Back to (Mass-)Square(d) One: The Neutrino Mass Ordering in Light of Recent Data
Kevin J. Kelly, Pedro A.N. Machado, Stephen J. Parke, Yuber F. Perez-Gonzalez, Renata Zukanovich Funchal
FFERMILAB-PUB-20-330-T
Back to (Mass-)Square(d) One:The Neutrino Mass Ordering in Light of Recent Data
Kevin J. Kelly, ∗ Pedro A. N. Machado, † Stephen J. Parke, ‡ Yuber F. Perez-Gonzalez,
1, 2, 3, § and Renata Zukanovich Funchal ¶ Theoretical Physics Department, Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, IL 60510, USA Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208, USA Colegio de F´ısica Fundamental e Interdisciplinaria de las Am´ericas (COFI),254 Norzagaray Street, San Juan, Puerto Rico 00901 Instituto de F´ısica, Universidade de S˜ao Paulo, C.P. 66.318, 05315-970 S˜ao Paulo, Brazil (Dated: July 20, 2020)We inspect recently updated neutrino oscillation data – specifically coming from the Tokai toKamioka and NuMI Off-axis ν e Appearance experiments – and how they are analyzed to determinewhether the neutrino mass ordering is normal ( m < m < m ) or inverted ( m < m < m ).We show that, despite previous results giving a strong preference for the normal ordering, with thenewest data from T2K and NOvA, this preference has all but vanished. Additionally, we highlightthe importance of this result for non-oscillation probes of neutrinos, including neutrinoless doublebeta decay and cosmology. Future experiments, including JUNO, DUNE, and T2HK will providevaluable information and determine the mass ordering at a high confidence level. Introduction. — By observing the phenomenon ofneutrino oscillations, we have determined a number oftheir properties fairly precisely. This information hascome from a wide variety of regimes, including atmo-spheric neutrinos, solar neutrinos, reactor antineutrinos,and long-baseline neutrino oscillation experiments. Cur-rent data allow us to understand, to a reasonable degree,how the neutrinos mix and that there are two non-zeromass scales. Because neutrinos in any oscillation environ-ment are highly relativistic, these experiments are onlysensitive to differences of masses squared, the so-calledmass-squared-splittings ∆ m ji ≡ m j − m i between thethree neutrino mass eigenstates ν i , with masses m i . Welabel the mass eigenstates by defining ν and ν as themass eigenstates with the largest and smallest admixtureof ν e , respectively.Among a combination of solar and reactor neutrino ex-periments, it has been determined that ∆ m ≈ +7 . × − eV > (cid:12)(cid:12) ∆ m (cid:12)(cid:12) ≈ . × − eV (cid:29) ∆ m but, in general, are not sensitive to the sign of∆ m – this is the crux of the neutrino mass ordering(MO) problem – whether nature prefers m < m < m ,the normal mass ordering (NO), or m < m < m , theinverted mass ordering (IO) [6].There are two straightforward ways to determine theMO, utilizing either interference or matter effects in neu-trino oscillations. The first relies on measuring neu-trino oscillations in a regime where both mass-squared-splittings ∆ m and ∆ m are relevant and an experi-ment that can measure ∆ m to a precision smaller thanthe magnitude of ∆ m – this is the strategy of the ∗ [email protected]; † [email protected]; ‡ [email protected]; § [email protected]; ¶ [email protected]; Barring additional new physics in the neutrino sector [1–3]. upcoming Jiangmen Underground Neutrino Observatory(JUNO) [7], a reactor antineutrino experiment operat-ing at L ≈
50 km and E ≈ m are dominant and matter effects (comingfrom coherent neutrino interactions with rock along thepath of propagation) are relevant, are also sensitive tothe MO. A combination of measuring oscillation proba-bilities for muon-neutrino disappearance P ( ν µ → ν µ ) andelectron-neutrino appearance P ( ν µ → ν e ) (as well as thecorresponding probabilities for antineutrinos) allows forlong-baseline experiments to measure the MO. However,challenging degeneracies exist between determining theMO, the atmospheric octant (whether sin θ is smalleror larger than 1 / δ CP .The latter strategy, where matter effects allow for sen-sitivity to the MO, octant, and δ CP is employed by thecurrently-operating Tokai to Kamioka (T2K) [8–10] andNuMI Off-axis ν e Appearance (NOvA) [11–13] experi-ments, which measure P ( ν µ → ν e ) and P ( ν µ → ν e ) atlong distances. Their ν e appearance measurements canbe well-approximated as measurements at a fixed lengthand energy – T2K operates at L = 295 km and E ≈ . L = 810 km and E ≈ . χ , IO) ≡ χ , IO − χ , NO ≈
10 as consistently determined by avariety of efforts to fit the global neutrino experimentaldata [15–17]. However, T2K, NOvA, and SK have eachrecently provided preliminary updated data [18–20]. Wewill demonstrate that this NO preference has all but van-ished due to interesting correlations between the data, aswell as the degeneracies between MO, octant, and δ CP . a r X i v : . [ h e p - ph ] J u l .
