diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzopby" "b/data_all_eng_slimpj/shuffled/split2/finalzzopby" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzopby" @@ -0,0 +1,5 @@ +{"text":"\\section{Survey motivation}\n\n\\subsection{A multiwavelength approach to galaxy evolution as a function of environment}\n\nThe precise role that environment plays in shaping galaxy evolution is\na hotly debated topic. Trends to passive and\/or more spheroidal\npopulations in dense environments are widely observed: galaxy\nmorphology \\citep{dressler80,dressler97,goto03,treu03},\ncolour \\citep{kodama01,blanton05,baldry06}, star-formation rate\n\\citep{gomez03,lewis02}, and stellar age and AGN fraction\n\\citep{kauffmann04} all correlate with measurements of the local\ngalaxy density. Furthermore, these relations persist over a wide\nrange of redshift \\citep{smith05, cooper07} and density\n\\citep{balogh04}.\n\nDisentangling the relative importance of internal and external\nphysical mechanisms responsible for these relations is challenging.\nIt is natural to expect that high density environments will\npreferentially host older stellar populations. Hierarchical models of\ngalaxy formation \\citep[e.g.][]{delucia06} suggest that galaxies in\nthe highest density peaks started forming stars and assembling mass\nearlier: in essence they have a head-start. Simultaneously, galaxies\nforming in high-density environments will have more time to experience\nthe {\\it external} influence of their local environment. Those\nprocesses will also act on infalling galaxies as they are continuously\naccreted into larger haloes. There are many plausible physical\nmechanisms by which a galaxy could be transformed by its environment:\nremoval of the hot \\citep{larson80} or cold \\citep{gunn72} gas supply\nthrough ram-pressure stripping; tidal effects leading to halo\ntruncation \\citep{bekki99} or triggered star formation through gas\ncompression \\citep{fujita98}; interactions between galaxies themselves\nvia low-speed major mergers \\citep{barnes92} or frequent impulsive\nencounters termed `harrassment' \\citep{moore98}.\n\n Though some of the above mechanisms are largely cluster-specific\n(e.g. ram-pressure stripping requires interaction with a hot\nintracluster medium), it is also increasingly clear that low density\nenvironments such as galaxy groups are important sites for galaxy\nevolution \\citep{balogh04,Zabludoff96}. Additionally, luminosity (or\nmore directly, mass) is also critical in regulating how susceptible a\ngalaxy is to external influences. For example, \\citet{haines06} find\nthat in low density environments in the SDSS the fraction of passive\ngalaxies is a strong function of luminosity. They find a complete\nabsence of passive dwarf galaxies in the lowest density regions (i.e.,\nwhile luminous passive galaxies can occur in all environments,\nlow-luminosity passive galaxies can only occur in dense environments).\n\nUnderstanding the full degree of transformation is further complicated\nby the amount of dust-obscured star formation that may or may not be\npresent. Many studies in the radio and MIR\n\\citep{millerowen03,coia05,gallazzi08} have shown that an optical\ncensus of star formation can underestimate the true\nrate. Cluster-cluster variations are strong, with induced star\nformation linked to dynamically-disturbed large-scale structure\n\\citep{geach06}. Nor are changes in morphology necessarily equivalent\nto changes in star formation. There is no guarantee that external\nprocesses causing an increase or decrease in the star-formation rate\nact on the same timescale, to the same degree, or in the same regime\nas those responsible for structural changes. A full census of\nstar-formation, AGN activity, and morphology therefore requires a\ncomprehensive view of galaxies, including multiwavelength coverage and\nhigh resolution imaging. These are the aims of the STAGES project\ndescribed in this paper, targetting the Abell 901(a,b)\/902 multiple\ncluster system (hearafter A901\/2) at $z\\sim 0.165$.\n\nIn addition to the STAGES coverage of A901\/2, there are several other\nmultiwavelength projects taking a similar approach to targeting\nlarge-scale structures. While we will argue below that STAGES\noccupies a particular niche, the following is a (non-exhaustive) list\nof surveys of large-scale structure including substantial HST imaging.\nAll are complementary to STAGES by way of the redshift range or\ndynamical state probed. The COSMOS survey has examined the evolution\nof the morphology-density relation to $z=1.2$ \\citep{capak07}, paying\nparticular attention to a large structure at $z=0.7$ \\citep{guzzo07}.\nRelevant to this work, in \\citet{Smolcic07} they identify a complex of\nsmall clusters at $z\\sim 0.2$ via a wide-angle tail\nradio galaxy. At intermediate redshift, an extensive comparison\nproject has been undertaken targeting the two contrasting clusters\nCL0024+17 and MS0451-03 at $z\\sim0.5$ to compare the low- and\nhigh-luminosity X-ray cluster environment \\citep{moran07,geach06}.\nLocally, the Coma cluster has also been extensively used as a\nlaboratory for galaxy evolution \\citep{poggianti04,\ncarter01,carter08}. There are many other examples of cluster-focused\nenvironmental studies covering a range of redshifts, including the\nlarge sample of EDisCS clusters at $z>0.5$\n\\citep{white05,poggianti06,desai07}; and the ACS GTO cluster program\nof 7 clusters at $z\\sim 1$\n\\citep{postman05,goto05,blakeslee06,homeier05}.\n\n\nWe summarize the motivation for our survey design as follows. In order\nto successfully penetrate the environmental processes at work in\nshaping galaxy evolution, several areas must be simultaneously\naddressed: a wide range of environments; a wide range in galaxy\nluminosity; and sensitivity to both obscured and unobscured star\nformation, stellar masses, AGN, and detailed morphologies.\nFurthermore, it is essential to use not just a single proxy for\n`environment' but to understand directly the relative influences of\nthe local galaxy density, the hot ICM and the dark matter on galaxy\ntransformation. A further advantage is given by examining systems\nthat are not simply massive clusters already in equilibrium. By\nincluding systems in the process of formation (when extensive mixing\nhas not yet erased the memory of early timescales), the various\nenvironmental proxies listed above might still be disentangled.\n\nTherefore, the goal of STAGES is to focus attention on a single\nlarge-scale structure to understand the detailed aspects of galaxy\nevolution as a function of environment. While no single study will\nprovide a definitive answer to the question of environment and galaxy\nevolution, we argue that STAGES occupies a unique vantage point in\nthis field, to be complemented by other studies locally and at higher\nredshift.\n\n\\subsection{Galaxy evolution as a function of redshift: STAGES and\nGEMS}\n\nIn addition to science focused on the narrow redshift slice containing\nthe multiple cluster system, the multiwavelength data presented here\nprovide a valuable resource for those wishing to study the evolution\nof the galaxy population since $z=1$. With the advent of the HST and\nmultiwavelength data for this field, it is possible to quantify better\nthe sample variance and investigate rare subsamples using the\ncombination of the STAGES field together with the Galaxy Evolution and\nMorphologies \\citep[GEMS;][]{rix04} coverage of the Extended Chandra\nDeep Field South (CDFS). In particular, the HST data were chosen to\nhave the same passband for both GEMS (F606W and 850LP) and STAGES\n(F606W only, to allow study at optimum S\/N of the cluster\nsubpopulation and to optimise the weak lensing analysis). While the\nchoice of F606W means that the data probe above the 4000{\\AA} break\nfor $z<0.5$ only, for a number of purposes the data can also be used\nat higher redshift (although in those cases one needs to be\nparticularly cognizant of the effects of bandpass shifting and surface\nbrightness dimming; such effects can be understood and calibrated\nusing the GEMS 850LP and GOODS 850LP data). Furthermore, the\n24{\\micron} observations (\\S\\ref{sec-mir}) are well-matched in depth\nwith the first Cycle GTO observations of the CDFS; analyses of the\nCDFS and A901\/2 fields have been presented by \\citet{zheng07} and\n\\citet{Bell07}. Several projects are already exploiting this\ncombined dataset (see \\S\\ref{sec-summary} for details), and with the\npublicly-available data in the CDFS, these samples provide a valuable\nstarting point for many investigations of galaxy evolution.\n\n\n\\subsection{The Abell 901(a,b)\/902 supercluster: a laboratory for galaxy evolution}\\label{sec-root2}\n\nThe A901\/2 system is an exceptional testing ground with which to\naddress environmental influences on galaxy evolution. Consisting of\nthree clusters and related groups at $z\\sim0.165$, all within\n$0.5\\degr\\times0.5\\degr$, this region has been the target of extensive\nground- and space-based observations. We have used the resulting\ndataset to build up a comprehensive view of each of the main\ncomponents of the large-scale structure: the galaxies, the dark\nmatter, and the hot X-ray gas. The moderate redshift is advantageous\nas it enables us to study a large number of galaxies, yet the\nstructure is contained within a tractable field-of-view and probes a\nvolume with more gas and more star formation in general than in the\nlocal universe.\n\nThe A901\/2 region, centred at $(\\alpha,\\delta)_{\\rm J2000}$ = $(9^{\\rm\nh}56^{\\rm m}17\\fs3$, $-10\\degr01\\arcmin11\\arcsec)$, was originally one\nof three fields targeted by the COMBO-17 survey \\citep{wolf03}. It\nwas specifically chosen as a known overdensity due to the multiple\nAbell clusters present. These included two clusters (A901a and A901b)\nwith X-ray luminosities sufficient to be included in the X-ray\nBrightest Abell-type Cluster Survey \\citep[XBACS;][]{ebeling96} of the\nROSAT All-Sky Survey, though pointed ROSAT HRI observations by\n\\cite{schindler00} subsequently revealed that the emission from A901a\nsuffers from confusion with several point sources in its vicinity.\nThe extended X-ray emission in the field is further resolved by our\ndeep XMM-Newton imaging (see \\S\\ref{sec-xray}). Additional structures at\n$z\\sim 0.165$ in the field include A902 and a collection of galaxies\nreferred to as the Southwest Group (SWG).\n\nThe five broad- and 12 medium-band observations from COMBO-17 provide\nhigh-quality photometric redshifts and spectral energy distributions\n(SEDs). Together with the high-quality imaging for ground-based\ngravitational lensing, the A901\/2 data have been used in a variety of\npapers to date. COMBO-17 derived results include 2D and 3D\nreconstructions of the mass distribution \\citep{gray02, taylor04}; the\nstar-formation--density relation \\citep{gray04}; the discovery of a\nsubstantial population of intermediate-age, dusty red cluster\ngalaxies \\citep[][here-after WGM05]{wgm05}; and the morphology-density\n\\citep{lane07} and morphology-age-density \\citep{wolf07} relations.\n\nFurther afield, the clusters are also known to be part of a larger\nstructure together with neighbouring clusters Abell 907 and Abell 868\n(1.5 degrees and 2.6 degrees away, respectively). Nowak et al. (in\nprep.) used a percolation (also called `friends-of-friends') algorithm\non the REFLEX cluster catalogue \\citep{boehringer04} to produce a\ncatalogue of 79 X-ray superclusters. Entry 33 is the\nA868\/A901a\/A901b\/A902\/A907 supercluster, which also contains an\nadditional, but not very bright, non-Abell cluster. Though not\nobserved as part of the STAGES study, these clusters are included in\nthe constrained N-body simulations used to understand the formation\nhistory of the large-scale structure (\\S\\ref{sec-mocks}).\n\nThe plan of this paper is as follows: in \\S\\ref{sec-hst} we outline\nthe observations taken to construct the 80-tile mosaic with the\nAdvanced Camera for Surveys on HST. We discuss data reduction, object\ndetection, and S\\'ersic\\ profile fitting. In \\S\\ref{sec-c17} we\npresent the COMBO-17 catalogue for the A901\/2 field and discuss how\nthe two catalogues are matched. In \\S\\ref{sec-multi} we present a\nsummary of the further multiwavelength data for the field and derived\nquantities such as stellar masses and star-formation rates. We finish\nwith describing ongoing science goals, future prospects, and\ninstructions for public access to the data and catalogues described\nwithin. Appendix~\\ref{app-notes} contains details on ten individual\nobjects of particular interest within the field.\n\nThroughout this paper we adopt a concordance cosmology with\n$\\Omega_m=0.3,\\Omega_\\lambda=0.7$, and $H_0=70$ km s$^{-1}$\nMpc$^{-1}$. In this cosmology, $1\\arcsec = 2.83$ kpc at the redshift\nof the supercluster ($z\\sim0.165$), and the COMBO-17 field-of-view\ncovers $5.3\\times5.1$ Mpc$^{2}$. Magnitudes derived from the HST\nimaging (\\S\\ref{sec-hst}) in the F606W ($V$-band) filter are on the AB\nsystem,\\footnote{For F606W, $m_{\\rm AB}-m_{\\rm Vega}=0.085$.} while\nmagnitudes from COMBO-17 (\\S\\ref{sec-c17}) in all filters are on the\nVega system.\n\n\n\n\n\\section{HST data}\\label{sec-hst}\n\n\\subsection{Observations}\n\nThe primary goal of the STAGES HST imaging was to obtain morphologies\nand structural parameters for all cluster galaxies down to $R=24$\n($M_V\\sim-16$ at $z\\sim0.165$). The full area of the COMBO-17\nobservations was targeted to sample a wide range of environments.\nSecondary goals included obtaining accurate shape measurements of\nfaint background galaxies for the purposes of weak lensing, and\nmeasuring morphologies and structural parameters for all remaining\nforeground and background galaxies to $R=24$. As discussed in\n\\S\\ref{sec-root2}, the survey design and filter was chosen to match\nthat of the GEMS survey \\citep{rix04} of the Chandra Deep Field South\n(CDFS). The CDFS is another field with both COMBO-17 and HST\ncoverage, but in contrast to the A901\/2 field is known to contain\nlittle significant large-scale structure. It will therefore serve as\na matched control sample for comparing cluster and field environments\nat similar epochs.\n\nTo this end we constructed an 80-tile mosaic with ACS in Cycle 13 to\ncover an area of roughly 29.5$\\arcmin\\times$29.5$\\arcmin$ in the F606W\nfilter, with a mean overlap of 100 pixels between tiles. Scheduling\nconstraints forced the roll angle to be 125 degrees for the majority\nof observations, and one gap in the northeast corner was imposed on\nthe otherwise contiguous region due to a bright ($V=9$) star. A\n4-point parallelogram-shaped dithering pattern was employed, with\nshifts of 2.5 pixels in each direction. An additional shift of 60.5\npixels in the y-direction was included between dithers two and three\nin order to bridge the chip gap.\n\nConcerns about a time-varying PSF and possible effects on the weak\nlensing measurements drove the requirement for the observations to be\ntaken in as short a time frame as possible. In practice this was\nlargely successful, with $>50\\%$ of tiles observed in a single\nfive-day period (Fig.~\\ref{fig-cumulative}), and $>90\\%$ within 21\ndays. Six tiles (29, 75, 76, 77, 79, 80) were unobservable in that cycle\nand were re-observed six months later, with a 180 degree rotation.\nFurthermore, tile 46 was also re-observed at this orientation as the\noriginal observation failed due to a lack of guide stars. These seven\ntiles were observed following the transition to two-gyro mode with no\nadverse consequences in image quality. \n\nDetails of the observations are listed in Table~\\ref{tab-obs}. A\nschematic of the field showing the ACS tiles and the multiwavelength\nobservations is shown in Fig.~\\ref{fig-field}. Additionally, four\nparallel observations with WFPC2 (F450W) and NICMOS3 (F110W and F160W)\nwere obtained simultaneously for each ACS pointing. Due to the\nseparation of different instruments on the HST focal plane, most but\nnot all parallel images overlap with the ACS mosaic (52\/10\/18\nWFPC images and 42\/9\/29 NICMOS3 images have full\/partial\/no overlap\nwith the ACS mosaic; most NICMOS3 images have partial overlap with a\nWFPC2 image). In this paper we restrict ourselves to a discussion of\nthe primary ACS data, analysis of the parallels will follow in a\nfuture publication.\n\n\n\n\\begin{table*}\n\\begin{tabular}{rrrrrrrr}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Tile} & \n\\multicolumn{1}{c}{Date} &\n\\multicolumn{1}{c}{$\\alpha$} & \n\\multicolumn{1}{c}{$\\delta$} &\n\\multicolumn{1}{c}{Exposure} & \n\\multicolumn{1}{c}{N$_{\\rm hot}$} &\n\\multicolumn{1}{c}{N$_{\\rm cold}$} & \n\\multicolumn{1}{c}{N$_{\\rm good}$}\\\\\n\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{[dd\/mm\/yyyy]} &\n\\multicolumn{1}{c}{[J2000]} & \n\\multicolumn{1}{c}{[J2000]} &\n\\multicolumn{1}{c}{[s]} & \n\\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{}\\\\\n\\hline\n 1 & 09 07 2005 & 09:55:22.8 & -10:14:01 & 1960 & 851 & 173 & 796 \\\\ \n 2 & 07 07 2005 & 09:55:44.5 & -10:13:54 & 1960 & 1082 & 209 & 982 \\\\ \n 3 & 08 07 2005 & 09:55:33.4 & -10:12:03 & 1960 & 1157 & 233 & 1008 \\\\ \n 4 & 07 07 2005 & 09:55:22.4 & -10:10:06 & 1960 & 1051 & 199 & 927 \\\\ \n 5 & 04 07 2005 & 09:56:09.5 & -10:14:26 & 1950 & 1069 & 195 & 973 \\\\ \n 6 & 03 07 2005 & 09:55:58.7 & -10:12:33 & 1950 & 1151 & 219 & 1027 \\\\ \n 7 & 04 07 2005 & 09:55:47.9 & -10:10:39 & 1950 & 1038 & 237 & 905 \\\\ \n 8 & 04 07 2005 & 09:55:37.0 & -10:08:45 & 1950 & 1095 & 262 & 938 \\\\ \n 9 & 04 07 2005 & 09:55:26.2 & -10:06:52 & 1950 & 1020 & 188 & 876 \\\\ \n10 & 05 07 2005 & 09:55:15.4 & -10:04:58 & 1950 & 1014 & 184 & 938 \\\\ \n11 & 07 07 2005 & 09:56:38.6 & -10:15:34 & 1960 & 989 & 219 & 876 \\\\ \n12 & 04 07 2005 & 09:56:27.8 & -10:13:40 & 1950 & 1020 & 226 & 885 \\\\ \n13 & 28 06 2005 & 09:56:16.9 & -10:11:46 & 2120 & 1193 & 256 & 1037 \\\\ \n14 & 28 06 2005 & 09:56:06.1 & -10:09:53 & 2120 & 1391 & 254 & 1111 \\\\ \n15 & 28 06 2005 & 09:55:55.3 & -10:07:59 & 2120 & 1182 & 253 & 1052 \\\\ \n16 & 29 06 2005 & 09:55:44.5 & -10:06:06 & 2120 & 1109 & 208 & 940 \\\\ \n17 & 29 06 2005 & 09:55:33.7 & -10:04:12 & 1960 & 1116 & 250 & 888 \\\\ \n18 & 04 07 2005 & 09:55:22.9 & -10:02:18 & 1950 & 995 & 178 & 868 \\\\ \n19 & 09 07 2005 & 09:56:57.3 & -10:14:25 & 1960 & 963 & 180 & 786 \\\\ \n20 & 07 07 2005 & 09:56:46.0 & -10:12:54 & 1960 & 979 & 222 & 829 \\\\ \n21 & 30 06 2005 & 09:56:35.2 & -10:11:00 & 1960 & 1166 & 288 & 1005 \\\\ \n22 & 28 06 2005 & 09:56:24.4 & -10:09:07 & 2120 & 1193 & 263 & 1012 \\\\ \n23 & 25 06 2005 & 09:56:13.6 & -10:07:13 & 2120 & 1143 & 241 & 1000 \\\\ \n24 & 25 06 2005 & 09:56:02.8 & -10:05:19 & 2120 & 1244 & 254 & 1128 \\\\ \n25 & 22 06 2005 & 09:55:52.0 & -10:03:26 & 2120 & 1274 & 248 & 1051 \\\\ \n26 & 29 06 2005 & 09:55:41.1 & -10:01:32 & 1960 & 1214 & 275 & 1063 \\\\ \n27 & 05 07 2005 & 09:55:30.3 & -09:59:39 & 1950 & 1258 & 279 & 1068 \\\\ \n28 & 08 07 2005 & 09:55:19.5 & -09:57:45 & 1960 & 1161 & 220 & 1052 \\\\ \n29 & 04 01 2006 & 09:57:10.7 & -10:14:08 & 2120 & 1274 & 272 & 1123 \\\\ \n30 & 09 07 2005 & 09:57:04.5 & -10:11:48 & 1960 & 943 & 209 & 781 \\\\ \n31 & 08 07 2005 & 09:56:53.5 & -10:10:14 & 1960 & 900 & 200 & 713 \\\\ \n32 & 03 07 2005 & 09:56:42.7 & -10:08:20 & 1950 & 1023 & 214 & 884 \\\\ \n33 & 28 06 2005 & 09:56:31.9 & -10:06:27 & 2120 & 1150 & 223 & 955 \\\\ \n34 & 22 06 2005 & 09:56:21.0 & -10:04:33 & 2120 & 1318 & 243 & 1111 \\\\ \n35 & 22 06 2005 & 09:56:10.2 & -10:02:40 & 2120 & 1220 & 244 & 1028 \\\\ \n36 & 24 06 2005 & 09:55:59.4 & -10:00:46 & 2120 & 1320 & 287 & 1101 \\\\ \n37 & 29 06 2005 & 09:55:48.6 & -09:58:53 & 1960 & 1150 & 239 & 974 \\\\ \n38 & 05 07 2005 & 09:55:37.8 & -09:56:59 & 1950 & 1123 & 205 & 951 \\\\ \n39 & 08 07 2005 & 09:55:27.0 & -09:55:05 & 1960 & 1094 & 210 & 965 \\\\ \n40 & 09 07 2005 & 09:57:12.5 & -10:09:14 & 1960 & 1062 & 198 & 916 \\\\ \n41 & 07 07 2005 & 09:57:00.9 & -10:07:34 & 1960 & 962 & 176 & 828 \\\\ \n42 & 03 07 2005 & 09:56:50.1 & -10:05:41 & 1950 & 1090 & 205 & 928 \\\\ \n43 & 27 06 2005 & 09:56:39.3 & -10:03:47 & 2120 & 1198 & 202 & 1052 \\\\ \n44 & 27 06 2005 & 09:56:28.5 & -10:01:54 & 2120 & 1266 & 230 & 1046 \\\\ \n45 & 23 06 2005 & 09:56:17.7 & -10:00:00 & 2120 & 1280 & 285 & 1064 \\\\ \n46 & 01 01 2006 & 09:56:05.4 & -09:57:47 & 2120 & 1438 & 355 & 1235 \\\\ \n47 & 01 07 2005 & 09:55:56.0 & -09:56:13 & 1960 & 1198 & 273 & 972 \\\\ \n48 & 06 07 2005 & 09:55:45.2 & -09:54:19 & 1950 & 989 & 176 & 852 \\\\ \n49 & 06 07 2005 & 09:55:34.4 & -09:52:26 & 1960 & 1054 & 223 & 901 \\\\ \n50 & 09 07 2005 & 09:55:24.4 & -09:50:31 & 1960 & 984 & 212 & 832 \\\\ \n51 & 07 07 2005 & 09:57:08.4 & -10:04:55 & 1960 & 1050 & 189 & 923 \\\\ \n52 & 03 07 2005 & 09:56:57.6 & -10:03:01 & 1960 & 1142 & 209 & 941 \\\\ \n53 & 03 07 2005 & 09:56:46.8 & -10:01:07 & 1950 & 1135 & 211 & 920 \\\\ \n54 & 02 07 2005 & 09:56:36.0 & -09:59:14 & 1950 & 1131 & 228 & 921 \\\\ \n55 & 02 07 2005 & 09:56:25.1 & -09:57:20 & 1960 & 1205 & 311 & 974 \\\\ \n56 & 02 07 2005 & 09:56:14.3 & -09:55:27 & 1960 & 1097 & 242 & 891 \\\\ \n57 & 01 07 2005 & 09:56:03.5 & -09:53:33 & 1960 & 1090 & 210 & 911 \\\\ \n58 & 06 07 2005 & 09:55:52.7 & -09:51:40 & 1950 & 1130 & 201 & 975 \\\\ \n59 & 08 07 2005 & 09:55:32.7 & -09:48:15 & 1960 & 1075 & 204 & 900 \\\\ \n60 & 07 07 2005 & 09:57:15.8 & -10:02:15 & 1950 & 1028 & 183 & 912 \\\\ \n\\hline\n\\end{tabular}\n\\caption{Details of STAGES HST\/ACS observations. Only the second\n (successful) acquisition of tile 46 is listed. `Hot',`cold', and\n `good' SExtractor configurations are described in\n \\S\\ref{sec-detect}. Tiles 29, 46, 75, 76, 77, 79, and 80 are\n oriented at 180\\degr\\ with respect to the rest of the mosaic. \n The exposure time varied according to the maximum window of\n visibility available in each orbit.}\\label{tab-obs}\n\\end{table*}\n\n\\begin{table*}\n\\contcaption{}\n\\begin{tabular}{rrrrrrrr}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Tile} & \n\\multicolumn{1}{c}{Date} &\n\\multicolumn{1}{c}{$\\alpha$} & \n\\multicolumn{1}{c}{$\\delta$} &\n\\multicolumn{1}{c}{Exposure} & \n\\multicolumn{1}{c}{N$_{\\rm hot}$} &\n\\multicolumn{1}{c}{N$_{\\rm cold}$} & \n\\multicolumn{1}{c}{N$_{\\rm good}$}\\\\\n\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{[dd\/mm\/yyyy]} &\n\\multicolumn{1}{c}{[J2000]} & \n\\multicolumn{1}{c}{[J2000]} &\n\\multicolumn{1}{c}{[s]} & \n\\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{} & \n\\multicolumn{1}{c}{}\\\\\n\\hline\n61 & 07 07 2005 & 09:57:05.0 & -10:00:21 & 1950 & 971 & 183 & 826 \\\\ \n62 & 07 07 2005 & 09:56:54.2 & -09:58:28 & 1950 & 1052 & 184 & 901 \\\\ \n63 & 06 07 2005 & 09:56:43.4 & -09:56:34 & 1950 & 1141 & 217 & 930 \\\\ \n64 & 06 07 2005 & 09:56:32.6 & -09:54:41 & 1950 & 1069 & 222 & 890 \\\\ \n65 & 06 07 2005 & 09:56:21.8 & -09:52:47 & 1950 & 1071 & 227 & 908 \\\\ \n66 & 06 07 2005 & 09:56:11.0 & -09:50:53 & 1950 & 1014 & 222 & 859 \\\\ \n67 & 06 07 2005 & 09:56:00.1 & -09:48:60 & 1950 & 1046 & 226 & 922 \\\\ \n68 & 08 07 2005 & 09:55:49.3 & -09:47:06 & 1960 & 967 & 179 & 851 \\\\ \n69 & 10 07 2005 & 09:57:12.5 & -09:57:42 & 1960 & 876 & 145 & 784 \\\\ \n70 & 09 07 2005 & 09:57:01.7 & -09:55:48 & 1960 & 934 & 183 & 798 \\\\ \n71 & 09 07 2005 & 09:56:50.9 & -09:53:54 & 1960 & 1032 & 182 & 888 \\\\ \n72 & 10 07 2005 & 09:56:40.0 & -09:52:01 & 1960 & 1118 & 212 & 950 \\\\ \n73 & 09 07 2005 & 09:56:29.2 & -09:50:07 & 1960 & 910 & 168 & 773 \\\\ \n74 & 08 07 2005 & 09:56:18.4 & -09:48:14 & 1960 & 907 & 192 & 822 \\\\ \n75 & 04 01 2006 & 09:57:11.0 & -09:53:30 & 2120 & 1708 & 260 & 1140 \\\\ \n76 & 05 01 2006 & 09:57:00.3 & -09:51:39 & 2120 & 1444 & 275 & 1134 \\\\ \n77 & 05 01 2006 & 09:56:49.5 & -09:49:48 & 2120 & 1324 & 287 & 1094 \\\\ \n78 & 05 07 2005 & 09:56:40.6 & -09:48:11 & 1960 & 1031 & 184 & 842 \\\\ \n79 & 05 01 2006 & 09:57:12.9 & -09:50:05 & 2120 & 1357 & 302 & 1019 \\\\ \n80 & 05 01 2006 & 09:57:02.8 & -09:48:36 & 2120 & 1255 & 246 & 973 \\\\ \n\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}\n\\centerline{\\psfig{file=cumulative.ps,width=0.75\\columnwidth}}\n\\caption{Cumulative plot of ACS data acquisition. In order to\n minimize the effects of a time-vary PSF on weak lensing\n applications, 50\\% of tiles were taken within 5 days and 90\\% within 21\n days. The remaining 7 tiles were observed 6 months\n later.}\\label{fig-cumulative}\n\\end{figure}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=field.ps,width=0.95\\textwidth,angle=270}}\n\\caption{Layout of multiwavelength observations of the A901\/2 field.\nThe numbered tiles represent the 80-orbit STAGES mosaic with HST\/ACS,\nwhich overlaps the 31.5$\\times$30 arcmin COMBO-17 field-of-view\n(long-dashed square). The seven shaded tiles were observed $\\sim$6\nmonths after the bulk of the observations and with a 180\\degr\nrotation. The centres of A901a\/A901b\/A902\/SWG are found in tiles\n55\/36\/21\/8 respectively. Interior to the STAGES region are the\nXMM-Newton coverage (heavy solid polygon) and the GMRT 1280 MHz\nobservations (short-dashed circle, indicating half-power beam\nwidth). The STAGES area is also overlapped by the field-of-view of the\nSpitzer 24$\\micron$ imaging (solid polygon), the GMRT 610 MHz\nobservations (long-dashed circle), and the GALEX imaging (dotted\ncircle).}\n\\label{fig-field}\n\\end{figure*}\n\n\n\n\\subsection{ACS data reduction}\n\nWe retrieve the reduced STAGES images processed by the CALNICA\npipeline of STScI, which corrects for bias subtraction and\nflat-fielding. However, as the ACS camera is located 6 arcmin off the\ncentre of the HST optical axis, the images from the telescope have a\nfield-of-view with a parallelogram keystone distortion. To produce a\nfinal science image from the reduced pipeline data, we therefore also\nhave to remove the geometric distortion before combining the\nindividual dithered sub-exposures. The removal of the image\ndistortion is now fairly routine through the use of the MULTIDRIZZLE\nsoftware \\citep{koekemoer07}. However, our particular science goals\nmotivated us to make several changes when optimizing the default\nsettings and combining the raw images. These changes are discussed\nbelow.\n\n\n\\subsubsection {Image Distortion Correction}\n\nIn STAGES, the science driver that demands the highest quality data\nreduction in terms of producing the most consistent and stable PSF\nfrom image to image, and across the field of view, is weak lensing\n\\citep{heymans08}. With this goal in mind, we benefit from the\nexperience of \\cite{rhodes07}, who conducted detailed studies of how\nthe pixel values are re-binned when the images are corrected for image\ndistortion. Briefly speaking, to transform an image that is sampled\non a geometrically distorted grid onto one that is a uniform Cartesian\ngrid fundamentally involves rebinning, i.e. interpolating, the\noriginal pixel values into the new grid. Doing so is not a\nstraightforward process since the original ACS pixel scale samples the\ntelescope diffraction limit below Nyquist frequency, i.e. the\ntelescope PSF is undersampled. When a PSF is undersampled, aliasing\nof the pixel fluxes occurs, the result of which is that the recorded\nstructure of the PSF appears to change with position, depending on the\nexact sub-pixel centroid of the PSF. This variability effectively\nproduces a change in the ellipticity of the PSF as a function of\nsub-pixel position, even if the PSF should be identical everywhere.\nBecause stellar PSFs are randomly centred about a pixel the\nintrinsic ellipticity one then measures has a non-zero scatter. So,\nas weak lensing relies heavily on measuring the ellipticities of\ngalaxies, which are convolved by the PSF, the scatter in the PSF\nellipticity contributes significant noise to weak lensing\nmeasurements.\n\nAn additional issue with non-Nyquist sampled images is that the process\nof interpolating pixel values necessarily degrades the original image\nresolution. While the intrinsic resolution can in principle be\nrecovered by dithering the images while making observations, strictly\nspeaking this inversion is only possible when the image is on a\nperfect Cartesian grid at the start, i.e. with no image distortion.\nOtherwise, there would be a residual ``beating frequency'' in the\nsampling of the reconstituted image, such that some pixels would be\nbetter sampled than others. Because of this, recovering the intrinsic\nresolution of the telescope when the field is distorted is not a well\nposed problem, and cannot easily be solved by a small number of image\ndithers. Some resolution loss will necessarily occur in some parts of\nthe image. This is especially true if the final images are combined\n{\\it after} having been geometrically corrected, as is currently the\nprocess in MULTIDRIZZLE. One last, unavoidable, side effect of\ninterpolating a non-Nyquist sampled image is that the pixel values\nbecome necessarily correlated. However, the degree of resolution loss\nand noise correlation can be balanced by a suitable choice of\ninterpolation kernels: whereas square top-hat kernels effectively\namounts to linear interpolation and correlates only the immediate\nneighbour pixels but cause high interpolation (pixellation) noise,\nbell-shaped kernels (e.g. Gaussian and Sinc) correlate more pixels\nbut better preserve the image resolution.\n\nIn light of these issues, it is clear that the goal of an optimal HST\ndata reduction should be a dataset where the PSF structure is stable\nacross the field of view and reproducible from image tile to tile.\nThe contribution to the PSF variation by the stochastic aliasing\nof the PSF that necessarily occurs during `drizzling' can be reduced\nby appropriate choices of drizzling kernel and output pixel scale.\n\\cite{rhodes07} characterize PSF stability in terms of the scatter in\nthe apparent ellipticity of the PSF in the ACS field of view. After\nexperimenting, they determine that the optimal set of parameters in\nMULTIDRIZZLE to use is a Gaussian drizzling kernel,\n\\texttt{pixfrac}=0.8, and an output pixel scale of $0\\farcs03$. We\nthus follow their approach by adopting those parameters for our own\nreduction, while keeping all the other default parameters unchanged.\nHowever, they note, as we do, that a Gaussian kernel causes more\ncorrelated pixels than tophat kernels. Nonetheless because the choice\nof interpolation kernel amounts effectively to a smoothing kernel,\ncorrelated noise should in principle not have an impact on photometry\nstatistics since the flux is conserved. Moreover, the same\ninterpolation (smoothing) kernel propagates into the PSF, thus the\nchoice of kernel should also not impact galaxy fitting analyses.\n\n\n\n\\subsubsection {Sky pedestal and further image flattening correction}\n\nThe images obtained from the HST archive have been bias subtracted and\nflatfielded. However, large-scale non-flatness on the order of 2-4\\%\nremains in the images, and there are slight but noticeable pedestal\noffsets that remain between the four quadrants. These large scale\npatterns and pedestals are both stationary and consistent in images\nthat are observed closely in time. And even though MULTIDRIZZLE tries\nto equalize the pedestals before combining the final images, the\ncorrection is not always perfect due to object contamination when\ncomputing the sky pedestal. These effects are small, and the sky\npedestal issue only affect large objects situated right on image\nboundaries, so that the effects on the entire survey itself may only\nbe cosmetic. Nevertheless, we try to correct for the effects by\nproducing a median image of data observed closely in time, after first\nrejecting the brightest 30\\% and faintest 20\\% of the images (to\navoid over-subtraction). Then, for each of the four CCD quadrants,\nwe fit a low order 2-D cubic-spline surface (IRAF\/imsurfit)\nindividually to model the large scale non-uniformity in the median sky\nimage, and to remove noise. The noiseless model of the sky is then\nsubtracted from all the data observed closely in time. After\ncorrection, the mean background in the four quadrants is essentially\nequal, and the residual non-flatness is $\\ll 1\\%$.\n\n\\subsection{Object Detection}\n\\label{sec-detect}\nObject detection and cataloguing were carried out automatically on the\nSTAGES F606W imaging data using the SExtractor V2.5.0 software\n\\citep{bertin96}. An optimized, dual (`cold' and `hot') configuration\nwas used, following the strategy developed for HST\/ACS data of similar\ndepth for GEMS \\citep{caldwell08}. The main challenge to extracting\nsources from the STAGES ACS data is the tradeoff between deblending\nhigh-surface brightness cluster members that are close on the sky in\nprojection, and avoiding spurious splitting (`shredding') of highly\nstructured spiral galaxies into multiple sources. In addition, we\ndesire high detection completeness for faint, and often low-surface\nbrightness, background galaxies. To optimize the detection\ncompleteness and deblending reliability for counterparts to $R_{\\rm\nap}\\leq24$ mag galaxies\\footnote{COMBO-17 redshifts are mostly useful\nat $R_{\\rm ap}\\leq24$ for reasons discussed in detail in \\S\\ref{sec-c17},\nand so we adopt this cut for our main science sample.} from the\nCOMBO-17 catalogue, we fine-tuned the combination of cold and hot\nconfiguration parameters using three representative STAGES tiles (21,\n39, and 55). For STAGES, we converged on the parameters given in\nTable \\ref{sex_config}, which successfully detected 99.5\\% (650\/653)\nof the $R_{\\rm ap}\\leq24$ mag COMBO-17 galaxies on these tiles, with\nreliable deblending for 98.0\\%.\n\nSExtractor produces a list of source positions and basic photometric\nparameters for each astrometrically\/photometrically calibrated image,\nand produces a segmentation map that parses the image into source and\nbackground pixels, which is necessary for subsequent galaxy fitting\nwith GALFIT \\citep{peng02} described in \\S\\ref{sec-fitting}. For both\nconfigurations, a weight map ($\\propto{\\rm variance}^{-1}$) and a\nthree-pixel (FWHM) top-hat filtering kernel were used. The former\nsuppresses spurious detections on low-weight pixels, and the latter\ndiscriminates against noise peaks, which statistically have smaller\nextent than real sources as convolved by the instrumental PSF. Our\nfinal catalogue contains 75\\,805 {\\it unique} F606W sources uniformly\nand automatically identified from 17\\,978 objects detected in the cold\nrun, and 89\\,464 `good' sources found in the hot run (before\nrejection of the unwanted hot detections that fell within the\nisophotal area of any cold detection). A total of 5\\,921 objects\nwere manually removed from the catalogue after the detection\nstage. These detections are mainly over-deblended galaxies or image\ndefects like cosmic rays. Another set of 658 detections were included\nin fitting the sample galaxies to ensure the accurate fitting of real\nobjects, but excluded from the final catalogue. These were also mainly\ncosmic ray hits or stellar diffraction spikes. Although the main\nanalysis was performed on a tile-by-tile basis, rather than\nmosaic-wise, the main catalogue only contains {\\it unique} sources.\nObjects detected on two tiles enter the catalogue only once. The most\ninterior-located was selected for entry into the catalogue. The\nbreakdown of cold, hot, and good sources per ACS frame is given in\nTable~\\ref{tab-obs}.\n\n\nIn Fig.~\\ref{fig-hist} we show a histogram of various object samples\nin the region of the HST-mosaic that overlaps with COMBO-17. The HST\ndata start to become incomplete at $V_{606}\\sim 26$ (solid\nline). Stars (hashed histogram) only make up a significant fraction of\nall detections at the brightest magnitudes. A histogram of\ncounterparts from a cross-correlation with COMBO-17 is shown in light\ngrey. When the match is restricted to extended objects with\n$R_{\\rm ap}<24$ (ie. the primary 'galaxy' sample for which we have\nreliable photometric redshifts), the HST sources largely have $V_{606}<24$.\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper1.eps,width=\\columnwidth}}\n\\caption{ Source detections in the HST mosaic (overlap region with\nSTAGES and COMBO-17 coverage). The solid line represents all\nSExtractor detected sources (74\\,534 objects). The grey histograms\nshows all objects with a corresponding match in the COMBO-17 catalogue\n(light grey; 50\\,701 sources) and extended sources with\n$R_{\\textrm{ap}}<24$ (dark grey; 12\\,748 sources). In addition, the\nhashed region indicates stars as defined by our star-galaxy separation\ncriterion (Equation~\\ref{eqn-stargal}; 4\\,969 stars in total). In the\ninset we highlight the bright magnitude end where the total number of\nstars dominates the source population.}\\label{fig-hist}\n\\end{figure}\n\nStar-galaxy separation is performed in the apparent magnitude -- \nsize plane spanned by the SExtractor parameters MAG\\_BEST ($V_{606}$) and\nFLUX\\_RADIUS ($r_f$). Objects with\n\n\\begin{equation}\\label{eqn-stargal}\n\\log(r_f) < {\\rm max}\\left( 0.35; 1.60-0.05 V_{606}; 5.10-0.22 V_{606} \\right)\n\\end{equation}\nare classified as point sources; sources above that line are\nidentified as extended sources (galaxies). This plane is shown in\nFig.~\\ref{fig-hstmorph}. The separation line clearly delineates\ncompact and extended sources, in particular when inspecting the\nCOMBO-17 sources only (crosses). Note that those AGN for which the\npoint source dominates are also found on the point-source locus and\ntherefore are removed from the galaxy sample by this selection.\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper7_msk_cmp.ps,width=0.8\\columnwidth}}\n\\caption{ Star-galaxy separation. We define a line in the\nmagnitude-size plane to separate stars and galaxies (solid line).\nObjects above this line are extended galaxies; objects below are other\ncompact objects (including most AGN). Grey pluses indicate all\ndetections; black crosses only those with a COMBO-17 cross-match and\n$R_{\\textrm{ap}}<24$ and a redshift $z>0$. Note, a significant number\nof mostly late-type stars are misidentified as galaxies by COMBO-17\nphotometry alone. The dashed line shows a line of constant surface\nbrightness, which is almost parallel to our selection line at the\nbright end. \n}\\label{fig-hstmorph}\n\\end{figure}\n\nIn the Fig.~\\ref{fig-xcorrcomp} we display the galaxy fraction as a\nfunction of $V_{606}$ magnitude (grey histogram). Out to $V_{606}\\sim\n22$ almost every galaxy detection on the HST images has a COMBO-17\ncounterpart; at the COMBO-17 sample limit $V_{606}\\sim 24$ the\nmatching completeness for STAGES objects is still $\\sim$90\\%. The\ncross-matching between COMBO-17 and the HST data is described in more\ndetail in \\S~\\ref{sec-xcorr}, where completeness is defined in\nreverse, i.e. maximizing HST counterparts for COMBO-17 objects.\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper2.eps,width=0.9\\columnwidth}}\n\\caption{Fraction of extended STAGES objects and COMBO-17\ncounterparts. The grey line shows the extended source fraction in\nSTAGES. At bright magnitudes most sources are compact, while at the\nfaint end almost all are extended. The black dotted line shows\nextended sources in STAGES with a COMBO-17 counterpart. At\n$V_{606}\\sim26$ the COMBO-17 completeness limit is reached. Almost no\nfainter sources are found in COMBO-17. The black solid line shows\nextended sources in STAGES with a COMBO-17 counterpart having\n$R_{\\textrm{ap}}<24$. Out to $V_{606}\\sim22$ almost every extended\nSTAGES object has a COMBO-17 counterpart: the cross-correlation\ncompleteness defined with respect to the STAGES catalogue is almost\n100\\% (i.e.~the ratio of black and grey lines); at $V_{606}\\sim24$ it\nis $\\sim$90\\%. See \\S\\ref{sec-xcorr} for further discussion.