Study of parameter degeneracy and hierarchy sensitivity of NOνA in presence of sterile neutrino
Monojit Ghosh, Shivani Gupta, Zachary M. Matthews, Pankaj Sharma, Anthony G. Williams
SStudy of parameter degeneracy and hierarchy sensitivity of NO ν A in presence of sterile neutrino
Monojit Ghosh, ∗ Shivani Gupta, † Zachary M. Matthews, ‡ Pankaj Sharma, § and Anthony G. Williams ¶ Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan Center of Excellence for Particle Physics at the Terascale (CoEPP), University of Adelaide, Adelaide SA 5005, Australia
The first hint of the neutrino mass hierarchy is believed to come from the long-baseline experiment NO ν A.Recent results from the NO ν A shows a mild preference towards the CP phase δ = − ◦ and normal hierar-chy. Fortunately this is the favorable area of the parameter space which does not suffer from the hierarchy- δ degeneracy and thus NO ν A can have good hierarchy sensitivity for this true combination of hierarchy and δ .Apart from the hierarchy- δ degeneracy there is also the octant- δ degeneracy. But this does not affect the fa-vorable parameter space of NO ν A as this degeneracy can be resolved with a balanced neutrino and antineutrinorun. However, ff we consider the existence of a light sterile neutrino then there may be additional degeneracieswhich can spoil the hierarchy sensitivity of NO ν A even in the favorable parameter space. In the present workwe find that apart from the degeneracies mentioned above, there are additional hierarchy and octant degenera-cies that appear with the new phase δ in the presence of a light sterile neutrino in the eV scale. In contrast tothe hierarchy and octant degeneracies appearing with δ , the parameter space for hierarchy- δ degeneracy isdifferent in neutrinos and antineutrinos though the octant- δ degeneracy behaves similarly in neutrinos and an-tineutrinos. We study the effect of these degeneracies on the hierarchy sensitivity of NO ν A for the true normalhierarchy.
I. INTRODUCTION
Neutrino oscillation physics has developed significantlysince its discovery, with precision measurements finally be-ing carried out for the mixing parameters. In the standardthree flavor scenario, neutrino oscillation is parametrized bythree mixing angles: θ , θ and θ , two mass squared dif-ferences: ∆ m and ∆ m and one Dirac type CP phase δ .Among these parameters the current unknowns are: (i) thesign of ∆ m which gives rise to two possible orderings ofthe neutrinos which are: normal ( ∆ m > ∆ m < or IH) (ii), two possible octants of themixing angle θ which are lower ( θ < ◦ or LO) andhigher ( θ > ◦ or HO), and (iii) finally the phase δ .The currently running experiments intending to discover theseunknowns are T2K [1] in Japan and NO ν A [2] at Fermilab.The main problem in determining the oscillation parametersin long-baseline experiments is the existence of parameter de-generacy [3, 4]. Parameter degeneracy implies same valueof oscillation probability for two different sets of oscillationparameters. In standard three flavor scenario, currently thereare two types of degeneracies: (i) hierarchy- δ degeneracy[5] and (ii) octant- δ degeneracy [6]. The dependence ofhierarchy- δ degeneracy is same in neutrinos and antineutri-nos but the octant- δ degeneracy behaves differently for neu-trinos and antineutrinos [7, 8]. Thus the octant- δ degeneracycan be resolved with a balanced run of neutrinos and antineu- ∗ Email Address: [email protected];ORCID ID: http://orcid.org/0000-0003-3540-6548 † Email Address: [email protected];ORCID ID: http://orcid.org/0000-0003-0540-3418 ‡ Email Address: [email protected];ORCID ID: http://orcid.org/0000-0001-8033-7225 § Email Address: [email protected];ORCID ID: http://orcid.org/0000-0003-1873-1349 ¶ Email Address: [email protected];ORCID ID: http://orcid.org/0000-0002-1472-1592 trinos but a similar method cannot remove the hierarchy- δ degeneracy. However, despite the hierarchy- δ degeneracybeing unremovable in general, the parameter space can be di-vided into a favorable region where it is completely