Precision neutrino experiments vs the Littlest Seesaw
Peter Ballett, Stephen F. King, Silvia Pascoli, Nick W. Prouse, TseChun Wang
IIPPP/16/113
Precision neutrino experiments vs the Littlest Seesaw
Peter Ballett, a Stephen F. King, b Silvia Pascoli, a Nick W. Prouse b,c and TseChunWang a a Institute for Particle Physics Phenomenology, Department of Physics, Durham University, SouthRoad, Durham DH1 3LE, United Kingdom. b School of Physics and Astronomy, University of Southampton, SO17 1BJ Southampton, UnitedKingdom. c Particle Physics Research Centre, School of Physics and Astronomy, Queen Mary University ofLondon, Mile End Road, London E1 4NS, United Kingdom.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We study to what extent upcoming precision neutrino oscillation experimentswill be able to exclude one of the most predictive models of neutrino mass and mixing:the Littlest Seesaw. We show that this model provides a good fit to current data, pre-dicting eight observables from two input parameters, and provide new assessments of itspredictions and their correlations. We then assess the ability to exclude this model usingsimulations of upcoming neutrino oscillation experiments including the medium-distancereactor experiments JUNO and RENO-50 and the long-baseline accelerator experimentsDUNE and T2HK. We find that an accurate determination of the currently least well mea-sured parameters, namely the atmospheric and solar angles and the CP phase δ , providecrucial independent tests of the model. For θ and the two mass-squared differences,however, the model’s exclusion requires a combination of measurements coming from avaried experimental programme. Our results show that the synergy and complementarityof future experiments will play a vital role in efficiently discriminating between predictivemodels of neutrino flavour, and hence, towards advancing our understanding of neutrinooscillations in the context of the flavour puzzle of the Standard Model. a r X i v : . [ h e p - ph ] D ec ontents η = ± π/ η as a free parameter 73.3 Fitting LS models to global fit data 9 The framework of neutrino masses and mixing for explaining neutrino oscillations — thefirst direct experimental evidence for physics beyond the Standard Model — is now firmlyestablished [1]. All three mixing angles together with the size of the two mass-squareddifferences have been measured, with experimental efforts now focused on determining thefinal few unknowns: the ordering and scale of the neutrino masses; the value of the Diracphase δ ; and a precision measurement of the angle θ including, if non-maximal, its octant.Although there is some as yet inconclusive evidence for δ in the third or fourth quadrant,as well as for normal ordering (NO) and non-maximal atmospheric mixing, we rely on thenext generation of oscillation experiments to set these issues to rest.On the theoretical side, however, the origin of neutrino masses and mixing remainsunknown with many possible models considered viable (for reviews see e.g. [2, 3]). Alarge proportion of these models are based on the classic seesaw mechanism, involvingheavy right-handed Majorana neutrinos [4], providing both a mechanism for generatingthe neutrino masses and a natural explanation for their smallness. However, in orderto make predictions that can be probed experimentally, seesaw models require additionalassumptions or constraints [5].To accommodate the three distinct light neutrino masses which drive the oscillationphenomenon, the seesaw mechanism requires at least two right-handed neutrinos [6]. In– 1 –rder to reduce the number of free parameters still further to the smallest number possi-ble, and hence increase predictivity, various approaches to the two right-handed neutrinoseesaw model have been suggested , such as postulating one [7] or two [8] texture zeroesin the Dirac mass matrix in the flavour basis (i.e. the basis of diagonal charged lep-ton and right-handed neutrino masses). However, such two texture zero models are nowphenomenologically excluded [9] for the case of a normal neutrino mass hierarchy. Theminimal two right-handed neutrino model with normal hierarchy which can accommodatethe known data of neutrino mixing involves a Dirac mass matrix with one texture zero anda characteristic form known as the Littlest Seesaw model [10]. The Littlest Seesaw modelmay be embedded in unified models of quarks and leptons in [11]. It leads to successfulleptogenesis where the sign of baryon asymmetry is determined by the ordering of theheavy right-handed neutrinos, and the only seesaw phase η is identified as the leptogenesisphase, linking violation of charge parity symmetry (CP) in the laboratory with that in theearly universe [12].The Littlest Seesaw model can be understood as an example of sequential dominance(SD) [13] in which one right-handed neutrino provides the dominant contribution to theatmospheric neutrino mass , leading to approximately maximal atmospheric mixing, whilethe other right-handed neutrino gives the solar neutrino mass and controls the solar andreactor mixing as well as the magnitude of CP violating effects via δ . SD generally leadsto normal ordering and a reactor angle which is bounded by θ (cid:46) m /m [7], proposeda decade before the reactor angle was measured [1]. Precise predictions for the reactor(and solar) angles result from applying further constraints to the Dirac mass matrix, anapproach known as constrained sequential dominance (CSD) [14]. For example, keeping thefirst column of the Dirac mass matrix proportional to (0 , , T , a class of CSD( n ) modelshas emerged [10, 14–17] corresponding to the second column proportional to (1 , n, ( n − T ,with a reactor angle approximately given by [18] θ ∼ ( n − √ m m . The Littlest Seesawmodel corresponds to n = 3 with a fixed seesaw phase η = 2 π/ , , T and seesaw phase η = − π/ S × U (1) symmetry, putting this version of the Littlest Seesaw model ona firm theoretical foundation [19] in which the required vacuum alignment emerges fromsymmetry as a semi-direct model [20]. In general the Littlest Seesaw model is an exampleof trimaximal TM mixing [21, 22], in which the first column of the tri-bimaximal mixingmatrix [23] is preserved, similar to the semi-direct model of trimaximal TM mixing thatwas developed in [24]. To fix the seesaw phase, one imposes a CP symmetry in the originaltheory which is spontaneously broken, where, unlike [25], there is no residual CP symmetryin either the charged lepton or neutrino sectors, but instead the phase η in the neutrinomass matrix is fixed to be one of the cube roots of unity due to a Z family symmetry, In seesaw models with two right-handed neutrinos, including those discussed in this paper, a hierarchicalspectrum of left-handed neutrino masses is obtained where the lightest left-handed neutrino is massless. With the lightest neutrino massless, m = 0, we refer to the two non-zero masses as the solar neutrinomass and the atmospheric neutrino mass , corresponding to the square roots of the experimentally measuredsolar and atmospheric neutrino mass splittings m = (cid:112) ∆ m and m = (cid:112) ∆ m respectively. – 2 –sing the mechanism proposed in [26].As explained in more detail later on, the Littlest Seesaw model predicts all neutrinomasses and mixing parameters in terms of two or three parameters, and it has been shownthat the model is in agreement with all existing data, for a suitable range of its internalparameters [17]. The model makes some key predictions about the neutrino mass spectrum,that the lightest neutrino is massless m = 0 and that normal ordering obtains ∆ m > m = 0 and NO), and study how the future long- andmedium-baseline oscillation programme will be able to test this model through the precisionmeasurement of the oscillation parameters.The