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+{"text":"\\section{Introduction}\n\\label{sec:intro}\nCircinus~X-1 exhibits a dramatic 16.55-d cycle of X-ray flaring which\nis believed to be the result of enhanced mass transfer occurring near\nperiastron of a highly eccentric binary orbit (\\cite{kaluzienski76};\n\\cite{murdin80}).  Although the identification of the orbital period\nseems to be secure, there is very little direct evidence concerning\nthe masses of the two components of the binary and the other orbital\nparameters. Nonetheless, the compact component is thought to be a\nweakly magnetized neutron star on the basis of three type~I X-ray\nbursts seen with {\\it EXOSAT\\,}\\ (\\cite{tennant86}).  Further type~I bursts\nhave not been observed from Cir X-1 since the {\\it EXOSAT\\,}\\ discovery,\npossibly because the source intensity has been higher during\nsubsequent observations.  No coherent pulsations, which would be\nexpected to be present if the compact star is a strongly magnetized\nneutron star, have been detected (\\cite{dower82}; \\cite{vaughan94}).\n\nQuasi-periodic oscillations (QPOs) were seen at 1.4~Hz, 5--20~Hz,\nand 100--200~Hz in {\\it EXOSAT\\,}\\ observations of Cir~X-1 in a bright state\n(\\cite{tennant87}; \\cite{tennant88}), but other observations at lower\nintensity showed no such QPOs (\\cite{oosterbroek95}). Based on these\ndata, it was suggested that Cir~X-1 is an atoll-source low-mass X-ray\nbinary (LMXB) that can uniquely reach the Eddington accretion rate and\nexhibit normal\/flaring branch QPOs (NBOs\/FBOs) at 5--20~Hz\n(\\cite{oosterbroek95}; \\cite{klis94}). Similar QPOs were observed in\n{\\it Ginga\\,}\\ observations of Cir~X-1 (\\cite{makino93}).\n\nObservations with the All-Sky Monitor (ASM) and Proportional Counter\nArray (PCA) on the {\\em Rossi X-ray Timing Explorer} ({\\it RXTE\\,}) have\nshown that since early 1996 (from the beginning of {\\it RXTE\\,}\\ monitoring)\nthe baseline intensity level of Cir~X-1 has remained bright\n($\\sim$1~Crab, 2--12~keV; $\\sim$1060~$\\mu$Jy at 5.2~keV) at all phases\nof the 16.55-d cycle, with both dips and flares associated with phase\nzero of the cycle (Shirey et~al.\\ 1996\\nocite{shirey96},\n1998\\nocite{shirey98:feb97}, hereafter Papers I \\& II;\n\\cite{shirey98:phd}).  During non-flaring phases (intensity $\\approx$\n1~Crab) of a cycle in 1996 March, the centroid frequency of a narrow\nQPO was observed to evolve between 1.3~Hz and 12~Hz and was strongly\ncorrelated with the cutoff frequency of low-frequency noise and with\nthe centroid frequency of a broad peak ranging from 20--100~Hz, at\n$\\sim$13 times the frequency of the lower frequency QPO\n(\\cite{shirey96})\\@. During portions of a more active cycle in 1997\nFebruary--March, a similar narrow QPO evolved between 6.8--32~Hz,\nwhile during other portions of the cycle, a broad QPO was observed at\n4~Hz (\\cite{shirey98:feb97})\\@.  By correlating the timing properties\nwith fragmented branches in hardness-intensity diagrams, we identified\nhorizontal, normal, and flaring branches. Thus, we interpreted the\nnarrow 1.3--32~Hz QPOs in Cir~X-1, including the 5--20~Hz QPOs\nobserved with {\\it EXOSAT\\,}, as horizontal-branch oscillations (HBOs), and\nthe broad 4~Hz QPOs as NBOs, thus calling into question the earlier\ninterpretation of the QPOs in Cir~X-1.\n\nIn this paper we present results from a 10-day set of high-efficiency\n{\\it RXTE\\,}\\ observations, including 56\\% coverage for 7~d starting 1.5~d\nbefore phase zero. Portions of these data show a complete Z-source\ntrack for Cir~X-1. These results confirm our previous interpretation\nbased on incomplete and shifted spectral tracks constructed from\nshorter and more widely spaced observations.  We show how both the\nFourier power spectra and the energy spectra evolve along the various\nspectral tracks.  We explore possible models for the energy spectrum\nand parameterize the evolution by fitting the data at various points\nalong the hardness-intensity track to a model consisting of a\nmulti-temperature ``disk blackbody'' and a higher-temperature\nisothermal blackbody.  \n\n\\section{Observations} \n\nThe PCA light curves and a hardness ratio (``broad color'') for the\n1997 June observations are shown in Figure~\\ref{fig:june97_10d}. These\ndata show only moderate variability for the first day of\nobservations. The source entered a phase of significant dipping during\nthe half day before phase zero (day 611.5), based on the radio\nephemeris (\\cite{stewart91}). The hardness ratio shows that dramatic\nspectral evolution, both hardening and softening, occurs during these\ndips.\n\nBy phase zero the main dipping episode ends, and the transition to the\nflaring state begins. While the intensity increases by more than a\nfactor of three in the lowest energy band (2--6.3~keV), the intensity\nbetween 6.3~keV and 13~keV does not climb at all, and above 13~keV it\ndecreases by a factor of about 10 over the first 1.5~days following\nphase zero. This anti-correlation of the low and high-energy intensity\nduring the transition results in a decreased hardness ratio after\nphase zero, as is observed in {\\it RXTE\\,}\\ ASM data (Papers\nI\\nocite{shirey96} \\& II\\nocite{shirey98:feb97};\n\\cite{shirey98:phd}). \n\nAfter a relatively smooth transition toward high total intensity\n(2--18~keV) during the first day following phase zero, the intensity\nbecomes highly variable (i.e., the ``active'' or ``flaring'' state)\nfor the remaining 7 days of the observation. The intensity in the hard\nband remains much lower than before phase zero but shows strong\nvariability with a peak value that gradually increases from about 24 to\n36 counts~s$^{-1}$~PCU$^{-1}$.  Based on the results in\n\\cite{shirey98:feb97} and those discussed below, we can identify\nshort-term variability within restricted limits as motion along\nbranches in hardness-intensity diagrams and the gradual evolution of\nthe limits as shifting of the branches.\n\n\n\\section{Complete Track in Hardness-intensity and Color-color Diagrams}\n\nThe color-color and hardness-intensity diagrams (CDs\/HIDs) for all\ndata in Figure~\\ref{fig:june97_10d} are shown in\nFigure~\\ref{fig:june97_cchid_all}. Only data from three of the five\nPCA detectors, PCUs 0, 1, and 2, were included in the diagrams, since\nPCUs 3 and 4 were not operating during portions of the observations.\nThese data cover a significant portion (10~d) of an entire 16.55~d\ncycle. Data from before day 612.5 generally had soft color ${\\stackrel{>}{_\\sim}}$\n1.5 and broad color ${\\stackrel{>}{_\\sim}}$ 0.325, while the data after day 612.5\ngenerally fell below those values.\n\nThe dips seen in Figure~\\ref{fig:june97_10d} appear as prominent but\nless dense tracks with two sharp bends in the CD (initially toward the\nright of the main arc-shaped locus) and one sharp bend in the HID\n($I<2.3$~kcounts s$^{-1}$ PCU$^{-1}$).  We show in another paper\n(Shirey, Levine, \\& Bradt 1999\\nocite{shirey99:dips}) that tracks with\nthese shapes are due to a variably absorbed bright spectral component\nplus an unobscured faint component.  Brandt et~al.\\\n(1996\\nocite{brandt96}) used a similar model to explain the spectral\nchanges during an intensity transition of Cir~X-1 observed with\n{\\it ASCA\\,}\\@.  Having identified absorption dip signatures, we now focus on\nnon-dip spectral behavior, which is presumably more directly related\nto the mechanisms of X-ray production.\n\nMost of the data in Figure~\\ref{fig:june97_cchid_all} fall along a\nsingle arc-shaped locus in the CD and a more complicated structure in\nthe HID\\@.  We use spectral bands with lower energies than those used\nfor the 1997 February--March observations in \\cite{shirey98:feb97};\nthis enhances the branch structure in the CD\\@.  When the diagrams for\nthe two observations are constructed in the same manner, with the same\nhardness ratios and only three PCUs rather than five, the data from\nthese two cycles cover approximately the same extent in both\ndiagrams. The high-efficiency coverage of the current observations\nresulted in tracks that are more complete than the tracks of the\nearlier observations. Unpublished data from several other cycles\nobserved with the PCA also fall in similar regions of the diagrams as\nthe data in Figure~\\ref{fig:june97_cchid_all}.\n\nThe detailed structure within the overall locus of CD\/HID points is\nrevealed in CD\/HID plots of data divided into shorter time segments\n(${\\stackrel{<}{_\\sim}}$12~h). In particular, four time segments labelled ``A'',\n``B'', ``C'', and ``D'' in Figure~\\ref{fig:june97_10d} have been\nselected for further study. The intensity from time segments A and B\n(days 609.93--610.16 and 610.66--610.90 respectively) shows minimal\nvariability in each energy channel, and thus these time segments\nproduce small CD\/HID clusters whose locations are indicated in\nFigure~\\ref{fig:june97_cchid_all}.\nThe source was more active during time segments C and D (days\n612.625--613.125 and 616.075--616.600 respectively), and the CD\/HID\ntracks from these times show several connected branches.  Enlarged\nviews of the CD and HID for these time segments are shown in\nFigure~\\ref{fig:june97_cchid_13_17}. The full ranges of the diagrams\nof Figure~\\ref{fig:june97_cchid_13_17} are indicated by dashed\nrectangular boxes in Figure~\\ref{fig:june97_cchid_all}.\nTracks of other time segments generally each resemble some portion of\nthe entire pattern shown in Figure~\\ref{fig:june97_cchid_13_17}, but\noften with a shifted position in the diagrams. \n\nThe data from segment~C in Figure~\\ref{fig:june97_cchid_13_17} include\nsome absorption dips, which result in tracks moving off the right side\nof the CD and the left side of the HID (and far beyond the limits of\nthe plot in both cases). The timing and spectral data associated with\nthese absorption dips will be omitted from analysis of the HID\nbranches.\n\nThe HID patterns reveal the shape of the full ``Z'' track as\npreviously inferred from the fragmented tracks in\n\\cite{shirey98:feb97}\\@.  Time segments C and D both show a horizontal\nbranch, a normal branch, and a flaring branch which turns above the\nnormal branch. In addition, segment~C exhibits a long nearly-vertical\nextension on the left end of the HB, while for segment~D, there is\nonly a small hint of an upward turn at the left end of the HB\\@. The\nHIDs show significant shifts of the HB and upper NB between the time\nsegments C and D, which were separated by several days.\n\nThe branches in the CDs of Figure~\\ref{fig:june97_cchid_13_17} are\nless well-separated than those in the HIDs. This was also the case for\nthe diagrams in \\cite{shirey98:feb97}\\@. However, the flaring branch\nclearly turns above the normal branch in the lower left part of the\ncurrent CDs, and the upturned left extension of the HID horizontal\nbranch of segment~C is marked by an increase in the slope in the upper\nright part of the associated CD.\n\nThe HID for segment~C is similar to that derived from\n{\\it RXTE\\,}\\ PCA observations of the Z~source Cyg~X-2 (\\cite{smale98}). The\nCyg~X-2 HID also shows a very prominent vertical extension of the\nHB\\@.  A similar upturned HB was reported in {\\it Ginga\\,}\\ and {\\it EXOSAT\\,}\\\nobservations of GX~5$-$1 (\\cite{lewin92:gx5-1};\n\\cite{kuulkers94:gx5-1}).\nThe upturned flaring branch is similar to the flaring branch observed\nin the CD for the Z~source GX~349+2 in recent {\\it RXTE\\,}\\ PCA observations\n(\\cite{zhang98}).\n\nThe HID track for time segment~C was divided into 20 regions which\nwere used to group data for further timing and spectral analysis.  The\n20 regions have been numbered as shown in\nFigure~\\ref{fig:june97_hid20reg}, with numbers increasing from the\nvertical HB, through the NB, to the FB\\@.  We show below that region~6\ndoes not adhere to the otherwise monotonic variation of\nspectral\/temporal characteristics with region number.  This region may\nbe an indication of an upward-shifted horizontal portion of the HB.\n\nDetails of the temporal variability of the intensity, hardness ratio,\nand HID region numbers for time segment~C are shown in\nFigure~\\ref{fig:june97_lc_hr_reg}.  During this half-day segment, the\nsource generally moves from lower to higher region number as the\nobservation progresses. Thus, the time series can be divided into four\nsub-segments which predominantly correspond to each portion of the HID\ntrack: the vertical and horizontal portions of the HB, the NB, and the\nFB\\@. Absorption dips occur in all but the flaring branch during this\nparticular data set; these are easily identified by brief intensity\ndips coupled with pronounced increases in broad color. No region\nnumber was assigned to most data points from dips since the HID\nregions in Figure~\\ref{fig:june97_hid20reg} were selected to avoid\ndips.\n\nThe light curves of the different branches\n(Fig.~\\ref{fig:june97_lc_hr_reg}) exhibit the following\ncharacteristics, excluding the behavior associated with the dips.\nWhen the source is on the vertical portion of the HB, the intensity\nevolves relatively smoothly, with a slight increase in the 2--6.3~keV\nband and a decrease of almost a factor of two in the 13--18~keV\nband. On the horizontal portion of the HB, the source shows a\nsubstantial increase in soft intensity and on average shows relatively\nsteady hard intensity. On the normal branch, the intensity is high in\nthe soft band while decreasing and highly variable in the hard band.\nThe NB\/FB transition occurs at lower intensity in all bands compared\nto most of the NB\\@. The flaring branch itself is then produced by\nhigh-variability ``mini-flares'' or bursts above the NB\/FB apex level\n(region~17).\n\nAlthough the HID regions were defined such that obvious absorption\ntracks were avoided, one brief dip, at day 612.98, occurred from\nregion 12 on the normal branch and placed a few points artificially\nacross regions 9, 7, and 5. These points are easily identified in\nFigure~\\ref{fig:june97_lc_hr_reg} (bottom plot) and are thus not\nincluded in subsequent timing and spectral analysis.\n\nLikewise, the highest mini-flares on the flaring branch actually\nextend beyond region~20 and cross regions 8 and 9. In fact a few such\npoints can even be seen above region~10 in\nFigure~\\ref{fig:june97_hid20reg}. The FB points that fell into HB\nregions can also be clearly identified as points with region numbers\nof 8 or 9 in the FB portion of\nFigure~\\ref{fig:june97_lc_hr_reg}. These are not included in\nsubsequent timing and spectral analysis.\n\n\\section{Evolution of the Power Density Spectrum}\n\nFourier power density spectra (PDSs) were computed for each 16~s of\ntime segment~C\\@.  Each transform used $2^{16}$ 244-$\\mu$s\n($2^{-12}$~s) time bins and covered the full 2--32~keV energy range.\nThe expected Poisson level, i.e., the level of white noise due to\ncounting statistics, was estimated taking into account the effects of\ndeadtime~(\\cite{morgan97}; \\cite{zhang95}; \\cite{zhang96}) and\nsubtracted from each PDS; this method tends to slightly underestimate\nthe actual Poisson level. For each of the 20 HID regions defined in\nFigure~\\ref{fig:june97_hid20reg}, an average PDS was calculated from\nthe power spectra corresponding to points in that region. The average\nPDSs were then logarithmically rebinned and are shown in\nFigure~\\ref{fig:june97_20pds}.\n\nThe general features of the power spectra are similar to those\nobserved in previous PCA observations (see Papers I\\nocite{shirey96}\n\\& II\\nocite{shirey98:feb97}).  In \\cite{shirey98:feb97}, based on\nfeatures of the power spectra associated with fragmented spectral\ntracks, we identified many of these features with those of a Z-source,\nand we discussed the properties of the QPOs on the horizontal and\nnormal branches. In the current observations, the characteristics of\nthe time variability are seen to evolve smoothly during a single 12-h\nobservation (segment~C).  The narrow QPO is observed to evolve from\n12~Hz in region~1 to 25~Hz in region~7 as the HID location moves down\nthe vertical extension of the HB (region 6 may be a shifted version of\n8). Across the horizontal portion of the HB (regions 8--11), the\nnarrow QPO fades into a knee close to 30~Hz, while the broad QPO\ngradually rises near 4~Hz. The broad QPO is present near 4~Hz along\nthe normal branch (regions 12--16). It is most prominently peaked in\nthe middle of the branch and weak at the bottom of the branch\n(region~17). On the flaring branch (regions 18--20), no QPOs are\npresent and the power spectrum shows only strong very low frequency\nnoise.\n\nThe PDS properties of the horizontal portion of the HB are somewhat\nsimilar to those of the upper normal branch of some Z sources, namely\nno significant evolution of the HBO frequency and weak NBOs.  However,\nwe will continue to refer to this branch as part of the the HB since\nother Z sources also show both vertical and horizontal portions of the\nHB (see above).\n\n\\section{Evolution of the Energy Spectrum}\n\\label{sec:june97_spec_qual}\n\nThe Standard2 data mode of the PCA instrument produces 129-channel\nenergy spectra every 16~s.  A parallel background file was constructed\nusing the ``pcabackest'' program\\footnote[1]{The background model was\ndefined in three files provided by the PCA instrument team at\nNASA\/GSFC: \\nl {\\tt pca\\_bkgd\\_q6\\_e03v01.mdl}, {\\tt\npca\\_bkgd\\_xray\\_e03v02.mdl}, and {\\tt pca\\_bkgd\\_activ\\_e03v03.mdl}.}\nprovided with the FTOOLS analysis package (version 4.0).  Average\npulse-height spectra (and background spectra) were constructed for\neach of the 20 HID regions, separately for each of the five\nPCUs. Version 2.2.1 response matrices were used in the analysis of\nthese spectra.  A 1\\% systematic error estimate was added in\nquadrature to the estimated statistical error (1~$\\sigma$) for each\nchannel of the spectra to account for calibration uncertainties.\nAlthough the instrument response matrix is imperfectly known, we can\nsafely assume that any spectral features that vary during the 12-hours\nspanned by time segment~C are due to evolution of the source spectrum.\nRepresentative spectra from the hard, bright, and soft extremes\n(regions 1, 11, and 17, respectively) of the evolution along the HID\ntrack are shown in Figure~\\ref{fig:june97_spec_branches}.\n\nThe evolution of the spectrum may be studied by inter-comparison of\nratios of pulse-height spectra from each region to that of a reference\nspectrum, from region~11 (Fig.~\\ref{fig:june97_pha_ratio}).\nThe spectrum is hardest in region~1, at the top of the vertical\nextension of the horizontal branch.  Motion down the branch\n(softening, regions~1--7) corresponds to pivoting of the spectrum\nabout $\\sim$7~keV, i.e., increasing intensity below $\\sim$7~keV and\ndecreasing intensity above that energy.\nMotion to the right across the horizontal portion of the HB\n(regions~8--11) corresponds to continued increasing low-energy\nintensity with modest softening, but with a nearly constant spectrum\nabove 12 keV\\@.  Note that in the hardness ratios of\nFigure~\\ref{fig:june97_hid20reg} the high-energy channel (6.3--13 keV)\nis dominated by photons near the lower bound of the interval.\n\nWhen the source moves down the normal branch (regions 11--17), the\nflux generally decreases across the entire 2.5--25~keV band but\ndecreases most significantly at high energy from region 11 to 15 (thus\nfurther softening).  Moving down the NB, the spectrum gradually\ndevelops a dip or step above $\\sim$9~keV and a peak slightly above\n10~keV.\nMotion up the FB (regions 18--20) is produced by increasing intensity\nat intermediate energies with a relatively constant spectrum at low\nenergy, thus hardening. The peaked feature near 10~keV becomes more\nprominent moving up the FB and will be discussed below.\n\n\\section{Selection of Spectral Models}\n\\label{sec:june97_spec_models}\n\nSpectral forms (e.g., blackbody emission, a power law, etc.) for use\nin fitting the spectra from the HID regions were explored by first\napplying them to high-quality spectra from time segments A and B (see\nFig.~\\ref{fig:june97_10d}). Variability in both of these segments was\nlimited to less than 10\\% in all energy bands between 2.5--18~keV. We\nthus constructed a single (averaged) pulse-height spectrum, known\nhereafter as spectrum~A or spectrum~B, for the entire 17--19~ks of\neach segment.  Errors in these spectra are dominated by the 1\\%\nsystematics at all energies up to $\\sim$20~keV.\n\nThe timing properties measured throughout data sets A and B indicate\nthat time segment~A falls on the vertical portion of the HB (strong\nnarrow QPO at 8.4--11.5~Hz) and time segment~B falls near the HB\/NB\napex (weak narrow QPO above 30~Hz and\/or the broad 4~Hz QPO).  The\nlocations of time segments A and B in\nFigure~\\ref{fig:june97_cchid_all} indicate that the branches are\nsignificantly shifted relative to those of segments C and D.\n\nRemillard et~al.\\ (1998\\nocite{remillard98}) studied version 2.2.1 PCA\nresponse matrices using Crab nebula data and found that the response\nmodel was most accurate for PCUs 0, 1, and 4 (of the five PCA\ndetectors) and for energies between 2.5~keV and 25~keV\\@. Thus, in\nfitting spectra we only include data from PCUs 0, 1, and 4 and from\nenergy channels corresponding to 2.5--25 keV\\@.  Spectra from each of\nthese detectors are fit separately. Fit parameters reported are the\naverage values for PCUs 0, 1, and, when appropriate, 4.  Errors are\nconservatively estimated as the entire range encompassed by the 90\\%\nconfidence intervals from each of the detectors. PCU~4 consistently\ngives lower normalizations for fitted spectral components, so fit\nparameters from that detector are not included when computing the\naverage normalizations and flux values and their errors.  Spectrum~B\nwas not constructed for PCU~4 since that detector was turned off\nduring part of time segment~B.\n\nInterstellar photoelectric absorption was included in all models. The\nabsorption model used solar abundances (\\cite{anders82}) and\ncross-sections given by Morrison \\& McCammon (1983\\nocite{morrison83}).\n\nSeveral single-component models were fit to spectra~A and\nB\\@. Blackbody and power-law models fit very poorly in both cases, as\ndid a multi-temperature ``disk blackbody'' spectrum (\\cite{mitsuda84};\n\\cite{makishima86}; model ``diskbb'' in XSPEC), with reduced $\\chi^2$\n($\\chi^2_r$) values of 22--545.  A thermal bremsstrahlung model\nprovided a better fit to spectrum~B ($\\chi^2_r=4.0$), but fit\nspectrum~A poorly ($\\chi^2_r=34$).\nA relatively good fit was achieved for both spectra with a modified\nbremsstrahlung model (see Table~\\ref{tab:june97_fitsAB}) which\nincludes the effects of Compton scattering of bremsstrahlung photons\nto higher energy in an optically thick plasma cloud (\\cite{compls};\nmodel ``compLS'' in XSPEC).\n\nA number of two-component models were also fit to these two spectra.\nA model using a disk blackbody and power law did not fit well\n($\\chi^2_r$=3--5), mainly because a single power-law slope does\nnot adequately describe the spectrum at high energy.\nA blackbody with $kT {\\stackrel{>}{_\\sim}}$ 2~keV is often included in models of the\nhard X-ray emission of LMXBs thought to contain a neutron star, where\nemission from or near the surface might produce high-temperature\nblackbody emission with a small effective area.