diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzdwua" "b/data_all_eng_slimpj/shuffled/split2/finalzzdwua" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzdwua" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe experiments CDF and D\\O, taking data at the Tevatron\nproton-antiproton collider located at the Fermi National Accelerator\nLaboratory, have made several direct experimental measurements of the\ntop-quark pole mass, \\ensuremath{M_{\\mathrm{t}}}. The pioneering measurements were based on about\n$100~\\ensuremath{\\mathrm{pb}^{-1}}$ of \\hbox{Run~I}\\ (1992-1996) data~\\cite{Mtop1-CDF-di-l-PRLa, \n Mtop1-CDF-di-l-PRLb,\n Mtop1-CDF-di-l-PRLb-E, Mtop1-D0-di-l-PRL, Mtop1-D0-di-l-PRD,\n Mtop1-CDF-l+j-PRL, Mtop1-CDF-l+j-PRD, Mtop1-D0-l+j-old-PRL,\n Mtop1-D0-l+j-old-PRD, Mtop1-D0-l+j-new1, Mtop1-CDF-all-j-PRL,\n Mtop1-D0-all-j-PRL} \nand include results from the \\ensuremath{\\ttbar\\rightarrow\\had}\\ (all-j), the \\ensuremath{\\ttbar\\rightarrow\\ljt}\\ (l+j), and the \n\\ensuremath{\\ttbar\\rightarrow\\dil}\\ (di-l) decay channels\\footnote{Here $\\ell=e$ or $\\mu$. Decay \nchannels with explicit tau lepton identification are presently under \nstudy and are not yet used for measurements of the top-quark mass.}. \nThe \\hbox{Run~II}\\ measurements summarized here are the most recent results in the \nl+j, di-l, and all-j channels using $1.9-2.8~\\ensuremath{\\mathrm{fb}^{-1}}$ of data and improved \nanalysis techniques~\\cite{\nMtop2-CDF-di-l-new,\nMtop2-CDF-l+j-new,\nMtop2-CDF-all-j-new, \nMtop2-CDF-trk-new,\nMtop2-D0-l+ja-final,\nMtop2-D0-l+j-new,\nMtop2-D0-di-l-jul08-1,\nMtop2-D0-di-l-jul08-2}. \n\\vspace*{0.10in}\n\nThis note reports the world average top-quark mass obtained by\ncombining five published\n\\hbox{Run~I}\\ measurements~\\cite{Mtop1-CDF-di-l-PRLb, Mtop1-CDF-di-l-PRLb-E,\n Mtop1-D0-di-l-PRD, Mtop1-CDF-l+j-PRD, Mtop1-D0-l+j-new1,\n Mtop1-CDF-all-j-PRL} with four preliminary \\hbox{Run~II}\\ CDF\nresults~\\cite{Mtop2-CDF-di-l-new, Mtop2-CDF-l+j-new, Mtop2-CDF-all-j-new,\nMtop2-CDF-trk-new} and three preliminary\n\\hbox{Run~II}\\ D\\O\\ results~\\cite{Mtop2-D0-l+ja-final, Mtop2-D0-l+j-new,\nMtop2-D0-di-l-jul08-1, Mtop2-D0-di-l-jul08-2}.\nThe combination takes into account the\nstatistical and systematic uncertainties and their correlations using\nthe method of references~\\cite{Lyons:1988, Valassi:2003} and\nsupersedes previous\ncombinations~\\cite{Mtop1-tevewwg04,Mtop-tevewwgSum05,\n Mtop-tevewwgWin06,Mtop-tevewwgSum06, Mtop-tevewwgWin07, Mtop-tevewwgWin08}.\n\\vspace*{0.10in}\n\nThe input measurements and error categories used in the combination are \ndetailed in Sections~\\ref{sec:inputs} and~\\ref{sec:errors}, respectively. \nThe correlations used in the combination are discussed in \nSection~\\ref{sec:corltns} and the resulting world average top-quark mass \nis given in Section~\\ref{sec:results}. A summary and outlook are presented\nin Section~\\ref{sec:summary}.\n \n\\section{Input Measurements}\n\\label{sec:inputs}\n\nFor this combination twelve measurements of \\ensuremath{M_{\\mathrm{t}}}\\ are used: five\npublished \\hbox{Run~I}\\ results, and seven preliminary\n\\hbox{Run~II}\\ results, all reported in Table~\\ref{tab:inputs}. In general,\nthe \\hbox{Run~I}\\ measurements all have relatively large statistical\nuncertainties and their systematic uncertainty is dominated by the\ntotal jet energy scale (JES) uncertainty. In \\hbox{Run~II}\\ both CDF and\nD\\O\\ take advantage of the larger \\ensuremath{t\\overline{t}}\\ samples available and employ\nnew analysis techniques to reduce both these uncertainties. In\nparticular the JES is constrained using an in-situ calibration based\non the invariant mass of $W\\rightarrow qq^{\\prime}$ decays in the l+j and\nall-j channels. The \\hbox{Run~II}\\ D\\O\\ analysis in the l+j channel\nconstrains the response of light-quark jets using the in-situ $W\\rightarrow\nqq^{\\prime}$ decays. Residual JES uncertainties associated with\n$\\eta$ and $p_{T}$ dependencies as well as uncertainties specific to\nthe response of $b$-jets are treated separately. Similarly, the\n\\hbox{Run~II}\\ CDF analyses in the l+j and all-j channels also constrain the\nJES using the in-situ $W\\rightarrow qq^{\\prime}$ decays. Small residual JES\nuncertainties arising from $\\eta$ and $p_{T}$ dependencies and the\nmodeling of $b$-jets are included in separate error categories. The\n\\hbox{Run~II}\\ CDF and D\\O\\ di-l measurements use a JES determined from external\ncalibration samples. Some parts of the associated uncertainty are\ncorrelated with the \\hbox{Run~I}\\ JES uncertainty as noted below.\n\\vspace*{0.10in}\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\small\n\\begin{tabular}{|l||rrr|rr||rrrr|rrr|}\n\\hline \n & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{2-13}\n & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & all-j & l+j & di-l & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l \\\\\n\\hline\n$\\int \\mathcal{L}\\;dt$ & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 2.7 & 1.9 & 2.1 & 1.9 & 1.0 & 1.2 & 2.8 \\\\\n\\hline\n\\hline \nResult & 186.0 & 176.1 & 167.4 & 180.1 & 168.4 & 172.2 & 171.2 & 176.9 & 175.3 & 171.5 & 173.0 & 174.4 \\\\\n\\hline \n\\hline \niJES & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.9 & 0.0 & 1.9 & 0.0 & 0.7 & 1.4 & 0.0 \\\\\naJES & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.8 & 0.8 & 1.2 \\\\\nbJES & 0.6 & 0.6 & 0.8 & 0.7 & 0.7 & 0.4 & 0.4 & 0.6 & 0.0 & 0.0 & 0.1 & 0.3 \\\\\ncJES & 3.0 & 2.7 & 2.6 & 2.0 & 2.0 & 0.3 & 1.7 & 0.6 & 0.6 & 0.0 & 0.0 & 0.0 \\\\\ndJES & 0.3 & 0.7 & 0.6 & 0.0 & 0.0 & 0.0 & 0.1 & 0.1 & 0.0 & 0.8 & 0.0 & 1.6 \\\\\nrJES & 4.0 & 3.4 & 2.7 & 2.5 & 1.1 & 0.4 & 1.9 & 0.6 & 0.1 & 0.0 & 0.0 & 0.0 \\\\\nlepPt & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.2 & 0.1 & 0.0 & 1.1 & 0.0 & 0.0 & 0.0 \\\\\nSignal & 1.8 & 2.6 & 2.8 & 1.1 & 1.8 & 0.3 & 0.8 & 0.5 & 1.6 & 0.5 & 0.5 & 0.5 \\\\\nBG & 1.7 & 1.3 & 0.3 & 1.0 & 1.1 & 0.4 & 0.4 & 1.0 & 1.6 & 0.4 & 0.4 & 0.6 \\\\\nFit & 0.6 & 0.0 & 0.7 & 0.6 & 1.1 & 0.2 & 0.6 & 0.5 & 1.4 & 0.3 & 0.2 & 0.3 \\\\\nMC & 0.8 & 0.1 & 0.6 & 0.0 & 0.0 & 0.5 & 0.9 & 0.5 & 0.6 & 0.0 & 0.0 & 0.0 \\\\\nUN\/MI & 0.0 & 0.0 & 0.0 & 1.3 & 1.3 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\\\\n\\hline \nSyst. & 5.7 & 5.3 & 4.9 & 3.9 & 3.6 & 1.3 & 2.9 & 2.6 & 3.0 & 1.5 & 1.7 & 2.1 \\\\\nStat. & 10.0 & 5.1 & 10.3 & 3.6 & 12.3 & 1.0 & 2.7 & 3.3 & 6.2 & 1.5 & 1.3 & 3.2 \\\\\n\\hline \n\\hline \nTotal & 11.5 & 7.3 & 11.4 & 5.3 & 12.8 & 1.7 & 4.0 & 4.2 & 6.9 & 2.1 & 2.2 & 3.9 \\\\ \n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption[Input measurements]{Summary of the measurements used to determine the\n world average $\\ensuremath{M_{\\mathrm{t}}}$. Integrated luminosity ($\\int \\mathcal{L}\\;dt$) is in\n \\ensuremath{\\mathrm{fb}^{-1}}, and all other numbers are in $\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$. The error categories and \n their correlations are described in the text. The total systematic uncertainty \n and the total uncertainty are obtained by adding the relevant contributions \n in quadrature.}\n\\label{tab:inputs}\n\\end{table}\n\n\nThe D\\O\\ Run~IIa l+j analysis is using the JES determined from the\nexternal calibration derived using $\\gamma$+jets events as an\nadditional Gaussian constraint to the in-situ calibration. Therefore\nthe total resulting JES uncertainty has been split into the part\ncoming solely from the in-situ calibration and the part coming from\nthe external calibration. To do that, the measurement without external\nJES constraint has been combined iteratively with a pseudo-measurement\nusing the BLUE method that would use only the external calibration so\nthat the combination gives the actual total JES uncertainty. The\nsplitting obtained in this way is used to assess the iJES and dJES\nuncertainties~\\cite{Mtop2-D0-comb}.\n\nA new analysis technique from CDF is included (trk).\nThis measurement uses both the mean\ndecay-length from B-tagged jets and the mean lepton transverse momentum\nto determine the top-quark mass in l+j candidate events.\nWhile the statistical sensitivity is not as good as the more\ntraditional methods, this technique has the advantage that since it\nuses primarily tracking information, it is almost entirely independent of\nJES uncertainties. As the statitistics of this sample continue to\ngrow, this method could offer a nice cross-check of the top-quark mass\nthat's largely independent of the dominant JES systematic uncertainty\nwhich plagues the other measurements. The statistical correlation\nbetween an earlier version of the trk analysis and a\ntraditional \\hbox{Run~II}\\ CDF l+j measurement was\nstudied using Monte Carlo signal-plus-background pseudo-experiments\nwhich correctly account for the sample overlap and was found to be\nconsistent with zero (to within $<1\\%$) independent of the assumed\ntop-quark mass.\n\\vspace*{0.10in}\n\nThe two D\\O\\ \\hbox{Run~II}\\ lepton+jets results~\\cite{Mtop2-D0-l+ja-final,\nMtop2-D0-l+j-new} are derived from Run~IIa and Run~IIb datasets,\nrespectively, and are labelled as such. The D\\O\\ \\hbox{Run~II}\\ dilepton\nresult is itself a combination of two results\nusing different techniques but partially overlapping dilepton data\nsets~\\cite{Mtop2-D0-di-l-jul08-1,Mtop2-D0-di-l-jul08-2}.\n\\vspace*{0.10in}\n\nTable~\\ref{tab:inputs} also lists the uncertainties of the results,\nsub-divided into the categories described in the next Section. The\ncorrelations between the inputs are described in\nSection~\\ref{sec:corltns}.\n\n\n\n\\section{Error Categories}\n\\label{sec:errors}\n\nWe employ the same error categories as used for the previous world\naverage~\\cite{Mtop-tevewwgWin08}, plus one new category (lepPt). They\ninclude a detailed breakdown of the various sources of uncertainty and\naim to lump together sources of systematic uncertainty that share the\nsame or similar origin. For example, the ``Signal'' category\ndiscussed below includes the uncertainties from ISR, FSR, and\nPDF---all of which affect the modeling of the \\ensuremath{t\\overline{t}}\\ signal. Some\nsystematic uncertainties have been broken down into multiple\ncategories in order to accommodate specific types of correlations.\nFor example, the jet energy scale (JES) uncertainty is sub-divided\ninto several components in order to more accurately accommodate our\nbest estimate of the relevant correlations. Each error category is\ndiscussed below.\n\\vspace*{0.10in}\n\n\\begin{description}\n \\item[Statistical:] The statistical uncertainty associated with the\n \\ensuremath{M_{\\mathrm{t}}}\\ determination.\n \\item[iJES:] That part of the JES uncertainty which originates from\n in-situ calibration procedures and is uncorrelated among the\n measurements. In the combination reported here it corresponds to\n the statistical uncertainty associated with the JES determination\n using the $W\\rightarrow qq^{\\prime}$ invariant mass in the CDF \\hbox{Run~II}\\\n l+j and all-h measurements and D\\O\\ Run~IIa and Run~IIb l+j\n measurements. Residual JES uncertainties, which arise\n from effects\n not considered in the in-situ calibration, are included in other\n categories.\n \\item[aJES:] That part of the JES uncertainty which originates from\n differences in detector $e\/h$ response between $b$-jets and light-quark\n jets. It is specific to the D\\O\\ \\hbox{Run~II}\\ measurements and is\n taken to be uncorrelated with the D\\O\\ \\hbox{Run~I}\\ and CDF measurements.\n \\item[bJES:] That part of the JES uncertainty which originates from\n uncertainties specific to the modeling of $b$-jets and which is correlated\n across all measurements. For both CDF and D\\O\\ this includes uncertainties \n arising from \n variations in the semi-leptonic branching fraction, $b$-fragmentation \n modeling, and differences in the color flow between $b$-jets and light-quark\n jets. These were determined from \\hbox{Run~II}\\ studies but back-propagated\n to the \\hbox{Run~I}\\ measurements, whose rJES uncertainties (see below) were \n then corrected in order to keep the total JES uncertainty constant.\n \\item[cJES:] That part of the JES uncertainty which originates from\n modeling uncertainties correlated across all measurements. Specifically\n it includes the modeling uncertainties associated with light-quark \n fragmentation and out-of-cone corrections.\n \\item[dJES:] That part of the JES uncertainty which originates from\n limitations in the calibration data samples used and which is\n correlated between measurements within the same data-taking\n period, such as \\hbox{Run~I}\\ or \\hbox{Run~II}, but not between\n experiments. For CDF this corresponds to uncertainties associated\n with the $\\eta$-dependent JES corrections which are estimated\n using di-jet data events. For D\\O\\ this includes uncertainties in the\n calorimeter response to light-quark jets, and $\\eta$- and\n $p_{T}$-dependent uncertainties\n constrained using \\hbox{Run~II}\\ $\\gamma+$jet data samples.\n \\item[rJES:] The remaining part of the JES uncertainty which is \n correlated between all measurements of the same experiment \n independent of data-taking period, but is uncorrelated between\n experiments. For CDF, this is dominated by uncertainties in the\n calorimeter response to light-quark jets, and also includes small \n uncertainties associated with the multiple interaction and underlying \n event corrections.\n \\item[lepPt:] The systematic uncertainty arising from uncertainties\n in the scale of lepton transverse momentum measurements. This is an\n important uncertainty for CDF's track-based measurement. It was not\n considered as a source of systematic uncertainty in the \\hbox{Run~I}\\\n measurements or in measurements at D\\O.\n \\item[Signal:] The systematic uncertainty arising from uncertainties\n in the modeling of the \\ensuremath{t\\overline{t}}\\ signal which is correlated across all\n measurements. This includes uncertainties from variations in the ISR,\n FSR, and PDF descriptions used to generate the \\ensuremath{t\\overline{t}}\\ Monte Carlo samples\n that calibrate each method. It also includes small uncertainties \n associated with biases associated with the identification of $b$-jets.\n \\item[Background:] The systematic uncertainty arising from uncertainties\n in modeling the dominant background sources and correlated across\n all measurements in the same channel. These\n include uncertainties on the background composition and shape. In\n particular uncertainties associated with the modeling of the QCD\n multi-jet background (all-j and l+j), uncertainties associated with the\n modeling of the Drell-Yan background (di-l), and uncertainties associated \n with variations of the fragmentation scale used to model W+jets \n background (all channels) are included.\n \\item[Fit:] The systematic uncertainty arising from any source specific\n to a particular fit method, including the finite Monte Carlo statistics \n available to calibrate each method.\n \\item[Monte Carlo:] The systematic uncertainty associated with variations\n of the physics model used to calibrate the fit methods and correlated\n across all measurements. For CDF it includes variations observed when \n substituting PYTHIA~\\cite{PYTHIA4,PYTHIA5,PYTHIA6} (\\hbox{Run~I}\\ and \\hbox{Run~II}) \n or ISAJET~\\cite{ISAJET} (\\hbox{Run~I}) for HERWIG~\\cite{HERWIG5,HERWIG6} when \n modeling the \\ensuremath{t\\overline{t}}\\ signal. Similar\n variations are included for the D\\O\\ \\hbox{Run~I}\\ measurements. The D\\O\\ \n \\hbox{Run~II}\\ measurements use ALPGEN~\\cite{ALPGEN} to model the \\ensuremath{t\\overline{t}}\\ signal and the\n variations considered are included in the Signal category above.\n \\item[UN\/MI:] This is specific to D\\O\\ and includes the uncertainty\n arising from uranium noise in the D\\O\\ calorimeter and from the\n multiple interaction corrections to the JES. For D\\O\\ \\hbox{Run~I}\\ these\n uncertainties were sizable, while for \\hbox{Run~II}, owing to the shorter\n integration time and in-situ JES determination, these uncertainties\n are negligible.\n\\end{description}\nThese categories represent the current preliminary understanding of the\nvarious sources of uncertainty and their correlations. We expect these to \nevolve as we continue to probe each method's sensitivity to the various \nsystematic sources with ever improving precision. Variations in the assignment\nof uncertainties to the error categories, in the back-propagation of the bJES\nuncertainties to \\hbox{Run~I}\\ measurements, in the approximations made to\nsymmetrize the uncertainties used in the combination, and in the assumed \nmagnitude of the correlations all negligibly effect ($\\ll 0.1\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$) the \ncombined \\ensuremath{M_{\\mathrm{t}}}\\ and total uncertainty.\n\n\\section{Correlations}\n\\label{sec:corltns}\n\nThe following correlations are used when making the combination:\n\\begin{itemize}\n \\item The uncertainties in the Statistical, Fit, and iJES\n categories are taken to be uncorrelated among the measurements.\n \\item The uncertainties in the aJES and dJES categories are taken\n to be 100\\% correlated among all \\hbox{Run~I}\\ and all \\hbox{Run~II}\\ measurements \n on the same experiment, but uncorrelated between \\hbox{Run~I}\\ and \\hbox{Run~II}\\\n and uncorrelated between the experiments.\n \\item The uncertainties in the rJES and UN\/MI categories are taken\n to be 100\\% correlated among all measurements on the same experiment.\n \\item The uncertainties in the Background category are taken to be\n 100\\% correlated among all measurements in the same channel.\n \\item The uncertainties in the bJES, cJES, Signal, and Generator\n categories are taken to be 100\\% correlated among all measurements.\n\\end{itemize}\nUsing the inputs from Table~\\ref{tab:inputs} and the correlations specified\nhere, the resulting matrix of total correlation co-efficients is given in\nTable~\\ref{tab:coeff}.\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{|ll||rrr|rr||rrrr|rrr|}\n\\hline \n & & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{3-14}\n & & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & & l+j & di-l & all-j & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l \\\\\n\\hline\n\\hline\nCDF-I & l+j & 1.00& & & & & & & & & & & \\\\\nCDF-I & di-l & 0.29& 1.00& & & & & & & & & & \\\\\nCDF-I & all-j & 0.32& 0.19& 1.00& & & & & & & & & \\\\\n\\hline\nD\\O-I & l+j & 0.26& 0.15& 0.14& 1.00& & & & & & & & \\\\\nD\\O-I & di-l & 0.11& 0.08& 0.07& 0.16& 1.00& & & & & & & \\\\\n\\hline\n\\hline\nCDF-II & l+j & 0.32& 0.18& 0.20& 0.19& 0.07& 1.00& & & & & & \\\\\nCDF-II & di-l & 0.45& 0.28& 0.33& 0.22& 0.11& 0.34& 1.00& & & & & \\\\\nCDF-II & all-j & 0.17& 0.11& 0.16& 0.10& 0.05& 0.15& 0.19& 1.00& & & & \\\\\nCDF-II & trk & 0.16& 0.08& 0.07& 0.13& 0.05& 0.17& 0.12& 0.05& 1.00& & & \\\\\n\\hline\nD\\O-II & l+j\/a & 0.11& 0.05& 0.03& 0.08& 0.03& 0.09& 0.04& 0.03& 0.09& 1.00& & \\\\\nD\\O-II & l+j\/b & 0.12& 0.06& 0.04& 0.09& 0.03& 0.10& 0.05& 0.03& 0.09& 0.24& 1.00 & \\\\\nD\\O-II & di-l & 0.05& 0.04& 0.02& 0.04& 0.04& 0.04& 0.05& 0.03& 0.03& 0.31& 0.16 & 1.00\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption[Global correlations between input measurements]{The resulting\n matrix of total correlation coefficients used to determined the\n world average top quark mass.}\n\\label{tab:coeff}\n\\end{table}\n\nThe measurements are combined using a program implementing a numerical\n$\\chi^2$ minimization as well as the analytic BLUE\nmethod~\\cite{Lyons:1988, Valassi:2003}. The two methods used are\nmathematically equivalent, and are also equivalent to the method used\nin an older combination~\\cite{TM-2084}, and give identical results for\nthe combination. In addition, the BLUE method yields the decomposition\nof the error on the average in terms of the error categories specified\nfor the input measurements~\\cite{Valassi:2003}.\n\n\\section{Results}\n\\label{sec:results}\n\nThe combined value for the top-quark mass is:\n\\begin{eqnarray}\n \\ensuremath{M_{\\mathrm{t}}} & = & 172.4 \\pm 1.2~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }\\,,\n\\end{eqnarray}\nwith a $\\chi^2$ of 6.9 for 11 degrees of freedom, which corresponds to\na probability of 81\\%, indicating good agreement among all the input\nmeasurements. The total uncertainty can be sub-divided into the \ncontributions from the various error categories as: Statistical ($\\pm0.7$),\ntotal JES ($\\pm0.8$), Lepton scale ($\\pm0.1$), Signal ($\\pm0.3$), Background ($\\pm0.3$), Fit\n($\\pm0.1$), Monte Carlo ($\\pm0.3$), and UN\/MI ($\\pm0.02$), for a total\nSystematic ($\\pm1.0$), where all numbers are in units of \\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }.\nThe pull and weight for each of the inputs are listed in Table~\\ref{tab:stat}.\nThe input measurements and the resulting world average mass of the top \nquark are summarized in Figure~\\ref{fig:summary}.\n\\vspace*{0.10in}\n\n\nThe weights of many of the \\hbox{Run~I}\\ measurements are negative. \nIn general, this situation can occur if the correlation between two measurements\nis larger than the ratio of their total uncertainties. This is indeed the case\nhere. In these instances the less precise measurement \nwill usually acquire a negative weight. While a weight of zero means that a\nparticular input is effectively ignored in the combination, a negative weight \nmeans that it affects the resulting central value and helps reduce the total\nuncertainty. See reference~\\cite{Lyons:1988} for further discussion of \nnegative weights.\n\n\\begin{figure}[p]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{topmass_tev0708.eps}\n\\end{center}\n\\caption[Summary plot for the world average top-quark mass]\n {A summary of the input measurements and resulting world average\n mass of the top quark.}\n\\label{fig:summary} \n\\end{figure}\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n{\\small\n\\begin{tabular}{|l||rrr|rr||rrrr|rrr|}\n\\hline \n & \\multicolumn{5}{|c||}{{\\hbox{Run~I}} published} & \\multicolumn{7}{|c|}{{\\hbox{Run~II}} preliminary} \\\\ \\cline{2-13}\n & \\multicolumn{3}{|c|}{ CDF } & \\multicolumn{2}{|c||}{ D\\O\\ }\n & \\multicolumn{4}{|c|}{ CDF } & \\multicolumn{3}{|c|}{ D\\O\\ } \\\\\n & l+j & di-l & all-j & l+j & di-l & l+j & di-l & all-j & trk & l+j\/a & l+j\/b & di-l\\\\\n\\hline\n\\hline\nPull & $+0.5$ & $-0.4$ & $+1.2$ & $+1.5$ & $-0.3$ & $-0.1$ & $-0.3$ & $+1.1$ & $+0.4$ \n & $-0.5$ & $+0.3$ & $+0.5$ \\\\\nWeight [\\%]\n & $- 3.4$ & $- 0.6$ & $- 0.6$ & $+ 1.2$ & $+ 0.2$ & $+46.1$ & $+ 3.7$ & $+ 5.2$ & $+ 0.02$ \n & $+23.7$ & $+21.5$ & $+ 3.0$ \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption[Pull and weight of each measurement]{The pull and weight for each of the\n inputs used to determine the world average mass of the top quark. See \n Reference~\\cite{Lyons:1988} for a discussion of negative weights.}\n\\label{tab:stat} \n\\end{table} \n\nAlthough the $\\chi^2$ from the combination of all measurements indicates\nthat there is good agreement among them, and no input has an anomalously\nlarge pull, it is still interesting to also fit for the top-quark mass\nin the all-j, l+j, and di-l channels separately. We use the same methodology,\ninputs, error categories, and correlations as described above, but fit for\nthe three physical observables, \\ensuremath{\\MT^{\\mathrm{all-j}}}, \\ensuremath{\\MT^{\\mathrm{l+j}}}, and \\ensuremath{\\MT^{\\mathrm{di-l}}}.\nThe results of this combination are shown in Table~\\ref{tab:three_observables}\nand have $\\chi^2$ of 4.9 for 9 degrees of freedom, which corresponds to a\nprobability of 84\\%.\nThese results differ from a naive combination, where\nonly the measurements in a given channel contribute to the \\ensuremath{M_{\\mathrm{t}}}\\ \ndetermination in that channel, since the combination here fully accounts\nfor all correlations, including those which cross-correlate the different\nchannels. Using the results of \nTable~\\ref{tab:three_observables} we calculate the chi-squared consistency\nbetween any two channels, including all correlations, as \n$\\chi^{2}(dil-lj)=0.08$, $\\chi^{2}(lj-allj)=1.7$, and \n$\\chi^{2}(allj-dil)=1.9$. These correspond to \nchi-squared probabilities of 78\\%, 19\\%, and 17\\%, respectively, and indicate \nthat the determinations of \\ensuremath{M_{\\mathrm{t}}}\\ from the three channels are consistent with \none another.\n\n\n\\begin{table}[t]\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.30}\n\\begin{tabular}{|l||c|rrr|}\n\\hline\nParameter & Value (\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }) & \\multicolumn{3}{|c|}{Correlations} \\\\\n\\hline\n\\hline\n$\\ensuremath{\\MT^{\\mathrm{all-j}}}$ & $177.5\\pm 4.0$ & 1.00 & & \\\\\n$\\ensuremath{\\MT^{\\mathrm{l+j}}}$ & $172.2\\pm 1.2$ & 0.14 & 1.00 & \\\\\n$\\ensuremath{\\MT^{\\mathrm{di-l}}}$ & $171.5\\pm 2.6$ & 0.17 & 0.32 & 1.00 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption[Mtop in each channel]{Summary of the combination of the 12\nmeasurements by CDF and D\\O\\ in terms of three physical quantities,\nthe mass of the top quark in the all-jets, lepton+jets, and di-lepton channels. }\n\\label{tab:three_observables}\n\\end{table}\n\n\\section{Summary}\n\\label{sec:summary}\n\nA preliminary combination of measurements of the mass of the top quark\nfrom the Tevatron experiments CDF and D\\O\\ is presented. The\ncombination includes five published \\hbox{Run~I}\\ measurements and \nseven preliminary \\hbox{Run~II}\\ measurements. Taking into\naccount the statistical and systematic uncertainties and their\ncorrelations, the preliminary world-average result is: $\\ensuremath{M_{\\mathrm{t}}}= 172.4 \\pm\n1.2~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$, where the total uncertainty is obtained assuming Gaussian\nsystematic uncertainties and adding them plus the statistical\nuncertainty in quadrature. While the central value is somewhat higher\nthan our 2007 average, the averages are compatible as appreciably more\nluminosity and refined analysis techniques are now used.\n\\vspace*{0.10in}\n\nThe mass of the top quark is now known with a relative precision of\n0.7\\%, limited by the systematic uncertainties, which are dominated by\nthe jet energy scale uncertainty. This systematic is expected to\nimprove as larger data sets are collected since new analysis\ntechniques constrain the jet energy scale using in-situ $W\\rightarrow\nqq^{\\prime}$ decays. It can be reasonably expected that with the full\n\\hbox{Run~II}\\ data set the top-quark mass will be known to better than\n0.7\\%. To reach this level of precision further work is required to\ndetermine more accurately the various correlations present, and to\nunderstand more precisely the $b$-jet modeling, Signal, and Background\nuncertainties which may limit the sensitivity at larger data sets.\nLimitations of the Monte Carlo generators used to calibrate each fit\nmethod also become more important as the precision reaches the\n$\\sim1~\\ensuremath{\\mathrm{ Ge\\kern -0.1em V }\\kern -0.2em \/c^2 }$ level; these warrant further study in the near future.\n\n\\clearpage\n\n\\bibliographystyle{tevewwg}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nQuantum field theory required the finest element of mathematical physics for its development, pushing at time the development of new aspects of mathematics. \nThe analyticity requirements have been instrumental in the theory of the holomorphic functions of many variables and their analyticity domains, \nwith such high points as the edge of the wedge theorem of Bogolubov or the Bros--Iagolnitzer transform used for the definition of the analytic \nwave front. Type III factors are everywhere and the instrument of their classification stems from the KMS condition, introduced for the study of \nfinite temperature theory, which was found to coincide with the Tomonoga condition. Perturbative expansions and their renormalization require \nsubtle combinatorics and the introduction of Hopf algebras allows one to clarify and make more explicit the classical proofs of renormalizability. \nUsing the formalism of groupoids may be useful to reduce the burden of controlling the effect of the symmetry factors. Evaluating Feynman \nintegrals requires numbers which can be periods, with the action of a motivic Galois group and links with many conjectures in algebraic geometry.\nConstructive theory has been able to show that some of these theories can be given a precise mathematical sense, but has failed to address \nthe most relevant ones for our understanding of the physical world, the four dimensional gauge theories. \n\nIn fact, due to the specificity of renormalization, the perturbative series is the richest source of information on quantum field theory.\nOther approaches deal necessarily with a regularized version of the theory which lacks many of the structural properties of the final theory. A precise \ndefinition will therefore seek to define the relevant Green functions from their perturbative series, but the procedure cannot be straightforward, \nsince it has been long recognized that quantum field theories (QFT for short) cannot depend holomorphically on the coupling parameter in a full \nneighborhood of the origin and therefore the perturbative series is at most asymptotic. A number of works have tended to show that the growth of \nthe terms of the formal perturbative series is slow enough to allow the definition of a Borel transform around the origin. This is however \nonly the first step in the definition of a sum for the perturbative series. One must also be able to analytically continue this Borel \ntransform up to infinity and furthermore verify that near infinity, this analytically continued Borel transform does not grow \nfaster than any exponential function.\n\nDefining an analytic \ncontinuation seems a formidable task, but in the cases where the functions obey equations, analog equations for their Borel transform can be \ndeduced and, with the help of alien calculus, used to constrain the possible singularities of the Borel transform. Singularities may appear on \nthe positive real axis, preventing a straightforward application of the Laplace transform. One could resort to lateral summation, using a \nshifted integration axis, but this means that the reality properties of the solution are lost. This can be an advantage in some situations, \nwhere the imaginary part of a perturbed energy signals the possible decay of a metastable state to the continuum, but unitarity could be at risk. However a suitable average of the analytic continuations circumventing the singularities on the real axis can be used to define a sum which both is real and respects the equations.\n\nIn this work, we will therefore argue that quantum field theory cannot dispense with the whole body of work on summation methods which has been \nthoroughly expanded by Jean Ecalle~\\cite{Ecalle81,Ecalle81b,Ecalle81c}. As in previous works~\\cite{BeCl14}, we will base our considerations on computations in a specific model, \nbut we think that they reveal phenomena at work in most exactly renormalizable field theories.\n\nAlthough the insight about these summation methods comes from the study of Borel transforms which form an algebra under convolutive products, \nmuch can be done while remaining in the domain of more or less formal series. In particular, formal transseries solutions allow to \nrecognize the possible forms of alien derivatives. It is then possible to express the solution in terms of transmonomials, special functions \nwith simple alien derivatives, so that we never have to explicitly refer to the Borel transforms in computations. The Borel-transformed functions are however \nwhat justify the different computations. \n\nThis article is divided as follows: first we give a short introduction to some concepts of resurgence theory that are of importance for our work. \nThen some previous results of \\cite{BeSc08,BeCl13,BeCl14} that are to be developed further are recalled. We then compute the leading terms \nfor every exponentially small terms in the anomalous dimension in the subsequent section. They are shown to sum up to a known analytic function of the \nnonperturbatively small quantity \\(e^{-r}\\). Alien calculus is then used to deduce from the computed terms the singularities of the Borel transform, while \nthe first singularity of the Borel transform is used to constrain a free parameter in the previous paragraphs.\nFinally, we apply the same process to the two-point function of the theory and see that the nonperturbative terms can get multiplied by powers of the momentum. The resulting function has a singularity for a timelike momentum, which fixes a nonperturbative mass scale, while the euclidean side is completely tame. The way such a result could appear in the process of Borel summation gives further indications on the analyticity domain of the theory.\n\n\n\\section{Elements of resurgence theory}\n\n\\subsection{Borel transform and Borel resummation}\n\nA very nice introduction to the subject of Borel transform and resurgence theory is \\cite{Sa14}. Here we will follow the presentation of \\cite{Bo11}, \nwith some additional material needed for our subject.\n\nA simple definition of the Borel transform is to say that it is a morphism between two rings of formal series, defined as being linear and its value on monomials:\n\\begin{eqnarray}\n \\mathcal{B}: a\\mathbb{C}[[a]] & \\longrightarrow & \\mathbb{C}[[\\xi]] \\\\\n\\tilde{f}(a) = a\\sum_{n=0}^{+\\infty}c_na^n & \\longrightarrow & \\hat{f}(\\xi) = \\sum_{n=0}^{+\\infty}\\frac{c_n}{n!}\\xi^n. \\nonumber\n\\end{eqnarray}\nThis definition is extended to noninteger powers by\n\\begin{equation*}\n \\mathcal{B}\\left(a^{\\alpha+1}\\right)(\\xi) := \\frac{\\xi^{\\alpha}}{\\Gamma(\\alpha+1)}.\n\\end{equation*}\nIn the case where $\\alpha$ is an integer, we find back the above definition.\n\nThen the point-like product of formal series becomes the convolution product of formal series, or of functions when the Borel transform is the \ngerm of an analytic function. It is therefore consistent to define the Borel transform of a constant as the formal unity of the convolution product (i.e.,\nthe Dirac ``function''):\n\\begin{equation}\n \\mathcal{B}(1) := \\delta.\n\\end{equation}\n\nIn the following, we will restrict ourselves to the cases where $\\hat f$ is the germ of an analytic function at the origin, endlessly continuable on \n$\\mathbb{C}$, i.e., that on any line $L$ of $\\mathbb{C}$ a representative of the germ $\\hat f$ has a countable number of singularities and is continuable along \nany path obtained by following $L$ and avoiding the singularities by going over or below them. Moreover, we will assume that these singularities are \nalgebraic, that is to say that they are not more singular than a pole. More precisely, if $\\xi_0$ is a singularity of $\\hat\\phi$, we shall have\n\\begin{equation}\n \\exists\\alpha\\in\\mathbb{R}\\mid |(\\xi-\\xi_0)^{-\\alpha}\\hat\\phi_r(\\xi)|\\underset{\\xi\\rightarrow\\xi_0}{\\longrightarrow}0.\n\\end{equation}\nThe supremum of the $\\alpha$ for which the above holds will be called the order of $\\hat\\phi$ at $\\xi_0$. Notice that if $\\alpha$ is positive, \nit is important that the condition is not directly on \\(\\hat\\phi\\) but on \\(\\hat\\phi_r\\), obtained from \\(\\hat\\phi\\) by subtracting a suitable polynomial in~\\(\\xi\\). {A typical example for \\(\\alpha\\) a positive integer is \\(\\hat\\phi_r(\\xi) \\simeq (\\xi-\\xi_0)^\\alpha \\ln(\\xi-\\xi_0) \\), but there are no reason for \\(\\hat\\phi\\) itself to have a zero at \\(\\xi_0\\). Alternatively, the singularity can be characterized by the behavior of the difference of the function and its analytic continuation after looping around \\(\\xi_0\\), but this does not work for poles.}\nIt is even possible to have singularities with an infinite order, but in our applications, the order will always be a finite rational number.\n\nLet us call $\\widehat{RES}$ the space of germs of analytic functions at the origin endlessly continuable on $\\mathbb{C}$ and $\\widetilde{RES}\\subset a\\mathbb{C}[[a]]$ the set of formal series whose image under the Borel transform is in\n$\\widehat{RES}$. When working with elements of $\\widehat{RES}$ we will say that we are in the convolutive model, while $\\widetilde{RES}$ will be called the \nformal model. We will also say that we are in the Borel plane and the physical plane, respectively.\n\nThere exists an inverse to the Borel transform: the Laplace transform. {For $\\phi\\in\\widehat{RES}$ we write $\\hat\\phi$ for the analytic continuation of the Borel transform of $\\phi$. The definition of the Laplace transform \non $\\hat\\phi$} involves a certain direction $\\theta$ in the complex plane:\n\\begin{equation}\n \\mathcal{L}_{\\theta}[\\hat\\phi] := \\int_0^{e^{i\\theta}\\infty}\\hat\\phi(\\zeta)e^{-\\zeta\/a}\\d\\zeta.\n\\end{equation}\nIt is well defined for at least some values of \\(a\\) if \\(\\hat\\phi\\) is smaller than some exponential in the direction \\(\\theta\\). The resummation operator in the direction $\\theta$ is the composition of the Borel transform and the Laplace transform in the direction \\(\\theta\\):\n\\begin{equation}\n S_{\\theta} := \\mathcal{L}_{\\theta}\\circ\\mathcal{B}.\n\\end{equation}\nIf $\\hat\\phi$ does not have any singularities in the directions \\(\\theta\\) between $\\theta'$ and $\\theta''$ included and satisfies suitable exponential bounds at infinity in this sector, different $\\mathcal{L}^{\\theta}[\\hat\\phi]$ coincide wherever they are both defined through Cauchy's theorem, so that they define a single analytic function on the sector delimited \nby $\\theta' -\\pi\/2$ and $\\theta''+ \\pi\/2$, which is a possible resummation of the formal series \\(\\tilde \\phi\\). We see that the different resummations are defined in sectors whose limits depend on the singularities of the Borel transform, but their definition domains have nontrivial intersections. \n\nIn the following, we will need more general objects than the elements of \\(\\widehat{RES}\\) and the corresponding $\\widetilde{RES}$. \nWe define simple resurgent symbols with an additional variable \\(\\omega \\in \\mathbb{C}\\) with the meaning that the corresponding object in the formal or geometric models get multiplied by \\(e^{-\\omega\/a}\\), so that: \n\\begin{equation*}\n \\dot\\phi^{\\omega} := e^{-\\omega\/a}\\phi\\in\\dot{\\widetilde{RES}} \\supset \\widetilde{RES}\n\\end{equation*}\nIn the formal model, linear combinations of simple resurgent symbols are simple examples of transseries. One can define more general transseries, but this would lead us away from our topic. {A very pleasant introduction to transseries, very accessible to physicists is} \\cite{Ed09}.\n\n\nFinally, the operator $S_{\\theta}$ extends to \\(\\dot{\\widehat{RES}}\\) through the formula\n\\begin{equation}\\label{SthetaDot}\n S_{\\theta}[\\hat\\phi^{\\omega}](a) := e^{-\\omega\/a}\\mathcal{L}_{\\theta}[\\hat\\phi](a)\n\\end{equation}\nand by linear extension.\n\n\\subsection{Stokes automorphism and Alien derivative}\n\nNow let $\\phi\\in\\widetilde{RES}$ be such that $\\hat\\phi\\in\\widehat{RES}$ has singularities in the direction $\\theta$. Then we define the lateral \nresummations $S_{\\theta\\pm}$ as the usual one but with the Laplace transform \\(\\mathcal{L}_{\\theta\\pm}\\) involving integrals avoiding the singularities by going above (for \n$\\mathcal{L}_{\\theta+}$) or below (for $\\mathcal{L}_{\\theta-}$) all of them. They correspond to the limit of \\(S_{\\theta'}\\) when \\(\\theta'\\) tends to \\(\\theta\\) either from above or from below.\\footnote{{\nIn most applications, the possible arguments of the positions of the singularities form a discrete set, so that the different \\(S_{\\theta'}\\) define the same analytic function for an open set of values of \\(\\theta'\\) and we do not really need to take the limit in the definition of \\(S_{\\theta\\pm}\\). However, it is possible to have singularities for example at the positions of all Gaussian integers \\(\\mathbb{Z} + i \\mathbb{Z}\\), in which case the limiting procedure is unavoidable.}}\n\\begin{equation}\n S_{\\theta\\pm}[\\phi](a) := \\int_0^{e^{i(\\theta\\pm\\varepsilon)}\\infty}\\hat\\phi(\\zeta)e^{-\\zeta\/a}\\d\\zeta.\n\\end{equation}\nThe extension of the lateral resummations to the elements of $\\dot{\\widetilde{RES}}$ is similar to the extension of the regular ones given by equation~(\\ref{SthetaDot}).\n\nNow, the key point is that the lateral resummation are linked by the so-called Stokes automorphism in the direction $\\theta$, written \n$\\mathfrak{G}_{\\theta}$.\n\\begin{equation} \\label{def_Stokes}\n \\mathcal{L}_{\\theta+} \\circ \\mathfrak{G}_{\\theta} = \\mathcal{L}_{\\theta-}.\n\\end{equation}\nWe clearly have $\\mathfrak{G}_{\\theta}:\\dot{\\widehat{RES}}\\longrightarrow\\dot{\\widehat{RES}}$. Since both \\(\\mathcal{L}_{\\theta+}\\) and \\(\\mathcal{L}_{\\theta-}\\) are algebra morphisms from the convolutive algebra in the Borel plane to the algebra of functions, \\(\\mathfrak{G}_{\\theta}\\) is an automorphism of the convolution algebra \\(\\dot{\\widehat{RES}}\\). This automorphism encodes how the \nfunction ``jumps'' when the direction of integration crosses a line of singularities (called a Stokes line), already in the extended convolutive model. It can be decomposed in homogeneous components that shift the exponents of the extended models by the complex numbers \\(\\omega\\) which belong to the direction \\(\\theta\\): they are called the lateral alien operators \\(\\Delta^+_\\omega\\). The action of the alien operator \\(\\Delta_\\omega^+\\) on \\({\\hat\\phi}^\\sigma\\) is linked to the singularity of \\({\\hat\\phi}^\\sigma\\) in \\(\\omega\\) and carries the index \\(\\omega+\\sigma\\). The fact that \\(\\mathfrak{G}_{\\theta}\\) is an automorphism translates in the following relations for its components:\n\\begin{equation} \\label{rel_lateral}\n\t\\Delta^+_\\omega (\\hat f \\star \\hat g) = \\sum_{\\omega' + \\omega'' = \\omega} \\Delta^+_{\\omega'} f \\star \\Delta^+_{\\omega''} g,\n\\end{equation}\nwhere the sum includes the cases where \\(\\omega'\\) or \\(\\omega''\\) is 0, and \\(\\Delta^+_0\\) is defined to be the identity.\n\nSince the relation \\eqref{rel_lateral} is not very simple, we use the logarithm of the Stokes automorphism, which is a derivation. The homogeneous components of this logarithm are therefore also derivations, which are called alien derivatives. More precisely\n\\begin{equation} \\label{def_alien}\n \\mathfrak{G}_{\\theta} = \\exp\\left(\\sum_{\\omega\\in\\Gamma_{\\theta}}\\Delta_{\\omega}\\right),\n\\end{equation}\nwith $\\Gamma_{\\theta}$ the singular locus of the function of interest in the direction $\\theta$. The alien derivatives and lateral alien \nderivatives are linked by the equivalent relations\n\\begin{align*}\n & \\Delta_{\\omega_n}^+ = \\sum_{p=1}^n\\sum_{\\omega_1+\\cdots+\\omega_p=\\omega_n}\\frac{1}{p!}\\Delta_{\\omega_1}\\cdots\\Delta_{\\omega_p} \\\\\n & \\Delta_{\\omega_n} = \\sum_{p=1}^n\\sum_{\\omega_1+\\cdots+\\omega_p=\\omega_n}\\frac{(-1)^{p-1}}{p}\\Delta_{\\omega_1}\\cdots\\Delta_{\\omega_p}\n\\end{align*}\nwith all the $\\omega_i$s on the same half-line from the origin to infinity.\n\nAs their names suggest, the alien derivatives are indeed derivations for the convolution product. We shall not give a proof here (we refer the \nreader to \\cite{Sa14} for such a proof), but this fact shall not come as a surprise. Indeed, it is well-known that the operator $\\mathcal{A}$ \nacting on smooth function as a translation\n\\begin{equation*}\n \\mathcal{A}[f](x) = f(x+1)\n\\end{equation*}\ninduces an automorphism on the space of functions. And we can write\n\\begin{equation*}\n \\mathcal{A} = \\exp\\left(\\frac{\\d}{\\d x}\\right)\n\\end{equation*}\nthanks to the Taylor expansion for analytic functions, so that it appears as the exponential of a derivation. Since the Stokes automorphism can be viewed as a translation across a Stokes line, it is understandable that its \nlogarithm is a derivative.\n\nThe alien operators are a priori defined in the convolutive model, but it is convenient to extend them to $\\dot{\\widetilde{RES}}$ by\n\\begin{equation}\n \\mathcal{B}[\\Delta_{\\omega}\\phi] := \\Delta_{\\omega}\\hat\\phi,\n\\end{equation}\nand similarly for $\\Delta_{\\omega}^+$. The alien derivatives, so extended to formal series, become derivatives for the point-like product, since the point-like product \nof functions becomes the convolution product in the Borel plane. \n\nThe alien derivatives have other useful properties. The most important one is \n\\begin{equation} \\label{commutation_alien_usuelle}\n \\Delta_{\\omega}\\partial_z = \\partial_z\\Delta_{\\omega}\n\\end{equation}\nin the formal model, with $z=1\/a$. A proof of this result\ncan be found in \\cite{Sa14}, and another in the general case (which is of interest for us) is in \\cite{Sa06}. The commutation with the ordinary derivative is not so simple if we consider the alien derivative to act in \\(\\widehat{RES}\\) and not in the dotted model. In fact, in many texts, what we denote simply as \\(\\Delta_\\omega\\) is denoted \\(\\dot\\Delta_\\omega\\).\n\n\\subsection{Real resummations}\n\nStokes operators can be a part of the study of the monodromy around singular points of a differential equation, but it may happen that the Borel transform $\\hat\\phi$ has singularities in the direction $\\theta$ in which we are interested to perform a \nresummation, typically the direction $\\theta=0$ for a series with only real coefficients. In this case, the lateral resummations get imaginary parts. This can be a nice feature when the imaginary part of \nthe energy corresponds to the decay probability through tunneling of a state, but generally, we would like to obtain a real solution for a physical \nproblem. The issue is that the simple mean of the two different lateral summations is real, but it is nevertheless not satisfying: we would like this \nreal resummation to satisfy the same equations as the formal solution and this can only be ensured if the convolution product of the means is the mean of the convolution products. \nThe only way to ensure this is to take a suitable combination of the analytic continuations of the function \nalong all the paths that can be taken, going above or below each of the singularities.\n\nIt has been known for a long time that such a real solution is given by the so-called median resummation, a fact that have been shown explicitly in \n \\cite{AnSc13}. A possible expression for this median resummation is\n\\begin{equation}\n S_{\\text{med}} := S_{\\theta-}\\circ\\mathfrak{G}_{\\theta}^{1\/2} = S_{\\theta+}\\circ\\mathfrak{G}_{\\theta}^{-1\/2}\n\\end{equation}\nwhere the power of the Stokes automorphism is defined from a natural extension of the definition \\eqref{def_alien}:\n\\begin{equation}\n \\mathfrak{G}_{\\theta}^{\\nu} := \\exp\\left(\\nu\\sum_{\\omega\\in\\Gamma_{\\theta}}\\Delta_{\\omega}\\right).\n\\end{equation}\nSince the Stokes operator \\(\\mathfrak{G}_{\\theta}\\) is an automorphism, its powers are also automorphisms and the median resummation respects products as a composition of operations preserving products. \n\nReturning to the definition of the Stokes automorphism, it can be seen that \\(\\Delta^+_\\omega\\) corresponds to taking the difference between two possible analytic continuation of the Borel transform beyond the point \\(\\omega\\). The different alien derivatives can also be computed as combinations of different analytic continuations of the Borel transform, going above or below the different singularities (but without ever going backwards). \nThe median summation likewise is a suitable average of the different possible analytic continuations of the Borel transform. \nWhen we go beyond a singularity \\(\\omega\\), we must take a different combination of analytic continuations of the Borel transform: the function we will integrate in the Laplace transform has therefore singularities at the points~\\(\\omega\\) that cannot be avoided and result in nonperturbative contributions to the resummed function.\n\nThe square root of the Stokes operator is simple, but since it gives quite an important weight to paths which cross the real axis a large number of times, the obtained average may grow faster than the lateral values. \nIn~\\cite{Ec92}, Ecalle shows how one could circumvent this problem through accelerations, which allow to reduce the ambiguity between lateral summations from \\(1{\/}\\!\\exp(z)\\) to \\(1{\/}\\!\\exp(\\exp(\\dots(z)\\dots)) \\), with theoretically any finite composition of exponentials, through the control of `emanated' resurgence.\nHowever, other averages are possible, the organic averages, still compatible with the convolution product, which are essentially no larger than the lateral determinations and therefore allow us to avoid this whole procedure. \nIn any cases, these averages still define from the Borel transform a function which is real on the real axis and has singularities on the real axis so that the sum is only defined in the positive half-plane, since it is impossible to relate this integral to others on different integration axis.\n\nAt the approximation level we will reach in the present work, such subtleties will not have a clear effect. However, they can become important if we are to improve on our treatment of these nonperturbative contributions.\n\n\\section{Rehearsal}\nWe are still working with the model used in our previous investigations~\\cite{BeSc08,BeCl13,BeCl14}, the massless supersymmetric Wess--Zumino \nmodel. Even if it is far from a realistic particle physics model, the fact that we only deal with two-point functions and their simple dependence on a unique kinetic \ninvariant gives a more tractable situation than more realistic theories. Nevertheless, the presence of singularities on all integer \npoints for the Borel transform of the renormalization group \\(\\beta\\)-function is probably the generic case in massless exactly \nrenormalizable QFTs, the kind we would like to better understand for their relevance in the description of our universe.