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+{"text":"\\section{ADIABATIC SPIKE AROUND THE CENTRAL BLACK HOLE}\n\nWe find the dark matter density profile in the region where the black hole\ndominates the gravitational potential. From the data in \\cite{ghe98,eck9697},\nthis is the region $r \\lesssim R_M \\simeq 0.2 \\, {\\rm pc}$. \n Other masses (the\ncentral star cluster, for example) also influence the dark matter distribution,\nbut since they make the gravitational potential deeper, their effect is to\nincrease the central dark matter density and the annihilation signals.\n\nWe work under the assumption that the growth of the black hole is\nadiabatic. This assumption is supported by the collisionless behavior of\nparticle dark matter.\nWe can find the final\ndensity after black hole formation from the final phase-space distribution\n$f'(E',L')$ as\n\\begin{equation}\n\\label{rhof}\n\\rho'(r) = \\int_{E'_{m}}^0 dE' \\,\n\\int_{L'_{c}}^{L'_{m}} dL' \\, \n\\frac{ 4 \\pi L' } { r^2 v_r } \\, f'(E',L') \\, ,\n\\end{equation}\nwith\n\\begin{eqnarray}\nv_r & = & \\left[ 2 \\left( E' + \\frac{GM}{r} - \\frac{L'^2}{2r^2} \\right)\n\\right]^{1\/2} , \\\\\nE'_{m} & = & - \\frac{GM}{r} \\, \\left( 1 - \\frac{ 4 R_{\\rm S}}{r} \\right), \\\\\nL'_{c} & = & 2cR_{\\rm S}, \\\\\nL'_{m} & = & \\left[ 2 r^2 \\left( E' + \\frac{GM}{r} \\right) \\right]^{1\/2} .\n\\end{eqnarray}\nWe have neglected the contribution from unbound orbits ($E'>0$).  The lower\nlimit of integration $L'_{c}$, and the second factor in $E'_{m}$, are\nintroduced to eliminate the particles with $L < 2cR_{\\rm S}$ which are captured\nby the black hole. ($R_{\\rm S}=2GM\/c^2$ is the Schwarzschild radius.) We relate\n$f'(E',L')$ to the initial phase-space distribution $f(E,L)$ through the\nrelations, valid under adiabatic conditions,\n$f'(E',L') = f(E,L)$, $L' = L$, $I'(E',L') = I(E,L)$,\nwhere the last two equations are the conservation of the angular momentum $L$\nand of the radial action $I(E,L)$.\n\nThe density slope in the spike depends not only on the slope of the inner halo\nbut also on the behavior of the initial phase-space density $f(E,L)$ as $E$\napproaches the potential at the center $\\phi(0)$~\\cite{qui95}. If $f(E,L)$\napproaches a constant, as in models with finite cores, the spike slope is\n$\\gamma_{\\rm sp} = 3\/2$.  If $f(E,L)$ diverges as $[E-\\phi(0)]^{-\\beta}$, as in\nmodels with an inner cusp, the spike slope is $\\gamma_{\\rm sp}>3\/2$. Models\nwith finite cores include~\\cite{models}: the non-singular and the modified\nisothermal sphere, the H\\'enon isochrone model, the Plummer model, the King\nmodels, the modified Hubble profile, the Evans power-law models with $R_c\\ne0$,\nand the Zhao $(\\alpha,\\beta,\\gamma)$ models, $\\rho \\sim r^{-\\gamma}\n(1+r^{1\/\\alpha})^{-\\beta\\alpha}$, with $\\gamma=0$ and $1\/(2\\alpha) = {\\rm\n  integer}$. Models with an inner cusp include~\\cite{models}: the models of\nJaffe, Hernquist, Navarro-Frenk-White, the $\\gamma\/\\eta$ models of Dehnen and\nTremaine et al., and the other Zhao models.\n\nAs an example of models with finite cores, we consider \nthe isothermal sphere. It has $f(E,L) = \\rho_0 \n(2\\pi\\sigma_v^2)^{-3\/2} \\allowbreak \\,\n\\exp(-E\/\\sigma_v^2)$. Close to the black hole, we have $E \\ll \\sigma_v^2$,\nand $f(E,L) \\simeq \\rho_0 (2\\pi\\sigma_v^2)^{-3\/2}$, a constant.  Then from\neq.~(\\ref{rhof}) we easily find\n\\begin{equation}\n\\label{rhof1}\n\\rho'_{\\rm iso}(r) = \n\\frac{4\\rho_0}{3\\sqrt{\\pi}} \\, \\left( \\frac{GM}{ r\\sigma_v^2} \n\\right)^{3\/2} \\,\n\\left( 1 - \\frac{4 R_{\\rm S}}{r} \\right)^{3\/2} ,\n\\end{equation}\nvalid for $r \\ll R_M \\simeq 0.2 $ pc.\n The last factor comes from the capture of particles by the black hole:\nthe density vanishes for $r<4R_{\\rm S}$. \n\nAs an example of models with an inner cusp, we consider \na single power law density profile, $\\rho(r) = \\rho_0 (r\/r_0)^{-\\gamma}$,\nwith $0 < \\gamma < 2$. Its\n phase-space distribution function is\n\\begin{equation}\nf(E,L) = \\frac{\\rho_0}{(2\\pi\\phi_0)^{3\/2}} \\,\n\\frac{\\Gamma(\\beta)}{\\Gamma(\\beta-\\case{3}{2})} \\,\n\\frac{\\phi_0^\\beta}{E^{\\beta}} \\, ,\n\\end{equation}\nwith $\\beta = (6-\\gamma)\/[2(2-\\gamma)] $ and $\\phi_0 = 4 \\pi G r_0^2\n\\rho_0\/[(3-\\gamma)(2-\\gamma)]$. To find $f'(E',L')$, we need to solve $\nI'(E',L') = I(E,L) $ for $E$ as a function of $E'$. In the field of\na point-like mass, we have \n$ \nI'(E',L') = 2 \\pi \\left[ - L' + GM\/\\sqrt{-2E'} \\right] \n$.\nIn the field of the power law profile, whose potential is proportional to\n$r^{2-\\gamma}$, the action integral cannot be performed exactly. We have\nfound an approximation good to better than 8\\% over all of phase space for\n$0 < \\gamma < 2$: \n\\begin{equation}\nI(E,L) = \\frac{2\\pi}{b} \\, \\left[ - \\frac{L}{\\lambda} + \n\\sqrt{2 r_0^2 \\phi_0} \\left( \\frac{E}{\\phi_0} \\right) ^ \n{ \\frac{4-\\gamma}{2(2-\\gamma)} } \\right],\n\\end{equation}\nwhere $\\lambda = [2\/(4-\\gamma)]^{1\/(2-\\gamma)} \\,\n[(2-\\gamma)\/(4-\\gamma)]^{1\/2}$ and $b = \\pi(2-\\gamma)\/{\\rm\n  B}(\\frac{1}{2-\\gamma},\\frac{3}{2}) $.  Expressing $E$ as a function of $E'$\nand integrating eq.~(\\ref{rhof}), we obtain\n\\begin{equation}\n\\label{rhoprime}\n\\rho'(r) = \\rho_R \\, g_{\\gamma}(r) \\,\n\\left( \\frac{R_{\\rm sp}}{r}\\right)^{\\gamma_{\\rm sp}} ,\n\\end{equation}\nwith \n$ \\rho_R = \\rho_0 \\left( {R_{\\rm sp}}\/{r_0} \\right)^{-\\gamma} $,\n$ \\gamma_{\\rm sp} = (9-2\\gamma)\/(4-\\gamma) $, and\n$ R_{\\rm sp} = \\alpha_{\\gamma} r_0 \\left( M\/\\rho_0 r_0^3\n\\right)^{1\/(3-\\gamma)} $.\nFor $0\\le \\gamma \\le 2$, the density slope in the spike, $\\gamma_{\\rm sp}$,\nvaries only between 2.25 and 2.5. \n\\begin{figure}[t]\n\\label{figprofile}\n\\epsfig{file=spikef1.eps,width=0.4\\textwidth}\n\\caption{\n  Examples of spike density profiles.}\n\\end{figure}\n While the exponent $\\gamma_{\\rm sp}$ can\nalso be obtained by scaling arguments~\\cite{qui95}, the normalization\n$\\alpha_{\\gamma}$ and the factor $g_{\\gamma}(r)$ accounting for capture must be\nobtained numerically. We find that $g_{\\gamma}(r) \\simeq ( 1 -4 R_{\\rm S}\/r)^3$\nover our range of $\\gamma$, and that $\\alpha_{\\gamma} \\simeq 0.293\n\\gamma^{4\/9}$ for $\\gamma \\ll 1$, and is $\\alpha_{\\gamma}=$ 0.00733, 0.120,\n0.140, 0.142, 0.135, 0.122, 0.103, 0.0818, 0.0177 at\n$\\gamma=0.05,0.2,0.4,\\ldots,1.4,2$.  The density falls rapidly to zero at $r\n\\lesssim 9.55 R_{\\rm S}$, vanishing for $r<4R_{\\rm S}$.\n\nAnnihilations in the inner regions of the spike set a maximal dark matter\ndensity $\\rho_{\\rm core} = m\/\\sigma v t_{\\rm bh}$, where $t_{\\rm bh}$ is the\nage of the black hole, conservatively $10^{10}$ yr, $m$ is the mass of the dark\nmatter particle, and $\\sigma v$ is its annihilation cross section times\nrelative velocity (notice that for non-relativistic particles $\\sigma v$ is\nindependent of $v$). Using $\\partial \\rho\/\\partial t = - \\sigma v \\rho^2\/m$,\nthe final spike profile is\n\\begin{equation}\n\\rho_{\\rm sp}(r) = \\frac{ \\rho'(r) \\rho_{\\rm core} } { \\rho'(r) + \\rho_{\\rm\n    core} } ,\n\\end{equation}\nwhich has a core of radius $R_{\\rm core} = R_{\\rm sp} \\left( \\rho_R\/\\rho_{\\rm\n    core} \\right) ^ {1\/\\gamma_{\\rm sp}} $. In the particle models we consider,\nnot more than the initial amount of dark matter within 300 pc is annihilated.\n\nExamples of spike density profiles are shown in Fig.~1.\n\nTo conclude this section, we derive a conservative estimate of the dark matter\ndensity near the galactic center.  Assume that the halo density is constant on\nconcentric ellipsoids.  Then the halo contribution $v_{h}(r)$ to the rotation\nspeed at distance $r$ fixes the halo mass within $r$. Assume further that the\nhalo density decreases monotonically with distance. Since at large radii it\ndecreases at least as fast as $r^{-2}$, and at small radii only as\n$r^{-\\gamma}$ with $\\gamma < 2$, the density profile becomes steeper with\ndistance. So to continue the $r^{-\\gamma}$ dependence to all radii keeping the\nsame halo mass interior to $r$ as given by the rotation speed, we must decrease\nthe density normalization. In this way we obtain an underestimate of the\ndensity near the center. Letting $\\rho_D = \\rho_0 (D\/r_0)^{-\\gamma} $, we have\n\\begin{equation}\n\\frac{ \\rho_D }{ 1 - \\gamma\/3} \\sim \n\\frac{ 3 v^2_h(D) }{ 4 \\pi G D^2 } \n\\simeq 0.0062 \\, \\frac{ M_{\\odot} }{ {\\rm pc}^{3} }\n\\simeq 0.24 \\, \\frac{ {\\rm GeV} }{ {\\rm cm}^{3} },\n\\end{equation}\nwhere we have taken $v_{h}(D) = 90 {\\rm \\,km\\,s^{-1}}$ at the Sun distance $D\n= 8.5$ kpc, as obtained after subtracting a (somewhat overestimated) luminous\nmatter contribution of 180 km s$^{-1}$ to the circular speed of $220 \\pm 20\n{\\rm \\,km\\,s^{-1}}$~\\cite{deh98}.\n\n\\section{SIGNALS FROM NEUTRALINO ANNIHILATIONS}\n\nOur analysis applies in general to a self-annihilating dark matter particle. To\nmake it concrete, we examine a case of supersymmetric dark matter, the lightest\nneutralino.  The minimal supersymmetric standard model privides a well-defined\ncalculational framework, but contains at least 106 yet-unmeasured\nparameters~\\cite{dim95}. Most of them control details of the squark and slepton\nsectors, and can safely be disregarded in dark matter studies. So we restrict\nthe number of parameters to 7, following Bergstr\\\"om and Gondolo~\\cite{ber96}.\nOut of the database of points in parameter space built in\nrefs.~\\cite{ber96,eds97,ber98}, we use the 35121 points in which the neutralino\nis a good cold dark matter candidate, in the sense that its relic density\nsatisfies $0.025 < \\Omega_\\chi h^2 < 1 $.  The upper limit comes from the age\nof the Universe, the lower one from requiring that neutralinos are a major\nfraction of galactic dark halos.\n\nGravitational interactions bring the cold neutralinos into our galactic halo\nand into the central spike, where neutralino pairs can annihilate and produce\nphotons, electrons, positrons, protons, antiprotons, and neutrinos. While most\nproducts are subject to absorption and\/or diffusion, the neutrinos escape\nthe spike and propagate to us undisturbed. We focus on the neutrinos and\npostpone the study of other signals.\n\nThe expected neutrino flux from neutralino annihilations in the direction\nof the galactic center can be divided into two components: emission from the\nhalo along the line of sight and emission from the central spike,\n\\begin{equation}\n\\Phi_{\\nu}^{\\rm neutralinos} =\n\\Phi_{\\nu}^{\\rm halo} + \\Phi_{\\nu}^{\\rm spike} .\n\\end{equation}\n\nThe halo flux from neutralino annihilations between us and the galactic center\ncan be estimated assuming that a single power law profile\n$\\rho(r) = \\rho_D (r\/D)^{-\\gamma} $ extends out to the Sun position $r=D$. \nThe integrated neutrino flux within an angle $\\Theta$ of the\ngalactic center is\n\\begin{equation}\n\\label{phihalo}\n\\Phi_{\\nu}^{\\rm halo} = \n\\frac{ \\rho_D^2 Y_{\\nu} \\sigma v D \\Omega(\\Theta) }{ m^2 } ,\n\\end{equation}\nwhere \n$m$ is the neutralino mass, $Y_{\\nu}$ is the number of neutrinos\nproduced per annihilation, either differential or integrated in energy, \n$\\sigma v $ is the\nneutralino--neutralino annihilation cross section times relative velocity,\nand\n\\begin{equation}\n\\Omega(\\Theta) =  \n\\frac{\\Theta^2 }{2(1-2\\gamma)} - \\frac{2^{\\gamma-1\/2}\n\\Theta^{3-2\\gamma}}{(3-2\\gamma)(1-2\\gamma)} -\n\\frac{ \\Theta_{\\rm min}^{3-2\\gamma}}{3-2\\gamma} .\n\\end{equation}\nHere $\\Theta$ is in radians and $\\Theta_{\\rm min} = \\max \\bigl[ 10R_{\\rm S}\/D ,\n$ $ (2\\gamma\/3)^{1\/(3-2\\gamma)} (\\rho_D\/\\rho_{\\rm\n  core})^{1\/\\gamma} \\bigr]$.\n\nWe evaluate the neutrino yield $Y_\\nu$ and the neutralino annihilation cross\nsections $\\sigma$ using the DarkSUSY package~\\cite{gon99}, which incorporates\nPythia simulations of the $\\nu$ continuum~\\cite{ber99}\nand the annihilation cross sections in~\\cite{eds97,annih}.\n\\begin{figure}[t]\n\\label{figenh}\n\\epsfig{file=spikef2.eps,width=0.4\\textwidth}\n\\caption{\n  Enhancement of annihilation signals from the\n  galactic center.}\n\\end{figure}\n\\begin{figure}[tbp]\n\\label{figphimu}\n\\epsfig{file=spikef3.eps,width=0.4\\textwidth}\n\\caption{For a Moore et al.\\ halo\n  profile, flux of neutrino-induced muons in a neutrino telescope from\n  neutralino dark matter annihilations in the direction of the galactic center,\n  with (upper panel) and without (lower panel) the central spike. The\n  horizontal line is the current upper limit.  }\n\\end{figure}\n\nTo the halo flux we need to add the contribution from the spike around the\nblack hole at the galactic center. \nFor an isothermal distribution \nwe find\n\\begin{equation}\n\\label{phispike1}\n\\Phi_{\\nu}^{\\rm spike} =\n\\frac{ \\rho_D^2 Y_{\\nu} \\sigma v D} {m^2} \\,\n\\left( \\frac{ R_M }{ D } \\right)^{3} \\,\n\\, \n\\ln \\left( \\frac{ R_M }{ 25 R_{\\rm S} } \\right) ,\n\\end{equation}\nthe factor of 25 serving to match the exact integration.  This flux is a factor\nof $\\sim 10^{-9}$ smaller than the flux from dark matter annihilations between\nus and the galactic center, and so the addition of the spike does not modify\nthe signal.  The same conclusion is reached in general for halo models with\nfinite cores.\n\nA strong enhancement results instead \nfor power law profiles. We find\n\\begin{equation}\n\\label{phispike2}\n\\Phi_{\\nu}^{\\rm spike} = \n\\frac{ \\rho_D^2 Y_{\\nu} \\sigma v D} {m^2} \\,\n\\left( \\frac{ R_{\\rm sp} }{ D } \\right)^{3-2\\gamma} \\,\n\\left( \\frac{ R_{\\rm sp} }{ R_{\\rm in} } \\right)^{2\\gamma_{\\rm sp}-3} ,\n\\end{equation}\nwhere $R_{\\rm sp}$ is given after eq.~(\\ref{rhoprime}) with $\\rho_0$\nreplaced by $\\rho_D$ and $r_0$ by $D$.\nWe fix $R_{\\rm in}$ so as to match the\nintegration of the numerically-calculated density profile including capture and\nannihilation: we\nfind that $R_{\\rm in} = 1.5 \\left[ (20 R_{\\rm S})^2+R_{\\rm core}^2\n\\right]^{1\/2} $ gives a good approximation to the flux\n(within 6\\% for our values of $\\gamma$).\n\nContrary to the case of finite cores, for cusped halos there is a huge increase\nin flux from the galactic center when the spike is included, typically 5 orders\nof magnitude or more, unless the inner halo slope $\\gamma$ is very small (see\nFig.~2, where $\\Theta=1.\\!\\!^\\circ 5$).  The enhancement is notable, for\nexample, for the profile of Moore et al.\\ which has $\\gamma=1.4$ (see\nFig.~3): including the spike around the black hole dramatically changes \nthe prospects of observing a neutrino flux.\n\nThe neutrino flux from the spike increases with the inner halo slope $\\gamma$.\nImposing that it does not exceed the observed upper limit of $\\sim 2 \\times\n10^4$ muons($>\\!1$GeV) km$^{-2}$ yr$^{-1}$ \\cite{imb-kgf-kamiokande} leads to\nan upper bound on $\\gamma$. There is a separate upper bound for each model.\nThey are plotted in Fig.~4a. (Plotted values of $\\gamma_{\\rm max}>2$ are\nunphysical extrapolations but are shown for completeness.) Present bounds are\nof the order of $\\gamma_{\\rm max} \\sim 1.5$.\n\nFuture neutrino telescopes situated in the Northern hemisphere may improve on\nthis bound or find a signal. For example, with a muon energy threshold of 25\nGeV, the neutrino flux from the spike after imposing the current constraints\ncould still be over 3 orders of magnitude above the atmospheric background\n(Fig.~5), allowing to probe $\\gamma$ as low as 0.05 (Fig.~4b).\n\\begin{figure}[tbp]\n\\label{figgmaxnu}\n\\epsfig{file=spikef4.eps,width=0.4\\textwidth}\n\\caption{Maximum inner slope\n  $\\gamma$ of the dark matter halo compatible with the upper limit on the\n  neutrino emission from the galactic center. (a) Current limit at 1 GeV;\n  (b) future reach at 25 GeV.}\n\\end{figure}\n\nIn conclusion, we have shown that if the galactic dark halo is cusped, as\nfavored in recent N-body simulations of galaxy formation, a bright dark matter\nspike would form around the black hole at the galactic center. A search of a \nneutrino signal from the spike could either set upper bounds on the\ndensity slope of the inner halo or clarify the nature of dark matter. \n\nThis research has been supported in part by grants from NASA and DOE.\n\n\\begin{figure}[tbp]\n\\label{figphimumax}\n\\epsfig{file=spikef5.eps,width=0.4\\textwidth}\n\\caption{Maximal flux of neutrino-induced muons in a neutrino telescope from\n  neutralino annihilations at the galactic center, after imposing the current\n  constraints on the neutrino emission.}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe structure of globular clusters is shaped by complex interaction between external (interaction with the host galaxy) and internal (for example, relaxation, core collapse, mass segregation) forces. \\cite{King1966} successfully described the star density of globular clusters with assuming no rotation.\n\\citet{Tiongco2017} have suggested that the internal relaxing processes dissipate all the angular momentum on a long term in every globular cluster. However, growing number of  studies \\citet{Lane2010, Bellazzini2012, Bianchini2013, Fabricius2014, Lardo2015, Kimmig2015, Boberg2017, Lee2017, Cordero2017, Kamann2018, Ferraro2018, Lanzoni2018, Bianchini2018} present evidence that a significant amount of internal rotation could still be observed in many globular clusters, however the observed rotational strengths are only a fraction of the initial ones \\citep{Kamann2018, Bianchini2018}. These studies mostly utilize recently recorded high quality radial velocity and proper motion data.\n\nThe presence of internal rotation could introduce some problems with the theory of globular clusters formation and evolution. Some studies indicate that the remaining rotation accelerates the evolution and shapes the morphology of the cluster \\citep{Einsel1999, Bianchini2013}. Others suggest that the present day rotation could be a remnant of a strong rotation during the early history of globular clusters  \\citep{Vesperini2014, Mapelli2017}. \n\nOrbital motions may be isotropic even during the formation of GCs, for example \\citet{Lahen2020} have argued that the massive clusters are isotropic already during their first 100 Myr after formation.\nMoreover, as \\citet{Lane2010} and \\citet{Tiongco2017} has shown, the interaction between the cluster and the galactic tidal field combined within the internal dynamics could produce complex kinematical features, e.g. radial variation in the orientation of the rotational axis, and anomalous velocity dispersions. The observational evidence of rotating globular clusters is based on high precision radial velocity data \\citep{Bellazzini2012, Bianchini2013, Lardo2015, Kimmig2015, Boberg2017,Lee2017, Cordero2017, Ferraro2018, Lanzoni2018,Cordoni2020a}, integral field unit (IFU) spectrograph  \\citep{Fabricius2014, Kamann2018}, and proper motion measurements \\citep{Massari2013, Bellini2017, Bianchini2018, Mastrobuono2020}. \n\nThe other aspect of the globular cluster rotation is the presence of multiple generations of stars and their rotational properties. \nIn the last decade, the existence of two or more distinct generations of stars in most globular clusters became well studied \\citet{Carretta2009a, Carretta2009b, Piotto2015, Milone2017, Milone2018, Meszaros2020}, however understanding the formation of multiple population is still an astrophysical challenge. Multiple populations manifest in light-element abundance variations, second generation stars (SG) are enhanced in N, Na, Ca and Al and depleted in C, O and Mg, while the first generation stars (FG) are the opposite. \nMost of the theories agree, that the second generation stars formed out of the first generation's ejecta mixed with the original intracluster medium \\citep{Decressin2007, Bekki2010, Denissenkov2014, Bekki2017}, but the exact process of this pollution is currently not known, and many observed processes can not be explained with this theory yet. \nThere are other alternative explanations for this phenomena and they are discussed in detail in \\citet{Henault2015, Bastian2018}. \n\nObservational evidence have showed differences in spatial distribution between FG and SG stars. For dynamically younger clusters, the SG is more concentrated than the FG \\citep{Sollima2007, Bellini2009, Lardo2011, Cordero2014, Boberg2016, Lee2017, Gerber2020}. On the other hand, the two generations are completely mixed in globular clusters with more advanced evolutionary stages \\citep{Dalessandro2014, Nardiello2015, Cordero2015, Gerber2018, Gerber2020}. In \\citet{Dalessandro2019}, the link between concentration differences and evolutionary stage was explored in detail based on observations and models.\nOther observational studies revealed differences in the kinematics between the multiple generations \\citep{Richer2013, Bellini2015, Bellini2018, Cordero2017, Milone2018, Libralato2019, Cordoni2020a, Cordoni2020b}, in other cases, the multiple generation of stars share similar kinematic properties \\citep{Pancino2007, Cordoni2020b}\nThese literature sources used different indicators of cluster kinematics, including rotation, velocity dispersion and anisotropy to show the different kinematical properties of multiple populations.\n\n\nOur main purpose is to investigate the rotational properties of the selected clusters using high precision and homogeneous radial velocity data. Our secondary goal is to identify potential differences in the cluster rotation properties between the multiple populations.\nFor this study, we use state-of-the-art data from the high-resolution spectroscopic survey Apache Point Observatory Galactic Evolution Experiment (APOGEE) \\citep{Majewski2016}. The APOGEE started as one component of the $3^{\\rm rd}$ Sloan Digital \nSky Survey (SDSS-III; \\cite{Eisenstein2011}) and continues as part of SDSS$-$IV \\citep{Blanton2017} as APOGEE-2\\footnote{http:\/\/www.sdss.org\/surveys\/apogee-2\/}. The goal of APOGEE-2 is to obtain high-resolution (R = 22500), high signal-to-noise, H-band spectra ($\\lambda$ = 1.51$-$1.70$\\mu$m) of more than 600,000 late-type stars in the Milky Way by the end of 2020, and to determine chemical abundances of $\\sim$26 elements in all observed stars. \nMost APOGEE targets are evolved red-giant branch (RGB) and asymptotic giant branch (AGB) stars from all major Galactic stellar populations.\n\n\n\n\n\\section{Data and reduction}\n\\subsection{Target Selection and Radial Velocities}\n \nThe data were gathered by the Sloan Foundation 2.5 m Richey-Chreritien altitude-azimuth telescope \\citep{Gunn2006} at Apache Point Observatory. The spectra were obtained via the APOGEE spectrograph \\citep{Wilson2019} with a resolution power of 22500. The stellar atmospheric parameters and chemical abundances are calculated from these spectra with APOGEE Stellar Parameters and Chemical Abundances Pipeline (ASPCAP) \\citep{Garcia2016}.    \nWe use the list of stars compiled by \\citet{Masseron2019}. The target selection is explained in detail in \\citet{Meszaros2015}, \\citet{Masseron2019} and \\citet{Meszaros2020}. In short, stars were selected based on their radial velocity, distance from cluster center and metallicity. In radial velocity, all studies required stars to be within three times the velocity dispersion of the mean cluster velocity, which were taken from \\citet{Baumgardt2019}.\n\nWe used the DR14 data release of APOGEE \\citep{Holtzman2018} for our study. Radial velocities are derived by the reduction pipeline \\citep{Nidever2015}, while details can be found in that paper, we provide a brief description of the algorithm. For almost all stars, observations are made in multiple visits to improve S\/N and to allow observations of faint objects, which are then combined together to provide the final spectrum of the star. The radial velocity is measured in multiple steps. First, we do an initial measurement for each star from the actual individual spectrum by cross correlating each spectrum with the best match in a template library. In the second step, all of the visits are combined, and relative radial velocities of each visit are iteratively refined by cross-correlating each visit spectrum with the combined spectrum. The final absolute radial velocity is set by cross-correlating the combined spectrum with the best match in a template library. The peak of the cross-correlation function is fitted with a Gaussian in order to determine the accurate spectral shift. \nBinary stars can distort the rotational velocity profile if included in the sample. We removed all the binaries from our sample using database from \\citet{PriceWhelan2020}. In \\citet{PriceWhelan2020}, they identified nearly 20000 binary candidates with high confidence in the 16th data release of APOGEE, out of these we have found 65 stars in common with our targets, only $\\approx~7\\%$ of the total number of stars.\n \nThe uncertainties of radial velocities depend on multiple factors, mainly the characteristic of spectra, the resolution and the S\/N ratio. For example, a star with lots of deep and thin lines in the spectra had a more precise RV than a star with wide and shallow lines. The typical uncertainty of the final radial velocity for stars in this study is 0.1 km\/s.\n\n\n\\subsection{Method}\n\n\\begin{table*}                                                                                 \n\\begin{tabular}{lccccccccc}                                                                                           \n\\toprule                                                                                                                \nGC name & N & [Fe\/H] & V$_{\\rm helio}$ & R$_{\\rm h}$ & d$_{\\rm avg}$ & PA & PA$_{\\rm err}$ & A$_{\\rm rot}$ & A$_{\\rm rot err}$  \\\\          \n&  &       &   [km\/s] & [arcmin]  &   [arcmin]   &       &            & [km\/s]        & [km\/s]                        \\\\    \n\n\\hline                                                                                                                \nM2 & 26 & -1.65 & -5.3 & 0.93 & 3.9 & 26 & 19 & 3.48 & 0.82            \t\\\\                        \n\nM3 & 145 & -1.5 & -148.6 & 1.12 & 6.3 & 164 & 13 & 1.19 & 0.28          \t\\\\\n\nM5 & 215 & -1.29 & 52.1 & 2.11 & 5.9 & 148 & 6 & 3.45 & 0.37           \t\\\\\n\nM12 & 65 & -1.37 & -42.1 & 2.16 & 5.8 & 56 & 93 & 0.24 & 0.19                \\\\\n\nM13 & 135 & -1.53 & -246.6 & 1.49 & 5.1 & 26 & 9 & 2.38 & 0.39  \t        \\\\                        \n\nM15 & 138 & -2.37 & -107.5 & 1.06 & 4.8 & 120 & 11 & 2.38 & 0.44      \t\\\\                        \n\nM53 & 40 & -1.86 & -79.1 & 1.11 & 4.8 & 98 & 27 & 1.54 & 0.57           \t\\\\                        \n\nM71 & 28 & -0.78 & -22.9 & 1.65 & 2.7 & ... & ... & ... & ...               \t\\\\                        \n \nM92 & 72 & -2.31 & -121.6 & 1.09 & 4.2 & 154 & 14 & 2.06 & 0.58       \t\\\\                       \n\nM107 & 67 & -1.02 & -33.8 & 2.70 & 4.3 & 168 & 30 & 0.72 & 0.27         \t\\\\  \n                        \n\n\\bottomrule \n\\end{tabular}  \n\\caption{The table contains the basic parameters of the targeted globular clusters and the results of this study. The second column represents the number of observed stars. The third and the fourth columns are the metallicity and the clusters heliocentric radial velocity \\citet{Harris1996, Miocchi2013}. The fifth is the halflight radii \\citet{Harris1996}. The sixth is the average distance from the cluster centers in our samples. The seventh and eighth columns represent the calculated position angle and its error, while the last two columns are the rotational velocity and its uncertainties. The position angles measured from North to East anti-clockwise direction.}\n\\end{table*}\n\nIn order to investigate the rotational velocity and the position angle of the rotational axes of the cluster, we follow the same method as \\cite{Cote1995, Bellazzini2012, Bianchini2013, Lardo2015, Kimmig2015, Boberg2017,Lee2017, Lanzoni2018}. \nFirst, each cluster was split into two halves along the cluster center. The position angle of separation was the independent variable of the analysis, varied between PA = 0 and 180 (PA = 90 is toward East) in 2 degree step-size.We ran the simulations with multiple step-sizes (e.g. 2, 5, 10, 20 degree) all providing the same end results within our uncertainties. At the end we choose the 2 degrees for all of our clusters to appropriately sample the densest areas.\nNext, the mean radial velocity of these sub-samples were calculated and the difference between the two sub-samples mean velocity were determined. If rotation is present in the system, the delta V$_{\\rm mean}$ draws a sinusoidal variation as a function of the position angle. The amplitude of the function is twice the rotational velocity (because the amplitude is the difference of the two hemispheres) and the min-max position (ideally the difference is 180 degree) is the PA of the rotational axis. \nThus, we caution the reader that rotational velocity values printed in Table 2 and 3, and all figures are twice as large as the real rotational velocity, in agreement to what has been used in the literature.\nOur results are listed in Table~1.\n\n\n\\subsection{Separating multiple populations in GCs}\n\nIn this study, first (FG) and second generation (SG) stars are separated from each other based on their [Al\/Fe] abundances, following the suggested cuts by \\citet{Meszaros2020} from APOGEE data. Previously, \\citet{Meszaros2015} used an extreme-deconvolution code for fitting a distribution with multiple Gaussian components to identify FG from SG stars in a mutli-dimensional abundance space, they showed that almost all of the SG stars have [Al\/Fe]$>$0.3~dex. For this reason we use this simplified criteria to set multiple populations apart using abundances from \\citet{Masseron2019}. For this analysis, we selected 4 clusters with the most observed stars, in which we have enough samples to properly fit the rotational curve.   \n\n\n\\subsection{Error estimation}\nWe checked the robustness of the method with a simple jackknife test. We randomly dropped more and more stars from the sample and calculated the rotational curves. The results indicate, we can get a good signal if we have at least 20-30 stars to work with. Fewer stars than this is insufficient for a robust measurement. \n\nIn order to calculate the final uncertainty, we randomly dropped 20 percent of the stars (dropping more than this may result in fewer than 20 stars for some clusters) and derived the position angle and the rotational velocity in the sub-samples. \nWe repeated this process 100 times, then the standard deviation of the sinusoidal fit was calculated for the position angles and rotational velocities for each of these sub-samples. The final uncertainties seen in Figure~1 are the average of the differences between the original fit and the sub-samples.\n\nWe were able to define the errors for all the selected clusters, however for M12, the combination of the small number of observed stars, low rotational signal, and the error estimation method produced a high uncertainty of the PA.   \nFor M71 we could not get a clear sinusoidal signal from the data at hand.  \nTable~1 contains our derived results. \n\nWe tested the robustness and the stability of the derived rotational amplitudes with a bootstrap analysis. For both populations in all clusters, 100 bootstrap distributions have been realised with redistributing the measured velocities randomly (with re-sampling allowed) among the field stars. These samples suffered a complete loss of any information on the rotation, and contained a \"null signal\". The distributions were then evaluated following exactly the same method as in the case of the observations, and we observed the best-fit amplitude of the inferred rotating model. This amplitude was considered as the upper limit of the rotation amplitudes if the clusters would not rotate, and the measured amplitudes were just a product of numerical fluctuations in the data distribution.\nThe standard distribution of the amplitudes in the bootstrap samples were in the range of 0.5--0.8 ~km\/s, proving that the detection of the rotation of all examined clusters is indeed significant.\n\n\n\n\\section{Discussion}\n\n\\subsection{Comparison with literature}\n\\begin{table*}\\centering\n\\begin{tabular}{@{}lcc@{}c@{}cc@{}c@{}cc@{}c@{}cc@{}c@{}cc@{}c@{}}\n\\toprule\n& \\multicolumn{2}{c}{M2} & \\phantom{abc}& \\multicolumn{2}{c}{M3} & \\phantom{abc} & \\multicolumn{2}{c}{M5} & \\phantom{abc} & \\multicolumn{2}{c}{M12} & \\phantom{abc} & \\multicolumn{2}{c}{M13} \\\\\n\\cmidrule{2-3} \\cmidrule{5-6} \\cmidrule{8-9} \\cmidrule{11-12} \\cmidrule{14-15} \n& PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$  \\\\ \n\\hline\n\\cite{Lane2010}       & ... & ... && ... & ... && ...  & ... && 40 & 0.15$\\scriptsize\\pm0.8$ && ...& ...       \\\\ \n\\cite{Bellazzini2012} & ... & ... && ... & ... && 157   & 2.6$\\scriptsize\\pm0.5$ && ... & ... && ...  & ...      \\\\\n\\cite{Fabricius2014}  & ... & ... && 192 $\\scriptsize\\pm11.8$& ... && 149$\\scriptsize\\pm5.6$ & ... && 89$\\scriptsize\\pm19.3$& ... && 17$\\scriptsize\\pm7.8$& ...       \\\\ \n\\citet{Kimmig2015}    & 53 & 4.7$\\scriptsize\\pm1.0$ && ... & 0.6$\\scriptsize\\pm1.0$ && ...  & 2.1$\\scriptsize\\pm0.7$ && ... & 0.2$\\scriptsize\\pm0.5$ && ... & ...       \\\\ \n\\citet{Lee2017}       & ... & ... && ... & ... && 128  & 3.36$\\scriptsize\\pm0.7$ && ... & ... && ... & ...       \\\\ \n\\cite{Cordero2017}    & ... & ... && ... & ... && ...  & ...  && ... & ... && 14$\\scriptsize\\pm19$  & 2.7$\\scriptsize\\pm0.9$   \t    \\\\\n\\cite{Kamann2018}     & 41.7$\\scriptsize\\pm2.7$ &... && ... & ... &&144$\\scriptsize\\pm20.3$ & ...  && ... & ... && ... & ...  \t    \\\\ \n\\cite{Ferraro2018}    & ... & ... && 151 & 1.0 && ... & ...   && ... & ... && ... & ...   \t    \\\\  \n\\cite{Lanzoni2018}    & ... & ... && ... & ... && 145 & 4.0   && ... & ... && ... & ...  \t    \\\\\n\\cite{Sollima2019}    & 14$\\scriptsize\\pm12.1$ & 3.01$\\scriptsize\\pm0.7$ && ... & 1.75$\\scriptsize\\pm0.4$ && 132$\\scriptsize\\pm6$ & 4.11$\\scriptsize\\pm0.4$   && ... & 0.93$\\scriptsize\\pm0.4$ && 15$\\scriptsize\\pm14.2$ & 1.53$\\scriptsize\\pm0.6$  \t    \\\\ \n\\hline\nthis work             & $26\\scriptsize\\pm19$ & $3.48\\scriptsize\\pm0.8$ && $164\\scriptsize\\pm15$ & $1.19\\scriptsize\\pm0.3$ && $148\\scriptsize\\pm6  $ & $3.45\\scriptsize\\pm0.4$   && $56\\scriptsize\\pm93$ & $0.24\\scriptsize\\pm0.2$ &&  $26\\scriptsize\\pm9$ & $2.38\\scriptsize\\pm0.4$ \\\\  \n\\bottomrule\n\\end{tabular}\n\\caption{Comparison with literature. Part~1. \\\\\nPosition angles and rotational amplitudes from earlier studies. Since different conventions were followed, we convert the published result to PA 90 = East, anti-clockwise system. Literature source which did not use the double rotational velocity were converted to our system. The first sub-column represent the position angle of the rotational axis and the second is the rotational velocity in [km\/s] } \n\\end{table*}\n\n\\begin{table*}\\centering\n\\begin{tabular}{@{}lcc@{}c@{}cc@{}ccc@{}c@{}cc@{}ccc@{}}\n\\toprule\n& \\multicolumn{2}{c}{M15} & \\phantom{abc}& \\multicolumn{2}{c}{M53} & \\phantom{abc} & \\multicolumn{2}{c}{M71} & \\phantom{abc} & \\multicolumn{2}{c}{M92} & \\phantom{abc} & \\multicolumn{2}{c}{M107} \\\\\n\\cmidrule{2-3} \\cmidrule{5-6} \\cmidrule{8-9} \\cmidrule{11-12} \\cmidrule{14-15} \n & PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$ && PA & A$_{\\rm rot}$  \\\\ \n\\hline\n\\cite{Lane2009}       & ... & ... && nf & nf && ...  & ... && ... & ... && ...& ...       \\\\ \n\\cite{Bellazzini2012} & 110 & 3.8$\\scriptsize\\pm0.5$ && ... & ... && 163   & 1.3$\\scriptsize\\pm0.5$ && ... & ... && 84  & 2.9$\\scriptsize\\pm1.0$ \\\\\n\\citet{Bianchini2013} & 106$\\scriptsize\\pm1$ & 2.84&& ... & ... && ...   & ... && ... & ... && ... & ...  \\\\\n\\cite{Fabricius2014}  & ... & ... &&113$\\scriptsize\\pm19.2$& ... && ...   & ... && 99$\\scriptsize\\pm12.0$& ... && ... & ...  \\\\ \n\\citet{Lardo2015}     & 120 & 3.63$\\scriptsize\\pm0.1$ && ... & ... && ...   & ... && ... & ... && ... & ...  \\\\\n\\citet{Kimmig2015}    & ... & 2.5$\\scriptsize\\pm0.8$ && ... & 0.4$\\scriptsize\\pm0.7$ && ...   & 0.4$\\scriptsize\\pm0.8$ && ... & 1.8$\\scriptsize\\pm0.8$ && ... & ...  \\\\ \n\\citet{Boberg2017}    & ... & ... && 74 & 2.8 && ...   & ... && ... & ... && ... & ...  \\\\    \n\\cite{Kamann2018}     &151$\\scriptsize\\pm$10.4& ... && ... & ... && ...   & ... && ... & ... && ... & ...  \\\\ \n\\cite{Ferraro2018}    & ... & ... && ... & ... && ...   & ... && ... & ... && 167 & 1.2  \\\\ \n\\cite{Sollima2019}    & 128$\\scriptsize\\pm28.8$ & 3.29$\\scriptsize\\pm0.5$ && ... & ... && ... & ...   && ... & 1.46$\\scriptsize\\pm0.6$ && ... & ...  \t    \\\\ \n\n\\hline\nthis work             & $120\\scriptsize\\pm11$ & $2.38\\scriptsize\\pm0.4$ && $98\\scriptsize\\pm27$ & $1.54\\scriptsize\\pm0.6$ && ...   &  ...&& $154\\scriptsize\\pm14$ & $2.06\\scriptsize\\pm0.6$ && $168\\scriptsize\\pm30$ & $0.72\\scriptsize\\pm0.3$ \n\\\\  \n\\bottomrule\n\\end{tabular}\n\\caption{Comparison with literature. nf = not found evidence of rotation. Part~2}\n\\end{table*}\n\nThe latest results available in the literature were collected in Table 2 and 3, which contain the calculated position angles of the rotational axis for clusters in common with our sample. These data also represented in Figure 3. There are multiple conventions used in the literature for angle and direction notations. We converted all these various approaches to the PA 90 = East convention. The last row contains our values.\n\nWe detected systematic rotation in almost all of the targeted globular clusters. We confidently could derive rotational velocity and position angles for nine out of the ten selected clusters. All nine clusters have been studied in the literature before, thus we are able to not only compare our results with previous studies, but also homogenize the rotational velocity as our radial velocities are from one homogeneous survey. \nWe caution the reader that the assumption of a constant rotation velocity as a function of distance to the cluster centre is a significant simplification. \nThe observations of \\citet{Boberg2017, Bianchini2018, Sollima2019} have shown that the peak of the rotational curve is located approximately at the cluster half-light radius, however this location is expected to change during the evolution of the cluster \\citep{Tiongco2017}.\nWe listed the halflight-radius and the average distance of our sample from the cluster centre in Table~1. In all cases, the average distance is at least 2-3 times larger than the halflight-radius suggesting that our assumption of a constant rotation velocity underestimates the magnitude of the rotational velocity.\n\nBefore such a comparison can be made one must transform the results from the literature to a common coordinate system (PA 90 = East, anti-clockwise). After the transformation we are able to conclude while for some of the cluster we have a good agreement within our uncertainties, other less studied clusters with fewer stars show larger than expected discrepancy between studies. In the next few sections we examine these differences for each cluster.   \n\n\\subsubsection{\\rm M5}\nM5 is a well observed cluster targeted by \\cite{Bellazzini2012, Fabricius2014, Kimmig2015, Lee2017, Kamann2018,Lanzoni2018}, therefore it is an excellent object to use it as a standard to compare our results to, especially because these literature sources used different measurement methods.\nThe position angle varies between 144.3 to 157 degree in the literature, our result of 148 degree fits nicely in this picture. \n\nAs mentioned before, our calculation technique is similar to many that studied M5 \\citep{Bellazzini2012, Lee2017, Lanzoni2018}, and by comparing our results to these studies, we can find a good agreement in all cases. \\citet{Fabricius2014} and \\citet{Kamann2018} used IFU spectrograph for the analysis. The advantage of this method that it is possible to measure crowded stellar fields in order to perform a detailed analysis of dispersion fields and central rotation.   \nIn \\cite{Bianchini2018} the rotational pattern was derived from proper motion data (GAIA). Despite the different observations and methodology we feel confident in our approach as it nicely reproduces the rotation velocity and angle reported by the mentioned studies for M5.\nIn \\citet{Sollima2019} the rotation of M5 (among other clusters) was investigated via radial velocity component from VLT and Keck instruments and proper motion data from GAIA 2nd data release. In \\citet{Sollima2019} the rotational velocity derived from all the three velocity components (taken into consideration the inclination of the rotational axis) while we were able to use only the line-of-sight part. Their derived results show a good agreement with ours in case of PA and the difference in rotational velocity can be explained by the fact that we observed line of sight velocity, while they were able to determine the inclination.  \n\n\\subsubsection{\\rm M2}\n\\cite{Kimmig2015} derived the rotational velocity as A$_{\\rm rot} = 4.5 ~\\rm km\/s$, which is slightly larger than our value at $3.48\\pm0.8$. Considering our uncertainty we conclude that these differences are not substantial. The position angle also  differs from \\citet{Kimmig2015, Kamann2018, Sollima2019}, but our limited sample size cause a 19 degree of uncertainty, which can explain the difference. \n\n\n\\subsubsection{\\rm M3, M12 and M13}\nWe have a good amount of observed stars in M3 and the derived position angles are within errors to \\cite{Ferraro2018}. The difference from \\cite{Fabricius2014} can be explained by their relatively large errors and the different analysis method applied. \n\nOur derived result for the rotational properties of M12 have high uncertainty, probably because either the amplitude of the rotation is too small, or the inclination is close to 90 degrees, so that we look close to the direction of rotational axis. We have a good agreement with the results \\cite{Lane2010} and considering the large uncertainty we agree well with \\cite{Fabricius2014} too.   \n\nOur results show a good agreement at the level of our uncertainties with \\cite{Fabricius2014}, \\cite{Cordero2017} and \\citet{Sollima2019} for M13. The rotational amplitude is also really similar to the one presented in \\cite{Cordero2017}. \n\n\\subsubsection{\\rm M15}\nIn case of M15 we have a slight deviation in the PA from one of the previous results, namely \\cite{Kamann2018}, however we are in a good agreement with the other four \\citep{Bellazzini2012, Bianchini2013, Lardo2015, Sollima2019}. The source of the misalignment with \\cite{Kamann2018}, which is 30 degrees, might be due to two different reasons. One is that \\cite{Kamann2018} used data from an IFU spectrograph and used Voronoi-binned maps of the mean velocity and velocity dispersion across the observed region.  The second reasons is that they focused on the cluster center region, while our sample contains fewer stars from the cluster's center and more from the outer region. There is a variation among the derived rotational amplitudes among all the literature sources, but there are on the level of what is expected from the uncertainties of the measurements and methods.  \n\n\\subsubsection{\\rm M53}\nFor M53, our PAs value lies between \\cite{Fabricius2014} and \\cite{Boberg2017} with a relatively high error of margin. The cause of this probably is the low sample size and the relatively small rotational signal originated from the cluster. The problem caused by the usage of fewer observed stars in the calculation is visible in Figure~1. If the cluster contains less stars for this method, the signal become more scattered, however the rotation is still detectable in this case. In a study of \\cite{Lane2009} a detectable rotational signal was not found for this cluster within the margin of error.  \n\n\\subsubsection{\\rm M71}\nIn case of M71, we did not find conclusive fit for the available data, which is interesting considering that we selected 28 stars from M71, similarly to M2, but such a small sample size was enough for a measurement in that cluster. The lack of measurement may suggest the possibility of this globular cluster not rotating or perhaps the inclination of the axis is close to 90 degree or the rotational signal simply too small to be detectable from our sample. \n\n\\subsubsection{\\rm M92}\nThe difference between our PA and that of \\citet{Fabricius2014} in M92 is significant. Since they have used a different calculation method, we might expect a large difference such as this, however in case of M5, M3 and M53 we have a good agreements with this source. For this reason we do not know why our PA differs so much from \\citet{Fabricius2014}, especially that our sample is large enough for a reliable measurement. Our derived rotational amplitude is similar the one derived in \\cite{Kimmig2015} and \\cite{Sollima2019}. \n\n\\subsubsection{\\rm M107}\nOur derived position angle is very close to \\cite{Ferraro2018}, however there is a 90 degree of deviation from \\cite{Bellazzini2012} and a significant difference between the rotational amplitudes too. We are not sure what causes this, because \\cite{Bellazzini2012} used the same method as we did and for other clusters we found an acceptable agreement. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6.5in,angle=0]{all_gc_curve_5.png}\n\\caption{ Global rotation of the globular clusters, having FG and SG samples unified. \\\\ \nPosition angle of the rotational axis vs. difference between the two sub-sample's mean value in case all of the studied globular clusters except M71 (we can not find conclusive fit). The line shows the best fit sin function and the actual rotational velocity is half of the amplitudes. } \n\\label{fig:m5radseb} \n\\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6.5in,angle=0]{all_2gen_curve7.png}\n\\caption{Comparison of rotation curves of FG (red) and SG (blue) stars in the 4 selected globular clusters.    \n} \n\\label{fig:2gen_rvcurve} \n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=6.5in,angle=0]{compare.png}\n\\caption{Comparison between values determined in other studies (Table 2 and 3) and our results. \n} \n\\label{fig:m5radseb} \n\\end{figure*}\n\n\\subsection{Rotation according to first and second generation stars}\n\nFrom theoretical studies \\citep{Decressin2007, Bekki2010, Denissenkov2014, Bekki2017} we can expect GCs to have higher cluster rotational speed when measured from the SG stars than from only the FG stars. At the same time, the velocity dispersion should be higher among the FG stars. This is because the FG stars formed from massive molecular clouds and their first supernovas expelled the remaining cold gas from the GC. In a next stage, the polluted gas from the FG stars accumulated in the GCs center and this was the origin of the SG stars. Numerical simulations based on this theory \\citep{Bekki2017} suggests higher rotational speed in case of SG than FG stars. We are able to test this idea in M3, M5, M13 and M15, clusters in which we have enough stars to sample both populations and are able to measure the cluster rotation based on the two generation of stars. Our results are listed in Table~4, and shown in Figure~2.  \nHowever, we have to mention that in case of other formation scenarios opposite rotational velocities might be observed for FG and SG stars, i.e. the first generation rotates faster and the second slower as suggested by \\citet{Henault2015}.    \nMany different literature studies have examined the rotation as a function of stellar populations, such as 47 Tuc \\citep{Milone2018}, Omega Centauri \\citep{Bellini2018}, NGC 6352 \\citep{Libralato2019}, M80 \\citep{Kamann2020} or M54 \\citep{Alfaro2020}, but none of these clusters are in our sample, thus no direct comparison is possible.\n\nFrom figure~2 we can conclude that in case of M5, M13, M15 the rotational velocity originated from the FG and SG stars do not differ significantly, the small discrepancies are all well within our derived uncertainties. We are not able to measure the predicted difference in these three clusters. The position angles of M5 and M13 are also very close for both generation of stars, but we observe a large deviation in case of M15. The position angle is $144\\pm56$ from the FG stars, and $98\\pm18$ from the SG stars, but the extremely large uncertainty of the first value does not allow us to conclude that this difference have an astrophysical origin. A larger sample of observed stars and supplemented by proper motion data may shed some light in this phenomena. \n\n\\citet{Cordoni2020a} examined the rotation of M5 according to FG and SG stars using GAIA proper motion data and line-of-sight velocities. The FG and SG stars present a significant difference in position angle in their study. We did not find these characteristics in our analysis, but it must be noted that our sample is much lower than theirs.\n\nIf we compare our M13 result (see in Table~4) to \\cite{Cordero2017} in which the 'extreme' population correspond to SG with PA = 7 and the remaining stars (\"normal\" + \"intermediate\") to FG with PA = 33, then we have a really good agreement with our results( PA = 12 for SG and PA = 34 for FG). Although the uncertainties in both studies can be considered high, the observed differences are well within these errors. Considering our errors, we do not believe the discrepancy between the PA of the FG and SG group has an astrophysical origin.   \n\nM3 is the peculiar object in our sample. Our observations prove that M3 does not appear to show any detectable global rotation when examined through only the FG group of stars, however in case of SG sample, we clearly see a strong rotation curves with an amplitude of $2.69\\pm0.6$ km\/s. The difference in cluster rotational velocities in the two population of stars is significant when compared to the uncertainties. \n\nIn order to estimate the upper limit for a possible rotation that could remain hidden in the numerical fluctuations we used a bootstrap analysis,  described in Section 2.4. In M3, the standard deviation of bootstrap amplitudes was 0.55~km\/s for FG stars, therefore the rotational amplitude is $<1.65$~km\/s with a 3-$\\sigma$ confidence. \n\nOur result appear to follow the theoretical predictions by \\cite{Bekki2017}; the cluster rotational velocity based on the SG stars is significantly higher than based on the FG stars. The behavior of the FG stars is also interesting, because our observations suggest a rotational velocity very close to zero without any hint to what the PA might be. The simplest explanation is that the rotational velocity is so small that it is not possible to detect within our precision. \n\n\\begin{table*}                                                                                 \n\\begin{tabular}{lcccccc}                                                                                    \n\\toprule\nGC  & Gen &  N & PA  & A$_{\\rm rot}$   \\\\          \n\\hline\nM3 & (fg) & 95 & ... & ...                       \\\\\n   & (sg) & 45 & 162 $\\pm$ 11 & 2.69 $\\pm$ 0.6         \\\\\n\nM5 & (fg) & 102 & 150 $\\pm$ 8 & 3.37 $\\pm$ 0.3  \t    \\\\\n   & (sg) & 92 & 150 $\\pm$ 6  & 4.36 $\\pm$ 0.5        \\\\\n   \nM13 & (fg) & 36 & 34 $\\pm$ 36  & 2.62 $\\pm$ 0.8        \\\\                        \n    & (sg) & 70 & 12 $\\pm$ 36  & 3.12 $\\pm$ 0.5       \\\\\n\nM15 & (fg) & 33 & 144 $\\pm$ 56  & 2.82 $\\pm$ 0.8     \t\\\\                        \n    & (sg) & 49 & 98 $\\pm$ 18   & 2.67 $\\pm$ 0.7         \\\\\n\\bottomrule\n\\end{tabular}  \n\\caption{The first and second generation's kinematic properties in case of the four selected clusters.}\n\\end{table*}\n  \n\n\\section{Conclusions}\n\nWe found evidence of rotation in M2, M3, M5, M12, M13, M15, M53, M92, and M107, but not in M71, supporting the theory that most globular cluster preserve significant amount of rotation during their lifetime. \nFor most clusters, these results show good agreement with other similar studies. With the precise radial velocity data of the APOGEE survey, we were able to provide homogeneous rotational velocities and PAs for several clusters for which such homogeneity did not exist in independent literature sources. \n\nWe successfully identified rotational signals originated from two different generation of stars in 4 selected clusters. \nIn M3, we discovered a significant difference between the rotational velocity of the cluster when it is examined through only the FG and SG stars. This is very much in agreement with the prediction of numerical simulations by several independent groups \\citep{Decressin2007, Bekki2010, Denissenkov2014, Bekki2017}. The cluster do not show a detectable rotational signal when selection the FG stars only,  while in case of the SG stars the cluster have a clear rotational signal at 2.69 km\/s. It is not clear what cause this phenomena, but a detailed analysis with supplement proper motion data might unfold this issue. \n\nIn M5 and M13 the FG rotational velocity is somewhat smaller than the SG velocity as the theory predicts, however the differences are well within the level or our uncertainties, thus we do not believe we see the prediction of the numerical simulations. In terms of PA, the deviations between the populations are really small or nonexistent and well within the derived uncertainties. \nFor M15 one can see a difference in PAs, but the relatively high uncertainty of the PA for the FG stars prevents us to draw a clear conclusion, therefore further analysis with a larger sample size is required. \n  \n\n\n\\section*{Acknowledgements} \n\nL. Sz. and Sz. M. has been supported by the Hungarian \nNKFI Grants K-119517 and GINOP-2.3.2-15-2016-00003 of the Hungarian National Research, Development and Innovation Office. Sz. M. has been supported by the J{\\'a}nos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the {\\'U}NKP-20-4 New National Excellence Program of the Ministry for Innovation and Technology.\n\nFunding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, \nthe U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges\nsupport and resources from the Center for High-Performance Computing at\nthe University of Utah. The SDSS web site is www.sdss.org.\n\nSDSS-IV is managed by the Astrophysical Research Consortium for the \nParticipating Institutions of the SDSS Collaboration including the \nBrazilian Participation Group, the Carnegie Institution for Science, \nCarnegie Mellon University, the Chilean Participation Group, the French Participation Group, \nHarvard-Smithsonian Center for Astrophysics, \nInstituto de Astrof\\'isica de Canarias, The Johns Hopkins University, \nKavli Institute for the Physics and Mathematics of the Universe (IPMU) \/ \nUniversity of Tokyo, Lawrence Berkeley National Laboratory, \nLeibniz Institut f\\\"ur Astrophysik Potsdam (AIP),  \nMax-Planck-Institut f\\\"ur Astronomie (MPIA Heidelberg), \nMax-Planck-Institut f\\\"ur Astrophysik (MPA Garching), \nMax-Planck-Institut f\\\"ur Extraterrestrische Physik (MPE), \nNational Astronomical Observatories of China, New Mexico State University, \nNew York University, University of Notre Dame, \nObservat\\'ario Nacional \/ MCTI, The Ohio State University, \nPennsylvania State University, Shanghai Astronomical Observatory, \nUnited Kingdom Participation Group,\nUniversidad Nacional Aut\\'onoma de M\\'exico, University of Arizona, \nUniversity of Colorado Boulder, University of Oxford, University of Portsmouth, \nUniversity of Utah, University of Virginia, University of Washington, University of Wisconsin, \nVanderbilt University, and Yale University.\n\n\\section*{Data availability} \nThe data used in this article is part of the 14th Data Release from SDSS-IV \/ APOGEE survey, and it is publicly available at \\url{https:\/\/www.sdss.org\/dr14\/data_access\/}. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nModern-day reasoning systems often have to react to real-time information about the real world\nprovided by e.g.~sensors.\nThis information is typically conceptualized as a data stream, which is accessed by the reasoning\nsystem.\nThe reasoning tasks associated to data streams -- usually called \\emph{continuous queries} -- are\nexpected to run continuously and produce results through another data stream in an online fashion,\nas new elements arrive.\n\nA data stream is a potentially unbounded sequence of data items generated by an active, uncontrolled\ndata source.\nElements arrive continuously at the system, potentially unordered, and at unpredictable rates.\nThus, reasoning over data streams requires dealing with incomplete or missing data, potentially\nstoring large amounts of data (in case it might be needed to answer future queries), and providing\nanswers in timely fashion -- among other problems, see\ne.g.~\\cite{Babcock2002,Stonebraker2005,DellAglio2017}.\n\nThe output stream is normally ordered by time, which implies that the system may have to\ndelay appending some answer because of uncertainty in possible answers relating to earlier\ntime points.\nThe length of this delay may be unpredictable (\\emph{unbound wait}) or infinite, for example if the\nquery uses operators that range over the whole input data stream (\\emph{blocking operations}).\nIn these cases, answers that have been computed may never be output.\nAn approach to avoid this problem is to restrict the language by forbidding blocking\noperations~\\cite{Zaniolo2012,Ronca2018}.\nAnother approach uses the concept of reasoning window~\\cite{Becketal2015,Ozcep2017}, which bounds\nthe size of the input that can be used for computing each output (either in time units or in number\nof events).\n\nIn several applications, it is useful to know that some answers are likely to be produced in the\nfuture, since there is already some information that might lead to their generation.\nThis is the case namely in prognosis systems (e.g., medical diagnosis, stock market prediction),\nwhere one can prepare for the possibility of something happening.\nTo this goal, we propose \\emph{hypothetical answers}: answers that are supported by information\nprovided by the input stream, but that still depend on other facts being true in the future.\nKnowledge about both the facts that support the answer and possible future facts that may make it\ntrue\ngives users the possibility to make timely, informed decisions\nin contexts where preemptive measures may have to be taken.\n\nMoreover, by giving such hypothetical answers to the user we cope with unbound wait in a\nconstructive way, since the system is no longer ``mute'' while waiting for an answer to become\ndefinitive.\n\nMany existing approaches to reasoning with data streams adapt and extend models, languages and\ntechniques used for querying databases and the semantic web~\\cite{Arasu2006,Barbieri2009}.\nWe develop our theory in line with the works\nof~\\cite{Zaniolo2012,Becketal2015,DaoTran2017,Ozcep2017,Motik}, where continuous queries are treated\nas rules of a logic program that reasons over facts arriving through a data stream.\n\n\\paragraph{Contribution.}\nWe present a declarative semantics for queries in Temporal Datalog, where we define the notions of\nhypothetical and supported answers.\nWe define an operational semantics based on SLD-resolution, and show that there is a natural\nconnection between the answers computed by this semantics and hypothetical and supported answers.\nBy refining SLD-resolution, we obtain an online algorithm for maintaining and updating the set\nof answers that are consistent with the currently available information.\nFinally, we show that our results extend to a language with negation.\n\n\\paragraph{Structure.}\nSection~\\ref{sec:backgr} revisits some fundamental background notions, namely the formalism\nfrom~\\cite{Motik}, which we extend in this paper, and introduces the running example that we use\nthroughout this article.\nSection~\\ref{sec:awp} introduces our declarative semantics for continuous queries, defining\nhypothetical and supported answers, and relates these concepts with the standard definitions of\nanswers.\nSection~\\ref{sec:sld} presents our operational semantics for continuous queries and relates it to\nthe declarative semantics. \nSection~\\ref{sec:algorithm} details our online algorithm to compute supported answers incrementally,\nas input facts arrive through the data stream, and proves it sound and complete.\nSection~\\ref{sec:neg} extends our framework to negation by failure.\nSection~\\ref{sec:rw} compares our proposal to similar ones in the literature, discussing its\nadvantages.\n\n\\section{Background}\n\\label{sec:backgr}\n\nIn this section, we review the most relevant concepts for our work.\n \n\\subsection{Continuous queries in Temporal Datalog}\n\\label{sec:motiks}\nWe use the framework from~\\cite{Motik} to write continuous queries over datastreams, slightly\nadapting some definitions.\nWe work in \\emph{Temporal Datalog}, the fragment of negation-free Datalog extended with the special\ntemporal sort from~\\cite{Chomicki1988}, which is isomorphic to the set of natural numbers equipped\nwith addition with arbitrary constants.\nIn Section~\\ref{sec:neg} we extend this language with negation.\n\n\\paragraph{Syntax of Temporal Datalog.}\nA vocabulary consists of constants (numbers or identifiers in lowercase), variables (single\nuppercase letters) and predicate symbols (identifiers beginning with an uppercase letter).\nAll these may be indexed if necessary; occurrences of predicates and variables are distinguished by\ncontext.\nIn examples, we use words in sans serif for concrete constants and predicates.\n\nConstants and variables have one of two sorts: \\emph{object} or \\emph{temporal}.\nAn \\emph{object term} is either an object (constant) or an object variable.\nA \\emph{time term} is either a natural number (called a \\emph{time point} or \\emph{temporal\n  constant}), a time variable, or an expression of the form $T+\\m{k}$ where $T$ is a time variable\nand $\\m{k}$ is an integer.\n\nPredicates can take at most\none temporal parameter, which we assume to be the last one (if present).\nA predicate with no temporal parameters is called \\emph{rigid}, otherwise it is called\n\\emph{temporal}.\nAn atom is an expression $P(t_1,\\ldots,t_n)$ where $P$ is a predicate and each $t_i$ is a term of\nthe expected sort.\n\nA rule has the form $\\wedge_i\\alpha_i\\rightarrow\\alpha$, where $\\alpha$ and each $\\alpha_i$ are\nrigid or temporal atoms.\nAtom $\\alpha$ is called the \\emph{head} of the rule, and $\\wedge_i\\alpha_i$ the \\emph{body}.\nRules are assumed to be \\emph{safe}: each variable in the head must occur in the body.\nA \\emph{program} is a set of rules.\n\nA predicate symbol that occurs in an atom in the head of a rule with non-empty body is called\n\\emph{intensional} (IDB predicate).\nPredicates that are defined only through rules with empty body are called extensional (EDB\npredicates).\nAn atom is extensional (EDB atom) or intensional (IDB atom) according to whether $P$ is extensional\nor intensional.\n\nA term, atom, rule, or program is \\emph{ground} if it contains no variables.\nWe write $\\var\\alpha$ for the set of variables occurring in an atom, and extend this function\nhomomorphically to rules and sets.\nA \\emph{fact} is a function-free ground atom; since Temporal Datalog does not allow function symbols\nexcept in temporal terms, every ground rigid atom is a fact.\n\nRules are instantiated by means of \\emph{substitutions}, which are functions mapping variables to\nterms of the expected sort.\nThe \\emph{support} of a substitution $\\theta$ is the set $\\supp\\theta=\\{X\\mid\\theta(X)\\neq X\\}$.\nWe consider only substitutions with finite support, and write\n$\\theta=\\subst{X_1:=t_1,\\ldots,X_n:=t_n}$ for the substition mapping each\nvariable $X_i$ to the term $t_i$, and leaving all remaining variables unchanged.\nA substitution is \\emph{ground} if every variable in its support is mapped to a constant.\nAn \\emph{instance} $r'=r\\theta$ of a rule $r$ is obtained by simultaneously replacing every variable\n$X$ in $r$ by $\\theta(X)$ and computing any additions of temporal constants.\n\nA \\emph{query} is a pair $Q=\\tuple{P,\\Pi}$ where $\\Pi$ is a program and $P$ is an IDB atom in\nthe language underlying $\\Pi$.\nQuery $Q$ is \\emph{temporal} (respectively, \\emph{rigid}) if the predicate in $P$ is a temporal\n(resp.\\ rigid) predicate.\n(Note that we do not require $P$ to be ground.)\n\nA \\emph{dataset} is a set of EDB facts (\\emph{input facts}), intuitively produced by a data stream.\nFor each dataset $D$ and time point $\\tau$, we consider $D$'s \\emph{$\\tau$-history}: the dataset\n$D_\\tau$ of the facts produced by $D$ whose temporal argument is at most $\\tau$.\nBy convention, $D_{-1}=\\emptyset$.\n\n\\paragraph{Semantics.}\nThe semantics of Temporal Datalog is a variant of the standard semantics based on Herbrand models.\nA Herbrand interpretation $I$ for Temporal Datalog is a set of facts.\nIf $\\alpha$ is an atom with no variables, then we define $\\bar\\alpha$ as the fact obtained from\n$\\alpha$ by evaluating each temporal term.\nIn particular, if $\\alpha$ is rigid, then $\\bar\\alpha=\\alpha$.\nWe say that $I$ satisfies $\\alpha$, $I\\models\\alpha$, if $\\bar\\alpha\\in I$.\nThe extension of the notion of satisfaction to the whole language follows the standard\nconstruction, and the definition of entailment is the standard one.\n\nAn \\emph{answer} to a query $Q=\\tuple{P,\\Pi}$ over a dataset $D$ is a ground substitution\n$\\theta$ whose domain is the set of variables in $P$, satisfying $\\Pi\\cup D\\models P\\theta$.\nIn the context of continuous query answering, we are interested in the case where $D$ is a\n$\\tau$-history of some data stream, which changes with time.\nWe denote the set of all answers to $Q$ over $D_\\tau$ as $\\ans QD\\tau$.\n\nWe illustrate the extension we propose with a Temporal Datalog program, which is a small variant of\nExample 1 in~\\cite{Motik}.\nThis will be the running example throughout our paper.\n\n\\begin{example}\n  \\label{ex:toy}\n  A set of wind turbines are scattered throughout the North Sea.\n  Each turbine has a sensor that sends temperature readings\n  $\\m{Temp}(\\mathit{Device},\\mathit{Level},\\mathit{Time})$ to a data centre.\n  The data centre tracks activation of cooling measures in each turbine, recording malfunctions and\n  shutdowns by means of the following program $\\ensuremath{\\Pi_E}$.\n  \\begin{align*}\n    \\m{Temp}(X,\\m{high},T) &\\rightarrow \\m{Flag}(X,T) \\\\\n    \\m{Flag}(X,T) \\land \\m{Flag}(X, T+1) &\\rightarrow \\m{Cool}(X,T+1) \\\\\n    \\m{Cool}(X,T) \\land \\m{Flag}(X, T+1) &\\rightarrow \\m{Shdn}(X,T+1) \\\\\n    \\m{Shdn}(X,T) &\\rightarrow \\m{Malf}(X,T-2)\n   \n   \n   \n  \\end{align*}\n\n  Consider the query $\\ensuremath{Q_E}=\\tuple{\\m{Malf}(X,T),\\ensuremath{\\Pi_E}}$.\n  If the history $D_0$ contains only the fact $\\m{Temp}(\\m{wt25},\\m{high},0)$, then at time\n  instant $0$ there is no output for $\\ensuremath{Q_E}$.\n  If $\\m{Temp}(\\m{wt25},\\m{high},1)$ arrives to $D$, then\n  $D_1=\\{\\m{Temp}(\\m{wt25},\\m{high},0),\\m{Temp}(\\m{wt25},\\m{high},1)\\}$,\n \n  and there still is no answer to $\\ensuremath{Q_E}$.\n  Finally, the arrival of $\\m{Temp}(\\m{wt25},\\m{high},2)$ to $D$ yields\n  $D_2=\\{\\m{Temp}(\\m{wt25},\\m{high},0),\\m{Temp}(\\m{wt25},\\m{high},1), \\m{Temp}(\\m{wt25},\\m{high},2)\\}$,\n \n  allowing us to infer $\\m{Malf}(\\m{wt25},0)$.\n  Then $\\{X:=\\m{wt25},T:=0\\}\\in\\ans\\ensuremath{Q_E} D2$.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nThroughout this work, we do not distinguish between the temporal argument in a fact (the timepoint\nwhere it is produced) and the instant when it arrives in $D$.\nIn other words, we assume that at each time point $\\tau$, the $\\tau$-history $D_\\tau$ contains all\nEDB facts about time instants $\\tau'<\\tau$.\n\n\\subsection{SLD-resolution}\n\\label{sec:SLD}\n\nWe now revisit some concepts from SLD-resolution.\n\nA \\emph{literal} is an atom or its negation.\nAtoms are also called \\emph{positive} literals, and a negated atom is a \\emph{negative} literal.\nA \\emph{definite clause} is a disjunction of literals containing at most one positive literal.\nIn the case where all literals are negative, the clause is a \\emph{goal}.\nWe use the standard rule notation for writing definite clauses.\n\n\\begin{definition}\n  Given two substitutions $\\theta$ and $\\sigma$, with\n  \\begin{align*}\n    \\theta&=\\subst{X_1:=t_1,\\ldots,X_m:=t_m}\\\\\n    \\mbox{and }\\sigma&=\\subst{Y_1:=u_1,\\ldots,Y_n:=u_n}\\,,\n  \\end{align*}\n  their \\emph{composition} $\\theta\\sigma$ is obtained from\n  $$\\subst{X_1:=t_1\\sigma,\\ldots,X_m:=t_m\\sigma,Y_1:=u_1,\\ldots,Y_n:=u_n}$$ by (i) deleting any\n  binding where $t_i\\sigma=X_i$ and (ii) deleting any binding $Y_j:=u_j$ where\n  $Y_j\\in\\{X_1,\\ldots,X_m\\}$.\n\\end{definition}\nFor every atom $\\alpha$, $\\alpha(\\theta\\sigma)=(\\alpha\\theta)\\sigma$.\n\n\\begin{definition}\n  Two atomic formulas $P(\\vec X)$ and $P(\\vec Y)$ are \\emph{unifiable} if there exists a\n  substitution $\\theta$ such that $P(\\vec x)\\theta=P(\\vec Y)\\theta$.\n\n  A unifier $\\theta$ of $P(\\vec X)$ and $P(\\vec Y)$ is called a \\emph{most general unifier (mgu)}\n  if for each unifier $\\sigma$ of $P(\\vec X)$ and $P(\\vec Y)$ there exists a substitution $\\gamma$\n  such that $\\sigma=\\theta\\gamma$.\n\\end{definition}\nIt is well known that there always exist several mgus of any two unifiable atoms, and that they are\nunique up to renaming of variables.\n\nRecall that a \\emph{goal} is a clause of the form $\\neg\\wedge_j\\beta_j$.\nIf $C$ is a rule $\\wedge_i\\alpha_i\\to\\alpha$, $G$ is a goal $\\neg\\wedge_j\\beta_j$\nwith $\\var G\\cap\\var C=\\emptyset$, and $\\theta$ is an mgu of $\\alpha$ and $\\beta_k$, then the\n\\emph{resolvent} of $G$ and $C$ is the goal\n$\\neg\\left(\\bigwedge_{j<k}\\beta_j\\wedge\\bigwedge_i\\alpha_i\\wedge\\bigwedge_{j>k}\\beta_j\\right)\\theta$.\n\nIf $P$ is a program and $G$ is a goal, an \\emph{SLD-derivation} of $P\\cup\\{G\\}$ is a (finite or\ninfinite) sequence $G_0,G_1,\\ldots$ of goals with $G=G_0$, a sequence $C_1,C_2,\\ldots$ of\n$\\alpha$-renamings of program clauses of $P$ and a sequence $\\theta_1,\\theta_2,\\ldots$ of\nsubstitutions such that $G_{i+1}$ is the resolvent of $G_i$ and $C_{i+1}$ using $\\theta_{i+1}$.\nA finite SLD-derivation of $P\\cup\\{G\\}$ where the last goal is a contradiction ($\\Box$) is called an\n\\emph{SLD-refutation} of $P\\cup\\{G\\}$ of length $n$, and the substitution obtained by restricting\nthe composition of $\\theta_1,\\ldots,\\theta_n$ to the variables occurring in $G$ is called a\n\\emph{computed answer} of $P\\cup\\{G\\}$.\n\n\\section{Hypothetical answers}\n\\label{sec:awp}\n\nIn our running example, $\\m{Temp(wt25,high,0)}$ being produced at time instant $0$ yields some\nevidence that $\\m{Malf(wt25,0)}$ may turn out to be true.\nAt time instant $1$, we may receive further evidence as in the example (the arrival of\n$\\m{Temp(wt25,high,1)}$), or we might find out that this fact will not be true (if\n$\\m{Temp(wt25,high,1)}$ does not arrive).\n\nWe propose a theory where such \\emph{hypothetical answers} to a continuous query are output: if some\nsubstitution can become an answer as long as some facts in the future are true, then we output this\ninformation.\nIn this way we can lessen the negative effects of unbound wait.\nHypothetical answers can also refer to future time points: in our example, \\subst{X:=\\m{wt25},T:=2}\nwould also be output at time point 0 as a substitution that may prove to be an answer to the query\n\\tuple{\\m{Shdn}(X,T),\\ensuremath{\\Pi_E}} when further information arrives.\n \nOur formalism uses ideas from multi-valued logic, where some substitutions correspond to answers\n(true), others are known not to be answers (false), and others are consistent with the available\ndata, but can not yet be shown to be true or false.\nIn our example, the fact $\\m{Malf(wt25,0)}$ is consistent with the data at time point $0$, and thus\n``possible''; it is also consistent with the data at time point $1$, and thus ``more possible''; and\nit finally becomes (known to be) true at time point 2.\n\nAs already motivated, we want answers to give us not only the substitutions that make the query goal\ntrue, but also ones that make the query goal possible in the following sense: they depend both on\npast and future facts, and the past facts are already known.\n\nFor the remainder of the article, we assume fixed a query $Q=\\tuple{P,\\Pi}$, a data stream $D$ and a\ntime instant $\\tau$.\n\\begin{definition}\n  \\label{defn:hypothetical}\n  A \\emph{hypothetical answer} to query $Q$ over $D_\\tau$ is a pair\n  $\\tuple{\\theta,H}$, where $\\theta$ is a substitution and $H$ is a finite set of ground EDB\n  temporal atoms (the hypotheses) such that:\n  \\begin{itemize}\n  \\item $\\supp\\theta=\\var P$;\n  \\item $H$ only contains atoms with time stamp $\\tau'>\\tau$;\n  \\item $\\Pi\\cup D_\\tau\\cup H\\models P\\theta$;\n  \\item $H$ is minimal with respect to set inclusion.\n  \\end{itemize}\n  $\\hans QD\\tau$ is the set of hypothetical answers to $Q$ over $D_\\tau$.\n\\end{definition}\n\nIntuitively, a hypothetical answer \\tuple{\\theta,H} states that $P\\theta$ holds if all facts\nin $H$ are ever produced by the data stream.\nThus, $P\\theta$ is currently backed up by the information available.\nIn particular, if $H=\\emptyset$ then $P\\theta$ is an answer in the standard sense (it is a known\nfact).\n\n\\begin{proposition}\n  \\label{prop:HA-answer}\n  Let $Q=\\tuple{P,\\Pi}$ be a query, $D$ be a data stream and $\\tau$ be a time instant.\n  If $\\tuple{\\theta,\\emptyset}\\in\\hans QD\\tau$, then $\\theta\\in\\ans QD\\tau$.\n\\end{proposition}\n\\begin{proof}\n  $\\tuple{\\theta,H}\\in\\hans QD\\tau$ if $\\Pi\\cup D_\\tau\\cup H\\models P\\theta$.\n  When $H=\\emptyset$, this reduces to $\\Pi\\cup D_\\tau\\models P\\theta$, which coincides with the\n  definition of answer.\n\\end{proof}\n\nWe can generalize this proposition, formalizing the intuition we gave for the definition of\nhypothetical answer.\n\n\\begin{proposition}\n  \\label{prop:HA-char}\n  Let $Q=\\tuple{P,\\Pi}$ be a query, $D$ be a data stream and $\\tau$ be a time instant.\n  If $\\tuple{\\theta,H}\\in\\hans QD\\tau$, then there exist a time point $\\tau'\\geq\\tau$ and\n  a data stream $D'$ such that $D_\\tau=D'_\\tau$ and $\\theta\\in\\ans Q{D'}{\\tau'}$.\n\\end{proposition}\n\\begin{proof}\n  Let $D'$ be the data stream $D\\cup H$ and $\\tau'$ be the highest timestamp occurring in $H$.\n  It is straightforward to verify that $D'$ satisfies the thesis.\n\\end{proof}\n\n\\begin{example}\n  \\label{ex:HA}\n  We illustrate these concepts in the context of Example~\\ref{ex:toy}.\n  Consider $\\theta=\\subst{X:=\\m{wt25},T:=0}$.\n  Then\n  $\\tuple{\\theta,\\{\\m{Temp}(\\m{wt25},\\m{high},1),\\m{Temp}(\\m{wt25},\\m{high},2)\\}}\\in\\hans\\ensuremath{Q_E} D0$.\n  Since $\\m{Temp}(\\m{wt25},\\m{high},1)\\in D_1$,\n  we get $\\tuple{\\theta,\\{\\m{Temp}(\\m{wt25},\\m{high},2)\\}}\\in\\hans\\ensuremath{Q_E} D1$.\n  Finally, $\\m{Temp}(\\m{wt25},\\m{high},2)\\in D_2$, so $\\tuple{\\theta,\\emptyset}\\in\\hans\\ensuremath{Q_E} D2$.\n  This answer has no hypotheses, and indeed $\\theta\\in\\ans\\ensuremath{Q_E} D2$.\n\n  Take $\\theta'=\\subst{X:=\\m{wt42},T:=1}$ for another constant $\\m{wt42}$.\n  Then also e.g. $\\tuple{\\theta',H'_0}\\in\\hans\\ensuremath{Q_E} D0$ with\n \n  $H'_0=\\{\\m{Temp}(\\m{wt42},\\m{high},1),\\m{Temp}(\\m{wt42},\\m{high},2),\\m{Temp}(\\m{wt42},\\m{high},3)\\}$,\n  but since $\\m{Temp}(\\m{wt42},\\m{high},1)\\notin D_1$ there is no element\n  $\\tuple{\\theta',H'}\\in\\hans\\ensuremath{Q_E} D\\tau$ for $\\tau\\geq1$.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nHypothetical answers $\\tuple{\\theta,H}\\in\\hans QD\\tau$ where $H\\neq\\emptyset$ can be further split into\ntwo kinds: those that are supported by some present or past true fact(s), and those for which there\nis no evidence whatsover -- they only depend on future, unknown facts.\nFor the former, $\\Pi\\cup H\\not\\models P\\theta$: they rely on some fact from $D_\\tau$.\nThis is the class of answers that interests us, as there is non-trivial information in\nsaying that they may become true.\n\n\\begin{definition}\n  \\label{def:supported}\n  Let $Q=\\tuple{P,\\Pi}$ be a query, $D$ be a data stream and $\\tau$ be a time instant.\n  A non-empty set of facts $E\\subseteq D_\\tau$ is \\emph{evidence} supporting\n  $\\tuple{\\theta,H}\\in\\hans QD\\tau$ if $E$ is a minimal set satisfying $\\Pi\\cup E\\cup H\\models P\\theta$.\n  A \\emph{supported answer} to $Q$ over $D_\\tau$ is a triple \\tuple{\\theta,H,E} where\n  $E$ is evidence supporting \\tuple{\\theta,H}.\n\n  $\\sans QD\\tau$ is the set of supported answers to $Q$ over $D_\\tau$.\n\\end{definition}\n\\longpaper{\\todo{Check that $E\\subseteq D_\\tau$ holds in all proofs.}}\n\nSince set inclusion is well-founded, if $\\tuple{\\theta,H}\\in\\hans QD\\tau$ and\n$\\Pi\\cup E\\cup H\\models P\\theta$, then there exists a set $E'$ such that \\tuple{\\theta,H,E'} is a\nsupported answer to $Q$ over $D_\\tau$.\nHowever, in general, several such sets $E'$ may exist.\nAs a consequence, Propositions~\\ref{prop:HA-answer} and~\\ref{prop:HA-char} generalize to supported\nanswers in the obvious way.\n\n\\begin{example}\n  \\label{ex:SA}\n  Consider the hypothetical answers from Example~\\ref{ex:HA}.\n \n  The hypothetical answer \\tuple{\\theta,H_0} is supported by the evidence\n  $$E_0=\\{\\m{Temp}(\\m{wt25},\\m{high},0)\\}\\,,$$\n \n  while \\tuple{\\theta,H_1} is supported by\n  $$E_1=\\{\\m{Temp}(\\m{wt25},\\m{high},0),\\m{Temp}(\\m{wt25},\\m{high},1)\\}\\,.$$\n \n  Since there is no evidence for \\tuple{\\theta',H'_0}, this answer is not supported.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nThis example illustrates that unsupported hypothetical answers are not very informative: it is the\nexistence of supporting evidence that distinguishes interesting hypothetical answers from any\narbitrary future fact.\n\nHowever, it is useful to consider even unsupported hypothetical answers in order to develop\nincremental algorithms to compute supported answers: the sequence of sets\n$\\Theta^E_\\tau=\\{\\theta\\mid\\tuple{\\theta,H,E}\\in\\sans QD\\tau\\mbox{ for some $H,E$}\\}$\nis non-monotonic, as at every time point new unsupported hypothetical answers may get evidence and\nsupported hypothetical answers may get rejected.\nThe sequence\n$\\Theta^H_\\tau=\\{\\theta\\mid\\tuple{\\theta,H}\\in\\hans QD\\tau\\mbox{ for some $H$}\\}$, on\nthe other hand, is anti-monotonic, as the following results state.\n\n\\begin{proposition}\n  \\label{prop:HA-split}\n  If $\\tuple{\\theta,H}\\in\\hans QD\\tau$, then there exists $H^0$ such that\n  $\\tuple{\\theta,H^0}\\in\\hans QD{-1}$ and $H=H^0\\setminus D_\\tau$.\n  Furthermore, if $H\\neq H^0$, then $\\tuple{\\theta,H,H^0\\setminus H}\\in\\sans QD\\tau$.\n\\end{proposition}\n\\begin{proof}\n  Recall that $D_{-1}=\\emptyset$ by convention.\n  If $\\tuple{\\theta,H}\\in\\hans QD\\tau$, then $\\Pi\\cup D_\\tau\\cup H\\models P\\theta$.\n  Since $D_\\tau$ is finite and set inclusion is well-founded, there is a minimal subset $H^0$ of\n  $D_\\tau\\cup H$ with the property that $H\\subseteq H^0$ and $\\Pi\\cup H^0\\models P\\theta$.\n  Clearly $H=H^0\\setminus D_\\tau$.\n\n  Assume that $H^-\\subseteq H^0$ is also such that $\\Pi\\cup H^-\\models P\\theta$.\n  Then $\\Pi\\cup D_\\tau\\cup(H^-\\setminus D_\\tau)\\models P\\theta$; but\n  $\\tuple{\\theta,H}\\in\\hans QD\\tau$, so $H\\subseteq H^-\\setminus D_\\tau$ and therefore also\n  $H\\subseteq H^-$.\n  By definition of $H^0$, this implies that $H^0\\subseteq H^-$, hence\n  $\\tuple{\\theta,H^0}\\in\\hans QD{-1}$.\n\n  Finally, if $E\\subseteq H^0\\setminus H$ is evidence supporting \\tuple{\\theta,H}, then\n  $\\Pi\\cup E\\cup H\\models P\\theta$, hence $H^0\\subseteq E\\cup H$, since\n  $\\tuple{\\theta,E\\cup H}\\in\\hans QD{-1}$.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{proposition}\n  \\label{prop:HA-split-general}\n  If $\\tuple{\\theta,H}\\in\\hans QD\\tau$ and $\\tau'<\\tau$, then there exists\n  $\\tuple{\\theta,H'}\\in\\hans QD{\\tau'}$ such that \n  $H=H'\\setminus\\left(D_\\tau\\setminus D_{\\tau'}\\right)$.\n\\end{proposition}\n\\begin{proof}\n  Just as the proof of the previous proposition, but dividing $\\Pi\\cup D_\\tau$ into\n  $\\Pi\\cup D_{\\tau'}$ and $D_\\tau\\setminus D_{\\tau'}$ instead of into $\\Pi$ and $D_\\tau$.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nExamples~\\ref{ex:HA} and~\\ref{ex:SA} also illustrate this property, with hypotheses turning into\nevidence as time progresses.\nSince $D_{-1}=\\emptyset$, Proposition~\\ref{prop:HA-split} is a particular case of\nProposition~\\ref{prop:HA-split-general}.\n\nIn the next sections we show how to compute hypothetical answers and the corresponding sets of\nevidence for a given continuous query.\n\n\\section{Operational semantics via SLD-resolution}\n\\label{sec:sld}\n\nThe definitions of hypothetical and supported answers are declarative.\nWe now show how SLD-resolution can be adapted to algorithms that compute these answers.\nWe use standard results about SLD-resolution, see for example~\\cite{Lloyd1984}.\n\nWe begin with a simple observation: since the only function symbol in our language is addition of\ntemporal parameters (which is invertible), we can always choose mgus that do not replace variables\nin the goal with new ones.\n\n\\begin{lemma}\n  \\label{lem:mgu-vars}\n  Let $\\neg\\wedge_i\\alpha_i$ be a goal and $\\wedge_j\\beta_j\\to\\beta$ be a rule such that\n  $\\beta$ is unifiable with $\\alpha_k$ for some $k$.\n  Then there is an mgu $\\theta=\\subst{X_1:=t_1,\\ldots,X_n:=t_n}$ of $\\alpha_k$ and $\\beta$ such that\n  all variables occurring in $t_1,\\ldots,t_n$ also occur in $\\alpha_k$.\n\\end{lemma}\n\\begin{proof}\n  Let $\\rho=\\subst{X_1:=t_1,\\ldots,X_n:=t_n}$ be an mgu of $\\alpha_k$ and $\\beta$.\n  For each $i$, $t_i$ can either be a variable $Y_i$ or a time expression $T_i+k_i$.\n  First, iteratively build a substitution $\\sigma$ as follows: for $i\\in[1,\\ldots,n]$, if $X_i$\n  occurs in $\\alpha_k$ but $t_i$ does not and $\\sigma$ does not yet include a replacement for the\n  variable in $t_i$, extend $\\sigma$ with $Y_i:=X_i$, if $t_i$ is $Y_i$, or $T_i:=X_i-k_i$, if $t_i$\n  is $T_i+k_i$.\n\n  We now show that $\\theta=\\rho\\sigma$ is an mgu of $\\alpha_k$ and $\\beta$ with the desired\n  property.\n  If $X:=t\\in\\theta$, then either (i)~$X$ is $X_i$ and $t$ is $t_i\\sigma$ for some $i$ or\n  (ii)~$X:=t\\in\\sigma$.\n  In case~(i), by construction of $\\sigma$ if $X_i$ occurs in $\\alpha_k$ but $t_i$ includes a\n  variable not in $\\alpha_k$, then $\\sigma$ replaces that variable with a term using only variables\n  in $\\alpha_k$. In case~(ii), by construction $X$ does not occur in $\\alpha_k$.\n\n  To show that $\\theta$ is an mgu of $\\alpha_k$ and $\\beta$ it suffices to observe that $\\sigma$ is\n  invertible, with\n  $$\\sigma^{-1}=\\subst{X:=Y \\mid Y:=X\\in\\sigma}\\subst{X:=T-k\\mid T:=X+k\\in\\sigma}\\,.$$\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nWithout loss of generality, we assume all mgus in SLD-derivations to have the property in\nLemma~\\ref{lem:mgu-vars}.\n\nOur intuition is as follows: classical SLD-resolution computes substitutions that make a conjunction\nof atoms a logical consequence of a program by constructing SLD-derivations that end in the empty\nclause.\nWe relax this by allowing derivations to end with a goal if: this goal only refers to EDB predicates\nand all the temporal terms in it refer to future instants (possibly after further instantiation).\nThis makes the notion of derivation also dependent on a time parameter.\n\n\\begin{definition}\n  An atom $P(t_1,\\ldots,t_n)$ is a \\emph{future atom wrt $\\tau$} if $P$ is a temporal predicate and\n  the time term $t_n$ either contains a temporal variable or is a time instant $t_n>\\tau$.