\nTwo blackbodies ($kT \\sim$ 1.1~keV and 2.2~keV) fit moderately well\n(see Table~\\ref{tab:june97_fitsAB}), but required negligible\ninterstellar absorption. The low absorption is inconsistent with\nprevious measurements from {\\it ASCA\\,}\\ and {\\it ROSAT\\,}\\ (both sensitive below\n2~keV where the absorption is most easily constrained) which were used\nto estimate the interstellar column density to be\n$N_H=$(1.8--2.4)$\\times 10^{22}$~cm$^{-2}$\n(\\cite{brandt96}; \\cite{predehl95}).\n\nTwo models commonly used to fit Z-source energy spectra are the\n``Western model'' and the ``Eastern model'' (\\cite{hasinger90};\n\\cite{asai94}). The Western model consists of blackbody emission from\nthe hot surface of the neutron star or from a boundary between the\naccretion disk and surface, plus a Boltzmann-Wien component due to\nunsaturated Comptonization of soft photons by hot electrons\n(\\cite{white86}; Schulz, Hasinger, \\& Tr\\\"{u}mper\n1989\\nocite{schulz89}; Langmeier, Hasinger, \\& Tr\\\"{u}mper\n1990\\nocite{langmeier90}; \\cite{schulz93}).  The Eastern model also\nincludes blackbody radiation emitted from or near the surface, plus\nemission from a multi-temperature accretion disk (\\cite{mitsuda84};\n\\cite{hoshi91}; \\cite{hirano95}).\n\nThe Western model, a power law with a high-energy exponential cutoff\nplus a blackbody, fit well (see Table~\\ref{tab:june97_fitsAB}), but\nthe best-fitting high-energy cutoff energy ($E_{cut}\\approx1.7$~keV)\nwas so low relative to the PCA bandpass (${\\stackrel{>}{_\\sim}} 2$~keV) that the\npower law photon index was not well constrained.\nThe Eastern model, a disk blackbody with temperatures at the inner\nedge of the disk of 1.5--1.8~keV, plus a $\\sim$2~keV blackbody, fit\nspectra A \\& B quite well (see Table~\\ref{tab:june97_fitsAB} and\nFig.~\\ref{fig:june97_specA_diskbb_bb}) and gave absorption column\ndensities roughly consistent with the {\\it ASCA\\,}\\ and {\\it ROSAT\\,}\\ values.\nAlthough a number of other two-component models also produce similar\nquality fits, the Eastern model is used below to provide a physically\nmotivated parameterization of the spectra from the HID regions.\n\nThe Eastern model fit to spectrum~A\n(Fig.~\\ref{fig:june97_specA_diskbb_bb}) shows peaked residuals at\n6--7~keV, suggesting the presence of an emission line, probably iron\nK$\\alpha$. Very similar residuals appear in most of the fits discussed\nabove for both spectra A and B\\@. Addition of a Gaussian line to the\nmodels does in fact improve the fits in almost all cases; however, the\nbest-fitting line often has an extremely large Gaussian width\n($\\sigma>1$~keV). The energy resolution of the PCA is about 1~keV FWHM\nat 6~keV; thus it is difficult to place reliable constraints on\nparameters such as the centroid and width of a narrow component.  We\nhave not included an emission line in the fits reported in\nTable~\\ref{tab:june97_fitsAB}. The presence of an emission line near\n6.4~keV is discussed in more detail in Shirey et~al.\\\n(1999\\nocite{shirey99:dips}) in conjunction with spectra of absorption\ndips, which show the line more prominently.\n\n\\section{Fits to Spectra from 20 HID Regions}\n\nA disk blackbody plus isothermal blackbody model was fit to the\naverage spectrum for each of the 20 HID regions.  Representative fits\nand residuals are shown in Figure~\\ref{fig:june97_spec_hidfits}.  The\nresulting fit parameters are listed in Table~\\ref{tab:june97_fitsHID}\nand plotted versus HID region number in\nFigure~\\ref{fig:june97_fit_params}. The distance to Cir~X-1 has been\nestimated to be about 6--10~kpc (\\cite{stewart91}; \\cite{goss77}), so\nwe adopt a value of 8~kpc in converting blackbody and disk blackbody\nnormalizations to radii.\n\nAlthough both spectra A and B are fit well by the Eastern model, the\nreduced chi-squared values in Table~\\ref{tab:june97_fitsHID} indicate\nthat none of the fits for the 20 regional spectra are formally\nacceptable.  The fit results must therefore be regarded as an\napproximate description of the spectrum and its evolution.  We\nemphasize that here, as in any case, caution is advised in drawing\nphysical conclusions from the best-fit model parameters.\n\nThe spectra along the horizontal branch (regions 1--11) were all fit\nrelatively well. The residuals for these fits (see\nFig.~\\ref{fig:june97_spec_hidfits}) are similar in structure to those\nfor spectra A and B above. Thus they also suggest the presence of an\nemission line from iron. These spectra all show column densities of\n1.8--2.3$\\times10^{22}$~cm$^{-2}$, consistent with the {\\it ASCA\\,}\\ and\n{\\it ROSAT\\,}\\ values discussed above.  \nThe temperature of the $\\sim$2.0~keV blackbody is relatively stable on\nthe HB\\@ (see Fig.~\\ref{fig:june97_fit_params}). The temperature of\nthe inner disk decreases from region 1 to region 5 (down the vertical\nportion of the branch) and then stabilizes at $\\sim$1.3 keV\\@. The\ncooling is at least in part responsible for the pivoting of the\nspectrum on the vertical portion of the HB.\n\nThe inner radius of the disk blackbody component, times a factor of\norder unity involving the inclination angle of the disk, increases\nfrom 19 to 33~km, while the radius of the blackbody remains\nbetween 3~and 4~km.  These size scales are consistent with the\nhypothesis that these components arise from emission close to a\nneutron star. In this model, an increasing inner radius of the\naccretion disk is the most significant factor in producing the HB\ntrack; however, one should use caution in interpreting this as an\nactual physical radius.  The inclination angle of the disk in Cir~X-1\nis unknown but might be high since absorption dips are observed.\n\nFrom region~1 to 11, the total 2.5--25~keV flux increases\nmonotonically, with the exception of region~6, from\n2.89$\\times10^{-8}$ to 4.35$\\times10^{-8}$~erg~cm$^{-2}$~s$^{-1}$,\ncorresponding to 1.2--1.8 times the Eddington luminosity limit for a\n1.4~${\\rm M}_\\odot$\\ neutron star at 8~kpc.\n\nAlong the normal branch, the quality of the fits decrease from\nregion~12 to region~17, as indicated by increasing $\\chi^2_r$ values\n(see Table~\\ref{tab:june97_fitsHID}). The absorption column density\ngradually decreases by a factor of two, but this may be related to the\ndecreasing fit quality.\nThe inner radius and temperature of the disk blackbody change only\nslightly on the normal branch. In contrast, the $\\sim$2~keV blackbody\nbegins to fade on the upper portion of the normal branch (regions\n12--14), as indicated by a decreasing radius for the emission area.\nThe fading blackbody is illustrated in\nFigure~\\ref{fig:june97_photon_spec}, which shows the modeled incident\nspectra and both model components for several spectral fits. By the\nmiddle of the normal branch, the $\\sim$2~keV blackbody has faded\nentirely and fits have lower $\\chi^2_r$ values without it. Thus, the\nblackbody is omitted from the fits for regions 15--20. The residuals\nbelow $\\sim$6~keV continue to appear similar to those on the HB (see\nFig.~\\ref{fig:june97_spec_hidfits}).  The peak at $\\sim$6.5~keV\nbecomes broader and more complicated than on the HB\\@, and the dip and\npeak above 8~keV become more prominent.\n\nOn the flaring branch (regions 18--20), the fit quality decreases\nfurther, accompanied by very low values for the absorption column\ndensity. A number of other spectral models were fit to the HID-region\nspectra, and all failed to satisfactorally fit the spectra from the\nlower portion of the HID track (region number 14 and greater). A\nsignificant contribution to the high $\\chi^2$ values on the flaring\nbranch is due to the feature near 10~keV.  Addition of a Gaussian line\nor an absorption edge at 9--11~keV does improve the fits somewhat, but\nthese components cannot account for all the residuals near that\nenergy.  A combination of a line {\\em and} an edge near 10~keV can\nadequately fit the residuals, but such features are difficult to\njustify physically at that energy.  Even hydrogen-like iron can be\nruled out as a possible cause due to the high energy of the feature.\nMany X-ray pulsars show cyclotron absorption features at high\nenergy. Inclusion of a cyclotron absorption component in the spectral\nmodel results in a fit similar in quality to that of an absorption\nedge. However, such features require magnetic fields of $\\sim\n10^{12}$~G, which would be expected to result in strong pulsations\nrather than Z or atoll behavior.\n\n\n\\section{Discussion}\n\nOur spectral and timing analysis of the current observations shows\nclear evidence for Z~source behavior in Cir~X-1. This is significant\nbecause Cir~X-1 was reported to exhibit atoll source behavior at lower\nintensity (\\cite{oosterbroek95}). Earlier {\\it RXTE\\,}\\ observations, each\nlasting about two hours and separated by about two days, showed\nfragments of one or two spectral branches in hardness-intensity\ndiagrams (\\cite{shirey98:feb97})\\@. In the much more extensive\nobservations presented in this paper, we have found longer 12-hour\nsegments (time intervals C and D) which clearly exhibit all three\nbranches of a Z~source.  We have demonstrated that these complete Z\ntracks shift in the HID, confirming the behavior we inferred from the\nfragmented tracks of the previous observations.\n\nThe current data also allow us to demonstrate how the timing\nproperties evolve along the complete HID track of Cir~X-1 and confirm\nour original identification, in \\cite{shirey98:feb97}\\@, of horizontal\nand normal branch QPOs. Fourier power spectra for different regions of\nthe complete HID track show continuous evolution from the narrow QPO\n(increasing in frequency from 12~Hz to 30~Hz in the current\nobservations) on the horizontal branch, to the broad 4~Hz QPO on the\nnormal branch, to only very low frequency noise on the flaring branch.\nProperties of the fast timing characteristics associated with spectral\nbranches in Cir~X-1 were discussed in \\cite{shirey98:feb97}\\@.  For\nthe remainder of this discussion we focus on the properties of the\nenergy spectrum.\n\nWe tried fitting energy spectra of Cir~X-1 with various simple models.\nThe spectra for time intervals A and B were well fit using the Western\nand Eastern models (see discussion below), but no simple spectral form\nwas found that fit the range of spectra seen during time interval~C\\@.\nWe have not attempted to go beyond simple parameterized spectral\nmodels, e.g., by computing model spectra based on the \"unified model\"\nof Lamb and collaborators (\\cite{lamb89}; Psaltis, Lamb, \\& Miller\n1995\\nocite{psaltis95}; \\cite{psaltis98}), which was proposed to\nexplain the X-ray spectra and rapid variability of Z~sources.  Such\nsophisticated models may be necessary to correctly interpret the\nspectral changes in Cir~X-1 and other Z~sources.\n\nThe fits of spectra A and B with the Western model yielded a\ncutoff energy of $\\sim$1.7 keV for the Comptonized (Boltzmann-Wien)\ncomponent.  The cutoff energy in GX 5-1 was found to be 1--3~keV\n(\\cite{asai94}), similar to our results for Cir~X-1, but was found\nto be higher, 4--6 keV, in Cyg~X-2 (\\cite{hasinger90}).  We did not\nuse the Western model in parameterizing evolution associated with the\n``Z'' track because the cutoff energy in Cir~X-1 is low relative to\nthe PCA bandpass, resulting in a poorly constrained power law index\nand absorption column density.\n\nParameters for the Eastern model were more well-constrained (see\nTable~\\ref{tab:june97_fitsAB}), and thus this model was used to\nparameterize the spectral variations associated with the\nhardness-intensity track.  In this model, motion along the HB is\nmainly associated with an increasing inner radius of the disk\n(increasing disk blackbody normalization) but also by a decreasing\ninner disk temperature. This would imply that, as the luminosity\nincreases across the HB, the inner edge of the disk is pushed further\naway from the surface. It is not clear how this is related to the\nincreasing QPO frequency, which would typically be expected to require\na {\\em decreasing} radius if the QPOs were related to Keplerian motion\nat the inner edge of the disk, e.g., through the magnetospheric beat\nfrequency model (\\cite{alpar85}; \\cite{lamb85}).  \n\nFits of the Eastern model to energy spectra from the normal branch\nindicate that the $\\sim$2~keV blackbody gradually fades away, leaving\nonly the disk blackbody. This is similar to the result obtained when\nthe Eastern model was fit to the spectrum of Cyg~X-2, where the\nblackbody luminosity decreases from the HB to the FB\n(\\cite{hasinger90}).  Furthermore, the FB of GX~5$-$1 is characterized\nby intensity dips which in the Eastern model can be explained by\ndisappearance of the blackbody component, suggesting that accretion\nflow onto the neutron-star surface is interrupted (\\cite{mitsuda84}).\n\nOn the lower NB, a feature in the spectrum develops above 10~keV. This\nfeature becomes more prominent on the flaring branch.  A very similar\nline-like feature at $\\sim$10~keV was reported in {\\it Ginga\\,}\\ observations\nof the Z~source GX~5$-$1 (\\cite{asai94}). In GX~5$-$1, as in Cir~X-1,\nthe feature was present on the lower NB and stronger on the FB; we\nthus suggest that these features of the two sources may be of similar\norigin.  Asai et~al.\\ showed that a peak near 11~keV occurs in the\ncorrelation coefficients of the time-series data of different energy\nbands versus the 1.7--4.0 keV band. This demonstrates a temporal\ncharacter in the narrow band at 11~keV different than that at adjacent\nenergies. In turn, this gives assurance that the line-like feature at\nthat energy is not the result of the continuum model used to fit the\nspectrum but is intrinsic to the source.\nWe carried out similar cross-correlation analysis for each of the 20\nHID regions in our study. The cross-correlation results, relative to\nthe 2.5--2.9~keV band, from three representative regions are shown in\nFigure~\\ref{fig:june97_cross_corr}.  We find a clear peak in the\ncross-correlation coefficient at about 11~keV in regions 19 and 20 of\nthe FB\\@, further confirming the similarity of the spectral features\nin Cir X-1 and GX~5$-$1.  Regions 16 and 17 of the lower NB show an\nabrupt drop in the cross-correlation coefficient above 8--10~keV.  The\ncross-correlation coefficients on the HB and upper NB, where the line-\nor edge-like 10~keV spectral feature is absent or weak, generally show\nmuch less remarkable behavior.\n\nAsai et al.\\ found that that the spectral feature near 10~keV in\nGX~5$-$1 was better fit with a Gaussian line rather than, e.g., an\nabsorption edge.  However, as we mentioned above, such a feature\ncannot be produced by even hydrogen-like iron. Asai et~al.\\ discuss\nseveral of the mechanisms that could possibly produce a line at\n$\\sim$10~keV. For example, emission from a heavy element such as Ni\ncould produce the line, but is unlikely since iron should be far more\nabundant than the heavier elements. Alternately, a line could be\nblue-shifted due to motion in a relativistic jet or rotation in the\naccretion disk, but extreme conditions would be required to boost the\nenergy up to $\\sim$10~keV and one might expect a red-shifted line to\nalso be observed in the X-ray spectrum.  As mentioned above, we find\nthat in Cir~X-1 an emission line component alone is insufficient to\nproduce the observed structure of the feature. Both line-like and\nedge-like components may be required to explain this unusual spectral\nfeature at high energy.\n\nCir X-1 exhibits a number of unusual features in its Z-source behavior\nin addition to the 10~keV spectral feature.  Its HBOs are observed at\nfrequencies as low as 1.3~Hz (\\cite{shirey96}), an order of magnitude\nlower than those of other Z~sources.  The highest frequency reached by\nHBOs in Cir X-1, at the HB\/NB apex, is 30--32~Hz\n(\\cite{shirey98:feb97}), about a factor of two below the extreme HBO\nfrequency in other Z sources.  Power spectra of Cir~X-1 show a broad\nhigh-frequency peak, centered at 20--200~Hz, which shifts in frequency\nmaintaining an almost constant ratio with the HBO frequency\n(\\cite{shirey96}; \\cite{tennant87}).  Cir X-1 shows a long vertical\nextension of the HB\\@.  The entire HID track shows very large\ncolor\/intensity shifts, possibly associated with the presumed\neccentric 16.55-d binary orbit (\\cite{shirey98:feb97}).  Atoll\nbehavior has been reported at lower intensity (\\cite{oosterbroek95}).\nIt is likely that some or many of these unusual features are related\nby some physical property of the system, indicating that Cir~X-1 may\nprovide important constraints on models of low-mass X-ray binaries.\n\n\\acknowledgements \n\nWe would like to acknowledge the {\\it RXTE\\,}\\ teams at MIT and GSFC for\ntheir support.  In particular we thank E.~Morgan, R.~Remillard,\nW.~Cui, and D.~Chakrabarty for useful discussions related to this\nwork.  We thank D.~Psaltis for useful discussions regarding Z-source\nspectral models. We also appreciate the detailed comments and\nsuggestions provided by the referee.  Support for this work was\nprovided through NASA Contract NAS5-30612.\n\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{...}\n\n\nAn important source of photons in nuclear collisions are ``prompt photons''. They are produced in initial hard parton collisions, through two mechanisms\\footnote{This division into two mechanisms is useful conceptually, although it can only be done rigorously at leading order.}\n: \n\t(i) processes such as Compton scattering ($q g \\to q \\gamma$) \n\tand quark annihilation ($q \\bar{q} \\to g \\gamma$), \n\twhere a photon is a direct final state of the hard parton interaction; and\n\t(ii) fragmentation, where a hard parton-parton collision (e.g. $q \\bar{q} \\to q \\bar{q}$) is followed by a photon being radiated from one of the final state partons.\n   \n\nPrompt photon production in proton-proton (p+p) collisions has been calculated in perturbation theory at next-to-leading order in $\\alpha_s$;\nagreement with p+p data is good\nin a wide range of center-of-mass energies~\\cite{Aurenche:2006vj}. \nPrompt photons remain an important source of direct photons in heavy ion collisions. They are generally said\nto scale with the number of binary nucleon collisions, up to modest corrections from isospin and nuclear effects on the parton distribution functions; this statement is supported by the good agreement of high $p_T^\\gamma$ photon spectra ($\\gtrsim 5-10$~GeV)  in heavy ion measurements with binary-scaled prompt photon calculations (or p+p photon data).\nAt low  $p_T^\\gamma$, the production of prompt photon in heavy ion collisions is more complex.\nThe fragmentation component\nof prompt photons, subdominant at high $p_T^\\gamma$, is the dominant source of prompt photons at low $p_T^\\gamma$. These fragmentation photons are affected by parton energy loss.\nMoreover prompt photons are not the only source of low $p_T^\\gamma$ direct photons in heavy ion collisions: additional photons originate from (i) blackbody radiation produced by the hot expanding plasma (``thermal photons''), and (ii) ``jet-medium photons'' produced in parton-plasma interactions (interactions which also lead to parton energy loss) \\cite{Turbide:2007mi}. \nMeasurements indicate that the low $p_T^\\gamma$ direct photon spectra measured in heavy ion collisions is considerably larger than binary-scaled photon spectra measurements from proton-proton collisions. This suggests that thermal and jet-medium photons more than compensate for the suppression of prompt photons resulting from parton energy loss. Additional simulations are still required to confirm this scenario\\footnote{Almost all recent comparisons with data rely on prompt photon calculations that neglect the effect of energy loss.\nUp-to-date calculations of jet-medium photons and of prompt photons that account for parton energy loss will be essential to clarify the status of model-to-data comparisons.\n}.\n\n\n\nRecent work by the PHENIX Collaboration brought the role of prompt photons back to the front of the discussion~\\cite{Adare:2018wgc}. They observed that the \\emph{centrality dependence} of the direct photon multiplicity is consistent with binary collisions scaling:\n\\begin{equation}\n\\left. d N_{direct}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx N_{binary}^{\\alpha} K\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right) \n\\label{eq:phenix_scaling}\n\\end{equation}\nwhere the number of binary collisions $N_{binary}$ is a \ncalculated\nfrom the Monte-Carlo Glauber model and the exponent $\\alpha$ was found to be consistent with $1$. The normalization factor $K(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma )$ depends on the center-of-mass energy $\\sqrt{s_{NN}}$, and the multiplicity cutoff $p_T^\\gamma$. However the centrality dependence itself of the multiplicity appears not to depend on the cutoff $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ (i.e. $\\alpha\\approx 1$ independent of $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$).\nThe same conclusions had already been obtained for one center-of-mass energy in Ref.~\\cite{Adare:2014fwh}.\n\nIn what follows we provide a theoretical perspective on this binary-scaling from the point of view of thermal photons, medium-modified prompt photons and jet-medium photons.\n\n\n\\paragraph{Medium-modified prompt photons and jet-medium photons}\nThe multiplicity of sufficiently high $p_T^\\gamma$ prompt and jet-medium photons in heavy ion collisions can be written schematically as\n\\begin{align}\n\\left. d N_{j}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} =\n\\frac{N_{binary}}{\\sigma^{inel}_{NN }} & \\int \\frac{d \\phi}{2 \\pi} \\int_{p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} d p_T p_T \\left[ \\sum_{a, b, c} f_{a\/A} \\left(x_a,Q\\right) \n\\otimes  f_{b\/A}\\left(x_b,Q \\right) \\otimes \\, d\\hat{\\sigma}_{a b \\to c \\gamma}  \\right. \\nonumber \\\\\n& \\left. +\\sum_{a, b, c, d} f_{a\/A} \\left(x_a,Q\\right) \n\\otimes  f_{b\/A}\\left(x_b,Q \\right) \\otimes \\, d\\hat{\\sigma}_{a b \\to c d}  \\otimes D^M_{\\gamma\/c} \\left(z_c,Q\\right)\n\\right]\n\\label{eq:prompt_jetmedium}\n\\end{align}\nwhere $f(x,Q)$ are nuclear parton distribution;\n$d\\hat{\\sigma}$ are perturbative parton cross-sections; and $\\otimes$ represents a convolution over the kinematic variables. All effects from the quark-gluon plasma --- parton energy loss and jet-medium photon production --- are absorbed into a medium-modified fragmentation function $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$. At a fixed collision energy, the centrality dependence of  Eq.~\\ref{eq:prompt_jetmedium} originates from the $N_{binary}$ pre-factor as well as from $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$. This medium-modified fragmentation function is assumed to encode the same non-perturbative fragmentation into photons as the vacuum function. However, the perturbative sector of the vacuum fragmentation function is modified to account for the presence of a medium: it includes\\footnote{We are not aware of numerical studies of photons that include both medium-modified high and low-virtuality showering. Simulations that include only the latter have been performed previously, however (e.g. Ref.~\\cite{Turbide:2005fk}).} (i) perturbative DGLAP-like photon emission between virtuality $Q$ and a lower virtuality $Q_0$; and (ii) perturbative low-virtuality showering, which simultaneously leads to medium-induced photon emission and medium-induced parton energy loss. In vacuum, only the  DGLAP-like photon emission is present.