{"text":"\\section{Introduction}\\label{sec:intro}\nThe majority of the 2922 known radio pulsars\\footnote{Pulsar catalogue v1.67, \\citet{manchesterATNFPulsarCatalogue2005}} have been discovered through searches for periodic emission, and the majority of these through incoherent Fourier-based methods~\\citep{lk12}. Although pulsars rotate in a reliable manner~\\citep{hobbs+2012} they do not emit identical pulses each rotation and there is a large variety to the pulse amplitude distributions observed~\\citep{burke-spolaorHighTimeResolution2011}. Rotating Radio Transient (RRAT) is the name given to any pulsar with a pulse amplitude distribution such that it is more significantly, or only, detectable through searches for individual bright pulses, as opposed to via periodicity-based searches~\\citep{mclaughlinTransientRadioBursts2006a,keaneRotatingRadioTransients2011b}. Fourier-based searches select against long-period pulsars; if erratic emission in pulsars increases with age, then that would be an \\textit{additional} compounding selection effect against long-period pulsars. Also, as real-world telescope data has a red noise characteristic~\\citep{keane+2018}, this further compounds the selection effect against pulsars with periods longer than $\\sim1$~s.\n\nUnderstanding the evolution of pulsars, and the conditions of how they emit electromagnetic radiation throughout their lives, is a complex but fundamental problem of astrophysics. Much effort has rightly been dedicated to this problem~\\citep{ridleyIsolatedPulsarSpin2010,vranesevicPulsarCurrentRevisited2011,johnstonPulsarBrakingPdotP2017} and work is ongoing, due to the large parameter space needed to describe the population. But in order to gain a complete understanding of the pulsar population, and its evolution, selection effects in the observed population must be understood and appropriately accounted for. To this end large-scale blind surveys across a broad spectral range are needed; these can counteract spectral selection effects which can be co-variant with other parameters~\\citep{bates+2013}. Further, a wide range of search techniques must be applied to these data with single pulse searches~\\citep{cm03}, and fast folding algorithms~\\citep{morelloOptimalPeriodicitySearching2020b} acting as vital ingredients in these efforts. In recent years this approach has been in action in many large-scale surveys undertaken with telescopes around the world~\\citep{sanidasLOFARTiedArrayAllSky2019b,morelloSUrveyPulsarsExtragalactic2020a}. \n\nMany of these efforts have resulted in the detection of new transient\/variable sources such as fast radio bursts (FRBs, \\citealt{Lorimer2018}), nulling pulsars and other neutron stars with extreme properties~\\citep{Ng2020}, a subsection of which fall into the definition of a RRAT. While the number of RRATs has been increasing in recent years, there has been a dearth of follow-up observations. Such follow-up timing is essential to properly characterise these sources, but due to the instantaneous sensitivity and observing time required to see sufficient individual pulses, only certain over-subscribed telescopes have been considered suitable. Consequently, as of early 2023, the vast majority of RRATs in the ATNF pulsar catalogue~\\citep{manchesterATNFPulsarCatalogue2005} lack a precise rotation period, and only a third have a measurement of the first derivative of the period. Given these sources are outliers to the general pulsar population, both in detectability and often in their underlying properties, the lack of characterisation for the majority of these neutrons stars may be severely hampering efforts to understand the true pulsar population and its evolutionary track(s). Therefore it might be expected that characterising RRATs would have a disproportionately large impact on disentangling of the pulsar evolution puzzle.\n\nIn this paper we present a systematic census of $113$ Northern Hemisphere RRATs using the Irish LOFAR station~\\citep[I--LOFAR;][]{murphyFirstResultsREALtime2021}, a component of the International LOFAR Telescope~\\citep[ILT;][]{haarlemLOFARLOwFrequencyARray2013a} with the aims of (i) determining if the sources can be profitably observed with a single international LOFAR station; and if so, (ii) to accurately quantify and characterise the nature of those sources that can be studied in this way through low-frequency observations and precision pulsar timing analyses. In~\\S\\ref{sec:sources} the source catalogues and selection constraints are described, while the observation and processing methodologies are described in~\\S\\ref{sec:obsback}. In~\\S\\ref{sec:results} the characteristics of the $29$ detected sources at \\SI{150}{\\mega\\hertz} at the sensitivity of an international LOFAR station, from \\SI{\\the\\numexpr495+\\detected*5\\relax}{\\hour} of initial census observations to determine the detectability of the sources and a further \\SI{\\the\\numexpr\\totalhours-\\surveyhours\\relax}{\\hour} follow-up observations, are presented. These are further discussed in~\\S\\ref{sec:discuss}. The work is then concluded, with observation plans described, in~\\S\\ref{sec:conclusions}.\n\n\\section{Source Selection}\\label{sec:sources}\n\nThere are a number of catalogues in the public domain which describe RRATs; each of these is detailed and updated heterogeneously. The sources chosen for this study come from three of these catalogues. (i) With a focus on RRATs detected in the Northern Hemisphere (at a central frequency of  \\SI{600}{\\mega\\hertz}), the catalogues provided by the CHIME\/FRB collaboration\\footnote{\\href{https:\/\/www.chime-frb.ca\/galactic}{https:\/\/www.chime-frb.ca\/galactic}} are referred to herein as the ``CHIME\/FRB catalogue''. These are discussed in detail in \\citealt{goodFirstDiscoveryNew2020} and \\citealt{dongCHIME2022} (herein \\citetalias{goodFirstDiscoveryNew2020} and \\citetalias{dongCHIME2022}); (ii) the catalogue of the Pushchino Radio Astronomy Observatory\\footnote{\\href{https:\/\/web.archive.org\/web\/20220708232514\/https:\/\/bsa-analytics.prao.ru\/en\/transients\/rrat\/}{https:\/\/bsa-analytics.prao.ru\/en\/transients\/rrat\/}} (at \\SI{111}{\\mega\\hertz}), is referred to herein as the ``PRAO catalogue''. This has most recently been updated in \\cite{samodurovDetectionStatisticsPulse2022}; and (iii) The West Virginia University RRATalog\\footnote{\\href{https:\/\/web.archive.org\/web\/20230210010446\/http:\/\/astro.phys.wvu.edu\/rratalog\/}{http:\/\/astro.phys.wvu.edu\/rratalog\/}} (at various sites and frequencies), an extensive catalogue of sources combined from the announcements from surveys using several more `traditional' dish telescopes that reported RRAT detections prior to 2017. Sources included in these catalogues as accessed on May 1st 2022 were considered for inclusion in the census.\n\nIt is worth noting that several sources in these catalogues have been classified differently following from further observations at different telescopes and observing frequencies (such as \\cite{luStudyThreeRotating2019a}) or independent re-discovery during surveys~\\citep[such as LOTAAS, described in][herein referred to as \\citetalias{michilliSinglepulseClassifierLOFAR2018} and \\citetalias{sanidasLOFARTiedArrayAllSky2019b}]{michilliSinglepulseClassifierLOFAR2018,sanidasLOFARTiedArrayAllSky2019b}. Given the `RRAT' label is an observation classification, and is not intrinsic to the source, it can initially be frequency or even telescope dependent due to the spectral and pulse-amplitude variability of pulsars. The extended family of reclassified pulsars were observed as a part of this census to determine if a single international LOFAR station has the sensitivity required to perform single-pulse detection and analysis of these extreme sources.\n\nSome constraints were placed on each catalogue to limit the number of sources included in the census, with a goal to maximise the scientific potential of the allocated observing time. However, some sources that did not meet the typical observing requirements, but which were present at a celestial longitude where there is a dearth of sources, were additionally observed in time that otherwise would not have been allocated on the telescope.\n\n\n\\subsection{The CHIME\/FRB Galactic Catalogue}\\label{sec:chime}\nThe CHIME\/FRB catalogue contains sources discovered with the Canadian Hydrogen Intensity Mapping Experiment \\citep{banduraCanadianHydrogenIntensity2014}, a zenith scanning array that it is focused on mapping hydrogen in the local universe. CHIME has additional back-ends for FRB and pulsar studies, which have helped it become a pivotal instrument in the recent developments in these fields \\citep{amiriCHIMEPulsarProject2021,collaborationFirstCHIMEFRB2021}. These back-ends have allowed it to quickly become one of the most productive telescopes for single-pulse detection and folded pulsar observations due to its sensitivity, field of view, large fractional bandwidth and high-cadence observing across the entire Northern Hemisphere.\n\nThis catalogue contained 38 sources at the time of this census, 4 of which were announced to be detected through periodic folding. All of these sources were observed as a part of this campaign.\n\nWhile the catalogue does not provide uncertainties on their measurements, they can be inferred from the results presented in \\citetalias{goodFirstDiscoveryNew2020}. Depending on the activity of the sources in this catalogue, their positional uncertainties are mostly on the order of minutes and arcminutes (for both timed and untimed sources, e.g. J1931+4229 did not have a known rotation period and has a similar uncertainty on its parameters to the other sources presented), while 2 of the periodic sources have been localised to sub-arcsecond positions. The dispersion measures (DMs) presented in the catalogue have uncertainties on the order of \\SIrange{0.1}{1}{\\parsec\\per\\cubic\\centi\\metre}, reduced from the underlying $2^i\\times$\\SI{1.62}{\\parsec\\per\\cubic\\centi\\metre} CHIME\/FRB trial sampling, and, where provided, the periods were found to be accurate.\n\n\\subsection{The Pushchino Radio Astronomy Observatory Catalogue}\\label{sec:push}\nThe PRAO catalogue focuses on sources detected with the Big Scanning Array of the Lebedev Physical Institute (BSA LPI, \\cite{tyulbashevDetectionNewPulsars2016}), a zenith scanning array at the Pushchino Radio Observatory outside Moscow, Russia.\n\nThe BSA LPI continually monitors declinations between \\SIrange{-6}{42}{\\degree} using 96 beams (48 of which are currently monitored), with \\SI{12.5}{\\milli\\second} sampling across 32 channels between \\SIrange{109}{111.5}{\\mega\\hertz}, which is contained within the bandwidth observed during each observation taken with I--LOFAR. While lacking the incredible \\SI{45000}{\\metre\\squared} collecting area of the BSA LPI, the relatively wide beam and \\SI{96}{\\mega\\hertz} usable bandwidth of an international LOFAR stations makes these arrays a strong candidate for follow-up observations on the brighter sources contained in the PRAO catalogue.\n\nWith the narrow bandwidth and relatively low sampling rate of the BSA LPI, the sources reported within the catalogue have wide uncertainties on their dispersion measure, typically from \\SIrange{2}{6}{\\parsec\\per\\centi\\metre\\cubed}, due to the coarse dispersion measure trials at these channels widths and sampling rates. Similarly, frequently detected sources have periods reported to $\\sim$\\SI{}{\\milli\\second} precision.\n\nAssuming that the sources in this catalogue have a bias towards having a steeper spectral index than typical pulsars, sources reported to have had their brightest pulse at a peak flux density below \\SI{5}{\\jansky} were not observed. Sources below this peak brightness are suspected to not be visible to I--LOFAR, as an international LOFAR station observing a $12.5$-ms\npulse, integrated over a \\SI{10}{\\mega\\hertz} bandwidth at \\SIrange{108}{118}{\\mega\\hertz} results in an expected signal-to-noise ratio (SNR) of only $3$. While the wider bandwidth available would improve the significance of such a pulse, it is unexpected that I--LOFAR would be able to detect such quiet pulses. It is expected that only sources with a reported peak flux densities of between \\SIrange{11}{60}{\\jansky} would be detectable, with the range depending on the sky temperature in the direction of the source (see~\\S\\ref{sec:sensitivity} for further details). Accounting for this, $41$ sources from this catalogue (and the four sources discussed in \\S~\\ref{sec:overlap}) were observed as a part of the census, with one source, J2018--07, not observed due to its low declination.\n\nSeveral sources from the PRAO catalogue were detected during the LOTAAS survey, and were reported to have both primarily periodic (J0317+1328, J1132+2513, J1404+1159) and single pulse emission (J0139+3336). These sources were observed as a part of the census to further gauge the capabilities of an international LOFAR station as compared to the 8 LOFAR core stations that were coherently beam-formed as a part of the LOTAAS survey~\\citep{sanidasLOFARTiedArrayAllSky2019b}.\n\n\\subsection{The RRATalog}\\label{sec:rratalog}\nThe RRATalog is a catalogue maintained by Cui and McLaughlin at West Virginia University, last updated in 2017, of previously announced RRAT sources from a wide variety of surveys. It contains sources detected during the multiple surveys performed over the past two decades using several telescopes, such as those at Arecibo, Parkes and Green Bank. Some sources detected using the Green Bank telescope have previously been followed up at the LOFAR core \\citep[][herein referred to as K15]{karako-argamanDiscoveryFollowupRotating2015}.\n\nWith the large number of `traditional' surveys consisting of tessellated pointings with narrow beams forming the foundation of this catalogue, it contains significantly lower uncertainties on positions and dispersion measures as compared to the catalogues discussed previously. Further, many of the sources have received full follow-up campaigns, allowing for their parameters to be constrained even further.\n\nA large fraction of the sources in the catalogue below a declination of \\SI{15}{\\degree} were not observed as a part of this census. The majority of these excluded sources are grouped around a small region of the sky near the Galactic plane, and have relatively low peak flux densities. This region of the sky already represents the location where the sensitivity of I--LOFAR is reduced due to the low peak elevations of the sources during their transits, and the scaling of the sky temperature of the Galactic disk to low frequencies (generally discussed in \\S~\\ref{sec:sensitivity}). It was concluded that it would be unlikely these sources would be detected in the several months of observations it would take to follow up the 18 sources that were not observed as a result of this constraint. \n\nAfter this filtering, 39 sources were observed from this catalogue as a part of this census.\n\nThis catalogue acts as a strong foundation for observers looking to dive into the world of RRATs, however there are several errata regarding entries in the catalogue, including the presence of sources that have since been re-classified as perytons~\\citep{10.1093\/mnras\/stv1242}, while some RRATs reported by the referenced surveys with discovery dates as far back as 2014 were found to be missing from the database. These were noted as this census progressed, with some additional sources from these surveys being added to the census under the ``RRATalog'' label. A full list of the updated sources is provided in~\\S\\ref{ap:extendedrratalog}.\n\n\\subsection{Catalogue Overlap}\\label{sec:overlap}\nSeveral sources are suspected to be duplicated in the aforementioned catalogues. \n\nJ1130+09 of the CHIME\/FRB catalogue and J1132+0921 of the PRAO catalogue have an angular separation of \\SI{0.2}{\\degree} with overlapping positional uncertainties, and have observed dispersion measures of 22.4 and \\SI{22(2)}{\\parsec\\per\\cubic\\centi\\metre}. Due to the reasonably large separation between the reported positions and low demands for telescope time at this source's right ascension, both pointings were observed, though neither resulted in a detection. \n\nJ1848+1518 of the PRAO catalogue and J1849+15 of the RRATalog (originally detected by the Green Bank Telescope) are suspected to be the same source for similar reasons \\citep[see further discussion in][]{tyulbashevDetection25New2018}. The ephemeris produced from a timing campaign performed at the LOFAR core (J1848+1516) was used for observing this source.\n\nDuring the later stages of the census, the PRAO catalogue was updated with results from \\cite{samodurovDetectionStatisticsPulse2022} to include two more sources, J1929+42 and J2214+45, that closely matched the location and dispersion measure properties of sources previously announced by CHIME, J1931+4229 and J2215+45. These sources had been monitored with I--LOFAR for several months at the time of the announcements using the original CHIME candidate parameters, consequently the PRAO reported positions were not specifically observed as a result.\n\n\\section{Observations and Methodology}\\label{sec:obsback}\n\n\\subsection{Census Strategy}\nEach source was observed for a minimum of \\SI{5}{\\hour}, typically in observations that ranged from \\SIrange{30}{90}{\\minute} centred near their transit time. When studying RRATs, longer total observation times increase the chance of brighter (detectable) pulses occurring, with the exact improvement dependent on the (typically unknown) pulse amplitude distributions. Five hours was chosen as a meaningful compromise between (a) probing sufficiently deep into the pulse-amplitude distribution of the sources given the sensitivity of the instrument, (b) investigating potentially low burst-rate sources emitting in a manner following Poisson statistics, and (c) the desire to complete the census in a reasonable time.\n\n\\subsection{Observing with I--LOFAR}\\label{sec:observing}\nI--LOFAR is one of 14 stations that make up the extended baselines of the ILT, which stretches from this westernmost site in Birr, Co. Offaly, Ireland to Irbene, Latvia in the East with a dense core of 38 stations with different configurations in the Netherlands \\citep{haarlemLOFARLOwFrequencyARray2013a}. \n\nFor \\SIrange{32}{48}{\\hour} per week, it is operated by the Irish LOFAR Consortium\\footnote{\\url{https:\/\/lofar.ie\/}} in `local mode', whereby astronomers can apply to use the station for standalone or coordinated observations. It is during this time that this census was performed, with data processed on the  Real-time Transient Acquisition compute cluster \\citep[REALTA;][]{murphyFirstResultsREALtime2021}.\n\nThe observations for this census were performed using the 96 High Band Antenna tiles (HBAs), operating in their lower frequency range of \\SIrange{102}{197}{\\mega\\hertz} with \\SI{195.3125}{\\kilo\\hertz} channel bandwidths (subbands 12 to 499 inclusive) and an underlying \\SI{5.12}{\\micro\\second} sampling frequency. Raw voltages were written to disk, which are then processed to the needs of the current project (the details for this census are discussed further in \\S\\ref{sec:processing}). This process allows for corrections in the dispersion measure used for coherent dedispersion~\\citep{1975MComP..14...55H} in the case that a source is detected with a significant deviation from the expected dispersion measure, re-sampling or extending the output data products in cases of particular interest and further analysis of unexpected features of the data.\n\nI--LOFAR is located in a relatively radio-quiet zone as compared to the rest of the stations spread across Europe. The low local population density alongside the lack of digital audio broadcast (DAB) radio in Ireland significantly reduced the radio frequency interference (RFI) present during observations, though typically 7--20 per cent of the total bandwidth is still flagged during observations due to transient RFI, the FM-radio band and to reduce noise contributions from filtered regions near the edge of the observed bandwidth.\n\n\n\\subsection{Sensitivity and Flux Calibration}\\label{sec:sensitivity}\nThe underlying sensitivity and resulting flux density measurements are calculated through a modified form of the methodology used by \\cite{kondratievLOFARCensusMillisecond2016a}, based on the radiometer equation and the additional factors needed to accurately characterise the properties of a low-frequency, wide-bandwidth interferometer. Further modifications to the methodology have been made for this work to account for the use of a single LOFAR station, resulting in a set of equations, with a focus on single-pulses in Eq.~\\ref{eq:radpulse}, and on periodic emission in Eq.~\\ref{eq:radperiodic}.\n\n\\begin{equation}\\label{eq:radpulse}\\centering\nS_\\text{pulse}(f, l, b) = \\text{S\/N}  \\frac{2k_b  m_\\text{II}\\left(T_\\text{sky} + T_\\text{ant}\\right)}{A_\\text{eff}\\beta \\sqrt{n_p  \\Delta \\nu_\\text{eff} w_\\text{pulse}}} \\hfill\\left[\\text{Jy}\\right]\n\\end{equation}\n\n\\begin{equation}\\label{eq:radperiodic}\\centering\nS_\\text{mean}(f, l, b) = \\text{S\/N} \\frac{2k_b  m_\\text{II}\\left(T_\\text{sky} + T_\\text{ant}\\right)}{A_\\text{eff}\\beta \\sqrt{n_p  \\Delta \\nu_\\text{eff} t_\\text{obs}}} \\sqrt{\\frac{W}{P - W}}\\hfill\\left[\\text{Jy}\\right]\n\\end{equation}\n\nThese equations differ only in the fraction of time integrated over, $\\sqrt{w_\\text{pulse}}$ verses $\\sqrt{t_\\text{obs}\\left(P - W\\right) \/ W}$, while the remaining components describe the sensitivity of the station for a given SNR.\n\nThe gain of the telescope ($A_\\text{eff} \/ \\left(2k_bm_\\text{II}\\right)$) depends on the effective area of the telescope, which can be found for an international station in \\cite{haarlemLOFARLOwFrequencyARray2013a}. The base value of \\SIrange{1150}{2400}{\\metre\\squared} was modified to account for 2 of the 96 HBA tiles not being used in the majority of the observations during this census due to hardware issues. As a LOFAR station is an interferometric array, the effective area varies both in frequency, with an upper limit of the area reached at \\SI{135}{\\mega\\hertz} due to mutual coupling of the densely packed antenna, and in pointing, which requires a correction in the form of a Mueller matrix, which is calculated from a Jones matrix generated by dreamBeam \\citep{carozzi2baOrNot2baDreamBeam2020} using the Hamaker beam model \\citep{hamaker2011mathematical} for each pulse or folded sub--integration.\n\nThe $\\beta$ correction factor of 0.94 in the single-pulse brightness equation is to account for the losses while processing the signal. The 8-bit digitisation of the signal is an extremely minor contribution, while the majority is to account for the loss in signal when attempting to integrate over Gaussian-like signals with a boxcar~\\citep{morelloOptimalPeriodicitySearching2020b}. While not all pulses take the shape of a Gaussian function, the majority do, with the remaining pulses having multiple components, or appear as scattered delta functions, both of which would see further losses beyond this correction factor (and are brighter than reported as a result).\n\nThe antenna and electronics temperatures ($T_\\text{ant}$) are sampled based upon measurements by \\cite{wijnholdsSituAntennaPerformance2011}, and are averaged across \\SI{5}{\\mega\\hertz} bandwidth blocks.\n\nThe sky temperatures ($T_\\text{sky}$) are calculated by performing a convolution between a 2D Gaussian function representing the approximated central component of the telescope beam, including a width correction factor of 1.02 \\citep[Table~B.1]{haarlemLOFARLOwFrequencyARray2013a}, with the region of the Low Frequency Sky Model \\citep[LFSM]{2017MNRAS.469.4537D}, as sampled using \\texttt{pygdsm}~\\citep{2016ascl.soft03013P}. The LOFAR beam width and sky temperature are sampled in \\SI{5}{\\mega\\hertz} blocks, which are then fit by a power law that is sampled to determine the sky temperature for the frequency-variable beam size. The $T_\\text{sky}$ variable was found to have a minimum, median, and maximum of \\SI{205}{\\kelvin}, \\SI{444}{\\kelvin} and \\SI{2006}{\\kelvin} at \\SI{150}{\\mega\\hertz} for the corpus of sources observed during the census. The corresponding sensitivity limits for the I--LOFAR telescope for single pulses and periodic emission at these temperatures are given in Table~\\ref{tab:sensitvity_pulse} and~\\ref{tab:sensitvity_pulsar}.\n\nWhile the recent work of \\cite{priceGlobalSkyModels2021} has indicated the advantages of using diffuse global sky models to determine more accurate measurements of $T_\\text{sky}$ for pulsar and FRB studies, Price notes that across the bandwidth used in this work the uncertainty is kept within a few per cent as the spectral index will only vary between $\\alpha = - 2.43, -2.53$ near the Galactic plane. However, it was noted that the extra Gaussian convolution step, introduced to account for the wide and variable beam width of a single international LOFAR station, had a significantly larger effect on the effective spectral index associated with $T_\\text{sky}$ than expected. After fitting the modelled sky temperatures to a power law, the index $\\alpha$ was found to vary significantly more than expected, with a maximum, median and minimum fit of $\\alpha = -2.29, -2.58, -2.83$ observed across the sampled sources. The difference in the $T_\\text{sky}$ measurement at \\SI{110}{\\mega\\hertz} of~$3$ per cent,~$1$ per cent,~$11$ per cent as compared to sampling the observed coordinate in the LFSM as a result of this modification.\n\nThis effect is due to the beam widening at lower frequencies which will amplify any anisotropic region in the field of view, causing $\\alpha$ to be more shallow where a source is located near a local maximum on the sky temperature max, and steeper in the case that is it near a local minimum. However, the median effect on the combined $T_\\text{sky} + T_\\text{ant}$ term is found to be below~$1$ per cent across the observed bandwidth, as a result of the large contributions of $T_\\text{ant}$ from each antenna that comprises the LOFAR interferometer.\n\nFor single pulses, SNRs were re-calculated in windows of \\SI{10}{\\mega\\hertz}, with the upper and lower components of the observation bandwidth discarded due to the sensitivity fall-off due to the pass-band of the electronics and the large amount of RFI due to the FM radio band near \\SI{100}{\\mega\\hertz}. Consequently, statistics are generated over the bandwidth between \\SIrange{116}{186}{\\mega\\hertz}, centred on \\SI{151}{\\mega\\hertz}. An equivalent flux density is then calculated for each segment of the bandwidth. These frequency dependant flux densities were fit to a power law to determine the spectral indices for detected sources, with the values presented (below) representing the best fit of a Gaussian function to a histogram of spectral indices.\n\nContinuing the method used by \\citeauthor{kondratievLOFARCensusMillisecond2016a}, a 50 per cent uncertainty is introduced on the results of these calculations, to account for compounding uncertainties on the measured station parameters, potential failures in the beam model, side-lobe contributions (which are particularly strong when using a single international LOFAR station) and the transient background sky due to ionospheric scintillation at low frequencies (see \\citealt[\\S~3.2]{kondratievLOFARCensusMillisecond2016a} for further details). \n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{llccc}\\hline\\hline\n    Samples & Width & S$_{T_\\text{sky, min}}$ & S$_{T_\\text{sky, median}}$ & S$_{T_\\text{sky, max}}$  \\\\\n     & \\multicolumn{1}{c}{\\SI{}{\\milli\\second}} & \\SI{}{\\jansky} & \\SI{}{\\jansky} & \\SI{}{\\jansky} \\\\\n   \\hline\n1 & 0.655 & 27 & 38 & 110 \\\\\n2 & 1.31 & 19 & 27 & 77 \\\\\n4 & 2.62 & 14 & 19 & 55 \\\\\n16 & 10.5 & 6.8 & 9.5 & 27 \\\\\n64 & 41.9 & 3.4 & 4.8 & 14 \\\\\n256 & 168 & 1.7 & 2.4 & 6.8 \\\\\n\n   \\hline\\hline\n    \\end{tabular}\n    \\caption{The bandwidth-averaged minimum flux density, in jansky, of a pulse of varying width required to be detected with an international LOFAR station, with a signal-to-noise ratio of 7.5.}\n    \\label{tab:sensitvity_pulse}\n\\end{table}\n\n\\begin{table}\n    \\centering\n    \\begin{tabular}{llccc}\\hline\\hline\n    \\multicolumn{2}{c}{Duty Cycle} & S$_{T_\\text{sky, min}}$ & S$_{T_\\text{sky, median}}$ & S$_{T_\\text{sky, max}}$  \\\\\n    & \\% & \\SI{}{\\milli\\jansky} & \\SI{}{\\milli\\jansky} & \\SI{}{\\milli\\jansky} \\\\\n   \\hline\n0.001 & 0.1 & 0.27 & 0.38 & 1.1 \\\\\n0.003 & 0.3 & 0.48 & 0.67 & 1.9 \\\\\n0.01 & 1 & 0.87 & 1.2 & 3.5 \\\\\n0.03 & 3 & 1.5 & 2.1 & 6.2 \\\\\n0.1 & 10 & 2.9 & 4.1 & 12 \\\\\n0.3 & 30 & 5.7 & 8 & 23 \\\\\n   \\hline\\hline\n    \\end{tabular}\n    \\caption{The folded average flux density, in millijansky, for an one hour observation of a pulsar of varying duty cycles required to be detected with an international LOFAR station, with a signal-to-noise ratio of 6.}\n    \\label{tab:sensitvity_pulsar}\n\\end{table}\n\n\\subsection{Observation Processing and Archival}\\label{sec:processing}\nThe raw data from the station is processed into Stokes $I$ SigProc filterbanks \\citep{lorimerSIGPROCPulsarSignal2011} using a modified version\\footnote{Modified to accept raw voltages from an international station rather than COBALT(2) correlated H5 files, see \\url{https:\/\/github.com\/David-McKenna\/cdmt}} of CDMT \\citep{bassaEnablingPulsarFast2017}, to perform coherent dedispersion, channelisation, temporal downsampling and detection to Stokes $I$ on the Nvidia Tesla V100s present in the REALTA compute cluster.\n\nCoherent dedispersion reduces the effect of temporal smearing of an incoming pulsar signal by performing a convolution between the station voltages and the predicted inverse transfer function for the interstellar medium at a given dispersion measure. Initially, the dispersion measure is taken from the source catalogue, but where observations with I--LOFAR constrained a source dispersion measure further, the dispersion measure is updated and used for re-processing voltages and future observations.\n\nChannelisation was performed to reduce the bandwidth of the subbands by a factor of 8, increasing the number of channels in the output data and allowing for a more accurate measurement of dispersion measure when a pulse is detected.\n\nThe data is down-sampled in time by a factor of 16 as a compute and storage saving procedure. This is not considered to have a negative effect on the data due to three main considerations, (a) the predicted underlying pulse widths, (b) the expected scattering timescales and (c) the data is coherently dedispersed prior to this step, causing any smearing to be on timescales far below the output time resolution. As discussed in~\\S\\ref{sec:intro}, RRATs typically have longer periods than the general pulsar population due to selection effects. As a result, given a typical duty cycle, the underlying width of a pulse can be expected to be measured on a scale of \\SI{}{\\milli\\second} to tens of \\SI{}{\\milli\\second}. Further, at low frequencies these pulses are heavily scattered, with scattering behaviour often scaling strongly with frequency, roughly following $\\nu^{-4}$ \\citep[see][]{langInterstellarScintillationPulsar1971,krishnakumarMultifrequencyScatterBroadening2017a}, further broadening the pulses. As a result, the sub-millisecond sampling available after this down-sampling step is still more than sufficient to detect and analyse the morphology of the pulses observed when a sufficiently bright signal is detected.\n\nThe resulting filterbank is a 32-bit floating point file with 3904 channels, with bandwidths of \\SI{24.41}{\\kilo\\hertz} each, and \\SI{655.36}{\\micro\\second} temporal sampling. However, in the case that a source has been observed as a part of the census and after the \\SI{5}{\\hour} of observations there has been no detections, either single pulse or periodic, the filterbank is re-sampled to reduce the number of channels by a factor of 8, returning them to the original resolution of the telescope, resulting in channel widths of \\SI{195.3126}{\\kilo\\hertz}. In all cases, the final filterbank is compressed using zstandard\\footnote{\\url{https:\/\/github.com\/facebook\/zstd}} for a further 5--10 per cent space-saving and archived.\n\nPrior to processing, the 32-bit filterbanks are re-sampled using DSPSR's \\texttt{digifil} tool~\\citep{vanstratenDSPSRDigitalSignal2011}, removing the bandpass contributions across the default $10$-s\nre-scale interval and reducing the data to 8--bit samples, which are computationally easier to analyse. The reduced bit depth further dampens any excess contributions of spurious RFI samples to dedispersed time series.\n\nAll observations undergo a single-pulse search using the GPU-accelerated \\texttt{heimdall} software~\\citep{barsdellAdvancedArchitecturesAstrophysical2012}, with a more sensitive search (with a 0.3 per cent loss of signal per dispersion measure trial) over a range of \\SIrange{10}{20}{\\parsec\\per\\cubic\\centi\\metre} centred on the known dispersion measure of the source, and a less sensitive search (with a 3 per cent loss of signal per trial) across the \\SIrange{10}{500}{\\parsec\\per\\cubic\\centi\\metre} range to search for unexpected signals in the data.\n\nIt is noted that the recent work of \\cite{quiFredda2023}\nhighlights some systemic errors and pitfalls in the \\texttt{heimdall} processing pipeline. The dense sampling of dispersion measure trials used for this work reduces the effect of the errors described by \\citeauthor{quiFredda2023}, however \nit seems that the spacing of the dispersion measure trials is insufficient for low--frequency observation, and will result in a true loss of signal higher than the intended 0.3 per cent and 3 per cent defined above between dispersion measure trials.