* [20] M. Di Mauro, R. De Luca, S. Esposito and A. Naddeo, _Some insight into Feynman’s approach to electromagnetism_ , Eur. J. Phys. 42 (2021) 025206.
  * [21] A. Mercier and M. Kervaire (eds.), Fünfzig Jahre Relativitätstheorie, Birkhäuser, Basel, 1956.
  * [22] H. Everett III, _“Relative state” formulation of quantum mechanics_ , Rev. Mod. Phys. 29 (1957) 454-462.
  * [23] C. DeWitt-Morette, _The pursuit of quantum gravity: Memoirs of Bryce DeWitt from 1946 to 2004_ , Springer, Berlin, 2011.
  * [24] B. S. DeWitt, _The Everett-Wheeler Interpretation of Quantum Mechanics_ , in C. M. DeWitt and J. A. Wheeler (eds.) _Battelle Rencontres: 1967 Lectures in Mathematics and Physics_ , W. A. Benjamin, New York, 1968, pp 318-332.
  * [25] B. S. DeWitt, _Quantum Mechanics and Reality_ , Phys. Today 23(9) (1970), 30-35.
  * [26] B. S. DeWitt, L. Ballentine, P. Pearle, E. H. Walker, M. Sachs, T. Koga, J. Gerver, _Quantum Mechanics Debate_ , Phys. Today 24(4) (1971), 36-44.
  * [27] B. S. DeWitt, _The Many-Universes Interpretation of Quantum Mechanics_ , in _Proceedings of the International School of Physics “Enrico Fermi” Course IL: Foundations of Quantum Mechanics_ , Academic Press, New York (1971), pp 211-262.
  * [28] B. S. DeWitt, N. Graham (eds.) _The Many-Worlds Interpretation of Quantum Mechanics_ , Princeton University Press, Princeton, 1973.
  * [29] D. Kaiser, _A $\psi$ is just a $\psi$? Pedagogy, Practice and the Reconstitution of General Relativity, 1942-1975_, Stud. Hist. Phil. Mod. Phys. 29 (1998), 321-338.
  * [30] D. Kennefick, _Traveling at the speed of thought: Einstein and the quest for gravitational waves_ , Princeton University Press, Princeton, 2007.
  * [31] A. Einstein, Näherungsweise Integration der Feldgleichungen der Gravitation (Approximate Integration of the Field Equations of Gravitation), Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916), 688-696.
  * [32] A. Einstein, Über Gravitationswellen (On Gravitational Waves), Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1918 (1918), 154-167.
  * [33] A. S. Eddington, The Mathematical Theory of Relativity, 2nd ed, Cambridge University Press, Cambridge, 1960.
  * [34] A. S. Eddington, _The propagation of gravitational waves_ , Proc. Roy. Soc. Lond. A 102 (1922), 268-282.
  * [35] A. Pais, Subtle is the Lord. The Science and Life of Albert Einstein, Oxford University Press, Oxford, 1982.
  * [36] A. Einstein and M. Born, The Born-Einstein Letters: Friendship, Politics, and Physics in Uncertain Times, MacMillan, New York, 2005.
  * [37] A. Einstein and N. Rosen, _On Gravitational waves_ , J. Frank. Inst. 223 (1937), 43-54.
  * [38] F. A. E. Pirani, _On the Physical significance of the Riemann tensor_ , Acta Phys. Pol. 15 (1956), 389-405; reprinted in Gen. Rel. Grav. 41 (2009), 1215-1232.
  * [39] F. A. E. Pirani, _Invariant formulation of gravitational radiation theory_ , Phys. Rev. 105 (1957), 1089-1098.
  * [40] J. Weber and J. A. Wheeler, _Reality of the Cylindrical Gravitational Waves of Einstein and Rosen_ , Rev. Mod. Phys. 29 (1957), 509-515.
  * [41] L. Blanchet, _Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries_ , Living Rev. Relativity 9 (2006) 4.
  * [42] T. Futamase and Y. Itoh, _The Post-Newtonian Approximation for Relativistic Compact Binaries_ , Living Rev. Relativity 10 (2007) 2.
  * [43] L. D. Landau and E. M. Lifshitz, _The Classical Theory of Fields,_ first English edition, Addison- Wesley, Cambridge, 1951\.
  * [44] R. P. Feynman, J. A. Wheeler, _Reaction of the Absorber as the Mechanism of Radiative Damping_ (Abstract only), Phys. Rev. 59 (1941), 692.
  * [45] J. A. Wheeler, R. P. Feynman, _Interaction with the absorber as the mechanism of radiation_ , Rev. Mod. Phys. 17(1945), 157-181.
  * [46] J. A. Wheeler, R. P. Feynman, _Classical electrodynamics in terms of direct interparticle action_ , Rev. Mod. Phys. 21 (1945), 425-433.
  * [47] R. P. Feynman, unpublished letter to Victor F. Weisskopf, February 1961; in Richard P. Feynman Papers, California Institute of Technology Archives, Box 3, Folder 8.
  * [48] H. Bondi, _Plane gravitational waves in general relativity_ , Nature 179 (1957), 1072-1073.
  * [49] H. Bondi, F. A. E. Pirani and I. Robinson, _Gravitational waves in general relativity. 3. Exact plane waves_ , Proc. Roy. Soc. Lond. A 251 (1959), 519-533.
  * [50] I. Robinson and A. Trautman, _Spherical Gravitational Waves_ , Phys. Rev. Lett. 4 (1960), 431-432.
  * [51] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, _Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems_ , Proc. Roy. Soc. Lond. A 269 (1962), 21-52.
  * [52] R. K. Sachs, _Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times_ , Proc. Roy. Soc. Lond. A 270 (1962), 103-126.
  * [53] R. K. Sachs, _Gravitational radiation_ , In _Relativity, Groups and Topology_ , C. DeWitt and B. S. DeWitt eds., Gordon and Breach, New York, 1964, pp. 521-562.
  * [54] E. Newman and R. Penrose, _An Approach to gravitational radiation by a method of spin coefficients_ , J. Math. Phys. 3 (1962), 566-578.
  * [55] R. Penrose, _Zero rest mass fields including gravitation: Asymptotic behavior_ , Proc. Roy. Soc. Lond. A 284 (1965), 159-203.
  * [56] J. Weber, _Detection and Generation of Gravitational Waves_ , Phys. Rev. 117 (1960), 306-313.
  * [57] J. Weber, _Observation of the Thermal Fluctuations of a Gravitational-Wave Detector_ , Phys. Rev. Lett. 17 (1966), 1228-1230.
  * [58] J. Weber, _Evidence for discovery of gravitational radiation_ , Phys. Rev. Lett. 22 (1969), 1320-1324.
  * [59] H. D. Zeh, _Feynman’s interpretation of quantum theory_ , Eur. Phys. J. H 36 (2011), 63-74.
  * [60] J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Mathematical foundations of quantum mechanics), Springer, Berlin, 1932 (Chap. 6).
  * [61] W. H. Zurek, _Decoherence, einselection, and the quantum origins of the classical_ , Rev. Mod. Phys. 75 (2003), 715-776.
  * [62] F. Károlyházy, _Gravitation and quantum mechanics of macroscopic objects_ , Il Nuovo Cimento A 42 (1966), 390-402.
  * [63] A. Frenkel, _Spontaneous localizations of the wave function and classical behavior_ , Found. Phys. 20 (1990), 159-188.
  * [64] L. Diosi, _Gravitation and quantum mechanical localization of macroobjects_ , Phys. Lett. A 105 (1984), 199-202.
  * [65] L. Diosi, _Models for universal reduction of macroscopic quantum fluctuations_ , Phys. Rev. A 40 (1989), 1165-1174.
  * [66] L. Diosi, _Notes on certain Newton gravity mechanisms of wavefunction localization and decoherence_ , J. Phys. A: Math. Gen. 40 (2007), 2989-2995.
  * [67] R. Penrose, _Gravity and state-vector reduction_ , in R. Penrose and C. J. Isham (Eds.), Quantum Concepts in Space and Time, Clarendon Press, Oxford, 1986 (p. 129).
  * [68] R. Penrose, _On gravity’s role in quantum state reduction_ Gen. Rel. Grav. 28 (1996), 581-600.
  * [69] R. Penrose, _Quantum computation, entanglement and state reduction_ , Phil. Trans. R. Soc. Lond. A 356 (1998), 1927-1939.
  * [70] R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Jonathan Cape, London, 2004.
  * [71] G. C. Ghirardi, R. Grassi and A. Rimini, _Continuous spontaneous reduction model involving gravity_ , Phys. Rev. A 42 (1990), 1057-1064.
  * [72] G. C. Ghirardi, A. Rimini and T. Weber, _Unified Dynamics for Microscopic and Macroscopic Systems_ , Phys. Rev. D 34 (1986), 470-491.
  * [73] P. Pearle, _Ways to describe dynamical state-vector reduction_ , Phys. Rev. A 48 (1993), 913-923.
  * [74] A. Bassi and G. C. Ghirardi, _Dynamical reduction models_ , Phys. Rept. 379 (2003), 257-426.
  * [75] S. Gerlich, S. Eibenberger, M. Tomandl, S. Nimmrichter, K. Hornberger, P. J. Fagan, J. Tuxen, M. Mayor and M. Arndt, _Quantum interference of large organic molecules_ , Nat. Commun. 2 (2011), 263.
  * [76] S. Eibenberger, S. Gerlich, M. Arndt, M. Mayor and J. Tuxen, _Matter-wave interference of particles selected from a molecular library with masses exceeding 10 000 amu_ , Phys. Chem. Chem. Phys. 15 (2013), 14696-14700.
  * [77] I. Pikovski, M. Zych, F. Costa and C. Brukner, _Universal decoherence due to gravitational time dilation_ , Nat. Phys. 11 (2015), 668-672.
  * [78] M. Zych, F. Costa, I. Pikovski and C. Brukner, _Quantum interferometric visibility as a witness of general relativistic proper time_ , Nat. Commun. 2 (2011), 505.
  * [79] Y. Margalit, Z. Zhou, S. Machluf, D. Rohrlich, Y. Japha and R. Folman, _A self-interfering clock as a “which path” witness_ , Science 349 (2015), 1205-1208.
  * [80] S. Bose, K. Jacobs and P. L. Knight, _Scheme to probe the decoherence of a macroscopic object_ , Phys. Rev. A 59 (1999), 3204-3210.
  * [81] W. Marshall, C. Simon, R. Penrose and D. Bouwmeester, _Towards quantum superpositions of a mirror_ Phys. Rev. Lett. 91 (2003), 130401.
  * [82] F. Maimone, G. Scelza, A. Naddeo and V. Pelino, _Quantum superpositions of a mirror for experimental tests for nonunitary Newtonian gravity_ , Phys. Rev. A 83 (2011), 062124.
  * [83] O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer and J. I. Cirac, _Large Quantum Superpositions and Interference of Massive Nanometer-Sized Objects_ , Phys. Rev. Lett. 107 (2011), 020405.
  * [84] H. Pino, J. Prat-Camps, K. Sinha, B. P. Venkatesh and O. Romero-Isart, _On-chip quantum interference of a superconducting microsphere_ , Quantum Sci. Technol. 3 (2018), 25001.
  * [85] S. Osnaghi, F. Freitas, O. Freire Jr., _The origin of the Everettian heresy_ , Stud. Hist. Phil. Mod. Phys. 40 (2009), 97-123.
  * [86] J. Preskill, _Feynman after 40_ , Talk given at APS April Meeting, 16 April 2018, available at http://theory.caltech.edu/~preskill/talks/APS-April-2018-Feynman-4-3.pdf, accessed on 22 May 2021.
  * [87] M. Gell-Mann and J. B. Hartle, _Classical equations for quantum systems_ , Phys. Rev. D 47 (1993), 3345-3382.
  * [88] R. B. Griffiths, Consistent Quantum Theory, Cambridge University Press, Cambridge, 2002.
  * [89] R. Omnes, Interpretation of Quantum Mechanics, Princeton University Press, Princeton, 1994.
  * [90] M. Gell-Mann and J. B. Hartle, _Quantum Mechanics in the Light of Quantum Cosmology_ , in W. H. Zurek (Ed.), Complexity, Entropy, and the Physics of Information, SFI Studies in the Sciences of Complexity, Vol. VIII, Addison Wesley, Reading, MA, 1990.
  * [91] P. C. Hohenberg, _An Introduction to consistent quantum theory_ , Rev. Mod. Phys. 82 (2010), 2835-2844.
  * [92] J. B. Hartle, _The Quantum Mechanics of Closed Systems_ , in B. L. Hu, M. P. Ryan, C. V. Vishveshwara (Eds.), Directions in General Relativity, Volume 1, Cambridge University Press, Cambridge, 1993.
  * [93] J. B. Hartle, _The Impact of Cosmology on Quantum Mechanics_ , [arXiv:1901.03933 [gr-qc]].
  * [94] R. H. Kraichnan, _Special-Relativistic Derivation of Generally Covariant Gravitation Theory_ , Phys. Rev. 98 (1955), 1118-1122.
  * [95] S. N. Gupta, _Gravitation and Electromagnetism_ , Phys. Rev. 96 (1954), 1683-1685.
  * [96] J. Preskill and K. S. Thorne, Foreword to Ref. [12].
  * [97] M. Fierz and W. Pauli, _On relativistic wave equations for particles of arbitrary spin in an electromagnetic field_ , Proc. Roy. Soc. Lond. A 173 (1939), 211-232.
  * [98] P. A. M. Dirac, _Relativistic wave equations_ , Proc. Roy. Soc. Lond. A 155 (1936), 447-459.
  * [99] T. Padmanabhan, _From gravitons to gravity: Myths and reality_ , Int. J. Mod. Phys. D 17 (2008), 367-398.
