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+{"text":"\\section{Introduction}\n\nThe binding energy of the nucleus, or its mass, contains\ninformation about interactions between its constituent protons and\nneutrons. Precision mass data as well as separation energies\nextracted from them can unveil much about the underlying nuclear\nstructure~\\cite{Blaum,Blaum2}. For instance, magic numbers can easily be seen from\nseparation energies which reveal a huge drop (depending on the\nsize of the closed shell), or a jump or a gap (depending on the way the\nseparation energy is constructed) after a magic number~\\cite{Novikov}. \nIn addition, separation energies can reflect\ncollective effects as we will discuss below. Such data are\ntherefore valuable in understanding the underlying shell structure\nin nuclei and the evolution of collective effects, and are\ntherefore essential to provide the basis for the development and\ntesting of a comprehensive theory of nuclei. In turn, a reliable\nnuclear theory is of utmost importance for calculating the\nproperties of unknown nuclei, which are needed, for example, in\nthe modelling of the rapid neutron capture process (r-process) of\nnucleosynthesis in stars~\\cite{Bertulani}. \nTherefore, new data on neutron-rich heavy\nnuclei are essential.\n\nHowever, such data are scarce mainly due to the complexity of\nproducing these exotic nuclei. This becomes evident if one\nglances at the chart of nuclides (see e.g. Ref.~\\cite{nchart}) where\nfor elements $Z\\sim70-80$ the number of observed neutron-rich\nnuclides is very limited. Very recently, several tens of new\nisotopes were discovered in projectile fragmentation of uranium\nbeams, though no spectroscopic information could yet be obtained for\nthese nuclei~\\cite{kurcewicz}. Their production rates are tiny, \nwhich requires very efficient measurement\ntechniques. One such technique for mass measurements is\nstorage-ring mass spectrometry~\\cite{FGM}.\n\nIn this paper we report on direct mass measurements of\nneutron-rich nuclei in the element range from lutetium to osmium.\nMasses for nine nuclei were measured for the first time, and for\nthree nuclei the mass uncertainty was improved. \nIt is known that nuclear collective effects can be seen in the behavior of nucleon separation energies~\\cite{Cakirli2}.\nHere, we investigate the relation between rather subtle effects in two-neutron separation energies, $S_{2n}$,\nand changes in both collectivity and neutron number.\nObserved irregularities in the smooth two-neutron separation\nenergies for Hf and W isotopes are linked to changes in\ncollective observables. The importance of the number of valence\nnucleons is discussed in the context of collective\ncontributions to binding energies calculated with the IBA\nmodel~\\cite{IBA}.\n\n\\section{Experiment}\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{fig1m.pdf}\n\\caption{Example of a Schottky frequency spectrum with the\ncorresponding isotope identification. This spectrum is\ncombined from eight independent injections acquired for the electron\ncooler voltage $U_c=209$~kV. Nuclides with known and previously\nunknown masses are indicated by different fonts (see legend).}\n\\label{id}\n\\end{figure*}\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2_u}\n\\caption{A zoom of the Schottky frequency spectrum illustrated in\nFig.~\\ref{id} on a quadruplet of $A=190$ isobars. The peaks\ncorresponding to nuclides in isomeric states are labeled with\n$m$.} \\label{spzoom}\n\\end{figure}\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=\\linewidth]{fig3}\n\\caption{(Color online) A part of the chart of nuclides indicating the\nnuclides measured in this work as well as the nuclides in the\nground and isomeric states identified in the other part of this\nexperiment devoted to the search for new K-isomers in this\nregion~\\cite{Matt,Matt2}.} \\label{nchart}\n\\end{figure*}\nThe experiment was conducted at GSI Helmholtzzentrum f\\\"ur\nSchwerionenforschung in Darmstadt. Heavy neutron-rich nuclei of\ninterest were produced in projectile fragmentation of $^{197}$Au\nprimary beams. The experiment described here was part of a\nlarger experimental campaign, some results of which are described\nin Refs.~\\cite{Matt,Matt2,Matt25,Matt3}. The $^{197}$Au beams were accelerated\nto the energy of 11.4~MeV\/u in the linear accelerator UNILAC and\nthen injected into the heavy ion synchrotron SIS-18~\\cite{sis},\nwhere they were further accelerated to an energy of 469.35~MeV\/u.\nThe $^{197}$Au$^{65+}$ beams were fast extracted (within about\n1~$\\mu$s) and focused on a production target located at the\nentrance of the fragment separator FRS~\\cite{frs,Geissel1992NIM}.\nAs target we used 1036~mg\/cm$^2$ thick $^9$Be with a 221~mg\/cm$^2$\nNb backing for more efficient electron stripping. The reaction\nproducts emerged from the target as highly-charged ions having\nmostly 0, 1, or 2 bound electrons. The nuclides of interest were\ntransported through the FRS, being operated as a pure magnetic\nrigidity ($B\\rho$) analyzer~\\cite{Geissel1992NIM}, and injected\ninto the cooler-storage ring ESR~\\cite{esr}. The transmission\nthrough the FRS and the injection into the ESR were optimized with\nthe primary beam, and the magnetic setting of FRS-ESR was fixed at\n$B\\rho=7.9$~Tm throughout the entire experiment. All ion species\nwithin the acceptance of the FRS-ESR of about $\\pm0.2$\\% were\ninjected and stored. Only $25$\\% of the ESR acceptance is filled\nat the injection. We note, that in contrast to the settings\ndescribed in Refs.~\\cite{Matt,Matt2}, in this experiment no\nenergy-loss degraders were employed in the FRS.\n\nThe relationship between relative revolution frequencies ($f$),\nrelative mass-over-charge ratios ($m\/q$) and velocities ($v$) of\nthe particles stored in a ring is given\nby~\\cite{FGM,Ra-PRL,Ra-NPA,LiBo}:\n\\begin{equation}\n\\label{sms1}\n\\frac{\\Delta f}{f}=-\\alpha_p\\frac{\\Delta \\frac{m}{q}}{\\frac{m}{q}}+(1-\\alpha_p\\gamma^2)\\frac{\\Delta v}{v},\n\\end{equation}\nwhere $\\gamma$ is the relativistic Lorentz factor, $\\alpha_p$\nis the momentum compaction factor, which characterizes\nthe relative variation of the orbit length of stored particles per\nrelative variation of their magnetic rigidity (for more details\nsee Refs.~\\cite{FGM,Ra-PRL,Ra-NPA,LiBo}). For the ESR, $\\alpha_p$\nis nearly constant for the entire revolution frequency acceptance\nand is $\\alpha_p\\approx0.179$. From Eq.~\\eqref{sms1} it becomes\nobvious that the revolution frequency is a measure of the\nmass-over-charge ratios of the stored ions provided that the\nsecond term on the right hand side, containing the velocity spread\n($\\Delta v\/v$), can be eliminated. The latter is achieved by\napplying electron cooling~\\cite{elcool}. For this purpose the\nstored ions are merged over a length of about 2.5~m with a\ncontinuous beam of electrons in the electron cooler device. The\nmean circumference of the ESR is 108.4~m and at our energies the\nions circulate with a frequency of about 2~MHz passing the\nelectron cooler at each revolution. The energy of the electrons is\nvery accurately defined by the applied acceleration potential.\nWithin a few seconds the mean velocity of the ions becomes equal\nto the mean velocity of the electrons. The velocity spread of the\nstored ions, which is $\\Delta v\/v\\approx4\\cdot10^{-3}$ at the injection,\nis thereby reduced to $\\Delta v\/v\\approx10^{-7}$~\\cite{elcool}.\n\nThe Schottky mass spectrometry (SMS) technique has been applied to\nthe electron cooled ions~\\cite{Ra-NPA,FGM}. In this technique,\nevery stored highly-charged ion at each revolution in the ESR\ninduces mirror charges on a couple of parallel electrostatic\ncopper plates, the Schottky pick-up installed inside the ring aperture.\nThe noise from the pick-up, which is dominated by the\nthermal noise, is amplified by a broad-band\nlow-noise amplifier~\\cite{Schaaf}. In the present experiment, we analyzed the\nnoise power at about 60~MHz, corresponding to the 30$^{th}$\nharmonic of the revolution frequency of the stored ions. The\npick-up signal was down-mixed using a $\\sim$60~MHz reference\nfrequency from an external frequency generator. The acceptance of\nthe ESR at the 30$^{th}$ harmonic corresponds to about\n320~kHz~\\cite{Li-2004,Li-NPA}. Therefore, to cover the entire ESR\nacceptance we digitized the signal with a sampling frequency of\n640~kHz using a commercial 16-bit ADC~\\cite{Kaza}. A Fourier\ntransform of the digitized data yielded the noise-power density\nspectrum, or the Schottky frequency spectrum~\\cite{FGM,Li-NPA}.\n\nNew ions were injected every few minutes. At the injection into\nthe ESR, the previously stored ions were removed. Several Schottky\nfrequency spectra were created for each injection. The parameters\nof the Fourier transform algorithm were optimized offline. A\nfrequency resolution of 4.77~Hz\/channel was chosen, which\ncorresponds to the time resolution of 0.21~s per spectrum.\nFurthermore, every 50 consecutive Schottky spectra were averaged\nto enhance the signal-to-noise ratio. Thus, Schottky spectra\nintegrated over 10~s were produced. The latter means that several\nindependent subsequent frequency spectra were obtained for each\ninjection of the ions into the ESR.\n\nThe electron cooling forces the ions to the same mean velocity\nthus filling the entire acceptance of the ESR of $\\Delta\nB\\rho\/B\\rho\\sim\\pm1.5\\%$ (see Ref.~\\cite{Ra-NPA}). Since $B\\rho=m\nv \\gamma \/ q$, by changing the velocity of the electrons in\ndifferent injections, ions with different $m\/q$ can be studied. In\nthe present experiment we varied the electron cooler voltage in\nthe range from 204~kV to 218~kV. On average eight injections were\nrecorded for each cooler setting. In order to facilitate the\nassignment of the revolution frequencies with the corresponding\nisotope identification, all spectra within each cooler setting\nwere combined together. In this case the maximum number of ion\nspecies present in each setting can be used for the\nidentification. The latter is done based on Eq.~\\eqref{sms1}.\nAs a starting point for the identification we used\nthe frequency of the stored $^{197}$Au$^{76+}$ primary ions. An\nexample of the combined Schottky frequency spectrum for an electron\ncooler voltage of $U_c=209$~kV is illustrated in Fig.~\\ref{id}.\nFig.~\\ref{spzoom} shows a zoom on a quadruplet of lines of\n$A=190$ isobars present in ground and\/or isomeric states. The\nlatter are indicated with a label $m$. The peak finding and the\nisotope identification were done automatically with a dedicated\nROOT-based~\\cite{ROOT} software~\\cite{Dasha}. The nuclides\nobserved in this experiment are illustrated on the chart of nuclides\nin Fig.~\\ref{nchart} together with the nuclides in the ground\nand isomeric states identified in the other part of this\nexperiment (see Refs.~\\cite{Matt,Matt2,Matt25,Matt3}).\n\n\\section{Data Analysis and Results}\n\\begin{table}[t!]\n\\caption{Nuclides with accurately known masses used as references\nto calibrate Schottky frequency spectra. Listed are the proton\n($Z$) and mass ($A$) numbers, the number of experimental settings \n($N_{set}$) in which this reference mass was observed, literature\nmass excess values from the Atomic-Mass Evaluation ~\\cite{AME} \n($ME_{AME}$) as well as the re-determined mass excess values\n($ME$) (see text) with the corresponding $\\sigma_{stat}$ uncertainty \nand its difference to the literature value\n($\\delta=ME-ME_{AME}$). \nNote that the systematic uncertainty of $\\sigma_{syst}=38$~keV (see text) is \nnot added here.\n\\label{references}}\n\\begin{center}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\nZ & A &  $N_{set}$ & $ME_{\\rm AME}$ & $ME$& $\\delta$ \\\\\n   &     &                     &  (keV)                      & (keV)                      &         (keV)     \\\\\n\\hline\n72 & 181 & 1 & -47412(2) & -47412(40) & 0(40)  \\\\\n\\hline\n73 & 181 & 1 & -48442(2) & -48383(40)& 59(40)  \\\\\n73 & 182 & 3 & -46433(2) & -46466(29)& -32(29)  \\\\\n73 & 183 & 3 & -45296(2) & -45276(16)& 20(16)  \\\\\n73 & 185 & 6 & -41396(14)& -41350(14)& 46(20)  \\\\\n\\hline\n74 & 184 & 5 & -45707(1) & -45663(17)& 44(17)  \\\\\n74 & 186 & 7 & -42510(2) & -42493(12)& 17(13)  \\\\\n74 & 187 & 8 & -39905(2) & -39863(8)& 41(8)  \\\\\n\\hline\n75 & 189 & 9 & -37978(8) & -38063(10)& -85(13)  \\\\\n75 & 191 & 9 & -34349(10)& -34364(3)& -15(11)  \\\\\n\\hline\n76 & 188 & 7 & -41136(1) & -41115(12)& 21(12)  \\\\\n76 & 190 & 7 & -38706(2) & -38637(15)& 69(15)  \\\\\n76 & 192 & 7 & -35881(3) & -35833(8)& 48(8)  \\\\\n76 & 193 & 7 & -33393(3) & -33329(8)& 63(8)  \\\\\n\\hline\n77 & 191 & 1 & -36706(2) & -36650(71)& 56(71) \\\\\n\\hline\n78 & 194 & 4 & -34763(1) & -34779(24)& -16(24)  \\\\\n78 & 196 & 9 & -32647(1) & -32655(4)& -7(4)  \\\\\n\\hline\n79 & 196 & 5 & -31140(3) & -31126(4)& 14(5)  \\\\\n\\hline\n\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\nIn order to determine the unknown mass-over-charge ratios, the\nSchottky frequency spectra have to be\ncalibrated~\\cite{Ra-NPA,Li-NPA}. For this purpose we selected the\nnuclides which were identified in our spectra and for which masses are\nknown experimentally according to the Atomic-Mass Evaluation 2003\n(AME)~\\cite{AME}. We note that an update of the AME was made\navailable in 2011~\\cite{AME11}, which however contains no new\ninformation in the mass region studied here. \nThe data of the present work were already included in the latest AME published very recently~\\cite{AME12}.\nFurthermore we\nrequired that the reference masses were obtained by more than one\nindependent measurement technique and that there must exist no\nother ionic species with a close mass-to-charge ratio which could simultaneously be stored in the ESR.\nAlso the peaks corresponding to long-lived isomeric states (we observed\n$^{182}$Hf$^{\\rm m1}$, $^{186}$W$^{\\rm m2}$, $^{190}$Os$^{\\rm m1}$ and $^{190}$Ir$^{\\rm m2}$ isomers)\nwere not used for calibration. The list of reference masses is\ngiven in Table~\\ref{references}.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\linewidth]{fig4} \n\\caption{Top: Mass-over-charge ratio $m\/q$ as a function of revolution\nfrequency. The solid line illustrates a straight line fit through\nthe calibration $m\/q$-values. Bottom: The same as top but with\nthe subtracted linear fit.} \\label{mqf}\n\\end{figure}\n\nThe momentum compaction factor $\\alpha_p$, although nearly\nconstant, is a complicated function of the revolution frequency\n$f$. If $\\alpha_p$ were exactly constant, then the $m\/q$ would linearly\ndepend on $f$. Fig.~\\ref{mqf} (top) shows an example of a linear\nfit through the calibration $m\/q$-values for one of the measured spectra. The\nresiduals of the fit (bottom panel of the same figure) clearly\nshow that the calibration function is more complicated than a\nlow-order polynomial function. Polynomials of up to 4th order (5\nfree coefficients) were employed, but different compared to the analyses\nperformed in Refs.~\\cite{Ra-NPA, Li-NPA,Chen2012}, the quality of\nthe fits was found unacceptable. \nA possible reason for the latter is the small number of reference masses in individual 10-s Schottky spectra.\nFurthermore, due to time variations of the storage ring and electron cooler parameters, \nsuch as, e.g., their magnetic fields, it is not possible to establish a universal calibration curve. \nTherefore, we employed an analysis procedure \nin which we used linear splines to approximate the calibration curve in each individual spectrum. \n\nChanges of the electron cooler voltage were done in steps of 0.5~kV\nso that for adjacent cooler settings the measured frequency spectra have a significant overlap.\nFurthermore, the same nuclide can be present in different charge states, which allows for a\nredundant analysis. The nuclides whose masses have been measured\nfor the first time or which mass accuracy was improved in this work are listed in Table~\\ref{newm}.\n\n\\begin{table}[t!]\n\\caption{Nuclides whose masses were determined for\nthe first time (in boldface) or whose mass uncertainty was improved in this work.\nListed are the proton ($Z$) and mass ($A$) numbers, the number of\nexperimental settings ($N_{set}$) in which this nuclide was\nobserved, and the obtained mass excess value ($ME$) with the\ncorresponding $1\\sigma$ total ($\\sqrt{\\sigma^2_{stat}+\\sigma^2_{syst}}$) uncertainty ($\\sigma(ME)$).\\label{newm}}\n\\begin{center}\n\\begin{tabular}{rrrrr}\n\\hline \\hline\nZ & A & $N_{set}$ &$ME$& $\\sigma(ME)$ \\\\\n   &     &                    & (keV) & (keV)                 \\\\\n\\hline\n{\\bf 71}&   {\\bf 181}& {\\bf 1} & {\\bf -44797}& {\\bf 126}\\\\\n{\\bf 71}&   {\\bf 183}&  {\\bf 1} & {\\bf -39716}& {\\bf 80}\\\\\n\\hline\n{\\bf 72}& {\\bf 185}& {\\bf 1} & {\\bf -38320}& {\\bf 64}\\\\\n{\\bf 72}& {\\bf 186}& {\\bf 1} & {\\bf -36424}&{\\bf 51}\\\\\n\\hline\n{\\bf 73}& {\\bf 187}& {\\bf 2} & {\\bf -36896}&{\\bf 56}\\\\\n{\\bf 73}&{\\bf 188}& {\\bf 2} & {\\bf -33612}&{\\bf 55}\\\\\n\\hline\n74& 189& 5 &    -35618& 40\\\\\n74& 190& 7 &    -34388& 41\\\\\n{\\bf 74}&{\\bf 191}&{\\bf 1} & {\\bf -31176}&{\\bf 42}\\\\\n\\hline\n{\\bf 75}&{\\bf 192}&{\\bf 1} & {\\bf -31589}&{\\bf 71}\\\\\n{\\bf 75}&{\\bf 193}&{\\bf 7} & {\\bf -30232}&{\\bf 39}\\\\\n\\hline\n76& 195& 1 &    -29512& 56\\\\\n\\hline\n\\hline\n\\end{tabular}\\end{center}\n\\end{table}\n\nSince the calibration curve is not known exactly and is approximated with linear splines, and\nsince the number of calibration points in each spectrum is small,\nthere is inevitably a systematic error introduced by the analysis method.\nWays to estimate systematic uncertainty have been described in our previous works~\\cite{Ra-NPA,Li-NPA,Chen2012}.\nFor this purpose in the present work, we re-determined the mass of each reference nuclide. \nThis was done consecutively by setting each of the references as ``no''-reference and\nobtaining its mass from the remaining 17 references. \nThe re-determined mass excess values are listed in Table~\\ref{references} along with their literature values~\\cite{AME}. \nThe systematic error $\\sigma_{syst}$ has been obtained from solving the following equation:\n\\begin{equation}\n\\sum_{i=1}^{N_{ref}}\\frac{(ME^{(i)}_{AME}-ME^{(i)})^2}{\\sigma_{AME(i)}^2+\\sigma_{(i)}^2+\\sigma_{syst}^2}=N_{ref},\n\\end{equation}\nwhere $N_{ref}=18$ is the number of reference nuclides, $ME^{(i)}$ ($\\sigma_{(i)}$) \nand $ME_{AME}^{(i)}$ ($\\sigma_{AME(i)}$) are the re-calculated and literature mass excess\nvalues (statistical uncertainties) of the $i$-th reference nuclide, respectively.\nThe systematic uncertainty of the present analysis amounts to $\\sigma_{syst}=38 $~keV. \nThe final uncertainties listed in Table~\\ref{newm} were obtained from a quadratic sum of the systematic and statistical uncertainties.\n\nWe note, that in contrast to Ref.~\\cite{Chen2012} we do not observe any significant systematic dependence \nof the re-calculated mass values versus their proton number and correspondingly do not reduce the systematic errors.\nA dedicated study should be performed to investigate the origin of this inconsistency.\n\n\\section{Discussion}\n\n\nThe new masses allow us to obtain interesting information on\nnuclear structure. Fig.~\\ref{s2n_e2} (left, middle) shows\ntwo-neutron separation energies ($S_{2n}$) as a function of\nneutron number for $Z=66-78$ in the $A\\sim180$ region.\nFig.~\\ref{s2n_e2} (left) is for even proton numbers while\nFig.~\\ref{s2n_e2} (middle) is for odd proton numbers. The new\n$S_{2n}$ values, for $^{181,183}$Lu, $^{185,186}$Hf,\n$^{187,188}$Ta, $^{191}$W, $^{192,193}$Re, calculated from the\nmasses measured in this study, are marked in red color. For known\nmasses whose values were improved in this experiment,\n$^{189,190}$W and $^{195}$Os, the literature $S_{2n}$ values are\nillustrated with black color.\n\nBy inspecting Fig.~\\ref{s2n_e2} (left), one can notice, that in\nthe $S_{2n}$ values of the even-$Z$ nuclei a flattening in Yb, Hf\nand W is seen at almost the last neutron numbers experimentally\nknown. The flattening in $S_{2n}$(W), using the improved W masses,\nis confirmed and $S_{2n}$ at $N=117$ continues with the same\nbehavior. In contrast to Hf and W, the new $S_{2n}$(Os) point,\nwhich is a bit lower at $N=119$ than in previous measurements,\nshows no flattening.\n\n\\begin{figure*}\n\\includegraphics[height=5.4cm]{fig5a}\\hspace{8mm}\n\\includegraphics[height=5.3cm]{fig5b}\n\\includegraphics[height=5.3cm]{fig5c}\n\\caption{(Color online) Data for the $A\\sim180$ region,\ntwo-neutron separation energies \\cite{AME} as a function of\nneutron number from $N=100$ to $N=124$ for even-$Z$ (left) from Dy\nto Pt and odd-$Z$ (middle) from Ho to Re. The new $S_{2n}$ values\nobtained from this work are shown in red color while the\nliterature $S_{2n}$ values are shown in black color. Right :\nEnergies of the first excited 2$^+$ states \\cite{NDC} for the same\neven-even nuclei as in the left panel.\\label{s2n_e2}}\n\\end{figure*}\n\nFig.~\\ref{s2n_e2} (middle) with the new measured masses does not\nshow similar effects in $S_{2n}$ as in Fig.~\\ref{s2n_e2} (left).\nHowever, there is a small change in slope (a more rapid fall-off)\nat $N=110$ compared to the lower-$N$ trend in $S_{2n}$($_{71}$Lu),\n$S_{2n}$($_{73}$Ta) and $S_{2n}$($_{75}$Re) (we also see this drop\nfor $_{72}$Hf and $_{74}$W in Fig.~\\ref{s2n_e2} (left)), and maybe\nat $N=115$ in $S_{2n}$($_{73}$Ta) as well. It is highly desirable\nto have more odd-$Z$ mass measurements in this region for a\ncomparison with even-$Z$ where more data are available. Thus, we\nconcentrate below on discussing even-$Z$ nuclei.\n\n\nFig.~\\ref{s2n_e2} (right) shows the energy of the first excited\n2$^+$ states against neutron number for $Z=70-78$ in the\n$A\\sim180$ region. This important, simple, observable has a high\nenergy (can be a few MeV) at magic numbers where nuclei are\nspherical and very low energies (less than 100 keV for well\ndeformed heavy nuclei) near mid-shell. The $E$(2$_1^+$) values\nusually decrease smoothly between the beginning and middle of a\nshell except when there is a sudden change in structure. Since\n$N\\sim104$ is mid-shell for the nuclei illustrated in\nFig.~\\ref{s2n_e2}, $E$(2$_1^+$) has a minimum at or close to\n$N=104$. After the mid-shell, the energy increases towards the\n$N=126$ magic number.\n\nLet us now focus on the W-Pt nuclei in Fig.~\\ref{s2n_e2} and\ncompare the behavior of $E$(2$_1^+$) and $S_{2n}$. In particular,\nwe look at $S_{2n}$ in isotopes where $E$(2$_1^+$) changes\nrapidly, indicating a sudden change in structure. In W,\n$E$(2$_1^+$) increases from $N=114$ to 116 by a considerably\nlarger amount compared to the other W isotopes (see\nRef.~\\cite{podolyak2000}). This jump signals a structural change\nfrom approximately constant deformation to decreasing deformation\nat $N=116$ (after $N=114$). Note that this neutron number is\nexactly where $S_{2n}$ exhibits flattening.\n\nSimilar to W, Os at $N=120$ (after $N=118$) has a jump in\n$E$(2$_1^+$). However, $S_{2n}$($^{196}$Os) does not reveal an\nobvious change at the same neutron number. At first glance, this\nseems inconsistent with the interpretation explained for W above\nbut, in fact, there might be an explanation for this different\nbehavior which would provide additional insight into the relation\nof binding to collectivity.\n\nReference~\\cite{Cakirli2} showed the structural sensitivity of\ncalculated collective contributions to binding. In addition\nRef.~\\cite{Cakirli2} stressed that collective binding is very\nsensitive to the number of valence nucleons.\nCalculated collective contributions to binding using the IBA-1\nmodel for boson numbers $N_{B}=5$ (left) and $N_{B}=16$ (right)\nare illustrated in the symmetry triangle \\cite{Cakirli2} in\nFig.~\\ref{triangle}. The three corners of the triangle describe\nthree dynamical symmetries, U(5) (vibrator), SU(3) (rotor) and\nO(6) ($\\gamma$-soft) (for more details, see Ref.~\\cite{Iachello}).\nThe color code in Fig.~\\ref{triangle} changes from yellow to red\nwhen the collective effects increase. Needless to say, nuclei have\nmore valence particles (so boson numbers) around the SU(3) corner\nthan the U(5) (and also O(6)) corner. One sees that the collective\nbinding energy (B.E.) rapidly increases for nuclei with axial\ndeformation, that is, near SU(3). Note that the triangles are\npresented for fixed boson numbers. In both, the color scale is\nkept the same to point out that the collective B.E.s are larger in\nN$_B$=16 than 5. As shown in Fig.~4 of Ref.~\\cite{Cakirli2}, the\ncollective binding energies vary approximately as the square of\nthe number of valence nucleons in the context of IBA calculations.\nTherefore, for a lower number of valence nucleons,\nFig.~\\ref{triangle} shows similar trends for both  $N_{B}=5$ and\n$N_{B}=16$, but the overall binding is considerably less (compare\n(left) and (right) of Fig.~\\ref{triangle}). We now suggest that\nthe behavior of $E(2_1^+)$ and $S_{2n}$ in Fig.~\\ref{s2n_e2} can\nbe understood in terms of this dual dependence of binding on\ncollectivity and valence nucleon number.\n\nIf $^{190}$W and $^{196}$Os are mapped in the symmetry triangle,\n$^{190}$W will likely be closer to the SU(3) corner than\n$^{196}$Os \\cite{Cakirli-pri}. One of the ways to understand this\nis from the $P$-factor \\cite{RCasten2}, defined as $P= N_p\\cdot N_n \/ (N_p +\nN_n)$, where $N_p$ denotes the number of valence protons (proton\nholes) and $N_n$ the number of valence neutrons (neutron holes).\nThus $P$ is a quantity that can provide a guide to structure. For\nexample, the onset of deformation in heavy nuclei corresponds to\nthe $P$-factor around 4 and 5. Generally, if $P$ is larger than 3,\ncollective effects increase. That is, nuclei become deformed and\napproach closer to the SU(3) corner.\n\nFig.~\\ref{Pfactor} shows color-coded values for the $P$-factor\nfor the $Z=50-82$, $N=82-126$ region, and indicates the\n$P$-factors for the nuclei relevant to this discussion. $^{190}$W\nhas 8 valence protons and 10 valence neutrons while $^{196}$Os has\n$N_p=6$ and $N_n=6$. Correspondingly, these nuclei have\n$P$-factors of 4.4 and 3, respectively. The greater collectivity\nof $^{190}$W compared to $^{196}$Os suggested by their $P$-factors\nis reflected in its lower 2$_1^+$ energies as seen in\nFig.~\\ref{s2n_e2} (right). Thus, for two reasons -- both greater\ncollectivity and more valence nucleons -- the collective binding\nshould be much greater in $^{190}$W than in $^{196}$Os and changes\nin binding energies ($S_{2n-coll}$ values) should be on a larger\nscale. We suggest that this accounts for the fact that we see a\nflattening in $^{190}$W clearly but not in $^{196}$Os in\nFig.~\\ref{s2n_e2} (left). Obviously, it is very important to have\nnew data, both masses and spectroscopic information, on even more\nneutron rich W isotopes although such experiments are difficult.\nEven the mass of $^{192}$W alone would be telling since the trend\nin $E$(2$_1^+$) is quite clear already.\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig6}\n\\caption{(Color online) The symmetry triangle of the IBA showing\nthe three dynamical symmetries at the vertices. The colors\nindicate calculated collective contributions in MeV to binding\nenergies for $N_B = 5$ (left) and $N_B=16$ (right). A similar\ntriangle for $N_B=16$ was presented in\nRef.~\\cite{Cakirli2}.\\label{triangle}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig7}\n\\caption{(Color online) $P$-factor values illustrated with a color\ncode for even-even nuclei in the $Z=50-82$ and $N=82-126$ shells.\nBlack points marked are for the key nuclei discussed, namely,\n$^{190}$W, $^{196}$Os, $^{188}$Pt,  $^{198}$Pt, and\n$^{152}$Sm.\\label{Pfactor}}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig8}\n\\caption{Calculated $E$(2$_1^+$) (top), $\\delta E$(2$_1^+$) (middle)\nand $\\delta S_{2n-coll}$ (bottom) values from IBA calculations as\na function of boson number $N_B$. The points for\n$\\delta E$(2$_1^+)$ and $\\delta S_{2n-coll}$  correspond to a set\nof schematic IBA calculations in which $\\kappa$, and $\\chi$ are\nconstant (at 0.02 and -1.32, respectively) while $\\epsilon$, and\n$N_B$ vary in a smooth way to simulate a spherical-to-deformed\ntransition region. The following equations are used for\n$\\delta E$(2$_1^+$) and $\\delta S_{2n-coll}$:\n$\\delta E(2_1^+)(Z,N)=[E(2_1^+)(Z,N) - E(2_1^+)(Z,N+2)] \/\nE(2_1^+)(Z,N)$ and\n$\\delta S_{2n-coll}=-[S_{2n-coll}(Z,N) -\nS_{2n-coll}(Z,N+2)$], respectively.\\label{dS2n-E2}}\n\\end{figure}\n\nExisting data on Pt nicely illustrate and support these ideas.\nFig.~\\ref{s2n_e2} (right) shows two jumps in $E$(2$_1^+$) for\nPt, around $N\\sim110$ and $N\\sim118-120$. Looking at $S_{2n}$ for\nPt, there is a kink near $N\\sim110$ but a smooth behavior near\n$N\\sim118$. For $^{188}$Pt, $N_p$ is 4 and $N_n$ is 16 so the\n$P$-factor is 3.2. This isotope, with 20 valence nucleons, is\nrelatively collective and once again one sees an anomaly in\n$S_{2n}$ as well. In contrast, for $^{198}$Pt$_{120}$ with only 10\nvalence nucleons, and a $P$-factor of only 2.4, the lower\ncollectivity (seen in the much higher 2$^+$ energy) and the lower\nnumber of valence nucleons are such that $S_{2n}$ shows no\nanomaly, but rather a nearly straight behavior.\n\nThis qualitative interpretation is supported by collective model\ncalculations. A thorough and detailed study of this or any\ntransition region requires a very careful and systematic\nassessment of all the data on energies, transition rates, and\nbinding energies, the choice for the specific terms to include in\nthe Hamiltonian and the optimum approach to fitting the data. We\nare undertaking such a study and will present the results in a\nfuture publication \\cite{Cakirli3}. Nevertheless, it is useful to\npresent an example of the model results here to validate the ideas\npresented above. To this end, we have carried out a schematic set\nof IBA calculations using the Hamiltonian \\cite{7,8}\n\n\\begin{equation}\n\\label{eqH}\nH = \\epsilon \\hat{n}_d - \\kappa{Q} \\cdot {Q}\n\\end{equation}\n\n\\noindent where $Q$ is a quadrupolar operator\n\n\\noindent\n\n\\begin{equation}\n\\label{eqQ} \n{Q}= (s^{\\dagger}\\tilde{d} +\nd^{\\dagger}s) + \\chi(d^{\\dagger}\\tilde{d})^{(2)}.\n\\end{equation}\n\n\\noindent The first term in Eq.~\\eqref{eqH} drives nuclei spherical while\nthe $Q\\cdot Q$ term induces collectivity and deformation. Therefore a\nspherical-deformed transition region involves a systematic change\nin the ratio of $\\epsilon$ to $\\kappa$. No generality is lost by\nkeeping $\\kappa$ constant (at 0.02 MeV). We follow a trajectory\nalong the bottom axis of the triangle corresponding to $\\chi\n=-1.3228$. Fig.~\\ref{dS2n-E2} illustrates the results, for $N_B\n= 6-16$ showing $E$(2$_1^+$) and the differentials of $E$(2$_1^+$) (for\n$N_B = 6-15$) and for the collective contributions to $S_{2n}$,\n$S_{2n-coll}$, (for $N_B = 6-14$). There is a clear change in\nstructure  at $N_B\\sim$10 which is seen in a change in trend of\n$E$(2$_1^+$). Between $N_B = 10$ and 11, R$_{4\/2}$ changes from 2.60\nto 3.13. This corresponds to a maximum in the normalized\ndifferential of $E$(2$_1^+$). Confirming our association of\nstructural changes with kinks in $S_{2n}$, the differential of the\ncollective part of $S_{2n}$ also shows an extremum at exactly the\nsame point. These ideas will be expanded in our future publication\n\\cite{Cakirli3}.\n\nBesides the experimental examples of a correlation of $E$(2$_1^+$) energies \nand $S_{2n}$ values discussed in the context of Fig.~\\ref{s2n_e2},  \nour interpretation can\neasily be illustrated with the Sm isotopes around $N=90$. As is well\nknown, there is a sudden onset of deformation for the rare earth\nnuclei from $N=88$ to 90. This effect is clear from various\nobservables. One example is seen in Fig.~\\ref{Sm-S2n-E2} which\nshows the experimental $E$(2$_1^+$) energies (top) and $S_{2n}$\n(bottom) as a function of neutron number. Note that we plot these\nagainst decreasing neutron number so that the deformed nuclei are\non the left and spherical ones on the right to make the comparison\nwith Fig.~\\ref{s2n_e2} easier. Note also that the overall trend in\n$S_{2n}$ is opposite from that in Fig.~\\ref{s2n_e2} since $S_{2n}$\nvalues decrease with increasing neutron number (going to the left in Fig.~\\ref{Sm-S2n-E2} (bottom) which simply \nreflects the filling of the shell model orbits. The noticeable\ndeviation occurs near $N\\sim90$ where there is\na distinct flattening. To correlate the trends in these two\nobservables in the $N=90$ region, one can use the same\ninterpretation as above for W at $N=116$, namely, if there is a\nvisible change at neutron number $N$ in $E$(2$_1^+$) and there are\nmany valence nucleons, we expect to see a change in the behavior\nof $S_{2n}$. In Fig.~\\ref{Sm-S2n-E2} (top), the $E$(2$_1^+$) change\noccurs at $N\\sim90$. The isotope $^{62}$Sm at $N=90$ has $N_p=12$ and\n$N_n=8$ so it has 10 bosons and its $P$-factor is 4.8 (see\nFig.~\\ref{Pfactor}). One therefore expects to see a change in\n$S_{2n}$. Fig.~\\ref{Sm-S2n-E2} (bottom) confirms this\nexpectation. The clear structural change at $N=90$ shown in\n$E$(2$_1^+$) is correlated with a larger binding compared to the\ngeneral trend in $S_{2n}$ as a function of $N$.\n\nSimilar correlations can be seen in some other nuclei as well.\nFurther details will be discussed in Ref.~\\cite{Cakirli3}.\nHowever, here, it is worth mentioning two more examples marked in\nFig.~\\ref{s2n_e2}. The case of Yb-isotopes is interesting. Yb at\n$N=107$ starts to change slope in $S_{2n}$ and a flattening occurs\nat $N=108$. The $P$-factor is $\\sim7$. With the interpretation\nabove, one would expect to see a change in $E$(2$_1^+$) after\n$N=106$, at $N=108$. However, there is no sudden change in\n$E$(2$_1^+$) in $^{178}$Yb. To understand Yb around $N\\sim108$\nbetter, it might therefore be useful to have additional $S_{2n}$\nvalues (mass measurements) and also more spectroscopic results for\nthe neutron-rich Yb isotopes.\n\n\nHf at $N=114$ has a $P$-factor 5.4 and one sees a flattening in\n$S_{2n}$. The corresponding 2$^+$ energies, however, are not\nknown. Thus, similarly as in Yb, we need more spectroscopic\nresults for Hf.\n\nTo summarize, we observed a correlation between the behavior of\n$S_{2n}$ obtained from our measured masses with the spectroscopic\ndata for $E(2_1^+)$, which could be related to nuclear\ncollectivity and valence nucleon number.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig9}\n\\caption{Experimental $E$(2$_1^+$) (top) and\n$S_{2n}$ (bottom) values against neutron number for $_{62}$Sm\n\\cite{AME11, NDC}. \\label{Sm-S2n-E2}}\n\\end{figure}\n\n\\section{Conclusion}\n\nDirect mass measurements of neutron-rich $^{197}$Au projectile\nfragments at the cooler-storage ring ESR yielded new mass data.\nMasses of nine nuclides were obtained for the first time and for\nthree nuclei the mass uncertainty was improved.\n\nWith the new masses, two-neutron separation energies, $S_{2n}$,\nare investigated. We showed that changes in structure, as\nindicated by changes in the collective observable $E$(2$^+_1$),\nare reflected in $S_{2n}$ values in nuclei such as $^{190}$W and\n$^{188}$Pt, which have large $P$-factors, are collective,  and\nhave large valence nucleon numbers. For nuclei with similar\nchanges in $E$(2$_1^+$), such as $^{196}$Os, and $^{198}$Pt, which\nhave lower collectivity and $P$-factors, and fewer valence\nnucleons, the sensitivity of collective binding to structure is\ngreatly reduced and smooth trends in $S_{2n}$ are observed. In Hf,\nthere are new $S_{2n}$ values at $N=113$, 114 where we see a\nflattening but there is no spectroscopic data at $N=114$. To\nconfirm the ideas discussed in this paper and also in\nRef.