\n\nOur former studies are all based on the same simple Schwinger--Dyson equation, solved through the combination of the extraction of the \nanomalous dimension of the field from the Schwinger--Dyson equation and the use of the renormalization group equation to obtain the full \npropagator from this anomalous dimension. We will limit ourselves to the simplest one of the Schwinger--Dyson equations, since it allows us to \nretain a degree of explicitness in the apparent explosion of different series appearing in the object named the {\\em display} by Jean Ecalle, \nan object which collects all information on the alien derivatives of a function. A proper extension of the arguments put forward \nin~\\cite{BeSc12} should prove that any higher order correction to this Schwinger--Dyson equation will only change higher order terms in the \nindividual components of this display, letting its main characteristic unchanged. The factorial growth of the number of high order terms \nbeyond the large \\(N\\), planar limit would present a further challenge, but we will see that there are plenty of questions to solve before.\n\nThe fundamental insight in~\\cite{BeCl14} is that it is in the Borel plane, where the alien derivatives \nhave a clear meaning as singular parts of a function at a given point, that general properties are easier to prove. \nHowever, most computations are easier to carry on in the form of \ntransseries, where the computations look like mechanical operations on formal objects. However, one important component in the \ncomputation scheme of~\\cite{BeCl14} was the contour integral representation of the propagator, with its characteristic property that the possible contours \nchange when considering different points in the Borel plane. It would be interesting to \ngive an interpretation of these contour integrals in the formal scheme and recover how computations can be done using the \nexpansion of the Mellin transform, with subtracted pole parts, at integer points. However, we will see that the simple approximations used for this \nwork do not need such a development.\n\nWe start with the renormalization group equation (RGE) for the two-point function\n\\begin{equation} \\label{renorm_G}\n \\partial_L G(a,L) = \\gamma(1 + 3a\\partial_a )G(a,L),\n\\end{equation}\nwhere we have used $\\beta=3\\gamma$, which can be proved by superspace \\cite{Piguet} or Hopf \\cite{CoKr00} techniques. {A derivation of this equation for the solution of a Schwinger--Dyson equation is detailed in~\\cite{BeSc08}, see also~\\cite{Cl15}.}\nHere $L=\\ln (p^2\/\\mu^2)$ is the kinematic parameter. Expanding $G$ in this parameter \\(L\\),\n\\begin{equation} \\label{rep_G_serie}\n G(a,L) = \\sum_{k=0}^{+\\infty}\\frac{\\gamma_k(a)}{k!}L^k\n\\end{equation}\n(with $\\gamma_1:=\\gamma$)\\footnote{{Be aware that other authors~\\cite{BrKr99,KrYe2006} use different conventions but the same $\\gamma_k$ notations.}} gives a simple recursion on the $\\gamma_k$s\n\\begin{equation} \\label{renorm_gamma_old}\n \\gamma_{k+1} = \\gamma(1+ 3a\\partial_a)\\gamma_k.\n\\end{equation}\nTherefore, at least in principle, it is enough to know $\\gamma$ to rebuild the two-point function.\n\nOn the other hand, we also have the (truncated) Schwinger--Dyson equation, graphically depicted as\n\\begin{equation}\\label{SDnlin}\n\\left(\n\\tikz \\node[prop]{} child[grow=east] child[grow=west];\n\\right)^{-1} = 1 - a \\;\\;\n\\begin{tikzpicture}[level distance = 5mm, node distance= 10mm,baseline=(x.base)]\n \\node (upnode) [style=prop]{};\n \\node (downnode) [below of=upnode,style=prop]{}; \n \\draw (upnode) to[out=180,in=180] \n \tnode[name=x,coordinate,midway] {} (downnode);\n\\draw\t(x)\tchild[grow=west] ;\n\\draw (upnode) to[out=0,in=0] \n \tnode[name=y,coordinate,midway] {} (downnode) ;\n\\draw\t(y) child[grow=east] ;\n\\end{tikzpicture}.\n\\end{equation}\n{The L.H.S. is the two-point function while the R.H.S. contains two dressed propagators, which are equal to the free propagator multiplied by the two-point function.} Computing the loop integral allows to write this equation as\n\\begin{equation} \\label{SDE_old}\n \\gamma(a) = a\\left.\\left(1+\\sum_{n=1}^{+\\infty}\\frac{\\gamma_n}{n!}\\frac{\\text{d}^n}{\\text{dx}^n}\\right)\\left(1+\\sum_{m=1}^{+\\infty}\\frac{\\gamma_m}{m!}\\frac{\\text{d}^m}{\\text{dy}^m}\\right)H(x,y)\\right|_{x=y=0} =: a\\mathcal{I}(H(x,y)).\n\\end{equation}\nwith $H$ known as the Mellin transform of the one-loop integral:\n\\begin{equation} \\label{def_H}\n H(x,y) := \\frac{\\Gamma(1-x-y)\\Gamma(1+x)\\Gamma(1+y)}{\\Gamma(2+x+y)\\Gamma(1-x)\\Gamma(1-y)}.\n\\end{equation}\nThe idea of \\cite{Be10a}, which was fully exploited in \\cite{BeCl13} is to replace the one-loop Mellin transform by a truncation \ncontaining its singularities. Let us define \n\\begin{equation} \\label{form_F}\n F_k := \\mathcal{I}\\left(\\frac1{k+x}\\right)=\\frac{1}{k}\\biggl(1+\\sum_{n=1}^{+\\infty}\\left(-\\frac{1}{k}\\right)^n\\gamma_n\\biggr).\n\\end{equation}\n(which gives the contributions of the poles $1\/(k+x)$ or $1\/(k+y)$ of $H$) and \n\\begin{equation} \\label{def_Lk}\n L_k:=\\mathcal{I}\\left(\\frac{Q_k(x,y)}{k-x-y}\\right)=\\sum_{n,m=0}^{+\\infty}\\frac{\\gamma_n\\gamma_m}{n!m!}\\left.\\frac{\\text{d}^n}{\\text{d}x^n}\\frac{\\text{d}^m}{\\text{d}y^m}\\frac{Q_k(x,y)}{k-x-y}\\right|_{x=0,y=0}\n\\end{equation}\nwhich contains the contributions from the {part of \\(H\\) singular on the line \\(k-x-y=0\\)}. Here $Q_k$ is a suitable expansion of \nthe residue of $H$ at this singularity, a polynomial in the product \\(xy\\).\n\nIt was shown in \\cite{Be10a} that these $F_k$ and $L_k$ obey renormalization group derived equations. Here we are only interested \nin the equations of $L_k$ which are\n\\begin{equation} \\label{equa_L}\n (k-2\\gamma - 3\\gamma a\\partial_a)L_k = Q_k(\\partial_{L_1}\\partial_{L_2})G(a,L_1)G(a,L_2)\\Bigr|_{L_1=L_2=0} = \\sum_{i=1}^kq_{k,i}\\gamma_i^2.\n\\end{equation}\nThe Schwinger--Dyson equations can also be written in terms of these functions. Since we are looking for a resummation of the two-point\nfunction along the positive real axis, and since it was shown in \\cite{BeCl14} that the $L_k$ are responsible for the singularities \nof the Borel transform of $\\gamma$ (and therefore of the two-point function) on the real axis, we will only take care of the terms \nof the Schwinger--Dyson equation involving $L_k$. They have the very simple form\n\\begin{equation} \\label{SDE_phys_plan}\n \\gamma(a) = a\\sum_{k=1}^{+\\infty}L_k(a) + (\\text{contributions from }F_k){=a+O(a^2)}.\n\\end{equation}\nIn order to simplify the results of \\cite{BeCl14}, we consider $\\gamma$ and all the other quantities as formal series in $r:=1\/(3a)$ rather than \nin $a$. We then perform a Borel transform in $r$, according to the conventions most used in the mathematical literature. The perturbative domain is then the one\nfor large values of \\(r\\) and typical nonperturbative contributions will be of the form \\(e^{-nr}\\) for some integer \\(n\\). The advantage being that the singularities of the Borel transform $\\hat\\gamma$ are now located in $\\mathbb{Z}^*$ rather than \n$\\mathbb{Z}^*\/3$. Let us notice that we now have $\\hat\\gamma(0)=1\/3$, but what is most relevant is that \\(\\hat\\beta(0) = 3 \\hat\\gamma(0) = 1\\).\n\nAs explained in \\cite{BeCl14}, a perturbative analysis (i.e., for $\\xi$ small) of the Borel-transformed renormalization group equation suggests to\nwrite the Borel transform $\\hat G:=\\mathcal{B}(G-1)$ as a loop integral\n\\begin{equation} \\label{param_G}\n \\hat{G}(\\xi,L) = \\oint_{\\mathcal{C}_{\\xi}}\\frac{f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta\n\\end{equation}\nwhere $\\mathcal{C}_{\\xi}$ is any contour enclosing $\\xi$ and the origin. Writing the \nrenormalisation group equation \\eqref{renorm_G} in the Borel plane and in term of the $f(\\xi,\\zeta)$ function we get\n\\begin{equation} \\label{renorm_f}\n (\\zeta-\\xi)f(\\xi,\\zeta) = \\hat{\\gamma}(\\xi) + \\int_0^{\\xi}\\hat{\\gamma}(\\xi-\\eta)f(\\eta,\\zeta)\\d\\eta + \\int_0^{\\xi}\\hat{\\beta}'(\\xi-\\eta)\\eta f(\\eta,\\zeta)\\d\\eta.\n\\end{equation}\n\n\\section{Resummations of the anomalous dimension}\n\nWe purposefully put a plural in the ``resummations'' of the title of this section to emphasize that two distinct resummations will be performed here.\nFirst the median resummation and its transseries analysis deliver exponentially small terms, then we sum the dominant terms of the obtained transseries.\n\n\\subsection{Transseries solution}\n\nWe want to compute the leading coefficient of \\(e^{-nr}\\) in the transseries expansion of $\\beta$. Let us start by writing \nthe Schwinger--Dyson equation \\eqref{SDE_phys_plan} and the renormalization group-like equation \\eqref{equa_L} with the variable \\(r\\).\nWe obtain, {while singling out the lowest order term coming from \\(F_1\\),} \n\\begin{equation} \\label{SDE_utile}\n \\beta =\\frac 1 r + \\frac 1 r \\sum_{k=1}^{+\\infty}L_k + (\\text{contributions from }F_k)\n\\end{equation}\nand \n\\begin{equation} \\label{eqLk}\nk {L}_k = \\frac23 {\\beta} {L}_k - r \\beta \\partial_r {L}_k\n \t+ \\sum_{i=1}^k q_{k,i}{\\gamma}_i^2.\n\\end{equation}\nWe are interested in the freedom in the solutions of this system of equations: the perturbative solution is uniquely defined, but since it is a system of differential equations, it must have a space of solutions. Using the fact that the two dominant terms of \\(\\beta\\) are \\(r^{-1} - 2\/3r^{-2}\\), the dominant orders of the linearized equation for \\(L_k\\) are:\n\\begin{equation}\n k L_k = - \\partial_r L_k + \\frac 2 3 r^{-1}(L_k + \\partial_r L_k) \n\\end{equation}\nThe dominant order of the solution is:\n\\begin{equation}\\label{dominant}\n \tL_k = m_k r^{\\frac 2 3 (1-k) } e^{-k r }\n\\end{equation}\nOne can check that the additional terms coming from substituting this value of \\(L_k\\) in the system of equations are smaller by at least \\(r^{-2}\\), so that they cannot change the exponent \\(\\frac23(1-k)\\) of this solution, but only multiply this solution by a power series in \\(r^{-1}\\).\nSince a possible deformation of the solution proportional to \\(e^{-kr}\\) signals the possibility of a nonzero \\(\\Delta_k\\), we recover the results of~\\cite{BeCl14} on the possible forms of the alien derivations of \\(\\beta\\), now written in the formal model instead of the convolutive one.\n\nThe computation of the \\(r^{-1}\\) corrections was carried out in~\\cite{BeCl13} in the case of \\(L_1\\) and involves summations over the effect of all the other \\(L_k\\) as well as over the \\(F_k\\). The language was different, but the resulting computations are totally equivalent to what would be the computation of the terms proportional to \\(m_1\\) in a full solution. \nSuch a computation nevertheless involves summations over \\(k\\) which give highly nontrivial combinations of multizeta values, some of which cancel and the others can be expressed as product of zeta values. In our following work~\\cite{BeCl14}, the introduction of the contour integral representation of the propagator of equation~(\\ref{param_G}) gave a simple interpretation of these results and a prospective way of carrying the computations up to larger orders.\nWe will not consider these corrections here, since the simple study of the dominant terms gives already quite interesting results.\n\nThe nonlinear nature of the system of equations~(\\ref{SDE_utile},\\ref{eqLk}) means that the general solution will have terms with any product of the \\(m_k\\) as coefficient. The behavior of exponentials under differentiation and multiplication ensures that such a term will also have a factor \\(e^{-nr}\\) with \\(n\\) the sum of the indices of the \\(m_k\\) in the coefficient.\nWe therefore have that \\(e^{-r}\\) only appears with the coefficient \\(m_1\\), but \\(e^{-2r}\\) can have the coefficients \\(m_2\\) or \\(m_1^2\\), \\(e^{-3r}\\), the coefficients \\(m_3\\), \\(m_2 m_1\\) or \\(m_1^3\\),\\dots\\ \nThe question then is to know which is the larger possible term for a given coefficient \\(\\prod m_k\\) in the evaluation of \\(\\beta\\). It turns out that the larger possible terms come from the product \\(r \\beta \\partial_r L_k\\) if \\(L_k\\) was the source of the largest term in \\(\\beta\\) with the same coefficient. \nThe end result is that the largest power of \\(r\\) coming in \\(L_k\\) for some product of \\(m_j\\) including \\(m_k\\) is \\(\\frac2 3 \\sum (1-j)\\), giving a term with an exponent less for \\(\\beta\\). The nice point is that the dominant term among the ones with the factor \\(e^{-nr}\\) is the one proportional to \\(m_1^n\\), which has no additional powers of \\(r\\) for \\(L_1\\) and just the factor \\(r^{-1}\\) for \\(\\beta\\).\n\nWe therefore can parameterize the sum of the dominant terms in \\(L_1\\) with\n\\begin{equation}\\label{l1trans}\n\tL_1 = \\sum_{n=1}^\\infty c_n m_1^n e^{-nr}\n\\end{equation}\nUsing that at this order, \\(\\beta\\) is \\(r^{-1} + r^{-1} L_1\\), the equation~(\\ref{eqLk}) for \\(k=1\\) gives the following recurrence relation for the \\(c_n\\):\n\\begin{equation}\\label{relation_c}\n\t(1-n) c_n = \\sum_{p=1}^{n-1} c_{n-p}\\, p\\, c_p\n\\end{equation}\nIf we define a formal series \\(S\\) by\n\\begin{equation} \\label{series_S}\n S(x) := \\sum_{n\\geq1}c_nx^n.\n\\end{equation}\nwe obtain that a first transseries solution for \\(L_1\\) is given by\n\\begin{equation}\n\tL_1 = S(m_1 e^{-r})\n\\end{equation}\n\n\n\\subsection{Summation of the transseries}\n\nThe previous subsection introduced the formal series \\(S\\), and we need to know its properties, in particular its radius of convergence.\nThe inductive formula \\eqref{relation_c} for the $c_n$'s implies a differential equation for $S(x)$ (seen as a formal series):\n\\begin{equation*}\n\t\\frac {S(x)} {x} - S'(x) = S(x)S'(x).\n\\end{equation*}\nDividing by \\(S(x)\\) and regrouping terms depending on \\(S\\), one obtains:\n\\begin{equation*}\n\t\\frac{S'(x)}{S(x)} + S'(x) = \\frac{1}{x}.\n\\end{equation*}\nThe left hand side is the logarithmic derivative of the function $F(x):=S(x)e^{S(x)}$ so that we have\n\\begin{equation*}\n\tS(x) e^{S(x)} = k x,\n\\end{equation*}\nfor some $k\\in\\mathbb{R}$. From the definition of \\(m_1\\) in~\\eqref{dominant} and the comparison with the formula~\\eqref{l1trans}, we see that \\(c_1=1\\), which also fixes \\(k=1\\). The presence of a minimum of the function \\(u \\rightarrow u e^u\\) for \\(u=-1\\) with the value \\(-1\/e\\) gives rise to \na singularity of \\(S(x)\\) for \\(x = -\\frac 1 {e}\\) of the square root type. Since it is the singularity nearest to the origin, it implies that the convergence radius of the series is \\(\\frac1 {e}\\).\n\nIn fact, the above function inversion problem has been studied and the solution of the case \\(k=1\\) is known as (the principal branch of) Lambert's \\(W\\)-function.\nUsing the initial condition $S'(0)=1$ and the fact that $W'(0)=1$ we find that the series $S(x)$ of \\eqref{series_S} is actually\n\\begin{equation}\n S(x)=W(x).\n\\end{equation}\nAn explicit series representation of the Lambert \\(W\\)-function is known :\n\\begin{equation*}\n W(x) = \\sum_{n\\geq1}\\frac{(-n)^{n-1}}{n!}x^n.\n\\end{equation*}\nThis formula can be deduced from the Lagrange inversion formula. The convergence radius $1\/e$ is then a simple consequence of the Stirling formula for the factorial.\n\nAll in all we have shown that the {sum of the lowest order terms in all nonperturbative sectors of the anomalous dimension} is\n\\begin{equation} \\label{result_anormal_dimension}\n \\gamma^\\mathrm{res}(r) = r^{-1} W( m_1 e^{-r}) + \\mathcal{O}(r^{-2})\n\\end{equation}\nand is defined in the region $|m_1 e^{-r}|<1\/e$ of the complex plane. \n\n{The construction presented here is a particular example of ``transasymptotic analysis'', which suggests that similar formulae exist at any order in $e^{-r}$. See for example \\cite{Costin2001} and references therein.} {An interesting aspect of the analysis in~\\cite{Costin2001} is that they show that the singularity of this lowest order resummed solution signals a singularity of the full solution in its vicinity.} \n\n\n\\subsection{Links with the alien calculus}\n\nIn the preceding sections, we studied a possible transseries deformation of the perturbative solution for the \\(\\beta\\)-function, but to what use can it be put for the evaluation of the function? In particular, could it be possible to have a determination of \\(m_1\\)? \nPart of the response comes from the idea of the bridge equations. Since the Stokes automorphism and its powers respect products and commutes with the derivation with respect to \\(r\\), the functions remain solution of the equations when transformed by these automorphisms. Since in the formal model, the alien derivation \\(\\Delta_n\\) gives rise to a factor \\(e^{-nr}\\), the solutions after the action of a Stokes automorphism will be in the form of a transseries.\n\nSince the most general transseries solution is a function of the parameters \\(m_k\\) appearing in the linear deformations of the solution~\\eqref{dominant}, \nalien derivatives can be expressed through derivations acting on these parameters, giving bridges between alien calculus and ordinary calculus. The alien derivation \\(\\Delta_n\\) is, a priori, any combination of operations which lower the weights by \\(n\\), so that for example, \\(\\Delta_1\\) has not only a term proportional to \\(\\partial\/\\partial m_1\\), but also \\(m_1 \\partial\/\\partial m_2\\) and many others. Nevertheless, the same reasons which made the terms proportional to \\(m_1^n\\) dominate imply that the coefficient \\(f\\) of \\(\\partial\/\\partial m_1\\) is the most important part of \\(\\Delta_1\\). A value for this coefficient \\(f\\) could be extracted in~\\cite{BeCl13} from the comparison of the asymptotic behavior of the perturbative series for \\(\\gamma\\) deduced from the singularities of the Borel transform and the known coefficients of this series. \n\nThe dominant term in the singularity at the point~\\(n\\) necessarily comes from the coefficient of \\(m_1^n\\) and can only be extracted by the \\(f^n (\\partial\/\\partial m_1)^n\\) term in \\(\\Delta_1^n\\). The \\(n!\\) coming from the iterated differentiations is compensated by the \\(1\/n!\\) factor in front of \\(\\Delta_1^n\\) in the definition of \\(\\Delta^+_n\\), so that we obtain, from the relation between alien derivatives and singularities of the Borel transform, that \\(\\hat L_1\\) has a pole in \\(n\\) with residue \\(c_n f^n\\), while the only divergent part for \\(\\hat \\beta\\) is proportional to \\(- c_n f^n \\log(|\\xi - n|) \\).\n\nThese singularities of the Borel transform are transmitted to the median average. In turn, these singularities of the integrand produce nonperturbative contributions to the result of resummation, so that the factors like \\(f\\), determined through alien calculus, can be used to fix the unknown coefficients in the transseries expansion. What is important is that the consistency of all the steps of the resummation procedure with the products and derivation ensures that the result respects the original equations and must therefore be of a form compatible with the transseries solution. The influence of the singularity at 1 will therefore be sufficient to obtain the dominant part for the singularities for all \\(n\\). \n \nAlthough we have been able to compute nonperturbative terms, with coefficients which could be computed from the perturbative expansion of the anomalous dimension, the situation seems quite complicated. \nIndeed the resurgent analysis seems to make the situation go from bad to worst: instead of a unique formal series, we end up with formal series multiplying \\(e^{-rn}\\) for each \\(n\\), and with furthermore coefficients which are polynomials in \\(r^{-2\/3}\\) and \\(\\log r\\) of degrees growing with \\(n\\), with many undetermined coefficients. \nMoreover, each of these series are actually divergent and need some form of resummation.\nHowever, we may remember that in many cases, divergent series are not so bad news, and as Poincar\\'e has put it, they are ``convergent in the sense of astronomers'': the first few terms give a fairly accurate approximation of the final result, as is the case for example in quantum electrodynamics.\nOur position will therefore be to use the information we have and forget for the time being about all the unknown quantities. We would of course prefer to have arguments proving that indeed what we neglect is negligible, but it is the best we can do at the moment.\n \n{We reshuffle the transseries and write them as series in $r$ whose terms are series in $e^{-r}$. This operation is inspired by a remark of Stingl \\cite{St02}, page 70 about\nphysical} considerations on what the ``true'' observables {are,} were put forward to provide a justification to this manipulation.\nThe take home message could be that it is important to keep all the terms of a convergent series but series with 0 convergence radius could be truncated without remorse. A physicist way of dealing with such a situation would be to look at how the results change when we add terms from the formal series, but we would need at least one more term.\n\nA more mathematical view could come from transmonomials~\\cite{Sa07}, which are special functions with simple properties under the action of alien derivatives.\nThis could lead to an expansion of the function where transmonomials get multiplied by ``alien constants'', functions on which all alien derivatives give zero and therefore easily computable from their power series expansion. The simplest transmonomial \\(\\mathcal U^1\\), with the only non zero alien derivative \\(\\Delta_1 {\\mathcal U}^1 =1 \\) would replace \\(e^{-r}\\) in all our transseries expressions and take care of the dominant terms at large orders neglected in the naive approach.\n\nTo conclude this short exposition of the idea of resumming the transseries, let us emphasize that in other contexts, methods using grouping of terms of nonsummable families have been put to good use to produce convergent expressions. \nArbitrary groupings can produce arbitrary results, but some well defined procedures have been shown to reproduce the results of a Borel summation.\nA prime example is the arborification procedure presented in Ecalle's work on mould calculus~\\cite{Ecalle1992}, which separates terms in smaller parts to be regrouped in other objects. {Our procedure might be seen as an \narborification where only ladder trees give nonzero contributions. The reader can be referred to \\cite{FaMe12}, Section 6} for a clear introduction of the arborification--coarborification \nin the context of linearization problems. Other cases appear in the study of Dulac's problem~\\cite{Ec92}. The main advantage of such procedures is that they allow practical computation. It is thus possible that the somewhat ad hoc computation presented above can also be justified.\n\n\\section{Properties of the Green function}\n\n\\subsection{Nonperturbative mass scale generation}\n\nThe mass of a particle is given by the position of a pole of the two-point function as a function of the invariant \\(p^2\\) of the momentum. In our case, there is always a pole for \\(p^2=0\\), reflecting the fact that we started from a massless theory, since the function \\(G\\) we are studying is a multiplication factor for the free propagator, the same for all states of the supermultiplet. We define a mass scale as the value of the external momentum $p$ for which the Green function has a singularity. The expansion of \\(G\\) in powers of the logarithm \\(L= \\log(p^2\/\\mu^2)\\) is ill suited for such an analysis. We expect the sign of \\(p^2\\) to fundamentally change the situation, the pole for a physical particle being for a timelike \\(p\\), while \\(L\\) only change by \\(i\\pi\\) when going from timelike to spacelike momenta.\n\nThis is why we will rather use the integral representation \\eqref{param_G}:\n\\begin{equation*}\n \\hat{G}(\\xi,L)=\\oint_{\\mathcal{C}_{\\xi}}\\frac{f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta.\n\\end{equation*}\nIt was shown in \\cite{BeCl14} that the function $\\zeta\\mapsto f(\\xi,\\zeta)$ has singularities at $\\zeta=\\xi$ and at \\(\\zeta=0\\). Therefore we can expand the contour \n$\\mathcal{C}_{\\xi}$ to infinity without changing the value of the integral. This being done, we can make the lateral alien derivative go \nthrough the integral and obtain:\n\\begin{equation*}\n \\Delta_n^+\\hat{G}(\\xi,L)=\\oint_{\\mathcal{C}}\\frac{\\Delta_n^+f(\\xi,\\zeta)}{\\zeta}e^{\\zeta L}\\d\\zeta,\n\\end{equation*}\nwhere the lateral alien derivative acts on the $\\xi$ variable. Taking the lateral alien derivative of the renormalisation group equation \n\\eqref{renorm_f} we get\n\\begin{equation} \\label{renorm_f_lat_alien_der}\n (\\zeta-(\\xi+n))\\Delta_n^+f(\\xi,\\zeta) = \\frac 1 3 \\Delta_n^+\\hat\\beta(\\xi) + \\frac 1 3 \\sum_{i=0}^n \\bigl(\\Delta_{n-i}^+\\hat{\\beta}\\star\\Delta_i^+ f \\bigr)(\\xi,\\zeta) + \\sum_{i=0}^n \\bigl(\\Delta_{n-i}^+\\hat{\\beta}'\\star \\text{Id}. \\Delta_i^+f\\bigr)(\\xi,\\zeta).\n\\end{equation}\nWe are looking for the dominant term in $\\Delta_n^+f(\\xi,\\zeta)$ seen as a function of $\\xi$. It will come from the dominant term in \\(\\Delta_n^+ \\hat\\beta\\), which is constant in \\(\\xi\\), while all other terms give higher powers of \\(\\xi\\). It will therefore be given by a term \\(f_{n,0}(\\zeta)\\), which satisfies\n\\begin{equation*}\n (\\zeta-n)f_{n,0}(\\zeta) = c_n f^n.\n\\end{equation*}\nPlugging this into the integral representation of $\\Delta_n^+\\hat{G}(\\xi,L)$ we get\n\\begin{equation*}\n \\Delta_n^+\\hat{G}(\\xi,L) = \\frac{c_n f^n}{n}\\left(e^{nL}-1\\right)+\\mathcal{O}(\\xi^{2\/3}).\n\\end{equation*}\nThe transseries expansion of \\(G\\) deduced from these singularities of \\(\\hat G\\) is then \n\\begin{equation*} \n G^\\mathrm{res}(r,L)= 1+\\frac 1 r \\sum_{n=1}^{+\\infty}\\frac {c_n} {n} (f e^{-r})^n \\Bigl(e^{nL} -1 \\Bigr) + \\text{higher orders}\n\\end{equation*}\nNow, as we have done \nfor the anomalous dimension in the previous section we can simply sum the above series, without worrying on the other terms. \nIf we neglect 1 with respect to \\(e^{nL} = (p^2\/\\mu^2)^n\\), we obtain a function of \\(f e^{L-r}\\) with Taylor coefficients \\(c_n\/n\\).\nSince the \\(c_n\\) are the Taylor coefficients of Lambert's \\(W\\)-function, these are the coefficients of the primitive of the \\(W\\)-function divided by \\(x\\).\nThis is a function which grows only logarithmically for positive arguments, but goes to 0 has a three half power of the variable at \\(-1\/e\\).\n$f$ was numerically computed in \\cite{BeCl13} to be $0.208143(4)$. This shows that for an euclidean momentum where \\(e^L\\) is positive, we are in a situation where the function grows really slowly and the large terms of the series, proportional to \\( (p^2)^n \\), combine to a simple logarithmic correction to the propagator. \nSince propagators can be Wick rotated to the euclidean domain in loop computations, this means that nothing prevents us, at this approximation level, from defining consistently the renormalized theory. \nOn the other hand, we have for a finite value of \\(p^2\\) in the timelike domain a singularity of the propagator which defines a mass scale\n\\begin{equation}\n M_{NP}(r)^2 = \\frac{\\mu^2}{f}e^{r-1}.\n\\end{equation}\nLet us notice that we find that the nonperturbative mass goes to infinity as $r$ goes to infinity, which corresponds to $a$ going to $0^+$. \nThis was to be expected. However this mass scale is not renormalization group invariant, since a renormalization group invariant mass scale should involve a factor \\(r^{2\/3}\\). We hope that a more careful analysis can give back this factor so that we obtain a fully consistent analysis of this nonperturbative mass scale.\n\nLet us finally remark that the detour by the contour integral representation of \\(G\\), which is very valuable if we wanted to consider all the corrections proportional to powers of \\(L\\) in the expansion of \\(G\\), is not really necessary at this level of approximation. We could have simply deduced that a term proportional to \\((p^2)^n\\) appears when considering a \\(e^{-nr}\\) term in the \\(\\beta\\)-function. \n\n\\subsection{Analyticity domain: a necessary acceleration?}\n\nWe want to see how the resummed two-points function $G^{\\text{res}}$ could be obtained through Laplace transforms, to better understand its analyticity domain. \nWe make the bold approximation that the size of the Borel transform at a point can be approximated by the contribution of the nearest singularity. We have seen that the singularity in the point \\(n\\) is dominated by the contribution \\(1\/n! \\Delta_1^n \\hat G\\) in the lateral derivative, which has the factor \\(c_n (p^2)^n\\). \nThe coefficients \\(c_n\\) have also a power like behavior so that the domain in \\(r\\) where the Laplace transform of \\(\\hat G\\) is well defined shrinks when \\(p^2\\) grows.\nSince we would like that our theory defines the two-point function for any values of the momentum, we cannot define it by a simple Laplace transform. {Indeed, even if the reformulation of the Schwinger--Dyson equation we use does not make it explicit, the proper definition of the two-point function is necessary to compute the loop integral appearing in the definition of the \\(\\beta\\) function. It is therefore important to have at all stages computation which are uniform in \\(p\\).}\n\nThe fact that the dominant terms in the transseries representation sum up to an analytic function of \\(p^2\\), with a well behaved extension to any positive values of \\(p^2\\) is of no relevance here: the Laplace transform is well defined only if \\(p^2\\) is small enough that we are in the convergence domain of the sum of the dominant terms. \nWe need\n \\begin{equation*}\n |e^{-r}|\\leq e^{\\kappa}=|c|^{-1} e^{-L+1},\n\\end{equation*}\nin other terms that the real part of \\(r\\) should be larger than \\(-\\kappa\\), which grows like \\(L\\).\n\nIn order to be able to do the Laplace integral, we could think of doing a Borel transform with respect to \\(p^2\\), but we cannot see how it could be possible to use the Borel-transformed two-point function as an ingredient of the Schwinger--Dyson equations. A possible way out is rather through acceleration. This formally corresponds to making a Laplace transform followed by the Borel transform of the function expressed in terms of a new variable, which is a growing function of the old one. If the Laplace transform was well defined, one sees that there exists a kernel \\(K(\\xi_1,\\xi)\\) such that the new Borel transform \\(\\hat f_1\\) of a function \\(f\\) is given by:\n\\begin{equation}\n\t\\hat f_1(\\xi_1) = \\int_0^\\infty K(\\xi_1,\\xi) \\hat f(\\xi) d\\xi.\n\\end{equation}\nFor fixed \\(\\xi_1\\), the kernel \\(K(\\xi_1,\\xi)\\) vanishes faster than any exponential when \\(\\xi\\) goes to infinity, allowing this acceleration transform to remain defined in cases where the Laplace transform in the original variable was not possible.\nIn fact, since the Borel transform remains of exponential growth, but only with a coefficient which can be arbitrarily large, the acceleration transform can be defined whenever its kernel has a slightly faster decay than the exponential. This is the case already for the kernel associated with the change of variable \\(r \\to r_1 = e^r\\), for which the kernel behaves like \\(\\exp(-\\xi \\log\\xi)\\) for large \\(\\xi\\) and fixed \\(\\xi_1\\).\n\nThe terms we kept of the transseries expression of the two-point function translate in the following value for its accelerated Borel transform in terms of the Borel transform \\(\\hat W\\) of the Lambert function:\n\\begin{equation}\n\t \\hat G_1(\\xi_1, p^2) \\simeq \\hat W ( \\xi_1 p^2 )\n\\end{equation}\nSince the Lambert function is holomorphic in a neighborhood of the origin, its Borel transform is an entire function, so that \\(\\hat G_1\\) is well defined {at this approximation level}. However, if the final Laplace transform is to be valid for any value of \\(p^2\\), it must be done in directions such that \\(\\hat W\\) is smaller than any exponential, and this in turn restrict the angular width of the domain in \\(r_1=e^r\\) where the field theory can be defined by resummation.\nIn turn, this implies that the imaginary part of \\(r\\) is bounded: the limits of the analyticity domain are lines of fixed imaginary part, which correspond to circle arcs tangent to the real axis in the original coupling \\(a\\).\nThe ensuing analyticity domain is quite similar to the one proposed by 't Hooft~\\cite{Ho79} for any sensible quantum field theory.\n\nIt remains to know whether for some functional of the two-point function singularities of the Borel transform in the variable \\(\\xi_1\\) could appear, ensuring that there are non trivial alien derivatives in this second Borel plane {and thus proving the unavoidable character of acceleration. We must remember that the approximation of the system of equations obeyed by the renormalization group function used in this work is a rather crude one. Conceptually, we do not have a clearly defined differential system, but a functional equation involving the two-point function with its dependence on the momentum. For the perturbative solution, we could transform it to an infinite system of differential equations by considering the function to be given by its Taylor series at the origin in the variable \\(L\\), but this approximation is not suitable for the computation of the properties of the analytic continuation of the Borel transform. We had to introduce terms associated with the poles of the Mellin transform to obtain deformation parameters naturally associated with each of the terms \\(e^{nr}\\), with \\(n\\) any nonzero integer. But our system of differential equations has other components, since we also need the coefficients \\(\\gamma_k\\). This suggests that the terms we discussed in~\\cite{BeCl14} do not exhaust the possible transseries deformations of the solution, but the change in the representation of the two-point function which would allow to pinpoint these additional possible terms goes far beyond the ambition of this work.}\n\n{In fact, this acceleration procedure should not be viewed with too much fear. It can be seen as a tool to transform the difficult problem of bounding the analytic continuation of the Borel transform into the much simpler problem of finding some kind of formal solutions which allow to characterize the possible alien derivatives in a second Borel plane. In any case, what happens in the different Borel planes are somehow unrelated, so that the process of analytic continuation, analysis of the singularities and eventually averaging of the Borel transform is totally independent of the fact that it will be followed by a Laplace transform or an acceleration transform.}\n\n\\section*{Conclusion}\n\nUsing to a bigger extent the power of alien calculus and transseries expansions we have been able to go much further than in our previous work~\\cite{BeCl14}. The dominant terms of the singularities of the Borel-transformed anomalous dimension of the theory has been computed.\nThis computation was carried out by using transseries expansions and considering the effect \nof lateral alien derivatives on the Schwinger--Dyson and the renormalization group equation of the theory. Then, using a suitable form \nof the median resummation, this gave the first order in every `instantonic' sectors of the anomalous dimension of our theory.\n\nFollowing a procedure suggested by Stingl, we have kept in this transseries solution the terms of a formal series in \n$e^{-r}$, forgetting the terms in (negative) powers of \\(r\\). This series of the dominant terms turned out to be convergent and its sum evaluated.\n\nThe same analysis could also be transferred to the two-point function of the theory. Indeed, the two-point function is \nessentially determined by the anomalous dimension through the renormalization group. \nThe important point is that the singularity of the anomalous dimension in the point \\(n\\) of the Borel plane, or a term \\(e^{-nr}\\) in the transseries expansion, gives rise to a term proportional to \\((p^2)^n\\) for the two-point function. What looked like negligible contributions to the anomalous dimension becomes dominant in the two-point function for large \\(p^2\\). \nWhile an explicit value of the factor multiplying \\(p^2\\) could not be \nobtained, the functional dependence can be obtained. In the euclidean domain, which corresponds to \\(p^2\\) positive with our conventions, the asymptotic behavior of the Lambert function means that {this series of powers of \n\\(p^2\\) has a finite radius of convergence.} These powers of \\(p^2\\) sum up to something of merely logarithmic growth. \nThis final asymptotic behavior may however appear only for so large values of the momentum that it would be totally invisible in usual nonperturbative studies, where the ratio between the largest and the smallest scales that can be studied is rather limited.\nNevertheless the two-point function has, at this approximation level, a quite regular behavior at the smallest scales, in contradistinction to the divergence for some finite scale obtained with finite order approximations of the \\(\\beta\\)-function: many general arguments for the triviality of such a quantum field theory\nwith a positive \\(\\beta\\)-function break down. We must however remain cautious, since we are just scratching the surface of the kind of beyond perturbative theory analysis resummation theories allow, and many new phenomena may be revealed by a more careful study.\n\nThe simple analytic dependences on \\(p^2\\) of all the terms in the expansion of the two-point function make it easy to study its analytic continuation to the timelike domain, with negative \\(p^2\\). In this case, a singularity appears which is of square root type. This singularity defines a nonperturbative mass scale for the theory, but our computation is not fully satisfying since this mass scale is not fully renormalization group invariant. \n\nThe fact that even our fairly simple computation revealed a nonperturbative mass scale for the theory is quite remarkable. \nAlso we did not have to choose the functional dependence of the two-point function, but it was provided by our computation. In our case, singularities of the Borel transform were associated with ultraviolet divergent contribution to the two-point function: \nin an asymptotically free theory like QCD, we would obtain infrared divergent contributions, but likewise it could be possible to sum this contributions to obtain well behaved propagators up to the lowest scales. Since we do not decide a priori their functional form, we may have signs of confinement in the form of states having singularities different than poles in the timelike domain. \n\nConsiderations about the way that the sum of the terms of the transseries could come from the Laplace integral give indications on the growth \nof the Borel transform, which is harder to check: changing the \nvariable on which the final Laplace transform is done corresponds to an integral transform (dubbed acceleration by Ecalle) with a kernel \nthat have faster than exponential decay at infinity. A suitable change of variables therefore allows to define a new germ of analytic function \nnear the origin, the singularities of which can be studied by a new set of alien derivations. If all such possible accelerations do not present \nsingularities, the first Borel transform should be suitable to directly define the sum, giving an indirect check on the growth of the Borel transform. \n\nLimits on the analyticity domain for complex values of the \ncoupling constant devised by 't Hooft~\\cite{Ho79} and Stingl~\\cite{St02} suggests, contrarily to what Stingl said in some papers, that at least \none such acceleration should be needed in the case of nonabelian gauge theories. \nThis question is linked to the shape of the analyticity domain in the coupling constant of the theory, that we did not study. However, in the case of the two-point function, if \\(r\\) is given an imaginary part of \\(i\\pi\\), \\(e^{-r}\\) changes sign and the singularity which was for timelike momenta enters the euclidean domain. \nThis should pose serious problems to the continuation of the theory to such values of the coupling, so that \\(r\\) should be limited to a band of finite extension in the imaginary direction, which converts to the horn-shaped domains proposed by\n't Hooft when converting back to the coupling proportional to \\(r^{-1}\\). \n\nIt should certainly be interesting to compute the higher order corrections to the transseries solutions of the system of equations studied here and try to deduce their full system of alien derivations. \nAn effective computation however appears to be quite a formidable task, but in what are certainly simpler cases, mould calculus has been shown to provide for quite explicit results, expressing results in terms of resurgent monomials~\\cite{Sa07}.\nWith this strategy, one could avoid as much as possible to work explicitly in the Borel plane, even if the analytic continuation of functions in the Borel planes (plural if acceleration is needed) is the ultimate justification of the computations one may attempt.\n\nMoreover, our study could also be carried out when including additional terms involving higher-loops primitively divergent diagrams \nin the Schwinger--Dyson equation. This should allow to expand the results of \\cite{BeSc12}, which only considered the asymptotic behavior of the perturbative series, that is, the singularities closest to the origin of the Borel transform.\n\nIn this work, we limited ourselves to the two-point functions, which have a simple dependence on a unique Lorentz invariant: a general study of quantum field theory would certainly benefit from a careful investigation of the analytic properties of the Borel-transformed Green functions in all their variables.\n\nFinally, let us notice that the probable usefulness of mould calculus in the realm of quantum field theory expands the \nlist of elements of Ecalle's theory of resurgence which should be used in physics: we have used alien calculus, median resummation \nand it seems very likely that acceleration will be needed in the next steps of our program. Bridge equations are nowadays a common tool of some \nphysicists (see e.g., \\cite{AnScVo12}) and we argued that we might also use mould calculus while resurgent monomials will come into the game. Proper use of these tools could well provide solutions to old questions in quantum field theory.\n\\bibliographystyle{unsrturl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCompressive Sensing (CS) is a method in signal processing which aims to reconstruct signals from a relatively small number of measurements. \nIt has been shown that sparse signals can be reconstructed with a sampling rate far less than the Nyquist rate by exploiting the sparsity \\cite{Donoho06}.\n\nIn this paper, we focus on Binary Compressive Sensing (BCS) which restricts the signals of interest to binary $\\{ 0, 1 \\}$-valued signals, which are widely used in engineering applications, such as fault detection \\cite{Bickson11}, single-pixel image reconstruction~\\cite{Duarte08}, and digital communications~\\cite{Wu11}.\nRelated works are as follows: Nakarmi and Rahnavard \\cite{Nakarmi12} designed a sensing matrix tailored for binary signal reconstruction. Wang et al.~\\cite{Wang13} combined $\\ell_1$ norm with $\\ell_\\infty$ norm to reconstruct sparse binary signals. Nagahara \\cite{Nagahara15} exploited the sum of weighted $\\ell_1$ norms to effectively reconstruct signals whose entries are integer-valued and, in particular, binary signals and bitonal images. Keiper et al.~\\cite{Keiper17} analyzed the phase transition of binary Basis Pursuit.\n\nWe note that most of the previous work on BCS are based on convex optimization. Indeed, convex optimization based algorithms allow performance guarantee via rich mathematical tools. However, they are found to be notoriously slow in large-scale applications compared to greedy methods such as the Orthogonal Matching Pursuit (OMP) \\cite{Donoho08}. On the other hand, greedy methods like OMP are fast but often have a worse recovery rate than convex optimization methods. In this work, we propose a fast BCS algorithm with a high recovery rate. Taking the binariness of signals into account, our algorithm is a gradient descent method based on the smoothed $\\ell_0$ norm \\cite{Mohimani09}. Through numerical experiments, we show that the proposed algorithm compares favorably against previously proposed CS and BCS algorithms in terms of recovery rate and speed.\n\nThe rest of the paper is organized as follows. We give a short review on CS\/BCS algorithms in Section \\ref{sec:BCS} and present our algorithm in Section \\ref{sec:BSSL0}. In Section \\ref{sec:Experiments}, we present experimental results which compare the performance of the proposed algorithm with other algorithms. We conclude this paper with some remarks in Section \\ref{sec:conclusion}.\n\n\n\\subsection*{Notations:}\n\nFor a vector $\\mathbf{v} = (v_1, \\cdots, v_N)^\\top$ and $1 \\leq p \\leq \\infty$, the $\\ell_p$ norm of $\\mathbf{v}$ is denoted by $\\| \\mathbf{v} \\|_p$. The number of non-zero entries in $\\mathbf{v}$ is denoted by $\\|\\mathbf{v}\\|_0$. The probability of an event $E$ is denoted by $\\mathbb{P}(E)$. Let $[N]=\\{1,\\cdots,N\\}$ for $N \\in \\mathbb{N}$. We denote by $\\textbf{1}_N$ the $N$-dimensional vector with all entries equal to $1$.\n\n\n\\section{Binary Compressive Sensing (BCS)}\n\\label{sec:BCS}\n\nIn the standard CS scheme, one aims to recover a sparse signal from its linear measurements. The constraints posed by the measurements can be formulated as\n\\begin{equation}\n\\Phi \\, \\mathbf{z} = \\mathbf{y},\n\\quad \\mathbf{z} \\in \\mathbb{R}^N,\n \\label{eq:measurement}\n\\end{equation}\nwhere $\\Phi \\in \\mathbb{R}^{m \\times N}$, $m \\ll N$, is the measurement matrix \nand $\\mathbf{y} = \\Phi \\mathbf{x} \\in \\mathbb{R}^m$ is the measurement of a \\emph{sparse} signal $\\mathbf{x} \\in \\mathbb{R}^N$. CS algorithms exploit the fact that $\\mathbf{x}$ is sparse and seek a sparse solution $\\mathbf{z}$ satisfying (\\ref{eq:measurement}).\n\n\nThe BCS scheme considers binary signals for $\\mathbf{x}$. Note that a binary signal $\\mathbf{x}$ is sparse if and only if its complementary binary signal $\\widetilde{\\mathbf{x}} := \\textbf{1}_N - \\mathbf{x}$ is dense, i.e., is almost fully supported. As the measurement matrix $\\Phi$ is known, the equation (\\ref{eq:measurement}) converts equivalently to\n\\begin{equation}\n\\Phi \\widetilde{\\mathbf{z}} = \\widetilde{\\mathbf{y}},\n \\label{eq:measurement_dense}\n\\end{equation}\nwhere $\\widetilde{\\mathbf{z}} := \\textbf{1}_N - \\mathbf{z}$ and $\\widetilde{\\mathbf{y}} := \\Phi \\widetilde{\\mathbf{x}} = \\Phi \\textbf{1}_N - \\mathbf{y}$. This shows that reconstructing a sparse signal $\\mathbf{z}$ under the constraint (\\ref{eq:measurement}) is equivalent to reconstructing a dense signal $\\widetilde{\\mathbf{z}}$ under the constraint (\\ref{eq:measurement_dense}). For this reason, in contrast to the case of generic signals, binary signals that are dense can be recovered as well as those that are sparse.\n\n\nTwo types of models for binary signals have been considered in the literature (e.g., \\cite{Donoho10,Wang13,Nagahara15}):\n(i) $\\mathbf{x}$ is a deterministic vector which is binary and sparse, i.e., most of its entries are $0$ and only few are $1$; (ii) $\\mathbf{x}$ is a random vector whose entries are independent and identically distributed (i.i.d.) with probability distribution $\\mathbb{P}(x_j = 1) = p$ for some fixed $0 \\leq p \\leq 1$. If $p$ is small, a realization of $\\mathbf{x}$ is likely a sparse binary signal.\n\nIn this work, we shall consider the second model which can accommodate dense binary signals as well as sparse binary signals.\n\nBelow we give a short review of CS\/BCS methods that are related to our work.\n\n\\subsection{$\\ell_0$ minimization (L0)}\n\nA naive approach to finding sparse solutions is the $\\ell_0$ minimization,\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_0 & & \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\tag{$P_0$}\n\\label{eq:P_0}\n\\end{equation}\nThis method works generally for continuous-valued signals that are sparse, i.e., signals whose entries are mostly zero. However, solving the $\\ell_0$ minimization requires a combinatorial search and is therefore NP-hard \\cite{Natarajan95}.\n\n\n\\subsection{Smoothed $\\ell_0$ minimization (SL0)}\n\nSmoothed $\\ell_0$ minimization (SL0) \\cite{Mohimani09} replaces the $\\ell_0$ norm in (\\ref{eq:P_0}) with a non-convex relaxation:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & \\sum_{i = 1}^N \\left ( 1 - \\exp \\left (\\frac{- z_i^2}{2 \\sigma^2} \\right ) \\right ) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\label{eq:SL0}\n\\end{equation*}\nThis is motivated by the observation \n\\[ \\lim_{\\sigma \\to 0} \\exp \\left( \\frac{-t^2}{2 \\sigma^2} \\right) = \\begin{cases}\n 1 & \\text{if~} t = 0 \\\\\n 0 & \\text{if~} t \\neq 0,\n \\end{cases}\n\\]\nwhich implies that for any $\\mathbf{z} = (z_1, \\dots, z_N )^\\top \\in \\mathbb{R}^N$,\n\\begin{equation}\n\\lim_{\\sigma \\to 0} \\sum_{i = 1}^N \\left( 1 - \\exp \\left( \\frac{-z_i^2}{2 \\sigma^2} \\right) \\right) = \\| \\mathbf{z} \\|_0 .\n\\label{eq:SL0convergence}\n\\end{equation}\nNoticing that $\\mathbf{z} \\mapsto \\sum_{i = 1}^N \\big( 1 - \\exp \\big( \\frac{-z_i^2}{2 \\sigma^2} \\big) \\big)$ is a smooth function for any fixed $\\sigma > 0$, Mohimani et al.~\\cite{Mohimani09} proposed an algorithm based on the gradient descent method. The algorithm iteratively obtains an approximate solution by decreasing $\\sigma$.\n\nMohammadi et al.~\\cite{Mohammadi14} adapted the SL0 algorithm particularly to non-negative signals. Their algorithm, called the Constrained Smoothed $\\ell_0$ method (CSL0), incorporates the non-negativity constraints by introducing some weight functions into the cost function. Empirically, CSL0 shows better performance than SL0 in the reconstruction of non-negative signals.\n\n\n\\subsection{Basis Pursuit (BP)}\n\nA well-known and by now standard relaxation of (\\ref{eq:P_0}) is the $\\ell_1$-minimization, also known as the \\emph{Basis Pursuit} (BP) \\cite{Chen01}:\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\tag{$P_1$}\n\\label{eq:P_q}\n\\end{equation}\nSimilar to (\\ref{eq:P_0}), this method works generally for continuous-valued signals $\\mathbf{x} \\in \\mathbb{R}^N$ that are sparse.\n\n\n\\subsection{Boxed Basis Pursuit (Boxed BP)}\n\nDonoho et al.