Probing Lorentz violation in 2νββ using single electron spectra and angular correlations
aa r X i v : . [ nu c l - t h ] D ec Probing Lorentz violation in νββ using single electron spectra and angularcorrelations O.V. Nit¸escu , , , S.A. Ghinescu , , , M. Mirea , † and S. Stoica , ∗ ”Horia Hulubei” National Institute of Physics and Nuclear Engineering,30 Reactorului, POB MG-6, RO-077125, Bucharest-M˘agurele, Romania International Centre for Advanced Training and Research in Physics, PO Box MG12, 077125-Magurele, Romania Faculty of Physics, University of Bucharest, 405 Atomi¸stilor,POB MG-11, RO-077125, Bucharest-M˘agurele, Romania (Dated: December 4, 2020)We show that the current search for Lorentz invariance violation (LIV) in the summed energyspectra of electrons in 2 νββ decay can be extended by investigating the single electron spectra andthe angular correlation between the emitted electrons. We derive and calculate the LIV contri-butions to these spectra associated with the anisotropic part of the countershaded operator andcontrolled through the coefficient ˚ a (3)of and discuss possible signatures that may be probed in ex-periments. First, we show that some distortion occurs in the single electron spectrum, maximalat small electron energies. Then, we show that other LIV effects may be highlighted by analysingthe angular correlation spectra and the ratio between the Standard Model Extension (SME) elec-tron spectra and their Standard Model (SM) forms. We found that these LIV signatures dependon the magnitude of ˚ a (3)of , manifest differently for positive and negative values of this coefficient,and become more pronounced as the electron energy approaches the Q -value. Finally, we proposean alternative, new method to constrain ˚ a (3)of through the measurement of the angular correlationcoefficient. Using this method, and considering only statistical uncertainties, we obtain bounds of˚ a (3)of at the level of present ones, obtained from summed energy spectra. We show that future experi-ments can improve these limits significantly. Our study is performed for Mo, but the results holdqualitatively for other nuclei that undergo a double-beta decay. We hope our results will provideadditional motivation for the LIV analyses performed in DBD experiments.
Keywords: 2 νββ -decay, Lorentz invariance violation, angular corelation
Introduction.
Searching for evidence to probe theLorentz invariance violation (LIV) is a very current topicthat joins the increasing effort to test the limits of theStandard Model (SM) [1, 2]. The theoretical basis ofthese searches is the SM extension (SME), an effectivefield theory including operators that break Lorentz in-variance for all the particles in the SM [2, 3]. In particu-lar, the neutrino sector of SME provides the theoreticalframework for a rich phenomenology for searching evi-dence of LIV, for example, those that can be proved inneutrino oscillations experiments [4–7]. However, thereare LIV signatures related to the so-called countershadedeffects associated with the oscillation-free operators ofmass dimension three, which cannot be investigated insuch experiments. The study of beta and double-betadecays offers the possibility to investigate the LIV ef-fects related to the time-like (isotropic) component ofthis oscillation-free operator whose size is controlled bythe coefficient ˚ a (3)of . In refs. [8, 9] the LIV effects in 2 νββ decay were calculated for the summed energy spectra ofelectrons, but employing a non-relativistic approximationfor the electron radial wave functions. At present, the ac-curacy required by the DBD experiments far exceeds thisapproximation. Recently, experiments like EXO-200 [10], † Deceased, 28.08.2020 ∗ Corresponding author:[email protected]
CUPID-0 [11], NEMO-3 [12], CUORE [13, 14], GERDA[15] have provided limits of the ˚ a (3)of parameter througha careful analysis of the summed energy spectra of elec-trons in 2 νββ decays, using theoretical spectra obtainedwith better but still approximate methods of calculation.In a recent paper [16], we examined the effects of LIVon summed energy spectra of electrons and quantitiesrelated to them using Fermi functions built with exactelectron wave functions. These were obtained by numer-ically solving the Dirac equation in a realistic Coulomb-type potential including the finite nuclear size, diffusenuclear surface, and screening effects [17, 18].In this work, we show that LIV signatures may also besearched in the single electron spectra and the angularcorrelation between the two electrons emitted in 2 νββ decay. First, we derive the LIV contributions to thesespectra and calculate them using an improved version ofour method described in refs. [16, 17]. Then, we discusspossible signatures that could be probed in experiments.We show that some distortion occurs in the single elec-tron spectrum, maximal at small electron energy. Then,we present other possible LIV effects that can be high-lighted by analysing the angular correlation spectra andthe ratio between the total electron spectra (SME) andtheir standard counterparts (SM). We found that theseLIV contributions manifest differently for positive andnegative values of the ˚ a (3)of coefficient, increasing in mag-nitude as the electron energy gets close to the Q -value.Finally, we propose an alternative, new method for con-straining the ˚ a (3)of coefficient, without a detailed analysisof the electron spectra, namely through the measurementof the angular correlation coefficient, k ν . With a roughestimation, considering only the statistical errors, we con-strained ˚ a (3)of at the level of the present bounds obtainedfrom summed energy electron spectra. We show that fu-ture experiments capable of measuring the angular corre-lation between electrons can significantly improve theselimits. Our study is performed for the nucleus Mo,but the results hold qualitatively for other nuclei thatundergo a double-beta decay.
