diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzacqy" "b/data_all_eng_slimpj/shuffled/split2/finalzzacqy" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzacqy" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\nBL Lacertae is a well known source which has been used to define a class of active\ngalactic nuclei (AGNs) that, together with flat spectrum radio quasars, make up the highly\nvariable objects called blazars. The BL Lacertae class is characterized by the absence or extreme\nweakness of emission lines (with equivalent width in the rest frame of the host galaxy of $< 5$ \\AA),\nintense flux and spectral variability across the\ncomplete electromagnetic spectrum on a wide variety of time-scales,\nand highly variable optical and radio polarization (e.g., Wagner \\& Witzel 1995). \nA relativistic plasma jet pointing close to our line of sight can account for the\nobserved properties of these objects.\nBL Lacertae is an optically bright blazar located at $z=0.0688 \\pm 0.0002$ (Miller \\&\nHawley 1977) hosted by a giant elliptical galaxy with R$=15.5$ (Scarpa et al.\\ 2000).\nBL Lacertae is an LBL (low frequency peaked blazar) as its low energy spectral component peaks at\nmillimeter to micron wavelengths while the high energy spectral component peaks in the MeV-GeV range.\nOn some occasions, BL Lacertae has shown broad $H \\alpha$ and $H \\beta$ emission lines\nin its spectrum, raising the issue of its membership in its eponymous class (Vermeulen et al. 1995). \n\nBL Lacertae was observed by several multi-wavelength campaigns carried out by the Whole Earth Blazar Telescope \n(WEBT\/GASP; B\\\"ottcher et al. 2003; Villata et al.\\ 2009; Raiteri et al. 2009, 2010, 2013 and references therein).\nBL Lacertae is well known for its intense optical variability on short and intra-day time-scales\n(e.g.\\ Massaro et al.\\ 1998; Tosti et al.\\ 1999; Clements \\& Carini 2001; Hagen-Thorn et al.\\ 2004)\nand strong polarization variability (Marscher et al.\\ 2008; Gaur et al.\\ 2014 and references therein).\nExtensive light curves for BL Lacertae have been presented by many authors and hence a number\nof investigations have been carried out to search for the flux variations, spectral changes \nand any possible periodicities in the light curves (e.g.\\ Racine 1970; Speziali \\& Natali 1998; \nB{\\\"o}ttcher et al.\\ 2003; Fan et al.\\ 2001; Hu et al.\\ 2006). Nesci et al.\\ (1998) found the \nsource to be variable with the amplitudes of flux\nvariations larger at shorter wavelengths. \nPapadakis et al.\\ (2003) studied the rise and decay time-scales of the source during the course of a single \nnight and found them to increase with decreasing frequency. They also studied the time-lags between the light\ncurves in different optical bands and found the B band to lead the I band by $\\sim$0.4 hours. Villata et al.\\ (2002) \ncarried out a campaign in 2000--2001 with exceptionally dense\ntemporal sampling, which was able to measure\nintra-night flux variations of this blazar. They found the optical spectrum to \nbe only weakly sensitive to the long term brightness trend and argued that this achromatic modulation of the flux base level \non long time-scales is due to variations of the jet Doppler factor. However, the short-term\n flux variations and especially the bluer-when-brighter trend indicate the importance of intrinsic processes related to the jet \n emission mechanism (Raiteri et al. 2013; Agarwal \\& Gupta 2015).\n\nA key motivation of this study is to look for intra-day flux and spectral variations\nin optical bands during the active state of BL Lacertae in 2010--2012. We also studied interband BVRI time delays\non intra-day time-scales of BL Lacertae. As BL Lacertae is a very well known LBL and the optical bands are located\nabove the first peak of the spectral energy distribution, the fast intra-day variability (IDV) properties\ncan yield rather direct implications for the nature of the acceleration and cooling mechanisms of the \nrelativistic electron populations.\nOver the course of 3 years, we performed quasi-simultaneous optical multi-band photometric monitoring of this source\nfrom various telescopes in Bulgaria, Greece, India and the USA on intra-day time-scales.\n\nThe paper is organized as follows. In Section 2 we briefly describe the observations and data reductions. Section 3 discusses\nthe methods of quantifying variability. We present our results in Section 4. Sections 5 \\& 6 contain a discussion and our conclusions,\nrespectively.\n\n\\section{Observations and Data Reduction}\n\nOur observations of BL Lacertae started on 10 June 2010 and ran through 26 October 2012.\n The entire observation log is presented in Table 1. The observations were carried out at \nseven telescopes in Bulgaria, Greece, India and the USA.\nThe telescopes in Bulgaria, Greece and India are described in detail in Gaur et al.\\ (2012, Table 1) \nand the standard data reduction methods we used at each telescope are given in Section 3 of that paper,\nso we will not repeat them here. During our observations, typical seeing vary\nbetween 1--3 arcsec.\nIn our observations of BL Lacertae, comparison stars are observed in the same field as the blazar \nand their magnitudes are taken from Villata et al.\\ (1998).\nWe used star C for calibration as it has both magnitude and colour close to those of BL Lacertae \nduring our observations.\n\nAt the MDM Observatory on the south-west ridge of Kitt Peak, Arizona, USA,\ndata were taken for limited periods during the nights of 3, 4, 5, 6, 7 and 8\nJuly 2010 with the 1.3 m McGraw-Hill Telescope, using the Templeton CCD with\nB, V, R, and I filters. CCD parameters are described in Table 2. The standard data reduction was performed using \\texttt{IRAF}\n\\footnote{IRAF is distributed by the National Optical Astronomy Observatories, which are operated\nby the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the\nNational Science Foundation.},\nincluding bias subtraction and flat-field division. Instrumental magnitudes of BL Lacertae\n plus four comparison stars in the field (Villata et al.\\ 1998) were extracted\nusing the \\texttt{IRAF} package \\texttt{DAOPHOT}\\footnote{Dominion Astrophysical\nObservatory Photometry software} with an aperture radius of 6 arcsec and\na sky annulus between 7.5 and 10 arcsec. \n\nThe host galaxy of BL Lacertae is relatively bright, so, in order to remove its contribution from the\nobserved magnitudes, we first dereddened the magnitudes using the Galactic extinction coefficient of Schlegel\net al.\\ (1998) and converted them into fluxes. We then subtracted the host galaxy contribution\nfrom the observed fluxes in R band by considering different aperture radii used by different observatories for the\nextraction of BL Lacertae magnitudes, using Nilsson et al.\\ (2007). We inferred the host \ngalaxy contribution in B, V and I bands\nby adopting the elliptical galaxy colours of V$-$R $ = 0.61$, B$-$V $= 0.96$ and R$-$I $= 0.70$ \nfrom Fukugita et al.\\ (1995).\nFinally, we subtracted the host galaxy contribution in the B, V and I bands in order to avoid host\ncontamination in the extraction of colour indices.\n\n\\section{Variability Detection Criterion}\n\n\\subsection{Power Enhanced F-test}\n\nThe F-test, as described by de Diego (2010), provides a standard criterion for testing for the presence of intra-night\nvariability. The F-statistic is defined as the ratio of two given sample variances such as s$_Q ^{2}$ for the blazar instrumental light \ncurve measurements and s$_* ^{2}$ for that of the standard star, i.e.,\n\\begin{equation}\nF=\\frac {s_Q ^{2}}{s_* ^{2}} .\n\\end{equation}\nUsually, two comparison stars in the blazar field are used to calculate $F_{1}$ and $F_{2}$, and evidence of variability is \nclaimed if both the F-tests simultaneously reject the null hypothesis at a specific significance level (usually 0.01 or 0.001) ( \nGaur et al. 2012 and references therein).\nIn this case, the number of degrees of freedom for each sample, $\\nu_Q$ and $\\nu_*$ will be the same, and equal to the number of\nmeasurements, $N$ minus 1 ($\\nu = N - 1$).\n\nRecently, de Diego (2014) called this procedure the ``Double Positive Test'' (DPT) and pointed out a problem with this \nprocedure that DPT has very low power and in practice its significance level can not be calculated. Also,\n a large brightness difference or some variability in one of the stars may lead to an underestimation of the \nsource's variability with respect to the dimmer or less variable star.\nTo avoid this issue we employed the power-enhanced F-test using the approach of de Diego (2014) and de Diego et al. (2015). \nIt consists of increasing the number of degrees of freedom in the denominator of the F-distribution by stacking \nall the light curves of the standard stars and it consists\nin transforming the comparison star differential light curves to have the same photometric noise as if their magnitudes\n matched exactly the mean magnitude of the target object. The mean brightness of both the comparison star and the target\nobject are matched to ensure that the photometric errors are equal. Including \nmultiple standard stars reduces the possibility of false detections of intra-night variability that can be produced by \none single peculiar comparison star light curve. More details are provided in de Diego et al. 2015. In the analysis,\n we used three standard stars B, C and H in the field of BL Lacertae whose brightnesses are very close to \nthe brightness of BL Lacertae. Thus, for the $i^{th}$ observation of the \nlight curve of standard star $j$, for which we have $N_j$ data points, we calculate the square deviation as: \n\\begin{equation}\n s_{j,i}^2 = (m_{j,i} - \\overline{m}_{j})^2.\n\\end{equation}\n\nBy stacking the results of all observations on the total of $k$ comparison stars, we can calculate the combined variance as:\n\\begin{equation}\n s_c^2 = \\frac{1}{(\\sum_{j=1}^k N_j) - k} \\sum_{j=1}^k \\sum_{i=1}^{N_j} s_{j,i}^2.\n\\end{equation}\nThen, we compare this combined variance with the blazar light curve variance to obtain the $F$ value\nwith $\\nu_Q = N - 1$ degrees of freedom in the numerator and $\\nu_* = k (N - 1)$ degrees of freedom in the denominator. \nThis value is then compared with the $F^{(\\alpha)}_{\\nu_Q,\\nu_*}$ critical value, where $\\alpha$ is the significance \nlevel set for the test.\nThe smaller the $\\alpha$ value, the more improbable it is that the result is produced by chance. If $F$ is larger\nthan the critical value, the null hypothesis (no variability) is discarded. We have performed the\n$F$-test at the $\\alpha = 0.001$ level.\n\n\\subsection{Discrete Correlation Function (DCF)}\n\nTo estimate the variability time-scales in the observed light curves of BL Lacertae and to determine the cross-correlations\nbetween different optical bands, we used the Discrete Correlation Function (Edelson \\& Krolik 1988; Hovatta et al. 2007).\n\n \nThe first step is to calculate the unbinned correlation (UDCF) using the given time series by:\n\\begin{equation}\nUDCF_{ij} = {\\frac{(a(i) - \\bar{a})(b(j) - \\bar{b})}{\\sqrt{\\sigma_a^2 \\sigma_b^2}}}.\n\\end{equation}\nHere $a(i)$ and $b(j)$ are the individual points in two time series $a$ and $b$, respectively, $\\bar{a}$\nand $\\bar{b}$ are respectively the means of the time series, and $\\sigma_a^2$ and $\\sigma_b^2$ are their variances.\nThe correlation function is binned after calculation of the UDCF. The DCF can be calculated by averaging the \nUDCF values ($M$ in number) for each time delay\n\\( \\Delta t_{ij} = (t_{yj}-t_{xi}) \\) lying in the range \\( \\tau - \\frac{\\Delta\\tau}{2} \\leq t_{ij} \n\\leq \\tau+ \\frac{\\Delta\\tau}{2} \\) via\n\n\\begin{equation}\nDCF(\\tau) = {\\frac{1}{n}} \\sum ~UDCF_{ij}(\\tau) .\n\\end{equation}\nwhere $\\tau$ is the centre of a time bin and $n$ are the number of points in each bin.\n DCF analysis is frequently used for finding the correlation and possible lags between multi-frequency\nAGN data. When the same data train is used, so $a$=$b$, there is obviously a peak at zero lag and is called Auto \nCorrelation Function (ACF), indicating that there \nis no time lag between the two but any other strong peaks in the ACF give indications of variability timescales. \n\n\n\n\\section{Results}\n\n\\subsection{Flux variations}\n\nWe extensively observed the source for a total of 38 nights during 2010--2012. During \n26 of those nights, we observed the source quasi-simultaneously in the B, V, R and I bands, providing a total of 113 light \ncurves in B, V, R and I. These light curves are displayed in Figs.\\ 1 and 2.\nThe lengths of the individual observations were usually between 2 and 6 hours.\nIt is clear from the figures that BL Lacertae was variable from day to day and also on hourly time-scales.\nWe searched for genuine flux variations on IDV time-scales and found 50 light curves to be variable in the whole set of filters\n during a total of 19 nights using the enhanced F-test. \nThe intranight variability amplitudes (in per cent) are given by Heidt \\& Wagner (1996):\n\\begin{eqnarray}\nAmp = 100\\times \\sqrt{{(A_{max}-A_{min}})^2 - 2\\sigma^2},\n\\end{eqnarray}\nwhere $A_{max}$ and $A_{min}$ are the maximum and minimum values in the calibrated light curves of the blazar, and $\\sigma$\nis the average measurement error. \nThe enhanced F-test values\nare presented in Table 1. \n\nThe duty cycle (in per cent) (DC) of BL Lacertae is computed following the definition of Romero et al.\\ (1999) that \nlater was used by many authors (e.g., Stalin et al.\\ 2004; Goyal et al.\\ 2012, and references therein),\n\n\\begin{equation} \nDC = 100\\frac{\\sum_{i=1}^n N_i(1\/\\Delta t_i)}{\\sum_{i=1}^n (1\/\\Delta t_i)} ,\n\\label{eqno1} \n\\end{equation}\nwhere $\\Delta t_i = \\Delta t_{i,obs}(1+z)^{-1}$ is the duration of the\nmonitoring session of a source on the $i^{th}$ night, corrected for\nits cosmological redshift, $z$. Since for a given source the monitoring durations on different nights are not always equal, \nthe computation of the DC is weighted by the actual monitoring duration\n$\\Delta t_i$ on the $i^{th}$ night. $N_i$ is set equal to 1 if intra-day variability \nwas detected by the F-test (given in Table 1), otherwise $N_i$ = 0.\nWe found the duty cycle of BL Lacertae to be around 44 per cent. \n\n\\subsection{Amplitude of Variability}\n\nWhen the variability is substantial the amplitude of variability usually is greater in higher energy bands (B) and smaller \nin lower energy bands (R) in most of the observations.\nIt has been found by many investigators that amplitude of variability is greater at higher frequencies for\nBL Lacertaes (Ghisellini et al. 1997; Massaro et al. 1998; Bonning et al. 2012) however, the amplitude\nof variability is not systemaically larger at higher frequencies is also found on some occasions (Ghosh et al. 2000;\nRamirez et al. 2004). In few cases, we found the amplitude of variability in lower energy bands \nto be comparable to or greater than the amplitude of variability in higher energy bands. Still, because \nthe errors in the B band are higher in these cases, while they are often lower in the lower frequency band, there are \nsome nights during which the statistical significance of the variations falls below our thresholds in \nB (also sometimes V), when clear variability is detected in R and I bands. Also, differences \nin the amplitude of variability in various optical bands change from one observation to another. \nThe highest fractional amplitude of variability was found to be $\\sim$38 per cent on 6 July 2010, where the \nsource showed a $\\sim$0.3 mag change in less than 3.5 hours in V band. \\\\\n\nWe searched for possible correlations between the variability amplitude and the duration of the observation, as displayed\nin Fig.\\ 3 (left panel). We found a significant positive correlation ($\\rho=0.3306$ with $p=0.0190$, where $\\rho$ and $p$ are\n the Spearman Correlation coefficient and its p-value, respectively) between them, i.e., the variability amplitude \nincreases with duration of the observation. We conclude that there is an enhanced likelihood of \nseeing higher amplitudes of variability in longer duration light curves, which is in agreement with Gupta \\& Joshi (2005) report\nfor a larger sample of blazars.\n\nNext we examined the possible correlation between the amplitude of variability with \nthe source flux (Fig.\\ 3, right panel). It can be seen that amplitude of variability decreases as the source flux increases \n($\\rho$=-0.4990 with $p$=0.0002). \nThis might be explained as, if the source attains a high flux state, the irregularities in the jet flow decreases \nparticularly if fewer \nnon-axisymmetric bubbles were carried outward in the relativistic magnetized jets (Gupta et al.\\ 2008).\nFor instance, the blazar 3C 279 was observed in 2006 during its outburst\/high state but did not show any genuine micro-variability, \nwhile other\nsources which were in their pre\/post outburst state showed significant micro-variability (Gupta et al.\\ 2008). \nThis anti-correlation between variability amplitude and flux could also be explained by a two component\nmodel where in more active phases or in an outburst state, the slowly varying jet component rises and dominates on the emission\nfrom the more variable regions, i.e., shocks\/knots, and therefore reduces the fractional amplitude. However, there is \n scatter in the plot, some of which could be due to an observational bias, as some observations were longer than others. \nDue to this, the observed variability amplitude at similar luminosities could be quite different as the\n amplitude of variability increases with the duration of the observation.\n\n\\subsection{R-band Autocorrelations}\n\nWe searched for the presence of a characteristic time-scale of variability in all of the nights by auto-correlating the\n R-band measurements, for which we had the most data. In most of the nights, the\n auto-correlation function (ACF) shows a sharp maximum at nearly zero lag, as expected, and it stays positively \nauto-correlated with itself for time lags of somewhat less than 1 hour followed by dropping to negative values. \n Hence, we conclude that we did not find any evidence for characteristic variability time-scales from this approach.\nOne example of R-band ACF is shown in the left panel of figure 4.\n\\begin{figure*}\n \\centering\n\\includegraphics[width=4cm , angle=0]{fig1a.eps}\n\\includegraphics[width=4cm , angle=0]{fig1b.eps}\n\\includegraphics[width=4cm , angle=0]{fig1c.eps}\n\\includegraphics[width=4cm , angle=0]{fig1d.eps}\n\\includegraphics[width=4cm , angle=0]{fig1e.eps}\n\\includegraphics[width=4cm , angle=0]{fig1f.eps}\n\\includegraphics[width=4cm , angle=0]{fig1g.eps}\n\\includegraphics[width=4cm , angle=0]{fig1h.eps}\n\\includegraphics[width=4cm , angle=0]{fig1i.eps}\n\\includegraphics[width=4cm , angle=0]{fig1j.eps}\n\\includegraphics[width=4cm , angle=0]{fig1k.eps}\n\\includegraphics[width=4cm , angle=0]{fig1l.eps}\n\\includegraphics[width=4cm , angle=0]{fig1m.eps}\n\\includegraphics[width=4cm , angle=0]{fig1n.eps}\n\\includegraphics[width=4cm , angle=0]{fig1o.eps}\n\\includegraphics[width=4cm , angle=0]{fig1p.eps}\n\\includegraphics[width=4cm , angle=0]{fig1q.eps}\n\\includegraphics[width=4cm , angle=0]{fig1r.eps}\n\\includegraphics[width=4cm , angle=0]{fig1s.eps}\n\\includegraphics[width=4cm , angle=0]{fig1t.eps}\n\n\\caption{IDV light curves of BL Lacertae during 2010 and early 2011 in the B (blue), V(green), R(red) and I(black) bands. The\nX-axis is JD (2455000+), and the Y-axis is the calibrated magnitudes in each of the panels. The B, V and I \nbands are shifted by\narbitrary offsets with respect to R-band light curve. Observations from observatory A are represented by squares; those from\nC are represented by triangles; D by filled circles; E by open circles and F by starred symbols.}\n \\end{figure*}\n\n\n\\begin{figure*}\n \\centering\n\\includegraphics[width=4cm , angle=0]{fig2a.eps}\n\\includegraphics[width=4cm , angle=0]{fig2b.eps}\n\\includegraphics[width=4cm , angle=0]{fig2c.eps}\n\\includegraphics[width=4cm , angle=0]{fig2d.eps}\n\\includegraphics[width=4cm , angle=0]{fig2e.eps}\n\\includegraphics[width=4cm , angle=0]{fig2f.eps}\n\\includegraphics[width=4cm , angle=0]{fig2g.eps}\n\\includegraphics[width=4cm , angle=0]{fig2h.eps}\n\\includegraphics[width=4cm , angle=0]{fig2i.eps}\n\\includegraphics[width=4cm , angle=0]{fig2j.eps}\n\\includegraphics[width=4cm , angle=0]{fig2k.eps}\n\\includegraphics[width=4cm , angle=0]{fig2l.eps}\n\\includegraphics[width=4cm , angle=0]{fig2m.eps}\n\\includegraphics[width=4cm , angle=0]{fig2n.eps}\n\\includegraphics[width=4cm , angle=0]{fig2o.eps}\n\\includegraphics[width=4cm , angle=0]{fig2p.eps}\n\\includegraphics[width=4cm , angle=0]{fig2q.eps}\n\\includegraphics[width=4cm , angle=0]{fig2r.eps}\n\\includegraphics[width=4cm , angle=0]{fig2s.eps}\n\n\\caption{As in Fig.\\ 1 for August 2011 through October 2012.}\n\\end{figure*}\n\n\\subsection{Inter-band cross-correlations}\n\nWe computed the DCFs to determine the cross-correlations and time delays between the B and I, V and I, and R and I bands. \nThe time delays are expected between emission in different energy bands, as the flare usually begin at higher frequencies\nand then propagates to lower frequencies in the inhomogeneous jet model. Injected high energy electrons\nemit synchrotron radiation first at higher frequencies and then cool, emitting at progressively lower frequencies, resulting\nin time lag between high and low frequencies.\nThe DCFs between the light curves in all bands show close correlations among the various bands in the nights\nwhere genuine variability is present. For the nights in which no genuine variability is present, we normally found\nmuch weaker correlations between the bands ($<$0.4). As the peak of the DCFs are broad, we fit the DCFs with\n Gaussian functions to determine the possible time delays; \nhowever, lags indicated by the DCFs are all consistent with zero. This is not surprising due to the closeness of the various optical \nbands we measured in frequency space, so if any lags are present they appear to be less than the resolution of our light curves.\nAn example of the DCF is shown in the right panel of fig.\\ 4.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm , angle=0]{fig3a.eps}\n\\includegraphics[width=8cm , angle=0]{fig3b.eps}\n\n\\caption{Dependence of amplitude of variability on duration (left panel) and flux (right panel) of the observations. \nHere, B band is represented by squares (blue), V by solid circles (green), R by triangles (red) and I by starred symbols (black) .}\n \\end{figure*}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=8cm , angle=0]{fig4a.eps}\n\\includegraphics[width=8cm , angle=0]{fig4b.eps}\n\n\\caption{One example of Auto-correlation (R band) is shown in the left figure and DCF (R versus I) is shown in the right figure.}\n \\end{figure*}\n\n\n\n\\subsection{Colour Indices}\n\nWe next investigated the existence of spectral variations by studying the behaviour of colour variations with respect \nto the brightness of BL Lacertae. The colour indices (CIs) are calculated by combining almost simultaneous (within 8 minutes)\n B, V, R and I magnitudes to yield CIs = B$-$V, V$-$R, R$-$I and B$-$I. For a particular night, we studied the \ncolour variability only when both light curves were identified as variable. Our data have the \nadvantage of being almost simultaneous in various optical bands. It is important to recall that the underlying host galaxy,\neffect of the accretion disk component and the Gravitational microlensing can also \nlead to apparent, but unreal, colour variations (Hawkins 2002). But, since Gravitational microlensing is important\non weeks to months time-scales and during our observations, BL Lacertae was in flaring state (Raiteri et al. 2013) where the Doppler\nboosting flux from the relativistic jet almost invariably swamps out the contribution of the accretion disk component, so we can\nrule out the contribution of these components. Also, the data we have used in calculating the colour indices are host \ngalaxy subtracted so we conclude that our results indicates \nvariability of the non-thermal continuum radiation. \n\nWe studied the variations of colour indices with respect to brightness in 13 of these observations. We fit all the \ncolour--magnitude diagrams with a linear model of the form CI = $m \\times $ mag $+$~c (where $m$ is the slope in the \nfit, V magnitude is taken as mag and $c$ is its respective intercept). The Pearson correlation coefficient (r), \nits p-value (null hypothesis probability; we consider a confirmed colour index correlation with the V magnitude when p $<$0.01) \nalong with the slopes and intercepts are presented in Table 3. The significant positive correlations between the \ncolour index and magnitude along with a change of slope $>$3 $\\sigma$ indicates that the source exhibits a bluer-when-brighter trend. \n\nIn five of the nights, which are marked by asterisks in Table 3, we found significant variations in colour indices at the 3-$\\sigma$ level.\nIn these observations, colour indices correlate with the source brightness and the overall correlation is positive, \nwhich indicates hardening of the spectrum as the source brightens. So, in these observations, BL Lacertae exhibits bluer-when-brighter\ntrend with different regression slopes. No significant negative correlations are found for the source.