The analogy between neutrons and atomic He isvery suggestive, for their fermionic character and for theform of their interaction, with a hard repulsive core anda small attractive well. The dineutron (with a critical en-hancement factor s ≈ . 1) is even closer to binding thanthe He dimer ( s ≈ . He dropbecomes bound beyond N ∼ 30 atoms [144]. On the otherhand, the nn P-waves are smaller and its centrifugal bar-rier is less peripheral than in the helium atomic case [85,145], and that could make a difference.The known as “helium anomaly”, i.e. the fact that Heis more bound than He, has long been a clue towards thesearch for light multineutrons, taken as a hint for addi-tional binding due to the increasing number of neutronson top of He. Less known but maybe more spectacularwould be the analog “hydrogen anomaly”, provided it isestablished. The , , H isotopes are unbound by severalMeV and exhibit broad resonances, but H ground stateis narrower than the other two, as demonstrated by recent ab initio calculations of these resonances [87]. Moreover, H is reported to lie almost at the 4 n threshold [146],although it has in common with the tetraneutron thatthe experimental signals to date are weak, ambiguous andsometimes contradictory.With the recent progress in the exact calculation of thefive-body problem, H is now accessible almost ab initio (treating the triton as a particle) for the theory and, as wehave seen at the end of Sec. 2, a related experiment thatshould provide very high statistics and resolution has beenalready undertaken. This super-heavy isotope of hydrogencould in this way be the key for the next steps in the field,both for the role of the tetraneutron at the t + 4 n thresh-old but even for the hexaneutron at the p + 6 n threshold.Going beyond A = 6 represents still a too important ob-stacle for experiments, and for the moment should be leftfor the theory, once the lighter multineutrons have helpedto provide a solid base for the models. In any event this domain will remain fascinating since,paraphrasing the authors of Ref. [128], it “would deeplyimpact our understanding of nuclear matter”. A Two neutrons in a trap In order to illustrate the kind of problems related to theextrapolation from bound states to the continuum, wepresent in this Appendix a detailed solution of the simplestcase, in the same spirit as it was considered in Ref. [135]:two neutrons confined in a trap.Let us consider two neutrons interacting via V nn andfurthermore submitted to a potential trap having, as inRefs. [105,127,138], a Woods-Saxon form: V T ( r i ) = V e ri − Ra (7)The total two-body Hamiltonian is: H = − ~ m n ∆ + V nn ( r − r ) + V T ( r ) + V T ( r ) (8)The particle coordinates r i can be expressed in terms ofthe relative ( r = r − r ) and center-of-mass (2 R = r + r ) coordinates as: r − r = 2 r − ( r + r ) ⇒ r = − r Rr − r = 2 r − ( r + r ) ⇒ r = + r R Assuming R ≡ H = − ~ m n ∆ + V nn ( r ) + 2 V T (cid:16) r (cid:17) (9)The numerical solution of this problem is trivial and canbe obtained with any required accuracy. For illustrativepurposes, we have considered a phenomenological CD MT13 nn interaction: V nn ( r ) = V r exp( − µ r r ) r − V a exp( − µ a r ) r (10)reproducing the nn scattering length a nn = − . 59 fmand effective range r nn = 2 . 94 fm values, with the param-eters V r = 1438 . 72 MeV, µ r = 3 . 11 fm, V a = 509 . 40 MeV, µ a = 1 . 55 fm, and ~ /m n = 41 . [83].The binding energies for two values of the trap size( R = 3 and 5 fm) and fixed diffusiveness a = 0 . 65 fm areplotted in Fig. 22 as a function of the strength V .At first glance it seems that both series of points canbe indeed linearly extrapolated (solid lines) towards a pos-itive value (empty square at V = 0), which could be inter-preted, as in Refs. [105,127,138], as a 2 n resonance withpositive energy. Furthermore, it is worth noting that evenin the absence of any interaction, the results of the linearextrapolation (dashed lines) display a similar trend and -4 -3 -2 -1 0 V (MeV) -5-4-3-2-1012 E ( M e V ) R=5 fmR=3 fm V nn =0 Fig. 22. Energy of 2 n in a trap (7) as a function of the strengthparameter V for two values of the trap size R . A linear extrap-olation (solid lines) converges towards a positive energy value.The free case V nn = 0 (dashed lines) displays a similar trend. i k E=kE=k k−plane E−surface E I E II E γ k γ ΓΓ E ’’ iff Fig. 23. Pole trajectory in the complex-momentum plane ( γ ∪ γ ′ ) and in the E -surface ( Γ ∪ Γ ′ ) of an artificially bound nn ( k i and E i ∈ E I ) evolving to the continuum ( k f and E f ∈ E II ). would lead to a very disturbing resonant state betweentwo non-interacting neutrons!One could argue that in the case of two neutrons in a S state, the extrapolated results of Fig. 22 correspondin fact to the nn virtual state with positive energy, as itis often considered. We remind, however, that the virtualstate, corresponding to a pole in the negative imaginaryaxis k = + iκ , has negative energy k = − κ . The poletrajectory in the complex k -plane and E -surface of an ar-tificially bound nn ( k i and E i ∈ E I ) into the continuum( k f and E f ∈ E II ) is summarized in Fig. 23. As one cansee, both bound and virtual (unbound) states correspondto negative-energy values in the E -surface, though livingin different Riemann sheets.If the extrapolation analysis sketched in Fig. 22 is donecarefully, in particular using much smaller binding ener-gies, one can see that a strong curvature in the E ( V )dependence starts at B ≈ . . Miguel Marqu´es, Jaume Carbonell: The quest for light multineutron systems 23 -0,5 -0,4 -0,3 -0,2 -0,1 0 V (MeV) -0,5-0,4-0,3-0,2-0,100,10,2 E ( M e V ) R=5 fmR=3 fm V V Fig. 24. Zoom of Fig. 22 showing the behavior of the energywhen approaching the continuum threshold at V c and E = 0. threshold E = 0 is reached at a critical value V c ( R ), whichincreases with R . For instance, V c ≈ . R = 5 fm(see the zoom displayed in Fig. 24).After the threshold, the V -dependence represented inFig. 