~\\cite{Donoho10} proposed the Boxed Basis Pursuit (Boxed BP) for the reconstruction of \\emph{k-simple bounded signals}:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in [0,1]^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\end{equation*}\nThe intuition behind Boxed BP is straightforward: the $\\ell_1$ norm minimization promotes sparsity of the solution while the restriction $\\mathbf{z} \\in [0,1]^N$ reduces the set of feasible solutions. Recently, Keiper et al.~\\cite{Keiper17} analyzed the performance of Boxed BP for reconstructing binary signals.\n\n\n\n\\subsection{Sum of Norms (SN)}\nWang et al. \\cite{Wang13} introduced the following optimization problem which combines the $\\ell_1$ and $\\ell_{\\infty}$ norms:\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in \\mathbb{R}^N}{\\text{min}}\n & & \\|\\mathbf{z}\\|_1 + \\lambda \\, \\| \\mathbf{z} - \\tfrac{1}{2} \\, \\textbf{1}_N \\|_{\\infty} && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}.\n\\end{aligned}\n\\end{equation*}\nMinimizing $\\|\\mathbf{z}\\|_1$ promotes sparsity of $\\mathbf{z}$ while minimizing $\\| \\mathbf{z} - \\frac{1}{2} \\, \\textbf{1}_N \\|_{\\infty}$ forces the entries $|z_i - \\frac{1}{2}|$ to be small and of equal magnitude (see Fig.~\\ref{fig:ell1-ellinfty}).\nThe two terms are balanced by a tuning parameter $\\lambda >0$.\n\n\n\n\n\n\n\n\\begin{figure}[t]\n \\centering\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\def8{8}\n\\filldraw [fill=blue!20, draw=blue!20] (4,0) -- (0,4) -- (-4,0) -- (0,-4) -- (4,0);\n\\draw[<->] (-9,0) -- (9,0) node[below] {$z_1$};\n\\draw[<->] (0,-9) -- (0,9) node[above] {$z_2$};\n\\draw[-, thick] (-0.5,-9) -- (8.5,9);\n\\draw (4,0) circle (2mm) [fill=black];\n\\node[below] at (5,-1) {\\scriptsize $(1,0)$};\n\\node[left] at (7,6) {\\scriptsize $\\Phi \\mathbf{z} = \\mathbf{y}$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\def8{8}\n\\filldraw [fill=blue!20, draw=blue!20] (4,0) -- (4,4) -- (0,4) -- (0,0) -- (4,0);\n\\draw[<->] (-9,0) -- (9,0) node[below] {$z_1$};\n\\draw[<->] (0,-9) -- (0,9) node[above] {$z_2$};\n\\draw[-, thick] (-0.5,-9) -- (8.5,9);\n\\draw (4,0) circle (2mm) [fill=black];\n\\node[below] at (5,-1) {\\scriptsize $(1,0)$};\n\\draw (2,2) circle (2mm) [fill=black];\n\\node[left] at (2.2,2) {\\scriptsize $(\\frac{1}{2},\\frac{1}{2})$};\n\\node[left] at (7,6) {\\scriptsize $\\Phi \\mathbf{z} = \\mathbf{y}$};\n\\end{tikzpicture}\n \\caption{Left: the minimization of $\\|\\mathbf{z}\\|_1$ finds sparse solutions. Right: the minimization of $\\| \\mathbf{z} - \\frac{1}{2} \\cdot \\textbf{1}_N \\|_{\\infty}$ forces the entries $|z_i - \\frac{1}{2}|$ to be small and of equal magnitude.}\n \\label{fig:ell1-ellinfty}\n\\end{figure}\n\n\n\\begin{figure}[t]\n \\centering\n\\begin{tikzpicture}[scale=0.2]\n\\def9{9}\n\\draw[<->] (-5,0) -- (9,0) node[below] {$t$};\n\\draw[<->] (0,-1) -- (0,9);\n\\draw[-, thick] (-5,6.5) -- (0,1.5);\n\\draw[-, thick] (0,1.5) -- (6,4.5);\n\\draw[-, thick] (6,4.5) -- (9,7.5);\n\\draw (6,4.5) circle (2mm) [fill=black];\n\\draw (0,1.5) circle (2mm) [fill=black];\n\\node[left] at (0,1.5) {\\scriptsize $(0,p)$};\n\\node[right] at (6,4.5) {\\scriptsize $(1,1-p)$};\n\\end{tikzpicture}\n\\caption{The function $f$ given in (\\ref{eq:ftn_f}).}\n \\label{fig:SAV}\n\\end{figure}\n\n\n\\subsection{Sum of Absolute Values (SAV)}\n\nNagahara \\cite{Nagahara15} proposed the following method for reconstruction of discrete signals whose entries are chosen independently from a set of finite alphabets $\\alpha = \\{\\alpha_1, \\alpha_2, \\dots, \\alpha_L\\}$ with a priori known probability distribution. In the special case $\\alpha = \\{0, 1\\}$ of binary signals, SAV is formulated as,\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & (1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1 && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y},\n\\end{aligned}\n\\end{equation*}\nwhere $p = \\mathbb{P}(x_j = 1)$, $j \\in [N]$, is the probability distribution of the entries of $\\mathbf{x}$.\nIf $p \\approx 0$, i.e., if $\\mathbf{x}$ is sparse, then $(1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1 \\approx \\| \\mathbf{z} \\|_1$ so that SAV performs similar to BP. We note that\n\\begin{align*}\n(1- p) \\, \\| \\mathbf{z} \\|_1 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_1\n= \\sum_{i=1}^N f(z_i),\n\\end{align*}\nwhere\n\\begin{align}\n\\label{eq:ftn_f}\nf(t) :=\n\\begin{cases}\n - t + p & \\text{if~} t < 0, \\\\\n (1-2p) \\, t + p & \\text{if~} 0 \\leq t < 1, \\\\\n t - p & \\text{if~} t \\geq 1 .\n \\end{cases}\n\\end{align}\n\n\n\\section{Box-Constrained Sum of Smoothed $\\ell_0$}\n\\label{sec:BSSL0}\nL0 and SL0 utilize the $\\ell_0$ norm and its smoothed version respectively, however, they do not take into account that $\\mathbf{x}$ is binary.\nOn the other hand, Boxed BP, SN, and SAV utilize the $\\ell_1$ norm in one way or another and are specifically adjusted to the binary setting.\nA natural question arises: Can we achieve a better recovery rate for binary signals by adjusting L0 and SL0 to the binary setting?\n\nWe note that Boxed BP takes into account the binariness of $\\mathbf{x}$ by imposing the restriction $\\mathbf{x} \\in [0,1]^N$. It is straightforward to apply this trick to L0 and SL0, and we will call the resulting algorithms \\emph{Boxed L0} and \\emph{Boxed SL0} respectively.\nBoxed L0 is still NP-hard like L0, but Boxed SL0 shows a clear improvement over SL0 while requiring a similar amount of run time (Fig.~\\ref{fig:Lt}). However, the recovery rate of Boxed SL0 is significantly worse than Boxed BP or SN.\n\nIn this paper, we aim to adapt the SAV method and the restriction $\\mathbf{x} \\in [0,1]^N$ to SL0, in order to achieve a better performance.\nA straightforward adaptation leads to the following formulation. For $\\sigma > 0$ small,\n\\begin{equation}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in [0,1]^N}{\\text{min}}\n & & F_{\\sigma}(\\mathbf{z}) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y}, \\label{eq:straightforward_adapt}\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\n& F_{\\sigma}(\\mathbf{z}) \\triangleq (1-p) \\sum_{i = 1}^N \\left (1 - e^{-z_i^2\/(2 \\sigma^2)} \\right ) \\\\\n& \\quad \\quad \\quad + p \\sum_{i = 1}^N \\left ( 1 - e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) \\\\\n& = \\sum_{i = 1}^N \\left ( 1 - (1-p) \\, e^{-z_i^2\/(2 \\sigma^2)} - p \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) \\label{eq:SWl0}\n\\end{split}\n\\end{equation}\nand $p = \\mathbb{P}(x_j = 1),~\\forall j \\in [N]$.\nNote that by (\\ref{eq:SL0convergence}), we have\n\\[ \\lim_{\\sigma \\to 0} F_{\\sigma}(\\mathbf{z}) =(1 - p) \\, \\| \\mathbf{z} \\|_0 + p \\, \\| \\mathbf{z} - \\textbf{1}_N \\|_0 \\] so that $F_0 (\\mathbf{z})$ can be approximated by $F_{\\sigma}(\\mathbf{z})$ with small $\\sigma > 0$.\n\n\nNext, we will use a weight function to incorporate the restriction $\\mathbf{z} \\in [0,1]^N$ into the function $F_{\\sigma}(\\mathbf{z})$. For integers $k \\geq 1$, let\n\\begin{align*}\n&\\begin{split}\nw_{k}(t) & \\triangleq \\begin{cases}\n 1 & \\text{if~} 0 \\leq t \\leq 1 \\\\\n k & \\text{otherwise}.\n\t\\end{cases}\n \t\\end{split} \\\\\n \\end{align*}\n\n \n \nFor $\\sigma > 0$ and integers $k \\geq 1$, we define\n\\begin{align*}\n& F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) \\\\\n&\\triangleq \\sum_{i = 1}^N w_{k}(z_i) \\left ( 1 - (1-p) \\, e^{-z_i^2\/(2 \\sigma^2)} - p \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right ) . \\nonumber\n\\end{align*}\nNote that since $1 - (1-p) \\, e^{-t^2\/(2 \\sigma^2)} - p \\, e^{-(t-1)^2\/(2 \\sigma^2)} > 0$ for all $t \\in \\mathbb{R}$, minimizing $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ forces $w_k(z_i)$ to be small so that all $z_i$'s lie within $[0,1]$.\nIn this way, the restriction $\\mathbf{z} \\in [0,1]^N$ is incorporated into the cost function.\nOur optimization problem now reads as follows:\nFor $\\sigma > 0$ small and $k \\in \\mathbb{N}$ large,\n\\begin{equation*}\n\\begin{aligned}\n& \\underset{\\mathbf{z} \\in {\\mathbb{R}}^N}{\\text{min}}\n & & F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) && \\text{subject to} & & \\Phi \\mathbf{z} = \\mathbf{y} .\n\\end{aligned}\n\\end{equation*}\nTo solve this problem, we propose an algorithm which is based on the gradient descent method and is implemented similarly as algorithms in \\cite{Mohimani09,Mohammadi14}.\nA major difference in our algorithm is that the cost function $\\sum_{i = 1}^N \\big( 1 - \\exp \\big(\\frac{- z_i^2}{2 \\sigma^2} \\big) \\big)$ of SL0 is replaced with $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ which is designed specifically for binary signals by adapting the formulation of SAV \\cite{Nagahara15}.\n\n\n\n\t\\begin{algorithm}\n\t\t\\caption{Box-Constrained Sum of Smoothed $\\ell_0$ (BSSL0)}\n\t\t\t\t\\begin{algorithmic}[1]\n\t\t\t\\State \\textbf{Data:} Measurement matrix $\\Phi \\in \\mathbb{R}^{m \\times N}$, observation $\\mathbf{y} \\in \\mathbb{R}^m$, probability distribution prior $p = \\mathbb{P}(x_j = 1)$.\n\t\t\t\\State \\textbf{Parameters:} \n \n Iters and $L$ are the number of iterations in the outer and inner loops respectively, $\\mu$ is a step-size parameter for gradient descent, and $d$ is a decreasing factor for $\\sigma$.\n \n\t\t\t\\State \\textbf{Initialization:} $\\hat{\\mathbf{x}}=\\Phi^\\top(\\Phi \\Phi^\\top)^{-1} \\mathbf{y}$, $\\sigma = 2 \\max |\\hat{\\mathbf{x}}|$, \\\\ \\quad \n \n \n $k = 1 + N p\/\\text{Iters}$;\n\t\t\t\\For{$1 : \\text{Iters}$}\n\t\t\t\\For{$1 : L$}\n\t\t\t\\State $\\hat{\\mathbf{x}} \\leftarrow \\hat{\\mathbf{x}} - \\sigma^2 \\mu \\nabla F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$;\n\\quad \\emph{\\% gradient descent}\n\t\t\t\\State $\\hat{\\mathbf{x}} \\leftarrow \\hat{\\mathbf{x}} - \\Phi^\\top(\\Phi \\Phi^\\top)^{-1}(\\Phi \\hat{\\mathbf{x}} - \\mathbf{y})$; \\quad \\emph{\\% projection}\n\t\t\t\\EndFor\n\t\t\t\\State $\\sigma = \\sigma \\times d$;\n\t\t\n \\State$k = k + N p\/\\text{Iters}$;\n\t\t\t\\EndFor\n\t\t\t \\State $\\hat{\\mathbf{x}} \\leftarrow \\textbf{round}(\\hat{\\mathbf{x}})$; \\quad \\emph{\\% round to a binary vector}\n\t\t\\end{algorithmic}\n\t\t\\label{algorighm:BSSL0}\n\t\\end{algorithm}\t\t\n\nThe proposed algorithm is comprised of two nested loops. In the outer loop, we slowly decrease $\\sigma$ and iteratively search for an optimal solution from a coarse to a fine scale by decreasing $\\sigma$ by a factor of $0 < d < 1$. As $\\sigma$ decreases, we also gradually increase $k$ so that a larger penalty is put on solutions that have entries outside the range $[0,1]$.\nThe inner loop performs a gradient descent of $L$ iterations for the function $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$, where $\\sigma$ and $k$ are given from the outer loop. In each iteration of the gradient descent, the solution is projected into the set of feasible solutions $\\{ \\mathbf{z} : \\Phi \\mathbf{z} = \\mathbf{y} \\}$.\n\t\n\nNumerical experiments in Section \\ref{sec:Experiments} show that for binary signals the proposed algorithm outperforms all other algorithms (BP, Boxed BP, SN, SAV, SL0, and Boxed SL0).\n\n\nAs already mentioned, our algorithm is implemented similarly as SL0 \\cite{Mohimani09,Mohammadi14}. The parameters used in our algorithm are exactly the same as in \\cite{Mohimani09} except $k$ and $p$. As justified in \\cite[Section IV-B]{Mohimani09}, we set the initial estimate of $\\mathbf{x}$ as the minimum $\\ell_2$ norm solution of $\\Phi \\mathbf{z} = \\mathbf{y}$, i.e., $\\hat{\\mathbf{x}}=\\Phi^\\top(\\Phi \\Phi^\\top)^{-1} \\mathbf{y}$.\nThe initialization value for $\\sigma$ is discussed in \\cite[Remark 5 in Section III]{Mohimani09}. Also, the choice of the step-size $\\sigma^2 \\mu$ for gradient descent is justified in \\cite[Remark 2 in Section III]{Mohimani09} and the choice of $k$ in \\cite[Lemma 1]{Mohammadi14}.\n\t\n\nThe gradient of $F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})$ used in Algorithm \\ref{algorighm:BSSL0} is given by\n\\begin{align*}\n\\begin{split}\n \\nabla F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z}) &= \\left( \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_1}, \\dots, \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_N} \\right)^\\top ,\n \\end{split}\n\\end{align*}\nwhere\n\\begin{align*}\n \\frac{\\partial F_{\\sigma, k}^{\\text{boxed}}(\\mathbf{z})}{\\partial z_i} &= \\frac{w_{k}(z_i)}{\\sigma^2} \\left ((1-p) \\, z_i \\, e^{-z_i^2\/(2 \\sigma^2) } \\right. \\\\\n & \\qquad \\left. + \\; p \\, (z_i -1) \\, e^{-(z_i-1)^2\/(2 \\sigma^2)} \\right )\n\\quad a.e.\n\\end{align*}\nThis is derived using the fact that $w_{k}'(t) = 0$ for all $t$ except $t = 0,1$; we have set $w_{k}'(0) = w_{k}'(1) = 0$ in the implementation.\nLet us point out that the discontinuity of $w_{k}(t)$ at $t = 0,1$ does not deteriorate the performance of gradient descent.\nOne can replace the function $w_{k}(t)$ with a smooth function, however, at the cost of increased run time.\n\n\n\\section{Numerical Experiments} \\label{sec:Experiments}\n\nIn this section, we compare the performance of our algorithm BSSL0 with other CS\/BCS algorithms described in Section \\ref{sec:BCS}.\nThe MATLAB codes for the experiments are available in \\cite{myCode}.\n\n\n\n\\subsection{Experiment 1: Binary Sparse Signal Reconstruction}\n\nIn this experiment, we tested BSSL0 with randomly generated binary signals and compared it with other CS\/BCS algorithms. \nRandom Gaussian matrices are considered for the measurement matrix $\\Phi \\in \\mathbb{R}^{40 \\times 100}$, that is, all entries of $\\Phi$ are drawn independently from the standard normal distribution.\nThe parameter $p$ is varied from $0$ to $1$ by step-size $0.05$, and a binary signal $\\mathbf{x} \\in \\{ 0 , 1 \\}^{100}$ is generated by drawing its entries independently with $\\mathbb{P}(x_i = 1) = p$ and $\\mathbb{P}(x_i = 0) = 1 -p$. For $\\Phi$ and $\\mathbf{x}$, we compute the measurement vector $\\mathbf{y} = \\Phi \\mathbf{x}$ and run the respective algorithms introduced in section II (BP, Boxed BP, SN, SAV, SL0, Boxed SL0, and BSSL0) to obtain a solution vector $\\mathbf{z}$ as a approximated reconstruction of $\\mathbf{x}$. Additionally, we consider the Orthogonal Matching Pursuit (OMP) \\cite{Tropp2007} which is a fast greedy algorithm for sparse signal reconstruction.\nThe following are considered for the performance evaluation: (i) \\textbf{Failure of Perfect Reconstruction (FPR)}: $0$ if $\\mathbf{z} = \\mathbf{x}$ (successfully recovered the signal perfectly) and $1$ if $\\mathbf{z} \\neq \\mathbf{x}$ (failed to recover perfectly); (ii) \\textbf{Noise Signal Ratio (NSR)}: NSR = $\\frac{\\| \\mathbf{x} - \\mathbf{z} \\|_2}{\\|\\mathbf{x}\\|_2}$; (iii) \\textbf{Run time}.\nFor each $p$, experiments are repeated $10,000$ times and the results are averaged. For SN, we set the parameter $\\lambda$ to be $100$ as fine-tuned in \\cite{Wang13}. For BSSL0, we set $\\sigma_{\\text{min}} = 0.1$, $d = 0.5$, $\\mu = 2$, and $L = 1000$.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=1\\linewidth]{result.eps}\n\\caption{Results for Experiment 1.}\n\\label{fig:Lt}\n\\end{figure}\nIn Fig.~\\ref{fig:Lt}, BSSL0 shows a better recovery rate than other CS\/BCS algorithms and also shows a run time comparable to SL0.\n\n\n\\subsection{Experiment 2: Bitonal Image Reconstruction}\n\nAs in \\cite{Nagahara15}, we considered reconstruction of the $37\\times 37$-pixel bitonal image given in Fig.~\\ref{fig:img_orig} (left).\nFollowing the same setup in \\cite{Nagahara15}, we added to each pixel a random Gaussian noise with mean-zero and standard deviation of $0.1$, as shown in Fig.~\\ref{fig:img_orig} (right).\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{img_orig.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_noise.eps}\n\\caption{Original image (left) and the image corrupted by Gaussian noise (right).}\n\\label{fig:img_orig}\n\\end{figure}\n\n\\noindent\nThe noisy image is represented by a real-valued $37 \\times 37$ matrix $X$\nand we apply the discrete Fourier transform (DFT) to obtain\n\\[\n\\hat{X} = WXW \\;\\; \\in \\mathbb{C}^{37 \\times 37} ,\n\\]\nequivalently,\n\\[\n\\mathrm{vec}(\\hat{X}) = (W\\otimes W) \\, \\mathrm{vec}(X) \\;\\; \\in \\mathbb{C}^{1369} ,\n\\]\nwhere $W = [ \\omega^{k, \\ell} ]_{k,\\ell =0}^{K-1}$ with $K=37$ and $\\omega = e^{-2 \\pi i \/ K}$ is the $K$-point DFT matrix.\nAs in \\cite{Nagahara15}, we randomly subsampled $\\mathrm{vec}(\\hat{X}) \\in \\mathbb{C}^{1369}$ to obtain a half-sized vector $\\mathbf{y} \\in \\mathbb{C}^{685}$ and set the measurement matrix $\\Phi$ as the corresponding $685 \\times 1369$ submatrix of $W\\otimes W$.\nFig.~\\ref{fig:reconstruction} shows the reconstructed images by BP, SN, SAV, and BSSL0, all with entrywise rounding off to $\\{ 0 , 1 \\}$.\nFor SN, an optimal tuning parameter $\\lambda$ was searched from $50$ to $1000$ by stepsize $50$ and the value $\\lambda = 800$ was chosen. For SAV and BSSL0, as in \\cite{Nagahara15}, we chose the parameter $p = \\mathbb{P}(x_j = 0) = 0.5$ as a rough estimate for the sparsity of the bitonal image (see \\cite{Nagahara15}).\nWe set $\\sigma_\\text{min} = 0.01$, $d = 0.9$, $\\mu = 2$, and $L = 3$ for the parameters of BSSL0.\nThe respective run time for BP, SN, SAV, and BSSL0 are also given in Tab.~\\ref{tab:timesonsumption}.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\linewidth]{img_BP.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_SN.eps}~ \\\\\n\\includegraphics[width=0.48\\linewidth]{img_SAV.eps}~\n\\includegraphics[width=0.48\\linewidth]{img_BSSL0.eps}\n\\caption{Reconstructed images by BP (upper left), SN (upper right), SAV (lower left), and the proposed method BSSL0 (lower right).}\n\\label{fig:reconstruction}\n\\end{figure}\n\\begin{table}[H]\n\\caption{The Run Time Comparison}\n\\begin{center}\n\\label{tab:timesonsumption}\n \\begin{tabular}{ | l | l | l | l | }\n \\hline\n \\text{Algorithm} & Run Time\\\\ \\hline\n Basis Pursuit & 185.2044 seconds \\\\ \\hline\n SN & 406.1007 seconds \\\\ \\hline\n SAV & 191.5366 seconds \\\\ \\hline\n \\textbf{BSSL0} (proposed) & \\textbf{0.92577 seconds} \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this work, we proposed a fast algorithm (BSSL0) for reconstruction of binary signals which is based on the gradient descent method and smooth relaxation techniques. We showed that for binary signals our algorithm outperforms other CS\/BCS methods in terms of the recovery rate and speed. Future work includes a detailed analysis of BSSL0 in stability\/robustness and extensions to ternary and finite alphabet signals.\n\n\n\\section*{Acknowledgment}\nT.~Liu and D.~G.~Lee acknowledge the support of the DFG Grant PF 450\/6-1. The authors are grateful to Robert Fischer and G\\\"otz E.~Pfander for their helpful suggestions. The authors thank anonymous reviewers for their comments.\n\n\n\n\n\n\n\n\\IEEEpeerreviewmaketitle\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\subsection{Different methods to determine the density}\nThe density sets a crucial scale for our problem. Its precise determination is mandatory for quantitative precision. We will discuss two different methods for its determination and show that the results agree within our precision. For $T=0$, we also find agreement with the Ward identity $n=\\rho_0$.\n\nThe first method is to derive flow equations for the density. This has the advantage that the occupation numbers for a given momentum $\\vec{p}$ are mainly sensitive to running couplings with $k^2=\\vec{p}^2$. In the grand canonical formalism, the density is defined by\n\\begin{equation}\nn=-\\frac{\\partial}{\\partial \\mu}\\frac{1}{\\Omega}\\Gamma[\\varphi]{\\Big |}_{\\varphi=\\varphi_0,\\mu=\\mu_0}\n\\end{equation}\nWe can formally define a $k$-dependent density $n_k$ by\n\\begin{equation}\nn_k=-\\frac{\\partial}{\\partial \\mu}\\frac{1}{\\Omega}\\Gamma_k[\\varphi]{\\Big |}_{\\varphi=\\varphi_0,\\mu=\\mu_0}=-(\\partial_\\mu U)(\\rho_0,\\mu_0).\n\\end{equation}\nThe flow equation for $n_k$ is given in Eq.\\ \\eqref{eqFlowprescriptionnk} and the physical density obtains for $k=0$. The term $\\partial_\\mu \\zeta \\big{|}_{\\rho_0,\\mu_0}$ that enters Eq.\\ \\eqref{eqFlowprescriptionnk} is the derivative of the flow equation \\eqref{eqFlowpotentialMatrix} for $U$ with respect to $\\mu$. To compute it, we need the $\\mu$-dependence of the propagator $G_k$ in the vicinity of $\\mu_0$. Within a systematic derivative expansion, we use the expansion of $U(\\rho,\\mu)$ and the kinetic coefficients $Z_1$ and $Z_2$ to linear order in $(\\mu-\\mu_0)$, as described in section \\ref{sec:Derivativeexpansionandwardidentities}. Here, $Z_1(\\rho,\\mu)$ and $Z_2(\\rho,\\mu)$ are the coefficient functions of the terms linear in the $\\tau$-derivative and linear in $\\Delta$, respectively. \nNo reasonable qualitative behavior is found, if the linear dependence of $Z_1$ and $Z_2$ on $(\\mu-\\mu_0)$ is neglected. Also, the scale dependence of $\\alpha$ and $\\beta$ are quite important. The flow equations for $\\alpha$ and $\\beta$ can be obtained directly by differentiating the flow equation of the effective potential with respect to $\\mu$ and $\\rho$, cf. Eq.\\ \\eqref{eqflowalphabeta}. The situation is more complicated for the kinetic coefficients $Z_1^{(\\mu)}=\\partial_\\mu Z_1(\\rho_0,\\mu_0)$ and $Z_2^{(\\mu)}=\\partial_\\mu Z_2(\\rho_0,\\mu_0)$. Their flow equations have to be determined by taking the $\\mu$-derivative of the flow equation for $Z_1(\\rho, \\mu)$ and $Z_2(\\rho,\\mu)$. As discussed in section \\ref{sec:Derivativeexpansionandwardidentities}, we use in this paper the approximation $Z_1^{(\\mu)}=Z_2^{(\\mu)}=2V=2V_1(\\rho_0,\\mu_0)$.\n\nAs a check of both our method and our numerics, we also use another way to determine the particle density. This second method is more robust with respect to shortcomings of the truncation, but less adequate for high precision calculations as needed e.g. to determine the condensate depletion. The second method determines the pressure $p=-U(\\rho_0,\\mu_0)$ as a function of the chemical potential $\\mu_0$. Here, the effective potential is normalized by $U(\\rho_0=0,\\mu_0)=0$ at $T=0$, $n=0$. The flow of the pressure can be read of directly from the flow equation of the effective potential and is independent of the couplings $\\alpha$ and $\\beta$. We calculate the pressure as a function of $\\mu$ and determine the density $n=\\frac{\\partial}{\\partial \\mu}p$ by taking the $\\mu$-derivative numerically. It turns out that $p$ is in very good approximation given by $p=c\\,\\mu^2$, where the constant $c$ can be determined from a numerical fit. The density is thus linear in $\\mu$. \n\nAt zero temperature and for $\\tilde{v}=0$, we can additionally use the Ward identities connected to Galilean symmetry, which yield $n=\\rho_0$. We compare our methods in figure \\ref{densitycompared} and find that they give numerically the same result. We stress again the importance of a reliable method to determine the density, since we often rescale variables by powers of the density to obtain dimensionless variables.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig5.eps}\n\\caption{Pressure and density as a function of the chemical potential at $T=0$. We use three different methods: $n=-\\partial_\\mu U_{\\text{min}}$ from the flow equation (triangles), $n=\\rho_0$ as implied by Galilean symmetry (stars) and $n=\\partial_\\mu p$, where the pressure $p=-U$ (boxes) was obtained from the flow equation and phenomenologically fitted by $p=56.5 \\mu^2$ (solid lines). Units are arbitrary and we use $a=3.4\\cdot 10^{-4}$, $\\Lambda=10^3$.}\n\\label{densitycompared}\n\\end{figure}\n\n\\subsection{Quantum depletion of condensate}\nWe want to split the density into a condensate part $n_c$ and a density for uncondensed particles or \"depletion density\" $n_d=n-n_c$. For our model the condensate density is given by the \"bare\" order parameter\n\\begin{equation}\nn_c=\\bar{\\rho}_0=\\bar{\\rho}_0(k=0).\n\\end{equation}\nIn order to show this, we introduce occupation numbers $n(\\vec{p})$ for the modes with momentum $\\vec{p}$ with normalization\n\\begin{equation}\n\\int_{\\vec{p}}n(\\vec{p})=n.\n\\end{equation}\nOne formally introduces a $\\vec{p}$ dependent chemical potential $\\mu(\\vec{p})$ in the grand canonical partition function\n\\begin{equation}\ne^{-\\Gamma_{\\text{min}}[\\mu]}=\\text{Tr}e^{-\\beta(H-\\Omega_3\\int_{\\vec{p}}\\mu(\\vec{p})n(\\vec{p}))},\n\\end{equation}\nwith three dimensional volume $\\Omega_3=\\int_{\\vec{x}}$. Then one can define the occupation numbers by\n\\begin{equation}\nn(\\vec{p})=-\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma[\\varphi,\\mu(\\vec{p})]{\\Big |}_{\\varphi=\\varphi_0,\\mu(\\vec{p})=\\mu_0}.\n\\end{equation}\nThis construction allows us to use $k$-dependent occupation numbers by the definition\n\\begin{equation}\nn_k(\\vec{p})=-\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma_k[\\varphi,\\mu]{\\Big |}_{\\varphi=\\varphi_0,\\mu(\\vec{p})=\\mu_0}.\n\\end{equation}\nOne can derive a flow equation for this occupation number $n_k(\\vec{p})$ \\cite{WetterichOccupationNumbers}:\n\\begin{eqnarray}\n\\nonumber\n\\partial_k n_k(\\vec{p}) &=& -\\frac{1}{2}\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\text{Tr}\\left\\{(\\Gamma^{(2)}+R_k)^{-1}\\partial_k R_k\\right\\}\\\\\n& & +\\frac{\\partial}{\\partial \\bar \\rho}\\frac{\\delta}{\\delta \\mu(\\vec{p})}\\frac{1}{\\beta \\Omega_3}\\Gamma[\\varphi,\\mu](\\partial_k \\bar\\rho_0).\n\\label{flowofnp}\n\\end{eqnarray}\n\nWe split the density occupation number into a $\\delta$-distribution like part and a depletion part, which is regular in the limit $\\vec{p}\\rightarrow0$\n\\begin{equation}\nn_k(\\vec{p})=n_{c,k}\\,\\delta(\\vec{p})+n_{d,k}(\\vec{p}).\n\\end{equation}\nOne can see from the flow equation for $n_k(\\vec{p})$ that the only contribution to $\\partial_k n_{c,k}$ comes from the second term in equation \\eqref{flowofnp}. Within a more detailed analysis \\cite{WetterichOccupationNumbers} one finds\n\\begin{equation}\n\\partial_k n_{c,k}=\\partial_k\\bar{\\rho}_{0,k}.\n\\end{equation}\nWe therefore identify the condensate density with the bare order parameter\n\\begin{equation}\nn_c=\\bar{\\rho}_0=\\frac{\\rho_0}{\\bar{A}}=\\bar{\\varphi}_0^2.\n\\end{equation}\nCorrespondingly, we define the $k$-dependent quantities\n\\begin{equation}\nn_{c,k}=\\bar{\\rho}_{0,k},\\quad n_k=n_{c,k}+n_{d,k}\n\\end{equation}\nand compute $n_d=n_d(k=0)$ by a solution of its flow equation. \n\nEven at zero temperature, the repulsive interaction connected with a positive scattering length $a$ causes a portion of the particle density to be outside the condensate. From dimensional reasons, it is clear, that $n_d\/n=(n-n_c)\/n$ should be a function of $an^{1\/3}$. The prediction of Bogoliubov theory or, equivalently, mean field theory, is $n_d\/n=\\frac{8}{3\\sqrt{\\pi}}(an^{1\/3})^{3\/2}$. We may determine the condensate depletion from the solution to the flow equation for the particle density, $n=n_{k=0}$, and $n_c=\\bar{\\rho}_0=\\bar{\\rho}_0(k=0)$. \n\nFrom Galilean invariance for $T=0$ and $\\tilde{v}=0$, it follows that\n\\begin{equation}\n\\frac{n_d}{n}=\\frac{\\rho_0-\\bar{\\rho}_0}{\\rho_0}=1-\\frac{1}{\\bar{A}},\n\\end{equation}\nwith $\\bar{A}=\\bar{A}(k=0)$. This gives an independent determination of $n_c$. \nIn figure \\ref{figDepletiond3} we plot the depletion density obtained from the flow of $n$ and $\\bar{\\rho}_0$ over several orders of magnitude. Apart from some numerical fluctuations for small $an^{1\/3}$, we find that our result is in full agreement with the Bogoliubov prediction.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig6.eps}\n\\caption{Condensate depletion $(n-n_c)\/n$ as a function of the dimensionless scattering length $a n^{1\/3}$. For the solid curve, we vary $a$ with fixed $n=1$, for the dashed curve we vary the density at fixed $a=10^{-4}$. The dotted line is the Bogoliubov-Result $(n-n_c)\/n=\\frac{8}{3\\sqrt{\\pi}}(a n^{1\/3})^{3\/2}$ for reference. We find perfect agreement of the three determinations. The fluctuations in the solid curve for small $a n^{1\/3}$ are due to numerical uncertainties. Their size demonstrates our numerical precision.}\n\\label{figDepletiond3}\n\\end{figure}\n\n\\subsection{Quantum phase transition}\n\\label{ssec:Quantumphasdiagram}\nFor $T=0$ a quantum phase transition separates the phases with $\\rho_0=0$ and $\\rho_0>0$.\nIn this section, we investigate the phase diagram at zero temperature in the cube spanned by the dimensionless parameters $\\tilde{\\mu}=\\frac{\\mu}{\\Lambda^2}$, $\\tilde{a}=a\\Lambda$ and $\\tilde{v}=\\frac{V_\\Lambda}{S_\\Lambda^2}\\Lambda^2$. This goes beyond the usual phase transition for nonrelativistic bosons, since we also include a microscopic second $\\tau$-derivative $\\sim\\tilde{v}$, and therefore models with a generalized microscopic dispersion relation.\nFor non-vanishing $\\tilde{v}$ (i.e. for a nonzero initial value of $V_1$ with $V_2=V_3=0$ in section \\ref{sec:Derivativeexpansionandwardidentities}), the Galilean invariance at zero temperature is broken explicitly. For large $\\tilde{v}$, we expect a crossover to the \"relativistic\" $O(2)$ model. If we send the initial value of the coefficient of the linear $\\tau$-derivative $S_\\Lambda$ to zero, we obtain the limiting case $\\tilde{v}\\rightarrow\\infty$. The symmetries of the model are now the same as those of the relativistic O(2) model in four dimensions. The space-time-rotations or Lorentz symmetry replace Galilean symmetry. \n\nIt is interesting to study the crossover between the two cases. Since our cutoff explicitly breaks Lorentz symmetry, we investigate in this paper only the regime $\\tilde{v}\\lesssim1$. Detailed investigations of the flow equations for $\\tilde{v}\\rightarrow\\infty$ can be found in the literature \\cite{Papenbrock:1994kf, Berges2000ew, PhysRevA.60.1442, PhysRevB.68.064421, PhysRevD.67.065004, Bervillier2007}. The phase diagram in the $\\tilde{\\mu}-\\tilde{v}$ plane with $\\tilde{a}=1$ is shown in figure \\ref{figQPTsigmava1}. The critical chemical potential first increases linearly with $\\tilde{v}$ and then saturates to a constant. The slope in the linear regime as well as the saturation value depend linearly on $\\tilde{a}$ for $\\tilde{a}<1$. \n\nAt $T=0$, the critical exponents are everywhere the mean field ones ($\\eta=0$, $\\nu=1\/2$). This is expected: It is the case for $\\tilde{v}=0$ \\cite{Wetterich:2007ba, Uzunov1981, SachdevQPT}, and for $\\tilde{v}=\\infty$ the theory is equivalent to a relativistic $O(2)$ model in $d=3+1$ dimensions. This is just the upper critical dimension of the Wilson-Fisher fixed point \\cite{PhysRevLett.28.240}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig7.eps}\n\\caption{Quantum phase diagram in the $\\tilde{\\mu}$-$\\tilde{v}$ plane for $\\tilde{a}=1$.}\n\\label{figQPTsigmava1}\n\\end{figure}\n\nFrom section 3 we know that for $\\tilde{v}=0$ the parameter $\\tilde{a}$ is limited to $\\tilde{a}<\\frac{3\\pi}{4}\\approx 2.356$. For $\\tilde{v}=0$ and a small scattering length $a\\rightarrow0$, a second order quantum phase transition divides the phases without spontaneous symmetry breaking for $\\mu<0$ from the phase with a finite order parameter $\\rho_0>0$ for $\\mu>0$. It is an interesting question, whether this quantum phase transition at $\\mu=0, \\tilde{v}=0$ also occurs for larger scattering length $a$. We find in our truncation that this is indeed the case for a large range of $a$, but not for $\\tilde{a}>1.55$. Here, the critical chemical potential suddenly increases to large positive values as shown in Fig. \\ref{figQPTsigmaofa}. For $\\tilde{v}>0$ this increase happens even earlier. (For a truncation with $V_1\\equiv0$, the phase transition would always occur at $\\mu=0$.) We plot the $\\tilde{\\mu}-\\tilde{a}$ plane of the phase diagram for different values of $\\tilde{v}$ in figure \\ref{figQPTsigmaofa}. The form of the critical line can be understood by considering the limits $\\tilde{v}\\rightarrow0$ as well as $\\tilde{a}\\rightarrow0$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig8.eps}\n\\caption{Quantum phase diagram in the $\\tilde{\\mu}$-$\\tilde{a}$ plane for $\\tilde{v}=1$ (dotted), $\\tilde{v}=0.01$ (dashed) and $\\tilde{v}=0$ (solid).}\n\\label{figQPTsigmaofa}\n\\end{figure}\n\nFor a fixed chemical potential, the order parameter $\\rho_0$ as a function of $a$ goes to zero at a critical value $a_c$ as shown in Fig. \\ref{figQPTrhoofa}. This happens in a continuous way and the phase transition is therefore of second order. For $\\mu\\rightarrow0$, we find $a_c=1.55 \\Lambda^{-1}$. We emphasize, however, that $a_c$ is of the order of the microscopic distance $\\Lambda^{-1}$. Universality may not be realized for such values, and the true phase transition may depend on the microphysics. For example, beyond a critical value for the repulsive interaction, the system may form a solid. Ultracold atom gases correspond to metastable states which may lose their relevance for $a\\rightarrow\\Lambda^{-1}$. For $\\tilde{v}>0$ and $\\mu\\ll\\Lambda^2$ the phase transition occurs for $a_c \\Lambda \\ll 1$ such that universal behavior is expected.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig9.eps}\n\\caption{Quantum phase transition for fixed chemical potential $\\mu=1$, with $\\Lambda=10^3$. The density $\\rho_0=n$ as a function of the scattering length $a$ goes to zero at a critical $a_c \\Lambda=1.55$, indicating a second order quantum phase transition at that point.}\n\\label{figQPTrhoofa}\n\\end{figure}\n\n\n\t\t\t\n\\subsection{Thermal depletion of condensate}\n\\label{ssec:Phasediagramatnonzerotemperature}\nSo far, we have only discussed the vacuum and the dense system at zero temperature. A non vanishing temperature $T$ will introduce an additional scale in our problem. For small $T\\ll n^{2\/3}$ we expect only small corrections. However, as $T$ increases the superfluid order will be destroyed. Near the phase transition for $T \\approx T_c$ and for the disordered phase for $T>T_c$, the characteristic behavior of the boson gas will be very different from the $T\\rightarrow0$ limit.\n\nFor $T>0$ the particle density $n$ receives a contribution from a thermal gas of bosonic (quasi-) particles. It is no longer uniquely determined by the superfluid density $\\rho_0$. We may write \n\\begin{equation}\nn=\\rho_0+n_T\n\\label{eqDensityatT}\n\\end{equation}\nand observe, that $n_T=0$ is enforced by Galilean symmetry only for $T=0, V_\\Lambda=0$. The heat bath singles out a reference frame, such that for $T>0$ Galilean symmetry no longer holds. In our formalism, the thermal contribution $n_T$ appears due to modifications of the flow equations for $T\\neq0$. We start for high $k$ with the same initial values as for $T=0$. As long as $k\\gg \\pi T$ the flow equations receive only minor modifications. For $k \\approx \\pi T$ or smaller, however, the discreteness of the Matsubara sum has important effects. We plot in Fig. \\ref{fignoftemperature} the density as a function of $T$ for fixed $\\mu=1$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig10.eps}\n\\caption{Density $n\/{\\mu^{2\/3}}$ (solid) and order parameter $\\rho_0\/{\\mu^{2\/3}}$ (dashed) as a function of the temperature $T\/\\mu$. The units are arbitrary with $a=2\\cdot 10^{-4}$ and $\\Lambda=10^3$. The plot covers only the superfluid phase. For higher temperatures, the density is given by the thermal contribution $n=n_T$ only.}\n\\label{fignoftemperature}\n\\end{figure}\n\nIn Fig. \\ref{fignofsigma} we show $n(\\mu)$, similar to Fig. \\ref{densitycompared}, but now for different $a$ and $T$. For $T=0$ the scattering length sets the only scale besides $n$ and $\\mu$, such that by dimensional arguments $a^2\\mu=f(a^3 n)$. Bogoliubov theory predicts \n\\begin{equation}\nf(x)=8\\pi x(1+\\frac{32}{3\\sqrt{\\pi}}x^{1\/2}).\n\\end{equation}\nThe first term on the r.h.s. gives the contribution of the ground state, while the second term is added by fluctuation effects. For small scattering length $a$, the ground state contribution dominates. We have then $\\mu\\sim a$ for $n=1$ and $\\mu\/n$ can be treated as a small quantity. For $T\\neq 0$ and small $a$ one expects $\\mu=g(T\/n^{2\/3})an$. The curves in Fig. \\ref{fignofsigma} for $T=1$ show that the density, as a function of $\\mu$, is below the curve obtained at $T=0$. This is reasonable, since the statistical fluctuations now drive the order parameter $\\rho_0$ to zero. At very small $\\mu$, the flow enters the symmetric phase. The density is always positive, but for simplicity, we show the density as a function of $\\mu$ in figure \\ref{fignofsigma} only in those cases, where the flow remains in the phase with spontaneous $U(1)$ symmetry breaking. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig11.eps}\n\\caption{Density $n$ for different temperatures and scattering length. We plot $n(\\mu)$ in arbitrary units, with $\\Lambda=10^3$, and for a scattering length $a=2\\cdot10^{-4}$ (solid and dotted), $a=10^{-4}$ (dashed and dashed-dotted). The temperature is $T=0$ (solid and dashed) and $T=1$ (dotted and dashed-dotted).}\n\\label{fignofsigma}\n\\end{figure}\n\nFor temperatures above the critical temperature, the order parameter $\\rho_0$ vanishes at the macroscopic scale and so does the condensate density $n_c=\\bar{\\rho_0}=\\frac{1}{\\bar{A}}\\rho_0$. The density is now given by a thermal distribution of particles with nonzero momenta. \nUp to small corrections from the interaction $\\sim aT$, it is described by a free Bose gas, \n\\begin{equation}\nn= \\frac{T^{3\/2}}{(4\\pi)^{3\/2}}g_{3\/2}(e^{\\beta \\mu}),\n\\end{equation}\nwith the \"Bose function\"\n\\begin{equation}\ng_p(z)=\\frac{1}{\\Gamma(p)}\\int_0^\\infty dx\\,x^{p-1}\\frac{1}{z^{-1}e^x-1}.\n\\end{equation}\n\nIn figure \\ref{figrhooftemperature} we show the dimensionless order parameter $\\rho_0\/n$ as a function of the dimensionless temperature $T\/n^{2\/3}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig12.eps}\n\\caption{Order parameter $\\rho_0\/n$ as a function of the dimensionless temperature $T\/(n^{2\/3})$ for scattering length $a=10^{-4}$. Here, we varied $T$ keeping $\\mu$ fixed. Numerically, this is equivalent to varying $\\mu$ with fixed $T$.\\label{figrhooftemperature}}\n\\end{figure}\nThis plot shows the second order phase transition from the phase with spontaneous $U(1)$ symmetry breaking at small temperatures to the symmetric phase at higher temperatures. The critical temperature $T_c$ is determined as the temperature, where the order parameter just vanishes - it is investigated in the next section. Since we find $(\\bar{A}-1)\\ll1$, the condensate fraction $n_c\/n=\\bar{\\rho_0}\/n=\\rho_0\/(\\bar{A}n)$ as a function of $T\/n^{2\/3}$ resembles the order parameter $\\rho_0\/n$. We plot $\\bar{A}$ as a function of $T\/n^{2\/3}$ in Fig. \\ref{Abaroftemperature}. Except for a narrow region around $T_c$, the deviations from one remain indeed small. Near $T_c$ the gradient coefficient $\\bar{A}$ diverges according to the anomalous dimension, $\\bar{A}\\sim \\xi^\\eta$, with $\\eta$ the anomalous dimension. The correlation length $\\xi$ diverges with the critical exponent $\\nu$, $\\xi\\sim |T-T_c|^{-\\nu}$, such that\n\\begin{equation}\n\\bar{A}\\sim |T-T_c|^{-\\eta\\nu}.\n\\end{equation}\nHere, $\\eta$ and $\\nu$ are the critical exponents for the Wilson Fisher fixed point of the classical three-dimensional $O(2)$ model, $\\eta=0.0380(4)$, $\\nu=0.67155(27)$ \\cite{Pelissetto2002549, Berges2000ew, PhysRevA.60.1442, PhysRevB.68.064421, PhysRevD.67.065004, Bervillier2007}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig13.eps}\n\\caption{Order parameter divided by the condensate density $\\bar{A}=\\rho_0\/n_c$, as a function of the dimensionless temperature $T\/(n^{2\/3})$, and for scattering length $a=10^{-4}$. Here, we varied $T$ keeping $\\mu$ fixed. Numerically, this is equivalent to varying $\\mu$ with fixed $T$. The plot covers only the phase with spontaneous symmetry breaking. For higher temperatures, the symmetric phase has $\\rho_0=n_c=0$. The divergence of $\\bar{A}$ for $T\\rightarrow T_c$ reflects the anomalous dimension $\\eta$ of the Wilson-Fisher fixed point.}\n\\label{Abaroftemperature}\n\\end{figure}\n\nIn figure \\ref{figrhoofsigma} we plot $\\rho_0\/n$ as a function of the chemical potential $\\mu$ for different temperatures and scattering lengths. We find, that $\\rho_0\/n=1$ is indeed approached in the limit $T\\rightarrow0$, as required by Galilean invariance. All figures of this section are for $\\tilde{v}=0$. The modifications for $\\tilde{v}\\neq0$ are mainly quantitative, not qualitative. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig14.eps}\n\\caption{Order parameter divided by the density, $\\rho_0\/n$, as a function of the chemical potential. We use arbitrary units with $\\Lambda=10^3$. The curves are given for a scattering length $a=2\\cdot10^{-4}$ (solid and dotted), $a=10^{-4}$ (dashed and dashed-dotted) and temperature $T=0.1$ (solid and dashed) and $T=1$ (dotted and dashed-dotted). At zero temperature, Galilean invariance implies $\\rho_0=n$, which we find within our numerical resolution.}\n\\label{figrhoofsigma}\n\\end{figure}\n\n\\subsection{Critical temperature}\nThe critical temperature $T_c$ for the phase transition between the superfluid phase at low temperature and the disordered or symmetric phase at high temperature depends on the scattering length $a$. By dimensional reasoning, the temperature shift $\\Delta T_c=T_c(a)-T_c(a=0)$ obeys $\\Delta T_c\/T_c\\sim an^{1\/3}$. The proportionality coefficient cannot be computed in perturbation theory \\cite{Andersen:2003qj}. It depends on $\\tilde{v}$ and we concentrate here on $\\tilde{v}=0$. Monte-Carlo simulations in the high temperature limit, where only the lowest Matsubara frequency is included, yield $\\Delta T_c\/T_c=1.3 \\,a n^{1\/3}$ \\cite{PhysRevLett.87.120401, PhysRevLett.87.120402}. Within the same setting, renormalization group studies \\cite{PhysRevLett.83.1703, blaizot:051116, blaizot:051117, PhysRevA.69.061601, PhysRevA.70.063621} yield a similar result, for composite bosons see \\cite{Diehl:2007th}. Near $T_c$, the long wavelength modes with momenta $\\vec{p}^2\\ll(\\pi T)^2$ dominate the \"long distance quantities\". Then a description in terms of a classical three dimensional system becomes valid. This \"dimensional reduction\" is achieved by \"integrating out\" the nonzero Matsubara frequencies. \nHowever, both $\\Delta T_c\/T_c$ and $n$ are dominated by modes with momenta $\\vec{p}^2\\approx(\\pi T_c)^2$ such that corrections to the classical result may be expected.\n\nWe have computed $T_c$ numerically by monitoring the zero of $\\rho_0$, as shown in Fig. \\ref{figrhooftemperature}, $\\rho_0(T\\rightarrow T_c)\\rightarrow0$. Our result is plotted in Fig. \\ref{Tcofa}. In the limit $a\\rightarrow0$ we find for the dimensionless critical temperature $T_c\/(n^{2\/3})=6.6248$, which is in good agreement with the expected result for the free theory $T_c\/(n^{2\/3})=\\frac{4\\pi}{\\zeta(3\/2)^{2\/3}}=6.6250$. For the shift in $T_c$ due to the finite interaction strength, we obtain\n\\begin{equation}\n\\frac{\\Delta T_c}{T_c}=\\kappa \\,a n^{1\/3}, \\quad \\kappa=2.1.\n\\end{equation}\nWe expect that the result for $\\kappa$ depends on the truncation and may change somewhat if additional higher order couplings are included.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig15.eps}\n\\caption{Dimensionless critical temperature $T_c\/(n^{2\/3})$ as a function of the dimensionless scattering length $an^{1\/3}$ (points). We also plot the linear fit $\\Delta T_c\/T_c=2.1\\, a n^{1\/3}$ (solid line).}\n\\label{Tcofa}\n\\end{figure}\n\n\\subsection{Zero temperature sound velocity}\nThe macroscopic sound velocity $v_S$ is a crucial quantity for the hydrodynamics of the gas or liquid. It is accessible to experiment. As a thermodynamic observable, the adiabatic sound velocity is defined as\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}\\bigg{|}_s\n\\end{equation}\nwhere $M$ is the particle mass (in our units $1\/M=2$), $p$ is the pressure, $n$ is the particle density, and $s$ is the entropy per particle. It is related to the isothermal sound velocity $v_T$ by\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\left(\\frac{\\partial p}{\\partial n}\\bigg{|}_T+\\frac{\\partial p}{\\partial T}\\bigg{|}_n\\frac{\\partial T}{\\partial n}\\bigg{|}_s\\right)=v_T^2+2\\frac{\\partial p}{\\partial T}\\bigg{|}_n \\frac{\\partial T}{\\partial n}\\bigg{|}_s\n\\label{eqSoundAdiabaticIsothermal}\n\\end{equation}\nwhere we use our units $2M=1$. One needs the \"equation of state\" $p(T,n)$ and\n\\begin{equation}\ns(T,n)=\\frac{S}{N}=\\frac{1}{n}\\frac{\\partial p}{\\partial T}\\bigg{|}_\\mu.\n\\end{equation}\nBy dimensional analysis, one has\n\\begin{equation}\np=n^{5\/3}{\\cal F}(t,c), \\quad t=\\frac{T}{n^{2\/3}}, \\quad c=a n^{1\/3},\n\\end{equation}\nwith ${\\cal F}(0,c)=4\\pi c$ (in Bogoliubov theory), and ${\\cal F}(t,0)=\\frac{\\zeta(5\/2)}{(4\\pi)^{3\/2}}t^{5\/2}$ (free theory), such that for small $c$\n\\begin{equation}\n{\\cal F}=\\frac{\\zeta(5\/2)}{(4\\pi)^{3\/2}}t^{5\/2}+g(t)c.\n\\end{equation}\n\nAt zero temperature the second term in Eq.\\ \\eqref{eqSoundAdiabaticIsothermal} vanishes, such that $v_S=v_T$. For the isothermal sound velocity one has \n\\begin{equation}\nv_T^2=2\\frac{\\partial p}{\\partial n}\\bigg{|}_{T}=2 \\frac{\\partial p}{\\partial \\mu}\\bigg{|}_T\\left(\\frac{\\partial n}{\\partial \\mu}\\bigg{|}_T\\right)^{-1}.\n\\end{equation}\nWe can now use \n\\begin{equation}\n\\frac{\\partial p}{\\partial \\mu}\\big{|}_T=-\\frac{dU_{\\text{min}}}{d\\mu}=-\\partial_\\mu U(\\rho_0)=n\n\\end{equation}\nand infer\n\\begin{equation}\nv_T^2=2\\left(\\frac{\\partial\\, \\text{ln}\\,n}{\\partial \\mu}\\right)^{-1}.\n\\end{equation}\n\nOne may also define a microscopic sound velocity $c_S$, which characterizes the propagation of (quasi-) particles. At zero temperature, where we can perform the analytic continuation to real time, we can calculate the microscopic sound velocity from the dispersion relation $\\omega(p)$ (with $p=|\\vec{p}|$). In turn, the dispersion relation is obtained from the effective action by setting $\\text{det}(G^{-1})=0$, where $G^{-1}$ is the full inverse propagator. We perform the calculation explicitly at the end of section \\ref{sec:Derivativeexpansionandwardidentities} and find\n\\begin{equation}\nc_S^{-2}=\\frac{S^2}{2\\lambda\\rho_0}+V.\n\\end{equation}\n\nThe Bogoliubov result for the sound velocity is in our units\n\\begin{equation}\nc_S^2=2\\lambda\\rho_0=16\\pi an.\n\\end{equation}\nIn three dimensions, the decrease of $S$ is very slow and the coupling $V$ remains comparatively small even on macroscopic scales, cf. Fig. \\ref{figFlowKinetic}. We thus do not expect measurable deviations from the Bogoliubov result for the sound velocity at $T=0$. In figure \\ref{figSound}, we plot our result over several orders of magnitude of the dimensionless scattering length and, indeed, find no deviations from Bogoliubovs result.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig16.eps}\n\\caption{Dimensionless sound velocity $c_s\/(n^{1\/3})$ at zero temperature, as a function of the scattering length $an^{1\/3}$. Within the plot resolution the curves obtained by varying $a$ with fixed $n$, by varying $n$ with fixed $a$, and the Bogoliubov result, $c_s=\\sqrt{16\\pi}(an)^{1\/2}$, coincide.}\n\\label{figSound}\n\\end{figure}\n\nWe finally show that for $T=0$ the macroscopic and microscopic sound velocities are equal, $v_S=v_T=c_S$. For this purpose, we use \n\\begin{equation}\n\\frac{\\partial n}{\\partial \\mu}\\big{|}_T=-\\frac{d}{d\\mu}\\left(\\partial_\\mu U(\\rho_0)\\right)=-\\partial_\\mu^2U(\\rho_0)-\\partial_\\rho\\partial_\\mu U(\\rho_0)\\frac{d\\rho_0}{d\\mu}.\n\\end{equation}\nFrom the minimum condition $\\partial_\\rho U=0$, it follows\n\\begin{equation}\n\\frac{d\\rho_0}{d\\mu}=-\\frac{\\partial_\\rho\\partial_\\mu U}{\\partial_\\rho^2 U}=-\\frac{\\alpha}{\\lambda}.\n\\end{equation}\nCombining this with the Ward identities from section \\ref{sec:Derivativeexpansionandwardidentities}, namely $\\partial_\\mu^2 U=-2V\\rho_0$ and $\\alpha=\\partial_\\rho\\partial_\\mu U=-S$, valid at $T=0$, it follows that the macroscopic sound velocity equals the microscopic sound velocity\n\\begin{equation}\nv_S^2(T=0)=c_S^2.\n\\end{equation}\n\n\n\n\\chapter{Introduction}\n\\label{ch:Introduction}\n\\pagenumbering{arabic}\n\\input{Introduction}\n\n\n\\section{Flow equations to solve an integral}\n\\label{ch:Flowequationstosolveanintegral}\n\\input{Flowequationstosolveanintegral}\n\n\n\\section{Functional integral representation of quantum field theory}\n\\label{ch:Functionalintegralrepresentationofquantumfieldtheory}\n\\input{Functionalintegralrepresentationofquantumfieldtheory}\n\n\t\n\\chapter{The Wetterich equation}\n\\label{ch:TheWetterichequation}\n\\input{TheWetterichequation}\n\n\t\t\n\\chapter{Generalized flow equation}\n\\label{ch:Generalizedflowequation}\n\\input{Generalizedflowequation}\n\n\\chapter{Truncations}\n\\label{ch:Truncations}\n\\input{Truncations}\n\t\t\n\\chapter{Cutoff choices}\n\\label{ch:Cutoffchoices}\n\\input{Cutoffchoices}\n\n\\chapter{Investigated models}\n\\label{ch:Variousphysicalsystems}\n\\input{Variousphysicalsystems}\n\n\\chapter{Symmetries}\n\\label{ch:Symmetries}\n\\input{Symmetries}\n\n\\chapter{Truncated flow equations}\n\\label{ch:Truncationsandflowequations}\n\\input{Truncationsandflowequations}\n\n\n\\chapter{Few-body physics}\n\\label{ch:Few-bodyphysics}\n\\input{Fewbodyphysics}\n\n\t\\section{Repulsive interacting bosons}\n\t\\label{sec:Repulsiveinteractingbosons}\n\t\\input{Repulsiveinteractingbosons}\n\t\n\n\t\\section{Two fermion species: Dimer formation}\n\t\\label{sec:Twofermionspecies:Dimerformation}\n\t\\input{TwofermionspeciesDimerformation}\n\n\t\\section{Three fermion species: Efimov effect}\n\t\\label{sec:Threefermionspecies:ThomasandEfimoveffect}\n\t\\input{ThreefermionspeciesThomasandEfimoveffect}\n\n\n\\chapter{Many-body physics}\n\\label{ch:Many-bodyphysics}\n\n\t\\section{Bose-Einstein Condensation in three dimensions}\n\t\\label{sec:Bose-EinsteinCondensationinthreedimensions}\n\t\\input{BoseEinsteinCondensationinthreedimensions}\n\t\\input{Thermodyn}\n\t\t\n\t\\section{Superfluid Bose gas in two dimensions}\n\t\\label{sec:SuperfluidBosegasintwodimensions}\n\t\\input{SuperfluidBose2d}\n\n\t\t\t\n\t\\section{Particle-hole fluctuations and the BCS-BEC Crossover}\n\t\\label{sec:Particle-holefluctuationsandtheBCS-BECCrossover}\n\t\\input{ParticleHole}\n\t\t\t\n\t\\section{BCS-Trion-BEC Transition}\n\t\\label{sec:BCS-Trion-BECTransitionlong}\n\t\\input{BCS-Trion-BECTransitionlong}\n\t\n\\chapter{Conclusions}\n\\label{ch:Conclusions}\n\\input{Conclusions}\n\n\\begin{appendix}\n\\chapter{Some ideas on functional integration and probability}\n\\label{ch:Foundationsofquantumtheory:someideas}\n\\input{functionalprobabilities}\n\n\\chapter{Technical additions}\n\\label{ch:Technicaladditions}\n\n\\section{Flow of the effective potential for Bose gas}\n\\label{sec:FlowoftheeffectivepotentialforBosegas}\n\\input{FlowoftheeffectivepotentialforBosegas}\n\n\\section{Flow of the effective potential for BCS-BEC Crossover}\n\\label{sec:FlowoftheeffectivepotentialforBCSBECcrossover}\n\\input{FlowoftheeffectivepotentialforBCSBECcrossover}\n\n\\section{Hierarchy of flow equations in vacuum}\n\\label{sec:Hierarchyofflowequationsinvacuum}\n\\input{Hierarchyofflowequationsinvacuum}\n\n\\end{appendix}\n\n\n\\input{dissSFbib}\n\n\\chapter*{Danken...