Maximising the DUNE early physics output with current experiments
aa r X i v : . [ h e p - ph ] M a r Maximising the DUNE early physics output with current experiments
Monojit Ghosh and Srubabati Goswami
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
Sushant K. Raut
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India andDepartment of Theoretical Physics, School of Engineering Sciences,KTH Royal Institute of Technology – AlbaNova University Center,Roslagstullsbacken 21, 106 91 Stockholm, Sweden
The Deep Underground Neutrino Experiment (DUNE) is a proposed next generation superbeamexperiment at Fermilab. Its aims include measuring the unknown neutrino oscillation parameters –the neutrino mass hierarchy, the octant of the mixing angle θ and the CP violating phase δ CP . Thecurrent and upcoming experiments T2K, NO ν A and ICAL@INO will also be collecting data for thesame measurements. In this paper, we explore the sensitivity reach of DUNE in combination withthese other experiments. We evaluate the least exposure required by DUNE to determine the abovethree unknown parameters with reasonable confidence. We find that for each case, the inclusion ofdata from T2K, NO ν A and ICAL@INO help to achieve the same sensitivity with a reduced exposurefrom DUNE thereby helping to economize the configuration. Further, we quantify the effect of theproposed near detector on systematic errors and study the consequent improvement in sensitivity.We also examine the role played by the second oscillation cycle in furthering the physics reach ofDUNE. Finally, we present an optimization study of the neutrino-antineutrino running of DUNE.
I. INTRODUCTION
The flavour mixing of neutrinos leading to neutrinooscillations was confirmed by the Super-Kamiokande ex-periment [1], more than a decade ago. In the years since,we have measured most of the neutrino oscillation param-eters to some precision. Solar neutrino experiments likeSNO [2, 3] and the reactor neutrino experiment Kam-LAND [4] have measured the solar oscillation parameters θ and ∆ (= m − m ) quite precisely. The atmo-spheric parameters θ and | ∆ | (= | m − m | ) havebeen measured by Super-Kamiokande, MINOS and T2K[5–7]. The smallest mixing angle θ has been measuredquite recently by the reactor neutrino experiments Dou-ble Chooz, Daya Bay and RENO [8–10]. The combinedfit to world neutrino data significantly constrains most ofthe oscillation parameters today [11–13].Some quantities however, still remain unmeasured.The sign of the atmospheric mass-squared difference∆ is currently unknown. The case with m > m ( m < m ) is called Normal (Inverted) hierarchy orNH(IH). The octant in which the atmospheric mixingangle lies is another unknown. If θ < ◦ ( θ > ◦ ),then θ is said to lie in the Lower (Higher) octant orLO(HO). Finally, the value of the CP-violating phase δ CP is completely undetermined with the whole rangefrom − ◦ to +180 ◦ being allowed at 3 σ C.L. However,recent hints point to a value of δ CP close to − ◦ [14].There are many other fundamental questions like the ab-solute masses of the neutrinos and their Dirac/Majorananature. However, these cannot be probed by neutrinooscillation experiments.The primary task before the current and next gener-ation of neutrino oscillation experiments is therefore, tomeasure the unknown parameters (mass hierarchy, oc- tant of θ and δ CP ) and to put more precise constraintson the values of the known ones. These can be achievedby experiments that probe the ν µ → ν e and ν µ → ν µ os-cillation channels at scales relevant to the atmosphericmass-squared difference. The superbeam experimentsT2K [15] and NO ν A [16] which are operational are thetwo current generation long-baseline experiments thatare likely to shed light on the above issues. The dis-covery of a non-zero θ and a precise measurement ofthis parameter have added a boost to the explorationsof the potential of these experiments toward measuringthe above unknowns [17–28]. Some of the earlier stud-ies on this topic can be found in [29–31]. Atmosphericneutrino experiments can also throw light on the aboveissues. One such project, ICAL@INO [32] is already ap-proved and will use a magnetized iron calorimeter detec-tor with charge sensitivity. The combined capabilities ofthe long-baseline experiments T2K and NO ν A and theatmospheric neutrino experiment ICAL have been dis-cussed extensively, e.g. see Refs. [22, 23, 26, 33–35].The main problem in determining the oscillation pa-rameters is the problem of parameter degeneracy [36–41],i.e. two different sets of oscillation parameters giving thesame value of probability. Therefore, in the degenerateparts of the parameter space, it is difficult for any oneexperiment to measure all the unknown parameters [42–47]. Depending on the values of the oscillation param-eters in nature, the current and upcoming experimentsmay be able to measure one or more of the unknownparameters over the next few years. However the ex-pected sensitivity even in favourable parameter space isin the range 2 − σ . For unfavourable values of parame-ters as well as for enhanced sensitivity in the favourableregion we will need next generation facilities. The LBNEexperiment [48] in the United States and the LBNO ex-periment [49] in Europe were two of the proposals forsuch a facility. Many studies have explored the physicsreach of these experiments [28, 50–61]. These differentproposals are now converging into a unified endeavour ofa long-baseline experiment using a high-intensity beamfrom Fermilab. The proposal outlines construction of adeep underground neutrino observatory at Sanford Un-derground Research Facility (SURF) in South Dakota.This was initially called Experiment at the Long Base-line Neutrino Facility (ELBNF) [62], now re-christenedas DUNE. The prospective detector is a modular 40 kilo-ton Liquid Argon Time Projection Chamber (LArTPC).One of the major goals of this facility as outlined in [63]is 3 σ CP sensitivity for 75% values of δ CP .There are also proposals for future atmospheric neu-trino experiments, such as HyperKamiokande [64, 65]which is a Water ˘Cerenkov detector and PINGU [66]which is a multi-Megaton ice detector using the˘Cerenkov technique. Some phenomenological stud-ies involving these experiments have been presented inRefs. [67–71].By the time the next generation experiments start col-lecting data, we will also have information from the cur-rent generation of experiments NO ν A, T2K and the up-coming ICAL experiment. It is therefore pertinent to askwhat the minimum amount of information needed fromthe future experiments in light of the information fromthis data is. This question was addressed in Ref. [54]in the context of the LBNO experiment. In that pa-per, three prospective baselines namely 2290km, 1540kmand 130 km were considered for the LBNO configuration.For the first two baselines the prototype detector was aLArTPC whereas for the 130 km baseline a ˘Cerenkovdetector was considered. It was shown that there existsa synergy between experiments and channels, because ofwhich the combined analysis of many experiments givesvery good sensitivity. Therefore the same physics goalscan be achieved with a lower exposure for LBNO. In thiswork, we carry out a similar analysis for DUNE with abaseline of 1300 km and taking a LArTPC detector. Wedetermine the most conservative specifications that thisexperiment needs, in order to measure the remaining un-known parameters to a specified level of precision. Thisearly physics reach of DUNE can be taken as the aim ofthe first stage, if the experiment is conducted in a stagedapproach. For this purpose, we use the latest experimen-tal specifications provided by the collaboration.In addition, we study the impact of the near detec-tor (ND) in reducing the systematic uncertainties,by ex-plicitly simulating events at both the near detector andthe far detector (FD). The role of the near detector andimproved systematics used for a superbeam experimenthave been considered in Ref. [72], specifically in the con-text of the precision measurement of δ CP . We show theeffect of systematics for all the three performance indica-tors - hierarchy, octant and δ CP considering the overallsignal and background normalization errors at both NDand FD. We also study the role of the second oscillation maxi-mum in improving the sensitivities, both for DUNE aloneand in conjunction with T2K, NO ν A and ICAL. Opti-mization of neutrino and antineutrino run has been stud-ied before in Refs.[51, 73]. In this work, the adequateexposure is obtained by assuming equal neutrino and an-tineutrino runs. Subsequently we also change the propor-tion of neutrino and antineutrino runs in the adequateexposure and study what is the optimal combination.This is determined for each of the three unknowns foronly DUNE as well as DUNE in conjunction with theLBL experiments T2K and NO ν Aand the atmosphericneutrino experiment [email protected] plan of the paper is as follows. In Section II,we discuss the configurations of the experiments consid-ered in this work. The next section explores the questionposed above – determining the minimal or ‘adequate’configuration required for DUNE in light of data fromT2K, NO ν A and ICAL, in order to determine the un-known parameters. We then discuss the effect of sys-tematics in Section IV and the significance of the secondoscillation maximum for DUNE baseline in Section V.Finally, in Section VI, we present an optimization studyof the neutrino-antineutrino running at DUNE to get thebest possible results.
