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+{"text":"\\section*{Introduction}\n\nFerromagnetic films with perpendicular magnetic anisotropy (PMA) are of wide interest for applications in established and nascent technologies such as ultrahigh density magnetic hard drives \\cite{doi:10.1063\/1.2750414}, MRAM\\cite{wong2015memory}, superconducting spintronics \\cite{linder2015superconducting}, and energy efficient spin-orbit torque memory \\cite{RevModPhys.91.035004}. PMA can be achieved in Pt\/Co\/Pt multilayer systems as a result of the interfacial anisotropy, however at a critical Co thickness, typically about 1~nm, the anisotropy will fall in-plane. Increasing the total ferromagnetic layer thickness further therefore involves adding additional interfaces to the multilayer. Having a multilayer structure introduces interfacial resistance and interfacial spin-flip scattering \\cite{bass2007spin}, which are disadvantageous for applications such as ours which require the transport current perpendicular-to-plane  \\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt}. Alternately, robust intrinsic PMA can be achieved in certain Co$_x$Pt$_{100-x}$ alloys and compounds at any thickness, without increasing the number of interfaces.   \n\nHere, we study the equiatomic Co$_{50}$Pt$_{50}$ alloy, hereafter referred to as CoPt. Through growth at elevated temperatures it is possible to form the $L1_1$ and $L1_0$ chemically ordered compounds of CoPt as epitaxial films. Previous experimental studies of such compounds tend to use MgO substrates as the basis for high temperature epitaxial growth. Visokay and Sinclair report $L1_0$ crystal structure on MgO [001] substrates for growth temperatures above 520$^{\\degree}$C \\cite{doi:10.1063\/1.113895}. Iwata \\textit{et al.} report growth of $L1_1$ crystal structure on MgO [111] substrates for a growth temperature of 300$^{\\degree}$C \\cite{619533}.\n\nEarly thin film studies of the chemically ordered CoPt (and the related FePt and FePd) compounds were motivated by the large out-of-plane anisotropy and narrow domain wall widths being candidate for high density storage mediums \\cite{619533, PhysRevB.50.3419, PhysRevB.52.13419, doi:10.1063\/1.362122, doi:10.1063\/1.361368, doi:10.1063\/1.113895, doi:10.1063\/1.364504, caro1998structure, doi:10.1063\/1.368158, doi:10.1063\/1.368831, doi:10.1063\/1.371397, doi:10.1126\/science.287.5460.1989, zeng2002orientation, PhysRevB.66.024413, chen2003effect, zhao2003promotion, doi:10.1021\/jp027314o, LAUGHLIN2005383,  doi:10.1063\/1.1991968, LIAO2006e243, doi:10.1063\/1.2176306, PhysRevB.76.174435, doi:10.1063\/1.2730568, Sato2008, chen2008review, 4492859, doi:10.1063\/1.3153513, Seemann_2010, yuan2010effect, Antje_Dannenberg_2010, doi:10.1063\/1.3561115,  doi:10.1063\/1.3672856, ohtake2012structure, sun2013formation, 6416998, ohtake2014structure, VARVARO2014415, doi:10.1063\/1.4960554, zygridou2017exploring, PhysRevApplied.10.054015, GAO2019406, doi:10.1021\/acsanm.9b01192, PhysRevMaterials.6.024403, spencer2022characterization}. Recently, renewed interest in these compounds has been driven by the discovery of self-induced spin-orbit torque switching in these materials, which can be used as the switching mechanism for a low-dissipation magnetic memory \\cite{PhysRevB.101.220402,https:\/\/doi.org\/10.1002\/adfm.202005201,doi:10.1063\/5.0028815,liu2021symmetry,9433557,DONG2022,doi:10.1063\/5.0077465, liu2022current}.\n\nOur motivation for studying [111] CoPt is to incorporate this PMA ferromagnet in an all epitaxial heterostructure suitable for superconducting Josephson devices\\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt} or MRAM\\cite{wong2015memory}. For these applications growth in the [111] direction is favourable. In Josephson junctions, the superconductor of choice is Nb, which can be grown epitaxially as a bcc structure in the [110] direction\\cite{WILDES20017}. In MRAM, the seed layer of choice is Ta, which has almost identical structural properties to Nb. On Ta or Nb [110] layers, we know that Pt and Co will grow with [111] orientation\\cite{PhysRevB.97.214509}. We therefore choose Al$_2$O$_3$ [0001] substrates for this study and use thin Pt [111] seed layers to grow CoPt [111]\\cite{yuan2010effect}. As far as we are aware, no previous works have reported the properties of [111] ordered films of $L1_0$ and $L1_1$ CoPt on Al$_2$O$_3$ [0001] substrates. We are also unaware of any measurements of the spin polarisation of CoPt. \n\n\nWe fabricate and report the properties of three sets of samples. The first set is designed to determine the optimal growth temperature. Therefore, we fix the thickness $d_{CoPt} = 40$~nm and vary the substrate heater temperature in the range from 27${\\degree}$C to 850${\\degree}$C. The next two sample sets are thickness series grown by varying the growth time at a fixed temperature and magnetron powers. The temperatures chosen for the thickness series are to produce either the $L1_1$ (350${\\degree}$C) or $L1_0$ (800${\\degree}$C) crystal structures. These temperatures are guided by the results of the first temperature series of samples. Thicknesses are varied over the range 1~nm$ \\leq d_\\text{CoPt} \\leq 128$~nm.\n\nOn each sample set we report systematically on the structural and magnetic properties of the CoPt. Additionally, on the thickest 128~nm $L1_1$ and $L1_0$ samples we perform point contact Andreev reflection (PCAR) measurements with a Nb tip at 4.2~K to determine the spin polarisation. The use of Nb as the tip, the temperature of this measurement, and the ballistic transport regime probed are relevant for our proposed application of the CoPt in Josephson devices\\cite{birge2018spin, doi:10.1063\/1.5140095 ,satchell2021pt}.   \n\n\n\n\n\\section*{Results and Discussion}\n\n\\subsection*{\\label{Growth}CoPt properties as a function of growth temperature}\n\n We expect that as the growth temperature is increased the CoPt will form initially a chemically disordered $A1$ alloy phase, followed by a chemically ordered $L1_1$ crystal structure, an intermediate temperature $A1$ phase, and finally a chemically ordered $L1_0$ crystal structure respectively. In order to map out the growth parameters we report on Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(40~nm)\/Pt(4~nm) sheet films grown at different set temperatures in the range from room temperature (RT) to 850$^{\\degree}$C. \n \n \n\\subsubsection*{Structure}\n\nIn order to understand the magnetic phases of sputter deposited CoPt, it is necessary to understand the underpinning structure and the influence of the growth temperature. Therefore, the structural characteristics and film quality were investigated using x-ray diffraction (XRD) as a function of growth temperature. Fig.~\\ref{fig:XRD} (a-c) shows the acquired XRD for selected growth temperatures around the main CoPt [111] structural peak. Further XRD data are available in the Supplemental Information online. The CoPt [111] structural peak is clearly observable for all growth temperatures. The presented data at growth temperatures of 350$^{\\degree}$C and 800$^{\\degree}$C show additional Pendellosung fringes which bracket the primary CoPt [111] peak. In Fig.~\\ref{fig:XRD} (a), for a growth temperature of 350$^{\\degree}$C, the additional feature at $2\\theta\\approx40^{\\degree}$ corresponds to the superimposed Pendellosung fringes and the Pt [111] structural peak (bulk Pt has a lattice constant of 3.92\\AA). This additional Pt structural peak is present for growths up to 550$^{\\degree}$C, see Supplementary Information online, which is within the expected optimal temperature range for sputter deposited Pt films on Al$_2$O$_3$ \\cite{4856395}.