}\n\\label{fig-xcorrcomp}\n\\end{figure}\n\n\n\\begin{table*}\n\\caption{Dual SExtractor parameter values for STAGES F606W object detection in `cold' and `hot' configurations. }\n\\label{sex_config}\n\\begin{tabular}{lrrl}\n\\hline\\hline\nParameter & Cold & Hot & Description\\\\\n\\hline\nDETECT\\_THRESH & 2.8 & 1.5 & detection threshold above background \\\\\nDETECT\\_MINAREA & 140 & 45 & minimum connected pixels above threshold \\\\\nDEBLEND\\_MINCONT & 0.02 & 0.25 & minimum flux\/peak contrast ratio\\\\\nDEBLEND\\_NTHRESH & 64 & 32 & number of deblending threshold steps\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\\subsection{S\\'ersic\\ profile fitting}\\label{sec-fitting}\n\nTo obtain S\\'ersic\\ model fits for each STAGES galaxy, the\nimaging data were processed with the data pipeline GALAPAGOS (Galaxy\nAnalysis over Large Areas: Parameter Assessment by GALFITting Objects\nfrom SExtractor; Barden et al., in prep.). GALAPAGOS performs all \ngalaxy fitting analysis steps from object detection to catalogue\ncreation automatically. This includes (i) source detection and\nextraction with SExtractor; (ii) preparing all detected objects for\nS\\'ersic\\ fitting with GALFIT \\citep{peng02}: i.e., constructing bad\npixel masks, measuring local background levels, and setting up\nstarting scripts with initial parameter estimates; (iii) running the\nS\\'ersic\\ model fits; and (iv) compiling all information into a final\ncatalogue. \n\nBased on a single startup script, GALAPAGOS first runs SExtractor in\nthe dual high dynamic range mode described in \\S\\ref{sec-detect}. As\nno SExtractor setup is ever 100\\% optimal, we manually inspected all\n80 tiles for unwanted detections or over-deblended objects. GALAPAGOS\nallows for the removal of such extraction failures automatically given\nan input coordinate list. Additionally, we also composed a list of\ndetections that are bright enough to influence the fitting of\nneighbouring astronomical sources (e.g. diffraction spikes from bright\nstars). Unlike the aforementioned bad detections these are not\nremoved instantly, but kept in the source catalogue throughout the\nfitting process and removed only from the final object\ncatalogue. Again, GALAPAGOS performs this operation automatically\ngiven a second list of coordinates. Further details on the\nprocess of manual fine-tuning of detection catalogues can be found in\nBarden et al. (in prep.).\n\nAfter the second run GALAPAGOS uses the cleaned output source list\n(described in \\S\\ref{sec-detect}) to cut postage stamps for every\nobject. Postage stamps are required for efficient S\\'ersic\\ profile\nfitting with GALFIT. The sizes of the postage stamps are based on a\nmultiple $m$ of the product of the SExtractor parameters KRON\\_RADIUS\nand A\\_IMAGE. We define a ``Kron-ellipse'' with semi-major axis\n$r_{\\textrm{K}}$ as\n\\begin{equation}\n r_{\\textrm{K}}=m \\times \\textrm{KRON\\_RADIUS} \\times \\textrm{A\\_IMAGE}.\n\\end{equation} \n\nThe sky level is calculated for each source individually by evaluating\na flux growth curve. GALAPAGOS uses the full science frame for this\npurpose in contrast to simply working on the postage stamp. Although\nin principle the background estimate provided by SExtractor could have\nbeen used, tests show that using the more elaborate GALAPAGOS scheme\nresults in more robust parameter fits \\citep{haeussler07}. For a\ndetailed description of the algorithm we refer to Barden et al. (in\nprep.). One might argue that GALFIT allows fitting the sky\nsimultaneously with the science object. However, this requires the\nsize of the postage stamp to be matched exactly to the size of the\nscience object. If the postage stamp is too small, the proper sky\nvalue cannot be found; if it is too big, computation takes\nunneccesarily long. Too many secondary sources would have to be\nincluded in the fit and the inferred sky value might be influenced by\ndistant sources. Additionally, galaxies may not be perfectly\nrepresented by a S\\'ersic\\ fit, and the sky may take on unrealistic\nvalues as a result. Although this method may be the easiest option\nfor manual fitting, in the general case of fitting large numbers of\nsources automatically the most robust option is to calculate the sky\nvalue beforehand and keep its value fixed when running GALFIT\n\\citep[as demonstrated in][]{haeussler07}.\n\nAnother crucial component for setting up GALFIT is determining which\ncompanion objects should be included in the fit. In particular, in\ncrowded regions with many closely neighbouring sources the fit quality\nof the primary galaxy improves dramatically when including\nsimultaneously fitting S\\'ersic\\ models to these neighbours rather than\nsimply masking them out. GALAPAGOS makes an educated guess as to\nwhich neighbours should be fitted or masked (see Barden et al., in\nprep. for further details). The decision is made by calculating\nwhether the Kron-ellipses of primary and neighbouring source\noverlap. This calculation is performed not only for sources on the\npostage stamp, but on all objects on the science frames surrounding\nthe current one, in order to take objects at frame edges into account\nproperly. Detections not identified as overlapping secondary sources\nare treated as well. Such non-overlapping companions are masked based\non their Kron-ellipse and thus excluded from fitting.\n\nAn additional requirement for fitting with GALFIT is an input PSF. We\nconstructed a general high S\/N PSF for STAGES by combining all stars\n(i.e.\\ classified by COMBO-17 photometry and having ACS \nSExtractor stellarity index $>0.85$) in the brightness interval\n$19.50\\fm2422$ error in the total magnitude MAG\\_BEST. As\na result we obtained a catalogue of 63\\,776 objects with positions,\nmorphology, total $R$-band magnitude and its error. The astrometric\naccuracy is better than $0\\farcs15$. Using our own aperture photometry\nwe reach a 5$\\sigma$ point source limit of $R\\approx 25.7$.\n\nWe obtained spectral energy distributions of all objects from\nphotometry in all 17 passbands by projecting the known object\ncoordinates into the frames of reference of each single exposure and\nmeasuring the object fluxes at the given locations. In order to\noptimize the signal-to-noise ratio, we measure the spectral shape in\nthe high surface brightness regions of the objects and ignore\npotential low surface brightness features at large distance from the\ncentre. However, this implies that for large galaxies at low redshifts\n$z<0.2$ we measure the SED of the central region and ignore colour\ngradients.\n\nAlso, we suppressed the propagation of variations in the seeing into\nthe photometry by making sure that we always probe the same physical\nfootprint outside the atmosphere of any object in all bands\nirrespective of the PSF. Here, the footprint $f(x,y)$ is the\nconvolution of the PSF $p(x,y)$ with the aperture weighting function\n$a(x,y)$. If all three are Gaussians, an identical physical footprint\ncan be probed even when the PSF changes, simply by adjusting the\nweighting function $a(x,y)$. We chose to measure fluxes on a footprint\nof $1\\farcs5$ FWHM outside the atmosphere ($\\sim 4.2$ kpc at $z\\sim\n0.165$). In detail, we use the package MPIAPHOT \\citep{mpiaphot93} to\nmeasure the PSF on each individual frame, choose the weighting\nfunction needed to conserve the footprint and obtain the flux on the\nfootprint. Fluxes from individual frames are averaged for each object\nand the flux error is derived from the scatter. Thus, it takes not\nonly photon noise into account, but also suboptimal flatfielding and\nuncorrected CCD artifacts.\n\nAll fluxes are finally calibrated by the tertiary standards in our\nfield. The aperture fluxes correspond to total fluxes for point\nsources, but underestimate them for extended sources. The difference\nbetween the total (SExtractor-based) and the aperture (MPIAPHOT-based)\nmagnitude is listed as an aperture correction and used to calculate\ne.g. luminosities. For further details on the observations and the\ndata processing, see W04.\n\nThe A901\/2 field is affected by substantial foreground dust reddening at\nthe level of $E(B-V)\\approx 0.06$, in contrast to the CDFS. Hence,\nany SED fitting and derivation of luminosities requires dereddened\nSEDs. Therefore, in the catalogue we list three sets of photometry:\n\n\\begin{enumerate}\n\\item $R$-band total and aperture magnitudes as observed for the definition \nof samples and completeness;\n\\item aperture fluxes $F_{\\rm phot}$ in 17 bands, dereddened using \n$A_V=0.18$ and $(A_U,A_B,A_R,A_I)=A_V \\times (1.63,1.24,0.82,0.6)$ with\nsimilar numbers for medium-band filters (rereddening with these \nnumbers would restore original measurements); and \n\\item aperture magnitudes (Vega) in all 17 bands, dereddened, on the\nAsinh system \\citep{lupton99} that can be used for logarithmic flux\nplots with no trouble arising from formally negative flux\nmeasurements.\n\\end{enumerate}\n\nFluxes are given as photon fluxes $F_\\mathrm{phot}$ in units of \nphotons\/m$^2$\/s\/nm, which are related to other flux definitions by\n\\begin{equation}\n \\nu F_\\nu = hcF_{\\rm phot} = \\lambda F_\\lambda ~ .\n\\end{equation}\nPhoton fluxes are practical units at the depth of current surveys. A\nmagnitude of $V=20$ corresponds to 1~photon\/m$^2$\/s\/nm in all systems\n(AB, Vega, ST), provided $V$ is centred on 548~nm. Flux values of an\nobject are missing in those bands where every exposure was saturated.\n\nThe final catalogue contains quality flags for all objects in an\ninteger column (`phot\\_flag'), holding the original SExtractor flags\nin bit 0 to 7, corresponding to values from 0 to 128, as well as some\nCOMBO-17 quality control flags in bits 9 to 11 (values from 512 to\n2048). We generally recommend that users ignore objects with flag\nvalues phot\\_flag $\\ge 8$ for any statistical analysis of the object\npopulation. If an object of particular interest shows bad flags, it\nmay still have accurate COMBO-17 photometry and could be used for some\npurposes. Often only the total magnitude was affected by bright\nneighbours, while the aperture SED is valid.\n\nWe then employ the usual COMBO-17 classification and redshift\nestimation by template fitting to libraries of stars, galaxies, QSOs\nand white dwarfs. There, the error rate increases very significantly\nat $R_{\\rm ap}>24$. We refer again to W04 for details of the libraries\nand known deficiencies of the process, but repeat here (and correct a\nmisprint in W04) the definition of the classifications (see\nTable~\\ref{tab-comboclasses}).\n\n\n\n\n\\begin{table}\n\\caption{Definition of entries for the `mc\\_class' column and\ncomparison of object numbers between the COMBO-17 data sets of the\nA901\/2 and CDFS field. The samples refer to a magnitude range of\n$R_{\\rm ap}=[16,24]$ and only objects with phot\\_flag$<8$. The A901\/2\nfield is richer in stars because of its galactic coordinates. It is\nalso richer in galaxies due to the cluster, while the CDFS is\nunderdense at $z=[0.2,0.4]$. We note that these definitions are based\non the COMBO-17 data SED and morphology; star-galaxy separation\nemploying morphological information from the HST imaging\n(Equation~\\ref{eqn-stargal}) is considered separately.\n\\label{tab-comboclasses}}\n\\begin{tabular}{llrr}\n\\hline\\hline\nClass entry & Meaning & $N\\_{\\rm A901\/2}$ & $N\\_{\\rm CDFS}$ \\\\\n\\hline\nStar & stars \t\t\t& 2096\t& 992\t\\\\\n & (only point sources) \\\\\nWDwarf & white dwarf\t\t\t& 14\t& 9\t\\\\\n & (only point sources) \\\\\nGalaxy & galaxies\t\t\t& 14555 & 11054\t \\\\\n & (shape irrelevant) \\\\\nGalaxy (Star?) & binary or low-z galaxy & 44\t& 46 \\\\\n & (star SED but extended; \\\\\n\t\t & ambiguous colour space) \\\\\nGalaxy (Uncl!) & SED fit undecided \t\t& 316\t& 243\t\\\\\n & (most often galaxy) \\\\\nQSO & QSOs \t\t\t& 73\t& 66\t\\\\\n & (only point sources) \\\\\nQSO (Gal?) & Seyfert-1 AGN or & 36\t& 31\t\\\\\n & interloping galaxy\t\\\\\n & (AGN SED but extended; \\\\\n\t\t & ambiguous colour space) \\\\\nStrange Object & unusual strange spectrum \t& 1\t& 3\t\\\\\n & ($\\chi ^2_{\\rm red}>30$) \\\\\n\\noalign{\\smallskip} \\hline\n\\end{tabular}\n\\end{table}\n\n\n\nWe also show in Table~\\ref{tab-comboclasses} a comparison of the sample\nsizes in different classes between the A901\/2 and the CDFS field of\nCOMBO-17. The main difference is that the A901\/2 field contains more\nthan twice the number of stars given its position at relatively low\ngalactic latitude ($+33.6\\deg$). Another difference is that it\ncontains 30\\% more galaxies than the CDFS, which is both a consequence\nof the cluster A901\/2 and the underdensity in the CDFS seen at\n$z\\sim[0.2,0.4]$. Fig.~\\ref{fig-CW_class} shows a colour-magnitude\ndiagram of the star and white dwarf sample as well as redshift-magnitude\ndiagrams for galaxies and QSOs.\n\n\\begin{figure*}\n\\includegraphics*[height=\\textwidth,angle=270]{CW_classes.ps}\n\\caption{ {\\it Left panel:} Stars (dots) and white dwarfs (crosses):\n$B-V$ colour vs. $R_{\\rm tot}$. The two reddest stars at $R\\approx 23$ and\n$B-V>2$ are M5-6 stars. {\\it Centre panel:} Red-sequence (black) and\nblue-cloud galaxies (green): MEV redshift vs. $R_{\\rm tot}$. {\\it Right\npanel:} QSOs: MEV redshift vs. $R_{\\rm tot}$.\n\\label{fig-CW_class}}\n\\end{figure*}\n\nRedshifts are given as Maximum-Likelihood values (the peak of the\nPDF), or as Minimum-Error-Variance values (the expectation value of\nthe PDF). MEV redshifts have smaller true errors, but are only given\nwhen the width of the PDF is lower than $\\sigma_z\/(1+z)< 0.125$. If\nPDFs are bimodal with modes of sufficiently small width, then both\nvalues are given with the preferred (larger-integral) mode providing\nthe primary redshift. Our team uses only MEV redshifts (with column\nname `mc\\_z') for their analyses.\n\nThe galaxy sample with MEV redshifts is $>90$\\% complete at all\nredshifts for $R_{\\rm ap}<23$. Near $z\\sim 1$, the MEV redshifts are\nthis complete even at $R_{\\rm ap}=24$. Below this cut, increasing\nphoton noise drives an expansion of the width of the PDF. The error\nlimit for MEV redshifts then makes the completeness of galaxy samples\nwith MEV redshifts drop. The 50\\% completeness is reached at $R\\sim\n24$ to $25$ depending on redshift. These results have been determined\nfrom simulations and are detailed in W04. Completeness maps are\nincluded in the data release and take the form of a 3-D map of\ncompleteness depending on aperture magnitude, redshift and restframe\n$U-V$ colour.\n\nTo date, the photo-z quality on the A901\/2 field has only been\ninvestigated with a comparison to spectroscopic redshifts at the\nbright end. W04 reported results from a sample of 404 bright galaxies\nwith $R<20$ and $z=[0,0.3]$, 351 of which were on the A901\/2 field,\nand 249 of which were members of the A901\/2 cluster complex\n(\\S~\\ref{sec-2df}). The other 53 objects were observed by the 2dFGRS\non the CDFS and S11 fields\n\\citep{Colless2001}. There we found that 77\\% of the sample had\nphoto-z deviations from the true redshift\n$|\\delta_z\/(1+z)|<0.01$. Three objects (less than 1\\%) deviate by more\nthan 0.04 from the true redshift.\n\nCurrently, we do not have faint spectroscopic samples on the A901\/2\nfield, however a spectroscopic dataset from VVDS exists on the\nCOMBO-17 CDFS field. From a sample of 420 high-quality redshifts that\nare reasonably complete to $R_{\\rm ap}<23$, we find a $1\\sigma$\nscatter in $\\delta_z\/(1+z)$ of 0.018, but also a mean bias of\n$-0.011$. Furthermore, the faint CDFS data show $\\sim 5$\\% outliers\nwith deviations of more than 0.06 \\citep{HWB08}. From a collection of\nspectroscopic samples we modelled the overall 1$\\sigma$ redshift\nerrors at $R\\la 24$ and $z\\la 1$ in W04 as\n\\begin{equation}\n \\sigma_z\/(1+z) \\approx 0.005 \\times \\sqrt{1+10^{0.6 (R_{\\rm ap}-20.5)}} ~ .\n \\label{dz_rel}\n\\end{equation}\nLater we use a variant of this approximation to estimate the\ncompleteness of photo-z based selection rules for cluster members.\n\nThe template fitting for galaxies produces three parameters,\ni.e. redshift as well as formal stellar age and dust reddening\nvalues. The age is encoded in a template number running from 0\n(youngest) to 59 (oldest), where we use the same PEGASE\n\\citep[see][for discussion of an earlier version of the model]{fioc97}\ntemplate grid as described in W04. The look back times to the onset of\nthe $\\tau =1$~Gyr exponential burst range from 50~Myr to 15~Gyr.\n\nRestframe properties are derived for all galaxies and QSOs as\ndescribed in W04. Table~\\ref{tab-rfvega} lists the restframe passbands\nwe calculate and gives conversion factors from Vega magnitudes to AB\nmagnitudes and to photon fluxes. The SED shape is defined by the\naperture photometry and the overall normalization is given by the\ntotal SExtractor photometry from the deep $R$-band. However, if a\ngalaxy has both a steep colour gradient {\\it and} a large aperture\ncorrection, then the restframe colours will be biased by the nuclear\nSED. \n\n\n\n\n\\begin{table}\n\\caption{The restframe passbands and their characteristics. \n\\label{tab-rfvega} }\n\\begin{tabular}{ll|cc}\n\\hline \\hline\n name & $\\lambda_\\mathrm{\\mathrm{cen}}$\/fwhm & \n mag of Vega & $F_\\mathrm{phot}$ of Vega \\\\\n & (nm) & (AB mags) & $(10^8~\\mathrm{phot\/m^2\/nm\/s})$ \\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \n(synthetic) & 145\/10 & $+2.33$ & 0.447\\\\ \n(synthetic) & 280\/40 & $+1.43$ & 0.529\\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \nJohnson $U$ & 365\/52 & $+0.65$ & 0.820\\\\ \nJohnson $B$ & 445\/101 & $-0.13$ & 1.407\\\\ \nJohnson $V$ & 550\/83 & $+0.00$ & 1.012\\\\ \n\\noalign{\\smallskip} \\hline \\noalign{\\smallskip} \nSDSS $u$ & 358\/56 & $+0.84$ & 0.704\\\\ \nSDSS $g$ & 473\/127 & $-0.11$ & 1.305\\\\\nSDSS $r$ & 620\/115 & $+0.14$ & 0.787\\\\ \n\\hline\n\\end{tabular}\n\\end{table}\n\nThe column `ApD\\_Rmag' contains the magnitude difference between the\ntotal object photometry and the point-source calibrated,\nseeing-adaptive aperture photometry:\n\\begin{equation}\n ApD\\_Rmag = Rmag - Ap\\_Rmag ~ .\n\\end{equation}\nOn average, this value is by calibration zero for point sources, and\nbecomes more negative for more extended sources.\n\n\n\\subsection{Cross-correlation of STAGES and COMBO-17 catalogues}\\label{sec-xcorr}\n\nHaving created separate catalogues from the STAGES\n(\\S\\ref{sec-detect},\\S\\ref{sec-fitting}) and COMBO-17\n(\\S\\ref{sec-c17cat}) datasets, we next wish to create a combined,\nmaster catalogue. In GEMS, this was accomplished by applying a\nnearest neighbour matching algorithm with a maximum matching radius of\n0\\farcs75. The choice of maximum radius is governed by the resolution\nof the two datasets (HST: 0\\farcs1; COMBO-17: 0\\farcs75).\n\nFor STAGES we have however chosen to improve over this approach. For most\ngalaxies, their measured centres do not change if the input image is\nsmoothed. For example, if the HST image of a normal spiral or\nelliptical galaxy is convolved with a Gaussian function to match the\nground-based seeing, the centre estimated from the high-resolution (in\nthis case STAGES) and the low-resolution (here COMBO-17) images should\ncoincide. For distorted galaxies or mergers, this may no longer be the\ncase. Instead, the brightest peak in the STAGES image, detected as\nthe object centre by SExtractor, may be relatively far from the centre\nin the COMBO-17 image.\n\n\\begin{figure*}\n\\centerline{\\psfig{file=data_paper3a_msk.ps,width=0.7\\columnwidth}\n\\hspace{0.1\\columnwidth}\\psfig{file=data_paper4a_msk_cmp.ps,width=0.75\\columnwidth}}\n\\caption{Cross-correlation of HST and COMBO-17 data. {\\it Left:} The\ndistance to the nearest neighbour within a search radius of 5\\arcsec\nis plotted as a function of HST magnitude. At the faint end galaxies\nare matched to uncorrelated neighbours. Resolving irregular\nstructures in the HST images results in detected galaxy centres being\nlocated farther from the COMBO-17 galaxy centre than a seeing\ndistance. Matching bright objects at large separations while removing\nrandom correlations at faint fluxes requires a cut as indicated by the\ndiagonal line. Objects within the box ($V_{606}<25$ and\n$1\\arcsec<$ match distance $<2.5\\arcsec$) were inspected by eye. {\\it\nRight:} Ratio of matching distance and Kron size as a function of HST\nmagnitude. Values larger than $\\sim1$ imply a matching radius larger\nthan the object size in the HST image. Sources with\n$R_{\\textrm{ap}}<24$ are shown as black symbols; objects with a match\nbelow the cut (diagonal line in left panel) are plotted in dark grey;\nthe remaining sources with a match within 5\\arcsec\\ are shown as light\ngrey symbols.\n}\\label{fig-xcorr2}\n\\end{figure*}\n\nIn order to maximise the number of good matches between STAGES and\nCOMBO-17, in particular at low redshift, i.e.\\ A901\/2 cluster\ndistance, we have devised the following scheme. For STAGES the average\nsource density corresponds to roughly two objects per 5\\arcsec-radius\ncircle. We cross-correlate the STAGES and COMBO-17 catalogues using a\nnearest neighbour matching algorithm as described above with a maximum\nmatching radius of 5\\arcsec. The resulting matches we plot in\nFig.~\\ref{fig-xcorr2} (left panel). In particular at faint\nmagnitudes many matches are found that appear unrelated. In\ncontrast, at brighter magnitudes several sources are correlated at\nradii much larger than the COMBO-17 seeing (0.75\\arcsec), which still\nidentify the same object. In Fig.~\\ref{fig-xcorr2} we also show a\nline that subdivides the plot into two regions:\n\\begin{equation}\nd_m=-0.3\\times\\left(V_{606}-29\\right),\n\\end{equation}\nwith the matching radius $d_m$ in arcsec and the STAGES SExtractor\nmagnitude $V_{606}$. Below the line, objects are considered to be\ncorrelated, while above they are not correlated. This division is\nempirically motivated by the requirement to match objects at the faint\nend out to the COMBO-17 resolution limit (0.5\\arcsec-1.0\\arcsec) while\nalso correlating sources at larger radii at the bright end. The slope\nof the curve was determined by visual inspection of the matches inside\nthe indicated box. Typically, the distance between centroids is\n$\\sim 0\\farcs1$ (Fig.~\\ref{fig-radhist}).\n\n\\begin{figure}\n\\centerline{\\psfig{file=data_paper5.eps,width=0.85\\columnwidth}}\n\\caption{Histogram of matching radii for all objects (outer\n histogram) and $R_{\\rm ap}<24$ objects (inner histogram). The\n typical angular separation between a COMBO-17 object with\n $R_{\\textrm{ap}}<24$ and its HST counterpart is $\\sim 0.12$\\arcsec\n $\\pm 0.08$\\arcsec.}\\label{fig-radhist}\n\\end{figure}\n\nAnother way of investigating this issue is by calculating whether the\nnearest matching neighbour falls within the area covered by the object\nin the STAGES image. If the projected COMBO-17 position is beyond\nthe optical extent of the source in STAGES, it is uncorrelated. From\nthe STAGES SExtractor data we estimate the `extent' of an object by\nits Kron size $K=r_\\textrm{K}\\times a$, from the Kron radius\n$r_\\textrm{K}$ and semi-major axis radius $a$. We limit the Kron\nsize to $K>0.75$\\arcsec. A ratio of $d_m\/K \\ga 1$ indicates that\nthe matched COMBO-17 source lies outside the region covered by the\nobject in the STAGES image. In Fig.~\\ref{fig-xcorr2} (right panel)\nwe overplot in grey all sources that were assigned a partner from the\nnearest neighbour matching. This provides further evidence for the\nimproved quality of our new cross-correlation method.\n\nIn summary, the combined catalogue contains 88\\,879 sources. Of these,\n$\\sim6\\,577$ objects with a COMBO-17 ID are not within the region\ncovered by the STAGES HST mosaic ($\\sim 1\\,664$ of these have\n$R_{\\textrm{ap}}<24$). Moreover, $\\sim 1\\,271$ STAGES detections are\noutside the COMBO-17 observation footprint.\\footnote{The observation\nfootprint for both STAGES and COMBO-17 is rather difficult to\ndetermine. Therefore, we provide only approximate numbers good to\n$\\sim50$ objects. A more elaborate scheme than the one used to produce\nthese numbers is well beyond the scope of this paper.} Inside the\nregion covered by both surveys, there are $\\sim 81\\,031$ sources. For\n50701 objects the method described above provides a match between\nCOMBO-17 and STAGES (15760 of these have $R_{\\textrm{ap}}<24$). $\\sim\n23\\,833$ sources detected in STAGES do not have counterparts in\nCOMBO-17; $\\sim 6\\,497$ sources from the COMBO-17 catalogue are not\nmatched to STAGES detections. Out of these, only $\\sim 79$ objects\nhave \\mbox{$R_{\\textrm{ap}}<24$}. We therefore emphasize that for\nour science sample of COMBO-17 objects, defined as having\n$R_{\\textrm{ap}}<24$, 99.9\\% have a STAGES counterpart. The majority\nof failures result from confusion by neighbouring objects or simply\nnon-detections.\n\n\n\\subsection{Selection of an A901\/2 cluster sample}\n\nWe wish to define a `cluster' galaxy sample of galaxies belonging to\nthe A901\/2 complex for various follow-up studies of our team that are\nin progress. These studies may have different requirements for the\n{\\em completeness} of cluster members and the {\\em contamination} by\nfield galaxies. We therefore quantified how these two key values vary\nwith both magnitude and width of the redshift interval in order to\ninform our choice of definition.\n\nThe photo-z distribution of cluster galaxies was assumed to follow a\nGaussian with a width given by the photo-z scatter in\nEquation~\\ref{dz_rel}. The distribution of field galaxies was assumed to\nbe consistent with the average galaxy counts $n(z,R)$ outside the\ncluster and varies smoothly with redshift and magnitude assuming no\nstructure in the field. Samples were then defined by redshift\nintervals $z_{\\rm phot}=[0.17-\\Delta z, 0.17+\\Delta z]$, where the\nhalf-width $\\Delta z$ was allowed to vary with the\nmagnitude.\\footnote{We use $z_{\\rm phot}=0.17$ for the mean cluster\nredshift here rather than the spectroscopically confirmed $z_{\\rm\nspec}\\sim0.165$ due to the known bias discussed in\n\\S\\ref{sec-c17cat}.} We calculated completeness and contamination at\nall magnitude points simply using the counts of our smooth models.\n\nWe found that as long as the half-width in redshift is not much larger\nthan a couple of Gaussian FWHMs, the contamination changes only\nlittle. The ratio of selected cluster to field galaxies is almost\ninvariant as shrinking widths cut into numbers for both origins. Only\nenlarging the width significantly over that of the Gaussian increases\ncontamination by field galaxies. On the contrary, such large widths do\nnot affect the completeness of the cluster sample much, while\nshrinking the width too far eats into the true cluster distribution\nand reduces completeness of the cluster sample.\n\nFor our purposes, we compromised on a photo-z width such that the\ncompleteness is $>90$\\% at any magnitude, just before further widening\nstarts to increase the contamination above its mag-dependent minimum\n(see Fig.~\\ref{fig-CW_clus} and Fig.~\\ref{fig-CW_comcon}, left\npanel). For this we chose a half-width of\n\n\\begin{equation}\\label{eqn-cluster}\n \\Delta z (R) = \\sqrt{0.015^2+0.0096525^2 \n (1+10^{0.6 (R_{\\rm tot}-20.5)}) } ~ .\n\\end{equation}\n\nThis equation defines a half-width that is limited to 0.015 at the\nbright end and expands as a constant multiple of the estimated photo-z\nerror at the faint end. The floor of the half-width is motivated by\nincluding the entire cluster member sample previously studied by\nWMG05. The completeness of this selection converges to nearly 100\\%\nfor bright galaxies, as a result of intentionally including the WGM05\nsample entirely.\n\n\n\\begin{figure}\n\\centerline{\\includegraphics*[height=0.85\\columnwidth,angle=270]{CW_cluster.ps}}\n\\caption{MEV redshift estimate vs. total $R$-band magnitude. The\n'galaxy' sample is shown in green, while the sample of `cluster'\ngalaxies defined by Equation~\\ref{eqn-cluster} is shown in black. The\nmagnitude-dependent redshift interval guarantees almost constant high\ncompleteness, while the field contamination increases towards faint\nlevels (Fig.~\\ref{fig-CW_comcon}). We note that at faint\nmagnitudes there is an apparent asymmetry towards lower redshift at\nfaint magnitudes within the cluster sample. The photometric redshifts\nmay be skewed by systematic effects but the average $-0.02$ offset at\n$R_{\\rm tot}\\sim22.5$ is within the $1\\sigma$ error envelope.\n\\label{fig-CW_clus}}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics*[height=\\columnwidth,angle=270]{CW_complcontam.ps}\n\\caption{{\\it Left panel:} Completeness of the cluster sample defined\nin Fig.~\\ref{fig-CW_clus} and designed to provide high completeness at all \nmagnitudes.\n{\\it Right panel:} The field contamination of the cluster sample \nincreases at faint levels due to photo-z dilution of the cluster. \nNarrowing the selected redshift interval would not reduce the \ncontamination. Contamination rates are estimated to be\n$(10,20,30,50,70)\\%$ at $R_{\\rm ap}=(20.3,21.65,22.3,23.2,24.0)$.\n\\label{fig-CW_comcon}}\n\\end{figure}\n\nThe right panel of Fig.~\\ref{fig-CW_comcon} shows that the\ndifferential contamination increases rapidly towards faint magnitudes,\nsimply as a result of the photo-z error-driven dilution of the cluster\nsample. Here, contamination means the fraction of galaxies that are\nfield members, as measured in a bin centred on the given magnitude\nwith width 0.1 mag. Contamination at a given apparent magnitude\ntranslates into contamination at a resulting luminosity at the cluster\ndistance (except that scatter in the aperture correction smears out\nthe contamination relation slightly).\n\nAlready at $R_{\\it ap}=23.2$ the sample contains as many cluster as\nfield members. This corresponds to $M_V \\approx -16.5$ for the average\ngalaxy, but scatters around that due to aperture corrections. As we\nprobe fainter this selection adds more field galaxies than cluster\nmembers. Follow-up studies can now determine an individual magnitude\nor luminosity limit given their maximum tolerance for field\ncontamination. For example, WGM05 selected cluster galaxies at $M_V -\n< -17.775$ ($M_V < -17$ for their adopted cosmology with $H_0=100$ km\ns$^{-1}$ Mpc$^{-1}$) for an earlier study of the A901\/2 system in\norder to keep the contamination at the faint end below 20\\%.\n\nThe cluster sample thus obtained covers quite a range of photo-z\nvalues at the faint end, and restframe properties are derived assuming\nthese redshifts to be correct. However, if we assume a priori that an\nobject is at the redshift of the cluster, then we may want to know\nthese properties assuming a fixed cluster redshift of\n$z=0.167$. Hence, the SED fits and restframe luminosities are\nrecalculated for this redshift and reported in additional columns of\nthe STAGES catalogue in Table~\\ref{tabcolumns} (with \\mbox{`\\_cl' }\nsuffix indicating cluster redshift). Of course, if the a-priori\nassumption is to believe the redshifts as derived, then the original\nset of columns for which we have derived the values is relevant.\n\n\n\\section{Further multiwavelength data and derived\n quantities}\\label{sec-multi}\n\nIn this section we describe further multiwavelength data for the\nA901\/2 region taken with other facilities (Fig.~\\ref{fig-field}). We\nalso present several resulting derived quantities (stellar masses and\nstar formation rates) that appear as entries in the STAGES master\ncatalogue.\n\n\\subsection{Spitzer}\\label{sec-mir}\n\nSpitzer observed a $1\\degr \\times 0{\\fdg}5$ field around the A901\/2\nsystem in December 2004 and June 2005 as part of Spitzer GO-3294 (PI:\nBell). The MIPS 24{\\micron} data were taken in slow scan-map mode,\nwith individual exposures of 10\\,s. We reduced the individual image\nframes using a custom data-analysis tool (DAT) developed by the GTOs\n\\citep{gordon05}. The reduced images were corrected for geometric\ndistortion and combined to form full mosaics; the reduction which we\ncurrently use does not mask out asteroids and other transients in the\nmosaicing.\\footnote{This only minimally affects our analyses because we\nmatch the IR detections to optical positions, and most of the bright\nasteroids are outside the COMBO-17 field.} The final mosaic has a\npixel scale of $1\\farcs25$~pixel$^{-1}$ and an image PSF FWHM of\n$\\simeq 6$\\arcsec. Source detection and photometry were performed\nusing techniques described in \\citet{papovich04}; based on the\nanalysis in that work, we estimate that our source detection is 80\\%\ncomplete at 97~$\\mu$Jy\\footnote{We note that for previous papers we\nused the catalogue to lower flux limits, down to 3$\\sigma$;\naccordingly, we have included such lower-significance (and more\ncontaminated) matches in the catalogue.} for a total exposure of\n$\\sim 1400$\\,s\\,pix$^{-1}$. By detecting artificially-inserted\nsources in the A901 24 image, we estimated the completeness of the\nA901 24 $\\mu$m catalog. The completeness is 80\\%, 50\\% and 30\\% at 5,\n4 and 3$\\sigma$, respectively.\n\n\nNote that there is a very bright star at 24{\\micron} near the centre\nof the field at coordinates $(\\alpha,\\delta)_{\\rm J2000}=(09^h56^m32\\fs 4,\n-10\\degr 01 \\arcmin 15\\arcsec)$ (see \\S\\ref{sec-mira} for\ndetails of this object). In our analysis of the 24{\\micron} data we\ndiscard all detections less than 4$'$ from this position in order to\nminimise contamination from spurious detections and problems with the\nbackground level in the wings of this bright star. It is to be noted\nthat there are a number of spurious detections in the wings of the\nvery brightest sources; while we endeavoured to minimise the incidence\nof these sources, they are difficult to completely eradicate without\nlosing substantial numbers of real sources at the flux limit of the\ndata.\n\nTo interpret the observed 24{\\micron} emission, we must match the\n24{\\micron} sources to galaxies for which we have redshift estimates\nfrom COMBO-17. We adopt a 1\\arcsec matching radius. In the areas of\nthe A901\/2 field where there is overlap between the COMBO-17 redshift\ndata and the full-depth MIPS mosaic, there are a total of 3506(5545)\n24{\\micron} sources with fluxes in excess of 97(58)$\\mu$Jy. Roughly\n62\\% of the 24{\\micron} sources with fluxes $> 58 \\mu$Jy are detected\nby COMBO-17 in at least the deep $R$-band, with $R \\la 26$. Some 50\\%\nof the 24{\\micron} sources have bright $R_{\\rm tot} < 24$ and have\nphotometric redshift $z<1$; these 50\\% of sources contain nearly 60\\%\nof the total 24{\\micron} flux in objects brighter than 58$\\mu$Jy.\nSources fainter than $R \\ga 24$ contain the rest of the $f_{24} >\n58\\mu$Jy 24{\\micron} sources; investigation of COMBO-17 lower\nconfidence photometric redshifts, their optical colours, and results\nfrom other studies lends weight to the argument that essentially all\nof these sources are at $z>0.8$, with the bulk lying at $z > 1$\n(e.g. \\citealt{lefloch2004}, \\citealt{papovich04}; see\n\\citealt{lefloch2005} for a further discussion of the completeness of\nredshift information in the CDFS COMBO-17 data).\n\nObservations with IRAC \\citep[Infrared Array Camera;][]{Fazio2004} at\n3.6, 4.5, 5.8 and 8.0{\\micron} were also taken as part of this Spitzer\ncampaign: those data are not discussed further here, and will be\ndescribed in full in a future publication.\n\n\n\\subsection{Star formation rates}\n\nWe provide estimates of star formation rate, determined using a\ncombination of 24{\\micron} data (to probe the obscured star formation)\nand COMBO-17 derived rest-frame 2800{\\AA} luminosities (to probe\nunobscured star formation). Ideally, we would have a measure of the\ntotal thermal IR flux from 8--1000{\\micron}; instead, we have an\nestimate of IR luminosity at one wavelength, 24{\\micron},\ncorresponding to rest-frame 22--12{\\micron} at the redshifts of\ninterest $z=0.1-1$. Local IR-luminous galaxies show a tight\ncorrelation between rest-frame 12--15{\\micron} luminosity and total IR\nluminosity \\citep[e.g.,][]{spi95,cha01,rou01,papovich02}, with a\nscatter of $\\sim 0.15$ dex.\\footnote{Star-forming regions in local\ngalaxies appear to follow a slightly non-linear relation between\nrest-frame 24{\\micron} emission and SFR, with SFR $\\propto L_{24\\mu\nm}^{0.9}$ \\citep{calzetti07}, although note that this calibration is\nbetween 24{\\micron} emission and SFR (not total IR luminosity).}\nFollowing \\citet{papovich02}, we choose to construct total IR\nluminosity from the observed-frame 24{\\micron} data. We use the Sbc\ntemplate from the \\citet{dev99} SED library to translate\nobserved-frame 24{\\micron} flux into the 8--1000{\\micron} total IR\nluminosity.\\footnote{Total 8--1000{\\micron} IR luminosities are $\\sim\n0.3$ dex higher than the 42.5--122.5{\\micron} luminosities defined by\n\\citet{Helou1988}, with an obvious dust temperature\ndependence.} The IR luminosity uncertainties are primarily\nsystematic. Firstly, there is a natural diversity of IR spectral\nshapes at a given galaxy IR luminosity, stellar mass, etc.; one can\ncrudely estimate the scale of this uncertainty by using the full range\nof templates from \\citet{dev99}, or by using templates from, e.g.,\n\\citet{dale01} instead. This uncertainty is $\\la$0.3\\,dex (this\nagrees roughly with the scatter seen between 24{\\micron} luminosity\nand SFR seen in \\citealp{calzetti07}). Secondly, it is possible that\na significant fraction of $0.1 11$ from their study.\n\n\n\\subsection{GALEX}\n\nThe Abell 901\/902 field was observed by GALEX in the far-UV ($f,\n\\lambda_{\\rm eff}\\sim 1528$\\AA ) and near-UV ($n, \\lambda_{\\rm\neff}\\sim 2271$\\AA) bands.\\footnote{Unlike all other datasets detailed\nhere, the GALEX observations were not led by members of the STAGES\nteam. We list the publicly archived data products here for\ncompleteness.} Individual observations (or single orbit `visits')\nbetween the dates 12 February 2005 and 25 February 2007 were coadded\nby the GALEX pipeline \\citep[GR4 version][]{Morrissey2007} to produce\nimages with net exposure times of 57.18 ks in $n$ (47 visits) and\n50.19 ks in $f$ (40 visits). The GALEX field of view in both bands is\na 0.6\\degr radius circle, and the average centre of the visits (the\nGALEX field centre) is $(\\alpha,\\delta)_{\\rm J2000} = (9^{\\rm\nh}56^{\\rm m}20\\fs7, -10\\degr6\\arcmin21\\farcs6)$. The GALEX PSF near\nthe field centre has $\\sim 4.2$\\arcsec\\ FWHM at $f$ and $\\sim\n5.3$\\arcsec\\ FWHM at $n$, both of which increase with distance from\nthe field centre (variations in the PSF that are not a function of\ndistance from the field centre are smoothed out by the distribution of\nroll angles of the visits). The astrometric accuracy is $\\sim\n0.7$\\arcsec, and $>97$\\% of catalogued source positions are within\n2\\arcsec~of their true positions. The photometric calibration is stable\nto $0.02$ mag in $n$ and $0.045$ mag in $f$ \\citep{Morrissey2007}.\n\nSource detection and photometry is via the GALEX pipeline code, which\nemploys a version of SExtractor \\citep{bertin96} modified for use with\nlow-background images. Magnitudes are measured both in fixed circular\napertures and in automatic Kron elliptical apertures, and in isophotal\napertures. The 5$\\sigma$ point-source sensitivities in the Abell\n901\/902 field are $f \\sim 24.7$ mag (AB) and $n \\sim 25.0$ mag (AB),\nthough there are spatial variations across field, especially a\nslightly decreasing sensitivity towards the edge of the field. At\nthese levels source confusion in the $n$ band becomes an issue, and\nthe $n$ band fluxes of faint objects ($n\\ga 23$ mag) are likely to\nbe overestimated. GALEX data products include intensity, background,\nand relative response (i.e., effective exposure time) maps in both\nbands as well as source catalogues in both bands and a band-merged\nsource catalogue.\n\n\n\\subsection{2dF spectroscopy}\\label{sec-2df}\n\nSpectra of cluster galaxies were obtained using the 2dF\ninstrument on the AAT in March 2002 and March 2003. A total of 86\ngalaxies were observed using the 1200B grating (spanning the observed\nwavelength range 4000--5100 \\AA) in a single fibre configuration during\nthe 2002 run. Three fibre configurations using the lower resolution\n600V grating (spanning 3800--5800 \\AA) were observed during the 2003\nrun: fibres were placed on 368 objects, with 47 repeated from 2002.\nThe primary selection function assigned higher priority to those\ngalaxies selected by photometric redshift to be within the\nsupercluster redshift slice and having $R<20$, with additional fibres\nbeing allocated to secondary targets (including fainter galaxies and a\nsmall number of white dwarfs and QSOs) when available. Data reduction\nwas performed with the standard {\\tt 2dfdr} (v2.3) pipeline package.\n\nIn total, spectra were obtained for 407 unique objects. Redshifts\nwere determined by two independent means: firstly by manual line\nprofile fitting of the Ca H and K features in absorption and secondly\nby cross-correlation with template spectra using the XCSAO task within\nIRAF (Kurtz \\& Mink 1998). Comparison of the two measurements showed\nno cause for concern, with $\\Delta_z=0.00149\\pm0.00006$. After\neliminating non-galaxy and poor quality spectra, we have redshifts for\n353 galaxies in total. \n\nThe 2dF spectroscopic data have previously been used to quantify the\nreliability of the COMBO-17 redshifts in W04 (see also \\S\\ref{sec-c17}),\nto verify cluster membership for the matched X-ray point sources\n\\citep{gilmour2007}, and to create composite spectra for three\nphotometric classes of cluster galaxies in WGM05. A dynamical\nanalysis of the the clusters using the 2dF redshifts will be presented\nin Gray et al. (in prep.).