\n\nAny candidates above a SNR of $7.5\\sigma$ are used to generate a candidate plot, which is inspected by eye to determine if a candidate is a true pulse detection. In the case of a detection, the candidate is processed with the methodology discussed in~\\S\\ref{sec:singlepulse} and~\\S\\ref{sec:timing}.\n\nGiven that \\texttt{heimdall} performs a search for single-pulses on a series of dispersed 1-dimensional time series, this methodology results in a reduced sensitivity towards pulses that are only visible in a fraction of the observed bandwidth. Consequently, pulses observed in a fraction of the bandwidth, either due to their intrinsic spectral index, scintillation, or other phenomena that cause sharp spectral variability are less likely to be reported in the results of this census, but may be detected in future processing of the census archive (further discussed in~\\S\\ref{sec:frbdiscuss}).\n\nIf the period of a source is known, the data is also folded with PRESTO's \\texttt{prepfold} tool \\citep{ransomPRESTOPulsaRExploration2011} at the given period and dispersion measure to search for periodic emission. Blind periodicity searches were not a component of the main processing pipeline for the census observations (but see~\\S\\ref{sec:J1329},~\\S\\ref{sec:conclusions}).\n\n\n\\subsection{Single Pulse Analysis}\\label{sec:singlepulse}\nSingle pulses are extracted and pre-processed using a combination of DSPSR and PSRCHIVE \\citep{hotanPSRCHIVEPSRFITSOpen2004}. DSPSR's \\texttt{digifil} is used to extract a small block of time around a candidate arrival time\\footnote{The amount of padding time varies depending on the source period due to the requirement of PSRCHIVE's \\texttt{pat} tool, which is used for producing time of arrival (TOA) measurements, whereby input time series must be powers of two in length, but is typically on the order of half a rotation period on each side of a pulse.}, and re-sampled from 32-bit floats to 8-bit unsigned chars, removing the bandpass offset in the process. \\texttt{dspsr} then dedisperses this filterbank, performs spectral kurtosis for RFI flagging and writes an output in the TIMER format. This is initially only performed for visually approved candidates with a SNR above 7.5, but once a source has been well constrained through detections in multiple observations, this criteria is reduced to a SNR of 7.\n\nPSRCHIVE's \\texttt{paz} tool is then used to perform further RFI zapping based on outlying channels in a windowed scan across the bandpass of the 32-bit data, to ensure no badly contaminated channels are included in the output pulse archive.\n\nThe overall pulse populations are then analysed to determine their properties. Namely, spectral brightness, pulse widths, burst rates and pulse amplitude distribution. Where noted, fitting is performed in Python using the \\texttt{lmfit} \\citep{newvilleLmfitLmfitpy2021} module. Any uncertainty on parameters fit through Gaussian distributions represent the uncertainty of the central point of the Gaussian, rather than the spread of the distribution. These results can be found in Table~\\ref{tab:obs_summary}.\n\nThe spectral properties of the pulses are calculated by fitting a power law to the spectral flux densities of each pulse determined using Eq.~\\ref{eq:radpulse}, and a Gaussian is fit to the overall distribution of power law exponents across the observed pulses.\n\nPulse amplitude distributions were analysed by fitting multiple models (power law, broken power law, log--normal, powerlaw--log--normal and log--normal--powerlaw) to the flux density distribution of pulses. From these models, the small-N corrected Akaike information criterion (AICc) was used to determine the best fit model. The fit values for these properties are presented accordingly in Fig.~\\ref{fig:spectralmodfits} and Table~\\ref{tab:pulseamplitudetab}.\n\nPulse widths were determined with PSRCHIVE's \\texttt{pdmp}, but were found to be not be well fit by a Gaussian distribution. As a result, the mean and standard deviation of the widths has been presented.\n\n\n\\subsection{Timing}\\label{sec:timing}\nUsing the single-pulse archive produced by \\texttt{dspsr} as described in the previous section, PSRCHIVE's \\texttt{pat} generates a per-pulse time of arrival (TOA) measurement using an analytical profile generated using the brightest pulse and the \\texttt{paas} tool as a standard reference. \n\n\\texttt{tempo2} \\citep{hobbsTEMPO2NewPulsartiming2006} is then used to model and update the source ephemeris using all times of arrival of a source. Some TOAs of low-significance pulses (SNR ranging from 7 to 7.5) are often rejected at this stage due to unexpectedly large uncertainties, large phase offsets or other unexpected phenomena. These cases are often caused by unflagged RFI or unexpected behaviour in the underlying dataset, most of which can be fixed through manually re-processing the pulse archives.\n\nWhere sources have not yet been published in \\texttt{psrcat}, the provided source catalogues were used to generate an ephemeris for each source, which was then used as a basis for timing. When a source with only a known period (and no derivative) was observed to have more than 3 pulses in an observation, the times of arrival were used to brute force a new period near the previously published result, which was then used for timing. When more than 6 pulses were seen for a source without a period, a similar approach was used to find the most likely period, which was then refined by using prior TOAs to find the period that represents the greatest common divisor, which was then used for timing\\footnote{The \\href{https:\/\/github.com\/evanocathain\/Useful_RRAT_stuff}{\\texttt{getper.py} script} from the SUPERB survey \\citep{keane+2018} was used for the brute-force periodicity search.}.\n\n\\subsection{Periodic Emission Analysis}\nIn the case that a source was found to emit periodic emission with a SNR greater than 7 in the previously described PRESTO \\texttt{prepfold} search, the observations of the source were folded and analysed using \\texttt{DSPSR} and \\texttt{PSRCHIVE} to prepare folded archives with integration times of \\SI{30}{\\second} and 256 bins. Observations are combined and integrated into a single-frequency time series for final analysis. To reduce the effects of RFI on the folded data and improve the accuracy of periodic emission SNRs, the data in each observation were flagged using \\texttt{clfd} \\citep{morelloHighTimeResolution2019} prior to analysis.\n\n\\subsection{Archival Data Follow-up}\nIn the case that a source was detected during the census, pointings from the LOTAAS survey~\\citep{sanidasLOFARTiedArrayAllSky2019b} were downloaded from the LOFAR Long Term Archive\\footnote{\\url{https:\/\/lta.lofar.eu\/Lofar}} (LTA) and processed using the same methodology described in this section to search for single-pulses and periodic emission. Due to the large positional uncertainty associated with many sources in the census, typically at least one `ring' of nearby beams were downloaded about the reported pointing of the observation.\n\nThis allowed for a potential detection of these sources at an earlier epoch and potentially allows for changes in source properties to be tracked from an earlier epoch.\n\n\\afterpage{\\begin{landscape}\n\\begin{table}\n\\centering\n    \\caption{A summary of the properties of the $29$ sources detected as a part of this census, based on both single-pulse and periodic emission. Data across \\SIrange{116}{186}{\\mega\\hertz}, centered on \\SI{151}{\\mega\\hertz}, are analysed and presented from observations that started in August 2020 and ended in September 2022. Each source notes their source catalogue, the reference to any previous work related to the source with LOFAR, the known rotation period  of the source, best-fit dispersion measure (from single-pulse emission, or periodic emission if single-pulses were not observed) and the number of hours each source was observed for. Basic parameters regarding the single-pulse emission and periodic emission during the observations are provided. Additionally, the single-pulse emission provides the number of pulses observed, the ratio between the brightest and quietest observed pulses (SR), and the effective burst rate for each source.}\n\t\\label{tab:obs_summary}\n\\begin{tabular}{lccccc|ccccccc|cccc}\n\\hline\\hline\n& & & & & & \\multicolumn{7}{c|}{Single Pulse} & \\multicolumn{4}{c}{Periodic Fold} \\\\\nSource & Cat. & Prev. & Period & DM & T\\textsubscript{obs} & N\\textsubscript{pulses} &  w$_{10}$ & Duty Cycle & S$_{150}^{\\text{peak}}$\\textsuperscript{a} & SR &$\\alpha$\\textsuperscript{b} & Burst Rate\\textsuperscript{c} &  w$_{10}$ & Duty Cycle & S$_{150}^{\\text{mean}}$\\textsuperscript{a} &  $\\alpha$\\textsuperscript{b}\\\\\n   & & & (\\SI{}{\\second}) & (\\SI{}{\\parsec\\per\\cubic\\centi\\metre})  & (\\SI{}{\\hour}) & & (\\SI{}{\\milli\\second}) & \\% & (\\SI{}{\\jansky}) & & & (\\SI{}{\\per\\hour}) & (\\SI{}{\\milli\\second}) & \\% & (\\SI{}{\\milli\\jansky}) & \\\\\n\\hline\nJ0054+6650 & R & -- & 1.3902 & 14.560(2) & 31.5 & 282 & 14(6) & 1.0(4) & 45 &  10 & -1.24(7) & 9.0(5) & -- & -- & -- & -- \\\\\nJ0102+5356 & R & \\citetalias{karako-argamanDiscoveryFollowupRotating2015} & 0.3543 & 55.6200(8) & 27.3 & 88 & 4.0(1.4) & 1.1(4) & 60 &   8 & 0.13(13) & 3.2(3) & -- & -- & -- & -- \\\\\nJ0139+3336 & P & \\citetalias{michilliSinglepulseClassifierLOFAR2018} & 1.2480 & 21.223(4) & 16.6 & 157 & 13(6) & 1.0(5) & 43 &   9 & -1.65(9) & 9.4(8) & -- & -- & -- & -- \\\\\nJ0201+7005 & R & \\citetalias{karako-argamanDiscoveryFollowupRotating2015} & 1.3492 & 21.047(2) & 10.9 & 9 & 2.6(8) & 0.19(6) & 37 &   3 & -- & 0.8(3) & 12 & 0.9 & 2.5 & -0.2(5) \\\\\nJ0209+5759\\textsuperscript{d} & C & -- & 1.0639 & 55.855(4) & 26.6 & 43 & 13(7) & 1.2(7) & 29 &   5 & -0.25(11) & 1.6(2) & 25 & 2.4 & 25.1 & -0.6(3) \\\\\nJ0226+3356 & C & -- & 1.2401 & 27.397(14) & 18.0 & -- & -- & -- & -- & -- & -- & -- & 43 & 3.5 & 3.3 & -0.9(3) \\\\\nJ0317+1328 & P & \\citetalias{michilliSinglepulseClassifierLOFAR2018} & 1.9742 & 12.7452(10) & 10.1 & 59 & 3.2(1.3) & 0.16(7) & 41 &   4 & -5.3(8) & 5.9(8) & 20 & 1.1 & 3.8 & -1.8(4) \\\\\nJ0332+79 & R & \\citetalias{karako-argamanDiscoveryFollowupRotating2015} & 2.0562 & 16.589(7) & 15.4 & 1 & 5.9 & 0.29 & 15 &   1 & -- & (0.065) & -- & -- & -- & -- \\\\\nJ0348+79 & C & -- & -- & 26.09(3) & 26.7 & 6 & 36(15) & -- & 9 &   3 & -- & 0.22(9) & -- & -- & -- & -- \\\\\nJ0746+5514 & C & -- & 2.8938 & 10.318(7) & 43.2 & 48 & 32(20) & 1.1(7) & 35 &  13 & -0.13(16) & 1.11(16) & -- & -- & -- & -- \\\\\nJ0854+5449 & C & -- & 1.2330 & 18.843(3) & 24.2 & 2 & 2.29(98) & 0.19(8) & 19 &   1 & -- & 0.08(6) & 11 & 0.9 & 1.1 & 0.7(2) \\\\\nJ0939+45 & P & -- & -- & 17.45(4) & 18.5 & 3 & 19(5) & -- & 7 &   2 & -- & 0.16(9) & -- & -- & -- & -- \\\\\nJ1006+3015 & P & -- & 3.0664 & 18.085(4) & 95.6 & 166 & 33(16) & 1.1(5) & 50 &  16 & -2.37(12) & 1.74(13) & -- & -- & -- & -- \\\\\nJ1132+2513 & P & \\citetalias{sanidasLOFARTiedArrayAllSky2019b} & 1.0021 & 23.716(7) & 11.2 & -- & -- & -- & -- & -- & -- & -- & 17 & 1.8 & 7.6 & -1.3(4) \\\\\nJ1218+47 & P & -- & -- & 20.144(11) & 34.7 & 13 & 16(9) & -- & 13 &   4 & -- & 0.37(10) & -- & -- & -- & -- \\\\\nJ1329+13 & P & -- & -- & 12.367(13) & 49.6 & 5 & 23(11) & -- & 11 &   1 & -- & 0.10(5) & -- & -- & -- & -- \\\\\nJ1336+3414 & P & -- & 1.5066 & 8.4688(11) & 79.0 & 132 & 5(3) & 0.31(18) & 48 &  11 & 0.12(18) & 1.67(15) & -- & -- & -- & -- \\\\\nJ1400+2125 & P & -- & 1.8555 & 11.214(3) & 86.7 & 43 & 17(6) & 0.9(3) & 29 &   4 & -3.3(6) & 0.50(8) & -- & -- & -- & -- \\\\\nJ1404+1159 & P & \\citetalias{michilliSinglepulseClassifierLOFAR2018} & 2.6504 & 18.53(3) & 5.7 & -- & -- & -- & -- & -- & -- & -- & 62 & 2.3 & 19.1 & -1.13(13) \\\\\nJ1538+2345 & R & \\citetalias{karako-argamanDiscoveryFollowupRotating2015} & 3.4494 & 14.89814(98) & 38.3 & 371 & 9(6) & 0.25(18) & 73 &  22 & -1.60(12) & 9.7(5) & 121 & 3.5 & 4.3 & -0.51(17) \\\\\nJ1848+1516 & R & \\citetalias{michilliSinglepulseClassifierLOFAR2018} & 2.2338 & 77.488(11) & 32.5 & 67 & 41(12) & 1.8(6) & 16 &   3 & -0.2(4) & 2.1(3) & 52 & 2.3 & 4.5 & 0.52(14) \\\\\nJ1931+4229 & C & -- & 3.9210 & 50.987(6) & 53.7 & 133 & 49(20) & 1.2(5) & 22 &   9 & -1.06(7) & 2.5(2) & -- & -- & -- & -- \\\\\nJ2105+19 & P & -- & 3.5298 & 34.47(3) & 7.7 & -- & -- & -- & -- & -- & -- & -- & 187 & 5.3 & 3.8 & -1.4(9) \\\\\nJ2108+4516\\textsuperscript{e} & C & -- & 0.5774 & 82.520(7) & 6.2 & 75 & 24(12) & 4(2) & 10 &   3 & 1.91(14) & 12.0(1.4) & 96 & 16.8 & 231.2 & -0.4(4) \\\\\nJ2138+69 & C & -- & -- & 46.530(3) & 30.8 & 9 & 7(2) & -- & 20 &   2 & -- & 0.292(97) & -- & -- & -- & -- \\\\\nJ2202+2134 & P & \\citetalias{sanidasLOFARTiedArrayAllSky2019b} & 1.3573 & 17.7473(19) & 18.1 & 11 & 3.0(7) & 0.22(5) & 25 &   3 & -- & 0.61(18) & 11 & 0.8 & 4.8 & -1.0(9) \\\\\nJ2215+4524 & C & -- & 2.7231 & 18.5917(16) & 37.2 & 121 & 9(5) & 0.32(18) & 19 &   4 & -1.76(17) & 3.3(3) & 21 & 0.8 & 1.7 & -1.45(12) \\\\\nJ2325-0530 & R & -- & 0.8687 & 14.9580(8) & 7.0 & 322 & 10(3) & 1.2(4) & 131 &  11 & -3.66(5) & 46(3) & 12 & 1.4 & 7.5 & -3.7(5) \\\\\nJ2355+1523 & C & -- & 1.0964 & 26.924(16) & 51.9 & 14 & 17(8) & 1.5(7) & 15 &   3 & -- & 0.27(7) & -- & -- & -- & -- \\\\\n\n\\hline\\hline\n\\end{tabular}\n\n\n\n\\bigskip\n\n\n{\\footnotesize \\textsuperscript{a} The standard uncertainty on all LOFAR flux density measurements is 50 per cent, see~\\S\\ref{sec:sensitivity}.}\n\n{\\footnotesize \\textsuperscript{b} Spectral indices are expected to be biased towards more negative values due to the positional uncertainty of the sources, as the telescope beam FWHM will increase the effective gain for emission at lower frequencies, see~\\S\\ref{sec:sensitivity}.}\n\t\n{\\footnotesize \\textsuperscript{c} Provided burst rate uncertainties are Poissonian.}\n\n{\\footnotesize \\textsuperscript{d} Periodic emission statistics for J0209+5759 are based on emission during the active phases of the observations, representing \\SI{1.3}{\\hour} of data, see~\\S\\ref{sec:periodicres}.}\n \n{\\footnotesize \\textsuperscript{e} Single pulse and periodic emission statistics for J2108+4516 are based on the single \\SI{1}{\\hour} observation where it was detectable.}\n\n\n\n\n\\end{table}\n\n\\end{landscape}}\n\n\\afterpage{\\begin{landscape}\n\\begin{table}\n\t\\centering\n    \\caption{Summary of the 8 new source ephemerides, and their derived quantities, determined through pulsar timing of single-pulse times of arrival at I--LOFAR.}\n\t\\label{tab:newephemerides}\n\\begin{tabular}{lcccccccc}\n\n\n\\hline\\hline\nSource & J0054+6650 & J0102+5356 & J0746+5514 & J1006+3015 & J1336+3414 & J1400+2125 & J1931+4229 & J2215+4524 \\\\\n\\\\\nCatalogue & RRATalog & RRATalog & CHIME\/FRB & PRAO & PRAO & PRAO & CHIME\/FRB & CHIME\/FRB \\\\\nCatalogue Source Name & J0054+66 & J0103+54 & J0746+55 & J1005+3015 & J1336+3346 & J1400+2127 & J1931+4229 & J2215+45 \\\\\n\\hline\nRight Ascension (hms) & 00\\textsuperscript{h}54\\textsuperscript{m}55\\textsuperscript{s}.412(14) & 01\\textsuperscript{h}02\\textsuperscript{m}57\\textsuperscript{s}.787(10) & 07\\textsuperscript{h}46\\textsuperscript{m}47\\textsuperscript{s}.4(2) & 10\\textsuperscript{h}06\\textsuperscript{m}34\\textsuperscript{s}.44(7) & 13\\textsuperscript{h}36\\textsuperscript{m}33\\textsuperscript{s}.953(18) & 14\\textsuperscript{h}00\\textsuperscript{m}14\\textsuperscript{s}.19(3) & 19\\textsuperscript{h}31\\textsuperscript{m}10\\textsuperscript{s}.88(7) & 22\\textsuperscript{h}15\\textsuperscript{m}46\\textsuperscript{s}.847(13) \\\\\nDeclination (dms) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+66;50;23.81}(8) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+53;56;11.91}(14) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+55;14;37}(2) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+30;15;46}(2) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+34;14;37.8}(2) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+21;25;40.0}(5) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+42;29;16.0}(1.0) & \\ang[angle-symbol-over-decimal,minimum-integer-digits = 2,]{+45;24;43.3}(2) \\\\\nGalactic Longitude ($^\\circ$) & \\ang[angle-symbol-over-decimal]{123.27546}(6) & \\ang[angle-symbol-over-decimal]{124.64828}(4) & \\ang[angle-symbol-over-decimal]{162.5359}(8) & \\ang[angle-symbol-over-decimal]{197.9536}(3) & \\ang[angle-symbol-over-decimal]{71.97329}(8) & \\ang[angle-symbol-over-decimal]{16.66008}(11) & \\ang[angle-symbol-over-decimal]{75.1545}(3) & \\ang[angle-symbol-over-decimal]{96.39171}(5) \\\\\nGalactic Latitude ($^\\circ$) & \\ang[angle-symbol-over-decimal]{3.97052}(2) & \\ang[angle-symbol-over-decimal]{-8.89669}(4) & \\ang[angle-symbol-over-decimal]{29.7344}(6) & \\ang[angle-symbol-over-decimal]{53.9052}(6) & \\ang[angle-symbol-over-decimal]{77.98183}(7) & \\ang[angle-symbol-over-decimal]{73.33655}(15) & \\ang[angle-symbol-over-decimal]{11.2299}(3) & \\ang[angle-symbol-over-decimal]{-9.28754}(6) \\\\\nDispersion Measure  (\\SI{}{\\parsec\\per\\centi\\metre\\cubed}) & 14.560(2) & 55.6200(8) & 10.318(7) & 18.085(4) & 8.4688(11) & 11.214(3) & 50.987(6) & 18.5917(16) \\\\\nDistance (\\SI{}{\\parsec}) & 878 & 2020 & 400 & 2120 & 715 & 990 & 4240 & 1140 \\\\\n\\\\\nPeriod (\\SI{}{\\second}) & 1.390218053458(12) & 0.35429906918(10) & 2.8936673467(8) & 3.0663650266(6) & 1.50660326546(10) & 1.85546489586(16) & 3.9210363832(19) & 2.72313592820(13) \\\\\nPeriod Derivative (\\SI{e-15}{\\second\\per\\second}) & 5.5532(8) & 0.5203(4) & 10.19(5) & 5.52(2) & 0.111(3) & 1.358(7) & 31.14(5) & 5.223(5) \\\\\nCharacteristic Age (\\SI{}{\\mega\\year}) & 3.97 & 10.8 & 4.5 & 8.8 & 216 & 21.7 & 2 & 8.27 \\\\\nMagnetic Field (\\SI{e12}{\\gauss}) & 2.81 & 0.434 & 5.49 & 4.16 & 0.413 & 1.61 & 11.2 & 3.82 \\\\\n\\\\\nTiming Start (MJD) & 59108 & 59240 & 59268 & 59123 & 59206 & 59142 & 59191 & 59184 \\\\\nTiming End (MJD) & 59849 & 59850 & 59848 & 59830 & 59848 & 59820 & 59849 & 59822 \\\\\nReference Epoch (MJD) & 59360 & 56657 & 59381 & 59142 & 59206 & 59234 & 59100 & 59247 \\\\\nN\\textsubscript{TOAs} & 279 & 87 & 48 & 163 & 123 & 43 & 133 & 183 \\\\\nModel Residuals (\\SI{}{\\micro\\second}) & 1663 & 1104 & 11702 & 10701 & 2844 & 2942 & 9261 & 2536 \\\\\n\\hline\\hline\n\n\n\n\n\\end{tabular}\n\t\n\t\t\\vspace{0.25cm}\n\t{\\footnotesize \\textsuperscript{a} The YWM16 model~\\citep{yaoDensityDistance2017} was used to determine the provided source distances, and has an underlying uncertainty of 10 per cent.}\n\t\n\\end{table}\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tabular}{cc}\n    \\begin{minipage}[t]{10.5cm}\n    \\centering\n    \\adjincludegraphics[height=5cm,trim={0 0 {0.85\\width} 0}]{Plots\/period_dist_0.pdf}\n    \\includegraphics[height=5cm]{Plots\/period_dist_no0.pdf}\n    \\caption{A histogram of source rotation periods, separating the sources that were detected (\\SIrange{0.22}{3.45}{\\second}) and not detected (\\SIrange{0.15}{6.4}{\\second}) as a part of this census. The group of sources without a known period has been placed on a separate axis on the left due to the order of magnitude difference between this bin and the remaining sources. The 0-period bin also contains sources that periods were determined for as a part of this census. See further discussion in~\\S\\ref{sec:discussproperties}.}\n    \\label{fig:rrat_periods}\n    \\end{minipage}\\hspace{0.9cm}\n    \\begin{minipage}[t]{10.5cm}\n    \\centering\n    \\includegraphics[height=5cm]{Plots\/dm_dist_wider.pdf}\n    \\caption{A histogram visualising the distribution of dispersion measures of the sources observed (\\SIrange{5}{280}{\\parsec\\per\\centi\\metre\\cubed}) and detected (\\SIrange{8.5}{82.5}{\\parsec\\per\\centi\\metre\\cubed}) as a part of the census. See further discussion in~\\S\\ref{sec:discussproperties}.}\n    \\label{fig:rrat_dms}\n    \\end{minipage}\n    \\end{tabular}\n\\end{figure}\n\\end{landscape}}\n\n\\section{Results}\\label{sec:results}\n\nFrom the $113$ sources in this census, $29$ sources were detected in some manner. $25$ sources were only detected through single-pulse emission, while $14$ produced sufficient emission to be detected as a periodic source in at least one observation during the census or further follow-up observations. $\\consistentperiodicdetected$ sources were consistently detected as sources of periodic emission.\n\nThe detected sources cover dispersion measures from \\SIrange{8.5}{82.5}{\\parsec\\per\\centi\\metre\\cubed} and rotation periods over the range of \\SIrange{0.354}{3.92}{\\second}. The majority of the sources do not present evidence of giant-pulse emission, though 6 source have a ratio of at least 10 between the maximum and minimum observed pulse brightness.\n\nThe source names used in this work may differ from their catalogue entries. The names used in this section and beyond will refer to their names in \\texttt{psrcat}, or the updated names as a result of timing with I--LOFAR (see Table~\\ref{tab:newephemerides}) or further cited publications. A map between names used can be found in Table~\\ref{tab:sourcemappings}.\n\nThe single-burst behaviour of detected sources is discussed in ~\\S\\ref{sec:singlepulseres} and periodic emission is discussed in ~\\S\\ref{sec:periodicres}. Novel source periods are noted in ~\\S\\ref{sec:newperiod}. An overview of new properties determined from performing pulsar timing on these sources to-date will be described in ~\\S\\ref{sec:timnigres}. Some notes regarding inspection of LOTAAS archival pointings are discussed within each relevant section.\n\n\n\\subsection{Sources with Single-Pulse Emission}\\label{sec:singlepulseres}\nFrom the $25$ sources with single-pulse emission, $18} % single pulse detected - 7 in \\citetalias{karako-argamanDiscoveryFollowupRotating2015}\/\\citetalias{michilliSinglepulseClassifierLOFAR2018}\/\\citetalias{sanidasLOFARTiedArrayAllSky2019b$ are novel to LOFAR detections, having not discussed in prior literature. \n\nSingle-pulses from one source previously reported to have single-pulse emission during the LOTAAS survey (\\citetalias{michilliSinglepulseClassifierLOFAR2018}), J1404+1159, was not detected as a part of the census. From the 7 sources detected with the full LOFAR core in \\citetalias{karako-argamanDiscoveryFollowupRotating2015}, only J0054+69 and J2105+6223 were not detected during this census.\n\n\n    \n\\subsubsection{LOTAAS Archive Reprocessing}\nWhile a number of the detected sources were previously detected and discussed in \\citetalias{karako-argamanDiscoveryFollowupRotating2015}, \\citetalias{michilliSinglepulseClassifierLOFAR2018} and \\citetalias{sanidasLOFARTiedArrayAllSky2019b}, all but one of the $18} % single pulse detected - 7 in \\citetalias{karako-argamanDiscoveryFollowupRotating2015}\/\\citetalias{michilliSinglepulseClassifierLOFAR2018}\/\\citetalias{sanidasLOFARTiedArrayAllSky2019b$ novel to LOFAR detected sources were within the field of the sky surveyed as a part of the LOTAAS survey (J2325-0530, too low in declination), while pointings could not be found for two sources (J1020+5356 and J2325-0530). The remainder of the \\SI{60}{\\minute} observations for each source were downloaded and re-analysed following the methodology described in~\\S\\ref{sec:processing}.\n\nFrom the novel detections, pointing near J0054+6650, J0746+5514, J1006+3015, J1336+3414, J1400+2125 and J2215+4524 were found to contain at least 1 pulse at the previously described detection criteria. Apart from J1400+2125, these sources were observed to have burst rates in excess of \\SI{1}{\\per\\hour} at I--LOFAR, making the probability of their detection more favourable during the LOTAAS survey. From the non-detected sources, only J0209+33 and J1931+4229 met the same burst rate criteria, with many of the other sources producing bursts extremely infrequently. \n\nNotably, while J2202+2134 was previously only reported as a source of periodic emission in \\citetalias{sanidasLOFARTiedArrayAllSky2019b}, 1 pulse was detected in the archival pointing.\n\n\\subsection{Sources With Periodic Emission}\\label{sec:periodicres}\nOf the $14$ sources that were detected through periodic emission, $7$ sources (J0209+5759, J0226+3356, J0854+5449, J2105+19, J2108+4516, J2215+4524, J2325-0530) have not previously been discussed in any LOFAR works, while J1538+2345 has only been discussed in a single-pulse context by \\citetalias{karako-argamanDiscoveryFollowupRotating2015} (this source is not discussed in \\citetalias{sanidasLOFARTiedArrayAllSky2019b} as the survey pointing had not been performed at the time of publication). The remaining 7 sources were reported as novel detections as a part of the LOTAAS survey, or re-detections of known sources in \\citetalias{sanidasLOFARTiedArrayAllSky2019b}.\n\nJ0209+5759 is a nulling pulsar reported by \\citetalias{goodFirstDiscoveryNew2020}, which placed a lower limit on the nulling fraction of the source of 0.21 due to the limited observing windows of the CHIME instrument. I--LOFAR is able to detect blocks of emission after periodic folding at the nominal frequency that align with observed single-pulses. Using PSRSALSA~\\citep{WeltevredePSRSALSA2016} to analyse 26 hours of data in 30-second sub-integrations, the nulling fraction can be estimated to be 0.953(4) for a 3-sigma detection threshold, or 0.971(3) for a 4-sigma detection threshold.\n\n\\subsubsection{LOTAAS Archive Reprocessing}\\label{sec:archivalmining}\nFrom the 6 novel to LOFAR detections with LOTAAS pointings, the LOTAAS survey pointings for J0226+3356, J0854+5449 and J2215+4524 and show evidence of weak periodic emission, though at signal-to-noise levels that were below the detection threshold for the survey. J2108+4516 was not detected in the LOTAAS pointing, which is to be expected given \\citetalias{goodFirstDiscoveryNew2020} report this source is an eclipsing binary pulsar, and is only visible during narrow windows of the system's orbit.\n\nThe LOTAAS pointing containing CHIME J0209+5759 showed similar behaviour to observations performed at I--LOFAR, with a nulling fraction of 0.95(2) for a 3-sigma cut-off.\n\n\\begin{table}\n\\centering\n    \\caption{Inspected LOTAAS pointing properties. In the case that emission is detected in some manner, the highest SNR beam has been noted, otherwise the closest sub-array-pointing is provided. Sources with a note of 'R' indicates that we detected the source in the pointing in a manner that was not previously reported, 'P' notes there were no nearby pointing in the LOFAR LTA, sources with 'E' were previously reported as detected in a given pointing, but that pointing was not available to be downloaded, and 'N' highlights non-detections.}\n\t\\label{tab:lotaas_summary}\n\\begin{tabular}{llccl}\\hline\\hline\nSource     & Pointing          & \\multicolumn{1}{l}{Bursts} & Periodic      & Note \\\\ \n & ID\/SAP\/BEAM & \\SI{}{\\per\\hour} & $\\sigma$ \\\\\n\\hline\nJ0054+6650 & L625106\/2\/18 & 12 & No & R \\\\\nJ0102+5356 & -- & -- & -- & \\hphantom{R}P \\\\\nJ0139+3336 & L526427\/0\/61 & 17 & No & \\\\\nJ0201+7005 & L642095\/1\/65 & 7 & 10.1 & R \\\\\nJ0209+5759 & L441006\/0\/72 & 0 & Nulling & R \\\\\nJ0226+3356 & L560221\/0\/59 & 0 & 5.4 & R \\\\\nJ0317+1328 & L215823\/1\/15 & -- & -- & \\hphantom{RP}E \\\\ \nJ0332+79 & L441036\/2 & 0 & No & \\hphantom{RPE}N \\\\\nJ0348+79 & -- & -- & -- & \\hphantom{R}P \\\\\nJ0746+5514 & L687604\/2\/67 & 4 & No & R \\\\\nJ0854+5449 & L769469\/2\/40 & 0 & 4.4 & R \\\\\nJ0939+45 & L468036\/0 & 0 & -- & \\hphantom{RPE}N\\\\\nJ1006+3015 & L452116\/1\/44 & 1 & No & R \\\\\nJ1132+2513 & L347106\/1\/47 & 0 & 18.1 & \\\\\nJ1218+47 & L644719\/1 & 0 & -- & \\hphantom{RPE}N\\\\\nJ1329+13 & L568513\/1 & 0 & -- & \\hphantom{RPE}N\\\\\nJ1336+3414 & L347300\/0\/28 & 3 & No & R \\\\\nJ1400+2125 & L646389\/2\/21 & 1 & No & R \\\\\nJ1404+1159 & L521602\/0\/54 & 8 & 76.8 & \\\\\nJ1538+2345 & L773549\/0\/32 & 47 & 12.5 & R \\\\\nJ1848+1516 & L337774\/0\/17 & 51 & 20.4 & \\\\\nJ1931+4229 & L543465\/1 & 0 & No & \\hphantom{RPE}N\\\\\nJ2105+19 & L640743\/1 & 0 & No & \\hphantom{RPE}N\\\\\nJ2108+4516 & L651504\/0 & 0 & No & \\hphantom{RPE}N\\\\\nJ2138+69 & L646349\/1 & 0 & -- & \\hphantom{RPE}N\\\\\nJ2202+2134 & L663798\/2\/70 & 1 & 21.6 & R \\\\\nJ2215+4524 & L599673\/0\/58 & 19 & 8.6 & R \\\\\nJ2325-0530 & -- & -- & -- & \\hphantom{R}P \\\\\nJ2355+1523 & L691558\/2 & 0 & No & \\hphantom{RPE}N\\\\ \\hline\\hline\n\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}\n    \\centering\n    \\begin{tabular}{cc}\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/spectralFitAxis.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/spectralFits\/fluxDensityAxis.pdf} &   \\includegraphics[width=3.62cm]{Plots\/spectralFits\/spectralFitAxis.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/spectralFits\/fluxDensityAxis.pdf} \\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0054+66_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0054+66_modindex_fit.pdf} &   \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0103+54_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0103+54_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(a) J0054+6650 & (b) J0102+5356 \\vspace{0.13cm}\\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0139+3310_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0139+3310_modindex_fit.pdf} &   \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0209+58_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0209+58_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(c) J0139+3336 & (d) J0209+5759 \\vspace{0.13cm}\\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0318+1341_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0318+1341_modindex_fit.pdf} & \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J0746+55_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J0746+55_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(f) J0317+1328 & (g) J0746+5514 \\vspace{0.13cm}\\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1005+3015_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1005+3015_modindex_fit.pdf} & \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1336+3346_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1336+3346_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(h) J1006+3015 & (i) J1336+3414 \\vspace{0.13cm}\\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1400+2127_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1400+2127_modindex_fit.pdf} & \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1538+2345_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1538+2345_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(j) J1400+2125 & (k) J1538+2345 \\vspace{0.13cm}\\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1849+15_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1849+15_modindex_fit.pdf} & \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J1931+4229_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J1931+4229_modindex_fit.pdf}\\vspace{-0.15cm}\\\\\n\n(l) J1848+1516 & (m) J1931+4229 \\vspace{0.13cm} \\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/J2215+45_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J2215+45_modindex_fit.pdf} & \\includegraphics[width=3.62cm]{Plots\/spectralFits\/J2325-0530_spectral_fit.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/modulationIndices\/J2325-0530_modindex_fit.pdf}\\vspace{-0.15cm} \\\\\n\n\\includegraphics[width=3.62cm]{Plots\/spectralFits\/spectralFitAxis.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/spectralFits\/fluxDensityAxis.pdf} &   \\includegraphics[width=3.62cm]{Plots\/spectralFits\/spectralFitAxis.pdf}\\hspace{0.1cm} \\includegraphics[width=3.62cm]{Plots\/spectralFits\/fluxDensityAxis.pdf} \\\\\n(n) J2215+4524 & (o) J2325-0530 \\\\\n    \\end{tabular}\n    \\vspace{-0.1cm}\n    \\caption{The (left columns) spectral flux density fit with a Gaussian function, and (right columns) burst pulse amplitude distribution fit with a function that maximised the AICc for the distribution, for 14 sources observed during the census. Each of these sources had at least 40 bursts detected in order to provide a reasonable sampling of the source properties. See the text (\\S~\\ref{sec:singlepulseres}) for further details on the methodology.}\n    \\label{fig:spectralmodfits}\n\\end{figure*}\n\n\n\\subsection{Novel Source Rotation Periods}\\label{sec:newperiod}\n\nFrom the $25$ sources with detectable single-pulse emission, 9 sources did not have a known period when initially observed during this census (J0348+79, J0939+45, J1006+3015, J1218+47, J1329+13, J1400+2125, J1931+4229, J2105+19 and J2138+69). Three of these sources, J1006+3015, J1400+2125 and J1931+4229, were found to have a sufficient burst rate to determine a rotation period, which can be found in Tables \\ref{tab:obs_summary} and \\ref{tab:newephemerides}. The reported period of J2105+19 is refined from the recent report of \\SI{3.5297}{\\second} from \\cite{tyulbashevWeakPulsar2022}, as determined from an observation with the LOFAR core in June 2021, which will be discussed in future work \\citep{McKennaInPrep}.\n\nIn the cases of J1006+3015 and J1931+4229, long enough observations of the sources in a single observing session were able to generate a sufficient number of times of arrival to determine the underlying source period through the standard brute force method. J1400+2125 required a brute force on the differences in times of arrival, generated from two sets of single--pulses observed across observations on two sequential days.\n\nThe rotation period of RRAT J1336+3414 was also found to differ by a factor of 2 from the published rotation period of \\SI{3.013}{\\second}, at \\SI{1.5066}{\\second}. This is discussed further in~\\S\\ref{sec:j1336}.\n\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[width = 0.45\\textwidth]{Plots\/ppdot_paper.pdf}\n    \\caption{A period-period derivative plot containing the sources of interest in this census, modified from the work of \\protect\\cite{psrqpy}. The background gray dots represent the normal pulsar population, while magnetars and x-ray isolated neutron stars (XINS) have been highlighted due to their regular comparison to RRAT populations. RRATs that were (1) not observed, (2) not detected, (3) detected and (4) newly timed as a part of this work have been separately highlighted. The provided radio pulsar deathline can be found as Eq.~4 in ~\\protect\\cite{bingDeathLine2000}. See~\\S\\ref{sec:discussproperties} for further details.