text:
\section{Introduction}
\label{Introduction}

Small-scale burst activity has been widely observed across the broad temperature range of the solar atmosphere, from Ellerman bombs (EBs) in the photosphere (see, for example, \citealt{Ellerman17, Vissers13, Nelson15}), through UV bursts in the transition region (as discussed by \citealt{Peter14, Judge15, Young18}), to the recently discovered campfires in the corona (e.g. \citealt{Berghmans21, Mandal21, Panesar21}). Each of these dynamic events is thought to be driven by localised magnetic reconnection between opposite polarity magnetic fields, with the properties of the reconnection (e.g. height in the atmosphere, amount of energy released) seemingly dictating which type of burst is formed. Recent numerical simulations have supported this assertion (see: \citealt{Danilovic17} for EBs; \citealt{Hansteen17} for UV bursts; and \citealt{Chen21} for campfires) providing us with a clearer understanding of the physical mechanisms responsible for these events. Despite this general understanding, there is still much we don't know about these features, including whether UV bursts are detected in all active regions (ARs) and whether their properties (potentially diagnosing the physics of the magnetic reconnection driving these events) change during the evolution of the host ARs.

A new sub-set of UV bursts, characterised by large increases in intensity in spectral lines sampling the transition region (including the \ion{Si}{IV} $1394$ \AA\ and $1403$ \AA\ lines), were identified recently in data from the Interface Region Imaging Spectrograph (IRIS; \citealt{DePontieu14}) by \citet{Peter14}. Those authors studied a raster sampling AR $11850$ and found four burst events, which were later named IRIS bursts in the literature, co-spatial to bi-poles in the photosphere. It was suggested that these events could be driven by magnetic reconnection heating the local plasma to temperatures close to $80$ kK. The presence of several chromospheric absorption lines (e.g. \ion{Ni}{I} $1393$ \AA) over-laid on the IRIS burst spectra, however, indicated that these events occurred deep in the solar atmosphere, potentially at heights as low as several hundred km. Subsequent research found that IRIS bursts could occur co-spatial to EBs (\citealt{Vissers15, Tian16}), which are known to have temperatures well below the typical formation temperatures of the \ion{Si}{IV} lines (see, for example, \citealt{Fang06, Hong14}), and cancelling bi-poles (e.g. \citealt{Nelson16}) in the photosphere. Dedicated 1D modelling efforts have so far been unable to reproduce co-spatial EB and IRIS burst spectra, in part due to their large temperature differences (e.g. \citealt{Reid17, Hong17}).

In order to better assess the overall impact of IRIS bursts on the global solar atmosphere, \citet{Kleint22} conducted an extensive statistical analysis of IRIS burst profiles using the more than $3500$ IRIS spectral datasets sampled during 2013 and 2014 (utilising the methods detailed in \citealt{Panos21a} and \citealt{Panos21b}). Those authors considered a \ion{Si}{IV} $1394$ \AA\ spectra to be IRIS burst-like if absorption lines were over-laid on it, agreeing with one of the key observational characteristics of these events as noted by \citet{Young18}. More than $100$ k IRIS spectra satisfied this criteria (within $287$ datasets), indicating that around $0.01$ \% of all \ion{Si}{IV} $1394$ \AA\ spectra recorded by IRIS are IRIS burst profiles. Of the sample reported, no IRIS burst profiles were detected in the quiet-Sun or close to the solar poles suggesting that these events occur exclusively in ARs. It is likely, therefore, that the potential quiet-Sun IRIS burst presented by \citet{Nelson17} would not have been classed as being IRIS burst-like by this algorithm (if it had been applied to that dataset) due to the lack of clear absorption lines over-laid on the relatively narrow \ion{Si}{IV} $1394$ \AA\ line. Finally, no systematic increase in emission co-spatial to IRIS burst profiles was found either for the IRIS \ion{Fe}{XXI} line or for the EUV lines sampled by the Solar Dynamics Observatory's Atmospheric Imaging Assembly (SDO/AIA; \citealt{Lemen12}) by \citet{Kleint22} suggesting that the magnetic reconnection hypothesised to be driving these events does not heat the local plasma to coronal (>10$^6$ K) temperatures.

Although we focus specifically on IRIS bursts in this article, it should be noted that a whole host of other burst events have also been detected in the solar atmosphere in spectral windows sampling similar temperature ranges. This wider family of events are collectively known as UV bursts, and include features such as explosive events (\citealt{Dere89}) and blinkers (\citealt{Harrison97}). Whilst one of the unique defining properties of IRIS bursts are the characteristic overlying absorption lines on the \ion{Si}{IV} $1394$ \AA\ spectra, other types of UV bursts have a plethora of other spectral shapes (some are extremely narrow whilst some are extremely broad) and properties (some are symmetric about the spectral line whilst some display broadenings in only the red or blue wings). This large variance in spectral signatures means other UV burst types will not be identified by the version of the algorithm developed by \citet{Kleint22}. The inclusion of more template spectra would be required if these other types of UV bursts were to be studied in an automated manner in the future. The spectral differences between distinct types of UV bursts also means that the defining properties of this family of events must be taken from other observable properties, with key signatures being that they are: i) compact (areas typically below $2$\arcsec); ii) short-lived (lifetimes of up to around one hour); iii) intense compared to the local background; iv) relatively stationary; and v) not linked to larger-scale flares. Given these properties it is, therefore, possible for an event to display an IRIS burst profile but not be a UV burst. For a comprehensive review of both IRIS bursts and the wider family of UV bursts see \citet{Young18}.

The algorithm developed by \citet{Kleint22} provides numerous opportunities for advancing our understanding of the importance of IRIS bursts in the solar atmosphere. In this article, we utilise a slightly modified version of this algorithm to investigate how the measurable properties of IRIS bursts (e.g. frequency, area, location, spectral characteristics) in large IRIS rasters vary during the evolution of seven ARs. The motivation behind this research is to identify whether there are preferential conditions for when and where magnetic energy is released in ARs as they evolve. We structure our work as follows: In Sect~\ref{Methods} we detail the data products analysed and outline the changes made to the IRIS burst detection algorithm; In Sect.~\ref{Results} we present the results of our research and a brief discussion; before in Sect.~\ref{Conclusions} we draw our conclusions.

\section{Methods}
\label{Methods}

\subsection{Observations}

The primary data products analysed in this research were sampled by IRIS, a highly versatile instrument which can be employed in a plethora of observing modes allowing the user to vary the spatial, temporal, and spectral resolutions of the returned data based on scientific need. This versatility creates a comprehensive data catalogue but also means we must filter the available datasets in order to isolate only those of interest for this particular study. Here, we focus our analysis on ARs that were sampled by a minimum of four separate $400$-step IRIS rasters in the time-period studied by \citet{Kleint22}. These observations must include the \ion{Si}{IV} $1394$ \AA\ line, and have time differences between the first and final rasters of at least 48 hours. Seven ARs (11850, 11856, 11871, 11909, 11916, 12104, and 12139) satisfied these criteria, forming the sample we analyse here. Of these, two ARs (11871 and 11916) emerged onto the solar disk between one and three days prior to their first IRIS raster scans, whilst the remaining five (11850, 11856, 11909, 12104, and 12139) rotated onto the solar disk as fully formed ARs between four and six days prior to the first IRIS raster scans which sampled them.

\begin{figure}
\center
\includegraphics[width=0.49\textwidth]{UV_fig1.eps}
\caption{Locations on the Sun of each of the IRIS burst profiles studied in this article, with different colours corresponding to different ARs. IRIS burst profiles are detected in multiple rasters sampled over multiple days for six of the seven ARs. Clearly, no IRIS burst profiles were detected at any time within AR 11871. General information about the identified IRIS burst profiles is presented in Table~\ref{Tab_overview}.}
\label{Fig_locations}
\end{figure}

\begin{figure*}
\center
\includegraphics[width=0.5\textwidth,trim={0.3cm 0.5cm 0 0.1cm,clip}]{UV_fig2a.eps}
\includegraphics[width=0.45\textwidth]{UV_fig2b.eps}
\caption{Example of IRIS burst profiles missed by the algorithm. (Left-hand panel) Intensity of the FOV during the fourth IRIS raster sampling AR $12104$ at the core wavelength of the \ion{Si}{IV} 1394 line (logarithmically scaled to enhance contrast). Our algorithm did not identify any bursts in this raster. The two times during which IRIS passed through the SAA, thereby causing noisy spectra, during this observation are marked by red arrows. The orange circles locate two pixels whose spectra are shown in the right-hand panels (black lines), one containing a manually detected burst and the other containing no burst spectra. (Right-hand panels) The two IRIS spectra selected to demonstrate how a small number of IRIS burst profiles may be missed by the algorithm. The top panel plots a manually detected IRIS burst with a clear, albeit shallow, absorption blend over-laid on the \ion{Si}{IV} $1394$ \AA\ line, while the bottom panel is a typical non-burst spectrum. Both spectra are normalised to their peak intensity. We over-plot the mathematically closest reference spectrum considered by the algorithm to each of these profiles (blue lines), which in both cases is a non-burst reference spectrum. Therefore, no IRIS burst profile is found for either pixel by the algorithm. The nearest IRIS burst reference spectrum to the manually detected IRIS burst profile is also over-plotted on the top panel for reference (orange line).}
\label{Fig_manual}
\end{figure*}

Overall, $42$ IRIS datasets were analysed in this study. Each of these datasets had a raster step size of $0.35$\arcsec\ and a pixel scale along the slit of $0.17$\arcsec, allowing us to consistently measure the properties of IRIS bursts. General information about each of these datasets is provided for reference in Table~\ref{Tab_overview}. From this sample, $35$ datasets were found to contain a combined total of $4019$ IRIS burst profiles using the modified IRIS burst detection algorithm. This accounts for $0.02$ \%\ of the total number of IRIS \ion{Si}{IV} $1394$ \AA\ profiles analysed here. To supplement this automated detection, we also conducted a manual analysis of the remaining seven datasets using CRISPEX (\citealt{Vissers12}) in order to identify whether any clear and obvious IRIS burst profiles were missed by the algorithm. IRIS burst profiles, identified through the presence of absorption lines on the \ion{Si}{IV} $1394$ \AA\ spectra for consistency, were detected in one further dataset through these checks. The remaining six datasets did not appear to contain any clear and obvious IRIS burst profiles (we cannot definitively state that a small number of pixels did not evade our manual search which will have been biased towards more intense pixels). The locations on the Sun of each of the IRIS burst profiles detected using automated methods are plotted in Fig.~\ref{Fig_locations}. The different coloured boxes correspond to different ARs, as detailed in the legend.

