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+{"text":"\\section{Introduction}\n\\label{intro}\nMassive stars contribute to the chemical composition of matter as we know it in the universe, and their deaths are accompanied by energetic core-collapse supernovae (SNe) that seed our universe with black holes (BHs) and neutron stars (NSs) -- the most exotic objects of the stellar graveyard. Large time-domain surveys of the sky \\citep[e.g.,][]{York2000,Drake2009,Law2009,Kaiser2010,Shappee2014,Abbott2016,Tonry2018,Bellm2019}, paired with targeted follow-up efforts, have greatly enriched our view on the final stages of massive star evolution. Yet, a lot remains to be understood about the diverse paths that bring massive stars toward\ntheir violent deaths \\citep{Langer2012}.  \n\nCore-collapse SNe can occur in stars with a hydrogen envelope\n(Type II) or in stars where hydrogen is almost or completely missing \\citep[Type Ib\/c, also referred to as stripped-envelope SNe;][]{Filippenko1997,Matheson2001,Li2011,Modjaz2014,Perley2020b,Frohmaier2021}. Type Ib\/c SNe constitute approximately 25\\% of all massive star explosions \\citep{Smith2011}, and their pre-SN progenitors are thought to be either massive ($M\\gtrsim 20-25$\\,M$_{\\odot}$) and metal-rich single stars that have been stripped through stellar mass loss; or the mass donors in close binary systems (at any metallicity) that have initial masses $\\gtrsim 8$\\,M$_{\\odot}$ \\citep[e.g.,][and references therein]{Langer2012}. \n\nA small fraction of Type Ib\/c SNe show velocities in their optical spectra that are systematically higher than those measured in ordinary SNe Ic at similar epochs. Hence, these explosions are referred to as SNe of Type Ic with broad lines  \\citep[hereafter, Ic-BL; e.g.,][]{Filippenko1997,Modjaz2016,GalYam2017}. Compared to Type Ib\/c SNe, broad-lined events are found to prefer environments with lower metallicity (in a single star scenario, mass loss mechanisms also remove angular momentum and are enhanced by higher metallicities), and in galaxies with higher star-formation rate density. Thus, it has been suggested that SN Ic-BL progenitors may be stars younger and more massive than those of normal Type Ic (more massive progenitors can lose their He-rich layers to winds at lower metallicity due to the higher luminosities driving the winds), and\/or tight massive binary systems that can form efficiently in dense stellar clusters \\citep[e.g.,][]{Kelly2014,Japelj2018,Modjaz2020}. \n\nThe spectroscopic and photometric properties used to classify core-collapse SNe are largely determined by the stars' outer envelopes \\citep[envelope mass, radius, and chemical composition;][]{Young2004}. On the other hand, quantities such as explosion energies, nickel masses, and ejecta geometries can be inferred via extensive multi-wavelength and multi-band observations. These quantities, in turn, can help constrain the properties of the stellar cores  \\citep[such as mass, density structure, spin, and magnetic fields; see e.g. ][and references therein]{Woosley2002,Burrows2007,Jerkstrand2015} that are key to determine the nature of the explosion. For example, based on nickel masses and ejecta masses derived from bolometric light curve analyses, \\citet{Taddia2019} found that $\\gtrsim 21\\%$ of Ic-BL  progenitors are compatible with massive ($\\gtrsim 28$\\,M$_{\\odot}$), possibly single stars, whereas $\\gtrsim 64\\%$ could be associated with less massive stars in close binary systems.  \n\nUnderstanding the progenitor scenario of SNe Ic-BL is particularly important as these SNe challenge greatly the standard explosion mechanism of massive stars \\citep[e.g.,][and references therein]{Mezzacappa1998,MacFadyen1999,Heger2003,WoosleyHeger2006,Janka2007,Janka2012,Smith2014,Foglizzo2015,Muller2020,Schneider2021}. The energies inferred from optical spectroscopic modeling of Ic-BL events are of order $\\approx 10^{52}\\,{\\rm erg}$, in excess of the $\\approx 10^{51}\\,{\\rm erg}$ inferred in typical SNe Ib\/c, while ejecta masses are comparable or somewhat higher \\citep{Taddia2019}. In the traditional core-collapse scenario, neutrino irradiation from the proto-NS revives the core-bounce shock, making the star explode. However, the neutrino mechanism cannot explain the more energetic SNe Ic-BL. Unveiling the nature of an engine powerful enough to account for the extreme energetics of SNe Ic-BL is key to understanding the physics behind massive stellar deaths, and remains as of today an open question. \n\nA compelling scenario invokes  the existence of a jet or a newly-born magnetar as the extra source of energy needed to explain SNe Ic-BL \\citep[e.g.,][]{Burrows2007,Papish2011,Gilkis2014,Mazzali2014,Lazzati2012,Gilkis2016,Soker2017,Barnes2018,Shankar2021}. The rapid rotation of a millisecond proto-NS formed in the collapse of a rotating massive star can amplify the NS magnetic field to $\\gtrsim 10^{15}$\\,G, creating a magnetar. The magnetar spins down quickly via magnetic braking, and in some cases magneto-rotational instabilities can launch a collimated jet that drills through the outer layers of the star producing a gamma-ray burst \\citep[GRB; e.g.,][]{Heger2003,Izzard2004,WoosleyHeger2006,Burrows2007,Bugli2020,Bugli2021}.  These jets can transfer sufficient energy to the surrounding stellar material to explode it into a SN.\n\nThe above scenario is particularly interesting in light of the fact that SNe Ic-BL are also the only type of core-collapse events that, observationally, have been unambiguously linked to GRBs \\citep[e.g.,][and references therein]{Woosley2006,Cano2017}. GRBs are characterized by bright afterglows that emit radiation from radio to X-rays, and are unique laboratories for studying relativistic particle acceleration and magnetic field amplification processes \\citep{Piran2004,Meszaros2006}. In between ordinary SNe Ic-BL and cosmological GRBs is a variety of transients that we still have to fully characterize. Among those are low-luminosity GRBs, of which the most famous example is GRB\\,980425, associated with the radio-bright Type  Ic-BL SN\\,1998bw \\citep{Galama1998,Kulkarni1998}.  \n\nRecently, \\citet{Shankar2021} used the jetted outflow model produced from a consistently formed proto-magnetar in a 3D core-collapse SN to extract a range of central engine parameters  (energy $E_{eng}$ and  opening  angle $\\theta_{eng}$)  that were then used as inputs to hydrodynamic models of  jet-driven explosions. The output of these models, in turn, were used to derive predicted SN light curves and spectra from different viewing angles, and found to be in agreement with SN Ic-BL optical observables \\citep[see also][]{Barnes2018}. It was also shown that additional energy from the engine can escape through the tunnel drilled in the star as an ultra-relativistic jet (GRB) with energy $\\approx 10^{51}$\\,erg. On the other hand, a SN Ic-BL can be triggered even if the jet engine fails to produce a successful GRB jet. The duration of the central engine, $t_{eng}$, together with $E_{eng}$ and $\\theta_{eng}$, are critical to determining the fate of the jet \\citep{Lazzati2012}. \n\nA more general scenario where the high velocity ejecta found in SNe Ic-BL originate from a cocoon driven by a relativistic jet (regardless of the nature of the central engine) is also receiving attention. In this scenario, cosmological long GRBs are explosions where the relativistic jet breaks out successfully from the stellar envelope, while low-luminosity GRBs and SNe Ic-BL that are not associated with GRBs represent cases where the jet is choked \\citep[see e.g. ][and references therein]{Piran2019,Eisenberg2022,Gottlieb2022,Pais2022}. \n\nOverall, the dividing line between successful GRB jets and failed ones is yet to be fully explored observationally, and observed jet outcomes in SNe Ic-BL have not yet been systematically compared to model outputs. While we know that SNe discovered by means of a GRB are all of Type Ic-BL, the question that remains open is whether all SNe Ic-BL make a GRB (jet), at least from some viewing angle, or if instead the jet-powered SNe Ic-BL are intrinsically different and rarer than ordinary SNe Ic-BL. Indeed, due to the collimation of GRB jets, it is challenging to understand whether all SNe Ic-BL are linked to  successful GRBs: a non-detection in $\\gamma$- or X-rays could simply be due to the explosion being directed away from us. \n Radio follow-up observations are needed to probe  the explosions' fastest-moving ejecta ($\\gtrsim 0.2c$) largely free of geometry and viewing angle constraints.  Determining observationally what is the fraction of Type Ic-BL explosions that output jets which successfully break out of the star (as mildly-relativistic or ultra-relativistic ejecta), and measuring their kinetic energy via radio calorimetry, can provide jet-engine  explosion models a direct test of their predictions. \n \n Using one of the largest sample of SNe Ic-BL with deep radio follow-up observations \\citep[which included 15 SNe Ic-BL discovered by the Palomar Transient Factory, PTF\/iPTF;][]{Law2009}, \\citet{Corsi2016}  already established that $<41\\%$ of SNe Ic-BL harbor relativistic ejecta similar to that of SN\\,1998bw. Here, we present the results of a systematic radio follow-up campaign of an additional 16 SNe Ic-BL (at $z\\lesssim 0.05$) detected independently of $\\gamma$-rays by the Zwicky Transient Facility \\citep[ZTF;][]{Bellm2019,Graham2019}. This study greatly expands our previous works on the subject \\citep[][]{Corsi2017,Corsi2016,Corsi2014}. Before the advent of PTF and ZTF, the comparison between jet-engine model outcomes and radio observables was severely limited by the rarity of SN Ic-BL discoveries \\citep[e.g.,][]{Berger2003,Soderberg2006} and\/or by selection effects \\citep[e.g.,][]{Woosley2006}---out of the thousands of jets identified, nearly  all were discovered at large distances via their high-energy emission (implying aligned jet geometry and ultra-relativistic speeds). In this work, we aim to provide a study free of these biases. \n \n Our paper is organized as follows. In Section \\ref{sec:discovery} we describe our multi-wavelength observations; in Section \\ref{section:sample} we describe in more details the SNe Ic-BL included in our sample; in Section \\ref{sec:modeling} we model the optical, X-ray, and radio properties of the SNe presented here and derive constraints on their progenitor and ejecta properties. Finally, in Section \\ref{sec:conclusion} we summarize and conclude. Hereafter we assume cosmological parameters  $H_0 = 69.6$\\,km\\,s$^{-1}$\\,Mpc$^{-1}$, $\\Omega_{\\rm M }= 0.286$, $\\Omega_{\\rm vac} = 0.714$ \\citep{Bennett2014}. All times are given in UT unless otherwise stated. \n\n\n\\begin{table*}\n\\begin{center}\n\\caption{The sample of 16 SNe Ic-BL  analyzed in this work. For each SN we provide the IAU name, the ZTF name, the position, redshift, and luminosity distance. \\label{tab:sample}}\n\\begin{tabular}{lccc}\n\\hline\n\\hline\nSN (ZTF name)  & RA, Dec (J2000) &  $z$& $d_L$  \\\\\n & (hh:mm:ss~~dd:mm:ss) & &(Mpc)\\\\\n\\hline\n2018etk (18abklarx) & 15:17:02.53 +03:56:38.7 & 0.044& 196 \\\\\n2018hom (18acbwxcc)& 22:59:22.96 +08:45:04.6 & 0.030 & 132   \\\\\n2018hxo (18acaimrb) &  21:09:05.80 +14:32:27.8 & 0.048 & 214  \\\\\n2018jex (18acpeekw) & 11:54:13.87 +20:44:02.4 & 0.094 & 434  \\\\\n2019hsx (19aawqcgy) & 18:12:56.22 +68:21:45.2 & 0.021 & 92  \\\\\n2019xcc (19adaiomg) & 11:01:12.39 +16:43:29.1 & 0.029 & 128  \\\\\n2020jqm (20aazkjfv) & 13:49:18.57 $-$03:46:10.4 & 0.037 & 164  \\\\\n2020lao (20abbplei) & 17:06:54.61 +30:16:17.3 & 0.031 & 137  \\\\\n2020rph (20abswdbg) & 03:15:17.83 +37:00:50.8 & 0.042 & 187\\\\\n2020tkx (20abzoeiw) &  18:40:09.01 +34:06:59.5 & 0.027 & 119 \\\\\n2021xv (21aadatfg) & 16:07:32.82 +36:46:46.2 & 0.041 & 182 \\\\\n2021aug (21aafnunh) & 01:14:04.81 +19:25:04.7 & 0.041 & 182 \\\\\n2021epp (21aaocrlm) & 08:10:55.27 $-$06:02:49.3 & 0.038 & 168\\\\\n2021htb (21aardvol) & 07:45:31.19 +46:40:01.3  & 0.035& 155 \\\\\n2021hyz (21aartgiv) &  \n09:27:36.51 +04:27:11.0 &  0.046& 205\\\\\n2021ywf (21acbnfos) & 05:14:10.99 +01:52:52.4 & 0.028 & 123 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{CorsiFig1_lcs_mag.png}\n    \\caption{P48 $r$- (top) and $g$-band (middle) light curves for the SNe Ic-BL in our sample, compared with the $R$- and $B$-band light curves of SN\\,1998bw, respectively. The bottom panel shows the corresponding color evolution. Observed AB magnitudes are corrected for Milky Way extinction. The archetypal SN\\,1998bw is shown in black solid points, and its Gaussian Process interpolation in black dashed lines. See also \\citet{Anand2022} and \\citet{Gokul2022}.}\n    \\label{fig:opt-lc-mag}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{CorsiFig2_lcs_flux.png}\n    \\caption{Top and middle panels: same as Figure \\ref{fig:opt-lc-mag} but in flux space and with fluxes  normalized to their Gaussian Process maximum. Bottom panel: bolometric light curves. We converted $g-r$ to bolometric magnitudes with the empirical relations by \\cite{Lyman2014,Lyman2016}. See also \\citet{Anand2022} and \\citet{Gokul2022}.\n    \\label{fig:opt-lc-flux}}\n\\end{figure*}\n\n\\section{Multi-wavelength observations}\n\\label{sec:discovery}\nWe have collected a sample of 16 SNe Ic-BL observed with the ZTF and with follow-up observations in the radio. The SNe Ic-BL included in our sample are listed in Table \\ref{tab:sample}. We selected these SNe largely based on the opportunistic availability of follow-up observing time on the Karl G. Jansky Very Large Array (VLA).  The sample of SNe presented here doubles the sample of SNe Ic-BL with deep VLA observations presented in \\citet{Corsi2016}. \n\nThe SNe considered in this work are generally closer than the PTF\/iPTF sample of SNe Ic-BL presented in \\citet{Taddia2019}. In fact, their median redshift ($\\approx 0.037$) is about twice as small as the median redshift of the PTF\/iPTF SN Ic-BL sample \\citep[$\\approx 0.076$;][]{Taddia2019}. However, the median redshift of the ZTF SNe in our sample is compatible with the median redshift ($\\approx 0.042$) of the full ZTF SN Ic-BL population presented in \\citet{Gokul2022}. A subset of the SNe Ic-BL presented here is also analyzed in a separate paper and in a different context \\citep[r-process nucleosynthesis;][]{Anand2022}. In this work, we report for the first time the results of our radio follow-up campaign of these events. We note that the ``Asteroid Terrestrial-impact Last Alert System''\\citep[ATLAS;][]{Tonry2018} has contributed to several of the SN detections considered here (see Section \\ref{section:sample}). Three of the Ic-BL in our sample were also reported in the recently released bright SN catalog by the All-Sky Automated Survey for Supernovae \\citep[ASAS-SN;][]{Neumann2022}.\n\n\nIn what follows, we describe the observations we carried for this work. In Section \\ref{section:sample} we give more details on each of the SNe Ic-BL in our sample. \n\n\\subsection{ZTF photometry}\nAll photometric observations presented in this work were conducted with the Palomar Schmidt 48-inch (P48) Samuel Oschin telescope as part of the ZTF survey \\citep{Bellm2019,Graham2019}, using the ZTF camera \\citep{dekany2020}. \nIn default observing mode, ZTF uses 30\\,s exposures, and survey observations are carried out in $r$ and $g$ band, down to a typical limiting magnitude of $\\approx 20.5$\\,mag. P48 light curves were derived using the ZTF pipeline \\citep{Masci2019}, and forced photometry \\cite[see][]{Yao2019}. Hereafter, all reported magnitudes are in the AB system. \nThe P48 light curves are shown in Figures~\\ref{fig:opt-lc-mag}-\\ref{fig:opt-lc-flux}. All the light curves presented in this work will be made public on the Weizmann Interactive Supernova Data Repository (WISeREP\\footnote{\\url{https:\/\/www.wiserep.org\/}}).\n\n\n\n\\begin{table}\n \\begin{center}\n\\caption{\\label{tab:xrt_summary}\\textit{Swift}\/XRT observations of 9 of the 16 SNe Ic-BL in our sample. We provide the MJD of the \\textit{Swift} observations, the XRT exposure time, and the 0.3-10\\,keV unabsorbed flux measurements (or $3\\sigma$ upper-limits for non detections).  \\label{tab:x-ray}}\n\\begin{tabular}{llccc}\n\\hline\n\\hline\nSN  & T$_{\\rm XRT}$  & Exp. & $F_{\\rm 0.3-10\\,keV}$ \\\\\n & (MJD)  & (ks) & ($10^{-14}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$)\\\\\n\\hline\n2018etk & 58377.85 & 4.8 & $< 4.2$ \\\\\n2018hom & 58426.02 & 4.3 & $< 6.4$ \\\\\n2019hsx & 58684.15 & 15 & $6.2_{-1.8}^{+2.3}$ \\\\\n2020jqm & 59002.09 & 7.4 & $< 3.3 $ \\\\\n2020lao & 59007.40 & 14 & $< 2.9 $ \\\\\n2020rph & 59088.89 & 7.5 & $< 3.6 $ \\\\\n2020tkx & 59125.38 & 8.1 & $< 3.3 $ \\\\\n2021hyz & 59373.09 & 4.7 & $< 3.5 $ \\\\\n2021ywf & 59487.60 & 7.2 & $5.3_{-3.3}^{+4.9} $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Optical Spectroscopy}\nPreliminary spectral type classifications of several of the SNe in our sample were obtained with the Spectral Energy Distribution Machine (SEDM) mounted on the Palomar 60-inch telescope (P60), and quickly reported to the Transient Name Server (TNS). The SEDM is a very low resolution ($R\\sim 100$) integral field unit spectrograph optimized for transient classification with high observing efficiency \\citep{Blagorodnova2018,Rigault2019}. \n\nAfter initial classification, typically further spectroscopic observations are carried out as part of the ZTF transient follow-up programs to confirm and\/or improve classification, and to characterize the time-evolving spectral properties of interesting events. Amongst the series of spectra obtained for each of the SNe presented in this work, we select one good quality photospheric phase spectrum  (Figures~\\ref{fig:spectra1}-\\ref{fig:spectra2}; grey) on which we run SNID \\citep{Blondin2007} to obtain the best match to a SN Ic-BL template (black), after clipping the host emission lines and fixing the redshift to that derived either from the SDSS host galaxy or from spectral line fitting (H$\\alpha$; see Section \\ref{section:sample} for further details).\nHence, in addition to the SEDM, in this work we also made use of the following instruments: the Double Spectrograph \\citep[DBSP;][]{Oke1995}, a low-to-medium resolution grating instrument for the Palomar 200-inch telescope Cassegrain focus that uses a dichroic to split light into separate red and blue channels observed simultaneously; the Low Resolution Imaging Spectrometer \\citep[LRIS;][]{Oke1982,Oke1995}, a visible-wavelength imaging and spectroscopy instrument operating at the Cassegrain focus of Keck-I; the Alhambra Faint Object Spectrograph and Camera (ALFOSC), a CCD camera and spectrograph installed at the Nordic Optical Telescope \\citep[NOT;][]{Djupvik2010}. All spectra presented in this work will be made public on WISeREP.\n\n\\subsection{X-ray follow up with \\textit{Swift}}\nFor 9 of the 16 SNe presented in this work we carried out follow-up observations in the X-rays using the X-Ray Telescope  \\citep[XRT;][]{Burrows+2005} on the \\textit{Neil Gehrels Swift Observatory} \\citep{Gehrels+2004}. \n\nWe analyzed these observations using the online XRT tools\\footnote{See \\url{https:\/\/www.swift.ac.uk\/user_objects\/}.}, as described in \\citet{Evans+2009}. \nWe correct for Galactic absorption, and adopt a power-law spectrum with photon index $\\Gamma = 2$  for count rate to flux conversion for non-detections, and for detections (two out of ninw events) where the number of photons collected is too small to enable a meaningful spectral fit (one out of two detections).  Table~\\ref{tab:xrt_summary} presents the results from co-adding all observations of each source.\n\n\n\\subsection{Radio follow up with the VLA}\n\\label{sec:radioobs}\nWe observed the fields of the SNe Ic-BL in our sample with the VLA via several of our programs using various array configurations and receiver settings (Table \\ref{tab:data}). \n\nThe VLA raw data were calibrated in \\texttt{CASA} \\citep{McMullin2007} using the automated VLA calibration pipeline. After manual inspection for further flagging, the calibrated data were imaged using the \\texttt{tclean} task. Peak fluxes were measured from the cleaned images using \\texttt{imstat} and circular regions centered on the optical positions of the SNe, with radius equal to the nominal width (FWHM) of the VLA synthesized beam (see Table \\ref{tab:data}). The RMS noise values were estimated with \\texttt{imstat} from the residual images. Errors on the measured peak flux densities in Table \\ref{tab:data} are calculated adding  a 5\\% error in quadrature to measured RMS values. This accounts for systematic uncertainties on the absolute flux calibration. \n\nWe checked all of our detections (peak flux density above $3\\sigma$) for the source morphology (extended versus point-like), and ratio between integrated and peak fluxes using the CASA task \\texttt{imfit}. All sources for which these checks provided evidence for extended or marginally resolved emission are marked accordingly in Table \\ref{tab:data}. For non detections, upper-limits on the radio flux densities are given at the $3\\sigma$ level unless otherwise noted.\n\n\\renewcommand\\arraystretch{1.5}\n\\setlength\\LTcapwidth{2\\linewidth}\n\\begin{longtable*}{ccclccc}\n\\caption{VLA follow-up observations of the 16 SN\\lowercase{e} I\\lowercase{c}-BL in our sample. For all of the observations of the SNe in our sample we report: the mid MJD of the VLA observation; the central observing frequency; the measured flux density (all flux density upper-limits are calculated at $3\\sigma$ unless otherwise noted); the VLA array configuration; the FWHM of the VLA nominal synthesized beam; the VLA project code under which the observations were conducted. See Sections \\ref{sec:radioobs} and \\ref{sec:radioanalysis} for discussion.} \\label{tab:data}\\\\\n\\toprule\n\\toprule\nSN & ${\\rm T_{VLA}}$$^{a}$    & $\\nu$  & $F_{\\nu}$ & Conf. & Nom.Syn.Beam  & Project \\\\\n & (MJD)  & (GHz) & ($\\mu$Jy) &  & (FWHM; arcsec) &  \\\\\n\\midrule\n2018etk & 58363.08  & 6.3&  $90.1\\pm8.7$$^{b}$   & D & 12&  VLA\/18A-176$^{d}$\\\\\n & 58374.09   &14 & $41\\pm11$ & D  & 4.6&  VLA\/18A-176$^{d}$ \\\\\n  & 58375.03   & 6.3& $89.7\\pm8.8$$^{b}$  & D  &$12$&  VLA\/18A-176$^{d}$ \\\\\n &  59362.27  &6.2 & $78.5\\pm6.3$$^{b}$  & D  &12 &  VLA\/20B-149$^{d}$\\\\\n\\midrule\n2018hom &  58428.04  & 6.6 &  $133\\pm11$  & D & 12&  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2018hxo  & 58484.73  & 6.4 & $\\lesssim 234$$^{c}$  & C & 3.5 &  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2018jex &  58479.38 &6.4 & $\\lesssim 28$ & C & 3.5&  VLA\/18A-176$^{d}$\\\\\n\\midrule\n2019hsx   & 58671.14 & 6.2 & $\\lesssim 19$ & BnA & 1.0&  VLA\/19B-230$^{d}$\\\\\n\\midrule\n2019xcc &  58841.43 &6.3 &$62.7\\pm 8.7$$^{b}$& D  & 12 &  VLA\/19B-230$^{d}$\\\\\n   & 58876.28  & 6.3  &$60.1\\pm8.5$$^{b}$ & D  & 12&  VLA\/19B-230$^{d}$  \\\\\n  &59363.00 & 6.3 & $50.4\\pm8.1$$^{b}$& D & 12&  VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2020jqm &  58997.03  & 5.6& $175\\pm13$& C  &  3.5&SG0117$^{d}$\\\\\n  &59004.03 & 5.6 & $310\\pm19$&C  & 3.5 &SG0117$^{d}$ \\\\\n &59028.48  & 5.5& $223\\pm18$&B & 1.0& VLA\/20A-568$^{d}$\\\\\n &59042.95  &5.7 & $202\\pm15$&B &1.0& VLA\/20A-568$^{d}$ \\\\\n  &59066.09  &5.5  &$136\\pm13$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n &59088.03  & 5.5  &$168\\pm13$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n &59114.74 & 5.5 & $620\\pm33$ &B  & 1.0& VLA\/20A-568$^{d}$ \\\\\n  &59240.37 & 5.5 & $720\\pm37$ &A &0.33& VLA\/20B-149$^{d}$\\\\\n\\midrule\n2020lao &  59006.21 & 5.2& $\\lesssim 33$ &C & 3.5& SG0117$^{d}$\\\\\n  & 59138.83 &5.5& $\\lesssim 21$&  B & 1.0& SG0117$^{d}$  \\\\\n\\midrule\n2020rph & 59089.59 &5.5 & $42.7\\pm7.4$&   B & 1.0& SG0117$^{d}$\\\\\n  & 59201.28 &  5.5 & $43.9\\pm7.0$&  A &0.33 & SG0117$^{d}$ \\\\\n\\midrule\n2020tkx &   59117.89 & 10 & $272\\pm 16$& B &0.6 & VLA\/20A-374$^{e}$\\\\\n & 59136.11  & 10  & $564\\pm29$& B  &0.6 & VLA\/20A-374$^{e}$ \\\\\n & 59206.92 & 5.5&$86.6\\pm7.3$& A & 0.33 & VLA\/20B-149$^{d}$\\\\\n\\midrule\n2021xv &  59242.42  & 5.5 & $\\lesssim 23$& A & 0.33& VLA\/20B-149$^{d}$\\\\\n & 59303.24 & 5.2& $\\lesssim 29$& D &12 & VLA\/20B-149$^{d}$ \\\\\n &59353.11 & 5.2 & $34.3\\pm8.1$$^{b}$& D & 12 & VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021aug &  59254.75  & 5.2& $\\lesssim 22$& A & 0.33& VLA\/20B-149$^{d}$\\\\\n  & 59303.62 &5.4 & $\\lesssim 45$& D & 12& VLA\/20B-149$^{d}$\\\\\n & 59353.48 & 5.4 & $\\lesssim 30$& D &12 & VLA\/20B-149$^{d}$\\\\\n\\midrule\n2021epp &  59297.06 &5.3 & $(2.62\\pm0.13)\\times10^3$$^{b}$ & D&12 & VLA\/20B-149$^{d}$\\\\\n &  59302.99 & 5.1 &$(2.82\\pm0.18)\\times10^3$$^{b}$ & D & 12& VLA\/20B-149$^{d}$ \\\\\n & 59352.83& 5.3  &$(2.75\\pm0.20)\\times10^3$$^{b}$ & D &12 & VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021htb  &59324.94 & 5.2& $50\\pm10$$^{b}$& D&12 & VLA\/20B-149$^{d}$\\\\\n  & 59352.87 & 5.2 &$59.4\\pm9.5$$^{b}$ &D  & 12& VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021hyz & 59326.08  & 5.2& $38\\pm11$&D &12 & VLA\/20B-149$^{d}$\\\\\n &  59352.99& 5.5 & $\\lesssim 30$& D & 12& VLA\/20B-149$^{d}$ \\\\\n\\midrule\n2021ywf &  59487.57  & 5.0  & $83\\pm10$ & B &1.0 &SH0105$^{d}$\\\\\n &  59646.12 & 5.4 &$19.8\\pm 6.3$ & A & 0.33& SH0105$^{d}$\\\\\n\\bottomrule\n\\multicolumn{7}{l}{$^{a}$ Mid MJD time of VLA observation (total time including calibration).}\\\\\n\\multicolumn{7}{l}{$^{b}$ Resolved or marginally resolved with emission likely dominated by the host galaxy.}\\\\\n\\multicolumn{7}{l}{$^{c}$ Image is dynamic range limited due to the presence of a bright source in the field.}\\\\\n\\multicolumn{7}{l}{$^{d}$ PI: Corsi.}\\\\\n\\multicolumn{7}{l}{$^{e}$ PI: Ho.}\n\\end{longtable*}\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=15cm]{CorsiFig3_IcBL_snid_spectra_part1.pdf}\n    \\caption{Photospheric phase spectra (grey) plotted along with their SNID best match templates (black) for the first half of the SNe Ic-BL in our sample. Spectra are labeled with their IAU name and spectroscopic phase (since $r$-band maximum; see Table \\ref{tab:opt_data}). We note that the spectra used to estimate the photospheric velocities of SN\\,2019xcc, SN\\,2020lao, and SN\\,2020jqm presented in Table \\ref{tab:opt_data} are different from the ones shown here for classification purposes. This is because for spectral classification we prefer later-time but higher-resolution spectra, while for velocity measurements we prefer earlier-time spectra even if taken with the lower-resolution SEDM. All spectra presented in this work will be made public on WISeREP. }\n    \\label{fig:spectra1}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=15cm]{CorsiFig4_IcBL_snid_spectra_part2.pdf}\n    \\caption{Same as Figure \\ref{fig:spectra1} but for the second half of the SNe Ic-BL in our sample. All spectra presented in this work will be made public on WISeREP. }\n    \\label{fig:spectra2}\n\\end{figure*}\n\n\n\\section{Sample description}\n\\label{section:sample} \n\\subsection{SN 2018etk}\nOur first ZTF photometry of SN\\,2018etk (ZTF18abklarx) was\nobtained on 2018 August 1 (MJD 58331.16) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $19.21\\pm0.12$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=15$^{\\rm h}$17$^{\\rm m}$02.53$^{\\rm s}$,\n$\\delta=+03^{\\circ}56'38\\farcs7$ (J2000). The object was reported to the TNS by ATLAS on 2018 August 8, who discovered it on 2018 August 6  \\citep{Tonry2018_ZTF18abklarx}.\nThe last ZTF non-detection prior to ZTF discovery was on 2018 July 16, and the last shallow ATLAS non-detection was on 2018 August 2, at $18.75\\,$mag.\nThe transient was classified as a Type Ic SN by \\citet{Fremling2018_ZTF18abklarx} based on a spectrum obtained on 2018 August 13 with the SEDM.\nWe re-classify this transient as a SN Type  Ic-BL most similar to SN\\,2006aj based on a P200 DBSP spectrum obtained on 2018 August 21 (see Figure~\\ref{fig:spectra1}).\nSN\\,2018etk exploded in a star-forming galaxy with a known\nredshift of $z= 0.044$ derived from SDSS data.\n\n\\subsection{SN 2018hom}\nOur first ZTF photometry of SN\\,2018hom (ZTF18acbwxcc) was\nobtained on 2018 November 1  (MJD 58423.54) with\nthe P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $16.60\\pm0.04$\\,mag (Figure \\ref{fig:opt-lc-mag}), at $\\alpha$=22$^{\\rm h}$59$^{\\rm m}$22.96$^{\\rm s}$,\n$\\delta=+08^{\\circ}45'04\\farcs6$ (J2000). The object was reported to the TNS by ATLAS on 2018 October 26, and discovered by ATLAS on 2018 October 24 at $o\\approx 17.3\\,$mag \\citep{Tonry2018_ZTF18acbwxcc}.\nThe last ZTF non-detection prior to ZTF discovery was on 2018 October 9 at $g>20.35\\,$mag, and the last ATLAS non-detection was on 2018 October 22 at $o> 18.25\\,$mag.\nThe transient was classified as a SN Type Ic-BL by \\citet{Fremling2018_ZTF18acbwxcc} based on a spectrum obtained on 2018 November 2 with the SEDM.\nSN\\,2018etk exploded in a galaxy with unknown redshift. We measure a redshift of $z = 0.030$ from star-forming emission lines in a Keck-I LRIS spectrum obtained on 2018 November 30. We plot this spectrum in Figure~\\ref{fig:spectra1}, along with its SNID template match to the Type Ic-BL SN\\,1997ef. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog \\citep[][]{Neumann2022}.\n\n\\subsection{SN 2018hxo}\nOur first ZTF photometry of SN\\,2018hxo (ZTF18acaimrb) was\nobtained on 2018 October 9  (MJD 58400.14) with the P48. This first detection was in the $g$ band, with a host-subtracted magnitude of $18.89\\pm0.09$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=21$^{\\rm h}$09$^{\\rm m}$05.80$^{\\rm s}$,\n$\\delta=+14^{\\circ}32'27\\farcs8$ (J2000). The object was first reported to the TNS by ATLAS on 2018 November 6, and first detected by ATLAS on 2018 September 25 at $o=18.36\\,$mag \\citep{Tonry2018_ZTF18acaimrb}.\nThe last ZTF non-detection prior to discovery was on 2018 September 27 at $r>20.12\\,$mag, and the last ATLAS non-detection was on 2018 September 24 at $o> 18.52\\,$mag.\nThe transient was classified as a SN Type Ic-BL by \\citet{Dahiwale2020_ZTF18acaimrb} based on a spectrum obtained on 2018 December 1 with Keck-I LRIS. In Figure \\ref{fig:spectra1} we plot this spectrum along with its SNID match to the Type Ic-BL SN\\,2002ap.