II. SIMULATION DETAILS
Among the current generation of neutrino oscillationexperiments, in this work we consider NO ν A, T2K andICAL@INO. NO ν A and T2K are currently operational,while ICAL@INO project has been approved. The pre-cise configuration of DUNE is still being worked on, andin this work we allow its specifications to be variable.For this work, we have simulated the long-baseline ex-periments using the GLoBES package [74, 75] along withits auxiliary data files [76, 77].The T2K experiment in Japan shoots a beam of muonneutrinos from J-PARC to the Super-Kamiokande detec-tor in Kamioka, through 295 km of earth. This experi-ment will run with a total integrated beam strength ofaround 8 × pot (protons on target). The specifica-tions used for this detector are as given in Ref. [15, 78–80]. We assume in our study that T2K will run only inthe neutrino mode with the above pot. The T2K col-laboration has started running in the antineutrino mode.For advantages of neutrino vis-`a-vis antineutrino runs werefer to [81, 82]. More discussions on the effect of antineu-trino data from T2K will follow in the relevant sections.The NO ν A experiment at Fermilab takes neutrinosfrom the NuMI beam, with a beam power of 0.7 MW.The planned run of this experiment is for 6 years, di-vided into 3 years of neutrino and 3 years of antineutrinomode. The neutrinos are intercepted at the TASD de-tector in Ash River, 812 km away and 14 milliradians offthe beam axis. The off-axis nature of these experimentshelps impose cuts to reduce the neutral current back-ground. After the measurement of the moderately largevalue of θ , the event selection criteria were re-optimizedwith the intention of exploiting higher statistics [19, 83].We have used this new configuration for the NO ν A ex-periment in our work.ICAL@INO is a magnetized iron detector for observ-ing atmospheric neutrinos [32]. Magnetization allows fora separation of µ + and µ − events, and hence a distinctionbetween neutrinos and antineutrinos. The total exposuretaken for this experiment is 500 kiloton yr, i.e. 10 yearsof data collection using a 50 kiloton (kt) detector. Weassume an energy resolution of 10% and angular resolu-tion of 10 ◦ for the neutrinos in the detector. These giveresults comparable to the muon analysis [35, 84] that hasbeen performed by the INO collaboration. The new ‘3-d analysis’ which also includes hadronic energy informa-tion [85] is expected to give better results. The statisticalprocedure followed in calculating the sensitivity of thisexperiment follows the treatment outlined in Ref. [86].DUNE is the next generation international neutrino os-cillation experiment proposed to be hosted at Fermilab.The beam of neutrinos, with a wide-band profile, willtravel 1300 km from Fermilab to a liquid argon detectorat SURF, South Dakota. The projected beam power is1.2 MW with the possibility of upgradation to 2.4 MW.The detailed design of this experiment including beam-line, detector and engineering aspects is expected to relyon the results of R&D work already carried out by ear-lier proposals including LBNE and LBNO. The higherenergy available, along with the long baseline means thatthe neutrinos will experience greater matter effects thanNO ν A or T2K. There are two options being consideredfor the proton beam – 80 GeV and 120 GeV. For a givenconfiguration of the beamline and beam power, protonenergy varies inversely with the number of protons in thebeam per unit time, and hence the neutrino flux. In thiswork, we have chosen the 120 GeV beam which gives usa lower flux of neutrinos and hence a conservative esti-mate of our results. The details of the LArTPC detectorresponse have been taken from Ref. [87]. In this workwe use the recently updated neutrino flux correspond-ing to 1.2 MW beam power [88]. However we give ourresults in terms of MW-kt-yr. This will enable one tointerpret the results in terms of varying detector volume,timescale and beam power. Note that although we usethe flux corresponding to 1.2 MW beam power, if the ac-celerator geometry remains the same, then the change inthe value of the beam power will proportionally changethe flux. Therefore, the flux for a different value of beampower can be obtained by simply scaling the ‘standard’flux file by the appropriate factor.Since DUNE is proposed to be an underground ob-servatory it will also be possible for it to observe atmo-spheric neutrinos. In this work, we have not consideredthis possibility. A detailed study on atmospheric neutri-nos for the DUNE experiment is presented in Ref. [55, 56].The sensitivity of DUNE to the mass hierarchy, octantof θ and δ CP comes primarily from the ν µ → ν e oscilla- tion probability P µe . An approximate analytical formulafor this probability can be derived perturbatively [89–91]in terms of the two small parameters α = ∆ / ∆ andsin θ . P µe = 4 sin θ sin θ sin [(1 − ˆ A )∆](1 − ˆ A ) + α sin 2 θ sin 2 θ sin 2 θ cos (∆ + δ CP ) × sin ˆ A ∆ˆ A sin [(1 − ˆ A )∆](1 − ˆ A ) + O ( α ) . (1)Here, ∆ = ∆ L/ E is the oscillating term, and the ef-fect of neutrinos interacting with matter in the earth isgiven by the matter term ˆ A = 2 √ G F n e E/ ∆ , where n e is the number density of electrons in the earth. Notethat this expression is valid in matter of constant density.This approximate formula is useful for understanding thephysics of neutrino oscillations. However in our simula-tions, we use the full numerical probability calculated byGLoBES.In the analyses that follow, we have evaluated the χ for determining the mass hierarchy, the octant of θ , and discovering CP violation using a combina-tion of DUNE and the current/upcoming experimentsT2K, NO ν A and ICAL. For each set of ‘true’ values as-sumed, we evaluate the χ marginalized over the ‘test’parameters. In our simulations, we have used the effec-tive atmospheric parameters corrected for three-flavoureffects [92–94]. The true values assumed for theparameters are sin θ = 0 . | ∆ | = 2 . × − eV , ∆ = 7 . × − eV and sin θ = 0 .
1. Thetrue value of δ CP is varied throughout the full range[ − ◦ , ◦ ). For true θ , we have considered threevalues – 39 ◦ , 45 ◦ and 51 ◦ which are within the cur-rent 3 σ allowed range. The test values of the parame-ters are varied in the following ranges: θ ∈ [35 ◦ , ◦ ],sin θ ∈ [0 . , . δ CP ∈ [ − ◦ , ◦ ). The testhierarchy is varied as well. The solar parameters are al-ready measured quite accurately, and their variation doesnot impact our results significantly. Therefore, we havenot marginalized over them. We have imposed a priorof σ (sin θ ) = 0 .