Given longer exposure times may lead to an increase in the number of saturated pixels (which are removed from the sample before the algorithm is applied), it is also important to investigate the numbers of potential IRIS burst profiles removed from each of these datasets before any additional analysis is conducted. In order to test this, we identified all pixels which were saturated (counts$>16180$) at one or more spectral positions in the wavelength range $1393.75\pm0.2$ \AA. In total $239$ pixels matched this criterion across all $42$ datasets with approximately $100$ of these appearing to be caused by cosmic ray spikes and another $100$ appearing to be some other kind of burst, with relatively narrow spectra and no absorption lines over-laid on the \ion{Si}{IV} $1394$ \AA\ spectra. This, therefore, left around $40$ potential IRIS burst profiles which were spread across four datasets. Around half of these saturated potential IRIS burst profiles were found in the first dataset sampling AR $11916$, which already contained $715$ IRIS burst profiles, whilst the others were detected in the seventh raster to sample AR $11916$, the first raster sampling AR $12104$, and the third raster sampling AR $12104$. Overall, the small numbers of saturated pixels displaying potential IRIS burst profiles (less than $1$ \%\ of the sample studied here) and the centralisation of these to only a small number of datasets indicates that the removal of saturated pixels does not overly influence the results obtained in this manuscript.

We also made use of data from SDO's Helioseismic and Magnetic Imager (SDO/HMI; \citealt{Scherrer12}) in order to study the line-of-sight (LOS) magnetic field co-spatial to the IRIS observations. Full-disk LOS magnetic field maps were downloaded with a cadence of one hour in the time-period spanning from six hours prior to the start time of the first IRIS raster to six hours after the end time of the last IRIS raster for each AR. A zoomed $180$\arcsec$\times180$\arcsec\ field-of-view (FOV) cut-out was then created co-temporal to the first IRIS scan centred on the pointing co-ordinates of that raster. All other SDO/HMI frames were then de-rotated to that time, before a co-spatial cut-out was constructed in order to create a consistent FOV through the entire time-period. Comparisons of these SDO/HMI maps with plage regions in the IRIS rasters indicated that these alignments were reasonable (estimated errors $<5$\arcsec). As we only study the overall behaviour of the magnetic field within the entire ARs and the general structure of the magnetic field co-spatial to the IRIS burst profiles through time, no further alignment between instruments was conducted. These SDO/HMI data have a post-reduction pixel scale of $0.6$\arcsec\ which is more than adequate for investigating the overall behaviour of these ARs.

\subsection{Modifications to the IRIS burst detection algorithm}
\label{Obs_mod}

To detect IRIS burst profiles, the algorithm of \citet{Kleint22} was used with slight modifications applied in order to both be able to study rasters that may have a small number of IRIS burst profiles and fix two minor bugs, which became apparent during this analysis. To tackle the first issue, the criterion of requiring at least 30 potential IRIS burst profiles applied by \citet{Kleint22} was dropped and any observation that had at least one candidate IRIS burst profile was analysed further. This increased the sample of IRIS burst profiles detected within these $42$ rasters from $2462$ to $3113$ (a $26.44$ \%\ increase). Following this adaption, the first bug, related to times when the IRIS orbit passes through the South Atlantic Anomaly (SAA) and hence returns noisy spectra, became apparent in the results. Specifically, the IRIS FITS SAA level2 keyword defaults to the value that is true for \textit{most} of the observation. Therefore, observations where more time was spent in the SAA than outside of the SAA (which is very rare) will have this keyword set to 1 at all times and were, thus, excluded in the first version of the pipeline. The first version of the pipeline correctly queried the level1 data, which includes the actual flag for SAA/no SAA and updated the corresponding header entry in the level2 data to 1 during times of SAA. However, it did not check if the header entry may already be set to 1 due to the convention of the level2 data for SAA dominant datasets, even though no SAA occurred at the specific time when the spectra were sampled. The new version of the pipeline does both checks.

The second bug is related to the fact that the \ion{Si}{iv} observing window width changed slightly with the evolution of the IRIS linelists, and between the different types of linelists. The original pipeline had a bug with extrapolations of spectra, leading to unintended values far away from the \ion{Si}{iv} lines. This did not affect strong bursts (since the signal is near the line center and the continuum is close to zero due to the normalization), which have always been classified properly, but it could mistakenly classify noisy burst spectra into a non-burst category. Extrapolated values are now properly set to zero and thus do not affect the classification. The classification in \citet{Kleint22} focused on minimizing false-positives and if in doubt, a spectrum was excluded from being of burst-type so this should not have influenced their results excessively. Applying the algorithm following these subsequent modifications increased the sample further from $3113$ IRIS burst profiles to a total of $4019$ IRIS burst profiles (a further increase of $29.10$ \%). These new IRIS burst profiles mostly have very weak absorption lines and/or are noisy, but do visually look like bursts so are included here for completeness. Notably, these events were detected across a total of $35$ datasets, up from $15$ using the algorithm before removal of the $30$ spectra criterion and the subsequent modifications.

\begin{figure*}
\center
\includegraphics[width=0.99\textwidth,trim={0 2cm 0 0}]{UV_fig3.eps}
\caption{Context plots for three of the ARs, namely ARs 11850, 11871, and 11916 from top to bottom, studied. (Left-hand column) Zoomed $100$\arcsec$\times100$\arcsec\ SDO/HMI magnetic context images. The plotted frames were selected as they contained the peak total unsigned magnetic flux for these regions during the time studied. (Middle column) The \ion{Si}{IV} $1394$ \AA\ line core intensity sampled from the IRIS raster with start time closest to the SDO/HMI images plotted in the left hand column. The pointing of the IRIS FOV was de-rotated to the SDO/HMI observation time. (Right-hand column) The evolution of the total unsigned (solid lines), positive (dashed lines), and negative (dotted lines) magnetic flux within the wider $180$\arcsec$\times180$\arcsec\ FOV through the time-period studied here for each AR. The red contours over-laid on the left-hand column outline the IRIS burst profiles returned using automated methods from the IRIS raster plotted in the respective middle column. The five ARs with the highest peak magnitudes of the unsigned flux were found to contain the majority of the IRIS burst profiles identified here.}
\label{Fig_context}
\end{figure*}

\begin{figure*}
\center
\includegraphics[width=0.99\textwidth]{UV_fig4.eps}
\caption{Structuring of IRIS burst profiles in these datasets. (Left-hand panel) Binary map showing the locations of IRIS burst profiles (black pixels) identified within a single raster sampling AR 11850. Four of the parent IRIS bursts within this raster correspond to the four events studied in \citet{Peter14}. The over-lying cut-out displays how connected objects are combined to form parent IRIS bursts, with nine events (all individually coloured) occurring in this small region. (Middle panel) Histogram of parent IRIS burst area against frequency constructed using all $42$ datasets, where the $y$-axis is plotted in logarithmic scale. Each bar is binned over a range of $0.3$ arcsec$^2$ (corresponding to five pixels). (Right-hand panel) Bar chart plotting the number of parent IRIS bursts by AR. The different coloured sections of each bar indicate the number of parent IRIS bursts identified in different rasters sampling that AR, with the light yellow at the bottom corresponding to the first raster within which IRIS bursts were detected using automated methods for that specific AR. The progressively darker colours up the bars correspond to the subsequent rasters within which IRIS bursts were detected. The physical number of bursts plotted in each bar segment can be found in Table~\ref{Tab_overview}.}
\label{Fig_stats}
\end{figure*}

It should be mentioned, however, that the variety of burst spectra makes it impossible to catch every single burst with a classification template of ``only'' 423 different spectral types. We are confident, though, that the most common types of burst spectra are identified reliably. An example of an IRIS burst profile which was not detected by the pipeline is displayed in Fig.~\ref{Fig_manual}. In the left-hand panel we plot the intensity of the \ion{Si}{IV} $1394$ \AA\ line core for the fourth raster sampling AR 12104 (i.e. the raster where IRIS bursts were detected manually but not automatically). The locations where IRIS passed through the SAA during this observation are marked by the red arrows. The orange circles outline two pixels, one containing a manually detected IRIS burst profile and one containing a non-burst profile, chosen for reference. In the right-hand panels, the black lines plot the observed spectra at each of these pixels with clear, albeit shallow, absorption blends indicative of an IRIS burst profile visible on the top panel. The blue lines plot the reference spectra deemed to be the closest mathematically to the observed spectra which in both cases is a non-burst reference spectra. The orange line included on the top panel plots the closest IRIS burst reference spectrum to this spectrum, which was mathematically `further' from the observed spectrum than the non-burst reference spectrum (distance of 0.83 compared to a distance of 0.52). The only way to improve detections of such bursts would be to increase the number of template reference spectra from the 423 currently used. Given we only found a very small number (around ten) of additional IRIS burst profiles through our manual checks, and adding further reference spectra risks increasing the rate of false detections, we did not deem this necessary for this study.

\section{Results and discussion}
\label{Results}

\subsection{General AR properties}

To begin our analysis, we investigated the general properties of each AR in order to better contextualise any subsequent inferences about IRIS bursts. Firstly, we identified the number of raster scans which displayed IRIS burst profiles for each AR finding varying behaviour. Of the seven ARs studied here, five (11850, 11909, 11916, 12104, and 12139) contained IRIS burst profiles in each raster scan, including both IRIS burst profiles detected using automated methods and those identified through the subsequent manual analysis. AR 11916 contained the most automatically detected IRIS burst profiles, with $2052$ ($51.06$ \%\ of the total sample studied here) being identified over the eight rasters analysed, whilst the other four ARs each contained between $1144$ and $168$ automatically detected IRIS burst profiles. On top of this, one AR (11856) was found to contain IRIS burst profiles in some but not all rasters and one AR (11871) was found to contain no IRIS burst profiles at any time. This analysis indicates that the age of the AR is not a defining factor in whether IRIS burst profiles are present with the two youngest ARs (11871 and 11916), which both emerged onto the solar disk in the three days prior to the IRIS scans, displaying sharply contrasting behaviour.

Secondly, we calculated the total unsigned magnetic flux within a zoomed $180$\arcsec$\times180$\arcsec\ FOV around each AR at each SDO/HMI time-step. Interestingly, the five ARs which contained IRIS burst profiles in each raster had the five highest peak fluxes ($1.44\times10^{21}$ Mx, $1.66\times10^{21}$ Mx, $1.91\times10^{21}$ Mx, $2.23\times10^{21}$ Mx, and $1.50\times10^{21}$ Mx for ARs 11850, 11909, 11916, 12104, and 12139, respectively) with each exceeding $10^{21}$ Mx. The remaining two ARs, which contained either a small number or zero IRIS burst profiles, both had peak fluxes of less than $10^{21}$ Mx ($8.60\times10^{20}$ Mx and $5.38\times10^{20}$ Mx for ARs 11856 and 11871, respectively). In the left-hand column of Fig.~\ref{Fig_context}, we plot $100$\arcsec$\times100$\arcsec\ zoomed SDO/HMI images sampled at the time at which the peak unsigned magnetic flux was measured for three ARs (11850, 11871, and 11916 from top to bottom, respectively). The middle column of Fig.~\ref{Fig_context} plots the IRIS \ion{Si}{IV} $1394$ \AA\ line core intensity across this FOV sampled at the closest raster to the SDO/HMI images, with the pointing corrected for solar rotation. The red contours over-laid on the SDO/HMI images outline the locations of the IRIS burst profiles identified in these rasters using automated methods. The right-hand column of Fig.~\ref{Fig_context} plots the evolution of the total unsigned magnetic flux (solid line), positive flux (dashed line), and negative flux (dotted line) calculated from the larger $180$\arcsec$\times180$\arcsec\ SDO/HMI FOV with time. Clearly, ARs 11916 and 11850 are more complex in their magnetic structuring (left-panels of Fig.~\ref{Fig_context}) and have total unsigned magnetic fluxes which initially increase before reaching a peak and then decreasing (right-panels of Fig.~\ref{Fig_context}). On the other hand, AR 11871 is less complex and has a total unsigned magnetic flux which slowly decreases with time. Our results indicate that the total unsigned magnetic flux and the complexity of the structuring of that flux within an AR may be a dominant factor determining whether IRIS bursts are widely detected or not, rather than the age of the AR.