Formalism.
The differential decay rate for the stan-dard 2 νββ process, 0 + → +1 transitions, can be ex-pressed as [19–22] d Γ ν = (cid:2) A ν + B ν cos θ (cid:3) w ν dω dε dε d (cos θ ) , (1)where ε , are the electron energies, ω , are the anti-neutrino energies, and θ is the angle between the twoemitted electrons. In what follows, we adopt the naturalunits ( ~ = c = 1). Within the SM framework the w ν quantity reads w ν SM = g A G F | V ud | π ω ω p p ε ε , (2)where g A is the axial vector constant, G F is the Fermicoupling constant, V ud is the first element of the Cabibbo-Kobayashi-Maskawa matrix and p , are the momenta ofthe electrons.The quantities A ν and B ν can be expressed to a goodapproximation by [21] A ν = 14 a ( ε , ε ) | M ν | ˜ A × (cid:20) ( h K N i + h L N i ) + 13 ( h K N i − h L N i ) (cid:21) , B ν = 14 b ( ε , ε ) | M ν | ˜ A × (cid:20) ( h K N i + h L N i ) −
19 ( h K N i − h L N i ) (cid:21) , (3)where M ν are the nuclear matrix elements (NMEs) and a ( ε , ε ) and b ( ε , ε ) are products of the radial wavefunctions of the emitted electrons. h K N i , h L N i arekinematic factors that depend on the electron and anti-neutrino energies, on the ground state energy E I of theparent nucleus and on an averaged energy h E N i of theexcited 1 + states in the intermediate nucleus (closure ap-proximation). The expressions for the kinematic factorsare given by [19] h K N i = 1 ε + ω + h E N i − E I + 1 ε + ω + h E N i − E I h L N i = 1 ε + ω + h E N i − E I + 1 ε + ω + h E N i − E I . (4) Here, the difference in energy in the denominator can beobtained from the approximation ˜ A = [ W / h E N i − E I ] , where ˜ A = 1 . A / (in MeV) gives the energyof the giant Gamow-Teller resonance in the intermediatenucleus. The energy W is defined as W = Q + 2 m e = E I − E F , (5)where Q is the kinetic energy available for the four lep-tons, m e is the rest energy of the electron, and E F is theground state energy of the final nucleus.The functions a ( ε , ε ) and b ( ε , ε ) are defined as a ( ε , ε ) = (cid:12)(cid:12) α − − (cid:12)(cid:12) + | α | + (cid:12)(cid:12) α − (cid:12)(cid:12) + (cid:12)(cid:12) α − (cid:12)(cid:12) ,b ( ε , ε ) = − ℜ{ α − − α ∗ + α − α − ∗ } , (6)with α − − = g − ( ε ) g − ( ε ) , α = f ( ε ) f ( ε ) ,α − = f ( ε ) g − ( ε ) , α − = g − ( ε ) f ( ε ) . (7)The functions f ( ε ) and g − ( ε ) are the electron radialwave functions evaluated on the surface of the daughternucleus as g − ( ε ) = Z ∞ g − ( ε, r ) δ ( r − R ) dr,f ( ε ) = Z ∞ f ( ε, r ) δ ( r − R ) dr, (8)where R = r A / , r = 1 . θ can be expressed as [20] d Γ ν SM d (cos θ ) = 12 Γ ν SM (cid:2) κ ν SM cos θ (cid:3) , (9)where κ ν SM is the angular correlation coefficient definedby κ ν SM = Λ ν SM Γ ν SM . (10)The decay rates Γ ν SM and Λ ν SM are obtained by inte-grating Eq. (1) over the lepton energies. In the closureapproximation their formulas can