Eﬀect of Matter Density in T2HK and DUNE Monojit Ghoshab1and Osamu Yasudac2 aSchool of Physics University of Hyderabad Hyderabad - 500046 India

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Eﬀect of Matter Density in T2HK and DUNE

Monojit Ghosh,a,b1and Osamu Yasudac,2

aSchool of Physics, University of Hyderabad, Hyderabad - 500046, India

bCenter of Excellence for Advanced Materials and Sensing Devices, Ruder Bošković

Institute, 10000 Zagreb, Croatia

cDepartment of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan

Abstract

CP phase determination for the near future long baseline experiments, T2HK and DUNE,

will require precise measurements of the oscillation probabilities. However, the uncertainty

in the Earth’s density must be considered in determining these oscillation probabilities.

Therefore, in this study, we update the individual sensitivities of these experiments for

determining the current unknowns in the standard three ﬂavor scenario considering the

latest conﬁguration and also the complementarity between them while considering the

uncertainty in the density. Our study showed that this uncertainty has a non-negligible

impact on the precision of the CP phase determination particularly for DUNE.

1E-mail: mghosh@irb.hr

2E-mail: yasuda@phys.se.tmu.ac.jp

arXiv:2210.09103v2 [hep-ph] 27 Feb 2023

1 Introduction

Many neutrino experiments over the last twenty years have successfully determined the prop-

erties of neutrino – such as mixing angles and mass squared diﬀerences – [1]. However, the

quantities yet to be determined include the mass ordering of the three neutrinos, the octant to

which θ23 belongs, and the leptonic CP phase. These unknown quantities are expected to be

determined by the near future long baseline experiments, which are T2HK [2] and DUNE [3].

The standard representation of leptonic mixing is given by a 3×3unitary matrix

U≡



1 0 0

0c23 s23

0−s23 c23 





c13 0s13e−iδCP

0 1 0

−s13eiδCP 0c13 





c12 s12 0

−s12 c12 0

0 0 1 



=



c12c13 s12c13 s13e−iδCP

−s12c23 −c12s23s13eiδCP c12c23 −s12s23s13eiδCP s23c13

s12s23 −c12c23s13eiδCP −c12s23 −s12c23s13eiδCP c23c13 

,(1.1)

where cjk ≡cos θjk,sjk ≡sin θjk,θ12,θ23,θ13 are three mixing angles, and δCP is a CP phase.

The oscillation probability is expressed based on the elements Uαj (α=e, µ, τ ;j= 1,2,3) of

this mixing matrix and the density of the matter through which the neutrino beam is traveling.

Eq. (1.1) shows that δCP appearance is always accompanied by sin θ13, which is small; therefore

determining δCP requires precise measurements of the neutrino oscillation probabilities.

In such precision measurements, we must further investigate the eﬀect of the uncertainty

in the Earth’s density. Refs. [4–9] studied the eﬀect of uncertainty in the density on the

oscillation probability measurement, although mainly in the context of a neutrino factory. In

this paper, we discuss the eﬀect of the uncertainty in the density on the measurements of

oscillation parameters at T2HK and DUNE and their combination. At these two experiments,

where the peak neutrino energy is less than 4 GeV, the matter eﬀect is smaller than that of the

mass squared diﬀerence term, and therefore the uncertainty in the density is not expected to

be as severe as it is at a neutrino factory where the neutrino energy can reach 12 GeV for which

causes the matter eﬀect to attain the same magnitude as that of the mass squared diﬀerence

term. Since the precision of the measured oscillation parameters has improved recently, it

is crucial to study the eﬀect of uncertainty in the density. Ref. [10] previously discussed the

sensitivity to the oscillation parameters of T2HK and DUNE and their combination1. Our

motivation for updating the ﬁndings of Ref. [10] is two fold as follows: (i) After Ref. [10] was

published, the setup of the T2HK and DUNE experiments were updated. Regarding T2HK,

only one talk at the far detector site is assumed in the ﬁrst phase; (ii) The uncertainty in the

Earth’s density was not considered in the analysis Ref.[10].2As we will observe in subsequent

sections, the uncertainty in the density gives a non-negligible contribution to the precision of

the the CP phase.