}\nm\\\"{o}chte ich ganz besonders Prof.\\ Dr.\\ Christof Wetterich f\\\"ur die hervorragende Betreuung und die ausgezeichnete Zusammenarbeit. Von den vielen Gespr\\\"achen und guten Diskussionen habe ich sehr profitiert.\\\\\n\nF\\\"ur tolle Zusammenarbeit und viele interessante Diskussionen danke ich auch ganz herzlich Michael Scherer, Richard Schmidt, Sergej Moroz, Dr.\\ Sebastian Diehl, Dr.\\ Hans-Christian Krahl, Dr.\\ Philipp Strack, Prof.\\ Dr.\\ Holger Gies, Prof.\\ Dr.\\ Jan Pawlowski, Prof.\\ Dr.\\ Markus Oberthaler, Prof.\\ Dr.\\ Selim Jochim, dessen Arbeitsgruppe sowie insbesondere auch allen Teilnehmern des Seminars ,,Kalter Quantenkaffee''. \\\\\n\nProf. Dr. Holger Gies danke ich auch f\\\"ur die bereitwillige \\\"{U}bernahme des Zweitgutachtens und der damit verbundenen M\\\"{u}hen.\\\\\n\nAnne Doster danke ich f\\\"ur die sehr sch\\\"{o}ne, gemeinsam verbrachte Zeit, willkommene und notwendige Ablenkung sowie viel Verst\\\"{a}ndniss und Unterst\\\"{u}tzung. Sehr dankbar f\\\"{u}r sehr Vieles bin ich auch meinen Eltern, Geschwistern und Freunden.\n\\end{document}\n\n\n\n\n\\subsubsection{From the lattice to field theory}\n\nOne way to approach the infinitely many degrees of freedom of a continuous field theory comes from a discrete lattice of space-time points. Consider a lattice of points\n\\begin{equation}\n\\vec x_{ijk} = a \\begin{pmatrix}i \\\\ j \\\\ k\\end{pmatrix}, \\quad i,j,k\\in \\mathbb{Z}\n\\end{equation}\nat times\n\\begin{equation}\nt_n = b\\, n, \\quad n\\in \\mathbb{Z}.\n\\end{equation}\nFor every set of indices $(ijk,n)$ the field $\\varphi(x_{ijk},t_n)$ has some value, e.~g. $\\varphi(x_{ijk},t_n)\\in \\mathbb{C}$ for a complex scalar. The partition sum, i.~e. the sum over all possible configurations weighted by the corresponding functional probability is given by\n\\begin{equation}\nZ = \\left( \\prod_{(ijk,n)\\in \\mathbb{Z}^4} \\int d \\varphi(\\vec x_{ijk},t_n)\\right)\\, e^{-S(\\varphi(\\vec x_{ijk},t_n))}.\n\\label{eq:PartitionfunctionSTlattice}\n\\end{equation}\nThe action $S$ depends on the values of $\\varphi(\\vec x_{ijk},t_n)$ for the different lattice points. Eq.\\ \\eqref{eq:PartitionfunctionSTlattice} describes a theory on a discrete space-time lattice. From the probabilities $e^{-S}$ we can calculate all sorts of expectation values, correlation functions and so on. \n\nOur theory becomes a continuum field theory in the limit where $a\\to 0$ and $b\\to 0$. The partition function reads then \n\\begin{equation}\nZ = \\lim_{a,b\\to0} \\left(\\prod_{(ijk,n)\\in \\mathbb{Z}^4}\\int d\\varphi(\\vec x_{ijk},t_n)\\right) e^{-S(\\varphi(\\vec x_{ijk},t_n))}.\n\\end{equation}\nThis can also be written as\n\\begin{equation}\nZ = \\int D\\varphi \\,e^{-S[\\varphi]}.\n\\end{equation}\nThe functional integral $\\int D\\varphi$ might be defined by the limiting procedure above. The microscopic action $S$ is now a functional of the field configuration $\\varphi(\\vec x, t)$, where space and time are now continuous, $(\\vec x, t)\\in \\mathbb{R}^4$. \n\n\n\\subsubsection{Expectation values, correlation functions}\n\nWith our formalism we aim for a statistical description of fields. Important concepts are expectation values of operators and correlation functions. For simplicity, we denote the field degrees of freedom by $\\tilde \\Phi_\\alpha$. The collective index $\\alpha$ labels both continuous degrees of freedom such as position or momentum and discrete variables such as spin, flavor or simply ``particle species''. The field $\\tilde \\Phi_\\alpha$ might consist of both bosonic and fermionic parts. The fermionic components are described by Grassmann numbers while the bosonic components correspond to ordinary ($\\mathbb{C}$) numbers. As an example we consider a theory with a complex scalar field $\\varphi$ and a fermionic complex two-component spinor $\\psi=(\\psi_1,\\psi_2)$. It is useful to decompose the complex scalar into real and imaginary parts\n\\begin{equation}\n\\varphi = \\frac{1}{\\sqrt{2}}(\\varphi_1+i \\varphi_2).\n\\end{equation}\nIn momentum space the Nambu spinor of fields reads\n\\begin{equation}\n\\Phi(q) = \\left( \\varphi_1(q), \\varphi_2(q), \\psi_1(q), \\psi_2(q), \\psi_1^*(-q), \\psi_2^*(-q)\\right)\n\\end{equation}\nThe index $\\alpha$ stands in this case for the momentum $q$ and the position in the Nambu-spinor, e.q. $\\Phi_\\alpha = \\psi_1(q)$ for $\\alpha=(q,3)$.\n\nThe field expectation value is given by\n\\begin{equation}\n\\Phi_\\alpha = \\langle \\tilde \\Phi_\\alpha\\rangle= \\frac{1}{Z}\\int D\\tilde \\Phi \\,\\tilde\\Phi_\\alpha \\, e^{-S[\\tilde\\Phi]},\n\\label{eq3:fieldexpectationvalue}\n\\end{equation}\nwith\n\\begin{equation}\nZ = \\int D\\tilde \\Phi\\, e^{-S[\\tilde \\Phi]}.\n\\end{equation}\nQuite similar one defines correlation functions as\n\\begin{equation}\nc_{\\alpha\\beta\\gamma\\dots} = \\langle\\tilde\\Phi_\\alpha\\tilde \\Phi_\\beta\\tilde\\Phi_\\gamma\\dots\\,\\rangle = \\frac{1}{Z}\\int D\\tilde\\Phi \\,\\tilde\\Phi_\\alpha \\tilde\\Phi_\\beta\\tilde\\Phi_\\gamma\\dots e^{-S[\\tilde\\Phi]}.\n\\end{equation}\nAs an example let us consider the two-point function. It is sensible to decompose it into a connected and a disconnected part like\n\\begin{equation}\n\\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle = \\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle_c + \\langle\\tilde\\Phi_\\alpha\\rangle \\langle\\tilde\\Phi_\\beta\\rangle.\n\\end{equation}\nThe connected part is the (full) propagator \n\\begin{equation}\nG_{\\alpha\\beta} = \\langle\\tilde\\Phi_\\alpha\\tilde\\Phi_\\beta\\rangle_c.\n\\end{equation}\nAlthough we discussed here the statistical formulation of the theory (``imaginary time'') the concepts of expectation values and correlation functions are also useful for the real-time formalism. Formally, the main difference is that the weighting factor $e^{-S[\\tilde\\Phi]}$ becomes complex after analytic continuation\n\\begin{equation}\ne^{-S[\\tilde\\Phi]}\\to e^{iS_t[\\tilde\\Phi]},\n\\end{equation} \nwhere $S_t[\\tilde\\Phi]$ is now the real-time action.\n\n\n\\subsubsection{Functional derivatives, generating functionals}\n\nTo calculate expectation values and correlation functions it is useful to work with sources, functional derivatives and generating functionals. We first explain what a functional derivative is. In some sense it is a natural generalization of the usual derivative to functionals, i.\\ e.\\ to objects that depend on an argument which is itself a function on some space. The basic axiom for the functional derivative is\n\\begin{equation}\n\\frac{\\delta}{\\delta f(x)} f(y) = \\delta(x-y) \\quad \\text{or} \\quad \\frac{\\delta}{\\delta f(x)}\\int_y f(y)g(y) = g(x).\n\\label{eq:axiomfunctionalderivative}\n\\end{equation} \nHere we use a notation where the precise meaning of $\\delta(x-y)$ and $\\int_x$ depends on the situation. For example when we consider a space with $3+1$ dimensions we have\n\\begin{equation}\n\\delta(x-y) = \\delta^{(4)}(x-y) = \\delta(x_0-y_0) \\delta^{(3)}(\\vec x-\\vec y)\n\\end{equation}\nand\n\\begin{equation}\n\\int_x = \\int dx_0 \\int d^3x.\n\\end{equation}\nIt should always be clear from the context what is meant. Eq.\\ \\eqref{eq:axiomfunctionalderivative} is the natural extension of the corresponding rule for vectors $x,y\\in \\mathbb{R}^n$\n\\begin{equation}\n\\frac{\\partial}{\\partial x_i} x_j = \\delta_{ij}\\quad \\text{or} \\quad \\frac{\\partial}{\\partial x_i}\\sum_j x_j y_j = y_i.\n\\end{equation}\nIn addition to Eq.\\ \\eqref{eq:axiomfunctionalderivative} the functional derivative should obey the usual derivative rules such as product rule, chain rule etc. Using the abstract index notation introduced before Eq.\\ \\eqref{eq3:fieldexpectationvalue} we write the axiom in Eq.\\ \\eqref{eq:axiomfunctionalderivative} as\n\\begin{equation}\n\\frac{\\delta}{\\delta f_\\alpha} f_\\beta = \\delta_{\\alpha\\beta} \\quad \\text{or}\\quad \\frac{\\delta}{\\delta f_\\alpha} \\sum_\\beta f_\\beta g_\\beta = g_\\alpha. \n\\end{equation}\n\nWith this formalism at hand we can now come back to our task of calculating expectation values and correlation functions. We introduce the source-dependent partition function by the definition\n\\begin{equation}\nZ[J]=\\int D\\tilde\\Phi \\, e^{-S[\\tilde \\Phi]+J_\\alpha \\tilde\\Phi_\\alpha}.\n\\label{eq3:sourcedeppartfunction}\n\\end{equation}\nExpectation values are obtained as functional derivatives\n\\begin{equation}\n\\Phi_\\alpha = \\langle\\tilde\\Phi_\\alpha\\rangle = \\frac{1}{Z} \\frac{\\delta}{\\delta J_\\alpha} Z[J],\n\\label{eq1:expectvalue}\n\\end{equation}\nand similarly correlation functions\n\\begin{equation}\nc_{\\alpha\\beta\\gamma\\dots} = \\frac{1}{Z} \\frac{\\delta}{\\delta J_\\alpha} \\frac{\\delta}{\\delta J_\\beta} \\frac{\\delta}{\\delta J_\\gamma}\\dots Z[J].\n\\end{equation}\nThe connected part of the correlation functions can be obtained more direct from the Schwinger functional\n\\begin{equation}\nW[J] = \\ln Z[J].\n\\label{eq3:Schwingerfunctaslog}\n\\end{equation}\nFor example the propagator $G$, the connected two-point function, is given by\n\\begin{equation}\nG_{\\alpha\\beta} = \\frac{\\delta}{\\delta J_\\alpha} \\frac{\\delta}{\\delta J_\\beta} W[J].\n\\end{equation}\nDue to these properties one calls $Z[J]$ ($W[J]$) the generating functional for the (connected) correlation functions. \n\n\\subsubsection{Microscopic actions in real time and analytic continuation}\n\nIn this subsection we discuss the relation between the real-time and the imaginary-time action as well as the analytic continuation in more detail. For concreteness we consider a nonrelativistic repulsive Bose gas in three-dimensional homogeneous space and in vacuum ($\\mu=0$). It is straightforward to transfer the discussion also to other cases. \n\nIn real time the microscopic action is given by\n\\begin{equation}\nS_t = -\\int dt \\int d^3 x \\left\\{\\varphi^*(-i\\partial_t-\\Delta-i\\epsilon)\\varphi+\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:microscopicactionrealtime}\n\\end{equation}\nThe overall minus sign is to match the standard convention. After Fourier transformation the term quadratic in $\\varphi$ that determines the propagator reads\n\\begin{equation}\nS_{t,2} = \\int \\frac{d \\omega}{2\\pi} \\int \\frac{d^3p}{(2\\pi)^3} \\, \\varphi^*\\left(\\omega-\\vec p^2+i\\epsilon\\right)\\varphi.\n\\label{eq3:quadraticpartrealtimeforier}\n\\end{equation}\nIn the basis with the complex fields $\\varphi$, $\\varphi^*$ the inverse microscopic propagator reads\n\\begin{equation}\nG(p)^{-1}\\,\\delta(p-p^\\prime) = \\frac{\\delta}{\\delta \\varphi^*(p)} \\frac{\\delta}{\\delta \\varphi(p^\\prime)} S_{t,2} = \\left(\\omega-\\vec p^2+i\\epsilon\\right)\\, \\delta (p-p^\\prime).\n\\label{eq3:propagatorrealtime}\n\\end{equation}\nFrom $\\det G^{-1}(p)=0$ we obtain for $\\epsilon\\to0$ the dispersion relation $\\omega=\\vec p^2$. \n\nFor the action in Eq.\\ \\eqref{eq3:microscopicactionrealtime} one can determine the field theoretic expectation values and correlation functions using the formalism described in the previous subsection with the complex weighting function\n\\begin{equation}\ne^{iS_t[\\varphi]}.\n\\label{eq3:weightingfactorrealtime}\n\\end{equation}\nIn Eqs. \\eqref{eq3:microscopicactionrealtime}, \\eqref{eq3:quadraticpartrealtimeforier} and \\eqref{eq3:propagatorrealtime} the small imaginary term $i\\epsilon$ is introduced to enforce the correct frequency integration contour (Feynman prescription). In Eq.\\ \\eqref{eq3:weightingfactorrealtime} it leads to a Gaussian suppression for large values of $\\varphi^*\\varphi$,\n\\begin{equation}\ne^{iS_t[\\varphi]} = e^{i \\text{Re} S_t[\\varphi]} e^{-\\epsilon \\int_x \\varphi^*\\varphi},\n\\end{equation}\nwhich makes the functional integral convergent. Let us now consider the analytic continuation to imaginary time\n\\begin{equation}\nt\\to e^{-i\\alpha}\\tau,\\quad 0\\leq \\alpha <\\pi\/2.\n\\end{equation}\nFor $\\alpha\\to \\pi\/2$ we have $t\\to - i \\tau$ and \n\\begin{equation}\n(-i\\frac{\\partial}{\\partial t}-i\\epsilon)\\to \\frac{\\partial}{\\partial \\tau}, \\quad \\int dt\\to -i\\int d\\tau.\n\\end{equation}\nThe weighting factor in Eq.\\ \\eqref{eq3:weightingfactorrealtime} becomes\n\\begin{equation}\ne^{-S[\\varphi]}\n\\end{equation}\nwith\n\\begin{equation}\nS[\\varphi] = \\int d\\tau \\int d^3 x \\left\\{\\varphi^*(\\partial_\\tau-\\Delta)\\varphi+\\frac{1}{2}\\lambda (\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:imaginarytimemicroscopicaction}\n\\end{equation}\n\n\n\\subsubsection{Matsubara formalism}\n\nIn statistical physics one is often interested in properties of the thermal (and chemical) equilibrium. For quantum field theories the thermal equilibrium is conveniently described using the Matsubara formalism. In this section we give a short account of the formalism and refer for a more detailed discussion to the literature \\cite{Mahan1981}. \n\nThe grand canonical partition function is defined as\n\\begin{equation}\nZ=\\text{Tr}\\, e^{-\\beta(H-\\mu N)}.\n\\label{eq3:grandcanparttrace}\n\\end{equation}\nHere we use $\\beta =\\frac{1}{T}$ and recall our units for temperature with $k_B=1$. The trace operation in Eq.\\ \\eqref{eq3:grandcanparttrace} sums over all possible states of the system, including varying particle number. The operator $H$ is the Hamiltonian and $N$ the particle number operator. The factor\n\\begin{equation}\ne^{-\\beta(H-\\mu N)}\n\\label{eq3:imaginarytimeevoloperator}\n\\end{equation}\nis quite similar to an unitary time evolution operator $e^{i\\Delta t H}$ evolving the system over some time interval $\\Delta t = t_2-t_1$. Indeed, we can define $\\tilde H = H-\\mu N$ and evolve the system from time $t_1=0$ to the imaginary time $t_2=-i\\beta$ with the operator in Eq.\\ \\eqref{eq3:imaginarytimeevoloperator}. If we take a (generalized) torus with circumference $\\beta$ in the imaginary time direction as our space-time manifold we can use the functional integral formulation of quantum field theory to write Eq.\\ \\eqref{eq3:grandcanparttrace} as\n\\begin{equation}\nZ = \\int D \\tilde\\varphi e^{-S[\\tilde\\varphi]}\n\\end{equation}\nwhere $S$ is an action with imaginary and periodic time. From the imaginary time action described in the last subsection it is obtained by replacing also the Hamiltonian $H$ by $H-\\mu N$. For our Bose gas example this results in\n\\begin{equation}\nS[\\varphi] = \\int_0^\\beta d\\tau \\int d^3 x \\left\\{\\varphi^*(\\partial_\\tau-\\Delta-\\mu)\\varphi+\\frac{1}{2}\\lambda (\\varphi^*\\varphi)^2\\right\\}.\n\\label{eq3:matsubaraaction}\n\\end{equation}\nSince time is now periodic, the Fourier transform leads to discrete frequencies. The quadratic part of $S$ in Eq.\\ \\eqref{eq3:matsubaraaction} reads in momentum space\n\\begin{equation}\nS_2[\\varphi] = T\\sum_{n=-\\infty}^\\infty \\int \\frac{d^3 p}{(2\\pi)^3} \\varphi^*(i\\omega_n+\\vec p^2-\\mu)\\varphi\n\\label{eq3:matsubactionquadraticpart}\n\\end{equation}\nwith the Matsubara frequency $\\omega_n=2\\pi T n$. In the limit $T\\to 0$ the summation over Matsubara frequencies becomes again an integration\n\\begin{equation}\nT \\sum_n \\to \\int \\frac{d\\omega}{2\\pi}.\n\\end{equation}\nFor the Fourier decomposition in Eq.\\ \\eqref{eq3:matsubactionquadraticpart} we used the boundary condition\n\\begin{eqnarray}\n\\nonumber\n\\varphi(\\tau=\\beta, \\vec x) &=& \\varphi(\\tau=0,\\vec x),\\\\\n\\varphi^*(\\tau=\\beta, \\vec x) &=& \\varphi^*(\\tau=0,\\vec x),\n\\end{eqnarray}\nas appropriate for bosonic fields. For fermionic or Grassmann-valued fields $\\psi$ a careful analysis (see e.g. \\cite{WegnerGrassmannVariable}) leads to the boundary conditions\n\\begin{eqnarray}\n\\nonumber\n\\psi(\\tau=\\beta,\\vec x) &=& -\\psi (\\tau=0,x)\\\\\n\\psi^*(\\tau=\\beta,\\vec x) &=& -\\psi^*(\\tau=0,\\vec x).\n\\end{eqnarray}\nIn this case the Matsubara frequencies appearing in Eq.\\ \\eqref{eq3:matsubactionquadraticpart} are of the form\n\\begin{equation}\n\\omega_n = 2\\pi T \\left(n+\\frac{1}{2}\\right), \\quad n\\in \\mathbb{Z}.\n\\end{equation}\n\\subsubsection{Functional integral with probability measure}\nIn this section we reconsider the functional integral formulation of quantum field theory and formulate an representation with a (quasi-) probability distribution. Let us start with a simple Gaussian theory\n\\begin{equation}\nS= \\sum_{\\alpha,\\beta} \\varphi_\\alpha^* \\left(P_{\\alpha\\beta}+i \\epsilon \\delta_{\\alpha\\beta} \\right) \\varphi_\\beta.\n\\label{eq:Gauusianaction}\n\\end{equation}\nWe use here an abstract index notation where e.g. $\\alpha$ stands for both continuous variables such as position or momentum and internal degrees of freedom. In principle, the field $\\varphi$ may have both bosonic and fermionic components, the latter are Grassmann-valued. For simplicity we assume in the following that $\\varphi$ is a bosonic field. The formalism can be extended to cover also the case of fermions with minor modifications. \n\nWe included in Eq.\\ \\eqref{eq:Gauusianaction} a small imaginary part $\\sim i\\epsilon$ to make the functional integral convergent and to enforce the correct frequency integration contour (Feynman prescription). Although $\\epsilon$ is usually taken to be infinitesimal, we work with an arbitrary positive value here and take the limit $\\epsilon \\to 0_+$ only at a later point in our investigation. \nFor simplicity, we will often drop the abstract index and use a short notation with e.~g.\n\\begin{equation}\nS=\\varphi^* (P+ i \\epsilon ) \\varphi.\n\\end{equation}\nThe operator $P$ is the real part of the inverse microscopic propagator. As an example we consider a relativistic theory for a scalar field where $P$ reads in position space\n\\begin{equation}\nP(x,y)=\\delta^{(4)}(x-y)\\left(-g^{\\mu\\nu}\\frac{\\partial}{\\partial y^\\mu}\\frac{\\partial}{\\partial y^\\nu}-m^2\\right).\n\\end{equation}\nAnother example is the nonrelativistic case with \n\\begin{equation}\nP(x,y)=\\delta^{(4)}(x-y)\\left(i\\frac{\\partial}{\\partial y_0} + \\frac{1}{2M}\\vec \\nabla_y^2\\right).\n\\end{equation}\n\nFor a Gaussian theory the microscopic propagator coincides with the full propagator. The latter is obtained for general actions $S$ from\n\\begin{eqnarray}\n\\nonumber\ni G_{\\alpha\\beta} &=& \\langle \\varphi_\\alpha \\varphi^*_\\beta\\rangle_c\\\\\n&=& (-i)^3 \\frac{\\delta}{\\delta J^*_\\alpha}\\frac{\\delta}{\\delta J_\\beta} W[J]\n\\end{eqnarray}\nwith\n\\begin{equation}\ne^{-i W[J]} = Z[J] = \\int D \\varphi e^{i S[\\varphi]+i\\int \\{J^*\\varphi+\\varphi^* J\\}}.\n\\label{eq:Schwingerfunctional}\n\\end{equation}\nFor $\\alpha=(x_0,\\vec x)$ and $\\beta=(y_0,\\vec y)$ the object $G_{\\alpha\\beta}$ can be interpreted as the {\\itshape probability amplitude} for a particle to propagate from the point $\\vec y$ at time $y_0$ to the point $\\vec x$ at time $x_0$. More general, one might label by $\\alpha$ some single-particle state $|\\varphi_\\alpha\\rangle$ at time $t_\\alpha$ and with $\\beta$ the state $|\\varphi_\\beta\\rangle$ at time $t_\\beta$. The propagator $G_{\\alpha\\beta}$ describes then the probability amplitude for the transition between the two states.\nHowever, the description of an actual physical experiment involves the transition probability given by the modulus square of the probability amplitude (no summation over $\\alpha$ and $\\beta$)\n\\begin{equation}\nq_{\\alpha\\beta} = |G_{\\alpha\\beta}|^2= G^*_{\\alpha\\beta} G_{\\alpha\\beta}.\n\\end{equation}\nThis transition probability can also directly be obtained from functional derivatives of\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_1,J_2]= \\int D\\varphi_1 \\int D \\varphi_2 e^{i S[\\varphi_1]} e^{-i S^*[\\varphi_2]} \\\\\ne^{i \\int \\{J_1^* \\varphi_1+\\varphi_1^* J_1\\}} e^{-i \\int \\{J_2^* \\varphi_2+\\varphi_2^* J_2\\}}.\n\\label{eq:tildedpartfunction}\n\\end{eqnarray}\nWe note that Eq.\\ \\eqref{eq:tildedpartfunction} contains twice the functional integral over the field configuration $\\varphi$. One may also write this as a single functional integral over fields that depend in addition to the position variable $\\vec x$ on the contour time $t_c$ which is integrated along the Keldysh contour \\cite{Keldysh}. For our purpose it will be more convenient to work directly with the expression in Eq.\\ \\eqref{eq:tildedpartfunction}.\n\nFor $\\langle \\varphi \\rangle = \\langle \\varphi^* \\rangle = 0$ we can write\n\\begin{equation}\nq_{\\alpha\\beta} =\\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_1)_\\beta} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\frac{\\delta}{\\delta (J_2^*)_\\beta} \\tilde Z[J_1,J_2].\n\\end{equation}\nThis is immediately clear since $\\tilde Z[J_1,J_2] = Z[J_1] Z^*[J_2]$ and\n\\begin{eqnarray}\n\\nonumber\ni G_{\\alpha\\beta} &=& (-i)^2 \\frac{1}{Z[J_1]} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_1)_\\beta} Z[J_1]\\\\\n-i G^*_{\\alpha\\beta} &=& i^2 \\frac{1}{Z^*[J_2]} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\frac{\\delta}{\\delta (J_2^*)_\\beta} Z^*[J_2].\n\\end{eqnarray}\nfor\n\\begin{equation}\n\\langle \\varphi \\rangle = \\frac{-i}{Z} \\frac{\\delta}{\\delta J^*} Z[J] = \\langle \\varphi^* \\rangle = \\frac{-i}{Z} \\frac{\\delta}{\\delta J} Z[J]=0.\n\\end{equation}\nUsually one obtains $q_{\\alpha\\beta}$ by first calculating $G_{\\alpha\\beta}$ and then taking the modulus square thereof. The way we go here seems to be more complicated from a technical point of view, but has the advantage that it will allow for an intuitive physical interpretation. \n\nWe first concentrate on $\\tilde Z[J_1,J_2]$. This object plays a similar role as the partition function in statistical field theory. In some sense it is a sum over microscopic states weighted with some ``probability''. However, in contrast to statistical physics, the summation does not go over states of a system at some fixed time $t$ but over field configurations that depend both on the position variable $\\vec x$ and the time variable $t$. The summation seems to go over a even larger space since the functional integral appears twice\n\\begin{equation}\n\\int D\\varphi_1 \\int D \\varphi_2\n\\end{equation}\nso that the configuration space seems to be the tensor product of twice the space that contains the field configurations in space-time $\\varphi(\\vec x,t)$. In addition the ``probability weight''\n\\begin{equation}\ne^{i S[\\varphi_1]} e^{-i S^*[\\varphi_2]}\n\\end{equation}\nis not positive semi-definite and has even complex values in general. This last two features (``doubled'' configuration space and missing positivity) prevent us from interpreting quantum field theory in a similar way as statistical field theory.ting quantum field theory in a similar way as statistical field theory.\n\nAn idea to overcome these difficulties is to partially perform the functional integral in Eq.\\ \\eqref{eq:tildedpartfunction}. For this purpose we make a change of variables of the form\n\\begin{eqnarray}\n\\nonumber\n\\varphi_1 = \\frac{1}{\\sqrt{2}}\\phi + \\frac{1}{\\sqrt{2}}\\chi, &\\quad& J_1 = \\frac{1}{\\sqrt{2}}J_\\phi +\\frac{1}{\\sqrt{2}} J_\\chi, \\\\\n\\varphi_2 = \\frac{1}{\\sqrt{2}}\\phi - \\frac{1}{\\sqrt{2}}\\chi, &\\quad& J_2 = -\\frac{1}{\\sqrt{2}}J_\\phi + \\frac{1}{\\sqrt{2}} J_\\chi.\n\\label{eq:12phichitranslation}\n\\end{eqnarray}\nFor $\\tilde Z$ this gives then\n\\begin{equation}\n\\tilde Z = \\int D\\phi\\,\\, v[\\phi,J_\\chi]\\,\\,e^{i\\int \\{J_\\phi^* \\phi+\\phi^* J_\\phi\\}}\n\\label{eq:ZwithJ1J2}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nv[\\phi, J_\\chi] &=& \\int D \\chi\\, e^{iS[(\\phi+\\chi)\/\\sqrt{2}]}\\, e^{-i S^*[(\\phi-\\chi)\/\\sqrt{2}]}\\\\\n&& \\times\\, e^{i \\int \\{J_\\chi^* \\chi+\\chi^* J_\\chi\\}}.\n\\label{eq:vphiJchigeneralcase}\n\\end{eqnarray}\nWe note that $v[\\phi,J_\\chi]$ as a functional of $\\phi$ and $J_\\chi$ is real. This follows from comparison with the complex conjugate together with the change of variables $\\chi\\to-\\chi$. If it is also positive, we can interpret this object as a probability for the field configurations $\\phi(x)$. We call $v$ the {\\itshape functional probability} for the field configuration $\\phi$. \n\nBefore we discuss the general properties of $v[\\phi,J_\\chi]$ in more detail, we consider it explicitly for a Gaussian action $S[\\varphi]$ as in Eq.\\ \\eqref{eq:Gauusianaction}. In that case we can perform the functional integral\n\\begin{eqnarray}\n\\nonumber\nv[\\phi,J_\\chi] &=& \\int D \\chi e^{-\\epsilon \\{\\phi^*\\phi+\\chi^*\\chi\\}} e^{i\\{\\phi^* P \\chi +\\chi^* P \\phi\\}}\\\\\n\\nonumber\n&& \\times e^{i\\{J_\\chi^* \\chi + \\chi^* J_\\chi\\}}\\\\\n&=& e^{-\\epsilon \\phi^*\\phi}\\, e^{-\\frac{1}{\\epsilon}(J_\\chi^*+\\phi^* P)(J_\\chi+P\\phi)}.\n\\label{eq:vphijGaussian}\n\\end{eqnarray}\nThe last line holds up to a multiplicative factor that is irrelevant for us here. For $\\tilde Z$ we are left with\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_\\phi,J_\\chi] &=& \\int D\\phi \\,\\,v[\\phi, J_\\chi]\\, \\,e^{i\\int\\{J_\\phi^*\\phi+\\phi^*J_\\phi\\}}\\\\\n\\nonumber\n&=& \\int D \\phi \\, e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi} \\, e^{i\\{J_\\phi^* \\phi +\\phi^* J_\\phi\\}}\\\\\n&&\\times \\, e^{-\\frac{1}{\\epsilon}\\{J_\\chi^* P \\phi+\\phi^* P J_\\chi\\}}\\, e^{-\\frac{1}{\\epsilon}J_\\chi^*J_\\chi}.\n\\label{eq:partitionfunctionphiJphiJchi}\n\\end{eqnarray}\nFor $J_\\phi=J_\\chi=0$ the integrand in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is strictly positive. Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} can therefore be interpreted in a similar way as the partition function in statistical field theory. The probability measure is\n\\begin{equation}\nv=e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}.\n\\label{eq:probmeasuregaussian}\n\\end{equation}\n\nWe can distinguish three different classes of field configurations $\\phi$. In the simplest case the norm vanishes,\n\\begin{equation}\n|\\phi|^2=\\phi^*\\phi=\\sum_\\alpha \\phi^*_\\alpha\\phi_\\alpha\\to 0.\n\\end{equation} \nThe functional probability for this case is of order $1$. The second class contains field configurations where the norm is nonzero, $\\phi^*\\phi\\neq 0$, but where $\\phi$ satisfies the on-shell condition, i.~e. $\\phi^*P^2\\phi=0$. The functional probability for this case is of order $e^{-\\epsilon}$ (for $\\phi^*\\phi\\sim1$). Finally, in the third class the norm is nonzero and the field configuration is off-shell, i.~e.\n\\begin{equation}\n\\phi^*\\phi\\sim1,\\quad \\phi^*P^2\\phi \\sim 1.\n\\end{equation}\nThe functional probability in Eq.\\ \\eqref{eq:probmeasuregaussian} for this case is only of the order $e^{-1\/\\epsilon}$. This shows that off-shell configurations are strongly suppressed in the limit $\\epsilon\\to0$ compared to the trivial case $|\\phi|=0$ and the on-shell fields with\n\\begin{equation}\n\\phi^* P^2\\phi = 0.\n\\label{eq:onshellcond}\n\\end{equation}\nHowever, for $\\epsilon>0$ the probability for off-shell configurations is not strictly zero and they give also contributions to $\\tilde Z$. This is in contrast to classical statistics where only states that fulfill the equation of motion are included. (At nonzero temperature states with different energies are weighted according to a thermal distribution.) \n\nThere are more differences between the partition function in classical statistics and the quantum partition function in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi}. In classical statistics the averaging over some phase space is directly linked to time averaging by the ergodic hypothesis. Indeed, this hypothesis says that a mean value calculated by taking the average of some quantity over the accessible phase space is equal to the average of that quantity over a -- sufficiently long -- time interval. \nIn classical statistics time plays an outstanding role. The formalism breaks space-time symmetries such as Lorentz- or Galilean symmetry explicitly. For the case of quantum field theory this point is different. The theory in the vacuum (zero temperature and density, $T=n=0$) is symmetric under Lorentz symmetry (or Galilean symmetry in the nonrelativistic case). \n\nThe summation over possible field configurations in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is not related to time averaging. We directly interpret it in the following way. Every physical experiment or ``measurement-like situation'' corresponds to a different microscopic field configuration $\\phi$. This field configuration does not necessarily have to fulfill the on-shell condition Eq.\\ \\eqref{eq:onshellcond} but has a probability $v$ that strongly favors on-shell fields. There is, however, a subtlety in this interpretation. When the action $S$ is given as an integral over the $3+1$ dimensional spacetime $\\Omega$, then $v$ describes the probability for a field configuration $\\phi(x_0,\\vec x)$ on this manifold $\\Omega$. Since we experience only one universe with one configuration one might ask why we should take the sum over different configurations weighted with some probability. To answer that question it is important to realize that our information about the field configuration $\\phi(x_0,\\vec x)$ is limited. \n\nFirst we can investigate only limited regions in space-time (around our own ``world-line''). Regions that are too far away either in the spatial or the temporal sense are not accessible. However, in the framework of a local field theory, the experiments in some region of space-time depend on the other (not accessible) regions only via the boundary conditions. Second, and more important, we have only access to the field configuration in some ``momentum range''. No experiment has an arbitrary large resolution and can resolve infinitely small wavelength. Therefore the true microscopic field configuration is inevitably hidden from our observation. In a Gaussian or non-interacting theory this issue seems to be not so important since different momentum modes decouple from each other. In a theory with interactions this is different, however. Modes with different momenta $p^\\mu$ (and different values of $p^\\mu p_\\mu$) are coupled via the interaction. The microscopic regime does influence the macroscopic states. \n\nOur interpretation of Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} is therefore that the functional integral sums over possible microscopic configurations $\\phi(x)$ with probability (up to a factor) given by\n\\begin{equation}\ne^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\n\\end{equation}\nfor the Gaussian theory considered above and more general (for $J_\\chi=0$) by\n\\begin{equation}\nv[\\phi]=v[\\phi,J_\\chi=0]=\\int D\\chi e^{iS[(\\phi+\\chi)\/\\sqrt{2}]}\\,e^{-i S^*[(\\phi-\\chi)\/\\sqrt{2}]}.\n\\end{equation}\nIn this general case, the functional probability for nonzero source terms is given by Eq.\\ \\eqref{eq:vphiJchigeneralcase}. As argued there, it is always real. This is made more explicit in the expression\n\\begin{eqnarray}\n\\nonumber\nv[\\phi,J_\\chi] &=& \\int D \\chi \\,\\text{cos}\\left(S_1[\\phi+\\chi]-S_1[\\phi-\\chi]\\right)\\\\\n&& \\times \\, \\text{exp}\\left(-S_2[\\phi+\\chi]-S_2[\\phi-\\chi]\\right)\n\\label{eq:vphiJchiexplrealform}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nS_1[\\varphi] &=& \\text{Re}\\, S[\\varphi\/\\sqrt{2}]+ \\frac{1}{\\sqrt{2}}\\left\\{J_\\chi^*\\varphi+\\varphi^*J_\\chi\\right\\}\n\\end{eqnarray}\nand\n\\begin{equation}\nS_2[\\varphi]=\\text{Im}\\, S[\\varphi\/\\sqrt{2}].\n\\end{equation}\nWe note that the functional integral in Eq.\\ \\eqref{eq:vphiJchiexplrealform} converges when $S_2[\\varphi]$ increases with $\\varphi^*\\varphi$ fast enough. For $S_2[\\varphi]\\sim \\varphi^*\\varphi$ as in our Gaussian example, the convergence is quite good. Although for arbitrary actions $S[\\varphi]$ the ``probability'' $v[\\phi,J_\\chi]$ does not have to be positive, this is expected to be the case for many choices of $S[\\varphi]$. \n\nWhen $v[\\phi,J_\\chi]$ is negative for some choices of $\\phi$, this indicates that different values for $\\phi$ do not directly correspond to independent physical configurations. One might come to positive definite probabilities when the space of possible fields is restricted to a physically subspace. However, $v[\\phi,J_\\chi]$ as defined above can in any case be seen an quasi-probability for $\\phi$. This is in some respect similar to Wigner's representation of density matrices \\cite{Wigner}. \n\nLet us make another comment on the case of non-Gaussian actions $S[\\varphi]$. When $S[\\varphi]$ contains terms of higher then quadratic order in the fields $\\varphi$ the form of the action is subject to renormalization group modifications. Usually the true microscopic action $S[\\varphi]$ is not known. Measurements have only access to the effective action $\\Gamma[\\varphi]$ which already includes the effect of quantum fluctuations. (Measurements at some momentum scale $k^2=|p_\\mu p^\\mu|$ might probe the average action or flowing action $\\Gamma_k[\\varphi]$ \\cite{Wetterich1993b}.) The microscopic action $S$ is connected to $\\Gamma$ by a renormalization group flow equation \\cite{Wetterich1993b}, however it is in most cases not possible to construct $S$ from the knowledge of $\\Gamma$. Typically many different microscopic actions $S$ lead to the same effective action $\\Gamma$. It may therefore often be possible that a microscopic action $S$ exists that is consistent with experiments and allows for a positive probability $v[\\phi,J_\\chi]$. \n\nFinally we comment on the general properties of $v[\\phi,J_\\chi]$. Since it is defined as a functional integral over a (local) complex action one expects that $v[\\phi,J_\\chi]$ is local to a similar degree as the the effective action $\\Gamma[\\varphi]$ or the Schwinger functional $W[J]$ defined in Eq.\\ \\eqref{eq:Schwingerfunctional}. For general non-Gaussian microscopic actions $S[\\varphi]$ the functional $v[\\phi,J_\\chi]$ may be quite complicated and not necessarily local in the sense that it can be written in the form\n\\begin{equation}\nv[\\phi,J_\\chi] = e^{-\\int_x{\\cal L}_v}\n\\end{equation}\nwhere ${\\cal L}_v$ is a local ``Lagrange density'' that depends only on $\\phi(x)$, $J_\\chi(x)$ and derivatives thereof at the space-time point $x$. \n\nSince $v[\\phi,J_\\chi]$ is similarly defined as the effective action $\\Gamma[\\varphi]$ or the Schwinger functional $W[J]$ we expect that it respects the same symmetries as the microscopic action $S[\\varphi]$ when no anomalies are present. For example, when $S[\\varphi]$ is invariant under some $U(1)$ symmetry transformation $\\varphi \\to e^{i\\alpha} \\varphi$, we expect that $v[\\phi,J_\\chi]$ has a corresponding symmetry under the transformation\n\\begin{equation}\n\\phi \\to e^{i\\alpha} \\phi, \\quad J_\\chi \\to e^{i\\alpha} J_\\chi.\n\\end{equation} \n\n\\subsubsection{Correlation functions from functional probabilities}\nIn this subsection we use the expression for $\\tilde Z$ in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} to derive functional integral representations of some correlation functions. In the following we denote by $\\langle \\cdot \\rangle$ the ``expectation value'' in the quantum field theoretic sense, e.~g. for an operator $A[\\varphi]$\n\\begin{equation}\n\\langle A[\\varphi] \\rangle = \\frac{1}{Z} \\int D \\varphi e^{i S[\\varphi]} \\,A[\\varphi].\n\\end{equation}\nIn contrast, we use $\\langle\\langle\\cdot\\rangle\\rangle$ to denote the expectation value with respect to the functional integral over $\\phi$, i.~e.\n\\begin{equation}\n\\langle\\langle A[\\phi]\\rangle\\rangle = \\frac{1}{\\tilde Z} \\int D \\phi \\,\\,e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi} \\,A[\\phi],\n\\label{eq:gaussiandistroperator}\n\\end{equation}\nor more general\n\\begin{equation}\n\\langle\\langle A[\\phi]\\rangle\\rangle = \\frac{1}{\\tilde Z} \\int D \\phi\\,\\, v[\\phi]\\, A[\\phi]. \n\\end{equation}\nFor the discussion of the correlation functions it is useful to express $\\tilde Z$ in Eq.\\ \\eqref{eq:partitionfunctionphiJphiJchi} again in terms of $J_1$ and $J_2$. Using Eq.\\ \\eqref{eq:12phichitranslation} we find\n\\begin{eqnarray}\n\\nonumber\n\\tilde Z[J_1,J_2] &=& \\int D\\phi \\,\\,e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&& \\times\\, e^{-\\frac{1}{\\sqrt{2}\\epsilon}\\left\\{J_1^*(P-i\\epsilon)\\phi+\\phi^*(P-i\\epsilon)J_1\\right\\}}\\\\\n\\nonumber\n&& \\times \\,e^{-\\frac{1}{\\sqrt{2}\\epsilon}\\left\\{J_2^*(P+i\\epsilon)\\phi+\\phi^*(P+i\\epsilon)J_2\\right\\}}\\\\\n&& \\times \\,e^{-\\frac{1}{2\\epsilon}\\left\\{J_1^*J_1+J_1^*J_2+J_2^*J_1+J_2^*J_2\\right\\}}.\n\\end{eqnarray}\nWe start with the modulus square of the quantum field theoretic one-point function (no summation convention used in the following)\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\tilde Z \\\\\n\\nonumber\n&=& \\frac{1}{\\tilde Z} \\int D\\phi \\,\\, e^{-\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&& \\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\beta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\beta}\\phi_\\beta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n\\nonumber\n&=& \\frac{1}{2\\epsilon} \\left[\\sum_{\\beta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\beta}\\langle\\langle\\phi_\\beta \\phi^*_\\gamma\\rangle\\rangle(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n&=& 0.\n\\label{eq:vanishingexpectationvalue}\n\\end{eqnarray}\nIn the last line of Eq.\\ \\eqref{eq:vanishingexpectationvalue} we used the standard property of the Gaussian distribution Eq.\\ \\eqref{eq:gaussiandistroperator}\n\\begin{equation}\n\\langle\\langle\\phi_\\beta\\phi^*_\\gamma\\rangle\\rangle=\\epsilon (P^2+\\epsilon^2)_{\\beta\\gamma}^{-1}.\n\\end{equation}\nNext we turn to the two-point function or ``transition probability''\n\\begin{eqnarray}\n\\nonumber\nq_{\\alpha\\beta} &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha}\\frac{\\delta}{\\delta (J_2)_\\alpha}\\frac{\\delta}{\\delta (J_1)_\\beta}\\frac{\\delta}{\\delta (J_2^*)_\\beta} \\tilde Z[J_1,J_2]\\\\\n\\nonumber\n&=& \\frac{1}{\\tilde Z}\\int D\\phi \\,\\, e^{\\frac{1}{\\epsilon}\\phi^*(P^2+\\epsilon^2)\\phi}\\\\\n\\nonumber\n&&\\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\eta,\\gamma}\\frac{1}{\\epsilon}(P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}-\\delta_{\\alpha\\alpha}\\right]\\\\\n\\nonumber\n&&\\times \\frac{1}{2\\epsilon}\\left[\\sum_{\\kappa,\\lambda}\\frac{1}{\\epsilon}(P+i\\epsilon)_{\\beta\\kappa}\\phi_\\kappa\\phi^*_\\lambda(P-i\\epsilon)_{\\lambda\\beta}-\\delta_{\\beta\\beta}\\right]\\\\\n&=& \\left\\langle\\langle\\rho_\\alpha\\rho_\\beta\\right\\rangle\\rangle-\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle.\n\\label{eq:qxyconnexptvalue}\n\\end{eqnarray}\nThis expression is the (connected) two-point correlation function of the operator\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2\\epsilon^2}\\sum_{\\gamma,\\eta}(P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta \\phi^*_\\gamma(P+i\\epsilon)_{\\gamma\\alpha}\n\\end{equation}\nwith respect to averaging over the possible field configurations $\\phi(x)$. Note that $\\rho_\\alpha$ is real and positive for all field configurations $\\phi$. Indeed, we can write with $P^\\dagger =P$\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2\\epsilon^2} \\left|\\sum_\\eta (P-i\\epsilon)_{\\alpha\\eta}\\phi_\\eta\\right|^2\n\\end{equation}\nshowing this more explicit. The multiplicative normalization of $\\rho$ is somewhat arbitrary and could be changed by rescaling the fields according to $\\phi\\to\\phi^\\prime=c\\phi$. \nNote that for on-shell modes with $\\phi^* P^2\\phi=0$ the operator $\\rho$ reads\n\\begin{equation}\n\\rho_\\alpha=\\frac{1}{2} \\phi^*_\\alpha\\phi_\\alpha.\n\\end{equation}\n\nAlthough the description of $q_{\\alpha\\beta}$ as a connected correlation function of the operators $\\rho_\\alpha$ and $\\rho_\\beta$ is appealing, its meaning as a transition probability is not yet completely clear. In a typical experiment one asks for the probability to find a particle both at the space-time point $y=(y_0,\\vec y)$ and at the space-time point $x=(x_0,\\vec x)$. We denote the probability for this by $p(x \\cap y)$. Quite generally, one would calculate this quantity as a sum over all field configurations $\\phi$ weighted by the product\n\\begin{equation}\np[\\phi]\\,p(x|\\phi] \\,p(y|\\phi].\n\\end{equation}\nHere $p(x|\\phi]$ gives the probability for the event ``particle measured at $x$'' under the condition that the field configuration $\\phi$ is realized. The expression $p[\\phi]$ is the probability for the field configuration $\\phi$. In combination, we find\n\\begin{eqnarray}\np(x \\cap y) &=& \\int D\\phi\\, p[\\phi]\\,p(x|\\phi] \\,p(y|\\phi].\n\\end{eqnarray}\nLet us now compare this to our expression for $q_{\\alpha\\beta}$ in Eq.\\ \\eqref{eq:qxyconnexptvalue}. If we identify $\\alpha=x=(x_0,\\vec x)$ and $\\beta=y=(y_0,\\vec y)$ and neglect for the moment the second term in the last line of Eq.\\ \\eqref{eq:qxyconnexptvalue}, we can write\n\\begin{equation}\nq(x,y)=\\int D\\phi\\, v[\\phi]\\, \\rho(x)\\, \\rho(y).\n\\end{equation}\nThe expressions for $p(x\\cap y)$ and $q(x,y)$ are proportional when the probability for the field configuration $\\phi$ is\n\\begin{equation}\np[\\phi] \\sim v[\\phi]\n\\end{equation}\nand the probability to find a particle at $x=(x_0,\\vec x)$ for the field configuration $\\phi$ is given by\n\\begin{equation}\np(x|\\phi] \\sim \\rho(x). \n\\end{equation}\nThe subtraction of the term $\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle$ in Eq.\\ \\eqref{eq:qxyconnexptvalue} provides for the two events ``particle measured at $y$'' and ``particle measured at $x$'' not to be in a coincidence. Instead, there has to be a ``causal connection'' between them. Only in that case would we speak of ``two measurements on the same particle''. Moreover, fluctuations at different space-time points that are uncorrelated would not show the characteristics of particles at all. Let us assume for definiteness that we use a cloud chamber as a particle detector. The vapor would only condense if neighboring points in space are stimulated during a small but nonzero period of time. Stimulations at random points in space-time would not lead to the detection of a particle. The disconnected part of the two point function $\\langle\\langle\\rho_\\alpha\\rangle\\rangle\\langle\\langle\\rho_\\beta\\rangle\\rangle$ should therefore be seen as part of the nontrivial vacuum structure in quantum field theory. \n\nTo end this subsection let us comment of the general, not necessary Gaussian case. We can obtain the quantum field theoretic one-point function from\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{\\tilde Z} \\frac{\\delta}{\\delta (J_1^*)_\\alpha} \\frac{\\delta}{\\delta (J_2)_\\alpha} \\tilde Z\\\\\n\\nonumber\n&=& \\frac{1}{2 \\tilde Z} \\left(\\frac{\\delta}{\\delta (J_\\phi^*)_\\alpha}+\\frac{\\delta}{\\delta (J_\\chi^*)_\\alpha}\\right)\\\\\n&&\\times \\left(-\\frac{\\delta}{\\delta (J_\\phi)_\\alpha}+\\frac{\\delta}{\\delta (J_\\chi)_\\alpha}\\right) \\tilde Z[J_\\phi,J_\\chi]. \n\\end{eqnarray}\nWith Eq.\\ \\eqref{eq:ZwithJ1J2} this gives\n\\begin{eqnarray}\n\\nonumber\n|\\langle\\phi_\\alpha\\rangle|^2 &=& \\frac{1}{2\\tilde Z} \\int D\\phi\\\\\n&& \\times\\, \\left[\\phi^*_\\alpha\\phi_\\alpha+\\frac{\\delta}{\\delta (J_\\chi^*)_\\alpha}\\frac{\\delta}{\\delta (J_\\chi)_\\alpha}\\right] \\, v[\\phi,J_\\chi]. \n\\end{eqnarray}\nHere we used that $v[\\phi,J_\\chi]$ and $|\\langle \\phi_\\alpha\\rangle|^2$ have to be real. The general expression for the two-point function $q_{\\alpha\\beta}$ is somewhat more complicated, but straightforward to obtain in an analogous way as the calculations above. \n\n\\subsubsection{Conservation laws for on-shell excitations}\nAlthough particles are created and annihilated in quantum field theory, these processes are constraint by several conservation laws. For example, in quantum electrodynamics, the electric charge is a conserved quantum number. Electrons and positrons can only be created in pairs such that the total charge remains constant. In a formalism where particles are described as excitations of fields, one must show that these excitations fulfill the usual conservation constraints. \n\nIn quantum field theory, conserved quantities such as charge or also energy are associated to a continuous symmetry via Noether's theorem. However, only the combination of a symmetry together with some field equation leads to a conservation law. For example, for a field that satisfies the on-shell condition\n\\begin{equation}\nP\\phi = (-\\partial_\\mu\\partial^\\mu-m^2)\\phi=0\n\\end{equation}\none can easily show that the current\n\\begin{equation}\nj^\\mu=i(\\partial^\\mu\\phi^*)\\phi-i\\phi^*(\\partial^\\mu\\phi)\n\\label{eq:currentrelativistic}\n\\end{equation}\nis conserved, i.~e. $\\partial_\\mu j^\\mu=0$. This current is directly linked to the symmetry of the action\n\\begin{equation}\nS[\\varphi]=\\int_x \\varphi^*(-\\partial_\\mu\\partial^\\mu-m^2)\\varphi\n\\end{equation}\nunder global U(1) transformations $\\varphi\\to e^{i\\alpha}\\varphi$, $\\varphi^*\\to e^{-i\\alpha}\\varphi^*$. As discussed in the last section, the functional probability $v[\\phi]$ is invariant under the same symmetries as the microscopic action $S[\\varphi]$ if no anomalies are present. This implies that there should be conservation laws associated with these symmetries for on-shell excitations, that fulfill a field equation as Eq.\\ \\eqref{eq:onshellcond}. \nWe emphasize again that e.~g. the current in Eq.\\ \\eqref{eq:currentrelativistic} is not conserved for general field configurations with $P\\phi\\neq0$. However, if particles correspond to on-shell field excitations, the usual conservation laws are indeed fulfilled. \n\n\\subsubsection{Conclusions}\nIn this appendix we discussed a (quasi-) probability representation of quantum field theory based on the functional integral. We showed for a Gaussian theory of bosonic fields that the functional integral can be reordered such that an interpretation in terms of real and positive probabilities for field configurations (``functional probabilities'') is possible. Our formalism is also applicable to the more general case of non-Gaussian microscopic actions where it may be necessary to work also with negative (quasi-) probabilities. We believe that a description using only positive probabilities is possible in many cases. However, it is not excluded that for some physical theories negative probabilities are needed. This would be highly interesting and demonstrate -- once again -- the extraordinariness of quantum theory. In any case the (quasi-) probability representation developed here might be useful as a theoretical tool, for example in studies of non-equilibrium quantum field dynamics. The formalism can be extended with minor modifications to fermionic or Grassmann valued fields.\n\nThe concept of functional probabilities addresses both classical field configurations and particles. The former are described by a nonzero expectation value $\\langle\\langle\\phi\\rangle\\rangle$ while particles correspond to on-shell excitations, described by the connected two-point function $\\langle\\langle\\phi\\phi\\rangle\\rangle-\\langle\\langle\\phi\\rangle\\rangle\\langle\\langle\\phi\\rangle\\rangle$. For quadratic microscopic actions as in Eq.\\ \\eqref{eq:Gauusianaction} the functional probability is local (Eq.\\ \\eqref{eq:probmeasuregaussian}). This does no longer have to be the case once interactions are included. For example in a perturbation theory for weak interactions it should be possible to derive explicit expressions beyond the Gaussian case. Higher order correlation functions can then be studied which might shed more light on the question of locality. Interesting features of quantum mechanics as entanglement and the implications of Bells inequalities \\cite{Bell} can then be studied in this framework. \n\n\\section{Scale-dependent Bosonization}\n\nLet us consider a scale-dependent Schwinger functional for a theory formulated in terms of the field $\\tilde \\psi$\n\\begin{equation}\ne^{W_k[\\eta]}=\\int D\\tilde\\psi\\, e^{-S_\\psi[\\tilde\\psi]-\\frac{1}{2}\\tilde\\psi_\\alpha(R_k^\\psi)_{\\alpha\\beta}\\tilde\\psi_\\beta+\\eta_\\alpha\\tilde\\psi_\\alpha}.\n\\label{eq:scaledepSF}\n\\end{equation}\nAgain we use the abstract index notation where e.g. $\\alpha$ stands for both continuous variables such as position or momentum and internal degrees of freedom. We now multiply the right hand side of Eq.\\ \\eqref{eq:scaledepSF} by a term that is for $R_k^\\varphi=0$ only a field independent constant. It has the form of the functional integral over the field $\\tilde\\varphi$ with a Gaussian weighting factor\n\\begin{equation}\n\\int D \\tilde\\varphi \\, e^{-S_\\text{pb}-\\frac{1}{2}\\tilde \\varphi_\\epsilon(R_k^\\varphi)_{\\epsilon\\sigma}\\tilde\\varphi_\\sigma+j_\\epsilon\\tilde\\varphi_\\epsilon},\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\nonumber\nS_\\text{pb} &=& \\frac{1}{2}\\left(\\tilde\\varphi_\\epsilon-\\chi_\\tau Q^{-1}_{\\tau\\epsilon}\\right)Q_{\\epsilon\\sigma}(\\tilde\\varphi_\\sigma-Q^{-1}_{\\sigma\\rho}\\chi_\\rho).\\\\\n&=& \\frac{1}{2}\\left(\\tilde\\varphi-\\chi Q^{-1}\\right)Q(\\tilde\\varphi-Q^{-1}\\chi),\n\\label{eq:Spb}\n\\end{eqnarray}\nand $\\chi$ depends on the ``fundamental field'' $\\tilde \\psi$. We will often suppress the abstract index as in the last line of Eq.