be written in a factor-ized form as followsΓ ν SM ln 2 = g A | m e M ν | G ν SM , Λ ν SM ln 2 = g A | m e M ν | H ν SM , (11)which essentially are products of NMEs and phase spacefactors (PSFs) G ν SM and H ν SM .Within the SME, the LIV effects in the neutrino sec-tor can also arise from the action of the so-called coun-tershaded operators, by changing each antineutrino 4-momentum from q α = ( ω, q ) to an effective 4-momentum˜ q α = ( ω, q + a (3)of − ˚ a (3)of ˆ q ) [9, 23, 24]. Here, ˚ a (3)of is relatedto the isotropic component of the ( a (3)of ) jm coefficient by˚ a (3)of = ( a (3)of ) / √ π [23]. Since in 2 νββ experiments thetwo antineutrinos are not measured, the integration overall neutrino orientations leaves only the isotropic coeffi-cient ˚ a (3)of , hence LIV effects related only to this contri-bution can be searched. This leads to a change in theform of the antineutrino differential phase space, fromthe standard one d q = 4 πω dω to the one containingthe LIV effects d q = 4 π ( ω + 2 ω ˚ a (3)of ) dω . To the firstorder in ˚ a (3)of , the term w ν in the differential decay rate,i.e. Eq. (1), acquires the form w ν SME = g A G F | V ud | π p p ε ε × h ω ω + 2˚ a (3)of ( ω ω + ω ω ) i . (12)In the above expression the first term represents the SMcontribution and the following two terms are the LIVcontributions. Following the same steps as in the case ofSM, we get the SME expression of the differential decayrate with respect to the cosine of the angle θ d Γ ν SME d (cos θ ) = 12 Γ ν SME (cid:2) κ ν SME cos θ (cid:3) , (13)where the angular correlation coefficient κ ν SME is definedby κ ν SME = Λ ν SME Γ ν SME . (14)The decay rates Λ ν SME and Γ ν SME can be expressed as sumof standard and LIV contributionsΓ ν SME = Γ ν + Γ ν + Γ ν , Λ ν SME = Λ ν + Λ ν + Λ ν , (15)where 00 stands for the SM contribution. The compo-nents of the decay rates can be also written in a goodapproximation in a factorized form as products NMEsand PSFs, as followsΓ νmn ln 2 = g A | m e M ν | G νmn , Λ νmn ln 2 = g A | m e M ν | H νmn , (16)with mn = { , , } and M ν the NMEs. The PSFexpressions can be written in the following compact form (cid:26) G νmn H νmn (cid:27) = (10˚ a (3)of ) m + n C mn m − m − ne × Z E I − E F − m e m e dε ε p Z E I − E F − ε m e dε ε p × Z E I − E F − ε − ε dω Ω mn × (cid:26) a ( ε , ε ) (cid:0) h K N i + h L N i + h K N ih L N i (cid:1) b ( ε , ε ) (cid:2) (cid:0) h K N i + h L N i (cid:1) + h K N ih L N i (cid:3)(cid:27) , (17) with Ω mn = ω − m ω − n and C = ˜ A G F | V ud | m e π ln 2 ,C = C = ˜ A G F | V ud | m e π ln 2 . (18)In this study, we consider that LIV influences only thePSFs. In the PSF definitions, the LIV parameter ˚ a (3)of isincluded in MeV and the energy ω of the antineutrino isdetermined as ω = E I − E F − ε − ε − ω . We observethat the first order LIV contributions (10) and (01) arefunctions of ω symmetric to