\nSN\\,2018etk exploded in a galaxy with unknown redshift. We measure a redshift of $z=0.048$ from star-forming emission lines in the Keck spectrum. \n\n\\subsection{SN 2018jex}\nOur first ZTF photometry of SN\\,2018jex (ZTF18acpeekw) was\nobtained on 2018 November 16 (MJD 58438.56) with\nthe P48. This first detection was in the $r$ band, with a host-subtracted magnitude of $20.07\\pm0.29$\\,mag, at $\\alpha$=11$^{\\rm h}$54$^{\\rm m}$13.87$^{\\rm s}$,\n$\\delta=+20^{\\circ}44'02\\farcs4$ (J2000). The object was reported to the TNS by AMPEL on November 28 \\citep{Nordin2018_ZTF18acpeekw}.\nThe last ZTF last non-detection prior to ZTF discovery was on 2018 November 16 at $r>19.85\\,$mag.\nThe transient was classified as a SN Type Ic-BL based on a spectrum obtained on 2018 December 4 with Keck-I LRIS. In Figure~\\ref{fig:spectra1} we show this spectrum plotted against the SNID template of the Type Ic-BL SN\\,1997ef.\nAT2018jex exploded in a galaxy with unknown redshift. We measure a redshift of $z=0.094$ from star-forming emission lines in the Keck spectrum.\n\n\\subsection{SN 2019hsx}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog  \\citep{Neumann2022}.\n\n\\subsection{SN 2019xcc}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}.\n\n\\subsection{SN 2020jqm}\nOur first ZTF photometry of SN\\,2020jqm (ZTF20aazkjfv) was\nobtained on 2020 May 11 (MJD 58980.27) with\nthe P48. This first detection was in the $r$ band, with a host-subtracted magnitude of $19.42\\pm0.13$\\,mag, at $\\alpha$=13$^{\\rm h}$49$^{\\rm m}$18.57$^{\\rm s}$,\n$\\delta=-03^{\\circ}46'10\\farcs4$ (J2000). The object was reported to the TNS by ALeRCE on May 11 \\citep{Forster2020_ZTF20aazkjfv}.\nThe last ZTF  non-detection prior to ZTF discovery was on 2020 May 08 at $g>17.63\\,$mag.\nThe transient was classified as a SN Type  Ic-BL based on a spectrum obtained on 2020 May 26 with the SEDM \\citep{Dahiwale2020_ZTF20aazkjfv}.\nSN\\,2020jqm exploded in a galaxy with unknown redshift. We measure a redshift of $z = 0.037$ from host-galaxy emission lines in a NOT ALFOSC spectrum obtained on 2020 June 6. We plot the ALFOSC spectrum along with its SNID match to the Type Ic-BL SN\\,1998bw in Figure~\\ref{fig:spectra1}.\n\n\\subsection{SN 2020lao}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra1}. We note that this SN was also reported in the recently released ASAS-SN bright SN catalog  \\citep{Neumann2022}.\n\n\\subsection{SN 2020rph}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2020tkx}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2021xv}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\\subsection{SN 2021aug}\nOur first ZTF photometry of SN\\,2021aug (ZTF21aafnunh) was\nobtained on 2021 January 18  (MJD 59232.11) with the P48. This first detection was in the $g$ band, with a host-subtracted magnitude of $18.73\\pm0.08$\\,mag, at $\\alpha$=01$^{\\rm h}$14$^{\\rm m}$04.81$^{\\rm s}$,\n$\\delta=+19^{\\circ}25'04\\farcs7$ (J2000).\nThe last ZTF non-detection prior to ZTF discovery was on 2021 January 16 at $g>20.12\\,$mag. The transient was publicly reported to the TNS by ALeRCE on 2021 January 18 \\citep{MunozArancibia2021}, and classified as a SN Type Ic-BL\nbased on a spectrum obtained on 2021 February 09 with the SEDM \\citep{Dahiwale2021_ZTF21aafnunh}.\nSN\\,2020jqm exploded in a galaxy with unknown redshift. We measure a redshift of $z= 0.041$  from star-forming emission lines in a P200 DBSP spectrum obtained on 2021 February 08. This spectrum is shown in Figure~\\ref{fig:spectra2} along with its template match to the Type Ic-BL SN\\,1997ef.\n\n\\subsection{SN 2021epp}\nOur first ZTF photometry of SN\\,2021epp (ZTF21aaocrlm) was\nobtained on 2021 March 5 (MJD 59278.19) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $19.61\\pm0.15$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=08$^{\\rm h}$10$^{\\rm m}$55.27$^{\\rm s}$, $\\delta=-06^{\\circ}02'49\\farcs3$ (J2000). \nThe transient was publicly reported to the TNS by ALeRCE on 2021 March 5 \\citep{MunozArancibia20210305}, and classified as a SN Type Ic-BL based on a spectrum obtained on 2021 March 13 by ePESSTO+ with the ESO Faint Object Spectrograph and Camera \\citep{Kankare2021}.\nThe last ZTF non-detection prior to discovery was on 2021 March 2 at $r>19.72\\,$mag.\nIn Figure~\\ref{fig:spectra2} we show the classification spectrum \nplotted against the SNID template of the Type Ic-BL SN\\,2002ap.\nSN\\,2021epp exploded in a galaxy with known redshift of $z = 0.038$.\n\n\\subsection{SN 2021htb}\nOur first ZTF photometry of SN\\,2021htb (ZTF21aardvol) was\nobtained on 2021 March 31 (MJD 59304.164) with the P48. This first ZTF detection was in the $r$ band, with a host-subtracted magnitude of $20.13\\pm0.21$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=07$^{\\rm h}$45$^{\\rm m}$31.19$^{\\rm s}$,\n$\\delta=46^{\\circ}40'01\\farcs4$ (J2000).\nThe transient was publicly reported to the TNS by SGLF on 2021 April 2 \\citep{Poidevin_2021}.\nThe last ZTF non-detection prior to ZTF discovery was on 2021 March 2, at $r>19.88\\,$mag.\nIn Figure~\\ref{fig:spectra2} we show a P200 DBSP spectrum taken on 2021 April 09   plotted against the SNID template of the Type Ic-BL SN\\,2002ap. SN\\, 2021htb exploded in a SDSS galaxy with redshift $z= 0.035$.\n\n\\subsection{SN 2021hyz}\nOur first ZTF photometry of SN\\,2021hyz (ZTF21aartgiv) was\nobtained on 2021 April 03  (MJD 59307.155) with the P48. This first ZTF detection was in the $g$ band, with a host-subtracted magnitude of $20.29\\pm0.17$\\,mag (Figure~\\ref{fig:opt-lc-mag}), at $\\alpha$=09$^{\\rm h}$27$^{\\rm m}$36.51$^{\\rm s}$,\n$\\delta=04^{\\circ}27'11\\farcs$ (J2000).\nThe transient was publicly reported to the TNS by ALeRCE on 2021 April 3 \\citep{Forster_2021}.\nThe last ZTF non-detection prior to ZTF discovery was on 2021 April 1, at $g>19.15\\,$mag. In Figure~\\ref{fig:spectra2} we show a P60 SEDM spectrum taken on 2021 April 30  plotted against the SNID template of the Type Ic-BL SN\\,1997ef.\nSN\\,2021hyz exploded in a galaxy with redshift $z= 0.046$.\n\n\\subsection{SN 2021ywf}\nWe refer the reader to \\citet{Anand2022} for details about this SN Ic-BL.  Its P48 light curves and the spectrum used for classification are shown in Figures \\ref{fig:opt-lc-mag} and \\ref{fig:spectra2}.\n\n\n\\begin{footnotesize}\n\\begin{longtable*}{lcccllllll}\n\\caption{Optical properties of the 16 SNe Ic-BL in our sample. We list the SN name; the MJD of maximum light in $r$ band; the absolute magnitude at $r$-band peak; the absolute magnitude at $g$-band peak; the explosion time estimated as days since $r$-band maximum; the estimated nickel mass; the characteristic timescale of the bolometric light curve; the photospheric velocity; the ejecta mass; and the kinetic energy of the explosion.  See Sections \\ref{sec:vphot} and \\ref{sec:optical_properties} for discussion. \\label{tab:opt_data}}\\\\\n\\toprule\n\\toprule\nSN & T$_{r,\\rm max}$ & M$^{\\rm peak}_{r}$ & M$^{\\rm peak}_{g}$  & ${\\rm T}_{\\rm exp}$-T$_{r,\\rm max}$  & $M_{\\rm Ni}$   & $\\tau_{m}$  & v$_{\\rm ph}$($^{a}$) & $M_{\\rm ej}$ & $E_{\\rm k}$ \\\\\n &  (MJD) & (AB mag)  & (AB mag) & (d) & (M$_{\\odot}$) & (d) & ($10^4$\\,km\/s) & (M$_{\\odot}$) & ($10^{51}$erg) \\\\\n\\midrule[0.5pt]\n2018etk & 58337.40 & $-18.31\\pm0.03$ & $-18.30\\pm0.02$ & $-9\\pm1$ & $0.13_{-0.02}^{+0.01}$ & $5.0_{-2}^{+2}$ &  $2.6\\pm0.2$ (5)\n& $0.7\\pm0.5$ & $3\\pm2$ \\\\\n2018hom & 58426.31 & $-19.30\\pm0.11$ & $-18.91\\pm0.01$ & $-9.3_{-0.4}^{+0.7}$ & $0.4\\pm0.1$ & $6.9\\pm0.2$ & $1.7\\pm0.2$ (27) & $> 0.7$ & $>1$ \\\\\n2018hxo & 58403.76 & $-18.68\\pm0.06$ & $-18.4\\pm0.1$ & $-28.6_{-0.3}^{+0.2}$ & $0.4\\pm0.2$ & $6\\pm2$ & $0.6\\pm0.1$ (48)\n& $>0.1$ & $>0.02$ \\\\\n2018jex & 58457.01 & $-19.06\\pm0.02$ & $-18.61\\pm0.04$ & $-18.49\\pm0.04$ & $0.53_{-0.06}^{+0.07}$ & $13_{-3}^{+2}$ & $1.8\\pm0.3$ (8)\n& $3\\pm1$ & $7\\pm3$ \\\\\n2019hsx & 58647.07 & $-17.08\\pm0.02$ & $-16.14\\pm0.04$ & $-15.6_{-0.5}^{+0.4}$ & $0.07_{-0.01}^{+0.01}$ & $12\\pm1$ & $1.0\\pm0.2$ (-0.2)\n& $1.6\\pm0.4$ & $1.0\\pm0.5$ \\\\\n2019xcc & 58844.59 & $-16.58\\pm0.06$ & $-15.6\\pm0.2$ & $-11\\pm2$ & $0.04\\pm0.01$ & $5.0_{-0.9}^{+1.4}$ & $2.4\\pm0.2$ (6)\n& $0.7\\pm0.3$ & $2\\pm1$ \\\\\n2020jqm & 58996.21 & $-18.26\\pm0.02$ & $-17.39\\pm0.04$ & $-17\\pm1$ & $0.29_{-0.04}^{+0.05}$ & $18\\pm2$ & $1.3\\pm0.3$ (-0.5) \n& $5\\pm1$ & $5\\pm3$ \\\\\n2020lao & 59003.92 & $-18.66\\pm0.02$ & $-18.55\\pm0.02$ & $-11\\pm1$ & $0.23\\pm0.01$ & $7.7\\pm0.2$ & $1.8\\pm0.2$ (9)\n& $1.2\\pm0.2$ & $2.5\\pm0.7$ \\\\\n2020rph & 59092.34 & $-17.48\\pm0.02$ & $-16.94\\pm0.03$ & $-19.88\\pm0.02$  & $0.07\\pm0.01$ & $17.23_{-0.9}^{+1.2}$ & $1.2\\pm0.5$ (-1)\n& $4\\pm2$ & $3\\pm3$ \\\\\n2020tkx & 59116.50 & $-18.49\\pm0.05$ & $-18.19\\pm0.03$ & $-13\\pm4$  & $0.22\\pm0.01$ & $10.9_{-0.8}^{+0.7}$ & $1.32\\pm0.09$ (53)\n& $> 1.5$ & $> 1.5$ \\\\\n2021xv & 59235.56 & $-18.92\\pm0.07$ & $-18.99\\pm0.05$ & $-12.8_{-0.3}^{+0.2}$ & $0.30_{-0.02}^{+0.01}$ & $7.7_{-0.5}^{+0.7}$ & $1.3\\pm0.1$ (3)\n& $0.9\\pm0.1$ & $1.0\\pm0.2$ \\\\\n2021aug & 59251.98 & $-19.42\\pm0.01$ & $-19.32\\pm0.06$ & $-24\\pm2$ & $0.7\\pm0.1$ & $17\\pm7$ & $0.8\\pm0.3$ (1) \n& $3\\pm2$ & $1\\pm1$ \\\\\n2021epp & 59291.83 & $-17.49\\pm0.03$ & $-17.12\\pm0.09$ & $-15\\pm1$ & $0.12\\pm0.02$ & $17_{-3}^{+4}$ & $1.4\\pm0.5$ (-4)\n& $5\\pm2$ & $6\\pm5$ \\\\\n2021htb & 59321.56 & $-16.55\\pm0.03$ & $-15.66\\pm0.07$ & $-19.38\\pm0.02$  & $0.04\\pm0.01$ & $13\\pm2$ & $1.0\\pm0.5$ (-6)\n& $1.8\\pm0.9$ & $1\\pm1$ \\\\\n2021hyz & 59319.10 & $-18.83\\pm0.05$ & $-18.81\\pm0.01$ & $-12.9\\pm0.9$  & $0.29_{-0.02}^{+0.01}$ & $7.7_{-0.4}^{+0.5}$ & $2.3\\pm0.3$ (16) & $> 1.3$ & $>4$ \\\\\n2021ywf & 59478.64 & $-17.10\\pm0.05$ & $-16.5\\pm0.1$ & $-10.7\\pm0.5$  & $0.06\\pm0.01$ & $8.9\\pm0.8$ & $1.2\\pm0.1$ (0.5) \n& $1.1\\pm0.2$ & $0.9\\pm0.3$ \\\\\n\\bottomrule\n\\multicolumn{9}{l}{$^{a}$ Rest-frame phase days of the spectrum that was used to measure the velocity.}\\\\\n\\end{longtable*}\n\\end{footnotesize}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[height=8cm]{CorsiFig5_speed.png}\n    \\caption{Photospheric velocities of the ZTF SNe in our sample (black) plotted as a function of (rest frame) time since explosion (see Table \\ref{tab:opt_data}).  Velocities are measured using Fe II (5169 $\\AA$); velocities quoted refer to 84\\% confidence and are measured relative to the Ic template velocity using the open source software \\texttt{SESNspectraLib} \\citep{Liu2016,Modjaz2016}. We compare our results with photospheric velocities derived from spectroscopic modeling for a number of Ib\/c SNe. Red symbols represent GRB-SNe \\citep{Iwamoto1998,Mazzali2003,Mazzali2006b}; magenta is used for XRF\/X-ray transients-SNe \\citep{Mazzali2006a,Pian2006,Modjaz2009}; blue represents SNe Ic-BL \\citep{Mazzali2000,Mazzali2002}; green is used for the ``normal'' Type Ic SN 1994I \\citep{Sauer2006}. Finally, for comparison we also plot the photospheric velocities for the SNe Ic-BL in the \\citet{Corsi2016} sample as measured by \\citet{Taddia2019} (see their Tables 2 and A1; yellow crosses). Errors on the times since explosion account for the uncertainties on T$_{\\rm exp}$ as reported in Table \\ref{tab:opt_data}. \\label{fig:velocities}}\n\\end{figure*}\n\n\n\\section{Multi-wavelength analysis}\n\\label{sec:modeling}\n\\subsection{Photospheric velocities}\n\\label{sec:vphot}\nWe confirm the SN Type Ic-BL classification of each object in our sample by measuring the photospheric velocities (${\\rm v_{ph}}$).\nSNe Ic-BL are characterized by high expansion velocities evident in the broadness of their spectral lines. A good proxy for the photospheric velocity is that derived from the maximum absorption position of the Fe II \\citep[$\\lambda 5169$; e.g.,][]{Modjaz2016}. We caution, however, that estimating this velocity is not easy given the strong line blending. We first pre-processed one high-quality spectrum per object using the IDL routine \\texttt{WOMBAT}, then smoothed the spectrum using the IDL routine \\texttt{SNspecFFTsmooth} \\citep{Liu2016}, and finally ran \\texttt{SESNSpectraLib} \\citep{Liu2016,Modjaz2016} to obtain final velocity estimates. \n\nIn  Figure~\\ref{fig:velocities} we show a comparison of the photospheric velocities estimated for the SNe in our sample with those derived from spectroscopic modeling for a number of SNe Ib\/c. The velocities measured for our sample are compatible, within the measurement errors, with what was observed for the PTF\/iPTF samples.\nMeasured values for the photospheric velocities with the corresponding rest-frame phase in days since maximum $r$-band light of the spectra that were used to measure them are also reported in  Table~\\ref{tab:opt_data}.  \n\nWe note that the spectra used to estimate the photospheric velocities of SN\\,2019xcc, SN\\,2020lao, and SN\\,2020jqm are different from those used for the classification of those events as SNe Ic-BL (see Section \\ref{section:sample} and Figure \\ref{fig:spectra1}). This is because for spectral classification we prefer later-time but higher-resolution spectra, while for velocity measurements we prefer earlier-time spectra even if taken with the lower-resolution SEDM. \n\n\\subsection{Bolometric light curve analysis}\n\\label{sec:optical_properties}\nIn our analysis we correct all ZTF photometry for Galactic extinction, using the Milky Way (MW) color excess $E(B-V)_{\\mathrm{MW}}$ toward the position of the SNe.\nThese are all obtained from \\cite{Schlafly2011}. All reddening corrections are applied using the \\cite{Cardelli1989} extinction law with $R_V=3.1$. \nAfter correcting for Milky Way extinction, we interpolate our P48 forced-photometry light curves \nusing a Gaussian Process via the {\\tt GEORGE}\\footnote{\\href{https:\/\/george.readthedocs.io}{https:\/\/george.readthedocs.io}} package with a stationary Matern32 kernel and the analytic functions of \\citet{Bazin2009} as mean for the flux form.\nAs shown in Figure~\\ref{fig:opt-lc-mag}, the colour evolution of the SNe  in our sample are not too dissimilar with one another, which implies that the amount of additional host extinction is small. Hence, we set the host extinction to zero.\nNext, we derive bolometric light curves calculating bolometric corrections from the $g$- and $r$-band data following the empirical relations by \\cite{Lyman2014,Lyman2016}. For SN\\,2018hxo, since there is only one $g$-band detection, we assume a constant bolometric correction to estimate its bolometric light curve. These bolometric light curves are shown in the bottom panel of Figure \\ref{fig:opt-lc-flux}. \n\nWe estimate the explosion time ${\\rm T}_{\\rm exp}$ of the SNe in our sample as follows.  For SN\\,2021aug, we fit the early ZTF light curve data following the method presented in \\citet{Miller2020}, where we fix the power-law index of the rising early-time temporal evolution to $\\alpha=2$, and derive an estimate of the explosion time from the fit. For most of the other SNe in our sample, the ZTF $r$- and $g$-band light curves lack enough early-time data to determine an estimate of the explosion time following the formulation of \\citet{Miller2020}. For all these SNe we instead set the explosion time to the mid-point between the last non-detection prior to discovery, and the first detection. \nResults on ${\\rm T}_{\\rm exp}$ are reported in Table  \\ref{tab:opt_data}.\n\nWe fit the bolometric light curves around peak ($-20$ to 60 rest-frame days relative to peak) to a model using the Arnett formalism \\citep{Arnett1982}, with the nickel mass ($M_{\\rm Ni}$) and characteristic time scale $\\tau_m$ as free parameters \\citep[see e.g. Equation A1 in][]{Valenti2008}. The derived values of $M_{\\rm Ni}$ (Table \\ref{tab:opt_data}) have a median of $\\approx 0.22$\\,M$_{\\odot}$, compatible with the median value found for SNe Ic-BL in the PTF sample by \\citet{Taddia2019}, somewhat lower than for SN\\,1998bw for which the estimated nickel mass values are in the range $(0.4-0.7)$\\,M$_{\\odot}$, but comparable to the $M_{\\rm Ni}\\approx 0.19-0.25$\\,M$_{\\odot}$ estimated for\nSN\\,2009bb \\citep[see e.g.,][]{Lyman2016,Afsariardchi2021}. We note that events such as SN\\,2019xcc and SN\\,2021htb have relatively low values of $M_{\\rm Ni}$, which are however compatible with the range of $0.02-0.05$\\,M$_{\\odot }$ expected for the nickel mass of magnetar-powered SNe Ic-BL \\citep{Nishimura2015,Chen2017,Suwa2015}. \n\nNext, from the measured  characteristic timescale $\\tau_m$ of the bolometric light curve, and the photospheric velocities estimated via spectral fitting  (see previous Section) we derive the ejecta mass ($M_{\\rm ej}$) and the kinetic energy ($E_{\\rm k}$) via the following relations \\citep[see e.g. Equations 1 and 2 in][]{Lyman2016}:\n\\begin{eqnarray}\n    \\tau^2_m {\\rm v_{ph, max}}=\\frac{2\\kappa}{13.8 c} M_{\\rm ej}~~~ & \n    ~~~{\\rm v}^2_{\\rm ph, max}=\\frac{5}{3}\\frac{2 E_{\\rm k}}{M_{\\rm ej}},\\label{eq:ejecta}\n\\end{eqnarray}\nwhere we assume a constant opacity of $\\kappa = 0.07$\\,g\\,cm$^{-2}$. \n\nWe note that to derive $M_{\\rm ej}$ and $E_{\\rm k}$ as described above we assume the photospheric velocity evolution is negligible within 15 days relative to peak epoch, and use the spectral velocities measured within this time frame to estimate ${\n\\rm v}_{\\rm ph, max}$ in Equation \\ref{eq:ejecta}. However, there are four objects in our sample (SN\\,2018hom, SN\\,2018hxo, SN\\,2020tkx, and SN\\,2021hyz) for which the spectroscopic analysis constrained the photospheric velocity only after day 15 relative to peak epoch. For these events, we only provide lower limits on the ejecta mass and kinetic energy (see Table \\ref{tab:opt_data}). \n \nConsidering only the SNe in our sample for which we are able to measure the photospheric velocity within 15\\,d since peak epoch, we derive median values for the ejecta masses and kinetic energies of  $1.7$\\,M$_{\\odot}$ and $2.2\\times10^{51}$\\,erg, respectively. These are both a factor of $\\approx 2$ smaller than the median values derived for the PTF\/iPTF sample of SNe Ic-BL \\citep{Taddia2019}. This could be due to either an intrinsic effect, or to uncertainties on the measured photospheric velocities. In fact, we note that the photospehric velocity is expected to decrease very quickly after maximum light (see e.g. Figure \\ref{fig:velocities}). Since the photospheric velocity in Equation (\\ref{eq:ejecta}) of the Arnett formulation is the one at peak, our estimates of ${\\rm v}_{\\rm ph,max}$ could easily underestimate that velocity by a factor of $\\approx 2$ for many of the SNe in our sample. This would in turn yield an underestimate of $M_{\\rm ej}$ by a factor of $\\approx 2$ (though the kinetic energy would be reduced by a larger factor). A more in-depth analysis of these trends and uncertainties will be presented in \\citet{Gokul2022}. \n \n\\subsection{Search for gamma-rays}\nBased on the explosion dates derived for each object in Section \\ref{sec:optical_properties} (Table \\ref{tab:opt_data}), we searched for potential GRB coincidences in several online archives. No potential counterparts were identified in both spatial and temporal coincidence with either the Burst Alert Telescope (BAT; \\citealt{Barthelmy2005}) on the \\textit{Neil Gehrels\nSwift Observatory}\\footnote{See \\url{https:\/\/swift.gsfc.nasa.gov\/results\/batgrbcat}.} or the Gamma-ray Burst Monitor  \\citep[GBM;][]{Meegan2009} on \\textit{Fermi}\\footnote{See \\url{https:\/\/heasarc.gsfc.nasa.gov\/W3Browse\/fermi\/fermigbrst.html}.}.\n\nSeveral candidate counterparts were found with \\textit{temporal} coincidence in the online catalog from the KONUS instrument on the \\textit{Wind} satellite (SN\\,2018etk, SN\\,2018hom, SN\\,2019xcc, SN\\,2020jqm, SN\\,2020lao, SN\\,2020tkx, SN\\,2021aug). However, given the relatively imprecise explosion date constraints for several of the events in our sample (see Table \\ref{tab:opt_data}), and the coarse localization information from the KONUS instrument, we cannot firmly associate any of these GRBs with the SNe Ic-BL. In fact, given the rate of GRB detections by KONUS ($\\sim 0.42$\\,d$^{-1}$) and the time window over which we searched for counterparts ($30$\\,d in total; derived from the explosion date constraints), the observed number of coincidences (13) is consistent with random fluctuations. Finally, none of the possible coincidences were identified in events with explosion date constraints more precise than 1\\,d. \n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=18cm]{CorsiFig6_GokulXray.png}\n    \\caption{\\textit{Swift}\/XRT upper-limits and detections (downward pointing triangles and filled circles with error bars, respectively) obtained for 9 of the 16 SNe Ic-BL in our sample. We plot the observed X-ray luminosity as a function of time since explosion. We compare these observations with the X-ray light curves of the low-luminosity GRB\\,980425 \\citep[red stars;][]{Kouveliotou2004}, GRB\\,060218 \\citep[red squares;][]{Campana2006},  GRB\\,100316D \\citep[red crosses;][]{Margutti2013}, and with the relativistic iPTF17cw \\citep[blue cross;][]{Corsi2017}. Dotted red lines connect the observed data points (some of which at early and late times are not shown in the plot) for these three low-luminosity GRBs. We also plot the observed  X-ray luminosity predicted by off-axis GRB models \\citep[black, green, and orange lines;][]{vanEerten2011, VanEerten2012}. We assume  $\\epsilon_B=\\epsilon_e=0.1$, a constant density ISM in the range $n = 1-10\\,{\\rm cm}^{-3}$, a top-hat jet of opening angle $\\theta_j=0.2$, and various observer's angles $\\theta_{\\rm obs}=(2.5-3)\\theta_j$.}\n    \\label{fig:X-raymodel}\n\\end{figure*}\n\n\\subsection{X-ray constraints}\n\\label{sec:X-raymodel}\nSeven of the 9 SNe Ic-BL observed with \\textit{Swift}-XRT did not result in a significant detection. In Table \\ref{tab:x-ray} we report the derived 90\\% confidence flux upper limits in the 0.3--10\\,keV band after correcting for Galactic absorption \\citep{Willingale+2013}. \n\nWhile observations of SN\\,2021ywf resulted in a  $\\approx 3.2\\sigma$  detection significance (Gaussian equivalent), the limited number of photons (8) precluded a meaningful spectral fit. Thus, a $\\Gamma = 2$ power-law spectrum was adopted for the flux conversion for this source as well. We note that because of the relatively poor spatial resolution of the \\textit{Swift}-XRT (estimated positional uncertainty of 11.7\\arcsec~radius  at 90\\% confidence), we cannot entirely rule out unrelated emission from the host galaxy of SN 2021ywf (e.g., AGN, X-ray binaries, diffuse host emission; see Figure \\ref{fig:hosts-det} for the host galaxy).\n\nFor SN\\,2019hsx we detected enough photons to perform a spectral fit for count rate to flux conversion. The spectrum is found to be relatively soft, with a best-fit power-law index of $\\Gamma = 3.9^{+3.0}_{-2.1}$. \nOur \\textit{Swift} observations of SN\\,2019hsx do not show significant evidence for variability of the source X-ray flux over the timescales of our follow up. While the lack of temporal variability is not particularly constraining given the low signal-to-noise ratio in individual epochs, we caution that also in this case the relatively poor spatial resolution of the \\textit{Swift}-XRT (7.4\\arcsec~radius position uncertainty at 90\\% confidence) implies that unrelated emission from the host galaxy cannot be excluded.\n\nThe constraints derived from \nthe \\textit{Swift}-XRT observations can be compared with the\nX-ray light curves of low-luminosity GRBs, or models of GRB afterglows \nobserved slightly off-axis. For the latter, we use\nthe numerical model by \\citet{vanEerten2011, VanEerten2012}. We assume equal energies in the electrons and magnetic fields ($\\epsilon_B= \\epsilon_e=0.1$), and an interstellar medium (ISM) of density $n = 1-10$\\,cm$^{-3}$. We note that a constant density ISM (rather than a wind profile) has been shown to fit the majority of GRB afterglow light curves, implying that most GRB progenitors might have relatively small wind termination-shock radii \\citep{Schulze2011}. We generate the model light curves for a nominal redshift of $z=0.05$ and then convert the predicted flux densities into X-ray luminosities by integrating over the 0.3-10\\,keV energy range and  neglecting the small redshift corrections. We plot the model light curves in Figure \\ref{fig:X-raymodel}, for various energies, different power-law indices $p$ of the electron energy distribution, and various off-axis angles (relative to a jet opening angle, set to $\\theta_j=0.2$). In the same Figure we also plot  the X-ray light curves of low-luminosity GRBs for comparison (neglecting redshift corrections). \nAs evident from this Figure, our \\textit{Swift}\/XRT upper limits (downward-pointing triangles) \nexclude X-ray afterglows associated with higher-energy GRBs observed slightly off-axis.  However, X-ray emission as faint as the afterglow of\nthe low-luminosity GRB\\,980425 cannot be excluded. As we discuss in the next Section, radio data collected with the VLA enable us to exclude GRB\\,980425\/SN\\,1998bw-like emission for most of the SNe in our sample. \n\nWe note that our two X-ray detections of SN\\,2019hsx and SN\\,2021ywf are consistent with several GRB off-axis light curve models and, in the case of SN\\,2021ywf, also with GRB\\,980425-like emission within the large errors. However, for this interpretation of their X-ray emission to be compatible with our radio observations (see Table \\ref{tab:data}), one needs to invoke a flattening of the radio-to-X-ray spectrum, similar to what has been invoked for other stripped-envelope SNe in the context of cosmic-ray dominated shocks \\citep{Ellison2000, Chevalier2006}.\n\n\\subsection{Radio constraints}\n\\label{sec:radioanalysis}\nAs evident from Table \\ref{tab:data}, we have obtained at least one radio detection for 11 of the 16 SNe in our sample.  None of these 11 radio sources were found to be coincident with known radio sources in the VLA FIRST \\citep{Becker1995} catalog (using a search radius of 30\\arcsec~around the optical SN positions). This is not surprising since the FIRST survey had a typical RMS sensitivity of $\\approx 0.15$\\,mJy at 1.4\\,GHz, much shallower than the deep VLA follow-up observations carried out within this follow-up program. We also checked the quick look images from the VLA Sky Survey (VLASS) which reach a typical RMS sensitivity of $\\approx 0.12$\\,mJy at 3\\,GHz \\citep{Hernandez2018,Law2018}. We could find images for all but one (SN\\,2021epp) of the fields containing the 16 SNe BL-Ic in our sample. The VLASS images did not provide any radio detection at the locations of the SNe in our sample.\n\nFive of the 11 SNe Ic-BL with radio detections are associated with extended or marginally resolved radio emission. Two other radio-detected events (SN\\,2020rph and SN\\,2021hyz) appear point-like in our images, but show no evidence for significant variability of the detected radio flux densities over the timescales of our observations. Thus, for a total of 7 out of 11 SNe Ic-BL with radio detections, we consider the measured flux densities as upper-limits corresponding to the brightness of their host galaxies, similarly to what was done in e.g. \\citet{Soderberg2006} and \\citet{Corsi2016}. The remaining 4 SNe Ic-BL with radio detections are compatible with point sources (SN\\,2018hom, SN\\,2020jqm, SN\\,2020tkx, and SN\\,2021ywf), and all but one (SN\\,2018hom) had more than one observation in the radio via which we were able to establish the presence of substantial variability of the radio flux density. Hereafter we consider these 4 detections as genuine radio SN counterparts, though we stress that with only one observation of SN\\,2018hom we cannot rule out a contribution from host galaxy emission, especially given that the radio follow up of this event was carried out with the VLA in its most compact (D) configuration with poorer angular resolution.\n\nIn summary, our radio follow-up campaign of 16 SNe Ic-BL resulted in 4 radio counterpart detections, and 12 deep upper-limits on associated radio counterparts.\n\n\\begin{figure*}\n    \\begin{center}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF18abklarx_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF19adaiomg_host_contour.png}\n    \\includegraphics[width=6.cm,height=5.6cm]{CorsiFig7_ZTF20abswdbg_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aadatfg_host_contour.png}\n    \\includegraphics[width=6cm,height=5.6cm]{CorsiFig7_ZTF21aaocrlm_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aardvol_host_contour.png}\n    \\includegraphics[width=6cm]{CorsiFig7_ZTF21aartgiv_host_contour.png}\n    \\caption{PanSTARRS-1 \\citep{Flewelling2020} reference $r$-band images of the fields of the SNe in our sample for which host galaxy light dominates the radio emission. Contours in magenta are 30\\%, 50\\%, and 90\\% of the radio peak flux reported in Table \\ref{tab:data} for the first radio detection of each field. The blue circles centered on the optical SN positions (not shown in the images) have sizes of 2\\arcsec  \\citep[comparable to the ZTF PSF at average seeing;][]{Bellm2019}. The red-dotted circles enclose the region in which we search for radio counterparts (radii equal to the nominal FWHM of the VLA synthesized beams;  Table \\ref{tab:data}). The sizes of the actual VLA synthesized beams  are shown as filled magenta ellipses. The red dots mark the locations of the radio peak fluxes measured in the radio search areas. \\label{fig:hosts}}\n    \\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n    \\begin{center}\n    \\hbox{\n    \\includegraphics[height=8.3cm]{CorsiFig8_ZTF18acbwxcc_contour.png}\n    \\hspace{0.2cm}\n    \\includegraphics[height=8.3cm]{CorsiFig8_ZTF20aazkjfv_contour.png}}\n    \\hbox{\n    \\includegraphics[height=7.8cm]{CorsiFig8_ZTF20abzoeiw_contour.png}\n    \\includegraphics[height=7.8cm]{CorsiFig8_ZTF21acbnfos_contour.png}}\n    \\caption{Same as Figure \\ref{fig:hosts} but for the fields containing the SNe in our sample for which we detected a SN radio counterpart.  