\n \n\nUsing the Scherrer equation and the full widths at half maxima (FWHM) of the Gaussian fits to the CoPt [111] structural peaks, an estimation of the CoPt crystallite sizes can be made. The FWHM for the main [111] structrual peak is given in Figure~\\ref{fig:XRD} (e). In the expected range of the ordered $L1_1$ crystal structure between 200$^{\\degree}$C and 400$^{\\degree}$C, the estimated CoPt crystallite size is 38~nm, which compared to the nominal thickness of 40~nm indicates that the CoPt has high crystallinity. On the other hand, at RT growth and intermediate temperatures between 400$^{\\degree}$C and 800$^{\\degree}$C, the estimated CoPt crystallite size is much lower, showing a minimum value of 22~nm at 700$^{\\degree}$C where we expect $A1$ growth. Interestingly, the disorder in the $A1$ growth appears to affect both the chemical disorder (random positions of the Co and Pt atoms in the unit cell) and a poorer crystallite size compared to the chemically ordered crystal structures. Finally, upon increasing the temperature further to 800$^{\\degree}$C and 850$^{\\degree}$C the crystallite size reaches a maximum of 52~nm. This value corresponds approximately to the entire thickness of the Pt\/CoPt\/Pt trilayer. \n\nFig.~\\ref{fig:XRD} (f) shows the CoPt $c$-plane space calculated from the center of the fitter Gaussian to the main [111] structural peak. At all temperatures the measured $c$-plane spacing is very close to the expected value based on single crystal studies \\cite{mccurrie1969magnetic}, indicating low out-of-plane strain. The trend with temperature, however, is non-monotonic and shows a discrete increase between the samples grown at 550$^{\\degree}$C and 650$^{\\degree}$C. The transition associated with the $L1_0$ crystal structure is expected to result in a transition from cubic to tetragonal crystal with a $c\/a=0.979$ in the single crystal \\cite{mccurrie1969magnetic}. In the single crystal study, however, the $c$-axis is in the [001] orientation. In our [111] orientated film, the small structural transition from cubic to tetragonal crystal structure is not expected to be visible in the [111] peak studied here. Nonetheless, Figure~\\ref{fig:XRD} (f) shows features consistent with the CoPt undergoing structural transitions.\n\n\n\n\n\n\n\n\\subsubsection*{Magnetic characterisation}\n\n\n\nThe magnetisation versus field data are shown in Fig. \\ref{fig:Magnetization_Growth_T} for Al$_2$O$_3$(sub)\/Pt(4 nm)\/CoPt(40 nm)\/Pt(4 nm) sheet film samples. Further magnetisation data for all samples in this study are available in the Supplemental Information online. The 350${\\degree}$C, 550${\\degree}$C, and 800${\\degree}$C samples are plotted here as they are representative of the magnetic response of the $L1_1$ crystal structure, the chemically disordered $A1$ phase, and $L1_0$ crystal structure respectively. Magnetisation is calculated from the measured total magnetic moments, the areas of the sample portions, and the nominal thicknesses of the CoPt layer. \n\nFor the chemically ordered $L1_1$ crystal structure shown in Fig. \\ref{fig:Magnetization_Growth_T} (a), the OOP hysteresis loop shows a wasp waisted behavior associated with the formation of magnetic domains at remanence. Such behavior is common in CoPt alloys and multilayer thin films \\cite{doi:10.1063\/1.113895}. The wasp waisted OOP hysteresis loop along with the low IP remanence and higher IP saturation field indicates that the 40~nm CoPt samples with the $L1_1$ crystal structure have strong PMA. We can estimate the uniaxial anisotropy using the expression $K_u = \\mu_0M_sH_s\/2$, where $\\mu_0$ is the vacuum permeability, $M_s$ the saturation magnetisation, and $H_s$ is the saturation magnetic field. From the data presented in Fig. \\ref{fig:Magnetization_Growth_T} (a) we find $K_u = 5\\pm1$ MJ\/m$^2$. \n\nFor samples grown at intermediate temperatures with the chemically disordered $A1$ structure, the magnetism favours IP anisotropy at 40~nm, shown for growth at 550${\\degree}$C in Fig. \\ref{fig:Magnetization_Growth_T} (b).\n\nFor the chemically ordered $L1_0$ crystal structure shown in Fig. \\ref{fig:Magnetization_Growth_T} (c), there is a significant increase in the coercivity and squareness ratio ($M_r\/M_s$) for both the IP and OOP field orientations. The increased coercive field suggests that the $L1_0$ CoPt is magnetically hard compared to the $L1_1$ and $A1$ samples. The magnetisation of the $L1_0$ 40~nm CoPt sample does not show clear IP or OOP anisotropy from these measurements. \n\nThe magnetisation vs growth temperature is shown in Fig. \\ref{fig:Magnetization_Growth_T} (d). At growth temperatures below 550${\\degree}$C the magnetisation remains approximately constant, however at higher temperatures the magnetisation begins to decrease with increasing temperature. The possible cause of this decrease is the higher growth temperature contributing to interdiffusion between the Pt and CoPt layers, creating magnetic dead layers.\n\nThe saturation field and squareness ratio vs growth temperature are shown in Fig. \\ref{fig:Magnetization_Growth_T} (e) and (f) respectively. The general trends can be seen in the differences observed in the hysteresis loops of Fig. \\ref{fig:Magnetization_Growth_T} (a-c). These trends allow us to characterise the growth temperatures corresponding to the different crystal growths in our samples, further supporting our conclusions from the XRD: the $L1_1$ crystal structure driving the magnetic response of samples grown from 200${\\degree}$C to 400${\\degree}$C, the $L1_0$ crystal structure for samples grown above 750${\\degree}$C, and the intermediate temperatures being the chemically disordered $A1$ phase. \n\n\nIn the $L1_0$ crystal structure, Fig. \\ref{fig:Magnetization_Growth_T} (c), the high squareness ratio for both field orientations suggests that the anisotropy axis of the material is neither parallel or perpendicular to the film. Instead, it is possible that the magnetic anisotropy is perpendicular to the layer planes, which are stacked in the [100] direction. \n\nTo further investigate the anisotropy, we pattern our $L1_1$ 350${\\degree}$C and $L1_0$ 800${\\degree}$C samples into Hall bars and perform angular dependent Hall resistivity, R$_\\text{xy}$($\\theta$), measurements, Fig~\\ref{fig:Hallbar}. The fabricated Hall bar and measurement geometry is shown in Fig~\\ref{fig:Hallbar} (a). R$_\\text{xy}$($\\theta$) for the $L1_1$ 350${\\degree}$C and $L1_0$ 800${\\degree}$C Hall bars are shown in  Fig~\\ref{fig:Hallbar} (b) and (c) respectively. For the $L1_1$ 350${\\degree}$C sample with out-of-plane anisotropy, R$_\\text{xy}$($\\theta$) shows a plateau close to out-of-plane field and a uniform response for angles in between. The plateau is interpreted as an angle forming between the magnetisation and applied field because of the anisotropy axis \\cite{doi:10.1063\/1.3262635}. In comparison, the $L1_0$ 800${\\degree}$C sample also shows a  R$_\\text{xy}$($\\theta$) plateau for out-of-plane applied field plus an additional plateau for applied field angles between about 45${\\degree}$ and 60${\\degree}$. We interpret the additional plateau in R$_\\text{xy}$($\\theta$) for the $L1_0$ 800${\\degree}$C CoPt sample as evidence for an additional anisotropy axis, which we propose is perpendicular the [100] direction. The [100] plane has a dihedral angle of 54.75${\\degree}$ with the [111] growth plane. Additional sources of anisotropy in our samples are interface anisotropy at the Pt\/CoPt interfaces, which would favour out-of-plane magnetisation for thin layers, and shape anisotropy, which for our thin films would favour in-plane magnetisation.  \n\nFigure \\ref{fig:IP_Rotation} shows the extracted coercive field as a function of in-plane rotator angle for the $L1_0$ 800${\\degree}$C sample. The data shows a clear six-fold symmetry. This is consistent with an easy axis for the plurality tetragonal $L1_0$ phase of $[001]$ when grown on $\\{111\\}$ planes - the $<100>$ directions of the parent cubic structure are inclined at $\\pm 45\\degree$ from the plane and are coupled with the three fold symmetry of $\\{111\\}$. The magnetometry data therefore strongly suggests that over the sample the $[001]_{L1_0}$ can be found in any of the three possible $<100>$ of the parent cubic structure without a strong preference for which of the possible twins grow.