Evidence against the Efimov effect in 12 C from spectroscopy and astrophysics
J. Bishop, G.V. Rogachev, S. Ahn, E. Aboud, M. Barbui, A. Bosh, J. Hooker, C. Hunt, J. Hooker, H. Jayatissa, E. Koshchiy, R. Malecek, S.T. Marley, M. Munch, E.C. Pollaco, C.D. Pruitt, B.T. Roeder, A. Saastamoinen, L.G. Sobotka, S. Upadhyayula
DDirect search for the Efimov effect in C J. Bishop,
1, 2, ∗ G.V. Rogachev,
1, 2, 3
S. Ahn,
1, 4
E. Aboud,
1, 2
M. Barbui, A. Bosh,
1, 2
J. Hooker, C. Hunt,
1, 2
H. Jayatissa, E. Koshchiy, R. Malecek, S.T. Marley, M. Munch, E.C. Pollacco, C.D. Pruitt,
10, 11
B.T. Roeder, A. Saastamoinen, L.G. Sobotka, and S. Upadhyayula Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA Department of Physics & Astronomy, Texas A&M University, College Station, TX 77843, USA Nuclear Solutions Institute, Texas A&M University, College Station, TX 77843, USA Center for Exotic Nuclear Studies, Institute for Basic Science, Daejeon, 34126, Korea Department of Physics & Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C, Denmark IRFU, CEA, Saclay, Gif-Sur-Ivette, France Department of Chemistry, Washington University, St. Louis, MO 63130, USA Lawrence Livermore National Laboratory, Livermore, CA 94550, USA TRIUMF, 4004 Westbrook Mall, Vancouver, British Columbia V6T 2A3, Canada (Dated: February 9, 2021)
Background:
The Efimov effect is a universal phenomenon in physics whereby three-body systems are stabilizedvia the interaction of an unbound two-body sub-systems. A hypothetical state in C at 7.458 MeV excitationenergy, comprising of a loose structure of three α -particles in mutual two-body resonance, has been suggested inthe literature to correspond to an Efimov state in nuclear physics. The existence of such a state has not beendemonstrated experimentally. Purpose:
Using a combination of γ -spectroscopy and charged-particle spectroscopy, strict experimental limitson the existence of such a state have been established here. Method:
Using the combined data sets from two recent experiments, one with the TexAT TPC to measure α -decay and the other with Gammasphere to measure γ -decay of states in C populated by N and B β -decayrespectively, we achieve high sensitivity to states in close-proximity to the α -threshold in C. Results:
No evidence of a state at 7.458 MeV is seen in either data set. Using a likelihood method, the 95% C.L. γ -decay branching ratio is determined as a function of the β -decay feeding strength relative to the Hoyle state. Conclusion:
At the 95% C.L., the Efimov state cannot exist at 7.458 MeV with any γ -decay branching ratiounless the β -strength is less than 0.7% of the Hoyle state. This limit is evaluated for a range of different excitationenergies and the results are not favorable for existence of the hypothetical Efimov state in C. I. INTRODUCTION
The Efimov effect is a universal quantum phenomenapresent in several areas of Physics. The effect is observedfor three-body systems that are comprised of subsystemswhere the sub-unit two-body systems are unbound buthave a large s-wave scattering length. Vitaly Efimovfound that under these conditions the long-range three-body attraction arises and that this attraction can sup-port a family of three-body states. A detailed review ofEfimov physics and experimental evidence for it can befound in Ref. [1]. The classical interpretation is that thethree-body force binds the system via the ‘shuttling’ backand forth of one of the particles. This then creates aninfinite series of states given by a universal scaling law.While predicted 50 years ago, this phenomenon took 35years to observe experimentally. The first clear evidencefor Efimov effect was reported for the system of ultracoldgas of caesium atoms in an external magnetic field [2]. Innuclear systems, which were the original focus of Efimov’s ∗ [email protected] investigation, the situation is more complicated. In prin-ciple, a J π = 0 + (corresponding to L=0) 3- α state in C,where the 2- α sub-systems are unbound but form a long-lived resonant state, can be seen as Efimov trimer. In hisoriginal paper Efimov argued that the 0 + excited stateat 7.65 MeV in C (Hoyle state) possibly originates dueto this interesting three-body quantum phenomena whichwe now call the Efimov effect [3]. Microscopic three-bodycontinuum calculations that utilize phenomenological α - α potential [4] indicate that the centrifugal (three-body)potential is small compared to the nuclear and Coulombpotentials and as a result the Hoyle state is probably notrelated to the Efimov effect. Faddeev calculations forthe 3 α system have previously been performed and donot predict an additional near-threshold 0 + that wouldcorrespond to an Efimov state [5]. More recently, exis-tence of an Efimov state in C at an excitation energythat corresponds to a mutual Be(g.s.) resonance for allthree α -particles was suggested in [6, 7]. This excitationenergy is given, in units of MeV, by the following simplerelation which takes into account that the narrow reso-nance of Be(g.s.) is unbound by 91.84 keV with respectto decay to two α -particles: a r X i v : . [ nu c l - e x ] F e b E = 23 (cid:88) i E rel i − Q = 2 × . . . . (1)The main goal of this paper is to search for any evi-dence of this hypothetical state or, if not, place experi-mental limits on its existence. We utilize the results oftwo recent experiments. The first one is the study thatis sensitive to three α -particles decay channel of the near α -threshold excited states in C populated in the β + de-cay of N [8, 9]. The second is the γ -spectroscopy studyof states in C populated in β − decay of B [10]. Com-bining these two data sets we demonstrate that thereis no evidence for a resonance at or near 7.458 MeV.Stringent experimental limits on its existence have beenestablished. We also examine the astrophysical implica-tions of such a state existing below the Hoyle state usinga simplistic model to calculate the triple-alpha reactionrate and demonstrate the incompatibility of this resultwith current astrophysical observations.