\nThe bluer-when-brighter tendency in BL Lacertae has been seen by several groups on long-term as well as short-term time-scales\n(e.g.\\ Papadakis et al.\\ 2003; Villata et al.\\ 2004; Stalin et al.\\ 2006; Larionov et al.\\ 2010; \nGu et al.\\ 2006; Gaur et al.\\ 2012; Wierzcholska et al.\\ 2015).\nVillata et al.\\ (2004) characterized the intra-day flares to be strong bluer-when-brighter chromatic events with a\nslope of $\\sim$0.4. In five observations, we found a bluer-when-brighter trend with a slope varying between 0.18--0.28 (in Table 3). \n\nIn the other eight observations, we did not find significant linear correlations between colour indices and magnitudes (Table 3). \nIn some of them significant colour variations are seen but are not well fitted by linear functions. \nDuring the observations on 6 July 2010 (Fig.\\ 1, fourth panel), we found a strong flare with magnitude variation of $\\sim$0.3\nin all the optical bands and the behaviour of the colour magnitude diagram varies according to the different flux states. \nBut, on intra-day timescales it is difficult to judge the variations of the colour indices with respect to different brightness states\n as the flare is on hours like timescales. \nAlso, in other observations, i.e, 8 July 2010, 24 and 25 August 2011, we saw small sub-flares superimposed on the long-term trend. \nIn these cases, the colour indices vary significantly within the individual observations, sometimes showing different branches in the \ncolour magnitude diagrams according to the flux states. Spectral steepening during the flux rise can be explained by the\npresence of two components, one variable with a flatter slope which dominates during the flaring states and another one that is\nmore stable and contributes to the long term achromatic emission (Villata et al.\\ 2004). So, it could be possible that during these \nobservations, the superposition of many distinct new variable components lead to the overall weaking of the colour-magnitude \ncorrelations. \nBonning et al.\\ (2012) studied a sample of FSRQs and BL Lacertaes and found that FSRQs follow redder-when-brighter trends while BL Lacertaes\nshow no such trends. They found complicated behaviour of the blazars on colour-magnitude diagrams: hysteresis tracks, and\nacromatic flares which depart from the trend suggesting different jet components becoming important at different times.\nTherefore, our colour variability results show that the intra-night \nflares between 2010 and early 2012 are chromatic but do not always follow simple bluer-when-brighter trends. \n\n\n\\begin{table*}\n\\center\n\\caption{ Results of Intra-day Variability of BL Lacertae}\n\\setlength{\\tabcolsep}{0.03in}\n\\begin{tabular}{lccccccr@{\\hskip 0.3in}lcccccc} \\cline{1-7} \\cline{9-15}\nDate &Telescope &Band &$F_{enh}$ &$F_{c}$(0.001) &$Amp\\%$ &Variable & & Date &Telescope &Band &$F_{enh}$ &$F_{c}$(0.001) &$Amp\\%$ &Variable \\\\ \\cline{1-7} \\cline{9-15}\n10.06.2010 &A &R &1.018 &2.386 &- & NV & & 24.08.2011 &D &B &4.035 &2.849 &13.55 &Var\\\\\n11.06.2010 &A &R &0.985 &2.008 &- & NV & & & &V &7.389 &2.849 &11.36 &Var \\\\\n12.06.2010 &A &R &1.034 &2.386 &- &NV & & & &R &16.187&2.849 &11.12 &Var \\\\\n14.06.2010 &A &R &1.054 &2.076 &- &NV & & & &I &13.665&2.849 &9.98 &Var \\\\\n &C &R &0.934 &2.033 &- &NV & & 25.08.2011 &D &B &11.247&2.790 &30.14 &Var \\\\\n18.06.2010 &E &R &51.328&1.940 &6.63 &Var & & & &V &56.7659&2.790 &26.30 &Var \\\\\n19.06.2010 &E &R &1.861 &1.940 &- &NV & & & &R &63.145&2.790 &25.55 &Var\\\\\n20.06.2010 &E &R &21.725&1.930 &3.48&Var & & & &I &17.878&2.790 &24.10 &Var\\\\\n21.06.2010 &E &R &1.227 &1.972 &- &NV & & 22.09.2011 &D &B &1.354 &2.639 &- &NV\\\\\n22.06.2010 &E &R &29.890 &1.94 &5.28 &Var & & & &V &1.565 &2.517 &- &NV\\\\\n04.07.2010 &F &B &2.164 &2.281 &- &NV & & & &R &1.329 &2.517 &- &NV\\\\\n & &V &2.155 &2.481 &- &NV & & & &I &0.930 &2.517 &- &NV\\\\\n & &R &0.146 &2.236 &- &NV & & 19.10.2011 &D &R &0.574 &3.239 &- &NV \\\\\n & &I &0.938 &2.281 &- &NV & & & &I &0.841 &4.142 &- &NV \\\\\n05.07.2010 &F &B &1.527 &2.305 &- &NV & & 06.07.2012 &E &V &25.590&2.596 &6.18 &Var \\\\\n & &V &0.618 &2.281 &- &NV & & & &R &21.632&2.596 &6.14 &Var \\\\\n & &R &0.109 &2.258 &- &NV & & & &I &9.829 &2.596 &5.30 &Var \\\\\n & &I &1.052 &2.281 &- &NV & & 10.07.2012 &E &B &16.174&2.983 &6.40 &Var\\\\\n06.07.2010 &F &B &32.4002&1.995 &35.85 &Var & & & &V &57.750&2.913 &6.27 &Var \\\\\n & &V &36.577 &1.995 &38.64 &Var & & & &R &28.700&2.913 &6.15 &Var\\\\\n & &R &45.931&2.047 &35.64 &Var & & & &I &11.878&2.913 &5.54 &Var\\\\\n & &I &55.757&1.995 &34.82 &Var & & 07.08.2012 &D &B &2.110 &3.753 &- &NV \\\\\n07.07.2010 &F &B &2.307 &2.481 &- &NV & & & &V &0.558 &3.932 &- &NV\\\\\n & &V &2.284 &2.305 &- &NV & & & &R &0.638 &3.932 &- &NV\\\\\n & &R &1.611 &2.258 &- &NV & & & &I &0.746 &3.932 &- &NV\\\\\n & &I &1.247 &2.258 & &NV & & 12.08.2012 &D &B &4.669 &3.598 &20.67 &Var \\\\\n08.07.2010 &F &B &17.271&2.305 &14.49 &Var & & & &V &18.835 &3.598 &17.35 &Var \\\\\n & &V &3.824 &2.281 &13.76 &Var & & & &R &36.030&3.598 &14.52 &Var \\\\\n & &R &6.864 &2.281 &12.40 &Var & & & &I &24.429&3.753 &12.43 &Var \\\\\n & &I &4.720 &2.331 &10.96 &Var & & 15.08.2012 &D &B &0.670 &3.932 &- &NV\\\\\n17.07.2011 &E &B &35.089&3.239 &8.23 &Var & & & &V &0.417 &3.932 &- &NV\\\\\n & &V &90.629&3.239 &7.28 &Var & & & &R &0.908 &3.932 &- &NV\\\\\n & &R &50.931&3.239 &7.20 &Var & & & &I &0.754 &3.932 &- &NV \\\\\n01.08.2011 &D &B &1.860 &3.932 &- &NV & & 16.08.2012 &D &B &0.908 &3.463 &- &NV\\\\\n & &V &2.774 &3.932 &- &NV & & & &V &1.726 &3.463 &- &NV\\\\\n & &R &2.094 &3.932 &- &NV & & & &R &1.622 &3.463 &- &NV\\\\\n & &I &1.862 &3.932 &- &NV & & & &I &0.764 &3.463 &- &NV \\\\\n02.08.2011 &D &B &1.584 &6.195 &- &NV & & 18.09.2012 &D &B &0.540 &3.345 &- &NV\\\\\n & &V &4.858 &6.195 &- &NV & & & &V &1.395 &3.345 &- &NV\\\\\n & &R &5.322&7.077 &- &NV & & & &R &1.034 &3.345 &- &NV\\\\\n & &I &2.540 &7.077 &- &NV & & & &I &0.678 &3.345 &- &NV \\\\\n04.08.2011 &D &B &0.907 &2.849 &- &NV & & 08.10.2012 &C &R &3.007 &1.961 &7.89 &Var\\\\\n & &V &0.552 &2.983 &- &NV & & 13.10.2012 &C &R &6.686 &3.239 &11.76 &Var\\\\\n & &R &1.507 &2.983 &- &NV & & 17.10.2012 &D &B &6.863 &3.932 &24.58 &Var \\\\\n & &I &1.346 &2.983 &- &NV & & & &V &27.535 &3.932 &19.40 &Var \\\\\n06.08.2011 &D &B &0.368 &2.736 &- &NV & & & &R &42.842&3.932 &16.02 &Var \\\\\n & &V &0.300 &2.736 &- &NV & & & &I &33.239&3.932 &13.18 &Var \\\\\n & &R &0.670 &2.790 &- &NV & & 22.10.2012 &D &B &1.383 &2.686 &- &NV \\\\\n & &I &0.859 &2.849 &- &NV & & & &V &1.462 &2.555 &- &NV\\\\\n07.08.2011 &D &B &0.997 &3.239 &- &NV & & & &R &10.608 &2.555 &13.80 &Var \\\\\n & &V &7.058&3.239 &10.63 &Var & & & &I &10.043 &2.555 &12.95 &Var \\\\\n & &R &7.773 &3.239 &8.23 &Var & & 23.10.2012 &D &B &0.606 &2.596 & &NV\\\\\n & &I &2.046 &3.239 & &NV & & & &V &3.585 &2.481 &9.29 &Var\\\\\n23.08.2011 &D &B &0.730 &2.448 &- &NV & & & &R &5.166 &2.517 &10.25 &Var \\\\\n & &V &4.960 &2.448 &12.49 &Var & & & &I &3.329 &2.481 &9.68 &Var \\\\\n & &R &8.400 &2.448 &9.47 &Var & & 26.10.2012 &C &R &4.330 &2.596 &9.42 &Var \\\\\n & &I &5.092 &2.448 &9.69 &Var \\\\\n\n\\cline{1-7} \\cline{9-15}\n\\end{tabular} \\\\\n\\footnotesize\n\\begin{tabbing}\nA: \\= 1.04 meter Samprnanand Telescope, ARIES, Nainital, India \\=\n\\hspace{3em} \\=\nC: \\= 50\/70-cm Schmidt Telescope at National Astronomical \\\\\nD: \\> 60-cm Cassegrain Telescope at Astronomical Observatory \\> \\> \\> Observatory, Rozhen, Bulgaria \\\\\n\\> Belogradchik, Bulgaria \\> \\>\nE: \\> 1.3-m Skinakas Observatory, Crete, Greece \\\\\nF: \\> 1.3 m McGraw-Hill Telescope, Arizona, USA\\\\\n$F_{enh}$: \\> \\hspace{1em} Enhanced F-test values \\> \\>\n$F_{c}$(0.001): \\> \\hspace{3em} Critical Values of F Distribution at 0.1\\%; \\\\\nAmp: \\> \\hspace{1em} Variability Amplitude \\> \\>\n Var \/ NV: \\> \\hspace{3em} Variable \/ Non-variable \\\\\n\\end{tabbing}\n\n\\end{table*}\n\n\n\\begin{table}\n\\caption{ Details of telescopes and instruments}\n\\textwidth=6.0in\n\\textheight=9.0in\n\\vspace*{0.2in}\n\\noindent\n\\begin{tabular}{ll} \\hline\nTelescope: &1.3 m McGraw-Hill Telescope \\\\\\hline\nChip size: & $4064\\times4064$ pixels \\\\\nPixel size: &$15\\times15$ $\\mu$m \\\\\nScale: &0.315\\arcsec\/pixel \\\\\nField: & $21.3\\arcmin\\times21.3\\arcmin$ \\\\\nGain: &2.2-2.4 $e^-$\/ADU \\\\\nRead Out Noise: &5 $e^-$ rms \\\\ \\hline\n\\end{tabular} \\\\\n\\noindent\n\\end{table}\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=5cm , angle=0]{fig5a.eps}\n\\includegraphics[width=5cm , angle=0]{fig5b.eps}\n\\includegraphics[width=5cm , angle=0]{fig5c.eps}\n\\includegraphics[width=5cm , angle=0]{fig5d.eps}\n\\includegraphics[width=5cm , angle=0]{fig5e.eps}\n\\includegraphics[width=5cm , angle=0]{fig5f.eps}\n\\includegraphics[width=5cm , angle=0]{fig5g.eps}\n\\includegraphics[width=5cm , angle=0]{fig5h.eps}\n\\includegraphics[width=5cm , angle=0]{fig5i.eps}\n\\includegraphics[width=5cm , angle=0]{fig5j.eps}\n\\includegraphics[width=5cm , angle=0]{fig5k.eps}\n\\includegraphics[width=5cm , angle=0]{fig5l.eps}\n\\includegraphics[width=5cm , angle=0]{fig5m.eps}\n\\caption{Colour index versus magnitude diagrams for BL Lacertae. The solid line is the best fit on the observations.}\n \\end{figure*}\n\n\n\\begin{table}\n\\caption{ Results of linear fits to Colour-Flux Diagrams.}\n\\setlength{\\tabcolsep}{0.03in}\n\\begin{tabular}{lccccc} \\hline \\hline\n\nDate of &Band &r &p &c$\\pm$$\\Delta$c &m$\\pm$$\\Delta$m \\\\ \nObservation & & & &Intercept &Slope \\\\\\hline\n06.07.2010 &(B-V) &-0.138 &0.339 & 1.038$\\pm$0.446 &-0.032$\\pm$0.032 \\\\\n &(V-R) &0.315 &0.026 &-1.055$\\pm$0.675 & 0.110$\\pm$0.048 \\\\\n &(R-I) &0.023 &0.873 & 0.459$\\pm$0.675 &0.008$\\pm$0.048 \\\\\n &(B-I) &0.316 &0.026 & 0.442$\\pm$0.534 &0.087$\\pm$0.038 \\\\\n08.07.2010 &(B-V) &-0.106 &0.537 & 1.500$\\pm$1.455 &-0.066$\\pm$0.105 \\\\\n &(V-R) &0.205 &0.230 &-1.389$\\pm$1.540 &0.136$\\pm$0.111 \\\\\n &(R-I) &0.110 &0.545 &-0.626$\\pm$1.836 &0.085$\\pm$0.133 \\\\\n &(B-I) &0.220 &0.198 &-0.516$\\pm$1.643 &0.156$\\pm$0.119 \\\\\n17.07.2011 &(B-V)* &0.629 &0.003 &-2.424$\\pm$0.879 &0.234$\\pm$0.068 \\\\\n &(V-R) &0.512 &0.021 &-1.150$\\pm$0.627 &0.123$\\pm$0.049 \\\\\n07.08.2011 &(V-R) &0.193 &0.415 &-0.434$\\pm$1.058 &0.067$\\pm$0.081 \\\\\n23.08.2011 &(V-R)* &0.714 &$<$0.001 &-3.423$\\pm$0.670 &0.289$\\pm$0.050 \\\\\n &(R-I) &0.280 &0.109 &-0.658$\\pm$0.779 &0.096$\\pm$0.058 \\\\\n24.08.2011 &(B-V) &0.057 &0.788 &0.113$\\pm$1.910 &0.038$\\pm$0.141 \\\\\n &(V-R) &0.181 &0.388 &-0.805$\\pm$1.446 &0.094$\\pm$0.107 \\\\\n &(R-I) &-0.022 &0.917 &0.787$\\pm$1.219 &-0.009$\\pm$0.090 \\\\\n &(B-I) &0.264 &0.202 &0.095$\\pm$1.269 &0.123$\\pm$0.094 \\\\\n25.08.2011 &(B-V) &0.087 &0.673 &0.235$\\pm$0.836 &0.027$\\pm$0.063 \\\\\n &(V-R) &0.345 &0.118 &-0.162$\\pm$0.366 &0.045$\\pm$0.028 \\\\\n &(R-I) &0.251 &0.217 &0.051$\\pm$0.439 &0.042$\\pm$0.033 \\\\\n &(B-I) &0.325 &0.105 &0.125$\\pm$0.894 &0.114$\\pm$0.067 \\\\\n06.07.2012 &(V-R)* &0.530 &0.003 &-1.643$\\pm$0.641 &0.159$\\pm$0.048 \\\\\n &(R-I)* &0.506 &0.004 &-2.429$\\pm$0.994 &0.231$\\pm$0.074 \\\\\n10.07.2012 &(B-V) &-0.084 &0.703 &1.019$\\pm$0.956 &-0.027$\\pm$0.071 \\\\\n &(V-R) &0.061 &0.781 & 0.332$\\pm$0.550 &0.011$\\pm$0.040 \\\\\n &(R-I) &0.487 &0.019 &-2.201$\\pm$1.125 &0.212$\\pm$0.083 \\\\\n &(B-I) &0.395 &0.062 &-0.850$\\pm$1.347 & 0.196$\\pm$0.099 \\\\\n12.08.2012 &(B-V) &0.310 &0.243 &-2.208$\\pm$2.320 &0.205$\\pm$0.168 \\\\\n &(V-R) &0.101 &0.709 &0.052$\\pm$1.131 &0.031$\\pm$0.082 \\\\\n &(R-I) &0.581 &0.011 &-1.618$\\pm$0.861 &0.166$\\pm$0.062 \\\\\n &(B-I) &0.517 &0.040 &-3.774$\\pm$2.458 &0.402$\\pm$0.178 \\\\\n17.10.2012 &(B-V) &-0.103 &0.716 &1.370$\\pm$1.927 &-0.051$\\pm$0.136 \\\\\n &(V-R)* &0.674 &0.006 &-1.526$\\pm$0.614 &0.142$\\pm$0.043 \\\\\n &(R-I)* &0.668 &0.006 &-1.945$\\pm$0.816 &0.186$\\pm$0.058 \\\\\n &(B-I) &0.542 &0.037 &-2.102$\\pm$1.694 &0.278$\\pm$0.120 \\\\\n22.10.2012 &(R-I)* &0.482 &0.006 &-2.203$\\pm$0.970 & 0.210$\\pm$0.071 \\\\\n23.10.2012 &(V-R) &0.217 &0.233 &-1.919$\\pm$1.973 & 0.170$\\pm$0.140 \\\\\n &(R-I) &0.163 &0.373 &-1.262$\\pm$2.147 & 0.137$\\pm$0.152 \\\\ \\hline\n\n\n\n\\end{tabular} \\\\\n$ $*: Significant variations are found in these observations. \\\\\nr \\& p: Pearson Correlation Coefficient and its probability values respectively. \\\\\n\\end{table}\n\n\\section{Discussion}\n\nWe performed photometric monitoring of BL Lacertae during the period 2010--2012 for a total of 38 nights \nin the B, V, R and I bands in order to study its flux and spectral variability. In 19 of those nights, we found \ngenuine IDV. The light curves often show gradual rises and decays, sometimes with smaller \nsub-flares superimposed. No evidence for periodicity or other characteristic time scales was found. \n We find the duty cycle of the source during this period to be $\\sim$44\\%.\nIn the earlier studies, it has been found that LBLs display stronger IDV than HBLs (high frequency peaked blazars)\nand the DC has been estimated to be $\\sim$70\\% for LBLs and $\\sim$30--50\\% for HBLs (Heidt \\& Wagner 1998; Romero et al. 2002;\nGopal-Krishna et al. 2003). Gopal-Krishna et al. (2011) studied a large sample of blazars and found that if variability\namplitude (Amp) $>$3\\% is considered, the DC is 22\\% for HBLs and 50\\% for LBLs. We found DC of $\\sim$44\\% for BL Lacertae \n(which is a well known LBL) is in accordance with the previous studies.\n\nIn the literature, there are various models which explains intra-day variability of blazars. Intrinsic ones focus on the\n evolution of the electron energy density distribution of the relativistic paricles leading to a variable synchrotron\nemission, with shocks accelerating turbulent particles in the plasma jet which then cools by synchrotron emission (e.g., \nMarscher, Gear \\& Travis 1992; Marscher 2014; Calafut \\& Wiita 2015). \nExtrinsic ones involve geometrical effects like swinging jets where the path of the relativistic moving blobs along the jet\ndeviated slightly from the line of sight, leading to a variable Doppler factor (e.g., Gopal-Krishna \\& Wiita 1992). The long term \nperiodic and acromatic BL Lacertae variability may be mostly explained by the geometrical scenarios where viewing angle\n variation can be due to the rotation of an inhomogeneous helical jet which causes variable Doppler boosting of the\n corresponding radiation (Villata et al 2002; Larionov et al. 2010 and references therein). As we are considering the \nfaster intra-night flux\nvariations that are associated with the colour variations, they are more likely to be associated with models\ninvolving shock propagating in a turbulent plasma jet. \n\nWhen variability is clearly detected, its amplitude is usually greater at higher frequencies, which is consistent with previous studies\n(Papadakis et al.\\ 2003; Hu et al.\\ 2006) and can be well explained by electrons that are accelerated at the shock front and then \nlose energy as they move away from the front. Higher-energy electrons lose energy faster through the production of synchrotron radiation, \n and are produced in a thin layer behind the shock front. In contrast, the lower-frequency emission is spread out over a larger volume \nbehind the shock front (Marscher \\& Gear 1985). This leads to time lags of the peak of the light curves toward lower frequencies and\namplitude of variability higher at higher frequencies which is clearly visible in the multi-frequency blazars light curves. \nDue to the closeness of the various optical bands, the starting time of a flare should be almost the same and hence \non short time-scales, it is difficult to detect the time lags between the optical bands.\nAlthough the amplitude of variability is an inherent property of the source, we had to examine whether\nit has any dependence on the duration of observation, and found a\nsignificant positive correlations between the observed amplitude of variability of the light curves and the duration of the observations.\nAs noted by Gupta \\& Joshi (2005), on intra-day\ntimescales, the probability of seeing a significant intra-day variability generally increases if the source is continuously observed for\nlong durations. In our observations, duration of monitoring varies between 1.5--5.8 hours. Also, we found that the \namplitude of variability decreases as the source flux increases which can be explained as the source flux increases, the irregularities\nin the turbulent jet (Marscher 2014) decrease and the jet flow becomes more uniform leading to a decrease in amplitude of variability. \n\nWe searched for the possible correlations between colour versus magnitude and found significant positive correlations\nbetween them in five of the observations out of total 13. So, BL Lacertae showed significant bluer-when-brighter trends\non these night with different regression slopes (in Table 3). This behaviour is very well known for BL Lacertaes \nand can be interpreted as resulting from rapid, impulsive injection\/acceleration of relativistic electrons, followed by\nsubsequent radiative cooling (e.g., B{\\\"o}ttcher \\& Chiang 2002). However, other observations do not show significant\n linear correlations and show complicated behaviour on colour-magnitude diagrams i.e different slopes according to the\ndifferent flux states or nearly zero slopes between colour-magnitude (Table 3).\nOf course it is possible that superposition of different spectral slopes from many variable components (standing shocks in\ndifferent parts of the jet) could lead to the overall weakening of the colour-magnitude correlations (Bonning et al.\\ 2012).\n\nHence, the behaviour of the colour--magnitude diagrams provides us with indirect information on the amplitude \ndifference and time-lags between these bands as Dai et al.\\ (2011) performed simulations to confirm that both the \namplitude differences and time delays between variations at different wavelengths result in \na hardening of the spectrum during the flare rise. They showed that if there is a difference in amplitude \nin two light curves, it leads to the evolution of the object along a diagonal path in the colour-magnitude diagram. \nIf there is a time-lag along with the amplitude change, counter-clockwise loop patterns on the colour index--magnitude \ndiagram arise (Dai et al.\\ 2011). However, any such lags are probably shorter than the sampling time of \nour observations, so we are not able to detect them through the DCFs we computed. \n\n\\section{Conclusions}\n \nOur conclusions are summarized as follows: \n\n\\begin{itemize}\n\\item During our observations in 2010-2012 BL Lacertae was highly variable in B, V, R and I bands. \nThe variations were well correlated in all four bands and were very smooth, with gradual rises\/decays.\nIn one of the observations, on 6 October 2010, we found a pronounced flare-like event, and the highest variability amplitude \nis found in the V band at $38$ per cent.\n\\item In the cases with significant variability, the amplitude of variability is highest in the highest energy band. \n\\item The amplitude of variability correlates positively with the duration of the observation and\n decreases as the flux of the source increases. \n\\item We searched for time delays between the B, V, R and I bands in our observations, but we did not find any\nsignificant lags. This implies that the variations are almost simultaneous in all of the bands and any time lags, if present, \nare less than our data sampling interval of $\\sim$8 minutes. \n\\item The flux variations are associated with spectral variations on intra-day time-scales. In 5 of the 13 observations, the\noptical spectrum showed the overall bluer-when-brighter trend which could well represent highly variable jet emission.\n\\item The colour vs.\\ magnitude diagrams show different behaviours which could represent the contribution of different variable \ncomponents during the flaring states.\n\\item We conclude that the acceleration and cooling timescales\nare very short for these optical variations and hence dense optical observations with even shorter cadence and higher sensitivity are required\nto better characterize them. \n\\end{itemize}\n \n\nWe thank the referee for useful and constructive comments.\nThis research was partially supported by Scientific Research Fund of the Bulgarian Ministry of Education and Sciences under \ngrant DO 02-137 (BIn 13\/09). The Skinakas Observatory is a collaborative project of the University of Crete, the Foundation \nfor Research and Technology -- Hellas, and the Max-Planck-Institut f\\\"ur Extraterrestrische Physik. \nH.G. is sponsored by the Chinese Academy of Sciences Visiting Fellowship for Researchers from Developing Countries\n (grant No. 2014FFJB0005), and supported by the NSFC Research Fund for International Young Scientists (grant No. 11450110398).\nACG is partially supported by the Chinese Academy of Sciences Visiting Fellowship for Researchers\nfrom Developing Countries (grant no. 2014FFJA0004).\nMB acknowledges support by the South African Department of Science and Technology through the National Research Foundation \nunder NRF SARChI Chair grant no. 64789. MFG acknowledges support from the National Science Foundation\nof China (grant 11473054) and the Science and Technology Commission of Shanghai Municipality (14ZR1447100).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Free energy of a binary mixture on a membrane}\n\\subsection{Free energy for single species membranes}\nFor a membrane comprising of a single specie, the free energy of a given shape $S$ can be described via the Helfrich hamiltonian \\cite{Lipowsky1991}:\n\\begin{align}\nH_{el} = \\int_{S} d\\mu_S \\: [\\kappa_m (C- C_{sp})^2 + \\kappa_g G], \\tag{SM1} \\label{Hel1} \n\\end{align}where $C \\equiv (c_1 + c_2)\/2$ is the local mean curvature, $G = c_1 c_2$ is the local gaussian curvature, $c_{1,2}$ are the local principal curvatures, $C_{sp}$ is the intrinsic mean curvature of the surface and $\\kappa_{m,g}$ are the mean and gaussian bending rigidities respectively. Furthermore, the Gauss-Bonnet theorem states that \\cite{Kamien02}:\n\\begin{equation}\n\\int_S d\\mu_S \\:G + \\oint_{\\partial S} dl \\:g_c = 2\\pi \\chi(S), \\tag{SM2 }\\label{Hel2}\n\\end{equation}where $\\partial S$ stands for the boundary of $S$, $g_c$ for the local geodesic curvature of the boundary \\cite{Kamien02} and $\\chi(S)$ for the Euler characteristic of $S$. The key consequence is that, if the boundary and the topologies of the problem are fixed, minimizing Eq. \\eqref{Hel1} is equivalent to minimizing only the integral over the mean curvature $C$. In the absence of intrinsic curvature, the lowest energy solutions correspond to surfaces with exactly $C = 0$ at all points: such surfaces are called {\\it minimal} surfaces \\cite{MinimalSurfaces}. \nHere we focus on the triply periodic surfaces which are known to be formed by lipid mixtures in water \\cite{Templer1998,Angelov2003,Tyler2015}.\nWe use standard notations P, D, and G for the {\\it primitive}, {\\it Diamond}, and {\\it Gyroid} surfaces respectively. Since they are periodic, we characterize their topology with their Euler characteristic per unit cell ({\\it e.g.} in Eq. \\eqref{Hel2}). \n\n\\subsection{Free energy for a binary mixture on a minimal surface}\nIf instead of a single species, the membrane comprises two different species, then different allowed mixture configurations may have different topologies and one cannot disregard anymore the gaussian curvature contribution to the energy. The simplest extension of Eq. \\eqref{Hel1} to a binary mixture would then read:\n\\begin{align}\n\\tilde{H}^S_{el}(f_A) = \\int_{S} d\\mu_S(x) \\: [\\kappa_g^A \\sigma_A(x) + \\kappa_g^B(1-\\sigma_A(x))]G(x) \\tag{SM3} \\label{Hel3}\n\\end{align}where $x$ denotes a point on $S$, $d\\mu_S(x)$ is the area measure on $S$ at $x$, $G(x)$ is the gaussian curvature at point $x$, $f_A$ is the imposed area fraction of species $A$ (with $f_B = 1-f_A$), the field $\\sigma_{A}(x) \\in [0,1]$ is the {\\it mean} occupation number of species $A$ at $x$ and $\\kappa_g^{A,B}$ are the gaussian bending rigidities associated to the species $A$ and $B$ respectively. We then choose the free energy $\\tilde{H}^S_{el}(f_A = 0)$ as reference so that, in practice, we look at the free energy $H^S_{el}(f_A ) \\equiv \\tilde{H}^S_{el}(f_A) - \\tilde{H}^S_{el}(f_A = 0)$ which yields:\n\\begin{align}\nH^S_{el}(f_A) = \\delta \\kappa \\int_{S} d\\mu_S(x) \\: \\sigma_A(x) G(x) \\tag{SM4} \\label{Hel4}\n\\end{align}where $\\delta \\kappa = \\kappa_g^A - \\kappa_g^B$. Eq. \\eqref{Hel4} is the starting point of our study.\n\n\n\\section{Weierstrass-Enneper representation}\n\\subsection{General formulation}\nIt can be shown that, locally, any minimal surface can be conformally mapped onto the complex plane via the Weierstrass-Enneper (W-E) representation \\cite{MinimalSurfaces}. More precisely, the W-E is a map from $\\mathbb{C}$ to $\\mathbb{R}^3$ which, to a point $(u,v)$ in an open subset of $\\mathbb{C}$, uniquely associates a point $(x(u,v), y(u,v), z(u,v))$ of $\\mathbb{R}^3$ that belongs to a minimal surface via:\n\n\\begin{align}\n&& x(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:f(w)(1-g(w)^2) \\right\\rbrace \\tag{SM5}\\label{WE1} \\\\\n&& y(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:if(w)(1+g(w)^2) \\right\\rbrace \\tag{SM6}\\label{WE2} \\\\\n&& z(u,v) = \\Re \\left\\lbrace \\int_{w_0}^{u+iv} dw \\:2f(w)g(w) \\right\\rbrace \\tag{SM7}\\label{WE3} \n\\end{align}where $g$ is a holomorphic function and $f$ is a meromorphic function such that $fg^2$ is analytic.