  * [100] D. Rickles, Covered with deep mist, Oxford University Press, Oxford, 2020.
  * [101] A. S. Blum and D. Rickles, Quantum Gravity in the First Half of the Twentieth Century: A Sourcebook, Edition Open Access, Berlin, 2018.
  * [102] A. Blum, T. Hartz, _The 1957 quantum gravity meeting in Copenhagen: An analysis of Bryce S. DeWitt’s report_ , Eur. Phys. J. H 42 (2017), 107-157.
  * [103] W. Heisenberg and W. Pauli, _Zur Quantenelektrodynamik der Wellenfelder (On the quantum electrodynamics of wave fields)_ , Zeitschrift fur Physik 56 (1929), 1-61.
  * [104] L. Rosenfeld, _Zur Quantelung der Wellenfelder (On the quantization of wave fields_ , Annalen der Physik 5 (1930) 1113-1152. English translation by D. C. Salisbury and K. Sundermeyer, Eur. Phys. J H 42 (2017), 63-94.
  * [105] L. Rosenfeld, _Über die Gravitationswirkungen des Lichtes (On the gravitational effects of light)_ , Zeitschrift fur Physik 65 (1930), 589-599.
  * [106] G. Peruzzi and A. Rocci, _Tales from the prehistory of Quantum Gravity: Léon Rosenfeld’s earliest contributions_ , Eur. Phys. J. H 43 (2018), 185-241.
  * [107] D. C. Salisbury, _Leon Rosenfeld’s pioneering steps toward a quantum theory of gravity_ , J. Phys. Conf. Ser. 222 (2010), 012052.
  * [108] D. C. Salisbury, K. Sundermeyer, _Leon Rosenfeld’s general theory of constrained Hamiltonian dynamics_ , Eur. J. Phys. H 42 (2017), 23-61.
  * [109] D. Salisbury, _Leon Rosenfeld and the challenge of the vanishing momentum in quantum electrodynamics_ , Stud. Hist. Phil. Sci. B 40 (2009), 363-373.
  * [110] J. Stachel, _The early history of quantum gravity (1916-1940)’_ , in B. R. Iyer and B. Bhawal (Eds.), Black Holes, Gravitational radiation and the Universe, Kluwer, The Netherlands, 1999.
  * [111] D. I. Blokhintsev and F. M. Gal’perin, _Gipoteza neitrino i zakon sokhraneniya energii (Neutrino hypothesis and energy conservation)_ , Pod znamenem marxisma 6 (1934), 147-157.
  * [112] M. P. Bronstein, _Quantentheories schwacher Gravitationsfeldern (Quantum theory of weak gravitational fields)_ , Physikalische Zeitschrift der Sowietunion 9 (1936), 140-157. English translation by M. A. Kurkov, edited by S. Deser, Gen. Relativ. Grav. 44 (2012), 267-283.
  * [113] G. E. Gorelik, _Matvei Bronstein and quantum gravity: 70th anniversary of the unsolved problem_ , Phys. Usp. 48 (2005), 1039-1053.
  * [114] G. E. Gorelik, V. Ya. Frenkel, _Matvei Petrovich Bronstein and Soviet theoretical physics in the sixties,_ Birkhäuser, 1994\.
  * [115] S. Deser, A. Starobinsky, _Editorial note to: Matvei P. Bronstein, Quantum theory of weak gravitational fields_ , Gen. Relativ. Grav. 44 (2012), 263-265.
  * [116] E. Fermi, _Quantum theory of radiation_ , Rev. Mod. Phys.4 (1932), 87-132.
  * [117] N. Bohr, L. Rosenfeld, _Zur Frage der Messbarkeit der elektromagnetischen Feldgrößen (On the question of measurability of electromagnetic field quantities)_ , Det Kgl. Danske Videnskabernes Selskab Mathematisk-fysiske Meddelelser 12 (1933), 3-65.
  * [118] J. Solomon, _Gravitation et Quanta_ , Journal de Physique et de Radium 9 (1938), 479-485.
  * [119] N. Rosen _Plane Polarised Waves in the General Theory of Relativity_ , Phys. Zeitsch. der Sowjetunion 12, 366-372.
  * [120] B. DeWitt, _I: The Theory of Gravitational Interactions. II: The Interaction of Gravitation with Light_ , Ph.D. thesis, Harvard, 1949\.
  * [121] J. Schwinger, _Quantum electrodynamics. I. A covariant formulation_ , Phys. Rev. 74 (1948), 1439-1461.
  * [122] P. G. Bergmann, _Non-linear Field Theories_ , Phys. Rev. 75 (1949), 680-685.
  * [123] P. G. Bergmann, J. H. M. Brunings, _Non-linear Field Theories II: Canonical Equations and Quantization_ , Rev. Mod. Phys. 21 (1949), 480-487.
  * [124] D. C. Salisbury, _Peter Bergmann and the invention of constrained Hamiltonian dynamics_ , Einstein Stud. 12 (2012), 247-257.
  * [125] D. C. Salisbury, _Rosenfeld, Bergmann, Dirac and the invention of constrained Hamiltonian dynamics_ , in H. Kleinert, R. T. Jantzen, R. Ruffini (eds.), _Proceedings of the eleventh Marcel Grossmann meeting on general relativity_ , World Scientific, Singapore, 2008.
  * [126] P. A. M. Dirac, _Generalized Hamiltonian dynamics_ , Canadian J. Math. 2 (1950), 129-148.
  * [127] P. A. M. Dirac, _Forms of Relativistic Dynamics_ , Rev. Mod. Phys. 21 (1949), 392-399.
  * [128] F. A. E. Pirani and A. Schild, _On the quantization of Einstein’s gravitational field equations_ , Phys. Rev. 79 (1950), 986-991.
  * [129] R. P. Feynman, _Problems in quantizing the gravitational field and the massless Yang-Mills field_ , in Ref. [141].
  * [130] S. Weinberg, _Derivation of gauge invariance and the equivalence principle from Lorentz invariance of the S- matrix_ , Phys. Lett. 9 (1964), 357-359.
  * [131] S. Weinberg, _Photons and gravitons in perturbation theory: Derivation of Maxwell’s and Einstein’s equations_ , Phys. Rev. 138 (1965), B988-B1002.
  * [132] S. Deser, _Self-interaction and gauge invariance_ , Gen. Rel. Grav. 1 (1970), 9-18.
  * [133] D. G. Boulware and S. Deser, _Classical General Relativity Derived from Quantum Gravity_ , Annals Phys. 89 (1975), 193-240.
  * [134] S. Deser, _Gravity From Self-interaction in a Curved Background_ , Class. Quant. Grav. 4 (1987), L99-L106.
  * [135] S. Deser, _Gravity from self-interaction redux_ , Gen. Rel. Grav. 42 (2010), 641-646.
  * [136] R. M. Wald, _Spin-2 Fields and General Covariance_ , Phys. Rev. D 33 (1986), 3613-3625
  * [137] K. S. Thorne, Black holes and time warps: Einstein’s outrageous legacy, Norton, New York, 1994.
  * [138] G. F. Chew, S-matrix theory of strong interactions: a lecture note and reprint volume, Frontiers in physics series, W.A. Benjamin, New York, 1961.
  * [139] W. R. Frazer, _Theory of Weak and Strong Interactions_ , Physics Today 14(12) (1961), 80-84.
  * [140] R. P. Feynman, _Closed loop and tree diagram_ , in Ref. [141].
  * [141] J. R. Klauder (ed.), Magic Without Magic - John Archibald Wheeler. A Collection Of Essays In Honor Of His 60th Birthday, Freeman, San Francisco, 1972.
  * [142] A. Einstein, _Notes on the Origin of the General Theory of Relativity_ , In _Ideas and Opinions_ , Translated by Sonja Bargmann, Crown, New York, 1954, pp. 285-290.
  * [143] G. Nordström, _Relativitätsprinzip und Gravo+itation (The principle of relativity and gravitation)_ , Phys. Z. 13 (1912), 1126-1129.
  * [144] J. D. Norton, _Einstein, Nordström and the early demise of scalar, Lorentz-covariant theories of gravitation_ , Arch. Hist. Exact Sci. 45 (1992), 17-94. Reprinted in J. Renn (ed.), _The Genesis of General Relativity, Vol. 3: Theories of Gravitation in the Twilight of Classical Physics. Part I._ , Kluwer, 2005, pp. 413-487.
  * [145] R. P. Feynman, Physics 2C: Miscellaneous notes, Folder 40.5, available online at https://www.feynmanlectures.caltech.edu/Notes.html
  * [146] A. Trautman, D. Salisbury, _Memories of my early career in relativity physics_ , Eur. Phys. J. H 44 (2019), 391-413.
  * [147] R. Britto, _Loop Amplitudes in Gauge Theories: Modern Analytic Approaches_ , J. Phys. A 44 (2011), 454006.
  * [148] I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, _Feynman’s Tree Theorem and Loop-Tree Dualities_ , PoS LC2010 (2010), 034.
  * [149] S. Caron-Huot, _Loops and trees_ , JHEP 05 (2011), 080.
  * [150] M. Maniatis, _Application of the Feynman-tree theorem together with BCFW recursion relations_ , Int. J. Mod. Phys. A 33 (2018) 1850042.
  * [151] B. S. DeWitt, _Theory of radiative corrections for non-abelian gauge fields_ , Phys. Rev. Lett. 12 (1964), 742-746.
  * [152] B. S. DeWitt, _Quantum Theory of Gravity. 2. The Manifestly Covariant Theory_ , Phys. Rev. 162 (1967), 1195-1239.
  * [153] B. S. DeWitt, _Quantum Theory of Gravity. 3. Applications of the Covariant Theory_ , Phys. Rev. 162 (1967), 1239-1256.
  * [154] L. D. Faddeev and V. N. Popov, _Feynman Diagrams for the Yang-Mills Field_ , Phys. Lett. B 25 (1967), 29-30.
  * [155] K. S. Stelle, _Renormalization of Higher Derivative Quantum Gravity_ , Phys. Rev. D 16 (1977), 953-969.
  * [156] D. Anselmi, _On the quantum field theory of the gravitational interactions_ , JHEP 06 (2017), 086.
  * [157] G. ’t Hooft, _Renormalization of Massless Yang-Mills Fields_ , Nucl. Phys. B 33 (1971), 173-199.
  * [158] G. ’t Hooft, _Renormalizable Lagrangians for Massive Yang-Mills Fields_ , Nucl. Phys. B 35 (1971), 167-188.
  * [159] G. ’t Hooft and M. J. G. Veltman, _Regularization and Renormalization of Gauge Fields_ , Nucl. Phys. B 44 (1972), 189-213.
  * [160] G. ’t Hooft and M. J. G. Veltman, _Combinatorics of gauge fields_ , Nucl. Phys. B 50 (1972), 318-353.
  * [161] G. ’t Hooft and M. J. G. Veltman, _One loop divergencies in the theory of gravitation_ , Ann. Inst. H. Poincare Phys. Theor. A 20 (1974), 69-94.
  * [162] M. H. Goroff and A. Sagnotti, _Quantum gravity at two loops_ , Phys. Lett. B 160 (1985), 81-86.
  * [163] M. H. Goroff and A. Sagnotti, _The Ultraviolet Behavior of Einstein Gravity_ , Nucl. Phys. B 266 (1986), 709-736.
  * [164] A. E. M. van de Ven, _Two loop quantum gravity_ , Nucl. Phys. B 378 (1992), 309-366.
  * [165] J. Mehra, _The Beat of a different drum: The Life and science of Richard Feynman_ , Oxford University Press, Oxford, 1994.
  * [166] S. Weinberg, _On the Development of Effective Field Theory_ , Eur. Phys. J. H 46 (2021), 6.
  * [167] S. Weinberg, _Effective field theory, past and future_ , Int. J. Mod. Phys. A 31 (2016), 1630007.
  * [168] C. P. Burgess, _Quantum gravity in everyday life: General relativity as an effective field theory_ , Living Rev. Rel. 7 (2004), 5-56
  * [169] F. Rohrlich, _Classical Charged Particles_ , 3rd ed., World Scientific, Singapore 2007.
  * [170] T. Fulton and F. Rohrlich, _Classical radiation from a uniformly accelerated charge_ , Ann. Phys. 9 (1960), 499-517.
  * [171] F. Rohrlich, _The principle of equivalence_ , Ann. Phys. 22 (1963), 169-191.
  * [172] B. S. DeWitt and R. W. Brehme, _Radiation damping in a gravitational field_ , Ann. Phys. 9 (1960), 220-259.
  * [173] C. M. DeWitt and B. S. DeWitt, _Falling charges_ , Physics 1 (1964), 3-20; err. ibid. 1 (1964), 145.
  * [174] D. G. Boulware, _Radiation from a Uniformly Accelerated Charge_ , Ann. Phys. 124 (1980), 169-188.
  * [175] C. de Almeida, and A. Saa, _The radiation of a uniformly accelerated charge is beyond the horizon: A simple derivation_ , Am. J. Phys. 74 (2006), 154-158.
  * [176] J. Weber, _General Relativity and Gravitational Waves_ , Interscience Publishers, New York, 1961.
  * [177] A. Einstein, _Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the influence of gravitation on the propagation of light)_ , Ann. Phys. 35 (1911), 898-908.
  * [178] A. Einstein, _Grundlage der allgemeinen Relativitätstheorie (The foundation of the general theory of relativity)_ , Ann. Phys. 49 (1916), 769-822.
  * [179] R.P. Feynman, _Working Notes and Calculations: Alternate Way to Handle Elctrodynamics_ ; December 13, 1963, Richard P. Feynman Papers, California Institute of Technology Archives, Box 62, Folder 8. Notes scanned and transcribed by M. A. Gottlieb, available online at http://www.feynmanlectures.caltech.edu/info/other/Alternate_Way_to_Handle_Electrodynamics.html, accessed on 16 May 2021.
  * [180] K. Krasnov, _Formulations of general relativity. Gravity, spinors and differential forms_ , Cambridge University Press, Cambridge, 2000.