~\\cite{Cakirli3}, it would be useful to measure the\n$E$(2$_1^+$) for Hf at $N=114$. Similarly, mass and spectroscopic\nmeasurements are suggested for nuclei such as Yb with $N\\sim108$.\nTo conclude, these new data illustrate subtle changes in structure\nand the correlation with $E$(2$_1^+$) reveals a valuable way to\ncorrelate changes in structure in terms of both masses and\nspectroscopic observables. Of course, to quantitatively test these\nideas requires a systematic collective model study of the\nmass-structure relationship in this region. Such a project has\nbeen initiated~\\cite{Cakirli3} and we illustrated some of the\nresults here.\n\n\n\n\n\\section{Acknowledgments}\n\n\nThe authors would like to thank the GSI accelerator team for the excellent technical support. \nThis work was supported by the BMBF Grant in the framework of the Internationale Zusammenarbeit in Bildung und Forschung Projekt-FKZ 01DO12012,\nby the Alliance Program of the Helmholtz Association (HA216\/EMMI), by the Max-Planck Society and the US DOE under Grant No. DE-FG02-91ER-40609.\nD.S. is supported by the International Max Planck Research School for Precision Tests of Fundamental Symmetries at MPIK. \nR.B.C. thanks the Humboldt Foundation for support.\nK.B. and Y.A.L. thank ESF for support within the EuroGENESIS program.\nK.B. acknowledge support by the Nuclear Astrophysics Virtual Institute (NAVI) of the Helmholtz Association. \nZ.P. would like to acknowledge the financial support by Narodowe Centrum Nauki (Poland) grant No. 2011\/01\/B\/ST2\/05131. \nM.S.S. acknowledges the support by the Helmholtz International Centre for FAIR within the framework of the LOEWE program launched by the State of Hesse. \nB.S. is partially supported by NCET, NSFC (Grants No. 10975008, 11105010 and 11035007).\nP.M.W. acknowledges the support by the UK STFC and AWE plc.\nT.Y. is grateful for a grant-in-aid for a scientific research No. A19204023 by the Japanese Ministry of Education, Science and Culture. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nDespite the fantastic successes of the Standard Model (SM)\nof electroweak interactions,\nits scalar sector remains largely untested~\\cite{hhg}.\nAn alternative to the single Higgs doublet of the SM\nis provided by the two-Higgs-doublet model (THDM),\nwhich can be supplemented by symmetry requirements\non the Higgs fields $\\Phi_1$ and $\\Phi_2$.\nSymmetries leaving the kinetic terms unchanged\\footnote{It has\nbeen argued by Ginsburg~\\cite{Gin} and by\nIvanov~\\cite{Ivanov1,Ivanov2} that one should also consider\nthe effect of non-unitary global symmetry transformations\nof the two Higgs fields, as the most general renormalizable\nHiggs Lagrangian allows for kinetic mixing of the two Higgs fields.\nIn this work, we study the possible\nglobal symmetries of the effective low-energy Higgs theory that arise\n\\textit{after} diagonalization of the Higgs kinetic energy terms.\nThe non-unitary transformations that diagonalize the Higgs kinetic\nmixing terms also transform the parameters of the Higgs potential,\nand thus can determine the structure of the remnant Higgs flavor symmetries\nof effective low-energy Higgs scalar potential.  It is the latter \nthat constitutes the main focus of this work.}\nmay be of two types.  On the one hand,\none may relate $\\Phi_a$ with some unitary transformation\nof $\\Phi_b$.\nThese are known as Higgs Family symmetries, or HF symmetries.\nOn the other hand,\none may relate $\\Phi_a$ with some unitary transformation\nof $\\Phi_b^\\ast$.\nThese are known as generalized CP symmetries,\nor GCP symmetries.\nIn this article we consider all such symmetries that are\npossible in the THDM,\naccording to their impact on the Higgs potential.\nWe identify three classes of GCP symmetries.\n\nThe study is complicated by the fact that one may perform a basis\ntransformation on the Higgs fields, thus hiding what might otherwise\nbe an easily identifiable symmetry. The need to seek basis invariant\nobservables in models with many Higgs was pointed out by Lavoura and\nSilva \\cite{LS}, and by Botella and Silva \\cite{BS}, stressing\napplications to CP violation. Refs.~\\cite{BS,BLS} indicate how to\nconstruct basis invariant quantities in a systematic fashion for any\nmodel, including multi-Higgs-doublet models. Work on basis\ninvariance in the THDM was much expanded upon\nby Davidson and Haber \\cite{DavHab},\nby Gunion and Haber \\cite{GunHab,Gun},\nby Haber and O'Neil \\cite{HabONe},\nand by other authors \\cite{others}.\nThe previous approaches highlight the role\nplayed by the Higgs fields. An alternative approach, spearheaded by\nNishi \\cite{Nishi1,Nishi2}, by Ivanov \\cite{Ivanov1,Ivanov2} and by\nManiatis {\\em et al}~\\cite{mani}, highlights the role played by\nfield bilinears, which is very useful for studies of the vacuum\nstructure of the model \\cite{Barroso,earlier_bilinears}. In this\npaper, we describe all classes of HF and GCP symmetries in both\nlanguages.\nOne problem with two classes of GCP identified here is that\nthey lead to an exceptional region of parameter space\n(ERPS) previously identified as problematic\nby Gunion and Haber \\cite{GunHab} and\nby Davidson and Haber \\cite{DavHab}.\nIndeed, no basis invariant quantity exists in the literature that\ndistinguishes between the $Z_2$ and $U(1)$ HF symmetries in\nthe ERPS.\n\nIf evidence for THDM physics is revealed in future experiments, then\nit will be critical to employ analysis techniques that are free\nfrom model-dependent assumptions.  It is for this reason that\na basis-independent formalism for the THDM is so powerful.\nNevertheless, current experimental data already impose significant\nconstraints on the most general THDM.   In particular, we know that\ncustodial symmetry breaking effects, flavor changing neutral\ncurrent (FCNC) constraints, and (to a lesser extent) CP-violating phenomena \nimpose some significant restrictions on the structure of the THDM\n(including the Higgs-fermion interactions).  For example,\nthe observed suppression of FCNCs implies that either the two heaviest\nneutral Higgs bosons of the THDM have masses above 1 TeV,\nor certain Higgs-fermion Yukawa couplings must be absent~\\cite{Pas}.\nThe latter can be achieved by imposing certain discrete symmetries on\nthe THDM.  Likewise, in the most general THDM, mass splittings between\ncharged and neutral Higgs bosons can yield custodial-symmetry breaking\neffects at one-loop that could be large enough to be in \nconflict with the precision electroweak data~\\cite{precision}.  \nOnce again, symmetries\ncan be imposed on the THDM to alleviate any potential disagreement\nwith data.  The implications of such symmetries for THDM phenomenology\nhas recently been explored by Gerard and collaborators~\\cite{Gerard}\nand by Haber and O'Neil~\\cite{custodial}.\n\nThus, if THDM physics is discovered, it will be important to \ndevelop experimental methods that can reveal the presence or absence\nof underlying symmetries of the most general THDM.  This requires two\nessential pieces of input.  First, one must identify all possible\nHiggs symmetries of interest.  Second, one must relate these\nsymmetries to basis-independent observables that can be probed by\nexperiment.  In this paper, we primarily address the first step,\nalthough we also provide basis-independent characterizations of these\nsymmetries.  Our analysis focuses the symmetries of the THDM scalar\npotential.  In principle, one can extend our study of these symmetries to the\nHiggs-fermion Yukawa interactions, although this lies beyond the scope\nof the present work.\n\n\nThis paper is organized as follows.\nIn section~\\ref{sec:notation} we\nintroduce our notation and define an invariant that does\ndistinguish the $Z_2$ and $U(1)$ HF symmetries in\nthe ERPS.\nIn section~\\ref{sec:vacuum} we explain the role\nplayed by the vacuum expectation values in\npreserving or breaking the $U(1)$ symmetry,\nand we comment briefly on renormalization.\nIn section~\\ref{sec:GCP} we introduce the GCP\ntransformations and explain why they are organized\ninto three classes.\nWe summarize our results and set them in the\ncontext of the existing literature in section~\\ref{sec:summary},\nand\nin section~\\ref{sec:allisCP} we prove a surprising result:\nmultiple applications of\nthe standard CP symmetry can be used to\nbuild all the models we identify,\nincluding those based on  HF symmetries.\nWe draw our conclusions in\nsection~\\ref{sec:conclusions}.\n\n\n\\section{\\label{sec:notation}The scalar sector of the THDM}\n\n\\subsection{Three common notations for the scalar potential}\n\nLet us consider a $SU(2) \\otimes U(1)$ gauge theory with\ntwo Higgs-doublets $\\Phi_a$,\nwith the same hypercharge $1\/2$,\nand with vacuum expectation values (vevs)\n\\begin{equation}\n\\langle \\Phi_a \\rangle\n=\n\\left(\n\\begin{array}{c}\n0\\\\\nv_a\/\\sqrt{2}\n\\end{array}\n\\right).\n\\label{vev}\n\\end{equation}\nThe index $a$ runs from $1$ to $2$,\nand we use the standard definition for the electric\ncharge,\nwhereby the upper components of the $SU(2)$ doublets are\ncharged and the lower components neutral.\n\nThe scalar potential may be written as\n\\begin{eqnarray}\nV_H\n&=&\nm_{11}^2 \\Phi_1^\\dagger \\Phi_1 + m_{22}^2 \\Phi_2^\\dagger \\Phi_2\n- \\left[ m_{12}^2 \\Phi_1^\\dagger \\Phi_2 + \\textrm{H.c.} \\right]\n\\nonumber\\\\[6pt]\n&&\n+ \\tfrac{1}{2} \\lambda_1 (\\Phi_1^\\dagger\\Phi_1)^2\n+ \\tfrac{1}{2} \\lambda_2 (\\Phi_2^\\dagger\\Phi_2)^2\n+ \\lambda_3 (\\Phi_1^\\dagger\\Phi_1) (\\Phi_2^\\dagger\\Phi_2)\n+ \\lambda_4 (\\Phi_1^\\dagger\\Phi_2) (\\Phi_2^\\dagger\\Phi_1)\n\\nonumber\\\\[6pt]\n&&\n+ \\left[\n\\tfrac{1}{2} \\lambda_5 (\\Phi_1^\\dagger\\Phi_2)^2\n+ \\lambda_6 (\\Phi_1^\\dagger\\Phi_1) (\\Phi_1^\\dagger\\Phi_2)\n+ \\lambda_7 (\\Phi_2^\\dagger\\Phi_2) (\\Phi_1^\\dagger\\Phi_2)\n+ \\textrm{H.c.}\n\\right],\n\\label{VH1}\n\\end{eqnarray}\nwhere $m_{11}^2$, $m_{22}^2$, and $\\lambda_1,\\cdots,\\lambda_4$\nare real parameters.\nIn general,\n$m_{12}^2$, $\\lambda_5$, $\\lambda_6$ and $\\lambda_7$\nare complex. ``H.c.''~stands for Hermitian conjugation.\n\nAn alternative notation,\nuseful for the construction of invariants\nand championed by Botella and Silva \\cite{BS} is\n\\begin{eqnarray}\nV_H\n&=&\nY_{ab} (\\Phi_a^\\dagger \\Phi_b) +\n\\tfrac{1}{2}\nZ_{ab,cd} (\\Phi_a^\\dagger \\Phi_b) (\\Phi_c^\\dagger \\Phi_d),\n\\label{VH2}\n\\end{eqnarray}\nwhere Hermiticity implies\n\\begin{eqnarray}\nY_{ab} &=& Y_{ba}^\\ast,\n\\nonumber\\\\\nZ_{ab,cd} \\equiv Z_{cd,ab} &=& Z_{ba,dc}^\\ast.\n\\label{hermiticity_coefficients}\n\\end{eqnarray}\nThe extremum conditions are\n\\begin{equation}\n\\left[ Y_{ab}\n+ Z_{ab,cd}\\, v_d^\\ast v_c \\right]\\ v_b = 0\n\\hspace{3cm}(\\textrm{for\\ } a = 1,2).\n\\label{stationarity_conditions}\n\\end{equation}\nMultiplying by $v_a^\\ast$ leads to\n\\begin{equation}\nY_{ab} (v_a^\\ast v_b) = - Z_{ab,cd}\\,\n(v_a^\\ast v_b)\\, (v_d^\\ast v_c).\n\\label{aux_1}\n\\end{equation}\n\nOne should be very careful when comparing Eqs.~(\\ref{VH1})\nand (\\ref{VH2}) among different authors,\nsince the same symbol may be used for quantities\nwhich differ by signs, factors of two, or complex conjugation.\nHere we follow the definitions of Davidson and Haber\n\\cite{DavHab}.\nWith these definitions:\n\\begin{eqnarray}\nY_{11}=m_{11}^2, &&\nY_{12}=-m_{12}^2,\n\\nonumber \\\\\nY_{21}=-(m_{12}^2)^\\ast && Y_{22}=m_{22}^2,\n\\label{ynum}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_{11,11}=\\lambda_1, && Z_{22,22}=\\lambda_2,\n\\nonumber\\\\\nZ_{11,22}=Z_{22,11}=\\lambda_3, && Z_{12,21}=Z_{21,12}=\\lambda_4,\n\\nonumber \\\\\nZ_{12,12}=\\lambda_5, && Z_{21,21}=\\lambda_5^\\ast,\n\\nonumber\\\\\nZ_{11,12}=Z_{12,11}=\\lambda_6, && Z_{11,21}=Z_{21,11}=\\lambda_6^\\ast,\n\\nonumber \\\\\nZ_{22,12}=Z_{12,22}=\\lambda_7, && Z_{22,21}=Z_{21,22}=\\lambda_7^\\ast.\n\\label{znum}\n\\end{eqnarray}\n\nThe previous two notations look at the Higgs fields $\\Phi_a$ individually.\nA third notation is used by Nishi \\cite{Nishi1,Nishi2} and\nIvanov \\cite{Ivanov1,Ivanov2},\nwho emphasize\nthe presence of field bilinears $(\\Phi_a^\\dagger \\Phi_b)$\n\\cite{earlier_bilinears}.\nFollowing Nishi \\cite{Nishi1} we write:\n\\begin{equation}\nV_H = M_\\mu r_\\mu + \\Lambda_{\\mu \\nu} r_\\mu r_\\nu,\n\\label{VH3}\n\\end{equation}\nwhere $\\mu = 0,1,2,3$ and\n\\begin{eqnarray}\nr_0 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_1) + (\\Phi_2^\\dagger \\Phi_2) \\right],\n\\nonumber\\\\\nr_1 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_2) + (\\Phi_2^\\dagger \\Phi_1) \\right]\n= \\textrm{Re}\\, (\\Phi_1^\\dagger \\Phi_2),\n\\nonumber\\\\\nr_2 &=&\n- \\frac{i}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_2) - (\\Phi_2^\\dagger \\Phi_1) \\right]\n= \\textrm{Im}\\, (\\Phi_1^\\dagger \\Phi_2),\n\\nonumber\\\\\nr_3 &=&\n\\frac{1}{2}\n\\left[(\\Phi_1^\\dagger \\Phi_1) - (\\Phi_2^\\dagger \\Phi_2) \\right].\n\\label{r_Ivanov}\n\\end{eqnarray}\nIn Eq.~(\\ref{VH3}),\nsummation of repeated indices is\nadopted with Euclidean metric.\nThis differs from Ivanov's notation \\cite{Ivanov1,Ivanov2},\nwho pointed out that $r_\\mu$ parametrizes the gauge orbits\nof the Higgs fields,\nin a space equipped with a Minkowski metric.\n\nIn terms of the parameters of Eq.~(\\ref{VH1}),\nthe $4$-vector $M_\\mu$ and $4 \\times 4$ matrix $\\Lambda_{\\mu\\nu}$ are written\nrespectively as:\n\\begin{equation}\nM_\\mu =\n\\left(\n\\begin{array}{cccc}\nm_{11}^2 + m_{22}^2,\n&\n-2\\, \\textrm{Re}\\, m_{12}^2,\n&\n2\\, \\textrm{Im}\\, m_{12}^2,\n&\nm_{11}^2 - m_{22}^2\n\\end{array}\n\\right),\n\\label{M_mu}\n\\end{equation}\nand\n\\begin{equation}\n\\Lambda_{\\mu \\nu} =\n\\left(\n\\begin{array}{cccc}\n(\\lambda_1+\\lambda_2)\/2 + \\lambda_3\\\n&\\,\\,\\,\n\\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 + \\lambda_7)\n&\\,\\,\\,\n(\\lambda_1 - \\lambda_2)\/2\n\\\\\n \\phantom{-}\\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\\\n&\\,\\,\\,\n\\lambda_4 + \\textrm{Re}\\, \\lambda_5\n&\\,\\,\\,\n- \\textrm{Im}\\, \\lambda_5\n&\\,\\,\\,\n \\phantom{-}\\textrm{Re}\\, (\\lambda_6 - \\lambda_7)\\\n\\\\\n- \\textrm{Im}\\, (\\lambda_6 + \\lambda_7)\\\n&\\,\\,\\,\n- \\textrm{Im}\\, \\lambda_5\n&\\,\\,\\,\n\\lambda_4 - \\textrm{Re}\\, \\lambda_5\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 - \\lambda_7)\\\n\\\\\n(\\lambda_1 - \\lambda_2)\/2\n&\\,\\,\\,\n\\textrm{Re}\\, (\\lambda_6 - \\lambda_7)\n&\\,\\,\\,\n- \\textrm{Im}\\, (\\lambda_6 - \\lambda_7)\n&\\,\\,\\,\n\\ (\\lambda_1+\\lambda_2)\/2 - \\lambda_3\n\\end{array}\n\\right).\n\\label{Lambda_munu}\n\\end{equation}\nEq.~(\\ref{VH3}) is related to Eq.~(\\ref{VH2}) through\n\\begin{eqnarray}\nM^\\mu &=&\n\\sigma^\\mu_{ab}\\, Y_{ba},\n\\label{M_vs_Y}\n\\\\\n\\Lambda^{\\mu \\nu} &=&\n\\tfrac{1}{2} Z_{ab,cd}\\, \\sigma^\\mu_{ba} \\sigma^\\nu_{dc},\n\\label{Lambda_vs_Z}\n\\end{eqnarray}\nwhere the matrices $\\sigma^i$ are the three Pauli matrices,\nand $\\sigma^0$ is the $2 \\times 2$ identity matrix.\n\n\n\n\\subsection{Basis transformations}\n\nWe may rewrite the potential in terms of new fields $\\Phi^\\prime_a$,\nobtained from the original ones by a simple \n(global) basis transformation\n\\begin{equation}\n\\Phi_a \\rightarrow \\Phi_a^\\prime = U_{ab} \\Phi_b,\n\\label{basis-transf}\n\\end{equation}\nwhere $U\\in U(2)$ is a $2 \\times 2$ unitary matrix.\nUnder this unitary basis transformation,\nthe gauge-kinetic terms are unchanged,\nbut the coefficients $Y_{ab}$ and $Z_{ab,cd}$ are transformed as\n\\begin{eqnarray}\nY_{ab} & \\rightarrow &\nY^\\prime_{ab} =\nU_{a \\alpha}\\, Y_{\\alpha \\beta}\\, U_{b \\beta}^\\ast ,\n\\label{Y-transf}\n\\\\\nZ_{ab,cd} & \\rightarrow &\nZ^\\prime_{ab,cd} =\nU_{a\\alpha}\\, U_{c \\gamma}\\,\nZ_{\\alpha \\beta,\\gamma \\delta}\\, U_{b \\beta}^\\ast \\, U_{d \\delta}^\\ast ,\n\\label{Z-transf}\n\\end{eqnarray}\nand the vevs are transformed as\n\\begin{equation}\nv_a \\rightarrow v_a^\\prime = U_{a b} v_b.\n\\label{vev-transf}\n\\end{equation}\nThus,\nthe basis transformations $U$ may be utilized in order to absorb\nsome of the degrees of freedom of $Y$ and\/or $Z$,\nwhich implies that not all parameters of Eq.~(\\ref{VH2})\nhave physical significance.\n\n\n\n\\subsection{\\label{subsec:HFsymmetry}Higgs Family symmetries}\n\nLet us assume that the scalar potential in\nEq.~(\\ref{VH2}) has some explicit internal symmetry.\nThat is,\nwe assume that the coefficients of $V_H$ stay\n\\textit{exactly the same} under a transformation\n\\begin{equation}\n\\Phi_a \\rightarrow \\Phi_a^S = S_{ab} \\Phi_b.\n\\label{S-transf-symmetry}\n\\end{equation}\n$S$ is a unitary matrix,\nso that the gauge-kinetic couplings\nare also left invariant by this Higgs Family symmetry\n(HF symmetry).\nAs a result of this symmetry,\n\\begin{eqnarray}\nY_{a b} & = &\nY^S_{a b} =\nS_{a \\alpha}\\, Y_{\\alpha \\beta}\\, S_{b \\beta}^\\ast ,\n\\label{Y-S}\n\\\\\nZ_{ab,cd} & = &\nZ^S_{ab,cd} =\nS_{a \\alpha}\\, S_{c \\gamma}\\,\nZ_{\\alpha \\beta, \\gamma \\delta}\\, S_{b \\beta}^\\ast \\, S_{d \\delta}^\\ast .\n\\label{Z-S}\n\\end{eqnarray}\nNotice that this is \\textit{not} the situation considered\nin Eqs.~(\\ref{basis-transf})--(\\ref{Z-transf}).\nThere,\nthe coefficients of the Lagrangian\n\\textit{do change}\n(although the quantities that are physically\nmeasurable are invariant with respect to any change of basis).\nIn contrast, Eqs.~(\\ref{S-transf-symmetry})--(\\ref{Z-S})\nimply the existence of a HF symmetry $S$ of the scalar potential\nthat leaves the coefficients of $V_H$ unchanged.\n\nThe Higgs Family symmetry group must be a subgroup of full $U(2)$\ntransformation group of $2\\times 2$ unitary \nmatrices employed in Eq.~(\\ref{basis-transf}).  Given the most\ngeneral THDM scalar potential, there is always a $U(1)$ subgroup\nof $U(2)$ under which the scalar potential is invariant.\nThis is the global hypercharge $U(1)_Y$ symmetry group:\n\\begin{equation}\nU(1)_Y:\n\\hspace{4ex}\n\\Phi_1 \\rightarrow e^{i \\theta} \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow e^{i \\theta} \\Phi_2,\n\\label{U1Y}\n\\end{equation}\nwhere $\\theta$ is an arbitrary angle (mod $2\\pi$).  The invariance\nunder the global $U(1)_Y$ is trivially guaranteed by the \ninvariance under the $SU(2)\\otimes U(1)$ electroweak gauge symmetry.\n\\textit{Since the  global hypercharge $U(1)_Y$ is always present, we shall \nhenceforth define the \nHF symmetries as those Higgs Family symmetries that are \northogonal to $U(1)_Y$.}\n\nWe now turn to the interplay between\nHF symmetries and basis transformations.\nLet us imagine that,\nwhen written in the basis of fields $\\Phi_a$,\n$V_H$ has a symmetry $S$.\nWe then perform a basis transformation from\nthe basis $\\Phi_a$ to the basis $\\Phi^\\prime_a$,\nas given by Eq.~(\\ref{basis-transf}).\nClearly,\nwhen written in the new basis,\n$V_H$ does \\textit{not} remain invariant under $S$.\nRather, it will be invariant under\n\\begin{equation}\nS^\\prime = U S U^\\dagger .\n\\label{S-prime}\n\\end{equation}\nAs we change basis,\nthe form of the potential changes\nin a way that may obscure the presence\nof a HF symmetry.   In particular, two HF symmetries\nthat naively look distinct\nwill actually yield precisely the same physical predictions\nif a unitary matrix $U$ exists such that Eq.~\\eqref{S-prime} is satisfied.\n\nHF symmetries in the two-Higgs-doublet model (THDM) have a long\nhistory.\nIn papers by Glashow and Weinberg and by Paschos~\\cite{Pas}, the discrete\n$Z_2$ symmetry was introduced,\n\\begin{equation}\nZ_2:\n\\hspace{4ex}\n\\Phi_1 \\rightarrow \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow - \\Phi_2,\n\\label{Z2}\n\\end{equation}\nin order to preclude flavour-changing neutral currents \\cite{Pas}.\nThis is just the interchange\n\\begin{equation}\n\\Pi_2:\n\\hspace{4ex}\n\\Phi_1 \\leftrightarrow \\Phi_2,\n\\label{Pi2}\n\\end{equation}\nseen in a different basis,\nas shown by applying Eq.~(\\ref{S-prime}) in the form\n\\begin{equation}\n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n1 & 0 \\\\\n\\end{array}\n\\right)\n=\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & -1 \\\\\n\\end{array}\n\\right)\n\\\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{array}\n\\right)\n\\\n\\frac{1}{\\sqrt{2}}\n\\left(\n\\begin{array}{cc}\n1 & 1 \\\\\n1 & -1 \\\\\n\\end{array}\n\\right).\n\\label{Z2ToPi2}\n\\end{equation}\nPeccei and Quinn~\\cite{PQ} introduced the continuous $U(1)$ symmetry\n\\begin{equation}\nU(1):\n\\hspace{4ex}\n\\Phi_1 \\rightarrow e^{-i \\theta} \\Phi_1,\n\\hspace{4ex}\n\\Phi_2 \\rightarrow e^{i \\theta} \\Phi_2,\n\\label{U1}\n\\end{equation}\ntrue for any value of $\\theta$,\nin connection with the strong CP problem.\nOf course,\na potential invariant under $U(1)$ is also invariant\nunder $Z_2$.\n\nFinally, we examine the largest possible Higgs Family symmetry group\nof the THDM, namely $U(2)$.  In this case, a basis transformation\nwould have no effect on the Higgs potential parameters.  Since\n$\\delta_{ab}$ is the only $U(2)$-invariant tensor, it follows that\n\\begin{eqnarray}\nY_{ab}&=& c_1 \\delta_{ab}\\,,\\label{u2y}\\\\\nZ_{ab,cd} &=& c_2 \\delta_{ab} \\delta_{cd} + c_3 \\delta_{ad} \\delta_{bc}\\,,\n\\label{u2z}\n\\end{eqnarray}\nwhere $c_1$, $c_2$ and $c_3$ are arbitrary real \nnumbers.~\\footnote{Note that there is no $\\delta_{ac} \\delta_{bd}$ term\ncontributing to $Z_{ab,cd}$, as such a term is not invariant under\nthe transformation of Eq.~(\\ref{Z-transf}).}\nOne can easily check from Eqs.~(\\ref{Y-transf}) and (\\ref{Z-transf})\nthat the unitarity of $U$ implies that $Y'=Y$ and $Z'=Z$ \nfor any choice of basis, as required\nby the $U(2)$-invariance of the scalar potential.\nEqs.~(\\ref{u2y}) and (\\ref{u2z}) impose the following constraints on\nthe parameters of the THDM scalar potential (independently of the\nchoice of basis):\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_1 = \\lambda_2=\\lambda_3+\\lambda_4\\,,\n& \\hspace{4ex} &\n\\lambda_5 = \\lambda_6 = \\lambda_7= 0\\,.\n\\label{u2pot}\n\\end{eqnarray}\nAs there are no non-zero potentially complex scalar potential parameters,\nthe $U(2)$-invariant THDM is clearly CP-invariant.\n\nAs previously noted, the\n$U(2)$ symmetry contains the global hypercharge\n$U(1)_Y$ as a subgroup.  Thus, in order to identify\nthe corresponding HF symmetry that is orthogonal to $U(1)_Y$,\nwe first observe that\n\\begin{equation}\nU(2)\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SU(2)\\otimes U(1)_Y\/Z_2 \\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SO(3)\\otimes U(1)_Y\\,.\n\\end{equation}\nTo prove the above isomorphism, simply note that any $U(2)$ matrix can\nbe written as $U=e^{i\\theta}\\hat{U}$, where $\\hat U\\in SU(2)$.\nTo cover the full $U(1)_Y$ group, we must take $0\\leq\\theta<2\\pi$.\nBut since both $\\hat U$ and $-\\hat U$ are elements of $SU(2)$\nwhereas $+1$ and $-1=e^{i\\pi}$ are elements of $U(1)_Y$, we must\nidentify $\\hat U$ and $-\\hat U$ as the same group element in order not to\ndouble cover the full $U(2)$ group.  The identification of $\\hat U$\nwith $-\\hat U$ in $SU(2)$ is isomorphic to $SO(3)$, using the well known\nisomorphism $SO(3)\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SU(2)\/Z_2$.  Consequently, we have identified\nSO(3) as the HF symmetry that constrains the scalar potential\nparameters as indicated in Eq.(\\ref{u2pot}).\n\n\n\nThe impact of these symmetries on the potential parameters\nin Eq.~(\\ref{VH1}) is shown in section~\\ref{sec:summary}.\nAs mentioned above,\nif one makes a basis change, the potential parameters\nchange and so does the explicit form of the symmetry and\nof its implications.\nFor example,\nEq.~(\\ref{Z2ToPi2}) shows that the symmetries $Z_2$ and $\\Pi_2$\nare related by a basis change.\nHowever,\nthey have a different impact on the parameters in their\nrespective basis.\nThis can be seen explicitly in Table~\\ref{master1}\nof section~\\ref{sec:summary}. One can also easily prove that\nthe existence of either the $Z_2$, $\\Pi_2$ or Peccei-Quinn $U(1)$\nsymmetry is sufficient to guarantee the existence of a basis choice in\nwhich all scalar potential parameters are real.  That is, \nthe corresponding scalar Higgs sectors are explicitly CP-conserving.\n\nBasis invariant signs of HF symmetries were discussed\nextensively in Ref.~\\cite{DavHab}.\nRecently,\nFerreira and Silva~\\cite{FS2} extended these methods to include\nHiggs models with more than two Higgs doublets.\n\nConsider first the THDM scalar potentials that are invariant under \nthe so-called \\textit{simple} HF symmetries of Ref.~\\cite{FS2}.\nWe define a simple HF symmetry to be a symmetry\ngroup $G$ with the following property: the requirement that the\nTHDM scalar potential is invariant under a particular element \n$g\\in G$ (where $g\\neq e$ and $e$ is the identity element)\nis sufficient to guarantee invariance under the entire\ngroup $G$.  The discrete cyclic group \n$Z_n=\\{e\\,,\\,g\\,,\\,g^2\\,,\\,\\ldots\\,,\\,g^{n-1}\\}$,\nwhere $g^n = e$,\nis an example of a possible simple HF symmetry group.  \nIf we restrict the TDHM scalar potential to include terms of\ndimension-four or less (e.g., the tree-level scalar potential of the THDM),\nthen one can show that the Peccei-Quinn $U(1)$ symmetry is also\na simple HF symmetry.  For example, consider the matrix\n\\begin{equation}\nS=\\left( \\begin{array}{cc}\ne^{- 2 i \\pi\/3} & 0\\\\\n0 & e^{2 i \\pi\/3}\n\\end{array} \\right)\\,.\n\\end{equation}\nNote that $S$ is an element of the cyclic sub-group\n$Z_3=\\{S\\,,\\,S^2\\,,\\,S^3=1\\}$ of the\nPeccei-Quinn $U(1)$ group.\nAs shown in Ref.~\\cite{FS2}, the \ninvariance of the tree-level THDM scalar potential\nunder $\\Phi_a\\to S_{ab}\\Phi_b$ automatically implies the\ninvariance of the scalar potential under the full Peccei-Quinn $U(1)$\ngroup.  In contrast, the maximal HF symmetry, $SO(3)$, introduced\nabove is not a simple HF symmetry, as there is no single element of\n$S\\in SO(3)$ such that invariance under  $\\Phi_a\\to S_{ab}\\Phi_b$ \nguarantees invariance of the tree-level THDM \nscalar potential under the\nfull SO(3) group of transformations.\n\n\nTypically, the simple HF\nsymmetries take on a simple form for a particular choice of basis\nfor the Higgs fields.\nWe summarize here a few of the results of Ref.~\\cite{FS2}:\n\\begin{enumerate}\n\\item In the THDM, there are only two\n\\textit{independent} classes of \\textit{simple} symmetries:\na discrete $Z_2$ flavor symmetry, and a continuous Peccei-Quinn $U(1)$\nflavor symmetry.\n\\item Other discrete flavor symmetry groups $G$ that are subgroups of $U(1)$\nare not considered independent.  That is, if $S\\in G$ (where\n$S\\neq e$), then invariance under the\nthe discrete symmetry $\\Phi\\to S\\Phi$\nmakes the scalar potential automatically invariant under the full\nPeccei-Quinn $U(1)$ group;\n\\item In most regions of parameter space,\none can build quantities invariant under basis transformations\nthat detect these symmetries;\n\\item There exists a so-called exceptional region of parameter space (ERPS)\ncharacterized by\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = - \\lambda_6.\n\\label{ERPS}\n\\end{eqnarray}\nAs shown by Davidson and Haber \\cite{DavHab},\na theory obeying these constraints does have a $Z_2$ symmetry,\nbut it may or not have a $U(1)$ symmetry.\nWithin the ERPS, the invariants in the literature cannot be\nused to distinguish the two cases.\n\\end{enumerate}\n\nThe last statement above is a result of the following considerations.\nIn order to distinguish between $Z_2$ and $U(1)$,\nDavidson and Haber \\cite{DavHab} construct two\ninvariant quantities given by Eqs.~(46) and (50) of Ref.~\\cite{DavHab}.\nOutside the ERPS,\nthese quantities are zero if and only if $U(1)$ holds.\nUnfortunately,\nin the ERPS these quantities vanish automatically\nindependently of whether or not $U(1)$ holds.\nSimilarly,\nFerreira and Silva \\cite{FS2} have\nconstructed invariants detecting HF symmetries.\nBut their use requires the existence of a matrix, obtained\nby combining $Y_{ab}$ and $Z_{ab,cd}$,\nthat has two distinct eigenvalues.\nThis does not occur when the ERPS is due to a symmetry.\nFinally, in the ERPS,\nIvanov \\cite{Ivanov1} states that the symmetry might be\n``$(Z_2)^2$ or $O(2)$''\n[our $Z_2$ \\textit{or} our $U(1)$]\nand does not provide a way to distinguish the\ntwo possible flavor symmetries \\cite{oversight}.\n\nGunion and Haber \\cite{GunHab} have shown that\nthe ERPS conditions of Eq.~(\\ref{ERPS})\nare basis independent;\nif they hold in one basis, then they hold in any basis.\nMoreover, for a model in the ERPS,\na basis may be chosen such that all parameters are \nreal.\\footnote{Given a scalar potential whose parameters satisfy\nthe ERPS conditions\nwith ${\\rm Im}(\\lambda_5^* \\lambda_6^2)\\neq 0$, the unitary matrix\nrequired to transform into a basis in which all the scalar potential\nparameters are real can be determined only by numerical means.}\nHaving achieved such a basis,\nDavidson and Haber \\cite{DavHab} demonstrate that\none may make one additional basis transformation\nsuch that\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\textrm{Im}\\, \\lambda_5 = 0.\n\\label{ERPS2}\n\\end{eqnarray}\nThese conditions express the ERPS for a specific basis choice.\n\nOne might think that this is such a special region of parameter\nspace that it lacks any relevance.\nHowever,\nthe fact that the conditions in Eq.~(\\ref{ERPS}) hold in\n\\textit{any} basis is a good indication that a\nsymmetry may lie behind this condition.\nIndeed,\nas pointed out by Davidson and Haber \\cite{DavHab},\ncombining the two symmetries $Z_2$ and $\\Pi_2$\n\\textit{in the same basis} one is lead immediately to\nthe ERPS in the basis of Eq.~(\\ref{ERPS2}).\nUp to now,\nwe considered the impact of imposing\non the Higgs potential only one symmetry.\nThis was dubbed a simple symmetry.\nNow we are considering the possibility that the\npotential must remain invariant under one symmetry\nand \\textit{also} under a second symmetry;\nthis implies further constraints on the parameters\nof the Higgs potential.\nWe refer to this possibility as a multiple symmetry.\nAs seen from Table~\\ref{master1} of section~\\ref{sec:summary},\nimposing $Z_2$ and $\\Pi_2$ in the same basis leads\nto the conditions in Eq.~(\\ref{ERPS2}).\nIncidentally,\nthis example shows that a model which lies in the\nERPS,\nis automatically invariant under $Z_2$.\n\nIn section~{\\ref{sec:GCP}} we will show that\nall classes of non-trivial CP transformations lead\ndirectly to the ERPS,\nreinforcing the importance of this particular region of\nparameter space.\n\n\n\\subsection{Requirements for $U(1)$ invariance}\n\nIn the basis in which the $U(1)$ symmetry takes the form\nof Eq.~(\\ref{U1}),\nthe coefficients of the potential must obey\n\\begin{equation}\nm^{\\prime\\,2}_{12}= 0,\n\\hspace{4ex}\n\\lambda^\\prime_5 = \\lambda^\\prime_6 = \\lambda^\\prime_7 = 0.\n\\label{U1_conditions}\n\\end{equation}\nImagine that we have a potential of Eq.~(\\ref{VH1})\nin the ERPS:\n$m_{11}^2 = m_{22}^2$,\n$m_{12}^2=0$,\n$\\lambda_2=\\lambda_1$,\nand $\\lambda_7 = - \\lambda_6$.\nWe now wish to know whether a transformation $U$ may be chosen\nsuch that the potential coefficients in the new basis\nsatisfy the $U(1)$ conditions in Eq.~(\\ref{U1_conditions}).\nUsing the transformation rules in Eqs.~(A13)-(A23) of Davidson and\nHaber \\cite{DavHab},\nwe find that such a choice of $U$ is possible if and only if the\ncoefficients in the original basis satisfy\n\\begin{equation}\n2 \\lambda_6^3 - \\lambda_5 \\lambda_6(\\lambda_1 - \\lambda_3 - \\lambda_4)\n- \\lambda_5^2 \\lambda_6^\\ast = 0,\n\\label{can_change_to_usual_U1}\n\\end{equation}\nsubject to the condition that $\\lambda_5^\\ast \\lambda_6^2$ is real.\n\n\n\\subsection{\\label{subsec:D}The D invariant}\n\nHaving established the importance of the ERPS\n(as it can arise from a symmetry),\nwe will now build a basis invariant quantity that\ncan be used to detect the presence of a U(1)\nsymmetry in this special case.\n\nThe quadratic terms of the Higgs potential are always\ninsensitive to the difference between $Z_2$ and $U(1)$.\nMoreover,\nthe matrix $Y$ is proportional to the unit matrix in the ERPS.\nOne must thus look at the quartic terms.\nWe were inspired by the expression of $\\Lambda_{\\mu \\nu}$\nin Eq.~(\\ref{Lambda_munu}),\nwhich appears in the works of Nishi \\cite{Nishi1,Nishi2} and\nIvanov \\cite{Ivanov1,Ivanov2}.\nIn the ERPS of Eq.~(\\ref{ERPS}),\n$\\Lambda_{\\mu \\nu}$ breaks into a $1 \\times 1$ block\n($\\Lambda_{00}$),\nand a $3 \\times 3$ block\n($\\tilde{\\Lambda} = \\left\\{\\Lambda_{ij}\\right\\}$; $i,j=1,2,3$).\nA basis transformation $U$ belonging to $SU(2)$ on the $\\Phi_a$ fields\ncorresponds to an orthogonal $SO(3)$ transformation\nin the $r_i$ bilinears,\ngiven by\n\\begin{equation}\nO_{ij} = \\hbox{$\\frac{1}{2}$}\\,\\textrm{Tr} \n\\left[ U^\\dagger \\sigma_i U \\sigma_j \\right].\n\\label{O}\n\\end{equation}\nAny matrix $O$ of $SO(3)$ can be obtained by considering an\nappropriate matrix $U$ of $SU(2)$\n(unfortunately this property does not generalize for\nmodels with more than two Higgs doublets).\nA suitable choice of $O$ can be made that diagonalizes\nthe $3 \\times 3$ matrix $\\tilde{\\Lambda}$,\nthus explaining Eq.~(\\ref{ERPS2}).\nIn this basis,\nthe difference between the usual choices for $U(1)$ and $Z_2$ corresponds\nto the possibility that $\\textrm{Re} \\lambda_5$ might\nvanish or not, respectively.\n\nWe will now show that,\nonce in the ERPS,\nthe condition for the existence of $U(1)$ is that\n$\\tilde{\\Lambda}$ has two eigenvalues which are equal.\nThe eigenvalues of a $3 \\times 3$ matrix are the\nsolutions to the secular equation\n\\begin{equation}\nx^3 + a_2 x^2 + a_1 x + a_0 = 0,\n\\label{secular}\n\\end{equation}\nwhere\n\\begin{eqnarray}\na_0 &=&\n\\det \\tilde{\\Lambda}\n= - \\tfrac{1}{3}  \\textrm{Tr\\,} (\\tilde{\\Lambda}^3)\n- \\tfrac{1}{6} ( \\textrm{Tr\\,} \\tilde{\\Lambda} )^3\n+ \\tfrac{1}{2} ( \\textrm{Tr\\,} \\tilde{\\Lambda} )\n\\textrm{Tr\\,} (\\tilde{\\Lambda}^2)\n\\nonumber\\\\\n&=&\n- \\tfrac{1}{3} Z_{ab,cd}\n\\left( Z_{dc,gh} Z_{hg,ba} - \\tfrac{3}{2} Z^{(2)}_{dc} Z^{(2)}_{ba} \\right)\n+ \\tfrac{1}{2} Z_{ab,cd} Z_{dc,ba}\n\\textrm{Tr\\,} \\left( Z^{(1)} - \\tfrac{1}{2} Z^{(2)} \\right)\n\\nonumber\\\\\n& &\n-\\tfrac{1}{6} \\left( \\textrm{Tr\\,} Z^{(1)}\\right)^3\n+\\tfrac{1}{4} \\left( \\textrm{Tr\\,} Z^{(1)}\\right)^2 \\textrm{Tr\\,} Z^{(2)}\n- \\tfrac{1}{2} \\textrm{Tr\\,} Z^{(1)} \\left( \\textrm{Tr\\,} Z^{(2)}\\right)^2,\n\\\\\na_1 &=&\n\\tfrac{1}{2}  ( \\textrm{Tr\\,} \\tilde{\\Lambda} )^2\n- \\tfrac{1}{2} \\textrm{Tr\\,} (\\tilde{\\Lambda}^2)\n\\nonumber\\\\\n&=&\n\\tfrac{1}{2}\n\\left[\n\\left( \\textrm{Tr\\,} Z^{(1)}\\right)^2\n- \\textrm{Tr\\,} Z^{(1)} \\textrm{Tr\\,} Z^{(2)}\n+ \\left( \\textrm{Tr\\,} Z^{(2)}\\right)^2\n- Z_{ab,cd} Z_{dc,ba}\n\\right],\n\\\\\na_2 &=&\n- \\textrm{Tr\\,} \\tilde{\\Lambda}\n\\nonumber\\\\\n&=&\n\\tfrac{1}{2} \\textrm{Tr\\,} Z^{(2)} - \\textrm{Tr\\,} Z^{(1)},\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nZ_{ab}^{(1)}\n&\\equiv&\nZ_{a \\alpha,\\alpha b} =\n\\left( \\begin{array}{cc}\n  \\lambda_1 + \\lambda_4 & \\quad \\lambda_6 + \\lambda_7 \\\\\n  \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad \\lambda_2 + \\lambda_4\n\\end{array} \\right),\n\\\\\nZ_{ab}^{(2)}\n&\\equiv&\nZ_{\\alpha \\alpha,a b} =\n\\left( \\begin{array}{cc}\n  \\lambda_1 + \\lambda_3 & \\quad \\lambda_6 + \\lambda_7 \\\\\n  \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad \\lambda_2 + \\lambda_3\n\\end{array} \\right).