Detailed β spectrum calculations of 214 Pb for new physics searches in liquid Xenon
DDetailed β spectrum calculations of Pb for new physics searches in liquid Xenon
L. Hayen,
1, 2, ∗ S. Simonucci,
3, 4 and S. Taioli
5, 6 Department of Physics, North Carolina State University, Raleigh, North Carolina 27695, USA Triangle Universities Nuclear Laboratory, Durham, North Carolina 27708, USA School of Science and Technology, Physics Division,University of Camerino, 62032 Camerino (MC), Italy INFN, Sezione di Perugia, 06123 Perugia (PG), Italy European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*-FBK)& Trento Institute for Fundamental Physics and Applications (TIFPA-INFN), Trento, Italy Institute of Physics, Nanotechnology and Telecommunications,Peter the Great St.Petersburg Polytechnic University, Russia (Dated: September 18, 2020)We present a critical assessment of the calculation and uncertainty of the Pb → Bi groundstate to ground state β decay, the dominant source of background in liquid Xenon dark matterdetectors, down to below 1 keV. We consider contributions from atomic exchange effects, nuclearstructure and radiative corrections. For each of these, we find changes much larger than previouslyestimated uncertainties and discuss shortcomings of the original calculation. Specifically, throughthe use of a self-consistent Dirac-Hartree-Fock-Slater calculation, we find that the atomic exchangeeffect increases the predicted flux by 10(3)% at 1 keV relative to previous exchange calculations.Further, using a shell model calculation of the nuclear structure contribution to the shape factor, wefind a strong disagreement with the allowed shape factor and discuss several sources of uncertainty.In the 1-200 keV window, the predicted flux is up to 20% lower. Finally, we discuss omissions anddetector effects in previously used QED radiative corrections, and find small changes in the slope atthe (cid:38)
1% MeV − level, up to 3% in magnitude due to omissions in O ( Zα , Z α ) corrections and3 .
5% uncertainty from the neglect of as of yet unavailable higher-order contributions. Combined,these give rise to an increase of at least a factor 2 of the uncertainty in the 1-200 keV window. Wecomment on possible experimental schemes of measuring this and related transitions.
I. INTRODUCTION
With many experiments looking for Beyond StandardModel (BSM) physics signatures at the expected level ofunavoidable background, an accurate calculation of thelatter is crucial. In order to mitigate obvious sourcesof background, most of these experiments are locatedsufficiently deep underground so as to have a significantamount of overburden and effectively shield it from mostcosmogenic radiation. As a consequence, the remain-ing background is typically related to naturally occurringradiation, either from the construction materials them-selves or the surrounding underground laboratory envi-ronment. In the case of dual-phase liquid Xenon (LXe)detectors, the main backgrounds consist of isotopes ofXenon itself, and β decay products of , Rn emanat-ing into the detector volume. Dark matter or axion-likeparticle searches in this type of geometry are expected toshow up in the keV range, placing stringent constraintson the calculation of β spectra.Recently, the XENON1T collaboration [1] observedan apparent excess of events relative to the backgroundmodel in the very lowest energy bins, between 1-5 keV,for which several possible BSM possibilities were inves-tigated and which have in turn inspired a flurry of phe-nomenological activity. At these low energies, the back- ∗ Corresponding author: [email protected] ground is dominated by the ground state to ground state β − decay of Pb, which has a 27 minute half-life andis continually produced as a byproduct of the Rn α decay chain present in the environment and which em-anates into the installation. The authors [1] stress theneed for a precise calculation of the Pb β spectrumnear low energies for which a number of corrections weretaken into account. Despite this, some of the approxima-tions turn out not to be valid, which underscores both thedifficulty and level of scrutiny required to make accuratepredictions in a regime where scarcely any high-qualitydata exist, particularly for the mass range of interest.In the following sections we discuss a number of correc-tions which were either not included or calculated toocrudely. We treat the atomic exchange correction in Sec.II B, nuclear structure effects in Sec. II C and finally ra-diative corrections in Sec. II D. Section III looks at thecumulative effect compared to the previous estimate andlooks ahead. All the other components of the β spectrumshape are based on a recent review [2] and its open-sourceimplementation [3]. II. SPECTRAL CORRECTIONSA. Preliminaries
