beta_shape_pnpi/.llm-wiki/raw/sources/SRC-2026-05-27-006/extracted.md

IL NUOVO CIMENTO Vol. ?, N. ? ? Shell-model treatment of the β decay of 99Tc M. Ramalho(1) and J. Suhonen(1)(2) (1) DepartmentofPhysics,UniversityofJyva¨skyla¨,P.O.Box35,FI-40014,Jyva¨skyl¨a,Finland (2) International Centre for Advanced Training and Research in Physics (CIFRA), P.O. Box MG12, 077125 Bucharest-Magurele, Romania Summary.— Inthepresentpaperwetreatthesecond-forbiddennon-unique(2nd- nu)ground-state-to-ground-stateβ−decay99Tc(9/2+)→ 99Ru(5/2+),witha100% branchingratio,withintheframeworkofthenuclearshellmodel(NSM).Theenergy spectrum of the electrons emitted in this β-decay transition (β-electron spectrum) is sensitive to the wave functions of the involved initial (9/2+) and final (5/2+) nuclear ground states through the many involved nuclear matrix elements (NME). The β-electron spectrum of this transition is potentially indicative of the effective value, geff, of the weak axial coupling, g , of crucial importance for extraction of A A information on beyond-the-standard-model physics from the results of the present and future rare-events experiments. We describe the β spectral shape of this de- cay by using a state-of-the-art β-decay formalism and compute the many involved NME using the well established NSM Hamiltonians jj45pnb and glekpn. We have found a strong dependence of the β spectral shape on the value of g making it a A good candidate for determination of the value of geff through comparison with the A corresponding experimental β spectral shape. We have also found an interesting dependence of the β spectral shape on the value of the so-called small relativistic vector NME, sNME, used to match the computed half-life with the measured one.

1. – Introduction Rare-eventsexperimentstypicallylookforbeyond-the-standard-model(BSM)physics by,e.g.,measuringrarenuclearβ decays,doubleβ decays,and(anti)neutrinoandWIMP (weakly interacting massive particle) scatterings off nuclei in searches for astrophysical neutrinos and cold dark matter of the Universe. In recent years, a booming interest in these experiments has concentrated on studies of β electrons (electrons emitted in β − decays) and their energy distributions, the so-called β-electron spectra. Experimental and/ortheoreticalinformationonthesespectraiscrucialfor,e.g.,resolvingtheanomalies related to the antineutrino flux from nuclear reactors and for resolving the common backgroundsintherare-eventsexperimentsthemselves. Also,pinningdowntheeffective 'Societ`aItalianadiFisica 1