\subsection{Spatial properties of IRIS burst profiles}

\begin{figure*}
\center
\includegraphics[width=0.99\textwidth]{UV_fig5.eps}
\caption{Spatial positions of IRIS burst profiles through time in two datasets. (Left-hand panel) Map of the \ion{Si}{IV} $1394$ \AA\ line core intensity (logarithmically scaled) within AR 11850 sampled during the raster starting at 05:59:43 UT on 26th September 2013. (Right-hand panel) Same as the left-hand panel but for AR 11916 sampled during the raster starting at 03:09:49 UT on 7th December 2013. The contours and crosses over-laid on both panels denote the IRIS burst profiles and their centres-of-mass, respectively, for each raster containing more than 100 automatically detected IRIS burst profiles. The colours indicate different rasters and are consistent with those used in the right-hand panel of Fig.~\ref{Fig_stats}.}
\label{Fig_loc}
\end{figure*}

We continued our analysis by grouping the $4019$ IRIS burst profiles identified here into separate, connected objects which we define as parent IRIS bursts. To achieve this, we created binary maps of the locations of IRIS burst profiles within each raster scan and then combined IRIS burst profiles which were adjacent in the $x-y$ plane together into parent IRIS bursts. If two parent IRIS bursts were separated by even one pixel, we considered that they were different events. Additionally, if the observational routine had more than one raster then only the middle raster was considered here in order to avoid longer-lived parent IRIS bursts, potentially present in repeated rasters, being counted multiple times. Importantly, our analysis of these parent UV bursts does not include any temporal component. This stipulation only removed $119$ IRIS burst profiles from our sample meaning it should not have a major influence on our results. Overall, this analysis returned $752$ parent IRIS bursts spread across the $35$ IRIS rasters. In the left-hand panel of Fig.~\ref{Fig_stats}, we plot an example of a binary map displaying the locations of numerous IRIS burst profiles from one specific raster within AR 11850. This plot is calculated from the dataset which was used by \citet{Peter14} to originally identify IRIS bursts, with the four largest parent IRIS bursts corresponding to the four events studied by those authors. This clearly shows the effectiveness of the algorithm developed by \citet{Kleint22} and modified here. The zoomed cut-out over-laid on this panel more clearly displays how adjacent IRIS burst profiles are combined to form separate parent IRIS bursts. Nine parent IRIS bursts are present within this small FOV, with each event being plotted in a different colour.

In the middle panel of Fig.~\ref{Fig_stats}, we plot a histogram of parent IRIS burst area against frequency for all of the $752$ identified events. The histogram is binned across $0.3$ arcsec$^2$ (five pixels) and the $y$-axis is plotted using a logarithmic scale. Clearly, the majority of the studied events are small, with $383$ of these parent IRIS bursts having areas of only one pixel. The mean area of this sample of parent IRIS bursts is $0.31$ arcsec$^2$ (corresponding to 5.19 pixels); however, the distribution is non-Gaussian and the standard deviation of this sample is $0.72$ arcsec$^2$ (12.18 pixels), more than double the mean, suggesting one should not interpret this as a precise estimate. Our results do, though, clearly indicate that the majority of parent IRIS bursts are extremely small-scale. Notably, several larger brightenings in the \ion{Si}{IV} $1394$ \AA\ line core are found to contain numerous parent IRIS bursts through our analysis. For example, one $6.30$\arcsec$\times2.72$\arcsec\ region in the second raster sampling AR 12104 (around $x_\mathrm{c}\approx200$\arcsec, $y_\mathrm{c}\approx-296$\arcsec) was found to contain $20$ parent IRIS bursts, with an average area of $0.12$ arcsec$^2$ (corresponding to 1.57 pixels). From the entire sample studied here, only 52 parent IRIS bursts (6.91 \%) have an area over $1$ arcsec$^2$ (corresponding to 17 pixels) with 28 of these occurring in AR 11916. The largest parent IRIS burst occurred in AR 11916 and had an area of $8.98$ arcsec$^2$ (corresponding to $151$ pixels), much larger than the typical sizes of UV bursts in general (see \citealt{Young18}).

\begin{figure*}
\center
\includegraphics[width=0.99\textwidth]{UV_fig6.eps}
\caption{Distributions of various spectral parameters for all IRIS burst profiles identified within datasets sampling ARs 11850 (left-hand panels) and 11916 (right-hand panels). Only datasets which returned more than $100$ IRIS burst profiles were considered. (Top panels) Logarithmic mean intensity from a $\pm0.075$ \AA\ window surrounding the \ion{Si}{IV} $1394$ \AA\ line core calculated for each IRIS burst profile. (Middle panels) Logarithmic maximum intensity for each IRIS burst profile. (Bottom panels) Spectral location of the maximum intensity of each IRIS burst profile. The diamonds indicate the mean for each raster, while the upper and lower crosses indicate the $75$th and $25$th percentiles, respectively. No systematic or monotonic changes in these parameters through time are apparent.}
\label{Fig_spec}
\end{figure*}

In the right-hand panel of Fig.~\ref{Fig_stats}, we plot a bar chart displaying the number of parent IRIS bursts identified per AR. The colours in each bar indicate parent IRIS bursts identified in different rasters, with the bottom (light yellow) portion corresponding to the first raster within which IRIS burst profiles were identified for that specific AR. A high proportion of the parent IRIS bursts were found within ARs 11850 and 11916 ($70.61$ \% of the total sample) whilst AR 11871 contains zero parent IRIS bursts. The highest number of parent IRIS bursts in a single raster was 65, which were identified in the first raster sampling AR 11916, with the mean number of parent IRIS bursts per raster being 17.90. The average area of the parent IRIS bursts was found to change in a seemingly random way from raster to raster (neither monotonically increasing or decreasing) as the host ARs evolved. In total, ten rasters within the sample analysed here contained more than $30$ parent IRIS bursts, with five of these occurring in AR 11916, three occurring in AR $11850$, and one occurring in both ARs $12104$ and $12139$. Overall, these results imply a parent IRIS burst spatial frequency of $0.00069$ arcsec$^{-2}$ in ARs, well below the spatial frequency of, for example, spicules in the chromosphere. See Table~\ref{Tab_overview} for a summary of these findings.

\subsection{Locations of IRIS burst profiles within ARs}

One of the main aims of this research is to identify whether IRIS burst profiles are detected randomly across ARs or whether they are confined to consistent localised regions through time. In order to analyse this, we studied the locations of IRIS burst profiles within the two ARs (namely ARs 11850 and 11916) for which more than 100 IRIS burst profiles were detected using the algorithm in more than one raster. We de-rotated each raster to a consistent pointing which we converted as the origin (0\arcsec, 0\arcsec), constructed binary maps of the locations of IRIS burst profiles, and then calculated the centres-of-mass of IRIS burst profiles for each raster. In Fig.~\ref{Fig_loc}, we plot a visualisation of this analysis on maps of the \ion{Si}{IV} $1394$ \AA\ line core intensity (logarithmically scaled) from one representative raster for each AR (11850 in the left-hand panel and 11916 in the right-hand panel). The over-laid contours denote the locations of IRIS burst profiles and the crosses indicate the centres-of-mass for each specific raster. The relationship between colour and raster number is consistent with the colours used in the right-hand panel of Fig.~\ref{Fig_stats}.

\begin{figure*}
\center
\includegraphics[width=0.96\textwidth,trim={0.7cm 4.0cm 0 0}]{UV_fig7.eps}
\caption{LOS magnetic field maps co-spatial to the $46$ parent IRIS bursts with measured areas between $1$ arcsec$^2$ and $4$ arcsec$^2$. Each plot covers a region of $5$\arcsec$\times5$\arcsec\ centred on the location of the parent IRIS burst from the closest SDO/HMI image in time, de-rotated to the time of IRIS burst profile detection. The red and blue contours outline regions of positive and negative magnetic fields saturated at $20$ G and $-20$ G, respectively, whilst the orange contours outline the approximate locations of the parent IRIS bursts. Opposite polarity magnetic fields are found within the FOV for $43$ ($93$ \%) of these parent IRIS bursts.}
\label{Fig_mag}
\end{figure*}

For both ARs, IRIS burst profiles were found to be predominantly localised to relatively small regions, with approximate areas of $100$\arcsec$\times60$\arcsec. The centres-of-mass of IRIS burst profiles also appeared to be consistent through time for both ARs, only varying by a maximum of $31.11$\arcsec\ in AR 11850 and $37.96$\arcsec\ in AR 11916. In general, the locations of IRIS burst profiles within the ARs appeared to be limited to the most pronounced polarity inversion lines, as would be expected for events hypothesised to be driven by magnetic reconnection. Interestingly, IRIS burst profiles in both of these ARs appeared to become more spatially separated through time, with the contours in Fig.~\ref{Fig_loc} appearing further away from the centres-of-mass for subsequent rasters. In order to investigate this, we calculated the average separation between the IRIS burst profiles and the centres-of-mass through time for the rasters containing more than $100$ IRIS burst profiles. For AR 11850, this separation increased from $13.00$\arcsec\ ($\pm10.42$\arcsec) to $45.46$\arcsec\ ($\pm9.20$\arcsec) over the five rasters studied here; whilst for AR 11916, the average separation increased from $9.51$\arcsec\ ($\pm10.32$\arcsec) to $32.10$\arcsec\ ($\pm15.07$\arcsec) over the first five rasters before decreasing again to $18.63$\arcsec\ ($\pm15.04$\arcsec) by the sixth raster. Although this analysis indicates there may be an increase in the separation of IRIS burst profiles within an AR as it evolves, we note that two ARs were studied and large standard deviations were calculated, meaning this result is still tentative and requires further analysis using a larger sample of ARs in the future.

\subsection{Spectral properties of IRIS burst profiles}

To continue our analysis, we next investigated whether several spectral properties of IRIS burst profiles varied systematically during the evolution of ARs or whether they remained relatively consistent. Once again, only rasters within ARs $11850$ (left-hand panels) and $11916$ (right-hand panels) which contained more than $100$ IRIS burst profiles were studied in order to provide appropriate statistics. In Fig.~\ref{Fig_spec}, we plot the distributions of three parameters calculated from the \ion{Si}{IV} $1394$ \AA\ spectra sampled at all pixels automatically identified as IRIS burst profiles, for the appropriate rasters. The top panels plot the mean intensity (logarithmically scaled) from a $\pm0.075$ \AA\ window around the \ion{Si}{IV} $1394$ \AA\ line core, the middle panels plot the logarithmically scaled peak intensity of the \ion{Si}{IV} $1394$ \AA\ line for the identified IRIS burst profiles, whilst the bottom panels plot the spectral positions where the maximum intensity was measured for each IRIS burst profile. The diamonds on each distribution indicate the mean value calculated for that raster, and the top and bottom crosses plot the $75$th and $25$th percentile values, respectively. Overall, although some changes in the distributions are observed between specific rasters, no systematic changes (e.g. monotonic intensity increases, transitions from blue to red shifts) in these parameters is detected as the ARs age.