\nThe final non-perturbative fragmentation into photons is performed after this perturbative showering. Unlike vacuum fragmentation functions, the medium-modified $D^M_{\\gamma\/c} \\left(z_c,Q\\right)$ is not universal: it must be evaluated dynamically to account for the exact profile of the medium.\n\n\nTypically the spectra of prompt and jet-medium photons fall off rapidly with $p_T^\\gamma$ and the photon multiplicity is dominated by the smallest $p_T^\\gamma$ regions of the integrand ($p_T^\\gamma \\sim p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$).\nThis presents a challenge for Eq.~\\ref{eq:prompt_jetmedium}.\nIntrinsically in collinear perturbative calculations, there is a lower scale $Q_0 \\sim 1-2$~GeV separating the perturbative and non-perturbative sectors. In Eq.~\\ref{eq:prompt_jetmedium}, the scale $Q>Q_0$ is expected to be of the order of the photon transverse momentum: $Q\\sim p_T^\\gamma$. This implies that multiplicities evaluated with small $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\sim Q_0$ may be dominated by $p_T^\\gamma$ regions where the perturbative calculations are the least reliable. Careful numerical simulations of medium-modified prompt and jet-medium photons will be essential to clarify the situation. Alternative approaches to calculate low momentum jet and photon production may also be necessary (see e.g. Ref.~\\cite{Hattori:2016jix} and references therein).\n\n\\paragraph{Thermal photons}\nThermal photons are calculated by convoluting a thermal emission rate $E d^3 \\Gamma\/d^3 p$ with a spacetime profile of the plasma obtained from hydrodynamics:\n\\begin{equation}\n\\left. d N_{th}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} =\n\\int d^4X \\int \\frac{d \\phi}{2 \\pi} \\int_{p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma} d p_T p_T  \\left [E \\frac{d^3 \\Gamma}{d^3 p}\\left(P \\cdot u(X),T(X),\\pi^{\\mu\\nu}(X),\\Pi(X)\\right) \\right]\n\\label{eq:thermal}\n\\end{equation}\nwhere $T$, $u$, $\\pi^{\\mu\\nu}(X)$,$\\Pi(X)$ are the temperature, flow velocity, shear tensor and bulk pressure profiles of the plasma, and $P$ is the photon four-momentum.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.38\\linewidth]{thermal_pTcuts_sqrts200_Nbin}\n\t\\hspace{2.cm}\n\t\\includegraphics[width=0.39\\linewidth]{thermal_pTcuts_sqrts5020_Nbin}\n\t\\caption{Thermal photon multiplicity as a function of the number of binary nucleon collisions, obtained by varying the centrality for (a) Au-Au at $\\sqrt{s_{NN}}=200$~GeV, and (b) Pb-Pb at $\\sqrt{s_{NN}}=5020$~GeV.}\n\t\\label{fig:thermal_vs_bin}\n\\end{figure}\n\nFigure~\\ref{fig:thermal_vs_bin} shows the power law scaling of the photon multiplicity as a function of $N_{binary}$\n, for Au-Au collisions at $\\sqrt{s_{NN}}=200$~GeV and Pb-Pb collisions at $\\sqrt{s_{NN}}=5020$~GeV. Three different multiplicity cutoff  $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ are shown: 0.5, 1 and 1.5~GeV. The results can be fitted reasonably well with a linear function; the slopes\nare indicated on the figures. Based on Figure~\\ref{fig:thermal_vs_bin}, we can write the thermal photon multiplicity as:\n\\begin{equation}\n\\left. d N_{th}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx N_{binary}^{\\alpha\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\right)} M\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right) \n\\label{eq:thermal_Nbin_scaling}\n\\end{equation}\nwhere $M(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma ) $ is a function that can be tabulated, and $\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ is an exponent for $N_{binary}$ that is approximately $1.1-1.3$. This exponent increases as the cutoff $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ is increased. The exact numerical values of $\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ depend on the details of the hydrodynamic simulation (initial conditions, viscosities, \\ldots) as well as the photon emission rates; this exact dependence will need to be studied in greater details\\footnote{In particular, we believe an observation made in Ref.~\\cite{Shen:2013vja} --- that an increase in the photon emission rate at low temperatures leads to smaller slopes for the centrality dependence --- may not as general as previously thought.}.\nNevertheless, the $p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma$ \\emph{dependence} of these values is a robust prediction: there is no expected scenario where the thermal photon multiplicity is independent from this cutoff.\n\n\\paragraph{Discussion \\& Outlook}\n\n\n   \n\nAny discussion of the multiplicity scaling must be made with other photon observables in mind, in particular the photon momentum anisotropy $v_2$. Thermal photons generally have a large $v_2$ at low $p_T^\\gamma$, while medium-modified prompt and jet-medium are generally understood to have a small $v_2$~\\cite{Turbide:2005bz}. The measured photon $v_2$ is large, and is interpreted as favoring thermal photons as the dominant source of low $p_T^\\gamma$ direct photons.\n\n\nSumming thermal, prompt and jet-medium photons, we write the direct photon multiplicity:\n\\begin{equation}\n\\left. d N_{direct}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n\\approx  \\left[ \n\\left. d N_{j}\/d y \\right|_{p_T^\\gamma>p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma}\n+ N_{binary}^{\\alpha\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma\\right)} M\\left(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma \\right)  \\right]\n\\end{equation}\n\nInevitably the centrality dependence of this prediction depends on the relative size of thermal photons as opposed to that of the medium-modified prompt photons and jet-medium photons. Given we do not currently have calculations of the latter two sources in a setting consistent with our thermal photon calculations, we limit our discussion to asymptotic scenarios.\n\nThe first limit is a multiplicity dominated by jet-medium and medium-modified prompt photons. To simultaneously describe the measured photon $v_2$, these jet-medium and prompt photons need to have a large $v_2$. This scenario is arguably not supported by older calculations~\\cite{Turbide:2005bz}. Whether calculations that include recent developments in hydrodynamics simulations (e.g. fluctuating initial conditions, initial flow and bulk viscosity) would produce a significantly larger momentum anisotropy for these photons will need to be studied numerically.\n\nIf thermal photons dominate the multiplicity, the centrality dependence of the photon multiplicity should be close to $N_{binary}^{\\alpha}$ with $\\alpha=\\alpha(\\sqrt{s_{NN}},p_{T, \\textrm{\\footnotesize cutoff}}^\\gamma)$ discussed above. Importantly measurements could rule out this possibility with improved constraints on the cutoff \\emph{independence} of the $\\alpha$ exponent.\n\nGiven that intermediate scenarios are also possible,\nadditional measurements and calculations are essential to determine if the observed centrality scaling can be explained by our current understanding of photon production, or if there is a new direct photon puzzle.\n\n\\paragraph{Acknowledgements} We are grateful to the ALICE and PHENIX photon working groups for insightful discussions. This work was supported by the U.S. Department of Energy\nunder Award Numbers DE-FG02-05ER41367 (JFP) and DE-SC0013460 (CS), and by the Natural Sciences and Engineering Research Council of Canada. SM acknowledges funding from The Fonds de recherche du Qu\\'ebec - Nature et technologies \n(FRQ-NT) \nthrough the Programme de Bourses d'Excellence pour \\'Etudiants \\'Etrangers.\nComputations were made \non the supercomputer Guillimin, managed by Calcul Qu\\'ebec and Compute Canada and funded by the Canada Foundation for Innovation (CFI), Minist\\`ere de l'\\'Economie, des Sciences et de l'Innovation du Qu\\'ebec (MESI) and FRQ-NT. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nA minor of a graph $G$ is a graph obtained from $G$ by a succession of\nedge deletions, edge contractions and vertex deletions.\nAll graphs we consider are simple, i.e. without loops or multiple edges.\nThe following theorem of Mader~\\cite{mader1} bounds the number of edges in a\n$K_r$-minor free graph.\n\\begin{thm}[Mader, 1968,~\\cite{mader1}]\nFor $3 \\leq r \\leq 7$, any $K_r$-minor free graph $G$ on $n\\ge r$\nvertices has at most $(r-2)n - {{r-1}\\choose{2}}$ edges.\n\\end{thm}\nNote that since $|E(G)| = \\frac{1}{2} \\sum_{u \\in  V(G)}\\,\\deg(u)$,\nthis theorem implies that every $K_r$-minor\nfree graph $G$, for $3 \\leq r \\leq 7$, is such that $\\delta(G)\\le\n2r-5$, where $\\delta(G)$ denotes the minimum degree of $G$.\nThis property will be of importance in the following. We are\ninterested in a sufficient condition for a graph to\nadmit a complete graph as a minor, dealing with the minimum number\nof triangles each edge belongs to. Nevo~\\cite{nevo1} already studied\nthis problem for small cliques. In the following, we assume\nthat every graph has at least one edge.\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nFor $3 \\leq r \\leq 5$, any $K_r$-minor free graph $G$ has an edge that\nbelongs to at most $r - 3$ triangles.\n\\label{th:nevoleq6}\n\\end{thm}\nHe also gave a weaker version for $K_6$-minor free graphs.\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nAny $K_6$-minor free graph $G$ has an edge that belongs to at\nmost $r - 3$ triangles, or is a clique-sum over $K_r$, $r \\leq 4$.\n\\label{th:nevok6}\n\\end{thm}\n\nNevo has conjectured that Theorem \\ref{th:nevok6} can be extended to\nthe case of $K_7$-minor free graphs. We improve\nTheorems~\\ref{th:nevoleq6} and \\ref{th:nevok6} in the following way.\n\n\\begin{thm}\\label{th:krtri}\nFor $3 \\leq r \\leq 7$, any $K_r$-minor free graph $G$ has\nan edge $uv$ such that $\\deg(u) \\leq 2r-5$ and $uv$ belongs to at\nmost $r-3$ triangles.\n\\end{thm}\n\nIn particular, this answers Nevo's conjecture about $K_7$-minor free graphs.\nAs pointed out by Nevo, Theorem~\\ref{th:nevok6} cannot be further extended\nto $K_8$-minor free graphs as such, since $K_{2,2,2,2,2}$\nis a $K_8$-minor free graph whose every edge belongs to $6$\ntriangles. Actually, one can obtain $K_8$-minor free graphs whose every edge\nbelongs to $6$ triangles by gluing copies of $K_{2,2,2,2,2}$ on cliques of\nany $K_8$-minor free graph. It is interesting to notice that $K_{2,2,2,2,2}$\nappears in a Mader-like theorem for $K_8$-minor free graphs~\\cite{jorg1}.\n\\begin{thm}[J\u00f8rgensen, 1994,~\\cite{jorg1}]\nEvery graph on $n \\geq 8$ vertices and at least $6n - 20$ edges either\nhas a $K_8$-minor, or is a $(K_{2,2,2,2,2}, 5)$-cockade (i.e. any\ngraph obtained from copies of $K_{2,2,2,2,2}$ by 5-clique sums).\n\\label{th:jorg}\n\\end{thm}\n\nAlthough Theorem~\\ref{th:krtri} cannot be extended to $K_8$-minor free\ngraphs, some similar conclusions can be reached by considering stronger\nhypotheses. By increasing the minimum degree of the graph or\nexcluding $K_{2,2,2,2,2}$ as an induced subgraph, we have the following\nthree theorems.\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with $\\delta(G)=11$ has an edge $uv$\nsuch that $u$ has degree 11 and $uv$ belongs to at most $5$\ntriangles.\n\\label{th:k8triweak}\n\\end{thm}\n\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with $\\delta(G)\\ge 9$ has an edge that\nbelongs to at most $5$ triangles.\n\\label{th:k8-deg9}\n\\end{thm}\n\n\\begin{thm}\nAny $K_8$-minor free graph $G$ with no $K_{2,2,2,2,2}$ as induced\nsubgraph has an edge that belongs to at most $5$ triangles.\n\\label{th:k8tri}\n\\end{thm}\n\nWe investigate applications of the previous results in the\nrest of the paper.\nIn Section~\\ref{sec:moytri}, we relax the hypothesis into a\nmore global condition on the overall number of triangles in\nthe graph. In particular, we prove that, for $3 \\leq k \\leq 7$\n(resp. $k=8$), if a graph has $m\\ge 1$ edges and at least\n$\\frac{k-3}{2}m$ triangles, then it has a $K_k$-minor (resp.\na $K_8$- or a $K_{2,2,2,2,2}$-minor).\nIn Section~\\ref{sec:stress}, we show applications to stress freeness\nof graphs, and settle some open problems of Nevo~\\cite{nevo1}.\nFinally, we show some applications to graph coloration\nin Section~\\ref{sec:double} and Section~\\ref{sec:coloration}.\nIn the former section, we show an application to double-critical\n$k$-chromatic graphs which settle a special case of a conjecture\nof Kawarabayashi, Toft and Pedersen~\\cite{kpt10}.\nIn the latter section, motivated by Hadwiger's conjecture,\nwe show that every $K_7$-minor free graph is $8$-colorable\nand that every $K_8$-minor free graph is $10$-colorable.\n\n\n\\section{Proof of Theorem~\\ref{th:krtri} for $r\\le 6$ : A slight improvement of Nevo's theorem}\\label{sec:proofk6}\n\nFirst note that the cases $r=3$ or $4$ are trivial.  The case $r=5$ is\nalso quite immediate, but we need a few definitions to prove it. A\n\\emph{separation} of a graph $G$ is a pair $(A,B)$ of subsets of\n$V(G)$ such that $A \\cup B = V(G)$, $A\\setminus B \\neq \\emptyset$,\n$B\\setminus A \\neq \\emptyset$, and no edge has one end in $A\n\\backslash B$ and the other in $B \\backslash A$. The \\emph{order} of a\nseparation is $|A \\cap B|$.  A separation of order $k$ will be denoted\nas a $k$-separation, and a separation of order at most $k$ as a $(\\leq\nk)$-separation. Given a vertex set $X\\subseteq V(G)$ (eventually $X$\nis a singleton) the sets $N(X)$ and $N[X]$ are respectively defined by\n$\\{y\\in V(G)\\setminus X \\ |\\ \\exists x\\in X\\ {\\rm s.t.}\\ xy\\in E(G)\\}$\nand $X\\cup N(X)$.\n\nLet us prove the case $r=5$. Consider any $K_5$-minor free graph\n$G$. According to Wagner's characterization of $K_5$-minor free\ngraphs~\\cite{w37}, $G$ is either the Wagner graph, a 4-connected\nplanar graph, or has a $(\\le 3)$-separation $(A,B)$ such that $H=G[A]$\nis either the Wagner graph or a 4-connected planar graph. If $G$ or\n$H$ is the Wagner graph, as this graph has only degree 3 vertices and\nno triangle, we are done. If $G$ (resp. $H$) is a 4-connected planar\ngraph, Euler's formula implies that  there is a vertex $v$\nof degree at most 5 in $V(G)$ (resp. in $A \\setminus B$). One can then observe\nthat, any edge around $v$ belongs to at most 2 triangles, as otherwise there would be\na separating triangle in $G$ (resp. $H$), contradicting its\n4-connectivity.\n\nLet us now focus on the case $r=6$ of Theorem~\\ref{th:krtri}.\nConsider by contradiction a $K_6$-minor free graph $G$ with at least\none edge, and such that every edge incident to a vertex of degree at\nmost $7$, belongs to at least $4$ triangles.  By Mader's theorem, we\nhave that $\\delta(G)\\le 7$.  We start by studying the properties of\n$G[N(u)]$, for the vertices $u$ of degree at most 7. First, it is\nclear that $G[N(u)]$ is $K_5$-minor free because otherwise there would\nbe a $K_6$-minor in $G$, contradicting the hypothesis.\n\n\\begin{lem}\n$\\delta(G)\\ge 6$, and for any vertex $u$ of degree at most 7, $\\delta(G[N(u)])\\ge 4$.\n\\label{lem:mindeg6}\n\\end{lem}\n\n\\begin{proof}\nFor any vertex $u$ of degree at most 7, and any vertex of $v\\in N(u)$\nthe edge $uv$ belongs to at least 4 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $4$ in $G[N(u)]$.\n\nSince for any vertex $u$ of degree at most $7$ we have\n$\\delta(G[N(u)])\\ge 4$, $|N(u)|\\ge 5$ (i.e. $\\deg(u)\\ge 5$).\nFurthermore if there was a vertex $u$ of degree 5, as\n$\\delta(G[N(u)])\\ge 4$, the graph $G[N(u)]$ would be isomorphic to\n$K_5$, contradicting the fact that $G[N(u)]$ is $K_5$-minor free. Thus\n$\\delta(G)\\ge 6$.\n\\end{proof}\n\nAs observed by Nevo (Proposition 3.3,~\\cite{nevo1}), since\n$|N(u)| \\leq 7$, $\\delta(G[N(u)]) \\geq 4$ and $N(u)$ is $K_5$-minor\nfree, then $G[N(u)]$ is $4$-connected. Note that by Wagner's\ncharacterization of $K_5$-minor free graphs, every $4$-connected\n$K_5$-minor free is planar. Chen and Kanevsky~\\cite{ck1} proved that\nevery $4$-connected graph can be obtained from $K_5$ and the\ndouble-axle wheel $W_4^2$ by operations involving vertex splitting and\nedge addition. Their result implies that the only two possibilities\nfor $G[N(u)]$ are the double-axle wheels on $4$ and $5$ vertices\ndepicted in Figure~\\ref{fig:doublewheel}.  Note that theses two graphs\nhave $3|N(u)| - 6$ edges, and hence are maximal $K_5$-minor free (by\nMader's theorem).\n\n\\begin{figure}[h]\n\\centering\n\\subfigure{\n\\includegraphics[scale=1]{doubleaxlewheel4.eps}\n}\n\\subfigure{\n\\includegraphics[scale=1]{doubleaxlewheel5.eps}\n}\n\n\\caption{The double-axle wheel on $4$ and $5$ vertices.}\n\\label{fig:doublewheel}\n\\end{figure}\n\nWe need the following lemmas on the neighborhood of the vertices with\nsmall degree.\n\n\\begin{lem}\nFor any vertex $u$ of degree at most 7, every vertex $v\\in N(u)$ has a\nneighbor in $G\\setminus N[u]$.\n\\label{lem:voisink6}\n\\end{lem}\n\\begin{proof}\nRecall that $G[N(u)]$ is a double-axle wheel. Note that in a\ndouble-axle wheel, every vertex has degree at most 5, and every edge\nbelongs to exactly 2 triangles.  Thus, every vertex of $N(u)$ has\ndegree at most 6 in $G[N[u]]$, and every edge of $G[N(u)]$ belongs to\nexactly 3 triangles in $G[N[u]]$. This implies that any vertex $v\\in\nN(u)$ has either degree $>8$ in $G$, and thus at least 2 neighbors in\n$G\\setminus N[u]$, or that any of its incident edges $vw$ in $G[N(u)]$\nis contained in a fourth triangle $vwx$, with $x\\in G\\setminus N[u]$.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $7$, and any connected component\n$C$ of $G\\setminus N[u]$, the graph $G[N(C)]$ is a clique on at most 3\nvertices.\n\\label{lem:voisin2k6}\n\\end{lem}\n\\begin{proof}\nAs $G[N(u)]$ has no clique on more than 3 vertices, let us show that\n$N(C)$ does not contain two non-adjacent vertices , say $v_1$ and\n$v_2$. There exists a path from $v_1$ to $v_2$ with inner vertices in\n$C$.  Since $G[N(u)]$ is maximal $K_5$-minor free, this path together\nwith $G[N[u]]$ induces a $K_6$ minor in $G$, a contradiction.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $7$, and any connected component\n$C$ of $G\\setminus N[u]$, there exists a vertex $u'\\in C$ of degree at\nmost 7 in $G$.\n\\label{lem:existsk6}\n\\end{lem}\n\\begin{proof}\nSuppose for contradiction that every vertex of $C$ has degree at least\n8 in $G$.  Note that by definition, every vertex in $N(C)$ has a\nneighbor in $C$.  Thus, as by Lemma~\\ref{lem:voisin2k6} $G[N(C)]$ is a\nclique on $k\\le 3$ vertices, the vertices in $N(C)$ have degree at\nleast $k$ in $G[N[C]]$.  Thus the number of edges of $G[N[C]]$ is at\nleast\n\\[|E(G[N[C]])| \\geq \\frac{1}{2}(8|C| + k^2)  > 4(|C|+k) - 10\\]\nand by Mader's theorem, there is a $K_6$-minor in $G[N[C]]$, a\ncontradiction.\n\\end{proof}\n\n\nNow choose a vertex $u$ of degree at most 7 and a connected component\n$C$ of $G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk6}, $C$ has a vertex $v$ of degree at most 7.\n\nLet $C_u$ be the connected component of $G\\setminus N[v]$ that\ncontains $u$, and let $x\\in N(v)\\setminus N(C_u)$.  By\nLemma~\\ref{lem:voisink6}, there is a connected component $C'$ of\n$G\\setminus N[v]$ such that $x\\in N(C')$.\n\nAs $N[u]\\subset N[C_u]$, it is clear that $G[C'\\cup\\{x,v\\}]$ is a\nconnected subgraph of $G\\setminus N[u]$. We thus have that\n$C'\\subsetneq C$ and thus that $|C'|<|C|$, contradicting the choice of\n$u$ and $C$.  This concludes the proof of the case $r=6$ of\nTheorem~\\ref{th:krtri}.\n\n\n\\section{Proof of Theorem~\\ref{th:krtri} for $r=7$ : the case of $5$ triangles}\\label{sec:proofk7}\n\nConsider by contradiction a $K_7$-minor free graph $G$ with at least\non edge, and such that every edge incident to a vertex of degree at\nmost $9$ belongs to at least $5$ triangles. By Mader's theorem,\n$|E(G)| \\leq 5|V(G)| - 15$, hence there are vertices $u$ such that\n$\\deg(u) \\leq 9$.\n\nWe start by studying the properties of $G[N(u)]$, for any vertex $u$\nof degree at most 9. First, it is clear that $G[N(u)]$ is $K_6$-minor\nfree because otherwise there would be a $K_7$-minor in $G$,\ncontradicting the hypothesis.\n\n\\begin{lem}\n$\\delta(G)\\ge 7$, and for any vertex $u$ of degree at most 9, $\\delta(G[N(u)])\\ge 5$.\n\\label{lem:mindeg7}\n\\end{lem}\n\n\\begin{proof}\nFor any vertex $u$ of degree at most 9, and any vertex of $v\\in N(u)$\nthe edge $uv$ belongs to at least 5 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $5$ in $G[N(u)]$.\n\nSince for any vertex $u$ of degree at most $9$ we have\n$\\delta(G[N(u)])\\ge 5$, $|N(u)|\\ge 6$ (i.e. $\\deg(u)\\ge 6$).\nFurthermore if there was a vertex $u$ of degree 6, as\n$\\delta(G[N(u)])\\ge 5$, the graph $G[N(u)]$ would be isomorphic to\n$K_6$, contradicting the fact that $G[N(u)]$ is $K_6$-minor free. Thus\n$\\delta(G)\\ge 7$.\n\\end{proof}\n\nThere is no appropriate theorem (contrarily to the previous case) to\ngenerate all possible neighbourhoods of the small degree vertices.\nInstead, we use a computer to generate all graphs with at most $9$\nvertices and minimum degree at least 5. Then we refine (by computer)\nour list of graphs, by removing the ones having a $K_6$-minor.  At the\nend, we end up with a list of $22$ graphs. A difference with the\nprevious case is that not all the $22$ graphs are maximal $K_6$-minor\nfree graphs.  We deduce two of the following lemmas from the study of\n$N(u)$ by computer~\\cite{ag1}.\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, any connected component $C$\nof $G\\setminus N[u]$ is such that $|N(C)| = k \\le 5$ and $|E(N(C))|\\ge\n{k \\choose 2} -3$ (i.e. $G[N[C]]$ has at most 3 non-edges).\n\\label{lem:compk7}\n\\end{lem}\n\n\\begin{proof}\nAs any connected component $C$ could be contracted into a single\nvertex, we prove the lemma by attaching a new vertex to all possible\ncombinations of $k$ vertices of $N[u]$ (as we know that $N(u)$ induces\none of the 22 graphs generated above), for any $k\\le 6$, and check\nwhen it induces a $K_7$-minor.\n\\end{proof}\n\nThis allows us to prove the following equivalent of\nLemma~\\ref{lem:existsk6}.\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, any connected component $C$\nof $G\\setminus N[u]$ has a vertex $u'$ of degree at most 9 in $G$.\n\\label{lem:existsk7}\n\\end{lem}\n\n\\begin{proof}\nLet $u$ be a vertex of $G$ of degree at most $9$ and let $C$ be a\nconnected component of $G\\setminus N[u]$ which vertices have degree at\nleast 10 in $G$. Note that by definition every vertex of $N(C)$ has at\nleast one neighbor in $C$. Lemma~\\ref{lem:compk7} implies that\n$|N(C)|=k \\leq 5$ and that $G[N(C)]$ has at most 3 non-edges. Thus,\ncontracting a conveniently choosen edge between $u$ and $N(C)$, one\nobtains that $G[N(C)]$ has at most 1 non-edge.  After this\ncontraction, we have:\n\\begin{align*}\n|E(N[C])| &\\geq \\frac{1}{2} \\Big[ 10|C| + k(k-1) - 2 + k \\Big] \\\\\n&= 5|C| + \\frac{k^2}{2} -1 > 5(|C|+k) - 15 .\n\\end{align*}\nThis contradicts the fact that $G[N[C]]$ is $K_7$-minor free, and thus\nconcludes the proof of the lemma.\n\\end{proof}\n\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$, at most one vertex $v$ of\n$N(u)$ is such that $N(v) \\subseteq N[u]$.\n\\label{lem:numberk7}\n\\end {lem}\n\n\\begin{proof}\nFor every such vertex $v$, as $\\deg(v)\\le \\deg(u)\\le 9$, the edges\nadjacent to $v$ with both ends in $N(u)$ belong to at least $5$\ntriangles in $G$ (i.e. belong to at least $4$ triangles in\n$G[N(u)]$). We checked that for every graph in the list at most one\nsuch vertex satisfies this condition.\n\\end{proof}\n\nThis allows us to prove the following lemma.\n\\begin{lem}\nFor any vertex $u$ of degree at most $9$ and any connected component\n$C$ of $G\\setminus N[u]$, there exists a connected component $C'$ of\n$G\\setminus N[u]$ such that $N(C')\\setminus N(C) \\neq \\emptyset$.