005 on the value of sin θ , whichis the expected precision from the reactor neutrino ex-periments [95]. We have included backgrounds arising The commonly used phrase ‘CP-violation discovery’ is taken tomean the distinguishing of a given value of δ CP from the CP-conserving cases 0, 180 ◦ . This should not be confused with thestatistical usage of the word ‘discovery’ that implies 5 σ evidence. The effective atmospheric mass-squared difference and atmo-spheric mixing angle are obtained by fitting oscillation data tothe effective two-flavour oscillation formula. From a physics pointof view, there is no advantage in using these effective parame-ters and then correcting for three-flavour effects. However froma computational point of view, we find that the use of these ef-fective parameters gives more precise results while scanning theparameter space, since the hierarchy and octant degeneracies areexact in these parameters. from NC events, mis-identified ν µ events, intrinsic beambackgrounds as well as wrong-sign backgrounds. The sys-tematic uncertainties are parameterized in terms of fournuisance parameters – signal normalization error 2.5%(7.5%), signal tilt error 2.5% (2.5%), background normal-ization error 10% (15%) and background tilt error 2.5%(2.5%) for the appearance (disappearance) channel [48].In section III, our aim is to economize the configura-tion of DUNE with the help of the current generationof experiments. We have done that by evaluating the‘adequate’ exposure for DUNE. The qualifier ‘adequate’,as defined in Ref. [54] in the context of LBNO, meansthe exposure required from the experiment to determinethe hierarchy and octant with χ = 25, and to detectCP violation with χ = 9. To do so, we have varied theexposure of DUNE, and determined the combined sensi-tivity of DUNE along with T2K, NO ν A and ICAL. Thevariation of total sensitivity with DUNE exposure tells uswhat the adequate exposure should be. In this work, wehave quantified the exposure for DUNE in units of MW-kt-yr. This is a product of the beam power (in MW),the runtime of the experiment (in years) and the detec-tor mass (in kilotons). As a phenomenological study, wewill only specify the total exposure in this paper. Thismay be interpreted experimentally as different combina-tions of beam power, runtime and detector mass whoseproduct quantifies the exposure. For example, an expo-sure of 40 MW-kt-yr could be achieved by using a 10 ktdetector for 2 years (in each, ν and ν mode), with a 1MW beam. We use events in the energy range 0.5 - 10GeV for DUNE which covers both first and second oscil-lation maxima. The relative contribution of the secondoscillation maximum is discussed in Section V. III. ADEQUATE EXPOSURE FOR DUNEA. Hierarchy sensitivity
In the left panel of Fig. 1, we have shown the combinedsensitivity of DUNE, NO ν A, T2K and ICAL for deter-mining the mass hierarchy, as the exposure for DUNE isvaried. The hierarchy sensitivity typically depends verystrongly on the true value of δ CP and θ . In this work,we are interested in finding out the least exposure neededfor DUNE, irrespective of the true values of the param-eters in nature. Therefore, we have evaluated the χ for various true values of these parameters as listed inSection II, and taken the most conservative case out ofthem. Thus, the exposure plotted here is for the mostunfavourable values of true δ CP and θ . Since hier-archy sensitivity of the P µe channel increases with θ , A runtime of n years is to be interpreted as n/ the worst case is usually found at the lowest value con-sidered – θ = 39 ◦ . The most unfavourable of δ CP isaround +( − )90 ◦ for NH(IH) [18]. Separate curves areshown for both hierarchies, but the results are almostthe same in both cases. We find that the adequate expo-sure for DUNE including T2K, NO ν A and ICAL data isaround 44 MW-kt-yr for both NH and IH. This is shownby the upper curves. The two intermediate curves showthe same sensitivity, but without including ICAL datain the analysis. In this case, the adequate exposure isaround 78 MW-kt-yr. Thus, in the absence of ICAL data,DUNE would have to increase its exposure by over 75%to achieve the same results. For the benchmark valuesof 1.2 MW power and 10 kt detector, the exposure of 44MW-kt-yr implies under 2 years of running in each modewhereas the adequate exposure 78 MW-kt-yr correspondsto about 3 years exposure in each mode.Finally, we show the sensitivity from DUNE alone, inthe lower most curves. For the range of exposures consid-ered, DUNE can achieve hierarchy sensitivity up to the χ = 16 level. The first row of Table I shows the ade-quate exposure required for hierarchy sensitivity reaching χ = 25 for only DUNE and also after adding the datafrom T2K, NO ν A and ICAL. The numbers in the paren-theses correspond to IH. With only DUNE, the exposurerequired to reach χ = 25 for the hierarchy sensitivity isseen to be much higher. B. Octant sensitivity
The mass hierarchy as well as the values of δ CP and θ in nature affect the octant sensitivity of experimentssignificantly. In our analysis, we have considered varioustrue values of δ CP across its full range, and two represen-tative true values of θ – 39 ◦ and 51 ◦ . Having evaluatedthe minimum χ for each of these cases, we have chosenthe lower value. Thus, we have ensured that the ade-quate exposure shown here holds, irrespective of the trueoctant of θ . Note that octant sensitivity reduces as wego more toward θ = 45 ◦ . Thus the above choice of true θ only corresponds to the more conservative value of θ out of 39 ◦ and 51 ◦ .The middle panel of Fig. 1 shows the combined octantsensitivity of the experiments, as a function of DUNEexposure. Around 70-74 MW-kt-yr for NH(IH) is therequired exposure for DUNE, to measure the octant withNO ν A, T2K and ICAL This implies a runtime of around3 years in each mode for the ‘standard’ configuration ofDUNE. Without information from ICAL however, DUNEwould have to increase its exposure to around 130(100)MW-kt-yr for NH(IH) to measure the octant with χ =25. For a 1.2 MW beam and a 10 kt detector this impliesabout 5(4) years for NH (IH) in each mode. DUNE-onlywould need a higher exposure of 168 (152) MW-kt-yr forNH(IH) corresponding to about 7(6) years in each mode.Thus including ICAL data reduces the exposure requiredfrom DUNE. This is summarized in the second row of χ Exposure(MW-kt-yr)Hierarchy sensitivity(120 GeV Proton Beam)DUNE+NO ν A+T2K+ICAL-NHDUNE+NO ν A+T2K+ICAL-IHDUNE+NO ν A+T2K-NHDUNE+NO ν A+T2K-IHDUNE-NHDUNE-IH 0 10 20 30 40 20 40 60 80 100 120 χ Exposure(MW-kt-yr)Octant sensitivity(120 GeV Proton Beam)DUNE+NO ν A+T2K+ICAL-NHDUNE+NO ν A+T2K+ICAL-IHDUNE+NO ν A+T2K-NHDUNE+NO ν A+T2K-IHDUNE-NHDUNE-IH 0 0.2 0.4 0.6 0.8 1 80 100 120 140 160 180 F r a c t i on o f δ C P ( σ C PV d i sc o v e r y ) Exposure(MW-kt-yr)CPV discovery (120 GeV Proton Beam)DUNE+NO ν A+T2K+ICAL-NHDUNE+NO ν A+T2K+ICAL-IHDUNE-NHDUNE-IH
FIG. 1.
Hierarchy (Octant) sensitivity χ vs DUNE exposure, for both hierarchies in the left (middle) panel. The value of exposureshown here is adequate to exclude the wrong hierarchy (octant) for all values of δ CP . Two additional sets of curves are shown to showthe fall in χ without data from ICAL, and the sensitivity of DUNE alone. The right panel shows the fraction of δ CP range for which itis possible to exclude the CP conserving cases of 0 and 180 ◦ , at the χ = 9 level. An additional set of curves is shown to show the the CPsensitivity of DUNE alone. Sensitivity DUNE+NO ν A+T2K+ICAL DUNE+NO ν A+T2K DUNEHierarchy( χ = 25) 44(44) 78(78) 190(212)Octant( χ = 25) 74(74) 130(100) 168(152)CP(40% at χ = 9) 130(72) 130(72) 228(180)TABLE I. Adequate exposures for hierarchy, octant and CP in units of MW-kt-yr for NH(IH) Table I.
