Baseline optimization for the measurement of $CP$ violation, mass hierarchy, and ${\theta }_{23}$ octant in a long-baseline neutrino oscillation experiment

Next-generation long-baseline electron neutrino appearance experiments will seek to discover $CP$ violation, determine the mass hierarchy and resolve the ${\theta }_{23}$ octant. In light of the recent precision measurements of ${\theta }_{13}$, we consider the sensitivity of these measurements in a study to determine the optimal baseline, including practical considerations regarding beam and detector performance. We conclude that a detector at a baseline of at least 1000 km in a wide-band muon neutrino beam is the optimal configuration.

Figure 1

The ${\nu }_{\mu }\to {\nu }_e$ oscillation probability as a function of baseline/neutrino energy ($L/E_{\nu }$) as given by Eq. (1). The top graph (a) shows the full probability for vacuum oscillations with ${\delta }_{CP}=0$ and shows the contribution from the ${\sin }^2(2{\theta }_{13})$ term and the ${\sin }^2(2{\theta }_{12})$ term (also called the solar term). The middle graph (b) shows the full probability in vacuum for different values of ${\delta }_{CP}$. The bottom graph (c) shows the full probability in matter, assuming constant matter density, for different baselines compared to the vacuum probability with ${\delta }_{CP}=0$. The oscillation parameter values used to draw these curves are given in Sec. 4.

Figure 2

Asymmetry vs baseline at the first (black line) and second (green line) oscillation maxima. The solid lines show the asymmetry due to the matter effect only (${\delta }_{CP}=0$) for the normal hierarchy (left plot) or inverted hierarchy (right plot). The dashed lines show the maximum $CP$ asymmetry in vacuum, with ${\delta }_{CP}=-\pi /2$ (left plot) or ${\delta }_{CP}=\pi /2$ (right plot). The oscillation parameter values used to draw these curves are given in Sec. 4.

Figure 3

Estimated ${\nu }_e$ appearance rates (with no detector effects) integrated over the first two oscillation maxima as a function of baseline for three conditions. The solid and dashed lines are the event rates obtained using a neutrino flux that is constant as a function of energy. The solid line assumes oscillations in vacuum and the dashed line assumes oscillations in a constant matter density with normal hierarchy. The dotted-dashed line is the event rate calculation using a neutrino-beam flux obtained from a perfect-focusing system with a 120-GeV primary beam and a fixed decay pipe length of 380 m assuming normal hierarchy.

Figure 4

Estimated ${\nu }_e$ (left) and ${\overline{\nu }}_e$ (right) appearance rates (with no detector effects) integrated over an energy region around the first oscillation maximum (solid line) or the second oscillation maximum (dashed line) assuming the flux obtained from a perfect-focusing system with a 120-GeV primary beam and a fixed decay pipe length of 380 m. The curves are shown for different values of ${\delta }_{CP}$. Matter effects are included assuming normal hierarchy. The rates for the second oscillation maximum have been increased by a factor of 10 for visibility.

Figure 5

The neutrino flux at 1 km from a 120-GeV proton beam incident on a target two interaction lengths in thickness. The black histogram is the flux from a perfectly focused hadron beam and a decay pipe 380 m in length. The gray histogram is the flux obtained using the NuMI focusing system operating at a current of 250 kA. (The focusing system works similarly for the antineutrino spectrum.)

Figure 6

Ratio of the ${\nu }_e$ appearance signal spectra used in this study to the appearance spectra obtained assuming perfect hadron focusing for baselines of 1000, 2000, and 3000 km. The solid horizontal lines show the energy region around the first maximum, where the limits are the energies above and below the first maximum at which the appearance probability is 50% of the maximum. The dashed horizontal lines show the energy region around the second maximum, where the limits are the energies above and below the second maximum at which the appearance probability is 10% of the maximum.

Figure 7

Neutrino spectra for normal hierarchy: Reconstructed energy distribution of selected ${\nu }_e$ CC-like events assuming a 175 kt-MW-yr exposure in the neutrino-beam mode at each baseline. The plots assume normal mass hierarchy. The signal contribution is shown for various values of ${\delta }_{CP}$.

Figure 8

Neutrino spectra for inverted hierarchy: Reconstructed energy distribution of selected ${\nu }_e$ CC-like events assuming a 175 kt-MW-yr exposure in the neutrino-beam mode at each baseline. The plots assume inverted mass hierarchy. The signal contribution is shown for various values of ${\delta }_{CP}$.

Figure 9

Antineutrino spectra for normal hierarchy: Reconstructed energy distribution of selected ${\nu }_e$ and ${\overline{\nu }}_e$ CC-like events assuming a 175 kt-MW-yr exposure in the antineutrino-beam mode at each baseline. The plots assume normal mass hierarchy. The signal contribution is shown for various values of ${\delta }_{CP}$.

Figure 10

Antineutrino spectra for inverted hierarchy: Reconstructed energy distribution of selected ${\nu }_e$ and ${\overline{\nu }}_e$ CC-like events assuming a 175 kt-MW-yr exposure in the antineutrino-beam mode at each baseline. The plots assume inverted mass hierarchy. The signal contribution is shown for various values of ${\delta }_{CP}$.

Figure 11

Event rates vs baseline: The number of signal and background events integrated in reconstructed energy over the first maximum (and second, if possible) for a 175 kt-MW-yr exposure at each baseline in neutrino (left) and antineutrino (right) modes. These rates assume normal hierarchy and ${\delta }_{CP}=0$. (Note that the sensitivities presented in this paper are not based solely on rates, but are calculated by fitting the spectra in bins of reconstructed energy.)