layout of the paper is as follows: in Section 2 we define the Littlest Seesaw modelsdiscussed above, and express some of the predictions in terms of exact sum rules of theneutrino oscillation parameters. In Section 3 the Littlest Seesaw models are confronted withexisting oscillation data and we show the precise predictions made once this data is takeninto account. Section 4 then covers how the predictions of the models could be probedat future experimental facilities, showing the sensitivities of experiments to exclude themodels and the combined measurements required to do so. We end with some concludingremarks in Section 5. Sequential dominance models of neutrinos arise from the proposal that, via the type-I see-saw mechanism, a dominant heavy right-handed (RH) neutrino is mainly responsible forthe atmospheric neutrino mass, a heavier subdominant RH neutrino for the solar neutrinomass, and a possible third largely decoupled RH neutrino for the lightest neutrino mass [13].This leads to the prediction of normal neutrino mass ordering and, in the minimal casecontaining just the dominant and subdominant right-handed neutrinos, the lightest neu-trino must be massless. Constrained sequential dominance (CSD) constrains these modelsfurther through the introduction of flavour symmetry, with the indirect approach used tofix the mass matrix from vacuum alignments of flavon fields [14]. A family of such mod-els, parameterized by n , either integer or real using the flavour symmetry groups S or A respectively, predicts the CSD( n ) mass matrix for left-handed neutrinos [10, 18]. Thismodel is also known as the Littlest Seesaw (LS) model since it provides a physically viableseesaw model with the fewest number of free parameters. After integrating out the heavyneutrinos, the resulting left-handed light effective Majorana neutrino mass matrix in the We follow the Majorana mass Lagrangian convention − ν L m ν ν cL . – 3 –harged-lepton flavour basis is given by m ν = m a + m b e iη n ( n − n n n ( n − n − n ( n −
2) ( n − , (2.1)where in addition to n there are three free real parameters: two parameters with thedimension of mass m a and m b which are proportional to the reciprocal of the masses ofthe dominant and subdominant right-handed neutrinos, and a relative phase η . A secondversion of this model has also been proposed, based on an S × U (1) symmetry, where thesecond and third rows and columns of the mass matrix are swapped [19]. In this paper, wediscuss both these versions for the case where n = 3, with the two versions of the modeldenoted as LSA and LSB; m ν LSA = m a + m b e iη , (2.2) m ν LSB = m a + m b e iη . (2.3)Although, in the most minimal set-up, the relative phase η is a free parameter, it hasbeen shown that in some models the presence of additional Z symmetries can fix thephase e iη to a cube root of unity [25], with η = 2 π/ η = − π/ η left free.Diagonalizing the mass matrices above leads to predictions for the neutrino masses aswell as the angles and phases of the unitary PMNS matrix, U PMNS , which describes themixing between the three left-handed neutrinos U T PMNS m ν U PMNS = m m
00 0 m , (2.4)where U PMNS is defined by U PMNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ c s s s − c s c e iδ − c s − s s c e iδ c c e i β e i β
00 0 1 (2.5)with s ij = sin θ ij and c ij = cos θ ij . All of the parameters in this decomposition are thereforepredicted in terms of the 2 (or 3) real parameters in Eqs. (2.2) and (2.3). Due to the minimalassumption of only two right-handed neutrinos, the lightest neutrino is massless m = 0and the mass-squared differences, which are the only combinations of masses accessible to– 4 –eutrino oscillation experiments, are predicted to be ∆ m = m and ∆ m = m . Of theremaining mixing parameters, θ , θ , θ and δ , are also experimentally accessible vianeutrino oscillation, while the Majorana phases β and β are not.As will be seen in more detail in the next section, due to their similar forms, LSAand LSB make similar predictions. However, the process of diagonalization reveals thatthe octant of θ is reversed, along with the sign of δ , while all other parameters areunchanged. Changing the sign of η , however, also reverses the sign of δ with no othereffect, and so with the sign of η not fixed by the model the only physical difference betweenLSA and LSB is the octant of θ . It has already been shown that, since the first column of the LS mixing matrix U PMNS isequal to that of the tri-bimaximal mixing matrix, LS (both LSA and LSB for all values of η ) obeys the TM1 sum rules [18, 19]tan θ = 1 √ (cid:113) − s , sin θ = 1 √ (cid:112) − s c , cos θ = (cid:114)
23 1 c , (2.6)cos δ = − cot 2 θ (1 − s )2 √ s (cid:112) − s , (2.7)where s ij = sin θ ij and c ij = cos θ ij , and the forms in Eq. (2.6) are equivalent.For LSA with η = π or LSB with η = − π , there are several additional sum rules,which we discuss here for the first time. A set of these additional sum rules can be derivedusing the fact that the only two remaining input parameters m a and m b have dimensionsof mass, so all the mixing angles and phases must depend only on the ratio r ≡ m b m a .Exact expressions for the mixing angles and Dirac phase as a function of r can be foundin Appendix A, along with new exact sum rules derived using these expressions. Theseresults make clear the difference between predictions of LSA and LSB; while θ and θ remain unchanged, cos 2 θ and cos δ differ by a change of sign.An exact expression for the Jarlskog invariant J was given as [18, 19] J = s c s c s c sin δ = ∓ m a m b ( n −
1) sin ηm m ∆ m . (2.8)with negative sign taken for LSA and positive for LSB. For both LSA with η = π , andLSB with η = − π we find the new relation m m = 6 m a m b . (2.9)Using this relation and inserting n = 3 into Eq. (2.8) leads to the new relation for theJarlskog invariant J J = − (cid:112) ∆ m ∆ m √ m (2.10)and hence the sum rule, sin δ = − (cid:112) ∆ m ∆ m √ m s c s c s c , (2.11)– 5 –hich is valid for both LSA with η = π and LSB with η = − π . Existing measurements of the neutrino mixing parameters have been shown to be in goodagreement for CSD( n ) for the n = 3 case [17]. The best-fit value of η is found to be closeto ± π , with the positive sign for LSA and the negative sign for LSB, which has beentheoretically motivated as one of the cube roots of unity required due to an additional Z symmetry as part of a larger GUT model [18]. In this section, we study both the casewhere η is fixed by symmetry and the case where it is left as a free parameter of the theory. η = ± π/ n = 3 case of LSA with η = π (or LSB with η = − π ), all neutrino masses, mixingangles and phases are fully determined from the two remaining parameters m a and m b andthe three most precisely measured of these parameters, θ , ∆ m and ∆ m , currentlyprovide the strongest test of the LS model. Figure 1 shows how these parameters varyin the m a − m b plane, along with the regions corresponding to the 1 σ and 3 σ ranges forthese parameters from the NuFit 3.0 (2016) global fit [28], assuming normal mass orderingand a lightest neutrino mass of m = 0. The SD proposal requires m a to be significantlylarger than m b and for this portion of the parameter space the approximate proportionalityrelations of m ∼ m b and m ∼ m a can be seen, verifying the approximations previouslyderived in [18].Even at 1 σ the three allowed regions coincide at a single point, as can be seen in Fig. 2,and so this benchmark point can be used to make predictions of the remaining angles θ and θ and the Dirac phase δ . As described in Section 2 these parameters, along with θ , depend only on the ratio r = m b /m a ; this dependence, given by the relations inEq. (A.1), is shown in Fig. 3, with the 1 σ and 3 σ NuFIT 3.0 ranges and reference pointat m b /m a = 0 .