C. Detecting CP violation
The CP detection ability of an experiment is defined asits ability to distinguish the true value of δ CP in naturefrom the CP-conserving cases of 0 and 180 ◦ . This obvi-ously depends on the true value of δ CP . If δ CP in natureis close to 0 or 180 ◦ , this ability will be poor, while if itis close to ± ◦ , it will be high. CP detection also de-pends on θ , and typically it is a decreasing function of θ [33]. Here, we have tried to determine the fraction ofthe entire δ CP range for which our setups can detect CPviolation with at least χ = 9. We have always chosenthe smallest fraction over various values of θ (39 ◦ , 45 ◦ and 51 ◦ ), so as to get a conservative estimate.We find in the right panel of Fig. 1 that for the rangeof exposures considered, the fraction of δ CP is between0.35 and 0.55. While the exposure increases by a factorof 2, the increase in the fraction of δ CP is very slow.In Ref. [23], it was shown that the addition of infor-mation from ICAL to NO ν A and T2K increases theirCP detection ability. This is because ICAL data breaksthe hierarchy- δ CP degeneracy that NO ν A and T2K suf-fer from. However, the DUNE experiment itself is also capable of lifting this degeneracy for most of the valuesof δ CP [53]. Therefore, the inclusion of ICAL data doesnot make any difference in this case. This combinationof experiments can detect CP violation over 40% of the δ CP range with an exposure of about 130 MW-kt-yr atDUNE for NH (i.e. a runtime of around 5.5 years in eachmode for DUNE with the initial 10 kt detector or around1.5 years in each mode with the final 40 kt detector).Without including T2K and NO ν A information the ex-posure required will be 228 MW-kt-yr for 40% coveragefor discovery of δ CP . As mentioned in the introduction,one of the mandates of DUNE is 3 σ CP coverage for 75%values of δ CP [63]. We find that an exposure of 300MW-kt-yr in neutrinos and 300 MW-kt-yr in antineutri-nos gives 69%(73%) CP coverage at 3 σ for θ = 39 ◦ and60%(65%) for 51 ◦ in NH(IH). We also find that additionof NO ν A and T2K data does not help much for such highvalues of exposure. The results are summarized in TableII.In the following sections, we fix the exposure in eachcase to be the adequate exposure as listed in Table I, forthe most conservative parameter values. χ δ CP (True)DUNE: 44 MW-kt-yrTrue IH DUNE(FD)DUNE(FD+ND)DUNE(FD)+OthersDUNE(FD+ND)+Others 0 10 20 30 40 50-180 -120 -60 0 60 120 180 χ δ CP (True)DUNE: 74 MW-kt-yrTrue IH DUNE(FD)DUNE(FD+ND)DUNE(FD)+OthersDUNE(FD+ND)+Others 0 5 10 15 20 25 30-180 -120 -60 0 60 120 180 χ δ CP (True)DUNE: 130 MW-kt-yrTrue NH DUNE(FD)DUNE(FD+ND)DUNE(FD)+OthersDUNE(FD+ND)+Others FIG. 2.
Hierarchy/Octant/CP violation discovery sensitivity χ vs true δ CP in the left/middle/right panel. The various curves show theeffect of including a near detector on the sensitivity of DUNE alone and DUNE combined with the other experiments. σ CPV coverage for θ DUNE DUNE+NO ν A+T2K39 o o σ for total 600 MW-Kt-yr exposure IV. ROLE OF THE NEAR DETECTOR INREDUCING SYSTEMATICS
The measurement of a relatively large value of θ makes the issue of systematic uncertainties more rele-vant. The role of the ND in long-baseline neutrino ex-periments has been well discussed in the literature; seefor example Refs. [96–98]. The measurement of events atthe ND and FD reduces the uncertainty associated withthe flux and cross-section of neutrinos. Thus the role ofthe near detector is to reduce systematic errors in the os-cillation experiment. It has recently been found that theND for the T2K experiment can bring about a spectac-ular reduction of systematic errors [99]. The impact ofND and systematic uncertainties in the context of a mea-surement of CP violation using appearance channels hasbeen studied in [72, 100] taking the T2HK experiment asan example.In this study, we have tried to quantify the improve-ment in results, once the ND is included. The conven-tional way of doing this is to assume that the existenceof the ND leads to a reduction of systematic effects, andtherefore input smaller systematic errors by hand in theanalysis. Instead of using this approach, we have explic-itly simulated the events at the ND using GLoBES. Thedesign for the ND is still being planned. For our simula-tions, we assume that the ND has a mass of 5 tons and isplaced 459 meters from the source. The flux at the NDsite has been provided by the DUNE collaboration [88]. The detector characteristics for the ND are as follows[101]. The muon(electron) detection efficiency is takento be 95%(50%). The NC background can be rejectedwith an efficiency of 20%. The energy resolution for elec-trons is 6% / p E (GeV), while that for muons is 37 MeVacross the entire energy range of interest. Therefore, forthe neutrinos, we use a (somewhat conservative) energyresolution of 20% / p E (GeV). The systematic errors thatthe ND setup suffers from are assumed to be the same asthose from the FD.In order to have equal runtime for both FD and ND,we fix the FD volume as 10 kt and consider both thedetectors to receive neutrinos from 1.2 MW beam. Thisfixes the runtime of FD which is then also used in thesimulation for ND. The run times used in this sectionare chosen corresponding to the adequate exposures fromthe previous section as given in the first column of TableI: 3.6 year for hierarchy sensitivity, 6.2 year for octantsensitivity and 10.8 year for CPV discovery sensitivity.In order to simulate the ND+FD setup for DUNE, weuse GLoBES to generate events at both detectors, treat-ing them as separate experiments. We then use thesetwo data sets to perform a correlated systematics analysisusing the method of pulls [102]. This gives us the com-bined sensitivity of DUNE using both ND and FD. (Wehave explained our methodology in A.) Thereafter, theprocedure of combining results with other experimentsand marginalizing over oscillation parameters continuesin the usual manner. The results are shown in Fig. 2.The effect of reduced systematic errors is felt most sig-nificantly in regions where the results are best. This isbecause for those values of δ CP , the experiment typicallyhas high enough statistics for systematic errors to playan important role.Next, we have tried to quantify the reduction in sys-tematic errors seen by the experiment, when the ND isincluded. To be more specific, if the systematic errorsseen by each detector setup are denoted by ~π , then wewonder what is the effective set of errors ~π eff for the FDsetup, once the ND is also included. In other words, forgiven systematic errors ~π , we have found the effectiveerrors ~π eff that satisfy the relation χ (FD( ~π eff )) ≡ χ (FD( ~π ) ⊕ ND( ~π )) , (2)where the right-hand side denotes the correlated combi-nation as described in A. The ~π eff thus computed can beused in future simulations as the reduced set of system-atic errors because of the presence of the ND. We havechosen typical values of systematic errors for the detec-tor: ν e appearance signal normalization error of 2.5%, ν µ disappearance signal normalization error of 7.5%, ν e ap-pearance background normalization error of 10% and ν µ disappearance background normalization error of 15%.The tilt error is taken as 2.5% in both appearance anddisappearance channels. The first four numbers consti-tute ~π , as labeled in the figure. We find that the tilterrors have a very small effect in this particular analy-sis, and we fix them to the value specified above. Theresult of the computation is shown in Fig. 3, for the caseof hierarchy determination. The sensitivity of FD+NDobtained using these numbers, are matched by an FDsetup with effective errors as follows: ν e appearance sig-nal normalization error of 1%, ν µ disappearance signalnormalization error of 1%, ν e appearance backgroundnormalization error of 5% and ν µ disappearance back-ground normalization error of 5%. Similar results areobtained in the case of octant and CP sensitivity also.Thus, inclusion of the ND brings the systematic errorsdown to 13-50% of their original value. These resultsare summarized in Table III. Note that the numbers pre-sented in Table III are indicative assuming the system-atic uncertainties are energy independent. However, theimprovement in the systematic uncertainties in the ac-tual analysis incorporate this energy dependence due toa full bin-by-bin analysis of the ND data. Systematic error only FD FD+ND ν e app signal norm error 2.5% 1% ν µ disapp signal norm error 7.5% 1% ν e app background norm error 10% 5% ν µ disapp background norm error 15% 5%TABLE III. Reduction in systematic errors with the addition of anear detector χ δ CP (True)DUNE: 44 MW-kt-yrTrue IH FD(2.5,7.5,10,15)FD+ND(2.5,7.5,10,15)FD(1,1,5,5) FIG. 3.