\n\\end{definition}\n\n\\begin{definition}\n  An \\emph{SLD-refutation with future premises} of $Q$ over $D_\\tau$ is a finite SLD-derivation of\n  $\\Pi\\cup D_\\tau\\cup\\{\\neg P\\}$ whose last goal only contains future EDB atoms wrt $\\tau$.\n\n  If $\\mathcal D$ is an SLD-refutation with future premises of $Q$ over $D_\\tau$ with last\n  goal $G=\\neg\\wedge_i\\alpha_i$ and $\\theta$ is the substitution obtained by restricting the\n  composition of the mgus in $\\mathcal D$ to $\\var P$, then \\tuple{\\theta,\\wedge_i\\alpha_i}\n  is a \\emph{computed answer with premises} to $Q$ over $D_\\tau$, denoted \\cans QD\\tau\\theta{\\wedge_i\\alpha_i}.\n\\end{definition}\n\n\\begin{example}\n  Consider the query $\\ensuremath{Q_E}$ from Example~\\ref{ex:toy} and let $\\tau=1$.\n  There is an SLD-derivation of $\\Pi\\cup D_1\\cup\\{\\neg\\m{Malf}(X,T)\\}$ ending with the goal\n  $\\m{Temp}(\\m{wt25},\\m{high},2)$, which is a future EDB atom with respect to $1$.\n  Thus, $\\cans\\ensuremath{Q_E} D1\\theta{\\m{Temp}(\\m{wt25},\\m{high},2)}$ with\n  $\\theta={\\subst{X:=\\m{wt25},T:=0}}$.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nAs this example illustrates,\ncomputed answers with premises are the operational counterpart to hypothetical answers, with two\ncaveats.\nFirst, a computed answer with premises need not be ground: there may be\nsome universally quantified variables in the last goal.\nSecond, $\\wedge_i\\alpha_i$ may contain redundant conjuncts, in the sense\nthat they might not be needed to establish the goal.\nWe briefly illustrate these two features.\n\n\\begin{example}\n  \\label{ex:SLD-problems}\n  In our running example, there is also an SLD-derivation of $\\Pi\\cup D_1\\cup\\{\\neg\\m{Malf}(X,T)\\}$\n  ending with the goal $\\neg\\bigwedge_{i=0}^2\\m{Temp}(X,\\m{high},T+i)$, which only contains future\n  EDB atoms wrt $1$.\n  Thus also\n  $\\cans{Q_E}D1\\emptyset{\\bigwedge_{i=0}^2\\m{Temp}(X,\\m{high},T+i)}$.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\n\\begin{example}\n  Consider the following program $\\Pi'$:\n \n \n  \\begin{align*}\n    \\m{P}(\\m{a},T) &\\rightarrow \\m{R}(\\m{a},T) \\\\\n    \\m{P}(\\m{a},T)\\wedge\\m{Q}(\\m{a},T) &\\rightarrow \\m{R}(\\m{a},T)\n  \\end{align*}\n  and the query $Q'=\\tuple{\\m{R}(X,T),\\Pi'}$.\n  \n  Let $D'_0=\\emptyset$.\n  There is an SLD-derivation of $\\Pi'\\cup D'_0\\cup\\{\\neg\\m{R}(X,T)\\}$ ending with goal\n  $\\neg\\left(\\m{P}(\\m{a},T)\\wedge\\m{Q}(\\m{a},T)\\right)$, which only contains future EDB atoms wrt\n  $\\tau$.\n  Thus $\\cans{Q'}{D'}0{\\subst{X:=\\m{a}}}{\\m{P}(\\m{a},T)\\wedge\\m{Q}(\\m{a},T)}$.\n  However, atom $\\m{Q}(\\m{a},T)$ is redundant, since $\\m{P}(\\m{a},T)$ alone suffices to make\n  $\\subst{X:=\\m{a}}$ an answer to $Q$ for any $T$.\n  \n  (Observe that also $\\cans{Q'}{D'}0{\\subst{X:=\\m{a}}}{\\m{P}(\\m{a},T)}$, but produced by a\n  different SLD-derivation.)\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nWe now look at the relationship between the operational definition of computed answer with premises\nand the notion of hypothetical answer.\nThe examples above show that these notions do not precisely correspond.\nHowever, we can show that computed answers with premises approximate hypothetical answers and,\nconversely, every hypothetical answer is a grounded instance of a computed answer with\npremises.\n\n\\begin{proposition}[Soundness]\n  \\label{prop:SLD-sound}\n  Let $Q=\\tuple{P,\\Pi}$ be a query, $D$ be a data stream and $\\tau$ be a time instant.\n  If $\\cans QD\\tau\\theta{\\wedge_i\\alpha_i}$ and $\\sigma$ is a ground substitution such that\n  $\\supp\\sigma=\\var{\\wedge_i\\alpha_i}\\cup(\\var P\\setminus\\supp\\theta)$ and\n  $t\\sigma>\\tau$ for every temporal term $t$ occurring in $\\wedge_i\\alpha_i$, then there is a set\n  $H\\subseteq\\{\\alpha_i\\sigma\\}_i$ such that $\\tuple{(\\theta\\sigma)|_{\\var P},H}\\in\\hans QD\\tau$.\n\\end{proposition}\n\\begin{proof}\n  Assume that there is some SLD-refutation with future premises of $Q$ over $D_\\tau$.\n  Then this is an SLD-derivation whose last goal $G=\\vee_i\\neg\\alpha_i$ only contains future EDB\n  atoms with respect to $\\tau$.\n  Let $\\sigma$ be any substitution in the conditions of the hypothesis.\n  Taking $H'=\\{\\alpha_i\\sigma\\}_i$, we can extend this SLD-derivation to a (standard) SLD-refutation\n  for $\\Pi\\cup D_\\tau\\cup H'\\cup\\{\\neg P\\}$, by resolving $G$ with each of the $\\alpha_i$ in\n  turn.\n  The computed answer is then the restriction of $\\theta\\sigma$ to $\\var P$.\n  By soundness of SLD-resolution, $\\Pi\\cup D_\\tau\\cup H'\\models P(\\theta\\sigma)|_{\\var P}$.\n  Since set inclusion is well-founded, we can find a minimal set $H\\subseteq H'$ with the latter\n  property.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{proposition}[Completeness]\n  \\label{prop:SLD-compl}\n  Let $Q=\\tuple{P,\\Pi}$ be a query, $D$ be a data stream and $\\tau$ be a time instant.\n  If $\\tuple{\\theta,H}\\in\\hans QD\\tau$, then there exist substitutions $\\rho$ and $\\sigma$ and a finite\n  set of atoms $\\{\\alpha_i\\}_i$ such that $\\theta=\\rho\\sigma$, $H=\\{\\alpha_i\\sigma\\}_i$ and\n  $\\cans QD\\tau\\rho{\\wedge_i\\alpha_i}$.\n\\end{proposition}\n\\begin{proof}\n  Suppose $\\tuple{\\theta,H}\\in\\hans QD\\tau$.\n  Then $\\Pi\\cup D_\\tau\\cup H\\models P\\theta$.\n  By completeness of SLD-resolution, there exist substitutions $\\gamma$ and $\\delta$ and an\n  SLD-derivation for $\\Pi\\cup D_\\tau\\cup H\\cup\\{\\neg P\\}$ with computed answer $\\gamma$ such that\n  $\\theta=\\gamma\\delta$.\n\n  By minimality of $H$, for each $\\alpha\\in H$ there must exist a step in this SLD-derivation where\n  the current goal is resolved with $\\alpha$.\n  Without loss of generality, we can assume that these are the last steps in the derivation (by\n  independence of the computation rule).\n  Let $\\mathcal D'$ be the derivation consisting only of these steps, and $\\mathcal D$ be the\n  original derivation without $\\mathcal D'$.\n  Let $\\rho$ be the answer computed by $\\mathcal D$ and $G=\\vee_i\\neg\\alpha_i$ be its last goal,\n  let $\\rho'$ be the answer computed by $\\mathcal D'$, and define $\\sigma=\\rho'\\delta$.\n  Then:\n  \\begin{itemize}\n  \\item Let $X\\in\\supp\\theta$; then $X$ occurs in $P$.\n    If $\\rho(X)$ occurs in $G$, then by construction $\\gamma(X)=(\\rho\\rho')(X)$.\n    If $\\rho(X)$ is a ground term or $\\rho(X)$ does not occur in $G$, then trivially\n    $\\gamma(X)=\\rho(X)=(\\rho\\rho')(X)$ since $\\rho'$ does not change $\\rho(X)$.\n    In either case, $\\theta=\\gamma\\delta=\\rho\\rho'\\delta=\\rho\\sigma$.\n  \\item $H=\\{\\alpha_i\\sigma\\}_i$: by construction of $\\mathcal D'$, $H=\\{\\alpha_i\\rho'\\}_i$, and\n    since $\\alpha_i\\rho'$ is ground for each $i$, it is also equal to\n    $\\alpha_i\\rho'\\delta=\\alpha_i\\sigma$.\n  \\end{itemize}\n  The derivation $\\mathcal D$ shows that $\\cans QD\\tau\\rho{\\wedge_i\\alpha_i}$.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nAll notions introduced in this section depend on the time parameter $\\tau$, and in particular on the\nhistory dataset $D_\\tau$.\nIn the next section, we explore the idea of ``organizing'' the SLD-derivation adequately to\npre-process $\\Pi$ independently of $D_\\tau$, so that the computation of (hypothetical) answers can\nbe split into an offline part and a less expensive online part.\n\n\\section{Incremental computation of hypothetical answers}\n\\label{sec:algorithm}\n\nProposition~\\ref{prop:HA-split-general} states that the set of hypothetical answers evolves as time\npasses, with hypothetical answers either gaining evidence and becoming query answers or being put\naside due to their dependence on facts that turn out not to be true.\n\nIn this section, we show how we can use this temporal evolution to compute supported answers\nincrementally.\nWe start by revisiting SLD-derivations and showing how they can reflect this temporal structure.\n\n\\begin{proposition}\n  \\label{prop:SLD-strat}\n  Let $Q=\\tuple{P,\\Pi}$ be a query and $D$ be a data stream.\n  For any time constant $\\tau$, if $\\cans QD\\tau\\theta{\\wedge_i\\alpha_i}$, then there exist an SLD-refutation with future premises\n  of $Q$ over $D_\\tau$ computing \\tuple{\\theta,\\wedge_i\\alpha_i} and a sequence\n  $k_{-1}\\leq k_0\\leq\\ldots\\leq k_\\tau$ such that:\n  \\begin{itemize}\n  \\item goals $G_1,\\ldots,G_{k_{-1}}$ are obtained by resolving with clauses from $\\Pi$;\n  \\item for $0\\leq i\\leq\\tau$, goals $G_{k_{i-1}+1},\\ldots,G_{k_i}$ are obtained by resolving with\n    clauses from $D_i\\setminus D_{i-1}$.\n  \\end{itemize}\n\\end{proposition}\n\\begin{proof}\n  Straightforward corollary of the independence of the computation rule.\n\\end{proof}\n\nAn SLD-refutation with future premises with the property guaranteed by\nProposition~\\ref{prop:SLD-strat} is called a \\emph{stratified} SLD-refutation with future\npremises.\nSince data stream $D$ only contains EDB atoms, it also follows that in a stratified SLD-refutation\nall goals after $G_{k_{-1}}$ are always resolved with EDB atoms.\nFurthermore, each \\longpaper{goal }$G_{k_i}$ contains only future EDB atoms with respect to $i$.\nLet $\\theta_i$ be the restriction of the composition of all substitutions in the SLD-derivation up\nto step $k_i$ to $\\var P$.\nThen $G_{k_i}=\\neg\\wedge_j\\alpha_j$ represents all hypothetical answers to $Q$ over $D_i$ of the\nform \\tuple{(\\theta_i\\sigma)|_{\\var P},\\wedge_j\\alpha_j} for some ground substitution $\\sigma$\n(cf.~Proposition~\\ref{prop:SLD-sound}).\n\nThis yields an online procedure to compute supported answers to continuous queries over data\nstreams.\nIn a pre-processing step, we calculate all computed answers with premises to $Q$ over $D_{-1}$, and\nkeep the ones with minimal set of formulas.\n(Note that Proposition~\\ref{prop:SLD-compl} guarantees that all minimal sets are generated by\nthis procedure, although some non-minimal sets may also appear as in Example~\\ref{ex:SLD-problems}.)\nThe online part of the procedure then performs SLD-resolution between each of these sets and the\nfacts produced by the data stream, adding the resulting resolvents to a set of\nschemata of supported answers (i.e.~where variables may still occur).\nBy Proposition~\\ref{prop:SLD-strat}, if there is at least one resolution step at this stage, then\nthe hypothetical answers represented by these schemata all have evidence, so they are indeed\nsupported.\n\nIn general, the pre-processing step of this procedure may not terminate, as the following example\nillustrates.\n\n\\begin{example}\n  \\label{ex:infinite}\n  Consider the following program $\\Pi''$, where $R$ is an extensional predicate and $S$ is an\n  intensional predicate.\n  \\begin{align*}\n    S(X,T) &\\rightarrow S(X,T+1) &\n    R(X,T) &\\rightarrow S(X,T)\n  \\end{align*}\n  If $R(a,t_0)$ is produced by the datastream, then $S(a,t)$ is true for every $t\\geq t_0$.\n  \n  Thus, $\\tuple{[X:=a],\\{R(a,T-k)\\}}\\in\\hans{\\tuple{S(X,T),\\Pi''}}D0$ for all $k$.\n  The pre-processing step needs to output this infinite set, so it cannot terminate.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\nWe establish termination of the pre-processing step for two different classes of queries.\nA query $Q=\\tuple{P,\\Pi}$ is \\emph{connected} if each rule in $P$ contains at most one temporal\nvariable, which occurs in the head whenever it occurs in the body; and it is \\emph{nonrecursive} if\nthe directed graph induced by its dependencies is acyclic, cf.~\\cite{Motik}.\n\\begin{proposition}\n  \\label{prop:termination}\n  Let $Q=\\tuple{P,\\Pi}$ be a nonrecursive and connected query.\n  Then the set of all computed answers with premises to $Q$ over $D_{-1}$ can be computed in finite\n  time.\n\\end{proposition}\n\\begin{proof}\n  Let $T$ be the (only) temporal variable in $P$.\n  Then all SLD-derivations for $\\Pi\\cup\\neg{P}$ have a maximum depth: if we associate to each\n  predicate the length of the maximum path in the dependency graph for $\\Pi$ starting from it and to\n  each goal the sorted sequence of such values for each of its atoms, then each resolution step\n  decreases this sequence with respect to the lexicographic ordering.\n  Since this ordering is well-founded, the SLD-derivation must terminate.\n\n  Furthermore, since $\\Pi$ is finite, there is a finite number of possible descendants for each\n  node.\n  Therefore, the tree containing all possible SLD-derivations for $\\Pi\\cup\\neg P$ is a finite\n  branching tree with finite height, and by K\u00f6nig's Lemma it is finite.\n\n  Since each resolution step terminates (possibly with failure) in finite time, this tree can be\n  built in finite time.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nThe algorithm implicit in the proof of Proposition~\\ref{prop:termination} can be improved by\nstandard techniques (e.g.~by keeping track of generated nodes to avoid duplicates).\nHowever, since it is a pre-processing step that is done offline and only once, we do not discuss\nsuch optimizations.\n\n\\cite{Motik} also formally define delay and window size as follows.\n$Q$ is a query with temporal variable $T$.\n\\begin{definition}\n  \\label{defn:delay}\n  A \\emph{delay} for $Q$ is a natural number $d$ such that: for every substitution $\\theta$ and\n  every $\\tau\\geq T\\theta+d$, $\\theta\\in\\ans QD\\tau$ iff $\\theta\\in\\ans QD{T\\theta+d}$.\n\n  \\label{defn:window}\n  A natural number $w$ is a \\emph{window size} for $Q$ if: each $\\theta$ is an answer to $Q$ over\n  $D_\\tau$ iff $\\theta$ is an answer to $Q$ over $D_\\tau\\setminus D_{\\tau-w}$.\n\\end{definition}\n\nTo show termination of the pre-processing step assuming existence of a delay and window size, we\napply a slightly modified version of SLD-resolution: we do not allow fresh non-temporal variables to\nbe added.\nSo if a rule includes new variables in its body, we generate a node for each of its possible\ninstances.\nWe start by proving an auxiliary lemma.\n\n\\begin{lemma}\n  \\label{lem:termination}\n  If $Q=\\tuple{P,\\Pi}$ has delay $d$ and window size $w$, then there exist two natural numbers $d'$\n  and $w'$ such that: if an atom in a goal in an SLD-derivation for $\\Pi\\cup\\{\\neg P\\}$ contains a\n  temporal argument $T+k$ with $k>d'$ or $k<{-w'}$, then the subtree with that goal as root does not\n  contain a leaf including an extensional atom with a temporal parameter dependent on $T$.\n\\end{lemma}\n\\begin{proof}\n  Assume that the thesis does not hold, i.e.~for any $d'$ and $w'$ there exists an atom in an\n  SLD-derivation for $\\Pi\\cup\\{\\neg P\\}$ whose subtree contains a leaf including an extensional atom\n  with temporal parameter dependent on $T$.\n  By the independence of the computation rule, this property must hold for any SLD-derivation for\n  $\\Pi\\cup\\{\\neg P\\}$.\n\n  Then there is a predicate symbol that occurs infinitely many times in the tree for $\\Pi\\cup\\{\\neg\n  P\\}$ with distinct instantiations of the temporal argument.\n  Since the number of distinct instantiations for the non-temporal arguments is finite (up to\n  $\\alpha$-equivalence), there must be at least one instantiation that occurs infinitely often.\n  If applying SLD-resolution to it generates an extensional literal whose temporal argument depends\n  on $T$, then the tree would have infinitely many leaves containing distinct occurrences of its\n  underlying predicate symbol.\n\n  For any $t$, it is then straightforward to construct an instance of the data stream and a\n  substitution $\\theta$ such that $\\theta\\not\\in\\ans QD{T\\theta}$ and $\\theta\\in\\ans QD{T\\theta+t}$,\n  so no number smaller than $t$ can be a delay for $Q$.\n  Similarly, we can construct an instance of the data stream and a substitution such that $\\theta$\n  is an answer to $Q$ over $D_{\\theta+t}$ but not over $D_{\\theta+t}\\setminus D_\\theta$, so $t$\n  cannot be a window size for $Q$.\n  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{proposition}\n  \\label{thm:termination-char}\n  If $Q$ has a delay $d$ and a window $w$, then the set of all computed answers with premises to $Q$\n  over $D_{-1}$ can be computed in finite time.\n\\end{proposition}\n\\begin{proof}\n  Let $w'$ and $d'$ be the two natural numbers guaranteed to exist by the previous lemma.\n  Then, if a goal contains a positive literal with temporal parameter $T+k$ with $k>d'$ or\n  $k<{-w'}$, we know that the subtree generated from that literal cannot yield leaves with temporal\n  parameter depending on $T$.\n  Hence, either this tree contains only failed branches, or it contains leaves that do not depend on\n  $T$, or it is infinite.\n\n  Although we do not know the value of $d'$ and $w'$, we can still use this knowledge by performing\n  the construction of the tree in a depth-first fashion.\n  Furthermore, we choose a computation rule that delays atoms with temporal variable other than $T$.\n  For each node, before expanding it we first check whether the atom we are processing has already\n  appeared before (modulo temporal variables), and if so we skip the construction of the\n  corresponding subtree by directly inserting the corresponding leaves.\n\n  If an atom contains a temporal variable other than $T$, all leaves in its subtree contain no\n  extensional predicates with non-constant temporal argument (otherwise there would not be a delay\n  for $T$).\n  Therefore these sub-goals can also be treated uniformly.\n\n  Finally, suppose that a branch is infinite.\n  Again using a finiteness argument, any of its branches must necessarily at some point include\n  repeated literals.\n  In this case we can immediately mark it as failed.\n  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nThe (offline) pre-processing step consisting of running the algorithm implicit in the proof of this\nproposition, allows us to compute a finite set $\\mathcal P_Q$ of \\emph{preconditions} for $Q$ that\nrepresents $\\hans QD{-1}$: for each computed answer \\tuple{\\theta,\\wedge_i\\alpha_i} with premises to\n$Q$ over $D_{-1}$ where $\\{\\alpha_i\\}_i$ is minimal, $\\mathcal P_Q$ contains an entry\n\\tuple{\\theta,M,\\{\\alpha_i\\}_i\\setminus M} where $M$ is the subset of the $\\alpha_i$ with minimal\ntimestamp\n(i.e.~those elements of $\\alpha_i$ whose temporal variable is $T+k$ with minimal $k$).\n\nEach tuple $\\tuple{\\theta,M,F}\\in\\mathcal P_Q$ represents the set of all hypothetical answers\n\\tuple{\\theta\\sigma,(M\\cup F)\\sigma} as in Proposition~\\ref{prop:SLD-sound}.\n\nWe now show that computing and updating the set $\\sans QD\\tau$ can be done efficiently.\nThis set is maintained again as a set $\\mathcal S_\\tau$ of schematic supported answers.\nWe assume that $Q$ is a connected query; we discuss how to remove this restriction later.\n\n\\begin{proposition}\n  \\label{prop:algorithm}\n  The following algorithm computes $\\mathcal S_{\\tau+1}$ from $\\mathcal P_Q$ and $\\mathcal S_\\tau$\n  in time polynomial in the size of $\\mathcal P_Q$, $\\mathcal S_\\tau$ and $D_{\\tau+1}\\setminus D_\\tau$.\n  \\begin{enumerate}\n  \\item For each $\\tuple{\\theta,M,F}\\in\\mathcal P_Q$ and each computed answer $\\sigma$ to\n    $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M\\}$, add\n    \\tuple{\\theta\\sigma,M\\sigma,F\\sigma} to $\\mathcal S_{\\tau+1}$.\n\n  \\item For each $\\tuple{\\theta,E,H}\\in\\mathcal S_\\tau$, compute the set $M\\subseteq H$ of\n    atoms with timestamp $\\tau+1$.\n    For each computed answer $\\sigma$ to\n    $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M\\}$, add\n    \\tuple{\\theta\\sigma,(E\\cup M)\\sigma,(H\\setminus M)\\sigma} to $\\mathcal S_{\\tau+1}$.\n  \\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n  By connectedness, all time variables in $M\\sigma\\cup F\\sigma$ are instantiated in $\\theta\\sigma$.\n\n  To show that this algorithm runs in polynomial time in the size of $\\mathcal P_Q$,\n  $\\mathcal S_\\tau$ and $D_{\\tau+1}\\setminus D_\\tau$, note that the size of every SLD-derivation\n  that needs to be constructed is bound by the number of atoms in the initial goal, since\n  $D_{\\tau+1}\\setminus D_\\tau$ only contains facts.\n  Furthermore, all unifiers can be constructed in time linear in the size of the formulas involved,\n  since the only function symbol available is addition of temporal terms.\n  Finally, the total number of SLD-derivations that needs to be considered is bound by the number of\n  elements of $\\mathcal P_Q\\times\\left(D_{\\tau+1}\\setminus D_\\tau\\right)$.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{example}\n  We illustrate this mechanism with our running example.\n  The set $\\mathcal P_Q$ contains\n  $$\\tuple{\\emptyset,\\{\\underbrace{\\m{Temp}(X,\\m{high},T)}_M\\},\n    \\{\\underbrace{\\m{Temp}(X,\\m{high},T+i)\\mid i=1,2}_F\\}}\\,.$$\n\n  From $\\m{Temp}(\\m{wt25},\\m{high},0)\\in D_0$, we obtain the substitution\n  $\\theta_0=\\subst{X:=\\m{wt25},T:=0}$ from SLD-resolution between $M$ and $D_0$ (step 1).\n  Therefore, $\\mathcal S_0$ contains\n  $$\\tuple{\\theta_0,\\{\\underbrace{\\m{Temp}(\\m{wt25},\\m{high},0)}_{E_0}\\},\n    \\{\\underbrace{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=1,2}_{H_0}\\}}\\,.$$\n\n  Next, $D_1\\setminus D_0=\\{\\m{Temp}(\\m{wt25},\\m{high},1)\\}$.\n  This is the only element of $H_0$ with timestamp $1$.\n  By step~2, $\\mathcal S_1$ contains\n  $$\\tuple{\\theta_0,\\{\\underbrace{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=0,1}_{E_1}\\},\n    \\{\\underbrace{\\m{Temp}(\\m{wt25},\\m{high},2)}_{H_1}\\}}\\,.$$\n  Furthermore, from $\\mathcal P_Q$ we also add (step 1)\n  $$\\tuple{\\theta_1,\\{\\m{Temp}(\\m{wt25},\\m{high},1)\\},\n    \\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=2,3\\}}$$\n  to $\\mathcal S_1$, with $\\theta_1=\\subst{X:=\\m{wt25},T:=1}$.\n\n  Next, $D_2\\setminus D_1=\\{\\m{Temp}(\\m{wt25},\\m{high},2)\\}$.\n  This is the only atom with timestamp $2$ in the premises of both elements of $\\mathcal S_1$,\n  so $\\mathcal S_2$ contains (step 2)\n  $$\\tuple{\\theta_0,\\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=0,1,2\\},\\emptyset}$$\n  and\n  $$\\tuple{\\theta_1,\\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=1,2\\},\\{\\m{Temp}(\\m{wt25},\\m{high},3)\\}}\\,.$$\n  From $\\mathcal P_Q$ we also get (step 1)\n  $$\\tuple{\\theta_2,\\{\\m{Temp}(\\m{wt25},\\m{high},2)\\},\\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=3,4\\}}$$\n  with $\\theta_2=\\subst{X:=\\m{wt25},T:=2}$.\n\n  If $D_3\\setminus D_2=\\emptyset$, then the premises for $\\theta_1$ and $\\theta_2$ become\n  unsatisfied, and no new supported answers are generated from $\\mathcal P_Q$.\n  Thus\n  $$\\mathcal S_3=\\{\\tuple{\\theta_0,\\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=0,1,2\\},\\emptyset}\\,.\\qquad\\hfill\\ensuremath{\\triangleleft}$$\n\\end{example}\n\nThe following example also illustrates that, by outputting hypothetical answers, we can answer\nqueries earlier than in other formalisms.\n\n\n\\begin{example}\n  \\label{ex:good}\n  Suppose that we extend the program $\\ensuremath{\\Pi_E}$ in our running example with the following rule (as in\n  Example~2 from~\\cite{Motik}).\n  \\[\n  \\m{Temp}(X,\\m{n\/a},T) \\to \\m{Malf}(X,T)\n  \\]\n  If\n  $D_1=\\{\\m{Temp}(\\m{wt25},\\m{high},0),\\m{Temp}(\\m{wt25},\\m{high},1),\\m{Temp}(\\m{wt42},\\m{n\/a},1)\\}$,\n  \\begin{align*}\n    \\mbox{then }\\mathcal S_1\n    &=\\{\\langle\\subst{T:=0,X:=\\m{wt25}},\\\\\n      &\\qquad\\{\\m{Temp}(\\m{wt25},\\m{high},i)\\mid i=0,1\\},\\\\\n      &\\qquad\\{\\m{Temp}(\\m{wt25},\\m{high},2)\\}\\rangle,\\\\\n    &\\quad\\langle\\subst{T:=1,X:=\\m{wt42}},\\{\\m{Temp}(\\m{wt42},\\m{n\/a},1)\\},\\emptyset\\rangle\\}\\,.\n  \\end{align*}\n  Thus, the answer \\subst{T:=1,X:=\\m{wt42}} is produced at timepoint 1, rather than being delayed\n  until it is known whether \\subst{T:=0,X:=\\m{wt25}} is an answer.\\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\n\\begin{proposition}[Soundness]\n  If $\\tuple{\\theta,E,H}\\in\\mathcal S_\\tau$ and $\\sigma$ instantiates all free variables in\n  $E\\cup H$, then $\\tuple{\\theta\\sigma,H\\sigma,E\\sigma}\\in\\sans QD\\tau$.\n\\end{proposition}\n\\begin{proof}\n  By induction on $\\tau$, we show that \\cans QD\\tau\\theta{\\wedge_i\\alpha_i}\\ with $H=\\{\\alpha_i\\}_i$.\n  If \\tuple{\\theta,E,H} is obtained from an element in $P_Q$ and $D_{\\tau+1}\\setminus D_\\tau$,\n  then this derivation is obtained by composing the derivation for generating the relevant element\n  of $P_Q$ with the one for \\tuple{\\theta,E,H}.\n  If \\tuple{\\theta,E,H} is obtained from an element of $\\mathcal S_\\tau$ and\n  $D_{\\tau+1}\\setminus D_\\tau$, then this derivation is obtained by composing the derivation\n  obtained by induction hypothesis to the one used for deriving \\tuple{\\theta,E,H}.\n\n  By applying Proposition~\\ref{prop:SLD-sound} to this SLD-derivation, we conclude that\n  $\\tuple{\\theta,H}\\in\\hans QD\\tau$.\n  Furthermore, $E\\neq\\emptyset$ and $E$ is evidence for this answer by construction.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\begin{proposition}[Completeness]\n  If $\\tuple{\\sigma,H,E}\\in\\sans QD\\tau$, then there exist a substitution $\\rho$ and a triple\n  $\\tuple{\\theta,E',H'}\\in\\mathcal S_\\tau$ such that $\\sigma=\\theta\\rho$, $H=H'\\rho$ and\n  $E=E'\\rho$.\n\\end{proposition}\n\\begin{proof}\n  By Proposition~\\ref{prop:SLD-compl}, \\cans QD\\tau\\theta{\\wedge_i\\alpha_i}\\ for some substitution $\\rho$ and set of atoms\n  $H'=\\{\\alpha_i\\}_i$ with $H=\\{\\alpha_i\\rho\\}_i$ and $\\sigma=\\theta\\rho$ for some $\\theta$.\n  By Proposition~\\ref{prop:SLD-strat}, there is a stratified SLD-derivation computing this answer.\n  The strata of this derivation correspond to an incremental proof that\n  $\\tuple{\\theta,E',H'}\\in\\mathcal S_\\tau$, where $E'$ is the set of facts from $D_\\tau$ that were\n  used in resolution steps in the derivation.\\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nIt also follows from our construction that, if $S_\\tau$ contains a triple \\tuple{\\theta,E,H} with\n$\\theta(T)\\leq\\tau$ and $H\\neq\\emptyset$ and $d$ is a delay for $Q$, then the time stamp of each\nelement of $H$ is at most $\\tau+d$.\nLikewise, if $w$ is a window size for $Q$, then all elements in $E$ must have time stamp at least\n$\\tau-w$.\n\n\\paragraph{Generalization.}\nThe hypotheses in Propositions~\\ref{prop:termination} and~\\ref{thm:termination-char} are not\nnecessary to guarantee termination of the algorithm presented for the pre-processing step.\nIndeed, consider the following example.\n\n\\begin{example}\n  \\label{ex:unbound}\n  In the context of our running example, we say that a turbine has a manufacturing defect if it\n  exhibits two specific failures during its lifetime: at some time it overheats, and at some\n  (different) time it does not send a temperature reading.\n  Since this is a manufacturing defect, it holds at timepoint $0$, regardless of when the failures\n  actually occur.\n  We can model this property by the rule\n  \\[\\m{Temp}(X,\\m{high},T_1),\\m{Temp}(X,\\m{n\/a},T_2) \\to \\m{Defective}(X,0)\\,.\\]\n  Let $\\Pi'_E$ be the program obtained from $\\Pi_E$ by adding this rule, and consider now the query\n  $Q'=\\tuple{\\m{Defective}(X,T),\\Pi'_E}$.\n\n  Performing SLD-resolution with $\\Pi'_E$ and $\\m{Defective}(X,0)$ yields (in one step) the goal\n  $$\\neg\\left(\\m{Temp}(X,\\m{high},T_1)\\wedge\\m{Temp}(X,\\m{n\/a},T_2)\\right)\\,,$$\n  which only contains future atoms with respect to $-1$.\n  The set of computed answers with premises to $Q'$ over $D_{-1}$ is indeed\n  $$\\{\\tuple{\\subst{T:=0},\\m{Temp}(X,\\m{high},T_1)\\wedge\\m{Temp}(X,\\m{n\/a},T_2)}\\}\\,.\\quad\\hfill\\ensuremath{\\triangleleft}$$\n\\end{example}\nAs this example shows, if a rule in the program includes different time variables, the query cannot\nhave a delay or window size (since no predicate can use both $T_1$ and $T_2$).\n\nWe can also adapt our algorithm to work in this situation, removing the hypothesis of connectedness\nin Proposition~\\ref{prop:algorithm} but sacrificing polynomial complexity.\nThe changes are the following.\n\\begin{itemize}\n\\item In the pre-processing step, set $\\mathcal P_Q$ now stores triples of the form\n  \\tuple{\\theta,\\{M_T\\}_T,\\{\\alpha_i\\}_i\\setminus M}, where $M$ is computed as before and $M_T$ is\n  the set of elements of $M$ whose temporal variable is $T$.\n  Recall that each predicate can only contain one temporal variable.\n\\item Step~1 of the algorithm in Proposition~\\ref{prop:algorithm} now reads:\n  for each $\\tuple{\\theta,\\{M_T\\}_T,F}\\in\\mathcal P_Q$ and each computed answer $\\sigma$ to\n  $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M_T\\}$, add\n  \\tuple{\\theta\\sigma,M_T\\sigma,(F\\cup(M\\setminus M_T))\\sigma} to $\\mathcal S_{\\tau+1}$.\n  Note that this step is now performed as many times as there are sets $M_T$, but its running time\n  is still polynomial on the size of $\\mathcal P_Q$.\n\\item After Step~2 of the algorithm in Proposition~\\ref{prop:algorithm}, we need to apply a fixpoint\n  construction to $\\mathcal S_{\\tau+1}$: for each temporal variable $T$ occurring in\n  $\\tuple{\\theta,E,H}\\in\\mathcal S_{\\tau+1}$, consider the set $M_T$ of the atoms in $H$ with time\n  variable $T$ and minimal timestamp.\n  For each computed answer $\\sigma$ to\n  $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M_T\\}$, add\n  \\tuple{\\theta\\sigma,(E\\cup M_T)\\sigma,(H\\setminus M_T)\\sigma} to $\\mathcal S_{\\tau+1}$.\n\\end{itemize}\nThe last step may add new elements to $\\mathcal S_{\\tau+1}$, but they will have fewer temporal\nvariables, so it always terminates.\nHowever, it may not run in polynomial time: each element $\\tuple{\\theta,E,H}\\in\\mathcal S_{\\tau+1}$\nbefore this construction can give rise to $2^v$ elements in the final $\\mathcal S_{\\tau+1}$, where\n$v$ is the number of temporal variables in $E$.\n\nThis modified algorithm allows us to deal with some situations of unbound wait, as we illustrate in\nthe following example.\n\\begin{example}\n  \\label{ex:unbound-cont}\n  Continuing with Example~\\ref{ex:unbound}, since $D_0$ contains $\\m{Temp}(\\m{wt25},\\m{high},0)$,\n  the set $\\mathcal S_0$ includes\n  \\[\\langle\\theta'=\\subst{X:=\\m{wt25},T:=0},\\{\\m{Temp}(\\m{wt25},\\m{high},0)\\},\\{\\m{Temp}(\\m{wt25},\\m{n\/a},T_2)\\}\\rangle\\,.\\]\n  Note that we do not know when (if ever) $\\theta'$ will become an answer to the original query, but\n  there is relevant information output to the user.\n  \\hfill\\ensuremath{\\triangleleft}\n\\end{example}\n\n\\section{Adding negation}\n\\label{sec:neg}\n\nWe now show how our framework extends naturally to programs including negated atoms in bodies of\nrules.\nWe make the usual assumptions that negation is safe (each variable in a negated atom in the body of\na rule occurs non-negated elsewhere in the rule).\n\nIn the pre-processing step, we compute $\\mathcal P_Q$ as before.\nHowever, the leaves in the SLD-derivations constructed may now contain negated (intensional) atoms\nas well as extensional atoms.\nFor each such negated atom we generate a fresh query $Q'$, replacing the time parameter with a\nvariable, and repeat the pre-processing step to compute $\\mathcal P_{Q'}$.\nWe iterate this construction until no fresh queries are generated.\nSince the number of queries that can be generated is finite, Propositions~\\ref{prop:termination} and\n~\\ref{thm:termination-char} still hold.\n\nThe online step of the algorithm is now more complicated, as it needs to keep track of all answers\nto the auxiliary queries generated by negated atoms.\nThe following algorithm computes $\\mathcal S_{\\tau+1}$ from $\\mathcal P$ and $\\mathcal S_\\tau$.\n\\begin{enumerate}\n\\item For each $Q$ and each $\\tuple{\\theta,M,F}\\in\\mathcal P_Q$: if there is a negated atom in $M$\n  with time parameter $t$, let $\\sigma$ be the substitution such that $t\\sigma=\\tau+1$; otherwise\n  let $\\sigma=\\emptyset$.\n  Let $M'$ be the set of positive literals in $M$.\n\n  For each computed answer $\\sigma'$ to\n  $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M'\\sigma\\}$\n  add \\tuple{\\theta\\sigma\\sigma',M'\\sigma\\sigma',((M\\setminus M')\\cup F)\\sigma\\sigma'} to\n  $\\mathcal S_{\\tau+1}(Q)$.\n\n  (Observe that all time variables in $M\\sigma\\sigma'\\cup F\\sigma\\sigma'$ are instantiated in\n  $\\theta\\sigma\\sigma'$.)\n\n\\item For each query $Q$ and each $\\tuple{\\theta,E,H}\\in\\mathcal S_\\tau(Q)$, compute the set\n  $M\\subseteq H$ of positive literals that have timestamp $\\tau+1$.\n  For each computed answer $\\sigma$ to\n  $\\left(D_{\\tau+1}\\setminus D_\\tau\\right)\\cup\\{\\neg\\bigwedge M\\}$, add the tuple\n  \\tuple{\\theta\\sigma,(E\\cup M)\\sigma,(H\\setminus M)\\sigma} to $\\mathcal S_{\\tau+1}(Q)$.\n\n\\item For each query $Q$ and each $\\tuple{\\theta,E,H}\\in\\mathcal S_{\\tau+1}(Q)$, compute the set\n  $M\\subseteq H$ of negative literals.\n  For each literal $\\m{not}\\ \\ell\\in M$ with timestamp $t\\leq\\tau+1$, let $\\ell'$ be the query on the\n  same predicate symbol as $\\ell$.\n    \n  If there is no tuple $\\tuple{\\theta',E',H'}\\in\\mathcal S_{\\tau+1}(\\ell')$ where $\\ell$\n  and $\\ell'\\theta'$ are unifiable, remove $\\m{not}\\ \\ell$ from $H$ and add it to $E$.\n\n  If there is a tuple $\\tuple{\\theta',E',\\emptyset}\\in\\mathcal S_{\\tau+1}(\\ell')$ such that\n  $\\ell$ and $\\ell'\\theta'$ are unifiable, then: (i)~remove \\tuple{\\theta,E,H} from $S_{\\tau+1}(Q)$,\n  (ii)~for each substitution $\\sigma$ ranging over the free variables in $\\ell$, if $\\ell\\sigma$\n  does not unify with $\\ell'\\theta'$, then add \\tuple{\\theta\\sigma,E\\sigma,H\\sigma} to\n  $S_{\\tau+1}(Q)$.\n\\end{enumerate}\n\nStep~3 makes the running time of this algorithm exponential, since it requires iterating over a set\nof substitutions.\nHowever, if negation is $T$-stratified (which we define below), we can still establish a polynomial\nbound.\n\n\\begin{definition}\n  Let $\\Pi$ be a program.\n  The \\emph{temporal closure} of a program $\\Pi$ is the program $\\ensuremath{\\Pi^\\downarrow}$ defined as follows.\n  For each $(n+1)$-ary predicate symbol $p$ with a temporal argument in the signature underlying\n  $\\Pi$, the signature for $\\Pi^\\downarrow$ contains a family of $n$-ary predicate symbols\n  $\\{p_t\\}_{t\\in\\mathbb N}$.\n  For each rule in $\\Pi$, $\\ensuremath{\\Pi^\\downarrow}$ contains all rules obtained by instantiating its temporal\n  parameter in all possible ways and replacing $p(x_1,\\ldots,x_n,t)$ by $p_t(x_1,\\ldots,x_n)$.\n  A program $\\Pi$ is $T$-stratified if $\\ensuremath{\\Pi^\\downarrow}$ is stratified in the usual sense.\n\\end{definition}\n\nSince the number of predicate symbols in $\\ensuremath{\\Pi^\\downarrow}$ is infinite, the usual procedure for deciding\nwhether a program is $T$-stratified does not necessarily terminate.\nHowever, this procedure can be adapted to our framework.\n\\begin{definition}\n  Let $\\Pi$ be a program.\n  The \\emph{temporal dependency graph} of $\\Pi$ is constructed as follows: for each predicate symbol\n  $p$ with a temporal argument, we create a node with label $p(T)$ (where $T$ is a formal temporal\n  variable); if $p$ does not have a temporal argument, we create a node with label $p$.\n\n  For each node with label $p(T)$ and each rule $r$ in the definition of $p$, we first compute the\n  substitution $\\theta$ that makes the temporal variable in the head of $r$ equal to $T$; for each\n  positive literal $q(x_1,\\ldots,x_n,T+k)$ in $\\mathsf{body}(r\\theta)$, we create an edge between $p(T)$ and\n  $q(T+k)$ (adding the node $q(T+k)$ if necessary) and label the edge with $+$.\n  We proceed likewise for negative literals, but label the edge with $-$.\n\n  We iterate this construction for any newly created nodes whose temporal variable is $T+k$ with\n  $k\\geq 0$.\n\\end{definition}\n\n\\begin{proposition}\n  \\label{prop:stratification-dec}\n  There is an algorithm that decides whether a program $\\Pi$ is $T$-stratified, and in the\n  affirmative case returns a finite representation of the strata.\n\\end{proposition}\n\\begin{proof}\n  We show that: if the temporal dependency graph of $\\Pi$ does not contain a path that passes\n  infinitely many times through (not necessarily distinct) edges labeled with $-$, then $\\Pi$ is\n  $T$-stratified.\n\n  Let $\\gr\\ensuremath{\\Pi^\\downarrow}$ be the graph of dependencies for $\\ensuremath{\\Pi^\\downarrow}$ and $\\gr\\Pi$ be the temporal dependency\n  graph for $\\Pi$.\n  For each predicate symbol $p_t$ in $\\ensuremath{\\Pi^\\downarrow}$, the graph $\\gr\\ensuremath{\\Pi^\\downarrow}$ contains a subgraph isomorphic to\n  a subgraph\\footnote{It may be a proper subgraph in case $t$ is low enough that some temporal\n    arguments would become negative.} of the connected component of $\\gr\\Pi$ containing $p(T)$.\n  In particular, $\\gr\\ensuremath{\\Pi^\\downarrow}$ contains infinitely many copies of $\\gr\\Pi$.\n\n  Suppose that $\\gr\\ensuremath{\\Pi^\\downarrow}$ contains a path that passes infinitely many times through edges labeled\n  with $-$, and let $p_t$ be a node in it with minimum value of $t$ (if $t$ occurs multiple times in\n  the path, choose any one of its occurrences).\n  Consider the isomorphic copy of $\\gr\\Pi$ embedded in $\\gr\\ensuremath{\\Pi^\\downarrow}$ where $p(T)$ is mapped to $p_t$.\n  The whole path starting from $p_t$ must be contained in this copy, since the timestamps of all\n  corresponding nodes in $\\gr\\Pi$ are greater or equal to $T$.\n\n  Using these two properties, we can construct a stratification of $\\ensuremath{\\Pi^\\downarrow}$ in the usual inductive\n  way.\n  \\begin{itemize}\n  \\item $\\mathcal G_0=\\gr\\ensuremath{\\Pi^\\downarrow}$\n  \\item For each $i\\geq 0$, $\\Pi_i$ contains all predicates labeling nodes $n$ in $\\mathcal G_i$\n    such that: all paths from $n$ contain only edges labeled with $+$.\n  \\item For each $i\\geq 0$, $\\mathcal G_{i+1}$ is obtained from $\\mathcal G_i$ by removing all nodes\n    with labels in $\\Pi_i$ and all edges to those nodes.\n  \\end{itemize}\n\n  The only non-trivial part of showing that $\\Pi_0,\\ldots,\\Pi_n,\\ldots$ is a stratification of\n  $\\ensuremath{\\Pi^\\downarrow}$ is guaranteeing that every predicate symbol $p_t$ is contained in $\\Pi_k$ for some $k$.\n  This is established by induction: if any path from $p_t$ contains at most $n$ edges labeled with\n  $-$, then $p_t\\in\\Pi_n$.\n  We now show that, for any $p_t$, there is such a bound on the number of edges labeled with $-$.\n\n  First observe that, in $\\gr\\Pi$, there is a bound $B$ on the number of edges labeled with $-$ in\n  any path starting from $p(T)$: (i)~if there is a path from $q(T+k_1)$ to $q(T+k_2)$ for some $q$\n  and $k_1\\leq k_2$ containing an edge labeled with $-$, then there would be a path with an infinite\n  number of such edges, and (ii)~since there is a finite number of predicate symbols in $\\Pi$ we\n  cannot build paths with an arbitrarily long number of edges labeled with $-$.\n\n  Now consider a path starting at $p_t$.\n  If this path has length greater than $B$, then it cannot be contained completely in the subgraph\n  of $\\gr\\ensuremath{\\Pi^\\downarrow}$ isomorphic to $\\gr\\Pi$ containing $p(T)$.\n  Therefore, it must at some point reach a node $q_{t'}$ with $t'<t$.\n  Likewise, the path starting at $q_{t'}$ corresponds to a path in a (different) subgraph of\n  $\\gr\\ensuremath{\\Pi^\\downarrow}$ isomorphic to $\\gr\\Pi$, and as such cannot contain more than $B$ edges labeled with $-$.\n  Since we can only decrease $t$ a finite number of times, the path cannot contain arbitrarily many\n  edges labeled with $-$.\n  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\nFurthermore, for $T$-stratified programs we can also give a complexity bound for the online step.\n\\begin{proposition}\n  \\label{prop:algorithm-negated}\n  Let $k$ be the highest arity of any predicate that occurs negated in $\\Pi$.\n  If $\\Pi$ is $T$-stratified, then the online algorithm runs in time polynomial in $k$, the size of\n  $\\mathcal P_Q$, $\\mathcal S_\\tau$, $D_{\\tau+1}\\setminus D_\\tau$ and the total number of queries.\n\\end{proposition}\n\\begin{proof}\n  Steps~1 and~2 are as in Proposition~\\ref{prop:algorithm}, but they are now iterating over all sets\n  of queries.\n  Therefore the previous time bounds apply, multiplied by the total number of queries.\n\n  Each item in Step~3 requires computing the set of negative literals in a hypothetical answer\n  (which can be done in time linear on the size of the answer) and checking whether a corresponding\n  tuple exists in $\\mathcal S_{\\tau+1}$.\n  This can be done in time linear in the size of $S_{\\tau+1}$, which is polynomial in the size of\n  $S_\\tau$ (since it was computed by a polynomial algorithm).\n  The number of substitutions $\\sigma$ that need to be considered is bound by the number of\n  constants to the power $k$, which is constant; thus any remaining answers can be added in\n  worst-case polynomial time.\n\n  Finally, if negation is stratified, then Step~3 needs to be done only once for each query, as long\n  as the order in which queries are processed respects the stratification.\n  \\hfill\\ensuremath{\\Box}\n\\end{proof}\n\n\\section{Related work and discussion}\n\\label{sec:rw}\n\n\\paragraph{Incremental evaluation.}\nComputing answers to a query over a data source that is continuously producing information, be it at\nslow or very fast rates, asks for techniques that allow for some kind of\n\\emph{incremental evaluation}, in order to avoid reevaluating the query from scratch each time a new\ntuple of information arrives.\nSeveral efforts have been made in that direction, capitalising on incremental algorithms based on\nseminaive evaluation~\\cite{Gupta1993,Abiteboul1995,Barbieri2010,Motik2015,Hu2018}, on truth\nmaintenance systems~\\cite{Beck2015}, window oriented~\\cite{Ghanem2007} among others.\nOur algorithm fits naturally in the first class, as it is an incremental variant of SLD-resolution.\n\n\\paragraph{Unbound wait and blocking queries.