II. EXPERIMENTAL LIMITS OF AN EFIMOVSTATE
There has not been a large amount of experimental ac-tivity to investigate if there is a resonance in C belowthe Hoyle state. One investigation claiming to observean Efimov state in a heavy-ion Zn+Zn/Ni+Ni collisionsat 35 MeV/u was recently published [11]. By examiningevents with 3 α -particles and measuring their relative en-ergies, potential Efimov states were found by looking at3 α -particle triplets where the relative energy betweenall α -particles is consistent with 92 keV. Due to the re-action mechanism used, there is a dominant contributionfrom uncorrelated α -particles which are required to beaccounted for via mixing. The remaining spectrum isthen accounted for using an arbitrary fit function on topof Breit Wigner peaks which show up around 0.1 MeVsuggested mutual Be resonances. Due to low statisticsand dependence on this fit function, it is therefore diffi-cult to definitely claim evidence of a peak. However, thisstudy represented a dedicated effort to experimentallyobserve such a state. While in-medium effects affordedby a heavy-ion reaction may enhance the production ofan Efimov state, the reaction complexity also introducesnumerous sources of background that are difficult to ac-count for. A cleaner population method is through the β -decay of N / B which populates 0 + , 1 + , and 2 + statesin accordance to the β -decay selection rules. Throughthe recently-developed β -delayed charged-particle decaytechnique using the TexAT TPC [8], one is afforded agood probe on any resonance existing at a low relativeenergy above the 3 α threshold. This has not been pre-viously identified due to the difficulties associated withlow-energy measurements using implantation in silicon detector arrays. Despite the advantages afforded usingthis technique, the measurement of a 3-particle final stateonly 180 keV above threshold is still challenging. A. Efimov state limits from β -delayedcharged-particle decays in a TPC To establish a limit on the population of the Efimovstate via β -decay of N, the TexAT TPC was used tostudy the β -delayed charged-particle decay of N [8]. Abeam of N with an energy of 24 MeV was stopped in-side of the active-area of the TexAT TPC by 20 Torrof CO gas. This was achieved on a single implantationdecay basis which allowed for the matching between theimplantation event and the subset of events where the β -decay populates states above the α -threshold in C.The trigger condition for the decay was that more thanone Micromegas pad fired. The trigger threshold is 10keV and therefore one is sensitive to decay events downto E x ∼ α -particlesarising from these events were then reconstructed and thetotal energy deposited was used to calculate the excita-tion energy. This is shown in Fig. 1 where the majorityof events seen are from the Gaussian tails of the Hoylestate at 7.654 MeV. One may also determine the energyof any 3 α decay by looking at the total length of thetracks and converting the range of the α -particles to anenergy and summing. Taking events which lay withinthe region of interest for both of these plots, one maythen manually check events to deduce the origin of thesecounts. The peak in Fig. 1 just above the threshold corre-sponds to events where the implanting beam is scattered(primarily off the entrance window) and the N beam isimplanted either on the cathode or anode. When it sub-sequently decays, part of the energy of the event is lost asone or two α -particles deposit their energy into the an-ode or cathode directly rather than liberating electronsin the gas volume. To determine the contribution fromstates between the absolute threshold at the Hoyle state,a Gaussian tail was fitted for the Hoyle state contribu-tion using the previously obtained experimental resolu-tion of 55 keV. To remain conservative, any backgroundabove this Gaussian component was counted as a poten-tial Efimov contribution: 137.1 counts in total. The 95%confidence limit (C.L.) was therefore taken as 160.5 for α -events from Poisson statistics.To convert this into a limit, the expected number of3 α Efimov-state decays is given by: N ES = N Hoyle β ES β Hoyle BR α (ES)BR α (Hoyle) , (2)where β i describes the β -decay feeding strength from N,BR α = Γ α Γ tot for the ES and the Hoyle state. Re-arrangingthe terms, one arrives at: β ES β Hoyle BR α (ES) = BR α (Hoyle) N ES N Hoyle . (3)The factor on the left-hand side of Eq. 3 allows us toplace a limit on the α branching ratio multiplied by the β -feeding strength relative to the Hoyle state. Insert-ing the values, BR α (Hoyle) = 99 . N Hoyle = 23,276and N ES < .