1. For θ and δ , the predictions of both LSA and LSB are shown. At thispoint it can be seen that while both θ and θ lie within their 1 σ ranges, θ lies justoutside its 1 σ range, and a prediction on the value of the Dirac phase is made of δ (cid:39) − ◦ .Combining these results for all parameters which have been experimentally measured,displayed together in Fig. 4, it is seen that the prediction for θ lies just within currentbounds. However, there is tension at the 1 σ level for θ , due to the allowed regions of LSparameter space requiring values close to maximal, while current data points towards largerdeviations from the maximal value. The experimental measurements of θ do not yet giveconsistent indications of its value; while the latest results from NO ν A disfavour maximalmixing at 2 . σ [30], results from T2K remain fully compatible with maximal θ [29]. Asa result, while the combined fit for θ is in tension with the LS models at 1 σ , the allowedrange at 2 σ is far wider, crossing both octants and the maximal value of 45 ◦ , including thevalues preferred by the LS model . For a more detailed discussion of the current status of experimental measurements of θ , see [28] – 6 – a ) m b [ m e V ]
15 20 25 30 35 m a [ meV ] ( b )
15 20 25 30 35 m a [ meV ] ( c )
15 20 25 30 35 m a [ meV ] θ [°] m [ meV ] m [ meV ]
30 35 40 45 50 55 60 65
Figure 1 : Predicted values from LSA with η = π (or LSB with η = − π ) of oscillationparameters depending on the input parameters m a and m b . Regions corresponding to theexperimentally determined 1 σ (solid lines) and 3 σ (dashed lines) ranges for each parameterare also shown. ★★
15 20 25 30 351.01.52.02.53.03.54.0 m a [ meV ] m b [ m e V ] θ Δ m Δ m Figure 2 : Regions in the m a - m b plane with fixed η = 2 π/ η = − π/
3) for LSA (LSB)corresponding to the experimentally determined 1 σ and 3 σ ranges for θ , ∆ m and ∆ m . η as a free parameter In the versions of the LS models with η as an additional free parameter, the mixing anglesand phases now depend on both the ratio r = m b /m a and η . The masses m and m depend on all three input parameters; however, their ratio m /m (and therefore the ratio∆ m / ∆ m ) will depend only on r and η . As previously, the strongest contraints comefrom the very precise measurements of θ and the mass-squared differences ∆ m and– 7 – .0 0.1 0.2 0.3 0.4 0.5010203040506070 m b / m a θ [ ° ] θ θ θ δ - - - m b / m a δ [ ° ] Figure 3 : Predicted values from LS with fixed η = 2 π/ η = − π/
3) for LSA (LSB) ofthe mixing angles and delta as a function of the ratio m b /m a . Horizontal bands show theexperimentally determined 1 σ and 3 σ ranges for each parameter. A reference point givinga good prediction for all parameters is shown at r = m b /m a = 0 . ★★ m b [ m e V ]
16 18 20 22 24 26 28 30 32 34 m a [ meV ] ★★
16 18 20 22 24 26 28 30 32 34 m a [ meV ] LSA LSB θ θ θ Δ m Δ m Figure 4 : Regions in the m a - m b plane with fixed η = 2 π/ η = − π/
3) for LSA (LSB)corresponding to the experimentally determined 1 σ ranges for solar and reactor mixingangles and mass-squared differences. The θ regions shown are in tension with othermeasurements, however, extending to 2 σ these regions become far larger, covering theentire parameter space shown in these plots.∆ m . Figure 5 shows the regions corresponding to the 1 σ ranges for all the mixingangles, δ and m /m , where we see that all the five regions come close to overlapping– 8 –round η = ± π/ η from θ and m /m , we still findsome tension with the value of θ even when allowing η to vary. As with the case with η fixed, this tension exists only at the 1 σ level, where close to maximal θ is excluded. ★★ - - η / π m b / m a ★★ m b / m a LSA LSB θ θ θ m / m δ Figure 5 : Regions in the m b /m a - η plane corresponding to the experimentally determined1 σ ranges for all mixing angles, δ and the ratio of neutrino masses m /m for LSA (leftpanel) and LSB (right panel). In order to provide a more concrete measure of the agreement between the predictions ofthe model and existing data, as well as to make further predictions of the less well measuredparameters, we have performed a χ fit to the four cases discussed above: LSA and LSBwith η fixed and free. As a proxy for the full data sets of previous experiments, our fits usethe results of the NuFIT 3.0 global analysis [28]. This analysis combines the latest results(as of fall 2016) of solar, atmospheric, long baseline accelerator, and long, medium andshort baseline reactor neutrino experiments, to obtain a combined fit to the six standardneutrino oscillation parameters. We use the χ data provided by NuFIT, for the casewhere normal mass ordering is assumed, combining both the 1D χ data for each mixingparameter with the 2D χ data to include correlations between parameter measurements χ (Θ) = (cid:88) θ i ∈ Θ χ ( θ i ) + (cid:88) θ i (cid:54) = θ j ∈ Θ (cid:0) χ ( θ i , θ j ) − χ ( θ i ) − χ ( θ j ) (cid:1) , (3.1)– 9 –here the first sum in this expression combines each of the 1D χ data into a first approx-imation of the full 6D χ while the second sum provides corrections to this coming fromthe 2D correlations between each pair of parameters.We then apply this result first to the standard mixing case, then to the LS model caseas follows: • For the case of standard mixing Θ = Θ