absent forlong-baseline experiments, and an unfavorable region where itis present. For NO ν A, the favorable parameter space is around { NH, δ = − ◦ } and { IH, δ = +90 ◦ } whereas the un-favorable parameter space is around { NH, δ = 90 ◦ } and { IH, δ = − ◦ } . The recent data from NO ν A shows amild preference towards δ = − ◦ and normal hierarchy[2]. From the above discussion we understand that for thesecombinations of true hierarchy and true δ , NO ν A can havegood hierarchy sensitivity and thus it is believed that the firstevidence for the neutrino mass hierarchy will come from theNO ν A experiment. However the understanding of degenera-cies can completely change in new physics scenarios. Thisoccurs for example if there exists a light sterile neutrino inaddition to the three active neutrinos (the 3+1 scenario).Sterile neutrinos are SU(2) singlets that do not interact withthe Standard Model (SM) particles but can take part in neu-trino oscillations. Recently there has been some experimentalevidence supporting the existence of a light sterile neutrino atthe eV scale. This has motivated re-examination of oscillationanalyses of the long-baseline experiments in the presence ofsterile neutrinos [9–19]. For details regarding the first hintsof the existence of sterile neutrinos and for the current statuswe refer to Refs [20–31]. In the presence of an extra sterileneutrino, there will be three new mixing angles namely θ , θ and θ , two new Dirac type CP phases δ , δ and onenew mass squared difference ∆ m . Thus in the presence ofthese new parameters there can be additional degeneracies in-volving the standard mixing parameters and sterile mixing pa-rameters. In this work we study the parameter degeneracy inthis increased parameter space in detail. From the probabilitylevel analysis we find that in 3+1 case, we have two new kindof degeneracies which are the (i) hierarchy- δ and (ii) octant- δ degeneracies. Our results also show that in this case thescenario is completely opposite to that of the hierarchy and a r X i v : . [ h e p - ph ] A p r octant degeneracy arising with δ . The hierarchy- δ degen-eracy is opposite for both neutrinos and antineutrinos but theoctant- δ degeneracy behaves similarly in neutrinos and an-tineutrinos. Thus unlike the octant- δ degeneracy, the octantdegeneracy in this case can not be resolved by a combinationof neutrino and antineutrino runs while the hierarchy degener-acy can be resolved with a balanced combination of neutrinoand antineutrinos which was not the case for the hierarchy- δ degeneracy. To show the degenerate parameter space interms of χ , we present our results in the θ (test) - δ (test) plane taking different values of δ . We do this for two valuesof θ (true): one in LO and one in HO and for the currentbest-fit of NO ν A i.e., δ = − ◦ and NH (favorable param-eter space). We show this for considering (i) NO ν A runningin pure neutrino mode and (ii) NO ν A running in equal neu-trino and equal antineutrino mode. Next we discuss the effectof these degeneracies on the hierarchy sensitivity of NO ν A.We find that because of the existence of hierarchy- δ andoctant- δ degeneracy, the hierarchy sensitivity of NO ν A ishighly compromised at the current best-fit value of NO ν A (i.e. δ = − ◦ and NH). To show this we plot hierarchy sensitiv-ity of NO ν A in the θ (true)- δ (true) plane taking differenttrue values of δ for NH. We also identify the values of δ for which the hierarchy sensitivity of NO ν A gets affected. Tothe best of our knowledge this is the first comprehensive anal-ysis of parameter degeneracies and their effect on hierarchysensitivity in presence of a sterile neutrino has been carriedout.The structure of the paper goes as follows. In Section II wediscuss the oscillatory behaviour in the 3+1 neutrino scheme.In Section III we give our experimental specification. In sec-tion IV we discuss the the various degeneracies both at prob-ability and event level. In Section V we give our results forhierarchy sensitivity and finally in Section VI we present ourconclusions.
