Comment on Daya Bay's definition and use of Delta m^2_ee

TL;DR

This paper discusses and critiques Daya Bay's definition of the effective neutrino mass-squared difference Delta m^2_ee used in reactor antineutrino experiments, aiming to clarify its implications.

Contribution

It provides a detailed commentary on the definition and application of Delta m^2_ee in reactor neutrino experiments, highlighting potential issues and clarifications.

Findings

Identifies ambiguities in Daya Bay's definition of Delta m^2_ee

Clarifies the proper use and interpretation of Delta m^2_ee in experiments

Suggests improvements for consistent application in future analyses

Abstract

We comment on Daya Bay's latest definition of the effective Delta m^2 for short baseline reactor electron antineutrino disappearance experiments used in arXiv:1809.02261 (Phys. Rev. Lett. 121, no. 24, 241805 (2018))

Figures

Equations

$$\Delta m_{ee}^2(DB2)$$

$$\Delta m_{ee}^2(\mathrm{NPZ})\equiv {\cos }^2{\theta }_{12}\mathrm{\Delta }{\mathrm{m}}_{31}^2+{\sin }^2{\theta }_{12}\mathrm{\Delta }{\mathrm{m}}_{32}^2,$$

$$=\Delta m_{31}^2-{\sin }^2{\theta }_{12}\Delta m_{21}^2=\Delta m_{32}^2+{\cos }^2{\theta }_{12}\Delta m_{21}^2.$$

$$\Delta m_{ee}^2(DB2)$$

Full text

††preprint: FERMILAB-Pub-19-078-T

In Adey:2018zwh , Daya Bay (DB) uses their latest definition of $\Delta m^{2}_{ee}$ which obfuscates the simple relationship between such an effective $\Delta m^{2}$ and the fundamental parameters of the neutrino sector. Furthermore, this definition of $\Delta m^{2}_{ee}$ is baseline divided by neutrino energy ($L/E$) dependent. Dependence on L/E implies dependence on the proper time between production and detection of the observed neutrinos, i.e. the proper age of the neutrinos propertime . This new definition is approximately constant for the Daya Bay experiment ($L/E<1$ km/MeV), unlike DB’s earlier definition, DB1 . However, for the JUNO experiment ($6<L/E<25$ km/MeV), currently under construction, this new definition has a $\sim$ 1% jump between smallest and largest $L/E$ values for the observed neutrinos. The expected precision on the measurement of $\Delta m^{2}_{ee}$ at JUNO is 0.5%, see An:2015jdp , and therefore comparable in size to this jump.

Daya Bay’s latest definition of $\Delta m^{2}_{ee}$ is given by

[TABLE]

using $\Delta_{jk}\equiv\Delta m^{2}_{jk}L/(4E)$, see An:2015rpe . $\Delta m^{2}_{ee}(\rm DB2)$ is clearly $L/E$ (proper time of the neutrino) dependent and it is far from transparent the relationship to the fundamental neutrino parameters.

The original definition of an effective $\Delta m^{2}$ for $\nu_{e}$ disappearance experiments, $\Delta m^{2}_{ee}$, given by Nunokawa, Parke and Zukanovich Funchal (NPZ) , is simply Nunokawa:2005nx

[TABLE]

The $L/E$ independence of this definition is manifest and so is the relationship to the fundamental parameters of the neutrino sector. $\Delta m^{2}_{ee}(\rm NPZ)$ is “the $\nu_{e}$ average of $\Delta m^{2}_{31}$ and $\Delta m^{2}_{32}$”. RENO uses this definition RENO:2015ksa .

Since there is no review of $\Delta m^{2}_{ee}$ in the PDG, clarification of the relationship between these different definitions of $\Delta m^{2}_{ee}$ is pertinent for understanding short baseline reactor neutrino oscillation physics. To start, consider the small and large $L/E$ limits of $\Delta m^{2}_{ee}(\rm DB2)$: for $\frac{L}{E}\ll\frac{2\pi}{\Delta m^{2}_{21}}\approx 15~{}{\rm km/MeV}$ ($\Delta_{21}\ll 1$),

[TABLE]

In Fig. 1, the ratio of $\Delta m^{2}_{ee}(\rm DB2)$ to $\Delta m^{2}_{ee}(\rm NPZ)$ is plotted as a function of $L/E$. Clearly, for the Daya Bay experiment, $L/E<1$ km/MeV, these two definitions are essentially identical as the second term in eqn. Comment on Daya Bay’s definition and use of $\Delta m^{2}_{ee}$ is always smaller than 1${}^{\text{o}}\mkern-5.0mu/\mkern-3.0mu_{\text{oo}}$  of $\Delta m^{2}_{21}$. Whereas for the JUNO experiment, $6<L/E<25$ km/MeV, there is a significant jump, $\sim$1%, in the value $\Delta m^{2}_{ee}(\rm DB2)$ between the smallest and largest $L/E$.

In summary, Daya Bay’s new definition of $\Delta m^{2}_{ee}$, eq. 1, does not manifestly show the simple relationship to the fundamental parameters of the neutrino sector for short baseline reactor experiments, such as Daya Bay and RENO. Nor is it useful for future medium baseline experiments like JUNO due to the $\sim$1% jump, precisely in the $L/E$ range of this experiment uniqueness . The original definition of $\Delta m^{2}_{ee}$, eq. 2, is clearly $L/E$ independent, the relationship to the fundamental parameters of the neutrinos sector is manifest and useful for both the short, Daya Bay and RENO, and medium baseline, JUNO, reactor neutrino experiments.