\n\n\\vspace{2mm}\n\nThe gaussian curvature $G(x,y,z)$ at any point of a surface represented by Eqs. \\eqref{WE1}, \\eqref{WE2} and \\eqref{WE3} can be expressed as a function of $w=u+iv$ via:\n\n\\begin{equation}\nG(w) = - \\left(\\frac{4|g'(w)|}{|f(w)|(1+|g(w)|^2)^2} \\right)^2 . \\tag{SM8}\\label{WE4} \n\\end{equation}\nThe negative sign is characteristic of minimal surfaces. Since the mean curvature $(c_1 + c_2)\/2 $ is zero everywhere, it implies that the gaussian curvature is always negative or zero. The W-E being a conformal map, it preserves the angles. Distances, however, are not conserved when mapping an infinitesimal segment from $\\mathbb{C}$ to $\\mathbb{R}^3$ and are scaled by a factor $\\Lambda(w)$ given by\n\\begin{equation}\n\\Lambda(w) = \\frac{|f(w)|(1+|g(w)|^2)}{2} \\tag{SM9} \\label{WE5}\n\\end{equation} \n\n\\subsection{Triply periodic Schwartz surfaces}\nThe three Schwartz surfaces {\\it P}, {\\it D} and {\\it G} can be obtained by choosing:\n\\begin{align}\n&& g(w) = w \\tag{SM10}\\label{WE6} \\\\\n&& f(w) = \\frac{e^{i \\theta_B}}{\\sqrt{w^8-14w^4+1}} \\tag{SM11}\\label{WE6bis}\n\\end{align}where $\\theta_B$ is the Bonnet angle such that $\\theta_B = 0 $ for the {\\it D} surface, $\\theta_B = \\pi\/2$ for the {\\it P} surface and $\\theta_B = \\mathrm{cotan}(K(1\/4)\/K(3\/4))$ for the {\\it G} surface. $K$ is a complete elliptic integral of the first kind. Since $e^{i \\theta_B}$ only changes the phase in the W-E representation, this means that the {\\it P}, {\\it D} and {\\it G} surfaces are simply related by an isometry called the Bonnet transformation and share many of their physical properties. \n\n\n\\subsection{Independence of the free energy on the member of the Bonnet family}\nIn particular, having the W-E map in mind, the whole integral in Eq. \\eqref{Hel4} can be thought of as an integral in the complex plane. Using the fact that $d\\mu_S(x(w)) =\\Lambda^2(w)dudv$, the Eq. \\eqref{Hel4} can be recast as:\n\\begin{equation}\nH^S_{el}(f_A) = \\delta \\kappa \\int_{\\mathcal{A}(S)} dudv \\: \\Lambda^2(w) \\: \\sigma_A(w) G(w) \\tag{SM12}\\label{WE8} \n\\end{equation}where $\\mathcal{A}(S)$ denotes the atlas used in $\\mathbb{C}$ to characterize $S$. It is worth noting that in the integrand of Eq. \\eqref{WE8}, the W-E functions $f$ and $g$ only appear via their complex modulus and therefore their contribution to the curvature energy would be unchanged by a phase factor. Thus, we concludethat the curvature energy of a binary mixture on a minimal surface is independent of which member of the Bonnet family is considered and, in particular, so is its ground state. In a similar fashion, a continuous model of the line tension contribution would read formally:\n\\begin{equation}\nH_{A-B}(f_A) = J \\int_{\\mathcal{I}(A-B)} d\\mu_l(x) \\tag{SM13}\\label{WE9}\n\\end{equation}where $\\mathcal{I}(A-B)$ denotes the set of points belonging to the $A-B$ interface on $S$ and $d\\mu_l(x)$ the length measure on $S$. Again, the integral can be thought as an integral on $\\mathbb{C}$ by virtue of the W-E map. Furthermore, the length measure on $S$ can be expressed in term of the length measure in $\\mathbb{C}$ via $d\\mu_l(x(w)) = \\Lambda(w) |dw|$. Eq. \\eqref{WE9} can thus be rewritten as:\n\\begin{equation}\nH_{A-B}(f_A) = J \\int_{WE^{-1}(\\mathcal{I}(A-B))} |dw| \\: \\Lambda(w) . \\tag{SM14}\\label{WE10}\n\\end{equation}\n\\begin{figure}\n\n \\vspace{2mm}\n \\includegraphics[width=0.49\\columnwidth]{SMFig1a.jpg} \n \\includegraphics[width=0.49\\columnwidth]{SMFig1b.jpg}\\\\\n \\includegraphics[width=0.49\\columnwidth]{SMFig1c.jpg} \n \\includegraphics[width=0.49\\columnwidth]{SMFig1d.jpg} \n \n \\caption{\\label{fig1} {\\it Constructing the P-surface}. Top left: complex domain of the fundamental patch. Top right: fundamental patch in three dimensions. Bottom left: hexagonal patch ($\\Sigma$ in the main article) made of 12 fundamental patches. Bottom right: Full P-surface per cubic unit cell $S$ made of 8 patches $\\Sigma$. }\n\\end{figure}\n\nAs before, the W-E functions $f$ and $g$ only contribute to the integral via their complex modulus and therefore $H_{A-B}$ is independent of the member of the Bonnet family under study.\nAs a consequence, the phenomenology of symmetry breaking and bridging-induced reentrant behaviour depicted in the phase diagrams of the P-surface in Fig. 2 of the main text hold in fact for all three Bonnet surfaces. The details of the phase diagram may slightly differ, however, as the cubic cells of the {\\it G} and {\\it D} surfaces do not contain the same surface area as the {\\it P} surface. This is outside the scope of this letter, and we will discuss these details in a separate publication.\n\n\\subsection{Angle of intersection between two geodesics}\nThe Euler characteristic of a compact domain $\\mathcal{D}$ of $A$ lipids that is not bridged to another domain on a neighbouring patch is $1$. Applying the Gauss-Bonnet theorem (Eq. \\eqref{Hel2}) to such a domain gives:\n\\begin{equation}\n\\int_{\\mathcal{D}} d\\mu_S(x) \\: G(x) + \\oint_{\\partial D} dl \\: g_c(x) = 2\\pi . \\tag{SM15} \\label{GB1} \n\\end{equation}Up to a factor the first term of the {\\it l.h.s} of Eq. \\eqref{GB1} is simply the curvature energy of the domain that we can denote $H^{\\Sigma_i}_{el}(f_i)$ in referring to notations introduced in the main text.\nBy using the fact that a patch has a 6-fold rotation symmetry, we can split the boundary integral of the geodesic curvature into 6 equivalent parts. If each piece of the boundary is almost everywhere a geodesic, then the second term of the {\\it l.h.s} of Eq. \\eqref{GB1} has zero integrand everywhere except at points where the geodesics meet. The total value of the contour integral becomes simply a sum over intersection angles that we call $\\theta$. We thus have:\n\\begin{equation}\n\\theta = \\frac{\\pi}{3} + \\frac{H^{\\Sigma_i}_{el}(f_A^i)}{6 |\\delta \\kappa|}. \\tag{SM16} \\label{GB2}\n\\end{equation}Finally, the actual interior angle $\\gamma$ between two geodesics making the hexagonal-like facetted domain is in fact the complementary angle of $\\theta$ and reads:\n\\begin{equation}\n\\gamma = \\frac{2\\pi}{3} - \\frac{H^{\\Sigma_i}_{el}(f_A^i)}{6 |\\delta \\kappa|}. \\tag{SM17} \\label{GB3}\n\\end{equation}\nNote that in the case where the curvature energy vanishes but the symmetry is still imposed, we retrieve the interior angle of a planar hexagon as expected.\n\n\\section{Numerical modelling of the P-surface}\n\\subsection{Fundamental patch}\nAll well behaved minimal surfaces admit a description in terms of a fundamental patch in $\\mathbb{R}^3$ that is repeated by using the symmetries of the surface. By the W-E representation, this fundamental patch is associated to a fundamental domain of the complex plane. For the {\\it P}, {\\it D} and {\\it G} surfaces, the fundamental domain is the set of complex points with positive real part bounded by the lines along the vectors $(1+i)\/\\sqrt{2}$ and $1$ and by the circle of radius $\\sqrt{2}$ whose center is located at the point $-(1+i)\/\\sqrt{2}$. \n\nThe top left of Fig. \\ref{fig1} shows the fundamental domain in $\\mathbb{C}$. The triangular tessellation is obtained by using the {\\it Surface Evolver} package \\cite{Brakke} and the images plus the management of the network structure have been performed with the {\\it Mathematica} software \\cite{Mathematica}. For the sake of illustration, Fig. \\ref{fig1} shows a coarse tessellation of the fundamental domain. The tessellations we used in the paper are typically 100 times finer. By using Eqs. \\eqref{WE1}-\\eqref{WE3} and \\eqref{WE6} and \\eqref{WE6bis}, we get a three dimensional realization of the fundamental patch that is represented in the top right of Fig. \\ref{fig1}. Then, following Ref. \\cite{Gandy-P}, we can generate first a full hexagonal patch of the P-surface ($\\Sigma$) by replicating and stitching together 12 fundamental patches as seen in the bottom left of Fig. \\ref{fig1}. The full cubic cell representation of the surface $S$ is then obtained by combining 8 such hexagonal patches with the right symmetry operations as illustrated on the bottom right of Fig. \\ref{fig1}. A similar procedure, albeit with different arrangements of the fundamental patches, can also be carried out for the D- and G-surfaces. \n\n\\subsection{Monte Carlo simulations}\nAs emphasized in Eq. \\eqref{WE8}, the curvature energy can be recast in terms of a sum over points on a euclidean (complex) plane of a curvature field that multiplies a scaling field. Moreover, the discretized surface $S$ on the bottom right of Fig. \\ref{fig1} is made of a network of cells $\\mathcal{N}(S)$ which are either an original version or a replica of a cell in the fundamental patch. Thus, the whole set of values of the curvature and scaling fields on the whole network is determined solely by that of the sub-network of cells in the fundamental patch. The particular topology of the P-surface (of genus 3 in a cubic cell) is then accounted for by the topology of the network {\\it i.e.} by assigning the right neighbours to each cell. If we add a species field $s$ into the picture such that $s_i= 1$ if cell $i \\:\\in \\: \\mathcal{N}(S)$ contains species $A$ and $s_i = 0$ otherwise, then the whole problem becomes that of paramagnetic spins on a network subject to an effective node-dependent magnetic field whose magnitude is $G(w)\\Lambda^2(w)\\Delta(w)$ and where $\\Delta(w)$ denotes the euclidean area of the triangular unit at point $w$ in the complex plane. Since the effective magnetic field is non uniform and non trivial, there is no simple explicit analytical expression for the thermodynamically favoured composition morphologies. By splitting the system into 8 equivalent patches, we could however suggest, as discussed in the main text, what would happen at low enough temperatures. In particular, a first approximation scheme neglecting the effect of curvature on the area measure and Taylor expanding the curvature field about its zero point $p_i$ suggested that the curvature energy $H^{\\Sigma_i}_{el}(f_A^i)$ in a patch $\\Sigma_i$, $i=1..8$, with an area fraction $f_A^i$ of $A$ lipids would go as $H^{\\Sigma_i}_{el}(f_A^i) \\sim (f_A^i)^{\\alpha}$ with $\\alpha = 2$. To test numerically this proposition, we performed Monte Carlo (MC) simulations of an Ising system whereby: \n\\begin{enumerate}\n\\item The total number of spins\/cells is fixed,\n\\item The total number of spins of value 1 is fixed,\n\\item A MC move consists then in:\n\\begin{enumerate}\n\\item picking at random a cell among those which have $s=1$, \n\\item picking at random a cell among those which have $s=0$,\n\\item swap the cells spin values,\n\\item accept the move with a probability satisfying the Metropolis criterion \\cite{Metropolis} $p_{acc} = \\min[1, e^{-\\beta \\Delta E}]$, where $\\Delta E = E_{final}-E_{initial}$ and the energy is in general given by Eqs. (4) and (5) of the main text article.\n\\end{enumerate}\n\\end{enumerate}\nIf we set the Ising parameter $J$ to zero, we can then probe the low energy curvature energy as a function of domain size for a single patch.\n\\begin{figure}\n\n \\vspace{2mm}\n \\includegraphics[width=0.90\\columnwidth]{SMFig2bis.jpeg} \n \n \\caption{\\label{fig2} {\\it Curvature energy as a function of domain size}. The red triangles are MC data points for the energy $H^{\\Sigma_i}_{el}(f_A^i)$ of an $A$ domain on a patch $\\Sigma_i$. The lines correspond to the best fit to these data points with a scaling law behaviour $H^{\\Sigma_i}_{el}(f_A^i) \\sim (f_A^i)^{\\alpha}$. The solid red line corresponds to the best fit exponent value $\\alpha \\approx 1.83$, while the green dashed line is the best fit to the data with imposed $\\alpha = 2$ which shows quite good agreement with the data.}\n\\end{figure}As we see in Fig. \\ref{fig2}, the proposition that the curvature energy goes as a power low is in very good agreement with MC data. In addition, the approximation that $\\alpha = 2$ is in remarkably quite good agreement with the data.\n\n\\section{Validity of the minimal surface assumption}\nIn the current study, it is assumed that the underlying surface remains minimal during segregation of the lipids\/species. However, in general domain formation can induce local membrane deformation which in turn can destabilise the entire morphology of the membrane itself. In this section we will have a closer look at how phase separation induced bud formation may affect the conclusions drawn in the manuscript.\n\n\\subsection{Budding on flat multicomponent membranes}\nBased on the work by Lipowsky \\cite{Lipowsky1992} on flat membranes, domain formation may lead to a budding phenomenon driven by the line tension between the two coexisting demixed phases. Budding occurs when the line tension energy cost overcomes the bending energy penalty. \n\nConsider a membrane domain of area $A = \\pi L^2 = 2\\pi R^2 (1-\\cos\\theta)$, where $R$ is the radius of curvature of the deformed membrane domain, $\\theta$ is the contact angle of the domain with respect to the horizontal plane, and\n$L$ is the domain radius if there is no deformation to the membrane. \nWe will now consider the competition between two energy terms: a) line tension energy that increases with the perimeter of the domain, $2\\pi J R \\sin \\theta$; and (ii) bending energy which depends on the domain area and curvature, $2 \\kappa_m A \/ R^2$.\nHere $J$ is the line tension and $\\kappa_m$ is the mean curvature bending rigidity. We have also assumed that there is no mean spontaneous curvature. \n\nFig. \\ref{fig7} shows the total energy (normalised by the line tension energy for a flat domain) as a function of the reduced membrane mean curvature $L\/R $ that plays the role of an order parameter. $L\/R = 0$ corresponds to a flat membrane, i.e. no deformation. $L\/R = 2$ corresponds to a complete bud formation. As we see, there are three regimes depending on the membrane domain size $L$: (i) For $0 < L < L^*= 4\\kappa_m\/J $, budding is unfavourable and $L\/R = 0$ is the global minimum configuration; (ii) For $ L^* \\le L < L^o = 8\\kappa_m\/J $, budding is favourable but there is an energy barrier for its formation; (iii) Finally, only for $ L \\ge L^o$ that the energy barrier for bud formation disappears. Here the flat membrane geometry is completely unstable. \n\n\n\\vspace{2mm}\n\\begin{figure}[h]\n\n \\vspace{2mm}\n \\includegraphics[width=0.99\\columnwidth]{SMFig7bis.jpeg} \n \\caption{\\label{fig7} {\\it Sum of the line tension and curvature energies of a membrane domain}. Normalised energy curves as a function of reduced curvature $L\/R$ for given values of the domain size $L$. From top to bottom the domain size $L$ is increased from $L^*\/2$ to $L^o$. }\n\\end{figure}\n\n\\subsection{Bud formation on a P-surface}\nSince the P-surface is a minimal surface with zero mean curvature everywhere, we will continue to assume that the spontaneous mean curvature is zero, as in the previous paragraph.\nThe presence of non-uniform gaussian curvature on the P-surface may alter the numerical prefactors in $L^*$ and $L^o$. \nHowever, since the domain formation occurs in the neighbourhood of very specific points on the P-surface {\\it i.e.} 8 zero-curvature points each at the centre of a patch in the cubic cell, to first approximation, it is reasonable to assume that the above results from \\cite{Lipowsky1992} to hold. As a result, we can approximate the two critical domain sizes as $L^* = 4\\kappa_m\/J$ and $L^o = 8 \\kappa_m\/J$.\nCorrespondingly, when $L < L^*$, we expect our assumption that the underlying surface remains minimal during phase separation to hold. Strong deviations to the results presented in the main text are only expected when $L \\ge L^*$.\n\n\\begin{figure}[h]\n\n \\vspace{2mm}\n \\includegraphics[width=0.99\\columnwidth]{SMFig8.jpeg} \n \n \\caption{\\label{fig8} Modification of the phase diagram of Fig. 2(a) in the main manuscript resulting from budding instability for $\\kappa_m\/|\\delta \\kappa| = 1\/4$. The diagonally hashed region corresponds to a region where budding is preferable but there is an energy barrier for its formation, while the horizontally hashed region corresponds to a fully unstable bud formation. }\n\\end{figure}\n\nTo see which part of the phase diagram in Fig. 2 of the main manuscript is affected by the budding instability, we relate the area fraction $f_A$ occupied by the A-lipids to the size $L$ of the domains depending on which configuration $\\binom 8 k $ they are in. For the purpose of this analysis, we will focus on cases where there are no bridge formations between lipid $A$ domains. The total area occupied by the A-lipids on a cubic cell of the P-surface is $f_A \\mu_S(S)$. If the A-lipids are partitioned in $k$ patches among the 8 available, then the typical area per domain is $f_A \\mu_S(S)\/k \\approx \\pi L^2$. It follows that requiring full stability against budding ($L < L^*$) is equivalent to requiring $f_A \\mu_S(S)\/k < \\pi 16 (\\kappa_m\/J)^2$. If the cubic cell bounding the P-surface has sides of length $\\ell$, then $\\mu_S(S) = 24 \\ell^2 K(1\/4)\/K(3\/4)$, where $K(x)$ is the complete elliptic integral of the first kind \\cite{Gandy-P}. Thus, the stability criterion becomes\n\\begin{equation}\nf_A^*\\left(\\frac{J \\ell}{|\\delta \\kappa|}; k\\right) = \\left( \\frac{\\kappa_m}{|\\delta \\kappa|} \\right)^2 \\frac{2 k \\pi K(3\/4)}{3K(1\/4)} \\left(\\frac{|\\delta \\kappa|}{J \\ell}\\right)^2 \\tag{SM21} \\label{criticallines1}\n\\end{equation}\nfor configuration $\\binom 8 k $.\nAs we see in Eq. \\eqref{criticallines1}, the set of stability lines depends on the ratio $\\kappa_m \/|\\delta \\kappa|$ which constitutes an additional parameter in our modelling. We have observed that for values $\\kappa_m\/|\\delta \\kappa| > 0.4$, there is no intersection between any of the stability lines and the lipid repartition phases they correspond to within the parameter ranges of the present study, $0 \\le J \\ell \/ |\\delta \\kappa| \\le 2$. \n\nIt is worth emphasizing that experimental and numerical estimates of the ratio $\\kappa_m\/|\\delta \\kappa|$ for various lipid systems \\cite{Hu12} show that it is rarely below 1. As an example, consider the mixture of DOPC, sphingomyelin, and cholesterol forming coexisting $L_o$ and $L_d$ domains reported in references \\cite{Semrau,Baumgart}. Here the difference in Gaussian bending moduli $|\\delta \\kappa| \\simeq 3 \\times 10^{-19}$ J and the mean curvature bending modulus for the $L_o$ phase $\\kappa_m \\simeq 8 \\times 10^{-19}$ J, which gives us $\\kappa_m\/|\\delta \\kappa| \\simeq 8\/3$. Even if we use the mean curvature bending modulus for the $L_d$ phase $\\kappa_m \\simeq 2 \\times 10^{-19}$ J, which may be more appropriate when bridges are formed such that the lipid B domains are now surrounded by A (see Fig. 2(b) and (c) in the main text), we still have $\\kappa_m\/|\\delta \\kappa| \\simeq 2\/3 > 0.4$. Thus, this strongly suggests that our conclusions in the main text will hold even when taking into account segregation induced membrane deformation.\n\nWe can redo a similar calculation to that leading to Eq. \\eqref{criticallines1} to determine the phase boundaries beyond which the membrane is completely unstable against bud formation. Not surprisingly we find that they satisfy a very similar equation:\n\\begin{equation}\nf_A^o\\left(\\frac{J \\ell}{|\\delta \\kappa|}; k\\right) = \\left( \\frac{\\kappa_m}{|\\delta \\kappa|} \\right)^2 \\frac{4 k \\pi K(3\/4)}{3K(1\/4)} \\left(\\frac{|\\delta \\kappa|}{J \\ell}\\right)^2. \\tag{SM22} \\label{criticallines2}\n\\end{equation}\n\nTo illustrate how the phase diagram is modified when budding instability is taken into account, we show the results for $\\kappa_m \/| \\delta \\kappa| = 1\/4$ in Fig. \\ref{fig8}. The parameter regime susceptible to budding is for large area fraction of $A$ lipids, $f_A$, and large line tension, $J \\ell\/|\\delta \\kappa|$. The hashed regions correspond to parameter regimes where budding is preferable. Energy barriers are present for budding to occur in the diagonally hashed region, while for the horizontally hashed region the P-surface is completely unstable against budding. \n\n\n\\bibliographystyle{apsrev4-1\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{SUPPLEMENTAL MATERIAL:\\\\ ``Quasiparticle lifetime of the repulsive Fermi polaron''}\n\\setcounter{page}{1}\n\\begin{center}\nHaydn S. Adlong,$^1$\nWeizhe Edward Liu,$^{1,2}$\nFrancesco Scazza,$^3$\nMatteo Zaccanti,$^3$\nNelson Darkwah Oppong,$^{4,5,6}$\nSimon~F\\\"olling,$^{4,5,6}$\nMeera M.~Parish,$^{1,2}$ and\nJesper~Levinsen$^{1,2}$, \\\\\n\\emph{\\small $^1$School of Physics and Astronomy, Monash University, Victoria 3800, Australia}\\\\\n\\emph{\\small $^2$ARC Centre of Excellence in Future Low-Energy Electronics Technologies, Monash University, Victoria 3800, Australia}\\\\\n\\emph{\\small $^3$Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO) and European Laboratory for Nonlinear Spectroscopy (LENS), 50019 Sesto Fiorentino, Italy}\\\\\n\\emph{\\small $^4$Ludwig-Maximilians-Universit{\\\"a}t, Schellingstra{\\ss}e 4, 80799 M{\\\"u}nchen, Germany}\\\\\n\\emph{\\small $^5$Max-Planck-Institut f{\\\"u}r Quantenoptik, Hans-Kopfermann-Stra{\\ss}e 1, 85748 Garching, Germany}\\\\\n\\emph{\\small $^6$Munich Center for Quantum Science and Technology (MCQST), Schellingstra{\\ss}e 4, 80799 M{\\\"u}nchen, Germany}\n\n\\end{center}\n\n\\section{Model and scattering parameters}\n\n\\subsection{Model}\n\nIn modelling the dynamics of impurities coupled to a Fermi sea, we use the following effective Hamiltonian\n\\begin{align} \\label{eq:effectiveHam}\n\\hat H=\\hat H_0+\\hat H_\\Omega+\\hat H_\\uparrow+\\hat H_\\downarrow,\n\\end{align}\nwhere\n\\begin{subequations}\n\\begin{align}\n \\hat H_0&=\\sum_\\k(\\epsilon_{\\k}-\\mu) \\hat f_\\k^\\dag\\hat f_\\k, \\label{eq:supHam1}\\\\\n \\hat H_\\Omega&=\\frac{\\Omega_0}{2} \\sum_{\\k} \\left(\n \\hat c^\\dag_{\\k \\downarrow} \\hat c_{\\k \\uparrow} +\n \\hat c^\\dag_{\\k \\uparrow} \\hat c_{\\k \\downarrow} \\right)+\\Delta\\omega\\,\\hat n_{\\downarrow}, \\label{eq:supHam2}\\\\\n \\hat{H}_{\\sigma} &= \\sum_{\\k} \\left[\\epsilon_{\\k} \\hat\n c^\\dag_{\\k\\sigma} \\hat c_{\\k\\sigma} +\n ( \\epsilon_{\\k}\/2 + \\nu_{\\sigma}) \\hat d^\\dag_{\\k\\sigma} \\hat d_{\\k\\sigma}\\right] \n+g_{\\sigma}\\sum_{\\k, {\\bf q}} \\left( \\hat d^\\dag_{{\\bf q}\\sigma}\\hat c_{{\\bf q}\/2 - \\k,\\sigma} \\hat f_{{\\bf q}\/2 + \\k}\n + \\hat f^\\dag_{{\\bf q}\/2 + \\k} \\hat c^\\dag_{{\\bf q}\/2\n -\\k, \\sigma}\\hat d_{{\\bf q}\\sigma} \\right).\\label{eq:supHam3}\n\\end{align}\n\\end{subequations}\nThe meaning of the various symbols is discussed in the main text.\n\n\\subsection{Relation between Hamiltonian operators and experimental atomic states}\n\nIn the 2D $^{173}$Yb experiment of Ref.~\\cite{Oppong2019}, the states of the atoms are defined by the electronic state, with ground state $^1S_0$ and long-lived excited state $^3P_0$ (i.e., the ``clock'' state), as well as the nuclear-spin state with $m_F \\in \\{-5\/2, -3\/2, \\ldots, +5\/2\\}$.\nThe majority $\\hat f_\\k^\\dag$ atoms exist in the $m_F=+5\/2$ ground state, which acts as a bath for the weakly interacting $\\hat c^\\dag_{\\k \\downarrow}$ and resonantly interacting $\\hat c^\\dag_{\\k \\uparrow}$ impurities in the $m_F=-3\/2$ ground state and $m_F=-5\/2$ `clock' state, respectively.\nInteractions between the $\\hat c^\\dag_{\\k \\uparrow}$ and $\\hat f_\\k^\\dag$ atoms are tunable through an orbital Feshbach resonance~\\cite{Zhang2015} as has been demonstrated experimentally~\\cite{Hofer2015,Pagano2015}.\n\nOn the other hand, the 3D experiment of Ref.~\\cite{Scazza2017} involves $^6$Li atoms in the three lowest Zeeman levels. The majority $\\hat f_\\k^\\dag$ atoms exist in the lowest Zeeman level, while the weakly interacting $\\hat c^\\dag_{\\k \\downarrow}$ and resonantly interacting $\\hat c^\\dag_{\\k \\uparrow}$ impurities occupy the second-lowest and third-lowest Zeeman levels respectively. Owing to two off-centered broad Feshbach resonances, the scattering between the $\\hat c^\\dag_{\\k \\uparrow}$ and $\\hat f_\\k^\\dag$ atoms can be resonantly enhanced while only moderately increasing the comparatively weak interactions between the $\\hat c^\\dag_{\\k \\downarrow}$ and $\\hat f_\\k^\\dag$ atoms~\\cite{Scazza2017}.\n\n\n\n\\subsection{Scattering parameters}\n\nWithin the model in Eq.~\\eqref{eq:effectiveHam}, fermions of the same spin do not interact and the two spin states of the impurity are only coupled through the light-field. Ignoring for the moment the light-field (i.e., setting $\\Omega_0=0$), we calculate the vacuum spin-dependent impurity-fermion $T$ matrix, which characterizes the interactions between the majority fermions and a particular spin state of the impurity. %\nThis yields\n\\begin{align}\n T_{\\sigma}(E)=\\left[\\frac{E - \\nu_\\sigma}{g^2_{\\sigma}}-\\sum_{\\k}^\\Lambda \\frac{1}{E-2 \\epsilon_{\\k}}\\right]^{-1}, %\n \\label{eq:Tmatrix}\n\\end{align}\nwhere $E$ is the collision energy and $\\Lambda$ is the ultraviolet cutoff.