Following on from this, we also studied which reference spectra were returned the most frequently as the `closest' fits to the IRIS burst profiles identified in each of these rasters. This analysis allowed us to infer whether any changes in the common spectral shapes of IRIS burst profiles are present through time. In these datasets, no single reference spectrum was found to be the most accurate mathematical match to more than $7.34$ \%\ of all of the IRIS burst profiles within a single raster. Additionally, unique reference spectra were found to make up, on average across these datasets, $47.57$ \%\ of all of the observed spectra indicating that a huge variety of spectral shapes of IRIS burst profiles exist for each dataset. These two combined facts suggest that no preferential reference spectrum is present in any given raster, with the identified IRIS burst profiles being randomly spread between different reference spectra.

\subsection{Relations to the local LOS magnetic field}

Our final analysis focused on the relationship between the parent IRIS bursts identified in this study and the co-spatial under-lying LOS magnetic field structure as inferred by the SDO/HMI instrument. It is well known that IRIS bursts often occur co-spatial to regions of opposite polarity magnetic field (see, for example, \citealt{Young18}) where magnetic reconnection is thought to be possible in the lower solar atmosphere. As the SDO/HMI instrument has a spatial resolution of approximately $1$\arcsec, we only studied parent IRIS bursts with areas greater than $1$ arcsec$^2$ (corresponding to approximately 17 pixels). Additionally, we removed the parent IRIS bursts with areas above $4$ arcsec$^2$ (corresponding to 70 pixels) as the foot-points of those events would be difficult to define in a consistent manner. These stipulations left $46$ parent IRIS bursts for analysis. In Fig.~\ref{Fig_mag}, we plot the LOS magnetic field saturated at $\pm100$ G sampled for a $5$\arcsec$\times5$\arcsec\ FOV surrounding the centre of each parent IRIS burst. The red and blue contours outline positive and negative polarity LOS magnetic fields at values of $20$ G and $-20$ G, respectively, while the orange contours outline the locations of the parent IRIS bursts. The magnetic field map for each event was constructed by de-rotating the closest temporal frame in the respective one hour cadence SDO/HMI dataset to the time at which the parent IRIS burst was detected, meaning the locations of the bursts with respect to the magnetic field are only an approximation. Although this method will have some inherent errors on small-scales, the alignment appears to work well for the full FOV (as can be seen in Fig.~\ref{Fig_context}) giving us confidence that a closer alignment is not required for this general analysis.

Overall, $43$ ($93$ \%) of the panels plotted in Fig.~\ref{Fig_mag} contain bi-poles in the LOS magnetic field within the $5$\arcsec$\times5$\arcsec\ surrounding the parent IRIS bursts. For the three remaining events, one occurs co-spatial to an apparently uni-polar region of positive polarity flux and the other two occur co-spatial to apparently uni-polar regions of negative polarity flux. As the magnetic field maps could have been measured up to $\pm30$ minutes around the detection of the parent IRIS burst, we note that these magnetic field maps may not be entirely representative of the exact magnetic field structure at the time of potential reconnection as, for example, cancellation could have occurred removing evidence of any bi-poles (see for example, \citealt{Nelson16}) for the three uni-polar events. Additionally, the spatial resolution of the SDO/HMI instrument may be insufficient for resolving all of the magnetic field elements at these locations (as discussed by \citealt{Reid16}) which could cause us to detect a uni-polar region when a bi-polar region was actually present. It is clear, however, that the majority of parent IRIS bursts with areas between $1$ arcsec$^2$ and $4$ arcsec$^2$ occur co-spatial to regions of opposite polarity magnetic field in the lower solar atmosphere, supporting the assertion that these events could be driven by magnetic reconnection.

\section{Conclusions}
\label{Conclusions}

In this article, we used a slightly modified (see Sect.~\ref{Obs_mod}) version of the algorithm originally applied in \citet{Kleint22} in order to investigate whether changes in the measurable properties of IRIS burst profiles occurred as their host ARs evolved. We studied $42$ dense IRIS rasters which each sampled one of seven host ARs, with each AR being sampled at least four times. We found:
\begin{itemize}
\item{The maximum total unsigned magnetic flux within an AR, and the complexity of the structuring of that flux, may be important as an indicator of whether IRIS burst profiles are present. The two ARs with the lowest peak total unsigned magnetic fluxes contained far fewer IRIS burst profiles than the five ARs with the highest unsigned magnetic fluxes. Specifically, AR $11871$ (with a peak unsigned magnetic flux of $5.38\times10^{20}$) was found to contain no IRIS burst profiles in the four rasters which sampled it, whilst AR $11856$ (with a peak unsigned magnetic flux of $8.60\times10^{20}$ was found to contain only $25$ IRIS burst profiles ($0.62$ \%\ of the total number of IRIS burst profiles studied here) across the five rasters which sampled it. The age of the AR was not found to influence the number of IRIS burst profiles returned by the algorithm, with the two youngest ARs ($11916$ and $11871$) being found to contain the most and least IRIS burst profiles, respectively.}
\item{In total, $4019$ IRIS burst profiles were returned by the algorithm from these $42$ rasters, with these being found to belong to $752$ parent IRIS bursts. These parent IRIS bursts had a mean area of $0.31$ arcsec$^2$, with $93.09$ \%\ of the identified events having areas below $1$ arcsec$^2$ (centre panel of Fig.~\ref{Fig_stats}). We note that the mean area should not be over-interpreted due to the large standard deviation from this dataset ($0.72$ arcsec$^2$) but it can be used to show that these events are typically small-scale. The frequencies and areas of these features did not monotonically increase or decrease as the host ARs evolved but instead varied in a seemingly randomly manner from raster to raster. The largest parent IRIS burst found through this analysis had an apparent area of $8.98$ arcsec$^2$ (corresponding to $151$ pixels). The overall spatial frequency of IRIS burst profiles across the studied ARs was calculated to be $0.00069$ arcsec$^{-2}$, well below the coverage of spicules, for example.}
\item{IRIS burst profiles are typically confined to localised regions (around $100$\arcsec$\times60$\arcsec\ for the two ARs studied here) close to the cores of ARs (as is displayed in Fig.~\ref{Fig_loc}). The average spatial separation between the centre-of-mass of all IRIS burst profiles within a raster and the individual IRIS burst profiles themselves appeared to increase with time in ARs $11850$ and $11916$ (the only ARs which contained more than $100$ IRIS burst profiles in multiple rasters), suggesting IRIS bursts occurred across a larger area as their host ARs evolved. This increase in area potentially corresponds to the growth of the ARs themselves. These results suggest that one must be very careful when selecting appropriate targets for co-observations of IRIS burst-Ellerman bomb pairs with ground-based telescopes with limited FOVs, e.g. with the Daniel K. Inouye Solar Telescope (DKIST; \citealt{Rimmele20, Rast21}).}
\item{The spectral properties (e.g. peak intensities, spectral locations of the peak intensities) of IRIS burst profiles varied in a non-monotonic way in datasets sampling ARs $11850$ and $11916$ which contained more than $100$ IRIS burst profiles, and then only slightly, through time. On top of this, the automatically detected IRIS burst profiles were found to be spread across a large number of the reference spectra used by the detection algorithm with, on average, unique reference spectra equalling $47.57$ \%\ of the total number of IRIS burst profiles in any given raster. Additionally, no individual reference spectrum was found to be the best fit for more than $7.34$ \%\ of the IRIS burst profiles returned for any given raster. These combined result indicate that no `typical' IRIS burst spectra exists for the rasters studied here.}
\item{$93$ \%\ of all parent IRIS bursts with areas between $1$ arcsec$^2$ and $4$ arcsec$^2$ occurred co-spatial to bi-poles in the solar photosphere. This result further supports the assertion that these events are driven by magnetic reconnection in the lower solar atmosphere (\citealt{Peter14, Vissers15, Hansteen17}).}
\end{itemize}
Expanding this research to include a larger number of ARs (i.e. those sampled after 2014) will allow us to make further inferences about IRIS bursts in the future.

\begin{acknowledgements}
CJN is thankful to ESA for support as an ESA Research Fellow. LK gratefully acknowledges funding via a SNSF PRIMA grant. IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. SDO/HMI data provided courtesy of NASA/SDO and the HMI science team. This research has made use of NASA's Astrophysics Data System Bibliographic Services.
\end{acknowledgements}

\bibliographystyle{aa}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
The plane-wave treatment of neutrino oscillations, in which the propagating neutrino is assigned a definitive momentum, is an excellent approximation for terrestial- and atmospheric-based neutrino experiments. This framework breaks down, however, in the case that the neutrino wavepacket is finite (see, e.g. Refs.~\cite{Kiers:1995zj,PhysRevD.24.110,Giunti:1997wq,Giunti:2003ax}). The different velocities of the propagating mass eigenstates leads to their separation and changes the oscillation probability, with the effect increasing for larger travel distances, lower energies, and larger mass splittings. All observed neutrino oscillation signatures, with the possible exception of solar and supernova neutrinos which are expected to be completely decoherent, are so far consistent with coherent neutrino oscillations due to the relatively small size of the observable contributions compared to experimental resolutions. However, the increasing capability of neutrino experiments makes gaining sensitivity to the finite size of the wavepacket a reasonable future possibility.

Originating inside of a nuclear reactor and with contributions from the beta decays of some $\sim$1000 isotopes~\cite{Huber:2011wv}, the typical size of an electron antineutrino wavepacket at creation, denoted by $\sigma_x$, is at present unknown. However, there is potential for performing this detailed and complex calculation in the future, ala Ref.~\cite{Jones:2014sfa}. For now, the distance scales of the decay, including the characteristic beta-decay-nucleus size ($\sim$$10^{-5}$\,nm) and the inverse of the antineutrino energy ($\sim$$10^{-4}$\,nm)~\cite{deGouvea:2021uvg}, considered alongside the current reactor-based limits ($\sigma_x>2.1\cdot 10^{-4}$~nm at 90\%~CL using a phenomenological combination of Daya Bay, KamLAND, and RENO data~\cite{deGouvea:2021uvg}, and $\sigma_x>1\cdot 10^{-4}$\,nm at 95\% CL from a dedicated measurement with Daya Bay~\cite{DayaBay:2016ouy}), motivate a study of the effects of $\sigma_x$ on experimental observables for values as low as $10^{-4}$\,nm. We note, however, that the other relevant distance scale in the decay, the inter-atomic spacing of the Uranium-based fuel ($\sim$0.1-1\,nm), is well above the level at which wavepacket effects would be discernible in any realistic future reactor-based experiment.