}\nThe problems of unbound wait and blocking queries have deserved much attention in the area of query\nanswering over data streams.\nThere have been efforts to identify the problematic issues~\\cite{Law2011} and varied proposals to\ncope with their negative effects, as in~\\cite{Zaniolo2012,Ozcep2017,Becketal2015,Ronca2018} among\nothers.\nOur framework deals with unbound wait by outputting at each time point all supported answers\n(including some that later may prove to be false), as illustrated in Examples~\\ref{ex:good}\nand~\\ref{ex:unbound-cont}.\nFurthermore, if we receive a supported answer whose time parameters are all instantiated, then we\nimmediately have a bound on how long we have to wait until the answer is definite (or rejected).\n\nBlocking queries may still be a problem in our framework, though: as seen in\nExample~\\ref{ex:infinite}, blocking operations (in the form of infinitely recursive predicates) may\nlead to infinite SLD-derivations, which cause the pre-processing step of our algorithm to diverge.\nWe showed that syntactic restrictions of the kind already considered by~\\cite{Motik} are guaranteed\nto avoid blocking queries.\n\nOur algorithm also implicitly embodies a forgetting algorithm, as the only elements of $D_\\tau$ kept\nin the $E$ sets are those that are still relevant to compute future answers to $Q$.\n\n\\cite{Ronca2018} proposes the \\emph{language of forward-propagating queries}, a\nvariant of Temporal Datalog that allows queries to be answered in polynomial time in the size of the\ninput data.\nThis is achieved at the cost of disallowing propagation of derived facts towards past time points --\nprecluding, e.g., rules like $\\m{Shdn}(X,T) \\rightarrow \\m{Malf}(X,T-2)$ in Example~\\ref{ex:toy}.\nThe authors present a generic algorithm to compute answers to a query, given a window size, and\ninvestigate methods for calculating a minimal window size.\n\n\\cite{Zaniolo2012}, working in another variant of Datalog called Streamlog, characterise\n\\emph{sequential rules and programs} with the purpose of avoiding blocking behaviour.\nAgain, rule $\\m{Shdn}(X,T) \\rightarrow \\m{Malf}(X,T-2)$ from Example~\\ref{ex:toy} is disallowed\nin this framework, since the timestamp in the head of a rule may never be smaller than the\ntimestamps of the atoms in the body.\n\n\\paragraph{Incomplete information.}\nWhen reasoning over sources with incomplete information, the concepts of certain and possible\nanswers, and granularities thereof, inevitably arise~\\cite{Grayetal2007,Lang2014,Razniewski2015} as\na way to assign confidence levels to the information output to the user.\nThese approaches, like ours, also compute answers with incomplete information.\n\nHowever, our proposal is substantially different from those works, since they focus on answers over\npast incomplete information.\nFirst, as mentioned in Section~\\ref{sec:backgr}, we assume that our time stream is complete, in the\nsense that whenever it produces a fact about a time instant $\\tau$, all EDB facts about time\ninstants $\\tau'<\\tau$ are already there (in line with the progressive closing world assumption of\n\\cite{Zaniolo2012}).\nSecondly, our hypothetical and supported answers are built over present facts and future, still\nunknown, hypotheses, and eventually either become effective answers or are discarded; in the\nscenario of past incomplete information, the confidence level of answers may never change.\n\n\\paragraph{Negation.}\nOther authors have considered languages using negation.\nOur definition of stratification is similar to the concept of temporal stratification\nfrom~\\cite{Zaniolo2015}.\nHowever, temporal stratification requires the strata to be also ordered according to time (i.e., if\nthe definition of $p_t$ is in $\\Pi_i$ and the definition of $p_{t+1}$ is in $\\Pi_j$, then $j\\geq i$).\nWe make no such assumption in this work.\n\n\\cite{Beck2015} also consider a notion of temporal stratification for stream reasoning.\nHowever, their framework also includes explicit temporal operators, making the whole\nformalization more complex.\n\n\\section{Conclusions and future work}\n\\label{sec:concl}\n\nWe introduced a formalism for dealing with answers to continuous queries that are consistent with\nthe information that is available, but may still need further confirmation from future data.\nWe defined the notions of hypothetical and supported answers declaratively, and showed how they can\nbe computed by means of a variant of SLD-resolution.\nBy refining this idea, we designed a two-phase algorithm whose online component maintains and\nupdates the set of supported answers.\n\nOur methodology avoids some of the typical problems with delay in continuous query answering.\nIn particular, hypothetical answers allow us to detect that we are not in a situation of unbound\nwait: if we receive a supported answer whose time parameters are all instantiated, then we\nimmediately have a bound on how long we have to wait until the answer is definite (or rejected).\nThe usefulness of this information is of course application-specific.\nIn our running example, we could for example take extra preventive measures to ensure that\n$\\m{Temp(wt25,high,2)}$ does not become true.\n\nThe offline step of our two-phase algorithm may diverge due to issues related to blocking queries.\nWe showed that adequate syntactic restrictions prevent this situation from arising, but that they\nare not necessary.\n\nFor nonrecursive and connected queries, the online step of our algorithm runs in polynomial time.\nHowever, it may still be computationally heavy if the sets of hypothetical answers that it needs to\ncompute are large.\nWe minimize the impact of this by representing hypothetical answers schematically (since we keep\nvariables uninstantiated), thus limiting the space and time requirements for this algorithm.\nThis is an empirical observation though, which we plan to measure more precisely in a future\npractical evaluation.\n\nThe sets of evidence for hypothetical answers can be used to define a partial order of relative\n``confidence'' of each hypothetical answer.\nIt would be interesting to explore the connection between such notions of confidence and other\nframeworks of reasoning with multiple truth values.\n\nWe also plan to implement our algorithm and evaluate its performance in practice.\n\n\\paragraph{Acknowledgements.}\nThis work was partially supported by the Independent Research Fund Denmark, grant DFF-7014-00041,\nand by UID\/MULTI\/04046\/2019 Research Unit grant from FCT, Portugal (to BioISI).\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:introduction}\n\nIn this paper we investigate graph-theoretic conditions on the links of vertices in certain simply-connected polygonal complexes, such that by recent work of Lazarovich \\cite{L}, local data specify these complexes uniquely.\n\nA $2$-dimensional CW-complex $X$ is called a \\emph{polygonal complex} if both of the following hold:\n\\begin{enumerate}\n\\item the attaching maps of $X$ are homeomorphisms; and\n\\item the intersection of any two closed cells of $X$ is either empty or exactly one closed cell.\n\\end{enumerate}\nThese conditions imply that the $1$-skeleton of $X$, that is, the graph given by the vertices and edges of $X$, is simple, that is, has no loops or multiple edges.  More details on these and subsequent topological terms will be provided in Section \\ref{s:algtop}. We will refer to the closed $1$-cells of $X$ as \\emph{edges} and the closed $2$-cells as \\emph{faces}.  The boundary of any face $P$ of $X$ is a cycle of $k$ edges, for some integer $k \\geq 3$, and so we may also refer to $P$ as a \\emph{$k$-gon}.  For each  vertex $x$ of $X$, the \\emph{link} of $x$ is the simple graph with vertex set the edges of $X$ containing $x$, edge set the faces of $X$ containing $x$, and two vertices of the link adjacent if and only if the corresponding edges in $X$ are contained in a common face.  We do not usually think of polygonal complexes as being embedded in any space.   Polygonal complexes are, however, often metrised so that each face is a regular Euclidean polygon of side length one.  In some cases the faces may be metrised as hyperbolic polygons instead.  \n\nPolygonal complexes play an important role in combinatorial and geometric group theory.  The Cayley $2$-complex of a group presentation is a polygonal complex, and many groups are investigated by considering their action on an associated polygonal complex of nonpositive or negative curvature (see Section \\ref{s:algtop} for the definitions of these curvature conditions).  From a slightly different point of view, if $X$ is a simply-connected, locally finite polygonal complex, then the automorphism group of $X$ is naturally a totally disconnected locally compact group.  As explained in the survey \\cite{FHT}, we may hope to extend to this setting results from the theory of Lie groups and their lattices.  \n\nGiven an integer $k \\geq 3$ and a simple graph $L$, a \\emph{$(k,L)$-complex} is a polygonal complex $X$ such that each face is a $k$-gon and the link of each vertex is $L$.    For example, if $C_n$ denotes the cycle graph on $n$ vertices, then the boundary of a tetrahedron is a $(3,C_3)$-complex, and the regular tiling of the Euclidean plane with squares is a $(4,C_4)$-complex.   The (cartesian) product of two trees of bivalency $\\{m,n\\}$ is a $(4,K_{m,n})$-complex, where $K_{m,n}$ is the complete bipartite graph on $m + n$ vertices.  If $L$ is the Heawood graph, that is, the point-line incidence graph of the projective plane $\\mathrm{PG}(2,2)$, then the buildings for both $\\ensuremath{\\operatorname{SL}}_3(\\mathbb{Q}_2)$ and $\\ensuremath{\\operatorname{SL}}_3(\\mathbb{F}_2((t)))$ are $(3,L)$-complexes, which are not isomorphic.  There is a close relationship between $(k,L)$-complexes and rank $3$ incidence geometries, which we discuss  in Section \\ref{s:algtop}.\n\nWe say that the pair $(k,L)$ satisfies the \\emph{Gromov Link Condition} if $k \\geq m$ and $\\operatorname{girth}(L) \\geq n$, where $(m,n) \\in \\{ (3,6), (4,4), (6,3) \\}$.  If this condition holds, then as we recall in Section \\ref{s:algtop}, a simply-connected $(k,L)$-complex $X$ is either nonpositively or negatively curved, as its faces are metrised as Euclidean or hyperbolic polygons respectively.  We assume throughout this paper that the graph $L$ is finite and connected, equivalently $X$ is locally finite and has no local cut-points.\n\nDenote by $\\mathcal{X}(k,L)$ the collection of all simply-connected $(k,L)$-complexes (up to isomorphism).  As described in \\cite{FHT}, simply-connected $(k,L)$-complexes are often constructed as universal covers of finite $(k,L)$-complexes.  Also, discrete groups which act upon simply-connected $(k,L)$-complexes are often constructed either as fundamental groups of finite $(k,L)$-complexes,  or as fundamental groups of complexes of finite groups over polygonal complexes which have all faces $k$-gons and the link of the local development at each vertex being the graph $L$.  In each of these cases, in order to identify the universal cover, which will be \\emph{some} simply-connected $(k,L)$-complex, it is essential to know whether $|\\mathcal{X}(k,L)| = 1$.\n\nThe question of when $|\\mathcal{X}(k,L)| = 1$ has been addressed by several authors, with the most complete answer to date given by Lazarovich~\\cite{L}.    Ballmann and Brin \\cite{BB} provided an inductive construction of simply-connected $(k,L)$-complexes whenever the pair $(k,L)$ satisfies the Gromov Link Condition and certain obvious obstructions do not occur.  (For example, if $k$ is odd and $L$ is bipartite of bivalency $\\{\\ell,r\\}$ with $\\ell \\neq r$, then there is no $(k,L)$-complex.)  Moreover, Ballmann and Brin and independently Haglund \\cite{H2} showed that there are uncountably many non-isomorphic simply-connected $(k,K_n)$-complexes when $k\\geq 6$ and $n\\geq 4$ (here, $K_n$ is the complete graph on $n$ vertices).   We discuss other results prior to \\cite{L}, on both uniqueness and non-uniqueness of $(k,L)$-complexes, in Section \\ref{s:examples}.  \n\nIn \\cite{L}, the graph-theoretic notions of \\emph{star-transitivity} and \\emph{st(edge)-transitivity} are introduced.  We recall the definition of these terms in Definitions~\\ref{def:star and iso} and \\ref{def:star trans} below, where we also define \\emph{$G$-star-transitivity} and \\emph{$G$-st(edge)-transitivity} for $G$ a subgroup of $\\ensuremath{\\operatorname{Aut}}(L)$.  The Uniqueness Theorem of \\cite{L}  states that if $k \\geq 4$, the pair $(k,L)$ satisfies the Gromov Link Condition and $L$ is both star-transitive and st(edge)-transitive, then $|\\mathcal{X}(k,L)| \\leq 1$.  Moreover, Theorem A of \\cite{L} states that for $k \\geq 4$ even and $(k,L)$ satisfying the Gromov Link Condition, $|\\mathcal{X}(k,L)| = 1$ if and only if $L$ is both star-transitive and st(edge)-transitive.  \n\nIn this paper we investigate star-transitivity and st(edge)-transitivity for graphs, expanding on the results and examples given in \\cite[Section 1.1]{L}.  Our main results are the following. \n(For the group notation and graph definitions used in these theorems, we refer to Section~\\ref{s:definitions}.)\nWe first consider vertex-transitive graphs, that is, graphs $L$ such that $\\ensuremath{\\operatorname{Aut}}(L)$ acts transitively on the set of vertices of $L$.\n\n\\begin{theorem}\\label{v-star-st(edge)-t}\nLet $L$ be a connected graph of valency $r\\ge3$, let $G\\le\\ensuremath{\\operatorname{Aut}}(L)$ be vertex-transitive and let $v$ be an arbitrary vertex of $L$.  Then $L$ is $G$-star-transitive and $G$-st(edge)-transitive if and only if\none of the following holds:\n\\begin{enumerate}[$(1)$]\n\\item $L$ is $(G,3)$-transitive, and $G_v=S_r\\times S_{r-1}$.\n\n\\item $L$ is cubic and $(G,4)$-arc-transitive, and $G_v=S_4$ or $S_4\\times S_2$.\n\n\\item $L$ is of valency $4$ and $(G,4)$-arc-transitive, and $G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$ or $[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$.\n\\end{enumerate}\n\\end{theorem}\n\nWe note that all cases in Theorem \\ref{v-star-st(edge)-t} give rise to examples. Case (1) is realised by the Odd graphs (see Section \\ref{s:odd}), Case (2) with $G_v=S_4$ is realised by the Heawood graph and Case (2) with $G_v=S_4\\times S_2$ is realised  by the generalised quadrangle associated with the symplectic group $\\ensuremath{\\operatorname{PSp}}(4,2)$. The groups $3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$ or $[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$ in part (3) of Theorem \\ref{v-star-st(edge)-t} are the parabolic subgroups of $\\ensuremath{\\operatorname{PGL}}(3,3)$ and the exceptional group of Lie type $G_2(3)$. Case (3) is then realised by the point-line incidence graph of the projective plane $\\ensuremath{\\operatorname{PG}}(2,3)$ and the generalised hexagon associated with $G_2(3)$.\n\n\nWe then consider graphs which are not vertex transitive.\n\n\\begin{theorem}\\label{vtx-intrans}\nLet $L$ be a connected graph with minimal valency at least three, let $G  \\le \\ensuremath{\\operatorname{Aut}}(L)$, and let $\\{ v,w \\}$ be an arbitrary edge of $L$. Assume that $L$ is $G$-star-transitive and $G$-st(edge)-transitive but not vertex-transitive. Then $L$ is a locally $(G,3)$-arc-transitive bipartite graph of bi-valency $\\{ \\ell,r\\}$ with $\\ell\\not=r$, and after possibly exchanging the role of $v$ and $w$,\none of the following holds:\n\\begin{enumerate}[$(1)$]\n\\item $G_{vw}^{[1]}$ acts nontrivially on both $\\Gamma_2(v)$ and $\\Gamma_2(w)$, and $\\{ \\ell,r \\}=\\{ 3,5 \\}$;\n\n\\item $G_{vw}^{[1]}=1$, $G_v=S_r\\times S_{\\ell-1}$, and $G_w=S_\\ell\\times S_{r-1}$;\n\n\\item $G_w^{[2]}=1$, and \n$(A_{r-1})^{\\ell-1}\\le G_{vw}^{[1]}\\le (S_{r-1})^{\\ell-1}$, \n\n\\[\\begin{array}{rllll}\n(A_r\\times (A_{r-1})^{\\ell-1}).2.S_{\\ell-1}&\\leqslant &G_v&\\leqslant&S_r\\times (S_{r-1}\\Wr S_{\\ell-1}), \\mbox{and}\\\\\n(A_{r-1})^{\\ell}.2.S_\\ell& \\leqslant& G_w &\\leqslant &S_{r-1}\\Wr S_\\ell;\n\\end{array}\\]\n\n\\item  $|\\Gamma(v)|=r\\le 5$.\n\\end{enumerate}\n\\end{theorem}\n\nWe provide examples in each of the first three cases. Case (1) is realised by the generalised quadrangle associated with the unitary group $\\ensuremath{\\operatorname{PSU}}(4,2)$ (Example \\ref{eg:hermitian}). Case (2) is realised by the vertex-maximal clique incidence graph of the Johnson graph (see Section \\ref{s:johnson}) while Case (3) is realised by the vertex-maximal clique incidence graph of the Hamming graph (Section \\ref{s:hamming}). We do not have any example for case (4) that is not also an example for one of the previous cases.\n\n\n\nWe also characterise graphs of small minimum valency which are star-transitive and\/or st(edge)-transitive.  In particular, we show the following.\n\n\\begin{theorem}\n\\label{thm:small valency}\nLet $L$ be a connected graph with minimal valency one or two. Then $L$ is star-transitive and st(edge)-transitive if and only if one of the following holds:\n\\begin{itemize}\n\\item[(1)] $L$ is a complete bipartite graph $K_{1,n}$ for some $n \\ge 1$;\n\\item[(2)] $L$ is a cycle of length $n$ for some $n \\ge 3$; or\n\\item[(3)] there exists a locally fully symmetric, arc-transitive graph $\\Sigma$ of\nvalency at least three such that $L$ can be obtained by subdividing each edge \nof $\\Sigma$ with a new vertex of valency two.\n\\end{itemize}\n\\end{theorem}\n\n In Section~\\ref{s:small}, we characterise star-transitive and st(edge)-transitive graphs of small girth and minimal valency and prove Theorem~\\ref{thm:small valency}. In Section~\\ref{s:observations}, we prove some preliminary general results about star-transitive and st(edge)-transitive graphs. The proofs of Theorems~\\ref{v-star-st(edge)-t} and \\ref{vtx-intrans} are given in Sections~\\ref{s:vertex trans} and \\ref{s:vertex intran}, respectively. Finally, in Section~\\ref{s:examples}, we describe many examples of graphs that are both star-transitive and st(edge)-transitive.\n\nWe expect that our results and examples will contribute to the program described in \\cite{FHT} of investigating the automorphism groups of nonpositively curved polygonal complexes and their lattices.  In particular, if $X$ is a simply-connected $(k,L)$-complex with $L$ star-transitive and st(edge)-transitive, then basic questions for which our results may prove useful include whether $\\ensuremath{\\operatorname{Aut}}(X)$ is discrete, and whether $\\ensuremath{\\operatorname{Aut}}(X)$ admits a lattice.\n\n\n\n\\subsection*{Acknowledgements}\n\nWe thank Nir Lazarovich and Michah Sageev for providing us with the preprint \\cite{L}.  We also thank Hendrik van Maldeghem for pointing us to the references for local actions of generalised polygons.\n\n\\section{Background on $(k,L)$-complexes}\\label{s:algtop}\n\nIn Section \\ref{sec:algtopggt} we briefly recall several key definitions from algebraic topology and geometric group theory, and apply these in the setting of $(k,L)$-complexes.  We then discuss in Section \\ref{sec:incidence} the relationship between $(k,L)$-complexes and rank $3$ incidence geometries.\n\n\\subsection{Definitions from algebraic topology and geometric group theory}\\label{sec:algtopggt}\n\nWe first recall the definition of a \\emph{$2$-dimensional CW-complex}, also known as a \\emph{$2$-dimensional cell complex}.  (A reference is, for example,  \\cite{Hatcher}.)   Denote by $D^1$ the closed interval $[-1,1]$ with boundary $\\partial D^1$ the points $S^0 = \\{-1,1\\}$, and by $D^2$ the closed unit disk in the Euclidean plane with boundary $\\partial D^2$ the unit circle $S^1$.\nA space $X$ is a $2$-dimensional CW-complex if it is constructed as follows:\n\\begin{enumerate}\n\\item Begin with a discrete set $X^{(0)}$, called the \\emph{$0$-skeleton}, whose points are the \\emph{$0$-cells}.\n\\item The \\emph{$1$-skeleton} $X^{(1)}$ is the quotient space obtained from the disjoint union $X^{(0)} \\sqcup_\\alpha D^1_\\alpha$, of $X^{(0)}$ with a collection of closed intervals $D^1_\\alpha$, by identifying each boundary point $x \\in \\partial D^1_\\alpha$ with a $0$-cell $\\varphi_\\alpha(x) \\in X^{(0)}$.  That is, each $\\varphi_\\alpha$ is a function from $S^0 = \\{-1,1\\}$ to $X^{(0)}$ (necessarily continuous).  We equip $X^{(1)}$ with the quotient topology. The images of the $D^1_\\alpha$ in $X^{(1)}$ are called the (closed) \\emph{$1$-cells}.  \n\\item The \\emph{$2$-skeleton} $X^{(2)}$ is the quotient space obtained from the disjoint union $X^{(1)} \\sqcup_\\beta D^2_\\beta$, of $X^{(1)}$ with a collection of closed disks $D^2_\\beta$, by identifying each boundary point $x \\in \\partial D^2_\\beta$ with a point $\\varphi_\\beta(x) \\in X^{(1)}$, where each $\\varphi_\\beta$ is a continuous function from the circle $\\partial D^2_\\beta = S^{1}$ to $X^{(1)}$.  We equip $X^{(2)}$ with the quotient topology. The images of the $D^2_\\beta$ in $X^{(2)}$ are called the (closed) \\emph{$2$-cells}.\n\\item Since $X$ is $2$-dimensional, $X$ is equal to its $2$-skeleton $X^{(2)}$. \n\\end{enumerate}\nThe maps $\\varphi_\\alpha$ and $\\varphi_\\beta$ are called the \\emph{attaching maps}.  The (closed) \\emph{cells} of $X$ are its (closed) $0$-, $1$- and $2$-cells.  The $1$-skeleton of $X$ may be thought of as a graph (not necessarily simple), with vertex set the $0$-skeleton and edges the $1$-cells.  The additional conditions required in order for a $2$-dimensional CW-complex $X$ to be a \\emph{polygonal complex} are stated in the introduction.\n\nWe next recall the definitions of geodesic metric spaces and the curvature conditions $\\operatorname{CAT}(0)$ and $\\operatorname{CAT}(-1)$.  For details on this material, see \\cite{BH}.  Let $(X,d_X)$ be a metric space.  A continuous function $\\gamma:[a,b] \\to X$ (for $a < b$ real numbers) is a \\emph{geodesic} if for all $a \\leq t < t' \\leq b$, we have $d_X(\\gamma(t),\\gamma(t')) = t' - t$.   The metric space $(X,d_X)$ is \\emph{geodesic} if for all $x,y \\in X$, there is a geodesic $\\gamma:[a,b]\\to X$ such that $\\gamma(a) = x$ and $\\gamma(b) = y$.  We may denote this geodesic by $[x,y]$.  Note that there may be more than one geodesic connecting $x$ and $y$.  For example, in Euclidean space, each geodesic is a straight line segment and there is a unique geodesic connecting each pair of points, while on the sphere $S^2$ with its usual metric, each geodesic is an arc of a great circle, and antipodal points are connected by infinitely many geodesics.\n\nLet $(X,d_X)$ be a geodesic metric space.  A \\emph{geodesic triangle} in $X$ is a triple of points $x,y,z$, together with a choice of geodesics $[x,y]$, $[y,z]$ and $[z,x]$.  Given a geodesic triangle $\\Delta = \\Delta(x,y,z)$, a \\emph{comparison triangle} in the Euclidean plane is a triple of points $\\bar{x}, \\bar{y}, \\bar{z}$ such that $d_X(x,y) = d(\\bar{x},\\bar{y})$, $d_X(y,z) = d(\\bar{y}, \\bar{z})$ and $d_X(z,x) = d(\\bar{z} , \\bar{x})$, where $d$ is the Euclidean metric.  For each point $p \\in [x,y]$, there is a \\emph{comparison point} denoted $\\bar{p}$ in the straight line segment $[\\bar{x}, \\bar{y}]$, with the comparison point $\\bar{p}$ defined by the equation $d_X(x,p) = d(\\bar{x} , \\bar{p})$.  Similarly, we define comparison points for $p$ in $[y,z]$ and $[z,x]$.  The space $X$ is said to be $\\operatorname{CAT}(0)$ if for every geodesic triangle $\\Delta = \\Delta(x,y,z)$, and every pair of points $p,q \\in [x,y] \\cup [y,z] \\cup [z,x]$, we have $d_X(p,q) \\leq d(\\bar{p} ,\\bar{q})$.  Roughly speaking, triangles in a $\\operatorname{CAT}(0)$ space are ``no fatter\" than Euclidean triangles.  A $\\operatorname{CAT}(0)$ space is sometimes said to be \\emph{nonpositively curved}.  \n\nWe may instead consider comparison triangles in the hyperbolic plane, and define $X$ to be $\\operatorname{CAT}(-1)$ or \\emph{negatively curved} if its triangles are ``no fatter\" than hyperbolic triangles.  Every $\\operatorname{CAT}(-1)$ space is also a $\\operatorname{CAT}(0)$ space (Theorem II.1.12 of \\cite{BH}).  Key properties of $\\operatorname{CAT}(0)$ spaces $X$ include that each pair of points in $X$ is connected by a unique geodesic, that $X$ is contractible hence simply-connected and that if a group acting by isometries on $X$ has a bounded orbit, then it fixes a point (see, respectively, Proposition II.1.4, Corollary II.1.5 and Corollary II.2.8  of \\cite{BH}).\n\nWe now consider the case of interest, when $X$ is a $(k,L)$-complex, as defined in the introduction.  Assume that each face of $X$ is metrised as a regular Euclidean $k$-gon, or that each face of $X$ is metrised as a regular hyperbolic $k$-gon.  Then by Theorem I.7.50 of \\cite{BH}, $X$ is a complete geodesic metric space when equipped with the ``taut string\" metric, in which each geodesic is a concatenation of a finite number of geodesics contained in faces. \n\nIf the faces of a $(k,L)$-complex $X$ are regular Euclidean $k$-gons and $(k,L)$ satisfies the Gromov Link Condition, then $X$ is locally $\\operatorname{CAT}(0)$ (see I.5.24 of \\cite{BH}).  Hence by the Cartan--Hadamard Theorem \\cite[II.4.1]{BH}, the universal cover of $X$ is a $\\operatorname{CAT}(0)$ space.  Similarly, if either $k > m$ and $\\operatorname{girth}(L) \\geq n$, or $k \\geq m$ and $\\operatorname{girth}(L) > n$,  for $(m,n) \\in \\{ (3,6), (4,4), (6,3) \\}$, then the faces of $X$ may be metrised as regular hyperbolic $k$-gons with vertex angles $2\\pi\/\\operatorname{girth}(L)$, the complex $X$ is locally $\\operatorname{CAT}(-1)$ and the universal cover of $X$ is thus $\\operatorname{CAT}(-1)$.  We will henceforth be considering simply-connected $(k,L)$-complexes $X$ which satisfy the Gromov Link Condition, and so  are $\\operatorname{CAT}(0)$. \n\n\n\\subsection{Relationship with incidence geometries}\\label{sec:incidence}\nA $(k,L)$-complex may be viewed combinatorially as a rank three incidence structure, namely, a geometry consisting of three types of objects, vertices, edges and faces, such that each edge is incident with exactly two vertices, each face is incident with exactly $k$ edges and $k$ vertices, and the graph with vertex set the edges incident with a given vertex and edges the faces is  isomorphic to $L$. In the notation of Buekenhout \\cite{Bu}, the geometry has diagram\n\\[\\xymatrix@R=0mm{\n{\\bullet}   \\ar@{-}[rr]^{(k)} &   & {\\bullet}   \\ar@{-}[rr]^{\\overline{L}} &  & {\\bullet} \n\\\\\n{\\mbox{vertices}} & & {\\mbox{edges}} & & {\\mbox{faces}} \n}\\]\nwhere the label $(k)$ denotes the vertex-edge incidence graph of a  $k$-cycle and $\\overline{L}$ denotes the vertex-edge incidence graph of $L$. \n\nWe call a $(k,L)$-complex \\emph{connected} if both the link graph and the vertex-edge incidence graph are connected. A \\emph{flag} in the geometry is an incident vertex-edge-face triple. All  connected $(k,L)$-complexes that can be embedded in $\\mathbb{E}^3$ and have a group of automorphisms acting regularly on flags were classified by Pellicer and Shulte \\cite{PS1,PS2}. The only finite ones were seen to be the eighteen finite regular polyhedra. Polyhedra are precisely the finite  $(k,L)$-complexes with $L$ a cycle.\n\n\n\n\n\n\n\n\n\n\\section{Graph- and group-theoretic definitions and notation}\n\\label{s:definitions}\n\nIn Section \\ref{sec:graphs} we discuss the main definitions from graph theory that we will require, and then in Section~\\ref{sec:groups} we define the group-theoretic notation that we will use.  For ease of comparison with the literature, we now switch to standard graph-theoretic notation.  We also assume some basic definitions from algebraic graph theory \\cite{GR} and from the theory of permutation groups \\cite{DM}. All graphs considered in this paper are finite. \n\n\\subsection{Graph theory}\\label{sec:graphs}\n\nLet $\\Gamma$ be a graph with vertex set $V(\\G)$ and edge set $E(\\G)$. If $\\Gamma$ is {\\em simple}, that is, a graph without loops or multiple edges, and $e \\in E(\\G)$ connects the vertices $u$ and $v$, then we identify the (undirected) edge $e$ with the set $\\{u,v\\}$, and we denote the {\\em arc} (directed edge) from $u$ to $v$ by $uv$.  For each vertex $v$ we denote by $\\Gamma(v)$ the set of neighbours of $v$, that is, the set of vertices adjacent to $v$. The set of all vertices at distance $i$ from $v$ will be denoted by $\\Gamma_i(v)$. In particular, $\\Gamma_1(v)=\\Gamma(v)$. For $X \\subseteq V(\\G)$, the {\\em restriction of $\\Gamma$ to $X$} is the graph $\\Gamma\\mid_X$ with vertex set $X$ and edge set consisting of those edges of $\\Gamma$ that have both endpoints in $X$. The {\\em girth} $\\operatorname{girth}(\\Gamma)$ is the length of the shortest cycle in $\\Gamma$.  The \\emph{valency} of a vertex is the number of neighbours that it has and the graph $\\Gamma$ is called {\\em $k$-regular} if each $v \\in V(\\G)$ has valency $k$. A $3$-regular graph is also called a \\emph{cubic graph}. If $\\Gamma$ is $k$-regular then we also say that $\\Gamma$ {\\em has valency $k$}. A graph is called \\emph{regular} if it is $k$-regular for some $k$. If $\\Gamma$ is bipartite and vertices in the two parts of the bipartition have valency $\\ell$ and $r$, respectively, then we say that $\\Gamma$ {\\em has bi-valency $\\{ \\ell,r \\}$}.\n\n\n\nIf $G$ is a group of automorphisms of $\\Gamma$ and $v \\in V(\\G)$ then $G_v$ denotes the stabiliser in $G$ of the vertex $v$. If $X \\subseteq V(\\G)$ is stabilised setwise by a subgroup $H \\le G$ then we denote by $H^X$ the permutation group induced by $H$ on $X$. In particular, $G_v^{\\Gamma(v)}$ is the group induced on $\\Gamma(v)$ by $G_v$. We say that $G$ is {\\em locally transitive}, {\\em locally primitive}, {\\em locally $2$-transitive}, or {\\em locally fully symmetric} if for each $v \\in V(\\G)$ the group $G_v^{\\Gamma(v)}$ is transitive, primitive, $2$-transitive, or the symmetric group on $\\Gamma(v)$, respectively. The graph $\\Gamma$ is locally transitive, locally primitive, locally $2$-transitive, or locally fully symmetric if there exists $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$ with the appropriate property (equivalently, as all four properties hold in overgroups, $\\ensuremath{\\operatorname{Aut}}(\\Gamma)$ has the appropriate property). \n\nFor an edge $\\{u,v\\}$ we define $G_{\\{u,v\\}}$ to be the setwise stabiliser of $\\{u,v\\}$, and for an arc $uv$, we define $G_{uv}:= G_u \\cap G_v$.  Let $d$ be the usual distance function on $\\Gamma$, so that each edge has length $1$.  Then for each natural number $n$ and each $v \\in V(\\G) $, we define \n\\[ G_v^{[n]} := \\{ g \\in G_v \\mid w^g = w \\, \\forall w \\in V(\\G) \\mbox{ such that } d(v,w) \\leq n \\}  \\]\nas the pointwise stabiliser of the ball of radius $n$ around $v$. For $\\{ u,v \\} \\in E(\\G)$, $G_{uv}^{[1]}:= G_u^{[1]} \\cap G_v^{[1]}$. \n\nAn \\emph{$s$-arc} in a graph $\\Gamma$ is an $(s+1)$-tuple $(v_0,v_1,\\ldots,v_s)$ of vertices such that $\\{ v_i,v_{i+1}\\} \\in E(\\G)$ and $v_{i-1}\\neq v_{i+1}$, that is, it is a walk of length $s$ that does not immediately turn back on itself. Let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$. We say that $\\Gamma$ is \\emph{locally $(G,s)$-arc transitive} if for each vertex $v$, the stabiliser $G_v$ acts transitively on the set of $s$-arcs of $\\Gamma$ starting at $v$. If $G$ is transitive on the set of all $s$-arcs in $\\Gamma$ then we say that $\\Gamma$ is \\emph{$(G,s)$-arc transitive}. If all vertices of $\\Gamma$ have valency at least two then locally $s$-arc transitive implies locally $(s-1)$-arc transitive. Moreover, $s$-arc transitive implies locally $s$-arc transitive. Conversely, if $G$ is transitive on $V(\\G)$ and $\\Gamma$ is locally $(G,s)$-arc transitive then $\\Gamma$ is $(G,s)$-arc transitive. We observe that a graph with all vertices having valency at least 2 is locally $(G,2)$-arc transitive if and only if $G_v^{\\Gamma(v)}$ is 2-transitive for all vertices $v$ (see for example \\cite[Lemma 3.2]{GLP1}). Moreover, if $\\Gamma$ is locally $G$-transitive then $G$ acts transitively on $E(\\G)$ and either $G$ is transitive on $V(\\G)$ or $\\Gamma$ is bipartite and $G$ acts transitively on both sets of the bipartition. If $\\Gamma$ is $(G,s)$-arc-transitive but not $(G,s+1)$-arc-transitive then we say that $\\Gamma$ is {\\em $(G,s)$-transitive}. Finally, if $G=\\ensuremath{\\operatorname{Aut}}(\\Gamma)$ then we drop the name $G$ from all notation introduced in this paragraph and say that $\\Gamma$ is \\emph{locally $s$-arc-transitive}, \\emph{$s$-arc-transitive}, and \\emph{$s$-transitive}, respectively. \n\nThe study of $s$-arc transitive graphs goes back to the  seminal work of Tutte \\cite{tutte47,tutte59} who showed that a cubic graph  is at most 5-arc transitive. This was later extended by Weiss \\cite{W2} to show that any graph of valency at least three is at most 7-arc transitive. Weiss \\cite{Weiss78} also showed that a cubic graph is at most locally 7-arc transitive while Stellmacher \\cite{sleq9} has announced that a graph of valency at least 3 is at most locally 9-arc transitive. In each case the upper bound is met. Note that a cycle is $s$-arc transitive for all values of $s$.\n\n\n\nThe following definitions, using a slightly different language, were introduced by Lazarovich \\cite{L}.\n\\begin{definition}  \n\\label{def:star and iso}\n{\\em Let $\\Gamma$ be a simple graph, let $v \\in V(\\G)$, and let $e=\\{ u,v\\} \\in E(\\G)$. \nThe \\emph{open star of $v$}, denoted $\\operatorname{st}(v)$, is the union of $\\{v \\}$ and the set $\\{ f \\in E(\\G) \\mid f \\mbox{ is incident to } v \\}$.  Similarly, the \\emph{open edge-star of $e$}, denoted $\\operatorname{st}(e)$ or $\\operatorname{st}(\\{u,v\\})$, is the union of the sets  $\\{ u \\}$, $\\{ v \\}$, and  $\\{ f \\in E(\\G) \\mid f \\mbox{ is incident to at least one of } u,v \\}$.\n\nGiven two open stars $\\operatorname{st}(v_1)$ and $\\operatorname{st}(v_2)$, a {\\em star isomorphism} is a bijection $\\varphi: \\operatorname{st}(v_1) \\to \\operatorname{st}(v_2)$ such that $\\varphi(v_1)=v_2$. \n\nGiven two open edge-stars $\\operatorname{st}(\\{u_1,v_1\\})$ and $\\operatorname{st}(\\{u_2,v_2\\})$, an {\\em edge-star isomorphism}  is a bijection $\\varphi: \\operatorname{st}(\\{u_1,v_1\\}) \\to \\operatorname{st}(\\{u_2,v_2\\})$ such that\n\\begin{itemize}\n\\item[(i)] $\\varphi(V(\\G) \\cap \\operatorname{st}(\\{u_1,v_1\\}))=V(\\G) \\cap \\operatorname{st}(\\{u_2,v_2\\})$, that is, the vertices $u_1,v_1$ are mapped (in some order) to the vertices $u_2,v_2$.\n\\item[(ii)] $\\varphi$ is incidence-preserving, that is, $f \\in E(\\G) \\cap \\operatorname{st}(\\{u_1,v_1\\})$ is incident to $u_1$ if and only if $\\varphi(f)$ is incident to $\\varphi(u_1)$ and \n$f \\in E(\\G) \\cap \\operatorname{st}(\\{u_1,v_1\\})$ is incident to $v_1$ if and only if $\\varphi(f)$ is incident to $\\varphi(v_1)$. In particular, $\\varphi(\\{ u_1,v_1 \\}) = \\{ u_2,v_2 \\}$. \n\\end{itemize}}\n\\end{definition}\n\n\\begin{definition}\n\\label{def:star trans}\n{\\em Let $\\Gamma$ be a simple graph, and let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$. Then $\\Gamma$ is called \n\\begin{itemize}\n\\item[(i)]  \\emph{$G$-star-transitive} if for all $v_1,v_2\\in V(\\G)$ and for all star isomorphisms $\\varphi: \\operatorname{st}(v_1) \\to \\operatorname{st}(v_2)$, there exists an automorphism $\\psi \\in G$\nsuch that $\\psi(v_1)=\\varphi(v_1)$ and for all $f \\in E(\\G) \\cap \\operatorname{st}(v_1)$ we have $\\psi(f)=\\varphi(f)$; and \n\\item[(ii)]  \\emph{$G$-st(edge)-transitive} if for all $\\{u_1,v_1\\}, \\{u_2,v_2\\} \\in E(\\G)$ and edge-star isomorphisms  $\\varphi: \\operatorname{st}(\\{u_1,v_1\\}) \\to \\operatorname{st}(\\{u_2,v_2\\})$,\nthere exists an automorphism $\\psi \\in G$\nsuch that $\\psi(u_1)=\\varphi(u_1)$,  $\\psi(v_1)=\\varphi(v_1)$, and for all $f \\in E(\\G) \\cap \\operatorname{st}(\\{u_1,v_1\\})$ we have $\\psi(f)=\\varphi(f)$.\n\\end{itemize}}\nIf $G =\\ensuremath{\\operatorname{Aut}}(\\Gamma)$ then we simply say that $\\Gamma$ is star-transitive or st(edge)-transitive, respectively. \n\\end{definition}\n\n\nA subtlety of Definition \\ref{def:star trans} is that if there is no star-isomorphism $\\operatorname{st}(v_1) \\to \\operatorname{st}(v_2)$ or edge-star isomorphism $\\operatorname{st}(\\{u_1,v_1\\}) \\to \\operatorname{st}(\\{u_2,v_2\\})$, then the required property of extending to a graph automorphism holds trivially. \nAnother subtlety is the introduction of the notions of star-transitivity and st(edge)-transitivity relative to subgroups $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$. Considering subgroups of $\\ensuremath{\\operatorname{Aut}}(\\Gamma)$ with certain transitivity properties is quite common in algebraic graph theory; the main reason is that there are examples where some $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$ extends to covers of $\\Gamma$ but the full automorphism group does not. For example, the icosahedron is a cover of the complete graph $K_6$ but not all of $\\ensuremath{\\operatorname{Aut}}(K_6)=S_6$ extends.\n\n\nThe reason for the somewhat cumbersome formulation of the definitions above is that in the case when $\\operatorname{girth}(\\Gamma) \\le 4$, the definition of a star isomorphism or edge-star isomorphism $\\varphi$ does {\\em not} \nrequire that $\\varphi$ preserves the possible adjacency relations among the neighbours of the vertices occurring in the open stars and edge-stars. However, the graph automorphisms defined in Definition~\\ref{def:star trans}, extending the star isomorphisms and edge-star isomorphisms, must preserve such adjacency relations.  The following result, whose proof is immediate, says that for large enough girth we can work with much simpler definitions.  Given a vertex $v$ we let  $X(v):=\\{ v \\} \\cup \\Gamma(v)$, and for an edge $\\{u,v\\}$ we let\n$X(\\{u,v\\}):=\\{ u \\} \\cup \\{ v \\} \\cup \\Gamma(u) \\cup \\Gamma(v)$.\n\n\\begin{prop}\n\\label{prop:equiv}\n$(i)$ If $\\operatorname{girth}(\\Gamma) \\ge 4$ then for $v \\in V(\\G)$, $\\operatorname{st}(v)$ can be identified with the restriction \n$\\Gamma \\mid_{X(v)}$. A star isomorphism is then \na graph isomorphism $\\varphi_1: \\Gamma \\mid_{X(v_1)} \\to \\Gamma \\mid_{X(v_2)} $ and \n$\\Gamma$ is star-transitive if and only if \nevery  star isomorphism extends to an automorphism of $\\Gamma$. \n\n$(ii)$ If $\\operatorname{girth}(\\Gamma) \\ge 5$ then for $\\{u,v\\} \\in E(\\G)$, $\\operatorname{st}(\\{u,v\\})$ can be identified with the restriction $\\Gamma \\mid_{X(\\{u,v\\})}$. An edge-star isomorphism is then a graph isomorphism \n$\\varphi_2: \\Gamma \\mid_{X(\\{u_1,v_1\\})} \\to \\Gamma \\mid_{X(\\{u_2,v_2\\})}$ and the graph $\\Gamma$ is st(edge)-transitive if and only if \nevery edge-star isomorphism extends to an automorphism of $\\Gamma$.\n\\end{prop}\n\n\\subsection{Group-theoretic notation}\\label{sec:groups}\n\nFor a natural number $k$, we denote by $S_k$ the symmetric group on $k$ letters, by $A_k$ the alternating group on $k$ letters, by $C_k$ the cyclic group of order $k$, and by $D_{2k}$ the dihedral group of order $2k$. The projective special linear group, projective general linear group, and projective semilinear group of dimension $d$ over a field of size $q$ is denoted by $\\ensuremath{\\operatorname{PSL}}(d,q)$, $\\ensuremath{\\operatorname{PGL}}(d,q)$, and $\\operatorname{P\\Gamma L}(d,q)$, respectively. Given a group $A$ and a natural number $k$, we denote by $A \\Wr S_k$ the following wreath product: let $B$ be the direct product of $k$ copies of $A$.  Then $S_k$ acts naturally on $B$ by permuting the $k$ copies of $A$, and $A \\Wr S_k$ is the semidirect product induced by this action. We denote the semidirect product of two groups $A$ and $B$ by $A{:}B$. A group that is a (not necessarily split) extension of a subgroup $A$ by a group $B$ will be denoted by $A.B$. Given a prime $p$, $p^m$ will be used to denote an elementary abelian group of order $p^m$, and we will use $[p^m]$ to denote a group of order $p^m$ when we do not wish to specify the isomorphism type.  The \\emph{socle} of a group $G$ is the subgroup generated by all the minimal normal subgroups of $G$, and is denoted by $\\ensuremath{\\operatorname{soc}}(G)$. When $n\\geq 5$ or $n=3$, we have that $\\ensuremath{\\operatorname{soc}}(S_n)=A_n$.\n\nWe refer to a triple of groups $(A,B,A\\cap B)$ as an \\emph{amalgam}. A \\emph{completion} of the amalgam $(A,B,A\\cap B)$ is a group $G$ together with group homomorphisms $\\phi_1:A\\rightarrow G$ and $\\phi_2:B\\rightarrow G$ such that $\\phi_1$ and $\\phi_2$ are one-to-one, $G=\\langle \\phi_1(A),\\phi_2(B)\\rangle$ and $\\phi_1(A)\\cap\\phi_2(B)=\\phi_1(A\\cap B)=\\phi_2(A\\cap B)$.\n\n\n\\section{Graphs with small girth or with small minimal valency}\n\\label{s:small}\n\nIn this section, we characterise star-transitive and st(edge)-transitive graphs of girth at most four, so that in the rest of the paper we can concentrate on the case $\\operatorname{girth}(\\Gamma) \\ge 5$ and use the simplified description of stars and edge-stars, as given in Proposition~\\ref{prop:equiv}. We also characterise star-transitive and st(edge)-transitive graphs of minimal valency one or two. \n\n\\begin{lemma}\n\\label{lem:girth 3}\nThe only connected star-transitive graphs of girth $3$ are the complete graphs $K_n$, for some $n \\ge 3$. The only connected st(edge)-transitive graph of girth $3$ is the triangle $K_3$. The only connected st(edge)-transitive graphs of girth $4$ are the complete bipartite graphs $K_{m,n}$, for some $m,n \\ge 2$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\{ u,v,w \\}$ be a cycle of length $3$ in $\\Gamma$, and suppose that $\\Gamma$ is star-transitive. Considering the extensions of all star isomorphisms $\\varphi: \\operatorname{st}(v) \\to \\operatorname{st}(v)$ to \nautomorphisms of $\\Gamma$, we obtain that any two vertices in $\\Gamma(v)$ are adjacent. Hence, for any $x \\in \\Gamma(v)$, $x$ is contained in a cycle of length $3$, and by the same argument as above\nall neighbours of $x$ are adjacent. In particular, all $y \\in \\Gamma(x) \\setminus \\{ v\\}$ are adjacent to $v$, and so $\\{ v \\} \\cup \\Gamma(v) = \\{ x\\} \\cup \\Gamma(x)$. \nAs $\\Gamma$ is connected, we obtain that $V(\\G) = \\{ v \\} \\cup \\Gamma(v)$ and $\\Gamma$ is a complete graph.