\n\n\\subsection{XMM-Newton} \\label{sec-xray}\n\nX-ray data for the A901\/2 region is desirous both to detect\npoint-source emission from cluster members (star-formation or AGN) and\nthe extended intracluster medium (ICM). A 90~ks XMM image of the\nA901\/2 field was taken on May 6\/7 2003 using the three EPIC cameras\n(MOS1, MOS2 and PN) and a thin filter, under program 14817 (PI: Gray).\nThe level 1 data were taken from the supplied pipeline products, and\nreduced with SAS v5.4 and the calibration files available in May 2003.\nFinal exposure times were $\\sim67$ ks for MOS and $\\sim$61 ks for PN\nfollowing the removal of time intervals suffering from soft proton\nflares. Four energy bands were used: 0.5-2 keV (soft band), 2-4.5 keV\n(medium band), 4.5-7.5 keV (hard band) and 0.5-7.5 keV (full band).\n\nThe creation of the point-source catalogue using wavelet detection\nmethods is described in detail elsewhere \\citep{gilmour2007}. A total\nof 139 significant sources were found. The presence of an X-ray\nluminous Type-I AGN near the centre of A901a (see\nAppendix~\\ref{sec-agnnote}) complicated the detection of the underlying\nextended cluster emission. A maximum-likelihood technique was used to\nmatch this catalogue to COMBO-17 resulting in 66 secure counterparts\nwith photometric redshifts. \\cite{gilmour2007} used these data to\nexamine the local environments of the cluster AGN and their host\nproperties.\n\nTo isolate the remaining extended emission coming from the clusters, a\nseparate conservative point-source catalogue was constructed. Care\nwas taken to remove both the cosmic background and spatial variations in\nthe non-cosmic background. The background subtracted images were\nweighted by appropriate energy conversion factors to create flux\nimages for each detector. These flux images were masked and summed\ntogether to create merged background-subtracted images in each band.\n\nPoint source regions were removed and replaced with the local\nbackground value selected randomly from a source free area within 10 pixels\n(or 20 pixels if there were not enough background pixels within the\nsmaller radius). Smoothed images were created in each band using a\nGaussian kernel of radius 4 pixels. Maps of the extended emission and\nan examination of the global X-ray properties of the clusters will be\npresented in Gray et al. (in prep.).\n\n\n\n\n\n\\subsection{GMRT}\n\n\nThe A901\/2 field was observed on 2007 March 25th and 26th March with\nthe Giant Metrewave Radio Telescope (GMRT, see\n\\citealt{Ananthakrishnan2005} for further details). The field was\ncentred at $(\\alpha,\\delta)_{\\rm J2000}=(09^{\\rm h} 56^{\\rm m} 17^{\\rm\ns}, -10\\degr 01\\arcmin 28\\arcsec)$ and observed at 610 and 1280~MHz on\nrespective nights. The GMRT is an interferometer, consisting of thirty\nantennas, each 45~m in diameter. The bright sources 3C147 and 3C286\nwere observed at the start and end of each observing session, in order\nto set the flux density scale. During the observations a nearby\ncompact source 0943$-$083 was observed for about 4 minutes at roughly\n30 minutes intervals, to monitor and correct any antenna-based\namplitude and phase variations.\n\nThe total integration time on the field was $\\sim$6.5 hours at\neach frequency. The observations covered two 16~MHz sidebands,\npositioned above and below the central frequency. Each sideband was\nobserved with 128 narrow channels, in order to allow narrow band\ninterference to be identified and efficiently removed. The observed\nvisibility data were edited and calibrated using standard tasks with\nthe AIPS package, and then groups of ten adjacent channels were\naveraged together, with some end channels discarded. This reduced the\nvolume of the visibility data, whilst retaining enough channels so\nthat chromatic aberration is not a problem \\citep[e.g., see][for further\ndetails of GMRT analysis]{Garn2007}. Given the relatively large field\nof view of the GMRT compared with its resolution, imaging in AIPS\nrequires several `facets' to be imaged simultaneously, and then be\ncombined. Preliminary imaging results, after several\niterations of self-calibration, have produced images with resolutions\nof about 5\\arcsec and 2\\farcs5 at 610 and 1280-MHz respectively, with\nr.m.s.\\ noises of approximately 25 and 20 $\\mu$Jy~beam$^{-1}$ in the\ncentre of the fields, before correction for the primary beam of the\nGMRT. The primary beam -- i.e.\\ the decreasing sensitivity away from\nthe field centres due to sensitivity of individual 45-m antennas -- is\napproximately Gaussian, with a half-power beam width (HPBW) of\napproximately 44\\arcmin and 26\\arcmin at 610 and 1280-MHz\nrespectively. These images are among the deepest images made at these\nfrequencies with the GMRT. Further analysis and the source catalogue\nwill be presented in Green et al.(in prep.).\n\n\\subsection{Simulations and mock galaxy catalogues}\\label{sec-mocks}\n\nIn order to facilitate the interpretation of the observational results\nand to study the physical processes of galaxy evolution, N-body,\nhydrodynamic, and semi-analytic simulations that closely mimic the\nA901\/2 system are being produced (van Kampen et al., in prep.). We\nconstrain initial conditions using the method of \\citet{hoffman1991}\nto take into account the gross properties of A901a, A901b, A902, the\nSW group, and the neighbouring clusters A868 and A907 (outside the\nobserved field). The simulations produce a range of mock large-scale\nstructures to test three basic formation scenarios: a 'stationary'\ncase, where A901(a,b) and A902 will not merge within a Hubble time,\nand a pre- as well as a post-merger scenario. When the likelihood of\neach scenario is understood, one can further test the models for the\ndetailed physical processes known to be operating on galaxies in and\naround such clusters.\n\n\n\\section{Summary and Data Access}\\label{sec-summary}\n\n\n\n\nWe have presented the multiwavelength data available for the A901\/2\nsupercluster field as part of the STAGES survey: high-resolution HST\nimaging over a wide area, extensive photometric redshifts from\nCOMBO-17, and further multiwavelength observations from X-ray to\nradio. These data have already been used to create a high resolution\nmass map of the system using weak gravitational lensing\n\\citep{heymans08}. \nFurther work by the STAGES team to study galaxy evolution and\nenvironment is ongoing and includes the following:\n\n\n\\begin{itemize}\n\n\\item \\citet{gallazzi08} explore the amount of obscured star-formation\nas a function of environment in the A901\/2 supercluster and associated\nfield sample by combining the UV\/optical SED from COMBO-17 with the\nSpitzer 24$\\mu$m photometry in galaxies with $M_*>10^{10} M_{\\sun}$.\nResults indicate that while there is an overall suppression in the\nfraction of star-forming galaxies with density, the small amount of\nstar formation surviving the cluster environment is to a large extent\nobscured.\n\n\\item Wolf et al. (MNRAS, accepted) investigate the properties of optically\npassive spiral and dusty red galaxies in the supercluster and find\nthat the two samples are largely equivalent. These galaxies form\nstars at a substantial rate that is only a factor of four times lower\nthan blue spirals at fixed mass, but their star formation is more\nobscured and has weak optical signatures. They constitute over half of\nthe star forming galaxies at masses above $\\log M_*\/M_{\\sun}=10$ and\nare thus a vital ingredient for understanding the overall picture of\nstar-formation quenching in cluster environments.\n\n\\item Marinova et al. (ApJ, submitted) identify and characterize bars in\nbright ($M_{V} \\le -18$) cluster galaxies through ellipse-fitting. The\nselection of moderately inclined disk galaxies via three commonly used\nmethods, visual classification, colour, and S\\'ersic\\ cuts, shows that\nthe latter two methods fail to pick up many red, bulge-dominated disk\ngalaxies in the clusters. However, all three methods of disk\nselection yields a similar global optical bar fractions ($f_{\\rm\nbar-opt}\\sim0.3)$, averaged over all galaxy types. When host galaxy\nproperties are considered, the optical bar fraction is found to be a\nstrong function of both the luminosity and morphological property\n(bulge-to-disk ratio) of the host galaxy, similar to trends recently\nreported in field galaxies. Furthermore, results indicate that the\nglobal optical bar fraction for bright galaxies is not a strong\nfunction of local environment.\n\n\\item Heiderman et al. (in prep.) identify interacting galaxies in the\nsupercluster using quantitative analysis and visual classifications.\nTheir findings include that $4.9\\pm 1.3\\%$ of bright ($M_{V} \\le\n-18$), intermediate mass ($M_{*} \\ge 1 \\times 10^{9} M_{\\sun}$)\ngalaxies are interacting. The interacting galaxies are found to lie\noutside the cluster cores and to be concentrated in the region between\nthe cores and virial radii of the clusters. Explanations for the\nobserved distribution include the large galaxy velocity dispersion in\nthe cluster cores and the possibility that the outer parts of the\nclusters are accreting groups, which are predicted to show a high\nprobability for mergers and strong interactions. The average star\nformation rate is enhanced only by a modest factor in interacting\ngalaxies compared to non-interacting galaxies, similar to conclusions\nreported in the field by \\citet{Jogee08}. Interacting galaxies only\ncontribute $\\sim$~20\\% of the total SFR density in the A901\/902\nclusters.\n\n\n\\item Boehm et al. (in prep.) are utilizing the stability of the PSF on\nthe STAGES images for a morphological comparison between the hosts of\n~20 type-1 AGN and ~200 inactive galaxies at an average redshift\n$\\left < z \\right >\\sim0.7$. This analysis includes extensive simulations of the\nimpact of a bright optical nucleus on quantitative galaxy morphologies\nin terms of the CAS indices and Gini\/$M_{20}$ space. We find that the\nmajority of the hosts cover parameters typical for disk+bulge systems\nand mildly disturbed galaxies, while evidence for strong gravitational\ninteractions is scarce.\n\n\\item Bacon et al. (in prep.) are examining the higher order lensing\nproperties of the STAGES data. They construct a shapelets catalogue\n\\citep{refregier03} for the STAGES galaxies; this is then used to\nestimate the gravitational flexion \\citep{bacon06} at each galaxy\nposition. Galaxy--galaxy flexion is measured, leading to estimates of\nconcentration and mass for STAGES galaxies; constraints on cosmic\nflexion are also found, showing very good containment of systematic\neffects. The ability of flexion to improve convergence maps is also\ndiscussed.\n\n\\item Robaina et al. (in prep.) make use of a combined GEMS and STAGES\nsample of $0.410^{10} M_{\\odot}$) show a modest\nenhancement of their star formation rate; in particular, less than\n15\\% of star formation at $0.4} J_{ij} S_i S_j - K \\sum_{\\ll k l \\gg} S_k S_l \\ ,\n\\label{model}\n\\ee \nwhere $$ denotes sum over all the first nearest neighbors pairs\n(distance 1) and $\\ll k l\\gg$ denotes sum over all the second nearest\nneighbors pairs (distance $\\sqrt 2$).The couplings $J_{ij}$ are quenched\nvariables with $J_{ij}=\\pm \\lambda$ with probability $1\/2$. $\\lambda$\nand $K$ are positive constants whose ratio determines the relative\nstrength between spin glass and ferromagnetic interactions. In the\nrest of the paper we fix $K=1$. In order to fix the notation we also\ndefine the variance of the $J_{ij}$ distribution as $\\sigma^2_J \\equiv\n\\overline{ J_{ij}^2}=\\lambda^2$.\n\nWe next describe two limiting cases of the model (for $K=1$):\n\\begin{itemize}\n\n\\item For $\\lambda \\to \\infty$ the system is the $2D$ spin glass for\nwhich is known that $T_c=0$\n\\cite{BOOK,review,bhatt,kawashima,rieger,mydosh}.\n\n\\item When $\\lambda=0$ the whole lattice decouples into two independent\nsub-lattices (that we will denote hereafter as sub-lattices 1 and 2,\nthe black and white sub-lattices in a chess-board. \nEach sub-lattice is itself a two\ndimensional ferromagnetic Ising model and will have a phase transition\n(paramagnetic-ferromagnetic) just at the Onsager temperature: $T_c\n=2\/\\log(1+\\sqrt 2) \\simeq 2.269$. The order parameter of the\nphase transition is the so-called staggered magnetization: i.e. the\nmagnetization of one of the two sub-lattices. Obviously, the\nprobability distribution of the total magnetization of the whole\nlattice for low temperatures will take into account the four possible\ndifferent magnetizations of the two independent sub-lattices.\n\n\\end{itemize}\n\nSo initially we have, when $\\lambda=0$, ferromagnetic order (in\nboth sub-lattices), that can be affected by the introduction of\nspin glass couplings. The effect of these couplings is to couple (by\nmeans of a spin glass interaction) both sub-lattices.\n\n\n\nIn the next section we will examine analytically the effect of the\nintroduction of random couplings (linking both sub-lattices) \nin the original stable (staggered) ferromagnetic order.\n\n\\section{\\label{A_RES}Analytic Results}\n\nWe will study in this section the analogy, suggested by \nLemke and Campbell~\\cite{lemke}, between the Hamiltonian defined by Eq.(\\ref{model})\nand the Random Field Ising Model (RFIM).\n\nWe can rewrite the original Hamiltonian Eq.(\\ref{model}) in the \nfollowing way\n\\be\n{\\cal H}= - \\sum_{} \\sigma_{i_1} \\sigma_{j_1} \n- \\sum_{} \\tau_{i_2} \\tau_{j_2} \n- \\sum_{i_1} \\sum_{j_2(i_1)} J_{i_1 j_2} \\sigma_{i_1} \\tau_{j_2} \\ ,\n\\label{ham_equiv}\n\\ee\nwhere the indices $i_1, j_1$ ($i_2, j_2$) run over the sub-lattice 1\n(respectively 2), $$ ($$) \ndenotes sum to first nearest neighbors\npairs in the sub-lattice 1 (respectively 2)\nand $j_2(i_1)$ denotes the sum over the two nearest neighbors ($j_2$'s) of\nthe site $i_1$ (in the two positive directions from $i_1$) \nin the whole lattice. \nMoreover we have denoted the variables of the\nsub-lattice 1 (2) as $\\sigma$'s (respectively $\\tau$'s).\n\nNow we fix the temperature to a small value and all the spins inside the\nsub-lattice 1 are fixed in the up state. \nNext we will examine the cost in energy to\ndo a compact droplet (in the sub-lattice 1) with all its spins flipped\ndown, with the spins of the sub-lattice 2 fixed to an\narbitrary configuration.\n\nThe Hamiltonian of the model with all the $\\tau$ spins\nfixed to an arbitrary configuration is, modulo a constant,\n\\begin{eqnarray}\n\\nonumber\n{\\cal H}[\\sigma | \\tau \\,\\,{\\rm fixed} ]&=& \n- \\sum_{} \\sigma_{i_1} \\sigma_{j_1} \n- \\sum_{i_1} \\left[\\sum_{j_2(i_1)} J_{i_1 j_2} \\tau_{j_2} \\right] \n\\sigma_{i_1} \\\\\n &\\equiv& - \\sum_{} \\sigma_{i_1} \\sigma_{j_1} \n- \\sum_{i_1} h_{i_1} \\sigma_{i_1} \\ ,\n\\end{eqnarray}\ni.e. a Random Field Ising Model (RFIM) where the magnetic field \n has zero mean and variance:\n\\be\n\\overline{h_i h_j}= \\left\\{ \n \\begin{tabular}{l l}\n $2 \\sigma^2_J$& if $i=j$ , \\\\\n $0$ & elsewhere.\n \\end{tabular} \n \\right.\n\\ee\n\nIn particular we can think that all the spins in the $\\tau$\nsub-lattice are fixed up. It is easy to reproduce all the steps of\nthe Imry-Ma argument~\\cite{ma,NATTERMANN} (by turning on the temperature)\nto show that one can find large regions where it is favorable\nenergetically to flip all the spins inside the region and hence to\ndestroy the long range order of the sub-lattice 1 independently of the\nconfiguration of the sub-lattice 2 (and vice-versa). \n\nThus we have\nshown that the ferromagnetic order (in either or in both sub-lattices)\nis unstable against an infinitesimal strength of the spin glass couplings:\ni.e. initially the sub-lattices 1 and 2 are fixed to up (staggered\nferromagnetic order) and we have found that for any configuration\n$\\tau$ the sub-lattice 1 disorders, and by redoing the same steps with\nsub-lattice 1 fixed (and disordered) we can see that the originally ordered\nsub-lattice 2 disorders too.\n\nMoreover, following Binder~\\cite{BINDER}, we should expect that there\nexists a crossover length $R_c$ such that for $RR_c$. The analytical expression for a RFIM with ferromagnetic\ncoupling $K$ and uncorrelated magnetic field with variance\n$\\sigma_h^2$ is~\\cite{BINDER}\n\\begin{equation}\nR_c \\propto \\exp \\left[ C\\left(\\frac{K}{\\sigma_h}\\right)^2\\right]\\ ,\n\\end{equation}\nwhere $C$ is a constant (O(1)), and so for our particular model \n(where $K=1$ and $\\sigma_h^2=2 \\sigma_J^2$), we finally obtain\n\\begin{equation}\nR_c \\simeq \\exp \\left[ \\frac{C}{2 \\sigma^2_J}\\right]=\n\\exp \\left[ \\frac{C}{2 \\lambda^2}\\right] \\ .\n\\end{equation}\n\nObviously when $\\lambda$ is infinitesimally small (i.e. we put an\ninfinitesimal amount of spin glass disorder in the model) the\ncrossover ratio is exponentially large and ferromagnetic order will\nonly be destabilized in extremely large systems. Nevertheless, in the\nthermodynamic limit the spin glass disorder is always relevant.\n\nFinally, we can estimate that for $\\lambda=0.5$ (the value that we\nhave used in our numerical simulations presented in this work) $R_c\n\\simeq 7$, assuming that $C$ is just 1.\n\nAt this point for $L > R_c$, where $L$ is the linear size of the\nsystem, the picture is the following: both sub-lattices\nare broken in clusters (inside of them all the spins, in average, point\nin the same direction) of size less than $R_c$ interacting between\nthem. From the Imry-Ma argument it is clear that the\n``effective'' interaction between these clusters is short range\n(i.e. it could be very large but not infinite). Consequently, we have a two\ndimensional spin glass with short range interactions. This phase can be\nthought as a frozen disordered phase (like the spin glass phase) where\nthe clusters play the role of the spin in the usual spin glass\nphase. \n\nBut we know that there exists no spin glass order at finite\ntemperature for short range spin glasses in two dimensions, and so we\nconclude that there must be a second crossover from the spin glass\nbehavior to paramagnetic behavior as the size of the system\nincreases. We need only a correlation length greater than the range of\nthe interaction between the clusters to return to the usual short\nrange spin glass in two dimensions that has no phase transition. In\nthe next sections we will try to put this fact in more quantitative\ngrounds.\n\n\\section{\\label{N_RES}Numerical Simulations and Observables}\n\nWe have simulated, using the Metropolis algorithm, systems with linear sizes\nranging from $L=4$ to $L=48$ and averaging over 200 to 10000 samples\ndepending on the size. The largest lattice size simulated in reference \n\\cite{lemke} used in the computation of their Binder cumulant was $L=12$.\n\nIn all the runs we have used an annealing procedure from higher\ntemperatures to the lower ones in order to thermalize the system. In\nTable \\ref{table:stat} we report the statistics we have used. We have\nperformed in the annealing procedure for all the temperatures the same\nnumber of thermalization steps ($N_T$), that we have written in Table\n\\ref{table:stat}. \n\nFor a given temperature we run $N_T$ steps for\nthermalization and we\nmeasure during $2 N_T$ and then we lower the temperature and we repeat\nthe process always with the same $N_T$. \nWe will return to the issue of the thermalization time at\nthe end of this section.\n\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|} \\hline\n$L$ & $T$'s & samples & Termalization time \\\\ \\hline\n4 & [1.5,4.0] & 10000 &10000 \\\\ \\hline\n6 & [1.5,4.0] & 10000 &10000 \\\\ \\hline\n8 & [1.5,4.0] & 4000 &30000 \\\\ \\hline\n16 & [1.5,3.0] & 1633 &30000 \\\\ \\hline\n24 & [1.5,3.0] & 700 &60000 \\\\ \\hline\n32 & [1.5,3.0] & 1576 &90000 \\\\ \\hline\n48 & [1.4,2.5] & 426(*) &900000 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ Description of our runs. The step in temperatures was $0.1$\nin all runs. (*) means that for the $L=48$ lattice in the measure of\nthe staggered overlap we only have simulated 100 samples.}\n\\label{table:stat}\n\\end{table}\n\nIn the next paragraphs we will describe the observables that we have\nmeasured in our numerical simulations.\n\nWe have measured the global ($m$) and staggered ($m_s$) magnetizations :\n\\be\nm\\equiv \\frac{1}{L^2} \\sum_i S_i \\,\\,\\, , \\,\\,\\, m_s\\equiv \\frac{2}{L^2}\n\\sum_{i_1} S_{i_1} \\ ,\n\\label{magnetization}\n\\ee\nwhere the sum runs over all the lattice sites ($i$) and over a sub-lattice \n($i_1$) respectively. \n\nIn order to calculate spin glass quantities we have simulated two\nreplicas $\\alpha$ and $\\beta$ in parallel with the same disorder. \nThe overlaps between the replicas, global ($q$) and staggered ($q_s$),\nare:\n\\be\nq \\equiv \\frac{1}{L^2} \\sum_i S_i^{\\alpha} S_i^{\\beta} \\,\\,\\, , \\,\\,\\,\nq_s \\equiv \\frac{2}{L^2} \\sum_{i_1} S_{i_1}^{\\alpha} S_{i_1}^{\\beta} \\ ,\n\\label{overlap}\n\\ee\nwhere, again, the sum runs over all the lattice sites ($i$) and over one of\nthe two sub-lattices ($i_1$) respectively.\n\nThe magnetic global and staggered susceptibilities (without the\n$\\beta$ factor) are defined as:\n\\be\n\\chi \\equiv L^2 \\left[\\overl{\\lan m^2 \\ran}- \\left(\\overl{\\lan |m|\n\\ran}\\right)^2\\right] \n\\,\\,\\, , \\,\\,\\,\n\\chi_s \\equiv \\frac{L^2}{2} \\left[\\overl{\\lan m_s^2 \\ran} - \\left(\\overl{\\lan |m_s|\n\\ran}\\right)^2\\right]\\ .\n\\label{sus_m}\n\\ee\n\nThe spin glass or overlap susceptibilities ($\\chi_q$, $\\chi^s_q$) \nare defined by:\n\\be\n\\chi_q \\equiv L^2 \\left[\\overl{\\lan q^2 \\ran} - \\left(\\overl{\\lan |q|\n\\ran}\\right)^2\\right] \\,\\,\\, , \\,\\,\\,\n\\chi_q^s \\equiv \\frac{L^2}{2}\\left[ \\overl{\\lan q_s^2 \\ran}\n- \\left(\\overl{\\lan |q_s|\n\\ran}\\right)^2\\right] \\ .\n\\label{sus_q}\n\\ee\n\nWe have measured also the Binder cumulants of the \nmagnetization: global $g_m$ and staggered $g_m^s$, \n\\be\ng_m \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan m^4 \\ran}}{\\left(\\overl{\\lan\nm^2 \\ran}\\right)^2} \\right] \n\\,\\,\\, , \\,\\,\\,\ng_m^s \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan m_s^4 \\ran}}{\\left(\\overl{\\lan\nm_s^2 \\ran}\\right)^2} \\right] \\ \n\\label{binder_m}\n\\ee\nand Binder parameters of the overlaps: global $g_q$ and staggered $g^s_q$,\n\\be\ng_q \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan q^4\n\\ran}}{\\left(\\overl{\\lan q^2 \\ran}\\right)^2} \\right] \n\\,\\,\\, , \\,\\,\\,\ng_q^s \\equiv \\frac{1}{2} \\left[ 3-\\frac{\\overl{\\lan q_s^4\n\\ran}}{\\left(\\overl{\\lan q_s^2 \\ran}\\right)^2} \\right] \\ .\n\\label{binder_q}\n\\ee\n\nFinally the specific heat is defined by:\n\\be\nC_V \\equiv \\frac{1}{L^2} \\left(\\overl{\\lan {\\cal H}^2 \\ran} - \\overline{\\lan\n{\\cal H}\n\\ran}^2 \\right) \\ .\n \\label{cesp}\n\\ee\n\nIn order to decide a safe thermalization time, $N_T$, (written in\nTable \\ref{table:stat}) we have used the method proposed by Bhatt and\nYoung~\\cite{bhatt} which consists in running, at the lowest\ntemperature, with an ordered (all spins up) and high temperature\ninitial configurations and monitoring the behavior of the\nsusceptibilities (in our case the non connected \noverlap susceptibility: $L^2 \\overline{\\lan q^2 \\ran}$) with the\nMonte Carlo time. When the two curves reach the same plateau we can\nsay that the system has thermalized. We show in Figures\n\\ref{fig:terma} and \\ref{fig:terma2} this procedure for two of our\nbiggest lattices and at the lower temperatures simulated (i.e. $L=32$\nand $T=1.5$ and $L=48$ and $T=1.4$ respectively).\n\nWe remark that we have used for {\\em all} temperatures of the\nannealing procedure the value that we have computed for the lower one\n(that we have reported in Table \\ref{table:stat}). \n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{terma_32_1.5.ps}\n\\end{center}\n\\caption{The (non connected) \noverlap susceptibility against the Monte Carlo time in a\ndouble logarithmic scale for one of the lower temperature that we have\nsimulated ($T=1.5$) and for $L=32$. The number of samples was 100. \nThe upper curve is from a configuration with all the spins up and the\nlower curve is with the starting configuration chosen random. One can\nsay that the system has thermalized when there is no difference between\nthe two curves and both stay on a plateau. We have chosen as\nthermalization time $t=90000$ (i.e. $\\log t=11.4$) .}\n\\label{fig:terma}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{terma_48_1.4.ps}\n\\end{center}\n\\caption{The (non connected) \noverlap susceptibility against the Monte Carlo time in a\ndouble logarithmic scale for the lowest temperature that we have\nsimulated ($T=1.4$) and for the largest lattice $L=48$. \nThe number of samples was 25. \nThe upper curve is from a configuration with all the spins up and the\nlower curve is with the starting configuration chosen random. \nWe have chosen as\nthermalization time $t=900000$ (i.e. $\\log t=13.7$) .}\n\\label{fig:terma2}\n\\end{figure}\n\n\n\\section{Numerical Results}\n\n\\subsection{Crossover Ferromagnetic-Spin Glass}\n\nIn this sub-section we will show numerical evidences of the first\ncrossover: from a staggered ferromagnetic phase to a ``spin glass'' phase. \n\nIt is natural to study this crossover examining firstly the\nsusceptibility and the Binder cumulant of the staggered magnetization,\nFigures \\ref{fig:sus_ms} and \\ref{fig:binder_ms} respectively. They\nare the observables that at $\\lambda=0$ describe the\nparamagnetic-staggered ferromagnetic phase transition.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_ms.ps}\n\\end{center}\n\\caption{Susceptibility of the staggered magnetization as a \nfunction of the temperature. The lattice sizes are (bottom to top): \n4, 6, 8, 16, 24, 32 and 48. Here and in the rest of the figures we have used\nthe following symbols for the lattices: \n$L=4$ triangle, $L=6$ square, $L=8$ pentagon,\n$L=16$ three-line star, $L=24$ four-line star, $L=32$ five-line star and \n$L=48$ six-line star.}\n\\label{fig:sus_ms}\n\\end{figure}\n\nIn Figure \\ref{fig:sus_ms} it is possible to see how the point where the\nstaggered susceptibility reaches the maximum drifts quickly to\nzero with increasing system size. \nMoreover in Figure \\ref{fig:binder_ms} there is no crossing of\nthe Binder cumulant: this is a stronger evidence that this parameter\n(the staggered magnetization) does not show a phase transition at\nfinite temperature, i.e. in the thermodynamic limit $\\lan |m_s|\n\\ran=0$ for all temperatures different from zero. \n \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nL & $T(\\chi_s^{\\rm max})$ &$\\chi_{s}^{\\rm max}$ \\\\ \\hline\n8 & 2.7(1) & 1.255(4) \\\\ \\hline\n16 & 2.3(1) & 4.21(2) \\\\ \\hline\n24 & 2.0(1) & 9.6(2) \\\\ \\hline\n32 & 1.8(2) & 19.9(4) \\\\ \\hline\n32 & 1.6(2) & 54(2) \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Maximum of $\\chi_s(L,T)$ in temperature\nand the temperature at which $\\chi_s(L,T)$\nreaches the maximum.}\n\\label{table:staggered}\n\\end{table}\n\nMore quantitatively, from the data of Table \\ref{table:staggered} it is\nclear that the temperatures in which $\\chi_s$ reaches the\nmaximum, that we denote as $T(\\chi_s^{\\rm max}) $, goes to zero following a\npower law: \n\\be\nT(\\chi_s^{\\rm max}) \\propto L^{-0.29(4)} ,\n\\ee \nusing only the data\nof the lattices: 8, 16, 24, 32 and 48. This fit has $\\chi^2\/{\\rm DF}=0.5\/3$,\nwhere DF means for the number of degrees of freedom in the fit.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_ms.ps}\n\\end{center}\n\\caption{Binder cumulant of the staggered magnetization as a \nfunction of the temperature. The lattice sizes are (bottom to top in\nthe right part of the plot): 48, 32, 24, 16, 8, 6 and 4.}\n\\label{fig:binder_ms}\n\\end{figure}\n\nThis numerical results support our previous analytic result\nthat the staggered ferromagnetic phase is unstable\nagainst an infinitesimal perturbation of the kind of a spin glass\ninteraction between the two sub-lattices.\n\n\nMoreover, we have predicted the presence of a crossover between the\nstaggered ferromagnetic \nphase and a ``spin-glass'' like phase for lattices of linear sizes of order\n$10$. To see this crossover we can analyze the specific heat. \nAll the lattice sizes show a maximum near the Onsager\ntemperature ($T_c=2.26$). \nWe will see that these maxima are the ``souvenir'' of the staggered\nphase transition. In particular we can examine the scaling of\nthe maxima, that we denote $C_{\\rm max}(L)$, against the lattice size. The\nresult is given in Figure \\ref{fig:c_l}. In this figure we have also\nplotted the finite size scaling prediction for the pure Ising model\n($\\lambda=0$) that is\n\\be\nC_{\\rm max}(L) \\propto \\log L \\ .\n\\ee\n\nIt is clear from Figure \\ref{fig:c_l} that up to $L=8$ the data follow\nin good agreement the prediction of the pure model (i.e. up to $L=8$\nthe low temperature region is staggered ferromagnetic). But between\n$L=8$ and $L=16$ the system crosses over to a different behavior where\nthere is no divergence of the specific heat, i.e. $\\alpha<0$. This\nresult is in very good agreement with our analytical estimate $R_c\n\\simeq 7$.\n\nHence, this plot shows us clearly a first crossover between the\nstaggered ferromagnetic phase and a ``spin glass'' like phase. In the first\npart of the crossover the specific heat diverges logarithmically\n($\\alpha=0$) whereas in the second part of the crossover the specific\nheat does not diverge ($\\alpha<0$).\n\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{c_L.ps}\n\\end{center}\n\\caption{The maximum specific heat as a function of $L$ for\n$L=4,6,8,16,24, 32$ and $48$. \nWe have marked the finite size scaling prediction\nof a logarithmic divergence for the pure model ($\\lambda=0$). \nThe straight line has been obtained from a fit\nto the formula $A \\log L +B$ using $L=4,6$ and $8$.}\n\\label{fig:c_l}\n\\end{figure}\n\nIt is clear that the task that remains is to check that effectively\nwhat we have named as a ``spin glass'' phase has really the properties\nof a spin glass phase.\n\nTo do this we can study the susceptibility and the Binder parameter of\nthe total overlap. In a spin glass phase the magnetization is zero but\nnot the overlap, that becomes the order parameter. If the low\ntemperature phase is spin glass the overlap susceptibility should peak\nnear the transition temperature and the value of the peak should grow with\nsome power of the lattice size (more precisely as\n$L^{\\gamma\/\\nu}$). Moreover, the analysis of the Binder cumulant should\nshow a clear crossing between curves of different lattice sizes.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_q.ps}\n\\end{center}\n\\caption{Susceptibility of the total overlap against the temperature. \nThe lattices sizes are (top to bottom): 48, 32, 24, 16, 8, 6 and 4. \nFrom this figure it is clear that $\\chi_q$ for $L=16, 24, 32$ and $48$\ndoes not show a maximum.}\n\\label{fig:sus_q}\n\\end{figure}\n\nIn Figures \\ref{fig:sus_q} and \\ref{fig:binder_q} we show the data for\nthe susceptibility and Binder parameter of the total overlap.\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_q.ps}\n\\end{center}\n\\caption{Binder parameter of the total overlap against the temperature. \nThe lattices sizes are (bottom to top in the right part of the plot): \n48, 32, 24, 16, 8, 6 and 4. We have marked with a vertical line the\nOnsager temperature.}\n\\label{fig:binder_q}\n\\end{figure}\nWe can see again the crossover staggered-spin glass in Figure\n\\ref{fig:binder_q}. The curves of the lattice sizes $4$, $6$ and $8$\ncross practically at the critical temperature of the pure model\n($\\lambda=0$, vertical line). If we examine the crossing point of\npairs of lattices for growing sizes, we observe that these crossing\npoints drift to lower temperatures. For instance, the crossing point\nof the $16$ and $32$ lattices is near $T=1.9$. This suggests that the\nlimit of this sequence of crossing points may be zero. This would\nimply that the spin glass phase is unstable and is crossing over to a\nparamagnetic phase.\n\nMoreover, the overlap susceptibility does not present a sharp maximum\nas a function of the temperature (see the $L=24, 32$ and $48$ curves). \nWe can extract one conclusions of this fact:\n\n\\begin{itemize}\n\n\\item As long as the spin glass is stable we will expect again that\n$\\chi_q(T)$ should show a sharp maximum as a function of $T$, or at\nleast that we have independent spin glass order in both sub-lattices\n(i.e. some sort of staggered spin glass order). From figure\n\\ref{fig:sus_q} it is clear that $\\chi_q$ does not show a sharp peak\nin the region that we have simulated (see, for instance, the $L=32$\nand $L=48$ data). Moreover the crossing point of the Binder cumulant of the\ntotal overlap of two different lattices is drifting to lower values of\nthe temperature, and so we pass to discus the second possible option:\nspin glass order on both sub-lattices. In this case the staggered\noverlap susceptibility should have a sharp peak at an intermediate\ntemperature which characterizes a phase transition between a\nparamagnetic phase and a spin glass one.\n\n\\end{itemize}\n\nTo study this issue in more detail, we will examine in the next\nsubsection the overlap defined only in one of the two sub-lattices.\n\n\n\\subsection{Crossover Spin Glass-Paramagnetic}\n\nIn Figures \\ref{fig:sus_qs} and \\ref{fig:binder_qs}\nwe show the susceptibility and the Binder cumulant of the staggered\noverlap (i.e. the overlap computed only in one of the two sub-lattices).\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{sus_qs.ps}\n\\end{center}\n\\caption{Susceptibility of the staggered overlap against the temperature. \nThe lattices sizes are (bottom to top): 48, 32, 24, 16, 8 and 4.}\n\\label{fig:sus_qs}\n\\end{figure}\n\nIt is clear that these two figures are similar to Figures \\ref{fig:sus_ms} and \n\\ref{fig:binder_ms}. For example, there is no crossing in the Binder\ncumulant. We remark that we have performed small statistic on the\n$L=48$ lattice and so we obtain large errors. Taking into account only the\nlattice sizes $L=4,6,8,16,24$ and 32 the effect is clear: the\nthermodynamical Binder cumulant goes to zero for all the temperatures\nsimulated.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nL & $T(\\chi_{qs}^{\\rm max})$ & $\\chi_{qs}^{\\rm max}$ \\\\ \\hline\n8 & 2.4(1) & 2.27(3) \\\\ \\hline\n16 & 2.1(1) & 7.2(2) \\\\ \\hline\n24 & 1.9(1) & 15.1(3) \\\\ \\hline\n32 & 1.6(2) & 28(2) \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Maximum in temperature of the staggered overlap \nsusceptibility $\\chi_q^s(L,T)$, \nand the temperature at which $\\chi_q^s(L,T)$ reaches the maximum. For\n$L=48$ the error bars do not permit a safe estimate of the maximum.}\n\\label{table:q-staggered}\n\\end{table}\nThe ``apparent'' critical temperature (defined as the value where\n$\\chi_q^s$ reaches the maximum) goes to zero following the law\n\\be\nT(\\chi_{qs}^{\\rm max}) \\propto L^{-0.23(5)}\n\\label{scaling}\n\\ee\nwhere we have used L = 8, 16, 24 and 32 in the fit that has\n$\\chi^2\/{\\rm DF}=0.8\/2$. The data used in this fit has been reported\nin Table \\ref{table:q-staggered}.\n\nWithin the statistical error the exponent is just the same as in the\nstaggered magnetization. We can compare this figure with that computed\nfor the 2D spin glass\\cite{bhatt}: $1\/\\nu=0.38(6)$. If we are seeing\nthe pure two dimensional spin glass transition which occurs at $T=0$,\nthen we will expect a behavior of the apparent critical temperature\nlike $T_{\\rm app} \\propto L^{-0.38(6)}$, which is, within the\nstatistical error, the law that we have found for the present model\n(Eq. (\\ref{scaling}))\\footnote{ The shift of the apparent critical\ntemperature follows a law: $T_c(L)-T_c \\propto L^{-1\/\\nu}$.}: the\ndifference between the two exponents is $0.15(8)$ (i.e. almost two\nstandard deviations). Obviously our simulations were done in a range of\ntemperatures far away of the critical point ($T=0$).\n\nBoth Figures \\ref{fig:sus_qs} and \\ref{fig:binder_qs} suggest that the\nphase transition is at $T=0$. In other words, there is not a spin\nglass phase in the sub-lattices 1 and 2. The whole system seems to be\ncrossing over to a paramagnetic phase.\n\n\\begin{figure}\n\\begin{center}\n\\addvspace{1 cm}\n\\leavevmode\n\\epsfysize=250pt\n\\epsffile{binder_qs.ps}\n\\end{center}\n\\caption{Binder parameter of the staggered overlap against the temperature. \nThe lattice sizes are (bottom to top): 48, 24, 32, 16, 8, and 4.}\n\\label{fig:binder_qs}\n\\end{figure}\n\n\n\n\\section{\\label{S_CONCLU}Conclusions}\n\nWe have studied the two dimensional Ising spin glass model with next\nnearest neighbor interactions both\nanalytical and numerically.\n\nWe have analytically obtained that the system should present two\ndifferent crossovers as the volume grows: the first one from a\nstaggered ferromagnetic phase to a spin glass phase for intermediate\nsizes, and a second crossover between this spin glass phase to a\nparamagnetic one that will dominate the physics in the thermodynamic\nlimit. Moreover we have obtained the dependence of the first crossover\nlength (staggered-spin glass) on the parameters of the system.\n\nThen we have checked this scenario by performing extensive numerical\nsimulations. We have clearly established the first crossover\n(staggered-spin glass) and have found strong evidences of the second (spin\nglass-paramagnetic). In particular, the exponent that governs the\nshift of the maxima of the spin glass susceptibility is compatible\nwith the known value for the pure two dimensional spin glass.\n\n\\vspace{2cm}\n{\\large \\bf {\\label{S_ACKNOWLEDGES}Acknowledgments}}\n\nWe have run mainly using ALPHA Workstations but the large lattices\nwere run in the RTNN parallel machine\\footnote{A parallel machine\ncomposed by 32 Pentium-Pro at Zaragoza University.}. \nWe acknowledge the RTNN project for\npermitting to us the use of the RTNN computer.\n\nJ.~J.~Ruiz-Lorenzo is supported by an EC HMC (ERBFMBICT950429)\ngrant and thanks to L.A. Fern\\'andez for interesting discussions. \nD. A. Stariolo is partially supported by Conselho Nacional de\nDesenvolvimento Cientifico e Tecnol\\'ogico (CNPq), Brazil.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe relation between confinement and chiral symmetry breaking is one of the long-standing puzzles in theoretical physics. \nRecently, strong interest on this issue revived in extreme environments especially at high temperatures and baryon densities, \nstimulated by the heavy-ion programs at GSI, CERN SPS, RHIC and LHC, see e.g., \\cite{KS04,YHM05} for a review.\nQuantum chromodynamics (QCD) for strong interactions is a fundamental theory for solving this problem. \n\n \nIn pure Yang-Mills theory, i.e., in the limit of infinitely heavy quark mass $m_q \\rightarrow \\infty$ of QCD, the Polyakov loop average $\\langle L \\rangle$, i.e., the vacuum expectation value of the Polyakov loop operator $L$, can be used as a criterion for quark confinement \\cite{Polyakov78}. \nThe Polyakov loop operator $L$ is a gauge invariant operator charged under the center group $Z(N_c)$ of the color gauge group $SU(N_c)$. The Polyakov loop average $\\langle L \\rangle$ vanishes $\\langle L \\rangle=0$ and quarks are confined at low temperatures $TT_d$ where the global center symmetry $Z(N_c)$ is spontaneously broken. \nThus, we can define $T_d$ as a critical temperature for confinement\/deconfinement phase transition. \n\nWhen dynamical quarks in the fundamental representation of the gauge group are added to the Yang-Mills theory, the center symmetry is no longer exact. \nOn the other hand, QCD with massless quarks $m_q \\rightarrow 0$ exhibits chiral symmetry $SU(N_f)_L \\times SU(N_f)_R$. \nThe chiral condensate $\\langle \\bar\\psi \\psi \\rangle$, i.e., the vacuum expectation value of a gauge-invariant composite operator $\\bar\\psi \\psi$, is used as an order parameter for chiral symmetry breaking.\nThe chiral condensate $\\langle \\bar\\psi \\psi \\rangle$ is nonzero $\\langle \\bar\\psi \\psi \\rangle \\not= 0$ at low temperatures $TT_\\chi$ where the chiral symmetry is restored. \nThus, we can define $T_\\chi$ as a critical temperature for chiral-symmetry breaking\/restoration phase transition. \n\n\nFor realistic quark mass (with finite and nonzero $m_q$: $0 < m_q <\\infty$), there are no exact symmetries directly related to the phase transitions, since both the center and chiral symmetries are explicitly broken, and $\\left< L \\right>$ and $\\left< \\bar\\psi \\psi \\right>$ are approximate order parameters. \nIn this case, there is no critical temperature $T_c$ in strict sense, and the transition can be a crossover transition for which the pseudo critical temperature $T_c^*$ is defined such that the susceptibility takes the maximal value at $T=T_c^*$.