Expecting first data in 2023, the ambitious JUNO project is at the forefront of what is possible with a reactor antineutrino experiment~\cite{JUNO:2015zny,JUNO_2022_physics_paper}. The experiment will feature a 20\,kton liquid scintillator far detector about 53\,km from a 26.6\,GWth total set of PWR reactor complexes. With 77.9\% photocoverage, JUNO can expect $\sim$$0.03/\sqrt{E_{\mathrm{vis}}~\mathrm{(MeV)}}$ positron visible energy resolution for characterizing the $\sim$100k IBD events ($\overline{\nu}_e p \rightarrow e^+ n$) expected in the data collection period (6~years). As has been shown in Refs.~\cite{deGouvea:2020hfl,JUNO:2021ydg}, the high statistics, long baseline, and strong energy resolution of JUNO translate to more than an order of magnitude better sensitivity to the wave packet effect as compared to previous reactor experiments, including Daya Bay~\cite{dayabay}, KamLAND~\cite{KamLAND:2008dgz}, and RENO~\cite{reno}.

The effect of a finite wavepacket size on the oscillation probability is easy to demonstrate by considering atmospheric-only electron-flavor disappearance (with normal mass ordering):
\begin{equation}
 P_{ee}\rightarrow 1- \frac{1}{2}\sin^2 2\theta_{13}\Big[1-\cos\frac{1.27\Delta m^2_{31}L}{E}\mathrm{exp}(-\frac{L^2(\Delta m^2_{31})^2}{32E^4\sigma^2_{x}})\Big] \label{simple_osc}
\end{equation}

We see that the dampening of the oscillation probability due to the wavepacket size ($\sigma_x$) increases with $L$, decreases with $E$, and increases with $\Delta m^2$. As such, the effect is larger for sterile ($\Delta m^2\sim 1~\mathrm{eV}^2$)~\cite{Arguelles:2022bvt} and atmospheric oscillations ($\Delta m^2\sim 3\cdot 10^{-3}~\mathrm{eV}^2$) as compared to solar oscillations ($\Delta m^2\sim 7\cdot 10^{-5}~\mathrm{eV}^2$). JUNO's large $L$ ($\sim$53\,km), low $E_{\overline{\nu}_e}$ ($\sim$2-10\,MeV), and sensitivity to the atmospheric mass splitting makes it an excellent testing ground for the effect of $\sigma_x$ on oscillation behavior. In general, one can expect JUNO to be more sensitive to wavepacket effects within the three neutrino oscillation paradigm~\cite{Arguelles:2022bvt}, at least, than any existing or even near-term-planned experiment.

In addition to the wavepacket effect, JUNO's extraordinary capability also makes it sensitive to the orientation of the neutrino mass hierarchy, among other oscillation parameters~\cite{JUNO:2015zny,JUNO_2022_physics_paper}. The three-neutrino oscillation probability in vacuum and without the wavepacket effect is given by $P_{ee}=1-P_{21}-P_{31}-P_{32}$, with
\begin{eqnarray}
P_{21}&& =\cos^4\theta_{13}\sin^2 2\theta_{12}\sin^2\Big(\frac{1.27\Delta m^2_{21}L}{E}\Big) \nonumber\\ \nonumber \\
P_{31}&& =\cos^2\theta_{12}\sin^2 2\theta_{13}\sin^2\Big(\frac{1.27\Delta m^2_{31}L}{E}\Big) \nonumber\\ \nonumber \\
P_{32}&& =\sin^2\theta_{12}\sin^2 2\theta_{13}\sin^2\Big(\frac{1.27\Delta m^2_{32}L}{E}\Big)~.
\label{Pee32}
\end{eqnarray}

The difference in oscillation probability between the normal hierarchy ($m_3>m_2>m_1$) and inverted hierarchy ($m_2>m_1>m_3$) for the JUNO experiment can be seen in Figure~\ref{rate_plot} in terms of IBD-based reconstructed antineutrino energy ($E_{\bar{\nu}_e}=E_{e^+}+0.78$\,MeV), with and without energy resolution smearing according to the expected positron energy resolution (see below).

\begin{figure*}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{osc_plot_wo_energy_res.pdf}
\includegraphics[width=0.49\textwidth]{osc_plot_w_energy_res.pdf}
\end{center}
\vspace{-.5cm}
\caption{The IBD event rate distribution shapes expected at JUNO, without (left) and with (right) energy smearing due to reconstruction resolution. The shape differences between the normal and inverted hierarchy are shown. In addition, the dampening effect of finite $\sigma_x$ ($10^{-12}$\,m) on the normal hierarchy oscillation probability can be seen.}
\label{rate_plot}
\end{figure*}

The discernible shape difference between the hierarchies is reduced with oscillation dampening due to a finite wavepacket size. The three-neutrino oscillation equation (Eq.~\ref{Pee32}) is modified to include wavepacket effects with
\begin{eqnarray}
P_{21}&& =\cos^4\theta_{13}\sin^2 2\theta_{12}\cdot \frac{1}{2}\Big[1-\cos\frac{1.27\Delta m^2_{21}L}{E}\cdot\nonumber\\&&\mathrm{exp}(-\frac{L^2(\Delta m^2_{21})^2}{32E^4\sigma^2_{x}})\Big]\nonumber \\
P_{31}&& =\cos^2\theta_{12}\sin^2 2\theta_{13}\cdot \frac{1}{2}\Big[1-\cos\frac{1.27\Delta m^2_{31}L}{E}\cdot\nonumber\\&&\mathrm{exp}(-\frac{L^2(\Delta m^2_{31})^2}{32E^4\sigma^2_{x}})\Big] \nonumber \\
P_{32}&& =\sin^2\theta_{12}\sin^2 2\theta_{13}\cdot \frac{1}{2}\Big[1-\cos\frac{1.27\Delta m^2_{32}L}{E}\cdot\nonumber\\&&\mathrm{exp}(-\frac{L^2(\Delta m^2_{32})^2}{32E^4\sigma^2_{x}})\Big] ~.
\label{Pee32_sigmax}
\end{eqnarray}

Figure~\ref{rate_plot} shows the dampening effect of $\sigma_x=10^{-12}$\,m, for example, on the oscillation probability in the case of a normal hierarchy. As can be seen, the fast-atmospheric oscillation probability, with frequency governed by $\Delta m^2_{3l}$~\footnote{We adopt the NuFIT notation here: $l=1$ corresponds to $\Delta m^2_{3l}>0$ (normal hierarchy) and $l=2$ corresponds to $\Delta m^2_{3l}<0$ (inverted hierarchy)}, is washed out and the power to distinguish the hierarchies is significantly reduced, especially at lower antineutrino energies where the effect is more pronounced. Fortunately, however, while the amplitude of oscillation decreases, the oscillation frequency and phase is left unchanged by the wavepacket size. Therefore, while the finite size of the wavepacket can lead to diminished, or even nullified (in the case that the fast atmospheric oscillations completely disappear), sensitivity to a hierarchy determination, it is unlikely to lead to a confused or systematically incorrect determination.

Previous work has considered the effect of the wavepacket size on hierarchy determination in generic medium baseline reactor experiments~\cite{Chan:2015mca}, and JUNO's sensitivity to the size of the wavepacket with a known hierarchy~\cite{deGouvea:2020hfl,JUNO:2021ydg}. Below, we report the JUNO experiment's ability to determine the hierarchy in the presence of a finite wavepacket, with an emphasis on \textit{experimentally allowed} values of this parameter, again noting that there is no reliable theoretical prediction, and consider JUNO's sensitivity to both the wavepacket size and the orientation of the hierarchy at the same time. The experimental assumptions and analysis methods are detailed in the next section, followed by a section presenting the results and discussion, and then conclusions.

\section{Analysis}
\label{analysis}

To model the JUNO experiment, we use the reactor electron antineutrino spectrum
as described in Ref.~\cite{Vogel:1989iv} with a PWR fuel mixture consistent with Ref.~\cite{JUNO:2015zny} ($\mathrm{^{235}U}:\mathrm{^{238}U}:\mathrm{^{239}Pu}:\mathrm{^{241}Pu}=0.577:0.076:0.295:0.052$). Nine relevant PWR nuclear reactor complexes create the antineutrino flux, and their powers and distances from the JUNO detector are modeled according to Ref.~\cite{JUNO_2022_physics_paper}. The largest contributions come from the Yangjiang (61.5\% of total flux) and Taishan (32.1\%) Nuclear Power Plant complexes at around a 53~km baseline.  Uncertainties from the reactor flux shape are expected to contribute at the sub-1\% level with the inclusion of data from a near detector called ``JUNO-TAO"~\cite{JUNO:2020ijm}. Construction of JUNO-TAO is ongoing and first data is expected in 2022. The existence of this near detector in constraining the flux shape uncertainty is implicitly assumed throughout this analysis. At the sub-1\% level, the reactor flux shape uncertainties are not expected to significantly affect either the hierarchy determination or sensitivity to $\sigma_x$. Similarly, the reactor flux normalization uncertainty contribution to the sensitivity can be considered negligible~\cite{JUNO:2015zny}. We ignore both sources of flux uncertainty in this analysis.

After creation, the electron antineutrinos are propagated from each reactor, appropriately weighted by distance and power, to the JUNO far detector while applying the three-neutrino oscillation equation, including the different baselines (Eq.~\ref{Pee32_sigmax}). The central-value three-neutrino oscillation parameters and uncertainties are taken from NuFIT\,5.0, including Super-K atmospheric data~\cite{Esteban:2020cvm}. Correlations among the parameters are ignored when sampling from the allowed ranges. Also, while terrestrial matter effects provide corrections to $\sin^2 \theta_{12}$ and $\Delta m^2_{21}$ at the 0.5-1.0\% level for these distances and energies, they are negligible for $\sin^2 \theta_{13}$ and $\Delta m^2_{3l}$, and the associated hierarchy and $\sigma_x$ discussion, and are ignored here for simplicity.