II. OSCILLATION THEORY
The PMNS matrix can be parametrized in many ways, themost common form with three neutrino flavors is: U ν PMNS = U ( θ , U ( θ , δ CP ) U ( θ , . (1)where U ( θ ij , δ ij ) contains a corresponding × mixing ma-trix: U × ( θ ij , δ ij ) = (cid:18) c ij s ij e iδ ij − s ij e iδ ij c ij (cid:19) (2)embedded in an n × n array in the i, j sub-block. Note theabbreviation of trigonometric terms: s ij = sin θ ij , (3) c ij = cos θ ij . (4)We also use the conventions for mass-squared differences ∆ m ij = m i − m j , (5) and we write the oscillation factors ∆ ij = ∆ m ij L E . (6)Extending to four flavors we use the parametrization: U ν PMNS = U ( θ , δ ) U ( θ , U ( θ , δ ) U ν PMNS . (7)Where the three new matrices introduce the new mixing an-gles: θ , θ , θ and phases: δ , δ . The final new oscil-lation parameter is the fourth independent mass-squared dif-ference which comes into the probability and is usually cho-sen to be ∆ m to remain consistent with the ν parameters.Assuming that ∆ m (cid:29) ∆ m , and that we are operatingnear the oscillation maximum where sin ∆ ≈ , then thesterile-induced oscillations from sin ∆ terms will be veryrapid. Hence the four flavor vacuum ν µ to ν e oscillation prob-ability can be averaged over the sterile oscillation factor ∆ i.e. (cid:104) sin ∆ (cid:105) = (cid:104) cos ∆ (cid:105) = 12 (8) (cid:104) sin ∆ (cid:105) = (cid:104) cos ∆ (cid:105) = 0 (9)this reflects the inherent averaging that the long-baseline de-tectors see due to the very short wavelength of the sterile in-duced oscillations and their limited energy resolution.Once the averaging has been done the probability expres-sion can be written using the conventions and approach from[32] as: P νµe = P ATM + P SOL + P STR (10) + P INTI + P INTII + P INTIII , where P ATM , P
SOL and P INTI are modified from the threeflavor probability terms by the factor (1 − s − s ) , i.e. P ATM = (1 − s − s ) P ATM3 ν , (11) P SOL = (1 − s − s ) P SOL3 ν , (12) P INTI = (1 − s − s ) P INT3 ν . (13)With the ν terms: P ATM3 ν ≈ s sin ∆ , (14) P SOL3 ν ≈ c s sin ∆ , (15) P INT3 ν ≈ s c s c sin ∆ sin ∆ cos(∆ + δ ) . (16)The new ν terms are P STR ≈ s , (17) P INTII ≈ s s s sin ∆ sin(∆ + δ − δ ) , (18) P INTIII ≈ − s c s c sin(∆ ) sin δ . (19)However, in the case of NO ν A we can simplify this with ap-proximations. Again, from [32],the constraints on the sterilemixing angles imply that the absolute values for P SOL , P
STR and P INTIII are less than 0.003 so can be neglected. Addition-ally, for simplicity we neglect the terms multiplied by s and s in P ATM and P INT , leaving: P νµe ≈ P ATM3 ν + P INT3 ν + P INTII . (20)which is: P νµe = 4s s sin ∆ (21) + 8s s c s c sin ∆ sin ∆ cos(∆ + δ )+ 4s s s s sin ∆ sin(∆ + δ − δ ) . From the ∆ , δ and δ dependent terms arise thehierarchy-CP degeneracies, due to the unconstrained sign of ∆ and the (mostly) unconstrained CP phases δ and δ ,which can compensate for sign changes in ∆ . The aboveformula is for neutrinos. The relevant formula for the antineu-trinos can be obtained by replacing δ by − δ and δ by − δ . Note that the above expression is for vacuum and freefrom the parameters θ and δ . III. EXPERIMENTAL SPECIFICATION
For our analysis we consider the currently running long-baseline experiment NO ν A. NO ν A is an 812 km baseline ex-periment using the NuMI beam line at Fermilab directing abeam of ν µ ’s through a near detector (also at Fermilab) ontothe NO ν A far detector located in Ash River Minnesota in theUSA. For NO ν A we assume (three years neutrino andthree years antineutrino running) unless specified otherwise.The detector is 14 kt liquid argon detector. Our experimentalspecification of coincides with [33]. To perform analysis weuse the GLoBES software package along with files for 3+1case PMNS matrices and probabilities [34–37].