the center of the integrationinterval [0 , E I − E F − ε − ε ], and hence they are equalin value. So, in what follows, we consider the correctionsto the standard PSFs as (cid:26) δG ν δH ν (cid:27) = G ν ˚ a (3)of H ν ˚ a (3)of . (19)We note that by making the approximation h K N i ≃ h L N i ≃ E I − h E N i − W / , (20)and integrating over the energy of the antineutrino ω ,one retrieves simplified expressions of the PSFs whichwere used in many previous works (see for example [25]and references therein) and also in the previous LIV an-alyzes [10–12]. Deriving the decay rate expression versusthe kinetic energy of one electron and to the total kineticenergy of the two electrons we get the single electronspectrum d Γ ν SME dε = C dG ν dε (cid:16) a (3)of χ (1) ( ε ) (cid:17) , (21)and the summed energy spectrum of electrons d Γ ν SME dK = C dG ν dK (cid:16) a (3)of χ (+) ( K ) (cid:17) , (22)where C is a constant including NME, K ≡ ε + ε − m e is the total kinetic energy of the two electrons and χ (1) ( ε ) = d ( δG ν ) dε / dG ν dε ,χ (+) ( K ) = d ( δG ν ) dK / dG ν dK , (23)incorporate the deviations of the electron spectra fromtheir SM forms. Deriving also the decay rate versus ε and cos( θ ), we get the expressions of the angular cor-relation and its deviation from the SM form due to LIV d Γ ν SME dε d (cos θ ) = C dG ν dε × (cid:20) a (3)of χ (1) ( ǫ ) + (cid:18) α ν SM + ˚ a (3)of d ( δH ν ) /dε dG ν /dε (cid:19) cos θ (cid:21) (24)where α SM ≡ ( dH ν /dε ) / ( dG ν /dε ) is the SM angularcorrelation while its SME expression is α SME = α SM + ˚ a (3)of d ( δH ν ) /dε dG ν /dε . (25)Deriving the decay rate expression versus cos( θ ) d Γ ν SME d (cos θ ) = CG ν × (cid:20) a (3)of δG ν G ν + (cid:18) κ ν SM + ˚ a (3)of δH ν G ν (cid:19) cos θ (cid:21) , (26)we can identify (in round brackets) the SME expres-sion of the angular correlation coefficient κ ν SME . For anindependent treatment with respect to ˚ a (3)of , we define ξ νLV ≡ δH ν /G ν in units of MeV − . Results and discussion.
We apply the formalism de-duced in the previous section to the
Mo nucleus. Forthis isotope, there is strong experimental evidence for thesingle state dominance (SSD) hypothesis [26, 27], namelythat only the first 1 + state in the intermediate odd-odd nucleus contribute to the decay [12]. Consequently,we adopt the SSD hypothesis in our calculations, whichamounts to replacing the averaged energy, h E N i , withthe energy of the dominant state, E + = − . SMLIV ( d N / d ϵ ) / N [ M e V - ] ϵ - m e [ MeV ] FIG. 1. (Color online) Normalized 2 νββ single electron spec-tra within SM with solid line and the first order contributionin ˚ a (3)of due to LIV with dashed line, for Mo in the SSDhypothesis.
The numerical results are obtained using the followingphysical constants: the electron mass m e = 0 . V ud = 0 . G F = 1 . × − MeV − , the fine structure constant α = 1 / .