\n\\end{eqnarray}\nThe cubic equation, Eq.~(\\ref{secular}), has at least two\ndegenerate solutions if \\cite{AbrSte}\n\\begin{equation}\nD \\equiv\n\\left[ \\tfrac{1}{3} a_1 - \\tfrac{1}{9} a_2^2 \\right]^3\n+  \\left[ \\tfrac{1}{6} (a_1 a_2 - 3 a_0) - \\tfrac{1}{27} a_2^3 \\right]^2\n\\end{equation}\nvanishes.\n\nThe expression of $D$ in terms of the parameters in Eq.~(\\ref{VH1})\nis rather complicated,\neven in the ERPS.\nBut one can show by direct computation that if the $U(1)$-symmetry\ncondition of Eq.~(\\ref{can_change_to_usual_U1}) holds\n(subject to $\\lambda_5^\\ast \\lambda_6^2$ being real),\nthen $D=0$.\nWe can simplify the expression for $D$ by changing to a\nbasis where all parameters are real \\cite{GunHab},\nwhere we get\n\\begin{equation}\nD =\n- \\tfrac{1}{27}\n\\left[ \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5)\n       - 2 \\lambda_6^2 \\right]^2\n\\left[ (\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)^2\n       + 16 \\lambda_6^2 \\right].\n       \\label{eq:D}\n\\end{equation}\nIf $\\lambda_6 \\neq 0$, then $D=0$ means\n\\begin{equation}\n2 \\lambda_6^2 = \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5).\n\\label{L6NEQ0}\n\\end{equation}\nIf $\\lambda_6 = 0$,\nthen $D=0$ corresponds to one of three possible conditions:\n\\begin{equation}\n\\lambda_5 = 0,\n\\hspace{4ex}\n\\lambda_5 = \\pm (\\lambda_1 - \\lambda_3 - \\lambda_4).\n\\label{L6EQ0}\n\\end{equation}\nNotice that Eqs.~(\\ref{L6NEQ0}) and (\\ref{L6EQ0}) are\nequivalent to Eq.~(\\ref{can_change_to_usual_U1})\nin any basis where the coefficients are real.\n\nAlthough $D$ can be defined outside the ERPS,\nthe condition $D=0$ only guarantees that the model is invariant under\n$U(1)$ inside the ERPS of Eq.~(\\ref{ERPS}).\nOutside this region one can detect the presence of a $U(1)$ symmetry\nwith the invariants proposed by Davidson and Haber \\cite{DavHab}.\nThis closes the last breach in the literature concerning basis-invariant\nsignals of discrete symmetries in the THDM.\nThus, in the ERPS $D=0$ is a necessary and sufficient condition for\nthe presence of a $U(1)$ symmetry.\n\n\n\\section{\\label{sec:vacuum}Vacuum structure and renormalization}\n\nThe presence of a $U(1)$ symmetry in the Higgs potential\nmay (or not) imply the existence of a massless scalar, the axion,\ndepending on whether (or not) the $U(1)$ is broken by the vevs.\nIn the previous section we related the basis-invariant condition\n$D=0$ in the ERPS with the presence of a $U(1)$ symmetry.\nIn this section we will show that,\nwhenever the basis-invariant condition $D=0$ is\nsatisfied in the ERPS,\nthere is always a stationary point for which a massless scalar,\nother than the usual Goldstone bosons, exists.\n\nWe start by writing the extremum conditions for the THDM in the\nERPS.\nFor simplicity, we will be working in a basis where all the\nparameters are real~\\cite{GunHab}.\nFrom Eqs.~\\eqref{stationarity_conditions} and~\\eqref{znum},\nwe obtain\n\\begin{align}\n0 = & \\;\\;\nY_{11}\\,v_1\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_1^3\\,+\n\\,\\lambda_{345}\\,v_1\\,v_2^2\\,+\\,\n\\lambda_6\\,(3\\,v_1^2\\,v_2\\,-\\,v_2^3)\\right],\n\\nonumber\\\\\n0 = & \\;\\;\nY_{11}\\,v_2\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_2^3\\,+\n\\,\\lambda_{345}\\,v_2\\,v_1^2\\,+\\,\n\\lambda_6\\,(v_1^3\\,-\\,3\\,v_2^2\\,v_1)\\right],\n\\label{eq:stat}\n\\end{align}\nwhere we have defined\n$\\lambda_{345}\\equiv\\lambda_3 + \\lambda_4 + \\lambda_5$.\nWe now compute the mass matrices.\nAs we will be considering only vacua with real vevs,\nthere will be no mixing between the real and imaginary\nparts of the doublets.\nAs such, we can define the mass matrix of the CP-even scalars as given by\n\\begin{equation}\n\\left[M^2_h\\right]_{ij}\\;=\\;\\frac{1}{2}\\,\\frac{\\partial^2\nV}{\\partial \\mbox{Re}(\\Phi_i^0)\\,\n\\partial \\mbox{Re}(\\Phi_j^0)}\n\\end{equation}\nwhere $\\Phi_i^0$ is the neutral (lower) component of the $\\Phi_i$\ndoublet. Thus, we obtain, for the entries of this matrix, the\nfollowing expressions:\n\\begin{align}\n\\left[M^2_h\\right]_{11}\\;=& \\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left(3\\,\\lambda_1\\,v_1^2\\,+\n\\,\\lambda_{345}\\,v_2^2\n\\,\n+\\,6\\,\\lambda_6\\,v_1\\,v_2\\right)\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_h\\right]_{22}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left(3\\,\\lambda_1\\,v_2^2\\,+\n\\,\\lambda_{345}\\,v_1^2\n\\,\n+\\,6\\,\\lambda_6\\,v_1\\,v_2\\right)\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_h\\right]_{12}\\;=& \\;\\lambda_{345}\\,v_1\\,v_2\\,\n\\,+\\,\\frac{3}{2}\\,\\lambda_6\\,(v_1^2\\,-\\,v_2^2)\\;\\;\\; .\n\\label{eq:mh}\n\\end{align}\nLikewise, the pseudoscalar mass matrix is defined as\n\\begin{equation}\n\\left[M^2_A\\right]_{ij}\\;=\\;\\frac{1}{2}\\,\\frac{\\partial^2\nV}{\\partial \\mbox{Im}(\\Phi_i^0)\n\\partial \\mbox{Im}(\\Phi_j^0)}\n\\end{equation}\nwhose entries are given by\n\\begin{align}\n\\left[M^2_A\\right]_{11}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_1^2\\,+\\,\n\\left(\\lambda_3\\,+\\,\\lambda_4\\,-\\,\\lambda_5\\right)\\,v_2^2\n\\,+\\,2\\,\\lambda_6\\,v_1\\,v_2 \\right]\n\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_A\\right]_{22}\\;=&\\;\nY_{11}\\,+\\,\\frac{1}{2}\\,\\left[\\lambda_1\\,v_2^2\\,+\\,\n\\left(\\lambda_3\\,+\\,\\lambda_4\\,-\\,\\lambda_5\\right)\\,v_1^2\n\\,-\\,2\\,\\lambda_6\\,v_1\\,v_2 \\right]\n\\nonumber \\vspace{0.2cm} \\\\\n\\left[M^2_A\\right]_{12}\\;=& \\;\\lambda_5\\,v_1\\,v_2 \\,+\\,\n\\frac{1}{2}\\,\\lambda_6\\,(v_1^2\\,-\\,v_2^2)\\;\\;\\; .\n\\label{eq:mA}\n\\end{align}\nThe expressions ~\\eqref{eq:mh} and~\\eqref{eq:mA} are valid for all\nthe particular cases we will now consider.\n\n\n\\subsection{Case $\\lambda_6\\,=\\,0$, $\\{v_1\\,,\\,v_2\\}\\,\\neq\\,0$}\n\nLet us first study the case $\\lambda_6\\,=\\,0$, wherein we may solve\nthe extremum conditions in an analytical manner. It is trivial\nto see that Eqs.~\\eqref{eq:stat} have three types of solutions: both\nvevs different from zero, one vev equal to zero (say, $v_2$) and\nboth vevs zero (trivial non-interesting solution). For a solution\nwith $\\{v_1\\,,\\,v_2\\}\\,\\neq\\,0$, a necessary condition must be\nobeyed so that there is a solution to Eqs.~\\eqref{eq:stat}:\n\\begin{equation}\n\\lambda_1^2\\,-\\,\\lambda_{345}^2 \\;\\neq\\;0\\;\\;\\; . \\label{eq:det}\n\\end{equation}\nIf we use the extremum conditions to evaluate\n$\\left[M^2_h\\right]$, we obtain\n\\begin{equation}\n\\left[M^2_h\\right]\\;=\\;\\begin{pmatrix} \\lambda_1\\,v_1^2 & \\quad\n\\lambda_{345}\\,v_1\\,v_2 \\\\ \\lambda_{345}\\,v_1\\,v_2 & \\quad\n\\lambda_1\\,v_2^2\n\\end{pmatrix}\n\\end{equation}\nwhich only has a zero eigenvalue if Eq.~\\eqref{eq:det} is broken.\nThus, there is no axion in this matrix in this case. As for\n$\\left[M^2_A\\right]$, we get\n\\begin{equation}\n\\left[M^2_A\\right]\\;=\\;-\\,\\lambda_5\\,\\begin{pmatrix}\nv_1^2 &\\quad  v_1\\,v_2 \\\\\nv_1\\,v_2 &\\quad v_2^2 \\end{pmatrix}\n\\end{equation}\nwhich clearly has a zero eigenvalue corresponding to the $Z$\nGoldstone boson. Further, this matrix will have an axion if\n$\\lambda_5\\,=\\,0$, which is the first condition of\nEq.~\\eqref{L6EQ0}.\n\n\\subsection{Case $\\lambda_6\\,=\\,0$, $\\{v_1\\,\\neq\\,0,\\,v_2\\,=\\,0\\}\\,$}\n\nReturning to Eq.~\\eqref{eq:stat}, this case gives us\n\\begin{equation}\nY_{11}\\,=\\,-\\,\\frac{1}{2}\\,\\lambda_1\\,v_1^2\\;\\;\\; ,\n\\end{equation}\nwhich implies $Y_{11}\\,<\\,0$. With this condition, the mass matrices\nbecome considerably simpler:\n\\begin{equation}\n\\left[M^2_h\\right]\\;=\\;\\begin{pmatrix} \\lambda_1\\,v_1^2 & \\quad 0 \\\\\n 0 & \\quad \\frac{1}{2}\\,(\\lambda_{345}\\,-\\,\\lambda_1)\\,v_1^2\n\\end{pmatrix}\n\\label{eq:mhll}\n\\end{equation}\nand\n\\begin{equation}\n\\left[M^2_A\\right]\\;=\\;\\frac{1}{2}\\,\\begin{pmatrix} 0 & \\quad 0 \\\\\n 0 & \\quad (\\lambda_3\\,+\\,\\lambda_4\\,-\\lambda_5\\,-\\,\\lambda_1)\\,v_1^2\n\\end{pmatrix} \\;\\;\\; .\n\\label{eq:mAll}\n\\end{equation}\nSo, we can have an axion in the matrix~\\eqref{eq:mhll} if\n\\begin{equation}\n\\lambda_{345}\\,-\\,\\lambda_1\\,=\\,0\\;\\;\\Leftrightarrow\\;\\;\\lambda_5\\,=\\,\n\\lambda_1\\,-\\,\\lambda_3\\,-\\,\\lambda_4 \\label{eq:c1}\n\\end{equation}\nor an axion in matrix~\\eqref{eq:mAll} if\n\\begin{equation}\n\\lambda_5\\,=\\, -\\lambda_1\\,+\\,\\lambda_3\\,+\\,\\lambda_4 \\;\\;\\; .\n\\label{eq:c2}\n\\end{equation}\nThat is, we have an axion if the second or third conditions of\nEq.~\\eqref{L6EQ0} are satisfied. The other possible case,\n$\\{v_1\\,=\\,0,\\,v_2\\,\\neq\\,0\\}\\,$, produces exactly the same\nconclusions.\n\n\\subsection{Case $\\lambda_6\\,\\neq\\,0$}\n\nThis is the hardest case to treat, since we cannot obtain analytical\nexpressions for the vevs. Nevertheless a full analytical treatment is\nstill possible. First, notice that with $\\lambda_6\\,\\neq\\,0$\nEqs.~\\eqref{eq:stat} imply that both vevs have to be non-zero. At\nthe stationary point of Eqs.~\\eqref{eq:stat}, the pseudoscalar mass\nmatrix has a Goldstone boson and an eigenvalue given by\n\\begin{equation}\n-\\lambda_5\\,(v_1^2\\,+\\,v_2^2)\\,-\n\\,\\lambda_6\\,\\frac{v_1^4\\,-v_2^4}{2\\,v_1\\,v_2}\n\\;\\;\\; .\n\\end{equation}\nSo, an axion exists if we have\n\\begin{equation}\n\\frac{v_1^2\\,-\\,v_2^2}{v_1\\,v_2}\\;=\\;-\n\\,\\frac{2\\,\\lambda_5}{\\lambda_6}\\;\\;\\;.\n\\label{eq:ves}\n\\end{equation}\nOn the other hand, after some algebraic manipulation, it is simple\nto obtain from~\\eqref{eq:stat} the following condition:\n\\begin{equation}\n\\lambda_1\\,-\\,\\lambda_{345}\\;=\\;\n\\lambda_6\\,\\left(\\frac{v_1^2\\,-\\,v_2^2}{v_1\\,v_2}\\,-\n\\,\\frac{4\\,v_1\\,v_2}{v_1^2\\,-\\,v_2^2}\\right)\n\\label{eq:les}\n\\end{equation}\nSubstituting Eq.~\\eqref{eq:ves} into~\\eqref{eq:les}, we obtain\n\\begin{equation}\n\\lambda_1\\,-\\,\\lambda_{345}\\,=\\,\n\\lambda_6\\,\\left(-\\,\\frac{2\\,\\lambda_5}{\\lambda_6}\\,+\n\\,\\frac{2\\,\\lambda_6}{\\lambda_5}\\right)\n\\;\\Longleftrightarrow\\; 2 \\lambda_6^2 \\,=\n\\, \\lambda_5 (\\lambda_1 -\n\\lambda_3 - \\lambda_4 + \\lambda_ 5).\n\\end{equation}\n\nThus, we have shown that all of the conditions stemming from the\nbasis-invariant condition $D=0$\nguarantee the existence of some stationary point for\nwhich the scalar potential yields an axion.\nNotice that, however,\nthis stationary point need not coincide with the global minimum\nof the potential.\n\n\n\\subsection{Renormalization group invariance}\n\nWe now briefly examine the renormalization group (RG) behavior of our\nbasis-invariant condition $D=0$. It would be meaningless to say that\n$D=0$ implies a $U(1)$ symmetry if that condition were only valid at\na given renormalization scale.  That is, it could well be that a numerical\naccident forces $D=0$ at only a given scale. To avoid such a\nconclusion, we must verify if\n$D=0$ is a RG-invariant condition (in addition to being\nbasis-invariant). For a given renormalization scale $\\mu$, the\n$\\beta$-function of a given parameter $x$ is defined as\n$\\beta_x\\,=\\,\\mu\\,\\partial x\/\\partial \\mu$. For simplicity, let us\nrewrite $D$ in Eq.~\\eqref{eq:D} as\n\\begin{equation}\nD\\;=\\;-\\,\\frac{1}{27}\\,D_1^2\\,D_2\\;\\;\\; ,\n\\end{equation}\nwith\n\\begin{align}\nD_1 &=\\; \\lambda_5 (\\lambda_1 - \\lambda_3 - \\lambda_4 + \\lambda_ 5)\n       - 2 \\lambda_6^2 \\nonumber \\\\\nD_2 &=\\; (\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)^2\n       + 16 \\lambda_6^2 \\;\\;\\;.\n\\end{align}\nIf we apply the operator $\\mu\\,\\partial \/\\partial \\mu$ to $D$, we\nobtain\n\\begin{equation}\n\\beta_D\\,=\\,-\\,\\frac{1}{27}\\,\n\\left(2\\,D_1\\,D_2\\,\\beta_{D_1}\\,+\\,D_1^2\\,\\beta_{D_2}\\right)\n\\;\\;\\;.\n\\end{equation}\n\nIf $D_1=0$ (which corresponds to three of the conditions presented\nin Eqs.~\\eqref{L6NEQ0} and~\\eqref{L6EQ0}) then we immediately have\n$\\beta_D\\,=\\,0$.  That is, if $D=0$ at a given scale, it is zero at\nall scales.\n\nIf $D_2=0$ and $D_1\\neq 0$ we will only have $\\beta_D\\,=\\,0$ if\n$\\beta_{D_2}\\,=\\,0$, or equivalently,\n\\begin{equation}\n2\\,(\\lambda_1 - \\lambda_3 - \\lambda_4 - \\lambda_ 5)\\,\n(\\beta_{\\lambda_1} - \\beta_{\\lambda_3} - \\beta_{\\lambda_4} -\n\\beta_{\\lambda_ 5}) \\,+\\,32\\,\\beta_{\\lambda_6}\\,\\lambda_6\\;=\\;0\n\\;\\;\\; .\n\\end{equation}\nGiven that $D_2=0$ implies that $\\lambda_6\\,=\\,0$ and\n$\\lambda_5\\,=\\,\\lambda_1 - \\lambda_3 - \\lambda_4$, we once\nagain obtain $\\beta_D\\,=\\,0$.\n\nThus, the condition $D=0$ is RG-invariant. A direct verification of\nthe RG invariance of Eqs.~\\eqref{L6NEQ0} and~\\eqref{L6EQ0}, and of\nthe conditions that define the ERPS itself, would require the\nexplicit form of the $\\beta$ functions of the THDM involving the\n$\\lambda_6$ coupling. That verification will be made\nelsewhere~\\cite{drtj}.\n\n\\section{\\label{sec:GCP}Generalized CP symmetries}\n\nIt is common to consider the standard CP transformation\nof the scalar fields as\n\\begin{equation}\n\\Phi_a (t, \\vec{x}) \\rightarrow\n\\Phi^{\\textrm{CP}}_a (t, \\vec{x}) = \\Phi_a^\\ast (t, - \\vec{x}),\n\\label{StandardCP}\n\\end{equation}\nwhere the reference to the time ($t$) and space ($\\vec{x}$)\ncoordinates will henceforth be suppressed.\nHowever,\nin the presence of several scalars with the same quantum numbers,\nbasis transformations can be included in the definition of the\nCP transformation.\nThis yields generalized CP transformations (GCP),\n\\begin{eqnarray}\n\\Phi^{\\textrm{GCP}}_a\n&=& X_{a \\alpha} \\Phi_\\alpha^\\ast\n\\equiv X_{a \\alpha} (\\Phi_\\alpha^\\dagger)^\\top,\n\\nonumber\\\\\n\\Phi^{\\dagger \\textrm{GCP}}_a\n&=& X_{a \\alpha}^\\ast \\Phi_\\alpha^\\top\n\\equiv X_{a \\alpha}^\\ast (\\Phi_\\alpha^\\dagger)^\\ast,\n\\label{GCP}\n\\end{eqnarray}\nwhere $X$ is an arbitrary unitary\nmatrix~\\cite{GCP1,GCP2}.\\footnote{Equivalently, one can \nconsider a generalized time-reversal\ntransformation proposed in Ref.~\\cite{Branco:1983tn}\nand considered further in Appendix A of Ref.~\\cite{GunHab}.}\n\n\nNote that the transformation\n$\\Phi_a\\to\\Phi^{\\rm GCP}_a$, where $\\Phi^{\\rm GCP}_a$ is given\nby Eq.~\\eqref{GCP},\nleaves the kinetic terms invariant.\nThe GCP transformation of a field bilinear yields\n\\begin{equation}\n\\Phi^{\\dagger \\textrm{GCP}}_a\n\\Phi^{\\textrm{GCP}}_b\n=\nX_{a \\alpha}^\\ast X_{b \\beta}\n(\\Phi_\\alpha \\Phi_\\beta^\\dagger)^\\top.\n\\end{equation}\nUnder this GCP transformation,\nthe quadratic terms of the potential may be written as\n\\begin{eqnarray}\nY_{ab} \\Phi^{\\dagger \\textrm{GCP}}_a\n\\Phi^{\\textrm{GCP}}_b\n&=&\nY_{ab} X_{a \\alpha}^\\ast X_{b \\beta}\n\\Phi_\\beta^\\dagger \\Phi_\\alpha\n\\nonumber\\\\\n&=&\nX_{b \\beta} Y_{ba}^\\ast X_{a \\alpha}^\\ast\n\\Phi_\\beta^\\dagger \\Phi_\\alpha\n\\nonumber\\\\\n&=&\nX_{\\alpha a} Y_{\\alpha \\beta}^\\ast X_{\\beta b}^\\ast\n\\Phi_a^\\dagger \\Phi_b\n=\n( X^\\dagger\\, Y\\, X )^\\ast_{ab}\n\\Phi_a^\\dagger \\Phi_b.\n\\end{eqnarray}\nWe have used the Hermiticity condition\n$Y_{ab}=Y_{ba}^\\ast$ in going to the second line;\nand changed the dummy indices $a \\leftrightarrow \\beta$\nand $b \\leftrightarrow \\alpha$ in going to the third line.\nA similar argument can be made for the quartic terms.\nWe conclude that the potential is invariant\nunder the GCP transformation\nof Eq.~\\eqref{GCP} if and only if the coefficients obey\n\\begin{eqnarray}\nY_{ab}^\\ast\n&=&\nX_{\\alpha a}^\\ast Y_{\\alpha \\beta} X_{\\beta b}\n= ( X^\\dagger\\, Y\\, X )_{ab},\n\\nonumber\\\\\nZ_{ab,cd}^\\ast\n&=&\nX_{\\alpha a}^\\ast X_{\\gamma c}^\\ast\nZ_{\\alpha \\beta, \\gamma \\delta} X_{\\beta b} X_{\\delta d}.\n\\label{YZ-CPtransf}\n\\end{eqnarray}\n\nIntroducing\n\\begin{eqnarray}\n\\Delta Y_{ab}\n&=&\nY_{ab} -\nX_{\\alpha a} Y_{\\alpha \\beta}^\\ast X_{\\beta b}^\\ast\n= \\left[Y - ( X^\\dagger\\, Y\\, X )^\\ast \\right]_{ab},\n\\nonumber\\\\\n\\Delta Z_{ab,cd}\n&=&\nZ_{ab,cd} -\nX_{\\alpha a} X_{\\gamma c}\nZ_{\\alpha \\beta, \\gamma \\delta}^\\ast X_{\\beta b}^\\ast X_{\\delta d}^\\ast.\n\\label{DY-DZ}\n\\end{eqnarray}\nwe may write the conditions for invariance under GCP as\n\\begin{eqnarray}\n\\Delta Y_{ab}\n&=&\n0,\n\\label{DY-GCP}\n\\\\\n\\Delta Z_{ab,cd}\n&=&\n0.\n\\label{DZ-GCP}\n\\end{eqnarray}\nGiven Eqs.~\\eqref{hermiticity_coefficients},\nit is easy to show that\n\\begin{eqnarray}\n\\Delta Y_{ab} &=& \\Delta Y_{ba}^\\ast,\n\\nonumber\\\\\n\\Delta Z_{ab,cd} \\equiv \\Delta Z_{cd,ab} &=& \\Delta Z_{ba,dc}^\\ast.\n\\label{DY-DZ-hermiticity}\n\\end{eqnarray}\nThus, we need only consider the real coefficients\n$\\Delta Y_{11}$, $\\Delta Y_{22}$,\n$\\Delta Z_{11,11}$, $\\Delta Z_{22,22}$,\n$\\Delta Z_{11,22}$, $\\Delta Z_{12,21}$,\nand the complex coefficients\n$\\Delta Y_{12}$, $\\Delta Z_{11,12}$,\n$\\Delta Z_{22,12}$, and $\\Delta Z_{12,12}$.\n\n\n\n\n\n\\subsection{GCP and basis transformations}\n\nWe now turn to the interplay between GCP transformations and basis\ntransformations.\nConsider the potential of Eq.~\\eqref{VH2} and call it\n$V(\\Phi)$.\nNow consider the potential obtained from $V(\\Phi)$\nby the basis transformation\n$\\Phi_a \\rightarrow \\Phi^\\prime_a = U_{ab} \\Phi_b$:\n\\begin{equation}\nV (\\Phi^\\prime) =\nY_{ab}^\\prime (\\Phi_a^{\\prime \\dagger} \\Phi^\\prime_b) +\n\\tfrac{1}{2}\nZ^\\prime_{ab,cd} (\\Phi_a^{\\prime \\dagger} \\Phi^\\prime_b)\n(\\Phi_c^{\\prime \\dagger} \\Phi^\\prime_d),\n\\end{equation}\nwhere the coefficients in the new basis are given by\nEqs.~(\\ref{Y-transf}) and (\\ref{Z-transf}).\nWe will now prove the following theorem: If $V(\\Phi)$ is invariant under\nthe GCP transformation of Eq.~(\\ref{GCP}) with the matrix $X$,\nthen $V (\\Phi^\\prime)$ is invariant under a new GCP transformation\nwith matrix\n\\begin{equation}\nX^\\prime = U X U^\\top.\n\\label{X-prime}\n\\end{equation}\nBy hypothesis $V(\\Phi)$ is invariant under\nthe GCP transformation of Eq.~(\\ref{GCP}) with the matrix $X$.\nEq.~(\\ref{YZ-CPtransf}) guarantees that $Y^\\ast = X^\\dagger Y X$.\nNow,\nEq.~(\\ref{Y-transf}) relates the coefficients in the two\nbasis through $Y = U^\\dagger Y^\\prime U$.\nSubstituting gives\n\\begin{equation}\nU^\\top Y^{\\prime \\ast} U^\\ast\n= X^\\dagger (U^\\dagger Y^\\prime U) X,\n\\end{equation}\nor\n\\begin{equation}\nY^{\\prime \\ast}\n= (U^\\ast X^\\dagger U^\\dagger) Y^\\prime (U X U^\\top)\n= X^{\\prime \\dagger} Y^\\prime X^\\prime,\n\\end{equation}\nas required.\nA similar argument holds for the quartic terms and the proof is complete.\n\nThe fact that the transpose $U^\\top$ appears in Eq.~(\\ref{X-prime})\nrather than $U^\\dagger$ is crucial.\nIn Eq.~(\\ref{S-prime}),\napplicable to HF symmetries,\n$U^\\dagger$ appears.\nConsequently,\na basis may be chosen where the HF symmetry is represented by\na diagonal matrix $S$.\nThe presence of $U^\\top$ in Eq.~(\\ref{X-prime}) implies\nthat, \ncontrary to popular belief,\n\\textit{it is not possible to reduce all GCP transformations\nto the standard CP transformation} of Eq.~(\\ref{StandardCP})\nby a basis transformation.\nWhat is possible,\nas we shall see below,\nis to reduce an invariance of the THDM potential under any GCP\ntransformation,\nto an invariance under the standard CP transformation\nplus some extra constraints.\n\nTo be more specific, the following result is easily established. If\nthe unitary matrix $X$ is symmetric, then it follows\nthat\\footnote{Here, we make use of a theorem in linear algebra that\nstates that for any unitary symmetric matrix $X$, a unitary matrix\n$V$ exists such that $X=VV^\\top$.  A proof of this result can be\nfound, e.g., in Appendix B of Ref.~\\cite{GunHab}.}\na unitary matrix\n$U$ exists such that $X'=UXU^\\top=1$, in which case\n$Y^{\\prime\\,*}=Y^\\prime$.  In this case, a basis exists in which the\nGCP is a standard CP transformation.\nIn contrast, if the unitary matrix $X$ is not symmetric, then no\nbasis exists in which $Y$ and $Z$ are real for generic values of the\nscalar potential parameters.\nNevertheless, as we shall demonstrate below, by \\textit{imposing}\nthe GCP symmetry on the scalar potential, the parameters of the\nscalar potential are constrained in such a way that for an\nappropriately chosen basis change, $Y^{\\prime\\,*}\n=X^{\\prime\\,\\dagger}Y'X'=Y'$ (with a similar result for $Z'$).\n\n\nGCP transformations were studied in \nRefs.~\\cite{GCP1,GCP2}. In\nparticular, Ecker, Grimus, and Neufeld \\cite{GCP2} proved that for\nevery matrix $X$ there exists a unitary matrix $U$ such that\n$X^\\prime$ can be reduced to the form\n\\begin{equation}\nX^\\prime = UXU^\\top=\n\\left(\n\\begin{array}{cc}\n   \\phantom{-}\\cos{\\theta} & \\quad \\sin{\\theta}\\\\\n   - \\sin{\\theta} & \\quad \\cos{\\theta}\n\\end{array}\n\\right),\n\\label{GCP-reduced}\n\\end{equation}\nwhere $0 \\leq \\theta \\leq \\pi\/2$. Notice the restricted range for\n$\\theta$. The value of $\\theta$ can be determined in either of two\nways: (i) the eigenvalues of $(X+X^\\top)^\\dagger(X+X^\\top)\/2$ are\n$\\cos{\\theta}$, each of which is twice degenerate; or (ii) $X\nX^\\ast$ has the eigenvalues $e^{\\pm 2 i \\theta}$.\n\n\n\\subsection{The three classes of GCP symmetries}\n\nHaving reached the special form of $X^\\prime$ in\nEq.~(\\ref{GCP-reduced}),\nwe will now follow the strategy adopted by Ferreira and\nSilva \\cite{FS2} in connection with HF symmetries.\nWe substitute Eq.~(\\ref{GCP-reduced}) for $X$ in\nEq.~(\\ref{YZ-CPtransf}),\nin order to identify the constraints imposed by this\nreduced form of the GCP transformations on\nthe quadratic and quartic couplings.\nFor each value of $\\theta$,\ncertain constraints will be forced upon the couplings.\nIf two different values of $\\theta$ enforce the same constraints,\nwe will say that they are in the same class\n(since no experimental distinction between the two will then be\npossible).\nWe will start by considering the special cases of $\\theta=0$\nand $\\theta=\\pi\/2$,\nand then turn our attention to $0 < \\theta < \\pi\/2$.\n\n\n\\subsubsection{CP1: $\\theta=0$}\n\nWhen $\\theta=0$,\n$X^\\prime$ is the unit matrix,\nand we obtain the standard CP transformation,\n\\begin{eqnarray}\n\\Phi_1 &\\rightarrow& \\Phi_1^\\ast,\n\\nonumber\\\\\n\\Phi_2 &\\rightarrow& \\Phi_2^\\ast, \\label{eq:cp1}\n\\end{eqnarray}\nunder which Eqs.~(\\ref{YZ-CPtransf}) take the very simple\nform\n\\begin{eqnarray}\nY_{ab}^\\ast\n&=&\n Y_{ab} ,\n\\nonumber\\\\\nZ_{ab,cd}^\\ast\n&=&\nZ_{ab,cd}.\n\\end{eqnarray}\nWe denote this CP transformation by CP1.\nIt forces all couplings to be real.\nSince most couplings are real by the Hermiticity of\nthe Higgs potential,\nthe only relevant constraints are\n$\\textrm{Im}\\, m_{12}^2 = \\textrm{Im}\\, \\lambda_5 =\n\\textrm{Im}\\, \\lambda_6 = \\textrm{Im}\\, \\lambda_7 = 0$.\n\n\n\\subsubsection{CP2: $\\theta=\\pi\/2$}\n\nWhen $\\theta=\\pi\/2$,\n\\begin{equation}\nX^\\prime =\n\\left(\n\\begin{array}{cc}\n   \\phantom{-} 0 & \\quad 1\\\\\n   - 1 & \\quad 0\n\\end{array}\n\\right), \\label{eq:cp2}\n\\end{equation}\nand we obtain the CP transformation,\n\\begin{eqnarray}\n\\Phi_1 &\\rightarrow& \\Phi_2^\\ast,\n\\nonumber\\\\\n\\Phi_2 &\\rightarrow& - \\Phi_1^\\ast,\n\\end{eqnarray}\nwhich we denote by CP2.\nThis was considered by Davidson and Haber \\cite{DavHab}\nin their Eq.~(37),\nwho noted that if this symmetry holds in one basis,\nit holds in \\textit{all} basis choices.\nUnder this transformation,\nEq.~(\\ref{DY-GCP}) forces the matrix of quadratic\ncouplings to obey\n\\begin{equation}\n0 =\n\\Delta Y =\n\\left(\n\\begin{array}{cc}\n   m_{11}^2 - m_{22}^2 & \\quad -2 m_{12}^2\\\\\n   - 2 m_{12}^{2 \\ast} & \\quad m_{22}^2 - m_{11}^2,\n\\end{array}\n\\right)\n\\end{equation}\nleading to $m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$.\nSimilarly,\nwe may construct a matrix of matrices containing\nall coefficients $\\Delta Z_{ab,cd}$.\nThe uppermost-leftmost matrix corresponds to $\\Delta Z_{11,cd}$.\nThe next matrix along the same line corresponds\nto $\\Delta Z_{12,cd}$, and so on.\nTo enforce invariance under CP2,\nwe equate it to zero,\n\\begin{equation}\n0 =\n\\left(\n\\begin{array}{cc}\n   \\left(\n   \\begin{array}{cc}\n      \\lambda_1 - \\lambda_2 & \\quad\n      \\lambda_6 + \\lambda_7 \\\\\n      \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n      0\n   \\end{array}\n   \\right)\n   &\n   \\left(\n   \\begin{array}{cc}\n      \\lambda_6 + \\lambda_7 & \\quad\n      0 \\\\\n      0 & \\quad\n      \\lambda_6 + \\lambda_7\n   \\end{array}\n   \\right)\n   \\\\*[7mm]\n   \\left(\n   \\begin{array}{cc}\n   \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n   0 \\\\\n   0 & \\quad\n   \\lambda_6^\\ast + \\lambda_7^\\ast\n   \\end{array}\n   \\right)\n   &\n   \\left(\n   \\begin{array}{cc}\n   0 & \\quad\n   \\lambda_6 + \\lambda_7 \\\\\n   \\lambda_6^\\ast + \\lambda_7^\\ast & \\quad\n   \\lambda_2 - \\lambda_1\n   \\end{array}\n   \\right)\n\\end{array}\n\\right).\n\\end{equation}\nWe learn that invariance under CP2 forces\n$m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$,\n$\\lambda_2=\\lambda_1$, and $\\lambda_7 = - \\lambda_6$,\nleading precisely to the ERPS of Eq.~(\\ref{ERPS}).\nRecall that Gunion and Haber \\cite{GunHab} found that,\nunder these conditions we can always find a basis where\nall parameters are real.\nAs a result,\nif the potential is invariant under CP2,\nthere is a basis where CP2 still holds and in which\nthe potential is also invariant under CP1.\n\n\n\n\\subsubsection{CP3: $0 < \\theta < \\pi\/2$}\n\nFinally we turn to the cases where $0 < \\theta < \\pi\/2$.\nImposing Eq.~(\\ref{DY-GCP}) yields\n\\begin{eqnarray}\n0 = \\Delta Y_{11} &=&\n\\left[ (m_{11}^2 - m_{22}^2)\\ s - 2\\ \\textrm{Re}\\, m_{12}^2\\ c \\right] s,\n\\nonumber\\\\\n0 = \\Delta Y_{22} &=&\n- \\Delta Y_{11},\n\\nonumber\\\\\n0 = \\Delta Y_{12} &=&\n\\textrm{Re}\\, m_{12}^2\\ ( c_2 - 1) - 2 i\\ \\textrm{Im}\\, m_{12}^2\n       + \\tfrac{1}{2} (m_{22}^2 - m_{11}^2)\\ s_2,\n\\end{eqnarray}\nwhere we have used $c=\\cos{\\theta}$, $s=\\sin{\\theta}$,\n$c_2=\\cos{2 \\theta}$, and $s_2=\\sin{2 \\theta}$.\nSince $\\theta \\neq 0, \\pi\/2$,\nthe conditions $m_{22}^2 = m_{11}^2$ and $m_{12}^2=0$ are imposed,\nas in CP2.\nSimilarly, Eq.~(\\ref{DZ-GCP}) yields\n\\begin{eqnarray}\n0 = \\Delta Z_{11,11} &=&\n\\lambda_1 (1-c^4) - \\lambda_2 s^4\n- \\tfrac{1}{2} \\lambda_{345} s_2^2\n+ 4\\ \\textrm{Re}\\, \\lambda_6 c^3 s + 4\\ \\textrm{Re}\\, \\lambda_7 c s^3,\n\\nonumber\\\\\n0 = \\Delta Z_{22,22} &=&\n\\lambda_2 (1-c^4) - \\lambda_1 s^4\n- \\tfrac{1}{2} \\lambda_{345} s_2^2\n- 4\\ \\textrm{Re}\\, \\lambda_7 c^3 s - 4\\ \\textrm{Re}\\, \\lambda_6 c s^3,\n\\nonumber\\\\\n0 = \\Delta Z_{11,22}\n&=&\n- \\tfrac{1}{4} s_2\n\\left[\n4 \\textrm{Re}\\, (\\lambda_6 - \\lambda_7) c_2\n+ (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345})s_ 2\n\\right],\n\\nonumber\\\\\n0 = \\Delta Z_{12,21} &=&\n\\Delta Z_{11,22}\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{11,12} &=&\n\\tfrac{1}{4} s \\left[\n(-3 \\lambda_1 + \\lambda_2 + 2 \\lambda_{345}) c\n- (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345}) c_3\n\\right.\n\\nonumber\\\\\n& & \\hspace{7mm}\n\\left.\n+ 4 \\textrm{Re}\\, \\lambda_6 (2 s + s_3)\n- 4 \\textrm{Re}\\, \\lambda_7 s_3\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{22,12} &=&\n\\tfrac{1}{4} s \\left[\n(- \\lambda_1 + 3 \\lambda_2 - 2 \\lambda_{345}) c\n+ (\\lambda_1 + \\lambda_2 - 2 \\lambda_{345}) c_3\n\\right.\n\\nonumber\\\\\n& & \\hspace{7mm}\n\\left.\n- 4 \\textrm{Re}\\, \\lambda_6 s_3\n+ 4 \\textrm{Re}\\, \\lambda_7 (2 s + s_3)\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Re}\\, \\Delta Z_{12,12} &=&\n\\Delta Z_{11,22}\n\\label{Real-5}\n\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{11,12} &=&\n\\tfrac{1}{2}\n\\left[\n\\textrm{Im}\\, \\lambda_6 (3+c_2)\n+ \\textrm{Im}\\, \\lambda_7 (1-c_2)\n- \\textrm{Im}\\, \\lambda_5 s_2\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{22,12} &=&\n\\tfrac{1}{2}\n\\left[\n\\textrm{Im}\\, \\lambda_6 (1-c_2)\n+ \\textrm{Im}\\, \\lambda_7 (3+c_2)\n+ \\textrm{Im}\\, \\lambda_5 s_2\n\\right],\n\\nonumber\\\\\n0 = \\textrm{Im}\\, \\Delta Z_{12,12} &=&\n2 c\n\\left[\n\\textrm{Im}\\, \\lambda_5 c + \\textrm{Im}\\,(\\lambda_6-\\lambda_7)s\n\\right],\n\\label{Im-3}\n\\end{eqnarray}\nwhere $\\lambda_{345}=\\lambda_3 + \\lambda_4 + \\textrm{Re}\\, \\lambda_5$,\n$c_3 = \\cos{3 \\theta}$, and $s_3 = \\sin{3 \\theta}$.\n\nThe last three equations may be written as\n\\begin{equation}\n0=\n\\left[\n\\begin{array}{ccc}\n-s_2 &\\quad (3+c_2) & \\quad (1-c_2)\\\\\n\\phantom{-}s_2 &\\quad  (1-c_2) &\\quad  (3+c_2)\\\\\n(1+c_2) &\\quad  s_2 &\\quad  -s_2\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\n\\textrm{Im}\\, \\lambda_5 \\\\\n\\textrm{Im}\\, \\lambda_6 \\\\\n\\textrm{Im}\\, \\lambda_7\n\\end{array}\n\\right].\n\\end{equation}\nThe determinant of this homogeneous system of three equations\nin three unknowns is $32 c^2$,\nwhich can never be zero since we are assuming that $\\theta \\neq \\pi\/2$.\nAs a result,\n$\\lambda_5$, $\\lambda_6$, and $\\lambda_7$ are real,\nwhatever the value of $0 < \\theta < \\pi\/2$ chosen for the GCP\ntransformation.\nSince $m_{12}^2=0$,\nall potentially complex parameters must be real.\nWe conclude that a potential invariant\nunder any GCP with $0 < \\theta < \\pi\/2$\nis automatically invariant under CP1.\nCombining this with what we learned from CP2,\nwe conclude the following:\nif a potential is invariant under some GCP transformation,\nthen a basis may be found in which it is also invariant\nunder the standard CP transformation,\nwith some added constraints on the parameters.\n\nThe other set of five independent homogeneous equations in\nfive unknowns has a determinant equal to zero,\nmeaning that not all parameters must vanish.\nWe find that\n\\begin{eqnarray}\n0=\n\\Delta Z_{11,11} - \\Delta Z_{22,22}\n&=&\n2s \\left[\ns\\ (\\lambda_1-\\lambda_2) + c\\ 2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\right],\n\\nonumber\\\\\n0=\n\\textrm{Re}\\, \\Delta Z_{11,12}\n- \\textrm{Re}\\, \\Delta Z_{22,12}\n&=&\ns \\left[\n-c\\ (\\lambda_1-\\lambda_2) + s\\ 2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\right].\n\\end{eqnarray}\nSince $s \\neq 0$,\nwe obtain the homogeneous system\n\\begin{equation}\n0=\n\\left[\n\\begin{array}{cc}\n\\phantom{-}s & \\quad c\\\\\n-c & \\quad s\n\\end{array}\n\\right]\n\\left[\n\\begin{array}{c}\n\\lambda_1 - \\lambda_2 \\\\\n2 \\textrm{Re}\\, (\\lambda_6 + \\lambda_7)\n\\end{array}\n\\right],\n\\end{equation}\nwhose determinant is unity.\nWe conclude that $\\lambda_2 = \\lambda_1$ and $\\lambda_7 = - \\lambda_6$.\nThus,\nGCP invariance with any value of $0 < \\theta \\leq \\pi\/2$ leads\nto the ERPS of Eq.~(\\ref{ERPS}).\nSubstituting back we obtain\n$\\Delta Z_{11,11} = \\Delta Z_{22,22} = - \\Delta Z_{11,22}$\nand\n$\\textrm{Re}\\, \\Delta Z_{11,12} = - \\textrm{Re}\\, \\Delta Z_{22,12}$,\nleaving only two independent equations:\n\\begin{eqnarray}\n0=\n\\Delta Z_{11,11}\n&=&\n\\tfrac{1}{2} s_2 \\left[\n(\\lambda_1 - \\lambda_{345}) s_2 + 4 \\lambda_6 c_2 \\right],\n\\nonumber\\\\\n0=\n\\textrm{Re}\\, \\Delta Z_{22,12}\n&=&\n\\tfrac{1}{2} s_2 \\left[\n(\\lambda_1 - \\lambda_{345}) c_2 - 4 \\lambda_6 s_2 \\right],\n\\end{eqnarray}\nwhere we have used $c + c_3 = 2 c c_2$ and $s + s_3 = 2 c s_2$.\nSince $s_2 \\neq 0$,\nthe determinant of the system does not vanish,\nforcing $\\lambda_1=\\lambda_{345}$ and $\\lambda_6=0$.\n\nNotice that our results do not depend on which exact\nvalue of $ 0 < \\theta < \\pi\/2$ in Eq.~(\\ref{GCP-reduced}) we have chosen.\nIf we require invariance of the potential under GCP with some\nparticular value of $ 0 < \\theta < \\pi\/2$,\nthen the potential is immediately invariant under GCP\nwith any other value of $ 0 < \\theta < \\pi\/2$.\nWe name this class of CP invariances, CP3.\nCombining everything,\nwe conclude that invariance under CP3 implies\n\\begin{eqnarray}\nm_{11}^2 = m_{22}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\nonumber\\\\\n\\textrm{Im}\\, \\lambda_5 = 0,\n& \\hspace{4ex} &\n\\textrm{Re}\\, \\lambda_5 = \\lambda_1 - \\lambda_3 -\\lambda_4.\n\\label{Region-CP3}\n\\end{eqnarray}\nThe results of this section are all summarized in Table~\\ref{master1}\nof section~\\ref{sec:summary}.\n\n\\subsection{The square of the GCP transformation}\n\nIf we apply a GCP transformation twice to the scalar fields, we will\nhave, from Eq.~\\eqref{GCP}, that\n\\begin{equation}\n\\left(\\Phi^{\\textrm{GCP}}_a\\right)^{\\textrm{GCP}}\n\\;=\\;\nX_{a \\alpha} \\left(\\Phi^{\\textrm{GCP}}_\\alpha\\right)^\\ast\n\\;=\\;\nX_{a \\alpha}\\, X_{\\alpha b}^\\ast\\ \\Phi_b \\;\\;\\; ,\n\\end{equation}\nso that the square of a GCP transformation is given by\n\\begin{equation}\n(GCP)^2 \\;=\\; XX^\\ast \\;\\;\\; .\n\\label{eq:cpq}\n\\end{equation}\nIn particular, for a generic unitary matrix $X$, $(GCP)^2$ is a Higgs Family \nsymmetry transformation.\n\nUsually, only GCP transformations with $(GCP)^2 = \\boldsymbol{1}$\n(where $\\boldsymbol{1}$ is the unit matrix)\nare considered in the literature.\nFor such a situation,\n$X=X^\\dagger=X^*$,\nand one can always find\na basis in which $X=\\boldsymbol{1}$.\nIn this case,\na GCP transformation is equivalent to a standard CP\ntransformation in the latter basis choice.\nFor example,\nthe restriction that $(GCP)^2 = \\boldsymbol{1}$\n(or equivalently, requiring the squared of the corresponding generalized \ntime-reversal transformation to equal the unit matrix)\nwas imposed in \nRef.~\\cite{GunHab} and more recently in Ref.~\\cite{mani}. \nHowever, as we have illustrated in this section, the invariance\nunder a GCP transformation, in which  $(GCP)^2 \\neq \\boldsymbol{1}$ \n(corresponding to a unitary matrix $X$ that is not symmetric)\nis a \\textit{stronger} restriction on the parameters of the \nscalar potential than the invariance under a standard CP transformation.   \n\nAs we see from the results in the previous sections,\n$X$ is {\\em not} symmetric for the symmetries CP2 and CP3.\nIn fact, this feature provides a strong distinction among the\nthree GCP symmetries previously introduced.  \nLet us briefly examine $(GCP)^2$ for the three possible\ncases $CP1$, $CP2$ and $CP3$.\n\n\\subsubsection{$(CP1)^2$}\n\nComparing Eqs.~\\eqref{GCP} and~\\eqref{eq:cp1}, we come to the\nimmediate conclusion that $X_{CP1}\\,=\\,\\boldsymbol{1}$, so that\nEq.~\\eqref{eq:cpq} yields\n\\begin{equation}\n(CP1)^2\\;=\\;\\boldsymbol{1}\\,.\n\\end{equation}\nThis implies that a CP1-invariant scalar potential \nis invariant under the symmetry group\n$Z_2=\\{\\boldsymbol{1}\\,,\\,CP1\\}$.\n\n\n\\subsubsection{$(CP2)^2$}\n\nThe matrix $X_{\\textrm{CP2}}$ is shown in Eq.~\\eqref{eq:cp2} so that, by\nEq.~\\eqref{eq:cpq}, we obtain\n\\begin{equation}\n(CP2)^2\\;=\\;-\\,\\boldsymbol{1}\\,.\n\\end{equation}\nAlthough this result significantly distinguished CP2 from CP1, \nthe authors of Ref.~\\cite{mani} noted (in considering their\n$CP_g^{(i)}$ symmetries) that the transformation law for $\\Phi_a$\nunder (CP2)$^2$ can be reduced to the identity by a global\nhypercharge transformation.  That is,\nif we start with the symmetry group $Z_4=\\{\\boldsymbol{1}\\,,\\,CP2\\,,\\,\n-\\boldsymbol{1}\\,,\\,-CP2\\}$, we can impose an equivalence relation\nby identifying two elements of $Z_4$ related by multiplication  \nby $-\\boldsymbol{1}$.  If we denote $(Z_2)_Y=\\{\\boldsymbol{1}\\,,\n-\\boldsymbol{1}\\}$ as the two-element\ndiscrete subgroup of the global hypercharge\n$U(1)_Y$, then the discrete symmetry group that is orthogonal to $U(1)_Y$\nis given by $Z_4\/(Z_2)_Y\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} Z_2$.   