# Updated evaluation of potential ultra-low Q value $\beta$ decay candidates

D. K. Keblbeck  Department of Physics, Central Michigan University, Mount
Pleasant, MI 48859, USA  R. Bhandari  Department of Physics, Central Michigan
University, Mount Pleasant, MI 48859, USA  N. D. Gamage  Department of
Physics, Central Michigan University, Mount Pleasant, MI 48859, USA  Facility
for Rare Isotope Beams, Michigan State University, East Lansing, Michigan,
48824 USA  M. Horana Gamage  Department of Physics, Central Michigan
University, Mount Pleasant, MI 48859, USA  K. G. Leach  Department of Physics,
Colorado School of Mines, Golden, Colorado 80401, USA  Facility for Rare
Isotope Beams, Michigan State University, East Lansing, Michigan, 48824 USA
X. Mougeot  Université Paris-Saclay, CEA, List, Laboratoire National Henri
Becquerel (LNE-LNHB), F-91120 Palaiseau, France  M. Redshaw  Department of
Physics, Central Michigan University, Mount Pleasant, MI 48859, USA  National
Superconducting Cyclotron Laboratory, East Lansing, Michigan, 48824 USA

(August 27, 2024)

###### Abstract

“Ultra-low” Q value $\beta$ decays are referred to as such due to their low
decay energies of less than $\approx$1 keV. Such a low energy decay is
possible when the parent nucleus decays to an excited state in the daughter,
with an energy close to that of the Q value. These decays are of interest as
potential new candidates for neutrino mass determination experiments and as a
testing ground for studies of atomic interference effects in the nuclear decay
process. In this paper, we provide an updated evaluation of atomic mass data
and nuclear energy level data to identify potential ultra-low Q value $\beta$
decay candidates. For many of these candidates, more precise and accurate
atomic mass data is needed to determine if the Q value of the potential ultra-
low decay branch is energetically allowed and in fact ultra-low. The relevant
precise atomic mass measurements can be achieved via Penning trap mass
spectrometry.

##  I Introduction

From the proposed existence of the neutrino, to the development of electroweak
theory, nuclear $\beta$-decay has played a central role in our understanding
and development of sub atomic physics. Experimental measurements and
theoretical descriptions of $\beta$-decay continue to impact nuclear and
particle physics, for example, via experiments that place constraints on the
neutrino mass Aseev  _et al._ (2011); Kraus  _et al._ (2005); Aker (2019);
Arnaboldi  _et al._ (2003); Springer  _et al._ (1987), searches for sterile
neutrinos Friedrich  _et al._ (2021); Martoff  _et al._ (2021) and
neutrinoless double $\beta$-decay Avignone  _et al._ (2008); Blaum  _et al._
(2020), and via experiments that test the validity of the Standard Model (SM)
and constrain physics beyond the SM Falkowski  _et al._ (2021); González-
Alonso  _et al._ (2019).

In the context of direct neutrino mass determination experiments, the approach
is to extract a value or upper limit for the (electron) neutrino mass from the
slight distortion of the $\beta$-decay or electron capture (EC) de-excitation
energy spectrum near the end-point due to a non-zero neutrino mass Formaggio
_et al._ (2021). The fraction of decays in an energy interval $\Delta E$, near
the end-point, goes as ($\Delta E/Q$)3 Ferri  _et al._ (2015) for
$\beta$-decay, and as ($\Delta E/Q$)2 for EC Ge  _et al._ (2021a) (but can be
enhanced when the binding energy of the captured orbital electron is close to
the Q value). Hence, a low Q value decay is an important characteristic for
the isotope to be studied. 3H, 187Re, and 163Ho, have the lowest known ground-
state to ground-state (gs-gs) Q values Myers  _et al._ (2015); Nesterenko  _et
al._ (2014); Eliseev  _et al._ (2015) and have provided the most stringent
upper limits on the neutrino and anti-neutrino masses to date. Nevertheless,
even lower Q value decays are possible for certain isotopes when the decay
occurs to an excited nuclear state in the daughter with an energy very close
to the gs-gs Q value, $Q_{gs}$. Such an isotope could be of interest as a
potential candidate for future direct neutrino mass determination experiments.

For $\beta^{-}$ and EC decay, $Q_{gs}$ is defined as the energy equivalent of
the mass difference between parent ($P$) and daughter ($D$) atoms:

$Q_{gs}^{\beta^{-}}=[M_{P}(^{A}_{Z}P)-M_{D}(^{\quad A}_{Z+1}D)]c^{2},$ (1)
$Q_{gs}^{EC}=[M_{P^{{}^{\prime}}}(^{A}_{Z}P^{{}^{\prime}})-M_{D^{{}^{\prime}}}(^{\quad
A}_{Z-1}D^{{}^{\prime}})]c^{2},$ (2)

where $M_{P^{(^{\prime})}}$ and $M_{D^{(^{\prime})}}$ are the masses of the
parent and daughter atoms, respectively. For $\beta^{+}$ decay, $Q_{gs}$ is
reduced due to the two additional electrons whose mass needs to be accounted
for:

$Q_{gs}^{\beta^{+}}=Q_{gs}^{EC}-2m_{e}c^{2},$ (3)

where $m_{e}$ is the electron mass. The Q value for a decay to an excited
state in the daughter with energy $E^{*}$ is given by

$Q_{es}=Q_{gs}-E^{*},$ (4)

and could potentially be very low. A decay with $Q_{es}<$ 1 keV, has been
dubbed as “ultra-low” Mustonen and Suhonen (2010a).

The identification of the first, and currently only known ultra-low Q value
decay was made by Cattadori et al. Cattadori  _et al._ (2005). The authors
inferred the existence of a weak and potentially very low Q value
$\beta$-decay branch in 115In to the first excited state in 115Sn via the
detection of a 497 keV $\gamma$-ray that was assumed to come from the
subsequent decay of the 115Sn(3/2+; 497.3 keV) state. A Q value of 2(4) keV
was deduced using available mass data for 115In and 115Sn and the energy of
the 497.334(22) keV 3/2+ state in 115Sn Blachot (2005). This decay was
confirmed to be energetically allowed by independent Penning trap measurements
of the 115In – 115Sn mass difference performed at Florida State University
Mount  _et al._ (2009) and with JYLFTRAP at the University of Jyväskylä
Wieslander  _et al._ (2009). The resulting $Q_{es}$ values were 0.155(24) keV
and 0.35(17) keV, respectively, indicating that the decay branch in question
is energetically allowed, and making it the lowest-known Q value
$\beta$-decay. Recent measurements of the 115In 3/2+ daughter state energy
provide results of 497.342(3) keV Zheltonozhsky  _et al._ (2018) and
497.316(7) keV Urban  _et al._ (2016). These results both agree with the
previous value Blachot (2005) and reduce the uncertainty, but disagree with
each other at the 3.4$\sigma$ level.

In addition to their relevance to direct neutrino mass determination
experiments, ultra-low Q value $\beta$-decays also serve as important test
cases for probing the boundaries between nuclear and atomic physics and
investigating atomic interference effects in nuclear $\beta$-decay.
Theoretical studies of the ultra-low Q value decay branch of 115In Mustonen
and Suhonen (2010a) revealed a discrepancy between the calculated and
experimental half-life for the ultra-low decay branch. Theoretical
calculations of the 115In gs-gs decay rate, however, were in reasonably good
agreement with the experimental result, suggesting that the discrepancy may be
due to atomic interference effects that become more significant when the Q
value is very small. However, more cases are required for study in order to
draw definite conclusions.

To push the sensitivity of $\beta$-decay-based neutrino mass measurements
below the 0.1 eV sensitivity level, a suitable ultra-low Q candidate is
required. Given the extremely small phase-space for these decays, an
experimentally suitable candidate needs to have a relatively large branching
ratio and short half-life to the state of interest in the daughter (ie. a
short partial half-life $t=\frac{T_{1/2}}{BR}$) to generate a signal in a
reasonable experimental time.

This paper is organized as follows: In section II, we review the literature on
theoretical and experimental studies of potential ultra-low Q value
$\beta$-decay candidates and discuss our updated evaluation of all potential
ultra-low Q value decays across the nuclear chart. In section III we present
our results for $\beta^{\pm}$ and EC decay candidates that could undergo an
ultra-low Q value decay. In section IV we provide a discussion of our results
and describe some of the experimental and theoretical challenges that would be
involved in using an ultra-low Q value decay in a direct neutrino mass
determination experiment. Section V then provides a brief conclusion.

##  II Evaluation of potential ultra-low Q value candidates

After the discovery of the 115In ultra-low Q value $\beta$-decay branch,
Suhonen, et al. identified a number of other potential ultra-low Q value decay
candidates, including 135Cs Mustonen and Suhonen (2011), 115Cd Haaranen and
Suhonen (2013), and several others Mustonen and Suhonen (2010b); Suhonen
(2014). The use of very low Q value decays in the context of neutrino mass
determination experiments was discussed earlier by Kopp and Merhl Kopp and
Merle (2010), who identified a number of potential candidates, and our
previous work provided a large survey of potential ultra-low Q value
candidates Gamage  _et al._ (2019).

Since the publication of these previous works, a number of precise Penning
trap mass measurements have been performed to investigate potential ultra-low
Q value decay candidates. Significantly, the JYFLTRAP group confirmed that
135Cs does potentially have an energetically allowed and ultra-low Q value
$\beta$-decay branch de Roubin  _et al._ (2020) with $Q_{es}$ = 0.44(31) keV.
They also determined that 159Dy Ge  _et al._ (2021a) and 111In Ge  _et al._
(2022a) have low energy EC decay branches (with $Q_{es}\sim$1 – 4 keV) to
excited states in their respective daughter nuclei, and that 131I has an
energetically allowed $\beta$-decay branch with $Q_{es}\approx$ 1 keV Eronen
_et al._ (2022). All of these transitions could be of interest for future
neutrino mass determination experiments. Measurements at JYFLTRAP Ramalho  _et
al._ (2022) and LEBIT Horana Gamage  _et al._ (2022) also determined that 75Se
has an energetically allowed EC decay branch to the 865.4(5) keV (3/2- or
5/2-) state in 75As with $Q_{es}\lesssim$ 1 keV. However, a more precise
determination of the 865.4 keV energy level is needed to further evaluate this
decay channel.

