Inference finds consistency between a neutrino flavor evolution model and Earth-based solar neutrino measurements

We continue examining statistical data assimilation, an inference methodology, to infer solutions to neutrino flavor evolution, for the first time using real—rather than simulated—data. The model represents neutrinos streaming from the Sun’s center and undergoing a Mikheyev-Smirnov-Wolfenstein resonance in flavor space, due to the radially varying electron number density. The model neutrino energies are chosen to correspond to experimental bins in the Sudbury Neutrino Observatory and Borexino experiments, which measure electron-flavor survival probability at Earth. The procedure successfully finds consistency between the observed fluxes and the model, if the Mikheyev-Smirnov-Wolfenstein resonance—that is, flavor evolution due to solar electrons—is included in the dynamical equations representing the model.

Figure 1

Schematic of the inference task. The constraints provided to the procedure are an assumed initial condition at the solar center (blue circle) and a measured survival probability at Earth (yellow circle). Meanwhile, the model consists of the potential $V(r)$ and flavor transformation dynamics [Eq. (2)] through the Sun. The inference task is to consider both the constraints and the model, to predict the flavor evolution within the Sun (red region), while also accounting for the transformation (grey wave) of the Earth-based measurement (yellow circle) to the solar surface (red circle).

Figure 2

Logarithmic plot of the action $A_0$ as a function of annealing parameter $\beta$. Left: $V(r)$ is taken from the standard solar model [65]. At $\beta =0$ the action begins to increase as model weight increases, then plateaus—at $\beta \sim 16–20$—as a solution is found that is consistent with both model and measurement (the increase in action beyond $\beta \sim 20$ is due to discretization error at high model weight). Right: $V(r)$ is set to zero. Now the action increases exponentially, indicating a failure to reconcile model with measurement. (See Ref. [37] for a detailed study of the action ($\beta$) plot).

Figure 3

True (blue) and predicted (red) state variable evolution, given model dynamics [Eqs. (2) and (21)] and Earth-based survival probability. Left: for matter potential $V(r)$ taken from the standard solar model; right: with $V(r)$ set to zero. From top: $P_{1,x}$, $P_{1,y}$, and $P_{1,z}$, for the first of three energy beams corresponding to the Borexino data. The units of distance are in percentage of solar radius $R_{\odot }$. Both left and right panels correspond to a value of annealing parameter $\beta$ of 20. For the result at left, $\beta =20$ lies on the “plateau” of Fig. 2, left panel—a solution consistent with both model and measurement. At right, a solution compatible with model dynamics is not found. These results are representative of all beams across both Borexino and SNO data sets, and for cases in which we added to the measurements the published maximum values of experimental error (not shown).

Figure 4

Details of the predictions shown in Fig. 3. Top: from left, three 1000-point segments of the $P_z$ evolution, beginning at the solar center (far left)—where the blue dot denotes the assumed initial condition, and ending at the solar surface (far right)—where the red dot denotes the location where $P_x$ and $P_z$ must be consistent with the measurement at Earth (Earth is not depicted). Bottom: Fast Fourier decomposition for the respective regions at top, showing that the oscillation frequency is well predicted throughout the Sun.

Figure 5

State predictions akin to Fig. 3, now with mixing angle $\theta$ recast as an unknown parameter to be estimated—where the Earth-Sun transformation has been omitted in the interest of easing computational expense. These predictions correspond to an estimate of $\theta$ of 0.59 radians (true: 0.58), where the permitted search range was [0.001,1.571]. The solution corresponds to a value of $\beta$ of 20.

Figure 6

State predictions akin to Fig. 3, again with mixing angle $\theta$ recast as an unknown parameter to be estimated, and including the Earth-Sun transformation. Importantly, in this version of the experiment, the measurements were taken to correspond not only to the ($P_x$ and $P_z$) pair at the solar surface $R_{\odot }$, but rather to the average of those two values over the final 1,000 discretized model locations in the Sun. See text for explanation. One out of four paths converged to this solution, which corresponds to a value of $\theta =0.56$, taken at a value of $\beta$ of 20.