IV. IDENTIFYING NEW DEGENERACIES IN THEPRESENCE OF A STERILE NEUTRINO
The information for the standard oscillation parameterscomes from the global analysis of the world neutrino data [38–40]. For the sterile neutrino parameters θ , θ and ∆ m ourbest-fit values are consistent with Refs. [30, 41–43]. We haveset θ and δ to zero throughout our analysis due to themnot appearing in the vacuum equation for P µe . Our choice ofthe neutrino oscillation parameters are listed in Table I. A. Identifying degeneracies at the probability level
In this section we will discuss parameter degeneracies in3+1 case at the probability level. In Fig. 1 we plot the ap-pearance channel probability P ( ν µ → ν e ) vs energy for theNO ν A baseline. For plotting the probabilities we have av-eraged the rapid oscillations due to ∆ m . The left columncorresponds to neutrinos and the right column corresponds to ν Parameters True Value Test Value Range sin θ .
304 N / Asin θ .
085 N / A θ LO23 ◦ (40 ◦ , ◦ ) θ HO23 ◦ (40 ◦ , ◦ )sin θ .
025 N / Asin θ .
025 N / A θ ◦ N / A δ − ◦ ( − ◦ , ◦ ) δ − ◦ , ◦ , ◦ ( − ◦ , ◦ ) δ ◦ N / A∆ m . × − eV N / A∆ m . × − eV (2 . , . × − eV ∆ m N / A TABLE I: Expanded ν parameter true values and testmarginalisation ranges, parameters with N/A are notmarginalised over.antineutrinos. In all the panels δ is taken as − ◦ and thebands are due to the variation of δ .The upper panels of Fig. 1 shows the hierarchy- δ degen-eracy. For these panels θ is taken as ◦ . NH (IH) cor-responds to ∆ m = +( − )2 . × − eV . In both thepanels the blue bands correspond to NH and the red bandscorrespond to IH. Note that in the neutrino probabilities, thegreen band is above the red band and it is opposite in the an-tineutrinos. This is because, the matter effect enhances theprobability for NH for neutrinos and IH for antineutrinos.For each given band, δ = − ◦ corresponds to the max-imum point in the probability and +90 ◦ corresponds to theminimum point in the probability, for both neutrinos and an-tineutrinos. These features in the probability can be under-stood in the following way. From Eq. 21, we see the neu-trino appearance channel probability depends on the phasesas: a + b cos(∆ + δ ) + c sin(∆ + δ − δ ) , where a , b and c are positive quantities. At the oscillation maximawe have ∆ = 90 ◦ . As our probability curves correspondto δ = − ◦ , for neutrinos we obtain a + b − c sin δ .Now it is easy to understand that the contribution to the prob-ability will be maximum for δ = − ◦ and minimum for δ = +90 ◦ . Now let us see what happens for antineutrinos.For antineutrinos, we change sign of δ and δ in Eq. 21and we obtain for δ = − ◦ as a − b − c sin δ . Thuseven for the antineutrinos, the probability is maximum for δ = − ◦ and minimum for δ = +90 ◦ . This is in starkcontrast to the behaviour of δ , as in the standard three flavorcase, δ = − ◦ corresponds to the maximum probabilitywhile δ = +90 ◦ corresponds the minimum probability forneutrinos (vice-versa for antineutrinos). From the plots wesee that there is overlap between { NH, δ = 90 ◦ } and { IH, δ = − ◦ } for the neutrinos and { NH, δ = − ◦ } and { IH, δ = +90 ◦ } for antineutrinos. Thus we understand thatunlike the nature of hierarchy- δ degeneracy, the hierarchy- δ degeneracy is different in neutrinos and antineutrinos so in P µ e ( neu t r i no s ) E [GeV] δ = − ° IHNH − ° °− ° ° P – µ – e ( an t i neu t r i no s ) E [GeV] δ = − ° IHNH − ° °− ° ° P µ e ( neu t r i no s ) E [GeV] δ = − ° LOHO − ° °− ° ° P – µ – e ( an t i neu t r i no s ) E [GeV] δ = − ° LOHO − ° °− ° ° FIG. 1: ν µ → ν e oscillation probability bands for δ = − ◦ . Left panels are for neutrinos and right panels are forantineutrinos. The upper panel shows the hierarchy- δ degeneracy and the lower