Hence,\nthe CP2-invariant scalar potential exhibits\na $Z_2$ symmetry orthogonal to the Higgs flavor symmetries\nof the potential. \n\n\n\\subsubsection{$(CP3)^2$}\n\nThe matrix $X_{\\textrm{CP3}}$ is given in Eq.~\\eqref{GCP-reduced}, with\n$0<\\theta<\\pi\/2$, so that, by Eq.~\\eqref{eq:cpq}, we obtain\n\\begin{equation}\n(CP3)^2\\;=\\;\\left(\n\\begin{array}{cc}\n  \\phantom{-} \\cos{2\\theta} & \\quad \\sin{2\\theta}\\\\\n   - \\sin{2\\theta} & \\quad \\cos{2\\theta}\n\\end{array}\n\\right)\\,,\n\\end{equation}\nwhich once again is {\\em not} the unit matrix.\nHowever, the transformation law for $\\Phi_a$ under (CP3)$^2$ \n\\textit{cannot} be reduced to the identity by a global\nhypercharge transformation. \nThis is the reason why Ref.~\\cite{mani} did not consider CP3.\nHowever, $(CP3)^2$ is a non-trivial HF symmetry of the CP3-invariant\nscalar potential.\\footnote{In Section~\\ref{sec:summary}B, we shall\nidentify $(CP3)^2$ with the Peccei Quinn U(1) symmetry defined as\nin Eq.~(\\ref{U1}) and then transformed to a new basis\naccording to the unitary matrix defined in Eq.~(\\ref{UPQ}).}\nThus, one can always reduce the square of \nCP3 to the identity by applying a suitable HF symmetry transformation.\nIn particular, a CP3-invariant scalar potential also exhibits a $Z_2$ symmetry\nthat is orthogonal to the Higgs flavor symmetries of the potential.\n\n\nIn this paper, we prove that there are three and only three\nclasses of GCP transformations.\nOf course, within each class,\none may change the explicit form of the\nscalar potential by a suitable basis transformation;\nbut that will not alter its physical consequences.\nSimilarly,\none can set some parameters to zero in some ad-hoc fashion,\nnot rooted in a symmetry requirement. \nBut, as we have shown, the constraints imposed on the scalar potential\nby a single GCP symmetry can be grouped into three classes:\nCP1, CP2, and CP3.\n\n\n\\section{\\label{sec:summary}Classification of the HF and\nGCP transformation classes in the THDM}\n\n\\subsection{Constraints on scalar potential parameters}\n\nSuppose that one is allowed one single symmetry requirement\nfor the potential in the THDM.\nOne can choose an invariance under one particular Higgs Family\nsymmetry.\nWe know that there are only two independent classes\nof such simple symmetries: $Z_2$ and Peccei-Quinn $U(1)$.\nOne can also choose an invariance under a particular\nGCP symmetry.  \nWe have proved that there are three classes of\nGCP symmetries, named CP1, CP2, and CP3.\nIf any of the above symmetries is imposed on the THDM scalar\npotential (in a specified basis), then the coefficients\nof the scalar potential are constrained, as summarized in\nTable~\\ref{master1}.  For completeness, we also exhibit\nthe constraints imposed by $SO(3)$,\nthe largest possible continuous HF symmetry that is orthogonal to\nthe global hypercharge $U(1)_Y$ transformation.\n\\begin{table}[ht!]\n\\caption{Impact of the symmetries on the coefficients\nof the Higgs potential in a specified basis.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\nsymmetry & $m_{11}^2$ & $m_{22}^2$ & $m_{12}^2$ &\n$\\lambda_1$ & $\\lambda_2$ & $\\lambda_3$ & $\\lambda_4$ &\n$\\lambda_5$ & $\\lambda_6$ & $\\lambda_7$ \\\\\n\\hline\n$Z_2$ &   &   & 0 &\n   &  &  &  &\n   & 0 & 0 \\\\\n$U(1)$ &  &  & 0 &\n &  & &  &\n0 & 0 & 0 \\\\\n$SO(3)$ &  & $ m_{11}^2$ & 0 &\n   & $\\lambda_1$ &  & $\\lambda_1 - \\lambda_3$ &\n0 & 0 & 0 \\\\\n\\hline\n$\\Pi_2$ &  & $ m_{11}^2$ & real &\n   & $ \\lambda_1$ &  &  &\nreal &  & $\\lambda_6^\\ast$\n\\\\\n\\hline\nCP1 &  &  & real &\n & &  &  &\nreal & real & real \\\\\nCP2 &  & $m_{11}^2$ & 0 &\n  & $\\lambda_1$ &  &  &\n   &  & $- \\lambda_6$ \\\\\nCP3 &  & $m_{11}^2$ & 0 &\n   & $\\lambda_1$ &  &  &\n$\\lambda_1 - \\lambda_3 - \\lambda_4$ (real) & 0 & 0 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{master1}\n\\end{table}\n\nEmpty entries in Table~\\ref{master1} correspond to a lack of constraints on the\ncorresponding parameters.\nTable~\\ref{master1}has been constructed for those basis choices in which\n$Z_2$ and $U(1)$ have the specific forms in Eqs.~(\\ref{Z2}) and\n(\\ref{U1}), respectively.\nIf, for example,\nthe basis is changed and $Z_2$ acquires the form $\\Pi_2$ in\nEqs.~(\\ref{Pi2}),\nthen the constraints on the coefficients are altered,\nas shown explicitly on the\nfourth line of Table~\\ref{master1}.\nHowever,\nthis does not correspond to a new model.\nAll physical predictions are the same since\nthe specific forms of $Z_2$ and $\\Pi_2$ differ only by\nthe basis change in Eq.~(\\ref{Z2ToPi2}).\nThe constraints for CP1, CP2, and CP3 shown in Table I\napply to the basis in which the GCP transformation\nof Eq.~(\\ref{GCP}) is used where $X$ has been transformed into $X^\\prime$\ngiven by Eq.~(\\ref{GCP-reduced}),\nwith $\\theta=0$, $\\theta = \\pi\/2$,\nand $0 < \\theta < \\pi\/2$, respectively.\n\n\n\n\n\\subsection{Multiple symmetries and GCP}\n\nWe now wish to consider the possibility of simultaneously imposing\nmore than one symmetry requirement on the Higgs potential.\nFor example, one can require that $Z_2$ and $\\Pi_2$ be enforced\n\\textit{within the same basis}.  In what follows, we shall indicate\nthat the two symmetries are enforced simultaneously by writing\n$Z_2\\oplus\\Pi_2$.\nCombining the constraints from the appropriate\nrows of Table~\\ref{master1},\nwe conclude that,\nunder these two simultaneous requirements\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\textrm{Im}\\, \\lambda_5 = 0.\n\\label{Z2+Pi2}\n\\end{eqnarray}\nThis coincides exactly with the conditions of the ERPS\nin a very special basis,\nas shown in Eq.~(\\ref{ERPS2}).\nSince CP2 leads to the ERPS of Eq.~(\\ref{ERPS}),\nwe conclude that\n\\begin{equation}\nZ_2 \\oplus \\Pi_2 \\equiv \\textrm{CP2 in some specific basis}.\n\\label{equiv-CP2}\n\\end{equation}\nThis was noted previously by Davidson and Haber \\cite{DavHab}.\nNow that we know what all classes of HF and CP symmetries can\nlook like,\nwe can ask whether all GCP symmetries can be written as\nthe result of some multiple HF symmetry.\n\nThis is clearly not possible for CP1 because of parameter counting.\nTable~\\ref{master1} shows that CP1 reduces the scalar potential to ten\nreal parameters.\nWe can still perform an orthogonal basis change while keeping\nall parameters real.\nThis freedom can be used to remove one further parameter;\nfor example, setting $m_{12}^2=0$ by diagonalizing the $Y$\nmatrix.\nNo further simplification is allowed.\nAs a result, CP1 leaves nine independent parameters.\nThe smallest HF symmetry is $Z_2$.\nTable~\\ref{master1} shows that $Z_2$ reduces the potential to six\nreal and one complex parameter.\nThe resulting eight parameters could never account for the\nnine needed to fully describe the most general model\nwith the standard CP invariance CP1.\\footnote{In\nIvanov's language, this is clear since CP1 corresponds to\na $Z_2$ transformation of the vector $\\vec{r}$,\nwhich is the simplest transformation on $\\vec{r}$ one\ncould possibly make.\nSee section~\\ref{more_multiple}.}\n\n\nBut one can utilize two HF symmetries in order\nto obtain the same constraints obtained by invariance under CP3.\nLet us impose \\textit{both} $U(1)$ and $\\Pi_2$\n\\textit{in the same basis}.\nFrom Table~\\ref{master1},\nwe conclude that,\nunder these two simultaneous requirements\n\\begin{eqnarray}\nm_{22}^2 = m_{11}^2,\n& \\hspace{4ex} &\nm_{12}^2=0,\n\\nonumber\\\\\n\\lambda_2 = \\lambda_1,\n& \\hspace{4ex} &\n\\lambda_7 = \\lambda_6 = 0,\n\\hspace{4ex} \\lambda_5 = 0.\n\\label{U1+Pi2}\n\\end{eqnarray}\nThis does not coincide with the\nconditions for invariance under CP3 shown in Eq.~(\\ref{Region-CP3}).\nHowever,\none can use the transformation rules in Eqs.~(A13)-(A23)\nof Davidson and Haber \\cite{DavHab},\nin order to show that a basis transformation, \n\\begin{equation} \\label{UPQ}\nU=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc} \\phantom{-}1 & \\quad -i \\\\\n-i & \\quad\\phantom{-}1\\end{array}\\right)\\,, \n\\end{equation}\nmay be chosen which takes us from Eqs.~(\\ref{Region-CP3}),\nwhere $\\textrm{Re}\\, \\lambda_5 = \\lambda_1 -\\lambda_3 -\\lambda_4$,\nto Eqs.~(\\ref{U1+Pi2}),\nwhere $\\lambda_5=0$ (while maintaining the other relations among\nthe scalar potential parameters).\nWe conclude that\n\\begin{equation}\nU(1) \\oplus \\Pi_2 \\equiv \\textrm{CP3 in some specific basis}.\n\\label{equiv-CP3}\n\\end{equation}\nNote that in the basis in which the CP3 relations of  Eq.~(\\ref{Region-CP3})\nare satisfied with $\\lambda_5\\neq 0$, the discrete HF symmetry\n$\\Pi_2$ is still respected.\nHowever, using Eq.~(\\ref{UPQ}), it follows that the U(1)-Peccei Quinn\nsymmetry corresponds to the invariance of the scalar potential under\n$\\Phi_a\\to \\mathcal{O}_{ab}\\Phi_b$, where $\\mathcal{O}$ is an arbitrary\n$SO(2)$ matrix.\n\nThe above results suggest that it should be possible to distinguish CP1,\nCP2, and CP3 in a basis invariant fashion.\nBotella and Silva \\cite{BS} have built three so-called $J$-invariants\nthat detect any signal of CP violation (either explicit or\nspontaneous) after the minimization\nof the scalar potential.  However, in this paper we are concerned\nabout the symmetries of the scalar potential independently of the\nchoice of vacuum.  Thus, we shall consider\nthe four so-called $I$-invariants built by\nGunion and Haber \\cite{GunHab} in order to detect any\nsignal of \\textit{explicit} CP violation present (before the vacuum state\nis determined).\nIf any of these invariants is nonzero, then CP is explicitly violated,\nand neither CP1, nor CP2, nor CP3 hold.\nConversely,\nif all $I$-invariants are zero, then CP is explicitly conserved, but we cannot\ntell a priori which GCP applies.\nEqs.~(\\ref{equiv-CP2}) and (\\ref{equiv-CP3}) provide the crucial hint.\nIf we have CP conservation, $Z_2\\oplus\\Pi_2$ holds,\nand $U(1)$ does not,\nthen we have CP2.\nAlternatively,\nif we have CP conservation, and $U(1)\\oplus\\Pi_2$ also holds,\nthen we have CP3.\nWe recall that both CP2 and CP3 lead to the ERPS,\nand that the general conditions for the ERPS in Eq.~(\\ref{ERPS})\nare basis independent.\nThis allows us to distinguish CP2 and CP3 from CP1.\nBut, prior to the present work,\nno basis-independent quantity had been identified in the literature\nthat could distinguish $Z_2$ and $U(1)$ in the ERPS.\nThe basis-independent quantity $D$ introduced\nin subsection~\\ref{subsec:D} is precisely the invariant required for\nthis task.  That is,\nin the ERPS $D\\neq0$ implies CP2, whereas $D=0$ implies CP3.\n\nOne further consequence of the results of Table~\\ref{master1}\ncan be seen by simultaneously imposing the U(1) Peccei-Quinn symmetry\nand the CP3 symmetry \\textit{in the same basis}.  The resulting\nconstraints on the scalar potential parameters are precisely those of\nthe SO(3) HF symmetry.  Thus, we conclude that\n\\begin{equation}\nU(1) \\oplus \\textrm{CP3} \\equiv SO(3).\n\\label{equiv-O3}\n\\end{equation}\nIn particular, $SO(3)$ is not a simple HF symmetry, as the invariance\nof the scalar potential under a single element of SO(3) is not\nsufficient to guarantee invariance under the full SO(3) group of\ntransformations. \n\n\n\n\\subsection{Maximal symmetry group of the scalar potential\northogonal to $U(1)_Y$}\n\nThe standard CP symmetry, CP1,\nis a discrete $Z_2$ symmetry that transforms the scalar\nfields into their complex conjugates, and hence \nis not a subgroup of the $U(2)$ transformation group\nof Eq.~\\ref{basis-transf}.  We have previously noted that\nTHDM scalar potentials that exhibit \\textit{any} non-trivial\nHF symmetry $G$ is automatically CP-conserving.  Thus, the actual\nsymmetry group of the scalar potential is in fact \nthe semidirect product\\footnote{In general, the non-trivial element of\n$Z_2$ will not commute with all elements of $G$, in which case the\nrelevant mathematical structure is that of a semidirect product.  In\ncases where the non-trivial element of $Z_2$ commutes with all\nelements of $G$, we denote the corresponding direct product as\n$G\\otimes Z_2$.}\nof $G$ and $Z_2$, which we write as $G\\rtimes Z_2$.\nNoting that $U(1)\\rtimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} SO(2)\\rtimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} O(2)$, and\n$SO(3)\\otimes Z_2\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} O(3)$, we conclude that the maximal \nsymmetry groups of the scalar potential orthogonal to $U(1)_Y$\nfor the possible choices of HF symmetries are given in \nTable~\\ref{maximal}.\\footnote{For ease of notation, we denote\n$Z_2\\otimes Z_2$ by $(Z_2)^2$ and $Z_2\\otimes Z_2\\otimes Z_2$\nby $(Z_2)^3$.}\n\n\n\n\\begin{table}[ht]\n\\caption{Maximal symmetry groups [orthogonal to global $U(1)_Y$\nhypercharge] of the scalar sector of the THDM.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\ndesignation &  HF symmetry group  & maximal symmetry group\\\\\n\\hline\n$Z_2$ & $Z_2$ & $(Z_2)^2$ \\\\\nPeccei-Quinn & $U(1)$ & $O(2)$ \\\\\n$SO(3)$ & $SO(3)$ & $O(3)$ \\\\\nCP1  & --- & $Z_2$ \\\\\nCP2  & $(Z_2)^2$ & $(Z_2)^3$ \\\\\nCP3 & $O(2)$ & $O(2)\\otimes Z_2$\n\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{maximal}\n\\end{table}\n\nFinally, we reconsider CP2 and CP3.  Eq.~(\\ref{equiv-CP2})\nimplies that the CP2 symmetry is equivalent to a $(Z_2)^2$ HF\nsymmetry.  To prove this statement, we note that in the\ntwo-dimensional flavor space of Higgs fields, the $Z_2$ and $\\Pi_2$\ndiscrete symmetries defined by Eqs.~(\\ref{Z2}) and (\\ref{Pi2})\nare given by:\n\\begin{equation}\nZ_2=\\{S_0\\,,\\,S_1\\}\\,,\\qquad\\qquad \\Pi_2=\\{S_0\\,,\\,S_2\\}\\,,\n\\end{equation}\nwhere $S_0\\equiv \\boldsymbol{1}$ is the $2\\times 2$ identity matrix and\n\\begin{equation}\nS_1=\\left(\n\\begin{array}{cc}\n1 & \\quad\\phantom{-} 0 \\\\\n0 & \\quad -1 \\\\\n\\end{array}\n\\right)\\,,\\qquad\\qquad\nS_2=\\left(\n\\begin{array}{cc}\n0 & \\quad 1 \\\\\n1 & \\quad 0 \\\\\n\\end{array}\n\\right)\\,.\n\\end{equation}\nIf we impose the $Z_2$ and $\\Pi_2$ symmetry in the same basis, then the\nscalar potential is invariant under the dihedral group of eight elements,\n\\begin{equation}\nD_4=\\{S_0\\,,\\,S_1\\,,\\,S_2\\,,\\,S_3\\,,\\,-S_0\\,,\\,-S_1\\,,\\,-S_2\\,,\\,-S_3\\}\\,,\n\\end{equation}\nwhere $S_3=S_1 S_2=-S_2 S_1$.  \nAs before, we identify $(Z_2)_Y\\equiv\\{S_0\\,,\\,-S_0\\}$ as the\ntwo-element discrete subgroup of the global hypercharge $U(1)_Y$.\nHowever, we have defined the HF symmetries to be orthogonal to $U(1)_Y$.\nThus, to determine the HF symmetry group of CP2, we identify as\nequivalent those elements of $D_4$ that are related by multiplication\nby $-S_0$.  Group theoretically, we identify the HF symmetry group\nof CP2 as\n\\begin{equation}\nD_4\/(Z_2)_Y\\mathchoice{\\cong}{\\cong}{\\isoS}{\\cong} Z_2\\otimes Z_2\\,.\n\\end{equation}\n\nThe HF symmetry group of CP2 is not the maximally allowed symmetry\ngroup.  In particular, the constraints of CP2 on the scalar potential\nimply the existence of a basis in which all scalar potential\nparameters are real.  Thus, the scalar potential is explicitly\nCP-conserving.  The $Z_2$ symmetry associated with this CP\ntransformation is orthogonal to the HF symmetry as previously noted.  \n(This is easily checked explicitly by employing a four-dimensional real\nrepresentation of the two complex scalar fields.)  Thus,\nthe maximal symmetry group of the CP2-symmetric scalar potential\nis $(Z_2)^3$.  Similarly, Eq.~(\\ref{equiv-CP3})\nimplies that the CP3 symmetry is equivalent to a $U(1)\\rtimes Z_2$ HF\nsymmetry. This is isomorphic to an $O(2)$ HF symmetry, which is\na subgroup of the maximally allowed $SO(3)$ HF symmetry group. \nHowever, the constraints of CP3 on the scalar potential\nimply the existence of a basis in which all scalar potential\nparameters are real.  Thus, the scalar potential is explicitly\nCP-conserving.  Once again, the $Z_2$ symmetry associated with this CP\ntransformation is orthogonal to the HF symmetry noted above.  Thus,\nthe maximal symmetry group of the CP3-symmetric scalar potential\nis $O(2)\\otimes Z_2$.  \n\nThe above results are also summarized in \nTable~\\ref{maximal}.  In all cases, the maximal symmetry group is\na direct product of the HF symmetry group and the $Z_2$ corresponding\nto the standard CP-transformation, whose square is the identity operator.\n\nOne may now ask whether Table~\\ref{maximal} exhausts all\npossible independent symmetry constraints that one\nmay place on the Higgs potential.\nPerhaps one can choose other combinations,\nor maybe one can combine three, four, or more\nsymmetries.\nWe know of no way to answer this problem\nbased only on the transformations of the scalar\nfields $\\Phi_a$.\nFortunately,\nIvanov has solved this problem~\\cite{Ivanov1} by looking\nat the transformation properties of field bilinears,\nthus obtaining for the first time the list of symmetries given in the last\ncolumn of Table~\\ref{maximal}.\n\n\n\\subsection{\\label{more_multiple}More on multiple symmetries}\n\n\nWe start by looking at the implications of the symmetries we have\nstudied so far on the vector $\\vec{r} = \\{ r_1, r_2, r_3\\}$,\nwhose components were introduced in Eq.~(\\ref{r_Ivanov}).\nNotice that a unitary transformation $U$ on\nthe fields $\\Phi_a$ induces an orthogonal\ntransformation $O$ on the vector of bilinears\n$\\vec{r}$,\ngiven by Eq.~(\\ref{O}).\nFor every pair of unitary transformations $\\pm U$\nof $SU(2)$,\none can find some corresponding transformation $O$\nof $SO(3)$,\nin a two-to-one correspondence.\nWe then see what these symmetries imply\nfor the coefficients of Eq.~(\\ref{VH3})\n(recall the $\\Lambda_{\\mu \\nu}$ is a symmetric matrix).\nBelow, we list the transformation of $\\vec{r}$ under which the\nscalar potential is invariant, followed by the \ncorresponding constraints on the\nquadratic and quartic scalar potential parameters, $M_\\mu$ and\n$\\Lambda_{\\mu\\nu}$.  \n\nUsing the results of Table I, we find that $Z_2$ implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n-r_1\\\\\n-r_2\\\\\n\\phantom{-}r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{03}\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, \\Lambda_{12} &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,\\Lambda_{12} &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, 0\\\\\n\\Lambda_{03} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-Z2}\n\\end{equation}\n$U(1)$ implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{ccc}\nc_2 & -s_2 & \\phantom{-}0\\\\\ns_2 &\\phantom{-}c_2 & \\phantom{-}0\\\\\n0 & \\phantom{-}0 & \\phantom{-}1\n\\end{array}\n\\right]\\ \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{03}\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{11} & \\,\\,\\,0\\\\\n\\Lambda_{03} & \\,\\,\\,0 & \\,\\,\\,0 &\\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-U1}\n\\end{equation}\nand SO(3) implies \n\\begin{equation}\n\\vec{r} \\rightarrow\n\\mathcal{O} \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & \\,\\,\\,0\\\\\n 0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n 0 & \\,\\,\\,0 & \\,\\,\\,\\Lambda_{11} & \\,\\,\\,0\\\\\n0 & \\,\\,\\,0 & \\,\\,\\,0 &\\,\\,\\, \\Lambda_{11}\n\\end{array}\n\\right],\n\\label{Iv-SO3}\n\\end{equation}\nwhere $\\mathcal{O}$ is an arbitrary $3\\times 3$ orthogonal \nmatrix of unit determinant.\nIn the language of bilinears, a basis invariant condition for the presence of\n$SO(3)$ is that the three eigenvalues of $\\tilde{\\Lambda}$ are equal.\n(Recall that $\\tilde{\\Lambda} = \\left\\{\\Lambda_{ij}\\right\\}$; $i,j=1,2,3$).\n\n\nAs for the GCP symmetries,\nCP1 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n\\phantom{-}r_1\\\\\n-r_2\\\\\n\\phantom{-}r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\nM_1\\\\\n0\\\\\nM_3\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} & \\,\\,\\,\\Lambda_{01}  & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{03}\\\\\n\\Lambda_{01} & \\,\\,\\, \\Lambda_{11} & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{13}\\\\\n 0 & \\,\\,\\, 0 & \\,\\,\\, \\Lambda_{22} &  \\,\\,\\,0\\\\\n\\Lambda_{03} &  \\,\\,\\,\\Lambda_{13} &  \\,\\,\\,0 & \\,\\,\\, \\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-CP1}\n\\end{equation}\nCP2 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{c}\n-r_1\\\\\n-r_2\\\\\n-r_3\n\\end{array}\n\\right],\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, \\Lambda_{11}\\,\\,\\, &\\,\\,\\, \\Lambda_{12} & \\,\\,\\,\\Lambda_{13}\\\\\n0 &\\,\\,\\, \\Lambda_{12} &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, \\Lambda_{23}\\\\\n0 &\\,\\,\\, \\Lambda_{13} & \\,\\,\\,\\Lambda_{23} & \\,\\,\\,\\Lambda_{33}\n\\end{array}\n\\right],\n\\label{Iv-CP2}\n\\end{equation}\nand CP3 implies\n\\begin{equation}\n\\vec{r} \\rightarrow\n\\left[\n\\begin{array}{ccc}\n\\phantom{-}c_2 & \\phantom{-}0 & \\phantom{-}s_2\\\\\n\\phantom{-}0 & -1 & \\phantom{-}0\\\\\n-s_2 & \\phantom{-}0 & \\phantom{-}c_2\n\\end{array}\n\\right]\\ \\vec{r},\n\\hspace{10mm}\n\\left[\n\\begin{array}{c}\nM_0\\\\\n0\\\\\n0\\\\\n0\n\\end{array}\n\\right],\n\\hspace{4ex}\n\\left[\n\\begin{array}{cccc}\n\\Lambda_{00} &\\,\\,\\, 0 & \\,\\,\\,0 & 0\\,\\,\\,\\\\\n0 &\\,\\,\\, \\Lambda_{11} &\\,\\,\\, 0 &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, 0 &\\,\\,\\, \\Lambda_{22} &\\,\\,\\, 0\\\\\n0 &\\,\\,\\, 0 &\\,\\,\\, 0 & \\,\\,\\,\\Lambda_{11}\n\\end{array}\n\\right].\n\\label{Iv-CP3}\n\\end{equation}\nNotice that in CP3 two of the eigenvalues of $\\Lambda$ are equal,\nin accordance with our observation that $D$ can be used\nto distinguish between CP2 and CP3.\n\nBecause each unitary transformation on the fields $\\Phi_a$\ninduces an $SO(3)$ transformation on the vector\nof bilinears $\\vec{r}$,\nand because the standard CP transformation\ncorresponds to an inversion of $r_2$\n(a $Z_2$ transformation on the vector $\\vec{r}$),\nIvanov \\cite{Ivanov1}\nconsiders all possible proper and improper transformations\nof $O(3)$ acting on $\\vec{r}$.  \nHe identifies the following six classes of transformations:\n(i) $Z_2$; (ii) $(Z_2)^2$; (iii) $(Z_2)^3$;\n(iv) $O(2)$; (v) $O(2) \\otimes Z_2$; and (vi) $O(3)$.\nNote that these symmetries are all orthogonal to the global $U(1)_Y$\nhypercharge symmetry, as the bilinears $r_0$ and $\\vec{r}$\nare all singlets under a $U(1)_Y$ transformation.\nThe six classes above identified by Ivanov\ncorrespond precisely to the six possible maximal\nsymmetry groups identified in Table~\\ref{maximal}.\nNo other independent symmetry transformations are possible.\n\n\nOur work permits one to identify the abstract transformation\nof field bilinears utilized by Ivanov in terms of\ntransformations on the scalar fields themselves,\nas needed for model building.\nCombining our work with Ivanov's,\nwe conclude that there is only one new type\nof symmetry requirement which one can place on\nthe Higgs potential via multiple symmetries.\nCombining this with our earlier results,\nwe conclude that all possible symmetries on the scalar\nsector of the THDM can be reduced to multiple HF symmetries,\nwith the exception of the standard CP transformation (CP1).\n\n\n\n\n\\section{\\label{sec:allisCP}Building all symmetries with the standard CP}\n\nWe have seen that there are only six independent\nsymmetry requirements, listed in Table~\\ref{maximal},\nthat  one can impose on the Higgs potential.\nWe have shown that all possible symmetries of the scalar\nsector of the THDM can be reduced to multiple HF symmetries,\nwith the exception of the standard CP transformation (CP1).\nNow we wish to show a dramatic result:\n\\textit{all possible symmetries on the scalar\nsector of the THDM can be reduced to multiple applications of\nthe standard CP symmetry.}\n\nUsing Eq.~(\\ref{X-prime}),\nwe see that the basis transformation of Eq.~(\\ref{basis-transf}),\nchanges the standard CP symmetry of Eq.~(\\ref{StandardCP})\ninto the GCP symmetry of Eq.~(\\ref{GCP}),\nwith\n\\begin{equation}\nX=U U^\\top.\n\\end{equation}\nIn particular,\nan orthogonal basis transformation does not affect the\nform of the standard CP transformation.\nSince we wish to generate $X \\neq 1$,\nwe will need complex matrices $U$.\n\nNow we wish to consider the following situation.\nWe have a basis (call it the original basis) and\nimpose the standard CP symmetry CP1 on that original basis.\nNext we consider the same model in a different basis\n(call it $M$) and impose the standard CP symmetry on that basis $M$.\nIn general, this procedure of imposing\nthe standard CP symmetry in the original basis \\textit{and also}\nin the rotated basis $M$ leads to two independent impositions.\nThe first imposition makes all parameters real in\nthe original basis.\nOne way to combine the second imposition with the first\nis to consider the basis transformation $U_M$ taking us\nfrom basis $M$ into the original basis.\nAs we have seen,\nthe standard CP symmetry in basis $M$ turns,\nwhen written in the original basis,\ninto a symmetry under\n\\begin{eqnarray}\n\\Phi^{\\textrm{CP}}_a\n&=& (X_M)_{a \\alpha} \\Phi_\\alpha^\\ast,\n\\nonumber\\\\\n\\Phi^{\\dagger \\textrm{CP}}_a\n&=&\n(X_M)^\\ast_{a \\alpha} (\\Phi_\\alpha^\\dagger)^\\ast,\n\\label{CGP-M}\n\\end{eqnarray}\nwith $X_M=U_M U_M^\\top$.\nNext we consider several such possibilities.\n\nWe start with\n\\begin{equation}\nU_{A} =\n\\left(\n\\begin{array}{cc}\n\\phantom{-}c_{\\pi\/4} & \\quad -i s_{\\pi\/4} \\\\\n- i s_{\\pi\/4} & \\quad \\phantom{-}c_{\\pi\/4}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{A} =\n\\left(\n\\begin{array}{cc}\n\\phantom{-}0 & \\quad -i  \\\\\n- i & \\quad \\phantom{-}0\n\\end{array}\n\\right).\n\\end{equation}\nHere and henceforth $c$ ($s$) with a subindex indicates the\ncosine (sine) of the angle given in the subindex.\nWe denote by CP1$_A$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_A$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=A$).\n\nNext we consider\n\\begin{equation}\nU_{B} =\n\\left(\n\\begin{array}{cc}\ne^{-i \\pi\/4} & \\quad 0 \\\\\n0 & \\quad e^{i \\pi\/4}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{B} =\n\\left(\n\\begin{array}{cc}\n-i & \\quad 0  \\\\\n\\phantom{-}0 & \\quad i\n\\end{array}\n\\right).\n\\end{equation}\nWe denote by CP1$_B$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_B$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=B$).\n\nA third possible choice is\n\\begin{equation}\nU_{C} =\n\\left(\n\\begin{array}{cc}\ne^{i \\delta\/2} & \\quad 0 \\\\\n0 & \\quad e^{-i \\delta\/2}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{C} =\n\\left(\n\\begin{array}{cc}\ne^{i \\delta} & \\quad 0 \\\\\n0 & \\quad e^{-i \\delta}\n\\end{array}\n\\right),\n\\end{equation}\nwhere $\\delta \\neq n \\pi\/2$ with $n$ integer.\nWe denote by CP1$_C$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_C$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=C$).\n\nFinally, we consider\n\\begin{equation}\nU_{D} =\n\\left(\n\\begin{array}{cc}\n\\phantom{i}c_{\\delta\/2} & \\quad i s_{\\delta\/2} \\\\\ni s_{\\delta\/2} & \\phantom{i}\\quad c_{\\delta\/2}\n\\end{array}\n\\right),\n\\hspace{3ex}\nX_{D} =\n\\left(\n\\begin{array}{cc}\n\\phantom{i}c_\\delta & \\quad i s_\\delta  \\\\\ni s_\\delta & \\quad \\phantom{i}c_\\delta\n\\end{array}\n\\right),\n\\end{equation}\nwhere $\\delta \\neq n \\pi\/2$ with $n$ integer.\nWe denote by CP1$_D$ the imposition of the CP symmetry\nin Eq.~(\\ref{CGP-M}) with $X_M=X_D$\n(which coincides with the imposition of the standard CP\nsymmetry in the basis $M=D$).\n\nThe impact of the first three symmetries on the coefficients of the\nHiggs potential are summarized in\nTable~\\ref{master3}.\n\\begin{table}[ht!]\n\\caption{Impact of the CP1$_M$ symmetries on the coefficients\nof the Higgs potential.\nThe notation ``imag'' means that the\ncorresponding entry is purely imaginary.\nCP1 in the original basis has been included for reference.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccccccc}\nsymmetry & $m_{11}^2$ & $m_{22}^2$ & $m_{12}^2$ &\n$\\lambda_1$ & $\\lambda_2$ & $\\lambda_3$ & $\\lambda_4$ &\n$\\lambda_5$ & $\\lambda_6$ & $\\lambda_7$ \\\\\n\\hline\nCP1 &  &  & real &\n &  & &  &\nreal & real & real\\\\\n\\hline\nCP1$_A$ &   & $m_{11}^2$  &  &\n   & $\\lambda_1$ &  &  &\n   &  & $\\lambda_6$ \\\\\nCP1$_B$ &  &  & imag &\n &  & &  &\nreal & imag & imag \\\\\nCP1$_C$ &  &  & $|m_{12}^2| e^{i \\delta}$ &\n & & & & $|\\lambda_5| e^{2 i \\delta}$  &\n$|\\lambda_6| e^{i \\delta}$ & $|\\lambda_7| e^{i \\delta}$\n\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{master3}\n\\end{table}\n\n\\noindent\nImposing CP1$_D$ on the Higgs potential leads to the more complicated\nset of equations:\n\\begin{eqnarray}\n2 \\textrm{Im} \\left( m_{12}^2 \\right)\\, c_\\delta\n+ (m_{22}^2 - m_{11}^2)\\, s_\\delta\n&=& 0,\n\\nonumber\\\\\n2 \\textrm{Im} \\left( \\lambda_6 - \\lambda_ 7 \\right)\\, c_{2 \\delta}\n+ \\lambda_{12345}\\, s_{2 \\delta}\n&=& 0,\n\\nonumber\\\\\n2 \\textrm{Im} \\left( \\lambda_6 + \\lambda_ 7 \\right)\\, c_\\delta\n+ \\left( \\lambda_1 - \\lambda_2 \\right)\\, s_\\delta\n&=& 0,\n\\nonumber\\\\\n\\textrm{Im} \\lambda_5\\, c_\\delta\n+ \\textrm{Re} \\left( \\lambda_6 - \\lambda_ 7 \\right)\\, s_\\delta\n&=& 0,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\lambda_{12345} = \\tfrac{1}{2} \\left( \\lambda_1 + \\lambda_ 2 \\right)\n- \\lambda_3 - \\lambda_4 + \\textrm{Re} \\lambda_5.\n\\end{equation}\n\nCombining these results with those in Table~\\ref{master1},\nwe have shown that\n\\begin{eqnarray}\n\\textrm{CP1} \\oplus \\textrm{CP1}_B\n&=& Z_2\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_C\n&=& U(1),\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B\n&=& \\textrm{CP2}\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_C\n&=& \\textrm{CP3}\\ \\  \\textrm{in some specific basis},\n\\nonumber\\\\\n\\textrm{CP1} \\oplus \\textrm{CP1}_C \\oplus \\textrm{CP1}_D\n&=& SO(3).\n\\label{incredible}\n\\end{eqnarray}\nLet us comment on the ``specific basis choices'' needed.\nImposing $\\textrm{CP1} \\oplus \\textrm{CP1}_B$ leads to\n$ m_{12}^2=\\lambda_6=\\lambda_7=0$ and $\\textrm{Im} \\lambda_5=0$,\nwhile imposing $Z_2$ leads to $ m_{12}^2=\\lambda_6=\\lambda_7=0$\nwith no restriction on $\\lambda_5$.\nHowever, when $Z_2$ holds one may rephase $\\Phi_2$\nby the exponential of $-i \\arg(\\lambda_5)\/2$,\nthus making $\\lambda_5$ real.\nIn this basis,\nthe restrictions of $Z_2$ coincide with the restrictions\nof $\\textrm{CP1} \\oplus \\textrm{CP1}_B$.\nSimilarly,\nimposing $\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_C$\nleads to $m_{12}^2=\\lambda_5=\\lambda_6=\\lambda_7=0$,\n$m_{22}^2=m_{11}^2$ and $\\lambda_2=\\lambda_1$.\nWe see from Table~\\ref{master1} that CP3 has these features,\nexcept that $\\lambda_5$ need not vanish; it is real and\n$\\textrm{Re} \\lambda_5 = \\lambda_1-\\lambda_3-\\lambda_4$.\nStarting from the CP3 conditions and\nusing the transformation rules in Eqs.~(A13)-(A23) of Davidson and\nHaber \\cite{DavHab},\nwe find that a basis choice is possible such that\n$\\textrm{Re} \\lambda_5=0$.\\footnote{Notice that, in the new basis,\n$\\lambda_1$ differs in general from $\\lambda_3 + \\lambda_4$;\notherwise the larger $SO(3)$ Higgs Family symmetry would hold.}\nPerhaps it is easier to prove the equality\n\\begin{equation}\n\\textrm{CP1} \\oplus \\textrm{CP1}_B \\oplus \\textrm{CP1}_D\n= \\textrm{CP3}\\ \\  \\textrm{in some specific basis}.\n\\end{equation}\nIn this case,\nthe only difference between the impositions from the\ntwo sides of the equality come from the sign of $\\textrm{Re} \\lambda_5$,\nwhich is trivial to flip through the basis change\n$\\Phi_2 \\rightarrow - \\Phi_2$.\nFinally,\nimposing $\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B$\nwe obtain $m_{12}^2=\\textrm{Im} \\lambda_5=\\lambda_6=\\lambda_7=0$,\n$m_{22}^2=m_{11}^2$ and $\\lambda_2=\\lambda_1$.\nThis does not coincide with the conditions of CP2 which\nlead to the ERPS of Eq.~(\\ref{ERPS}).\nFortunately,\nand as we mentioned before,\nDavidson and Haber \\cite{DavHab} proved that\none may make a further basis transformation\nsuch that Eq.~(\\ref{ERPS2}) holds,\nthus coinciding with the conditions imposed by\n$\\textrm{CP1} \\oplus \\textrm{CP1}_A \\oplus \\textrm{CP1}_B$.\n\nNotice that our description of CP2 in terms of several\nCP1 symmetries is in agreement with the results found by the\nauthors of Ref.~\\cite{mani}.\nThese authors also showed a very interesting\nresult, concerning  spontaneous symmetry breaking in 2HDM models\npossessing a CP2 symmetry.\nNamely, they prove (their Theorem 4)\nthat electroweak symmetry breaking will {\\em necessarily}\nspontaneously break CP2.\nHowever, they also show that the vacuum\nwill respect at least one of the CP1 symmetries which compose\nCP2.\nWhich is to say, in a model which has a CP2 symmetry,\nspontaneous symmetry breaking necessarily respect the CP1\nsymmetry.\n\nIn summary,\nwe have proved that all possible symmetries on the scalar\nsector of the THDM,\nincluding Higgs Family symmetries,\ncan be reduced to multiple applications of\nthe standard CP symmetry.\n\n\n\\section{\\label{sec:conclusions}Conclusions}\n\nWe have studied the application of generalized CP symmetries\nto the THDM,\nand found that there are only two independent classes\n(CP2 and CP3),\nin addition to the standard CP symmetry (CP1).\nThese two classes lead to an exceptional region of parameter,\nwhich exhibits either a $Z_2$ discrete symmetry or\na larger $U(1)$ Peccei-Quinn symmetry.\nWe have succeeded in\nidentifying a basis-independent invariant quantity that can\ndistinguish between the $Z_2$ and $U(1)$ symmetries.\nIn particular, such an invariant is required\nin order to distinguish between CP2 and CP3,\nand completes the description of all symmetries in the THDM\nin terms of basis-invariant quantities.\nMoreover, CP2 and CP3 can be obtained by combining\ntwo Higgs Family symmetries and that this is not possible\nfor CP1.  \n\nWe have shown that all symmetries of the THDM previously identified\nby Ivanov \\cite{Ivanov1} can be achieved through simple symmetries.\nwith the exception of $SO(3)$.\nHowever, the $SO(3)$ Higgs Family symmetry\ncan be achieved by imposing a $U(1)$ Peccei-Quinn  \nsymmetry and the CP3-symmetry in the same basis.\nFinally, we have demonstrated that\nall possible symmetries of the scalar\nsector of the THDM can be reduced to multiple applications of\nthe standard CP symmetry.\nOur complete description of the symmetries on the scalar fields\ncan be combined with symmetries in the quark and lepton sectors,\nto aid in model building.\n\n\n\\begin{acknowledgments}\nWe would like to thank Igor Ivanov and Celso Nishi for their\nhelpful comments on the first version of this manuscript.\nThe work of P.M.F. is supported in part by the Portuguese\n\\textit{Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia} (FCT)\nunder contract PTDC\/FIS\/70156\/2006. The work of H.E.H. is\nsupported in part by the U.S. Department of Energy, under grant\nnumber DE-FG02-04ER41268. The work of J.P.S. is supported in\npart by FCT under contract CFTP-Plurianual (U777).\n\nH.E.H. is most grateful for the kind hospitality and support of the\nCentro de F\\'{\\i}sica Te\\'orica e Computacional at Universidade de\nLisboa\n(sponsored by the Portuguese FCT and\nFunda\\c{c}\\~{a}o Luso-Americana para o Desenvolvimento)\nand the Centro de F\\'{\\i}sica Te\\'orica de Part\\'{\\i}culas at\nInstituto Superior T\\'ecnico during his visit to Lisbon. This work\nwas initiated during a conference in honor of Prof. Augusto Barroso,\nto whom we dedicate this article.