Figure 12

The fraction of all possible ${\delta }_{CP}$ values for which we can determine normal (left) or inverted (right) mass hierarchy with a minimum value of $\overline{\mathrm{\Delta }{\chi }^2}=9$ as a function of baseline. An expected average value of $\overline{\mathrm{\Delta }{\chi }^2}=9$ corresponds to a 93.32% probability of determining the correct mass hierarchy according to the analysis in [34]. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background. The shaded band shows the possible range in the fraction due to the uncertainty in the other oscillation parameters and considers both octant solutions for ${\theta }_{23}$.

Figure 13

The fraction of all possible ${\delta }_{CP}$ values for which we can determine normal (left) or inverted (right) mass hierarchy with a minimum value of $\overline{\mathrm{\Delta }{\chi }^2}=25$ as a function of baseline. An expected average value of $\overline{\mathrm{\Delta }{\chi }^2}=25$ corresponds to a 99.38% probability of determining the correct mass hierarchy according to the analysis in [34]. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background. The shaded band shows the possible range in the fraction due to the uncertainty in the other oscillation parameters and considers both octant solutions for ${\theta }_{23}$.

Figure 14

The exposure required to determine normal (left) or inverted (right) mass hierarchy with a significance of $\overline{\mathrm{\Delta }{\chi }^2}=25$ for all possible values of ${\delta }_{CP}$ as a function of baseline. We assume no ${\nu }_{\tau }$ CC background.

Figure 15

The exposure required to determine normal (left) or inverted (right) mass hierarchy with a significance of $\overline{\mathrm{\Delta }{\chi }^2}=9$ for all possible values of ${\delta }_{CP}$ as a function of baseline. We assume no ${\nu }_{\tau }$ CC background.

Figure 16

The fraction of all possible ${\delta }_{CP}$ values for which we can observe $CP$ violation with a sensitivity of at least $3\sigma$ ($\mathrm{\Delta }{\chi }^2=9$) for normal (left) and inverted (right) mass hierarchy as a function of baseline. The true mass hierarchy is assumed to be unknown. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background. The shaded band shows the possible range in the fraction due to the uncertainty in the other oscillation parameters and considers both octant solutions for ${\theta }_{23}$.

Figure 17

The fraction of all possible ${\delta }_{CP}$ values for which we can observe $CP$ violation with a sensitivity of at least $3\sigma$ ($\mathrm{\Delta }{\chi }^2=9$) for normal (left) and inverted (right) mass hierarchy as a function of baseline. The true mass hierarchy is assumed to be perfectly known. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background. The shaded band shows the possible range in the fraction due to the uncertainty in the other oscillation parameters and considers both octant solutions for ${\theta }_{23}$.

Figure 18

The significance ($\sigma =\sqrt{\mathrm{\Delta }{\chi }^2}$) at which $CP$ violation can be observed as a function of the true value of ${\delta }_{CP}$ assuming normal hierarchy for 300 km, 750 km, and 1300 km baselines. The significance is shown assuming the hierarchy is perfectly known (black-solid) and assuming both hierarchy solutions are considered (red-dashed). Knowing the hierarchy improves the $CP$ sensitivity at 300 km and 750 km, but has no effect at 1300 km (or baselines $>1300\mathrm{km}$).

Figure 19

The exposure required to observe $CP$ violation with a significance of $5\sigma$ for 50% of all possible values of ${\delta }_{CP}$, assuming normal (left) and inverted (right) mass hierarchy, as a function of baseline. The true mass hierarchy is assumed to be unknown, and we assume no ${\nu }_{\tau }$ CC background.

Figure 20

The exposure required to observe $CP$ violation with a significance of $3\sigma$ for 75% of all possible values of ${\delta }_{CP}$, assuming normal (left) and inverted (right) mass hierarchy, as a function of baseline. The true mass hierarchy is assumed to be unknown, and we assume no ${\nu }_{\tau }$ CC background.

Figure 21

The $1\sigma$ uncertainty in ${\delta }_{CP}$ as a function of baseline assuming the true value of ${\delta }_{CP}$ is 0° (gray solid), $+90°$ (black solid), and $-90°$ (black dashed) for normal (left) and inverted (right) mass hierarchy. The true mass hierarchy is assumed to be perfectly known.

Figure 22

The $1\sigma$ uncertainty in ${\delta }_{CP}$ as a function of the true value of ${\delta }_{CP}$, assuming the true hierarchy is known to be normal, for baselines of 300 km, 1300 km, and 3000 km.

Figure 23

The $1\sigma$ uncertainty in ${\delta }_{CP}$ as a function of baseline, assuming the true value of ${\delta }_{CP}$ is 0°, for normal (left) and inverted (right) mass hierarchy. The true mass hierarchy is assumed to be perfectly known. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background. The shaded band shows the possible range in the resolution due to the uncertainty in the other oscillation parameters and considers both octant solutions for ${\theta }_{23}$.

Figure 24

The fraction of all possible ${\delta }_{CP}$ values for which we can determine the ${\theta }_{23}$ octant with a sensitivity of at least $5\sigma$ ($\mathrm{\Delta }{\chi }^2=25$) for normal (left) and inverted (right) mass hierarchy as a function of baseline. The true mass hierarchy is assumed to be unknown. The solid black (red dashed) line shows the result including zero (maximum) ${\nu }_{\tau }$ CC background.

Figure 25

The $1\sigma$ uncertainty in ${\sin }^2(2{\theta }_{13})$ as a function of baseline assuming ${\sin }^2(2{\theta }_{13})=0.094$ and ${\delta }_{CP}=0$ for normal (left) and inverted (right) mass hierarchy. The true mass hierarchy is assumed to be perfectly known, and we assume no ${\nu }_{\tau }$ CC background.

Figure 26

The $1\sigma$ uncertainty in ${\sin }^2({\theta }_{23})$ as a function of baseline assuming ${\sin }^2({\theta }_{23})=0.39$.