\n\nSuppose now that  $\\{ u,v,w \\}$ is a cycle of length $3$ in $\\Gamma$, and that $\\Gamma$ is st(edge)-transitive. If the vertex $u$ has valency greater than $2$ then there exists an edge-star isomorphism \n$\\varphi: \\operatorname{st}(\\{u,v\\}) \\to \\operatorname{st}(\\{u,v\\}) $ such that $\\varphi(u)=u$, $\\varphi(v)=v$, $\\varphi(\\{u,w\\}) \\ne \\{ u,w\\}$, and  $\\varphi(\\{v,w\\}) = \\{ v,w\\}$. However, $\\varphi$ cannot be\nextended to an automorphism of $\\Gamma$, a contradiction. Similarly, $v$ and $w$ also must be of valency $2$ and so, as $\\Gamma$ is connected, $V(\\G) = \\{ u,v,w \\}$.\n\nFinally, suppose that $\\operatorname{girth}(\\Gamma)=4$, $\\Gamma$ is st(edge)-transitive, and let $\\{ u,v,w,z \\}$ be a $4$-cycle. Considering the extensions of all edge-star isomorphisms $\\varphi: \\operatorname{st}(\\{u,v\\}) \\to \\operatorname{st}(\\{ u,v\\})$ that fix $u$ and $v$, we obtain\nthat, as an image of the edge $\\{ w,z\\}$, every pair $\\{ w_1,z_1 \\}$ with \n$w_1 \\in \\Gamma(v)$ and $z_1 \\in \\Gamma(u)$ is in $E(\\G)$. Therefore, for every \n$w_1 \\in \\Gamma(v)$ we have $\\Gamma(w_1) \\supseteq \\Gamma(u)$. Repeating the same argument\nwith edge-star isomorphisms $\\varphi: \\operatorname{st}(\\{w_1,v\\}) \\to \\operatorname{st}(\\{ w_1,v\\})$ and \na four-cycle containing $\\{ w_1,z \\}$, we obtain that for every \n$w_2 \\in \\Gamma(v)$ we have $\\Gamma(w_2) \\supseteq \\Gamma(w_1)$. In particular, for $w_2=u$,\n$\\Gamma(w_1)= \\Gamma(u)$. As $w_1$ was an arbitrary element of $\\Gamma(v)$, all vertices in $\\Gamma(v)$ have the same neighbours. Similarly, all vertices in $\\Gamma(u)$ have the same neighbours, $V(\\G)= \\Gamma(u) \\cup \\Gamma(v)$, and $\\Gamma$ is a complete bipartite graph. \n\\end{proof}\n\nNext, we characterise the star-transitive and st(edge)-transitive graphs with a vertex of valency one. For $n \\ge 3$, we define the {\\em spider graph $T_n$} as a graph with $2n+1$ vertices  \n$V(T_n)=\\{ x,y_1,\\ldots,y_n,z_1,\\ldots,z_n \\}$ and $2n$ edges \n$E(T_n)= \\{ \\{ x,y_i \\}, \\{ y_i,z_i \\} \\mid 1 \\le i \\le n \\}$. \n\n\\begin{lemma}\n\\label{lem:valency one}\nThe only connected star-transitive graphs with a vertex of valency one are the\ncomplete bipartite graphs $K_{1,n}$, for some $n \\ge 1$. \nThe only connected st(edge)-transitive graphs with a vertex of valency one are: the complete bipartite graphs $K_{1,n}$, for some $n \\ge 1$; the path $P_4$ with four vertices; and the spider graphs $T_n$, for some $n \\ge 3$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\Gamma$ be a simple graph, let $v \\in V(\\G)$ have valency one, and let $u$ be the unique neighbour of $v$. If $\\Gamma$ is star-transitive then, considering the star-isomorphisms $\\varphi: \\operatorname{st}(u) \\to \\operatorname{st}(u)$ mapping $v$ to other neighbours of $u$, we obtain that all neighbours of $u$ have valency one. As $\\Gamma$ is connected, we obtain $\\Gamma \\cong K_{1,n}$, where $n$ is the valency of $u$. \n\nSuppose now that $\\Gamma$ is st(edge)-transitive. We distinguish three cases, according to the valency of $u$. If $u$ has valency at least three then let $w,z$ be neighbours of $u$ that are different from $v$. Considering edge-star isomorphisms $\\varphi: \\operatorname{st}(\\{u,w\\}) \\to \\operatorname{st}(\\{u,w\\})$ that fix $u$ and map $v$ to neighbours of $u$ different from $w$, we obtain that all neighbours of $u$ different from $w$ have valency one. Repeating the same process with edge-star isomorphisms \n$\\varphi: \\operatorname{st}(\\{u,z\\}) \\to \\operatorname{st}(\\{u,z\\})$, we deduce that $w$ also has valency one and $\\Gamma \\cong K_{1,n}$, where $n$ is the valency of $u$.\n\nIf $u$ has valency one then $\\Gamma \\cong K_{1,1}$. If $u$ has valency two then let $x$ be the neighbour of $u$ different from $v$. We distinguish three subcases, according to the valency of $x$. If $x$ has valency one then $\\Gamma \\cong K_{1,2}$. If $x$ has valency two then, from the edge-star isomorphism \n$\\varphi: \\operatorname{st}(\\{u,x\\}) \\to \\operatorname{st}(\\{u,x\\})$ that exchanges $u$ and $x$, we obtain that $\\Gamma \\cong P_4$. Finally, if the valency of $x$ is at least three then let \n$w,z$ be neighbours of $x$ that are different from $u$. Considering edge-star isomorphisms $\\varphi: \\operatorname{st}(\\{x,w\\}) \\to \\operatorname{st}(\\{x,w\\})$ that fix $x$ and map $u$ to neighbours of $x$ different from $w$, we obtain that all neighbours of $x$ different from $w$ have valency two and they are adjacent to a vertex of valency one. \nRepeating the argument with edge-star isomorphisms $\\varphi: \\operatorname{st}(\\{x,z\\}) \\to \\operatorname{st}(\\{x,z\\})$, we see that the neighbour $w$ also has this property and so $\\Gamma \\cong T_n$, where $n$ is the valency of $x$. \n\\end{proof}\n\nFinally, we handle the case of minimal valency two. For any $n \\ge 3$, the cycle $C_n$ is $2$-regular, star-transitive, and st(edge)-transitive. We obtain further examples by the following constructions. \n\nLet $\\Sigma$ be a simple graph of minimal valency at least three. We construct\nthe {\\em $1$-subdivision} $\\Gamma$ of $\\Sigma$ by replacing each edge by a path of length two. Formally, we define $V(\\G)=V(\\Sigma) \\cup E(\\Sigma)$. The sets \n$V(\\Sigma)$ and $ E(\\Sigma)$ are independent in $\\Gamma$, and $v \\in V(\\Sigma)$\nis connected to $e \\in E(\\Sigma)$ in $\\Gamma$ if and only if $v$ and $e$ are\nincident in $\\Sigma$.\nSimilarly, we construct the {\\em $2$-subdivision} of $\\Sigma$ by replacing each edge by a path of length three. The following proposition is easy to verify.\n\n\\begin{prop}\n\\label{min two examples}\nLet  $\\Sigma$ be an arc-transitive graph of minimal valency at least three which is locally fully symmetric. Then the $1$-subdivision of $\\Sigma$ is both star-transitive and st(edge)-transitive. The $2$-subdivision of $\\Sigma$ is st(edge)-transitive, but not star-transitive. \n\\end{prop}\n\n\\begin{lemma}\n\\label{min two star}\nSuppose that $\\Gamma$ is star-transitive and the minimal valency in $\\Gamma$ is two, but $\\Gamma$ is not $2$-regular. Then there exists an arc-transitive graph $\\Sigma$ of valency at least three which is locally fully symmetric such that $\\Gamma$ is isomorphic to the $1$-subdivision of $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Gamma$ is not 2-regular, there exists $\\{ v,w \\} \\in E(\\G)$ with $v$ of valency $k>2$ and $w$ with valency $2$. Lazarovich~\\cite[Lemma 1.1]{L} proved that this implies $\\Gamma$ is edge-transitive; consequently, all edges of $\\Gamma$ connect valency $2$ vertices with vertices of valency $k$. Hence $\\Gamma$ is bipartite and $\\Gamma$ is a $1$-subdivision of a graph $\\Sigma$ with minimal valency at least $3$.\n\nAutomorphisms of $\\Gamma$, restricted to the vertices of valency $k$, naturally define automorphisms of $\\Sigma$. Star isomorphisms $\\operatorname{st}(v) \\to \\operatorname{st}(v)$, with $v \\in V(\\G)$ and $v$ of valency $k$, show that $\\Sigma$ is locally fully symmetric, and consequently $\\Sigma$ is edge-transitive. Finally, star isomorphisms $\\operatorname{st}(w) \\to \\operatorname{st}(w)$ of $\\Gamma$, with $w \\in V(\\G)$ of valency two, show that edges in $\\Sigma$ can be turned around by automorphisms, and so $\\Sigma$ is arc-transitive.  \n\\end{proof}\n\n\\begin{lemma}\n\\label{min two edge}\nSuppose that $\\Gamma$ is st(edge)-transitive, the minimal valency in $\\Gamma$ is two, but $\\Gamma$ is not $2$-regular. Then there exists an edge-transitive graph $\\Sigma$ of minimal valency at least three which is locally fully symmetric such that one of the following holds. \n\\begin{itemize}\n\\item[$(1)$] $\\Sigma$ is non-regular and $\\Gamma$ is isomorphic to the $1$-subdivision of $\\Sigma$.\n\\item[$(2)$] $\\Sigma$ is arc-transitive, and $\\Gamma$ is isomorphic to the $1$- or $2$-subdivision of $\\Sigma$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nWe claim that there are no three vertices $u,v,w$, all of valency two, such that $\\{ u,v\\} \\in E(\\G)$ and $\\{ v,w\\} \\in E(\\G)$. Indeed, if $u$, $v$, $w$ are such vertices then there is a unique path in $\\Gamma$ starting with the edge $\\{ u,v\\}$, consisting of vertices of valency two, such that the endpoint $x$ of the path has a neighbour $z$ of valency greater than two. Then, for the last two vertices $x$ and $y$ of this path, the edge-star isomorphism $\\varphi: \\operatorname{st}(\\{ x,y \\}) \\to \\operatorname{st}(\\{ x,y \\})$ that exchanges $x$ and $y$ has no extension to an automorphism of $\\Gamma$, a contradiction.\n\nLet $\\{ v,w \\} \\in E(\\G)$ with $v$ of valency greater than $2$ and $w$ with valency $2$, let $x,y$ be two further neighbours of $v$, and let $m$ be the maximal number of vertices on a path starting at $w$ and consisting of vertices of valency $2$. By the claim in the previous paragraph, $m \\in \\{ 1,2\\}$. Considering the edge-star isomorphisms \n\\begin{equation}\n\\label{eq:xyv}\n\\operatorname{st}(\\{ x,v \\}) \\to \\operatorname{st}(\\{ x,v \\}) \\mbox{ and } \n\\operatorname{st}(\\{ y,v \\}) \\to \\operatorname{st}(\\{ y,v \\})\n\\end{equation}\nthat fix the vertex $v$, we obtain that\nall neighbours of $v$ have valency $2$ and for each neighbour $z$, the maximal length of a path starting at $z$ and consisting of vertices of valency $2$ is $m$. Then, by induction on the distance from $v$, we get that all vertices $v'$ of valency greater than $2$ have this property, and so $\\Gamma$ is the $m$-subdivision of a graph $\\Sigma$ of minimal valency at least $3$.\n\nAutomorphisms of $\\Gamma$, restricted to the vertices of valency greater than two, naturally define automorphisms of $\\Sigma$. \nThe edge-star isomorphisms in \\eqref{eq:xyv} show that $\\Sigma$ is locally fully symmetric, and consequently $\\Sigma$ is edge-transitive. If $m=2$ and $vwab$ is a path in $\\Gamma$ connecting the vertices $v,b$ of valency greater than $2$ then the \nedge-star isomorphism  $\\operatorname{st}(\\{ w,a \\}) \\to \\operatorname{st}(\\{ w,a \\})$ that exchanges $w$ and $a$ shows that $\\Sigma$ is arc-transitive and we are in case $(2)$ of the lemma. If $m=1$ and $\\Sigma$ is regular then let $vwb$ a path in $\\Gamma$ connecting \nthe vertices $v,b$ of valency greater than $2$. The edge-star isomorphism  $\\operatorname{st}(\\{ w,v \\}) \\to \\operatorname{st}(\\{ w,b\\})$ shows that $\\Sigma$ is arc-transitive, and again we are in case $(2)$. Finally, if $\\Sigma$ is non-regular then we are in case $(1)$.\n\\end{proof}\n\nCombining the results of this section, we obtain Theorem~\\ref{thm:small valency}.\n\n\n\\section{Connections among star-transitivity, st(edge)-transitivity, and arc-transitivity}\\label{s:observations}\n\nThis section contains preliminary results used for the proofs of Theorem \\ref{v-star-st(edge)-t} and \\ref{vtx-intrans}.\nWe begin by recording the following result of Lazarovich \\cite[Lemma 1.1]{L}:\n\n\\begin{lemma}\\label{l:lazarovich} If $\\Gamma$ is a connected star-transitive graph then either:\n\\begin{enumerate}\n\\item\\label{c:vt} $\\Gamma$ is 2-arc-transitive; or\n\\item\\label{c:et} $\\Gamma$ is edge-transitive and bipartite, with $V(\\G) = A_1 \\sqcup A_2$, and there exist $d_1, d_2 \\in \\mathbb{N}$ so that for all $v \\in A_i$, the vertex $v$ has valency $d_i$ ($i = 1,2$).\n\\end{enumerate}\n\\end{lemma}\n\n\\noindent It is noted in the proof of \\cite[Lemma 1.1]{L} that in Case \\eqref{c:et}, $d_1 \\neq d_2$.  We will discuss both cases of Lemma \\ref{l:lazarovich} further below.\nOur first observations are as follows.   \n\n\\begin{lemma}\\label{l:vertex transitive}  Let $\\Gamma$ be a $G$-star-transitive graph.  If $\\Gamma$ is $k$-regular then $\\Gamma$ is $G$-vertex-transitive.\n\\end{lemma}\n\n\\begin{lemma}\\label{l:locally fully symmetric}  Let $\\Gamma$ be a $G$-star-transitive graph.   Then $G_v^{\\Gamma(v)} = S_{|\\Gamma(v)|}$ for all $v \\in V(\\G)$, that is, $\\Gamma$ is locally fully symmetric.\n\\end{lemma}\n\nThe converse of Lemma \\ref{l:locally fully symmetric} does not hold, as there are graphs which are locally fully symmetric but are not star-transitive.  In fact, there are regular graphs which are locally fully symmetric but are not star-transitive. The following example was first described by Lipschutz and Xu \\cite{LX}.  Let $G = \\ensuremath{\\operatorname{PGL}}(2,p)$ for $p$ prime, $p \\equiv \\pm1 \\pmod{24}$.  Then $G$ is generated by subgroups $H \\cong D_{24}$ and $K \\cong S_4$, such that $H \\cap K \\cong D_8$.  The graph $\\Gamma$ is defined to be the bipartite graph with vertex set $G\/H \\sqcup G\/K$ and edge set $G\/(H \\cap K)$, so that the edge $g(H \\cap K)$, for $g \\in G$, connects the vertices $gH$ and $gK$.  Then $\\Gamma$ is cubic and locally fully symmetric, since the natural left-action of $G$ induces $S_3$ at each vertex.  However, $\\Gamma$ is not vertex-transitive.\n\nThe following sufficient conditions for star-transitivity are easily verified.\n\n\\begin{lemma}\\label{l:suff star}  If there exists a subgroup $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$ such that either:\n\\begin{enumerate}\n\\item\\label{suff1} $G$ is locally fully symmetric and vertex-transitive; or \n\\item\\label{suff2} $G$ is locally fully symmetric and edge-transitive, and there are \nnatural numbers $k \\neq \\ell$ such that each vertex has valency either $k$ or $\\ell$;\n\\end{enumerate}\nthen $\\Gamma$ is $G$-star-transitive.\n\\end{lemma}\n\nWe now consider st(edge)-transitivity.  \nLazarovich's main results concern graphs which are both star-transitive and st(edge)-transitive. We shall prove that, with the exception of the small-valency cases handled in Section~\\ref{s:small}, st(edge)-transitivity implies star-transitivity.\nRecall that for $v \\in V(\\G)$ and $\\{ u,v \\} \\in E(\\G)$, we defined \n$X(v) = \\{ v \\} \\cup \\Gamma(v)$ and $X(\\{ u,v\\})=\\{ u \\} \\cup \\{ v\\} \\cup \n\\Gamma(u) \\cup \\Gamma(v)$.\n\n\\begin{lemma}\\label{lem:stedge}\\label{lem:edgeaction} A connected graph $\\Gamma$ with $G = \\ensuremath{\\operatorname{Aut}}(\\Gamma)$, minimal valency at least three and girth at least four is st(edge)-transitive if and only if it is edge-transitive and either:\n\\begin{enumerate}\n\\item there is a $k \\in \\mathbb{N}$ so that for all edges $\\{ u,v\\}$, $G_{\\{u,v\\}}^{X(\\{u,v\\})} = S_{k-1} \\Wr S_2$, in which case $\\Gamma$ is $k$-regular; or \n\\item there are $k,\\ell \\in \\mathbb{N}$ with $k \\neq \\ell$ so that for all edges $\\{ u,v\\}$, $G_{\\{u,v\\}}^{X(\\{u,v\\})} = S_{k-1} \\times S_{\\ell-1}$, in which case $\\Gamma$ is $(k,\\ell)$-biregular.\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nObserve first that since the minimum valency of $\\Gamma$ is at least three then $\\Gamma$ is not a tree. If $\\Gamma$ has girth  four and is st(edge)-transitive then Lemma \\ref{lem:girth 3} implies that $\\Gamma$ is complete bipartite. Thus $\\Gamma$ is edge-transitive and (1) holds if $\\Gamma$ is regular while (2) holds in $\\Gamma$ is biregular. Conversely, assume that  $\\Gamma$ has girth four, is edge-transitive and either (1) or (2) hold.  Let $\\{u,v,w,z\\}$ be a 4-cycle. Then $z^{G_{u,v,w}}=\\Gamma(u)\\backslash\\{v\\}$ and so $\\Gamma(u)=\\Gamma(w)$. Similarly we see that $\\Gamma(v)=\\Gamma(z)$ and so $\\Gamma$ is complete bipartite and hence st(edge)-transitive.\n\nIf $\\operatorname{girth}(\\Gamma) \\ge 5$ then clearly $\\ensuremath{\\operatorname{Aut}}(\\Gamma \\mid_{X(\\{u,v\\})}) \\cong S_{k-1} \\Wr S_2$ or\n$S_{k-1} \\times S_{\\ell-1}$ in the cases $k=\\ell$ and $k \\ne \\ell$, respectively. Moreover, by Proposition~\\ref{prop:equiv}(ii), every edge-star isomorphism is a graph isomorphism, and  $\\Gamma$ is st(edge)-transiitive if and only if $\\Gamma$ is edge-transitive and every $\\varphi \\in \\ensuremath{\\operatorname{Aut}}(\\Gamma \\mid_{X(\\{u,v\\})})$ extends to an automorphism in\n$G_{\\{u,v\\}}$.  Since  the restriction of any \n$\\psi \\in G_{\\{u,v\\}}$ to $X(\\{u,v\\})$ is in $\\ensuremath{\\operatorname{Aut}}(\\Gamma \\mid_{X(\\{u,v\\})})$ the result follows.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:stimpliesstar}\nLet $\\Gamma$ be a $G$-st(edge)-transitive graph of minimum valency at least three. Then $\\Gamma$ is $G$-star-transitive.\n\\end{lemma}\n\n\\begin{proof}\nIf $\\operatorname{girth}(\\Gamma) \\le 4$ then $\\Gamma \\cong K_{m,n}$ by Lemma~\\ref{lem:girth 3} and the statement of this lemma holds. Suppose that $\\operatorname{girth}(\\Gamma) \\ge 5$, let \n$\\{u,v\\}$ be an edge of $\\Gamma$, let $k$ be the valency of $v$ and let $G=\\ensuremath{\\operatorname{Aut}}(\\Gamma)$.  By Lemma \\ref{lem:edgeaction}, $G_v^{\\Gamma(v)\\backslash \\{u\\}}=S_{k-1}$. Let $w\\in\\Gamma(v)\\backslash\\{u\\}$. Then again by Lemma \\ref{lem:edgeaction}, $G_v^{\\Gamma(v)\\backslash \\{w\\}}=S_{k-1}$. As $k \\ge 3$, it follows that $G_v^{\\Gamma(v)}=S_k$ and in particular $G_x^{\\Gamma(x)}=S_{|\\Gamma(x)|}$ for each vertex $x$. Thus $\\Gamma$ is locally fully symmetric, hence locally 2-arc transitive and edge-transitive. \n\nIf $u$ has valency $\\ell\\neq k$ then Lemma \\ref{l:suff star}(\\ref{suff2}) implies that $\\Gamma$ is star-transitive.\nIf $u$ also has valency $k$ then, since $\\Gamma$ is st(edge)-transitive, Lemma \\ref{lem:edgeaction}(1) implies that there is an element of $G$ interchanging $u$ and $v$. Hence $G$ is arc-transitive and in particular vertex-transitive. Thus by Lemma \\ref{l:suff star}(\\ref{suff1}), $\\Gamma$ is star-transitive.\n\\end{proof}\n\n\n\n\n\nWe now consider actions on arcs.\n\n\\begin{lemma}\\label{l:locally 2}  If $\\Gamma$ is $G$-star-transitive then $G$ is locally 2-transitive on $\\Gamma$ and thus $\\Gamma$ is locally $(G,2)$-arc transitive.  \n\\end{lemma}\n\n\\begin{proof}  Since $G_v^{\\Gamma(v)} = S_{|\\Gamma(v)|}$ which is $2$-transitive, the graph $\\Gamma$ is locally $2$-transitive.\n\\end{proof}\n\nIt follows that if a connected star-transitive graph $\\Gamma$ is vertex-transitive then $\\Gamma$ is $2$-arc transitive, as was given in Case \\eqref{c:vt} of Lemma \\ref{l:lazarovich} above.\n\n\\begin{lemma} \n\\label{lem:3 arc}\nIf $\\Gamma$ has minimal valency at least three and $\\Gamma$ is $G$-st(edge)-transitive then $\\Gamma$ is locally $(G,3)$-arc transitive.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lem:stimpliesstar}, $\\Gamma$ is $G$-star-transitive and so by Lemma \\ref{l:locally 2}, $\\Gamma$ is locally $(G,2)$-arc transitive. Let $(u,v,w)$ be a 2-arc of $\\Gamma$. Since $\\Gamma$ is $G$-st(edge)-transitive, Lemma \\ref{lem:edgeaction} applied to the edge $\\{v,w\\}$ implies that $G_{uvw}^{\\Gamma(w)\\backslash\\{v\\}}=S_{|\\Gamma(w)|-1}$. Hence $G_{uvw}$ acts transitively on the set of $3$-arcs starting with $(u,v,w)$. Thus $\\Gamma$ is locally $(G,3)$-arc transitive.\n\\end{proof}\n\nBy Lemmas~\\ref{lem:3 arc}, \\ref{lem:stimpliesstar} and \\ref{l:vertex transitive}, we have the following corollary. \n\n\\begin{corollary}\\label{c:3-arc transitive} Suppose $\\Gamma$ is a connected $k$-regular graph.  If $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive, then $\\Gamma$ is $(G,3)$-arc transitive.\n\\end{corollary}\n\n\\section{The vertex-transitive case}\n\\label{s:vertex trans}\n\nIn this section we prove Theorem \\ref{v-star-st(edge)-t}, that is, we conduct a local analysis of vertex-transitive and st(edge)-transitive graphs. \nWe recall that a group is called \\emph{$p$-local} for some prime $p$ if it contains a normal $p$-subgroup. Part (1) of the following fundamental theorem was proven by \nGardiner~\\cite[Corollary 2.3]{G} and was also established by Weiss in \\cite{W1}.  Part (2) of the theorem is due to Weiss~\\cite{W2}. With the hypothesis as in Theorem~\\ref{t:arc kernel}(2),\nWeiss~\\cite{W2} proved additional results on the structure of the point stabiliser $G_u$, which we shall recall as needed in the proofs of Lemmas~\\ref{val=4} and \\ref{val>4}.\n\n\\begin{theorem}\\label{t:arc kernel} Let $\\Gamma$ be a connected graph and let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$ be vertex-transitive and locally primitive.  Then there exists a prime $p$ such that for all arcs $uv$:\n\\begin{enumerate}\n\\item $G_{uv}^{[1]}$ is a $p$-group; and \n\\item if in addition $G_{uv}^{[1]} \\ne 1$ then \n$G_{uv}^{\\Gamma(u)}$ is $p$-local.\n\\end{enumerate}\n\\end{theorem}\n\nLet $\\Gamma$ be a connected graph, and let $G\\le\\ensuremath{\\operatorname{Aut}}(\\Gamma)$ be vertex-transitive.\nRecall that a $(G,s)$-arc-transitive graph is called {\\em $(G,s)$-transitive} if it is not $(G,s+1)$-arc-transitive.\nFor small valencies, the explicit structure of a vertex stabiliser is known. For example, in the cubic case we have the following result due to Tutte \\cite{tutte47}, and Djokovi\\v{c} and Miller \\cite{DM2}.\n\n\\begin{theorem}\nLet $\\Gamma$ be a  cubic $(G,s)$-transitive graph. Then one of the following hold:\n\\begin{enumerate}\n \\item $s=1$ and $G_v=C_3$;\n\\item $s=2$ and $G_v=S_3$;\n\\item $s=3$ and $G_v=S_3\\times C_2$;\n\\item $s=4$ and $G_v=S_4$;\n\\item $s=5$ and $G_v=S_4\\times C_2$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{corollary}\\label{cor:cubic}\nLet $\\Gamma$ be a cubic $(G,s)$-transitive graph. Then $\\Gamma$ is $G$-star-transitive if and only if $s\\geq 2$, while $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive if and only if $s\\geq 3$.\n\\end{corollary}\n\n\nIn the 4-regular case, a complete determination of the vertex stabilisers for 4-regular 2-arc transitive graphs was given by Potocnik \\cite{P}, building on earlier work of Weiss \\cite{W4}.\n\n\n\\begin{lemma}\\label{val=4}\nLet $\\Gamma$ be $4$-regular, let $v \\in V(\\G)$, and let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$. Then $\\Gamma$ is $G$-star-transitive if and only if\none of the following statements holds:\n\\begin{enumerate}\n\\item $\\Gamma$ is $(G,2)$-transitive, and $G_v=S_4$;\n\n\\item $\\Gamma$ is $(G,3)$-transitive, and $G_v=S_4\\times S_3$ or $G_v=(A_4\\times C_3).2$ with the element of order 2 inducing a nontrivial automorphism of both $C_3$ and $A_4$;\n\n\\item $\\Gamma$ is $(G,4)$-transitive, and $G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$;\n\n\\item $\\Gamma$ is $(G,7)$-transitive, and $G_v=[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nSuppose first that $G$ satisfies one of the conditions $(1)$--$(4)$. Then $G$ is locally $2$-transitive, so \n$G_v^{\\Gamma(v)}\\cong A_4$ or $S_4$. None of the listed point stabilisers have a quotient group isomorphic to \n$A_4$, so $G_v^{\\Gamma(v)}\\cong S_4$. Consequently, by Lemma~\\ref{l:suff star},  $\\Gamma$ is $G$-star-transitive.\n\nConversely, suppose that $\\Gamma$ is $G$-star-transitive. \nBy Lemma~\\ref{l:locally fully symmetric}, $G_v^{\\Gamma(v)}=S_4\\cong\\ensuremath{\\operatorname{PGL}}(2,3)$. If $G_v^{[1]}=1$, then $G_v\\cong G_v^{\\Gamma(v)}=S_4$ and so the stabiliser $G_{uvw}$ of the 2-arc $uvw$ is isomorphic to $S_2$.  Hence $G_{uvw}$ is not transitive on the set of three 3-arcs beginning with $uvw$ and so $\\Gamma$ is $(G,2)$-transitive.\n\nSuppose that $G_v^{[1]}\\not=1$ and let $\\{ v,w\\} \\in E(\\G)$.\nIf $G_{vw}^{[1]}=1$, then \n\\[1\\not=G_v^{[1]}\\cong G_v^{[1]}\/G_{vw}^{[1]}\\cong (G_v^{[1]})^{\\Gamma(w)}\\lhd G_{vw}^{\\Gamma(w)}\\cong S_3.\\]\nThus, $G_v^{[1]}=C_3$ or $S_3$, and so for $u\\in\\Gamma(v)\\backslash\\{w\\}$ we have that $G_{uvw}$ induces either $C_3$ or $S_3$ on the set of three 3-arcs beginning with $uvw$. Hence $\\Gamma$ is $(G,3)$-arc-transitive.   Moreover, since $G_{vw}^{[1]}=1$, it follows from \\cite{W2} that  $\\Gamma$ is not $(G,4)$-arc-transitive.  Since $S_3$ has no outer automorphisms, if $G_v^{[1]}=S_3$ then we must have $G_v=S_3\\times S_4$. If $G_v^{[1]}=C_3$ then $G_v=C_3 \\times S_4$ or $(C_3\\times A_4).2$ with the element of order 2 inducing a nontrivial automorphism of both $C_3$ and $A_4$. However, the first case does not occur (see for example \\cite[p.1330]{P}). Thus $G_v=S_3\\times S_4$ or $(C_3\\times A_4).2$ and in both cases $\\Gamma$ is $(G,3)$-transitive.\n\nFinally, assume that $G_{vw}^{[1]}\\not=1$.\nThen by \\cite{W2}, $G_{vw}^{[1]}$ is a 3-group,\n\\[G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3),\\ \\mbox{or}\\ [3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3),\\]\nand $\\Gamma$ is $(G,4)$-transitive or $(G,7)$-transitive, respectively.\n\\end{proof}\n\n\\vskip0.1in\n\\noindent{\\bf Remarks:}\n\\begin{enumerate}\n \\item In case (2) of Lemma \\ref{val=4} we have $G_{vw}=S_3\\times S_3$ or $G_{vw}=(C_3\\times C_3).2$ with respectively $G_v^{[1]}=S_3$ or $C_3$.\n \\item The stabiliser $G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$ is a parabolic subgroup of $\\ensuremath{\\operatorname{PGL}}(3,3)$, while\nthe stabiliser $G_v=[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$ is a parabolic subgroup of the exceptional group $G_2(3)$ of Lie type. In both cases $G_v^{\\Gamma(v)}\\cong\\ensuremath{\\operatorname{PGL}}(2,3)\\cong S_4$ and $(G_v^{[1]})^{\\Gamma(w)}\\cong S_3$.\n\\end{enumerate}\n\n\n\n\\vskip0.1in\n\n\\begin{lemma}\\label{val=4-2}\nAssume that $\\Gamma$ is of valency $4$, let $v \\in V(\\G)$, and let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$.\nThen $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive if and only if \none of the following is true.\n\\begin{itemize}\n\\item[(1)] $\\Gamma$ is $(G,3)$-transitive, and $G_v=S_4\\times S_3$;\n\n\\item[(2)] $\\Gamma$ is $(G,4)$-transitive, and $G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$;\n\n\\item[(3)] $\\Gamma$ is $(G,7)$-transitive, and $G_v=[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nBy Corollary \\ref{c:3-arc transitive} and Lemma \\ref{lem:edgeaction}, if $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive  then $\\Gamma$ is $(G,3)$-arc transitive and $G_{vw}$ induces $S_3\\times S_3$ on $(\\Gamma(v)\\cup\\Gamma(w))\\backslash \\{v,w\\}$. This rules out case (1) of Lemma \\ref{val=4} and case (2) where $G_v=(A_4\\times C_3).2$.  By Lemma \\ref{lem:stedge} and  the remarks following Lemma \\ref{val=4}, it follows that the case where $G_v=S_4\\times S_3$ is $G$-st(edge)-transitive. It remains to prove that the $(G,4)$-arc-transitive graphs given in Lemma~\\ref{val=4} $(3)$ and $(4)$ are $G$-st(edge)-transitive.\n\nSuppose that $G_v=3^2{:}\\ensuremath{\\operatorname{GL}}(2,3)$. Since $G_v^{\\Gamma(v)}\\cong G_v\/G_v^{[1]}$ is a transitive subgroup of $S_4$, the normal subgroup structure of $G_v$ implies that $G_v^{[1]}=3^2{:}2$, and $G_{vw}=3^2{:}2.S_3$.  Since $G_v^{[1]} \\neq 1$,  it follows that there exists $u\\in\\Gamma(v)$ such that  $G_v^{[1]}$ acts on  $\\Gamma(u)$ non-trivially; otherwise $G_v^{[1}=G_v^{[2]}$, and it follows from the connectivity of $\\Gamma$ that $G_v^{[1]}$ fixes all vertices, contradicting $G_v^{[1]}\\neq 1$. Since $G_v$ is transitive on $\\Gamma(v)$ we deduce that  $G_v^{[1]}$ acts non-trivially on $\\Gamma(w)$. Hence $1 \\neq (G_v^{[1]})^{\\Gamma(w)} \\lhd G_{vw}^{\\Gamma(w)} = S_3$. Recall also that in this case $G_{vw}^{[1]}$ is a 3-group and so $(G_v^{[1]})^{\\Gamma(w)}$ has even order.  Thus $G_v^{[1]}\/G_{vw}^{[1]}=S_3$ and so  $G_{vw}^{[1]}=C_3$.\nThen we have\n\\[3.2.S_3\\cong G_{vw}\/G_{vw}^{[1]}\\cong G_{vw}^{\\Gamma(v)\\cup\\Gamma(w)}\\le\nG_{vw}^{\\Gamma(v)}\\times G_{vw}^{\\Gamma(w)}\\cong S_3\\times S_3.\\]\nTherefore, $G_{vw}^{\\Gamma(v)\\cup\\Gamma(w)}\\cong S_3\\times S_3$, and so by Lemma \\ref{lem:stedge},\n$\\Gamma$ is $G$-st(edge)-transitive.\n\nNext suppose that  $G_v=[3^5]{:}\\ensuremath{\\operatorname{GL}}(2,3)$.  Then the normal subgroup structure of $G_v$ implies that $G_v^{[1]}=[3^5].2$ and $G_{vw}=[3^5].2.S_3$. Arguing as in the previous case we deduce again that $G_v^{[1]}\/G_{vw}^{[1]}\\cong S_3$, and so $G_{vw}^{[1]}=[3^4]$. A similar argument to the previous case  reveals that  $G_{vw}^{\\Gamma(v)\\cup\\Gamma(w)}\\cong S_3\\times S_3$ and so by Lemma \\ref{lem:stedge}, $\\Gamma$ is st(edge)-transitive.\n\\end{proof}\n\nNext, we consider the case of valency at least 5.\n\n\\begin{lemma}\\label{val>4}\nSuppose that $\\Gamma$ is of valency $r\\geq 5$, let $v \\in V(\\G)$, and let $G \\le \\ensuremath{\\operatorname{Aut}}(\\Gamma)$. Then $\\Gamma$ is $G$-star-transitive if and only if one of the following holds.\n\n\\begin{enumerate}[$(1)$]\n\\item $\\Gamma$ is $(G,2)$-transitive and $G_v=S_r$;\n\n\\item $\\Gamma$ is $(G,3)$-transitive, $r=7$,  $G_v=S_7$ and $G_{\\{u,v\\}}=\\ensuremath{\\operatorname{Aut}}(A_6)$;\n\n\\item $\\Gamma$ is $(G,3)$-transitive, and $G_v=S_r\\times S_{r-1}$ or $(A_r\\times A_{r-1}).2$ with the element of order 2 inducing a nontrivial outer automorphism of both $A_r$ and $A_{r-1}$;\n\n\\item $r=5$, $\\Gamma$ is $(G,4)$-transitive and $G_v=[4^2]{:}{\\rm \\Gamma L}(2,4)$;\n\n\\item $r=5$, $\\Gamma$ is $(G,5)$-transitive and  $G_v=[4^3]{:}{\\rm \\Gamma L}(2,4)$.\n\\end{enumerate}\nMoreover, $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive if and only if\n$\\Gamma$ is $(G,3)$-transitive and $G_v=S_r\\times S_{r-1}$.\n\\end{lemma}\n\\begin{proof}\nSuppose that $\\Gamma$ and $G$ satisfy one of $(1)$  - $(5)$. Then $G_v^{\\Gamma(v)}$ is 2-transitive. Except when both $r=6$ and case (3) holds, the only 2-transitive quotient group of $G_v$ on $r$ points is $S_r$ and so $G$ is locally fully symmetric. Thus by Lemma \\ref{l:suff star}, $\\Gamma$ is $G$-star-transitive. It remains to consider case (3) when $r=6$. Here $G_v$ has $S_6$ and $S_5$ as 2-transitive factor groups of degree 6. If $G_v^{\\Gamma(v)}=S_5$ acting 2-transitively on 6 points it follows that for a 2-arc $uvw$ we have  $G_{uvw}=S_6\\times C_4$, which does not have a transitive action of degree 5, contradicting $G$ being 3-arc transitive. Thus $G_v^{\\Gamma(v)}=S_6$ and so $\\Gamma$ is $G$-star-transitive in this case as well.\n\n\n\n\n\nConversely, suppose that $\\Gamma$ is $G$-star-transitive.\nIf $G_v^{[1]}=1$, then $G_v\\cong G_v^{\\Gamma(v)}\\cong S_r$ and so $\\Gamma$ is $(G,2)$-arc-transitive. For $u\\in\\Gamma(v)$ we have $G_{uv}=S_{r-1}$ which acts faithfully on both $\\Gamma(u)\\backslash\\{v\\}$ and $\\Gamma(v)\\backslash\\{u\\}$. For $r-1\\neq 6$ this implies that the actions are equivalent, that is, the stabiliser of a vertex in one action fixes a vertex in the other. So for a 2-arc $wvu$ we have that $G_{wvu}$ fixes an element of $\\Gamma(u)\\backslash\\{v\\}$ and hence $\\Gamma$ is not $(G,3)$-arc-transitive.  For $r=7$, $S_{r-1}$ has two inequivalent actions of degree 6 that are interchanged by an outer automorphism. The stabiliser of a point in one action is transitive in the other action and so $\\Gamma$ will be $(G,3)$-arc-transitive if and only if $G_{\\{u,v\\}}=\\ensuremath{\\operatorname{Aut}}(S_6)$. Moreover, $\\Gamma$ is not $(G,4)$-arc-transitive in this case as the stabiliser of a 3-arc is $C_5\\rtimes C_4$, which does not have a transitive action of degree 6.\n\nIf $G_v^{[1]}\\not=1$ and $G_{vw}^{[1]}=1$, then we have\n\\[1\\not=G_v^{[1]}\\cong G_v^{[1]}\/G_{vw}^{[1]}\\cong(G_v^{[1]})^{\\Gamma(w)}\\lhd G_{vw}^{\\Gamma(w)}\\cong S_{r-1}.\\]\nThus either $G_v^{[1]}=A_{r-1}$ or $S_{r-1}$, or $r=5$ and $G_v^{[1]}=2^2$. The last case is eliminated by \\cite[Lemma 5.3]{M}. Thus for  $u\\in\\Gamma(v)\\backslash\\{w\\}$ we have that $G_{uvw}$ induces either $A_{r-1}$ or $S_{r-1}$ on the set of $r-1$ 3-arcs beginning with $uvw$. Hence $\\Gamma$ is $(G,3)$-arc-transitive.   Moreover, since $G_{vw}^{[1]}=1$, it follows from \\cite{W2} that  $\\Gamma$ is not $(G,4)$-arc-transitive.  If $G_v^{[1]}=S_{r-1}$, since $S_{r-1}$ has no outer automorphisms for $r\\neq 7$, it follows that either $G_v=S_r\\times S_{r-1}$ or $r=7$ and $G_v=(A_7\\times S_6).2$ with elements of  $G_v\\backslash (A_7\\times S_6)$ inducing an outer automorphism of both $S_6$ and $A_7$.  Suppose that we have the latter case. Then $G_{vw}=(A_6\\times S_6).2$. However, in such a group $G_v^{[1]}$ is a characteristic subgroup of $G_{vw}$ as it is the only normal subgroup isomorphic to $S_6$. Thus $G_v^{[1]}$ is normalised in $G_{\\{v,w\\}}$ and hence normal in $\\langle G_v,G_{\\{v,w\\}}\\rangle=G$. Hence $G_v$ contains a nontrivial normal subgroup of $G$,  contradicting the action on $V\\Gamma$ being faithful. Thus if $G_v^{[1]}=S_{r-1}$ then $G_v=S_r\\times S_{r-1}$.\n\n If $G_v^{[1]}=A_{r-1}$ then $G_v=S_r\\times A_{r-1}$ or $(A_r\\times A_{r-1}).2$.  However, in the first case we can argue as above to show that $G_v^{[1]}\\lhd G$ again yielding a contradiction. (In this case $G_v^{[1]}$ is characteristic in $G_{vw}$ as it is the only normal subgroup isomorphic to $A_{r-1}$ not contained in an $S_{r-1}$.) Similarly in the second case, elements of $G_{vw}\\backslash (A_{r-1}\\times A_{r-1})$ must induce nontrivial outer automorphisms of both normal subgroups isomorphic to $A_{r-1}$ and so $G_v= (A_r\\times A_{r-1}).2$ with $G_v\/A_r\\cong S_{r-1}$ and $G_v\/A_{r-1}\\cong S_r$.\n\nNext, assume that $G_{vw}^{[1]}\\not=1$.\nBy Theorem~\\ref{t:arc kernel}, $G_{vw}^{\\Gamma(v)} \\cong  S_{r-1}$ is $p$-local.\nThus, $r=5$, and $p=2$.\nFurther, by \\cite{W2}, either $\\Gamma$ is $(G,4)$-transitive, and\n$G_v=[4^2]{:}\\ensuremath{\\operatorname{GL}}(2,4)$ or $[4^2]{:}{\\rm \\Gamma L}(2,4)$ (that is, the maximal parabolics in $\\ensuremath{\\operatorname{PGL}}(3,4)$ or $\\operatorname{P\\Gamma L}(3,4)$ respectively),\nor $\\Gamma$ is $(G,5)$-transitive, and \n$G_v=[4^3]{:}\\ensuremath{\\operatorname{GL}}(2,4)$ or $[4^3]{:}{\\rm \\Gamma L}(2,4)$ (that is, the maximal parabolics in $\\ensuremath{\\operatorname{PSp}}(4,4)$ or  $\\operatorname{P\\Gamma Sp}(4,4)$ respectively). Since neither $[4^2]{:}\\ensuremath{\\operatorname{GL}}(2,4)$ nor $[4^3]{:}\\ensuremath{\\operatorname{GL}}(2,4)$ have $S_5$ as a quotient group they cannot occur.\n\nWe also have to establish which graphs in the list are st(edge)-transitive. \nIf $G_v=S_r\\times S_{r-1}$ in case $(3)$ \nthen $G_{vw}=S_{r-1}\\times S_{r-1}$ acts faithfully on $(\\Gamma(v)\\cup\\Gamma(w))\\backslash\\{v,w\\}$ and so the arc-transitivity of $\\Gamma$ together with Lemma \\ref{lem:stedge} implies that $\\Gamma$ is $G$-st(edge)-transitive. In all other graphs in cases $(1)$, $(2)$ and $(3)$, $G_v$ is too small to satisfy the necessary condition in Lemma~\\ref{lem:edgeaction}. In cases (3) and (4), Lemma \\ref{lem:stedge} implies that to be $G$-st(edge)-transitive we must have that $G_v^{[1]}$ has $S_4$ as a quotient group. However, a simple \\textsf{GAP} \\cite{gap} computation shows that this is not the case.\n\\end{proof}\n\n\\vskip0.1in\n\\noindent{\\bf Remark:} \nThe stabiliser $G_v=[4^2]{:}{\\rm \\Gamma L}(2,4)$ is a parabolic subgroup in $\\operatorname{P\\Gamma L}(3,4)$ while the stabiliser $G_v=[4^3]{:}{\\rm \\Gamma L}(2,4)$ is a parabolic subgroup in $\\operatorname{P\\Gamma Sp}(4,4)$. In both cases $G_v^{\\Gamma(v)}\\cong \\operatorname{P\\Gamma L}(2,4)\\cong S_5$.\n\n\\vskip0.1in\n\\noindent{\\bf Remark:} The Hoffman--Singleton graph with automorphism group $\\ensuremath{\\operatorname{PSU}}(3,5).2$ on 50 vertices is an example of a graph in case (2).\n\n\\vskip0.1in\nCombining the lemmas in this section, we obtain a characterisation of graphs which are vertex-transitive,\nstar-transitive and st(edge)-transitive, as stated in Theorem~\\ref{v-star-st(edge)-t}.\n\n\n\\section{The vertex intransitive case}\n\\label{s:vertex intran}\n\nIn this section, we study connected star-transitive and st(edge)-transitive graphs $\\Gamma$ with vertex-intransitive automorphism groups. Such graphs must be bipartite, of bivalency $\\{\\ell,r\\}$ for some $\\ell \\ne r$. \nThe analogue of Theorem~\\ref{t:arc kernel} was proved by  van Bon~\\cite{vB1}.\n\n\\begin{theorem}\n\\label{t:vB}\nLet $\\Gamma$ be a connected graph and $G\\leqslant \\ensuremath{\\operatorname{Aut}}(\\Gamma)$ such that $G_v^{\\Gamma(v)}$ is primitive for all vertices $v$.  Then for an edge $\\{v,w\\}$ either:\n\\begin{itemize}\n\\item[(i)] $G_{vw}^{[1]}$ is a $p$-group; or\n\\item[(ii)] (possibly after interchanging $v$ and $w$), $G_{vw}^{[1]}=G_v^{[2]}$, and $G_w^{[2]}=G_w^{[3]}$ is a $p$-group.\n\\end{itemize}\n\\end{theorem}\n\nAssume that $G\\le\\ensuremath{\\operatorname{Aut}}(\\Gamma)$, and $\\Gamma$ is $G$-star-transitive and $G$-st(edge)-transitive.\nLet $uvwx$ be a $3$-arc of $\\Gamma$, and suppose that $|\\Gamma(v)|=r$ and $|\\Gamma(w)|=\\ell$. We note that $G_{vw}^{[1]}$ acts on both $\\Gamma(u)$ and $\\Gamma(x)$.\n\n\\begin{lemma}\\label{3,5}\nAssume that $G_{vw}^{[1]}$ acts non-trivially on both $\\Gamma(u)$ and $\\Gamma(x)$.\nThen $\\{\\ell,r\\}=\\{3,5\\}$.\n\\end{lemma}\n\\begin{proof}\nSince $G_{vw}^{[1]}$ acts non-trivially on $\\Gamma(u)$ it follows that $G_{vw}^{[1]}\\neq G_v^{[2]}$ and so Theorem~\\ref{t:vB} implies that $G_{vw}^{[1]}$ is a $p$-group. Since \n$G_{vw}^{[1]}\\lhd G_w^{[1]}\\lhd G_{wx}$ and $G_{vw}^{[1]}$ acts non-trivially on $\\Gamma(x)$, we have\n\\[1\\not=(G_{vw}^{[1]})^{\\Gamma(x)}\\lhd (G_w^{[1]})^{\\Gamma(x)}\\lhd G_{wx}^{\\Gamma(x)}.\\]\nThus, $G_{wx}^{\\Gamma(x)}\\cong S_{r-1}$ has a subnormal $p$-subgroup, and similarly,  so does $G_{uv}^{\\Gamma(u)}\\cong S_{\\ell-1}$.\nHence $r,\\ell\\le 5$.\nIf $r=4$, then $G_{vw}^{\\Gamma(v)}\\cong S_3$ and $p=3$.\nAs $\\ell\\not=r$, we have $\\ell=3$ or 5, and thus $G_{vw}^{\\Gamma(w)}=S_2$ or $S_4$,\nwhich do not have subnormal $3$-subgroups. This is a contradiction.\nThus, $r\\not=4$, and similarly, $\\ell\\not=4$. Hence $\\{\\ell,r\\}=\\{3,5\\}$.\n\\end{proof}\n\n\n\n\\begin{example}\n\\label{eg:hermitian}\nLet $\\Gamma$ be the point-line incidence graph of the generalised quadrangle of order $(2,4)$ arising from a nondegenerate Hermitian form on a 4-dimensional vector space over $\\ensuremath{\\operatorname{GF}}(4)$. That is, the points are the totally isotropic 1-spaces, the lines are the totally isotropic 2-spaces, and a 1-space and 2-space are incident if one is contained in the other. Let $G=\\operatorname{P\\Gamma U}(4,2)=\\ensuremath{\\operatorname{PSU}}(4,2).2$, the full automorphism group of $\\Gamma$, and let $(w,v)$ be an incident point-line pair. Then $G_v=2^4{:}S_5$, and $G_w=2.(A_4\\times A_4).2^2$\nwith $G_v\\cap G_w=2^4.S_4$.\nThen $|G_v:G_v\\cap G_w|=5$ and $|G_w:G_v\\cap G_w|=3$, and $\\Gamma$ is locally $(G,4)$-arc-transitive\nof bivalency $\\{3,5\\}$. It follows from Lemma \\ref{l:suff star} that $\\Gamma$ is star-transitive.\nWe have $G_v^{[1]}=2^4$, and $G_{vw}^{[1]}=2^3$. Thus $G_{vw}\/G_{vw}^{[1]}=S_4\\times S_2$, and so by Lemma \\ref{lem:stedge}, $\\Gamma$ is st(edge)-transitive.\n\\end{example}\n\\vskip0.1in\n\n\\begin{lemma}\\label{[2]=1}\nAssume that $G_w^{[1]}\\not=1$ and $G_w^{[2]}=1$.\nThen one of the following holds:\n\\begin{itemize}\n\\item[(i)] $G_{vw}^{[1]}=1$, $G_v=S_r\\times S_{\\ell-1}$ and $G_w=S_{\\ell}\\times S_{r-1}$;\n\n\\item[(ii)] $(A_{r-1})^{\\ell-1}\\leqslant G_{vw}^{[1]}\\leqslant (S_{r-1})^{\\ell-1}$,  \n\\[\\begin{array}{rllll}\n(A_r\\times (A_{r-1})^{\\ell-1}).2.S_{\\ell-1}&\\leqslant &G_v&\\leqslant&S_r\\times (S_{r-1}\\Wr S_{\\ell-1}), \\mbox{and}\\\\\n\n(A_{r-1})^{\\ell}.2.S_\\ell& \\leqslant& G_w &\\leqslant &S_{r-1}\\Wr S_\\ell; \\textrm{ or}\n\\end{array}\\]\n\n\\item[(iii)]\n$|\\Gamma(v)|\\le 5$.\n\n\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nSince $G_w^{[2]}=1$, we have\n\\[\\begin{array}{l}\nG_w\\cong G_w\/G_w^{[2]}\\cong G_w^{\\Gamma_1(w)\\cup\\Gamma_2(w)}\\le \nG_{vw}^{\\Gamma(v)}\\Wr G_w^{\\Gamma(w)}\\cong S_{r-1}\\Wr S_{\\ell}, \\text{ and }\\\\\n\nG_{vw}\\cong G_{vw}\/G_w^{[2]}\\cong G_{vw}^{\\Gamma_1(w)\\cup\\Gamma_2(w)}\\le \n(S_{r-1}\\Wr S_{\\ell-1})\\times S_{r-1}\\\\\n\\end{array}\\]\nLet $\\Gamma(w)=\\{v_1,\\ldots,v_\\ell\\}$ with $v_1=v$. For each $i\\in\\{1,\\ldots, \\ell\\}$, let $B_i$ be the subgroup of $\\ensuremath{\\operatorname{Sym}}(\\Gamma_1(w)\\cup\\Gamma_2(w))$ consisting of all permutations that fix each vertex\nin $\\Gamma(w)$, act trivially on $\\Gamma(v_j)$ for $j\\neq i$ and induce an element of $S_{r-1}$ on $\\Gamma(v_i)\\backslash\\{w\\}$. Then $B_i\\cong S_{r-1}$. Also, let $C$ be a subgroup of $\\ensuremath{\\operatorname{Sym}}(\\Gamma_1(w)\\cup\\Gamma_2(w))$ isomorphic to $S_\\ell$ and such that $C$ acts faithfully on $\\Gamma(w)$. \nIn particular, if $g\\in C$ and $v_i^g=v_j$ then $B_i^g=B_j$.\nThen we can identify $G_w$ with a subgroup of $X:=(B_1\\times B_2\\times\\cdots\\times B_\\ell)\\rtimes C$.\nThus $G_{vw}=G_w\\cap ( B_1\\times ((B_2\\times \\cdots \\times B_\\ell)\\rtimes C_v))$, where $C_v\\cong S_{\\ell-1}$ and $G_w^{[1]}=G_w\\cap (B_1\\times\\cdots\\times B_\\ell)$. \n\nLet $\\rho$ be the projection of $X$ onto $C$ and for each $i=1,2,\\ldots, \\ell$, let $\\pi_i$ be the projection of $B_1\\times\\cdots\\times B_\\ell$ onto $B_i$. Since $G$ is st(edge)-transitive, by Lemma \\ref{lem:edgeaction} we have that $G_w^{\\Gamma(w)}=S_\\ell$ and so $\\rho(G_w)=C$. Moreover, \n$G_{vw}^{\\Gamma(v)\\cup\\Gamma(w)}\\cong S_{r-1}\\times S_{\\ell-1}$ and so $\\rho(G_{vw})=C_v\\cong S_{\\ell-1}$ and $\\pi_1( \\ker(\\rho)\\cap G_{vw})=B_1\\cong S_{r-1}$. Since $\\ker\\rho\\cap G_w=G_w^{[1]}$ it follows that $\\pi_1(G_w^{[1]})\\cong S_{r-1}$. Since $G_w$ acts transitively on the $\\ell$ factors $B_1,\\ldots,B_{\\ell}$ it follows that $\\pi_i(G_w^{[1]})\\cong\\pi_j(G_w^{[1]})\\cong S_{r-1}$ for distinct $i$ and $j$. \n\nSuppose that $r\\geq 6$. Then $A_{r-1}$ is a nonabelian simple group and since $G_w$ acts primitively on the $\\ell$ factors $B_1,\\ldots,B_{\\ell}$ it follows from \\cite[p. 328]{scott} that either $G_{w}^{[1]}\\cong \\pi_i(G_{w}^{[1]})$ for each $i$, or $(A_{r-1})^\\ell\\cong \\ensuremath{\\operatorname{soc}}(B_1)\\times\\cdots\\times\\ensuremath{\\operatorname{soc}}(B_\\ell)\\leqslant G_w^{[1]}$.  Since $G_{vw}^{[1]}\\leqslant G_w^{[1]}$ and $\\pi_1(G_{vw}^{[1]})=1$ these two cases correspond to $G_{vw}^{[1]}=1$ and $G_{vw}^{[1]}\\neq 1$ respectively.