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nYoung high-mass stars (M $ \\geq \\,$ 8 M$_\\odot$) probed in cm-wavelength interferometric studies typically appear as fairly bright (flux densities of $\\sim$ few mJy to Jy)  regions of ionized gas  that are classified according to their size and emission measure, e.g., compact, ultracompact (UC), and hypercompact (HC) H{\\small II} regions \\citep[e.g.,][]{2005IAUS..227..111K}. It is generally thought that once  nuclear burning  has  begun the star  produces enough UV radiation to photoionize the surrounding gas. However, theories of the earliest stages remain poorly constrained by observations mainly due to the characteristics of the regions where they are born, which are highly dust-obscured, distant ($\\gtrsim$ 1 kpc) regions that undergo rapid evolution, and they reach the zero-age main sequence (ZAMS) while still heavily accreting. In fact, an evolutionary sequence for high-mass stars has not yet been established \\citep[e.g.,][]{2013A&A...550A..21S, 2014prpl.conf..149T}, although significant progress has been achieved on both observational and theoretical fronts \\citep[e.g.,][]{2018ARA&A..56...41M}. The identification and study of\nobjects in the early stages of their evolution will help us to\n  discriminate among proposed mechanisms for their formation; the two main\nscenarios being core accretion (i.e., scaled-up version of low-mass\nstar formation) and competitive accretion (i.e., in which stars in a cluster attract each other while they accrete from a shared reservoir of gas; see \\citealp{2014prpl.conf..149T}). The low-mass star formation process is modeled by\naccretion via a circumstellar disk and a collimated jet\/outflow that removes angular momentum and allows accretion to proceed \\citep[e.g.,][]{1988ApJ...328L..19S}. The jet\/outflow system is powered  magnetohydrodynamically by rotating magnetic fields coupled to either the disk (disk winds: e.g., \\citealp{2000prpl.conf..759K}) and\/or the protostar (X-winds: e.g., \\citealp{1987ARA&A..25...23S}).  Additionally, protostellar collisions have been  proposed as an alternative mechanism for the formation of high-mass stars \\citep{1998MNRAS.298...93B, 2005AJ....129.2281B}. \\\\\n\nMassive molecular outflows are a common phenomenon in high-mass star forming regions (e.g., \\citealp{1996ApJ...457..267S, 2002A&A...383..892B}); hence accretion disks and ionized jets similar to those found towards low-mass protostars are also expected. \n In addition, several surveys toward high-mass star forming regions in the NIR spectral lines of H$_2$ have detected a large number of molecular jets \\citep[e.g.,][]{2017ApJ...844...38W, 2015MNRAS.450.4364N}.\nHowever, the current sample of known high-mass protostars associated with disks \\citep[see review by][]{2016A&ARv..24....6B} and collimated jets \\citep[e.g.,][]{1995ApJ...449..184M, 1998ApJ...502..337M, 2006ApJ...638..878C, 2008AJ....135.2370R} is inadequate to draw conclusions about the entire population.\nThe detection of sources at the onset of high-mass star formation and the measurement of their physical properties is essential  to test theoretical models of high-mass star formation \\citep[e.g.,][]{2014prpl.conf..149T}. Furthermore, the  most sensitive instruments are necessary  to place significant constraints on the occurrence rate and parameters of these detections.\\\\\n\nIn  \\citet[][hereafter Paper I]{2016ApJS..227...25R} we described our high sensitivity ($\\sim$3 -- 10 $\\mu$Jy beam$^{-1}$) continuum survey, which aimed to identify candidates in early evolutionary phases of high-mass star formation and to study their centimeter continuum emission. We observed  58 high-mass star forming region candidates using the Karl G. Jansky Very Large Array (VLA)\\footnote{The National Radio Astronomy \nObservatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.} at 1.3 and 6 cm wavelengths at an angular resolution ${\\scriptstyle <}\\,$0\\rlap.$^{\\prime \\prime}$6.  The 58 targets were grouped into three categories based on their mid and far-IR luminosity as well as the temperature of the cores: 25 hot molecular cores (HMCs), 15 cold molecular cores with mid-IR point source association (CMC--IRs), and 18 cold molecular cores (CMCs) devoid of IR point source associations. The cores in our sample cover a wide range of parameters such as bolometric luminosity and distance. They have similar masses and densities, however, the latter two types of cores---mainly found within infrared dark clouds (IRDCs)---have lower temperatures (T$\\sim$ 10--20 K) than HMCs (T${\\scriptstyle >}\\,$50 K; depending on the probe and scale). In \\citetalias{2016ApJS..227...25R} we reported  detection rates of 1\/18 (6$\\%$) CMCs, 8\/15 (53$\\%$) CMC-IRs and 25\/25 (100$\\%$) HMCs.\nIn several cases, we detected multiple sources within a region, which resulted in a total detection of 70 radio sources associated with 1.2 mm dust clumps.\nThe 100$\\%$ detection rate of centimeter emission in the HMCs  is a higher fraction than previously reported. This suggests that radio continuum may be present, albeit weak, in {\\it all} HMCs although in many cases it is only detectable with the superior sensitivity now available with the upgraded VLA.   Our results show further evidence for an evolutionary sequence in the formation of high-mass stars, from  very early stage cold cores (i.e., CMCs) to relatively more evolved ones (i.e., HMCs).\\\\\n  \n\n\nA number of physical processes can cause centimeter continuum emission associated with high-mass star forming regions (see \\citealt{2012ApJ...755..152R} and \\citealt{2013ApJ...766..114S} summaries of thermal and non-thermal emission detected at centimeter wavelengths from YSOs). Recently, \\citet*[][hereafter TTZ16]{2016ApJ...818...52T}  developed a model to predict the radio emission from high-mass stars forming via core accretion. The \\citetalias{2016ApJ...818...52T} model predicts that during the first stages of ionization the H{\\small II} region is initially confined to the vertical (or outflow) axis and produces free-free emission with  similar features and parameters as observed towards ionized jets. Ionized jets are detected as weak and  compact centimeter continuum sources. At subarcsecond resolutions, they usually show a string-like morphology, often aligned  with a large-scale molecular outflow  of size up to a few parsecs. Ionized jets trace outflows on  smaller scales, providing the location of the driving protostar, that otherwise are deeply embedded in the natal clump and generally remain undetected at other wavelengths  due to the high extinction in the region \\citep{1998AJ....116.2953A}.  However, less extincted sources may\nhave molecular jet counterparts visible in H$_2$ line emission from shocked gas.\nThese sources are also called `thermal radio jets' due to their characteristic  rising spectrum which is consistent with free-free radiation from ionized gas.  \nThe  ionization mechanism of these jets has been proposed to be \\emph{shock-induced ionization} when the wind from the central protostar ionizes itself through  shocks due to variations in velocity of the flow or variations of the mass loss rate  (\\citealt*{1987RMxAA..14..595C}; \\citealt{1989ApL&C..27..299C}).  Unlike the simple model of a  uniform electron density H{\\small II} region,  ionized jets and winds have a radial density gradient and thus  are partially optically thick.  \\citet{1986ApJ...304..713R} discussed the behavior of  collimated jets and the dependency  of their physical parameters (such as temperature, velocity, density and ionization fraction) on  morphology, independently of the mechanism of ionization, and showed that the spectral index of a partially ionized jet  ranges  between $-0.1 \\leq \\alpha \\leq 1.1$.\\\\\n\nThe detection of ionized jets toward  high-mass stars at their early stages, as predicted by \\citetalias{2016ApJ...818...52T}, can help to distinguish between accretion scenarios (highly organized outflows are expected from core accretion but not from competitive accretion scenarios; \\citealt{2016ApJ...821L...3T}),  and ultimately will give us insight about accretion disks around high-mass stars. Several systematic studies searching for ionized jets have been reported in the literature. \\citet{2012ApJ...753...51G},  from a sample of 33 IR luminous objects, detected 2 ionized jets using the Australia Telescope Compact Array (ATCA) with a 4$\\sigma$ detection limit and an image rms ($\\sigma$) of $\\sim$0.1-0.2 mJy beam$^{-1}$  at 4.8 and 8.6 GHz.  \\citet{2016A&A...585A..71M} observed  11 high-mass YSOs using the Jansky VLA and detected 5 collimated ionized jets and 6 ionized wind candidates with a 3$\\sigma$ detection limit and an rms $\\sim$ 11 $\\mu$Jy beam$^{-1}$ at $\\sim$ 6.2 GHz. \\citet{2016MNRAS.460.1039P} observed 49 high-mass YSOs using the ATCA and detected 16 ionized jets and 12 jet candidates  with a 3$\\sigma$ detection limit and an rms $\\sim$ 17 $\\mu$Jy beam$^{-1}$ at $\\sim$ 5.5 GHz. Additionally, the Protostellar Outflow at the EarliesT Stage (POETS) survey is undertaking a search of radio-jets  using the VLA with an angular resolution of $\\sim$0\\rlap.$^{\\prime \\prime}$1 and an image rms of $\\sim$10 $\\mu$Jy beam$^{-1}$ \\citep{2018A&A...619A.107S, 2019A&A...623L...3S}. In our radio continuum Jansky VLA survey we observed 58 high-mass star forming regions and detected 70 radio sources with a \n5$\\sigma$ detection limit and an rms $\\sim$ 5 $\\mu$Jy beam$^{-1}$ at $\\sim$ 6 GHz \\citep{2016ApJS..227...25R}. \\citealt*{2018A&ARv..26....3A} is a recent comprehensive review of ionized jets in star forming regions. \\\\\n\nThe main goal of this  paper is to investigate the nature of the 70 detected radio sources \nreported in \\citetalias{2016ApJS..227...25R}. The observations along with the complete list of targets, coordinates, radio  detections and derived observational parameters are presented in \\citetalias{2016ApJS..227...25R}. In Section \\ref{sec:models} we examine several scenarios to explain the origin of the ionized gas emission and we study the physical properties of the detected sources.  Section \\ref{discussion_paperII}  contains a discussion of the viability of the different scenarios. In Section \\ref{conclusions_paperII}\nwe summarize our findings. Additionally, Appendix \\ref{app:lum} shows the bolometric luminosity estimates for these high-mass star forming regions using {\\it Herschel}\/Hi--GAL data and Appendix \\ref{mom_rate_appe} shows a study of the momentum rate of ionized jets.\n\n\n\n\n\n\n\n\n\n\n\\section[Models  for the Radio Emission]{ Models Considered for the Radio Emission} \\label{sec:models}\n\n\\subsection{Low-mass Young Stellar Objects}\\label{yso}\n\nThe main goal of our high sensitivity continuum survey presented in  \\citetalias{2016ApJS..227...25R} \nwas to detect radio emission from high-mass protostars. However, there exists a variety of sources that could also appear as radio detections in our images.\nIn \\citetalias{2016ApJS..227...25R}  we considered  contamination by extragalactic radio\nsources, and found that only a small number of extragalactic sources are expected to be observed within\nthe typical dust clump size of $\\sim$30$^{\\prime \\prime}$ (8 and 2 sources in the 6 and $1.3\\,$cm bands, respectively for the entire sample). \nA more likely source of contamination would be the presence of low-mass YSOs which are expected\nto be present in regions of high-mass star formation  \\citep[e.g.,][]{2013A&A...554A..48R}. We are thus interested in identifying possible low-mass class 0 -- class III YSOs  that could have \nbeen detected in our survey toward high-mass star forming regions. \n\nA large sample of low-mass YSOs has been  observed with the VLA at 4.5 and 7.5 GHz as part of  the Gould Belt survey (i.e.,  Ophiuchus at a distance of 120 pc:  \n\\citealp{2013ApJ...775...63D}; Orion at 414 pc: \\citealp{2014ApJ...790...49K}; Serpens at 415 pc: \\citealp{2015ApJ...805....9O}; Taurus-Auriga at 140 pc: \\citealp{2015ApJ...801...91D}, and\nPerseus at 235 pc: \\citealp{2016ApJ...818..116P}). \nThe brightest low-mass YSO in the entire Gould Belt survey (excluding the Orion region) was found in the Ophiuchus region (source J162749.85--242540.5,\na class III YSO, i.e., weak-lined T-Tauri star, with S$_{7.5\\,GHz} = 8.51\\,$mJy, \\citealt{2013ApJ...775...63D}).  To determine whether such an object would have been detected in our survey,\nwe scaled its flux density to the assumed distance\\footnote{Distances were taken from the literature, and are listed in Table \\ref{SED_Parameters}. \nMost distances are kinematic; only a few regions have trigonometric parallax measurements.}\nof each of our targets, and compared its scaled flux density to our adopted detection limit of  $\\geq$ 5 times the image rms\nat 7.4 GHz for each of our regions. We found that such a YSO would not be detected in any of our targets located at distances beyond $2\\,$kpc.\nSince the majority\nof our targets exceed this distance (see Figure \\ref{T_Tauri_hist}), we conclude that for most of our observed regions the detected radio sources are not low-mass YSOs.\\\\\n\n\nThere are 10 regions in our survey that are located at distances $\\leq$ 2 kpc. However, given the 7.4 GHz image rms for these regions, only in 7 of them would we have detected the brightest low-mass YSO of Ophiuchus.\nThese regions are  five  HMCs: 18517$+$0437, 20126$+$4104, 20293$+$3952, 20343$+$4129, G34.43$+$00.24mm1, and  two CMC--IRs: LDN1657A$-$3  and\nUYSO1. In these 7 regions we detected a total of 13 radio sources within the FWHM of the mm clumps: 10 towards HMCs and 3 towards CMC--IRs. That some of these \nsources are  possibly low-mass YSOs can be seen in the case of  IRAS 20126$+$4104: Besides the well-studied high-mass protostars associated\nwith radio sources 20126$+$4104 A and 20126$+$4104 B, the radio source G$78.121+3.632$ in this region (see  \\citetalias{2016ApJS..227...25R} , Table 4) corresponds to the source I20var, which was\n discussed by \\citet{2007A&A...465..197H}. This is a highly \nvariable radio source and  has observational properties consistent with a flaring T-Tauri star. In the same region, we have also detected a new object of similar characteristics. Radio source 20126$+$4104 C \nwas detected for the first time in our survey although several high sensitivity observations of this region have been made in the past \\citep[][]{2007A&A...465..197H}.\n\nHence, 20126$+$4104 C is clearly \nvariable in the radio regime, and is a candidate for a low-mass pre-main sequence star.\nAdditionally, the radio source LDN1657A$-$3 A, which has a negative spectral index ($\\alpha=-1.2$), is also a candidate for a variable radio source, where the emission is probably\ncaused by non-thermal processes on the surface of a T-Tauri star.  While the observational properties of these sources are consistent with low mass YSOs, we note that alternative explanations are possible \\citep[e.g.,][]{2018A&A...612A.103C}.\n\nIn summary, while some degree of contamination by low-mass YSOs probably exists in our survey for the nearest sources,\nfor the majority of our targets the detected radio sources are \nvery likely not contaminated by emission from low-mass YSOs. \n\n\n\\begin{figure}[!h]%\n    \\centering\n\n    \\includegraphics[width=0.7\\linewidth, clip=True]{Regions_T_Tauri}%\n     \\caption[The distance distribution of our targets is shown as a black line]{\\small{The distance distribution of our targets is shown as a black line. The color histogram shows the number of targets where we expect the detection of T-Tauri \n     stars which are as bright as J162749.85--242540.5 in Ophiuchus. This is the case for 5 HMCs and 2 CMC-IRs.}\n}%\n    \\label{T_Tauri_hist}%\n\\end{figure}\n\n\\subsection{H{\\small II} Regions}\nIn \\citetalias{2016ApJS..227...25R}  we reported the detection of 70 radio continuum sources associated with three different types of mm clumps and we calculated their 5--25 GHz spectral index ($\\alpha$) using power-law fits of the form $S_{\\nu}$ $\\propto$ $\\nu^{\\alpha}$.  The spectral index values and the fits to the data for all the radio detections are reported in \\citetalias{2016ApJS..227...25R} in Table 4 (electronic version) and in Figure 4, respectively. The range of spectral indices found was $-$1.2 to 1.8 (see Figure 5 in \\citetalias{2016ApJS..227...25R}). Based on their radio spectra, we classify  these sources as flat spectral index ($-$0.25$<\\alpha<$0.2),  positive spectral index ($\\alpha \\geq$ 0.2), and negative spectral index ($\\alpha <$-0.25). Thus, we have 10 sources with flat, 44 sources with positive, and 9 sources with negative spectral index. For the remaining 7, there is not a clear estimate of the spectral index.\n\nThe radio sources have a variety of morphologies. Excluding the sources without spectral index information, there are 6 extended sources, 8 sources with elongated structures,\nand the majority of sources (49) are compact with respect to our synthesized beam. In this section we consider whether a family of H{\\small II} region models could  explain sources with flat and positive spectral index.\n\n\n\\subsubsection{Extended Sources}\\label{ext_sources}\n\nAmong the sources detected in our survey associated with $1.2\\,$mm dust emission, there are six sources that are clearly extended at cm wavelengths with\nrespect to the $\\sim$0\\rlap.$^{\\prime \\prime}$4 resolution of the maps, hence they are candidates for  H{\\small II} regions,\ni.e. photoionized gas.\n These sources are relatively bright (S$_{25.5\\,GHz} \\approx$ 1\\,mJy), and are found mostly toward HMCs.\nMoreover, they generally show a flat spectral index, indicative of optically thin free-free emission. \nFor five of these sources, we calculate the physical properties from the $25.5\\,$GHz continuum flux using the formulae from \\citet{1994ApJS...91..659K},\nwhich assume spherical symmetry, and optically thin emission from a uniform density plasma with T$_{e}= 10^{4}$\\,K. The results are listed in Table \\ref{HII_Parameters}, where column \n1 is the region name, column 2 is the \nspecific radio source, and columns 3 and 4 are the frequency ($\\nu$) and radio flux (S$_{\\nu}$), respectively.  Column 5 is the observed linear size (diameter)\nof the radio source ($\\Delta s$) at 3$\\sigma$ rms level in the image, column 6 is the emission measure (EM), column 7 is the electron density (n$_e$),\ncolumn 8 is the \nexcitation parameter (U) and column 9 is the logarithm of the Lyman continuum  flux (N$^{\\prime}_{Ly}$)  required for ionization.   We use log\\,N$^{\\prime}_{Ly}$ to \nestimate \nthe spectral type of the ionizing star (listed in column 10) using the tabulation in \\citet{1973AJ.....78..929P}, further assuming that a single ZAMS star is photoionizing the nebula and \nproducing the \nLyman continuum flux. The distances used for these calculations are listed in Table \\ref{SED_Parameters}, and the near kinematic distance is adopted  when the region has a distance ambiguity.\n\n\n\n\n\\begin{deluxetable}{l c c c c c c c c c}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Extended Sources:  Parameters  from Radio Continuum  \\label{HII_Parameters}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio }  &\n\\colhead{$\\nu$}   &\n\\colhead{S$_{\\nu}$ }               & \n\\colhead{$\\Delta s$}  & \n\\colhead{EM\/$10^{5}$ }             & \n\\colhead{n$_{e}\/10^{3}$ } & \n\\colhead{U } & \n\\colhead{log\\,N$^{\\prime}_{Ly}$ } &\n\\colhead{Spectral } \n \\\\[2pt]\n\\colhead{}                            & \n\\colhead{Source}                      & \n\\colhead{(GHz)}                      & \n\\colhead{($\\mu$Jy)}                      & \n\\colhead{(pc)}                      & \n\\colhead{(pc\\,cm$^{-6}$)}                      &  \n\\colhead{(cm$^{-3}$)}                      & \n\\colhead{(pc\\,cm$^{-2}$)}                     & \n\\colhead{(s$^{-2}$)}  &\n\\colhead{Type\\tablenotemark{a}}                         \\\\[-20pt]\\\\}\n\\startdata\n\\input{latex_classic_HII_table_mod.txt}\n\\enddata\n\\tablenotetext{\\text{a}}{Using the tabulation in  \\citet{1973AJ.....78..929P}.}\n\\tablenotetext{\\text{b}}{Includes radio source 20293$+$3952 B (see Figure 2 in \\citetalias{2016ApJS..227...25R}).}\n\\end{deluxetable}\n\nThe measured sizes of the five sources listed in Table \\ref{HII_Parameters} are all below $0.1\\,$pc, which according to  \\citet{2005IAUS..227..111K} would suggest a classification as ultra (UC)- or hypercompact (HC)\nH{\\small II} regions. However, the calculated emission measures and electron densities are an order of magnitude or more  smaller than  typical values for such H{\\small II} regions. The typical values of the emission measure and electron density for UCH{\\small II} regions are EM$\\gtrsim 10^{7}$ pc cm$^{-6}$ and n$_{e}\\gtrsim 10^{4}$ cm$^{-3}$ and for HCH{\\small II} regions are EM$\\gtrsim 10^{10}$ pc cm$^{-6}$ and n$_{e}\\gtrsim 10^{6}$ cm$^{-3}$ \\citep{2005IAUS..227..111K}.\nWe note that four of these resolved extended radio sources (18470--0044 A, 18521$+$0134 B, 19035$+$0641 B  and 20293$+$3952 C) are offset $\\sim$ 2\/3 of the radius from the center of the mm clumps,\nand are thus located at their outskirts. A plausible explanation for the lower electron densities and emission measures is that early B-type stars have\nformed near the edge of the dust clumps where the density of the surrounding medium is much lower than in the center.\n\nWe will comment on two additional resolved and extended radio sources detected in this survey. The first one is  18470--0044 A, which was previously observed by \\citet{2011ApJ...739L..17H} at 25.5 GHz using the VLA in the C-configuration.  This radio source has an offset of 7\\rlap.$^{\\prime \\prime}$2 with respect to the peak of the mm clump associated with IRAS 18470--0044  reported by \\citet{2002ApJ...566..945B}. Our image towards this region is affected by sidelobes due to the large flux and extended emission of a \nnearby radio source, and we were not able to accurately measure the radio flux at 1.3 cm. This source is among the detections without a clear estimate of its spectral index. However, based on the flux \nreported by \\citet{2011ApJ...739L..17H},  and our measured flux at 6 cm (see \\citetalias{2016ApJS..227...25R}), this radio source has a flat spectrum, and it is likely an H{\\small II} region ionized by a \nB2 ZAMS star. \n\nThe second bright, and resolved, radio source is G34.43$+$00.24mm2 A, which is the only extended source detected towards an IRDC clump in our survey. Interestingly, this radio source \nhas a spectral index of $\\alpha= -0.5$ (see \\citetalias{2016ApJS..227...25R}) and is associated with a 24 $\\mu$m point source, and  at least two molecular outflows \\citep{2007ApJ...669..464S}. This radio \nsource was originally detected at 6 cm by \\citet[labeled by them as Mol 74]{1998A&A...336..339M}  and \\citet{2004ApJ...602..850S} using the VLA (FWHM $\\sim$6\\rlap.$^{\\prime \\prime}$$\\times\n$3\\rlap.$^{\\prime \\prime}$). The  flux densities at 6 cm reported in those studies are consistent with our data.  This radio source has an offset of 7\\rlap.$^{\\prime \\prime}$8 with respect to the \ncenter of the $1.3\\,$mm clump G34.43$+$00.24mm2  detected by \\citet{2006ApJ...641..389R}.\\\\\n\nWe found that at least 36$\\%$ of the HMCs have an extended radio source within the 1\\rlap.$^{\\prime}$8 FWHM primary beam at 25.5 GHz. However, most of them are located slightly outside the FWHM of the mm clump, and are not discussed in this study. These extended radio sources are: 18089$-$1732   G12.890$+$0.495, 18182$-$1433   G16.584$-$0.053, 18470$-$0044  G32.113$+$0.097, 18521$+$0134  G34.749$+$0.021, 19012$+$0536  \nG39.389$-$0.143 and  19266$+$1745 G53.037$+$0.115.    All these sources and their radio continuum parameters are reported in \\citetalias{2016ApJS..227...25R}. \n\n\n\\subsubsection{Compact Sources}\\label{compact_sources_sect}\n\nIn \\citetalias{2016ApJS..227...25R} we characterized the change in flux density with frequency using a power law. \nWe reported the detection of  36 compact radio sources  (51$\\%$ of total detections) with a rising spectrum ($\\alpha >$0.2), of which \n7 were not detected at 6 cm, thus a lower limit for their spectral index was estimated. We also detected 5 compact radio sources with a flat cm spectrum.\nIn this section we investigate whether a uniform density UC\/HC H{\\small II} region can explain the observed fluxes and spectral indices for compact sources with rising spectra.\n\nWhile optically thin free-free emission results in a flat spectral index ($\\alpha = -0.1$), a rising spectrum implies appreciable optical depth in the emitting gas.\nTo fit our spectra, we thus will require a turnover, (i.e. $\\tau_{\\nu} \\sim 1$) near the intermediate frequency of our observing bands, around $\\nu_{t}=$14.7 GHz,\nwhich in turn requires an emission measure near 9$\\times$10$^{8}$ pc\\,cm$^{-6}$. For fitting the data we use a uniform density, spherical H{\\small II} region model\nwith electron temperature of $T_{e}=$ 10$^{4}$\\,K\nas shown in equation (11) from \\citet{1975A&A....39..217O}. We find that \nfor all compact, rising spectrum sources detected in our survey, within the given uncertainties, a uniform density H{\\small II} region spectrum can be reasonably fit to the radio continuum data.\nExamples of the fits are shown by the continuous blue line in Figure \\ref{HII_fit_examples}. \n The fits for all the 36 compact radio sources with rising spectral index is shown in Appendix \\ref{all_HII_fit} Figure \\ref{HII_fit_app}.\n\n\n\n\\begin{figure}[!h]%\n    \\centering\n  \\hspace*{-0.5cm} \n    \\includegraphics[width=0.35\\linewidth, clip]{g23_01_alpha_HII_A}%\n        \\includegraphics[width=0.35\\linewidth, clip]{18440_alpha_HII_A}%\n        \\includegraphics[width=0.35\\linewidth, clip]{20343_alpha_HII_B}%\n     \\caption[Spectra of the compact radio sources G23.01--0.41 A  (left), 18440--0148 A (center) and 20343$+$4129 B  (right)]{\\small{Spectra of the compact radio sources G23.01--0.41 A  (left), 18440--0148 A (center) and 20343$+$4129 B  (right). Error bars are an assumed uncertainty of 10$\\%$ from the flux densities added in quadrature with an assumed 10$\\%$ error in calibration. The  continuous blue line is the H{\\small II} region fit using a spherical, constant density model. The red arrow indicates the frequency where $\\tau_{\\nu}=$1 in each model fit. The dashed line is the best fit to the data from a power-law of the form $S_{\\nu}$ $\\propto$ $\\nu^{\\alpha}$.}}\\label{HII_fit_examples}%\n\\end{figure}\n\n\nOur fitting results show that the generally quite weak emission from these very compact, rising spectrum sources implies a very small size\nfor the emitting regions. The sizes are much smaller than our angular resolution, and are on the order of the initial Str\\\"{o}mgren sphere radius.\nThe initial Str\\\"{o}mgren sphere radius ($R_{s}$) depends on the Lyman continuum (N$_{Ly}$) flux and the ambient molecular density (n$_{H_{2}}$) as\nstated in equation 1 from \\citet{1996ApJ...473L.131X}:\n\n\\begin{equation}\nR_{s}= 4104.7 \\left( \\frac{N_{Ly}}{10^{49}s^{-1}}\\right)^{1\/3}\\left(\\frac{n_{H_{2}}}{10^{5}cm^{-3}}\\right)^{-2\/3}  \\text{[au]}  .\n\\end{equation}\n\nWe show the relation R$_{s}$ versus N$_{Ly}$, represented by the solid lines, for n$_{H_{2}}=$ 10$^{5}$, 10$^{6}$,  10$^{7}$ and 10$^{8}$ cm$^{-3}$ in Figure \n\\ref{Stromgren_sphere}. To place our sources in this diagram, we estimated the Lyman continuum flux (N$_{Ly}$) from our $25.5\\,$GHz flux density, with\nthe formulae of \\citet{1994ApJS...91..659K}, and use the radii ($\\Delta\\,s\/2$) derived from the spectral fitting.  These data are listed in Table~\\ref{tab:fig3}.\nOur sources, represented as solid purple dots\nin Figure \\ref{Stromgren_sphere}, cluster around the expected Str\\\"{o}mgren radius for an initial density of $10^{6}$\\,cm$^{-3}$. \n\n\\begin{figure}[!h]%\n    \\vspace{1.2cm}\n    \\centering\n    \\includegraphics[width=0.5\\linewidth, clip]{R_strongrem}%\n     \\caption{\\small{Initial Str\\\"{o}mgren sphere  radius as a function of the Lyman continuum for compact sources with rising spectra. The solid lines represent the  ambient molecular density for n$_{H_{2}}=$ 10$^{5}$ --  10$^{8}$ cm$^{-3}$. The solid purple dots represent the radius of the\n     H{\\small II} regions ($\\Delta\\,s\/2$) as implied by the spherical, constant density H{\\small II} region model fits.  The dashed green line represents the lower limit of  $R_{Turb}$ (for $\\xi =$1) and the  dashed blue line presents $R_{Th}$ if the initial ambient molecular density for both cases is n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$. Shaded areas  represent the path from their initial  radius up to their final Str\\\"{o}mgren sphere radius if the sources were born at a density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ (green and blue shaded areas for turbulence and thermal pressure confinement, respectively).\nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}%\n    \\label{Stromgren_sphere}%\n\\end{figure}\n\n\n\nWhen nuclear burning begins and a high-mass star produces enough UV photons to photo-ionize the surrounding material, the initial Str\\\"{o}mgren sphere radius is reached (to within a few percent) in a recombination timescale, $t_{r}= $(n$_{H_{2}}$$\\beta_{2}$)$^{-1}$[s], where $\\beta_{2}$= 2.6$\\times$10$^{-13}$\\,cm$^{3}$s$^{-1}$ is the recombination coefficient \\citep{1980pim..book.....D}. For any reasonable initial density this time scale is extremely short ($< 1\\,$yr), and the initial\nStr\\\"{o}mgren sphere radius is reached almost instantaneously. The highly over-pressured ionized region will then begin to expand, and hence the initial\nStr\\\"{o}mgren sphere is a very short-lived configuration, and therefore it is unlikely that the large number of sources detected represent this evolutionary stage.\n\n\n\n\nAfter  formation of the initial Str\\\"{o}mgren sphere around a star, the UC H{\\small II} region is highly overpressured,  and as a result, it expands approximately at the sound speed until approaching pressure equilibrium with the ambient medium. \n\\citet{1995RMxAA..31...39D} and \\citet{1996ApJ...473L.131X} have studied  the confinement of UC H{\\small II} regions in a molecular core by thermal, and thermal plus turbulent pressure, respectively. \nAssuming pressure equilibrium between ionized and surrounding molecular gas, \\citet{1996ApJ...473L.131X}  gives the final radius of the ionized region (R) as :\n\n\n\\begin{equation}\nR = R_{s} \\left(   \\frac{2k\\xi T_{H^{+}}}{m_{H_{2}}\\sigma_{v}^{2} + kT_{k}}\\right)^{2\/3}  ,\n\\end{equation}\n\n\\noindent\nwhere $T_{H^{+}}$ is the temperature of the ionized region, $\\xi$ is a turbulence factor ($>$ 1) that takes into account the pressure due to stellar winds and turbulence in the ionized gas, $\\sigma_{v}$ is the velocity dispersion produced by turbulence and $T_{k}$ is the kinetic temperature of the surrounding molecular gas. \nUsing typical values for the physical conditions in regions where high-mass stars form, we can  test whether the sources discussed in this section could be ionized regions in pressure equilibrium with the surrounding molecular gas.\nWhile molecular line observations with single dish instruments indicate average densities of n$_{H_{2}}=$ 10$^{5}$\\,cm$^{-3}$ over the $1\\,$pc clump sizes \\citep[e.g.,][]{2000ApJ...536..393H}, interferometric measurements of high-mass star forming cores\nhave revealed central densities of n$_{H_{2}}= 10^7 - 10^{10} $\\,cm$^{-3}$ on scales $< 0.1\\,$pc \\citep[e.g.,][]{1990ApJ...362..191G, 2015A&A...573A.108G}. Following \\citet{1996ApJ...473L.131X}, we adopt values of $T_{H^{+}}=$ 10$^{4}$\\,K, $\\sigma_{v}=$ 2 km\\,s\n$^{-1}$ (FWHM$\\,\\sim$\\,5 km\\,s$^{-1}$) and $T_{k}=$ 100 K. Assuming $\\xi = 1$, and evaluating the above equation with these numbers we get $R_{turb} = 11.2 R_{s} $ for the case of thermal plus turbulent pressure, and\n$R_{th} = 54.3 R_{s}$ for thermal pressure only. These relations are shown in Figure \\ref{Stromgren_sphere} for n$_{H_{2}}= 10^7$\\,cm$^{-3}$ as green and blue dashed lines, respectively. \nConsidering  the location of our data points in Figure \\ref{Stromgren_sphere}, we can exclude the extremely high densities of $10^{10} $\\,cm$^{-3}$ as found by \\citet{2015A&A...573A.108G}, which would predict\nmuch smaller source sizes. On the other hand, our data points are located within the  shaded areas that represent the path from their initial Str\\\"omgren radius up to their final radius in pressure equilibrium,\nif the sources were born at a density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ (green and blue shaded areas for turbulence and thermal pressure confinement, respectively).\n\n\nAn estimate of the expansion time $\\tau_{expansion}$ for an ionized region can be obtained assuming that it  expands at its sound speed ($C_{s} \\sim$ 10 km\\,s$^{-1}$). To expand to $\\sim$\\,200 au,\nthen $\\tau_{expansion} \\sim R\/C_{s} \\sim $ 100 yr. Thus, an initial Str\\\"omgren sphere will expand fairly quickly and, as suggested by \\citet{1995RMxAA..31...39D} and \\citet{1996ApJ...473L.131X}, \nthe ionized regions can remain compact as long as the molecular core provides the outside pressure. Observations of  UCH{\\small II} regions and HMCs suggest that this time is on order $10^5\\,$yr\n\\citep{1989ApJ...340..265W,2001ApJ...550L..81W}, and it hence appears that  our sources could be ionized regions around newly formed stars in pressure equilibrium in their\nmolecular cores. While our calculations have not been fitted to a particular source, the observed scatter in Figure \\ref{Stromgren_sphere} can be accounted for with a varying amount of turbulence in the molecular gas,\ni.e. if the molecular line FWHM varies between $\\sim$ 7 -- 20 km\\,s$^{-1}$ for the case of an  ambient molecular density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$.\n\n\nIt is  interesting to note that we found that the radius of the extended sources discussed above are within the pressure equilibrium zone for an initial density of n$_{H_{2}}=$ 10$^{5}$\\,cm$^{-3}$. \nThese sources are located on the outskirts of the mm core, and one might ask whether they have migrated out of the molecular core center, or if they were born in their current location. Assuming stellar velocities between 2 and 12 km\\,s$^{-1}$ \n\\citep{2007ApJ...660.1296F} in order for them to travel to the half power point of the cores (FWHM median angular size for HMC $=$18\\rlap.$^{\\prime \\prime}$ at a distance of 4 kpc), times between\naround 10$^{5}$ yr and 10$^{4}$ yr, respectively, are needed. While migration toward lower density regions is thus possible, we note that we do not find any strong evidence for cometary regions which would be predicted due to\nbow shocks between molecular and ionized gas \\citep{1990ApJ...353..570V}.\n\n\\subsubsection{Lyman Continuum}\n\nAn additional point to consider to understand the nature of our detections is the Lyman continuum photon rate  as a function of the bolometric luminosity. We analyze this relation for all the sources \nwith a flat or a rising spectrum (including extended, elongated structure, as well as compact morphology) as shown in Figure \\ref{Lyman_cont_plot}. The Lyman continuum photon rate is estimated from the radio \ncontinuum flux at 25.5 GHz and the bolometric luminosities for our regions are estimated  from  {\\it Herschel}\/Hi--GAL fluxes, and from ancillary data (see \\S \\ref{app:lum}).  We list these data in Table~\\ref{tab:fig4}.