Practically all β − decays, while a weak quark-level pro-cess, occur not simply in a nucleus but typically inside a r X i v : . [ nu c l - t h ] S e p an atom or molecule. As a consequence, the total Hamil-tonian one must deal with is then H = H nucl + H e − e + H weak , (1)where H nucl also contains the Coulomb potential of initialand final nuclear states . Because of the smallness of theweak interaction strength near zero momentum transferappropriate to β decay (the Fermi coupling constant is G F ≈ − GeV − ), it is sufficient for all practical pur-poses to treat it only to first order. Scattering states areeigenfunctions of the remaining part of Eq. (1).Since the energy transfer in nuclear β decay is muchsmaller than the W boson mass, we can up to zeroth-order in QED effects write the weak Hamiltonian as alocal current-current interaction H weak = G F √ V ud H µ L µ + h . c . (2)where V ud is the up-down quark mixing matrix element, L µ is the lepton current L µ = ¯ u e γ µ (1 − γ ) v ν (3)and H µ a hadronic operator. While we hold off on a de-scription of the latter until Sec. II C, Lorentz-invariancerequires it to be a combination of vector, V µ , and axialvector, A µ , parts. Using this information we can writean initial Fock state | i (cid:105) = | J i , M i , T i , T i , α i (cid:105) nucl | j i , m i ; n b · · · n bk (cid:105) e − e (4)where J ( T ) denotes (iso)spin, α are additional quantumnumbers and n bi are bound electrons with suppressedquantum numbers combining to total angular momen-tum j i . The final state can be constructed analogouslywith a continuum electron and antineutrino.The β spectrum shape can then be decomposed d Γ dW e = G F V ud π F ( W e ) C ( W e ) R ( W e ) X ( W e ) K ( W e ) × p e W e ( W − W e ) (5)where F is the usual Fermi function which takes into ac-count the Coulomb interaction between outgoing β par-ticle and the final nucleus, X is the exchange correctiondiscussed in Sec. II B, C is the so-called shape factorwhich takes into account the nuclear structure effectsdiscussed in Sec. II C, R are QED radiative correctionsdiscussed further in Sec. II D, and K is a collection ofsmaller corrections [2]. Equation (5) was written assum-ing (cid:126) = c = m e = 1 while W e = E kin /m e + 1 is thetotal β particle energy, W is its maximum value, and p e = (cid:112) W e − β momentum. We have assumed here the Born-Oppenheimer approximation toseparate the nuclear and electronic response.
B. Atomic exchange effect
From Eq. (1) it is clear that initial and final electroniceigenstates are non-orthogonal due to the charge-change.This generates a richness as multiple decay channels openup. The most evident of these are atomic final state exci-tations (shake-up and shake-off), and while this processtakes away small amounts of energy it does not appre-ciably change the spectrum shape at low energies [2].In the case of β − decay, however, there is an additionalpossibility due to the indistinguishability of the emergingelectron with the surrounding atomic electrons. Since thefinal state consist simply of a continuum electron, the β electron can be thought to decay directly into a boundstate in the daughter atom with the subsequent expulsionof the previously bound electron, i.e. an atomic exchange[4]. This is a single-step process which interferes linearlywith the tree-level β decay amplitude and is found toincrease the decay rate substantially near low energies[5]. The reason for the latter is intuitively clear, as theprobability for decay into a bound state and the creationof a continuum electron state depends on the overlap be-tween the wave functions. As the β energy decreases, thespatial extension of its wave function increases, and theoverlap with bound state wave functions (with an extenton the order of the Bohr radius) increases.The calculation of the exchange correction requiresan accurate knowledge of both bound and continuumwave functions. It’s well-known that orbital angular mo-mentum, L , doesn’t commute with the Dirac Hamil-tonian and the solution to the central equation con-tains a mixture of states categorized instead accordingto K = β ( σ · L + 1), with eigenvalues κ = − s / ( p / ) states. The solution to the Dirac equation ina central potential is then commonly written as φ µκ ( r ) = (cid:18) sign( κ ) f k ( r ) χ µ − κ (ˆ r ) g κ ( r ) χ µκ (ˆ r ) (cid:19) (6)where χ µκ (ˆ r ) is a 2 × g, f are purelyradial functions.In the easily tractable case for an allowed β transi-tion (∆ π = no, ∆ J = 0 , j = 1 /
