Tensions between the appearance data of T2K and NOvA
Mohammad Nizam, Suman Bharti, Suprabh Prakash, Ushak Rahaman, S. Uma Sankar
TTensions between the appearance data of T2K and NO ν A Mohammad Nizam,
1, 2, ∗ Suman Bharti, † SuprabhPrakash, ‡ Ushak Rahaman,
3, 5, § and S. Uma Sankar ¶ Tata Institute of Fundamental Research, Mumbai 400005, India Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India Instituto de F´ısica Gleb Wataghin - UNICAMP, 13083-859, Campinas, SP, Brazil Centre for Astro-Particle Physics (CAPP) and Department of Physics,University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa (Dated: March 13, 2020)
Abstract
The long baseline neutrino experiments, T2K and NO ν A, have taken significant amount of datain each of the four channels: (a) ν µ disappearance, (b) ¯ ν µ disappearance (c) ν e appearance and(d) ¯ ν e appearance. There is a mild tension between the disappearance and the appearance datasets of T2K. A more serious tension exists between the ν e appearance data of T2K and the ν e / ¯ ν e appearance data of NO ν A. This tension is significant enough that T2K rules out the best-fit pointof NO ν A at 95% confidence level whereas NO ν A rules out T2K best-fit point at 90% confidencelevel. We explain the reason why these tensions arise. We also do a combined fit of T2K andNO ν A data and comment on the results of this fit.
PACS numbers: 14.60.Pq,14.60.Lm,13.15.+gKeywords: Neutrino Mass Hierarchy, Long Baseline Experiments ∗ Email Address: [email protected] † Email Address: [email protected] ‡ Email Address: sprakash@ifi.unicamp.br § Email Address: [email protected] ¶ Email Address: [email protected] a r X i v : . [ h e p - ph ] M a r . INTRODUCTION Recently T2K [1] and NO ν A [2] collaborations have published their data on both neu-trino and anti-neutrino runs. The best-fit values of various neutrino oscillations parametersdetermined by each of these experiments are listed in table I below. In the table, NH refersto normal hierarchy (∆ >
0) and IH refers to inverted hierarchy (∆ < Parameter NO ν A T2K∆ / − eV (NH) 2.51 2.463∆ / − eV (IH) -2.56 -2.506sin θ (NH) 0.58 0.526sin θ (IH) 0.58 0.530 δ CP (NH) 30 . ◦ − . ◦ δ CP (IH) − . ◦ − . ◦ TABLE I:
Best-fit points of T2K [1] and NO ν A [2] data. The fit is based on the following numbers ofprotons on target (POT). For NO ν A POT are 8 . × (6 . × ) in neutrino (anti-neutrino) modes.For T2K they are 14 . × (7 . × ) in neutrino (anti-neutrino) modes. The best-fit values of ∆ , both for NH and for IH, for the two experiments are veryclose. These two experiments are sensitive to all the three unknown parameters of neutrinooscillations: the neutrino mass hierarchy, the octant of θ and the CP violating phase δ CP .Both experiments prefer NH over IH and the higher octant for θ . However, T2K preferssin θ close to the maximal value of 0 . ν A prefers a higher value which is closerto 0 .