\\ \\eqref{eq:Spb}. We assume that the field $\\tilde\\varphi$ and the operator $\\chi$ are bosonic. Without further loss of generality we can then also assume that $Q$ and $R_k^\\varphi$ are $k$-dependent symmetric matrices.\n\nAs an example, we consider an operator $\\chi$ which is quadratic in the original field $\\tilde\\psi$,\n\\begin{equation}\n\\chi_\\epsilon = H_{\\epsilon\\alpha\\beta}\\tilde\\psi_\\alpha\\tilde\\psi_\\beta.\n\\end{equation}\nThe Schwinger functional reads now\n\\begin{equation}\ne^{W_k[\\eta,j]} = \\int D\\tilde \\psi\\,D\\tilde\\varphi \\, e^{-S_k[\\tilde \\psi, \\tilde\\varphi]+\\eta\\tilde\\psi+j\\tilde\\varphi}\n\\label{eq:SFwithbosonfi}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\nonumber\nS_k[\\tilde\\psi,\\tilde\\varphi] &=& S_\\psi[\\tilde\\psi]+\\frac{1}{2}\\tilde\\psi R_k^\\psi\\tilde \\psi + \\frac{1}{2}\\tilde\\varphi (Q+R_k^\\varphi)\\tilde\\varphi\\\\\n&&+ \\frac{1}{2}\\chi Q^{-1}\\chi -\\tilde\\varphi \\chi. \n\\label{eq:actionferbos}\n\\end{eqnarray}\nIn the integration over $\\tilde\\varphi$, we can easily shift the variables to obtain\n\\begin{eqnarray}\n\\nonumber\ne^{W_k[\\eta,j]} &=& \\int D \\tilde\\psi \\, e^{-S_\\psi[\\tilde \\psi]-\\frac{1}{2}\\tilde\\psi R_k^\\psi\\tilde\\psi+\\eta \\tilde\\psi}\\\\\n\\nonumber\n&& \\times e^{\\frac{1}{2}(j+\\chi)(Q+R_k^\\varphi)^{-1}(j+\\chi)-\\frac{1}{2}\\chi Q^{-1}\\chi}\\\\\n&&\\times \\int D\\tilde\\varphi\\, e^{-\\frac{1}{2}\\tilde\\varphi(Q+R_k^\\varphi)\\tilde\\varphi}.\n\\label{eq:Schingerfas}\n\\end{eqnarray}\nThe remaining integral over $\\tilde\\varphi$ gives only a ($k$-dependent) constant. For $R_k^\\varphi=0$ and $j=0$ we note that $W_k[\\eta,j]$ coincides with $W_k[\\eta]$ in Eq.\\ \\eqref{eq:scaledepSF}.\n\nWe next derive identities for correlation functions of composite operators which follow from the equivalence of the equations\\eqref{eq:SFwithbosonfi} and \\eqref{eq:Schingerfas}. Taking the derivative with respect to $j$ we can calculate the expectation value for $\\tilde \\varphi$\n\\begin{eqnarray}\n\\nonumber\n\\varphi_\\epsilon &=& \\langle\\tilde\\varphi_\\epsilon\\rangle = \\frac{\\delta}{\\delta j_\\epsilon} W_k[\\eta,j]\\\\\n&=& (Q+R_k^\\varphi)^{-1}_{\\epsilon\\sigma}\\,\\left(j_\\sigma+H_{\\sigma\\alpha\\beta}\\langle\\tilde\\psi_\\alpha\\tilde\\psi_\\beta\\rangle\\right).\n\\label{eq:varphiintermsofpsi}\n\\end{eqnarray}\nThis can also be written as\n\\begin{equation}\n\\langle\\chi\\rangle = Q\\varphi - l\n\\label{eq:expvchi}\n\\end{equation}\nwith the modified source $l$\n\\begin{equation}\nl_\\epsilon = j_\\epsilon-(R_k^\\varphi)_{\\epsilon\\sigma}\\varphi_\\sigma.\n\\end{equation}\nFor the connected two-point function\n\\begin{equation}\n(\\delta_j\\delta_j W_k)_{\\epsilon\\sigma}= \\frac{\\delta^2}{\\delta j_\\epsilon \\delta j_\\sigma} W_k =\\langle\\tilde\\varphi_\\epsilon\\tilde\\varphi_\\sigma\\rangle_c\n\\end{equation}\nwe obtain from Eq.\\ \\eqref{eq:Schingerfas} \n\\begin{eqnarray}\n\\nonumber\n&&(Q+R_k)(\\delta_j\\delta_j W_k)(Q+R_k) \\\\\n\\nonumber\n&&= \\langle(j+\\chi)(j+\\chi)\\rangle -\\langle(j+\\chi)\\rangle\\langle(j+\\chi)\\rangle+(Q+R_k^\\varphi)\\\\\n&&= \\langle\\chi\\chi\\rangle-\\langle\\chi\\rangle\\langle\\chi\\rangle+(Q+R_k^\\varphi)\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n\\nonumber\n\\langle\\chi_\\epsilon\\chi_\\sigma\\rangle &=& \\left[(Q+R_k^\\varphi) (\\delta_j\\delta_j W_k)(Q+R_k^\\varphi)\\right]_{\\epsilon\\sigma} \\\\\n&&+ (Q\\varphi-l)_\\epsilon(Q\\varphi-l)_\\sigma -(Q+R_k^\\varphi)_{\\epsilon\\sigma}.\n\\label{eq:idk12}\n\\end{eqnarray}\nSimilarly, the derivative of Eq.\\ \\eqref{eq:expvchi} with respect to $j$ yields\n\\begin{eqnarray}\n\\langle\\tilde\\varphi_\\epsilon\\chi_\\sigma\\rangle = \\langle\\tilde\\varphi_\\epsilon\\tilde\\varphi_\\tau\\rangle (Q+R_k^\\varphi)_{\\tau\\sigma}-\\varphi_\\epsilon j_\\sigma -\\delta_{\\epsilon\\sigma}\\\\\n\\nonumber\n= \\varphi_\\epsilon (Q\\varphi)_\\sigma + \\left[(\\delta_j\\delta_j W_k)(Q+R_k^\\varphi)\\right]_{\\epsilon\\sigma}-\\varphi_\\epsilon l_\\sigma-\\delta_{\\epsilon\\sigma}.\n\\label{eq:idk13}\n\\end{eqnarray}\n\nWe now turn to the scale-dependence of $W_k[\\eta,j]$. In addition to $R_k^\\psi$ and $R_k^\\varphi$ also $Q$ and $H$ are $k$-dependent. For $H$ we assume\n\\begin{equation}\n\\partial_k H_{\\epsilon\\alpha\\beta} = (\\partial_k F_{\\epsilon\\rho}) H_{\\rho\\alpha\\beta}\n\\end{equation}\nwhere we take the dimensionless matrix $F$ to be symmetric for simplicity. For the operator $\\chi$ this gives\n\\begin{equation}\n\\partial_k \\chi_\\epsilon = \\partial_k H_{\\epsilon\\alpha\\beta}\\tilde\\psi_\\alpha\\tilde\\psi_\\beta = \\partial_k F_{\\epsilon\\rho} \\chi_\\rho.\n\\end{equation}\nFrom Eqs. \\eqref{eq:SFwithbosonfi} and \\eqref{eq:actionferbos} we can derive (for fixed $\\eta$, $j$)\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\langle\\tilde\\psi(\\partial_k R_k^\\psi)\\tilde\\psi\\rangle - \\frac{1}{2}\\langle\\tilde\\varphi(\\partial_k R_k^\\varphi+\\partial_k Q)\\tilde\\varphi\\rangle\\\\\n\\nonumber\n&&-\\frac{1}{2}\\langle\\chi\\left(\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_k F)Q^{-1}\\right)\\chi\\rangle\\\\\n&&+\\langle\\tilde\\varphi(\\partial_k F)\\chi\\rangle.\n\\end{eqnarray}\nNow we insert Eqs. \\eqref{eq:idk12} and \\eqref{eq:idk13}\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\psi (\\partial_k R_k^\\psi)\\psi - \\frac{1}{2}\\varphi (\\partial_k R_k^\\varphi) \\varphi\\\\\n\\nonumber\n&&-\\frac{1}{2} \\text{STr}\\,\\{ (\\delta_\\eta\\delta_\\eta W_k)(\\partial_k R_k^\\psi) \\}\\\\\n\\nonumber\n&&- \\frac{1}{2}\\text{Tr}{\\big \\{} (\\delta_j\\delta_j W_k)(\\partial_k R_k^\\varphi){\\big \\}}\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{Tr} {\\big \\{}{\\big [}Q(\\partial_k Q^{-1})R_k^\\varphi+R_k^\\varphi(\\partial_k Q^{-1})Q\\\\\n\\nonumber\n&&\\,\\,\\,+R_k^\\varphi(\\partial_k Q^{-1})R_k^\\varphi+ R_k^\\varphi Q^{-1}(\\partial_k F)(Q+R_k)\\\\\n\\nonumber\n&&\\,\\,\\,+(Q+R_k^\\varphi)(\\partial_kF)Q^{-1}R_k^\\varphi{\\big ]}(\\delta_j\\delta_j W_k){\\big \\}}\\\\\n\\nonumber\n&&+\\frac{1}{2}l\\left[(\\partial_k Q^{-1})Q+Q^{-1}(\\partial_k F)Q\\right]\\varphi\\\\\n\\nonumber\n&&+\\frac{1}{2}\\varphi \\left[Q(\\partial_k Q^{-1})+Q(\\partial_k F)Q^{-1}\\right]l\\\\\n\\nonumber\n&&-\\frac{1}{2}l \\left[\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_kF)Q^{-1}\\right]l\\\\\n\\nonumber\n&&+\\frac{1}{2} \\text{Tr} \\left\\{\\left[\\partial_k Q^{-1}+Q^{-1}(\\partial_k F)+(\\partial_k F)Q^{-1}\\right]R_k^\\varphi\\right\\}\\\\\n&&+\\frac{1}{2}\\text{Tr} \\left\\{Q\\partial_k Q^{-1}\\right\\}.\n\\label{eq:longflowW}\n\\end{eqnarray}\nThe supertrace $\\text{STr}$ contains the appropriate minus sign in the case that $\\psi_\\alpha$ are fermionic Grassmann variables.\n\nEquation \\eqref{eq:longflowW} can be simplified substantially when we restrict the $k$-dependence of $F$ and $Q$ such that\n\\begin{equation}\n\\partial_k F = -Q(\\partial_k Q^{-1}) = -(\\partial_k Q^{-1})Q.\n\\label{eq:restrSQ}\n\\end{equation}\nIn fact, one can show that the freedom to choose $F$ and $Q$ independent from each other that is lost by this restriction, is equivalent to the freedom to make a linear change in the source $j$, or at a later stage of the flow equation in the expectation value $\\varphi$. With the choice in Eq.\\ \\eqref{eq:restrSQ} we obtain\n\\begin{eqnarray}\n\\nonumber\n\\partial_k W_k &=& -\\frac{1}{2}\\psi (\\partial_k R_k^\\psi)\\psi -\\frac{1}{2}\\varphi(\\partial_k R_k^\\varphi)\\varphi\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{STr}{\\big \\{}(\\partial_k R_k^\\psi)(\\delta_\\eta\\delta_\\eta W_k){\\big \\}}\\\\\n\\nonumber\n&&-\\frac{1}{2}\\text{Tr}{\\big \\{} \\left[\\partial_k R_k^\\varphi-R_k^\\varphi(\\partial_k Q^{-1})R_k^\\varphi\\right](\\delta_j\\delta_j W_k){\\big \\}}\\\\\n\\nonumber\n&&+\\frac{1}{2}l(\\partial_k Q^{-1})l+\\frac{1}{2}\\text{Tr}{\\{}\\partial_kQ^{-1}(Q-R_k^\\varphi){\\}}.\n\\label{eq:shortflowW}\n\\end{eqnarray}\nThe last term is independent of the sources $\\eta$ and $j$ and is therefore irrelevant for many purposes.\n\n\n\\section{Flowing action}\n\nThe average action or flowing action is defined by subtracting from the Legendre transform\n\\begin{equation}\n\\tilde\\Gamma_k[\\psi,\\varphi] = \\eta \\psi + j \\varphi - W_k[\\eta,j]\n\\end{equation}\nthe cutoff terms\n\\begin{equation}\n\\Gamma_k[\\psi,\\varphi]=\\tilde\\Gamma_k[\\psi,\\varphi]-\\frac{1}{2}\\psi R_k^\\psi\\psi -\\frac{1}{2}\\varphi R_k^\\varphi\\varphi.\n\\label{eq:defflowingaction}\n\\end{equation}\nAs usual, the arguments of the effective action are given by\n\\begin{equation}\n\\psi_\\alpha=\\frac{\\delta}{\\delta \\eta_\\alpha}W_k \\quad \\text{and} \\quad \\varphi_\\epsilon=\\frac{\\delta}{\\delta j_\\epsilon} W_k.\n\\end{equation}\nBy taking the derivative of Eq.\\ \\eqref{eq:defflowingaction} it follows\n\\begin{equation}\n\\frac{\\delta}{\\delta \\psi_\\alpha}\\Gamma_k = \\pm \\eta_\\alpha - (R_k^\\psi)_{\\alpha\\beta} \\psi_\\beta,\n\\end{equation}\nwhere the upper (lower) sign is for a bosonic (fermionic) field $\\psi$. Similarly,\n\\begin{equation}\n\\frac{\\delta}{\\delta \\varphi_\\epsilon}\\Gamma_k = j_\\epsilon - (R_k^\\varphi)_{\\epsilon\\sigma} \\varphi_\\sigma=l_\\epsilon.\n\\end{equation}\nIn the matrix notation\n\\begin{eqnarray}\n\\nonumber\nW_k^{(2)} &=& \\begin{pmatrix}\\delta_\\eta\\delta_\\eta W_k, && \\delta_\\eta\\delta_j W_k \\\\ \\delta_j\\delta_\\eta W_k, && \\delta_j\\delta_j W_k\\end{pmatrix},\\\\\n\\nonumber\n\\Gamma_k^{(2)} &=& \\begin{pmatrix}\\delta_\\psi\\delta_\\psi \\Gamma_k, && \\delta_\\psi\\delta_\\varphi \\Gamma_k \\\\\\delta_\\varphi\\delta_\\psi \\Gamma_k, && \\delta_\\varphi\\delta_\\varphi \\Gamma_k\\end{pmatrix},\\\\\nR_k&=&\\begin{pmatrix} R_k^\\psi, && 0 \\\\ 0,&& R_k^\\varphi \\end{pmatrix},\n\\end{eqnarray}\nit is straight forward to establish\n\\begin{equation}\nW_k^{(2)} \\,\\tilde\\Gamma_k^{(2)} =1,\\quad\\quad W_k^{(2)} = (\\Gamma_k^{(2)}+R_k)^{-1}.\n\\end{equation}\n\nIn order to derive the exact flow equation for the average action we use the identity\n\\begin{equation}\n\\partial_k \\tilde \\Gamma_k{\\big |}_{\\psi,\\varphi} = -\\partial_k W_k{\\big |}_{\\eta,j}.\n\\end{equation}\nThis yields the central result of this chapter\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\Gamma_k &=& \\frac{1}{2}\\text{STr}\\, \\left\\{(\\Gamma_k^{(2)}+R_k)^{-1}\\left(\\partial_k R_k-R_k(\\partial_k Q^{-1})R_k\\right)\\right\\}\\\\\n&&-\\frac{1}{2}\\Gamma_k^{(1)} \\left(\\partial_k Q^{-1}\\right)\\Gamma_k^{(1)}+\\gamma_k\n\\label{eq:flowequationGamma}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\gamma_k=-\\frac{1}{2} \\text{Tr}\\left\\{ (\\partial_k Q^{-1})(Q-R_k)\\right\\}.\n\\end{equation}\nAs it should be, this reduces to the standard flow equation for a framework with fixed partial bosonization in the limit $\\partial_k Q^{-1}=0$. The additional term is quadratic in the first derivative of $\\Gamma_k$ with respect to $\\varphi$ -- we recall that $\\partial_k Q^{-1}$ has non-zero entries only in the $\\varphi$-$\\varphi$ block. Furthermore there is a field independent term $\\gamma_k$ that can be neglected for many purposes. At this point a few remarks are in order.\n\n(i) For $k\\to 0$ the cutoffs $R_k^\\psi$, $R_k^\\varphi$ should vanish. This ensures that the correlation functions of the partially bosonized theory are simply related to the original correlation functions generated by $W_0[\\eta]$, Eq.\\ \\eqref{eq:scaledepSF}, namely\n\\begin{eqnarray}\n\\nonumber\nW_0[\\eta, j] &=& \\ln \\left(\\int D\\tilde \\psi \\, e^{-S_\\psi[\\tilde \\psi]+\\eta\\tilde\\psi+jQ^{-1}\\chi}\\right)+\\frac{1}{2}j\\, Q^{-1} \\,j+\\text{const.},\\\\\nW_0[\\eta,j=0] &=& W_0[\\eta] +\\text{const.}\n\\end{eqnarray}\nKnowledge of the dependence on $j$ permits the straightforward computation of correlation functions for composite operators $\\chi$.\n\\newline\n\n(ii) For solutions of the flow equation one needs a well known ``initial value'' which describes the microscopic physics. This can be achieved by letting the cutoffs $R_k^\\psi$, $R_k^\\varphi$ diverge for $k\\to\\Lambda$ (or $k\\to\\infty$). In this limit the functional integral in Eqs. \\eqref{eq:SFwithbosonfi}, \\eqref{eq:actionferbos} can be solved exactly and one finds\n\\begin{equation}\n\\Gamma_\\Lambda[\\psi,\\varphi] = S_\\psi[\\psi] +\\frac{1}{2}\\varphi Q_\\Lambda \\varphi +\\frac{1}{2}\\chi[\\psi]Q_\\Lambda^{-1} \\chi[\\psi]-\\varphi \\chi[\\psi].\n\\label{eq:averageactionmicrosc}\n\\end{equation}\nThis equals the ``classical action'' obtained from a Hubbard-Stratonovich transformation, with $\\chi$ expressed in terms of $\\psi$. \n\n(iii) In our derivation we did not use that $\\chi$ is quadratic in $\\psi$. We may therefore take for $\\chi$ an arbitrary bosonic functional of $\\psi$. It is straightforward to adapt our formalism such that also fermionic composite operators can be considered.\n\nThe flow equation \\eqref{eq:flowequationGamma} has a simple structure of a one loop expression with a cutoff insertion -- $\\text{STr}$ contains the appropriate integration over the loop momentum -- supplemented by a ``tree-contribution'' $\\sim \\left(\\Gamma_k^{(1)}\\right)^2$. Nevertheless, it is an exact equation, containing all orders of perturbation theory as well as non-perturbative effects. The simple form of the tree contributions will allow for easy implementations of a scale dependent partial bosonization. The details of this can be found in \\cite{FloerchingerWetterich2009exact}.\n\n\\section{General coordinate transformations}\n\nIt is sometimes useful to perform a change of coordinates in the space of fields during the renormalization flow. In this section we discuss the transformation behavior of the Wetterich equation \\eqref{eq4:Wettericheqn} under such a change of the basis for the fields. We follow here the calculation in \\cite{Wetterich1996, Gies:2001nw} For simplicity we restrict the discussion to bosonic fields. It is straightforward to transfer this to fermions as well \\cite{Wetterich1996, Gies:2001nw}. Similarly, one might also consider a general coordinate transformation for the generalized flow equation \\eqref{eq:flowequationGamma}. \n\nLet us consider a transformation of the form\n\\begin{equation}\n\\Phi \\to \\Psi[\\Phi].\n\\label{eq2:Fieldmap}\n\\end{equation}\nHere we denote by $\\Phi$ the original fields. The functional $\\Psi[\\Phi]$ is a $k$-dependent map of the old coordinates to the new ones. We assume that the map in Eq.\\ \\eqref{eq2:Fieldmap} is invertible and write the inverse\n\\begin{equation}\n\\Phi[\\Psi].\n\\end{equation}\nIn terms of the fields $\\Psi$ the definition of the flowing action reads\n\\begin{equation}\n\\Gamma_k[\\Psi] = J_\\alpha \\Phi_\\alpha [\\Psi]-W_k[J]-\\frac{1}{2} \\Phi[\\Psi] R_k \\Phi[\\Psi]\n\\end{equation}\nwhere $J$ is determined by\n\\begin{equation}\n\\Phi_\\alpha[\\Psi]=\\frac{\\delta W_k}{\\delta J_\\alpha}.\n\\end{equation}\nIn the limit $k\\to0$ the flowing action is a Legendre transform with respect to the old fields $\\Phi$ but not with respect to the new fields $\\Psi$. This implies for example that $\\Gamma[\\Psi]$ is not necessarily convex with respect to the fields $\\Psi$. In addition, only the fields $\\Phi$ are expectation values of fields as in Eq.\\ \\eqref{eq1:expectvalue}. The relation of the fields $\\Psi$ to the microscopic fields $\\tilde \\Phi$ is more complicated. The field equation reads in terms of the new fields\n\\begin{equation}\n\\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} = J_\\beta \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\alpha} - \\frac{\\delta}{\\delta \\Psi_\\alpha} \\Delta S_k.\n\\end{equation}\nNote that the cutoff term\n\\begin{equation}\n\\Delta S_k = \\frac{1}{2}\\Phi_\\alpha[\\Psi](R_k)_{\\alpha\\beta} \\Phi_\\beta[\\psi]\n\\end{equation}\nis not necessarily quadratic in the fields $\\Psi$. The matrix $R_k$ obtains from\n\\begin{eqnarray}\n\\nonumber\n(R_k)_{\\alpha\\beta} &=& \\frac{\\delta}{\\delta \\Phi_\\alpha}\\frac{\\delta }{\\delta \\Phi_\\beta} \\Delta S_k\\\\\n&=& \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\alpha} \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\mu}\\frac{\\delta }{\\delta \\Psi_\\nu} \\Delta S_k\\right) + \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu}\\Delta S_k\\right).\n\\label{eq3:defRk}\n\\end{eqnarray}\nSimilarly we obtain for the matrix $\\Gamma_k^{(2)}$\n\\begin{eqnarray}\n\\nonumber\n(\\Gamma_k^{(2)})_{\\alpha\\beta} &=& \\frac{\\delta}{\\delta \\Phi_\\alpha}\\frac{\\delta }{\\delta \\Phi_\\beta} \\Gamma_k\\\\\n&=& \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\alpha} \\frac{\\delta \\Psi_\\mu}{\\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\mu}\\frac{\\delta }{\\delta \\Psi_\\nu} \\Gamma_k\\right) + \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu}\\Gamma_k\\right).\n\\label{eq3:defGamma2}\n\\end{eqnarray}\nWe now come to the scale dependence of $\\Gamma_k$. It is given by\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi]=\\partial_k \\Gamma_k[\\Psi]{\\big |}_\\Phi - \\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\label{eq2:scaledepGammaphipsi}\n\\end{equation}\nFor the first term on the right hand side of Eq.\\ \\eqref{eq2:scaledepGammaphipsi} we can use the Wetterich equation\\eqref{eq4:Wettericheqn} and obtain\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi] = \\frac{1}{2} \\text{Tr} (\\Gamma_k^{(2)}+R_k)^{-1} \\partial_k R_k-\\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\label{eq3:Wettericheqwithcoordtransf}\n\\end{equation}\nWe emphasize that $\\Gamma_k^{(2)}$ and $R_k$ are now somewhat more complicated objects then usually. They are defined by Eqs. \\eqref{eq3:defRk} and \\eqref{eq3:defGamma2}. One might also define the transformed matrices\n\\begin{eqnarray}\n\\nonumber\n(\\widehat \\Gamma_k^{(2)})_{\\mu\\nu} &=& \\frac{\\delta}{\\delta \\Psi_\\mu} \\frac{\\delta}{\\delta \\Psi_\\nu} \\Gamma_k + \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu} \\Gamma_k\\right),\\\\\n(\\widehat R_k)_{\\mu\\nu} &=& \\frac{\\delta}{\\delta \\Psi_\\mu} \\frac{\\delta}{\\delta \\Psi_\\nu} \\Delta S_k + \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} \\frac{\\delta^2 \\Psi_\\nu}{\\delta \\Phi_\\alpha \\delta \\Phi_\\beta}\\left(\\frac{\\delta}{\\delta \\Psi_\\nu} \\Delta S_k\\right),\n\\label{eq3:hatteddef}\n\\end{eqnarray}\nand similar\n\\begin{eqnarray}\n(\\widehat{\\partial_k R_k})_{\\mu\\nu} &=& \\frac{\\delta \\Phi_\\alpha}{\\delta \\Psi_\\mu} \\frac{\\delta \\Phi_\\beta}{\\delta \\Psi_\\nu} (\\partial_k R_k)_{\\alpha\\beta}.\n\\label{eq3:hatted2}\n\\end{eqnarray}\nIn Eq.\\ \\eqref{eq3:hatteddef} the second functional derivatives are supplemented by connection terms as appropriate for general (non-linear) coordinate systems. With Eqs.\\ \\eqref{eq3:hatteddef} \\eqref{eq3:hatted2} the flow equation for $\\Gamma_k$ reads\n\\begin{equation}\n\\partial_k \\Gamma_k[\\Psi] = \\frac{1}{2} \\text{Tr} (\\widehat \\Gamma_k+\\widehat R_k)^{-1} \\widehat{\\partial_k R_k} - \\frac{\\delta \\Gamma_k}{\\delta \\Psi_\\alpha} \\partial_k \\Psi_\\alpha{\\big |}_\\Phi.\n\\end{equation}\nUnfortunately this equation has lost its one-loop structure due to the connection terms in Eq.\\ \\eqref{eq3:hatteddef}. An important exception is a linear coordinate transformation\n\\begin{equation}\n\\Psi_\\alpha[\\Phi] = \\Xi_\\alpha + M_{\\alpha\\beta} \\Phi_\\beta.\n\\end{equation}\nIn that case the terms $\\frac{\\delta^2 \\Psi}{\\delta \\Phi\\delta \\Phi}$ vanish and the one-loop structure is preserved.\n\n\n\\subsection{Particle-hole fluctuations}\n\\label{sec:ParticleHole}\nThe BCS theory of superfluidity in a Fermi gas of atoms is valid for a small attractive interaction between the fermions \\cite{PhysRev.104.1189, Bardeen:1957kj, Bardeen:1957mv}. In a renormalization group setting, the features of BCS theory can be described in a purely fermionic language. The only scale dependent object is the fermion interaction vertex $\\lambda_\\psi$. The flow depends on the temperature and the chemical potential. \nFor positive chemical potential ($\\mu>0$) and small temperatures $T$, the appearance of pairing is indicated by the divergence of $\\lambda_\\psi$.\n\nIn general, the interaction vertex is momentum dependent and represented by a term\n\\begin{eqnarray}\n\\Gamma_{\\lambda_\\psi}&=&\\int_{p_1,p_2,p_1^\\prime,p_2^\\prime}\\lambda_{\\psi}(p_1^\\prime,p_1,p_2^\\prime,p_2)\\nonumber\\\\\n& &\\times\\psi_1^{\\ast}(p_1^\\prime)\\psi_1(p_1)\\psi_2^{\\ast}(p_2^\\prime)\\psi_2(p_2)\n\\label{eq:momentumdepvertex}\n\\end{eqnarray}\nin the effective action. In a homogeneous situation, momentum conservation restricts the expression in Eq.\\ \\eqref{eq:momentumdepvertex} to three independent momenta, $\\lambda_{\\psi}\\sim \\delta(p_1^\\prime+p_2^\\prime-p_1-p_2)$. The flow of $\\lambda_\\psi$ has two contributions which are depicted in Fig. \\ref{fig:lambdaflow}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{PHfigure1.eps}\n\\caption{Running of the momentum dependent vertex $\\lambda_{\\psi}$. Here $\\tilde{\\partial}_k$ indicates derivatives with respect to the cutoff terms in the propagators and does not act on the vertices in the depicted diagrams. We will refer to the first loop as the particle-particle loop (pp-loop) and to the second one as the particle-hole loop (ph-loop).}\n\\label{fig:lambdaflow}\n\\end{figure}\nThe first diagram describes particle-particle fluctuations. For $\\mu>0$ its effect increases as the temperature $T$ is lowered. For small temperatures $T\\leq T_{c,\\text{BCS}}$ the logarithmic divergence leads to the appearance of pairing, as $\\lambda_\\psi\\to \\infty$. \n\nIn the purely fermionic formulation the flow equation for $\\lambda_{\\psi}$ has the general form \\cite{Ellwanger1994137, Aoki2000, PTPS.160.58, PTP.105.1}\n\n\\begin{equation}\\label{eq:lambdapsi2}\n\\partial_k \\lambda_{\\psi}^{\\alpha}=A^{\\alpha}_{\\beta\\gamma}\\lambda_{\\psi}^{\\beta}\\lambda_{\\psi}^{\\gamma}\\,, \n\\end{equation}\nwith $\\alpha, \\beta, \\gamma$ denoting momentum as well as spin labels. A numerical solution of this equation is rather involved due to the rich momentum structure. The case of the attractive Hubbard model in two dimensions, which is close to our problem, has recently been discussed in \\cite{strack:014522}.\nThe BCS approach concentrates on the pointlike coupling, evaluated by setting all momenta to zero. For $k \\rightarrow 0,\\ \\mu_0 \\rightarrow 0,\\ T \\rightarrow 0$ and $n \\rightarrow 0$ this coupling is related the scattering length, $a= \\frac{1}{8\\pi} \\lambda_{\\psi}(p_i=0)$. In the BCS approximation only the first diagram in Fig. \\ref{fig:lambdaflow} is kept, and the momentum dependence of the couplings on the right-hand side of Eq.\\ \\eqref{eq:lambdapsi2} is neglected, by replacing $\\lambda_{\\psi}^{\\alpha}$ by the pointlike coupling evaluated at zero momentum. In terms of the scattering length $a$, Fermi momentum $k_F$ and Fermi temperature $T_F$, the critical temperature is found to be \n\\begin{equation}\n \\frac{T_c}{T_F}\\approx 0.61 e^{\\pi\/(2 a k_F)}\\,.\n\\end{equation}\nThis is the result of the original BCS theory. However, it is obtained by entirely neglecting the second loop in Fig. \\ref{fig:lambdaflow}, which describes particle-hole fluctuations. At zero temperature the expression for this second diagram vanishes if it is evaluated for vanishing external momenta. Indeed, the two poles of the frequency integration are always either in the upper or lower half of the complex plane and the contour of the frequency integration can be closed in the half plane without poles. \n\nThe dominant part of the scattering in a fermion gas occurs, however, for momenta on the Fermi surface rather than for zero momentum. For non-zero momenta of the \"external particles\" the second diagram in Fig. \\ref{fig:lambdaflow} - the particle-hole channel - makes an important contribution. \n\nSetting the external frequencies to zero, we find that the inverse propagators in the particle-hole loop are \n\\begin{equation}\\label{eq:loopmom1}\nP_\\psi(q)=i q_0 +(\\vec{q}-\\vec{p}_1)^2-\\mu\\,,\n\\end{equation}\nand \n\\begin{equation}\\label{eq:loopmom2}\nP_\\psi(q)=i q_0 +(\\vec{q}-\\vec{p}_2^{\\,\\prime})^2-\\mu.\n\\end{equation}\nDepending on the value of the momenta $\\vec{p}_1$ and $\\vec p_2^{\\,\\prime}$, there are now values of the loop momentum $\\vec q$ for which the poles of the frequency integration are in different half planes so that there is a nonzero contribution even for $T=0$.\n\nTo include the effect of particle-hole fluctuations one could try to take the full momentum dependence of the vertex $\\lambda_\\psi$ into account. However, this leads to complicated expressions which are hard to solve even numerically. \nOne therefore often restricts the flow to the running of a single coupling $\\lambda_\\psi$ by choosing an appropriate projection prescription to determine the flow equation. In the purely fermionic description with a single running coupling $\\lambda_\\psi$, this flow equation has a simple structure. The solution for $\\lambda_{\\psi}^{-1}$ can be written as a contribution from the particle-particle (first diagram in Fig. \\ref{fig:lambdaflow}, pp-loop) and the particle-hole (second diagram, ph-loop) channels \n\\begin{equation}\\label{PHComp}\n \\frac{1}{\\lambda_{\\psi}(k=0)}=\\frac{1}{\\lambda_{\\psi}(k=\\Lambda)} + \\mbox{pp-loop} + \\mbox{ph-loop}\\,.\n\\end{equation}\nSince the ph-loop depends only weakly on the temperature, one can evaluate it at $T=0$ and add it to the initial value $\\lambda_\\psi(k=\\Lambda)^{-1}$. Since $T_c$ depends exponentially on the \"effective microscopic coupling\"\n\\begin{equation}\n \\left(\\lambda_{\\psi,\\Lambda}^{\\text{eff}}\\right)^{-1}=\\lambda_{\\psi}(k=\\Lambda)^{-1} + \\text{ph-loop}\\,,\n\\end{equation}\nany shift in $\\left(\\lambda_{\\psi,\\Lambda}^{\\text{eff}}\\right)^{-1}$ results in a multiplicative factor for $T_c$. The numerical value of the ph-loop and therefore of the correction factor for $T_c\/T_F$ depends on the precise projection description.\n\nLet us now choose the appropriate momentum configuration. For the formation of Cooper pairs, the relevant momenta lie on the Fermi surface, \n\n\\begin{equation}\\label{absmom}\n\\vec{p}^2_1=\\vec{p}^2_2=\\vec{p}^{{\\,\\prime}2}_1=\\vec{p}^{{\\,\\prime}2}_2=\\mu\\,,\n\\end{equation}\nand point in opposite directions\n\n\\begin{equation}\\label{oppmom}\n \\vec{p}_1=-\\vec{p}_2,\\ \\vec{p}^{\\,\\prime}_1=-\\vec{p}^{\\,\\prime}_2\\,.\n\\end{equation}\nThis still leaves the angle between $\\vec{p}_1$ and $\\vec{p}^{\\,\\prime}_1$ unspecified. Gorkov's approximation uses Eqs. \\eqref{absmom} and \\eqref{oppmom} and projects on the $s$-wave by averaging over the angle between $\\vec{p}_1$ and $\\vec{p}^{\\,\\prime}_1$. One can shift the loop momentum such that the internal propagators depend on $\\vec{q}^2$ and $(\\vec{q}+\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2$. In terms of spherical coordinates the first propagator depends only on the magnitude of the loop momentum $q^2=\\vec{q}^2$, while the second depends additionally on the transfer momentum $\\tilde{p}^2=\\frac{1}{4}(\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2$ and the angle $\\alpha$ between $\\vec{q}$ and $(\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)$,\n\n\\begin{equation}\n (\\vec{q}+\\vec{p}_1-\\vec{p}^{\\,\\prime}_1)^2=q^2+4\\tilde{p}^2+4\\,q\\, \\tilde{p}\\,\\text{cos}(\\alpha)\\,.\n\\end{equation}\nPerforming the loop integration involves the integration over $q^2$ and the angle $\\alpha$. The averaging over the angle between $\\vec{p}_1$ and $\\vec{p}_1^{\\,\\prime}$ translates to an averaging over $\\tilde{p}^2$. Both can be done analytically \\cite{Heiselberg} for the fermionic particle-hole diagram and the result gives the well-known Gorkov correction to BCS theory, resulting in\n\n\\begin{equation}\nT_c=\\frac{1}{(4e)^{1\/3}}T_{c,\\text{BCS}}\\approx \\frac{1}{2.2} T_{c,\\text{BCS}}\\,.\n\\end{equation}\n\nIn our treatment we will use a numerically simpler projection by choosing $\\vec{p}^{\\,\\prime}_1=\\vec{p}_1$, and $\\vec{p}_2=\\vec{p}^{\\,\\prime}_2$, without an averaging over the angle between $\\vec{p}^{\\,\\prime}_1$ and $\\vec{p}_1$. The size of $\\tilde p^2 = \\vec{p}^2_1$ is chosen such that the one-loop result reproduces exactly the result of the Gorkov correction, namely $\\tilde p = 0.7326 \\sqrt{\\mu}$. Choosing different values of $\\tilde p$ demonstrates the dependence of $T_c$ on the projection procedure and may be taken as an estimate for the error that arises from the limitation to one single coupling $\\lambda_{\\psi}$ instead of a momentum dependent function.\n\n\\subsection{Bosonization}\nIn Sec. \\ref{sec:BCS-BECCrossover} we describe an effective four-fermion interaction by the exchange of a boson. In this picture the phase transition to the superfluid phase is indicated by the vanishing of the bosonic ``mass term'' $m^2 = 0$. Negative $m^2$ leads to the spontaneous breaking of U(1)-symmetry, since the minimum of the effective potential occurs for a nonvanishing superfluid density $\\rho_0>0$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{PHfigure2.eps}\n\\caption{Flow of the boson propagator.}\n\\label{fig:bosonexchangeloop}\n\\end{figure}\nFor $m^2 \\geq 0$ we can solve the field equation for the boson $\\varphi$ as a functional of $\\psi$ and insert the solution into the effective action. This leads to an effective four-fermion vertex describing the scattering $\\psi_1(p_1)\\psi_2(p_2)\\to \\psi_1(p_1^{\\,\\prime})\\psi_2(p_2^{\\,\\prime})$\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=\\frac{-h^2}{i(p_1+p_2)_0+\\frac{1}{2}(\\vec p_1+\\vec p_2)^2+m^2}.\n\\label{eq:lambdapsieff}\n\\end{equation}\nTo investigate the breaking of U(1) symmetry and the onset of superfluidity, we first consider the flow of the bosonic propagator, which is mainly driven by the fermionic loop diagram. For the effective four-fermion interaction this accounts for the particle-particle loop (see r.h.s. of Fig. \\ref{fig:bosonexchangeloop}). In the BCS limit of a large microscopic $m_\\Lambda^2$ the running of $m^2$ for $k\\to0$ reproduces the BCS result \\cite{PhysRev.104.1189, Bardeen:1957kj, Bardeen:1957mv}.\n\nThe particle-hole fluctuations are not accounted for by the renormalization of the boson propagator. Indeed, we have neglected so far that a term\n\n\\begin{equation}\\label{eq:fourfermionvertex}\n \\int_{\\tau,\\vec{x}}\\lambda_{\\psi}\\psi_1^{\\ast}\\psi_1\\psi_2^{\\ast}\\psi_2\\,,\n\\end{equation}\nin the effective action is generated by the flow. This holds even if the microscopic pointlike interaction is absorbed by a Hubbard-Stratonovich transformation into an effective boson exchange such that $\\lambda_\\psi(\\Lambda)=0$. The strength of the total interaction between fermions\n\n\\begin{equation}\\label{eq:lambdapsieff2}\n\\lambda_{\\psi,\\text{eff}}=\\frac{-h^2}{i(p_1+p_2)_0+\\frac{1}{2}(\\vec p_1+\\vec p_2)^2+m^2} + \\lambda_{\\psi}\n\\end{equation}\nadds $\\lambda_\\psi$ to the piece generated by boson exchange. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{PHfigure3.eps}\n\\caption{Box diagram for the flow of the four-fermion interaction.}\n\\label{fig:boxes}\n\\end{figure}\nIn the partially bosonized formulation, the flow of $\\lambda_\\psi$ is generated by the box-diagrams depicted in Fig. \\ref{fig:boxes}. We may interpret these diagrams and establish a direct connection to the particle-hole diagrams depicted in Fig. \\ref{fig:lambdaflow} on the BCS side of the crossover and in the microscopic regime. There the boson gap $m^2$ is large. In this case, the effective fermion interaction in Eq.\\ \\eqref{eq:lambdapsieff2} becomes momentum independent. Diagrammatically, this is represented by contracting the bosonic propagator. One can see, that the box-diagram in Fig. \\ref{fig:boxes} is then equivalent to the particle-hole loop investigated in Sec. \\ref{sec:ParticleHole} with the pointlike approximation $\\lambda_{\\psi,\\text{eff}}\\to-\\frac{h^2}{m^2}$ for the fermion interaction vertex. As mentioned above, these contributions vanish for $T=0$, $\\mu<0$ for arbitrary $\\vec p_i$. Indeed, at zero temperature, the summation over the Matsubara frequencies becomes an integral. All the poles of this integration are in the upper half of the complex plane and the integration contour can be closed in the lower half plane. We will evaluate $\\partial_k \\lambda_\\psi$ for $\\vec p_1=\\vec p_1^{\\,\\prime}=-\\vec p_2=-\\vec p_2^{\\,\\prime}$, $|\\vec p_1|=\\tilde p = 0.7326 \\sqrt{\\mu}$, as discussed in the Sec. \\ref{sec:ParticleHole}. For $\\mu>0$ this yields a nonvanishing flow even for $T=0$.\n\nAnother simplification concerns the temperature dependence. While the contribution of particle-particle diagrams becomes very large for small temperatures, this is not the case for particle-hole diagrams. For nonvanishing density and small temperatures, the large effect of particle-particle fluctuations leads to the spontaneous breaking of the U(1) symmetry and the associated superfluidity. In contrast, the particle-hole fluctuations lead only to quantitative corrections and depend only weakly on temperature. This can be checked explicitly in the pointlike approximation, and holds not only in the BCS regime where $T\/\\mu \\ll 1$, but also for moderate $T\/\\mu$ as realized at the critical temperature in the unitary regime. We can therefore evaluate the box-diagrams in Fig. \\ref{fig:lambdaflow} for zero temperature. We note that an implicit temperature dependence, resulting from the couplings parameterizing the boson propagator, is taken into account.\n\nAfter these preliminaries, we can now incorporate the effect of particle-hole fluctuations in the renormalization group flow. A first idea might be to include the additional term \\eqref{eq:fourfermionvertex} in the truncation and to study the effects of $\\lambda_{\\psi}$ on the remaining flow equations. On the initial or microscopic scale one would have $\\lambda_{\\psi}=0$, but it would then be generated by the flow. This procedure, however, has several shortcomings. First, the appearance of a local condensate would now be indicated by the divergence of the effective four-fermion interaction\n\n\\begin{equation}\n \\lambda_{\\psi,\\text{eff}}=-\\frac{h^2}{m^2}+\\lambda_{\\psi}\\,.\n\\end{equation}\nThis might lead to numerical instabilities for large or diverging $\\lambda_{\\psi}$. The simple picture that the divergence of $\\lambda_{\\psi,\\text{eff}}$ is connected to the onset of a nonvanishing expectation value for the bosonic field $\\varphi_0$, at least on intermediate scales, would not hold anymore. Furthermore, the dependence of the box-diagrams on the center of mass momentum would be neglected completely by this procedure. Close to the resonance the momentum dependence of the effective four-fermion interaction in the bosonized language as in Eq.\\ (\\ref{eq:lambdapsieff2}) is crucial, and this might also be the case for the particle-hole contribution.\n\nAnother, much more elegant way to incorporate the effect of particle-hole fluctuations is provided by the method of bosonization \\cite{Gies:2001nw, Gies:2002kd, Pawlowski2007a}, see also chapter \\ref{ch:Generalizedflowequation}. For this purpose, we use scale dependent fields in the average action. The scale dependence of $\\Gamma_k[\\chi_k]$ is modified by a term reflecting the $k$-dependence of the argument $\\chi_k$ \\cite{Gies:2001nw, Gies:2002kd}\n\n\\begin{equation}\\label{eq:scalefield}\n \\partial_k \\Gamma_k[\\chi_k]=\\int\\frac{\\delta \\Gamma_k}{\\delta \\chi_k}\\partial_k\\chi_k+\\frac{1}{2}\\mathrm{STr}\\left[ \\left(\\Gamma_k^{(2)}+R_k \\right)^{-1} \\partial_k R_k\\right] \\,.\n\\end{equation}\n\nFor our purpose it is sufficient to work with scale dependent bosonic fields $\\bar\\varphi$ and keep the fermionic field $\\psi$ scale independent. In practice, we employ bosonic fields $\\bar\\varphi_k^*$, and $\\bar\\varphi_k$ with an explicit \nscale dependence which reads in momentum space\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\bar \\varphi_k(q) & = & (\\psi_1\\psi_2)(q) \\partial_k \\upsilon\\,,\\\\\n\\partial_k \\bar \\varphi_k^*(q) & = & (\\psi_2^\\dagger\\psi_1^\\dagger)(q) \\partial_k \\upsilon.\n\\label{eq:scaledependenceoffields}\n\\end{eqnarray}\nIn consequence, the flow equations in the symmetric regime get modified\n\n\\begin{eqnarray}\n \\partial_k \\bar{h} &=& \\partial_k \\bar{h}{\\big |}_{\\bar \\varphi_k}-\\bar{P}_{\\varphi}(q)\\partial_k \\upsilon\\,,\\\\\n \\partial_k \\lambda_{\\psi} &=& \\partial_k \\lambda_{\\psi}{\\big |}_{\\bar \\varphi_k}-2\\bar{h}\\partial_k \\upsilon.\n\\label{eq:modifiedflowequations}\n\\end{eqnarray}\nHere $q$ is the center of mass momentum of the scattering fermions. In the notation of Eq.\\ \\eqref{eq:lambdapsieff} we have $q=p_1+p_2$ and we will take $\\vec q=0$, and $q_0=0$. The first term on the right hand side in Eq.\\ \\eqref{eq:modifiedflowequations} gives the contribution of the flow equation which is valid for fixed field $\\bar \\varphi_k$. The second term comes from the explicit scale dependence of $\\bar \\varphi_k$. The inverse propagator of the complex boson field $\\bar{\\varphi}$ is denoted by $\\bar{P}_{\\varphi}(q)=\\bar A_\\varphi P_\\varphi(q)=\\bar A_\\varphi (m^2+i Z_\\varphi q_0+\\vec q^2\/2)$, cf. Eq.\\ \\eqref{eq:Bosonpropagator}. \n\nWe can choose $\\partial_k \\upsilon$ such that the flow of the coupling $\\lambda_{\\psi}$ vanishes, i.e. that we have $\\lambda_{\\psi}=0$ on all scales. This modifies the flow equation for the renormalized Yukawa coupling according to\n\n\\begin{equation}\n \\partial_k h = \\partial_k h{\\big |}_{\\bar \\varphi_k}-\\frac{m^2}{2h}\\partial_k \\lambda_{\\psi}{\\big |}_{\\bar \\varphi_k}\\,,\n \\label{eq:modfiedflowofh}\n\\end{equation}\nwith $\\partial_k h{\\big |}_{\\bar \\varphi_k}$ the contribution without bosonization and $\\partial_k \\lambda_\\psi{\\big |}_{\\bar \\varphi_k}$ given by the box diagram in Fig. \\ref{fig:boxes}. Since $\\lambda_\\psi$ remains zero during the flow, the effective four-fermion interaction $\\lambda_{\\psi,\\text{eff}}$ is now purely given by the boson exchange. However, the contribution of the particle-hole exchange diagrams is incorporated via the second term in Eq.\\ \\eqref{eq:modfiedflowofh}. \n\nIn the regime with spontaneously broken symmetry we use a real basis for the bosonic field\n\\begin{equation}\n \\bar{\\varphi}=\\bar{\\varphi}_0+\\frac{1}{\\sqrt{2}}(\\bar{\\varphi}_1+i\\bar{\\varphi}_2),\n\\end{equation}\nwhere the expectation value $\\bar{\\varphi}_0$ is chosen to be real without loss of generality. The real fields $\\bar \\varphi_1$ and $\\bar \\varphi_2$ then describe the radial and the Goldstone mode, respectively. To determine the flow equation of $\\bar{h}$, we use the projection description\n\n\\begin{equation}\\label{eq:projectiononh}\n \\partial_k \\bar{h}=i\\sqrt{2}\\Omega^{-1}\\frac{\\delta}{\\delta\\varphi_2(0)}\\frac{\\delta}{\\delta\\psi_1(0)}\\frac{\\delta}{\\delta\\psi_2(0)}\\partial_k \\Gamma_k\\,,\n\\end{equation}\nwith the four volume $\\Omega=\\frac{1}{T}\\int_{\\vec{x}}$. Since the Goldstone mode has vanishing ``mass'', the flow of the Yukawa coupling is not modified by the box diagram (Fig. \\ref{fig:boxes}) in the regime with spontaneous symmetry breaking.\n\nWe emphasize that the non-perturbative nature of the flow equations for the various couplings provides for a resummation similar to the one in Eq.\\ \\eqref{PHComp}, and thus goes beyond the treatment by Gorkov and Melik-Barkhudarov \\cite{Gorkov} which includes the particle-hole diagrams only in a perturbative way. Furthermore, the inner bosonic lines $h^2\/P_\\varphi(q)$ in the box-diagrams represent the center of mass momentum dependence of the four-fermion vertex. This center of mass momentum dependence is neglected in Gorkov's pointlike treatment, and thus represents a further improvement of the classic calculation. Actually, this momentum dependence becomes substantial -- and should not be neglected in a consistent treatment -- away from the BCS regime where the physics of the bosonic bound state sets in. Finally, we note that the truncation \\eqref{eq:truncation} supplemented with \\eqref{eq:fourfermionvertex} closes the truncation to fourth order in the fields except for a fermion-boson vertex $\\lambda_{\\psi\\varphi}\\psi^\\dagger\\psi\\varphi^*\\varphi$ which plays a role for the scattering physics deep in the BEC regime \\cite{DKS} but is not expected to have a very important impact on the critical temperature in the unitarity and BCS regimes.\n\n\n\\subsection{Critical temperature}\nTo obtain the flow equations for the running couplings of our truncation Eq.\\ \\eqref{eq:truncation} we use projection prescriptions similar to Eq.\\ \\eqref{eq:projectiononh}. The resulting system of ordinary coupled differential equations is then solved numerically for different chemical potentials $\\mu$ and temperatures $T$. For temperatures sufficiently small compared to the Fermi temperature $T_F=(3\\pi^2n)^{2\/3}$, $T\/T_F\\ll 1$ we find that the effective potential $U$ at the macroscopic scale $k=0$ develops a minimum at a nonzero field value $\\rho_0>0$, $\\partial_\\rho U(\\rho_0)=0$. The system is then in the superfluid phase. For larger temperatures we find that the minimum is at $\\rho_0=0$ and that the ``mass parameter'' $m^2$ is positive, $m^2=\\partial_\\rho U(0)>0$. The critical temperature $T_c$ of this phase transition between the superfluid and the normal phase is then defined as the temperature where one has\n\\begin{equation}\n\\rho_0=0,\\quad \\partial_\\rho U(0)=0\\quad \\text{at} \\quad k=0.\n\\end{equation}\nThroughout the whole crossover the transition $\\rho_0\\to0$ is continuous as a function of $T$ demonstrating that the phase transition is of second order.\n\nIn Fig. \\ref{fig:tcrit2} we plot our result obtained for the critical temperature $T_c$ and the Fermi temperature $T_F$ as a function of the chemical potential $\\mu$ at the unitarity point with $a^{-1}=0$. From dimensional analysis it is clear that both dependencies are linear, $T_c, T_F\\sim \\mu$, provided that non-universal effects involving the ultraviolet cutoff scale $\\Lambda$ can be neglected. That this is indeed found numerically can be seen as a nontrivial test of our approximation scheme and the numerical procedures as well as the universality of the system. Dividing the slope of both lines gives $T_c\/T_F=0.264$, a result that will be discussed in more detail below. \n\\begin{figure}\n \\centering\n\t\\includegraphics[width=0.45\\textwidth]{PHfigure4.eps}\n\t\\caption{Critical temperature $T_c$ (boxes) and Fermi temperature $T_F=(3\\pi^2 n)^{2\/3}$ (triangles) as a function of the chemical potential $\\mu$. For convenience the Fermi temperature is scaled by a factor 1\/5. We also plot the linear fits $T_c=0.39\\mu$ and $T_F=1.48\\mu$. The units are arbitrary and we use $\\Lambda=e^7$.}\n\t\\label{fig:tcrit2}\n\\end{figure}\nWe emphasize that part of the potential error in this estimates is due to uncertainties in the precise quantitative determination of the density or $T_F$. \n\n\n\\subsection{Phase diagram}\nThe effect of the particle-hole fluctuations shows most prominently in the result for the critical temperature. With our approach we can compute the critical temperature for the phase transition to superfluidity throughout the crossover. The results are shown in Fig. \\ref{fig:tcrit}. We plot the critical temperature in units of the Fermi temperature $T_c\/T_F$ as a function of the scattering length measured in units of the inverse Fermi momentum, i.~e. the concentration $c=a k_F$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{PHfigure5.eps}\n\\caption{Dimensionless critical temperature $T_c\/T_F$ as a function of the inverse concentration $c^{-1}=(a k_F)^{-1}$. The black solid line includes the effect of particle-hole fluctuations. We also show the result obtained when particle-hole fluctuations are neglected (dot-dashed line). For comparison, we plot the BCS result without (left dotted line) and with Gorkov's correction (left dashed). On the BEC side with $c^{-1}>1$ we show the critical temperature for a gas of free bosonic molecules (horizontal dashed line) and a fit to the shift in $T_c$ for interacting bosons, $\\Delta T_c\\sim c$ (dotted line on the right). The black solid dot gives the QMC results \\cite{bulgac:090404, bulgac:023625, burovski:160402}.}\n\\label{fig:tcrit}\n\\end{figure}\nWe can roughly distinguish three different regimes. On the left side, where $c^{-1}\\lesssim-1$, the interaction is weakly attractive. Mean field or BCS theory is qualitatively valid here. In Fig. \\ref{fig:tcrit} we denote the BCS result by the dotted line on the left ($c^{-1}<0$). However, the BCS approximation has to be corrected by the effect of particle-hole fluctuations, which lower the value for the critical temperature by a factor of $2.2$. This is the Gorkov correction (dashed line on the left side in Fig. \\ref{fig:tcrit}). The second regime is found on the far right side, where the interaction again is weak, but now we find a bound state of two atoms. In this regime the system exhibits Bose-Einstein condensation of molecules as the temperature is decreased. The dashed horizontal line on the right side shows the critical temperature of a free Bose-Einstein condensate of molecules. In-between there is the unitarity regime, where the two-atom scattering length diverges ($c^{-1} \\rightarrow 0$) and we deal with a system of strongly interacting fermions.\n\nOur result including the particle-hole fluctuations is given by the solid line. This may be compared with a functional renormalization flow investigation without including particle-hole fluctuations (dot-dashed line) \\cite{Diehl:2007th}. For $c\\to 0_-$ the solid line of our result matches the BCS theory including the correction by Gorkov and Melik-Barkhudarov \\cite{Gorkov},\n\\begin{equation}\n\\frac{T_c}{T_F}=\\frac{e^C}{\\pi}\\left(\\frac{2}{e}\\right)^{7\/3} e^{\\pi\/(2c)}\\approx 0.28 e^{\\pi\/(2c)}.\n\\end{equation}\nIn the regime $c^{-1}>-2$ we see that the non-perturbative result given by our RG analysis deviates from Gorkov's result, which is derived in a perturbative setting. \n\nOn the BEC-side for very large and positive $c^{-1}$ our result approaches the critical temperature of a free Bose gas where the bosons have twice the mass of the fermions $M_B=2M$. In our units the critical temperature is then\n\\begin{equation}\n\\frac{T_{c,\\text{BEC}}}{T_F}=\\frac{2\\pi}{\\left[6\\pi^2 \\zeta(3\/2)\\right]^{2\/3}}\\approx 0.218.\n\\end{equation} \nFor $c\\to 0_+$ this value is approached in the form\n\\begin{equation}\n\\frac{T_c-T_{c,\\text{BEC}}}{T_{c,\\text{BEC}}}=\\kappa a_M n_M^{1\/3}=\\kappa \\frac{a_M}{a}\\frac{c}{(6\\pi^2)^{1\/3}}.\n\\end{equation}\nHere, $n_M=n\/2$ is the density of molecules and $a_M$ is the scattering length between them. For the ratio $a_M\/a$ we use our result $a_M\/a=0.718$ obtained from solving the flow equations in vacuum, i.~e. at $T=n=0$, see section \\ref{DimerDimer}. For the coefficients determining the shift in $T_c$ compared to the free Bose gas we find $\\kappa=1.55$. \n\nFor $c^{-1}\\gtrsim0.5$ the effect of the particle-hole fluctuations vanishes. This is expected since the chemical potential is now negative $\\mu<0$ and there is no Fermi surface any more. Because of that there is no difference between the new curve with particle-hole fluctuations (solid in Fig. \\ref{fig:tcrit}) and the one obtained when particle-hole contributions are neglected (dot-dashed in Fig. \\ref{fig:tcrit}). Due to the use of an optimized cutoff scheme and a different computation of the density our results differ slightly from the ones obtained in \\cite{Diehl:2007th}.\n\nIn the unitary regime ($c^{-1}\\approx 0$) the particle-hole fluctuations still have a quantitative effect. We can give an improved estimate for the critical temperature at the resonance ($c^{-1}=0$) where we find $T_c\/T_F=0.264$. Results from quantum Monte Carlo simulations are $T_c\/T_F = 0.15$ \\cite{bulgac:090404, bulgac:023625, burovski:160402} and $T_c\/T_F = 0.245$ \\cite{akkineni:165116}. The measurement by Luo \\textit{et al.} \\cite{Luo2007} in an optical trap gives $T_c\/T_F = 0.29 (+0.03\/-0.02)$, which is a result based on the study of the specific heat of the system.\n\n\n\\subsection{Crossover to narrow resonances}\nSince we use a two channel model (Eq.\\ \\eqref{eqMicroscopicAction}) we can not only describe broad resonances with $h_\\Lambda^2\\to \\infty$ but also narrow ones with $h_\\Lambda^2\\to0$. This corresponds to a nontrivial limit of the theory which can be treated exactly \\cite{Diehl:2005an, Gurarie2007}. In the limit $h_\\Lambda\\to 0$ the microscopic action Eq.\\ \\eqref{eqMicroscopicAction} describes free fermions and bosons. The essential feature is, that they are in thermodynamic equilibrium so that they have equal chemical potential. (There is a factor 2 for the bosons since they consist of two fermions.) For vanishing Yukawa coupling $h_\\Lambda$ the theory is Gaussian and the macroscopic propagator equals the microscopic propagator. There is no normalization of the ``mass''-term $m^2$ so that the detuning parameter in Eq.\\ \\eqref{eqMicroscopicAction} is $\\nu=\\mu_M(B-B_0)$ and\n\\begin{equation}\nm^2=\\mu_M(B-B_0)-2\\mu.\n\\end{equation}\n\nTo determine the critical density for fixed temperature, we have to adjust the chemical potential $\\mu$ such that the bosons are just at the border to the superfluid phase. For free bosons this implies $m^2=0$ and thus\n\\begin{equation}\n\\mu=\\frac{1}{2}\\mu_M (B-B_0)=-\\frac{1}{16\\pi}h^2 a^{-1}.\n\\label{eq:ChemicalpotetialatTc}\n\\end{equation}\nIn the last equation we use the relation between the detuning and the scattering length\n\\begin{equation}\\label{eq:detuningandscatlenth}\na=-\\frac{h^2}{8\\pi \\mu_M(B-B_0)}.\n\\end{equation}\nThe critical temperature $T_c$ is now determined from the implicit equation\n\\begin{equation}\n\\int \\frac{d^3 p}{(2\\pi)^3}\\left\\{\\frac{2}{e^{\\frac{1}{T_c}(\\vec p^2-\\mu)}+1}+\\frac{2}{e^{\\frac{1}{2T_c}\\vec p^2}-1}\\right\\}=n.\n\\label{eq:TcNarrowresonanceimplicit}\n\\end{equation}\n\nWhile the BCS-BEC crossover can be studied as a function of $B-B_0$ or $\\mu$, Eq.