2. In this case the exchange effect under someapproximations reduces to X ≡ η where [5] η = f s (2 T s + T s ) + (1 − f s )(2 T p + T p ) (7)with f s the proportional effect of s / states f s = g c − ( R ) g c − ( R ) + f c ( R ) , (8)where { g, f } c are continuum radial functions evaluatedat the nuclear radius, R . Finally, T l = − (cid:88) nl ∈ γ (cid:104) El | nl (cid:105) { g, f } bn,κ l ( R ) { g, f } cκ l ( R ) (9)where { g, f } is the large component radial wave functiondepending on κ l , n is the principal quantum number ofthe bound state, and (cid:104) El | nl (cid:105) represents the overlap be-tween a continuum state of energy E and bound state | nl (cid:105) . Finally, γ is the electron configuration of occupiedstates, which corresponds to the parent configuration inthe sudden approximation.The calculations quoted in Ref. [1] are based on Eq.(7) and use fits to numerically calculated atomic electricpotentials by Salvat et al. [6] with a free fit parameterfor every orbital to reach agreement with single-electronbinding energy calculations. The second term in the rhsof Eq. (7), i.e. p / contributions, was neglected togetherwith higher κ contributions for forbidden transitions. Ofall the | ns (cid:105) states, the exchange effect in the very low-est energy range predominantly occurs with the highestoccupied | ns (cid:105) states. As a consequence, the overlap in-tegral in Eq. (9) is extremely sensitive to the positionof its n − Pb performed in Ref. [1], this poses a number of is-sues: ( i ) The fitted potentials of Ref. [6] are reported asa sum of three Yukawa functions over the entire range ofthe atom, and do not capture the oscillatory behaviour ofthe charge density; ( ii ) Only the electronic ground statewas taken into account; ( iii ) Electron wave functions inEq. (8) are evaluated only at the nuclear radius insteadof integrated over the nuclear volume; ( iv ) In Eq. (9)initial and final bound states are assumed orthonormal,i.e. (cid:104) n (cid:48) l (cid:48) | nl (cid:105) = δ nn (cid:48) δ ll (cid:48) .Further, high-quality β spectroscopy data at very lowenergies are extremely scarce and theoretical calculationsof Mougeot et al. [7] on which Ref. [1] is based reachedagreement only through the introduction of a nonphysicalscreening correction [2] which enhanced the β spectrumshape by >
5% in the lowest energy range rather thandiminish it by about − p / orbitals can make up for thisdifference. Using the same methods, however, this con-tribution was estimated to be 0 .
3% at 1 keV [2]. To ourknowledge, unfortunately no agreement from the sameauthors has been published. An independent study of Ni could also not reach agreement with measured spec-tra below 5 keV [8]. It is hypothesized that a downturnin the exchange contribution of one or more orbitals nearvery low energies could be the cause, as this feature isnot replicated in a more direct numerical calculation [9].This downturn is also present in the exchange correctionof Ref. [1].We have performed a direct numerical calculationwhich does not require the approximations specifiedabove. Specifically, the electron density is calculated self-consistently in a Dirac-Hartree-Fock-Slater (DHFS) fash-ion [9]. The integration is performed numerically over thefull nuclear volume, and takes into account all κ values.The latter is a straightforward generalization and can,e.g., be found in Ref. [9]. We note that a further gener-alization of the latter using the Behrens-B¨uhring machi- nary in a molecular geometry is in preparation. Evenso, the contributions of | κ | > a priori be estimated to be negligible for spin 0 ↔ p / orbitals [10] g b − ( r ) /g b − ( r ) ≈ r αZ ) / [3 − ( αZ ) ] / (10)in natural units, which evaluates to about 0 .
3% at thenuclear radius so that its contribution is sub-percent evenwhen the overlap integrals are of the same order as the | ns (cid:105) orbitals.Our results for the exchange correction are shown inFig. 1, together with those reported in Ref. [1]. Ourresults point towards a substantially larger exchange ef-fect in the very lowest energy range. Specifically, our ex-change correction reaches 13% at 2 keV, to be comparedto 4% from Ref. [1], with the effect rising to 27% at0.1 keV. Even though the latter lies under the detectionthreshold of most experiments, a finite energy resolutionwill propagate some of this effect further. We note thatwe do not recover the downturn in our calculation. Asthis is a multiplicative term to the β spectrum shape, theincrease reported here translates directly into a decreasedstatistical significance of the reported XENON1T excess.We will come to back to this in Sec. III. E x c h a n g e c o rr e c t i o n Downturn
This workRef. 1
Figure 1. Comparison of our calculated exchange correctionand that as reported by Ref. [1]. The latter was reported tohave a flat 1% uncertainty, shown in the blue band. We havechosen a 30% relative error, shown in the orange band.