6. Regarding δ CP , T2K demands that it should be in the lower half plane (LHP) with apreference for a value close to − ◦ . The best-fit value of δ CP for NO ν A is in the upper halfplane (UHP) and it disfavours values close to − ◦ . The results of these experiments areusually given in the form of contours of allowed regions in δ CP − sin θ plane, for NH andfor IH. Overall, T2K rules out the best-fit point of NO ν A in this plane at 95% confidencelevel and NO ν A rules out the T2K best-fit point at 90% confidence level. This disagreementis the result of the tension between the appearance data of these two experiments. Thoughboth the experiments prefer NH, the possibility of IH is not ruled out. The IH best-fit point2f T2K is allowed only at 2 σ while the corresponding point for NO ν A is allowed at 1 σ . Itis interesting to note that the IH best-fit points of the two experiments are reasonably closeto each other.Long baseline accelerator neutrino experiments take data in the following four channels: • ν µ disappearance The number of events in this channel are the largest because thesurvival probability is moderately large over a wide energy range and also becausethe neutrino flux and the cross section are larger. Hence this channel has the higheststatistical weight. • ¯ ν µ disappearance This channel has the second highest statistical weight. The survivalprobability is moderately large but the neutrino flux and the cross section are smaller. • ν e appearance The statistical weight of this channel is not very high because the os-cillation probability is rather small ( ≤ . • ¯ ν e appearance This channel has the least statistical weight because the oscillationprobability is low and the anti-neutrino flux and the cross section are smaller.The tension between the data of the two experiments may well be the result of low statisticsand may disappear with more data. However, it is possible to make some predictionsregarding the trend along which the tension is likely to be resolved by doing a combined fit.In this article we study the neutrino and anti-neutrino appearance data of these twoexperiments using the expressions for P ( ν µ → ν e ) and P (¯ ν µ → ¯ ν e ) in three flavour oscilla-tions including matter effect. The disappearance data is related to the survival probabilities P ( ν µ → ν µ ) and P (¯ ν µ → ¯ ν µ ). For the energies and the baselines of T2K and NO ν A, thesurvival probabilities can be approximated by a two flavour expression with an effectivemass-squared difference ∆ µµ [3] and sin 2 θ . We find that there are mild tensions betweenthe neutrino and anti-neutrino disappearance data of NO ν A [2] and between the neutrinoappearance and disappearance data of T2K. These tensions are likely to be resolved withmore data. There is, however, a serious tension between the appearance data of the twoexperiments, which is leading to each experiment ruling out the best-fit point of the otherexperiment. We explain the cause of this tension. We also do a combined fit of the datafrom both the experiments and compare the best-fit point to the current best-fit points ofthe two experiments. 3 I. P µe AND P ¯ µ ¯ e FOR T2K AND NO ν A For both T2K and NO ν A experiments, the expression for P ( ν µ → ν e ) is [4, 5] P ( ν µ → ν e ) = P µe = sin θ sin θ sin ˆ∆(1 − ˆ A )(1 − ˆ A ) + α cos θ sin 2 θ sin 2 θ sin 2 θ cos( ˆ∆ + δ CP ) sin ˆ∆ ˆ A ˆ A sin ˆ∆(1 − ˆ A )1 − ˆ A , (1)where ˆ∆ = 1 . L/E , ˆ A = A/ ∆ and α = ∆ / ∆ . The Wolfenstein matter term A is[6] A (in eV ) = 0 . × − ρ (in gm / cc) E (in GeV) , (2)where E is the energy of the neutrino and ρ is the density of the matter. For anti-neutrinos, P (¯ ν µ → ¯ ν e ) = P ¯ µ ¯ e is given by a similar expression with δ CP → − δ CP and A → − A . Since α ≈ .
03, the term proportional to α in P µe is neglected.The best-fit points of T2K and NO ν A, for both NH and IH, can be understood byconsidering the changes induced in P µe and P ¯ µ ¯ e by the change in each of the unknownsrelative to a common reference set of parameter values. We take this reference set to bevacuum oscillations with θ = 45 ◦ and δ CP = 0. • Inclusion of matter effect increases P µe for NH and decreases it for IH. The effect isopposite for P ¯ µ ¯ e . • Both P µe and P ¯ µ ¯ e increase if θ is in the higher octant (HO) and decrease if it is inthe lower octant (LO). • If δ CP is in the lower half plane (LHP) P µe increases whereas it decreases for δ CP inthe upper half plane (UHP). Here