\\ \\eqref{eq:detuningandscatlenth} implies that for $h_\\Lambda^2\\to 0$ a finite scattering length $a$ requires $B\\to B_0$. For all $c\\neq 0$ the narrow resonance limit implies for the phase transition $B=B_0$ and therefore $\\mu=0$. (A different concentration variable $c_\\text{med}$ was used in \\cite{Diehl:2005an, Diehl:2005ae}, such that the crossover could be studied as a function of $c_\\text{med}$ in the narrow resonance limit, see the discussion at the end of this section.) For $\\mu=0$ Eq.\\ \\eqref{eq:TcNarrowresonanceimplicit} can be solved analytically and gives\n\\begin{equation}\n\\frac{T_c}{T_F}=\\left(\\frac{4\\sqrt{2}}{3(3+\\sqrt{2})\\pi^{1\/2}\\zeta(3\/2)}\\right)^{2\/3}\\approx 0.204.\n\\end{equation} \nThis result is confirmed numerically by solving the flow equations for different microscopic Yukawa couplings $h_\\Lambda$ and taking the limit $h_\\Lambda\\to 0$. In Fig. \\ref{fig:narrowbroad}, we show the critical temperature $T_c\/T_F$ as a function of the dimensionless Yukawa coupling $h_\\Lambda\/\\sqrt{k_F}$ in the ``unitarity limit'' $c^{-1}=0$ (solid line). For small values of the Yukawa coupling, $h_\\Lambda\/\\sqrt{k_F} \\lesssim 2$ we enter the regime of the narrow resonance limit and the critical temperature is independent of the precise value of $h_\\Lambda$. The numerical value matches the analytical result $T_c\/T_F\\approx 0.204$ (dotted line in Fig. \\ref{fig:narrowbroad}). For large Yukawa couplings, $h_\\Lambda\/\\sqrt{k_F} \\gtrsim 40$, we recover the result of the broad resonance limit as expected. In between there is a smooth crossover of the critical temperature from narrow to broad resonances.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{PHfigure6.eps}\n\\caption{The critical temperature divided by the Fermi temperature $T_c\/T_F$ as a function of the dimensionless Yukawa coupling $h_\\Lambda\/\\sqrt{k_F}$ for $c^{-1}=0$ (solid line). One can clearly see the plateaus in the narrow resonance limit ($T_c\/T_F \\approx 0.204$, dotted line) and in the broad resonance limit ($T_c\/T_F \\approx 0.264$, dashed line).}\n\\label{fig:narrowbroad}\n\\end{figure}\n\nWe use here a definition of the concentration $c=a k_F$ in terms of the vacuum scattering length $a$. This has the advantage of a straightforward comparison with experiment since $a^{-1}$ is directly related to the detuning of the magnetic field $B-B_0$, and the ``unitarity limit'' $c^{-1}=0$ precisely corresponds to the peak of the resonance $B=B_0$. However, for a nonvanishing density other definitions of the concentration parameter are possible, since the effective fermion interaction $\\lambda_{\\psi,\\text{eff}}$ depends on the density. For example, one could define for $n\\neq 0$ a ``in medium scattering length'' $\\bar a=\\lambda_{\\psi,\\text{eff}}\/(8\\pi)$, with $\\lambda_{\\psi,\\text{eff}}=-h^2\/m^2$ evaluated for $T=0$ but $n\\neq 0$ \\cite{Diehl:2005an}. The corresponding ``in medium concentration'' $c_\\text{med}=\\bar a k_F$ would differ from our definition by a term involving the chemical potential, resulting in a shift of the location of the unitarity limit if the latter is defined as $c_\\text{med}^{-1}=0$. While for broad resonances both definitions effectively coincide, for narrow resonances a precise statement how the concentration is defined is mandatory when aiming for a precision comparison with experiment and numerical simulations for quantities as $T_c\/T_F$ at the unitarity limit. For example, defining the unitarity limit by $c_\\text{med}^{-1}=0$ would shift the critical temperature in the narrow resonance limit to $T_c\/T_F=0.185$ \\cite{Diehl:2005an}.\n\\subsubsection{Vacuum flow equations and their solution for $d=3$}\n\nThe vacuum is defined to have zero temperature $T=0$ and vanishing density $n=0$, which also implies $\\rho_0=0$. The interaction strength $\\lambda$ at the scale $k=0$ determines the four point vertex at zero momentum. It is directly related to the scattering length $a$ for the scattering of two particles in vacuum, which is experimentally observable. We therefore want to replace the microscopic coupling $\\lambda_\\Lambda$ by the renormalized coupling $a$. In our units ($2M=1$), one has the relation \n\\begin{equation}\na=\\frac{1}{8\\pi}\\lambda(k=0, T=0, n=0).\n\\end{equation}\nThe vacuum properties can be computed by taking for $T=0$ the limit $n\\rightarrow0$. We may also perform an equivalent and technically simple computation in the symmetric phase by choosing $m^2(k=\\Lambda)$ such, that $m^2(k\\rightarrow0)=0$. This guarantees that the boson field $\\varphi$ is a gap-less propagating degree of freedom. \n\nWe first investigate the model with a linear $\\tau$-derivative, $S_\\Lambda=1$, $V_\\Lambda=0$. Projecting the flow equation \\eqref{eq4:Wettericheqn} to the truncation in Eqs. \\eqref{eqSimpleTruncation}, \\eqref{eq10:truncationU}, we find the following equations:\n\\begin{eqnarray}\n\\nonumber \\partial_t m^2 & = & 0\\\\\n\\partial_t \\lambda & = & \\left(\\frac{\\lambda^2}{6}\\right)\\frac{{\\left( k^2 - m^2 \\right) }^{3\/2}}{k^2\\,\n {\\pi }^2\\,S}\\,\\Theta(k^2-m^2).\n\\label{eqflowvacuummlambda}\n\\end{eqnarray}\nThe propagator is not renormalized, $\\partial_t S=\\partial_tV=\\partial_t \\bar{A}=0$, $\\eta=0$, $\\partial_t\\alpha=0$, and one finds $\\partial_tn_k=0$. The coupling $\\beta$ is running according to\n\\begin{equation}\n\\partial_t \\beta = \\left(\\frac{1}{3}\\alpha\\lambda^2-\\frac{1}{3}k^2\\beta\\lambda\\right)\\frac{\\left(k^2 - m^2\\right)^{3\/2}}{k^4\\,\\pi^2 \\,S}\\Theta(k^2 - m^2).\n\\label{eqflowvacuumbeta}\n\\end{equation}\nSince $\\beta$ appears only in its own flow equation, it is of no further relevance in the vacuum. Also, no coupling $V$ is generated by the flow and we have therefore set $V=0$ on the r.h.s. of Eqs. \\eqref{eqflowvacuummlambda} and \\eqref{eqflowvacuumbeta}. \n\nInserting in Eq. \\eqref{eqflowvacuummlambda} the vacuum values $m^2=0$ and $S=1$, we find\n\\begin{equation}\n\\partial_t\\lambda=\\frac{k}{6\\pi^2}\\lambda^2.\n\\end{equation}\nThe solution \n\\begin{equation}\n\\lambda(k)=\\frac{1}{\\frac{1}{\\lambda_\\Lambda}+\\frac{1}{6\\pi^2}(\\Lambda-k)}\n\\end{equation}\ntends to a constant for $k\\rightarrow0$, $\\lambda_0=\\lambda(k=0)$. The dimensionless variable $\\tilde{\\lambda}=\\frac{\\lambda k}{S}$ goes to zero, when $k$ goes to zero. This shows the infrared freedom of the theory. For fixed ultraviolet cutoff, the scattering length\n\\begin{equation}\na=\\frac{\\lambda_0}{8\\pi}=\\frac{1}{\\frac{8\\pi}{\\lambda_\\Lambda}+\\frac{4}{3\\pi}\\Lambda},\n\\end{equation}\nas a function of the initial value $\\lambda_\\Lambda$, has an asymptotic maximum\n\\begin{equation}\na_{\\text{max}}=\\frac{3\\pi}{4\\Lambda}.\n\\label{eqscatteringbound}\n\\end{equation}\nThe relation between $a$ and $\\lambda_\\Lambda$ is shown in fig. \\ref{figscatteringbound}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{FRforBECfig4.eps}\n\\caption{Scatt\\-er\\-ing length $a$ in dependence on the microscopic interaction strength $\\lambda_\\Lambda$ (solid). The asymptotic maximum $a_{\\text{max}}=\\frac{3 \\pi}{4\\Lambda}$ is also shown (dashed).}\n\\label{figscatteringbound}\n\\end{figure}\n\nAs a consequence of Eq.\\ \\eqref{eqscatteringbound}, the nonrelativistic bosons in $d=3$ are a \"trivial theory\" in the sense that the bosons become noninteracting in the limit $\\Lambda\\rightarrow\\infty$, where $a\\rightarrow0$. The upper bound \\eqref{eqscatteringbound} has important practical consequences. It tells us, that whenever the \"macrophysical length scales\" are substantially larger than the microscopic length $\\Lambda^{-1}$, we deal with a weakly interacting theory. As an example, consider a boson gas with a typical inter-particle distance substantially larger than $\\Lambda^{-1}$. (For atom gases $\\Lambda^{-1}$may be associated with the range of the Van der Waals force.) We may set the units in terms of the particle density $n$, $n=1$. In these units $\\Lambda$ is large, say $\\Lambda=10^3$. This implies a very weak interaction, $a\\lesssim 2.5\\cdot10^{-3}$. In other words, the scattering length cannot be much larger than the microscopic length $\\Lambda^{-1}$. For such systems, perturbation theory will be valid in many circumstances. We will find that the Bogoliubov theory indeed gives a reliable account of many properties. Even for an arbitrary large microphysical coupling $(\\lambda_\\Lambda\\rightarrow\\infty)$, the renormalized physical scattering length $a$ remains finite.\n\nLet us mention, however, that the weak interaction strength does not guarantee the validity of perturbation theory in all circumstances. For example, near the critical temperature of the phase transition between the superfluid and the normal state, the running of $\\lambda(k)$ will be different from the vacuum. As a consequence, the coupling will vanish proportional to the inverse correlation length $\\xi^{-1}$ as $T$ approaches $T_c$, $\\lambda \\sim T^{-2}\\xi^{-1}$. Indeed, the phase transition will be characterized by the non-perturbative critical exponents of the Wilson-Fisher fixed point. Also for lower dimensional systems, the upper bound \\eqref{eqscatteringbound} for $\\lambda_0$ is no longer valid - for example the running of $\\lambda$ is logarithmic for $d=2$. For our models with $V_\\Lambda\\neq0$, the upper bound becomes dependent on $V_\\Lambda$. It increases for $V_\\Lambda>0$. In the limit $S_\\Lambda\\rightarrow0$, it is replaced by the well known \"triviality bound\" of the four dimensional relativistic model, which depends only logarithmically on $\\Lambda$. Finally, for superfluid liquids, as $^4\\text{He}$, one has $n\\sim\\Lambda^3$, such that for $a\\sim \\Lambda^{-1}$ one finds a large concentration $c$.\n\nThe situation for dilute bosons seems to contrast with ultracold fermion gases in the unitary limit of a Feshbach resonance, where $a$ diverges. One may also think about a Feshbach resonance for bosonic atoms, where one would expect a large scattering length for a tuning close to resonance. In this case, however, the effective action does not remain local. It is best described by the exchange of molecules. The scale of nonlocality is then given by the gap for the molecules, $m_M$. Only for momenta $\\vec{q}^20$. At the microscopic scale $k=\\Lambda$ the minimum of the effective potential $U$ is then at $\\rho_{0,\\Lambda}=\\mu\/\\lambda>0$. \n\nThe superfluid density $\\rho_0$ is connected to a nonvanishing ``renormalized order parameter'' $\\varphi_0$, with $\\rho_0=\\varphi_0^*\\varphi_0$. It is responsible for an effective spontaneous breaking of the U(1)-symmetry. Indeed, the expectation value $\\varphi_0$ points out a direction in the complex plane so that the global U(1)-symmetry of phase rotations is broken by the ground state of the system. Goldstone's theorem implies the presence of a gapless Goldstone mode, and the associated linear dispersion relation $\\omega\\sim|\\vec{q}|$ accounts for superfluidity. The Goldstone physics is best described by using a real basis in field space by decomposing the complex field $\\varphi=\\varphi_0+\\frac{1}{\\sqrt{2}}(\\varphi_1+i\\varphi_2)$. Without loss of generality we can choose the expectation value $\\varphi_0$ to be real. The real fields $\\varphi_1$ and $\\varphi_2$ describe then the radial and Goldstone mode. respectively. For $\\mu=\\mu_0$ the inverse propagator reads in our truncation\n\\begin{equation}\nG^{-1}=\\bar{A}\\begin{pmatrix} \\vec{p}^2+V p_0^2+U^\\prime+2\\rho U^{\\prime\\prime}, & -S p_0 \\\\ S p_0, & \\vec{p}^2+V q_0^2+U^\\prime \\end{pmatrix}.\n\\label{eqprop}\n\\end{equation}\nHere $\\vec{p}$ is the momentum of the collective excitation, and for $T=0$ the frequency obeys $\\omega=-ip_0$.\nIn the regime with spontaneous symmetry breaking, $\\rho_0(k)\\neq 0$, the propagator for $\\rho=\\rho_0$ has $U^\\prime=0$, $2\\rho U^{\\prime\\prime}=2\\lambda \\rho_0\\neq 0$, giving rise to the linear dispersion relation characteristic for superfluidity. This strongly modifies the flow equations as compared to the vacuum flow equations once $k^2 \\ll 2\\lambda \\rho_0$. For $n\\neq0$ the flow is typically in the regime with $\\rho_0(k)\\neq0$. In practice, we have to adapt the initial value $\\rho_{0,\\Lambda}$ such that the flow ends at a given density $\\rho_0(k_\\text{ph})=n$. For $k_\\text{ph}\\ll n^{1\/2}$ one finds that $\\rho_0(k_\\text{ph})$ depends only very little on $k_\\text{ph}$. As mentioned above, we will often choose the density to be unity such that effectively all length scales are measured in units of the interparticle distance $n^{-1\/2}$. \n\nIn contrast to the vacuum with $T=\\rho_0=0$, the flow of the propagator is nontrivial in the phase with $\\rho_0>0$ and spontaneous U(1) symmetry breaking. In Fig. \\ref{figFlowKinetic} we show the flow of the kinetic coefficients $\\bar{A}$, $V$, $S$ for a renormalized or macroscopic interaction strength $\\lambda=1$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig3.eps}\n\\caption{Flow of the kinetic coefficients $\\bar{A}$ (solid), $S$ (dashed), and $V$ (dashed-dotted) at zero temperature $T=0$, density $n=1$, and vacuum interaction strength $\\lambda=1$.}\n\\label{figFlowKinetic}\n\\end{figure}\nThe wavefunction renormalization $\\bar{A}$ increases only a little at scales where $k\\approx n^{1\/2}$ and saturates then to a value $\\bar{A}>1$. As will be explained below, we can directly infer the condensate depletion from the value of $\\bar{A}$ at macroscopic scales. The coefficient of the linear $\\tau$-derivative $S$ goes to zero for $k\\rightarrow 0$. The frequency dependence is then governed by the quadratic $\\tau$-derivative with coefficient $V$, which is generated by the flow and saturates to a finite value for $k\\rightarrow 0$. \n\nThe flow of the interaction strength $\\lambda(k)$ for different values of $\\lambda=\\lambda(k_\\text{ph})$ is shown in Fig. \\ref{figFlowLambda}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig4.eps}\n\\caption{Flow of the interaction strength $\\lambda(k)$ at zero temperature $T=0$, density $n=1$, for different initial values $\\lambda_\\Lambda$. The dotted lines are the corresponding graphs in the vacuum $n=0$. The vertical line labels our choice of $k_\\text{ph}$. The lower plot shows $\\lambda(k)\/k$ for the same parameters, demonstrating that $\\lambda(k)\\sim k$ for small $k$.}\n\\label{figFlowLambda}\n\\end{figure}\nWhile the decrease with the scale $k$ is only logarithmic in vacuum, it becomes now linear $\\lambda(k)\\sim k$ for $k\\ll n^{1\/2}$. It is interesting that the ratio $\\lambda(k)\/k$ reaches larger values for smaller values of $\\lambda_\\Lambda$. \n\n\\subsection{Quantum depletion of condensate}\n\nAs $k$ is lowered from $\\Lambda$ to $k_\\text{ph}$, the renormalized order parameter or the superfluid density $\\rho_0$ increases first and then saturates to $\\rho_0=n=1$. This is expected since the superfluid density equals the total density at zero temperature. In contrast, the bare order parameter $\\bar{\\rho}_0=\\rho_0\/\\bar{A}$ flows to a smaller value $\\bar{\\rho}_0<\\rho_0$. As argued in section \\ref{sec:Bose-EinsteinCondensationinthreedimensions}, the bare order parameter is just the condensate density, such that\n\\begin{equation}\nn-n_C=\\rho_0-\\bar{\\rho}_0=\\rho_0(1-\\frac{1}{\\bar{A}})\n\\end{equation}\nis the condensate depletion. Its dependence on the interaction strength $\\lambda$ is shown in Fig. \\ref{figDepletion}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig5.eps}\n\\caption{Condensate depletion $(n-n_c)\/n$ as a function of the vacuum interaction strength $\\lambda$. The dashed line is the Bogoliubov result $(n-n_c)\/n=\\frac{\\lambda}{8\\pi}$ for reference.}\n\\label{figDepletion}\n\\end{figure}\nFor small interaction strength $\\lambda$ the condensate depletion follows roughly the Bogoliubov form\n\\begin{equation}\n\\frac{n-n_c}{n}=\\frac{\\lambda}{8\\pi}.\n\\end{equation}\nHowever, we find small deviations due to the running of $\\lambda$ which is absent in Bogoliubov theory. For large interaction strength $\\lambda\\approx 1$ the deviation from the Bogoliubov result is quite substantial, since the running of $\\lambda$ with the scale $k$ is more important. \n\n\\subsection{Dispersion relation and sound velocity}\n\nWe also investigate the dispersion relation at zero temperature. The dispersion relation $\\omega(p)$ follows from the condition\n\\begin{equation}\n\\text{det}\\, G^{-1}(\\omega(p),p)=0\n\\label{eqdispersionfromprop}\n\\end{equation}\nwhere $G^{-1}$ is the inverse propagator after analytic continuation to real time $p_0\\rightarrow i\\omega$. As was shown at the end of section\\ref{sec:Derivativeexpansionandwardidentities} the generation of the kinetic coefficient $V$ by the flow leads to the emergence of a second branch of solutions of Eq.\\ \\eqref{eqdispersionfromprop}. In our truncation the dispersion relation for the two branches $\\omega_+(\\vec{p})$ and $\\omega_-(\\vec{p})$ are\n\\begin{eqnarray}\n\\nonumber\n\\omega_\\pm(\\vec{p})&=&{\\Bigg (}\\frac{1}{V}(\\vec{p}^2+\\lambda \\rho_0)+\\frac{S^2}{2V^2}\\\\\n&&\\pm{\\Bigg (}\\left(\\frac{1}{V}(\\vec{p}^2+\\lambda\\rho_0)+\\frac{S^2}{2V^2}\\right)^2-\\frac{1}{V^2}\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0){\\Bigg )}^{1\/2}{\\Bigg )}^{1\/2}.\n\\label{eqdispersionrelation}\n\\end{eqnarray}\nIn the limit $V\\rightarrow 0$, $S\\rightarrow 1$ we find that the lower branch approaches the Bogoliubov result $\\omega_-\\rightarrow \\sqrt{\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0)}$ while the upper branch diverges $\\omega_+\\rightarrow \\infty$ and thus disappears from the spectrum. The lower branch is dominated by phase changes (Goldstone mode), while the upper branch reflects waves in the size of $\\rho_0$ (radial mode). \n\nIn principle, the coupling constants on the right-hand side of Eq.\\ \\eqref{eqdispersionrelation} also depend on the momentum $p=|\\vec{p}|$. Since an external momentum provides an infrared cutoff of order $k\\approx |\\vec{p}|$ we can approximate the $|\\vec{p}|$-dependence by using on the right-hand side of Eq.\\ \\eqref{eqdispersionrelation} the $k$-dependent couplings with the identification $k=|\\vec{p}|$. Our result for the lower branch of the dispersion relation is shown in Fig. \\ref{figDispersionlinear}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig6.eps}\n\\caption{Lower branch of the dispersion relation $\\omega_{-}(p)$ at temperature $T=0$ and for the vacuum interaction strength $\\lambda=1$ (solid curve), $\\lambda=0.5$ (upper dashed curve), and $\\lambda=0.1$ (lower dashed curve). The units are set by the density $n=1$. We also show the Bogoliubov result for $\\lambda=1$ (upper dotted curve) and $\\lambda=0.5$ (lower dotted curve). For $\\lambda=0.1$ the Bogoliubov result is identical to our result within the plot resolution.}\n\\label{figDispersionlinear}\n\\end{figure}\nWe also plot the Bogoliubov result $\\omega=\\sqrt{\\vec{p}^2(\\vec{p}^2+2\\lambda \\rho_0)}$ for comparison. For small $\\lambda$ our result is in agreement with the Bogoliubov result, while we find substantial deviations for large $\\lambda$. Both branches $\\omega_+$ and $\\omega_-$ are shown in Fig. \\ref{figDispersion} on a logarithmic scale. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig7.eps}\n\\caption{Dispersion relation $\\omega_{-}(p)$, $\\omega_{+}(p)$ at temperature $T=0$ and for vacuum interaction strength $\\lambda=1$ (solid), $\\lambda=0.5$ (long dashed), and $\\lambda=0.1$ (short dashed). The units are set by the density $n=1$.}\n\\label{figDispersion}\n\\end{figure}\nSince we start with $V=0$ at the microscopic scale $\\Lambda$ we find $\\omega_+(\\vec{p})\\rightarrow\\infty$ for $|\\vec{p}|\\rightarrow \\Lambda$.\n\nThe sound velocity $c_S$ can be extracted from the dispersion relation. More precisely, we compute the microscopic sound velocity for the lower branch $\\omega_-(\\vec{p})$ as $c_S=\\frac{\\partial \\omega}{\\partial p}$ at $p=0$. In our truncation we find\n\\begin{equation}\nc_S^2=\\frac{2\\lambda\\rho_0}{S^2+2\\lambda\\rho_0 V}.\n\\end{equation}\nOur result for $c_S$ at $T=0$ is shown in Fig. \\ref{figsoundvelocity} as a function of the interaction strength $\\lambda$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig8.eps}\n\\caption{Dimensionless sound velocity $c_S\/n^{1\/2}$ as a function of the vacuum interaction strength (solid). We also show the Bogoliubov result $c_S=\\sqrt{2\\lambda \\rho_0}$ for reference (dashed).}\n\\label{figsoundvelocity}\n\\end{figure}\nFor a large range of small $\\lambda$ we find good agreement with the Bogoliubov result $c_S^2=2\\lambda \\rho_0$. However, for large $\\lambda$ or result for $c_S$ exceeds the Bogoliubov result by a factor up to 2. \n\n\n\\subsection{Kosterlitz-Thouless physics}\n\n\\subsubsection{Superfluidity and order parameter}\nAt nonzero temperature and for infinite volume, long range order is forbidden in two spatial dimensions by the Mermin-Wagner theorem. Because of that, no proper Bose-Einstein condensation is possible in a two-dimensional homogeneous Bose gas at nonvanishing temperature. However, even if the order parameter vanishes in the thermodynamic limit of infinite volume, one still finds a nonzero superfluid density for low enough temperature. The superfluid density can be considered as the square of a renormalized order parameter $\\rho_0=|\\varphi_0|^2$ and the particular features of the low-temperature phase can be well understood by the physics of the Goldstone boson for a phase with effective spontaneous symmetry breaking \\cite{Wetterich:1991be}. The renormalized order parameter $\\varphi_0$ is related to the expectation value of the bosonic field $\\bar{\\varphi}_0$ and therefore to the condensate density $\\bar{\\rho}_0=\\bar{\\varphi}_0^2$ by a wave function renormalization, defined by the behavior of the bare propagator $\\bar{G}$ at zero frequency for vanishing momentum\n\\begin{equation}\n\\varphi_0=\\bar{A}^{1\/2}\\bar{\\varphi}_0,\\quad \\rho_0=\\bar{A}\\bar{\\rho}_0,\\quad \\bar{G}^{-1}(\\vec{p}\\rightarrow 0)=\\bar{A}\\vec{p}^2.\n\\end{equation}\nWhile the renormalized order parameter $\\rho_0(k)$ remains nonzero for $k\\rightarrow 0$ if $T0$ for nonzero temperature $00$.\n\n\\subsubsection{Critical temperature}\nThe flow equations permit a straightforward computation of $\\rho_0(T)$ for arbitrary $T$, once the interaction strength of the system has been fixed at zero temperature and density. We have extracted the critical temperature as a function of $\\lambda=\\lambda(k_\\text{ph})$ for different values of $k_\\text{ph}$. The behavior for small $\\lambda$,\n\\begin{equation}\n\\frac{T_c}{n}=\\frac{4\\pi}{\\text{ln}(\\zeta\/\\lambda)}\n\\label{Tcperturbative}\n\\end{equation}\nis compatible with the free theory where $T_c$ vanishes for $k_\\text{ph}\\rightarrow0$ and with the perturbative analysis in Ref. \\cite{Popov1983, PhysRevB.37.4936, MarkusHolzmann01302007}. We find that the value of $\\zeta$ depends on the choice of $k_\\text{ph}$. For $k_\\text{ph}=10^{-2}$ we find $\\zeta=100$, while $k_\\text{ph}=10^{-4}$ corresponds to $\\zeta=225$ and $k_\\text{ph}=10^{-6}$ to $\\zeta=424$. In Fig. \\ref{figtcoflambda} we show or result for $T_c\/n$ as a function of $\\lambda$ for these choices. We also plot the curve in Eq.\\ \\eqref{Tcperturbative} with the Monte-Carlo result $\\zeta=380$ from Ref. \\cite{PhysRevLett.87.270402, PhysRevA.66.043608, PhysRevB.69.144504, PhysRevB.59.14054}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig9.eps}\n\\caption{Critical temperature $T_c\/n$ as a function of the interaction strength $\\lambda$. We choose here $k_\\text{ph}=10^{-2}$ (circles), $k_\\text{ph}=10^{-4}$ (boxes) and $k_\\text{ph}=10^{-6}$ (diamonds). For the last case the bound on the scattering length is $\\lambda<\\frac{4\\pi}{\\text{ln}(\\Lambda\/k_\\text{ph})}\\approx 0.78$. We also show the curve $\\frac{T_c}{n}=\\frac{4\\pi}{\\text{ln}(\\zeta\/\\lambda)}$ (dashed) with the Monte-Carlo result $\\zeta=380$ \\cite{PhysRevLett.87.270402, PhysRevA.66.043608, PhysRevB.69.144504, PhysRevB.59.14054} for reference.}\n\\label{figtcoflambda}\n\\end{figure}\nWe find that $T_c$ vanishes for $k_\\text{ph}\\to0$ in the interacting theory as well. This is due to the increase of $\\zeta$ and, for a fixed microscopical interaction, to the decrease of $\\lambda(k_\\text{ph})$. Since the vanishing of $T_c\/n$ is only logarithmic in $k_\\text{ph}$, a phase transition can be observed in practice. We find agreement with Monte-Carlo results \\cite{PhysRevLett.87.270402} for small $\\lambda$ if $k_\\text{ph}\/\\Lambda\\approx 10^{-7}$. The dependence of $T_c\/n$ on the size of the system $k_\\text{ph}^{-1}$ remains to be established for the Monte-Carlo computations.\n\nThe critical behavior of the system is governed by a Kosterlitz-Thouless phase transition. Usually this is described by considering the thermodynamics of vortices. In Refs. \\cite{Grater:1994qx, VonGersdorff:2000kp} it was shown that functional renormalization group can account for this ``nonperturbative'' physics without explicitly taking vortices into account. The correlation length in the low-temperature phase is infinite. In our picture, this arises due to the presence of a Goldstone mode if $\\rho_0>0$. The system is superfluid for $T0$) the transition is smoothened. In order to see the jump, as well as essential scaling for $T$ approaching $T_c$ from above, our truncation is insufficient. These features become visible only in extended truncations that we will briefly describe next. \n\nFor very small scales $\\frac{k^2}{T}\\ll 1$, the contribution of Matsubara modes with frequency $q_0=2\\pi T n$, $n\\neq 0$, is suppressed since nonzero Matsubara frequencies act as an infrared cutoff. In this limit a dimensionally reduced theory becomes valid. The long distance physics is dominated by classical two-dimensional statistics, and the time dimension parametrized by $\\tau$ no longer plays a role. \n\nThe flow equations simplify considerably if only the zero Matsubara frequency is included, and one can use more involved truncations. Such an improved truncation is indeed needed to account for the jump in the superfluid density. In Ref. \\cite{VonGersdorff:2000kp} the next to leading order in a systematic derivative expansion was investigated. It was found that for $k\\ll T$ the flow equation for $\\rho_0$ can be well approximated by\n\\begin{equation}\n\\partial_t \\rho_0=2.54\\,T^{-1\/2}(0.248\\, T-\\rho_0)^{3\/2}\\,\\theta(0.248 \\,T-\\rho_0).\n\\label{eq:improvedflowofrho}\n\\end{equation}\nWe switch from the flow equation in our more simple truncation to the improved flow equation \\eqref{eq:improvedflowofrho} for scales $k$ with $k^2\/T<10^{-3}$. We keep all other flow equations unchanged. A similar procedure was also used in Ref. \\cite{DrHCK}.\n\nIn Fig. \\ref{figFlowofnrho} we show the flow of the density $n$, the superfluid density $\\rho_0$ and the condensate density $\\bar{\\rho}_0$ for different temperatures.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig10.eps}\n\\caption{Flow of the density $n$ (solid), the superfluid density $\\rho_0$ (dashed), and the condensate density $\\bar{\\rho}_0$ (dotted) for chemical potential $\\mu=1$, vacuum interaction strength $\\lambda=0.5$ and temperatures $T=0$ (top), $T=2.4$ (middle) and $T=2.8$ (bottom). The vertical line marks our choice of $k_\\text{ph}$. We recall $n=\\rho_0$ for $T=0$ such that the upper dashed and solid lines coincide.}\n\\label{figFlowofnrho}\n\\end{figure}\nIn Fig. \\ref{figrhodnoftemperature} we plot our result for the superfluid fraction of the density as a function of the temperature for different scales $k_\\text{ph}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig11.eps}\n\\caption{Superfluid fraction of the density $\\rho_0\/n$ as a function of the dimensionless temperature $T\/n$ for interaction strength $\\lambda=0.5$ at different macroscopic scales $k_\\text{ph}=1$ (upper curve), $k_\\text{ph}=10^{-0.5}$, $k_\\text{ph}=10^{-1}$, $k_\\text{ph}=10^{-1.5}$, $k_\\text{ph}=10^{-2}$, $k_\\text{ph}=10^{-2.5}$ (bottom curve). We plot the result obtained with the improved truncation for small scales (solid) as well as the result obtained with our more simple truncation (dotted). (The truncations differ only for the three lowest lines.)}\n\\label{figrhodnoftemperature}\n\\end{figure}\nOne can see that with the improved truncation the jump in the superfluid density is indeed found in the limit $k_\\text{ph}\\rightarrow 0$. Fig. \\ref{figcondensatedensity} \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{SB2dfig12.eps}\n\\caption{Condensate fraction of the density $\\bar{\\rho}_0\/n$ as a function of the dimensionless temperature $T\/n$ for interaction strength $\\lambda=0.5$ at macroscopic scale $k_\\text{ph}=10^{-2}$ (solid curve) and $k_\\text{ph}=10^{-4}$ (dashed curve). For comparison, we also plot the superfluid density $\\rho_0\/n$ at $k_\\text{ph}=10^{-2}$ (dotted). These results are obtained with the improved truncation.}\n\\label{figcondensatedensity}\n\\end{figure}\nshows the condensate fraction $\\bar{\\rho}_0\/n$ and the superfluid density fraction $\\rho_0\/n$ as a function of $T\/n$. We observe the substantial $k_\\text{ph}$ dependence of the condensate fraction, as well as an effective jump at $T_c$ for small $k_\\text{ph}$. We recall that the infinite volume limit $k_\\text{ph}=0$ amounts to $\\bar{\\rho}_0=0$ for $T>0$. \n\nThe Kosterlitz-Thouless description is only valid if the zero Matsubara frequency mode ($n=0$) dominates. For a given nonzero $T$ this is always the case if the the characteristic length scale goes to infinity. In the infinite volume limit the characteristic length scale is given by the correlation length $\\xi$. The description in terms of a classical two dimensional system with U(1) symmetry is the key ingredient of the Kosterlitz-Thouless description and holds for $\\xi^2 T\\gg 1$. In the infinite volume limit this always holds for $Tk_\\text{ph}^2$. \n\nFor very small temperatures $Tk_\\text{ph}^2$ and $T\\Lambda_\\text{UV}^2$. \n\nFor bosons with a pointlike repulsive interaction we found in section \\ref{sec:Repulsiveinteractingbosons} that the scattering length is bounded by the ultraviolet scale $a<3\\pi\/(4\\Lambda)$. This is an effect due to quantum fluctuations similar to the ``triviality bound'' for the Higgs scalar in the standard model of elementary particle physics. For a given value of the dimensionless combination $an^{1\/3}$ we cannot choose $\\Lambda\/n^{1\/3}$ larger then $3\\pi\/(4 an^{1\/3})$. For our numerical calculations we use $\\Lambda\/n^{1\/3}\\approx 10$. Other momentum scales are set by the temperature and the chemical potential. The lowest nonzero Matsubara frequency gives the momentum scale $\\Lambda_T^2=2\\pi T$. For a Bose gas with $a=0$ one has $T_c\/n^{2\/3}\\approx 6.625$ such that $\\Lambda_{T_c}\/n^{1\/3}\\approx 6.45$. The momentum scale associated to the chemical potential is $\\Lambda_\\mu^2=\\mu$. For small temperatures and scattering length one finds $\\mu\\approx 8\\pi a n$ and thus $\\Lambda_\\mu\/n^{1\/3}\\approx \\sqrt{8\\pi a n^{1\/3}}$.\n\nWe finally note that the thermodynamic relations for intensive quantities can only involve dimensionless ratios. We may set the unit of momentum by $n^{1\/3}$. The thermodynamic variables are then $T\/n^{2\/3}$ and $\\mu\/n^{2\/3}$. The thermodynamic relations will depend on the strength of the repulsive interaction $\\lambda$ or the scattering length $a$, and therefore on a ``concentration'' type parameter $a n^{1\/3}$.\n\n\n\\subsubsection{Density, superfluid density, condensate and correlation length}\nLet us start our discussion with the density. In the grand canonical formalism it is obtained by taking the derivative of the thermodynamic potential with respect to $\\mu$\n\\begin{equation}\nn=-\\frac{1}{V}\\frac{\\partial}{\\partial \\mu}\\Omega_G = \\frac{\\partial p}{\\partial \\mu}{\\big |}_T.\n\\end{equation}\nWe could compute the $\\mu$-derivative of $p$ numerically by solving the flow equation for U with neighboring values of $\\mu$. As desribed above, we use also another method which employs a flow equation directly for $n$. Since we often express dimensionful quantities in units of the interparticle distance $n^{-1\/3}$, it is crucial to have an accurate value for the density $n$. Comparison of the numerical evaluation and the solution of a separate flow equation for $n$ shows higher precision for the latter method and we will therefore employ the flow equation. We plot in Fig. \\ref{fig:densityofmu} the density in units of the scattering length, $na^3$, as a function of the dimensionless combination $\\mu a^2$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig1.eps}\n\\caption{Density in units of the scattering length $n a^3$ as a function of the (rescaled) chemical potential $\\mu a^2$. We choose for the temperatures $T a^2=2\\cdot 10^{-4}$ (solid curve), $T a^2=4 \\cdot 10^{-4}$ (dashed-dotted curve) and $T a^2= 6 \\cdot 10^{-4}$ (dashed curve). For all three curves we use $a\\Lambda=0.1$.}\n\\label{fig:densityofmu}\n\\end{figure}\n\nFor a comparison with experimentally accessible quantities we have to replace the interaction parameter $\\lambda$ in the microscopic action \\eqref{microscopicaction} by a scattering length $a$ which is a macroscopic quantity. For this purpose we start the flow at the UV-scale $\\Lambda_\\text{UV}$ with a given $\\lambda$, and then compute the scattering length in vacuum ($T=n=0$) by following the flow to $k=0$, see section \\ref{sec:Repulsiveinteractingbosons}. This is a standard procedure in quantum field theory, where a ``bare coupling'' ($\\lambda$) is replaced by a renormalized coupling ($a$). For an investigation of the role of the strength of the interaction we may consider different values of the ``concentration'' $c=an^{1\/3}$ or of the product $\\mu a^2$. While the concentration is easier to access for observation, it is also numerically more demanding since for every value of the parameters one has to tune $\\mu$ in order to obtain the appropriate density. For this reason we rather present results for three values of $\\mu a^2$, i.~e. $\\mu a^2= 2.6\\times 10^{-5}$ (case I), $\\mu a^2=0.0040$ (case II) and $\\mu a^2=0.044$ (case III). The prize for the numerical simplicity is a week temperature dependence of the concentration $c=a n^{1\/3}$ for the three different cases, as shown in Fig. \\ref{fig:an13}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig2.eps}\n\\caption{Concentration $c=an^{1\/3}$ as a function of temperature $T\/(n^{2\/3})$ for the three cases investigated in this paper. Case I corresponds to $an^{1\/3}\\approx 0.01$ (crosses), case II to $an^{1\/3}\\approx 0.05$ (dots) and case III has $an^{1\/3}\\approx 0.01$ (stars).}\n\\label{fig:an13}\n\\end{figure}\nHere and in the following figures case I, which corresponds to $an^{1\/3}\\approx 0.01$, is represented by the little crosses, case II with $an^{1\/3}\\approx 0.05$ by the dots and case III with $an^{1\/3}\\approx 0.1$ by the stars. It is well known that the critical temperature depends on the concentration $c=an^{1\/3}$. From our calculation we find $T_c\/(n^{2\/3})=6.74$ with $c=0.0083$ at $T=T_c$ in case I, $T_c\/(n^{2\/3})=7.16$ with $c=0.044$ at $T=T_c$ in case II and finally $T_c\/(n^{2\/3})=7.75$ with $c=0.088$ at $T=T_c$ in case III.\n\nThis values can are obtained by following the superfluid fraction of the density $n_S\/n$, or equivalently the condensate part of the density $n_C\/n$ as a function of temperature. For small temperatures $T\\to0$ all of the density is superfluid, which is a consequence of Galilean symmetry. However, in contrast to the ideal gas, not all particles are in the condensate. For $T=0$ this condensate depletion is completely due to quantum fluctuations. With increasing temperature both the superfluid density and the condensate decrease and vanish eventually at the critical temperature $T=T_c$. That the melting of the condensate is continuous shows that the phase transition is of second order. We plot our results for the superfluid fraction in Fig. \\ref{fig:superfluidfraction} and for the condensate in Fig. \\ref{fig:condensatefraction}. For small temperatures, we also show the corresponding result obtained in the framework of Bogoliubov theory \\cite{Bogoliubov} (dashed lines). This approximation assumes a gas of non-interacting quasiparticles (phonons) with dispersion relation\n\\begin{equation}\n\\epsilon(p)=\\sqrt{2\\lambda n \\vec p^2+\\vec p^4}.\n\\end{equation} \nIt is is valid in the regime with small temperatures $T\\ll T_c$ and small interaction strength $an^{1\/3}\\ll 1$. For a detailed discussion of Bogoliubov theory and the calculation of thermodynamic observables in this framework we refer to ref. \\cite{PitaevsikiiStringari2003}. Our curves for the superfluid fraction match the Bogoliubov result for temperatures $T\/n^{2\/3}\\lesssim 1$ in all three cases I, II, and III. For larger temperatures there are deviations as expected. For the condensate density, there is already notable a deviation at small temperatures for case III with $an^{1\/3}\\approx 0.1$. This is also expected, since Bogoliubov theory gives only the first order contribution to the condensate depletion in a perturbative expansion for small $an^{1\/3}$. \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig3.eps}\n\\caption{Superfluid fraction of the density $n_S\/n$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III. For small $T\/n^{2\/3}$ we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines).}\n\\label{fig:superfluidfraction}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig4.eps}\n\\caption{Condensate fraction of the density $n_C\/n$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III. For small $T\/n^{2\/3}$ we also show the corresponding curves obtained in the Bogoliubov approximation (dashed lines).}\n\\label{fig:condensatefraction}\n\\end{figure}\nFor temperatures slightly smaller than the critical temperature $T_c$ one expects that the condensate density behaves like \n\\begin{equation}\nn_c(T)=B^2 \\left(\\frac{T_c-T}{T_c}\\right)^{2\\beta}\n\\label{eq:scalingnc}\n\\end{equation}\nwith $\\beta=0.3485$ the critical exponent of the three-dimensional XY-universality class \\cite{Pelissetto2002549}. Indeed, the condensate density is given by $n_C=\\bar\\varphi_0^*\\bar\\varphi_0$\nwhere $\\bar \\varphi_0$ is the expectation value of the boson field which serves as an order parameter in close analogy to e.~g. the magnetization $\\vec M$ in a ferromagnet. Eq.\\ \\eqref{eq:scalingnc} is compatible with our findings, although our numerical resolution does not allow for a precise determination of the exponent $\\beta$. \n\nWith our method we can also calculate the correlation length $\\xi$. For temperatures $TT_c$ there is only one correlation length $\\xi^{-1}=m=\\frac{1}{\\bar A}\\frac{\\partial U}{\\partial \\bar \\rho}$, which also diverges for $T\\to T_c$. From the theory of critical phenomena one expects close to $T_c$ the behavior\n\\begin{eqnarray}\n\\nonumber\n\\xi_R &=& f_R^- \\left(\\frac{T_c-T}{T_c}\\right)^{-\\nu} \\quad \\text{for}\\quad TT_c$ as a function of the temperature $T\/n^{2\/3}$ in Fig. \\ref{fig:correlationlength}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig5.eps}\n\\caption{Correlation length $\\xi_R n^{1\/3}$ for $TT_c$ as a function of the temperature $T\/n^{2\/3}$ for the cases I, II, and III.}\n\\label{fig:correlationlength}\n\\end{figure}\n\n\n\\subsubsection{Entropy density, energy density, and specific heat}\nThe next thermodynamic quantity we investigate is the entropy density $s$ and the entropy per particle $s\/n$. We can obtain the entropy as\n\\begin{equation}\ns=\\frac{\\partial p}{\\partial T}{\\big |}_\\mu.\n\\end{equation}\nWe compute the temperature derivative by numerical differentiation, using flows with neighboring values of $T$ and show the result in Fig. \\ref{fig:entropypp}. For small temperatures our result coincides with the entropy of free quasiparticles in the Bogoliubov approximation (dashed lines in Fig. \\ref{fig:entropypp}). \nAs it should be, the entropy per particle increases with the temperature. For small temperatures, the slope of this increase is smaller for larger concentration $c$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig6.eps}\n\\caption{Entropy per particle $s\/n$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I, II, and III. For $T\/n^{2\/3}<5$ we also plot the results obtained within the Bogoliubov approximation (dashed lines).}\n\\label{fig:entropypp}\n\\end{figure}\n\nFrom the entropy density $s$ we infer the specific heat per particle,\n\\begin{equation}\nc_v=\\frac{T}{n}\\frac{\\partial s}{\\partial T}{\\bigg |}_n,\n\\end{equation}\nas the temperature derivative of the entropy density at constant particle density.\nUsing the Jacobian, we can write\n\\begin{equation}\n\\frac{\\partial s}{\\partial T}{\\big |}_n = \\frac{\\partial(s,n)}{\\partial(T,n)}=\\frac{\\partial(s,n)}{\\partial(T,\\mu)}\\frac{\\partial(T,\\mu)}{\\partial(T,n)}.\n\\end{equation}\nFor the specific heat this gives\n\\begin{equation}\nc_v=\\frac{T}{n}\\left(\\frac{\\partial s}{\\partial T}{\\big |}_\\mu -\\frac{\\partial s}{\\partial \\mu}{\\big |}_T\\frac{\\partial n}{\\partial T}{\\big |}_\\mu \\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right)^{-1} \\right).\n\\end{equation}\nOur result for the specific heat per particle is shown for different scattering lengths in Fig. \\ref{fig:specificheatpp}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig7.eps}\n\\caption{Specific heat per particle $c_v$ as a function of the dimensionless temperature $T\/n^{2\/3}$. The dashed lines show the Bogoliubov result for $c_v$ which coincides with our findings for small temperature. However, the characteristic cusp behavior cannot be seen in a mean-field theory.}\n\\label{fig:specificheatpp}\n\\end{figure}\nWhile this quantity is positive in the whole range of investigated temperatures, it is interesting to observe the cusp at the critical temperature $T_c$ which is characteristic for a second order phase transition. This behavior cannot be seen in a mean-field approximation, where fluctuations are taken into account only to second order in the fields. Only for small temperatures, our curve is close to the Bogoliubov approximation, shown by the dashed lines in Fig. \\ref{fig:specificheatpp}. \n\nIn fact, close to $T_c$ the specific heat is expected to behave like\n\\begin{eqnarray}\n\\nonumber\nc_v &\\approx& b_1-b_2^- \\left(\\frac{T_c-T}{T_c}\\right)^{-\\alpha} \\quad \\text{for} \\quad TT_c,\n\\end{eqnarray}\nwith the universal critical exponent $\\alpha$ of the $3$-dimensional $XY$ universality class, $\\alpha=-0.0146(8)$ \\cite{Pelissetto2002549}. The critical region, where the law $c_v~\\sim |T-T_c|^{-\\alpha}$ holds, may be quite small.\nOur numerical differentiation procedure cannot resolve the details of the cusp. \n\nIn the grand canonical formalism, the energy density $\\epsilon$ is obtained as\n\\begin{equation}\n\\epsilon = -p +Ts+\\mu n.\n\\end{equation}\nWe plot $p\/(n^{5\/3})$ as a function of temperature in Fig. \\ref{fig:pressure} and the energy density $\\epsilon\/(n^{5\/3})$ is plotted in Fig. \\ref{fig:energy}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig8.eps}\n\\caption{Pressure in units of the density $p\/n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines).}\n\\label{fig:pressure}\n\\end{figure}\nWe have normalized the pressure such that it vanishes for $T=\\mu=0$. Technically we subtract from the flow equation of the pressure the corresponding expression in the limit $T=\\mu=0$. This procedure has to be handled with care and leads to an uncertainty in the offset of the pressure, i.~e. the part that is independent of $T\/n^{2\/3}$ and $\\mu\/n^{2\/3}$. \n\nFor zero temperature, the pressure is completely due to the repulsive interaction between the particles. For nonzero temperature, the pressure is increased by the thermal kinetic energy, of course.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig9.eps}\n\\caption{Energy per particle $\\epsilon\/n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the curves obtained in the Bogoliubov approximation for small temperatures (dashed lines).}\n\\label{fig:energy}\n\\end{figure}\nFor the energy and the pressure we find some deviations from the Bogoliubov result already for small temperatures in cases II and III. These deviations may be partly due to the uncertainty in the normalization process described above. For weak interactions $an^{1\/3}=0.01$ as in case I, the Bogoliubov prediction coincides with our result. \n\n\n\\subsubsection{Compressibility}\nThe isothermal compressibility is defined as the relative volume change at fixed temperature $T$ and particle number $N$ when some pressure is applied\n\\begin{equation}\n\\kappa_T=-\\frac{1}{V}\\frac{\\partial V}{\\partial p}{\\big |}_{T,N}=\\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_T.\n\\label{eq:isothermalkomp}\n\\end{equation}\nVery similar, the adiabatic compressibility is\n\\begin{equation}\n\\kappa_S=-\\frac{1}{V}\\frac{\\partial V}{\\partial p}{\\big |}_{S,N}=\\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_{s\/n}\n\\end{equation}\nwhere now the entropy $S$ and the particle number $N$ are fixed. Let us first concentrate on the isothermal compressibility $\\kappa_T$. To evaluate it in the grand canonical formalism, we have to change variables to $T$ and $\\mu$. \nWith $\\partial p\/\\partial \\mu{\\big |}_{n,T}=n$ and $\\partial p\/\\partial n{\\big |}_T=n \\partial \\mu\/\\partial n{\\big |}_T$ one obtains\n\\begin{equation}\n\\kappa_T=\\frac{1}{n^2}\\frac{\\partial n}{\\partial \\mu}{\\big |}_{T}.\n\\end{equation}\nThis expression can be directly evaluated in our formalism by numerical differentiation with respect to $\\mu$. \n\nThe approach to the adiabatic compressibility is similar. Using again the Jacobian we have\n\\begin{eqnarray}\n\\nonumber\n\\kappa_S &=& \\frac{1}{n}\\frac{\\partial n}{\\partial p}{\\big |}_{s\/n}=\\frac{1}{n}\\frac{\\partial(n,s\/n)}{\\partial(p,s\/n)}\\\\\n&=& \\frac{1}{n}\\frac{\\partial (n,s\/n)}{\\partial (\\mu,T)}\\frac{\\partial(\\mu,T)}{\\partial(p,s\/n)}.\n\\end{eqnarray}\nWe need therefore\n\\begin{equation}\n\\frac{\\partial(n,s\/n)}{\\partial(\\mu,T)}=\\frac{1}{n}\\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_{T}\\frac{\\partial s}{\\partial T}{\\big |}_{\\mu}-\\frac{\\partial n}{\\partial T}{\\big |}_{\\mu}\\frac{\\partial s}{\\partial \\mu}{\\big |}_{T}\\right)\n\\end{equation}\nand also\n\\begin{eqnarray}\n\\nonumber\n\\frac{\\partial(p,s\/n)}{\\partial(\\mu,T)} &=& \\frac{1}{n}{\\bigg (}\\frac{\\partial p}{\\partial \\mu}{\\big |}_{T}\\frac{\\partial s}{\\partial T}{\\big |}_\\mu - \\frac{\\partial p}{\\partial \\mu}{\\big |}_{T} \\frac{s}{n}\\frac{\\partial n}{\\partial T}{\\big |}_\\mu\\\\\n&& -\\frac{\\partial p}{\\partial T}{\\big |}_{\\mu} \\frac{\\partial s}{\\partial \\mu}{\\big |}_T+\\frac{\\partial p}{\\partial T}{\\big |}_\\mu \\frac{s}{n} \\frac{\\partial n}{\\partial \\mu}{\\big |}_T {\\bigg )}\\\\\n\\nonumber\n&=& \\left(\\frac{\\partial s}{\\partial T}{\\big |}_\\mu-2\\frac{s}{n}\\frac{\\partial n}{\\partial T}{\\big |}_\\mu+\\frac{s^2}{n^2}\\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right).\n\\end{eqnarray}\nIn the last equations we used the Maxwell identity $\\frac{\\partial n}{\\partial T}{\\big |}_\\mu=\\frac{\\partial s}{\\partial \\mu}{\\big |}_T$. Combining this we find\n\\begin{equation}\n\\kappa_S=\\frac{\\left(\\frac{\\partial n}{\\partial \\mu}{\\big |}_T \\frac{\\partial s}{\\partial T}{\\big |}_\\mu-\\left(\\frac{\\partial n}{\\partial T}{\\big |}_\\mu\\right)^2\\right)}{\\left(n^2 \\frac{\\partial s}{\\partial T}{\\big |}_\\mu-2 s n \\frac{\\partial n}{\\partial T}{\\big |}_\\mu+s^2 \\frac{\\partial n}{\\partial \\mu}{\\big |}_T\\right)}.\n\\end{equation}\nSince $\\partial s\/\\partial T{\\big |}_\\mu=(\\partial^2 p\/\\partial T^2){\\big |}_\\mu$ we need to evaluate a second derivative numerically.\nWe plot the isothermal and the adiabatic compressibility in Figs. \\ref{fig:Isothermalcompressibility} and \\ref{fig:Adiabaticcompressibility}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig10.eps}\n\\caption{Isothermal compressibility $\\kappa_T\\, n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:Isothermalcompressibility}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig11.eps}\n\\caption{Adiabatic compressibility $\\kappa_S\\, n^{5\/3}$ as a function of temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:Adiabaticcompressibility}\n\\end{figure}\n\nFor the isothermal compressibility the temperature dependence is qualitatively different than in Bogoliubov theory already for small temperatures, while there seem to be only quantitative differences for the adiabatic compressibility. The perturbative calculation of the compressibility is difficult since it is diverging in the non-interacting limit $an^{1\/3}\\to 0$. \n\n\n\\subsubsection{Isothermal and adiabatic sound velocity}\nThe sound velocity of a normal fluid under isothermal conditions, i.~e. for constant temperature $T$ is given by\n\\begin{equation}\nv_T^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}{\\big |}_T.\n\\label{eq:singlefluidisothermalsound}\n\\end{equation} \nWe can obtain this directly from the isothermal compressibility\n\\begin{equation}\nM v_T^2=(n \\kappa_T)^{-1}\n\\end{equation}\nas follows from Eq.\\ \\eqref{eq:isothermalkomp}. We plot our result for $v_T^2$ in Fig. \\ref{fig:SingleFluidisothermalsound}, recalling our units $2M=1$ such that $v_T^2$ stands for $2Mv_T^2$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig12.eps}\n\\caption{Isothermal velocity of sound as appropriate for single fluid $v_T^2\/n^{2\/3}=1\/(\\kappa_T\\, n^{5\/3})$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:SingleFluidisothermalsound}\n\\end{figure}\nThis plot also covers the superfluid phase where the physical meaning of $v_T^2$ is partly lost. This comes since the sound propagation there has to be described by more complicated two-fluid hydrodynamics. In addition to the normal gas there is now also a superfluid fraction allowing for an additional oscillation mode. We will describe the consequences of this in the next section.\n\nFor most applications the adiabatic sound velocity is more important then the isothermal sound velocity. Keeping the entropy per particle fixed, we obtain\n\\begin{equation}\nv_S^2=\\frac{1}{M}\\frac{\\partial p}{\\partial n}{\\big |}_{s\/n}\n\\label{eq:singlefluidadiabaticsound}\n\\end{equation}\nand therefore\n\\begin{equation}\nM v_S^2 = (n\\kappa_S)^{-1}.