}\n    \\label{fig:rrat_ppdot}\n\\end{figure}\n\n\\subsection{Timing}\\label{sec:timnigres}\nFrom the $29$ detected sources, $\\timeable$ were detected at a high enough burst rate or periodic brightness to allow for pulsar timing with I--LOFAR\\footnote{We consider an average burst rate of \\SI{0.5}{\\per\\hour} for a single-pulse source, or a SNR greater than 10 across a typical \\SI{2}{\\hour} observation for a more standard pulsar, to be the minimum criteria for a source to be timed.}. From these $\\timeable$, $\\wecouldtime$ sources either did not have a full set of pulsar timing parameters available in \\texttt{psrcat}. \n\n\nTo date, $8$ of these sources have had sufficient follow-up to produce stable timing ephemerides, which are presented in Table~\\ref{tab:newephemerides}. Each of these sources can be considered as RRATs. While J2215+4524 has shown intermittent periodic emission, it is infrequent not detectable with I--LOFAR, and consequently is not considered to be a reliable method for timing the source. These sources cover the length and breath of standard RRAT properties, covering an order of magnitude in rotation period, from \\SIrange{0.35}{3.92}{\\second}, period derivatives across three orders of magnitude, \\SIrange{1.1e-16}{3.1e-14}{\\second\\per\\second} and dispersion measures from \\SIrange{8}{55}{\\parsec\\per\\cubed\\centi\\metre}.\n\nOne further source, J0317+1328, has been timed using the LOFAR core (results unpublished), while further observations are ongoing to time the remaining $\\tobetimed$ sources at I--LOFAR.\n\n\\subsection{Blind Pulse Search}\\label{sec:blindresult}\nIn addition to performing narrow dispersion measure range searches for single pulses, all observations that were taken as a part of the census underwent a wide dispersion measure scan for single pulses, from \\SIrange{10}{500}{\\parsec\\per\\cubic\\centi\\metre}. This can be considered as an \\SI{1408}{\\hour} directionally biased survey for fast transients (other RRATs, pulsars with giant pulses and fast radio bursts) as a by-product of this work.\n\nWhile a number of pulses have been detected at dispersion measures that differ significantly from the observed source, verifying the methodology can detect such pulses, these were easily tied to well known pulsars within the beam's field of view, such as B0301+19 or B1842+14.\n\nAs of October 2022, this work has resulted in one novel detection of a source at a dispersion measure of \\SI{18.6}{\\parsec\\per\\centi\\metre\\cubed} near CHIME source PSR J2119+49. No sources near this dispersion measure have been reported within \\SI{10}{\\deg} of the origin beam pointing. The source was only detectable in the bottom \\SI{30}{\\mega\\hertz} of the bandwidth, indicating that it was either off-axis of the beam pointing (up to \\SI{3}{\\deg}) or spectrally steep. 6 hours of follow-up observations to attempt to re-detect and localise this source through the use of 4 \\SI{24}{\\mega\\hertz} beams separated by \\SI{2}{\\deg} from the original pointing have not resulted in a re-detection of this source as of November 2022.\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[width = 0.48\\textwidth]{Plots\/phase_space-crop.pdf}\n    \\caption{A phase space plot of astronomical transients signals (duration against luminosity), modified from the work of \\protect\\cite{pietkaPhase2015} and \\protect\\cite{hurleyWalkerTransient2022} to include the brightest pulses from each of the sources detected as a part of this work. This plot includes a fix to correctly consider the frequency of data in the RRATalog, where previous plots assumed all data was taken at L--band.}\n    \\label{fig:phase_space}\n\\end{figure}\n\n\\section{Discussion}\\label{sec:discuss}\n\n\n\n\\subsection{Source Properties and Detectability}\\label{sec:discussproperties}\n\n\n\\subsubsection{Source Dispersion Measures}\\label{sec:discussdm}\n\nAs seen in Fig.~\\ref{fig:rrat_dms}, both the RRAT population as a whole, and the subgroup detected by the census is heavily biased towards lower dispersion measures, with only 6 of the detected source having a dispersion measure above \\SI{40}{\\parsec\\per\\centi\\metre\\cubed}. \n \nThis is likely a result of the scattering of the (an-)isotropic medium along the line of sight between the observing telescope and the neutron star, causing the pulse emission to be dispersed in time (reducing the sensitivity of the instrument by $\\sqrt{\\tau}$) and fall below the noise floor of the instrument as a result. \n\nScattering effects are particularly strong at the lower frequencies used by many of the telescopes behind the catalogues used for this census. As described in \\cite{cordesPropgation2002}, the scattering timescale of a radio pulsar is dependent on the dispersion measure ($\\log\\tau_\\text{sc}\\sim\\log\\text{DM} + \\left(\\log\\text{DM}\\right)^2$) and observing frequency ($\\tau\\sim\\nu^{-4}$). While lower dispersion measure sources are still detectable, high dispersion measures cause the pulses to be broadened by an order of magnitude or even more in time. Using Eq.~10 of \\citeauthor{cordesPropgation2002}, we can estimate the scatter broadening of a RRAT with a dispersion measure of 10, 30, 100 and \\SI{300}{\\parsec\\per\\centi\\metre\\cubed} observed at \\SI{150}{\\mega\\hertz} to be 0.02, 0.29, 24 and \\SI{4100}{\\milli\\second}. With the described frequency scaling, these are expected to be more than three orders of magnitude larger at \\SI{150}{\\mega\\hertz} as compared to L-band, and nearly 30 times larger than at P-band.\n\nGiven the typical brightness of these sources, the sensitivity of I--LOFAR, and typical observed pulse widths from these sources at higher frequencies, the potential to detect a RRAT with a dispersion measure beyond \\SI{100}{\\parsec\\per\\centi\\metre\\cubed} is highly unlikely.\n\nSimilarly, these high dispersion measure sources must be close to the Galactic plane due to the sharp falloff of free electrons away from the Galactic disk. As previously described, the disk also resulted in significantly higher $T_{\\text{sky}}$ measurements, which can severely hamper the instantaneous sensitivity of I--LOFAR towards these sources, which is a further compounding effect that reduces the detectability of these sources.\n\n\\subsubsection{Source Rotation Periods}\n\nAs seen in Fig.~\\ref{fig:rrat_periods}, the majority of the sources observed but undetected in this census do not have a known rotation period. This is to be expected as in order for the period of a RRAT to be determined it must produce pulses that are sufficiently bright to meet the detectability criteria of the observing telescope, and have a sufficient burst rate at that sensitivity for multiple time-of-arrival measurements in a short window of time to allow for a rotation period to be brute-forced. Observations with I--LOFAR over a 5 hour window may not have been sufficient time to detect a single pulse in the case that the pulse-amplitude distribution has a sufficiently long, but low-probability tail to produce pulses within our sensitivity range.\n\nWhile longer period sources may have been able to offset the effects of scattering discussed in~\\S\\ref{sec:discussdm} due to wider intrinsic pulse widths at typical duty cycles, fewer rotations of the pulsar are observed, reducing the number of samples taken from the tail of the pulse amplitude distribution.\n\n\\subsubsection{Pulse Amplitude Distributions}\nThe pulse amplitude distributions are visualised in Fig.~\\ref{fig:spectralmodfits}, with the underlying fitted data in Table~\\ref{tab:pulseamplitudetab}. The majority of these distributions are well fit by a combination of powerlaw or log-normal distribution, with the exceptions J1848+1516 and J0746+5514.\n\nIn the case of J1848+1516, this is expected to be a result of the detected pulse brightness being extremely close to the sensitivity limits of the I--LOFAR instrument. From the 68 detected pulses, only 40 per cent were above a SNR of 8, while only 13 per cent were above a SNR of 9. This low sample size, taken from the tail of the pulse amplitude distribution, results in a skewed distribution which should not be considered to be an accurate model of the underlying distribution (which should be visible with higher sensitivity instruments).\n\nJ0746+5514 appears to have a flat pulse amplitude distribution. While this may be in part due to the first bin causing a positive shift on the fitted function, the raw data for this source is extremely erratic. The observed pulses are at an abnormally high significance, with only 16 per cent of the 48 bursts below a SNR of 9 and more than half of the observed pulses have a SNR above 13, making the pulses from this source extremely bright in the data. As presented in Table~\\ref{tab:obs_summary}, the standard deviation of the pulse width is more than 60 per cent of the mean pulse width (the observed pulses have widths varying from \\SIrange{3}{77}{\\milli\\second}). The flat distribution is likely a consequence of the relatively low sample of pulses from such a diverse population, and will require further follow-up to accurately characterise the pulse amplitude distribution.\n\n\\subsubsection{Spectral Indices}\nWhile spectral indices of RRATs have been shown to be more extreme than the normal pulsar population \\citep{shapiro-albertRadioPropertiesRotating2018}, it is expected that the single-pulse spectral indices presented in Table~\\ref{tab:obs_summary} and Fig.~\\ref{fig:spectralmodfits} are biased towards being steeper than the actual emission properties. Multiple different factors contribute to this, including (a) the position uncertainty on a number of the sources (discussed in~\\S\\ref{sec:sources}) as the beam sensitivity is reduced for off-axis sources as the beam narrows at higher frequencies, and (b) reduced SNRs after splitting the bandwidth into \\SI{10}{\\mega\\hertz} segment for low significance pulses in order to better sample the spectral behaviour of the sources. (b) will have a particularly strong effect on sources with more extreme spectral indices (both positive and negative), as the gain of I--LOFAR is relatively flat across the observed bandwidth, potentially causing the dimmer emission to fall below the noise floor near the edges of the bandwidth.\n\nWe can identify sources that have these biases when they have been detected through both single-pulse and periodic, as after folding the data to produce a folded archive there will likely be sufficient data to have a strong SNR across the entire sampled bandwidth. Consequently, the effects of (a) can be seen for the untimed source J0317+1328 ($\\alpha_{\\text{sp}}=-5.3(8)$ and $\\alpha_{\\text{f}}=-1.8(4)$), while (b) is likely contributing to the spectral index disparity of J1538+2345 ($\\alpha_{\\text{sp}}=-1.6(12)$ and $\\alpha_{\\text{f}}=-0.51(17)$) and, despite the scattering causing a strong positive bias in the single pulse data, J1848+1516 ($\\alpha_{\\text{sp}}=-0.2(4)$ and $\\alpha_{\\text{f}}=0.52(14)$).\n\nJ2108+4516, a binary pulsar, is a case where the spectral index is significantly more positive in the single-pulse data as opposed to the periodic emission ($\\alpha_{\\text{sp}}=1.91(14)$ and $\\alpha_{\\text{f}}=-0.4(4)$). This is likely as a result of the source showing strong signs of scattering (in the folded profile visible in Fig.~\\ref{fig:periodicprofiles}(k)), with the resulting diminished SNRs at lower frequencies causing the source to appear to have a positive spectral index in the single-pulse data.\n\n\\subsubsection{Further Limits on Detectability}\n\nSome sources in the PRAO catalogue may have not been detected due to limitations in the reported positions of sources. Due to the fixed declination and wide full-width-half-max (FWHM) of their beams, position uncertainty may be present in their initial reported positions which can only be reduced with the detection of further pulses (to report a better mean right ascension, and more accurate declination). \n\nThis effect is noted specifically for RRAT J1336+3414~(\\S\\ref{sec:j1336}), where the reported position and timed position differed by \\SI{0.48}{\\degree}, (between a third and a fifth of the beam FWHM depending on the observing frequency). Correcting for the position of this source is also correlated with the detection of off-pulse emission that leads the main pulse train, demonstrating even these small changes can have an effect on detectability of pulses from these sources. Quieter sources on the edge of detectability would be unlikely to be detected with such a large offset between the true sky position and the observed sky position.\n\n\\subsubsection{Phase Space}\n\nThe transient phase space plot in Fig.~\\ref{fig:phase_space} shows that the pulses detected as a part of this census have been brighter but have shorter spectral duration than emission previous reported for RRATs.\n\nIn terms of luminosity, the baseline data for RRATs is generated from entries in the RRATalog, which has been described in~\\S\\ref{sec:rratalog}, where sources are primarily described at either L-band or P-band observations. With the expected brightening of these sources at lower frequencies, combined with the selection effects from the lower relative sensitivity of I--LOFAR compared to other instruments at their respective frequencies, it is not a surprise that the results from this work tend to be at a higher brightness than the previous RRAT population on this plot.\n\nWhile the multiple surveys cited as the data sources for the RRATalog data provide the peak pulse brightness for each source, it is unclear what form of sampling is used to determine the width values, with different works using a combination of median, mean and brightest pulse sampling to provide the pulse width. For this work, the width of the brightest pulse for each source was used to generate the data for the plot.\n\nRegardless, a clear separation is visible between the sources detected as a part of this work and the general RRAT population. This is likely due to the selection effects previously mentioned that result in the telescope detecting strong, minimally scattered pulses with similar widths to those presented in the RRATalog, resulting in an order of magnitude increase in the luminosity, and order of magnitude reduction in the frequency-duration samples from the lower observing frequency.\n\n\\subsubsection{Comparisons to Previous LOFAR Work}\n\nDuring the work of \\citetalias{karako-argamanDiscoveryFollowupRotating2015}, 5 of the sources characterised as a part of this work were detected using 20--22 stations of the LOFAR core, while our reprocessing of the LOTAAS survey (\\S\\ref{sec:archivalmining}, \\citetalias{michilliSinglepulseClassifierLOFAR2018}, \\citetalias{sanidasLOFARTiedArrayAllSky2019b}) in Table~\\ref{tab:lotaas_summary} lead to 12 detected sources, with 2 overlapping. \n\nComparing the observed burst rates of these 15 detected sources, there are two distinct categories of sources: those with similar observed burst rates at both telescopes, and those with significant increases in observed burst rate as compared to this work.\n\nThe former group of 8 sources (J0054+6650, J0102+5356, J0139+3336, J0746+5514, J1006+3015, J1336+3414, J1400+2125, J2325-0530) can be considered a group of sources with the potential to be monitored with an international LOFAR station, with any observing time offering a relatively high completeness as compared to significantly more sensitivity instruments. These sources may have a discontinuity in their pulse amplitude distributions between the emission during the majority of rotations and the observable giant pulses detected during this work.\n\nIt is notable that out of the 7 sources that fall in the latter group (J0201+7005, J0332+79, J1404+1159, J1538+2345, J1848+1510, J2202+2134, J2215+4524), all but one source, J0332+79, were found to produce detectable periodic emission in at least one observation during the census and further follow-up observations. While none of these sources were consistently detectable at I--LOFAR, J1538+2345 produced consistent periodic emission when observed with FAST~\\citep{luStudyThreeRotating2019a}, and J1848+1510 can be found to produce persistent periodic emission during follow-up observations present in the LOFAR LTA. Consequently, there is an argument to re-classify these sources as highly variable pulsars rather than RRATs, given an increase in sensitivity allows for consistent detection of periodic emission at some sites.\n\nExcept for J1404+1159 and J0854+5449, the sources with periodic emission detected during the LOTAAS survey and presented in Table~\\ref{tab:lotaas_summary} have shown significant variability during their observations at I--LOFAR. Consequently, a comparison of the emission properties between these epochs is unlikely to offer any further insight beyond what is presented elsewhere within this work.\n\n\\subsection{Sources of Interest}\n\n\n\\subsubsection{PSR J0209+5759}\n\nJ0209+5759 is a nulling pulsar reported by the CHIME\/FRB collaboration, and it's properties at \\SI{600}{\\mega\\hertz} have been discussed in \\cite{goodFirstDiscoveryNew2020}. It has a high level of nulling, with \\citeauthor{goodFirstDiscoveryNew2020} reporting a lower limit on the nulling fraction of 21 per cent, though while active, they typically can observe \\SI{21.4(5)}{\\per\\hour}. They reported two estimates of the dispersion measure of the source, of \\SI{55.282}{\\parsec\\per\\centi\\metre\\cubed}, though this was calculated through PRESTO's \\texttt{prepfold} rather than through analysis of single pulses, and \\SI{55.3(6)}{\\parsec\\per\\centi\\metre\\cubed} from the CHIME\/FRB system metadata.\n\nAt I--LOFAR, this source has been detected through both single-pulse and period emission. Analysis of the nulling behaviour across \\SI{26.6}{\\hour} of observations indicates an upper bound for the nulling fraction of 95.3(4) per cent for\na 3-sigma detection threshold of emission on \\SI{30}{\\second} windows, though given the source likely has a lower SNR at I--LOFAR, and the granularity of the data, this could be an overestimate of the true nulling fraction.\n\nThe source has a significantly lower single-burst rate at I--LOFAR as compared to CHIME, with only 1.6(2) bursts observed per hour on average, though this peaked at \\SI{7(1)}{\\per\\hour} during a \\SI{40}{\\minute} observation in November 2021.\n\nThe dispersion measure of the source at I--LOFAR was measured as \\SI{55.855(4)}{\\parsec\\per\\centi\\metre\\cubed}, within the confidence interval of the CHIME\/FRB measurement, but offset from the \\texttt{prepfold} estimates. At \\SI{151}{\\mega\\hertz}, the lower bound of their reported measurements results in a distinct\n\nInspecting the LOTAAS archival pointing of the source indicates that the nulling emission was detectable during the survey (using the same methodology as I--LOFAR indicated a nulling fraction of 95(2) per cent during the hour-long observation), however no single pulses were detected during the observation. This may be a consequence of the relatively flat spectral index combined with the lower observing frequencies of the LOTAAS survey (typically \\SIrange{119}{151}{\\mega\\hertz}) causing pulses to be below the sensitivity limit of the instrument, or the source may not have produced pulses bright enough to be detected by either I--LOFAR or the 8-core stations used during the LOTAAS survey (41 per cent of observations at I--LOFAR did not result in a detection of J0209+5759).\n\n\\subsubsection{PSR J0226+3356}\nWhile initially announced as RRAT J0226+3356 in the CHIME catalogue, the recent work of \\citetalias{dongCHIME2022} reclassifies the source as a pulsar due to persistent periodic emission that has been detected with the CHIME telescope.\n\nObservations at I--LOFAR have yet to detect a burst from this source, however periodic emission has been detected in all but one observation taken to date. \n\n\\subsubsection{RRATs J0939+45, J1218+47 and J2138+69}\nRRATs J0939+45 and J1218+47 are recent discoveries by the Pushchino telescope, while RRAT J2138+69 was reported by the CHIME\/FRB collaboration, which have been detected on a number of occasions by I--LOFAR, despite their lack of known period. While these sources have low burst rates compared to other sources in this work (0.16(9), 0.37(10) and \\SI{0.292(97)}{\\per\\hour}), they have been detected on a regular basis due to a number of extended observations over recent months. It is hoped that a period will be able to be determined for these sources with the in-development spare-sampled period solver \\texttt{altris} software~\\citep{morelloPrivateComms} after further pulse detections, and phase-coherent ephemerides produced following regular observations with I--LOFAR.\n\n\\subsubsection{RRAT J1006+3015}\nJ1006+3015 was reported by the PRAO catalogue, and was found to have a period of \\SI{3.066}{\\second} and further timed as a part of this work. While an uncertainty was not provided, this result is similar to the multi-epoch period estimate of \\SI{3.069}{\\second} presented in \\cite{smirnovRRATVariability2022}.\n\nWhile relatively infrequent as compared to J2325-0530, this source has consistently produced extremely high significance sources observed during this work, with 8 of the 10 highest SNR bursts during the census observed during follow-up observations of this source, and an overall brightness ratio between the brightest and dimmest pulse of 16. The brightest pulses have demonstrated interesting morphologies, such as significant swings in amplitude (as visible in the best-pulse plot, Fig.~\\ref{fig:singleprofiles}) or multi-peaked pulses. Further follow-up with a more sensitive instrument may help determine if these are unique traits to these bright pulses, or an underlying variability that falls below the noise floor for observations with I--LOFAR.\n\n\n\\subsubsection{PSR J1132+2513}\nJ1132+2513 is reported by the PRAO catalogue, and, as a source of RRAT--like bursts, though was previously only reported as a periodic source during the GBT350 Drift Scan Survey~\\footnote{While reported on the \\href{https:\/\/web.archive.org\/web\/20220626134940\/http:\/\/astro.phys.wvu.edu\/GBTdrift350\/}{survey webpage}, the source is never discussed in any publications regarding the survey, has an unknown discovery date and discovery plot with half the current period.} (J1132+25) and LOFAR (J1134+25). During this work, no single pulses were detected, though the periodic emission reported during the LOTAAS survey was detected.\n\nThis behaviour appears to align with the previous report from \\cite{smirnovRRATVariability2022} which indicates the source is an intermittent source of single-pulse emission, at a low single-to-noise ratio. Given this pulsar currently does not have a full timing ephemeris in the pulsar catalogue (entry as J1132+25), future observations performed at I--LOFAR with the goal of timing the source through the detectable periodic emission may result in the detection of single pulse emission from the source.\n\n\\subsubsection{RRAT J1329+13}\\label{sec:J1329}\nJ1329+13 is a source from the PRAO catalogue with an extremely low observed burst rate \\citep{tyulbashevDetection25New2018}, with the current online version of the PRAO catalogue only reporting a single pulse being detected from the source with the BSA LPI. \n\nFrom \\SI{50}{\\hour} of observations with I--LOFAR, only 5 pulses have been detected from the source. The first two pulses were observed within the space of a week in February 2021, and it was not detected again until two further pulses were detected in November and early December 2021. Assuming there is no bias on the temporal distribution of pulses (due to nulling, presence in binary system or intermittency), this would give the source a burst rate of \\SI{0.10(5)}{\\per\\hour} for an ILT station, and \\SI{0.008(8)}{\\per\\hour} at the BSA LPI. \n\nIf the time between pulses were to form a standard Poisson distribution, unless the source has a strongly positive spectral index (which is unlikely given the properties of the few observed pulses), the BSA LPI burst rate should be expected to be closer to that of the I--LOFAR measurement, if not higher due to the increased sensitivity of the instrument. As a result, it can be inferred that this source likely acts similarly to an intermittent neutron star.\n\nA blind periodicity search across the 5 observations with any observed pulses using riptide \\citep{morelloOptimalPeriodicitySearching2020b} did not result in a credible detection of periodic emission from the source, with 7$\\sigma$ upper limits of between \\SI{1.6}{\\milli\\jansky} and \\SI{1.0}{\\milli\\jansky} assuming a 1 per cent duty cycle and that the source was active for the entirety of the observations.\n\n\\subsubsection{RRAT J1336+3414}\\label{sec:j1336}\nThe reported timing ephemeris for BSA RRAT J1336+3414 differs significantly from the original parameters published in \\cite{tyulbashevDetection25New2018} and subsequent papers. Pulsar timing indicates it is offset by \\SI{0.48}{\\deg} from the reported position, and the rotation period was found to be half of the originally reported \\SI{3.013}{\\second}. \n\nThe confusion in the reported source period may have been the result of an infrequent off-pulse component visible in the pulsar timing residuals. After updating the source pointing to match the fit position from pulsar timing, 11 pulses (10 per cent) have been observed to have a 0.044 phase (roughly \\SI{66}{\\milli\\second}) offset from the majority of observed pulses. Off-pulse components can cause issues for most algorithms used to brute-force RRAT periods and may have been biased to a harmonic rotation frequency, which was then reported.\n\nJ1336+3414 has both the slowest spin down rate and the oldest characteristic age of any RRAT, with \\SI{1.11(3)e-16}{\\second\\per\\second} and \\SI{216}{\\mega\\year}, replacing J1647-3607 (\\SI{1.29(2)e-16}{\\second\\per\\second}) and J1739-2521 (\\SI{120}{\\mega\\year}). It's rotation period and derivative result in it sitting extremely close to the standard radio pulsar deathline~\\citep[equation 4]{bingDeathLine2000}, making it an interesting source for modelling the emission properties of ageing pulsars.\n\n\\subsubsection{RRAT J1538+2345}\nJ1538+2345 is described as initially being reported as part of the GBT \\SI{350}{\\mega\\hertz} drift-scan survey, though did not appear in any publications until it received follow-up with both the GBT and LOFAR core to characterise and time it prior to this work, with the results discussed in \\citetalias{karako-argamanDiscoveryFollowupRotating2015}.\n\nOn several occasions it has been detected through periodic folding at I--LOFAR, with the folded profile in Fig.~\\ref{fig:periodicprofiles} showing a dual peaked source, with a quieter leading peak compared to the lagging peak, with weak emission between the two peaks. Single-pulses are detectable from both peaks, and form two distinct pulse trains with a separation of \\SI{0.024}{} rotations when times of arrival are analysed. 30\\% of the pulses align with the first peak, while the remainder align with the second. No pulses at I--LOFAR have been detected offset from these two groups.\n\n\n\\subsubsection{J1848+1516}\nJ1848+1516 is a source reported by several surveys. The RRATalog cites the discovery to the GBNCC survey as J1849+15 (no publications discuss this, survey pages do not mention the source), it was detected as a part of the LOTAAS survey in \\citetalias{michilliSinglepulseClassifierLOFAR2018}, and by the PRAO as J1848+1518 in \\cite{tyulbashevDetection25New2018}.\n\n\\cite{michilliLOFARTiedarrayAllsky2020} discuss the source at a higher sensitivity to the observations taken at I--LOFAR and note the source appear to show both strong pulse-to-pulse variability, and some nulling behaviour, with emission turning on and off on the order of tens of rotations. They also note that the three peaks visible in a folded profile of the source (as seen from I--LOFAR in Fig.~\\ref{fig:periodicprofiles}(i)) are extremely time variable, with the main peak being both lead and lagged by secondary and tertiary peaks intermittently. The data taken with I--LOFAR does not have the sensitivity required to allow us to further comment on this behaviour.\n\nSingle-pulse times of arrival from this source at I--LOFAR have only been detected to align with the main peak of the folded profile.\n\n\\subsubsection{RRAT J1931+4229}\nJ1931+4229 was one of the earliest source detected by CHIME\/FRB and announced in the CHIME\/FRB Galactic Source catalogue. It was initially observed to have a high burst rate at low frequencies and as a result a rotation period of \\SI{3.921}{\\second} was determined in December 2020 through the detection of 13 pulses within an observation. This result was also confirmed as a result of 9 and 11 pulses during single observations in April and June 2021.\n\nThis was the slowest source detected (and timed) as a part of this census, and the resulting ephemeris indicates it falls into a region period-period derivative phase space (see Fig.~\\ref{fig:rrat_ppdot}) where sources have properties that border those of x-ray isolated neutron stars (XINS) and magnetars. It is unknown whether these sources represent a sub-set of neutron stars separate from the remainder of the RRAT population, only associated by emission properties, or are just a more extreme group of RRATs.\n\nThis source was also detected during an expanded search using the BSA LPI. In \\cite{samodurovDetectionStatisticsPulse2022}, the authors discuss it and report that it has a rotation period of \\SI{3.6375}{\\second}. Our data has not been able to reproduce this result, and attempting to process times of arrival at I--LOFAR using this result as a trial period does not result in a coherent timing solution.\n\nThis source was not detected during re-processing of the LOTAAS archive, which was unexpected given the burst rate of \\SI{2.5(2)}{\\per\\hour} at I--LOFAR, which indicates it was likely to be detected within an hour-long pointing, and the negative spectral index, of $-1.06(7)$, that would cause it to have a higher SNR when observed with the 8-core station in the LOTAAS survey's configuration.\n\n\\subsubsection{RRAT J2215+4524}\nRRAT J2215+4524 was an early source announced by the CHIME\/FRB collaboration. An ephemeris and overall source characteristics at~\\SI{600}{\\mega\\hertz} are described in the recent work of \\citetalias{dongCHIME2022}. We note that the position and period parameters in the ephemeris provided Table~\\ref{tab:newephemerides} have lower limits than provided in their work.\n\n\\subsubsection{RRAT J2355+1523}\nRRAT J2355+1523 was announced by the CHIME\/FRB collaboration, and a timing ephemeris has been published in the recent work of \\citetalias{dongCHIME2022}. \n\nWe are highlighting the source as while it was detected on a regular basis at I--LOFAR prior to July 2021, with a burst rate of \\SI{0.49(16)}{\\per\\hour}, since then it was only been detected 5 times at a rate of \\SI{0.12(6)}{\\per\\hour}. Prior to July 2021, the source was detectable at a reasonable significance, with 8 of the 10 detected pulses having SNRs above 8, and two being above 11. However, none of the 5 pulses observed since July 2021 have been above a SNR of 8. \n\nThe CHIME\/FRB collaboration provides a plot of detected epochs for a number of sources, including this source\\footnote{\\href{https:\/\/storage.googleapis.com\/chimefrb-dev.appspot.com\/J2355+1523\/J2355+1523_prepfold.png}{Direct link for J2355+1523}, and an \\href{http:\/\/web.archive.org\/web\/20230213164601\/https:\/\/storage.googleapis.com\/chimefrb-dev.appspot.com\/J2355+1523\/J2355+1523_prepfold.png}{archival copy}}, with their catalogue. Inspecting the plot indicating this source might show signs of intermittency, but on shorter time scales than the delay between detections at I--LOFAR. Further coordinated observations may be required to determine the behaviour of this source, and if it is variable depending on observing frequency.\n\n\\subsection{Viability of Single LOFAR stations for RRAT Monitoring}\nThis work has shown that nearly a third of known Northern Hemisphere RRATs can be detected with an international LOFAR station working as a standalone instrument. Given the lack of follow-up and systemic analysis of a large amount of these sources, ongoing monitoring with the 14 international stations could allow for more details of these elusive neutron stars to be unearthed, especially in the case of sources with significantly low burst rates that can't be easily monitored.\n\nThe planned scheduling improvements being rolled out for LOFAR 2.0 will give rise to the LOFAR Mega Mode\\footnote{\\href{https:\/\/web.archive.org\/web\/20220703090811\/https:\/\/www.astron.nl\/what-we-look-forward-to-in-lofar-2-0-a-brain-transplant-for-lofar\/}{ASTRON.nl: What we look forward to in LOFAR 2.0: A brain transplant for LOFAR}}. This automated scheduler allows for the international stations to be scheduled to be controlled independent of the core and other station while the telescope operates in `international mode', and is expected to provide a wealth of new data for pulsar timing and other monitoring campaigns. \n\nThis work shows that RRATs (and other similarly bright radio transients) could be another pool of sources that can be sampled for monitoring efforts with the new scheduler. Several sources have been shown to have low-burst rates (J0348+79, J2355+1523), or appear extremely intermittent (J1329+13), when observed with a single station, such that they are detected too infrequently to be reliably monitored alongside other observation campaigns at a single station during `local mode' observations. Mega Mode would allow for these sources to be monitored during what would otherwise be downtime for international stations while observations are taken with the core, and help further investigate these elusive sources.