again the effect is opposite for P ¯ µ ¯ e .If the hierarchy is NH, the octant is HO and δ CP is in LHP, all the three unknowns boost P µe and we expect a large excess of ν e appearance events. This combination, however, leadsto a moderate suppression of P ¯ µ ¯ e because it is reduced due to hierarchy and δ CP and increaseddue to octant. A moderate increase in P µe , and hence in ν e appearance events, occurs whentwo unknowns boost it and the third suppresses it. Three different combinations can causethis possibility. They are: 4 (A) Hierarchy is NH, octant is HO and δ CP is in UHP. For this combination, the effectsof hierarchy and δ CP are opposite for both P µe and P ¯ µ ¯ e . Both these probabilities receivea modest boost due to HO. • (B) Hierarchy is NH, octant is LO and δ CP is in LHP. Here hierarchy and δ CP bothboost P µe and LO lowers it. All three parameters lower P ¯ µ ¯ e leading to the lowestexpected number of ¯ ν e appearance events. • (C) Hierarchy is IH, octant is HO and δ CP is in LHP. Here again, the effects of hierarchyand δ CP are opposite for both P µe and P ¯ µ ¯ e and both these probabilities receive a modestboost due to HO.For the T2K experiment, the peak flux occurs for E ν ≈ . P µe and P ¯ µ ¯ e . Maximal values of δ CP can change the probability by about 20%. The change induced by the octant, of coursedepends on the value of sin θ . For the NO ν A experiment, the flux peaks at E ν ≈ ν A had the followingfeatures: (a) ν µ disappearance preferred non-maximal θ and (b) ν e appearance showed amodest increase relative to the expectation from the reference point [7]. The non-maximal θ values also induced a 20% change in P µe and P ¯ µ ¯ e . Thus, each of the three unknownsinduced change of similar magnitude which lead to three degenerate solutions of the forms(A), (B) and (C) listed above [8]. III. CURRENT ACCELERATOR NEUTRINO DATAA. T2K
T2K experiment observed maximal disappearance in both ν µ and ¯ ν µ channels. Thisimplies that sin θ is close to 0 .
5. We did separate analyses of the disappearance data andthe appearance data of T2K. In these analyses, the theoretical expectations are calculatedusing the software GLoBES [9, 10]. We matched the GLoBES predictions for the expectedbin-wise event numbers with those given by the Monte-Carlo simulations of the experiments,quoted in refs. [1] and [2], for the same input parameters. In calculating the theoretical5xpectations, the values of ∆ = 7 . × − eV and sin θ = 0 .
307 are held fixed.The other mixing angles are varied in the following ranges: sin θ = (0 . ± × . θ = (0 . , . δ CP =( − ◦ , ◦ ). The effective mass-squared difference, ∆ µµ [3], is varied in the range (2 . ± × . × − eV . The value of ∆ is determined from the equation [3]∆ = ∆ µµ + ∆ (cos θ − cos δ sin θ sin 2 θ tan θ ) . (3)For NH, ∆ µµ is positive and for IH it is negative.The results of the disappearance analysis give the best-fit value of sin θ to be 0 .
51. Thedata from both neutrino and anti-neutrino channels are included in this analysis. We alsofind that this data constrains sin θ to be in the range (0 . , .
6) at 3 σ . The disappearancedata has no sensitivity to δ CP . Therefore, the constraints on sin θ are valid for all valuesof δ CP .From table-2 of reference [1] we can estimate that the ν e appearance events for thereference point is about 60, for the given neutrino run of T2K. Inclusion of matter effectschanges this number by 4 and inclusion of maximal CP violating effects changes it by about11. Thus, for T2K, the change induced by CP violation is much larger than the changeinduced by the matter effects. The combined effect of matter effects (assuming NH) andCP violation effects (assuming δ CP = − ◦ ) increases the estimated number to 80 [1]. Anyfurther increase must necessarily require a value of sin θ larger than 0 .
5. T2K observes 89 ν e appearance events. Hence, the ν e appearance data of T2K pulls sin θ to larger values.An analysis of the ν e appearance data is shown in fig 1. It gives the best-fit value of sin θ as 0 .
63, though 0 . σ range. Thus, we see that there is amild tension between T2K disappearance data and its ν e appearance data. This tensionmay be due to the limited statistics and may go away with more data. However, the finalvalue of sin θ is likely to be determined by the disappearance data because of its largerstatistical weight. The combined analysis done by the T2K collaboration gives the best-fitvalue sin θ = 0 .