We stress that with only one observation of SN\\,2018hom we cannot rule out a contribution from host galaxy emission, especially given that the radio follow up of this event was carried out with the VLA in its D configuration. \\label{fig:hosts-det}}\n    \\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=18cm]{CorsiFig9_RadioLight.png}\n    \\caption{Radio ($\\approx 6$\\,GHz) observations of the 16 SNe Ic-BL in our sample (filled circles and downward pointing triangles in shades of pink, purple, and blue). Upper-limits associated with non-detections ($3\\sigma$ or brightness of the host galaxy at the optical location of the SN) are plotted with downward-pointing triangles; detections are plotted with filled circles. We compare these observations with the radio light curves of GRB-SNe (red); of relativistic-to-mildly relativistic SNe Ic-BL discovered independently of a $\\gamma$-ray trigger (cyan); and with PTF11qcj \\citep{Corsi2014}, an example of a radio-loud non-relativistic and CSM-interacting SN Ic-BL (yellow). As evident from this Figure, our observations exclude SN\\,1998bw-like radio emission for all but one (SN\\,2021epp) of the events in our sample. This doubles the sample of SNe Ic-BL for which radio emission observationally similar to SN\\,1998bw was previously excluded \\citep{Corsi2016}, bringing the upper limit on the fraction of SNe compatible with SN\\,1998bw down to $< 19\\%$ \\citep[compared to $< 41\\%$ previously reported in][]{Corsi2016}.\n    For 10 of the 16 SNe presented here we also exclude relativistic ejecta with radio luminosity densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d, similar to SNe associated with low-luminosity GRBs such as SN\\,1998bw \\citep{Kulkarni1998}, SN\\,2003lw \\citep{Soderberg2004}, SN\\,2010dh \\citep{Margutti2013}, or to the relativistic SN\\,2009bb \\citep{Soderberg2010} and iPTF17cw \\citep{Corsi2017}.  None of our observations exclude radio emission similar to that of SN\\,2006aj. }\n    \\label{fig:radiolight}\n\\end{figure*}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.6cm]{CorsiFig10_RadioSpeed.png}\n    \\caption{Properties of the radio-emitting ejecta of the SNe in our sample for which we detect a radio counterpart (magenta dots), compared with those of GRB-SNe (red stars) and of relativistic-to-mildly\nrelativistic SNe Ic-BL discovered independently of a $\\gamma$-ray trigger (cyan stars). As evident from this Figure, only SN\\,2018hom is compatible with an ejecta speed $\\gtrsim 0.3c$, though with the caveat that we only have one radio observation for this SN. None of the other ZTF SNe Ic-BL in our sample shows evidence for ejecta faster than $0.3c$. We also note that SN\\,2020jqm lies in the region of the parameter space occupied by radio-loud CSM-interacting SNe similar to PTF11qcj. See Section \\ref{sec:radio_properties} for discussion.}\n    \\label{fig:radiospeed}\n\\end{figure}\n\n\\subsubsection{Fraction of SN\\,1998bw-like SNe Ic-BL}\nThe local rate of SNe Ic-BL is estimated to be $\\approx 5\\%$ of the core-collapse SN rate \\citep{Li2011,Shivvers2017,Perley2020b} or $\\approx 5\\times10^3$\\,Gpc$^{-3}$\\,yr$^{-1}$ assuming a core-collapse SN rate of $\\approx 10^5$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep{Perley2020b}.  Observationally, we know that cosmological long GRBs are characterized by ultra-relativistic jets observed on-axis, and have an \\textit{intrinsic} (corrected for beaming angle) local volumetric rate of $79^{+57}_{-33}$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep[e.g.,][and references therein]{Ghirlanda2022}. Hence, only ${\\cal O}(1)\\%$ of SNe Ic-BL can make long GRBs. For low-luminosity GRBs, the \\textit{observed} local rate is affected by large errors, $230^{+490}_{-190}$\\,Gpc$^{-3}$\\,yr$^{-1}$ \\citep[see][and references therein]{Bromberg2011}, and their typical beaming angles are largely unconstrained. Hence, the question of what fraction of SNe Ic-BL can make low-luminosity GRBs remains to be answered. \n\nRadio observations of SNe Ic-BL are a powerful way to constrain this fraction independently of relativistic beaming effects that preclude observations of jets in X-rays and $\\gamma$-rays for off-axis observers. However, observational efforts aimed at constraining the fraction of SNe Ic-BL harboring low-luminosity GRBs independently of $\\gamma$-ray observations have long been challenged by the rarity of the SN Ic-BL optical detections (compared to other core-collapse events), coupled with the small number of these rare SNe for which the community has been able to collect deep radio follow-up observations within 1\\,yr since explosion \\citep[see e.g.,][]{Soderberg2006catalog}. Progress in this respect has been made since the advent of the PTF, and more generally with synoptic optical surveys that have greatly boosted the rate of stripped-envelope core-collapse SN discoveries \\citep[e.g.,][]{Shappee2014,Sand2018,Tonry2018}. \n\nIn our previous work \\citep{Corsi2016}, we presented one of the most extensive samples of SNe Ic-BL with deep VLA observations, largely composed of events detected by the PTF\/iPTF. Combining our sample with the SN Ic-BL\\,2002ap \\citep{GalYam2002,Mazzali2002} and SN\\,2002bl \\citep{Armstrong2002,Berger2003}, and the CSM-interacting SN Ic-BL\\,2007bg \\citep{Salas2013}, we had overall 16 SNe Ic-BL for which radio emission observationally similar to SN\\,1998bw was excluded, constraining the rate of SNe Ic-BL observationally similar to SN\\,1998bw to $< 6.61\/16\\approx 41\\%$, where we have used the fact that the Poisson 99.865\\% confidence (or $3\\sigma$ Gaussian equivalent for a single-sided distribution) upper-limit on zero SNe compatible with SN\\,1998bw is $\\approx 6.61$.\n\nWith the addition of the 16 ZTF SNe Ic-BL presented in this work, we now have doubled the sample of SNe Ic-BL with deep VLA observations presented in \\citet{Corsi2016}, providing evidence for additional 15 SNe Ic-BL (all but SN\\,2021epp; see Figure \\ref{fig:radiolight}) that are observationally different from SN\\,1998bw in the radio. Adding to our sample also SN\\,2018bvw \\citep{Ho2020_ZTF18aaqjovh}, AT\\,2018gep \\citep{Ho2019_ZTF18abukavn}, and SN\\,2020bvc \\citep{Ho2020_ZTF20aalxlis}, whose radio observations exclude SN\\,1998bw-like emission, we are now at 34 SNe Ic-BL that are observationally different from SN\\,1998bw. Hence, we can tighten our constraint on the fraction of 1998bw-like SNe Ic-BL to $< 6.61\/34\\approx 19\\%$ (99.865\\% confidence). This upper-limit implies that the \\textit{intrinsic} rate of 1998bw-like GRBs is $\\lesssim 950$\\,Gpc$^{-3}$\\,yr$^{-1}$. Combining this constraint with the rate of low-luminosity GRBs derived from their high-energy emission, we conclude that low-luminosity GRBs have inverse beaming factors $2\/\\theta^2\\lesssim 4^{+20}_{-3}$, corresponding to  jet half-opening angles $\\theta \\gtrsim 40^{+40}_{-24}$\\,deg. \n\nWe note that for 10 of the SNe in the sample presented here we also exclude relativistic ejecta with radio luminosity densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d, pointing to the fact that SNe Ic-BL similar to those associated with low-luminosity GRBs, such as SN\\,1998bw \\citep{Kulkarni1998}, SN\\,2003lw \\citep{Soderberg2004}, SN\\,2010dh \\citep{Margutti2013}, or to the relativistic SN\\,2009bb \\citep{Soderberg2010} and iPTF17cw \\citep{Corsi2017}, are intrinsically rare.  However, none of our observations exclude radio emission similar to that of SN\\,2006aj. This is not surprising since the afterglow of this low-luminosity GRB faded on timescales much faster than the $20-30$ days since explosion that our VLA monitoring campaign allowed us to target. To enable progress, obtaining prompt ($\\lesssim 5$\\,d since explosion) and accurate spectral classification paired with deep radio follow-up observations of SNe Ic-BL should be a major focus of future studies. At the same time, as discussed in \\citet{Ho2020_ZTF20aalxlis}, high-cadence optical surveys can provide an alternative way to measure the rate of SNe Ic-BL that are similar to SN\\,2006aj independently of $\\gamma$-ray and radio observations, by catching potential optical signatures of shock-cooling emission at early times.  Based on an analysis of ZTF SNe with early high-cadence light curves, \\citet{Ho2020_ZTF20aalxlis} concluded that it appears that SN\\,2006aj-like events are uncommon, but more events will be needed to measure a robust rate.\n\n\\subsubsection{Properties of the radio-emitting ejecta}\n\\label{sec:radio_properties}\nGiven that none of the SNe in our sample shows evidence for relativistic ejecta, hereafter we consider their radio properties within the synchrotron self-absorption (SSA) model for radio SNe \\citep{Chevalier1998}. Within this model, constraining the radio peak frequency and peak flux can provide information on the size of the radio emitting material. We start from Equations (11) and (13) of \\citet{Chevalier1998}:\n\\begin{eqnarray}\n\\nonumber R_{p}\\approx 8.8\\times10^{15}\\,{\\rm cm}\\left(\\frac{\\eta}{2\\alpha}\\right)^{1\/(2p+13)} \\left(\\frac{F_p}{\\rm Jy}\\right)^{(p+6)\/(2p+13)}\\times\\\\\\left(\\frac{d_L}{\\rm Mpc}\\right)^{(2p+12)\/(2p+13)}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\right)^{-1},~~~\n\\label{eq:ejspeed}\n\\end{eqnarray}\nwhere $\\alpha \\approx 1$ is the ratio of relativistic electron energy density to magnetic energy density, $F_{p}$  is the flux density at the time of SSA peak, $\\nu_{p}$ is the SSA frequency, and where $R\/\\eta$ is the thickness of the radiating electron shell. The normalization of the above Equation has a small dependence on $p$ and in the above we assume $p\\approx 3$ for the power-law index of the electron energy distribution. Setting $R_p \\approx {\\rm v_s} t_p$ in Equation (\\ref{eq:ejspeed}), and considering that $L_p \\approx 4\\pi d^2_L F_p$ (neglecting redshift effects), we get:\n\\begin{eqnarray}\n \\nonumber  \\left(\\frac{L_p}{\\rm erg\\,s^{-1}\\,Hz^{-1}}\\right) \\approx 1.2\\times10^{27} \\left(\\frac{\\beta_s}{3.4}\\right)^{(2p+13)\/(p+6)} \\\\\\times\\left(\\frac{\\eta}{2\\alpha}\\right)^{-1\/(p+6)} \\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\frac{t_p}{\\rm 1\\,d}\\right)^{(2p+13)\/(p+6)}\n\\end{eqnarray}\nwhere we have set $\\beta_s = {\\rm v_s}\/c$. We plot in Figure \\ref{fig:radiospeed} with blue-dotted lines the relationship above for various values of $\\beta_s$ (and for $p=3$, $\\eta=2$, $\\alpha=1$). As evident from this Figure, relativistic events such as SN\\,1998bw (for which the non-relativistic approximation used in the above Equations breaks down) are located at $\\beta_s\\gtrsim 1$. None of the ZTF SNe Ic-BL in our sample for which we obtained a radio counterpart detection shows evidence for ejecta faster than $0.3c$, except possibly for SN\\,2018hom. However, for this event we only have one radio observation and hence contamination from the host galaxy cannot be excluded. We also note that SN\\,2020jqm lies in the region of the parameter space occupied by radio-loud CSM interacting SNe similar to PTF\\,11qcj. \n\nThe magnetic field can be expressed as \\citep[see Equations (12) and (14) in][]{Chevalier1998}:\n\\begin{eqnarray}\n\\nonumber B_p \\approx 0.58\\rm\\,G \\left(\\frac{\\eta}{2\\alpha}\\right)^{4\/(2p+13)}\\left(\\frac{F_{p}}{\\rm Jy}\\right)^{-2\/(2p+13)}\\times\\\\\\left(\\frac{d_{L}}{\\rm Mpc}\\right)^{-4\/(2p+13)}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\right).\n\\label{eq2}\n\\end{eqnarray} \nConsider a SN shock expanding in a circumstellar medium (CSM) of density:\n\\begin{equation}\n\\rho\\approx 5 \\times10^{11} \\,{\\rm g\\,cm}^{-1}A_{*}R^{-2}\n\\end{equation}\nwhere:\n\\begin{equation}\n  A_{*}=  \\frac{\\dot{M}\/(10^{-5}M_{\\odot}\/{\\rm yr})}{4\\pi {\\rm v}_w\/(10^3{\\rm km\/s})}.\n\\label{eq:rho}\n\\end{equation}\nAssuming that a fraction $\\epsilon_B$ of the energy density $\\rho{\\rm v}^2_s$  goes into magnetic fields:\n\\begin{equation}\n \\frac{B_p^2}{8\\pi} = \\epsilon_B \\rho {\\rm v_s}^2 = \\epsilon_B \\rho R_p^2 t_p^{-2},\n \\label{eq3}\n\\end{equation}\none can write:\n\\begin{eqnarray}\n\\nonumber \\left(\\frac{L_p}{\\rm erg\\,s^{-1}\\,Hz^{-1}}\\right) \\approx 1.2\\times10^{27} \\left(\\frac{\\eta}{2\\alpha}\\right)^{2}\\left(\\frac{\\nu_p}{5\\,\\rm GHz}\\frac{t_p}{1\\,\\rm d}\\right)^{(2p+13)\/2}\\\\\\times \\left(5\\times10^3\\epsilon_B A_*\\right)^{-(2p+13)\/4},~~~~~~\n\\end{eqnarray}\nwhere we have used Equations  (\\ref{eq2}), (\\ref{eq:rho}), and (\\ref{eq3}). We plot in Figure \\ref{fig:radiospeed} with yellow-dashed lines the relationship above for various values of $\\dot{M}$ (and for $p=3$, $\\eta=2$, $\\alpha=1$, $\\epsilon_B =0.33$, $v_w=1000$\\,km\\,s$^{-1}$).  As evident from this Figure, relativistic events such as SN\\,1998bw show a preference for smaller mass-loss rates. We note that while the above relationship depends strongly on the assumed values of $\\eta$, $\\epsilon_B$, and $v_w$, this trend for $\\dot{M}$ remains true regardless of the specific values of these (uncertain) parameters. We also note that the above analysis assumes mass-loss in the form of a steady wind. While this is generally considered to be the case for relativistic SNe Ic-BL, binary interaction or eruptive mass loss in core-collapse SNe can produce denser CSM with more complex profiles \\citep[e.g.][]{Montes1998,Soderberg2006,Salas2013,Corsi2014,Margutti2017,Balasubramanian2021,Maeda2021,Stroh2021}.\n\nFinally, the total energy coupled to the fastest (radio emitting) ejecta can be expressed as \\citep[e.g.,][]{Soderberg2006}:\n\\begin{equation}\nE_r\\approx \\frac{4\\pi R_p^3}{\\eta}\\frac{B_p^2}{8\\pi\\epsilon_B}=\\frac{R_p^3}{\\eta}\\frac{B_p^2}{2\\epsilon_B}.\n\\label{eq4}\n\\end{equation}\n In Table \\ref{tab:radioproperties} we summarize the properties of the radio ejecta derived for the four SNe for which we detect a radio counterpart. These values can be compared with  $\\dot{M}\\approx2.5\\times10^{-7}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx (1-10)\\times10^{49}$\\,erg estimated for SN\\,1998bw by \\citet{Li1999}, with $\\dot{M}\\approx2\\times10^{-6}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx 1.3\\times10^{49}$\\,erg estimated for SN\\,2009bb by \\citet{Soderberg2010}, and with  with $\\dot{M}\\approx(0.4-1)\\times10^{-5}M_{\\odot}{\\rm yr}^{-1}$ and $E_r\\approx (0.3-4)\\times10^{49}$\\,erg estimated for GRB\\,100316D by \\citet{Margutti2013}. \n\n\\begin{center}\n\\begin{table}\n\\caption{Properties of the radio ejecta of the SNe in our sample for which we detect a radio counterpart. We report the SN name, the estimated SN shock speed normalized to the speed of light ($\\beta_s$), the mass-loss rate of the pre-SN progenitor ($\\dot{M}$), the energy coupled to the fastest (radio-emitting) ejecta ($E_r$), and the ratio between the last and the kinetic energy of the explosion (estimated from the optical light curve modeling, $E_k$). See Section \\ref{sec:radioanalysis} for discussion. \\label{tab:radioproperties}}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\nSN  & $\\beta_s$ &  $\\dot{M}$ & $E_r$ & $E_r\/E_k$ \\\\\n & & (M$_{\\odot}$\\,yr$^{-1}$) & (erg) & \\\\\n\\hline\n2018hom &  0.35 & $1.1\\times10^{-6}$ & $3.6\\times10^{47}$ & $<0.04$\\%\\\\\n2020jqm &  0.048 & $2.7\\times10^{-4}$ & $5.7\\times10^{48}$ & 0.1\\% \\\\\n2020tkx &  0.14 & $1.7\\times10^{-5}$ & $1.1\\times10^{48}$ & $<0.07$\\% \\\\\n2021ywf & 0.19 &  $2.2\\times10^{-6}$ & $2.3\\times10^{47}$ & 0.02\\%\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=16cm]{CorsiFig11_OffAxisRadioLight.png}\n    \\caption{Approximate radio luminosity density for GRBs observed largely off-axis during the sub-relativistic phase (black solid, dotted, dashed, and dash-dotted lines) compared with the radio detections and upper-limits of the SNe Ic-BL in our sample. Most of our observations exclude fireballs with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. However, our observations become less constraining for smaller energy and ISM density values. For example, most of our radio data cannot exclude off-axis jets with energies $E\\sim 10^{49}$\\,erg and $n\\sim 0.1$\\,cm$^{-3}$. See Section \\ref{sec:off-axis-radio} for discussion.}\n    \\label{fig:off-axis-radio}\n\\end{figure*}\n\n\\subsubsection{Off-axis GRB radio afterglow constraints}\n\\label{sec:off-axis-radio}\nWe finally consider what type of constraints our radio observations put on a scenario where the SNe Ic-BL in our sample could be accompanied by relativistic ejecta from \na largely (close to 90\\,deg) off-axis GRB afterglow that would become visible in the radio band when the relativistic fireball enters the sub-relativistic phase and approaches spherical symmetry.\nBecause our radio observations do not extend past 100-200 days since explosion, we can put only limited constraints on this scenario. Hence, hereafter we present some general order-of-magnitude considerations rather than a detailed event-by-event modeling. \n\nFollowing \\citet{Corsi2016}, we can model approximately the late-time radio emission from an off-axis GRB based on the results by \\citet{Livio2000}, \\citet{Waxman2004b}, \\citet{Zhang2009}, and \\citet{VanEerten2012}. For fireballs expanding in an interstellar medium (ISM) of constant\ndensity $n$ (in units of cm$^{-3}$), at timescales $t$ such that:\n\\begin{equation}\n    t \\gtrsim (1+z) \\times t_{\\rm SNT}\/2\n\\end{equation}\nwhere the transition time to the spherical Sedov\u2013Neumann\u2013Taylor (SNT) blast wave, $t_{\\rm SNT}$,  reads:\n\\begin{equation}\n    t_{\\rm SNT}\\approx 92\\,{\\rm d}\\left(E_{51}\/n\\right)^{1\/3},\n\\end{equation}\nthe luminosity density can be approximated analytically via the following formula \\citep[see Equation (23) in][ where we neglect redshift corrections and assume $p=2$]{Zhang2009}:\n\\begin{eqnarray}\n\\nonumber L_{\\nu}({\\rm t}) \\approx 4\\pi d^{2}_{\\rm L} F_{\\nu}({\\rm t}) \\approx 2\\times10^{30}\\left(\\frac{\\epsilon_e}{0.1}\\right)\\left(\\frac{\\epsilon_B}{0.1}\\right)^{3\/4}n^{9\/20}\\\\\\times E^{13\/10}_{51}\\left(\\frac{\\nu}{\\rm 1\\,GHz}\\right)^{-1\/2}\\left(\\frac{t}{92\\,{\\rm d}}\\right)^{-9\/10}\\,{\\rm erg\\,s^{-1}\\,{\\rm Hz}^{-1}}.~~~~~~~\n\\end{eqnarray}\nIn the above Equations, $E_{51}$ is the beaming-corrected ejecta energy in units of $10^{51}$\\,erg. We note that here we assume again a constant density ISM in agreement with the majority of GRB afterglow observtions \\citep[e.g.,][]{Schulze2011}. \n\nWe plot the above luminosity in Figure \\ref{fig:off-axis-radio} together with our radio observations and upper-limits, assuming $\\epsilon_e=0.1$, $\\epsilon_B=0.1$, and for representative values of low-luminosity GRB energies and typical values of long GRB ISM densities $n$. As evident from this Figure, our observations exclude fireballs with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. However, our observations become less constraining for smaller energy and ISM density values. \n\n\\section{Summary and Conclusion}\\label{sec:conclusion}\nWe have presented deep radio follow-up observations of 16 SNe Ic-BL that are part of the ZTF sample.  \nOur campaign resulted in 4 radio counterpart detections and 12 deep radio upper-limits. For 9 of these 16 events we have also carried out X-ray observations with \\textit{Swift}\/XRT. All together, these results constrain the fraction of SN\\,1998bw-like explosions to $< 19\\%$ (3$\\sigma$ Gaussian equivalent), tightening previous constraints by a factor of $\\approx 2$. Moreover, our results exclude relativistic ejecta with radio luminosities densities in between $\\approx 5\\times10^{27}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ and  $\\approx 10^{29}$\\,erg\\,s$^{-1}$\\,Hz$^{-1}$ at $t\\gtrsim 20$\\,d since explosion for $\\approx 60\\%$ of the events in our sample, pointing to the fact that SNe Ic-BL similar to low-luminosity-GRB-SN such as SN\\,1998bw, SN\\,2003lw, SN\\,2010dh, or to the relativistic SN\\,2009bb and iPTF17cw, are intrinsically rare. This result is in line with numerical simulations that suggest that a SN Ic-BL can be triggered even if a jet\nengine fails to produce a successful GRB jet.\n\nWe showed that our radio observations exclude an association of the SNe Ic-BL in our sample with largely off-axis GRB afterglows with energies $E\\gtrsim 10^{50}$\\,erg expanding in ISM media with densities $\\gtrsim 1$\\,cm$^{-3}$. On the other hand, our radio observations are less constraining for smaller energy and ISM density values, and cannot exclude off-axis jets with energies $E\\sim 10^{49}$\\,erg.\n\nWe noted that the main conclusion of our work is subject to the caveat that the parameter space of SN\\,2006aj-like explosions (with faint radio emission peaking only a few days after explosion) is left largely unconstrained by current systematic radio follow-up efforts like the one presented here. In other words, we cannot exclude that a larger fraction of SNe Ic-BL harbors GRB\\,060218\/SN\\,2006aj-like emission. In the future, obtaining fast and accurate spectral classification of SNe Ic-BL paired with deep radio follow-up observations executed within $5$\\,d since explosion would overcome this limitation. While high-cadence optical surveys can provide an alternative way to measure the rate of SNe Ic-BL that are similar to SN\\,2006aj via shock-cooling emission at early times, more optical detections are also needed to measure a robust rate. \n\nThe Legacy Survey of Space and Time on the Vera C. Rubin Observatory \\citep[LSST;][]{Rubin2019} promises to provide numerous discoveries of even the rarest type of explosive transients, such as the SNe Ic-BL discussed here. The challenge will be to recognize and classify these explosions promptly \\citep[e.g.,][]{Villar2019,Villar2020}, so that they can be followed up in the radio with current and next generation radio facilities. Indeed, Rubin, paired with the increased sensitivity of the next generation VLA \\citep[ngVLA;][]{ngVLA}, could provide a unique opportunity for building a large statistical sample of SNe Ic-BL with deep radio observations that may be used to guide theoretical modeling in a more systematic fashion, beyond what has been achievable over the last $\\approx 25$ years (i.e., since the discovery of GRB-SN\\,1998bw).  In addition, the Square Kilometer Array (SKA) will enable discoveries of radio SNe and other transients in an untargeted and optically-unbiased way \\citep{Lien2011}. Hence, one can envision that the Rubin-LSST+ngVLA and SKA samples will, together, provide crucial information on\nmassive star evolution, as well as SNe Ic-BL physics and CSM properties. \n\nWe conclude by noting that understanding the evolution of single and stripped binary stars up to core collapse is of special interest in the new era of time-domain multi-messenger (gravitational-wave and neutrino) astronomy \\citep[see e.g., ][for recent reviews]{Murase2018,Scholber2012,Ernazar2020,Guepin2022}. Gravitational waves from nearby core-collapse SNe, in particular, represent an exciting prospect for expanding multi-messenger studies beyond the current realm of compact binary coalescences. While they may come into reach with the current LIGO \\citep{Aasi2015} and Virgo \\citep{Acernese2015} detectors, it is more likely that next generation gravitational-wave observatories, such as the Einstein Telescope \\citep{Maggiore2020} and the Cosmic Explorer \\citep{Evans2021}, will enable painting the first detailed multi-messenger picture of a core-collapse explosion. The physics behind massive stars' evolution and deaths also impacts the estimated rates and mass distribution of compact object mergers \\citep[e.g.,][]{Schneider2021} which, in turn, are current primary sources for LIGO and Virgo, and will be detected in much large numbers by next generation gravitational-wave detectors. Hence, continued and coordinated efforts dedicated to understanding massive stars' deaths and the link between pre-SN progenitors and properties of SN explosions, using multiple messengers, undoubtedly represent an exciting path forward.\n\n\\begin{acknowledgements}\n\\small\nA.C. and A.B. acknowledge support from NASA \\textit{Swift} Guest investigator programs (Cycles 16 and 17 via grants \\#80NSSC20K1482 and \\#80NSSC22K0203). S.A. gratefully acknowledges support from the National Science Foundation GROWTH PIRE grant No. 1545949. S.Y. has been supported by the research project grant ``Understanding the Dynamic Universe'' funded by the Knut and Alice Wallenberg Foundation under Dnr KAW 2018.0067, and the G.R.E.A.T research environment, funded by {\\em Vetenskapsr\\aa det}, the Swedish Research Council, project number 2016-06012.\nBased on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar\nObservatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grants\nNo. AST-1440341 and No. AST-2034437, and a collaboration including Caltech, IPAC, the Weizmann Institute for Science, the Oskar Klein Center at\nStockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, Los Alamos\nNational Laboratories, the TANGO\nConsortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Berkeley\nNational Laboratories, Lawrence Livermore National\nLaboratories, and IN2P3, France. Operations are conducted by COO, IPAC, and UW. The SED Machine is based upon work supported by the National Science Foundation under Grant No. 1106171. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant \\#12540303 (PI: Graham).\nThe National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.\nThe Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation. Based in part on observations made with the Nordic Optical Telescope, owned in collaboration by the University of Turku and Aarhus University, and operated jointly by Aarhus University, the University of Turku and the University of Oslo, representing Denmark, Finland and Norway, the University of Iceland and Stockholm University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\n\\end{acknowledgements}\n\n\\bibliographystyle{aasjournal}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMelanoma is the most dangerous type of skin cancer which cause almost 60,000 deaths annually. In order to improve the efficiency, effectiveness, and accuracy of melanoma diagnosis, International Skin Imaging Collaboration (ISIC) provides over 2,000 dermoscopic images of various skin problems for lesion segmentation, disease classification and other relative research. \n\nLesion segmentation aims to extract the lesion segmentation boundaries from dermoscopic images to assist exports in diagnosis. In recent years, U-Net, FCN and other deep learning methods are widely used for medical image segmentation. However, the existing algorithms have the restriction in lesion segmentation because of various appearance of lesion caused by the diversity of persons and different collection environment. FCN, U-Net and other one-stage segmentation methods are sensitive to the size of the lesion. Too large or too small size of lesion decreasing the accuracy of these one-stage methods. Two-stage method could reduce the negetive influence of diverse size of lesion. Mask R-CNN can be viewed as a two-stage method which has an outstanding performance in COCO. However, Mask R-CNN still has some brawbacks in lesion segmentation. Unlike clear boundary between different objects in COCO data, the boundary between lesion and healthy skin is vague in this challenge. The vague boundary reduces the accuracy of RPN part in Mask R-CNN which may cause the negative influence in following segmentation part. Furthermore, the low resolution of input image in the segmentation of Mask R-CNN also reduce the accuracy of segmentation.\n\nIn this report, we propose a method for lesion segmentation. Our method is a two-stage process including detection and segmentation. Detection part is used to detect the location of the lesion and crop the lesion from images. Following the detection, segmentation part segments the cropped image and predict the region of lesion. Furthermore, we also propose an optimised processed for cropping image. Instead of cropping the image by bounding box exactly, in training, we crop the image with a random expansion and contraction to increase robustness of neural networks. Finally, image augmentation and other ensemble methods are used in our method. Our method is based on deep convolutional networks and trained by the dataset provided by ISIC  \\cite{Tschandl2018_HAM10000} \\cite{DBLP:journals\/corr\/abs-1710-05006}. \n\n\\section{Materials and Methods}\n\n\\begin{figure*}\n \\centering\n  \\includegraphics[height=6cm,width=0.75\\textwidth]{process.png}\n  \\caption{Process}\n  \\label{fig:Process}\n\\end{figure*}\n\n\\subsection{Database and Metrics}\nFor Lesion Segmentation, ISIC 2018 provides 2594 images with corresponding ground truth for training. There are 100 images for validation and 1000 images for testing without ground truth. The number of testing images in this year is 400 more than which in 2017. The size and aspect ratio of images are various. The lesion in images has different appearances and appear in different parts of people. There are three criterions for ground truth which are a fully-automated algorithm, a semi-automated flood-fill algorithm and manual polygon tracing. The ground truth labelled under different criterions has different shape of boundary which is a challenge in this task. We split the whole training set into two sets with ratio 10:1.\n\nThe Evaluation of this challenge is Jaccard index which is shown in Equation \\ref{eq:jaccard}. In 2018, the organiser adds a penalty when the Jaccard index is below 0.65.\n\\begin{equation}\nJ(A,B) = \n\\begin{cases}\n\\frac{|A \\bigcap B|}{|A \\bigcup B|}  & J(A,B) \\geq 0.65 \\\\\n0 & \\text{otherwise}\\\\\n\\end{cases}\n\\label{eq:jaccard}\n\\end{equation}\n\n\n\n\n\\subsection{Methods}\n\nWe design a two-stage process combining detection and segmentation. Figure \\ref{fig:Process} shows our process. Firstly, the detection part detects the location of the lesion with the highest probability. According to the bounding box, the lesion is cropped from the original image and the cropped image is normalised to 512*512 which is the size of the input image of segmentation part. A fine segmentation boundray of the cropped image is provided by the segmentation part.\n\n\\subsubsection{Detection Part}\nIn the detection process, we use the detection part in MaskR-CNN \\cite{DBLP:journals\/corr\/HeGDG17}. We also use the segmentation branch of Mask R-CNN to supervise the training of neural networks.