02 0 .
04 0 .
06 0 . P ( ν µ → ν e ) . . . . P ( ν µ → ν e ) NOvA
Pre − δ CP = 0 δ CP = 0 .
02 0 .
04 0 .
06 0 . P ( ν µ → ν e ) T2K
NO (solid) , IO (dashed)NOvA / T2K Best − fit points Pre − δ CP = 0 δ CP = 0 FIG. 1. Bi-probability plots depicting the oscillation probability for neutrinos (x-axes) and antineutrinos (y-axes) at the baselinelength/neutrino energy for NOvA (left panel) and T2K (right panel) while varying δ CP . Black (grey) crosses indicate extractedmeasurements with statistical uncertainty only for the two experiments using their 2020 (pre-2020) results. Different ellipsescorrespond to best-fit points according to NOvA (blue) and T2K (red) fits under the assumption of the Normal (solid) orInverted (dashed) mass ordering. The dots denote probabilities for δ CP = 0, with the arrow indicating increasing δ CP values.See text for more detail. This letter is organized as follows. First, we explain howthe long-baseline experiments are sensitive to the MO,as well as the degeneracies with the atmospheric octantand δ CP . We show how previous data, driven likely byfortuitous statistical fluctuations, provided the previousstrong preference for NO over IO, as well as how the up-dated data drive this preference back to being marginalat best. Finally, we discuss the ramifications of this resultand provide some outlook for the future. Mass ordering sensitivity at Long-Baseline Os-cillation Experiments. — In long-baseline experi-ments like T2K and NOvA (and the planned T2HK [21]and DUNE [22, 23] experiments), oscillations due to thesmaller mass-squared splitting ∆ m have yet to develop,so the expansion parameter ∆ m L/ E can be consid-ered to be perturbatively small. Assuming neutrinospropagate through constant-density matter, the oscilla-tion probability of a muon neutrino into an electron neu-trino with energy E and after travelling a distance L canbe approximated as [24] P µe ≡ P ( ν µ → ν e ) ≈ s s c sin (∆ − aL )(∆ − aL ) ∆ + 8 J sin δ CP sin (∆ − aL )(∆ − aL ) ∆ sin ( aL )( aL ) ∆ cos (∆ + δ CP )+ 4 s c c c sin ( aL )( aL ) ∆ , (1)where ∆ j ≡ ∆ m j L/ E , s ij ≡ sin θ ij , c ij ≡ cos θ ij and J ≡ s c s c s c sin δ CP is the Jarlskog invari-ant [25]. Effects of propagation through matter are givenby the matter potential [26] as follows: a = G F n e √ ≈ (cid:32) ρ . / cm (cid:33) , (2) where ρ is the assumed-constant density along the pathof propagation. For current and planned ν e appearanceoscillation experiments, the parameter ( aL ) can be deter-mined to be aL = .