Reduction in systematics due to inclusion of the neardetector. The numbers in brackets denote ν e appearance signalnormalization error, ν µ disappearance signal normalization error, ν e appearance background normalization error and ν µ disappear-ance background normalization error. V. SIGNIFICANCE OF THE SECONDOSCILLATION MAXIMUM
For a baseline of 1300 km, the oscillation probability P µe has its first oscillation maximum around 2-2.5 GeV.This is easy to explain from the formula∆ ( m )31 L E = π , where ∆ ( m )31 is the matter-modified atmospheric mass-squared difference. In the limit ∆ →
0, it is given by∆ ( m )31 = ∆ q (1 − ˆ A ) + sin θ . The second oscillation maximum, for which the oscillat-ing term takes the value 3 π/
2, occurs at an energy ofaround 0.6-1.0. Studies have discussed the advantages ofusing the second oscillation maximum to get informationon the oscillation parameters [73, 103]. In fact, one of themain aims of the proposed ESSnuSB project [104, 105]is to study neutrino oscillations at the second oscillationmaximum.The neutrino flux that DUNE will use has a wide-bandprofile, which can extract physics from both, the first andsecond maxima. Figure 4 shows P µe for the DUNE base-line, superimposed on the ν µ flux. This is in contrastwith NO ν A, which uses a narrow-band off-axis beam con-centrating on its first oscillation maximum, in order toreduce the π background at higher energies.In order to understand the impact of the second oscil-lation maximum, we have considered two different energyranges. Above 1.1 GeV, only the first oscillation cycle isrelevant. However, if we also include the energy rangefrom 0.5 to 1.1 GeV, we also get information from the E (GeV)P µ e (DUNE)P µ e (NOvA) DUNE fluxNOvA flux δ CP = -90 o δ CP = 0 o δ CP = 90 o δ CP = 180 o FIG. 4.
Neutrino oscillation probability P µe for various representa-tive values of δ CP and normal hierarchy, for the NO ν A and DUNEbaselines. Also shown as shaded profiles in the background arethe ν µ flux for both these experiments (on independent, arbitraryscales). second oscillation maximum. Figure 5 compares the sen-sitivity to the hierarchy, octant and CP violation onlyfrom the first oscillation cycle, and from both the oscil-lation cycles assuming the adequate exposures obtainedin the previous section. We see that inclusion of datafrom the second oscillation maximum only increases the χ by a small amount. This increase is visible only forhierarchy sensitivity. The effect is seen to be more pro-nounced in the region δ CP ∼ − ◦ . This is because theprobability for { IH, δ CP = − ◦ } is closer to that for { NH, δ CP = +90 ◦ } at the first oscillation maximum, asreflected in the first panel of Fig. 6. But at the secondoscillation maximum the separation between the proba-bilities for these two sets is higher. Therefore adding thesecond oscillation maximum aids the hierarchy sensitiv-ity.In the right panel of Fig. 6 we show how the expo-sure for hierarchy sensitivity depends on the inclusionof the second oscillation maximum for only DUNE andDUNE+T2K+NO ν A+ICAL. It is clear from the figurethat the second maximum plays a more significant rolefor higher exposure. For the combined case, 5 σ sensitiv-ity is reached at a relatively lower exposure and hencethe second maximum does not play a major role. This isalso seen in the figure 5. However only for DUNE since5 σ sensitivity is reached for a relatively higher exposurethe inclusion of the second oscillation maximum is seento play an important role. This feature is reflected inTable IV. VI. OPTIMIZING THENEUTRINO-ANTINEUTRINO RUNS
One of the main questions while planning any beam-based neutrino experiment is the ratio of neutrino to an-tineutrino run. Since the dependence of the oscillationparameters on the neutrino and antineutrino probabili-ties are different, an antineutrino run can provide a dif-ferent set of data which may be useful in determinationof the parameters. However, the interaction cross-sectionfor antineutrinos in the detectors is smaller by a factorof 2.5-3 than the neutrino cross-sections. Therefore, anantineutrino run typically has lower statistics. Thus, thechoice of neutrino-antineutrino ratio is often a compro-mise between new information and statistics.It is now well known that neutrino and antineutrinooscillation probabilities suffer from the same form ofhierarchy- δ CP degeneracy [18]. However, the octant- δ CP degeneracy has the opposite form for neutrinos and an-tineutrinos [20, 24]. Thus, inclusion of an antineutrinorun helps in lifting this degeneracy for most of the valuesof δ CP [53]. For measurement of δ CP , it has been shownfor T2K that the antineutrino run is required only forthose true hierarchy-octant- δ CP combination for whichoctant degeneracy is present [81]. Once this degeneracyis lifted by including some amount of antineutrino data,further antineutrino run does not help much in CP dis-covery; in fact it is then better to run with neutrinos togain in statistics [81]. But this conclusion may changefor a different baseline and matter effect. From Fig. 4we see that for NO ν A the oscillation peak does not co-incide with the flux peak. Around the energy where theflux peaks, the probability spectra with δ CP = ± , ◦ are not equidistant from the δ CP = ± ◦ spectra. Forantineutrino mode the curves for ± ◦ switch position.Hence for neutrinos δ CP = 0 ◦ is closer to δ CP = − ◦ and δ CP = 180 ◦ is closer to δ CP = 90 ◦ , while the op-posite is true for antineutrinos. This gives a synergyand hence running in both neutrino and antineutrinomodes can be helpful. For T2K the energy where theflux peak occurs coincides with the oscillation peak. Atthis point the curves for δ CP = 0 , ◦ are equidistantfrom δ CP = ± ◦ and hence this synergy is not present.Thus, the role of antineutrino run is only to lift the oc-tant degeneracy. The recent hint of δ CP from T2K [14]already gives us some evidence of the octant (see TableXXVIII in Ref. [106], or Ref. [81]). Moreover, NO ν A andDUNE will collect far more data with antineutrinos thanT2K. Thus, the inclusion of antineutrino run at T2K doesnot make much difference to our results. In the followingwe have varied the proportion of neutrino and antineu-trino runs at DUNE to ascertain what is the optimalcombination. The adequate exposure is split into variouscombinations of neutrinos and antineutrinos – 1/6 ν +5/6 ν , 2/6 ν + 4/6 ν , ... 6/6 ν + 0/6 ν . The intermediateconfiguration 3/6 ν + 3/6 ν corresponds to the equal-runconfiguration used in the other sections. For convenienceof notation, these configurations are referred to simply as χ δ CP (True)DUNE: 44 MW-kt-yrTrue IH DUNE(1st+2nd maxima)DUNE(Only 1st maxima)DUNE(1st+2nd maxima)+OthersDUNE(only 1st maxima)+Others 0 10 20 30 40 50-180 -120 -60 0 60 120 180 χ δ CP (True)DUNE: 74 MW-kt-yrTrue IH DUNE(1st+2nd maxima)DUNE(Only 1st maxima)DUNE(1st+2nd maxima)+OthersDUNE(only 1st maxima)+Others 0 5 10 15 20-180 -120 -60 0 60 120 180 χ δ CP (True)DUNE: 130 MW-kt-yrTrue NHDUNE(1st+2nd maxima)DUNE(Only 1st maxima)DUNE(1st+2nd maxima)+OthersDUNE(only 1st maxima)+Others FIG. 5.
Hierarchy/Octant/CP violation discovery sensitivity χ vs true δ CP in the left/middle/right panel. The various curves show theeffect of data from the second oscillation maximum on the sensitivity of DUNE alone and DUNE combined with the other experiments. χ Exposure(MW kt yr)Hierarchy sensitivity(120 GeV Proton Beam)DUNE+others(1st+2nd)DUNE+others(1st)DUNE(1st+2nd)DUNE(1st)
FIG. 6.