\n\n\n\n\n\n\\subsection*{CoPt properties as a function of thickness for $L1_1$ and $L1_0$ crystal structures}\n\nHaving obtained optimal growth conditions for CoPt with chemically ordered $L1_1$ and $L1_0$ crystal structures, in this section, we report on the properties of samples Al$_2$O$_3$ (sub)\/Pt (4~nm)\/CoPt ($d_\\text{CoPt}$)\/Pt (4~nm) with $d_\\text{CoPt}$ varied between 1~nm and 128~nm for both the $L1_1$ and $L1_0$ crystal structures.\n\n\n\n\n\\subsubsection*{\\label{MagL11}Magnetisation of $L1_1$ and $L1_0$ CoPt}\n\nThe hysteresis loops for Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(4~nm)\/Pt(4~nm) sheet films is shown in Fig.~\\ref{fig:Magnetization_d} (a) and (b) for the $L1_1$ (350${\\degree}$C) and $L1_0$ (800${\\degree}$C) crystal structures respectively. Hysteresis loops over the full thickness range are given in the Supplementary Information online. The moment\/area at saturation (or 6~T) verses nominal CoPt thickness are presented in Fig~\\ref{fig:Magnetization_d} (c) and (d).\n\nWe calculate the magnetisation ($M$) by fitting the moment\/area versus nominal CoPt thickness data. In order to account for interfacial contributions to the magnetisation of the CoPt, we model the system as a magnetic slab with possible magnetic dead layers and\/or polarised adjacent layers. Magnetic dead layers can form as a result of interdiffusion, oxidation, or at certain interfaces with non-ferromagnetic layers. At some ferromagnet\/non-ferromagnet interfaces, the ferromagnetic layer can create a polarisation inside the non-ferromagnetic layer by the magnetic proximity effect. Polarisation is particularly common at interfaces with Pt \\cite{doi:10.1063\/1.344903, PhysRevB.65.020405, PhysRevB.72.054430,  rowan2017interfacial, PhysRevB.100.174418}. To take these into account, we fit to the expression,\n\\begin{equation}\n\\label{M2}\n    \\text{moment\/area}= M  (d_\\text{CoPt} - d_i),\n\\end{equation}\n\\noindent where $d_i$ is the contribution to $M$ from any magnetic dead layers or polarisation. The resulting best fit and the moment\/area versus the nominal CoPt thickness is shown in Fig.~\\ref{fig:Magnetization_d}.\n\nFor $L1_1$ growth at 350${\\degree}$C, the result of fitting Eqn.~\\ref{M2} shown in Fig.~\\ref{fig:Magnetization_d} (c) gives $M=750\\pm 50$~emu\/cm$^3$. For $L1_0$ growth at 800${\\degree}$C, the thinnest 1~nm and 2~nm films did not display any magnetic response and are excluded from the analysis in Fig.~\\ref{fig:Magnetization_d}. The result of fitting Eqn.~\\ref{M2} shown in Fig.~\\ref{fig:Magnetization_d} (d) for the samples thicker than 2~nm gives $M=520\\pm 50$~emu\/cm$^3$. Interestingly, we find a significant difference in $M$ between the two crystal structures. It is possible that the differences in $M$ corresponds to a true difference in the saturation magnetisation of the two crystal structures. An alternative scenario is that the 6~T applied field is not large enough to fully saturate the $L1_0$ samples, leading to a reduced measured $M$.\n\nThe thickness dependence of the magnetic switching of the $L1_1$ samples is well summarised by the squareness ratio shown in Fig~\\ref{fig:Magnetization_d} (e). The thickest 40 and 128~nm samples are wasp waisted, as presented in Fig.~\\ref{fig:Magnetization_Growth_T}. At reduced thicknesses, between 2 and 8~nm, the $L1_1$ CoPt no longer displays the wasp waisted switching for out-of-plane field orientation, and now has a square loop shown in Fig.~\\ref{fig:Magnetization_d} (a). The 16~nm sample showed an intermediate behaviour. At 1~nm, the magnetic switching showed \"S\" shaped hysteresis loops for both in- and out-of-plane applied fields with small remanent magnetisation, see Supplementary Information online. \n\nThe thickness dependence of the $L1_0$ crystal structure samples is significantly different to the $L1_1$. In the thinnest films of 1 and 2~nm there is no evidence of ferromagnetic ordering in the hystersis loops, see Supplementary Information online. For $L1_0$ growth at 800${\\degree}$C, the high growth temperature may cause interdiffusion at the Pt\/CoPt interfaces. The interdiffusion may account for magnetic dead layers, which in the thinnest samples may prevent ferromagnetic ordering. Upon increasing the thickness to 4~nm, a ferromagnetic response was recovered, however the hysteresis loops and extracted squareness ratio (Fig.~\\ref{fig:Magnetization_d}) (f)) indicate that the 4~nm and 8~nm $L1_0$ samples have in-plane magnetisation. The in-plane magnetisation in the thinner films suggests that the long-range $L1_0$ ordering may have not established at those thicknesses. \n\nThe thickness dependence in both crystal structures suggest that the Pt\/CoPt\/Pt trilayers grown on Al$_2$O$_3$ substrates are not suitable for applications where ultrathin magnetic layers are required. To improve the magnetic properties of the thinnest samples in this study our future work will focus on replacing the Pt layers with seed and capping layers where interdiffusion may be less. \n\n\n\n\n\n\\subsection*{Spin polarisation}\n\nTo estimate the spin polarisation in the chemically ordered $L1_0$ and $L1_1$ CoPt samples, we perform Point Contact Andreev Reflection (PCAR) spectroscopy  experiments \\cite{doi:10.1126\/science.282.5386.85, baltz2009conductance, PhysRevB.85.064410, Seemann_2010, 6971749, PhysRevB.76.174435}. In the PCAR technique, spin polarisation in the ballistic transport regime can be determined from fitting the bias dependence of the conductance with the a modified Blonder-Tinkham-Klapwijk (BTK) model \\cite{PhysRevB.63.104510}. \n\nWe measure the Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt(128~nm)\/Pt(4~nm) samples grown at 350${\\degree}$C, corresponding to $L1_1$ crystal structure, and 800${\\degree}$C, corresponding to $L1_0$ crystal structure. The PCAR experiment was performed with a Nb wire tip at $4.2\\,$K. Exemplar conductance spectra with fits to the BTK model are given in Figure \\ref{fig:PCAR} (a). The interpretation of PCAR data is rife with difficulties\\cite{Yates2018} and a common issue with the PCAR technique is the presence of degenerate local fitting minima. To ensure that a global best fit is obtained, the fitting code makes use of a differential-evolution algorithm and we then consider the spin polarisation and barrier strength parameter for a large number of independent contacts to the same sample.\n\nFigure \\ref{fig:PCAR} (c) shows the dependence of the polarisation as a function of the square of the barrier strength, Z$^2$. The dashed lines in Figure \\ref{fig:PCAR} are linear fits to the data. The value of the true spin polarisation is is often taken to correspond to $Z=0$, however this is strictly nonphysical. Nevertheless, in an all metal system it is possible to produce contacts approaching an ideal case and extrapolating to $Z=0$ is close to the (finite) minimum. We find that P = 47$\\pm3$\\% for both $L1_1$ and $L1_0$ CoPt samples. This compares to $\\approx42$\\% for $L1_0$ FePt \\cite{PhysRevB.76.174435} and $\\approx50$\\% for $L1_0$ FePd \\cite{Seemann_2010}.