Because of the scarcity of experimental data to com-pare to, it is hard to rigorously define a theoretical un-certainty. We remark that using the same methods, how-ever, previously acquired theoretical results [9] show ex-cellent agreement with the Ni and
Pu spectra downto the lowest energy bins. Configuration interaction (i.e.beyond Hartree-Fock) estimates for lanthanum isotopesshowed changes of O (5%) near zero energy. As such, wemake a conservative estimate of a 30% relative error, andnote that more data is needed to rigorously quantify it.This translates into a factor 4 larger uncertainty estimatecompared to that of Ref. [1] at 2 keV. C. Forbidden shape factor
We go into some more depth of the hadronic struc-ture of the weak interaction Hamiltonian of Eq. (2), thematrix element of which to zeroth order in QED is nucl (cid:104) J f , M f , T f , T f , α f | H µ | J i , M i , T i , T i , α i (cid:105) nucl ≡ (cid:104) f | V µ + A µ | i (cid:105) (11)where the notation is obvious. We use the Behrens-B¨uhring formalism, which consists of a spherical harmon-ics expansion of the currents and encodes all nuclear in-formation into model-independent form factors [11]. Asan example, the timelike component can then be writtenas H = (cid:88) LM C J i J f ; Lm i m f ; M Y ML (ˆ q ) ( qR ) L (2 l + 1)!! F L ( q ) , (12)where C denotes a Wigner-3 j symbol, Y LM is a sphericalharmonic, q = p f − p i and R is the nuclear radius so that qR (cid:28)
1. All information is now encoded in the formfactors F L ( q ), and if one performs a similar harmonicexpansion of the leptonic current, Eq. (3), the number ofcontributing form factors and their prefactors are singu-larly determined through angular momentum conserva-tion. This is the so-called elementary particle method ina nutshell [11–13]. The shape factor can then be writtenfor a non-unique first-forbidden β decay as C ( W e ) = M (1 ,
1) + m (1 , − γ µ W e M (1 , m (1 , M (1 ,
2) + λ M (2 ,
1) (13)where the { M , m } ( κ e , κ ν ) are a linear combination ofform factors, and κ are the angular momentum eigen-values from the lepton spherical wave expansion. Fi-nally, γ = (cid:112) − ( αZ ) and µ and λ are O{ αZ ) } Coulomb functions [14].In practice, the form factors are reduced to calculablenuclear matrix elements through the impulse approxima-tion. The latter assumes that the weak nuclear responsecan be written as a sum of responses of individual freenucleons moving in a mean-field potential. This allowsone to write (cid:104) f | O K | i (cid:105) = (cid:88) α,β (cid:104) β | O K | α (cid:105)(cid:104) f | [ a † β a α ] K | i (cid:105) (14)where O K is an order K operator and α, β are singleparticle states. The first term is a single particle ma-trix element and can, e.g., be calculated analytically forspherical harmonic oscillator functions. The second termis a reduced one-body transition density (ROBTD), and is calculated using a many-body code such as the nuclearshell model.The ground state to ground state decay of Pb[0 + ] to Bi [1 − ] is a so-called first-forbidden β decay(∆ π = yes, ∆ J = 0 , , π )in the π h / and neutrons ( ν ) in the ν g / orbitals assuggested in Ref. [1]. The 1 − ground state of Bi,however, shows that even if this is the dominant configu-ration, an effective interaction must be present to arriveto a 1 − rather than 0 − ground state. The latter impliesthat the average occupation of a ( ν g / π h / ) config-uration is necessarily less than 1. Further, the matrixelement (cid:104) h / | O KLs | g / (cid:105) has only a single radial nodefrom the initial state, so that the position of the nodedetermines the partial cancellation between positive andnegative contributions. In this respect, it is similar to thedecay of Bi for which a large deviation in the shape fac-tor is well-known [15]. Unlike
Bi, however, the partialhalf-life of the ground state to ground state
Pb decayis not particularly hindered which would otherwise pointtowards significant cancellation.Because of spin-parity requirements, only rank 1 op-erators can contribute to the decay, which to first orderreduce down to three different non-relativistic matrix el-ements to be calculated O : g V p , g V r , g A ( σ × r ) (15)in the classic cartesian notation, with g V ( A ) the vec-tor (axial vector) coupling constant. This is onlythe simplest picture, however, as the interaction ofEq. (2) must be integrated over the full nuclearvolume, which then folds in the spatial dependenceof the leptonic wave functions. Once again thento first order the shape factor is determined by fiveform factors: V F , V F , A F , V F (1 , , ,
1) and A F (1 , , , F KLs ( ρ, m, n, k ) ∝ (cid:90) d rφ † f ( r ) O KLs I ( ρ, m, n, k ; r ) φ i ( r )(16)in impulse approximation, with I a function originatingfrom the expansion of the radial lepton wave functions.For the leading order terms this is I (1 , , , r ) = 32 − (cid:16) rR (cid:17) for 0 ≤ r ≤ RRr − (cid:16) rR (cid:17) for R ≤ r (17)Finally, we note that the two vector form factors canbe related to each other through the conserved vectorcurrent hypothesis, so that [17, 18] V F = − √ T,T − R V F (18)where ∆ T,T − ≈ .
