Constraining CPT violation with Hyper-Kamiokande and ESSnuSB

CPT invariance is one of the most fundamental symmetries in nature and it plays a major role in the formulation of Quantum Field Theory. Although no definitive signal of CPT violation has been observed so far, there are many reasons to carefully investigate various low-energy phenomena that can provide better probes to test CPT symmetry. In this context, neutrino experiments are expected to provide more stringent bounds on CPT invariance violation when compared to the existing bounds from the Kaon system. In this work, we investigate the sensitivity of the upcoming long-baseline experiments: Hyper Kamiokande (T2HK, T2HKK), ESSnuSB and DUNE to constrain the CPT violating parameters $\Delta(\delta_{CP})$, $\Delta(m^2_{31})$ and $\Delta(\sin^2 \theta_{23})$, which characterize the difference between neutrino and antineutrino oscillation parameters. Further, we analyse neutrino and antineutrino data independently and constrain the oscillation parameters governing them by considering the combination of these experiments (DUNE+T2HKK and DUNE+ESSnuSB). In addition, assuming CPT symmetry is violated in nature, we study the individual ability of the aforementioned experiments to establish CPT violation. We found that the experiments Hyper-K (T2HK, T2HKK) and ESSnuSB, along with DUNE, will be able to establish CPT violation in their proposed run-times.


I. INTRODUCTION
Understanding the physics beyond Standard Model (BSM) is one of the prime objectives of present-day particle physics research. With the non-observation of any new heavy BSM particle through direct detection at LHC, the focus has been shifted to other frontiers, e.g., Intensity and Cosmic. In the Intensity frontier, neutrinos provide a promising avenue for revealing new physics. The compelling evidence of neutrino oscillations from various experiments already indicates that the minimal Standard Model (SM) of particle physics is not exhaustive and requires modification. In general, SM is considered as a low-energy effective theory originating from the unified theory of Quantum Gravity at the Planck scale.
Hence, understanding the true nature of the Planck scale physics through experimental signatures is of great importance, albeit extremely challenging to identify. It is expected that the long-baseline experiments will provide the ideal platform to look for tiny violations of Lorentz invariance or CPT symmetry that may exist as the low-energy remnants of Planck scale physics.
It is well-established that local relativistic quantum field theories, including the Standard Model, are invariant under Lorentz and CPT transformations. CPT theorem [1] states that "Any quantum theory formulated on flat space time is symmetric under the combined action of CPT transformations, provided the theory respects (i) Locality (ii) Unitarity and (iii) Lorentz invariance". One of the phenomenological consequences of CPT symmetry is that particles and antiparticles will have the same masses and lifetimes. If any discrepancy is found either in their masses or lifetimes, it would be a clear sign of CPT violation. The results from numerous experiments are consistent with the predictions of this symmetry. Although no conclusive evidence of CPT violation has been observed so far, there are many reasons to perform a careful investigation of possible mechanisms and descriptions of Lorentz and CPT violations. One of the ambitious motivations is that the Lorentz and CPT violations might arise from a fundamental theory at the Planck scale, but nonetheless may leave their footprints in some low-energy observables which can be detected in the current or upcoming experiments of exceptional sensitivity.
There exist experimental limits on CPT violating parameters from kaon and the lepton sectors. However, the current neutrino oscillation data provides the most stringent constraints on various oscillation parameters [17]: Further, in Ref. [3] it has been shown that DUNE will test the CPT violation in atmospheric mass difference to an unprecedented level and provide the most stringent limit as ∆m 2 31 − ∆m 2 31 < 8.
Analogously, the corresponding states for the antineutrinos are related as Denoting the neutrino and antineutrino masses by m i and m i (i = 1, 2, 3), the mass-squared differences of neutrinos are represented as ∆m 2 ij ≡ m 2 i − m 2 j and that of antineutrinos Consequently the oscillation probabilities of neutrinos and antineutrinos are functions of the oscillation parameters (θ 12 , θ 13 , θ 23 , ∆m 2 21 , ∆m 2 31 , δ CP ) and (θ 12 , θ 13 , θ 23 , ∆m 2 21 , ∆m 2 31 , δ CP ) respectively. In principle, neutrino oscillation experiments will be able to place bounds on the predictions of CPT symmetry violation. In this work, we investigate the ability of the future long-baseline experiments: T2HK, T2HKK, ESSnuSB and DUNE to constrain the CPT violating parameters, such as δ CP − δ CP , |∆m 2 31 − ∆m 2 31 | and sin 2 θ 23 − sin 2 θ 23 . We further, analyse neutrino and antineutrino data independently and constrain the oscillation parameters by considering the combination of the experiments DUNE+T2HKK and DUNE+ESSnuSB. In addition, assuming CPT symmetry is violated in nature, we study the individual ability of T2HK, T2HKK, DUNE and ESSnuSB experiments to establish CPT violation.
The outline of the paper is as follows. In section II, we give a brief overview of the experimental and simulation details of T2HK, T2HKK, ESSnuSB and DUNE. In section III, we determine the bounds placed by these experiments on the parameters ∆(δ CP ), ∆(m 2 31 ) and ∆(sin 2 θ 23 ) by assuming CPT symmetry exists in nature. Further, in section III A we analyse the combined data of DUNE+T2HKK, DUNE+ESSnuSB and how well they can measure neutrino and antineutrino oscillation parameters independently. Additionally, in section IV we assume that CPT symmetry is violated in nature and estimate the sensitivity of T2HK, T2HKK, ESSnuSB and DUNE to establish CPT invariance violation individually.
Finally, our results are summarized in section V.