5. This 95% confidence limit is repre-sented in Fig. 2 (slanted-right blue hash region) where β ES β Hoyle is plotted against BR γ (ES) = 1 − BR α (ES). Theproduct of the β -strengths and the gamma branching-ratio ( β ES β Hoyle BR α (ES)) is < .
69% from Eq. 3 thereforeconstraining x (1 − y ) < .
69% in the exclusion plot. C oun t s / k e V FIG. 1. Excitation function obtained from 3- α decay energydeposition. The dotted red line corresponds to a Gaussianfit for the Hoyle state contribution. The yield above this isconservatively taken as possibly arising from additional statessuch as the Efimov state (ES). The peak around 7.3 MeVcorresponds to deposition events on the cathode. B. Efimov state limits from β -delayedgamma-spectroscopy Aside from observation in the 3 α channel, there havebeen studies of the γ -decay spectrum associated with β -delayed population of C using B [10]. By using Gam-masphere to study for coincident gamma rays with the4.44 MeV first-excited state in C, one may examine anypossible contribution from C(ES) → C(2 +1 ) whichshould correspond to E γ = (7 . − . . < . ±
9. Following the same prescription above,the number of γ rays from Efimov state decays is givenby: N ES = N Hoyle β ES β Hoyle BR γ (ES)BR γ (Hoyle) . (4) - - - -
10 1Ratio of beta-strengths - - -
10 110 G a mm a B R ( % ) Gamma limitAlpha limit
FIG. 2. 95% C.L. exclusion plot for a 7.458 MeV Efimov state.The gamma and alpha experimental limits exclude regions ofthe plot relating to the strength of the β -feeding strength rela-tive to the Hoyle state (abscissa) and the gamma-ray branch-ing ratio (ordinate). With a particular range of the gammabranching ratio, the existence of a state cannot be conclu-sively excluded if the β -feeding strength ratio is < × − . Separating the unknowns onto the left-hand side fromthose known on the right-hand side, one gets: One maytherefore re-arrange this equation as: β ES β Hoyle BR γ (ES) = BR γ (Hoyle) N ES N Hoyle . (5)The known values are N ES < . N Hoyle = 58 ± γ (Hoyle) = 0 . β -strengths andthe gamma branching-ratio of < . β -feeding strength is < × − the strength of the Hoylestate and has a specific decay γ -decay branch. Requiringthe gamma branching ratio to be ≈ < − .The existence of an Efimov state at an energy otherthan 7.458 MeV may also be examined using the same ap-proach as detailed above. The product of the β -strengthsand the gamma branching-ratio at the 95% C.L. does notexceed 0.016%. The technique used for the alpha-limitis robust up until the separation of the Hoyle peak fromany potential second peak is roughly 2.35 σ , i.e. up until E x = 7.525 MeV where the gamma- and alpha-branchingratios are comparable. If one applies the condition thatthe beta-feeding strength is the same for the Efimov andHoyle states, the gamma-limit is therefore highly appli-cable until the gamma branching ratio is such that thepredicted number of Efimov gamma decays is less thanthe 95% C.L. Naturally, this implies an excitation energy g E010203040506070 C oun t s / k e V FIG. 3. Gammasphere energy spectrum gated on 4439 keVtransition. The peak at 3214 keV is from the Hoyle → +1 transition. The yield in the 3014 keV region (red arrow) showsno sign of a peak from the Efimov → +1 transition. Data from[10]. exceeding the Hoyle state and as such, this scenario maybe definitely excluded. III. ASTROPHYSICAL LIMITATIONS ONLOW-LYING STATES