\n\n\n\\subsection{Fast Radio Burst Search}\\label{sec:frbdiscuss}\n\nAs mentioned in~\\S\\ref{sec:processing}, each observation taken as a part of this census was additionally searched for single-pulse candidates between \\SIrange{10}{500}{\\parsec\\per\\centi\\metre\\cubed} to contribute to a directionally biased search for FRBs as a part of this census. However, no credible candidates (apart from the Galactic sources discussed in \\ref{sec:blindresult}) were detected as a part of this work.\n\n\nA blind FRB search at \\SI{145}{\\mega\\hertz} has been performed using the ARTEMIS backend at the Rawlings (UK) and Nan\u00e7ay (FR) LOFAR stations \\citep{karastergioArtemis2015}, using $8\\times$\\SI{ 6}{\\mega\\hertz} bandwidth beams to increase the observed sky area from a normal HBA beam field of view of \\SI{3.6}{\\square\\deg} to \\SI{28.8}{\\square\\deg} for a similar amount of time to this census (\\SI{1446}{\\hour}). From this, they determined an upper limit on the fast radio burst rate of \\SI{29}{\\per\\sky\\per\\day} in the \\SIrange{143}{149}{MHz} range, and a dispersion measure search for pulses at up to \\SI{300}{\\parsec\\per\\centi\\metre\\cubed}.\n\nThis work consisted of a similar amount of observing time, but this work has used a wider bandwidth, with an upper limit of \\SI{89}{\\mega\\hertz} after RFI flagging, rather than the wider field of view used in  the work of \\citeauthor{karastergioArtemis2015}. Considering the overall volume of sky probed, the sky sampled as a part of this work was probed to a depth 7.6 times greater than their work, however as a consequence of the reduced field of view, we only reach an upper limit on the low--frequency fast radio burst rate that is 92\\% ($\\sim$\\SI{29}{\\per\\sky\\per\\day}) of that presented by \\citeauthor{karastergioArtemis2015}.\nIt should additionally be noted that a fraction of time spent observing was near the Galactic plane, which can be considered occulting due to the high sky temperatures severely decreasing the telescope's sensitivity (\\S\\ref{sec:sensitivity}).\n\n\nIt is worth highlighting that the \\texttt{heimdall} search software used for the census is optimised to detect broadband, spectrally flat sources, which are words not often associated with the complex burst structures of many types of fast transients. Such behaviour has been clearly documented for FRB 20180916B at the LOFAR core, see~\\cite[Fig.~2 and Table~1 of][]{pleunisLOFARR32021}. As a result, reprocessing the data across subsections of the bandwidth is planned for the future, and may result in previously missed candidates being detected.\n\n\\section{Conclusions and Future Work}\\label{sec:conclusions}\nWe have shown that a reasonable number of Northern Hemisphere RRATs can be both detected and monitored with a single international LOFAR station. From the 29 detected sources, 4 rotation periods have been updated and 8 new phase-coherent pulsar timing ephemerides have been presented. 15 sources had sufficient detected bursts that their emission properties could be studied.\n\nObservations are ongoing at I--LOFAR to continue the census for newly announced sources and slowly observe the sources that were not initially observed from the RRATalog (as described in~\\S\\ref{sec:rratalog}), while also monitoring and timing a number of the sources mentioned in this work. Plans are in place to start extracting full Stokes $IQUV$ data to characterise the polarisation properties of pulses from these sources in the near future.\n\nPlans were already made during the census to inspect some sources detected in early 2021 and 2022 at other frequencies and higher sensitivities at \\SI{150}{\\mega\\hertz}. As a result, observations and initial analysis have been completed for a sub-set of the discussed sources at some of the most sensitive instrument at their respective observing bandwidths. These include observations at FAST (L-band), the LOFAR core (24 stations, \\SI{150}{\\mega\\hertz}) and NenuFAR (72 mini-arrays, \\SI{50}{\\mega\\hertz}).\n\nThe RRAT census dataset itself also has a high legacy value, which is likely to yield more scientific results in the future. For instance, blind periodicity searches could yet be performed on the census data; these were computationally infeasible at the time of data collection, but this is becoming ever more tractable with time. To aid in such investigations, a publicly accessible I--LOFAR data archive is currently being created, the details of which will be reported when it is complete. Such an archive will facilitate archival analysis as new ideas arise and\/or when improved\/faster algorithms allow for deeper studies of extant data.\n\n\\section*{Acknowledgements}\\label{sec:acknowledgements}\nThe Irish LOFAR Consortium consists of Trinity College Dublin, Armagh Observatory and Planetarium, University College Dublin, Dublin City University, University College Cork, University of Galway, Dublin Institute for Advance Studies and Technological University of the Shannon Athlone. It receives generous funding from Science Foundation Ireland and the Department of Further and Higher Education, Research, Innovation and Science.\n\nThe REALTA compute cluster was funded by Science Foundation Ireland.\n\n\nWe acknowledge use of the BSA-Analytics project catalogues, provided at \\url{https:\/\/bsa-analytics.prao.ru\/en\/}, the CHIME\/FRB Public Database, provided at \\url{https:\/\/www.chime-frb.ca\/} by the CHIME\/FRB Collaboration and the RRATalog, provided at \\url{http:\/\/astro.phys.wvu.edu\/rratalog\/} by Cui and McLaughlin.\n\nD.J.McK is receiving funding under the Government of Ireland Postgraduate Scholarship (GOIPG\/2019\/2798) administered by the Irish Research Council (IRC).\n\n\n\\section*{Data Availability}\\label{sec:data}\nThe data of single pulses, pulse profiles and timing solutions, can be found on Zenodo~\\citep{dmckennaCensusData2023}\\footnote{\\url{https:\/\/zenodo.org\/deposit\/4438541}}. The code used to generate the final data products can be found on GitHub~\\citep{dmckennaCensusScripts2023}\\footnote{\\url{https:\/\/github.com\/David-McKenna\/RRATCensusScripts}}.\n\nAccess to the archived datasets described in~\\S\\ref{sec:processing} can be made available upon request to the authors.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nSeemingly unrelated regressions (SUR) are multivariate regression\nmodels with correlated response (or dependent) variables that follow a\njoint Gaussian distribution. Usually different regressions contain\ndifferent covariates (or independent variables) and seem\n``unrelated.''  However, due to the correlated response variables the\nregressions are only ``seemingly unrelated'' and contain valuable\ninformation about each other \\citep{zellner:62}. SUR play ``a central\nrole in contemporary econometrics'' \\cite[p.~323]{goldberger:91} but\nalso appear in other contexts\n\\citep{rochon:jrss,rochon:bio,verbyla:1988}.  Moreover, SUR arise in\nthe context of Gaussian graphical models\n(\\citealp[\\S5]{amp:cgmp}; \\citealp[\\S8.5]{rich:annals}).  \n\nThe parameters of a SUR model can be estimated efficiently, i.e.\\ with\nsmall variance, by maximizing the likelihood function, which maps the\nparameters to the likelihood of observing the given data.\n\\cite{oberhofer:74} and \\cite{telser:64} give two popular algorithms\nfor this maximization.  In general, however, these algorithms will not\nglobally maximize the likelihood function, which indeed may be\nmultimodal; a fact neglected in the literature \\cite[\\S6]{drton:2004}.\n\\cite{drton:2004} demonstrated the possibility of multimodality in a\nstudy of a bivariate SUR model that may have a likelihood function\nwith five stationary points. In this paper, we use algebraic geometry\nto apply the approach of \\cite{drton:2004} to more general SUR models.\nIn Sections \\ref{sec:sur} and \\ref{sec:mlepoly} we give an\nintroduction to SUR and show how maximum likelihood estimation can be\nperformed by solving a polynomial optimization problem, opening the\ndoor for tools from algebraic geometry.  With\nthese tools, we first revisit the work by \\cite{drton:2004}, see\nSection \\ref{sec:revisit}, and then obtain new results on more general\nSUR models (Section \\ref{sec:groebner}).  In particular, we identify\nexamples of SUR models, for which all stationary points of the\nlikelihood function can be computed.\n\n\n\\section{Seemingly unrelated regressions}\n\\label{sec:sur}\n\nIn SUR a family of response variables, indexed by a finite set $R$, is\nstochastically modeled using a family of covariates, indexed by a finite\nset $C$.  All response variables and all covariates are observed on a\nfinite set of subjects $N$. We denote the cardinalities of the three sets\nalso by $R$, $C$ and $N$, respectively. The\nobservations can be represented by two matrices $X$ and $Y$.  The matrix\n$Y=(Y_{rm})\\in\\mathbb{R}^{R\\times N}$ has the $(r,m)$-entry equal to the\nobservation of response variable $r\\in R$ on subject $m\\in N$, and the\nmatrix $X=(X_{cm})\\in \\mathbb{R}^{C\\times N}$ has the $(c,m)$-entry equal to the\nobservation of covariate $c\\in C$ on subject $m\\in N$.  For $c\\in C$ and\n$r\\in R$, $X_c\\in\\mathbb{R}^N$ and $Y_r\\in \\mathbb{R}^N$ denote the $c$-th and $r$-th\nrow of $X$ and $Y$, respectively.  Similarly, $X^m$ and $Y^m$, $m\\in N$,\ndenote the $m$-th column of $X$ and $Y$, respectively. Clearly, $X_c$ and\n$Y_r$ comprise all observations of the $c$-th covariate and the $r$-th\nresponse variable; $X^m$ and $Y^m$ comprise all covariate and response\nvariable observations on the $m$-th subject.\n\nIn this regression setting, the matrix $X$ is assumed to be\ndeterministic and fixed but the matrix $Y$ is modeled to follow a\nmultivariate normal distribution, where the mean vector of $Y_r$, $r\\in R$,\nis a linear combination of some $X_c$, $c\\in C_r\\subseteq C$,\n\\begin{equation}\n  \\label{eq:betadef}\n  \\mathrm{E}[Y_r] = \\sum_{c\\in C_r} \\beta_{rc} X_c\\in \\mathbb{R}^N, \\quad r\\in R. \n\\end{equation}\nHere $(C_r\\mid r\\in R)$ is a fixed family of subsets of\n$C$ indexing the covariates involved in each one of the $R$ regressions. The\nweights $\\beta_{rc}$ in (\\ref{eq:betadef}) are called {\\em regression\n  coefficients\\\/}.  Setting $\\beta_{rc}=0$ if $c\\not\\in C_r$, \nwe can define a matrix of regression coefficients\n$B=(\\beta_{rc})\\in \\mathbb{R}^{R\\times C}$. The random vectors $Y^m$, $m\\in\nN$, are assumed to be independent with common positive definite covariance\nmatrix  \n\\begin{equation}\n  \\label{eq:sigmadef}\n  \\mathrm{Var}[Y^m]=\\Sigma\\in \\mathbb{R}^{R\\times   R}, \\quad m\\in N.\n\\end{equation}\nLetting\n\\begin{equation}\n  \\label{eq:CRdef}\n  \\mathcal{C_R}=\\cup (\\{r\\}\\times C_r \\mid r\\in R) \\subseteq R\\times C,\n\\end{equation}\nthe {\\em seemingly unrelated regressions model\\\/} is the family\nof normal distributions \n\\begin{equation}\n  \\label{eq:surdef}\n  \\mathbf{N}(\\CRfam) = \\big( \\mathcal{N}_{R\\times N}(BX,\\Sigma\\otimes I_N) \\,\\big|\\,\n  (B,\\Sigma)\\in \\mathbb{B}(\\CRfam)\\times \\mathbb{P} \\big). \n\\end{equation}\nHere $\\mathcal{N}_{R\\times N}$ is the multivariate normal distribution on\n$\\mathbb{R}^{R\\times N}$; $I_N$ is the $N\\times N$ identity matrix; $\\otimes$ is\nthe Kronecker product; $B$ and $\\Sigma$ are the mean and the variance\nparameters; and {\\em the parameter space\\\/} $\\mathbb{B}(\\CRfam)\\times \\mathbb{P}$ is the\nCartesian product of the linear space\n\\begin{equation}\n  \\label{eq:bspacedef}\n  \\mathbb{B}(\\CRfam) = \\big\\{ B\\in \\mathbb{R}^{R\\times C} \\,\\big|\\,\n  B=(\\beta_{rc}),\\; \\beta_{rc}=0 \\;\\forall (r,c)\\not\\in \\mathcal{C_R} \\big\\}\n\\end{equation}\nand the cone $\\mathbb{P}$ of all positive definite real $R\\times R$ matrices.\nThe response matrix $Y$ is then an observation from some (unknown) distribution\nin the model,\n\\begin{equation*}\n  Y \\sim \\mathcal{N}(BX, \\Sigma \\otimes I_N), \\quad (B,\\Sigma)\\in\n  \\mathbb{B}(\\CRfam)\\times \\mathbb{P}. \n\\end{equation*}\nIf $N\\ge R+C$ and $X$ is a matrix of full rank, then with probability\none the $(R+C)\\times N$ matrix obtained by stacking $X$ and $Y$ has\nfull rank,  \n\\begin{equation}\n  \\label{eq:fullrank}\n  \\text{rank} \n  \\begin{pmatrix}\n    Y\\\\X\n  \\end{pmatrix} = R+C.\n\\end{equation}\nWe assume (\\ref{eq:fullrank}) to hold throughout the paper.\n\n\\section{Maximum likelihood estimation by polynomial optimization}\n\\label{sec:mlepoly}\n\nThe {\\em probability density function\\\/} $f_{(B,\\Sigma)}: \\mathbb{R}^{R\\times N} \n\\to (0,\\infty)$ of the distribution\n$\\mathcal{N}(BX,\\Sigma\\otimes I_n)$ can be written as \n\\begin{equation*}\n  f_{(B,\\Sigma)}(Y) = \\frac{1}{\\sqrt{(2\\pi)^{RN}\n      |\\Sigma|^N}}\\exp\\left\\{-\\frac{1}{2}\n    \\mathrm{tr}\\big[\\Sigma^{-1}(Y-BX)(Y-BX)'\\big] \\right\\}. \n\\end{equation*}\nFor data $Y$, the {\\em likelihood function\\\/} $L: \\mathbb{B}(\\CRfam)\\times \\mathbb{P}\n\\to (0,\\infty)$ of the model $\\mathbf{N}(\\CRfam)$ is \ndefined as\n\\begin{equation*}\n  L(B,\\Sigma) = f_{(B,\\Sigma)}(Y). \n\\end{equation*}\nIn maximum likelihood estimation the parameters $(B,\\Sigma)$ are\nestimated by\n\\begin{equation}\n  \\label{eq:maxL}\n  (\\hat B, \\hat\\Sigma) = \\arg\\max \\{ L(B,\\Sigma) \\mid (B,\\Sigma)\\in\n  \\mathbb{B}(\\CRfam)\\times \\mathbb{P} \\} .\n\\end{equation}\nIt follows from (\\ref{eq:fullrank}) that the maximum of the\nlikelihood function exists. \n\nWe can parameterize $\\mathbb{B}(\\CRfam)$ by mapping a vector\n\\begin{equation*}\n  \\beta=(\\beta_{rc}\\mid (r,c)\\in \\mathcal{C_R}) \\in \\mathbb{R}^{\\mathcal{C_R}},\n\\end{equation*}\nto the matrix $B(\\beta)\\in \\mathbb{B}(\\CRfam)$ with entry $B(\\beta)_{rc}=\\beta_{rc}$ if\n$(r,c)\\in\\mathcal{C_R}$ and $B(\\beta)_{rc}=0$ otherwise.  \nDefine $\\ell: \\mathbb{R}^{\\mathcal{C_R}}\\times \\mathbb{P} \\to \\mathbb{R}$ by \n  \\begin{equation}\n    \\label{eq:elldef}\n    \\begin{split}\n      \\ell(\\beta,\\Sigma) &= \\log L(B(\\beta),\\Sigma) \n      \\\\ &\n      \\propto\n      -\\frac{N}{2}\\log |\\Sigma| - \\frac{1}{2}\n      \\mathrm{tr}\\big[\\Sigma^{-1}\n      \\big(Y-B(\\beta)X\\big)\\big(Y-B(\\beta)X\\big)'\\big].    \n    \\end{split}\n  \\end{equation}\nClearly we can solve (\\ref{eq:maxL}) by finding\n\\begin{equation}\n  \\label{eq:maxell}\n  (\\hat \\beta, \\hat\\Sigma) = \\arg\\max \\{ \\ell(\\beta,\\Sigma) \\mid\n  (\\beta,\\Sigma)\\in \\mathbb{R}^{\\mathcal{C_R}}\\times \\mathbb{P} \\},\n\\end{equation}\nand setting $\\hat B=B(\\hat \\beta)$.  \nThe standard approach to solve (\\ref{eq:maxell}) is to solve the \n{\\em likelihood equations\\\/}  \n\\begin{equation}\n  \\label{eq:likeqn}\n  \\left(\\frac{\\partial \\ell(\\beta,\\Sigma)}{\\partial \\beta},\n  \\frac{\\partial \\ell(\\beta,\\Sigma)}{\\partial \\Sigma}\\right) =0. \n\\end{equation}\nIt can be shown that (\\ref{eq:likeqn}) holds if and only if\n\\begin{equation}\n  \\label{eq:sigmamax}\n  \\Sigma = \\frac{1}{N} \\big(Y-B(\\beta)X\\big)\\big(Y-B(\\beta)X\\big)' \n\\end{equation}\nand \n\\begin{equation}\n  \\label{eq:betamax}\n  \\beta = \\left[ A'(XX'\\otimes \\Sigma^{-1}) A \\right]^{-1}\n  A' \\mathrm{vec}(\\Sigma^{-1} YX'),\n\\end{equation}\nwhere $A$ is a matrix of zeroes and ones that satisfies\n$\\mathrm{vec}(B(\\beta))=A\\beta$.  In fact, each column of $A$ has precisely\none entry equal to one and the remaining entries equal to zero.\n\\cite{oberhofer:74} show how one solution to the likelihood equations\ncan be obtained by alternating between solving\n(\\ref{eq:sigmamax}) for fixed $\\beta$ and solving (\\ref{eq:betamax})\nfor fixed $\\Sigma$.\nHere, we take a different approach that, for certain SUR models,\nallows us to compute all solutions to the likelihood equations.\n\n\nFrom (\\ref{eq:fullrank}) and (\\ref{eq:elldef}), it follows that for\nfixed $\\beta\\in\\mathbb{R}^{\\mathcal{C_R}}$ the function $\\ell_\\beta: \\Sigma \\mapsto\n\\ell(\\beta,\\Sigma)$ is strictly concave with maximizer\n(\\ref{eq:sigmamax}).   Thus the {\\em profile \nlog-likelihood function\\\/} $\\ell_{\\mathrm{prof}}: \\mathbb{R}^{\\mathcal{C_R}} \\to \\mathbb{R}$\ndefined as \n\\begin{equation}\n  \\label{eq:profell}\n   \\ell_{\\mathrm{prof}} (\\beta) = \\max\\{\\ell(\\beta,\\Sigma) \\mid\n  \\Sigma\\in\\mathbb{P}\\}\n\\end{equation}\ntakes on the form \n\\begin{equation}\n  \\label{eq:profellsimple}\n   \\ell_{\\mathrm{prof}} (\\beta) \\propto -\\frac{N}{2}\\log \\big| \\frac{1}{N}\n  \\big(Y-B(\\beta) X\\big)\\big(Y-B(\\beta)X\\big)' \\big| - \\frac{RN}{2}.\n\\end{equation}\nBy the strict con-cavity of $\\ell_\\beta$, $(\\beta,\\Sigma)$ is a stationary\npoint of $\\ell(\\beta, \\Sigma)$ if and only if $\\beta$ is a\nstationary point of $ \\ell_{\\mathrm{prof}}(\\beta)$ and $\\Sigma$ satisfies\n(\\ref{eq:sigmamax}); compare \\citet[Lemma 1]{drton:2004}.  \nThe same holds for\n\\begin{equation}\n  \\label{eq:defG}\n  G(\\beta)= \\left| \\big(Y-B(\\beta)X\\big)\\big(Y-B(\\beta)X\\big)' \\right|,\n\\end{equation}\nwhich conveniently is a polynomial in $\\beta$.  Thus we can solve\n(\\ref{eq:maxell}) by using (\\ref{eq:sigmamax}) and solving the\nunconstrained polynomial program \n\\begin{equation}\n  \\label{eq:minG}\n  \\hat\\beta = \\arg\\min \\{ G(\\beta) \\mid \\beta\\in\\mathbb{R}^{\\mathcal{C_R}} \\}.\n\\end{equation}\n\n\nWe try to solve (\\ref{eq:minG}) by\ncomputing the stationary points of $G$, i.e. by solving the equations\n\\begin{equation}\n  \\label{eq:defgrc}\n  g_{rc} = \\frac{\\partial G(\\beta)}{\\partial\\beta_{rc}} = 0, \\quad\n  (r,c)\\in\\mathcal{C_R}.\n\\end{equation} \nIn practice the observations\n$Y$ and $X$ are available only in finite accuracy and the partial\nderivatives $g_{rc}$, $(r,c)\\in\\mathcal{C_R}$, are elements of the ring $\\mathbb{Q}[\\beta]$\nof polynomials in $\\beta$ with rational\ncoefficients. \nIn an algebraic approach to solving polynomial equations \n\\citep{coxlittleoshea:1997,coxlittleoshea:1998,sturmfels:2002} we\nallow the indeterminants in the polynomial equation system\n(\\ref{eq:defgrc}) to be complex, i.e.\\ \n$\\beta\\in\\mathbb{C}^{\\mathcal{C_R}}$, where $\\mathbb{C}$ is the field of complex numbers.\nWe define the {\\em maximum likelihood ideal\\\/} $I_G$ to be the ideal\nof $\\mathbb{Q}[\\beta]$ that is generated by the partial derivatives\n$g_{rc}$, $(r,c)\\in\\mathcal{C_R}$, i.e.\\\n\\begin{equation}\n  \\label{eq:defIG}\n  I_{G} = \\langle g_{rc} \\mid (r,c)\\in\\mathcal{C_R} \\rangle;\n\\end{equation}\ncompare \\citet[\\S8.4]{sturmfels:2002} who defines maximum likelihood ideals\nin a different statistical context.  Software like {\\tt Macaulay\n  2}\\footnote{\\tt http:\/\/www.math.uiuc.edu\/Macaulay2\/} and {\\tt Singular}\n\\citep{singular} permits us to check whether $I_G$ is a zero-dimensional\nideal.  If $\\dim(I_G)=0$, then the variety $V_\\mathbb{C}(I_G)$, i.e.\\ the set of\ncommon complex zeroes of the partial derivatives $g_{rc}$, is a finite set\nand all its elements can be computed using, for example, \n{\\tt Singular} or also {\\tt PHCpack}\\footnote{\\tt\n  http:\/\/www.math.uic.edu\/\\~{}jan\/}. The real points   \n$V_\\mathbb{R}(I_G)=V_\\mathbb{C}(I_G)\\cap \\mathbb{R}^\\mathcal{C_R}$ can then be identified and yield\nthe stationary points of $G$.\n\n\\section{Revisiting the multimodal bivariate seemingly unrelated\n  regressions with two covariates}\n\\label{sec:revisit}\n\n\\cite{drton:2004} study a SUR model with two response variables and two\ncovariates, in which response variable 1 is regressed only on covariate 1,\nand response variable 2 only on covariate 2.  Hence, $R=\\{1,2\\}$, \n$C=\\{1,2\\}$, $C_1=\\{1\\}$, and $C_2= \\{2\\}$.  Therefore, $\\mathcal{C_R}=\\{(1,1), \n(2,2)\\}$, and $B\\in\\mathbb{B}(\\CRfam)$ if $B$ is of the form \n\\begin{equation*}\n  B= \\begin{pmatrix}\n    \\beta_{11} & 0\\\\\n    0 & \\beta_{22}\n  \\end{pmatrix} \\in \\mathbb{R}^{2\\times 2}.\n\\end{equation*}\nUsing {\\tt Singular} and the data\nin \\citet[Table 1]{drton:2004},  we can solve (\\ref{eq:minG}) as shown in\nTable \\ref{tab:singularcode}.\n\\begin{table}[tb]\n\\begin{center}\n\\hrule\n\\vspace{0.05cm}\n\\hrule\n{ \n\\medskip\n\\begin{verbatim}\n> ring R=0,(b(1..2)), lp;\n> matrix X[2][8] = 188,22,-46,77,-103,74,83,101,   \n.                  55,-216,116,-30,131,195,-311,-239;\n> matrix Y[2][8] = 234,-5,6,182,-193,278,62,-68,   \n.                  497,-326,266,-3,93,558,-584,-224;\n> matrix B[2][2] = b(1),0, 0,b(2);\n> poly G = det((Y-B*X)*transpose(Y-B*X));\n> ideal IG =jacob(G);  \n> ideal J = groebner(IG);\n> dim(J); vdim(J);\n0\n5\n> LIB \"solve.lib\"; solve(J,6);\n[1]:\n   [1]:  0.778796\n   [2]:  1.538029\n[2]:\n   [1]:  1.622609\n   [2]:  2.034745\n[3]:\n   [1]:  (1.480687-i*1.547274)\n   [2]:  (2.16845+i*0.765283)\n[4]:\n   [1]:  (1.480687+i*1.547274)\n   [2]:  (2.16845-i*0.765283)\n[5]:\n   [1]:  2.764418\n   [2]:  2.504006\n\\end{verbatim} \n}\n\\medskip\n\\hrule\n\\end{center}\n\\medskip\n\\caption{{\\tt Singular}-session for the model in\n\\cite{drton:2004}.\\label{tab:singularcode} }\n\\end{table}\n\nAs computed by {\\tt dim} and {\\tt vdim}, the maximum likelihood ideal\n$I_G={\\tt IG}$ is zero-dimensional and of degree five. The five points\nin the variety $V_\\mathbb{C}(I_G)$ are computed \nby {\\tt solve}, which lists $\\beta_{11}={\\tt b(1)}$ as first component\nand $\\beta_{22}={\\tt b(2)}$ as second component. There are three real\npoints in $V_\\mathbb{R}(I_G)$, which yield the \nstationary points of the likelihood function of the model $\\mathbf{N}(\\CRfam)$. \nNote that we confirm the values\nstated in \\citet[Table 2 with $\\beta_{11}=\\beta_1$ and \n$\\beta_{22}=\\beta_2$]{drton:2004}. \nThe Gr\\\"obner basis computed by the command {\\tt \ngroebner(IG)} has two elements that are (i)\na quintic in $\\beta_{22}={\\tt b(2)}$ and (ii) a sum of a linear\nfunction in $\\beta_{11}={\\tt b(1)}$ and a quartic in $\\beta_{22}={\\tt\nb(2)}$. Thus it follows immediately that the stationary\npoints of $G$ can be found from solving a quintic \\cite[cf.][Thm.\\\n2]{drton:2004}.\n\n\n\\section{Dimensions and degrees of maximum likelihood ideals}\n\\label{sec:groebner}\n\n\n\\subsection{Seemingly unrelated regressions} \n\\label{sec:dimdegsur}\n\nThe algebraic approach can also be applied to more\ngeneral models.  Here we focus on SUR models $\\mathbf{N}(\\CRfam)$ for which\n$(C_r\\mid r\\in R)$ consists of disjoint sets; in other models \ninclusion relations among the sets $C_r$ may be\nexploited \\citep[cf.][]{ap:ims}.  \nMore precisely, we consider models $\\mathbf{N}(\\CRfam)$ in\nwhich $r_1< r_2$,  $r_1,r_2\\in R$, implies that $c_1< c_2$ for all \n$c_1\\in C_{r_1}$ and $c_2\\in C_{r_2}$.  Then $\\mathbb{B}(\\CRfam)$\nis a linear space of block-diagonal matrices.\n\n\nTable\n\\ref{tab:gensurdimdegree} states the dimension and degree of the\nmaximum likelihood ideal for seven examples including the\none from Section \\ref{sec:revisit}.\n\\begin{table}[p]\n  \\begin{center}\n    \\begin{tabular}[h]{cccc} \n      $\\mathcal{C_R}$ & $\\mathbb{B}(\\CRfam)$ & $\\dim(I_G)$ & $\\mathrm{degree}(I_G)$\\\\\n      \\hline\n      \\hline\n      \\raisebox{-0.25cm}{\n      $\\{(1,1),(2,2)\\}$} & \n      \\raisebox{-0.25cm}{\n        $\\left(\\begin{smallmatrix}\n        \\beta_{11} & 0\\\\\n        0 & \\beta_{22}\n      \\end{smallmatrix}\\right)$} & \\raisebox{-0.25cm}{0} &\n      \\raisebox{-0.25cm}{5}\\\\[0.75cm] \n      $\\{(1,1),(1,2),(2,3)\\}$ & \n        $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & 0\\\\\n        0 & 0 & \\beta_{23}\n      \\end{smallmatrix}\\right)$ & 0 & 9\\\\[0.5cm] \n    $\\{(1,1),(2,2),(3,3)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} &  0 & 0\\\\\n        0 & \\beta_{22} & 0\\\\\n        0 & 0 & \\beta_{33}\n      \\end{smallmatrix}\\right)$ & 0 & 29\\\\[0.5cm]\n    $\\{(1,1),(1,2),(1,3),(2,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & \\beta_{13} & 0\\\\\n        0 & 0 & 0 & \\beta_{24}\n      \\end{smallmatrix}\\right)$ & 1 & 4\\\\[0.5cm]\n    $\\{(1,1),(1,2),(2,3),(2,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & 0 & 0\\\\\n        0 & 0 & \\beta_{23} & \\beta_{24}\n      \\end{smallmatrix}\\right)$ & 1 & 8\\\\[0.5cm]\n    $\\{(1,1),(2,2),(3,3),(4,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & 0 & 0 & 0\\\\\n        0 & \\beta_{22} & 0 & 0\\\\\n        0 & 0 & \\beta_{33} & 0\\\\\n        0 & 0 & 0 & \\beta_{44}\n      \\end{smallmatrix}\\right)$ & 1 & 32\\\\[0.65cm]\n    $\\{(1,1),(2,2),(3,3),(4,4),(5,5)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & 0 & 0 & 0 & 0\\\\\n        0 & \\beta_{22} & 0 & 0 & 0\\\\\n        0 & 0 & \\beta_{33} & 0 & 0\\\\\n        0 & 0 & 0 & \\beta_{44} & 0\\\\\n        0 & 0 & 0& 0 & \\beta_{55}\n      \\end{smallmatrix}\\right)$ & 2 & 80\\\\[0.75cm]\n        \\hline  \n    \\end{tabular}\n  \\end{center}\n  \\smallskip\n  \\caption{Dimension and degree of maximum likelihood ideals.}\n  \\label{tab:gensurdimdegree}\n\\end{table}\nFor the models with zero-dimensional maximum likelihood ideal\n$I_G$, we can find all stationary points of the likelihood function\nby computations analogous \nto the ones demonstrated in Table \\ref{tab:singularcode}.  \nThe likelihood functions of these models may be multimodal and\nit would be interesting to find, for each model, reference data for which \nthe cardinality of $V_\\mathbb{R}(I_G)$ is large.  \nFor example, let $\\mathcal{C_R}=\\{(1,1),(2,2)\\}$ and choose\n\\begin{equation}\n  \\label{eq:xytrimod}\n  \\begin{split}\n  X = &\n  \\left(\\begin{array}{r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r} \n    -0.65 & -0.80 & \\phantom{-}1.34  & -1.03 & -1.08 \\\\\n    -0.04 & -1.18 & \\phantom{-}1.98  & -2.42 & -3.75 \n  \\end{array}\\right), \\\\\n  Y = &\n  \\left(\\begin{array}{r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r@{\\hspace{0.25cm}}r} \n    \\phantom{-}0.14  & -0.73 & \\phantom{-}1.40  & -2.29 & -3.30 \\\\\n    \\phantom{-}0.52 & -1.93 & 3.02 & -6.67 & -9.94\n \\end{array}\\right),\n \\end{split}\n\\end{equation}\nthen the variety of the maximum likelihood ideal of $\\mathbf{N}(\\CRfam)$ is purely\nreal, i.e.\\  \n$V_\\mathbb{R}(I_G)=V_\\mathbb{C}(I_G)=5$. \nFigure \\ref{wireframe.3M} shows a three-dimensional plot and a\ncontour plot of the profile log-likelihood function for these\nobservations. \n\\begin{figure}[p] \n\\centering\n\\begin{minipage}[t]{6.5cm}\n\\begin{center} \n\\includegraphics[width=6.5cm]{wireframe.3M.eps} \n\\end{center} \n\\end{minipage}\n\\hfill \n\\begin{minipage}[t]{6.5cm}\n\\begin{center} \n\\includegraphics[width=6cm]{contour.3M.2.eps} \n\\end{center} \n\\end{minipage}\n\\caption{\\label{wireframe.3M} Three-dimensional plot and contour plot\n  of profile log-likelihood function. \n}\n\\end{figure}\nWe conjecture \nthat data with $V_\\mathbb{R}(I_G)=V_\\mathbb{C}(I_G)$ exist for all \nthree models in Table \\ref{tab:gensurdimdegree} that have\nzero-dimensional maximum likelihood ideal. For \nmodels with  maximum likelihood ideal of dimension one or higher,\nit is not clear whether $V_\\mathbb{R}(I_G)=\\infty$, i.e.\\ a likelihood\nfunction with an infinite number of stationary points, can occur with\nnon-zero probability.\n\n\n\n\\subsection{Submodels of seemingly unrelated regressions} \n\\label{sec:submodels}\n\nIt is obvious that the algebraic approach developed in Section\n\\ref{sec:mlepoly} immediately carries over to\nthe submodels of SUR that are of interest\nin testing equality of regression coefficients.  In the model $\\mathbf{N}(\\CRfam)$ with\n$\\mathcal{C_R}=\\{(1,1),(2,2)\\}$, for example, we may be interested in testing\nwhether $\\beta_{11}=\\beta_{22}$.  If this is done using a likelihood ratio\ntest, then the likelihood function of the submodel in which\n$\\beta_{11}=\\beta_{22}$ is imposed has to be maximized.  More precisely,\nthe submodel has the restricted parameter space\n\\begin{equation}\n  \\label{eq:subspace}\n  \\{ B\\in \\mathbb{B}(\\CRfam)\\mid \\beta_{11}=\\beta_{22}\\} \\times \\mathbb{P}.\n\\end{equation}\nTable \\ref{tab:submoddimdeg} lists similarly obtained submodels of the\nmodels in Table \\ref{tab:gensurdimdegree}, for which the maximum likelihood\nideal is zero-dimensional and the variety $V_\\mathbb{C}(I_G)$ can be computed.\n\\begin{table}[t]\n  \\begin{center}\n    \\begin{tabular}[h]{cccc} \n      $\\mathcal{C_R}$ & Subspace of $\\mathbb{B}(\\CRfam)$ & $\\dim(I_G)$ & $\\mathrm{degree}(I_G)$\\\\\n      \\hline\n      \\hline\n      \\raisebox{-0.25cm}{\n      $\\{(1,1),(2,2)\\}$} & \n      \\raisebox{-0.25cm}{\n        $\\left(\\begin{smallmatrix}\n        \\beta_{11} & 0\\\\\n        0 & \\beta_{11}\n      \\end{smallmatrix}\\right)$} & \\raisebox{-0.25cm}{0} &\n      \\raisebox{-0.25cm}{3}\\\\[0.75cm] \n      $\\{(1,1),(1,2),(2,3)\\}$ & \n        $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & 0\\\\\n        0 & 0 & \\beta_{12}\n      \\end{smallmatrix}\\right)$ & 0 & 7\\\\[0.5cm] \n    $\\{(1,1),(2,2),(3,3)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} &  0 & 0\\\\\n        0 & \\beta_{11} & 0\\\\\n        0 & 0 & \\beta_{33}\n      \\end{smallmatrix}\\right)$ & 0 & 11\\\\[0.5cm]\n    $\\{(1,1),(1,2),(1,3),(2,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & \\beta_{13} & 0\\\\\n        0 & 0 & 0 & \\beta_{13}\n      \\end{smallmatrix}\\right)$ & 0 & 11\\\\[0.5cm]\n    $\\{(1,1),(1,2),(2,3),(2,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & \\beta_{12} & 0 & 0\\\\\n        0 & 0 & \\beta_{12} & \\beta_{24}\n      \\end{smallmatrix}\\right)$ & 0 & 23\\\\[0.5cm]\n    $\\{(1,1),(2,2),(3,3),(4,4)\\}$ & \n    $\\left(\\begin{smallmatrix}\n        \\beta_{11} & 0 & 0 & 0\\\\\n        0 & \\beta_{11} & 0 & 0\\\\\n        0 & 0 & \\beta_{33} & 0\\\\\n        0 & 0 & 0 & \\beta_{44}\n      \\end{smallmatrix}\\right)$ & 0 & 63\\\\[0.65cm]\n    \\hline\n   \\end{tabular}\n  \\end{center}\n  \\smallskip\n  \\caption{Dimension and degree of maximum likelihood ideals of submodels.}\n  \\label{tab:submoddimdeg}\n\\end{table}\n\nIt should also be noted that submodels of SUR need not inherit\nunimodal likelihood \nfunctions from their parent model.  For example, the bivariate SUR\nmodel $\\mathbf{N}(\\CRfam)$ with $\\mathcal{C_R}=\\{(1,1),(2,1),(2,2)\\}$ is monotone, \ni.e.\\ the family $(C_r\\mid r\\in R)$ is totally ordered by inclusion, which\nguarantees that the likelihood function has precisely one\nstationary point corresponding to the global maximum\n\\citep{ap:ims,drton:2003}. However, the submodel induced by the\nrestriction $\\beta_{11}=\\beta_{21}$ can be reexpressed in the form of\nthe model studied in Section \\ref{sec:revisit} by means of the linear\ntransformation that changes response $Y_2$ into $Y_2-Y_1$. Hence, the\nsubmodel does not always have a unimodal likelihood function.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nThe presented algebraic approach to maximum likelihood estimation in\nSUR permits us to \ncompute all stationary points of the \nlikelihood function if the maximum likelihood ideal is\nzero-dimensional.   This is the case for three seemingly unrelated\nregressions models considered in this paper (cf.\\ Table\n\\ref{tab:gensurdimdegree}): (i) the previously studied model based on\n$\\mathcal{C_R}^{(1)}=\\{(1,1),(2,2)\\}$, (ii) the model with\n$\\mathcal{C_R}^{(2)}=\\{(1,1),(1,2),(2,3)\\}$, and (iii) the model with\n$\\mathcal{C_R}^{(3)}=\\{(1,1),(2,2),(3,3)\\}$.   Additionally, \ninteresting submodels of SUR may have a\nzero-dimensional maximum likelihood ideal (cf.\\ Table\n\\ref{tab:submoddimdeg}).  The computations in {\\tt Singular} that\nfind all stationary points of the likelihood\nfunctions of the models with zero-dimensional maximum likelihood ideal are\ninstantaneous for all but the model in Table \\ref{tab:submoddimdeg}\nthat has a maximum likelihood ideal of degree 63.  \nThus we advocate the use of {\\tt Singular} or similarly capable\nsoftware in statistical data analysis.  \n\nIn future work it would be interesting to find \nreference data sets leading to likelihood functions with a large\nnumber of stationary points.  Moreover, the algebraic approach\npresented herein could be \ncombined with regression approaches \\citep[e.g.][]{ap:ims,drton:2003} in\norder to identify larger classes of SUR models for which all stationary\npoints of the likelihood function can be computed.  \nFinally, it could be explored whether\nmethods for global minimization of polynomials\n\\citep{parrilo:2001} can be used to find the global maximum of SUR\nlikelihood functions.\n\n\\begin{ack}\nI would like to thank Michael Perlman, Thomas Richardson, and Bernd\nSturmfels for their help and support, and an anonymous referee for helpful\ncomments on the presentation.\n\\end{ack}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Submission of conference papers to ICLR 2022}\n\nICLR requires electronic submissions, processed by\n\\url{https:\/\/openreview.net\/}. See ICLR's website for more instructions.\n\nIf your paper is ultimately accepted, the statement {\\tt\n  {\\textbackslash}iclrfinalcopy} should be inserted to adjust the\nformat to the camera ready requirements.\n\nThe format for the submissions is a variant of the NeurIPS format.\nPlease read carefully the instructions below, and follow them\nfaithfully.\n\n\\subsection{Style}\n\nPapers to be submitted to ICLR 2022 must be prepared according to the\ninstructions presented here.\n\n\nAuthors are required to use the ICLR \\LaTeX{} style files obtainable at the\nICLR website. Please make sure you use the current files and\nnot previous versions. 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The table number and title always appear before the table. See\nTable~\\ref{sample-table}.\n\nPlace one line space before the table title, one line space after the table\ntitle, and one line space after the table. The table title must be lower case\n(except for first word and proper nouns); tables are numbered consecutively.\n\n\\begin{table}[t]\n\\caption{Sample table title}\n\\label{sample-table}\n\\begin{center}\n\\begin{tabular}{ll}\n\\multicolumn{1}{c}{\\bf PART}  &\\multicolumn{1}{c}{\\bf DESCRIPTION}\n\\\\ \\hline \\\\\nDendrite         &Input terminal \\\\\nAxon             &Output terminal \\\\\nSoma             &Cell body (contains cell nucleus) \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Default Notation}\n\nIn an attempt to encourage standardized notation, we have included the\nnotation file from the textbook, \\textit{Deep Learning}\n\\cite{goodfellow2016deep} available at\n\\url{https:\/\/github.com\/goodfeli\/dlbook_notation\/}.  