\n\\label{lem:numberk7bis}\n\\end {lem}\n\\begin{proof}\nAs $\\deg(u)\\ge 7$ (by Lemma~\\ref{lem:mindeg7}) and $|N(C)|\\le 5$ (by\nLemma~\\ref{lem:compk7}), there are at least 2 vertices in\n$N(u)\\setminus N(C)$. By Lemma~\\ref{lem:numberk7}, one of these 2\nvertices has a neighbor $x$ out of $N[u]$.  Thus the component of\n$G\\setminus N[u]$ containing $x$ fulfills the requirements of the\nlemma.\n\\end{proof}\n\nNow choose a vertex $u$ of degree at most 9 and a connected component\n$C$ of $G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk7}, $C$ has a vertex $v$ of degree at most 9.\nLet $C_u$ be the connected component of $G\\setminus N[v]$ that\ncontains $u$. By Lemma~\\ref{lem:numberk7bis} there exists a connected\ncomponent $C'$ of $G\\setminus N[v]$ such that $N(C')\\setminus N(C_u)\n\\neq \\emptyset$, and let $x\\in N(C')\\setminus N(C_u)$.  As\n$N[u]\\subset N[C_u]$, it is clear that $G[C'\\cup\\{x,v\\}]$ is a\nconnected subgraph of $G\\setminus N[u]$. We thus have that\n$C'\\subsetneq C$ and thus that $|C'|<|C|$, contradicting the choice of\n$u$ and $C$.  This concludes the proof of case $r=7$ of\nTheorem~\\ref{th:krtri}\n\n\\section{Proof of Theorem~\\ref{th:k8triweak}, \\ref{th:k8-deg9} and \\ref{th:k8tri} : the case of $6$ triangles}\\label{sec:proofk8}\n\nAs in the previous sections, we will consider vertices of small degree\n(and their neighborhoods) in $K_8$-minor free graphs.  We thus need the\nfollowing technical lemma that has been proven by computer~\\cite{ag1}.\n\n\\begin{lem}\nEvery $K_7$-minors free graph $H$ distinct from $K_{2,2,2,2}$,\n$K_{3,3,3}$ and $\\overline{P_{10}}$ (the complement of the Petersen\ngraph), and such that $8\\le |V(H)| \\le 11$ and $\\delta(H)\\ge 6$,\nverifies:\n\\begin{itemize}\n\\item $H$ is 5-connected.\n\\item $H$ has at most one vertex $v$ such that each of its\n  incident edges belongs to 5 triangles.\n\\item For any subset $Y\\subsetneq V(H)$ of size 7, the graph obtained\n  from $H$ by adding two vertices $x$ and $y$ such that $N(x)=V(H)$\n  and $N(y)=Y$, has a $K_8$-minor.\n\\end{itemize}\n\\label{lem:compk8}\n\\end{lem}\nNote that the second property also holds for $K_{2,2,2,2}$,\n$K_{3,3,3}$ and $\\overline{P_{10}}$. Actually any edge of these 3\ngraphs belongs to less than 5 triangles.\n\nBy Theorem~\\ref{th:jorg}, any $K_8$-minor free graph has minimum\ndegree at most 11.  Theorem~\\ref{th:k8triweak} considers the case\nwhere the minimum degree is exactly 11.  It will be used in\nSection~\\ref{sec:coloration} to color $K_8$-minor free graphs.\n\n\\begin{proofof}{Theorem~\\ref{th:k8triweak}}\nWe prove this using the same technique as in\nSection~\\ref{sec:proofk7}.  Consider by contradiction a $K_8$-minor\nfree graph $G$ with $\\delta(G)=11$, and such that every edge adjacent\nto a degree 11 vertex belongs to at least $6$ triangles.  We start by\nstudying the properties of $G[N(u)]$, for any degree 11 vertex $u$.\nFirst, it is clear that $G[N(u)]$ is $K_7$-minor free because\notherwise there would be a $K_8$-minor in $G$, contradicting the\nhypothesis.\n\n\\begin{lem}\nFor any degree 11 vertex $u$, $\\delta(G[N(u)])\\ge 6$.\n\\label{lem:mindegk8}\n\\end{lem}\n\\begin{proof}\nFor any degree 11 vertex $u$ and any vertex of $v\\in N(u)$,\nthe edge $uv$ belongs to at least 6 triangles. The third vertex of\neach triangle clearly belongs to $N(u)$ and is adjacent to $v$.  Thus\n$v$ has degree at least $6$ in $G[N(u)]$.\n\\end{proof}\n\n\\begin{lem}\nFor any degree 11 vertex $u$, any connected component $C$\nof $G\\setminus N[u]$ has a vertex $u'$ of degree at most 11 in $G$.\n\\label{lem:existsk8}\n\\end{lem}\n\\begin{proof}\nLet $u$ be a degree 11 vertex of $G$ and let $C$ be any connected\ncomponent of $G\\setminus N[u]$ which vertices have degree at least 12\nin $G$. Lemma~\\ref{lem:compk8} implies that $G[N(u)]$ is 5-connected\nand that $|N(C)|=k \\leq 6$.  Thus the lemma holds by considering the\ngraph $G[N[u]\\cup C]$ in the following Lemma~\\ref{lem:existsk8bis}.\n\\end{proof}\n\n\\begin{lem}\nA graph $H$ with a degree 11 vertex $u\\in V(H)$ and such that:\n\\begin{itemize}\n\\item[(A)] $H[N(u)]$ is 5-connected,\n\\item[(B)] $\\delta(H[N(u)])\\ge 6$,\n\\item[(C)] the set $C=V(H)\\setminus N[u]$ is non-empty, and all its vertices have degree at least 12, and\n\\item[(D)] the set $N(C) \\subseteq N(u)$ has size $k\\le 6$,\n\\end{itemize}\nhas a $K_8$-minor.\n\\label{lem:existsk8bis}\n\\end{lem}\n\\begin{proof}\nConsider a minimal counter-example $H$, that is a $K_8$-minor free\ngraph $H$ fulfilling conditions (A), (B) (C) and (D), and minimizing\n$|V(H)|$.  Note that by definition every vertex of $N(C) \\subseteq\nN(u)$ has at least one neighbor in $C$. Let us prove that actually\nevery vertex of $N(C)$ has at least 2 neighbors in $C$.  If $x\\in\nN(C)$ has only one neighbor $y$ in $C$, contract the edge $xy$ and\ndenote $G'$ the obtained graph. It is clear that $G'$ is $K_8$-minor\nfree, and fulfills conditions (A), (B) and (D). Moreover, $C\\setminus\n\\{y\\}$ is non-empty as it contains at least 6 vertices of $N(y)\\cap C$\n(as $\\deg(y)\\ge 12$ and $|N(C)|=k\\le 6$), and every vertex of\n$C\\setminus \\{y\\}$ has degree at least 12 in $H'$ as none of these\nvertices are adjacent to $x$ in $H$. So $G'$ also fulfills condition\n(C), and this contradicts the minimality of $G$. Thus every vertex of\n$N(C)$ has at least 2 neighbors in $C$.\n\nOne can easily see that every $(K_{2,2,2,2,2}, 5)$-cockade has at\nleast 10 degree 8 vertices. Thus the graph $H[N[C]]$, and any graph\nobtained from $H[N[C]]$ by adding edges, cannot be a\n$(K_{2,2,2,2,2}, 5)$-cockade as it has at most $6$ vertices of\ndegree $8$. Thus as $H[N[C]]$ has at least $\\frac{1}{2}(12|C| + 2k)$\nedges and as this is at least $6(|C|+k)-20$ for $k \\leq 4$, by\nTheorem~\\ref{th:jorg} we have that $5\\le k\\le 6$.\n\nNow suppose that $k = 5,6$. Let $v_1$ and $v_2$ be two vertices of\nsmallest degree in $H[N(C)]$. Denote $\\delta_1$ and $\\delta_2$ their\nrespective degree in $H[N(C)]$. Note that if $k=6$ then $\\delta_1\\ge\n1$ as $v_1$ has at least $6$ neighbors in $N(u)$ and as there are only\n$5$ vertices in $N(u)\\setminus N(C)$. By contracting the edge $uv_1$,\nwe have $k - 1 -\\delta_1$ additionnal edges in $H[N[C]]$. Moreover\nsince $H[N(u)]$ is $5$-connected and since $|N(C)| \\leq 6$, for every\nvertex $x\\neq v_2$ of $N(C)$ we have $|N(C)\\setminus\\{x,v_2\\}|=4$ and\nthus the graph $H[N(u)]\\setminus (N(C)\\setminus\\{x,v_2\\})$ is\nconnected.  Thus, iteratively contracting all the edges between $v_2$\nand $N(u)\\setminus N(C)$ we add at least $k - 2 - \\delta_2$ edges in\n$H[N[C]]$ (as we have potentially already added the edge $v_1v_2$ in\nthe previous step). The number of edges in the obtained graph is at\nleast\n\\[\\frac{1}{2}[(\\delta_1 +2) + (\\delta_2\n  +2)(k-1)) + 12|C|] + (k - 1 -\\delta_1) + (k - 2 -\\delta_2)\\]\nwhich is more than $6(|C|+k)-20$ (as $k\\le 6$ and as if $k=6$ then\n$\\delta_1 \\ge 1$). Thus this graph has a $K_8$-minor, and so does $H$.\nThis completes the proof of the lemma.\n\\end{proof}\n\n\n\\begin{lem}\nFor any degree 11 vertex $u$ and any connected component\n$C$ of $G\\setminus N[u]$, there exists a connected component $C'$ of\n$G\\setminus N[u]$ such that $N(C')\\setminus N(C) \\neq \\emptyset$.\n\\label{lem:numberk8bis}\n\\end {lem}\n\\begin{proof}\nAs $\\deg(u)=11$ and $|N(C)|\\le 6$ (by Lemma~\\ref{lem:compk8}), there\nare at least 5 vertices in $N(u)\\setminus N(C)$.  As $\\delta(G)=11$\none can easily derive from Lemma~\\ref{lem:compk8} that one (actually,\nat least 4) of these vertices has a neighbor $x$ out of $N[u]$.  Thus\nthe component of $G\\setminus N[u]$ containing $x$ fulfills the\nrequirements of the lemma.\n\\end{proof}\n\nNow choose a degree 11 vertex $u$ and a connected component $C$ of\n$G\\setminus N[u]$, in such a way that $|C|$ is minimum.  By\nLemma~\\ref{lem:existsk8}, $C$ has a degree 11 vertex $v$.  Let $C_u$\nbe the connected component of $G\\setminus N[v]$ that contains $u$. By\nLemma~\\ref{lem:numberk8bis} there exists a connected component $C'$ of\n$G\\setminus N[v]$ such that $N(C')\\setminus N(C_u) \\neq \\emptyset$,\nand let $x\\in N(C')\\setminus N(C_u)$.  As $N[u]\\subset N[C_u]$, it is\nclear that $G[C'\\cup\\{x,v\\}]$ is a connected subgraph of $G\\setminus\nN[u]$. We thus have that $C'\\subsetneq C$ and thus that $|C'|<|C|$,\ncontradicting the choice of $u$ and $C$.  This concludes the proof of\nTheorem~\\ref{th:k8triweak}\n\\end{proofof}\n\nlet us now prove Theorem~\\ref{th:k8tri}.  Given a counter-exemple $G$\nof Theorem~\\ref{th:k8tri}, note that adding a vertex $s$ to $G$,\nadjacent to a single vertex of $G$, one obtains a counter-exemple of\nthe following theorem, thus Theorem~\\ref{th:k8tri} is a corollary of\nthe following theorem.\n\n\\begin{thm}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7 and such that $N[s] \\subsetneq V(G)$.  If every edge\n$e \\in E(G) \\setminus E(G[N[s]])$ belongs to at least 6 triangles, then\n$G$ contains an induced $K_{2,2,2,2,2}$.\n\\label{th:k8triwiths}\n\\end{thm}\nNote that as $K_{2,2,2,2,2}$ is maximal $K_8$-minor free, any\n$K_8$-minor free graph $G$ containing a copy of $K_{2,2,2,2,2}=G[X]$,\nfor some vertex set $X\\subseteq V(G)$, is such that any connected\ncomponent $C$ of $G\\setminus X$ verifies that $N(C)$ induces a clique\nin $G[X]$.\n\\begin{proof}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7 such that $N[s] \\subsetneq V(G)$, such that $G$ does not\ncontain an induced $K_{2,2,2,2,2}$, and such that every\nedge $e \\in E(G) \\setminus E(G[N[s]])$ belongs to at least 6\ntriangles.  Assume also that $G$ minimizes the number of vertices.\nThis property implies that $G\\setminus N[s]$ is connected. Indeed,\notherwise one could delete one of the connected components in\n$G\\setminus N[s]$ and obtain a smaller counter-example.  The graph $G$\nis almost 8-connected as observed in the following lemma.\n\\begin{lem}\nFor any separation $(A,B)$ of $G$ (denote $S = A \\cap B$), we have\neither:\n\\begin{itemize}\n\\item $|S| \\geq 8$, or\n\\item $s \\notin S$ and $A\\setminus B = \\{s\\}$ (i.e. $B = V(G) \\setminus\n  \\{s\\}$), or\n\\item $s \\in S$ and $|S| \\geq 6$.\n\\end{itemize}\n\\label{lem:connectk8withs}\n\\end{lem}\n\\begin{proof}\nSuppose there exists a separation $(A,B)$ contradicting the lemma.\nNote that $|S| < 8$ and let us assume that $s\\in A$.\n\nConsider first the case where $s\\notin S = A\\cap B$, that is the case\nwhere $\\{s\\} \\subsetneq A\\setminus B$.  Assume that among all such\ncounter-examples, $(A,B)$ minimizes $|S|$.  In this case, if the\nconnected component of $A\\setminus B$ containing $s$ has more vertices\nthen, contracting this component into $s$, one obtains a proper minor\n$G'$ of $G$ such that $N[s]\\subsetneq V(G')$ (as $B\\setminus A \\neq\n\\emptyset$) and such that every edge not in $E(N[s])$ belongs to $6$\ntriangles.  This would contradict the minimality of $G$, and we thus\nassume the existence of a component $C_0=\\{s\\}$ in $G\\setminus B$.  As\n$\\{s\\} \\subsetneq A\\setminus B$, let $C_1\\neq \\{s\\}$ be some connected\ncomponent of $G\\setminus B$. Let also $C_2$ be some component of\n$G\\setminus A$.  Note that for any of these components $C_i$,\n$N(C)\\subsetneq S$. Otherwise one could contract (if needed) all the\ncomponent into a single vertex $s'$ and the graph induced by\n$\\{s\\}\\cup N[C_1]$ or by $\\{s\\}\\cup N[C_2]$ (a proper minor of $G$)\nwould be a smaller counter-example. Note now that $S = N(s)\\cup\nN(C_i)$ for $i=1$ or 2.  Indeed, otherwise the separation $(N[s]\\cup\nN[C_i], V(G)\\setminus (C_i\\cup \\{s\\}))$ would be a counter-example\ncontradicting the minimality of $S$. Finally note that\n$N(C_2)\\not\\subseteq N(C_1)$, as otherwise contracting $C_1$ into a\nsingle vertex $s'$ and considering the graph induced by\n$C_2\\cup\\{s'\\}$ one would obtain a smaller counter-example. Thus there\nexists a vertex $x\\in N(s)\\cap N(C_2)$ such that $x\\notin\nN(C_1)$. Contracting the edge $xs$ and contracting the whole component\n$C_2$ into $x$, and considering the graph induced by $C_1\\cup \\{x\\}$\none obtains a smaller counter-example (where $x$ plays the role of\n$s$).\n \nConsider now the case where $s\\in S=A\\cap B$ and note that $|S|< 6$.\nAssume that among all the separations containing $s$, $(A,B)$\nminimizes $|S|$. Note that every connected component $C$ of $G\n\\setminus S$ is such that $s\\in N(C)$. Indeed, we have seen above that\notherwise $C$ would be such that $|N(C)|\\ge 8$ (by considering the\nseparation $(V(G)\\setminus C,N[C])$ and noting that $\\{s\\}\\subsetneq\nV(G)\\setminus N[C]$), and this would contradict the fact that $|S|<\n6$. This implies that every connected component $C$ of $G \\setminus S$\nis such that $N(C)=S$. Otherwise $(V(G)\\setminus C, N[C])$ would be a\nseparation containing $s$ contradicting the minimality of $S$. We can\nassume without loss of generality that $s$ has at most as many\nneighbors in $B\\setminus A$ than in $A\\setminus B$. In particular,\nsince $\\deg(s)\\le 7$, $s$ has at most 3 neighbors in $B\\setminus\nA$. Note that $B \\not\\subseteq N[s]$ as otherwise $G \\setminus\n(B\\setminus A)$ would be a smaller counter-example. Thus there is an\nedge in $G[B\\setminus N[s]$ that belongs to at least 6 triangles, and\nthus $|B|\\ge 9$ ($s$ and the 6 triangles). Thus contracting every\ncomponent of $A \\setminus B$ on $s$, results in a proper minor $G'$\nof $G$ such that $\\deg(s) \\leq 7$ (at most 4 in $S$ and 3 in\n$B\\setminus A$), such that $N[s]\\subsetneq V(G')$ (as\n$|V(G')|=|B|\\ge 9$), and such that every edge not in $E(N[s])$\nbelongs to $6$ triangles, contradicting the minimality of $G$. This\nconcludes the proof of the lemma.\n\\end{proof}\n\nBy Theorem~\\ref{th:jorg} $G$ has at most $6n-20$ edges, and thus there\nare several vertices in $G$ with degree at most 11.  Let us prove that\nthere are such vertices out of $N[s]$.\n\n\\begin{lem}\nThere are at least 2 vertices in $V(G) \\setminus N[s]$ with degree at\nmost 11.\n\\label{lem:2small_ink8}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that every vertex of $V(G) \\setminus N[s]$\nbut one, say $x$, has degree at least 12, and recall that such vertex\nhas degree at least 8. Note that every vertex $v\\in N(s)$ has a\nneighbor in $V(G) \\setminus N[s]$, as otherwise $G\\setminus v$ would\nbe a smaller counter-exemple. Thus every vertex $v\\in N(s)$ has an\nincident edge that belongs to at least 6 triangles (without using the\nedge $sv$), which implies that $\\deg(v)\\ge 8$. This implies that the\nnumber of edges in $G$ verifies :\n\\[ 12n-42 \\ge 2|E(G)| = \\sum_{v\\in V(G)} \\deg(v) \\ge 8 + k + 8k + 12(n-k-2) \\] \nwhere $k=\\deg(s)$. This implies that $3k\\ge 26$ which contradicts\nthe fact that $k=\\deg(s) \\le 7$. This concludes the proof of the\nlemma.\n\\end{proof}\n\nAs for any vertex $u\\in V(G)\\setminus N[s]$ each of its incident edges\nbelongs to 6 triangles, the graph $G[N(u)]$ has minimum degree at\nleast 6. As $G$ does not contain $K_8$ as subgraph, this also implies\nthat $\\deg(u)\\ge 8$.  So there are at least two vertices in\n$V(G)\\setminus N[s]$ with degree between 8 and 11.  The next lemma\ntells us more on the neighborhood of these small degree vertices.\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, $G[N(u)]$ is isomorphic to $K_{2,2,2,2}$, $K_{3,3,3}$ or\n$\\overline{P_{10}}$.\n\\label{lem:NminDeg}\n\\end{lem}\n\\begin{proof}\nLet $u$ be any vertex of $V(G) \\setminus N[s]$ with degree at most 11\nin $G$.  As observed earlier $8\\le \\deg(u)\\le 11$ and\n$\\delta(G[N(u)])\\ge 6$.  Assume for contradiction that $N(u)$, is not\nisomorphic to $K_{2,2,2,2}$, $K_{3,3,3}$ or $\\overline{P_{10}}$. Note\nthat $|N(u)\\cap N(s)| \\le 6$, as otherwise Lemma~\\ref{lem:compk8}\nwould contradict the $K_8$-minor freeness of $G$.\n\nBy Lemma~\\ref{lem:compk8} one of the (at least two) vertices in\n$N(u)\\setminus N(s)$, say $x$, has an incident edge in $G[N(u)]$ that\nbelongs to at most 5 triangles in $G[N[u]]$.  Thus the sixth triangle\ncontaining this edge goes through a vertex $v$ of $V(G) \\setminus\n(N[u] \\cup \\{s\\})$.\n\nLemma~\\ref{lem:connectk8withs} implies that the connected component\n$C$ of $v$ in $V(G) \\setminus N[u]$ is such that $N(C)\\ge 8$. The\ngraph obtained by contracting $C$ into a single vertex has a\n$K_8$-minor (by Lemma~\\ref{lem:compk8}), a contradiction.\n\\end{proof}\n\nA $K_3$-minor rooted at $\\{a, b, c\\}$, or a $\\{a, b, c\\}$-minor, is a\n$K_3$-minor in which you can contract edges incident to $a$, $b$ or\n$c$, to obtain a $K_3$ with vertex set $\\{a, b, c\\}$.  For the rest of\nthe proof we need the following characterization of rooted $K_3$-minor.\n\n\\begin{thm}[D. R. Wood and S. Linusson, Lemma 5 of~\\cite{wl1}]\nFor distinct vertices a, b, c in a graph G, either:\n\\begin{itemize}\n\\item $G$ contains an $\\{a, b, c\\}$-minor, or\n\\item for some vertex $v \\in V(G)$ at most one of $a, b, c$ are in each component of $G \\setminus v$.\n\\end{itemize}\n\\label{th:rootedtri}\n\\end{thm}\n\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, the graph $G[N(u)]$ is not isomorphic to $K_{3,3,3}$.\n\\label{lem:noK333}\n\\end{lem}\n\\begin{proof}\nObserve that adding two vertex disjoint edges or three edges of a\ntriangle in $K_{3,3,3}$ yields a $K_7$-minor.  Now assume for\ncontradiction that there exists some vertex $u\\in V(G) \\setminus N[s]$\nsuch that $G[N(u)]$ is isomorphic to $K_{3,3,3}$.\n\nAs the set $N(u) \\setminus N[s]$ is non-empty (it has size at least\n$9-7$) and as every vertex $v$ in $N(u) \\setminus N[s]$ has degree at\nleast $8$, and thus has a neighbor out of $N[u]$, $G \\setminus N[u]$\nhas a connected component $C\\neq \\{s\\}$.  By\nLemma~\\ref{lem:connectk8withs} $|N(C)| \\geq 8$.\n\nIf $G \\setminus N[u]$ has another connected component $C'$ such that\n$|N(C')| \\geq 6$, one can create two vertex disjoint edges in\n$K_{3,3,3}$ by contracting two vertex disjoint paths with non-adjacent\nends in $N(u)$, one living in each component. This would contradict\nthe $K_8$-minor freeness of $G$. Thus if there is a component $C'$,\nwe should have $C'=\\{s\\}$ and $\\deg(s)\\le 5$, as by\nLemma~\\ref{lem:connectk8withs} a component $C'\\neq \\{s\\}$ would be\nsuch that $|N(C')| \\geq 8$. In the following we consider the graph\n$G' = G[N[u] \\cup C]$ (which is $G$ or $G \\setminus s$).\n\nLet $\\{a_1,a_2,a_3\\}$, $\\{b_1,b_2,b_3\\}$ and $\\{c_1,c_2,c_3\\}$ be the\nthree disjoint stables of $N(u) = K_{3,3,3}$. Without loss of\ngenerality we can assume that $\\{a_1,a_2,a_3\\}\\subset N(C)$, and that\n$a_1\\notin N(s)$. As the edges of $N(u)$ incident to $a_1$ belong to\nat least 6 triangles, $a_1$ has at least two neighbors in $G'\n\\setminus N[u]$. By Theorem~\\ref{th:rootedtri} (applied to\n$\\{a_1,a_2,a_3\\}$ in the graph $G''= G'\\setminus\n\\{u,b_1,b_2,b_3,c_1,c_2,c_3\\}$), there is a vertex $v \\in V(G'')$ such\nthat at most one of $a_1, a_2, a_3$ are in each component of $G''\n\\setminus v$. Note that since $a_1$, $a_2$ and $a_3\\in N(C)$, all the\nsets $C\\cup\\{a_i,a_j\\}$ induce a connected graph, and thus $v\\neq\na_1$, $a_2$ or $a_3$. Equivalently we have that $v\\in V(G') \\setminus\nN[u]$.  Hence $G'' \\setminus \\{v\\}$ contains at least $3$ components\n$C_1$, $C_2$ and $C_3$ with $a_i \\in C_i$, for $1 \\leq i \\leq\n3$. Since $a_1$ has at least two neighbors in $G' \\setminus N[u]$, one\nof them is distinct from $v$ and we can define $C'_1$ as a connected\ncomponent of $C_1\\setminus \\{a_1\\}$. Note that by construction\n$N(C'_1) \\subset N(u)\\cup\\{v\\}$.  Since $C'_1\\neq\\{s\\}$ (as $a_1\\notin\nN(s)$) and as we might have $v=s$, Lemma~\\ref{lem:connectk8withs}\nimplies that $N(C'_1)\\ge 6$ (including $v$ and $a_1$). So $C'_1$ has\nat least 4 neighbors in $\\{b_1,b_2,b_3,c_1,c_2,c_3\\}$ and there is a\npath with interior vertices in $C'_1$ between two vertices $b_i$ and\n$b_j$, or between two vertices $c_i$ and $c_j$. Furthermore, there is\na path with interior vertices in $C_2 \\cup \\{v\\} \\cup C_3$ between the\nvertices $a_2$ and $a_3$. This contradicts the $K_8$-minor freeness of\n$G$, and thus concludes the proof of the lemma.\n\\end{proof}\n\n\\begin{lem}\nFor every vertex $u \\in V(G) \\setminus N[s]$ with degree at most 11 in\n$G$, the graph $G[N(u)]$ is not isomorphic to $K_{2,2,2,2}$.\n\\label{lem:noK2222}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that there exists some vertex $u\\in V(G)\n\\setminus N[s]$ such that $G[N(u)]$ is isomorphic to\n$K_{2,2,2,2}$. One can check that adding two edges in $K_{2,2,2,2}$\ncreates a $K_7$-minor. Thus as $G$ is $K_8$-minor free it should not\nbe possible to add (by edge contractions) two new edges in $N(u)$.\n\n\\begin{claim}\nA vertex $v\\in V(G) \\setminus N[u]$ has at most six neighbors in\n$N(u)$.\n\\label{claim:k8-v6neighbors}\n\\end{claim}\n\\begin{proof}\nIf there was a vertex $v$ with 8 neighbors in $N(u)$, $N[u]\\cup \\{v\\}$\nwould induce a $K_{2,2,2,2,2}$, a contradiction to the definition of\n$G$. We thus assume for contradiction that there is a vertex $v$ with\nexactly 7 neighbors in $N(u)$. Note that eventually $v=s$.  Let us\ndenote $x$ the only vertex in $N(u)\\setminus N(v)$. Note that among\nthe 4 non-edges of $G[N(u)]$, only one cannot be created by\ncontracting an edge incident to $v$. So if there is a path whose ends\nare non-adjacent in $N(u)$ and whose inner vertices belong to\n$V(G)\\setminus (N[u]\\cup \\{v\\})$, then we have a $K_8$-minor, a\ncontradiction. There is clearly such path if $s\\neq v$ and if $s$ has\n5 neighbors in $N(u)$, we thus have that either $s=v$ or $s$ has at\nmost 4 neighbors in $N(u)$. Both cases imply that some edge $xy$\n(incident to $x$) does not belong to $G[N[s]]$, and thus $xy$ belongs\nto at least 6 triangles. As $xy$ belongs to only 5 triangles in\n$G[N[u]]$, this implies the existence of a vertex $w\\in V(G) \\setminus\nN[u]$ adjacent to $x$ such that $w\\neq s, v$. Let $C$ be the connected\ncomponent of $w$ in $G\\setminus (N[u]\\cup \\{v\\})$. As $C\\neq \\{s\\}$,\nLemma~\\ref{lem:connectk8withs} implies that $N(C)$ has size at least\n6. Thus $C$ has at least 5 neighbors in $N(u)$ and one can link two\nnon-adjacent vertices of $N(u)$ by a path going through $C$, a\ncontradiction.\n\\end{proof}\n\nBy Lemma~\\ref{lem:2small_ink8} there exists another vertex $u'\\in V(G)\n\\setminus N[s]$ such that $\\deg(u')\\le 11$. By Lemma~\\ref{lem:NminDeg}\nand Lemma~\\ref{lem:noK333}, $G[N(u')]$ is isomorphic to $K_{2,2,2,2}$\nor $\\overline{P_{10}}$.\n\n\\begin{claim}\nThe vertices $u$ and $u'$ are non-adjacent.\n\\end{claim}\n\\begin{proof}\nWe assume for contradiction that $u$ and $u'$ are adjacent and we\nfirst consider the case where $G[N(u')]$ is isomorphic to\n$K_{2,2,2,2}$. In this case, as $u'$ has allready 7 neighbors in\n$N[u]$, $u'$ has a exactly one neighbor $v$ in $G\\setminus N[u]$.  As\n$v$ has 7 neighbors in $N(u')$, we have that $|N(u)\\cap N(v)|\\ge 7$, a\ncontradiction to Claim~\\ref{claim:k8-v6neighbors}.\n\nIf $G[N(u')]$ is isomorphic to $\\overline{P_{10}}$, this implies that\n$G[N(u)\\cap N(u')]$ is isomorphic to $\\overline{C_6}$ (the complement\nof the 6-cycle). This is not compatible with $G[N(u)]$ being\nisomorphic to $K_{2,2,2,2}$, as this in turn implies that $G[N(u)\\cap\nN(v)]$ is isomorphic to $K_{2,2,2}$.\n\\end{proof}\n\nAs by Lemma~\\ref{lem:connectk8withs} there is no $(\\le 5)$-separator\n$(A,B)$ with $u\\in A\\setminus B$ and $u'\\in B\\setminus A$, Menger's\nTheorem implies the existence of 6 vertex disjoint paths between $u$\nand $u'$.  