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{detection.png}\n  \\caption{Detection}\n  \\label{fig:Detection}\n\\end{figure}\n\n\\subsubsection{Segmentation Part}\nIn the segmentation part, we design an encode-decode architecture of network inspired by DeepLab \\cite{DBLP:journals\/corr\/ChenPK0Y16}, PSPNet \\cite{DBLP:journals\/corr\/ZhaoSQWJ16}, DenseASPP \\cite{Yang_2018_CVPR} and Context Contrasted Local \\cite{Ding_2018_CVPR}. Our architecture is shown in Figure \\ref{fig:Segmentation}. The features are extracted by extended ResNet 101 with three cascading blocks. After ResNet, a modified ASPP is used to compose various scale features. We also use a skip connection to transfer detailed information.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{segmentation.png}\n  \\caption{Segmentation}\n  \\label{fig:Segmentation}\n\\end{figure}\n\nFigure \\ref{fig:ASPP} shows the structure of the modified ASPP block. A 1x1 convolutional layer is used to compose the feature extracted by ResNet and reduce the number of feature maps. After that, we use a modified ASPP block to extract information in various scales. The modified ASPP has three parts which are dense ASPP, standard convolution layers and pooling layers. Dense ASPP is proposed by \\cite{Yang_2018_CVPR} which reduce the influence of margin in ASPP. Considering the vague boundary and low contrast appearance of the lesion, we add standard convolution layers to enhance the ability of neural networks in distinguishing the boundary. The aim of pooling layers is to let the networks consider the surrounding area of the low contrast lesion. The modified ASPP includes three dilated convolutions with rate 3, 6, 12 respectively, three standard convolution layers with size 3, 5, 7 respectively and four pooling layers with size 5, 9, 13, 17 respectively.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[height=13cm,width=0.4\\textwidth]{ASPP.png}\n  \\caption{Modified ASPP}\n  \\label{fig:ASPP}\n\\end{figure}\n\n\n\n\\subsection{Pre-processing}\nInstead of only using RBG channels, we combine the SV channels in Hue-Saturation-Value colour space and lab channels in CIELAB space with the RGB channels. These 8 channels are the input of segmentation part. Figure \\ref{fig:hsvlab} shows different channels of HSV and CIELAB colour space.\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.4\\textwidth]{hsvlab.png}\n  \\caption{Single channel in HSV and CIELAB colour space}\n  \\label{fig:hsvlab}\n\\end{figure}\n\n\\subsection{Post-processing}\nThe ensemble is used as the post-processing to increase the performance of segmentation. The input image of the segmentation part is rotated 90, 180 degrees and flipped to generate the other three images. Each image has a result predicted by the segmentation part. The results of the rotated and flipped image need to rotate and flip back to the original image. The final mask is the average of four results of these images.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{ensemable.png}\n  \\caption{Ensamble}\n  \\label{fig:Ensamble}\n\\end{figure}\n\n\\subsection{Training}\nImage augmentation is used in training of detection and segmentation. Rotation, colour jitter, flip, crop and shear is operated in each image. Each channel of images is scaled to 0 to 1. In segmentation part the size of input images is set to 512x512. Some examples are shown in Figure \\ref{fig:aug1}. \n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}{.25\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{fliplr.png}\n  \\caption{Flip Left to Right}\n  \\label{fig:aug1-1}\n\\end{subfigure}%\n\\begin{subfigure}{.25\\textwidth}\n  \\centering\n  \\includegraphics[width=0.4\\linewidth]{rotate45.png}\n  \\caption{Roatate 45$^\\circ$}\n  \\label{fig:aug1-2}\n\\end{subfigure}\n\\caption{Examples for image augmentation}\n\\label{fig:aug1}\n\\end{figure}\n\nWe use the Adam optimiser and set the learning rate to 0.001. The learning rate will be set at 92\\% of the previous after each epoch. The batch size is 8. We early stop the training when the net start overfitting. We use the dice loss which is shown in Equation \\ref{eq:diceloss}, where $p_{i,j}$ is the prediction in pixel $(i,j)$ and $g_{i,j}$ is the ground truth in pixel $(i,j)$. \n\n\\begin{equation}\n\\label{eq:diceloss}\nL =  -\\frac{\\sum_{i,j}(p_{i,j} g_{i,j})}{\\sum_{i,j}p_{i,j} + \\sum_{i,j}g_{i,j}-\\sum_{i,j}(p_{i,j} g_{i,j})}\n\\end{equation}\n\nThe input image of the segmentation part is crop randomly in a range near the bounding box predicted by detection part. In order to improve the diversity of input images and provide more information about background context. In training, the input image of the segmentation part is cropped from 81\\% to 121\\% of the bounding box randomly which is shown in Figure \\ref{fig:crop}.\n\n\\begin{figure}[H]\n  \\centering\n  \\includegraphics[width=0.3\\textwidth]{crop.png}\n  \\caption{Crop image}\n  \\label{fig:crop}\n\\end{figure}\n\n\\begin{figure*}\n \\centering\n  \\includegraphics[width=0.9\\textwidth]{results.png}\n  \\caption{Predicted masks of different segmentation methods}\n  \\label{fig:result}\n\\end{figure*}\n\n\\subsection{Implementation}\nThe detection part of our method is implemented by using Pytorch 0.4 in Ubuntu 14.04. The framework is from \\url{https:\/\/github.com\/roytseng-tw\/Detectron.pytorch}. The segmentation part is implemented by using Pytorch 0.3.1 in Ubuntu 14.04. The neural networks are trained by two Nvidia 1080 Ti with 60 GB of RAM.\n\n\\section{Results}\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{llll}\n\\hline\nMethod & Jaccard & Jaccard(>0.65)\\\\\n\\hline\nMask R-CNN & 0.825 & 0.787\\\\\nOne-stage Segmentation & 0.820 & 0.783\\\\\nTwo-stage Segmentation & 0.846 & 0.816\\\\\n\\hline\n\\end{tabular}\n\\label{table:results}\n\\caption{Evaluation metrics of different segmentation methods}\n\\end{table}\n\n\nThe evaluation metrics on 257 images of our two-stage segmentation method with MaskR-CNN and our one-stage segmentation method is shown in Table 1. The thresholded Jaccard of our two-stage method on the official testing set in 0.802. Figure \\ref{fig:result} shows the outputs of different segmentation methods. Compared with other methods, The results of our two-stage method has a better location and more smooth edge.\n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:I}\n\nNeutron stars, which are produced through core-collapse supernovae, are one of the most suitable environments for probing physics under extreme conditions, which are quite difficult to realize on the Earth. The density inside the star significantly exceeds the standard nuclear density, while the gravitational and magnetic fields become much stronger than those observed in our solar system. So, by observing the neutron star itself and\/or the phenomena associated with neutron stars, inversely one would be able to extract the information for the extreme conditions. For example, the discovery of the $2M_\\odot$ neutron stars excludes some of the soft equations of state (EOSs) \\cite{D10, A13, C20}. Namely, if the maximum mass of the neutron star constructed with an EOS does not reach the observed mass, such an EOS is not good for the EOS describing neutron stars. The light bending due to the strong gravitational field, which is one of the relativistic effects, may also tell us the neutron star properties. That is, owing to the strong gravitational field induced by the neutron star, the light curve from the neutron star would be modulated. Thus, by carefully observing the pulsar light curve, one could mainly constrain the stellar compactness, i.e., the ratio of the stellar mass to the radius (e.g., \\cite{PFC83, LL95, PG03, PO14, SM18, Sotani20a}). In practice, the mass and radius of the neutron stars, i.e., PSR J0030+0451 \\cite{Riley19, Miller19} and PSR J0740+6620 \\cite{Riley21, Miller21}, are successfully constrained through the observations with the Neutron star Interior Composition Explorer (NICER) on the International Space Station. The current uncertainty in the mass and radius constraints is still large, but this type of constraint must help us to know the true EOS when the uncertainty is reduced through future observations. \n\n\nIn addition to the direct observations of the stellar mass and radius, the oscillation frequency from the neutron star is another important observable. Since the oscillation frequency strongly depends on the interior properties of the object, as an inverse problem, one can extract the stellar properties by observing the frequency. This technique is known as asteroseismology, which is similar to seismology on the Earth and helioseismology on the Sun. In fact, by identifying the quasi-periodic oscillations observed in the afterglow following the giant flares with the neutron star crustal oscillations, one could constrain the crust properties (e.g., Refs. \\cite{GNHL2011, SNIO2012, SIO2016}). Similarly, one may know the neutron star mass, radius, and EOS by observing the gravitational wave frequencies from neutron stars and by identifying them with specific oscillation modes  (e.g., Refs. \\cite{AK1996,AK1998,STM2001,SH2003,SYMT2011,PA2012,DGKK2013,Sotani20b,Sotani20c,Sotani21,SD22}). Furthermore, this technique is recently adopted for understanding the gravitational wave signals appearing in the numerical simulations for core-collapse supernova (e.g., Refs. \\cite{FMP2003,FKAO2015,ST2016,SKTK2017,MRBV2018,SKTK2019,TCPOF19,SS2019,ST2020,STT2021}).\n\n\nIn general, the gravitational wave frequencies from the neutron stars are complex numbers, where the real and imaginary parts respectively correspond to an oscillation frequency and damping rate, and the corresponding eigenmodes are called quasi-normal modes. So, to determine the quasi-normal modes of the neutron stars, one has to somehow solve the eigenvalue problem with respect to the eigenvalue in two-dimensional parameter space with the real and imaginary parts. Since this solution may be a bother, one sometimes adopts the approximation to estimate the frequency of gravitational waves. The simple approximation is the relativistic Cowling approximation, where the metric perturbations are neglected during the fluid oscillations, i.e., the frequencies of fluid oscillations are determined with the fixed metric. One can qualitatively discuss the behavior of frequencies with the Cowling approximation, even though the accuracy of the determined frequencies is within $\\sim 20\\%$ \\cite{ST2020, YK97}.  Another approximation adopted so far is the zero-damping approximation, where one takes into account the metric perturbations as well as the fluid perturbations, but the imaginary part of the eigenvalue is assumed to be zero. This is because the damping rate, i.e, the imaginary part of the eigenvalue, for the gravitational waves induced by the fluid oscillations is generally much smaller than the oscillation frequency, i.e., the real part of the eigenvalue. This approximation is considered to estimate the frequency of the gravitational waves well, but it is not discussed how well this approximation works. So, in this study, we first discuss how well the zero-damping approximation works by comparing the frequencies determined with the approximation to those determined through the proper eigenvalue problem. Anyway, with either the Cowling approximation or the zero-damping approximation, one can discuss only the frequency of the gravitational waves induced by the fluid oscillations, such as the fundamental, pressure, and gravity modes, but one cannot discuss the gravitational waves associated with the oscillations of the spacetime itself, i.e., the $w$-modes. \n\n\nIn order to estimate the $w_1$-mode frequency, in this study, we find the empirical relation for the ratio of the damping rate of the $w_1$-mode gravitational wave, i.e., the imaginary part of the quasi-normal mode, to its frequency, i.e., the corresponding real part, as a function of the stellar compactness almost independently of the adopted EOSs. With this empirical relation, we newly propose a one-dimensional approximation for estimating the $w_1$-mode frequency and show how well this approximation works. Unless otherwise mentioned, we adopt geometric units in the following, $c=G=1$, where $c$ and $G$ denote the speed of light and the gravitational constant, respectively, and the metric signature is $(-,+,+,+)$.\n\n\n\\section{Neutron star models}\n\\label{sec:EOS}\n\nIn this study, we simply consider the static, spherically symmetric stars, where the metric describing the system is given by \n\\begin{equation} \n  ds^2 = -e^{2\\Phi}dt^2 + e^{2\\Lambda}dr^2 + r^2\\left(d\\theta^2 + \\sin^2\\theta d\\phi^2\\right).  \\label{eq:metric}\n\\end{equation}\nThe metric functions $\\Phi$ and $\\Lambda$ in Eq. (\\ref{eq:metric}) are functions only of $r$, while $e^{2\\Lambda}$ is directly related to the mass function, $m(r)$, as $e^{-2\\Lambda}=1-2m\/r$. The stellar structure is determined by integrating the Tolman-Oppenheimer-Volkoff equation with an appropriate EOS for neutron star matter. In this study, we adopt the same EOSs as in Refs. \\cite{Sotani20c, Sotani21}, which are listed in Table \\ref{tab:EOS} together with the EOS parameters and maximum mass of the neutron star. Here, $K_0$ and $L$ are the incompressibility for symmetric nuclear matter and the density dependence of the nuclear symmetry energy. We note that any EOSs can be characterized by these nuclear saturation parameters, which are constrained via terrestrial nuclear experiments, such as $K_0= 230\\pm 40$ MeV \\cite{KM13} and $L\\simeq 58.9\\pm 16$ MeV \\cite{Li19}. Comparing these fiducial values of $K_0$ and $L$ to those for the adopted EOSs, some of the EOSs listed in Table \\ref{tab:EOS} seem to be excluded, but we consider even such EOSs in this study to examine the EOS dependence in the wide parameter space. Additionally, $\\eta$ in Table \\ref{tab:EOS} is the specific combination of $K_0$ and $L$ as $\\eta\\equiv \\left(K_0 L^2\\right)^{1\/3}$ \\cite{SIOO14}, which is a suitable parameter not only for expressing the properties of low-mass neutron stars but also for discussing the maximum mass of neutron stars \\cite{SSB16,Sotani17,SK17}. The mass and radius relations for the neutron star models constructed with the EOS listed in Table \\ref{tab:EOS} are shown in Fig. \\ref{fig:MR}, where the filled and open marks respectively correspond to the EOSs constructed with the relativistic framework and with the Skyrme-type effective interaction, while the double-square is the EOS constructed with a variational method. As in Ref. \\cite{Sotani20c}, we consider the stellar models denoted with marks in this figure. \n\n\n\\begin{table}\n\\caption{EOS parameters adopted in this study, $K_0$, $L$, and $\\eta$, and the maximum mass, $M_{\\rm max}$, of the neutron star constructed with each EOS.} \n\\label{tab:EOS}\n\\begin {center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\nEOS & $K_0$ (MeV) & $L$ (MeV) & $\\eta$ (MeV) & $M_{\\rm max}\/M_\\odot$   \\\\\n\\hline\nDD2\n & 243 & 55.0  & 90.2  & 2.41  \\\\ \nMiyatsu\n & 274 &  77.1 & 118  & 1.95  \\\\\nShen\n\\ & 281 & 111  & 151  &  2.17  \\\\  \nFPS\n & 261 & 34.9 & 68.2  & 1.80  \\\\  \nSKa\n & 263 & 74.6 & 114 & 2.22  \\\\ \nSLy4\n & 230 & 45.9 &  78.5 & 2.05  \\\\ \nSLy9\n & 230 & 54.9 &  88.4 & 2.16  \\\\  \nTogashi\n & 245  & 38.7  & 71.6 & 2.21  \\\\ \n\\hline \\hline\n\\end{tabular}\n\\end {center}\n\\end{table}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{MR1}\n\\vspace{0.5cm}\n\\caption\nFor the neutron star models constructed with various EOSs, the mass is shown as a function of the radius. Figure is taken from Ref. \\cite{Sotani20c}.\n\n\\label{fig:MR}\n\\end{figure}\n\n\n\n\n\\section{Quasi-normal modes}\n\\label{sec:QNMs}\n\nIn order to determine the quasi-normal modes of gravitational waves from compact objects, one has to solve the eigenvalue problem. The perturbation equations are derived from the linearized Einstein equation by adding the metric and fluid perturbations on the background stellar models. By imposing the appropriate boundary conditions at the stellar center, surface, and spacial infinity, the problem to solve becomes the eigenvalue problem with respect to the eigenvalue, $\\omega$. As a practical matter, how to numerically deal with the boundary condition at the spatial infinity may become a problem. In this study, we especially adopt the continued fraction method, the so-called Leaver's method \\cite{Leaver85,Leaver90}. That is, the perturbation variable outside the star is expressed as a power series around the stellar surface, which also satisfies the boundary condition at the infinity. By substituting this expansion into the Regge-Wheeler equation, one can get a three-term recurrence relation including $\\omega$, which is rewritten in the form of a continued fraction. So, one has to find the eigenvalue, $\\omega$, which satisfies the continued fraction. Here, we symbolically express this condition as $f(\\omega)=0$. The resulatant $\\omega$ is generally a complex value, where the real and imaginary parts correspond to the oscillation frequency,  Re($\\omega$)\/$2\\pi$, and damping rate of the corresponding gravitational waves. The concrete perturbation equations, boundary conditions, and functional form of $f(\\omega)$ are shown in Refs. \\cite{STM2001,ST2020}. \n\nIn practice, since $f(\\omega)$ is also a complex value, we try to find the value of $\\omega$ numerically, with which the absolute value of $f(\\omega)$ becomes the local minimum. In this study, we especially focus on the fundamental ($f$-), 1st pressure ($p_1$-), and 1st spacetime ($w_1$-) modes for the various neutron star models constructed with the EOSs listed in Table \\ref{tab:EOS}. We note that the $f$- and $p_1$-modes are the quasi-normal modes induced by the fluid oscillations, while the $w_1$-mode is the quasi-normal mode associated with the oscillations of spacetime itself. \n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ratio-ffDf}\n\\vspace{0.5cm}\n\\caption\nIn the top panel, the ratio of Im($\\omega$) to Re($\\omega$) for the $f$-mode is shown as a function of the stellar compactness for various neutron star models, where the solid line denotes the fitting formula given by Eq. (\\ref{eq:fitting_ff}). In the bottom pane, the relative deviation of the value of Im($\\omega$)\/Re($\\omega$) estimated with the empirical formula from that calculated via the eigenvalue problem is shown. \n\n\\label{fig:ratio-ff}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ratio-p1}\n\\vspace{0.5cm}\n\\caption\nThe ratio of Im($\\omega$) to Re($\\omega$) for the $p_1$-mode is shown as a function of the stellar compactness for various neutron star models.\n\n\\label{fig:ratio-p1}\n\\end{figure}\n\nWith the resultant $f$-mode frequency, we show the ratio of Im$(\\omega)$ to Re$(\\omega)$ in the top panel of Fig. \\ref{fig:ratio-ff}. From this figure, one can observe that the values of Im$(\\omega)$\/Re$(\\omega)$ are well expressed as a function of the stellar compactness, $M\/R$, almost independently of the EOS. In fact, we can derive the empirical relation for Im$(\\omega)$\/Re$(\\omega)\\, \\raisebox{-0.8ex}{$\\stackrel{\\textstyle >}{\\sim}$ } 10^{-5}$, i.e., except for the quite low-mass neutron stars, given by \n\\begin{equation}\n  \\frac{{\\rm Im}(\\omega_f)}{{\\rm Re}(\\omega_f)} = \\left[0.13193 -4.4754\\left(\\frac{M}{R}\\right)\n        +290.9\\left(\\frac{M}{R}\\right)^2 -756.14 \\left(\\frac{M}{R}\\right)^3\\right]\\times10^{-4},\\label{eq:fitting_ff}\n\\end{equation}\nwith which the expected values are shown with the thick-solid line in the top panel. In the bottom panel of Fig. \\ref{fig:ratio-ff}, we also show the relative deviation calculated with\n\\begin{equation}\n  \\Delta = \\frac{{\\rm abs}[{\\cal R}-{\\cal R}_{\\rm fit}]}{{\\cal R}}, \\label{eq:Delta}\n\\end{equation}\nwhere ${\\cal R}$ and ${\\cal R_{\\rm fit}}$ denote the ratio of Im$(\\omega)$ to Re$(\\omega)$ detemined through the eigenvalue problem and estimated with Eq. (\\ref{eq:fitting_ff}) for each stellar model, respectively. From this figure, we find that the values of Im$(\\omega)$\/Re$(\\omega)$ can be usually estimated within $\\sim10\\%$ accuracy by using the empirical relation. In a similar way, we show Im$(\\omega)$\/Re$(\\omega)$ for the $p_1$-modes in Fig. \\ref{fig:ratio-p1}. But, for the case of the $p_1$-mode we can not derive the empirical relation as a function of $M\/R$, unlike the case of the $f$-mode. \n\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{ReIm-MR1}\n\\vspace{0.5cm}\n\\caption\nThe ratio of Im$(\\omega)$ to Re$(\\omega)$ of the $w_1$-mode for various EOSs is shown as a function of the corresponding stellar compactness (top panel), where the solid line denotes the fitting line given by Eq. (\\ref{eq:fitting_w1}). The bottom panel denotes the relative deviation of the estimation with the fitting formula from the values of Im$(\\omega)$\/Re$(\\omega)$ calculated via the eigenvalue problem.\n\n\\label{fig:ratio}\n\\end{figure}\n\nOn the other hand, we also find that one can express the value of Im(${\\omega}$)\/Re($\\omega$) for the $w_1$-mode as a function of $M\/R$ almost independently of the adopted EOS. In the top panel of Fig. \\ref{fig:ratio}, we show the ratio of Im$(\\omega)$ to Re$(\\omega)$ for the $w_1$-mode determined through the eivenvlue problem for each stellar model as a function of the corresponding stellar compactness, where the solid line denotes the fitting formula given by\n\\begin{equation}\n  \\frac{{\\rm Im}(\\omega_{w_1})}{{\\rm Re}(\\omega_{w_1})} = 1.0659 -4.1598\\left(\\frac{M}{R}\\right)\n        +16.4565\\left(\\frac{M}{R}\\right)^2 -39.5369\\left(\\frac{M}{R}\\right)^3.  \\label{eq:fitting_w1}\n\\end{equation}\nIn the bottom panel, we show the relative deviation calculated with Eq. (\\ref{eq:Delta}) for the $w_1$-mode, where ${\\cal R}_{\\rm fit}$ is estimated with Eq. (\\ref{eq:fitting_w1}). Considering to the fact that $M\/R=0.172$ (0.207) for the canonical neutron star model with $M=1.4M_\\odot$ and $R=12$ km (10 km), one can observe that the fitting formula given by Eq. (\\ref{eq:fitting_w1}) works better for the low-mass neturon star models. \n\n\n\n\n\\section{One-dimensional approximation}\n\\label{sec:Approximation0}\n\nIn order to determine the quasi-normal modes, one has to somehow search the solution of $\\omega$ in two-dimensional parameter space with the real and imaginary parts, and this procedure may be trouble. One may sometimes adopt a suitable approximation to get out of this trouble, even if one can estimate only the frequency of quasi-normal modes. In this study, we especially consider the approximation, where the eigenvalue belongs to one-dimensional parameter space, depending only on the real part of the eigenvalue. We refer to this approximation as the one-dimensional approximation in this study. For example, since the imaginary part of the quasi-normal modes induced by the fluid oscillations is much smaller than the real part of them, as shown in Figs. \\ref{fig:ratio-ff} and \\ref{fig:ratio-p1}, one may assume that Im$(\\omega)=0$. This approximation, referred to as zero-damping approximation, is a special case of one-dimensional approximation. In fact, this approximation has been adopted in some previous studies, but it has not been discussed how well this approximation works. So, in the next subsections, we will see the accuracy of the zero-damping approximation for the $f$- and $p_1$-mode frequencies, and then we also propose the one-dimensional approximation for estimating the $w_1$-mode frequency.\n\n\n\\subsection{Zero-damping approximation}\n\\label{sec:Approximation1}\n\nThe zero-damping approximation, neglecting the imaginary part of the eigenvalue, is the simplest one-dimensional approximation, i.e., the eigenvalue, $\\omega$, is assumed to be \n\\begin{equation}\n  \\omega_{{\\rm 1D},f} = \\omega_{r}, \\label{eq:omega_1D0}\n\\end{equation}\nwhere $\\omega_r$ is some real number and $\\omega_{{\\rm 1D},f}$ denotes the eigenvalue with the approximation for the gravitational wave induced by the fluid oscillations. With the zero-damping approximation, one can estimate the frequency, with which the absolute value of $f(\\omega_{{\\rm 1D},f})$ becomes the local minimum. Once the value of $\\omega_r$ would be determined, the frequency of a gravitational wave is given by $\\omega_r\/2\\pi$. As an example, we show the absolute value of $f(\\omega_{{\\rm 1D},f})$ as a function of the frequency for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS in Fig. \\ref{fig:SLy4-13}, where the vertical dashed lines denote the frequencies of the $f$- and $p_1$-modes determined through the proper eigenvalue problem without the approximation and the inserted panel is an enlarged drawing in the vicinity of the $p_1$-mode frequency. \n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{SLy4-13fp1}\n\\vspace{0.5cm}\n\\caption\nFor the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS, we show the absolute value of $f(\\omega_{{\\rm 1D},f})$ as a function of frequency. The vertical dashed lines denote the Re$(\\omega)\/2\\pi$ for the $f$- and $p_1$-modes determined through the proper eigenvalue problem without the approximation. The panel inserted in the figure is an enlarged drawing in the vicinity of the $p_1$-mode frequency. \n\n\\label{fig:SLy4-13}\n\\end{figure}\n\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{Dfp1}\n\\vspace{0.5cm}\n\\caption\nThe relative deviation of the value of $\\omega_r$ determined with the zero-damping approximation from the value of Re$(\\omega)$ determined through the proper eigenvalue problem without the approximation, which is calculated with Eq. (\\ref{eq:DRef}), is shown as a function of $M\/R$ for various neutron star models. The top and bottom panels correspond to the results for the $f$- and $p_1$-modes, respectively. \n\n\\label{fig:Dfp1}\n\\end{figure}\n\nIn order to check the accuracy of the frequencies of the $f$- and $p_1$-modes with the zero-damping approximation, we estimate the relative deviation of them from the corresponding frequencies determined through the proper eigenvalue problem, which is calculated by\n\\begin{equation}\n  \\Delta {\\rm Re}(\\omega_i) = \\frac{{\\rm abs}[{\\rm Re}(\\omega_i)-\\omega_{r,i}]}{{\\rm Re}(\\omega_i)}, \n      \\label{eq:DRef}\n\\end{equation}\nwhere ${\\rm Re}(\\omega_i)$ and $\\omega_{r,i}$ respectively denote the real part of $\\omega$ determined through the proper eivenvalue problem without the approximation and the value of $\\omega_r$ determined with the zero-damping approximation for the $f$- ($i=f$) and $p_1$-modes ($i=p_1$). For various neutron star models, we show the values of $\\Delta {\\rm Re}(\\omega_i)$ for the $f$-mode ($p_1$-mode) in the top (bottom) panel of Fig. \\ref{fig:Dfp1}. From this figure, one can observe that the zero-damping approximation works significantly well.\n\n\n\n\n\\subsection{One-dimensional approximation for the $w_1$-mode}\n\\label{sec:Approximation2}\n\nUnlike the gravitational waves induced by fluid oscillations, such as the $f$- and $p_1$-modes, the imaginary part of the spacetime modes ($w$-modes) becomes generally comparable to the real part of them, as shown in Fig. \\ref{fig:ratio}. So, one cannot estimate the $w_1$-mode frequency with the zero-damping approximation. However, owing to the finding of the emprical relation for Im($\\omega$)\/Re($\\omega$) as shown in Fig. \\ref{fig:ratio}, we propose the one-dimensional approximation for the $w_1$-mode, i.e., the eigenvalue with the one-dimensional approximation is assumed to be\n\\begin{equation}\n   \\omega_{{\\rm 1D},w}=\\omega_r\\left(1+{\\rm i}{\\cal R}_{\\rm fit}\\right), \\label{eq:omega_1D1}\n\\end{equation}\nwhere $\\omega_r$ is some real value, while ${\\cal R}_{\\rm fit}$ denotes the ratio of Im$(\\omega_{w_1})$ to Re$(\\omega_{w_1})$ estimated with Eq. (\\ref{eq:fitting_w1}). Then, one should find the suitable value of $\\omega_r$, with which the absolute value of $f(\\omega_{{\\rm 1D},w})$ becomes local minimum. As an example, in Fig. \\ref{fig:SLy4-08} we show the absolute value of $f(\\omega_{{\\rm 1D},w})$ as a function of the frequency given by $\\omega_r\/2\\pi$ for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS. In this figure, for reference we also show the $w_1$-mode frequency determined through the proper eigenvalue problem with the dashed vartical line. \n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[scale=0.5]{SLy4-08w1}\n\\vspace{0.5cm}\n\\caption\nThe absolute value of $f(\\omega_{{\\rm 1D},w})$ is shown as a function of the frequency for the neutron star model with $1.46M_\\odot$ constructed with SLy4 EOS, where the vertical dashed line denotes the $w_1$-mode frequency determined though the proper eigenvalue problem without the approximation. \n\n\\label{fig:SLy4-08}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[scale=0.5]{Dw1Im}\n\\end{center}\n\\vspace{0.5cm}\n\\caption\nThe relative deviation of the value of $\\omega_r$ determined with the one-dimensional approximation from that of Re$(\\omega)$ determined through the proper eigenvalue problem is shown as a function of stellar compactness for various neutron star models in the top panel. In the bottom panel, we show the relative deviation of the imaginary part of the eigenvalue estimated with the one-dimensional approximation from that of Im$(\\omega)$ determined through the proper eigenvalue problem, calculated with Eq. (\\ref{eq:DIm})\n\n\\label{fig:Dw1}\n\\end{figure}\n\nTo check the accuracy of the one-dimensional approximation, in the top panel of Fig. \\ref{fig:Dw1} we show the relative deviation of the value of $\\omega_r$ determined with the one-dimensional approximation from that of Re$(\\omega)$ determined through the proper eigenvalue problem, calculated with Eq. (\\ref{eq:DRef}) for the $w_1$-mode, as a function of $M\/R$ for various neutron star models. In the bottom panel of Fig. \\ref{fig:Dw1}, we also show the relative deviation, $\\Delta {\\rm Im}(\\omega_{w_1})$, of the imaginary part of the eigenvalue estimated with the one-dimensional approximation, i.e., $\\omega_r{\\cal R}_{\\rm fit}$, from Im$(\\omega)$ determined through the proper eigenvalue problem, which is calculated with\n\\begin{equation}\n  \\Delta {\\rm Im}(\\omega_{w_1}) \n    = \\frac{{\\rm abs}[{\\rm Im}(\\omega_{w_1})- \\omega_r{\\cal R}_{\\rm fit}]}{{\\rm Im}(\\omega_{w_1})}.