2 M.RAMALHOand J.SUHONEN values of weak couplings is a considerable incentive for those present and future β-decay experiments able to tackle the spectral shapes of β electrons, see the review [1]. Nuclearβdecaysvaryincomplexity: fromallowedtohighlyforbiddenones[2,3]. Like in the case of allowed β decays, also in the case of forbidden unique β decays the lepton phase space can be separated from the nuclear part, resulting in a universal β spectral shape,independentofnuclear-structuredetails[2]. Ofspecialinterestfortherare-events experiments are the forbidden non-unique β decays for which the β spectral shapes can bestronglynuclear-structuredependentthroughseveralnuclearmatrixelements(NME). ThevaluesoftheseNMEarenon-triviallydeterminedbythewavefunctionsoftheinitial and final states of a β-decay transition. In addition to the many NME, the (partial) half-life of a forbidden non-unique β transition depends on the values of the weak vector and axial-vector couplings, g and V g [3]. In atomic nuclei, the CVC (conserved vector current) hypothesis sets the value A g =1.0, whereas the PCAC (partially conserved axial-vector current) hypothesis leads V only to the so-called effective value of g , recently discussed in the reviews [1, 4, 5]. A On the other hand, the so-called bare-nucleon value g = 1.27 stems from data on the A decay of an isolated neutron. Typically, the effective value of g is quenched relative A to the bare-nucleon value which can have drastic effects on the sensitivity estimates of rare-events experiments trying to detect the neutrinoless double beta decay [6, 7], of crucial importance in the search for the BSM physics. Only in rare cases there is an enhancement of g present [8]. A As discussed in [9, 10], information on the value of g can be gained by using the A so-called spectrum-shape method (SSM). Use of SSM requires a β-electron spectrum with a notable g dependence in its shape. In this case information on the effective A value of g can be gained through the comparison of computed template β spectra, A for different g values, with the measured one in the aim to find a match. Such SSM A analyses of β-spectral shapes of individual β transitions have been done recently for − the fourth-forbidden non-unique β decays of 113Cd and 115In in [11, 12, 13]. Anadditionalingredientinthetheoreticalanalysesofβspectralshapesistheso-called small relativistic vector NME, the sNME, used to fix the measured (partial) half-life in the so-called enhanced SSM [14, 15]. In spite of its smallness, sNME can influence the (partial) half-lives and shapes of β-electron spectra quite strongly, see [12, 16, 17]. The sNME gathers its major contributions outside the proton (neutron) valence major shell thatcontainstheproton(neutron)Fermisurfacemakingithardtobecalculatedinmany nuclear models. In the present paper we investigate the g and sNME dependence of the second- A forbidden non-unique (2nd-nu) β decay of the 9/2+ ground state of 99Tc to the 5/2+ − ground state of 99Ru, with a branching ratio of 100%. 2. – Theoretical Background . 21. NSM Hamiltonians. – The NSM calculations were performed using the software KSHELL [18] with the interactions jj45pnb [19] and glekpn [20] with the set of single- particle energies fit for the mass region of A = 94 98. Their model spaces consist of − the proton 0f (2)-1p -1p -1g orbitals and the neutron 0g -1d -1d -2s - 5/2 3/2 1/2 9/2 7/2 5/2 3/2 1/2 0h orbitals for jj45pnb, as well as 0f (1)-0f (2)-1p -1p -1g proton orbitals 11/2 7/2 5/2 3/2 1/2 9/2 and 0g (1)-0g -1d -1d -2s neutron orbitals for glekpn. The restrictions in the 9/2 7/2 5/2 3/2 1/2 orbitaloccupationsareimposedinordertoreducethecomputationalburden. Webelieve,

SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 3 99Tc 0 0 .7 . 5 5 ( 3 2 5 2 3 2 7 2 − − − )+ ( ( 3 2 9 2 7 2 5 2 ) ) + + + + 1 2 1+ 1 2 5 2 3 2 1 2 3 − − + + 1 2 1+ 3 2 7 2 − + 1 9 2 2 1 + − 9 2− 0.25 7 2 + 1 2 5 2 − + 9 5 2 + + 1 2− 7 2− 2 5+ 2 0 9 2 + 7 2 + 5 2− Exp. jj45pnb glekpn )VeM(ygrenEnoitaticxE 99Ru 1 2 3 2+ + 5 2 + 9 2 + 5 3 2+ + 9+ (5 2)+ 9 2 + 2 2 7+ (1 2)+ (5 2 7 2 ) + + 2 1 2 1 2 1+ + 5 2 + (3 2)+ 1 2 + 7+ 3 2 + 2 7 2 + 3 2 + 3+ 2 3+ 2 5+ 7+ 5+ 2 3+ 5+ 2 2 2 2 Exp. jj45pnb glekpn Fig.1.: Energylevelschemesfor99Tcand99RuwiththeircorrespondingNSM-computed values using the interactions jj45pnb and glekpn. Experimental data are taken from the ENDSF [21] evaluation. Levels in blue are the initial and final states relevant for this work. however,thattheseposedlimitationsdonotaffectmuchthewavefunctionsoftheinitial and final ground states, relevant for the present work. With the considered truncations we have calculated the excitation spectra for both interactions and compare them with the experimental spectrum in Fig 1. It can be seen that the jj45pnb Hamiltonian works, overall, much better than the glekpn one, in particular for the ground state of 99Tc. To further assess the reliability of the Hamiltonians employed, we have analyzed the magnetic dipole and electric quadrupole moments of the 99Tc and 99Ru ground states. The analysis can be seen in Table I, where once again the jj45pnb Hamiltonian does better overall than the glekpn Hamiltonian. . 22. β spectral shapes and half-lives. – The half-life of a β transition can be obtained fromt =κ/C˜,whereκisaconstant[14,15]andC˜ istheintegratedshapefunctionC, 1/2 (1) Orbital fixed with maximum amount of nucleons, e.g, 8 for 0f . 7/2 (2) Orbital with at least 4 particles. TableI.: Comparisonofexperimentalandtheoretical(jj45pnbandglekpnHamiltonians) nuclear magnetic dipole (in units of nuclear magneton) and electric quadrupole (in units of barns) moments for 99Tc and 99Ru, for the experimental ground states and their correspondents in theory. 99Tc(9/2+) 99Ru(5/2+) Exp. jj45pnb glekpn Exp. jj45pnb glekpn µ (µ ) +5.678(2) +5.662 +5.4160 -0.641(5) -0.2875 +1.0826 N Q (b) -0.129(6) +0.0218 +0.2428 +0.079(4) -0.0018 -0.3222