II. EXPERIMENTAL AND SIMULATION DETAILS
In this section, we discuss the detailed experimental features of the long-baseline experiments T2HK, T2HKK, ESSnuSB and DUNE.
T2HK : Tokai to Hyper Kamiokande (T2HK) is an upgradation proposed to the existing T2K facility in Japan. In this plan, the JPARC beam will produce a 1.3 MW powered beam and the far detector (FD) will have two identical water Cherenkov detectors of 187 kt (2 × 187 = 374 kt) fiducial volume to be placed at 295 km baseline, 2.5 • off from the beam axis.
T2HKK : T2HKK is an alternative choice to T2HK, where the proposed FD is placed in Korea, which is 1100 km away from the JPARC facility. One of the two tanks (187 kt) proposed in the T2HK experiment will be placed at 1100 km with an off-axis angle (OAA) of 1.5 • or 2 • or 2.5 • . Basing on the study in [25], we consider the OAA 1.5 • as it provides maximum sensitivity to the oscillation parameters. We consider the proposed run time ratio of (1ν : 3ν) years corresponding to a total exposure of 27 × 10 21 protons on target (POT).
The detector systematics are taken as per the [25,26].
ESSnuSB : The major objective of the European Spallation Source ν-Beam (ESSnuSB) project [28] is to measure the leptonic CP violation. A neutrino beam with a peak energy of 0.25 GeV is produced at the ESS facility in Lund, Sweden. This beam is made to travel 540 km to encounter a water Cherenkov detector of 500 kt to be placed at a mine in Garpenberg.
The proposed runtime is (2ν + 8ν) years with a total POT of 27 × 10 22 corresponding to a 5 MW proton beam. We have performed the numerical analysis using the GLoBES package [29,30]. The experimental specifications along with the signal and background normalisation errors, are listed in Table I. The statistical χ 2 is obtained using where N test  Table I. Here, χ 2 pull accounts for these errors and acts as a penalty term to the total χ 2 (χ 2 tot ).    values for oscillation parameters used are given in Table II In the test values, we marginalize over all the oscillation parameters for both neutrinos and antineutrinos except x,x and the solar parameters (since T2HK, T2HKK, ESSnuSB and DUNE have no sensitivity to these parameters). After marginalization, we calculate the χ 2 value for the CPT violating observable through the relation The coloured curves blue, red, green and magenta show the sensitivities of T2HK, T2HKK, ESSnuSB and DUNE respectively. The black dash-dot line represents 3σ confidence limit.
The sensitivity for ∆(sin 2 θ 13 ) is not very significant, for which we have not shown the corresponding result here.
From Fig. 1, one can estimate the best bound on the parameter ∆(∆m 2 31 ) at 3σ C.L. by T2HK experiment (blue curve) as ∆(∆m 2 31 ) < 3.32 × 10 −5 eV 2 . It can be seen from all the columns of Fig. 1 that T2HK provides a better bound compared to T2HKK, ESSnuSB  Table-III. From Fig. 3, the best ever bounds on ∆(δ CP ) can be extracted from T2HKK (red curve) for CPT violation which is ∆(δ CP ) < 100 0 at 3σ confidence level. The next best bound on ∆(δ CP ) is obtained by ESSnuSB experiments. This is because both T2HKK and ESSnuSB experiments are planned at the second oscillation maxima to meet their primary goal of measuring the CP phase δ CP . The list of the bounds at 3σ C.L. on ∆(δ CP ) are given in the   third row of Table-III. A. Constraining CPT violation with combination of DUNE+T2HKK and