PMNS ≡ (cid:8) θ , θ , θ , ∆ m , ∆ m , δ (cid:9) and wesimply combine the NuFIT 3.0 results as shown above, in order to include correlations,and use it to calculate χ (Θ PMNS ) ≡ χ (Θ) for this case. • For the LS model we use instead Θ = Θ LS ≡ { m a , m b , η } (or Θ LS = { m a , m b } whenfitting with η fixed), which is then minimised over the LS parameter space using theanalytic relations to calculate standard mixing parameters from LS parameters, andhence calculate χ (Θ LS ) ≡ χ (Θ) for this case.Our test statistic for a particular LS model is then given by: (cid:112) ∆ χ = (cid:114) min Θ LS [ χ (Θ LS )] − min Θ PMNS [ χ (Θ PMNS )] . (3.2)We have verified through Monte-Carlo calculations that Wilk’s theorem holds for thisstatistic, i.e. it is approximately distributed according to a chi-squared distribution.The best fit LSA and LSB points for fits with η left free or with η fixed at π aregiven in Table 1. The number of degrees of freedom (d.o.f.) is either 3 or 4, which is justthe difference between the number of observables (which we take to be the parameters inΘ PMNS ) and the number of LS parameters (namely the parameters in Θ LS , which is either3 or 2, depending on whether η is free or fixed). For LSA we find a best fit with ∆ χ = 4 . η free and ∆ χ = 5 . η = π ,while for LSB we find better fits, with ∆ χ = 3 . χ = 4 . η free and η = − π respectively.Figure 6 shows the best fit points with 1 σ and 3 σ contours of the fits in the m a − m b plane for fixed η and in the r − η plane for free η . The significance at which a LS model isallowed is determined from the distribution of the ∆ χ test statistic, where N σ has beencalculated assuming the that Wilks’ theorem applies. Note that despite LSA predictingvalues of θ which lie outside its individual 1 σ range reported by NuFIT 3.0, there arestill regions not excluded at 1 σ . This is due to the high predictivity of the model; bypredicting many parameters from few input parameters there is a greater chance that oneof these may lie outside its experimentally determined range. Statistically, this comes fromthe increased number of degrees of freedom of the χ -distribution which approximates ourtest statistic ∆ χ .Our fit can also be used to identify the regions of standard neutrino mixing parameterspace predicted by LS, once existing data has been taken into account. This correspondsto mapping the regions of LS input parameter space allowed by our fit onto the standardmixing parameter space. Figure 7 shows the predictions of LS (for the fixed η case) inthe planes made from each pair of mixing angles and δ . Since these values all depend– 10 –SA LSB NuFIT 3.0 η free η fixed η free η fixed global fit m a [meV] 27.19 26.74 26.95 26.75 m b [meV] 2.654 2.682 2.668 2.684 — η [rad] 0 . π π/ − . π − π/ θ [ ◦ ] 34.36 34.33 34.35 34.33 33 . +0 . − . θ [ ◦ ] 8.46 8.60 8.54 8.60 8 . +0 . − . θ [ ◦ ] 45.03 45.71 44.64 44.28 41 . +1 . − . δ [ ◦ ] -89.9 -86.9 -91.6 -93.1 − +38 − ∆ m [10 − eV ] 7.499 7.379 7.447 7.390 7 . +0 . − . ∆ m [10 − eV ] 2.500 2.510 2.500 2.512 2 . +0 . − . ∆ χ / d.o.f 4.1 / 3 5.6 / 4 3.9 / 3 4.5 / 4 — Table 1 : Results of our fit of existing data to LSA and LSB with η left free and for η = π for LSA and η = − π for LSB. The results of the NuFIT 3.0 (2016) global fit to standardneutrino mixing are shown for the normal ordering case for comparison.only on the single parameter r , the predictions of LS form lines of allowed solutions ineach plane, corresponding to sum-rules between the oscillation parameters. For example,Fig. 7a corresponds to the TM1 sum rule in Eq. (2.6), while Figs. 7b to 7f correspond tothose in Eq. (A.6) or to combinations of these sum rules. It can be seen that very strongrestrictions are placed on the allowed values of the less well measured parameters, θ , θ and δ . For the remaining angle, θ , around two thirds of the NuFIT 3.0 range remainsviable in LS.Figure 8 shows the allowed regions of parameter space for pairs of variables includingthe mass-squared differences. In these plots, as the mass-squared differences can depend onboth m a and m b independently, we see regions of allowed values instead of lines. For eachof these planes, any point will fully determine both input parameters m a and m b , and sothese contours correspond exactly to the equivalent regions shown in Fig. 6. In addition tothe tight constraints on θ , θ and δ already mentioned, in Figs. 8b and 8e it can be seenthat the allowed range of θ is correlated with that of both ∆ m and ∆ m , suggestingthat combining future measurements of these parameters could provide a better probe ofLS than the individual parameter measurements alone. The ability of future experiment toexclude the model then depends on both the predictions of the model seen here, combinedwith the sensitivity of experiments to measurements of the parameters in the region ofinterest predicted by LS, which is the focus of the next section. In order to understand the potential for future experiments to exclude the LS models,we have performed simulations of a combination of accelerator and reactor experiments,– 11 – SA ★ Best fit 1 σ σ LSB ★ Best fit 1 σ σ ★★★★ m b [ m e V ] m a [ meV ] ★★ ★★ - - - - - - η / π m b / m a Figure 6 : Results of the fits to LS of the NuFIT 3.0 (2016) global neutrino oscillationdata. Left: LS fit with fixed η = 2 π/ η = − π/
3) for LSA (LSB). Right: LS fit with η as a free parameter.modelling the experimental data expected over the next two decades. We have used theGeneral Long Baseline Experiment Simulator (GLoBES) libraries [48, 49] to simulate futureexperiments and to fit the simulated data to both standard mixing and the LS models. Inall our simulations we assume that the mass ordering is known to be normal ordering, as thisis a requirement of the LS models; a measurement of inverted ordering would immediatelyexclude the models. Our combination of experiments include detailed simulations of the T2HK and DUNE long-baseline accelerator experiments, which aim to provide precision measurements of ∆ m , θ and δ , together with basic constraints on θ from the Daya Bay short baseline reactorexperiment and on θ and ∆ m from the JUNO and RENO-50 medium baseline reactorexperiments. We will now briefly recap the salient features of these experiments and ourtreatment of them. T2HK