85 MeV is the excitation energy ofisobaric analogoue state from which we extract V F .The full calculation taking into account all higher-order terms consists rather of 23 terms [15], even thoughthe calculation is dominated by the first 5 we have men-tioned above. We perform a shell model calculationwith the NuShellX@MSU code [19], using the uncon-strained jj67pn model space and the khpe interaction[20]. The latter was used in particular for a study of Bi. Since this is the only interaction available for thisregion, our one-body transition densities are identical tothose of another very recent study [21]. We find goodagreement with the experimental lifetime for an effective g A = 0 . ± .
15, in accordance with their results. S h a p e f a c t o r C HO g A = 0.60 ± 0.10 KHPE g A = 0.80 ± 0.15 Figure 2. Calculated shape factor using the khpe shell modelinteraction using g A = 0 . ± .
15, for which we find almostidentical results to Ref. [21]. We show also the single parti-cle estimate ν g / → π h / for g A = 0 . ± .
10, and findgood agreement as discussed in the text. The dashed linescorrespond to changing the CVC prediction of Eq. (18) by10%. The inset shows the influence of the λ Coulomb func-tion, which becomes very large at low energies. All shapefactors are normalized to unity at 1/3 of the endpoint energyfor visual aid.
Our results are shown in Fig. 2. We have additionallycalculated the expected shape factor using a single parti-cle transition ν g / → π h / , and find good agreementfor the partial half-life using g A ≈ .
60. This is not sur-prising since a more severe truncation of the model spacegenerally requires stronger quenching to reach agreement.The shape factor, however, depends only the ratio of ma-trix elements and tends to be less sensitive to the specificvalue of g A . In this case, since the shape factor is dom-inated by the A F form factor, the dependence on g A is small. We also show the resulting effect of changingEq. (18) by 10%, as this relation is only approximateand carries model dependence [11, 18]. This has theeffect of inflating the uncertainties by about 50%, andso has a reasonable influence on the shape factor. Fi-nally, the khpe shape factor lies very close to harmonicoscillator estimate, which is reflected in the dominanceof the ( ν g / π h / ) configuration in the shell model ROBTDs, with only minor contributions from π f / and ν i / , ν g / orbitals.It is interesting, however, that at very low energies asignificant enhancement occurs because of the λ contri-bution in Eq. (13). In this regime, the j = 3 / , Xe isotopes, for whichthe constraints on their activity are wide enough to ac-commodate such a shift. In the 1-30 keV range, however,the calculated shape factor is essentially flat and cannotinfluence the observed excess for a normalized spectrum,contrary to the claims of Ref. [21]. The λ contribu-tion in our calculation is an exception here, however, assome of its effect will be observable through the energyresolution of the experiment. D. Radiative corrections
Besides the Coulomb interaction between the emitted β particle and the final state which is largely taken intoaccount through the Fermi function, additional QED cor-rections contribute to O ( α ) and beyond. The majorityof this correction stems from real bremsstrahlung emis-sion, with contributions from virtual photon exchangemainly responsible for removing the infrared divergencein a gauge-invariant way. These consist then of the well-known outer radiative corrections [22], which are largelymodel-independent. The requirements for which theseexpressions were derived, however, typically do not holdin calorimetric detection systems. Specifically, one as-sumes that the real photon that is emitted from the de-cay process goes either undetected or can be completelydisentangled in energy from the β particle. Up until someenergy, however, this is typically not possible.In the case of LXe detectors, the path length of keVphotons is on the order of micrometers, whereas the de-tector resolution is on the order of 1 cm [23]. As a conse-quence, all photons which cannot be distinguished from a β particle are counted only in the total energy deposited.Below some threshold then, the sum E tot = E β + E γ iscounted, whereas above this threshold the analysis proce-dure decides what happens. There has been limited ana-lytical work regarding this situation, and results are onlyavailable for O ( α ) corrections [24–26]. Figure 3 showsresults for the standard O ( α ) radiative corrections [22],the finite energy resolution (FER) results for a range ofsoft photon thresholds between 10 keV and infinity [26],and the full radiative corrections up to O ( α Z ) [2, 3]which we discuss below. R a d i a t i v e c o rr e c t i o n ( ) RC ( ) FERFull RC