\n\n\nSuppose first that $G_{vw}^{[1]}=1$. Then $G_{vw}\\cong S_{r-1}\\times S_{\\ell-1}$. As $\\ker(\\rho)\\cap G_w\\leqslant G_{vw}$ we have $\\ker(\\rho)\\cap G_w\\cong S_{r-1}$. Moreover, as $\\rho(G_w)=S_\\ell$ we have $G_w\/(\\ker(\\rho)\\cap G_w)\\cong S_{\\ell}$ with $\\rho(G_{vw})=S_{\\ell -1}$ and $G_{vw}\\cap \\ker\\rho\\cong S_{r-1}$. Thus either $G_w\\cong S_{r-1}\\times S_\\ell$, or $G_w\\cong (S_{r-1}\\times A_{\\ell}).2$ with each element of $G_w$ not in $S_{r-1}\\times A_{\\ell}$ inducing a nontrivial automorphism of both $S_{r-1}$ and $A_\\ell$. Since $G_w$ contains  $G_{vw}\\cong S_{r-1}\\times S_{\\ell-1}$, which has an element of $S_{\\ell-1}$ not in $A_\\ell$ that centralises $S_{r-1}$, the second case is not possible. Thus $G_w\\cong S_{r-1}\\times S_\\ell$.  Similarly, $G_v^{[1]}=S_{\\ell -1}$ and $G_v^{\\Gamma(v)}\\cong S_r$ and arguing as in the $G_w$ case we deduce that $G_v=S_r\\times S_{\\ell-1}$. Hence we are in case (i).\n\n\n\n\nAssume now that $G_{vw}^{[1]}\\neq 1$. Then, as we have already seen, $\\ensuremath{\\operatorname{soc}}(B_1)\\times\\cdots\\times\\ensuremath{\\operatorname{soc}}(B_\\ell)\\leqslant G_w^{[1]}$.  Since $\\pi_1(G_w^{[1]})=S_{r-1}$ and $\\pi_i(G_w^{[1]})\\cong \\pi_j(G_w^{[1]})$ for all distinct $i$ and $j$, it follows that $G_w^{[1]}$ contains a subgroup $N$ isomorphic to $(A_{r-1})^\\ell.2$ such that $\\pi_i(N)\\cong S_{r-1}$ for all $i$. Thus\n$$(A_{r-1}^{\\ell}).2.S_\\ell \\leqslant G_w \\leqslant S_{r-1}\\Wr S_\\ell$$\nMoreover, for the arc stabiliser we have\n$$(A_{r-1})^{\\ell}.2.S_{\\ell-1} \\leqslant G_{vw}\\leqslant S_{r-1}\\times (S_{r-1}\\Wr S_{\\ell-1})$$\nwith $(A_{r-1})^{\\ell-1}\\leqslant G_{vw}^{[1]}\\leqslant (S_{r-1})^{\\ell-1}$.\n\nConsider the group $\\overline{G}_w=G_w\/(\\ensuremath{\\operatorname{soc}}(B_1)\\times\\cdots\\times\\ensuremath{\\operatorname{soc}}(B_\\ell))$. Then $2.S_{\\ell}\\leqslant \\overline{G}_w\\leqslant 2^\\ell.S_\\ell$. Hence $\\overline{G}_w=2^m.S_\\ell$ where the $2^m$ is a submodule of the permutation module for $S_\\ell$. This implies that $G_{w}=(A_{r-1})^\\ell.2^m.S_\\ell$. Hence $G_{vw}=(A_{r-1})^\\ell.2^m.S_{\\ell-1}$ and $G_v^{[1]}=(A_{r-1})^{\\ell-1}.2^{m-1}.S_{\\ell-1}$, as $G_{vw}\/G_{v}^{[1]}\\cong G_{vw}^{\\Gamma(v)}\\cong S_{r-1}$. Observe that $G_v^{[1]}$ has a unique minimal normal subgroup $N$ and $N\\cong (A_{r-1})^{\\ell-1}$. Thus $N\\lhd G_v$. Since $G_v\/G_v^{[1]}\\cong S_r$ and $G_v^{[1]}$ already induces $S_{\\ell-1}$ on the $\\ell-1$ distinct simple factors of $N$, it follows that $G_v$ contains a normal subgroup isomorphic to  $A_r\\times (A_{r-1})^{\\ell-1}$. We then deduce that $G_v=(A_r\\times (A_{r-1})^{\\ell-1}).2^m.S_\\ell$ and in particular part (ii) holds.\n\\end{proof}\n\nBy Lemma \\ref{[2]=1}  we may thus assume that $G_v^{[2]}\\neq 1$ and $G_w^{[2]}\\neq 1$. Moreover, by Lemma \\ref{3,5} we may assume that $G_{vw}^{[1]}$ acts trivially on $\\Gamma(u)$ for some neighbour $u$ of $v$. Since $G_{vw}$ is transitive on $\\Gamma(v)\\setminus\\{w\\}$ and normalises $G_{vw}^{[1]}$,\nwe conclude that $G_{vw}^{[1]}$ fixes all vertices in $\\Gamma_2(v)$. In particular, $G_{vw}^{[1]}=G_v^{[2]}$. \n\n\n\\begin{lemma}\\label{[2]not=1}\nAssume that $G_v^{[2]}\\neq 1\\neq G_w^{[2]}$ and $G_{vw}^{[1]}=G_v^{[2]}$ fixes all vertices in $\\Gamma_2(v)$. Then $r\\leq 5$ and either $G_w^{[2]}$ or $G_v^{[2]}$ is a $p$-group with $p=2$ or $3$.\n\\end{lemma}\n\\begin{proof}\nBy Theorem~\\ref{t:vB}, either $G_v^{[2]}$ is a $p$-group, or  $G_w^{[2]}=G_w^{[3]}$ is a $p$-group.\n\nSuppose first that $G_v^{[2]}$ is a $p$-group. Then \n$$1\\neq G_v^{[2]}\\lhd G_w^{[1]}\\lhd G_{wx}$$\nSuppose that  $G_v^{[2]}$ acts trivially on $\\Gamma(x)$.\nThen as $G_v^{[2]}\\lhd G_{vw}$ and $G_{vw}$ acts transitively on $\\Gamma(w)$ \nit follows that $G_v^{[2]}$ acts trivially on $\\Gamma_2(w)$.  \nThus, $G_v^{[2]}\\le G_w^{[2]}\\le G_{vw}^{[1]}=G_v^{[2]}$, and so $G_v^{[2]}=G_w^{[2]}$.\nSince $G$ is edge-transitive on $\\Gamma$, we conclude that $G_v^{[2]}$ fixes all vertices of $\\Gamma$, which is a contradiction.\n\n Hence $G_v^{[2]}$ acts nontrivially on $\\Gamma(x)$ and so $G_{wx}^{\\Gamma(x)}\\cong S_{r-1}$ has a nontrivial normal $p$-subgroup. Thus $r\\leq 5$ and $p=2$ or $3$.\n\n\nNext suppose that $G_w^{[2]}=G_w^{[3]}$ is a $p$-group.\nPick $y\\in\\Gamma(u)\\setminus\\{v\\}$.\nThen \n\\[1\\not=G_w^{[2]}=G_w^{[3]}\\lhd G_{uv}^{[1]}\\lhd G_u^{[1]}\\lhd G_{uy}.\\]\nSuppose that $G_w^{[3]}$ acts trivially on $\\Gamma(y)$.\nSince $G_{wvu}$ normalises $G_w^{[3]}$ and is transitive on $\\Gamma(u)\\backslash\\{v\\}$,\nit follows that $G_w^{[3]}$ is trivial on $\\Gamma(x)$ for all $x\\in \\Gamma(u)$,\nand hence on $\\Gamma_2(u)$.\nFurther, $G_{wv}$ normalises $G_w^{[3]}$ and is transitive on $\\Gamma(v)\\setminus\\{w\\}$,\nand it follows that $G_w^{[3]}$ acts trivially on $\\Gamma_2(z)$ for all $z\\in\\Gamma(v)$.\nThus, $G_w^{[3]}$ fixes all vertices in $\\Gamma_3(v)$, and $G_w^{[3]}\\le G_v^{[3]}$.\nSince $G$ is edge-transitive on $\\Gamma$, we conclude that $G_w^{[3]}=G_w^{[4]}$.\nHence we have \n\\[1\\not=(G_w^{[3]})^{\\Gamma(y)}\\lhd (G_{uv}^{[1]})^{\\Gamma(y)}\\lhd\n(G_u^{[1]})^{\\Gamma(y)}\\lhd G_{uy}^{\\Gamma(y)}\\cong S_{r-1}.\\]\nThus, $S_{r-1}$ has a subnormal $p$-subgroup and so $r\\le 5$, and $p=2$ or 3.\n\\end{proof}\n\nCombining the lemmas of this section, we obtain the characterisation of vertex-intransitive,\nstar-transitive and st(edge)-transitive graphs, as stated in Theorem~\\ref{vtx-intrans}.\n\n\n\\section{Examples}\\label{s:examples}\n\nLazarovich \\cite{L} provides a list of examples of graphs $L$ which are star-transitive and st(edge)-transitive, as well as a discussion of earlier work on uniqueness of $(k,L)$-complexes.  In this section we use our results above to expand on the examples and discussion in \\cite{L}, and to give new infinite families of examples in both the vertex-transitive  and vertex-intransitive cases.\n\nRecall that we denote by $\\mathcal{X}(k,L)$ the collection of all simply-connected $(k,L)$-complexes and that we assume $L$ is finite and connected.\n\n\n\n\n\n\n\n\\subsection{Special cases}\n\n\\subsubsection{Cycles}\n\nThe only finite, connected, $2$-regular graph is the cycle on $n$ vertices $C_n$.  For $n \\geq 3$, as observed in \\cite{L}, the graph $C_n$ is star-transitive and st(edge)-transitive.  If $(k,n) \\in \\{ (3,6), (4,4), (6,3)\\}$ and the polygons are metrised as regular Euclidean $k$-gons, then the unique simply-connected $(k,C_n)$-complex is the tessellation of the Euclidean plane by regular Euclidean $k$-gons.  If $k \\geq k'$ and $n > n'$, or $k > k'$ and $n \\geq n'$, where $(k',n') \\in \\{ (3,6), (4,4), (6,3)\\}$, then the unique simply-connected $(k,C_n)$-complex is combinatorially isomorphic to the tessellation of the hyperbolic plane by regular hyperbolic $k$-gons with vertex angles $\\frac{2\\pi}{n}$. \n\n\\subsubsection{Cubic graphs}\n\nLet $k \\geq 3$ be an integer and let $L$ be a finite, connected, cubic graph, such that the pair $(k,L)$ satisfies the Gromov Link Condition.  In \\cite{Sw}, \\'Swi{\\polhk{a}}tkowski\\ proved that if $k \\geq 4$ and $L$ is $3$-arc transitive, then there exists a unique simply-connected $(k,L)$-complex, while if $k = 3$ and $L$ is $3$-arc transitive, or $k \\geq 3$ and $L$ is not $3$-arc transitive, $|\\mathcal{X}(k,L)| > 1$.  (In fact, \\'Swi{\\polhk{a}}tkowski\\ proved that in these latter cases $\\mathcal{X}(k,L)$ is uncountable; compare \\cite[Theorem B]{L}.)\n\nIt follows from results of Tutte \\cite{tutte47, tutte59} that, as observed in \\cite{L}, a connected cubic graph which is $3$-arc transitive is both star-transitive and st(edge)-transitive.  Conversely, if a connected cubic graph $L$ is star-transitive and st(edge)-transitive then $L$ is $3$-arc transitive, by Corollary \\ref{c:3-arc transitive} above.  Together with the main results of \\cite{L}, this recovers the results of \\'Swi{\\polhk{a}}tkowski\\  from \\cite{Sw} when $k \\geq 4$ is even.\n\nThe collection of $3$-arc transitive cubic graphs is too large to classify.  For examples of such graphs, see for instance \\cite{GR}.\n\n\n\n\\subsubsection{Complete bipartite graphs}\n\nIt is observed in \\cite{L} that the complete bipartite graph $K_{m,n}$ is star-transitive and st(edge)-transitive.   The unique simply-connected $(4,K_{m,n})$-complex is the product of an $m$-valent and an $n$-valent tree \\cite{Wise96}.  If $k > 4$, $m,n \\geq 2$ and either $k$ is even or $m = n$ the unique simply-connected $(k,K_{m,n})$-complex is isomorphic to Bourdon's building, which is a $2$-dimensional hyperbolic building (see \\cite{B1}).  If $k$ is odd and $m \\ne n$, there is no $(k,L)$-complex.\n\nBy Lemma \\ref{lem:valency one} above, the only finite, connected graphs with a vertex of valency one that are star-transitive and st(edge)-transitive are the complete bipartite graphs $K_{1,n}$.  If $n \\geq 2$ and $k \\geq 4$ is even then the unique simply-connected $(k,K_{1,n})$-complex $X$ is a ``tree\" of $k$-gons, with alternating edges of each $k$-gon contained in either a unique $k$-gon or $n$ distinct $k$-gons.\n\n\n\\subsubsection{Odd graphs}\\label{s:odd}\n\nFor $n \\geq 2$ the \\emph{Odd graph} $O_{n+1}$ is defined to have vertex set the $n$-element subsets of $\\{ 1, 2, \\ldots, 2n+1\\}$, with two vertices being adjacent if the corresponding $n$-sets are disjoint.  Thus the graph is $(n+1)$-valent.  The Petersen graph is the case $n = 2$.  \n\nAs noted in \\cite{L}, the Odd graphs are star-transitive and st(edge)-transitive.  Their vertex stabilisers are $S_{n} \\times S_{n+1}$, and their edge-stabilisers are $S_n \\Wr S_2$.  The girth of the Petersen graph $O_3$ is $5$ and for all $n \\geq 3$ the girth of $O_{n+1}$ is $6$. \n\nIt was proved by Praeger  \\cite[Theorem 4]{CEP} that if $\\Gamma$ is a graph of valency $r$ with  $A_r\\leqslant G_v^{\\Gamma(v)}$ and  $G$ primitive on $V\\Gamma$ then either $|\\Gamma_3(v)|=r(r-1)^2$, that is girth at least 7, or $\\Gamma$ is the Odd graph on $2r-1$ points.  Thus by Lemma \\ref{l:locally fully symmetric}, if $\\Gamma$ is $G$-star-transitive with $G$ primitive on vertices then either $\\Gamma$ has girth at least 7 or is an Odd graph.  Vertex-transitive graphs of girth 5 and with $G_v^{\\Gamma(v)}=S_r$ were investigated by Ivanov, who showed that  that such graphs are either the Petersen graph, the Hoffman--Singleton graph, the double cover of the Clebsch graph or each 2-arc in $\\Gamma$ is contained in a unique cycle of length 5 \\cite[Lemma 5.6]{I}. Combining \\cite[Lemma 5.4]{I} and Theorem \\ref{v-star-st(edge)-t} we deduce that the only vertex-transitive graph of girth 5 that is both star-transitive and st(edge)-transitive is the Petersen graph.\n\nWe note that any finite cover of an Odd graph for which all automorphisms lift will also be star-transitive and st(edge)-transitive.  Thus for every valency $n+1 \\geq 3$ we obtain an infinite family of $(n+1)$-regular graphs which are star-transitive and st(edge)-transitive.  \n\n\n\n\n\n\n\\subsubsection{Spherical buildings}\n\nLet $\\mathfrak{L}$ be a simple Lie algebra of rank $2$ over $\\mathbb{C}$.  So\n$\\mathfrak{L}$ is of type $A_2$, $B_2 = C_2$ or $G_2$.  We\ndenote by $\\mathfrak{L}(q)$ the (untwisted) Chevalley group of type\n$\\mathfrak{L}$ over the field $\\ensuremath{\\operatorname{GF}}(q)$.  Denote by $L$ the spherical\nbuilding associated to the group $\\mathfrak{L}(q)$.  Assume that $k\n\\geq 5$.  Then by results\nof Haglund~\\cite{H}, in the following cases,\nthere is a unique simply-connected $(k,L)$-complex:\n\\begin{enumerate}\n\\item $\\mathfrak{L}$ is of type $A_2$ and $q \\in \\{2,3\\}$;\n\\item $\\mathfrak{L}$ is of type $B_2$ and $q = 2$;\n\\item $\\mathfrak{L}$ is of type $G_2$ and $q = 3$.\n\\end{enumerate}\nIn each of these cases, the $(k,L)$-complex obtained may be metrised\nas a $2$-dimensional hyperbolic building.\n(For larger values of $q$, Haglund also established the uniqueness of\n$(k,L)$-complexes which satisfy some additional local conditions,\ncalled\nlocal reflexivity and constant holonomy.)\n\n\n\n\n\n\nThe local actions for the rank 2 buildings arising from finite groups of Lie type are given in  the first five columns of Table \\ref{tab:localbuild}.  The full automorphism group $G$ in each case can be found in \\cite[Chapter 4]{vM} and the local actions of $\\ensuremath{\\operatorname{soc}}(G)$ are given in \\cite[Theorem 8.4.1 and Table 8.1]{vM}. The local actions for $G$ can then be deduced. By determining the values of $q$ for which the local action induces the full symmetric group we can deduce the information in the locally fully symmetric column. Moreover, since the rank 2 buildings for $\\ensuremath{\\operatorname{PSp}}(4,q)$ are vertex-transitive if and only if $q$ is even while the rank 2 buildings for $G_2(q)$ are vertex-transitive if and only if $q$ is a power of 3, the star-transitive column follows from Lemma \\ref{l:suff star}. The case where $G=\\operatorname{P\\Gamma U}(4,2)$ was analysed in Example \\ref{eg:hermitian}. The fact that the buildings for $\\ensuremath{\\operatorname{PSL}}(3,2)$ and $\\ensuremath{\\operatorname{PSp}}(4,2)$ are st(edge)-transitive follow from Corollary \\ref{cor:cubic} and the fact that they are 4-arc transitive and 5-arc transitive respectively. The fact that the building for $G_2(3)$ is st(edge)-transitive follows from part (3) of Lemma \\ref{val=4-2} and the preceding remark.\n\n Finally, the fact that the buildings for $\\ensuremath{\\operatorname{PSL}}(3,4)$ and $\\ensuremath{\\operatorname{PSp}}(4,4)$ are not st(edge)-transitive follows from the fact that for $g\\in G_v$ to induce an odd permutation on  $\\Gamma(v)$ it must induce a nontrivial field automorphism of the simple group associated with  $G$.  Thus  it is not possible for an element of $G_{vw}$ to induce an odd permutation of $\\Gamma(v)$ and an even permutation of $\\Gamma(w)$, and so the necessary condition of Lemma \\ref{lem:stedge} does not hold. \n\n\n\n\\begin{center}\n \\begin{table}\n  \\begin{tabular}{l|llllp{1.9cm}ll}\n$G$             & $|\\Gamma(v)|$ & $G_v^{\\Gamma(v)}$ & $|\\Gamma(w)|$ &$G_w^{\\Gamma(w)}$ & locally fully symmetric & star-transitive& st(edge)-transitive\\\\ \n\\hline \n$\\operatorname{P\\Gamma L}(3,q)$ & $q+1$ & $\\operatorname{P\\Gamma L}(2,q)$ & $q+1$ &$\\operatorname{P\\Gamma L}(2,q)$ & $q=2,3,4$ &  $q=2,3,4$ &$q=2,3$ \\\\\n$\\operatorname{P\\Gamma Sp}(4,q)$ & $q+1$ & $\\operatorname{P\\Gamma L}(2,q)$ & $q+1$ &  $\\operatorname{P\\Gamma L}(2,q)$ & $q=2,3,4$ &  $q=2,4$ & $q=2$\\\\\n$\\operatorname{P\\Gamma U}(4,q)$ & $q^2+1$ &  $\\operatorname{P\\Gamma L}(2,q^2)$ & $q+1$ & $\\operatorname{P\\Gamma L}(2,q)$ & $q=2$& $q=2$& $q=2$\\\\\n$\\operatorname{P\\Gamma U}(5,q)$ & $q^2+1$ &  $\\operatorname{P\\Gamma L}(2,q^2)$ & $q^3+1$& $\\operatorname{P\\Gamma U}(3,q)$ & never\\\\\n$\\ensuremath{\\operatorname{Aut}}(G_2(q))$        &  $q+1$ &  $\\operatorname{P\\Gamma L}(2,q)$  & $q+1$  & $\\operatorname{P\\Gamma L}(2,q)$  & $q=2,3,4$ & $q=3$ & $q=3$   \\\\\n$\\ensuremath{\\operatorname{Aut}}({}^3D_4(q))$    & $q+1$ & $\\operatorname{P\\Gamma L}(2,q)$ & $q^3+1$ & $\\operatorname{P\\Gamma L}(2,q^3)$ & never \\\\\n$\\ensuremath{\\operatorname{Aut}}({}^2F_4(q))$    & $q+1$ & $\\operatorname{P\\Gamma L}(2,q)$ & $q^2+1$ & $\\ensuremath{\\operatorname{Aut}}(Sz(q))$ & never \\\\ \\\\\n\n\\end{tabular}\n\n\\caption{Local actions of rank 2 buildings}\n\\label{tab:localbuild}\n \\end{table}\n\n\\end{center}\n\n\n\n\\subsection{New vertex-transitive examples}\nOne way to frame the search for examples is in the context of amalgams. Let $G$ be a group with subgroup $H$ and element $g\\in G$ that does not normalise $H$ such that $g^2\\in H$ and $\\langle H,g\\rangle=G$. Following Sabidussi \\cite{Sa}, we can construct a connected graph $\\mathrm{Cos}(G,H,g)$ with vertices the right cosets of $H$ in $G$ and $Hx$ being adjacent to $Hy$ if and only if $xy^{-1}\\in HgH$. The group $G$ acts by right multiplication as an arc-transitive group of automorphisms of $\\mathrm{Cos}(G,H,g)$ with the stabiliser in $G$ of the vertex corresponding to $H$ being $H$. Conversely, if $\\Gamma$ is a connected graph with an arc-transitive group $G$ of automorphisms then for an edge $\\{v,w\\}$ and $g\\in G_{\\{v,w\\}}\\backslash G_{vw}$ we have that $\\Gamma$ is isomorphic to $\\mathrm{Cos}(G,G_v,g)$. Moreover, note that $G=\\langle G_v,G_{\\{v,w\\}}\\rangle$. Thus by knowing the possible $G_v$ and $G_{\\{v,w\\}}$ for a vertex-transitive, star-transitive and st(edge)-transitive graph, finding examples becomes a search for completions of the amalgam $(G_v,G_{\\{v,w\\}},G_{vw})$.  \n\nTaking $(G_v,G_{\\{v,w\\}},G_{vw})=(S_r\\times S_{r-1},S_{r-1}\\Wr S_2,S_{r-1}\\times S_2)$ as in case (1) of Theorem \\ref{v-star-st(edge)-t}, the Odd graphs arise when we take $G=S_{2r-1}$ as a completion while the complete bipartite graphs arise when we take $G=S_r\\Wr S_2$ as a completion. \n\nAnother example can be constructed with $G=S_{(r-1)^2}$. Let $n=(r-1)^2$ and note that $n=\\binom{r}{2}+\\binom{r-1}{2}$. Thus letting $\\Omega=\\{1,\\ldots,n\\}$ we can identify $\\Omega$ with the disjoint union of the set $\\Omega_1$ of $2$-subsets of a set $A$ of size $r$ and the set $\\Omega_2$ of 2-subsets of a set $B$ of size $r-1$. Then let $H$ be the subgroup of $G$ isomorphic to $S_r\\times S_{r-1}$ that has $\\Omega_1$ and $\\Omega_2$ as its orbits on $\\Omega$. Choose $\\overline{B}$ to be a subset of $A$ of size $r-1$ and let $\\overline{\\Omega}_2$ be the subset of $\\Omega_1$ consisting of all 2-subsets of $\\overline{B}$. Note that $H_{\\overline{\\Omega}_2}\\cong S_{r-1}\\times S_{r-1}$ and choosing $g\\in G$ to be an element of order 2 that interchanges $\\overline{\\Omega}_2$ and $\\Omega_2$, we have that $\\langle H_{\\overline{\\Omega}_2},g\\rangle \\cong S_{r-1}\\Wr S_2$. Let  $\\Gamma=\\mathrm{Cos}(G,H,g)$ and let $v$ be the vertex corresponding to the coset $H$ and $w$ the vertex corresponding to the coset $Hg$. Then $G_v=H\\cong S_r\\times S_{r-1}$, $G_{vw}=H_{\\overline{\\Omega}_2}\\cong S_{r-1}\\times S_{r-1}$  and $G_{\\{v,w\\}}=\\langle H_{\\overline{\\Omega}_2},g\\rangle \\cong S_{r-1}\\Wr S_2$. Thus $G_v^{\\Gamma(v)}\\cong S_r$ and since $G_{\\{v,w\\}}$ acts faithfully on $\\Gamma(v)\\cup\\Gamma(w)$ it follows from Lemmas \\ref{l:suff star} and \\ref{lem:stedge} that $\\Gamma$ is \nstar-transitive and st(edge)-transitive. It remains to show that $\\Gamma$ is connected, that is, we need to show that $\\langle H,g\\rangle=G$. Let $X=\\langle H,g\\rangle$. Then $X$ is transitive on $\\Omega$ and since $H$ is primitive on each of its orbits it follows that $X$ is primitive on $\\Omega$. A transposition on $B$ induces $r-3$ transpositions on $\\Omega_2$ and so $X$ contains an element $\\sigma$ that moves only $2(r-3)$ elements of $\\Omega$. Since $2(r-3) < 2(\\sqrt{n}-1)$, \\cite[Corollary 3]{LS} implies that $X=A_n$ or $S_n$. If $r$ is even then $\\sigma$ is an odd permutation and so $X=S_n=G$. If $r$ is odd then a transposition of $B$ induces $r-2$ transpositions of $\\Omega$, and so $H$ also contains an odd permutation in this case.  Thus $X=S_n=G$ for all $r$. We conclude that $\\Gamma$ is connected.\n\n\n\n\n\\subsection{New vertex-intransitive examples}\n\nWe have already seen that the complete bipartite graphs $K_{n,m}$ with $n\\neq m$ are examples in the vertex-intransitive case, as is the generalised quadrangle associated with $\\operatorname{P\\Gamma U}(4,2)$.\n\nFor all natural numbers $m > n \\geq 3$, we will construct examples of connected star-transitive and st(edge)-transitive which are $(m,n)$-biregular. (The restriction to $n \\geq 3$ is justified by the results of Section \\ref{s:small} above on graphs of minimal valency $2$.)\n\n\n\nAs in the vertex-transitive case, the search for examples can be framed in terms of amalgams. Given a group $G$ and subgroups $L$ and $R$ such that $G=\\langle L,R\\rangle$ we can construct the bipartite graph $\\mathrm{Cos}(G,L,R)$ with vertices the right cosets of $L$ in $G$ and the right cosets of $R$ in $G$ such that $Lx$ is adjacent to $Ry$ if and only if $Lx\\cap Ry\\neq \\varnothing$. Then $G$ acts by right multiplication as an edge-transitive group of automorphisms of $\\mathrm{Cos}(G,L,R)$. Conversely, if $\\Gamma$ is a connected graph with a group $G$ of automorphisms that is edge-transitive but not vertex-transitive then $\\Gamma$ is isomorphic to $\\mathrm{Cos}(G,G_v,G_w)$ for some edge $\\{v,w\\}$. Also $G=\\langle G_v,G_w\\rangle$. Thus examples of vertex-intransitive, star-transitive and st(edge)-transitive graphs can be found by finding completions of the amalgam $(G_v,G_w,G_v\\cap G_w)$.\n\n\n\n\n\\subsubsection{Construction from Johnson graphs}\\label{s:johnson}\n\nLet $\\Gamma=\\Gamma_{m,n}$ be the bipartite graph whose vertices are the $m$-subsets and $(m-1)$-subsets of an $n$-set, with two vertices being adjacent if one is contained in the other. Then $G=S_n\\leqslant \\ensuremath{\\operatorname{Aut}}(\\Gamma_{m,n})$. When $n=2m+1$, $\\Gamma$ is called the \\emph{doubled Odd graph}.  Let $v$ be an $m$-subset. Then $\\Gamma(v)$ is the set of $(m-1)$-subsets contained in $v$ and $G_v=S_m\\times S_{n-m}$ with $G_v^{\\Gamma(v)}=S_m$. Given $w\\in\\Gamma(v)$ we have $G_w=S_{m-1}\\times S_{n-m+1}$ and $\\Gamma(w)$ is the set of $m$-subsets containing $w$. Thus $G_w^{\\Gamma(w)}=S_{n-m+1}$ and $G_{vw}=S_{m-1}\\times S_{n-m}$. Hence by Lemma \\ref{l:suff star}, $\\Gamma$ is $G$-star-transitive when $m\\neq n-m+1$. Moreover, $G_{vw}$ acts faithfully on $\\Gamma(v)\\cup \\Gamma(w)$ so Lemma \\ref{lem:edgeaction} implies that $\\Gamma$ is also $G$-st(edge)-transitive when $m\\neq n-m+1$.\n\nThe \\emph{Johnson graph} $J(n,m)$ is the graph with vertex set the set of $m$-subsets of an $n$-set such that two $m$-subsets are adjacent if and only if their intersection has size $m-1$. Note that an $(m-1)$-subset defines a maximal clique of $J(n,m)$, namely the set of all $m$-sets containing the given $(m-1)$-set. Let $\\mathcal{C}$ be the set of all such maximal cliques and define the graph whose vertices are the vertices of $J(n,m)$ and the maximal cliques in $\\mathcal{C}$, with adjacency being the natural inclusion. Then the new graph is isomorphic to $\\Gamma_{m,n}$.\n\n\n\n\\subsubsection{Construction from Hamming graphs}\\label{s:hamming}\n\nAnother new family of examples are as follows.  This is from Example 4.3 of Giudici--Li--Praeger \\cite{GLP1}.  Let $H(k,n)$ be the Hamming graph whose vertex set is the set of ordered $k$-tuples (possibly with repeats) from a set $\\Omega$ of size $n$, with two vertices being adjacent if and only if they differ in exactly one coordinate. Let $\\Gamma$ be the bipartite graph with vertex set $\\Delta_1\\cup\\Delta_2$, where $\\Delta_1$ is the set of vertices of $H(m,n)$ and $\\Delta_2$ is the set of maximal cliques of $H(m,n)$. Adjacency is given by inclusion. The group $G=S_n\\Wr S_k$ is a group of automorphisms of both $H(m,n)$ and $\\Gamma$, and acts transitively on the edges of $\\Gamma$ with orbits $\\Delta_1$ and $\\Delta_2$ on vertices.\n\nLet $\\omega\\in\\Omega$ and $w=(\\omega,\\ldots,\\omega)\\in\\Delta_1$. Then $G_w=S_{n-1}\\Wr S_k$ and the maximal cliques of $H(m,n)$ containing $w$ are $$\\{(\\alpha,\\omega,\\ldots,\\omega)\\mid \\alpha\\in \\Omega\\}, \\{(\\omega,\\alpha,\\omega,\\ldots,\\omega)\\mid \\alpha\\in \\Omega\\},\\,\\, \\ldots, \\,\\, \\{(\\omega,\\ldots,\\omega,\\alpha)\\mid \\alpha\\in\\Omega\\}$$ Hence $G_w^{\\Gamma(w)}=S_k$. Moreover, letting $v=\\{(\\alpha,\\omega,\\ldots,\\omega)\\mid \\alpha\\in \\Omega\\}$ we have that $G_v= S_n\\times (S_n\\Wr S_{k-1})$ and $G_v^{\\Gamma(v)}=S_n$. Thus by Lemma \\ref{l:suff star}, $\\Gamma$ is star-transitive when $k\\neq n$. Now $G_{vw}=S_{n-1}\\times (S_{n-1}\\Wr S_{k-1})$ and this induces $S_{k-1}$ on $\\Gamma(w)\\backslash\\{v\\}$ and independently induces $S_{n-1}$ on $\\Gamma(v)\\backslash\\{w\\}$. Hence by Lemma \\ref{lem:edgeaction}, $\\Gamma$ is st(edge)-transitive when $k\\neq n$. This provides an example for case (3) of Theorem \\ref{vtx-intrans} with $G_v$ and $G_w$ being as large as possible.\n\nLet $\\sigma\\in S_n$ such that $\\omega^\\sigma=\\omega$ and $S_n=\\langle A_n,\\sigma\\rangle$. Then for $H=\\langle A_n^k,(\\sigma,\\ldots,\\sigma)\\rangle\\rtimes S_k\\cong (A_n^k).2.S_k$,  we have \n$$H_w=\\langle (A_{n-1})^k,(\\sigma,\\ldots,\\sigma)\\rangle \\rtimes S_k\\cong (A_{n-1})^k.2.S_k$$ \n$$H_v=\\langle A_n\\times (A_{n-1})^{k-1},(\\sigma,\\ldots,\\sigma)\\rangle\\rtimes S_{k-1}\\cong (A_n\\times (A_{n-1})^{k-1}).2.S_{k-1}$$ and \n$$H_{vw}= \\langle A_{n-1}^k,(\\sigma,\\ldots,\\sigma)\\rangle S_{k-1}$$\n Moreover, we still have that $H_w^{\\Gamma(w)}\\cong S_{k}$, $H_v^{\\Gamma(v)}\\cong S_n$ and $H_{vw}$ induces $S_{k-1}$ on $\\Gamma(w)\\backslash\\{v\\}$ and independently induces $S_{n-1}$ on $\\Gamma(v)\\backslash\\{w\\}$. Thus $\\Gamma$ is also $H$-star-transitive and $H$-st(edge)-transitive. This gives an example for case (3) of Theorem \\ref{vtx-intrans} with the vertex stabilisers being as small as possible.\n\n\\subsubsection{}\n\nThis example is a specialisation of \\cite[Example 4.6]{GLP2}. Let $n$ be a positive integer coprime to 3, let $V=\\ensuremath{\\operatorname{GF}}(3)^n$ and $W$ be the subspace of $V$ of all vectors $(v_1,\\ldots,v_n)$ such that $\\sum v_i=0$.  Let $G_0=S_n\\times Z$, where $Z$ is the group of scalar transformations and $V$ is the natural permutation module for $S_n$. Then $G_0$ fixes the subspace $W$. Let $v=(1,\\ldots,1,-n+1)\\in W$ and let $\\Delta$ be the set of all translates of images of $\\langle v\\rangle$ under $G_0$, that is, $\\Delta=\\{\\langle v\\rangle^g +w\\mid w\\in W, g\\in G_0\\}$. Note that $|\\langle v\\rangle^{G_0}|=n$. Let $\\Gamma$ be the bipartite graph with vertex set $W\\cup \\Delta$ and adjacency given by inclusion. Then $\\Gamma$ is biregular with bivalency $\\{n,3\\}$ and $G=W\\rtimes G_0\\leqslant \\ensuremath{\\operatorname{Aut}}(\\Gamma)$.\n\nNote that the stabiliser in $G$ of the zero vector is $G_0$ and $\\Gamma(0)=\\langle v\\rangle^{G_0}$. Thus $G_0^{\\Gamma(0)}=S_n$. Moreover, $\\Gamma(\\langle v\\rangle)=\\{\\lambda v\\mid \\lambda\\in\\ensuremath{\\operatorname{GF}}(3)\\}$ and $G_{\\langle v\\rangle}=(\\langle v\\rangle\\rtimes Z)\\times S_{n-1}$. Thus $G_{\\langle v\\rangle}^{\\Gamma(\\langle v\\rangle)}=S_3$. Hence by Lemma \\ref{l:suff star}, $\\Gamma$ is $G$-star-transitive. Moreover, $G_{0,\\langle v\\rangle}=S_{n-1}\\times Z$ acting faithfully on $\\Gamma(0)\\cup \\Gamma(\\langle v\\rangle)$. Since $Z\\cong S_2$, Lemma \\ref{lem:edgeaction} implies that $\\Gamma$ is st(edge)-transitive. We note that $\\Gamma$ belongs to case (i) of Lemma \\ref{[2]=1}.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\setcounter{equation}{0}\n\nWhilst there is little doubt that Quantum Chromodynamics [QCD] is the theory \nof the strong interaction, despite four decades of intense effort the \ngenuine solution of the confinement puzzle and the hadron spectrum remains \nelusive.  That is not to say that no progress has been made -- our \nunderstanding of QCD is being steadily augmented in many ways: for example, \nwith lattice Monte-Carlo techniques \\cite\n{Greensite:2003bk,Chandrasekharan:2004cn} and effective theories (\\cite\n{Bernard:2006gx} and references therein).  One way to understand the problem \nof confinement and the hadron spectrum from \\emph{ab initio} principles is \nto study the Dyson--Schwinger  equations.  These equations are the central equations to \nthe Lagrange formulation of a field theory.  They are in the continuum and \nembody all symmetries of the system at hand.\n\nDyson--Schwinger  studies of QCD in Landau gauge have enjoyed a renaissance in the last \nten years.  A consistent picture of how the important degrees of freedom in \nthe infrared (i.e., those responsible for confinement and the hadron \nspectrum) stem from the ghost sector of the theory has emerged \n\\cite{vonSmekal:1997isa,vonSmekal:1997is,Watson:2001yv} and this has led to \nincreasingly sophisticated calculations of QCD properties, culminating \nrecently in hadron observables \\cite{Fischer:2005en} and possible \nexplanations of confinement (see the recent review \\cite{Fischer:2006ub} for \na discussion of this topic).  Landau gauge, in addition to having the \nappealing property of covariance, has a distinct advantage when searching \nfor practical approximation schemes that allow one to extract information \nabout the infrared behaviour of QCD Green's functions, namely, that the \nghost-gluon vertex remains UV finite to all orders in perturbation theory \n\\cite{Taylor:1971ff}.  For this reason, it has been possible to extract \nunambiguous information about the system \\cite{Watson:2001yv}.\n\nDespite the calculational advantages enjoyed by Landau gauge, it is perhaps \nnot the best choice of gauge to study the infrared physics of QCD.  In this \nrespect, Coulomb gauge is perhaps more advantageous.  \nThere exists a natural picture (though not a proof) of \nconfinement in Coulomb gauge \\cite{Zwanziger:1998ez} and phenomenological \napplications guide the way to understanding the spectrum of hadrons, for \nexample in \\cite{Ligterink:2003hd,Szczepaniak:2006nx} (as indeed they have \ndone in Landau gauge \\cite{Fischer:2006ub,Maris:2003vk}).  This is not to \nsay that one choice of gauge is better than another -- it is crucial to our \nunderstanding of the problem that more than one gauge is considered: firstly \nbecause the physical observables are gauge invariant and it is a test of our \napproximations that the results respect this and secondly because whilst \nconfinement is a gauge invariant reality, its mechanism may be manifested \ndifferently in different gauges such that we will learn far more by studying \nthe different gauges.\n\nRecently, progress has been made in studying Yang-Mills theory in Coulomb \ngauge within the Hamiltonian approach \n\\cite{Szczepaniak:2001rg,\nSzczepaniak:2003ve,Feuchter:2004mk,Reinhardt:2004mm}.  \nHere, the advantage is that Gau\\ss' law can be explicitly resolved (such \nthat, in principle, gauge invariance is fully accounted for) and this \nresults in an explicit expression for the static potential between color-\ncharges.  In \n\\cite{Szczepaniak:2001rg,\nSzczepaniak:2003ve,Feuchter:2004mk,Reinhardt:2004mm}, \nthe Yang-Mills Schroedinger equation was solved variationally for the vacuum \nstate using Gaussian type ans\\\"{a}tze for the wave-functional.  Minimizing \nthe energy density results in a coupled set of Dyson--Schwinger  equations which have been \nsolved analytically in the infrared \\cite{Schleifenbaum:2006bq} and \nnumerically in the entire momentum regime.  If the geometric structure of \nthe space of gauge orbits reflected by the non-trivial Faddeev-Popov \ndeterminant is properly included \\cite{Feuchter:2004mk}, one finds an \ninfrared divergent gluon energy and a linear rising static quark potential \n-- both signals of confinement.  Furthermore, these confinement properties \nhave been shown to be not dependent on the specific ansatz for the vacuum \nwave-functional but result from the geometric structure of the space of \ngauge orbits \\cite{Reinhardt:2004mm}.  However, in spite of this success, \none should bear in mind that an ansatz for the wave-functional is always \nrequired and so the approach does not {\\it a priori} provide a systematic \nexpansion or truncation scheme as, for example, the loop expansion scheme \nused in the common Dyson--Schwinger  approach.  A study of the Dyson--Schwinger  equations in Coulomb \ngauge will hopefully shed some light on the problem.\n\nGiven the appealing properties of Coulomb gauge, it is perhaps surprising \nthat no pure Dyson--Schwinger  study exists in the literature.  However, there is a good \nreason for this: in Coulomb gauge, closed ghost-loops give rise to \nunregulated divergences -- the energy divergence problem.  It has been found \nonly relatively recently how one may circumvent this problem such that a Dyson--Schwinger  \nstudy may be attempted \\cite{Zwanziger:1998ez}.  The key lies in using the \nfirst order formalism.  There is not yet a complete proof that the local \nformulation of Coulomb gauge Yang-Mills theory within the first order \nformalism is renormalisable but significant progress has been made \n\\cite{Zwanziger:1998ez,Baulieu:1998kx}.  There is one undesirable feature to \nthe first order formalism and this is that the number of fields \nproliferates.  As will be seen in this paper, this does have serious \nimplications for the Dyson--Schwinger  equations.\n\nThe purpose of the present work is to derive the Dyson--Schwinger  equations for Coulomb \ngauge Yang-Mills theory within the first order formalism.  These equations \nwill form the basis for an extended program studying QCD in Coulomb gauge.  \nThe paper is organised as follows.  We begin in Section~2 by introducing \nYang-Mills theory in Coulomb gauge and the first order formalism.  In \nparticular, we consider the BRS invariance of the system.  Having introduced \nthe first order formalism, we then motivate the reasons for considering it \n(the cancellation of the energy divergent sector and the reduction to \nphysical degrees of freedom) in Section~3.  Section~4 is then concerned with \nthe derivation of the equations of motion and the equations that stem from \nthe BRS invariance.  There exists certain relationships that give rise to \nexact statements about the Green's functions that enter the system and these \nare detailed in Section~5.  The Feynman rules and the general decomposition \nof the two-point Green's functions are derived and discussed in Sections~6~\n\\&~7.  In Section~8, the Dyson--Schwinger  equations are derived in some detail and are \ndiscussed.  Finally, we summarize and give an outlook of future work in \nSection~9.\n\n\n\\section{First Order Formalism and BRS Invariance}\n\\setcounter{equation}{0}\nThroughout this work, we work in Minkowski space and with the following \nconventions.  The metric is $g_{\\mu\\nu}=\\mathrm{diag}(1,-\\vec{1})$.  Greek \nletters ($\\mu$, $\\nu$, $\\ldots$) denote Lorentz indices, roman subscripts \n($i$, $j$, $\\ldots$) denote spatial indices and superscripts ($a$, $b$, \n$\\ldots$) denote color indices.  We will sometimes also write configuration \nspace coordinates ($x$, $y$, $\\ldots$) as subscripts where no confusion \narises.\n\nThe Yang-Mills action is defined as\n\\begin{equation}\n{\\cal S}_{YM}=\\int\\dx{x}\\left[-\\frac{1}{4}F_{\\mu\\nu}^aF^{a\\mu\\nu}\\right]\n\\end{equation}\nwhere the (antisymmetric) field strength tensor $F$ is given in terms of the \ngauge field $A_{\\mu}^a$:\n\\begin{equation}\nF_{\\mu\\nu}^a=\\partial_{\\mu}A_{\\nu}^a-\\partial_{\\nu}A_{\\mu}^a\n+gf^{abc}A_{\\mu}^bA_{\\nu}^c.\n\\end{equation}\nIn the above, the $f^{abc}$ are the structure constants of the $SU(N_c)$ \ngroup whose generators obey $\\left[T^a,T^b\\right]=\\imath f^{abc}T^c$.  The \nYang-Mills action is invariant under a local $SU(N_c)$ gauge transform \ncharacterised by the parameter $\\th_x^a$:\n\\begin{equation}\nU_x=\\exp{\\left\\{-\\imath\\th_x^aT^a\\right\\}}.\n\\end{equation}\nThe field strength tensor can be expressed in terms of the chromo-electric \nand -magnetic fields ($\\sigma=A^0$)\n\\begin{equation}\n\\vec{E}^a=-\\partial^0\\vec{A}^a-\\vec{\\nabla}\\sigma^a+gf^{abc}\\vec{A}^b\\sigma^c,\\;\\;\\;\\;\nB_i^a=\\epsilon_{ijk}\\left[\\nabla_jA_k^a-\\frac{1}{2} gf^{abc}A_j^bA_k^c\\right]\n\\end{equation}\nsuch that ${\\cal S}_{YM}=\\int(E^2-B^2)\/2$.  The electric and magnetic terms in \nthe action do not mix under the gauge transform which for the gauge fields \nis written\n\\begin{equation}\nA_\\mu\\rightarrow A'_\\mu=U_xA_\\mu U_x^\\dag\n-\\frac{\\imath}{g}(\\partial_\\mu U_x)U_x^\\dag.\n\\end{equation}\nGiven an infinitessimal transform $U_x=1-\\imath\\th_x^aT^a$ the variation of \nthe gauge field is\n\\begin{equation}\n\\delta A_{\\mu}^a=-\\frac{1}{g}\\hat{D}_{\\mu}^{ac}\\th^c\n\\end{equation}\nwhere the covariant derivative in the adjoint representation is given by\n\\begin{equation}\n\\hat{D}_{\\mu}^{ac}=\\delta^{ac}\\partial_{\\mu}+gf^{abc}A_{\\mu}^b.\n\\end{equation}\n\nLet us consider the functional integral\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\exp{\\left\\{\\imath{\\cal S}_{YM}\\right\\}}\n\\end{equation}\nwhere $\\Phi$ denotes the collection of all fields.  Since the action is \ninvariant under gauge transformations, $Z$ is divergent by virtue of the \nzero mode.  To overcome this problem we use the Faddeev-Popov technique and \nintroduce a gauge-fixing term along with an associated ghost term \n\\cite{IZ}.  Using a Lagrange multiplier field to implement the gauge-fixing, \nin Coulomb gauge ($\\s{\\div}{\\vec{A}}=0$) we can then write\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\exp{\\left\\{\\imath{\\cal S}_{YM}+\\imath{\\cal S}_{fp}\\right\\}},\\;\\;\\;\\;\n{\\cal S}_{fp}=\\int d^4x\\left[-\\lambda^a\\s{\\vec{\\nabla}}{\\vec{A}^a}\n-\\ov{c}^a\\s{\\vec{\\nabla}}{\\vec{D}^{ab}}c^b\\right].\n\\end{equation}\nThe new term in the action is invariant under the standard BRS transform \nwhereby the infinitessimal parameter $\\th^a$ is factorised into two \nGrassmann-valued components $\\th^a=c^a\\delta\\lambda$ where $\\delta\\lambda$ is the \ninfinitessimal variation (not to be confused with the colored Lagrange \nmultiplier field $\\lambda^a$).  The BRS transform of the new fields reads\n\\begin{eqnarray}\n\\delta\\ov{c}^a&=&\\frac{1}{g}\\lambda^a\\delta\\lambda\\nonumber\\\\\n\\delta c^a&=&-\\frac{1}{2} f^{abc}c^bc^c\\delta\\lambda\\nonumber\\\\\n\\delta\\lambda^a&=&0.\n\\end{eqnarray}\nFor reasons that will become clear in the next section (expounded in \n\\cite{Zwanziger:1998ez}) we convert to the first order (or phase space) \nformalism by splitting the Yang-Mills action into chromo-electric and \n-magnetic terms and introducing an auxiliary field ($\\vec{\\pi}$) via the \nfollowing identity\n\\begin{equation}\n\\exp{\\left\\{\\imath\\int d^4x\\frac{1}{2}\\s{\\vec{E}^a}{\\vec{E}^a}\\right\\}}\n=\\int{\\cal D}\\vec{\\pi}\\exp{\\left\\{\\imath\\int d^4x\\left[-\\frac{1}{2}\n\\s{\\vec{\\pi}^a}{\\vec{\\pi}^a}-\\s{\\vec{\\pi}^a}{\\vec{E}^a}\\right]\\right\\}}.\n\\end{equation}\nClassically, the $\\vec{\\pi}$-field would be the momentum conjugate to \n$\\vec{A}$.  In order to maintain BRS-invariance, we require that\n\\begin{equation}\n\\int d^4x\\left[\\s{\\delta\\vec{\\pi}^a}{\\left(\\vec{\\pi}^a+\\vec{E}^a\\right)}\n+\\s{\\vec{\\pi}^a}{\\delta\\vec{E}^a}\\right]=0.\n\\label{eq:inv0}\n\\end{equation}\nGiven that the variation of $\\vec{E}$ under the infinitessimal gauge \ntransformation is $\\delta\\vec{E}^a=f^{abc}\\vec{E}^c\\th^b$, then the general \nsolution to \\eq{eq:inv0} is\n\\begin{equation}\n\\delta\\vec{\\pi}^a=f^{abc}\\th^b\\left[(1-\\alpha)\\vec{\\pi}^c-\\alpha\\vec{E}^c\\right]\n\\end{equation}\nwhere $\\alpha$ is some non-colored constant, but which in general could be some \nfunction of position $x$.  The $\\vec{\\pi}$-field is split into transverse \nand longitudinal components using the identity\n\\begin{eqnarray}\n\\mathrm{const}&=&\\int{\\cal D}\\phi\\delta\\left(\\s{\\vec{\\nabla}}{\\vec{\\pi}}\n+\\nabla^2\\phi\\right)\\nonumber\\\\\n&=&\\int{\\cal D}\\left\\{\\phi,\\tau\\right\\}\\exp{\\left\\{-\\imath\\int \nd^4x\\tau^a\\left(\\s{\\vec{\\nabla}}{\\vec{\\pi}^a}+\\nabla^2\\phi^a\\right)\\right\\}}.\n\\end{eqnarray}\nThis constant is gauge invariant and this means that the new fields $\\phi$ \nand $\\tau$ must transform as\n\\begin{equation}\n\\delta\\phi^a=\\s{\\frac{\\vec{\\nabla}}{\\left(-\\nabla^2\\right)}}{\\delta\\vec{\\pi}^a},\n\\;\\;\\delta\\tau^a=0.\n\\end{equation}\nIf we make the change of variables \n$\\vec{\\pi}\\rightarrow\\vec{\\pi}-\\vec{\\nabla}\\phi$ then collecting together \nall the parts of $Z$ that contain $\\vec{\\pi}$, we can write\n\\begin{equation}\nZ_{\\pi}=\\int{\\cal D}\\left\\{\\vec{\\pi},\\phi,\\tau\\right\\}\\exp{\\left\\{\\imath\\int \nd^4x\\left[-\\tau^a\\s{\\vec{\\nabla}}{\\vec{\\pi}^a}\n-\\frac{1}{2}\\s{(\\vec{\\pi}^a-\\div\\phi^a)}{(\\vec{\\pi}^a-\\div\\phi^a)}\n-\\s{\\left(\\vec{\\pi}^a-\\div\\phi^a\\right)}{\\vec{E}^a}\\right]\\right\\}}\n\\end{equation}\nwhich is now invariant under\n\\begin{eqnarray}\n\\delta\\vec{E}^a&=&f^{abc}\\vec{E}^c\\th^b,\\nonumber\\\\\n\\delta\\vec{\\pi}^a&=&f^{abc}\\th^b\\left[(1-\\alpha)\\left(\\vec{\\pi}^c\n-\\vec{\\nabla}\\phi^c\\right)-\\alpha\\vec{E}^c\\right]+\\vec{\\nabla}\\delta\\phi^a,\n\\nonumber\\\\\n\\delta\\phi^a&=&f^{abc}\\left\\{\\s{\\frac{\\vec{\\nabla}}{\\left(-\\nabla^2\\right)}}\n{\\left[(1-\\alpha)\\left(\\vec{\\pi}^c-\\vec{\\nabla}\\phi^c\\right)\n-\\alpha\\vec{E}^c\\right]}\\th^b\\right\\},\\nonumber\\\\\n\\delta\\tau^a&=&0.\n\\end{eqnarray}\nWe notice that the parts of the transform that are proportional to $\\alpha$ are \nindependent of the rest of the BRS transform and can thus be regarded as a \nseparate invariance.  In particular, since it is independent of the \nFaddeev--Popov components, then we may regard it quite generally as a local \ntransform parameterised by $\\th^a$.  This new invariance stems from the \narbitrariness in introducing the $\\vec{\\pi}$-field.  If we expand the \nchromo-electric field into its component form then in summary we can write \nour full functional integral as\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\exp{\\left\\{\\imath{\\cal S}_B+\\imath{\\cal S}_{fp}+\\imath{\\cal S}_{\\pi}\\right\\}}\n\\label{eq:func}\n\\end{equation}\nwith\n\\begin{eqnarray}\n{\\cal S}_B&=&\\int d^4x\\left[-\\frac{1}{2}\\s{\\vec{B}^a}{\\vec{B}^a}\\right],\\nonumber\\\\\n{\\cal S}_{fp}&=&\\int d^4x\\left[-\\lambda^a\\s{\\vec{\\nabla}}{\\vec{A}^a}\n-\\ov{c}^a\\s{\\vec{\\nabla}}{\\vec{D}^{ab}}c^b\\right],\\nonumber\\\\\n{\\cal S}_{\\pi}&=&\\int d^4x\\left[-\\tau^a\\s{\\vec{\\nabla}}{\\vec{\\pi}^a}\n-\\frac{1}{2}\\s{(\\vec{\\pi}^a-\\div\\phi^a)}{(\\vec{\\pi}^a-\\div\\phi^a)}\n+\\s{(\\vec{\\pi}^a-\\div\\phi^a)}{\\left(\\partial^0\\vec{A}^a+\\vec{D}^{ab}\\sigma^b\n\\right)}\\right]\n\\label{eq:act}\n\\end{eqnarray}\nand which is invariant under \\emph{two} sets of transforms: the BRS\n\\begin{eqnarray}\n\\delta\\vec{A}^a&=&\\frac{1}{g}\\vec{D}^{ac}c^c\\delta\\lambda,\\nonumber\\\\\n\\delta\\sigma^a&=&-\\frac{1}{g}D^{0ac}c^c\\delta\\lambda,\\nonumber\\\\\n\\delta\\ov{c}^a&=&\\frac{1}{g}\\lambda^a\\delta\\lambda,\\nonumber\\\\\n\\delta c^a&=&-\\frac{1}{2} f^{abc}c^bc^c\\delta\\lambda,\\nonumber\\\\\n\\delta\\vec{\\pi}^a&=&f^{abc}c^b\\delta\\lambda\\left(\\vec{\\pi}^c\n-\\vec{\\nabla}\\phi^c\\right)+\\vec{\\nabla}\\delta\\phi^a,\\nonumber\\\\\n\\delta\\phi^a&=&f^{abc}\\left\\{\\s{\\frac{\\vec{\\nabla}}{\\left(-\\nabla^2\\right)}}\n{\\left(\\vec{\\pi}^c-\\vec{\\nabla}\\phi^c\\right)}c^b\\delta\\lambda\\right\\},\\nonumber\\\\\n\\delta\\lambda^a&=&0,\\nonumber\\\\\n\\delta\\tau^a&=&0,\n\\label{eq:brstrans}\n\\end{eqnarray}\nand the new transform, which we denote the $\\alpha$-transform,\n\\begin{eqnarray}\n\\delta\\vec{\\pi}^a&=&f^{abc}\\th^b\\left(\\vec{\\pi}^c-\\vec{\\nabla}\\phi^c\n-\\partial^0\\vec{A}-\\vec{D}^{cd}\\sigma^d\\right)+\\vec{\\nabla}\\delta\\phi^a,\\nonumber\\\\\n\\delta\\phi^a&=&f^{abc}\\left\\{\\s{\\frac{\\vec{\\nabla}}{\\left(-\\nabla^2\\right)}}\n{\\left(\\vec{\\pi}^c-\\vec{\\nabla}\\phi^c-\\partial^0\\vec{A}^c-\\vec{D}^{cd}\\sigma^d\n\\right)}\\th^b\\right\\}\n\\label{eq:altrans}\n\\end{eqnarray}\n(all other fields being unchanged).  