\n\n\n\n\n \n\n\n\n\\section*{Conclusions}\n\nThe major conclusions of this work may be summarised as follows. On $c$-plane Al$_2$O$_3$ with thin Pt [111] buffer layers, the Co$_{50}$Pt$_{50}$ grows following the [111] ordering. Through growth at elevated temperatures, Co$_{50}$Pt$_{50}$ is grown epitaxially in the chemically ordered $L1_1$ and $L1_0$ crystal structures or the chemically disordered $A1$ phase. The $L1_1$ Co$_{50}$Pt$_{50}$ grown between 200$^\\circ$C and 400$^\\circ$C shows perpendicular magnetic anisotropy for thicknesses $\\geq2$~nm. The $L1_0$ Co$_{50}$Pt$_{50}$ grown above 800$^\\circ$C shows significantly harder magnetic anisotropy, which is perpendicular to the [100] direction for thicknesses $\\geq40$~nm. For growth at intermediate temperatures, $450<800^\\circ$C, the Co$_{50}$Pt$_{50}$ shows a disordered structure and in-plane magnetic anisotropy associated with the $A1$ phase. The spin-polarisation of the $L1_1$ and $L1_0$ Co$_{50}$Pt$_{50}$ is determined by the PCAR technique to be 47$\\pm3$\\%.\n\n\n\n\n\\section*{Methods}\n\n\n\n\n\\subsection*{Epitaxial growth}\n\nSamples are dc sputter deposited in the Royce Deposition System \\cite{Royce}. The magnetrons are mounted below, and confocal to, the substrate with a source-substrate distance of 134~mm. The base pressure of the vacuum chamber is $1\\times10^{-9}$~mbar. The samples are deposited at elevated temperatures, with an Ar (6N purity) gas pressure of $4.8\\times10^{-3}$~mbar.\n\nFor alloy growth, we use the co-sputtering technique. To achieve as close to a Co$_{50}$Pt$_{50}$ stoichiometry as possible, first, single layer samples of Co or Pt are grown at room temperature on 10x10~mm thermally oxidised Si substrates varying the magnetron power. From this initial study, it is found that a growth rate of 0.05~nm s$^{-1}$ is achieved for a Co power of 45W and a Pt power of 25W. These growth powers are fixed for the rest of the study.\n\nFor the growth of the CoPt samples, 20x20~mm $c$-plane sapphire substrates are used. The substrates are heated by a ceramic substrate heater mounted directly above the substrate holder. The measured substrate heater temperature is reported. We note that the temperature on the substrate surface is most likely to be below the reported heater temperature. The substrate heater is ramped up from room temperature to the set temperature at a rate of 3-5${\\degree}$C min$^{-1}$. Once at the set temperature, the system is given 30~min to reach equilibrium before starting the sample growth.\n\nOnce the system is ready for growth, 4~nm Pt seed layer is deposited. The seed layer is immediately followed by the  CoPt layer, which is deposited at a rate of 0.1~nm s$^{-1}$ by co-sputtering from the two targets at the determined powers. Finally, a 4~nm Pt capping layer is deposited to prevent the samples from oxidising. The final sample structure is Al$_2$O$_3$(sub)\/Pt(4~nm)\/CoPt($d_{CoPt}$)\/Pt(4~nm). Following deposition, the samples are post growth annealed for 10~min at the growth temperature before the substrate heater is ramped down to room temperature at 10${\\degree}$C min$^{-1}$.\n\n\n\n\\subsection*{Characterisation}\n\n\n Magnetisation loops are measured using a Quantum Design MPMS 3 magnetometer. Angular dependent magnetization measurements are performed using the Quantum Design Horizontal Rotator option. X-ray diffraction and reflectivity is performed on a Bruker D8 diffractometer with an additional four-bounce monochromator to isolate Cu K-$\\alpha$ at a wavelength of 1.5406~\\AA. Sheet films are patterned into Hall bars of 5~$\\mu$m width using conventional photolithography and Ar ion milling. Resulting devices are measured using 4-point-probe transport to measure the Hall resistance of the films using a combined Keithley 6221-2182A current source and nano-voltmeter.\n\n\\subsection*{PCAR}\n\nOur experimental setup for performing PCAR measurements is described elsewhere \\cite{baltz2009conductance, PhysRevB.85.064410, Seemann_2010, 6971749, PhysRevB.76.174435}. The Nb tips are prepared from commercial 99.9\\% pure Nb wires with a diameter of 0.5~mm. An AC lock-in detection technique using Stanford Research Systems SR830 lock-in amplifiers is used for the differential conductance measurements. The tip position is mechanically adjusted by a spring-loaded rod driven by a micrometer screw. The experiment is carried out in liquid He at a fixed temperature of $4.2\\,$K and at zero applied magnetic field.\n\n\n\n\\section*{Data Availability}\n\nThe datasets generated during the current study are available in the University of Leeds repository, https:\/\/doi.org\/xx.xxxx\/xxx.\n\n\n\n\\section*{X-ray Diffraction Data}\n\nFigures \\ref{fig:XRD_ALL_1}, \\ref{fig:XRD_ALL_2}, \\ref{fig:XRD_ALL_3} show a complete set of X-ray diffraction scans for samples grown at each temperature. The very sharp peak at $2\\theta\\approx42\\degree$ is from the c-plane sapphire $(0\\,0\\,0\\,6)$ plane with $\\{1\\,1\\,1\\}$ peaks from the CoPt phase clearly visible to left. The substrate $(0\\,0\\,0\\,12)$ and CoPt $\\{2\\,2\\,2\\}$ peaks are also present close to $2\\theta$ of $90\\degree$. Samples grown at $200-450\\degree\\,$C and also at $800\\degree\\,$C clearly exhibit Pendellosung fringes, indicative of good crystal ordering from the L$1_1$ and L$1_0$ phases. The peak intensity and FWHM shown in the main text are extracted from a Gaussian peak fit to the CoPt $\\{1\\,1\\,1\\}$ peak data, although  similar data could be obtained from the CoPt $\\{2\\,2\\,2\\}$ data. \n\n\\section*{Magnetometry}\n\nMagnetic hysteresis loops were measured for a set of samples of common thickness grown at temperatures from room temperature to $850\\degree\\,$C with the field applied in the plane of the substrate and perpendicular to the substrate plane - shown in Figure \\ref{fig:stream}. For the data shown here, a diamagnetic background resulting from the substrate and sample stick has been removed. The saturation magnetisation is determine by fitting a line to the outer section of the upper and lower branches and taking the difference in intercepts with the $H=0$ axis. The uncertainty is dominated by the determination of the sample area. The saturation field is determined from the point at which the measured data has deviated from the saturation magnetisation by more than twice the scatter in the data in the saturated regions of the loop. Figures \\ref{fig:stream2} and \\ref{fig:stream3} show similar magnetic hysteresis loops for samples of increasing thickness grown at $350\\degree\\,$C and $800\\degree\\,$C respectively.\n\n\\section*{PCAR}\n\nThe Point Contact Andreev Reflection spectra are measured by an AC lockin technique where the contact is subjected to a controlled DC bias with a small AC modulation which is sensed across the contact and across a control resistor to allow a differential conductance to be determined. The DC bias is separately monitored as it is adjusted to give a tip bias typically in the range of $\\approx\\pm30\\,$meV. The DC bias is swept over 6 quarters from 0 to maximum and then to minimum and back to maximum and finally to zero. We only analyse the parts of the trace from maximum to minimum and back to maximum. The measured DC bias is subject to a small instrumentation offset which is removed at this first stage of data processing. Since all the features of interest in the spectra should be symmetrical about zero bias, we decompose the data into symmetric and anti-symmetric parts and retain only the symmetric part. The BTK model is usually considered in terms of the normalised conductance of the contact -- normalised to the conductance at high bias where transfer between electron\/hole states in the ferromagnetic and electron\/hole-like quasiparticles in the superconductor. In this high bias regime it is possible in some contact to get parabolic conductance due to tunnelling or Joule heating effects\\cite{baltz2009conductance} and so we normalise the experimental data by dividing through a parabola fitted to the outer 20\\% of the spectra.