Left panel: The probability vs energy showing the hierarchy- δ CP degeneracy affecting hierarchy sensitivity at the two oscillationmaxima. Right panel: Hierarchy exclusion χ vs exposure, with and without the second oscillation maximum. θ = 39 ◦ have been as-sumed as the true parameters. For DUNE, we have cho-sen a total exposure of 44 MW-kt-yr which was found tobe the adequate exposure in Section III assuming equalneutrino and antineutrino runs. In the left panel, we seethe results for DUNE alone. the figure shows that in thefavourable region of δ CP ∈ [ − ◦ ,
0] the best sensitivitycomes from the combination 3+3 or 4+2. Although thestatistics is more for neutrinos, the antineutrino run is required to remove the wrong-octant regions. For NH, δ CP ∈ [0 , ◦ ] is the unfavourable region for hierarchydetermination [18], as is evident from the figure. In thisregion, we see that the results are worst for pure neutrinorun. The best sensitivity comes for the case 5+1. Thisamount of antineutrino run is required to remove the oc-tant degeneracy. The higher proportion of neutrino runensures better statistics. In the right panel, along withDUNE we have also combined data from NO ν A, T2Kand ICAL. With the inclusion of these data the hierarchysensitivity increases further and even in the unfavourableregion χ = 25 sensitivity is possible with only neutrinorun from DUNE. This is because NO ν A, which will run in0 χ true δ CP True NH, 39 o DUNE (44 MW-kt-yr) 1 + 52 + 43 + 34 + 25 + 16 + 0 0 10 20 30 40 50 60 70 80-180 -120 -60 0 60 120 180 χ true δ CP True NH, 39 o DUNE (44 MW-kt-yr) +NOvA+T2K+ICAL 1 + 52 + 43 + 34 + 25 + 16 + 0 0 10 20 30 40 50 60-180 -120 -60 0 60 120 180 χ true δ CP True IH, 39 o DUNE (74 MW-kt-yr) 1 + 52 + 43 + 34 + 25 + 16 + 0 0 10 20 30 40 50 60-180 -120 -60 0 60 120 180 χ true δ CP True IH, 39 o DUNE (74 MW-kt-yr) +NOvA+T2K+ICAL 1 + 52 + 43 + 34 + 25 + 16 + 0 0 4 8 12 16 20 24-180 -120 -60 0 60 120 180 χ true δ CP True NH, 51 o DUNE (130 MW-kt-yr) 1 + 52 + 43 + 34 + 25 + 16 + 0 0 4 8 12 16 20 24-180 -120 -60 0 60 120 180 χ true δ CP True NH, 51 o DUNE (130 MW-kt-yr) +NOvA+T2K+ICAL 1 + 52 + 43 + 34 + 25 + 16 + 0
FIG. 7. Sensitivity for DUNE for various combinations of neutrino and antineutrino run by itself (left panel) and in conjunctionwith T2K, NO ν A and ICAL (right panel). The top/middle/bottom row shows the sensitivity to hierarchy/octant/CP violationdetection. The total exposure has been divided into 6 equal parts and distributed between neutrinos and antineutrinos. Forexample, for hierarchy sensitivity, 6+0 corresponds to 44 MW-kt-yr in only neutrino; 3+3 correspond to 22 MW-kt-yr in eachneutrino and antineutrino mode. Sensitivity DUNE+NO ν A+T2K+ICAL(MW-kt-yr) Only DUNE(MW-kt-yr)1st + 2nd osc. cycles Only 1st osc. cycle 1st + 2nd osc. cycles Only 1st osc. cycleHierarchy ( χ = 25) 44 56 212 436Octant ( χ = 25) 74 78 168 190CP (40% coverage at χ = 9) 130 140 228 256TABLE IV. Effect of the second oscillation maximum on the sensitivity of DUNE. The numbers indicate the adequate exposure(in MW-kt-yr) required by DUNE for determining the oscillation parameters, with and without the contribution from thesecond oscillation maximum. For each of the three unknowns, the true parameters (including hierarchy) are taken to be onesfor which we get the most conservative sensitivity. antineutrino mode for 3 years and the antineutrino com-ponent in the atmospheric neutrino flux at ICAL, willprovide the necessary amount of information to lift theparameter degeneracies that reduce hierarchy sensitivity.Therefore, the best option for DUNE is to run only inneutrino mode, which will have the added advantage ofincreased statistics. In the favourable region also the sen-sitivity is now better for 6+0 and 5+1 i.e. less amountof antineutrinos from DUNE is required because of theantineutrino information coming from NO ν A. Note thatoverall, the amount of antineutrino run depends on thevalue of δ CP . However combining information from allthe experiments 4+2 seems to be the best option overthe largest fraction of δ CP values.In the middle row of Fig. 7, we have shown the octantsensitivity of DUNE alone (left panel) and in combinationwith the current experiments (right panel). For DUNEwe have used an exposure of 74 MW-kt-yr. We have fixedthe true hierarchy to be inverted, and θ = 39 ◦ i.e. inthe lower octant. For this case the probability for neutri-nos is maximum for δ CP ∼ − ◦ and overlaps with thehigher octant probabilities. Thus the octant sensitivityin neutrino channel is very poor. This the worst resultsfor these values of δ CP come from only neutrino runs. Forantineutrino channel because of the flip in δ CP the prob-ability for δ CP = − ◦ is well separated from those forHO. Therefore the octant sensitivity comes mainly fromantineutrino channel [20]. Thus, addition of antineutrinoruns help in enhancing octant sensitivity. Therefore at − ◦ the best sensitivity is from 1+5 i.e 1 / th neutrino +5 / th antineutrino combination. On the other hand theneutrino probability is minimum for δ CP = +90 ◦ and LOand therefore there is octant sensitivity in the neutrinochannel. However since we are considering IH the an-tineutrino probabilities are enhanced due to matter effectand for a broadband beam some sensitivity comes fromthe antineutrino channel also. Therefore there is slightincrease in octant sensitivity by adding antineutrino dataas can be seen. Overall, the best compromise is seen to bereached for 2+4 i.e 1 / rd neutrino and 2 / rd antineutrinocombination, which gives the best results over the widestrange of δ CP values. Addition of NO ν A, T2K and ICALdata increases the octant sensitivity. The octant sensitiv-ity is best for combinations having more antineutrinos.For δ CP ∼ +90 ◦ all combinations give almost the samesensitivity. We have not presented the results for NH inthis case. For this case after adding T2K+NO ν A+ICAL to DUNE requires at least 4+2 to reach χ = 25 for δ CP ∈ [ − ◦ ,
0] while for δ CP ∈ [0 , ◦ ] the octantsensitivity almost crosses χ = 25 for all combinationsof neutrino and antineutrino run. Therefore, the exactcombination chosen does not make much difference to thefinal result.The left and right panels of the bottom row in Fig. 7show the ability of DUNE (by itself, and in conjunctionwith the current generation of experiments, respectively)to detect CP violation. Here the true hierarchy is NHand true θ is 51 ◦ . Although this true combination doesnot suffer from any octant degeneracy, we see in the leftpanel that 6+0 is not the best combination. This is dueto the synergy between neutrino-antineutrino runs forlarger baselines as discussed earlier. In both cases, wefind that the best option is to run DUNE with antineu-trinos for around a third of the total exposure. On addinginformation from T2K and NO ν A, we find great improve-ment in the CP sensitivity. From the right panel, we seethat the range of δ CP for which χ = 9 detection of CPis possible is almost the same for most combinations ofneutrino and antineutrino run. Therefore, as in the caseof octant determination, the exact choice of combinationis not very important. VII. SUMMARY