These paths induces $6$ disjoint paths $P_1 \\ldots P_6$\nbetween $N(u)$ and $N(u')$. Note that every vertex in $N(u) \\cap\nN(u')$ can be seen as a path of length $0$.\n\nTherefore, since $N(u)$ is isomorphic to $K_{2,2,2,2}$, there are two\nnon-edges $a_1a_2$ and $a_3a_4$ of $G[N(u)]$ such that each $a_i$ is\nthe end of the path $P_i$. We denote by $b_i$, $1 \\leq i \\leq 4$ the\nend in $N(u')$ of the path $P_i$.  Note that if $a_i \\in N(u) \\cap\nN(u')$ then $a_i = b_i$.  Moreover we can suppose that the choice of\n$a_1a_2$ and $a_3a_4$ maximizes the size of $\\{a_1,a_2,a_3,a_4\\} \\cap\nN(u')$. Since $N(u)$ is isomorphic to $K_{2,2,2,2}$ and since $|N(u)\n\\cap N(u')| \\leq 6$ (by Claim~\\ref{claim:k8-v6neighbors}), there are\nat most two vertices in $N(u) \\cap N(u')$ distinct from $a_1,a_2,a_3$,\nand $a_4$. Let $X= (N(u) \\cap N(u')) \\setminus \\{a_1,a_2,a_3,a_4\\}$.\n\nSince both $K_{2,2,2,2}$ and $\\overline{P_{10}}$ are $6$-connected\nthen $N[u']$ is $7$-connected and so $G[N[u'] \\setminus X]$ is\n$5$-connected.  Moreover $G[N[u'] \\setminus X]$ has too many edges to\nbe planar. Indeed, it has $9-|X|$ vertices and at least $32-7|X|$\nedges, which is more than $3(9-|X|)-6$ for $0 \\le |X| \\le 2$. We now\nneed the following theorem of Robertson and Seymour about vertex\ndisjoint pairs of paths.\n\n\\begin{thm}[Robertson and Seymour~\\cite{rs1}]\n\\label{th:disjointpath}\nLet $v_1 , \\ldots, v_k$ be distinct vertices of a graph $H$. Then either\n\\begin{itemize}\n\\item[(i)] there are disjoint paths of $H$ with ends $p_1$ $p_2$ and\n  $q_1$ $q_2$ respectively, so that $p_1$, $q_1$, $p_2$, $q_2$ occur\n  in the sequence $v_1, \\ldots, v_k$ in order, or\n\\item[(ii)] there is a $(\\le 3)$-separation $(A,B)$ of $H$ with $v_1,\n  \\ldots, v_k \\in A$ and $|B \\setminus A| \\geq 2$, or\n\\item[(iii)] $H$ can be drawn in a disc with $v_1 , \\ldots, v_k$ on\n  the boundary in order.\n\\end{itemize}\n\\end{thm}\n\nApplying this theorem to the graph $G[N[u'] \\setminus X]$ with\n$(v_1,\\ldots v_k) = (b_1,b_3,b_2,b_4)$ one obtains that there are two\nvertex disjoint paths in $N[u'] \\setminus X$, a path $P_{1,2}$ between\n$b_1$ and $b_2$, and a path $P_{3,4}$ between $b_3$ and $b_4$.  Theses\npaths are disjoint from $N[u]$ by construction, except possibly at\ntheir ends.  Finally, since the paths $P_i$, for $1 \\leq i \\leq 4$,\nconstructed above are disjoint from $N[u]$ and from $N[u'] \\setminus\nX$, except at their ends, there exists two disjoint paths respectively\nlinking $a_1$ with $a_2$ (through $P_1$, $P_{1,2}$ and $P_2$), and\n$a_3$ with $a_4$ (through $P_3$, $P_{3,4}$ and $P_4$). This\ncontradicts the $K_8$-minor freeness of $G$ and thus concludes the\nproof of the lemma.\n\\end{proof}\n\n\nBy Lemma~\\ref{lem:2small_ink8} there exists at least two vertices\n$u$ and $u'\\in V(G) \\setminus N[s]$ with degree at most $11$. By\nLemma~\\ref{lem:NminDeg}, Lemma~\\ref{lem:noK333}, and\nLemma~\\ref{lem:noK2222}, both $G[N(u)]$ and $G[N(u')]$ are isomorphic\nto $\\overline{P_{10}}$. The two graphs induced by $N[u]$ and $N[u']$\nare close to a $K_8$-minor as observed in the following claim.\n\\begin{claim}\n\\label{cm:2edgeP10}\nIn $\\overline{P_{10}}$, adding two edges $ab$ $cd$, such that $ab$,\n$bc$ and $cd \\notin E(\\overline{P_{10}})$, creates a $K_7$-minor.\nFurthermore adding three edges $e_1$ $e_2$ and $e_3$, such that $e_1\n\\cap e_2 \\cap e_3 =\\emptyset$ in $\\overline{P_{10}}$, creates a\n$K_7$-minor.\n\\end{claim}\n\\begin{proof}\nOne can easily check the accuracy of the first statement, by noting\nthat adding any such pair of edges $ab$ and $cd$, yields the same\ngraph, and by noting that adding the edges $u_1u_2$ and $u_3u_4$ in\n$\\overline{P_{10}}$ (notations come from Figure~\\ref{fig:P10}) the\npartition $\\{\\{0,2\\},\\{1\\},\\{3\\},\\{4\\},\\{5\\}, \\{6,7\\}, \\{8,9\\}\\}$\ninduces a $K_7$-minor.\n\nFor the second statement, we can assume that the three added edges are\nsuch that they pairwise do not correspond to the first statement.\nWithout loss of generality, assume that one of the three edges is\n$u_0u_5$, and note that the other added edges are distinct from\n$u_1u_2$, $u_1u_6$, $u_3u_4$, $u_4u_9$, $u_2u_7$, $u_7u_9$, $u_3u_8$\nand $u_6u_8$. Consider the case where one of the other added edges is\nincident to $u_0u_5$. By symmetry one can assume that this edge is\n$u_0u_1$, but this implies that the third added edge is distinct from\n$u_0u_4$ (as the three edges would intersect), and from $u_3u_4$,\n$u_6u_9$, $u_5u_7$ and $u_5u_8$ (by the first statement). There is\nthus no remaining candidate for the third edge.\nThis implies that it is sufficient to consider the case where the\nedges $u_0u_5$, $u_2u_3$ and $u_6u_9$ are added in\n$\\overline{P_{10}}$. In this case the partition $\\{\\{1\\},\\{ 4\\},\\{\n7\\},\\{ 8\\},\\{ 0,5\\},\\{ 6,9\\},\\{ 2,3\\} \\}$ induces a $K_7$-minor.\n\\end{proof}\n\\begin{figure}[h]\n\\centering\n\\includegraphics{petersen.eps}\n\\caption{The Petersen graph $P_{10}$.}\n\\label{fig:P10}\n\\end{figure}\n\nLet us list the induced subgraphs of $\\overline{P_{10}}$ of size 6.\n\\begin{claim}\n\\label{cm:k8-P10-6subgraphs}\nThere are exactly 6 distinct induced subgraphs of size 6 in\n$\\overline{P_{10}}$, including $K_{2,2,2}$. The complements of these\ngraphs are represented in Figure~\\ref{fig:P10-6subgraphs}.\nFurthermore note that every induced subgraphs of $\\overline{P_{10}}$\nof size at least 7, has a subgraph of size 6 distinct from\n$K_{2,2,2}$.\n\\end{claim}\nWe do not prove the claim here as one can easily check its accuracy.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=390px]{p10-sub.eps}\n\\caption{The complements of the subgraphs of $\\overline{P_{10}}$ of\n  size 6 (i.e. the subgraphs of $P_{10}$ of size 6).}\n\\label{fig:P10-6subgraphs}\n\\end{figure}\n\n\\begin{lem}\nThe vertices of $N(u)\\setminus N(s)$ (resp. of $N(u')\\setminus N(s)$)\nhave degree at least 12. Thus in particular, $u$ and $u'$ are\nnon-adjacent.\n\\label{lem:k8-P10-deg12}\n\\end{lem}\n\\begin{proof}\nWe assume for contradiction that $u$ has a neighbor $v$ of degree at\nmost 11.  By Lemma~\\ref{lem:NminDeg}, Lemma~\\ref{lem:noK2222}, and\nLemma~\\ref{lem:noK333}, the graph $G[N(v)]$ is isomorphic to\n$\\overline{P_{10}}$.\n\nAssume $v=u_0$ in Figure~\\ref{fig:P10}. Since $N(u_0) \\supset\n\\{u,u_2,u_3,u_6,u_7,u_8,u_9\\}$, the adjacencies in\n$G[\\{u,u_2,u_3,u_6,u_7,u_8,u_9,\\}]$ allow us to denote $u$ by $u'_0$,\nand denote $u'_1$, $u'_4$ and $u'_5$ the vertices in $N(u_0)\\setminus\nN[u]$, in such a way that these indices again correspond to\nFigure~\\ref{fig:P10}. It is now easy to see that contracting one edge\nin each of the paths $(u_2,u'_4,u_7)$ and $(u_6,u'_5,u_9)$ creates the\nedges $u_2u_7$ and $u_6u_9$ in $G[N[u]]$ and thus yields a $K_8$-minor\n(by Claim~\\ref{cm:2edgeP10} as $u_7u_9$ is a non-edge of\n$\\overline{P_{10}}$), a contradiction.\n\\end{proof}\n\nThe vertices $u$ and $u'$ are non-adjacent, however they can share\nneighbors.  Let us prove that they cannot share more than 7 neighbors.\n\\begin{lem}\n$|N(u) \\cap N(u')|\\le 7$.\n\\label{lem:k8-P10->=7common}\n\\end{lem}\n\\begin{proof}\nAssume for contradiction that $|N(u) \\cap N(u')|\\ge 8$, that is\nequivalently that $|N(u) \\setminus N(u')|\\le 2$ and $|N(u') \\setminus\nN(u)|\\le 2$.  Note that as $\\deg(s)\\le 7$ the set $(N(u) \\cap N(u'))\n\\setminus N(s)$ is non-empty, and denote $x$ one of its vertices.  By\nLemma~\\ref{lem:k8-P10-deg12}, this vertex $x$ as degree at least $12$.\nAs it has exactly 6 neighbors in $N(u)$, at most 2 neighbors in $N(u')\n\\setminus N(u)$, and as it is adjacent to both $u$ and $u'$, $x$ has\nat least two neighbors in $V(G) \\setminus (N[u] \\cup N[u'])$. Thus\nthere exists a component $C\\neq\\{s\\}$ in $G \\setminus (N[u] \\cup\nN[u'])$. As $C\\neq\\{s\\}$ and $N(C)\\subseteq N(u)\\cup N(u')$,\nLemma~\\ref{lem:connectk8withs} implies that $|N(C)|\\ge 8$. Therefore,\nas $|N(u') \\setminus N(u)|\\le 2$, $|N(C) \\cap N(u)| \\ge 6$ and there\nexist a path $P$ with inner vertices in $C$ and with non-adjacent ends\nin $N(u)$ (by Claim~\\ref{cm:k8-P10-6subgraphs}). Let us denote $x$ and\n$y$ the ends of $P$. As $|N(u) \\cap N(u')|\\ge 8$ and by\nClaim~\\ref{cm:k8-P10-6subgraphs}, there exists a vertex $z\\in N(u)\n\\cap N(u')$ such that $z\\neq x$ or $y$, and such that contracting the\nedge $zu'$ creates at least two edges in $N(u)$. As these three added\nedges ($xy$ and the edges adjacent to $z$) do not intersect,\nClaim~\\ref{cm:2edgeP10} implies that there is a $K_8$-minor, a\ncontradiction.\n\\end{proof}\n\nAs by Lemma~\\ref{lem:connectk8withs} there is no $(\\le 5)$-separator\n$(A,B)$ with $u\\in A\\setminus B$ and $u'\\in B\\setminus A$, Menger's\nTheorem implies the existence of 6 vertex disjoint paths $P_1 \\ldots\nP_6$ between $u$ and $u'$.\nBy minimizing the total length of these paths we can assume that\neach vertex in $N(u)\\cap N(u')$ corresponds to one of these paths,\nand that any of these paths intersect $N(u)$ (resp. $N(u')$) in\nonly one vertex. Contracting the inner edges (those non-incident to\n$u$ or $u'$) of these paths, and considering the graph induced by\n$N[u]\\cup N[u']$ one obtains a graph $H$ such that:\n\\begin{itemize}\n\\item $u$ and $u'$ are nonadjacent and $|N_H(u)\\cap N_H(u')| = 6$ or\n  $7$.\n\\item $\\deg_H(u)=10$, and $H[N(u)]$ contains $\\overline{P_{10}}$ as a\n  subgraph.\n\\item $\\deg_H(u')=10$, and $H[N(u')]$ contains $\\overline{P_{10}}$ as\n  a subgraph.\n\\end{itemize}\n\nIf the graph induced by $N_H(u)\\cap N_H(u')$ is isomorphic to\n$K_{2,2,2}$, then one can assume without loss of generality that\n$N(u)=\\{u_0,\\ldots,u_9\\}$ and that $N(u')=\\{u_0,u'_1, u_2,u_3,\nu'_4,u_5,u_6, u'_7,u'_8,u_9\\}$, where the indices correspond to\nFigure~\\ref{fig:P10}. Now observe that contracting the edge $u_0u'$,\nthe path $(u_6,u'_7,u'_8)$, and the path $(u_2,u'_4,u'_1)$,\nrespectively create the edges $u_0u_5$, $u_6u_9$, and $u_2u_3$.  This\nimplies by Claim~\\ref{cm:2edgeP10} that $N[u]$ contains a $K_8$-minor,\na contradiction. We can thus assume by\nClaim~\\ref{cm:k8-P10-6subgraphs} that the complement of $N_H(u)\\cap\nN_H(u')$ contains a path $(a,b,c,d)$.  As $\\overline{P_{10}}$ is\n6-connected, the graph induced by $\\{a,b\\}\\cup (N_H(u')\\setminus\nN(u))$ is connected, and thus contains a path from $a$ to $b$.  By\nClaim~\\ref{cm:2edgeP10}, this path with the path $(c,u',d)$, imply\nthat $H$ (which is a minor of $G$) contains a $K_8$-minor, a\ncontradiction. Thus there is no counter-example $G$, and this\nconcludes the proof of the theorem.\n\\end{proof}\n\nThe proof Theorem~\\ref{th:k8-deg9} is very similar. To do this one can\nprove the following variant of Theorem~\\ref{th:k8triwiths}.\n\\begin{thm}\nConsider a connected $K_8$-minor free graph $G$ with a vertex $s$ of\ndegree at most 7, such that $N[s] \\subsetneq V(G)$ and such that\n$\\min_{v\\in V(G)\\setminus N[s]}\\ge 9$. Then $G$ has an edge\n$e \\in E(G) \\setminus E(G[N[s]])$ that belongs to at most 5 triangles.\n\\end{thm}\nThe proof of this theorem is as the proof of\nTheorem~\\ref{th:k8triwiths}, except that one does not need to consider\nthe case where some vertex $u$ is such that $N(u)$ induces a\n$K_{2,2,2,2}$.\n\n\\section{Global density of triangles}\\label{sec:moytri}\n\nIn this section, we investigate the relation between the number of\ntriangles and the number of edge of a graph. Denotes by $\\rho =\n\\frac{t}{m}$ the ratio between the number of triangles $t$ and the\nnumber of edges $m$ of a graph $G$.  For each $k$, what is the minimum\nnumber $f(k)$ such that for all graph $G$ with $\\rho \\ge f(k)$, $G$\ncontains a $K_k$ minor ?\n\nIt is easy to notice that 2-trees on $n \\ge 2$ vertices have exactly\n$1 + 2(n-2)$ edges and $n-2$ triangles.  Furthermore, for $k \\ge 3$\none can notice that $k$-trees on $n \\ge k$ vertices have exactly\n$\\frac{k(k-1)}{2} + k(n-k)$ edges and $\\frac{k(k-1)(k-2)}{6} +\n(n-k)\\frac{k(k-1)}{2}$ triangles. Thus any $k$-tree, for $k \\ge 2$,\nverifies\n\\[ t = \\frac{k-1}{2} m -\\frac{1}{2}{{k+1}\\choose{3}} .\\]\nSince $k$-trees are $K_{k+2}$-minor free, for all $k \\ge 4$ there\nexists $K_k$-minor free graphs with $\\frac{k-3}{2} m\n-\\frac{1}{2}{{k-1}\\choose{3}}$ triangles.\n\nWe deduce that for all $k \\ge 4$, $f(k) \\geq \\frac{k-3}{2}$.  Indeed\nfor every $\\epsilon > 0$, there exists a number $m$ and a $K_k$-minor\nfree graph with $m$ edges such that $\\frac{k-3}{2} - \\epsilon \\leq\n\\rho < \\frac{k-3}{2}$.  In fact, for $4 \\le k \\le 7$, the following\ntheorem proves that this lower bound is best possible, so we have\n$f(k) = \\frac{k-3}{2}$.\n\n\\begin{thm}\nFor $4 \\le k \\le 7$ (resp. $k = 8$), every graph with $m \\ge 1$ edges\nand $t \\ge m(k-3)\/2$ triangles has a $K_k$-minor (resp. a $K_8$- or a\n$K_{2,2,2,2,2}$-minor).\n\\label{thm:global}\n\\end{thm}\n\n\\begin{proof}\nConsider by contradiction, a non-trivial $K_k$-minor free\n(resp. $K_8$- and $K_{2,2,2,2,2}$-minor free) graph $G$ with $t\n\\ge m(k-3)\/2$ triangles.  Among the possible graphs $G$, consider one\nthat minimizes $m$ (given that $m \\ge 1$).\n\nGiven any edge $uv \\in E(G)$ let $H_{uv} = G[N(u) \\cap N(v)]$ and\ndenote $n'$ and $m'$ its number of vertices and edges respectively.\nContracting $uv$ yields a proper minor of $G$, with exactly $1 + n'$\nedges less, and with at most $n' + m'$ triangles less. Thus by\nminimality of $G$, for every edge $uv$\n\\[ n' + m'  >  \\frac{k-3}{2} (1 + n') \\]\nwhich implies that\n\\[ m'  >  \\frac{k-3}{2} + \\frac{k-5}{2} n' .\\]\nOn the other hand we have that $\\frac{n'(n' - 1)}{2} \\ge m'$, and this\nimplies that $n'$ should verify $(n' + 1)(n' + 3 - k) > 0$, that is\nthat $n' \\ge k - 2$. In other words, every edge $uv$ of $G$ belongs to\nat least $k-2$ triangles. By Theorems~\\ref{th:krtri},\n(resp. Theorem~\\ref{th:k8tri}), this contradicts the $K_k$-minor\nfreeness (resp. $K_8$- and $K_{2,2,2,2,2}$-minor freeness) of $G$.\n\\end{proof}\n\n\n\\section{Application to stress freeness of graphs}\\label{sec:stress}\n\nThe motivation of this application is a problem that arises from the\nstudy of tension and compression forces applied on frameworks in the\nEuclidian space $\\mathbb{R}^d$. A $d$-framework is a graph $G=(V,E)$\nand an embedding $\\rho$ of $G$ in $\\mathbb{R}^d$. The reader should\nthink of a framework as an actual physical system where edges are\neither straight bars or cables and vertices are articulated joints.  A\n\\textit{stress} on a framework $(G,\\rho)$ is a function $\\omega:\nE(G)\\, \\rightarrow\\; \\mathbb{R}$ such that $\\forall v \\in V$,\n\\[\\underset{\\{u,v\\}\\in E}{\\sum}\\:\\omega(\\{u,v\\})(\\rho(v) - \\rho(u)) = 0 .\\]\nStress corresponds to some notion of equilibrium for the associated\nphysical system.  Each vertex is affected by tension and compression\nforces created by the bars and cables.  $\\omega(\\{u,v\\})$ can be\nthought of as the magnitude of such force per unit length, with\n$\\omega(\\{u,v\\}) < 0$ for a cable tension and $\\omega(\\{u,v\\}) > 0$\nfor a bar compression.  A stress is a state of the system where these\nforces cancel each other at every vertex.  We can see that every\nframework admits a \\textit{trivial} stress where $\\omega$ is\nidentically zero.  A $d$-framework admitting only the trivial stress\nis called \\textit{$d$-stress free}.\n\nTo make this notion independent of the embedding of $G$, the following\nwas introduced.  A graph $G$ is \\textit{generically $d$-stress free}\nif the set of all $d$-stress free embeddings of $G$ in $\\mathbb{R}^d$\nis open and dense in the set of all its embeddings (i.e. every\nstressed embedding of $G$ is arbitrary close to a stress free\nembedding).\n\nThis notion has been first used on graphs coming from $1$-skeletons of\n$3$-dimensional polytopes\n\\cite{cauchy-13,maxwell-64,cw-93,whiteley-84}, which are planar by\nSteiniz's theorem.  Gluck generalized the results on $3$-dimensional\npolytopes to the whole class of planar graphs.\n\\begin{thm}[Gluck, 1975,~\\cite{gluck-75}]\nPlanar graphs are generically $3$-stress free.\n\\label{th:gluckstress}\n\\end{thm}\nNevo proved that we can generalize Theorem~\\ref{th:gluckstress} for\n$K_5$-minor free graphs, and extended the result as follows.\n\n\\begin{thm}[Nevo, 2007,~\\cite{nevo1}]\nFor $2 \\leq r \\leq 6$, every $K_r$-minor free graph is\ngenerically $(r-2)$-stress free.\n\\label{th:nevostress}\n\\end{thm}\n\nHe conjectured this to hold also for $r = 7$ and noticed that the\ngraph $K_{2,2,2,2,2}$ is an obstruction for the case $r=8$. Indeed,\n$K_{2,2,2,2,2}$ is $K_8$-minor free and has too many edges to be\ngenerically $6$-stress free (a generically $\\ell$-stress free graph\nhas at most $\\ell n - {\\ell+1 \\choose 2}$ edges~\\cite{nevo1}).  We\nanswer positively to Nevo's conjecture and we give a variant for the\ngenerically $6$-stress freeness.\n\n\\begin{thm}\nEvery $K_7$-minor free graph (resp. $K_8$- and $K_{2,2,2,2,2}$-minor\nfree graph) is generically $5$-stress free (resp. $6$-stress free).\n\\label{th:genstressk78}\n\\end{thm}\n\nThe following result of Whiteley~\\cite{whiteley-89} is used to derive\nTheorem~\\ref{th:genstressk78}.\n\\begin{thm}[Whiteley, 1989,~\\cite{whiteley-89}]\nLet $G'$ be obtained from a graph $G$ by contracting an edge $\\{u,v\\}$.\nIf $u$, $v$ have at most $d - 1$ common neighbors and $G'$ is generically\n$d$-stress free, then $G$ is generically $d$-stress free.\n\\label{th:whitcontract}\n\\end{thm}\n\nNow, we prove Theorem~\\ref{th:genstressk78}.\n\\begin{proof}\nAssume that $G$ is a $K_7$-minor free graph (resp. a $K_8$- and\n$K_{2,2,2,2,2}$-minor free graph). Without loss of generality, we can\nalso assume that $G$ is connected.  Now, contract edges belonging to\nat most $4$ (resp. 5) triangles as long as it is possible and we\ndenotes by $G'$ the graph obtained. Note that by construction, every\nedge of $G'$ belongs to 5 (resp. 6) triangles. Note also that $G'$ is\na minor of $G$, and is thus $K_7$-minor free (resp.  $K_8$- and\n$K_{2,2,2,2,2}$-minor free). Theorem~\\ref{th:krtri}\n(resp. Theorem~\\ref{th:k8tri}) thus implies that $G'$ is the trivial\ngraph without any edge and with one vertex. This graph is trivially\ngenerically $5$-stress free (resp. $6$-stress free), and so by\nTheorem~\\ref{th:whitcontract}, $G$ also is generically $5$-stress free\n(resp. $6$-stress free).\n\\end{proof}\n\nWe denote by $\\mu(G)$ the Colin de Verdi\u00e8re parameter of a graph $G$.  A\nresult of Colin de Verdi\u00e8re~\\cite{cdv1} is that a graph $G$ is planar\nif and only if $\\mu(G) \\leq 3$. Lov\\'asz and Schrijver~\\cite{ls1}\nproved that $G$ is linklessy embeddable if and only if $\\mu(G) \\leq\n4$.  Nevo conjectured that the following holds.\n\n\\begin{conj}[Nevo, 2007,~\\cite{nevo1}]\nLet $G$ be a graph and let $k$ be a positive integer. If $\\mu(G) \\leq k$\nthen $G$ is generically $k$-stress free.\n\\label{conj:munevo}\n\\end{conj}\n\nThis conjecture holds for the cases $k = 5$ and $k = 6$ as a\nconsequence of Theorem~\\ref{th:genstressk78}.\n\n\\begin{cor}\nIf $G$ is a graph such that $\\mu(G) \\leq 5$ (resp. $\\mu(G) \\leq 6$) then\n$G$ is generically $5$-stress free (resp. $6$-stress free).\n\\end{cor}\n\n\\begin{proof}\nNote that $\\mu(K_r) = r - 1$ and that if the complement of an\n$n$-vertex graph $G$ is a linear forest, then $\\mu(G) \\geq n -\n3$~\\cite{lsv1}. So we have that $\\mu(K_7) = 6$, $\\mu(K_8) = 7$,\nand $\\mu(K_{2,2,2,2,2}) \\geq 7$.\n\nAs the parameter $\\mu$ is minor-monotone~\\cite{cdv1}, the graph $K_7$\n(resp. $K_8$ and $K_{2,2,2,2,2}$) is an excluded minor for the class\nof graphs defined by $\\mu(G) \\leq 5$ (resp. $\\mu(G) \\leq 6$). Hence by\nTheorem~\\ref{th:genstressk78}, these graphs are generically $5$-stress\nfree (resp. $6$-stress free).\n\\end{proof}\n\n\n\\section{Application to double-critical $k$-chromatic graphs}\\label{sec:double}\n\nA connected $k$-chromatic graph is said to be double-critical is for\nall edge $uv$ of $G$, $\\chi(G \\setminus\\{u,v\\}) = \\chi(G) - 2$. It is\nclear that the clique $K_k$ is such a graph.  The following\nconjecture, known has the Double-Critical Graph Conjecture, due to\nErd\u0151s and Lov\u00e1sz states that they are the only ones.\n\n\\begin{conj}[Erd\u0151s and Lov\u00e1sz, 1968,~\\cite{e68}]\nIf $G$ is a double-critical $k$-chromatic graph, then $G$ is\nisomorphic to $K_k$.\n\\label{conj:dcgraph}\n\\end{conj}\n\nThis conjecture has been proved for $k \\leq 5$ but remains open for $k\n\\geq 6$.  Kawarabayashi, Pedersen and Toft have formulated a relaxed\nversion of both Conjecture~\\ref{conj:dcgraph} and the Hadwiger's\nconjecture, called the Double-Critical Hadwiger Conjecture.\n\n\\begin{conj}[Kawarabayashi, Pedersen, and Toft, 2010,~\\cite{kpt10}]\nIf $G$ is a double-critical $k$-chromatic graph, then $G$ contains a\n$K_k$-minor.\n\\label{conj:dchadwiger}\n\\end{conj}\n\nThe same authors proved this conjecture for $k \\leq 7$~\\cite{kpt10},\nbut the case $k = 8$ is left as an open problem. Pedersen proved that\nevery $8$-chromatic double-critical contains $K_8^-$ as a\nminor~\\cite{ped11}. Below we prove that the conjecture also holds for\n$k=8$.\n\nThe following proposition lists some interesting properties about\n$k$-chromatic double-critical graphs :\n\\begin{prop}[Kawarabayashi, Pedersen, and Toft, 2010,~\\cite{kpt10}]\nLet $G \\neq K_k$ be a double-critical $k$-chromatic graph, then\n\\begin{itemize}\n\\item The graph $G$ does not contains $K_{k-1}$ as a subgraph,\n\\item The graph $G$ has minimum degree at least $k + 1$,\n\\item For all edges $uv \\in E(G)$ and all $(k-2)$-coloring of $G - u -\n  v$, the set of common neighbors of $u$ and $v$ in $G$ contains\n  vertices from every color class.\n\\end{itemize}\n\\label{prop:kpt}\n\\end{prop}\n\nIn particular, the last item implies that every edge belongs to at\nleast $k - 2$ triangles.\n\n\\begin{thm}\nEvery double-critical $k$-chromatic graph, for $k\\le 8$, contains\n$K_k$ as a minor.\n\\end{thm}\n\\begin{proof}\nConsider for contradiction a $K_k$-minor free graph $G$ that is\ndouble-critical $k$-chromatic. By the second item of\nProposition~\\ref{prop:kpt}, $\\delta(G)\\ge k+1$. By\nTheorem~\\ref{th:krtri} and Theorem~\\ref{th:k8-deg9}, this graph\nhas an edge that belongs to at most $k-3$ triangles. This\ncontradicts the last item of Proposition~\\ref{prop:kpt}.\n\\end{proof}\n\nLet us now give an alternative proof of the case $k=8$ that does not\nneed Theorem~\\ref{th:k8-deg9}, but uses Theorem~\\ref{th:k8tri}\ninstead. This might be usefull to prove the next case of\nConjecture~\\ref{conj:dchadwiger}.\n\nConsider for contradiction a $K_8$-minor free graph $G$ that is\ndouble-critical $8$-chromatic.  By Theorem~\\ref{th:k8tri} this graph\nhas an edge that belongs to at most $5$ triangles or contains\n$K_{2,2,2,2,2}$ as an induced subgraph.  By Proposition~\\ref{prop:kpt}\nevery edge of $G$ belongs to at least 6 triangles, thus $G$ contains\n$K_{2,2,2,2,2}$ as an induced subgraph.  Let us denote $K\\subseteq\nV(G)$ the vertex set of a copy of $K_{2,2,2,2,2}$ in $G$. As\n$K_{2,2,2,2,2}$ is maximal $K_8$-minor free, any connected component\n$C$ of $G\\setminus K$ is such that $N(C)\\subset K$ induces a clique.\nAs $G$ is double-critical $8$-chromatic, there exists a $6$-coloring\nof $G[N[C]]$, and a $6$-coloring of $G\\setminus C$. As these two\ngraphs intersect on a clique one can combine their colorings and thus\nobtain a 6-coloring of $G$, a contradiction.\n\n\n\\section{Application for coloration of $K_d$-minor free graphs}\\label{sec:coloration}\n\nHadwiger's conjecture says that every $t$-chromatic graph $G$\n(i.e. $\\chi(G) =t$) contains $K_t$ has a minor. This conjecture has\nbeen proved for $t \\leq 6$, where the case $t = 5$ is equivalent to\nthe Four Color Theorem by Wagner's structure theorem of $K_5$-minor\nfree graph, and the case $t = 6$ has been proved by Robertson, Seymour\nand Thomas~\\cite{rst1}. The conjecture remains open for $t \\geq 7$.