036 [30] and the Q-value of
Mo 2 νββ decay Q = 3 . a (3)of limits, the one reported bythe EXO-200 collaboration ( − . × − MeV ≤ ˚ a (3)of ≤ . × − MeV) [10] and the one reported by the NEMO-3collaboration ( − . × − MeV ≤ ˚ a (3)of ≤ . × − MeV)[12], as they represent the least and most stringent limits,respectively, reported until now.In Fig.1 we illustrate the normalized single electronenergy spectrum and its deviation due to LIV. Our pre-dictions indicate some distortion of the SM spectrum,maximal at very small electron energy, unlike a similareffect that was predicted in the summed energy electronspectra [10, 16], and which is maximal at larger electronenergy. We note that the maximum at very small elec-tron energy is due to the use in calculation of the SSDprescription, namely for the nuclei where SSD is valid, asimilar distortion of the single electron spectra at smallenergies, due to LIV, occurs as well. The calculationshows that this LIV effect is larger in the single electronspectra than in the summed energy ones, but the statis-tics for its observation is smaller than in the last case.However, it is worth to mention that such LIV effectsin the electron spectra can be presently detected only asglobal distortions of these spectra. With higher statisticsplanned for future DBD experiments, such perturbationsdue to LIV might become detectable, and accurate the-oretical predictions of the electron spectra and angularcorrelation are essential to interpret the data and con-strain the ˚ a (3)of coefficient. Our results indicate that bothsingle and summed energy spectra of electrons can inves-tigated to detect similar distortions due to LIV. FIG. 2. (Color online) The quantity 1 +˚ a (3)of χ (1) ( ε ) depictedfor current limits of ˚ a (3)of (dashed for upper limit and dot-dashed for lower limit). The solid line represents the SMprediction. Then, we analyse the ratio between the total singleelectron spectrum (including LIV contribution) and itsSM form, namely 1 + ˚ a (3)of χ (1) ( ε ). We plot this quan-tity in Fig. 2 where other LIV effects are observed. Withsolid line we plotted the standard (SM) spectrum, withdashed line the total spectrum for positive values of ˚ a (3)of and with dot-dashed line the total spectrum for nega-tive values of ˚ a (3)of . One observes that the curves are veryclose to each other, but they diverge at electron ener-gies approaching Q -value. This divergence is due to aslower descent (in absolute value) of the spectrum withrespect to the SM one at the end of the energy interval.This effect is quite visible for the ˚ a (3)of limits reported byEXO-200, while for the more stringent ones reported byNEMO-3, it becomes practically un-observable. We notethat searching for such LIV effect just in the vicinity ofthe Q -value is difficult, due to the current small exper-imental statistics but, as we already mentioned, lookingfor it as a global distortion of the ratio (1 +˚ a (3)of χ (1) ( ε )),could be possible.Next, we perform the same study for the summedenergy spectrum of the electrons, plotting the quantity1 + ˚ a (3)of χ (+) ( K ). As seen in Fig. 3 (we use the same no-tations as in Fig. 2), an effect similar to that of the singleelectron spectrum also occurs and has the same explana-tion, namely the LIV contribution becomes higher thanthe standard spectrum itself at energies close to the Q -value. It is also seen that for negative values of ˚ a (3)of , thequantity 1 +˚ a (3)of χ (+) cut the K -axis at a certain summedenergy of electrons, K . This feature is illustrated in Fig. 3and is most visible only for ˚ a (3)of limits provided by EXO-200 collaboration. Beyond this K value, the differentialrate becomes negative, and thus the DBD process be-comes forbidden. Conversely, for positive ˚ a (3)of , the decayrate is non-zero for some range K > Q . The general ac-cepted interpretation of this behavior is a possible shift inthe Q -value of the process [8]. The same considerationsalso hold for the LIV effects described above, in singleelectron spectra.The deviations from the standard spectra presented inFig. 2 and Fig. 3 may be investigated experimentally bydividing the measured spectra by the theoretical predic-tion calculated within SM. The analysis will include theuncertainties for each energy bin measured. The best fitof the data can conclude if the quantity 1 + ˚ a (3)of χ (1) ( ε )or 1 + ˚ a (3)of χ (+) ( K ) is increasing or decreasing, and so ifthe ˚ a (3)of coefficient is positive or negative, respectively.Further, we discuss the implications of the LIV con-tributions on the angular correlation spectrum, α ν , andthe coefficient k ν SME . They appear in Eqs. (25) and (26)as additional terms to the standard quantities, controlledby the ˚ a (3)of coefficient.Fig. 4 depicts the angular correlation spectrum for thesame sets of the ˚ a (3)of limits. We note the total angularcorrelation spectrum for negative values of ˚ a (3)of exceedsthe SM spectrum since δH is also negative, making theLIV contribution positive. As seen, there is similar, di- FIG. 3. (Color online)The quantity 1 +˚ a (3)of χ (+) ( K ) depictedfor current limits of ˚ a (3)of .The same conventions as in Fig. 2are used.FIG. 4. (Color online) The angular correlation spectrum plot-ted for the current limits of ˚ a (3)of . The same conventions as inFig. 2 are used. vergent behavior of the curves (even more pronouncedthan in the case of the electron spectra) as the electronenergy increases, this time because δH ν decreases slowerthan G ν . For the ˚ a (3)of limits provided by EXO-200, thiseffect is quite large even over a large energy region, whileusing the NEMO-3 limits, the effect becomes negligible.For completeness, we also calculate the LIV effect onthe angular correlation coefficient, k ν . Its value is re-lated to the PSF expressions from Eq. (17), which wecomputed using our method described in [17, 18], withinthe SSD hypothesis. First, we computed its SM value k ν SM = − . . We note that this value differs from that reported in [32],but the difference may stem from the fact that we usedexact electronic wave functions instead of analytic, ap-proximate ones, in the construction of the Fermi func-tions. Using the same Fermi functions as in [32], weobtained k ν SM = − . ξ νLV = − .