065 T2K/T2HK [21]0 .
22 NOvA [13]0 .
29 T2HKK [27]0 .
35 DUNE [22] (3)while, by design, | ∆ | ≈ π/ P µe ≡ P ( ν µ → ν e ) can be determined by taking Eq. (1) andreplacing δ CP → − δ CP as well as ( aL ) → − ( aL ). InFig. 1 we display how the oscillation probabilities P µe and P µe vary at NOvA (left panel) and T2K (right) base-lines/energies. We assumed fixed L = 810 km (left)and 295 km (right), as well as E = 1 . . θ = 0 . θ = 0 . m = 7 . × − eV [15]. The colored ellipses aregenerated by varying δ CP for different combinations of (cid:0) sin θ , ∆ m (cid:1) . These combinations are determinedby obtaining the best-fit parameters according to a fitto NOvA (blue ellipses) or T2K (red), assuming the MOis normal (solid) or inverted (dashed) – we discuss howthese points are obtained in the “results” section. Fig. 1also displays measured oscillation probabilities (with sta-tistical uncertainty) as black (current data [18, 19]) andgrey (pre-2020 data [8, 13]) crosses. Comparing olderresults to the current ones, we immediately see that themeasured oscillation probabilities are trending toward the“IO” region of this space, where P µe > P µe .We find it instructive to analyze the sums and differ-ences of the neutrino and antineutrino oscillation proba-bilities, Σ P µe ≡ P µe + P µe and ∆ P µe ≡ P µe − P µe . Nearthe first oscillation maximum, | ∆ | ≈ π/
2, and under the (appropriate) approximation ( aL ) (cid:28) ∆ ,Σ P µe → s c s − s c s c s c sin δ CP ( aL ) ∆ m | ∆ m | sign (cid:0) ∆ m (cid:1) , Σ P µe ≈ . s − . aL ) s c sin δ CP sign (cid:0) ∆ m (cid:1) , (4a)∆ P µe → aL ) π s c s sign (cid:0) ∆ m (cid:1) − πs c s c s c sin δ CP ∆ m | ∆ m | , ∆ P µe ≈ . aL ) s sign (cid:0) ∆ m (cid:1) − . s c sin δ CP . (4b)where in Eqs. (4a) and (4b) we have used the current best-fit measurements of θ , θ , ∆ m , and (cid:12)(cid:12) ∆ m (cid:12)(cid:12) . Analyz-ing Eq. (4a), it is apparent that measurements of the sumof oscillation probabilities are beneficial for extracting s (giving the atmospheric octant), while the effects of CPviolation and the MO have an impact at a small level.On the other hand, according to Eq. (4b), measurementsof ∆ P µe can allow for extraction of the MO, octant, andCP violation, however these are all comparable and com-peting effects.We show the sums and differences of oscillation prob-abilities at NOvA and T2K in Fig. 2, presenting Σ P µe (∆ P µe ) in the top (bottom) panel. We show the extractedmeasurements of these sums/differences as black (cur-rent) and grey (pre-2020) crosses, assuming statistically-independent measurements of P µe and P µe at each ex-periment, adding uncertainties in quadrature. The redand blue ellipses are again generated fixing all parame-ters except δ CP to the same combinations as in Fig. 1.Here, specifically in the bottom panel, the impact of themass ordering is abundantly clear – even while varying δ CP , NOvA requires ∆ P µe > P µe < aL ) is a factor of ∼ P µe . We also note that, in the toppanel, NOvA’s NO best-fit point predicts a much smallervalue of Σ P µe at T2K than what is observed, so this com-bination of parameters is slightly disfavored by T2K data.As with Fig. 1, we see that current data have moved ina direction that begins to favor IO for both T2K andNOvA. In what follows, we quantify all of these effects,performing fits to T2K and NOvA individually, as well asa joint fit, to determine their individual and joint prefer-ences for the neutrino mass ordering. Analysis. — In the case of T2K, we consider the lat-est results from data collection equivalent to 1 . . × protons-on-target (POT) in neutrino (antineutrino)mode [18]. T2K observes a total of 108 (16) ν e ( ν e ) likeevents, and 318 (137) ν µ ( ν µ ) like events. We classifythe data in the same five categories as the collabora-tion, muon-ring (1 Rµ ) and electron-ring (1 Re ) events inboth neutrino and antineutrino modes, plus ν e − CC π events in neutrino mode. We perform our simulation bydefining a loglikelihood function comparing the expectedand observed events, including pull parameters related to .