\nIf quarks are in the fundamental representation, deconfinement (a rise in the Polyakov loop average) happens at the temperature where the chiral symmetry is restored (chiral condensate decreases rapidly). \nThe chiral and deconfinement transitions seems to coincide, \n$T_d^*=T_\\chi^* \\simeq T_c$ \\cite{Karsch02,KL94}, \nalthough the property of the phase transition, e.g., the critical temperature and the order of the transition depend on numbers of color $N_c$ and flavor $N_f$. \nWhereas, for quarks in the adjoint representation, deconfinement and chiral-symmetry restoration do not happen at the same temperature, rather $T_\\chi^* \\gg T_d^*$ \\cite{KL98}. \nAlthough there exist theoretical considerations on the interplay between chiral symmetry breaking and confinement at zero baryon density \\cite{Casher79}, the underlying reasons for the coincidence are still unknown and uncertain at nonzero baryon density \\cite{KST99,MP07,HKS10}. \n\nThe hadronic properties, especially, chiral dynamics at low energy have been successfully described by chiral effective models such as the linear sigma model \\cite{Lee72}, the Nambu-Jona-Lasinio (NJL) model \\cite{NJL61,Klevansky92,HK94}, the chiral random matrix model \\cite{SV93}, chiral perturbation theory \\cite{GL84} and so on. \nHowever, those models based on chiral symmetry lack any dynamics coming from confinement dictated by the Polyakov loop, although there are some efforts to clarify the interplay between chiral dynamics and the Polyakov loop \n\\cite{MO95,MST03}.\n\nRecent chiral effective models with the Polyakov loop degrees of freedom augmented called the Polyakov loop--extended NJL (PNJL) model or quark-meson (PQM) model \\cite{Fukushima04,MRAS06,RTW05,SFR06,KKMY07,HRCW08,BBRV07,SPW07} are successful from a phenomenological point of view to incorporate a coupling between the chiral condensate and the Polyakov loop. \nHowever, these PNJL\/PQM models are still far from treating the chiral condensate and the Polyakov loop on an equal footing, except for the work \\cite{SPW07} where the backcoupling of the matter sector to the glue sector was discussed by changing the phase transition parameter.\nIn fact, the gluonic part in these models has several fitting parameters which are determined only from lattice QCD data.\n\nHere we must mention a preceding work for a first-principle derivation of confinement\/deconfinement and chiral-symmetry-breaking\/restoration crossover phase transition based on the flow equation \\cite{Wetterich93} of the functional renormalization group \\cite{BTW00,Gies06} given by Braun, Haas, Marhauser and Pawlowski \\cite{BHMP09} for the full dynamical QCD with 2 massless flavors (at zero and imaginary chemical potential). In this work, the Yang-Mills theory is fully coupled to the matter sector by taking into account the Polyakov-loop effective potential \\cite{Weiss81} obtained in a nonperturbative way put forward by Braun, Gies and Pawlowski \\cite{BGP07} and Marhauser and Pawlowski \\cite{MP08}.\n\n\nThe main purpose of this paper is to provide a theoretical framework (a reformulation of QCD) which enables one to describe in a unified way the chiral dynamics and confinement signaled by the Polyakov loop.\nWe give an important step towards a first-principle derivation of confinement\/deconfinement and chiral-symmetry breaking\/restoration crossover transition. \nIn fact, we demonstrate that a low-energy effective theory of QCD obtained in simple but non-trivial approximations within this framework enables one to treat both transitions simultaneously on equal footing. \n\n\nThe basic ingredients in this paper are a reformulation of QCD based on new variables \\cite{Cho80,DG79,FN99,Shabanov99,KMS06,KMS05,Kondo06,KSM08,KKMSSI05,IKKMSS06,SKKMSI07,SKKMSI07b,KSSMKI08,SKS09,Cho00,Kondo08,KS08,Kondo08b} and the flow equation of the Wetterich type in the Wilsonian renormalization group \\cite{Wetterich93,BTW00,Gies06}. \nThe reformulation was used to confirm quark confinement in pure Yang-Mills theory at zero temperature and zero density based on a dual superconductor picture \\cite{dualsuper}. \nIn this paper, it is extended to QCD at finite temperature and density. \nIn principle, our framework can be applied to any color gauge group and arbitrary number of flavors. For technical reasons, however, we study two color QCD with two flavors in this paper. The three color and\/or three flavor case will be studied in a subsequent paper. \nIn future publications, this framework will be applied to investigate the QCD phase diagram at finite density. \nWe hope that this paper will give an insight into this issue complementary to other works, e.g., \\cite{BHMP09}.\n\nIn sec. II, we give a reformulation of QCD written in terms of new variables and explain why the reformulated QCD is efficient to study the interplay between confinement and chiral-symmetry breaking. \n\nIn sec. III, we choose a specific gauge (modified Polyakov gauge) to simplify the representation of the Polyakov loop.\nWe can choose any gauge to calculate the Polyakov loop average and the chiral condensate, since both are gauge-invariant quantities and should not depend on the gauge chosen. \n\nIn sec. IV, we give a definition of the Polyakov loop operator and examine how the Polyakov loop average is related to the average of the time-component of the gauge field. \n\nIn sec.~V and sec.~VI, we study the confinement\/deconfinement phase transition in pure $SU(2)$ Yang-Mills theory at finite temperature.\nWe exploit the Wilsonian renormalization group in our framework to obtain the effective potential $V_{\\rm eff}$ of the Polyakov loop $L$, whose minimum gives the Polyakov loop average $\\langle L \\rangle$. \nIt is known that the Weiss potential $V_{\\rm W}$ \\cite{Weiss81} calculated in the perturbation theory to one loop exhibits spontaneous center-symmetry breaking, i.e., deconfinement, irrespective of the temperature $T$. \nThis result can be used at high temperature where the perturbation theory will be trustworthy due to asymptotic freedom, while nonperturbative approach is necessary to treat the low-temperature case. \nThe Weiss potential can be improved according to the Wilsonian renormalization group to obtain a nonperturbative effective potential which is valid even at low temperature.\n\n\nIn sec. V, we write down the flow equation of the Wetterich type for the effective potential of the Polyakov loop in our framework. \nIn fact, the effective potential obtained by solving the flow equation in a numerical way shows the existence of confinement phase below a certain temperature $T_d$. \nThis solution was shown for the first time by Marhauser and Pawlowski \\cite{MP08} and by Braun, Gies and Pawlowski \\cite{BGP07}, see \\cite{MO97} for the previous works. \nIn this sense, this section is nothing but the translation of their results \\cite{MP08,BGP07} into our framework.\n\n\n\n\nIn sec. VI, we give a qualitative understanding for the confinement\/deconfinement transition given in sec. V based on the Landau-Ginzburg argument. We answer a question why the center-symmetry restoration occurs as the temperature is decreased, by observing the flow equation for the coefficient of the effective potential. \n\nIn sec. VII, we describe the low-energy effective interaction among quarks by a nonlocal version of the (gauged) NJL model in which the effect of confinement is explicitly incorporated through the Polyakov loop dependent nonlocal interaction. \nThe resulting effective theory is regarded as a modified and improved version of nonlocal Polyakov-loop extended Nambu-Jona-Lasinio (PNJL) models proposed recently by Hell, R\\\"ossner, Cristoforetti and Weise \\cite{HRCW08}, Sasaki, Friman and Redlich \\cite{SFR06}, and Blaschke, Buballa, Radzhabov and Volkov \\cite{BBRV07}, extending the original (local) PNJL model by Fukushima \\cite{Fukushima04}. \nThe nonlocal (gauged) NJL model can be converted to the nonlocal (gauged) Yukawa model to be bosonized to study the chiral dynamics. \n\n\nIn sec. VIII, we show that the nonlocal NJL interaction among quarks becomes temperature dependent through the coupling to the Polyakov loop. \nThis is a first nontrivial indication for the entanglement between the chiral symmetry breaking and confinement. \nThis feature was overlooked in conventional PNJL models. \n\nIn sec. IX, we consider how to understand the entanglement between confinement and chiral symmetry breaking in our framework. \nThis is just a short sketch for our strategy following the line given in the preceding sections. \n\nThe final section is used to summarize the results and give some perspective in the future works. \nSome technical materials are collected in Appendices. \n\n\n\n\\section{Reformulation of QCD}\\label{sec:reformulation}\n\nTo fix the notation, we write the action of QCD in terms of the gluon field $\\mathscr{A}_\\mu$ and the quark field $\\psi$:\n\\begin{align}\n\\label{eq:QCDaction}\n S_{\\rm QCD} =& S_{\\rm q} + S_{\\rm YM} ,\n\\nonumber \\\\\n S_{\\rm q} :=& \\int d^Dx \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{A}] -\\hat{m}_q + \\mu_q \\gamma^0) \\psi ,\n\\nonumber\\\\\n S_{\\rm YM} :=& \\int d^Dx \\frac{-1}{2} {\\rm tr}(\\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{A}]) \n ,\n\\end{align}\nwhere $\\psi$ is the quark field, $\\mathscr{A}_\\mu=\\mathscr{A}_\\mu^A T_A$ is the gluon field with $su(N_c)$ generators $T_A$ for the gauge group $G=SU(N_c)$ ($A=1, \\cdots, {\\rm dim}SU(N_c)=N_c^2-1$), $\\hat{m}_q$ is the quark mass matrix, $\\mu_q$ is the quark chemical potential, $\\gamma^\\mu$ are the Dirac gamma matrices ($\\mu=0, \\cdots, D-1$), $\\mathcal{D}_\\mu[\\mathscr{A}]:=\\partial_\\mu - ig \\mathscr{A}_\\mu$ is the covariant derivative in the fundamental representation, $\\mathscr{F}_{\\mu\\nu}[\\mathscr{A}]:=\\partial_\\mu \\mathscr{A}_\\nu - \\partial_\\nu \\mathscr{A}_\\mu -i g [\\mathscr{A}_\\mu , \\mathscr{A}_\\nu]$ is the field strength and $g$ is the QCD coupling constant. \nIn what follows, we suppress the spinor, color and flavor indices. \n\nThe main purpose of this paper is to give a theoretical framework for extracting a low-energy effective theory which enables one to discuss the confinement\/deconfinement and chiral-symmetry breaking\/restoration (crossover) transition simultaneously on an equal footing. \nWe reformulate QCD in terms of new variables which are efficient for this purpose. \nWe start with decomposing the original $SU(N)$ Yang-Mills field $\\mathscr{A}_\\mu(x) =\\mathscr{A}_\\mu^A(x) T_A$ into two pieces $\\mathscr{V}_\\mu=\\mathscr{V}_\\mu^A(x) T_A$ and $\\mathscr{X}_\\mu=\\mathscr{X}_\\mu^A(x) T_A$:\n\\begin{equation}\n \\mathscr{A}_\\mu(x) = \\mathscr{V}_\\mu(x) + \\mathscr{X}_\\mu(x) ,\n \\label{decomp}\n\\end{equation} \nto rewrite the original QCD action into a new form:\n\\begin{align}\n S_{\\rm q} =& \\int d^Dx \\Big\\{ \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] -\\hat{m}_q + \\mu_q \\gamma^0) \\psi \n + g\\mathscr{J}^{\\mu} \\cdot \\mathscr{X}_\\mu \\Big\\} ,\n\\nonumber\\\\\n S_{\\rm YM} =& \\int d^Dx \\Big\\{ \\frac{-1}{4} (\\mathscr{F}_{\\mu\\nu}^A[\\mathscr{V}])^2\n- \\frac{1}{2} \\mathscr{X}^{\\mu A} Q_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \\Big\\}\n + S_{\\rm FP},\n\\label{QCDaction2}\n\\end{align}\nwhere \n$\\mathscr{J}^{\\mu A} := g\\bar{\\psi} \\gamma^\\mu T_A \\psi $ is the color current, \n$D_\\mu[\\mathscr{V}] := \\partial_\\mu - ig[\\mathscr{V}_\\mu , \\cdot ]$ is the covariant derivative in the adjoint representation and \n\\begin{align} \nQ_{\\mu\\nu}^{AB}[\\mathscr{V}] :=& G^{AB} [\\mathscr{V}] g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] , \n\\nonumber\\\\\nG^{AB}[\\mathscr{V}] :=& - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} \n\\nonumber\\\\\n=& -(\\partial_\\rho \\delta^{AC}+gf^{AEC}\\mathscr{V}_\\rho^E) (\\partial^\\rho \\delta^{CB}+gf^{CFB}\\mathscr{V}^{\\rho F}) \n\\nonumber\\\\\n=& - \\partial_\\rho^2 \\delta^{AB} + g^2 f^{AEC}f^{BFC} \\mathscr{V}_\\rho^E \\mathscr{V}^{\\rho F} \n\\nonumber\\\\\n&+ 2g f^{ABE} \\mathscr{V}_\\rho^E \\partial^\\rho + gf^{ABE} \\partial^\\rho \\mathscr{V}_\\rho^E \n .\n\\label{Q}\n\\end{align}\nIn what follows we use the notation $\\mathscr{A} \\cdot \\mathscr{B}$ for two Lie-algebra valued functions $\\mathscr{A}=\\mathscr{A}^A T_A$ and $\\mathscr{B}=\\mathscr{B}^A T_A$ in the sense that \n$\\mathscr{A} \\cdot \\mathscr{B} := \\mathscr{A}^A \\mathscr{B}^A = 2 {\\rm tr}(\\mathscr{A} \\mathscr{B})$\nand especially $\\mathscr{A}^2:=\\mathscr{A} \\cdot \\mathscr{A}= \\mathscr{A}^A \\mathscr{A}^A$.\n\n\nHistorically, the decomposition of Yang-Mills theory into new variables has been proposed by Cho \\cite{Cho80} and Duan and Ge \\cite{DG79} independently, and readdressed later by Faddeev and Niemi \\cite{FN99}. \nThe decomposition was further developed by Shabanov \\cite{Shabanov99}. \n\n\n\n\n \nThe decomposition (\\ref{decomp}) is performed such that \n$\\mathscr{V}_\\mu$ transforms under the gauge transformation just like the original gauge field $\\mathscr{A}_\\mu$: \n\\begin{equation}\n \\mathscr{V}_\\mu(x) \\rightarrow \\mathscr{V}_\\mu^\\prime(x) = \\Omega(x) (\\mathscr{V}_\\mu(x) + ig^{-1} \\partial_\\mu) \\Omega^{-1}(x) \n ,\n \\label{V-ctransf}\n\\end{equation}\nwhile $\\mathscr{X}_\\mu$ transforms like an adjoint matter field:\\begin{equation}\n \\mathscr{X}_\\mu(x) \\rightarrow \\mathscr{X}_\\mu^\\prime(x) = \\Omega(x) \\mathscr{X}_\\mu(x) \\Omega^{-1}(x) \n \\label{X-ctransf}\n . \n\\end{equation}\n\n\nIn the decomposition (\\ref{decomp}), we introduce a new field \n\\begin{equation}\n \\bm{n}(x)=n^A(x)T_A , \n\\end{equation}\nwith a unit length in the sense that $n^A(x)n^A(x)=1$, which we call the \\textit{color field}. \nIn the decomposition (\\ref{decomp}), the color field $\\bm{n}(x)$ plays a crucial role as follows.\nThe color field is defined by the following property.\nIt must be a functional or composite operator of the original Yang-Mills field $\\mathscr{A}_\\mu(x)$ such that it transforms according to the adjoint representation under the gauge transformation: \n\\begin{equation}\n \\bm{n}(x) \\rightarrow \\bm{n}^\\prime(x) = \\Omega(x) \\bm{n}(x) \\Omega^{-1}(x) \n .\n \\label{n-ctransf}\n\\end{equation}\n\n\n\nThe color field plays the key role in the reformulation. Once a color field is given, the decomposition is uniquely determined by solving a set of defining equations and hence $\\mathscr{V}_\\mu(x)$ and $\\mathscr{X}_\\mu(x)$ are written in terms of $\\mathscr{A}_\\mu(x)$ and $\\bm{n}(x)$.\nFor $G=SU(2)$, the defining equations are given by\n\n(I) covariant constantness of color field $\\bm{n}(x)$ in $\\mathscr{V}_\\mu(x)$:\n\\begin{equation}\n 0 = D_\\mu[\\mathscr{V}]\\bm{n}(x) ,\n\\end{equation}\n\n\\text{(II) orthogonality of $\\mathscr{X}_\\mu(x)$ to $\\bm{n}(x)$:}\n\\begin{equation}\n 0 = \\mathscr{X}_\\mu(x) \\cdot \\bm{n}(x) .\n\\end{equation}\nThen the decomposition for $G=SU(2)$ is uniquely determined as\n\\begin{align}\n & \\mathscr{V}_\\mu(x)= c_\\mu(x)\\bm{n}(x) + ig^{-1} [ \\bm{n}(x) , \\partial_\\mu \\bm{n}(x) ] ,\n\\nonumber\\\\\n & \\quad\\quad\\quad\\quad c_\\mu(x) := \\mathscr{A}_\\mu(x) \\cdot \\bm{n}(x) ,\n\\nonumber\\\\\n & \\mathscr{X}_\\mu(x) = i g^{-1} [ D_\\mu[\\mathscr{A}] \\bm{n}(x) , \\bm{n}(x) ] ,\n\\end{align}\n\nTo arrive at the result (\\ref{QCDaction2}), we have used the following facts. See Appendix~\\ref{app:reformulation} for the details.\n\n(i) \nThe $O(\\mathscr{X})$ terms vanish, \n$\\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)=0$,\nfrom the property of the new variables as shown using the \\textit{defining equations of the decomposition} (\\ref{decomp}). \nThis is somewhat similar to the usual background field method in which $O(\\mathscr{X})$ terms in the quantum fluctuation field $\\mathscr{X}_{\\mu}$ are eliminated by requiring that the background field $\\mathscr{V}_\\mu$ satisfies the classical Yang-Mills equation of motion, i.e., $D_\\mu[\\mathscr{V}]\\mathscr{F}^{\\mu\\nu}[\\mathscr{V}]=0$. \nIn our case, however, $\\mathscr{V}_\\mu$ do not necessarily satisfy the classical equation of motion.\n\n \n(ii) To obtain $Q_{\\mu\\nu}^{AB}[\\mathscr{V}]$ in (\\ref{Q}), an $O(\\mathscr{X}^2)$ term is eliminated, $-\\frac{1}{2} \\mathscr{X}^{\\mu A} D_\\mu^{AC}[\\mathscr{V}]D_\\nu^{CB}[\\mathscr{V}] \\mathscr{X}^{\\nu B}=0$,\nby imposing the condition:\n\\begin{equation}\n D_\\mu[\\mathscr{V}] \\mathscr{X}^\\mu=0 .\n \\label{nMAG}\n\\end{equation}\nFor the reformulated QCD to be equivalent to the original QCD, we must impose such a constraint to avoid mismatch in the independent degrees of freedom, which is called the \\textit{reduction condition} \\cite{KMS06,KSM08}. \n\n(iii) The $O(\\mathscr{X}^3)$ term vanishes, \n$\n\\frac{1}{2} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ] = 0 \n$, since $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$ is orthogonal to $[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]$.\n\n\nFor $G=SU(2)$, $\\mathscr{V}$ can be chosen in such a way that the field strength $\\mathscr{F}[\\mathscr{V}]$ of the field $\\mathscr{V}$ is proportional to $\\bm{n}$:\n\\begin{align}\n \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}](x) \n:=& \\partial_\\mu \\mathscr{V}_\\nu(x) - \\partial_\\nu \\mathscr{V}_\\mu(x) - ig [\\mathscr{V}_\\mu(x) , \\mathscr{V}_\\nu(x)]\n\\nonumber\\\\\n=& \\bm{n}(x) G_{\\mu\\nu}(x) ,\n\\label{F=nG}\n\\end{align}\nwhere $G_{\\mu\\nu}$ is a gauge--invariant antisymmetric tensor of rank 2, i.e., \n$\\mathscr{F}_{\\mu\\nu}^\\prime[\\mathscr{V}](x)=\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}^\\prime](x)=\\Omega(x) \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}](x) \\Omega^{-1}(x) = \\bm{n}^\\prime(x) G_{\\mu\\nu}(x) \n$. The explicit form of $G_{\\mu\\nu}$ is written in term of $\\mathscr{A}_\\mu(x)$ and $\\bm{n}(x)$ as\n\\begin{align}\n G_{\\mu\\nu}(x) &= \\partial_\\mu [ \\bm{n}(x) \\cdot \\mathscr{A}_\\nu(x)] - \\partial_\\nu [ \\bm{n}(x) \\cdot \\mathscr{A}_\\mu(x)] \n\\nonumber\\\\&\n+ i g^{-1} \\bm{n}(x) \\cdot [\\partial_\\mu \\bm{n}(x) , \\partial_\\nu \\bm{n}(x) ] .\n\\end{align}\n\n\nIn the present approach, we wish to regard the field decomposition as a change of variable from the original gluon field to new variables describing a reformulated Yang-Mills theory in the quantum level \\cite{KMS06,Kondo06,KSM08} (see \\cite{KKMSS05,KKMSSI05,IKKMSS06,SKKMSI07,SKKMSI07b,KSSMKI08,SKKISF09,SKS09} for the corresponding lattice gauge formulation). To achieve this goal, first of all, $\\bm{n}(x)$ must be written as a functional of $\\mathscr{A}_\\mu(x)$ and thereby all new fields are written in terms of the original gluon field $\\mathscr{A}_\\mu(x)$. \nSuch a required relationship between $\\mathscr{A}_\\mu(x)$ and $\\bm{n}(x)$ is given by the reduction condition which is given as a variational problem of obtaining an absolute minimum of a given functional. \nThe condition for local minima is given in the form of a differential equation. For $G=SU(2)$, \n\\begin{equation}\n [ \\bm{n}(x) , D_\\mu[\\mathscr{A}]D_\\mu[\\mathscr{A}]\\bm{n}(x) ] =0 .\n \\label{dRed}\n\\end{equation}\nThis is another form of (\\ref{nMAG}). \nSee \\cite{KMS06} in $SU(2)$ case and \\cite{KSM08} in $SU(N)$ case for the full details. \n\nRemarkable properties of new variables are as follows. \nFirst, we remind you of the role played by the field $\\mathscr{V}$.\n\n$\\bullet$ The variable $\\mathscr{V}_\\mu$ alone is responsible for the Wilson loop operator $W_C[\\mathscr{A}]$ and the Polyakov loop operator $L[\\mathscr{A}]$ in the sense that \n\\begin{equation}\nW_C[\\mathscr{A}]=W_C[\\mathscr{V}] , \\quad\nL[\\mathscr{A}]=L[\\mathscr{V}] .\n\\end{equation}\nwhere the Wilson loop operator is defined by\n\\begin{equation}\n W_C[A] := \\mathcal{N}^{-1} {\\rm tr} \\left[ \\mathscr{P} \\exp \\left\\{ ig \\oint_C dx^\\mu \\mathscr{A}_\\mu(x) \\right\\} \\right] ,\n\\end{equation}\nwhere $\\mathscr{P}$ denotes the path ordering and the normalization factor $\\mathcal{N}$ is the dimension of the representation $R$, in which the Wilson loop is considered, i.e., \n$\n \\mathscr{N}:=d_R = {\\rm dim}({\\bf 1}_R) = {\\rm tr}({\\bf 1}_R) \n$.\nThe Polyakov loop operator will be defined later. \nIn other words, $\\mathscr{X}_\\mu$ do not contribute to the Wilson loop and the Polyakov loop in the operator level. \nThis is because the defining equation for the decomposition is a (necessary and) sufficient condition for a \\textit{gauge-invariant Abelian dominance} (or $\\mathscr{V}$ dominance) in the operator level. \nThis proposition was first proved in \\cite{Cho00} for $SU(2)$ and for $SU(N)$ in the continuum \\cite{Kondo08} and for $SU(N)$ on a lattice \\cite{KS08}. On the lattice, the equality does not exactly hold due to non-zero lattice spacing $\\epsilon$, but the deviation vanishes in the continuum limit of the lattice spacing $\\epsilon$ going to zero, $\\epsilon \\rightarrow 0$.\nIt should be remarked that both the Wilson loop operator and the Polyakov loop operator are gauge-invariant quantities and that their average do not depend on the gauge fixing condition adopted in the calculation. \n \n $\\bullet$ We can introduce a gauge-invariant magnetic monopole current $k$ in Yang-Mills theory (without matter fields) where $k$ is the ($D-3$)-form. For $D=4$ and $G=SU(2)$, \n\\begin{align}\n k_\\mu(x) :=& \\partial_\\nu {}^*G_{\\mu\\nu}(x) ,\n\\\\\n G_{\\mu\\nu} :=& \\bm{n} \\cdot \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]\n = \\partial_\\mu c_\\nu - \\partial_\\nu c_\\mu +i g^{-1} \\bm{n} \\cdot [\\partial_\\mu \\bm{n} , \\partial_\\nu \\bm{n} ] ,\n \\nonumber\n\\end{align}\nwhere $f_{\\mu\\nu}$ is gauge-invariant field strength. \nThis is because the field strength $\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]:= \\partial_\\mu \\mathscr{V}_\\nu - \\partial_\\nu \\mathscr{V}_\\mu -ig [\\mathscr{V}_\\mu , \\mathscr{V}_\\nu ]$ is proportional to $\\bm{n}$:\n\\begin{equation}\n \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]\n = \\bm{n} \\{ \\partial_\\mu c_\\nu - \\partial_\\nu c_\\mu +i g^{-1} \\bm{n} \\cdot [\\partial_\\mu \\bm{n} , \\partial_\\nu \\bm{n} ] \\} .\n\\end{equation}\n $\\bullet$ The gauge-invariant ``Abelian\" dominance (or $\\mathscr{V}$ dominance) and magnetic monopole dominance (constructed from $\\mathscr{V}$) in quark confinement have been confirmed at $T=0$ (and $\\mu_q=0$) by comparing string tensions calculated from the Wilson loop average by numerical simulations by \\cite{IKKMSS06} for SU(2) and by \\cite{SKKMSI07b} for SU(3). \nHere it should be remarked that the ``Abelian\" dominance is the dominance for the vacuum expectation value (or average):\n\\begin{equation}\n\\langle W_C[\\mathscr{A}] \\rangle \\simeq \\langle W_C[\\mathscr{V}] \\rangle \n, \\quad \n\\langle L[\\mathscr{A}] \\rangle \\simeq \\langle L[\\mathscr{V}] \\rangle.\n\\end{equation}\n\n\nNext, we pay attention to the role played by the remaining field $\\mathscr{X}$. \n\\\\\n$\\bullet$ In the absence of dynamical quarks (corresponding to the limit $m_q=\\infty$ of QCD, i.e., gluodynamics), \n$\\mathscr{X}_\\mu^A$ decouple in the low-energy regime as the correlator $\\left< \\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(y) \\right>$ behaves like a massive correlator with mass $M_X$. \nIn fact, numerical simulations demonstrate for $G=SU(2)$ and $D=4$ \\cite{SKKMSI07} \n\\begin{equation}\nM_X=1.2 \\sim 1.3 \\ {\\rm GeV}.\n\\label{M_X}\n\\end{equation}\nWe can understand this result as follows.\nThe field $\\mathscr{X}_\\mu$ can acquire the (gauge-invariant) mass dynamically. This comes from a fact that, in sharp contrast to the field $\\mathscr{A}_\\mu$, \n a ``gauge-invariant mass term\" for $\\mathscr{X}_\\mu$ can be introduced\n\\begin{equation}\n\\frac12 M_X^2 \\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(x) ,\n\\end{equation}\nsince $\\mathscr{X}_\\mu^A(x) \\mathscr{X}_\\mu^A(x)$ is a gauge-invariant operator.\nMoreover, this mass term can originate from a vacuum condensation of ``mass dimension-2\", $\\left< \\mathscr{X}_\\nu^B(x) \\mathscr{X}_\\nu^B(x) \\right> \\not= 0$ as proposed in \\cite{Kondo01}.\nIn fact, this condensation can be generated through self-interactions $O(\\mathscr{X}^4)$ among $\\mathscr{X}_\\mu$ gluons, $M_X^2 \\simeq \\left< \\mathscr{X}_\\nu^B(x) \\mathscr{X}_\\nu^B(x) \\right>$, as examined in \\cite{Kondo06,KKMSS05}.\nIt is instructive to remark that the value (\\ref{M_X}) agrees with the earlier result of the off-diagonal ``gluon mass'' $M_A$ in the Maximally Abelian (MA) gauge \\cite{AS99} for $SU(2)$ case, $M_A \\simeq 1.2$ GeV. See \\cite{SAIIMT02} for $SU(3)$ case, $M_A \\simeq 1.1$ GeV.\nIn MA gauge, it was shown that even at finite temperatures Abelian dominance (diagonal part dominance and off-diagonal part suppression) holds for the spatial propagation of gluons in the long distance greater than 0.4fm. It was observed that the diagonal gluon correlator largely changes between the confinement and the deconfinement phase, while the off-diagonal gluon correlator is almost the same even in the deconfinement phase \\cite{AS00}. \nAlthough the similar results are expected to hold in our formulation, this observation must be checked directly, as will be confirmed in \\cite{KSSK10}.\n\n\n\n \n\n$\\bullet$ In the presence of dynamical quarks ($m_q<\\infty$), $\\mathscr{X}_\\mu^A$ is responsible for chiral-symmetry breaking in the following sense. \nWe consider to integrate out the field $\\mathscr{X}_\\mu^A$ in a naive way. \nThis helps us to obtain an intuitive and qualitative understanding for the interplay between the chiral symmetry breaking and confinement. \nLater, this integration procedure will be reconsidered from the viewpoint of the renormalization group to obtain a systematic improvement of the result. \n\nHere we neglect $O(\\mathscr{X}^3)$ and $O(\\mathscr{X}^4)$ terms, which will be taken into account later. Then the integration over $\\mathscr{X}_\\mu^A$ can be achieved by the Gaussian integration according to \\cite{Kondo06}. \nConsequently, a nonlocal 4 fermion-interaction is generated:\n\\begin{align}\n S_{\\rm eff}^{\\rm QCD} =& S_{\\rm eff}^{\\rm glue} + S_{\\rm eff}^{\\rm gNJL} ,\n\\nonumber\\\\\n S_{\\rm eff}^{\\rm glue} :=& \\int d^Dx \\frac{-1}{4} (\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}])^2 \n\\nonumber\\\\&\n+ \\frac{i}{2} \\ln \\det Q[\\mathscr{V}]_{\\mu\\nu}^{AB} - i \\ln \\det G[\\mathscr{V}]^{AB} \n ,\n\\nonumber\\\\\nS_{\\rm eff}^{\\rm gNJL} :=& \\int d^Dx \\ \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] -\\hat{m}_q + i \\gamma^0 \\mu) \\psi \n\\nonumber\\\\ \n +& \\int d^Dx \\int d^Dy \\frac{g^2}{2} \\mathscr{J}^{\\mu}_{A}(x) Q^{-1}[\\mathscr{V}]_{\\mu\\nu}^{AB}(x,y) \\mathscr{J}^{\\nu}_{B}(y) \n ,\n\\end{align}\nwhere the last term $- \\ln \\det G^{AB}$ in $S_{\\rm eff}^{\\rm glue}$ comes from the Faddeev-Popov determinant associated with the reduction condition (\\ref{nMAG}) (see \\cite{KMS05} for the precise form). \n\n\nThis is a nonlocal version of a gauged NJL model (realized after Fierz transformation). \nThe chiral-symmetry breaking\/restoration transition and the phase structure of a local version of gauge NJL models were first studied by solving the Schwinger-Dyson equation in the ladder approximation for QED-like \\cite{Miransky85,KMY89,KKM89} (see \\cite{Kondo91} for a review) and QCD-like \\cite{KSY91} running gauge coupling constant. \nThey are confirmed later by a systematic approach of the renormalization group \\cite{Aoki-etal99}.\n\n\nThe range of the nonlocality is determined by the correlation length $\\xi$, which is characteristic of the color exchange through gluon fields. \nTherefore, this correlation length $\\xi$ is identified with the inverse of the effective mass $M_X$, i.e., $\\xi \\simeq M_X^{-1}$. In fact, $(Q^{-1})_{\\mu\\nu}^{AB}(x,y)$ is the $\\mathscr{X}$ field correlator, see (\\ref{QCDaction2}).\n\nIn other words, $M_X$ is identified with the ultraviolet cutoff $\\Lambda$ below which the effective NJL model appears and works well. Interesting enough, $M_X$ is nearly equal to the ultraviolet cutoff adopted in the NJL model \n\\begin{equation}\n\\sqrt{p^2} \\lesssim \\Lambda_4=1.4 {\\rm GeV} , \\quad |\\bm{p}| \\lesssim \\Lambda_3=0.6 {\\rm GeV} , \n\\end{equation}\nsee \\cite{HK94}.\n\n\nWe can decompose the gauge field $\\mathscr{A}_\\mu$ into the the low-energy (light) mode $pM_X$: \n\\begin{align}\n \\mathscr{A}_\\mu(p)\n=&\\mathscr{A}_\\mu(p) \\theta(M_X^2-p^2)+\\mathscr{A}_\\mu(x) \\theta(p^2-M_X^2)\n .\n \\label{decomp2}\n\\end{align}\nIn the above treatment, $\\mathscr{X}_\\mu(p)$ is supposed to have only the high-energy mode. \nThe low-energy mode, if any, will be responsible for the vacuum condensation \\cite{Kondo06}. \nFor the precise understanding, we need the renormalization group treatment as given later and the implications for the nonlocal NJL model will be discussed there. \n\n\n\\section{Gluon sector and gauge fixing}\\label{sec:gauge}\n\n\nThe Polyakov loop operator $L$ and the chiral operator $\\bar\\psi \\psi$ are gauge-invariant quantities. Therefore, their average do not depend on the gauge-fixing procedure adopted in the calculation. \nWe can choose a gauge in which the actual calculation becomes easier than other gauges. \n\nIn what follows, we treat the time-component $\\mathscr{V}_0$ and space-component $\\mathscr{V}_j$ of $\\mathscr{V}_\\mu$ differently to consider the finite-temperature case. \nWe consider the following Polyakov gauge modified for new variables in our reformulation. \nIf the color field $n^A(x)$ is uniform in time, \n\\begin{equation}\n \\partial_0 n^A(x) = 0 \\Leftrightarrow\nn^A(x) = n^A (\\bm{x})\n , \n \\label{my-gauge-0}\n\\end{equation}\nthen $\\mathscr{V}_0$ reduces to\n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(x)n^A (\\bm{x}) \\quad (A=1,2,3)\n .\n \\label{my-gauge-1}\n\\end{equation}\nMoreover, if $c_0(x)$ is uniform in time, \n\\begin{equation}\n \\partial_0 c_0(x) = 0 \\Leftrightarrow\nc_0(x) = c_0(\\bm{x})\n , \n\\end{equation}\nthen $\\mathscr{V}_0$ reduces to\n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(\\bm{x})n^A (\\bm{x}) \\quad (A=1,2,3)\n ,\n \\label{my-gauge-2}\n\\end{equation}\nwhich satisfies \n\\begin{equation}\n \\partial_0 \\mathscr{V}_0^A(x) = 0 \\quad (A=1,2,3)\n .\n \\label{my-gauge-3}\n\\end{equation}\nIn this setting, $\\mathscr{V}_j$ are given by\n\\begin{equation}\n \\mathscr{V}_j^A(x) = c_j(x)n^A(\\bm{x}) + g^{-1} \\epsilon^{ABC} \\partial_j n^B(\\bm{x}) n^C(\\bm{x}) \n . \n\\end{equation}\n\n\n\n(1) In order to simplify the calculation of the Polyakov line, we adopt the Polyakov gauge in which the gauge field is diagonal and time-independent: \nfor the background field $\\mathscr{V}_0^A(x)$, \n\\begin{equation}\n \\mathscr{V}_0^A(x) = c_0(\\bm{x}) \\delta^{A3} ,\n\\end{equation}\nwhich leads to\n\\begin{equation}\n \\partial_0 \\mathscr{V}_0^A(x) = 0 \n .\n\\end{equation}\nThis is realized, if we take the gauge\n\\footnote{This is an oversimplified choice for the color field $\\bm{n}(x)$. \nBy this choice, we can not separate the non-perturbative contribution coming from topological configurations such as magnetic monopole.\nIt is desirable to take into account color field degrees of freedom explicitly to see the effect of magnetic monopole in the confinement\/deconfinement transition. \n}\n\\begin{equation}\n n^A(\\bm{x}) = \\delta^{A3}\n . \n\\end{equation}\nIn this gauge, the space-component reads\n\\begin{equation}\n \\mathscr{V}_j^A(x) = c_j(x)\\delta^{A3} \n . \n\\end{equation}\nwhich is not time-independent, $\\partial_0 \\mathscr{V}_j^A(x) \\not= 0$.\n\n(2) We expand the theory around the non-trivial uniform background $g^{-1}T\\varphi \\delta^{A3} $ for the time-component $\\mathscr{V}_0^A$, while the trivial background for space-components $\\mathscr{V}_j^A$:\\footnote{\nI have assumed that the spatial component of $\\mathscr{V}_\\mu$ has a trivial background. \nIn view of logical consistency, one must expand the spatial and temporal components around non-trivial backgrounds, and then one must search for the minima of the effective potential calculated as a function of two variables, i.e., the temporal and spatial backgrounds. In this paper, it is assumed that a minimum is realized at vanishing spatial background and that the neglection of the spatial background does not so much affect the confinement\/deconfinement transition temperature. \nIndeed, it must be checked whether this assumption is good or not.\n}\n\\begin{align}\n \\mathscr{V}_0^A(x) =& c_0(\\bm{x})\\delta^{A3} , \\ c_0(\\bm{x}) = g^{-1}T\\varphi + v_0(\\bm{x}) \n ,\n\\nonumber\\\\\n \\mathscr{V}_j^A(x) =& 0+ v_j^A(x)\n , \n\\end{align}\nsuch that \n$\\langle c_0(\\bm{x}) \\rangle = g^{-1}T \\langle \\varphi \\rangle + \\langle v_0(\\bm{x}) \\rangle=g^{-1}T \\langle \\varphi \\rangle $ with $\\langle v_0(\\bm{x}) \\rangle=0$ and $\\langle v_j^A(x) \\rangle=0$. \nHere the prefactor $g^{-1}T=(g\\beta)^{-1}$ was introduced just for the purpose of simplifying the expression of the Polyakov loop, see (\\ref{P}). \n\n(3) We take into account the expansion up to quadratic in the fluctuation fields $v_0$ and $v_j$, which we call the quadratic approximation. \n\n\nIn the calculation of $Q_{\\mu\\nu}^{AB}$, if we neglect all fluctuation fields $v_0$ and $v_j$, namely, \n$\\mathscr{V}_0^A (x) = g^{-1}T \\varphi \\delta^{A3}$ and $\\mathscr{V}_j^A (x) = 0$, then we can put \n$\\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] = 0$ in $Q_{\\mu\\nu}^{AB}$, and $Q_{\\mu\\nu}^{AB}$ is diagonal in the Lorentz indices:\n\\begin{align} \nQ_{\\mu\\nu}^{AB} &= G^{AB} g_{\\mu\\nu} , \n \\label{approx0}\n\\\\ \nG^{AB} \n& = - \\delta^{AB} \\partial_\\mu^2 + (\\delta^{AB} - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 + 2 \\epsilon^{AB3} T \\varphi \\partial_0 \n .\n\\nonumber\n\\end{align}\nIn this approximation, we have\n\\begin{align}\n & G^{AB} = - \\delta^{AB} \\partial_\\ell^2 - D_0^{AC}[\\mathscr{V}]D_0^{CB}[\\mathscr{V}] ,\n \\label{approx}\n\\end{align}\nwith\n\\begin{align}\n - D_0^{AC}[\\mathscr{V}]D_0^{CB}[\\mathscr{V}] \n=& - \\delta^{AB} \\partial_0^2\n+ 2 \\epsilon^{AB3} T \\varphi \\partial_0 \n \\nonumber\\\\&\n+ (\\delta^{AB} - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 \n .\n\\end{align}\nThus we rewrite the gluon part $S_{\\rm eff}^{\\rm glue}[\\mathscr{V}]$ as\n\\begin{align}\n & S_{\\rm eff}^{\\rm glue}[\\mathscr{V}] \n \\nonumber\\\\ \n&= \\frac12 \\beta \\int d^3x \\mathscr{V}_0(\\bm{x}) (-\\partial_j \\partial_j) \\mathscr{V}_0(\\bm{x}) \n \\nonumber\\\\&\n + \\frac12 \\int d^4x \\mathscr{V}_{T}^A(x) \\{ \n - \\delta^{AB} \\partial_\\ell^2 - \\delta^{AB} \\partial_0^2\n \\} \\mathscr{V}_{T}^B(x) \n \\nonumber\\\\&\n + \\frac12 \\int d^4x \\mathscr{V}_{L}^A(x) \\{ - \\delta^{AB} \\partial_0^2 \\} \\mathscr{V}_{L}^B(x) \n ,\n\\nonumber\\\\&\n+ \\frac{i}{2} \\ln \\det Q[\\mathscr{V}]_{\\mu\\nu}^{AB} - i \\ln \\det G[\\mathscr{V}]^{AB} \n ,\n \\label{mode-decomp1}\n\\end{align}\nwhere $\\beta$ is the inverse temperature $\\beta:=1\/T$, and $\\mathscr{V}_{T}$ and $\\mathscr{V}_{L}$ denote the transverse and longitudinal components of $\\mathscr{V}_\\mu$ respectively.\n\n\n\n\n\\section{Polyakov loop}\\label{sec:Polyakov}\n\nFor $G=SU(2)$, the Polyakov loop operator $L(\\bm{x})=L[\\mathscr{V}_0(\\bm{x}, \\cdot)]$ is defined by\n\\begin{align}\nL(\\bm{x}) :=& \\frac12 {\\rm tr}(P) ,\n\\nonumber\\\\\n P(\\bm{x}) :=& \\mathscr{P} \\exp \\left[ ig \\int_{0}^{\\beta=1\/T} dx_0 \\mathscr{V}_0^A(\\bm{x},x_0) \\frac{\\sigma_A}{2} \\right] \n, \n\\label{P}\n\\end{align}\nwhere \n$P P^\\dagger = \\mathbf{1}$\nand\n$\\det P =1$.\nIn the above gauge choice, \n\\begin{equation}\n P(\\bm{x}) = \\exp \\left[ ig \\beta c_0(\\bm{x}) n^A(\\bm{x}) \\frac{\\sigma_A}{2} \\right] . \n \\nonumber\n\\end{equation}\nAfter a suitable ($t$-independent) gauge transformation, the color field $n^A(\\bm{x})$ is eliminated:\n\\begin{equation}\n L(\\bm{x}) \n= \\frac12 {\\rm tr}(\\exp \\left[ ig \\beta c_0(\\bm{x}) \\frac{\\sigma_3}{2} \\right] )\n= \\cos \\left( \\frac{g \\beta c_0(\\bm{x})}{2} \\right) . \n\\label{LL}\n\\end{equation}\n\nOwing to periodicity and center symmetry, we can restrict the Polyakov loop average to $\\langle L \\rangle \\ge 0$ for $G=SU(2)$. Then the Polyakov loop average $\\langle L[\\mathscr{V}] \\rangle$ is bounded from above by $L[ \\langle \\mathscr{V}_0(\\bm{x}, \\cdot) \\rangle]$:\n\\begin{equation}\n 0 \\le \\langle L[\\mathscr{V}_0(\\bm{x}, \\cdot)] \\rangle\n\\le L[ \\langle \\mathscr{V}_0(\\bm{x}, \\cdot) \\rangle] \n= \\cos \\left( \\frac{\\langle \\varphi \\rangle}{2} \\right) ,\n\\end{equation}\nwhere we have only to consider the range $0 \\le \\varphi \\le \\pi$.\nThis inequality follows from the Jensen inequality, since $\\cos (x)$ is concave for $0 \\le x\\le \\pi\/2$, see \\cite{MP08}.\n\n\nIn the case of $m_q=\\infty$, if the center-symmetry is broken $\\langle L \\rangle >0$, namely, deconfinement takes place, then the vacuum (as a minimum of the effective potential $V_{\\rm eff}(\\varphi)$) is realized at $\\langle \\varphi \\rangle < \\pi$.\nIf the vacuum is realized at $\\langle \\varphi \\rangle=\\pi$, then the center-symmetry is restored $\\langle L \\rangle =0$, namely, confinement occurs. The relation (\\ref{LL}) yields the relationship for the average between the gauge field and the Polyakov loop operator: \n\\begin{equation}\n \\langle \\arccos L(\\bm{x}) \\rangle = \\frac{g \\beta \\langle c_0(\\bm{x}) \\rangle }{2} = \\frac{\\langle \\varphi \\rangle}{2} ,\n\\end{equation}\nwhere the left-hand side is the average of an gauge-invariant object (since $L$ is gauge invariant) and happens to agree with the average $\\langle \\mathscr{V}_0^3 \\rangle$ of the gauge field in the Polyakov gauge. \nIt is also shown \\cite{MP08} that the converse is true: In the center-symmetry-restored phase, $\\langle \\varphi \\rangle=\\pi$, since \n\\begin{equation}\n \\frac{\\langle \\varphi \\rangle}{2} = \\langle \\arccos L(\\bm{x}) \\rangle = \\arccos \\langle L(\\bm{x}) \\rangle = \\frac{\\pi}{2} .\n\\end{equation}\n\n\nTherefore, $\\langle \\mathscr{V}_0 \\rangle$ or $\\langle \\varphi \\rangle$ in the Polyakov gauge gives a direct physical interpretation as an order parameter for the confinement\/deconfinement (order-disorder) phase transition. \nThe effective potential $U_{\\rm eff}(\\langle L \\rangle)$ of the Polyakov loop average $\\langle L \\rangle$ could be different from the effective potential $V_{\\rm eff}(\\langle \\mathscr{V}_0 \\rangle)$ of the gauge field average $\\langle \\mathscr{V}_0 \\rangle$ in the following sense.\nAlthough both potentials give the same critical temperature $T_d$ as a boundary between $\\langle L \\rangle=0$ and $\\langle L \\rangle \\not= 0$, \nthe value of the effective potential $U_{\\rm eff}(\\langle L[\\mathscr{V}_0 ] \\rangle)$ does not necessarily agree with $U_{\\rm eff}(L[ \\langle \\mathscr{V}_0 \\rangle])=V_{\\rm eff}(\\langle \\mathscr{V}_0 \\rangle)$ at a given temperature $T$, since we have only an inequality \n$\n\\langle L[\\mathscr{V}_0 ] \\rangle \\le L[ \\langle \\mathscr{V}_0 \\rangle]\n$.\nThis difference could affect the critical exponent and other physical quantities of interest. Therefore, the result obtained from $V_{\\rm eff}(\\langle \\mathscr{V}_0 \\rangle)$ must be carefully examined. \n\n\n\\section{Deriving the confinement\/deconfinement transition}\\label{sec:deconfinement}\n\nIn this section, we restrict our consideration to the pure glue case. \nWe show that the pure gluon part $S_{eff}^{\\rm glue}$ can describe confinement\/deconfinement transition signaled by the Polyakov loop average $\\langle L \\rangle$. \nIn this section, we completely follow two remarkable papers by Marhauser and Pawlowski \\cite{MP08} and by Braun, Gies and Pawlowski \\cite{BGP07}, which succeeded to show the transition for the first time based on the functional renormalization group (FRG).\nIn the next section, we explain how these results are understood from the Landau-Ginzburg argument. \n\nWe consider the flow equation called the Wetterich equation \\cite{Wetterich93} for the $k$(RG scale)-dependent effective action $\\Gamma_k$:\n\\begin{align}\n\\partial_t \\Gamma_k[\\Phi] \n=& \\frac12 {\\rm STr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\Phi^\\dagger} \\Gamma_k[\\Phi] \\frac{\\overleftarrow{\\delta}}{\\delta \\Phi} + R_{\\Phi,k} \\right]^{-1} \\cdot \\partial_t R_{\\Phi,k} \\right\\}\n ,\n\\end{align}\nwhere $t$ is the RG time \n$\n t := \\ln \\frac{k}{\\Lambda}\n$, \n$\n \\partial_t := \\frac{\\partial}{\\partial t} = k \\frac{d}{dk} \n$\nfor some reference scale (UV cutoff) $\\Lambda$ \nand $R_{\\Phi,k}$ is the regulator function for the field $\\Phi$. \nHere ${\\rm STr}$ denotes the super-trace introduced to include both commutative field (gluon) and anticommutative field (quark, ghost). \nSee \\cite{BTW00,Gies06} for reviews of the functional renormalization group. \n\nIf we restrict our consideration to the pure glue case $S_{\\rm YM}$ under the gauge $n^A(x)=\\delta^{A3}$, then the relevant fields $\\Phi$ are $\\mathscr{V}_\\mu^A(x)$ , $\\mathscr{X}_\\mu^A(x)$ and FP ghosts (ghost and antighost) $\\mathscr{C}^A(x), \\bar{\\mathscr{C}}^A(x)$, i.e., $\\Phi^\\dagger = (\\mathscr{V}_\\mu^A,\\mathscr{X}_\\mu^A,\\mathscr{C}^A,\\bar{\\mathscr{C}}^A)$. \nIn this section, we use the Euclidean formulation. \nIn the modified Polyakov gauge and within the quadratic approximation adopted in sec.