Figure 3. Comparison of O ( α ) radiative corrections with aperfectly distinguishable photon, O ( α ) RC using a finite en-ergy resolution between 10 keV and infinity, and the full radia-tive corrections as in Ref. [2, 3], the latter assuming perfectlydistinguishable photons. We make a crude estimate of a soft photon thresholdenergy from the position resolution of the XENON1Texperiment [27] and X-ray stopping power tables [28].We take the conservative estimate that two vertices aredistinguishable on average when separated by one full-width at half-maximum, and consider the effect after oneabsorption length. This crude estimate translates into asoft photon threshold of ∼
400 keV, which correspondsto the full FER line in Fig. 3. The FER results are notvery sensitive to the energy cut above a few tens of keV,with the results quickly approaching those of an infinitethreshold. While a normalized spectrum will change theamplitude at 0 keV by up to a percent, the change inslope is only on the order of 0 .
3% within a 0-200 keVwindow, but up to 3% over the entire spectrum.This brings us to higher order radiative corrections of O ( Z n − α n ) ≡ δ n . The development of radiative correc-tions has a storied history and has been at the forefrontof BSM physics searches for many decades [2, 29, 30].Higher-order O ( Zα ) and O ( Z α ) terms are knownwithin reasonable approximations, for which an exhaus-tive overview can be found in, e.g., Ref. [2]. The radiativecorrections employed in Ref. [1] are energy-independent We note for posterity that the last term in Eq. (9) of Ref. [26]should instead read Θ( E max e − (∆ E + E e )) I (∆ E, E e ). by Mougeot [31] δ m = 1 . Zα ln (cid:18) m p m e (cid:19) (19) δ m = Z α π (cid:18) −
32 + π (cid:19) ln (cid:18) m p m e (cid:19) , (20)where m p is the proton mass and we have corrected whatis assumed to be a typo for the leading log. These ap-pear to be inspired by the works of Jaus and Rasche[32, 33], although it is unclear how one arrived to, e.g.,Eq. (19) from the original results in Ref. [32]. Regard-less, throughout the decades following those results in-consistencies were resolved and missing terms identified[34, 35]. The full results can be found elsewhere [2]. Asan example we write the ‘model-independent’ parts of δ δ MI2 = Zα (cid:20) ln (cid:18) m p m e (cid:19) −
53 ln(2 W ) + 4318 (cid:21) (21)in the extremely relativistic approximation, while themodel-dependent part requires knowledge of the chargedistribution and is energy-independent. Similarly, δ hasseveral energy-dependent terms, writing only the leading-log terms δ ( W ) ≈ Z α (cid:20) .
63 ln( W ) − .
09 ln (2 W ) − .
65 ln(2 W ) (cid:21) (22)where we have numerically evaluated the prefactors forconvenience. This translates into the curve shown inFig. 3, which was calculated using the BSG library [3]and where all terms were included. Relative to the or-der α results, this introduces a slope on the order of 1%MeV − , and dominates in the absolute magnitude . Re-sults up to this order have only been calculated for theleading-log terms, however, and an estimate of their un-certainty is not settled. Historically, a 100% uncertaintywas attached in the superallowed Fermi f t analysis [30],which we adopt here. Additionally, no analytical resultsare available to take into account the soft photon thresh-old due to the computational complexity, so we simplyconservatively estimate the uncertainty at 1% from thebehaviour of the O ( α ) terms.Finally, because αZ ≈ . Pb transition,even higher contributions become non-negligible. We arenot aware of any calculations that have been performed,other than a heuristic estimate by Wilkinson [36]. Thelatter describes an estimated geometric series summa-tion of δ n for n = 4 to infinity using only the leading logresults [2], which for Z = 83 results in a total contribu-tion of 1 . We note that the proper radiative corrections also result in a2.5% lower magnitude than what is obtained from Eq. (19) and(20). This is not relevant for the spectral shape, but is directlyproportional to the ft value. not corroborated through more detailed calculations. Assuch, we attach a 200% uncertainty to this value to allowfor a sign change.In summary, because of the contribution of correctedN (2 , LO QED results, a finite soft photon threshold incalorimetric systems, and heuristic estimate of higher-order results, changes on the few percent level are ob-tained. If we take estimated uncertainties in quadrature,this results in an uncertainty of 3 .