\n    \\label{eq:DIm}\n\\end{equation}\nFrom this figure, one can observe that the $w_1$-mode frequency can be estimated with the one-dimensional approximation within $\\sim 1\\%$ accuracy independently of the adopted EOSs. On the other hand, the damping rate of the $w_1$-mode can be estimated $\\sim 30\\%$ accuracy, which seems to strongly depend on the accuracy of the empirical relation for Im$(\\omega_{w_1})$\/Re$(\\omega_{w_1})$ given by Eq. (\\ref{eq:fitting_w1}).\n\n\n\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nQuasi-normal modes are one of the important properties characterizing compact objects. In this study, first, we show that the ratio of the imaginary part to the real part of the quasi-normal mode for the $f$- and $w_1$-modes can be expressed as a function of the stellar compactness almost independently of the adopted EOSs, and we derive the corresponding empirical relations. Then, focusing on the $f$- and $p_1$-modes, which are gravitational wave frequencies induced by the stellar fluid oscillations, we examine the accuracy of the zero-damping approximation to estimate the corresponding frequencies of gravitational waves, where the damping rate, i.e., the imaginary part of the quasi-normal mode, is neglected. As a result, we show that one can estimate the frequencies of the $f$- and $p_1$-mode with the zero-damping approximation with considerable accuracy. In addition, we newly propose the one-dimensional approximation for estimating the $w_1$-mode frequency by adopting the empirical relation (that we find in this study) for the ratio of the imaginary part to the real part of the $w_1$-modes, and show that this approximation works well, where one can estimate the frequency within $\\sim 1\\%$ accuracy. \n\n\n\n\n\\bmhead{Acknowledgments}\nThis work is supported in part by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers \nJP19KK0354, \nJP20H04753,  and\nJP21H01088, \nand by Pioneering Program of RIKEN for Evolution of Matter in the Universe (r-EMU).\n\n\n\n\\section*{Declarations}\n\nThe author has no relevant financial or non-financial interests to disclose.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\setcounter{equation}{0} \\ALTsect{\\setcounter{equation}{0} \\ALTsect}\n\\def{\\arabic{section}}.{\\arabic{equation}}{{\\arabic{section}}.{\\arabic{equation}}}\n\\begin{document}\n\\sloppy\n\\begin{center}\n{\\Large \\bf\nBody Fixed Frame, Rigid Gauge Rotations \\\\\nand  Large N Random Fields in QCD}\\\\\n\\vskip .5cm\n{\\large{\\bf Shimon Levit}\\\\}\n\\vskip .5cm\n{\\it Department of Physics}\\\\\n{\\it Weizmann Institute of Science}\\\\\n{\\it Rehovot 76100 Israel\\ \\footnote{ Permanent address. Supported in\npart by\nUS -- Israel Binational Science Foundation grant no. 89--00393}}\\\\\n{\\it and}\\\\\n{\\it Max-Planck-Institut f\\\"ur Kernphysik}\\\\\n{\\it D-6900 Heidelberg, Germany\\ \\footnote{ Supported by Humboldt Award.}}\\\\\n\\end{center}\n\\vskip .5cm\n\\centerline{\\Large{\\bf Abstract}}\n\n\n The \"body fixed frame\" with respect to  local gauge transformations is\n introduced.  Rigid gauge \"rotations\" in QCD and their Schr\\\"odinger  equation\n are studied for static and dynamic quarks.\nPossible choices of the rigid gauge field configuration\n corresponding to a nonvanishing\nstatic colormagnetic field in the \"body fixed\" frame are discussed.\nA gauge invariant variational equation is derived in this frame.\nFor large number  N of colors\n the rigid gauge field configuration is regarded as random\nwith maximally random probability distribution under constraints on\nmacroscopic--like quantities. For the uniform magnetic field\nthe joint probability distribution of the field components\nis determined  by maximizing the\nappropriate entropy under the area law constraint for the Wilson loop.\nIn the quark sector the gauge invariance requires\nthe  rigid gauge field configuration to appear\nnot only as a background  but  also  as inducing\nan instantaneous quark-quark interaction.  Both are random in the large N\nlimit.\n\\vskip .5cm\n\\setcounter{equation}{0} \\ALTsect{Introduction}\n\nStudies of  non perturbative aspects of dynamics\nof non abelian gauge fields will continue to remain one of\nthe focuses of  theoretical activities. These fields appear\nat all levels of the \"elementary\" interactions and even begin to\nenter at a more phenomenological macroscopic level in condensed matter\nsystems. Quantum Chromodynamics represents a prime example of a\nstrongly coupled theory with non abelian gauge fields.  Despite many\nefforts, e.g. instantons \\cite{ins}, large N expansion\n\\cite{lar,fra,wit,mgm,egk,ind},\nlattice gauge theory and  strong coupling\nexpansion \\cite{lat,str}, topological considerations \\cite{pol,top},\n QCD sum rules \\cite{svz}, \"spaghetti\" vacuum \\cite{cop},  light cone\napproach \\cite{bro}, explicit color projection \\cite{jon}, and others\n\\cite{adl,yaf,dua}, the quantitative understanding\nof the basic QCD features is still far from satisfactory. A sustained\neffort with different angles of attack is clearly in order with the hope\nthat accumulated qualitative experience will finally lead to the\ndevelopment of quantitative calculational tools. This paper is a\ncontribution to this effort.\n\n Invariance under local gauge transformations is the most important\nfeature of a non abelian gauge theory. In the framework of the Hamiltonian\nformulation of QCD I wish to explore the consequences of this\ninvariance using some\ngeneral methods common to molecular and nuclear physics. I wish to\ndefine an appropriate generalization of the body-fixed\n (intrinsic, rotating) frame formalism in the context of\nthe local gauge transformations. After doing so one can attempt to\nseparately investigate the\ndynamics of the gauge \"rotations\" of the frame and the intrinsic frame\ndynamics. These would be the analogs of rigid rotations and\nintrinsic vibrations in molecular and nuclear physics. Most of my\ninterest in this paper will concentrate on the study of the \"rigid\ngauge rotations\".\nThe study of the couplings between the \"rotations\" and\nthe \"vibrations\" of the gauge field is deferred to future work.\nTo avoid misunderstanding I wish to stress that by \"gauge rotations\" or\n\"rotations of the gauge field\"\nin this paper I will always mean Eq.(\\ref{rgd}) below, which  includes\nthe proper SU(N) rotation  as well as the inhomogeneous \"shift\" term.\n\nPerhaps the most important conceptual advantage of using the body-fixed\nframe associated with a given symmetry\nlies in the fact that one can freely approximate the dynamics in this\nframe without fears to violate the symmetry.\nIn particular the use of this  formalism appears to be\nfruitful provided there exist such a body-fixed,  intrinsic\nframe in which the \"rotational - vibrational\" coupling can be considered\nas small. This in turn generically\nhappens when the \"rotational inertia\" is much\nlarger than the \"inertia\" associated with the intrinsic motion\nso that a variant of the Born-Oppenheimer approximation is valid.  A typical\nsituation is when the system's ground state is strongly \"deformed\" away\nfrom a symmetric state.\nBy deformation I mean absence of symmetry with respect to\n\"intrinsic transformations\",\ni.e. transformations in the body-fixed frame  and not\nwith respect to the transformations in \"laboratory\". Absence of symmetry\nin \"laboratory\" would correspond to the symmetry breakdown\nwhich can not occur for a local gauge symmetry.\nQuantum mechanical examples of deformed bodies are e.g. non spherical\nmolecules, deformed nuclei, etc.\n\nI do not have a priori arguments that the QCD vacuum\nis strongly \"deformed\" in the above sense. Appearance of various QCD\ncondensates, Ref. \\cite{svz}, suggests  that this may be true. The\ncondensate wavefunction should then play a role of the strongly deformed\nconfiguration. Another positive indication is the large N master field concept,\nRef. \\cite{wit}, according to which a special gauge field  configuration\nshould exist which dominates the vacuum wave function or the\ncorresponding functional integral. It is expected,  however, that the\nmaster  field is not\nsimply a fixed classical configuration. It should rather be regarded as\na statistical distribution allowing to calculate quantities which are\nanalogous to macroscopic\nthermodynamic quantities in statistical physics, i.e. such that their\nfluctuations are suppressed\nin the large N limit, Refs. \\cite{mat,ran}. Glimpses of the meaning of\nthese vague notions were found in various matrix models,\ncf. Refs. \\cite{mgm,egk,mat,ran},\nand, e.g. in 1+1 dimensional QCD, Ref. \\cite{bor}.  If this view point\nis correct then suitably chosen rigidly rotating \"deformed\" gauge field\ncould play a role of the master field provided one understands\nin which sense it should also be statistical.\nThe following formalism will clarify some of these issues and\nprovide a general framework in which they could be further discussed.\n\nWorks in the spirit of our study have already appeared in the\npast,cf. Refs. \\cite{lee,jac,sim,dos} and the analogy with various\ntypes of rotational motion  is frequently used in QCD.\nIn this sense the present study is a continuation of these works.\n\nThis paper is organized as following. In Section 2\nI introduce the transformation to the body-fixed frame in the context\nof  the simplest model of the gauge rotational motion  --\nthe rigid gauge rotor.             Giving a  natural definition\nof the rigid gauge \"rotations\" I  proceed to determine\nthe appropriate generalization of the standard space rigid rotor\n results --  the moment of inertia tensor,\nthe \"body-fixed\"\nframe, the generators of the \"body-fixed\" gauge group\nin terms of which the character of \"deformation\" can be classified, etc.\nDespite severe limitation on the set of the allowed gauge field\nconfigurations the model is gauge invariant.\n I work out in Section 3\nthe quantum mechanics of the  model. As with the\nspace rotor the generators of the \"laboratory\" and \"body-fixed\"\ngauge transformations provide a complete set of quantum numbers\nfor the wave functionals of the model. The vacuum has zero energy and\nis the most disordered state. Higher states correspond to the presence\nof very heavy, i.e. static quarks and antiquarks in the system.\nAs an important example I consider the wave function and the\ncorresponding Schr\\\"odinger   equation  for  a pair of static quark and antiquark.\nThis and other similar equations in the model are\nsimple matrix equation with the inverse \"moment of inertia\" determining\nthe interaction between the color sources and depending on the assumed\nrigid gauge configuration which plays the role of the \"free parameter\".\n\nIn Section 4 I discuss the meaning of the results obtained so far and\npossible choices of the rigid gauge field which physically represents\na non vanishing  colormagnetic field\nin the body-fixed  frame. In the ground state this frame\ndoes not \"rotate\" but has random orientations in local\ncolor spaces at every space point. Introduction of static quarks forces\nthe frame to \"rotate\" quantum mechanically\nat the points where the quarks are situated.  The energy eigenvalues\nof these \"rotations\" are the energies of the quantum states of the\ncolorelectric field generated by the  quarks. The propagator of\n this field\nis the moment of inertia of the model and depends explicitly on the\nassumed configuration of the rigid static colormagnetic field.\nFor zero field the propagator is a simple Coulomb while for\na uniform field diagonal in color the\npropagator  behaves asymptotically as a decaying Gaussian.\n  The so called dual Meissner effect picture of the confinement,\n Ref. \\cite{man}, could be implemented  if a configuration\nof the rigid colormagnetic field is found which \"channels\" the\ncolorelectric field and makes its propagator effectively one dimensional.\nIt turns out that the creation of such a magnetic \"wave-guide\"\nis connected with existance of a zero eigenvalue of a certain\noperator in the model.\n\nSince  the quark color degrees of freedom are treated\n quantum mechanically\nthe model allows for a possibility that confinement of fundamental\nrepresentations  does not automatically mean confinement of higher\nrepresentations.   I discuss this\npossibility and derive a variational equation for the rigid field.\nThis equation is fully gauge invariant.\n\nIn Section 5\nexpecting that  rigid gauge rotations should be relevant for the\nmaster field concept I study the model in the large N limit.\nAny candidate for the master field must be allowed to undergo free\n\"gauge rotations\" which can not be frozen by this limit\nand should induce an interaction between the quarks.\nGoing to the body fixed frame of these rotations I regard the\nrigid gauge field configuration as random and introduce a natural\nrequirement that it is least biased under  constraints that it\nshould reproduce\ngauge invariant quantities which can be regarded macroscopic-like in the\nlarge N limit. This means that it should be maximally random under these\nconstraints. In order to make these ideas explicit I discuss in some\ndetail the case of the uniform\ncolormagnetic field. Such configuration in QCD was already discussed in the\npast, Refs. \\cite{sav,cop} but it seems that its appearance in the interaction\nis a novel feature\nof the model. The detailed form of  this interaction depends on the\ndifferences of the color components of the magnetic field. It is not\nconfining for any finite number of colors. For $N \\rightarrow \\infty$\n I assume  that the form of the density of the color components of\n  the field is known.\nIn 2+1 dimensions I choose  it such that it gives area\nlaw for space oriented Wilson loops.\nI treat then the entire distribution of these components as\n a joint distribution of their probabilities and regard\n the adopted \"single component\" density as an analog of a\nmacroscopic quantity that must be reproduced by this\njoint distribution. I postulate that it must otherwise be maximally random,\n i.e.  must have the maximum entropy (minimum\ninformation content) under suitable constraints. In this way I\nderive the maximally random distribution for this model. I discuss its relation\nto the large N limit of the Schr\\\"odinger   equation for the static quark--antiquark\nsystem. I also give possible generalizations of this development\nto  3+1 dimensions.\n\nIn Section 6 I include dynamical quarks\nand show that the rigid gauge rotor limit corresponds exactly to the\nlimit in QED in which only the instantaneous Coulomb interaction\nbetween the charges is retained. The major difference in QCD is that\nin this limit   the quarks not only interact instantaneously via a more\ncomplicated interaction  controlled by the rigid gauge field configuration,\nbut at the same time are also found  in a static  colormagnetic field\ninduced by this configuration. This dual appearance of the rigid field is\n a consequence of the gauge invariance and in the large N limit\nis apparently the way the\nmaster field should enter the quark sector of the theory. According to the\nideology developed in Section 5 both the field in which the quarks\nmove and their interaction should be considered as random in the large N\nlimit. The random interaction between the quarks  opens interesting\npossibilities to discuss the relationship between confinement and\nlocalization.\n\nThe body fixed  Hamiltonian with dynamical quarks is gauge invariant.\nIts invariance  with  respect to global symmetries however is not\nguaranteed for an arbitrary choice of the rigid gauge configuration.\nI discuss possible variational approaches to determine this configuration\nand derive an analogue of the Hartree-Fock equations for the model.\n\nIn the rest of the Introduction I will establish my notations, cf.,\n Ref.\\cite{bjo}.  I\nconsider the QCD Hamiltonian in d=3 space dimensions in the $A^{0} = 0$\ngauge,\n\\begin{equation}\nH=\\frac{1}{2}\\int d^{3}x [(E_{a}^{i}(x))^{2}+(B_{a}^{i}(x))^{2}]\n+ \\int d^{3}x q_{\\gamma}^{+}(x)[\\alpha^{i} \\left(p^{i}-g\nA_{a}^{i}(x) \\frac{ \\lambda_{\\gamma \\delta}^{a}}{2} \\right)\n+\\beta m]q_{\\delta}(x).\n\\label{ham}\\end{equation}\nwith\n\\begin{equation}\nB_{a}^{i}(x)=\\epsilon_{ijk}(\\partial_{j}A_{a}^{k}+gf_{abc}A_{b}^{j}A_{c}^{k}),\n\\end{equation}\n$i,j,k=1,...,3$;  $\\gamma,\\delta$=1,...,N;  $a=1,...,N^2 - 1$ for SU(N)\ngauge group and  $f_{abc}$ -- the structure constants of the SU(N).\nDirac and flavor indices are omitted and the summation\nconvention for all repeated indices is employed here and in the following.\nThe gluon vector potential  $A_{a}^{i}(x)$ and  minus the electric\nfield  $-E_{a}^i (x)$ are canonically conjugate variables,\n\\begin{equation}\n[E_{a}^{i}(x),A_{b}^{j}(y)]=i\\delta_{ab}\\delta_{ij}\\delta (x-y),\n\\end{equation}\nand the quark fields obey the standard anticommutation relations.\n\nThe Hamiltonian (\\ref{ham}) is invariant under\nthe time independent gauge transformations. Using the matrix valued\nhermitian fields\n\\begin{equation}\nA_{\\alpha\\beta}^{i}(x)=A_{a}^{i}(x) \\frac{ \\lambda_{\\alpha\\beta}^{a}}{2},\n\\; \\; \\; \\; \\;\nE_{\\alpha\\beta}^{i}(x)=E_{a}^{i}(x) \\frac{\\lambda_{\\alpha\\beta}^{a}}{2},\n\\end{equation}\nwhere $\\lambda^{a}$ are the SU(N) generators with the properties\n\\begin{eqnarray}\n[\\lambda^{a},\\lambda^{b}]&=&2if_{abc}\\lambda^{c};\\; \\; \\;\n\\{\\lambda^a,\\lambda^b\\} = \\frac{4}{N}\n\\delta_{ab} + 2d_{abc} \\lambda^c;\\nonumber \\\\\nTr\\lambda^{a}\\lambda^{b}&=&2\\delta_{ab};\\; \\; \\; \\; \\;\n\\lambda_{\\alpha \\beta}^{a} \\lambda_{\\gamma\\delta}^{a} = 2[\\delta_{\\alpha\\delta}\n\\delta_{\\beta\\gamma}-\\frac{1}{N}\\delta_{\\alpha\\beta}\\delta_{\\gamma\\delta}]\n\\end{eqnarray}\none can write the gauge transformation as\n\\begin{equation}\n A^i \\rightarrow SA^{i}S^{+}+\\frac{i}{g}S\\partial^{i}S^{+};\\;\\; \\;\nE^i\\rightarrow SE^{i}S^{+};\\; \\; \\;  q\\rightarrow Sq  \\end{equation}\nwhere S(x) are time independent but x - dependent unitary $N\\times N$\nmatrices, elements of the SU(N) group. The generators of this\n transformation\n\\begin{equation}\n G_{a}(y) \\equiv G_a^A (y) + G_a^q (y),  \\nonumber \\end{equation}\n\\begin{equation}\nG_{a}^A (y) =  \\partial_{i}E_{a}^{i}(y)+gf_{abc}A_{b}^{i}(y)E_{c}^{i}(y),\n\\; \\; G_{a}^q (y) = -gq^{+}(y)\n\\frac{\\lambda^{a}}{2} q(y) \\label{gen} \\end{equation}\nare conserved,\n\\begin{equation}\n\\frac{\\partial G_{a}(x)}{\\partial t}=i[H,G_{a}(x)]=0\\end{equation}\nand it is consistent to impose the Gauss law constraints\n\\begin{equation}\nG_{a}(x)|\\Psi>=0 \\label{gl} \\end{equation}\nfor all physical states.\nAlthough $G_{a}(x)$ do not commute, their commutators\n\\begin{equation}\n[G_{a}(x),G_{b}(y)]=gf_{abc}\\delta (x-y)G_{c}(x)\\end{equation}\nallow to set  them all simultaneously zero.\n\n\n\\setcounter{equation}{0} \\ALTsect{Rigid Gauge Rotor.}\n\nIn this section I will discuss the rigid gauge \"rotations\". Classically\nI define them as gauge field configurations of the type\n\\begin{equation}\n A^i(x,t) = U(x,t)a^{i}(x)U^{+}(x,t)+\\frac{i}{g}U(x,t)\\partial^{i}U^{+}(x,t)\n\\label{rgd}\\end{equation}\nwhere $a^i(x)$ are t-independent, fixed as far as their x-dependence is\nconcerned, \"rigid\" fields which I do not\nspecify and leave them arbitrary for the moment. Eq.(\\ref{rgd}) is\nthe simplest example of the transformation to the \"body fixed\" frame\nof the local gauge symmetry in which I have assumed that the dynamics\nof the field in this frame is  very stiff so that     the field\ncan  be approximately replaced by\nits static  average. In general $a^i$ is of course dynamical but should\nbe viewed as constrained since $U(x,t)$ already contains a third of the\ndegrees of freedom. For non rigid  $a^i$  there is no obvious choice of\nthe body-fixed frame and it can be constrained in a variety of ways,\nsay, $a^3 = 0$ (axial gauge), $\\partial_i a^i = 0$ (Coulomb gauge),\netc. In our language\nthese different gauge fixings correspond to different \"rotating\" frames.\nSince they are \"non inertial\" the dynamics will look very differently\ndepending on the choice of the frame and  different fictitious\nforces, the analogue of  Coriolis and centrifugal forces,\nwill be present.  I am planning to discuss these issues elsewhere.\n\nWith the anzatz (\\ref{rgd}) the covariant derivatives are\n\\begin{equation}\nD^i \\equiv \\partial ^i - igA^i = U(x,t) d^i(x) U^{+}(x,t),\\end{equation}\nwith fixed, rigid\n\\begin{equation}\nd^i(x) \\equiv \\partial ^i - iga^i(x). \\end{equation}\nInserting (\\ref{rgd}) in the Hamiltonian (\\ref{ham})\none finds that the gauge invariant potential term\n$\\sum_{i,a}(B_{a}^{i}(x))^{2} \\sim \\sum_{i,j}Tr[D^i,D^j]^2 = \\sum_{i,j}\nTr[d^i,d^j]^2$ is\nindependent of the $U$'s, i.e. it is fixed, nondynamical in this model.\nThe dynamics of the gauge field\nis governed by the kinetic energy, i.e. the term with\nthe electric field  in (\\ref{ham}).  Using\n$\\partial_0(U\\partial_iU^{+}) = (i\/2)U(\\partial_i\\omega)U^{+}$ with\n$\\omega = 2i U^{+}\\partial_0 U$ one finds\n\\begin{eqnarray}\n-E^{i} = \\partial_0 A^{i} =  \\frac{1}{2g}U [\\omega,d^i] U^{+},\n\\label{dai}\\end{eqnarray}\nand therefore the kinetic energy in (\\ref{ham}) is\n\\begin{equation}\n\\frac{1}{4}\\int d^3 x Tr(\\partial_0 A^i)^2 =\n - \\frac{1}{16g^2}\\int d^3 x Tr\\left( \\omega[d^i,[d^i,\\omega]]\\right)\n \\label{ke} \\end{equation}\nwhere I have disregarded surface terms, ignoring for the moment\npossible non vanishing fields at infinity, non trivial topologies and\nother global issues (cf.  below).\n\nThe double commutator in (\\ref{ke}) with the summation over all indices,\n$x,i$ and the color is the straightforward generalization of the\nfamiliar double vector product summed over all particles indices\n in the moment of inertia tensor\nappearing in the kinetic energy of rigid space rotations of system of\nparticles with fixed relative positions.\n Following this analogy  the energy (\\ref{ke}) of the rigid gauge\n rotations can be written\n\\begin{equation}\nE_{rot} = \\frac{1}{4}\\int d^3 x Tr(\\omega I \\omega) \\label{en} \\end{equation}\nwhere the moment of inertia is defined as a differential matrix operator\nsuch that\n\\begin{eqnarray}\nI\\omega &\\equiv& - \\frac{1}{4g^2} \\left[ d^i,[d^i,\\omega]\\right] = \\nonumber \\\\\n&=& - \\frac{1}{4g^2}\\left(\\partial_i^2 - ig[\\partial_i a^i,\\omega] -\n2ig[a^i,\\partial_i\\omega] - g^2 \\left[ a^i,[a^i,\\omega]\\right]\\right) .\n \\label{mom}\n\\end{eqnarray}\n To obtain the corresponding\nHamiltonian one can use the gauge field part $G^A$ of the generators\n(\\ref{gen}). Using  Eqs. (\\ref{rgd}), (\\ref{dai}) and the\ndefinition (\\ref{mom}) one finds\n\\begin{equation}\nG^A = [\\partial^i - ig A^i,E^i] =\n\\frac{1}{2g}U \\left[d^i,[d^i,\\omega]\\right] U^{+} = -2g U (I\\omega) U^{+}.\n\\label{omg} \\end{equation}\nDefining the gauge generators in the rotating frame\n$\\hat{G} = U^{+} G^A U$, expressing $\\omega = - I^{-1} \\hat{G} \/2g$ from (\\ref{omg}) and\nsubstituting in (\\ref{en}) one finds the Hamiltonian of the rigid gauge\nrotor\n\\begin{equation}\nH_{rot}^A = \\frac{1}{16g^2}\\int d^3 x d^3 y  Tr \\hat{G} (x) I^{-1}(x,y)\n\\hat{G} (y) = \\frac{1}{4g^2}\\int d^3 x d^3 y \\hat{G}_a(x) I_{ab}^{-1}(x,y)\n\\hat{G}_b(y).\n\\label{hrot} \\end{equation}\nwhere $I_{ab}^{-1}(x,y) = (1\/4) Tr(\\lambda^a I^{-1}(x,y)\n\\lambda^b)$ is proportional to the inverse of the operator $-d^i_{ac}\nd^i_{cb}$ with $d^i_{ab} = \\partial_i \\delta_{ab} - g f_{abc} a^i_{c}$\nand I assumed  that this operator  does not have zero eigenvalues.\nIn a more careful way of handling fields at infinity one should avoid\nthe integration by parts in (\\ref{ke}). The  inverse \"moment\nof inertia\" operator is then replaced by a less transparent\n\\begin{equation}\nK_{aa'}(x,x') = \\int d^3 y \\left[ d^i_{bc}(y) I^{-1}_{ca}(y,x)\\right]\n\\left[ d^i_{bc'}(y) I^{-1}_{c'a'}(y,x')\\right].\\end{equation}\nMost of the following results remain valid for both forms of this\noperator.\n\nThe meaning of the preceeding expressions is quite obvious. They are the\nfield-theoretic generalization of the standard rigid rotor results.\nThe unbroken\nlocal gauge symmetry of QCD means that there are free SU(N) color\ngauge \"rotations\" at\nevery space point. Expression (\\ref{hrot}) shows that the \"rotations\"\nat different  points as well as around different color axes are coupled\nvia the non diagonal\nelements of the moment of inertia \"tensor\" $I_{ab}(x,y)$  in the manner\nsimilar to the coupling between rigid rotations around different space axes in\nsystems of particles.\n\nIt does not seem to be useful to diagonalize the operator $I^{-1}$ in\n(\\ref{hrot}).\nThe standard diagonal form of the rigid rotor Hamiltonian , i.e., $H = \\frac{1}{2}\\sum_a\nL_a^2\/I_a$  can be usefully achieved only in the case of\nrotations corresponding to a single SU(2) group to which the familiar\nrigid space rotations  belong. Diagonalizing the moment of inertia\nin the case of higher groups will introduce combinations\nof the generators multiplied by  matrices of orthogonal rotations.\nThese in general will not have\nthe group commutation relations.  Already for a single\nSU(3) the group O(8) of\nrotations in the adjoint space needed in order to diagonalize the\nmoment of inertia is much larger than SU(3).\n\n The actual values of the moment of inertia depend on the rigid\nconfiguration $a^i (x)$ of the gauge field via the\nexpression (\\ref{mom}). This comprises the \"free parameter\" of the rigid\ngauge rotor model. For abelian theory or alternatively\nin the limit $g \\rightarrow 0$ the inverse of $I(x,y)$\nappearing in Eq.(\\ref{hrot})\nis just the Coulomb propagator.\n In the opposite large $g$ or long wavelength limit\n$I(x,y)$ becomes a local tensor  given by the last term in (\\ref{mom})\nwhich is obviously the SU(N)\ngeneralization of the moment of inertia expression.\n\n An important feature of\nthe Hamiltonian  (\\ref{hrot}) is that despite the severe limitation of the\nallowed gauge field configurations imposed by  (\\ref{rgd}) it remaines\ngauge invariant. This is because (\\ref{hrot}) depends on $\\hat{G}$\nrather than $G^A$. Under a gauge transformation $U \\rightarrow SU$,\n$G^A$ transforms as $SG^A S^{+}$ so that  $\\hat{G}(x)$ and therefore\n$H_{rot}^A$ stay invariant. The gauge invariance of (\\ref{hrot}) is the\nsimplest illustration of the usefulness of the introduction of the\nbody fixed frame. One can freely approximate the dynamics in this frame\nwithout fears of violating the symmetry with respect to which the frame\n has been defined, i.e. the local gauge symmetry in the present case.\n\nConsider another transformation, $U \\rightarrow US$. Referring to\nEq.(\\ref{rgd}) one can interpret this transformation\neither as the change of $U$ i.e. the transformation of the intrinsic\nframe with respect to the rigid \"shape\" $a^i$ or\nas the change of $a^i$ ,\n $a^i \\rightarrow Sa^{i}S^{+}+\\frac{i}{g}S\\partial^{i}S^{+}$, i.e.\nthe  transformation of the intrinsic \"shape\" with respect\nto the intrinsic frame.\nSuch transformations obviously form a group of local SU(N) gauge\ntransformations  which I will call\n the intrinsic or \"body fixed\"\n  gauge group to distinguish it from the \"laboratory\" gauge\ngroup of the ordinary gauge transformations. According to two different\ninterpretations of the intrinsic gauge transformations\ngiven above one has two options. One option is to\n regard  $\\hat{G}$'s as the generators of the intrinsic group.\nThey act on  the dynamical variables $U$ but they have a disadvantage in\nthat  the \"laboratory\"\ngroup is not completely independent of such an  intrinsic group, e.g.\nthey both have identical Casimir operators. Another option is to formally\nintroduce operators which gauge transform  the intrinsic variables $a^i$.\nThey will have the same form as $G^A$'s but with $a^i$ replacing $A^i$.\nDefined in this way the intrinsic group will be completely independent\nof the \"laboratory\" gauge group but will act on nondynamical variables\nwhich do not appear in the wavefunctions. Convenience should dictate\nwhich one to use.\n\nThe above introduction of the intrinsic vs \"laboratory\" gauge groups\nis obviously quite general with e.g. the definition of $\\hat G(x)$ being\nindependent of the rigid rotor restrictions set by fixing $a^i$\nto be nondynamical in (\\ref{rgd}).\nUnlike the local gauge symmetry in \"laboratory\", the symmetry in the\n\"body fixed\" frame can be broken.\nE.g., the gauge invariant Hamiltonian  (\\ref{hrot})\n is in general not invariant under the transformations of the intrinsic\ngauge group. This is a simple example of the situation to which I\nreferred earlier as a possible existence of \"deformation\"\nvs impossibility of the symmetry breakdown in the context of non\nabelian local\ngauge theory. The character of the deformation can be classified\nusing the intrinsic gauge group, e.g. in classification of\npossible \"deformed shapes\" of the rigid gauge rotor (\\ref{hrot})\nby the  transformation properties of the moment of inertia\n$I_{ab}(x,y)$ under this group. Here I obviously adopt the second\ninterpretation of the intrinsic group. The invariance of\n$I_{ab}(x,y)$ under all\nintrinsic transformations would be analogous\nto the spherical rotor limit  in the space rotation case.\nThe invariance under a continuous subgroup of the intrinsic group is the\nanalog of the axial symmetric rotor, etc.  Discrete intrinsic\nsubgroups should also be considered.\n\nConsider a rigid gauge configuration $a^{i'}$ which is a gauge\ntransform of $a^i$, $a^{i'} = S(a^i +(i\/g)\\partial_i)S^{+}$.\n  The Hamiltonian\n(\\ref{hrot}) will have the same form with the same moment of inertia\nbut with $\\hat{G}$ replaced by $S^{+}\\hat{G} S$. The eigenvalues of this\ntransformed Hamiltonian will not change and will therefore depend only\non gauge invariant combinations of the rigid $a^i$, i.e. on the Wilson\nloop variables $Tr P exp (ig\\oint a^i dx_i)$.\n\\setcounter{equation}{0} \\ALTsect{Static Quarks}\n\nSo far I have discussed the rigid gauge rotor limit\nof only the first term in (\\ref{ham}). The resulting $H_{rot}^A$ is relevant\nfor the discussion of very heavy quarks.