In addition to the standard oscillations, $\sigma_x$ values from $10^{-13}$-$10^{-9}$\,m are considered as contributing to the effective oscillation probability. The lower end of this range roughly corresponds to the current experimental limit ($\sigma_x>2.1\cdot 10^{-4}$~nm at 90\% CL~\cite{deGouvea:2021uvg}) and the upper end roughly corresponds to the largest relevant distance scale of the antineutrino-producing beta decay(s), the upper end of the inter-atomic spacing of the Uranium-based fuel. We consider a single $\sigma_x$ value as affecting the oscillation probability of all $\overline{\nu}_e$ from all reactor complexes considered here noting that a, likely small, range of relevant $\sigma_x$ values will contribute in reality given the different environmental conditions affecting the size of the wavepacket in each reactor core and even individual isotope. For simulating electron antineutrino induced IBD events, we use the IBD cross-section from Ref.~\cite{Vogel:1999zy}. The resulting positron energy spectrum is smeared by the following energy resolution to simulate JUNO's energy reconstruction capability~\cite{JUNO_2022_physics_paper}:
\begin{equation}
\frac{\sigma_{E_{\mathrm{vis}}}}{E_{\mathrm{vis}}}=\sqrt{\Big(\frac{a}{\sqrt{E_{\mathrm{vis}}}}\Big)^2+b^2+\Big(\frac{c}{E_{\mathrm{vis}}}\Big)^2}~,
\label{resolution}
\end{equation}
with $a=2.61\%,~b=0.82\%,~c=1.23\%$ and $E_{\mathrm{vis}}$ in MeV. Notably, the non-linear response of the liquid scintillator due to the ``quenching effect" is not accounted for in this resolution estimate. The reader is referred to Refs.~\cite{JUNO_2022_physics_paper,DayaBay:2019fje} for more on this relevant issue, which is ignored here for simplicity.

\begin{table} [t]
\begin{center}
\begin{tabular}{|c|c|}
\hline
Experimental parameters &   \\ \hline
Total IBD events ($\sim$6 years) &  $10^5$ (post-oscillations)  \\
IBD cross section shape &  Vogel \& Beacom~\cite{Vogel:1999zy}  \\
$e^+$ visible energy resolution &See Eq.~\ref{resolution} \\
Near detector?  &  Yes, JUNO-TAO~\cite{JUNO:2020ijm}  \\  \hline
Reactor flux shape  &  Vogel \& Engel~\cite{Vogel:1989iv}  \\
Fuel composition  &    \\
$\mathrm{^{235}U}:\mathrm{^{238}U}:\mathrm{^{239}Pu}:\mathrm{^{241}Pu}$  &   $0.577:0.076:0.295:0.052$   \\
\hline
9 reactor complexes  &  [4.6, 52.77], [4.6, 52.64],    \\
{[}power (GWth), baseline (km){]} &  [2.9, 52.74], [2.9, 52.82],   \\
  &  [2.9, 52.41], [2.9, 52.49],   \\
    &  [2.9, 52.11], [2.9, 52.19],   \\
        &  [17.4, 215]   \\\hline \hline
Oscillation parameters (3$\nu$) &   \\ \hline
$\theta_{12}$ ($^\circ$)&  $33.44^{+0.77}_{-0.74}$  \\
$\theta_{23}$ ($^\circ$)&  $49.2^{+1.0}_{-1.3}$  \\
$\theta_{13}$ ($^\circ$)&  $8.57^{+0.13}_{-0.12}$  \\
$\Delta m^2_{21}$ ($\mathrm{eV}^2$)&  $(7.42^{+0.21}_{-0.20})\cdot 10^{-5}$  \\
$\Delta m^2_{31_\mathrm{NH}}$ ($\mathrm{eV}^2$)&  $(2.515\pm0.028)\cdot 10^{-3}$  \\
$\Delta m^2_{32_\mathrm{IH}}$ ($\mathrm{eV}^2$)&  $(-2.498^{+0.028}_{-0.029})\cdot 10^{-3}$  \\  \hline
\end{tabular} \caption{Summary of the relevant experimental parameter assumptions.}\label{table:values}
\end{center}
\end{table}

The representative JUNO dataset size is assumed to be 100k electron antineutrino IBD events after standard oscillations~\cite{JUNO_2022_physics_paper}, from 6~years of running, and we consider this sample in 200~equally-spaced reconstructed antineutrino energy bins from 1.8 to 8\,MeV. The statistical and shape uncertainties from all backgrounds (mainly originating from geo-neutrinos, fast neutrons, accidentals, and the decays of cosmogenic-induced isotopes) are expected to have a minimal impact on hierarchy sensitivity~\cite{JUNO:2015zny} and are ignored here. A summary of the experimental parameter assumptions is shown in Table~\ref{table:values}.

With the goal of differentiating one hierarchy from another, we follow Ref.~\cite{JUNO:2015zny} in forming a $\chi^2$ test statistic, comparing the simulated ``observed" events ($O$) in each reconstructed electron antineutrino energy bin ($i$) to a ``prediction" ($P$) under a particular oscillation+wavepacket scenario:
\begin{equation}
\chi^2=\sum^{200}_{i=1}  \frac{[O_i-P_i(1+\sum_k \alpha_{ik}\epsilon_{k})]^2}{P_i}+\sum_{k}\frac{\epsilon_{k}^2}{\delta_{k}^2}~.
\end{equation}

A ``fake data set" observed spectrum ($O_i$) is created with a particular hierarchy and $\sigma_x$ value, along with oscillation parameters sampled according to their central values and uncertainties shown in Table~\ref{table:values}, and the statistics expected in the experiment. The prediction ($P_i$) is varied to find the $\chi^2_{min}$, while pull terms on the oscillation parameters ($\epsilon_k$, with an $\alpha_{ik}$ fractional contribution to $P_i$) and associated uncertainties ($\delta_k$) (shown in Table~\ref{table:values}) are used to constrain the prediction in the minimization procedure, and the $\sigma_x$ parameter is left unconstrained.

Sensitivity to the mass hierarchy determination can be quantified by comparing the $\chi^2_{\mathrm{min}}$ with a normal hierarchy based prediction to the $\chi^2_{\mathrm{min}}$ with an inverted hierarchy based prediction:
\begin{equation}
\Delta\chi^{2}_{\mathrm{MH}}=\chi^2_{\mathrm{min}}(\mathrm{NH})-\chi^2_{\mathrm{min}}(\mathrm{IH})
\end{equation}

$\Delta\chi^{2}_{\mathrm{MH}}$ will strongly depend on whether the observed spectrum ($O_i$) was produced with an underlying normal hierarchy or an underlying inverted hierarchy. That is, the sensitivity to hierarchy determination can be dependent on the true underlying hierarchy. We use a subscript on $\Delta \chi^2$ to denote the underlying ``true" orientation used to produce the observed sample (i.e. either $\Delta \chi^2_{\mathrm{NH}}$ or $\Delta \chi^2_{\mathrm{IH}}$).

To summarize, we test JUNO's ability to resolve the mass hierarchy for an unknown
$\sigma_{x}$ by considering many different underlying ``true" $\sigma_{x}$ values with a known hierarchy. At each scanned true value of $\sigma_{x}$, a JUNO fake dataset ``observed spectrum" is produced. A fit is then done to that simulated dataset under both a normal hierarchy hypothesis and an inverted hierarchy hypothesis, with all neutrino mixing parameters allowed to vary as part of the fit according to Table~\ref{table:values}. In addition, a hypothesized $\sigma_{x}$ value is varied as part of the fit. The best-fit $\chi^{2}$ values for the normal-hierarchy fit and the inverted-hierarchy fit are then compared to produce a $\Delta \chi^{2}_{\mathrm{MH}}$ for finding the level of statistical separation JUNO will be able to achieve between the two hierarchies. This process is then repeated for each true $\sigma_{x}$ value many times to create multiple ``universes".
The distribution of results across these simulated universes characterizes JUNO's ability to measure
the mass hierachy and $\sigma_{x}$ accounting for both experimental uncertainty and uncertainty on the
mixing parameters.

\section{Results and Discussion} \label{results}
The $\Delta\chi^2_{\mathrm{NH}}$ and $\Delta\chi^2_{\mathrm{IH}}$ distributions of universes are shown with two example $\sigma_x$ values, $10^{-12}$\,m and $5\cdot10^{-10}$\,m, in Figure~\ref{figure_example_gaussians}. As can be seen, the distinction between the hierarchies is significantly degraded in the case of $\sigma_x=10^{-12}$\,m ($|<\Delta \chi^2_{\mathrm{NH}}>|\sim3.2$ and $|<\Delta \chi^2_{\mathrm{IH}}>|\sim3.4$). In contrast, for $\sigma_x=5\cdot10^{-10}$\,m, the wavepacket effect on oscillation probability is negligible, and therefore consistent with the plane-wave treatment, with $|<\Delta \chi^2_{\mathrm{NH}}>|\sim10$ and $|<\Delta \chi^2_{\mathrm{IH}}>|\sim12$. The results across a large range of $\sigma_x$ values are shown in Figure~\ref{figure_hierarchy} (left) in terms of both $\Delta\chi^2_{\mathrm{NH}}$ and $\Delta\chi^2_{\mathrm{IH}}$ as a function of true $\sigma_x$. The error bar on each point represents the RMS of the distribution of universes while the central value represents the mean.
For $\sigma_x>10^{-11}$\,m, the nominal hierarchy determination capability of JUNO is largely unaffected by the finite size of the wavepacket. However, the sensitivity to the hierarchy rapidly degrades for $\sigma_x<10^{-11}$\,m, until it disappears around $\sigma_x\sim10^{-12}$\,m.

\begin{figure*}[t]
\centering
    \centering
     \includegraphics[width=.49\textwidth]{mh_gaussians_small_sigma.pdf}
     \includegraphics[width=.49\textwidth]{mh_gaussians_big_sigma.pdf}

     \label{figure_small_big_gaussians}
 \caption{Distribution of $\Delta\chi^{2}_{\mathrm{NH}}$ (black) and $\Delta\chi^{2}_{\mathrm{IH}}$ (red) for two different values of $\sigma_{x}$, $10^{-12}$\,m~(left) and $5\cdot10^{-10}$\,m~(right). For JUNO, $\sigma_{x} = 5\cdot10^{-10}$\,m is effectively the same as plane-wave oscillations.}
 \label{figure_example_gaussians}
\end{figure*}

\begin{figure*}[t]
\begin{centering}
\includegraphics[width=.49\textwidth]{mh_vs_true_sigma_x.pdf}
\includegraphics[width=.49\textwidth]{mh_vs_sigma_x.pdf}
\vspace{0cm}
\caption{(Left) JUNO's ability to determine the neutrino mass hierarchy as a function of true $\sigma_x$ for the case of an underlying normal hierarchy (black; $\Delta\chi^{2}_{\mathrm{NH}}$) and an underlying inverted hierarchy (red; $\Delta\chi^{2}_{\mathrm{IH}}$). (Right) The same plot, but now showing JUNO's simultaneous ability to measure both $\sigma_x$ and determine the hierarchy as a function of true $\sigma_x$ value. For $\sigma_x$ values above about $4\cdot 10^{-12}$~m, only a lower limit can be obtained. The horizontal error bars represent the bounds of the 68\% most central fit results.}