06 0 .
08 0 .
10 0 . P µe + P µe (NOvA) . . . . P µ e + P µ e ( T K ) δ CP = 0 δ CP = 0 Pre − − . − .
03 0 0 .
03 0 . P µe − P µe (NOvA) − . − . . . P µ e − P µ e ( T K ) δ CP = 0 δ CP = 0 NO (solid) , IO (dashed)NOvA / T2K Best − fit points Pre − FIG. 2. Sums (top) and differences (bottom) of oscillationprobabilities at NOvA (x-axes) and T2K (y-axes) at fixedbaseline lengths and energies as described in the text. Ellipsesare generated by varying δ CP while fixing the other oscillationparameters. The dots denote Σ P µe (top) and ∆ P µe (bottom)for δ CP = 0, with arrows indicating increasing δ CP values.Crosses display extracted sums/differences of oscillation prob-abilities assuming statistically independent measurements ateach experiment for current (black) and pre-2020 (grey) re-sults. the systematic uncertainties on the normalization of eachchannel. We consider the uncertainties for 1 Re events tobe 4.7% (5.9%), while for 1 Rµ we assume a 3.0% (4.0%)in neutrino (antineutrino) modes. For the ν e − CC π wetake a 14.3% of uncertainty.When analyzing any combination of T2K, NOvA, andSK data, we include external experimental informationin our analysis to provide constraints on the oscillationparameters sin θ , sin θ , and ∆ m . The informa-tion included is as follows: from the SNO and SK experi-ments, we include priors on the effective oscillation prob-ability P ee = c s + s = 0 . ± . m =(6 . ± . × − eV [20]. From reactor antineutrinoexperiments, we include priors on ∆ m = (7 . ± . × − eV from KamLAND [5] and sin (2 θ ) = 0 . ± . (cid:12)(cid:12) ∆ m (cid:12)(cid:12) = (2 . ± . × − eV fromDaya Bay [28]. Additionally, we include the ∆ χ mapfrom Ref. [14] which we refer to as “SK18” henceforth.For NOvA, we include information from the muon neu-trino disappearance channels in the following way. Wefind that we are best able to reproduce their results ifwe include Gaussian priors on the parameters (cid:12)(cid:12) ∆ m (cid:12)(cid:12) =(2 . ± . × − eV and 4 | U µ | (1 − | U µ | ) =4 c s (1 − c s ) = 0 . ± .
02 [12, 13]. For electron(anti)neutrino appearance, we assume that NOvA mea-sures an event rate of ν e and ν e at a fixed L = 810 kmand E = 1 . ρ = 2 .
84 g/cm [12]. Underthis assumption, we treat NOvA as a counting experimentand approximate the number of events observed as [19] n NOvA ν e = 1202 . × P ( ν µ → ν e ) + 29 . , (5) n NOvA ν e = 438 . × P ( ν µ → ν e ) + 16 . , (6)where the factors 1202 . . . . ν e ( ν e ) events, and we include these fac-tors using a Poissonian likelihood function, incorporatingonly statistical uncertainties (due to the relatively smallnumber of events). See Refs. [29, 30] for further details. Results. — We perform three different analyses andcompare their results. First, we perform a joint analysisof T2K/NOvA/SK18, including the priors on other pa-rameters discussed above. As we have observed before, cfFig. 1, without SK18, this fit results in a mild preferencefor the IO (∆ χ , IO) = − .