5% because of the ex-tremely high Z value and unknown detector response. Inthe original work [1] no uncertainty was included. III. SUMMARY AND OUTLOOK
An accurate description of β spectra down to very lowenergies, i.e. the first few keV, is an extremely challeng-ing task because of the multitude of effects that becomeprominent there. Since almost all of the latter are elec-tromagnetic in origin, their importance and complexityscales roughly with αZ . For LXe detectors whose mainbackground at these energies arises from high-mass decayproducts of natural radioactivity, accurate calculationsrequire heightened attention and scrutiny. We have com-mented here on three elements of the calculation of Ref.[1] for the ground state to ground state Pb to Bi β decay and described significant improvements relativeto their initial implementation. All of these give riseto changes much larger than the estimated uncertaintyin Ref. [1]. Specifically, we have shown an increase of > .
5% was added. Our resultsare summarized in Fig. 4.Significant changes occur throughout the entire spec-trum and in the first few keV in particular. In the latterportion a steep change arises both from the λ Coulombfunction effect of Sec. II C and the significantly enhancedexchange effect of Sec. II B. For the majority of the spec-trum, the largest change is a result of a calculated shape S p e c t r u m [ a r b . u . ]
1e 30 20
Ref. 1This work R a t i o Finalw/o exchange
Figure 4. Overview of the spectral changes and uncertain-ties for the ground state to ground state Pb β decay, thedominant background at low energies in LXe detectors. Top:difference in spectral shape between our work and Ref. [1],with dashed lines showing results without the exchange ef-fect. The common spectrum shape was calculated using BSG [3] based on Ref. [2]. Bottom: Ratio between the spectralshapes. The gray band shows the quoted uncertainty of Ref.[1], whereas the red band shows our uncertainty. factor with a positive slope significantly different fromzero, discussed in Sec. II C. In the 1-200 keV region of in-terest of the XENON1T experiment [1], this correspondsto a downward shift of almost 20%. This implies a lowercontribution of the other backgrounds, mainly above 100keV. Here the background model is dominated by
Xeand
Xe, whose amplitude constraints can easily acco-modate such a shift. Since the shape factor varies onlyvery slowly on a keV scale, however, this large slope doesnot directly contribute to the observed excess, with theexception of our inclusion of the λ contribution. Finally,we have estimated a β spectrum shape uncertainty basedon physical arguments which translates into at least adoubling of the uncertainty at low energies compared toRef. [1]. As a consequence, the statistical significanceof the observed excess will be reduced, although the fi-nal result depends on the complex analysis chain of theXENON1T experiment.In the future, measurements of the Pb ground stateto ground state β spectrum and close neighbors will bevital in assessing the uncertainty budget of its inclusion,particularly because of its expected strong deviation froman allowed shape. Both the effects of the atomic exchangecorrection and the shape factor give rise to changes ofseveral percents over modest energy ranges, although thelowest energies are extremely challenging to determine tovery high precision. An interesting path forward in thisendeavor is through the use of cyclotron radiation emis-sion spectrometry (CRES). It is currently used for highprecision measurements of the tritium endpoint energyby Project 8 [37] and development work is being done touse CRES for, e.g., a full He β spectrum at the Uni-versity of Washington [38] with the possibility of mea-suring gaseous isotopes up to the highest masses. Thiswould provide invaluable and clean tests for the atomiccalculations that go into the first tens of keV in β de-cays throughout the nuclear chart. A measurement ofthe shape factor of Pb is particularly challenging be-cause of the very similar lifetime of the
Bi final state,and its production chain from
Rn. A possibility thatis being investigated uses an ion trap [39] to load puri- fied
Pb and measure its decay through β - γ coincidencewhile using the α -particle from the Bi chain as a tag.