Use of this style\nis not required and can be disabled by commenting out\n\\texttt{math\\_commands.tex}.\n\n\n\\centerline{\\bf Numbers and Arrays}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1in}p{3.25in}}\n$\\displaystyle a$ & A scalar (integer or real)\\\\\n$\\displaystyle {\\bm{a}}$ & A vector\\\\\n$\\displaystyle {\\bm{A}}$ & A matrix\\\\\n$\\displaystyle {\\tens{A}}$ & A tensor\\\\\n$\\displaystyle {\\bm{I}}_n$ & Identity matrix with $n$ rows and $n$ columns\\\\\n$\\displaystyle {\\bm{I}}$ & Identity matrix with dimensionality implied by context\\\\\n$\\displaystyle {\\bm{e}}^{(i)}$ & Standard basis vector $[0,\\dots,0,1,0,\\dots,0]$ with a 1 at position $i$\\\\\n$\\displaystyle \\text{diag}({\\bm{a}})$ & A square, diagonal matrix with diagonal entries given by ${\\bm{a}}$\\\\\n$\\displaystyle {\\textnormal{a}}$ & A scalar random variable\\\\\n$\\displaystyle {\\mathbf{a}}$ & A vector-valued random variable\\\\\n$\\displaystyle {\\mathbf{A}}$ & A matrix-valued random variable\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Sets and Graphs}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {\\mathbb{A}}$ & A set\\\\\n$\\displaystyle \\mathbb{R}$ & The set of real numbers \\\\\n$\\displaystyle \\{0, 1\\}$ & The set containing 0 and 1 \\\\\n$\\displaystyle \\{0, 1, \\dots, n \\}$ & The set of all integers between $0$ and $n$\\\\\n$\\displaystyle [a, b]$ & The real interval including $a$ and $b$\\\\\n$\\displaystyle (a, b]$ & The real interval excluding $a$ but including $b$\\\\\n$\\displaystyle {\\mathbb{A}} \\backslash {\\mathbb{B}}$ & Set subtraction, i.e., the set containing the elements of ${\\mathbb{A}}$ that are not in ${\\mathbb{B}}$\\\\\n$\\displaystyle {\\mathcal{G}}$ & A graph\\\\\n$\\displaystyle \\parents_{\\mathcal{G}}({\\textnormal{x}}_i)$ & The parents of ${\\textnormal{x}}_i$ in ${\\mathcal{G}}$\n\\end{tabular}\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Indexing}\n\\bgroup\n\\def1.5{1.5}\n\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle {a}_i$ & Element $i$ of vector ${\\bm{a}}$, with indexing starting at 1 \\\\\n$\\displaystyle {a}_{-i}$ & All elements of vector ${\\bm{a}}$ except for element $i$ \\\\\n$\\displaystyle {A}_{i,j}$ & Element $i, j$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{i, :}$ & Row $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\bm{A}}_{:, i}$ & Column $i$ of matrix ${\\bm{A}}$ \\\\\n$\\displaystyle {\\etens{A}}_{i, j, k}$ & Element $(i, j, k)$ of a 3-D tensor ${\\tens{A}}$\\\\\n$\\displaystyle {\\tens{A}}_{:, :, i}$ & 2-D slice of a 3-D tensor\\\\\n$\\displaystyle {\\textnormal{a}}_i$ & Element $i$ of the random vector ${\\mathbf{a}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\\centerline{\\bf Calculus}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle\\frac{d y} {d x}$ & Derivative of $y$ with respect to $x$\\\\ [2ex]\n$\\displaystyle \\frac{\\partial y} {\\partial x} $ & Partial derivative of $y$ with respect to $x$ \\\\\n$\\displaystyle \\nabla_{\\bm{x}} y $ & Gradient of $y$ with respect to ${\\bm{x}}$ \\\\\n$\\displaystyle \\nabla_{\\bm{X}} y $ & Matrix derivatives of $y$ with respect to ${\\bm{X}}$ \\\\\n$\\displaystyle \\nabla_{\\tens{X}} y $ & Tensor containing derivatives of $y$ with respect to ${\\tens{X}}$ \\\\\n$\\displaystyle \\frac{\\partial f}{\\partial {\\bm{x}}} $ & Jacobian matrix ${\\bm{J}} \\in \\mathbb{R}^{m\\times n}$ of $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$\\\\\n$\\displaystyle \\nabla_{\\bm{x}}^2 f({\\bm{x}})\\text{ or }{\\bm{H}}( f)({\\bm{x}})$ & The Hessian matrix of $f$ at input point ${\\bm{x}}$\\\\\n$\\displaystyle \\int f({\\bm{x}}) d{\\bm{x}} $ & Definite integral over the entire domain of ${\\bm{x}}$ \\\\\n$\\displaystyle \\int_{\\mathbb{S}} f({\\bm{x}}) d{\\bm{x}}$ & Definite integral with respect to ${\\bm{x}}$ over the set ${\\mathbb{S}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Probability and Information Theory}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle P({\\textnormal{a}})$ & A probability distribution over a discrete variable\\\\\n$\\displaystyle p({\\textnormal{a}})$ & A probability distribution over a continuous variable, or over\na variable whose type has not been specified\\\\\n$\\displaystyle {\\textnormal{a}} \\sim P$ & Random variable ${\\textnormal{a}}$ has distribution $P$\\\\% so thing on left of \\sim should always be a random variable, with name beginning with \\r\n$\\displaystyle  \\mathbb{E}_{{\\textnormal{x}}\\sim P} [ f(x) ]\\text{ or } \\mathbb{E} f(x)$ & Expectation of $f(x)$ with respect to $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Var}(f(x)) $ &  Variance of $f(x)$ under $P({\\textnormal{x}})$ \\\\\n$\\displaystyle \\mathrm{Cov}(f(x),g(x)) $ & Covariance of $f(x)$ and $g(x)$ under $P({\\textnormal{x}})$\\\\\n$\\displaystyle H({\\textnormal{x}}) $ & Shannon entropy of the random variable ${\\textnormal{x}}$\\\\\n$\\displaystyle D_{\\mathrm{KL}} ( P \\Vert Q ) $ & Kullback-Leibler divergence of P and Q \\\\\n$\\displaystyle \\mathcal{N} ( {\\bm{x}} ; {\\bm{\\mu}} , {\\bm{\\Sigma}})$ & Gaussian distribution %\nover ${\\bm{x}}$ with mean ${\\bm{\\mu}}$ and covariance ${\\bm{\\Sigma}}$ \\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\\centerline{\\bf Functions}\n\\bgroup\n\\def1.5{1.5}\n\\begin{tabular}{p{1.25in}p{3.25in}}\n$\\displaystyle f: {\\mathbb{A}} \\rightarrow {\\mathbb{B}}$ & The function $f$ with domain ${\\mathbb{A}}$ and range ${\\mathbb{B}}$\\\\\n$\\displaystyle f \\circ g $ & Composition of the functions $f$ and $g$ \\\\\n  $\\displaystyle f({\\bm{x}} ; {\\bm{\\theta}}) $ & A function of ${\\bm{x}}$ parametrized by ${\\bm{\\theta}}$.\n  (Sometimes we write $f({\\bm{x}})$ and omit the argument ${\\bm{\\theta}}$ to lighten notation) \\\\\n$\\displaystyle \\log x$ & Natural logarithm of $x$ \\\\\n$\\displaystyle \\sigma(x)$ & Logistic sigmoid, $\\displaystyle \\frac{1} {1 + \\exp(-x)}$ \\\\\n$\\displaystyle \\zeta(x)$ & Softplus, $\\log(1 + \\exp(x))$ \\\\\n$\\displaystyle || {\\bm{x}} ||_p $ & $L^p$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle || {\\bm{x}} || $ & $L^2$ norm of ${\\bm{x}}$ \\\\\n$\\displaystyle x^+$ & Positive part of $x$, i.e., $\\max(0,x)$\\\\\n$\\displaystyle \\bm{1}_\\mathrm{condition}$ & is 1 if the condition is true, 0 otherwise\\\\\n\\end{tabular}\n\\egroup\n\\vspace{0.25cm}\n\n\n\n\\section{Final instructions}\nDo not change any aspects of the formatting parameters in the style files.\nIn particular, do not modify the width or length of the rectangle the text\nshould fit into, and do not change font sizes (except perhaps in the\n\\textsc{References} section; see below). Please note that pages should be\nnumbered.\n\n\\section{Preparing PostScript or PDF files}\n\nPlease prepare PostScript or PDF files with paper size ``US Letter'', and\nnot, for example, ``A4''. The -t\nletter option on dvips will produce US Letter files.\n\nConsider directly generating PDF files using \\verb+pdflatex+\n(especially if you are a MiKTeX user).\nPDF figures must be substituted for EPS figures, however.\n\nOtherwise, please generate your PostScript and PDF files with the following commands:\n\\begin{verbatim}\ndvips mypaper.dvi -t letter -Ppdf -G0 -o mypaper.ps\nps2pdf mypaper.ps mypaper.pdf\n\\end{verbatim}\n\n\\subsection{Margins in LaTeX}\n\nMost of the margin problems come from figures positioned by hand using\n\\verb+\\special+ or other commands. We suggest using the command\n\\verb+\\includegraphics+\nfrom the graphicx package. Always specify the figure width as a multiple of\nthe line width as in the example below using .eps graphics\n\\begin{verbatim}\n   \\usepackage[dvips]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.eps}\n\\end{verbatim}\nor\n\\begin{verbatim}\n   \\usepackage[pdftex]{graphicx} ...\n   \\includegraphics[width=0.8\\linewidth]{myfile.pdf}\n\\end{verbatim}\nfor .pdf graphics.\nSee section~4.4 in the graphics bundle documentation (\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/required\/graphics\/grfguide.ps})\n\nA number of width problems arise when LaTeX cannot properly hyphenate a\nline. Please give LaTeX hyphenation hints using the \\verb+\\-+ command.\n\n\\subsubsection*{Author Contributions}\nIf you'd like to, you may include  a section for author contributions as is done\nin many journals. This is optional and at the discretion of the authors.\n\n\\subsubsection*{Acknowledgments}\nUse unnumbered third level headings for the acknowledgments. All\nacknowledgments, including those to funding agencies, go at the end of the paper.\n\n\n\n\\section{Introduction}\n\t\n\n\tAncient languages such as COBOL still underpin much of the financial industry and government services. Their outdated structures and thinning developer bases induce costs and severely slow down development, prompting businesses to modernize their codebases. For instance, the Commonwealth Bank of Australia spent around \\$750 million over 5 years to migrate its COBOL codebase to a more recent language. More generally, most large companies own code written in several programming languages, which can hinder interoperability and make programmers less efficient.\n\t\\br{Automatic translation systems could make codebase migrations faster and cheaper, and help programmers learn new languages or understand existing code.}\n\n\n\t\n\n\tHowever, translation systems are not as effective for source code as for natural languages.\n\tRule-based systems are still commonly used, but they are never exhaustive due to the considerable number of translation rules that should be written to translate every function and object from every standard library. Unlike in natural languages, there is little to no parallel data available for source code, making it impossible to train standard machine translation models.\n\tRecently, TransCoder\\xspace~\\citep{roziere2020unsupervised} showed that unsupervised methods can be used to translate source code. \n\tHowever, it is trained without any supervised signal and only learns the semantics of tokens from their contexts. As shown in Figure~\\ref{fig:teaser_examples}, it can confuse tokens that have different semantics in different languages, for instance the float division in Python and integer division in C++ and Java which use the token \\texttt{\/} or more subtle operator priority differences (e.g. Java prioritizes \\texttt{==} over \\texttt{\\&} unlike Python and C++).\n\tWhile small inaccuracies often merely hinder comprehension in natural languages, they \\br{often} make the entire translation erroneous in the context of programming languages.\n\n\n\n\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1.\\textwidth,center}\n\t\t\t\\begin{tabular}{l l l}\n\t\t\t\t\\small{Input function} & \\small{TransCoder\\xspace} & \\small{TransCoder-ST\\xspace}\\\\\n\t\t\t\t\\midrule\n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{python}\n\t\t\t\t\tdef is_odd(x):\n\t\t\t\t\treturn x & 1 == 1\n\t\t\t\t\t\\end{minted}\n\t\t\t\t\\end{minipage} &\n\t\t\t\t\\begin{minipage}[t]{0.32\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{java}\n\t\t\t\t\tstatic boolean isOdd(int x) {\n\t\t\t\t\treturn x & 1 == 1;\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage} & \n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{java}\n\t\t\t\t\tstatic boolean isOdd(int x) { \n\t\t\t\t\treturn ( x & 1 ) == 1;\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage}\\\\\n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{java}\n\t\t\t\t\tstatic void printb(int x){ \n\t\t\t\t\twhile (x > 0){ \n\t\t\t\t\tSystem.out.println(x\n\t\t\t\t\tx \/= 2; \n\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted}\n\t\t\t\t\\end{minipage} &\n\t\t\t\t\\begin{minipage}[t]{0.32\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{python}\n\t\t\t\t\tdef printb(x):\n\t\t\t\t\twhile x > 0:\n\t\t\t\t\tprint(x\n\t\t\t\t\tx \/= 2\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage} & \n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{python}\n\t\t\t\t\tdef printb(x):\n\t\t\t\t\twhile x > 0:\n\t\t\t\t\tprint(x\n\t\t\t\t\tx \/\/= 2\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage}\\\\\n\t\t\t\t\\begin{minipage}[t]{0.4\\textwidth}\n\t\t\t\t\t\\begin{minted}{java}\n\t\t\t\t\tstatic String reverse(char[] str){\n\t\t\t\t\tStack <Character> st = new Stack<>();\n\t\t\t\t\tfor(int i = 0; i<str.length; i++) \n\t\t\t\t\tst.push(str[i]);\n\t\t\t\t\tfor(int i = 0; i<str.length; i++){\n\t\t\t\t\tstr[i] = st.peek();\n\t\t\t\t\tst.pop();\n\t\t\t\t\t}\n\t\t\t\t\treturn String.valueOf(str);\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted}\n\t\t\t\t\\end{minipage} &\n\t\t\t\t\\begin{minipage}[t]{0.30\\textwidth}\n\t\t\t\t\t\\begin{minted}{python}\n\t\t\t\t\tdef reverse(str):\n\t\t\t\t\tst = Stack()\n\t\t\t\t\tfor i in range(len(str)):\n\t\t\t\t\tst.push(str[i])\n\t\t\t\t\tfor i in range(len(str)):\n\t\t\t\t\tstr[i] = st.pop()\n\t\t\t\t\tst.push(str[i])\n\t\t\t\t\treturn str\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage} & \n\t\t\t\t\\begin{minipage}[t]{0.3\\textwidth}\n\t\t\t\t\t\\begin{minted}{python}\n\t\t\t\t\tdef reverse(data):\n\t\t\t\t\tst = []\n\t\t\t\t\tfor c in data :\n\t\t\t\t\tst.append(c)\n\t\t\t\t\tfor i in range(len(data)):\n\t\t\t\t\tdata[i] = st[-1]\n\t\t\t\t\tst.pop()\n\t\t\t\t\treturn ''.join(data)\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage}\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\caption{\\small{\\textbf{Improvements over TransCoder\\xspace.} The first function returns whether an input integer is odd and is translated from Python to Java. \n\t\t\t\tThe translation of TransCoder\\xspace does not compile because the \\texttt{==} operator has precedence over \\texttt{\\&} in Java, and parentheses are required unlike in Python.\n\t\t\t\tThe second example is a function that prints an integer in base two, which is translated from Java to Python. \n\t\t\t\tTransCoder\\xspace translates does not modify the expression \\texttt{x\/=2}, even though it corresponds to the integer division in Java and to the float division in Python. \n\t\t\t\tIn the third example, a function reversing a char array, TransCoder\\xspace does not manage to translate the Java Stack object into the right Python object and uses the unsafe \\texttt{str} parameter name.\n\t\t\t\tIn all three cases, TransCoder-ST\\xspace manages to leverage the semantics contained in unit tests to translate the function correctly.}}\n\t\t\\label{fig:teaser_examples}\n\t\t\\vspace{-0.5cm}\n\t\\end{figure}\n\t\n\t\n\tTransCoder\\xspace leverages back-translation \\citep{sennrich2015improving}, an effective data-augmentation scheme where the model translates source sequences to generate training data for the target-to-source direction, and vice versa. Although being highly effective in low-resource translation, back-translation also has issues, as the model is trained on potentially invalid input-output pairs. Neural machine translation models being highly sensitive to input noise~\\citep{belinkov2017synthetic, khayrallah2018impact}, this can severely deteriorate the performance.\n\n\tFortunately, many programming languages come with relatively mature tools and technologies for automated test data generation.\n\tIn this paper, we propose to leverage these tools to guide the translation process, weeding out unsuccessful translations, thereby increasing the overall confidence in the machine translation process.\n\t\n\n\n\n\n\t\n\n\tThe topic of automated test data generation has been active for over three decades in the software engineering research community \\citep{myers:testing,miller76_floating_pt_test_data}.\n\tThere are now many existing mature tools for test data generation, both open source research tools \\citep{fraser:evosuite,kletal:austin-ist,cadar2008klee}, and  production testing systems \\citep{mhetal:ssbse18-keynote,tillman:ase14}.\n\tBecause of its pivotal impact on practical software engineering, automated testing remains a highly active research area \\citep{anandetal:orchestrated}, with the result that future automated testing advances will lead to ongoing improvement in automated translation.\n\t\n\tWe use one such open source automated test generation tool, EvoSuite\\xspace \\citep{fraser:evosuite}, in this paper.\n\tEvoSuite\\xspace is a well-established test generation tool for Java which uses coverage metrics \\citep{mike:icse17} and mutation scores \\citep{yjmh:analysis} to generate high-quality tests.\n\tIt has been widely used in the Software Testing research literature for test data generation although it has  not, hitherto, been used as part of an automated translation approach guided by mutation, the topic of the present paper. \n\t\n\n\n\t\n\n\t\n\t\n\t\n\t\n\n\t\n\n\n\t\n\tMore generally, software testing tools have been largely ignored by the machine learning community.\n\tIn this paper, we propose to use automatically created unit tests to guide unsupervised translation models for programming languages. \n\tMore precisely, we create unit tests automatically for a large number of functions from the source dataset. Since the unit tests are composed of simple inputs and asserts, they can easily be translated to semantically equivalent tests in the target languages using simple scripts. \n\tUsing our unit-tests and a pre-trained unsupervised translation model, we create parallel datasets by translating functions and selecting the translations that have the same semantics as the original function for the tested inputs.\n\n\n\n\n\n\tOverall, we make the following contributions: \n\t\\begin{itemize}\n\t\t\\item We introduce a novel approach, TransCoder-ST\\xspace~\\br{(for Self-Trained)}, that leverages an automated unit test generation pipeline to filter out invalid translations and reduce the noise coming from the back-translation process in unsupervised machine translation.\n\t\t\\item We present two implementations of this approach (online and offline), and show that it significantly outperforms the previous state of the art in code translation on all the language pairs we considered. In particular, we improve the state of the art for translating between Java, Python and C++ by an average of 12.6\\% Computational Accuracy (CA@1), corresponding to an average relative improvement of 25.5\\%. For Python~$\\rightarrow$~C++, we improve the CA@1 by 24\\%, reducing the error rate by 35.7\\% compared to previous models. \n\t\n\t\n\t\n\t\t\\item We generate multilingual unit tests for hundreds of thousands of Java functions and create a large parallel dataset of 135,000 parallel functions between Java, Python, and C++.\n\t\t\\item Our method is completely unsupervised and could easily be generalized to other programming languages and unit test creation tools.\n\t\\end{itemize}\n\t\n\t\n\t\n\t\n\t\n\n\t\n\t\\section{Related work}\n\t\\label{sec:relatedwork}\n\t\\paragraph{Unit Test Generation.} \n\n\n\t\n\tSoftware testing is challenging due to the large number of possibilities to be tested, and the inherent cost of covering reasonable representative sample \\citep{myers:testing}. \n\tWhen test design is performed by humans, the cost can be prohibitive so much research over the last three decades has focused on automating the process of test generation \\citep{anandetal:orchestrated}.\n\tAlthough automated test generation has been studied since the mid-1970s \\citep{miller76_floating_pt_test_data}, \n\tit was only in the last decade that industrial-strength tools have become widely available.\n\tThere are now several test data generation tools for  languages, including C \\citep{cadar2008klee,kletal:austin-ist} and Java \\citep{fraser:evosuite}.\n\tPopular test data generation techniques include symbolic execution of the code \\citep{cadar:three-decades}, dynamic execution guided by a fitness function \\citep{mh:icst15-keynote}, and hybrids of these two techniques \\citep{abetal:ase11}. Recently, neural networks have also been used successfully to generate unit tests \\citep{tufano2020unit}.\n\t\n\tOne of the most well-established and widely-used open source tools for test data generation is the EvoSuite\\xspace system \\citep{fraser:evosuite}.\n\tEvoSuite\\xspace uses search based software  engineering (SBSE) \\citep{mhamyz:acm-surveys} to generate test cases.\n\tLike all SBSE techniques, EvoSuite\\xspace is guided by fitness functions, in this case aimed at capturing the test suite's coverage of the code being tested. \n\tWe use EvoSuite\\xspace in our work  for three reasons: it is publicly available in open source (thereby facilitating replication), \n\tit is under current active development (thereby supporting future work), and it is widely used by other researchers (thereby enabling interoperability).\n\t\n\tIn order to assess the effectiveness of the test suites generated, we use mutation testing, a topic also widely-studied since the 1970s \\citep{demillo78}.\n\tA mutant is a version of the program into which a fault is deliberately inserted, thereby assessing the test suite's ability to detect this fault \\citep{yjmh:analysis}. \n\tFor a given set of mutants and a test suite, the mutation score is defined to be the proportion of mutants for which the test suite distinguishes the behavior of the mutant from that of the original program. \n\tThe mutant score is thus a proxy for the fault-revealing power of the test suite on a set of simulated faults (the mutants).\n\tMutation scores have been empirically demonstrated to be correlated to real fault revelation \\citep{mike:icse17}, motivating our adoption of this\n\tapproach.\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\paragraph{Machine Learning for Programming Languages.} In recent years, deep learning methods have been used to tackle various tasks in software engineering, with a particular interest in bug detection and repair~\\citep{wang2017dynamic,chen2019sequencer,Allamanis2018LearningTR,tarlow2019learning,murali2020industry,dinella2020hoppity,yasunaga2020graph,tufano2019empirical} and code completion~\\citep{li2017code,liu2020self,kim2020code,svyatkovskoy2020fast}. Unsupervised pre-training methods for code based on BERT~\\citep{kanade2020learning,feng2020codebert}, BART~\\citep{ahmad2021unified} or other objectives tailored to source code~\\citep{guo2020graphcodebert,roziere2021dobf} have shown strong results on benchmarks such as CodeXGLUE~\\citep{CodeXGLUE}. \n\t\n\tRecently, \\citet{hendrycks2021measuring} evaluated the competence of several language models for solving coding challenges. \\citet{chen2021evaluating} trained a large trained a large model to generate programs from docstrings and are able to solve 28.8\\% of the problems in their HumanEval dataset. They generate code solving problem statements written in natural language. The goal of code translation is also to generate code solving a specific problem, but the input (code written in a different programming language) is more precise and often more concise.\n\n\t\n\t\\paragraph{Translation of Programming Languages}\n\tSeveral studies used statistical methods to translate between programming languages. Early methods extracted parallel datasets and trained phrase-based models to translate between C\\# and Java~\\citep{nguyen2013lexical, karaivanov2014phrase} or from Python~2 to Python~3~\\citep{aggarwal2015using}.\n\tLater, \\cite{chen2018tree} proposed a tree-to-tree neural network to translate between CoffeeScript and JavaScript and between C\\# and Java using the dataset created by \\cite{nguyen2013lexical}.\n\n\tHowever, these approaches are limited to a few language pairs for which small parallel datasets were created manually (e.g. C\\#-Java) or can be created with rule-based tools (e.g. Python~2-Python~3 and CoffeeScript-JavaScript). \n\t\n\tInstead, \\cite{roziere2020unsupervised} proposed TransCoder\\xspace, an unsupervised model that leverages the principles of unsupervised machine translation \\citep{lample2018phrase}, to translate between Python, Java and C++. They showed that their method outperforms well-established rule-based baselines, does not require any parallel data or expert knowledge, and can easily be generalized to other languages. They pre-trained their model with the Masked Language Modeling (MLM) objective of \\cite{devlin2018bert}, and trained it with the denoising auto-encoding (DAE)~\\citep{vincent2008extracting} and the back-translation (BT)~\\citep{sennrich2015improving} objectives. \n\tLater, \\cite{roziere2021dobf} showed that\n\taugmenting MLM with a deobfuscation objective (dubbed DOBF\\xspace) can substantially improve the performance of TransCoder\\xspace.\n\n\tIn the rest of the paper, we will refer to their model as DOBF\\xspace.\n\t\n\t\n\tEven though unsupervised methods can be trained on large amounts of data, they sometimes lack the signal needed to differentiate between semantically different tokens that often occur in similar contexts (see Figure~\\ref{fig:teaser_examples}). There is a need for a method providing supervised signal directly related to the semantics of the code without manually crafted parallel datasets.\n\t\n\n\n\n\n\n\t\n\t\\section{Method}\n\n\n\n\n\n\n\n\t\n\t\\begin{figure}[t]\n\t\t\\centering\n\\makebox[\\textwidth][c]{\\includegraphics[width=1.2\\textwidth]{images\/unittests_diagram.jpg}}\n\t\t\\caption{\\small\\textbf{Our iterative self-training method.} Using EvoSuite\\xspace, we generate unit tests in Java, Python and C++ corresponding to several input Java functions. With a machine translation model (e.g.~TransCoder\\xspace), we generate several candidate translations of the the Java function in Python and C++.\n\t\t\tGenerated translations that pass the unit tests are used to create a parallel dataset on which we fine-tune the model. Discarding translations that fail the unit tests reduces the noise of data coming from the back-translation process, and significantly improves the overall performance of the model\n\t\t\n\t\t\\label{fig:method}\n\t\t\\vspace{-0.4cm}\n\t\\end{figure}\n\t\n\n\t\\subsection{Parallel data creation}\n\t\\label{sec:data_creation}\n\n\n\n\t\n\n\t\\paragraph{Parallel unit test generation:} We use EvoSuite\\xspace to automatically generate unit tests for Java functions.\n\tEvoSuite\\xspace is a well-established open source tool for automated test generation in Java, which is still under active development and frequently used. \n\n\tIt is designed for Java programs but its search-based technique is general and could be used for any programming language.\n\tUnit tests can be thought of as lists of inputs and asserts testing the semantics of a program (e.g., the output of the function, the side effects on its arguments such as sorting the input list).\n\tEvoSuite\\xspace uses evolutionary methods to derive tests that maximize a criteria such as code coverage or mutation score.\n\tDuring its search, each candidate solution in EvoSuite\\xspace is a test input.\n\tThe candidate inputs are evolved using crossover and mutation, and filtered by a fitness function (e.g., mutation score).\n\tWith each generation the fitness improves until it reaches a plateau or the budget is exhausted.\n\tThe final test inputs are wrapped up as test cases. Each program is associated to a test suite containing a series of test cases. \n\tFigure~\\ref{fig:unit_test_example} shows an example of a test case generated by EvoSuite\\xspace.\n\n\t\n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1.\\textwidth,center}\n\t\t\t\\begin{tabular}{l l}\n\t\t\t\t\\small{Java function} & \\small{A generated unit test}\\\\\n\t\t\t\t\\midrule\n\t\t\t\t\\begin{minipage}[t]{0.5\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{Java}\n\t\t\t\t\tpublic class CLAMP_CLASS{\n\t\t\t\t\tpublic static double clamp(\n\t\t\t\t\tdouble a, double min, double max){\n\t\t\t\t\treturn a<min?min:(a>max?max:a);\n\t\t\t\t\t}\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted}\n\t\t\t\t\\end{minipage} &\n\t\t\t\t\\begin{minipage}[t]{0.5\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{Java}\n\t\t\t\t\t@Test(timeout = 4000)\n\t\t\t\t\tpublic void test0()  throws Throwable  {\n\t\t\t\t\tdouble double0 = Example.clamp(\n\t\t\t\t\t742.0, 0.0, 0.0);\n\t\t\t\t\tassertEquals(0.0, double0, 0.01);\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage}\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\caption{\\small\\textbf{A unit test generated by EvoSuite\\xspace.} The Java function clamps the given value $a$ between the given $min$ and $max$. This test case is not sufficient to test the semantics of the function thoroughly but could be part of a suitable test suite. See Figure~\\ref{fig:good_unit_tests_evosuite} in the Appendix for a generated test suite with a high mutation score.}\n\t\t\\label{fig:unit_test_example}\n\t\t\\vspace{-0.4cm}\n\t\\end{figure}\n\t\n\n\t\\paragraph{Parallel test suites selection:}\n\n\tSome test suites created by EvoSuite\\xspace only cover a few parts of the semantics of functions. We only trust the translations verified by test suites which examine the function semantics thoroughly.\n\tWe use the mutation score, which is the most effective test assessment metric in the literature~\\citep{yjmh:analysis}, to pick out these test suites.\n\n\tThe mutation score is computed through mutation testing, in which mutants (i.e., program variants with syntactic changes) are generated from the original program based on a set of transformation rules (more details in Appendix~\\ref{appendix:mutation_score}). \n\tA mutant is said to be killed if at least one test from the test suite has different results on the mutant and the original program. Otherwise,\n\tthe mutant is said to survive. \n\tThe mutation score is the ratio of killed mutants.\n\tA test suite with a higher mutation score checks the code semantics more thoroughly. \n\tWe adopt a strict strategy in test suite selection: we keep only the Unit test suites with a mutation score larger than 90\\% for building the parallel dataset.\n\t\n\t\n\t\\paragraph{Parallel dataset building:}\n\n\tThe generated test suites can be used to test the semantics of  programs written in any programming language as long as there is a clear mapping between the types of the output and parameters in the original language and the language of the translated unit tests. \n\tWe transform the generated Java tests into C++ and Python tests with identical inputs and expected outputs and side effects (i.e., assertions).\n\tIn practice, we selected the Java functions which can be compiled and run in isolation and with simple output and parameter types. These types are the Java primitive types (e.g. \\texttt{int}, \\texttt{long}, \\texttt{bool}, \\texttt{float}\\dots), standard data types (e.g. \\texttt{Integer}, \\texttt{Double}, \\texttt{String}\\dots), array and \\texttt{List} or \\texttt{ArrayList} types of elements of supported types (e.g.~\\texttt{double[]}, \\texttt{List<Integer>}\\dots). \n\t\n\t\n\tWe use the best unsupervised translation models available for Java to Python and Java to C++ translation, namely TransCoder\\xspace~\\citep{roziere2020unsupervised} for Java to C++ and DOBF\\xspace~\\citep{roziere2021dobf} for Java to Python. For each Java function, we generate 20 Python and C++ translations with beam search and select the first element in the beam that passes the unit tests. The created tests are executed against the translated functions. \n\tIf all the tests pass, the Python and C++ functions have the same semantics assessed by the generated tests.\n\tOur method is illustrated in Figure~\\ref{fig:method}.\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\t\n\t\n\n\n\t\n\t\n\t\n\t\\subsection{Training method}\n\t\n\n\n\tOur parallel data generation method relies on a pre-existing model to translate from Java to Python and C++. There is little parallel data for these tasks and the best performing published models are unsupervised. TransCoder\\xspace~\\citep{roziere2020unsupervised} is trained using the MLM, denoising and back-translation objectives and is able to translate between Java, C++ and Python. DOBF\\xspace~{\\citep{roziere2021dobf}} provides clear improvements over TransCoder\\xspace for translating between Java and Python but was not trained on C++. Therefore, we use DOBF\\xspace to translate from Java to Python and TransCoder\\xspace to translate from Java to C++. When fine-tuning, we also reload these models. For DOBF\\xspace, we initialize the C++ language embeddings with those of Java.\n\t\n\tThe parallel examples we generate can be used to improve the performance of pre-existing translation models. Since the number of examples we generate also depends on the performance of the translation model, it creates a positive feedback loop where improving the model allows to improve the parallel dataset which in turn can be used to improve the model again. We propose offline and online approaches to use our method to maximize the unsupervised translation performance.\n\t\n\t\\paragraph{Offline training.} With the offline training method, we use the method described in Section~\\ref{sec:data_creation} to create parallel Java $\\leftrightarrow$ Python, Java $\\leftrightarrow$ C++ and Python $\\leftrightarrow$ C++ datasets using every input Java function we selected. For the first iteration, we fine-tune the model on these parallel examples until convergence. \n\tWe can iterate this process by selecting the best checkpoints for Java~$\\rightarrow$~Python and Java~$\\rightarrow$~C++ using the validation dataset and using them to generate new parallel datasets, which can in turn be used to train a better model.\n\tWe iterate this process until convergence, i.e. when we see no significant improvements on the validation set.\n\t\n\t\\begin{table}\n\t\t\\centering\n\t\t\\caption{\n\t\t\t\\small\n\t\t\t\\textbf{Size of the parallel datasets generated offline at each iteration.}\n\t\t}\n\t\t\\vspace{0.2cm}\n\t\t\\begin{tabular}{llll}\n\t\t\t\\toprule\n\t\t\tLanguages & First iteration & Second iteration & Third iteration\\\\\n\t\t\t\\midrule\n\t\t\tJava $\\leftrightarrow$ C++\n\t\t\t& 37,769 & 47,729 & 60,495\\\\\n\t\t\tJava $\\leftrightarrow$ Python\n\t\t\t&  43,194 & 43,956 & 45,311\\\\ \n\t\t\tC++ $\\leftrightarrow$ Python \n\t\t\t& 21,026 & 27,080 & 32,869\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:offline_data_size}\n\t\\end{table}\n\t\n\t\\paragraph{Online training.