\nFor data taken from the literature, care was taken that the  Lyman continuum flux and bolometric luminosities refer to the same distance. For sources with distance ambiguity, we use the near kinematic distance.\nIn Figure \\ref{Lyman_cont_plot}, compact and elongated sources are represented by filled circles if the bolometric luminosities are estimated in this work  (see \\S \\ref{app:lum}), or open circles if the luminosity\nis taken from the literature. The extended sources from \\S \\ref{ext_sources} are represented by the  $\\color{blue} \\times $  symbol. \nThe continuous black line is the expected Lyman continuum photon rate from a single zero-age main-sequence (ZAMS) star at a given luminosity,  and the shaded area\nbounded by the solid black line shows the expected Lyman continuum from a stellar cluster of the same N$_{Ly}$. For more details on these curves see \\citet{2013A&A...550A..21S}.\nThus, H{\\small II} regions ionized by stellar UV photons from a single early-type star are expected to lie on the black line. If,  on the other hand, the Lyman continuum comes from a cluster of stars (a likely scenario for high-mass \nstars) rather than from a single ZAMS star, the expected  N$_{Ly}$ is lower, and should be located within the shaded area \\citep{2015A&A...579A..71C}.\n\n\n\n As seen in Figure \\ref{Lyman_cont_plot}, only a small fraction of our sources fall in the shaded area of the plot  indicating direct stellar photoionization. Most of the HMC sources (red open\/filled circles) lie\n below the curve of the expected Lyman continuum flux, and hence are underluminous at radio wavelengths.\n On the other hand, the majority of the sources detected towards CMCs and CMC--IRs are located in the so-called ``forbidden area'' above the Lyman continuum line, showing  an excess of Lyman continuum compared \n to the expected value based \non their luminosities. This is true even if the sources are corrected by the distance, i.e., when there is ambiguity in the kinematic distance of the source or the value of the distance is incorrect. For reference, the arrow in the plot indicates the amount \nthat a point will move if the distance increases by a factor of 2. If the distance changes by any other factor the point will move parallel to the arrow. Additionally, there is  a possibility that some  bolometric luminosities  are underestimated (see \\S \\ref{app:lum}), however if this is the case we believe that the luminosities will shift to the right by less than 0.5 dex.\n\n\\begin{figure}[!h]\n    \n    \\centering\n    \\includegraphics[width=0.5\\linewidth, clip]{Lyman_plot_v3}\n     \\caption{\\small{Lyman continuum measured at 25.5 GHz as a \n     function of the bolometric luminosity for all detected sources with flat or rising spectra in our sample. The bolometric luminosity is mainly estimated from  {\\it Herschel}\/Hi--GAL data (except for the open circles for\n     whose the bolometric luminosity  is from the literature). The circles  represent the compact sources with flat or rising spectra, while the blue  $\\color{blue} \\times $ symbol represents the flat spectrum extended \n     sources from \\S \\ref{ext_sources}. UC H{\\small II} regions from \\citet{1994ApJS...91..659K} are represented by the gray $\\color{gray} \\times $ symbol. The continuous black line is the expected Lyman continuum photon rate of a \n     single  ZAMS star at a given luminosity, and the shaded area gives these quantities for the case of a cluster \\citep{2013A&A...550A..21S}. The arrow indicates how much a point would move if the distance were increased by \n     a factor of 2. \nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}\n    \\label{Lyman_cont_plot}\n\n\\end{figure}    \n    \n\n\n\n\\citet{2013A&A...550A..21S} have reported Lyman continuum excess for several sources in an 18 and 22.8 GHz survey of high-mass star forming regions with the Australian Telescope Compact Array (ATCA).\nInterestingly, $\\sim$70$\\%$ of their H{\\small II} regions with Lyman excess are associated with molecular clumps belonging to two types of sources that are in the earliest evolutionary stages of high-mass stars based on their classification (equivalent to our CMCs and CMC-IRs clumps).  \nAdditionally, \\citet{2015A&A...579A..71C}  found  Lyman  continuum excess for about 1\/3 of their  sample of 200 compact and UC H{\\small II} regions selected from the CORNISH survey \\citep{2013ApJS..205....1P}.  Their sources with Lyman continuum excess are also in an earlier evolutionary phase within their sample. Both studies argued that the Lyman excess is not easily justified, leaving room for two possible scenarios, invoking additional\nsources of  UV photons from an  ionized jet, or from an accretion shock in the protostar\/disk system. \\citet{2016A&A...588L...5C} suggested that the Lyman excess is  produced by  accretion shocks, based on outflow (SiO) and infall (HCO$^{+}$) tracer observations towards the 200 H{\\small II} regions studied in \\citet{2015A&A...579A..71C}. \n\nIt is important to mention that due to our selection criteria (see \\citetalias{2016ApJS..227...25R}) the sources studied by \\citet{2013A&A...550A..21S} and \\citet{2015A&A...579A..71C} are much brighter at radio wavelengths than the ones from our work, with radio luminosities at 5 GHz of $\\sim$ 10$^{2}$--10$^{6}$ mJy kpc$^{2}$ vs 10$^{-2}$--10 mJy kpc$^{2}$ in our sample. In Figure \\ref{Lyman_cont_plot} we also show several UC H{\\small II} regions from  \\citet{1994ApJS...91..659K}, denoted by the $\\color{gray} \\times $ symbol. These sources seem to be produced by higher free-free emission compared with our sample, suggesting that our sources represent a different population of radio sources. \\citet{1999RMxAA..35...97C} based on selection criteria similar to ours,  detected sources with low radio luminosities like the ones in this work. Furthermore, these low radio luminosities are typical of thermal jets, with UV photons that are produced by shocks from collimated winds from the protostar  with the surrounding material \\citep[e.g.,][]{1996ASPC...93....3A}. Thus, while the above analysis of the cm SEDs suggests a model of pressure confined H{\\small II} regions for our compact sources, the Lyman continuum photon rate  as a function of the bolometric luminosity\nshown in Figure \\ref{Lyman_cont_plot} does not lend strong support to this model. A further possible explanation for the compact sources with rising spectra, as well as for several elongated sources\ndetected in our survey is that they arise from thermal jets. We explore this scenario in the following section.   \n\n\\subsection{Ionized Jets}\\label{ionized_jet_sect}\n\nBased on the low radio luminosities (S$_{5\\,GHz}\\,$d$^{2}$ $\\sim$ 10$^{-2}$ -- 10 mJy kpc$^{2}$) of our detected sources, we need to consider the possibility that the source of ionization is not a ZAMS star, but rather \nthat their nature is that of a thermal, ionized jet produced by shock ionization as described in \\S \\ref{sec:intro}.\nSupport for this hypothesis comes from a subset of resolved sources from  our survey. We have characterized 12 jet candidates based on their elongated, or string-like morphology in conjunction with an\nassociation with a molecular outflow. These sources are listed in Table \\ref{jet_cand_list}, where column 1 is the name of the region, column 2 are the radio sources that are thought to be part of the ionized jet, and \ncolumn 3 lists the approximate direction of the ionized jet. Column 4  shows the approximate direction of the molecular outflows associated with the centimeter continuum emission as found in the literature. Column 5 \nindicates if the centimeter continuum emission is a new detection or if it has been  detected in previous studies.  Column 6 lists the references for the molecular outflow detections and previous centimeter continuum \ndetections if any.  Examples for these sources are shown in Figures \\ref{fig:spitzer_examples}a, \\ref{fig:spitzer_examples}b, and \\ref{fig:UKIDSS_examples}a, \\ref{fig:UKIDSS_examples}b. To our knowledge, 6 of these ionized jet candidates are new detections. In the cases of previous detections of centimeter continuum emission towards the listed regions, our high-sensitivity observations described in Paper I\ngenerally show the elongation or string-like morphology of a jet for the first time \\citep[e.g.,][]{2017ApJ...843...99H}. Furthermore, this subset of resolved jet candidates have the expected spectral index ( $0.2 \\leq \\alpha \\leq 1.2$) for ionized jets,  several of them are associated with  6.7 GHz CH$_{3}$OH masers and H$_{2}$O \nmasers, and they have excess emission at 4.5 $\\mu$m,  which may trace shocked gas via H$_{2}$ emission in outflows or  scattered continuum from an outflow cavity \\citep[e.g.,][]{2011ApJ...729..124C, \n2013ApJS..208...23L}. In some cases, like towards the ionized jet in 18182$-$1433, some of the radio sources  have  negative spectral indices, consistent with non-thermal lobes, since it is thought that when very strong shock waves from a fast jet move through a magnetized medium, some of the electrons are accelerated to relativistic velocities producing synchrotron emission \\citep{2003ApJ...587..739G, 2010Sci...330.1209C}. \\citet{2016MNRAS.460.1039P} and \\citep{2018A&A...619A.107S, 2019A&A...623L...3S} also reported the detection of  ionized jets with non-thermal lobes (see also review by \\citealt{2018A&ARv..26....3A}). \n\n\n\n\n\\begin{deluxetable}{l c c c c c c}\n\\tabletypesize{\\scriptsize}\n \\renewcommand*{\\arraystretch}{1.5}\n\\tablecaption{Ionized Jets  \\label{jet_cand_list}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio Source}        &\n\\colhead{Jet Direction}          &\n\\colhead{Outflow Direction}   &\n\\colhead{ H$_{2}-$Jet Direction}   &\n\\colhead{New Detection}             &\n \\colhead{Reference}      \\\\[-20pt]\\\\}\n\\startdata\n\\setcounter{iso}{0}\t\nG11.11$-$0.12P1     &    A, C, D  &   NE$-$SW   &   E$-$W, NE$-$SW\\tablenotemark{a}   &  E$-$W   &    y   & \\rxn \\label{rxn:Wang2014}  \\rxn \\label{rxn:Rosero2014} \\rxn \\label{rxn:Lee2013}  \\\\\n18089$-$1732         &    A           &   N$-$S        &    N$-$S                 &   no\/very weak\\tablenotemark{b}     & n          & \\rxn \\label{rxn:Beuther2004} \\rxn \\label{rxn:Beuther2010} \\rxn \\label{rxn:Zapata2006}      \\\\\n18151$-$1208         &    B          &   NE$-$SW   &   NW$-$SE\\tablenotemark{c}    &  NW-SE  &    n     &  \\rxn \\label{rxn:Fallscheer2011} \\rxn \\label{rxn:Hofner2011}  \\rxn \\label{rxn:Davis2004}   \\rxn \\label{rxn:Varricat2010} \\\\\n18182$-$1433         &    A$-$C\\tablenotemark{d}   &   E$-$W        &   NE$-$SW, NW$-$SE     &   E$-$W   &    n       &  \\rxn \\label{rxn:Beuther2006}   \\rxn \\label{rxn:Moscadelli2013}  \\rxn \\label{rxn:Lee2012}        \\\\\nIRDC18223$-$3      &    A$-$B\\tablenotemark{e}       &    NE$-$SW  &  NW$-$SE\\tablenotemark{f}      &  SE-NW  &  y     &  \\rxn \\label{rxn:Fallscheer2009}  \\rxn \\label{rxn:Beuther2005}   \\\\   \n G23.01$-$0.41      &    A    &    NE$-$SW         &  NE$-$SW  &  non-detection   & n & \\rxn \\label{rxn:Sanna2016} \\rxn \\label{rxn:Araya2008} \\rxn \\label{rxn:Sanna2018} \\osref{rxn:Lee2013}\\\\\n18440$-$0148         &    A          &    NW-SE       &  \\nodata\\tablenotemark{g}        &  non-detection  &    y    &     \\rxn \\label{rxn:Navarete2015}      \\\\\n18566$+$0408        &    A$-$D\\tablenotemark{h}  &    E$-$W        &   NW$-$SE   &  non-detection  &    n    &    \\rxn \\label{rxn:Zhang2007}\\rxn \\label{rxn:Araya2007}  \\rxn \\label{rxn:Hofner2017} \\osref{rxn:Lee2013}                                   \\\\\n19035$+$0641        &    A          &    NE$-$SW   &   NW$-$SE  &  no\/very weak\\tablenotemark{b}    &  y   &    \\rxn \\label{rxn:Lopez_Sep2010}       \\\\\n19411$+$2306        &    A          &    NE$-$SW   &   NE$-$SW  &   detection\\tablenotemark{b}    &  y   &    \\rxn \\label{rxn:Beuther2002bb}            \\\\\n20126$+$4104        &   A$-$B   &    NW$-$SE   &   NW$-$SE, S$-$N   &   NW$-$SE  &   n  &  \\rxn \\label{rxn:Su2007}  \\rxn \\label{rxn:Shepherd2000} \\rxn \\label{rxn:Hofner2007} \\rxn \\label{rxn:Cesaroni1999}  \\rxn \\label{rxn:Cesaroni2013}  \\\\\n20216$+$4107        &    A          &    NE$-$SW   &  NE$-$SW   &   NE$-$SW   &    y   &   \\osref{rxn:Lopez_Sep2010}  \\osref{rxn:Navarete2015}     \\\\\n\\enddata\n\\tablenotetext{\\text{a}}{ALMA unpublished data (Rosero et al. in prep).}\n\\tablenotetext{\\text{b}}{ T. Stanke and H. Beuther (private communication).}\n\\tablenotetext{\\text{c}}{A blue-shifted component of a molecular outflow going in the direction of 18151$-$1208 B is seen in Figure 4 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors.}\n\\tablenotetext{\\text{d}}{Radio source B has a negative spectral index and radio source A has an upper limit value in its spectral index. Their fluxes are not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablenotetext{\\text{e}}{Radio source B has an upper limit value for the flux at 4.9 GHz and its value is not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablenotetext{\\text{f}}{A blue-shifted component of a molecular outflow going in the direction of IRDC18223$-$3  is seen in Figure 5 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors.}\n\n\\tablenotetext{\\text{g}}{\\citet{2002ApJ...566..931S} report the presence of CO (2--1) wings towards this region, but contour maps of the molecular outflow are not available.}\n\\tablenotetext{\\text{h}}{Radio sources C and D have  upper limit spectral indices that are consistent with being negative and their fluxes were not included in \n Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel). \n}\n\\tablecomments{Generally there are multiple molecular outflows in each of these high-mass star forming region. We reference the ones that are located closest to the centimeter continuum emission. The  `y' and `n'  indicates if the centimeter radio continuum detection  is new or if it has been detected in a previous study, respectively.\\\\\n \\osref{rxn:Wang2014}  \\citet{2014MNRAS.439.3275W};  \\osref{rxn:Rosero2014}  \\citet{2014ApJ...796..130R}; \\osref{rxn:Lee2013} \\citet{2013ApJS..208...23L}; \\osref{rxn:Beuther2004} \\citet{2004ApJ...616L..23B}; \\osref{rxn:Beuther2010} \\citet{2010ApJ...724L.113B};   \\osref{rxn:Zapata2006} \\citet{2006AJ....131..939Z}; \\osref{rxn:Fallscheer2011}  \\citet{2011ApJ...729...66F}; \\osref{rxn:Hofner2011}  \\citet{2011ApJ...739L..17H}; \\osref{rxn:Davis2004}  \\citet{2004A\\string&A...425..981D}; \\osref{rxn:Varricat2010}  \\citet{2010MNRAS.404..661V}; \\osref{rxn:Beuther2006}  \\citet{2006A\\string&A...454..221B};  \\osref{rxn:Moscadelli2013}  \\citet{2013A\\string&A...558A.145M};  \\osref{rxn:Lee2012}  \\citet{2012ApJS..200....2L}; \\osref{rxn:Fallscheer2009} \\citet{2009A\\string&A...504..127F}; \\osref{rxn:Beuther2005} \\citet{2005ApJ...634L.185B}; \\osref{rxn:Sanna2016} \\citet{2016A\\string&A...596L...2S}; \\osref{rxn:Araya2008} \\citet{2008ApJS..178..330A}; ; \\osref{rxn:Sanna2018} \\citet{2019A\\string&A...623A..77S}; \\osref{rxn:Navarete2015}  \\citet{2015MNRAS.450.4364N}; \\osref{rxn:Zhang2007}  \\citet{2007A\\string&A...470..269Z}; \\osref{rxn:Araya2007}  \\citet{2007ApJ...669.1050A}; \\osref{rxn:Hofner2017} \\citet{2017ApJ...843...99H}; \\osref{rxn:Lopez_Sep2010}  \\citet{2010A\\string&A...517A..66L}; \\osref{rxn:Beuther2002bb}  \\citet{2002A\\string&A...383..892B}; \\osref{rxn:Su2007} \\citet{2007ApJ...671..571S}; \\osref{rxn:Shepherd2000}  \\citet{2000ApJ...535..833S}; \\osref{rxn:Hofner2007}  \\citet{2007A\\string&A...465..197H}; \\osref{rxn:Cesaroni1999} \\citet{1999A\\string&A...345..949C};  \\osref{rxn:Cesaroni2013} \\citet{2013A\\string&A...549A.146C}\n }\n\\end{deluxetable}\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{c}\n\n\\includegraphics[width=0.7\\textwidth,clip=true, trim = 10 240 10 80, clip, angle = 0]{18089}  \\\\\n\\includegraphics[width=0.68\\textwidth,clip=true, trim = 10 240 10 60, clip, angle = 0]{19411}   \n\\end{tabular}\n\\caption{\\small{\\emph{Spitzer} IRAC GLIMPSE three-color (3.6$\\mu m$-blue, 4.5$\\mu m$-green and 8.0$\\mu m$-red )\nimages of two ionized jet candidates, overlayed with VLA 6 cm continuum emission contours. \nNote that both regions show 4.5 $\\mu$m excess emission. In the right panel we show\nan enlarged version of the radio continuum from \\citetalias{2016ApJS..227...25R}.\n{\\bf Top: 18089-1732 A:}  The arrows represent the direction of a the north-south bipolar SiO outflow detected by  \\citet{2004ApJ...616L..23B, 2010ApJ...724L.113B}. The black circle and the square are the 6.7 GHz CH$_{3}$OH and H$_{2}$O masers reported in \\citep{2002A&A...390..289B}, respectively.\nVLA 6 cm contour levels are ($-$2.0, 3.0, 10.0, 25.0, 40.0) $\\times $6 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$1.5, 3.0, 5.0, 7.5, 15.0, 25.0, 45.0, 95.0) $\\times$  10 $\\mu$Jy beam$^{-1}$.\n{\\bf Bottom: 19411+2306 A:} The arrows represent the direction of the detected CO  outflow by \\citet{2002A&A...383..892B}. \\citet{2002ApJ...566..931S} reported that 6.7 GHz CH$_{3}$OH and H$_{2}$O\n masers were not detected for this source in their survey.  VLA 6 cm contour levels are ($-$2.0, 2.0, 3.0, 6.0, 10.0, 13.0, 15.0) $\\times$ 5.5 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 2.0, 3.0, 6.0, 8.0, 10.0, 12.0) $\\times$  8 $\\mu$Jy beam$^{-1}$.  }}\n \\label{fig:spitzer_examples}\n\\end{figure}\n\n\nAs listed in Table~\\ref{jet_cand_list}, at least 5 of the ionized jet candidates are aligned in the same direction as a large scale molecular outflow (see Figures \\ref{fig:spitzer_examples} for examples). The other \nionized jet candidates appear to be associated with molecular outflows where the directions are approximately perpendicular. In Figure \\ref{fig:UKIDSS_examples} we present the examples of 18151$-$1208 B and 19035+0641 \nA where we show VLA~6$\\,$cm continuum emission contours overlayed on a UKIDSS\\footnote{United Kingdom Infrared Telescope (UKIRT) Infrared Deep Sky Survey (UKIDSS) Galactic Plane Survey \\citep{2007MNRAS.379.1599L}.} {\\emph K}-band (2.2 $\\mu m$) image. It is interesting to note that the putative ionized jets and the \nUKIDSS {\\emph K}-band emission in both cases are elongated in the same direction. This together with the fact that the ionized jets are located nearly at the peak of the UKIDSS {\\emph K}-band emission could indicate \nthat the latter is tracing scattered light from the central protostar that is escaping from an outflow cavity \\citep{2013ApJS..208...23L}. The observed misalignment between cm continuum emission and  the dominating molecular flow in the region could be explained by the existence of two flows, where the molecular outflow associated\nwith the jet is weaker, and hence undetected. This could in fact be the case for 18151$-\n$1208 B, where a blue-shifted component of a CO molecular outflow observed with the Submillimeter Array appears to be aligned in the direction of the ionized jet\n(see Figure 4 of \\citealt{2011ApJ...729...66F}), although this outflow component is not discussed by the authors. Another possible explanation for the misalignment in the directions of the ionized jet and the molecular outflow is that they are subjected to precession, where \nthe flow axis changes from the small to the large scale as suggested by e.g., \\citet{2000ApJ...535..833S} and \\citet{2005A&A...434.1039C} for the case of 20126$+$4104,  \\citet{2013A&A...558A.145M} to explain the \ncase of  18182$-$1433 and \\citet{2007ApJ...669.1050A} for 18566$+$0408. \n\nFor a further test of their jet nature, we have also attempted to estimate the deconvolved sizes of the central jet components using the CASA task  {\\tt imfit}. This was possible for 3 sources within the subsample of jet candidates listed in\nTable~\\ref{jet_cand_list}. Figure \\ref{fig:size_freq} shows the deconvolved major axis as a function of frequency for 18151$-$1208 B and 18440$-$0148 A (the case of 18566$+$0408 B is reported in \\citealt{2017ApJ...843...99H}). Within the uncertainties these radio sources follow the relation $\\theta_{maj} \\propto \\nu^{\\gamma}$, where a major axis index of $\\gamma = -0.7$ is expected for a biconical ionized wind or jet \\citep{1986ApJ...304..713R}. Therefore, at least in these 3 cases, we have further evidence for the jet nature of these specific radio sources. \n\nIn summary, for the subsample of elongated continuum sources listed in Table~\\ref{jet_cand_list} it is very likely that the nature of these sources are ionized jets at the base of a molecular outflow.\n\n\n\n\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{c}\n\\includegraphics[width=0.7\\textwidth,clip=true, trim = 10 200 10 80, clip, angle = 0]{18151} \\\\\n\\includegraphics[width=0.68\\textwidth, clip=true, trim = 10 150 10 80, clip, angle = 0]{19035}  \n\\end{tabular}\n\\caption{\\small{UKIDSS {\\emph K}-band images of two ionized jet candidates, overlayed with VLA 6 cm continuum emission contours. In the right panel we show\nan enlarged version of the radio continuum from \\citetalias{2016ApJS..227...25R}. \n{\\bf Top: 18151$-$1208 B:} The arrows represent the direction of the two nearly perpendicular  CO  outflows detected by \\citet{2011ApJ...729...66F}. A blue-shifted component of a molecular outflow going in the direction\nof 18151$-$1208 B is seen in Figure 4 of \\citet{2011ApJ...729...66F} but it is not discussed by the authors. The black circle  is the 6.7 GHz CH$_{3}$OH  maser from \\citet{2002A&A...390..289B}. The x symbol represents the position of an \nadditional radio source detected at 1.3 cm reported in \\citet{2016ApJS..227...25R}. VLA 6 cm contour levels are ($-$2.0, 3.0, 9.0, 15.0) $\\times$ 6 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 3.0, 6.0, 8.5, 20.0, 40.0, 60.0)   $\\times$  8 $\\mu$Jy beam$^{-1}$.\n{\\bf Bottom: 19035+0641 A:} The arrows represent the direction of the detected CO and HCO$^{+}$ outflows \\citep{2002A&A...383..892B, 2010A&A...517A..66L}. The black circle  and the square are the 6.7 GHz CH$_{3}$OH   and  \nH$_{2}$O masers from \\citet{2002A&A...390..289B}, respectively. VLA 6 cm contour levels are ($-$2.5, 3.0, 10.0, 25.0, 120.0, 280.0, 380.0) $\\times$  4 $\\mu$Jy beam$^{-1}$, and 1.3 cm contour levels ($-$2.0, 5.0, 10.0, 20.0, 30.0, 50.0, 90.0, 170.0) $\\times$  8 $\\mu$Jy beam$^{-1}$.}}\n \\label{fig:UKIDSS_examples}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}[!h]\n\\centering\n\\begin{tabular}{cc}\n\n\\includegraphics[width=0.34\\textwidth,  clip, angle = 0]{18151alpha_size_freq}  &\n\\includegraphics[width=0.34\\textwidth,  clip, angle = 0]{18440alpha_size_freq}  \\\\\n \\vspace{-1.cm} \n\\end{tabular}\n\\caption{\\small{Deconvolved major angular axis as a function of frequency for the ionized jet candidates 18151$-$1208 B, 18440$-$0148 A and 18566$+$0408 B. The arrows represent the size limit value from the synthesized beam of the map at the given frequency. The dashed line is the power law fit of the form $\\theta_{maj} \\propto \\nu^{\\gamma}$.}}\n \\label{fig:size_freq}\n\\end{figure}\n\n\n\\defcitealias{2011MNRAS.415..893A}{AMI Consortium: Scaife et al. (2011}\n\\defcitealias{2012MNRAS.420.1019A}{2012)}\n\nAs mentioned above, most of our detected radio sources with a rising spectrum are compact, i.e., spatially unresolved, or marginally resolved. Several of these sources are associated with molecular \noutflows, and 6.7 GHz CH$_{3}$OH  and 22 GHz H$_{2}$O masers as found in the literature. More precisely, only   6 of the 25 regions where we detected a radio source with a rising spectrum, are not \nassociated with molecular outflows, or adequate data that would trace such outflows do not seem to exist.\nFurthermore, after taking into account the shape of the synthesized beam of our VLA observations  some of \nthese radio sources appear slightly elongated in a certain direction. Examples are 18264$-$1152 F and G53.25$+$00.04mm2 A (see \\citetalias{2016ApJS..227...25R}, Figure~2). Therefore, we now investigate the \npossibility that  the compact radio continuum sources with a rising spectrum represent ionized jets. \n\n\\begin{figure}[h]\n\\centering\n\\begin{tabular}{ccc}\n\\hspace*{\\fill}%\n\\includegraphics[width=0.5\\textwidth, clip=true, angle = 0]{Rad_LumVSBol_Lum_paper}\n\\end{tabular}\n\\caption{\\small{Radio luminosity at 4.9 GHz  as a function of the bolometric luminosity. The red stars and octagons are our ionized jet and jet candidates listed in Table \\ref{jet_cand_list} and \\ref{candidates}  towards HMCs and CMC-IRs, respectively.  The bolometric luminosity for the red symbols is mainly estimated from our {\\it Herschel}\/Hi--GAL data (except for the open symbols whose bolometric luminosity information is from the literature).  The green circles represent ionized jets associated with low-mass protostars  ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$)   from \\citet{2018A&ARv..26....3A} and the yellow circles are the very low luminosity objects (VeLLOs) and  low-mass protostars from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe purple triangles represent ionized jets from high-mass stars as found in the literature,\nfrom \\citet{2008AJ....135.2370R} and \\citet{2016A&A...585A..71M}. The  $\\times$ symbols are UC and HC H{\\small II} regions from \\citet{1994ApJS...91..659K}. \nThe dashed line relation shows the positive correlation found by \\citet{2015aska.confE.121A} derived for jets from low-mass stars. The red dotted line is our best fit to the data including ionized jets from low, intermediate and high-mass YSOs, but excluding the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty.\n}}\n \\label{fig:rad_bol_lum}\n\\end{figure}\n\n\nA  statistical way of investigating the nature of our compact sources is to study the energy contained in the ionized gas. In Figure \\ref{fig:rad_bol_lum} we show the \nradio luminosity S$_\\nu\\,$d$^2$ of all the components of the ionized jet (or the jet candidate) as a function of the bolometric luminosity of the region. As in Figure \\ref{Lyman_cont_plot} above, the black line is the radio luminosity expected from the Lyman continuum\nflux at a given bolometric luminosity if it arises from photoionization of a single ZAMS star. In addition to the compact, rising spectra sources from our survey we also show in Figure \\ref{fig:rad_bol_lum} as green circles\nthe radio luminosity from low-mass stars ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$) associated with ionized jets from \\citet{2018A&ARv..26....3A} and as yellow circles the radio luminosity of very low luminosity objects (VeLLOs) and  low-mass protostars detected at 1.8 cm, and reported in \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}. In order to compare the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} and \\citet{2018A&ARv..26....3A} with our $4.9\\,$GHz data, we scaled their\nfluxes using a factor of 0.48 and 0.76, respectively, assuming that those sources have a spectral index $\\alpha=0.6$, which is the canonical value of ionized jets. \n The scaling factors are calculated using \n$\\frac{S_{\\lambda_{1}}}{S_{\\lambda_{2}}}= \\left(\\frac{\\lambda_{2}}{\\lambda_{1}} \\right)^{\\alpha}$. \n\nIt is well known that for low mass YSOs the radio luminosities are correlated with the bolometric luminosity, and we show the correlation\n$\\frac{S_{\\nu}d^{2}}{\\text{mJy kpc}^{2}}= 8.7 \\times 10^{-3} \\left( \\frac{L_{\\text{bol}}}{L_{\\odot}} \\right)^{0.54}$ first found by \\citet{1995RMxAC...1...67A} and recently updated by \\citet{2018A&ARv..26....3A}. The black dashed line is the best fit to the green circles in Figure \\ref{fig:rad_bol_lum}, which are  the low-mass ionized jets presented by \\citet{2018A&ARv..26....3A}.\nIt is clear from Figure \\ref{fig:rad_bol_lum} that the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} also follow this relation, although their data were observed at low resolution ($\\sim$ 30\\rlap.$^{\\prime \\prime}$) and there is not enough information that proves that they correspond to ionized jets. \n\\citet{1995RMxAC...1...67A} used this observed correlation to explain the apparent excess ionization levels from low mass YSOs by shock induced ionization from jets,\nas modeled by \\citet*{1987RMxAA..14..595C} and \\citet{1989ApL&C..27..299C}. \nA handful of detections of ionized jets towards high-mass stars  in recent years suggested that this correlation appears to also hold for stars with luminosities up to $\\sim$ 10$^{5}$ L$_{\\odot}$ (see \n\\citealt{2016MNRAS.460.1039P}). We have added these objects from \\citet{2008AJ....135.2370R} and  \\citet{2016A&A...585A..71M} as purple triangles in Figure \\ref{fig:rad_bol_lum} and the data has been properly scaled to our frequency of 4.9~GHz assuming $\\alpha=0.6$. The data from our survey (\\citetalias{2016ApJS..227...25R}) in conjunction with improved estimates of the \nluminosities based on Herschel data (see \\S \\ref{app:lum}) allow us to further populate this plot and test if a correlation exists. In Figure \\ref{fig:rad_bol_lum} the red stars and octagons are our radio sources with rising spectrum detected\ntowards HMCs and CMC-IRs, respectively, and we see that most sources are located very close to the relation found by \\citet{1995RMxAC...1...67A} up to luminosities of $\\sim$ 10$^{5}$ L$_{\\odot}$. In fact, a fit of the data \nincluding low, intermediate and high-mass YSOs is shown as a red dotted line and the result is similar to what was found by  \\citet{1995RMxAC...1...67A}. We excluded the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A} from our fit since it is unclear if those source are indeed  ionized jets.\n We take this result as a strong indication that the weak, and compact radio sources which we found in our survey are caused by the same mechanism which causes the radio emission the low mass YSOs, namely it is caused by ionized jets.  We also note that of the 6 compact radio sources where currently no observational association with molecular flows is known,\n 5 match our fit (red dotted line in Fig 8) of the $S_\\nu\\,d^2$ vs $L_{bol}$ relationship.\n \n \n\\defcitealias{2011MNRAS.415..893A}{AMI Consortium: Scaife et al. (2011}\n\\defcitealias{2012MNRAS.420.1019A}{2012)} \n \nFurthermore, in Figure \\ref{fig:Curiel_plot} we show the momentum rate ($\\dot{P}$) of the molecular outflow as a function of the radio luminosity (S$_\\nu\\,$d$^2$) of the ionized jet estimated from our flux values at 4.9 GHz (symbols and colors are the same as  in Figure  \\ref{fig:rad_bol_lum}). The momentum rate of the molecular outflows comes from information from the literature for our ionized jets (and jet candidates), if available, and the values are in most cases from single dish observations.  For consistency, we have scaled the physical values, so that they are based on the same distance. However,\nmany uncertainties remain due to the inhomogeneity of the data set. In particular, the values for the momentum rate come from observations taken\nby different authors, using different spectral lines, as well as different telescopes. Hence, the large scatter in Figure \\ref{fig:Curiel_plot} is not unexpected,\nand the creation of a homogenous data set for the $\\dot{P}$ versus $S_\\nu\\,d^2$ relation will be an important future task.\n\nIn spite of the large scatter, the correlation seen in  Figure \\ref{fig:Curiel_plot} indicates that the more radio luminous the protostar are the more powerful they are in pushing outflowing material. This correlation, which has been studied by several authors (e.g., \\citealt{1995RMxAC...1...67A}, \\citealt{2008AJ....135.2370R}, \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}, \\citealt{2018A&ARv..26....3A}), follows the shocked-induced ionization model introduced by \\citet{1987RMxAA..14..595C, 1989ApL&C..27..299C}, suggesting that the ionization of thermal jets is due to shocks. The  shocked-induced ionization model implies $\\left(\\frac{S_{\\nu}d^{2}}{mJy\\,kpc^{2}}\\right) = 10^{3.5} \\eta \\left(\\frac{\\dot{P}}{M_{\\odot}\\,yr^{-1}\\,km\\,s^{-1}}\\right)$ at $\\nu=5$ GHz where $\\eta$ is the shock efficiency fraction or the fraction of material that gets ionized by the shocks, which for low-mass protostars has been observationally found to be around 10$\\%$ (or $\\eta=0.1$).  \\citet{2018A&ARv..26....3A} suggested that the ionization fraction of jets in general is low ($\\sim 1 - 10\\%$). With the current data, and due to the large scatter seen in the correlation of  $\\dot{P}$ vs S$_\\nu\\,$d$^2$, we cannot yet properly quantify how the efficiency fraction changes with the luminosity of the protostar (e.g., if the ionization in thermal jets associated with high-mass protostars is higher than for low-mass protostars). Therefore, a uniform survey to measure the momentum rate of the molecular outflows associated with  ionized jets (ideally with  comparable resolutions) will be fundamental to further constrain this model. \\citet{2018A&ARv..26....3A} discussed both  correlations shown in Figure~\\ref{fig:rad_bol_lum} and Figure~\\ref{fig:Curiel_plot}  in great detail  and they interpreted them as an indication that the mechanism of ionization, accretion and ejection of outflows associated with protostars do not depend on their luminosities.\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth, clip=true, angle = 0]{Momentum_Rate_Survey}\n\\caption{\\small{Momentum rate of the molecular outflow  as a function of the radio luminosity at 4.9 GHz. The red stars and octagons are our ionized jets and jet candidates listed in Table \\ref{jet_cand_list} and \\ref{candidates}  towards HMCs and CMC-IRs, respectively, symbols for which there is information of the momentum rate.  The momentum rate values of the molecular outflow for all the sources including our data are collected from the literature.  The green circles represent ionized jets associated with low-mass protostars  ($1\\,L_{\\odot} \\leq L_{bol} \\leq 1000\\,L_{\\odot}$)   from \\citet{2018A&ARv..26....3A}  and the yellow circles are the very low luminosity objects (VeLLOs) and  low-mass protostars from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}.\nThe purple triangles represent ionized jets from high-mass stars as found in the literature,\nfrom \\citet{2008AJ....135.2370R} and \\citet{2016A&A...585A..71M}. The  $\\times$ symbols are UC and HC H{\\small II} regions from \\citet{1994ApJS...91..659K}. \nThe dashed line relation shows the positive correlation found by \\citet{1995RMxAC...1...67A} derived for jets from low-mass stars. The red dotted line is our best fit to the data including ionized jets from low, intermediate and high-mass YSOs, but excluding the sources from \\citetalias{2011MNRAS.415..893A, 2012MNRAS.420.1019A}. The gray shaded area corresponds to the momentum rate as predicted by the shock ionization model from \\citet*{1987RMxAA..14..595C} for values of the shock efficiency fraction of $\\eta = 0.1$ and $\\eta = 1.0$. \n The error bar in the middle right  corresponds to a 20$\\%$ calibration uncertainty. The error in $\\dot{P}$  is not represented in the figure because it \nvaries widely, and depends strongly on how different authors have gathered the data.