4								M.RAMALHOand
J.SUHONEN
Table
II.: Values of the small relativistic NME emerging from the fitting procedure for
the 99Tc(9/2+) Ru(5/2+) decay. The sNME values (1) and (2) shown reproduce the
→
experimental		half-life	for	their	given geff.
------------	---	---------	---	-----	-----------	---	---	---	---	---
A
	jj45pnb		- CVC =−0.1705				glekpn	- CVC	=−0.0080
----	-------	-------	--------------	-------	---	----	------	-------	--------	-------
geff		sNME(1)		sNME(2)		geff		sNME(1)		sNME(2)
A						A
0.8		0.0496		-0.0856		0.8		0.1042		-0.0311
0.9		0.0582		-0.0772		0.9		0.1091		-0.0257
1.0		0.0666		-0.0687		1.0		0.1140		-0.0202
1.1		0.0750		-0.0601		1.1		0.1189		-0.0148
1.2		0.0833		-0.0515		1.2		0.1237		-0.0093
fullofphase-spacefactorsandNMEsinthenext-to-leading-orderexpansion,asdiscussed
in detail in [9, 10]. Our calculations employ screening, radiative, and atomic exchange
corrections. The most essential to the current study, due to its small β-decay Q value,
is the atomic exchange correction which was originally derived for allowed β decays [22];
the effects of this correction are the most important for electron energies below 50 keV.
The complexity of the shape function C can, however, be cast in a very simple de-
pendence		on the	weak couplings		by writing
--------	---	------	--------------	-----	----------	----------	---	---------	----	---
				)=g2C	)+g2C
(1)			C(w e		V (w e	A (w e )+g	V g	A C VA (w	e ),
				V		A
where w is the total (mass plus kinetic) energy of the emitted electron. The variation
e
of the shape of the β-electron spectrum with the value of g comes from the subtle
A
interference of the combined vector C (w ) and axial C (w ) parts with the mixed
					V	e		A e
------------	---	----	----	------	---	---	---	---	---	---
vector-axial		part	C (w	) [9].
			VA e
Forthecurrentlystudied2nd-nudecaytransition99Tc(9/2+) 99Ru(5/2+)wetake
→
theCVC-compatiblevalueg =1.0fortheweakvectorcoupling,andexploretheimpact
V
ofthequenchedgeff
			values,rangingfrom0.8			1.2(seeTableII),ontheβ			spectralshape.
---	---	---	---------------------	---	---	----------------------	---	---	--------------	---
			A			−
.
23.SmallrelativisticNuclearElements.–Animportantcontributortotheβ spectral
shape and the half-life is the small relativistic NME (sNME) [12, 15, 16, 17]. In these
works, by fitting the value of the sNME, alongside with geff, one is able to reproduce
A
both the β spectral shape and the half-life of a β transition. The computation of the
value of the sNME is a challenge for the ISM. This is why its CVC value (also presented
in Table II) can be used as a reference [3]. It should be noted, though, that the CVC
value represents an ”ideal”, pertaining to a perfect many-body theory, which is not the
case for NSM, mainly due to its restricted proton and neutron valence spaces.
Following the philosophy adopted in [12, 15, 16, 17], we search for the values of the
sNME that reproduce the experimental half-life of the 99Tc(9/2+) Ru(5/2+) transi-
→
tion. Sincetheβ-decayratehasaquadraticdependencyonthesNME,twosolutionsare
geff,
usually	found	for	each value	of	as presented	in	Table	II.
-------	-----	---	----------	---	------------	---	-----	---	---	---
A
3. –	Results	for	the β	spectral	shapes
----	-------	---	-----	--------	------	---	---	---	---	---
Theβ-decaytransitionofinterest,99Tc(9/2+) Ru(5/2+),is2nd-nuandissensitive
→
to the Q value of the decay. Thus, to ensure reliability of our β spectral-shape calcula-