} With the online method, we create parallel examples on the fly while training the model. \n\tCompared to the offline method, it allows to always use the last model to generate new examples and it is much more convenient to automate.\n\n\tHowever, this process can be unstable if done naively. For instance, the model can start over-fitting only a few examples and stop generating anything that passes the unit tests for any other example. In order to stabilize the training, we follow \\cite{likhomanenko2020slimipl} and implement a cache mechanism storing the previous examples that passed the unit tests. At each step, the model can either train on parallel functions sampled from the cache or create new parallel functions to add to the cache. When an example is sampled, we remove it from the cache with a given probability. The online training allows the model to always benefit from the performance of the latest model and the cache mechanism ensures that the model does not forget the correct examples that it was able to generate at previous time steps.\n\n\t\n\n\t\n\t\\subsection{Evaluation}\n\tIn the context of natural languages, machine translation models are generally benchmarked against a reference solution using the BLEU score~\\citep{koehn2009statistical,bahdanau2014neural,vaswani2017attention}. Early studies on source code translation used the same metric to evaluate the quality of the generated functions~\\citep{nguyen2013lexical,karaivanov2014phrase,aggarwal2015using, barone2017parallel}, or the exact match score which requires the translation to be exactly equal to the ground truth~\\citep{chen2018tree}. \n\tHowever, these metrics fail to capture the semantics of the code and typically correlate poorly with the correctness of the generated function, prompting the use of new metrics checking if the generated solution passes series of test cases~\\citep{kulal2019spoc,roziere2020unsupervised,hendrycks2021measuring,chen2021evaluating}. \n\t\n\tWe evaluate our models on the validation and test sets of TransCoder\\xspace. It contains a few hundreds of parallel functions extracted from \\gfg along with associated unit tests.\n\tAs our TransCoder\\xspace and DOBF\\xspace baselines, we evaluate our models with the CA@N metric, which checks if any of the top-N solutions proposed by the model passes all the corresponding unit tests. This metric can be computed independently of the beam size (as long as the beam size is greater or equal to N). \n\n\t\n\n\n\t\n\t\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\section{Experiments}\n\t\\subsection{Training details}\n\t\\label{sec:training_details}\n\t\\paragraph{Model architecture.} We use a sequence-to-sequence model with attention composed of an encoder and a decoder model with a transformer architecture~\\citep{vaswani2017attention}. In order to provide fair comparisons, we use the exact same architecture as TransCoder\\xspace: an encoder and a decoder of 6 layers each, a hidden dimension of 1024 and 8 attention heads. We limit the size of the input to 512 tokens. \\citet{roziere2021dobf} train models with two different architectures. For Java $\\leftrightarrow$ Python, we compare ourselves to the version of DOBF\\xspace using the same architecture as TransCoder\\xspace. We initialize our models with either the best TransCoder\\xspace checkpoint for Java~$\\rightarrow$~C++ or the best DOBF\\xspace checkpoint for Java~$\\rightarrow$~Python  with C++ language embeddings initialized with those of Java.\n\t\n\t\\paragraph{Datasets.} As TransCoder\\xspace and DOBF\\xspace, we use the GitHub public dataset available on Google BigQuery filtered to keep only projects with permissive licences. As our unit test creation tool can only be used on Java code, we only use the Java files and we select only the functions that can be compiled in isolation. We obtain a dataset containing 333,542 Java functions. \n\tWe run EvoSuite\\xspace with a budget of 20 seconds and a criterion including the line, branch, cbranch and output coverages, as well as the weak and strong mutation scores. We set the maximum absolute value of integers that can be generated as an input to $\\sqrt{2^{31}-1}$ to limit the number of overflows.\n\tWe manage to obtain high-quality (mutation score $>$ 0.9 and at least two asserts) test cases for 103,488 functions. See Figures~\\ref{fig:unit_test_example} and~\\ref{fig:good_unit_tests_evosuite},~\\ref{fig:uninteresting_unit_tests_evosuite} in the appendix for examples of selected and filtered out test suites.\n\t\n\t\\paragraph{Training details.} During the training, we alternate between batches for every source and target language so that language pairs for which we managed to create more parallel examples are not overrepresented in our training batches. \n\tFor the online version, we set a cache warm-up parameter to ensure that we always generate new parallel examples if there are less than 500 examples in the cache for any language pair. \n\tOtherwise, we sample from the cache with probability $0.5$, or generate new examples, train on them once and put them in the cache also with probability $0.5$.\n\tThe sampled elements are removed from the cache with probability $0.3$, so that each element we create is trained on about 4 times in average before being removed from the cache. \n\tWe initialize the cache with parallel examples created offline.\n\t\n\tDuring beam decoding, we compute the score of generated sequences by dividing the sum of token log-probabilities by $l^{\\alpha}$ where $l$ is the sequence length. We found that taking $\\alpha=0.5$ (and penalizing long generations) leads to the best performance on the validation set.\n\n\n\n\t\n\n\t\\begin{table}[t]\n\t\t\\centering\n\t\t\\caption{\n\t\t\t\\small\n\t\t\t\\textbf{Computational accuracy scores for our methods and baselines.} We show the CA@1 metric computed with beam size 10. For the baselines, we ran the evaluations again and reported the best result between those reported in the original paper and those we obtained. Both the offline and online self-training methods lead to significant improvements over our baselines for every language pair and direction. Online self-training outperforms offline self-training, even after several iterations.\n\t\t\n\t\t}\n\t\t\\vspace{0.3cm}\n\t\t\\begin{adjustbox}{width=1.\\textwidth,center}\n\t\t\t\\begin{tabular}{l|cccccc|c}\n\t\t\t\t\\toprule\n\t\t\t\t& \\small{C++~$\\rightarrow$~Ja} & \\small{C++~$\\rightarrow$~Py} & \\small{Ja~$\\rightarrow$~C++} & \\small{Ja~$\\rightarrow$~Py }&  \\small{Py~$\\rightarrow$~C++ } & \\small{Py~$\\rightarrow$~Ja }&  \\small{AVG}\\\\\n\t\t\t\n\t\t\t\t\\midrule\n\t\t\t\tTransCoder\\xspace &\n\t\t\t\t\\compacc{65.1} &\n\t\t\t\t\\compacc{47.08} &\n\t\t\t\t\\compacc{79.8} &\n\t\t\t\t\\compacc{49.0} &\n\t\t\t\t\\compacc{32.62} &\n\t\t\t\t\\compacc{36.6} &\n\t\t\t\t\\compacc{51.7}\n\t\t\t\t\\\\\n\t\t\t\tDOBF\\xspace &\n\t\t\t\t- &\n\t\t\t\t- &\n\t\t\t\t- &\n\t\t\t\t\\compacc{52.7} &\n\t\t\t\t- &\n\t\t\t\t\\compacc{45.7} &\n\t\t\t\t-\n\t\t\t\t\\\\\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\\midrule\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\tOffline ST 1 &\n\t\t\t\t\\compacc{65.49} &\n\t\t\t\t\\compacc{58.32} &\n\t\t\t\t\\compacc{83.69} &\n\t\t\t\t\\compacc{63.28} &\n\t\t\t\t\\compacc{46.35} &\n\t\t\t\t\\compacc{52.18} &\n\t\t\t\t\\compacc{61.55}\n\t\t\t\t\\\\\n\t\t\t\tOffline ST 2 &\n\t\t\t\t\\compacc{66.53} &\n\t\t\t\t\\compacc{56.16} &\n\t\t\t\t\\textbf{\\compacc{85.19}} &\n\t\t\t\t\\compacc{66.31} &\n\t\t\t\t\\compacc{48.07} &\n\t\t\t\t\\compacc{56.55} &\n\t\t\t\t\\compacc{63.135}\n\t\t\t\t\\\\\n\t\t\t\tOffline ST 3 &\n\t\t\t\t\\compacc{65.28} &\n\t\t\t\t\\compacc{48.1648} &\n\t\t\t\t\\compacc{81.12} &\n\t\t\t\t\\compacc{58.1} &\n\t\t\t\t\\compacc{48.93} &\n\t\t\t\t\\compacc{54.68} &\n\t\t\t\t\\compacc{59.37833333}\n\t\t\t\t\\\\\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\tOnline ST &\n\t\t\t\t\\textbf{\\compacc{67.98}} &\n\t\t\t\t\\textbf{\\compacc{61.34}} &\n\t\t\t\t\\compacc{84.55} &\n\t\t\t\t\\textbf{\\compacc{68.9}} &\n\t\t\t\t\\textbf{\\compacc{56.65}} &\n\t\t\t\t\\textbf{\\compacc{58.21}} &\n\t\t\t\t\\textbf{\\compacc{66.27166667}}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\label{tab:results_beam}\n\t\\end{table}\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\t\\subsection{Results and discussion}\n\n\t\n\t\n\t\\begin{table}[t]\n\t\t\\centering\n\t\t\\caption{\n\t\t\t\\small\n\t\t\t\\textbf{CA@n metric for several beam sizes averaged on all language pairs.} The value k corresponds to the beam size. For instance, CA@1 k=10 means that we use beam decoding to generate 10 translations, and select the one with the highest score. The best baseline corresponds to taking the best model between TransCoder\\xspace and DOBF\\xspace for every language pair and direction.\n\t\t\tThe error rate reduction of the offline and online self-training methods over the best baseline are high ($>20\\%$) across all CA@N metrics and beam sizes.\n\t\t\n\t\t\t\\vspace{0.2cm}\n\t\t\t\\label{tab:results_average}\n\t\t}\n\t\n\t\t\\begin{tabular}{l|ccccc}\n\t\t\t\\toprule\n\t\t\t& \\small{CA@1 k=1} & \\small{CA@1 k=10} & \\small{CA@1 k=20} & \\small{CA@10 k=10}&  \\small{CA@20 k=20}\\\\\n\t\t\n\t\t\t\\midrule\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\tBest baseline &\n\t\t\t\\compacc{52.24833333} &\n\t\t\t\\compacc{53.7} &\n\t\t\t\\compacc{53.38} &\n\t\t\t\\compacc{67.27666667} &\n\t\t\t\\compacc{70.54333333}\n\t\t\t\\\\\n\t\t\t\\midrule\n\t\t\tOffline ST 1 &\n\t\t\t\\compacc{61.37166667} &\n\t\t\t\\compacc{61.55166667} &\n\t\t\t\\compacc{61.37166667} &\n\t\t\t\\compacc{73.26666667} &\n\t\t\t\\compacc{75.825}\n\t\t\t\\\\\n\t\t\tOffline ST 2 &\n\t\t\t\\compacc{61.698} &\n\t\t\t\\compacc{63.135} &\n\t\t\t\\compacc{63.02666667} &\n\t\t\t\\compacc{73.255} &\n\t\t\t\\compacc{75.76833333}\n\t\t\t\\\\\n\t\t\tOffline ST 3 &\n\t\t\t\\compacc{58.50666667} &\n\t\t\t\\compacc{59.37833333} &\n\t\t\t\\compacc{59.235} &\n\t\t\t\\compacc{70.825\n\t\t\t} &\n\t\t\t\\compacc{73.555}\n\t\t\t\\\\\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\tOnline ST &\n\t\t\t\\textbf{\\compacc{64.68}} &\n\t\t\t\\textbf{\\compacc{66.27166667}} &\n\t\t\t\\textbf{\\compacc{66.30833333}} &\n\t\t\t\\textbf{\\compacc{75.35}} &\n\t\t\t\\textbf{\\compacc{77.19}}\n\t\t\t\\\\\n\t\t\t\n\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\n\t\\end{table}\n\t\\begin{table}[t]\n\t\t\\centering\n\t\t\\caption{\n\t\t\t\\small\n\t\t\t\\textbf{Ablation study.} We show the CA@1 metric computed with greedy decoding at evaluation time except for the last line where the beam size is set to 10. We evaluate models trained with no cache system, without initializing the cache (with or without selecting the tests with a minimum mutation score of 0.9), and a beam size of 1 when generating examples. We also compare the CA@1 score of our full model when evaluating with greedy decoding and with beam size 10.\n\t\t\n\t\t\tUsing a pre-filled cache and selecting only the tests with a high mutation score lead to substantially better performance, although these steps are not necessary to outperform our baseline. The online method already performs well with greedy decoding at generation time, but generating with beam size 20 further improves the results.}\n\t\t\\vspace{0.3cm}\n\t\t\n\t\t\\begin{adjustbox}{width=1.\\textwidth,center}\n\t\t\t\\begin{tabular}{p{3.5cm}|cccccc|c}\n\t\t\t\t\\toprule\n\t\t\t\t& \\small{C++~$\\rightarrow$~Ja} & \\small{C++~$\\rightarrow$~Py} & \\small{Ja~$\\rightarrow$~C++} & \\small{Ja~$\\rightarrow$~Py }&  \\small{Py~$\\rightarrow$~C++ } & \\small{Py~$\\rightarrow$~Ja }&  \\small{AVG}\\\\\n\t\t\t\n\t\t\t\t\\midrule\n\t\t\t\tNo cache &\n\t\t\t\t\\compacc{66.53} &\n\t\t\t\t\\compacc{52.7} &\n\t\t\t\t\\compacc{83.69} &\n\t\t\t\t\\compacc{60.26} &\n\t\t\t\t\\compacc{41.2} &\n\t\t\t\t\\compacc{51.77} &\n\t\t\t\t\\compacc{59.35833333}\n\t\t\t\t\\\\\n\t\t\t\tCache not initialized &\n\t\t\t\t\\compacc{64.86} &\n\t\t\t\t\\compacc{51.62} &\n\t\t\t\t\\compacc{82.4} &\n\t\t\t\t\\compacc{62.42} &\n\t\t\t\t\\compacc{46.57} &\n\t\t\t\t\\compacc{52.6} &\n\t\t\t\t\\compacc{60.07833333}\n\t\t\t\t\\\\\n\t\t\t\n\t\t\t\t~~+ No min mut. score &\n\t\t\t\t\\compacc{64.03} &\n\t\t\t\t\\compacc{50.11} &\n\t\t\t\t\\compacc{82.62} &\n\t\t\t\t\\compacc{60.91} &\n\t\t\t\t\\compacc{47.42} &\n\t\t\t\t\\compacc{46.99} &\n\t\t\t\t\\compacc{58.68}\n\t\t\t\t\\\\\n\t\t\t\tST greedy decoding &\n\t\t\t\t\\compacc{65.9} &\n\t\t\t\t\\compacc{54.21} &\n\t\t\t\t\\compacc{82.19} &\n\t\t\t\t\\compacc{60.91} &\n\t\t\t\t\\compacc{56.22} &\n\t\t\t\t\\compacc{56.55} &\n\t\t\t\t\\compacc{62.66333333}\n\t\t\t\t\\\\\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\tFull model (ST beam 20) &\n\t\t\t\t\\compacc{66.74} &\n\t\t\t\t\\compacc{61.12} &\n\t\t\t\t\\compacc{84.12} &\n\t\t\t\t\\compacc{67.82} &\n\t\t\t\t\\compacc{52.15} &\n\t\t\t\t\\compacc{56.65} &\n\t\t\t\t\\compacc{64.68} \\\\\n\t\t\t\t~~+ Eval beam 10&\n\t\t\t\t\\textbf{\\compacc{67.98}} &\n\t\t\t\t\\textbf{\\compacc{61.34}} &\n\t\t\t\t\\textbf{\\compacc{84.55}} &\n\t\t\t\t\\textbf{\\compacc{68.9}} &\n\t\t\t\t\\textbf{\\compacc{56.65}} &\n\t\t\t\t\\textbf{\\compacc{58.21}} &\n\t\t\t\t\\textbf{\\compacc{66.27166667}} \\\\\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\t\\bottomrule\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\label{tab:ablation}\n\t\n\t\\end{table}\n\t\\paragraph{Results.}\n\tIn Tables~\\ref{tab:results_beam}~and~\\ref{tab:results_average}, we compare the results of our offline and online training methods with those of TransCoder\\xspace and DOBF\\xspace. DOBF\\xspace outperforms TransCoder\\xspace for the Java $\\leftrightarrow$ Python pair.\n\tWe compare our models against the best baseline for each language pair and direction.\n\t\n\n\n\tTraining on the generated parallel examples brings substantial improvements for every language pair, direction, and metric. \n\tOffline training already provides clear improvements over the baseline after one iteration. \n\tThe computational accuracy (CA@1) computed with beam size 10 is higher for every direction and it is substantially higher for the language pairs involving Python. It allows to reduce the error rate of the best baseline by 22.4\\% for Java~$\\rightarrow$~Python. In average, it increases the CA@1 by 7.9\\% over the best previous models, and reduces the error rate by 17.0\\%. In the second iteration, the model is trained on more than 25\\% more examples for Java $\\leftrightarrow$ Python and C++ $\\leftrightarrow$ Python and a few more examples for Java $\\leftrightarrow$ Python (see Table~\\ref{tab:offline_data_size}). It results in improvements in every direction except for C++~$\\rightarrow$~Java where there is no significant improvement and C++~$\\rightarrow$~Python where we observe a small regression. The average improvement over the first iteration is 1.5\\% point.\n\tAlthough the model for the third iteration is trained on more parallel samples, its performance on the test set of TransCoder\\xspace is actually worse than after the second iteration. \n\tAfter two iterations, the model learned to generate more samples that pass the unit tests but some of them are actually incompatible with the types of translations expected by TransCoder\\xspace (e.g. example with overflows in Figure~\\ref{fig:limitation_overflow}), causing the computational accuracy score to go down.\n\n\t\n\tThe online self-training method provides further improvements over training on the pseudo-labeled examples offline. It outperforms every other method in every case except the second iteration of offline training for Java~$\\rightarrow$~C++. \n\tIn average, this model outperforms the baseline by 12.6\\% points, corresponding to an error rate reduction of 25.5\\%. For Python~$\\rightarrow$~C++, it improves previous performance by more than 24\\% points, which corresponds to reducing the error rate by 35.7\\%.\n\tExamples of avoided errors can be found in Figure~\\ref{fig:teaser_examples} and Appendix~\\ref{appendix:translation_examples}.\n\n\tOverall, all our models significantly improve previous results.\n\tAs shown in Table~\\ref{tab:results_average}, these improvements are stable across several beam sizes and CA@n metrics. \n\tThe CA@20 metric shows that the number of examples for which none of the 20 elements in the beam are correct is reduced by more than 22\\% with online self-training. \n\tIt indicates that, even though we train only on the output of the model, our method does much more than reordering the elements in the beam and allows the model to find correct solutions that were not assigned a high probability by the baseline model. \n\t\\br{See Table~\\ref{tab:full_results_tab} in the appendix for more results.}\n\t\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\paragraph{Ablation study.} The results of our ablation study are shown in Table~\\ref{tab:ablation}. Training online with no cache makes the training much less stable. The model improves at the beginning of training and we can select a few checkpoints where it performs well, but it ends up over-fitting a few examples it generated and the performance drops after a few epochs. \n\tStarting with an empty cache slows down the training and hinders generalization, leading to a clear drop in performance.\n\tWe also try removing the minimum mutation score requirement for the model with no initial cache, which leads to even lower scores as the model is trained partly on lower-quality parallel data. \n\t\n\tAll these models were trained using a self-training beam size of 20 when generating new examples. Training with greedy decoding is much faster since computing the results for all the 20 elements of the beam is costly.\n\tHowever, generating new examples with greedy decoding leads to a loss of about two percentage points in average compared to our full model using beams of size 20.\n\n\n\tIt shows that initializing the cache of the model with beam size 20 is not sufficient and creating new examples with beam search is necessary to reach our best performance. \n\tOur full model provides some improvements over the ablated versions for every language pair and direction, except over the model trained with greedy decoding for Python~$\\rightarrow$~C++ translation. Evaluating with beam size 10 (still returning only the first element) leads to some improvements for every language pair. \n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\t\\paragraph{Limitations.} We found that the unit tests we create with this method are sometimes incompatible with those of the test set of TransCoder\\xspace, and that the capacity of a model to generate functions that pass these unit tests is not perfectly correlated to its score on the test set.\n\tIt raises the deeper issue of defining what constitutes a correct translation. For instance, most programmers would translate a factorial function implemented with \\texttt{long} integers into a factorial function implemented with Python's integer type. However, these functions are not semantically equivalent since the Java implementation would return a negative number for the input \\texttt{21} due to integer overflow while the Python implementation would return $21!$ correctly. The human developers who wrote the parallel functions in the test set of TransCoder\\xspace often assumed that these functions would only be used on a limited domain where no overflow occurs (see Figure~\\ref{fig:limitation_overflow}). However, the test cases of EvoSuite\\xspace and TransCoder\\xspace are not limited to this domain and they sometimes assert different semantics. By using the test suites from EvoSuite\\xspace as source of truth, we sometimes train the model to generate translations that are more rigorous but also less natural. \n\t\\begin{figure}[ht]\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1.\\textwidth,center}\n\t\t\t\\begin{tabular}{l l l}\n\t\t\t\t\\small{Input Java function} & \\small{Gold translation} & \\small{Translation passing multilingual tests}\\\\\n\t\t\t\t\\midrule\n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{java}\n\t\t\t\t\tstatic int factorial(int n){\n\t\t\t\t\tif (n < 2) return 1;\n\t\t\t\t\treturn n * factorial(n - 1);\n\t\t\t\t\t}\n\t\t\t\t\t\\end{minted}\n\t\t\t\t\\end{minipage} &\n\t\t\t\t\\begin{minipage}[t]{0.32\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{python}\n\t\t\t\t\tdef factorial(n):\n\t\t\t\t\tif n < 2:\n\t\t\t\t\treturn 1\n\t\t\t\t\treturn n * factorial(n-1)\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage} & \n\t\t\t\t\\begin{minipage}[t]{0.33\\textwidth}\n\t\t\t\t\t\\begin{minted}[escapeinside=||]{python}\n\t\t\t\t\tdef factorial(n):\n\t\t\t\t\tn = np.int32(n)\n\t\t\t\t\tif n < 2:\n\t\t\t\t\treturn np.int32(1)\n\t\t\t\t\treturn n * factorial(n - 1)\n\t\t\t\t\t\\end{minted} \n\t\t\t\t\\end{minipage}\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\caption{\\small{\\textbf{Example of disagreement between our multilingual tests and the test set of TransCoder\\xspace.} \n\t\t\t\n\t\t\t\tThe gold translation is only equivalent to the input Java function on a small domain where there is no integer overflow and does not pass our multilingual unit tests. The version that passes the unit tests casts the Python integers to \\texttt{np.int32}, reproducing the behaviour of the original Java code but causing it to fail some of the unit tests of TransCoder\\xspace.}}\n\t\t\\vspace{-0.5cm}\n\t\t\\label{fig:limitation_overflow}\n\t\\end{figure}\n\t\n\t\n\t\\section{Conclusion}\n\tIn this paper, we introduced a novel method to grow a parallel corpus for automated code translation, from completely monolingual data.\n\tWe leverage multilingual unit tests to filter good pseudo-labels, improving the model, and in turn the candidate translations.\n\tWe show that both offline and online methods substantially improve the state of the art in unsupervised code translation, with an average improvement of 12.6\\% points in computational accuracy, and up to 24\\% points for Python~$\\rightarrow$~C++, corresponding to translation error rate reductions of 25.5\\% and 35.7\\% respectively.\n\t\n\n\n\t\n\tOur method would automatically gain from improvements of automatic unit test generation tools. \n\tWe could also increase the size of the dataset we generate by using test creation tools written for other languages than Java, or by generating tests with EvoSuite\\xspace on translated examples. \n\tSimilarly, we could also extract the semantics of human-written unit tests found in open-source projects to obtain larger, and possibly higher-quality datasets. \n\tIn this paper, we focused on translation correctness and our parallel example validation criterion was only based on semantics. It could be supplemented with other requirements, such as a specific code formatting or the output of linters to generate code verifying arbitrary criteria. \n\n\tFinally, the approach presented in this paper could easily be transferred to natural languages.\n\tAlthough there is no concept of unit tests in natural language, traditional grammar and syntax checkers could be used to filter out some incorrect generations, and reduce the noise coming from the back-translation process.\n\tNeural machine translation systems being highly sensitive to noise coming from parallel data \\citep{belinkov2017synthetic, khayrallah2018impact}, this may improve the performance in low-resource machine translation significantly.\n\t\n\t\n\n\n\n\n\t\n\n\n\n\n\t\n\n\n\n\n\n\n\n\t\n\t\n\t\n\n\t\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\n\\section{Introduction}\n\nAmong the decay channels of the $\\tau$ lepton, $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$\nhas the largest branching fraction. The decay is dominated by\nintermediate resonances and thus provides information on the \nproperties of the $\\rho(770)$, $\\rho^{\\prime}(1450)$, and \n$\\rho^{\\prime\\prime}(1700)$ mesons and their interference.\nSince leptons do not participate in the strong interaction,\nhadronic $\\tau$ decays provide a clean environment for  \nstudying the dynamics of hadronic states\nin an interesting energy range dominated by resonances.\n\nUnder the Conserved Vector Current~(CVC) theorem,\nthe  $\\pi^{-}\\pi^{0}$ mass spectrum in this range\ncan be used to improve the theoretical error \non the anomalous magnetic moment of the muon \n$a_{\\mu}=(g_{\\mu}-2)\/2$. A recent review of the \ncalculations of $a_{\\mu}$ is given in Ref.~\\cite{CM}.   \nIt is known that the theoretical error on $a_{\\mu}$ is dominated by \nthe (leading-order) hadronic contribution \n$a_{\\mu}^{\\rm had,LO}$, given by the hadronic vacuum polarization.\nThis contribution cannot be evaluated within the framework of\nperturbative QCD; however, it can be evaluated from a\nmeasurement of the cross section for $e^+e^-$ annihilation \nto hadrons~\\cite{DEHZ,HMNT}. Alternatively, \nCVC \nrelates the properties of \nthe $\\pi^{+}\\pi^{-}$ system produced \nin $e^+e^-\\rightarrow\\pi^+\\pi^{-}$ to those of the \n$\\pi^{-}\\pi^{0}$ system produced in $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ decay; \nthus, using CVC and correcting for \nisospin-violating effects, \n$\\tau$ data have also been used to obtain a\n more precise\n prediction for $a^{\\rm had,LO}_{\\mu}$~\\cite{DH98,DEHZ}.\n\n\nRecently,  data on $e^+e^-\\!\\rightarrow\\!\\pi^+\\pi^-$  \nhas become available from the CMD-2, KLOE, \nand SND experiments~\\cite{CMD03,CMD2002,KLOE2005,SND2005}. \nData on $\\tau$ decays is available from the\nALEPH~\\cite{ALEPH97,DATAU02}, CLEO~\\cite{CLEO2000}, \nand OPAL~\\cite{OPAL1999} experiments. The most recent evaluation \nof the hadronic contribution to $a_{\\mu}$ using\n$e^+e^-$ data gives~\\cite{DAV2005}\n$ a^{\\rm exp}_{\\mu} - a^{\\rm th}_{\\mu} = (25.2\\pm 9.2  )\\times 10^{-10}$,\n while that using the  $\\tau$ lepton data \nwhere applicable gives\n$ a^{\\rm exp}_{\\mu} - a^{\\rm th}_{\\mu} = (9.4\\pm 10.5  )\\times 10^{-10}$.\nThe experimental value $a_\\mu^{\\rm exp}$\nis dominated by the BNL E821 measurement~\\cite{BNL2004}\n$( 11\\ 659\\ 208 \\pm 5.8)\\times 10^{-10}$.\nThese differences correspond to 2.7 and 0.9 standard \ndeviations, respectively, and thus \nthere is a significant difference \nbetween the $e^+e^-$-based and \n$\\tau$-based predictions. \nTo clarify the situation, \nmore data for $e^+e^-\\!\\rightarrow\\!\\pi^-\\pi^+$ and for\n$\\tau^-\\!\\rightarrow\\!\\pi^-\\pi^0\\nu_{\\tau}$\n decays are needed.\nIn this paper we present a high-statistics measurement \nof  the $\\pi^{-}\\pi^{0}$ mass spectrum produced in\n$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ decays~\\cite{CC} using \ndata collected by the Belle experiment at the \nKEKB $e^+e^-$ collider operating at a \ncenter-of-mass (CM) energy of~10.6~GeV. \nThe data sample is about 50 times larger \nthan those of previous experiments.\n\n\\section{Basic formulae}\n\nThe differential decay rate for\n$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ can be expressed as\n\\begin{eqnarray}\n\\frac{1}{\\Gamma}\\frac{d\\Gamma}{ds}\n(\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}) =\n\\frac { 6\\pi |V_{ud}|^{2} S_{EW} } {m_{\\tau}^{2}} \n\\left(\n\\frac{\\mathcal{B}_{e}} { \\mathcal{B}_{\\pi\\pi} } \\right)\n     \\left( 1 - \\frac{s}{m_{\\tau}^{2}} \\right)^{2}\n     \\left( 1 + \\frac{2s}{m_{\\tau}^{2}} \\right)\n\\,v_{-}(s),\n\\label{eq:tauspec} \n\\end{eqnarray}\nwhere $s$ is the invariant-mass-squared of the $\\pi^{-}\\pi^{0}$ system,\n$v_{-}(s)$ is the vector spectral function\ncharacterizing the $\\pi^{-}\\pi^{0}$ system,   \n$|V_{ud}|$ denotes the CKM mixing matrix element, and \n$S_{EW}$ accounts for electroweak radiative corrections. \n$\\mathcal{B}_{e}$ \nand $\\mathcal{B}_{\\pi\\pi}$ \nare the branching fractions for \n$\\tau^{-}\\rightarrow e^{-}\\nu_{\\tau}\\bar{\\nu}_{e}$ and\n$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$, respectively. \n\nThe corresponding $\\pi^{+}\\pi^{-}$spectral function $v_{0}(s)$ can\nbe obtained from the $e^+e^-\\rightarrow \\pi^+\\pi^-$ cross section:\n\\begin{eqnarray}\n\\sigma(e^+e^-\\rightarrow\\pi^+\\pi^-)  = \n\\frac{4\\pi\\alpha_{0}^{2}}{s}\\,v_{0}(s),\n\\label{eq:eepipi}\n\\end{eqnarray}\nwhere $s$ is the $e^+e^-$ CM energy squared and\n$\\alpha_{0}$ is the fine-structure constant  at $s=0$.\nUp to isospin-violating effects, CVC allows one to \nrelate the spectral function from $\\tau$ decays\nto the isovector part of the $e^+e^-$ spectral function\n\\cite{Weakf}:\n\\begin{eqnarray}\nv_{-}(s) & = & v_{0}^{I=1}(s)\\,.\n\\label{eq:cvc}\n\\end{eqnarray}\n\nThe mass spectrum of the two-pion system is typically expressed \nin terms of pion form factors; these are useful for comparing  \nresonance shapes in the charged and neutral two-pion systems. \nThe spectral function $v_{j}(s)\\,(j=-,0)$ is related to the \nform factor $F^{j}_{\\pi}(s)$ via\n\\begin{equation}\nv_{j}(s) = \\frac{\\beta_{j}^{3}(s)}{12\\pi}|F_{\\pi}^{j}(s)|^{2},\n\\label{eq:pionform}\n\\end{equation} \nwhere $\\beta_{-}(s)\\, (\\beta_{0}(s))$ \nis the pion velocity in the $\\pi^-\\pi^{0}$\\, ($\\pi^{+}\\pi^{-}$) rest-system.\nThe velocities $\\beta_{j}(s)$ are explicitly given by\\\n$\\beta^2_{-}(s) = \n\\left[ 1 - (m_{\\pi^{-}} - m_{\\pi^{0}})^2\/s\\right]\n\\left[ 1 - (m_{\\pi^{-}} + m_{\\pi^{0}})^2\/s\\right]$\\, \nand\\, $\\beta^2_{0}(s) = \n\\left[ 1 - 4m_{\\pi^{-}}^2 \/s\\right]$ \\,.\n\n\n\n\nThe leading-order hadronic contribution to the muon anomalous \nmagnetic moment ($a_{\\mu}^{\\rm had,LO}$) is related to the\n$e^+e^-$ annihilation cross section via the dispersion integral\n\\begin{eqnarray}\na^{\\rm had,LO}_{\\mu}\n&=& \\left( \\frac{\\alpha_{0} m_{\\mu}}{3\\pi} \\right)^{2}\n\\int_{4m_{\\pi}^{2}}^{\\infty} \\frac{R(s)}{s^{2}} \\hat{K}(s)\\,ds\\,,\n\\label{eq:amu}\n\\end{eqnarray}  \nwhere $s$ is the invariant-mass-squared of the two-pion system, and \n$R(s) = \\sigma(e^+e^-\\!\\rightarrow\\!{\\rm hadrons})\/(4\\pi\\alpha_{0}^{2}\/3s)$.\nThe kernel $\\hat{K}(s)$  is a smooth function increasing from 0.63 at\nthe threshold $s=4m_{\\pi}^{2}$ to unity at $s\\!=\\!\\infty$~\\cite{KERNEL}.\nDue to the $1\/s^{2}$ dependence, hadronic final states at low energy \ndominate the contribution to $a_{\\mu}^{\\rm had,LO}$; \nin fact about 70\\% of $a_{\\mu}^{\\rm had,LO}$ is due to the two-pion \nstate having $4m_{\\pi}^{2}\\le s\\le 0.8$  $({\\rm GeV}\/c^{2})^{2}$.\nConsequently, the $2\\pi$ spectral function in \n$\\tau$ data is \nuseful to obtain predictions for $a_\\mu^{\\rm had,LO}$.\nUsing Eqs.~(\\ref{eq:eepipi}) and (\\ref{eq:cvc})\nto evaluate (\\ref{eq:amu}) we obtain\n\\begin{eqnarray}\na^{\\pi\\pi}_{\\mu}\n&=& \\left( \\frac{\\alpha_{0} m_{\\mu}}{3\\pi} \\right)^{2}\n\\int_{4m_{\\pi}^{2}}^{m_{\\tau}^{2}}\n\\frac{3\\, v_{-}(s)}{  s^{2}} \\hat{K}(s)\\,ds\\,\n+ ... \\,,\n\\label{eq:amu2pi}\n\\end{eqnarray}  \n\\noindent\nwhere ''$...$'' \nindicates the integral above the $m_{\\tau}^{2}$ region.\nTo determine $v_-(s)$, \none must measure both the branching fraction for\n$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ and the $\\pi^-\\pi^0$ mass spectrum $(1\/N)(dN\/ds)$. \nHere we report new measurements for both of these.\n\n\n\n\\section{Data Sample and Selection Criteria }\n\n\nThe data sample used was collected by the Belle detector at \nthe KEKB energy-asymmetric $e^{+}e^{-}$ collider~\\cite{KEKB}. \nIt is based on an integrated luminosity of $72.2~{\\rm fb}^{-1}$ \nrecorded at a CM energy of 10.6~GeV.\nThe Belle detector is a large-solid-angle magnetic spectrometer\nconsisting of a three-layer silicon-vertex detector (SVD), a 50-layer \ncentral drift chamber (CDC) for charged particle tracking\nand specific ionization measurement~($dE\/dx$),\nan array of aerogel threshold Cerenkov counters~(ACC),\na barrel-like arrangement of time-of-flight scintillation counters~(TOF),\nand an electromagnetic calorimeter~(ECL) comprised of CsI(Tl) crystals\nlocated inside a superconducting solenoid coil that provides a \n1.5 T magnetic field. An iron flux-return located outside of \nthe coil is instrumented\nto identify muons and to detect $K_{L}^{0}$ mesons (KLM).\nThe detector is described in detail elsewhere~\\cite{Belle}.\n\n\nTo study backgrounds and determine selection criteria,\nwe perform Monte Carlo~(MC) simulation studies \nfor various processes. The KORALB\/TAUOLA\nprogram~\\cite{TAUOLA,TAUOLA2004} is used for $\\tau^+\\tau^-$-pair\n generation,\nthe QQ generator~\\cite{QQ} for ${\\bar B B}$ and ${\\bar q q}$ continuum\nprocesses, the BHLUMI~\\cite{BHLUMI} program for radiative Bhabha events,\nthe KKMC~\\cite{KKMC} program for radiative $\\mu^+\\mu^-$-pair\n events, and the\nAAFHB~\\cite{AAFHB} program for two-photon processes.\nThe BHLUMI and KKMC programs include higher-order radiative \ncorrections and are among the most accurate programs available. \nThe detector response is simulated by a GEANT-based program~\\cite{GEANT}. \nIn order to realistically simulate beam-induced background, \ndetector hits taken from randomly-triggered data are added \nto wire hits in the CDC and to energy deposits in the ECL.\n\n\n\\subsection{$\\tau^+\\tau^-$ pair selection}\n\nThe event selection consists of two steps. \nInitially, a sample of generic\n$e^+e^-\\rightarrow \\tau^+\\tau^-(\\gamma)$ events are\nselected with relatively loose criteria. \nFrom this sample\n$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ decays are then selected.\nThe number of  generic $\\tau^+\\tau^-$ events is used to determine\nthe $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ branching fraction.\n\nGeneric $\\tau^+\\tau^-$ events are selected by requiring that\nthe number of charged tracks in an event  \nbe two or four with zero net charge; \nthat each track have a momentum\ntransverse to the beam axis ($p_T$)\nof greater than 0.1~GeV\/$c$; and that\neach track extrapolate to the interaction point (IP) \nwithin ${\\pm1}$ cm transversely \nand within ${\\pm5}$ cm along the beam direction.\nTo suppress background from Bhabha and $\\mu^+\\mu^-$ events, \nthe reconstructed CM energies and\nthe sum of the momenta of \nthe two leading tracks are required to be less \nthan 9.0 GeV\/$c$. The maximum $p_T$ among the tracks is \nrequired to be greater than 0.5~GeV\/$c$.