The DUNE experiment at Fermilab has a promisingphysics potential. Its baseline is long enough to see mat-ter effects which will help it to break the δ CP -relateddegeneracies and determine the neutrino mass hierarchyand the octant of θ . This experiment is also known tobe good for detecting CP violation in the neutrino sector.The current and upcoming experiments T2K, NO ν A andICAL@INO will also provide some indications for the val-ues of the unknown parameters. In this work, we haveexplored the physics reach of DUNE, given the data thatthese other experiments will collect. We have evaluatedthe adequate exposure for DUNE (in units of MW-kt-yr), i.e. the minimum exposure for DUNE to determinethe unknown parameters in combination with the otherexperiments, for all values of the oscillation parameters.The threshold for determination is taken to be χ = 25for the mass hierarchy and octant, and χ = 9 for detect-ing CP violation. The results are summarized in TableI. We find that adding information from NO ν A and T2K2helps in reducing the exposure required by only DUNEfor determination of all the three unknowns– hierarchy,octant and δ CP . Adding ICAL data to this combinationfurther help in achieving the same level of sensitivity witha reduction in exposure of DUNE (apart from δ CP ). Thusthe synergy between various experiments can be helpfulin economizing the DUNE configuration. We have alsoprobed the role of the ND in improving the results byreducing systematic errors. We have simulated events atthe near and far detectors and performed a correlatedsystematics analysis of both sets of events. We find animprovement in the physics reach of DUNE when the NDis included. We have also evaluated the drop in system-atics because of the near detector. Our results are shownin Table III.Further we have checked the role of information fromthe lowest energy bins which are affected by the secondoscillation maximum of the probability. We find thatinclusion of these bins enhances the the hierarchy sensi-tivity since the hierarchy- δ CP degeneracy has a comple-mentary behaviour at the two oscillation maxima. Thusthe increase in sensitivity is most significant in regions ofparameter space where the degeneracies reduce the sensi-tivity. We find that the effect is more prominent when agreater exposure is required. For the combined analysisto reach χ = 25 one needs, respectively, 44(56) MW-kt-yr including(excluding) the second oscillation maximum.However for only DUNE the same sensitivity requires436 MW-kt-yr but including the second oscillation max-imum the exposure is reduced to 212 MW-kt-yr to reach χ = 25.Finally, we have done an optimization study of theneutrino-antineutrino run for DUNE. The amount of an-tineutrino run required depends on the true value of δ CP .It helps in achieving two objectives – (i) reduction in oc-tant degeneracy and (ii) synergy between neutrino andantineutrino data for octant and CP sensitivity. For a hi-erarchy determination using a total exposure of 44 MW-kt-yr the optimal combination for only DUNE is (3+3)which corresponds to 22 MW-kt-yr in neutrino and an-tineutrino mode each, for δ CP in the lower half-plane[ − ◦ ,
0] and true NH-LO. For δ CP in the upper half-plane ([0 , ◦ ]) the optimal ratio is 5 / th of the totalexposure in neutrinos and + 1 / th of the total exposurein antineutrinos. Adding information from T2K, NO ν Aand ICAL the best combination for DUNE is 2 / rd neu-trino + 1 / rd antineutrino for δ CP in the lower half-plane. In the upper half-plane, pure neutrino run givesthe best sensitivity. In the latter case, the antineutrinocomponent coming from NO ν A and ICAL helps in re-ducing the required antineutrino run from DUNE. Foroctant sensitivity the best result from the combined ex-periments comes from the proportion (1 / th + 5 / th ) ex-cept for δ CP = +90 ◦ where all combinations give almostthe same sensitivity. For δ CP all combinations give sim-ilar results when all data are added together, with equalneutrino and antineutrino or 2 / rd neutrino + 1 / rd an-tineutrino combination faring slightly better. To conclude, the DUNE experiment can measure masshierarchy, octant and δ CP with considerable precision.Inclusion of the data from the experiments like T2K,NO ν A and ICAL can help DUNE to attain the samelevel of precision with a reduced exposure. Thus the syn-ergistic aspects between different experiments can helpin the planning of a more economized configuration forDUNE.
ACKNOWLEDGMENTS
We would like to thank Daniel Cherdack, Raj Gandhi,Newton Nath and Robert Wilson for useful discussions.
Appendix A: Computing the effect of the neardetector on systematics
In this appendix, we discuss briefly the simple proce-dure that we have used to combine results from the NDand FD, with correlated systematics. This procedure isbased on the method of pulls [102]. The (1 σ ) system-atic errors are given by a set of numbers ~π . These errorscan be normalization errors (which affect the scaling ofevents) or tilt errors (which affect the energy dependenceof the events). The ‘experimental’ data N det ( ex ) i are sim-ulated using the ‘true’ oscillation parameters ~p ex , whilethe ‘theoretical’ events N det ( th ) i are generated using the‘test’ oscillation parameters ~p th . The subscript i hereruns over all the energy bins. The superscript det cantake values ND or FD. The theoretical events get modi-fied due to systematic errors as M det ( th ) i ( ~p th ) = N det ( th ) i ( ~p th ) " X k ξ k π k + X l ξ l π l E i − E av E max − E min , where the index k ( l ) runs over the relevant normaliza-tion(tilt) systematic errors for a given experimental ob-servable. All the pull variables { ξ j } take values in therange ( − , − σ to+3 σ . Here, E i is the mean energy of the i th energy bin, E min and E max are the limits of the full energy range,and E av is their average.The Poissonian χ is calculated for each detector as χ det ( ~p ex , ~p th ; { ξ j } ) = X i " M det ( th ) i ( ~p th ) − N det ( ex ) i ( ~p ex )+ N det ( ex ) i ( ~p ex ) ln N det ( ex ) i ( ~p ex ) M det ( th ) i ( ~p th ) ! . The results from the two detector setups are then com-bined, along with a penalty for each source of systematic3error. The final χ is then calculated by minimizing overall combinations of ξ j , as χ ( ~p ex , ~p th ) = min { ξ j } (cid:20) χ F D ( ~p ex , ~p th ; { ξ j } )+ χ ND ( ~p ex , ~p th ; { ξ j } )+ X j ξ j (cid:21) ≡ χ (FD ⊕ ND) . Usually, for two experiments with uncorrelated system-atics, the adding of penalties and minimizing over thepull variables is done independently, and the resulting χ values are added. In contrast, here we add the samepulls to both detector setups, and then minimize over thepull variables. This takes care of correlations between thesystematic effects of the two setups. [1] Super-Kamiokande, S. Fukuda et al., Phys. Lett. B539(2002) 179, hep-ex/0205075.