In the triple-alpha process, there is a strong contribu-tion from low-lying 0 + states [12, 13]. It is for this reasonthat the Hoyle state so successfully enhances the triple-alpha reaction rate by 7 orders of magnitude, overcomingthe A=5, 8 bottleneck in helium-burning stars. Any ad-ditional low-lying state therefore must also contribute toa large degree, particularly at lower temperatures. Tounderstand the role that the Efimov state may have, onecan examine the expected astrophysical reaction rate in-cluding such a resonance in addition to the Hoyle state.A low-lying s-wave state has a reaction rate, R , propor-tional to: R ∝ Γ α Γ rad Γ tot T − / exp (cid:18) − QkT (cid:19) , (6)where Γ α is the α -decay width, Γ rad is the radiativewidth, Γ tot is the total width, T is the temperature, k is the Boltzmann constant and Q is the energy abovethreshold. For a 0 + state lying at Q=186 keV (corre-sponding to the Efimov state at 7.458 MeV), the ratelimits can be used to place a restriction of the α andradiative widths. For such a state, in Table I, na¨ıve esti-mates of the widths are calculated via assuming the sameunderlying reduced width and radiative width for the Efi-mov state as the Hoyle state when scaled appropriatelyby their energies (for penetrability and E γ for the α and γ widths accordingly). For the Efimov state, Γ rad (cid:29) Γ α - - - - - - - - - - - - R ea c t i on r a t e HoyleEfimov
FIG. 4. Reaction rates (arb. units) for the triple-alpha pro-cess for the Hoyle state (magenta squares) against the Efimovstate (blue circles) as a function of the astrophysical temper-ature. At low temperatures ( < .
15 GK), the Efimov stateexceeds the Hoyle state contribution. Such an effect woulddrastically alter the dynamics of the triple-alpha process andthe life of stars. so Γ rad ≈ Γ tot . This is verified by more developed in-vestigations into the properties of an Efimov state [14].Therefore, the rate is driven by Γ α . Using the parame-ters for the Efimov width from Table I, Fig. 4 shows thereaction rate for both the Hoyle and the Efimov state. TABLE I. Na¨ıve estimates of the Efimov decay mechanism.The α widths, Γ α , assume the same reduced width and theradiative width assumes the dominance of the E2 transitionto the 4.44 MeV 2 + state and is scaled as E γ .Parameter Hoyle state Efimov state E x P (cid:96) =0 ( E x ) 3 . × − . × − Γ α rad α / Γ tot > .
9% 0 . While the Hoyle state contribution clearly dominates(due to the suppression of the Efimov width due to thepenetrability factor) at higher temperatures, at lowertemperatures, the Efimov state contribution starts togreatly exceed that of the Hoyle state. The effect this hasin the triple-alpha rate is dramatic and has a large impacton the ignition temperatures of the triple-alpha reactionand also on the final C/O and C/Ne ratios [15]. Thislow-energy enhancement affords an additional preferencein the triple-alpha reaction over the C( α , γ ) which pro-duces O in a competiting process. The reduced oxygenabundance also affects the overall neon (and heavier iso-topic abundances) as a larger carbon abundance meansthe C( C , α ) Ne has a higher reaction rate. Addi-tionally, the CNO cycle is also afforded a similar boostin the reaction rate. The influence this has on a star’sstructure is therefore extreme. Given the current suc-cess of nucleosynthesis codes in replicating the observedC/O abundances and other aspects of a stars evolution,any pertubation to the triple-alpha reaction rate from thepreviously understood level must be extremely small. Assuch, one can use the astrophysical observables to limitany modification to the triple-alpha reaction rate. There-fore, this can be used to place limits of the Efimov stateswidth or even its existence.While detailed analysis of the constraints on the Efi-mov state imposed by He-burning is outside the scope ofthis paper, it would be very interesting to pursue this inthe future.