Other potential ultra-low Q value decays branches in 115Cd, identified in
Haaranen and Suhonen (2013), and 75Ge, 89Sr, 139Ba, and 112,113Ag, identified
in Ref. Gamage  _et al._ (2019), have been investigated by the LEBIT group at
the National Superconducting Cyclotron Facility Sandler  _et al._ (2019) and
the CPT group at Argonne National Laboratory Gamage  _et al._ (2022) and have
been ruled out as not being ultra-low. The ISOLTRAP group performed a precise
measurement of the 131Cs EC Q value, indicating that it does not have an
ultra-low Q value decay branch Karthein  _et al._ (2019), and JYLFTRAP
recently investigated and ruled out potential ultra-low Q value decay branches
in 72,76As, 75Ge, and 155Tb Ge  _et al._ (2021b); Ramalho  _et al._ (2022); Ge
_et al._ (2022b).

In this work, we provide an update to our previous survey of potential ultra-
low Q value decay candidates using the recently published update to the Atomic
Mass Evaluation (AME2020) Wang  _et al._ (2021), and most recent nuclear
energy level data compiled by the National Nuclear Data Center nnd . In our
evaluation we have identified candidates with a half-life $\geq$ 1 d, that
could potentially undergo $\beta^{\pm}$ or EC decay to an excited state in the
daughter nucleus. The main criteria we used to select candidates was that the
difference between the gs-gs Q value and the excited state in the daughter
nucleus be $\sim\pm$10 keV (see Fig. 1). This range was chosen to account for
the typical maximum size of observed shifts in atomic masses of isotopes close
to stability after precise measurements with a Penning trap compared to values
obtained from indirect methods see e.g. Refs. Shi  _et al._ (2005); Redshaw
_et al._ (2012); Fink  _et al._ (2012).

Figure 1: Decay scheme for potential ultra-low Q value candidates showing the
energy range of the excited state of the daughter nucleus around the gs-gs Q
value.

For this updated survey, we have taken into account binding energies of K, L,
or M shell electrons for the EC decays, which modifies the Q value of an EC
decay to an excited state in the daughter as follows:

$Q_{es}^{EC}=Q_{gs}^{EC}-E^{*}-E_{B},$ (5)

where $E_{B}$ is the binding energy of the K, L, or M shell electron in the
daughter, obtained from Ref. Larkins (1977). For L and M shell capture, an
average of the L1–L3 and M1–M5 energies, respectively, from Ref. Larkins
(1977) were used 111The spread of the L1, L2, L3 and M1, M2, M3, M4, and M5
energies is much less than our criteria that $Q^{EC}_{es}-E_{B}$ be within
$\pm$10 keV of $E^{*}$..

The list of potential UL Q value candidates was further refined based on the
uncertainties in $Q_{gs}$ and $E^{*}$: candidates with $\sigma(Q_{gs})\gtrsim$
10 keV and $\sigma(E^{*})\gtrsim$ 1 keV were generally not included. Selection
was also determined by whether or not the masses had been previously measured
via Penning trap mass spectrometry. Decays with a forbiddenness greater than
fourth-forbidden were also not considered since no such decays have been
observed Quarati  _et al._ (2022).

##  III Results

Our updated analysis provides a list of 77 potential ultra-low Q value $\beta$
and EC decay candidate isotopes, involving around 100 transitions to distinct
energy levels in their respective daughter nuclei. These are listed in Tables
1, 2 and 4. This analysis shows a decrease in the number of candidate isotopes
compared to our previous evaluation Gamage  _et al._ (2019), primarily due to
the more stringent minimum half-life cutoff of one day that we imposed in this
evaluation as compared to one hr in our previous one. This condition was
chosen to provide a more realistic list of candidates that could be used for
experimental study. Some additional candidates have been included in this
evaluation due to changes in atomic masses between the 2016 and 2020 atomic
mass evaluations, and due to the inclusion of electron binding energies in the
$Q_{es}^{EC}$ calculation. The isotopes 72,76As, 75Ge, 89Sr, 112Ag, 115Cd,
131Cs, and 155Tb whose Q values have been precisely measured by Penning trap
mass spectrometry in the last few years, ruling them out as not having ultra-
low Q value transitions, have been omitted from these tables. 139Ba and 113Ag,
that both do still have potential UL Q value decay transitions have also been
left out of the current tables because they have half-lives of $<$ 1 d and the
uncertainties in the energies of the relevant daughter states is $\approx$10
keV. Isotopes whose Q values have been measured with a Penning trap and have
been shown to have a low or potentially ultra-low Q value decay branch e.g.
75Se, 111In, 131I, 135Cs, and 159Dy have been included.

In the tables of results, Column I lists the parent isotope along with its
ground state spin and parity assignment, and Column II its half-life. Column
III lists the daughter isotope with spin and parity assignment for the excited
state to which the potential ultra-low Q value decay would occur, and Column V
the energy of that daughter state. Column IV provides the gs-gs Q value from
Eqns. (1), (2) or (3) for $\beta^{-}$, EC, or $\beta^{+}$ decay, respectively.
In Table 2, Column VII lists the electron binding energy for K, L, and M shell
electrons (as listed in Column VI) in the daughter atom. Table 2 is divided
into three sections, upper for K shell capture, middle for L shell capture,
and lower for M shell capture. The final two columns in the tables list the
forbiddenness of the decay (see e.g. Ref. Suhonen (2007) for details and For
for a forbiddenness calculator tool) and the Q value for a decay to the
excited state, which can be considered as ultra-low if it is energetically
allowed and $<$ 1 keV. The uncertainty in the ultra-low Q value includes the
uncertainty from $Q_{gs}$ and $E^{*}$ (if available) added in quadrature.

Table 1: Potential ultra-low Q value $\beta^{-}$ decays. Columns I and III list parent and daughter isotopes, respectively, and their spin and parity ($J^{\pi}$) assignments. $J^{\pi}$ assignments enclosed by braces indicates uncertain assignments, resulting in uncertainty in the decay type, indicated by a {?} in Column VI. For uncertain ($J^{\pi}$) assignments, we have listed the $J^{\pi}$ that results in the least-forbidden decay. The parent half-life is listed in Column II. Column IV lists the gs-gs Q value (Ref. Wang  _et al._ (2021)); Column V, the excited state energy of the daughter nucleus (Ref. nnd ) to which the potential ultra-low Q value decay would occur; and Column VII, the Q value of the decay to the excited state. Parent | T1/2 | Daughter |  $Q_{gs}$ (keV) |  $E^{*}$ (keV) | Decay type |  $Q_{es}$ (keV)
---|---|---|---|---|---|---
47Ca(7/2-) | 5 d |  47Sc(3/2+) | 1992.2(29) | 2002.30(30) | 1st FU | -10.1(30)
74As(2-) | 18 d |  74Se(4+) | 1353.1(17) | 1363.17(7) | 1st FU | -10.0(17)
77As(3/2-) | 2 d |  77Se(5/2+) | 683.2(17) | 680.10(20) | 1st FNU | 3.1(17)
98Tc{6+} | 4.2 Myr |  98Ru(3+) | 1792.7(73) | 1797.03(6) | 2nd FU {?} | -4.4(73)
124Sb(3-) | 60 d |  124Te(2+) | 2905.1(19) | 2897.3(10) | 1st FNU | 7.8(22)
|  |  124Te{7-} | 2905.1(19) | 2911.18(14) | 4th FNU {?} | -6.1(19)
127Sb(7/2+) | 4 d |  127Te(5/2+) | 1582.2(53) | 1568.13(11) | Allowed | 14.1(53)
131I(7/2+) | 8 d |  131Xe{9/2+} | 972.25(19) | 971.22(13) | Allowed {?} | 1.03(23)222Ref. Eronen  _et al._ (2022).
133Xe(3/2+) | 5 d |  133Cs(1/2+) | 427.4(24) | 437.01(1) | Allowed | -9.7(24)
135Cs(7/2+) | 2.3 Myr |  135Ba(11/2-) | 268.66(30) | 268.218(20) | 1st FU | 0.44(31)333Ref. de Roubin  _et al._ (2020).
136Cs(5+) | 13 d |  136Ba(4+) | 2548.2(19) | 2544.48(24) | Allowed | 3.7(19)
140Ba(0+) | 13 d |  140La(4-) | 1044.2(79) | 1035.63(17) | 3rd FU | 8.5(79)
|  |  140La(3-) | 1044.2(79) | 1037.60(50) | 3${\rm{rd}}$ FNU | 6.6(79)
140La(3-) | 2 d |  140Ce{3+} | 3762.2(14) | 3767.97(10) | 1st FNU {?} | -5.8(14)
148Pm(1-) | 5 d |  148Sm{3-} | 2470.2(58) | 2467.38(8) | 2nd FNU {?} | 2.8(58)
151Pm(5/2+) | 1 d |  151Sm(5/2+) | 1190.2(47) | 1188.0(20) | Allowed | 2.2(51)
154Eu(3-) | 9 yr |  154Gd{2+} | 1968.0(15) | 1964.05(12) | 1st FNU {?} | 3.9(16)
|  |  154Gd(2+) | 1968.0(15) | 1973.07(17) | 1st FNU | -5.1(16)
155Eu(5/2+) | 5 yr |  155Gd(9/2-) | 252.0(16) | 251.71(1) | 1st FU | 0.3(16)
156Eu(0+) | 15 d |  156Gd{2+} | 2452.5(37) | 2451.5444An uncertainty for this energy level is not provided in Ref. nnd . | 2nd FNU {?} | 1.0(37)
|  |  156Gd(1-) | 2452.5(37) | 2449.7${}^{\textrm{c}}$ | 1st FNU | 2.8(37)
|  |  156Gd(2+) | 2452.5(37) | 2446.16(3) | 2nd FNU | 6.3(37)
161Tb(3/2+) | 7 d |  161Dy(1/2+) | 593.7(14) | 607.58(2) | Allowed | -13.9(14)
166Dy(0+) | 3 d |  166Ho(3+) | 485.9(11) | 481.846(4) | 2nd FU | 4.0(11)
169Er(1/2-) | 9 d |  169Tm(1/2-) | 353.49(80) | 341.94(4) | Allowed | 11.55(80)
|  |  169Tm(5/2-) | 353.49(80) | 345.028(3) | 2nd FNU | 8.46(80)
171Tm(1/2+) | 2 yr |  171Yb(7/2+) | 96.55(97) | 95.282(2) | 2nd FU | 1.26(97)
182Ta(3-) | 115 d |  182W(5-) | 1815.5(17) | 1809.64(7) | 2nd FNU | 5.8(17)
|  |  182W(6-) | 1815.5(17) | 1810.85(4) | 2nd FU | 4.6(17)
183Ta(7/2+) | 5 d |  183W(7/2-) | 1072.1(18) | 1069.42(9) | 1st FNU | 2.7(18)
186Re(1-) | 4 d |  186Os(4+) | 1072.7(11) | 1070.48(3) | 3rd FNU | 2.2(11)
188W(0+) | 70 d |  188Re{1+} | 349.0(32) | 353.57(1) | Allowed {?} | -4.6(32)
|  |  188Re(4+) | 349.0(32) | 342.59(2) | 4th FNU | 6.4(32)
189Re(5/2+) | 1 d |  189Os(3/2+) | 1007.7(82) | 996.40(40) | Allowed | 11.3(82)
193Os(3/2-) | 1 d |  193Ir{9/2-} | 1141.9(27) | 1145.61(10) | 2nd FU {?} | -3.7(27)
|  |  193Ir(5/2-) | 1141.9(27) | 1131.17(11) | Allowed | 10.7(27)
199Au(3/2+) | 3 d |  199Hg(3/2-) | 452.31(76) | 455.46(2) | 1st FNU | -3.15(76)
Table 2: Potential ultra-low Q value EC decays. See Table I for descriptions
of Columns I – V. Column VI lists the shell type. Column VII lists the binding
energy of the K shell, L shell, or M shell electron of the daughter atom.
Column VIII lists the decay type and Column IX the Q value of the decay to the
excited state.