\n\\end{equation}\nOur numerical result is plotted in Fig. \\ref{fig:SingleFluidadiabaticsound}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TDfig13.eps}\n\\caption{Adiabatic velocity of sound as appropriate for single fluid $v_S^2\/n^{2\/3}=1\/(\\kappa_S \\,n^{5\/3})$ as a function of the dimensionless temperature $T\/n^{2\/3}$ for the cases I (crosses), II (dots), and III (stars). We also show the Bogoliubov result for small temperatures (dashed lines).}\n\\label{fig:SingleFluidadiabaticsound}\n\\end{figure}\nAgain the plot covers both the superfluid and the normal part, but only in the normal phase the object $v_S^2$ has its physical meaning as a sound velocity.\n\n\n\\subsubsection{First and second velocity of sound}\nFor temperatures $00$ (dashed). The inset is a magnification of the little box and shows the energy of the first excited Efimov state.}\n\\label{fig:Efimov}\n\\end{figure}\nFor small Yukawa couplings $\\bar h^2\/\\Lambda\\ll 1$, or narrow resonances we find that the range of scattering length where the trimer is the lowest excitation of the vacuum increases linear with $\\bar h^2$ \\cite{PhysRevLett.93.143201, gogolin:140404}. More explicit, we find $a^{-1}_{c1}=-0.0015\\,\\bar h^2$, $a^{-1}_{c2}=0.0079\\,\\bar h^2$. However, for very broad Feshbach resonances $\\bar h^2\/\\Lambda\\gg1$ the range depends on the ultraviolet scale $a^{-1}_{c1},a^{-1}_{c2}\\sim \\Lambda$. We show this behavior in Fig. \\ref{fig:ascaling}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{TFfig3.eps}\n\\caption{Interval of scattering length $a^{-1}_{c1}0$, while the other half has formally $\\bar g^2(k)<0$. We may use the mapping discussed after Eq.\\ \\eqref{eq:subst} to obtain an equivalent picture with positive $\\bar g^2$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\linewidth]{TFfig4.eps}\n\\caption{Limit cycle in the renormalization group flow at the unitarity point $a^{-1}=0$, and for energy at the fermion threshold $\\tilde{\\mu}=\\mu+E=0$. We plot the rescaled gap parameter of the trimer $\\tilde m^2(t)$ (solid) and the rescaled Yukawa coupling $\\tilde g^2(t)$ (dashed). The dotted curves would be obtained from naive continuation of the flow after the point where $\\tilde g^2=0$.}\n\\label{fig:limitcycla}\n\\end{figure}\n\nWe may understand the repetition of states by the following qualitative picture. The coupling $\\tilde m^2$, that is proportional to the energy gap of the trimer, starts on the ultraviolet scale with some positive value. The precise initial value is not important. The Yukawa-type coupling $\\bar g$ vanishes initially so that the trion field $\\chi$ is simply an auxiliary field which decouples from the other fields and is not propagating. However, quantum fluctuations lead to the emergence of a scattering amplitude between the original fermions $\\psi$ and the bosons $\\varphi$. We describe this by the exchange of a composite fermion $\\chi$. \nThis leads to an increase of the coupling $\\tilde g^2$ and a decrease of the trion gap $\\tilde m^2$. At some scale $t_1=\\text{ln}(k_1\/\\Lambda)$ with $k_1^2\\approx -\\mu$ the coupling $\\tilde m^2$ crosses zero which indicates that a trion state $\\chi$ becomes the lowest energy excitation of the vacuum. Indeed, would we consider the flow without modifying the chemical potential, this would set an infrared cutoff that stops the flow at the scale $k_1\\approx\\sqrt{|\\mu|}$ and the trion $\\chi$ would be the gapless propagating particle while the original fermions $\\psi$ and the bosons $\\varphi$ are gapped since they have higher energy. \n\nFollowing the flow further to the infrared, we find that the Yukawa coupling $\\tilde g^2$ decreases again until it reaches the point $\\tilde g^2=0$ at $t=t_1^\\prime$ (see Fig. \\ref{fig:limitcycla}). Naive continuation of the flow below that scale would lead to $\\tilde g^2<0$ and therefore imaginary Yukawa coupling $\\tilde g$. However, since the trion field $\\chi$ decouples from the other fields for $\\tilde g=0$, we are not forced to use the same field $\\chi$ as before. We can simply use another auxiliary field $\\chi_2$ with very large gap $m^2_{\\chi_2}=\\bar m_{\\chi_2}^2\/\\bar A_{\\chi_2}$ to describe the scattering between fermions and bosons on scales $t2\\, l_\\textrm{vdW}$. Therefore the zero-range approximation might be questionable. Since the precise value of $h$ is not known, we use the dependence of our results on $h$ as an estimate of their uncertainty within our truncation \\eqref{eq:action}.\nThe initial values of the couplings $m_\\chi^2$ and $g_i$ are parameters in addition to the scattering lengths which have to be fixed from experimental observation. For equal interaction between atoms $\\psi$ and bosons $\\varphi$ in the UV, the parameter to be fixed is\n\\begin{equation}\n\\lambda^{(3)}=-\\frac{g^2(\\Lambda)}{m_\\chi^2(\\Lambda)}\n\\end{equation}\nwith $g=g_1=g_2=g_3$. Pointlike interactions at the microscopic scale may be realized by $m_\\chi^2(\\Lambda)\\to \\infty$. \n\n\nWe solve the flow equations \\eqref{eq:flowbosonprop}, \\eqref{eq:flowofh}, \\eqref{eq:flowofm} and \\eqref{eq:flowofg} numerically. We find $m_\\chi^2=0$ at $k=0$\nfor some range of $\\lambda^{(3)}$ and $\\mu\\leq 0$ for large enough values of the scattering lengths $a_{12}$, $a_{23}$ and $a_{31}$. This indicates the presence of a bound state of three atoms $\\chi\\widehat{=}\\psi_1\\psi_2\\psi_3$. The binding energy $E_T$ of this bound state is given by the chemical potential, $E_T=3|\\mu|$ with $\\mu$ fixed such that $m_\\chi^2=0$. To compare with the recently performed experimental investigations of $^6$Li \\cite{ottenstein:203202, Huckans2008}, we adapt the initial value $\\lambda^{(3)}$ such that the appearance of this bound state corresponds to a magnetic field $B=125\\, \\text{G}$, the point where strong three-body losses have been observed. Using the same initial value of $\\lambda^{(3)}$ also for other values of the magnetic field, all microscopic parameters are fixed. We can now proceed to the predictions of our model.\n\nFirst we find that the bound state of three atoms exists in the magnetic field region from $B=125\\, \\text{G}$ to $B=498\\, \\text{G}$. The binding energy $E_T$ is plotted as the solid line in the lower panel of Fig. \\ref{fig:EnergiesLi}. We choose here $h^2=100\\, a_0^{-1}$, as appropriate for $^6$Li in the (1,2)-channel close to the resonance, while the shaded region corresponds to $h^2\\in(20\\, a_0^{-1},300\\, a_0^{-1})$. If one includes the contribution to the flow of the atom-dimer interaction arising from box-diagrams by means of a refermionization procedure (see section \\ref{ssect:SU3symmetricmodel}), the flow equations for the Yukawa couplings $g_i$, as in Eq.\\ \\eqref{eq:flowofg}, receive an additionial contribution\n\\begin{eqnarray}\nm_\\chi^2\\left(-\\frac{\\partial_t \\lambda_{ij}^{(3)}}{2 g_j}-\\frac{\\partial_t \\lambda_{il}^{(3)}}{2 g_l}+\\frac{g_i\\partial_t \\lambda_{jl}^{(3)}}{2 g_j g_l}\\right).\n\\end{eqnarray}\nHere we define $(i,j,l)=(1,2,3)$ and permutations thereof and we find\n\\begin{eqnarray}\n\\partial_t\\lambda_{ij}^{(3)}=\\frac{k^5 h_1 h_2 h_3 h_l(9k^2-7\\mu+\\frac{4 m_{\\varphi l}^2}{A_{\\varphi l}})}{6\\pi^2 A_{\\varphi l}(k^2-\\mu)^3(3k^2-2\\mu+\\frac{2 m_{\\varphi l}^2}{A_{\\varphi l}})^2}.\n\\end{eqnarray}\nThis leads to a reduction of the trion binding energy $E_T$ and the result is shown as the dashed line in Fig. \\ref{fig:EnergiesLi}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Lifig3.eps}\n\\caption{{\\itshape Upper panel:} Scattering length $a_{12}$ (solid), $a_{23}$ (dashed) and $a_{31}$ (dotted) as a function of the magnetic field $B$ for $^6$Li. These curves were calculated by P.~S.~Julienne\nand taken from Ref.\n{\\itshape Lower panel:} Binding energy $E_T$ of the three-body bound state $\\chi\\widehat{=}\\psi_1\\psi_2\\psi_3$. The solid line shows the result without the inclusion of box diagrams contributing to the atom-boson interaction and it corresponds to the initial value $h^2=100\\, a_0^{-1}$, while the shaded region gives the result in the range $h^2=20\\, a_0^{-1}$ (upper border) to $h^2=300\\, a_0^{-1}$ (lower border). The dashed line corresponds to the calculated binding energy $E_T$ when the refermionization of the atom-boson interaction is taken into account.}\n\\label{fig:EnergiesLi}\n\\end{figure}\n\n\nAs a second prediction, we present an estimate of the three-body loss coefficient $K_3$ that has been measured in the experiments by Jochim {\\itshape et al.} \\cite{ottenstein:203202} and O'Hara {\\itshape et al.} \\cite{Huckans2008}. For this purpose it is important to note that the fermionic bound state particle $\\chi$ might decay into states with lower energies. These may be some deeply bound molecules not included in our calculation here. In order to include these decay processes we introduce a decay width $\\Gamma_\\chi$ of the trion. We first assume that such a loss process does not depend strongly on the magnetic field $B$ and therefore work with a constant decay width $\\Gamma_\\chi$. \nThe decay width $\\Gamma_\\chi$ appears as an imaginary part of the trion propagator when continued to real time\n\\begin{equation}\nG_\\chi^{-1}=\\omega-\\frac{\\vec p^2}{3}-m_\\chi^2+i\\frac{\\Gamma_\\chi}{2}.\n\\end{equation}\nWe now perform the calculation of the loss for the fermionic energy gap $\\mu=0$, which corresponds to the open channel energy level. In the region from $B=125\\, \\text{G}$ to $B=498\\, \\text{G}$ the energy gap of the trion is then negative $m_\\chi^2<0$.\nThe three-body loss coefficient $K_3$ for arbitrary $\\Gamma_\\chi$ is obtained as follows. The amplitude to form a trion out of three fermions with vanishing momentum and energy is given by $\\sum_{i=1}^3 h_i g_i\/m_{\\varphi i}^2$.\nThe amplitude for the transition from an initial state of three atoms to a final state of the trion decay products (cf. Fig \\ref{fig:treeLossProcess}) further involves the trion propagator that we evaluate in the limit of small momentum $\\vec p^2=(\\sum_i \\vec p_i)^2\\to0$, and small on-shell atom energies $\\omega_i=\\vec p_i^2$, $\\omega=\\sum_i \\omega_i\\to 0$. A thermal distribution of the initial momenta will induce some corrections. Finally, the loss coefficient involves the unknown vertices and phase space factors of the trion decay -- for this reason our computation contains an unknown multiplicative factor $c_K$. In terms of $p$ given by Eq.\\ \\eqref{eq:decayprob} we obtain the three-body loss coefficient\n\\begin{equation}\nK_3=c_K \\, p.\n\\end{equation}\n\nOur result as well as the experimental data points \\cite{ottenstein:203202} are shown in Fig. \\ref{fig:Losscoefficient}. The agreement between the form of the two curves is already quite remarkable.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Lifig4.eps}\n\\caption{Loss coefficient $K_3$ in dependence on the magnetic field $B$ as measured in \\cite{ottenstein:203202} (dots). The solid line is the fit of our model to the experimental curve. We use here a decay width $\\Gamma_\\chi$ that is independent of the magnetic field $B$.}\n\\label{fig:Losscoefficient}\n\\end{figure}\n\nWe have used three parameters, the location of the resonance at $B_0=125\\,\\text{G}$ which we translate into $\\lambda^{(3)}$, the overall amplitude $c_K$ and the decay width $\\Gamma_\\chi$. They are essentially fixed by the peak at $B_0=125\\,\\text{G}$. The extension of the loss rate away from the peak involves then no further parameter. Our estimated decay width $\\Gamma_\\chi$ corresponds to a short lifetime of the trion of the order of $10^{-8}\\,\\textrm{s}$.\n\nOur simple prediction involves a rather narrow second peak around $B_1\\approx500\\,\\text{G}$, where the trion energy becomes again degenerate with the open channel, cf. Fig. \\ref{fig:EnergiesLi}. The width of this peak is fixed so far by the assumption that the decay width $\\Gamma_\\chi$ is independent of the magnetic field. This may be questionable in view of the close-by Feshbach resonance and the fact that the trion may actually decay into the associated molecule-like bound states which have lower energy. We have tested several reasonable approximations, which indeed lead to a broadening or even disappearance of the second peak, without much effect on the intermediate range of fields $150\\,\\text{G}0, \\nonumber\\\\\n\\partial_k \\lambda &=& \\partial_kU''_k{\\big |}_{\\rho_0}. \n\\end{eqnarray}\nTaking a derivative of Eq.\\ \\eqref{2J} with respect to $\\rho$ one obtains for $\\tilde T=0$\n\\begin{eqnarray}\n\\nonumber\nk\\partial_k U_k^\\prime \\!&=&\\! \\eta_{A_\\varphi}(U_k^\\prime-\\rho U_k^{\\prime\\prime})+\\frac{\\sqrt{2}k}{3\\pi^2 Z_\\varphi}\\left(1-\\frac{2}{d+2}\\eta_{A_\\varphi}\\right)\\\\\n\\nonumber\n&&\\!\\!\\times\\! \\left[ 2\\rho (U_k^{\\prime\\prime})^2\\left(s_{\\text{B,Q}}^{(1,0)}\n +3 s_{\\text{B,Q}}^{(0,1)}\\right)+4\\rho^2 U_k^{\\prime\\prime} U_k^{(3)}s_{\\text{B,Q}}^{(0,1)}\\right]\\\\\n&&+\\frac{k}{3\\pi^2}h_\\varphi^2\\,l(\\tilde \\mu)\\, s_{\\text{F,Q}}^{(1)}.\n\\label{eq:Flowu1}\n\\end{eqnarray}\nThe threshold functions $s_{\\text{B,Q}}^{(0,1)}$, $s_{\\text{B,Q}}^{(1,0)}$, and\n$s_{\\text{F,Q}}^{(1)}$ are defined in App. \\ref{sec:FlowoftheeffectivepotentialforBCSBECcrossover} and\ndescribe again the decoupling of the heavy modes. They can be obtained\nfrom $\\rho$ derivatives of $s^{(0)}_{\\text{B}}$ and $s^{(0)}_{\\text{F}}$. Setting $\\rho=0$\nand $\\tilde T\\to0$, we can immediately infer from Eq.\\ \\eqref{eq:Flowu1} the\nrunning of $m^2$ in the symmetric regime.\n\\begin{equation}\nk\\partial_k m^2=k\\partial_k U_k^\\prime =\\eta_{A_\\varphi}m^2 \n+\\frac{k}{3\\pi^2} h^2\\,l(\\tilde \\mu) \\,s^{(1)}_{\\text{F,Q}}(w_3=0).\n\\label{eq:flowmphi}\n\\end{equation}\nOne can see from Eq.\\ \\eqref{eq:flowmphi} that fermionic fluctuations lead to a\nstrong renormalization of the bosonic ``mass term'' $m^2$. In the course\nof the renormalization group flow from large scale parameters $k$\n(ultraviolet) to small $k$ (infrared) the parameter $m^2$\ndecreases strongly. When it becomes zero at some scale $k>0$ the flow\nenters the regime where the minimum of the effective potential $U_k$ is at\nsome nonzero value $\\rho_0$. This is directly related to spontaneous\nbreaking of the $U(1)$ symmetry and to local order. If $\\rho_0\\neq0$ persists\nfor $k\\to0$ this indicates superfluidity.\n\nFor given $\\bar A_\\varphi,Z_\\varphi,h_\\varphi$, Eq.\\ \\eqref{2J} is a nonlinear\ndifferential equation for $U_k$, which depends on two variables $k$\nand $\\rho$. It has to be supplemented by flow equations for\n$\\bar A_\\varphi,Z_\\varphi,h$. The flow equations for the wave\nfunction renormalization $Z_\\varphi$ and the gradient coefficient\n$\\bar A_\\varphi$ cannot be extracted from the effective potential, but are\nobtained from the following projection prescriptions,\n\\begin{eqnarray}\n\\nonumber\n\\partial_t \\bar Z_\\varphi &=& -\\partial_t \n\\frac{\\partial}{\\partial q_0} (\\bar P_\\varphi)_{12}(q_0,0){\\big |}_{q_0=0} ,\\\\\n\\partial_t \\bar A_\\varphi &=& \\partial_t 2 \\frac{\\partial}{\n \\partial \\vec q\\,^2} (\\bar P_\\varphi)_{22}(0,\\vec q){\\big |}_{\\vec q=0},\n\\end{eqnarray} \nwhere the momentum dependent part of the propagator is defined by\n\\begin{equation}\n\\frac{\\delta^2 \\Gamma_k}{\\delta\\bar\\varphi_a(q)\n\\delta\\bar\\varphi_b(q^\\prime)}\\Big|_{\\varphi_1 =\n\\sqrt{2\\rho_0}, \\varphi_2=0} = (\\bar P_\\varphi)_{ab}(q)\\delta(q+q^\\prime). \n\\end{equation}\nThe computation of the flow of the gradient coefficient is rather involved,\nsince the loop depends on terms of different type, $\\sim (\\vec q \\cdot\\vec\np)^2,~ \\vec q\\,^2$, where $\\vec p$ is the loop momentum. An outline of the calculation and explicit expressions can be found in \\cite{Diehl:2008}.\n\n\n\\section{BCS-Trion-BEC Transition}\n\nNow we turn to the truncation used to investigate the model with three fermion species in Eq.\\ \\eqref{eq8:microscopicactiontrionmodel}. For this model the focus will be on the few-body problem where the approximation scheme can be simpler in some respects. We use the following truncation for the average action\n\\begin{eqnarray}\n\\nonumber\n\\Gamma_k&=&\\int_x {\\bigg \\{} \\psi^\\dagger\\left(\\partial_\\tau-\\Delta-\\mu\\right)\\psi+\\varphi^\\dagger\\left(\\partial_\\tau-\\Delta\/2+m_\\varphi^2\\right)\\varphi\\\\\n\\nonumber\n&&+h\\,\\epsilon_{ijk}\\,\\left(\\varphi_i^*\\psi_j\\psi_k-\\varphi_i\\psi_j^*\\psi_k^*\\right)\/2+\\lambda_\\varphi\\left(\\varphi^\\dagger \\varphi\\right)^2\/2\\\\\n\\nonumber\n&&+\\chi^*\\left(\\partial_\\tau-\\Delta\/3+m_\\chi^2\\right)\\chi+g\\left(\\varphi_i^*\\psi_i^*\\chi-\\varphi_i\\psi_i\\chi^*\\right)\\\\\n&&+\\lambda_{\\varphi\\psi}\\left(\\varphi_i^*\\psi_i^*\\varphi_j\\psi_j\\right){\\bigg \\}}.\n\\label{eq:triontruncation}\n\\end{eqnarray}\nHere we use as always natural nonrelativistic units with $\\hbar=k_B=2M=1$, where $M$ is the mass of the original fermions.\nThe integral in Eq.\\ \\eqref{eq:triontruncation} goes over homogeneous space and over imaginary time as appropriate for the Matsubara formalism $\\int_x=\\int d^3x \\int_0^{1\/T}d\\tau$. On the level of the three-body sector, the symmetry of the problem would allow also for a term $\\sim \\psi^\\dagger \\psi\\varphi^\\dagger \\varphi$ in Eq.\\ \\eqref{eq:triontruncation}. This term plays a similar role as for the case of two fermion species, where it was investigated in \\cite{DKS}. The qualitative features of the three-body scattering are dominated by the term $\\sim\\lambda_{\\varphi\\psi}$ in Eq.\\ \\eqref{eq:triontruncation}. The quantitative influence of a term $\\sim \\psi^\\dagger \\psi \\varphi^\\dagger \\varphi$ on the flow equations was also investigated in \\cite{Moroz2008}.\n\nAt the microscopic scale $k=\\Lambda$, we use the initial values of the couplings in Eq.\\ \\eqref{eq:triontruncation} $g=\\lambda_\\varphi=\\lambda_{\\varphi\\psi}=0$ and $m^2_\\chi\\to\\infty$. Then the fermionic field $\\chi$ decouples from the other fields and is only an auxiliary field which is not propagating. However, depending on the parameters of our model we will find that $\\chi$, which describes a composite bound state of three original fermions $\\chi=\\psi_1\\psi_2\\psi_3$, becomes a propagating degree of freedom in the infrared. The initial values of the boson energy gap $\\nu_\\varphi$ and the Yukawa coupling $h$ will determine the scattering length $a$ between fermions and the width of the resonance, see below. The pointlike limit (broad resonance) corresponds to $\\nu_\\varphi\\to\\infty$, $h^2\\to\\infty$ where the limits are taken such that the effective renormalized four fermion interaction remains fixed. In Eq.\\ \\eqref{eq:triontruncation} we use renormalized fields $\\varphi=\\bar A_\\varphi^{1\/2}(k)\\, \\bar{\\varphi}$, $\\psi=\\bar A_\\psi^{1\/2}(k)\\,\\bar \\psi$, $\\chi = \\bar A_\\chi^{1\/2}(k)\\,\\bar \\chi$, with $\\bar A_\\varphi(\\Lambda)=\\bar A_\\psi(\\Lambda)=\\bar A_\\chi(\\Lambda)=1$, and renormalized couplings $m_\\varphi^2=\\bar{m}_\\varphi^2\/\\bar{A}_\\varphi$, $h=\\bar{h}\/(\\bar{A}_\\varphi^{1\/2}\\bar A_\\psi)$, $\\lambda_\\varphi=\\bar \\lambda_\\varphi\/\\bar A_\\varphi^2$, $m_\\chi^2=\\bar m_\\chi^2\/\\bar A_\\chi$, $g=\\bar g \/(\\bar A_\\chi^{1\/2}\\bar A_\\varphi^{1\/2}\\bar A_\\chi^{1\/2})$, and $\\lambda_{\\varphi\\psi}=\\bar \\lambda_{\\varphi\\psi}\/(\\bar A_\\varphi \\bar A_\\psi)$.\n\nTo derive the flow equations for the couplings in Eq.\\ \\eqref{eq:triontruncation} we have to specify an infrared regulator function $R_k$. Here we use the particularly simple function\n\\begin{equation}\nR_{k}=r(k^2-\\vec p^2)\\theta(k^2-\\vec p^2),\n\\label{eq:cutofftrions}\n\\end{equation}\nwhere $r=1$ for the fermions $\\psi$, $r=1\/2$ for the bosons $\\varphi$, and $r=1\/3$ for the composite fermionic field $\\chi$. This choice has the advantage that we can derive analytic expressions for the flow equations and that it is optimized in the sense of \\cite{Litim2000b}.\n\\subsubsection{Symmetries as a guiding principles}\n\nHow should one choose a truncation? The choice of the appropriate ansatz for the flowing action is certainly one of the most important points for someone who wants to work with the flow equation method in praxis. Besides the necessary physical insight there is one major guiding principle: symmetries. As will be discussed in chapter \\ref{ch:Symmetries} the flowing action $\\Gamma_k$ respects the same symmetries as the microscopic action $S$ if no anomalies of the functional integral measure are present and if the cutoff term $\\Delta S_k$ is also invariant. In the notation of Eq.\\ \\eqref{eq6:truncationexpansion} this implies that the coefficient $g_i$ of an operator ${\\mathcal O}_i[\\Phi]$ that is not invariant under all symmetries will not be generated by the flow equation such that $g_i=0$ for all $k$. As an example we consider the microscopic action of a Bose gas\n\\begin{equation}\nS=\\int \\varphi^* (\\partial_\\tau-\\Delta-\\mu)\\varphi +\\frac{1}{2}\\lambda_\\varphi (\\varphi^*\\varphi)^2.\n\\label{eq6:actionBosegas}\n\\end{equation}\nIt is invariant under the global U(1) symmetry\n\\begin{eqnarray}\n\\nonumber\n\\varphi &\\to& e^{i\\alpha} \\varphi \\\\\n\\varphi^* &\\to& e^{-i\\alpha} \\varphi^*. \n\\end{eqnarray}\nThis implies that only operators that are invariant under this transformation may appear in the flowing action $\\Gamma_k[\\Phi]$. For example, the part that describes homogeneous fields, the effective potential is of the form\n\\begin{equation}\n\\Gamma_k = \\ldots + \\int_x U(\\varphi^*\\varphi)\n\\end{equation}\nwhere $U(\\rho)$ is a function of the U(1)-invariant combination $\\rho=\\varphi^*\\varphi$, only. The action in Eq.\\ \\eqref{eq6:actionBosegas} has more symmetries such as translation, rotation or, at zero temperature, Galilean invariance. \n\nA useful strategy to find a sensible truncation is to start from the microscopic action $S$ or an effective action $\\Gamma$ calculated in some (perturbative) approximation scheme such as for example mean-field theory. One now renders the appearing coefficients to become $k$-dependent ``running couplings'' and adds also additional terms after checking that they are allowed by the symmetries of the microscopic action. \n \n\\subsubsection{Separation of scales}\n\nSome symmetries are realized only in some range of the renormalization group flow. For example, Galilean symmetry is broken explicitly by the thermal heat bath for $T>0$. Nevertheless, for $k^2\\gg T$ the flowing action $\\Gamma_k$ (or its real-time version obtained from analytic continuation) will still be invariant under Galilean boost transformations. This comes since the scale parameter $k$ sets the infrared scale on the right hand side of the flow equation. As long as $k^2\/T$ is large, the flow equations are essentially the same as for $T\\to0$. In other words, the flow only ``feels'' the temperature once the scale $k$ is of the same order of magnitude $k^2\\approx T$. On the other side, for $k^2\\ll T$ the flow equations may simplify again. Now they are equivalent to those obtained in the large temperature limit $T\\to \\infty$. Different symmetries may apply to the action in this limit. This separation of scales is often very useful for practical purposes. In different regimes of the flow different terms are important, while others might be neglected. For example, the universal critical properties such as the critical exponents or amplitude ratios can be calculated in the framework of the classical theory, i.~e. in the large temperature limit $T\\to\\infty$. The flow equations in this limit are much simpler then the ones obtained for arbitrary temperature $T$. \n\nThe scale-separation is also useful for the fixing of the initial coupling constants at the initial scale $k=\\Lambda$. If this scale is much larger then the temperature $\\Lambda^2\\gg T$ and the relevant momentum scale for the density, the inverse interparticle distance $\\Lambda\\gg n^{1\/3}$, the initial flow is the same as in vacuum where $T=n=0$. One can then also use the same initial values for most of the couplings and only change the temperature and the chemical potential appropriately to describe points in the phase diagram that correspond to $T>0$ and $n>0$. \n\n\\subsubsection{Derivative expansion}\n\nA central part of a truncation is the form of the propagator. It follows from the second functional derivative of the flowing action. For the example of a Bose gas one has in the normal phase\n\\begin{equation}\nG_k^{-1}(p) \\,\\delta(p-p^\\prime) = \\frac{\\delta}{\\delta \\varphi^*(p)} \\frac{\\delta}{\\delta \\varphi(p^\\prime)} \\Gamma_k[\\Phi].\n\\label{eqn6:derexpansion}\n\\end{equation}\nThe inverse propagator $G_k^{-1}$ may be a quite complicated function of the momentum $p$ which consists of the spatial momentum and the (Matsubara-) frequency, $p=(p_0, \\vec p)$. From rotational invariance it follows that $G_k^{-1}$ depends on the spatial momentum only in the invariant combination $\\vec p^2$. At zero temperature it follows from Galilean invariance that $G_k^{-1}$ is a (analytic) function of the combination $ip_0+\\vec p^2$, provided that Galilean invariance is not broken by the cutoff. \n\nUsing a derivative expansion, one truncates the flowing action in the form\n\\begin{equation}\n\\Gamma_k = \\int_x \\varphi^* (Z\\partial_\\tau-A\\vec \\nabla^2 - V \\partial_\\tau^2+\\ldots)\\varphi + U(\\varphi^*\\varphi).\n\\end{equation}\nThe ``kinetic coefficients'' $Z$, $A$, $V$ etc.\\ depend on the scale parameter $k$ and for more advanced approximations also on the U(1) invariant combination $\\rho=\\varphi^* \\varphi$. One can improve the expansion in Eq.\\ \\eqref{eqn6:derexpansion} by promoting the coefficients $Z$, $A$, $V$ etc.\\ to functions of $p_0$ and $\\vec p^2$. \n\nIn praxis one usually neglects terms higher then quadratic in the momenta. Nevertheless, derivative expansion often leads to quite good results. The reason is the following. On the right hand side of the flow equation the cutoff insertion $R_k$ in the propagator $(\\Gamma^{(2)}+R_k)^{-1}$ suppresses the contribution of the modes with small momenta. On the other side, the cutoff derivative $\\partial_k R_k$ suppresses the contribution of very large momenta provided that $R_k(q)$ falls of sufficiently fast for large $q$. Effectively mainly modes with momenta of the order $k^2$ contribute. It would therefore be sensible to use on the right hand side of the flow equations the coefficients\n\\begin{equation}\nZ(p_0 = k^2,\\vec p^2=k^2), A(p_0=k^2,\\vec p^2=k^2), \\ldots.\n\\end{equation}\nOne main effect of the external frequencies and momenta in $Z(p_0,\\vec p^2)$ etc.\\ is to provide an infrared cutoff scale of order $\\text{Max}(p_0,\\vec p^2)$. Such an infrared cutoff scale is of course also provided by $R_k$ itself and one might therefore also work with the $k$-dependent couplings\n\\begin{equation}\nZ(p_0=0,\\vec p^2=0), A(p_0=0,\\vec p^2=0),\\ldots.\n\\end{equation}\nWe emphasize that it is important that the cutoff $R_k(q)$ falls off sufficiently fast for large $q$. If this is not the case, the derivative expansion might lead to erroneous results since the kinetic coefficients as appropriate for small momenta and frequencies are then also used for large momenta and frequencies. Only when the scale derivative $\\partial_k R_k$ provides for a sufficient ultraviolet cutoff does the derivative expansion work properly.\n\\subsubsection{Two-body problem}\n\nThe two-body problem is best solved in terms of the bare couplings. Their flow\nequations read\n\\begin{eqnarray}\n\\nonumber\n\\partial_k \\bar{m}_\\varphi^2 &=& \\frac{\\bar{h}_\\varphi^2}{6\\pi^2\n k^3}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\nonumber\n\\partial_k \\bar Z_\\varphi &=& -\\frac{\\bar{h}_\\varphi^2}{6\\pi^2\n k^5}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\nonumber\n\\partial_k \\bar A_\\varphi &=& -\\frac{\\bar{h}_\\varphi^2}{6\\pi^2 k^5}\\theta(k^2+\\mu)(k^2+\\mu)^{3\/2}\\\\ \n\\partial_k \\bar{h}_\\varphi &=&0.\n\\label{eq:vacuumflow}\n\\end{eqnarray}\nThe flow in the two-body sector is driven by fermionic diagrams only. There\nis no renormalization of the Yukawa coupling $\\bar h$. The equations \\eqref{eq:vacuumflow} are solved by direct\nintegration with the result\n\\begin{eqnarray}\n\\nonumber\n\\bar{m}_\\varphi^2(k) &=&\n\\bar{m}_\\varphi^2(\\Lambda)\\\\\n\\nonumber\n&&-\\theta(\\Lambda^2+\\mu)\\frac{\\bar{h}_\\varphi^2}{6\\pi^2}{\\bigg\n [}\\sqrt{\\Lambda^2+\\mu}\\,\\left(1-\\frac{\\mu}{2\\Lambda^2}\\right)\n-\\frac{3}{2}\\sqrt{-\\mu}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]} \\\\\n\\nonumber\n&&+\\theta(k^2+\\mu)\\frac{\\bar{h}_\\varphi^2}{6\\pi^2}{\\bigg\n [}\\sqrt{k^2+\\mu}\\,\\left(1-\\frac{\\mu}{2k^2}\\right) \n-\\frac{3}{2}\\sqrt{-\\mu}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\ \n\\nonumber\n\\bar Z_\\varphi(k) &=&\n\\bar Z_\\varphi(\\Lambda)\\\\\n\\nonumber\n&& -\\theta(\\Lambda^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{\\Lambda^2+\\mu}\n\\,\\frac{\\left(5\\Lambda^2+2\\mu\\right)}{\\Lambda^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber\n&& +\\theta(k^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{k^2+\\mu}\n\\,\\frac{\\left(5k^2+2\\mu\\right)}{k^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber \n\\bar A_\\varphi(k) &=& \\bar A_\\varphi(\\Lambda)\\\\\n\\nonumber\n&& -\\theta(\\Lambda^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{\\Lambda^2+\\mu}\\,\n\\frac{\\left(5\\Lambda^2+2\\mu\\right)}{\\Lambda^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}\\\\\n\\nonumber\n&& +\\theta(k^2+\\mu)\n\\frac{\\bar{h}_\\varphi^2}{48\\pi^2}{\\bigg [}\\sqrt{k^2+\\mu}\\,\n\\frac{\\left(5k^2+2\\mu\\right)}{k^4}\n-\\frac{3}{\\sqrt{-\\mu}}\\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{k^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\bigg ]}.\\\\\n\\label{eq:vacuumsolutions}\n\\end{eqnarray}\nHere, $\\Lambda$ is the initial ultraviolet scale. Let us discuss the initial value for the boson mass. It is given by\n\\begin{equation}\n\\bar{m}_\\varphi^2(\\Lambda)=\\nu(B) -2\\mu+\\delta\n\\nu(\\Lambda). \\label{eq:new41} \n\\end{equation}\nThe detuning $\\nu(B)= \\mu_{\\text{M}} (B-B_0)$ describes the energy level of the microscopic state represented by the field $\\varphi$ with respect to the fermionic state $\\psi$. At a Feshbach resonance, this energy shift can be tuned by the magnetic field $B$, $\\mu_{\\text{M}}$ denotes the magnetic moment of the field $\\varphi$, and $B_0$ is the resonance position. Physical observables such as the scattering length and the binding energy are obtained from the effective action and are therefore related to the coupling constants at the infrared scale $k=0$. The quantity $\\delta\\nu(\\Lambda)$ denotes a renormalization counter term that has to be adjusted conveniently, see below.\n\n\n\\subsubsection{Renormalization}\n\nWe next show that close to a Feshbach resonance the microscopic parameters \n$\\bar m_{\\varphi,\\Lambda}\\equiv\\bar m_\\varphi^2(k=\\Lambda)$ and $\\bar\nh_{\\varphi,\\Lambda}^2\\equiv\\bar h_\\varphi^2(k=\\Lambda)$\nare related to $B-B_0$ and $a$ by two simple relations\n\\begin{equation}\n\\bar m_{\\varphi,\\Lambda}^2=\\mu_\\text{M}(B-B_0)-2\\mu+\\frac{\\bar h_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda\n\\label{eq:mvarphiLambda}\n\\end{equation}\nand \n\\begin{equation}\na=-\\frac{\\bar h_{\\varphi, \\Lambda}^2}{8\\pi \\mu_\\text{M}(B-B_0)}.\n\\label{eq:hvarphiLambda}\n\\end{equation}\nAway from the Feshbach resonance the Yukawa coupling may depend on $B$, $\\bar\nh_\\varphi^2(B)=\\bar h_\\varphi^2+c_1(B-B_0)+\\dots$. Also the microscopic\ndifference of energy levels between the open and closed channel may show\ncorrections to the linear $B$-dependence,\n$\\nu(B)=\\mu_\\text{M}(B-B_0)+c_2(B-B_0)^2+\\dots$ or $\\mu_\\text{M}\\to\n\\mu_\\text{M}+c_2(B-B_0)+\\dots$. Using $\\bar h_\\varphi^2(B)$ and\n$\\mu_\\text{M}(B)$ our formalism can easily be adapted to a more general\nexperimental situation away from the Feshbach resonance. The relations in\nEqs. \\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} hold for all\nchemical potentials $\\mu$ and temperatures $T$. For a different choice of the\ncutoff function the coefficient $\\delta\\nu(\\Lambda)$ being the term linear in $\\Lambda$ in Eq.\\ \\eqref{eq:mvarphiLambda} might be modified.\n\nWe want to connect the bare parameters $\\bar m_{\\varphi,\\Lambda}^2$ and $\\bar\nh_{\\varphi,\\Lambda}^2$ with the magnetic field $B$ and the scattering length $a$ for\nfermionic atoms as renormalized parameters. In our units, $a$ is related to\nthe effective interaction $\\lambda_{\\psi,\\text{eff}}$ by\n\\begin{equation}\na=\\frac{\\lambda_{\\psi,\\text{eff}}}{8\\pi}.\n\\end{equation}\nThe fermion interaction\n$\\lambda_{\\psi,\\text{eff}}$ is determined by the molecule exchange process in\nthe limit of vanishing spatial momentum\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{\\bar\n h_{\\varphi,\\Lambda}^2}{\\bar{P}_\\varphi(\\omega,\\vec{p}^2=0,\\mu)}. \n\\label{eqlambdaeffmoleculeexchange}\n\\end{equation}\nEven though \\eqref{eqlambdaeffmoleculeexchange} is a tree-level process, it is not an approximation, since $\\bar{P}_\\varphi\\equiv\\bar P_{\\varphi}|_{k\\to0}$ denotes the full bosonic propagator which includes all fluctuation effects. The frequency in Eq.\\ \\eqref{eqlambdaeffmoleculeexchange} is the sum of the frequency of the incoming fermions which in turn is determined from the on-shell condition\n\\begin{equation}\n\\omega=2\\omega_\\psi=-2\\mu.\n\\label{eqinitialvalueofmass}\n\\end{equation} \n\nOn the BCS side we have\n$\\mu=0$ and find with \n\\begin{equation}\n\\bar P_\\varphi(\\omega=0,\\vec q=0)=\\bar m_\\varphi^2(k=0)\\equiv \\bar m^2_{\\varphi,0}\n\\end{equation}\nthe relation\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{\\bar m_{\\varphi,0}^2},\n\\end{equation}\nwhere $\\bar m_{\\varphi,0}^2=\\bar m_\\varphi^2(k=0)$. For the bosonic mass terms at $\\mu=0$, we can read off from\nEqs. \\eqref{eq:vacuumsolutions} and \\eqref{eq:new41} that \n\\begin{equation}\n \\bar{m}_{\\varphi,0}^2=\\bar{m}_{\\varphi,\\Lambda}^2\n -\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2} \\Lambda \n = \\mu_{\\text{M}}(B-B_0)+\\delta \\nu(\\Lambda)\n -\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda.\n\\end{equation}\nTo fulfill the resonance condition $a\\to\\pm\\infty$ for $B=B_0$, $\\mu=0$,\nwe choose\n\\begin{equation}\n\\delta \\nu(\\Lambda)=\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{6\\pi^2}\\Lambda.\n\\end{equation}\nThe shift $\\delta\\nu (\\Lambda)$ provides for the additive UV renormalization\nof $\\bar{m}_\\varphi^2$ as a relevant coupling. It is exactly canceled by the\nfluctuation contributions to the flow of the mass. This yields the general\nrelation \\eqref{eq:mvarphiLambda} (valid for all $\\mu$) between the bare mass\nterm $\\bar m_{\\varphi,\\Lambda}^2$ and the magnetic field. On the BCS side we\nfind the simple vacuum relation\n\\begin{equation}\n\\bar m_{\\varphi,0}^2=\\mu_\\text{M}(B-B_0).\n\\end{equation}\nFurthermore, we obtain for the fermionic\nscattering length\n\\begin{equation}\na=-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{8\\pi \\mu_{\\text{M}}(B-B_0)}.\n\\label{eqscatterinlengthandmageticfield}\n\\end{equation}\nThis equation establishes Eq.\\ \\eqref{eq:hvarphiLambda} and shows that\n$\\bar{h}_{\\varphi,\\Lambda}^2$ determines the width of the resonance. We have thereby\nfixed all parameters of our model and can express $\\bar m_{\\varphi,\\Lambda}^2$\nand $\\bar h_{\\varphi,\\Lambda}^2$ by $B-B_0$ and $a$. The relations\n\\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} remain valid also at\nnonzero density and temperature. They fix the ``initial values'' of the flow\n($\\bar h_\\varphi^2\\to \\bar h_{\\varphi,\\Lambda}^2$) at the microscopic scale\n$\\Lambda$ in terms of experimentally accessible quantities, namely $B-B_0$ and\n$a$.\n\nOn the BEC side, we encounter $\\mu<0$ and thus $\\omega>0$. We therefore need\nthe bosonic propagator for $\\omega\\neq 0$. Even though we have computed\ndirectly only quantities related to $\\bar P_\\varphi$ at $\\omega=0$ and\nderivatives with respect to $\\omega$ ($Z_\\varphi$), we can obtain information\nabout the boson propagator for nonvanishing frequency by using the semilocal\n$U(1)$ invariance described in section \\ref{sec:Derivativeexpansionandwardidentities}. In momentum space,\nthis symmetry transformation results in a shift of energy levels\n\\begin{eqnarray}\n\\nonumber\n\\psi(\\omega, \\vec{p}) &\\to& \\psi(\\omega-\\delta,\\vec{p})\\\\\n\\nonumber\n\\varphi(\\omega,\\vec{p}) &\\to& \\varphi(\\omega-2\\delta,\\vec{p})\\\\\n\\mu &\\to& \\mu+\\delta.\n\\end{eqnarray}\nSince the effective action is invariant under this symmetry, it follows for\nthe bosonic propagator that\n\\begin{equation}\n\\bar{P}_\\varphi(\\omega,\\vec{p},\\mu)\n=\\bar{P}_\\varphi(\\omega-2\\delta,\\vec{p},\\mu+\\delta).\n\\end{equation}\nTo obtain the propagator needed in Eq.\\ \\eqref{eqlambdaeffmoleculeexchange}, we\ncan use $\\delta=-\\mu$ and find as in Eq.\\ \\eqref{eqscatterinlengthandmageticfield}\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}\n=-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2}{\\bar{P}_\\varphi(\\omega=0,\\vec{p}^2=0,\\mu=0)}\n=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{\\mu_\\text{M}(B-B_0)}.\n\\end{equation}\nThus the relations \\eqref{eq:mvarphiLambda} and \\eqref{eq:hvarphiLambda} for\nthe initial values $\\bar m_{\\varphi,\\Lambda}$ and $\\bar h_{\\varphi,\\Lambda}^2$\nin terms of $B-B_0$ and $a$ hold for both the BEC and the BCS side of the\ncrossover.\n\n\n\\subsubsection{Binding energy}\n\nWe next establish the relation between the molecular binding energy\n$\\epsilon_\\text{M}$, the scattering length $a$, and the Yukawa coupling $\\bar\nh_{\\varphi,\\Lambda}^2$. From Eq.\\ \\eqref{eq:vacuumsolutions}, we obtain for\n$k=0$ and $\\mu\\leq 0$\n\\begin{eqnarray}\\label{mphiFinal}\n\\nonumber\n\\bar{m}_{\\varphi,0}^2 &=& \\mu_{\\text{M}} (B-B_0)-2\\mu\\\\\n\\nonumber\n&& +\\frac{\\bar h_{\\varphi,\\Lambda}^2}{6\\pi^2} {\\Bigg [}\\Lambda-\\sqrt{\\Lambda^2+\\mu}\n\\left(1-\\frac{\\mu}{2\\Lambda^2}\\right)\\\\\n&& +\\frac{3}{2}\\sqrt{-\\mu} \\,\\,\\text{arctan}\n\\left(\\frac{\\sqrt{\\Lambda^2+\\mu}}{\\sqrt{-\\mu}}\\right){\\Bigg ]}.\n\\end{eqnarray}\nIn the limit $\\Lambda\/\\sqrt{-\\mu}\\to\\infty$ this yields\n\\begin{equation}\n\\bar m_{\\varphi,0}^2 = \\mu_{\\text{M}} (B-B_0)-2\\mu+\\frac{\\bar{h}_{\\varphi,\\Lambda}^2\\sqrt{-\\mu}}{8\\pi}.\n\\end{equation}\nTogether with Eq.\\ \\eqref{eqscatterinlengthandmageticfield}, we can deduce\n\\begin{equation}\na=-\\frac{\\bar h_{\\varphi,\\Lambda}^2}{8\\pi \\left( \\bar m_{\\varphi,0}^2 +2\\mu \n-\\frac{\\bar{h}_{\\varphi,\\Lambda}^2\\sqrt{-\\mu}}{8\\pi}\\right)},\n\\end{equation}\nwhich holds in the vacuum for all $\\mu$. On the BEC side where $\\bar\nm_{\\varphi,0}^2=0$ this yields\n\\begin{equation}\n a=\\frac{1}{\\sqrt{-\\mu}\\left(1+\\frac{16 \\pi}{\\bar h_{\\varphi,\\Lambda}^2}\\sqrt{-\\mu}\\right)}.\n\\label{ScattLength}\n\\end{equation}\nThe binding energy of the bosons is given by the difference between the\nenergy for a boson ${\\bar{m}_\\varphi^2}\/{\\bar{Z}_\\varphi}$ and the energy\nfor two fermions $-2\\mu$. On the BEC side, we can use $\\bar{m}_{\\varphi,0}^2=0$ and\nobtain\n\\begin{equation}\n\\epsilon_{\\text{M}}=\\frac{\\bar{m}_\\varphi^2}{\\bar{Z}_\\varphi}+2\\mu\\Big|_{k\\to0}=2\\mu.\n\\label{eq:bindingenergy}\n\\end{equation}\nFrom Eqs.\\ \\eqref{ScattLength} and \\eqref{eq:bindingenergy} we find a relation\nbetween the scattering length $a$ and the binding energy $\\epsilon_\\text{M}$\n\\begin{equation}\n\\frac{1}{a^2}=\\frac{-\\epsilon_\\text{M}}{2}\n+(-\\epsilon_\\text{M})^{3\/2} \\frac{4\\sqrt{2}\\pi}{\\bar h_{\\varphi,\\Lambda}^2}\n+(-\\epsilon_\\text{M})^2\\frac{(8\\pi)^2}{\\bar h_{\\varphi,\\Lambda}^4}.\n\\label{eq:scatteringlengthandbindingenergy}\n\\end{equation}\nIn the broad resonance limit $\\bar{h}_{\\varphi,\\Lambda}^2\\to\\infty$, this is\njust the well-known relation between the scattering length $a$ and the binding\nenergy $\\epsilon_{\\text{M}}$ of a dimer (see for example \\cite{BraatenHammer})\n\\begin{equation}\n\\epsilon_{\\text{M}}=-\\frac{2}{a^2}=-\\frac{1}{M a^2}.\n\\label{eq:scatteringlengthbroad}\n\\end{equation}\nThe last two terms in Eq.\\ \\eqref{eq:scatteringlengthandbindingenergy} give\ncorrections to Eq.\\ \\eqref{eq:scatteringlengthbroad} for more narrow\nresonances.\n\nThe solution of the two-body problem turns out to be exact as expected. In our\nformalism, this is reflected by the fact that the two-body sector decouples\nfrom the flow equations of the higher-order vertices: no higher-order\ncouplings such as $\\lambda_\\varphi$ enter the set of equations\n(\\ref{eq:vacuumflow}). Extending the truncation to even higher order vertices or\nby including a boson-fermion vertex $\\psi^\\dagger \\psi \\varphi^\\ast\\varphi$\ndoes not change the situation.\n\n\\subsubsection{Dimer-Dimer Scattering}\n\\label{DimerDimer}\n\nSo far we have considered the sector of the theory up to order\n$\\varphi^\\ast\\psi\\psi$, which is equivalent to the fermionic two-body problem\nwith pointlike interaction in the limit of broad resonances. Higher-order\ncouplings, in particular the four-boson coupling\n$\\lambda_\\varphi(\\varphi^\\ast\\varphi)^2$, do not couple to the two-body\nsector. Nevertheless, a four-boson coupling emerges dynamically from the\nrenormalization group flow. \nIn vacuum we have $\\rho_0=0$ and $\\lambda_\\varphi$ is defined as $\\lambda_\\varphi=U^{\\prime\\prime}_k(0)$. The flow equation for $\\lambda_\\varphi$ can be found by taking the $\\rho$-derivative of Eq.\\ \\eqref{eq:Flowu1}\n\\begin{eqnarray}\n\\nonumber\nk \\partial_k \\lambda_\\varphi &=& 2 \\eta_{A_\\varphi} U_k^{\\prime\\prime} - \\frac{\\sqrt{2} k^3}{3\\pi^2 S_\\varphi}\\left(1-\\frac{2}{d+2}\\eta_{A_\\varphi}\\right)\\\\\n\\nonumber\n&&\\times 2 (U_k^{\\prime\\prime})^2 \\left(s_{B,Q}^{(1,0)}+3 s_{B,Q}^{(0,1)}\\right)+\\frac{h_\\varphi^4}{3\\pi^2 k^3} s_{F,Q}^{(2)}\\\\\n\\nonumber\n&=& 2\\eta_{A_\\varphi} \\lambda_\\varphi + \\frac{\\sqrt{2} k^5 \\lambda_\\varphi^2}{3\\pi^2\\, S_\\varphi\\, (m_\\varphi^2+k^2)^2}(1-2\\eta_{A_\\varphi}\/5)\\\\\n&&-\\frac{h_\\varphi^4\\,\\theta(\\mu+k^2)\\,(\\mu+k^2)^{3\/2}}{4\\pi^2k^6}.\n\\end{eqnarray} \nThere are contributions from fermionic and bosonic vacuum fluctuations, but no contribution from higher $\\rho$ derivatives of $U$. The fermionic diagram generates a four-boson coupling even for zero initial value. This coupling then feeds back into the flow equation via the bosonic diagram.\n\nThe scattering lengths are related to the corresponding couplings by the relation (cf. \\cite{Diehl:2007th})\n\\begin{eqnarray}\n \\frac{a_{\\text{M}}}{a}=2\\,\n \\frac{\\lambda_\\varphi}{\\lambda_{\\psi,\\text{eff}}}, \n \\quad \\lambda_{\\psi,\\text{eff}}= 8\\pi a.\n\\end{eqnarray}\nOmitting the bosonic fluctuations, a direct integration yields the mean field result $a_{\\text{M}}\/a=2$. This value is lowered when the bosonic fluctuations are taken into account. With our truncation and choice of cutoff one finds $a_{\\text{M}}\/a =0.718$. The calculation can be improved by extending the truncation to\ninclude a boson-fermion vertex $\\lambda_{\\varphi\\psi}$ which describes the\nscattering of a dimer with a fermion \\cite{DKS}. Inspection of the diagrammatic structure\nshows that this vertex indeed couples into the flow equation for $\\lambda_\\varphi$.\n\nThe ratio $a_M\/a$ has been computed by other methods. Diagrammatic approaches give $a_{\\text{M}}\/a =0.75(4)$\n\\cite{PhysRevB.61.15370}, whereas the solution of the 4-body Schr\\\"{o}dinger\nequation yields $a_{\\text{M}}\/a =0.6$ \\cite{PhysRevLett.93.090404}, confirmed in QMC\nsimulations \\cite{PhysRevLett.93.200404} and with diagrammatic techniques \\cite{Brodsky2005}.\n\\section{Bose gas in three dimensions}\n\\label{sec:Bosegasinthreedimensions}\nA gas of non-relativistic bosons with a repulsive pointlike interaction is one of the simplest interacting statistical systems. Since the first experimental realization \\cite{Andersonetal1995, PhysRevLett.75.1687, PhysRevLett.75.3969} of Bose-Einstein condensation (BEC) \\cite{Einstein1924, Einstein1925, Bose1924} with ultracold gases of bosonic atoms, important experimental advances have been achieved, for reviews see \\cite{RevModPhys.71.463, RevModPhys.73.307, morsch:179, bloch:885, PethickSmith2002, PitaevsikiiStringari2003}. Thermodynamic observables like the specific heat \\cite{1367-2630-8-9-189} or properties of the phase transition like the critical exponent $\\nu$ \\cite{Donneretal2007} have been measured in harmonic traps. Still, the theoretical description of these apparently simple systems is far from being complete.\n\nFor ultracold dilute non-relativistic bosons in three dimensions, Bogoliubov theory gives a successful description of most quantities of interest \\cite{Bogoliubov}. This approximation breaks down, however, near the critical temperature for the phase transition, as well as for the low temperature phase in lower dimensional systems, due to the importance of fluctuations. One would therefore like to have a systematic extension beyond the Bogoliubov theory, which includes the fluctuation effects beyond the lowest order in a perturbative expansion in the scattering length. Such extensions have encountered obstacles in the form of infrared divergences in various expansions \\cite{Beliaev1958, Gavoret1964, Nepomnyashchii1975}. Only recently, a satisfactory framework has been found to cure these problems \\cite{PhysRevLett.78.1612, PhysRevB.69.024513, Wetterich:2007ba}.\nIn this thesis, we extend this formalism to a nonvanishing temperature. We present a quantitative rather accurate picture of Bose-Einstein condensation in three dimensions and find that the Bogoliubov approximation is indeed valid for many quantities. The same method is also applied for two spatial dimensions (see section \\ref{sec:Bosegasintwodimensions}) and can also be applied for one dimension.\n\nFor dilute non-relativistic bosons in three dimensions with repulsive interaction we find an upper bound on the scattering length $a$. This is similar to the \"triviality bound\" for the Higgs scalar in the standard model of elementary particle physics. As a consequence, the scattering length is at most of the order of the inverse effective ultraviolet cutoff $\\Lambda^{-1}$, which indicates the breakdown of the pointlike approximation for the interaction at short distances. Typically, $\\Lambda^{-1}$ is of the order of the range of the Van der Waals interaction. For dilute gases, where the interparticle distance $n^{-1\/3}$ is much larger than $\\Lambda^{-1}$, we therefore always find a small concentration $c=a n^{1\/3}$. This provides for a small dimensionless parameter, and perturbation theory in $c$ becomes rather accurate for most quantities. For typical experiments with ultracold bosonic alkali atoms one has $\\Lambda^{-1}\\approx 10^{-7} \\,\\text{cm}$, $n^{1\/3}\\approx 10^4 \\,\\text{cm}^{-1}$, such that $c\\lesssim 10^{-3}$ is really quite small.\n\nBosons with pointlike interactions can also be employed for an effective description of many quantum phase transitions at zero temperature, or phase transitions at low temperature $T$. In this case, they correspond to quasi-particles, and their dispersion relation may differ from the one of non-relativistic bosons, $\\omega=\\frac{\\vec{p}^2}{2M}$. We describe the quantum phase transitions for a general microscopic dispersion relation, where the inverse classical propagator in momentum and frequency space takes the form $G_0^{-1}=-S\\omega-V\\omega^2+\\vec{p}^2$ (in units where the particle mass $M$ is set to $1\/2$). We present the quantum phase diagram at $T=0$ in dependence on the scattering length $a$ and a dimensionless parameter $\\tilde{v}\\sim V\/S^2$, which measures the relative strength of the term quadratic in $\\omega$ in $G_0^{-1}$. In the limit $S\\rightarrow 0$ ($\\tilde{v}\\rightarrow\\infty$) our model describes relativistic bosons.\n\n\\subsubsection{Lagrangian}\n\nOur microscopic action describes nonrelativistic bosons, with an effective interaction between two particles given by a contact potential. It is assumed to be valid on length scales where the microscopic details of the interaction are irrelevant and the scattering length is sufficient to characterize the interaction. The microscopic action reads\n\\begin{equation}\nS[\\varphi]=\\int_x \\,{\\Big \\{}\\varphi^*\\,(S\\partial_\\tau-V\\partial_\\tau^2-\\Delta-\\mu)\\,\\varphi\\,+\\,\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2{\\Big \\}},\n\\label{microscopicaction}\n\\end{equation}\nwith \n\\begin{equation}\nx=(\\tau,\\vec{x}), \\,\\,\\int_x=\\int_0^{\\frac{1}{T}}d\\tau\\int d^3x.\n\\end{equation}\nThe integration goes over the whole space as well as over the imaginary time $\\tau$, which at finite temperature is integrated on a circle of circumference $\\beta=1\/T$ according to the Matsubara formalism. We use natural units $\\hbar=k_B=1$. We also scale time and energy units with appropriate powers of $2M$, with $M$ the particle mass. In other words, our time units are set such that effectively $2M=1$. In these units time has the dimension of length squared. For standard non-relativistic bosons one has $V=0$ and $S=1$, but we also consider quasiparticles with a more general dispersion relation described by nonzero $V$.\n\nAfter Fourier transformation, the kinetic term reads\n\\begin{equation}\n\\int_q \\varphi^*(q)(i S q_0+V q_0^2+\\vec{q}^2)\\varphi(q),\n\\label{eqMicroscopicFourier}\n\\end{equation}\nwith\n\\begin{eqnarray}\nq=(q_0,\\vec{q}),\\quad \\int_q=\\int_{q_0}\\int_{\\vec{q}},\\quad \\int_{\\vec{q}}=\\frac{1}{(2\\pi)^3}\\int d^3q.\n\\end{eqnarray}\nAt nonzero temperature, the frequency $q_0=2\\pi T n$ is discrete, with\n\\begin{equation}\n\\int_{q_0}=T \\sum_{n=-\\infty}^\\infty,\n\\end{equation}\nwhile at zero temperature this becomes\n\\begin{equation}\n\\int_{q_0}=\\frac{1}{2\\pi}\\int_{-\\infty}^\\infty dq_0.\n\\end{equation}\nThe dispersion relation encoded in Eq.\\ \\eqref{eqMicroscopicFourier} obtains by analytic continuation\n\\begin{equation}\nS\\omega+V\\omega^2=\\vec{q}^2\/2M.\n\\end{equation}\n\nIn this thesis, we consider homogeneous situations, i.e. an infinitely large volume without a trapping potential. Many of our results can be translated to the inhomogeneous case in the framework of the local density approximation. One assumes that the length scale relevant for the quantum and statistical fluctuations is much smaller than the characteristic length scale of the trap. In this case, our results can be transferred by taking the chemical potential position dependent in the form $\\mu\\left(\\vec{x})=2M(\\mu-V_t(\\vec{x}\\right))$, where $V_t(\\vec{x})$ is the trapping potential.