\nFor $t = 7$ (resp. $t = 8$) the conjecture asks $K_7$-minor free graphs\n(resp. $K_8$-minor free graphs) to be $6$-colorable (resp. $7$-colorable).\nUsing Claim~\\ref{claim-alpha-Nv} and the $9$-degeneracy\n(resp. $11$-degeneracy) of these graphs, one can prove that they are\n$9$-colorable (resp. 11-colorable). We improve these bounds by one.\n\nA graph $G$ is said to be \\emph{$t$-minor-critical} if $\\chi(G) = t$\nand $\\chi(H) < t$ whenever $H$ is a strict minor of $G$.  Hadwiger's\nconjecture can thus be reformulated as follows : Every\n$t$-minor-critical graph contains $K_t$ has a minor.  In the following\n$\\alpha(S)$ means $\\alpha(G[S])$,the independence number of\n$G[S]$. The following is a folklore claim, here for completeness.\n\n\\begin{claim}\n\\label{claim-alpha-Nv}\nGiven a $k$-minor critical graph $G$, for every vertex $v\\in V(G)$ we\nhave that $\\deg(v) + 2 - \\alpha(N(v)) \\ge k$.\n\\end{claim}\n\n\\begin{proof}\nGiven a vertex $v$ and a stable set $S$ of $N(v)$, consider the graph\n$G'$ obtained from $G$ by contracting the edges between $v$ and\n$S$. Since $G'$ is a strict minor of $G$ it is\n$(k-1)$-colorable. Given such coloring of $G'$, one can $(k-1)$-color\n$G\\setminus\\{v\\}$ in such a way that all the vertices of $S$ have the\nsame color assigned. In this coloring at most $\\deg(v) +1 - |S|$\ncolors are used in $N(v)$, thus $\\deg(v) +2 - |S|$ colors\nare sufficient to color $G$, and thus $\\deg(v) + 2 - \\alpha(N(v)) \\ge k$.\n\\end{proof}\n\nA \\emph{split graph} is a graph which vertices can be partionned into\none set inducing a clique, and one set inducing an independent set.\nThese graphs are the graphs that do not contain $C_4$, $C_5$ or $2K_2$\nas induced subgraphs~\\cite{fh1}.\n\n\\begin{claim}\n\\label{claim-no-split}\nGiven a $k$-minor critical graph $G$, every separator $(A,B)$ of $G$\nis such that $G[A \\cap B]$ is not a split graph (i.e $G[A \\cap B]$\ncontains $C_4$, $C_5$ or $2K_2$ as an induced subgraph).\n\\end{claim}\n\n\\begin{proof}\nAssume by contradiction that there exists such separator $(A',B')$.\nThis implies the existence of a separator $(A,B)$ such that $S = A\\cap\nB \\subseteq A'\\cap B'$ , and such that each $G[A\\setminus S]$ and\n$G[B\\setminus S]$ have a connected component, $C_{A}$ and $C_{B}$ such\nthat $N(C_{A})=N(C_{B})=S$. Note that $G[S]$ is a split graph and let\n$I$ be one of its maximum independent sets and let $K=S \\setminus I$\nbe a clique. Let $G_A$ and $G_B$ be the graphs respectively obtained\nfrom $G[A]$ and $G[B]$ by identifying the vertices of $I$ into a\nsingle vertex $i$. By maximality of $I$, in both graphs the vertex set\n$K\\cup \\{i\\}$ induces a clique.  Furthermore, these graphs are strict\nminors of $G$ as the identification of the vertices in $I$ can be done\nby contracting edges incident to $C_B$ or $C_A$ respectively.  Thus,\nthese graphs are $(k-1)$-colorable and these colorings imply the\nexistence of compatible $(k-1)$-colorings of $G[A]$ and $G[B]$, since\nin both colorings the vertices of $I$ use the same color, and each\nvertex of $K$ uses a distinct color. This yields in a $(k-1)$-coloring\nof $G$, a contradiction.\n\\end{proof}\n\n\\begin{thm}\n$K_7$-minor free graphs are $8$-colorable.\n$K_8$-minor free graphs are $10$-colorable.\n\\end{thm}\n\n\\begin{proof}\nConsider by contradiction that there is a $K_7$-minor free graph $G$\nnon-8-colorable (resp. a $K_8$-minor free graph $G$ non-10-colorable).\nThis graph is chosen such that $|E(G)|$ is minimal, this graph is\nthus 9-minor-critical (resp. 11-minor-critical).\n\nFor any vertex $v$, since $\\alpha(N(v))$ is at least 1,\nClaim~\\ref{claim-alpha-Nv} implies that $\\deg(v) > 7$ (resp. $\\deg(v)\n> 9$).  If $\\deg(v) = 8$ (resp. $\\deg(v) = 10$), since $G$ is\n$K_7$-minor free (resp. $K_8$-minor free), we have $\\alpha(N(v)) \\ge\n2$, contradicting Claim~\\ref{claim-alpha-Nv}. Finally if $\\deg(v) = 9$\n(resp. $\\deg(v) = 11$), Claim~\\ref{claim-alpha-Nv} implies that\n$3>\\alpha(N(v))$, and since $N(v)$ cannot be a clique, $\\alpha(N(v)) =\n2$. Thus with Mader's theorem we have that $\\delta(G)= 9$\n(resp. $\\delta(G)= 11$), and that for every vertex $v$ of degree 9\n(resp. of degree 11), $\\alpha(N(v)) = 2$.  By Theorem~\\ref{th:krtri}\n(resp. Theorem~\\ref{th:k8triweak}), we consider a vertex $u$ of degree\n9 (resp. 11) such that there is an edge $uv$ which belongs to at most\n$4$ (resp. $5$) triangles. Let $H = G[N(u)]$, and recall that\n$\\alpha(H) = 2$.\n\n\\begin{claim}\n\\label{claim-no-K5}\nThe graph $H = G[N(u)]$ does not contain a $K_5$ (resp. a $K_6$).\n\\end{claim}\n\n\\begin{proof}\nAssume by contradiction that $H$ contains a $K_t$ with vertices\n$x_1,\\ldots,x_t$, for $t=5$ (resp. for $t=6$). Assume first that the\ngraph induced by $Y = N(u) \\setminus \\{x_1,\\ldots,x_t\\}$ is\nconnected. Since $\\delta(G) \\ge 9$ every vertex $x_i$ has a neighbor\nin $Y$ or a neighbor $w_i$ in $G \\setminus N[u]$. In the latter case,\ndenote $C_i$ the connected component of $w_i$ in $G\\setminus N[u]$.\nSince by Claim~\\ref{claim-no-split} (for the partition $(N[C_i], V(G)\n\\setminus C_i)$) $N(C_i)$ intersects $Y$, one can contract $Y\\cup\n(V(G) \\setminus N[u])$ into a single vertex and form a $K_{t+2}$\ntogether with vertices $u,x_1,\\ldots,x_t$, a contradiction.\n\nAssume now that the graph induced by $Y$ is not connected and let\n$y_1, y_2 \\in Y$ be non-adjacent vertices. Since $G$ is $(2t-1)$-minor\ncritical, consider a $(2t-2)$-coloring of the graph $G'$ obtained from\n$G$ by contracting $uy_1$ and $uy_2$. This coloring implies the\nexistence of a $(2t-2)$-coloring $c$ of $G\\setminus u$ such that\n$c(y_1) = c(y_2)$. As this coloring does not extends to $G$, the\n$2t-1$ vertices in $N(u)$ use all the $(2t-2)$ colors. This implies\nthat the colors used for the $x_i$ are used only once in $N(u)$, and\nthat there exists a vertex $z \\in Y$ which color is used only once in\n$N(u)$. Assume $c(x_i)=i$ and $c(z)=7$. Given two colors $a$, $b$ and\na vertex $v$ colored $a$, the \\emph{$(a,b)$-component of $v$} is the\nthe connected component of $v$ in the graph induced by $a$- or\n$b$-colored vertices. For any $1\\le i \\le t$, suppose we switch colors\nin the $(i,7)$-component of $z$. As this cannot lead to a coloring\nwhich does not use all the colors in $N(u)$, there exists a\n$(7,i)$-bicolored path from $z$ to $x_i$. This is impossible as\ncontracting these paths on $z$ would lead to a $K_{t+2}$ (with vertex\nset $\\{u,z,x_1,\\ldots,x_t\\}$). This concludes the proof of the claim.\n\\end{proof}\n\nLet $v$ be a vertex of $H$ with minimum degree in $H$.  By the choice\nof $u$ and Theorem~\\ref{th:krtri} (resp. Theorem~\\ref{th:k8triweak}),\n$\\deg_{H}(v)\\le 4$ (resp. $\\deg_{H}(v)\\le 5$).\n\n\\begin{claim}\n$\\delta(H) = \\deg_H(v) = 4$ (resp. $\\delta(H) = \\deg_H(v) = 5$).\n\\end{claim}\n\\begin{proof}\nSince $\\alpha(H)=2$, the non-neighbors of $v$ in $H$ form a\nclique. Furthermore since $H$ does not contain a $K_5$ (resp. a $K_6$)\nwe have that $9 - 1 - \\deg_{H}(v) < 5$ (resp. that $11 - 1 -\n\\deg_{H}(v) < 6$), and hence $\\deg_{H}(v) = 4$ (resp. $\\deg_{H}(v) =\n5$).\n\\end{proof}\n\nLet $y_1,\\ldots,y_t$ with $t = 4$ (resp. $t = 5$) be the neighbors of\n$v$ in $H$, and let $K$ be the $t$-clique formed by its non-neigbors.\nBy Claim~\\ref{claim-no-K5} we can assume that $y_1$ and $y_2$ are\nnon-adjacent. Note that since $\\alpha(G[N(u)]) = 2$ every vertex of\n$K$ is adjacent to $y_1$ or $y_2$. Since $G$ is $(2t+1)$-minor\ncritical, consider a $2t$-coloring of the graph $G'$ obtained from $G$\nby contracting $uy_1$ and $uy_2$. This coloring implies the existence\nof a $2t$-coloring $c$ of $G \\setminus u$ such that $c(y_1) = c(y_2)$. As\nthis coloring does not extends to $G$, the $2t + 1$ vertices in $N(u)$\nuse all the $2t$ colors. In particular, the colors used by $K$ (say\n$1,\\ldots t$) and $y_3$ (say 6) are thus used only once in $N(u)$.\nFor any $1\\le i\\le t$, suppose we switch colors in the $(i,6)$-component\nof $y_3$. As this cannot lead to a coloring which does not use all the\ncolors in $N(u)$, there exists a $(i,6)$-bicolored path from $y_3$ to\nthe $i$-colored vertex of $K$. This is impossible as contracting these\npaths on $y_3$, and contracting the edges $vy_1$ and $vy_2$ on $v$\nwould lead to a $K_{t+2}$ with vertex set $\\{u,v,y_3\\} \\cup K$. This\nconcludes the proof of the theorem.\n\\end{proof}\n\n\n\\section{Conclusion}\\label{sec:concl}\n\nTheorem~\\ref{thm:global} gives a sufficient condition for a graph to\nhave a $K_k$-minor. We wonder whether this condition is stronger than\nMader's Theorem : Is there a graph $G$ with a $K_k$-minor, for $4\\le\nk\\le 7$, that has $m\\le (k-2)n - {k-1 \\choose 2}$ edges and $t\\ge\nm(k-3)\/2$ triangles ?\n\nWe believe that our work can be extended to the next case.  Song and\nThomas~\\cite{st1} proved a Mader-like theorem, similar to\nTheorem~\\ref{th:jorg} in the case of $K_9$-minor free graphs.\n\n\\begin{thm}[Song and Thomas, 2006,~\\cite{st1}]\nEvery graph on $n \\geq 9$ vertices and at least $7n - 27$ edges either\nhas a $K_9$-minor or is a $(K_{1,2,2,2,2,2}, 6)$-cockade or is isomorphic to\n$K_{2,2,2,3,3}$.\n\\end{thm}\n\nNote that $K_{2,2,2,3,3}$ has edges that belong to exactly $6$ triangles\nand contains $K_{2,2,2,2,2,1}$ as a minor. We conjecture that we\ncan extend our main theorem as follows.\n\\begin{conj}\nLet $G$ a graph such that every edge belongs to at least $7$ triangles\nthen either $G$ has a $K_9$-minor or contains $K_{1,2,2,2,2,2}$\nas an induced subgraph.\n\\label{conj:extendk9}\n\\end{conj}\n\n\nProving this conjecture would have several consequences.\nThis would extend Theorem~\\ref{thm:global}\nas follows : Every graph $G$ with $m\\ge 1$ edges and $t\\ge 3m$ triangles\nhas a $K_9$ or $K_{1,2,2,2,2,2}$-minor.\nIt would also imply Conjecture~\\ref{conj:munevo} for the case $k = 7$,\ni.e.  $\\mu(G)\\le 7$ implies that $G$ is generically $7$-stress free.\nFinally, it would imply Conjecture~\\ref{conj:dchadwiger} for $k = 9$,\ni.e. double-critical $9$-chromatic graphs have a $K_9$-minor.\nWe also conjecture that the following holds. In particular, it would\nimply that $K_9$-minor free graphs are $12$-colorable (using the same\narguments as in Section \\ref{sec:coloration}).\n\\begin{conj}\nAny $K_9$-minor free graph $G$ with $\\delta(G)=13$ has an edge $uv$\nsuch that $u$ has degree $13$ and $uv$ belongs to at most $6$\ntriangles.\n\\label{conj:extendk9weak}\n\\end{conj}\n\nWe also believe that these structural properties on graph with edges\nbelonging to many triangles can actually be extended to\nmatroids. Graph minors can be studied in the more general context of\nmatroid minors~\\cite{o1}. A triangle is then a circuit of size\n$3$. Contrary to graphs, the case when every element of the matroid\nbelongs to $3$ triangles is already intricate. There are three\nwell-known matroids for which each element belongs to $3$ triangles :\nthe Fano matroid $F_7$, the uniform matroid $\\mathcal{U}_{2,4}$, and\nthe graphical matroid $\\mathcal{M}(K_5)$.  We conjecture that the\nfollowing holds.\n\n\\begin{conj}\nLet $\\mathcal{M}$ be a matroid where each element is contained in $3$\ntriangles, then $\\mathcal{M}$ admits $\\mathcal{M}(K_5)$, $F_7$ or\n$\\mathcal{U}_{2,4}$ as a minor.\n\\end{conj}\n\n\\bibliographystyle{plain}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nManipulating the molecular rotational degrees of freedom in gas phase by means of laser fields remains a very attractive topic in quantum control \\cite{reviewQC1,reviewQC2} with a wide range of applications in photochemistry extending from chemical reactivity \\cite{Warren1993, Stapelfeldt2003} to nanoscale design \\cite{Seideman1997,Stapelfeldt1997}, stereochemistry \\cite{Rakitzis2004}, surface processing \\cite{Seideman1997,reuter2008}, catalysis \\cite{Bulthuis1991}, and attosecond molecular dynamics \\cite{Krausz2009}. Such phenomena play also a role in quantum computing \\cite{shapiro2003} and high-order harmonic generation \\cite{Atabek2003,Ramakrishna2007,Ramakrishna2010,houzet2013}. In this setting, molecular alignment and orientation can be identified as crucial prerequisites before exploring more complex control scenarios \\cite{reviewseideman,Stapelfeldt2003}. The alignment process \\cite{Friedrich1995} is by now a well-established concept from both the experimental and theoretical points of view with recent extensions ranging from the deflection of aligned molecules \\cite{gershnabel2010}, the introduction of planar alignment \\cite{hoque2011}, the control of molecular unidirectional motion \\cite{korech2013,steinitz2014,karras2015}, the study of molecular superrotors \\cite{korobenko2016,korobenko2014,milner2015} or the analysis of dissipation effects due to molecular collisions \\cite{ramakrishna2005,viellard1,viellard2,milner2014}. The description and the control of molecular orientation are  not currently at the same degree of improvement, in particular from the experimental point of view. Molecular orientation was achieved in the adiabatic regime \\cite{Goban2008,Ghafur2009,Filsinger2009,Takei2016,Muramatsu2009,Holmegaard2009}. If no static field is used, a rapid turn-off of the laser field allows to get orientation under field-free conditions. At low temperature, a very high degree of orientation can be obtained using such control strategies and a molecular quantum-state selection \\cite{Filsinger2009,Takei2016,Muramatsu2009,Holmegaard2009}. Control schemes in the sudden regime, where the duration of the control field is short with respect to the rotational period, have been also developed \\cite{averbukh2001,daems2005,dion2001,dion2005,ortigoso2012,Atabek2003,sugny2005,sugny2014,Lapert2012,machholm2001,henriksen1999,prasad2013,shu2013,Tehini08,zhang,Tehini2012,wu,Kanai2001}.\nAmong other techniques, we can mention the interaction of the molecule with a terahertz (THz) laser pulse and the $(\\omega,2\\omega)$- scheme. Note that a larger degree of orientation can be achieved with the two-color  mechanisms through ionization depletion \\cite{Spanner2012}. In this second regime, molecular orientation has been recently addressed experimentally for linear molecules using only a THz field \\cite{Nelson2011} or its combination with a laser field \\cite{Kitano2011}, and in \\cite{Znakovskaya2009,Kraus2014} for an excitation process with two-color laser fields.\n\nIn this work, we complement the previous experimental and theoretical works on the orientation dynamics produced by THz fields by exploring the orientation at high temperature (typically room temperature) of a symmetric top molecule, CH$_3$I. This molecule is a good candidate to achieve a high degree of orientation at room temperature due to its large permanent dipole moment and its relatively small rotational constant \\cite{Lapert2012}. The THz pulses are obtained from the excitation of a plasma by a two-color femtosecond laser field \\cite{Cook2000}, while the detection process is based on the free-induction decay (FID) emitted by the molecular sample after the THz excitation \\cite{hardprl,hard1991,hard1994,bigourd2008}. We extend the previous studies on the subject by considering the case of a symmetric top molecule at room temperature. We show that a noticeable degree of orientation can be reached. A theoretical description of the propagation of a THz field in the sample shows that the FID is not proportional to the degree of orientation but to its time derivative. A complete analytical derivation of this result is given in this paper. Note that this dependency has been already mentioned in \\cite{Nelson2011}. The relation between the FID and the degree of orientation allows us to quantitatively compare the experimental observations with the numerical simulations. A very good match is found for the first two orientation revivals. This agreement has been improved by accounting for the centrifugal distorsion and the relaxation effects in the computations \\cite{ramakrishna2005,viellard1,viellard2}. We also use this theoretical description to explore the influence of the laser parameters on the orientation dynamics.\n\nThe paper is organized as follows. The experimental setup is described in Sec.~\\ref{sec3}, with a special emphasis on the generation of THz pulses and on the detection process. We show in Sec.~\\ref{sec2p} that the FID is given at first order by the time derivative of the degree of orientation. The model system is introduced in Sec.~\\ref{sec2}. The numerical and the experimental results are discussed in Sec.~\\ref{sec4}. Conclusions and prospective views are given in Sec.~\\ref{sec5}.\n\n\\section{Experimental setup for producing molecular orientation \\label{sec3}}\nThe experimental set-up for producing and measuring molecular orientation is shown in Fig.\\ref{Set_Up}. It is based on a THz pulsed source and an electro-optical sampling device for the detection. The THz pulses are produced through plasma generation in gases with two-color femtosecond pulses \\cite{Cook2000,Kress2004,Kim2007,Kim2008,Vvedenskii2014}. A fraction of the output of a chirped pulse amplifier (CPA, 800 nm wavelength, 100 fs pulse duration, 7.5 mJ energy, 100 Hz repetition rate) is focused in dry nitrogen. A type I phase matching $\\beta$-Barium Borate (BBO) crystal  is inserted on the beam path at a given distance from the focusing point to produce the second harmonic at 400 nm. The generation of THz pulses is optimized by adjusting the phase matching angle of the frequency doubling crystal and its longitudinal position using a micrometric stage. The THz source provides pulses of typically few hundred femtoseconds pulse duration with only one cycle (see Fig.~\\ref{fig2} below for details) and covers a spectral range from 0 to 4 THz, with a maximum at about 1.5 THz. After collimation by an off-axis parabolic mirror, the two incoming beams at 800 and 400 nm are filtered out by means of a 2-mm thick plate from polytetrafluoroethylene and a thin black polyethylene sheet. The THz pulse is then focused in a cell (40 mm optical pathway) containing the sample at room temperature. The CH$_{3}$I sample (\\textit{Sigma Aldrich}, batch reference \\emph{BCBK 1300 V}, reagent grade chemicals) is initially stored in the liquid phase and vaporized by expansion into the cell under vacuum, just before the experiment so as to avoid water pollution of the sample.\nThe THz electric field in the gas sample is less than 100 kVcm$^{-1}$. The typical pressure used in this work is around 0.2-0.3 bar (\\textit{e.g.} below the saturation vapour pressure of CH$_{3}$I). For a good transmission of the THz beam through the gas cell, the windows are in polymethylpentene polymer (TPX, 38.1 mm diameter, 2.0 mm  thickness). The transmitted THz radiation is collected from the sample by another off-axis parabolic mirror and sent to the detection device. This latter is based on an electro-optical sampling of the THz pulse shape in the time domain \\cite{Winnewisser1997,Cai1998,Gallot1999,Planken2001}. The weak probe beam (typically $\\approx 5$ nJ) derived from the output of the CPA is focused and spatially overlapped with the THz radiation in a ZnTe (110) crystal. The polarization of the probe beam is modified through the Pockels effect induced by the THz beam (electro-optic detection process). This allows us to sample the THz electric field by changing the time delay between the two pulses with a motorized delay line stage. The change of the polarization state of the probe beam is measured from the combination of a quarter wave plate (QWP), a Wollaston Prism (WP), and two head-to-tail connected photodiodes so that the difference of their signals is directly obtained  (see Fig.~\\ref{Set_Up}). The difference signal is then amplified and sent to a lock-in amplifier synchronized with the laser repetition rate. The quarter wave plate is oriented so as to get a circular polarization without THz pulse and equivalent signals for the two photodiodes. The THz pathway is included in a box continuously purged (relative humidity $\\lesssim$ 7 $\\%$) with dry nitrogen to avoid absorption by water vapor.\n\\begin{figure*} [ht]\n                \\centering\n                \\includegraphics[width=12cm]{fig1}\n                \\caption{Experimental set-up. CPA: Chirped Pulse Amplifier (100 Hz repetition rate, $\\tau_L  \\approx 100~\\textrm{fs}$, $\\lambda_0 = 800~\\textrm{nm}$, $E_{100~\\textrm{Hz}}  \\approx 7.5~\\textrm{mJ}$), QWP: Quarter Wave Plate, HWP: Half Wave Plate, P: Polarizer, L: Lens, BBO:  Beta Barium Borate crystal, Filter: see the text, M: off-axis parabolic Mirror, BS: indium tin oxide Beam Splitter, ZnTe: electro-optical crystal, DL: motorized Delay Line stage,  WP: Wollaston Prism, PhD1 and PhD2: balanced photodiodes, A: integrated amplifier, PB: purged box.}\n                \\label{Set_Up}\n\\end{figure*}\nThe signal $\\Delta{V}$ recorded by the lock-in amplifier is proportional to $\\Delta{I}$, the difference between the intensities measured by the two head-to-tail connected photodiodes. $\\Delta{I}$ is given by \\cite{Planken2001}:\n\\begin{equation}\\label{firsteq}\n\\Delta{V}\\propto\\Delta{I}=I_{\\textrm{probe}} \\omega n^{3} {E(t) } r_{41} \\frac{L}{{c}},\n\\end{equation}\nwhere $I_{\\textrm{probe}}$ is the probe intensity, $\\omega$ the probe angular frequency, $n$ the refraction index at the probe frequency, $E(t)$ the THz electric field, $r_{41}$ the electro-optic coefficient of the ZnTe crystal (Pockels effect, $r_{41}$=4 pm~V$^{-1}$), $L$ the crystal length (200 $\\mu$m), and $c$ the speed of light in vacuum. The typical ratio $\\frac{{\\Delta V_{\\textrm{THz}} }}{{2V_{\\textrm{PhD}} }}=\\frac{{\\Delta{I}}}{I_{\\textrm{probe}}}$ measured in the experiment is $\\cong$ 1 - 5 $\\%$, $\\Delta V_{\\textrm{THz}}$ being the signal produced by the THz pulse  and $V_{\\textrm{PhD}}$ the signal delivered by each photodiode. The electric field $E(t)$ can be directly evaluated by using Eq.~(\\ref{firsteq}) and taking into account the transmission and reflection coefficients of the different optical components. The estimated electric field in the gas sample is typically within the range 6~-~30 kV~cm$^{-1}$. Note that $E(t)$ in Eq.~(\\ref{firsteq}) is the total electric field including the transmitted THz pulse $E_{0}(x,t)$ and the FID electric field, as discussed below. A typical recording of the electro-optical sampling signal is depicted in  Fig.~\\ref{Typical experimental signal}. It exhibits the transmitted THz pulse at zero delay and the two first orientational revivals at 67 and 134 ps. The goal of Sec.~\\ref{sec2p} is to interpret this experimental trace in terms of orientation efficiency.\n\\begin{figure} [ht]\n\\centering\n\\includegraphics[width=10cm]{fig2}\n\\caption{Experimental electro-optical sampling signal as a function of the delay between probe and THz pulses in CH$_{3}$I. The pressure in the gas cell was 0.26 bar. The transmitted THz pulse around zero delay and two first orientational revivals at 67 and 134 ps are shown.}\n\\label{Typical experimental signal}\n\\end{figure}\n\\section{Propagation of a THz field in a gaseous sample \\label{sec2p}}\nThis section is aimed at describing the production of FID and its propagation in a gaseous sample. We follow here the formalism and the approximations used in \\cite{hard1991,hard1994,hardprl,bigourd2008}.\n\nTo be more concrete, we consider a THz pulse linearly polarized along the $z$- direction and propagating along the $x$- one. This field is of the form $E_{0}(0, t)$ at $x = 0$.