DUNE+ESSnuSB
In this subsection, we continue to assume that CPT is a conserved symmetry in nature. We analyse the neutrino and antineutrino data independently and determine whether the corresponding oscillation parameters in both cases are the same as predicted by CPT symmetry. The true oscillation parameters are considered in the analysis are provided in Table-II and for the test scenario, we take the six oscillation parameters for both neutrino (∆m 2 31 , θ 23 ,δ CP ) and antineutrino (∆m 2 31 , θ 23 , δ CP ) in their allowed ranges as given in Table  II. Marginalisation is done over the remaining four parameters while showing the effect of the rest two oscillation parameters. The results are shown in Fig. 4, where the axes can be visualised for both neutrino and antineutrino parameters. It is shown in Ref. [24] that, while DUNE and T2HK can resolve the octant degeneracy assuming CPT conservation, the combination of DUNE+T2HK cannot resolve this degeneracy while treating neutrino and antineutrino parameters individually. In this subsection, we explore the same by considering However, this degeneracy disappears when we considered the neutrino beams of DUNE and ESSnuSB. Overall, from all the plots of Fig. 4, we can observe that neutrino oscillation data constrains the parameters better than the antineutrino data.

IV. DISCOVERING CPT VIOLATION
In this section, we assume that CPT is violated in nature. We generate the simulated data for the experiments T2HK, T2HKK, ESSnuSB and DUNE by assuming different neutrino and antineutrino oscillation parameters. In particular, we only consider the case where the CP-violating phases δ CP andδ CP are not equal 1 . We further consider θ 23 = θ 23 , ∆m 2 31 = ∆m 2 31 , θ 13 = θ 13 and their true values are taken from Table-II. In Fig. 5, we plot the allowed regions of δ CP (test) andδ CP (test) as obtained from the experiments T2HK (blue), T2HKK (red), ESSnuSB (green) and DUNE (magenta). The solid and dotted contours in all the figures correspond to 68% and 99% C.L. and the dashed black lines correspond to CPT conserving values. Top panel of Figure 5 shows that both the configurations of Hyper-Kamiokande experiment -T2HK (blue) and T2HKK (red) will be able to establish CPT violation with 99% C.L. by showing that δ CP =δ CP in their proposed run-time.
This can be inferred from the fact that there are no degenerate solutions obtained in the 1 We consider the variation in δ CP andδ CP as these parameters are poorly constrained. In this paper, we have studied the sensitivity reach of the upcoming long-baseline experiments, T2HK, T2HKK, ESSnuSB and DUNE to explore the CPT violation in the neutrino sector. Our findings are summarized below: • We obtained the sensitivity limits on the CPT violating parameters ∆(∆m 2 31 ), ∆(sin 2 θ 23 ) and ∆(δ CP ). We found that the T2HKK and ESSnuSB experiments are quite sensitive to the CP violating phase δ CP , whereas T2HK, T2HKK and DUNE are sensitive to the atmospheric mixing parameters. The most stringent limits on ∆(∆m 2 31 ) and ∆(sin 2 θ 23 ) come from T2HK experiment whereas T2HKK will provide the best bound on ∆(δ CP ).
• Next, we obtained the constraint on CPT violation with the combination In conclusion, we found that the upcoming experiments T2HK, T2HKK, ESSnuSB and DUNE have great potential to establish CPT violation in neutrino oscillation and provide stringent limits on the CPT violating parameters ∆(∆m 2 31 ) and ∆(sin 2 θ 23 ).