~\\ref{sec:gauge}, \n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 {\\rm Tr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}} + R_{k} \\right]^{-1} \\cdot \\partial_t R_{k} \\right\\}\n\\nonumber\\\\&\n + \\partial_t \\frac12 {\\rm Tr} \\{ \\ln [Q_{\\mu\\nu}^{AB}+\\delta^{AB}\\delta_{\\mu\\nu}R_{k} ] \\} \n\\nonumber\\\\&\n - \\partial_t {\\rm Tr} \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} \n .\n\\end{align}\nwhere the second contribution in the right-hand side comes from the $\\mathscr{X}$ field and the last one from the ghosts fields \\cite{KMS05}, and we have used the same regulator function $R_k$ for the gluon and ghost up to the difference due to the tensor structure. \n\n\n\n\n\nWe neglect back-reactions of the $\\mathscr{V}_{0}$ potential on the other gauge fields $\\mathscr{V}_{j}$, as in the treatment \\cite{MP08}. \nAssuming an expansion around $\\mathscr{V}_{j}=0$, $\\Gamma_{k}^{(2)}:=\\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}}$ is block-diagonal like the regulators, and the flow equation can be decomposed into a sum of two contributions:\nunder the approximation (\\ref{approx0}),\n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 {\\rm Tr} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}+R_{k}} \\right)_{\\mu\\nu} \\cdot \\partial_t R_{k,\\mu\\nu} \\right]\n\\nonumber\\\\&\n + \\partial_t {\\rm Tr} \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\}\n ,\n\\end{align}\nwhere the gluon regulator $R_{k,\\mu\\nu}$ is a block-diagonal matrix in field space,\n\\begin{align}\n R_{k,00} =& R_{0,k} = Z_0 R_{\\rm opt,k}(\\bm{p}^2), \\\n R_{k,0j} = 0 = R_{k,j0} ,\n\\nonumber\\\\\n R_{k,j\\ell} =& R_{T,k} T_{j\\ell}(\\bm{p})\n= Z_j T_{j\\ell}(\\bm{p}) R_{\\rm opt, k_T}(\\bm{p}^2) ,\n\\end{align}\nwhere $T_{j\\ell} := \\delta_{j\\ell} - \\frac{p_jp_\\ell}{p_m^2}$ is the transverse projection operator and \n$R_{\\rm opt,k}(\\bm{p}^2)$ is the (3 dim.) optimized choice \\cite{Litim00}:\n\\begin{equation}\n R_{\\rm opt,k}(\\bm{p}^2) = (k^2-\\bm{p}^2) \\theta(k^2-\\bm{p}^2) .\n\\end{equation} \n\n\n\nThe first term in the right-hand side encodes the quantum fluctuations of $\\mathscr{V}_0$, while the second one encodes those of the other components of the gauge field and ghosts. \nIn the present truncation, the second term is a total derivative with respect to $t$, and does not receive contributions from the first term. Therefore, we can evaluate the flow of the second contribution, and use its output $V_{T,k}(\\mathscr{V}_0)$ as an input for the remaining flow. \n\\begin{align}\n \\partial_t \\Gamma_k\n =& \\frac12 \\beta \\int \\frac{d^3p}{(2\\pi)^3} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}\n+R_\\mathscr{V}} \\right)_{00} \\partial_t R_{0,k} \\right]\n\\nonumber\\\\&\n + \\partial_t V_{T,k} \n ,\n\\end{align}\nwhere for $\\omega=2\\pi Tn$\n\\begin{align}\n& V_{T,k} \n := {\\rm Tr} \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} \n \\nonumber\\\\\n=& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr}\\ln [\\tilde{G}^{AB}(\\omega, \\bm{p})\n+\\delta^{AB}(k_T^2-\\bm{p}^2) \\theta (k_T^2-\\bm{p}^2)]\n \\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr}\\ln [ \\delta^{AB}\\bm{p}^2-D_0^2 \n+ \\delta^{AB}(k_T^2-\\bm{p}^2) \\theta (k_T^2-\\bm{p}^2) ] \n \\nonumber\\\\\n =& T \\sum_{n \\in \\mathbb{Z}} 4\\pi \\int_{0}^{k_T} \\frac{dp p^2}{(2\\pi)^3} {\\rm tr}\\ln [ \\delta^{AB}k_T^2 -D_0^2 ] \n \\nonumber\\\\&\n- T \\sum_{n \\in \\mathbb{Z}} 4\\pi \\int_{0}^{k_T} \\frac{dp p^2}{(2\\pi)^3} {\\rm tr}\\ln [ \\delta^{AB}\\bm{p}^2-D_0^2 ] \n + V_W\n .\n \\label{V_{T,k}}\n\\end{align}\nHere we have introduced the Weiss potential $V_W$ which was obtained by one-loop calculation \\cite{Weiss81}:\n\\begin{align}\nV_W =& \n {\\rm Tr} \\ln [G^{AB}] \n \\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr} \\ln [\\tilde{G}^{AB}(p_0=\\omega, \\bm{p})]\n \\nonumber\\\\ \n=& \n T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr}\\ln [\\bm{p}^2+(\\omega + T\\varphi)^2] \n \\nonumber\\\\&\n + T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr}\\ln [\\bm{p}^2+(\\omega - T\\varphi)^2]\n ,\n\\end{align}\nwhere we have neglected the $\\varphi$-independent (or $\\mathscr{V}_0$ independent) contributions. \n\n\n\n\\begin{figure}[ptb\n\\begin{center}\n \\includegraphics[width=5.5cm]{Fig-ep-184\/V_w-SU2.eps}\n \\includegraphics[width=5.5cm]{Fig-ep-184\/L-SU2.eps}\n\\end{center}\n \\caption{\\small \n(The upper panel) (normalized) SU(2) Weiss potential $\\hat V_{W}$ as a function of $\\varphi$.\n(The lower panel) SU(2) Polyakov loop $L$ as a function of $\\varphi$,\n$L = \\cos \\left( \\frac{\\varphi}{2} \\right) \n$.\n}\n \\label{fig:Weiss}\n\\end{figure}\n\n\nThe closed form of the Weiss potential is obtained after summing up the Matsubara frequencies:\n\\begin{align}\n V_W(\\varphi)\n=& T^4 \\left[ - \\frac{1}{6} (\\varphi-\\pi)^2 + \\frac{1}{12\\pi^2} (\\varphi-\\pi)^4 + \\frac{\\pi^2}{12} \\right]\n\\nonumber\\\\\n ({\\rm mod} \\ 2\\pi)\n .\n\\end{align}\nThe Weiss potential $V_W$ is $g^2$ independent and the overall curve scales as $T^4$.\n $V_W$ has symmetries: \n$V_W(-\\varphi)=V_W(\\varphi)$ and $V_W(\\varphi+2\\pi n)=V_W(\\varphi)$.\n $V_W(\\varphi)$ has minima at $\\varphi=2\\pi n$, and the Polyakov loop has the nonvanishing value $L=\\cos \\frac{\\varphi}{2}=(- 1)^n$, implying deconfinement. See Fig.~\\ref{fig:Weiss}.\n Therefore, $V_W(\\varphi)$ is considered to be valid at very high temperature where the perturbation theory is trustworthy. \n In Fig.~\\ref{fig:VPreWeiss3D}, we observe\n\\begin{equation}\n\\lim_{k \\downarrow 0} V_{T,k} = V_{T,0} = V_{W} \n ,\n\\quad\n\\lim_{k \\uparrow \\infty} V_{T,k} = 0 .\n\\end{equation}\n\n\n\n\n\\begin{figure}[ptb\n\\begin{center}\n \\includegraphics[width=6cm]{Fig-ep-184\/PreWeiss.eps}\n\\end{center} \n \\caption{$\\hat V_{T,k}$ for different values of $\\hat k$ [reprinted from \\cite{MP08}].}\n \\label{fig:VPreWeiss3D}\n\\end{figure}\n\n\n\n\nAfter integrating over the fields other than $\\mathscr{V}_0$, we are lead to the effective action of $\\mathscr{V}_0$, \n\\begin{align}\n \\Gamma_{k}[\\mathscr{V}_0] \n=& \\beta \\int d^3x \\left\\{ - \\frac{1}{2}Z_0 \\mathscr{V}_0(\\bm{x}) \\partial_j \\partial_j \\mathscr{V}_0(\\bm{x}) + V_{{\\rm eff},k}^{\\rm glue}[\\mathscr{V}_0] \\right\\} ,\n \\nonumber\\\\ \n V_{{\\rm eff},k}^{\\rm glue}[\\mathscr{V}_0] =& V_{T,k}[\\mathscr{V}_0] + \\Delta V_k[\\mathscr{V}_0] .\n \\label{V_g}\n\\end{align}\nThen the flow equation is reformulated for $\\Delta V_k$ with the external input $V_{T,k}$:\n\\begin{equation}\n \\beta \\partial_{t} (\\Delta V_k[\\mathscr{V}_0])\n = \\frac12 \\beta \\int \\frac{d^3p}{(2\\pi)^3} \\left[ \\left( \\frac{1}{\\Gamma_k^{(2)}+R_{k}} \\right)_{00} \\partial_t R_{0,k} \\right]\n ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\Gamma_{k}^{(2)}[\\mathscr{V}_0] \n= \\beta \\left\\{ Z_0 \\bm{p}^2 + \\partial_{\\mathscr{V}_0}^2 V_{k}[\\mathscr{V}_0] \\right\\} \n .\n\\end{equation}\n\n\n\n\n\nUsing the specific regulator, \n$\n R_{0,k}\n= Z_0 (k^2-\\bm{p}^2)\\theta(k^2-\\bm{p}^2) \n$, \nwhich yields\n\\begin{align}\n \\partial_t R_{0,k}\n= \\left[ \\partial_{t} Z_0 (k^2-\\bm{p}^2) + 2 Z_0 k^2 \\right] \\theta(k^2-\\bm{p}^2) \n ,\n\\end{align}\nwe can perform the momentum integration analytically. \n\\begin{align}\n & \\beta \\partial_{t} (\\Delta V_k[\\mathscr{V}_0]) \n\\nonumber\\\\\n=& \\frac23 \\frac{1}{(2\\pi)^2} \\frac{ (\\eta_k\/5+1) k^5 }{ Z_k^{-1} g^2 \\beta^2 \\partial_{\\varphi}^2 (V_{T,k}[\\mathscr{V}_0] + \\Delta V_k[\\mathscr{V}_0] ) + k^2 }\n ,\n\\end{align}\nwhere we have introduced \n the running coupling $\\alpha_k$ defined by \n\\begin{equation} \n g_k^2 := Z_k^{-1} g^2, \\quad \\alpha_k := \\frac{g_k^2}{4\\pi} = Z_k^{-1} \\frac{g^2}{4\\pi}\n ,\n\\end{equation}\nand the anomalous dimension $\\eta_k$ defined by\n\\begin{equation}\n \\eta_k := \\partial_{t} \\ln Z_k = - \\partial_{t} \\ln \\alpha_k .\n\\end{equation}\n\n\\begin{figure}[ptb\n\\begin{center}\n \\includegraphics[width=7.5cm]{Fig-ep-184\/coupling.eps}\n\\end{center} \n \\caption{The running gauge coupling constant $\\alpha_s$ for temperatures $T=0,150,300,600$ MeV [reprinted from \\cite{MP08}].}\n \\label{fig:alpha}\n\\end{figure}\n\n\\begin{figure}[ptb\n\\begin{center}\n \\includegraphics[width=7.5cm]{Fig-ep-184\/VeffforT.eps}\n\\end{center} \n \\caption{Full effective potential $\\hat V_{{\\rm eff}}^{\\rm glue}$, \n normalized to 0 at $\\varphi =0$ [reprinted from \\cite{MP08}].}\n \\label{fig:Veff}\n\\end{figure}\n\n\n\nBy introducing the dimensionless RG scale $\\hat{k}$ and the dimensionless effective potential $\\hat{V}$ defined by\n\\begin{equation}\n \\hat{k} := \\beta k = k\/T ,\n \\quad \n \\hat{V} := \\beta^4 V =V\/T^4 \n ,\n \\label{rescaling}\n\\end{equation}\nthe flow equation is simplified as\n\\begin{equation}\n \\partial_{\\hat k} \\Delta \\hat V_{\\hat k}[\\mathscr{V}_0] \n= \\frac{1}{6\\pi^2} \\frac{ (1+\\eta_k\/5 ) \\hat k^2 }{ 1+\\frac{4\\pi\\alpha_k}{\\hat k^2} \\partial_{\\varphi}^2 (\\hat V_{T,\\hat k}[\\mathscr{V}_0] + \\Delta \\hat V_{\\hat k}[\\mathscr{V}_0] ) }\n ,\n \\label{flow-eq}\n\\end{equation}\nwhere all scales are measured in units of temperature. \nIt turns out that the input in solving the flow equation is just a running gauge coupling constant $\\alpha_k$. A specific choice for the running gauge coupling constant is given in Fig.~\\ref{fig:alpha}. For the derivation from the renormalization group, see \\cite{BG05}. \nThe flow is initialized in the broken phase at any temperature. \nBy solving the flow equation in a numerical way with an input for the running gauge coupling given in Fig.~\\ref{fig:alpha}, the full effective potential $\\hat V_{\\rm eff}$ (normalized to 0 at $\\varphi =0$) is obtained in Fig.~\\ref{fig:Veff} for various temperature.\n\n\n\n\nAccording to \\cite{MP08}, a second order phase transition occurs at a critical temperature\n\\begin{equation}\n T_d = 305_{-55}^{+40} \\text{MeV}, \\quad\nT_d\/\\sqrt{\\sigma}=0.69_{-.12}^{+.04}\n ,\n\\end{equation}\nwith the string tension $\\sqrt{\\sigma}=440$ MeV.\nThis agrees within errors with the lattice result\n$T_d\/\\sqrt{\\sigma}=0.709$.\nMoreover, these results were confirmed by considering another gauge \\cite{BGP07}.\n\n\n\n\n\\section{Understanding the existence of confinement transition according to the Landau-Ginzburg argument}\\label{sec:Landau}\n\nIn this section, we show that some qualitative aspects of the deconfinement\/confinement transition found in the previous section can be understood without detailed numerical works, although the precise value of the transition temperature $T_d$ cannot be determined without them. \n\nFor $G=SU(2)$ in the pure Yang-Mills limit $m_q \\rightarrow \\infty$, the effective potential $V_{\\rm glue}(L)$ for the Polyakov loop $L$ must be invariant under the center symmetry $Z(2)$. Therefore, $V_{\\rm glue}(L)$ is an even function of $L$, i.e., $V_{\\rm glue}(L)=V_{\\rm glue}(-L)$ where $L$ is real-valued $L=L^*$. \nThus the Landau-Ginzburg argument suggests that the effective potential $V_{\\rm eff}^{\\rm glue}(L)$ for $G=SU(2)$ has the power-series expansion in $L$ near the transition point $L=0$: \n\\begin{equation}\n V_{\\rm eff}^{\\rm glue}(L) = c_0 + \\frac{c_2}{2} L^2 + \\frac{c_4}{4} L^4 + O(L^6)\n .\n\\end{equation} \nAs the vacuum is specified as minima of the effective potential, the confinement\/deconfinement transition temperature $T_d$ is determined from the condition $c_2(T_d)=0$ so that the low-temperature ($T0$, while the high-temperature ($T>T_d$) deconfinement phase $\\langle L \\rangle \\not=0$ is realized for $c_2(T)<0$, provided that the positivity $c_4(T)>0$ is maintained across the transition temperature. \nConsequently, the transition is of the 2nd order. \n\nIndeed, we confirm that the Landau-Ginzburg description is correct and valid for the confinement\/deconfinement transition, by making use of the flow equation given in the previous section. \nThis is a microscopic justification of the Landau-Ginzburg argument for the confinement\/deconfinement transition.\nIn our treatment, however, it is more convenient to write the effective potential $V_{\\rm eff}^{\\rm glue}$ in terms of the angle variable $\\varphi$ (rather than $L$) around the transition point $\\varphi=\\pi$ (instead of $L=0$).\nDefining $\\tilde\\varphi :=\\varphi-\\pi$, we find that \n$V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi)$ must be an even function $V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi)=V_{\\rm eff}^{\\rm glue}(-\\tilde\\varphi)$ due to the center symmetry and hence odd terms (e.g., $\\tilde\\varphi$, $\\tilde\\varphi^3$) do not appear: \n\\begin{equation}\n V_{\\rm eff}^{\\rm glue}(\\tilde\\varphi) = C_{0} + \\frac{C_{2}}{2} \\tilde \\varphi^2 + \\frac{C_{4}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n .\n\\end{equation}\nAt sufficiently high temperature, we observe that $C_2(T)<0$ and hence $V_{\\rm eff}^{\\rm glue}$ has the minimum at $\\tilde \\varphi \\not=0$ ($\\Longleftrightarrow L\\not=0$) leading to deconfinement. \nIn order to show the existence of the confinement\/deconfinement transition at $T=T_d$, $C_2(T)$ must change the signature $C_2(T)>0$ below this temperature $T0$ is maintained.\n\n\nFor this purpose, we study the scale dependent effective potential $V_{\\rm eff,k}^{\\rm glue}$ at $k>0$\n\\begin{equation}\n V_{\\rm eff,k}^{\\rm glue} = C_{0,k} + \\frac{C_{2,k}}{2} \\tilde \\varphi^2 + \\frac{C_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n ,\n\\end{equation}\nand see how it evolves towards the limit $k\\rightarrow 0$ according to the flow equation to obtain the physical effective potential $V_{\\rm eff}^{\\rm glue}:=V_{\\rm eff,k=0}^{\\rm glue}$.\n\nAs in (\\ref{V_g}), $V_{\\rm eff,k}^{\\rm glue}$ is decomposed into two pieces:\n\\begin{equation}\n V_{\\rm eff,k}^{\\rm glue} = \\hat V_{T,\\hat k} + \\Delta \\hat V_{\\hat k} \n ,\n \\label{Veff}\n\\end{equation}\nwhere we have defined the dimensionless potential according to the rescaling (\\ref{rescaling}).\nThe first part $\\hat V_{T,\\hat k}$ is the ($k$-dependent) perturbative part (\\ref{V_{T,k}}) obtained essentially by the one-loop calculation with the regulator function $R_k$ being included. For this part, the closed analytical form can be obtained, see Appendix~\\ref{app:coefficient}.\nWhile the second part $\\Delta \\hat V_{\\hat k}$ represents the non-pertubative part which is initially zero $\\Delta \\hat V_{\\hat k}|_{k=\\Lambda}=0$ and is generated in the evolution of the renormalization group. This part is obtained only by solving the flow equation (\\ref{flow-eq}) and its analytical form is not available (at this moment).\n\n\nWe expand $\\hat V_{T,\\hat k}$ in powers of $\\tilde\\varphi=\\varphi-\\pi$: \n\\begin{equation}\n \\hat V_{T,\\hat k}\n= A_{0,k} + \\frac{A_{2,k}}{2} \\tilde \\varphi^2 + \\frac{A_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n ,\n \\label{VTcoeff}\n\\end{equation}\nwhere coefficients are drawn as functions of $k$ in Fig.~\\ref{fig:A_{2,k}}, see Appendix~\\ref{app:coefficient} for their closed analytical forms. \n \n \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/A2_k.eps}\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/A4_k.eps}\n\\caption{$A_{2,k}$ and $A_{4,k}$ as functions of $\\hat k$.}\n\\label{fig:A_{2,k}}\n\\end{figure}\n\n\nSuppose that $\\Delta \\hat V_{\\hat k}$ is of the form:\n\\begin{equation}\n \\Delta \\hat V_{\\hat k} \n= a_{0,k} + \\frac{a_{2,k}}{2} \\tilde \\varphi^2 + \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n .\n \\label{V_T}\n\\end{equation}\nA flow equation (\\ref{flow-eq}) for the effective potential \n(\\ref{V_g}) is reduced to a set of coupled flow equations for coefficients in the effective potential (\\ref{Veff}) with (\\ref{VTcoeff}) and (\\ref{V_T}):\n\\begin{align}\n \\partial_{\\hat k} a_{2,k} \n=& - \\frac{(1+\\frac15 \\eta_k ) \\hat k^2 }{6\\pi^2} \\frac{ \\frac{4\\pi\\alpha_k}{\\hat k^2}(A_{4,k}+a_{4,k}) }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2} (A_{2,k}+a_{2,k})]^2} \n ,\n\\nonumber\\\\\n\\partial_{\\hat k} a_{4,k} \n=& + \\frac{(1+\\frac15 \\eta_k ) \\hat k^2 }{6\\pi^2} \\frac{6 [\\frac{4\\pi\\alpha_k}{\\hat k^2}(A_{4,k}+a_{4,k})]^2 }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2} (A_{2,k}+a_{2,k})]^3}\n ,\n\\nonumber\\\\\n\\vdots \n\\end{align}\nwhich are coupled first-order ordinary but nonlinear differential equations for coefficients.\nIn Appendix~\\ref{app:flow-equation}, we see that this form (\\ref{V_T}) is justified as a solution of the flow equation. In fact, it is easy to see that \n$\\partial_{\\hat k} a_{1,k} =0$ and\n$\\partial_{\\hat k} a_{3,k}=0$ are guaranteed from the flow equation, if the effective potential has no odd terms at arbitrary $k$.\nTherefore, if an initial condition, $a_{1,k} =0=a_{3,k}$ at $k=\\Lambda$ is imposed, then \n$a_{1,k} \\equiv 0$ and $a_{3,k} \\equiv 0$ \nare maintained for $0 \\le k \\le \\Lambda$ by solving the flow equation.\nIn performing numerical calculations, however, one must truncate the infinite series of differential equations up to some finite order to obtain manageable set of equations. \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/coupling1_k.eps}\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/coupling2_k.eps}\n\\caption{The running gauge coupling constant $\\alpha_k$ \nat $T=0.001, 0.125, 0.25, 0.50, 1.0$ GeV, from the top at the lowest temperature $T=0.001$GeV to the bottom at the highest temperature $T=1.0$ GeV.\n(The upper panel) $\\alpha_k$ as functions of $k$.\n(The lower panel) $\\alpha_k$ as functions of $\\hat{k}$.\nFor a given temperature, there is a critical value $\\hat k_c$ separating the deep IR region (\\ref{coupling-2}) from the higher momentum region (\\ref{coupling-1}).\nThe discontinuity of the derivative seen at $\\hat k_c$ comes from a crude approximation in which we have taken into account just the first linear term (i.e., $c_1=c_2=\\cdots=0$) in the expansion (\\ref{coupling-2}), and can be avoided if we take into account higher order terms as explained below (\\ref{coupling-2}). However, this is not essential to see qualitative behaviors of the solution of the flow equation.\n}\n\\label{fig:alpha_k}\n\\end{figure}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=3.0in,height=1.5in]{Fig-ep-184\/anomalous_k.eps}\n\\caption{The anomalous dimension $\\eta_k$ as functions of $\\hat k$\nat $T=0.001, 0.125, 0.25, 0.50, 1.0$ GeV.\nIn each graph for a given temperature, there is a critical value $\\hat k_c$ of $\\hat k$ separating the deep IR region $\\eta_k \\simeq -1$ from the higher momentum (intermediate and UV) region $\\eta_k >0$.\nThe temperature is distinguished by $\\hat k_c$ ranging from the smallest value at the highest temperature $T=1.0$GeV to the largest value at the lowest temperature $T=0.001$ GeV where $\\eta_k \\simeq -1$ for $\\hat k<\\hat k_c$ and $\\eta_k \\simeq 0$ for $\\hat k>\\hat k_c$.\nThe discontinuity of the derivative seen at $\\hat k_c$ is due to the same reason as that explained in Fig.~\\ref{fig:alpha_k} and is not essential to see qualitative behaviors of the solution of the flow equation.\n}\n\\label{fig:eta_k}\n\\end{figure}\n\n\n\nWe can understand qualitatively why a 2nd order phase transition from the deconfinement phase to the confinement phase can occur by lowering the temperature.\n\n\nThe flow starts from $a_{2,k}=0$ and hence $C_{2,k}=A_{2,k}+a_{2,k}<0$ (because of $A_{2,k}<0$) at $k=\\Lambda \\gg 1$.\nWe assume $C_{4,k}=A_{4,k}+a_{4,k}>0$ for $0 \\le k \\le \\Lambda$, as a necessary condition for realizing a 2nd order transition. Otherwise, we must consider the higher-order terms, e.g. $O(\\varphi^6)$.\n (This assumption is assured to be true by numerical calculations of the full effective potential \\cite{MP08,BGP07}, as reproduced in the previous section.) \nThis assumption allows us to analyze just one differential equation for obtaining qualitative understanding:\n\\begin{align}\n \\partial_{\\hat k} a_{2,k} \n=& - \\frac{(1+\\frac15 \\eta_k ) }{6\\pi^2} \\frac{ 4\\pi\\alpha_k(A_{4,k}+a_{4,k}) }{[1+\\frac{4\\pi\\alpha_k}{\\hat k^2} (A_{2,k}+a_{2,k})]^2} \n .\n \\label{flow-2}\n\\end{align}\nThen the right-hand side of (\\ref{flow-2}) is negative, since the running coupling constant $\\alpha_k$ is positive and $1+\\frac15 \\eta_k$ is positive, see Fig.~\\ref{fig:alpha_k} and Fig.~\\ref{fig:eta_k}. \nConsequently, $a_{2,k}$ started at zero becomes positive $a_{2,k}>0$ just below $\\Lambda$ and increases (monotonically) as $k$ decreases.\nSee Fig.~\\ref{fig:mA_k}.\n\n\n\nNote that the denominator can vanish \n$1+\\frac{4\\pi\\alpha_k}{\\hat k^2} (A_{2,k}+a_{2,k})=0$ at some $k^*$ (since $C_{2,k}=A_{2,k}+a_{2,k}<0$ or $00$ or $a_{2,k}>-A_{2,k}$, the flow equation reads\n\\begin{align}\n \\partial_{\\hat k} a_{2,k} \n\\simeq - \\frac{(1+\\frac15 \\eta_k ) }{6\\pi^2} \\frac{ (A_{4,k}+a_{4,k}) \\hat k^4 }{4\\pi\\alpha_k (A_{2,k}+a_{2,k})^2} \n ,\n\\end{align}\nthe right-hand-side gets small negative, and $a_{2,k}$ becomes flat near the IR limit. See Fig.~\\ref{fig:mA_k}.\nFinally, $a_{2,k}$ reaches the value realizing $C_{2,k}=A_{2,k}+a_{2,k}>0$ or $a_{2,k}>-A_{2,k}$ at $k=0$. \nThis leads to the recovery of the center symmetry.\nThe difference is clearly seen from the second figure of Fig.~\\ref{fig:alpha_k} where the running gauge coupling $\\alpha_k$ is drawn as a function of $\\hat k$ for various temperatures. \n\nIn our treatment, the difference between the three-dimensional RG scale $k_T$ and the four-dimensional one $k$ is neglected by equating two scales $k_T=k$ just for simplifying the analysis, since it is enough for obtaining a qualitative understanding for the transition. This is not be the case for obtaining quantitative results, see Appendix C of \\cite{MP08} for the precise treatment on this issue. \n\n\n\n\n\\section{Quark part and gauged nonlocal NJL model}\\label{sec:NJL}\n\nWe examine the quark self-interaction part\n$S_{\\rm int} = \\int d^Dx \\int d^Dy \\frac{1}{2} \\mathscr{J}^{\\mu A}(x) g^2(Q^{-1}[\\mathscr{V}])_{\\mu\\nu}^{AB}(x,y) \\mathscr{J}^{\\nu B}(y)\n$.\nIn estimating the effect of $Q^{-1}[\\mathscr{V}]$, we take the same approximation as the above. \nConsequently, the inverse $(Q^{-1})_{\\mu\\nu}^{AB}[\\mathscr{V}]$ is diagonal in the Lorentz indices:\n\\begin{align} \n& (Q^{-1})_{\\mu\\nu}^{AB}[\\mathscr{V}] \n= g^{\\mu\\nu} (G^{-1})^{AB}[\\mathscr{V}] \n\\nonumber\\\\\n=& g^{\\mu\\nu}\n \\begin{pmatrix}\n \\frac12 [F_{\\varphi}+F_{-\\varphi}] & -\\frac{1}{2i} [F_{\\varphi}-F_{-\\varphi}] & 0 \\cr\n \\frac{1}{2i} [F_{\\varphi}-F_{-\\varphi}] & \\frac12 [F_{\\varphi}+F_{-\\varphi}] & 0 \\cr\n 0 & 0 & F_{0} \\cr\n \\end{pmatrix} \n ,\n\\end{align}\nwhere $F_{\\varphi}$ is defined by\n\\begin{align} \nF_{\\varphi}(i\\partial) :=& \\frac{1}{(i\\partial_\\ell)^2+(i\\partial_0+T\\varphi)^2}\n\\nonumber\\\\ \n=& \\frac{1}{(i\\partial_\\mu)^2+(T\\varphi)^2+2T\\varphi i\\partial_0}\n . \n\\end{align}\nIn what follows, we consider only the diagonal parts of $(G^{-1})^{AB}[\\mathscr{V}]$. \nThis is achieved by the procedure\n\\begin{equation}\n \\frac{g^2}{2} (Q^{-1})_{\\mu\\nu}^{AB}(x,y) = g^{\\mu\\nu} \\delta^{AB} \\mathcal{G}(x-y)\n ,\n\\end{equation}\nwhich yields\n\\begin{equation}\n \\mathcal{G}(x-y) = \\frac{g^2}{2} (Q^{-1})_{\\mu\\nu}^{AB}(x,y) \\frac{g_{\\mu\\nu}}{D} \\frac{\\delta^{AB}}{N_c^2-1} \n .\n\\end{equation}\nThen the nonlocal interaction is obtained as\n\\begin{equation}\n \\mathcal{G}(x-y) \n= \\frac{g^2}{2} \\frac{{\\rm tr}(G^{-1})}{N_c^2-1} \n= \\frac{g^2}{2} \\frac{F_{\\varphi}+F_{-\\varphi}+F_{0}}{3}\n .\n\\end{equation}\nThis approximation is used just for simplifying the Fierz transformation performed below and hence it can be improved by taking into account the off-diagonal parts of $G^{-1}$ if it is necessary to do so. \n\n\nFor $D=4$, we use the Fierz identity \\cite{Fierz} to rewrite the nonlocal current-current interaction as\n\\begin{align}\n S_{\\rm int} \n =& \\int d^4x \\int d^4y \\mathscr{J}^{\\mu A}(x) \\mathcal{G}(x-y) \\mathscr{J}^{\\mu A}(y)\n \\nonumber\\\\\n=& \\int d^4x \\int d^4y \\mathcal{G}(x-y) \\sum_{\\alpha} c_\\alpha (\\bar{\\psi}(x) \\Gamma_\\alpha \\psi(y) ) \n\\nonumber\\\\& \n \\times (\\bar{\\psi}(y) \\Gamma_\\alpha \\psi(x) ) \n \\nonumber\\\\\n =& \\int d^4x \\int d^4z \\mathcal{G}(z) \\sum_{\\alpha} c_\\alpha \\{ \\bar{\\psi}(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2) \\} \n \\nonumber\\\\&\n\\times \\{ \\bar{\\psi}(x-z\/2) \\Gamma_\\alpha \\psi(x+z\/2) \\} ,\n\\end{align}\nwhere the $\\Gamma_\\alpha$ are a set of Dirac spinor, color and flavor matrices, resulting from the Fierz transform, with the property $\\gamma_0 \\Gamma_\\alpha^\\dagger \\gamma_0=\\Gamma_\\alpha$. \nAlthough the Fierz transformation induces mixings and recombinations among operators, the resulting theory must maintain the symmetries of the original QCD Lagrangian. \nA minimal subset of operators satisfying the global chiral symmetry $SU(2)_L \\times SU(2)_R$ which governs low-energy QCD with two-flavors is the color-singlet of scalar-isoscalar and pseudoscalar-isovector operators \nThus, by restricting $\\Gamma_\\alpha$ hereafter to \n\\begin{equation}\n\\Gamma_\\alpha:=(\\mathbf{1}, i\\gamma_5 \\vec{\\tau})\n\\end{equation}\nand ignoring other less relevant operators (vector and axial-vector terms in color singlet and color octet channels), \n we arrive at a nonlocal gauged NJL model \n\\begin{align}\n S_{\\rm eff}^{\\rm gNJL} =& \\int d^4x \\bar\\psi(x) ( i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] - \\hat m_q+i\\gamma^0 \\mu_q) \\psi(x) + S_{\\rm int} ,\n \\nonumber\\\\\n S_{\\rm int} =& \\int d^4x \\int d^4z \\mathcal{G}(z) [\\bar \\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2) \n \\nonumber\\\\&\n\\times \\bar \\psi(x-z\/2) \\Gamma_\\alpha \\psi(x+z\/2) ] \n .\n \\label{gNJL}\n\\end{align}\nThis form is regarded as a gauged version of the nonlocal NJL model proposed in \\cite{HRCW08}. \nThe function $\\mathcal{G}(z)$ is replaced by a coupling constant $G$ times a normalized distribution $\\mathcal{C}(z)$:\n\\begin{equation}\n \\mathcal{G}(z) := \\frac{G}{2} \\mathcal{C}(z) ,\n \\quad\n \\int d^4z \\mathcal{C}(z) = 1 \n .\n\\end{equation}\nThe standard (local) gauged NJL model follows for the limiting case $\\mathcal{C}(z)=\\delta^4(z)$ \nwith\n$\\int d^4z \\mathcal{C}(z)=1$.\n\n\n\nIn contrast to \\cite{HRCW08}, however, $G$ and $\\mathcal{C}$ are determined in conjunction with the behavior of the Polyakov loop $L$ or $\\varphi$ at temperature $T$:\nusing the Fourier transform $\\tilde{\\mathcal{G}}(p)$ of $\\mathcal{G}$, they are expressed as\n\\begin{equation}\n \\frac{G}{2} \n= \\tilde{\\mathcal{G}}(p=0) ,\n \\quad\n \\tilde{\\mathcal{C}}(p) = \\tilde{\\mathcal{G}}(p)\/\\tilde{\\mathcal{G}}(p=0)\n ,\n \\label{G}\n\\end{equation}\nwhere \n\\begin{align}\n \\tilde{\\mathcal{G}}(p)\n=& \\frac{g^2}{2} \\frac{\\tilde{F}_{\\varphi}(p)+\\tilde{F}_{-\\varphi}(p)+\\tilde{F}_{0}(p)}{3} , \n\\\\\n\\tilde{F}_{\\varphi}(p)=& \\frac{1}{p^2+(T\\varphi)^2+2T\\varphi p_0} .\n\\nonumber\n\\end{align}\nNote that $\\tilde{F}_{\\varphi}(p=0)$ and hence $G$ diverge at $T=0$.\nThis comes from an improper treatment of the $T=0$ part.\nTo avoid this IR divergence at $T=0$, we add the $T=0$ contribution $M_0^2 \\simeq M_X^2$ and replace $F_{\\varphi}(i\\partial)$ by \n\\begin{align} \nF_{\\varphi}(i\\partial) \n&= \\frac{1}{(i\\partial_\\ell)^2+(i\\partial_0+T\\varphi)^2+M_0^2}\n\\nonumber\\\\\n=& \\frac{1}{(i\\partial_\\mu)^2+(T\\varphi)^2+2T\\varphi i\\partial_0+M_0^2}\n , \n\\end{align}\nand\n\\begin{align}\n\\tilde{F}_{\\varphi}(p)\n&= \\frac{1}{\\bm{p}^2+(p_0+T\\varphi)^2+M_0^2}\n\\nonumber\\\\\n=& \\frac{1}{p^2+(T\\varphi)^2+2T\\varphi p_0+M_0^2} .\n\\end{align}\nIn fact, such a contribution $\\frac12 M_0^2 $ comes in $G^{AB}$ as an additional term $M_0^2 \\delta^{AB}$ from the $O(\\mathscr{X}^4)$ terms (Note that $O(\\mathscr{X}^3)$ terms are absent for $G=SU(2)$), as already mentioned in the above. \n\n\nAnother way to avoid this IR divergence is to introduce the regulator term which is needed to improve the one-loop perturbative result and obtain a nonperturbative one according to the Wilsonian renormalization group:\n\\begin{align}\n \\Delta S_k =& \\int d^Dx \\frac12 \\mathscr{X}_\\mu^A(x) [\\delta^{AB} g_{\\mu\\nu} R_k(i\\partial)] \\mathscr{X}_\\nu^B(x) \n \\nonumber\\\\\n =& \\int \\frac{d^Dp}{(2\\pi)^D} \\frac12 \\tilde{\\mathscr{X}}_\\mu^A(-p) [\\delta^{AB} g_{\\mu\\nu} \\tilde{R}_k(p)] \\tilde{\\mathscr{X}}_\\nu^B(p) \n ,\n\\end{align}\nwhere $k$ is the RG scale and $\\tilde{R}_k(p)$ is the Fourier transform of $R_k(i\\partial)$. \nThe regulator function $R_k$ introduces a mass proportional to $k^2$, which plays a similar role to $M_0^2$ in the above, as long as $k >0$.\n\n \n\n\n\n\nThe NJL model \\cite{NJL61} is well known as a low-energy effective theory of QCD to describe the dynamical breaking of chiral symmetry in QCD (at least in the confinement phase), see e.g. \\cite{Klevansky92,HK94}. \nThe theory given above by $S_{\\rm eff}^{\\rm QCD}=S_{\\rm eff}^{\\rm glue}+S_{\\rm eff}^{\\rm gNJL}$ is able to describe chiral-symmetry breaking\/restoration and quark confinement\/deconfinement on an equal footing where \nthe pure gluon part $S_{\\rm eff}^{\\rm glue}$ describes confinement\/deconfinement transition signaled by the Polyakov loop average. \nWe can incorporate the information on confinement\/deconfinement transition into the quark sector through the covariant derivative $\\mathcal{D}[\\mathscr{V}]$ and the nonlocal NJL interaction $\\mathcal{G}$ ($G$ and $\\mathcal{C}$), in sharp contrast to the conventional PNJL model where the entanglement between chiral-symmetry breaking\/restoration and confinement\/deconfinement was incorporated through the covariant derivative $\\mathcal{D}[\\mathscr{V}]$ alone and the nonlocal NJL interaction $\\mathcal{G}$ is fixed to the zero-temperature case. \nIn our theory, the nonlocal NJL interaction $\\mathcal{G}$ ($G$ and $\\mathcal{C}$) is automatically determined through the information of confinement\/deconfinement dictated by the Polyakov loop $L$ (non-trivial gluon background), while in the nonlocal PNJL model \\cite{HRCW08} the low-momentum (non-perturbative) behavior of $\\mathcal{C}$ was not controlled by first principles and was provided by the instanton model. \n\n\n\n\nTo study chiral dynamics, it is convenient to bosonize the gauged nonlocal NJL model as done \\cite{HRCW08}.\nThe nonlocal gauged NJL model (\\ref{gNJL}) can be bosonised as follows. \nDefine\n\\begin{equation}\n\\Phi_\\alpha(x) :=(\\sigma(x), \\vec\\pi(x)) .\n\\end{equation}\nTo eliminate the quadratic term in the nonlocal currents, \nwe insert the unity:\n\\begin{align}\n 1=& \\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \\exp \\Big\\{ -\\int d^4z \\mathcal{C}(z) \n\\nonumber\\\\&\n \\times \\int d^4x \\frac{1}{2G} [\\Phi_\\alpha(x) + G\\bar\\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2)]\n\\nonumber\\\\&\n \\times [\\Phi_\\alpha(x) + G\\bar\\psi(x+z\/2) \\Gamma_\\alpha \\psi(x-z\/2)]^* \\Big\\} \n ,\n\\end{align}\nwhere we have used\n$\\int d^4z \\mathcal{C}(z)=1 $.\nThen we have the gauged Yukawa model: \n\\begin{align}\n& \\int \\mathcal{D} \\bar\\psi \\mathcal{D} \\psi e^{-S_{\\rm eff}^{\\rm gNJL} }\n\\nonumber\\\\\n =& \n \\int \\mathcal{D} \\bar\\psi \\mathcal{D} \\psi \n\\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \n\\exp \\{ - S_{\\rm eff}^{\\rm gY}\n \\} \n ,\n\\end{align}\nwhere with $x^\\prime:=x+z\/2$, $y^\\prime:=x-z\/2$, \n\\begin{align}\n S_{\\rm eff}^{\\rm gY} \n =& \n \\int d^4x^\\prime \\int d^4y^\\prime \\bar\\psi(x^\\prime) \\Big[ \\nonumber\\\\\n& \\delta^4(x^\\prime-y^\\prime) \n(-i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] + \\hat m_q+i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\frac12 \\mathcal{C}(x^\\prime-y^\\prime) \\Gamma_\\alpha [\\Phi_\\alpha(\\frac{x^\\prime+y^\\prime}{2}) + \\Phi_\\alpha^*(\\frac{x^\\prime+y^\\prime}{2})] \n\\Big] \\psi(y^\\prime)\n\\nonumber\\\\\n&+ \\int d^4x \\frac{1}{2G} \\Phi_\\alpha(x)\\Phi_\\alpha^*(x) \n ,\n\\end{align}\nor\n\\begin{align}\n& S_{\\rm eff}^{\\rm gY} \n\\nonumber\\\\\n =& \n \\int \\frac{d^4p}{(2\\pi)^4} \\frac{d^4p^\\prime}{(2\\pi)^4} \\bar\\psi(p) \\Big[ \n\\nonumber\\\\\n& (2\\pi)^4 \\delta^4(p-p^\\prime) (- \\gamma^\\mu (p_\\mu + g \\tilde{\\mathscr{V}}_\\mu(p)) + \\hat m_q+i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\frac12 \\tilde{\\mathcal{C}}(\\frac{p+p^\\prime}{2}) \\Gamma_\\alpha [\\Phi_\\alpha(p-p^\\prime) + \\Phi_\\alpha^*(p-p^\\prime)] \\Big] \\psi(p^\\prime)\n\\nonumber\\\\&\n + \\int \\frac{d^4p}{(2\\pi)^4} \\frac{1}{2G} \\Phi_\\alpha(p)\\Phi_\\alpha^*(p)\n .\n\\end{align}\n\nFinally, the bosonized theory of the gauged NJL model is obtained by way of the gauged Yukawa model by integrating out quark fields as\n\\begin{align}\n& \\int \\mathcal{D} \\bar\\psi \\mathcal{D} \\psi e^{-S_{\\rm eff}^{\\rm gNJL} }\n\\nonumber\\\\\n =& \n\\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \n\\int \\mathcal{D} \\bar\\psi \\mathcal{D} \\psi \n\\exp \\{ -S_{\\rm eff}^{\\rm gY}\n \\} \n\\nonumber\\\\\n =& \\int \\mathcal{D} \\sigma \\mathcal{D} \\vec{\\pi} \\exp \\{ - S_{\\rm eff}^{\\rm boson} \\}\n ,\n\\end{align}\nwhere the bosonised action $S_{\\rm eff}^{\\rm boson}$ is \n\\begin{align}\n S_{\\rm eff}^{\\rm boson}\n =& \n - {\\rm Tr} \\ln \\Big\\{ \n \\delta^4(x^\\prime-y^\\prime) \n(-i\\gamma^\\mu (\\partial_\\mu -ig \\mathscr{V}_\\mu) +i\\gamma^4 \\mu_q)\n\\nonumber\\\\\n&+ \\hat m_q + \\frac12 \\mathcal{C}(x^\\prime-y^\\prime) \\Gamma_\\alpha [\\Phi_\\alpha(\\frac{x^\\prime+y^\\prime}{2}) + \\Phi_\\alpha^*(\\frac{x^\\prime+y^\\prime}{2})] \\Big\\} \n\\nonumber\\\\\n&+ \\int d^4x \\frac{1}{2G} \\Phi_\\alpha(x)\\Phi_\\alpha^*(x) \n ,\n\\end{align}\nor\n\\begin{align}\n S_{\\rm eff}^{\\rm boson}\n =& \n - {\\rm Tr} \\ln \\Big\\{ \n(2\\pi)^4 \\delta^4(p-p^\\prime) [- \\gamma^\\mu (p_\\mu + g \\tilde{\\mathscr{V}}_\\mu) +i\\gamma^4 \\mu_q]\n\\nonumber\\\\&\n+ \\hat m_q + \\frac12 \\tilde{\\mathcal{C}}(\\frac{p+p^\\prime}{2}) \\Gamma_\\alpha [\\Phi_\\alpha(p-p^\\prime) + \\Phi_\\alpha^*(p-p^\\prime)] \\Big\\} \n\\nonumber\\\\&\n+ \\int \\frac{d^4p}{(2\\pi)^4} \\frac{1}{2G} \\Phi_\\alpha(p)\\Phi_\\alpha^*(p)\n .\n\\end{align}\n\n\n\n\n\n\n\\section{Implications of the Polyakov loop for chiral-symmetry breaking at finite temperature}\\label{sec:}\n\nThe thermodynamics of QCD can be studied based on our effective theory derived in this paper in the similar way to the nonlocal PNJL model \\cite{HRCW08}. \nBut this must be done by including the effect of gluon properly. \nIn the PNJL model, the effect of the gluon was introduced by the standard minimal gauge coupling procedure, i.e., replacing the normal derivative $\\partial_\\mu$ by the covariant derivative $D_\\mu[\\mathscr{A}]:=\\partial_\\mu -ig\\mathscr{A}_\\mu$. \nIn our effective theory, the effect of the gluon is introduced through the NJL coupling constant $G$ and the nonlocality function $\\mathcal{C}$, in addition to the minimal coupling $D_\\mu[\\mathscr{V}]$. \nThe nonlocal NJL interaction among quarks are mediated by gluons at finite temperature in QCD. \nTherefore, both $G$ and $\\mathcal{C}$ characterizing nonlocal NJL interaction inevitably have temperature dependence, which could be different depending on whether quarks are in confinement or deconfinement phases. \n \nAt $T=0$, QCD must be in the hadron phase where the chiral symmetry is spontaneously broken, which means that the NJL coupling constant $G(0)$ at zero temperature must be greater than the critical NJL coupling constant $G_c$:\n\\begin{equation}\n G(0) = g^2 \\frac{1}{M_0^2} > G_c .\n\\end{equation}\nThe nonlocality function or the form factor $\\tilde{\\mathcal{C}}(p)$ at $T=0$ behaves\n\\begin{equation}\n \\tilde{\\mathcal{G}}(p)\n= \\frac{g^2}{2} \\frac{1}{p^2 +M_0^2 }, \\ \n \\tilde{\\mathcal{G}}(0)\n= \\frac{g^2}{2} \\frac{1}{M_0^2 } ,\n\\end{equation}\nand\n\\begin{equation}\n\\tilde{\\mathcal{C}}(p) = \\frac{M_0^2}{p^2 +M_0^2 } \n .\n\\end{equation}\n\nAs an immediate outcome of our effective theory, this determines the temperature-dependence of the coupling constant $G$ of nonlocal NJL model.\nUsing (\\ref{G}), we have\n\\begin{equation}\n G(T) = \\frac13 g^2 \\left[ \\frac{2}{(T\\varphi)^2+M_0^2}+\\frac{1}{M_0^2} \\right] , \\\n\\end{equation}\nwhich lead to the NJL coupling constant normalized at $T=0$:\n\\begin{equation}\n G(T)\/G(0) = \\frac13 \\left[ \\frac{2M_0^2}{(T\\varphi)^2+M_0^2}+1 \\right] .\n\\end{equation}\n\n\nIn the presence of the dynamical quark $m_q < \\infty$, the Polyakov loop is not an exact order parameter and does not show a sharp charge with discontinuous derivatives. Even in this case, we can introduce the pseudo critical temperature $T_d^*$ as a temperature achieving the peak of the susceptibility. \nBelow the deconfinement temperature $T_d^*$, i.e., $T < T_d^*$, therefore, $L \\simeq 0$ or $\\varphi \\simeq \\pi$, the NJL coupling constant $G$ has the temperature-dependence\n\\begin{equation}\nG(T)\/G(0) \\simeq \\frac13 \\left[ \\frac{2M_0^2}{\\pi^2 T^2+M_0^2}+1 \\right] \\ (T < T_d^*) .\n\\end{equation}\n\n\nThis naive estimation gives a qualitative understanding for the existence of chiral phase transition. Since $G(T)$ is (monotonically) decreasing as the temperature $T$ increases, it becomes smaller than the critical NJL coupling constant \n\\begin{equation}\n G(0) = g^2 \\frac{1}{M_0^2} > G_c ,\n \\quad \n T \\uparrow \\infty \\Longrightarrow \nG \\downarrow 0 .\n\\end{equation}\nThus, the chiral transition temperature $T_\\chi$ will be determined (if the chiral-symmetry restoration and confinement coexist or the chiral symmetry is restored in the confinement environment before deconfinement takes place, i.e., $T_\\chi \\le T_d^*$ ) by solving\n\\begin{equation}\n G(T_\\chi) \\equiv \\frac{G(0)}{3} \\left[ \\frac{2M_0^2}{T_\\chi^2 \\pi^2 +M_0^2}+ 1\\right] = G_c \n .\n\\end{equation}\nHere we have assumed that the nonlocality function $\\tilde{\\mathcal{C}}(p)$ gives the dominant contribution at $p=0$, namely, $\\tilde{\\mathcal{C}}(p) \\le \\tilde{\\mathcal{C}}(0)=\\int d^4z \\mathcal{C}(z)=1$ and that the occurrence of the chiral transition is determined by the NJL coupling constant alone. \n\nAt finite temperature $T$, the form factor reads\n\\begin{align}\n& \\mathcal{C}(\\bm{x}-\\bm{y}) \n\\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} \n \\tilde{\\mathcal{C}}(p_0=\\omega, \\bm{p}) e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n\\nonumber\\\\ \n =& T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} \n\\frac{M_0^2}{3} \\Big[ \\frac{2}{\\bm{p}^2+(\\omega+T\\varphi)^2+M_0^2} \n\\nonumber\\\\&\n+ \\frac{1}{\\bm{p}^2+\\omega^2+M_0^2} \\Big] e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n\\nonumber\\\\ \n =& \\int \\frac{d^3p}{(2\\pi)^3} \n\\frac{M_0^2}{6\\epsilon_p} \\Big[ 2\\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-\\cos(\\varphi)} \n\\nonumber\\\\&\n+ \\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-1} \\Big] e^{i\\bm{p} \\cdot (\\bm{x}-\\bm{y})} \n ,\n\\end{align}\nwhere we have defined \n$\\epsilon_p:=\\sqrt{\\bm{p}^2+M_0^2}$\nand used\n\\begin{align}\n T \\sum_{n \\in \\mathbb{Z}} \\frac{1}{(\\omega+C)^2+\\epsilon_p^2} \n = \\frac{1}{2\\epsilon_p} \\frac{\\sinh (\\epsilon_p\/T)}{\\cosh(\\epsilon_p\/T)-\\cos(C\/T)} \n .\n\\end{align}\nThe form factor $\\mathcal{C}$ does not change so much around the deconfinement temperature $T \\sim T_d^*$ (or $\\varphi \\sim \\pi$). \nThis is reasonable since the form factor is nearly equal to the $\\mathscr{X}$ correlator $Q^{-1}$, as already mentioned in sec. II. \n\nFor more precise treatment, we must obtain the full effective potential $V_{\\rm eff}(\\sigma, \\varphi)$ as a function of two order parameters $\\sigma$ (or $\\langle \\bar\\psi \\psi \\rangle$) and $\\varphi$ (or $\\langle L \\rangle$), and look for a set of values $(\\sigma, \\varphi)= (\\sigma_0, \\varphi_0)$ at which the minimum $V_{\\rm eff}(\\sigma_0, \\varphi_0)$ of $V_{\\rm eff}(\\sigma, \\varphi)$ is realized. \nThen $\\varphi$ must be replaced by $\\varphi_0$ in the above consideration. \nFor this goal, we must develop the RG treatment for the full theory. \nThis issue will be studied in a subsequent paper. \n\n\\section{How to understand the entanglement between confinement and chiral symmetry breaking}\\label{sec:entanglement}\n\n\nTo discuss the entanglement between confinement and chiral symmetry breaking, we wish to obtain the total effective potential $V_{}^{\\rm QCD}$ of QCD written in terms of two order parameters, i.e., the Polyakov loop average $\\langle L \\rangle$ and chiral condensate $\\langle \\bar \\psi \\psi \\rangle$, so that its minima determine the vacuum for a given set of parameters $m_q$, $T$ and $\\mu_q$ when $N_c$ and $N_f$ are fixed. \nHere $m_q \\uparrow \\infty$ is the pure Yang-Mills limit and $m_q \\downarrow 0$ is the chiral limit.