\nThey can be considered static as far as their translational motion is\nconcerned. They still have a wavefunction describing the motion of their color\ndegrees of freedom. Because of (\\ref{gl}) this motion is coupled to\nthe \"rotations\" of the gauge field which I will treat\nusing (\\ref{hrot}).\n\nIn the limit of $m\\rightarrow \\infty$  the quark kinetic energy\nterm $q^{+}\\vec{\\alpha}\\vec{p}q$  and the quark color current coupling\n$q^{+}\\vec{\\alpha}\\lambda^{a}q$ in Eq.(\\ref{ham}) can be neglected\nand the resulting Hamiltonian decouples into a part containing the gauge field\nand another  containing the quarks, $H=H_{A}+H_{q}$, where\n$H_{A}$ is the first term in (\\ref{ham})\nand $H_q = m\\int d^3x q^{+} (x)\\beta q(x)$.\nThe coupling  appears only via the Gauss law constraint, Eq.(\\ref{gl}).\nThe wave function can not be taken\nas a product $\\Psi=\\Psi(q)\\Psi(A)$ but should be a local color singlet.\nIn the representation  in which\n\\[ \\beta=\\left( \\begin{array}{cc}  1 & 0 \\\\  0 & -1\\end{array}\\right) \\]\n\\[ q_{\\alpha}(x)=a_{\\alpha}(x) \\left( \\begin{array}{cc} 1 \\\\ 0\\end{array}\n\\right)   + b_{\\alpha}^{+}(x) \\left( \\begin{array}{cc} 0 \\\\ 1\\end{array}\n\\right)  \\]\nwith\n\\begin{eqnarray}\n\\{a_{\\alpha}(x),a_{\\beta}^{+}(y)\\}&=&\\delta_{\\alpha\\beta}\\delta (x-y)\n\\nonumber \\\\\n\\{b_{\\alpha}(x),b_{\\beta}^{+}(y)\\}&=&\\delta_{\\alpha\\beta}\n\\delta (x-y), etc...\\nonumber\n\\end{eqnarray}\nthe eigenfunctions of $H_{q}$ are trivially written down.\nConsider, e.g.,\n\\begin{eqnarray}\n|\\Psi(q)>\\equiv |vac(q)> &=& |0>, \\label{vac} \\\\\n|\\Psi(q)>\\equiv |x_{0},\\alpha> &=& a_{\\alpha}^{+}(x_{0})|0> \\label{1qu} \\\\\n|\\Psi(q)>\\equiv |x_{0},\\alpha ;y_{0},\\beta > &=& a_{\\alpha}^{+}(x_{0})b_{\\beta}\n^{+}(y_{0})|0> \\label{2qu} \\end{eqnarray}\nThese wave functions describe respectively  zero quarks, one\nstatic quark at $x_{0}$ with color component $\\alpha$ and a static\nquark - antiquark pair at $x_0$ and $y_0$. It is easy to form local\ncolor singlets with these wave functions.\n For e.g. the quark-antiquark pair it is\n\\begin{equation}\n |\\Psi>=\\sum_{\\alpha\\beta} \\Psi_{x_{0},\\alpha ;y_{0},\\beta}(A)|x_{0},\n\\alpha ;y_{0},\\beta> \\end{equation}\nwith the wave functional\n$\\Psi_{x_{0},\\alpha ;y_{0},\\beta}(A)$ satisfying\n\\begin{eqnarray}G_{a}^{A}(x)\\Psi_{x_{0},\\alpha ;y_{0},\\beta }(A) =\ng\\delta(x-x_{0})\\frac{\\lambda_{\\alpha\\alpha^{'}}^{a}}{2}\n\\Psi_{x_{0},\\alpha^{'} ;y_{0},\\beta}(A)+g\\delta(x-y_{0})\n\\frac{\\bar{\\lambda}_{\\beta\n\\beta^{'}}^{a}}{2}\\Psi_{x_{0},\\alpha ;y_{0},\\beta^{'}}(A)\\label{cnd}\n\\end{eqnarray}\nwhere $\\bar{\\lambda}_{\\alpha\\beta}^{a}=-\\lambda_{\\alpha\\beta}^{a*}$.\nThe wave functional of the\ngauge field should be a singlet at every point in space except\nat the position of the quarks where it should transform as  N and $\\bar{N}$\nmultiplets of SU(N).  This constraint together with the\nSchr\\\"odinger   equation\n\\begin{equation}\nH_{A}\\Psi_{x_{0},\\alpha;y_{0},\\beta}(A)=E\\Psi_{x_{0},\\alpha;y_{0},\\beta}(A)\n\\end{equation}\ncompletely defines the problem for the gauge field.\n\n In the rigid gauge rotor limit $H_A$ is given by Eq.(\\ref{hrot}).\nThe wavefunctions of this Hamiltonian  are general functionals\n $\\Psi [U(x)]$ of the SU(N) matrices $U_{\\alpha \\beta} (x)$.\nTheir scalar product is determined by functional integration over the\n$U$'s with the corresponding group invariant measure.\nThe vacuum wave functional must obey $G_a^A (x)\\Psi_{vac}[U] = 0$.\nThis means that it is a constant independent of $U_{\\alpha \\beta}(x)$.\nSince also $\\hat{G}_a (x)\\Psi_{vac}[U] = 0$ the vacuum energy is zero according\nto (\\ref{hrot}).\nRegarding the parametrization of the $U$'s in terms of the appropriate Euler\nangles of the SU(N) rotations at every space point, the constant\n$\\Psi_{vac}[U(x)]$ means that all the \"orientations\" of $U(x)$ at all points\nare equally\nprobable, i.e. there are no correlations between the \"orientations\"\nof the rigid gauge rotor at different points. This is as \"random\" as\nthe distribution of the $U$'s can get. The absence of correlations\nis the property only of the vacuum. For other states the \"orientations\"\nof the gauge fields at different space points are correlated via the\n\"moment of inertia\" operator.\n\nIn order to discuss\nthe  wave functions with non zero number of quarks it is sufficient\nto know some simple\nproperties of the gauge generators $G_a^A(x)$ and $\\hat{G}_a(x)$. Since\n$\\hat{G}_a(x)$ are gauge scalars, they commute with $G_a^A (x)$,\n\\begin{equation}\n[G_a^A(x),\\hat{G}_b(y)] = 0 \\end{equation}\nwhich means that together the generators of the \"laboratory\" and the\nintrinsic gauge groups provide a complete set of commuting\nquantum numbers for the wave functionals $\\Psi [U_{\\alpha \\beta }(x)]$.\nIndeed, since the Casimir operators for $G^A$'s and $\\hat{G}$'s coincide one has\ne.g., for the SU(2) the $(G_a^A(x))^2$, $G_3^A(x)$ and $\\hat{G}_3(x)$,\ni.e. three local commuting operator fields  for the three fields of the\nEuler angles needed to specify the $U(x)$.\nIn the SU(3) one has eight fields of the \"Euler angles\" and eight local\ncommuting generators made off $G^A$'s and $\\hat{G}$'s --\n the two group Casimir operators, one Casimir operator\nof an SU(2) subgroup for $G^A$'s, say $\\sum_{a=1}^{3} (G_a^A)^2(x)$ and the\ncorresponding one for the $\\hat{G}$'s and respectively two\npairs of the Cartan generators -- $G_3^A(x)$, $G_8^A(x)$ and $\\hat{G}_3(x)$ and\n $\\hat{G}_8(x)$. This counting continues correctly for any N, i.e.\n$N-1$ for the SU(N) Casimir operators, $2((N-1)+(N-2)+... +1)$\nfor the  Casimir operators of pairs of SU$(N-1)$...SU(2) subgroups and $2(N-1)$\nfor the Cartan generators. Altogether there are $N^2 - 1$ local commuting\noperators as needed. An eigenfunction\nof this complete set of operators is  the Wigner function\n$D_{K K^{\\prime}}^L(U(x))$ of U at a certain space point. $K$ and $K^{\\prime}$\nare the quantum numbers of the \"laboratory\" and the intrinsic groups and $L$\ndetermines the representation.\n\nSince under an infinitesimal gauge transformations $U\\rightarrow\n(1+i\\epsilon_a(x) \\frac{\\lambda_a}{2})U$  one can easily verify that\n\\begin{eqnarray}\n\\left[ G_{a} (x),U_{\\alpha \\beta}(y) \\right] &=& g \\delta (x - y)\n\\frac{\\lambda^a_{\\alpha \\gamma}}{2} U_{\\gamma \\beta}(x) \\nonumber \\\\\n\\left[ G_{a} (x),U_{\\alpha \\beta}^{+}(y) \\right] &=& - g \\delta (x -y)\nU_{\\alpha \\gamma}^{+}(x)\\frac{\\lambda^a_{\\gamma \\beta}}{2} \\label{com} \\\\\n\\left[ \\hat{G}_{a} (x),U_{\\alpha \\beta}(y) \\right] &=& g \\delta (x - y)\nU_{\\alpha \\gamma}(x) \\frac{\\lambda^a_{\\gamma \\beta}}{2} \\nonumber \\\\\n\\left[ \\hat{G}_{a} (x),U_{\\alpha \\beta}^{+}(y) \\right] &=& - g \\delta (x -y)\n\\frac{\\lambda^a_{\\alpha \\gamma}}{2} U_{\\gamma \\beta}^{+}(x)\n \\nonumber \\end{eqnarray}\n All the operators in the rigid gauge rotor model are functions of the\n$G$'s and $U$'s. E.g. consider the electric field operator. According\nto Eq.(\\ref{dai}) it is\n\\begin{equation}\nE^i =  -\\frac{1}{4g^2} U [d^i,I^{-1} \\hat{G}] U^{+} \\label{elf} \\end{equation}\nwhere I have expressed $\\omega$ in terms of $\\hat{G}$ using (\\ref{omg}).\n\nUsing the relations (\\ref{com}) it is easy to write the general form of\nthe wave functions for a single quark and for a quark--antiquark pair,\n\\begin{eqnarray}\n\\Psi_{x_0,\\alpha}[U] = U_{\\alpha \\gamma}(x_0)c_{\\gamma} \\nonumber \\\\\n\\Psi_{x_0,\\alpha;y_0,\\beta}[U] = U_{\\alpha \\gamma}(x_0)U_{\\delta\\beta}\n^{+}(y_0)c_{\\gamma \\delta} \\label{wfn}\\end{eqnarray}\nThey satisfy the conditions (\\ref{cnd}) following from the\nGauss law with constant coefficients\n $c_{\\gamma}$ and $c_{\\gamma \\delta}$\nwhich give  the probability amplitudes of the intrinsic quantum\nnumbers $\\gamma$ and $\\delta$. They should be normalized,\n$\\sum |c_{\\gamma}|^2 = 1 ; \\sum |c_{\\gamma \\delta}|^2 = 1$ to assure\nthe normalization $\\int d[U(x)] |\\Psi [U]|^2 = 1$\n\nThese amplitudes\n must be found by solving the corresponding Schr\\\"odinger  equations but before\ndescribing this I wish to remark that the above form of\nthe wave functions is valid also when the limitation of\nthe rigid gauge rotations is relaxed and the most general gauge\nconfigurations are allowed. The parametrization (\\ref{rgd}) is still very\nuseful but now with fully dynamical fields $a^i$\n the variation of which should be limited only by a \"gauge fixing\"\ncondition to avoid overcounting as described above.\n  The dynamics will of course be\nthat of the full QCD but  the wave functions\nof the static quark and the quark--antiquark pair will have the same form\n(\\ref{wfn}).  The difference will be that amplitudes $c_{\\gamma}$ and\n$c_{\\gamma \\delta}$ will be functionals of $a^i(x)$  describing the\nspace and\ncolor fluctuations of the \"string\" attached to the quark or between the\nquark and the antiquark. In the rigid gauge rotation case there are only\ncolor fluctuations described by constant amplitudes.\n\nFor quarks in higher representations the wave functions have the same\nform with $U$ replaced by the appropriate Wigner D-function. E.g. in\nthe adjoint representation\n\\begin{eqnarray}\n\\Psi_{x_0,a}[U] = Tr(U(x_0)\\lambda^aU^{+}(x_0)\\lambda^b)c_b,\n\\nonumber \\end{eqnarray}\netc.\n\nI will now derive the Schr\\\"odinger   equation for the string amplitudes\n$c_{\\gamma}$ and $c_{\\gamma\\delta}$.\n Acting with the Hamiltonian  (\\ref{hrot}) on (\\ref{wfn}), using (\\ref{com})\nand the orthogonality of $U$'s  with respect to the integration over the\ngroup, $\\int dU U_{\\alpha \\beta}^{*} U_{\\mu \\nu} = \\delta_{\\alpha \\mu}\n\\delta_{\\beta \\nu}$,  I find\n\\begin{eqnarray}\nQ_{\\alpha \\gamma}(x_0)c_{\\gamma} =  E c_{\\alpha}, \\nonumber \\\\\nQ_{\\alpha \\gamma}(x_0)c_{\\gamma \\beta} + Q^{*}_{\\beta \\mu}(y_0)c_{\\alpha \\mu}\n- P_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0)c_{\\gamma \\mu} =\nE c_{\\alpha \\beta}, \\label{sch1} \\end{eqnarray}\nwhere I denoted\n\\begin{eqnarray}\nQ_{\\alpha \\gamma}(x_0) = \\frac{1}{4} I_{ab}^{-1}(x_0,x_0)(\\lambda^a\n\\lambda^b)_{\\alpha \\gamma}, \\\\\nP_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0) =  \\frac{1}{2} I_{ab}^{-1}(x_0,y_0)\n\\lambda^a_{\\alpha \\gamma} \\lambda^b_{\\mu \\beta}.\\end{eqnarray}\nIn SU(2) $Q_{\\alpha \\gamma}$ takes a particularly simple diagonal form,\n$Q_{\\alpha \\gamma} = \\delta_{\\alpha \\gamma}(1\/4)I_{aa}^{-1}(x_0,x_0)$.\nand is the eigenvalue for a single quark. For quarks in e.g. adjoint\nrepresentation the lambda matrices in the expressions above are replaced\nby the corresponding group generators $if_{abc}$.\nThe first two terms in the second line of\n(\\ref{sch1}) are the quark and the antiquark self\nenergies whereas the last term is their interaction. In QCD one expects\nthat terms like $Q$ are inflicted by the long and short distance\ndivergences and should be properly regularized which I will assume for the\nrest of the paper. I will further assume the translational invariance of\n$Q$, i.e. its independence of $x_0$. One can then  rewrite Eq.(\\ref{sch1})\nby transforming it to the basis in\nwhich $Q$ is diagonal. Defining its eigenvectors\n$Qb^{(n)} = \\epsilon_n b^{(n)}$ and expanding\n$c_{\\gamma \\beta} = d_{mn} b_{\\gamma}^{(m)} b_{\\beta}^{(n)*}$ one finds\n\\begin{equation}\n<kl| P (x_0,y_0) |mn> d_{mn} = (E - \\epsilon^k - \\epsilon^l) d_{kl}\n\\;\\;\\; {\\rm(no\\;\\;sum\\;\\;over\\;\\;k\\;\\;and\\;\\;l)} \\label{peq} \\end{equation}\nwhere $<kl| P |mn> = P_{\\alpha \\beta , \\gamma \\mu } b_\\gamma^{(m)}\nb_\\mu^{(n)*} b_\\alpha^{(k)} b_\\beta^{(l)*}$. The Schr\\\"odinger  equation (\\ref{peq})\nis $N^2 \\times N^2$ matrix equation and the most interesting question of course\nconcerns the dependence of its eigenvalues on the distance $|x_0 - y_0|$\nfor various possible choices of the rigid gauge field configuration\n$a^i(x)$ on which the matrix $P$ depends. I will address this question\nin the next section.\n\n\\setcounter{equation}{0} \\ALTsect{Choices of The Rigid Field. Mean Field Equations.}\n\nThe rigid configuration if it exists in QCD must reflect the\nproperties of the gluon condensate of the  vacuum. One of the more\naccepted views of the QCD vacuum  is that this is a condensate of non\ntrivial topological configurations -- the Z(N) vortices,\nc.f.,\\cite{top}.  Although\nsuch configurations are easily incorporated in the above formalism\nI was not able to overcome\ntechnical difficulties in working out a theory of their condensation.\n\nOn a heuristic level each Z(N) vortex carries\n a unit of flux of the colormagnetic\nfield. Condensation of the vortices presumably\nmeans that there is a non zero average of this field in  the vacuum.\n Of course due to unbroken local gauge symmetry  it\nmust undergo free \"gauge rotations\" at each space point. In the ground\nstate this means that there are\nequal probabilities of all the \"orientations\"   yielding  zero\naverage value in the laboratory. The finite   average\nvalue of the condensate field\ncan only be \"seen\" in the \"body fixed\" frame and should appear in this\npicture in the manner similar to\n $a^i$ in the expression (\\ref{rgd}) for our rigid gauge\nrotations. The field strength\n\\begin{equation}\nB^i(x) = U(x)b^i(x)U^{+}(x),\\,\\,\\, \\  b^i = \\frac{i}{g} \\epsilon_{ijk}\n [d^j,d^k], \\end{equation}\nalso averages to zero in the ground state but has a non zero value $b^i$\nin the \"body fixed\" frame.\n\nVia the dynamics of $U(x)$ the anzatz (\\ref{rgd})  leads to colorelectric\nfield  (\\ref{elf}) which propagates away from points\nwhere $\\hat{G}(x)$ is non zero, i.e. from  the location of static quarks.\nThe propagator of this field is controlled by the condensate\nfield $a^i$ which enters the expressions for $I^{-1}$ and $d^i$.\nThis propagator is a long range Coulomb potential for zero $a^i$ and\n  is a Gaussian for\n$a^i$ corresponding to a uniform colormagnetic $b^i$ (cf., below).\nThe screening of the propagation range of the colorelectric field\nin the presence of the colormagnetic \"condensate\"  $b^i$ is\nreminiscent of the dual to the Meissner effect of screening of\na magnetic field by the electric condensate of a superconductor.\nThis possibility of the dual Meissner effect is  of course a\nstandard scenario for confinement in QCD. It is expected that\ntubes of flux of the colorelectric field are formed which\nconnect quarks and make their energy depend linearly on the distance.\n\nIn the present formalism a way to attempt to model the formation\nof a confining string is to look for  such a configuration of\nthe rigid field $a^i(x)$ for which the propagator $I^{-1}$ behaves\nroughly speaking  as\none dimensional for large separations along some  given  line in space\nat the end of which quarks can be placed. This means that\na sort of magnetic \"wave guide\" should be constructed so that\nthe Green's function of the operator $-d^2_{ab} =\n-d^i_{ac} d^i_{cb} =\n -(\\partial_i \\delta_{ac} - g f_{acc'} a^i_{c'})(\\partial_i - g f_{cbb'}\n a^i_{b'})$ is\nasymptotically  $\\propto |x-x'|$ along, say, one of the coordinate\naxes. In order to see the difficulties in finding such a configuration\nconsider for simplicity 2 space dimensions and choose\n$a^1 = c(y)$ and $a^2 = 0$ with an arbitrary $c(y)$. This choice\ncorresponds to the colormagnetic field $b(y) = \\partial_y c(y)$\ndepending only on one coordinate $y$. The operator to invert is then\n\\begin{equation}\n-(\\partial_x -ig c(y) \\cdot F)^2- \\partial_y^2   \\end{equation}\nwhere I denoted the color spin matrices $F^a_{bc} = if_{bac}$\nand $c(y) \\cdot F = c_a (y) F^a$. The propagator  is then\n\\begin{equation}\n\\int_{-\\infty}^{\\infty}\n dk e^{ik (x - x')} \\sum_{n} \\frac{\\chi_n(k,y) \\chi_n (k,y')}\n{\\epsilon_n (k)}, \\label{prop} \\end{equation}\nwhere $\\chi_n (k,x)$ and $\\epsilon_n (k)$ are solutions of\n\\begin{equation}\n [-\\partial_y^2 + (k +c(y) \\cdot F)^2 ] \\chi_n (k,y) = \\epsilon_n (k)\n\\chi_n (k,y). \\label{lan} \\end{equation}\nIn order to achieve the desired confining behaviour of the propagator\nthe sum in (\\ref{prop}) must be\n$\\sim k^2$ for $k \\rightarrow 0$. The simplest\nis to assume that the lowest eigenvalue of (\\ref{lan}),\n$\\epsilon_0 (k)$ should vanish  as $k^2$ for small $k$. However\nthe operator in (\\ref{lan}) is a sum of squares and does not have zero\neigenvalues for non trivial regular $c(y)$. It is also not symmetric\nin $k$ for small values of $k$ but this seems to be less of  a problem.\nThe same conclusions seem to hold in 3 space dimensions. It is quite\npossible that perhaps a singular configuration $a^i$ exists\nwhich leads to  zero eigenvalue in (\\ref{lan}) at zero $k$\nbut I was not able to find it.\n\nThe strong coupling limit of lattice QCD suggests\n that  quarks in the fundamental reperesentation are confined\nwhereas they are only\nscreened if put  in the adjoint representation. This\ncrucial difference  comes from fairly simple quantum mechanics\nof color degrees of freedom related to  matching of group\nrepresentations in neighboring lattice points.\n In our rigid gauge rotor model a similar\nsimple quantum mechanics of  colors is retained. As a result the\neigenenergies of a systems of static quarks will be determined by\ndifferent\ncombinations of the color components of $I^{-1}$ depending on the\nrepresentation of the quarks.  E.g.,  as already mentioned\nin Section 3  when the quarks are taken in\nthe adjoint representation the $\\lambda$ matrices in the expressions\nfor $P$ and $Q$ in the Schr\\\"odinger  equations (\\ref{sch1})\nare replaced by their adjoint counterpartners $F$.\n\nIn order to find the optimal $a^i$ in a systematic way  one can\nfollow a variational approach and  minimize the\nground state energy of the rigid gauge rotations for  fixed positions\nof static quarks.\n This energy is given by a sum\nof the lowest eigenenergy of $H^A_{rot}$, Eq.(\\ref{hrot}) and the\ncolormagnetic energy given by the second term in (\\ref{ham}) with\nrigidly \"rotating\" $A(x)$, Eq.(\\ref{rgd}), i.e.\n\\begin{equation}\nE[a^i] = E_{rot}[a^i]  - \\frac{1}{2g^2}\\int d^3x Tr [d^i,d^j]^2 \\end{equation}\nVariation of this expression gives\n\\begin{equation}\n\\partial_i f^{ij} - ig [a^i,f^{ij}] = \\frac{1}{2}\\frac{\\delta E_{rot}}\n{\\delta a^i} , \\label{mfld} \\end{equation}\nwhere $f^{ij} = (i\/g) [d^i,d^j]$ .\n Eq.(\\ref{mfld}) is obviously gauge invariant.\n In the vacuum $E_{rot}$ is zero and the minimization of the second\nterm  simply gives the classical equation for $a^i$ in the vacuum.\nFor a quark- antiquark system $E_{rot}$ is non trivial and\ndepends on the distance between the quarks.  I plan to discuss\nthe solutions of the equation (\\ref{mfld}) and their relation\nto confinement elsewhere.\n\nIn the rest of this Section  as an illustration of a simple\n choice for the rigid field $a^i$ which allows\nto obtain some analytic results I consider it  to be diagonal,\n$a^i_{\\alpha \\beta} (x) = \\delta_{\\alpha \\beta} a^i_\\alpha (x)$.\nThe moment of inertia operator with such $a^i$ is\n\\begin{equation}\n-\\frac{1}{4g^2}\\left[d^i ,[d^i ,\\omega ] \\right] _{\\alpha \\beta} =\n-\\frac{1}{4g^2} \\left[ \\partial ^i - ig(a^i_\\alpha - a^i_\\beta )\\right] ^{2}\n\\omega_{\\alpha \\beta} . \\label{dmi} \\end{equation}\nUsing Green's function satisfying\n\\begin{equation}\n\\left(\\partial ^i - ig(a^i_\\alpha - a^i_\\beta )\\right) ^{2}\n J_{\\alpha \\beta }(x,y) = - \\delta (x-y),\\label{grf} \\end{equation}\nand following the procedure leading to Eq. (\\ref{hrot}) one finds the rigid\ngauge rotor Hamiltonian in this case\n\\begin{equation}\nH = \\frac{1}{4} \\int d^2 x d^2 y \\hat{G}_{\\alpha \\beta}(x) J_{\\alpha \\beta}(x,y)\n\\hat{G}_{\\beta \\alpha}(y). \\end{equation}\nThe Schr\\\"odinger   equation for the static quark--antiquark wave function\n(\\ref{wfn}) has the form (\\ref{peq}) with\n\\begin{equation}\nQ_{\\alpha \\gamma}(x_0) = \\frac{1}{4} \\delta_{\\alpha \\gamma}\\left[\n\\sum_\\beta J_{\\alpha \\beta}(x_0,x_0) - \\frac{1}{N}\n\\left(2 J_{\\alpha \\alpha}(x_0,x_0) -\n\\frac{1}{N}\\sum_\\beta J_{\\beta \\beta}(x_0,x_0)\\right)\\right] \\end{equation}\n\\begin{eqnarray}\nP_{\\alpha \\beta , \\gamma \\mu }(x_0,y_0) &=&- \\frac{1}{2}\n \\delta_{\\gamma \\mu} \\delta_{\\alpha \\beta} J_{\\alpha \\gamma}(x_0,y_0) + \\\\\n&+& \\frac{1}{2N}\\delta_{\\alpha \\gamma} \\delta_{\\mu \\beta}\\left[J_{\\beta \\beta}\n(x_0,y_0) + J_{\\gamma \\gamma}(x_0,y_0)\n - \\frac{1}{N} \\sum_\\nu J_{\\nu \\nu}(x_0,y_0)\\right]. \\nonumber \\end{eqnarray}\nThe diagonal components of $J$ are simple Coulomb propagators independent\nof the color so that the expressions for $Q$ and $P$ can be simplified\n further but I will not go into the details of this. Instead\nI will now consider the choice of $a^i$\nwhich corresponds to a much discussed in the\nliterature situation of a uniform colormagnetic field.\nI emphasize that in the present model  this field is uniform in the\nintrinsic, \"body fixed\" frame.   For simplicity I will\nfirst work in 2+1 dimensions  and will  try to extend to 3+1 in the\nnext section. I set\n\\begin{equation}\n a^i_\\alpha (x) =\n \\frac{1}{2}b_{\\alpha} \\epsilon_{ij}x^j\n\\label{cb1} \\end{equation}\nwhere the space indices $i,j$ presently run over the values 1 and 2.\nIn two space dimensions one can take $b$ diagonal in  color\nsince the transformation diagonalizing it is a part of $U$'s in\n (\\ref{rgd}).\nExplicit expression for $J$ is easily obtained\nin this case from the known  Green's function\nof a Schr\\\"odinger   equation in a constant magnetic field, cf. Ref.\\cite{fey},\n\\begin{equation}\nJ_{\\alpha \\beta}(x,y) = \\frac{1}{4\\pi} e^{i(gb_{\\alpha \\beta}\/2)\n\\epsilon_{ij} x^i y^j} \\int_0^{\\infty} \\frac{ds}{\\sinh s}\ne^{-(|g b_{\\alpha \\beta}|\/4)(x-y)^2 \\coth s} \\end{equation}\nwhere $b_{\\alpha \\beta} = b_\\alpha - b_\\beta$. For $x = y$ this expression\nis independent of color indices. It must be regulated to prevent the\ndivergence, e.g.\n\\begin{equation}\n \\frac{g^2 N}{2\\pi} \\int_{s_0}^{\\infty} \\frac{ds}{\\sinh s},\\end{equation}\nwhere $s_0$ is a regularization\ncutoff. Although $J(x,y)$ does not depend only on\nthe distance $|x-y|$ the Schr\\\"odinger  equation\n(\\ref{peq}) with $P$ and $Q$ based on such $J$\nis translationally invariant. Shifting  the coordinates by say a vector\n$h$ and simultaneously performing a gauge transformation of the wave function\n$c_{\\alpha \\alpha} \\rightarrow c_{\\alpha \\alpha}\\exp\\left[i(gb_{\\alpha}\/2)\n\\epsilon_{ij}h^i (x_0^j - y_0^j)\\right]$ leaves  Eq.(\\ref{peq})\ninvariant.  The integral in  the expression for $J(x,y)$\ncan be expressed in terms of the Bessel function\n$K_0(|g(b_{\\alpha} - b_{\\beta}|z^2\/4)$  with $z = x - y$ and for\n $|g (b_{\\alpha} - b_{\\beta})| z^2 \\rightarrow \\infty$\nit has the following asymptotic form\n\\begin{equation}\n\\frac{1}{4\\sqrt{\\pi}}(2|g(b_\\alpha - b_\\beta)|)^{-1\/4} \\exp\\left[ -\n |g(b_\\alpha - b_\\beta)| z^2\/4\\right]. \\end{equation}\nFor finite values of  $g|b_{\\alpha} - b_{\\beta}|$\nit decreases as a Gaussian at large separations\n$z$. This should lead to a similar decrease of\nthe eigenvalues of (\\ref{peq})  -- an entirely unsatisfactory behavior\nas far as the confinement is concerned.\nIn the next Section it will be seen that the situation may be\ndifferent in the large N limit.\n\n\\setcounter{equation}{0} \\ALTsect{ Large N Random Colormagnetic Fields.}\n\nAs mentioned in the Introduction\nrigid gauge field rotations should be relevant\nfor QCD in the large N limit  where it is expected that a\nmaster field configuration dominates the vacuum, \\cite{wit}.\nAs in the case of a condensate\nsuch a configuration can not be just some fixed\ngauge field potential $A^i_a(x)$.\nIt must be allowed to undergo free gauge \"rotations\"\nexactly as $a^i$ in\nEq.(\\ref{rgd}) since the gauge invariance is not expected to be  broken\nin the large N limit. The dynamics of these rotations can not be \"frozen\"\nand must be described by the gauge rotor Hamiltonian considered in\n Section 2. These \"rotations\" induce an interaction\nbetween quarks as was shown in Section 3 for static quarks and will be\ndemonstrated for dynamical quarks in Section 6 below where it will\n also be shown that in addition\n$a^i$ appears as a background field in the Dirac operator.\n\nAnother  important consideration is that for large N there is a large\nnumber of degrees of freedom operating at each space point which\nintroduces statistical elements in the theory, cf. Refs.\n\\cite{mat,ran,bor}.\nExperience with this limit for simple systems indicates that\ntwo types of gauge invariant physical operators  should exist, analogoes to\nmacroscopic and microscopic observables in thermodynamics. The former\ndepend on finite (relative to N) number of dynamical variables\nand involve sums over all labels of the degrees of freedom,\ni.e. the color indices. A simple example is\n$a^i_{\\alpha\\beta}a^i_{\\beta\\alpha}$, etc. Operators without such\nsummations, e.g., $a^i_{\\alpha\\beta}$ with fixed $\\alpha$ and $\\beta$\nbelong to the second  type which must be regarded as\nmicroscopic observables\n like ,e.g., a coordinate of a particle or a single\nspin variable in thermodynamic systems. The fluctuations of the macroscopic\noperators are suppressed and expectations of their products factorize at\n$N = \\infty$. This is not so for microscopic observables.\n\nOn the basis of these considerations one can adopt the following point of view.\nAfter allowing for free gauge rotations according to (\\ref{rgd}), i.e.\nafter transformation to the  body-fixed frame, one should\n consider $a^i(x)$ as  static\n random matrix functions described by a probability\ndistribution $P[a^i(x)]$. This distribution can be determined following\nthe ideas of the random matrix theory, cf.\n Ref.\\cite{ent}. To this end one should introduce the amount  of information\n (negative entropy)\n\\begin{equation}\nI\\left\\{P\\left[ a \\right]\\right\\} =\n \\int D\\mu[a^i(x)] P[a^i(x)] ln P[a^i(x)]  \\label{inf}\n\\end{equation}\nassociated with the $P[a^i(x)]$.\nMinimizing $I\\{P[a]\\}$ subject to suitably chosen constraints on\nmacroscopic-like variables should determine the least biased distribution\n$P[a^i(x)]$. As in statistical mechanics\nthe large N factorization should   then simply appear as a consequence of\nthe central limit theorem.\n\nThere are two crucial questions which need to be answered\nin following this procedure - what is the appropriate measure in\nthe integral (\\ref{inf}) and  what are the variables which should\nbe constrained. I hope to address the general answer to\nthese questions in the future work. Presently I will illustrate\nhow the procedure can be put to work for a uniform colormagnetic field,\nEq.(\\ref{cb1}).\n\n   In the limit of  large N\nonly the first terms in the expressions for $P$ and $Q$ above should\nbe retained and the Schr\\\"odinger   equation (\\ref{peq}) for diagonal components\nof the string amplitude becomes\n\\begin{equation}\n-2g^2\\sum_\\beta J_{\\alpha \\beta} (x_0,y_0) c_{\\beta \\beta} =\n(E - E^0_\\alpha) c_{\\alpha \\alpha}, \\label{meq} \\end{equation}\nwhere $E^0_\\alpha = 2g^2 \\sum_\\mu J_{\\alpha \\mu}(x_0,x_0)$.\nThe non diagonal string amplitudes decouple and satisfy a trivial equation\n$(E^0_\\mu + E^0_\\nu)c_{\\mu \\nu} = 2 E c_{\\mu \\nu}$ the eigenvalues of which are\nsimply the sums of the selfenergies. Without careful treatment of long and\nshort distance regularization in the large N limit one can not reliably\ndiscuss these eigenvalues taken separately and I will concentrate\non Eq.(\\ref{meq}).  Using translational invariance and\nwriting this equation for $x_0 = 0$ and $y_0 = z$ one obtains\n\\begin{equation}\n\\sum_{\\beta=1}^N \\int_0^{\\infty} \\frac{ds}{4\\pi \\sinh s}\ne^{-(|g (b_{\\alpha} - b_{\\beta})|\/4) z^2 \\coth s} c_{\\beta \\beta} =\n- \\frac{E}{2g^2} c_{\\alpha \\alpha}.\\label{teq}  \\end{equation}\nas the large N limit of the Schr\\\"odinger  equation for a static quark --\nantiquark pair in the rigid  gauge\nconfiguration corresponding to a uniform colormagnetic field in 2+1\ndimensions. One must still specify the large N scaling of various quantities\nwhich enter this equation. Provided each term in the sum on the left hand\nside is of the same order of magnitude  I get the standard\nscaling of the coupling constant  requiring that $g^2 N$ is held fixed.\nIn the exponential of the integrand one can then extract the finite combination\n$\\bar g = g\\sqrt{N}$. The problem is then to determine the\nscaling and in general the entire distribution of\nthe field components  $b_{\\alpha}\/\\sqrt{N}$. Regarding the behavior at\nlarge separations $z$ one notes that if\nthe limit of $N \\rightarrow \\infty$ is taken first in such a way that\nthe differences $|b_{\\alpha} - b_{\\beta}|\/\\sqrt{N}$ decrease then the Gaussian\ndecay can possibly be prevented.\n\nI use this example to  demonstrate how the ideas about the\nstatistical nature of the large N limit can be used to determine the\ndistribution of the components $b_\\alpha\/\\sqrt{N}$.\nI consider what happens with the Wilson loop\n$W(C) = \\frac{1}{N}<Tr P \\exp(ig\\oint_C A^idx_i)>$ in the present theory.\nChoosing the loop perpendicular to the time axis, inserting (\\ref{rgd})\nin $W(C)$, using its  gauge invariance and the explicit form (\\ref{cb1}) of\n$a^i$  one finds\n\\begin{equation}\n W(C) = \\frac{1}{N}\\sum_\\alpha e^{ig b_\\alpha S} = \\int_{-\\infty}^\\infty\n db \\rho (b) e^{igbS}\n\\end{equation}\nwhere $S$ is the area of the loop and $\\rho (b) =\n(1\/N)\\sum_{\\alpha=1}^N \\delta (b - b_\\alpha)$ is the density of the field\ncomponents. In the large N limit $\\rho(b)$ can be approximated\nby a smooth function provided the range of variations of b does\nnot grow with N. Assuming this one easily finds simple expressions for\n$\\rho (b)$ , e.g. Lorenzian\n\\begin{equation}\n\\rho (b) = \\frac{b_0 \\sqrt{N}}{\\pi (b^2 + Nb_0^2)} \\label{lor} \\end{equation}\nwhich lead to the area law dependence of the Wilson loop,\n$W(C) = \\exp (-\\bar{g}b_0 S)$. The combination $\\bar{g} b_0$ plays the role of\nthe string constant. The placing of $N$'s in (\\ref{lor}) was chosen in such\na way as to have this constant finite for $N \\rightarrow \\infty$.