83) and a strong preferencefor δ CP ≈ − π/
2, maximal CP violation. When SK18is included, this preference changes to ∆ χ , IO) = 2 . δ CP ≈ − χ as a function of δ CP (after marginaliz-ing over the other five oscillation parameters) when wefix ourselves to be in the NO (solid black line) or IO(dashed black line). The middle (bottom) panel presentstwo-dimensional measurement contours (at 68.3% CL,dashed, and 90% CL, solid) of δ CP vs. sin θ , assumingNO (IO). The best-fit point, sin θ ≈ . δ CP ≈ − σ σ σ σ ∆ χ NO (solid) , IO (dashed) . . . . . s i n θ NO .
3% CL (dashed) ,
90% CL (solid) − π − π/ π/ π δ CP . . . . . s i n θ IO T2K+NOvA+SK18 , ∆ χ , IO) = +2 . , ∆ χ , IO) = +0 . , ∆ χ , IO) = +1 . FIG. 3. Results of our fit of the oscillation parameters δ CP and sin θ . In the top panel we show ∆ χ as a function of δ CP for a fixed MO. In the middle (bottom) panel we show δ CP versus sin θ for the NO(IO). Except for the top panel, allcontours are 68.3% CL (dashed) and 90% CL (solid) – blacklines indicate a joint fit of T2K/NOvA/SK18, where blue (red)indicate a fit to NOvA (T2K) alone. The corresponding starsindicate the best-fit point of each fit, and the text indicates therelative preference for mass ordering by each fit (∆ χ , IO) ≡ χ , IO) − χ , NO) ). we find that the combined results are consistent with thehypothesis that CP is conserved ( δ CP = 0 or ± π ) at < σ CL.The other two fits we perform are with only T2K oronly NOvA data. In each case, information from so-lar/reactor neutrino experiments are included indepen-dently. The results of these two fits are shown in Fig. 3for NOvA (blue) and T2K (red). Both of these fits re-sult in a small preference for NO over IO, again, as dis-cussed cf. Fig. 1, with T2K giving ∆ χ , IO) = 1 . χ , IO) = 0 .
13. The red and blue
TABLE I. MO preference by an experiment or combination ofexperiments. The plus (minus) sign indicates preference forNO (IO). For completeness, we also present the exclusion ∆ χ for CP conservation.Experiment(s) ∆ χ , IO) ∆ χ T2K +1 . .
13 0.49SK18/SK20 +3 . . − . . . . stars in the middle panel of Fig. 3 represent the best-fit points of these two fits – T2K prefers δ CP ≈ − π/ θ ≈ .
55. On the other hand, NOvA prefers δ CP ≈ .
47 and sin θ ≈ .
46. The two best-fit regionsare somewhat in tension, leading to a joint T2K/NOvAfit preferring maximal CP violation, but inverted massordering. We show a version of Fig. 3 without SK18 inAppendix A.Table I summarizes the MO preference by each experi-ment or combination of experiments we consider, as wellas the updated SK 2020 result presented in Ref. [20] –as these results are not yet published, we do not have a∆ χ map for this to perform a complete fit with T2K andNOvA. We comment on what may happen with a jointT2K/NOvA/SK20 fit in the following section. Discussion & Conclusions. — The neutrino massordering remains one of the largest outstanding mysteriesin the Standard Model of particle physics. Prior to thisSummer, experimental data seemed to be preferring thenormal mass ordering, m < m < m , corresponding tothe same ordering that the charged fermions of the Stan-dard Model obey. However, as we have shown, this evi-dence is waning given the updated results from the long-baseline oscillation experiments T2K and NOvA, specif-ically when the two are combined in a joint fit. WithSK18, a mild preference for the normal ordering is ob-tained. However, preliminary updated results from SKare likely to reduce this preference even further – Ref. [20]showed that with updated data, SK no longer has asstrong of a preference for NO as it did with Ref. [14].Additionally, the updated results prefer the lower octant, s < /