\n}}\n \\label{fig:Curiel_plot}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{Discussion}\\label{discussion_paperII}\nResults reported in \\citetalias{2016ApJS..227...25R} of detection rates  of CMC (6$\\%$), CMC-IR (53$\\%$) and HMCs (100$\\%$) provide further evidence for an evolutionary sequence in the formation of high-mass stars, from a very early stage type of cores (i.e., CMCs) to relatively more evolved ones (i.e., HMCs). The fraction of centimeter wavelength sources detected towards HMCs is  higher  than previously expected towards this type of cores and suggests that radio continuum may be detectable at weak levels in all HMCs. The lack of radio detections for some objects in the sample (including most CMCs) provides interesting constraints and are ideal follow up candidates for studies of the earliest stages of high-mass stars.\nIt is important to note that it is likely that the ionized material from  jets or  HC H{\\small II} regions associated with these type of cores remains undetected at our sensitivity, thus in order to rule out these regions as pre-stellar cores deeper observations are required or alternative tracers for ongoing star formation in these cores need to be identified.\n\nHere we consider some constraints on the nature of the  centimeter continuum emission detected in these cores and clumps towards high-mass star forming regions. As described in \\S \\ref{yso}, most of our radio detections arise from high-mass YSOs and at least for 7 regions some of the radio detections could potentially arise from (solar-like mass) T-Tauri stars. Also, we detected at least 10 radio sources associated with the mm cores\/clumps with a flat spectral index, most of them resolved sources, which are most likely UC H{\\small II} regions.  Understanding the nature of the rising spectral index sources has proven to be more challenging. These compact radio sources appear to be well fitted (within the uncertainties) when using either a homogeneous H{\\small II} region or a power-law fit  (as shown in Figure \\ref{HII_fit_examples}).  \nTherefore, we will discuss two plausible scenarios, that the radio sources are either  UC\/HC H{\\small II} regions or that the emission arises from shock ionized jets. \n\n\n\\subsection{H{\\small II} regions}\n\nFor the first scenario, when fitting the sources in terms of a homogeneous H{\\small II} region, the solutions required a significantly smaller size (several of them an order of magnitude smaller) for the H{\\small II} region than the upper limit given by the FWHM synthesized beam. However, since these calculations assumed a pure hydrogen nebula, we must consider whether internal dust absorption can make the regions as small as the H{\\small II} region model is predicting.   \\citet{1989ApJS...69..831W} suggested that, even if the dust absorbs 90$\\%$ of the UV photons, the radius of the H{\\small II} region is reduced by only a factor of 0.46. Thus, dust absorption alone appears insufficient to explain the small region sizes predicted by the H{\\small II} region model used to fit our data. As shown in Figure \\ref{Stromgren_sphere}, these sources could be explained as turbulence-pressured confined  H{\\small II} regions if they are born in a clump with density of n$_{H_{2}}=$ 10$^{7}$\\,cm$^{-3}$ and assuming  velocity dispersions of $\\sigma \\sim$ 3--8 km~s$^{-1}$ (FWHM$\\sim$7--20 km~s$^{-1}$). However, it is not clear if high velocity dispersions are common towards  the dense clumps harboring high-mass stars, since   $\\sigma \\sim$ 2 km s$^{-1}$ seems to be more typical. Measuring the line width of an optically thin tracer on $\\sim$100 au scale would provide a decisive constraint on the velocity dispersion. We also found that the sources could be consistent with having been born in a denser environment of $n_{H_{2}}\\approx$ 10$^{8}$\\,cm$^{-3}$. Arguably, \\citetalias{2016ApJ...818...52T}  predicts  that such a density for the ionized region is already too high for a protostar of  8 M$_{\\odot}$ to 24 M$_{\\odot}$.  Recently,  \\citet{2016A&A...585A..71M} detected  compact radio sources towards high-mass YSOs with similar physical characteristics to the ones found in this survey, where the Lyman continuum derived from the bolometric luminosities always exceeds the one obtained from the radio luminosities (as seen in Figure \\ref{fig:rad_bol_lum}). From their analysis they conclude that those sources cannot be HC or UC H{\\small II} regions, unless the ionized gas has a density gradient (e.g., model IV of \\citealt{1975A&A....39..217O}). \n\n\nAdditionally, for the extended UC H{\\small II} regions detected at the outskirts of the mm cores there are two scenarios: either they were born in a low density clump of $n_{H_{2}}\\approx$ 10$^{5}$\\,cm$^{-3}$ or  they were born at a higher density and have migrated out of the center potential.  The latter scenario requires large stellar dispersion velocities ($\\gtrsim$10 km~s$^{-1}$), which are not typical unless the source is a runaway OB star that has been dynamically ejected.   Observed stellar dispersion velocities, for instance for Orion's brightest  population is  only $\\lesssim$3 km~s$^{-1}$  \\citep[e.g.,][]{2005AJ....129..363S,2009ApJ...697.1103T}, which makes the former scenario more plausible. However, \\citet{2007ApJ...660.1296F} predicted that stellar velocities up to $\\lesssim$13 km~s$^{-1}$ are likely for core densities of 10$^{7}$\\,cm$^{-3}$, and  that these high stellar velocities carry the star to lower density regions of the core\/clump, where the H{\\small II} region is free to expand. We are leaning to favor the scenario  where the sources have migrated since it allows \nto explain the occurrence of both compact and extended emission in the same protocluster (e.g., \\citealt{2007prpl.conf..181H}). \n\\newpage         \n\\subsection{Radio Jets}\n         \nNow we discuss the second scenario where the radio emission of the radio compact sources with rising spectrum is due to shock ionization. \nThe observable properties of several of our radio detections indicate that they likely have a jet nature: one can argue that the low centimeter emission from the majority of the sources detected in this survey, their free-free\nspectral index being in the range $0.2 \\leq \\alpha \\leq 1.8$  and their association with molecular outflows indicate that even those sources without an elongated radio morphology are also ionized jets or stellar winds that are conical, accelerating and\/or recombining.  From the analysis in \\S\\ref{compact_sources_sect} and \\S \\ref{ionized_jet_sect} we inferred that from the 44 sources with rising spectral index, approximately 12 of them are ionized jets (see Table \\ref{jet_cand_list}) and 13 are jet\/wind candidates (see Table \\ref{candidates}). \nIn fact, half of the jet candidates in Table \\ref{candidates} have a spectral index \n$\\alpha \\approx 0.6$ and all but two of them (UYSO1 A and 18521$+$0134 A) have a spectral index $\\alpha \\leq 1.0$, which is consistent with the expected value of a spherical, isothermal and constant velocity ionized wind \\citep[e.g.,][]{1975A&A....39....1P}. As stated before, the deviation from the value $\\alpha = 0.6$ could be due to acceleration or recombination within the flow. \n\n\n\n\\begin{deluxetable}{l c c c c}\n\\tabletypesize{\\scriptsize}\n \\renewcommand*{\\arraystretch}{1.5}\n\\tablecaption{Ionized Jet\/Wind Candidates  \\label{candidates}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Region}                  & \n\\colhead{Radio Source}        &\n\\colhead{Outflow Direction}   &\n\\colhead{ H$_{2}-$Jet Direction}   &\n \\colhead{Reference}      \\\\}\n\\startdata\n\\setcounter{iso}{0}\t\nUYSO1                    &    A   &   NW$-$SE    &   \\nodata  & \\rxn \\label{rxn:Forbrich04}   \\\\\n18264$-$1152         &    F  &   NW$-$SE     &  E$-$W  & \\rxn \\label{rxn:Sanchez-Monge2013} \\rxn \\label{rxn:Navarete2015}    \\\\\n18345$-$0641        &    A   &    NW$-$SE    &   very weak &\\rxn \\label{rxn:Beuther2002} \\rxn \\label{rxn:Varricat2013}  \\rxn \\label{rxn:Varricat2010}  \\\\\n18470$-$0044        &    B   &   E$-$W          &   no\/very weak\\tablenotemark{a}      &  \\osref{rxn:Beuther2002}   \\\\\n18517$+$0437       &    A    &   N$-$S          &   very weak   & \\rxn \\label{rxn:Lopez_Sep10} \\osref{rxn:Varricat2010} \\\\\n18521$+$0134       &    A   &    \\nodata\\tablenotemark{b}     &     non-detection       &   \\rxn \\label{rxn:Cooper13}   \\\\\n G35.39$-$00.33mm2        &    A   &  \\nodata            &  \\nodata   &    \\nodata   \\\\\n18553$+$0414       &    A   &    \\nodata\\tablenotemark{c}     &   non-detection         &    \\osref{rxn:Navarete2015}   \\\\\n19012$+$0536       &    A   &    NE$-$SW      &   non-detection   &   \\osref{rxn:Beuther2002} \\osref{rxn:Navarete2015}   \\\\\nG53.25$+$00.04mm2        &    A   & \\nodata    &     \\nodata       &    \\nodata   \\\\\n 19413$+$2332      &    A   &      \\nodata\\tablenotemark{d}     &     \\nodata      &   \\osref{rxn:Beuther2002}  \\\\\n20293$+$3952      &    E\\tablenotemark{e}    &     NE$-$SW    &     detection    &  \\rxn \\label{rxn:Beuther04} \\rxn \\label{rxn:Palau07_a} \\osref{rxn:Varricat2010} \\\\        \n 20343$+$4129      &    B   &     E$-$W      &   non-detection     &  \\rxn \\label{rxn:Palau07}  \\osref{rxn:Cooper13}\\\\        \n\\enddata\n\\tablenotetext{\\text{a}}{ T. Stanke and H. Beuther (private communication).}\n\\tablenotetext{\\text{b}}{\\citet{2002ApJ...566..931S} reports non-detection of CO (2--1) wings towards this region, although an outflow  could be present at an inclination angle of $< 10^{\\circ}$ to the plane of the sky.}\n\\tablenotetext{\\text{c}}{\\citet{2002ApJ...566..931S} reports the presence of CO (2--1) wings towards this region, but contour maps of the molecular outflow are not available.}\n\\tablenotetext{\\text{d}}{CO outflow is detected in the region, but the data does not show a clear bipolar structure.}\n\n\\tablenotetext{\\text{e}}{Radio source E has an upper limit value for the flux at 4.9 GHz and its value is not included in Figures \\ref{fig:rad_bol_lum} and \\ref{fig:Tanaka_tracks} (right panel).}\n\\tablecomments{The dots indicates that there is not enough information available about observations of the molecular outflow in the literature.\\\\\n\\osref{rxn:Forbrich04} \\citet{2004ApJ...602..843F}; \\osref{rxn:Sanchez-Monge2013}  \\citet{2013A\\string&A...557A..94S};  \\osref{rxn:Navarete2015}  \\citet{2015MNRAS.450.4364N}; \\osref{rxn:Beuther2002}  \\citet{2002A\\string&A...383..892B};  \\osref{rxn:Varricat2013}  \\citet{2013A\\string&A...554A...9V}; \\osref{rxn:Varricat2010}  \\citet{2010MNRAS.404..661V}; \\osref{rxn:Lopez_Sep10} \\citet{2010A\\string&A...517A..66L}; \\osref{rxn:Cooper13} \\citet{2013MNRAS.430.1125C}, \\osref{rxn:Beuther04} \\citet{2004ApJ...608..330B}; \\osref{rxn:Palau07_a} \\citet{2007A\\string&A...465..219P}; \\osref{rxn:Palau07} \\citet{2007A\\string&A...474..911P}.}\n\n\\end{deluxetable}\n\n\n\\subsection{H{\\small II} regions vs Radio Jets}\n\nIn Figure \\ref{fig:rad_bol_lum} we compared the radio luminosity with the bolometric luminosity using the radio flux at 4.9 GHz.  When fitting the ionized jets (and jet candidates) from low, intermediate and high-mass protostars using a power-law (represented by the dotted red line) we find an index of 0.63$\\pm$0.04 with a correlation coefficient  of $r=$0.89 which yields the relation $S_{\\nu}d^{2}$ [mJy kpc$^{-2}$]=$ 6.5 \\times 10^{-3}$ (L$_{bol}$\/L$_{\\odot}$)$^{0.63}$.  This result is comparable with the index found by \\citet{2016MNRAS.460.1039P} of 0.64$\\pm$0.04 for jets spanning luminosities from $\\sim 10^{-1}$ to $10^{5}$ L$_{\\odot}$, although their fit has a lower correlation coefficient ($r=$0.73). Their estimates for bolometric luminosities, which include {\\it Herschel}\/Hi--GAL data for most of their sources, are similar to ours. Therefore, the scatter in their data may come from the radio fluxes. \\citet{2016MNRAS.460.1039P} have stated that some of their jets have high flux densities probably because those objects represent a transition  between a jet and H{\\small II} region stages. \n Further, it is important to note that a similar relation between the bolometric luminosity and the luminosity of shocked H$_2$ emission from molecular jets\nhas been reported by \\citet{2015A&A...573A..82C} for sources with a wide range of bolometric luminosities. These studies together with our\nrefined relation point to a common flow mechanism from YSOs of any luminosity.\n\nUntil very recently, the stellar evolutionary models that have been used to analyze this type of sources  correspond to more evolved objects (i.e., ZAMS star). However, the recently introduced \\citetalias{2016ApJ...818...52T} model predicts the ionizing luminosity of a protostar which will allow us to compare our data with a more appropriate part of the evolutionary track.  These evolutionary stellar models mainly depend on the accretion history, this is the mass of the core (M$_{c}$) and the mass surface density of the ambient clump ($\\Sigma_{cl}$). Figure \\ref{fig:Tanaka_tracks} shows the same relations as those in Figs. \\ref{Lyman_cont_plot} and \\ref{fig:rad_bol_lum}, but now we also consider an evolutionary track for a YSO which is represented by the cyan continuous line for an initial core mass of M$_{c}=$ 60 M$_{\\odot}$ and a mass surface density of ambient clump of $\\Sigma_{cl}=$~1~g~cm$^{-2}$ (\\citetalias{2016ApJ...818...52T} fiducial case). This cyan track shows the evolutionary sequence of the ionizing photon luminosity as a function of the protostellar luminosity. Its shape shows each of the physical stages in the evolution of the protostar: accretion stage, swelling stage (as seen with the decrease in the ionizing luminosity as the temperature decreases), contraction stage (increase of the ionizing luminosity as the temperature also increases) and nuclear burning stage when the protostar reaches the ZAMS (represented by the black continuous line; for more discussion on this evolutionary track see \\citealt{2014ApJ...788..166Z}; \\citetalias{2016ApJ...818...52T}).  The left panel of Figure \\ref{fig:Tanaka_tracks} shows that\n for the majority of the radio sources detected  towards CMCs and CMC--IRs, the  Lyman continuum excess (for the fiducial case L$_{bol} \\sim$10$^{2}$--10$^{3}$) is still evident and it is not likely due to photoionization. Additionally, the evolutionary  track for a YSO shows how the ionizing luminosity decreases as the protostar swells while accreting its mass and before it enters the Kelvin-Helmholz contraction (for the fiducial case L$_{bol} \\sim$10$^{3}$--10$^{4}$). This further indicates that the measured radio flux for most of our radio sources detected towards HMCs are also very unlikely to be photoionized by the central object.\n \nModel calculations presented by \\citet{2002ApJ...568..754K,2003ApJ...599.1196K,2007ApJ...666..976K}  predict that high accretion rates on the order of  $10^{-4} - 10^{-3}$ M$_{\\odot}$yr$^{-1}$ can choke off the H{\\small II} region to very small sizes producing very low radio continuum; see also Section 5 of \\citet{1995RMxAC...1..137W}. This might be a possible scenario for some of our more compact sources, but additional evidence is necessary such as high-resolution mm observations of infall tracers to determine mass infall rates for these sources. Based on the analysis and discussion presented above, we are inclined to favor the scenario that most of our compact sources (see Table \\ref{candidates}) with rising spectrum are ionized jets.  However, the confirmation of these radio sources as shocked ionized gas requires further observational and theoretical work. Additional  observations and tests are necessary in  order to have conclusive information of the nature of these detections. Higher resolution data ($\\lesssim$ 0\\rlap.$^{\\prime \\prime}$1) of the radio continuum is required to resolve the ionized jets and estimate their degree of  collimation. Additionally, high resolution millimeter data will help us to disentangle multiple outflows, possibly being driven by protostellar clusters as expected toward high-mass star forming regions and to study the  kinematics of the outflow material. Masers arise from the hot core regions and their  association with ionized material is very important. They indicate the evolutionary stage \\citep[e.g.,][]{2018A&A...619A.107S} of the exciting object and allow detailed studies of the kinematics at a smaller scale, very close to the powering high-mass YSO and the disk\/jet interface.   For resolved sources, long-term monitoring of the ionized jet is necessary in order to estimate proper motions,\nvelocities of the radio jets and evolution in the morphology of the jet. \n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{tabular}{cc}\n\\vspace{-0.7cm}\n\\includegraphics[width=0.46\\textwidth, clip=true, angle = 0]{Lyman_plot_Tanaka} &\\hspace{-1.6em} \n\\includegraphics[width=0.46\\textwidth,  clip=true, angle = 0]{Tanaka_M60}  \n\\end{tabular}\n\n\\caption{\\small{Lyman continuum (left) and radio luminosity (right) as a function of the bolometric luminosity. Symbols and colors are the same as used in Figures \\ref{Lyman_cont_plot} and \\ref{fig:rad_bol_lum}, except that now we also show the estimated Lyman continuum from the \\citetalias{2016ApJ...818...52T} model for an optically thin H{\\small II} region based on the ionization of a protostar (cyan continuous line). The stellar model evolution  starts with a  core mass of M$_{c}=$ 60 M$_{\\odot}$ and a mass surface density of ambient clump of $\\Sigma_{cl}=$~1~g~cm$^{-2}$. The black continuous line is the Lyman continuum from a ZAMS star.\n The error bars in the bottom right corner correspond to a 20$\\%$ calibration uncertainty. \n}}\n \\label{fig:Tanaka_tracks}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\section{Summary and Conclusions}\\label{conclusions_paperII}\nIn this work we investigate the nature of the 70  radio sources\nreported in \\citetalias{2016ApJS..227...25R}.  These radio sources were observed using the VLA at 6 and 1.3 cm towards a sample of high-mass star forming region candidates having either no previous radio continuum detection or a relatively weak detection at the 1 mJy level. We have explored several scenarios such as pressure confined H{\\small II} regions and ionized jets to explain the origin of the ionized gas emission and we have studied the physical properties of the detected sources. Based on our results we favor the scenario that $\\sim 30 - 50 \\%$ of our radio detections are ionized jets and\/or jet knots. These sources, listed in Tables \\ref{jet_cand_list} and \\ref{candidates}, have observational properties that are not expected towards regular H{\\small II} regions such as the correlation of their radio luminosity and bolometric luminosity and the correlation of the momentum rate of the molecular outflow with the radio luminosity of the ionized jet. Such correlations have been found observationally towards ionized jets associated with high-mass protostars of different luminosities and are also predicted in recent theoretical models such as the \\citetalias{2016ApJ...818...52T} model. However, for the most compact radio continuum detections we cannot rule out the scenario that they correspond to pressure confined H{\\small II} regions.  Our main results from this survey are summarized below: \n\n\\begin{itemize}\n\\item We detected centimeter wavelength sources in 100$\\%$ of our HMCs, which is a higher fraction than previously expected and suggests that radio continuum may be detectable at weak levels in all HMCs.  The lack of radio detections for some objects in the sample (including most CMCs) contributes  evidence that these clumps are in an earlier evolutionary stage than HMCs, providing interesting constraints and ideal follow up candidates for studies of the earliest stages of high-mass stars.\n\n\\item At least 10$\\%$ of our detected radio sources are consistent with non-thermal emission and likely due to either  active magnetospheres in T-Tauri stars (possibly for the few regions located at a distance $<$ 2 kpc) or synchrotron emission from fast shocks in disks or jets.\n\n\\item For the most compact radio detections, the sources are consistent with being small pressure confined H{\\small II} regions. Also, we cannot completely exclude the possibility that these sources are gravitationally trapped H{\\small II} regions. \n\n\\item The majority of our detected radio continuum sources ($\\sim$80$\\%$) have spectral indices ($-$0.1$<\\alpha<$2) that are consistent with thermal (free-free) emission from ionized gas. \n\n\\item Most of the radio sources with a rising spectrum detected towards clumps at an earlier evolutionary stage (i.e., CMCs and CMC--IRs) show Lyman continuum excess, consistent with previous results.  This can be explained either by  UV photons from shocks producing an ionized jet or shocks in an accretion flow onto the disk.\n\n\\item For most of the radio sources with a rising spectrum detected towards HMCs, the estimated Lyman continuum is lower than  expected if the radio flux comes from a single ZAMS star. This could indicate that the origin of the measured radio flux is not from HC\/UC H{\\small II} regions but  shock ionized jets.\n\n\\item We detected at least 12 ionized jets (6 of them are new detections to the\nknowledge of the authors) based on their spectral index, morphology and molecular outflow associations. For several of the previously detected jets, we detected additional knots or lobes that are part of the collimated structure. Additionally, we detected at least 13 jet\/wind candidates. \n\n\\item We found that ionized jets from low and high-mass stars are very well correlated. This is consistent with previous studies and is further evidence of a common origin for jets of any luminosity.\n\n\n\n\\end{itemize}\n\n\n\n\n\\acknowledgments\n We thank the anonymous referee, whose comments improved this manuscript.\nSupport for this work was provided by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc.\/National Radio Astronomy Observatory.  P. H. acknowledges support from NSF grant AST--1814011. C. C-G. acknowledges support from UNAM DGAPA--PAPIIT grant number IA102816, IN10818. E. D. A. is partially supported by NSF grant AST--1814063. We thank K. E. I. Tanaka for providing the stellar model evolutionary tracks predicted by  the \\citetalias{2016ApJ...818...52T} model. We thank K. E. I. Tanaka, K. Johnston and J. Marvil for  useful discussions. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. This work is based in part on observations made with the Spitzer Space telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. P.H. acknowledges support from NSF grant AST$-$0908901 for this project.  Some of the data reported here were obtained as part of the UKIRT Service Program. The United Kingdom Infrared Telescope is operated by the Joint Astronomy Centre on behalf of the UK Particle Physics and Astronomy Research Council.\nThis research made use of APLpy, an open-source plotting package for Python hosted at http:\/\/aplpy.github.com.\n\n\\vspace{5mm}\n\n\\software{CASA \\citep{2007ASPC..376..127M}, APLpy \\citep{2012ascl.soft08017R}.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction }\nThe Batalin--Vilkovisky formalism (BV--formalism) is the most\ngeneral method of quantization\nof gauge field theories~\\cite{bat1},~\\cite{bat2}.\nIn recent years the interest\nin studying  its geometrical nature has increased.\nIt was stimulated by  Witten's paper \\cite{witten},\nwhere the necessity of such an investigation is pointed out and\nparticularly for the\nformulation of the background independent open-string\nfield theory on the base of the BV-formalism.\nThe realization of this program  began in \\cite{witten1}, \\cite{witten2}.\n\nIt is known that the BV-formalism uses unusual structures --  odd\nPoisson brackets (antibrackets) and the operator $\\Delta$.\nAs one of the main obstacles to the construction of\na background independent open\nstring field theory,  indicated by Witten \\cite{witten},\n was the nonexistence  of an invariant\ndefinition of the operator $\\Delta$, and of a\nnaturally defined integral measure.\nHowever, such a definition of the operator $\\Delta$\nwas shown by one of us (O. K.) in\n\\cite{khud} before the cited Witten's paper  appeared (see also \\cite{km}).\nIts realization on K\\\"ahlerian supermanifolds  \\cite{ners} and its\nsimplest properties\n\\cite{knjinr} were considered.\n\nIn the present paper we study the operator $\\Delta$ in more detail.\nWe propose an invariant definition of this operator and show that\nthe condition of its nilpotency defines (in some sense) the choice of the\nintegration measure.\n\nIn Section 2 we propose an invariant definition of the operator $\\Delta$\n on supermanifolds, given by the odd symplectic structure\nand by the volume form as the divergence of the Hamiltonian vector field.\n We show that  all the relations  between the antibrackets and the operator\n$\\Delta$, which  are satisfied for canonical ones in the BV-formalism, are\nsatisfied for\nany generalized operator $\\Delta$  and the corresponding odd brackets.\nHowever\nthe nilpotency condition for such an operator holds only for\na certain class of the integral density.\n\nIn Section 3 we consider the realization of the operator $\\Delta $ on\nsupermanifolds, given by the odd (and even) K\\\"ahlerian structure,\nand show that it is\nnilpotent, if the integral density is the\ncharacteristic class of the basic K\\\"ahlerian manifold (the function from\nChern classes). Then it corresponds to the\ndivergence operator  $\\delta =* d *$\nof the basic manifold.\n\nIn Section 3 we discuss the geometrical nature of\nthe Bat\\-al\\-in-Vil\\-kov\\-isky formalism.\\\\\n\nWhen this paper was in preparation, we received two very important papers of\nA.S. Schwarz \\cite {schwarz}, \\cite{schwarz2}  where the geometry of\nthe BV-formalism is analyzed in detail and in particular the same definition\nof the operator $\\Delta$, as in \\cite{khud}, \\cite{ners}, \\cite{knjinr} and\nin the present paper is given.\n       \\setcounter{equation}0\n\\section{Odd Poisson Brackets and Operator $\\Delta$}\nThe  odd Poisson bracket (odd bracket, antibracket, Buttin bracket)\n  of the functions $f$ and $g$ on the supermanifold ${\\cal M}$ is\ndefined by the following  conditions \\cite{ber}, \\cite{leit} :\n\\begin{equation}\n\\begin{array}{l}\n\\{ f, ga +hb \\}_{1} = \\{ f, g\\}_{1}a + \\{ f, h \\}_{1}b,\n         \\quad {\\rm where}\\quad a,b =  const   \\\\[3mm]\np(\\{ f, g \\}_{1} )= p(f)+ p(g) + 1\n         \\quad {\\rm (grading \\ condition)}  \\label{eq:bgrad} \\\\[3mm]\n\\{ f, g \\}_1 = -(-1)^{(p(f)+1)(p(g)+1)}\\{ g, f \\}_1\n         \\quad {\\rm ( \"antisymmetricity\"\\ condition )} \\label{eq:anti} \\\\[3mm]\n\\{ f, gh \\}_1 =\\{ f, g \\}_{1}h +(-1)^{(p(f)+1)p(g)} g\\{ f, h \\}_1\n         \\quad {\\rm ( Leibnitz \\ rule )} \\label{eq:bLieb} \\\\[3mm]\n ( -1)^{(p(f)+1)(p(h)+1)}\\{ f,\\{ g, h \\}_{1}\\}_1\n                  +{\\rm {cycl. perm.}}( f,g,h) = 0\n         \\quad{\\rm {(Jacobi\\ id.)}}\n\\end{array}\n\\label{eq:bjac}\n\\end{equation}\n  Locally, the  odd bracket can be written as:\n\\begin{equation}\n\\{ f, g \\}_1 = \\frac{\\partial ^R f}{\\partial x^A} \\Omega^{AB}\n            \\frac{\\partial  ^L g}{\\partial x^B}\n\\label{eq:bloc}\n\\end{equation}\nwhere $\\Omega^{AB}$ satisfies to conditions\n\\begin{eqnarray}\n& & p(\\Omega^{AB} )= p(A)+ p(B) + 1\n\\quad  {\\rm (grading\\ condition)} \\nonumber  \\\\[3mm]\n& & \\Omega^{AB}  = -(-1)^{(p(A)+1)(p(B)+1)} \\Omega^{BA}\n\\quad {\\rm ( \"antisymmetricity\" \\ condition )} \\nonumber \\\\[3mm]\n& & ( -1)^{(p(A)+1)(p(C)+1)}\\frac{\\partial^{R}\n \\Omega^{AB}}{\\partial x^D }\\Omega^{DC}\n   +{\\rm {cycl.\\  perm.}}(A,B,C ) = 0\n  \\quad{\\rm {(Jacobi\\ id.)}}\n\\nonumber\n\\end{eqnarray}\nwhere\n$x^A$ are the local coordinates of ${\\cal M}$, $p_A \\equiv p( x^A ) $.\n$ \\frac{\\partial ^R }{\\partial x^A}$ and\n$\\frac{\\partial ^L}{\\partial x^A} $ denote\ncorresponding right and left derivatives.\nThey are connected with each other by:\n$$\\frac{\\partial ^R f}{\\partial x^A}=\n  (-1)^{(p(f)+1)p_A}\\frac{\\partial ^l f}{\\partial x^A}. $$\nIf ${\\cal M}$ has an equal number of\neven and odd coordinates, the odd bracket\ncan be nondegenerate. Then one can associate with it the\nodd symplectic structure\n\\begin{equation}\n\\Omega =dx^A \\Omega^{AB}dx^B   \\label{eq:symp}\n\\end{equation}\n where $\\Omega_{AB}\\Omega^{BC} =\\delta_{A}^{C}$.\nThis form is closed because of the Jacobi identities (\\ref{eq:bjac}).\nLocally, one can reduce  (\\ref{eq:symp}) to the canonical form \\cite{leit}:\n\\begin{equation}\n\\Omega^{\\rm can} =\\sum_{i=1}^{N} dx^i \\wedge d \\theta_i\n   \\label{eq:sympcan}\n\\end{equation}\nwhere $( x^i,\\theta _i) $ are some local coordinates (Darboux coordinates)\n($p( x^i )= 0, p(\\theta_i)= 1 $).\nThe corresponding odd bracket takes the form\n\\begin{equation}\n\t\t  \\{f,g\\}_1 =\n\t\t\t\\sum_{i=1}^{N}\\left(\n\t\t\\frac{\\partial^{R} f}{\\partial  x^i}\n\t\t\\frac{\\partial^{L} g}{\\partial\\theta_i}\n\t\t\t +\n              \\frac{\\partial^{R} f}{\\partial  \\theta_i}\n\t\t\\frac{\\partial^{L} g}{\\partial  x^i}\n\t\t   \\right)       \\label{eq:bcan}\n\\end{equation}\n Transformations preserving odd symplectic structures (or odd brackets)\n have (locally) the hamiltonian form:\n  \\begin{equation}\n{\\cal L}_{{\\bf V}}\\Omega = O\n\\quad {\\rm iff}\\quad {\\bf V}=\\{.,H\\}_1 \\equiv {\\bf D}_{H}     \\label{ham}\n\\end{equation}\n where $ H$ is an arbitrary function (Hamiltonian) on ${\\cal M}$,\n${\\cal L}_{{\\bf V}}$ denotes the Lie derivative\nalong the vector field {\\bf V}.\n\nIt is well known that any supermanifold  can be associated with some vector\n bundle \\cite{ber}. The odd symplectic structure\ncan be globally defined on the supermanifolds\n which are associated with the cotangent bundles of manifolds.\n\n Let $T^* M$ be the cotangent bundle of the manifold $M$.\n  $x^{i}$ are local coordinates on $M$\n  and $( x^{i}, v_{i} )$ are the corresponding\n  local coordinates on $T^* M$.\n From map to map\n  \\begin{equation}\nx^{i} \\rightarrow {\\tilde x^{i}} = {\\tilde x^{i}} ( x),\\quad\n      v_{i} \\rightarrow {\\tilde v_{i}} = \\sum_{i=1}^N\n  \\frac{\\partial x^{j}}{\\partial {\\tilde x^{i}}} v_{j}.\n     \\label{eq:trans}\n\\end{equation}\n Considering for every map the superalgebra generated by\n  $(x^{i}, \\theta _{i})$, where the $x^{i}$ are even and\nthe  $\\theta_{i}$ are odd coordinates,  transforming from map\n to map like $(x^{i}, v_{i})$ in ref{eq:trans})\n  ($v\\leftrightarrow \\theta$), we come to the\n supermanifold ${\\cal M}$ which is associated with $T^*M$ in the\n  coordinates $(x^{i}, \\theta_{i})$.\n\n Obviously, on this supermanifold, in the coordinates $(x^i, \\theta_{i})$,\none can globally define the\n canonical odd symplectic structure (\\ref{eq:sympcan}) \\cite{leit}.\n\n\nFor the coordinates $(x^i, \\theta_{i})$ on ${\\cal M}$,\none can admit a more general class  of transformations:\n                           $$\nx^i\\rightarrow {\\tilde x}^i(x,\\theta) \\quad\n\t\\theta_{i}\\rightarrow {\\tilde \\theta}_{i}(x,\\theta )    $$\n   which do not correspond to (\\ref{eq:trans}).\n In particular,\nif $\\theta_i \\to {\\tilde \\theta}^i = \\omega^{ij}\\theta _j $,\n  where $\\omega _{ij}$ is the matrix of some nondegenerate Poisson bracket\n on $M$, then the supermanifold\n  ${\\cal M}$ in the coordinates $(x^i,{\\tilde \\theta}^i )$ is\n  associated with the tangent bundle $TM$ of $M$,\ne.g. ${\\tilde \\theta^i}$ transform\n  under (\\ref{eq:trans}) as $dx^i$.\n\n On the supermanifolds which can be associated in some\n  coordinates with the tangent or cotangent bundle the\n  superstructures are evidently reduced to  standard geometrical objects.\n\n   Concerning the integration, the\nproperties of the odd brackets strongly\ndiffer from the properties of odd brackets \\cite {khud}, \\cite {km},\nsuch as:\\\\\n\n -- the odd bracket hasn't an invariant volume form and invariant\n   integral densities;\\\\\n\n -- it has semidensities, which depend on higher order derivatives.\\\\\n\nThe first of this properties plays an essential role in\nthe Batalin--Vilkovisky quantization\nformalism.\nUsing this property, we can construct on ${\\cal M}$\nan invariant generalization of the\nimportant object of the BV-formalism  -- the operator $\\Delta$.\n\nLet the supermanifold ${\\cal M}$ be\n   provided with the odd symplectic structure (\\ref{eq:symp})  and\n the volume form\n\\begin{equation}\ndv=\\rho(x, \\theta)d^{N}x d^{N}\\theta.  \\label{eq:vol}\n\\end{equation}\nHere $\\rho (x,\\theta)$\nis some integral density. Under coordinate transformation\n${\\tilde x}^{A} = {\\tilde x}^{A}(x)$,\nit  transforms as:\n\\begin{equation}\n   {\\tilde \\rho}({\\tilde x}) = \\rho (x ({\\tilde x})) {\\rm Ber}\n \\frac{\\partial^{R} x^A}{\\partial {\\tilde x}^B}  \\label{eq:denstrans}\n\\end{equation}\nOn this supermanifold  one can invariantly define a second order odd\ndifferential operator, which we call the \"generalized operator $ \\Delta $\",\nand which is invariant\n under the transformations preserving the symplectic structure\n  and the volume form \\cite {khud}. Its action on a function $f(x,\\theta)$\n  is the divergence  of the Hamiltonian\n  vector field ${\\bf D}_{f}$\n  with the volume form $dv$:\n\\begin{equation}\n\\Delta _{\\rho} f =\\frac{1}{2}div_{\\rho} {\\bf D}_{f}\n\\equiv \\frac{1}{2}\\frac{{\\cal L}_{{\\bf D}_f} dv}{dv}, \\label{eq:delta}\n\\end{equation}\nwhere  ${\\cal L}_{{\\bf D}_f}$ is the Lie derivative\nalong ${{\\bf D}_f}$ \\cite {ber}, \\cite {voronov}.\nIn coordinate form:\n\\begin{equation}\n           \\Delta f=\\frac{1}{2\\rho}\n     \\frac{\\partial^R}{\\partial x^A}\n\\left(\\rho\\{x^A,f\\}_1\\right)   \\label{eq:deltaloc}\n\\end{equation}\n   It has no analog within even symplectic structures.\nThe oddness of the Poisson bracket (\\ref{eq:bloc})  forces\n the nontrivial grading of $\\Delta $,\nand the \"antisymmetricity\" condition (\\ref{eq:anti}) forces\nits dependence on second derivatives.\n\nIf the Poisson bracket  in (\\ref{eq:delta}) is canonical,\nand $\\rho =constant$,\nthe generalized operator $\\Delta$ takes the canonical  form\n\\begin{equation}\n     \\Delta^{\\rm can} = \\frac{\\partial^R}{\\partial x^i}\n\\frac{\\partial ^L}{\\partial \\theta_i} \\label{eq:deltacan}\n\\end{equation}\nused in the BV-formalism.\n\n From the Leibnitz rule (\\ref{eq:bLieb}) and the\ndefinition (\\ref{eq:delta}) follows\n\\begin{equation}\n(-1)^{p(g)}\\{f,g \\}_1 = \\Delta(fg) - f\\Delta g\n -(-1)^{p(g)}(\\Delta f)g  \\label{eq:Liebdelta}\n\\end{equation}\n\n From the Jacobi identity (\\ref{eq:bjac}) and the\ndefinition (\\ref{eq:delta}) follows\n\\begin{equation}\n\\Delta \\{f,g \\}_1 = \\{f,\\Delta g \\}_1\n+(-1)^{p(g)+1}\\{\\Delta f ,g \\}_1     \\label{eq:jacdelta}\n\\end{equation}\nThe density transformation rule (\\ref{eq:denstrans}) implies for\nthe generalized operator $\\Delta$\nthe following transformation rule\nunder canonical transformations:\n\\begin{equation}\n \\Delta'f = \\Delta f +\\frac{1}{2}\n\\{\\log {\\cal J} ,f \\}_1 , \\label{eq:transdelta}\n\\end{equation}\nwhere ${\\cal J}$ is the Jacobian of the canonical transformation of\nthe odd bracket, $\\Delta'$ is the\ngeneralized operator $\\Delta$ in the new coordinates.\nFor example, let us demonstrate the derivation of (\\ref{eq:jacdelta}):\n\\begin{eqnarray}\n&&\\Delta \\{f,g \\}_{1}dv = {\\cal L}_{{\\bf D}_{\\{f,g \\}_1}}dv =\\nonumber\\\\[3mm]\n&&=\\left ( {\\cal L}_{{\\bf D}_f}{\\cal L}_{{\\bf D}_g} -\n(-1)^{(p(f)+1)(p(g)+1)}{\\cal L}_{{\\bf D}_g}{\\cal L}_{{\\bf D}_f} \\right )dv\n= \\nonumber\\\\[3mm]\n&&={\\cal L}_{{\\bf D}_f} {\\Delta g}dv\n-(-1)^{(p(f)+1)(p(g)+1)}{\\cal L}_{{\\bf D}_g} {\\Delta f}dv = \\nonumber\\\\[3mm]\n&& = \\left ( \\{f,\\Delta g \\}_{1}\n+(-1)^{p(g)+1}\\{\\Delta f ,g \\}_1 \\right ) dv \\nonumber\n\\end{eqnarray}\nLet us write the following useful expressions, too:\n\\begin{equation}\n \\Delta f(g)=f'(g) \\Delta g + \\frac{1}{2}f''(g)\n\\{g,g \\}_1,  \\label{eq:difdelta}\n\\end{equation}\n where $f(g)$ is an even complete function, and $g$ is\nan even function on ${\\cal M}$.