\nBeam-related background is rejected by requiring that the position \nof the reconstructed event vertex be less than 0.5~cm from the IP\nin the transverse direction and less than 2.5~cm from the IP\nalong the beam direction.\nThe polar angle of the leading particle with respect to the\nbeam axis ($\\theta^*$) in the CM frame is required\nto be in the fiducial region of the detector:\n${35^{\\circ} < \\theta^* < 145^{\\circ} }$. \n\nTo reduce remaining background from Bhabha, $\\mu^+\\mu^-\\gamma$,\nand two-photon events,\na cut is applied in the two-dimensional plane of the\nmissing-mass $MM$ and the direction of missing momentum \nin CM ${\\theta^{*}_{\\rm miss}}$, where\n$MM$ is evaluated from the four-momenta\nof the measured tracks and  photons:\n${(MM)^{2} = (P_{\\rm ini} - P_{\\rm tracks} \n- P_{\\gamma s})^{2} }$. In this expression $P_{\\rm ini}$\nis the four-momentum of the initial $e^+e^-$ system.\nEach photon (reconstructed from clusters in the calorimeter) must \nbe separated \nfrom the nearest track projection by at least 20~cm\nand have an energy greater than 0.05~GeV in the \ncentral part \n($-0.63\\le\\cos\\theta< 0.85$),\\, and 0.1~GeV in the endcap \npart ($-0.90\\le\\cos\\theta< -0.62$ and $0.85\\le\\cos\\theta<0.95$).\nPhotons measured at the \ndetector edge are rejected. A scatterplot of \n$MM$ vs.\\ $\\theta_{\\rm miss}$ \nfor data is shown in Fig.~\\ref{data_mm}.\nIn this plot, events at $MM\\approx 0$ are due to radiative Bhabha\nevents and $e^+e^-\\rightarrow \\mu^+\\mu^-(\\gamma)$, \nwhile events in the high-$MM$ region are from two-photon \nprocesses.  \nEvents within the octagonal region\nare selected as $\\tau^+\\tau^-$candidates.\n\n\n\\begin{figure}[t]\n\\rotatebox{0}{\\includegraphics*[width=0.410\\textwidth,clip]\n {.\/Fig_hep-ex\/Fig-1-a.eps}} \n\\rotatebox{0}{\\includegraphics*[width=0.45\\textwidth,clip]\n  {.\/Fig_hep-ex\/Fig-1-b.eps}} \n\\caption\n{Missing mass ($MM$) versus  the polar \nangle for the direction \nof the missing momentum ($\\theta^{*}_{\\rm miss}$).\nThe left plot shows MC \n$e^+e^-\\rightarrow \\tau^+\\tau^-$ events, and the right plot \nshows the data. Events inside the octagonal region are selected \nas $\\tau^+\\tau^-$-pair candidates.}\n\\label{data_mm}\n\\end{figure}\n\n\nCandidate events are divided into two hemispheres in the \nCM frame with respect to the highest momentum particle, and \nthe remaining background from $e^{+}e^{-}$ annihilation \nprocesses is suppressed by selecting events with low \nmultiplicity as characterized by the quantity\n$X_{\\rm part} \\equiv \n  (n_{\\rm tr} + n_{\\gamma})_{1} \\times \n  (n_{\\rm tr} + n_{\\gamma})_{2}$,\nwhere $n_{{\\rm tr},j}$ and  $n_{\\gamma,j}$ are the numbers \nof tracks and photons in hemisphere~$j$. We require\n$X_{\\rm part}\\leq 25$. \nFinally, in order to eliminate\nBhabha events in which \none or both electrons  produce a shower in \nmaterial near the interaction region,\nthe acoplanarity angle $\\xi$ between\nthe first and second highest momentum tracks is required \nto be $\\xi>1^{\\circ}$,\nwhere \n$\\xi\\equiv||\\phi_{1} -\\phi_{2}|- \\pi|$ \nis defined as the two-track acollinearity in azimuth.\n\nAfter applying all selection criteria,\n$22.71\\times 10^{6}$  $\\tau^+\\tau^-$-pairs survive.\nThe background is estimated using MC simulation.\nThe dominant source is from continuum processes \n$e^+e^-\\rightarrow q\\bar{q}\\,(q=u,d,s,c)$ and amounts to~5.5\\%.\nBackgrounds from Bhabha \nevents, $\\mu^+\\mu^-(\\gamma)$, and two-photon \n$e^+e^-\\rightarrow e^+e^- e^+e^-(\\mu^+\\mu^-)$ \nevents are estimated to be \n0.6\\%, 0.4\\%, and 0.8\\%, respectively.\nOther sources are found to be small. These background estimates are \nchecked by comparing the number of events in control samples.\nThe control samples for continuum, \nBhabha\\,+\\,$\\mu^+\\mu^-$, and two-photon processes are \nhigh multiplicity events having $25<X_{\\rm part}<30$ or\n$|MM|< 0.5~{\\rm GeV}\/c^{2}$ or $|MM|>8.0~{\\rm GeV}\/c^{2}$, respectively. The differences\nin event yields for these control samples and the\nMC predictions (5-10\\%) are included as \nsystematic errors for the results discussed in latter sections.\n\n\n\\subsection{$\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ candidate selection}\n\nWithin  the $\\tau^+\\tau^-$-pair sample, $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ decays are reconstructed \nby requiring that there be both one charged track\nand one ${\\pi ^{0}}$ in one hemisphere. \nThe $\\pi^{0}$ candidate is selected based on the normalized\ninvariant mass\n$S_{\\gamma\\gamma}\\equiv \n(m_{\\gamma\\gamma} - m^{}_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$,\nwhere \n$\\sigma_{\\gamma\\gamma}$ is the mass resolution of the $\\gamma\\gamma$ system.\nThe value of $\\sigma_{\\gamma\\gamma}$ ranges from 0.005 GeV to 0.008 GeV,\ndepending on the $\\pi^{0}$ momentum and polar angle.\nPairs of photons with $|S_{\\gamma\\gamma}|<9.0$ are considered as $\\pi^{0}$\ncandidates.\nTo keep beam-related background at a negligible level, \nwe require that the CM momentum of the ${\\pi^0}$ be greater than \n0.25~GeV\/$c$ and the photon CM energy be greater than 0.08~GeV.\n\n\nThe distribution in the normalized di-photon invariant mass $S_{\\gamma\\gamma}$\nfor the selected $\\pi^{-}\\pi^{0}$ sample, where there are one charged track and one $\\pi^{0}$ candidate \nin one\nhemisphere, \nis shown in Fig.~\\ref{mresol}.\nThe lower-side tail of the $S_{\\gamma\\gamma}$ distribution is \nprimarily due to\nrear and transverse leakage of electromagnetic showers out of the \nCsI(Tl) crystals and the conversion \nof the photons in the material located in front of  the crystals. \nGood agreement between data and Monte-Carlo indicates that\nthese effects are properly \nmodeled by the Monte-Carlo simulation. \nWe define the interval ${-6.0 < S_{\\gamma \\gamma} < 5.0}$ as the\n$\\pi^{0}$ signal region.\nSpurious $\\pi^{0}$ background \nis small and estimated from the sideband regions\n${7 < \\left|S_{\\gamma \\gamma}\\right| < 9}$.\nTo reduce feed-down background from multi-$\\pi^{0}$\ndecays such as \n$\\tau^-\\rightarrow\\pi^- (n\\pi^{0})\\nu_{\\tau}$\n( $n\\ge 2$),\nsignal candidates (in a hemisphere) are rejected \nif there are additional $\\gamma$'s \nin the same hemisphere with energy greater than 0.2~GeV.  \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-2.eps}}\n\\caption{\n Normalized $\\gamma\\gamma$ invariant mass ($S_{\\gamma \\gamma}$)\nspectrum in the data(points) and the $\\tau^+\\tau^-$ MC event(histogram),\nfor the sample described in the text.\n The plotted data correspond to 6.1\\% of the full data used in this analysis.\nThe arrows indicate the signal region\n$-6 < S_{\\gamma \\gamma} < 5$ and the sideband regions  \n$9 < \\left|S_{\\gamma \\gamma}\\right| < 7$. The sideband regions \nare used to subtract fake-$\\pi^{0}$ background.\n}\n\\label{mresol}\n\\end{center}\n\\end{figure}\n\n\nThe $\\pi^{-}\\pi^{0}$ invariant-mass-squared ($M^{2}_{\\pi\\pi^{0}}$) spectrum\nis obtained assuming the pion mass for the charged track; it is shown\nin Fig.~\\ref{pipi0_log} along with the MC prediction. To improve the\nenergy resolution of the $\\pi^{0}$, a $\\pi^{0}$ mass constraint is imposed.\nThe amount of a spurious $\\pi^{0}$ background \ndepends on the $M^{2}_{\\pi\\pi^{0}}$ region, varying from 4\\% to 15\\%.\n(This is subtracted using $S_{\\gamma\\gamma}$ sidebands.) \nThe final sample contains $5.55\\times 10^{6}$ $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$ candidates\nafter the ${\\pi^{0}}$ background subtraction, where $h^{-}$ denotes\n$\\pi$ or $K$. This sample is 50 times larger than those of previous \nstudies.\n\nFeed-down background arises mainly from multi-$\\pi^{0}$ modes \nsuch as $\\tau^{-}\\rightarrow \\pi^{-}(n\\pi^{0})\\nu_{\\tau}$ (5.5\\%)\nand $\\tau\\rightarrow K^{-}\\pi^{0}\\nu_{\\tau}$ (0.48\\%).\nIncluding other modes, the total feed-down background is\n$(6.0\\pm 0.1)$\\%.\nThe error listed includes statistical uncertainty \nas well as the uncertainty in relevant branching fractions.\nBackground from non-$\\tau$ processes is negligible,\nexcept that from continuum processes. \nThe amount of continuum background is \nestimated from MC simulation\nto be $(2.45\\pm 0.05)\\%$. The normalization \nof the continuum MC is checked using data\nin the high-mass region $M^{2}_{\\pi\\pi^{0}}>M^2_\\tau$.\n\nThe MC simulation of $\\tau$ decays is based on the TAUOLA\nprogram~\\cite{TAUOLA2004}. A small difference observed \nbetween data and MC \nin Fig.3 for $M^{2}_{\\pi\\pi^{0}} \\ge 2.0 $ $\\rm{(GeV\/c^{2})^{2}}$ \nis attributed to the $\\rho^{\\prime\\prime}(1700)$ resonance, which\nis not included in the current TAUOLA program.\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n {.\/Fig_hep-ex\/Fig-3.eps}}\n\\caption\n{\nInvariant-mass-squared ($M^{2}_{\\pi\\pi^{0}}$) distribution for $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$.\nThe solid circles with error bars \nrepresent the data, and\nthe histogram represents MC simulation (signal\\,+\\,background).\nThe open area shows the contribution from $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$; \nthe narrow cross-hatched \narea shows that from ${\\tau^{-}\\!\\rightarrow\\! K^{-} \\pi^{0} \\nu_{\\tau}}$; \nthe wide cross-hatched \narea shows that from ${\\tau^{-}\\!\\rightarrow\\! h^{-}(n\\pi^{0})\\nu_{\\tau}}$;\nand the striped area\nshows that from $q\\bar{q}$ continuum \nand other non-$\\tau$ processes.\n}\n\\label{pipi0_log}\n\\end{center}\n\\end{figure}\n\n\n\\section{Measurement of the Branching Fraction}\n\\subsection{Formula}\n\nThe branching fraction for $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$ \n($\\mathcal{B}_{h\\pi^0}$) is determined \nby dividing the signal yield $N_{h\\pi^{0}}$ by the number of\nselected generic \n$\\tau^+\\tau^-$-pairs $N_{\\tau\\tau}$:  \n\\begin{eqnarray}\n\\mathcal{B}_{h\\pi^{0}}   &=& \\frac{N_{h\\pi^{0}}}{2 N_{\\tau\\tau}}\\cdot\n\t \\frac{ (1 - b^{{\\rm feed}\\mbox{-}{\\rm down}}-\n\t   b^{{\\rm non}\\mbox{-}\\tau})}\n\t      { (1 - b_{\\tau\\tau})}  \\cdot\n  \\left(       \\frac{\\epsilon_{\\tau\\tau}}\n\t     {\\epsilon^{\\tau}_{h\\pi^{0}}}\n   \\right)\n   \\cdot \\frac{1}{\\epsilon^{ID}_{h\\pi^{0}} }\\,.\n\\label{br}\n\\end{eqnarray}\n\\noindent\nIn this formula, $b_{\\tau\\tau}$ is the background fraction \nin the $\\tau^+\\tau^-$ sample,\n$\\epsilon_{\\tau\\tau}$ is the efficiency of the $\\tau^+\\tau^-$-pair selection,\n$\\epsilon^{\\tau}_{h\\pi^{0}}$ is the efficiency for\n$\\tau^-\\!\\rightarrow\\!h^-\\pi^{0}\\nu$ decays to pass \nthe $\\tau^+\\tau^-$-pair selection,\nand $\\epsilon^{ID}_{h\\pi^{0}}$ is the efficiency for\n$\\tau^-\\!\\rightarrow\\!h^-\\pi^{0}\\nu$ decays\nsatisfying the $\\tau^+\\tau^-$-pair selection \nto pass the $h^-\\pi^{0}$ selection.\nThe product $\\epsilon^{\\tau}_{h\\pi^{0}}\\cdot \\epsilon^{ID}_{h\\pi^{0}}$ \nis the overall detection efficiency for the $h^-\\pi^{0}\\nu$ final state.\nThe parameter $b^{{\\rm feed}\\mbox{-}{\\rm down}}$ is the fraction of \n$h^{-}\\pi^0\\nu$ candidates coming from other $\\tau$ decay modes, and \n$b^{{\\rm non}\\mbox{-}\\tau}$ is the fraction coming from non-$\\tau$ \nprocesses. In this formula, several common uncertainties such as \nthat in the luminosity, that in the cross section for \n$\\tau^+\\tau^-$-pair production, that in the trigger efficiency, and that\nin the $\\tau^+\\tau^-$ selection efficiency cancel in the ratio.\nThe value for each factor is listed in Table~\\ref{tab:br} along\nwith the statistical error.\n\n\n\\begin{table}[!htb]\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{center}                      \n\\begin{tabular}{l|c}  \\hline\n\\hline\nParameter &\\hspace{1.5cm} Values \\hspace{1.5cm}   \\\\\n\\hline \n\\hline    \n${\\varepsilon_{\\tau\\tau}}$  &  \n ${30.81 \\pm 0.05~\\%}$  \\\\ \n${\\varepsilon^{\\tau}_{h\\pi^{0}}}$ &  ${34.26\\pm 0.07 ~\\%}$  \\\\ \n${\\displaystyle f_{b} = \n\t     \\frac { \\varepsilon^{\\tau}_{h\\pi^{0}} }\n\t\t\t { \\varepsilon_{\\tau\\tau} }   }$ &\n\t      ${1.112 \\pm 0.003 }$  \\\\ \n${\\varepsilon_{h\\pi^{0}}^{ID}}$ &\n\t      ${42.62 \\pm 0.13 ~\\%}$ \\\\ \n${b_{\\tau \\tau}}$ &\n\t      ${7.66 \\pm 0.03~\\%}$  \\\\ \n${ b^{{\\rm feed}\\mbox{-}{\\rm down}}_{h\\pi^{0}} }$ & \n\t      ${5.98 \\pm 0.08 ~\\% }$  \\\\\n${ b^{{\\rm non}\\mbox{-}\\tau}_{h\\pi^{0}} }$  &\n\t      ${2.45 \\pm 0.06 ~\\% }$  \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{ Values of parameters used for the \nbranching fraction measurement along with statistical errors.}\n\\label{tab:br}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Systematic uncertainty}\n\nThe sources of systematic uncertainty are listed in Table~\\ref{tab:br_sys}. \nThe uncertainty in the tracking efficiency is estimated using  \n$D^{*\\,+}\\rightarrow D^0\\pi^+\\rightarrow K^-\\pi^+\\pi^+$\ndecays to be \n1\\% per track.\nA large part of this uncertainty cancels in the ratio  of\nEq.(\\ref{br}); the resulting uncertainty  from this source is\n$\\Delta \\mathcal{B}_{h\\pi^{0}}=0.12$ \\%.\nThe $\\gamma\/\\pi^{0}$ detection efficiency is  \nobtained from the ratio of\n$D^0\\rightarrow K^-\\pi^+\\pi^{0}$\nto $D^0\\rightarrow K^-\\pi^+$ decays, in \nwhich the branching fractions are known\nprecisely. The uncertainty is estimated to be\n$\\pm 1.7\\%$ for a $\\pi^{0}$ momentum less than 1.0~GeV\/$c$.\nAs a consistency check, the branching fraction is re-measured \nafter changing the photon threshold from \n0.05~GeV to 0.10~GeV; the difference \nin $\\mathcal{B}^{}_{h\\pi^0}$ is only \n0.10\\%.\nThe uncertainty in background in \nthe non-$\\tau$ sample\n$\\delta b_{h\\pi^{0}}^{\\rm non-\\tau}$\nis estimated from the control sample\nabove the $\\tau$ mass region,\n while the uncertainty\nin  feed-down\nbackground $\\delta b_{h\\pi^{0}}^{\\rm feed-down}$\n is obtained \nfrom the uncertainty in \n$\\tau^{-}\\!\\rightarrow\\!h^{-}(n\\pi^{0})\\nu_{\\tau}$ and\n$\\tau^-\\!\\rightarrow\\!K^{-}\\pi^{0}\\nu_{\\tau}$ \nbranching fractions. \n\n\\begin{table}[htbp!]\n\\begin{center}\n\\begin{tabular}{l|c} \n\\hline \n\\hline\nSource of  uncertainty & $\\Delta \\mathcal{B}_{h\\pi^{0}}$ (\\%)  \\\\\n\\hline\n\\hline\nTracking efficiency & 0.12  \\\\\n$\\pi^0\/\\gamma$ efficiency & 0.25  \\\\\nBackground for $\\tau^+\\tau^-$  &   0.09 \\\\\nFeed-down background for $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$   & 0.04  \\\\\nNon-$\\tau$ background for $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$  & 0.05   \\\\\n$\\gamma$ veto  &  0.05    \\\\\nTrigger            & 0.08 \\\\\nMC statistics &   0.04  \\\\\n\\hline\n\\hline\nTotal &  0.31 \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ Systematic uncertainties for the $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$ branching fraction.\n}\n\\label{tab:br_sys}\n\\end{table}\n\nThe veto of additional $\\gamma$'s is required in the event\nselection to reduce background from multi-$\\pi^0$ decay channels. \nHowever, it also vetoes signal if there are photons radiated in \nthe initial or final state. In addition, photon candidates\n can \nalso appear due to electromagnetic shower fragments and\/or \nmis-reconstructed of electrons.\nThe uncertainty from these sources is estimated by \nchanging the veto threshold by \n$\\pm 0.1$~GeV; the resulting change in \n$\\mathcal{B}^{}_{h\\pi^0}$ \nis only\n$\\pm 0.05$\\%.\nSignal events are flagged by several trigger conditions\nthat require two or more CDC tracks with associated TOF hits, \nECL clusters, or a significant sum of energy in the ECL. \nThis redundancy \nallows one to monitor the efficiency of each trigger requirement.\nThe uncertainty arising from the trigger is estimated by\nassuming there is a $\\pm 3$\\% uncertainty in the track and \nenergy trigger efficiencies, which is the maximum variation \nmeasured during experimental running. The resulting uncertainty \non $\\mathcal{B}_{h\\pi^{0}}$ is small (0.08\\%) since the \n$\\tau^+\\tau^-$ trigger efficiency is high (97\\%).\n\n\n\\subsection{Results}\n\nInserting all values into Eq.~(\\ref{br}) gives\n\\begin{eqnarray}\n\\mathcal{B}_{h\\pi^{0}} & = & (25.60\\,\\pm \\,0.04\\,\\pm\\,0.31)\\%\\,,  \n\\end{eqnarray}\n\\normalsize\nwhere the first error is statistical and the second is systematic.\nSubtracting the small kaon-channel branching fraction listed in the \nPDG~\\cite{PDG2004} [$\\mathcal{B}_{ K^-\\pi^0}=(0.45\\pm 0.03)\\%$] \ngives a $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ branching fraction of\n\\begin{eqnarray}\n\\mathcal{B}_{\\pi\\pi^{0}} & = & (25.15\\,\\pm\\,0.04\\,\\pm\\,0.31)\\%\\,.\n\\end{eqnarray}\nThis result is in good agreement with previous measurements,\nas shown in Table~\\ref{tab:br_comp}. Our statistical error is \nsignificantly lower than those of the other measurements,\nwhile our systematic error is similar to those of the others \n(except for the ALEPH result).\n\n\\begin{table}[htbp!]\n\\begin{center}\n\\begin{tabular}{l|c |c} \n\\hline \n\\hline \nExperiment & $\\mathcal{B}_{h\\pi^{0}}(\\%)$ & Reference \\\\ \n\\hline\n\\hline\nOPAL & $25.89 \\pm 0.17 \\pm 0.29$  & \\cite{OPAL98M}   \\\\\nALEPH & $25.924 \\pm 0.097 \\pm 0.085$ & \\cite{ALEPH05}   \\\\\nL3 & $25.05 \\pm 0.35 \\pm 0.50$   &\\cite{L395}       \\\\\nCLEO & $25.87 \\pm 0.12 \\pm 0.42$ & \\cite{CLEO94} \\\\\n\\hline\nThis work & $25.60 \\pm 0.04 \\pm 0.31$       \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{ \nBranching fractions for $\\tau^{-}\\!\\ra h^{-}\\pi^{0}\\,\\nu_{\\tau}$ measured \nby various experiments. \n}\n\\label{tab:br_comp}\n\\end{table}\n\n\n\n\\section{Measurement of the Mass Spectrum}\n\nIn order to obtain the true $\\pi^{-}\\pi^{0}$ mass spectrum,\none must correct for (1) background, (2) smearing due to finite \nresolution and radiative effects, and (3) mass-dependent acceptance.\n\n\n\\subsection{Background Correction}\n\nAs noted earlier, background entering the $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ sample \nis small. The sidebands of the $M_{\\gamma\\gamma}$ distribution \nare used to estimate the fake $\\pi^{0}$ contribution.\nThis background dominates at values of $M^{2}_{\\pi\\pi^{0}}$ less \nthan about $0.25~({\\rm GeV}\/c^{2})^{2}$.\n\nAs seen in Fig.~\\ref{pipi0_log}, feed-down background arises \nfrom $\\tau^{-}\\rightarrow h^{-}(n\\pi^{0})\\nu_{\\tau}$ \nand $\\tau^{-}\\rightarrow K^{-}\\pi^{0}\\nu_{\\tau}$ decays;\nboth backgrounds dominate at low values of $M^{2}_{\\pi\\pi^{0}}$.\nIn the high mass region, continuum background dominates.\nFor this analysis we did not use information \nfrom particle identification~(PID) detectors \nto separate charged pions from kaons, as the\nfeed-down background is dominated by \n$\\tau^{-}\\rightarrow h^{-}(n\\pi^{0})\\nu_{\\tau}$ \nrather than $\\tau^{-}\\rightarrow K^{-}\\pi^{0}\\nu_{\\tau}$.\nThe $M^{2}_{\\pi\\pi^{0}}$ distribution after subtracting this\nbackground is shown in Fig.~\\ref{pipi0_aftsub}. \n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-4.eps}}\n\\caption\n{\nInvariant-mass-squared ($M^{2}_{\\pi\\pi^{0}}$) distribution for $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$\nafter background subtraction.\n}\n\\label{pipi0_aftsub}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Acceptance  Corrections}\n\nThe detector effects include $M^{2}_{\\pi\\pi^{0}}$-dependent \nacceptance and bin-by-bin migration caused by the finite mass resolution. \nWe correct for these effects by performing an unfolding procedure. The \nunfolding program used is that employed by the ALEPH experiment~\\cite{SVD}.\nIn this program, the unfolding is based on the\nSingular-Value-Decomposition (SVD) method~\\cite{SVD}, \nin which the acceptance matrix is inverted \nby limiting the number of singular values to\nonly those that are statistically significant.\nThe output of the program is the unfolded distribution and \nits covariance matrix.\n\nThe correlation between the generated quantity \n$M^{2}_{\\rm gen}$ and the measured one $M^{2}_{\\rm obs}$ \nis shown in Fig.~\\ref{cor_matrix}. The figure \nshows a clear correlation between \n$M^{2}_{\\rm gen}$ and $M^{2}_{\\rm obs}$.\nThe resolution in $M^{2}_{\\pi\\pi^{0}}$ is 0.005~$({\\rm GeV}\/c^{2})^{2}$\nin the low-mass region and \n0.030~$({\\rm GeV}\/c^{2})^{2}$ in the \nhigh-mass region; thus the bin size chosen is\n$\\Delta M^{2} = 0.050~({\\rm GeV}\/c^{2})^{2}$ so that the off-diagonal \ncomponents of the acceptance matrix are small.\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n {.\/Fig_hep-ex\/Fig-5.eps}}\n\\caption\n{\nCorrelation between $M^{2}_{\\rm gen}$ (vertical axis) and \n$M^{2}_{\\rm obs}$ (horizontal axis), the generated and\nobserved invariant masses squared of the $\\pi^-\\pi^{0}$ \nsystem in $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ decay.\n}\n\\label{cor_matrix}\n\\end{center}\n\\end{figure}\n\nThe acceptance as a function of $M^{2}_{\\rm gen}$ is shown in \nFig.~\\ref{acceptance}. The acceptance varies smoothly \nand its average value is~17\\%. It decreases at low values\nof $M^{2}_{\\rm gen}$ due to the overlap of $\\gamma$ clusters\nwith the $\\pi^{-}$ track projection at the calorimeter.\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n  {.\/Fig_hep-ex\/Fig-6.eps}}\n\\caption\n{Acceptance as a function of $M^{2}_{\\rm gen}$, as determined\nfrom $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ MC simulation.  }\n\\label{acceptance}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Results}\n\nThe unfolded $s= M^2_{(\\pi\\pi^0\\,{\\rm unf.})}$ spectrum\n$dN\/ds$ is shown in Fig.~\\ref{unfold_pipi0}. \nThe square roots of the diagonal components of the covariance\nmatrix are used as the errors.\nThe $\\rho$ peak and the shoulder due to the $\\rho'(1450)$\nare clearly visible. The dip at $s\\approx 2.5~({\\rm GeV}\/c^{2})^{2}$ is\ncaused by destructive interference between the $\\rho'(1450)$ \nand $\\rho''(1700)$ resonances.\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-7.eps}}\n\\end{center}\n\\caption\n{ Fully-corrected $M^2_{\\pi\\pi^0}$ distribution\nfor $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$. \nThe solid curve is the \nresult of a fit to the Gounaris-Sakurai model with \n$\\rho(770)$, $\\rho'(1450)$, and $\\rho''(1700)$ resonances.\nAll parameters are floated.\n}\n\\label{unfold_pipi0}\n\\end{figure}\n\n\n\nTo obtain parameters for the $\\rho$, $\\rho'$ and $\\rho^{''}$ \nresonances, a $\\chi^{2}$ fit using Breit-Wigner functions \nis performed. Since the unfolded mass spectrum has bin-by-bin \ncorrelations, the off-diagonal components of the covariance matrix\n$X$ are included in the $\\chi^{2}$ evaluation:\n\\begin{equation}\n\\chi^{2}= \\sum_{i,j} \\left( y _{i} - f(s_{i}; \\alpha) \\right)\n\t(X^{-1})_{ij}\n\t\\left( y _{j} - f(s_{j}; \\alpha) \\right)\\,,\n\\label{eq:chi2def} \n\\end{equation}\nwhere $y_{i}$ is the measured value at the $i$-th bin,\n$f(s; \\alpha)$ is the value of the function for parameters $\\alpha$,\nand $(X^{-1})_{ij}$ is the inverse of the covariance matrix. \n\nIn the fit, the $s$ dependence of the decay rate is given by \nEq.~(\\ref{eq:tauspec}). The pion form factor in Eq.~(\\ref{eq:pionform})   \nis parametrized with Breit-Wigner functions corresponding to the \n$\\rho$, $\\rho^{\\prime}(1450)$, and $\\rho^{\\prime\\prime}$(1700) \nresonances:\n\\begin{equation}\nF_{\\pi}(s) = \\frac{1}{1 + \\beta +\\gamma } \n\t  (BW_{\\rho} + \\beta \\cdot  BW_{\\rho^{\\prime}} \n +\\gamma \\cdot BW_{\\rho^{\\prime\\prime}})\\,,\n\\end{equation}\nwhere the parameters $\\beta$ and $\\gamma$\n(denoting the relative size of the two resonances) are \nin general complex.\nWe use the Gounaris-Sakurai\\,(GS) model~\\cite{GS} for the\nBreit-Wigner shape:\n\\begin{equation}\nBW_{i}^{GS} = \\frac {M_{i}^2  + d \\cdot M_{i}\\Gamma_{i}(s) }\n\t   {(M_{i}^{2} - s) + f(s) - i \\sqrt{s} \\Gamma_{i}(s)}\\,,\n\\end{equation}\n\\noindent\nwith an energy-dependent width\n\\begin{equation}\n\\Gamma_{i}(s) = \\Gamma_{i}  \\left( \\frac{M_{i}^{2}}{s}\\right)\n\\left( \\frac{k(s)}{k(M_{i}^2)}\\right)^{3}\\,.\n\\end{equation}\nHere, $ k(s)  =  \\frac{1}{2} \\sqrt{s} \\beta_{-}(s)$ is the pion\nmomentum in the $\\pi^{-}\\pi^{0}$ rest frame.\nThe functions $f(s)$ and $h(s)$ are defined as \n\n\n\\begin{eqnarray}\nf(s) & = & \\Gamma^{}_i\\,\\frac{M^2_i}{k^3(M^2_i)} \n  \\left[\\, \n     k^2(s) \\left( h(s) -h(M_{i}^2) \\right)\n     + (M_{i}^2 - s) k^2(M_{i}^2) \n\t\\left.\\frac{dh}{ds}\\right|_{s=M^2_{i}} \n  \\,\\right]  \n\\label{eq:dif} \\\\\n& & \\nonumber \\\\  \nh(s) & = & \\frac{2}{\\pi} \\frac{k(s)}{\\sqrt{s}} \nln \\frac{\\sqrt{s} + 2k(s)}{2m_{\\pi}}\\,,\n\\end{eqnarray}\n\\noindent\nwith  \n$ \\left. dh\/ds\\right|_{M_{i}^{2}} = \nh(M_{i}^{2})  \\left [  \n\t\t \\left( 8k^2(M_{i}^{2}) \\right)^{-1}\n\t\t  - (2 M_{i}^{2})^{-1}    \\right ]\n+ (2 \\pi M_{i}^{2})^{-1}\n$\nand  \n\n\\begin{equation}\n    d = \\frac{3}{\\pi} \\frac{m_{\\pi}^2} {k^2(M_{i}^2)} \n   ln \\frac{M_{i} + 2 k(M_{i}^2)} {2 m_{\\pi}}\n\t\t +\n  \\frac{M^{}_i} {2 \\pi k(M^2_{i})} -\n  \\frac{m_{\\pi}^2 M^{}_i} {\\pi k^3(M^2_{i})}\\,.          \n\\end{equation}\n\\noindent\nThere are ten parameters in this formula: \nthe masses~($M_{i}$) and the widths~($\\Gamma_{i}$) for the \n$\\rho,\\,\\rho^{\\prime}$,\\, and $\\rho^{\\prime\\prime}$ resonances,\ntheir relative amplitudes $|\\beta|$,\\,$|\\gamma|$,\\, and their phases\n$\\phi_{\\beta}$ and $\\phi_{\\gamma}$.\n\n\n\\begin{table}\n\\begin{tabular}{l|c|c}  \n\\hline \n\\hline \nParameter &  Fit result & Fit result\\\\\n  &  (all free)  & (fixed $\\phi_{\\gamma}$)   \\\\\n\\hline \n\\hline \n$M_{\\rho}$~~$({\\rm MeV\/c^{2}})$    &  $774.6\\pm 0.2\\pm 0.3$  \n&  $774.3\\pm 0.2\\pm 0.3$   \\\\\n$\\Gamma_{\\rho}$~~$({\\rm MeV})$ &  $150.6\\pm 0.3\\pm 0.5$   \n& $150.0\\pm 0.3\\pm 0.5$     \\\\\n$M_{\\rho^{\\prime}}$ ~~$({\\rm MeV\/c^{2}})$  & $1336\\pm 12\\pm 23$  \n&   $1436\\pm 15\\pm 23$    \\\\\n$\\Gamma_{\\rho^{\\prime}}$ ~~$({\\rm MeV})$  & $471\\pm 29\\pm 21$   \n   & $553\\pm 31\\pm 21$     \\\\\n$|\\beta|$ & $0.090 \\pm 0.009 \\pm   0.013$\n   &  $0.161 \\pm 0.020 \\pm   0.013$ \\\\\n$\\phi_{\\beta}$ ~~(degree)  & $123.7 \\pm 5.0\\pm  7.0$  \n  & $149.1 \\pm 2.4\\pm  7.0$  \\\\\n$M_{\\rho^{\\prime\\prime}}$ ~~$({\\rm MeV\/c^{2}})$  & $1600 \\pm 13\\pm 4$  \n  & $1804 \\pm 16\\pm 4$  \\\\\n$\\Gamma_{\\rho^{\\prime\\prime}}$ ~~$({\\rm MeV})$  & $255\\pm 19\\pm 79$  \n&  $567\\pm 81\\pm 79$  \\\\\n$|\\gamma|$ & $0.062 \\pm 0.015 \\pm 0.015$  \n& $0.136 \\pm 0.024 \\pm 0.015$   \\\\\n$\\phi_{\\gamma}$ ~~(degree) & $-64.1\\pm 7.9$ \n& $[0] $  \\\\\n\\hline\n$\\chi^{2}\/{\\rm d.o.f}$ & 55\/51 &  94\/52   \n\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\nResults of fitting the $M^2_{\\pi\\pi^0}$ distribution \nfor $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ to the Gounaris-Sakurai model with \nthe $\\rho(770)$, $\\rho'(1450)$, and $\\rho''(1700)$ resonances.\nThe results for two cases, all parameters floated (the second column) \nand fixed $\\phi_{\\gamma}$ (the third column) are shown.\nFor both cases,\nthe first error is statistical and the second one is systematic. \nThe systematic errors include the uncertainty of the backgrounds, \nunfolding, as well as the uncertainty of the photon energy scale.\n}\n\\label{tab:fit_param}\n\\end{table}\n\n\nAll parameters are floated in the fit. When evaluating the $\\chi^{2}$, \nthe 1\\% systematic uncertainty resulting from the unfolding procedure \nis included in the diagonal part of the covariance matrix.\nThis uncertainty is estimated by applying the same unfolding procedure \nto MC events and comparing the unfolded spectrum with the original.  \nThe result of the fit is shown in Fig.~\\ref{unfold_pipi0}\nas the solid line; the values obtained for the parameters \nare listed in Table~\\ref{tab:fit_param}.\nThe results are compared with the previous ALEPH measurements in\nTable~\\ref{tab:fit_result}.\n\\begin{table*}\n\\begin{tabular}{l|c|c|c}  \n\\hline \\hline\nParameter & \nThis work  & This work & ALEPH($\\tau$)    \\\\\n &\n(fixed $\\phi_{\\gamma}$) &\n(fixed $M_{\\rho^{\\prime\\prime}},\n \\Gamma_{\\rho^{\\prime\\prime}},\\phi_{\\gamma}$)    \\\\\n\\hline\n\\hline \n${M_{\\rho^{-}} ~~({\\rm MeV}\/c^{2})}$   &\n${774.3 ~ \\pm 0.2}$  & $773.9~\\pm 0.1$ \n& ${775.5 ~ \\pm 0.7}$    \\\\\n${\\Gamma_{\\rho^{-}} ~~({\\rm MeV})}$   & \n${150.0 ~ \\pm 0.3}$  &   $ 150.8~\\pm 0.3 $\n&${149.0 ~\\pm 1.2}$  \\\\\n${M_{\\rho^{\\prime}} ~~({\\rm MeV}\/c^{2})}$   & \n${1436 ~ \\pm 15}$   &  $ 1395 ~\\pm 4 $\n&${ 1328 ~ \\pm 15 }$   \\\\\n${\\Gamma_{\\rho^{\\prime}} ~~({\\rm MeV})}$   & \n${553 ~ \\pm 31}$   &  $411 ~\\pm 9$\n&${468 ~\\pm 41}$   \\\\\n${|\\beta|}$  &\n${0.161 ~ \\pm 0.020}$ &   $ 0.095~\\pm 0.02$\n&${0.120 ~\\pm 0.008}$     \\\\\n${\\phi_{\\beta}}$ ~~(degree)  &\n${149.1 ~ \\pm 2.4}$   &$161~\\pm 2.0$\n&${153 ~\\pm 7}$             \\\\\n${M_{\\rho^{\\prime\\prime}} ~~({\\rm MeV}\/c^{2})}$   & \n$1804 \\pm 16$   & [1713]\n&  [1713]    \\\\\n${\\Gamma_{\\rho^{\\prime\\prime}} ~~({\\rm MeV})}$   & \n${567 ~ \\pm 81}$  & [235]  \n& [235]       \\\\ \n${|\\gamma|}$  &\n${0.136 ~ \\pm 0.024}$  & $ 0.045 \\pm 0.002$  \n&${0.023 ~\\pm 0.008}$     \\\\\n${\\phi_{\\gamma}}$ ~~(degree)  &\n [0]   &   [0]\n& [0]            \\\\\n${\\chi ^{2} \/{\\rm (d.o.f)}}$   &\n94 \/ 52      &  134\/54\n   &    119 \/ 110       \\\\\nReference         &  &\n& ~\\cite{ALEPH05}   \\\\\n\\hline \n\\hline\n\\end{tabular}\n\\caption{ Comparison of our fit results for \nthe $\\rho(770)$, $\\rho'(1450)$, and $\\rho''(1700)$ parameters with those\nobtained by the ALEPH  experiment\\protect{\\cite{ALEPH05}}. \nThe numbers in brackets indicate\nthe values fixed in the fit.} \n\\label{tab:fit_result}\n\\end{table*}\nIn the table, the first error is statistical and the second one is\nsystematic.\n\nThe main sources of systematic uncertainty are \nthe photon energy scale, the unfolding procedure, \nand the background subtraction.\nThe uncertainty in the $\\rho$ mass (0.3~MeV) is \nmainly due to the uncertainty in the photon energy scale.  \nThe uncertainty in background dominates for the $\\rho''$ parameters. \nOur result for the mass of the\n$\\rho$ resonance agrees well \nwith the \nALEPH~\\cite{ALEPH05} and CLEO\\cite{CLEO2000} results.\n\n\n\nAs can be seen from the second and third columns \nof Table~\\ref{tab:fit_result},\n where \nthe interference angle $\\phi_{\\gamma}$ or three parameters\n $M_{\\rho^{\\prime\\prime}},\\Gamma_{\\rho^{\\prime\\prime}},\\phi_{\\gamma}$\nare fixed, respectively,\n  as in the previous  ALEPH~\\cite{ALEPH05} fit,\nthe values for $\\rho^{\\prime}$ and $\\rho^{\\prime\\prime}$  resonance parameters\nare quite sensitive to the values of other parameters fixed in the fit.\n\n\nThe results are shown in terms of the pion form factor squared\n($|F_{\\pi}(s)|^{2}$) in Figs.~\\ref{Fpi} and~\\ref{FpiALEPH}.\nA dip caused by destructive interference between the $\\rho'(1450)$ and \n$\\rho''(1700)$ is clearly visible.\nFor the first time production of the $\\rho''(1700)$ \nin $\\tau^-$ decays has been unambiguously demonstrated \nand its parameters determined.\nFor comparison, the figures also show results from the \nCLEO~\\cite{CLEO2000} and ALEPH~\\cite{ALEPH05} experiments, \nrespectively; there is good agreement with both data sets.\nFigure \\ref{Fpi_comp} shows our data and that of\nCLEO for the mass range 0.2--2.2~${\\rm GeV}\/c^{2}$, where the\ncontribution to $a^{\\rm had,LO}_\\mu$ \nis largest.\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics*[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-8.eps}}\n\\caption\n{ Pion form factor for $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$.\nThe solid circles show the Belle result and the\nopen squares show the CLEO result~\\protect{\\cite{CLEO2000}}.\nThe error bars for the Belle data include both\nstatistical and systematic errors added in quadrature.\nThe solid curve is the \nresult of a fit to the Gounaris-Sakurai model with \nthe\n$\\rho(770)$, $\\rho'(1450)$, and $\\rho''(1700)$ resonances,\nwhere all parameters are floated.\n}\n\\label{Fpi}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics*[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-9.eps}}\n\\caption\n{ Pion form factor for $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$.\nThe solid circles show the Belle result and the\nopen squares show the ALEPH result~\\protect{\\cite{ALEPH05}}. \nThe error bars for the Belle data include both\nstatistical and systematic errors added in quadrature.\nThe solid curve is the \nresult of a fit to the Gounaris-Sakurai model, where\nall parameters are floated.\n}\n\\label{FpiALEPH}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\rotatebox{0}{\\includegraphics*[width=0.6\\textwidth,clip]\n{.\/Fig_hep-ex\/Fig-10.eps}}\n\\caption\n{ \nComparison of the pion form factor squared $|F_{\\pi}(s)|^{2}$ \nmeasured  \nby Belle to that measured by CLEO~\\protect{\\cite{CLEO2000}}\n and ALEPH~\\protect{\\cite{ALEPH05}} \nexperiments\n in the $\\rho(770)$ \nand $\\rho^{\\prime}(1430)$ mass region.  Difference from the fit of  \n the Belle $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ data divided by the fit value is plotted.\n}\n\\label{Fpi_comp}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Evaluation of $a_{\\mu}^{\\pi\\pi}$}\n\nUsing the unfolded $s= M^2_{(\\pi\\pi^0\\,{\\rm unf.})}$ distribution\n$(1\/N)(dN\/ds)$, the spectral function $v^{-}$ is obtained\nby taking the inverse of Eq.~(\\ref{eq:tauspec}):\n\\begin{eqnarray}\nv_{-} & = & \n\\frac {m_{\\tau}^{2}} { 6\\pi |V_{ud}|^{2} S_{EW} } \n\\left(\n\\frac{ \\mathcal{B}_{\\pi\\pi} } {\\mathcal{B}_{e}} \\right)\n\\left[     \\left( 1 - \\frac{s}{m_{\\tau}^{2}} \\right)^{2}\n     \\left( 1 + \\frac{2s}{m_{\\tau}^{2}} \\right)\n\\right]^{-1}\n\\frac{1}{N}\\frac{dN}{ds}\\,.\n\\label{eq:specmeas} \n\\end{eqnarray}\nThe resulting function can be inserted into Eq.~(\\ref{eq:amu2pi})\nto obtain \nthe dominant low-mass contribution to\nthe hadronic part of the anomalous magnetic moment, \n$a_{\\mu}^{\\pi\\pi}$. This assumes the CVC relation (\\ref{eq:cvc})\nholds.\n\nThere are several external parameters in these equations; \nthe values used for these are listed in Table~\\ref{tab:amuext}.\nFor $m_{\\tau}$, $V_{ud}$, and $\\mathcal{B}_{e}$, \nPDG~\\cite{PDG2004} values are used.\nFor the electroweak  radiative correction $S_{\\rm EW}$,\nwe use the recent value $1.0233\\pm 0.0006$, which is \nbased on a consistent treatment of \nthe isospin-breaking correction~\\cite{ISB2001,DEHZ}.