\n\nTo proceed, we compare the scattering $T$ matrix with the 2D and 3D scattering amplitudes using $f_{2{\\rm D}\\sigma}(k)=mT_\\sigma(E)$ and $f_{3{\\rm D}\\sigma}(k)=-\\frac{m}{4\\pi}T_\\sigma(E)$, with $E=k^2\/m$. The scattering amplitudes at low energy are known to take the forms\n\\begin{subequations}\n\\begin{align} \n f_{\\text{2D}\\sigma}(k) &\\simeq \\frac{4 \\pi}{ - \\ln (k^2 a^2_{\\text{2D}\\sigma}) + R^2_{\\text{2D}\\sigma}k^2+i \\pi},\n \\label{eq:f2D}\\\\\n f_{\\text{3D}\\sigma}(k) &\\simeq -\\frac{1}{ a_{\\text{3D}\\sigma}^{-1} + R_{\\text{3D}\\sigma} k^2 + ik }.\n\\end{align}\n\\end{subequations}\nBy comparing Eq.~\\eqref{eq:Tmatrix} with the low-energy scattering amplitudes, we obtain the renormalized scattering parameters. In 2D, this results in\n\\begin{align}\n\\frac{\\nu_\\sigma+E_{2\\sigma}}{g^2_{\\sigma}} = \\sum_{\\k}^\\Lambda \\frac{1}{E_{2\\sigma}+2 \\epsilon_{\\k}}, \\qquad R_{\\text{2D}\\sigma}^2 = \\frac{4\\pi}{m^2 g^2_\\sigma},\n\\end{align}\nwhere $a_{{\\rm 2D}\\sigma}$ is the scattering length, $R_{2{\\rm D}\\sigma}$ is the effective range~\\cite{Kirk2017}, and $E_{2\\sigma}$ is the binding energy of the two-body bound state that exists for all interactions in 2D:\n\\begin{align}\n E_{2 \\sigma} = \\frac{1}{m R^2_{\\text{2D}\\sigma}} W\\left( \\frac{R^2_{\\text{2D}\\sigma}}{a^2_{\\text{2D}\\sigma}} \\right),\n\\end{align}\nwith $W$ the Lambert $W$ function.\nIn 3D, we have\n\\begin{align}\n \\frac{m}{4 \\pi a_{\\text{3D}\\sigma}} = - \\frac{\\nu_\\sigma}{g^2_\\sigma} + \\sum_{\\k}^\\Lambda \\frac{1}{ 2 \\epsilon_{\\k}}, \\qquad R_{\\text{3D} \\sigma} = \\frac{4 \\pi}{m^2 g_\\sigma^2},\n \\label{eq:renorm3D}\n\\end{align}\nwhere $a_{{\\rm 3D}\\sigma}$ is the scattering length and $R_{3{\\rm D}\\sigma}$ is a range parameter~\\cite{Gurarie2007}. In this case, we only have a bound state when $a_{{\\rm 3D}\\sigma}>0$, with corresponding binding energy\n\\begin{align}\n E_{3\\sigma}=\\frac{\\left[\\sqrt{1+4R_{3{\\rm D}\\sigma}\/a_{3{\\rm D}\\sigma}}-1\\right]^2}{4m{R_{3{\\rm D}\\sigma}^{2}}}.\n\\end{align}\nThrough this procedure, we have related the bare interaction parameters $g$, $\\Lambda$, and $\\nu$ to the physical parameters, the scattering length $a$ and effective range $R$, which characterises the relevant Feshbach resonance. For the broad Feshbach resonances in ${}^6$Li we use $R_{\\text{3D}\\sigma}=0$ for both spin states. For the orbital Feshbach resonance in quasi-2D, the description of the effective range is somewhat more complicated, as discussed in the following.\n\n\n\n\n\\subsection{Effective model for scattering at an orbital Feshbach resonance in the presence of confinement}\n\nWe now discuss how the scattering parameters, $a_{\\text{2D}\\sigma}$ and $R_{\\text{2D}\\sigma}$, of the $^{173}$Yb experiment are determined. The orbital Feshbach resonance is known to lead to a strongly energy dependent scattering~\\cite{Zhang2015}, which is well approximated by the introduction of a 3D effective range~\\cite{Xu2016}. Furthermore, quasi-2D confinement generally leads to a non-trivial energy dependence of the effective 2D scattering amplitude even for a broad Feshbach resonance~\\cite{Petrov2001}; for strong confinement, this energy dependence can be modelled by the introduction of a 2D effective range~\\cite{Levinsen2013}. Thus, we use the effective range in our model to provide the simplest possible description of both the orbital Feshbach resonance and the confinement. This greatly simplifies the numerical simulations of Rabi oscillations, since it drastically reduces the possible degrees of freedom in the problem.\n\nWe now describe the possible scattering channels in ${}^{173}$Yb atoms close to an orbital Feshbach resonance, following the analysis in Ref.~\\cite{Zhang2015}. We focus on the two electronic orbitals described above (denoted here by $\\ket{g}$ and $\\ket{e}$) as well as two particular nuclear spin states (denoted here by $\\ket{\\Downarrow}$ and $\\ket{\\Uparrow}$). These states form the open channel $\\ket{o} \\equiv \\ket{g \\Uparrow, e \\Downarrow}$ and the closed channel $\\ket{c} \\equiv \\ket{e \\Uparrow, g \\Downarrow}$, which are detuned by $\\delta = \\Delta \\mu B$, where $\\Delta \\mu = h\\times 554\\,\\mathrm{Hz}\/\\mathrm{G}$~\\cite{Oppong2019} is the differential Zeeman shift and $B$ is the magnetic field strength in Gauss.\nThe interactions in this system are not diagonal in the open- and closed-channel basis, but instead proceed via the triplet $(+)$ and singlet $(-)$ channels, with $\\ket{\\pm} \\equiv \\frac{1}{\\sqrt{2}}(\\ket{o} \\pm \\ket{c})$. Associated with the singlet and triplet interactions are the singlet and triplet scattering lengths $a_\\pm$~\\cite{Zhang2015} and effective ranges $r_\\pm$~\\cite{Hofer2015}. \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.7\\linewidth]{a2D_R2D_Figure.pdf}\n \\caption{The open channel 2D scattering length (a) and 2D effective range (b) as a function of magnetic field strength (in Gauss). These parameters are used to create an effective 2D two-channel model description of scattering at an orbital Feshbach resonance in the presence of confinement. To match Ref.~\\cite{Oppong2019} we use $\\omega_z = 2\\pi \\times 37.1\\,\\mathrm{kHz}$.}\n \\label{fig:a2DandR2D}\n\\end{figure}\n\nIn the experiment~\\cite{Oppong2019}, the ${}^{173}$Yb atoms were confined to move in a 2D plane with a harmonic potential $V(z) = \\frac{1}{2} m \\omega_z^2 z^2$ acting in the transverse direction. Reference~\\cite{Oppong2019} used this to extend the theoretical analysis of quasi-2D scattering in Refs.~\\cite{Petrov2001,Bloch2008mbp} to derive an effective scattering amplitude of the form in Eq.~\\eqref{eq:f2D}. For the weakly interacting $\\downarrow$ state, it was found that $\\ln (k_F a_{\\text{2D}\\downarrow}) \\simeq -4.9 (1)$~\\cite{Oppong2019}. In this case, the dependence on $R_{\\text{2D} \\downarrow}$ is strongly suppressed, and we simply take $R_{\\text{2D} \\downarrow}=0$. For the case of strong interactions, the open channel scattering amplitude was found to be given by~\\cite{Oppong2019}\n\\begin{align}\n f_{\\text{q2D}}(E) &=2 \\sqrt{2 \\pi } \\frac{ l_z \\left(\\Tilde{a}^{-1}_- - \\frac{m r_- E}{2} \\right)+ l_z \\left(\\Tilde{a}^{-1}_+ - \\frac{m r_+ E}{2} \\right)-2 \\mathcal{F}\\left(\\frac{-E + \\delta}{\\omega_z}\\right)}{ \\left\\{\\splitfrac{ -\\mathcal{F}\\left(\\frac{-E + \\delta}{\\omega_z}\\right) \\left[l_z (\\Tilde{a}^{-1}_- - \\frac{m r_- E }{2})+l_z (\\Tilde{a}^{-1}_+ - \\frac{m r_+ E }{2})-2\n \\mathcal{F}\\left(-\\frac{E}{\\omega_z}\\right) \\right]}{-\\mathcal{F}\\left(-\\frac{E}{\\omega_z}\\right) \\left[l_z (\\Tilde{a}^{-1}_- - \\frac{m r_- E }{2})+l_z (\\Tilde{a}^{-1}_+ - \\frac{m r_+ E}{2}) \\right] +2 l_z^2 (\\Tilde{a}^{-1}_- - \\frac{m r_- E}{2}) (\\Tilde{a}^{-1}_+ - \\frac{ m r_+ E }{2}) } \\right\\}}, \\label{eq:fopen}\n\\end{align}\nwhere $l_z = 1\/\\sqrt{m \\omega_z}$ and $\\Tilde{a}_\\pm^{-1} = a^{-1}_\\pm - \\frac{r_\\pm}{4 l_z^2} \\left( 1- \\delta\/\\omega_z \\right)$. Here, $\\mathcal{F}$ is a transcendental function defined by~\\cite{Bloch2008mbp}\n\\begin{align}\n \\mathcal{F}(x) = \\int_0^\\infty \\frac{du}{\\sqrt{4 \\pi u^3}} \\left[ 1 - \\frac{e^{-xu}}{\\sqrt{(1-e^{-2u})\/2u}} \\right],\n\\end{align}\nand the energy is measured with respect to the quasi-2D zero point energy.\n\nTo extract an effective low-energy open-channel 2D scattering length and effective range for the strongly interacting spin-$\\uparrow$ impurity case, we perform a low energy expansion of Eq.~\\eqref{eq:fopen} and compare it with the standard form of the low-energy scattering amplitude, Eq.~\\eqref{eq:f2D}. In the following, we use the notation $a_{\\text{2D}}\\equiv a_{\\text{2D}\\uparrow}$ and $R_{\\text{2D}}\\equiv R_{\\text{2D}\\uparrow}$, as in the main text. Assuming that $\\delta \\gg |E|$ (i.e., that we are not close to $B=0$) we find the 2D scattering length \\cite{Oppong2019}\n\\begin{align}\n a_{\\text{2D}} = l_z \\sqrt{\\frac{\\pi}{D}} \\exp[-\\sqrt{2 \\pi} \\frac{ l_z^2 ( \\Tilde{a}_{-} \\Tilde{a}_{+})^{-1}- \\frac{1}{2}(l_z \\Tilde{a}_{-}^{-1}+ l_z \\Tilde{a}_{+}^{-1}) \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z} \\right) }{l_z \\Tilde{a}_{-}^{-1}+l_z \\Tilde{a}_{+}^{-1}-2 \\mathcal{F} \\left(\\frac{\\delta}{\\omega_z}\n \\right)} ],\n\\end{align}\nwhere $D \\simeq 0.905$~\\cite{Petrov2001}. The 2D effective range takes the form\n\\begin{align}\n \\left(\\frac{R_{\\text{2D}}}{l_z}\\right)^2 &= \\ln 2 %\n -\\frac{\\sqrt{2 \\pi } \\left\\{ l_z^{-1} r_+ \\left[ l_z \\Tilde{a}_-^{-1} - \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z}\n \\right) \\right]^2 + l_z^{-1} r_- \\left[l_z \\Tilde{a}_+^{-1} - \\mathcal{F}\\left(\\frac{\\delta}{\\omega_z}\n \\right) \\right]^2 + \\left[l_z\\Tilde{a}_-^{-1} - l_z \\Tilde{a}_+^{-1} \\right]^2 \\mathcal{F}'\\left( \\frac{\\delta}{\\omega_z} \\right)\\right\\}}{ \\left(l_z \\Tilde{a}_-^{-1}+ l_z\\Tilde{a}_+^{-1} -2 \\mathcal{F}\\left( \\frac{\\delta}{\\omega_z} \\right) \\right)^2},\n\\end{align}\nwhere $\\mathcal{F}'$ is the first derivative of $\\mathcal{F}$. \n\nFigure~\\ref{fig:a2DandR2D} shows the extracted scattering parameters using the experimentally determined values for all parameters~\\cite{Oppong2019} (see also Ref.~\\cite{Hofer2015}). We see that indeed the change of the magnetic field provides a means to tune the interactions via $a_{{\\rm 2D}}$. On the other hand, the effective range is not strongly dependent on magnetic field --- indeed, the effective range is approximately constant ($k_F R_{\\text{2D}} \\sim 1$) within the magnetic field strengths of interest to the present work.\n\n\\jfl{\nIn general, we find that the 2D effective range is able to highly accurately capture the physics of the orbital Feshbach resonance and the confinement in the domain of interaction strengths used in Ref.~\\cite{Oppong2019}. This is shown in Fig.~\\ref{fig:fulltheoryvs2channel}, where we show the near perfect agreement between the full theory model \\cite{Oppong2019} and our effective model in calculating the energy of the Fermi polaron.}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.35\\linewidth]{FullTheory_vs_2Channel.pdf}\n \\caption{\\jfl{\n The attractive (orange) and repulsive (blue) Fermi polaron energy as calculated through our effective two-channel model description of an orbital Feshbach resonance in the presence of confinement. The energies are highly consistent with the full theory calculation of the polaron energies (black) from Ref. \\cite{Oppong2019}. The curves are calculated at temperature $T\/T_F=0.16$.}}\n \\label{fig:fulltheoryvs2channel}\n\\end{figure}\n\n\\section{Finite temperature variational approach for impurity dynamics}\nIn order to model the dynamics of the Rabi oscillations and the impurity spectral response, we use the finite-temperature Truncated Basis Method (TBM) developed in Refs.~\\cite{Parish2016,Liu2019}. The basic details of this method are discussed in the main text. Our ansatz for the impurity operator is\n\\begin{align} \\label{eq:OpApprox}\n \\hat{c}(t)= &\n\\sum_{\\sigma}\\Bigg[\\alpha_{0}^\\sigma(t)\\hat\n c_{{\\bf 0}\\sigma}+\\sum_\\k\\alpha_{\\k}^\\sigma(t)\\hat f^\\dag_\\k\\dh_{\\k\\sigma}\n+\\sum_{\\k,{\\bf q}} \\alpha_{\\k{\\bf q}}^\\sigma(t) \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\sigma}\\Bigg],\n\\end{align}\nwhere, for simplicity, we consider a vanishing total\nmomentum. As discussed in the main text, we may quantify the error incurred in the Heisenberg equation of motion by introducing the error operator $\\hat\\epsilon(t)\\equiv i\\partial_t \\hat{c}(t)- \\comm*{\\hat{c}(t)}{\\hat H}$ and the associated error quantity $\\Delta(t)\\equiv\\Tr[\\hat \\rho_0 \\hat\\epsilon(t) \\hat\\epsilon^\\dag(t)]$. Using the minimization condition $\\pdv*{\\Delta(t)}{\\dot\\alpha^{\\sigma*}_j(t)}=0$ with respect to the variational coefficients $\\{\\alpha_j\\}$,\nwe arrive at \n\\begin{subequations} \\label{eq:VariationalEqs}\n\\begin{align}\n E \\alpha^{\\uparrow}_{0} &= g_\\uparrow \\sum_{{\\bf q}} \\alpha^{\\uparrow}_{{\\bf q}} \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }] + \\frac{\\Omega_0}{2} \\alpha^{\\downarrow}_{0}\\\\\n (E - \\varepsilon_{{\\bf q} \\uparrow}) \\alpha^{\\uparrow}_{{\\bf q}} &= g_\\uparrow \\alpha^{\\uparrow}_{0} + g_\\uparrow \\sum_{\\k} \\alpha^{\\uparrow}_{\\k {\\bf q}} \\Tr[\\hat \\rho_0 \\hat f_{\\k}\\hat f^\\dag_{\\k}]\\\\\n (E - \\varepsilon_{\\k {\\bf q}}) \\alpha^{\\uparrow}_{\\k {\\bf q}} &= g_\\uparrow \\alpha^{\\uparrow}_{{\\bf q}} + \\frac{\\Omega_0}{2} \\alpha^{\\downarrow}_{\\k {\\bf q}}\\\\\n (E - \\Delta \\omega) \\alpha^{\\downarrow}_{0} &=\n g_\\downarrow \\sum_{{\\bf q}} \\alpha^{\\downarrow}_{{\\bf q}} \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }] + \\frac{\\Omega_0}{2} \\alpha^{\\uparrow}_{0} \\\\\n (E - \\varepsilon_{{\\bf q} \\downarrow} - \\Delta \\omega) \\alpha^{\\downarrow}_{{\\bf q}} &= g_\\downarrow \\alpha^{\\downarrow}_{0} + g_\\downarrow \\sum_{\\k} \\alpha^{\\downarrow}_{\\k {\\bf q}} \\Tr[\\hat \\rho_0 \\hat f_{\\k}\\hat f^\\dag_{\\k}]\\\\\n (E - \\varepsilon_{\\k {\\bf q}} - \\Delta \\omega) \\alpha^{\\downarrow}_{\\k {\\bf q}} &= g_\\downarrow \\alpha^{\\downarrow}_{{\\bf q}} + \\frac{\\Omega_0}{2} \\alpha^{\\uparrow}_{\\k {\\bf q}}.\n\\end{align}\n\\end{subequations}\nHere we have taken the stationary condition since our Hamiltonian is time-independent,\n\\begin{align}\n \\alpha^\\sigma_{j}(t)=\\alpha^\\sigma_{j}(0)e^{-iEt}\\equiv\\alpha^\\sigma_{j}e^{-iEt},\n\\end{align}\nand defined $\\varepsilon_{{\\bf q} \\sigma} \\equiv \\nu_{\\sigma}-\\epsilon_{{\\bf q}}\/2$ and $\\varepsilon_{\\k {\\bf q}} \\equiv \\epsilon_{{\\bf q} - \\k} + \\epsilon_{\\k} - \\epsilon_{{\\bf q}}$.\n\nEquation~\\eqref{eq:VariationalEqs} represents a set of linear integral equations that define the time dependence of the variational coefficients $\\{ \\alpha_j \\}$. It is quite similar to the corresponding set of equations derived in Refs.~\\cite{Parish2016,Liu2019}. However, \nhere we account for temperature, initial-state interactions, and the Rabi coupling between the two impurity spin states. \nDiagonalizing Eq.~\\eqref{eq:VariationalEqs} yields eigenvectors $\\{ \\alpha_j^{(l)} \\}$ and corresponding eigenvalues $E_l$. In what follows it is useful to write the eigenvectors as the union of the spin-$\\uparrow$ and spin-$\\downarrow$ components, i.e., $\\{ \\alpha_j ^{(l)}\\} = \\{ \\alpha^{ \\uparrow (l)}_j \\} \\cup \\{ \\alpha^{ \\downarrow (l)}_j \\}.$\n\nThe solutions of Eq.~\\eqref{eq:VariationalEqs} allow us to obtain stationary impurity operators\n\\begin{align}\n \\hat{\\phi}^{(l)} \\equiv \\sum_j \\alpha_j^{(l)} \\hat{O}_j,\n\\end{align}\nwhere the impurity basis operators $\\{ \\hat{O}_j \\}=\\{\\hat c_{{\\bf 0}\\uparrow}, \\hat f^\\dag_\\k\\dh_{\\k\\uparrow}, \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\uparrow},\\hat c_{{\\bf 0}\\downarrow}, \\hat f^\\dag_\\k\\dh_{\\k\\downarrow}, \\hat f^\\dag_{{\\bf q}} \\hat f_{\\k} \\hat c_{{\\bf q}-\\k \\downarrow}\\}$ are those introduced in Eq.~\\eqref{eq:OpApprox}. These all satisfy $\\Tr[\\hat\\rho_0\\hat{O}_j \\hat{O}^\\fix{\\dag}_k]=0$ when $j\\neq k$, since the trace is over medium-only states. This in turn allows us to normalize the stationary solutions according to\n\\begin{align}\n \\Tr[\\hat \\rho_0 \\hat \\phi^{(l)}\\hat \\phi^{(m)\\dag}]=\\delta_{lm}.\n\\end{align}\n Using this, the impurity annihilation operator in Eq.~\\eqref{eq:OpApprox} can be expressed as\n\\begin{align}\n \\hat{c}(t) = \\sum_l \\Tr[\\hat \\rho_0\\hat c(0) \\Hat{\\phi}^{(l)\\dagger}] \\Hat{\\phi}^{(l)} e^{-i E_l t}=\\sum_l \\alpha_0^{\\fix{\\downarrow}(l)*} \\Hat{\\phi}^{(l)} e^{-i E_l t},\n\\end{align}\nwhere the initial impurity operator $\\hat c(0) = \\hat c_{{\\bf 0}\\downarrow}$. We take a bare impurity as the initial state even though there are initial-state interactions, since the simulations yield essentially the same result as when we use a weakly interacting polaron. We presume this is because the weakly interacting polaron forms quickly (on a time scale set by $a_{\\mathrm{3D}\\downarrow}^2$ in 3D) once we start the dynamics.\n\n\nFinally, we discuss the character of the medium-only states appearing in Eq.~\\eqref{eq:VariationalEqs}. Assuming that the interactions between the initial $\\downarrow$ impurity and the medium particles are negligible (as is the case in the experiments~\\cite{Scazza2017,Oppong2019}), we may take the medium states to be thermal eigenstates at temperature $T$. Therefore, we have\n\\begin{align}\n \\Tr[\\hat \\rho_0 \\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }]\\equiv \\expval{\\hat f^\\dag_{{\\bf q} } \\hat f_{{\\bf q} }}_\\beta = n_F(\\epsilon_{\\q}) \n\\end{align}\nwhere $\\beta$ is the inverse temperature and $n_F$ is the Fermi-Dirac distribution function:\n\\begin{align}\n n_F(\\epsilon_{\\q})\\equiv \\Tr[\\hat \\rho_0 \\hat f^\\dag_{\\bf q} \\hat f_{\\bf q}]=\\frac1{e^{\\beta(\\epsilon_{\\q}-\\mu)}+1}.\n\\end{align}\nThe chemical potential $\\mu$ is related to the medium density $n$ via\n\\begin{align}\n n=\\sum_{\\bf q} n_F(\\epsilon_{\\q})=\\begin{cases}\n -\\left(\\frac{m}{2\\pi \\beta}\\right)^{3\/2}{\\rm Li}_{3\/2}(-e^{\\beta\\mu}) & \\rm(3D)\\\\[0.5em] \\frac{m}{2 \\pi \\beta} \\ln(1+ e^{\\beta \\mu}) & \n \\mbox{(2D)}\\end{cases}\n\\end{align}\nwhere ${\\rm Li}$ is the polylogarithm. The density is related to the Fermi energy via\n\\begin{align}\n E_F=\\frac{k_F^2}{2m}=\\begin{cases}\n \\frac{(6\\pi^2n)^{2\/3}}{2m} & \\rm(3D)\\\\ \\frac{4\\pi n}{2m} & \\rm{(2D)}\\end{cases}\n\\end{align}\n\n\\subsection{Impurity spectral function}\n\nIn the limit of a weak Rabi coupling, we can obtain the spin-$\\uparrow$ impurity spectral function within linear response:\n\\begin{align}\n A_\\uparrow(\\omega) \\simeq \\sum_l |\\alpha^{\\uparrow (l)}_0|^2 \\delta(\\omega - E_l),\n \\label{eq:spec}\n\\end{align}\nwhere the variational equations in Eq.~\\eqref{eq:VariationalEqs} are solved at $\\Omega_0=0$.\nIn practice, the spin-$\\downarrow$ impurity interacts weakly with the medium, and thus the spectral function can be measured by driving transitions from the initial (nearly) non-interacting spin-$\\downarrow$ impurity state into the spin-$\\uparrow$ state. Indeed, this has been done in both experiments~\\cite{Scazza2017,Oppong2019}.\n\nSince the solution of Eq.~\\eqref{eq:VariationalEqs} are discrete, we convolve the resulting spectrum with a Gaussian to yield\n\\begin{align}\n I(\\omega) &= \\sum_l |\\alpha^{\\uparrow (l)}_0|^2 g(\\omega - E_l).\n\\end{align}\nConvolution of the spectrum in this case has the added benefit of enabling one to approximately model the finite duration of the pulses used in experiment.\n\n\\subsection{Simulating Rabi oscillations}\n\nThe Rabi oscillations are defined by\n\\begin{align}\n{\\cal N}_\\downarrow(t)=\\expval{\\hat c(t)\\hat n_\\downarrow \\hat c^\\dag(t)}_\\beta.\n\\end{align}\nWe will take as our initial condition that $\\hat c(\\fix{t=0})=\\hat c_{{\\bf 0}\\downarrow}$, i.e., the impurity is initially in a bare spin-$\\downarrow$ state, which means that the impurity number is ${\\cal N}_\\downarrow+{\\cal N}_\\uparrow=1$ at all times. The Rabi oscillations are then given by\n\\begin{align}\n{\\cal N}_\\downarrow(t)& %\n\\simeq \\Trace[\\hat\\rho_0\\hat{c}_\\downarrow(t)\\hat n_\\downarrow\\hat{c}^\\dag_\\downarrow(t)]\n= \\sum_{j} \\fix{\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta} \\bigg|\\sum_l \\alpha_{0}^{\\downarrow(l)^*} e^{-i E_l t} \\alpha_j^{\\downarrow(l)} \\bigg|^2 .\\label{eq:RabiOscEq}\n\\end{align}\n\n\nWith the exception of detuning, all of the parameters used to define the Rabi oscillations are provided from the relevant experiment. The detuning in experiment is set to address the repulsive polaron peak, and to match this procedure, we use a calculated detuning such that the Rabi oscillations address the theoretically obtained spin-$\\uparrow$ repulsive polaron. Due to small but finite initial state interactions, this detuning must take into account the energy of the spin-$\\downarrow$ impurities. This is achieved by assuming the spin-$\\downarrow$ impurities exist as zero-momentum repulsive polarons with a narrow spectral width. Owing to this assumption, combined with thermal fluctuations, a finite density of impurities and experimental limitations, we place an uncertainty on the detuning (see Fig.~\\ref{fig:DetuningUncertaintyCalc}). This requires that we simulate Rabi oscillations over the range of possible detunings.\nThe parameters used in the simulations of the Rabi oscillations in Figs.~\\ref{fig:Rabi} and \\ref{fig:ResidueAndDamping} can be found in Table~\\ref{tab:params}.\n\nWe find that the inclusion of initial state interactions leads to a small reduction in the damping of the Rabi oscillations.\nThis can be understood from the fact that in the limit in which the spin states have equal interactions, spin-symmetry implies that the Rabi oscillations would be undamped and oscillate at the bare Rabi frequency.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\linewidth]{DetuningCalc.pdf}\n \\caption{Example of the calculation of the uncertainty in the detuning of Rabi oscillations onto the repulsive polaron. We show the spectral function of the repulsive polaron in the 2D ${}^{173}$Yb experiment of Ref.~\\cite{Oppong2019} (solid line), with the dashed lines indicating the full-width half-maxima. The detuning is set %\n between these half-maxima, leading to the uncertainty shown in Fig.~\\ref{fig:Rabi} in the main text. The shown spectral function is for $\\ln(1\/k_F a_{\\text{2D}}) = 0.41$, $k_F R_{\\text{2D}} = 0.69$ and $T\/T_F = 0.16$.}\n \\label{fig:DetuningUncertaintyCalc}\n\\end{figure}\n\n\n\\begin{figure}[th]\n \\centering\n \\includegraphics[width=\\linewidth]{RabiSamples2D_FiniteOffset.pdf}\n \\caption{\\jfl{Comparison of theory (blue solid lines) of Fig.~2(a)-(c) in the manuscript and the experimental data (black circles) with a constant offset of $0.2$ removed and normalized to the value at time $t=0$.\n The light-gray circles correspond to the unmodified data in the main text.}}\n \\label{fig:RabiSamples2D_finiteoffset}\n\\end{figure}\n\n\n\\jfl{\n\\subsection{Finite offset in the Rabi oscillation measurement of the 2D experiment}\nAs noted in the main text, there is a slight disagreement in the amplitude of the Rabi oscillations in the 2D experiment \\cite{Oppong2019} and our variational approach.\nThis disagreement can be explained with a constant offset $\\epsilon > 0$ of the $\\mathcal{N}_\\downarrow$ measurement in the experiment.\nIn our work, the relative population $\\mathcal{N}_\\downarrow\/(\\mathcal{N}_\\downarrow + \\mathcal{N}_\\uparrow)$ of the 2D experiment is inferred from the sole measurement of $\\mathcal{N}_\\downarrow(t)$ and\n\\begin{align}\n\\mathcal{N}_\\downarrow\\left\/(\\mathcal{N}_\\downarrow + \\mathcal{N}_\\uparrow)\\right. \\approx \\mathcal{N}_\\downarrow\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\mathcal{N}_\\uparrow(t = 0)]\\right. \\approx \\mathcal{N}_\\downarrow\/[\\mathcal{N}_\\downarrow(t = 0) + 0] = \\mathcal{N}_\\downarrow\\left\/\\mathcal{N}_\\downarrow(t = 0)\\right..\n\\end{align}\nThus, a constant offset in the measurement of $\\mathcal{N}_\\downarrow$ changes the relative population as\n\\begin{align}\n\\mathcal{N}_\\downarrow\\left\/\\mathcal{N}_\\downarrow(t = 0)\\right. \\rightarrow (\\mathcal{N}_\\downarrow + \\epsilon)\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right. = \\mathcal{N}_\\downarrow\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right. + \\epsilon\\left\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]\\right.,\n\\end{align}\nwhich directly shows how the mean of the relative population is artificially increased by $\\epsilon\/[\\mathcal{N}_\\downarrow(t = 0) + \\epsilon]$ and the amplitude is artificially reduced by the additional term $\\epsilon$ in the denominator.\n\nThe offset $\\epsilon$ in the experimental data originates from the detection method, for which the majority Fermi sea is removed, but a finite number of remaining majority atoms contributes a positive spurious signal to the measurement of $\\mathcal{N}_\\downarrow$.\nThis contribution is independent of $t$ and we estimate $\\epsilon\\lesssim 0.2\\, \\mathcal{N}_\\downarrow$ from a reevaluation of the existing data of Ref.~\\cite{Oppong2019}.\nIn Fig.~\\ref{fig:RabiSamples2D_finiteoffset}, we illustrate how the removal of such an offset considerably\nimproves the agreement between experiment and the variational approach.\nHowever, we have chosen not to subtract $\\epsilon$ from the data in Fig.~2 of the main text as we lack a precise number for each data set.}\n\n\\heavyrulewidth=.08em\n\\lightrulewidth=.05em\n\\cmidrulewidth=.03em\n\\belowrulesep=.65ex\n\\belowbottomsep=0pt\n\\aboverulesep=.4ex\n\\abovetopsep=0pt\n\\cmidrulesep=\\doublerulesep\n\\cmidrulekern=.5em\n\\defaultaddspace=.5em\n\n\\begin{table}[h]\n \\caption{\\label{tab:params}%\n\t Parameters used for obtaining the theoretical curves and simulations shown in Figs.~2~and~3, matching the experimental ones from Refs.~\\cite{Oppong2019} and \\cite{Scazza2017}. Experimental uncertainties are given in parenthesis were applicable. Note that the 3D effective range is always zero.