Parent | T1/2 | Daughter |  $Q_{gs}$ (keV) |  $E^{*}$ (keV) | Shell |  $E_{B}$ (keV) | Decay type |  $Q_{es}$ (keV)
---|---|---|---|---|---|---|---|---
56Co(4+) | 78 d |  56Fe(4+) | 4566.64(55) | 4554.77(9) | K | 7.1 | Allowed | 4.76(55)
57Ni(3/2-) | 1 d |  57Co{3/2-} | 3261.70(77) | 3262.70(70) | K | 7.7 | Allowed {?} | -8.7(10)
77Br(3/2-) | 2 d |  77Se{11/2-} | 1364.7(28) | 1351.58(12) | K | 12.7 | 4th FNU {?} | 0.4(28)
81Kr(7/2+) | 229 kyr |  81Br(5/2-) | 280.9(15) | 275.98(1) | K | 13.5 | 1st FNU | -8.6(15)
56Co(4+) | 78 d |  56Fe(4+) | 4566.69(55) | 4554.77(9) | L | 0.8 | Allowed | 11.12(55)
57Ni(3/2-) | 1 d |  57Co{3/2-} | 3261.66(77) | 3262.70(70) | L | 0.8 | Allowed {?} | -1.8(10)
|  |  57Co{3/2-} | 3261.66(77) | 3272.2(11) | L | 0.8 | Allowed {?} | -11.3(13)
73As(3/2-) | 80 d |  73Ge{5/2} | 344.7(39) | 353.7(16) | L | 1.3 | Allowed {?} | -10.3(42)
74As(2-) | 18 d |  74Ge{6+} | 2562.3(17) | 2569.33(14) | L | 1.3 | 3rd FU {?} | -8.3(17)
77Br(3/2-) | 2 d |  77Se{3/2-} | 1364.7(28) | 1364.27(4) | L | 1.5 | Allowed {?} | -1.1(28)
81Kr(7/2+) | 229 kyr |  81Br(5/2-) | 280.9(15) | 275.98(1) | L | 1.6 | 1st FNU | 3.2(15)
96Tc(7+) | 4 d |  96Mo(5+) | 2973.2(51) | 2975.28(7) | L | 2.7 | 2nd FNU | -4.7(51)
|  |  96Mo(8+) | 2973.2(51) | 2978.37(8) | L | 2.7 | Allowed | -7.8(51)
97Tc(9/2+) | 4.2 Myr |  97Mo(9/2+) | 320.3(41) | 320.0(10) | L | 2.7 | Allowed | -2.4(42)
101Rh(1/2-) | 3 yr |  101Ru(7/2+) | 545.7(59) | 545.11(1) | L | 3.0 | 3rd FNU | -2.4(59)
113Sn(1/2+) | 115 d |  113In(1/2+) | 1039.0(16) | 1029.65(5) | L | 4.0 | Allowed | 5.4(16)
124I(2-) | 4 d |  124Te(2+) | 3159.6(27) | 3162.92(17) | L | 4.6 | 1st FNU | -8.0(27)
129Cs(1/2+) | 1 d |  129Xe{5/2+} | 1197.0(46) | 1197.11(21) | L | 5.1 | 2nd FNU {?} | -5.2(46)
|  |  129Xe{9/2+} | 1197.0(46) | 1194.60(30) | L | 5.1 | 4th FNU {?} | -2.7(46)
132Cs(2+) | 7 d |  132Xe(6+) | 2126.3(10) | 2111.88(16) | L | 5.1 | 4th FNU | 9.3(10)
146Pm(3-) | 6 yr |  146Nd(2+) | 1471.6(45) | 1470.63(6) | L | 6.7 | 1st FNU | -5.8(45)
147Eu(5/2+) | 24 d |  147Sm(9/2-) | 1721.4(29) | 1717.30(40) | L | 7.3 | 1st FU | -3.1(29)
156Tb(3-) | 5 d |  156Gd(2+) | 2444.3(39) | 2446.16(3) | L | 7.8 | 1st FNU | -9.7(39)
|  |  156Gd{2+} | 2444.3(39) | 2436.95(10) | L | 7.8 | 1st FNU {?} | -0.5(39)
|  |  156Gd{2+} | 2444.3(39) | 2434.70(1) | L | 7.8 | 1st FNU {?} | 1.8(39)
157Tb(3/2+) | 71 yr |  157Gd(5/2-) | 60.0(14) | 54.54(1) | L | 7.8 | 1st FNU | -2.3(14)
167Tm(1/2+) | 9 d |  167Er(7/2-) | 746.1(13) | 745.32(10) | L | 9.1 | 3rd FNU | -8.3(13)
169Yb(7/2+) | 32 d |  169Tm(11/2-) | 899.12(76) | 884.62(20) | L | 9.5 | 1st FU | 5.04(79)
169Lu(7/2+) | 1 d |  169Yb(5/2+) | 2293.0(30) | 2286.2(12) | L | 9.8 | Allowed | -3.0(32)
|  |  169Yb(7/2-) | 2293.0(30) | 2287.23(5) | L | 9.8 | 1st FNU | -4.0(30)
173Lu(7/2+) | 1.5 yr |  173Yb(9/2-) | 670.2(16) | 659.40(90) | L | 9.8 | 1st FNU | 1.0(18)
175Hf{5/2-} | 70 d |  175Lu{7/2-} | 683.9(26) | 672.83(15) | L | 10.2 | Allowed {?} | 1.0(26)
177Ta(7/2+) | 2 d |  177Hf{11/2-} | 1166.0(36) | 1156.90(90) | L | 10.5 | 1st FU {?} | -1.4(37)
183Re(5/2+) | 70 d |  183W(9/2-) | 556.0(81) | 551.24(3) | L | 11.3 | 1st FU | -6.5(81)
184Re{3-} | 35 d |  184W(6+) | 1485.6(43) | 1479.90(50) | L | 11.3 | 3rd FNU {?} | -2.6(44)
190Ir(4-) | 12 d |  190Os(2+) | 1954.2(15) | 1943.50(40) | L | 12.1 | 1st FU | -1.4(16)
195Au(3/2+) | 186 d |  195Pt(1/2-) | 226.8(12) | 222.23(4) | L | 12.9 | 1st FNU | -8.3(12)
200Tl(2-) | 1 d |  200Hg(1-) | 2456.0(58) | 2442.70(30) | L | 13.8 | Allowed | -0.4(58)
74As(2-) | 18 d |  74Ge(4+) | 2562.4(17) | 2569.33(14) | M | 0.1 | 1st FU | -7.0(17)
75Se(5/2+) | 120 d |  75As{5/2-} | 866.041(81) | 865.40(50) | M | 0.1 | 1st FNU {?} | 0.54(51)555Q value from Ref. Ramalho  _et al._ (2022), see also Ref. Horana Gamage  _et al._ (2022).
77Br(3/2-) | 2 d |  77Se{3/2-} | 1364.7(28) | 1364.27(4) | M | 0.1 | Allowed {?} | 0.3(28)
81Kr(7/2+) | 229 kyr |  81Br(5/2-) | 280.9(15) | 275.98(1) | M | 0.2 | 1st FNU | 4.7(15)
96Tc(7+) | 4 d |  96Mo(2+) | 2973.2(51) | 2975.28(7) | M | 0.4 | 4th FU | -2.4(51)
|  |  96Mo(2+) | 2973.2(51) | 2978.37(8) | M | 0.4 | 4th FU | -5.5(51)

Table 3: Continued.

Parent | T1/2 | Daughter |  $Q_{gs}$ (keV) |  $E^{*}$ (keV) | Shell |  $E_{B}$ (keV) | Decay type |  $Q_{es}$ (keV)
---|---|---|---|---|---|---|---|---
97Tc(9/2+) | 2.6 Myr |  97Mo(9/2+) | 320.3(41) | 320.0(10) | M | 0.4 | Allowed | -0.1(42)
101Rh(1/2-) | 3 yr |  101Ru(7/2+) | 545.7(59) | 545.11(1) | M | 0.4 | 3rd FNU | 0.1(59)
111In(9/2+) | 3 d |  111Cd(7/2+) | 857.63(17) | 853.94(7) | M | 0.6 | Allowed | 3.09(18)666Ref. Ge  _et al._ (2022a).
|  |  111Cd(3/2+) | 857.63(17) | 855.6(10) | M | 0.6 | 2nd FU | 1.4(10)${}^{\textrm{a}}$
113Sn(1/2+) | 115 d |  113In(1/2+) | 1039.0(16) | 1029.65(5) | M | 0.6 | Allowed | 8.7(16)
124I(2-) | 4 d |  124Te(2+) | 3159.6(27) | 3149.50(70) | M | 0.8 | 1st FNU | 9.3(28)
|  |  124Te(2+) | 3159.6(27) | 3154.37(3) | M | 0.8 | 1st FNU | 4.4(27)
|  |  124Te(2+) | 3159.6(27) | 3162.92(17) | M | 0.8 | 1st FNU | -4.1(27)
|  |  124Te(2+) | 3159.6(27) | 3167.94(8) | M | 0.8 | 1st FNU | -9.1(27)
129Cs(1/2+) | 1 d |  129Xe{5/2+} | 1197.0(46) | 1194.60(30) | M | 0.9 | 2nd FNU {?} | 1.5(46)
|  |  129Xe{5/2+} | 1197.0(46) | 1197.11(21) | M | 0.9 | 2nd FNU {?} | -1.0(46)
146Pm(3-) | 5 yr |  146Nd(2+) | 1471.6.6(46) | 1470.63(6) | M | 1.3 | 1st FNU | -0.3(45)
146Eu(4-) | 5 d |  146Sm(0+) | 3878.8(67) | 3869.7(10) | M | 1.4 | 3rd FU | 7.7(68)
156Tb(3-) | 5 d |  156Gd{2+} | 2444.3(39) | 2436.95(10) | M | 1.5 | 1st FNU {?} | 5.9(39)
|  |  156Gd{2+} | 2444.3(39) | 2442.41(10) | M | 1.5 | 1st FNU {?} | 0.4(39)
|  |  156Gd{2+} | 2444.3(39) | 2446.16(3) | M | 1.5 | 1st FNU {?} | -3.3(39)
159Dy(3/2-) | 144 d |  159Tb(5/2-) | 364.73(19) | 363.545(1) | M | 1.6 | Allowed | -0.42(19)777Note that the transition to the 159Tb(5/2${}^{-})$ state is energetically allowed for N and O shell capture, see Ref. Ge  _et al._ (2021a).
|  |  159Tb(11/2+) | 364.73(19) | 362.05(4) | M | 1.6 | 3rd FU | 1.08(19)${}^{\textrm{b}}$
167Tm(1/2+) | 9 d |  167Er{3/2-} | 746.1(13) | 745.32(10) | M | 1.8 | 1st FNU {?} | -1.0(13)
|  |  167Er{3/2-} | 746.1(13) | 752.72(8) | M | 1.8 | 1st FNU {?} | -8.4(13)
173Lu(7/2+) | 1.4 yr |  173Yb(9/2-) | 670.2(16) | 659.40(90) | M | 1.9 | 1st FNU | 8.9(18)
174Lu{1-} | 3.3 yr |  174Yb(3-) | 1374.23(16) | 1382.01(6) | M | 1.9 | 2nd FNU {?} | -9.7(16)
177Ta(7/2+) | 2 d |  177Hf{11/2-} | 1166(36) | 1156.90(90) | M | 2.1 | 1st FU {?} | 7.0(37)
183Re(5/2+) | 70 d |  183W{5/2-} | 556.0(81) | 551.24(3) | M | 2.3 | 1st FNU {?} | 2.5(81)
|  |  183W{5/2-} | 556.0(81) | 558.50(70) | M | 2.3 | 1st FNU {?} | -3.8(81)
184Re{3-} | 38 d |  184W{5-} | 1485.6(43) | 1476.90(50) | M | 2.3 | 2st FNU {?} | 6.5(44)
190Ir(4-) | 12 d |  190Os{2+} | 1954.2(15) | 1943.50(40) | M | 2.5 | 1st FU {?} | 8.3(16)
|  |  190Os{2+} | 1954.2(15) | 1956.60(40) | M | 2.5 | 1st FU {?} | -4.8(16)
|  |  190Os{2+} | 1954.2(15) | 1958.10(30) | M | 2.5 | 1st FU {?} | -6.3(15)
194Hg(0+) | 440 yr |  194Au{2-} | 28.0(36) | 35.19(7) | M | 2.8 | 1st FU {?} | -10.0(36)
195Au(3/2+) | 186 d |  195Pt(5/2-) | 226.8(12) | 222.23(4) | M | 2.7 | 1st FNU | 1.9(12)
200Tl(2-) | 1 d |  200Hg{1+} | 2456.0(58) | 2461.83(4) | M | 2.9 | 1st FNU {?} | -8.7(58)
205Bi(9/2-) | 15 d |  205Pb(9/2+) | 2704.6(49) | 2707.72(11) | M | 3.1 | 1st FNU | -6.2(49)

Table 4: Potential ultra-low Q value $\beta^{+}$ decays. See Table I for
descriptions of Columns I – VII.