53 [1]. In doing the analysis of the appearance data we have not includedthe data from the anti-neutrino channel. The number of events observed in this channel aretoo small for any meaningful analysis.The observed large excess of ν e events requires that δ CP must be in the neighbourhoodof − ◦ . For values of δ CP in the UHP, the expected number of ν e appearance events is6 s i n θ ( t e s t ) δ cp (test) 1 σ σ σ best fit (-80 o , 0.63) 0.25 0.35 0.45 0.55 0.65-180 -90 0 90 180IH test s i n θ ( t e s t ) δ cp (test) 1 σ σ σ IH best fit point (-90 o , 0.74) FIG. 1:
Expected allowed regions in sin θ − δ CP plane from the current appearance data of T2K in theneutrino channel. In the left panel, the hierarchy is assumed to be NH and in the right panel, the hierarchyis assumed to be IH. The best-fit point has minimum χ = 29 for both NH and IH for 24 energy bins. smaller than the estimated number for the reference point. Since the observed number ismuch higher, δ CP in UHP is strongly disfavoured. This data also disfavours IH because thecorresponding matter effects lead to a lower prediction for the number of events. IH with δ CP = − ◦ is barely allowed at 2 σ [1].T2K has also observed 7 ¯ ν e appearance events. For δ CP = 0 and NH, they expect toobserve 9 events. This number is expected to go down to 8 if δ CP = − ◦ . Therefore, thenumber of ν e and ¯ ν e appearance events are consistent with each other. Due to the largestatistical error in the ¯ ν e appearance events, it is not possible to make any strong commenton the value of δ CP preferred by this data. B. NO ν A The recent neutrino disappearance data of NO ν A is consistent with maximal mix-ing (sin θ = 0 .
5) whereas the anti-neutrino disappearance data prefers a non-maximalvalue [2]. Here again there is a mild tension between the different data sets of the same7 s i n θ ( t e s t ) δ cp (test) 1 σ σ best fit (120 o , 0.65) 0.3 0.4 0.5 0.6 0.7-180 -90 0 90 180IH test s i n θ ( t e s t ) δ cp (test) 1 σ σ σ IH best fit point (-50 o , 0.67) FIG. 2:
Expected allowed regions in sin θ − δ CP plane from the appearance data of NO ν A in bothneutrino and anti-neutrino channels, as given in [2]. In the left panel, the hierarchy is assumed to be NHand in the right panel, the hierarchy is assumed to be IH. The best-fit point occurs for NH with aminimum χ = 6 for 12 data points. The best-fit point of IH has ∆ χ = 0 . experiment. This tension also is not statistically significant because of the limited statisticsof the anti-neutrino data. Regarding the appearance events, we expect 39 ν e events and15 . ν e events for the reference point [8]. NO ν A experiment observed 58 and 18 eventsrespectively in these channels. That is, there is a moderate excess in both these channels.As mentioned above, each of the three unknowns induce a change of about 20% in theappearance events of NO ν A. A moderate excess in both ν e and ¯ ν e appearance events ispossible only if the changes induced by the hierarchy and δ CP cancel each other and theincrease in both channels occurs because θ is in HO. That is, the combination of theunknowns must have either form (A) or form (C) listed in section-2. We performed ananalysis of NO ν A appearance data in a manner similar to the analysis we did for T2K data.8he results are shown in fig. 2. There are two nearly degenerate best-fit solutions, withthe unknown parameter values (NH, sin θ = 0 . δ CP = 120 ◦ ) and (IH, sin θ = 0 . δ CP = − ◦ ) respectively. The first solution is in the form (A) and the second in the form(C).The analysis done by the NO ν A collaboration, of both their disappearance and ap-pearance data, also finds two nearly degenerate best-fit points [2]: (NH, sin θ = 0 . δ CP = 30 . ◦ ) and (IH, sin θ = 0 . δ CP = − ◦ ). Here again, we find two solutions, onein form (A) and the other in form (C). Since the disappearance data also is included inthe NO ν A analysis, a smaller value of sin θ is obtained compared to the values shown infig. 