\n\nThe properties (\\ref{eq:Liebdelta}) -\n(\\ref{eq:difdelta}) are satisfied for any\n$\\rho$ and in the same manner as\nthe relations between canonical Poisson brackets\n(\\ref{eq:bcan}) and (\\ref{eq:deltacan}) in the\nBV-formalism \\cite{bat2}, \\cite{witten}.\nThis can br derived in the same way as (\\ref{eq:jacdelta}).\n\nHowever (\\ref{eq:deltacan}) satisfies the\n nilpotency  condition\n\\begin{equation}\n\t       \\Delta^2=0     \\label{eq:nilp}\n\\end{equation}\nwhich is very important in the BV-formalism.\n\nThe latter condition is violated for arbitrary $\\rho (x,\\theta)$.\n Indeed, if we have two densities $\\rho$ and ${\\tilde \\rho}$,\nand ${\\tilde \\rho} = \\lambda\\rho,\\ p(\\lambda )=0$,\nthen the corresponding operators $\\Delta$ are related by\n\\begin{equation}\n    \\Delta_{\\tilde \\rho} f =\\Delta_{\\rho} f +\n\\frac{1}{2} \\{\\log \\lambda, f\\}_{1} \\label{eq:deltacon}\n\\end{equation}\nIt is easy to see that\n \\begin{equation}\n \\Delta^{2}_{\\tilde \\rho} f =\\Delta^{2}_{\\rho} f\n+ \\{ \\Gamma_{\\lambda}, f \\}_{1} ,  \\label{eq:deltasqcon}\n\\end{equation}\nwhere\n\\begin{equation}\n  \\Gamma_{\\lambda}\n=\\lambda^{-\\frac{1}{2}}\\Delta_{\\rho} \\lambda^{\\frac{1}{2}},\n\\quad p(\\Gamma_{\\lambda}) = 1\n\\end{equation}\nIf for some $\\Delta_{\\rho}$  the nilpotency condition\n(\\ref{eq:nilp}) is satisfied,\nthen it is also satisfied for  $\\Delta_{\\tilde \\rho}$ (\\ref{eq:deltacon})\nif  $ \\Gamma_{\\lambda} =$odd constant$=0$.\n\nFor example, if the  symplectic structure is canonical,\n(\\ref{eq:nilp}) holds if\n$\\rho (x,\\theta)$ satisfies the equation\n\\begin{equation}\n\\Delta^{{\\rm can}} \\sqrt \\rho =0.\\label{eq:sqrt}\n\\end{equation}\nBut this is the master equation of the\nBV-formalism for the action $ S= -i\\frac{1}{2}\\log\\rho $.\n\n  The geometrical meaning of the\nnilpotency condition (and correspondingly of the master equation)\nwill be illustrated on a simple example\nin the next Section.\n\\setcounter{equation}0\n\\section{Example: The operator $\\Delta$ on K\\\"ahlerian Supermanifolds}\n\nAs we saw in the previous section, in contrary to the case of an even\nsymplectic structure, on supermanifolds with an odd symplectic\nstricture there arises\na nontrivial differential geometry.\n\n It is sufficient to show\nthe correspondence between the generalized operator\n$ \\Delta$ and geometrical objects on basic manifolds in the case of a\nK\\\"ahlerian basic manifold,\nbecause\non K\\\"ahlerian manifolds the symplectic structure corresponds to a\nRiemannian one,\nand a Riemmannian structure has a rich differential geometry.\nMoreover, in this case there also  exists\n on ${\\cal M}$ an even K\\\"ahlerian\nstructure, and, using it, we can construct a natural\nintegral density \\cite {ners}, \\cite{knjinr}.\n\nLet ${\\cal M}$ be a\ncomplex supermanifold, and $z^A$ local complex coordinates\non ${\\cal M}$.\nA symplectic structure $\\Omega^{\\kappa}$ -- here and further\n$\\kappa =0(1)$ if\nthe symplectic structure is even (odd) --  on ${\\cal M}$ is\ncalled K\\\"ahlerian,\nif in local coordinates $z^{A}$ it takes the following form:\n\\begin{equation}\n\\Omega^{\\kappa}=i(-1)^{p(A)(p(B)+\\kappa+1)}g^\\kappa_{A {\\bar B}}\n           dz^A \\wedge d{\\bar z}^B,\n\\end{equation}\n      where\n$$  g^\\kappa_{A {\\bar B}}  =\n     (-1)^{(p(A)+\\kappa+1)(p(B)+\\kappa+1)+\\kappa +1}\n         \\overline {g^\\kappa _{B {\\bar A}}},\n           \\quad p(g^\\kappa _{A\\bar B})=p_A +p_B+\\kappa$$\n\nThen there exists a local real even (odd) function\n$K^\\kappa(z,{\\bar z})$\n    (K\\\"ahlerian potential), such that\n\\begin{equation}\n\t        g^\\kappa_{A {\\bar B}}  =\n              \\frac{\\partial ^L}{\\partial z^A}\n\t\t\\frac{\\partial ^R}{\\partial {\\bar z}^B}\n\t\t\t       K^\\kappa  (z,{\\bar z})\n\\end{equation}\n         To  $\\Omega^{\\kappa}$ there corresponds\n    the  Poisson bracket\n   \\begin{equation}\n\t\t     \\{ f,g\\}_\\kappa\n                        =\n\t\t     i\\left(\n\t     \\frac{\\partial ^R f}{\\partial \\bar z^A}\n\t\t\t   g^{{\\bar A}B}_\\kappa\n\t        \\frac{\\partial  ^L g}{\\partial z^B}\n\t\t           -\n            (-1)^{(p(A)+\\kappa)(p(B)+\\kappa)}\n      \t\t \\frac{\\partial  ^R f}{\\partial z^A}\n\t\t     g^{{\\bar A}B}_\\kappa\n\t\t \\frac{\\partial ^L g }{\\partial \\bar z^B}\n\t\t       \\right),\n\\end{equation}\n where\n$$g^{{\\bar A}B}_\\kappa g_{B{\\bar C}}^\\kappa=\n  \\delta^{\\bar A}_{\\bar C} \\;\\;,\\;\\;\\;\\; \\overline{g^{{\\bar A}B}_\\kappa}\n = (-1)^{(p(A)+\\kappa)(p(B)+\\kappa)}g^{{\\bar B}A}_\\kappa.$$\n Its  satisfies the conditions of reality and \"antisymmetricity\"\n       \\begin{equation}\n \\overline{\\{ f, g\\}_\\kappa }\n     =\\{\\bar f,\\bar g \\}_\\kappa,\\;\\;\\;\\{ f, g \\}_\\kappa = -\n(-1)^{(p(f)+\\kappa)(p(g)+\\kappa)}\\{ g, f \\}_\\kappa,\n\\end{equation}\nand the Jacobi identities :\n\\begin{equation}\n( -1)^{(p(f)+\\kappa)(p(h)+\\kappa)}\n\\{ f,\\{ g, h \\}_{\\kappa}\\}_\\kappa +{\\rm {cycl. perm.}}(f,g,h) = 0\n\\end{equation}\nLet ${\\cal M}$ be associated with the tangent bundle $TM$ of the\nK\\\"ahlerian manifold $M$,\nand $z^A = (w^{a}, \\theta^a )$  local coordinates on it,\n$\\theta^a$ transforming from map to map\nlike $dw^a$.\nLet\n\\begin{equation}\ng_{a{\\bar b}}(w,{\\bar w})\n =\\frac{\\partial^{2} K(w,{\\bar w})}\n{{\\partial \\omega^{a}}{\\partial {\\bar\\omega^b}}}\n\\end{equation}\nbe a K\\\"ahlerian metric on $M$, with $K$ its K\\\"ahlerian\npotential \\cite{kobnom}.\nThen  the local functions\n\\begin{eqnarray}\nK_0(w,{\\bar w},\\sigma,{\\bar \\sigma})&=& K(w,{\\bar w})+\nig_{a{\\bar b}}(w,{\\bar w})\\sigma^a{\\bar \\sigma}^b,\n\\quad p( K_0 )=0 \\label{eq:evenpot}\\\\[3mm]\n  K_{1}(w,{\\bar w},\\sigma,{\\bar \\sigma})&=&\n    \\epsilon\\frac{\\partial K(w,{\\bar w})}{\\partial w^a}\\sigma^a+\n  {\\bar \\epsilon}\\frac{\\partial K(w,{\\bar w})}\n   {\\partial {\\bar w^a}}{\\bar \\sigma^a} \\quad p( K_1 )=1\n \\label{eq:oddpot}\n   \\end{eqnarray}\n(where  $\\epsilon$ is\nan arbitrary complex constant) correctly define an even and an\nodd symplectic structures on ${\\cal M}$\n(this is not the most general form of K\\\"ahlerian potentials\non such supermanifolds \\cite {ners})\n\nThe odd K\\\"ahlerian potential (\\ref{eq:oddpot})\ndefines on ${\\cal M}$ the following odd bracket:\n\\begin{equation}\n  \\{ f,g\\}_1 = \\frac{i}{\\epsilon}\\left (\n\t     \\frac{\\partial ^R f}{\\partial \\theta^a} \\nabla^{a}g -\n   \\nabla^{a}f \\frac{\\partial^L g }{\\partial\n\\theta^a } \\right ) +{\\rm c.c}  \\label {eq:oddbk}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\overline {\\nabla^a}}=g^{{\\bar a}b}\\nabla_b \\quad \\nabla_a=\n\\frac{\\partial}{\\partial w^a} -\n        \\Gamma^c_{ab}\\theta^b\\frac{\\partial^L}{\\partial\\theta^c},\n\\end{equation}\nand $\\Gamma^c_{ab}=g^{\\bar d c}g_{a \\bar d,b} $ are\nthe Christoffel symbols of the K\\\"ahlerian metric  on $M$.\n\nIt is easy to see, that in the coordinates\n$(w^a, \\theta_a =i g_{a\\bar b}{\\bar \\sigma}^b )$,\nin which ${\\cal M}$ is associated with $ T^* M$, the\nodd Poisson bracket takes the canonical form.\n\nThe generalized operator $\\Delta$ corresponding to (\\ref{eq:oddbk}) takes the\nform\n    \\begin{equation}\n \\Delta f = \\left(\\frac{1}{\\epsilon}\\nabla^a\n\\frac{\\partial^L}{\\partial\\theta^a} +\n\t\t\\frac{1}{\\bar \\epsilon}\n{\\overline {\\nabla^a}}\\frac{\\partial^L}{\\partial{\\bar \\theta}^a}\\right)f +\n       \\frac{1}{2}\\{\\log \\rho,f \\}_{1}\n\\end{equation}\nIf $\\nabla_{a} \\rho =0$ (or, in fact, if $\\rho$ is a characteristic class\nof $M$) then\n   \\begin{equation}\n\t    \\Delta f = \\frac{1}{\\sqrt \\rho}\n  \\left(\\frac{1}{\\epsilon}\\nabla^a \\frac{\\partial^L}{\\partial\\theta^a} +\n   \\frac{1}{\\bar \\epsilon} {\\overline {\\nabla^a}}\\frac{\\partial^L}\n    {\\partial{\\bar \\theta}^a}\\right)(\\sqrt\\rho f),  \\label{eq:deltan}\n\\end{equation}\nis obviously nilpotent.\n\nThe invariant density which corresponds to (\\ref{eq:evenpot}) is\n    \\begin{equation}\n\t     \\rho=  det(\\delta^a_b+i\n\t      R^a_{bc{\\bar d}}\\theta^c \\bar\\theta^d)\n\\end{equation}\nwhere $ R^a_{bc\\bar d}=(\\Gamma^a_{bc})_{,\\bar d}$ is\nthe curvature tensor on $M$.\nIt is associated with the\ngenerating functions of the Chern classes of the underlying\n K\\\"ahlerian manifold \\cite{kobnom}.\n\nObviously (\\ref {eq:deltan}) corresponds  to the operator\nof covariant divergence $\\delta = \\ast d \\ast$ on $M$ with some effective\nweight.\n\n \\setcounter{equation}0\n\\section { Discussion}\n\nAs we have seen in the previous Sections, the operator $\\Delta$ has\na simple geometrical\nnature on the  supermanifolds associated with the\ncotangent bundles of manifolds.\nObviously, the same construction holds, if we replace the basic\nmanifold $M$ in the previous Sections by some\nsupermanifold ${\\cal M}_{0}$. Then if $x^i$ are local coordinates on\n${\\cal M}_{0}$ ($p( x^i)\\neq 0$), then, on the supermanifold ${\\cal M}$ which\nis associated with the cotangent bundle $T^*{\\cal M}_{0}$,\none can naturally define\nthe odd Poisson bracket (\\ref{eq:bcan})\n where $\\theta_i$ corresponds to coordinates of the bundle, $p(\\theta_i) =\n p( x^i ) +1$.\nThese coordinates are the  analogs of the antifields of the BV-formalism.\n\nBut what is the reason for the introduction of antifields\n(and, correspondingly,\nfor the structure of supermanifolds with an odd symplectic structure)\n in the BV-formalism ?\n\nIn our opinion, this is connected to the\npeculiarity of the integration on supermanifolds.\nIndeed, if we have some differential form $\\omega ( x^i, dx^i )$\non the supermanifold ${\\cal M}_{0}$, its integral over ${\\cal M}_{0}$\ndefined in the following way \\cite{bernleit}, \\cite{voronov}:\n\\begin{equation}\n\\int_{{\\cal M}_{0}}\\omega \\equiv \\int_{\\hat{\\cal M}_{0}}\n\\omega (x^i, \\theta^i ) D(x,\\theta)[dxd\\theta]   \\label{eq:blint}\n\\end{equation}\nwhere $ p(\\theta^{i})=p(x^{i}) + 1$, and $\\hat{\\cal M}_{0}$\ndenotes the supermanifold\nassociated with the tangent bundle of (supermanifold) ${\\cal M}_{0}$\n(i.e. $\\theta^{i}$ transforms\nlike $dx^{i}$), then $D(x,\\theta)$ is\nthe natural density on $\\hat{\\cal M}$ \\cite{bernleit2}.\n\nTransiting from the description on $\\hat{\\cal M}_{0}$ to that on\n${\\cal M} ={\\cal T}_{*}{\\cal M}_{0}$ --\nthe supermanifold associated with the cotangent bundle of\nthe supermanifold ${\\cal M}_{0}$ -- we saw that\nthe integral (4.2) takes the form of the partition function of the\nBV-formalism.\n\nCorrespondingly, the master-equation of the BV-formalism\ncorresponds to the closeness of the\ninitial differential form. This is clearly seen in the case where\nthe  basic manifold,\nis K\\\"ahlerian (in the general case this proposition was\nstrongly proved in \\cite{schwarz}.\nThen the gauge invariance of the partition function in the BV-formalism\nfollows from  Stokes' theorem \\cite{bernleit2}, \\cite{voronov}.\n\n\\section {Acknowledgments}\nWe are very indebted to I. A. Batalin for valuable discussions.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{intro}\n\nRecent discovery of charge density waves (CDW) in hole-doped cuprates has raised a new wave of interest to the physics of these materials. The CDWs have been detected in underdoped samples of YBCO \\cit\n{y123REXS-3,y123REXS-4,y123REXS-5,y123REXS-6,y123XRD-3,y123XRD-4\n, Bi-2201 \\cite{bi2201STM-1,bi2201STM-2}, Bi-2212 \\cit\n{Bi2212REXS-2,Bi2212STM-2,Bi2212STM-3}\nand Hg-1201 \\cite{HgREXS,HgXRD} by direct methods such as resonant X-ray\nscattering, hard X-ray diffraction and scanning tunneling microscopy(STM).\nA wide variety of techniques, that can be sensitive to a CDW indirectly, also confirm its presence, among them transport measurements \\cite{y123Transp,HgTransp}, nuclear magnetic resonance \\cite{y123NMR-2}, ultrasound propagation \\cite{y123US} and pump-probe experiments \\cite{y123Refl}.\n\n\nSeveral common properties of this CDW state in hole-doped cuprates have been identified. The transition temperature $T_{CDW}$ is higher than $T_c$ but lower or equal to the pseudogap temperature $T^*$. The temperature and magnetic filed dependence of the CDW amplitude({\\it e.g.} \\cite{y123XRD-3}) is consistent with the CDW state competing with the superconductivity.\n\nThe CDW wave vectors seen in the experiments \\cite{y123REXS-5,y123XRD-3,HgXRD,Bi2212STM-0} are directed along the $Cu-O-Cu$ bonds of the $CuO_2$ plane (axes of the Brillouin zone, axial CDW). The CDW period is approximately equal along both the axes and increases with doping \\cite{y123XRD-3,bi2201STM-1,Bi2212REXS-2}.\n\n\nRecent studies have also revealed important information about the distribution of the modulated charge inside the unit cell, {\\it i.e.} the CDW form factor. It has been found for Bi-2212 \\cite{Bi2212STM-2} and YBCO \\cite{y123REXS-5} that\nthe charge is modulated approximately in antiphase at two oxygen sites of the unit cell with the charge at $Cu$ site being constant. In other words, the CDW form factor is characterized by a dominant d- component. The properties mentioned up to now are quite different from the stripe state of the La-based compounds \\cite{StripeRev,y123REXS-6}.\n\n\nConsiderable attention has been also drawn to the nanoscale structure of the CDW. Quantum resistance oscillation experiments \\cite{y123QO-1,y123QO-2,HgQO} have been interpreted \\cite{seb1} as being due to a checkerboard modulation, where CDWs with two orientations uniformly coexist throughout the sample. Results of studies \\cite{Bi2212STM-0,y123REXS-4} suggest, however, show that the charge ordered state consists of domains where CDW is unidirectional.\n\n\nThere have been a number of attempts to obtain the CDW state with the properties discussed above from microscopic calculations. In a model of fermions interacting with antiferromagnetic critical spin fluctuations \\cite{SFREV} (spin fermion (SF) model) a charge order appears in perturbation theory as a subleading instability \\cite{MetSach} hindered by the curvature of the Fermi surface. This order is a checkerboard CDW with d-form factor and wave vectors directed along the diagonals of the BZ \\cite{Efetov2013,Sach2013}. The nearest-neighbor Coulomb interaction can, in principle, make this state leading as has been shown in Ref. \\cite{SachSau}. Moreover, thermal fluctuations between this charge order and SC have been shown to be able to destroy both the orders while pertaining a single-particle gap \\cite{Efetov2013}, which can explain the pseudogap phase. Qualitative aspects of CDW-SC competition are also well-captured in the SF model: moderate magnetic fields suppressing the superconductivity have been shown to favor CDW \\cite{MEPE} resembling the experiment \\cite{y123US}. The vortex cores in the SC state have been shown to contain CDW \\cite{EMPE} which is seen in STM \\cite{hoffman,hamidian}. The diagonal direction of modulation wavevectors contrasting the experiments, however, has proved to be quite robust.\n\n\nSome proposals have been put forward to overcome this contradiction. A CDW with the correct wavevector direction has been\nobtained in Refs. \\cite{cascade,WangChub2014}, however the form factor has been found to lack the dominant d-symmetry with a large s- component. In Ref. \\cite{pepin} a mixture of the states proposed in Ref. \\cite{Efetov2013} and Ref. \\cite{WangChub2014} has been considered, which should contain either the diagonal modulation or an axial CDW with a non d-form factor. CDW considerations using other models \\cite{Punk2015,DavisDHLee2013,Kampf2013,yamakawa2015,Kampf2014,ChowdSach2014,ThomSach}also do not seem to explain the robustness of the axial d-form factor CDW in the cuprates.\n\nIn this contribution we review the treatment of the SF model allowing the neighboring hot spots to overlap, such that eight hot spots merge into two hot regions entirely covering the antinodal portions of the Fermi\nsurface. This corresponds to sufficiently small values $|\\varepsilon\n(\\pi ,0)-E_{F}|\\lesssim \\Gamma $, where $E_{F}$ is the Fermi energy, $\\varepsilon \\left( \\pi ,0\\right) $ is the energy in the middle of the Brillouin zone edge, and $\\Gamma $ is a characteristic energy of the\nfermion-fermion interaction due to the antiferromagnetic fluctuations.\nConsideration of this limit is motivated by ARPES data \\cite{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1,Bi2212ARPES-2} showing that the energy separation between the hot spots and $(\\pi ,0);(0,\\pi )$ is actually\nquite small. In addition to the electron-electron interaction via\nparamagnons, we consider also the effects of low-energy (low-momentum) part of\nthe Coulomb interaction, which should not contradict the philosophy of the\nlow energy SF model. A detailed derivation and discussion of the results can\nbe found in our paper \\cite{preprint}.\n\n\\section{Model and main equations}\n\\label{sec1}\n\nWe consider a single band of fermions interacting through critical antiferromagnetic (AF) fluctuations (paramagnons) represented by a spinful bosonic field as well as the Coulomb force. As the AF fluctuations peak at momentum transfer $(\\pi,\\pi)$ we restrict our model to two regions of the Fermi surface connected with this vector represented in Fig. \\ref{fig1}. Inside these regions we do not specify individual hot spots, {\\it i.e.} points on the FS connected by $(\\pi,\\pi)$ as we assume the interaction to be important in all the whole region. This assumption is supported by ARPES experiments \\cite{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1,Bi2212ARPES-2} showing that $|\\varepsilon (\\pi ,0)-E_{F}|$ is actually smaller than the pseudogap energy, which can be taken as the interaction scale.\n\\begin{figure}[tbp]\n\\includegraphics[width=0.5\\linewidth]{1.eps}\n\\centering\n\\caption{A typical cuprate Fermi Surface with two regions connected\nby the antiferromagnetic wavevector $(\\pi,\\pi)$ }\n\\label{fig1}\n\\end{figure}\n\nThe fermion-paramagnon part of the Lagrangian takes the form:\n\\begin{equation}\n\\begin{split}\nL_{\\mathrm{SF}} =\\sum_{\\mathbf{p,}\\nu =1,2} \\chi\n_{\\mathbf{p}}^{\\nu \\dagger } \\left[\\partial _{\\tau }+\\varepsilon _{\\nu}\\left(\\mathbf{p}\\right) -\\mu_0 \\right] \\chi _{\\mathbf{p}}^{\\nu }+\n\\\\\n+\\sum_{q}\\vec{\\varphi}_{-\\mathbf{q}}(-v_{s}^{-2}\\partial _{\\tau }^{2}+\n\\mathbf{q}^{2}+\\xi^{-2})\\vec{\\varphi}_{\\mathbf{q}} \\\\ +\\lambda\n^{2}\\sum_{\\mathbf{p,q}}\\left[ \\chi _{\\mathbf{p+q}}^{1\\dagger }\n\\vec{\\varphi}_{\\mathbf{q}}\\vec{\\sigma}\\chi _{\\mathbf{p}}^{2}+\\chi _{\\mathbf{\np+q}}^{2\\dagger }\\vec{\\varphi}_{\\mathbf{q}}\\vec{\\sigma}\\chi\n_{\\mathbf{p}}^{1} \\right].\n\\end{split}\n\\label{sf_h}\n\\end{equation}\nwhere $\\varepsilon _{\\nu}(\\mathbf{p})$ is the electron dispersion in region $\\nu=1,\\;2$ (including the chemical potential), $v_s$ is the velocity of spin waves and $\\xi$ is the magnetic correlation length. We shall not write explicitly the terms corresponding the Coulomb interaction as we will take their effect into account qualitatively.\n\nAssuming that the regions 1 and 2 occupy a small portion of the BZ we expand $\\varepsilon _{1(2)}(\\mathbf{p})$ around $[\\pi,0]([0,\\pi])$ resulting in $\\varepsilon _{p}^{1}=\\alpha p_{x}^{2}-\\beta p_{y}^{2}-\\mu_0,\\;\\varepsilon_{p}^{2}=\\alpha p_{y}^{2}-\\beta p_{x}^{2}-\\mu_0$, where $\\mu _{0}$ is the chemical potential counted from $\\varepsilon (\\pi ,0)=\\varepsilon (0,\\pi )$. Moreover, we will average the curvature term (the one with $\\beta$) inside each region leading to the final form:\n\\begin{equation}\n\\varepsilon _{p}^{1}=\\alpha p_{x}^{2}-\\mu ,\\quad \\varepsilon _{p}^{2}=\\alpha\np_{y}^{2}-\\mu ,  \\label{mod_disp}\n\\end{equation}\nwhere $\\mu =\\mu _{0}+\\langle \\beta p_{\\parallel }^{2}\\rangle $. To study particle-hole instabilities we define the order parameter:\n\\begin{equation}\nW_{\\mathbf{Q}}(\\tau-\\tau',\\mathbf{k})=\\langle \\chi_{\\mathbf{k}-\\mathbf{Q}\/2,\\sigma}^{\\dagger}(\\tau')\n\\chi_{\\mathbf{k}+\\mathbf{Q}\/2,\\sigma }(\\tau)\\rangle  \\label{mod_op}\n\\end{equation}\nAs has been shown in \\cite{preprint} this order parameter is related to density modulations at the three atoms of the unit cell in the following way:\n\\begin{equation}\n\\begin{split}\n\\delta n_{Cu}(\\mathbf{r}) =2e^{i\\mathbf{Q}\\mathbf{r}}\\sum_{\\mathbf{k}}W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c,  \\\\\n\\delta n_{O_{x}}(\\mathbf{r}) =\\frac{p}{4}e^{i\\mathbf{Q}\\mathbf{r}}\\sum_{\\mathbf{k}}\\cos(k_{x}a_{0})W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c.,  \\\\\n\\delta n_{O_{y}}(\\mathbf{r}) =\\frac{p}{4}e^{i\\mathbf{Q}\\mathbf{r}}\\sum_\n\\mathbf{k}}\\cos (k_{y}a_{0})W_{\\mathbf{Q}}(0,\\mathbf{k})+c.c.\n\\end{split}\n\\label{mod_dens}\n\\end{equation}\nAs both the regions we consider yield approximately $\\cos (k_{x}a_{0})+\\cos (k_{y}a_{0})\\approx 0$ we have $\\delta n_{O_{x}}(\\mathbf{r}) + \\delta n_{O_{y}}(\\mathbf{r})\\approx 0$ in or model, {i.e.} charge is modulated in antiphase at the two oxygen sites of the unit cell.\n\n\nNow we can discuss the qualitative effects of the Coulomb interaction in the $CuO_2$ plane. The strong on-site repulsion prohibits any real charge modulations on the $Cu$ sites leading to the constraint: $\\delta n_{Cu}=0$ for the order parameter. Together with $\\delta n_{O_{x}}(\\mathbf{r}) + \\delta n_{O_{y}}(\\mathbf{r})\\approx 0$ discussed above this leads to the conclusion that the charge modulations obtained in our model will have the d-form factor in accord with the experiments \\cite{Bi2212STM-2,y123REXS-5}.\n\nThe nearest-neighbor Coulomb interaction has been shown in \\cite{SachSau} to suppress superconductivity and support charge ordering, explaining $T_{CDW}>T_c$. This allows one to consider the particle-hole channel of the model separately from the particle-particle one.\n\n\n\\section{Pomeranchuk instability and intra-cell charge modulation.}\nOur main finding is that for sufficiently small $\\mu$ the leading particle-hole instability is the one with ${\\bf Q}=0$. The ordered state is then characterized not by a CDW, but rather a deformation of the FS (this type of transition is known as Pomeranchuk instability \\cite{pomeranchuk}, \\cite{yamase2005}). Moreover, it follows from (\\ref{mod_dens}) that such a deformation leads to a redistribution of charge between the oxygen sites of the unit cell (see Fig.\\ref{figpom}).\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\linewidth]{6.eps}\n\\caption{Pictorial representation of two possible shapes of the Fermi\nsurface below the Pomeranchuk transition and the corresponding\nintra-unit-cell charge redistributions. Grey arrows mark the emergent nesting vectors for regions $1$ and $2$.}\n\\label{figpom}\n\\end{figure}\n\nOne can obtain this result analytically for a simplified BCS-like model where the paramagnon propagator is replaced by a constant. For that case a mean-field analysis yields that if $\\mu\/T_{Pom}\\leq1.1$ then it is the leading instability. $T_{Pom}$ is given in this case by $\\frac{1}{2\\alpha }\\left( \\frac{\\lambda _{0}\\Lambda }{4\\pi ^{2}}\\right) ^{2}$ where $\\lambda_0$ is the dimensionless coupling constant and $\\Lambda$ is the size of a single region in the momentum space. This expression contrasts the usual exponential dependence obtained in BCS-like theories. A detailed account on this simplified case is presented in \\cite{preprint}.\n\n\nNow let us turn to the model presented here. As a starting approximation we will use the self-consistent equations represented by diagrams in Fig. \\ref{figFeyn}\n\\begin{figure}[h]\n\\includegraphics[trim = 100 150 0 0,width=0.8\\linewidth]{7.eps}\n\\caption{Feynman diagrams for fermionic and bosonic propagators illustrating\nthe approximations used.}\n\\label{figFeyn}\n\\end{figure}\nThe integral over momentum in the fermionic self-energy can be greatly simplified provided $\\mu v_{s}^{2}\/\\alpha \\ll (v_{s}\/\\xi )^{2}$, i.e. that the correlation length is not too large. Then the self-energy and polarization operator do not depend on the momentum. To analyze the FS deformation we distinguish the 'even' $(\\Sigma_1+\\Sigma_2)\/2\\equiv i\\varepsilon _{n}-if(\\varepsilon _{n})$ and 'odd' $(\\Sigma_1-\\Sigma_2)\/2\\equiv P$ contributions to self-energy, with the latter being zero in the normal state. After the momentum integration one can introduce an energy scale $\\Gamma =\\left( \\frac{\\lambda ^{2}v_{s}}{\\sqrt{\\alpha }\\hbar ^{2}}\\right)^{2\/3}$ and write the self-consistency equations in the dimensionless form (see Eq. \\ref{sf_Pom}), where $\\bar{a}$ denotes $\\overline{(v_s\/\\xi)}$.\n\\begin{figure*}\n\\begin{centering}\n\\begin{equation}\n\\begin{gathered}\n\\bar{f}(\\bar{\\varepsilon}_{n})-\\bar{\\varepsilon}_{n}=\n0.75\\bar{T}\\sum_{\\bar{\\varepsilon}_{n}^{\\prime }}\n\\frac{1}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon}_{n}^{\\prime})+\\bar{a}}}\n\\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })])}{2}\n\\left[\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_{n}^{\\prime })}}\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}-\\bar{P}(\\bar\n\\varepsilon}_{n}^{\\prime })}}\n\\right]\n\\\\\n\\bar{P}(\\bar{\\varepsilon}_{n})=i\\cdot 0.75\\bar{T}\\sum_{\\bar{\\varepsilon\n_{n}^{\\prime }}\\frac{1}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar\n\\varepsilon}_{n}^{\\prime })+\\bar{a}}}\\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar\n\\varepsilon}_{n}^{\\prime })])}{2}\\left[ \\frac{1}{\\sqrt{i\\bar{f}(\\bar\n\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_{n}^{\\prime })}}\n\\frac{1}{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}+\\bar{P}(\\bar\n\\varepsilon}_{n}^{\\prime })}}\\right]\n\\\\\n\\bar{\\Omega}(\\bar{\\omega}_{n})-\\bar{\\omega}_{n}^{2}=-\\frac{\\bar{T}}{2}\\sqrt{\\frac{v_{s}^{2}\/\\alpha }{\\Gamma }}\n\\sum_{\\bar{\\varepsilon}_{n}}\n\\left[\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon_{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_n)+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_n)}}\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon _{n}+\\omega _{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}+\\omega _{n})+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_{n}+\\omega _{n})}}\n+\n\\right.\n\\\\\n\\left.\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon_{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n})+\\bar{\\mu}-\\bar{P}(\\bar{\\varepsilon}_n)}}\n\\frac\n{\\mathrm{sgn}(\\mathrm{Re}[f(\\varepsilon _{n}+\\omega _{n})])}\n{\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}+\\omega _{n})+\\bar{\\mu}+\\bar{P}(\\bar{\\varepsilon}_{n}+\\omega _{n})}}\n\\right]\n.\n\\end{gathered}\n\\label{sf_Pom}\n\\end{equation}\n\\end{centering}\n\\end{figure*}\nNote that the polarization operator $\\bar{\\Omega}(\\bar{\\omega}_{n})-\\bar{\\omega}_{n}^{2}$ contains a factor $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}$ absent in the fermionic self-energy part. This factor will also arise if one calculates the vertex correction, as there one has to integrate a product of fermionic Green's functions like in the polarization operator. This allows us to use $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}$ as a small parameter to justify the Eliashberg-like approximation given by Fig. \\ref{figFeyn}. We shall not neglect, however, the polarization operator, as it behaves linearly at low frequencies and might outpower the initial quadratic dispersion.\n\n\n\nThe equations (\\ref{sf_Pom}) have been numerically solved by an iteration scheme, yielding the transition temperature $T_{Pom}$ where the 'odd' self-energy $P$ becomes non-zero. To show that this transition can be indeed leading we have also computed the transition temperature for a CDW with wavevector along the BZ diagonal. This instability has been found to be universally leading in previous studies. The transition temperature can be found from the linearized equation for the CDW order parameter $W_{diag}(\\varepsilon_{n})$:\n\\begin{equation}\n\\begin{gathered}\n\\bar{W}_{diag}(\\bar{\\varepsilon}_{n})\n=\\frac{0.75\\bar{T}}{2}\\sum_{\\varepsilon _{n}^{\\prime\n}}\\frac{\\bar{W}_{diag}(\\bar{\\varepsilon}_{n}^{\\prime\n})}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon _{n}^{\\prime\n}})+\\bar{a}}}\n\\\\\n\\times \\frac{\\mathrm{sgn}\\left(\n\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })]\\right)\n}{\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime\n})\\sqrt{i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{\\mu}}}.\n\\end{gathered}\n\\label{sf_tdiag}\n\\end{equation}\nThe results of the numerical solutions are presented in Fig.\\ref{figpomdiag}.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=\\linewidth]{9.eps}\n\\caption{$\\bar{T}_{Pom}(\\bar{\\protect\\mu})$ (dashed line) and $\\bar{T}_{diag}(\\bar{\\protec\n\\mu})$ (dotted line) for $\\overline{(v_s\/\\xi)}=0.1$, $\\sqrt{\\frac{v_{s}^{2}\/\\alpha }\n\\Gamma }}=0.5(a),\\;0.1(b)$.}\n\\label{figpomdiag}\n\\end{figure}\nOne can clearly see that for $\\bar{\\mu}$ less than a certain value Pomeranchuk instability is the leading one. Note that the ratio $\\mu\/T_{Pom}$ can be as high as $12$ for $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}=0.5$ and $9$ for $\\sqrt{v_{s}^{2}\/\\alpha\\Gamma}=0.1$.\n\nAs the Fermi Surface seen in ARPES experiments is universally found to be $C_4$-symmetric and in the light of the domained CDW structure \\cite{Bi2212STM-3,y123REXS-4}, we assume that Pomeranchuk order should also be organized in domains with different sign of the order parameter. This constitutes a way of 'masking' a $C_4$ breaking alternative to the one proposed in \\cite{yamase2009}.\n\\section{Incommensurate charge modulation.}\nThe deformed Fermi surface of Fig.\\ref{figpom} can be unstable to CDW formation at lower temperatures. The direction and the magnitude of the wavevector are directly related to the sign and the magnitude of $P$. We assume that the CDW wavevector should yield nesting in the region where the FS 'expands' due to the FS deformation. Then one has:\n\\begin{equation}\nQ^{SF}(T)=2\\sqrt{(\\mu +0.5\\left\\vert P(-\\pi T)+P(\\pi T)\\right\\vert )\/\\alpha }.  \\label{sf_q}\n\\end{equation}\nIn our model the FS in the second region 'closes' moving out of the considered region for $P>\\mu$. However, as is seen from Fig.\\ref{figpom}, in reality such a deformation can lead to emergent nesting in this region with the same vector direction as in the first one. The best-case scenario is that the nesting vectors in both regions coincide also in magnitude, {\\it i.e.} $Q_1=Q_2$ (see Fig.\\ref{figpom}). We shall assume that this is indeed the case, thus providing an upper limit on the $T_{CDW}$. In this case the equation for the CDW transition is:\n\\begin{eqnarray}\n&&\\bar{W}(\\bar{\\varepsilon}_{n})=0.75\\;i\\frac{\\bar{T}_{CDW}}{2\n\\sum_{\\varepsilon _{n}^{\\prime }}\\frac{\\bar{W}(\\bar{\\varepsilon}_{n}^{\\prime\n})}{\\sqrt{\\bar{\\Omega}(\\bar{\\varepsilon}_{n}-\\bar{\\varepsilon _{n}^{\\prime }\n)+\\bar{a}}}  \\notag \\\\\n&&\\times \\frac{\\mathrm{sgn}(\\mathrm{Re}[f(\\bar{\\varepsilon}_{n}^{\\prime })]\n}{(\\left[ i\\bar{f}(\\bar{\\varepsilon}_{n}^{\\prime })+\\bar{P}(\\bar{\\varepsilon\n_{n}^{\\prime })-P(0)\\right] g\\left( \\bar{\\varepsilon}^{\\prime }\\right) }.\n\\label{c11}\n\\end{eqnarray\nThe results of numerical calculations are presented in Fig.\\ref{figCDW}. It turns out that the CDW transition can closely follow the onset of the FS deformation.\n\\begin{figure}[tbp]\n\\includegraphics[width=\\linewidth]{12.eps}\n\\caption{$T_{Pom}(\\bar{\\protect\\mu})$ (dashed line) and $\\bar{T}_{CDW}(\\bar\n\\protect\\mu})$ (dotted line) determined from Eq. (\\protect\\ref{c11}) for $\\overline{(v_s\/\\xi)}=0.1$, $\\protect\\sqrt{\\frac{v_{s}^{2}\/\\protect\\alpha }{\\Gamma }\n=0.5(a),\\;0.1(b)$.}\n\\label{figCDW}\n\\end{figure}\n\n\\section{Comparison with experiments and conclusions.}\n\n\\label{sec4}\n\nMotivated by the existing ARPES data \\cit\n{Bi2201ARPES-1,Bi2201ARPES-2,Bi2212ARPES-1} we have considered the SF model\nwith overlapping hotspots and demonstrated that the d-wave Fermi\nsurface distortion can be the leading instability. The\ntransition is further followed at a lower temperature by a transition into\na state with a d-form factor CDW directed along one of the BZ axes. The corresponding transition temperatures $T_{Pom}$ and $T_{CDW}$ can be not far away from each other.\n\nThe results obtained allow us to draw the following qualitative picture of\nthe charge order formation:\n\n$\\bullet $ At $T_{Pom}\\geq T^{\\ast }$ $C_{4}$ symmetry is broken by a Pomeranchuk\ntransition. The Fermi surface is deformed(see Fig. \\ref{figpom}) and doped holes\nare redistributed between the oxygen orbitals of the unit cell. The sample consists\nof domains with different signs of the order parameter corresponding to two alternatives presented in Fig.\\ref{figpom}.\n\n$\\bullet $ At $T_{CDW}<T_{Pom}$ a uniaxial d-form factor CDW forms in each domain.  CDW wave vector\nis along one of the BZ axes dependinig on the sign of the Pomeranchuk order parameter inside each domain (see Fig.\\ref{figpom}). The CDW period generally exceeds the one corresponding to antinodal nesting and is determined\nself-consistently by the interaction and parameters of the Fermi surface. Qualitatively, the CDW wavevector tracks the FS and should decrease with hole doping (thus the CDW period should increase).\n\n\nOur findings help us in understanding the results of recent experiments. The\nPomeranchuk deformation explains well why the C$_{4}$-symmetry at\ncommensurate peaks in Fourier transformed STM data \\cite{Bi2212STM-0}. Formation of domains with different directions of the C$_{4}$-symmetry breaking is seen in STM experiments \\cite{Bi2212STM-3} and can\nalso help explaining results of the transport measurements in YBCO \\cit\n{y123Transp-2}. It is important to note, though, that the effects of the\ndeformation of the Fermi surface on transport can be masked by existence of\nthe domains. This may also resolve the apparent contradiction to the ARPES\ndata \\cite{Bi2201ARPES-1,Bi2212ARPES-1} always showing a C$_{4}$-symmetric\nFermi surface.\n\nThe most important aspect of the Pomeranchuk order is that it explains the\nrobustness of the axial d-form factor CDW in the cuprates. We also note that\nthe organization of the CDW phase in the unidirectional domains is indeed\nseen in STM \\cite{Bi2212STM-3} and XRD \\cite{y123REXS-4,HgXRD} measurements.\nThe coexistence of the unidirectional CDW and Pomeranchuk order also allows one to resolve\na seeming contradiction to results obtained in experiments on quantum\noscillations \\cite{seb1}. Although the unidirectional CDW leads\nto an open Fermi surface that does not support quantum oscillations, the\nsimultaneous presence of a C$_{4}$-symmetry breaking can indeed \\cit\n{kivelson2011} close the Fermi surface leading to quantum oscillations in\nhigh magnetic fields.\n\n\\begin{acknowledgements}\nThe authors gratefully acknowledge the financial support of the Ministry of\nEducation and Science of the Russian Federation in the framework of Increase\nCompetitiveness Program of NUST~\\textquotedblleft MISiS\\textquotedblright\\\n(Nr.~K2-2014-015).\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}