Parent T1/2 Daughter $Q_{gs}$ (keV) $E^{*}$ (keV) Decay type $Q_{es}$ (keV)
48V(4+) 16 d 48Ti(0+) 2992.95(97) 2997.22(16) 4th FNU -4.27(99) 56Ni(0+) 6 d
56Co(3+) 1110.87(62) 1114.51(5) 2nd FU -3.64(62) 79Kr(1/2-) 1 d 79Br(3/2-)
603.8(36) 606.03(6) Allowed -2.2(36) 83Sr(7/2+) 1 d 83Rb{5/2+} 1251.0(72)
1242.90(40) Allowed {?} 8.1(72) 105Ag(1/2-) 41 d 105Pd(5/2+) 325.1(47)
319.38(22) 1st FU 5.7(47) 144Pm(5-) 1 yr 144Nd(4+) 1309.9(32) 1314.67(13) 1st
FNU -4.8(32) 145Eu(5/2+) 6 d 145Sm(3/2+) 1637.9(34) 1627.74(4) Allowed
10.1(34) 146Pm(3-) 6 yr 146Nd(2+) 449.6(45) 453.84(3) 1st FNU -4.3(45)
146Eu(4-) 5 d 146Sm(2+) 2856.8(67) 2859.0(10) 1st FU -2.2(68) 148Eu(5-) 55 d
148Sm(4-) 2016.6(100) 2031.4(13) Allowed -14.8(101) 153Tb(5/2+) 2 d
153Gd(5/2-) 547.3(41) 548.77(2) 1st FNU -1.4(41) 153Gd{5/2-} 547.3(41)
551.09(2) 1st FNU {?} -3.8(41) 169Lu(7/2+) 1 d 169Yb{7/2-} 1271.0(30)
1261.89(17) 1st FNU {?} 9.1(30) 169Yb{1/2-} 1271.0(30) 1270.74(8) 3rd FNU {?}
0.3(30) 169Yb{7/2+} 1271.0(30) 1285.13(8) Allowed {?} -14.1(30) 171Lu(7/2+) 8
d 171Yb(11/2-) 456.4(19) 449.60(2) 1st FU 6.8(19) 188Ir(1-) 2 d 188Os(0+)
1770.3(95) 1765.40(40) 1st FNU 4.9(95)

##  IV Discussion

###  IV.1 Verification or elimination of potential candidates

The goal of identifying additional ultra-low Q value $\beta$-decay candidates
requires more precise $Q_{es}$ value determinations for many of the isotopes
listed in Table I – III. In most cases this will involve more precise
measurements of the gs-gs Q value, $Q_{gs}$, determined from the mass
difference between parent and daughter isotopes. This quantity can be
accurately and precisely determined using Penning trap mass spectrometry, see
e.g. Ref. Brown and Gabrielse (1986); Blaum (2006). In some cases, the
uncertainty in the energy of the final state in the daughter isotope may also
need to be determined more precisely, particularly if it is found that
$Q_{es}$ $<$ 1 keV, with an uncertainty dominated by the uncertainty in
$E^{*}$. Although all of the candidates listed in these tables are of
potential interest, the allowed, first-forbidden unique, and first-forbidden
non-unique decays are of most interest for neutrino mass determination
experiments and for theoretical and experimental comparison for studies of
atomic interference effects. As such, we suggest a short list of candidates
from these tables to be prioritized in future studies. These are allowed and
first-forbidden decays with $Q_{es}$ values typically within a few keV of zero
and with correspondingly small uncertainties in $Q_{es}$ that are dominated by
the uncertainty in $Q_{gs}$. These are listed in Table 5. We note that the
evaluation of partial half-lives for these decays would require dedicated
calculations to be performed on a case-by-case basis, as is discussed in more
detail in section IV. C.

###  IV.2 Prospects for decay measurements with UL Q value candidates

The primary challenge for experiment in using these UL Q value cases to
perform neutrino mass measurements is the very long partial half-lives,
$t=\frac{T_{1/2}}{BR}$, for decay to the state(s) of interest in the daughter.
The observed 115In(9/2+, gs) $\rightarrow$ 115Sn(3/2+) UL Q value decay, for
example, has a branching ratio of $\sim$10-6 Wieslander  _et al._ (2009) and a
parent half-life of $>10^{14}$ years which yields a partial half-life to that
state of $t\geq 10^{20}$ years – roughly on the same order as double
$\beta$-decay. These issues can be addressed by a careful choice of the UL Q
value candidate. To maximize the branching ratio and decay rate and simplify
the analysis of the $\beta$-spectrum, an allowed or first forbidden transition
is strongly preferred. An UL Q value decay to an excited daughter state that
is not an isomer and has a single $\gamma$-transition or simple multiple
$\gamma$-cascade to the ground state would provide also a convenient
$\gamma$-ray or $\gamma$-rays to gate on to identify the $\beta$-decay signal.
Finally, a parent isotope with a sufficiently long total half-life, $T_{1/2}$,
would be required to allow for isotope harvesting to enable a controlled, off-
line experiment with sufficient statistics to be performed. A large fraction
of the isotopes listed in Table 5 will be able to be produced at the Facility
for Rare Isotope Beams (FRIB) with rates of $\sim$107 – 109 particles/s 888The
quoted rates are stopped beam rates, so that ions could be implanted into a
detector, for example. They also represent the “ultimate FRIB yields” FRI .,
and for some cases even higher at ISOL-type facilities such as TRIUMF and
CERN-ISOLDE. Hence, one day of isotope accumulation at FRIB could provide
$\sim$1012 – 1014 parent nuclides. Even with partial half-lives of $t\geq
10^{14}$ s or longer, this would allow for enough statistics to perform a
high-precision $\beta$ or EC spectrum measurement given a sufficient running
period, and multiple loading of the radioactive sample into the measurement
setup.

Assuming a suitable UL Q value candidate is identified, among the chief
obstacles of such a case for neutrino mass measurements is the tremendous
challenge it provides for experiment, both on the sub-keV energy-scale
sensitivity needed and the precision required for a measurement of the
effective neutrino mass below the $\sim 0.1$ eV/$c^{2}$ level. Recent
advancements in low-energy quantum sensing have opened sensitive avenues for
performing neutrino mass studies using the kinetic energy of the nuclear
recoil and atomic relaxation following EC decay (c.f. Refs. Gastaldo (2017);
Alpert  _et al._ (2015); Friedrich  _et al._ (2021)). These methods can also
be extended to precision measurements of nuclear $\beta^{\pm}$ decay.

The main experimental criteria for exploiting UL Q value decays with such
energy resolved detectors is sub-keV sensitivity to the endpoint of the decay
radiation with sub-eV energy resolution. Not only is this challenging from a
detector standpoint, it also requires complete knowledge of molecular and
atomic broadening effects of the initial and final states – something that
poses a number of challenges Fretwell  _et al._ (2020); Samanta  _et al._
(2023).

Direct momentum measurements of the decay products, however, have
significantly less strict experimental requirements than those that use energy
resolving sensors since the momentum carried by secondary atomic particles
emitted following the decay is much smaller than the nuclear recoil momentum.
Such measurements also avoid the atomic smearing that is inherent in the
nuclear recoil energy discussed above. In this respect, UL Q value cases are
very attractive, and large increases in sensitivity to the neutrino mass can
be achieved by searching for eV-level Q values that are experimentally
accessible. Although challenging, recent work has suggested that such
measurements could be made using optically levitated nanospheres as mechanical
quantum sensors Carney  _et al._ (2022). In fact, the authors of Ref. Carney
_et al._ (2022) explicitly state that,

> Realizing an optomechanical measurement of the light $\nu$ masses would
> likely require a transition with sufficiently low Q value, experimentally
> manageable $T_{1/2}$, and high branching ratio to be identified through such
> measurements. While speculative, the existence of such a transition is
> plausible, and we strongly encourage the precision atomic mass measurement
> community to continue their work in this area.

As these techniques mature, the search for suitable UL Q value decays with the
criteria outlined above becomes increasingly important, and sets the bar for
the evaluation presented here.

###  IV.3 Challenges for theoretical calculations

Table 5:  List of some of the most promising allowed and first forbidden potential ultra-low Q value $\beta^{\pm}$ and EC decay candidates to investigate. Candidates were selected from Tables I – III for having a relatively long total half-life—making an experimental search for such a decay more feasible, and a $Q_{es}$ value around zero with relatively small uncertainty—making it more likely that the $Q_{es}$ value is in fact $\lesssim$ 1 keV. Isotope | Decay | Forbiddenness | Half-life |  $Q_{es}$ (keV)
---|---|---|---|---
136Cs | $\beta^{-}$ | Allowed | 13 d | 3.7(19)
188W | $\beta^{-}$ | Allowed | 70 d | -4.6(32)
155Eu | $\beta^{-}$ | 1${}^{\textrm{st}}$ Forbidden | 5 yr | 0.3(16)
156Eu | $\beta^{-}$ | 1${}^{\textrm{st}}$ Forbidden | 15 d | 1.0(37)
56Co | EC | Allowed | 78 d | 4.76(55)
97Tc | EC | Allowed | 4.2 Myr | -0.1(42)
175Hf | EC | Allowed | 70 d | 1.0(26)
81Kr | EC | 1${}^{\textrm{st}}$ Forbidden | 229 kyr | 3.2(15)
146Pm | EC | 1${}^{\textrm{st}}$ Forbidden | 6 yr | -0.3(45)
157Tb | EC | 1${}^{\textrm{st}}$ Forbidden | 71 yr | -2.3(14)
173Lu | EC | 1${}^{\textrm{st}}$ Forbidden | 1.5 yr | 1.0(18)
183Re | EC | 1${}^{\textrm{st}}$ Forbidden | 70 d | 2.5(81)
195Au | EC | 1${}^{\textrm{st}}$ Forbidden | 186 d | 1.9(12)
148Eu | $\beta^{+}$ | Allowed | 55 d | -15(10)
105Ag | $\beta^{+}$ | 1${}^{\textrm{st}}$ Forbidden | 41 d | 5.7(47)
144Pm | $\beta^{+}$ | 1${}^{\textrm{st}}$ Forbidden | 1 yr | -4.8(32)
146Pm | $\beta^{+}$ | 1${}^{\textrm{st}}$ Forbidden | 6 yr | -4.3(45)

Theoretical predictions of allowed $\beta$-decays are nowadays achievable at
the 0.1% precision level in the vast majority of the cases Hayen  _et al._
(2018). However, the calculation of a $\beta$-spectrum with an ultra-low end-
point energy, and of the related integrated quantities like the $f$-value, the
decay constant, the partial half-life and the branching ratio, still remains
extremely challenging. The usual approximations reach their limits and every
correction that is generally neglected, or assumed to be sufficiently well-
known, can have a major influence Mustonen and Suhonen (2010a).

The most complete $\beta$-decay formalism is probably from Behrens and Bühring
Behrens and Bühring (1982), in which the Hamiltonian density is defined from
the nuclear and lepton currents. The core of the method is a double multipole
expansion of these currents. Depending on the change of angular momentum and
parity between the initial and final nuclear states, rules emerge for the
selection of the terms that dominantly contribute to the transition
probability. In addition, the lepton wave functions are expanded in powers of
electron mass and energy, nuclear radius, fine structure constant and atomic
number, i.e. ($m_{e}R$), ($WR$) and ($\alpha Z$), and only the first orders
are kept. These usual prescriptions have been shown not to be sufficient to
accurately describe $\beta$-decay observables hindered because of large
angular momentum change or low transition energy ($\lesssim$ 100 keV), or both
(see references in Haaranen  _et al._ (2017)). The extension to next-to-
leading order terms in the lepton current has been studied in Haaranen  _et
al._ (2017). However, the method is still based on the assumption that
$\beta$-particles feel only the Coulomb potential of an extended nucleus.
Taking into account a more elaborated potential for the screening effect due
to the atomic electrons prevents any expansion of the lepton wave functions. A
possibility can be a full numerical calculation of the lepton current, thus
virtually including all terms of the expansion. The computational effort is
much more significant and has recently been investigated in the 151Sm low-
energy transitions Kossert  _et al._ (2022).

Atomic corrections have been demonstrated to have a significant influence on
the $\beta$-spectrum shape, especially at very low energies ($\lesssim$ 10
keV), both directly Mougeot  _et al._ (2012); Loidl  _et al._ (2014); Mougeot
and Bisch (2014); Haselschwardt  _et al._ (2020) and indirectly Kossert and
Mougeot (2015); Kossert  _et al._ (2018); Kossert and Mougeot (2021).
Screening has already been discussed above. Another correction is due to the
overlap effect that comes from the mismatch of the initial and final atomic
wave functions due to the sudden change in the number of protons, and that
thus accounts for shake-up and shake-off effects. This effect is usually
negligible but should play a more important role in transitions with ultra-low
end-point energy as it tends to reduce the end-point energy by tens to
hundreds of eV. The current available correction is only first order and
depends on binding energies from an approximate fit Hayen  _et al._ (2018). It
should be revised as its precision is questionable in the present context.
Most importantly, the exchange effect has the strongest impact on the spectrum
shape and the emission probabilities at very-low energies. Precise modeling
exists for allowed transitions and has recently been extended to forbidden
unique transitions Haselschwardt  _et al._ (2020). At ultra-low energies, the
precision of the formalism used to generate the atomic wave functions becomes
of high importance. Deep studies would be necessary with a comparison of
different many-body approaches and different electron correlation treatments
in order to quantify their influence on the $\beta$-decay observables. In
addition, the subtle interplay between the atomic and nuclear matrix elements
is neglected but might have an influence in the case of forbidden non-unique
transitions.