Unlike in Ref. [10], we did not include the atmospheric neutrino measurement at Hyper-

kamiokande, because a treatment of the uncertainty in the density is quite complicated for

the analysis of atmospheric neutrinos. For simplicity, we assume that the true mass ordering

is normal ordering.

This paper is organized as follows. In sect. 2, we brieﬂy introduce the parameter degen-

eracy. In sect. 3, we describe basic parameters of the T2HK and DUNE experiments, and

1For other phenomenological studies in the context of T2HK and DUNE in the standard three ﬂavor

scenario, we refer to Refs. [11–20]

2The eﬀect of the uncertainty in the Earth’s density on the oscillation probabilities at T2HK and DUNE

was studied in Refs. [21–23]. But they do not study the eﬀect of the Earth’s density on the measurement of

the unknown neutrino oscillation parameters.

comprehensively discuss our simulation details, and in sect. 4, we present the results of our

analysis. Furthermore, we summarize our conclusions in sect. 5. Moreover, in the appendix

A, we also provide a discussion on the octant degeneracy which appears at DUNE.

2 Parameter degeneracy

In research of neutrino oscillation with the standard three ﬂavor scenario, determination of the

CP phase δCP is regarded as the important goal. It has been known that so-called parameter

degeneracy makes it diﬃcult for us to determine δCP uniquely, even if we are given the values

of the appearance oscillation probabilities P(νµ→νe)and P(¯νµ→¯νe)at a ﬁxed neutrino

energy and at a ﬁxed baseline length.

Before θ13 was precisely measured by the reactor neutrino experiments [1], three kinds of

parameter degeneracy existed. The ﬁrst was the intrinsic degeneracy [5], which occurs since

the appearance oscillation probabilities are approximately quadratic in sin 2θ13 for small θ13;

thus it yields the two solutions (θ13, δCP)and (θ0

13, δ0

CP). The second is the sign degeneracy[24].

The value of δCP depends on whether the true mass ordering is normal or inverted; therefore,

determining mass hierarchy is crucial. The third one is the octant degeneracy [25]. The

primary contribution to the probabilities 1−P(νµ→νµ)and 1−P(¯νµ→¯νµ)is proportional

to sin22θ23, and we can only determine sin22θ23 from the disappearance channel; subsequently,

if θ23 is not maximal, then we have two solutions (θ23, δCP)and (90◦−θ23, δ0

CP).

These three types of degeneracies together created an eight fold degeneracy [26] when

the precise value of θ13 was unknown. After the precise value of θ13 was determined by the

reactor neutrino experiments [1], some of the eight fold degeneracy is resolved. However, in

a practical experimental situation, the formal discussions in Ref. [5, 24, 25] −in which the

probabilities P(νµ→νe)and P(¯νµ→¯νe)are assumed to be determined precisely −may not

necessarily apply to the T2HK or DUNE experiments because of experimental errors. Since

the cross sections for neutrinos and antineutrinos vary, the number of events (and therefore

the statistical errors) for neutrinos and antineutrinos can diﬀer. Previously, some authors

discussed parameter degeneracy by investigating only the neutrino mode, and degeneracy

which occurs in the neutrino oscillation probability is known as the “generalized hierarchy-

octant-δCP degeneracy” [27]. This generalized degeneracy comprises hierarchy-δCP degeneracy

[28] and octant-δCP degeneracy [29]. When the number of events for antineutrinos is not

suﬃciently large, “generalized hierarchy-octant-δCP degeneracy” becomes relevant. Note that

when T2HK and DUNE will be running, the medium baseline reactor experiment JUNO

[30] may provide a clear answer of the neutrino mass ordering. In that case the degeneracy

associated with the mass ordering will be resolved.