\n\n\n\n\n\nThe effective potential for the quark part is obtained by integrating out quark degrees of freedom.\nThe simplest form is obtained e.g., from the bosonized model as\n\\begin{align}\n & V_{}^{\\rm quark}\n\\nonumber\\\\\n =& - {\\rm Tr} \\ln \\Big\\{ i\\gamma^\\mu \\partial_\\mu \n+ m_q + \\mathcal{C} \\sigma \n - g \\mathscr{A}_4 \\gamma^4 \n + i \\mu_q \\gamma^4 \\Big\\} \n + \\frac{1}{2G} \\sigma^2 \n .\n\\label{V^q}\n\\end{align}\nThen the RG-scale $k$ dependent effective potential $V_{k}^{\\rm quark}$ for the quark part must be given as the solution of the flow equation. \nIn the same approximation as the above, it is written in terms of two order parameters $\\sigma$ and $\\varphi$: \n\\begin{align}\n & V_{k}^{\\rm quark}(\\sigma,\\varphi)_{m_q,T,\\mu_q}\n\\nonumber\\\\ \n=& \n - T \\sum_{n \\in \\mathbb{Z}} \\int \\frac{d^3p}{(2\\pi)^3} {\\rm tr} \\ln \\Big[ i \\omega_n \\gamma^0- p_j \\gamma^j + m_q + \\mathcal{C}(p) \\sigma\n\\nonumber\\\\&\n\\quad\\quad\\quad\\quad \\quad\\quad\\quad\\quad\\quad\\quad\n- T\\varphi T_3 \\gamma^4 + i \\mu_q \\gamma^4 + R_k^{\\rm quark} \\Big] \n\\nonumber\\\\&\n + \\frac{1}{2G} \\sigma^2 \n ,\n \\label{V^q_k}\n\\end{align}\nwhere $T_3=\\sigma_3\/2$ and $R_k^{\\rm quark}$ is the regulator function for quarks.\nIn the limit $k \\downarrow 0$, indeed, $V_{k}^{\\rm quark}$ (\\ref{V^q_k}) reduces to $V_{}^{\\rm quark}$ (\\ref{V^q}). \nThe effective potential $V_{k}^{\\rm quark}$ (\\ref{V^q_k}) depends on $m_q$, $T$ and $\\mu_q$ when $Nc$ and $N_f$ are fixed. \nDue to the $p_0$ dependence of the ``mass'' function $M(p)$ which is an immediate consequence of the nonlocality of the present NJL model, it is difficult to obtain the closed analytical form by performing the summation over the Matsubara frequencies.\n\n\n\n\nIn our strategy, a full effective potential $V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)$ of QCD is given by summing three parts:\n\\begin{equation} \n V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)\n = V_{k}^{\\rm glue}(\\varphi) + V_{k}^{\\rm quark}(\\sigma,\\varphi) + \\Delta V_{k}^{\\rm QCD}(\\sigma,\\varphi) \n ,\n\\end{equation}\nwith the pure gluon part $V_{k}^{\\rm glue}(\\varphi) = V_{T,k}$ (\\ref{V_g}), \n\\begin{align}\n V_{k}^{\\rm glue}(\\varphi) \n=& {\\rm Tr} \\{ \\ln [G^{AB}+\\delta^{AB}R_{k} ] \\} ,\n\\nonumber\\\\\n =& {\\rm Tr} \\{ \\ln [\n - \\delta^{AB} \\partial_\\mu^2 + (\\delta^{AB} - \\delta^{A3}\\delta^{B3})(T \\varphi)^2 \n\\nonumber\\\\&\n\\quad\\quad\\quad\\quad\n+ 2 \\epsilon^{AB3} T \\varphi \\partial_0 \n+\\delta^{AB}R_{k} ] \\} ,\n\\end{align}\n the quark part (\\ref{V^q_k}), \n\\begin{align}\n V_{k}^{\\rm quark}(\\sigma,\\varphi)\n =& \\frac{1}{2G} \\sigma^2\n - {\\rm Tr} \\ln \\{ \n i\\gamma^\\mu \\partial_\\mu + m_q + \\mathcal{C} \\sigma \n- T\\varphi T_3 \\gamma^4 \n\\nonumber\\\\& \n\\quad\\quad\\quad\\quad\\quad \n + i \\mu_q \\gamma^4 + R_k^{\\rm quark}\n\\} ,\n\\end{align}\nand a non-perturbative part $\\Delta V_{k}^{\\rm QCD}(\\sigma,\\varphi)$ induced in the RG evolution according to a flow equation.\nWe assume that the total effective action of QCD obtained after integrating out the fields other than those relevant to chiral symmetry and confinement is the form\n\\begin{align}\n \\Gamma_k =& \\int_{0}^{1\/T}dx_4 \\int d^3x \\Big\\{ \n \\frac{1}{2}Z_0 [ \\partial_j \\mathscr{V}_0(\\bm{x})]^2\n+ \\frac{1}{2} Z_\\sigma [\\partial_j \\sigma(x)]^2 \n\\nonumber\\\\&\n+ V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi) \\Big\\} \n ,\n\\end{align}\nand obeys the flow equation:\n\\begin{align}\n\\partial_t \\Gamma_k \n=& \\frac12 {\\rm Tr} \\left\\{ \n\\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\sigma^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\sigma} + R_{k} \\right]^{-1} \\cdot \n\\partial_t R_{k} \\right\\}\n\\nonumber\\\\\n &+ \\frac12 {\\rm Tr} \\left\\{ \\left[ \\frac{\\overrightarrow{\\delta}}{\\delta \\mathscr{V}^\\dagger} \\Gamma_k \\frac{\\overleftarrow{\\delta}}{\\delta \\mathscr{V}} + R_{k} \\right]^{-1} \\cdot \\partial_t R_{k} \\right\\} \n .\n\\end{align}\n\nIf the flow equation was solved, we would have obtained the effective potential of QCD, $V_{{\\rm eff},k}^{\\rm QCD}(\\sigma,\\varphi)$ which has the following power-series expansion with respect to two variables $\\sigma$ and $\\tilde\\varphi$ in the neighborhood of the transition point where $\\sigma=0=L$ according to the Landau argument (as demonstrated in the pure glue case). \n\\begin{align}\n V_{{\\rm eff}}^{\\rm QCD}(\\sigma,\\tilde\\varphi)\n =& V^{\\rm g}(\\tilde\\varphi) + V^{\\rm q}(\\sigma)\n+ V^{\\rm c}(\\sigma, \\tilde\\varphi) ,\n\\nonumber\\\\\n V^{\\rm g}(\\tilde\\varphi) =& C_0 + C_1 \\tilde\\varphi + \\frac{C_2}{2} \\tilde\\varphi^2 + \\frac{C_3}{3} \\tilde\\varphi^3 + \\frac{C_4}{4} \\tilde\\varphi^4 + O(\\tilde\\varphi^6)\n ,\n\\nonumber\\\\\n V^{\\rm q}(\\sigma) =& \\frac{E_2}{2} \\sigma^2 + \\frac{E_4}{4} \\sigma^4 + O(\\sigma^6)\n ,\n\\nonumber\\\\\n V^{\\rm c}(\\sigma, \\tilde\\varphi) =& \n F_1 \\sigma^2 \\tilde\\varphi + \\cdots \n ,\n\\end{align}\nwhere \n$V^{\\rm g}(\\hat\\varphi)$ denotes a part written in terms of $\\hat\\varphi$ alone, \nand $V^{\\rm q}(\\sigma)$ denotes a part written in terms of $\\sigma$ alone, \nwhile $V^{\\rm c}(\\sigma, \\hat\\varphi)$ denotes the cross term between $\\sigma$ and $\\hat\\varphi$. \n\n\n\nOnce dynamical quarks are introduced, the exact center symmetry in pure Yang-Mills theory is no longer intact. \nTherefore, the QCD effective potential includes the explicitly center-symmetry-breaking term.\nFor $G=SU(2)$, the center symmetry $\\tilde\\varphi \\rightarrow - \\tilde\\varphi$ is explicitly broken as $C_1 \\not=0$, $C_3 \\not=0$ in $V^{\\rm g}(\\varphi)$ and $F_1 \\not= 0$ in $V^{\\rm c}(\\sigma, \\varphi)$. \nThe existence of the cross term is important to understand the entanglement between center symmetry and chiral symmetry, as pointed out by \\cite{Fukushima04}.\nIn fact, the one-loop calculation leads to \n$C_1=0.97434 N_f>0$ and \n$F_1=- 0.106103 N_f<0$ ($\\mu_q=0$ case) which appears to be a good indication for this purpose and serves as the initial condition in solving the flow equation.\n\n\n\nIn the paper by Schaefer, Pawlowski and Wambach \\cite{SPW07}, \na sort of back-reaction from quarks has been introduced to improve the effective potential of the Polyakov loop, while the NJL coupling remains local. \nIn contrast, this paper introduces a back-reaction from gluons to improve the NJL interaction, leading to the nonlocal NJL coupling. \nHowever, this does not mean that two treatments are considered to be alternative. \nIn the presence of dynamical quarks, the running coupling $\\alpha$ is changed due to fermionic contributions. In \\cite{SPW07}, this effect has been taken into account as a modification of the expansion coefficient in the effective potential of the Polyakov loop, resulting in e.g., the $N_f$ flavor-dependent deconfinement temperature $T_d(N_f)$.\nRemembering that the input of our analysis is just a running coupling, a sort of back-reaction from quarks considered in \\cite{SPW07} is easily included into our framework by using the running coupling modified by quark contributions. \nThus, the treatment in this paper is already able to take into account back-reactions from quarks and gluons mentioned above. \n\n\n\nThis section is a sketch of our strategy of understanding the entanglement between center symmetry and chiral symmetry.\nThe detailed analysis will be given in a subsequent paper. \n\\footnote{\nIt is known that appearance of a mixed-term $\\sigma^2\\tilde\\varphi$ plays an essential role in the chiral-confinement\nentanglement. Such a term appears in the original PNJL model and leads to the 2 crossovers happening almost simultaneously. \nIn the following paper posted to the archive after this paper was submitted for publication, it has been shown that an effective Polyakov loop-dependent four-quark interaction derived by this paper yields stronger correlation between the chiral and deconfinement transitions, making $T_\\chi \\sim T_d$ more tightly, than the usual PNJL model. \n\\\\\nY. Sakai, T. Sasaki, H. Kouno and M. Yahiro,\nEntanglement between deconfinement transition and chiral symmetry restoration, \ne-Print: arXiv:1006.3648 [hep-ph]. \n}\n\n \n\n\n\n\n\n\n\\section{Conclusion and discussion}\\label{sec:conclusion}\n\n\nIn this paper, we have presented a reformulation of QCD and suggested a framework for deriving a low-energy effective theory of QCD which enables one to study the deconfinement\/confinement and chiral-symmetry restoration\/breaking crossover transition simultaneously on an equal footing.\nA resulting low-energy effective theory based on this framework can be regarded as a modified (improved) version of the nonlocal PNJL model \\cite{HRCW08}. \nIn our framework, the basic ingredients are a reformulation of QCD based on new variables and the flow equation of the Wetterich type for the Wilsonian renormalization group.\n\nA lesson we learned in this study is that a perturbative (one-loop) result can be a good initial condition for solving the flow equation of the renormalization group to obtain the non-perturbative result. \nIn gluodynamics, recently, it has been demonstrated \\cite{MP08,BGP07} that the existence of confinement transition, i.e., recovery of the center symmetry signaled by the vanishing Polyakov loop average can be shown by approaching the phase transition point from the high-temperature deconfinement phase in which the center symmetry is spontaneously broken.\nIndeed, the effective potential for the Polyakov loop obtained in the one-loop calculation which we call the Weiss potential leads to the non-vanishing Polyakov loop average, i.e., spontaneous breaking of the center symmetry. \n \n\n\nFor gluon sector, to understand the existence of confinement transition by approaching from the deconfinement side, we have given the Landau-Ginzburg description in the neighborhood of the (crossover) phase transition point by analyzing the flow equation of the functional renormalization group. \nThe deconfinement\/confinement phase transition is consistent with the second order transition for $G=SU(2)$, while the first order transition is expected for $G=SU(3)$.\nThe detailed study of the $SU(3)$ case will be given in a subsequent paper. \n\n\n\nThe input for solving the flow equation was just a running gauge coupling constant, in sharp contrast to the PNJL model including several parameters. \nFrom the viewpoint of a first-principle derivation, this is superior to phenomenological models with many input parameters.\n \n\nFor quark sector, it is possible to obtain the chiral-symmetry breaking\/restoration transition from the first principle.\nHowever, we need more hard works, especially, to discuss the QCD phase diagram at finite density and the critical endpoint. \nA possibility in this direction from the first principle of QCD was demonstrated in one-flavor QCD based on the FRG \\cite{Braun09}. \nIt will be possible to treat chiral dynamics and confinement on an equal footing based on our framework along this line \\cite{Kondo10}. \nStill, however, we must overcome some technical issues to achieve the goal of understanding full phase structures of QCD.\nThe detailed studies will be hopefully given in a subsequent paper. \n\n\n\n\n\n\n\n\n\n\n\n\n{\\em Acknowledgements --} \nThe author would like to thank the Yukawa Institute for Theoretical Physics at Kyoto University where the YITP workshop ``New Frontiers in QCD 2010 (NFQCD10)'' was held. \nHe thanks the organizers and the participants, especially, Hideo Suganuma, Kenji Fukushima, Akira Ohnishi, Hiroshi Toki, Wolfram Weise and Akihiro Shibata for discussions and comments on his talk, which were useful to complete this work. \nThanks are due to Jan Pawlowski for discussions and correspondences on papers \\cite{MP08,BGP07}, and Pengming Zhang for translating the original paper \\cite{DG79} from Chinese into English. \nHe is grateful to High Energy Physics Theory Group and Theoretical Hadron Physics Group in the University of Tokyo, especially, Prof. Tetsuo Hatsuda for kind hospitality extended to him on sabbatical leave between April 2009 and March 2010.\nThis work is financially supported by Grant-in-Aid for Scientific Research (C) 21540256 from Japan Society for the Promotion of Science\n(JSPS).\n \n\n\\begin{appendix}\n\\section{Reformulation of QCD}\\label{app:reformulation}\n\n\nWe apply the decomposition (\\ref{decomp}) to QCD Lagrangian. \n\nThe quark part is decomposed according to (\\ref{decomp}) as \n\\begin{align}\n \\mathscr{L}_{q} \n:=& \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{A}] -\\hat{m}_0 + i \\mu \\gamma^0) \\psi\n\\nonumber\\\\\n=& \\bar{\\psi} (i\\gamma^\\mu \\mathcal{D}_\\mu[\\mathscr{V}] -\\hat{m}_0 + i \\mu \\gamma^0) \\psi + g\\bar{\\psi} \\gamma^\\mu \\mathscr{X}_\\mu \\psi\n ,\n\\end{align}\nwhere the covariant derivative $\\mathcal{D}_\\mu[\\mathscr{V}]$ is defined by\n\\begin{equation}\n \\mathcal{D}_\\mu[\\mathscr{V}] := \\partial_\\mu - ig \\mathscr{V}_\\mu .\n\\end{equation}\n\n\nThe Yang-Mills part is treated as follows. \nFor the general decomposition $\\mathscr{A}_\\mu(x)=\\mathscr{V}_\\mu(x)+\\mathscr{X}_\\mu(x)$, the field strength $\\mathscr{F}_{\\mu\\nu}$ is decomposed as \n\\begin{align}\n \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] :=& \\partial_\\mu \\mathscr{A}_\\nu - \\partial_\\nu \\mathscr{A}_\\mu -i g [ \\mathscr{A}_\\mu , \\mathscr{A}_\\nu ]\n\\nonumber\\\\\n=& \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] + \\partial_\\mu \\mathscr{X}_\\nu - \\partial_\\nu \\mathscr{X}_\\mu -i g [ \\mathscr{V}_\\mu , \\mathscr{X}_\\nu ] \n\\nonumber\\\\&\n-i g [ \\mathscr{X}_\\mu , \\mathscr{V}_\\nu ]\n-i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ]\n\\nonumber\\\\\n=& \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] + D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu \n\\nonumber\\\\&\n-i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ] ,\n\\end{align}\nwhere the covariant derivative $D_\\mu[\\mathscr{V}]$ in the background field $\\mathscr{V}_\\nu$ is defined by\n\\begin{align}\nD_\\mu[\\mathscr{V}] := \\partial_\\mu \\mathbf{1} -ig [ \\mathscr{V}_\\mu , \\cdot ] ,\n\\end{align}\nor, equivalently, \n\\begin{equation}\n D_\\mu[\\mathscr{V}]^{AC} := \\partial_\\mu \\delta^{AC} + g f^{ABC} \\mathscr{V}_\\mu^B .\n\\end{equation}\nThe Lagrangian density \n$\\mathscr{L}_{YM} = -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}] \\cdot \\mathscr{F}^{\\mu\\nu}[\\mathscr{A}]$ \nof the Yang-Mills theory is decomposed as\n\\begin{align}\n \\mathscr{L}_{YM} \n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{A}]^2\n\\\\\n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]^2\n\\nonumber\\\\&\n- \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\&\n- \\frac{1}{4} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)^2 \n\\nonumber\\\\&\n+ \\frac{1}{2} \\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n+ \\frac{1}{2} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2\n.\n\\end{align}\nHere the third term on the right-hand side of the above equation is rewritten using integration by parts (or up to total derivatives) as \n\\begin{align}\n & \\frac{1}{4} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)^2\n\\nonumber\\\\\n=& \\frac{1}{2} (- \\mathscr{X}_\\mu \\cdot D_\\nu[\\mathscr{V}]D^\\nu[\\mathscr{V}] \\mathscr{X}^\\mu + \\mathscr{X}_\\mu \\cdot D_\\nu[\\mathscr{V}]D^\\mu[\\mathscr{V}] \\mathscr{X}^\\nu ) \n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^\\mu \\cdot \\{ - D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}] g_{\\mu\\nu} + D_\\nu[\\mathscr{V}]D_\\mu[\\mathscr{V}] \\} \\mathscr{X}^\\nu\n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^{\\mu A} \\{ - (D_\\rho[\\mathscr{V}]D_\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n- [ D_\\mu[\\mathscr{V}], D_\\nu[\\mathscr{V}]]^{AB}\n\\nonumber\\\\&\n+ (D_\\mu[\\mathscr{V}] D_\\nu[\\mathscr{V}])^{AB} \\} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\\n=& \\frac{1}{2} \\mathscr{X}^{\\mu A} \\{ - (D_\\rho[\\mathscr{V}]D_\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] \n\\nonumber\\\\&\n+ D_\\mu[\\mathscr{V}]^{AC} D_\\nu[\\mathscr{V}]^{CB} \\} \\mathscr{X}^{\\nu B} ,\n\\end{align}\nwhere we have used\n\\begin{align}\n[ D_\\mu[\\mathscr{V}], D_\\nu[\\mathscr{V}]]^{AB}\n=& [ D_\\mu[\\mathscr{V}]^{AC}, D_\\nu[\\mathscr{V}]^{CB}] \n\\nonumber\\\\ \n=& - gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] .\n\\end{align}\nThus we obtain\n\\begin{align}\n \\mathscr{L}_{YM} \n=& - \\frac{1}{4} \\mathscr{F}_{\\mu\\nu}[\\mathscr{V40}]^2\n\\nonumber\\\\&\n- \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\&\n- \\frac{1}{2} \\mathscr{X}^{\\mu A} W_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n+ \\frac{1}{2} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]\n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \n ,\n\\end{align}\nwhere we have defined\n\\begin{align}\nW_{\\mu\\nu}^{AB} :=& - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] \n\\nonumber\\\\&\n+ D_\\mu[\\mathscr{V}]^{AC} D_\\nu[\\mathscr{V}]^{CB} . \n\\label{W2}\n\\end{align}\n\nIn the usual background field method, the $\\mathcal{O}(\\mathscr{X})$ term is eliminated by requiring that the back ground field $\\mathscr{V}$ satisfies the equation of motion $D_\\mu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}]=0$:\n\\begin{align}\n& \\frac{1}{2} \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu)\n\\nonumber\\\\\n=& - \\frac{1}{2} ( D_\\mu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot \\mathscr{X}_\\mu) = 0 .\n\\end{align}\nIn our framework, $\\mathscr{V}$ do not necessarily satisfy the equation of motion. Nevertheless, the $\\mathcal{O}(\\mathscr{X})$ term vanishes from the defining equations which specify the decomposition. For $G=SU(2)$, $D_\\mu[\\mathscr{V}]\\bm{n}=0$ and $\\mathscr{X}_\\mu \\cdot \\bm{n}=0$ lead to\n\\begin{align}\n& \\mathscr{F}^{\\mu\\nu}[\\mathscr{V}] \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu) \n= G^{\\mu\\nu} \\bm{n} \\cdot (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu)\n\\nonumber\\\\\n=& G^{\\mu\\nu} [\\partial_\\mu (\\mathscr{X}_\\nu \\cdot \\bm{n}) - \\mathscr{X}_\\nu \\cdot D_\\mu[\\mathscr{V}]\\bm{n}] = 0\n .\n\\end{align}\nIn order for the reformulated theory written in terms of new variables to be equivalent to the original QCD, we must impose the reduction condition \\cite{KMS06}:\n\\begin{equation}\n D_\\mu[\\mathscr{V}] \\mathscr{X}^\\mu = 0 .\n \\label{reduction-cond}\n\\end{equation}\nThis eliminate the last term of $W_{\\mu\\nu}^{AB}$ in (\\ref{W2}). \n\nMoreover, the $\\mathcal{O}(\\mathscr{X}^3)$ term is absent, i.e., \n\\begin{equation}\n\\frac{1}{2} (D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu) \\cdot ig[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ] = 0 \n ,\n\\end{equation}\nsince $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$ is orthogonal to $[ \\mathscr{X}^\\mu , \\mathscr{X}^\\nu ]$. See \\cite{KMS06,KSM08,Kondo08}.\n\n \nThus, the Yang-Mills Lagrangian density reads\n\\begin{align}\n \\mathscr{L}_{YM} \n=& -\\frac{1}{4} \\mathscr{F}_{\\mu\\nu}^A[\\mathscr{V}]^2\n- \\frac{1}{2} \\mathscr{X}^{\\mu A} Q_{\\mu\\nu}^{AB} \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2\n ,\n\\end{align}\nwhere we have defined \n\\begin{equation}\nQ_{\\mu\\nu}^{AB} := - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2gf^{ABC} \\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}] .\n\\end{equation}\n\nFor $G= SU(2)$, the $\\mathcal{O}(\\mathscr{X}^3)$ term is absent, because $\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}]$ and $-ig [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu]$ \nare parallel to $\\bm{n}$ (this is also the case for the sum\n$\\mathscr{F}_{\\mu\\nu}[\\mathscr{V}] -i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu]$),\nwhile $D_\\mu[\\mathscr{V}] \\mathscr{X}_\\nu - D_\\nu[\\mathscr{V}] \\mathscr{X}_\\mu$\n is orthogonal to $\\bm{n}$ (which follows from the fact $\\bm{n} \\cdot \\mathscr{X}_\\mu=0$). \n For $G=SU(2)$, therefore, we have\n\\begin{equation}\n\\mathscr{F}_{\\mu\\nu}^{C}[\\mathscr{V}]=n^C G_{\\mu\\nu}[\\mathscr{V}] .\n\\end{equation}\nThen the $SU(2)$ gluon part is rewritten into \n\\begin{align}\n \\mathscr{L}_{YM} \n=& -\\frac{1}{4} (G_{\\mu\\nu}[\\mathscr{V}])^2\n- \\frac{1}{2} \\mathscr{X}^{\\mu A} Q_{\\mu\\nu}^{AB}[\\mathscr{V}] \\mathscr{X}^{\\nu B} \n\\nonumber\\\\&\n- \\frac14 (i g [ \\mathscr{X}_\\mu , \\mathscr{X}_\\nu ])^2 \n,\n\\end{align}\nwhere\n\\begin{equation}\nQ_{\\mu\\nu}^{AB}[\\mathscr{V}] = - (D_\\rho[\\mathscr{V}]D^\\rho[\\mathscr{V}])^{AB} g_{\\mu\\nu} \n+ 2g\\epsilon^{ABC} n^C G_{\\mu\\nu}[\\mathscr{V}] .\n\\label{W}\n\\end{equation}\n\n\n\n\n\\section{Coefficients in the effective potential}\\label{app:coefficient}\n\nWe expand $\\hat V_{T,\\hat k}$ defined by\n\\begin{align}\n \\hat V_{T,\\hat k}\n =& \\hat V_{W} + 4 \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{(2\\pi)^2} \\{ \\ln (1- 2 e^{-\\hat k_{T}} \\cos \\varphi + e^{-2\\hat k_{T}}) \n\\nonumber\\\\&\n- \\ln (1- 2 e^{-\\hat p} \\cos \\varphi + e^{-2\\hat p}) \\} \n ,\n\\end{align}\nin power series of $\\tilde \\varphi$ by using the expansion \n$\n - \\cos \\varphi = - \\cos (\\pi+\\tilde \\varphi) = \\cos ( \\tilde \\varphi)\n = 1 - \\frac12 \\tilde \\varphi^2 + \\frac{1}{24} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n$\nas follows.\n\\begin{widetext}\n\\begin{align}\n \\hat V_{T,\\hat k}\n =& \\hat V_{W} + \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln \\left[ 1+ 2 e^{-\\hat k_{T}} - e^{-\\hat k_{T}} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat k_{T}} \\tilde \\varphi^4 + e^{-2\\hat k_{T}} + O(\\tilde \\varphi^6) \\right]\n \\nonumber\\\\\n & - \\ln \\left[ 1+ 2 e^{-\\hat p} - e^{-\\hat p} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat p} \\tilde \\varphi^4 + e^{-2\\hat p} + O(\\tilde \\varphi^6) \\right] \\Big\\} \n \\nonumber\\\\\n =& \\hat V_{W} + \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln \\left[ (1+ e^{-\\hat k_{T}} )^2 - e^{-\\hat k_{T}} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat k_{T}} \\tilde \\varphi^4 + O(\\tilde \\varphi^6) \\right]\n \\nonumber\\\\\n & - \\ln \\left[ (1+ e^{-\\hat p})^2 - e^{-\\hat p} \\tilde \\varphi^2 + \\frac{1}{12} e^{-\\hat p} \\tilde \\varphi^4 + O(\\tilde \\varphi^6) \\right] \\Big\\} \n \\nonumber\\\\\n =& \\hat V_{W} + \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln (1+ e^{-\\hat k_{T}} )^2 + \\ln \\left[ 1 - \\frac{e^{-\\hat k_{T}}}{(1+ e^{-\\hat k_{T}} )^2} \\tilde \\varphi^2 + \\frac{e^{-\\hat k_{T}}}{12(1+ e^{-\\hat k_{T}} )^2} \\tilde \\varphi^4 + O(\\tilde \\varphi^6) \\right] \n \\nonumber\\\\\n & - \\ln (1+ e^{-\\hat p})^2 - \\ln \\left[ 1 - \\frac{e^{-\\hat p}}{(1+ e^{-\\hat p})^2} \\tilde \\varphi^2 + \\frac{e^{-\\hat p}}{12(1+ e^{-\\hat p})^2} \\tilde \\varphi^4 + O(\\tilde \\varphi^6) \\right] \\Big\\} \n .\n\\end{align}\nBy using \n$\\log(1+x)=x-\\frac12 x^2+O(x^3)$, therefore, $\\hat V_{T,\\hat k}$ has the polynomial expansion:\n\\begin{align}\n \\hat V_{T,\\hat k}\n = A_{0,k} + \\frac{A_{2,k}}{2} \\tilde \\varphi^2 + \\frac{A_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n ,\n\\end{align}\nwhere the coefficient is given by the integral form:\n\\begin{align}\n\\frac{A_{2,k}}{2}\n=& - \\frac16 + \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \n\\left[ \\frac{e^{-\\hat p}}{(1+ e^{-\\hat p})^2} - \\frac{e^{-\\hat k_{T}}}{(1+ e^{-\\hat k_{T}} )^2} \\right]\n ,\n\\nonumber\\\\\n \\frac{A_{4,k}}{4!} =& \\frac{1}{12\\pi^2} - \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \n\\left[ \\frac{-6e^{-2\\hat p}+e^{-\\hat p}(1+ e^{-\\hat p})^2}{12(1+ e^{-\\hat p})^4} - \\frac{-6e^{-2\\hat k_{T}}+e^{-\\hat k_{T}}(1+ e^{-\\hat k_{T}} )^2}{12(1+ e^{-\\hat k_{T}} )^4} \\right]\n ,\n\\nonumber\\\\\nA_{0,k} =& \\int_{0}^{\\hat k_{T}} \\frac{d\\hat p \\hat p^2}{\\pi^2} \\Big\\{ \\ln (1+ e^{-\\hat k_{T}} )^2 - \\ln (1+ e^{-\\hat p})^2 \\Big\\}\n .\n\\end{align}\nThe integration can be performed analytically and the coefficient has the closed form:\n\\begin{align}\n\\frac{A_{2,k}}{2}\n=& - \\frac16 + \\frac{1}{\\pi^2} \\left[ -\\frac{e^s s^3}{3 \\left(1+e^s\\right)^2}+\\frac{e^s\n s^2}{1+e^s}-2 \\log \\left(1+e^s\\right) s-2\n \\text{Li}_2\\left(-e^s\\right)-\\frac{\\pi ^2}{6} \\right] \\Big|_{s=\\hat k_{T}}\n ,\n\\nonumber\\\\\n \\frac{A_{4,k}}{4!} =& \\frac{1}{12\\pi^2} \n + \\frac{e^{2 s} \\left(-2 s^3+\\left(s^2+6\\right) \\cosh (s)\n s+6 s+3 \\left(s^2-2\\right) \\sinh (s)-3 \\sinh (2\n s)\\right)}{18 \\left(1+e^s\\right)^4 \\pi ^2} \\Big|_{s=\\hat k_{T}}\n ,\n\\end{align}\n\\end{widetext}\nwhere ${\\rm Li}_n(z)={\\rm PolyLog}[n,z]$ is the polylogarithm function, and in particular, the dilogarithm satisfies ${\\rm Li}_2(z) = \\int_{z}^{0} \\frac{\\log(1-t)}{t} dt$ which is known as the Spence integral.\n\nNote that the function $\\frac{e^{-\\hat x}}{(1+ e^{-\\hat x})^2}$ is monotonically decreasing in $x$ and hence the second term in $A_{2,k}$ is positive (non-negative). \nThe coefficient $A_{2,k}$ is negative and monotonically increasing in $k$ and approaches zero for $k \\rightarrow \\infty$. \n\\begin{equation}\n\\frac{A_{2,k}}{2} = -\\frac16, \\quad\n-\\frac16 \\le \\frac{A_{2,k}}{2} < 0 \\quad {\\rm for} \\quad k \\in [0, \\infty) ,\n\\end{equation}\nor\n\\begin{equation}\n-\\frac13 \\le \\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\hat V_{T,\\hat k} \\Big|_{\\tilde\\varphi=0} < 0 \\quad {\\rm for} \\quad k \\in [0, \\infty).\n\\end{equation}\nThis is because\n\\begin{align}\n \\int_{0}^{\\hat k_{T}} d\\hat p \\hat p^2 \n \\frac{e^{-\\hat p}}{(1+ e^{-\\hat p})^2} \n&\\rightarrow \\int_{0}^{\\infty} d\\hat p \\hat p^2 \n \\frac{e^{-\\hat p}}{(1+ e^{-\\hat p})^2} \n= \\frac16 \\pi^2 \n ,\n\\\\ \n \\int_{0}^{\\hat k_{T}} d\\hat p \\hat p^2 \n \\frac{e^{-\\hat k_{T}}}{(1+ e^{-\\hat k_{T}} )^2} \n&= \\frac13 \\hat k_{T}^3 \\frac{e^{-\\hat k_{T}}}{(1+ e^{-\\hat k_{T}} )^2} \\rightarrow 0\n.\n\\end{align}\n\n\n\nThe coefficient $A_{4,k}$ is positive and approaches $0$ for $k \\rightarrow \\infty$, although $A_{4,k}$ is not monotonically decreasing in $k$. \n\\begin{equation}\n\\frac{A_{4,0}}{4!}= \\frac{1}{12\\pi^2} , \\quad\n\\frac{A_{4,k}}{4!} > 0 \\quad {\\rm for} \\quad k \\in [0, \\infty).\n\\end{equation}\nThis is because\n\\begin{align}\n & \\int_{0}^{\\hat k_{T}} d\\hat p \\hat p^2 \n\\left[ \\frac{-6e^{-2\\hat p}+e^{-\\hat p}(1+ e^{-\\hat p})^2}{12(1+ e^{-\\hat p})^4} \\right]\n \\rightarrow \\frac{1}{12} \n ,\n\\nonumber\\\\\n & \\int_{0}^{\\hat k_{T}} d\\hat p \\hat p^2 \n\\left[ \\frac{-6e^{-2\\hat k_{T}}+e^{-\\hat k_{T}}(1+ e^{-\\hat k_{T}} )^2}{12(1+ e^{-\\hat k_{T}} )^4} \\right]\n\\\\&\n= \\frac13 \\hat k_{T}^3 \\left[ \\frac{-6e^{-2\\hat k_{T}}+e^{-\\hat k_{T}}(1+ e^{-\\hat k_{T}} )^2}{12(1+ e^{-\\hat k_{T}} )^4} \\right] \\rightarrow 0 \n.\n\\end{align}\n\n\n\\section{Flow equation for the coefficient}\\label{app:flow-equation}\n\nSuppose that $\\Delta \\hat V_{\\hat k}$ is of the form:\n\\begin{equation}\n \\Delta \\hat V_{\\hat k} \n= a_{0,k} + a_{1,k} \\tilde\\varphi + \\frac{a_{2,k}}{2} \\tilde \\varphi^2 \n+ \\frac{a_{3,k}}{3!} \\tilde \\varphi^3 \n+ \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 + O(\\tilde \\varphi^6)\n .\n\\end{equation}\nThe left-hand side of the flow equation reads\n\\begin{align}\n \\partial_{\\hat k} \\Delta \\hat V_{\\hat k} \n=& \\partial_{\\hat k} a_{0,k} + \\partial_{\\hat k} a_{1,k} \\tilde \\varphi + \\partial_{\\hat k} \\frac{a_{2,k}}{2} \\tilde \\varphi^2 \n+ \\partial_{\\hat k} \\frac{a_{3,k}}{3!} \\tilde \\varphi^3 \n\\nonumber\\\\&\n+ \\partial_{\\hat k} \\frac{a_{4,k}}{4!} \\tilde \\varphi^4 \n+ O(\\tilde \\varphi^6)\n .\n\\end{align}\nThe flow equation $\\partial_{\\hat k} a_{n,k}$ for the coefficient of $\\tilde\\varphi^n$ is extracted by differentiating both sides of the flow equation $n$ times and by putting $\\tilde\\varphi=0$.\nThe left-hand side is \n\\begin{align}\n\\partial_{\\hat k} a_{n,k} = \\frac{\\partial^n}{\\partial \\tilde\\varphi^n} \\partial_{\\hat k} \\Delta \\hat V_{\\hat k} \\Big|_{\\tilde\\varphi=0} \n .\n\\end{align}\nDefine \n\\begin{equation}\n f(\\varphi):= \\frac{4\\pi\\alpha_k}{\\hat k^2} (\\hat V_{T,\\hat k} + \\Delta \\hat V_{\\hat k} ) .\n\\end{equation}\nThe right-hand sides of the flow equation $\\partial_{\\hat k} a_{n,k}$ are calculated from\n\\begin{align}\n \\frac{\\partial}{\\partial \\tilde\\varphi} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \n=& \\frac{-\\partial_\\varphi^3 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2}\n ,\n\\end{align}\n\\begin{align}\n \\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \n=& \\frac{-\\partial_\\varphi^4 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2 \\frac{-[\\partial_\\varphi^3 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3}\n ,\n\\end{align}\n\\begin{align}\n \\frac{\\partial^3}{\\partial \\tilde\\varphi^3} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \n=& \\frac{-\\partial_\\varphi^5 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2 \\frac{-3\\partial_\\varphi^4 f(\\varphi) \\partial_\\varphi^3 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^3} \n\\nonumber\\\\&\n+ 6 \\frac{-[\\partial_\\varphi^3 f(\\varphi)]^3}{[1+\\partial_\\varphi^2 f(\\varphi)]^4}\n ,\n\\end{align}\n\\begin{align} \n \\frac{\\partial^4}{\\partial \\tilde\\varphi^4} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \n=& \\frac{-\\partial_\\varphi^6 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \n\\nonumber\\\\&\n-2 \\frac{-4\\partial_\\varphi^5 f(\\varphi) \\partial_\\varphi^3 f(\\varphi)\n-3[\\partial_\\varphi^4 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3} \n\\nonumber\\\\&\n+ 6 \\frac{-6\\partial_\\varphi^4 f(\\varphi)[\\partial_\\varphi^3 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^4}\n\\nonumber\\\\&\n-24 \\frac{- [\\partial_\\varphi^3 f(\\varphi)]^4}{[1+\\partial_\\varphi^2 f(\\varphi)]^5}\n , \\cdots\n\\end{align}\n\nIf $f$ is an even polynomial in $\\tilde\\varphi$, then the flow equation is simplified:\n\\begin{align}\n\\partial_{\\hat k} a_{1,k} &\\simeq \n\\frac{\\partial}{\\partial \\tilde\\varphi} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \\Big|_{\\tilde\\varphi=0} \n= 0\n ,\n\\\\ \n\\partial_{\\hat k} a_{2,k} &\\simeq \n\\frac{\\partial^2}{\\partial \\tilde\\varphi^2} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\&\n= \\frac{-\\partial_\\varphi^4 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \\Big|_{\\tilde\\varphi=0} \n ,\n\\\\ \n \\partial_{\\hat k} a_{3,k} &\\simeq\n \\frac{\\partial^3}{\\partial \\tilde\\varphi^3} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \\Big|_{\\tilde\\varphi=0} \n= 0\n ,\n\\\\ \n\\partial_{\\hat k} a_{4,k} &\\simeq\n \\frac{\\partial^4}{\\partial \\tilde\\varphi^4} \\left[ \\frac{1}{1+\\partial_\\varphi^2 f(\\varphi)} \\right] \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\ \n&= \\frac{-\\partial_\\varphi^6 f(\\varphi)}{[1+\\partial_\\varphi^2 f(\\varphi)]^2} \\Big|_{\\tilde\\varphi=0} \n\\nonumber\\\\&\n+6 \\frac{[\\partial_\\varphi^4 f(\\varphi)]^2}{[1+\\partial_\\varphi^2 f(\\varphi)]^3} \\Big|_{\\tilde\\varphi=0} \n , ...\n\\end{align}\nTherefore, with an initial condition, $a_{1,k} =0=a_{3,k}$ at $k=\\Lambda$, the flow equations in the above\n\\begin{equation}\n\\partial_{\\hat k} a_{1,k} =0, \\quad \\partial_{\\hat k} a_{3,k}=0 .\n\\end{equation}\nguarantee the solution \n\\begin{equation}\n a_{1,k} \\equiv 0, \\quad a_{3,k} \\equiv 0 \\quad (0 \\le k \\le \\Lambda) .\n\\end{equation}\n\n\n\n\n\n\\end{appendix}\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum resource theory is an essential tool to understand the nature of physical systems from the perspective of nonlocality and provides an operational interpretation of various quantum effects \\cite{QCbook}. Quantum coherence, superposition, entanglement and quantum correlations beyond entanglement are few notable resources which can prove to be advantageous over the classical algorithms. Understanding these resources in physical systems continue to be a formidable and interesting task even today. The quantum information processing is usually studied within the framework of entanglement-versus-separability . The entanglement is considered to be the only early version of nonlocal aspects of quantum systems \\cite{EPR,Schrodinger1,Schrodinger2} and is demonstrated through the violation of Bell inequality \\cite{Bell}. The seminal work of Werner and Almedia et al. reveals that the entanglement is an incomplete manifestation of nonlocality implying that quantum correlations cannot only be limited to entanglement \\cite{Werner,Almedia} and the separable states can also come in handy in the implementation of some specific quantum tasks. To study this quantum correlation beyond entanglement, Ollivier and W. H. Zurek introduced a measure called quantum discord which captures the nonlocality of separable (unentangled) states\\cite{Ollivier}. \n \n \n Other than entanglement, many other types of quantum correlations have been discovered in recent years \\cite{Ollivier,Luo2011,GoD}. As a result, a number of quantum correlation concepts have emerged with each being motivated by specific applications in quantum information science competing for recognition as the most accurate quantumness metric. For example, the skew-information and quantum Fisher information play an important role in parameter estimation and bring out the limitations on the variance of the observable. As stated, each quantum correlation captures different quantumness in quantum systems due to their distinct type of measurement techniques. Among them, skew information based quantum correlation measures such as local quantum uncertainty (LQU) and uncertainty-induced nonlocality (UIN) are widely used as a quantumness measure to characterize quantum\/physical systems. Girolami et al. \\cite{GirolamiPRL2013} have introduced a realizable discord-like nonclassical correlation measure for bipartite systems known as local quantum uncertainty. Further, this measure quantifies the uncertainty in a quantum state arising due to its noncommutativity with the measured local observables. The LQU is defined in terms of the skew information. The UIN provides an alternative to capture the nonlocal effects via local measurements. Both LQU and UIN possess the closed formula for qubit\u2013qudit ($2\\otimes n$ dimensional) systems. Apart from the correlation quantifiers, the quantum coherence is a faithful nonclassicality indicator even for a single qubit system and is a consequence of superposition of quantum states. Like quantum correlations, coherence is also considered to be critical in the development of quantum technology \\cite{Giovannetti,Demkowicz,Lloyd,Huelga,Lambert,Aberg,Lostaglio,Buffoni}. The quantification of quantum coherence has been recently developed by Baumgratz et al \\cite{Bau}. Recently, the coherence measure based on the $l_1$-norm as a valid measure has been introduced and characterized in different contexts \\cite{Rana,Chen,Wang}. Further, the interplay between the quantum coherence measure and correlations is also studied \\cite{Tancoh,Hucoh1}.\n \nIn relativistic field theory, exploiting the formulation of the detector models, two well-known effects such as Unruh and Hawking effects have been studied. In this formulation, idealized particles are considered as detectors similar to a two-level system following the classical worldline and whose internal states are coupled to the field. In recent times, Unruh-deWitt (UdW) detector model has been devised as a computational element. The intervention of the environment on the system may cause decoherence and degrades the unique features of quantum system. Assuming that the quantum fluctuations of the background field plays the role of an environment, the Unruh temperature $T_U = a\/2\\pi $($a$ is acceleration) plays the role of decoherence and destroys the properties of coupled UdW detectors. Here, the pair of detectors is considered as an open system and its dynamics is governed by the master equation \\cite{Op1}. Recently, different approaches have been employed to study the quantum effects like entanglement, quantum correlations and coherence \\cite{Entg6,Entg8,Entg9,Huang1,Huang2,Huang3}. Moreover, the quantum correlation and entropic uncertainty relation are also investigated in a multipartite scenario\\cite{multiparty1,multiparty2}. \n\nIn this article, we focus on the generation and retention of quantumness in two-UdW detectors interacting with the scalar field. To quantify the quantumness of the detectors, we use the entanglement, local quantum uncertainty, uncertainty-induced nonlocality and $l_1$-norm of coherence. We show that the entanglement decreases monotonically and vanishes at a finite temperature. The coherence and LQU exhibit a remarkable difference namely, revival of quantum correlation after vanishing at a finite time. Further, we study the dynamics of a product and an incoherent state and the investigation reveals that the dynamics generates the correlation and coherence from the product and incoherent state respectively. \n\n\nThe paper is structured as follows. In Sec. \\ref{corr}, we review the quantifier of quantum correlation and coherence measures . In Sec. \\ref{Sec3}, we introduce the physical model under our consideration and thermal state of two-UdW detectors. The investigations on quantum correlation and coherence are presented in Sec. \\ref{Res}. Finally, the conclusions are drawn in Sec. \\ref{conc}.\n \n\\section{Quantum Correlation Measures}\n\\label{corr}\n\nIn this section, we review some of the popular quantum correlation measures to be investigated in this article. For this purpose, we consider an arbitrary bipartite state $\\rho$ shared by the subsystems $a$ and $b$ in the separable Hilbert space $\\mathcal{H}^a \\otimes \\mathcal{H}^b$.\n\n\\subsection{Entanglement}\nEntanglement is a nonnegative real function of a state $\\rho$ which cannot increase under the local operations and classical communication (LOCC) and is zero for unentangled states. To quantify the amount of entanglement associated with the two-qubit physical system $\\rho$ under consideration, various measures have been introduced. The concurrence is the most popular measure of entanglement for mixed bipartite systems and is defined as \\cite{Hill}\n\\begin{align}\nC(\\rho)= \\text{max}\\{0,~ \\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4\\},\n\\end{align}\nwhere $\\lambda_i$ are eigenvalues of the matrix $R=\\sqrt{\\sqrt{\\rho}\\tilde{\\rho}\\sqrt{\\rho} }$ $(\\lambda_1\\geq \\lambda_2\\geq \\lambda_3\\geq \\lambda_4)$. Here, $\\tilde{\\rho}=(\\sigma_y \\otimes \\sigma_y) \\rho^* (\\sigma_y \\otimes \\sigma_y)$ is a spin flipped matrix with * denoting the complex conjugate in the computational basis. The function $C(\\rho)$ ranges from 0 to 1 and its minimal and maximal values correspond to separable and entangled states respectively.\n\n \\subsection{Local quantum uncertainty}\n\nThe local quantum uncertainty (LQU) is a more reliable measure of the quantumness of bipartite states which goes beyond entanglement. In recent times, researchers have paid wide attention to this discord-like measure. This is essentially due to its easy computation and the fact that it enjoys all necessary properties of being a faithful measure of quantum correlation. It is shown that LQU is non-zero for the separable state even in the absence of entanglement. For a bipartite state $\\rho$, the LQU is defined as the minimal skew information attainable with a single local measurement. Mathematically, it is defined as \\cite{GirolamiPRL2013},\n\\begin{align}\n\\mathcal{Q}_{\\mathcal{I}}(\\rho)=~^{\\text{min}}_{H^{a}} ~~\\mathcal{I}(\\rho, H^a\\otimes\\mathds{1}^b),\n\\end{align}\nwhere $H^a$ is anylocal observable on the subsystem $a$, $\\mathds{1}^b$ is the $2\\times 2$ identity operator acting on the system $b$ and\n\\begin{align}\n\\mathcal{I}(\\rho)=-\\frac{1}{2}\\text{Tr}\\left( [\\sqrt{\\rho}, H^a\\otimes\\mathds{1}^b]^2 \\right)\n\\end{align}\nis the skew information which provides an analytical tool to quantify the information content in the state $\\rho$ with respect to the observable $H^a$, $[\\cdot, \\cdot]$ is the commutator operator. Here, the information content of $\\rho$ about $H^a$ is quantified by how much the measurement of $H^a$ on the state is uncertain. The measurement outcome is certain if only if the state is an eigenvector of $H^a$. On the other hand, if it is a mixture of eigenvectors of $H^a$, the uncertainty is only due to the imperfect knowledge of the state. For pure bipartite states, the local quantum uncertainty reduces to the linear entropy of entanglement and vanishes for classically correlated states. For any $2\\times n$ dimensional bipartite system, the closed formula of LQU is computed as\n\\begin{align}\n\\mathcal{U}(\\rho)=~ 1- \\text{max}\\{\\omega_1,\\omega_2,\\omega_3 \\}.