The Tokai to Hyper-Kamiokande (T2HK) experiment is a proposed long-baseline acceler-ator neutrino experiment using the Hyper-Kamiokande detector, a megatonne scale waterCherenkov detector to be constructed near to the Super-Kamiokande detector in Kamioka,Japan [41]. The standard design is for two tanks to be built, each with 258 kt (187 kt) oftotal (fiducial) volume. The tanks are to be built in a staged process with the second tankconstructed and commissioned after the first, such that the second begins to take data– 12 – a ) s i n θ ( b ) s i n θ ( c )( d ) - - - - - - δ [ ° ] sin θ ( e ) sin θ ( f ) sin θ LSA1 σ AllLSB1 σ AllNuFIT1 σ σ Figure 7 : Allowed values for LSA (red) and LSB (blue) with η = 2 π/ η = − π/ σ range (solid). These lines ofallowed solutions correspond to the sum rules in Eqs. (2.6) and (A.6), or combinationsthereof. Also shown are the 1 σ (solid) and 3 σ (dashed) regions from the NuFIT 3.0 2016global fit (grey).six years after the first. The water Cherenkov technique is capable of detecting the (anti-)muons and electrons (positrons) produced in (anti-)neutrino interactions, with the abilityto distinguish the charged leptons’ flavours but not their charge. The detector would beused to observe neutrinos from (amongst other sources) an upgraded version of the T2Kneutrino beam produced at J-PARC in Tokai, L = 295 km from the detector. The 1.3 MWbeam, produced from a 30 GeV protons, is directed 2.5 ◦ away from the detector in order toprovide a narrow energy spectrum at the far detector peaked around the first atmosphericneutrino oscillation maximum for ∆ m ∼ . × − eV and E = 0 . ν µ or ¯ ν µ can be produced as the principle component of the beam, such that the oscillationprobabilities P ( ν µ → ν e ), P (¯ ν µ → ¯ ν e ), P ( ν µ → ν µ ) and P (¯ ν µ → ¯ ν µ ) can all be measured.While the main goal of T2HK is to search for CP symmetry violation by observing a non-CP conserving value of δ , precision measurements of θ and the magnitude of ∆ m willalso be made [42]. – 13 – SA 1 σ σ LSB 1 σ σ NuFIT 1 σ σ ( a ) Δ m [ - e V ] ( b ) ( c )( d ) Δ m [ - e V ] sin θ ( e ) sin θ ( f ) sin θ ( g ) - - - - - - δ [ ° ] Δ m [ - eV ] ( h ) Δ m [ - eV ] ( i ) Δ m [ - e V ] Δ m [ - eV ] Figure 8 : Allowed 1 σ (solid) and 3 σ (dashed) regions for LSA (red) and LSB (blue) with η = 2 π/ η = − π/ DUNE
The Deep Underground Neutrino Experiment [43] (DUNE) is a proposed long-baselineaccelerator experiment, which differs from the T2HK experiment through its longer baselineand higher energy wide-band beam. The experiment will use a new neutrino beam sourcedat Fermilab, directed towards a large liquid argon detector in Sanford, L =1300 km fromthe beam source. The 40 kt LArTPC detector is able to detect both the charged leptonsand the hadrons produced from muon and electron (anti-)neutrino interactions, with strongparticle identification and energy reconstruction capabilities. The standard design is fora 1.07 MW ν µ or ¯ ν µ beam produced from 80 GeV protons, with an on-axis design toproduce a wide energy spectrum spanning E =0.5 to 5 GeV, allowing observations of the ν e appearance spectrum around the first atmospheric neutrino oscillation maximum for∆ m ∼ . × − eV . While measuring the same oscillation channels as T2HK, thewider band beam with longer baseline provides complementary information on the value– 14 –f δ as well as measurements of θ and, due to the matter effects from the longer baseline,both the sign and the magnitude of ∆ m [44]. Short baseline reactor experiments
By observing the oscillations of the ¯ ν e produced in nuclear reactors, short baseline reactorneutrino experiments are able to measure the mixing angle θ with particularly highaccuracy. The Daya Bay experiment [45] currently has the most precise measurement ofthis parameter with the aim to achieve a precision on sin θ of better than 3% [46]. Theexperiment measures anti-neutrinos produced in six nuclear reactors in south China. Atotal of eight 20 t liquid scintillator detectors are used; two are located at each of twonear detector sites and four at a far detector site L =1.5 to 1.9 km from the reactors nearthe first atmospheric neutrino oscillation maximum for ∆ m ∼ . × − eV , giventhe low nuclear energy of the neutrino beam E ∼ few MeV. Results of the Double Chooz[52] and RENO [51, 53] short baseline reactor experiments also contribute to the precisionobtained on θ combined with the Daya Bay result. Although DUNE and T2HK willalso measure this parameter with high precision, the measurement of the short baselinereactor programme by that time is expected to be at least as precise, and will provide ameasurement independent of the other parameters which influence the appearance channelat long-baseline accelerator experiments. Medium baseline reactor experiments
The Jiangmen Underground Neutrino Observatory [47] (JUNO) and the future plans of theReactor Experiment for Neutrino Oscillation (RENO-50) [51] are medium baseline reactorneutrino experiments which, like the Daya Bay experiment, will observe the oscillationsof electron anti-neutrinos produced in nuclear reactors. The JUNO experiment will usea 20 kt liquid scintillator detector approximately L =53 km from two planned nuclearreactors in southern China, while RENO-50 will use an 18 kt liquid scintillator detectorapproximately L =50 km from a nuclear reactor in South Korea. Given the low nuclearenergy of the neutrino beam E ∼ few MeV, these longer baselines correspond to the firstsolar neutrino oscillation maximum for ∆ m ∼ . × − eV , where the higher frequencyatmospheric oscillations appear as wiggles. Thus the longer baseline than at Daya Baygives greatest sensitivity to a different set of oscillation parameters, in particular θ and∆ m . The precision on the measurements of both sin θ and ∆ m is expected to reach0.5% [47, 51]. Details of experimental simulation
We have used complete simulations of the latest designs for both DUNE and T2HK wherewe have assumed both experiments run for 10 years. Full details of the GLoBES imple-mentations we have used can be found in [50]. For the short and medium baseline reactorexperiments, we have included basic constraints on the values of sin θ , sin θ and ∆ m .Since these measurements are expected to be approximately independent of other parame-ters we have implemented these constraints as simple Gaussian measurements with a meanof the true simulated value and error as given in Table 2.– 15 –xperiment Parameter PrecisionShort baseline reactor sin θ θ m Table 2 : Precision of oscillation parameter measurements made by reactor experimentswhich we have used as constraints in our simulations.