SHELL-MODELTREATMENTOFTHEβ DECAYOF99TC 5 10 3 − × 8 6 4 2 0 0 50 100 150 200 250 300 Electronenergy(keV) ).mron(Ed/Nd jj45pnb glekpn 10 3 − × 12 8 4 0 0 50 100 150 200 250 300 Electronenergy(keV) Fig. 2.: Computed β spectral shapes for the decay transition 99Tc(9/2+) Ru(5/2+) → using the Hamiltonians jj45pnb (left panel) and glekpn (right panel). The crossed-blue curvescorrespondtothefittedsNMEclosesttotheCVC-valueandthecircled-redtothe farthest(seeTableII),withgeff =1.0. Thegray-hatched(gray-crossed)areacorresponds A to the range geff = 0.8 1.2 with all the fitted sNME values chosen closest to (farthest A − from)thecorrespondingCVCvalues. Allcurvesarenormalizedtounitareaunderthem. tions, we have fixed the energy difference of the initial 9/2+ and final 5/2+ state by the experimental Q value. We plot our results for the β spectral shapes of this transition in Fig. 2. There the crossed-blue (circled-red) curves correspond to a typical moderately quenched value geff =1.0 [4] and the fitted sNME value closer to (farther from) its CVC A value, see Table II. The gray-hatched (gray-crossed) area, in turn, corresponds to the range geff =0.8 1.2 with all the sNME values chosen closest to (farthest from) the cor- A − respondingCVCvalues,thesNMEvaluesbeingshowninTableII. Interestinglyenough, thetwodifferentoptionsforthesNMEvalue(thetwodifferenthatchedregionsinFig.2) sometimes produce drastically different shapes, as can be seen for glekpn, a pattern also recorded in [17]. As for the interaction jj45pnb, both hatched regions overlap nicely and there seems not to be any strong dependency of the β spectral shape on the choice of the value of sNME. Since the β spectral shapes produced by the two interactions clearly disagree, it is hard to judge which one of the shapes is the more realistic one. Based on both the level schemesinFig. 1andthemagneticandelectricmomentsofTableIitistemptingtosay that the jj45pnb Hamiltonian is better suited for these particular nuclei. Furthermore, looking at a recently measured experimental β spectral shape for this transition in [23], especially in their Fig. 5, one can see a striking resemblance of it to the spectral shapes forbothfittedsNME(1,2) ofthejj45pnb Hamiltonian(seeTableII)andabitlessstriking forthesolutionsNME(2) fortheglekpn Hamiltonian. However,onlyapropercomparison with data will allow us to draw any definite conclusions concerning the effective value of the axial coupling and the proper value of sNME. This we leave as a future exercise.

6					M.RAMALHOand
J.SUHONEN
4. – Summary	and	conclusions
------------	---	-----------	---	---	---	---
We have performed shell-model calculations of the β spectral shape related to the
second-forbidden non-unique β -decay transition 99Tc(9/2+) 99Ru(5/2+), with a
−
→
100% branching ratio. We use a state-of-the-art β-decay formalism and compute the
many involved NME using the well established NSM Hamiltonians jj45pnb and glekpn.
When compared with future available data, the β-electron spectrum of this transition
is potentially indicative of the effective value of the weak axial coupling, g , of crucial
A
importance for extraction of information on beyond-the-standard-model physics from
the results of the present and future rare-events experiments. We also point out the
important role played by the small relativistic vector matrix element, used to reproduce
the measured half-life of the transition, in shaping the β-electron spectrum.
5. – Acknowledgments
WeacknowledgethesupportbyCSC–ITCenterforScience,Finland,forthegenerous
computational	resources,	which	were	essential to	this project.
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