\n\\end{align}\nHere, $\\omega_i$ are the eigenvalues of matrix $W$ and the matrix elements are defined as\n\\begin{align}\n\\omega_{ij}=\\text{Tr}[\\sqrt{\\rho}(\\sigma_i^a\\otimes\\mathds{1}^b)\\sqrt{\\rho}(\\sigma_j^a\\otimes\\mathds{1}^b)],~~~~~\\text{with}~~~ i,j=1,2,3,\n\\end{align}\nwhere $\\sigma_i$ represents the Pauli spin matrices.\n\n\\subsection{Uncertainty-induced nonlocality}\n\nNext, we employ another important skew information based measure ,namely uncertainty-induced nonlocality (UIN) and it can be considered as an updated version of measurement-induced nonlocality (MIN) \\cite{Luo2011}. It is defined as \\cite{UIN}\n\\begin{align}\n\\mathcal{Q}(\\rho)=~^{\\text{max}}_{H^{a}} ~~\\mathcal{I}(\\rho, H^a\\otimes\\mathds{1}^b).\n\\end{align}\nThis measure also satisfies all the necessary axioms of a valid measure of bipartite quantum correlation and is reduced to entanglement monotone for $2\\times n$ dimensional pure states.\nIt also possesses a closed formula \\cite{UIN}.\n\\subsection{$C_{l_1}$-norm coherence}\nQuantum coherence is an important resource for information processing and a manifestation of the quantum superposition principle. Recently, its quantification has been formulated and a set of conditions to be satisfied by any proper measure of coherence has been identified \\cite{Rana,Chen,Wang}. The distance based quantum coherence measure\n\\begin{align}\n C(\\rho)=~^{\\text{min}}_{\\delta\\in \\mathcal{I}} ~d(\\rho, \\delta)\n\\end{align}\nis the minimal distance between $\\rho$ and a set of incoherent quantum states $\\delta\\in \\mathcal{I}$. \n Recently, a faithful measure of coherence using $l_1$-norm has been identified and is defined as the sum of absolute values of all off-diagonal elements of $\\rho$ \\cite{Bau}\n\\begin{align}\nC_{{l_1}}(\\rho)=\\sum_{i\\neq j}|\\rho_{ij}|.\n\\end{align}\nThe above definition is a basis dependent measure and is crucial in the identification of phase transition in physical models \\cite{LPT1,LPT2,LPT3}.\n\n\n\\section{Physical Model}\\label{Sec3}\nTo understand the behaviors of quantum correlations in two accelerating UdW detectors in $3+1$-dimensional Minkowski space-time \\cite{Entg6}, we first introduce the Hamiltonian of the physical system under consideration. Considering the detector as a two-level atomic system and a valid qubit system, we consider the two accelerating UdW detectors in $3+1$-dimensional Minkowski space time as an open quantum system for our investigation. The total Hamiltonian of the combined system becomes \n\\begin{align}\nH=\\frac{\\omega}{2}\\Sigma_3+H_\\Phi+\\mu H_I \\label{eq1}\n\\end{align}\nwhere $\\omega$ is the energy level spacing of the atom and $\\Sigma_3$ is one of the symmetrized bipartite operators. $\\Sigma_i\\equiv\\sigma_i^a\\otimes\\mathbf{1}^b+\\mathbf{1}^a\\otimes\\sigma_i^b$ is defined by Pauli matrices $\\{\\sigma^{(\\alpha)}_i|i=1,2,3\\}$ with superscripts $\\{\\alpha=a, b\\}$ labelling distinct atoms. $H_\\Phi$ is the Hamiltonian of free massless scalar fields $\\Phi(t,\\mathbf{x})$ satisfying standard Klein-Gordon relativistic equation. The interaction Hamiltonian between the atoms and the fluctuating field bath in a dipole form can be written as $H_I= (\\sigma_2^{(a)}\\otimes\\mathbf{1}^{(b)})\\Phi(t,\\mathbf{ x}_1)+(\\mathbf{1}^{(a)}\\otimes\\sigma_2^{(b)})\\Phi(t,\\mathbf{ x}_2)$ \\cite{Op2}.\n\nHere, we consider the coupling between the detectors and the environment ($\\mu \\leq 1$) and the initial state of the composite system is $\\rho_{tot}(0)=\\rho_{ab}(0)\\otimes |0\\rangle \\langle 0|$, with $\\rho_{ab}(0)$ being the initial state of the detectors and $|0\\rangle$ is the field vacuum (environment). The time evolution of $\\rho_{tot}(0)$ is unitary governed by von Neumann equation $\\dot{\\rho}_{tot}(\\tau)=-i[H,\\rho_{tot}(\\tau)]$, where $\\tau$ is the proper time of the atom. Due to the environment decoherence or dissipation on the system $\\rho_{ab}$, the density matrix is governed by a Lindblad master equation and evolves non-unitarily in the following form \\cite{Op3,Op4}\n\\begin{align}\n\\frac{\\partial\\rho_{ab}(t)}{\\partial t}=-i[H_{\\tiny\\mbox{eff}},\\rho_{ab}(t) ]+\\mathcal{L}\\left[\\rho_{ab}(t)\\right] \n\\label{eq2}\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{L}\\left[\\rho\\right]=\\sum_{\\substack{i,j=1,2,3\\\\ \\alpha,\\;\\beta=a,b}}\\frac{C_{ij}}{2}\\big[2\\sigma_j^{(\\beta)}\\rho_{ab}\\sigma_i^{(\\alpha)}-\\{\\sigma_i^{(\\alpha)}\\sigma_j^{(\\beta)},\\rho_{ab}\\}\\big] \\label{eq3}\n\\end{align}\ndescribes the evolution due to the interaction between the detectors and external field. \n\nAfter introducing the Wightman function of scalar field $G^{+}\\left(x, x^{\\prime}\\right)=\\left\\langle 0\\left|\\Phi(x) \\Phi\\left({x}^{\\prime}\\right)\\right| 0\\right\\rangle$, its Fourier transform\n\\begin{align}\n\\mathcal{G}(\\lambda)=\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}G^+(\\tau)= \\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\left\\langle\\Phi(\\tau)\\Phi(0)\\right\\rangle \n\\label{eq4}\n\\end{align}\ndetermines the coefficients $C_{ij}$ by a decomposition\n\\begin{align}\nC_{ij}=\\frac{\\gamma_+}{2}\\delta_{ij}-i\\frac{\\gamma_-}{2}\\epsilon_{ijk} \\delta_{3,k}+\\gamma_0\\delta_{3,i}\\delta_{3,j} \\label{eq5}\n\\end{align}\nwhere\n\\begin{align}\n\\gamma_\\pm= \\mathcal{G}(\\omega)\\pm \\mathcal{G}(-\\omega),~~~\\gamma_0=\\mathcal{G}(0)-\\gamma_+\/2. \\label{eq6}\n\\end{align}\nMoreover, the interaction with external scalar field would also induce a Lamb shift contribution for the effective Hamiltonian of the detector $H_{\\mbox{\\tiny eff}}=\\frac{1}{2}\\tilde{\\omega}\\sigma_3$ in terms of a renormalized frequency $\\tilde{\\omega}=\\omega+i[\\mathcal{K}(-\\omega)-\\mathcal{K}(\\omega)]$ where $\\mathcal{K}(\\lambda)=\\frac{1}{i\\pi}\\mbox{P}\\int_{-\\infty}^{\\infty}d\\omega\\frac{\\mathcal{G}(\\omega)}{\\omega-\\lambda}$ is a Hilbert transform of Wightman function. \n\nFollowing a trajectory of the accelerating detectors, one can find that the field Wightman function fulfills the Kubo-Martin-Schwinger (KMS) condition, i.e., $G^{+}(\\tau)=G^{+}(\\tau+i \\beta)$, where $\\beta\\equiv1\/T_U=2\\pi\/a$. Translating it into frequency space, one finds that\n\\begin{align}\n\\mathcal{G}(\\lambda)=e^{\\beta\\omega}\\mathcal{G}(-\\lambda). \\label{eq7}\n\\end{align}\nUsing translation invariance $\\langle 0|\\Phi(x(0)) \\Phi(x(\\tau))| 0\\rangle=\\langle 0|\\Phi(x(-\\tau)) \\Phi(x(0))| 0\\rangle$ and after some algebraic manipulations, we find that eq.(\\ref{eq6}) can be resolved as\n\n\\begin{align}\n\\gamma_+=&\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\langle 0|\\left\\{\\Phi(\\tau),\\Phi(0)\\right\\}|0\\rangle =\\left(1+ e^{-\\beta\\omega}\\right) \\mathcal{G}(\\omega), ~~~~~~~~\\\\\n\\gamma_-=&\\int_{-\\infty}^{\\infty}d\\tau~e^{i\\lambda\\tau}\\langle 0|\\left[\\Phi(\\tau),\\Phi(0)\\right]|0\\rangle =\\left(1- e^{-\\beta\\omega}\\right) \\mathcal{G}(\\omega).\n\\label{eq8} \n\\end{align}\nIt should be be noted that eq. (\\ref{eq8}) holds true for generic interacting fields. Considering the two-atom state in Bloch representation, we compute t he final equilibrium state of two-UdW detectors asymptotically as \n\\begin{align}\n\\rho_{ab}=\\left(\\begin{array}{cccc}\n\\varrho_{11} & 0 & 0 & 0 \\\\\n0 & \\varrho_{22} & \\varrho_{23} & 0 \\\\\n0 & \\varrho_{23} & \\varrho_{22} & 0 \\\\\n0 & 0 & 0 & \\varrho_{44} \n\\end{array}\\right)\\label{eq11}\n\\end{align}\nwhere the matrix elements are \n\\begin{eqnarray}\n \\displaystyle \\varrho_{11} &=&\\frac{(3+\\Delta_0)(\\gamma-1)^{2}}{4\\left(3+\\gamma^{2}\\right)},~~~~~~~~\n\\displaystyle \\varrho_{44}=\\frac{(3+\\Delta_0)(\\gamma+1)^{2}}{4\\left(3+\\gamma^{2}\\right)}, \\nonumber\\\\\n\\displaystyle \\varrho_{22}&=& \\frac{3-\\Delta_0-(\\Delta_0+1) \\gamma^{2}}{4\\left(3+\\gamma^{2}\\right)},~~~~\n\\displaystyle \\varrho_{23}= \\frac{\\Delta_0-\\gamma^{2}}{2\\left(3+\\gamma^{2}\\right)}, \\label{eq12}\n\\end{eqnarray}\nwith the parameter \n\\begin{align}\n\\gamma\\equiv\\gamma_-\/\\gamma_+=\\frac{1- e^{-\\beta\\omega}}{1+ e^{-\\beta\\omega}}=\\tanh(\\beta\\omega \/ 2). \\label{eq9}\n\\end{align}\nIt should be mentioned that the parameter depends solely on the Unruh temperature $T_U$ and characterizes the thermal nature of Unruh effect. Further, the dimensionless parameter $\\Delta_0=\\sum_i\\text{Tr}[\\rho_{ab}(0)\\sigma^{a}_i\\otimes\\sigma^{b}_i]$ provides the choice of initial state and ranges as $-3\\leqslant\\Delta_0\\leqslant1$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig1.pdf}\n\\includegraphics[width=.45\\textwidth]{fig2.pdf}\n\\includegraphics[width=.45\\textwidth]{fig3.pdf}\n\\includegraphics[width=.45\\textwidth]{fig3n.pdf}\n\\caption{Thermal quantum correlation quantified by (a) Entanglement, (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function of Unruh temperature $T_U$ for different initial states for $\\omega=1$.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\\section{Results and Discussions}\n\\label{Res}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig5.pdf}\n\\includegraphics[width=.45\\textwidth]{fig6.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4n.pdf}\n\\caption{Behaviors of quantum correlation measures (a) Entanglement (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UdW detector as a function of Unruh temperature $T_U$ for $\\Delta_0=0$.}\n\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig5a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig6a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4a.pdf}\n\\includegraphics[width=.45\\textwidth]{fig4an.pdf}\n\\caption{Behaviors of quantum correlation measures (a) Entanglement (b) $l_1$ norm of coherence, (c) LQU and (d) UIN of UDW detector as a function of Unruh temperature $T_U$ for different initial states for $\\Delta_0=0.5$.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\nWe now study the dynamics of quantum correlations measures quantified in terms of entanglement, $C_{l_1}$-norm of coherence , local quantum uncertainty and uncertainity induced nonlocality. \n\nIn general, Unruh effect is recognized as an environment decoherence. Hence, we study the quantum correlations as a function of Unruh temperature $T_U$. In Fig. (1), we plot the quantum correlation measures as a function of Unruh temperature for different choices of initial conditions $\\Delta_0$ with a fixed value of $\\omega$. For a given initial state, the entanglement is maximum at $T=0$. As temperature increases, the concurrence is a monotonically decreasing function of Unruh temperature $T_U$ and abruptly vanishes . The entanglement completely vanishes for $\\Delta_0=1$. Further, one can observe that the increase of $\\Delta_0$ also reduces the quantum of entanglement between the detectors. In order to compare the entanglement with other correlation measures such as coherence, LQU and UIN, we plot these measures in Fig (1b \\& c) as a function of Unruh temperature. As Unruh temperature $T_U$ increases, the quantum correlation measures decrease initially and reach a dark point (zero). While increasing the temperature $T_U$ further, both LQU and coherence start to grow from the dark point and then correlations increase with the Unruh temperature $T_U$ unconventionally. It is quite obvious that both the measures are non-monotonic functions of $T_U$. In general, if we immerse any static two-qubit in a thermal bath, the quantumness of the system degrades with the temperature of the thermal bath. But here, on the contrary, we observe the revival of quantum correlations and coherence due to the acceleration of the qubits. In addition, the dark point is shifted towards higher temperatures with the increase of $\\Delta_0$. Similar to entanglement, UIN decreases from the maximum values with the increase of temperature and saturates at a nonzero constant value. A similar observation is noticed using trace distance correlation measure \\cite{Huang1}.\n\n We observe the nonlocal features between the detectors even in the absence of entanglement which are captured through the $C_{l_1}$- norm of coherence and skew information based measures. The entanglement fails to completely quantify the nonlocality of the UdW detector for $\\Delta_0=1$. On the other hand, the correlation measures beyond entanglement are encapsulated in UdW detector even in the absence of entanglement. \n\nNext, we analyze the role of energy spacing $\\omega$ on the quantum correlation measures. For this purpose, we plot them as a function of Unruh temperature $T_U$ with specific energy spacing of detectors $\\omega=1,3,5$. It is obvious that entanglement always degrades monotonously with increasing Unruh temperature, i.e., no revival of entanglement can happen for any initial state preparation. This clearly demonstrates the distinction between the entanglement and coherence. In Fig. (\\ref{fig1})a, we observe that the entanglement vanishes at low temperatures $T_U=0.5$ for $\\Delta_0=0$. For the same initial state $(\\Delta_0=0)$, comparing Fig. (\\ref{fig1})a with Fig. (\\ref{fig2})a, we observe that the entanglement is induced in the higher temperature regime while increasing the values of energy spacing of detectors from $\\omega=1$ to $\\omega=3$. If we increase the value of $\\omega$ further, one observes similar monotonic decreasing behavior of the correlation measures while the entanglement survives for sufficiently higher temperatures compared to lower values of $\\omega$. On the other hand, the other correlation measures also increase with the increase of energy spacing. In Fig. (\\ref{fig3}), we have plotted the correlation measures as a function of $T_U$ for another initial state such as $\\Delta_0=0.5$. Here again, the entanglement decreases monotonously while one sees the revival of LQU and coherence measures. \n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{fig8.pdf}\n\\includegraphics[width=.45\\textwidth]{fig82.pdf}\n\\includegraphics[width=.45\\textwidth]{fig81.pdf}\n\\includegraphics[width=.45\\textwidth]{fig8n.pdf}\n\\caption{The Density of quantum correlation measures (a) Coherence, (b) LQU, (c) Entanglement and (d) UIN of UDW detector as a function of $T_U$ and $\\Delta_0$ for $\\omega=3$}\n\\label{fig5}\n\\end{center}\n\\end{figure}\n\nNow, we illustrate our results with an example. If we choose the initial state of two-detector in a product form \n\\begin{align}\n\\rho_{\\text{in}}=\\rho_a(0)\\otimes\\rho_b(0), \\label{eq19}\n\\end{align}\n correlations of the initial state become zero. \n The state of each detector can be written in Bloch form as\n\\begin{align}\n\\rho_{a}(0)=\\frac{1}{2}\\left(I+n \\cdot \\sigma \\right),~~~\\rho_{b}(0)=\\frac{1}{2}\\left(I+m \\cdot \\sigma \\right) \\label{eq20}\n\\end{align}\nwhere $\\mathbf{n}$ and $\\mathbf{m}$ are two unit Bloch vectors. Without loss of generality, taking $\\mathbf{n}=(0,0,1)$ and $\\mathbf{m}=(0,\\sin\\theta,\\cos\\theta)$, we have $\\Delta_0=\\mathbf{n}\\cdot\\mathbf{m}=\\cos\\theta$ where $\\theta\\in[0,\\pi]$ is an angle between two vectors giving $\\Delta_0\\in[-1,1]$. \n\nIn Fig. (\\ref{fig5}), we have plotted the density of quantum correlation measures as a function of $T_U$ and $\\Delta_0$ with $\\omega=3$. Again, it is obvious that the entanglement is generated during time evolution of the detectors for $\\Delta_0=(-1,0)$ and there is no entanglement induced in the range $\\Delta_0=(0,1)$. Here again, we notice that the entanglement vanishes at low temperatures and is unable to capture the complete manifestation of nonlocal attributes of UD detectors. On the other hand, the quantum correlation measures quantify more nonlocal aspects of UD detectors compared to entanglement. It can be noticed that the generation of entanglement and quantum correlations in the initial product state arises due to the acceleration of the system. \n\n\n\n\n\n\\section{Conclusions}\n\\label{conc}\nIn this paper, we have studied the quantum coherence and correlations of two accelerating Unruh-\nDdeWitt detectors coupled to a scalar background in 3 + 1 Minkowski spacetime. We have employed the entanglement, local quantum uncertainty, uncertainty-induced nonlocality and $l_1$-norm of coherence as the quantumness quantifiers. Results of the paper indicate the revival of LQU and coherence while entanglement does not. We have also shown that the energy spacing of detectors induces the nonlocality between the detectors. The distinction between the static and accelerating qubit systems is observed in terms of generation of quantumness in a product state and revival mechanism of quantum correlations.\n\nFurther, the results of our manuscript can have wider ramifications in understanding the relativistic quantum information processing from the perspective of quantum correlation measures.\n\\noindent\n\n\\section*{Acknowledgment}\n\nAuthors are indebted to the referees for their critical comments to improve the contents of the manuscript. SB and RR wish to acknowledge the financial support received from the Council of Scientific and Industrial Research (CSIR), Government of India under Grant No. 03(1456)\/19\/EMR-II. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nEarly 1990s were marked by several major developments in knowledge \nrepresentation and nonmonotonic reasoning. One of the most important\namong them was the introduction of \\emph{disjunctive logic programs \nwith classical negation} by Michael Gelfond and Vladimir Lifschitz \n\\cite{gl90b}. The language of the formalism allowed for rules\n\\[\nH_1 \\vee \\ldots \\vee H_k \\leftarrow B_1,\\ldots, B_m,\\mathit{not\\;} B_{m+1},\\ldots,\\mathit{not\\;} B_n,\n\\]\nwhere $H_i$ and $B_i$ are classical literals, that is, atoms and\nclassical or \\emph{strong} negations ($\\neg$) of atoms. In the\npaper, we will write ``strong'' rather than ``classical'' negation,\nas it reflects more accurately the role and the behavior of the \noperator. The \\emph{answer-set} semantics for programs consisting of \nsuch rules, introduced in the same paper, generalized the stable-model \nsemantics of normal logic programs proposed a couple of years earlier \nalso by Gelfond and Lifschitz \\cite{gl88}. The proposed extensions of \nthe language of normal logic programs were motivated by knowledge \nrepresentation considerations. With two negation operators it was\nstraightforward to distinguish between $P$ being \\emph{false by default}\n(there is no justification for adopting $P$), and $P$ being \n\\emph{strongly false} (there is evidence for $\\neg P$). The former\nwould be written as $\\mathit{not\\;} P$ while the latter as $\\neg P$. And with the\ndisjunction in the head of rules one could model ``indefinite'' rules \nwhich, when applied, provide partial information only (one of the \nalternatives in the head holds, but no preference to any of them is \ngiven).\n\nSoon after disjunctive logic programs with strong negation were \nintroduced, Michael Gelfond proposed an additional important\nextension, this time with a modal operator \\cite{Gelfond91}. He called the\nresulting formalism the language of \\emph{epistemic specifications}. \nThe motivation came again from knowledge representation. The goal \nwas to provide means for the ``correct representation of incomplete \ninformation in the presence of multiple extensions'' \\cite{Gelfond91}. \n\nSurprisingly, despite their evident relevance to the theory of\nnonmonotonic\nreasoning as well as to the practice of knowledge representation,\nepistemic specifications have received relatively little attention \nso far. This\nstate of affairs may soon change. Recent work by Faber and Woltran\non \\emph{meta-reasoning} \nwith answer-set programming \\cite{FaberW09} shows the need for languages,\nin which one could express properties holding across all answer sets\nof a program, something Michael Gelfond foresaw already two decades\nago.\n\nOur goal in this paper is to revisit the formalism of epistemic \nspecifications and show that they deserve a second look, in fact,\na place in the forefront of knowledge representation research. We will \nestablish a general semantic framework for the formalism, and identify \nin it the precise location of Gelfond's epistemic specifications. We \nwill derive several complexity results. We will also show that the \noriginal idea \nof Gelfond to use a modal operator to model ``what is known to a \nreasoner'' has a broader scope of applicability. In particular, we will \nshow that it can also be used in combination with the classical logic. \n\nComplexity results presented in this paper provide an additional \nmotivation to study epistemic specifications. Even though programs \nwith strong negation often look ``more natural'' as they more directly \nalign with the natural language description of knowledge \nspecifications, the extension of the language of normal logic \nprograms with the strong negation operator does not actually increase\nthe expressive power of the formalism. This point was made already by \nGelfond and Lifschitz, who observed that there is a simple and concise \nway to compile the strong negation away. On the other hand, the \nextension allowing the disjunction operator in the heads of rules is \nan essential one. As the complexity results show \\cite{mt88,eg95}, \nthe class of problems that can be represented by means of disjunctive \nlogic programs is strictly larger (assuming no collapse of the \npolynomial hierarchy) than the class of problems that can be modeled by \nnormal logic programs. In the same vein, extension by the modal operator\nalong the lines proposed by Gelfond is essential, too. It does\nlead to an additional jump in the complexity.\n\n\n\\section{Epistemic Specifications}\n\nTo motivate epistemic specifications, Gelfond discussed the following\nexample. A certain college has these rules to determine the eligibility\nof a student for a scholarship:\n\\begin{enumerate}\n\\item Students with high GPA are eligible\n\\item Students from underrepresented groups and with fair GPA are eligible\n\\item Students with low GPA are not eligible\n\\item When these rules are insufficient to determine eligibility,\nthe student should be interviewed by the scholarship committee.\n\\end{enumerate}\nGelfond argued that there is no simple way to represent these rules\nas a disjunctive logic program with strong negation. There is no problem\nwith the first three rules. They are modeled correctly by the following\nthree logic program rules (in the language with both the default and strong\nnegation operators):\n\\begin{enumerate}\n\\item $eligible(X) \\leftarrow \\mathit{highGPA}(X)$\n\\item $eligible(X) \\leftarrow underrep(X), \\mathit{fairGPA}(X)$\n\\item $\\neg eligible(X) \\leftarrow \\mathit{lowGPA}(X)$.\n\\end{enumerate}\nThe problem is with the fourth rule, as it has a clear meta-reasoning\nflavor. It should apply when the possible worlds (answer sets)\ndetermined by the first three rules do not fully specify the status of \neligibility of a student $a$: neither \\emph{all} of them contain \n$eligible(a)$ nor \\emph{all} of them contain $\\neg eligible(a)$. An \nobvious attempt at a formalization:\n\\begin{enumerate}\n\\item[4.] $interview(X) \\leftarrow \\mathit{not\\;} eligible(X), \\mathit{not\\;} \\neg eligible(X)$ \n\\end{enumerate}\nfails. It is just another rule to be added to the program. Thus,\nwhen the answer-set semantics is used, the rule is interpreted with\nrespect to individual answer sets and not with respect to collections\nof answer-sets, as required for this application. For a concrete \nexample, let us assume that all we know about a certain student named \nMike is that Mike's GPA is fair or high. Clearly, we do not have \nenough information to determine Mike's eligibility and so we must \ninterview Mike. But the program consisting of rules (1)-(4) and the \nstatement \n\\begin{enumerate}\n\\item[5.] $fairGPA(mike) \\vee highGPA(mike)$\n\\end{enumerate}\nabout Mike's GPA, has two answer sets:\n\\begin{quote}\n$\\{highGPA(mike), eligible(mike)\\}$\\\\\n$\\{fairGPA(mike), interview(mike)\\}$. \n\\end{quote}\nThus, the query $?interview(mike)$ has the answer ``unknown.'' To\naddress the problem, Gelfond proposed to extend the language with a\nmodal operator $K$ and, speaking informally, interpret premises \n$K\\varphi$ as ``$\\varphi$ is known to the program'' (the original phrase\nused by Gelfond was ``known to the reasoner''), that is, true in all \nanswer-sets. With this language extension, the \nfourth rule can be encoded as \n\\begin{enumerate}\n\\item[4$'$.] $interview(X) \\leftarrow \\mathit{not\\;} K\\, eligible(X), \\mathit{not\\;} K \\neg eligible(X)$\n\\end{enumerate}\nwhich, intuitively, stands for ``\\emph{interview} if neither the \neligibility nor the non-eligibility is known.'' \n\nThe way in which Gelfond \\cite{Gelfond91} proposed to formalize this \nintuition is strikingly elegant. We will now discuss it. We start with\nthe syntax of \\emph{epistemic specifications}. As elsewhere in the \npaper, we restrict attention to the propositional case. We assume a \nfixed infinite countable set $\\mathit{At}$ of \\emph{atoms} and the corresponding\nlanguage $\\mathcal{L}$ of propositional logic. A \\emph{literal} is an atom, say\n$A$, or its \\emph{strong} negation $\\neg A$. A \\emph{simple modal atom}\nis an expression $K\\varphi$, where $\\varphi\\in \\mathcal{L}$, and a \\emph{simple modal \nliteral} is defined accordingly. An \\emph{epistemic premise} is an \nexpression (conjunction)\n\\[\nE_1,\\ldots, E_s,\\mathit{not\\;} E_{s+1},\\ldots, \\mathit{not\\;} E_t,\n\\]\nwhere every $E_i$, $1\\leq i\\leq t$, is a simple modal literal. \nAn \\emph{epistemic rule} is an expression of the form\n\\[\nL_1 \\vee \\ldots\\vee L_k \\leftarrow L_{k+1},\\ldots, L_m,\\mathit{not\\;} L_{m+1},\\ldots, \\mathit{not\\;} L_n, E,\n\\]\nwhere every $L_i$, $1\\leq i\\leq k$, is a literal, and $E$ is an epistemic \npremise.\nCollections of\nepistemic rules are \\emph{epistemic programs}. It is clear that (ground\nversions of) rules (1)-(5) and (4$'$) are examples of epistemic rules, \nwith rule (4$'$) being an example of an epistemic rule that actually \ntakes advantage of the extended syntax. Rules such as\n\\begin{quote}\n$a\\vee \\neg d \\leftarrow b, \\mathit{not\\;} \\neg c, \\neg K(d\\lor \\neg c)$\\\\\n$\\neg a \\leftarrow \\neg c, \\mathit{not\\;} \\neg K(\\neg (a\\land c)\\rightarrow b)$\n\\end{quote}\nare also examples of epistemic rules. We note that the language of \nepistemic programs is only a fragment of the language of epistemic \nspecifications by Gelfond. However, it is still expressive enough to \ncover all examples discussed by Gelfond and, more generally, a broad\nrange of practical applications, as natural-language formulations of \ndomain knowledge typically assume a rule-based pattern.\n\nWe move on to the semantics, which is in terms of \\emph{world \nviews}. The definition of a world view consists of several steps. \nFirst, let $W$ be \na consistent set of literals from $\\mathcal{L}$. We regard $W$ as a three-valued\ninterpretation of $\\mathcal{L}$ (we will also use the term \\emph{three-valued \npossible world}), assigning to each atom one of the three logical values\n$\\mathbf{t}$, $\\mathbf{f}$ and $\\mathbf{u}$. The interpretation extends by recursion to all \nformulas in $\\mathcal{L}$, according to the following truth tables\n\\begin{figure}[h]\n\\begin{minipage}[t]{2.1cm}\n\\ \n\\end{minipage}\n\\begin{minipage}[t]{1.5cm}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline\n$\\neg$ & \\\\\n\\hline\n$\\mathbf{f}$ & $\\mathbf{t}$ \\\\\n$\\mathbf{t}$ & $\\mathbf{f}$\\\\\n$\\mathbf{u}$ & $\\mathbf{u}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\lor$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{u}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\land$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{u}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{f}$ & $\\mathbf{f}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[t]{2.0cm}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|}\n\\hline\n$\\rightarrow$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n\\hline\n\\hline\n$\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{f}$ \\\\\n$\\mathbf{u}$ & $\\mathbf{t}$ & $\\mathbf{u}$ & $\\mathbf{u}$ \\\\\n$\\mathbf{f}$ & $\\mathbf{t}$ & $\\mathbf{t}$ & $\\mathbf{t}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{minipage}\\\\ \\\\ \\mbox{}\n\\caption{Truth tables for the 3-valued logic of Kleene.}\n\\label{t.1}\n\\end{figure}\n\nBy a \\emph{three-valued possible-world structure} we mean a non-empty\nfamily of consistent sets of literals (three-valued possible worlds). \nLet $\\mathcal{A}$ be a three-valued possible-world structure and let $W$ be\na consistent set of literals. For every formula $\\varphi\\in\\mathcal{L}$, we define \n\\begin{enumerate}\n\\item $\\langle\\mathcal{A},W\\rangle\\models\\varphi$, if $v_W(\\varphi)=\\mathbf{t}$\n\\item $\\langle\\mathcal{A},W\\rangle\\models K\\varphi$, if for every $V\\in\\mathcal{A}$, $v_V(\\varphi)=\\mathbf{t}$\n\\item $\\langle\\mathcal{A},W\\rangle\\models \\neg K\\varphi$, if there is $V\\in\\mathcal{A}$ such\nthat $v_V(\\varphi)=\\mathbf{f}$.\n\\end{enumerate}\nNext, for every literal or simple modal literal $L$, we define\n\\begin{enumerate}\n\\item[4.] $\\langle\\mathcal{A},W\\rangle \\models\\mathit{not\\;} L$ if $\\langle \\mathcal{A},W\\rangle \\not\\models L$.\n\\end{enumerate}\nWe note that neither $\\langle\\mathcal{A},W\\rangle\\models K\\varphi$ nor $\\langle\\mathcal{A},W\\rangle\\models \n\\neg K\\varphi$ depend on $W$. Thus, we will often write $\\mathcal{A}\\models F$, when \n$F$ is a simple modal literal or its default negation.\n\nIn the next step, we introduce the notion of the \\emph{G-reduct} of an \nepistemic program.\n\n\\begin{definition}\n\\label{def1}\nLet $P$ be an epistemic program, $\\mathcal{A}$ a three-valued possible-world\nstructure and $W$ a consistent set of literals. The \\emph{G-reduct}\nof $P$ with respect to $\\langle\\mathcal{A},W\\rangle$, in symbols $P^{\\langle\\mathcal{A},W\\rangle}$, \nconsists of the heads of all\nrules $r\\in P$ such that $\\langle\\mathcal{A},W\\rangle\\models \\alpha$, for every \nconjunct $\\alpha$ occurring in the body of $r$. \n\\end{definition}\n\nLet $H$ be a set of disjunctions of literals from $\\mathcal{L}$. A set $W$ of \nliterals is \\emph{closed} with respect to $H$ if $W$ is consistent\nand contains at least one literal in common with every disjunction in \n$H$. \nWe denote by $\\mathit{Min}(H)$ the family of all minimal sets of literals that \nare closed with respect to $H$.\nWith the notation $\\mathit{Min}(H)$ in hand, we are finally ready to\ndefine the concept of a world view of an epistemic program $P$. \n\n\\begin{definition}\n\\label{def2}\nA three-valued possible-world structure $\\mathcal{A}$ is a \\emph{world view} \nof an epistemic program $P$ if $\\mathcal{A}= \\{W\\,|\\; W\\in \\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})\\}$.\n\\end{definition}\n\n\\begin{remark}\nThe $G$-reduct of an epistemic program consists of disjunctions of literals.\nThus, the concept of a world view is well defined.\n\\end{remark}\n\\begin{remark}\nWe note that Gelfond considered also inconsistent sets of literals as \nminimal sets closed under disjunctions. However, the only such set \nhe allowed consisted of \\emph{all} literals. Consequently, the \ndifference between the Gelfond's semantics and the one we described \nabove is that some programs have a world view in the Gelfond's approach\nthat consists of a single set of all literals, while in our approach \nthese programs do not have a world view. But in all other cases, the \ntwo semantics behave in the same way. \n\\end{remark}\n\nLet us consider the ground program, say $P$, corresponding to the \nscholarship eligibility example (rule (5), and rules (1)-(3) and \n(4$'$), grounded with respect to the Herbrand universe $\\{mike\\}$).\nThe only rule involving simple modal literals is\n\\begin{quote}\n$interview(mike) \\leftarrow \\mathit{not\\;} K\\, eligible(mike), \\mathit{not\\;} K \\neg eligible(mike)$.\n\\end{quote}\nLet $\\mathcal{A}$ be a world view of $P$. Being a three-valued possible-world \nstructure, $\\mathcal{A}\\not=\\emptyset$. No\nmatter what $W$ we consider, no minimal set closed with respect to \n$P^{\\langle\\mathcal{A},W\\rangle}$ contains $\\mathit{lowGPA}(mike)$ and, consequently,\nno minimal set closed with respect to $P^{\\langle\\mathcal{A},W\\rangle}$ contains \n$\\neg eligible(mike)$. It follows that $\\mathcal{A}\\not\\models K \\neg \neligible(mike)$. \n\nLet us assume that $\\mathcal{A}\\models K\\, eligible(mike)$. Then, no reduct\n$P^{\\langle\\mathcal{A},W\\rangle}$ contains $interview(mike)$. Let\n$W=\\{\\mathit{fairGP}(mike)\\}$. It follows that $P^{\\langle\\mathcal{A},W\\rangle}$\nconsists only of $\\mathit{fairGPA}(mike) \\vee \\mathit{highGPA}(mike)$.\nClearly, $W\\in\\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})$ and, consequently, $W\\in \\mathcal{A}$.\nThus, $\\mathcal{A}\\not\\models K\\, eligible(mike)$, a contradiction.\n\nIt must be then that $\\mathcal{A}\\models\\mathit{not\\;} K\\, eligible(mike)$ and \n$\\mathcal{A}\\models \\mathit{not\\;} K \\neg eligible(mike)$. Let $W$ be an arbitrary \nconsistent set of literals. Clearly, the reduct $P^{\\langle\\mathcal{A},W\\rangle}$ \ncontains $interview(mike)$ and\n$\\mathit{fairGPA}(mike) \\vee \\mathit{highGPA}(mike)$. If \n$\\mathit{highGPA}(mike)\\in W$, the reduct also contains $eligible(mike)$. \nThus, $W\\in\\mathit{Min}(P^{\\langle\\mathcal{A},W\\rangle})$ if and only if \n\\begin{quote}\n$W=\\{\\mathit{fairGPA}(mike),interview(mike)\\}$, or\\\\\n$W=\\{\\mathit{highGPA}(mike),eligible(mike),interview(mike)\\}$.\n\\end{quote}\nIt follows that if $\\mathcal{A}$ is a world view for $P$ then it consists of \nthese two possible worlds. Conversely, it is easy to check that a \npossible-world structure consisting of these two possible worlds is a \nworld view for $P$. Thus, $interview(mike)$ holds in $\\mathcal{A}$, and so \nour representation of the example as an epistemic program has the\ndesired behavior.\n\n\\section{Epistemic Specifications --- a Broader Perspective}\n\nThe discussion in the previous section demonstrates the usefulness of\nformalisms such as that of epistemic specifications for knowledge\nrepresentation and reasoning. We will now present a simpler yet, in \nmany respects, more general framework for epistemic specifications. \nThe key to our approach is that we consider the semantics given by \n\\emph{two-valued} interpretations (sets of atoms), and standard \n\\emph{two-valued} possible-world structures (nonempty collections of \ntwo-valued interpretations). We also work within a rather standard\nversion of the language of modal propositional logic and so, in \nparticular, we allow only for one negation operator. Later in the paper\nwe show that epistemic specifications by Gelfond can be encoded in a \nrather direct way in our formalism. Thus, the restrictions we impose \nare not essential even though, admittedly, not having two kinds of \nnegation in the language in some cases may make the modeling task \nharder. \n\nWe start by making precise the syntax of the language we will be \nusing. As we stated earlier, we assume a fixed infinite countable \nset of atoms $\\mathit{At}$. The language we consider is determined by the \nset $\\mathit{At}$, the modal operator $K$, and by the \\emph{boolean connectives}\n$\\bot$ (0-place), and $\\land$, $\\lor$, and $\\rightarrow$ (binary). The BNF \nexpression \n\\begin{quote}\n$\\varphi::= \\bot\\,|\\,A\\,|\\,(\\varphi\\land \\varphi)\\,|\\,(\\varphi \\lor \\varphi)\\,|\\,\n(\\varphi\\rightarrow\\varphi)\\,|\\ K\\varphi$,\n\\end{quote}\nwhere $A\\in \\mathit{At}$, provides a concise definition of a formula. The \nparentheses are used only to disambiguate the order of binary \nconnectives. Whenever possible, we omit them. We define the unary\n\\emph{negation} connective $\\neg$ and the 0-place connective $\\top$\nas abbreviations:\n\\begin{quote}\n$\\neg\\varphi::= \\varphi\\rightarrow\\bot$\\\\\n$\\top::=\\neg\\bot$.\n\\end{quote}\nWe call formulas $K\\varphi$, where $\\varphi\\in\\mathcal{L}_K$, \\emph{modal atoms} \n(simple modal atoms that we considered earlier and will consider below\nare special modal atoms with $K$-depth equal to 1). We denote this \nlanguage by $\\mathcal{L}_K$ and refer to subsets of $\\mathcal{L}_K$ as \\emph{epistemic \ntheories}. We denote the modal-free fragment of $\\mathcal{L}_K$ by $\\mathcal{L}$. \n\nWhile we will eventually describe the semantics (in fact, several of \nthem) for arbitrary epistemic theories, we start with an important\nspecial case. Due to close analogies between the concepts we define \nbelow and the corresponding ones defined earlier in the context of the \nformalism of Gelfond, we ``reuse'' the terms used there. Specifically, \nby an \\emph{epistemic premise} we mean a conjunction of simple modal \nliterals. Similarly, by an \\emph{epistemic rule} we understand an \nexpression of the form\n\\begin{equation}\n\\label{eq10}\nE\\land L_1\\land\\ldots\\land L_m \\rightarrow A_1\\lor\\ldots\\lor A_n,\n\\end{equation}\nwhere $E$ is an epistemic premise, $L_i$'s are literals (in $\\mathcal{L}$) and \n$A_i$'s are atoms. Finally, we call a collection of epistemic rules an \n\\emph{epistemic program}. It will always be clear from the context, in \nwhich sense these terms are to be understood.\n\nWe stress that $\\neg$ is not a primary connective in the language but a\nderived one (it is a shorthand for some particular formulas involving\nthe rule symbol). Even though under some semantics we propose below this\nnegation operator has features of default negation, under some others it\ndoes not. Thus, we selected for it the standard negation symbol $\\neg$\nrather than the ``loaded'' $\\mathit{not\\;}$. \n \nA (two-valued) \\emph{possible-world structure} is any nonempty family\n$\\mathcal{A}$ of subsets of $\\mathit{At}$ (two-valued interpretations). In the remainder\nof the paper, when we use terms ``interpretation'' and ``possible-world \nstructure'' without any additional modifiers, we always mean a two-valued\ninterpretation and a two-valued possible-world structure.\n\nLet $\\mathcal{A}$ be a possible-world structure and $\\varphi\\in \\mathcal{L}$. We recall \nthat $\\mathcal{A}\\models K\\varphi$ precisely when $W\\models \\varphi$, for every \n$W\\in\\mathcal{A}$, and $\\mathcal{A}\\models \\neg K\\varphi$, otherwise. We will now define \nthe \\emph{epistemic reduct} of an epistemic program with respect to a\npossible-world structure. \n\n\\begin{definition}\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program and let $\\mathcal{A}$ be a\npossible-world structure. The \\emph{epistemic reduct} of $P$ with \nrespect to $\\mathcal{A}$, $\\gl{P}{\\mathcal{A}}$ in symbols, is the theory obtained \nfrom $P$ as follows: eliminate every rule with an epistemic premise $E$\nsuch that $\\mathcal{A}\\not\\models E$; drop the epistemic premise from every \nremaining rule. \n\\end{definition}\nIt is clear that $\\gl{P}{\\mathcal{A}}\\subseteq\\mathcal{L}$, and that it consists of \nrules of the form \n\\begin{equation}\n\\label{eq11}\nL_1\\land\\ldots\\land L_m \\rightarrow A_1\\lor\\ldots\\lor A_n,\n\\end{equation}\nwhere $L_i$'s are literals (in $\\mathcal{L}$) and $A_i$'s are atoms. \n\nLet $P$ be a collection of rules (\\ref{eq11}). Then, $P$ is a \npropositional theory. Thus, it can be interpreted by the standard \npropositional logic semantics. However, $P$ can also be regarded as a \ndisjunctive logic program (if we write rules from right to left rather \nthan from left to right). Consequently, $P$ can also be interpreted by\nthe stable-model semantics \\cite{gl88,gl90b} and the supported-model \nsemantics \\cite{abw87,bg93,bradix96jlp1,Inoue98-JLP}. (For normal logic\nprograms, the supported-model semantics was introduced by Apt et al. \n\\cite{abw87}. The notion was extended to disjunctive logic programs by\nBaral and Gelfond \\cite{bg93}. We refer to papers by Brass and Dix\n\\cite{bradix96jlp1}, Definition 2.4, and Inoue and Sakama \n\\cite{Inoue98-JLP}, Section 5, for more details).\nWe write $\\mathcal{M}(P)$, $\\mathcal{ST}(P)$ and \n$\\mathcal{SP}(P)$ for the sets of models, stable models and supported models \nof $P$, respectively. An important observation is that \\emph{each} of \nthese semantics gives rise to the corresponding notion of an epistemic\nextension.\n\n\\begin{definition}\n\\label{def11}\nLet $P\\subseteq \\mathcal{L}_K$ be an epistemic program. A possible-world \nstructure $\\mathcal{A}$ is an \\emph{epistemic model} (respectively, an\n\\emph{epistemic stable model}, or an \\emph{epistemic supported model})\nof $P$, if $\\mathcal{A} = \\mathcal{M}(\\gl{P}{\\mathcal{A}})$ (respectively, $\\mathcal{A} = \n\\mathcal{ST}(\\gl{P}{\\mathcal{A}})$ or $\\mathcal{A} = \\mathcal{SP}(\\gl{P}{\\mathcal{A}})$).