To determine the statistical significance with which the LS model could be excluded basedon simulated data, we perform a minimum- χ fit to both standard three neutrino mixingand to the LS model. As in section 3.3, for the case of standard mixing we use Θ =Θ PMNS ≡ (cid:8) θ , θ , θ , ∆ m , ∆ m , δ (cid:9) , while for LS we use Θ = Θ LS ≡ { m a , m b , η } (orΘ LS = { m a , m b } when fitting with η fixed). Our test statistic for the significance to excludethe LS model is then given by (cid:112) ∆ χ = (cid:114) min Θ LS [ χ (Θ LS )] − min Θ PMNS [ χ (Θ PMNS )] . (4.1)The significance at which LS is excluded is then determined from the distribution of the∆ χ test statistic; where we give sensitivities in terms of N σ , this quantity has beencalculated assuming the that Wilks’ theorem applies. Wilks’ theorem states that whencomparing nested models, the ∆ χ test statistic is a random variable asymptotically dis-tributed according to the χ -distribution with the number of degrees of freedom equal tothe difference in number of free parameters in the models. In this case we treat the LSmodels, with two or three free parameters, as sub-models of standard neutrino mixing withsix free parameters, leading to a χ -distribution with 4 degrees of freedom when η is keptfixed or 3 degrees of freedom when η is left as a free parameter. We have verified via Monte-Carlo simulations that the distribution of our ∆ χ test statistic is well approximated bythese distributions.In applying the above formula, the χ (Θ) is minimised over the parameters Θ in ourfits and is built from three parts; χ (Θ) = χ (Θ) + χ (Θ) + P (Θ) , (4.2)with χ (Θ) for the full simulations of the long-baseline experiments DUNE and T2HK, χ (Θ) for the constraints from reactor experiments Daya Bay and JUNO, and P (Θ) fora prior intended to include information from the results of existing experimental measure-ments.For the long-baseline experiments we use the statistical model of the GLoBES library[48, 49], where the χ LB (Θ) is a sum of contributions from each of the experiments’ channels.The individual contributions are constructed as χ c (Θ) = min ξ = { ξ s ,ξ b } (cid:34) (cid:88) i (cid:18) η i (Θ , ξ ) − n i + n i ln n i η i (Θ , ξ ) (cid:19) + p ( ξ, σ ) (cid:35) , (4.3)– 16 –here χ c denotes the contribution from a given channel of a given experiment. The sum inthis expression is over the i energy bins of the experimental configuration, with simulatedtrue event rates of n i and simulated event rates η i (Θ , ξ ) for the hypothesis parameters Θ andsystematic error parameters ξ . The systematic errors of the experiments are treated usingthe method of pulls, parameterized as ξ s for the signal error and ξ b for the background error.These parameters are given Gaussian priors which form the term p ( ξ, σ ) = ξ s /σ s + ξ b /σ b ,where σ = { σ s , σ b } are the sizes of the systematic errors given by the experiment.For the reactor experiments we simply assume independent Gaussian measurementssuch that χ = (cid:0) sin θ − sin θ (cid:1) σ θ + (cid:0) sin θ − sin θ (cid:1) σ θ + (cid:16) ∆ m − ∆ m (cid:17) σ m , (4.4)where θ , θ and ∆ m are the true parameter values and σ θ , σ θ and σ ∆ m thecorresponding experimental measurement uncertainties.The prior P (Θ) provides information from existing experimental measurements andis calculated using the results of the NuFIT 3.0 global fit in the same way as our fit inSection 3.3, so that P (Θ) = χ (Θ) as defined in Eq. (3.1).In all our simulations, the true parameters are taken to be the best-fit values from theappropriate LS fit results given in Table 1, except where stated otherwise. The sensitivity to exclude either version of the LS model is shown as a function of thetrue value of each parameter in Fig. 9, for true values, with the range selected along thehorizontal axes to be that given by the currently allowed at 3 σ by the latest NuFIT 3.0global fit. In each case, the parameters not shown are assumed to take their best-fit valuesfrom the fit to LS described in Section 3.3.From the upper panels in Fig. 9, we see that θ , θ and δ provide the strongest testsof the model, with there only being a relatively small portion of the presently allowed trueparameter space where the model would not be excluded. This is due to the strong predic-tions of these parameters by the LS models, as discussed in Section 3.1. Note that theseparameters are those that will be measured most precisely by the three next-generationexperiments used in our simulations, JUNO, DUNE and T2HK. For these three param-eters, the effect of allowing η to vary does not much change the sensitivity, other thanthe additional solution (currently disfavoured by experiment) with δ = +90 ◦ which occurswhen changing the sign of η . For θ in particular there is no effect of allowing η to vary.This is due to the sum rule in Eq. (2.6) which relates θ with θ independently from thevalue of η ; the precise measurement of θ then fixes the value of θ to a narrow range suchthat a measurement of θ outside of this would exclude the LS model regardless of theLS parameter values. Similarly the precise measurements of θ , ∆ m and ∆ m stronglyconstrain the magnitude (but not sign) of η , so that the LS allowed regions of the othervariables are not significantly changed when η is allowed to vary, with the noted exceptionthat changing the sign of η allows the sign of δ to also change.– 17 – SA fixed η = π / η LSB fixed η =- π / η ( a ) N σ t oe xc l udeL S True sin θ ( b ) True sin θ ( c ) - -
90 0 90 180