\n\nThe modified BTK model used in the analysis of the PCAR data has 4 fitting parameters. Firstly, an interfacial barrier strength $Z$ that is dimensionless. This effectively determines a scattering potential at the interface and so even in a perfect contact will take a small, but finite, value. Secondly, the spin polarisation $P=\\frac{N_\\uparrow v_{f\\uparrow}-N_\\downarrow v_{f\\downarrow}}{N_\\uparrow v_{f\\uparrow}+N_\\downarrow v_{f\\downarrow}}$ which thus ranges from 0 for non-spin-polarised materials to 1 for fully spin-polarised materials. Thirdly, the superconducting energy gap $\\Delta$ which might reasonably be expected not to be fully bulk like at the tip of the contact. Finally there is a `smearing parameter` $\\omega$ that encompasses both thermal effects and athermal scattering processes not otherwise accounted for in the model.\n\nThe modified BTK model is then fitted to the now normalised data. We start with an initial set of fitting values of $Z=0.5$, $\\omega=0.5\\,$meV (c.f. $4.2\\,$K $\\equiv 0.36\\,$meV), $\\Delta=1.5\\,$meV and $P=0.5$. We use a differential-evolution algorithm to identify a global minimum $\\chi^2$ value to locate the best fit. This best-fit location is then used to seed a non-linar least squares fit using the Levenberg\u2013Marquardt algorithm, allowing us to determine estimated standard errors for the fitting parameters.\n\nWe then discard fits with non-physical values of $\\Delta$, excessibley large $\\omega$ -- both of which would indicate an unaccounted spreading resistance, before considering all the fitted $(Z,P)$ combinations. As noted in the main text, the extrapolation back to $Z^2=0$ is not strictly physical, however, in practice in an all-metal system, very low values of $Z$ with uncertainties that encompass $Z=0$ can be obtained.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_1}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_2}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{XRD_ALL_3}\n\\caption{X-ray diffraction for the growth temperature set of samples reported in the main text. Left column: Full data range. Right column: Cropped region around the 111 structural peak.}\n\\label{fig:XRD_ALL_3}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_Growth_T_ALL}\n\\caption{Volume magnetisation for the growth temperature set of samples reported in the main text. The thickness of all samples is 40~nm.}\n\\label{fig:stream}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_350_ALL}\n\\caption{Volume magnetisation for the thickness series set of samples reported in the main text for growth at 350$^\\circ$C.}\n\\label{fig:stream2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{Magnetization_800_ALL}\n\\caption{Volume magnetisation for the thickness series set of samples reported in the main text for growth at 800$^\\circ$C. For the 1~nm and 2~nm samples which did not show ferromagnetism, the uncorrected moment vs field is presented.}\n\\label{fig:stream3}\n\\end{figure}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith the establishment of the standard model (SM) by the discovery of the Higgs boson, searching for physics \nbeyond the SM (BSM) and understanding the electroweak phase transition have become main topics in  particle physics.\nAs one scenario of BSM, the gauge-Higgs unification (GHU) scenario has been studied\n\\cite{Manton,Hosotani1983,Hosotani1988,\nDavies:1987,Davies:1988,\nHIL,Hatanaka1999,Kubo:2001zc,Burdman:2002se,\nCsaki:2002,\nScrucca2003,\nAgashe,\nCacciapaglia2006,\nMedina,\nFalkowski}.\nIn GHU, the Higgs boson is a part of the extra-dimensional \ncomponent of gauge potentials, appearing as a fluctuation mode of an \nAharonov-Bohm (AB) phase $\\theta_H$ in the fifth dimension.\nMany GHU models are proposed for electroweak unification \\cite{\nHosotani:2007qw,\nHosotani:2008by,\nFunatsu:2013ni,\nFunatsu:2014fda,\nFunatsu:2016uvi,\nFunatsu:2017nfm,\nHatanaka:2013iya,\nFunatsu:2019xwr,\nFunatsu:2019fry,\nFunatsu:2020haj,\nFunatsu2019a,\nFunatsu-FT,\nFunatsu:2021yrh,\nYoon2017,Yoon2018,Yoon2019,\nKurahashi:2014jca,\nMatsumoto:2016okl,\nHasegawa:2018jze,\nKakizaki:2021kof\n},\nand  GHU models for grand unification are also proposed \\cite{\nLim2007,\nHosotani:2015hoa,\nFurui,\nHosotani:2017edv,\nEnglert:2019xhz,\nEnglert:2020eep,\nKakizaki:2013eba,\nKojima:2017qbt,\nMaru:2019bjr,\nAngelescu:2021nbp\n}. \nAmong them, two types of $\\mathrm{SU}(3)_{C} \\times \\mathrm{SO}(5) \\times \\mathrm{U}(1)_{X}$ GHU models,\nthe A-model\\cite{\nHosotani:2007qw,\nHosotani:2008by,\nFunatsu:2013ni,\nFunatsu:2014fda,\nFunatsu:2016uvi,\nFunatsu:2017nfm,\nHatanaka:2013iya}\nand B-model\\cite{Funatsu:2019xwr,Funatsu:2019fry,Funatsu:2020haj,\nFunatsu2019a,\nFunatsu-FT,\nFunatsu:2021yrh},  in warped spacetime have been extensively studied. \nAt low energies below the electroweak scale the mass spectrum and gauge and Higgs couplings of\nSM particles are nearly the same as in the SM.\n\nCouplings of the first Kaluza-Klein (KK) neutral gauge bosons to right-handed SM fermions are large in the A-model,\nwhereas those to left-handed SM fermions become large in the B-model.\nIn proton-proton collisions KK bosons of photons and $Z$ bosons appear as huge broad resonances\nof $Z'$ bosons in the Drell-Yan process, and can be seen in current and future hadron collider experiments\n \\cite{Funatsu:2014fda,Funatsu:2016uvi,Funatsu:2021yrh}.\nThe KK-excited states of the $W$ boson are also seen as resonances of $W'$ bosons. \nIn the A-model the couplings of the first KK $W$ boson to \nthe SM fermions are small.\nIn the B-model the couplings of the first KK $W$ boson to right-handed fermions are negligible, while couplings to left-handed fermions are much larger than the $W$ boson couplings.\nTherefore at the LHC the first KK $W$ boson appears as a  narrow resonance of $W'$ boson in the A-model, \nbut appears as a broad resonance in the B-model\\cite{Funatsu:2016uvi,Funatsu:2021yrh}.  \n\n\nIn  $\\mathrm{e}^{+}\\mathrm{e}^{-}$ collider experiments,   effects of GHU can be examined\nby exploring interference effects among photon, $Z$ boson and $Z'$ bosons.\nIn the previous papers we have studied  effects of new physics (NP) on\nsuch observable quantities  as cross section, forward-backward asymmetry and left-right asymmetry \nin $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) with polarized and unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams\n\\cite{Funatsu:2017nfm,Funatsu:2020haj,\nYoon2018,\nYoon2019,\nBilokin2017,\nIrles2019,\nIrles2020}.\nIn Ref.\\ \\cite{Funatsu:2017nfm} we compared such observable quantities of GHU with those of the SM in LEP experiments\nat $\\sqrt{s} = M_Z$, and LEP2 experiments for $130 \\text{GeV} \\le \\sqrt{s} \\le 207 \\,\\text{GeV}$\n\\cite{ALEPH:2005ab,ALEPH:2013dgf}.\nIn Refs.~\\cite{Funatsu:2017nfm,Funatsu:2020haj} we also gave predictions of several  signals of $Z'$ bosons in \nGHU in  $\\mathrm{e}^{+}\\mathrm{e}^{-} $ collider experiments \ndesigned  for future  with collision energies $\\sqrt{s} \\ge 250 \\text{ GeV}$ with polarized electron and positron beams.\nIn the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) modes,\nthe deviations of total cross sections become large for\nright-polarized electrons in the A-model, whereas\nin the B-model the deviations are large for left-polarized electrons.\n\nDeviations from the SM can be seen in the Higgs couplings as well.\n$HWW$, $HZZ$ and Yukawa couplings deviate from those in the SM in a universal manner\\cite{Hosotani:2007qw,Hosotani:2008by,Kurahashi:2014jca}.\nThey are suppressed by a common factor\n\\begin{eqnarray}\n\\frac{g_{HWW}^{\\text{GHU}}}{g_{HWW}^{{\\rm SM}}},\\,\n\\frac{g_{HZZ}^{\\rm \\text{GHU}}}{g_{HZZ}^{{\\rm SM}}}\n\\simeq\n\\cos(\\theta_H) \n\\end{eqnarray}\nfor $W$ and $Z$ bosons, and \n\\begin{eqnarray}\n\\frac{y_{\\bar{f}f}^{\\text{GHU}}}{y_{\\bar{f}f}^{{\\rm SM}}}\n\\simeq\n\\begin{cases}\n\\cos(\\theta_{H}) & \\text{A-model \\cite{Hosotani:2008by}} \\\\\n\\cos^{2}(\\theta_{H}\/2) & \\text{B-model \\cite{Funatsu:2019fry}} \n\\end{cases}\n\\label{Hcoupling1}\n\\end{eqnarray}\nfor SM fermions $f$.