\n\nThe choice (\\ref{lor}) is the simplest possible.\nAny meromorfic function $\\rho(b)$ with poles in the upper plane\nwill give the area law with the string tension controlled by the\nposition of the pole closest to the real axis. One can also take functions\nwith other type of singularities in the upper complex plane, etc. The simple\nchoice (\\ref{lor}) gives area law for any S,\nmissing entirely the asymptotic freedom behavior\nat small S. One can attempt to correct this by choosing more\ninvolved expressions for $\\rho$. A  much more serious problem\nis that the space oriented Wilson loop may not be a good measure\nof confining properties in the model where one has\nnever worried about the  Lorenz invariance.\n\nAdopting any  form of the \"single component\"  density $\\rho(b)$\nstill leaves the distribution of the values of $b_\\alpha$ needed\nin, e.g., Eq. (\\ref{teq}) largely undetermined. Using\n statistical concepts described in the beginning of this\nSection one should  view $b_\\alpha$'s as random quantities\nand introduce their joint probability distribution\n$P(b_1, b_2,...,b_N)$ which should be such\nthat $\\rho(b)$ is reproduced but is\notherwise  maximally random, i.e. contains least amount of  information.\nThe question immediately arises as to whether $\\rho(b)$ is the only\nquantity which should constrain $P(b_1,...,b_N)$ and what is the complete\nset of such constraints. In the absence of  general answers\nI take  $\\rho(b)$ controlling $W(C)$ as an example\nand determine the distribution $P(b_1,...,b_N)$ by\nminimizing the appropriate negative entropy (information)\nwith this constraint.\n\nThe quantities $b_\\alpha$'s are eigenvalues of a\nhermitian, in general complex matrix. The information\ncontent of a probability distribution $P(b_1,...,b_N)$ of such eigenvalues\nis a well studied question, cf. Ref.\\cite{ent}. It is\n\\begin{equation} \\int d\\mu[b] P(b_1,...,b_N) ln P(b_1,...,b_N)\\label{ent}\\end{equation}\nwhere the measure is $d\\mu[b] = const \\prod_{\\alpha > \\beta} |b_\\alpha\n- b_\\beta|^2\ndb_1 db_2...db_N$,  reflecting the repulsion of the eigenvalues. Minimizing\n(\\ref{ent}) under the condition of a given $\\rho (b) =\n(1\/N)\\sum_\\alpha (b - b_\\alpha)$ one finds\n\\begin{equation}\nP(b_1,...,b_N) = const \\;\\exp \\left(\\sum_{\\alpha\\neq\\beta}^N ln|b_\\alpha -\nb_\\beta| - 2N\\sum_\\alpha^N \\int_{-\\infty}^\\infty ln|b_\\alpha - b^\\prime|\n\\rho (b^\\prime ) db^\\prime\\right).\\label{dis} \\end{equation}\n Using, e.g. Eq. (\\ref{lor}) for $\\rho (b)$ this expression becomes explicitly\n\\begin{equation}\nP(b_1,...,b_N) = const\\; \\exp \\left(\\sum_{\\alpha\\neq\\beta}^N ln|b_\\alpha -\nb_\\beta| - N\\sum_\\alpha^N ln (b_\\alpha^2 +Nb_0^2) \\right).\\label{pbb} \\end{equation}\nThe constant in front of this expression must assure the normalization of\n$P$ and can be calculated by the methods described in Ref.\\cite{meh}.\nUsing the standard  interpretation  of $P(b_1,...,b_N)$  as a partition\nfunction of a fictitious Coulomb gas\n one can say that the \"particles\" $b_\\alpha$\nare \"repelled\" from each other by the first\nterm in its exponential but are kept within the interval  $b_0$ by the second\nterm representing the interaction with the background \"charge\" distributed\naccording to $\\rho(b)$. The average distance $|b_\\alpha - b_\\beta| \\sim b_0\/N$\nbecomes very small in the large N limit. The Schr\\\"odinger   equation (\\ref{teq}) is now\na random matrix equation with the probability distribution of its elements\ncontrolled by the $P(b_1,...,b_N)$ above. The actual numerical\nsolution of this equation is now in progress.\n\nIn a similar way one can consider rigid gauge configuration representing\nuniform colormagnetic field in $3 + 1$ dimensions. This field in the intrinsic\nframe corresponds to the choice\n\\begin{equation}\na^i(x) = \\frac{1}{2} \\epsilon_{ijk} b^j x^k \\label{cb2} \\end{equation}\nwhere  now however  the three color matrices $b^i$ in general cannot be\nassumed diagonal. For such non-diagonal colormagnetic field the inversion\nof the moment of inertia operator  $-(1\/4g^2)[d^i,[d^i,\\omega]]$\nrequires a solution of a matrix differential equation. This equation\nsimplifies considerably if $b^i$'s are nonetheless restricted to be diagonal,\n$b^i_{\\alpha \\beta} = \\delta_{\\alpha \\beta} b^i_\\alpha$. Then\nthe equation (\\ref{dmi}) is still valid with the index $i$ now running from\n1 to 3. In the following equation (\\ref{grf}) for the Green's function  one\nshould just replace $(b_\\alpha - b_\\beta)\\epsilon_{ij}x^j$ by\n$\\epsilon_{ijk}(b^j_\\alpha -\n b^j_\\beta)x^k$. The expression for this Green's function is known\n and one can repeat all the\nsteps leading to the  static quark--antiquark\nSchr\\\"odinger  equation which is the analog of Eq.(\\ref{teq}) in 3+1 dimensions.\n\nTurning again to the Wilson loop one finds in this case\n $\\oint_C a^i dx_i =  b^j S_j$ with $S_j = (1\/2) \\epsilon_{jki}\n\\oint_C x^k dx_i$ so that $\\left( \\oint_c a^i dx_i \\right)_\\alpha =\nb_\\alpha S \\cos \\theta _\\alpha$ (no sum over $\\alpha$) where\n$S = \\sqrt{\\sum_i (S^i)^2},  b_\\alpha = \\sqrt {\\sum_i (b^i_\\alpha)^2}$\nand $\\theta_\\alpha$ -- the angle between the vectors $b^i_\\alpha$ and $S^i$\nat a given $\\alpha$. S is the area of the loop when it is planar and\nis related to the minimal area in general.\nThe Wilson loop is\n\\begin{eqnarray}\nW(C) & = & \\frac{1}{N}<\\sum_\\alpha e^{igSb_\\alpha \\cos \\theta_\\alpha}>\n = \\nonumber \\\\\n   & = & \\frac{1}{N}\\sum_\\alpha 2\\pi \\int _{-1}^{1} d(\\cos \\theta_{\\alpha})\ne^{igSb_\\alpha \\cos \\theta_\\alpha} = \\label{wl3} \\\\\n    & = & \\frac{4\\pi}{gS} \\int_0^{\\infty} \\frac{db}{b} \\rho(b) \\sin (gbS),\n \\nonumber \\end{eqnarray}\nwhere $\\rho (b) = (1\/N) \\sum_\\alpha \\delta (b - b_\\alpha)$ is the density\nof the positive lengths of the color components of the vector\n$b^i$. In (\\ref{wl3}) I have performed the angle averaging which must be\npresent in the vacuum wavefunction.\nOne can easily choose $\\rho (b)$, e.g. the square of Lorentzian\n\\begin{equation}\n\\rho(b) = \\frac{4 b_0 b^2\\sqrt{N}}{\\pi (b^2 + Nb_0^2)^2} \\label{rh2} \\end{equation}\nwhich gives  the area law $W(C) = 4\\pi\\exp (-\\bar{g}b_0S)$. This choice is\nagain not unique  and gives the area law for any $S$. It has a powerlike\ntale as opposed to the perturbative Gaussian.\n\nThe statistical arguments for finding the entire distribution\nof $b^i_\\alpha$  can be used in $3+1$ dimensional\ncase as well with the difference that in this case the density of the lengths\nof the vectors $b^i_\\alpha$ is fixed by, e.g. Eq.(\\ref{rh2}) and their directions\nare distributed isotropically.\n\n\\setcounter{equation}{0} \\ALTsect{Dynamic Quarks}\n\nDynamic quarks can be easily included in the rigid gauge rotation model.\nFor this I define quark fields in the \"rotating frame\", $q = U\\hat{q}$,\nuse Eq. (\\ref{rgd}) in the second term of the QCD Hamiltonian  (\\ref{ham})\nand replace the first term in it by (\\ref{hrot}). Using moreover\n the Gauss law constraint (\\ref{gl}) I can write the original\nQCD Hamiltonian (\\ref{ham}) in the rigid gauge rotor limit as\nexpressed in terms of the quark fields only,\n\\begin{equation}\nH_{rot} = \\frac{1}{2}\\int d^3 x d^3 y\n\\hat{\\rho}_a (x) I_{ab}^{-1}(x,y,[a^i]) \\hat{\\rho}_b (y)\n+\\int d^3 x \\hat{q}^{+}(x)[\\alpha^{i}(p^{i}-g\na^{i}(x))+\\beta m]\\hat{q}(x), \\label{hrt1} \\end{equation}\nwhere $\\hat{\\rho}_a = \\hat{q}^{+}\\lambda ^a\\hat{q}$\nare the color quark densities in the rotating frame.\nThe Hamiltonian  $H_{rot}$  describes quarks with gauge strings attached to them,\ni.e. $q(x)$  are multiplied by $U^{+}(x)$. They  move in an external\ncolormagnetic field described by the vector potential $a^i(x)$ and\n interact\nvia an instantaneous  interaction  $I^{-1}_{ab}(x,y,[a^i])$ also\ndepending on $a^i$ via Eq.(\\ref{mom}). The simultaneous appearance of\nthe rigid gauge field configuration both as a background and as \"inducing\"\nthe quark-quark interaction is ultimately a consequence of the\ngauge invariance which requires that non dynamical rigid gauge fields\nappear only in the form (\\ref{rgd}).\n\n$H_{rot}$ is gauge invariant since the operators $\\hat{q}$ are. Moreover\nthis Hamiltonian should only be used in the color singlet sector of the\ntheory since I have used the Gauss law to derive it.\n\nFor the vanishing $a^i$ the Hamiltonian  $H_{rot}$ describes free quarks interacting\nCoulombically. Also for a general non zero $a^i$\n$H_{rot}$ should be regarded as the QCD analogue\nof the QED Hamiltonian  in which only the instantaneous Coulomb\ninteraction between the charges has been retained.\nIndeed the analogue of the rigid gauge \"rotations\"  (\\ref{rgd}) in QED\nis $A^i(x,t) = a^i (x) + \\partial ^i \\chi(x,t)$\nwith abelian $U = \\exp (ig\\chi(x,t))$, fixed rigid $a^i (x)$ and dynamical\n$\\chi (x,t)$. Repeating the steps leading to (\\ref{hrt1}) one will\nderive in the QED case the Coulomb interaction between the charge densities.\n\nRegarding the possible role of $a^i(x)$ as the master field in the large N\nlimit\none has in $H_{rot}$ a way in which this field should enter the quark\nsector of the theory, i.e. serving both as a background field and\nperhaps somewhat surprisingly also  controlling\nthe quark interaction. Following the developments of Section 5 this\nfield should be regarded as random. The appearance of a random interaction\nbetween the quarks means that a possible mechanism for confinement\nof dynamical quarks in the large N limit could be\nrelated to the localization of their relative distances.\nThe possible connection between\n confinement and localization has already been mentioned\nin the past but usually in the context of a random background field and not\nwith random interactions as appear in the present model.\n\nThe Hamiltonian  (\\ref{hrt1}) takes exact account of the gauge symmetry.\nOne must however also worry about global symmetries.\nFor any N the Hamiltonian  $H_{rot}$ may serve as a possible basis for various\nphenomenological\ndevelopments. Both for this matter and conceptually  one must face the\nissue that allowing for an arbitrary x-dependence of various color components\nof $a^i$  in (\\ref{hrt1}) leads to breaking\nof important symmetries such as translational, rotational, Lorenz,\ntime reversal, and various discrete symmetries. Of course the breaking\nof continuous space symmetries is not uncommon in phenomenology,\ne.g. the bag model, the quark potential model, the Skyrme model, etc.\n Symmetries can be restored by\nconsidering all configurations translated by the symmetry and integrating\nover them using collective coordinates. This of course\napplies to both continuous and discrete symmetries.\nIn the absence of the guidance from the\nsymmetries a more dynamical criterion for fixing\n$a^i$ seems to be the condition of lowest  energy.\nThis leads naturally to a  generalization of the variational approach\nof Section 4 in which the  variational energy\nshould be replaced by the ground state energy  $E_0[a^i(x)]$\nof (the suitably regularized)  $H_{rot}$  found for a given $a^i (x)$.\nShould the solution $a^i$\n break a global symmetry, the symmetry \"images\" of this $a^i$\nwill also be solutions and one should \"sum\" over all of them in a standard\nway thereby restoring the symmetry.\nThis variational approach may be combined with the Hartree-Fock\nmethod which should allow to calculate $E_0 [a^i(x)]$ approximately. The\nHartree-Fock approximation for fermions was shown to be consistent with\nthe large N approximation, c.f., Ref.\\cite{sal}.\nIn a combined approach one should form for fixed $a^i$\nan expectation value of\n$H_{rot}$ with respect to a trial state of a chosen\ncolor singlet  configuration of quarks\n(e.g., vacuum, baryon, etc) which must be a product state, i.e. such that\nthe expectation values with respect to this state have a non interacting\nfactorized form,i.e.,\n\\begin{eqnarray}\n <\\hat{\\rho}_a (x) \\hat{\\rho}_b (y)> &=& \\lambda^a_{\\alpha \\beta}\n\\lambda^b_{\\gamma \\delta}(\n<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\beta(x)><\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\delta(y)> -\\nonumber\n\\\\\n&-&<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\delta(y)><\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\beta(x)>).\n\\label{hf}  \\end{eqnarray}\nFor  a global color singlet state\n$<\\hat{q}_\\alpha^{+}(x)\\hat{q}_\\beta(y)> = \\delta_{\\alpha \\beta}\\rho (y,x)$, with\n$\\rho(x,y) = \\frac {1}{N}\\sum_{\\gamma}<\\hat{q}_\\gamma^{+}(y)\\hat{q}_\\gamma(x)>$\nand therefore\n\\begin{equation}\n<\\hat{\\rho}_a (x) \\hat{\\rho}_b (y)> = -2\\delta_{ab}\\rho(x,y)\\rho(y,x)\n\\label{fck} \\end{equation}\n\nFollowing the standard Hartree-Fock routine, cf., Ref.\\cite{hfr},\nthe single quark density matrix can be expanded\nin terms of a complete set of functions, i.e.,  $\\rho (x,y) =\n\\sum_n f_n \\psi_n (x) \\psi_n^* (y)$. At this stage it is customary\nto add the so called Slater determinant condition\nwhich means that the single quark states $\\psi_n$ have sharp occupations\n $f_n = 0$ or $1$.  This condition and its\ncompatibility with the large N limit and Lorenz invariance were discussed in\nRef.\\cite{bor}. Adopting this condition, using (\\ref{fck}) in forming the\n expectation\n $E_{HF}[\\rho(x,y),a^i(x)] = <H_{rot}>$ and varying with the respect to\n$\\psi_n(x)$'s with constraints on their\nnormalization one obtains the Hartree-Fock equation\n\\begin{equation}\n[\\alpha^{i}p^{i}+\\beta m]\\psi_n(x)\n- 2\\int d^3 y I_{aa}^{-1}(x,y,[a^i]) \\rho (x,y)\\psi_n(y)\n = \\epsilon_n \\psi_n(x), \\label{feq} \\end{equation}\nwhich appears here as a selfconsistent\nDirac equation for the quark wave functions $\\psi_n(x)$.\nNote that  in this equation $a^i$ disappeared from the\nDirac operator and enters only the interaction $I_{aa}^{-1}(x,y)$.\n Solutions of (\\ref{feq}) determine the optimum $\\rho$ and thus\n$E_{HF}$ for a given $a^i$.\n\n Hartree-Fock equation similar to\n(\\ref{feq}) has  been investigated in the  1+1 dimensional QCD, \\cite{bor}\n although\nthere for obvious reasons the field $a^i$ was absent and $I^{-1}(x,y)$ was\nsimply $|x - y|$.   Both\nthe t'Hooft meson spectrum and  the baryon soliton solutions\nwere found in this approach\nin 1+1 dimensions. For small quark masses the baryon was the realization of the\nskyrmion described in the quark language. If\nsuccessful the approach based on Eq.(\\ref{feq}) can possibly lead to similar\nresults in 2+1 and 3+1 dimensions.\n\n\n\\setcounter{equation}{0} \\ALTsect*{Acknowledgement}\nUseful discussions with S. Elizur, S. Finkelstein, J. Goldstone, K. Johnson,\nA. Kerman,  F. Lenz, M. Milgrom, J. Negele, J. Polonyi, E. Rabinovici,\nS. Solomon and H. Weidenm\\\"uller are acknowledged gratefully. Special\nacknowledgement is to A. Schwimmer for patient teaching and\nanswering my questions and for critical reading of the manuscript. Parts of\nthis work were done during visits in Max-Planck-Institut fur Kernphysik\nat Heidelberg and I wish to thank H. Weidenm\\\"uller for warm hospitality\n there.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe hadronic decays of heavy quarkonia below the threshold for heavy \nmeson pair production are understood to proceed predominantly via\nthree intermediate gluons. One of the gluons can be replaced by a \nphoton with a penalty of order the ratio of coupling constants, \n$\\alpha \/ \\alpha_s$. Such exclusive radiative decays of the heavy vector\nmesons $J\/\\psi$\\ and $\\Upsilon$\\ have been the subject of many experimental and\ntheoretical studies. For the experimenter, the final states from\nradiative decays are relatively easy to identify as they have a high\nenergy photon, a low multiplicity of other particles, and low\nbackground. Theoretically, the radiative decays of heavy quarkonia\ninto a single light hadron provide a particularly clean environment to \nstudy the conversion of gluons into hadrons, and thus their study is a\ndirect test of QCD. $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is one such candidate channel. This\ndecay channel has been observed to be produced in the $J\/\\psi$\\\ncharmonium system (the $1^3\\text{S}_1$ state of $c \\bar{c}$) with\n$\\mathcal{B}(J\/\\psi\\to\\gamma \\eta^{\\prime}) = (4.71\\pm0.27)\\times10^{-3}$~\\cite{PDG}.  \nNaive scaling predicts that decay rates for radiative $\\Upsilon(1\\text{S})$ decays\nare suppressed by the factor \n$(q_{b}m_{c}\/q_{c}m_{b})^{2}$ $\\approx 1\/40$ \nwith respect to the corresponding $J\/\\psi$\\ radiative decays.\nThis factor arises because the quark-photon coupling is proportional\nto the electric charge, and the quark propagator is roughly\nproportional to $1\/m$ for low momentum quarks. Taking into account the\ntotal widths~\\cite{PDG} of $J\/\\psi$\\ and $\\Upsilon(1\\text{S})$, the branching fraction\nof a particular $\\Upsilon(1\\text{S})$ radiative decay mode is expected to be around\n0.04 of the corresponding $J\/\\psi$\\ branching fraction. However, the CLEO \nsearch~\\cite{CleoEtaPrimeStudy} for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ in $61.3\\,\\rm pb^{-1}$ of data\ncollected with the CLEO~II detector found no signal in this mode, and\nresulted in a 90\\% confidence level \nupper limit of $1.6\\times10^{-5}$ for the branching\nfraction $\\Uos \\to \\gamma \\eta^{\\prime}$, an order of magnitude smaller than this\nexpectation.\n\nThe two-body decay \\UpsToGammaFTwo\\ has been\nobserved~\\cite{CleoF2_1270} in the older CLEO~II $\\Upsilon(1\\text{S})$ analysis,\nand this observation has been confirmed~\\cite{LuisAnalysis,Holger}, with much\ngreater statistics, in CLEO~III data. \nThe measurement $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma f_2(1270)) = (10.2\\pm1.0) \\times10^{-5}$, \nfrom the combination of the two CLEO~III measurements, \nis $0.074\\pm0.010$ times the corresponding $J\/\\psi$\\\ndecay mode, showing a deviation of roughly a factor of two from the\nnaive scaling estimates.\nIn radiative $J\/\\psi$\\ decays the\nratio of $\\eta^{\\prime}$\\ to \\fTwo\\ production is $3.4\\pm0.4$. If the same ratio\nheld in $\\Upsilon(1\\text{S})$, the $\\eta^{\\prime}$\\ channel would be clearly visible.\nThe channel $\\Uos \\to \\gamma \\eta$\\ has\nreceived significant theoretical attention. \nThis channel has been observed in $J\/\\psi$\\ decays~\\cite{PDG}\nwith the branching fraction of $(9.8\\pm1.0)\\times 10^{-4}$, a value\nsmaller by a factor of \nfive\nthan $\\mathcal{B}(J\/\\psi\\to\\gamma \\eta^{\\prime})$. \nThe previous CLEO search of $\\Upsilon(1\\text{S})$ decays produced an upper limit of\n$2.1\\times10^{-5}$ at the 90\\% confidence level \nfor this mode~\\cite{CleoEtaStudy}.\n\n\nSeveral authors have tried to explain the lack of signals in radiative \n$\\Upsilon(1\\text{S})$ decays into pseudoscalar mesons, using a variety of models\nwhich produce branching fraction predictions of \n$10^{-6}\\ \\text{to}\\ 10^{-4}$.\nEmploying the Vector \nMeson Dominance Model (VDM), Intemann~\\cite{Intemann} predicts the\nbranching fractions for the heavy vector meson radiative decay into\nlight pseudoscalar mesons. Using  the mixing mechanism of $\\eta$,\n$\\eta^{\\prime}$\\ with the as-yet-unobserved pseudoscalar resonance $\\eta_{b}$,\nChao~\\cite{KTChao} first calculated the mixing angle\n$\\lambda_{\\eta\\eta_{b}}$ in order to estimate the radiative branching\nfractions. Baier and Grozin~\\cite{BaierGrozin} showed that for light\nvector mesons (such as $J\/\\psi$) there might be an additional ``anomaly''\ndiagram that contributes significantly to the radiative decays. Noting\nthat VDM has no direct relation to QCD as the fundamental theory of\nstrong interactions, and referring to~\\cite{Intemann},\nMa tries to address the problem by using factorization at tree level\nwith NRQCD matrix elements to describe the heavy vector meson portion\nmultiplied by a set of twist-2 and twist-3 gluonic distribution\namplitudes~\\cite{JPMa}.\n\n\n\\section{Detector and Data Sample}\nThis study is based upon data collected by the CLEO III detector\nat the Cornell Electron Storage Ring (CESR). CLEO III is\na versatile multi-purpose particle detector described fully\nelsewhere~\\cite{cleoiii-detector}. Centered on the $e^+e^-$ interaction\nregion of CESR, the inner detector consists of a silicon strip vertex\ndetector and a wire drift chamber measuring the momentum vectors and\nthe ionization energy losses ($dE\/dx$) of charged tracks based on their\ntrajectories in the presence of a 1.5T solenoidal magnetic field. The\nsilicon vertex detector and the drift chamber tracking system together\nachieve a charged particle momentum resolution of 0.35\\% (1\\%) at\n1\\,\\rm GeV\/$c$\\ (5\\,\\rm GeV\/$c$)\nand a fractional $dE\/dx$\\ resolution of 6\\% for hadrons and 5\\% for electrons. \nBeyond the drift chamber is a Ring Imaging Cherenkov Detector, RICH,\nwhich covers 80\\% of the solid angle \nand is used to further identify charged particles by giving for each\nmass hypothesis the fit likelihood to the measured Cherenkov radiation\npattern. After the RICH is a CsI crystal calorimeter that covers 93\\%\nof the solid angle, allowing both photon detection and electron\nsuppression. The calorimeter provides an energy \nresolution of 2.2\\% (1.5\\%) for 1\\ \\rm GeV\\ (5\\ \\rm GeV) photons. Beyond the calorimeter \nis a superconducting solenoidal coil providing the magnetic field,\nfollowed by iron flux return plates with wire chambers interspersed \nat 3, 5, and 7 hadronic interaction lengths (at normal \nincidence) to provide\nmuon identification.\n\nThe data sample has an integrated luminosity of $1.13\\,\\rm fb^{-1}$ taken at the\n$\\Upsilon(1\\text{S})$ energy $\\sqrt{s} = 9.46\\ \\rm GeV$, which corresponds to \n$N_{\\Upsilon(1\\text{S})} = 21.2\\pm0.2$ million $\\Upsilon(1\\text{S})$ decays~\\cite{CLEO-III-NUPS}. \nThe efficiencies for decay chain reconstruction were obtained from\nMonte \nCarlo simulated radiative events generated with the \n($1+\\cos^{2}\\theta$) angular distribution expected for decays \n$\\Upsilon(1\\text{S}) \\to \\gamma+\\text{pseudoscalar}$. The Monte Carlo simulation of\nthe detector response was based upon GEANT~\\cite{GEANT}, and\nsimulation  events were processed in an identical fashion to data.\n\n\n\n\\newcommand{$\\sigma_{\\pi,i}$}{$\\sigma_{\\pi,i}$}\n\\newcommand{$\\sigma^{2}_{\\pi}$}{$\\sigma^{2}_{\\pi}$}\n\\newcommand{$\\sum_{i}^{3}\\sigma^{2}_{\\pi,i}$}{$\\sum_{i}^{3}\\sigma^{2}_{\\pi,i}$}\n\n\\newcommand{$S_{\\pi,i}$}{$S_{\\pi,i}$}\n\\newcommand{$S^{2}_{\\pi}$}{$S^{2}_{\\pi}$}\n\\newcommand{$\\sum_{i}^{3}S^{2}_{\\pi,i}$}{$\\sum_{i}^{3}S^{2}_{\\pi,i}$}\n\n\\newcommand{$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$}{$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$}\n\\newcommand{$((M_{reconstructed}-M_{\\pi^0})\/resolution)$}{$((M_{reconstructed}-M_{\\pi^0})\/resolution)$}\n\\newcommand{\\SDeDxDefn}{\\text{S}_{dE\/dx} \\equiv\n(dE\/dx(\\text{measured}) - dE\/dx(\\text{expected}))\/\\sigma_{dE\/dx}}\n\\newcommand{\\text{S}_{dE\/dx}}{\\text{S}_{dE\/dx}}\n\\newcommand{$\\text{S}_{dE\/dx}^{2}$}{$\\text{S}_{dE\/dx}^{2}$}\n\\section{Event Selection and Results}\\label{sec:event-selection}\nIn our search for $\\Uos \\to \\gamma \\eta$\\ and $\\Uos \\to \\gamma \\eta^{\\prime}$, we reconstruct $\\eta$\nmesons in the modes $\\eta \\to \\gamma \\gamma$, $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$, and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$; the\nlatter two will collectively be referred to as $\\eta\\to3\\pi$. We\nreconstruct the\n$\\eta^{\\prime}$\\ meson in the mode $\\eta \\pi^+ \\pi^-$ with\n$\\eta$ decaying in any of the above modes, and in addition, \nthe mode $\\eta^{\\prime} \\to \\gamma \\rho^0$, where $\\rho^0 \\to \\pi^+\\pi^-$. \nFrom the CLEO~II studies~\\cite{CleoEtaPrimeStudy,CleoEtaStudy} we\nexpected five out of the seven modes under investigation to be relatively\nbackground free and so we employ minimal selection\ncriteria to maximize sensitivity and minimize possible systematic\nbiases. \nThe other two, $\\eta \\to \\gamma \\gamma$ and $\\eta^{\\prime} \\to \\gamma \\rho^0$, have large branching\nfractions, but also large backgrounds, and so our event selection for these\nmodes aims to decrease the background with a corresponding loss of\nefficiency.  \n\nOur general analysis strategy is to reconstruct the complete decay\nchain ensuring that none of the constituent tracks or showers have\nbeen used more than once, then kinematically constrain the intermediate\n$\\pi^{0}$\\ and $\\eta$ meson candidates  to their nominal masses~\\cite{PDG},\nand finally require the event to be consistent with having the 4-momentum of the\ninitial $e^+e^-$ system. Multiply-reconstructed $\\Upsilon(1\\text{S})$ candidates in an\nevent, a problem of varying severity from mode to mode, is dealt with \nby selecting the combination with lowest $\\chi^{2}_{\\mathrm{Total}}$, the sum of\nchi-squared of the 4-momentum constraint ($\\chi^{2}_{\\text{P}4}$) and chi-squared of\nall the mass-constraints involved in a particular decay chain. For\nexample, there are four mass-constraints involved in the decay chain\n$\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, three $\\pi^{0}$\\ mass-constraints and one $\\eta$\nmass-constraint. \nThe mode $ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ is an exception in which we\npreferred to accept the $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ candidate having the lowest\n$S^{2}_{\\pi}$\\ $\\equiv$ $\\sum_{i}^{3}S^{2}_{\\pi,i}$, with $S_{\\pi,i}$\\ $\\equiv$\n$(m_{\\gamma\\gamma} - m_{\\pi^0})\/\\sigma_{\\gamma\\gamma}$\\ of the \\textit{i}th $\\pi^{0}$ candidate.  \nThe yield is obtained by counting the number of final state $\\eta$ or\n$\\eta^{\\prime}$\\ candidates within our acceptance mass window defined as the\ninvariant mass region centered around the mean value and providing\n98\\% signal acceptance as determined from signal Monte Carlo. Whenever\npossible, an event vertex is calculated using the information from\nthe charged tracks, and the 4-momentum of the photon candidates is then\nrecalculated, assuming that the showers originate from the event\nvertex rather than the origin of the CLEO coordinate system. This produces\nan improvement in the $\\eta$ and $\\eta^{\\prime}$\\\ncandidates' invariant mass resolution of roughly 10\\%, leading to a slight\nincrease in the sensitivity of the measurement.\n\nThe CLEO~III trigger~\\cite{cleoiii-trigger} relies upon two components:\n(1) the tracking-based ``axial'' and ``stereo'' triggers derived from\nthe signals on the 16 axial layers of the drift chamber, and  the signals\nregistered on the chamber's 31 stereo layers, and (2) the calorimeter-based\ntrigger derived from the energy deposition in the CsI crystal\ncalorimeter. The events for the ``all neutral'' modes $\\Uos \\to \\gamma \\eta; \\EtaGG$\\ and\n$ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ are collected by the calorimeter-based trigger\ncondition requiring two high energy back-to-back showers.\nWe demand that triggered events meet the\nfollowing analysis requirements: (a) a high energy calorimeter shower not\nassociated with a charged track, having a lateral profile consistent\nwith being a photon, and  having a measured energy greater than\n4.0\\ \\rm GeV\\ must be present; \n(b) there must be the correct\nnumber of pairs of oppositely charged, \ngood quality tracks with usable $dE\/dx$\\ information. \nThe efficiency of these requirements is more than 60\\% in\nmodes involving charged tracks and approximately 54\\% and 45\\% \nfor cases where $\\eta \\to \\gamma \\gamma$ and $\\eta\\to 3\\pi^{0}$, respectively.\n\n\nThe photon candidates we use in forming $\\pi^{0}$\\ and $\\eta \\to \\gamma \\gamma$ candidates\nhave minimum energy depositions of 30\\ \\rm MeV\\ and 50\\ \\rm MeV,\nrespectively. All photon candidates are required to be not associated\nto charged tracks, and at least one of the photon candidates of \neach pair must have\na lateral profile consistent with that expected for a photon. The\nphoton  candidates we use in reconstructing the $\\eta$ meson in the\n$\\gamma \\gamma$\\ mode must be detected either in the fiducial barrel or\nthe fiducial endcap\\footnote{The fiducial regions of the barrel and \nendcap are defined by \n$|\\cos(\\theta)|<0.78$ and $0.85<|\\cos(\\theta)|<0.95$,\nrespectively; the region between the barrel fiducial region and the\nendcap fiducial region is not used due to its relatively poor\nresolution. \n} calorimeter region only. \nThese candidates are then kinematically\nconstrained to the nominal meson mass, the exception being $\\Uos \\to \\gamma \\eta; \\EtaGG$,\nwhere no mass-constraining was done to the $\\eta$ candidate, because\nwe examine $m_{\\gamma\\gamma}$ in this mode to determine our yield.\n\nThe $\\eta$ candidates in the mode $\\pi^+\\pi^-\\pi^0$ are built by first\nforcing pairs of oppositely charged quality tracks to originate from a\ncommon vertex. The $\\pi^{0}$\\ candidate having invariant mass within\n$7\\sigma_{\\gamma\\gamma}$ is then added to complete the reconstruction\nof $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ candidates. The charged tracks are required to be\nconsistent with being pions by adding the pion hypothesis $\\SDeDxDefn$\nin quadrature for two tracks and requiring the sum of $\\text{S}_{dE\/dx}^{2}$\\ to be\nless than 16. \n\nIn the case of $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, the $\\eta$ candidate is simply built by\nadding three different $\\pi^{0}$\\ candidates, where no constituent photon\ncandidate contributes more than once in a candidate $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$\nreconstruction. The $\\pi^{0}$\\ candidates are selected by requiring\n$S_{\\pi}<10.0$. In order to increase the efficiency in this mode, \nan exception was made to the fiducial region requirement, and\nphotons in the gap between the barrel and endcap fiducial regions\nwere allowed.