[2] SNO, Q.R. Ahmad et al., Phys. Rev. Lett. 89 (2002)011301, nucl-ex/0204008.[3] SNO, B. Aharmim et al., Phys. Rev. C72 (2005) 055502,nucl-ex/0502021.[4] KamLAND, K. Eguchi et al., Phys. Rev. Lett. 90 (2003)021802, hep-ex/0212021.[5] Super-Kamiokande Collaboration, R. Wendell et al.,Phys.Rev. D81 (2010) 092004, 1002.3471.[6] MINOS Collaboration, P. Adamson et al.,Phys.Rev.Lett. 112 (2014) 191801, 1403.0867.[7] T2K Collaboration, K. Abe et al., Phys.Rev.Lett. 111(2013) 211803, 1308.0465.[8] Double Chooz Collaboration, Y. Abe et al., JHEP 1410(2014) 86, 1406.7763.[9] Daya Bay Collaboration, F. An et al., Phys.Rev.Lett.112 (2014) 061801, 1310.6732.[10] RENO collaboration, J. Ahn et al., Phys.Rev.Lett. 108(2012) 191802, 1204.0626.[11] F. Capozzi et al., Phys.Rev. D89 (2014) 093018,1312.2878.[12] D.V. Forero, M. Tortola and J.W.F. Valle, Phys. Rev.D90 (2014) 093006, 1405.7540.[13] M.C. Gonzalez-Garcia, M. Maltoni and T. Schwetz,JHEP 11 (2014) 052, 1409.5439.[14] T2K, K. Abe et al., Phys. Rev. Lett. 112 (2014) 061802,1311.4750.[15] T2K, Y. Itow et al., (2001), hep-ex/0106019.[16] NOvA Collaboration, D. Ayres et al., (2004), hep-ex/0503053.[17] P. Huber et al., JHEP 11 (2009) 044, 0907.1896.[18] S. Prakash, S.K. Raut and S.U. Sankar, Phys.Rev. D86(2012) 033012, 1201.6485.[19] S.K. Agarwalla et al., JHEP 1212 (2012) 075, 1208.3644.[20] S.K. Agarwalla, S. Prakash and S.U. Sankar, JHEP1307 (2013) 131, 1301.2574.[21] S. Prakash, U. Rahaman and S.U. Sankar, JHEP 07(2014) 070, 1306.4125.[22] A. Chatterjee et al., JHEP 1306 (2013) 010, 1302.1370.[23] M. Ghosh et al., Phys.Rev. D89 (2014) 011301,1306.2500.[24] P.A.N. Machado et al., JHEP 05 (2014) 109, 1307.3248.[25] P. Coloma, H. Minakata and S.J. Parke, Phys. Rev.D90 (2014) 093003, 1406.2551.[26] M. Ghosh et al., Phys. Rev. D93 (2016) 013013,1504.06283. [27] J. Elevant and T. Schwetz, JHEP 09 (2015) 016,1506.07685.[28] A.M. Ankowski et al., Phys. Rev. D92 (2015) 091301,1507.08561.[29] P. Huber, M. Lindner and W. Winter, Nucl.Phys. B654(2003) 3, hep-ph/0211300.[30] H. Minakata and H. Sugiyama, Phys.Lett. B580 (2004)216, hep-ph/0309323.[31] O. Mena and S.J. Parke, Phys.Rev. D70 (2004) 093011,hep-ph/0408070.[32] ICAL, S. Ahmed et al., (2015), 1505.07380.[33] M. Ghosh et al., Nucl.Phys. B884 (2014) 274, 1401.7243.[34] M. Blennow and T. Schwetz, JHEP 1208 (2012) 058,1203.3388.[35] A. Ghosh, T. Thakore and S. Choubey, JHEP 1304(2013) 009, 1212.1305.[36] V. Barger, D. Marfatia and K. Whisnant, Phys.Rev.D65 (2002) 073023, hep-ph/0112119.[37] V. Barger, D. Marfatia and K. Whisnant, Phys. Rev.D66 (2002) 053007, hep-ph/0206038.[38] H. Minakata and H. Nunokawa, JHEP 0110 (2001) 001,hep-ph/0108085.[39] J. Burguet-Castell et al., Nucl.Phys. B646 (2002) 301,hep-ph/0207080.[40] H. Minakata, H. Nunokawa and S.J. Parke, Phys.Rev.D66 (2002) 093012, hep-ph/0208163.[41] G.L. Fogli and E. Lisi, Phys.Rev. D54 (1996) 3667, hep-ph/9604415.[42] M. Ishitsuka et al., Phys. Rev. D72 (2005) 033003, hep-ph/0504026.[43] T. Kajita et al., Phys.Rev. D75 (2007) 013006, hep-ph/0609286.[44] K. Hagiwara, N. Okamura and K. ichi Senda, Phys.Lett.B637 (2006) 266, hep-ph/0504061.[45] O. Mena Requejo, S. Palomares-Ruiz and S. Pascoli,Phys.Rev. D72 (2005) 053002, hep-ph/0504015.[46] O. Mena, S. Palomares-Ruiz and S. Pascoli, Phys.Rev.D73 (2006) 073007, hep-ph/0510182.[47] V. Barger, D. Marfatia and K. Whisnant, Phys.Lett.B560 (2003) 75, hep-ph/0210428.[48] LBNE Collaboration, C. Adams et al., (2013),1307.7335.[49] A. Stahl et al., (2012), CERN-SPSC-2012-021, SPSC-EOI-007.[50] P. Coloma, T. Li and S. Pascoli, (2012), 1206.4038.[51] S.K. Agarwalla, T. Li and A. Rubbia, JHEP 1205 (2012)154, 1109.6526. [52] M. Blennow et al., JHEP 1307 (2013) 159, 1303.0003.[53] S.K. Agarwalla, S. Prakash and S. Uma Sankar, JHEP1403 (2014) 087, 1304.3251.[54] M. Ghosh et al., JHEP 1403 (2014) 094, 1308.5979.[55] V. Barger et al., Phys.Rev. D89 (2014) 011302,1307.2519.[56] V. Barger et al., (2014), 1405.1054.[57] D. Dutta and k. Bora, Mod. Phys. Lett. A30 (2015)1550017, 1409.8248.[58] M. Blennow et al., JHEP 03 (2014) 028, 1311.1822.[59] M. Blennow, P. Coloma and E. Fernandez-Martinez,JHEP 03 (2015) 005, 1407.3274.[60] K.N. Deepthi, C. Soumya and R. Mohanta, New J.Phys. 17 (2015) 023035, 1409.2343.[61] K. Bora, D. Dutta and P. Ghoshal, Mod. Phys. Lett.A30 (2015) 1550066, 1405.7482.[62] LBNF, J. Strait, 2014, Talk given at NuFact2014, August 25-30, 2014, University of Glasgow, .[63] DUNE, R. Acciarri et al., (2015), 1512.06148.[64] K. Nakamura, Front. Phys. 35 (2000) 359.[65] Hyper-Kamiokande Proto-Collaboration, K. Abe et al.,PTEP 2015 (2015) 053C02, 1502.05199.[66] IceCube-PINGU Collaboration, M. Aartsen et al.,(2014), 1401.2046.[67] W. Winter, Phys.Rev. D88 (2013) 013013, 1305.5539.[68] S. Choubey and A. Ghosh, JHEP 1311 (2013) 166,1309.5760.[69] S.F. Ge and K. Hagiwara, JHEP 09 (2014) 024,1312.0457.[70] M. Blennow and T. Schwetz, JHEP 09 (2013) 089,1306.3988.[71] E.K. Akhmedov, S. Razzaque and A.Yu. Smirnov,JHEP 02 (2013) 082, 1205.7071, [Erratum:JHEP07,026(2013)].[72] P. Coloma et al., Phys. Rev. D87 (2013) 033004,1209.5973.[73] P. Huber and J. Kopp, JHEP 1103 (2011) 013,1010.3706.[74] P. Huber, M. Lindner and W. Winter, Comput. Phys.Commun. 167 (2005) 195, hep-ph/0407333.[75] P. Huber et al., Comput. Phys. Commun. 177 (2007)432, hep-ph/0701187.[76] M.D. Messier, (1999), Ph.D. Thesis (Advisor: JamesL. Stone).[77] E. Paschos and J. Yu, Phys.Rev. D65 (2002) 033002,hep-ph/0107261.[78] P. Huber, M. Lindner and W. Winter, Nucl.Phys. B645(2002) 3, hep-ph/0204352.[79] M. Fechner, Ph.D. Thesis DAPNIA-2006-01-T. [80] T2K Collaboration, I. Kato, J.Phys.Conf.Ser. 136(2008) 022018.[81] M. Ghosh, S. Goswami and S.K. Raut, (2014),1409.5046.[82] T2K Collaboration, K. Abe et al., (2014), 1409.7469.[83] NO ν A, R. Patterson, 2012, Talk given at the Neu-trino 2012 Conference, June 3-9, 2012, Kyoto, Japan, http://neu2012.kek.jp/http://neu2012.kek.jp/