24 changes drastically the curvature, and turns down-wards toward negative values of E as can be see in Fig. 25,and as expected from the location of a virtual state in thecontinuum illustrated in Fig. 23. The highly non-trivialtrajectory in the ( E, V ) plane displayed in Fig. 25, re-sults from the ACCC extrapolation given by equation (5).To be faithful, a set of bound state values ( V n , E n ) mustbe computed with very high accuracy, as accurate shouldbe the critical values of the potential strength at threshold V c , when the system enters the continuum. As an exam-ple of the accuracy required, the values used for the case R = 5 fm are listed in Table 1.The trajectory in the ( E, V ) plane is represented inthe complex k -plane in the left panel of Fig. 23. It is thestraight line formed by the union of γ ∪ { } ∪ γ ′ . Thecorresponding path in the E -surface is Γ ∪{ } ∪ Γ ′ , shownin the right panel, where E i belongs to E I and E f to E II .The transition from bound state to continuum is at k = 0and coincides with the transition E I → E II . Note thatboth initial ( k i ) and final ( k f ) states have negative energy( E i and E f respectively), despite the fact that E f is inthe continuum, but belong to different Riemann sheets.From the above results it is clear that a simple poly-nomial extrapolation cannot describe the non-trivial evo-lution of a bound state into the continuum displayed inFig. 25. This is in fact the main conclusion of this Ap-pendix. ACCCM can do it accurately by taking into ac-count the analytical properties related to the thresholds.However, the method requires the previous computationof several bound state values with high accuracy and veryclose to E = 0. This can be easily done in the two-bodycase we have considered above, although it becomes moreand more involved when increasing the number of par-ticles. This level of accuracy is not accessible to all themethods used to solve the few-nucleon problem, but it -0,4 -0,3 -0,2 -0,1 0 V (MeV) -0,2-0,100,1 E ( M e V ) R=5 fmR=3 fm nn virtual state Fig. 25. Energy of 2 n in a trap, from bound state to contin-uum. The corresponding trajectory in the complex-momentumplane ( γ ∪ γ ′ ) and the E -surface ( Γ ∪ Γ ′ ) is displayed in Fig. 23. V E Table 1. Energy values of 2 n in a trap used in the ACCCextrapolation (5), as a function of the potential strength V for R = 5 fm. The last point corresponds to the thresholdextrapolated value. has been successfully applied to A = 3 , , n and4 n P-wave resonances displayed in Figs. 14,16. They havehowever in common that the trajectories of the corre-sponding pole positions end at negative energy, Re( E ) < E II , asillustrated in Fig. 23. Of course, it is clear that a polyno-mial extrapolation of the results displayed in Fig. 22 (orFig. 15) is forced to end at positive energy values, and istherefore unable to reproduce the realistic behavior of the3 n and 4 n resonant states.Furthermore, it is worth noting that even in the case ofpositive energy P-wave resonances, with Re( E ) > E ) < nn interaction (10) butsuitably scaled in order to generate a P-wave nn resonantstate. For instance, with a scaling factor s = 3 . E = 4 . − . i MeV, that is a statewith Γ ≈ . nn S-wave, we have -5 -4 -3 -2 -1 0 V (MeV) -6-5-4-3-2-1012345 E ( M e V ) R=5.0 fmR=3.0 fm CD MT13 s=3.0 E=4.50-3.32i MeV Fig. 26. Energy of 2 n P-wave resonant state in a trap (7)as a function of the strength parameter V for two values ofthe trap size R . Quadratic extrapolations (dashed lines) of thecomputed binding energies in the trap converge at V = 0 to-wards positive energy values, sensibly different from the exactresult (blue circle). computed the energies of this P-wave resonant state in thetrap, as a function of V for two different parameters ofthe trap size R , and performed a quadratic extrapolationtowards the ‘free’ case V = 0. Results are displayed inFig. 26. They show a significant disagreement between theextrapolated values and the physical resonance energy, in-dicated by a blue solid circle. We cannot exclude, as notedin Ref. [135], that for very narrow P-wave resonances thedifference could be small, but the polynomial extrapola-tion is not reliable in a general case. This could be thecase in the unphysical example (P-wave resonance in anS-wave two-Gaussian potential) considered in Ref. [127] inorder to justify their extrapolation method.The exact solution of two neutrons in a trap can bestraightforwardly extended to the n case. The Jacobi co-ordinates of the 3 n system are expressed in terms of theparticle coordinates r i as: y = 2 √ (cid:18) r − r + r (cid:19) = 2 √ (cid:18) r + r − r + r + r (cid:19) Since in the center of mass r + r + r = 0, one has: r = y √ r i .The three-body intrinsic Hamiltonian takes the form: H = H + V nn ( x ) + V nn ( x ) + V nn ( x ) + W ( y , y , y )with: W ( y , y , y ) = V T (cid:18) y √ (cid:19) + V T (cid:18) y √ (cid:19) + V T (cid:18) y √ (cid:19) Since the different sets of Jacobi coordinates are re-lated by the linear transformations: y = c x + s y y = c x + s y the problem is equivalent to adding a three-body force: W ( x , y ) = W [ y , y ( x , y ) , y ( x , y )]which in the case of three identical particles can be easilysolved with a single Faddeev equation. 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