It useful for later to denote the \ncombination of fields and differential operators occuring in \\eq{eq:altrans} \nas\n\\begin{equation}\n\\vec{X}^c=\\vec{\\pi}^c-\\vec{\\nabla}\\phi^c-\\partial^0\\vec{A}-\\vec{D}^{cd}\\sigma^d.\n\\label{eq:X}\n\\end{equation}\n\n\\section{Formal Reduction to ``Physical\" Degrees of Freedom}\n\\setcounter{equation}{0}\nThere are two factors that motivate our use of the first order formalism.  \nThe first lies in the ability, albeit formally, to reduce the functional \nintegral previously considered (and hence the generating functional) to \n``physical\" degrees of freedom \\cite{Zwanziger:1998ez}.  These are the \ntransverse gluon and transverse $\\vec{\\pi}$ fields which in classical terms \nwould be the configuration variables and their momentum conjugates.  We keep \nthe term ``physical\" in quotation marks because it is realised that in \nYang-Mills theory, the true physical objects would be the color singlet \nglueballs, their observables being the mass spectrum and the decay widths.  \nThe second factor concerns the well-known energy divergence problem of \nCoulomb gauge QCD \\cite{Heinrich:1999ak,Andrasi:2005xu,Doust:1987yd}.  In \nCoulomb gauge, the Faddeev-Popov operator involves only spatial derivatives \nand the spatial components of the gauge fields, but these fields are \nthemselves dependent on the spacetime position.  This leads to the ghost \npropagator and ghost-gluon vertex being independent of the energy whereas \nloops involving pure ghost components are integrated over both 3-momentum \n{\\it and} energy which gives an ill-defined integration.  In the usual, \nsecond order, formulation of the theory these energy divergences do in \nprinciple cancel order by order in perturbation theory (tested up to \ntwo-loops \\cite{Heinrich:1999ak}) but this cancellation is difficult to \nisolate.  Within the first order formalism the cancellation is made manifest \nsuch that the problem of ill-defined integrals can be circumvented.\n\nGiven the functional integral, \\eq{eq:func}, and the action, \\eq{eq:act}, we \nrewrite the Lagrange multiplier terms as $\\delta$-function constraints and the \nghost terms as the original Faddeev-Popov determinant.  Since the \n$\\delta$-function constraints are now exact we can automatically eliminate any \n$\\s{\\div}{\\vec{A}}$ and $\\s{\\div}{\\vec{\\pi}}$ terms in the action.  This is \nclearly at the expense of a local formulation and the BRS invariance of the \ntheory is no longer manifest.  The functional integral is now\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\mathrm{Det}\\left[-\\s{\\vec{\\nabla}}{\\vec{D}}\\delta^4(x-y)\\right]\n\\delta\\left(\\s{\\div}{\\vec{A}}\\right)\\delta\\left(\\s{\\div}{\\vec{\\pi}}\\right)\n\\exp{\\left\\{\\imath{\\cal S}\\right\\}}\n\\end{equation}\nwith\n\\begin{equation}\n{\\cal S}=\\int d^4x\\left[-\\frac{1}{2}\\s{\\vec{B}^a}{\\vec{B}^a}\n-\\frac{1}{2}\\s{\\vec{\\pi}^a}{\\vec{\\pi}^a}+\\frac{1}{2}\\phi^a\\nabla^2\\phi^a\n+\\s{\\vec{\\pi}^a}{\\partial^0\\vec{A}^a}+\\sigma^a\\left(\\s{\\div}{\\vec{D}^{ab}}\\phi^b\n+g\\hat{\\rho}^a\\right)\\right].\n\\end{equation}\nwhere we have defined an effective charge \n$\\hat{\\rho}^a=f^{ade}\\s{\\vec{A}^d}{\\vec{\\pi}^e}$.  The integral over $\\sigma$ \ncan also be written as a $\\delta$-function constraint and is the implementation \nof the chromo-dynamical equivalent of Gau\\ss' law giving\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\mathrm{Det}\\left[-\\s{\\vec{\\nabla}}{\\vec{D}}\\delta^4(x-y)\\right]\n\\delta\\left(\\s{\\div}{\\vec{A}}\\right)\\delta\\left(\\s{\\div}{\\vec{\\pi}}\\right)\n\\delta\\left(-\\s{\\div}{\\vec{D}^{ab}}\\phi^b-g\\hat{\\rho}^a\\right)\n\\exp{\\left\\{\\imath{\\cal S}\\right\\}}\n\\end{equation}\nwith\n\\begin{equation}\n{\\cal S}=\\int d^4x\\left[-\\frac{1}{2}\\s{\\vec{B}^a}{\\vec{B}^a}\n-\\frac{1}{2}\\s{\\vec{\\pi}^a}{\\vec{\\pi}^a}+\\frac{1}{2}\\phi^a\\nabla^2\\phi^a\n+\\s{\\vec{\\pi}^a}{\\partial^0\\vec{A}^a}\\right].\n\\end{equation}\nLet us define the inverse Faddeev-Popov operator $M$:\n\\begin{equation}\n\\left[-\\s{\\div}{\\vec{D}^{ab}}\\right]M^{bc}=\\delta^{ac}.\n\\end{equation}\nWith this definition we can factorise the Gau{\\ss} law $\\delta$-function \nconstraint as\n\\begin{equation}\n\\delta\\left(-\\s{\\div}{\\vec{D}^{ab}}\\phi^b-g\\hat{\\rho}^a\\right)\n=\\mathrm{Det}\\left[-\\s{\\vec{\\nabla}}{\\vec{D}}\\delta^4(x-y)\\right]^{-1}\n\\delta\\left(\\phi^a-M^{ab}g\\hat{\\rho}^b\\right).\n\\end{equation}\nCrucially, the inverse functional determinant cancels the original \nFaddeev-Popov determinant, leaving us with\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\delta\\left(\\s{\\div}{\\vec{A}}\\right)\n\\delta\\left(\\s{\\div}{\\vec{\\pi}}\\right)\\delta\\left(\\phi^a-M^{ab}g\\hat{\\rho}^b\\right)\n\\exp{\\left\\{\\imath{\\cal S}\\right\\}}.\n\\end{equation}\nWe now use the $\\delta$-function constraint to eliminate the $\\phi$-field.  \nRecognising the Hermitian nature of the inverse Faddeev-Popov operator $M$ \nwe can reorder the operators in the action to give us\n\\begin{equation}\nZ=\\int{\\cal D}\\Phi\\delta\\left(\\s{\\div}{\\vec{A}}\\right)\n\\delta\\left(\\s{\\div}{\\vec{\\pi}}\\right)\\exp{\\left\\{\\imath{\\cal S}\\right\\}}\n\\end{equation}\nwith\n\\begin{equation}\n{\\cal S}=\\int d^4x\\left[-\\frac{1}{2}\\s{\\vec{B}^a}{\\vec{B}^a}\n-\\frac{1}{2}\\s{\\vec{\\pi}^a}{\\vec{\\pi}^a}\n-\\frac{1}{2} g\\hat{\\rho}^bM^{ba}(-\\nabla^2)M^{ac}g\\hat{\\rho}^c\n+\\s{\\vec{\\pi}^a}{\\partial^0\\vec{A}^a}\\right].\n\\end{equation}\nThe above action is our desired form, with only transverse $\\vec{A}$ and \n$\\vec{\\pi}$ fields present.  All other fields, especially those responsible \nfor the Faddeev-Popov determinant (i.e., the certainly unphysical ghosts) \nhave been formally eliminated.  However, the appearance of the functional \n$\\delta$-functions and the inverse Faddeev-Popov operator $M$ have led to a \nnon-local formalism.  It is not known how to use forms such as the above in \ncalculational schemes.  The issue of renormalisability is certainly unclear \nand one does not have a Ward identity in the usual sense.\n\nThe non-local nature of the above result may not lend itself to \ncalculational devices but does serve as a guide to the local formulation.  \nIn particular, it is evident that decomposition of degrees of freedom, both \nphysical and unphysical, inherent to the first order formalism leads more \nnaturally to the cancellation of the unphysical components in the \ndescription of physical phenomena than perhaps other choices such as Landau \ngauge.  The task ahead is to identify, within the local formulation, how \nthese cancellations arise and to ensure that approximation schemes respect \nsuch cancellations.  For example, the cancellation of the Faddeev-Popov \ndeterminant and the appearance of the inverse Faddeev-Popov operator should \nlead to the separation of the physical gluon dynamics contained within the \nghost sector and the unphysical ghosts themselves, i.e., the unphysical \nghost loop of the gluon polarisation should be cancelled whilst another loop \ncontaining only physical information will take its place.  Also, it should \nbe evident that the energy divergences associated with ghost loops are \nexplicitly cancelled such that ill-defined integrals do not occur.\n\n\\section{Field Equations of Motion and Continuous Symmetries}\n\\setcounter{equation}{0}\nThe generating functional of the theory is given by our previously \nconsidered functional integral in the presence of sources.  Explicitly, \ngiven the action, \\eq{eq:act}, we have\n\\begin{equation}\nZ[J]=\\int{\\cal D}\\Phi\\exp{\\left\\{\\imath{\\cal S}_B+\\imath{\\cal S}_{fp}\n+\\imath{\\cal S}_{\\pi}+\\imath{\\cal S}_s\\right\\}}\n\\label{eq:gen0}\n\\end{equation}\nwith sources defined by\n\\begin{equation}\n{\\cal S}_s=\\int\\dx{x}\\left[\\rho^a\\sigma^a+\\s{\\vec{J}^a}{\\vec{A}^a}+\\ov{c}^a\\eta^a\n+\\ov{\\eta}^ac^a+\\kappa^a\\phi^a+\\s{\\vec{K}^a}{\\vec{\\pi}^a}+\\xi_\\lambda^a\\lambda^a\n+\\xi_\\tau^a\\tau^a\\right].\n\\label{eq:source0}\n\\end{equation}\nIt is useful to introduce a compact notation for the sources and fields and \nwe denote a generic field $\\Phi_\\alpha$ with source $J_\\alpha$ such that the index \n$\\alpha$ stands for all attributes of the field in question (including its \ntype) such that, for instance, we could write\n\\begin{equation}\n{\\cal S}_s=J_\\alpha\\Phi_\\alpha\n\\end{equation}\nwhere summation over all discrete indices and integration over all \ncontinuous arguments is implicitly understood.\n\nThe field equations of motion are derived from the observation that the \nintegral of a total derivative vanishes up to boundary terms.  The boundary \nterms vanish but this is not so trivial in the light of the Gribov problem.  \nPerturbatively (expanding around the free-field), there are certainly no \nboundary terms to be considered, however, nonperturbatively the presence of \nso-called Gribov copies does complicate the picture somewhat.\n\nIn \\cite{Gribov:1977wm}, Gribov showed that the Faddeev-Popov technique does \nnot uniquely fix the gauge and showed that even after gauge-fixing there are \n(physically equivalent) gauge configurations related by finite gauge \ntransforms still present.  It was proposed to restrict the space of gauge \nfield configurations ($A$) to the so-called Gribov region $\\Omega$ defined by\n\\begin{equation}\n\\Omega\\equiv\\left\\{A:\\s{\\div}{\\vec{A}}=0;-\\s{\\div}{\\vec{D}}\\geq0\\right\\}.\n\\end{equation}\n$\\Omega$ is a region where the Coulomb gauge condition holds and \nfurthermore, the eigenvalues of the Faddeev-Popov operator are all \npositive.  It contains the element $\\vec{A}=0$ and is bounded in every \ndirection \\cite{Zwanziger:1998yf}.  However, as explained in \n\\cite{Zwanziger:2003cf} and references therein, the Gribov region is not \nentirely free of Gribov copies and one should more correctly consider the \nfundamental modular region $\\Lambda$ which is defined as the region free of \nGribov copies.  It turns out though that the functional integral is \ndominated by configurations on the common boundary of $\\Lambda$ and $\\Omega$ \nso that, in practise, restriction to the Gribov region is sufficient.\n\nGiven that non-trivial boundary conditions are being imposed (i.e., \nrestricting to the Gribov region $\\Omega$), the question of the boundary \nterms in the derivation of the field equations of motion now becomes \nextremely relevant.  However, by definition on the boundary of $\\Omega$, the \nFaddeev-Popov determinant {\\it vanishes} such that the boundary terms are \nidentically zero.  The form of the field equations of motion is therefore \nequivalent as if we had extended the integration region to the full \nconfiguration space \\cite{Zwanziger:2003cf}.\n\nWriting ${\\cal S}={\\cal S}_B+{\\cal S}_{fp}+{\\cal S}_{\\pi}$ we have that\n\\begin{equation}\n0=\\int{\\cal D}\\Phi\\frac{\\delta}{\\delta\\imath\\Phi_\\alpha}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}\n=\\int{\\cal D}\\Phi\\left\\{\\frac{\\delta{\\cal S}}{\\delta\\Phi_\\alpha}\n+\\frac{\\delta{\\cal S}_s}{\\delta\\Phi_\\alpha}\\right\\}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}\n\\end{equation}\nand so, taking advantage of the linearity in the fields of the source term \nof the action we have\n\\begin{equation}\nJ_{\\alpha}Z=-\\int{\\cal D}\\Phi\\left\\{\\frac{\\delta{\\cal S}}{\\delta\\Phi_\\alpha}\\right\\}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}.\n\\label{eq:eom}\n\\end{equation}\nWe use the convention that all Grassmann-valued derivatives are \nleft-derivatives and so in the above there will be an additional minus sign \non the left-hand side when $\\alpha$ refers to derivatives with respect to \neither the $c$-field or the $\\eta$-source.  The explicit form of the various \nfield equations of motion are given in Appendix~\\ref{app:eom}.\n\nContinuous transforms, under which the action is invariant, can be regarded \nas changes of variable and providing that the Jacobian is trivial, one is \nleft with an equation relating the variations of the source terms in the \naction.  We consider the two invariances derived explicitly in the previous \nsection and that the Jacobian factors are trivial is shown in \nAppendix~\\ref{app:jac}.  In the case of the BRS transform, \\eq{eq:brstrans}, \nwe have\n\\begin{eqnarray}\n0&=&\\int{\\cal D}\\Phi\\frac{\\delta}{\\delta\\imath\\delta\\lambda}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s+\\imath\\delta{\\cal S}_s\\right\\}}_{\\delta\\lambda=0}\n\\nonumber\\\\\n&=&\\int{\\cal D}\\Phi\\int\\dx{x}\\left\\{-\\frac{1}{g}\\rho^aD^{0ab}c^b\n+\\frac{1}{g}\\s{\\vec{J}^a}{\\vec{D}^{ab}}c^b-\\frac{1}{g}\\lambda^a\\eta^a\n-\\frac{1}{2} f^{abc}\\ov{\\eta}^ac^bc^c\n\\right.\\nonumber\\\\&&\\left.\n+f^{abc}c^b\\s{(\\vec{\\pi}^c-\\div\\phi^c)}{\\left[\\vec{K}^a-\\frac{\\div}{\n(-\\nabla^2)}(\\kappa^a-\\s{\\div}{\\vec{K}^a})\\right]}\\right\\}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}.\n\\end{eqnarray}\nNotice that the infinitessimal variation $\\delta\\lambda$ that parameterises the BRS \ntransform is a global quantity, leading to the overall integral over $x$.  \nThe equation for the $\\alpha$-transform is:\n\\begin{eqnarray}\n0&=&\\int{\\cal D}\\Phi\\frac{\\delta}{\\delta\\imath\\th_x^a}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s+\\imath\\delta{\\cal S}_s\\right\\}}_{\\th=0}\\nonumber\\\\\n&=&\\int{\\cal D}\\Phi f^{abc}\\s{\\vec{X}_x^c}{\\left[\\vec{K}_x^a-\\frac{\\div_x}{\n(-\\nabla_x^2)}(\\kappa_x^a-\\s{\\div_x}{\\vec{K}_x^a})\\right]}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}\n\\label{eq:alinv}\n\\end{eqnarray}\nwhere $\\vec{X}$ is given by \\eq{eq:X}.  Constraints imposed by the discrete \nsymmetries of time-reversal and parity will be discussed later.\n\nThe above equations of motion and symmetries refer to functional derivatives \nof the full generating functional.  In practise we are concerned with \nconnected two-point (propagator) and one-particle irreducible $n$-point \n(proper) Green's functions since these comprise the least sophisticated \ncommon building blocks from which all other amplitudes may be constructed.  \nThe generating functional of connected Green's functions is $W[J]$ where\n\\begin{equation}\nZ[J]=e^{W[J]}.\n\\end{equation}\nWe introduce a bracket notation for functional derivatives of $W$ such that\n\\begin{equation}\n\\ev{\\imath J_1}=\\frac{\\delta W}{\\delta\\imath J_1}.\n\\end{equation}\nThe classical field $\\Phi_\\alpha$ is defined as\n\\begin{equation}\n\\Phi_\\alpha=\\frac{1}{Z}\\int{\\cal D}\\Phi\\Phi_\\alpha\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}\n=\\frac{1}{Z}\\frac{\\delta Z}{\\delta\\imath J_\\alpha}\n\\end{equation}\n(the classical field is distinct from the quantum fields which are \nfunctionally integrated over, but for convenience we use the same \nnotation).  The generating functional of proper Green's functions is the \neffective action $\\Gamma$, which is a function of the classical fields and is \ndefined via a Legendre transform of $W$:\n\\begin{equation}\n\\Gamma[\\Phi]=W[J]-\\imath J_\\alpha\\Phi_\\alpha.\n\\end{equation}\nWe use the same bracket notation to denote derivatives of $\\Gamma$ with respect \nto fields -- no confusion arises since we never mix derivatives with respect \nto sources and fields.\n\nLet us now present the equations of motion in terms of proper functions \n(from which we will derive the Dyson--Schwinger  equations).  Using the  equations of \nmotion listed in Appendix~\\ref{app:eom} we have the following equations:\n\\begin{itemize}\n\\item\n$\\sigma$-based.  This is the functional form of Gau\\ss' law.\n\\begin{eqnarray}\n\\ev{\\imath\\sigma_x^a}-\\ev{\\imath\\tau_x^a}&=&\\nabla_x^2\\phi_x^a\n+gf^{abc}A_{ix}^b\\pi_{ix}^c-gf^{abc}A_{ix}^b\\nabla_{ix}\\phi_x^c\n+gf^{abc}\\ev{\\imath J_{ix}^b\\imath K_{ix}^c}\n\\nonumber\\\\&&\n-gf^{abc}\\int d^4y\\delta(x-y)\\nabla_{ix}\\ev{\\imath J_{iy}^b\\imath\\kappa_x^c}.\n\\label{eq:sidse0}\n\\end{eqnarray}\nNote that we have implicitly used the $\\tau$ equation of motion, \n\\eq{eq:tadse1}, in order to eliminate terms involving $\\s{\\div}{\\vec{\\pi}}$ \nin favour of the source $\\xi_{\\tau}$.\n\\item\n$\\vec{A}$-based.  We write this in such a way as to factorise the functional \nderivatives and the kinematical factors.  The equation reads:\n\\begin{eqnarray}\n\\ev{\\imath A_{ix}^a}&=&\\nabla_{ix}\\lambda_x^a-\\partial_x^0\\pi_{ix}^a\n+\\partial_x^0\\nabla_{ix}\\phi_x^a+\\left[\\delta_{ij}\\nabla_x^2\n-\\nabla_{ix}\\nabla_{jx}\\right]A_{jx}^a\n\\nonumber\\\\&&\n+gf^{abc}\\int\\dx{y}\\dx{z}\\delta(y-x)\\delta(z-x)\\left[\\nabla_{iz}\\ov{c}_z^bc_y^c\n+\\pi_{iz}^b\\sigma_y^c-\\nabla_{iz}\\phi_z^b\\sigma_y^c\\right]\n\\nonumber\\\\&&\n+gf^{abc}\\int\\dx{y}\\dx{z}\\delta(y-x)\\delta(z-x)\n\\left[\\nabla_{iz}\\ev{\\imath\\ov{\\eta}_y^c\\imath\\eta_z^b}\n+\\ev{\\imath K_{iz}^b\\imath\\rho_y^c}\n-\\nabla_{iz}\\ev{\\imath\\kappa_z^b\\imath\\rho_y^c}\\right]\n\\nonumber\\\\&&\n+gf^{abc}\\int\\dx{y}\\dx{z}\\delta(y-x)\\delta(z-x)\\left\\{\\delta_{jk}\\nabla_{iz}\n+2\\delta_{ij}\\nabla_{ky}-\\delta_{ik}\\nabla_{jy}\\right\\}\n\\left[\\ev{\\imath J_{jy}^b\\imath J_{kz}^c}+A_{jy}^bA_{kz}^c\\right]\n\\nonumber\\\\&&\n-\\frac{1}{4}g^2f^{fbc}f^{fde}\\delta_{jk}\\delta_{li}\n\\left[\\delta^{cg}\\delta^{eh}(\\delta^{ab}\\delta^{di}+\\delta^{ad}\\delta^{bi})\n+\\delta^{bg}\\delta^{dh}(\\delta^{ac}\\delta^{ie}+\\delta^{ae}\\delta^{ic})\\right]\\times\n\\nonumber\\\\&&\n\\left[\\ev{\\imath J_{jx}^g\\imath J_{kx}^h\\imath J_{lx}^i}\n+A_{jx}^g\\ev{\\imath J_{kx}^h\\imath J_{lx}^i}\n+A_{kx}^h\\ev{\\imath J_{jx}^g\\imath J_{lx}^i}\n+A_{lx}^i\\ev{\\imath J_{jx}^g\\imath J_{kx}^h}+A_{jx}^gA_{kx}^hA_{lx}^i\\right].\n\\label{eq:adse0}\n\\end{eqnarray}\n\\item\nghost-based.  The ghost and the antighost equations provide the same \ninformation.  The two fields are complimentary and derivatives must come in \npairs if the expression is to survive when sources are set to zero.  The \nantighost equation is\n\\begin{equation}\n\\ev{\\imath\\ov{c}_x^a}=-\\nabla_x^2c_x^a\n-gf^{abc}\\nabla_{ix}\n\\left[\\ev{\\imath\\ov{\\eta}_x^b\\imath J_{ix}^c}+c_x^bA_{ix}^a\\right].\n\\label{eq:ghost0}\n\\end{equation}\n\\item\n$\\vec{\\pi}$-based.\n\\begin{equation}\n\\ev{\\imath\\pi_{ix}^a}=\\nabla_{ix}\\tau_x^a-\\pi_{ix}^a+\\nabla_{ix}\\phi_x^a\n+\\partial_x^0A_{ix}^a+\\nabla_{ix}\\sigma_x^a\n+gf^{abc}\\left[\\ev{\\imath\\rho_x^b\\imath J_{ix}^c}+\\sigma_x^bA_{ix}^c\\right].\n\\label{eq:pidse0}\n\\end{equation}\n\\item\n$\\phi$-based.  We notice that the interaction terms in the equation of \nmotion for the $\\phi$-field, \\eq{eq:phdse1} are, up to a derivative, \n\\emph{identical} to those of the $\\vec{\\pi}$-based equation, \\eq{eq:pidse1}.  \nThis arises since $-\\div\\phi$ is nothing more than the longitudinal part of \n$\\vec{\\pi}$ and means that there is a redundancy in the formalism that can \nbe exploited to simplify proceedings.  We can write\n\\begin{equation}\n\\left(\\s{\\div_x}{\\vec{K}_x^a}-\\kappa_x^a\\right)Z\n=-\\int{\\cal D}\\Phi\\nabla_x^2\\tau_x^a\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}},\n\\label{eq:phidse1}\n\\end{equation}\nfrom which it follows that\n\\begin{eqnarray}\n\\label{eq:phidse}\\s{\\div_x}{\\vec{K}_x^a}-\\kappa_x^a&=&\n-\\nabla_x^2\\ev{\\imath\\xi_{\\tau x}^a},\\\\\n\\label{eq:phidse0}\\ev{\\imath\\phi_x^a}\n-\\s{\\div_x}{\\ev{\\imath\\vec{\\pi}_x^a}}&=&-\\nabla_x^2\\tau_x^a.\n\\end{eqnarray}\n\\item\n$\\lambda$- and $\\tau$-based.  It will be useful to have these equations written \nin terms of both connected and proper Green's functions.\n\\begin{eqnarray}\n\\label{eq:ladse}\\xi_{\\lambda x}^a=\\s{\\div_x}{\\ev{\\imath\\vec{J}_x^a}},&\n\\;\\;\\;\\;&\\ev{\\imath\\lambda_x^a}=-\\s{\\div_x}{\\vec{A}_x^a},\\\\\n\\label{eq:tadse}\\xi_{\\tau x}^a=\\s{\\div_x}{\\ev{\\imath\\vec{K}_x^a}},&\n\\;\\;\\;\\;&\\ev{\\imath\\tau_x^a}=-\\s{\\div_x}{\\vec{\\pi}_x^a}.\n\\end{eqnarray}\n\\end{itemize}\nThe BRS transform gives rise to the following equation (the Ward--Takahashi \nidentity):\n\\begin{eqnarray}\n0&=&\\int d^4x\\left\\{\\frac{1}{g}(\\partial_x^0\\rho_x^a)\\ev{\\imath\\ov{\\eta}_x^a}\n-f^{acb}\\rho_x^a\\left[\\ev{\\imath\\rho_x^c\\imath\\ov{\\eta}_x^b}\n+\\ev{\\imath\\rho_x^c}\\ev{\\imath\\ov{\\eta}_x^b}\\right]\n-\\frac{1}{g}(\\nabla_{ix}J_{ix}^a)\\ev{\\imath\\ov{\\eta}_x^a}\n\\right.\\nonumber\\\\&&\n-f^{acb}J_{ix}^a\\left[\\ev{\\imath J_{ix}^c\\imath\\ov{\\eta}_x^b}\n+\\ev{\\imath J_{ix}^c}\\ev{\\imath\\ov{\\eta}_x^b}\\right]\n-\\frac{1}{g}\\eta_x^a\\ev{\\imath\\xi_{\\lambda x}^a}\n-\\frac{1}{2} f^{abc}\\ov{\\eta}_x^a\\left[\\ev{\\imath\\ov{\\eta}_x^b\\imath\\ov{\\eta}_x^c}\n+\\ev{\\imath\\ov{\\eta}_x^b}\\ev{\\imath\\ov{\\eta}_x^c}\\right]\n\\nonumber\\\\&&\n+f^{abc}\\left[K_{ix}^a-\\frac{\\nabla_{ix}}{(-\\nabla_x^2)}\n(\\kappa_x^a-\\nabla_{jx}K_{jx}^a)\\right]\\times\n\\nonumber\\\\&&\\left.\n\\left[\\ev{\\imath K_{ix}^c\\imath\\ov{\\eta}_x^b}\n-\\int\\dx{y}\\delta(x-y)\\nabla_{ix}\\ev{\\imath\\kappa_x^c\\imath\\ov{\\eta}_y^b}\n+\\ev{\\imath K_{ix}^c}\\ev{\\imath\\ov{\\eta}_x^b}\n-\\ev{\\imath\\ov{\\eta}_x^b}\\nabla_{ix}\\ev{\\imath\\kappa_x^c}\\right]\\right\\}.\n\\label{eq:stid0}\n\\end{eqnarray}\nWe consider for now only the form of the equation relating connected Green's \nfunctions.  As will be seen in the next section, it will not be necessary to \nconsider the equation generated by the invariance under the $\\alpha$-transform.\n\n\\section{Exact Relations for Green's Functions}\n\\setcounter{equation}{0}\nGiven the set of `master' field equations of motion and symmetries, it is \npertinent to find out if any of the constraints can be combined to give \nunambiguous information about the eventual Green's functions of the theory.  \nWe find that such simplifications do in fact exist.\n\nLet us start by discussing the functional equation generated by \n$\\alpha$-invariance, \\eq{eq:alinv}.  It is not necessary here to consider \nfunctional derivatives of either the generating functional of connected \nGreen's functions ($W$) or the effective action ($\\Gamma$) since the derivation \napplies to the functional integrals directly.  From Appendix~\\ref{app:eom}, \nthe $\\vec{\\pi}$-based field equation of motion, \\eq{eq:pidse1} is\n\\begin{equation}\nK_{ix}^aZ[J]=-\\int{\\cal D}\\Phi\\left\\{\\nabla_{ix}\\tau_x^a-X_{ix}^a\\right\\}\n\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}},\\label{eq:pidse2}\n\\end{equation}\nUsing \\eq{eq:pidse2} we can rewrite \\eq{eq:alinv} as\n\\begin{equation}\n0=f^{abc}\\int{\\cal D}\\Phi\\s{\\left[\\vec{K}_x^a-\\frac{\\div_x}{(-\\nabla_x^2)}\n(\\kappa_x^a-\\s{\\div_x}{\\vec{K}_x^a})\\right]}{\\left[\\vec{K}_x^c\n+\\div_x\\tau_x^c\\right]}\\exp{\\left\\{\\imath{\\cal S}+\\imath{\\cal S}_s\\right\\}}.\n\\label{eq:alinv1}\n\\end{equation}\nSince $f^{abc}$ is antisymmetric and noting \\eq{eq:phidse1}, the above is \nnow an almost trivial identity.  We have thus shown that the \n$\\vec{\\pi}$- and $\\phi$-based equations of motion, \\eq{eq:pidse2} and \n\\eq{eq:phidse1}, guarantee that $\\alpha$-invariance is respected.  Conversely, \napproximations to the equations of motion will destroy the symmetry.  This \nis a concrete example of a general feature of any physical field theory -- \nthe full solutions of the field equations of motion (and the subsequent \nfunctional derivatives which comprise the Dyson--Schwinger  equations) contain all the \ninformation given by the symmetry considerations.  In this case, we have the \nambiguity associated with introducing the $\\vec{\\pi}$-field and assigning \nits properties under the BRS transform encoded within the invariance under \nthe $\\alpha$-transform and the field equations of motion are `aware' of this.  \nWhat is unusual about this, however, is that the equivalence of the field \nequations of motion and the equations generated by invariance under a \nsymmetry is invariably impossible to show (except order by order in \nperturbation theory) -- full gauge invariance being the archetypal example.\n\nLet us now continue the discussion by considering those equations of motion \nwhich do not contain interaction terms.  In the absence of interactions, the \nsolutions to these equations can be written down without difficulty.  In \nterms of connected Green's function, the only non-zero functional derivative \nof the $\\lambda$-equation, \\eq{eq:ladse} is\n\\begin{equation}\n\\s{\\div_x}{\\ev{\\imath\\xi_{\\lambda y}^b\\imath\\vec{J}_x^a}}\n=-\\imath\\delta^{ba}\\delta(y-x),\n\\label{eq:ladse2}\n\\end{equation}\nthe right-hand side vanishing for all other derivatives.  Separating the \nconfiguration space arguments and setting our conventions for the Fourier \ntransform, we have for a general two-point function (connected or proper) \nwhich obeys translational invariance:\n\\begin{equation}\n\\ev{\\imath J_{\\alpha}(y)\\imath J_{\\beta}(x)}\n=\\ev{\\imath J_{\\alpha}(y-x)\\imath J_{\\beta}(0)}\n=\\int\\dk{k}W_{\\alpha\\beta}(k)e^{-\\imath k\\cdot(y-x)}\n\\end{equation}\nwhere $\\dk{k}=d^4k\/(2\\pi)^4$ and it is implicitly understood that the \nrelevant prescription to avoid integration over poles is present such that \nthe analytic continuation to Euclidean space may be performed.  We can \nimmediately write down the functional derivatives of $\\ev{\\imath J_{ix}^q}$ \nusing \\eq{eq:ladse}:\n\\begin{eqnarray}\n\\ev{\\imath J_{jy}^b\\imath J_{ix}^a}&=&\n\\int\\dk{k}W_{AAji}^{ba}(k)t_{ji}(\\vec{k})e^{-\\imath k\\cdot(y-x)},\\nonumber\\\\\n\\ev{\\imath K_{jy}^b\\imath J_{ix}^a}&=&\n\\int\\dk{k}W_{\\pi Aji}^{ba}(k)t_{ji}(\\vec{k})e^{-\\imath k\\cdot(y-x)},\n\\nonumber\\\\\n\\ev{\\imath \\xi_{\\lambda y}^b\\imath J_{ix}^a}&=&\n\\int\\dk{k}\\delta^{ba}\\frac{k_i}{\\vec{k}^2}e^{-\\imath k\\cdot(y-x)},\\nonumber\\\\\n\\ev{\\imath \\xi_{\\tau y}^b\\imath J_{ix}^a}&=&\n\\ev{\\imath \\rho_y^b\\imath J_{ix}^a}=\\ev{\\imath \\kappa_y^b\\imath J_{ix}^a}=0.\n\\end{eqnarray}\nSimilarly, for the $\\tau$-equation, \\eq{eq:tadse}, we have\n\\begin{eqnarray}\n\\ev{\\imath J_{jy}^b\\imath K_{ix}^a}&=&\n\\int\\dk{k}W_{A\\pi ji}^{ba}(k)t_{ji}(\\vec{k})e^{-\\imath k\\cdot(y-x)}\n,\\nonumber\\\\\n\\ev{\\imath K_{jy}^b\\imath K_{ix}^a}&=&\n\\int\\dk{k}W_{\\pi\\pi ji}^{ba}(k)t_{ji}(\\vec{k})e^{-\\imath k\\cdot(y-x)},\n\\nonumber\\\\\n\\ev{\\imath \\xi_{\\tau y}^b\\imath K_{ix}^a}&=&\n\\int\\dk{k}\\delta^{ba}\\frac{k_i}{\\vec{k}^2}e^{-\\imath k\\cdot(y-x)},\\nonumber\\\\\n\\ev{\\imath \\rho_y^b\\imath K_{ix}^a}&=&\\ev{\\imath \\kappa_y^b\\imath K_{ix}^a}\n=\\ev{\\imath \\xi_{\\lambda y}^b\\imath K_{ix}^a}=0.\n\\end{eqnarray}\nWe see that, as expected, the propagators involving only the vector fields \nare transverse and the only other contributions relate to the Lagrange \nmultiplier fields and are purely kinematical in nature.  There is one \nsubtlety to the above and that is that whilst the equations for \n$\\ev{\\imath\\xi_{\\lambda}\\imath J}$ and $\\ev{\\imath\\xi_{\\tau}\\imath K}$ are exact \nin the presence of sources, all other equations refer implicitly to the case \nwhere the sources are set to zero and the discrete parity symmetry has been \napplied (see later for a more complete discussion).\n\nIn the same fashion, let us consider the $\\lambda$-equation, \\eq{eq:ladse}, (and \nsimilarly the $\\tau$-equation, \\eq{eq:tadse}) in terms of proper Green's \nfunctions.  This equation does contain the same information as its \ncounterpart for connected Green's functions, but clearly has a different \ncharacter.  The only non-zero functional derivative is\n\\begin{equation}\n\\ev{\\imath A_{iy}^b\\imath\\lambda_x^a}\n=\\int\\dk{k}\\delta^{ba}k_ie^{-\\imath k\\cdot(y-x)},\n\\end{equation}\nall others vanishing, \\emph{even in the presence of sources}.  This applies \nto all proper $n$-point functions involving the $\\lambda$-field.  Similarly, we \nhave the only non-vanishing proper function involving the $\\tau$-field:\n\\begin{equation}\n\\ev{\\imath \\pi_{iy}^b\\imath\\tau_x^a}\n=\\int\\dk{k}\\delta^{ba}k_ie^{-\\imath k\\cdot(y-x)}.\n\\end{equation}\nThat there are no proper $n$-point functions involving functional \nderivatives with respect to the Lagrange multiplier fields apart from the \ntwo special cases above leads to an important facet concerning the Dyson--Schwinger  \nequations -- there will be no self-energy terms involving derivatives with \nrespect to the $\\lambda$- or $\\tau$-fields since they have no proper vertices, \ndespite the fact that the propagators associated with these fields may be \nnon-trivial.\n\nNext, let us turn to the $\\phi$-based equation of motion in terms of \nconnected Green's functions, \\eq{eq:phidse}.  There are only two \nnon-vanishing functional derivatives and we can write down the solutions as \nbefore\n\\begin{eqnarray}\n\\ev{\\imath K_{iy}^b\\imath\\xi_{\\tau x}^a}&=&\n\\int\\dk{k}\\delta^{ba}\\frac{(-k_i)}{\\vec{k}^2}e^{-\\imath k\\cdot(y-x)}\n,\\nonumber\\\\\n\\ev{\\imath\\kappa_y^b\\imath\\xi_{\\tau x}^a}&=&\n\\int\\dk{k}\\delta^{ba}\\frac{\\imath}{\\vec{k}^2}e^{-\\imath k\\cdot(y-x)}\n,\\nonumber\\\\\n\\ev{\\imath J_{iy}^b\\imath\\xi_{\\tau x}^a}&=&\n\\ev{\\imath\\rho_y^b\\imath\\xi_{\\tau x}^a}=\n\\ev{\\imath\\xi_{\\lambda x}^b\\imath\\xi_{\\tau x}^a}=\n\\ev{\\imath\\xi_{\\tau x}^b\\imath\\xi_{\\tau x}^a}=0.\n\\end{eqnarray}\nNotice that \\emph{all} of the connected Green's functions involving the \n$\\tau$-field are now known and are purely kinematical in nature.  In terms of \nproper Green's functions, we consider the $\\phi$-based equation of motion, \n\\eq{eq:phidse0}.  Recognising that functional derivatives with respect to \nthe $\\lambda$- and $\\tau$-fields yield no more information, we can omit them from \nthe current discussion.  The equation tells us that given a proper Green's \nfunction involving $\\pi$, we can immediately construct the corresponding \nfunctional derivative with respect to $\\phi$.  We can thus conclude that as \nfar as the proper Green's functions are concerned, derivatives with repect \nto the $\\phi$-field are redundant.\n\nFinally, let us consider the equation derived from the BRS transform in \nterms of connected Green's functions, \\eq{eq:stid0}.  Since the \nghost\/antighost fields must come in pairs, we may take the functional \nderivative of this with respect to $\\imath\\eta_z^d$ and subsequently set the \nghost sources to zero whilst considering only the rest.  We get:\n\\begin{eqnarray}\n\\lefteqn{\\frac{\\imath}{g}\\ev{\\imath\\xi_{\\lambda z}^d}=\n\\int\\dx{x}\\left\\{\n\\frac{1}{g}(\\partial_x^0\\rho_x^a)\\ev{\\imath\\ov{\\eta}_x^a\\imath\\eta_z^d}\n-f^{acb}\\rho_x^a\\left[\\ev{\\imath\\rho_x^c\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\n+\\ev{\\imath\\rho_x^c}\\ev{\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\\right]\\right.}\n\\nonumber\\\\&&\n-\\frac{1}{g}(\\nabla_{ix}J_{ix}^a)\\ev{\\imath\\ov{\\eta}_x^a\\imath\\eta_z^d}\n-f^{acb}J_{ix}^a\\left[\\ev{\\imath J_{ix}^c\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\n+\\ev{\\imath J_{ix}^c}\\ev{\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\\right]\n\\nonumber\\\\&&\n+f^{abc}\\left[K_{ix}^a-\\frac{\\nabla_{ix}}{(-\\nabla_x^2)}\n(\\kappa_x^a-\\nabla_{jx}K_{jx}^a)\\right]\\times\n\\nonumber\\\\&&\\left.\n\\left[\\ev{\\imath K_{ix}^c\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\n-\\int\\dx{y}\\delta(x-y)\\nabla_{ix}\\ev{\\imath\\kappa_x^c\\imath\\ov{\\eta}_y^b\n\\imath\\eta_z^d}+\\ev{\\imath K_{ix}^c}\\ev{\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\n-\\ev{\\imath\\ov{\\eta}_x^b\\imath\\eta_z^d}\\nabla_{ix}\\ev{\\imath\\kappa_x^c}\\right]\n\\right\\}.\n\\label{eq:stid1}\n\\end{eqnarray}\nFor now, the pertinent information from this identity comes from taking the \nfunctional derivative with respect to the source $\\xi_{\\lambda}$ and setting all \nsources to zero.  The result is\n\\begin{equation}\n\\ev{\\imath\\xi_{\\lambda\\omega}^e\\imath\\xi_{\\lambda z}^d}=0.\n\\end{equation}\nWe notice that all other functional derivatives lead to non-trivial \nrelations involving interaction terms.  This includes both the \n$\\ev{\\imath\\rho\\imath\\xi_\\lambda}$ and the $\\ev{\\imath\\kappa\\imath\\xi_\\lambda}$ \nconnected Green's functions and we conclude that these functions are not \nmerely kinematical factors as one might expect from quantities involving \nLagrange multiplier fields.  We shall return to this topic at a later stage.\n\n\\section{Feynman (and Other) Rules}\n\\setcounter{equation}{0}\nWhilst it is entirely possible to deduce the complete set of Feynman rules \ndirectly from the action, we shall follow a slightly less obvious path \nhere.  We derive not only the basic Feynman rules but collect all the \ntree-level (and incidently primitively divergent) quantities that will be of \ninterest.  This means that in addition to the tree-level propagators (i.e., \nconnected two-point Green's functions) and proper vertices (i.e., proper \nthree- and four-point functions) we derive also the proper two-point \nfunctions.  The reason for this is that (as will be discussed in some detail \nlater) the connected and proper two-point functions are not related in the \nusual way as inverses of one another.  The tree-level quantities of interest \ncan be easily derived from the respective equations of motion.  Indeed, \nrecalling the previous section, some are already known exactly and we will \nnot need to discuss them further.\n\nBefore beginning, let us highlight a basic feature of the Fourier transform \nto momentum space.  We know the commutation or anti-commutation rules for \nour fields\/sources and this will lead to the simplification that we need only \nconsider combinations of fields\/sources and let the commutation rules take \ncare of the permutations.  However, momentum assignments must be uniformly \napplied and this leads to some non-trivial relations.  Consider firstly the \ngeneric proper two-point function $\\ev{\\imath\\Phi_\\alpha(x)\\imath\\Phi_\\beta(y)}$ \nwhere we have $\\Phi_\\beta(y)\\Phi_\\alpha(x)=\\eta\\Phi_\\alpha(x)\\Phi_\\beta(y)$ with \n$\\eta=\\pm1$.  We then have\n\\begin{equation}\n\\ev{\\imath\\Phi_\\alpha(x)\\imath\\Phi_\\beta(y)}\n=\\eta\\ev{\\imath\\Phi_\\beta(y)\\imath\\Phi_\\alpha(x)}\n\\end{equation}\nsuch that in momentum space\n\\begin{equation}\n\\Gamma_{\\alpha\\beta}(k)=\\eta\\Gamma_{\\beta\\alpha}(-k).\n\\end{equation}\nA similar argument applies for connected two-point functions.  The situation \nfor proper three-point functions is slightly less complicated since all \nmomenta are defined as incoming.  Indeed, we have (the $\\delta$-function \nexpressing momentum conservation comes about because of translational \ninvariance)\n\\begin{equation}\n\\ev{\\imath\\Phi_\\alpha\\imath\\Phi_\\beta\\imath\\Phi_\\gamma}\n=\\int\\dk{k_\\alpha}\\dk{k_\\beta}\\dk{k_\\gamma}(2\\pi)^4\\delta(k_\\alpha+k_\\beta+k_\\gamma)\n\\Gamma_{\\alpha\\beta\\gamma}(k_\\alpha,k_\\beta,k_\\gamma)\ne^{-\\imath k_\\alpha\\cdot x_\\alpha-\\imath k_\\beta\\cdot x_\\beta-\\imath k_\\gamma\\cdot x_\\gamma}\n\\end{equation}\nsuch that, for example\n\\begin{equation}\n\\Gamma_{\\beta\\alpha\\gamma}(k_\\beta,k_\\alpha,k_\\gamma)\n=\\eta_{\\alpha\\beta}\\Gamma_{\\alpha\\beta\\gamma}(k_\\alpha,k_\\beta,k_\\gamma)\n\\end{equation}\nwhere $\\eta_{\\alpha\\beta}$ refers to the sign incurred when swapping $\\alpha$ and \n$\\beta$.\n\nLet us now consider the connected two-point functions.  Setting the coupling \nto zero in the equations of motion (listed in Appendix~\\ref{app:eom}) that \ninvolve interaction terms gives us the following non-trivial relations (the \nsuperscript $\\ev{}^{(0)}$ denotes the tree-level quantity)\n\\begin{eqnarray}\n\\rho_x^a-\\xi_{\\tau x}^a&=&-\\nabla_x^2\\ev{\\imath\\kappa_x^a}^{(0)},\\nonumber\\\\\nJ_{ix}^a&=&-\\nabla_{ix}\\ev{\\imath\\xi_{\\lambda x}^a}^{(0)}\n+\\partial_x^0\\ev{\\imath K_{ix}^a}^{(0)}\n-\\partial_x^0\\nabla_{ix}\\ev{\\imath\\kappa_x^a}^{(0)}\n-\\left[\\delta_{ij}\\nabla_x^2+\\nabla_{ix}\\nabla_{jx}\\right]\n\\ev{\\imath J_{jx}^a}^{(0)},\\nonumber\\\\\n\\eta_x^a&=&\\nabla_x^2\\ev{\\imath\\ov{\\eta}_x^a}^{(0)},\\nonumber\\\\\nK_{ix}^a&=&-\\nabla_{ix}\\ev{\\imath\\xi_{\\tau x}^a}^{(0)}\n+\\ev{\\imath K_{ix}^a}^{(0)}-\\nabla_{ix}\\ev{\\imath\\kappa_x^a}^{(0)}\n-\\partial_x^0\\ev{\\imath J_{ix}^a}^{(0)}-\\nabla_{ix}\\ev{\\imath\\rho_x^a}^{(0)}.\n\\end{eqnarray}\nClearly, the ghost propagator is distinct from the rest, since the ghost \nfield must appear with its antighost counterpart.  The treel-level ghost \npropagator is\n\\begin{equation}\nW_{\\ov{c}c}^{(0)ab}(k)=-\\delta^{ab}\\frac{\\imath}{\\vec{k}^2}.\n\\end{equation}\nThe remaining tree-level propagators in momentum space (without the common \ncolor factor $\\delta^{ab}$) are summarised in Table~\\ref{tab:w0}.  Those \nentries that are underlined are the exact relations considered previously.\n\n\\begin{table}\n\\begin{tabular}{|c||c|c||c|c||c|c|}\\hline\n$W$&$A_j$&$\\pi_j$&$\\sigma$&$\\phi$&$\\lambda$&$\\tau$\\\\\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$A_i$&$t_{ij}(k)\\frac{i}{(k_0^2-\\vec{k}^2)}$&\n$t_{ij}(k)\\frac{(-k^0)}{(k_0^2-\\vec{k}^2)}$&$\\underline{0}$&$\\underline{0}$&\n$\\underline{\\frac{(-k_i)}{\\vec{k}^2}}$&$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\pi_i$&$t_{ij}(k)\\frac{k^0}{(k_0^2-\\vec{k}^2)}$&\n$t_{ij}(k)\\frac{\\imath\\vec{k}^2}{(k_0^2-\\vec{k}^2)}$&$\\underline{0}$&\n$\\underline{0}$&$\\underline{0}$&$\\underline{\\frac{(-k_i)}{\\vec{k}^2}}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\sigma$&$\\underline{0}$&$\\underline{0}$&$\\frac{\\imath}{\\vec{k}^2}$&\n$\\frac{(-\\imath)}{\\vec{k}^2}$&$\\frac{(-k^0)}{\\vec{k}^2}$&$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\phi$&$\\underline{0}$&$\\underline{0}$&$\\frac{(-\\imath)}{\\vec{k}^2}$&0&0&\n$\\underline{\\frac{\\imath}{\\vec{k}^2}}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\lambda$&$\\underline{\\frac{k_j}{\\vec{k}^2}}$&$\\underline{0}$&\n$\\frac{k^0}{\\vec{k}^2}$&0&$\\underline{0}$&$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\tau$&$\\underline{0}$&$\\underline{\\frac{k_j}{\\vec{k}^2}}$&$\\underline{0}$&\n$\\underline{\\frac{\\imath}{\\vec{k}^2}}$&$\\underline{0}$&$\\underline{0}$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:w0}Tree-level propagators (without color factors) in \nmomentum space.  Underlined entries denote exact results.}\n\\end{table}\n\nWe can repeat the analysis for the tree-level proper two-point functions.  \nThe relevant equations are:\n\\begin{eqnarray}\n\\ev{\\imath\\sigma_x^a}^{(0)}-\\ev{\\imath\\tau_x^a}^{(0)}&=&\n\\nabla_x^2\\phi_x^a,\\nonumber\\\\\n\\ev{\\imath A_{ix}^a}^{(0)}&=&\\nabla_{ix}\\lambda_x^a-\\partial_x^0\\pi_{ix}^a\n+\\partial_x^0\\nabla_{ix}\\phi_x^a+\\left[\\delta_{ij}\\nabla_x^2\n-\\nabla_{ix}\\nabla_{jx}\\right]A_{jx}^a,\\nonumber\\\\\n\\ev{\\imath\\ov{c}_x^a}^{(0)}&=&-\\nabla_x^2c_x^a,\\nonumber\\\\\n\\ev{\\imath\\pi_{ix}^a}^{(0)}&=&\\nabla_{ix}\\tau_x^a-\\pi_{ix}^a\n+\\nabla_{ix}\\phi_x^a+\\partial_x^0A_{ix}^a+\\nabla_{ix}\\sigma_x^a.\n\\end{eqnarray}\nThe ghost proper two-point function is\n\\begin{equation}\n\\Gamma_{\\ov{c}c}^{(0)ab}(k)=\\delta^{ab}\\imath\\vec{k}^2.\n\\end{equation}\nThe remaining proper two-point functions are summarised in \nTable~\\ref{tab:g0} where the reader is reminded that all proper functions \ninvolving derivatives with respect to the $\\phi$-field can be constructed \nfrom the corresponding $\\pi$ derivative.\n\n\\begin{table}\n\\begin{tabular}{|c||c|c||c|c||c|c|}\\hline\n$\\Gamma$&$A_j$&$\\pi_j$&$\\sigma$&$\\phi$&$\\lambda$&$\\tau$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$A_i$&$t_{ij}(k)\\imath\\vec{k}^2$&$\\delta_{ij}k^0$&0&\n$\\left\\{-\\imath k^0k_i\\right\\}$&$\\underline{k_i}$&$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\pi_i$&$-k^0\\delta_{ij}$&$\\imath\\delta_{ij}$&$k_i$&$\\left\\{k_i\\right\\}$&\n$\\underline{0}$&$\\underline{k_i}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\sigma$&0&$-k_j$&0&$\\left\\{\\imath\\vec{k}^2\\right\\}$&$\\underline{0}$&\n$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\phi$&$\\left\\{-\\imath k^0k_j\\right\\}$&$\\left\\{-k_j\\right\\}$&\n$\\left\\{\\imath\\vec{k}^2\\right\\}$&$\\left\\{\\imath\\vec{k}^2\\right\\}$&\n$\\underline{0}$&$\\underline{0}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\lambda$&$\\underline{-k_j}$&$\\underline{0}$&$\\underline{0}$&$\\underline{0}$&\n$\\underline{0}$&$\\underline{0}$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\tau$&$\\underline{0}$&$\\underline{-k_j}$&$\\underline{0}$&$\\underline{0}$&\n$\\underline{0}$&$\\underline{0}$\\\\\\hline\n\\end{tabular}\n\\caption{\\label{tab:g0}Tree-level proper two-point functions (without color \nfactors) in momentum space.  Underlined entries denote exact results.  \nBracketed quantities refer to functions that are fully determined by others.}\n\\end{table}\n\nDetermining the tree-level vertices (three- and four-point proper Green's \nfunctions) follows the same pattern as for the two-point functions.  They \nfollow by isolating the parts of the equations of motion that have explicit \nfactors of the coupling $g$ and functionally differentiating.  In momentum \nspace (defining all momenta to be incoming), we have\n\\begin{eqnarray}\n\\Gamma_{\\pi\\sigma A ij}^{(0)abc}&=&-gf^{abc}\\delta_{ij},\\nonumber\\\\\n\\Gamma_{3A ijk}^{(0)abc}(p_a,p_b,p_c)&=&\n-\\imath gf^{abc}\\left[\\delta_{ij}(p_a-p_b)_k+\\delta_{jk}(p_b-p_c)_i\n+\\delta_{ki}(p_c-p_a)_j\\right],\\nonumber\\\\\n\\Gamma_{4A ijkl}^{(0)abcd}&=&-\\imath g^2\\left\\{\\delta_{ij}\\delta_{kl}\n\\left[f^{ace}f^{bde}-f^{ade}f^{cbe}\\right]\n+\\delta_{ik}\\delta_{jl}\\left[f^{abe}f^{cde}-f^{ade}f^{bce}\\right]\n+\\delta_{il}\\delta_{jk}\\left[f^{ace}f^{dbe}f^{abe}f^{cde}\\right]\\right\\}\n,\\nonumber\\\\\n\\Gamma_{\\ov{c}cA i}^{(0)abc}(p_{\\ov{c}},p_c,p_A)&=&-\\imath gf^{abc}p_{\\ov{c}i}.