True δ [°]( d ) N σ t oe xc l udeL S True sin θ ( e ) True Δ m [ - eV ] ( f ) True Δ m [ - eV ] Figure 9 : The predicted sensitivity of future experiments to excluding LSA (red) and LSB(blue), shown as a function of the true value of each parameter. Solid curves correspondto the case with η fixed at η = π for LSA or η = − π for LSB, while dashed curvescorrespond to the case with η left free. The ranges of true parameters shown in the plotscorresponds to the current three sigma allowed NuFIT 3.0 regions.From the lower panels in Fig. 9, we see that the sensitivity to exclude LS from mea-surements of θ , ∆ m or ∆ m is much less than for the other three parameters and thesensitivity is also significantly reduced when allowing η to vary. By the converse argumentto that used above, this is due to these three parameter measurements driving the fit to m a and m b (and η ), and so a measurement of these parameters will tend to move the fittedLS parameter values rather than exclude the model, particularly when fitting the extrafree parameter η . However, a particularly small measurement of θ or particularly largemeasurement of ∆ m , relative to their current allowed range of values, may still excludethe fixed η version of the models.The results shown in Fig. 9 show only the dependence of the significance to excludeLS on the true value of each variable individually. However, the sensitivity will generally– 18 –ave a strong dependence on the true values of the other parameters. The significance toexclude the LS models depending on the true values of each pair of variables, for the caseswhere η is kept fixed, is shown in Figures 10 and 12 for LSA and in Figs. 11 and 13 forLSB.Each panel of Figs. 10 and 11 includes two dimensionless variables (i.e. angle or phase)which both depend only on the ratio of LS input parameters r = m b /m a , and so, in a LSmodel, a measurement of any one of these parameters corresponds to a measurement of r = m b /m a (see Fig. 3). Combining two of these parameter measurement therefore give twomeasurements of r = m b /m a , with any conflict between them providing strong evidence toexclude the model. For this reason the significance to exclude the models is close to beingsimply the combined significance from individual measurements implied by Fig. 9.By contrast, each panel of Figures 12 and 13 shows the results for the pairs of variablesincluding at least one dimensionful mass-squared difference. Here we can see in Figs. 12b,12e and 12i for LSA, and in Figs. 13b, 13e and 13i for LSB, there is a strong correlationbetween the measurements of θ , ∆ m and ∆ m . This shows clearly that, althoughindividual measurements of these parameters cannot exclude a LS model (since the pa-rameters of the LS model could be adjusted to accommodate any of them individually)a combined measurement of two of them could serve to exclude the model. This is thereason for presenting these combined sensitivity plots. Of the three parameters for whichsuch combined measurements provide the strongest test of the model, each pair includesmeasurements from different experiments, with θ coming mainly from the short-baselinereactor measurement such as Daya Bay, ∆ m from the medium-baseline reactor measure-ment such as JUNO, and ∆ m from the long-baseline accelerator measurement such asDUNE and T2HK. This demonstrates a strong synergy between all these experiments inattempts to exclude the LS models. In this paper, we have investigated the ability to probe one of the most predictive viableneutrino mass and mixing models with future neutrino oscillation experiments: the LittlestSeesaw. The LS models work within the framework of the Type I seesaw mechanism,using two right-handed neutrinos to generate the left-handed neutrino masses. Combinedwith constraints from flavour symmetries, the neutrino mixing angles and phases can bepredicted from a small number of parameters; in its most constrained form all neutrinomasses, angles, and phases are determined from just two input parameters. In fact, we haveshown that while the neutrino masses depend on the two mass parameters independently,the mixing angles and phases depend only on a single dimensionless quantity, the ratio ofthese two input parameters.We have studied two versions of this model (LSA and LSB) which use different flavoursymmetries to enforce constraints which result in different permutations of the second andthird rows and columns of the neutrino mass matrix, leading to different predictions forthe octant of θ . Using the results of a recent global fit of neutrino oscillation experi-ments, we have found that both versions can well accommodate the parameter values as– 19 – a ) s i n θ ( b ) s i n θ ( c )( d ) - - - - δ [ ° ] sin θ ( e ) sin θ ( f ) sin θ N σ to exclude LSA Figure 10 : The predicted sensitivity of future experiments to excluding LSA, with η fixed at η = π , shown as a function of each pair of true parameters. The ranges of trueparameters shown in the plots corresponds to the current three sigma allowed NuFIT 3.0regions.measured by experiment, with the greatest tension on the value of θ at the 1 σ level. Theprediction of LS is very close to the maximal mixing value with experimental results fromNO ν A suggesting a more non-maximal value, while results from T2K still consistent witha maximal value of θ . We find that the LSB version, predicting a value of θ in the loweroctant, to be slightly preferred.The ability of future experiments to exclude these models then comes from a convolu-tion of the strength of the predictions of the model with the sensitivity of the experimentsin measuring those parameters. Through our fit of the models to current global neutrinooscillation data, we have seen that the LS models make strong predictions for the valuesof θ , θ , and δ , the three parameters for which current measurements are weakest. Inaddition we find that, for certain combinations of the remaining observables, θ , ∆ m and ∆ m , the LS models predict strong correlations.With future experiments expected to improve precision on all six parameters measuredthrough oscillations, our simulations have shown that the LS models can be thoroughlytested through future precise individual measurements of θ , θ , and δ . This can be readily– 20 – a ) s i n θ ( b ) s i n θ ( c )( d ) - - - - δ [ ° ] sin θ ( e ) sin θ ( f ) sin θ N σ to exclude LSB Figure 11 : The predicted sensitivity of future experiments to excluding LSB, with η fixedat η = − π , shown as a function of each pair of true parameters. The ranges of trueparameters shown in the plots corresponds to the current three sigma allowed NuFIT 3.0regions.understood since the free parameters of the LS models are currently most constrained bythe precise measurements of θ , ∆ m and ∆ m , leading to predictions for the currentlyless well determined parameters θ , θ , and δ .The predictivity of the LS models means that an even