\nIn the analysis of both $Z'$ and $W'$ bosons in hadron colliders\n\\cite{Funatsu:2014fda,Funatsu:2016uvi,\nFunatsu:2021yrh},\nit is found that the AB phase is constrained as $\\theta_{H} \\lesssim 0.1$.\nFor $\\theta_H = \\mathcal{O}(0.1)$,  probable values in the model,    \nthe deviation of the couplings amounts to $(1-\\cos\\theta_H) = \\mathcal{O}(0.005)$, and is small.\nAt the International Linear Collider (ILC) at $\\sqrt{s} = 250\\text{GeV}$, the $ZZH$ coupling can be measured \nin the $0.6 \\%$ accuracy with $2\\text{ ab}^{-1}$ data \\cite{Barklow:2017suo}.\nSince the masses and Higgs couplings of the SM fields in the GHU models are very close to those in the SM, \nthe electroweak phase transition (EWPT) occurs\nat $T_{C} \\sim 160 \\text{GeV}$ and appears very weak first order\\cite{Hatanaka:2013iya,Funatsu-FT}\nin both A- and B-models, which is very similar to EWPT in the SM\\cite{Senaha:2020mop}.\n\nIn this paper we study  effect of $Z'$ bosons in GHU models\non the $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-}$ Bhabha scattering.\nMeasurements of  Bhabha scattering at  linear colliders have\ncontributed to the establishment of the SM\\cite{\nAbe:1994sh,\nAbe:1994wx,\nAbe:1996nj\n}.\nBhabha scattering is also useful to explore NP \\cite{\nRichard:2018zhl,\nDas:2021esm\n}.\nUnlike $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to f\\bar{f}$ ($f\\ne \\mathrm{e}$) scattering, \nin Bhabha scattering not only $s$-channel but also  $t$-channel contribution enter the process.\nSince $t$-channel contribution of photon exchange dominates in   forward scattering amplitudes,\nthe cross section becomes very large for  small scattering angles,\nwhich improves the statistics of experiments.\nIt will be seen below that effects of $Z'$ bosons on cross sections can be measured \nwith large significances.\n\nBhabha scattering at very small scattering angle\nis used for the determination of the luminosity of the $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams.\nSince cross sections of  all other scattering processes depend on the luminosity, \none needs to know whether or not effects of $Z'$ bosons on the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ cross section\nare sufficiently small.\nForward-backward asymmetry $A_{FB}$ of the cross section in Bhabha scattering  is no longer a good quantity \nfor searching NP, since the backward scattering cross section is much smaller than the forward scattering cross section.\nWe will propose a new quantity $A_X$ to measure with polarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams,\nwhich can be used for seeing NP effects instead of $A_{FB}$.\n\nIn Section 2 we briefly review the GHU A- and B-models  and \ndiscuss the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering \nin both the SM and GHU models. In Section 3, we show the formulas of\n$\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering cross sections for longitudinally polarized $\\mathrm{e}^{\\pm}$ beams, \nand numerically evaluate the effects\nof $Z'$ bosons in GHU models on  differential cross sections and left-right asymmetries.\nWe also show that effects of $Z'$ bosons on the cross section\nare very small at the very small scattering angle.\n\n\\section{Gauge-Higgs unification}\n\nIn GHU A- and B-models the electroweak $\\mathrm{SU}(2)\\times\\mathrm{U}(1)$ symmetry\nis embedded in $\\mathrm{SO}(5)\\times\\mathrm{U}(1)_{X}$ symmetry in the Randall-Sundrum warped space \\cite{Randall:1999ee},\nwhose metric is given by\n\\begin{align}\nds^{2} &= \\frac{1}{z^{2}} \\left[\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu} \n+ \\frac{dz^{2}}{k^{2}}\n\\right],\n\\quad\n1 \\le z \\le z_{L} = e^{kL}\n\\end{align}\nwhere $\\eta_{\\mu\\nu} = \\diag(-1,+1,+1,+1)$ and $k$ is the AdS-curvature.\n We refer two 4D hyperplanes at $z=1$ and $z=z_{L}$ as the UV and IR branes, respectively.\nThe $\\mathrm{SO}(5)$ symmetry is broken to $\\mathrm{SO}(4)\\simeq \\mathrm{SU}(2)_{L} \\times \\mathrm{SU}(2)_{R}$ by the boundary conditions at $z=1,\\,z_{L}$ and\nthe $\\mathrm{SU}(2)_{R} \\times \\mathrm{U}(1)_{X}$ symmetry is broken to $\\mathrm{U}(1)_{Y}$ by a scalar field \nlocalized on the UV brane.\n$\\mathrm{SU}(2)_{L}\\times\\mathrm{U}(1)_{Y}$ symmetry is broken to the \nelectromagnetic $\\mathrm{U}(1)_{\\rm EM}$ symmetry by\nthe VEV of the $z$-component gauge fields of $\\mathrm{SO}(5)\/\\mathrm{SO}(4)$. \nThe VEV is related to the gauge-invariant AB phase $\\theta_{H}$. \n\nThe difference between the A- and B-models lies in the content of fermions.\nIn the A-model, quarks and leptons in the SM are embedded in the $\\mathrm{SO}(5)$-vector representation. \nIn the B-model quarks and leptons are embedded in the $\\mathrm{SO}(5)$-spinor, vector and singlet representations.\nThese fields are naturally derived from spinor and vector multiplets in the $\\mathrm{SO}(11)$ \ngauge-Higgs grand unification\\cite{Hosotani:2015hoa, Furui}. \n\nInteractions of the electron and gauge bosons are given by\n\\begin{align}\n\\int d^{4}x \\int_{1}^{z_{L}} \\frac{dz}{k}\n\\biggl\\{\n\\bar{\\check{\\Psi}} \\gamma^{\\mu} (\\partial_{\\mu} - i g A_{\\mu}) \\check{\\Psi}\n\\biggr\\}\n\\end{align}\nwhere $A_{\\mu}(x^{\\mu},z)$ is a four-dimensional component of the 5D gauge field and\n$\\Psi(x^{\\mu},z) = z^2 \\check{\\Psi}$ is the 5D electron field.\nThe electron corresponds to the zero mode of the $\\check{\\Psi}(x^{\\mu},z)$.\nIn the A-model  the left-handed electron is localized in the vicinity of the UV brane and the right-handed \ncomponent is localized near the IR brane.\nIn the B-model the right-handed electron is localized in the vicinity of the UV brane and the left-handed \ncomponent is localized near the IR brane.\n$A_{\\mu}(x^{\\mu},z)$ has a KK expansion which contains the photon, $Z$ boson and their KK excited modes. \nThe wave function of the photon is constant in the fifth dimension coordinate $z$.\nThe wave function of  the $Z$ boson is almost constant in $z$, but has nontrivial dependence near the IR brane. \nCouplings of the electron to  $Z$ boson are very close to those in the SM.\nWave functions of the first KK-excited modes of gauge bosons  are localized near the IR brane so that \nthe first KK-excited gauge bosons  couple  strongly with   fermions localized near the IR brane.\nIn the A-model  right-handed electrons have large couplings with first KK-excited gauge bosons \nwhereas in the B-model left-handed electrons couple strongly with the first KK-excited gauge bosons. \n\nIn Tables~\\ref{tbl:model} and \\ref{tbl:couplings}, parameters and couplings in the A- and B-models are tabulated.\nHere, model parameters ($\\theta_H,m_{{\\rm KK}}$ and $z_{L}$),\nmasses, widths and couplings of $Z'$-bosons\nare referred from Refs.~\\cite{Funatsu2019a,Funatsu:2020haj}.\n\\begin{table}[t]\n\\caption{Parameters in GHU models.}\\label{tbl:model}\n\\vspace{5mm}\n\\centering\n\\begin{tabular}{c|ccc|cccccc}\n\\hline\\hline\nModel & $\\theta_{H}$ & $m_{{\\rm KK}}$ & $z_{L}$ \n& $m_{\\gamma^{(1)}}$ & $\\Gamma_{\\gamma^{(1)}}$  \n& $m_{Z^{(1)}}$ & $\\Gamma_{Z^{(1)}}$  \n& $m_{Z_{R}^{(1)}}$ & $\\Gamma_{Z_{R}^{(1)}}$   \n\\\\\n& [rad] & [TeV] & & [TeV] & [TeV] & [TeV] & [TeV] & [TeV] & [TeV] \n\\\\\n\\hline\nA & $0.08$ & $9.54$ & $1.01\\times10^{4}$\n& $7.86$ & $0.99$\n& $7.86$ & $0.53$\n& $7.31$ & $1.01$\n\\\\\nB & $0.10$ & $13.0$ & $3.87\\times10^{11}$ \n& $10.2$ & $3.25$ \n& $10.2$ & $7.84$ \n& $9.95$ & $0.816$  \n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nThe difference in the magnitude  of $z_{L}$  in the A- and B-models\noriginates from the formulas of top-quark mass.\nIn the A-model \n$\nm_{\\rm top}^{\\rm A} \\simeq (m_{{\\rm KK}}\/(\\sqrt{2}\\pi)) \\sqrt{1 - 4c_{\\rm top}^{2}} \\sin\\theta_{H}\n$ \\cite{Funatsu:2013ni}\nwhereas in the B-model \n$\nm_{\\rm top}^{\\rm B} \\simeq (m_{{\\rm KK}}\/\\pi) \\sqrt{1-4c_{\\rm top}^{2}} \n\\sin\\frac{1}{2}\\theta_{H}$ \\cite{Funatsu:2019xwr}.