\n\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta ,\\eta\\to 3\\pi$}\nThe $\\Upsilon$ candidate in the mode $\\gamma \\eta$ is\nformed by combining a high-energy photon ($E>4\\ \\rm GeV$) with the $\\eta$\ncandidate, requiring that this photon is not a daughter of the $\\eta$\ncandidate. The $\\Upsilon$ candidate is then subjected to the\n4-momentum constraint of the initial $e^+e^-$ system. In the case of\n$\\eta\\to3\\pi$, multiply reconstructed $\\Upsilon$ candidates were\nrestricted by selecting only one candidate. \nFor $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$, we select the candidate with the lowest $\\chi^{2}_{\\mathrm{Total}}$,\nthe sum of chi-squared of the 4-momentum constraint\nand chi-squared of the mass-constraint to the $\\pi^{0}$\\ candidate.\nFor $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, we select the candidate with the smallest $S^{2}_{\\pi}$.\nThe selected $\\Upsilon$ candidate is further required\nto satisfy the 4-momentum consistency criterion, restricting \n$\\chi^{2}_{\\text{P}4}$\\ $<100$ for $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and a less stringent cut of 200\nfor $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ measurements.\nIn addition, we limit the number of reconstructed calorimeter showers \nfor the mode $ \\Uos \\to \\gamma \\eta; \\EtaThreePZ $ to minimize backgrounds such as\n$e^{+} e^{-} \\to \\gamma \\phi$\\ where \\phiKsKl\\ without jeopardizing the signal efficiency.\n\nFrom Monte Carlo simulations, the overall reconstruction efficiencies,\n$\\epsilon_i$, for each channel are determined to be $(28.5\\pm4.3)\\%$\nand $(11.8\\pm1.9)\\%$ for the decay chains $\\Upsilon\\to\\gamma\\eta,\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and\n$\\Upsilon\\to\\gamma\\eta, \\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, respectively. \nThe uncertainties in the efficiency\ninclude the Monte Carlo samples' statistical uncertainty and our\nestimate of possible systematic biases, which are discussed further\nin Section~\\ref{sec:sys-lim}.\n\nWe find no candidate events within our acceptance invariant mass\nwindow for the search $\\Uos \\to \\gamma \\eta$, $\\eta\\to3\\pi$. The invariant mass\ndistributions for candidate $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$, after\nimposing all the selection criteria are shown in Figure~\\ref{fig:eta-3pi}.\n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-001.eps}\n\\caption{Candidate $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ (top) and $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ (bottom)\ninvariant mass distributions from $\\Upsilon(1\\text{S})$ data.\nThe large number of events near 780\\,\\rm MeV\/$c^2$\\ (top) is due to the \nabundant process $e^{+} e^{-} \\to \\gamma \\omega$. No events are observed in our acceptance\nregion, bounded by the arrows.}  \n\\label{fig:eta-3pi}\n\\end{figure}\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta ,\\eta\\to\\gamma\\gamma$}\nThe 3-photon final state resulting from $\\Uos \\to \\gamma \\eta; \\EtaGG$\\ is dominated\nby the QED process $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$. Our selection criteria of loosely\nreconstructing an $\\eta \\to \\gamma \\gamma$ meson and requiring the \\chisq\\ of\n4-momentum constraint on the $\\Upsilon(1\\text{S})$ meson formed by adding a\nhard-photon to be $<200$ are not sufficient to suppress this\nbackground. The QED background, however has a distinct feature - the\ntwo photons having energies \n$E_{hi}$ and $E_{lo}$ used in reconstructing the $\\eta$ candidate have\na large energy asymmetry, where asymmetry is defined as\n$(E_{hi}-E_{lo})\/(E_{hi}+E_{lo})$.  \nReal $\\eta$ mesons are expected to have an approximately\nuniform distribution of\nasymmetry in the range (0,1). We require the asymmetry to be less than\n0.8. To further discriminate between the signal and the background, we\nused a neural net approach. \n\nThe input to the neural net is a vector of six variables, namely the\nmeasured energy and the polar angle $\\theta$ of each of the three calorimeter\nshowers used in the reconstruction chain. The training sample is\ncomprised of 20,000 simulated signal and background events in equal\nproportion. The simulated $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\ background events have a\nhigh-energy photon ($E>4\\ \\rm GeV$), $\\gamma \\gamma$\\ invariant mass for the two\nlower-energy photons in the range 0.4-0.7\\,\\rm GeV\/$c^2$, and energy asymmetry\nless than 0.8.\n\n\nFor our final selection, we choose neural-net output with $51\\%$\nefficiency while rejecting $86\\%$ of the background.\nThe combined efficiency of our\nselection criteria for this mode is $(23.8\\pm2.4)\\%$, which includes\npossible systematic biases and statistical uncertainties from the\nsimulation. The resulting $\\gamma \\gamma$\\ invariant mass distribution from\n$\\Upsilon(1\\text{S})$ data is fit, as shown in Figure~\\ref{fig:eta-gg},\nto a double Gaussian function, whose mass and widths are fixed to values\nfound from signal Monte Carlo data, along with a second order polynomial\nbackground function. From this likelihood fit, we obtain $-2.3\\pm8.7$\nevents; consistent with zero. We then perform the same likelihood fit\nmultiple times fixing the signal area to different values, assigning\neach of the fits a probability proportional to  $e^{-{\\chi^2}\/2}$,\nwhere \\chisq\\ is obtained from the likelihood fit.\nThe resulting probability distribution is normalized and numerically\nintegrated up to 90\\% of the area to obtain the yield at 90\\%\nconfidence level. Our limit thus obtained is 14.5 events at 90\\%\nconfidence level. \n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-002.eps}\n\\caption{Invariant mass distribution of $\\gamma \\gamma$\\ candidates in $\\Upsilon(1\\text{S})$\n  data for the mode $\\Uos \\to \\gamma \\eta; \\EtaGG$, overlaid with fits using a) floating area\n  (solid red) yielding $-2.3\\pm8.7$ events, and b) area fixed to 14.5\n  events (dashed blue), the upper limit corresponding to 90\\% C.L.}\n\\label{fig:eta-gg}\n\\end{figure}\n\n\\subsection{The Decay  $\\Upsilon\\to\\gamma\\eta^{\\prime} ,\\eta^{\\prime}\\to\\eta\\pi^+\\pi^-$}\nReconstruction of the decay chains $\\Uos \\to \\gamma \\eta^{\\prime}$, where \n$\\eta^{\\prime} \\to \\eta \\pi^+ \\pi^-$, builds on the search $\\Uos \\to \\gamma \\eta$\\. \nThe reconstructed $\\eta$ candidate is\nconstrained to the nominal $\\eta$ mass. The mass-constrained $\\eta$\ncandidate is further combined with a pair of oppositely charged\nquality tracks by forcing the tracks and the $\\eta$ candidate to\noriginate from a common vertex. In reconstruction of $\\eta^{\\prime}; \\EtaPMZ$,\ncare is exercised to ensure that no track is used more than once in the\ndecay chain. The high energy photon is combined with the $\\eta^{\\prime}$\\\ncandidate to build an $\\Upsilon$ candidate which is further\nconstrained to the 4-momentum of the initial $e^+e^-$ system. \nIn the reconstruction chain $\\eta^{\\prime}; \\EtaGG$, the $\\Upsilon$ candidate with\nthe lowest sum of chi-squared to the 4-momentum constraint ($\\chi^{2}_{\\text{P}4}$)\ncombined with the chi-squared of the mass-constraint to the $\\eta$\ncandidate (\\EtaMassFitChisq) is accepted as\nthe representative $\\Upsilon$ candidate in the reconstructed event.\nIn the modes where $\\eta\\to3\\pi$, \nthe $\\pi^{0}$\\ mass-constraint chi-squared, $\\chi^2_{\\pi^0}$, also contributes to the \n$\\chi^{2}_{\\mathrm{Total}}$ . \n\n\nTo ensure that only good quality $\\eta$ candidates participate in the decay\nchain, the \\EtaMassFitChisq\\ values of ``$\\eta\\to \\text{all neutral}$''\ncandidates are required to be less than 200. Owing to the better\nmeasurements of charged track momenta, this criterion is more\nstringent (\\EtaMassFitChisq $<100$) in the case of $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$. The \ntargeted efficiency (around 99\\%) of this requirement is achieved in\nall three cases.\n \nThe charged tracks used in reconstructing $\\eta^{\\prime}$\\ candidates have to be\nconsistent with the pion hypothesis.\nWe again require the sum of squared $\\text{S}_{dE\/dx}$ added in quadrature to\nbe less than 16 for both the two track and four track cases. The\nefficiency of this requirement alone is around 99\\%.\n\nThe selected $\\Upsilon$ candidate is further required to satisfy the\n4-momentum consistency criterion, restricting $\\chi^{2}_{\\text{P}4}$\\ $<100$\nin the $\\eta \\to \\gamma \\gamma$ case and a less stringent value of 200 for\n$\\eta\\to3\\pi$. The overall reconstruction efficiencies of our\nselection criteria as determined from signal Monte Carlo simulations are\n$(35.3\\pm5.2)\\%$, $(24.5\\pm2.2)\\%$ and $(14.4\\pm2.9)\\%$ for $\\eta$\ndecays to $\\gamma \\gamma$, $\\pi^+\\pi^-\\pi^0$ and $3\\pi^0$, respectively.\n\nAfter these selection criteria, we find\nno candidate events in the modes \\ModeEtapGG\\ and $\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, as\nshown in Figure~\\ref{fig:etap-eta-pi-pi}. However, in the mode\n\\ModeEtapPMZ, we find two good candidate events passing our selection\ncriteria as shown in Figure~\\ref{fig:etap-eta-pi-pi}.\nThese two events have been looked at in detail and appear to be\ngood signal events. However, they are insufficient to allow\nus to claim a positive signal, as no candidate events are observed \nin the modes \\ModeEtapGG\\ and $\\Uos \\to \\gamma \\eta^{\\prime};\\EtaThreePZ$, each providing higher\nsensitivity than the decay chain \\ModeEtapPMZ.\n\\begin{figure}\n\\includegraphics*[width=3.44in]{0990307-003.eps}\n\\caption{Invariant mass distributions of $\\eta\\pi^+\\pi^-$ candidates from $\\Upsilon(1\\text{S})$\ndata. The $\\eta$ candidate is constrained to the nominal $\\eta$ meson\nmass. No events are observed in the signal box for $\\eta \\to \\gamma \\gamma$ (top) and\n$\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ (bottom); two signal events are observed for\n$\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ (middle).}\n\\label{fig:etap-eta-pi-pi}\n\\end{figure}\n\n\\subsection{The Decay $\\Upsilon\\to\\gamma\\eta^{\\prime} ,\\eta^{\\prime}\\to\\gamma\\rho^0$}\nThe reconstruction scheme for the decay chain \\ModeEtapGR\\ is slightly\ndifferent from those previously described. We first build $\\rho^0$\ncandidates by forcing pairs of oppositely charged tracks to originate\nfrom a common vertex. Next, we add a photon candidate \n(which we refer to as the ``soft shower'' having energy $E_s$ in\ncontrast with the high energy radiative photon)  \nnot associated with charged tracks, and having a lateral profile\nconsistent with being a photon, to build $\\eta^{\\prime}$\\ candidates. To obtain\nthe maximum yield, we neither restrict the energy $E_s$ of the\nphoton nor the invariant mass of the $\\rho^0$ candidate at this stage. \nA high energy photon is then added, ensuring that the soft shower and\nhigh energy photon are distinct, to build the $\\Upsilon$ candidate. The\n$\\Upsilon$ candidate is then constrained to the 4-momentum of the\ninitial $e^+e^-$ system and the candidate with the lowest\n$\\chi^{2}_{\\text{P}4}$\\ value is selected.\n\nThe candidate $\\eta^{\\prime}$\\ invariant mass resolution is vastly improved due\nto the mass-constraints on the candidate $\\pi^{0}$\\ and $\\eta$ mesons in\n$\\eta^{\\prime} \\to \\eta \\pi^+ \\pi^-$ decays. In reconstruction of\n$\\eta^{\\prime} \\to \\gamma \\rho^0$, a significant improvement in candidate $\\eta^{\\prime}$\\ invariant mass\nresolution ($\\approx 30\\%$) as well as the energy resolution of the\nsoft shower is achieved by performing the 4-momentum constraint on\nthe $\\Upsilon$ candidate.\n\nParticle identification in the channel $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ is achieved by\ndemanding the combined RICH and $dE\/dx$\\ likelihood for the pion\nhypothesis be greater than the combined likelihood for each of the\nelectron, kaon and proton hypotheses. Copiously produced QED processes\nsuch as $e^+e^- \\to \\gamma \\gamma e^+e^-$ are\nsuppressed by imposing an electron veto, requiring that\n$|E\/p-1.0|>0.05$, where $p$ is the measured momentum and $E$ is the\nassociated calorimeter energy of the charged track. QED events of the type \n$e^+e^- \\to \\gamma \\gamma \\mu^+\\mu^-$ are suppressed by requiring that\nneither track registers a hit five hadronic interaction lengths deep\ninto the muon detector system. Continuum background of the type \n$e^+e^- \\to \\gamma \\gamma \\rho^0$ is suppressed by demanding\n$E_s>100$\\ \\rm MeV. Finally, the event is ensured to be complete by\ndemanding $\\chi^{2}_{\\text{P}4}$\\ $< 100$. \nThe overall efficiency of the selection criteria for this mode is\n$(40.1\\pm2.1)\\%$, including possible systematic uncertainties and\nthe statistical uncertainty of the Monte Carlo sample.\n\n\\begin{figure}\n\\includegraphics*[width=3.25in]{0990307-004.eps}\n\\caption{Invariant mass distribution of $\\gamma\\rho^0$ candidates in $\\Upsilon(1\\text{S})$\n  data for the mode \\ModeEtapGR\\ overlaid with fits using a) floating area\n  (solid red) yielding $-3.1\\pm5.3$ events, and b) area fixed to\n  8.6 events (dashed blue), corresponding to the upper limit at 90\\% C.L.}\n\\label{fig:etap-gr}\n\\end{figure}\n\nAlthough highly efficient, our selection criteria are not sufficient to\nsuppress the smooth continuum background from the reaction\n$e^+e^- \\to \\gamma \\gamma \\rho^0$.\nThe candidate $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ invariant mass distribution after our selection\ncriteria, shown in Figure~\\ref{fig:etap-gr}, is fit to\na double Gaussian function over a floating polynomial background\nfunction of order one. The parameters of the double Gaussian function\nare fixed to the values obtained from a fit to signal Monte\nCarlo and the area is left to float. The likelihood fit yields\n$-3.1\\pm5.3$ events, which is consistent with zero. In the absence of a\nclear signal, we determine the upper limit yield\nas we do in the case of $\\Uos \\to \\gamma \\eta; \\EtaGG$, and find an upper limit at 90\\%\nconfidence level of 8.6\nevents. \n\n\n\n\n\\section{Systematic Uncertainties and Combined Upper Limits}\\label{sec:sys-lim}\nSince we do not have a signal in any of the modes, and since the\nkinematic efficiency is near-maximal, statistical uncertainties\ndominate over systematic uncertainties. \nBy comparison of the expected yield of the QED process $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\\nwith the calculated cross-section for this process, we estimate the\nuncertainty on the trigger simulation for ``all neutral'' modes to be\n4.5\\%. For modes with only two charged tracks, we have studied the \nQED processes $e^+e^- \\to \\gamma \\rho^0$ and $e^+e^- \\to \\gamma \\phi$, \nand assign a 13\\% uncertainty\non the efficiency due to possible trigger mismodeling. For events with\nmany charged tracks, we assign a systematic\nuncertainty of 1\\% as the relevant trigger lines are very well understood,\nredundant, and\nvery efficient.\nWe assign 1\\% uncertainty per track in charged track\nreconstruction based upon CLEO studies~\\cite{CLEOSystematics}\nof low-multiplicity events, and\n2.5\\% systematic uncertainty per photon from mismodeling of\ncalorimeter response which translates to 5\\% uncertainty per meson\n($\\pi^{0}$\\ and $\\eta$) decaying into $\\gamma \\gamma$, \nagain based upon CLEO studies~\\cite{CLEOSystematics}.\nThe systematic uncertainty in $\\text{S}_{dE\/dx}$ for two\ntracks added in quadrature (as in $ \\Uos \\to \\gamma \\eta; \\EtaPMZ $) was evaluated to be 4\\%\nby considering the efficiency difference of this requirement in\nMonte Carlo and data samples of\n$e^{+} e^{-} \\to \\gamma \\omega$. Consequently, we assign 4\\% and 5.7\\% uncertainty to the\nreconstruction efficiencies of modes involving two and four charged\ntracks, respectively, excepting $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ where this requirement was not\nimposed. For the mode $\\eta^{\\prime} \\to \\gamma \\rho^0$, the systematic uncertainty in the\nefficiency of analysis cuts, found to be 3.9\\%, was evaluated by\ncomparing the efficiency difference in Monte Carlo and data by\nstudying the $\\rho^0$ signal due to the QED processes. \nFor the neural-net cut in the mode $\\Uos \\to \\gamma \\eta; \\EtaGG$, we studied the\nefficiency in QED $e^{+} e^{-} \\to \\gamma \\gamma \\gamma$\\ simulated events and the real data\ndominated by the same QED process for a wide range of neural-net\noutput values. We find a maximum difference of 7\\% in these two\nnumbers, which we take as a conservative estimate of the\nassociated systematic uncertainty.\nThe systematic uncertainties for various $\\eta$ and $\\eta^{\\prime}$\\ decay modes\nare listed in Table~\\ref{tab:sys-errs}. \nThese uncertainties were added in quadrature, along with the\nstatistical error due to the limited size of Monte Carlo samples, to\nobtain the overall systematic uncertainties in the efficiencies.\n\n\\begin{small}\n\\begingroup\n\\squeezetable\n\\begin{table*}[!]\n\\begin{center}\n\\caption{\\label{tab:sys-errs}Contributions to systematic\nuncertainties in the efficiencies for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (upper half) and\n$\\Uos \\to \\gamma \\eta$\\ (lower half). The uncertainties are expressed as relative\npercentages and combined in quadrature.}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\nUncertainty source & ~~$\\eta^{\\prime}; \\EtaGG$ & ~~$\\eta^{\\prime}; \\EtaPMZ$ & ~~$\\eta^{\\prime}; \\EtaThreePZ$\n& $\\eta^{\\prime} \\to \\gamma \\rho^0$ \\\\  \n\n\\hline\nTrigger mismodeling    & 13   & 1   & 13    & 1   \\\\\nTrack reconstruction   & 2   & 4   & 2    & 2   \\\\\nCalorimeter response   & 5   & 5   & 15   & 2.5 \\\\\nAnalysis cuts          & 4   & 5.7 & 4    & 3.9 \\\\\nMonte Carlo statistics & 1.0 & 1.6 & 2.4  & 1.0 \\\\\n\\hline\nCombined uncertainty   & 14.7 & 8.8 & 20.4 & 5.2 \\\\\n\\hline\n& & & & \\\\\nUncertainty source &   $\\eta \\to \\gamma \\gamma$ & $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ & $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ & \\\\\n\\hline\nTrigger mismodeling    & 4.5   & 13   & 4.5    & \\\\\nTrack reconstruction   & -   & 2   & -    & \\\\\nCalorimeter response   & 5   & 5   & 15   & \\\\\nAnalysis cuts          & 7   & 4   & -    & \\\\\nMonte Carlo statistics & 1.3 & 1.2 & 1.7  & \\\\\n\\hline\nCombined uncertainty   & 9.8 & 15.2 & 16.0 & \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{center}\n\\end{table*}\n\\endgroup\n\\end{small}\n\n\n\\begin{small}\n\\begingroup\n\\squeezetable\n\\begin{table*}[ht]\n\\begin{center}\n\\caption{Results of the search for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ and $\\Uos \\to \\gamma \\eta$. Results include\nstatistical and systematic uncertainties, as described in the\ntext. The combined limit is obtained after including the systematic\nuncertainties.}\n\\label{tab:limits}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\n& ~~$\\eta^{\\prime}; \\EtaGG$ & ~~$\\eta^{\\prime}; \\EtaPMZ$ & ~~$\\eta^{\\prime}; \\EtaThreePZ$ & ~~$\\eta^{\\prime} \\to \\gamma \\rho^0$ \\\\\n\\hline\n\nObserved events & 0 & 2 & 0 & $-3.1\\pm5.3$ \\\\\n\n$\\mathcal{B}_{\\eta^{\\prime},i}\\%$ & \n$17.5\\pm0.6$ & $10.0\\pm0.4$ & $14.4\\pm0.5$ & $29.5\\pm1.0$ \\\\\n\nReconstruction efficiency (\\%) & \n$35.2\\pm5.2$ & $24.5\\pm2.2$ & $14.4\\pm2.9$ & $40.1\\pm2.1$ \\\\ \n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})(90\\%~\\text{C.L.})$\\footnotemark[1] &\n\\ulbr{1.8} & \\ulbr{10.3} & \\ulbr{5.2} & \\ulbr{3.4} \\\\ \n\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})(90\\%~\\text{C.L.})$\\footnotemark[2] &\n\\ulbr{1.9} & \\ulbr{10.4} & \\ulbr{5.8} & \\ulbr{3.4} \\\\ \n\n\\hline\n\n\\multicolumn{2}{l}{Combined limit on \n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})$} &\n\\multicolumn{2}{c}{\\ulbr{1.9}} \\\\\n\n\\hline\n\n& & & & \\\\\n\n& $\\eta \\to \\gamma \\gamma$ & $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ & $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$ & \\\\\n\n\\hline\n\nObserved events & $-2.3\\pm8.7$ & 0 & 0 & \\\\\n\n$\\mathcal{B}_{\\eta,i}\\%$ & $39.4\\pm0.3$ & $22.6\\pm0.4$ & $32.5\\pm0.3$ & \\\\\n\nReconstruction efficiency (\\%) & $23.8\\pm2.4$ & $28.5\\pm2.9$ &\n$11.8\\pm1.9$ & \\\\\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)(90\\%~\\text{C.L.})$\\footnotemark[1]\n& \\ulbr{7.3} & \\ulbr{1.7} & \\ulbr{2.8} & \\\\\n\n\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)(90\\%~\\text{C.L.})$\\footnotemark[2] & \n\\ulbr{7.4} & \\ulbr{1.8} & \\ulbr{2.9} & \\\\\n\\hline\n\nCombined limit on $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)$ &\n\\multicolumn{3}{c}{\\ulbr{1.0}} &\\\\\n\n\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{excluding systematic uncertainties}\n\\footnotetext[2]{including systematic uncertainties}\n\\end{center}\n\\end{table*}\n\\endgroup\n\\end{small}\n\n\n\\begin{figure*}[!]\n\\mbox{\\includegraphics*[width=6.3in]{0990307-005.eps}}\n\\caption{Likelihood distributions as a function of branching fraction\nfor the decay mode $\\Uos \\to \\gamma \\eta$\\ (left) and $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (right). All\ndistributions are smeared by respective systematic uncertainties and\nnormalized to the same area. The solid black curve denotes the combined\nlikelihood distribution.} \n\\label{fig:limit-plots}\n\\end{figure*}\nThe systematic uncertainties in efficiencies, uncertainties in the\nproduct branching ratios, and the statistical uncertainty in the number\nof $\\Upsilon(1\\text{S})$ decays, $N_{\\Uos}$, are incorporated~\\cite{Cousins}\nby a ``toy'' Monte Carlo procedure to obtain smeared likelihood\ndistributions for the branching fraction in each mode,\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\text{P}) = N_{\\text{P}}\/(\\epsilon_{i}\\cdot\\mathcal{B}_{\\text{P},i}\\cdot N_{\\Uos})$, \nwhere $\\text{P} = \\eta , \\eta^{\\prime}$, and $\\epsilon_i$ and\n$\\mathcal{B}_{\\text{P},i}$ denote the efficiency and branching\nfractions of the $i$th mode. \nTo obtain the smeared likelihood distribution $\\mathcal{L}_{\\text{P},i}$, the\nexperiment is performed multiple times, randomly selecting\n$N_{\\text{P}}$  from the likelihood function appropriate for each\nmode\\footnote{For modes with zero or few observed events, the\n  appropriate likelihood function is generated from Poisson statistics.\n  For the background limited modes $\\eta \\to \\gamma \\gamma$ and $\\eta^{\\prime} \\to \\gamma \\rho^0$, we \n  already have the likelihood function which we used in calculating\n  the upper limit of the observed number of events at 90\\% CL.} and then\ndividing by the sensitivity factor $\\epsilon_{i}\\cdot\\mathcal{B}_{\\text{P},i}\\cdot N_{\\Uos}$, where each term is picked\nfrom a Gaussian distribution about their mean values with the\nappropriate standard deviation. \n\nThe combined likelihood distribution for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\text{P})$ is derived as\n$\\mathcal{L}_{\\text{P}} = \\prod_{i}{\\mathcal{L}_{\\text{P},i}}$ which\nis summed up to 90\\% of the area in the physically allowed region to obtain\nthe upper limit branching fraction for $\\Upsilon(1\\text{S})\\to\\gamma\\text{P}$.\nFrom the constituent $\\mathcal{L}_{\\text{P},i}$ and the combined\n$\\mathcal{L}_{\\text{P}}$ as shown in \nFigure~\\ref{fig:limit-plots}, \nwe obtain upper limits on\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta)$ of \\br{7.4}, \\br{1.8}, \\br{2.9}, \nand \\br{1.0} for $\\eta$ decaying into $\\gamma \\gamma$,\n$\\pi^+\\pi^-\\pi^0$, $\\pi^0\\pi^0\\pi^0$, and all three combined,\nrespectively. We obtain upper limits for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime})$ \nof \\br{1.9}, \\br{10.4}, \\br{5.8}, and \\br{3.4} for $\\eta$\ndecaying into $\\gamma \\gamma$, $\\pi^+\\pi^-\\pi^0$, $\\pi^0\\pi^0\\pi^0$,\nand $\\eta^{\\prime} \\to \\gamma \\rho^0$, respectively. The combined upper limit for $\\mathcal{B}$(\\ModeEtap)\\ is\n$1.9\\times10^{-6}$, a value larger than one of the sub-modes\n(\\ModeEtapGG), due to the two candidate events in \\ModeEtapPMZ.\nThe numbers of observed events, detection efficiencies and upper\nlimits are listed in Table~\\ref{tab:limits}. \n\n\n\\section{Summary and Conclusion}\\label{sec: summary}\nWe report on a new search for the radiative decay of $\\Upsilon(1\\text{S})$ to the \npseudoscalar mesons $\\eta$ and $\\eta^{\\prime}$\\ in $21.2\\times10^{6}$ $\\Upsilon(1\\text{S})$ \ndecays collected with the CLEO~III detector. The $\\eta$ meson was reconstructed\nin the three modes $\\eta \\to \\gamma \\gamma$, $\\eta \\to \\pi^{+} \\pi^{-} \\pi^{0}$ or $\\eta \\to \\pi^{0} \\pi^{0} \\pi^{0}$.\nThe $\\eta^{\\prime}$\\ meson was reconstructed either in the mode $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ or \n$\\eta^{\\prime} \\to \\pi^{+} \\pi^{-} \\eta$ with $\\eta$ decaying through any\nof the above three modes. All these modes except for $\\eta^{\\prime} \\to \\gamma \\rho^0$\\ had earlier\nbeen investigated in CLEO~II data amounting to \n$N_{\\Uos}$ $ = 1.45\\times 10^6$ $\\Upsilon(1\\text{S})$ mesons and resulted in\nprevious upper limits $\\mathcal{B}$(\\ModeEtap)\\ $< 1.6 \\times 10^{-5}$ and \n$\\mathcal{B}$(\\ModeEta)\\ $< 2.1 \\times 10^{-5}$  \nat 90\\% C.L. These limits were already smaller than \nthe naive predictions based upon the scaling of the decay rate for the\ncorresponding $J\/\\psi$\\ radiative decay mode by the factor \n$(q_{b}m_{c}\/q_{c}m_{b})^{2}$,\nand also the model of K\\\"orner {\\it et al.,}~\\cite{KKKS}, \nwhose perturbative QCD approach predictions\nfor $\\mathcal{B}(J\/\\psi\\to\\gamma X)$ where $X = \\eta,\n\\eta^{\\prime}, f_2$ as well as $\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma f_2)$ agree\nwith experimental results.\n\nWith a CLEO~III data sample 14.6 times as large as the CLEO~II data\nsample, we find no convincing signal in any of the modes. Based purely\nupon the luminosities, we would expect the new upper limits to be scaled \ndown by a factor of between 14.6 (in background-free modes) and\n$\\sqrt{14.6}$ in background dominated modes if the two CLEO detectors \n(CLEO~II and CLEO~III) offered similar particle detection efficiencies. \nIn the search for $\\Uos \\to \\gamma \\eta$\\ we find no hint of \na signal, and manage to reduce the limit by an even larger factor. In\nthe search for $\\Uos \\to \\gamma \\eta^{\\prime}$, however, we find two clean candidate events\nin the channel \\ModeEtapPMZ, which, though we\ncannot claim them as signal, do indicate the possibility that we are\nclose to the sensitivity necessary to obtain a positive result. \nBecause of these two events, our combined limit for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is not \nreduced by as large a factor as the luminosity ratio, and in fact is \nlooser than that which would be obtained if we analyzed \none sub-mode (\\ModeEtapGG) alone.\nIn this analysis we found upper limits which we\nreport at 90\\% confidence level as\n\\begin{displaymath}\n\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta) < 1.0 \\times 10^{-6},\n\\end{displaymath}\n\\begin{displaymath}\n\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta^{\\prime}) < 1.9 \\times 10^{-6}.\n\\end{displaymath}\n\nOur results are sensitive enough to test the appropriateness of the\npseudoscalar mixing approach as pursued by Chao~\\cite{KTChao}, where\nmixing angles among various pseudoscalars including $\\eta_{b}$\\ are\ncalculated. Then, using a calculation for the M1 transition\n$\\Upsilon\\to\\gamma\\eta_{b}$, he predicts \n$\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta) = 1\\times 10^{-6}$ and\n$\\mathcal{B}(\\Upsilon(1\\text{S}) \\to \\gamma \\eta^{\\prime}) = 6\\times 10^{-5}$. Our\nlimit for $\\Uos \\to \\gamma \\eta^{\\prime}$\\ is significantly smaller than Chao's prediction\nand does not support his approach.\n\nThe sensitivity challenge posed by both the extended vector dominance\nmodel and the higher twist approach of Ma are beyond our reach. \nIn extended VDM, Intemann predicts \n$1.3\\times 10^{-7} < \\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta) < 6.3\\times 10^{-7}$ \nand \n$5.3\\times 10^{-7} < \\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime}) <\n2.5\\times 10^{-6}$, where the two limits are determined by having\neither destructive or constructive interference, respectively, between\nthe terms involving $\\Upsilon(1\\text{S})$ and $\\Upsilon(2\\text{S})$. Even if it is determined that \nthe amplitudes are added constructively, our limit remains higher than\nthe VDM prediction for $\\Upsilon(1\\text{S})\\to\\gamma\\eta$. \n\nMa's prediction of \n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta^{\\prime}) \\approx 1.7\\times 10^{-6}$ is\nconsistent with our result. However, his prediction for\n$\\mathcal{B}(\\Upsilon(1\\text{S})\\to\\gamma\\eta) \\approx 3.3\\times 10^{-7}$ is a factor of\n$\\sim 3$ smaller than our limit.\n\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n\\includegraphics*[bb = 35 150 535 610, width=3.25in]{\\etalimitplotps}\n\\caption{Likelihood distribution as function of branching\nratio for the decay mode $\\Uos \\to \\gamma \\eta$. Black curve denotes the combined\ndistribution.}\n\\label{fig:eta log limits}\n\\end{figure}\n\\begin{figure}\n\\includegraphics*[bb = 35 150 535 610, width=3.25in]{\\etaplimitplotps}\n\\caption{Likelihood distribution as function of branching\nratio for the decay mode $\\Uos \\to \\gamma \\eta^{\\prime}$. Black curve denotes the combined\ndistribution.}\n\\label{fig:etap log limits}\n\\end{figure}\n\\end{comment}\n\\begin{comment}\n\\begin{figure*}\n\\mbox{\\includegraphics*[bb = 55 150 535 610, width=3.3in]{\\etalimitplotps}}\n\\mbox{\\includegraphics*[bb = 55 150 535 610, width=3.3in]{\\etaplimitplotps}}\n\\caption{Likelihood distributions as function of branching fraction\nfor the decay mode $\\Uos \\to \\gamma \\eta$\\ (left) and $\\Uos \\to \\gamma \\eta^{\\prime}$\\ (right). All\ndistributions are smeared by respective systematic uncertainties and\nnormalized to unit area. Black curve denotes the combined likelihood\ndistribution.}\n\\label{fig:limit-plots}\n\\end{figure*}\n\\end{comment}\n\n\nWe gratefully acknowledge the effort of the CESR staff \nin providing us with excellent luminosity and running conditions. \nD.~Cronin-Hennessy and A.~Ryd thank the A.P.~Sloan Foundation. \nThis work was supported by the National Science Foundation,\nthe U.S. Department of Energy, and \nthe Natural Sciences and Engineering Research Council of Canada.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}