panels shows the octant- δ degeneracy.principle a balanced combination of neutrino and antineutrinoshould be able to resolve this degeneracy.In the lower panels of Fig. 1, we depict the octant- δ de-generacy. In these panels LO corresponds to θ = 40 ◦ andHO corresponds to ◦ . Here the hierarchy is chosen to benormal with ∆ m = +2 . × − eV . In both the panels,the blue band correspond to LO and the red band correspondto HO. Note that in both the panels, the red band is abovethe blue band. This is because the appearance channel oscil-lation probability increases with increasing θ for both neu-trinos and antineutrinos. As already explained in the aboveparagraph, for each given band, δ = − ◦ corresponds tothe maximum value in the probability and δ = +90 ◦ to theminimum point in the probability for both neutrinos and an-tineutrinos. From the panels we see that (LO, δ = − ◦ ) isdegenerate with (HO, δ = +90 ◦ ). It is interesting to notethat this degeneracy is same in both neutrinos and antineu-trinos [12]. This is a remarkable difference compared to theoctant- δ degeneracy which is different for neutrinos and an-tineutrinos. Thus we understand that in the 3+1 scenario, itis impossible to remove the octant degeneracy by combiningneutrino and antineutrino runs. B. Identifying degeneracies at the event level
Now we analyze the relevant degeneracies at the χ level.In Fig. 2 we have given the contours in the θ (test) - δ (test) plane for three different values of δ at C.L. The firstand second column correspond to the case when NO ν A runs in pure neutrino mode and the third and fourth column corre-spond to the case when NO ν A runs in equal neutrino and an-tineutrino mode. Note that though the current plan for NO ν Ais to run in the equal neutrino and antineutrino mode, wehave produced plots corresponding to the pure neutrino runof NO ν A to understand the role of antineutrinos in resolvingthe degeneracies. We have chosen the true parameter space tocoincide with the latest best-fit of NO ν A i.e. δ = − ◦ and NH. While generating the plots we have marginalizedover δ , | ∆ m | in the test parameters while all the otherrelevant parameters are kept fixed in both the true and testspectrum. The top, middle and bottom rows correspond to δ = − ◦ , ◦ and +90 ◦ respectively. In each row the firstand third panel correspond to LO ( θ = 40 ◦ ) and the sec-ond and fourth panel correspond to HO ( θ = 50 ◦ ). Thesevalues of θ are the closest to the current best-fit accordingthe latest global analysis. For comparison we also have giventhe contours for the standard three generation case. Note thatbecause of the existence of hierarchy- δ and octant- δ de-generacies, there will be three spurious solutions in additionto the true solution which are the: (i) right hierarchy-wrongoctant (RH-WO), (ii) wrong hierarchy-right octant (WH-RO)and (iii) wrong hierarchy-wrong octant (WH-WO) solutions.As the hierarchy- δ and octant- δ degeneracy occurs for anygiven value of δ (which is − ◦ in this case), all the abovementioned three spurious solutions should appear at the cor-rect value of δ (test) = − ◦ . Below we discuss the appear-ance of these spurious solutions in detail.Let us start with the three generation case. The red con-tour is for RH solutions and the purple contour is for WH -180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] = - 90True Pt. NO A[6 + 0] = - 90True Pt. NO A[6 + 0]90% CL (4 ) NH-NH90% CL (3 ) NH-NH90% CL (3 ) NH-IH90% CL (4 ) NH-IH-180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] = 0True Pt. NO A[6 + 0] = 0True Pt. NO A[6 + 0]35 40 45 50 55 (test) [degree] -180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] = 90True Pt. NO A[6 + 0] 35 40 45 50 55 (test) [degree] = 90True Pt. NO A[6 + 0] -180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] =-90True Pt. NH-NHNO A[3 +3] =-90True Pt. NH-NHNO A[3 +3]-180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] =0True Pt. NH-NHNO A[3 +3] =0True Pt. NH-NHNO A[3 +3]35 40 45 50 55 (test) [degree] -180-135-90-4504590135180 ( t e s t ) [ d e g r ee ] =90True Pt. NH-NHNO A[3 +3]90% CL (4 )90% CL (3 )90% CL NH-IH 35 40 45 50 55 (test) [degree] =90True Pt. NH-NHNO A[3 +3] FIG. 2: Contour plots in the θ (test) vs δ (test) plane for two different true values of θ = 40 ◦ (first and third column) and ◦ (second and fourth column) for NO ν A (6 + ¯0) (first and second column) and ( ) (third and fourth column). The first,second and third rows are for δ = − ◦ , ◦ and ◦ respectively. The true value for the δ is taken to be − ◦ . The truehierarchy is NH. We marginalize over the test values of δ . Also shown is the contours for the ν flavor scenario.solutions. For NO ν A (6 + ¯0) and LO (first column), we seethat apart from correct solution (the contour around the truepoint), there is a RH-WO solution around δ (test) = +90 ◦ and a WH-WO solution around δ (test) = − ◦ . Note thatboth of these wrong solutions vanish in the NO ν A (3 + ¯3) case (third column). This is because as we mentioned ear-lier, the octant degeneracy in the standard three flavor scenariobehaves differently for neutrinos and antineutrinos and a bal-anced combination of them can resolve this degeneracy. Onthe other hand for NO ν A (6 + ¯0) and HO (second column),there are no wrong solutions apart from the true solution butin NO ν A (3 + ¯3) (fourth column), a small RH-WO solutionappears around δ (test) = − ◦ . This can be understoodin the following way. The addition of antineutrinos helps inthe sensitivity only if there is degeneracy in the pure neutrinomode. But if there is no degeneracy, then replacing neutri-nos with antineutrinos causes a reduction in the statistics asthe neutrino cross section is almost three times higher thanthe antineutrino cross section. As { δ = − ◦ , NH, HO } does not suffer from degeneracy in the pure neutrino mode,addition of antineutrinos makes the precision of θ worse as compared to NO ν A ( ) and a WO solution appears forNO ν A (3 + ¯3) .Now let us discuss the case for the 3+1 scenario for δ = − ◦ (first row). In these figures the blue contours correspondto the RH solution and the green contours correspond to theWH solutions. For NO ν A (6 + ¯0) and LO (first panel), wesee that there is a RH-WO solution for the entire range of δ (test) . Note that NH and δ = − ◦ don’t suffer fromthe hierarchy- δ degeneracy but we find a WH solution ap-pears with WO around δ (test) = − ◦ which disappears inthe NO ν A (3 + ¯3) case (third panel). The RH-WO solutionaround δ (test) = − ◦ on the other hand, remains unre-solved even in the NO ν A (3 + ¯3) case. This is because thatthe octant - δ degeneracy is same for neutrinos and antineu-trinos. This is one of the major new features of the 3+1 casewhen compared to the three generation case. In the three gen-eration case, NO ν A (3 + ¯3) is free from all the degeneraciesfor δ = − ◦ in NH and LO but if we introduce a sterileneutrino, then there will be an additional WO solution even at90% C.L. For HO, we see that (6 + ¯0) configuration is almostfree from any degeneracies except for a small RH-WO solu-tion (second panel). For NO ν A (3 + ¯3) , the lack of statisticsdecrease the θ precision and there is a growth in the WOregion (fourth panel).Next let us discuss the case for δ = +90 ◦ (third row).For and LO (first panel) we see that there is a WH-RO solution around δ (test) = − ◦ , a WH-WO solutionfor the entire range of δ (test) and RH-WO solution around δ (test) = +90 ◦ . In this case the inclusion of the