An important challenge comes obviously from the accuracy of the description of
the nuclear structure and of its inclusion in the $\beta$-decay formalism. The
latter must be totally relativistic due to the small rest mass of the $\beta$
particle. This implies the existence of relativistic transition matrix
elements that couple small and large components of the nucleon wave functions.
Such matrix elements are strictly speaking null with nuclear wave functions
from the most common non-relativistic formalism, while they can be dominant
e.g. in forbidden non-unique transitions. A long-standing approach is to
estimate them by applying either the conserved vector current (CVC) hypothesis
for the vector matrix elements, or the partially conserved axial-vector
current (PCAC) hypothesis for the axial-vector matrix elements Behrens and
Bühring (1982). The former depends on the Coulomb displacement energy between
the initial and final nuclear states, for which it is not always easy to get
an accurate estimate as it can be sensitive to the mismatch of the nucleon
wave functions Damgaard and Winter (1966). Another effect is weak magnetism,
which emerges as an induced interaction when considering nucleons of finite
size. In the context of ultra-low end-point energies, its accuracy should not
be a constraint as it appears to first order as a correction on the spectrum
shape linear in energy (see e.g. in Hayen  _et al._ (2018)). However, it has
never been studied in detail in the case of forbidden transitions.

Regarding nuclear structure, different approaches exist depending on the mass
region, such as the realistic shell model with fitted Hamiltonian from the
NushellX code Brown and Rae (2014) or the microscopic quasiparticle-phonon
model Toivanen and Suhonen (1998). Nucleus deformation can also be
incorporated Martini  _et al._ (2016). It has been shown that the spectrum
shape can be strongly influenced by the value of the axial-vector coupling
constant $g_{A}$ Kostensalo and Suhonen (2017); Kostensalo  _et al._ (2017).
In fact, an effective renormalization of $g_{A}$ may be necessary to account
for nuclear medium effects and limitations of the many-body treatment, the
constant appearing usually as quenched compared to its free-nucleon value. As
an example, one can mention core polarization effects due to the restricted
nucleon valence space employed Warburton (1990, 1991). Thanks to the excellent
review from Suhonen Suhonen (2017), an effective $g_{A}$ value can be
estimated from the relationship given between infinite nuclear matter and
finite nuclei. However, one cannot expect nowadays a high-precision predictive
power and it is still necessary to have a case-by-case approach comparing
theory and experiment. On the other hand, tremendous progress has been made in
the past decade in nuclear theory, in particular by bridging the gap between
low-energy quantum chromodynamics and microscopic nuclear forces Epelbaum  _et
al._ (2009). It has led to the framework of the chiral effective field theory
that considers nucleons and pions as effective degrees of freedom instead of
quarks and gluons, and in which many-body nucleon forces naturally emerge
Machleidt and Entem (2011). Applied with ab initio many-body methods, high-
precision nuclear structure calculations become possible, controlling the
precision and quantifying the theoretical uncertainties Carlsson  _et al._
(2016). Such an approach has been applied successfully to the determination of
transition rates of allowed $\beta$-decays from 3H to 100Sn Gysbers  _et al._
(2019), and should make useless any effective $g_{A}$ constant.

The formalism of Behrens and Bühring is a low-energy effective theory, not
renormalizable, and only includes the electrostatic part of the Coulomb
interaction that influences the $\beta$-decay observables. Non-static Coulomb
corrections, called radiative corrections, have to be determined in the
framework of quantum electrodynamics. Historically, they have been split into
inner and outer corrections. The former can be calculated with high-precision
and typically appears as a renormalization of the coupling constants. The
latter have an influence on the $\beta$-spectrum shape with a dependency on
both the lepton and nuclear terms. Analytical high-precision formulation has
been established over the years, including the influence of nuclear structure,
but only in the case of (super-)allowed decays Towner and Hardy (2008); Sirlin
and Ferroglia (2013). There is still a need for a detailed theoretical study
in the case of the forbidden $\beta$ transitions. In addition, the outer
corrections include the effect of internal bremsstrahlung in which a real
photon is created in the final state, reducing the kinetic energy of the
$\beta$ particle. If photons can be emitted up to the transition energy, the
process is much more probable at low energy. These photons can be (partially)
re-absorbed in the detection system, depending on the actual configuration.
High-precision measurements of ultra-low end-point energy spectra would thus
require a careful experimental analysis, for which a precise knowledge of the
photon spectrum is needed.

To summarize, several challenges subsist in atomic physics, nuclear physics
and their inclusion in the weak interaction decay formalism for a high-
precision description of the $\beta$-spectrum and integrated quantities in the
case of ultra-low transition energies. Different high-level expertise would
have to be gathered to tackle this problem.

##  V Conclusions

In summary, so-called ultra-low Q value $\beta$-decays (with $Q$ $<$ 1 keV) to
excited nuclear states in the daughter isotope present interesting
possibilities for future direct neutrino mass determination experiments and
also to test and develop theoretical understanding of the role of atomic
interference effects in nuclear $\beta$-decay. Over the last decade or so,
interest in these decays has grown, and the $Q_{gs}$ values for 16 candidates
have so far been investigated via Penning trap mass spectrometry at five
facilities world-wide. In addition to the verification of the ultra-low Q
value decay of 115In, four of these additional candidates have been shown to
have energetically allowed decay branches to excited nuclear states in the
daughter with $Q_{es}$ values of around 1 keV or less, indicating the
existence of new potential ultra-low Q value decay candidates. Other potential
decays have been ruled out. Our updated evaluation lists $\approx$80 isotopes
that could potentially have ultra-low Q value decay branches, but whose gs-gs
Q values and in some cases excited daughter state energies need to be
determined more precisely. The Q values can be readily determined to the
necessary precision via precise mass measurements of the parent and/or
daughter isotopes using Penning trap mass spectrometry. A short list of these
candidates that are of particular interest to neutrino physics and nuclear
$\beta$-decay studies due to the fact that they have allowed or first
forbidden decays and relatively long life-times, is provided in Table 5. These
candidates also have $Q_{es}$ values that are close to zero with relatively
small uncertainties, increasing the likelihood that a precise Penning trap
measurement would reveal a Q value of $\approx$1 keV.

## Acknowledgments

This material is based upon work supported by the US Department of Energy,
Office of Science, Office of Nuclear Physics under Award No. DE-SC0015927 and
DE-FG02-93ER40789, and by the National Science Foundation under Contract No.
2111302. K.G.L also acknowledges support from the Gordon and Betty Moore
Foundation and the Facility for Rare Isotope Beams (FRIB), which is a DOE
Office of Science User Facility under Award No. DE-SC0000661. Support was also
provided by Central Michigan University.