2. The wide difference in the values of δ CP preferred by T2K and by NO ν A shows thetension between the data of NO ν A and T2K. T2K prefers a large boost of P µe by all thethree unknowns whereas NO ν A prefers a moderate boost of both P µe and P ¯ µ ¯ e due to thecombinations mentioned above. This tension is also visible in the fact that T2K rules outthe best-fit point of NO ν A at 95% C.L. whereas NO ν A rules out the best-fit point of T2K at90% C.L. The appearance events of NO ν A, especially in the ¯ ν e channel, are limited. Withmore statistics it is possible that the present tension may go away. But the likely resolutionof this tension will have δ CP close to − ◦ because of the very large excess of ν e appearanceevents observed by T2K. IV. COMBINED FIT TO T2K AND NO ν A DATA
In this section we present our results of combined fit of the disappearance and the ap-pearance data of T2K and NO ν A in both neutrino and anti-neutrino channels. The data ofT2K is taken from ref. [1] and that of NO ν A from ref. [2]. The theoretical expectations forthe two experiments are calculated using the software GLoBES [9, 10], using the procedurethat was described in section-3.1 for T2K analyses. There are a total of 182 energy bins,half in neutrino channel and half in anti-neutrino channel. In each case, there are 42 indisappearance data and 24 in appearance data for T2K and 19 in disappearance data and6 in appearance data for NO ν A. In computing the χ between the data and the theoreticalexpectations, prior is added for sin θ and ∆ µµ . We have also included a 10% overallsystematic error for each channel of both the experiments. The results of our fit are shownin figure 3. The best-fit point for NH is (sin θ = 0 . δ CP = − ◦ ) with a χ of 219 and9 s i n θ ( t e s t ) δ cp (test) 1 σ σ σ best fit (-130 o , 0.56) 0.3 0.4 0.5 0.6 0.7-180 -90 0 90 180IH test s i n θ ( t e s t ) δ cp (test) 1 σ σ σ IH best fit point (-90 o , 0.56) FIG. 3:
Expected allowed regions in δ CP − sin θ plane from the combined fit of the neutrino andanti-neutrino data of T2K and NO ν A, as of July 2018. In the left panel, the hierarchy is assumed to beNH and in the right panel, the hierarchy is assumed to be IH. The χ for NH best-fit point is 219 and thatfor IH best-fit point is 220 .
5, for 182 data points. the best-fit point for IH is (sin θ = 0 . δ CP = − ◦ ) with a χ of 220 . ν A, especially with regard to choice of sin θ = 0 .
56. On the other hand, the choiceof δ CP = − ◦ as its best-fit value is enforced by the large excess of ν e events observed byT2K. For NH, values of δ CP in UHP are essentially ruled out at 2 σ . For IH, the best-fitpoint in our fit is close to the IH best-fit points of T2K and NO ν A. This is not surprisingbecause those two points are close to each other. For IH, the whole region of δ CP in upperhalf plane is ruled out at 3 σ because it is disfavoured by both T2K and NO ν A.Recently NO ν A collaboration published their results with increased anti-neutrino run [11].The data in this analysis is based on 8 . × POT in neutrino mode (which is the samefor the previous analysis also) and 12 . × POT in anti-neutrino mode (which is doublethat of the previous analysis). They have observed 27 ¯ ν e appearance events. The resultsof this analysis give the best-fit point as (NH, sin θ = 0 . δ CP = 0). The invertedhierarchy is disfavored with its best-fit point (IH, sin θ = 0 . δ CP = − ◦ ) being allowed10 s i n θ ( t e s t ) δ cp (test) 1 σ σ σ best fit (-120 o , 0.55) 0.3 0.4 0.5 0.6 0.7-180 -90 0 90 180IH test s i n θ ( t e s t ) δ cp (test) 2 σ σ IH best fit point (-90 o , 0.55) FIG. 4:
Expected allowed regions in δ CP − sin θ plane from the combined fit of the neutrino andanti-neutrino data of T2K and NO ν A, as of July 2019. In the left panel, the hierarchy is assumed to beNH and in the right panel, the hierarchy is assumed to be IH. The χ for NH best-fit point is 209 and thatfor IH best-fit point is 211 .