\nIn both models $W$ boson mass is given by\n$m_{W} \\simeq  m_{{\\rm KK}}\/(\\pi \\sqrt{kL}) \\sin\\theta_{H}$\nso that the lower bounds of $z_{L}$ becomes\n$z_{L} \\gtrsim 8\\times10^{3}$ in the A-model  and $z_{L} \\gtrsim 7 \\times 10^{7}$ in the B-model.\n\\begin{table}[t]\n\\caption{\nLeft-handed and right-handed couplings of the electron, \n$\\ell_{V},r_{V}$ ($V = Z,Z^{(1)}$, $Z_{R}^{(1)}$ and $\\gamma^{(1)}$), in unit of $g_{w}\\equiv g_{A}\/\\sqrt{L}$ (see text).\nRatios of $e$ and $g_{w}$ are shown at the second column.}\\label{tbl:couplings}\n\\vspace{5mm}\n\\centering\\small\n\\begin{tabular}{c|cccccccccc}\n\\hline\\hline\nModel & $(e\/g_{w})^{2}$ \n & $\\ell_{Z}$ & $r_{Z}$ \n & $\\ell_{Z^{(1)}}$ & $r_{Z^{(1)}}$ \n & $\\ell_{Z_{R}^{(1)}}$ & $r_{Z_{R}^{(1)}}$ \n & $\\ell_{\\gamma^{(1)}}$ & $r_{\\gamma^{(1)}}$ & \n\\\\\n\\hline\nA & $0.2312$\n& $-0.3066$ & $0.2638$\n& $0.1195$ & $0.9986$\n& $0.0000$ & $-1.3762$ \n& $0.1879$ & $-1.8171$ \\\\\nB  & $0.2306$ \n& $-0.3058$ & $0.2629$ \n& $-1.7621$ & $-0.0584$ \n& $-1.0444$ & $0.0000$ \n& $-2.7587$ & $0.1071$\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nIn Table~\\ref{tbl:couplings}, the left- and right-handed electron couplings to $Z'$ bosons, $r_{V}, \\ell_{V}$ ($V = Z,Z^{(1)},Z_{R}^{(1)},\\gamma^{(1)}$), are tabulated.\nIn the table $g_{w}\\equiv g_{A}\/\\sqrt{L}$ is the 4D gauge coupling of the $\\mathrm{SO}(5)$ where\n$g_{A}$ is the 5D $\\mathrm{SO}(5)$ coupling.\nIn terms of $g_{A}$ and the 5D $\\mathrm{U}(1)_{X}$ coupling $g_{B}$, a mixing parameter is defined as \\cite{Funatsu:2014fda,Funatsu:2020haj}\n\\begin{align}\ne\/g_{w} = \\sin\\theta_{W}^{0} &\\equiv \n\\frac{s_{\\phi}}{\\sqrt{1 + s_{\\phi}^{2}}},\n\\quad\ns_{\\phi} \\equiv g_{B}\/\\sqrt{ g_{A}^{2} + g_{B}^{2}}.\n\\end{align}\nThe value of $\\sin\\theta_{W}^{0}$ is determined so as to reproduce \nthe experimental value of the forward-backward asymmetry in \n$\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mu^{+}\\mu^{-}$ scattering at the $Z$-pole.\nIn the A-model electron's right-handed couplings to $Z'$-bosons are larger than left-handed couplings.\nIn the B-model electron's left-handed couplings to $Z'$-bosons are larger than right-handed couplings.\n\n\n\\section{Bhabha scattering in $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders}\n\nWe consider the $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-}$ scattering in the center-of-mass frame.\nIn this frame, the Mandelstam variables $(s,t,u)$ are given by\n\\begin{align}\ns &= 4 E^{2},\n\\nonumber \\\\\nt &=-\\frac{s}{2}(1-\\cos\\theta) = -s \\sin^{2}\\frac{\\theta}{2},\n\\nonumber \\\\\nu &=-\\frac{s}{2}(1+\\cos\\theta) = -s \\cos^{2} \\frac{\\theta}{2}\n\\end{align}\nwhere $E$  is the energy of initial electron and positron, and $\\theta$ is the scattering angle of the electron.\nSince the $\\mathrm{e}^{+}\\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ scattering process consists\nboth $s-$ and $t-$channel processes,\nthe scattering amplitude is written in terms of the following six building blocks:\n\\begin{align}\nS_{LL} = S_{LL}(s) \n&\\equiv \n\\sum_{i} \\frac{\\ell_{V_{i}}^{2}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nS_{RR}= S_{RR}(s) \n&\\equiv\n\\sum_{i} \\frac{r_{V_{i}}^{2}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nS_{LR} = S_{LR}(s) &\\equiv\n\\sum_{i} \\frac{\\ell_{V_{i}} r_{V_{i}}}{s - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{LL} = T_{LL}(s,\\theta) &\\equiv\n\\sum_{i} \\frac{\\ell_{V_{i}}^{2}}{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{RR} = T_{RR}(s,\\theta) &\\equiv\n\\sum_{i} \\frac{r_{V_{i}}^{2}}{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\nonumber \\\\\nT_{LR} = T_{LR}(s,\\theta) &\\equiv \n\\sum_{i} \\frac{\\ell_{V_{i}} r_{V_{i}} }{t - M_{V_{i}}^{2} + iM_{V_{i}}\\Gamma_{V_{i}}},\n\\label{eq:blocks}\n\\end{align}\nwhere $M_{V_{i}}$ and $\\Gamma_{V_{i}}$ are mass and width of the vector boson $V_{i}$.\n$\\ell_{V_{i}}$ and $r_{V_{i}}$ are left- and right-handed couplings of electrons to the vector boson $V_{i}$ ($V_{0} =\\gamma$, $V_{1} = Z$), respectively. \nIn particular, we have\n$\\ell_{\\gamma} = r_{\\gamma} = Q_{\\mathrm{e}} e$, $Q_{\\mathrm{e}} = -1$,\n$\\ell_{Z} = \\frac{e}{\\sin\\theta_{W}\\cos\\theta_{W}} [I_{\\mathrm{e}}^{3} - Q_{e}\\sin^{2}\\theta_{W}]$,\n$r_{Z} = \\frac{e}{\\sin\\theta_{W}\\cos\\theta_{W}} [- Q_{e}\\sin^{2}\\theta_{W}]$,\n $I^{3}_{\\mathrm{e}} = -\\frac{1}{2}$ in the SM. \nHere $e$, $I_{\\mathrm{e}}^{3}$ and $\\theta_{W}$ are the electromagnetic coupling, weak isospin of the electron and Weinberg angle, respectively.\n\nWhen  initial state electrons and positrons are longitudinally polarized, \nthe differential cross section is given by\n\\begin{align}\n\\frac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}}, P_{\\mathrm{e}^{+}})\n&= \\frac{1}{4} \\biggl\\{\n(1+ P_{\\mathrm{e}^{-}}) (1+P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R} }}{d\\cos\\theta}\n+\n(1 - P_{\\mathrm{e}^{-}}) (1 - P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L} }}{d\\cos\\theta}\n\\nonumber\\\\& \\qquad\n+\n(1 + P_{\\mathrm{e}^{-}}) (1 - P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L} }}{d\\cos\\theta}\n+\n(1 - P_{\\mathrm{e}^{-}}) (1 + P_{\\mathrm{e}^{+}}) \n\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R} }}{d\\cos\\theta}\n\\biggr\\},\n\\end{align}\nwhere $P_{\\mathrm{e}^{-}}$ and $P_{\\mathrm{e}^{+}}$ are the polarization of the electron and positron beam, respectively. $P_{\\mathrm{e}^{-}}=+1$ ($P_{\\mathrm{e}^{+}}=+1$) denotes purely right-handed electrons (positrons) \\cite{MoortgatPick:2005cw,Arbuzov:2020ghr}.\n$\\sigma_{\\mathrm{e}_{X}^{-} \\mathrm{e}_{Y}^{+}}$ ($X,Y=L,R$) denotes the cross section \nfor left-handed or right-handed electron and positron. \nWhen the electron mass is neglected, these cross sections \nare given by\n\\begin{align}\n\\frac{d\\sigma_{\\mathrm{e}^-_L \\mathrm{e}^+_R}}{d\\cos\\theta}\n&= \\frac{1}{8\\pi s}\n\\left(  u^2 |S_{LL} + T_{LL}|^2 + t^2 |S_{LR}|^2 \\right),\n\\nonumber \\\\\n\\frac{d\\sigma_{\\mathrm{e}^-_R \\mathrm{e}^+_L}}{d\\cos\\theta}\n&= \\frac{1}{8\\pi s}\n\\left(  u^2 |S_{RR} + T_{RR}|^2 + t^2 |S_{LR}|^2 \\right),\n\\nonumber \\\\\n\\frac{d\\sigma_{\\mathrm{e}^-_L \\mathrm{e}^+_L}}{d\\cos\\theta}\n&= \\frac{d\\sigma_{\\mathrm{e}^-_R \\mathrm{e}^+_R}}{d\\cos\\theta}\n= \\frac{1}{8\\pi s} \\cdot \\left(s^2 |T_{LR}|^2 \\right).\n\\end{align}\n\nWhen $s,t \\ll M_{Z}^{2}$, \nthe cross section is  approximated by the one at the QED level,\nwhere we obtain $S_{LL} = S_{RR} = S_{LR} = e^2\/s$ and\n$T_{LL} = T_{RR} = T_{LR} = e^2\/t$, and \n\\begin{align}\n\\frac{d\\sigma_{\\rm QED}}{d\\cos\\theta}(P_{e^-},P_{e^+})\n&= \n\\frac{e^{4}}{16\\pi s}\\left\\{\n(1 - P_{\\mathrm{e}^{-}}P_{\\mathrm{e}^{+}} ) \\frac{t^{4}+u^{4}}{s^{2}t^{2}}\n+ (1 + P_{\\mathrm{e}^{-}} P_{\\mathrm{e}^{+}}) \\frac{s^{2}}{t^{2}}\n\\right\\}.\n\\end{align}\nFor unpolarized electron or positron beams, the above expression reduces to\na familiar formula\n\\begin{align}\n\\frac{d\\sigma^{\\rm unpolarized}_{\\rm QED}}{ d\\cos\\theta }\n&=\n \\frac{e^4}{16\\pi s} \\frac{s^4 + t^4 + u^4}{s^2 t^2}.