\label{figure_hierarchy}
\end{centering}
\end{figure*}

As might be expected from Figure~\ref{rate_plot}, there is a range of true $\sigma_x$ values for which JUNO is sensitive to both $\sigma_x$ and the orientation of the hierarchy at the same time. Below this range, JUNO is \text{only} sensitive to $\sigma_x$, and above this range, JUNO is \text{only} sensitive to the hierarchy orientation. Figure~\ref{figure_hierarchy} (right) again shows the behavior of  $\Delta\chi^2_{\mathrm{NH}}$ and $\Delta\chi^2_{\mathrm{IH}}$ as a function of true $\sigma_x$, but with the $\sigma_x$ measurement resolution achievable displayed as well. JUNO can produce a two-sided constraint on $\sigma_x$ for $\sigma_x<3\cdot10^{-12}$\,m. This result is reasonably consistent with Refs.~\cite{deGouvea:2020hfl,JUNO:2021ydg}. Figure~\ref{figure_ninefive_confidence} shows the probability that JUNO will be able to resolve the neutrino mass hierarchy with 95\% confidence at different
$\sigma_{x}$ values for the cases of a true and inverted hierarchy. As can be seen, wavepacket effects detract from JUNO's hierarchy determination capability as far as nearly two orders of magnitude above the current experimental lower limit, up to about $\sigma_x=10^{-11}$\,m. Fig.~\ref{figure_sigmax_resolution} shows the 1$\sigma$ confidence
interval on $\sigma_{x}$ that JUNO is likely to produce at different
$\sigma_{x}$ values. At $\sigma_x=5\cdot10^{-13}$\,m, for example, the expected achievable measurement resolution is $\delta(\sigma_x)=1.7\cdot10^{-13}$\,m. These figures show that for $\sigma_{x}$ values between $1\cdot10^{-12}$\,m and $3\cdot 10^{-12}$\,m JUNO will likely be able to resolve the neutrino mass hierarchy at 95\% confidence and produce a measurement of $\sigma_{x}$.

\begin{figure}[h]
\begin{centering}
\includegraphics[width=9.3cm]{mh_ninetyfive_confidence_plot.pdf}
\vspace{-.6cm}
\caption{The probability that JUNO will reject the incorrect hierarchy
with at least 95\% confidence as a function of $\sigma_{x}$ for a true normal hierarchy (black) and a true inverted hierarchy (red).}
\label{figure_ninefive_confidence}
\end{centering}
\end{figure}

\begin{figure}[t]
\begin{centering}
\includegraphics[width=9.3cm]{sigma_x_resolution.pdf}
\vspace{-.6cm}
\caption{The 1$\sigma$ measurement interval JUNO will be able to place on $\sigma_{x}$ as a function of true $\sigma_{x}$.
For the points with error bars extending above $\sigma_{x}=1\cdot10^{-9}$~m, an achievable lower limit measurement is depicted.}
\label{figure_sigmax_resolution}
\end{centering}
\end{figure}

While sensitivity to the hierarchy can be reduced by finite $\sigma_x$, JUNO's expected leading measurements of both $\sin^2\theta_{12}$ and $\Delta m^2_{21}$, represented by the depth and energy of the IBD event rate dip around $E_{\overline{\nu}_e}=3$\,MeV in Figure~\ref{rate_plot}, respectively, are not significantly changed for any experimentally allowed value of the parameter ($\sigma_x>2\cdot10^{-13}$\,m)~\cite{deGouvea:2021uvg} since, as previously discussed (see Eq.~\ref{simple_osc}), the slower, solar oscillations, with frequency governed by $\Delta m^2_{21}$, are less affected by $\sigma_{x}$.

Similar to $\Delta m^2_{3l}$, a measurement of $\theta_{13}$ in JUNO, represented by the amplitude of the small, atmospheric oscillations in Figure~\ref{rate_plot}, would be severely affected by $\sigma_x$. Fortunately, $\theta_{13}$ has been precisely reported by the Daya Bay experiment~\cite{DayaBay:2018yms} and, since JUNO cannot expect to improve upon this measurement, even in the absence of wavepacket effects~\cite{JUNO:2015zny,JUNO_2022_physics_paper}, it will rely on this external result. Notably, however, JUNO's sensitivity to both $\sigma_x$ and the hierarchy, considered individually or together, can be greatly improved with additional, future constraints on $\Delta m^2_{3l}$ from long-baseline accelerator and atmospheric experiments~\cite{T2K:2011qtm,NOvA:2021nfi,DUNE:2020jqi,Super-Kamiokande:2019gzr,Hyper-Kamiokande:2018ofw,IceCube:2016zyt}. These experiments, which primarily rely on pion decay-in-flight neutrinos traveling 100s-10000s of km, are insensitive to the wavepacket effect, in the sense that it won't alter their oscillation parameter measurements~\cite{Jones:2014sfa}. Figure~\ref{figure_better_precision} again shows $\Delta\chi^2_{\mathrm{NH}}$ and $\Delta\chi^2_{\mathrm{IH}}$ as a function of true $\sigma_x$, but with a 50\% reduction in the uncertainty on $\Delta m^2_{3l}$ compared to the current value(s) shown in Table~\ref{table:values} [$\delta(\Delta m^2_{3l})=(0.028\rightarrow0.014)\cdot 10^{-3}~\mathrm{eV}^2$], consistent with projections from a low-exposure DUNE measurement (300\,kton$\cdot$MW$\cdot$years)~\cite{DUNE:2020jqi}, at least, but also roughly representative of expected near-term global improvements informed by T2K~\cite{T2K:2011qtm}, NOvA~\cite{NOvA:2021nfi}, and IceCube~\cite{IceCube:2016zyt}.

Of course, almost any new neutrino physics relevant at these baselines and energies, such as (e.g.) non-standard neutrino interactions, neutrino decay, and/or oscillations involving a sterile flavor, seriously encumbers the use of long-baseline $\nu_\mu \rightarrow \nu_\mu$ and $\nu_\mu \rightarrow \nu_e$ results as inputs for performing medium-baseline reactor-based measurements of $\overline{\nu}_e \rightarrow \overline{\nu}_e$. Indeed, any deviation to the three-neutrino paradigm, or standard neutrino interaction paradigm insofar as it is used to extract oscillation measurements, will lead us to revisit many existing results and future sensitivities. While a $\sigma_x$ measurement and/or hierarchy determination may be confused in such a situation, JUNO's plethora of high precision results will certainly form an essential global input for either probing this new physics or overly constraining the three neutrino oscillation paradigm.

Aside from improved external measurements of $\Delta m^2_{3l}$ (and/or the orientation of the hierarchy itself), realistic future sensitivity enhancements to JUNO in terms of $\sigma_x$, the hierarchy, or $\sigma_x$ and the hierarchy considered simultaneously, other than ``more data" given the statistics-limited measurement, are somewhat difficult to envision. One could perhaps consider a smaller spread in reactor baselines, far-detector upgrades, and/or a reliable theoretical prediction for $\sigma_x$, as potential sources of sensitivity improvements. The spread in source-to-detector distances leads to oscillated spectrum smearing, with the largest bin migration coming from the currently operating 17.4~GWth Daya Bay reactor complex at 215~km (6.4\% of the expected IBD events~\cite{JUNO_2022_physics_paper}). Although it is hard to imagine that this complex stops producing power for any significant length of time, which would enhance JUNO's sensitivity, perhaps the JUNO experiment will start taking data before the other long(er)-baseline complex at Huizhou (17.4~GWth at 265~km) comes online in 2025. Improvements to the JUNO energy resolution, primarily driven by photon statistics, would strengthen sensitivity, but JUNO is already at the cutting edge of what is possible with a multi-kiloton-scale detector (77.9\% photocoverage, with 17612 20'' PMTs @ 34\% quantum efficiency~\cite{JUNO:2015zny,JUNO_2022_physics_paper}). A reliable theoretical prediction for $\sigma_x$ could also offer improved experimental sensitivity to the value and the orientation of the hierarchy, but only in and near the ``transition region" presented above ($\sim10^{-12}<\sigma_x<10^{-11}$\,m), in which the experiment has some sensitivity to both. Even in the case that $\sigma_x$ is perfectly predicted, sensitivity to the hierarchy cannot be achieved by JUNO significantly below this range.

In terms of JUNO ``upgrades" towards improved sensitivity, it is worth emphasizing the need for a near detector, which is assumed for this analysis, especially in consideration of the possible reactor flux substructure and uncertainties associated with the $5<E_{\bar{\nu}_e}<7$~MeV ``reactor bump"~\cite{prospect2,stereo,NEOS:2016wee,reno,dayabay,doublechooz}. Measurements from the near detector JUNO-TAO (2.8\,tons, 30\,m from a 4.6\,GWth reactor core of the Taishan Nuclear Power Plant; 2000~IBD events per day; sub-1\% energy resolution) are expected to constrain the reactor flux bin-to-bin shape uncertainties at the sub-1\% level and thereby improve hierarchy sensitivity by $\Delta \chi^2\sim1.5$ compared to without~\cite{JUNO:2020ijm}. In the absence of this near detector, the reactor flux shape uncertainties are expected at the 2\% bin-to-bin level using the Daya Bay reference spectrum, noting Daya Bay's energy resolution of $0.08/\sqrt{E_{\mathrm{vis}}~\mathrm{(MeV)}}$~\cite{JUNO:2020ijm}. However, even with a precision near detector (proximal to a \textit{single} core), uncertainties in the bin-to-bin migration due to the different fission fractions among the many contributing reactor cores, and associated time dependence of these fractions, may be significant.
Indeed, reactor flux predictions and their uncertainties are still evolving substantially in time (for a recent example, see Ref.~\cite{Kopeikin:2021ugh}), with the reliant relevant oscillation parameter sensitivities and measurements, including in short- and medium-baseline experiments, following suit (see, e.g., Refs.~\cite{Berryman:2020agd,Giunti:2021kab}). In general, these issues point to the continued need for theoretical and experimental work towards understanding reactor antineutrino production.

\begin{figure}[h]
\begin{centering}
\includegraphics[width=9.3cm]{improved_dm_mh_plot.pdf}
\vspace{-.6cm}
\caption{In the case that the worldwide precision on $\Delta m^2_{3l}$ improves by a factor of two [$\delta(\Delta m^2_{3l})=(0.028\rightarrow0.014)\cdot 10^{-3}~\mathrm{eV}^2$], JUNO's ability to determine the hierarchy as a function of true $\sigma_x$ value, for both the case of an underlying normal hierarchy (black) and an underlying inverted hierarchy (red).}
\label{figure_better_precision}
\end{centering}
\end{figure}

\section{Conclusion}
We have presented the JUNO sensitivity to both the orientation of the neutrino mass hierarchy in the presence of wavepacket effects and the size of the wavepacket itself. We find that wavepacket effects will detract from the hierarchy determination up to nearly two orders of magnitude above the current experimental lower limit, and that JUNO will be highly capable of precisely measuring the size of the wavepacket in large regions of currently allowed parameter space. In addition, external experimental measurements, in particular from long-baseline oscillations, will significantly enhance these physics capabilities of JUNO. Given the demonstrated importance of this non-exotic, ``standard" quantum mechanics effect on JUNO, and the experiment's expected contributions to the worldwide neutrino oscillation program, without even mentioning the relevance of the hierarchy determination to cosmological- and terrestrial-based probes of neutrino mass, we would like to encourage more study of the electron antineutrino wavepacket, in particular towards forming a reliable theoretical prediction of its characteristic size at creation inside a nuclear reactor.

\section{Acknowledgements}
We thank C.A.~Arg{\"u}elles and B.J.P. Jones for useful discussions. This work is supported by the Department of Energy, Office of Science, under Award No. DE-SC0007859.