\\bigskip}\n \\begin{tabular}{l l c c c c c c c c c c c} \n \\midrule\\midrule\n && \\multicolumn{7}{c}{Fig.~2} & \\hspace{1.75em} & \\multicolumn{3}{c}{Fig.~3} \\\\ \\cmidrule{3-9} \\cmidrule{11-13}\n && (a) & (b) & (c) & \\hspace{0.5em} & (d) & (e) & (f) & & (a), (c) & \\hspace{0.5em} & (b), (d) \\\\ \\midrule \n Interaction parameter & $\\ln(1\/k_F a_{\\mathrm{2D}\\uparrow})$ \\hspace{0.75em} & 0.73(4) & 0.57(5) & 0.25(5) & & \\multicolumn{3}{c}{---} & & 0.07--0.91 & & --- \\\\\n & $\\ln(1\/k_F a_{\\mathrm{2D}\\downarrow})$ & \\multicolumn{3}{c}{4.9(1)} & & \\multicolumn{3}{c}{---} & & 4.9(1) & & --- \\\\\n & $1\/k_F a_{\\mathrm{3D}\\uparrow}$ & \\multicolumn{3}{c}{---} & & 2.63(4) & 1.27(2) & 0.22(1) & & --- & & 0.22--4.23 \\\\\n & $1\/k_F a_{\\mathrm{3D}\\downarrow}$ & \\multicolumn{3}{c}{---} & & 9.20(15) & 6.94(11) & 5.03(8) & & --- & & 5.03--11.43 \\\\\n Range parameter & $k_F R_{\\text{2D}\\uparrow}$ \\hspace{1.5em} & 0.71 & 0.67 & 0.68 & & \\multicolumn{3}{c}{---} & & 0.67--0.76 & & --- \\\\\n %\n Rabi coupling & $\\Omega_0\/E_F$ & 0.95(11) & 0.96(11) & 0.94(11) && 0.68(1) & 0.69(1) & 0.67(1) & & 1.08 && 0.70 \\\\\n Reduced temperature & $T\/T_F$ & \\multicolumn{3}{c}{0.16(4)} && \\multicolumn{3}{c}{0.13(2)} & & 0.16(4) && 0.13--0.14 \\\\\n \\addlinespace[0.5em]\n Rep. polaron energy & $E_{+ \\uparrow}\/E_F$ & 0.71 & 0.76 & 0.89 && 0.19 & 0.42 & 1.08 & & 0.64--1.02 && 0.11--1.08 \\\\\n & $E_{+ \\downarrow}\/E_F$ & \\multicolumn{3}{c}{0.19} && 0.05 & 0.07 & 0.09 & & 0.19 && 0.04--0.09 \\\\\n Detuning uncertainty & $\\delta E_+\/E_F$ & 0.24 & 0.31 & 0.47 && 0.02 & 0.05 & 0.42 & & 0.19--0.65 && 0.02--0.42 \\\\\n \\midrule\\midrule\n \\end{tabular}\n\\end{table}\n\n\n\\section{Green's function approach for quasiparticle properties}\nA key alternative to our study of the impurity dynamics and properties in the TBM is provided through a Green's function approach. It has been shown that a variational approach using a single particle-hole excitation is equivalent to a Green's function approach calculated with non-self consistent $T$ matrix theory --- see Ref.~\\cite{Combescot2007} for a zero-temperature treatment, or Ref.~\\cite{Liu2019} at finite temperature. %\n\nIn this section, we take $\\Omega_0=0$ in the variational equations~\\eqref{eq:VariationalEqs}, while we consider the more general case in the next section. This allows us to derive a finite temperature impurity self energy $\\Sigma_\\sigma(E)$ separately within each of the impurity subspaces. Solving these equations for the energy then yields the expression\n\\begin{align} \\label{eq:definingSelfE}\n E = \\sum_{{\\bf q}} n_F(\\epsilon_{\\q}) \\left[ \\frac{E - \\varepsilon_{{\\bf q} \\sigma}}{g_\\sigma^2} - \\sum_{\\k} \\frac{1 - n_F(\\epsilon_{\\k})}{E - \\varepsilon_{\\k {\\bf q}}} \\right]^{-1}.\n\\end{align}\nThe right hand side of this expression is precisely the impurity self energy at zero momentum using ladder diagrams at finite temperature~\\cite{Liu2019}. The self energy is then related to the impurity (single-particle) Green's function through Dyson's equation,\n\\begin{align} \\label{dyson}\n G_\\sigma(E) = \\frac{1}{E - \\Sigma_\\sigma(E)}.\n\\end{align}\n\nThe relevant properties of the repulsive polaron can now be defined in terms of the impurity self energy. In particular, the repulsive polaron energy $E_{+\\sigma}$ is a (positive) solution to the implicit equation\n\\begin{align}\n \\Re \\left[\\Sigma_\\sigma (E) \\right] = E.\n\\end{align}\nExpanding the Green's function around this pole, the repulsive polaron quasiparticle residue is\n\\begin{align}\n Z_\\sigma = \\left( 1 - \\left. \\pdv{\\Re\\left(\\Sigma_\\sigma(E) \\right)}{E} \\right|_{E = E_{+\\sigma}} \\right)^{-1},\n\\end{align}\nand the quasiparticle width is\n\\begin{align} \\label{Eq:quasiparticlewidth}\n \\Gamma_\\sigma = - Z_\\sigma \\Im \\left[ \\Sigma_\\sigma(E_{+\\sigma}) \\right].\n\\end{align}\nIn addition to these properties, the impurity spectral function in Eq.~\\eqref{eq:spec} is given by\n\\begin{align}\n A_\\sigma(E) = -\\frac{1}{\\pi} \\Im[G_\\sigma(E)].\n\\end{align}\n\n\\subsection{Repulsive polaron width at weak interactions}\n\nHere we provide details of the calculation of the approximate width of the repulsive polaron peak at weak interactions, Eq.~\\eqref{eq:GammaPT}. To perform this calculation, we specialize to three dimensions, zero temperature, and to a broad Feshbach resonance, i.e., $R_{\\rm 3D}=0$. In fact, the arguments in the following are valid as long as the thermal wavelength exceeds the scattering length while $R_{\\rm 3D}\\lesssim 1\/(na_{\\rm 3D}^2)$. \n\nAssuming zero impurity momentum, the spin-$\\uparrow$ impurity self-energy in Eq.~\\eqref{eq:definingSelfE} is given by (for simplicity, in this section we suppress all spin indices as well as ``3D'' subscripts)\n\\begin{align}\n \\Sigma(E) = \\sum_{\\vectorbold{q}} \\Theta(k_F - q) \\left[ \\frac{m}{4 \\pi a} - \\left( \\sum_{\\k} \\frac{1}{2 \\epsilon_{\\k}} + \\sum_{\\k} \\frac{1-\\Theta(k_F - k)}{E - \\varepsilon_{\\k \\vectorbold{q}}+i0} \\right) \\right]^{-1}.\n\\end{align}\nwhere $k\\equiv |\\k|$ and $q \\equiv |\\vectorbold{q}|$ and we take the limit of $R_{\\text{3D}}\\to 0$, which according to Eq.~\\eqref{eq:renorm3D} is equivalent to taking the limit of $\\nu,g\\to\\infty$ in such a way that\n\\begin{align}\n \\frac{\\nu}{g^2} %\n = - \\frac{m }{4 \\pi a} + \\sum_{\\vb{k}}^\\Lambda \\frac{1}{2\\epsilon_{\\vb{k}} }.\n\\end{align}\nWe have also introduced a convergence factor $+i0$ which shifts the energy poles by an infinitesimal amount into the lower half of the complex plane.\n\nIn order to find the approximate form of the self-energy (and thereby the width) in the limit of weak interactions, we perturbatively expand the self-energy in scattering length (up to order $a^2$):\n\\begin{align}\n \\Sigma(E) = \\sum_{\\vectorbold{q}} \\Theta(k_F - q) \\left[ \\frac{4 \\pi a}{m} + \\frac{16 \\pi^2 a^2}{m^2} \\left( \\sum_{\\k} \\frac{1}{2 \\epsilon_{\\k}} + \\sum_{\\k} \\frac{1-\\Theta(k_F - k)}{E - \\varepsilon_{\\k \\vectorbold{q}} + i 0} \\right) \\right].\n\\end{align}\nIn the limit of weak repulsive interactions, the repulsive polaron will have residue $Z \\simeq 1$ and the width is thus given by\n\\begin{align}\n \\Gamma \\simeq - \\Im[\\Sigma(E_+)].\n\\end{align}\nWe can extract the imaginary component of the self-energy through the symbolic identity, which is valid for all real $\\alpha$:\n\\begin{align}\n \\frac{1}{\\alpha + i 0} = \\mathscr{P} \\frac{1}{\\alpha} - i\\pi \\delta(\\alpha),\n\\end{align}\nwhere $\\mathscr{P}$ denotes the Cauchy principal value. We thus have,\n\\begin{align}\n - \\Im[\\Sigma(E)] = \\frac{16 \\pi^3 a^2}{m^2 E_F} \\sum_{\\vectorbold{q}, \\k } \\Theta(k_F - q) (1 - \\Theta(k_F - k)) \\delta(E\/E_F - \\varepsilon_{ \\k \\vectorbold{q}}\/E_F).\n\\end{align}\nIn the thermodynamic limit, these sums reduce to integrals that can be solved analytically in spherical coordinates:\n\n\\begin{align}\n - \\Im[\\Sigma(E)] = & \\frac{k_F^4 a^2}{16 \\pi m E_F^2} \\Bigg[ 4E^2 \\ln \\left(\\frac{2E_F}{\\sqrt{E_F (2 E+E_F)}+E_F}\\right) %\n +3E^2-4 E E_F-2\n E_F^2+2 (E+E_F)\n \\sqrt{E_F (2 E+E_F)}\n \\Bigg].\n\\end{align}\n\n\nImportantly, the imaginary part of the self energy is zero at zero energy, and we must therefore consider finite energy.\nIn the limit of weak interactions, the energy of the repulsive polaron is given by the mean-field approximation~\\cite{Bishop1973}:\n\\begin{align}\n E_+ = \\frac{2 k_F^3 a}{3 \\pi m}.\n\\end{align}\nUsing this energy and only retaining terms of order $a^4$ (the lowest non-zero contribution) the width is given by\n\\begin{align} \\label{PertbExpDamp}\n \\frac{\\Gamma}{E_F}= \\frac{8 }{9 \\pi^3} (k_F a)^4.\n\\end{align}\nSince we originally expanded the self-energy up to order $a^2$ and have ended with a result that is of order $a^4$ we justify this result numerically in Fig.~\\ref{fig:PertAndNumerical}. Furthermore, we have checked that all contributions with multiple particle-hole excitations vanish at order $a^4$.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.4\\linewidth]{LogLogNumericalVsPert.pdf}\n \\caption{Comparison of the perturbative expression for the repulsive polaron quasiparticle width in Eq.~\\eqref{PertbExpDamp} against the numerically exact calculation from Eq.~\\eqref{Eq:quasiparticlewidth}.} %\n \\label{fig:PertAndNumerical}\n\\end{figure}\n\n\n\n\\section{Approximate Rabi oscillations based on Green's functions}\n\nWe finally turn to the arguments that led to Eq.~\\eqref{eq:TheoryFit} in the main text, which allowed us to link the repulsive polaron width to the damping of the Rabi oscillations. This will allow us to extend the standard approximation for extracting polaron properties from impurity Rabi oscillations \\cite{Kohstall2012}. \nIt is useful to introduce a spectral decomposition of the Rabi oscillations, which is provided by the Fourier transform of Eq.~\\eqref{eq:RabiOscEq}: %\n\\begin{align} \\label{eq:RabiSpectrum}\n \\mathcal{R}_\\downarrow (\\omega)\\equiv \\int dt\\,e^{i\\omega t}{\\cal N}_\\downarrow(t) \\simeq \\sum_{j, l,l'} \\fix{\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta} \\, \\alpha^{\\downarrow (l)^*}_{0} \\alpha^{\\downarrow(l)}_{j}\\alpha^{ \\downarrow(l')}_{0} \\alpha^{\\downarrow(l')^*}_{j} \\delta(\\omega - E_l + E_{l'}).\n\\end{align}\nIf the interactions are not too strong, the sum over intermediate states is dominated by the ``bare'' $\\alpha_0$ component. \\fix{In this case, $\\expval*{\\hat{O}_j \\hat{n}_\\downarrow \\hat{O}^\\dag_j}_\\beta \\simeq \\expval*{\\hat{O}_j \\hat{c}^\\dag_{{\\bf 0} \\downarrow} \\hat{c}_{{\\bf 0} \\downarrow} \\hat{O}^\\dag_j}_\\beta$} and thus\n\\begin{align} %\n \\mathcal{R}_\\downarrow (\\omega) \\simeq \\sum_{l,l'} \\abs*{\\alpha^{\\downarrow (l)}_{0}}^2 \\abs*{\\alpha^{ \\downarrow(l')}_{0}}^2 %\n \\delta(\\omega - E_l + E_{l'}).\n\\end{align}\nWe thus recognize the Rabi spectrum as developing according to a convolution of spectral functions \\textit{in the presence of} Rabi coupling:\n\\begin{align}\n{\\cal N}_\\downarrow(t)& \\simeq \n\\int d \\omega\\, d \\omega' \\, \\Tilde{A}_\\downarrow (\\omega) \\Tilde{A}_\\downarrow (\\omega') e^{-i (\\omega - \\omega')t}.\n\\end{align}\nHere \n\\begin{align}\n \\Tilde{A}_\\downarrow (\\omega) = -\\frac{1}{\\pi} \\Im[\\Tilde{G}_\\downarrow (\\omega)],\n\\end{align}\nis calculated from the impurity Green's function including Rabi coupling.\n\n\nWe \\fix{can approximate} the Rabi coupled Green's function $\\tilde G$ via the relation\n\\begin{align} \\label{eq:ApproxGreenRelation}\n \\Tilde{G}(\\omega) \\fix{\\simeq} \\mqty( G_\\uparrow^{-1}(\\omega) & \\Omega_0\/2 \\\\ \\Omega_0\/2 & G_\\downarrow^{-1}(\\omega))^{-1}\\fix{,}\n\\end{align}\n\\fix{where, for ease of notation,}\nwe define $\\Tilde{G}_\\downarrow (\\omega) \\equiv \\Tilde{G}_{22}(\\omega)$. \\fix{We point out that in using Eq.~\\eqref{eq:ApproxGreenRelation} to calculate $\\Tilde{G}_\\downarrow (\\omega)$, we are ignoring the coexistence of the spin-$\\downarrow$ impurity with excitations of the Fermi gas.}\nApproximating the decoupled Green's functions as\n\\begin{align} \n G_\\sigma (\\omega) \\simeq \\frac{Z_\\sigma}{\\omega - E_{+\\sigma} - \\delta_{\\sigma \\downarrow} \\Delta \\omega + i \\Gamma_\\sigma},\n\\end{align}\nwe find that\n\\begin{align} \\label{eq:ApproxRabi}\n {\\cal N}_\\downarrow(t) &\\simeq Z_\\downarrow^2 e^{- \\left(\\Gamma _{\\downarrow}+\\Gamma _{\\uparrow}\\right) t} \\left[\\frac{\\Gamma\n _{\\uparrow}-\\Gamma _{\\downarrow} }{\\sqrt{\\Omega _0^2 Z_{\\downarrow}\n Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n _{\\downarrow}\\right)^2}} \\sin \\left(t \\sqrt{\\Omega _0^2 Z_{\\downarrow}\n Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n _{\\downarrow}\\right)^2}\\right) \\right. \\nonumber\\\\\n &{}\\hspace{2em}+ \\left. \\left(1-\\frac{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}}{2\n \\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}-2 \\left(\\Gamma _{\\uparrow}-\\Gamma\n _{\\downarrow}\\right)^2}\\right) \\cos \\left(t \\sqrt{\\Omega _0^2 Z_{\\downarrow}\n Z_{\\uparrow}-\\left(\\Gamma _{\\uparrow}-\\Gamma\n _{\\downarrow}\\right)^2}\\right)+\\frac{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}}{2\n \\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}-2 \\left(\\Gamma _{\\uparrow}-\\Gamma\n _{\\downarrow}\\right)^2}\\right].\n\\end{align}\nHere, we have taken $ \\Delta \\omega = E_{+\\uparrow} -E_{+\\downarrow}$\nfor simplicity (i.e., on resonance Rabi oscillations). In the cases of interest where $Z_\\downarrow$ is slightly below 1, the Rabi oscillations are not normalised. However, this is simply an artefact of our approximation of $G_\\downarrow(\\omega)$ and is overcome by dividing Eq.~\\eqref{eq:ApproxRabi} by $Z_\\downarrow^2$.\n\nEquation~\\eqref{eq:ApproxRabi} allows us to immediately identify the Rabi frequency and damping\n\\begin{align}\n \\Omega &\\simeq \\sqrt{\\Omega _0^2 Z_{\\downarrow} Z_{\\uparrow}- \\left(\\Gamma _{\\uparrow}-\\Gamma_\\downarrow\\right)^2},\n \\label{eq:OmegaRabi}\\\\\n \\Gamma_R & \\simeq \\Gamma_\\downarrow + \\Gamma_\\uparrow.\n\\end{align}\nThese reduce to $\\Omega\\simeq \\sqrt{\\Omega _0^2 Z_{\\uparrow}-\\Gamma _{\\uparrow}^2}$ and $\\Gamma_R\\simeq \\Gamma_\\uparrow$ in the case of weak initial state interactions. Equation~\\eqref{eq:OmegaRabi} illustrates why a strong Rabi coupling is necessary in order to drive coherent oscillations once $\\Gamma_\\uparrow$ becomes appreciable, which is why the oscillations are strongly suppressed for the strongest repulsive interactions in the 2D case~(see Fig.~\\ref{fig:Rabi}). It also implies that we can estimate the quasiparticle residue by\n\\begin{align}\n Z_\\uparrow \\simeq \\frac{ \\Omega ^2 + \\Gamma_\\uparrow^2}{\\Omega _0^2}.\n\\end{align}\nFinally, assuming weak initial state interactions such that $Z_\\downarrow=1$ and $\\Gamma_\\downarrow=0$, and taking $\\Omega_0 Z_\\uparrow\\gtrsim\\Gamma_\\uparrow$, we arrive at the form in Eq.~\\eqref{eq:MainApproxRabi} of the main text:\n\\begin{align} \n {\\cal N}_\\downarrow(t) & \\simeq e^{- \\Gamma_\\uparrow t} \n \\left[\\frac12+\\frac12 \\cos \\left(t \\sqrt{\\Omega_0^2\n Z_\\uparrow-\\Gamma_\\uparrow^2}\\right)\\right].\n\\end{align}\n\n\nThe limiting feature of this effective model comes from our approximation of $\\hat n_\\downarrow \\simeq \\hat c_{{\\bf 0} \\downarrow}^\\dag \\hat c_{{\\bf 0} \\downarrow}$. For any $\\Gamma_\\uparrow, \\Gamma_\\downarrow >0$, this approximation will always lead to $\\mathcal{N}_\\downarrow(t) \\to 0$ for large $t$, which does not match the behaviour of Rabi oscillations in experiment or the TBM. However, at the intermediate times considered in Fig.~\\ref{fig:Rabi} it provides a good model of the actual oscillations.\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe non-decoupling of the relevant scales on a wide and continuous range of\nmagnitudes in many areas of physics has led to the invention (discovery) of\nthe renormalisation group (RG) \\cite{206}. Whereas they have been discovered\nin the framework of the perturbative (quantum field) theory, the RG\ntechniques tackle a nonperturbative physical phenomenon \\cite{425}.\nNonperturbative approaches are difficult to implement and to control, and\nduring a long time one has essentially carried on perturbative RG techniques\n(see, e.g., \\cite{4948}). Nowadays, the huge growth of the computing\ncapacity has greatly modified this behaviour pattern and, already since the\nbeginning of the ninety's, one has considered \\cite{4374} with a greater\nacuteness the exact RG equations (ERGEs) originally introduced by Wilson \n\\cite{Irvine}, Wegner and Houghton \\cite{414} in the seventy's and slightly\nreformulated by Polchinski \\cite{354} in the eighty's (for some reviews on\nthe ERGEs see \\cite{4595}).\n\nInitially, the ERGEs are integro-differential equations for the running\naction $S\\left[ \\phi ,t\\right] $ [assuming that $\\phi \\left( x\\right) $\ngenerically stands for some field with as many indices as necessary and $%\nt=-\\ln \\left( \\Lambda \/\\Lambda _{0}\\right) $ the logarithm of a running\nmomentum scale $\\Lambda $]. They have been extended to the running (average)\neffective action $\\Gamma \\left[ \\varphi ,t\\right] $ \\cite{4281,4374}. Such\ngeneral equations cannot be studied without the recourse to approximations\nor truncations. One of the most promising approximations is a systematic\nexpansion in powers of the derivative of the field (derivative expansion) \n\\cite{212} which yields a set of coupled nonlinear partial differential\nequations the number of which grows quickly with the order of the expansion.\nIn the simplest cases (e.g., for the scalar field), the determination of\nfixed points\\ (and of their stability) amounts to study ordinary\ndifferential equations (ODEs) with a two-point boundary value problem that\nmay be carried out numerically via a shooting (or a relaxation) method.\n\nA pure numerical study is in general not easy to implement and to control.\nFor example, in the shooting method, the discovery of the right adjustment\nof the parameters at the boundaries requires a good knowledge a priori of\ntheir orders of magnitude (initial guesses). It is thus interesting to\ndevelop concurrently some substitute analytical methods. A popular\nsubstitute to the ODEs of the derivative expansion is provided by an\nadditionnal expansion in powers of the field which yields a set of coupled\nalgebraic equations which may be solved analytically, at least with the help\nof a symbolic computation software. Various field expansions have been\nimplemented with more or less success \\cite{3478,3642,4192,3553}.\nUnfortunately, the methods proposed up to now, if they are easy to\nimplement, do not work in all cases and especially in the most famous and\nsimplest case of the Wilson-Polchinski ERGE \\cite{Irvine,354} (equation for\nthe running action $S\\left[ \\phi ,t\\right] $ with a smooth cutoff).\n\nThe object of this paper is to present two new substitute analytical methods\nfor studying ODEs which, at least in\\ the local potential approximation of\nthe derivative expansion (LPA), works for the Wilson-Polchinski ERGE. One of\nthe methods, recently proposed in \\cite{6110}, is a genuine analytical\napproximation scheme to two-point boundary value problems of ODEs. The other\nmethod is new. It is based on approximations of the solution looked for by\ngeneralized hypergeometric functions. It has a certain similarity with\nanother new and interesting method based on the representation of the\nsolution by Pad\\'{e} approximants just proposed in \\cite{6201} by P. Amore\nand F. M. Fernandez independantly from the present work. We illustrate the\neffectiveness of the two methods with the explicit consideration of two\nERGEs in the local potential approximation: the Wilson-Polchinski equation\nand the Litim optimized RG equation \\cite{5020} for the running effective\naction (named the Litim equation in the following).\\ Following a conjecture\nfirst stated in \\cite{5049,5902}, the equivalence of these two equations (in\nthe LPA) has been proven by Morris \\cite{5911} and recently been numerically\nillustrated \\cite{6137} with an unprecedented accuracy for the scalar field\nin three dimensions ($d=3$). This particular situation provides us with the\nopportunity of testing efficiently the various methods of study at hand.\n\nThe following of the paper is divided in five sections. In section \\ref%\n{Calculations}, we briefly present the direct numerical integration of the\nODEs for the scalar model\\ using the shooting method: determinations of the\nfixed point and the critical exponents for both the Wilson-Polchinski and\nLitim equations in the LPA (distinguishing between the even and odd\nsymmetries). A brief presentation of the currently used field expansion is\ngiven in section \\ref{WPE}. In section \\ref{BFG}, we analyse several aspects\nof the method of \\cite{6110} applying it to the study of the two equations.\nWe calculate this way the fixed point locations with high precision and\ncompare the results with the estimates obtained in section \\ref{Calculations}%\n. We show how the leading and the subleading critical exponents may be\nestimated using this recent method. In section \\ref{Ratios} we present a new\napproximate analytical method for ODEs which is based on the definition of\nthe generalized hypergeometric functions. We show that it is well adapted to\ntreat the Wilson-Polchinski case whereas the Litim case is less easily\ntreated. We relate these effects to the convergence properties of the series\nin powers of the field. Finally we summarize this work and conclude in\nsection \\ref{Conc}.\n\n\\section{Two-point boundary value problem in the LPA}\n\n\\label{Calculations}\n\nIn this section we briefly present the two-point boundary value problem to\nbe solved in the LPA of the ERGE. The Wilson-Polchinski equation is first\nchosen as a paradigm in section \\ref{WilPol}. The principal numerical\nresults obtained from the numerical integration of the ODE using the\nshooting method are given. In section (\\ref{Litim}), the Litim equation is\nalso studied.\n\n\\subsection{Wilson-Polchinski's flow equation for the scalar-field\\label%\n{WilPol}}\n\nThe original Wilson-Polchinski ERGE in the LPA expresses the evolution of\nthe potential $U\\left( \\phi ,t\\right) $ as varying the logarithm of the\nmomentum scale of reference $t=-\\ln \\left( \\Lambda \/\\Lambda _{0}\\right) $\n(with $\\phi \\in \n\\mathbb{R}\n$). In three dimensions, it reads:%\n\\begin{equation}\n\\dot{U}=U^{\\prime \\prime }-\\left( U^{\\prime }\\right) ^{2}-\\frac{1}{2}\\phi\nU^{\\prime }+3U\\,, \\label{eq:LPAV}\n\\end{equation}%\nin which $\\dot{U}\\equiv \\partial U\\left( \\phi ,t\\right) \/\\partial t$, $%\nU^{\\prime }\\equiv \\partial U\\left( \\phi ,t\\right) \/\\partial \\phi $, $%\nU^{\\prime \\prime }\\equiv \\partial ^{2}U\\left( \\phi ,t\\right) \/\\partial \\phi\n^{2}$.\n\n\\subsubsection{Fixed point equation\\label{FP1}}\n\nThe fixed point equation corresponds to $\\dot{U}=0$. It is a second order\nODE for the function $U\\left( \\phi \\right) $:%\n\\begin{equation}\nU^{\\prime \\prime }-\\left( U^{\\prime }\\right) ^{2}-\\frac{1}{2}\\phi U^{\\prime\n}+3U=0\\,, \\label{eq:FPF}\n\\end{equation}%\nthe solution of which (denoted $U^{\\ast }\\left( \\phi \\right) $ below)\ndepends on two integration constants which are fixed by two conditions. The\nfirst one comes from a property of symmetry assumed to be\\footnote{%\nThe other possibility $U^{\\ast }\\left( -\\phi \\right) =-U^{\\ast }\\left( \\phi\n\\right) $ gives only singular solutions at finite $\\phi $.} $U^{\\ast }\\left(\n-\\phi \\right) =U^{\\ast }\\left( \\phi \\right) $ which provides the following\ncondition at the origin for $U^{\\ast }\\left( \\phi \\right) $:%\n\\begin{equation}\nU^{\\ast \\prime }\\left( 0\\right) =0\\,. \\label{eq:fori}\n\\end{equation}%\nThe second condition is the requirement that the solution we are interested\nin must be non singular in the entire range $\\phi \\in \\left[ 0,\\infty \\right[\n$. Actually, the general solution of (\\ref{eq:FPF}) involves a moving\nsingularity \\cite{2080} of the form:%\n\\begin{equation}\nU_{\\text{sing}}=-\\ln \\left\\vert \\phi _{0}-\\phi \\right\\vert \\,,\n\\label{eq:sing}\n\\end{equation}%\ndepending on the arbitrary constant $\\phi _{0}$. Pushing $\\phi _{0}$ to\ninfinity allows to get a non-singular potential since, in addition to the\ntwo trivial fixed points $U^{\\ast }\\equiv 0$ (Gaussian fixed point) and $%\nU^{\\ast }\\equiv -\\frac{1}{3}+\\frac{{\\phi }^{2}}{2}$ (high temperature fixed\npoint), eq.(\\ref{eq:FPF}) admits a non-singular solution which, for $\\phi\n\\rightarrow \\infty $, has the form:%\n\\begin{equation}\nU_{\\text{asy}}(\\phi )=\\frac{{\\phi }^{2}}{2}+b\\,{\\phi }^{\\frac{6}{5}}+\\frac{%\n18\\,b^{2}\\,{\\phi }^{\\frac{2}{5}}}{25}-\\frac{1}{3}+\\frac{108\\,b^{3}}{625\\,{%\n\\phi }^{\\frac{2}{5}}}+O\\left( \\phi ^{-4\/5}\\right) \\,, \\label{eq:fasy}\n\\end{equation}%\nin which $b$ is the only remaining arbitrary integration constant. The non\ntrivial (Wilson-Fisher \\cite{439}) fixed point solution which we are\ninterested in must interpolate between eqs. (\\ref{eq:fori}) and (\\ref%\n{eq:fasy}). Imposing these conditions fixes uniquely the value $b^{\\ast }$\nof $b$ which corresponds to the fixed point solution we are looking for$.$\n\nWe have determined $b^{\\ast }$ by using the shooting method \\cite{5465}:\nstarting from a value $\\phi _{a}$ supposed to be large where the condition (%\n\\ref{eq:fasy}) is imposed (with a guess, or trying, value of $b\\simeq\nb^{\\ast }$), we integrate the differential equation (\\ref{eq:FPF}) toward\nthe origin where the condition (\\ref{eq:fori}) is checked (shooting to the\norigin), we adjust the value of $b$ to $b^{\\ast }$ so as the latter\ncondition\\ is satisfied with a required accuracy. A study of the stability\nof the estimate of $b^{\\ast }$ so obtained on varying the value $\\phi _{a}$\nprovides some information on the accuracy of the calculation.