\n\nThe microscopic action \\eqref{microscopicaction} is invariant under the global $U(1)$ symmetry which is associated to the conserved particle number,\n\\begin{equation}\n\\varphi\\rightarrow e^{i\\alpha}\\varphi.\n\\end{equation}\nOn the classical level, this symmetry is broken spontaneously when the chemical potential $\\mu$ is positive. In this case, the minimum of $-\\mu\\varphi^*\\varphi+\\frac{1}{2}\\lambda(\\varphi^*\\varphi)^2$ is situated at $\\varphi^*\\varphi=\\frac{\\mu}{\\lambda}$. The ground state of the system is then characterized by a macroscopic field $\\varphi_0$, with $\\varphi_0^*\\varphi_0=\\rho_0=\\frac{\\mu}{\\lambda}$. It singles out a direction in the complex plane and thus breaks the $U(1)$ symmetry. Nevertheless, the action itself and all modifications due to quantum and statistical fluctuations respect the symmetry. For $V=0$ and $S=1$, the situation is similar for Galilean invariance. At zero temperature, we can perform an analytic continuation to real time and the microscopic action \\eqref{microscopicaction} is then invariant under transformations that correspond to a change of the reference frame in the sense of a Galilean boost. It is easy to see that in the phase with spontaneous $U(1)$ symmetry breaking also the Galilean symmetry is broken spontaneously: A condensate wave function, that is homogeneous in space and time, would be represented in momentum space by\n\\begin{equation} \n\\varphi(\\omega,\\vec{p})=\\varphi_0 \\,(2\\pi)^4\\, \\delta^{(3)}(\\vec{p})\\delta(\\omega).\n\\end{equation}\nUnder a Galilean boost transformation with a boost velocity $2\\vec{q}$, this would transform according to\n\\begin{eqnarray}\n\\nonumber\n\\varphi(\\omega,\\vec{p})\\rightarrow&&\\varphi(\\omega-\\vec{q}^2,\\vec{p}-\\vec{q})\\\\\n&&=\\varphi_0\\,(2\\pi)^4\\,\\delta^{(3)}(\\vec{p}-\\vec{q})\\delta(\\omega-\\vec{q}^2).\n\\end{eqnarray} \nThis shows that the ground state is not invariant under such a change of reference frame. This situation is in contrast to the case of a relativistic Bose-Einstein condensate, like the Higgs boson field after electroweak symmetry breaking. A relativistic scalar transforms under Lorentz boost transformations according to\n\\begin{equation}\n\\varphi(p^\\mu)\\rightarrow\\varphi((\\Lambda^{-1})^\\mu_{\\,\\,\\nu}\\,p^\\nu),\n\\end{equation}\nsuch that a condensate wave function\n\\begin{eqnarray}\n\\nonumber\n\\varphi_0\\,(2\\pi)^4 \\,\\delta^{(4)}(p^\\mu)\\rightarrow&&\\varphi_0\\,(2\\pi)^4\\,\\delta^{(4)}((\\Lambda^{-1})^\\mu_{\\,\\,\\nu}\\,p^\\nu)\\\\\n&&=\\varphi_0\\,(2\\pi)^4\\, \\delta^{(4)}(p^\\mu)\n\\end{eqnarray}\ntransforms into itself. We will investigate the implications of Galilean symmetry for the form of the effective action in chapter \\ref{ch:Symmetries}. An analysis of general coordinate invariance in nonrelativistic field theory can be found in \\cite{Son2006}.\n\n\n\\section{Bose gas in two dimensions}\n\\label{sec:Bosegasintwodimensions}\n\nBose-Einstein condensation and superfluidity for cold nonrelativistic atoms can be experimentally investigated in systems of various dimensions \\cite{morsch:179, bloch:885, PethickSmith2002}. Two dimensional systems can be achieved by building asymmetric traps, resulting in different characteristic sizes for one ``transverse extension'' $l_T$ and two ``longitudinal extensions'' $l$ of the atom cloud \\cite{PhysRevLett.87.130402, PhysRevLett.92.173003, 0953-4075-38-3-007, 0295-5075-57-1-001, Koehl2005, C.Orzel03232001, spielman:080404, PhysRevLett.93.180403, Hazibabic2006}. For $l \\gg l_T$ the system behaves effectively two-dimensional for all modes with momenta $\\vec{q}^2\\lesssim l_T^{-2}$. From the two-dimensional point of view, $l_T$ sets the length scale for microphysics -- it may be as small as a characteristic molecular scale. On the other hand, the effective size of the probe $l$ sets the scale for macrophysics, in particular for the thermodynamic observables.\n\nTwo-dimensional superfluidity shows particular features. In the vacuum, the interaction strength $\\lambda$ is dimensionless such that the scale dependence of $\\lambda$ is logarithmic \\cite{lapidus:459}. The Bogoliubov theory with a fixed small $\\lambda$ predicts at zero temperature a divergence of the occupation numbers for small $q=|\\vec{q}|$, $n(\\vec{q})\\sim n_C\\, \\delta^{(2)}(\\vec{q})$ \\cite{Bogoliubov}. In the infinite volume limit, a nonvanishing condensate $n_c=\\bar{\\rho}_0$ is allowed only for $T=0$, while it must vanish for $T>0$ due to the Mermin-Wagner theorem \\cite{PhysRevLett.17.1133, PhysRev.158.383}. On the other hand, one expects a critical temperature $T_c$ where the superfluid density $\\rho_0$ jumps by a finite amount according to the behavior for a Kosterlitz-Thouless phase transition \\cite{Berezinskii1971, Berezinskii1972, 0022-3719-6-7-010, PhysRevLett.39.1201}. We will see that $T_c\/n$ (with $n$ the atom-density) vanishes in the infinite volume limit $l\\to \\infty$. Experimentally, however, a Bose-Einstein condensate can be observed for temperatures below a nonvanishing critical temperature $T_c$ -- at first sight in contradiction to the theoretical predictions for the infinite volume limit.\n\nA resolution of these puzzles is related to the simple observation that for all practical purposes the macroscopic size $l$ remains finite. Typically, there will be a dependence of the characteristic dimensionless quantities as $\\bar{\\rho}_0\/n$, $T_c\/n$ or $\\lambda$ on the scale $l$. This dependence is only logarithmic. While $\\lambda(n=T=0, l\\to \\infty)=0$, $(\\bar{\\rho}_0\/n)(T\\neq0, l\\to \\infty)=0$, $(T_c\/n)(l\\to0)=0$, in accordance with general theorems, even a large finite $l$ still leads to nonzero values of these quantities, as observed in experiment. \n\nThe description within a two-dimensional renormalization group context starts with a given microphysical or classical action at the ultraviolet momentum scale $\\Lambda_\\text{UV}\\sim l_T^{-1}$. When the scale parameter $k$ reaches the scale $k_\\text{ph}\\sim l^{-1}$, all fluctuations are included since no larger wavelength are present in a finite size system. The experimentally relevant quantities and the dependence on $l$ can be obtained from $\\Gamma_{k_\\text{ph}}$. For a system with finite size $l$ we are interested in $\\Gamma_{k_\\text{ph}}$, $k_\\text{ph}=l^{-1}$. If statistical quantities for finite size systems depend only weakly on $l$, they can be evaluated from $\\Gamma_{k_\\text{ph}}$ in the same way as their thermodynamic infinite volume limit follows from $\\Gamma$. Details of the geometry etc. essentially concern the appropriate factor between $k_\\text{ph}$ and $l^{-1}$. \n\nThe microscopic model we use for the two-dimensional Bose gas is basically the one for the three-dimensional case in Eq.\\ \\eqref{microscopicaction}. The difference is that now $\\vec x$ and the space-integral are two-dimensional\n\\begin{equation}\nx=(\\tau,\\vec{x}), \\,\\,\\int_x=\\int_0^{\\frac{1}{T}}d\\tau\\int d^2x,\n\\end{equation}\nand similarly in momentum space. The dimensionless interaction parameter $\\lambda$ in \\eqref{microscopicaction} describes now a reduced two-dimensional interaction strength and is directly related to the scattering length in units of the transverse extension $a\/l_T$. The few-body physics and the logarithmic scale-dependence of $\\lambda$ is discussed in section \\ref{sec:Repulsiveinteractingbosons}. \n\n\n\\section{BCS-BEC Crossover}\n\\label{sec:BCS-BECCrossover}\n\nBesides the bosons we also investigate systems with ultracold fermions. A qualitative new feature for fermions in comparison to bosons is the antisymmetry of the wavefunction and the tightly connected ``Pauli blocking''. Due to the antisymmetry of the wavefunction it is not possible to have two identical fermions in the same state. This feature has many interesting consequences. For example, a $s$-wave interaction between two identical fermions is not possible. This in turn implies that a gas of fermions in the same spin- (and hyperfine-) state has many properties of a free Fermi gas provided the $p$-wave and higher interactions are suppressed. The situation changes for a Fermi gas with two spin or hyperfine states. $S$-wave interactions and pairing are now possible. In the simplest case the densities of the two components are equal. Depending on the microscopic interaction the system has different properties. For a repulsive interaction one expects Landau Fermi liquid behavior (for not too small temperature) where many qualitative properties are as for the free Fermi gas \\cite{Landau1957}. For weak attractive interaction the theory of Baarden, Cooper and Schrieffer (BCS) \\cite{Bardeen:1957kj, Bardeen:1957mv} is valid. Cooper-pairs are expected to form at small temperatures and the system is then superfluid. On the other hand, for strong attractive interaction one expects the formation of bound states of two fermions. These bound states are then bosons and undergo Bose-Einstein condensation (BEC) at small temperatures. Again, the system shows superfluidity. As first pointed out by Eagles \\cite{PhysRev.186.456} and Leggett \\cite{Leggett1980} there is a smooth and continuous crossover (BCS-BEC crossover) between the two limits described above. \n\nExperimental realizations of this crossover can be realized using Feshbach resonances. The detailed mechanism how these resonances work can be found in the literature, e.~g. \\cite{PitaevsikiiStringari2003, PethickSmith2002}. It is important that the scattering length $a$ which serves as a measure for the $s$-wave interaction can be tuned to arbitrary values. As an example we consider the case of $^6$Li where the resonance was investigated in Refs. \\cite{PhysRevA.66.041401, PhysRevLett.94.103201} and is shown in Fig.\\ \\ref{fig:Feshbach}. For magnetic fields in the range around $B=1200\\, \\text{G}$ the scattering length is relatively small and negative. In this regime the many-body ground state is of the BCS-type. Fermions with different spin and with momenta on opposite points on the Fermi surface form pairs. These Cooper pairs are (hyperfine-) spin singlets and have small or vanishing momentum. They are condensed in a Bose-Einstein condensate (BEC). The system is superfluid and the U(1) symmetry connected with particle number conservation is spontaneously broken. The macroscopic wavefunction of the BEC can be seen as an order parameter which is quadratic in the fermion field $\\varphi_0\\sim \\langle\\psi_1\\psi_2\\rangle$. Increasing the temperature, the system will at some point undergo a second order phase transition to a normal state where the order parameter vanishes, $\\varphi_0=0$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.35\\textwidth]{Feshbach.eps}\n\\caption{Scattering length $a$ in units of the Bohr radius $a_0$ as a function of the magnetic field $B$ for the lowest hyperfine states of $^6$Li \\cite{PhysRevA.66.041401, PhysRevLett.94.103201}.}\n\\label{fig:Feshbach}\n\\end{figure}\n\n\nIn the magnetic field range around $B=600 \\,\\text{G}$ in Fig. \\ref{fig:Feshbach} the scattering length $a$ is small and positive. There is now a bound state of two fermions in the spectrum and the ground state of the many-body system is BEC-like. Pairs of fermions with different spin constitute bound states (dimers) which are pairs in position space. The interaction between these dimers is repulsive and proportional to the scattering length between fermions. When this repulsive interaction is weak the dimers are completely condensed in a BEC at zero temperature (no quantum depletion of the condensate). Again the order parameter is the macroscopic wavefunction of this condensate which is quadratic in the fermion fields $\\varphi_0\\sim\\langle\\psi_1\\psi_2\\rangle$. The phase transition between the superfluid state at small temperatures and the normal state is of second order, again. \n\nNow we come to the magnetic field in the intermediate crossover regime $700\\, \\text{G} \\lesssim B \\lesssim 1000 \\,\\text{G}$. The scattering length is now large and positive or large and negative with a divergence at $B\\approx 834\\, \\text{G}$ \\cite{PhysRevLett.94.103201}. Since the two-body scattering properties are solely governed by the requirement of unitarity of the scattering matrix for $a\\to\\pm \\infty$, the point $B=834\\, \\text{G}$ is also called the ``unitarity point''. Due to the divergent scattering length one speaks of strongly interacting fermions. Perturbative methods for small coupling constants fail in the crossover regime. Non-perturbative methods show that the ground state is superfluid and governed by a order parameter $\\varphi_0\\sim \\langle \\psi_1\\psi_2\\rangle$ as before. \n\nThe crossover from the BCS- to the BEC-like ground state is conveniently parameterized by the inverse scattering length in units of the Fermi momentum $c^{-1}=(a k_F)^{-1}$ where the Fermi momentum is determined by the density $n=\\frac{1}{3\\pi^2}k_F^3$ (in units with $\\hbar=k_B=2M=1$). The dimensionless parameter $c^{-1}$ varies from large negative values on the BCS side to large positive values on the BEC side of the crossover. It crosses zero at the unitarity point. We will also use the Fermi energy which equals the Fermi temperature in our units $E_F=T_F=k_F^2$.\n\nThe quantitatively precise understanding of BCS-BEC crossover physics is a challenge for theory. Experimental breakthroughs as the realization of molecule condensates and the subsequent crossover to a BCS-like state of weakly attractively interacting fermions have been achieved \\cite{PhysRevLett.92.040403, PhysRevLett.92.120403, PhysRevLett.92.150402, PhysRevLett.93.050401, C.Chin08202004, PhysRevLett.95.020404}. Future experimental precision measurements could provide a testing ground for non-perturbative methods. An attempt in this direction are the recently published measurements of the critical temperature \\cite{Luo2007} and collective dynamics \\cite{altmeyer:040401, wright:150403}.\n\nA wide range of qualitative features of the BCS-BEC crossover is already well described by extended mean-field theories which account for the contribution of both fermionic and bosonic degrees of freedom \\cite{Nozieres1985, PhysRevLett.71.3202}. In the limit of narrow Feshbach resonances mean-field theory becomes exact \\cite{Diehl:2005an, Gurarie2007}. Around this limit perturbative methods for small Yukawa couplings \\cite{Diehl:2005an} can be applied. Using $\\epsilon$-expansion \\cite{nussinov:053622, nishida:050403, nishida:063617, nishida:063618, arnold:043605, chen:043620} or $1\/N$-expansion \\cite{Sachdev06} techniques one can go beyond the case of small Yukawa couplings.\n\nQuantitative understanding of the crossover at and near the resonance has been developed through numerical calculations using various quantum Monte-Carlo (QMC) methods \\cite{PhysRevLett.91.050401, PhysRevLett.93.200404, bulgac:090404, bulgac:023625, burovski:160402, akkineni:165116}. Computations of the complete phase diagram have been performed from functional field-theoretical techniques, in particular from $t$-matrix approaches \\cite{Haussmann1993, PhysRevLett.85.2801, PhysRevB.61.15370, PhysRevLett.92.220404, PhysRevB.70.094508}, Dyson-Schwinger equations \\cite{Diehl:2005an,Diehl:2005ae}, 2-partice irreducible (2-PI) methods \\cite{haussmann:023610}, and renormalization-group flow equations \\cite{Birse2005,Diehl:2007th,Diehl:2007ri,Gubbels:2008zz}. These unified pictures of the whole phase diagram \\cite{Sachdev06, Haussmann1993, PhysRevLett.85.2801, PhysRevB.61.15370, PhysRevLett.92.220404, PhysRevB.70.094508, Diehl:2005an, Diehl:2005ae, haussmann:023610, Diehl:2007th, Diehl:2007ri, Gubbels:2008zz}, however, do not yet reach a similar quantitative precision as the QMC calculations.\n\nIn this thesis we discuss mainly the limit of broad Fesh\\-bach resonances for which all thermodynamic quantities can be expressed in terms of two dimensionless parameters, namely the temperature in units of the Fermi temperature $T\/T_F$ and the concentration $c=ak_F$. In the broad resonance regime, macroscopic observables are to a large extent independent of the concrete microscopic physical realization, a property referred to as universality \\cite{Diehl:2005an, Sachdev06, Diehl:2007th}. This universality includes the unitarity regime where the scattering length diverges, $a^{-1}=0$ \\cite{PhysRevLett.92.090402}, however it is not restricted to that region. Macroscopic quantities are independent of the microscopic details and can be expressed in terms of only a few parameters. In our case this is the two-body scattering length $a$ or, at finite density, the concentration $c=ak_F$. At nonzero temperature, an additional parameter is given by $T\/T_F$. \n\nFor small and negative scattering length $c^{-1}<0, |c|\\ll 1$ (BCS side), the system can be treated with perturbative methods. However, there is a significant decrease in the critical temperature as compared to the original BCS result. This was first recognized by Gorkov and Melik-Barkhudarov \\cite{Gorkov}. The reason for this correction is a screening effect of particle-hole fluctuations in the medium \\cite{Heiselberg}. There has been no systematic analysis of this effect in approaches encompassing the full BCS-BEC crossover so far.\n\nIn section \\ref{sec:Particle-holefluctuationsandtheBCS-BECCrossover}, we present an approach using the flow equation described in chapter \\ref{ch:TheWetterichequation}. We include the effect of particle-hole fluctuations and recover the Gorkov correction on the BCS side. We calculate the critical temperature for the second-order phase transition between the normal and the superfluid phase throughout the whole crossover.\n\nWe also calculate the critical temperature at the point $a^{-1}=0$ for different resonance widths $\\Delta B$. As a function of the microscopic Yukawa coupling $h_\\Lambda$, we find a smooth crossover between the exact narrow resonance limit and the broad resonance result. The resonance width is connected to the Yukawa coupling via $\\Delta B=h_\\Lambda^2\/(8\\pi\\mu_M a_b)$ where $\\mu_M$ is the magnetic moment of the bosonic bound state and $a_b$ is the background scattering length.\n\n\n\\subsubsection{Lagrangian}\n\nWe start with a microscopic action including a two-component Grassmann field $\\psi=(\\psi_1,\\psi_2)$, describing fermions in two hyperfine states. Additionally, we introduce a complex scalar field $\\varphi$ as the bosonic degrees of freedom. In different regimes of the crossover, it can be seen as a field describing molecules, Cooper pairs or simply an auxiliary field. Using the resulting two-channel model we can describe both narrow and broad Feshbach resonances in a unified setting. Explicitly, the microscopic action at the ultraviolet scale $\\Lambda$ reads\n\n\\begin{eqnarray}\n\\nonumber\nS[\\psi, \\varphi] & = & \\int_0^{1\/T} d\\tau \\int d^3x{\\Big \\{}\\psi^\\dagger(\\partial_\\tau-\\Delta-\\mu)\\psi\\\\\n\\nonumber\n& & +\\varphi^*(\\partial_\\tau-\\frac{1}{2}\\Delta-2\\mu+ \\nu_\\Lambda)\\varphi\\\\\n& & - h_\\Lambda(\\varphi^*\\psi_1\\psi_2+h.c.){\\Big \\}}\\,,\n\\label{eqMicroscopicAction}\n\\end{eqnarray}\nwhere we choose nonrelativistic natural units with $\\hbar=k_B=2M=1$, with $M$ the mass of the atoms.\nThe system is assumed to be in thermal equilibrium, which we describe using the Matsubara formalism. In addition to the position variable $\\vec{x}$, the fields depend on the imaginary time variable $\\tau$ which parameterizes a torus with circumference $1\/T$. The variable $\\mu$ is the chemical potential. The Yukawa coupling $h$ couples the fermionic and bosonic fields. It is directly related to the width of the Feshbach resonance. The parameter $\\nu$ depends on the magnetic field and determines the detuning from the Feshbach resonance. Both $h$ and $\\nu$ get renormalized by fluctuations, and the microscopic values $h_\\Lambda$, and $\\nu_\\Lambda$ have to be determined by the properties of two body scattering in vacuum. For details, we refer to \\cite{Diehl:2007th} and section \\ref{sec:Twofermionspecies:Dimerformation}. \n\nMore formally, the bosonic field $\\varphi$ appears quadratically in the microscopic action in Eq.\\ \\eqref{eqMicroscopicAction}. The functional integral over $\\varphi$ can be carried out. This shows that our model is equivalent to a purely fermionic theory with an interaction term\n\\begin{eqnarray}\n\\nonumber\nS_{\\text{int}} & = & \\int_{p_1,p_2,p_1^\\prime,p_2^\\prime} \\left\\{-\\frac{h^2}{P_\\varphi(p_1+p_2)}\\right\\}\\psi_1^{\\ast}(p_1^\\prime){\\psi_1}(p_1)\\\\\n&& \\times \\psi_2^{\\ast}(p_2^\\prime){\\psi_2}(p_2)\\,\\delta(p_1+p_2-p_1^\\prime-p_2^\\prime),\n\\label{lambdapsieff}\n\\end{eqnarray}\nwhere $p = (p_0,\\vec p)$ and the classical inverse boson propagator is given by\n\\begin{equation}\n\tP_{\\varphi}(q)= i q_0 + \\frac{\\vec{q}^2}{2}+ \\nu_\\Lambda-2\\mu\\,.\n\\label{eq:Bosonpropagator}\n\\end{equation}\n\nOn the microscopic level the interaction between the fermions is described by the tree level expression\n\\begin{equation}\n\\lambda_{\\psi,\\text{eff}}=-\\frac{h^2}{-\\omega+\\frac{1}{2}\\vec q^2+\\nu_\\Lambda-2\\mu}.\n\\end{equation}\nHere, $\\omega$ is the real-time frequency of the exchanged boson $\\varphi$. It is connected to the Matsubara frequency $q_0$ via analytic continuation $\\omega=-iq_0$. Similarly, $\\vec q=\\vec p_1+\\vec p_2$ is the center of mass momentum of the scattering fermions $\\psi_1$ and $\\psi_2$ with momenta $\\vec p_1$ and $\\vec p_2$, respectively.\n\nThe limit of broad Feshbach resonances, which is realized in current experiments, e.g. with $\\mathrm{^6Li}$ and $\\mathrm{^{40}K}$ corresponds to the limit $h\\to\\infty$, for which the microscopic interaction becomes pointlike, with strength $-h^2\/\\nu_\\Lambda$. \n\n\n\\section{BCS-Trion-BEC Transition}\n\\label{sec:BCS-Trion-BECTransitionshort}\n\nIn the last section we discussed the interesting BCS-BEC crossover that is realized in a system consisting of two fermion species. We restricted ourselves to the case where the density for both components is equal. Interesting physics is also found if this constraint is released. The phase diagram of the imbalanced Fermi gas shows also first order phase transitions and phase separation, see \\cite{KetterleZwierlein2007, Chevy2007} and references therein. \n\nAnother interesting generalization is to take a third fermion species into account. A very rich phase diagram can be expected for the general case where the total density is arbitrarily distributed to the different components. Even the simpler case where the densities for all three components are equal is far less understood as the analogous two-component case. For simplicity we restrict much of the discussion to the case where all properties of the three components apart from the hyperfine-spin are the same. In particular, we assume that they have equal mass, chemical potential and scattering properties. We label the different hyperfine states by 1, 2 and 3. The $s$-wave scattering length $a_{12}$ for scattering between fermions of components 1 and 2 is the same as for scattering between fermions of species 2 and 3 or 3 and 1, $a_{12}=a_{23}=a_{31}=a$. \n\nClose to a common resonance where $a\\to\\pm\\infty$ one expects the three-body problem to be dominated by the Efimov effect \\cite{Efimov1970, Efimov1973}. This implies the formation of a three-body bound state (the ``trion''). Directly at the resonance an infinite tower of three-body bound states, the Efimov-trimers, exists. We refer to the Efimov trimer with the lowest lying energy as trion. The few-body physics is discussed in more detail in section \\ref{sec:Threefermionspecies:ThomasandEfimoveffect}. \n\nThe many-body phase diagram is far less understood. Not too close to the resonance one expects a superfluid ground state which is similar to the BCS ground state for $a<0$ or a BEC-like ground state for $a>0$. However, there are also some important differences. While in the two-component case the order parameter is a singlet under the corresponding SU(2) spin symmetry, the order parameter for the three component case with SU(3) spin symmetry is a (conjugate) triplet. In the superfluid phase the spin symmetry is therefore broken spontaneously. Due to some similarities with QCD this was called color superfluidity \\cite{PhysRevB.70.094521, paananen:053606, paananen:023622, cherng:130406, zhai:031603, Bedaque2007}.\n\nBetween the extended BCS and BEC phase one can expect the ground state to be dominated by trions. Since trions are SU(3) singlets, the spin symmetry is unbroken in this regime such that there have to be true quantum phase transitions at the border to the BCS and BEC regimes. Such a trion phase has first been proposed for fermions in an optical lattice by Rapp, Zarand, Honerkamp and Hofstetter \\cite{rapp:160405, rapp:144520}, see also \\cite{Wilczek2007}. We will further discuss the many-body physics in section \\ref{sec:BCS-Trion-BECTransitionlong}. To the knowledge of the author, there have been no experiments addressing the many-body issues so far. Only recently, experiments with $^6$Li probing the few-body physics found interesting phenomena \\cite{ottenstein:203202, Huckans2008}. For the case of $^6$Li the assumption of equal scattering properties for the three different species are not fulfilled. We will present a more general model where SU(3) symmetry is broken explicitly and where the parameters can be chosen to describe $^6$Li in section \\ref{sec:Threefermionspecies:ThomasandEfimoveffect}. We also discuss the experiments and show how their results can be explained in our framework. The remainder of this section is devoted to the discussion of the microscopic model in the SU(3) symmetric case. \n\n\\subsubsection{Lagrangian}\n\nAs our microscopic model we use an action similar to the one for the BCS-BEC crossover in Eq.\\ \\eqref{eqMicroscopicAction}\n\\begin{eqnarray}\n\\nonumber\nS&=&\\int_x {\\bigg \\{} \\psi^\\dagger\\partial_\\tau-\\Delta-\\mu)\\psi+\\varphi^\\dagger(\\partial_\\tau-\\frac{1}{2}\\Delta-2\\mu+\\nu_\\varphi)\\varphi\\\\\n\\nonumber\n&&+\\chi^*(\\partial_\\tau-\\frac{1}{3}\\Delta-3\\mu+\\nu_\\chi)\\chi\\\\\n\\nonumber\n&&+\\frac{1}{2} h\\,\\epsilon_{ijk}\\,\\left(\\varphi_i^*\\psi_j\\psi_k-\\varphi_i\\psi_j^*\\psi_k^*\\right)\\\\\n\\nonumber\n&&+g\\left(\\varphi_i^*\\psi_i^*\\chi-\\varphi_i\\psi_i\\chi^*\\right){\\bigg \\}}.\n\\label{eq8:microscopicactiontrionmodel}\n\\end{eqnarray}\nThe (Grassmann valued) fermion field has now three components $\\psi=(\\psi_1,\\psi_2,\\psi_3)$ and similar the boson field $\\varphi=(\\varphi_1,\\varphi_2,\\varphi_3)\\hat{=} (\\psi_1\\psi_2,\\psi_2\\psi_3,\\psi_3\\psi_1)$. In addition we also include a single component fermion field $\\chi$. This trion field represents the totally antisymmetric combination $\\psi_1\\psi_2\\psi_3$. One choose the parameters such that $g=0$ and $\\nu_\\chi\\to\\infty$ at the microscopic scale. The trion field $\\chi$ is then only an non-dynamical auxiliary field. \n\nWe assumed in Eq.\\ \\eqref{eq8:microscopicactiontrionmodel} that the fermions $\\psi_1$, $\\psi_2$, and $\\psi_3$ have equal mass $M$ and chemical potential $\\mu$. We also assume that the interactions are independent of the spin (or hyperspin) so that our microscopic model is invariant under a global SU(3) symmetry transforming the fermion species into each other. While the fermion field $\\psi=(\\psi_1,\\psi_2,\\psi_3)$ transforms as a triplet ${\\bf 3}$, the boson field $\\varphi=(\\varphi_1,\\varphi_2,\\varphi_3)$ transforms as a conjugate triplet $\\bar {\\bf 3}$. The trion field $\\chi$ is a singlet under SU(3). In concrete experiments, for example with $^6\\text{Li}$ \\cite{ottenstein:203202}, the SU(3) symmetry may be broken explicitly since the Feshbach resonances of the different channels occur for different magnetic field values and have different widths. In addition to the SU(3) spin symmetry our model is also invariant under a global U(1) symmetry $\\psi\\to e^{i\\alpha}\\psi$, $\\varphi\\to e^{2i\\alpha}\\varphi$, and $\\chi\\to e^{3i\\alpha}\\chi$. The conserved charge related to this symmetry is the total particle number. Since we do not expect any anomalies the quantum effective action $\\Gamma=\\Gamma_{k=0}$ will also be invariant under $\\text{SU(3)}\\times \\text{U(1)}$.\n\nApart from the terms quadratic in the fields that determine the propagators, Eq.\\ \\eqref{eq8:microscopicactiontrionmodel} contains the Yukawa-type interactions $\\sim h$ and $\\sim g$. The energy gap parameters $\\nu_\\varphi$ for the bosons and $\\nu_\\chi$ for the trions are sometimes written as $m_\\varphi^2=\\nu_\\varphi-2\\mu$, $m_\\chi^2=\\nu_\\chi-3\\mu$, absorbing an explicit dependence on the chemical potential $\\mu$.\n\nIn Eq.\\ \\eqref{eq8:microscopicactiontrionmodel}, the fermion field $\\chi$ can be ``integrated out'' by inserting the $(\\psi,\\varphi)$-dependent solution of its field equation into $\\Gamma_k$. For $m_\\chi^2\\rightarrow \\infty$ this results in a contribution to a local three-body interaction, $\\lambda_{\\varphi\\psi}=-g^2\/m_\\chi^2$. Furthermore one may integrate out the boson field $\\varphi$, such that (for large $m_\\varphi^2$) one replaces the parts containing $\\varphi$ and $\\chi$ in $\\Gamma_k$ by an effective pointlike fermionic interaction\n\\begin{equation}\n\\Gamma_{k,\\text{int}}=\\int_x\\frac{1}{2}\\lambda_\\psi(\\psi^\\dagger\\psi)^2+\\frac{1}{3!}\\lambda_3 \\left(\\psi^\\dagger \\psi\\right)^3,\n\\end{equation}\nwith\n\\begin{equation}\n\\lambda_\\psi= -\\frac{h^2}{m_\\varphi^2}, \\quad \\quad \\lambda_3=-\\frac{h^2 g^2}{m_\\varphi^4 m_\\chi^2}.\n\\label{eq:subst}\n\\end{equation}\nWe note that the contribution of trion exchange to $\\lambda_{\\varphi\\psi}$ or $\\lambda_3$ depends only on the combination $g^2\/m_\\chi^2$. The sign of $g$ can be changed by $\\chi\\rightarrow - \\chi$, and the sign of $g^2$ can be reversed by a sign flip of the term quadratic in $\\chi$. Keeping the possible reinterpretation by this mapping in mind, we will formally also admit negative $g^2$ (imaginary $g$).","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe are in the age of renaissance of hadron\nspectroscopy, initiated by the announcement of the pentaquark baryon\n\\cite{nakano}, which is followed by the discovery of \nmany other possible exotic hadrons with a mass larger than 2 GeV \ncontaining heavy quarks\\cite{Quigg:2005tv}.\nThese experimental developments prompted the intensive theoretical\nstudies of QCD dynamics with new as well as old ideas on the structure\nand dynamics of the exotic hadrons, such as\nchiral dynamics\\cite{Nowak:1992um},\nmulti-quark states with diquark correlations or\nmolecular states and hybrids\\cite{Quigg:2005tv}. \n\nSuch a controversy on the structure of hadrons is\nalso the case for the scalar mesons below 1 GeV:\nthe existence of the $I=0$ and $J^{PC}=0^{++}$ meson, i.e.,\nthe $\\sigma(400-600)$, has been reconfirmed \\cite{PDG,leutwyler} \nafter around twenty years not only in $\\pi$$\\pi\\ $ scattering \nbut also in various decay\nprocesses from heavy-quark systems, {\\it e.g.} , \nD $\\to \\pi \\pi \\pi$ and $\\Upsilon(3S) \\to \\Upsilon \\pi \\pi$ \n\\cite{KEK,E791decay,Ishida,Bugg}.\nMoreover, the resonance of a scalar meson with $I=1\/2$\nis also reported to exist in the K-$\\pi$ system \nwith a mass $m_{\\kappa}$ of about \n800 MeV \\cite{Bugg,E791,BES}.\nThis meson is called the $\\kappa$ meson and may constitute\nthe nonet scalar state together with the $\\sigma$ meson.\nSee Fig.\\ref{fig:nonet}.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{figure1.eps}\n\\caption{\nScalar meson nonet. \nThe $\\sigma$ and $f_0(980)$ mesons may be ideal mixing\nstates of singlet state \n$\\frac{1}{\\sqrt{3}} ( u \\bar{u} + d \\bar{d} + s \\bar{s} )$\nand octet state \n$\\frac{1}{\\sqrt{6}} ( u \\bar{u} + d \\bar{d} - 2 s \\bar{s} )$. \nThere is , however, experimental evidence that the $\\sigma$ meson\nconsist of only $u \\bar{u}$\nand $d \\bar{d}$ components. Hence, we take the $\\sigma$ wave function given\nin the figure.\n}\n\\label{fig:nonet}\n\\end{center}\n\\end{figure}\n\nThe problem is the nature of these low-lying scalar mesons\n\\cite{close-tornqvist}:\nthey cannot be ordinary $q\\bar{q}$ mesons as described in the \nnon-relativistic constituent quark model\nsince in such a quark model, \nthe $J^{PC}$=$0^{++}$ meson is realized in the $^{3}P_{0}$ state, \nwhich implies that the mass of the $\\sigma$ meson must be as high as 1.2\n$\\sim$ 1.6 GeV.\nThus, the low-lying scalar mesons below 1 GeV \nhave been a source of various ideas of exotic structures, as mentioned above:\nthey may be four-quark states such as $qq\\bar{q}\\bar{q}$ \\cite{Alfold},\nor $\\pi\\pi$ or K$\\pi$ molecules\nas the recent high-lying exotic hadrons can be.\nThese mesons may be {\\em collective} $q\\bar{q}$ states described\nas a superposition of many {\\em atomic} $q\\bar{q}$ \nstates \\cite{NJL,HK85}.\nA mixing with glueball states is also possible\n\\cite{Lee,McNeile,Narison,giacosa}. \n\nIn the previous work\\cite{ScalarSIGMA1,ScalarSIGMA2,ScalarSIGMA3,ScalarSIGMA4},\nwe have presented a lattice calculation for the $\\sigma$ meson,\nby full lattice QCD simulation\non the $8^{3}\\times16$ lattice using the plaquette action and\nWilson fermions: We have shown that the disconnected diagram \nplays an essential role in order to make the $\\sigma$ meson mass \nlight. \nThe importance of the disconnected diagram suggests that the\nwave function of the $\\sigma$ meson may have a significant \nfour-quark, a collective $q$-$\\bar{q}$\nor an even glueball component, although the smallness of the \nlattice requires caution in giving a definite conclusion.\nIn contrast to the $\\sigma$ meson, the $\\kappa$ is a flavor non-singlet state\nwith which a glueball state cannot mix.\nIn previous reports \\cite{ScalarSIGMA4,ScalarKAPPA},\nwe reported also a preliminary analysis on the $\\kappa$ meson \nusing the dynamical fermion for \nthe $u(d)$ quark but using the valence approximation for the $s$ quark,\nwhich shows that \nthe $I=1\/2$ scalar meson has a mass as large as about 1.8 GeV and \ncannot be identified with the $\\kappa$ meson observed in experiments.\n\nThe lattice volume in the previous investigations was admittedly \ntoo small to yield a definite conclusion at all, and \n the lattice cutoff was not appropriately chosen to accommodate\nlarge masses : $m_{\\kappa}a>1$, where $a$ is the lattice spacing.\nHence, we present a simulation with weaker couplings\non a larger lattice than any other previous simulations\nalthough in the quenched level.\nWe perform quenched level simulations on the \n$\\kappa$ meson so as {\\it to clarify the structure of the mysterious \nscalar meson rather than to reproduce the experimental\nvalue of the mass}; a precise quenched-level simulation should\ngive a rather clear perspective on whether\nthe system can fit with the simple constituent-quark model\npicture or not. \n\n\\section{Simulation}\n\nWe perform a quenched QCD calculation using the Wilson fermions,\nwith the plaquette gauge action, on a relatively large lattice\n($20^3 \\times 24$). \n\nThe values of the hopping parameter for the $u\/d$ quark are \n$h_{u\/d} = 0.1589, 0.1583$ and 0.1574, while \n$h_s = 0.1566$ and 0.1557 for the $s$ quark.\nUsing these hopping parameters except for $h_s=0.1557$, \nCP-PACS collaboration performed a quenched QCD calculation of \nthe light meson spectrum \nwith a larger lattice ($32^3 \\times 56$) \\cite{CP-PACS}, which we refer to for comparison.\nThe gauge configurations are generated \nby the heat bath algorithm at $\\beta = 5.9$. \nAfter 20000 thermalization iterations, we start to calculate \nthe meson propagators. On every 2000 configurations,\n80 configurations are used for the ensemble average.\n\nWe emply the point-like source and sink for the $\\kappa^{+}$ meson\n\\begin{equation}\n\\hat{\\kappa}(x) \\equiv \\sum_{c=1}^3\\sum_{\\alpha=1}^4 \n{\\bar{s}_\\alpha^c(x) u_\\alpha^c(x)} \\ \\ ,\n\\label{eq:kappa_operator}\n\\end{equation}\nwhere $u(x)$ and $s(x)$ are the Dirac operators \nfor the $u\/d$ and $s$ quarks, and \nthe indices $c$ and $\\alpha$ \ndenote the color and Dirac-spinor indices, respectively.\nThe point source and sink in Eq.(\\ref{eq:kappa_operator}) lead a positive spectral\nfunction $\\rho(m^2)$ in the correlation function\n $ \\langle \\hat{\\kappa}(t) \\hat{\\kappa}(0) \\rangle =\n\\int dm \\rho(m^2){\\rm exp}(-mt) $.\nThe result obtained here is thus an upper bound of $\\kappa$ mass,\nbecause our result should include excited states.\n\n\n\nFirst, we check finite lattice volume effects by comparing\nour results for \nthe $\\pi$ and $\\rho$ masses as well as the mass ratio $m_{\\pi}\/m_{\\rho}$ \nwith those of the CP-PACS group. \nThe results are summarized in Table \\ref{table:pi_rho}. \nOur result for the $\\rho$ meson mass\nis only slightly ($<$ 5 $\\%$) larger than the CP-PACS's result.\nThe resulting larger value is reasonable because\nthe smaller lattice size gives rise to a mixture of higher mass states.\nWe rather emphasize that the deviation between our results and the\nlarger lattice result (CP-PACS) is so small in spite of the large difference \nin the lattice size. \n\\begin{center}\n\\begin{table*}[h]\n\\caption{Summary of results for $\\bar{q}q$ type mesons. }\n\\label{table:pi_rho}\n\\begin{tabular}{c|c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & 0.1589 & 0.1583 & 0.1574 & 0.1566 & 0.1557 \\\\ \\hline\n$m_{\\pi}$ & 0.2064(62) & 0.2691(36) & 0.3401(29) & 0.3935(28) & 0.4478(28) \\\\\n$m_{\\rho}$ & 0.442(13) & 0.461(06) & 0.496(05) & 0.527(04) & 0.563(03) \\\\\n$m_{\\pi}\/m_{\\rho}$ & 0.467(21) & 0.584(10) & 0.686(05) & 0.746(03) & 0.796(03) \\\\ \\hline\n$m_{\\sigma_v}$ & 1.12(74) & 0.84(23) & 0.886(98) & 0.857(52) & 0.897(35) \\\\\n\\hline\n\\multicolumn{6}{c}{CP-PACS}\\cite{CP-PACS} \\\\ \\hline \\hline\n$m_{\\pi}$ & 0.20827(33) & 0.26411(28) & 0.33114(26) & 0.38255(25) & $-$ \\\\\n$m_{\\rho}$ & 0.42391(132) & 0.44514(96) & 0.47862(71) & 0.50900(60) & $-$ \\\\\n$m_{\\pi}\/m_{\\rho}$ & 0.491(2) & 0.593(1) & 0.692(1) & 0.752(1) & $-$ \\\\ \\hline\n\\end{tabular}\n\\label{table:qqbar}\n\\end{table*}\n\\end{center}\nIn Fig.~\\ref{fig:extrapolation}, \nwe show $m_{\\pi}^2$, $m_{\\rho}$ and $m_{\\sigma_v}$ in the lattice unit\nas a function of the inverse hopping parameter $1\/h_{u\/d}$ for the $u\/d$ quark. \nThe chiral limit ($m_{\\pi}^2 = 0$) is obtained\nat $h_{u\/d}=0.1598(1)\\equiv h_{\\rm crit}$ ($1\/h_{\\rm crit}$=6.2581). \nWe find the lattice spacing $a$ = 0.1038(33) [fm] in the chiral limit\nfrom the value $m_{\\rho}a$ = 0.406(13) at this point \nwith the physical $\\rho$ meson mass being used for $m_{\\rho}$.\nNote that these values are consistent with\nthe CP-PACS's result, $h_{\\rm crit}$ = 0.1598315(68) and $a$ = 0.1020(8) [fm], \nwithin the error bars. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure2.eps}\n\\caption{$m_{\\pi}^2$, $m_{\\rho}$ and $m_{\\sigma_v}$ in the lattice\nunit as a function of the inverse $h_{u\/d}$. \nThe chiral limit is obtained at $h_{\\rm crit}$ = 0.1598(1).}\n\\label{fig:extrapolation}\n\\end{center}\n\\end{figure}\n\nIn Table \\ref{table:pi_rho}, the mass of the \nvalence $\\sigma$ for each\nhopping parameter is shown;\nthe valence $\\sigma$, which is denoted as $\\sigma_v$,\nis defined as the scalar \nmeson described solely with the connected propagator. \nThe mass ratio $m_{\\sigma_v}\/m_\\rho$ varies from 2.5 ($h_{u\/d}=0.1589$) \nto 1.6 ($h_{u\/d}=0.1557$), which is consistent with our previous \nresults \\cite{ScalarSIGMA4}. \nIn other words, without the disconnected part of \nthe propagator the ``$\\sigma$\" mass becomes heavy. \n\nThe propagators of the $K$, $K^*$ and $\\kappa$ mesons \nare calculated with the same configurations \nusing the $s$-quark hopping parameter, $h_s$ = 0.1566 and 0.1557.\nFor $h_s$ = 0.1557, the effective mass plots of the $K^*$ and $\\kappa$ \nmesons are shown in Figs.~\\ref{fig:K*} and \\ref{fig:kappa}. \nThe masses of the $K$, $K^{*}$ and $\\kappa$ mesons, which are \nextracted from the effective mass plots \\cite{DeGrand}\n, are \nsummarized in Tables \\ref{table:ratio1566} and \n\\ref{table:ratio1557}.\nErrors are estimated by jack-knife method.\nWe find that the effective masses of the $K$ and $K^*$ mesons have \nonly small errors and are taken to be reliable, \nwhile that of the $\\kappa$ meson suffers from large errors, especially at\nlarger time regions.\nTo avoid possible large errors coming from the data at large $t$, \nwe fit the effective mass of the $\\kappa$ meson\nonly in the time range $5 \\le t \\le 7, 8$ \nwhere the effective masses are almost constant with small errors.\nSince the effective mass of the $K^*$ meson is reliable,\nwe show the mass of the $\\kappa$ in terms of the ratio to $m_{K^*}$:\nTable \\ref{table:mass_ratio} gives the mass ratios $m_{K}\/m_{K^{*}}$\nand $m_{\\kappa}\/m_{K^{*}}$ at the chiral \nlimit together with $m_{\\phi}\/m_{K^*}$ for $h_s=0.1566$ and 0.1577. \nFor example, $m_\\kappa\/m_{K^*}=0.89(29)\/0.4649(69)=1.92(61)$ at \n$h_s=0.1566$ in Table \\ref{table:mass_ratio}. \nThese calculated mass ratios are shown in Fig.~\\ref{fig:mass_ratio}.\nAll the mass ratios are almost independent of $h_s$.\nAlthough the error bar for $m_\\kappa\/m_{K^*}$ is \nlarge, the behavior as a function of $h_s$ is reasonable. \n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure3.eps}\n\\caption{Effective mass plots \nof $K^*$ for $s$ quark hopping parameter $h_s$ = 0.1557.}\n\\label{fig:K*}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure4.eps}\n\\caption{Effective mass plots \nof the $\\kappa$ meson for the $s$ quark \nhopping parameter $h_s$ = 0.1557.}\n\\label{fig:kappa}\n\\end{center}\n\\end{figure}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the $K$, $K^{*}$ and $\\kappa$ mesons at\n$h_s$ = 0.1566.}\n\\label{table:ratio1566}\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & $h_{\\rm crit}^{1)}$ & 0.1589 & 0.1583 & 0.1574 \\\\ \\hline\n$m_{K}$ & 0.2829(23) & 0.3138(33) & 0.3368(30) & 0.3677(29) \\\\\n$m_{K^{*}}$ & 0.4649(69) & 0.4821(57) & 0.4941(49) & 0.5117(42) \\\\\n$m_{\\kappa}$ & 0.89(29) & 0.88(23) & 0.81(12) & 0.814(81) \\\\\n\\hline\n\\multicolumn{5}{c}{CP-PACS} \\cite{CP-PACS} \\\\ \\hline \\hline\n$m_{K}$ & $-$ & 0.30769(28) & 0.32833(26) & $-$ \\\\\n$m_{K^{*}}$ & $-$ & 0.46724(84) & 0.47749(74) & $-$ \\\\ \\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ $h_{\\rm crit}$ = 0.1598(1).\n\\end{table}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the $K$, $K^{*}$ and $\\kappa$ mesons at\n$h_s$ = 0.1557.}\n\\label{table:ratio1557}\n\\begin{tabular}{c|c|c|c|c}\n\\hline\n\\hline\n$h_{u\/d}$ & $h_{\\rm crit}^{1)}$ & 0.1589 & 0.1583 & 0.1574 \\\\ \\hline\n$m_{K}$ & 0.3188(25) & 0.3474(31) & 0.3684(29) & 0.3971(28) \\\\\n$m_{K^{*}}$ & 0.4835(61) & 0.5006(52) & 0.5126(44) & 0.5299(37) \\\\\n$m_{\\kappa}$ & 0.89(21) & 0.88(16) & 0.828(96) & 0.833(72) \\\\\n \\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ $h_{\\rm crit}$ = 0.1598(1). \\\\\n\\end{table}\n\\begin{table}[htb]\n\\begin{center}\n\\caption{Summary of results for the mass ratios\n$m_K\/m_{K^*}$ and $m_\\kappa\/m_{K^*}$\ntogether with $m_\\phi\/m_{K^*}$ at chiral limit for $u\/d$ quarks. \n}\n\\label{table:mass_ratio}\n\\begin{tabular}{c|c|c||c|c}\n\\hline\n\\hline\n$h_s$ & 0.1566 & 0.1557 & 0.1563(3) & 0.1576(2) \\\\\n$1\/h_s$ & 6.3857 & 6.4226 & 6.396(13) & 6.3452(80) \\\\ \\hline\n$m_{\\phi}\/m_{K^*}$ & 1.135(10) & 1.164(10) & 1.143$^{1)}$ & $-$ \\\\\n$m_K\/m_{K^{*}}$ & 0.6086(79) & 0.6593(63)& 0.623(11) & 0.5556$^{1)}$ \\\\\n$m_{\\kappa}\/m_{K^{*}}$ & 1.92(61) & 1.84(43) & 1.89(55) & 2.00(80) \\\\\n\\hline\n\\end{tabular} \\\\\n\\end{center}\n$^{1)}$ inputs for calculation of physical value of $h_s$. See the text. \n\\end{table}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=\\linewidth]{figure5.eps}\n\\caption{The ratios $m_K\/m_{K^*}$ and $m_{\\kappa}\/m_{K^*}$\nat chiral limit, and $m_{\\phi}\/m_{K^*}$ for $s$ quark hopping parameters \n$h_s$ = 0.1566 and 0.1557.}\n\\label{fig:mass_ratio}\n\\end{center}\n\\end{figure}\n\nWe have searched for the physical value of the $s$ quark hopping \nparameter $h_s$ in the following two ways, both of \nwhich are found to give similar results:\n1)~ By tracing a regression line \nfor $m_{\\phi}\/m_{K^*}$ (Fig.~\\ref{fig:mass_ratio}), we have $h_s$ = 0.1563(3)\n(or $1\/h_s$ = 6.396(13)) for $m_{\\phi}\/m_{K*}$=1019[MeV]\/892.0[MeV]=1.143 \n(input), taken from the PDG \\cite{PDG}.\nThis hopping parameter gives the mass ratio \n$m_{\\kappa}\/m_{K^*}$ = $1.89(55)$.\n2)~ We have also determined the hopping parameter\nso as to reproduce the mass ratio \n$m_K\/m_{K^*}$ = 495.6[MeV]\/892[MeV] = 0.5556, with\n$m_K= 495.6$ [MeV] being the average value of the Kaon masses given in the PDG \n\\cite{PDG}.\nThe resulting value is found to be $h_s$ = 0.1576(2) \n(or $1\/h_s$ = 6.3452(80)), which in turn gives the mass ratio \n$m_{\\kappa}\/m_{K^*}$ = 2.00(80).\nThe mass ratios obtained using methods 1) and 2) are \nalso presented in Table \\ref{table:mass_ratio}. \nBoth methods give almost identical results for the masses \nof the $\\kappa$, that are about twice that of the $K^*$.\n\n\\section{Concluding remarks}\n\nThe motivation of our lattice study is \nto reveal the nature of the scalar meson nonet,\nand the results should be important especially \nin clarifying how the $\\kappa$ meson with\na reported low mass $\\sim 800$ MeV obtained from experiments\ncan be compatible with the \nvalence or constituent quark model:\nthe $\\kappa$ is a\n$P$-wave $q\\bar{q}$ bound state in the non-relativistic\nquark model, and the $\\kappa$ meson constitutes\na nonet together with the $\\sigma$ meson and the $a_0$ mesons.\n\nThere have not been many lattice studies of $\\kappa$ meson.\nRecently, estimations of the $\\kappa$ meson have been reported\nby two groups.\nPrelovsek {\\it et al.} \\cite{Sasa2} \nhave presented a rough estimate of the mass of\nthe $\\kappa$ as $1.6$ GeV, which is obtained using the average quark mass\nof the $u$ and $s$ quarks from the dynamical simulations\nwith the degenerate $N_f=2$ quarks on a $16^3\\times 32$ lattice.\nMathur {\\it et al.} have studied $u\\bar{s}$ meson with the overlap fermion in the quenched approximation\n and obtained a mass of the $u\\bar{s}$ scalar meson to be 1.41 $\\pm$ 0.12 GeV \\cite{Mathur}.\nThe UKQCD Collaboration has\nstudied to some extent the $\\kappa$ meson using the dynamical\n$N_{f}$=2 sea quarks and a valence strange quark\non a $16^3\\times 32$ lattice \\cite{UKQCDkappa};\nthey estimated the $\\kappa $ mass as about 1.1 GeV,\nwhich is much smaller than those in \\cite{ScalarSIGMA4,ScalarKAPPA,Sasa2} \nbut still far from the experimental value $\\sim$800 MeV.\n\nIn this paper, we have presented the lattice simulation results\nin the quenched approximation\nfor the $\\kappa$ meson; the results on the $\\pi$, $\\rho$, $K$\nand $K^*$ mesons are also shown for comparison.\n\nWe have first checked that \nthe masses of the $\\pi$, $\\rho$, $K$ and $K^*$ mesons \nobtained in our simulation are in good agreement\nwith those on a larger lattice ($32^3\\times 56$) \\cite{CP-PACS}; \nour results are only within five percent larger than the latter.\nOur estimated value of the mass of the $\\kappa$ is $\\sim$ 1.7 GeV, which is \nlarger than twice the experimental mass $\\sim 800$ MeV. \nThis result was expected on the basis of our experience \nin calculating the $\\sigma$ meson.\nThe relatively heavy mass of the $\\kappa$ may\nbe at least partly attributed to the absence of the disconnected diagram in \nthe $\\kappa$ propagator; the $\\kappa$ propagator is composed of only \na connected diagram.\nWhile the disconnected diagram was\nessential for realizing the low-mass $\\sigma$ \\cite{ScalarSIGMA4}, \nit does not exist for the $\\kappa$; therefore, the mass of the $\\kappa$\nis not made lighter by the disconnected diagram.\nIndeed, the mass of the valence \n$\\sigma_v$ described solely with the connected propagator \nis far larger than the experimental value\n$\\sim 500$-$600$ MeV, as seen in Table \\ref{table:qqbar}. \n\nOur lattice study and the quark model analysis\\cite{QM85} suggest \nthat the simple two-body constituent-quark picture \nof the $\\kappa$ meson does not agree well with \nthe experimentally observed $\\kappa$.\nNote that the quench simulation is a clean theoretical experiment\nin which a virtual intermediate like $qq\\bar{q}\\bar{q}$ is\nhighly suppressed \\cite{Alfold}.\nTherefore, \nif its existence with the reported low mass is experimentally established, \nthe dynamical quarks may play an essential role for making\nthe $\\kappa$ mass so lighter\nor \nthe $\\kappa$ may contain\nan unconventional state such as\na $qq\\bar{q}\\bar{q}$\\cite{tetra} or $K\\pi$ molecular\nstate\\cite{torn-phys-rep}, which are missing in the \ncalculation here. \n\nIn order to establish this possible scenario, \nthe systematic errors should be much reduced \nin future simulations. \nOur statistics here is reasonably high \n(80 configurations separated by 2000 sweeps), \nand the standard meson masses have small error bars; see Fig.\\ref{fig:K*}. \nOn the contrary, as seen in Fig.\\ref{fig:kappa},\nthe effective mass of $\\kappa$ suffers from large errors \nfor large $t$,\nwhich may be due to a small overlap of the physical states.\nThis is not surprising because $\\kappa$ is a P-wave meson, and\nexpected to be extended.\nChoosing more adequate\nextrapolation operators and with much higher statistics,\nwe can study the dynamics of\nhadrons by comparing results in \nthe quenched lattice QCD,\nfull lattice QCD and various effective theories\/models that include \nthe constituent quark models with and without \nthe tetra-quark structure, chiral effective theories.\n\n{\\bf Acknowledgment}\nT.K. is supported by Grants-in-Aid for Scientific Research from \nthe Ministry of Education, Culture, Sports, Science and Technology\n(No. 17540250) and \nfor the 21st century COE ``Center for Diversity and Universality in\nPhysics'' program of Kyoto University.\nThe work is partially supported by \nGrants-in-Aid for Scientific Research from\nthe Ministry of Education, Culture, Sports, Science and Technology\nNos. 13135216 and 17340080.\nThe calculation was carried out on SX-5 at RCNP, Osaka University and\non SR-8000 at KEK.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}