285 MeV − , which can be plugged back into Eq. (26) to obtain itsSME value k ν SME = − . − . × ˚ a (3)of , where ˚ a (3)of must be taken in units of MeV. On the otherhand, experimentally, the angular correlation coefficientcan be determined via the forward-backward asymmetrydefined by [33] A ν ≡ (cid:18)Z − d Γ ν dx dx − Z d Γ ν dx dx (cid:19) / Γ= N + − N − N + + N − = 12 k ν SM , (27)where x = cos θ and N − ( N + ) are the 2 νββ events withthe angle θ smaller (larger) than π/
2. For a number of N = 5 × events at NEMO-3 [12] and considering onlythe statistical errors, the angular correlation coefficient ismeasurable with the uncertainty k ν SM = 0 . ± . | ˚ a (3)of | . . × − MeVat 90% CL. This is only a rough estimation, and dedi-cated experimental analysis, including the systematic un-certainties, is necessary for a better one. We note thatthis estimation lies between the ˚ a (3)of limits reported byNEMO-3 and EXO-200, which were obtained from theanalysis of the summed energy spectra of electrons. Wenote here that if in a future experiment the number of2 νββ events would increase by three orders of magnitude(as planned for example in the SuperNEMO experiment),our estimation yields | ˚ a (3)of | . . × − MeV at 90%CL, which is comparable with the limits obtained fromtritium decay experiments [8]. Thus, we predict goodperspectives for searching for LIV effects in future DBDexperiments, due to the significant increase of statistics.
Conclusions.
Concluding, we propose that the inves-tigation of LIV effects in 2 νββ to be extended to sin-gle electron spectra and the angular correlation between the two electrons. We derived the LIV contributions tothe standard spectra and provided theoretical predictionsfor them to be used in the experimental analyses. Ourstudy refers to the case of
Mo, but similar results areexpected for other nuclei. We found some distortion inthe single electron spectrum introduced by LIV (differ-ent from the distortion reported in the literature for thesummed energy spectra). Then, we presented some otherLIV signatures that may be experimentally investigated.For example, if the angular correlation spectrum and theratios between the total electron spectra and their stan-dard forms are analysed, distortions may occur. We showthat they depend on the sign and magnitude of the ˚ a (3)of coefficient and increase in magnitude as the electron en-ergies approach the Q -value. Moreover, the LIV effectsmay shift the end-point of the allowed energies.Finally, we propose an alternative, new method forconstraining ˚ a (3)of , namely by analysing the LIV effectson the angular correlation coefficient. This coefficient iscalculated from PSFs of the 2 νββ decay and can be de-termined experimentally via the forward-backward asym-metry of emitted electrons. Using the present NEMO-3sensitivity and taking into account only statistical er-rors, we obtained bounds for ˚ a (3)of situated between thelimits reported by NEMO-3 and EXO-200. Consider-ing the expected performances of the future DBD exper-iments (high statistics, very low backgrounds, improvedmethods of measurement), we appreciate promising per-spectives of these experiments to perform new relevantLIV investigations and to significantly improve the ˚ a (3)of constrains. We hope that our study will contribute ad-ditional, useful information to motivate the LIV inves-tigations in DBD. In this respect, we mention that wecan provide, at request, the numerical data for the con-struction of the theoretical spectra necessary in the LIVanalyses. Acknowledgments.
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