\nFor the $\\pi^{-}\\pi^{0}$ branching fraction, \nour measurement is consistent with the world average given\nin Ref.~\\cite{DATAU02}. \n\nIncluding our result and the recent ALEPH $\\mathcal{B}_{\\pi\\pi^{0}}$\nmeasurement, the new world average is \n\\begin{equation}\n\\mathcal{B}_{\\pi\\pi^{0}}= (25.42\\pm 0.11)\\%.\n\\end{equation}\nWe use this new world average for the evaluation of\n$a_{\\mu}^{\\pi\\pi}$.\n\n\n\\begin{table*}[!ht]\n\\begin{center}\n\\begin{tabular}{l|c|c|c|c}\n\\hline \\hline\nSource & Value & Relative error   &  $\\Delta a_{\\mu}^{\\pi\\pi}$ & Reference \\\\\n&       &       (\\%)    &   $(10^{-10})$             &       \\\\\n\\hline\n\\hline\n$S_{EW}$    &  $1.0233 \\pm 0.0006$     &  0.06   &  $\\pm$ 0.32   & \n ~\\cite{DEHZ},\\cite{ISB2001} \\\\\n$V_{ud}$    &  $0.9734 \\pm 0.0008$      & 0.08  &   $\\pm$ 0.42   & \n\\cite{PDG2004}  \\\\\n$\\mathcal{B}_{e}$    & ($17.84\\pm\\,0.06$)\\%  &  0.34 &  $\\pm$ 1.82   & \n\\cite{PDG2004}  \\\\\n$\\mathcal{B}_{\\pi\\pi^{0}}$ & ($25.42\\pm\\,0.11$)\\% & 0.43 &$\\pm$ 2.30   &\n\\\\\n\\hline\nTotal external\\ \\ &                       &           & $\\pm$ 3.0 &    \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption {  Values of the external parameters and the \nsystematic errors for $a_\\mu^{\\pi\\pi}$ arising from \nthese sources.}\n\\label{tab:amuext}\n\\end{center}\n\\end{table*}\n\nThe integration in Eq.~(\\ref{eq:amu2pi}) is carried out numerically \nby taking the sum of the integrand evaluated at the center of each bin.\nThe statistical error in $a_{\\mu}^{\\pi\\pi}$ is calculated\nincluding the off-diagonal elements of the covariance \nmatrix $X_{ij}$:\n\n\\begin{eqnarray}\n\\delta a^{\\pi\\pi}_\\mu & = & \\sum_{i,j} \n\\left( \\frac{\\partial a_{\\mu}}{\\partial\\alpha_{i}} \\right)\nX_{ij}                             \n\\left( \\frac{\\partial a_{\\mu}}{\\partial\\alpha_{j}} \\right).\n\\end{eqnarray}\n\nBecause of uncertainties associated with the background \nestimate and with the acceptance correction in the \nlowest\nmass region, the integration is carried out over the \nmass range $\\sqrt{s}=$ 0.50--1.80~${\\rm GeV}\/c^{2}$.\n\n\n\\subsection{Systematic uncertainty}\n\nSystematic uncertainty in $a_{\\mu}^{\\pi\\pi}$ arises\nfrom both external and internal sources. The errors\narising from external parameters are\nsummarized in Table \\ref{tab:amuext}; the total systematic\nerror from these sources is $\\pm 3.0\\times 10^{-10}$\n(dominated by $\\delta \\mathcal{B}_{\\pi\\pi^{0}}$).\n\n\nThe systematic error arising from internal sources (specific \nto this measurement) are listed in Table~\\ref{tab:amuerror2}.\nThere are two sources of background in the $\\pi^{-}\\pi^{0}$ sample:\n(i) feed-down from  $\\tau^{-}\\rightarrow h^{-} (n\\pi^{0})\\nu_{\\tau}$ and\n$\\tau^{-}\\rightarrow K^{-}\\pi^{0}\\nu_{\\tau}$, and\n(ii) non-$\\tau$ background.\nIn the first case, the uncertainty in the branching fraction\nis used to estimate the error.\nIn the second case,\nthe uncertainty in the background as estimated from the control\nsamples is assigned as the error.\nAs mentioned earlier, the fake-$\\pi^{0}$ background \nis subtracted using sideband events; the uncertainty \nis determined by varying the signal and sideband regions.\n\n\\begin{table}[!ht]\n\\begin{tabular}{l|c}\n\\hline \\hline\nSource    &     $\\Delta a_{\\mu}^{\\pi\\pi} \\times 10^{10}$   \\\\\n      &    (0.50--1.80~${\\rm GeV}\/c^{2}$) \\\\\n\\hline\\hline\nBackground:                      &        \\\\\n\\ \\ \\  non-$\\tau$\\,($e^+e^-\\!\\rightarrow\\!\\bar{q}q$)   &    $\\pm 0.11$    \\\\\n\\ \\ \\ feed-down $h(n\\pi^{0})\\nu$     &  $\\pm 0.09$   \\\\\n\\ \\ \\ feed-down $K^{-}\\pi^{0}\\nu$       & $\\pm 0.15$  \\\\\nEnergy scale                     &  $\\pm 0.10 $     \\\\\n$\\pi^{0}\/\\gamma$ selection       &   $\\pm 0.24$    \\\\\n$\\gamma$ veto                         & $\\pm 0.93 $   \\\\\nEfficiency:                      &       \\\\\n\\ \\ \\  $\\pi^{0}\/\\gamma$                 &  $\\pm 0.35 $     \\\\\n\\ \\ \\ \\ charged track                   &    $ <0.10$      \\\\ \nIntegration procedure\\ \\            &    $<0.10$     \\\\\n\\hline\nTotal internal\\ \\                                 & \n$\\pm$ 1.04    \\\\\n\\hline\\hline\n\\end{tabular}\n\\begin{center}\n\\caption { Systematic errors for $a_\\mu^{\\pi\\pi}$ arising from\ninternal sources (specific to this measurement).}\n\\label{tab:amuerror2}\n\\end{center}\n\\end{table}\n\n\n\nThe ratio of the branching fractions for the decays \n$D^{0}\\rightarrow K^{-}\\pi^{-}\\pi^{0}$ and \n$D^{0}\\rightarrow K^{-}\\pi^{+}$ is used to monitor \nthe $\\pi^{0}$ efficiency. \nIt is found that the shape of the mass spectrum is insensitive \nto uncertainty in the $\\pi^{0}$ efficiency, as it is only\nat the few \\% level. \nAdding all individual errors in quadrature gives a total\nerror on $a^{\\pi\\pi}_{\\mu}$ arising from \ninternal sources of \n$\\pm 1.0\\times 10^{-10}$. \n\n\nTo check the stability of $a_{\\mu}^{\\pi\\pi}$, we perform \nthe following tests:\n\\begin{enumerate}\n\\item \nThe sample is divided into subsamples\nbased on the tag-side topology, i.e., \none electron, one-prong, or three-prong.\nThe values of $a^{\\pi\\pi}_{\\mu}$ obtained from these \nsubsamples are consistent within the statistical errors.\n\\item \nThe sample is divided into subsamples based on the\nrunning period, e.g., years 2000, 2001, or 2002.\nAgain, the values of $a^{\\pi\\pi}_\\mu$ obtained are \nconsistent within the statistical errors.\n\\item The sample might be  sensitive to the \nrequirement on the overlap region between\nthe projection of the charged track and $\\gamma$ clusters. To \nestimate this sensitivity, we select events with a tighter isolation \nrequirement on $\\gamma$'s and on the track extrapolation: \n50~cm instead of 20~cm.\n\nThe resulting variation in $a^{\\pi\\pi}_{\\mu}$ is small \nand is included as an additional systematic error.\n\\end{enumerate}  \n\n\n\\section{Results} \n\nThe result for $a_{\\mu}^{\\pi\\pi}$ integrated over the mass range\n$\\sqrt{s}=$0.50--1.80~${\\rm GeV}\/c^{2}$ is\n\\begin{eqnarray}\na_{\\mu}^{\\pi\\pi}[0.50, 1.80] = ( 464.4 \\pm 0.6\\,{\\rm (stat.)}\n\\pm 1.0\\,{\\rm (sys.)} \\pm 3.0\\,{\\rm (sys.\\,ext.)} \\times 10^{-10},\n\\nonumber \n\\end{eqnarray}\nwhere the first error is statistical and the second and third errors\nare systematic errors arising from internal and external sources, \nrespectively.\nIn addition, there is a systematic uncertainty \ncaused by isospin violation effects arising \nfrom $\\rho$-$\\omega$ interference, \nfrom the $\\pi^-$ and $\\pi^{0}$ mass difference, and from\nradiative corrections (see~Ref.~\\cite{ISB2001}). \nThe overall correction is estimated to be \n$(-1.8\\pm 2.3) \\times 10^{-10}$,\nwhere the central value is taken from Ref.~\\cite{CLEO2000}\nand we enlarged the error according to the value in Table 5 \nof Ref.~\\cite{DEHZ};\nthis correction is small because the threshold region is not included.\nApplying this correction gives\n\\begin{eqnarray}\na_{\\mu}^{\\pi\\pi}[0.50,1.80] = ( 462.6 \n\\pm 0.6\\,{\\rm (stat.)} \\pm 3.2\\,{\\rm (sys.)}\n  \\pm 2.3\\,{\\rm (isospin)} ) \\times 10^{-10}\\,,\n\\nonumber \n\\end{eqnarray}\nwhere the first error is statistical, the second is \nsystematic, and the third arises from isospin violation.\n\nThis result can be compared to those from previous $\\tau$~\\cite{DEHZ} \nand $e^+e^-$ experiments~\\cite{DAV2003}:\n\\begin{eqnarray*}\n a_{\\mu}^{\\pi\\pi}[0.50,1.80]& = &\n ( 464.0 \\pm 3.0\\,{\\rm (exp.)} \\pm 2.3\\,{\\rm (isospin)})\n \\times 10^{-10} \\quad (\\tau:{\\rm  ALEPH,CLEO}) \\\\\n a_{\\mu}^{\\pi\\pi}[0.50,1.80]& = &\n ( 448.3 \\pm 4.1\\,{\\rm (exp.)} \\pm 1.6\\,{\\rm (rad.)})\n \\times 10^{-10} \\quad\\quad (e^+e^-:{\\rm CMD2,KLOE})\\,.\n\\end{eqnarray*}\nThe first error includes both statistical and experimental systematic errors added in  quadrature.\nThe second error in the $e^+e^-$ result is due\nto radiative corrections.\nOur result agrees well with the $\\tau$-based result but is \nnoticeably higher than the $e^+e^-$ result. This supports the\nhypothesis that there is a difference between the mass spectra \nof the $2\\pi$ systems produced in $\\tau$-decay and $e^+e^-\\rightarrow\\pi^+\\pi^-$ reactions.\n\n\n\\vspace*{1cm}\n\nIn summary, we have studied the decay $\\tau^{-}\\!\\ra\\pi^{-}\\pi^{0}\\,\\nu_{\\tau}$ using high \nstatistics data\ntaken with the Belle detector at the KEKB $e^+e^-$ collider. \nThe branching fraction is measured with 1.2\\% accuracy,\nwhich is better than that in the previous experiments\n(except for the ALEPH result).\nIn the unfolded $\\pi^-\\pi^0$ mass spectrum, in addition to the \n $\\rho(770)$ and $\\rho^{\\prime}(1450)$ mesons, \nthe production of the $\\rho^{\\prime\\prime}(1700)$ in $\\tau^{-}$ \ndecays has been unambiguously demonstrated and its parameters determined. \nThe unfolded spectrum is used to evaluate  \nthe 2$\\pi$ contribution to the muon anomalous magnetic\nmoment $a_{\\mu}^{\\pi\\pi}$  in the region \n$\\sqrt{s}=0.50-1.80~\\rm{GeV}\/c^{2}$.\nOur results agree well \nwith the previous $\\tau$ based results but \nare higher than the $e^+e^-$ results.\n\n\n\\section*{Acknowledgments}\n\nWe thank M. Davier and J. H. K\\\"{u}hn for\ntheir advice and encouragement during this analysis. \nWe thank the KEKB accelerator group for the excellent\noperation of the KEKB accelerator.\nWe acknowledge support from the Ministry of Education,\nCulture, Sports, Science, and Technology of Japan\nand the Japan Society for the Promotion of Science;\nthe Australian Research Council\nand the Australian Department of Industry, Science and Resources;\nthe National Science Foundation of China under contract No.~10175071;\nthe Department of Science and Technology of India;\nthe BK21 program of the Ministry of Education of Korea\nand the CHEP SRC program of the Korea Science and Engineering\nFoundation;\nthe Polish State Committee for Scientific Research\nunder contract No.~2P03B 01324;\nthe Ministry of Science and Technology of the Russian Federation;\nthe Ministry of Education, Science and Sport of the Republic of\nSlovenia;\nthe National Science Council and the Ministry of Education of Taiwan;\nand the U.S.\\ Department of Energy.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nThe recent discovery of fairly high-temperature and possibly unconventional superconductivity in doped LaOFeAs provoked intensive investigation into Fe-pnictogen containing compounds\\cite{Hosono:2008, Norman:2008, Wang:2008, Yin:2008}. Superconductivity has since been found in many related materials, all of which share the common structural motif of a Fe-pnictogen sandwich, with the surrounding layers acting as spacers and\ncharge donors\/acceptors. Experiment and theory both point to the FeAs layers as the locus for understanding superconductivity\\cite{Hosonoreview:2009, Eremin:2008, Maier:2008, Yildrim:2008}. Although the pairing \nmechanism is not yet known, many structural and electronic properties of the superconducting family have been characterized. The phase diagram as a function\nof doping has been intensely investigated and many competing structural and magnetic orderings identified\\cite{Dai:2008}; of particular interest is the antiferromagnetic phase which lies next to the\n superconducting region as a function of both doping and pressure. Several correlations have been suggested between the superconducting transition temperature and structural parameters of the FeAs layer such as the pnictogen height and the tetrahedral bond angle, showing the importance of understanding the structure-property relationships in this class of materials\\cite{Arita:2009, Lee:2008}.\n\nFeAs-based compounds had previously been widely studied for their potential ``spintronic'' functionalities\\cite{Prinz:1990, Munekata:1989, Ohno:1996, Shirai:2001, Rahman:2006}. These are magnetic semiconducting materials in which the charge degree of freedom is augmented by a spin component, making it desirable for magnetic storage.\n\nDensity functional theory (DFT) has been utilised successfully to address many questions regarding the properties of the iron-pnicitide superconductors and the related Fe-As spintronic compounds\\cite{Singh:2008, Cao:2008, Dong:2008, Griffin\/Spaldin:2012}. The ground-state structural and magnetic properties have been correctly reproduced with DFT, provided due care is taken with the choice of exchange-correlational functional. \n Band structure calculations have confirmed the semimetallic nature of the compounds,\n finding hole and electron pockets at the Fermi level, suggesting that the pairing mechanism may be \nrelated to Fermi surface nesting. DFT has also confirmed the ground state as striped antiferromagnetic ordering for all of the parent compounds of the superconducting pnictides\\cite{Kroll:2008}. \n\n\nDespite its many achievements, there have been some problems with using DFT to model the Fe-pnictide materials. The choice of the exchange-correlation functional poses particular difficulties in these compounds since they lie between the weakly- and strongly-correlated limits. Studies comparing the local density approximation (LDA) with the generalized gradient approximation (GGA) have concluded that, while GGA gives better structures, the two give similar results for magnetic and electronic properties\\cite{Mazin:2008}. The theoretically-determined magnetic moment on the Fe sites is greatly overestimated within both the LDA and GGA, \nwhich is an unusual failing of DFT\\cite{Sushko:2008}. In addition, the pnictogen height with respect to the Fe planes is consistently underestimated compared to experiment\\cite{Singh:2009}. For this reason, many electronic structure calculations are now carried out using the experimentally-determined structure.\n\n\nWhile the Fe environment is different in iron monoarsenide from that in the Fe-pnictide superconductors (the former has a bulk octahedral network while the latter consists of tetrahedrally coordinated layers), both compounds lie at the boundary between itinerant and localized magnetism. Since many of the computational issues for the pnictides are related to this dichotomy -- DFT exchange-correlation functionals exist which successfully describe localized-moment insulators and simple metals, but how to best treat the meeting of these extremes is an ongoing question -- a methodology that describes well the structural and magnetic features of bulk FeAs will likely also be appropriate for the more complex Fe-pnictide superconductors.\n\n\nIn this work we perform a systematic investigation of the effects of the choice of DFT exchange-correlation functional on the calculated properties of the parent iron pnictide compound FeAs. We study the bulk ground state, MnP-type FeAs, and calculate the crystal structure, magnetic\nproperties and electronic structure using both the well-established LDA and GGA functionals and their ``+U'' extensions, as well as the recently introduced hybrid functional\\cite{HSE}. Our goal is to identify the most appropriate functional for describing FeAs, and in turn the pnictide superconductors, and to understand the fundamental physics underlying the choice. \n\n\n\\section{Existing Literature}\nThe ground state of bulk FeAs is the orthorhombic MnP-type structure which it adopts up to its melting temperature of 1300 K (Fig. 1). The primitive unit cell contains eight atoms, with the Fe ions, coordinated by distorted As octahedra, forming zig-zag chains along the $a$-axis. \n\nThe space group of bulk FeAs is still controversial. The first experimental characterization of the structural and magnetic properties was performed in 1969 by Selte et al.\\cite{Selte:1969}  using X-ray diffraction. They found the ground state to have $Pnma$ symmetry, adopting the same structure as MnP. Next, Lyman and Prewitt suggested the space group $Pna2_{1}$ based on a comparison of x-ray refinements in both structures\\cite{Lyman:1984}. More recently, Rodriguez et al.\\cite{Rodriguez:2011} performed both powder and single-crystal neutron diffraction experiments again finding $Pnma$ symmetry. \n\nFurthermore, the magnetic properites of this metallic antiferromagnetic are not fully understood. The first magnetic study using powder neutron diffraction indicated a simple incommensurate spin spiral of wavevector \\textbf{q}=0.375\\textbf{c*} with the moments lying in the $ab-$plane\\cite{Selte:1972}. This was later disputed by transport experiments which revealed highly anistropic magnetic properties, where the susceptibility along the $a$ and $b$ axis was found to differ greatly\\cite{Segawa:2009}. More recently, Rodriguez et al.\\cite{Rodriguez:2011} performed single-crystal neutron-diffraction experiments to elucidate the nature of the magnetic ordering. They proposed incommensurate modulated magnetism, however were unable to distinguish between the previously proposed simple spiral structure\\cite{Selte:1972}, or a collinear spin-density wave structure. They confirmed the highly anistropic magnetism obtaining ~15\\% greater spin polarization in the $b$-plane compared to the $a$-direction.  Further confirmation of the anistropy also came from M\\\"{o}ssbauer measurements from Blachowski et al.\\cite{Blachowski:2014}. However, neither Refs. 28 or 30 were able to give a precise conclusion for the spin structure. \n\nVery little theoretical work has been carried out on bulk MnP-type FeAs. First principles calculations performed by Parker and Mazin\\cite{Parker\/Mazin:2011} confirmed the antiferromagnetic ground-state magnetic ordering. However, unlike the Fe-based superconductors, this could not be explained by Fermi-surface nesting. They also found a 3-dimensional Fermi surface, lending support to the anistropic magnetic behavior from experiments.\n\n\\section{Calculation Details}\n\nWe performed density functional calculations as implemented in the Vienna ab initio Simulation Package (VASP)\\cite{VASP1, VASP2} and wavefunctions expanded in plane waves to an energy cutoff of 500 eV. We used the projector augmented wave (PAW) methods for the electron-core interactions with Fe(3d, 4s) and As(4s, 4p) shells treated as valence. Since MnP-type FeAs is close to semimetallic, a dense Brillouin zone\n sampling scheme of 10x10x10 Monkhorst-Pack grid was used\\cite{Monkhorst_Pack}. The internal coordinates were relaxed in all cases until the Hellmann-Feynmann\n forces were less than 1 meV\/\\AA\\  on each atom. Equations of state were fitted with the Murnaghan-Birch equation\\cite{EOS:1983}.\n\n\nComparisons were made between spin-polarised LDA, PBE\\cite{PBE1, PBE2}, LDA+U and GGA+U, in addition to hybrid functional calculations.\nDFT+U calculations were performed in the Dudarev scheme with an effective $U_{eff}=U-J$\nwhere $U$ represents the electron-electron correlation term and $J$ is the electron exchange energy\\cite{Dudarev}. The value of $U_{eff}$ was varied between -2 eV and 4 eV. In the hybrid functional calculations, the standard HSE functional with 75\\% PBE and 25\\% exact Hartree-Fock exchange was used\\cite{HSE}. \n\n\\section{Results}\n\n\\subsection{Ground State Structure}\nFirst, we address the question of the ground state structure by comparing the calculated energies of the $Pnma$ and $Pna2_{1}$ structures. In both cases, the initial structure was taken from experiments and then a full relaxation of the lattice parameters and internal coordinates was performed with symmetry constrained. The ground state structure was found to be the $Pnma$ structure, with an energy $4$ meV per formula unit (f.u.) and $17$ meV\/f.u. lower than the $Pna2_{1}$ for the LDA and GGA calculations respectively. For the rest of this work we focus therefore on the $Pnma$ structure.\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=8cm, keepaspectratio=true]{.\/structuremnp.png}\n\n \\caption{(i) The MnP-type structure of bulk FeAs where each Fe is 6-fold coordinated by As in a network of edge-sharing polyhedra. (ii) View in the [010] direction of the MnP unit cell. The Fe sites are labelled for later discussion of bond lengths, likewise the shortest (a) and longest (b) Fe-As bondlengths are shown. The black arrows indicate the lowest-energy collinear AFM order.}\n \\label{Fig. 3.}\n\\end{figure}\n\nNext we compare our cell parameters and atomic positions calculated with the five considered functionals with experiment (Table 1). All results are for collinear AFM ordering where the nearest neighbours are antiferromagnetically coupled, and the next-nearest are ferromagnetically coupled (Fig. 1 (ii)). We later show this to be the lowest energy commensurate arrangement. We see that LDA greatly \nunderestimates the experimental unit cell volume of $110.64$ \\AA$^3$ by around 9\\%, which is larger than the usual LDA underestimation. However, both GGA and HSE volumes agree well with experiment, with values of $109.3$ \\AA$^3$ and $108.4$ \\AA$^3$ respectively. The usual behavior of GGA is to overestimate the volume, but here the calculated value is \n~1\\% less than experiment. Interestingly the inclusion of the Hartree-Fock exact exchange term does not significantly alter the GGA structure.\n\n\n\\begin{center}\n\\begin{table*}[ht]\n\\caption{\\label{1} Calculated lattice parameters and atomic fractional coordinates obtained using the LDA, GGA and HSE functionals as well as the experimental values from Ref.[\\onlinecite{Selte:1969}]}.\n\\begin{ruledtabular}\n\\begin{tabular}{l|c|ccc|cccc|cc|cc}\n & Vol (\\AA$^{3}$) & a(\\AA) & b(\\AA) & c(\\AA) & Fe\\textit{(x)} & Fe\\textit{(z)} & As\\textit{(x)} & As\\textit{(z)} & Fe-Fe (a) & Fe-Fe (b) & Fe-As (1) & Fe-As (2) \\\\ \\hline\n LDA & 100.33 & 5.313 & 3.194 & 5.912 & 0.0017 & 0.2017 & 0.2000 & 0.5728 & 2.717 & 2.871 & 2.281 & 2.434  \\\\ \n GGA & 108.46 & 5.468 & 3.277 & 6.051 & 0.0017 & 0.2016 & 0.2007 & 0.5728 & 2.796 & 2.939 & 2.338 & 2.496\\\\ \n HSE+PBE & 108.40 & 5.470 & 3.276 & 6.050 & 0.0019 & 0.2016 & 0.2007 & 0.5727 & 2.797 & 2.938 & 2.338 & 2.495 \\\\ \n Exp\\cite{Selte:1969} & 110.64 & 5.442 & 3.373 & 6.028 & 0.0027 & 0.1994 & 0.1992 & 0.5773 & 2.788 & 2.937 & 2.347 & 2.516 \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\\end{center}\n\n\nIn Fig. 2 (upper panel) we show the effect on the calculated volume of adding a U to the GGA functional. Interestingly, the volume increases by over 20\\% as U is increased from 0 eV to 4 eV  leading to unphysically large volumes for the higher U values. The volume does not change significantly for a negative U, remaining close to the U=0 eV and experimental value.\n\nThe calculated shortest Fe-As bond distance was found to be $2.28$ \\AA\\ for LDA, $2.34$\\AA\\  for GGA, and $2.34$ \\AA\\  for the hybrid functional, compared to the experimental value of $2.35$ \\AA. Here again, the LDA performs poorly with a substantional underestimation of the bond distance. GGA, GGA+U (for U$<$1 eV) and the hybrid functional values are very close to the experimental value.  \n\nIn conclusion, the hybrid functionals along with GGA and GGA+U for small values of U give the best match to the experimental lattice parameters and internal coordinates.\n\n\n\\subsection{Magnetic Properties}\n\n\n\\begin{figure}\n \\includegraphics[width=10cm, keepaspectratio=true]{mnp_comp_new.png}\n\n \\caption{\\label{2} Calculated unit cell volume, magnetic energy, and magnetic moment within the GGA+U method as a function of U. The top panel shows the volume of the unit cell containing four formula units; the middle panel shows the energy difference between the ferromagnetic and antiferromagnetic orderings per eight atoms, and the bottom shows the magnetic moment of one Fe site. The experimental values of the Fe magnetic moment and cell volume are indicated by the dashed orange lines. The results using the hybrid functional are shown with the dotted purple lines.}\n \\label{Fig. 2}\n\\end{figure}\n\nNext we compare the calculated magnetic ordering and magnetic moment size obtained with different functionals. To isolate the effects of magnetic ordering, we use the same structure -- that obtained from optimized hybrid functional calculations -- for all calculations. We obtain an antiferromagnetic ground state with nearest-neighbour Fe ions coupled antiferromagnetically, and second-nearest neighbours ferromagnetically coupled, for all functionals and all values of U.\n\n\\begin{figure}\n \\includegraphics[width=7cm, keepaspectratio=true]{spiralnewcop.png}\n\n \\caption{\\label{2} Calculated energies for a spiral wavevector propagating along the \\textbf{c}-axis. In our units, $\\xi=0$ corresponds to FM ordering, and $\\xi=1$ to nearest-neighbour AFM. The experimental result of Selte et al. [23] of $\\textbf{q}=0.375\\times \\mathbf{c*}$ is shown by the dashed orange line, and of Rodriquez et al. [21] of $\\textbf{q}=0.395\\times \\mathbf{c*}$ by the dotted purple line.}\n \\label{Fig. 6}\n\\end{figure}\n\nFig. 2 (middle panel) shows our calculated total energy difference between the FM ordering and the ground-state AFM ordering as a function of U.\nNote that increasing the value of U in the GGA+U method increases the relative stabilization energy for AFM. This shows that increasing U moves the system away from any frustration caused by competition between AFM and FM ordering. The bottom panels shows the calculated magnetic moment as a function of U. As expected, the moment steadily increases with higher values of U as a result of the increased localization of the bands.\nAs in the case of the Fe-pnictide superconductors, all three functionals overestimate the value of the magnetic moment. The deviation scales with U in the GGA+U method, and the experimental value of magnetic moment can be attained for a negative $U_{eff}=-1$eV.\n\n\n\nAs discussed above, the experimental bulk structure has an incommensurate antiferromagnetic ordering. Such incommensurate magnetic ground states often result either from electronic instabilities such as nesting, or from competition between ferromagnetic and antiferromagnetic exchange interactions. To investigate a possible electronic origin, we perform spin-spiral calculations as implemented in the VASP code. This allows us to impose helimagnetism (a spin spiral in which neighboring spins tilt by a fixed angle with constant amplitude) by modifying the periodic boundary conditions in the system. Spin-orbit coupling is not considered and so the role of the electronic structure is isolated from that of the lattice.\n\nIn Fig. 3 we plot our calculated total energy as a function of the propagation wavevector $\\xi$. The overall energy minimum is for $\\xi=1$, which corresponds to a commensurate antiferromagnetic coupling of the nearest-neighbour Fe ions. The spiral wavevectors reported by Selte et al. and Rodriguez et al. are shown by the vertical orange and purple lines; both occur at higher energies.\n\nWe next calculated the magnetocrystalline anistropy energy (MAE) by including spin-orbit coupling and calculating the energy difference for spins aligned along the x, y and z axes. Our calculations show a preference for spins to lie along the x-axis with energy differences of 0.07 meV\/f.u. and 0.08 meV\/f.u. compared to the y-axis and z-axis respectively.\n\nWe conclude, therefore, that the reported incommensurate magnetism is unlikely drive by an electronic instability causing a helimagnetic spiral, and that coupling to the lattice is important when considering the origin of magnetism in this compound.\n\n\\subsection{Electronic Properties}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=18cm, keepaspectratio=true]{combined.png}\n\n \\caption{\\label{2} Calculated band structures for MnP-type FeAs using the GGA+U method with U=-1,0,2 eV. The density of states for U=2 eV is also shown next to the U=2 eV band structure. The orbitally-projected Fe-\\textit{d} states are indicated by the dashed orange line and region, the As-\\textit{p} by the shaded green region. In all plots the Fermi level has been set to 0 eV and is marked by the dashed line.}\n \\label{Fig. 8}\n\\end{figure*}\n\nFig. 4 shows the calculated band structures of FeAs for values of U ranging from -1 eV to 2 eV, again for the optimized structure calculated with the hybrid functional. The density of states for U=0 eV was also calculated and is shown in the figure. In each case the region around the Fermi level is composed of a broad band of Fe-\\textit{d} states with a small contribution from As-{p} states. Compared with the U=0 eV case, the negative-U band structure has more delocalized bands, consistent with the lower value of magnetic moment on the Fe-{d} states. Increasing the U value has the expected result of pushing bands away from the Fermi level, increasing any gaps between bands at E$_{F}$. \n\nHowever, depsite this general trend of band localization, the main features of the electronic structure do not change significantly on varying the U parameter. In each case we find an electron pocket at the $\\Gamma$ point (shown in turquoise in Fig. 4), whose size and dispersion are slightly U-dependent. The curvature of the pocket increases with U giving higher electron effective masses. We also find two hole pockets for each case (marked in purple), however their location in the Brillouin Zone changes as a function of U. Along the S to $\\Gamma$ direction, a Fermi surface pocket is present for all U values. However, for U=-1 eV, a second pocket appears in the X to S direction which is not present in the U=0 eV or U=2 eV cases. In the latter two calculations, we instead find the second pocket in the $\\Delta$ to D direction. As U is increased, this first X to S pocket as seen in the U=-1 eV case is pushed further away from the Fermi level.\nIn summary, the character of the bands at E$_{F}$ changes modestly with increasing U, however these subtle changes could have a big effect on Fermi surface nesting since the location of the pockets changes as a function of U.\n\n\\section{Discussion}\n\n\nPerhaps surprisingly for this ostensibly simple ferropnictide compound, we encountered similar problems with using density functional theory to calculate its structural and magnetic properties to those reported for the pnictide superconductors. As in the case of the Fe-pnictide parent compounds, simple LDA calculations do not reproduce the  measured structural parameters, with a 10\\% LDA underestimation of the cell volume in this case. We also find that the Fe-As distance is underestimated in the LDA, which is a common failing in the Fe-pnictide literature. However, both the GGA and hybrid functionals give structures that are much closer to experiment. Subsequent addition of a Hubbard-U to the GGA calculations has little effect on the volume for U$<2$eV (including for negative U), but larger U values cause a strong divergence from the experimental volume. The conclusion for structural calculations echoes that previously made for the ferropnictide superconductors; that GGA best reproduces the structural parameters while LDA does extremely badly.\n\nNext we examined the magnetic properties using LDA, LDA+U, GGA, GGA+U and the hybrid functional. The lowest energy magnetic ordering was found to be antiferromagnetic, consistent with the spin spiral structure found in experiments where the nearest neighbour spins are antiferromagnetically coupled. The simple spin spiral as proposed by Selte and later Rodriguez  was not found to be the ground state when helimagnetic calculations were performed. However, magnetocrystalline anistropy calculations compare well with experiment -- there is a preference for the spins to lie along the x-axis. The most likely magnetic ordering from our calculations is some modulated antiferromagnetism as proposed by Rodriguez in which a noncollinear spin-density wave traces out an ellipse (rather than a circle in the case of a simple spiral).\n\nAs in the case of the pnictide superconductors, the magnetic moment on the Fe ions is strongly overestimated compared with experiment; for example the calculated moment using hybrid functionals and GGA+U is 2.2 times the experimental value. Spin fluctuations have been proposed to account for this discrepancy between theory and experiment\\cite{Johannes:2009}; since DFT is a mean field theory it does not include the results of temporal fluctuations. One possible solution we considered was to venture into a \\textit{negative}-Hubbard-U regime. However a physical rationale for doing so is not so obvious, despite its match with experiment in both the magnetic moment and structural properties. \n\nWhile spin fluctuations are a plausible cause to the overestimation of the moment in density functional theory, the impact of the covalent bonding between Fe and As in these materials should not be overlooked. This was considered recently for the cuprate class of superconductors, where `missing' neutron intensities were found by reinterpreting the neutron data with the strong covalency of the materials\\cite{Walters:2009}. In order to fully understand the magnetism of these ferropnictide materials, the extent to which spin fluctuations and\/or covalency influence the magnetic moments must be further examined.\n\n\\section{Summary}\nWe performed a thorough investigation of the effect of exchange-correlation functional (LDA, LDA+U, GGA, GGA+U, HSE) on the calculated structural, magnetic and electronic properties of MnP-type FeAs. The hybrid functional and GGA best reproduced the experimental structures and ground-state magnetic ordering in the collinear limit. As is also found in the Fe-pnictide superconductors, LDA performs poorly for the structural calculations, and all functionals overestimate the Fe magnetic moment by at least a factor of 2. Only a \\textit{negative}-U regime correctly reproduces the experimental value of the magnetic moment.\n\nTo investigate the observed modulated noncollinear magnetic structure, we performed helimagnetic calculations, and found a simple spin-spiral is not the calculated ground state. This suggests a different origin for the incommensurate magnetism such as the elliptical spin-density wave as proposed by Rodriguez et al.\\cite{Rodriguez:2011}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}