2. In combination with T2K and NOvA, this willlikely result in the IO, upper or lower octant, and maxi-mal CP violation δ CP being the overall favored solution.The importance of this result cannot be understated. If neutrinos do follow the inverted ordering, this can havefar-reaching consequences. If in addition neutrinos areMajorana fermions, there exists some minimum mass rel-evant for neutrino-less double beta decay. If the invertedordering is true and neutrino-less double beta decay re-mains unobserved by upgraded experiments, then we candetermine that neutrinos are Dirac fermions. Moreover,measurements of the cosmic microwave background andthe matter power spectrum allow us to infer the sum ofthe neutrino masses. If neutrinos follow the inverted or-dering, their sum is at least ∼
100 meV, while for nor-mal ordering (cid:80) m i (cid:38)
60 meV. The lower limit for theinverted ordering is attainable by next-generation experi-ments. Furthermore, experiments that measure neutrinomasses via kinematic effects, such as KATRIN [31], couldalso be impacted by the mass ordering, as the minimumeffective electron neutrino mass is about 50 meV for theinverted ordering as opposed to 9 meV for normal order-ing. Finally, the mass ordering may also play an impor-tant role in the potential observation of relic neutrinosfrom the early universe by the proposed PTOLEMY ex-periment [32].It could well be that statistical fluctuations in the data,moving in the same direction in the bi-probability planesof Fig. 1, are the cause for this vanishing preference fornormal ordering. This highlights the importance of twothings. First, the accumulation of more data. As T2Kand NOvA continue to run, their statistical uncertain-ties will decrease, and thus will become more robustagainst statistical fluctuations. Second, this shows theneed for the future experiments that will definitively pindown the neutrino mass ordering. Between JUNO’s long-baseline reactor antineutrino measurements, and DUNEand T2HK’s long-baseline high-energy oscillation and at-mospheric oscillation measurements (which take differentapproaches to determine the ordering, see, e.g., [33]), wewill be able to determine the neutrino mass ordering ab-solutely.
Acknowledgments
KJK, PANM, SJP, and YFPG are supported byFermi Research Alliance, LLC under contract DE-AC02-07CH11359 with the U.S. Department of Energy. RZF issupported by CNPq and FAPESP. This project has re-ceived support from the European Unions Horizon 2020research and innovation programme under the MarieSklodowska-Curie grant agreement No 690575 and No674896. In generating Figs. 1 and 2 we used the preferred points of theNOvA-only and T2K-only fits in each MO. In NO (IO), T2Kprefers s = 0 .
55, ∆ m = +2 . × − eV , δ CP = − .
98 (0 . , − . × − eV , − . s = 0 . m = 2 . × − eV and δ CP = 0 .
473 (0 . − . × − eV , − .
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In Fig. 3 we presented the results of fits to NOvA andT2K’s recently-updated data, as well as a combined fit toT2K, NOvA, and Super-Kamiokande’s published resultfrom Ref. [14], using the ∆ χ map from that publication.In this appendix, we repeat the exercise of Fig. 3 with ajoint T2K/NOvA fit without Super-Kamiokande. This isshown in Fig. 4.As discussed in the main text, the combination of T2Kand NOvA prefer the inverted mass ordering over the nor-mal at the ∆ χ , IO) = − . δ CP ≈ − π/
2, and the up-per octant s > /
2. The marginalized one-dimensional∆ χ lines in the top panel of Fig. 4 allow us to determineT2K on its own (with the included priors) can excludea small interval of δ CP ≈ π/ > σ CL. However,once NOvA is included, the interval shrinks (note thatnear δ CP ≈ π/
2, the exclusion of the red solid line ishigher than that of the black solid line). According toour results, the combination of T2K and NOvA can onlyexclude the hypothesis that CP is conserved ( δ = 0 or δ = ± π ) at roughly 1 − σ CL. σ σ σ σ ∆ χ NO (solid) , IO (dashed) . . . . . s i n θ NO .
3% CL (dashed) ,