antineu-trino run of NO ν A (third panel) almost resolves all the degen-erate solutions but a small WH solution remains unresolved.This indicates that in this case the statistics of the antineu-trino run are not sufficient to remove the RH-WO solution.For HO, we have the RH-WO and WH-RO solutions, bothat δ (test) = − ◦ for NO ν A( ) (second panel). ForNO ν A( ) we see that the WH solution gets removed butthe WO solution remains unresolved (fourth panel).For δ = 0 ◦ (middle row), we see that there is a RH-WOsolution in the entire range of δ (test) and WH-WO solu-tion around δ (test) = − ◦ for NO ν A (6 + ¯0) in LO(first panel). By the inclusion of antineutrino run, the WH-WO region gets resolved but the RH-WO solution remainsunresolved at δ (test) = − ◦ (third panel). Apart fromthat, there is also the emergence of a WH-RO solution at δ (test) = − ◦ . For the HO, we see that apart from the truesolution, there is a RH-WO region for both NO ν A (6 + ¯0) and (3 + ¯3) configurations around at δ (test) = − ◦ (secondand fourth panel respectively). V. RESULTS FOR HIERARCHY SENSITIVITY
We now discuss the hierarchy sensitivity of NO ν A (3 + ¯3) in the presence of a sterile neutrino. In the Fig. 3 we havegiven the σ hierarchy contours in the δ (true) - θ (true) plane for three values of δ . The red contours are for stan-dard three flavor case and the blue contours are for 3+1 case.For the region inside the contours one can exclude the wronghierarchy at σ . Here the true hierarchy is NH. While gener-ating these plots we have marginalized over test values of δ , δ and | ∆ m | . We have assumed the octant to be unknownand known in the left and right panels respectively. The top,middle and bottom rows corresponds to δ = − ◦ , ◦ and ◦ respectively.For the standard three flavor scenario we see NO ν A has σ hierarchy sensitivity around − ◦ for all the values of θ ranging from ◦ to ◦ . This is irrespective of the in-formation of the octant. This is because for NO ν A (3 + ¯3) , δ = − ◦ do not suffer from hierarchy degeneracy in NH.This can be understood from Fig. 2 by noting the absence ofpurple contour in NO ν A (3 + ¯3) for both LO and HO.In the 3+1 case, if δ is − ◦ then the hierarchy sensitivityis lost when θ is less than ◦ in the known octant case (topleft panel). Note that though NO ν A (3 + ¯3) does not have aWH solution at 90%, the loss of hierarchy sensitivity impliesthat this degeneracy re-appears at σ . If the octant is knownthen the sensitivity of 3+1 coincides with the standard 3 flavorcase (top right panel). This signifies that the loss of sensitivityin the 3+1 case for the value of δ = − ◦ is mainly due -180-135-90-4504590135180 ( T r u e ) [ d e g r ee ] =-90 NH-IH
Octant Unknown NO A2 (3 )2 (4 ) =-90 NH-IH
Octant Known NO A -180-135-90-4504590135180 ( T r u e ) [ d e g r ee ] =0 NH-IH
Octant Unknown NO A =0 NH-IH
Octant Known NO A
35 40 45 50 55 (True) [degree] -180-135-90-4504590135180 ( T r u e ) [ d e g r ee ] =90 NH-IH
Octant Unknown NO A
35 40 45 50 55 (True) [degree] =90 NH-IH
Octant Known NO A
FIG. 3: Contour plots at σ C.L. in the θ (true) vs δ (true) plane for Octant Unknown (left panel) and OctantKnown (right panel) scenarios for NO ν A ( ). The first,second and third rows are for δ = − ◦ , ◦ and ◦ respectively. The true and test hierarchies are chosen to benormal (NH) and inverted hierarchy (IH) respectively. Alsoshown contours for the ν flavor scenario.to the WH-WO solution. In the middle row we see that inthe 3+1 case, one cannot have hierarchy sensitivity at σ fortrue δ = 0 ◦ if θ is less than ◦ ( ◦ ) when the octantis unknown (known) as can be seen from the middle panels.This implies that for this value of true δ the hierarchy sensi-tivity of NO ν A is affected by the WH solution occurring withboth right and wrong octant. But the most remarkable result isfound for δ = 90 ◦ (bottom panels). For this value of δ wesee that the hierarchy sensitivity of NO ν A is completely lost.This is mainly due to the WH-RO solution. Thus we under-stand that if there exists a ∼ sterile neutrino in additionto the three active neutrinos and the value of δ chosen bynature is +90 ◦ , then NO ν A can not have even a σ hierarchysensitivity for δ = − ◦ and NH which is present best fit ofNO ν A. VI. CONCLUSION