\nThis THz field experiences an instantaneous dipole $d(t)$ along the $z$- direction which satisfies:\n\\begin{equation}\nd(t) = \\mu_{0} \\langle\\cos\\theta\\rangle(t),\n\\end{equation}\nwhere $\\mu_0$ is the permanent dipole moment, $\\mu_0=1.6406$~D for the CH$_{3}$I molecule \\cite{gachi1989}, and $\\theta$ stands for the angle between the molecular axis defined by the C-I bond and the field polarization direction. The induced dipole $d(t)$ generates a contribution to the THz field which also propagates within the sample.\nWe introduce the polarization $p(\\omega)$ of the sample given in the frequency domain by:\n\\begin{equation}\np(\\omega) =\\frac{N}{V}d(\\omega)\n\\end{equation}\nwhere $N$ is the number of molecules and $V$ the corresponding volume. This polarization is also related to the electric field through the susceptibility parameter $\\chi(\\omega)$:\n\\begin{equation}\np(\\omega) = \\epsilon_{0}\\chi(\\omega)E(\\omega)\n\\end{equation}\nwhere $\\varepsilon_0$ is the vacuum permittivity. The spectral distribution of the field $E(\\omega)$ is defined as:\n\\begin{equation}\nE(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} E_0(0, t) e^{i\\omega t} dt,\n\\end{equation}\nwhich leads to:\n\\begin{equation}\n\\chi(\\omega) = \\frac{Nd(\\omega)}{\\epsilon_{0}V E(\\omega)}.\n\\end{equation}\nIn a linear approximation framework where $\\chi$ does not depend on the amplitude of the field, the resolution of the Maxwell equations gives that the propagation of the field can be written as follows \\cite{hard1991,hard1994}:\n\\begin{equation}\nE(x, t) = \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i[k(\\omega)x-\\omega t]}d\\omega,\n\\end{equation}\nwith $k(\\omega) = n(\\omega)\\omega\/c$, $n^{2} = 1+\\chi(\\omega)$. In the case of a dilute medium, the complex refractive index is given by $n(\\omega) \\simeq 1+\\frac{\\chi(\\omega)}{2}$. Expanding the exponential term $\\exp[\\frac{i\\omega x\\chi}{2c}]$ in Taylor series up to the second order, we arrive at:\n\\begin{eqnarray*}\nE(x, t)\n&= \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i( \\omega \\frac{x}{c}-\\omega t)}d\\omega \\\\\n&+ \\int_{-\\infty}^{+\\infty}E(\\omega)e^{i( \\omega\\frac{x}{c}-\\omega t)}i\\frac{\\omega x}{c}\\frac{\\chi}{2}d\\omega ,\n\\end{eqnarray*}\nwhich leads to:\n\\begin{equation}\nE(x, t) = E_{0}(x, t) + \\int_{-\\infty}^{+\\infty} E(\\omega)e^{i( \\omega\\frac{x}{c}-\\omega t)}i \\frac{\\omega x}{c} \\frac{Nd(\\omega)}{2\\epsilon_{0}V E(0, \\omega)} d\\omega\n\\end{equation}\nwhere $E_{0}(x, t)$ is the initial THz pulse.\nWe then get:\n\\begin{equation}\nE(x,t) = E_{0}(x,t) - \\frac{xN}{2c\\epsilon_{0}V}\\frac{d}{dt}d(t-\\frac{x}{c}) ,\n\\end{equation}\nwhere we have used the fact that the time derivative of a Fourier transform ($FT$) of a function $f$ is given by: $FT[\\frac{d}{dt}f(t)] = -i\\omega FT[f(t)]$. The THz electric field can finally be written as follows\n\\begin{equation}\nE(x,t) = E_{0}(x,t) - \\alpha(x) \\frac{d}{dt} \\langle \\cos\\theta\\rangle(t-\\frac{x}{c})\n\\label{eq:A10}\n\\end{equation}\nwhere $\\alpha(x)$ is a positive scalar factor depending upon the propagation coordinate $x$. The second term of the right hand side of Eq. (\\ref{eq:A10}) is the FID electric field emitted by the molecules of the sample. This contribution is used in the detection process as described above.\n\nFrom the experimental point of view, we point out that the FID manifests itself as recurrent THz echos launched by the molecular sample after its interaction with the THz pulse. It originates from transient orientation revivals of molecules inducing, under field-free conditions, a non-zero dipole. Another way of interpreting this phenomenon is to consider that the THz pulse experiences spectral shaping during its propagation. Absorption related to $J\\to J+1$  transitions produces periodic holes in the spectrum with a spectral separation in angular frequency $\\Delta \\omega= 4 \\pi c B_e$, with $B_e$ the rotational constant (neglecting the centrifugal distortion).  A periodic modulation in the frequency domain leads to periodic replica in the time domain every $\\Delta t=2\\pi\\Delta \\omega$. We emphasize that the first term of Eq.~(\\ref{eq:A10}) does not reflect the absorption experienced by the incident pulse because of the use of first order expansion.\n\n\\section{Model system \\label{sec2}}\nWe describe in this section the model used in the numerical computations to study the control of molecular orientation of the symmetric top molecule of methyl-iodide CH$_{3}$I. We consider the rotational dynamics of this molecular system, which is assumed to be in its ground vibronic state, in interaction with a linear polarized THz field. The Hamiltonian of the system can be written as:\n\\begin{equation}\n\tH(t) = H_{0} + H_{\\textrm{int}}(t),\n\t\\label{eq:01}\n\\end{equation}\nwhere $H_0$ and $H_{\\textrm{int}}$ describe respectively the field-free Hamiltonian and the interaction with the laser field.\nThe Hamiltonian of the molecular system is given by \\cite{zare}:\n\\begin{equation}\n\tH_{0} = B_{e}J^2 + (A_{e} - B_{e})J_{Z}^{2} - D_{J}J^{4} - D_{JK}J^2J_{Z}^{2} - D_{K}J_{Z}^{4}\n\t\\label{eq:02}\n\\end{equation}\nwhere $J$ is the angular momentum operator and $J_{Z}$ the component of $J$ along the body-fixed $Z$- axis defined by the C-I bond. The energy eigenvalues $E_{JK}$ of the operator $H_{0}$ in the Wigner basis $|JKM\\rangle$ for a prolate symmetric top can be expressed as follows:\n\\begin{eqnarray}\n\tE_{JK}\n\t&= B_{e}J(J+1) + (A_{e} - B_{e})K^{2} - D_{J}J^{2}(J+1)^{2}\\nonumber \\\\\n\t& - D_{JK}J(J+1)K^{2} - D_{K}K^{4},\n\\label{eq:03}\n\\end{eqnarray}\nwhere $A_{e}$ and $B_{e}$ are the rotational constants. The states $|JKM\\rangle$ are the eigenstates of the square of the angular momentum operator $J^2$ and of its projections, $J_Z$ and $J_z$ on the body-fixed $Z$- axis and space-fixed $z$- axis, respectively \\cite{zare}. The molecular parameters of the CH$_{3}$I molecule used in the numerical computations are given in Tab.~\\ref{tab:01} \\cite{Carocci1998}.\n\\begin{table}[!htp]\n\\centering\n\\begin{tabular}{p{3cm} p{3cm}}\n\t\\hline\n\tParameters & Values in cm$^{-1}$ \\\\ \\hline\\hline\n\t$B_{e}$    & $0.25098$                 \\\\\n\t$A_{e}$    & $5.173949$                 \\\\\n\t$D_{J}$    & $2.1040012\\times 10^{-7}$   \\\\\n\t$D_{JK}$   & $3.2944780\\times 10^{-6}$   \\\\\n\t$D_{K}$    & $8.7632195\\times 10^{-5}$   \\\\ \\hline\n\\end{tabular}\n\\caption{Values of the rotational and of the centrifugal constants of the CH$_{3}$I molecule used in the numerical computations.}\n\\label{tab:01}\n\\end{table}\nThe interaction between the molecular system and the external electromagnetic field reads:\n\\begin{equation}\n\tH_{\\textrm{int}}(t) = -\\mu_{0} E(t) \\cos\\theta,\n\t\\label{eq:04}\n\\end{equation}\nwhere the function $E(t)$ represents here the amplitude of the THz electric field. We neglect in this paper the effect of the polarizability components since the maximum intensity of the electric field remains moderate. The units used are atomic units unless otherwise specified.\n\nAt room temperature, the system is described by a density matrix $\\rho(t)$ whose dynamics is\ngoverned by the Liouville-von Neumann equation \\cite{Shapiro2012}:\n\\begin{equation}\n\ti\\frac{\\partial{\\rho(t)}}{\\partial{t}} = [H(t),\\rho(t)],\n\t\\label{eq:05}\n\\end{equation}\nwhere the initial condition $\\rho(0)$ is given by the canonical density operator at thermal equilibrium\n\\begin{equation}\n\t\\rho (0) = \\frac{1}{Z} \\sum_{J=0}^{\\infty}\\sum_{M,K=-J}^{J}  e^{-E_{JK}\/(k_{B}T)} |JKM\\rangle \\langle JKM|,\n\t\\label{eq:06}\n\\end{equation}\nwhere $Z = \\sum_{J=0}^{\\infty}\\sum_{M,K=-J}^{J} e^{-E_{JK}\/(k_{B}T)}$ is the partition function, with $T$ the temperature fixed to $T=298$~K and $k_{B}$ the Boltzmann constant.\n\nThe degree of orientation of the molecular system is given by the expectation value:\n\\begin{equation}\n\t\\langle\\cos\\theta\\rangle (t) = \\textrm{tr}[\\rho(t) \\cos\\theta].\n\t\\label{eq:07}\n\\end{equation}\nFurthermore, in order to simulate more realistic experimental conditions (see Sec.~\\ref{sec4}), we add to the model system the dissipative effects due to molecular collisions \\cite{Seideman2005,viellard1}. To limit the complexity of the numerical computations, we consider in this paper the effective approach proposed in \\cite{Seideman2005} to account for coherence relaxation described by the time $T_2\/P$, where $P$ is the pressure of the sample. We approximate the decoherence by an exponential decay such that the final degree of orientation is given by:\n\\begin{equation}\n\\langle\\cos\\theta\\rangle (t) = \\langle \\cos\\theta\\rangle _{\\textrm{ND}} e^{-\\frac{tP}{T_{2}}},\n\\end{equation}\nwhere the non-dissipative orientation, $\\langle \\cos\\theta\\rangle _{\\textrm{ND}}$, is computed from the Liouville-von Neumann equation (\\ref{eq:05}).\n\n\\section{Numerical and experimental results \\label{sec4}}\nIn this paragraph, we first investigate theoretically the degree of orientation that can be achieved with a THz laser pulse in the experimental conditions of the set-up. The Hilbert space is spanned by the Wigner's functions $|J,K,M\\rangle$, with $0\\leq J$, $-J\\leq K\\leq J$ and $-J\\leq M\\leq J$. $M$ and $K$ being good quantum numbers, the Hamiltonian of the system only depends on the angle $\\theta$. Numerically, we consider a finite dimensional Hilbert space with $J\\leq J_{\\textrm{max}}$, $J_{\\textrm{max}}=90$. From a physical point of view, this reduction can be justified by the fact that the THz excitation only transfers a finite amount of energy to the system, which thus stays in a finite dimensional subspace.\n\nThe control pulse can be approximated by a set of Hermite polynomials as follows:\n\\begin{equation}\\label{eqherm}\nE_0(t)\n= \\frac{E_1}{2}  e^{ -\\frac{t^2}{2\\sigma^2}} [ -3 D_{3} \\hat{H}_{2}(\\frac{t}{\\sigma}) + D_{1} \\hat{H}_{0}(\\frac{t}{\\sigma})]\n\\end{equation}\nwhere $D_{n}=(2^{n}n!\\pi^{1\/2})^{-\\frac{1}{2}}$, and $\\hat{H}_{n}(t)$ stands for the Hermite polynomials of order $n$, $\\hat{H}_2(t)=4 t^2-2$ and $\\hat{H}_0(t)=1$.\nThe parameter $\\sigma$ is given by the relation $16\\log(2)\\sigma^2 = \\tau^2$ with $\\tau = 1$~ps. The reasonable match between the theoretical THz pulse and the experimental one shown in Fig.~\\ref{fig2} justifies the choice made for the electric field in Eq.~(\\ref{eqherm}). Note that the peak-to-peak amplitude is experimentally of 9.4~kV~cm$^{-1}$ in this case.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig3}\n\t\t\\caption{(Color online) Experimental (red or dark gray) and theoretical (black), given by Eq.~(\\ref{eqherm}), THz pulses at delay zero. The amplitude $E_1$ has been adjusted to get the best match between the two pulses.}\n\t\t\\label{fig2}\n\\end{figure}\n\nWe start the analysis of the dynamics by giving a global picture of the time evolution of the molecular orientation as displayed in Fig.~\\ref{fig3}, where the relaxation effects are taken into account with $T_{2}= 23$~ps~atm \\cite{Hennequin1987,Roberts1968}. The pressure of the cell is $P=0.35~\\textrm{bar}$. Here, we fix the amplitude $E_1$ to 100 kV~cm$^{-1}$, which is the maximum experimental available amplitude. Revivals are observed at times multiples of the rotational period. The maximum of orientation is respectively of the order of $5\\times 10^{-4}$ and $2\\times 10^{-4}$ for the first and second revivals. Note that this maximum is larger than $10^{-3}$ in the non-dissipative case.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig4}\n\t\t\\caption{(Color online) Numerical orientation dynamics of the CH$_{3}$I molecule after an excitation with a $1~\\textrm{ps}$ pulse centered at $t=0~\\textrm{ps}$. The small insert displays the numerical evolution of the maximum degree of orientation at delay zero (blue, circle) and for the first (red, rectangle) and second (yellow, diamond) revivals as a function of the amplitude $E_1$ of the THz field.}\\label{fig3}\n\\end{figure}\nIn Fig.~\\ref{fig3}, we also study how the amplitude of the initial THz field affects the behavior of the orientation dynamics. Due to the low intensity of the field, we observe that the maximum values of each transient evolve linearly with respect to the amplitude. The effect of the pulse duration on the degree of orientation is displayed in Fig.~\\ref{fig4new}. More precisely, we consider the role of the parameter $\\tau$ as defined in Eq.~(\\ref{eqherm}). Note that the overall structure of the field is not modified by this parameter, the field is only compressed or extended in time. This modification corresponds to the currently available shaping techniques of THz pulses. In Fig.~\\ref{fig4new}, we observe a nonlinear behavior of the degree of orientation, which is maximum for the two revivals for a value of $\\tau$ of the order of 2 ps.\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig5}\n\t\t\\caption{(Color online) Maximum of $|\\langle\\cos\\theta\\rangle |$ for the delay zero (blue-circle) and the first (orange-square) and the second (yellow-diamond) revivals as a function of the parameter $\\tau$ of the pulse given by Eq.~(\\ref{eqherm}). In the experiment, the parameter $\\tau$ is of the order of 1 ps.}\\label{fig4new}\n\\end{figure}\n\nAs shown in Sec.~\\ref{sec2p}, the detection process is sensitive to the time derivative of the degree of orientation, i.e. $\\frac{d[\\langle\\cos\\theta\\rangle]}{dt}$ and not directly to the orientation. In Fig.~\\ref{var_1tr}, we study the effect of the amplitude of the field on the first revival. We observe that the shape of the transient does not change when the amplitude is varied. Similar results are obtained for the second revival.\n\n\n\\begin{figure} [ht]\n\t\t\\centering\n\t\t\\includegraphics[width=8cm]{fig6}\n\t\t\\caption{(Color online) Numerical evolution of the time derivative of $\\langle \\cos\\theta\\rangle$ of the first revival as a function of the peak amplitude of the electric field which is set to 20, 40, 60, 80 and 100 kV~cm$^{-1}$. The $\\alpha$ parameter is set to 1~kV~cm$^{-1}$~ps.}\n\t\t\\label{var_1tr}\n\\end{figure}\nAfter this complete theoretical description of the orientation dynamics, the goal is now to make a full comparison of the experimental and theoretical results. A scaling factor and a shift parameter along the vertical axis are determined to get the best match between the two sets of data. As shown in Fig.~\\ref{fig:Figs}, the numerical simulation reproduces quite well the experimental signal for the two first transients. The signal is due to a large number of molecules within the volume of the sample, each molecule being excited by a field of different amplitude. The fact that the shape of the revivals does not depend on the intensity of the electric field explains the good agreement observed in Fig.~\\ref{fig:Figs}. The signal is too weak after the second revival to pursue this comparison.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.6]{fig7}\n\\caption{(Color online) Time derivative of the degree of molecular orientation of the (a) first and (b) second revivals. The red lines (light gray) represent the experimental data while the black ones correspond to the numerical results. A filtered experimental signal is plotted in blue (dark gray) to ease the comparison with the simulated dynamics. The parameter $\\alpha$ is set to 5.75~kV~cm$^{-1}$~ps.}\n\\label{fig:Figs}\n\\end{figure}\n\n\\section{Conclusion \\label{sec5}}\nIn this article we have investigated the orientation of a symmetric top molecule, namely CH$_{3}$I. We have shown the efficiency of the full-optical ultrafast set-up resolved in time to generate THz pulses and to produce molecular orientation. We provide a detailed description of the experimental set-up used and a complete numerical study of the corresponding dynamics. The analysis of the detection process shows that the degree of orientation is indirectly measured via the time derivative of the expectation value of $\\cos\\theta$.  The theoretical model reproduces accurately the experimental results up to the second orientation revival. Additional numerical simulations reveal that the orientation dynamics induced by this THz pulse is qualitatively similar for the linear molecule, OCS. It will be interesting to consider also asymmetric top molecules which have a more complex and non-periodic field-free evolution than linear or symmetric top molecules.\n\nThe results of this work can be viewed as an important step forward for the control of molecular orientation. The good match between theory and experiment will allow us to explore the efficiency of more complex strategies using for instance a pre-alignment by a laser field. Such approaches are necessary to increase the degree of orientation and reach efficiencies where molecular orientation could be useful in practice.\\\\ \\\\\n\n\n\n\\noindent\\textbf{ACKNOWLEDGMENT}\\\\\nD. Sugny acknowledges the support from the ANR-DFG research programs Explosys (ANR-14-CE35-0013-01) and Coqs (ANR-15-CE30-0023-01). This work was supported by the Conseil R\\'egional de Bourgogne under the Photcom Pari program as well as the Labex ACTION program (ANR-11-LABX-01-01) and the CoConiCs program (Contract No. ANR-13-BS08-0013). This work has been done with the support of the Technical University of Munich, Institute for Advanced\nStudy, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement 291763.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is generally agreed that two-dimensional field-theory models may provide an excellent and rich framework to test ideas in gauge theories. In fact, the interest in studying these models is basically connected to the possibility of obtaining exact solutions, which are believed to be shared by their more realistic counterparts in four dimensions.\nOf these, the Schwinger model, also known as Quantum Electrodynamics in $(1+1)$-space-time dimensions, or ${QED}_2$\\cite{Schwinger:1962tn,Coleman:1975pw} has probably enjoyed the greatest popularity due to some special features that it possesses. For example, the energy spectrum contains a massive mode in spite of the gauge invariance of the original Lagrangian, the charge is screened and confinement is enforced by the explicit occurrence of a rising Coulomb potential. To our mind, these special features represent the essential ingredients of a mechanism by which one hopes to understand the phenomenon of quark-binding into physical hadrons. These issues were first analyzed in ${QED}_2$ in Refs. \\cite{Schwinger:1962tn,Schwinger:1962tp, Casher:1973uf,Coleman:1975pw}.\n\nUnfortunately, against this suggestive two-dimensional perspective, it seems to us that a convincing analytical proof of color confinement in quantum chromodynamics (QCD) still eludes us. The root of the problem is well known: while asymptotic freedom is a well established property of the perturbative dynamics of QCD, the transition to infrared slavery is problematic because of non-perturbative effects that dominate in the large distance limit of the theory. Once this ?large distance limit? is defined in terms of some phenomenological scale of distance, the immediate problem is that of identifying the dynamical variables that operate in that limit. A hint about the nature of those hidden dynamical variables comes from the phenomenological bag models of hadrons: the partial success of those models indicate that, in the large distance limit of QCD, the spatial extension of hadrons and the bag degrees of freedom must somehow be included among those new dynamical variables. It is clear that, in order to speak meaningfully of a ?QCD-solution? of the confinement problem, one would expect that such variables should arise from the very dynamics of QCD and control the mechanisms of color confinement \\cite{Luscher:1978rn}. \nThis is where the extrapolation of results from two to four spacetime dimensions may play a significant role in the understanding of the confinement mechanism in QCD. For instance, the correspondence between the colorless topological sector of QCD and the zero-charge sector of ${QED}_2$ was noted long ago in Ref.\\cite{Aurilia:1979dw} but never fully exploited;  The extrapolation from two to four dimensions, at least for the bosonized version of the Schwinger model, was considered in \\cite{  Aurilia:1980jz} while a general \"gauge mixing mechanism for the generation of mass\" was proposed in \\cite{Aurilia:1981xg}. \n\nMotivated by these observations, the general purpose of the present discussion is to communicate a deeper understanding  of the physical content of the $(3+1)$-dimensional generalization of the Schwinger model. The many avenues of research that are open to us were outlined in a research proposal by the authors \\cite{Aurilia:2015qia}. However, it seems clear that the first line of inquiry is to explore in more detail the role of the Abelian $3$-form field among the physical observables of the model. It has long been known that this $3$-form field does not support any propagating degree of freedom, its sole physical effect consisting of a static interaction between two probe charges.This remarkable property is entirely analogous to the two dimensional case where in ${QED}_2$ there are no \"photons\" associated with the electromagnetic field \\cite{Aurilia:1979dw}. Then, if the Schwinger model has any relevance in the issue of confinement in four dimensions, then the static potential induced by the Abelian $3$-form field must also exhibit the same behavior found in the two-dimensional case. We find that this reasonable expectation is fully supported by the explicit calculation of the interaction energy between two external test charges.\n\nA second objective of this work is to elucidate the remarkable interplay between guge invariance and the appearance of mass in the physical spectrum of the Schwinger model. With hindsight, the emergence of this massive mode can be traced back directly to the dimensionality of the coupling constant in ${QED}_2$ which sets a mass scale in the model. Evidently this is not the case in ${QED}_4$ but a similar phenomenon takes place, at least in the bosonized version of the Schwinger model in $(3+1)$  dimensions. We illustrate how this same generalization of the S-model basically amounts to a Stueckelberg-like formulation of a massive gauge theory characterized by the mixing between a $U(1)$ potential and an Abelian $3$-form field.\n\nOur work is organized according to the following outline: in Section II, we recall the salient features of dualization in terms of two simple Lagrangian systems and show their equivalence to different representations of a massive Proca field. In Section III,  using a path-integral approach, we compute the interaction energy, and hence the analytic form of the static potential in the bosonized version of the Scwinger model in four spacetime dimensions. Finally, some Concluding Remarks are cast in Sec. IV. \n\nThroughout the following discussion, the signature of the metric is ($+1,-1,-1,-1$).