\n\nRather than (\\ref{eq:fasy}), it is more usual to characterize the fixed\npoint solution from its small field behaviour:\n\n\\begin{equation}\nU(\\phi )=k-\\frac{3\\,k\\,}{2}\\phi ^{2}+\\frac{k\\,\\left( 1+3\\,k\\right) \\,}{4}%\n\\phi ^{4}-\\frac{k\\,\\left( 1+3\\,k\\right) \\,\\left( 1+24\\,k\\right) \\,}{120}\\phi\n^{6}+O\\left( \\phi ^{8}\\right) \\,, \\label{eq:Uori}\n\\end{equation}%\nand to provide the value of either of the two (related) quantities: \n\\begin{eqnarray}\nk^{\\ast } &=&U^{\\ast }\\left( 0\\right) \\,, \\label{eq:kstarWP} \\\\\nr^{\\ast } &=&U^{\\ast \\prime \\prime }\\left( 0\\right) =-3k^{\\ast }\\,.\n\\label{eq:rstar}\n\\end{eqnarray}\n\nIn the shooting-to-origin method, the determination of $r^{\\ast }$ (or $%\nk^{\\ast }$)\\ is a byproduct of the adjustment of $b^{\\ast }$.\n\nThe adjustment of $b^{\\ast }$ may be bypassed by shooting \\emph{from} the\norigin toward $\\phi _{a}$, then $r^{\\ast }$ is adjusted in such a way as to\nreach the largest possible value of $\\phi _{a}$. In that case $b^{\\ast }$ is\na byproduct of the adjustment.\n\nBecause the boundary condition at $\\phi _{a}$ is under control, the shooting-%\n\\emph{to}-origin method provides a better determination of $r^{\\ast }$ than\nthe shooting-\\emph{from}-origin method. However, this latter method is more\nflexible and may easily yield a rough estimate on $r^{\\ast }$ which can be\nused as a guess in a more demanding management of the method. Notice that,\ndue to the increase of the number of adjustable parameters, this way of\ndetermining a guess is no longer possible in a study involving several\ncoupled EDOs. Consequently, the development of other methods as, for\nexample, those two presented below is useful to this purpose (see also \\cite%\n{6201}).\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$r^{\\ast }$ & & $b^{\\ast }$ & & $\\phi _{a}$ \\\\ \\hline\n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,022\\,0$} & & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 296\\,3$} & & $10$ \\\\ \n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,021\\,9$} & & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 311\\,6$} & & $20$ \\\\ \n\\multicolumn{1}{l}{$-0.228\\,598\\,202\\,437\\,021\\,9$} & & \\multicolumn{1}{l}{$%\n\\allowbreak -2.\\,\\allowbreak 316\\,2$} & & $40$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ The fixed point parameter $r^{\\ast }$ is already well determined\nfor rather small values of $\\protect\\phi _{a}$ whereas $b^{\\ast }$ [fixed\npoint value of $b$ in (\\protect\\ref{eq:fasy})] still is not. }\n\\label{Table 1}\n\\end{table}\n\nTable \\ref{Table 1} displays the determinations of $r^{\\ast }$ and $b^{\\ast\n} $ for three values of $\\phi _{a}$. One may observe that a high accuracy on \n$r^{\\ast }$ is required to reach a yet small value of $\\phi _{a}$ whereas $%\nb^{\\ast }$ is only poorly determined. Obviously, considering higher values\nof $\\phi _{a}$ and\/or higher order terms in eq. (\\ref{eq:fasy}) allows to\nbetter determine $b^{\\ast }$, one more term in (\\ref{eq:fasy}) and $\\phi\n_{a}=1000$ yields:%\n\\begin{equation}\nb^{\\ast }=-2.318\\,29\\,, \\label{eq:bstar}\n\\end{equation}%\nbut the estimate of $r^{\\ast }$ is not improved compared to the values given\nin table \\ref{Table 1} (the machine-precision was already reached). We\nfinally extract from table \\ref{Table 1} our best estimate of $r^{\\ast }$\n(or $k^{\\ast }$) as obtained from the study of the fixed point equation (\\ref%\n{eq:FPF}) alone:%\n\\begin{eqnarray}\nr^{\\ast } &=&-0.228\\,598\\,202\\,437\\,022\\pm 10^{-15}\\,\\,, \\label{eq:rstar1}\n\\\\\nk^{\\ast } &=&0.076\\,199\\,400\\,812\\,340\\,7\\pm 10^{-16}\\,. \\label{eq:kstar1}\n\\end{eqnarray}\n\nIndividually, these values do not define the potential function $U^{\\ast\n}\\left( \\phi \\right) $ the knowledge of which requires the numerical\nintegration explicitly performed in the shooting method.\n\n\\subsubsection{Eigenvalue equation}\n\nThe critical exponents are obtained by linearizing the flow equation (\\ref%\n{eq:LPAV}) near the fixed point solution $U^{\\ast }\\left( \\phi \\right) $. If\none inserts:%\n\\begin{equation*}\nU\\left( \\phi ,t\\right) =U^{\\ast }\\left( \\phi \\right) +\\epsilon \\,e^{\\lambda\nt}g\\left( \\phi \\right) \\,,\n\\end{equation*}%\ninto the flow equation and keeps the linear terms in $\\epsilon $, one\nobtains the eigenvalue equation:%\n\\begin{equation}\ng^{\\prime \\prime }-2\\,g^{\\prime }U^{\\ast \\prime }-\\frac{\\phi }{2}\\,g^{\\prime\n}+\\left( 3-\\lambda \\right) \\,g=0\\,. \\label{eq:VPeq}\n\\end{equation}\n\nAgain it is a second order ODE the solutions of which are characterized by\ntwo integration constants.\n\nSince $U^{\\ast }\\left( \\phi \\right) $ is an even function of $\\phi $, eq. (%\n\\ref{eq:VPeq}) is invariant under a parity change. Then one of the\nintegration constants is fixed by looking for either an even or an odd\neigenfunction $g\\left( \\phi \\right) $ which implies either $g^{\\prime\n}\\left( 0\\right) =0$ (even) or $g\\left( 0\\right) =0$ (odd). The second\nintegration constant is fixed at will due to the arbitrariness of the\nnormalisation of an eigenfunction. Thus, assuming either $g\\left( 0\\right)\n=1 $ (even) or $g^{\\prime }\\left( 0\\right) =1$ (odd), the solutions of (\\ref%\n{eq:VPeq}) depend only on $\\lambda $ and on the fixed point parameter $%\nk^{\\ast }$. For example, these solutions have the following expansions about\nthe origin $\\phi =0$ :\n\n\\begin{eqnarray*}\n&&g_{\\text{even}}\\left( \\phi \\right) =1+\\frac{\\left( \\lambda -3\\right) }{2}%\n\\phi ^{2}\\left[ 1+\\frac{\\,\\,\\left( \\lambda -2-12\\,k^{\\ast }\\right) }{12}\\phi\n^{2}\\right] +O\\left( \\phi ^{6}\\right) \\,, \\\\\n&&g_{\\text{odd}}\\left( \\phi \\right) =\\phi +\\frac{\\,\\left( 2\\,\\lambda\n-5-12\\,k^{\\ast }\\right) }{12}\\phi ^{3}+O\\left( \\phi ^{5}\\right) \\,.\n\\end{eqnarray*}%\nWhen the fixed point solution $U^{\\ast }$ is known, the values of $\\lambda $\n[the only remaining unknown parameter in (\\ref{eq:VPeq})] are determined by\nlooking for the solutions which interpolate between either $g^{\\prime\n}\\left( 0\\right) =0$ (even) or $g\\left( 0\\right) =0$ (odd) and the regular\nsolution of (\\ref{eq:VPeq}) \\bigskip which, for $\\phi \\rightarrow \\infty $,\nis:%\n\\begin{equation}\ng_{\\text{asy}}(\\phi )=S_{0}{\\phi }^{\\frac{2\\,\\left( 3-\\lambda \\right) }{5}%\n}\\left\\{ 1+\\left( 3-\\lambda \\right) \\left[ \\frac{12\\,b^{\\ast }\\,}{25\\,{\\phi }%\n^{\\frac{4}{5}}}-\\frac{36\\,b^{\\ast 2}\\,\\left( 2\\,\\lambda -3\\right) }{625\\,{%\n\\phi }^{\\frac{8}{5}}}+\\frac{2\\,\\,\\left( 2\\,\\lambda -1\\right) }{125\\,{\\phi }%\n^{2}}+O\\left( {\\phi }^{-\\frac{12}{5}}\\right) \\right] \\right\\} \\,,\n\\label{eq:gasy}\n\\end{equation}%\nin which $b^{\\ast }$ is given by (\\ref{eq:bstar}). The value of $S_{0}$ is\nrelated to the choice of the normalisation of the eigenfunction at the\norigin, it is a byproduct of the adjustment in a shooting-\\emph{from}-origin\nprocedure.\n\nIn the even case, it is known that the first nontrivial positive eigenvalue $%\n\\lambda _{1}$ (there is also the trivial value $\\lambda _{0}=d=3$), is\nrelated to the critical exponent $\\nu $ which characterizes the Ising-like\ncritical scaling of the correlation length $\\xi $. One has $\\nu =1\/\\lambda\n_{1}$ and the first negative eigenvalue, $\\lambda _{2}$, is minus the\nIsing-like first correction-to-scaling exponent $\\omega _{1}$ ($\\omega\n_{1}=-\\lambda _{2}$) and so on.\n\nIn the odd case, the two first (positive) eigenvalues are trivial in the\nLPA. One has:%\n\\begin{eqnarray}\n\\breve{\\lambda}_{1} &=&\\frac{d+2-\\eta }{2}\\,, \\label{eq:lambdac1} \\\\\n\\breve{\\lambda}_{2} &=&\\frac{d-2+\\eta }{2}\\,, \\label{eq:lambdac2}\n\\end{eqnarray}%\nin which $\\eta $ is the critical exponent which governs the large distance\nbehaviour of the correlation functions\\ right at the critical point, it\nvanishes in the LPA. With the dimension $d=3$ and the approximation (LPA)\npresently considered, (\\ref{eq:lambdac1}) and (\\ref{eq:lambdac2}) reduce to $%\n\\breve{\\lambda}_{1}=2.5$ and $\\breve{\\lambda}_{2}=0.5$. Consequently the\nfirst non-trivial eigenvalue is negative and defines the subcritical\nexponent $\\theta _{5}=\\breve{\\omega}_{1}=-\\breve{\\lambda}_{3}$ sometimes\nconsidered to characterize the deviation of the critical behaviour of fluids\nfrom the pure Ising-like critical behaviour.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$\\nu $ & & $b^{\\ast }$ & & $\\phi _{a}$ \\\\ \\hline\n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,11$} & & \\multicolumn{1}{l}{$%\n-2.\\,\\allowbreak 318\\,145$} & & \\multicolumn{1}{l}{$12$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,80$} & & \\multicolumn{1}{l}{$%\n-2.318\\,257$} & & \\multicolumn{1}{l}{$22$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,65$} & & \\multicolumn{1}{l}{$%\n-2.318\\,280$} & & \\multicolumn{1}{l}{$32$} \\\\ \n\\multicolumn{1}{l}{$0.649\\,561\\,773\\,880\\,65$} & & \\multicolumn{1}{l}{$%\n-2.318\\,285$} & & \\multicolumn{1}{l}{$40$} \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Values of the critical exponent $\\protect\\nu $ determined together\nwith $b^{\\ast }$ (and thus $r^{\\ast }$) whereas $\\protect\\phi _{a}$ is\nvaried. Compared to table \\protect\\ref{Table 1}, a better determination of $%\nb^{\\ast }$ is obtained [see the best value of $b^{\\ast }$ \\ given by eq. (%\n\\protect\\ref{eq:bstar})].}\n\\label{Table 2}\n\\end{table}\n\nTo determine the eigenvalues we use again the shooting-to-origin method with\nthe two equations (\\ref{eq:FPF}, \\ref{eq:VPeq}). However, in addition to $%\n\\lambda $, we leave also $b^{\\ast }$ adjustable instead of fixing it to the\nvalue given in (\\ref{eq:bstar}).\n\nIn the even case, the values we obtain\\ for $\\nu $ and $b^{\\ast }$ are shown\nin table \\ref{Table 2} for four values of $\\phi _{a}$. Comparing with the\nvalues displayed in table \\ref{Table 1} one observes a better convergence of \n$b^{\\ast }$ to the best value (\\ref{eq:bstar}) whereas $r^{\\ast }$ remains\nunchanged compared to (\\ref{eq:rstar1}). As for the best estimate of $\\nu $,\nit is:%\n\\begin{equation}\n\\nu _{\\mathrm{best}}=0.649\\,561\\,773\\,880\\pm 10^{-12}\\,, \\label{eq:nubest}\n\\end{equation}%\nthat is to say:%\n\\begin{equation}\n\\lambda _{1\\mathrm{best}}=1.539\\,499\\,459\\,808\\pm 10^{-12}\\,.\n\\label{eq:l1best}\n\\end{equation}\n\nWe have proceeded similarly to determine the Ising-like subcritical exponent\nvalues displayed in table \\ref{Table 3}.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccccccccc}\n\\hline\\hline\n$\\omega _{1}$ & & $\\omega _{2}$ & & $\\omega _{3}$ & & $\\omega _{4}$ & & $%\n\\omega _{5}$ & & $\\omega _{6}$ \\\\ \\hline\n$0.655\\,745\\,939\\,193$ & & $3.180\\,006\\,512\\,059$ & & $5.912\\,230\\,612$ & \n& $8.796\\,092\\,825$ & & $11.798\\,087\\,66$ & & $14.896\\,053\\,176$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Best estimates of the six first subcritical exponents for the\nIsing-like scalar model (i.e. even case), all digits are significant.}\n\\label{Table 3}\n\\end{table}\n\nIn the odd case, we obtain: \n\\begin{equation}\n\\breve{\\omega}_{1}=1.886\\,703\\,838\\,091\\pm 10^{-12}\\,. \\label{eq:omegac}\n\\end{equation}\n\nTable \\ref{Table 4} displays the values of the other subcritical exponents\nof the same family as $\\breve{\\omega}$ but with a lower accuracy. Of course,\nthe values presently obtained are in agreement with the previous estimates \n\\cite{3491,6137}.\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline\\hline\n$\\breve{\\omega}_{2}$ & & $\\breve{\\omega}_{3}$ & & $\\breve{\\omega}_{4}$ \\\\ \n\\hline\n$4.524\\,390\\,734$ & & $7.337\\,650\\,643$ & & $10.283\\,900\\,73$ \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\caption{ Best estimates of the odd-case subcritical exponents other than $%\n\\breve{\\protect\\omega}_{1}$ for the scalar model.}\n\\label{Table 4}\n\\end{table}\n\n\\subsection{ Litim's flow equation for the scalar field \\label{Litim}}\n\nFollowing a conjecture first stated in \\cite{5049,5902}, the equivalence in\nthe LPA between the Wilson-Polchinski flow (\\ref{eq:LPAV}) and the Litim\noptimized ERGE \\cite{5020} for the running effective action $\\Gamma \\left[\n\\varphi ,t\\right] $ has been proven by Morris \\cite{5911}. The Litim flow\nequation for the potential $V\\left( \\varphi ,t\\right) $ reads in three\ndimensions (compared to \\cite{5911} an unimportant shift $V\\rightarrow V-1\/3$\nis performed):%\n\\begin{equation}\n\\dot{V}=1-\\frac{1}{1+V^{\\prime \\prime }}-\\frac{\\varphi }{2}V^{\\prime }+3V\\,.\n\\label{eq:LPAVLitim}\n\\end{equation}\n\nIt is related to (\\ref{eq:LPAV}) via the following Legendre transformation:%\n\\begin{equation}\n\\left. \n\\begin{array}{l}\n\\left[ \\frac{1}{2}\\phi ^{2}-U\\left( \\phi ,t\\right) \\right] +\\left[ \\frac{1}{2%\n}\\varphi ^{2}+V\\left( \\varphi ,t\\right) \\right] =\\varphi \\phi \\\\ \n\\varphi =\\phi -U^{\\prime }\\left( \\phi ,t\\right)%\n\\end{array}%\n\\right\\} \\,. \\label{eq:Legendre}\n\\end{equation}\n\nThe general solution of the fixed point equation ($\\dot{V}=0$) involves the\nfollowing moving \\textquotedblleft singularity\\textquotedblright\\ ($%\nV^{\\prime \\prime }$ is singular)\\ at the arbitrary point $\\varphi _{0}$:%\n\\begin{equation}\nV_{\\text{sing}}\\left( \\varphi \\right) =-\\frac{1}{3}+\\frac{4}{3\\sqrt{\\varphi\n_{0}}}\\left\\vert \\varphi _{0}-\\varphi \\right\\vert ^{3\/2}\\,. \\label{eq:singL}\n\\end{equation}\n\n\\subsubsection{Fixed point solution}\n\nThe numerical study of the fixed point solution of (\\ref{eq:LPAVLitim})\nfollows the lines described in the preceding sections. This may be done\nindependently, but due to (\\ref{eq:Legendre}), one may already deduce from\nthe previous study the expected results. Similarly to (\\ref{eq:fasy}), the\nasymptotic behaviour of the non trivial fixed point potential is\ncharacterized by the integration constant $b_{L}$ in the following\nexpression [deduced from (\\ref{eq:LPAVLitim})]:%\n\\begin{equation}\nV_{\\mathrm{asy}}(\\varphi )=b_{L}\\,{\\varphi }^{6}-\\frac{1}{3}+\\frac{1}{%\n150\\,b_{L}\\,{\\varphi }^{4}}-\\frac{1}{6300\\,b_{L}^{2}\\,{\\varphi }^{8}}%\n+O\\left( {\\varphi }^{-12}\\right) \\,. \\label{eq:asyv}\n\\end{equation}\n\nIt is easy to show from (\\ref{eq:fasy}) and (\\ref{eq:Legendre}) that the\nvalue $b_{L}^{\\ast }$ we are looking for is related to $b^{\\ast }$ as\nfollows:%\n\\begin{equation*}\nb_{L}^{\\ast }\\,=-\\frac{1}{6^{6}}\\left( \\frac{5}{b^{\\ast }}\\right) ^{5},\n\\end{equation*}%\nthen, from the previous result (\\ref{eq:bstar}) we get:%\n\\begin{equation}\nb_{L}^{\\ast }\\simeq 0.001\\,000\\,25\\,. \\label{eq:bstarL}\n\\end{equation}\n\nSimilarly for the potential parameters \n\\begin{eqnarray*}\nk_{L}^{\\ast } &=&V^{\\ast }\\left( 0\\right) \\,, \\\\\nr_{L}^{\\ast } &=&V^{\\ast \\prime \\prime }\\left( 0\\right) \\,,\n\\end{eqnarray*}%\nwhich correspond to $b_{L}^{\\ast }$, they are related to the\nWilson-Polchinski counterparts $k^{\\ast }$ and $r^{\\ast }$ as follows:%\n\\begin{eqnarray}\nk_{L}^{\\ast } &=&k^{\\ast }\\,, \\label{eq:kstarL0} \\\\\nr_{L}^{\\ast } &=&\\frac{r^{\\ast }}{1-r^{\\ast }}\\,. \\label{eq:rstarL0}\n\\end{eqnarray}%\nThis latter relation, using (\\ref{eq:rstar1}), gives:%\n\\begin{equation}\nr_{L}^{\\ast }\\simeq -0.186\\,064\\,249\\,470\\,314\\pm 10^{-15}\\,.\n\\label{eq:rstarL}\n\\end{equation}\n\nAs precedingly, those values do not provide the potential function $V^{\\ast\n}\\left( \\varphi \\right) $ the knowledge of which requires an explicit\nnumerical integration.\n\n\\subsubsection{Eigenvalue equation}\n\nA linearization of the flow equation (\\ref{eq:LPAVLitim}) near the fixed\npoint solution $V^{\\ast }\\left( \\varphi \\right) $:%\n\\begin{equation*}\nV\\left( \\varphi ,t\\right) =V^{\\ast }\\left( \\varphi \\right) +\\epsilon\n\\,e^{\\lambda t}h\\left( \\varphi \\right) \\,,\n\\end{equation*}%\nprovides the Litim eigenvalue equation:%\n\\begin{equation}\n\\left( 3-\\lambda \\right) \\,h-\\frac{\\varphi \\,h^{\\prime }}{2}+\\frac{h^{\\prime\n\\prime }}{{\\left( 1+{V^{\\ast }}^{\\prime \\prime }\\right) }^{2}}=0\\,.\n\\label{eq:VPeqLitim}\n\\end{equation}\n\nTaking into account (\\ref{eq:asyv}), one can show that (\\ref{eq:VPeqLitim})\nadmits a regular solution which, for $\\varphi \\rightarrow \\infty $, has the\nform:%\n\\begin{equation}\nh_{\\mathrm{asy}}\\left( \\varphi \\right) =S_{1}{\\varphi }^{2\\,\\left( 3-\\lambda\n\\right) }\\left\\{ 1-\\left( \\lambda -3\\right) \\,\\left( 2\\,\\lambda -5\\right) %\n\\left[ \\frac{1}{2250\\,b_{L}^{\\ast 2}\\,{\\varphi }^{10}}-\\frac{1}{%\n47250\\,b_{L}^{\\ast 3}\\,{\\varphi }^{14}}+O\\left( \\varphi ^{-18}\\right) \\right]\n\\right\\} \\,, \\label{eq:hasy}\n\\end{equation}%\nin which $b_{L}^{\\ast }$ is given by (\\ref{eq:bstarL}). In the following we\nmay set $S_{1}=1$ since the normalisation of the eigenfunction may be chosen\nat will.\n\nAs precedingly, we must distinguish between the odd and even eigenfunction $%\nh\\left( \\varphi \\right) $. The shooting method gives the same values as in\nthe Wilson-Polchinski case (see \\cite{5049,5252,5625,6137}) and we do not\npresent them again.\n\n\\section{Expansion in powers of the field}\n\n\\label{WPE}\n\nIn advanced studies of the derivative expansion \\cite{5469} or other\nefficient approximations of the ERGE \\cite{5903} and in the consideration of\ncomplex systems via the ERGEs \\cite{5677}, a supplementary truncation in\npowers of the field is currently used (see also \\cite{4595}). With a scalar\nfield, this expansion transforms the partial differential flow equations\ninto ODEs whereas the fixed point or eigenvalue ODEs are transformed into\nalgebraic equations. Provided auxiliary conditions are chosen, the latter\nequations are easy to solve analytically using a symbolic computation\nsoftware. Actually the auxiliary conditions currently chosen are extremely\nsimple: they consist in setting equal to zero the highest terms of the\nexpansion so as to get a balanced system of equations.\n\nA first kind of expansion, about the zero field --referred to as the\nexpansion I in the following, has been proposed by Margaritis et al \\cite%\n{3478} and applied to the LPA of Wegner-Houghton's ERGE \\cite{414} (the hard\ncutoff version of the Wilson-Polchinski equation). A second kind of\nexpansion, relative to the (running) minimum of the potential (expansion\nII), has been proposed by Tetradis and Wetterich \\cite{3642} and more\nparticularly presented by Alford \\cite{4192} using it, again, with the sharp\ncutoff version of the ERGE.\n\nIt is known that, for the Wegner-Houghton equation in the LPA, expansion I\ndoes not converge due to the presence of singularities in the complex plane\nof the expansion variable \\cite{3358}. Expansions I and II have been more\nconcretely studied and compared to each other by Aoki et al in \\cite{3553}\nwho also propose a variant to II (expansion III) by letting the expansion\npoint adjustable. They showed, again on the LPA of\\ the Wegner-Houghton\nequation, that expansion II is much more efficient than expansion I although\nit finally does not converge and expansion III is the most efficient one.\nExpansions II and III work well also on the ERGE expressed on the running\neffective action (effective average action, see the review by Berges et al\nin \\cite{4595}). The convergence of those expansions have also been studied\nin \\cite{5252} according to the regularisation scheme chosen and in\nparticular for the Litim equation (\\ref{eq:LPAVLitim}). In this latter study\nit is concluded that both expansions I and II seem to converge although II\nconverges faster than I.\n\nA striking fact emerges from those studies, the Wilson-Polchinski equation\nin the LPA, the simplest equation, is never studied using the field\nexpansion method. The reason is simple: none of the expansions currently\nused works in that case.\n\nActually the strategy of these methods, which consists in arbitrarily\nsetting equal to zero one coefficient for the expansion I and two for the\nexpansions II and III, is probably too simple. With regards to this kind of\nauxiliary conditions, the failure observed with the Wilson-Polchinski\nequation is not surprising and, most certainly, there should be many other\ncircumstances where such simple auxiliary conditions would not solve\ncorrectly the derivative expansion of an ERGE.\n\nIn the following sections we examine two alternative methods with more\nsophisticated auxiliary conditions. We show that they yield the correct\nsolution for the Wilson-Polchinski and its Legendre transformed (Litim)\nequations. Both methods are associated to expansion I (about the\nzero-field). The first one has recently been proposed in \\cite{6110} as a\nmethod to treat the two point boundary value problem of ODEs. It relies upon\nan efficient account for the large field behaviour of the solution looked\nfor. An attempt of accounting for this kind of behaviour within the field\nexpansion had already been done by Tetradis and Wetterich via their eq.\n(7.11) of \\cite{3642}. In the present work, a much more sophisticated\nprocedure is used. It relies upon the construction of an added auxiliary\ndifferential equation (ADE). We refer to it in the following as the ADE\nmethod. The second method is new. It relies upon the approximation of the\nsolution looked for by a generalized hypergeometric function. We refer to it\nin the following as the\\ hypergeometric function approximation (HFA) method.\n\n\\section{Auxiliary differential equation method}\n\n\\label{BFG}\n\nLet us first illustrate the auxiliary differential equation (ADE) method on\nthe search for the non trivial fixed point in the LPA for both the\nWilson-Polchinski equation (\\ref{eq:FPF}) and the Litim optimized equation (%\n\\ref{eq:LPAVLitim}). Since there are two boundaries (the origin and the\n\"point at\" infinity), we distinguish between two strategies.\n\n\\begin{itemize}\n\\item An expansion about the origin in the equations (small field expansion)\nand the account for the leading high field behaviour of the regular solution\nwhich we are looking for. This determines the value of $r^{\\ast }$ or $%\nr_{L}^{\\ast }$.\n\n\\item A change of variable $\\phi \\rightarrow 1\/\\phi $ or $\\varphi\n\\rightarrow 1\/\\varphi $ which reverses the problem: an expansion about\ninfinity (new origin) in the equations (high field expansion) and the\naccount for the leading small field behaviour of the regular solution which\nwe are looking for. This determines the value of $b^{\\ast }$ or $b_{L}^{\\ast\n}$.\n\\end{itemize}\n\n\\subsection{Wilson-Polchinski's fixed point}\n\n\\subsubsection{ Small field expansion and leading high field behaviour\\label%\n{WPsmallField}}\n\nFor practical and custom reasons\\footnote{%\nThe change $x=\\phi ^{2}$ is useful in practice to avoid some degeneracies\nobserved in \\cite{6110} when forming the auxiliary differential equation.\nTaking the derivative $f=U^{\\prime }$ is only a question of habit.}, instead\nof (\\ref{eq:FPF}) we consider the equation satisfied by the function $%\nw\\left( x\\right) $ related to the derivative of the potential $U^{\\prime\n}\\left( \\phi \\right) $ as follows:\n\n\\begin{equation}\nU^{\\prime }\\left( \\phi \\right) =\\phi \\,w\\left( \\phi ^{2}\\right) \\,,\n\\label{eq:chgt1}\n\\end{equation}%\nso that, with $x=\\phi ^{2}$, the fixed point equation (\\ref{eq:FPF}) reads:\n\n\\begin{equation}\n4\\,x\\,w^{\\prime \\prime }-2\\,{w}^{2}-4\\,x\\,w\\,w^{\\prime }+\\left( 6-\\,x\\right)\n\\,\\,w^{\\prime }+2\\,w=0\\,, \\label{eq:FPw}\n\\end{equation}%\nin which a prime indicates a derivative with respect to $x$.\n\nThis second order ODE has a singular point at the origin and, by analyticity\nrequirement, the solution we are looking for depends on a single unknown\nintegration-constant (noted $r$ below).\n\nLet us first introduce the expansion I of Margaritis et al \\cite{3478}. The\nfunction $w\\left( x\\right) $ is expanded up to order $M$ in powers of $x$:%\n\\begin{equation}\nw_{M}\\left( x\\right) =r+\\sum\\limits_{n=1}^{M}a_{n}x^{n}\\,, \\label{eq:wm}\n\\end{equation}%\nand inserted into the fixed point equation (\\ref{eq:FPw}).\n\nRequiring that (\\ref{eq:FPw}) be satisfied order by order in powers of $x$\nprovides an unbalanced system of $M$ algebraic equations with $M+1$ unknown\nquantities $\\left\\{ r,a_{1},\\cdots ,a_{M}\\right\\} $ [eq. (\\ref{eq:FPw}) is\nthen satisfied up to order $M-1$ in powers of $x$]. With a view to balancing\nthe system, $a_{M}=0$ is simply set equal to zero and if the solution\ninvolves a stable value $r_{M}^{\\ast }$ as $M$ grows, then it constitutes\nthe estimate at order $M$ of the fixed point location corresponding to\nexpansion I. As already mentioned, in the case of the Wilson-Polchinski\nequation (\\ref{eq:FPw}) under study, the method fails: all the values\nobtained for $r_{M}^{\\ast }$ are positive whatever the value of $M$ whereas\nthe correct value should be negative as shown in section \\ref{FP1}.\n\nIn the ADE method, the condition $a_{M}=0$ is not imposed. The previous\nalgebraic system is first solved in terms of the unknown parameter $r$ \\ so\nas to get the generic solution of (\\ref{eq:FPw}) at order $M$ in powers of $%\nx $:%\n\\begin{equation}\nw_{M}\\left( r;x\\right) =r+\\sum\\limits_{n=1}^{M}a_{n}\\left( r\\right) x^{n}\\,.