3 Analysis

3.1 T2HK and DUNE

T2HK is the long baseline experiment which is planned in Japan, and its baseline length and

peak energy is L=295 km, E∼0.6GeV, respectively. In the simulation of T2HK, we follow

the conﬁguration as given in Ref. [2]. We consider one water-Cerenkov detector tank having

ﬁducial volume of 187 kt located at Kamioka which is 295 km from the neutrino source at

J-PARC having a beam power of 1.3 MW with a total exposure of 27×1021 protons on target,

corresponding to 10 years of running. We have divided the total run-time into 5 years in

neutrino mode and 5 years in anti-neutrino mode. This assumption diﬀers from the original

setup in Ref.[2] where a 3:1 ratio of antineutrino mode to neutrino mode operation is assumed.3

The reference value for the Earth’s density is 2.70 g/cm3, as given in Ref. [2].

DUNE is another long baseline experiment which is planned in USA, and its baseline

length and peak energy is L=1300 km, E∼3GeV, respectively. In the case of DUNE, we

have used the oﬃcial GLoBES ﬁles of the DUNE technical design report [3] which reproduces

the results presented in Ref. [32]. A 40 kt liquid argon time-projection chamber detector is

placed 1300 km from the source having a power of 1.2 MW delivering 1.1×1021 protons on

target per year with a running time of 7 years. We have divided the run-time into 3.5 years

in neutrino mode and 3.5 years in antineutrino mode. The 1:1 ratio of antineutrino mode

to neutrino mode is the same setup as that assumed in Ref. [3]. The neutrino source will

be located at Fermilab, USA and the detector will be located at South Dakota, USA. The

reference value for the Earth’s density is 2.848 g/cm3, as given in Ref. [3].

This work is an update of the previous work [10] and adopted here the latest conﬁguration

of each experiment, including the detector volume, beam power, event selection, among others.

In this study, we employ the conﬁguration of T2HK from Ref. [2] and that of DUNE from

Ref. [32], whereas in the previous work [10] we took the T2HK conﬁguration from Ref. [33]

and that of DUNE from Ref. [34]. Some preliminary results from this work were presented by

an author at Neutrino 2022 [35]. After then, we made an additional eﬀort to match our event

spectrum with the one in Ref. [2] by tweaking the systematics a little to reproduce the results

in Ref. [2] more accurately.

3.2 Simulation Details

The experiments T2HK and DUNE are simulated using the software GLoBES [36,37]. We per-

form our analysis with the Poisson log-likelihood function χ2, which depends on the two sets of

the oscillation parameters and the Earth’s density ρ ~pex = (∆m2

21,∆m2

31, θ12, θ13, θ23, δCP, ρ)true

for the "true" parameters and ~pth = (∆m2

21,∆m2

31, θ12, θ13, θ23, δCP, ρ)test for the "test" ones:

χ2(~pex, ~pth) = min

{ξ(α)

j}χ2(~pex, ~pth;{ξ(α)

j}) + X

α=e,µ

j=1 ξ(α)

π(α)

j!2,(3.1)

where {ξ(α)

j}are the pull variables, π(α)

jare the (1σ) systematic errors for the pull variable

ξ(α)

χ2(~pex, ~pth;{ξ(α)

j})=2X

α=e,µ X

i"M(α)

i(~pth)−N(α)

i(~pex)

+N(α)

i(~pex) ln N(α)

i(~pex)

M(α)

i(~pth)!# (3.2)

is a χ2function deﬁned from the Poisson statistics, the "experimental" α-like data N(α)

i(~pex) (α=

e, µ)for the ith energy bin are deﬁned as the sum of the numbers of events for signals (S(α)

and for backgrounds (B(α)