\n\n\\section{Dualization, gauge invariance and mass generation}\n\nLet us start our considerations by recalling that the study of duality symmetry in gauge theories has been of considerable importance in \norder to provide an equivalent description of physical phenomena by distinct theories. As well-known, duality refers to a physical equivalence \nbetween two field theories which formulated in terms of different dynamical variables \\cite{Hjelmeland:1997eg}. \n\nIn order to put our discussion into context, we also recall that the dualization of Stueckelberg-like massive gauge theories \nand $ B \\wedge F$ models follows from a general $p$ dualization of interacting theories in $d$ spacetime \ndimensions \\cite{Ansoldi:1999wi,Smailagic:1999qw,Ansoldi:2000qs,Smailagic:2000hr,Smailagic:2001ch}. \nIn particular, in the case of $(3+1)$ dimensions, the following $ B \\wedge F$ models are found:\n\\begin{equation}\n{\\cal L}^{(1)} =  - \\frac{1}{4}F_{\\mu \\nu }^2\\left( A \\right) + \\frac{1}{{12}}H_{\\mu \\nu \\rho }^2\\left( B \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{B_{\\mu \\nu }}{\\partial _{[\\rho }}{A_{\\sigma ]}},  \\label{Dual05}\n\\end{equation}\n\n\\begin{equation}\n{\\cal L}^{(2)} =  - \\frac{1}{4}H_{\\mu \\nu }^2\\left( B \\right) + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2\\left( A \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{B_\\mu }{\\partial _{[\\nu }}{A_{\\rho \\sigma ]}}, \\label{Dual10}\n\\end{equation}\n\n\\begin{equation}\n{\\cal L}^{(3)} = \\frac{1}{2}{\\left( {{\\partial _\\mu }\\varphi } \\right)^2} - \\frac{1}{{48}}F_{\\mu \\nu \\rho \\sigma }^2\\left( A \\right) \n+ \\frac{m}{{24}}{\\varepsilon ^{\\mu \\nu \\rho \\sigma }}\\varphi {\\partial _{[\\mu }}{A_{\\nu \\rho \\sigma ]}}. \\label{Dual15}\n\\end{equation}\n\nAt this point, it is instructive to make a brief re-examination of equations (\\ref{Dual05}) and (\\ref{Dual10}). For this purpose, we observe \nthat the Lagrangian density (\\ref{Dual05}) may be rewritten as\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}F_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma } \n- \\frac{m}{6}{\\tilde H^\\sigma }{A_\\sigma }, \\label{Dual25}\n\\end{equation}\nwhere we have made use of ${\\tilde H^\\mu } = \n{\\raise0.5ex\\hbox{$\\scriptstyle 1$}\\kern-0.1em\/\\kern-0.15em\\lower0.25ex\\hbox{$\\scriptstyle 2$}}{\\varepsilon ^{\\mu \\nu \\lambda \\rho }}{\\partial _\\nu }{B_{\\lambda \\rho }}$.\n\nNext, in order to eliminate the dual-field $H^{\\sigma}$ care must be taken, for it satisfies the constraint \n${\\partial _\\mu }{\\tilde H^\\mu } = 0$ (Bianchi identity). Thus, to take into account the constraint, we shall introduce a Lagrange \nmultiplier $\\chi$. In such a case, the corresponding effective Lagrangian density (\\ref{Dual25}) reads \n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} = - \\frac{1}{4}F_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma }\n- \\frac{m}{6}{\\tilde H^\\sigma }{A_\\sigma } + \\chi {\\partial _\\sigma }{\\tilde H^\\sigma }   .\\label{Dual30}\n\\end{equation}\nBy defining ${Z_\\sigma } \\equiv {A_\\sigma } + \\frac{6}{m}{\\partial _\\sigma }\\chi$, with $  {Z_{\\mu \\nu }} \n= {F_{\\mu \\nu }}$, we readily verify that\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}Z_{\\mu \\nu }^2 - \\frac{1}{2}{\\tilde H_\\sigma }{\\tilde H^\\sigma } \n- \\frac{m}{6}{\\tilde H^\\sigma }{Z_\\sigma }.\\label{Dual35} \n\\end{equation}\nBy a further definition of the fields, ${W_\\sigma } \\equiv {\\tilde H_\\sigma } + \\frac{m}{6}{Z_\\sigma }$, we find that the Lagrangian \ndensity (\\ref{Dual05}) can be brought to the form\n\\begin{equation}\n{{\\cal L}^{\\left( 1 \\right)}} =  - \\frac{1}{4}Z_{\\mu \\nu }^2 + \\frac{1}{2}{\\mu ^2}Z_\\mu ^2, \\label{Dual40}\n\\end{equation}\nwith ${\\mu ^2} \\equiv {\\raise0.5ex\\hbox{$\\scriptstyle {{m^2}}$}\n\t\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle {36}$}}$. We immediately see that the Lagrangian density (\\ref{Dual40}) exhibits a Proca-type mass term.\n\nWe now turn our attention to the Lagrangian density (\\ref{Dual10}). It is convenient to rewrite this equation in the alternative form \n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{\\tilde H^{\\rho \\sigma }}{A_{\\rho \\sigma }}, \\label{Dual45}\n\\end{equation}\nwhere ${\\tilde H^{\\mu \\nu }} = {\\raise0.5ex\\hbox{$\\scriptstyle 1$}\n\t\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle 2$}}{\\varepsilon ^{\\mu \\nu \\lambda \\rho }}{H_{\\lambda \\rho }}$.\n\nIt is worthy to notice that the $B^{\\mu}$- field appears only through ${\\tilde H^{\\mu\\nu}}$.\nAgain, in order to eliminate the dual-field $\\tilde H^{\\mu\\nu}$ care must be taken, for it satisfies the \nconstraint ${\\partial _\\mu }{\\tilde H^{\\mu\\nu} } = 0$. As before, we shall introduce a Lagrange multiplier $\\chi_{\\nu}$. \nIt gives rise to the following Lagrangian density,\n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{\\tilde H^{\\mu \\nu }}{A_{\\mu \\nu }} - \\frac{1}{2}{\\tilde H^{\\mu \\nu }}{\\chi _{\\mu \\nu }},\\label{Dual50}\n\\end{equation}\nwhere ${\\chi _{\\mu \\nu }} = {\\partial _\\mu }{\\chi _\\nu } - {\\partial _\\nu }{\\chi _\\mu }$.\nNow, letting ${Z_{\\mu \\nu }} = {A_{\\mu \\nu }} - \\frac{{12}}{m}{\\chi _{\\mu \\nu }}$, we obtain\n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{4}\\tilde H_{\\mu \\nu }^2 + \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 \n+ \\frac{m}{{24}}{{\\tilde H}^{\\mu \\nu }}{Z_{\\mu \\nu }}. \\label{Dual55}\n\\end{equation}\nIt should be further noted that, by defining ${W_{\\mu \\nu }} = {{\\tilde H}_{\\mu \\nu }} \n+ \\frac{m}{{12}}{Z_{\\mu \\nu }}$, equation (\\ref{Dual55}) reduces to \n\\begin{equation}\n{{\\cal L}^{\\left( 2 \\right)}} = \\frac{1}{{12}}F_{\\mu \\nu \\rho }^2 - \\frac{1}{2}{\\mu ^2}Z_{\\mu \\nu }^2, \\label{Dual60} \n\\end{equation}\nwhere we have written, ${\\mu ^2} = \\frac{{{m^2}}}{{288}}$, and \n$F_{\\mu \\nu \\rho }^2 = Z_{\\mu \\nu \\rho }^2$. Thus ${{\\cal L}^{\\left( 2 \\right)}}$ describes a massive field of spin $1$, exactly a \nProca equation, although ${Z_{\\mu \\nu }} \\in \\left[ {\\left( {1,0} \\right) \\oplus \\left( {0,1} \\right)} \\right]$. Actually, a massive\nrank-two skew-symmetric tensor field is, on-shell, equivalent to a Proca field.\n\nIn short, equations (\\ref{Dual05}) and (\\ref{Dual10}) are equivalent; both of these equations describe a Proca field.\n\nConsidering, finally, equation (\\ref{Dual15}), we find that this model reduces to a  massless Schwinger model in $(3+1)$ dimensions, \nas we shall indicate it below.\n\n\\section{Interaction energy}\n\nInspired by the preceding observation, we shall now consider the $(3+1)$-dimensional generalization of the Schwinger model, as originally \nintroduced in Ref.\\cite{Aurilia:1979dw}. As we have already noticed, we will work out the static potential for this $(3+1)$ generalization, \nvia a path-integral approach. To this end, we consider the bosonized form of the Schwinger model in D=$(3+1)$, that is,\n\\begin{equation}\n{\\cal L} = \\frac{1}{2}{\\left( {{\\partial _\\mu }\\phi } \\right)^2} + \\frac{1}{2}m_\\phi ^2{\\phi ^2} \n+ \\frac{g}{{6\\sqrt \\pi  }}{\\partial _\\mu }\\phi\\ {\\varepsilon ^{\\mu \\nu \\rho \\sigma }}{A_{\\nu \\rho \\sigma }} \n- \\frac{1}{{48}}F_{\\mu \\nu \\rho \\sigma }^2, \\label{Scwhin3-05}\n\\end{equation}\nwhere $g$ is a coupling constant and $m_\\phi$ refers to the mass of the scalar field $\\phi$.\n\nWe readily verify that when, ${m_\\phi } \\to 0$,\nequation (\\ref{Scwhin3-05}) reduces to equation (\\ref{Dual15}).\n\nAccording to usual procedure, integrating out the $\\phi$ field induces an effective theory for the $A_{\\nu \\rho \\sigma }$ field.  \nIt is now important to recall that the $ A_{\\nu \\rho \\sigma }$ field  can also be written as ${A_{\\nu \\rho \\sigma }} \n= {\\varepsilon _{\\nu \\rho \\sigma \\lambda }}{\\partial ^\\lambda }\\xi$  \\cite{Aurilia:2004cb, Aurilia:2004fz}, \nwhere $\\xi$ refers to an another scalar field. This then leads to the following effective theory for the model under consideration:\n\\begin{equation}\n{\\cal L} = \\frac{1}{2}\\left[ {\\xi \\ \\Delta \\left( {1 + \\frac{{{\\raise0.7ex\\hbox{${{g^2}}$} \\!\\mathord{\\left\/\n\t\t\t\t\t\t{\\vphantom {{{g^2}} \\pi }}\\right.\\kern-\\nulldelimiterspace}\n\t\t\t\t\t\\!\\lower0.7ex\\hbox{$\\pi $}}}}{{\\left( {\\Delta  - m_\\phi ^2} \\right)}}} \\right)\\Delta \\ \\xi } \\right], \\label{Scwhin3-10}\n\\end{equation}\nwhere $\\Delta  = {\\partial _\\mu }{\\partial ^\\mu }$.\n\nWe are now ready to compute the interaction energy between static pointlike sources. We start off our analysis by writing down the functional \ngenerator of the Green's functions, that is,\n\\begin{equation}\nZ\\left[ J \\right] = \\exp \\left( { - \\frac{i}{2}\\int {{d^4}x{d^4}yJ(x)D(x,y)J(y)} } \\right), \\label{Scwhin3-15}\n\\end{equation}\nwhere, $D(x,y) = \\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}D(k){e^{ - ikx}}}$, is the propagator. In this case, the corresponding \npropagator is given by\n\\begin{equation}\nD(k) = \\left( {1 - \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}} \\right)\\frac{1}{{{k^2}\\left( {{k^2} + {{\\cal M}^2}} \\right)}} \n+ \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}\\frac{1}{{{k^4}}}, \\label{Scwhin3-20}\n\\end{equation}\nwhere ${{\\cal M}^2} = m_\\phi ^2 - {\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\\kern-0.1em\/\\kern-0.15em\n\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}$.\n\nBy means of expression $Z = {e^{iW\\left[ J \\right]}}$ and employing Eq. (\\ref{Scwhin3-15}), ${W\\left[ J \\right]}$ takes the form \n\\begin{eqnarray}\nW\\left[ J \\right] &=&  - \\frac{1}{2}\\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}} {J^ * }\\left( k \\right)\n\\frac{{\\left( {1 - \\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}} \\right)}}{{{k^2}\\left( {{k^2} + {{\\cal M}^2}} \\right)}}\nJ\\left( k \\right) \\nonumber\\\\\n&-& \\frac{1}{2}\\int {\\frac{{{d^4}k}}{{{{\\left( {2\\pi } \\right)}^4}}}} {J^ * }\\left( k \\right)\\frac{{m_\\phi ^2}}{{{{\\cal M}^2}}}\n\\frac{1}{{{k^4}}}J\\left( k \\right).\n\\label{Scwhin3-25}\n\\end{eqnarray}\n\nNext, for $J({\\bf x}) = \\left[ {Q{\\delta ^{\\left( 3 \\right)}}\\left( {{\\bf x} - {{\\bf x}^{\\left( 1 \\right)}}} \\right) \n\t+ {Q^ \\prime }{\\delta ^{\\left( 3 \\right)}}\\left( {{\\bf x} - {{\\bf x}^{\\left( 2 \\right)}}} \\right)} \\right]$, we obtain that the interaction \nenergy of the system is given by\n\\begin{eqnarray}\nV &=&  - Q{Q^ * }\\int {\\frac{{{d^3}k}}{{{{\\left( {2\\pi } \\right)}^3}}}} \n\\frac{{\\left( {\\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2}}} \\right)}}{{\\left( {{{\\bf k}^2} + {\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}}\n{e^{i{\\bf k} \\cdot {\\bf r}}} \\nonumber\\\\\n&+& Q{Q^ * }\\int {\\frac{{{d^3}k}}{{{{\\left( {2\\pi } \\right)}^3}}}} \\left( {\\frac{{m_\\phi ^2}}{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2}}} \\right)\\frac{1}{{{{\\bf k}^4}}}{e^{i{\\bf k} \\cdot {\\bf r}}}, \\label{Scwhin3-30}\n\\end{eqnarray}\nwhere ${\\bf r} = {{\\bf x}^{\\left( 1 \\right)}} - {{\\bf x}^{\\left( 2 \\right)}}$.\n\nThis, together with ${Q^ \\prime }=-Q$, yields finally\n\\begin{eqnarray}\nV &=& \\frac{{{Q^2}}}{{4\\pi }}\\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{{\\left( {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}^2}}}\\frac{1}{L}\\left( {1 - {e^{ - \\sqrt {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} L}}} \\right)  \\nonumber\\\\\n&+& \\frac{{{Q^2}}}{{4\\pi }}\\frac{{m_\\phi ^2}}{{2\\left( {{\\raise0.5ex\\hbox{$\\scriptstyle {{g^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}} - m_\\phi ^2} \\right)}}L, \\label{Scwhin3-35}\n\\end{eqnarray}\nwhere $L = |{\\bf r}|$. One immediately sees that the above static potential profile is analogous to that encountered in the two-dimensional \nSchwinger model. Incidentally, in order to put our discussion into context it is useful to summarize the relevant aspects of the \ntwo-dimensional Schwinger model. In such a case, we begin by recalling the bosonized form of the model under consideration \\cite{Gross:1995bp}:\n\\begin{eqnarray}\n{\\cal L} &=&  - \\frac{1}{4}F_{\\mu \\nu }^2 + \\frac{1}{2}{\\left( {{\\partial _\\mu }\\phi } \\right)^2} \n- \\frac{e}{{2\\sqrt \\pi  }}{\\varepsilon ^{\\mu \\nu }}{F_{\\mu \\nu }}\\phi \\nonumber\\\\\n&+&m\\sum \\left( {\\cos \\left( {2\\pi \\phi  + \\theta } \\right) - 1} \\right), \\label{Scwhin3-40}\n\\end{eqnarray}\nwhere $\\sum  = \\left( {\\frac{e}{{2{\\pi ^{{\\raise0.5ex\\hbox{$\\scriptstyle 3$}\n\t\t\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle 2$}}}}}}} \\right){e^{{\\gamma _E}}}$ with ${\\gamma _E}$ the Euler-Mascheroni constant and $\\theta$ refers to \nthe $\\theta$-vacuum. \n\nConsequently, by using the gauge-invariant but path-dependent variables formalism which provides a physically-based alternative to the Wilson \nloop approach \\cite{Gaete:1999iy, Gaete:2001wh}, the static potential reduces to\n\\begin{equation}\nV = \\frac{{{Q^2}}}{2}\\frac{{\\sqrt \\pi  }}{e}\\left( {1 - {e^{ - \\frac{e}{{\\sqrt \\pi  }}L}}} \\right), \\label{Scwhin3-45}\n\\end{equation}\nfor the massless case. On the other hand, for the massive case ($\\theta=0$), the static potential then becomes\n\\begin{equation}\nV = \\frac{{{Q^2}}}{{2\\lambda }}\\left( {1 + \\frac{{4\\pi m\\sum }}{{{\\lambda ^2}}}} \\right)\\left( {1 - {e^{ - \\lambda L}}} \\right) \n+ \\frac{{{q^2}}}{2}\\left( {1 - \\frac{{{\\raise0.5ex\\hbox{$\\scriptstyle {{e^2}}$}\n\t\t\t\t\\kern-0.1em\/\\kern-0.15em\n\t\t\t\t\\lower0.25ex\\hbox{$\\scriptstyle \\pi $}}}}{{{\\lambda ^2}}}} \\right)L, \\label{Scwhin3-50}\n\\end{equation}\nwhere ${\\lambda ^2} = \\frac{{{e^2}}}{\\pi } + 4\\pi m\\sum$. The above results clearly show that the $(3+1)$-D generalization of the Schwinger \nmodel is structurally identical to the $(1+1)$-D Schwinger model.\n\nIn this perspective it is worth recalling that there is an alternative way of obtaining the Lagrangian density  (\\ref{Scwhin3-10}), \nwhich provides a complementary view into the physics of confinement. In fact, we refer to a theory of antisymmetric tensor fields that \nresults from the condensation of topological defects as a consequence of the Julia-Toulouse mechanism. \nMore precisely, the Julia-Toulouse mechanism is a condensation process dual to the Higgs mechanism proposed in \\cite{Quevedo:1996uu}. \nThis mechanism describes phenomenologically the electromagnetic behavior of antisymmetric tensors in the presence of magnetic-branes \n(topological defects) that eventually condensate due to thermal and quantum fluctuations. Using this phenomenology we have discussed \nin \\cite {Gaete:2004dn,Gaete:2005am} the dynamics of the extended charges (p-branes) inside the new vacuum provided by the condensate. \nActually, in \\cite {Gaete:2004dn} we have considered the topological defects coupled both longitudinally and \ntransversally to two different tensor potentials, $A_p$ and $B_q$, such that $p+q+2=D$, where $D=d+1$ space-time dimensions.\n\nWe skip all the technical details and refer to \\cite{Gaete:2004dn} for them. Thus, after the condensation, the Lagrangian density turns out to be\n\\begin{eqnarray}\n{\\cal L} &=& \\frac{{{{\\left( { - 1} \\right)}^q}}}{{2\\left( {q + 1} \\right)!}}{\\left[ {{H_{q + 1}}\\left( {{B_q}} \\right)} \\right]^2} \n+ e{B_q}{\\varepsilon ^{q,\\alpha ,p + 1}}{\\partial _\\alpha }{\\Lambda _{p + 1}} \\nonumber\\\\\n&+& \\frac{{{{\\left( { - 1} \\right)}^{p + 1}}}}{{2\\left( {p + 2} \\right)!}}{\\left[ {{F_{p + 2}}\\left( {{\\Lambda _{p + 1}}} \\right)} \\right]^2} \n\\nonumber\\\\\n&+& \\frac{{{{\\left( { - 1} \\right)}^{p + 1}}\\left( {p + 1} \\right)!}}{2}{m^2}\\Lambda _{p + 1}^2,  \\label{Scwhin3-55}\n\\end{eqnarray}\nshowing a $B$$\\wedge$$F$ type of coupling between the $B_q$ potential with the tensor $\\Lambda_{p+1}$ carrying the degrees of freedom of the \ncondensate. Following our earlier procedure \\cite{Gaete:2004dn}, the effective theory that results from integrating out the fields representing \nthe vacuum condensate, is given by\n\\begin{equation}\n{\\cal L} = \\frac{{{{\\left( { - 1} \\right)}^{q + 1}}}}{{2\\left( {q + 1} \\right)!}}{H_{q + 1}}\\left( {{B_q}} \\right)\n\\left( {1 + \\frac{{{e^2}}}{{\\Delta  - {m^2}}}} \\right){H^{q + 1}}\\left( {{B_q}} \\right). \\nonumber\\\\\n\\label{Scwhin3-60}\n\\end{equation}  \nHence we see that this expression with $p=-1$ and $q=3$ becomes\n\\begin{equation}\n{\\cal L} = \\frac{1}{{2 \\times 4!}}{F_{\\mu \\nu \\rho \\lambda }}\\left( A \\right)\\left( {1 + \\frac{{{e^2}}}{{\\Delta  - {m^2}}}} \\right)\n{F^{\\mu \\nu \\rho \\lambda }}\\left( A \\right). \\label{Scwhin3-65}\n\\end{equation}\nIt is straightforward to verify that Eq. (\\ref{Scwhin3-65}) reduces to Eq. (\\ref{Scwhin3-10}).\n\nIn this way, we establish a new connection among different effective theories. It must be clear from this discussion that the above connections \nare of interest from the point of view of providing unifications among diverse models as well as exploiting their equivalence in explicit \ncalculations. \n\n\n\\section{Concluding Remarks}\n\nFinally, the point we wish to emphasize is that there are two generic features that are common in the four-dimensional case and their \nupper\/lower extensions, as we shall show below. First, the existence of a linear potential, leading to the confinement of static charges. \nThe second point is related to the correspondence among diverse effective theories. To see this, it should be noted that by using the \nmethodology illustrated in \\cite{  Smailagic:1999qw}, we have that one of the $ B \\wedge F$ models in $(4+1)$ dimensions is given by the \nmixing between a $U(1)$ potential and an Abelian $3$-form field by means of a topological mass term, that is,\n\\begin{eqnarray}\n{\\cal L}^{\\left( {4 + 1} \\right)} &=&  - \\frac{1}{4}{F_{\\mu \\nu }}\\left( A \\right){F^{\\mu \\nu }}\\left( A \\right) \n+ \\alpha {H_{\\mu \\nu \\kappa \\lambda }}\\left( C \\right){H^{\\mu \\nu \\kappa \\lambda }}\\left( C \\right) \\nonumber\\\\\n&+& \\beta {\\varepsilon ^{\\mu \\nu \\kappa \\lambda \\rho }}{A_\\mu }{\\partial _\\nu }{C_{\\kappa \\lambda \\rho }}, \\label{CR-05}\n\\end{eqnarray} \nwith $\\alpha  =  - \\frac{1}{{48}}$ and $ \\beta  = \\frac{\\sigma }{6}$, where the parameter $\\beta$ has mass dimension. This model was considered \nin \\cite{Cocuroci:2013bga}, and the main motivation to consider this model is based on the possible connection with dark energy.\n\nHowever, we shall start from the five-dimensional spacetime model\n\\begin{eqnarray}\n{\\cal L}^{\\left( {4 + 1} \\right)} &=&  - \\frac{1}{4}{F_{\\hat \\mu \\hat \\nu }}{F^{\\hat \\mu \\hat \\nu }} \n+ \\alpha {H_{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda }}{H^{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda }} \\nonumber\\\\\n&+& \\beta {\\varepsilon ^{\\hat \\mu \\hat \\nu \\hat \\kappa \\hat \\lambda \\hat \\rho }}{A_\\mu }{\\partial _\\nu }{C_{\\hat \\kappa \\hat \\lambda \\hat \\rho }} + \\frac{1}{{12}}m_C^2{C_{\\hat \\mu \\hat \\nu \\hat \\rho }}{C^{\\hat \\mu \\hat \\nu \\hat \\rho }}, \\nonumber\\\\\n\\label{CR-10}\n\\end{eqnarray}\nwith the additional presence of a mass term $m_C$ for the Abelian $3$-form field; this explicit mass term makes a difference: if it were not introduced, the model\ncould be reduced to nothing but a Proca-type model in $(4+1)$ dimensions. Next, we perform its dimensional reduction along the \nlines of \\cite{Cocuroci:2013bga,Gaete:2012yu}: \n${A^{\\hat \\mu }} \\to \\left( {{A^{\\bar \\mu }},{A^4}} \\right)$, ${A^4} = \\phi$, ${\\partial _4}\\left( {everything} \\right) = 0$, \n${C^{\\hat \\mu \\hat \\nu \\hat \\kappa }} = \\left( {{C^{\\bar \\mu \\bar \\nu \\bar \\kappa }},{C^{\\bar \\mu \\bar \\nu 4}}} \\right)$ \nand ${C^{\\bar \\mu \\bar \\nu 4}} = {B^{\\bar \\mu \\bar \\nu }}$. Carrying out this prescription in equation (\\ref{CR-10}), we then obtain\n\\begin{eqnarray} \n{{\\cal L}^{\\left( {3 + 1} \\right)}} &=&  - \\frac{1}{4}{F_{\\bar \\mu \\bar \\nu }}{F^{\\mu \\nu }} \n+ \\frac{1}{2}{\\left( {{\\partial _{\\bar \\mu} }\\phi } \\right)^2} + \\alpha {H_{\\bar \\mu \\bar \\nu \\bar \\kappa \\bar \\lambda }}\n{H^{\\bar \\mu \\bar \\nu \\bar \\kappa \\bar \\lambda }} \\nonumber\\\\\n&-& 4\\alpha {G_{\\bar \\mu \\bar \\nu \\bar \\kappa }}{G^{\\bar \\mu \\bar \\nu \\bar \\kappa }} - 3\\beta {\\varepsilon ^{4 \\bar \\mu \\bar \\nu \\bar \\kappa \n\t\t\\bar \\lambda }}{A_{\\bar \\mu} }{\\partial _{\\bar \\nu} }{B_{\\bar \\kappa \\bar \\lambda }} \\nonumber\\\\\n&-& \\beta {\\varepsilon ^{4 \\bar \\nu \\bar \\kappa \\bar \\lambda \\bar \\rho }}\\phi {\\partial _{\\bar \\nu} }{C_{\\bar \\kappa \\bar \\lambda \\bar \\rho }} \n+ \\frac{{m_C^2}}{{12}}{C_{\\bar \\mu \\bar \\nu \\bar \\rho }}{C^{\\bar \\mu \\bar \\nu \\bar \\rho }} \\nonumber\\\\\n&-& \\frac{{m_C^2}}{4}{B_{\\bar \\mu \\bar \\nu }}{B^{\\bar \\mu \\bar \\nu }}, \\label{CR-15}\n\\end{eqnarray}\nwhere $\\bar \\mu ,\\bar \\nu ,\\bar \\kappa ,\\bar \\lambda ,\\bar \\rho  = 0,1,2,3$. Making use of an additional dimensional reduction, that is, \n$ {A^{\\bar \\mu }} \\to \\left( {{A^\\mu },{A^3}} \\right)$, ${\\partial _3}\\left( {everything} \\right) = 0$, ${B^{\\bar \\mu \\bar \\nu }} \n= \\left( {{B^{\\mu \\nu }},{C^{\\mu }}} \\right)$\n\\begin{eqnarray}\n{{\\cal L}^{\\left( {2 + 1} \\right)}} &=&  - \\frac{1}{4}{F_{\\mu \\nu }}{F^{\\mu \\nu }} + 12\\alpha {G_{\\mu \\nu }}{G^{\\mu \\nu }} \\nonumber\\\\\n&-& 6\\beta {\\varepsilon ^{\\mu \\nu \\kappa }}{A_\\mu }{\\partial _\\nu }{C_\\kappa } \n+ \\frac{{m_C^2}}{2}{C_\\mu }{C^\\mu}, \\label{CR-20}\n\\end{eqnarray}\nwhere ${G_{\\mu \\nu }} = {\\partial _\\mu }{C_\\nu } - {\\partial _\\nu }{C_\\mu }$. Next, after performing the integration over $C_{\\mu}$, \nthe induced effective Lagrangian density is given by\n\\begin{equation}\n{{\\cal L}^{\\left( {2 + 1} \\right)}} =  - \\frac{1}{4}{F_{\\mu \\nu }}\\left( {1 + \\frac{\\sigma }{{\\left( {\\Delta  + m_C^2} \\right)}}} \\right)\n{F^{\\mu \\nu }}. \\label{CR-25}\n\\end{equation}\nAgain, by applying the gauge-invariant formalism, the corresponding static potential for two opposite charges located at ${\\bf y}$ and \n${\\bf y}\\prime$ turns out to be\n\\begin{equation}\nV =  - \\frac{{{q^2}}}{{2\\pi }}{K_0}\\left( {ML} \\right) + \\frac{{{q^2}m_C^2}}{{4M}}L, \\label {CR-30}\n\\end{equation}\nwhere $L = |{\\bf y} - {{\\bf y}^ \\prime }|$ and ${M^2} = {\\sigma ^2} + m_C^2$. In summary, then: this potential displays the conventional \nscreening part, encoded in the Bessel function, and the linear confining potential. As expected, confinement disappears whenever ${m_C} \\to 0$\nand also in the case $m_{C}$ is non-trivial, but much smaller than the topological mass parameter, $\\sigma$.\n\n\nA final consideration we would like to raise concerns the presence of some sort \nof fundamental mechanism that endows one of the gauge potentials, the $p$- or the\n$(p+1)$-form, with a Proca-type mass term: if only the usual field-strength squared and\nthe topological mass terms are present, a field reshuffling is always possible to be done \nand one of the gauge potentials can be integrated over yielding, at the end, a Proca-like \n$p$-form or $(p+1)$-form massive model; exactly like we have worked out for the Lagrangians\n(\\ref{Dual05}) and (\\ref{Dual10}). However, if a more fundamental mechanism is at work\n(like the Higgs mechanism, for example) that gives an explicit (non-topological) mass term\nto one of the gauge fields, then the simple equivalence to a $p$-form\nProca field is no longer true and a confining contribution to the static interparticle potential\nshows. We would like to conclude our work by pointing out the relationship between the\ngeneration of a non-topological mass and the confinig profile of the interparticle potential.\n\n\\section{Acknowledgments}\n\nOne of us (P. G.) wishes to thank the Abdus Salam ICTP for hospitality, the Field Theory Group of the COSMO\/CBPF for the pleasant visit with the PCI-BEV\/MCTIC support. P. G. was partially supported by Fondecyt (Chile) grant 1130426 and by Proyecto Basal FB0821.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}