\n\\label{eq:wmrx}\n\\end{equation}\n\nIn order to get a definite value for $r$, instead of arbitrarily imposing $%\na_{M}\\left( r\\right) =0$, an auxiliary condition is formed which explicitly\naccounts for the behaviour at large $\\phi $ given by (\\ref{eq:fasy}). With $%\nw\\left( x\\right) $, this behaviour corresponds to:%\n\\begin{eqnarray}\n&&w_{\\text{asy}}(x)\\underset{x\\rightarrow \\infty }{=}1\\,, \\label{eq:asyw} \\\\\n&&w_{\\text{asy}}^{\\prime }(x)\\underset{x\\rightarrow \\infty }{=}0\\,.\n\\label{eq:asyw'}\n\\end{eqnarray}%\nThe auxiliary condition is obtained via the introduction of an auxiliary\ndifferential equation:\n\n\\begin{itemize}\n\\item Consider a first order differential equation for $w\\left( x\\right) $\nconstructed as a polynomial of degree $s$ (eventually incomplete) in powers\nof the pair $\\left( w,w^{\\prime }\\right) $: \n\\begin{equation}\nG_{1}+G_{2}\\,w+G_{3}\\,w^{\\prime }+G_{4}\\,w^{2}+G_{5}\\,w\\,w^{\\prime\n}+G_{6}w^{\\prime 2}+\\cdots +G_{n}\\,w^{s-q}\\,w^{\\prime q}=0\\,,\n\\label{eq:auxil1}\n\\end{equation}%\nin which, when the degree $s$ of the polynomial is saturated then $q=s$ and\nthe number $n$ of coefficients $G_{i}$ is equal to $\\left( s+1\\right) \\left(\ns+2\\right) \/2$, conversely when it is not\\ then $0\\leq q-2$ in the even case and $\\lambda >1\/2$ in the odd case. Hence one\ncould expect that, with the simple condition at infinity: $\\mathrm{v}=%\n\\mathrm{v}^{\\prime }=0$ imposed in the auxiliary differential equation, the\nADE procedure will, at best, allow the determination of exclusively the\nleading ($\\lambda _{1}=1\/\\nu $) and first subleading ($\\lambda _{2}=-\\omega\n_{1}$) eigenvalues in the even case and of only the trivial eigenvalue $%\n\\breve{\\lambda}_{1}=-\\breve{\\omega}_{1}$ in the odd case [see the values of\nthese quantities in eqs. (\\ref{eq:l1best}, \\ref{eq:omegac}) and tables (\\ref%\n{Table 3}, \\ref{Table 4})]. Actually it is better than that since, as $M$\ngrows, we observe\\ among the real roots of the auxiliary condition for $%\n\\lambda $ that a hierarchy of successive accumulations takes place about the\nright values of the leading and subsequent eigenvalues [see figure \\ref{fig4}%\n]. \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure4.eps}\n\\end{center}\n\\caption{Accumulations of real roots (open circles)\\ of the auxiliary\ncondition (\\protect\\ref{eq:auxilCond}) about eigenvalues as the order $M$\\\nof the series varies in the Wilson-Polchinski even case. From top to bottom: \n$\\protect\\lambda _{1}=1\/\\protect\\nu $ (second horizontal line), $\\protect%\n\\lambda _{2}=-\\protect\\omega _{1}$ (third h. line) and $\\protect\\lambda %\n_{3}=-\\protect\\omega _{2}$ (fourth h. line). A simple criterion of choice\nallows to determine their estimates at $M=20$, see the values in eqs. (%\n\\protect\\ref{eq:WPvpnu}--\\protect\\ref{eq:WPvpomega2}). An accumulation also\noccurs about the spurious value 5.8 (first h. line).}\n\\label{fig4}\n\\end{figure}\n\nWithin each of these accumulations of real roots, we have been able to\nfollow without ambiguity a convergent sequence to the right estimate. At\norder $M=20$ with the ADE pair $\\left( \\mathrm{v},\\mathrm{v}^{\\prime\n}\\right) $ supposed to vanish at infinity, and $r^{\\ast }$ fixed to the\nvalue given in (\\ref{eq:rstar2}), we have obtained the following estimates\nin the even case\n\n\\begin{eqnarray}\n\\nu &=&0.649\\,561\\,773\\,86\\,, \\label{eq:WPvpnu} \\\\\n\\omega _{1} &=&0.655\\,745\\,92\\,, \\label{eq:WPvpomega1} \\\\\n\\omega _{2} &=&3.178\\,, \\label{eq:WPvpomega2}\n\\end{eqnarray}%\nwhere the number of digits has been truncated with regard to the accuracy of\nthe estimates obtained [by comparison with (\\ref{eq:nubest}) and table \\ref%\n{Table 3}]. We see that the accuracy decreases as the order of the\neigenvalue grows but also that we obtain an estimate of $\\omega _{2}$\nwhereas for that value $\\mathrm{v}$ does not vanish at infinity.\n\nThe same kind of observations stands in the odd case. We take the\nopportunity to indicate that choosing the ADE pair $\\left( f,f^{\\prime\n}\\right) $ with $f=$ $\\frac{\\left( 1-2\\lambda \\right) }{10}\\mathrm{v}$-$x%\n\\mathrm{v}^{\\prime }$ makes \\ $f$ vanish for $\\lambda >-3\/2$ and the\nprocedure gives a better accuracy on $\\breve{\\omega}_{1}$ than with the pair \n$\\left( \\mathrm{v},\\mathrm{v}^{\\prime }\\right) $. This way we obtain the\nfollowing estimate [at order $M=20,$ compare with (\\ref{eq:omegac})]%\n\\begin{equation*}\n\\breve{\\omega}_{1}=1.886\\,718\\,.\n\\end{equation*}\n\nWe have also noted the presence of accumulations of real roots about\nspurious positive values of order 5.8 in the even case and 3.77 in the odd\ncase.\n\n\\subsubsection{Litim's eigenvalues}\n\nThe determination using the ADE method of the eigenvalues from the Litim\nflow equation follows the same lines as previously for the Wilson-Polchinski\nflow equation. We limit ourselves in this section to a brief presentation of\nthe main differences encountered.\n\n\\paragraph{ Small field expansion and leading high field behaviour}\n\nCompared to (\\ref{eq:VPeqLitim}), we perform a change of eigenfunction, $%\nh\\rightarrow \\,\\mathrm{v}_{L}$, according to the symmetry considered:\n\n\\begin{itemize}\n\\item in the even case:%\n\\begin{equation*}\nh\\left( \\varphi \\right) =\\,\\mathrm{v}_{L}\\left( \\varphi ^{2}\\right) \\,,\n\\end{equation*}\n\nthen eq. (\\ref{eq:hasy}) yields the following behaviour at large $\\bar{x}%\n=\\varphi ^{2}$:%\n\\begin{equation*}\n\\mathrm{v}_{L\\text{asy}}\\left( \\bar{x}\\right) =S_{1}{\\bar{x}}^{\\,\\left(\n3-\\lambda \\right) }\\left[ 1+O\\left( \\bar{x}^{-5}\\right) \\right] \\,.\n\\end{equation*}\n\n\\item in the odd case:%\n\\begin{equation*}\nh\\left( \\varphi \\right) =\\varphi \\,\\mathrm{v}_{L}\\left( \\varphi ^{2}\\right)\n\\,,\n\\end{equation*}%\nand eq. (\\ref{eq:hasy}) gives:%\n\\begin{equation*}\n\\mathrm{v}_{L\\text{asy}}\\left( \\bar{x}\\right) =S_{1}{\\bar{x}}^{\\,\\left(\n5\/2-\\lambda \\right) }\\left[ 1+O\\left( \\bar{x}^{-5}\\right) \\right] \\,.\n\\end{equation*}\n\\end{itemize}\n\nSo defined, the two functions $\\mathrm{v}_{L}\\left( \\bar{x}\\right) $ and $%\n\\mathrm{v}_{L}^{\\prime }\\left( \\bar{x}\\right) $ vanish at infinity provided\nthat $\\lambda >3$ in the even case and $\\lambda >5\/2$ in the odd case\n(whereas the arbitrary global normalisation of the eigenfunctions allows to\nchoose $\\mathrm{v}_{L}\\left( 0\\right) =1$ (even) and $\\mathrm{v}_{L}^{\\prime\n}\\left( 0\\right) =1$ (odd) corresponding respectively to specific values of $%\nS_{1}$).\n\nAlthough it works, the original ADE pair $\\left( \\mathrm{v}_{L},\\mathrm{v}%\n_{L}^{\\prime }\\right) $ is not the most efficient choice to obtain estimates\nof the first nontrivial eigenvalues. A better choice appears to be the pairs \n$\\left( f\\left( \\bar{x}\\right) ,f^{\\prime }\\left( \\bar{x}\\right) \\right) $\nwith $f\\left( \\bar{x}\\right) =\\left( 3-\\lambda \\right) \\mathrm{v}_{L}\\left( \n\\bar{x}\\right) -\\bar{x}\\mathrm{v}_{L}^{\\prime }\\left( \\bar{x}\\right) $ in\nthe even case and $f\\left( \\bar{x}\\right) =$ $\\left( 5\/2-\\lambda \\right) \n\\mathrm{v}_{L}\\left( \\bar{x}\\right) -\\bar{x}\\mathrm{v}_{L}^{\\prime }\\left( \n\\bar{x}\\right) $ in the odd case (they correspond to eigenfunctions which\nvanish as $\\bar{x}\\rightarrow \\infty $ for more negative values of $\\lambda $%\n). With these choices and $\\bar{k}^{\\ast }=0.409\\,532\\,734\\,145\\,7$ we\nidentify immediately the trivial eigenvalues $\\lambda _{0}=3$ in the even\ncase and $\\breve{\\lambda}_{1}=2.5$, $\\breve{\\lambda}_{2}=0.5$ in the odd\ncase but also, for $M=20$, we obtain good estimates of the nontrivial\nleading and first subleading eigenvalues:%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =\\allowbreak 0.649\\,561\\,774,\\quad & \\omega _{1}=0.655\\,745\\,5,\\quad & \n\\omega _{2}=3.180\\,008,\\quad & \\omega _{3}=5.896, \\\\ \n\\breve{\\omega}_{1}=1.886\\,703\\,7, & \\breve{\\omega}_{2}=4.524\\,1\\,, & & \n\\end{array}%\n\\end{equation*}%\nwhere the numbers of digits have been limited with respect to the estimated\naccuracy [compare with (\\ref{eq:nubest}), table \\ref{Table 3} (even) and (%\n\\ref{eq:omegac}), table \\ref{Table 4} (odd)]. For each eigenvalue, the\nsuccessive estimates may be followed unambiguously step by step when $M$\ngrows so that the right values may be easily selected following the rules\ndefined\\ precedently.\n\nWe notice also the presence of spurious convergences and especially in the\neven case to the value about 5.8 already encountered with the\nWilson-Polchinski case.\n\n\\section{Approximating by hypergeometric functions (HFA)}\n\n\\label{Ratios}\n\nThe ADE method is most certainly efficient in many cases but it is\nrelatively heavy regarding the computing time whereas the current methods,\nwhen they work, are lighter. In addition, none of these methods provides a\nglobal solution to the ODE studied: they yield an approximate value of the\nintegration constant but not a function as global approximation of the\nsolution looked for.\n\nWe propose in this section an alternative method which is lighter than the\nADE method and which provides a global approximation of the solution of\ninterest. This new method is based on the definition property of the\ngeneralized hypergeometric functions. Let us first review the definition and\nmain properties of these functions.\n\n\\subsection{Generalized hypergeometric functions\\label{Brief}}\n\nFor $x\\in \n\\mathbb{C}\n$, a series $S=\\sum_{n=0}^{\\infty }a_{n}x^{n}$ is hypergeometric (see for\nexample \\cite{6181}) if the ratio $a_{n+1}\/a_{n}$ is a rational function of $%\nn$, i.e.%\n\\begin{equation*}\n\\frac{a_{n+1}}{a_{n}}=\\frac{P\\left( n\\right) }{Q\\left( n\\right) }\\,,\n\\end{equation*}%\nfor some polynomials $P\\left( n\\right) $ and $Q\\left( n\\right) $.\n\nIf we factorize the polynomials, we can write:%\n\\begin{equation}\n\\frac{a_{n+1}}{a_{n}}=\\alpha _{0}\\frac{\\left( n+\\alpha _{1}\\right) \\left(\nn+\\alpha _{2}\\right) \\cdots \\left( n+\\alpha _{p}\\right) }{\\left( n+\\beta\n_{1}\\right) \\left( n+\\beta _{2}\\right) \\cdots \\left( n+\\beta _{q}\\right)\n\\left( n+1\\right) }\\,. \\label{eq:PolyFac}\n\\end{equation}%\nThe factor $\\left( n+1\\right) $ in the denominator may or may not result\nfrom the factorization. If not, we add it along with the compensating factor\nin the numerator. Usually, the global factor $\\alpha _{0}$ is set equal to 1.\n\nIf the set $\\left\\{ \\alpha _{i}\\right\\} $ includes negative integers, then $%\nS $ degenerates into a polynomial in $x.$\n\nWhen it is not a polynomial, the series $S$ converges absolutely for all $x$\nif $p\\leq q$ and for $\\left\\vert x\\right\\vert <1\/\\left\\vert \\alpha\n_{0}\\right\\vert $ if $p=q+1$. It diverges for all $x\\neq 0$ if $p>q+1.$\n\nThe analytic continuation of the hypergeometric series $S$ with a non-zero\nradius of convergence is called a generalized hypergeometric function and is\nnoted:%\n\\begin{equation*}\n_{p}F_{q}\\left( \\alpha _{1},\\cdots ,\\alpha _{p};\\beta _{1},\\cdots ,\\beta\n_{q};\\alpha _{0}x\\right) =\\frac{1}{a_{0}}S\\,.\n\\end{equation*}\n\n$_{p}F_{q}\\left( x\\right) $ is a solution of the following differential\nequation (for $\\alpha _{0}=1$):%\n\\begin{equation}\n\\left[ \\theta \\left( \\theta +\\beta _{1}-1\\right) \\cdots \\left( \\theta +\\beta\n_{q}-1\\right) -x\\left( \\theta +\\alpha _{1}\\right) \\cdots \\left( \\theta\n+\\alpha _{p}\\right) \\right] \\,_{p}F_{q}\\left( x\\right) =0\\,,\n\\label{eq:EDOhyper}\n\\end{equation}%\nwhere%\n\\begin{equation*}\n\\theta =x\\frac{d}{dx}.\n\\end{equation*}\n\nWhen $p>2$ or $q>1$, the differential equation (\\ref{eq:EDOhyper}) is of\norder $\\max \\left( p,q+1\\right) >2$. It is of second order when $q=1$ and $%\np=0$, $1$ or $2$. It is of first order when $q=0$ and $p=1$\n\n$_{2}F_{1}$ is currently named the hypergeometric function. A number of\ngeneralized hypergeometric functions have also special names: $_{0}F_{1}$ is\ncalled confluent hypergeometric limit function and $_{1}F_{1}$ confluent\nhypergeometric function.\n\nIn the cases $p\\leq q$ for fixed $\\left\\{ \\alpha _{i}\\right\\} $ and $\\left\\{\n\\beta _{i}\\right\\} $, $_{p}F_{q}\\left( x\\right) $ is an entire function of $%\nx $ and has only one (essential) singular point at $x=\\infty $.\n\nFor $p=q+1$ and fixed $\\left\\{ \\alpha _{i}\\right\\} $ and $\\left\\{ \\beta\n_{i}\\right\\} $ in non-polynomial cases\\textbf{, }$_{p}F_{q}\\left( x\\right) $\ndoes not have pole nor essential singularity. It is a single-valued function\non the $x$-plane cut along the interval $\\left[ 1,\\infty \\right] $, i.e. it\nhas two branch points at $x=1$ and at $x=\\infty $.\n\nConsidered as a function of $\\left\\{ \\beta _{i};i=1,\\cdots ,q\\right\\} $, $%\n_{p}F_{q}\\left( x\\right) $ has an infinite set of singular points:\n\n\\begin{enumerate}\n\\item $\\beta _{i}=-m$, $m\\in \n\\mathbb{N}\n$ which are simple poles\n\n\\item $\\beta _{i}=\\infty $\\ which is an essential singular point (the point\nof accumulation of the poles).\n\\end{enumerate}\n\nAs a function of $\\left\\{ \\alpha _{i};i=1,\\cdots ,p\\right\\} $, $%\n_{p}F_{q}\\left( x\\right) $ has one essential singularity at each $\\alpha\n_{i}=\\infty $.\n\nThe elementary functions and several other important functions in\nmathematics and physics are expressible in terms of hypergeometric functions\n(for more detail see \\cite{6181}).\n\nThe wide spread of this family of functions suggests trying to represent the\nsolution of the ODEs presently of interest in this article, under the form\nof a generalized hypergeometric function.\n\n\\subsection{\\textbf{\\ }The HFA method\\label{HFA}}\n\nFor the sake of the introduction of the new method, let us first consider\nthe Wilson-Polchinski fixed point equation (\\ref{eq:FPw}) and the truncated\nexpansion (\\ref{eq:wmrx}) in which the coefficients $a_{n}\\left( r\\right) $ $%\n(n=1,\\cdots ,M)$ are already determined as function of $r$ via a generic\nsolution of (\\ref{eq:FPw}) truncated at order $M$ (in powers of $x)$. The\nquestion is again to construct an auxiliary condition to be imposed\\ with a\nview to determining the fixed point value $r^{\\ast }$. To this end, by\nanalogy with the generalized hypergeometric property definition recalled in\nsection \\ref{Brief}, we construct the ratio of two polynomials in $n$:%\n\\begin{equation}\n\\frac{P_{m_{1}}\\left( n\\right) }{Q_{m_{2}}\\left( n\\right) }=\\frac{%\n\\sum_{i=1}^{m_{1}}c_{i}\\,n^{i-1}}{\\sum_{i=1}^{m_{2}}d_{i}\\,n^{i-1}}\\,,\n\\label{eq:hyper0}\n\\end{equation}%\nso that $P_{m_{1}}\\left( n\\right) \/Q_{m_{2}}\\left( n\\right) $ \\ match the $%\nM-2$ ratios $a_{n+1}\\left( r\\right) \/a_{n}\\left( r\\right) $ \\ for $%\nn=1,\\cdots ,M-2$. Hence, accounting for the arbitrariness of the global\nnormalisation of (\\ref{eq:hyper0}), the complete determination of the two\nsets of coefficients $\\left\\{ c_{i};i=1,\\cdots ,m_{1}\\right\\} $ and $\\left\\{\nd_{i};i=1,\\cdots ,m_{2}\\right\\} $ as functions of $r$ implies $%\nm_{1}+m_{2}=M-1$.\\ Finally, the auxiliary condition on $r$ is obtained by\nrequiring that the last (still unused) ratio $a_{M}\\left( r\\right)\n\/a_{M-1}\\left( r\\right) $ satisfies again the $n$-dependency satisfied by\nits predecessors, namely that:%\n\\begin{equation}\n\\frac{\\sum_{i=1}^{m_{1}}c_{i}\\left( r\\right) \\left( M-1\\right) ^{i-1}}{%\n\\sum_{i=1}^{m_{2}}d_{i}\\left( r\\right) \\left( M-1\\right) ^{i-1}}=\\frac{%\na_{M}\\left( r\\right) }{a_{M-1}\\left( r\\right) }\\,. \\label{eq:hyperauxil}\n\\end{equation}\n\nThe auxiliary condition so obtained is a polynomial in $r$, the roots of\nwhich are candidates to give an estimate at order $M$ of $r^{\\ast }$ (noted\nbelow $r_{M}^{\\ast }$). Notice that, to obtain faster this auxiliary\ncondition, one may avoid the calculation of the coefficients $c_{i}\\left(\nr\\right) $\\ and $d_{i}\\left( r\\right) $\\ by following the same\nconsiderations as those leading to (\\ref{eq:detF}) with the ADE method.\n\nAt this point, the method potentially reaches the same goal as the ADE and\nother preceding methods. However, according to section \\ref{Brief}, in\ndetermining the ratio of polynomials (\\ref{eq:hyper0}) we have also\nexplicitly constructed the function \n\\begin{equation}\nF_{M}\\left( x\\right) =r_{M}^{\\ast }\\cdot \\,_{m_{1}+1}F_{m_{2}}\\left( \\alpha\n_{1},\\cdots ,\\alpha _{m_{1}},1;\\beta _{1},\\cdots ,\\beta _{m_{2}};\\alpha\n_{0}x\\right) \\,, \\label{eq:FM}\n\\end{equation}%\nin which $r_{M}^{\\ast }$ is the selected estimate of $r^{\\ast }$, the sets $%\n\\left\\{ -\\alpha _{i}\\right\\} $ and $\\left\\{ -\\beta _{i}\\right\\} $ are the\nroots of the two polynomials $P_{m_{1}}\\left( n\\right) $ and $%\nQ_{m_{2}}\\left( n\\right) $ when $r=r_{M}^{\\ast }$ whereas: \n\\begin{equation}\n\\alpha _{0}=\\frac{c_{m_{1}}\\left( r_{M}^{\\ast }\\right) }{d_{m_{2}}\\left(\nr_{M}^{\\ast }\\right) }\\,. \\label{eq:alpha0}\n\\end{equation}\n\nNow, by construction, $F_{M}\\left( x\\right) $, has the same truncated series\nin $x$ as the solution of (\\ref{eq:FPw}) we are looking for. This function\nis thus a candidate for an approximate representation of this solution.\n\nIt is worth noticing that, contrary to the ADE method, the HFA method does\nnot make an explicit use of the conditions at infinity (large $x$) to\ndetermine $r^{\\ast }$. Only a local information, in the neighbourhood of the\norigin $x=0$, is explicitly employed.\n\nLet us apply the method to the two equations of interest in this paper.\n\n\\subsection{Wilson-Polchinski's equation}\n\n\\subsubsection{Fixed point}\n\nWe know that the absolute value of the ratio $a_{n}\\left( r^{\\ast }\\right)\n\/a_{n+1}\\left( r^{\\ast }\\right) $ has a definite value $R_{WP}$ [given by\neq. (\\ref{eq:RWP})] as $n\\rightarrow \\infty $. Consequently, we must\nconsider the ratio (\\ref{eq:hyper0}) with $m_{1}=m_{2}$ (this implies also\nthat $M$ be odd). In this circumstance, according to section \\ref{Brief},\nthe relevant hypergeometric functions have a branch cut on the positive real\naxis (as functions of $\\alpha _{0}x$). Consequently the analytic\ncontinuation to large positive values of $x$ is only possible if $\\alpha\n_{0}<0$. We note also that, according to (\\ref{eq:RWP}), $\\left\\vert \\alpha\n_{0}\\right\\vert $ should converge to $1\/R_{WP}=0.174774$. Finally by\nconsidering the large $x$ behaviour directly on (\\ref{eq:EDOhyper}), it is\neasy to convince oneself that the leading power is given by one of the\nparameters $\\left\\{ -\\alpha _{i}\\right\\} $, consequently we expect to\nobserve a stable convergent value among the $\\alpha _{i}$'s toward the\nopposite of the leading power at large $x$ of the solution looked for. For\nthis reason, instead of the function $w\\left( x\\right) $ of section \\ref%\n{WPsmallField} the limit of which is 1 as $x\\rightarrow \\infty $ [see (\\ref%\n{eq:asyw})], we have considered the translated function $w_{t}\\left(\nx\\right) =w\\left( x\\right) -1$ which, according to eqs (\\ref{eq:fasy}) and (%\n\\ref{eq:chgt1}, \\ref{eq:astar}), tends to $A^{\\ast }x^{-2\/5}$. In this case\nwe\\ thus expect to observe a stable value among the $\\alpha _{i}$'s about $%\n0.4$ with the eventual possibility of estimating $A^{\\ast }$.\n\nWhen looking at the roots of the auxiliary condition (\\ref{eq:hyperauxil})\nas $M$ varies, we obtain the same kind of accumulations about the expected\nfixed point value $r^{\\ast }$ as shown in figure \\ref{fig2} (with much less\npoints however). We can also easily select the right nontrivial solution\nusing the procedure described just above (\\ref{eq:rstar2}). We get precisely\nthis excellent estimate with $M=25$ and a reduced computing time compare to\nthe ADE method. Figure (\\ref{fig6}) shows the accuracies obtained on $%\nr^{\\ast }$ (crosses) compared to the ADE method (open circles). \n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics*[width=10cm]{figure5.eps}\n\\end{center}\n\\caption{Respective approximate number of digits (defined in the caption of\nfigure \\protect\\ref{fig5}) obtained for r$_{M}^{\\ast }$ with the HFA method\n(crosses) and the ADE method (open circles) for the Wilson-Polchinski fixed\npoint equation. Whereas at a given order $M$ the accuracy is similar, a\nsmaller time computing is necessary with the HFA method.}\n\\label{fig6}\n\\end{figure}\n\nFurthermore, the sets of parameters of the successive hypergeometric\nfunctions involve two stable quantities the values of which at $M=25$ are:%\n\\begin{eqnarray}\n\\alpha _{0} &=&-0.174\\,775\\,, \\label{eq:alpha0estim} \\\\\n\\alpha _{1} &=&0.396\\,2\\,. \\label{eq:alpha1estim}\n\\end{eqnarray}\n\nThose two results are quantitatively and qualitatively very close to the\nexpected values (respectively $-0.174774$ and $0.4$ as given just above).\n\nThis clearly shows that the hypergeometric function determined this way\nprovides us with a really correct (but approximate) global representation of\nthe fixed point function. This contrasts strongly with the numerical\nintegration of the ODE which, due to the presence of the moving singularity,\nnever provides us with such an approximate global representation of the\nsolution looked for.\n\nFrom (\\ref{eq:alpha1estim}) we have obtained a rough estimate of $A^{\\ast }$\n($=6b^{\\ast }\/5$) by a direct consideration of the value of the\ncorresponding function\\ $F_{M}\\left( x\\right) $ defined in (\\ref{eq:FM}) for\nsome relatively large value of $x$ and we obtain $A^{\\ast }\\simeq -2.6$ what\nis a sufficiently accurate estimate to serve as a guess in the shooting\nmethod.\n\nWe have also tried to determine, using the HFA method, the value $A^{\\ast }$\ndirectly from the \\textquotedblleft reverse side\\textquotedblright\\\ncorresponding to (\\ref{eq:WFPasy}). We have not improved the previous\\\nbiased estimate obtained by ADE (about $A^{\\ast }=-2.735$). We do not\nunderstand the significance of this coincidence. We recall, however, that\nthe radius of convergence of the Taylor series of $u^{\\ast }\\left( y\\right) $\nabout $y=0$ probably vanishes. This biased result shows again that the\nproperty of convergence of the Taylor series is crucial for the accuracy of\nthe two methods.\n\n\\subsubsection{Eigenvalues}\n\nWe have also applied the HFA method to the determination of the eigenvalues.\nWith $M=17$, we have easily and without ambiguity obtained the following\nexcellent estimates [compare with (\\ref{eq:nubest}, \\ref{eq:omegac}) and\ntables \\ref{Table 3} and \\ref{Table 4}]:%\n\\begin{equation*}\n\\begin{array}{lll}\n\\nu =\\allowbreak \\allowbreak 0.649\\,561\\,774\\,,\\quad & \\omega\n_{1}=0.655\\,745\\,939\\,3\\,,\\quad & \\omega _{2}=3.180\\,006\\,53\\,, \\\\ \n\\omega _{3}=5.912\\,229\\,4\\,,\\quad & \\omega _{4}=8.796\\,045\\,,\\quad & \\omega\n_{5}=11.800\\,4\\,, \\\\ \n\\breve{\\omega}_{1}=1.886\\,703\\,839\\,,\\quad & \\breve{\\omega}%\n_{2}=4.524\\,390\\,3\\,,\\quad & \\breve{\\omega}_{3}=7.337\\,635\\,.%\n\\end{array}%\n\\end{equation*}\n\nThese results show a greater efficiency than with the ADE method especially\nin the determination of the subleading eigenvalues.\n\nIt is worth indicating also that, surprisingly enough, we observe again\n(i.e. as with the ADE method) the presence of convergences to the same\nspurious eigenvalues: 5.8 and 3.8 in the even and odd cases respectively.\n\n\\subsection{Litim's equation\\label{GeoLitim}}\n\n\\subsubsection{Fixed point}\n\nApplying the HFA method with the ratio of two successive coefficients $%\na_{n}\\left( \\bar{k}\\right) $ provides again an accumulation of roots about\nthe right value of $\\bar{k}^{\\ast }$ given in (\\ref{eq:kstar}). However,\nthis time, we have encountered some difficulties in defining a process of\nselection of the right root. We obtain the following estimate for $M=21$:%\n\\begin{equation*}\n\\bar{k}^{\\ast }\\simeq 0.409\\,531\\,,\n\\end{equation*}%\nwhich is not bad [compare with (\\ref{eq:kstar})] but not as satisfactory as\nin the preceding Wilson-Polchinski's case.\n\nWith regard to the transformation (\\ref{eq:Legendre}) and the preceding\nsuccess of the HFA method, it is not amazing that the representation of the\nsolution in the Litim case be more complicated than in the Wilson-Polchinski\ncase.\n\nWe have already mentioned that, instead of the ratio of two successive terms\nof the series $a_{n}\\left( \\bar{k}\\right) $, it is a shifted ratio that\nroughly converges to the finite radius of convergence (\\ref{eq:radiusL}).\\\nAs a matter of fact, if we use the ratios%\n\\begin{equation*}\n\\frac{a_{n+3}\\left( \\bar{k}\\right) }{a_{n}\\left( \\bar{k}\\right) }\\,,\n\\end{equation*}%\ninstead of the ratio $a_{n+1}\/a_{n}$ without changing the procedure\\footnote{%\nNotice that the procedure does not define some generalized hypergeometric\nfunction of $\\bar{x}^{3}.$ This would have been obtained by considering\nseparately three series in the original series. Then a combination of three\ngeneralized hypergeometric functions would have represented the solution\nlooked for.} described in section \\ref{HFA}, then we get a better estimate\nfor $M=21$ [compare with (\\ref{eq:kstar})]:%\n\\begin{equation*}\n\\bar{k}^{\\ast }\\simeq 0.409\\,532\\,737\\,,\n\\end{equation*}%\nalthough the convergence properties are not substantially modified.\n\nBecause the case is apparently more complicated than precedently, we do not\npursued further the discussion of the global representation of the fixed\npoint solution by generalized hypergeometric functions.