ACKNOWLEDGMENTS
One of the authors (L.H.) would like to thank Alejan-dro Garcia for the inspiration to look into this work, andAlbert Young and Helena Almaz´an for a careful read-ing of this manuscript. L.H. would additionally like tothank Albert Young, Alejandro Garcia and Guy Savardfor stimulating discussions on experimental possibilities.L.H. acknowledges support from the U.S. National Sci-ence Foundation (PHY-1914133) and U.S. Departmentof Energy (DE-FG02-ER41042). S.S. and S.T. acknowl-edge the National Institute of Nuclear Physics (INFN)under the grant PANDORA. [1] E. Aprile, J. Aalbers, F. Agostini, M. Alfonsi, L. Al-thueser, F. D. Amaro, V. C. Antochi, E. Angelino,J. R. Angevaare, F. Arneodo, D. Barge, L. Baudis,B. Bauermeister, L. Bellagamba, M. L. Benabderrah-mane, T. Berger, A. Brown, E. Brown, S. Bruenner,G. Bruno, R. Budnik, C. Capelli, J. M. R. Cardoso,D. Cichon, B. Cimmino, M. Clark, D. Coderre, A. P.Colijn, J. Conrad, J. P. Cussonneau, M. P. Decowski,A. Depoian, P. Di Gangi, A. Di Giovanni, R. Di Ste-fano, S. Diglio, A. Elykov, G. Eurin, A. D. Ferella,W. Fulgione, P. Gaemers, R. Gaior, M. Galloway, F. Gao,L. Grandi, C. Hasterok, C. Hils, K. Hiraide, L. Hoetzsch,J. Howlett, M. Iacovacci, Y. Itow, F. Joerg, N. Kato,S. Kazama, M. Kobayashi, G. Koltman, A. Kopec,H. Landsman, R. F. Lang, L. Levinson, Q. Lin, S. Lin-demann, M. Lindner, F. Lombardi, J. Long, J. A. M.Lopes, E. L. Fune, C. Macolino, J. Mahlstedt, A. Man-cuso, L. Manenti, A. Manfredini, F. Marignetti, T. M.Undagoitia, K. Martens, J. Masbou, D. Masson, S. Mas-troianni, M. Messina, K. Miuchi, K. Mizukoshi, A. Moli-nario, K. Mor˚a, S. Moriyama, Y. Mosbacher, M. Murra,J. Naganoma, K. Ni, U. Oberlack, K. Odgers, J. Palacio,B. Pelssers, R. Peres, J. Pienaar, V. Pizzella, G. Plante,J. Qin, H. Qiu, D. R. Garc´ıa, S. Reichard, A. Rocchetti,N. Rupp, J. M. F. dos Santos, G. Sartorelli, N. ˇSarˇcevi´c,M. Scheibelhut, J. Schreiner, D. Schulte, M. Schumann,L. S. Lavina, M. Selvi, F. Semeria, P. Shagin, E. Shockley,M. Silva, H. Simgen, A. Takeda, C. Therreau, D. Thers,F. Toschi, G. Trinchero, C. Tunnell, M. Vargas, G. Volta,H. Wang, Y. Wei, C. Weinheimer, M. Weiss, D. Wenz,C. Wittweg, Z. Xu, M. Yamashita, J. Ye, G. Zavattini,Y. Zhang, T. Zhu, J. P. Zopounidis, and X. Mougeot, ,1 (2020), arXiv:2006.09721.[2] L. Hayen, N. Severijns, K. Bodek, D. Rozpedzik, andX. Mougeot, Reviews of Modern Physics , 015008(2018), arXiv:1709.07530.[3] L. Hayen and N. Severijns, Computer Physics Commu-nications , 152 (2019), arXiv:1803.00525.[4] J. Bahcall, Physical Review , 2683 (1963).[5] M. R. Harston and N. C. Pyper, Physical Review A ,6282 (1992). [6] F. Salvat, J. D. Martinez, R. Mayol, and J. Parellada,Physical Review A , 467 (1987).[7] X. Mougeot and C. Bisch, Physical Review A , 012501(2014).[8] S. Vanlangendonck, Analysis of the β spectrum shape of63Ni (MSc Thesis, KU Leuven, 2018).[9] T. Morresi, S. Taioli, and S. Simonucci, Advanced The-ory and Simulations , 1800086 (2018).[10] M. E. Rose, Relativistic Electron Theory (Wiley-VCH,1961).[11] H. Behrens and W. B¨uhring,
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