\n\\end{eqnarray}\nWe notice that all the tree-level vertices are independent of the energy.  \nIn addition, there is a tree-level vertex involving $\\phi$ that can be \nconstructed from its counterpart involving $\\vec{\\pi}$ and that reads:\n\\begin{equation}\n\\Gamma_{\\phi\\sigma Ai}^{(0)abc}(p_\\phi,p_\\sigma,p_A)\n=\\imath p_{\\phi j}\\Gamma_{\\pi\\sigma Aji}^{(0)abc}=-\\imath gf^{abc}p_{\\phi i}.\n\\end{equation}\nThis vertex has exactly the same form as the ghost-gluon vertex with the \nincoming $\\phi$-momentum playing the same role as the incoming \n$\\ov{c}$-momentum.  It is worth mentioning that the ghost-gluon, three- and \nfour-gluon vertices are identical to the Landau gauge forms except that only \nthe spatial components of the vectors are present.\n\nLet us now discuss the cancellation of the ghost (energy divergent) sector.  \nIn any Feynman diagram containing a closed ghost loop, there will be an \nassociated energy divergence.  It is a general result that associated with \nany closed loop involving Grassmann-valued fields (ghosts or fermions) there \nwill be a factor of $(-1)$.  However, Green's functions are given by the sum \nof all possible contributing Feynman diagrams.  Since the Feynman rules for \n$W_{\\sigma\\phi}$ and $\\Gamma_{\\phi\\sigma A}$ are identical to $W_{\\ov{c}c}$ and \n$\\Gamma_{\\ov{c}cA}$ we will have, for each closed ghost loop, another loop \ninvolving scalar fields without the factor $(-1)$.  Even before performing \nthe loop integration (and regularisation) the integrands of the two diagrams \nwill cancel exactly.  In this way we see that the energy divergences coming \nfrom the ghost sector will be eliminated, as expected given that the \nFaddeev-Popov determinant can formally be cancelled.  There is one caveat to \nthis.  Whilst we have shown that the energy divergences coming from the \nghost sector have been eliminated, we have not shown that the remaining \nloops involving scalar fields are free of energy divergences (although a \nquick glance at the form of the Dyson--Schwinger  equations later will suffice to see that \nthis is the case at leading order).  We propose to look further into this in \na future publication.\n\n\\section{Decomposition of Two-Point Functions}\n\\setcounter{equation}{0}\nIn order to constrain the possible form of the two-point functions under \ninvestigation we can utilise information about discrete symmetries.  We \nconsider time-reversal and parity and we know that Yang-Mills theory \nrespects both.  Under time-reversal the generic field \n$\\Phi_\\alpha(x^0,\\vec{x})$ is transformed as follows:\n\\begin{equation}\n\\Phi_\\alpha(x^0,\\vec{x})=\\eta_\\alpha\\Phi_\\alpha(-x^0,\\vec{x})\n\\end{equation}\nwhere $\\eta_\\alpha=\\pm1$.  Since the action, \\eq{eq:act}, is invariant under \ntime-reversal (it is a pure number) then by considering each term in turn, \nwe deduce that\n\\begin{equation}\n\\eta_A=\\eta_\\lambda=1,\\;\\;\\;\\;\\eta_\\pi=\\eta_\\tau=\\eta_\\phi=\\eta_\\sigma=-1,\\;\\;\\;\\;\n\\eta_{\\ov{c}}=\\eta_c=\\pm1.\n\\end{equation}\nThe sources have the same transformation properties as the field.  These \nproperties allow us to extract information about the energy dependence of \nGreen's functions.  For instance, we have that\n\\begin{equation}\n\\Gamma_{A\\pi ij}^{ab}(k^0,\\vec{k})=-\\Gamma_{A\\pi ij}^{ab}(-k^0,\\vec{k}),\n\\end{equation}\nfrom which one can infer that\n\\begin{equation}\n\\Gamma_{A\\pi ij}^{ab}(k^0,\\vec{k})=\\delta^{ab}k^0\\Gamma_{A\\pi ij}(k_0^2,\\vec{k})\n\\end{equation}\n(the sign convention is chosen to match the perturbative results).  Aside \nfrom $\\Gamma_{A\\phi}$ which is unambiguously related to $\\Gamma_{A\\pi}$, the only \nother proper two-point function that carries the external factor $k^0$ is\n\\begin{equation}\n\\Gamma_{A\\sigma i}^{ab}(k^0,\\vec{k})=\\delta^{ab}k^0\\Gamma_{A\\sigma i}(k_0^2,\\vec{k}).\n\\end{equation}\nHaving extracted the explicit factors of $k^0$ in the proper two-point \nfuncitons, the (as yet) unknown functions that multiply them are functions \nof $k_0^2$.  Turning to the propagators, we assign the factor $-k^0$ to \n$W_{A\\pi}$, $W_{\\sigma\\lambda}$ and $W_{\\phi\\lambda}$.\n\nThe second discrete symmetry of interest is parity whereby\n\\begin{equation}\n\\Phi_\\alpha(x^0,\\vec{x})=\\eta_\\alpha\\Phi_\\alpha(x^0,-\\vec{x})\n\\end{equation}\nwhere again, $\\eta_\\alpha=\\pm1$.  Again the action is invariant and we deduce \nthat\n\\begin{equation}\n\\eta_A=\\eta_\\pi=-1,\\;\\;\\;\\;\\eta_\\sigma=\\eta_\\phi=\\eta_\\lambda=\\eta_\\tau=1,\\;\\;\\;\\;\n\\eta_{\\ov{c}}=\\eta_c=\\pm1\n\\end{equation}\nwith the sources transforming as the fields.  This symmetry is rather more \nobvious than time-reversal.  The physical sense is that for every vector \nfield (and with an associated spatial index) we have some explicit vector \nfactor (again with the associated spatial index).  Where the vector fields \n$\\vec{A}$ and $\\vec{\\pi}$ occur in the propagators, we use the \ntransversality conditions from earlier to see that the vector-scalar \npropagators must vanish, except those involving the apropriate Lagrange \nmultiplier field.\n\nWhat the above tells us is how to construct the most general allowed forms \nof the two-point functions.  The dressing functions are scalar functions of \nthe positive, scalar arguments $k_0^2$ and $\\vec{k}^2$.  We summarise the \nresults in Tables~\\ref{tab:w1} and \\ref{tab:g1}.  The ghost propagator is \nwritten $W_{\\ov{c}c}^{ab}(k)=-\\delta^{ab}\\imath D_c\/\\vec{k}^2$.  We have nine \nunknown propagator dressing functions.  Inlcuding the proper ghost two-point \nfunction $\\Gamma_{\\ov{c}c}^{ab}(k)=\\delta^{ab}\\imath\\vec{k^2}\\Gamma_c$ we see that \nthere are ten proper two-point dressing functions.  The extra functions \ncomes about because we have used only the propagator form of the identity \n\\eq{eq:stid1} to eleminate $W_{\\lambda\\la}$.\n\n\\begin{table}\n\\begin{tabular}{|c||c|c||c|c||c|c|}\\hline\n$W$&$A_j$&$\\pi_j$&$\\sigma$&$\\phi$&$\\lambda$&$\\tau$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$A_i$&$t_{ij}(k)\\frac{\\imath D_{AA}}{(k_0^2-\\vec{k}^2)}$&\n$t_{ij}(k)\\frac{(-k^0)D_{A\\pi}}{(k_0^2-\\vec{k}^2)}$&0&0&\n$\\frac{(-k_i)}{\\vec{k}^2}$&0\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\pi_i$&$t_{ij}(k)\\frac{k^0D_{A\\pi}}{(k_0^2-\\vec{k}^2)}$&\n$t_{ij}(k)\\frac{\\imath\\vec{k}^2D_{\\pi\\pi}}{(k_0^2-\\vec{k}^2)}$&0&0&0&\n$\\frac{(-k_i)}{\\vec{k}^2}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\sigma$&0&0&$\\frac{\\imath D_{\\sigma\\si}}{\\vec{k}^2}$&\n$\\frac{-\\imath D_{\\sigma\\phi}}{\\vec{k}^2}$&\n$\\frac{(-k^0)D_{\\sigma\\lambda}}{\\vec{k}^2}$&0\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\phi$&0&0&$\\frac{-\\imath D_{\\sigma\\phi}}{\\vec{k}^2}$&\n$\\frac{-\\imath D_{\\phi\\phi}}{\\vec{k}^2}$&\n$\\frac{(-k^0)D_{\\phi\\lambda}}{\\vec{k}^2}$&$\\frac{\\imath}{\\vec{k}^2}$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\lambda$&$\\frac{k_j}{\\vec{k}^2}$&0&$\\frac{k^0D_{\\sigma\\lambda}}{\\vec{k}^2}$&\n$\\frac{k^0D_{\\phi\\lambda}}{\\vec{k}^2}$&0&0\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\tau$&0&$\\frac{k_j}{\\vec{k}^2}$&0&$\\frac{\\imath}{\\vec{k}^2}$&0&0\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:w1}General form of propagators in momentum space.  The \nglobal color factor $\\delta^{ab}$ has been extracted.  All unknown functions \n$D_{\\alpha\\beta}$ are dimensionless, scalar functions of $k_0^2$ and $\\vec{k}^2$.}\n\\end{table}\n\n\\begin{table}\n\\begin{tabular}{|c||c|c||c|c||c|c|}\\hline\n$\\Gamma$&$A_j$&$\\pi_j$&$\\sigma$&$\\phi$&$\\lambda$&$\\tau$\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$A_i$&$t_{ij}(\\vec{k})\\imath\\vec{k}^2\\Gamma_{AA}+\\imath k_ik_j\\ov{\\Gamma}_{AA}$&\n$k^0\\left(\\delta_{ij}\\Gamma_{A\\pi}+l_{ij}(\\vec{k})\\ov{\\Gamma}_{A\\pi}\\right)$&\n$-\\imath k^0k_i\\Gamma_{A\\sigma}$&$\\left\\{-\\imath k^0k_i\\left(\\Gamma_{A\\pi}\n+\\ov{\\Gamma}_{A\\pi}\\right)\\right\\}$&$k_i$&0\\\\\n\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\pi_i$&$-k^0\\left(\\delta_{ij}\\Gamma_{A\\pi}+l_{ij}(\\vec{k})\\ov{\\Gamma}_{A\\pi}\\right)$&\n$\\imath\\delta_{ij}\\Gamma_{\\pi\\pi}+\\imath l_{ij}(\\vec{k})\\ov{\\Gamma}_{\\pi\\pi}$&\n$k_i\\Gamma_{\\pi\\sigma}$&\n$\\left\\{k_i\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)\\right\\}$&0&$k_i$\\\\\n\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\sigma$&$-\\imath k^0k_j\\Gamma_{A\\sigma}$&$-k_j\\Gamma_{\\pi\\sigma}$&\n$\\imath\\vec{k}^2\\Gamma_{\\sigma\\si}$&$\\left\\{\\imath\\vec{k}^2\\Gamma_{\\pi\\sigma}\\right\\}$&0&\n0\\\\\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\phi$&$\\left\\{-\\imath k^0k_j\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)\\right\\}$&\n$\\left\\{-k_j\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)\\right\\}$&\n$\\left\\{\\imath\\vec{k}^2\\Gamma_{\\pi\\sigma}\\right\\}$&\n$\\left\\{-\\imath\\vec{k}^2\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)\n\\right\\}$&0&0\\\\\\hline\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\lambda$&$-k_j$&0&0&0&0&0\\\\\\hline\\rule[-2.4ex]{0ex}{5.5ex}\n$\\tau$&0&$-k_j$&0&0&0&0\\\\\\hline\n\\end{tabular}\n\\caption{\\label{tab:g1}General form of the proper two-point functions in \nmomentum space.   The global color factor $\\delta^{ab}$ has been extracted.  \nAll unknown functions $D_{\\alpha\\beta}$ are dimensionless, scalar functions of \n$k_0^2$ and $\\vec{k}^2$. Bracketed quantities refer to functions that are \nfully determined by others.}\n\\end{table}\n\nObviously the propagator and proper two-point dressing functions are related \nvia the Legendre transform.  Whereas in covariant gauges this relationship \nis merely an inversion, in our case there is considerably more detail.  The \nconnection between the connected and proper two-point functions stems from \nthe observation that\n\\begin{equation}\n\\frac{\\delta\\imath J_\\beta}{\\delta\\imath J_\\alpha}=\\delta_{\\alpha\\beta}\n=-\\imath\\frac{\\delta}{\\delta\\imath J_\\alpha}\\ev{\\imath\\Phi_\\beta}\n=\\frac{\\delta\\Phi_\\gamma}{\\delta\\imath J_\\alpha}\\ev{\\imath\\Phi_\\gamma\\imath\\Phi_\\beta}\n=\\ev{\\imath J_\\alpha\\imath J_\\gamma}\\ev{\\imath\\Phi_\\gamma\\imath\\Phi_\\beta}.\n\\label{eq:leg}\n\\end{equation}\n(Recall here that there is an implicit summation over all discrete indices \nand integration over continuous variables labelled by $\\gamma$.)  The ghost \ntwo-point functions are somewhat special in that once sources are set to \nzero, only ghost-antighost pairs need be considered.  The above relation \nbecomes\n\\begin{equation}\n\\int d^4z\\ev{\\imath\\ov{\\eta}_x^a\\imath\\eta_z^c}\n\\ev{\\imath\\ov{c}_z^c\\imath c_y^b}=\\delta^{ab}\\delta(x-y).\n\\label{eq:ghleg}\n\\end{equation}\nFourier transforming to momentum space and using the decomposition from \nabove, we get that\n\\begin{equation}\nD_c(k_0^2,\\vec{k}^2)\\Gamma_c(k_0^2,\\vec{k}^2)=1\n\\end{equation}\nshowing that the ghost propagator dressing function is simply the inverse of \nthe ghost proper two-point function.  Turning to the rest, we are faced with \na problem akin to matrix inversion in order to see the connection since the \nsum over all the different possible sources\/fields labelled by $\\gamma$ is \nnon-trivial in the above general formula.  The decompositions of the \ntwo-point function do however mitigate the complexity somewhat.  We tabulate \nthe possible combinations of terms in Table~\\ref{tab:leg0}.\n\n\\begin{table}[t]\n\\begin{tabular}{|c||c|c||c|c||c|c|}\\hline\n$\\alpha$,$\\beta$&$\\vec{A}$&$\\vec{\\pi}$&$\\sigma$&$\\phi$&$\\lambda$&$\\tau$\\\\\n\\hline\\hline\n$\\vec{A}$&$\\vec{A}$,$\\vec{\\pi}$,$\\lambda$&$\\vec{A}$,$\\vec{\\pi}$&\n$\\vec{A}$,$\\vec{\\pi}$&$\\vec{A}$,$\\vec{\\pi}$&$\\vec{A}$&$\\vec{\\pi}$\\\\\n\\hline\n$\\vec{\\pi}$&$\\vec{A}$,$\\vec{\\pi}$&$\\vec{A}$,$\\vec{\\pi}$,$\\tau$&\n$\\vec{A}$,$\\vec{\\pi}$&$\\vec{A}$,$\\vec{\\pi}$&$\\vec{A}$&$\\vec{\\pi}$\\\\\n\\hline\\hline\n$\\sigma$&$\\sigma$,$\\phi$,$\\lambda$&$\\sigma$,$\\phi$&$\\sigma$,$\\phi$&$\\sigma$,$\\phi$&---&---\\\\\n\\hline\n$\\phi$&$\\sigma$,$\\phi$,$\\lambda$&$\\sigma$,$\\phi$,$\\tau$&$\\sigma$,$\\phi$&$\\sigma$,$\\phi$&\n---&---\\\\\\hline\\hline\n$\\lambda$&$\\vec{A}$,$\\sigma$,$\\phi$&$\\vec{A}$,$\\sigma$,$\\phi$&$\\vec{A}$,$\\sigma$,$\\phi$&\n$\\vec{A}$,$\\sigma$,$\\phi$&$\\vec{A}$&---\\\\\\hline\n$\\tau$&$\\vec{\\pi}$,$\\phi$&$\\vec{\\pi}$,$\\phi$&$\\vec{\\pi}$,$\\phi$&\n$\\vec{\\pi}$,$\\phi$&---&$\\vec{\\pi}$\\\\\\hline\n\\end{tabular}\n\\caption{\\label{tab:leg0}Possible terms for the equations relating \npropagator and proper two-point functions stemming from the Legendre \ntransform.  Entries denote the allowed field types $\\gamma$ in \\eq{eq:leg}.}\n\\end{table}\n\nWe start by considering the top left components of Table~\\ref{tab:leg0} \ninvolving only $\\vec{A}$, $\\vec{\\pi}$ and the known functions with $\\lambda$ and \n$\\tau$.  After decomposition, we have (suppressing the common argument $k$)\n\\begin{eqnarray}\nk_0^2D_{A\\pi}\\Gamma_{A\\pi}-\\vec{k}^2D_{AA}\\Gamma_{AA}&=&k_0^2-\\vec{k}^2,\\nonumber\\\\\nk_0^2D_{A\\pi}\\Gamma_{A\\pi}-\\vec{k}^2D_{\\pi\\pi}\\Gamma_{\\pi\\pi}&=&k_0^2-\\vec{k}^2\n,\\nonumber\\\\\nD_{AA}\\Gamma_{A\\pi}-D_{A\\pi}\\Gamma_{\\pi\\pi}&=&0,\\nonumber\\\\\nD_{A\\pi}\\Gamma_{AA}-D_{\\pi\\pi}\\Gamma_{A\\pi}&=&0.\n\\end{eqnarray}\nWe can thus express the propagator functions $D$ in terms of the proper \ntwo-point functions $\\Gamma$ and we have\n\\begin{eqnarray}\nD_{AA}&=&\\frac{\\left(k_0^2-\\vec{k}^2\\right)\\Gamma_{\\pi\\pi}}{\n\\left(k_0^2\\Gamma_{A\\pi}^2-\\vec{k}^2\\Gamma_{AA}\\Gamma_{\\pi\\pi}\\right)},\\nonumber\\\\\nD_{\\pi\\pi}&=&\\frac{\\left(k_0^2-\\vec{k}^2\\right)\\Gamma_{AA}}{\n\\left(k_0^2\\Gamma_{A\\pi}^2-\\vec{k}^2\\Gamma_{AA}\\Gamma_{\\pi\\pi}\\right)},\\nonumber\\\\\nD_{A\\pi}&=&\\frac{\\left(k_0^2-\\vec{k}^2\\right)\\Gamma_{A\\pi}}{\n\\left(k_0^2\\Gamma_{A\\pi}^2-\\vec{k}^2\\Gamma_{AA}\\Gamma_{\\pi\\pi}\\right)}.\n\\end{eqnarray}\nClearly, these expressions can be inverted to give the functions $\\Gamma$ in \nterms of the functions $D$.  Next, let us consider the central components of \nTable~\\ref{tab:leg0} involving only $\\sigma$ and $\\phi$.  We get the following \nequations:\n\\begin{eqnarray}\n-D_{\\sigma\\si}\\Gamma_{\\sigma\\si}+D_{\\sigma\\phi}\\Gamma_{\\pi\\sigma}&=&1,\\nonumber\\\\\nD_{\\sigma\\phi}\\Gamma_{\\pi\\sigma}+D_{\\phi\\phi}\\left(\\Gamma_{\\pi\\pi}\n+\\ov{\\Gamma}_{\\pi\\pi}\\right)&=&1,\\nonumber\\\\\n-D_{\\sigma\\si}\\Gamma_{\\pi\\sigma}+D_{\\sigma\\phi}\\left(\\Gamma_{\\pi\\pi}\n+\\ov{\\Gamma}_{\\pi\\pi}\\right)&=&0,\\nonumber\\\\\nD_{\\sigma\\phi}\\Gamma_{\\sigma\\si}+D_{\\phi\\phi}\\Gamma_{\\pi\\sigma}&=&0.\n\\end{eqnarray}\nThe propagator functions in terms of the proper two-point functions are then\n\\begin{eqnarray}\nD_{\\sigma\\si}&=&\\frac{\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)}{\n\\Gamma_{\\pi\\sigma}^2-\\Gamma_{\\sigma\\si}\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)}\n,\\nonumber\\\\\nD_{\\phi\\phi}&=&-\\frac{\\Gamma_{\\sigma\\si}}{\\Gamma_{\\pi\\sigma}^2-\\Gamma_{\\sigma\\si}\n\\left(\\Gamma_{\\pi\\pi}+\\ov{\\Gamma}_{\\pi\\pi}\\right)},\\nonumber\\\\\nD_{\\sigma\\phi}&=&\\frac{\\Gamma_{\\pi\\sigma}}{\\Gamma_{\\pi\\sigma}^2-\\Gamma_{\\sigma\\si}\\left(\\Gamma_{\\pi\\pi}\n+\\ov{\\Gamma}_{\\pi\\pi}\\right)}.\n\\end{eqnarray}\nThere are three more equations that are of interest.  These are the $\\sigma-A$, \n$\\phi-A$ and $\\lambda-A$ entries of Table~\\ref{tab:leg0} and they read:\n\\begin{eqnarray}\nD_{\\sigma\\si}\\Gamma_{A\\sigma}-D_{\\sigma\\phi}\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)\n+D_{\\sigma\\lambda}&=&0,\\\\\n-D_{\\sigma\\phi}\\Gamma_{A\\sigma}-D_{\\phi\\phi}\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)\n+D_{\\phi\\lambda}&=&0,\\\\\n\\ov{\\Gamma}_{AA}-\\frac{k_0^2}{\\vec{k}^2}\\left[D_{\\sigma\\lambda}\\Gamma_{A\\sigma}\n+D_{\\phi\\lambda}\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)\\right]&=&0.\n\\end{eqnarray}\nWhat these equations tell us is that $D_{\\sigma\\lambda}$ and $D_{\\phi\\lambda}$ are \nrelated to $\\ov{\\Gamma}_{A\\pi}$ and $\\Gamma_{A\\sigma}$ with all other coefficients \nbeing determined.  $\\ov{\\Gamma}_{AA}$ is then given as a specific combination \nand is the `extra' proper two-point function alluded to earlier.  However, \nthese functions will not be of any real concern since $D_{\\sigma\\lambda}$ and \n$D_{\\phi\\lambda}$ do not enter any loop diagrams of the Dyson--Schwinger  equations.  In \neffect, $\\ov{\\Gamma}_{A\\pi}$, $\\Gamma_{A\\sigma}$ and $\\ov{\\Gamma}_{AA}$ form a consistency \ncheck on the truncation of the Dyson--Schwinger  equations since we have that\n\\begin{equation}\n\\ov{\\Gamma}_{AA}=\\frac{k_0^2}{\\vec{k}^2}\\left[-D_{\\sigma\\si}\\Gamma_{A\\sigma}^2\n+2D_{\\sigma\\phi}\\Gamma_{A\\sigma}\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)\n+D_{\\phi\\phi}\\left(\\Gamma_{A\\pi}+\\ov{\\Gamma}_{A\\pi}\\right)^2\\right].\n\\end{equation}\nIt is apparent that unlike covariant gauges, the proper two-point function \nfor the gluon is not necessarily transverse.\n\nIn summary, leaving the problem of the vertices aside, in order to solve the \ntwo-point Dyson--Schwinger  equations we need to calculate seven proper two-point \nfunctions:\n\\begin{equation}\n\\Gamma_c,\\;\\;\\Gamma_{AA},\\;\\;\\Gamma_{A\\pi},\\;\\;\\Gamma_{\\pi\\pi},\\;\\;\\ov{\\Gamma}_{\\pi\\pi},\\;\\;\n\\Gamma_{\\sigma\\si},\\;\\;\\Gamma_{\\sigma\\pi},\n\\end{equation}\nwhich will give us the required propagator functions:\n\\begin{equation}\nD_c,\\;\\;D_{AA},\\;\\;D_{A\\pi},\\;\\;D_{\\pi\\pi},\\;\\;D_{\\sigma\\si},\\;\\;D_{\\sigma\\phi}\n,\\;\\;D_{\\phi\\phi}.\n\\end{equation}\nThe three proper two-point functions $\\ov{\\Gamma}_{AA}$, $\\ov{\\Gamma}_{A\\pi}$ and \n$\\Gamma_{A\\sigma}$ give a consistency check on any truncation scheme but do not \ndirectly contribute further.\n\n\\section{Derivation of the Propagator Dyson--Schwinger  Equations}\n\\setcounter{equation}{0}\nIn this section, we present the explicit derivation of the relevant Dyson--Schwinger  \nequations for proper two-point functions.\n\n\\subsection{Ghost Equations}\nAs will be shown in this subsection, the ghost sector of the theory plays a \nrather special role.  We will begin by deriving the ghost Dyson--Schwinger  equation (this \nwill serve as a template for the derivation of the other Dyson--Schwinger  equations).  \nWith this it is possible to point out two particular features of the ghost \nsector: that the ghost-gluon vertex is UV finite and that the energy ($k^0$ \ncomponent) argument of any ghost line is irrelevant, i.e., that any proper \nfunction involving ghost fields is independent of the ghost energy.\n\nThe derivation of the Dyson--Schwinger  equation for the ghost proper two-point function \nbegins with \\eq{eq:ghost0}.  Taking the functional derivative with respect \nto $\\imath c_w^d$, using the configuration space definition of the \ntree-level ghost-gluon vertex and omitting terms which will eventually \nvanish when sources are set to zero, we have\n\\begin{equation}\n\\ev{\\imath c_w^d\\imath\\ov{c}_x^a}=\\imath\\delta^{da}\\nabla_x^2\\delta(w-x)\n+\\int\\dx{y}\\dx{z}\\Gamma_{\\ov{c}cAi}^{(0)abc}(x,y,z)\\frac{\\delta}{\\delta\\imath c_w^d}\n\\ev{\\imath\\ov{\\eta}_y^b\\imath J_{iz}^c}.\n\\end{equation}\nUsing partial differentiation we see that\n\\begin{equation}\n\\frac{\\delta}{\\delta\\imath c_w^d}\\ev{\\imath\\ov{\\eta}_y^b\\imath J_{iz}^c}\n=-\\imath\\int\\dx{v}\\ev{\\imath J_{iz}^c\\imath\\ov{\\eta}_y^b\\imath\\eta_v^e}\n\\ev{\\imath\\ov{c}_v^e\\imath c_w^d}.\n\\end{equation}\nIn the above we have again used the fact that when sources are set to zero, \nthe only ghost functions that survive are those with pairs of \nghost-antighost fields.  Since the ghost fields anticommute, we get that\n\\begin{equation}\n\\ev{\\imath\\ov{c}_x^a\\imath c_w^d}=-\\imath\\delta^{ad}\\nabla_x^2\\delta(x-w)\n+\\imath\\int\\dx{y}\\dx{z}\\dx{v}\\Gamma_{\\ov{c}cAi}^{(0)abc}(x,y,z)\n\\ev{\\imath J_{iz}^c\\imath\\ov{\\eta}_y^b\\imath\\eta_v^e}\n\\ev{\\imath\\ov{c}_v^e\\imath c_w^d}\n\\end{equation}\nTaking the partial derivative of \\eq{eq:ghleg} with respect to \n$\\imath J_{iz}^c$ we have (notice that when using partial derivatives here, \nwe must include all possible contributions which for clarity are included \nexplicitly here):\n\\begin{equation}\n\\int\\dx{v}\\ev{\\imath J_{iz}^c\\imath\\ov{\\eta}_y^b\\imath\\eta_v^e}\n\\ev{\\imath\\ov{c}_v^e\\imath c_w^d}=-\\imath\\int\\dx{v}\\dx{u}\n\\ev{\\imath\\ov{\\eta}_y^b\\imath\\eta_v^e}\\left\\{\n\\ev{\\imath J_{iz}^c\\imath J_{ju}^f}\n\\ev{\\imath A_{ju}^f\\imath\\ov{c}_v^e\\imath c_w^d}\n+\\ev{\\imath J_{iz}^c\\imath K_{ju}^f}\n\\ev{\\imath\\pi_{ju}^f\\imath\\ov{c}_v^e\\imath c_w^d}\\right\\}.\n\\end{equation}\nOur ghost Dyson--Schwinger  equation in configuration space is thus\n\\begin{eqnarray}\n\\lefteqn{\\ev{\\imath\\ov{c}_x^a\\imath c_w^d}\n=-\\imath\\delta^{ad}\\nabla_x^2\\delta(x-w)}\\nonumber\\\\&&\n+\\int\\dx{y}\\dx{z}\\dx{v}\\dx{u}\\Gamma_{\\ov{c}cAi}^{(0)abc}(x,y,z)\n\\ev{\\imath\\ov{\\eta}_y^b\\imath\\eta_v^e}\\left\\{\n\\ev{\\imath J_{iz}^c\\imath J_{ju}^f}\n\\ev{\\imath A_{ju}^f\\imath\\ov{c}_v^e\\imath c_w^d}\n+\\ev{\\imath J_{iz}^c\\imath K_{ju}^f}\n\\ev{\\imath\\pi_{ju}^f\\imath\\ov{c}_v^e\\imath c_w^d}\\right\\}.\n\\end{eqnarray}\nWe Fourier transform this result to get the Dyson--Schwinger  equation for the proper \ntwo-point ghost function in momentum space:\n\\begin{eqnarray}\n\\lefteqn{\\Gamma_c^{ad}(k)=\\delta^{ad}\\imath\\vec{k}^2}\\nonumber\\\\\n&&-\\int\\left(-\\dk{\\omega}\\right)\\Gamma_{\\ov{c}cAi}^{(0)abc}(k,-\\omega,\\omega-k)\nW_c^{be}(\\omega)\\left\\{W_{AAij}^{cf}(k-\\omega)\\Gamma_{\\ov{c}cAj}^{edf}(\\omega,-k,k-\\omega)\n+W_{A\\pi ij}^{cf}(k-\\omega)\\Gamma_{\\ov{c}c\\pi j}^{edf}(\\omega,-k,k-\\omega)\\right\\}\n.\\nonumber\\\\\n\\label{eq:ghdse0}\n\\end{eqnarray}\nWith the convention that the self-energy term on the right-hand side has an \noverall minus sign, we identify $\\left(-\\dk{\\omega}\\right)$ as the loop \nintegration measure in momentum space.\n\nWith any two-point Dyson--Schwinger  equation, it is clear that there are two orderings \nfor the functional derivatives on the left-hand side.  In the same way, \nthere are three orderings for three-point functions and so on.  This means \nthat there are $n$ different equations for the $n$-point proper Green's \nfunctions, although obviously they all have the same solution and must be \nrelated in some way.  It is therefore instructive to consider also the \nequation generated by the reverse ordering to see if this will have any \nconsequence.  In the ghost case, this means repeating the above analysis but \nstarting with the second ghost equation of motion, \\eq{eq:ghdse2}.  The \ncorresponding Dyson--Schwinger  equation in momentum space is\n\\begin{eqnarray}\n\\lefteqn{\\Gamma_c^{ad}(k)=\\delta^{ad}\\imath\\vec{k}^2}\\nonumber\\\\&&\n-\\int\\left(-\\dk{\\omega}\\right)\\left\\{W_{AAij}^{fc}(\\omega)\n\\Gamma_{\\ov{c}cAj}^{abc}(k,-k-\\omega,\\omega)+W_{A\\pi ij}^{fc}(\\omega)\n\\Gamma_{\\ov{c}c\\pi j}^{abc}(k,-k-\\omega,\\omega)\\right\\}W_c^{be}(k+\\omega)\n\\Gamma_{\\ov{c}cAi}^{(0)edf}(k+\\omega,-k,-\\omega).\\nonumber\\\\\n\\label{eq:ghdse1}\n\\end{eqnarray}\nThis equation is formally equivalent to \\eq{eq:ghdse0} but we notice that \nthe ordering of the dressed vertices is different.  (It is useful to check \nthat the two equations are the same by taking both vertices to be bare such \nthat the equivalence is manifest.)\n\nNotice that one of the vertices that form the loop term(s) must be bare.  \nThis arises naturally through the derivation above and if one considers a \nperturbative expansion it is crucial to avoid overcounting of graphs.  The \nchoice of which vertex is bare is arbitrary and related to the fact that \nthere are $n$ ways of writing the equation for an $n$-point function.  Given \nthat for any loop term we can extract a single bare vertex, for any \nthree-point function involving a ghost-antighost pair we will have a loop \nterm with the following structure (see also Figure~\\ref{fig:ghost}):\n\\begin{equation}\n\\int\\dk{\\omega}\\Gamma_{\\alpha\\beta\\ov{c}cj}^{dgac}(\\omega,p,k-p,-k-\\omega)W_c^{ce}(k+\\omega)\nW_{A\\alpha ij}^{fd}(\\omega)\\Gamma_{\\ov{c}cAi}^{(0)ebf}(k+\\omega,-k,-\\omega).\n\\end{equation}\nNow, since the only propagators involving $A$ are transverse (the $W_{A\\lambda}$ \npropagator is disallowed since no proper vertex function with \n$\\lambda$-derivative exists) the loop term must vanish as $k\\rightarrow0$ for \nfinite $p$\\footnote{The possibility that \n$\\Gamma_{\\Phi_{\\alpha}\\Phi_{\\beta}\\ov{c}c}^{dgac}(\\omega,p,k-p,-k-\\omega)$ is also singular \nin this limit such that the loop remains finite is discounted since at \nfinite $\\omega$ and $p$ this would imply that some colored bound state \nexists.}.  Since the loop term vanishes under some finite, kinematical \nconfiguration, an UV divergence (which is independent of the kinematical \nconfiguration) cannot occur and we can can say that this vertex is UV \nfinite.  It is tempting to think that such an argument applies to the \ntwo-point ghost equation, however this is false since whilst the loop term \nvanishes, so does the $\\vec{k}^2$ factor that multiplies the rest of the \nequation.\n\n\\begin{figure}[t]\n\\includegraphics{ghostvert.ps}\n\\caption{\\label{fig:ghost} A diagrammatical representation of the \n$\\Gamma_{\\ov{c}c\\beta}(k-p,-k,p)$ proper vertex dressing.  Because of the form of \nthe tree-level ghost-gluon vertex and the tranversality of the vector \npropagator, the dressing function vanishes in the limit $k\\rightarrow0$.  \nFilled blobs denote dressed propagators and empty circles denote dressed \nproper vertex functions.  Wavy lines denote proper functions, springs deonte \nconnected (propagator) functions and dashed lines denote the ghost \npropagator.}\n\\end{figure}\n\nLet us now show that any Green's function involving a ghost-antighost pair \nis independent of the ghost and antighost energies.  The proof of this is \nperturbative in nature.  We notice that both the tree-level ghost propagator \nand the ghost-gluon vertex are independent of the energy.  This means that \nin any one-loop diagram which has at least one internal ghost propagator \n(and hence at least two ghost-gluon vertices) the energy scale associated \nwith the ghost propagator is absent.  Using energy conservation another \nenergy scale can be eliminated and we choose this to be the antighost \nenergy.  At two-loops, we now have the situation whereby the dressed \ninternal ghost propagator is again independent of the energy and the dressed \nghost-gluon vertex only depends on the gluon energy and so the argument can \nbe repeated.  This can be applied to all orders in the perturbative \nexpansion which completes the proof.  We thus have that in particular\n\\begin{eqnarray}\nD_c(k_0^2,\\vec{k}^2)&=&D_c(\\vec{k}^2)\\nonumber\\\\\n\\Gamma_{A\\ov{c}ci}(k_1,k_2,k_3)&=&\\Gamma_{A\\ov{c}ci}(k_1,\\vec{k}_2,\\vec{k}_3).\n\\end{eqnarray}\n\n\\subsection{$\\sigma$-based Equation}\nGiven the discussion in the previous section about which proper two-point \nfunctions are relevant, there are only two proper two-point functions \ninvolving derivatives with respect to $\\sigma$ to consider --- \n$\\ev{\\imath\\sigma\\imath\\sigma}$ and $\\ev{\\imath\\sigma\\imath\\pi}$.  Since the \n$\\sigma$-based equation of motion, \\eq{eq:sidse0}, involves two interaction terms \nwhereas the $\\pi$-based equation, \\eq{eq:pidse0}, has only one, we use the \nderivatives of \\eq{eq:pidse0} to derive the Dyson--Schwinger  equation for \n$\\ev{\\imath\\sigma\\imath\\pi}$ (see next subsection).  We therefore consider the \nfunctional derivative of \\eq{eq:sidse0} with respect to $\\imath\\sigma_w^d$, \nafter which the sources will be set to zero.  We have, again identifying the \ntree-level vertices,\n\\begin{equation}\n\\ev{\\imath\\sigma_w^d\\imath\\sigma_x^a}=-\\int\\dx{y}\\dx{z}\n\\Gamma_{\\pi\\sigma Aij}^{(0)cab}(z,x,y)\\frac{\\delta}{\\delta\\imath\\sigma_w^d}\n\\ev{\\imath J_{jy}^b\\imath K_{iz}^c}-\\int\\dx{y}\\dx{z}\n\\Gamma_{\\phi\\sigma Ai}^{(0)cab}(z,x,y)\\frac{\\delta}{\\delta\\imath\\sigma_w^d}\n\\ev{\\imath J_{iy}^b\\imath\\kappa_z^c}.\n\\end{equation}\nUsing partial differentiation, and with compact notation,\n\\begin{equation}\n\\frac{\\delta}{\\delta\\imath\\sigma_w^d}\\ev{\\imath J_{jy}^b\\imath K_{iz}^c}\n=-\\ev{\\imath K_{iz}^c\\imath J_\\alpha}\\ev{\\imath J_{jy}^b\\imath J_\\beta}\n\\ev{\\imath\\Phi_\\beta\\imath\\Phi_\\alpha\\imath\\sigma_w^d}\n\\end{equation}\n(similarly for the second term).  This gives the Dyson--Schwinger  equation in \nconfiguration space:\n\\begin{eqnarray}\n\\ev{\\imath\\sigma_w^d\\imath\\sigma_x^a}&=&\\int\\dx{y}\\dx{z}\n\\Gamma_{\\pi\\sigma Aij}^{(0)cab}(z,x,y)\\ev{\\imath K_{iz}^c\\imath J_\\alpha}\n\\ev{\\imath J_{jy}^b\\imath J_\\beta}\n\\ev{\\imath\\Phi_\\beta\\imath\\Phi_\\alpha\\imath\\sigma_w^d}\\nonumber\\\\&&\n+\\int\\dx{y}\\dx{z}\\Gamma_{\\phi\\sigma Ai}^{(0)cab}(z,x,y)\n\\ev{\\imath\\kappa_z^c\\imath J_\\alpha}\\ev{\\imath J_{iy}^b\\imath J_\\beta}\n\\ev{\\imath\\Phi_\\beta\\imath\\Phi_\\alpha\\imath\\sigma_w^d}\n\\end{eqnarray}\nTaking the Fourier transform and tidying-up indices, the Dyson--Schwinger  equation in \nmomentum space is thus\n\\begin{eqnarray}\n\\Gamma_{\\sigma\\si}^{ad}(k)&=&-\\int\\left(-\\dk{\\omega}\\right)\n\\Gamma_{\\pi\\sigma Aij}^{(0)cab}(\\omega-k,k,-\\omega)W_{A\\beta jl}^{be}(\\omega)\n\\Gamma_{\\beta\\alpha\\sigma lk}^{efd}(\\omega,k-\\omega,-k)W_{\\alpha\\pi ki}^{fc}(\\omega-k)\\nonumber\\\\\n&&-\\int\\left(-\\dk{\\omega}\\right)\\Gamma_{\\phi\\sigma Ai}^{(0)cab}(\\omega-k,k,-\\omega)\nW_{A\\beta ij}^{be}(\\omega)\\Gamma_{\\beta\\alpha\\sigma j}^{efd}(\\omega,k-\\omega,-k)\nW_{\\alpha\\phi}^{fc}(\\omega-k).\n\\label{eq:sidse1}\n\\end{eqnarray}\nA couple of remarks are in order here.  Firstly, there is no bare term on \nthe right-hand side because the action, under the first order formalism, is \nlinear in $\\sigma$.  Secondly, the implicit summation over the terms labelled \nby $\\alpha$ and $\\beta$ means that in fact there are eight possible loop terms \ncomprising the self-energy.  However, only two of these involves a \nprimitively divergent vertex.  It is an uncomfortable truth that the formal, \nnon-local delta function constraint arising from the linearity of the action \nin $\\sigma$ blossoms into a large set of local self-energy integrals.\n\n\\subsection{$\\pi$-based equations}\nSince the $\\pi$-based equation of motion, \\eq{eq:pidse0}, contains only a \nsingle interaction term, we favor it to calculate the \n$\\ev{\\imath A\\imath\\pi}$ and $\\ev{\\imath\\pi\\imath\\sigma}$ proper two-point \nfunctions (as well as $\\ev{\\imath\\pi\\imath\\pi}$).  As discussed previously, \nwe could in principle calculate these from the $A$-based and $\\sigma$-based \nequations as well in order to check the veracity of any truncations used and \nin fact, this connection may serve useful in elucidating constraints on the \nform of the truncated vertices used.  Perturbatively, all equations will \nprovide the same result at any given order.\n\nUsing the same techniques as in the last subsection, we get the following \nDyson--Schwinger  equations in momentum space:\n\\begin{eqnarray}\n\\Gamma_{\\pi\\sigma i}^{ad}(k)&=&\\delta^{ad}k_i-\\int(-\\dk{\\omega})\n\\Gamma_{\\pi\\sigma Aij}^{(0)abc}(k,-\\omega,\\omega-k)W_{\\sigma\\beta}^{be}(\\omega)\n\\Gamma_{\\beta\\alpha\\sigma l}^{efd}(\\omega,k-\\omega,-k)W_{\\alpha Alj}^{fc}(\\omega-k),\\\\\n\\Gamma_{\\pi Aik}^{ad}(k)&=&-\\delta^{ad}k^0\\delta_{ik}-\\int(-\\dk{\\omega})\n\\Gamma_{\\pi\\sigma Aij}^{(0)abc}(k,-\\omega,\\omega-k)W_{\\sigma\\beta}^{be}(\\omega)\n\\Gamma_{\\beta\\alpha Alk}^{efd}(\\omega,k-\\omega,-k)W_{\\alpha Alj}^{fc}(\\omega-k),\\\\\n\\Gamma_{\\pi\\pi ik}^{ad}(k)&=&\\imath\\delta^{ad}\\delta_{ik}-\\int(-\\dk{\\omega})\n\\Gamma_{\\pi\\sigma Aij}^{(0)abc}(k,-\\omega,\\omega-k)W_{\\sigma\\beta}^{be}(\\omega)\n\\Gamma_{\\beta\\alpha\\pi lk}^{efd}(\\omega,k-\\omega,-k)W_{\\alpha Alj}^{fc}(\\omega-k).\n\\end{eqnarray}\nAgain, notice that the summation over the allowed types of fields indicated \nby $\\alpha$ and $\\beta$ leads to multiple possibilities.\n\n\\subsection{$A$-based equation}\nUsing the tree-level forms for the vertices and discarding those terms which \nwill eventually vanish when sources are set to zero, it is possible to \nrewrite \\eq{eq:adse0} as\n\\begin{eqnarray}\n\\ev{\\imath A_{ix}^a}&=&\n\\left[\\delta_{ij}\\nabla_x^2-\\nabla_{ix}\\nabla_{jx}\\right]A_{jx}^a\n+\\int\\dx{y}\\dx{z}\\Gamma_{\\ov{c}cAi}^{(0)bca}(z,y,x)\n\\ev{\\imath\\ov{\\eta}_y^c\\imath\\eta_z^b}-\\int\\dx{y}\\dx{z}\n\\Gamma_{\\phi\\sigma Ai}^{(0)bca}(z,y,x)\\ev{\\imath\\rho_y^c\\imath\\kappa_z^b}\n\\nonumber\\\\&&\n-\\int\\dx{y}\\dx{z}\\Gamma_{\\pi\\sigma Aji}^{(0)bca}(z,y,x)\n\\ev{\\imath\\rho_y^c\\imath K_{jz}^b}\n-\\int\\dx{y}\\dx{z}\\frac{1}{2}\\Gamma_{3Akji}^{(0)bca}(z,y,x)\n\\ev{\\imath J_{jy}^c\\imath J_{kz}^b}\n\\nonumber\\\\&&\n-\\int\\dx{y}\\dx{z}\\dx{w}\\frac{1}{6}\\Gamma_{4Alkji}^{(0)dcba}(w,z,y,x)\n\\left[3\\imath A_{jy}^b\\ev{\\imath J_{kz}^c\\imath J_{lw}^d}\n+\\imath\\ev{\\imath J_{jy}^b\\imath J_{kz}^c\\imath J_{lw}^d}\\right].\n\\end{eqnarray}\nFunctionally differentiating this with respect to $A$ and proceeding as \nbefore, noting the following for the four-gluon connected vertex\n\\begin{eqnarray}\n\\imath\\frac{\\delta}{\\delta\\imath A_{mv}^e}\n\\ev{\\imath J_{jy}^b\\imath J_{kz}^c\\imath J_{lw}^d}&=&\n-\\ev{\\imath J_{kz}^c\\imath J_\\nu}\n\\ev{\\imath A_{mv}^e\\imath\\Phi_\\nu\\imath\\Phi_\\mu}\n\\ev{\\imath J_\\mu\\imath J_\\gamma}\\ev{\\imath J_{jy}^b\\imath J_\\lambda}\n\\ev{\\imath\\Phi_\\lambda\\imath\\Phi_\\gamma\\imath\\Phi_\\delta}\n\\ev{\\imath J_\\delta\\imath J_{lw}^d}\n\\nonumber\\\\&&\n-\\ev{\\imath J_{kz}^c\\imath J_\\gamma}\\ev{\\imath J_{jy}^b\\imath J_\\nu}\n\\ev{\\imath A_{mv}^e\\imath\\Phi_\\nu\\imath\\Phi_\\mu}\n\\ev{\\imath J_\\mu\\imath J_\\lambda}\n\\ev{\\imath\\Phi_\\lambda\\imath\\Phi_\\gamma\\imath\\Phi_\\delta}\n\\ev{\\imath J_\\delta\\imath J_{lw}^d}\n\\nonumber\\\\&&\n-\\ev{\\imath J_{kz}^c\\imath J_\\gamma}\\ev{\\imath J_{jy}^b\\imath J_\\lambda}\n\\ev{\\imath\\Phi_\\lambda\\imath\\Phi_\\gamma\\imath\\Phi_\\delta}\n\\ev{\\imath J_\\delta\\imath J_\\mu}\n\\ev{\\imath A_{mv}^e\\imath\\Phi_\\mu\\imath\\Phi_\\nu}\n\\ev{\\imath J_\\nu\\imath J_{lw}^d}\n\\nonumber\\\\&&\n+\\ev{\\imath J_{kz}^c\\imath J_\\gamma}\\ev{\\imath J_{jy}^b\\imath J_\\lambda}\n\\ev{\\imath A_{mv}^e\\imath\\Phi_\\lambda\\imath\\Phi_\\gamma\\imath\\Phi_\\delta}\n\\ev{\\imath J_\\delta\\imath J_{lw}^d}\n\\end{eqnarray}\ngives the gluon Dyson--Schwinger  equation which in momentum space reads:\n\\begin{eqnarray}\n\\Gamma_{AAim}^{ae}(k)&=&\n\\imath\\delta^{ae}\\left[\\vec{k}^2\\delta_{im}-k_ik_m\\right]\n+\\int(-\\dk{\\omega})\\Gamma_{\\ov{c}cA i}^{(0)bca}(\\omega-k,-\\omega,k)W_c^{cd}(\\omega)\n\\Gamma_{\\ov{c}cAm}^{dfe}(\\omega,k-\\omega,-k)W_c^{fb}(\\omega-k)\n\\nonumber\\\\&&\n-\\int(-\\dk{\\omega})\\Gamma_{\\phi\\sigma Ai}^{(0)bca}(\\omega-k,-\\omega,k)W_{\\sigma\\beta}^{cd}(\\omega)\n\\Gamma_{\\beta\\alpha Am}^{dfe}(\\omega,k-\\omega,-k)W_{\\alpha\\phi}^{fb}(\\omega-k)\n\\nonumber\\\\&&\n-\\int(-\\dk{\\omega})\\Gamma_{A\\pi\\sigma ij}^{(0)bca}(\\omega-k,-\\omega,k)W_{\\sigma\\beta}^{cd}(\\omega)\n\\Gamma_{\\beta\\alpha A km}^{dfe}(\\omega,k-\\omega,-k)W_{\\alpha\\pi kj}^{fb}(\\omega-k)\n\\nonumber\\\\&&\n-\\frac{1}{2}\\int(-\\dk{\\omega})\\Gamma_{3Akji}^{(0)bca}(\\omega-k,-\\omega,k)\nW_{A\\beta jl}^{cd}(\\omega)\\Gamma_{\\beta\\alpha Alnm}^{dfe}(\\omega,k-\\omega,-k)W_{\\alpha Ank}^{fb}(\\omega-k)\n\\nonumber\\\\&&\n-\\frac{1}{6}\\int(-\\dk{\\omega})(-\\dk{v})\n\\Gamma_{4Alkji}^{(0)dcba}(-v,-\\omega,v+\\omega-k,k)W_{A\\lambda jn}^{bf}(k-v-\\omega)\nW_{A\\gamma ko}^{cg}(\\omega)W_{A\\delta lp}^{dh}(v)\\times\n\\nonumber\\\\&&\n\\Gamma_{\\lambda\\gamma\\delta Anopm}^{fghe}(k-\\omega-v,\\omega,v,-k)\n\\nonumber\\\\&&\n+\\frac{1}{2}\\int(-\\dk{\\omega})\\Gamma_{4Aimlk}^{(0)aecd}(k,-k,\\omega,-\\omega)\nW_{AAkl}^{cd}(-\\omega)\n\\nonumber\\\\&&\n+\\frac{1}{2}\\int(-\\dk{\\omega})(-\\dk{v})\\Gamma_{4Alkji}^{(0)dcba}(-v,-\\omega,v+\\omega-k,k)\nW_{A\\delta ln}^{df}(v)W_{A\\gamma ko}^{cg}(\\omega)\\Gamma_{\\delta\\gamma\\lambda nop}^{fgh}(v,\\omega,-v-\\omega)\n\\times\n\\nonumber\\\\&&\nW_{\\lambda\\mu pq}^{hi}(v+\\omega)\\Gamma_{\\mu\\nu Aqrm}^{ije}(v+\\omega,k-v-\\omega,-k)\nW_{\\nu Arj}^{jd}(\\omega+v-k).\n\\end{eqnarray}\nAgain, the occurence of the summation over $\\alpha,\\ldots,\\lambda$ leads to many \ndifferent possible loop terms.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\linewidth]{dse.ps}\n\\caption{\\label{fig:dses} A diagrammatical representation of the coupled \nsystem of Dyson--Schwinger  equations.  Filled blobs denote dressed propagators and empty \ncircles denote dressed proper vertex functions.  Wavy lines denote proper \nfunctions, springs denote connected (propagator) functions and dashed lines \ndenote the ghost propagator.  Labels indicate the various possible \npropagator and vertex combinations that comprise the self-energy terms.}\n\\end{figure*}\n\nWe present the complete set of Dyson--Schwinger  equations in Figure~\\ref{fig:dses}.  \n\n\\section{Summary and Outlook}\n\nIn this work, we have derived the Dyson--Schwinger  equations for Coulomb gauge Yang-Mills \ntheory within the first order formalism.  In discussing the first order \nformalism it was noted that the standard BRS transform is supplemented by a \nsecond transform which arises from the ambiguity in setting the gauge \ntransform \nproperties of the $\\pi$ and $\\phi$ fields.  The motivation behind the use of \nthe first order formalism is two-fold: the energy divergent ghost sector can \nbe formally eliminated and the system can be formally reduced to physical \ndegrees of freedom, formal here meaning that the resulting expressions are \nnon-local and not useful for practical studies.  The cancellation of the \nghost sector is seen within the context of the Dyson--Schwinger  equations and the Green's \nfunctions stemming from the local action.  It remains to be seen how the \nphysical degrees of freedom emerge.\n\nGiven that the boundary conditions imposed by considering the Gribov problem \nand that the Jacobians of both the standard BRS transform and its \nsupplemental transform within the first order formalism remain trivial, the \nfield equations of motion and the Ward-Takahashi identity have been \nexplicitly derived.  The supplemental part of the BRS transform has been \nshown to be equivalent to the equations of motion at the level of the \nfunctional integral and as such is more or less trivial.  Certain exact \n(i.e., not containing interaction terms) relations for the Green's functions \nof the theory have been discussed and their solutions presented.  These \nrelations serve to simplify the framework considerably.  The propagators \npertaining to vector fields are shown to be transverse, the proper functions \ninvolving Lagrange multiplier fields reduce to kinematical factors or vanish \nand the proper functions involving functional derivatives with respect to \nthe $\\phi$-field can be explicitly derived from those involving the \ncorresponding $\\pi$-field derivatives.\n\nThe full set of Feynman rules for the system has been derived along with the \ntree-level proper two-point functions and the general form of the two-point \nfunctions (connected and proper) has been discussed.  The relationship \nbetween the (connected) propagators and the proper two-point functions, \nstemming from the Legendre transform, has been studied.  The resulting \nequations show that within the first order formalism the dressing functions \nof the two types of two-point Green's functions are non-trivially related to \neach other.  In addition, given that there are no vertices involving \nderivatives with respect to the Lagrange multiplier fields, the set of Dyson--Schwinger  \nequations needed to study the two-point functions of the theory is reduced.\n\nThe relevant Dyson--Schwinger  equations for the system have been derived in some detail.  \nIt is shown how the number of self-energy terms is considerably amplified by \nthe introduction of the various fields inherent to the first order \nformalism.  The Dyson--Schwinger  equations arising from the ghost fields are shown to be \nindependent of the ghost energy and the vertices involving the ghost fields \nare UV finite.\n\nDespite the complexity of dealing with a non-covariant system with many \ndegrees of freedom, the outlook is positive and the rich structure of the \nDyson--Schwinger  equations is not as intimidating as it might initially appear.  The \nnon-covariance of the setting means that all the dressing functions that \nmust be calculated are generally functions of two variables.  However, as \nseen from the Feynman rules, the energy dependence of the theory stems from \nthe tree-level propagators alone and not from vertices (a consequence of the \nfact that the ony explicit time derivative in the action occurs within a \nkinetic term).  The time-dependence of the integral kernels will therefore \nbe significantly less complicated than perhaps would otherwise occur.  Given \nthe experience in Landau gauge adapting the techniques, both analytical and \nnumerical, to solve the Dyson--Schwinger  equations in Coulomb gauge seems eminently \npossible though certainly challenging.  The results of such a study should \nprovide a better understanding of the issues of confinement, and with the \ninclusion of quarks, the hadron spectrum.\n\n\n\\begin{acknowledgments}\nIt is a pleasure to thank R.~Alkofer and D.~Zwanziger for useful and \ninspiring discussions.  This work has been supported by the Deutsche \nForschungsgemeinschaft (DFG) under contracts no. Re856\/6-1 and Re856\/6-2.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}