5, for 182 data points. only at 1 . σ . We see that the tension between the T2K data and the NO ν A data persistsbecause the preferred values of δ CP are widely different. We did a reanalysis of the datafrom T2K [1] and NO ν A [11]. The results are plotted in fig. 4. Comparing it with fig. 3, wenote that the allowed regions have become more constrained though they are very similarto the previously allowed regions. We also note that the best-fit value of δ CP changed from − ◦ to − ◦ and IH is allowed only at 2 σ . Thus we see that the additional anti-neutrinodata of NO ν A leads only to small changes in the combined fit.11 . CONCLUSION
The two long baseline accelerator neutrino experiments, T2K and NO ν A, have taken sig-nificant amount of data both in the neutrino channel as well as in the anti-neutrino channel.The disappearance data of T2K prefers sin θ close to 0 . ν A preferssin θ to be non-maximal. T2K has observed 89 ν e appearance events but the numberof ¯ ν e appearance events is not statistically significant. NO ν A has observed 58 ν e and 18 ¯ ν e appearance events and has established ¯ ν e appearance at 4 σ .To understand the constraints imposed by the appearance data on the three unknownparameters of neutrino oscillations, we define a reference point: no matter effects, sin θ =0 . δ CP = 0. We consider the change induced in P µe and P ¯ µ ¯ e by the inclusion of mattereffect due to NH/IH, by the change of sin θ to HO/LO and by the effect of CP-violationwith δ CP in LHP/UHP. Both matter effects and non-zero δ CP induced opposite deviationsin P µe and in P ¯ µ ¯ e . But the octant of θ changes both the probabilities the same way.The observed ν e appearance events in T2K are about 50% more than what is expected forthe reference point. Such a large excess is possible only if the change in P µe is positivedue to the changes in all the three unknowns. That is if hierarchy is NH, θ is in HOand δ CP ≈ − ◦ . The best-fit point of T2K finds the unknowns to be: hierarchy is NH,sin θ = 0 .
53 and δ CP = − ◦ . The value of sin θ is a compromise value of the best-fitvalues of the disappearance and the appearance data. T2K appearance data requires δ CP to be in the neighbourhood of − ◦ quite strongly. In the case of NO ν A, the observed ν e and ¯ ν e appearance events are in moderate excess relative to the reference point. Suchan observation can be explained only if the changes induced by hierarchy and δ CP in P µe and P ¯ µ ¯ e nearly cancel each other and the increase is due to θ being in HO. This is whyNO ν A obtains two nearly degenerate solutions: hierarchy is NH, θ in HO and δ CP ≈ ◦ and hierarchy is IH, θ in HO and δ CP ≈ − ◦ . The large excess of ν e appearance eventsin T2K rules out both these points at 95% C.L. On the other hand, the moderate excess of ν e and ¯ ν e appearance events in NO ν A disfavours enhancement of P µe due to both hierarchyand δ CP .The analysis of NO ν A data picks sin θ = 0 .
58 as the best-fit value [2]. In the combinedanalysis of the appearance and the disappearance data of the two experiments the best-fitvalue of sin θ is pulled a little lower by the disappearance data of T2K. The best-fit value12f δ CP for the NH solution is in the LHP at − ◦ . This value is the result of the large excessof ν e appearance events seen by T2K which force δ CP to take a large value in LHP. Values of δ CP in UHP, for NH, are ruled out at 2 σ , even though the best-fit point of NO ν A is in thisregion. This also is a result of the large excess of ν e appearance events observed by T2K.Values of δ CP in UHP predict the number of ν e appearance events for T2K to be close to orbelow that of the reference point. Such values are strongly disfavoured by T2K because theobserved number of events is significantly larger. Even though this region is preferred byNO ν A appearance data, the conflict between its predictions and T2K data is ruling it outat 2 σ in the combined fit.Even though T2K barely allows an IH solution at 2 σ , the combined fit has a nearlydegenerate IH solution which is the common IH solution of each experiment, with δ CP = − ◦ and sin θ = 0 .
56. If the hierarchy is IH and δ CP is in UHP P µe is doubly suppressed bymatter effects and by δ CP . There is a corresponding double enhancement of P ¯ µ ¯ e . Such afeature is not seen by either experiment hence this possibility is ruled out at 3 σ .We have redone our analysis where we have included the latest NO ν A data [11]. Thecombined analysis of T2K plus NO ν A data leads to slightly smaller allowed regions withslightly shifted best-fit parameters. The tension between the T2K data and the NO ν A datastill persists because the former prefers δ CP close to maximal CP violation whereas the latterprefers δ CP = 0. Our best-fit points, for both NH and IH, are close to the correspondingbest-fit points obtained by the Nu-fit collaboration [12], in their global neutrino oscillationdata analysis, which includes the latest long baseline accelerator neutrino data. Therefore,a simple analysis of the data of T2K and NO ν A experiments leads to reliable informationon the values of the unknown parameters.
ACKNOWLEDGEMENTS
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