Sea Levels rise when arctic ice melts.
You might not think this is the case, but - after watching Paul Beckwith on YouTube - I have no doubt this is so.
The melting ice sheets from Greenland & Antarctica and the feedback systems changing. "We need to declare a Climate Change Emergency" - quote from the video ... by Paul Beckwith - Climate System Scientist.
Last Messages, the 20.. Is Strange series, which provides video of catastrophic events worldwide in sequential time periods by month on a year by year basis. But the channel is not climate change specific but just a wake up call to all of us to take notice at the forces of nature and how insane and deadly they can actually be on all points of the globe.
Here is a link to Last Messages on YouTube, for you to take a look at.
Here is another link to Collapse Chronicles on YouTube, that talks about all kinds of things that are taking down this planet from greed, ignorance or plain stupidity. With many videos of interviews and introspection from a regular guy like the rest of us.
And if this doesn't keep you up at night, you are not paying attention.
There are other resources besides these but the ones I have listed here are good to look at. They are direct, informative and give a voice to inform others who might pass global abrupt climate change as a simple speculation. Look first, then think again.
One more link. How Do We Stop global dimming? Forbes online.
Yes, how anyone can think that climate change is not happening NOW, I do not know!


Using a circular warp makes it possible to make a piece twice as long as your frame - very convenient! It helps if the crossbars of your frame are thick, to keep the front and back layers separate.
Wrap the warp around and around the entire frame. Work the sprang only on the front layer - the half of the warp that is on the front of the frame. Push the linkings around to the back. They won't undo each other- a bit like magic! It helps to use something smooth like a knitting needle in each shed to slide it around to the back. It's a bit difficult to get the tension exactly right so that there aren't any little loops at the central join. Work until you run out of room, pushing the loops back around to the growing edges of the fabric. When done, you need to actually cut the warp and finish it somehow, unlike sprang done in one layer.
My large sprang frame has an adjustable top, so that I can change the size of a circular warp. This baldric was done on a circular warp. The embroidery covers the central join - I finally decided it was too much hassle to completely eliminate all the tensioning irregularities, and in a fabric that fine and tight, the loops were bothersome.
My large sprang frame was designed for use with both flat and circular sprang waps. It's free-standing, and about 4 feet tall. When using a flat warp, the two cords are adjusted to be the appropriate distance apart and the warp run around them. The third cord, hanging down in the photo, can be put thru the current shed and tied to the opposite bar if I need to stop before finishing the fabric.
For a circular warp, the upper crossbar can be slid down the posts to position it at the right height. The upper and lower crossbars are made from 2x4 lumber. The outer edges were rounded and smoothed so the warp would slide over them. There are two bolts, not shown, that tighten the upper crossbar to the sidepieces to hold it in place.


Each year Rush Soccer will hold Regional Centers for the U14/U15 age groups to continue to promote the Rush Way to both coaches and players around the country. At these Regional Centers, the top performers will be identified and asked to join Regional Select teams. Players and teams are invited in to train and play games over the three day period, all while being evaluated by the national Rush Soccer staff. These Centers are a great way for all Rush teams to start the season off with pre-season friendlies, learn more about the Rush Way, and get the opportunity to be identified for a higher level team. Join today in your Region!


text:
Willards man convicted of sex abuse in 1982 incidents
By Brian Shane;
SNOW HILL – A Willards man has been sentenced to three years imprisonment on charges that he molested his 7-year-old nephew 30 years ago.
David Wayne Tatman, 57, pleaded guilty to a third-degree sex offense and custodial child abuse, stemming from allegations occurring from October 1982 to May 1983, at his former residence on Cedar Hall Road in Pocomoke City. Prosecutors said the uncle made his nephew shower with him and inappropriately touched the child, and later forced the child to touch his uncle inappropriately in a bedroom.
"I felt like the abuse was my fault," the nephew, who is now 38 and makes his home in Colorado, said in Worcester County Circuit Court. "David Tatman stole my innocence and corrupted my understanding of myself. I've been steeped in emotional disarray."
The Daily Times does not identify victims of sexual abuse.
The nephew told Judge Richard R. Bloxom that his uncle's abuse has led to a lifetime of low self-esteem, drug abuse and suicidal thoughts. He kept the two incidents of abuse hidden from his immediate family all his life because he was feared no one would believe him. Now in therapy, the man has since been diagnosed with post-traumatic stress disorder.
When offered his chance to speak in court, Tatman made extensive remarks about his family, his work life and the incident. He turned and faced his nephew to offer an apology.
"I should have been the uncle he needed me to be. I am sorry for whatever happened, for what has happened," Tatman said.
Authorities became aware of the abuse after the victim placed a phone call to his uncle in September to speak about and verify the incident. The two had not spoken in more than 20 years, prosecutors said.
Tatman's attorney, Robert Fine, told the judge his client could have hung up the phone, or denied the allegations, but he chose not to and, instead, decided to face up to the accusations.
"In the years since," Fine said, "he's tried to atone. When confronted with it, he acknowledged it."
Tatman's ex-wife of 18 years said "there's never been anything inappropriate" that she knew of that had happened to her or their two children.
The judge sentenced Tatman to a 10-year sentence with all but three years suspended. In a plea deal, prosecutors dropped additional sex offense charges.
"You're not an evil person," the judge told Tatman, "but what you did 30 years ago was evil by any measure. Think about how it's affected (your nephew's) life, in lasting and serious ways."
Tatman will not be required to register as a sex offender because his crimes occurred before the 1995 establishment of the state's sex offender registry, according to prosecutor Diane Cuilhe.

redpajama_set_name: RedPajamaCommonCrawl


For the first time in a good week, the Nashville Predators come away with zero points at Bridgestone Arena with a 4-3 regulation loss to the Calgary Flames. Even though it nearly had the same late heroics as Tuesday's thriller, the boys couldn't get more magic in the waning seconds.
This would be a contest where the Predators would see two leads. They opened up the scoring at 13:45 into the first period off a back door tip in from Ryan Johansen. Nashville 1-0.
Those pesky Flames don't know how to go away and stay away. Matthew Tkachuk, on the power play where the Flames kept chipping away around Rinne, would nod the score 1-1 off a backhand shot 19:16 in the opening period.
The Preds would go up 2-1 9:30 into the second off a redirect from the stick of a not oft-goal scorer Yannick Weber. It would not take the Flames long to answer, as on yet another power play, Dougie Hamilton buried a shot to tie the game at two.
It was all Calgary in the third, two tough shots past Rinne, one back door from Mark Janikowski (3:24, 3rd) and one hard slap shot from Curtis Lazar (10:14, 3rd) put Calgary up 4-2. The attempt for late game heroics began on another late Nashville goal scored off a steal and a rebound by Nick Bonino at 12:19. Preds now trail 4-3.
In a situation that seemed like destiny, the Preds spent nearly the last 2:13 of regulation with a man advantage, which they later went 6-on-4 pulling Rinne. Kevin Fiala nearly knotted the game up at four, but the strong glove of Rittich barely kept the puck out upon further review.
It was a game that maybe third place Calgary needed more. The boys never say die spirit is still inspiring to us all.
Oh well. Wings on Saturday, y'all.


text:
Iodwasser ist die wässrige Lösung des Halogens Iod.

Dabei tritt folgende Gleichgewichtsreaktion auf:

Iod ist in Wasser fast unlöslich. In einem Liter Wasser lösen sich unter Standardbedingungen 0,29 g Iod. Dabei hängt der Iodgehalt vom pH-Wert ab. Die Hypoiodige Säure im Iodwasser dissoziiert in H+− Ionen und IO−-Hypoiodit-Ionen, welche als Desinfektionsmittel verwendet werden können.

Da Iod ein Radikalinhibitor ist, kann Iodwasser zur Unterbrechung einer laufenden radikalischen Substitution verwendet werden.

Zusammen mit Kaliumiodid ergibt sich die Lugolsche Lösung.

Siehe auch
 Bromwasser
 Chlorwasser (siehe unter Chlor)

Einzelnachweise

Chemische Lösung
Iod

redpajama_set_name: RedPajamaWikipedia


This vintage 25-unit apartment building has a mix of full-size one bedroom and two bedroom apartments. In the heart of the Wrigleyville Neighborhood and Southport Corridor, it is close to the Brown Line Elevated stops as well as bus transportation.
Apartments feature generous room sizes, hardwood floors, new baths and kitchens, as well as laundry and storage in the basement. There is also a spacious rear courtyard for summer use. Heat is included in the rent. One bedrooms rent for around $1,225-$1,375 and two bedrooms are around $1,475-$1,675 per month.


One of our new luxury ships will be docked in Split harbour, where you can board at any time during the day. Relax in your luxury cabin before attending the Welcome meeting from 1900hrs. The Cruise Director will introduce you to the Captain and crew and you will also meet your fellow passengers and enjoy your sumptuous Welcome Dinner.
Enjoy breakfast as we cruise towards Mljet Island. For a magical back to nature experience, join our optional excursion to the stunning Mljet National Park. Stroll around the small and large saltwater lakes that dominate the natural beauty spot, and take a boat trip to St. Mary Isle to explore the ancient Benedictine monastery, old abbey and 12th century church that are situated there. To navigate the vast 5400 hectares of the park, and to spot its endemic wildlife and beautiful plants, you can hire a scooter, bicycle or kayak. Enjoy free time exploring Mljet Island at your own pace, before returning to the ship. Captain's dinner tonight with live entertainment.


Press Release: The country has been facing a rise in the number of underweight and overweight Filipinos for more than 20 years. The World Health Organization (WHO) defined this phenomenon as the "Double Burden of Malnutrition" or the "prevalence of both undernutrition and overweight in the same community, nation, or region." To address this problem,… Continue reading Do you fall into #MalnutritionTraps?
If you think Anda is just a long stretch of white sandy beach, perfect for relaxation, romance, or finding inner peace, you're absolutely wrong. This piece of heavenly-made paradise is so much more than that – more than you can imagine. In doubt? Read on.
Dining at Quinale Beach Bar, you are not only guaranteed of festive food served by the bay. Quinale Beach Bar also hosts local acoustic bands to fill your evenings with music. From rythmn and blues to retro, to the latest top hits, an ordinary breezy night at this well-maintained public beach becomes extraordinarily romantic.
Like most Boholanos, people from Anda are welcoming. A simple gesture of hanging a ribbon around a visitor's neck proves their hospitality. Their smiles give warmth and engaging in conversations with them will make you feel at home, like you've known each other for a long time.
They say, we cannot talk when our heart is full. So when I'm silent, I must have a lot of things to say. I just don't know where to start, like words are trapped behind bars.


WhiteMag is an elegant looking premium blogger template was designed to work with every device (Desktops, Tablets and Mobiles) and every modern web browsers. Check it using Google's mobile friendly testing tool. This theme is very user friendly. WhiteMag blogger theme is compatible with "Blogger Template Designer". So background image and every color and every font of this theme can be changed within few seconds without having a knowledge about HTML, CSS and Blogger XML. WhiteMag theme contains search engine friendly structured data. You can check it using Google's structured data testing tool.
Here is how WhiteMag theme displays on different devices.
This theme is designed by NewBloggerThemes.com. You can see more details about this blogger template on their theme info page.