\n\\end{definition}\n\nIt is clear that Definition \\ref{def11} can easily be adjusted also to\nother semantics of propositional theories and programs. We briefly \nmention two such semantics in the last section of the paper.\n\nWe will now show that epistemic programs with the semantics of \nepistemic stable models can provide an adequate representation to\nthe scholarship eligibility example for Mike. The available information\ncan be represented by the following program $P(mike) \\subseteq\\mathcal{L}_K$:\n\n\\begin{enumerate}\n\\item $eligible(mike)\\land neligible(mike) \\rightarrow \\bot$\n\\item $fairGPA(mike) \\vee highGPA(mike)$\n\\item $\\mathit{highGPA}(mike) \\rightarrow eligible(mike)$\n\\item $underrep(mike) \\land \\mathit{fairGPA}(mike)\\rightarrow eligible(mike)$\n\\item $\\mathit{lowGPA}(mike) \\rightarrow neligible(mike)$\n\\item $\\neg K\\, eligible(mike), \\neg K\\, neligible(mike)\\rightarrow interview(mike)$.\n\\end{enumerate}\nWe use the predicate \\emph{neligible} to model the strong negation \nof the predicate $eligible$ that appears in the representation in terms \nof epistemic programs by Gelfond (thus, in particular, the presence of \nthe first clause, which precludes the facts $eligible(mike)$ and \n$neligible(mike)$\nto be true together). This extension of the language and an extra\nrule in the representation is the price we pay\nfor eliminating one negation operator. \n\nLet $\\mathcal{A}$ consist of the interpretations \n\\begin{quote}\n$W_1=\\{\\mathit{fairGPA}(mike),interview(mike)\\}$\\\\\n$W_2=\\{\\mathit{highGPA}(mike),eligible(mike),interview(mike)\\}$.\n\\end{quote}\nThen the reduct $\\gl{[P(mike)]}{\\mathcal{A}}$ consists of rules (1)-(5),\nwhich are unaffected by the reduct operation, and of the fact \n$interview(mike)$, resulting from rule (6) when the reduct operation\nis performed (as in logic programming, when a rule has the empty \nantecedent, we drop the implication symbol from the \nnotation). One can check that \n$\\mathcal{A}=\\{W_1,W_2\\}=\\mathcal{ST}(\\gl{[P(mike)]}{\\mathcal{A}})$. Thus, $\\mathcal{A}$ is\nan epistemic stable model of $P$ (in fact, the only one). Clearly, \n$interview(mike)$ holds in the model (as we would expect it to), as it \nholds in each of its possible-worlds. We note that in this particular \ncase, the semantics of epistemic supported models yields exactly the \nsame solution.\n\n\\section{Complexity}\n\nWe will now study the complexity of reasoning with epistemic (stable, \nsupported) models. We provide details for the case of epistemic stable\nmodels, and only present the results for the other two semantics, as the \ntechniques to prove them are very similar to those we develop for the\ncase of epistemic stable models.\n\nFirst, we note that epistemic stable models of an epistemic program $P$\ncan be represented by partitions of the set of all modal atoms of $P$. \nThis is important as \\emph{a priori} the size of possible-world \nstructures one needs to consider as candidates for epistemic stable\nmodels may be exponential in the size of a program. Thus, to obtain \ngood complexity bounds alternative polynomial-size representations \nof epistemic stable models are needed.\n\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program and $(\\Phi,\\Psi)$ be the \nset of modal atoms of $P$ (all these modal atoms are, in fact, simple).\nWe write $P_{|\\Phi,\\Psi}$ for the program\nobtained from $P$ by eliminating every rule whose epistemic premise\ncontains a conjunct $K\\psi$, where $K\\psi\\in\\Psi$, or a conjunct\n$\\neg K\\varphi$, where $K\\varphi\\in\\Phi$ (these rules are ```blocked'' by\n$(\\Phi,\\Psi)$), and by eliminating the epistemic premise from every\nother rule of $P$.\n\n\\begin{proposition}\n\\label{prop:char}\nLet $P\\subseteq\\mathcal{L}_K$ be an epistemic program. If a possible-world\nstructure $\\mathcal{A}$ is an epistemic stable model of $P$, then there\nis a partition $(\\Phi,\\Psi)$ of the set of modal atoms of $P$ such\nthat\n\\begin{enumerate}\n\\item $\\mathcal{ST}(P_{|\\Phi,\\Psi})\\not=\\emptyset$\n\\item For every $K\\varphi\\in\\Phi$, $\\varphi$ holds in every stable model\nof $P_{|\\Phi,\\Psi}$ \n\\item For every $K\\psi\\in\\Psi$, $\\psi$ does not hold in at least one\nstable model of $P_{|\\Phi,\\Psi}$. \n\\end{enumerate}\nConversely, if there are such partitions, $P$ has epistemic stable \nmodels.\n\\end{proposition} \n\nIt follows that epistemic stable models can be represented by partitions\n$(\\Phi,\\Psi)$ satisfying conditions (1)-(3) from the proposition above. \n\nWe observe that deciding whether a partition $(\\Phi,\\Psi)$ satisfies \nconditions (1)-(3) from Proposition \\ref{prop:char}, can be accomplished\nby polynomially many calls to an $\\Sigma_2^P$-oracle and, if we restrict\nattention to non-disjunctive epistemic programs, by polynomially many \ncalls to an $\\mathit{NP}$-oracle. \n\n\\begin{remark}\n\\label{rem1}\nIf we adjust Proposition \\ref{prop:char} by replacing the term ``stable''\nwith the term ``supported,'' and replacing $\\mathcal{ST}()$ with $\\mathcal{SP}()$, we\nobtain a characterization of epistemic supported models. Similarly, \nomitting the term ``stable,'' and replacing $\\mathcal{ST}()$ with $\\mathcal{M}()$ \nyields a characterization of epistemic models. In each case, one can \ndecide whether a partition $(\\Phi,\\Psi)$ satisfies conditions (1)-(3)\nby polynomially many calls to an $\\mathit{NP}$-oracle (this claim is evident \nfor the case of epistemic models; for the case of epistemic supported\nmodels, it follows from the fact that supported models semantics does\nnot get harder when we allow disjunctions in the heads or rules). \n\\end{remark}\n\n\\begin{theorem}\nThe problem to decide whether a non-disjunctive epistemic program has \nan epistemic stable model is $\\Sigma_2^P$-complete. \n\\end{theorem}\nProof: Our comments above imply that the problem is in the class \n$\\Sigma_2^P$. Let $F=\\exists Y \\forall Z \\Theta$, where $\\Theta$\nis a DNF formula. The problem to decide whether $F$ is true is \n$\\Sigma_2^P$-complete. We will reduce it to the problem in question \nand, consequently, demonstrate its $\\Sigma_2^P$-hardness. To this\nend, we construct an epistemic program $Q\\subseteq\\mathcal{L}_K$ by including \ninto $Q$ the following clauses (atoms $w$, $y'$, $y\\in Y$, and $z'$, \n$z\\in Z$ are fresh):\n\\begin{enumerate}\n\\item $Ky\\rightarrow y\\;$; and $Ky' \\rightarrow y'$, for every $y\\in Y$\n\\item $y\\land y'\\rightarrow\\;$; and $\\neg y\\land \\neg y'\\rightarrow\\;$, for every $y\\in Y$\n\\item $\\neg z'\\rightarrow z\\;$; and $\\neg z\\rightarrow z'$, for $z\\in Z$\n\\item $\\sigma(u_1)\\land \\ldots\\land \\sigma(u_k)\\rightarrow w\\;$, where \n$u_1\\wedge\\ldots\\wedge u_k$ is a disjunct of $\\Theta$, and \n$\\sigma(\\neg a)=a'$ and $\\sigma(a)=a$, for every $a\\in Y\\cup Z$\n\\item $\\neg Kw\\rightarrow\\;$.\n\\end{enumerate}\n\nLet us assume that $\\mathcal{A}$ is an epistemic stable model of $Q$. In particular,\n$\\mathcal{A}\\not=\\emptyset$. It must \nbe that $\\mathcal{A}\\models Kw$ (otherwise, $\\gl{Q}{\\mathcal{A}}$ has no stable models, \nthat is, $\\mathcal{A}=\\emptyset$). Let us define $A= \\{y\\in Y\\,|\\; \\mathcal{A} \\models \nKy\\}$, and $B=\\{y\\in Y\\,|\\;\\mathcal{A}\\models Ky'\\}$. It follows that $\\gl{Q}{\\mathcal{A}}$\nconsists of the following rules:\n\\begin{enumerate}\n\\item $y$, for $y\\in A$, and $y'$, for $y\\in B$\n\\item $y\\land y'\\rightarrow\\;$; and $\\neg y\\land \\neg y'\\rightarrow \\;$, for every $y\\in Y$\n\\item $\\neg z'\\rightarrow z\\;$; and $\\neg z\\rightarrow z'$, for $z\\in Z$\n\\item $\\sigma(u_1)\\land \\ldots\\land \\sigma(u_k)\\rightarrow w\\;$, where\n$u_1\\wedge\\ldots\\wedge u_k$ is a disjunct of $\\Theta$, and\n$\\sigma(\\neg a)=a'$ and $\\sigma(a)=a$, for every $a\\in Y\\cup Z$.\n\\end{enumerate}\nSince $\\mathcal{A}=\\mathcal{ST}(\\gl{Q}{\\mathcal{A}})$ and $\\mathcal{A}\\not=\\emptyset$, $B=Y\\setminus A$\n(due to clauses of type (2)).\nIt is clear that the program $\\gl{Q}{\\mathcal{A}}$ has stable models\nand that they are of the form $A\\cup \\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}$, if that set does not imply $w$ through\na rule of type (4), or $A\\cup \\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, otherwise, where $D$ is any\nsubset of $Z$. As $\\mathcal{A}\\models Kw$,\nthere are no stable models of the first type. Thus, the family of stable \nmodels\nof $\\gl{Q}{\\mathcal{A}}$ consists of all sets $A\\cup \\{y'\\,|\\; y\\in Y\\setminus \nA\\} \\cup\nD \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, where $D$ is an arbitrary\nsubset of $Z$. It follows that for every $D\\subseteq Z$, the set $A\\cup \n\\{y'\\,|\\; y\\in Y\\setminus A\\} \\cup D \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}$\nsatisfies the body of at least one rule of type (4). By the construction, \nfor every $D\\subseteq Z$, the valuation of $Y\\cup Z$ determined by $A$ and \n$D$ satisfies the corresponding disjunct in $\\Theta$ and so, also $\\Theta$.\nIn other words, $\\exists Y\\forall Z \\Theta$ is true.\n\nConversely, let $\\exists Y\\forall Z\\Theta$ be true. Let $A$ be a subset\nof $Y$ such that $\\Theta_{|Y\/A}$ holds for every truth assignment of $Z$ \n(by $\\Theta_{|Y\/A}$, we mean the formula obtained by simplifying the \nformula $Q$ with respect to the truth assignment of $Y$ determined by \n$A$). Let \n$\\mathcal{A}$ consist of all sets of the form $A\\cup \\{y'\\,|\\; y\\in Y\\setminus \nA\\} \\cup D \\cup \\{z'\\,|\\; z\\in Z\\setminus D\\}\\cup \\{w\\}$, where $D\\subseteq\nZ$. It follows that $\\gl{Q}{\\mathcal{A}}$ consists of clauses (1)-(4) above,\nwith $B=Y\\setminus A$.\nSince $\\forall Z \\Theta_{|A\/Y}$ holds, it follows that $\\mathcal{A}$ is precisely \nthe set of stable models of $\\gl{Q}{\\mathcal{A}}$. Thus, $\\mathcal{A}$ is an epistemic\nstable model of $Q$. \n\\hfill$\\Box$\n\nIn the general case, the complexity goes one level up.\n\n\\begin{theorem}\n\\label{thm:2}\nThe problem to decide whether an epistemic program $P\\subseteq\\mathcal{L}_K$\nhas an epistemic stable model is $\\Sigma_3^P$-complete.\n\\end{theorem}\nProof: The membership follows from the earlier remarks. To prove the \nhardness part, we consider a QBF formula $F=\n\\exists X \\forall Y \\exists Z \\Theta$, where $\\Theta$ is a 3-CNF \nformula. For each atom $x\\in X$ ($y\\in Y$ and $z\\in Z$, respectively), we \nintroduce a fresh atom $x'$ ($y'$ and $z'$, respectively). Finally, \nwe introduce three additional fresh atoms, $w$, $f$ and $g$.\n\nWe now construct a disjunctive epistemic program $Q$ by including into\nit the following clauses:\n\\begin{enumerate}\n\\item $Kx\\rightarrow x$; and $Kx' \\rightarrow x'$, for every $x\\in X$\n\\item $x\\land x'\\rightarrow$; and $\\neg x\\land \\neg x'\\rightarrow$, for every $x\\in X$\n\\item $\\neg g\\rightarrow f$; and $\\neg f\\rightarrow g$\n\\item $f\\rightarrow y \\vee y'$; and $f\\rightarrow z \\vee z'$, for every $y\\in Y$ \nand $z\\in Z$\n\\item $f\\land w\\rightarrow z$; and $f\\land w\\rightarrow z'$, for every $z\\in Z$\n\\item $f\\land \\sigma(u_1)\\land \\sigma(u_2)\\land \\sigma(u_3)\\rightarrow w$, \nfor every clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Theta$, where \n$\\sigma(a)=a'$ and \n$\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$ \n\\item $f\\land \\neg w \\rightarrow w$\n\\item $\\neg K\\neg w \\rightarrow$\n\\end{enumerate}\nLet us assume that $\\exists X \\forall Y\\exists Z \\Theta$ is true. Let\n$A\\subseteq\nX$ describe the truth assignment on $X$ so that $\\forall Y\n\\exists Z \\Phi_{X\/A}$ holds (we define $\\Phi_{X\/A}$ in the proof\nof the previous result). We will show that $Q$ has an epistemic \nstable model\n$\\mathcal{A}=\\{A\\cup \\{a'\\,|\\; a\\in X\\setminus A\\}\\cup\\{g\\}\\}$. Clearly, $Kx$, $x\\in A$,\nand $Kx'$, $x\\in X\\setminus A$, are true in $\\mathcal{A}$. Also, $K\\neg w$ is \ntrue in $\\mathcal{A}$. All other modal atoms in $Q$ are false in $\\mathcal{A}$.\nThus, $\\gl{Q}{\\mathcal{A}}$ consists of rules $x$, for $x\\in A$, $x'$, for $x\\in\nX\\setminus A$ and of rules (2)-(7) above. Let $M$ be a stable model of\n$\\gl{Q}{\\mathcal{A}}$ containing $f$. It follows that $w\\in M$ and so, $Z\\cup Z'\n\\subseteq M$. Moreover, the Gelfond-Lifschitz reduct of $\\gl{Q}{\\mathcal{A}}$\nwith respect to $M$ consists of rules $x$, for $x\\in A$, $x'$, for $x\\in\nX\\setminus A$, all $\\neg$-free constraints of type (2), rule $f$, and \nrules (4)-(6) above, and $M$ is a minimal model of this program.\n\nLet $B=Y\\cap M$. By the minimality of $M$, $M=A\\cup \\{x'\\,|\\; x\\in X\n\\setminus A\\}\\cup B\\cup\\{y'\\,|\\; y\\in Y\\setminus B\\} \\cup Z\\cup Z'\\cup \n\\{f,w\\}$. Since $\\forall Y \\exists Z \\Phi_{X\/A}$ holds, $\\exists Z \n\\Phi_{X\/A,Y\/B}$ holds, too. Thus, let $D\\subseteq Z$ be a subset of $Z$\nsuch that $\\Phi_{X\/A,Y\/B,Z\/D}$ is true. It follows that $M'=A\\cup \\{x'\n\\,|\\; x\\in X \\setminus A\\}\\cup B\\cup\\{y'\\,|\\; y\\in Y\\setminus B\\} \\cup \nD\\cup\\{z'\\,|\\; z\\in Z\\setminus D\\} \\cup \\{f\\}$ is also a model of the\nGelfond-Lifschitz reduct of $\\gl{Q}{\\mathcal{A}}$ with respect to $M$, \ncontradicting the minimality of $M$.\n\nThus, if $M$ is an answer set of $\\gl{Q}{\\mathcal{A}}$, it must contain $g$. \nConsequently, it does not contain $f$ and so no rules of type (4)-(7) \ncontribute to it. It follows that $M=A\\cup \\{a'\\,|\\; a\\in \nX\\setminus A\\} \\cup\\{g\\}$ and, as it indeed is an answer set of \n$\\gl{Q}{\\mathcal{A}}$, $\\mathcal{A}=\\mathcal{ST}(\\gl{Q}{\\mathcal{A}})$. Thus, $\\mathcal{A}$ is a epistemic\nstable model, as claimed.\n\nConversely, let as assume that $Q$ has an epistemic stable model, say,\n$\\mathcal{A}$. It must be that $\\mathcal{A}\\models K\\neg w$ (otherwise, $\\gl{Q}{\\mathcal{A}}$ \ncontains a contradiction and has no stable models). Let us define \n$A=\\{x\\in X\\,|\\; \\mathcal{A}\\models Kx\\}$ and $B=\\{x\\in X\\,|\\; \\mathcal{A}\\models Kx'\\}$. \nIt follows that $\\gl{Q}{\\mathcal{A}}$ consists of the clauses: \n\\begin{enumerate}\n\\item $x$, for $x\\in A$ and $x'$, for $x\\in B$\n\\item $x\\land x'\\rightarrow $; and $\\neg x\\land \\neg x'\\rightarrow$, for every $x\\in X$\n\\item $\\neg g\\rightarrow f$; and $\\neg f\\rightarrow g$\n\\item $f \\rightarrow y \\vee y'$; and $f\\rightarrow z \\vee z'$, for every $y\\in Y$\nand $z\\in Z$\n\\item $f\\land w \\rightarrow z$; and $f\\land w \\rightarrow z'$, for every $z\\in Z$\n\\item $f\\land \\sigma(u_1)\\land\\sigma(u_2)\\land\\sigma(u_3)\\rightarrow w$, for \nevery clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Phi$, where $\\sigma(a)=a'$\nand $\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$.\n\\item $f, \\neg w \\rightarrow w$\n\\end{enumerate}\nWe have that $\\mathcal{A}$ is precisely the set of stable models of this \nprogram. Since $\\mathcal{A}\\not=\\emptyset$, $B=X\\setminus A$. If $M$ is \na stable model of $\\gl{Q}{\\mathcal{A}}$ and contains $f$, then it contains \n$w$. But then, as $M\\in\\mathcal{A}$, $\\mathcal{A}\\not\\models K\\neg w$, a \ncontradiction. It follows that there is no stable model containing\n$f$. That is, the program consisting of the following rules has no\nstable model:\n\\begin{enumerate}\n\\item $x$, for $x\\in A$ and $x'$, for $x\\in X\\setminus A$\n\\item $y \\vee y'$; and $z \\vee z'$, for every $y\\in Y$ and $z\\in Z$\n\\item $w \\rightarrow z$; and $w \\rightarrow z'$, for every $z\\in Z$\n\\item $\\sigma(u_1)\\land\\sigma(u_2)\\land\\sigma(u_3)\\rightarrow w$, for\nevery clause $C= u_1\\vee u_2 \\vee u_3$ of $\\Theta$, where $\\sigma(a)=a'$\nand $\\sigma(\\neg a)= a$, for every $a\\in X\\cup Y \\cup Z$.\n\\item $\\neg w \\rightarrow w$\n\\end{enumerate}\nBut then, the formula $\\forall Y\\exists Z \\Theta_{|X\/A}$ is true and,\nconsequently, the formula $\\exists X \\forall Y\\exists Z \\Theta$ is\ntrue, too. \\hfill$\\Box$\n\nFor the other two epistemic semantics, Remark 1 implies that the problem\nof the existence of an epistemic model (epistemic supported model) is\nin the class $\\Sigma_2^P$. The $\\Sigma_2^P$-hardness of the problem\ncan be proved by similar techniques as those we used for the case of\nepistemic stable models. Thus, we have the following result.\n\n\\begin{theorem}\n\\label{thm:3}\nThe problem to decide whether an epistemic program $P\\subseteq\\mathcal{L}_K$\nhas an epistemic model (epistemic supported model, respectively) is\n$\\Sigma_2^P$-complete.\n\\end{theorem}\n\n\\section{Modeling with Epistemic Programs}\n\nWe will now present several problems which illustrate the \nadvantages offered by the language of epistemic programs we developed\nin the previous two sections. Whenever we use predicate programs, we \nunderstand that their semantics is that of the corresponding ground \nprograms.\n\nFirst, we consider two graph problems related to the existence of \nHamiltonian cycles. Let $G$ be a directed graph. An edge in $G$ is \n\\emph{critical} if it belongs to every hamiltonian cycle in $G$. \nThe following problems are of interest:\n\\begin{enumerate}\n\\item Given a directed graph $G$, find the set of all critical \nedges of $G$\n\\item Given a directed graph $G$, and integers $p$ and $k$, find a set\n$R$ of no more than $p$ new edges such that $G\\cup R$ has no more than\n$k$ critical edges.\n\\end{enumerate}\n\nLet $HC(vtx,edge)$ be any standard ASP encoding of the Hamiltonian \ncycle problem, in which predicates $vtx$ and $edge$ represent $G$, \nand a predicate $hc$ represents edges of a candidate hamiltonian \ncycle. We assume the rules of $HC(vtx,edge)$ are written from left\nto right so that they can be regarded as elements of $\\mathcal{L}$. Then, \nsimply adding to $HC(vtx,edge)$ the rule:\n\\begin{quote}\n$K hc(X,Y) \\rightarrow critical(X,Y)$\n\\end{quote}\nyields a correct representation of the first problem. We write\n$HC_{cr}(vtx,edge)$ to denote this program.\nAlso, for a directed graph $G=(V,E)$,\nwe define \n\\begin{quote}\n$D=\\{vtx(v)\\,|\\; v\\in V\\} \\cup \\{edge(v,w)\\,|\\; (v,w)\\in E\\}$.\n\\end{quote}\nWe have the following result.\n\n\\begin{theorem}\nLet $G=(V,E)$ be a directed graph. If $HC_{cr}(vtx,edge) \\cup D$ has \nno epistemic stable models, then every edge in $G$ is critical \n(trivially). Otherwise, the epistemic program\n$HC_{cr}(vtx,edge)\\cup D$ has a unique \nepistemic stable model $\\mathcal{A}$ and the set $\\{(v,w)\\,|\\; \\mathcal{A}\\models \ncritical(u,v) \\}$ is the set of critical edges in $G$.\n\\end{theorem}\nProof (Sketch): Let $H$ be the grounding of $HC_{cr}(vtx,edge) \\cup D$.\nIf $H$ has no epistemic stable models, it follows that the ``non-epistemic''\npart $H'$ of $H$ has no stable models (as no atom of the form \n$critical(x,y)$ appears in it). As $H'$ encodes the existence of a hamiltonian\ncycle in $G$, it follows that $G$ has no Hamiltonian cycles. Thus, trivially,\nevery edge of $G$ belongs to every Hamiltonian cycle of $G$ and so, every\nedge of $G$ is critical.\n\nThus, let us assume that $\\mathcal{A}$ is an epistemic stable model of $H$. \nAlso, let $S$ be the set of all stable models of $H'$ (they correspond \nto Hamiltonian cycles of $G$; each model contains, in particular, atoms \nof the form $hc(x,y)$, where $(x,y)$ ranges over the edges of the \ncorresponding Hamiltonian cycle). The reduct $\\gl{H}{\\mathcal{A}}$ consists of \n$H'$ (non-epistemic part of $H$ is unaffected by the reduct operation) \nand of $C'$, a set of some facts of the form $critical(x,y)$. Thus, the \nstable models of the reduct are of the form $M\\cup C'$, where $M\\in S$.\nThat is, $\\mathcal{A}=\\{M\\cup C'\\,|\\; M\\in S\\}$. Let us denote by $C$ the set\nof the atoms $critical(x,y)$, where $(x,y)$ belongs to every hamiltonian \ncycle of $G$ (is critical). One can compute now that $\\gl{H}{\\mathcal{A}} = \nH'\\cup C$. Since $\\mathcal{A}=\\mathcal{ST}(\\gl{H}{\\mathcal{A}})$, $\\mathcal{A}= \\{M\\cup C\\,|\\; M\\in S\\}$. \nThus, $HC_{cr}(vtx,edge)\\cup D$ has a unique epistemic stable model, as \nclaimed. It also follows that the set $\\{(v,w)\\,|\\; \\mathcal{A}\\models \ncritical(u,v) \\}$ is the set of critical edges in $G$.\n\\hfill$\\Box$\n\nTo represent the second problem, we proceed as follows. First, we\n``select'' new edges to be added to the graph and impose constraints\nthat guarantee that all new edges are indeed new, and that no more than $p$\nnew edges are selected (we use here \\emph{lparse} syntax for brevity; the\nconstraint can be encoded strictly in the language $\\mathcal{L}_K$).\n\\begin{quote}\n$vtx(X)\\land vtx(Y) \\rightarrow newEdge(X,Y)$\\\\\n$newEdge(X,Y)\\land edge(X,Y)\\rightarrow\\bot$\\\\\n$(p+1) \\{newEdge(X,Y): vtx(X), vtx(Y)\\} \\rightarrow \\bot$.\n\\end{quote}\nNext, we define the set of edges of the extended graph, using a predicate\n$edgeEG$:\n\\begin{quote}\n$edge(X,Y)\\rightarrow edgeEG(X,Y)$\\\\\n$newEdge(X,Y)\\rightarrow edgeEG(X,Y)$\n\\end{quote}\nFinally, we define critical edges and impose a constraint on their \nnumber (again, exploiting the \\emph{lparse} syntax for brevity sake):\n\\begin{quote}\n$edgeEG(X,Y)\\land K hc(X,Y) \\rightarrow critical(X,Y)$\\\\\n$(k+1)\\{critical(X,Y): edgeEG(X,Y)\\}\\rightarrow \\bot$.\n\\end{quote}\nWe define $Q$ to consist of all these rules together with all the rules\nof the program $HC(vtx,edgeEG)$. We now have the following theorem. The\nproof is similar to that above and so we omit it.\n \n\\begin{theorem}\nLet $G$ be a directed graph. There is an extension of $G$ with no more\nthan $p$ new edges so that the resulting graph has no more than $k$ \ncritical edges if and only if the program $Q\\cup D$ has an epistemic \nstable model.\n\\end{theorem}\n \nFor another example we consider the unique model problem: given a CNF \nformula $F$, the goal is to decide whether $F$ has a unique minimal \nmodel. The unique model problem was also considered by Faber and Woltran\n\\cite{FaberW09}. We will show two encodings of the problem by means of\nepistemic programs. The first one uses the semantics of epistemic \nmodels and is especially direct. The other one uses the semantics of\nepistemic stable models. \n\nLet $F$ be a propositional theory consisting of constraints $L_1\\land\n\\ldots\\land L_k\\rightarrow \\bot$, where $L_i$'s are literals. Any propositional\ntheory can be rewritten into an equivalent theory of such form. We \ndenote by $F^K$ the formula obtained from $F$ by replacing every atom \n$x$ with the modal atom $Kx$. \n\n\\begin{theorem}\nFor every theory $F\\subseteq\\mathcal{L}$ consisting of constraints, $F$ has a \nleast model if and only if the epistemic program $F\\cup F^K$ has an\nepistemic model.\n\\end{theorem}\nProof: Let us assume that $F$ has a least model. We define $\\mathcal{A}$ to\nconsist of all models of $F$, and we denote the least model of $F$ by \n$M$. We will show that $\\mathcal{A}$ is an epistemic model of $F\\cup F^K$. \nClearly, for every $x\\in M$, $\\mathcal{A}\\models Kx$. Similarly, for every \n$x\\not\\in M$, $\\mathcal{A}\\models \\neg Kx$. Thus, $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$.\nConsequently, $\\gl{[F\\cup F^K]}{\\mathcal{A}} = F$ and so, $\\mathcal{A}$ is precisely \nthe set of all models of $\\gl{[F\\cup F^K]}{\\mathcal{A}}$. Thus, $\\mathcal{A}$ is an \nepistemic model. \n\nConversely, let $\\mathcal{A}$ be an epistemic model of $F\\cup F^K$. It follows\nthat $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$ (otherwise, $\\gl{[F\\cup F^K]}{\\mathcal{A}}$ \ncontains $\\bot$ and $\\mathcal{A}$ would have to be empty, contradicting the definition\nof an epistemic model). Thus, $\\gl{[F\\cup F^K]}{\\mathcal{A}}=F$ and consequently, \n$\\mathcal{A}$ is the set of all\nmodels of $F$. Let $M=\\{x\\in\\mathit{At}\\,|\\; \\mathcal{A}\\models Kx\\}$ and let\n\\begin{equation}\n\\label{eq17}\na_1\\land\\ldots\\land a_m \\land\\neg b_1\\land\\ldots\\land\\neg b_n \\rightarrow \\bot\n\\end{equation}\nbe a rule in $F$. Then, \n\\[\nK a_1\\land\\ldots\\land K a_m \\land\\neg K b_1\\land\\ldots\\land\\neg K b_n\n\\rightarrow \\bot\n\\]\nis a rule in $F^K$. As $\\gl{[F^K]}{\\mathcal{A}}=\\emptyset$,\n\\[\n\\mathcal{A}\\not\\models K a_1\\land\\ldots\\land K a_m \\land\\neg K b_1\\land\\ldots\\land\n\\neg K b_n.\n\\]\nThus, for some $i$, $1\\leq i\\leq m$, $\\mathcal{A}\\not\\models K a_i$, or for \nsome $j$, $1\\leq j\\leq n$, $\\mathcal{A}\\models K b_j$. In the first case, $a_i\n\\notin M$, in the latter, $b_j\\in M$. In either case, $M$ is a model\nof rule (\\ref{eq17}). It follows that $M$ is a model of $F$. Let $M'$\nbe a model of $F$. Then $M'\\in\\mathcal{A}$ and, by the definition of $M$, $M\n\\subseteq M'$. Thus, $M$ is a least model of $F$.\n\\hfill$\\Box$\n \nNext, we will encode the same problem as an epistemic program under the\nepistemic stable model semantics. The idea is quite similar. We\nonly need to add rules to generate all candidate models.\n\n\\begin{theorem}\nFor every theory $F\\subseteq\\mathcal{L}$ consisting of constraints, $F$ has a\nleast model if and only if the epistemic program \n\\[\nF\\cup F^K \\cup \\{\\neg x\\rightarrow x'\\,|\\; x\\in \\mathit{At}\\}\\cup \\{\\neg x'\\rightarrow x\\,|\\; \nx\\in \\mathit{At}\\}\n\\]\nhas an epistemic stable model.\n\\end{theorem}\n\nWe note that an even simpler encoding can be obtained if we use \n\\emph{lparse} choice rules. In this case, we can replace \n$\\{\\neg x\\rightarrow x'\\,|\\; x\\in \\mathit{At}\\}\\cup \\{\\neg x'\\rightarrow x\\,|\\; x\\in \\mathit{At}\\}$ with\n$\\{\\{x\\}\\,|\\; x\\in\\mathit{At}\\}$.\n\n\\section{Connection to Gelfond's Epistemic Programs}\n\nWe will now return to the original formalism of epistemic specifications\nproposed by Gelfond \\cite{Gelfond91} (under the restriction to epistemic\nprograms we discussed here). We will show that it can be expressed in a\nrather direct way in terms of our epistemic programs in the two-valued\nsetting and under the epistemic supported-model semantics.\n \nThe reduction we are about to describe is similar to the well-known\none used to eliminate the ``strong'' negation from disjunctive logic\nprograms with strong negation. In particular, it requires an extension \nto the language $\\mathcal{L}$. Specifically, for every atom $x\\in \\mathit{At}$ we \nintroduce a fresh atom $x'$ and we denote the extended language by \n$\\mathcal{L}'$. The intended role of $x'$ is to represent in $\\mathcal{L}'$ the literal \n$\\neg x$ from $\\mathcal{L}$. Building on this idea, we assign to each set $W$ of\nliterals in $\\mathcal{L}$ the set\n\\[\nW'=(W\\cap \\mathit{At}) \\cup \\{x'\\,|\\; \\neg x\\in W\\}.\n\\]\nIn this way, sets of literals from $\\mathcal{L}$ (in particular, three-valued\ninterpretations of $\\mathcal{L}$) are represented as sets of atoms from $\\mathcal{L}'$\n(two-valued interpretations of $\\mathcal{L}'$).\n\nWe now note that the truth and falsity of a formula form $\\mathcal{L}$ under\na three-valued interpretation can be expressed as the truth and\nfalsity of certain formulas from $\\mathcal{L}'$ in the two-valued setting.\nThe following result is well known.\n\n\\begin{proposition}\n\\label{prop:2}\nFor every formula $\\varphi\\in\\mathcal{L}$ there are formulas\n$\\varphi^-, \\varphi^+\\in\\mathcal{L}'$ such that for every set of literals $W$\n(in $\\mathcal{L}$)\n\\begin{enumerate}\n\\item $v_W(\\varphi)=\\mathbf{t}$ if and only if $u_{W'}(\\varphi^+)=\\mathbf{t}$\n\\item $v_W(\\varphi)=\\mathbf{f}$ if and only if $u_{W'}(\\varphi^-)=\\mathbf{f}$\n\\end{enumerate}\nMoreover, the formulas $\\varphi^-$ and $\\varphi^+$ can be constructed in \npolynomial time with respect to the size of $\\varphi$.\n\\end{proposition}\nProof: This a folklore result. We provide a sketch of a proof for the\ncompleteness sake. We define $\\varphi^+$ and $\\varphi^-$ by recursively as\nfollows:\n\\begin{enumerate}\n\\item $x^+ = x$ and $x^- = \\neg x'$, if $x\\in\\mathit{At}$\n\\item $(\\neg \\varphi)^+ = \\neg\\varphi^-$ and $(\\neg \\varphi)^- = \\neg\\varphi^+$\n\\item $(\\varphi\\lor \\psi)^+=\\varphi^+\\lor \\psi^+$ and \n$(\\varphi\\lor \\psi)^-=\\varphi^-\\lor \\psi^-$; the case of the conjunction is dealt\nwith analogously\n\\item $(\\varphi\\rightarrow\\psi)^+ = \\varphi^-\\rightarrow\\psi^+$ and \n$(\\varphi\\rightarrow\\psi)^- = \\varphi^+\\rightarrow\\psi^-$.\n\\end{enumerate}\nOne can check that formulas $\\varphi^+$ and $\\varphi^-$ defined in this way satisfy\nthe assertion. \\hfill$\\Box$\n\n\\smallskip\nWe will now define the transformation $\\sigma$ that allows us to eliminate\nstrong negation. First, for a literal $L\\in \\mathcal{L}$, we now define\n\\[\n\\sigma(L) = \\left\\{ \\begin{array}{ll}\n x & \\mbox{if $L=x$}\\\\\n x' & \\mbox{if $L=\\neg x$}\n \\end{array}\n \\right.\n\\]\nFurthermore, if $E$ is a simple modal literal or its default negation, \nwe define\n\\[\n\\sigma(E) = \\left\\{ \\begin{array}{ll}\n K\\varphi^+ & \\mbox{if $E=K\\varphi$}\\\\\n \\neg K\\varphi^{-} & \\mbox{if $E=\\neg K\\varphi$}\\\\\n \\neg K\\varphi^{+} & \\mbox{if $E=\\mathit{not\\;} K\\varphi$}\\\\\n K\\varphi^{-} & \\mbox{if $E=\\mathit{not\\;} \\neg K\\varphi$}\n \\end{array}\n \\right.\n\\]\nand for an epistemic premise $E = E_1,\\ldots, E_t$\n(where each $E_i$ is a simple modal literal or its default negation) we set\n\\[\n\\sigma(E) = \\sigma(E_1)\\land\\ldots\\land \\sigma(E_t).\n\\]\nNext, if $r$ is an epistemic rule\n\\[\nL_1 \\vee \\ldots\\vee L_k \\leftarrow F_1,\\ldots, F_m,\\mathit{not\\;} F_{m+1},\\ldots, \\mathit{not\\;} F_n, E\n\\]\nwe define\n\\[\n\\sigma(r) = \\sigma(E)\\land \\sigma(F_1)\\land\\ldots\\land \\sigma(F_m)\\land \\neg\n\\sigma(F_{m+1})\\land\\ldots\\land \\neg \\sigma(F_n) \\rightarrow \n\\sigma(L_1) \\vee \\ldots\\vee \\sigma(L_k).\n\\]\nFinally, for an epistemic program $P$, we set \n\\[\n\\sigma(P)=\\{\\sigma(r)\\,|\\; r\\in P\\}) \\cup\\{x\\land x' \\rightarrow \\bot\\}.\n\\]\nWe note that $\\sigma(P)$ is indeed an epistemic program\nin the language $\\mathcal{L}_K$ (according to our definition of epistemic \nprograms). The role of the rules $x\\land x' \\rightarrow \\bot$\nis to ensure that sets forming epistemic (stable, supported) models \nof $\\sigma(P)$ correspond to consistent sets of literals (the only type \nof set of literals allowed in world views).\n\nGiven a three-valued possible structure $\\mathcal{A}$, we define $\\mathcal{A}'=\\{W'\\,|\\;\nW\\in \\mathcal{A}\\}$, and we regard $\\mathcal{A}'$ as a two-valued possible-world\nstructure. We now have the following theorem.\n\n\\begin{theorem}\nLet $P$ be an epistemic program according to Gelfond. Then a three-valued\npossible-world structure $\\mathcal{A}$ is a world view of $P$ if and only if \na two-valued possible-world structure $\\mathcal{A}'$ is an epistemic supported\nmodel of $\\sigma(P)$.\n\\end{theorem}\nProof (Sketch): Let $P$ be an epistemic program according to Gelfond, $\\mathcal{A}$\na possible-world structure and $W$ a set of literals. We first \nobserve that the G-reduct $P^{\\langle \\mathcal{A},W\\rangle}$ can be described as the\nresult of a certain two-step process. Namely, we define the \n\\emph{epistemic reduct} of $P$ with respect to $\\mathcal{A}$ to be the disjunctive\nlogic program $P^\\mathcal{A}$ obtained from $P$ by removing every rule whose\nepistemic premise $E$ satisfies $\\mathcal{A}\\not\\models E$, \nand by removing the epistemic\npremise from every other rule in $P$. This construction is the three-valued\ncounterpart to the one we employ in our approach.\nIt is clear that the epistemic reduct of $P$ with respect to $\\mathcal{A}$, with \nsome abuse of notation we will denote it by $\\gl{P}{\\mathcal{A}}$, is a disjunctive\nlogic program with strong negation. \n\nLet $Q$ be a disjunctive program with strong negation and $W$ a set \nof literals. By the \n\\emph{supp-reduct} of $Q$ with respect to $W$, $R^{sp}(Q,W)$, we mean \nthe set of the heads of all rules whose bodies are satisfied by $W$ \n(which in the three-valued setting means that every literal in the body \nnot in the scope of $\\mathit{not\\;}$ is in $W$, and every literal in the body in \nthe scope of $\\mathit{not\\;}$ is not in $W$). A consistent set $W$ of literals is \na supported answer set of $Q$ if $W\\in\\mathit{Min}(R^{sp}(Q,W))$ (this is a \nnatural extension of the definition of a supported model \\cite{abw87,bg93}\nto the case of disjunctive logic programs with strong negation; again,\nwe do not regard inconsistent sets of literals as supported answer sets).\n\nClearly, $P^{\\langle \\mathcal{A},W\\rangle} = R^{sp}(\\gl{P}{\\mathcal{A}},W)$. Thus, $\\mathcal{A}$ is a\nworld view of $P$ according to the definition by Gelfond if and only if\n$\\mathcal{A}$ is a collection of all supported answer sets of $\\gl{P}{\\mathcal{A}}$. \n\nWe also note that by Proposition \\ref{prop:2}, if $E$ is an epistemic\npremise, then $\\mathcal{A}\\models E$ if and only if $\\mathcal{A}'\\models \\sigma(E)$.\nIt follows that $\\sigma(\\gl{P}{\\mathcal{A}}) = \\gl{\\sigma(P)}{\\mathcal{A}'}$. In other \nwords, constructing\nthe epistemic reduct of $P$ with respect to $\\mathcal{A}$ and then translating \nthe resulting disjunctive logic program with strong negation into the\ncorresponding disjunctive logic program without strong negation yields \nthe same result as first translating the epistemic program (in the\nGelfond's system) into our language of epistemic programs and then computing\nthe reduct with respect to $\\mathcal{A}'$. We note that there is a one-to-one\ncorrespondence between supported answer sets of $\\gl{P}{\\mathcal{A}}$ and supported\nmodels of $\\sigma(\\gl{P}{\\mathcal{A}})$ ($\\sigma$, when restricted to programs\nconsisting of rules without epistemic premises, is the standard transformation\neliminating strong negation and preserving the stable and supported \nsemantics). Consequently, there is a one-to-one\ncorrespondence between supported answer sets of $\\gl{P}{\\mathcal{A}}$ and supported\nmodels of $ \\gl{\\sigma(P)}{\\mathcal{A}'}$ (cf. our observation above). Thus, \n$\\mathcal{A}$ consists of supported answer \nsets of $\\gl{P}{\\mathcal{A}}$ if and only if $\\mathcal{A}'$ consists of supported models\nof $\\gl{\\sigma(P)}{\\mathcal{A}'}$. Consequently, $\\mathcal{A}$ is a world view of $P$\nif and only if $\\mathcal{A}'$ is an epistemic supported model of $\\sigma(P)$.\n\\hfill$\\Box$ \n\n\\section{Epistemic Models of Arbitrary Theories}\n\nSo far, we defined the notions of epistemic models, epistemic stable\nmodels and epistemic supported models only for the case of epistemic \nprograms. However, this restriction is not essential. We recall that \nthe definition of these three epistemic semantics consists of two \nsteps. The first step produces the reduct of an epistemic program $P$\nwith respect to a possible-world structure, say $\\mathcal{A}$. This reduct \nhappens to be (modulo a trivial syntactic transformation) a standard \ndisjunctive logic program in the language $\\mathcal{L}$ (no modal atoms \nanymore). If the set of models (respectively, stable models, supported \nmodels) of the reduct program coincides with $\\mathcal{A}$, $\\mathcal{A}$ is an \nepistemic model (respectively, epistemic stable or supported model) \nof $P$. However, the concepts of a model, stable model and supported \nmodel are defined for \\emph{arbitrary} theories in $\\mathcal{L}$. This is \nobviously well known for the semantics of models. The stable-model \nsemantics was extended to the full language $\\mathcal{L}$ by Ferraris \n\\cite{fer05} and the supported-model semantics by Truszczynski\n\\cite{tr10}. Thus, there is no reason precluding the extension of the\ndefinition of the corresponding epistemic types of models to the general\ncase. We start be generalizing the concept of the reduct.\n\n\\begin{definition}\nLet $T$ be an arbitrary theory in $\\mathcal{L}_K$ and let $\\mathcal{A}$ be a\npossible-world structure. The \\emph{epistemic reduct} of $T$ with\nrespect to $\\mathcal{A}$, $\\gl{T}{\\mathcal{A}}$ in symbols, is the theory obtained\nfrom $T$ by replacing each maximal modal atom $K\\varphi$ with $\\top$,\nif $\\mathcal{A}\\models K\\varphi$, and with $\\bot$, otherwise.\n\\end{definition}\n\nWe note that if $T$ is an epistemic program, this notion of the reduct\ndoes not coincide with the one we discussed before. Indeed, now no rule \nis dropped and no modal literals are dropped; rather modal atoms are \nreplaced with $\\top$ and $\\bot$. However, the replacements are executed\nin such a way as to ensure the same behavior. Specifically, one can show\nthat models, stable models and supported models of the two reducts \ncoincide.\n\nNext, we generalize the concepts of the three types of epistemic models.\n\n\\begin{definition}\n\\label{def14}\nLet $T$ be an arbitrary theory in $\\mathcal{L}_K$. A possible-world\nstructure $\\mathcal{A}$ is an \\emph{epistemic model} (respectively, an\n\\emph{epistemic stable model}, or an \\emph{epistemic supported model})\nof $P$, if $\\mathcal{A}$ is the set of models (respectively, stable models or\nsupported models) of $\\mathcal{M}(\\gl{P}{\\mathcal{A}})$. \n\\end{definition}\n\nFrom the comments we made above, it follows that if $T$ is an epistemic \nprogram, this more general definition yields the came notions of epistemic\nmodels of the three types as the earlier one. \n\nWe note that even in the more general setting the complexity of reasoning\nwith epistemic (stable, supported) models remains unchanged. Specifically,\nwe have the following result.\n\n\\begin{theorem}\n\\label{thm:4}\nThe problem to decide whether an epistemic theory $T\\subseteq\\mathcal{L}_K$\nhas an epistemic stable model is $\\Sigma_3^P$-complete. The problem to \ndecide whether an epistemic theory $T\\subseteq\\mathcal{L}_K$ has an epistemic \nmodel (epistemic supported model, respectively) is $\\Sigma_2^P$-complete.\n\\end{theorem}\nProof(Sketch): The hardness part follows from our earlier results concerning\nepistemic programs. To prove membership, we modify Proposition\n\\ref{prop:char}, and show a polynomial time algorithm with a \n$\\Sigma_2^P$ oracle (NP oracle for the last two problems) that\ndecides, given a propositional theory $S$ and a modal formula $K\\varphi$\n(with $\\varphi\\in\\mathcal{L}_K$ and not necessarily in $\\mathcal{L}$) whether $\\mathcal{ST}(S)\n\\models K\\varphi$ (respectively, $\\mathcal{M}(S)\\models K\\varphi$, or $\\mathcal{SP}(S)\n\\models K\\varphi$). \\hfill$\\Box$\n\n\\section{Discussion}\n\nIn this paper, we proposed a two-valued formalism of epistemic \ntheories --- subsets of the language of modal propositional logic.\nWe proposed a uniform way, in which semantics of propositional \ntheories (the classical one as well as nonmonotonic ones: stable \nand supported) can be extended to the case of epistemic theories.\nWe showed that the semantics of epistemic supported models is closely\nrelated to the original semantics of epistemic specifications proposed\nby Gelfond. Specifically we showed that the original formalism of Gelfond\ncan be expressed in a straightforward way by means of epistemic programs\nin our sense under the semantics of epistemic supported models. Essentially\nall that is needed is to use fresh symbols $x'$ to represent strong \nnegation $\\neg x$, and use the negation operator of our formalism, \n$\\varphi \\rightarrow \\bot$ or, in the shorthand, $\\neg \\varphi$, to model the default \nnegation $\\mathit{not\\;} \\varphi$. \n\nWe considered in more detail the three semantics mentioned above. \nHowever, other semantics may also yield interesting epistemic\ncounterparts. In particular, it is clear that Definition \\ref{def14}\ncan be used also with the minimal model semantics or with the \nFaber-Leone-Pfeifer semantics \\cite{flp04}. Each semantics gives rise\nto an interesting epistemic formalism that warrants further studies.\n\nIn logic programming, eliminating strong negation does not result in \nany loss of the expressive power but, at least for the semantics of\nstable models, disjunctions cannot be compiled away in any concise way \n(unless the polynomial hierarchy collapses). In the setting of \nepistemic programs, the situation is similar. The strong negation can be \ncompiled away. But the availability of disjunctions in the heads and \nthe availability of epistemic premises in the bodies of rules are \nessential. Each of these factors separately brings the complexity one \nlevel up. Moreover, when used together under the semantics of epistemic \nstable models they bring the complexity two levels up. This points to \nthe intrinsic importance of having in a knowledge representation \nlanguage means to represent indefiniteness in terms of disjunctions, \nand what is known to a program (theory) --- in terms of a modal operator \n$K$.\n\n\n\\section*{Acknowledgments}\n\\vspace*{-0.1in}\nThis work was partially supported by the NSF grant IIS-0913459.\n\n{\\small\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}