In this work we have studied the parameter degeneracy andhierarchy sensitivity of NO ν A in the presence of a sterile neu-trino. Apart from the hierarchy- δ and octant- δ degeneracyin the standard three flavor scenario, we have identified twonew degeneracies appearing with the new phase δ whichoccur for every value of δ . These are hierarchy- δ degen-eracy and octant- δ degeneracy. Unlike the standard threegeneration case, here the octant degeneracy behaves similarlyfor neutrinos and antineutrinos and the hierarchy degeneracybehaves differently. Thus a combination of neutrinos and an-tineutrinos are unable to resolve the octant- δ degeneracy butcan resolve the hierarchy- δ degeneracy. To identify the de-generate parameter space we present our results in θ (test) - δ (test) plane for three values of δ (true) assuming (i)NO ν A runs in pure neutrino mode and (ii) NO ν A runs inequal neutrino and antineutrino mode. We have chosen nor-mal hierarchy and δ = − ◦ motivated by the latest fit fromNO ν A data. In those plots we find that there are different RH-WO, WH-RO and WH-WO regions depending on the true na-ture of the octant of θ and true value of δ . From theseplots we find that the addition of antineutrinos helps to re-solve the WH solutions but fails to remove the WO solutionsappearing at δ (test) = − ◦ . However we find that for δ (true) = 90 ◦ and LO, the antineutrino run of NO ν A isunable to resolve the WH solution appearing with right octantat 90% C.L. While for δ (true) = 0 ◦ , the WH-RO solu-tion grows in size for NO ν A (3 + ¯3) as compared to NO ν A (6 + ¯0) . Comparing these with that of standard three flavorcase we find that apart from the small RH-WO regions for thetrue higher octant, there are no other degenerate allowed re-gions for this choice of δ (true) and hierarchy in the three flavor case for NO ν A (3 + ¯3) . Note the region δ = − ◦ and NH is the favorable parameter space of NO ν A which doesnot suffer from hierarchy- δ degeneracy in the standard threeflavor scenario where NO ν A can have good hierarchy sen-sitivity. But now in the 3+1 case, the hierarchy sensitivityof NO ν A for this parameter value can suffer due to the exis-tence of the new degeneracies. To study that we plot the σ hierarchy contours in the θ (true) - δ (true) plane for threevalues of true δ in NH. While in the standard three flavorcase one can have σ hierarchy sensitivity for all the values of θ ranging from ◦ to ◦ , in the 3+1 case we find that for δ = − ◦ and θ = 43 ◦ the hierarchy sensitivity of NO ν Ais lost. For the value of δ = 0 ◦ , the hierarchy sensitivityof NO ν A is also compromised if θ is less than ◦ . Butthe most serious deterioration in hierarchy sensitivity occursif the value of δ chosen by nature is +90 ◦ . At this value of δ , NO ν A suffers from hierarchy degeneracy and thus it hasno hierarchy sensitivity for any value of θ . Therefore if: (i)the hint of δ = − ◦ persists; (ii) the data begins to show apreference towards LO; and (iii) the observed hierarchy sen-sitivity is less than the expected sensitivity, then this can be asignal from NO ν A towards existence of a sterile neutrino with δ (cid:54) = − ◦ . VII. ACKNOWLEDGEMENT
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