## References

  * Aseev  _et al._ (2011) V. N. Aseev, A. I. Belesev, A. I. Berlev, E. V. Geraskin, A. A. Golubev, N. A. Likhovid, V. M. Lobashev, A. A. Nozik, V. S. Pantuev, V. I. Parfenov, A. K. Skasyrskaya, F. V. Tkachov, and S. V. Zadorozhny, Phys. Rev. D 84, 112003 (2011).
  * Kraus  _et al._ (2005) C. Kraus, B. Bornschein, L. Bornschein, J. Bonn, B. Flatt, A. Kovalik, B. Ostrick, E. Otten, J. Schall, T. Thümmler, _et al._ , Eur. Phys. J. C 40, 447 (2005).
  * Aker (2019) K. C. Aker, M. et al. (KATRIN Collaboration), Phys. Rev. Lett. 123, 221802 (2019).
  * Arnaboldi  _et al._ (2003) C. Arnaboldi, C. Brofferio, O. Cremonesi, E. Fiorini, C. Lo Bianco, L. Martensson, A. Nucciotti, M. Pavan, G. Pessina, S. Pirro, E. Previtali, M. Sisti, A. Giuliani, B. Margesin, and M. Zen, Phys. Rev. Lett. 91, 161802 (2003).
  * Springer  _et al._ (1987) P. T. Springer, C. L. Bennett, and P. A. Baisden, Phys. Rev. A 35, 679 (1987).
  * Friedrich  _et al._ (2021) S. Friedrich, G. B. Kim, C. Bray, R. Cantor, J. Dilling, S. Fretwell, J. A. Hall, A. Lennarz, V. Lordi, P. Machule, D. McKeen, X. Mougeot, F. Ponce, C. Ruiz, A. Samanta, W. K. Warburton, and K. G. Leach, Phys. Rev. Lett. 126, 021803 (2021).
  * Martoff  _et al._ (2021) C. J. Martoff, F. Granato, V. Palmaccio, X. Yu, P. F. Smith, E. R. Hudson, P. Hamilton, C. Schneider, E. Chang, A. Renshaw, F. Malatino, P. D. Meyers, and B. Lamichhane, Quantum Science and Technology 6, 024008 (2021).
  * Avignone  _et al._ (2008) F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod. Phys. 80, 481 (2008).
  * Blaum  _et al._ (2020) K. Blaum, S. Eliseev, F. A. Danevich, V. I. Tretyak, S. Kovalenko, M. I. Krivoruchenko, Y. N. Novikov, and J. Suhonen, Rev. Mod. Phys. 92, 045007 (2020).
  * Falkowski  _et al._ (2021) A. Falkowski, M. González-Alonso, and O. Naviliat-Cuncic, Journal of High Energy Physics 2021, 126 (2021).
  * González-Alonso  _et al._ (2019) M. González-Alonso, O. Naviliat-Cuncic, and N. Severijns, Progress in Particle and Nuclear Physics 104, 165 (2019).
  * Formaggio  _et al._ (2021) J. A. Formaggio, A. L. C. de Gouvêa, and R. H. Robertson, Physics Reports 914, 1 (2021).
  * Ferri  _et al._ (2015) E. Ferri, D. Bagliani, M. Biasotti, G. Ceruti, D. Corsini, M. Faverzani, F. Gatti, A. Giachero, C. Gotti, C. Kilbourne, _et al._ , Phys. Procedia 61, 227 (2015).
  * Ge  _et al._ (2021a) Z. Ge, T. Eronen, K. S. Tyrin, J. Kotila, J. Kostensalo, D. A. Nesterenko, O. Beliuskina, R. de Groote, A. de Roubin, S. Geldhof, W. Gins, M. Hukkanen, A. Jokinen, A. Kankainen, A. Koszorús, M. I. Krivoruchenko, S. Kujanpää, I. D. Moore, A. Raggio, S. Rinta-Antila, J. Suhonen, V. Virtanen, A. P. Weaver, and A. Zadvornaya, Phys. Rev. Lett. 127, 272301 (2021a).
  * Myers  _et al._ (2015) E. G. Myers, A. Wagner, H. Kracke, and B. A. Wesson, Phys. Rev. Lett. 114, 013003 (2015).
  * Nesterenko  _et al._ (2014) D. A. Nesterenko, S. Eliseev, K. Blaum, M. Block, S. Chenmarev, A. Dörr, C. Droese, P. E. Filianin, M. Goncharov, E. Minaya Ramirez, Y. N. Novikov, L. Schweikhard, and V. V. Simon, Phys. Rev. C 90, 042501 (2014).
  * Eliseev  _et al._ (2015) S. Eliseev, K. Blaum, M. Block, S. Chenmarev, H. Dorrer, C. E. Düllmann, C. Enss, P. E. Filianin, L. Gastaldo, M. Goncharov, U. Köster, F. Lautenschläger, Y. N. Novikov, A. Rischka, R. X. Schüssler, L. Schweikhard, and A. Türler, Phys. Rev. Lett. 115, 062501 (2015).
  * Mustonen and Suhonen (2010a) M. Mustonen and J. Suhonen, J. Phys. G: Nucl. Part. Phys. 37, 064008 (2010a).
  * Cattadori  _et al._ (2005) C. M. Cattadori, M. De Deo, M. Laubenstein, L. Pandola, and V. I. Tretyak, Nucl. Phys. A 748, 333 (2005).
  * Blachot (2005) J. Blachot, Nuclear Data Sheets 104, 967 (2005).
  * Mount  _et al._ (2009) B. J. Mount, M. Redshaw, and E. G. Myers, Phys. Rev. Lett. 103, 122502 (2009).
  * Wieslander  _et al._ (2009) J. S. E. Wieslander, J. Suhonen, T. Eronen, M. Hult, V.-V. Elomaa, A. Jokinen, G. Marissens, M. Misiaszek, M. T. Mustonen, S. Rahaman, C. Weber, and J. Äystö, Phys. Rev. Lett. 103, 122501 (2009).
  * Zheltonozhsky  _et al._ (2018) V. A. Zheltonozhsky, A. M. Savrasov, N. V. Strilchuk, and V. I. Tretyak, Europhys. Lett. 121, 12001 (2018).
  * Urban  _et al._ (2016) W. Urban, U. Köster, M. Jentschel, P. Mutti, B. Märkisch, T. Rza̧ca-Urban, C. Bernards, C. Fransen, J. Jolie, T. Thomas, and G. S. Simpson, Phys. Rev. C 94, 011302 (2016).
  * Mustonen and Suhonen (2011) M. Mustonen and J. Suhonen, Physics Letters B 703, 370 (2011).
  * Haaranen and Suhonen (2013) M. Haaranen and J. Suhonen, The European Physical Journal A 49, 93 (2013).
  * Mustonen and Suhonen (2010b) M. T. Mustonen and J. Suhonen, AIP Conference Proceedings 1304, 401 (2010b).
  * Suhonen (2014) J. Suhonen, Physica Scripta 89, 054032 (2014).
  * Kopp and Merle (2010) J. Kopp and A. Merle, Phys. Rev. C 81, 045501 (2010).
  * Gamage  _et al._ (2019) N. D. Gamage, R. Bhandari, M. Horana Gamage, R. Sandler, and M. Redshaw, Hyperfine Interactions 240, 43 (2019).
  * de Roubin  _et al._ (2020) A. de Roubin, J. Kostensalo, T. Eronen, L. Canete, R. P. de Groote, A. Jokinen, A. Kankainen, D. A. Nesterenko, I. D. Moore, S. Rinta-Antila, J. Suhonen, and M. Vilén, Phys. Rev. Lett. 124, 222503 (2020).
  * Ge  _et al._ (2022a) Z. Ge, T. Eronen, A. de Roubin, K. Tyrin, L. Canete, S. Geldhof, A. Jokinen, A. Kankainen, J. Kostensalo, J. Kotila, M. Krivoruchenko, I. Moore, D. Nesterenko, J. Suhonen, and M. Vilén, Physics Letters B 832, 137226 (2022a).
  * Eronen  _et al._ (2022) T. Eronen, Z. Ge, A. de Roubin, M. Ramalho, J. Kostensalo, J. Kotila, O. Beliushkina, C. Delafosse, S. Geldhof, W. Gins, M. Hukkanen, A. Jokinen, A. Kankainen, I. Moore, D. Nesterenko, M. Stryjczyk, and J. Suhonen, Physics Letters B 830, 137135 (2022).
  * Ramalho  _et al._ (2022) M. Ramalho, Z. Ge, T. Eronen, D. A. Nesterenko, J. Jaatinen, A. Jokinen, A. Kankainen, J. Kostensalo, J. Kotila, M. I. Krivoruchenko, J. Suhonen, K. S. Tyrin, and V. Virtanen, Phys. Rev. C 106, 015501 (2022).
  * Horana Gamage  _et al._ (2022) M. Horana Gamage, R. Bhandari, G. Bollen, N. D. Gamage, A. Hamaker, D. Puentes, M. Redshaw, R. Ringle, S. Schwarz, C. S. Sumithrarachchi, and I. Yandow, Phys. Rev. C 106, 065503 (2022).
  * Sandler  _et al._ (2019) R. Sandler, G. Bollen, N. D. Gamage, A. Hamaker, C. Izzo, D. Puentes, M. Redshaw, R. Ringle, and I. Yandow, Phys. Rev. C 100, 024309 (2019).
  * Gamage  _et al._ (2022) N. D. Gamage, R. Sandler, F. Buchinger, J. A. Clark, D. Ray, R. Orford, W. S. Porter, M. Redshaw, G. Savard, K. S. Sharma, and A. A. Valverde, Phys. Rev. C 106, 045503 (2022).
  * Karthein  _et al._ (2019) J. Karthein, D. Atanasov, K. Blaum, S. Eliseev, P. Filianin, D. Lunney, V. Manea, M. Mougeot, D. Neidherr, Y. Novikov, L. Schweikhard, A. Welker, F. Wienholtz, and K. Zuber, Hyperfine Interactions 240, 61 (2019).
  * Ge  _et al._ (2021b) Z. Ge, T. Eronen, A. de Roubin, D. A. Nesterenko, M. Hukkanen, O. Beliuskina, R. de Groote, S. Geldhof, W. Gins, A. Kankainen, A. Koszorús, J. Kotila, J. Kostensalo, I. D. Moore, A. Raggio, S. Rinta-Antila, J. Suhonen, V. Virtanen, A. P. Weaver, A. Zadvornaya, and A. Jokinen, Phys. Rev. C 103, 065502 (2021b).
  * Ge  _et al._ (2022b) Z. Ge, T. Eronen, A. de Roubin, J. Kostensalo, J. Suhonen, D. A. Nesterenko, O. Beliuskina, R. de Groote, C. Delafosse, S. Geldhof, W. Gins, M. Hukkanen, A. Jokinen, A. Kankainen, J. Kotila, A. Koszorús, I. D. Moore, A. Raggio, S. Rinta-Antila, V. Virtanen, A. P. Weaver, and A. Zadvornaya, Phys. Rev. C 106, 015502 (2022b).
  * Wang  _et al._ (2021) M. Wang, W. Huang, F. Kondev, G. Audi, and S. Naimi, Chinese Physics C 45, 030003 (2021).
  * (42) https://www.nndc.bnl.gov/.
  * Shi  _et al._ (2005) W. Shi, M. Redshaw, and E. G. Myers, Phys. Rev. A 72, 022510 (2005).
  * Redshaw  _et al._ (2012) M. Redshaw, G. Bollen, M. Brodeur, S. Bustabad, D. L. Lincoln, S. J. Novario, R. Ringle, and S. Schwarz, Phys. Rev. C 86, 041306(R) (2012).
  * Fink  _et al._ (2012) D. Fink, J. Barea, D. Beck, K. Blaum, C. Böhm, C. Borgmann, M. Breitenfeldt, F. Herfurth, A. Herlert, J. Kotila, M. Kowalska, S. Kreim, D. Lunney, S. Naimi, M. Rosenbusch, S. Schwarz, L. Schweikhard, F. Šimkovic, J. Stanja, and K. Zuber, Phys. Rev. Lett. 108, 062502 (2012).
  * Larkins (1977) F. Larkins, Atomic Data and Nuclear Data Tables 20, 311 (1977).
  * Quarati  _et al._ (2022) F. G. A. Quarati, G. Bollen, P. Dorenbos, M. Eibach, K. Gulyuz, A. Hamaker, C. Izzo, D. K. Keblbeck, X. Mougeot, D. Puentes, M. Redshaw, R. Ringle, R. Sandler, J. Surbrook, and I. Yandow, arXiv:2207.14195 (2022).
  * Suhonen (2007) J. Suhonen,  _From Nucleons to Nucleus_ (Springer Berlin, Heidelberg, 2007).
  * (49) http://people.se.cmich.edu/redsh1m/forbidcalc.html.
  * Brown and Gabrielse (1986) L. S. Brown and G. Gabrielse, Rev. Mod. Phys. 58, 233 (1986).
  * Blaum (2006) K. Blaum, Physics Reports 425, 1 (2006).
  * (52) https://groups.nscl.msu.edu/frib/rates/fribrates.html.
  * Gastaldo (2017) E. C. Gastaldo, L. et al. (ECHo Collaboration), The European Physical Journal Special Topics 226, 1623 (2017).
  * Alpert  _et al._ (2015) B. Alpert, M. Balata, D. Bennett, M. Biasotti, C. Boragno, C. Brofferio, V. Ceriale, D. Corsini, P. K. Day, M. De Gerone, R. Dressler, M. Faverzani, E. Ferri, J. Fowler, F. Gatti, A. Giachero, J. Hays-Wehle, S. Heinitz, G. Hilton, U. Köster, M. Lusignoli, M. Maino, J. Mates, S. Nisi, R. Nizzolo, A. Nucciotti, G. Pessina, G. Pizzigoni, A. Puiu, S. Ragazzi, M. R. Reintsema, C.and Gomes, D. Schmidt, M. Schumann, D.and Sisti, D. Swetz, F. Terranova, and J. Ullom, The European Physical Journal C 75, 112 (2015).
  * Fretwell  _et al._ (2020) S. Fretwell, K. G. Leach, C. Bray, G. B. Kim, J. Dilling, A. Lennarz, X. Mougeot, F. Ponce, C. Ruiz, J. Stackhouse, and S. Friedrich, Phys. Rev. Lett. 125, 032701 (2020).
  * Samanta  _et al._ (2023) A. Samanta, S. Friedrich, K. G. Leach, and V. Lordi, Phys. Rev. Appl. 19, 014032 (2023).
  * Carney  _et al._ (2022) D. Carney, K. G. Leach, and D. C. Moore, arXiv:2207.05883 (2022).
  * Hayen  _et al._ (2018) L. Hayen, N. Severijns, K. Bodek, D. Rozpedzik, and X. Mougeot, Rev. Mod. Phys. 90, 015008 (2018).
  * Behrens and Bühring (1982) H. Behrens and W. Bühring,  _Electron Radial Wave Functions and Nuclear Beta Decay_ (Clarendon, Oxford, 1982).
  * Haaranen  _et al._ (2017) M. Haaranen, J. Kotila, and J. Suhonen, Phys. Rev. C 95, 024327 (2017).
  * Kossert  _et al._ (2022) K. Kossert, M. Loidl, X. Mougeot, M. Paulsen, P. Ranitzsch, and M. Rodrigues, Applied Radiation and Isotopes 185, 110237 (2022).
  * Mougeot  _et al._ (2012) X. Mougeot, M.-M. Bé, C. Bisch, and M. Loidl, Phys. Rev. A 86, 042506 (2012).
  * Loidl  _et al._ (2014) M. Loidl, M. Rodrigues, C. Le-Bret, and X. Mougeot, Applied Radiation and Isotopes 87, 302 (2014).
  * Mougeot and Bisch (2014) X. Mougeot and C. Bisch, Phys. Rev. A 90, 012501 (2014).
  * Haselschwardt  _et al._ (2020) S. J. Haselschwardt, J. Kostensalo, X. Mougeot, and J. Suhonen, Phys. Rev. C 102, 065501 (2020).
  * Kossert and Mougeot (2015) K. Kossert and X. Mougeot, Applied Radiation and Isotopes 101, 40 (2015).
  * Kossert  _et al._ (2018) K. Kossert, J. Marganiec-Gałazka, X. Mougeot, and O. Nähle, Applied Radiation and Isotopes 134, 212 (2018).
  * Kossert and Mougeot (2021) K. Kossert and X. Mougeot, Applied Radiation and Isotopes 168, 109478 (2021).
  * Damgaard and Winter (1966) J. Damgaard and A. Winter, Physics Letters 23, 345 (1966).
  * Brown and Rae (2014) B. Brown and W. Rae, Nuclear Data Sheets 120, 115 (2014).
  * Toivanen and Suhonen (1998) J. Toivanen and J. Suhonen, Phys. Rev. C 57, 1237 (1998).
  * Martini  _et al._ (2016) M. Martini, S. Péru, S. Hilaire, S. Goriely, and F. Lechaftois, Phys. Rev. C 94, 014304 (2016).
  * Kostensalo and Suhonen (2017) J. Kostensalo and J. Suhonen, Phys. Rev. C 96, 024317 (2017).
  * Kostensalo  _et al._ (2017) J. Kostensalo, M. Haaranen, and J. Suhonen, Phys. Rev. C 95, 044313 (2017).
  * Warburton (1990) E. K. Warburton, Phys. Rev. C 42, 2479 (1990).
  * Warburton (1991) E. K. Warburton, Phys. Rev. Lett. 66, 1823 (1991).
  * Suhonen (2017) J. T. Suhonen, Frontiers in Physics 5, 55 (2017).
  * Epelbaum  _et al._ (2009) E. Epelbaum, H.-W. Hammer, and U.-G. Meißner, Rev. Mod. Phys. 81, 1773 (2009).
  * Machleidt and Entem (2011) R. Machleidt and D. Entem, Phys. Rep. 503, 1 (2011).
  * Carlsson  _et al._ (2016) B. D. Carlsson, A. Ekstrkm, C. Forssén, D. F. Strömberg, G. R. Jansen, O. Lilja, M. Lindby, B. A. Mattsson, and K. A. Wendt, Phys. Rev. X 6, 011019 (2016).
  * Gysbers  _et al._ (2019) P. Gysbers, G. Hagen, J. D. Holt, G. R. Jansen, T. D. Morris, P. Navrátil, T. Papenbrock, S. Quaglioni, A. Schwenk, S. R. Stroberg, and K. A. Wendt, Nat. Phys. 15, 428 (2019).
  * Towner and Hardy (2008) I. S. Towner and J. C. Hardy, Phys. Rev. C 77, 025501 (2008).
  * Sirlin and Ferroglia (2013) A. Sirlin and A. Ferroglia, Rev. Mod. Phys. 85, 263 (2013).