higher precision measurement ofthose parameters which currently drive the fit of the input parameters, namely θ , ∆ m and ∆ m , could still exclude the LS models when considered in combination with eachother. For example, the combination of any two of them could require a region of LSparameter space already excluded by the third.These above results all highlight the strong complementarity between different classesof oscillation experiment. While the long baseline accelerator experiments DUNE andT2HK are expected to provide the strongest measurements of θ and δ (two of thosethat can individually test the model’s viability) the third, θ , will come from mediumbaseline reactor experiments such as JUNO and RENO-50. The strongest complementarity,however, comes from combining precision measurements of ∆ m , ∆ m and θ , whereany pair of these measurements relies on the results from all the different experiments:– 21 – σ to exclude LSA ( a ) Δ m [ - e V ] ( b ) ( c )( d ) Δ m [ - e V ] sin θ ( e ) sin θ ( f ) sin θ ( g ) - - - - δ [ ° ] Δ m [ - eV ] ( h ) Δ m [ - eV ] ( i ) Δ m [ - e V ] Δ m [ - eV ] Figure 12 : The predicted sensitivity of future experiments to excluding LSA, with η fixed at η = π , shown as a function of each pair of true parameters. The ranges of trueparameters shown in the plots corresponds to the current three sigma allowed NuFIT 3.0regions.long-baseline accelerator experiments for ∆ m , medium-baseline reactor experiments for∆ m , and short-baseline reactor experiments for θ .In summary, the work presented in this paper shows that the most straightforwardway to exclude the LS model is to provide a better individual determination of the threecurrently less precisely measured parameters θ , θ , and δ , which requires both mediumbaseline experiments such as JUNO and RENO-50, and long baseline experiments such asDUNE and T2HK, where the synergy between the latter two experiments is thoroughlyexplored in [50]. In addition, the LS model could be constrained by combined measurementsof the three remaining parameters ∆ m , ∆ m and θ , where an even higher precision ofthe latter reactor parameter at the short baseline Daya Bay experiment can also play animportant role. – 22 – σ to exclude LSB ( a ) Δ m [ - e V ] ( b ) ( c )( d ) Δ m [ - e V ] sin θ ( e ) sin θ ( f ) sin θ ( g ) - - - - δ [ ° ] Δ m [ - eV ] ( h ) Δ m [ - eV ] ( i ) Δ m [ - e V ] Δ m [ - eV ] Figure 13 : The predicted sensitivity of future experiments to excluding LSB, with η fixedat η = − π , shown as a function of each pair of true parameters. The ranges of trueparameters shown in the plots corresponds to the current three sigma allowed NuFIT 3.0regions.We remark that, although the above conclusions have been established for the LSAand LSB models, similar arguments can be expected to apply to any highly predictiveflavour models which determine the oscillation parameters from a smaller number of inputmodel parameters. In any such model, the input parameters will tend to be tuned to fit thestrong constraints from the most precisely measured parameters, leading to predictions ofthe other parameters. If the models can accommodate individual measurements in this way,distinguishing between them using those parameters which drive the fit is still possible, ifthose models are highly constrained, but this requires the parameter measurements to beconsidered in combination.In conclusion, the need for future reactor and accelerator experiments to measureindividually θ , θ and δ , plus combinations of θ , ∆ m and ∆ m , may be considered– 23 –o be general requirements in order to probe predictive flavour symmetry models. Thereforea broad programme of such precision experiments seems to be essential in order to takethe next step in understanding neutrino oscillations in the context of the flavour puzzle ofthe Standard Model. Acknowledgments
We would like to thank Michel Sorel, Alan Bross and Ao Liu for providing experimentalinformation for use in our simulation of DUNE, and also the Hyper-Kamiokande proto-collaboration collaboration for information used in our simulations for Hyper-Kamiokande.PB, SP and TC acknowledge partial support from the European Research Coun-cil under ERC Grant “NuMass” (FP7-IDEAS-ERC ERC-CG 617143). We all acknowl-edge partial support from ELUSIVES ITN (H2020-MSCA-ITN-2015, GA-2015-674896-ELUSIVES), and InvisiblesPlus RISE (H2020-MSCARISE-2015, GA-2015-690575-InvisiblesPlus).SP gratefully acknowledges partial support from the Wolfson Foundation and the RoyalSociety.
AppendixA Exact expressions for LS sum rules
The angles and Dirac phase can then be written assin θ = s ( r ) , tan θ = t ( r ) , cos 2 θ = ± c ( r ) , cos δ = ± d ( r ) , (A.1)with positive signs taken for LSA and negative for LSB and where s ( r ) = 16 (cid:32) − r + 4(1 − r ) (cid:112) ((11 r ) + 4(1 − r )) ((11 r ) + 4(1 − r )) (cid:33) (A.2) t ( r ) = 14 (cid:32) r + 4(1 − r ) (cid:112) ((11 r ) + 4(1 − r )) ((11 r ) + 4(1 − r )) (cid:33) (A.3) c ( r ) = 2 r (11 r − (cid:16) r − r + 4 − (cid:112) ((11 r ) + 4(1 − r )) ((11 r ) + 4(1 − r )) (cid:17) ((11 r ) + 4(1 − r )) ((11 r ) + 4(1 − r )) + 4 r ((11 r ) + 2(2 − r )) (A.4) d ( r ) = − c ( r )(1 − s ( r ))2 (cid:112) s ( r )(1 − c ( r ) )(1 − s ( r )) . (A.5)Similar expressions for the Majorana phases also possible. Combining these, expressionsrelating any two of the angles and/or phases can be found. The first such relation, relating θ and θ , is the same as Eq. (2.6), which is general for all CSD( n ). New exact relationsbetween θ and θ or θ and θ , as well as the relation between δ and θ , true for LSAwith η = π or LSB with η = − π , are found of the form f ± ( θ , θ ) = 0 , g ± ( θ , θ ) = 0 , h ± ( δ, θ ) = 0 , (A.6)– 24 –here again the positive (negative) sign is used in the functions valid for LSA (LSB). Exactexpressions are given as f ± ( θ , θ ) = 44 s (cid:112) − s − s ) ∓ c cos 2 θ ± c cos 2 θ (cid:112) − s − (cid:115) s − c cos θ − s ) , (A.7) g ± ( θ , θ ) = 22 s (cid:112) − s s − ∓ cos 2 θ ± cos 2 θ (cid:112) − s − (cid:115) s − cos θ − s ) , (A.8) h ± ( δ, θ ) = 5 s − s (cid:112) − s ± √ δ (cid:113) − s (1 − s ) sin δ + 11 (cid:113) − s (1 − s ) sin δ s −
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