IV. CONCLUSION
Using combined data from particle and gamma-decayfrom N / B, one may place incredibly restrictive lim-its on the beta-feeding strength and therefore existenceof an Efimov state below the Hoyle state in carbon-12. An Efimov state in such close proximity to the alpha-threshold would decay almost solely via gamma-decayand therefore the beta-feeding strength relative to theHoyle state is expected to be < − at the 95% C.L.Furthermore, it can clearly be seen that the Efimov statehas a tremendous impact on stellar evolution. Sufficeto say, a low-lying state 0 + (whether Efimov or merelyan α -clustered state) would have a significant impact onstellar abundances that the existence of such a resonanceseems incompatible with current astrophysical models. V. ACKNOWLEDGEMENTS
J.B. acknowledges and thanks A. Bonasera for discus-sions on the Efimov state. This work was supportedby the U.S. Department of Energy, Office of Science,Office of Nuclear Science, under award no. DE-FG02-93ER40773 and by National Nuclear Security Admin-istration through the Center for Excellence in NuclearTraining and University Based Research (CENTAUR)under grant number DE-NA0003841. G.V.R. also ac-knowledges the support of the Nuclear Solutions Insti-tute. [1] P. Naidon and S. Endo, Reports on Progress in Physics , 056001 (2017).[2] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl,C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola,H.-C. N¨agerl, and R. Grimm, Nature , 315 (2006).[3] V. Efimov, Phys. Lett. B , 563 (1970).[4] H. Suno, Y. Suzuki, and P. Descouvemont, Phys. Rev.C , 014004 (2015).[5] O. Shinsho and K. Hiroyuki, Nuclear Physics A , 91(1989).[6] H. Zheng, A. Bonasera, M. Huang, and S. Zhang, Phys.Lett. B , 460 (2018).[7] H. Zheng and A. Bonasera, Journal of Physics Commu-nications , 085011 (2020).[8] J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Bar-bui, P. Baron, A. Bosh, E. Delagnes, J. Hooker, C. Hunt,H. Jayatissa, E. Koshchiy, R. Malecek, S. Marley,R. O’Dwyer, E. Pollacco, C. Pruitt, B. Roeder, A. Saas-tamoinen, L. Sobotka, and S. Upadhyayula, Nuclear In-struments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and AssociatedEquipment , 163773 (2020).[9] J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Barbui,A. Bosh, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek,S. T. Marley, E. C. Pollacco, C. D. Pruitt, B. T. Roeder,A. Saastamoinen, L. G. Sobotka, and S. Upadhyayula, Phys. Rev. C , 041303(R) (2020).[10] M. Munch, M. Alcorta, H. O. U. Fynbo, M. Albers,S. Almaraz-Calderon, M. L. Avila, A. D. Ayangeakaa,B. B. Back, P. F. Bertone, P. F. F. Carnelli, M. P. Car-penter, C. J. Chiara, J. A. Clark, B. DiGiovine, J. P.Greene, J. L. Harker, C. R. Hoffman, N. J. Hubbard,C. L. Jiang, O. S. Kirsebom, T. Lauritsen, K. L. Laursen,S. T. Marley, C. Nair, O. Nusair, D. Santiago-Gonzalez,J. Sethi, D. Seweryniak, R. Talwar, C. Ugalde, andS. Zhu, Phys. Rev. C , 065803 (2016).[11] S. Zhang, A. Bonasera, M. Huang, H. Zheng, D. X. Wang,J. C. Wang, L. Lu, G. Zhang, Z. Kohley, Y. G. Ma, andS. J. Yennello, Phys. Rev. C , 044605 (2019).[12] S. Ishikawa, Phys. Rev. C , 055804 (2013).[13] N. B. Nguyen, F. M. Nunes, I. J. Thompson, and E. F.Brown, Phys. Rev. Lett. , 141101 (2012).[14] H. Zheng, A. Bonasera, M. Huang, and S. Zhang, PhysicsLetters B , 460 (2018).[15] H. O. U. Fynbo, C. A. Diget, U. C. Bergmann, M. J. G.Borge, J. Cederk¨all, P. Dendooven, L. M. Fraile, S. Fran-choo, V. N. Fedosseev, B. R. Fulton, W. Huang,J. Huikari, H. B. Jeppesen, A. S. Jokinen, P. Jones,B. Jonson, U. K¨oster, K. Langanke, M. Meister, T. Nils-son, G. Nyman, Y. Prezado, K. Riisager, S. Rinta-Antila,O. Tengblad, M. Turrion, Y. Wang, L. Weissman, K. Wil-helmsen, J. ¨Ayst¨o, and T. I. Collaboration, Nature433