N(α)

i(~pex) = S(α)

i(~pex) + B(α)

i(~pex),

3The reason why we work with 1:1 ratio rather than 3:1 is because it is more advantageous for T2K to

run in the dominant neutrino mode than to run with the ratio of 3:1, where the antineutrino mode is required

only in resolving the octant degeneracy, as was described by an author in Ref. [31].

the "theoretical" α-like events M(α)

i(~pth)are deﬁned as

M(α)

i(~pth) = 1 + ξ(α)

1+ξ(α)

Ei−Eav

Emax −Emin S(α)

i(~pth)

+1 + ξ(α)

2+ξ(α)

Ei−Eav

Emax −Emin B(α)

i(~pth)(3.3)

using the "test" oscillation parameters ~pth together with the systematic uncertainty ξ(α)

j. In

Eq. (3.2) the subscript iruns over all the energy bins in both the appearance (e-like) and

disappearance (µ-like) channels. Table 1 presents the systematic errors used in our calcula-

tions for the two experiments. π(α)

j(j= 1,2) (π(α)

j(j= 3,4)) corresponds to the relevant

normalization (tilt) systematic errors for a given experimental observable. For our simulation

we have ﬁxed the tilt error π(α)

j(j= 3,4) to a constant value which is 10% for T2HK4and

2.5% for DUNE corresponding to all the channels. In Eq. (3.3) Eiis the mean energy of the

ith energy bin, Emin and Emax are the limits of the full energy range, and Eav is their average.

The normalization errors aﬀect the scaling of events and the tilt errors inﬂuence the energy

dependence of the events. All the pull variables {ξ(α)

j}take values in the range (−3π(α)

j,3π(α)

j),

so that the errors can vary from −3σto +3σ.χ2(~pex, ~pth)is then calculated by minimizing

over all combinations of ξ(α)

Systematics T2HK DUNE

Sg-norm νe4.71% (4.47%) 2% (2%)

Sg-norm νµ4.13% (4.15%) 5% (5%)

Bg-norm νe4.71% (4.47%) 5% to 20% (5% to 20%)

Bg-norm νµ4.13% (4.15%) 5% to 20% (5% to 20%)

Table 1: The values of systematic errors that we considered in our analysis. “norm" denotes

the normalization error, “Sg" represents the signal and “Bg" signiﬁes the background. The

numbers without (with) parenthesis are for neutrinos (antineutrinos). For T2HK, the system-

atics are the same for signal and background whereas for DUNE the systematics errors are the

same for neutrinos and antineutrinos.

To obtain sensitivity to an oscillation parameter including δCP, we marginalize χ2(~pex, ~pth)

in Eq. (3.1), i.e., we minimize it with respect to the test parameters ~pth including the Earth’s

density. The true values of the oscillation parameters and the range of the test ones are

taken from Nuﬁt v5.1 [38] and they are presented together with the values of the density in

Table 2. As for the the uncertainty in the Earth’s density, the reasonable reference value for

the uncertainty in the density is 5% based on Ref. [39]. However, we also analyzed a more

conservative case for which the the uncertainty is 10%. In our analysis, we do not introduce

priors for the test oscillation parameters or for the test variable for the Earth’s density, over

which we marginalize.

4The values for the tilt variables ξ(α)

j(j= 3,4) are chosen such that our simulation results for T2HK

match with the sensitivities reported in the collaboration study.

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EectofMatterDensityinT2HKandDUNEMonojitGhosh,a;b1andOsamuYasudac;2aSchoolofPhysics,UniversityofHyderabad,Hyderabad-500046,IndiabCenterofExcellenceforAdvancedMaterialsandSensingDevices,RuderBo²kovi¢Institute,10000Zagreb,CroatiacDepartmentofPhysics,TokyoMetropolitanUniversity,Hachioji,Tokyo192-0397,J...

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Eﬀect of Matter Density in T2HK and DUNE Monojit Ghoshab1and Osamu Yasudac2 aSchool of Physics University of Hyderabad Hyderabad - 500046 India

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