\n\n\\subsubsection{Eigenvalues}\n\nFor the eigenvalue problem, a similar difficulty occurs where the right\nvalues do not appear as clear convergent series of roots. At order $M=17$,\nwe get the following estimates:$\\allowbreak $%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =0.649\\,55\\,,\\quad & \\omega _{1}=0.657\\,6\\,,\\quad & \\omega\n_{2}=3.20\\,,\\quad & \\omega _{3}=5.8\\,, \\\\ \n\\breve{\\omega}_{1}=1.89\\,,\\quad & \\breve{\\omega}_{2}=4.5\\,. & & \n\\end{array}%\n\\end{equation*}\n\nAs in the case of the fixed point determination, if instead of applying the\nmethod with the ratio of two successive terms of the series $a_{n}\\left( \n\\bar{k}\\right) $ we consider the ratios%\n\\begin{equation*}\n\\frac{a_{n+3}\\left( \\bar{k}\\right) }{a_{n}\\left( \\bar{k}\\right) }\\,,\n\\end{equation*}%\nthen we get better estimates for $M=19$:%\n\\begin{equation*}\n\\begin{array}{llll}\n\\nu =0.649\\,561\\,774\\,,\\quad & \\omega _{1}=0.655\\,75\\,,\\quad & \\omega\n_{2}=3.180\\,7\\,,\\quad & \\omega _{3}=5.905\\,, \\\\ \n\\breve{\\omega}_{1}=1.886\\,71\\,,\\quad & \\breve{\\omega}_{2}=4.524\\,. & & \n\\end{array}%\n\\end{equation*}%\nwhere the numbers of digits have been limited having regard to the estimated\naccuracies [compare with (\\ref{eq:nubest}), table \\ref{Table 3} (even) and (%\n\\ref{eq:omegac}), table \\ref{Table 4} (odd)].\n\n\\section{Summary and conclusions}\n\n\\label{Conc}\n\nWe have presented the details of a highly accurate determination of the\nfixed point and the eigenvalues for two equivalent ERGEs in the local\npotential approximation. First, we have made use of a standard numerical\n(shooting) method to integrate the ODEs concerned. Beyond the test of the\nequivalence between the two equations, already published in \\cite{6137}, the\nresulting numerics have been used to concretely test the efficiency of two\nnew approximate analytic methods for solving two point boundary value\nproblems of ODEs based on the expansion about the origin of the solution\nlooked for (field expansion).\n\nWe have considered explicitly those two methods applied to the study of the\ntwo equivalent ODEs. We have shown that they yield estimates as accurate as\nthose obtained with the shooting method provided that the Taylor series\nabout the origin of the function looked for has a non-zero radius of\nconvergence.\n\nThis is an important new result since, up to now, no such approximate\nanalytical method was known to work in the simplest case of the\nWilson-Polchinski equation. In the case of the Litim equation the two\nmethods converge better than the currently used expansions (usually referred\nto as I and II in the literature, see e.g. [\\cite{3553}]). Our results\nsupport concretely the conclusions of \\cite{5902} which indicated that the\nhigh field contributions were important in the Wilson-Polchinski case\nwhereas they were less important in the Litim case.\n\nThe first of the two methods relies upon the construction of an auxiliary\ndifferential equation (ADE) satisfied by the Taylor series at the origin and\nto which is imposed the condition of the second boundary (at infinity) \\cite%\n{6110}.\n\nThe second method (HFA) is new. It consists in defining a global\nrepresentation of the solution of the ODE via a generalized hypergeometric\nfunction. The HFA method provides the advantage of yielding a global\n(approximate) representation of the solution via an explicit hypergeometric\nfunction.\n\nIn both cases it is possible to obtain easily (with few terms in the field\nexpansion) rough estimates of the solution which may be used as guesses in a\nsubsequent shooting method.\n\nThe procedures may be applied to several coupled ODEs as shown in \\cite{6110}\nfor the ADE method. Hence, we hope that the present work will\\ make easier\nand more efficient future explicit (and ambitious) considerations of the\nderivative expansion of exact renormalisation group equations.\n\n\\section{Acknowledgements}\n\nWe thank D. Litim for comments on an earlier version of this article.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\nIn the standard model (SM) of particle physics, \\ensuremath{C\\!P}\\xspace violation in the quark sector of weak interactions arises from a single\nirreducible phase in the Cabibbo-Kobayashi-Maskawa (CKM) that describes the mixing of quarks~\\cite{ref:CKM}. \nThe unitarity of the CKM \\mbox{matrix $V$} defines a unitarity triangle (UT) in the complex plane. \n\\ensuremath{C\\!P}\\xspace violation measurements and semileptonic decay rates (and other methods)\ncan be conveniently displayed and compared as constraints on the angles and sides, respectively, of this triangle.\nInconsistencies between all these (in general) precise and redundant constraints can be used to search for new physics (NP). \nAs today, there is an impressive overall agreement between all measurements~\\cite{ref:globalCKMfits}.\nAmong these the angle $\\gamma$, defined as the phase of $V_{ub}$ in the Wolfenstein parametrization~\\cite{ref:CKM},\nis particularly relevant since it is the only \\ensuremath{C\\!P}\\xspace-violating measurement that,\ntogether with the determination of the \\ensuremath{C\\!P}\\xspace-conserving magnitude of $V_{ub}$,\nselects a region of the UT apex independently of most types of NP, and thus constitutes a SM candle type of measurement.\nCurrent constraints, provided by the \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ and Belle experiments, make use of \n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays, and are still weak ($\\sim 15^\\circ$).\nNeutral \\ensuremath{B}\\xspace decays have also been proposed, although \ndo not yet provide significant constraints.\n\n\nThe angle $\\gamma$ from $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays is determined\nmeasuring the interference between the amplitudes $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{c}\\xspace$,\nwhen the neutral \\ensuremath{D}\\xspace meson is reconstructed in a final state accessible from both \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace decays. \nSince both amplitudes are tree level, the interference is unaffected by NP appearing in the loops, making the theoretical \ninterpretation of observables in terms of $\\gamma$ very clean. \nThe disadvantage is that the branching fractions of the involved decays are small due to CKM suppression ($10^{-5}-10^{-7}$), \nand the size of the interference, given by the ratio $r_\\ensuremath{B}\\xspace$ between the magnitudes of the $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{c}\\xspace$ amplitudes, \nis small due to further CKM and color suppressions ($\\sim 10\\%$). As a consequence, the measurements are\nstatistically limited and one has to combine complementary methods applied on the same \\ensuremath{B}\\xspace decay modes sharing the\nhadronic parameters ($r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$, i.e. the relative magnitude and phase of the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ transitions) and $\\gamma$, \nand use as many as possible different \\ensuremath{B}\\xspace decay modes to improve the overall sensitivity to $\\gamma$.\n\nIn this talk we present the most recent determinations of $\\gamma$ obtained by \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}, based on the full\ndata sample of charged \\ensuremath{B}\\xspace meson decays produced in $e^+e^- \\ensuremath{\\rightarrow}\\xspace \\Y4S \\ensuremath{\\rightarrow}\\xspace \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace$ and \nrecorded in the years 1999-2007, about $468\\times10^6$ \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace pairs.\nWe have studied $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ decays,\nwith the neutral \\ensuremath{D}\\xspace mesons reconstructed in a number of different final states: $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace \\KS h^+ h^-$, with $h=\\pi,\\ensuremath{K}\\xspace$\n(Dalitz plot method); $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{K^\\pm}\\xspace\\ensuremath{\\pi^\\mp}\\xspace$ (ADS method); \nand $\\ensuremath{D}\\xspace \\ensuremath{\\rightarrow}\\xspace f_{\\ensuremath{C\\!P}\\xspace}$, with $f_{\\ensuremath{C\\!P}\\xspace}$ a \\ensuremath{C\\!P}\\xspace-eigenstate (GLW method)~\\cite{ref:dalitz_ads_glw}.\n\n\n\n\n\n\nOne of the charged \\ensuremath{B}\\xspace mesons produced in the \\Y4S decay is fully reconstructed, with efficiencies ranging \nbetween 40\\% (for low-multiplicity decays with no neutrals) \nand 10\\% (for high-multiplicity decays with neutrals). The selection is optimized to maximize the statistical sensitivity. \nThe reconstruction efficiencies have substantially improved (20\\% to 60\\% relative) with respect to our previous measurements \nbased on $384\\times10^6$ \\ensuremath{\\Bu {\\kern -0.16em \\Bub}}\\xspace pairs, reflecting improvements in tracking and particle identification,\nand optimization of analysis procedures.\nSignal \\ensuremath{B}\\xspace decays are characterized by means of two nearly independent kinematic variables exploiting the constraint from the\nknown beam energies: \nthe beam-energy $\\mbox{$m_{\\rm ES}$}\\xspace \\equiv \\sqrt{E^{*2}_{\\rm beam}-|p^{*}_{\\ensuremath{B}\\xspace}|^2}$\nand the energy-difference $\\mbox{$\\Delta E$}\\xspace \\equiv E^*_\\ensuremath{B}\\xspace - E^*_{\\rm beam}$.\nSince the main source of background comes from \\ensuremath{q\\overline q}\\xspace continuum production, additional discrimination\nis achieved using multivariate analysis tools, from the combination (either a linear Fisher discriminant \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace,\nor a non-linear neural network $NN$) of several event-shape quantities. These variables distinguish between\nspherical \\BB events from more \njet-like continuum events and exploit the different angular correlations in the two event categories. \nThe signal is finally separated from background through \nunbinned maximum likelihood (UML) fits to the\n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ data using \\mbox{$m_{\\rm ES}$}\\xspace, \\mbox{$\\Delta E$}\\xspace, and \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace or $NN$. \n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{\\pi^\\pm}\\xspace$ decays, \nwhich are\nabout 12 times more abundant than $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^{(*)} \\ensuremath{K^\\pm}\\xspace$, have a similar topology but are \ndiscriminated by means of excellent pion and kaon identification provided by $dE\/dx$ and Cerenkov measurements,\nand show negligible \\ensuremath{C\\!P}\\xspace-violating effects ($r_\\ensuremath{B}\\xspace \\sim 1\\%$). Therefere, these decays provide powerful calibration\nand control samples for negative tests of \\ensuremath{C\\!P}\\xspace violation.\n\n\n\n\nIn the Dalitz plot (DP) method the amplitude for a \\ensuremath{\\Bub}\\xspace decay has for the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace$ transition the DP of the \\ensuremath{D^0}\\xspace decay, \nwhile for the $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ transition the DP is the corresponding to the \\ensuremath{\\Dbar^0}\\xspace decay. If we assume no \\ensuremath{D}\\xspace mixing nor \\ensuremath{C\\!P}\\xspace violation \nin the \\ensuremath{D}\\xspace decay, and use as independent kinematic variables $s_\\pm=m^2(\\KS\\ensuremath{\\pi^\\pm}\\xspace)$,\nthen the two DPs are one rotated $90^\\circ$ to each other.\nThis is of critical importance since allows to determine directly from data the strong charm phase variation \nfor \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace, as well as well as the hadronic parameters $r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$, and the weak phase $\\gamma$, provided\nthat a \\ensuremath{D}\\xspace decay amplitude model is assumed. For \\ensuremath{\\Bu}\\xspace decays one has to interchange the \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace DPs, \nand change the sign of $\\gamma$. This results in an interference term proportional to our observables \n$x_\\pm \\equiv r_\\ensuremath{B}\\xspace \\cos(\\delta_\\ensuremath{B}\\xspace \\pm \\gamma)$ \nand\n$y_\\pm \\equiv r_\\ensuremath{B}\\xspace \\sin(\\delta_\\ensuremath{B}\\xspace \\pm \\gamma)$, i.e.\nthe real and imaginary parts of the ratio of $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{u}\\xspace$ and $\\ensuremath{b}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace$ amplitudes for \\ensuremath{B^\\pm}\\xspace decays.\nWe reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$, $D^* \\ensuremath{K^\\pm}\\xspace$ with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace,D\\gamma$, and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^{*\\pm}}\\xspace$ with $\\ensuremath{K^{*\\pm}}\\xspace \\ensuremath{\\rightarrow}\\xspace \\KS\\ensuremath{\\pi^\\pm}\\xspace$ decays,\nfollowed by neutral \\ensuremath{D}\\xspace meson decays to the 3-body self-conjugate final states $\\KS h^+ h^-$, with $h=\\pi,\\ensuremath{K}\\xspace$. \nFrom the UML fit we determine the signal and background yields in each of the eight different final states for each \\ensuremath{B}\\xspace charge, \nalong with the \\ensuremath{C\\!P}\\xspace-violating \nparameters $x_\\pm$ and $y_\\pm$~\\cite{ref:babar_dalitzpub2010}.\nWe find 1507 \\ensuremath{B^\\pm}\\xspace signal candidates with $\\KS\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$, and 268 with $\\KS\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$. \nPrior to the \\ensuremath{C\\!P}\\xspace fit, we model the \\ensuremath{D^0}\\xspace and \\ensuremath{\\Dbar^0}\\xspace decay amplitudes as a coherent sum of S-, P-, and D-waves,\nand determine their amplitudes and phases (along with other relevant parameters)\nrelative to the dominant two-body \\ensuremath{C\\!P}\\xspace-eigenstates $\\KS \\rho(770)$ (for $\\KS\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$) and $\\KS a_0(980)$ (for $\\KS\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$),\nusing a large ($\\approx 6.2\\times10^5$) and very pure ($\\approx 99\\%$) signal sample of flavor tagged \nneutral \\ensuremath{D}\\xspace mesons from $\\ensuremath{D^{*+}}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{D^0}\\xspace\\ensuremath{\\pi^+}\\xspace$ decays produced in $e^+e^- \\ensuremath{\\rightarrow}\\xspace\\ensuremath{c}\\xspace\\ensuremath{\\overline c}\\xspace$ events~\\cite{ref:dmixing-kshh}.\n\\begin{figure}[htb!]\n\\begin{center}\n\\vskip-0.1cm\n\\begin{tabular}{ccc}\n\\includegraphics[width=0.32\\textwidth]{d0k_Dalitz-ubl-Contours-DzK-zoom.eps}&\n\\includegraphics[width=0.32\\textwidth]{d0k_Dalitz-ubl-Contours-DstK-zoom.eps} &\n\\includegraphics[width=0.32\\textwidth]{scan-gammat-Dalitz.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{fig:dalitz} $1\\sigma$ and $2\\sigma$ contours in the $(x_\\pm,y_\\pm)$ planes for\n(a) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ and (b) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^* \\ensuremath{K^\\pm}\\xspace$, for \\ensuremath{\\Bub}\\xspace (solid lines) and \\ensuremath{\\Bu}\\xspace (dotted lines) decays.\n(c) $1 - {\\rm CL}$ as a function of $\\gamma$ for $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{D}\\xspace\\ensuremath{K^\\pm}\\xspace, \\ensuremath{D^*}\\xspace\\ensuremath{K^\\pm}\\xspace, \\ensuremath{D}\\xspace\\ensuremath{K^{*\\pm}}\\xspace$ decays.\nThe dashed (upper) and dotted (lower) horizontal lines correspond to the $1\\sigma$ and $2\\sigma$ intervals, respectively.\n}\n\\end{figure}\nFrom the $(x_\\pm,y_\\pm)$ confidence regions for each of the 3 different \\ensuremath{B}\\xspace decay modes --Fig.~\\ref{fig:dalitz}.(a)(b)-- we determine, \nusing a frequentist procedure, $1\\sigma$ $[2\\sigma]$ intervals for $\\gamma$ --Fig.~\\ref{fig:dalitz}.(c)--. \nWe obtain $\\gamma~({\\rm mod}~180^\\circ) = (68\\pm14\\pm4\\pm3)^\\circ$ $[39^\\circ,98^\\circ]$, where the three uncertainties are statistical,\nexperimental systematic, and amplitude model systematic.\nWe also determine the hadronic parameters\n$r_\\ensuremath{B}\\xspace^{DK^\\pm}=(9.6\\pm2.9)\\%$ $[3.7,15.5]\\%$, \n$r_\\ensuremath{B}\\xspace^{D^*K^\\pm}=(13.3^{+4.2}_{-3.9})\\%$ $[4.9,21.5]\\%$, \n$\\kappa r_\\ensuremath{B}\\xspace^{DK^{*\\pm}}=(14.9^{+6.6}_{-6.2})\\%$ $[0,28.0]\\%$ ($\\kappa = 0.9\\pm0.1$ takes into account \nthe \\ensuremath{K^*}\\xspace intrinsic width),\nand the strong phases \n$\\delta_B^{DK^\\pm}$, \n$\\delta_B^{D^*K^\\pm}$, \nand $\\delta_B^{DK^{*\\pm}}$~\\cite{ref:babar_dalitzpub2010}. \nA $3.5\\sigma$ evidence of direct \\ensuremath{C\\!P}\\xspace violation ($\\gamma \\ne 0$) is found from the combination of the 3 channels,\nwhich corresponds to the significance of the separation between the $(x_+,y_+)$ and $(x_-,y_-)$ solutions in Fig.~\\ref{fig:dalitz}.(a)(b).\n\n\n\n\n\n\nIn the ADS method, we reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$, $D^* \\ensuremath{K^\\pm}\\xspace$ with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace,D\\gamma$, followed by \\ensuremath{D}\\xspace decays to both the doubly-Cabibbo-suppressed (DCS)\n\\ensuremath{D^0}\\xspace final state $K^+\\pi^-$ and the Cabibbo-favored (CF) $K^-\\pi^+$, which is used as normalization and control sample. \nFinal states with opposite-sign kaons arise \neither from the CKM favored \\ensuremath{B}\\xspace decay followed by the DCS \\ensuremath{D}\\xspace decay\nor from the CKM- and color-suppressed \\ensuremath{B}\\xspace decay followed by the CF \\ensuremath{D}\\xspace decay,\nproducing an interference which can be potentially large since the magnitudes of the interfering amplitudes are similar.\nHowever, their overall branching ratios are very small ($\\sim 10^{-7}$) and background suppression becomes crucial.\nThe UML fit directly determines the three branching fraction ratios $R_{ADS}$ between \\ensuremath{B}\\xspace decays with opposite-sign \nand same-sign kaons, and the three yields of \\ensuremath{B}\\xspace decays with same-sign kaons, using \\mbox{$m_{\\rm ES}$}\\xspace and $NN$.\nThe three \\ensuremath{C\\!P}\\xspace asymmetries $A_{ADS}$ are inferred from all these. \nWe obtain first indications of signals for the\n$\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ and $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^* \\ensuremath{K^\\pm}\\xspace$ (with $D^* \\ensuremath{\\rightarrow}\\xspace D\\ensuremath{\\pi^0}\\xspace$) opposite-sign modes --Fig.~\\ref{fig:ads}--, \nwith significances of $2.1\\sigma$ and $2.2\\sigma$, respectively~\\cite{ref:babar_ADS-DK-DstarK}.\nThe measured branching fraction ratios are \n$R_{ADS}^{DK}=(1.1\\pm0.5\\pm0.2)\\times10^{-2}$, \n$R_{ADS}^{[D\\ensuremath{\\pi^0}\\xspace]K}=(1.8\\pm0.9\\pm0.4)\\times10^{-2}$, and\n$R_{ADS}^{[D\\gamma]K}=(1.3\\pm1.4\\pm0.8)\\times10^{-2}$, and the \\ensuremath{C\\!P}\\xspace asymmetries are\n$A_{ADS}^{DK}=-0.86\\pm0.47^{+0.12}_{-0.16}$, \n$A_{ADS}^{[D\\ensuremath{\\pi^0}\\xspace]K}=0.77\\pm0.35\\pm0.12$, and\n$A_{ADS}^{[D\\gamma]K}=0.36\\pm0.94^{+0.25}_{-0.41}$.\nFrom these results and external measurements of the relative amplitude and phase of \\ensuremath{\\Dbar^0}\\xspace to \\ensuremath{D^0}\\xspace mesons \ndecaying into the $K^- \\ensuremath{\\pi^+}\\xspace$ final state~\\cite{ref:hfag} we infer, using a frequentist procedure similar to that used in\nthe DP method,\n$r_\\ensuremath{B}\\xspace^{DK^\\pm}=(9.5^{+5.1}_{-4.1})\\%$ $[0,16.7]\\%$, $r_\\ensuremath{B}\\xspace^{D^*K^\\pm}=(9.6^{+3.5}_{-5.1})\\%$ $[0,15.0]\\%$,\nand the strong phases $\\delta_B^{DK^\\pm}$, $\\delta_B^{D^*K^\\pm}$, in good agreement with those obtained with the DP technique.\n\\begin{figure}[htb]\n\\begin{center}\n\\vskip-0.1cm\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{fig6a.eps} &\n\\includegraphics[width=0.45\\textwidth]{fig6b.eps} \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{fig:ads}\nProjections on \\mbox{$m_{\\rm ES}$}\\xspace for (a) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace \\ensuremath{D}\\xspace\\ensuremath{K^\\pm}\\xspace$ and (b) $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D^*[\\ensuremath{D}\\xspace\\ensuremath{\\pi^0}\\xspace]\\ensuremath{K^\\pm}\\xspace$, $\\ensuremath{D}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^\\mp}\\xspace\\ensuremath{\\pi^\\pm}\\xspace$ \nopposite-sign decays, for ADS samples enriched in signal ($NN>0.94$). \nThe points with error bars are data while the curves represent the fit projections for\nsignal plus background (solid), the sum of all background components (dashed), and $q\\bar q$ background only (dotted).\n}\n\\end{figure}\n\n\n\n\n\n\n\nIn the GLW method, we reconstruct $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays, followed by \\ensuremath{D}\\xspace decays to non-\\ensuremath{C\\!P}\\xspace ($\\ensuremath{D^0}\\xspace\\ensuremath{\\rightarrow}\\xspace\\ensuremath{K^-}\\xspace\\ensuremath{\\pi^+}\\xspace$), \\ensuremath{C\\!P}\\xspace-even ($\\ensuremath{K^+}\\xspace\\ensuremath{K^-}\\xspace$, $\\ensuremath{\\pi^+}\\xspace\\ensuremath{\\pi^-}\\xspace$), \nand \\ensuremath{C\\!P}\\xspace-odd ($\\KS\\ensuremath{\\pi^0}\\xspace$, $\\KS\\phi$, $\\KS\\omega$) eigenstates. The partial decay rate charge asymmetries $A_{\\ensuremath{C\\!P}\\xspace\\pm}$ for\n\\ensuremath{C\\!P}\\xspace-even and \\ensuremath{C\\!P}\\xspace-odd \\ensuremath{D}\\xspace final states and the ratios $R_{\\ensuremath{C\\!P}\\xspace\\pm}$ of the charged-averaged \\ensuremath{B}\\xspace meson partial decay\nrates in \\ensuremath{C\\!P}\\xspace ($R_{K\/\\pi}^\\pm$) and non-\\ensuremath{C\\!P}\\xspace ($R_{K\/\\pi}$) decays (normalized to the corresponding $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{\\pi^\\pm}\\xspace$ decays, \nto reduce systematic uncertainties)\nprovide four observables from which the three \nunknowns $\\gamma$, $r_\\ensuremath{B}\\xspace$ and $\\delta_\\ensuremath{B}\\xspace$ can be extracted (up to an 8-fold ambiguity for the phases). \nThe signal yields, expressed in terms of $A_{\\ensuremath{C\\!P}\\xspace\\pm}$, $R_{K\/\\pi}^\\pm$ and $R_{K\/\\pi}$ are extracted from UML fits \nto \\mbox{$m_{\\rm ES}$}\\xspace, \\mbox{$\\Delta E$}\\xspace, and \\ensuremath{\\mbox{$\\mathcal{F}$}}\\xspace. We identify about 500 $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays with \\ensuremath{C\\!P}\\xspace-even \\ensuremath{D}\\xspace final states and a similar\namount for \\ensuremath{C\\!P}\\xspace-odd \\ensuremath{D}\\xspace final states, and measure~\\cite{ref:babar_GLW-DK} \n$A_{\\ensuremath{C\\!P}\\xspace+}=0.25\\pm0.06\\pm0.02$, \n$A_{\\ensuremath{C\\!P}\\xspace-}=-0.09\\pm0.07\\pm0.02$,\n$R_{\\ensuremath{C\\!P}\\xspace+}=1.18\\pm0.09\\pm0.05$, and\n$R_{\\ensuremath{C\\!P}\\xspace-}=1.07\\pm0.08\\pm0.04$. \nThe parameter $A_{\\ensuremath{C\\!P}\\xspace+}$ is different from zero with a significance of $3.6\\sigma$, and constitutes evidence for\ndirect \\ensuremath{C\\!P}\\xspace violation in $\\ensuremath{B^\\pm}\\xspace \\ensuremath{\\rightarrow}\\xspace D \\ensuremath{K^\\pm}\\xspace$ decays. \nThese results can be written in terms of the observables\n$x_\\pm$ using the relationship $x_\\pm = [R_{\\ensuremath{C\\!P}\\xspace+}(1\\mp A_{\\ensuremath{C\\!P}\\xspace+}) - R_{\\ensuremath{C\\!P}\\xspace-}(1\\mp A_{\\ensuremath{C\\!P}\\xspace-})]\/4$.\nExcluding the $D\\ensuremath{\\rightarrow}\\xspace\\KS\\phi$, $\\phi\\ensuremath{\\rightarrow}\\xspace K^+K^-$ channel to facilitate the combination with the DP method,\nwe find $x_+=-0.057\\pm0.039\\pm0.015$ and $x_-=0.132\\pm0.042\\pm0.018$, which are consistent (and of similar\nprecision) with the DP method.\nFrom these results and using a frequentist procedure similar to that used previously we infer\n$24\\% < r_\\ensuremath{B}\\xspace < 45\\%$ $[6,51]\\%$, and\nmod $180^\\circ$, \n$11^\\circ < \\gamma < 23^\\circ$ or $81^\\circ < \\gamma < 99^\\circ$ or $157^\\circ < \\gamma < 169^\\circ$\n$[7^\\circ,173^\\circ]$.\n\n\n\nWe have reported the recent progress in the determination of the CKM angle $\\gamma$, using the complete \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ data sample and three different\nand complementary methods (DP, ADS, and GLW). \nA coherent and consistent set of results on $\\gamma$ and the hadronic parameters characterizing the \\ensuremath{B}\\xspace decays has been\nobtained. The central value for $\\gamma$, around $70^\\circ$ with a precision around $15^\\circ$, is consistent with indirect determinations from\nCKM fits~\\cite{ref:globalCKMfits}. \nA proper average of all the three methods using the full \\mbox{\\sl B\\hspace{-0.4em} {\\small\\sl A}\\hspace{-0.37em} \\sl B\\hspace{-0.4em} {\\small\\sl A\\hspace{-0.02em}R}}\\ sample of $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D^{(*)} K^{\\pm}$, $D K^{*\\pm}$ decays is foreseen.\nWe obtain $x_- - x_+ = 0.175\\pm0.040$ by combining the $x_\\pm$ measurements from the DP and GLW methods for $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D K^{\\pm}$ decays,\nwhich is different from zero with a significance of $4.4\\sigma$,\nthus constitutes strong evidence for direct \\ensuremath{C\\!P}\\xspace violation in these charged \\ensuremath{B}\\xspace decays.\nFinally, we have the first sign of an ADS signal in $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D K^{\\pm}$ and $\\ensuremath{B^\\pm}\\xspace\\ensuremath{\\rightarrow}\\xspace D^{(*)} K^{\\pm}$ decays.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}