\n\\end{align}\nWe also note that in terms of building blocks \\eqref{eq:blocks} we \ncan write down components of s-, t-channels, and interference terms as\n\\begin{align}\n\\frac{d\\sigma}{d\\cos\\theta}\n&=\\frac{d\\sigma^{\\text{s-channel}}}{d\\cos\\theta}\n+ \\frac{d\\sigma^{\\text{t-channel}}}{d\\cos\\theta}\n+ \\frac{d\\sigma^{\\text{interference}}}{d\\cos\\theta},\n\\end{align}\nwhere each component is given by\n\\begin{align}\n\\frac{d\\sigma^{\\text{s-channel}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{32\\pi s} \\biggl\\{\n  (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) \\left[ u^{2} |S_{RR}|^{2} + t^{2} |S_{LR}|^{2} \\right]\n\\nonumber\\\\& \\qquad\n+ (1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) \\left[ u^{2} |S_{LL}|^{2} + t^{2} |S_{LR}|^{2} \\right]\n\\biggr\\},\n\\nonumber \\\\\n\\frac{d\\sigma^{\\text{t-channel}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{32\\pi s}\\biggl\\{\n  (1+P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) s^{2} |T_{LR}|^{2}\n+ (1-P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) s^{2} |T_{LR}|^{2}\n\\nonumber\\\\& \\qquad\n+ (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) u^{2} |T_{RR}|^{2}\n+ (1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) u^{2} |T_{LL}|^{2}\n\\biggr\\},\n\\nonumber \\\\\n\\frac{d\\sigma^{\\text{interference}}}{d\\cos\\theta}(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}})\n&= \\frac{1}{16\\pi s} u^{2} \\biggl\\{\n (1+P_{\\mathrm{e}^{-}})(1-P_{\\mathrm{e}^{+}}) \\Re(S_{RR} T_{RR}^{*})\n\\nonumber\\\\& \\qquad\n+(1-P_{\\mathrm{e}^{-}})(1+P_{\\mathrm{e}^{+}}) \\Re(S_{LL} T_{LL}^{*})\n\\biggr\\}.\n\\end{align}\n\n\nWhen initial electrons and\/or positrons are longitudinally polarized, one can measure left-right asymmetries.\nThe left-right asymmetry of polarized cross sections is given by\n\\begin{align}\nA_{\\text{LR}}(P_{-},P_{+})\n&\\equiv \\frac{\n\\sigma(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n- \\sigma(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}{\n\\sigma(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n+ \\sigma(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}\n\\nonumber \\\\\n&= (P_{-} -  P_{+}) \\cdot \n\\frac{\n\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}} - \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}\n}{\n(1 + P_{-} P_{+}) (\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}} + \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}})\n+\n(1 - P_{-} P_{+}) (\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}} + \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}})\n}\n,\n\\nonumber \\\\ \n& 1\\ge P_{-} \\ge 0,\\quad 1 \\ge P_{+} \\ge -1,\n\\label{eq:ALR}\n\\end{align}\nwhere the cross section in a given bin $[\\theta_{1},\\theta_{2}]$ is given by\n$\n\\sigma \\equiv \\int_{\\cos\\theta_{1}}^{\\cos\\theta_{2}} \\frac{d\\sigma}{d\\cos\\theta} d\\cos\\theta\n$.\nWe have used $\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}} = \\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}$\nbecause $\\frac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} - \n \\frac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} = 0$.\nWe can also define the left-right asymmetry of the differential cross section as\n\\begin{align}\n\\lefteqn{A_{\\text{LR}}(P_{-},P_{+},\\cos\\theta)}\n\\quad\n\\nonumber\\\\\n&\\equiv\n\\frac{\n\\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n- \\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}{\n\\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = -P_{-}, P_{\\mathrm{e}^{+}}= - P_{+})\n+ \\dfrac{d\\sigma}{d\\cos\\theta}(P_{\\mathrm{e}^{-}} = +P_{-}, P_{\\mathrm{e}^{+}}= + P_{+})\n}\n\\nonumber \\\\\n&= \\frac{\n(P_{-} +  P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \n- \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} \\right)\n+\n(P_{-} -  P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta}\n - \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \\right)\n}{\n(1 + P_{-} P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta}\n + \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta} \\right)\n+\n(1 - P_{-} P_{+}) \\left(\\dfrac{d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{R}}}{d\\cos\\theta}\n + \\dfrac{d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{L}}}{d\\cos\\theta} \\right)\n}\n\\nonumber \\\\\n&= (P_{-} - P_{+}) \\cdot\n\\frac{\n \\Sigma_{LR-RL}\n}{(1 + P_{-} P_{+}) \\Sigma_{LL+RR} + (1-P_{-} P_{+}) \\Sigma_{LR+RL}},\n\\end{align}\nwhere we have used $d\\sigma_{\\mathrm{e}^{-}_{L} \\mathrm{e}^{+}_{L}}\/d\\cos\\theta =d\\sigma_{\\mathrm{e}^{-}_{R} \\mathrm{e}^{+}_{R}}\/d\\cos\\theta  $ and defined\n\\begin{align}\n\\Sigma_{LL+RR} &\\equiv 2s^{2} |T_{LR}|^{2},\n\\nonumber \\\\\n\\Sigma_{LR+RL} &\\equiv u^{2} (|S_{LL} + T_{LL}|^{2} + |S_{RR} + T_{RR}|^{2}) + 2t^{2} |S_{LR}|^{2},\n\\nonumber \\\\\n\\Sigma_{LR-RL} &\\equiv  u^{2} ( |S_{LL} + T_{LL}|^{2} - |S_{RR} + T_{RR}|^{2}  ).\n\\end{align}\n\nIn $\\mathrm{e}^+ \\mathrm{e}^- \\to \\mathrm{e}^+ \\mathrm{e}^-$ scatterings we have\n$A_{LR}(P_{-},+P_{+},\\cos\\theta)$ and $A_{LR}(P_{-},-P_{+},\\cos\\theta)$ as independent observables\nand one may define the following non-trivial quantity:\n\\begin{align}\n&A_{X}(\\cos\\theta) \\equiv \n\\frac{\\Sigma_{LL+RR} - \\Sigma_{LR+RL}}{\\Sigma_{LL+RR} + \\Sigma_{LR+RL}}\n\\nonumber \\\\\n&\\quad\n= \\frac{1}{P_{-} P_{+}} \\cdot \\dfrac{\n (P_{-} - P_{+}) A_{\\text{LR}}(P_{-},-P_{+},\\cos\\theta) - (P_{-} + P_{+}) A_{\\text{LR}}(P_{-}, +P_{+}\\cos\\theta)\n}{\n (P_{-} - P_{+}) A_{\\text{LR}}(P_{-},-P_{+},\\cos\\theta) + (P_{-} + P_{+}) A_{\\text{LR}}(P_{-}, +P_{+},\\cos\\theta)\n}.\\label{eq:AX}\n\\end{align} \nThis quantity may be used to explore NP beyond the SM as discussed below.\n\nSince $\\mathrm{e}^+ \\mathrm{e}^- \\to \\mathrm{e}^+ \\mathrm{e}^-$ scattering contains $t$-channel processes,\nforward scatterings dominate. Therefore unlike the $\\mathrm{e}^+\\mathrm{e}^- \\to f\\bar{f}$ ($f\\ne \\mathrm{e}^{-}$) scattering, \nthe forward-backward asymmetry of $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+} \\mathrm{e}^{-} $ scattering is a less-meaningful quantity.\n\nWe note that all of the above formulas can be applied\nto $\\ell^{+}\\ell^{-} \\to \\ell^{+}\\ell^{-}$  ($\\ell = \\mu,\\tau$) scatterings.\n\n\n\n\\section{Numerical Study}\n\nIn the followings we calculate $\\mathrm{e}^{+} \\mathrm{e}^{-} \\to \\mathrm{e}^{+}\\mathrm{e}^{-}$\ncross sections both in the SM and GHU models,\nand we evaluate  effects of $Z'$ bosons in GHU models on observables\ngiven in the previous section.\nAs benchmark points, we have chosen typical parameters of the A- and B-models\nin Tables \\ref{tbl:model} and \\ref{tbl:couplings}. \nFor experimental parameters we choose\n$\\sqrt{s}=250$ GeV and $L_{\\mathrm{int}} = 250$ fb$^{-1}$ as typical value of linear $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders like ILC\\cite{ILC}. \nWe also choose $L_{\\mathrm{int}} = 2$ ab$^{-1}$, which will be achieved\nat circular $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders like FCC-ee\\cite{Blondel:2021ema} and CEPC\\cite{CEPCStudyGroup:2018ghi}. \nFor the new asymmetry $A_{X}(\\cos\\theta)$ in \\eqref{eq:AX},\n we consider $\\sqrt{s} = 3$ TeV for future linear colliders like  CLIC\\cite{CLICdesign}.\nAs for the longitudinal polarization, we set the parameter ranges\n$- 0.8 \\le P_{\\mathrm{e}^{-}} \\le +0.8$ and $-0.3 \\le P_{\\mathrm{e}^{+}} \\le 0.3$,\nwhich can be achieved at ILC\\cite{ILC}.\n\nIn Figure~\\ref{fig:dsigma}   $\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ differential cross sections in the SM are plotted.\nIn the forward-scattering region ($\\cos\\theta>0$),\n the magnitudes of cross sections of $t$-channel and the interference\nparts are much larger than those of the $s$-channel part.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{dsigma-ee-SM-P00.pdf\n\\includegraphics[width=0.47\\textwidth]{dsigma-ee-SM-P00-Linear.pdf}\n\\caption{%\nDifferential cross sections for unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ initial\nstates in the SM.\n(a) Log-scale plot with $-0.9 \\le \\cos\\theta \\le 0.9$.\n(b) Linear plot with $-0.7 \\le \\cos\\theta \\le 0.7$.\nIn both plots, red-solid lines indicate the total of $s$-, $t$-channels and interferences. \nThe $s$-channel and $t$-channel contributions are drawn as blue-dashed and purple-dotted lines, respectively. \nIn (b), the negative contribution from interferences is plotted with the green-dashed line.\n}\\label{fig:dsigma}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P00.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P00.pdf}\n\\caption{\nDeviations of differential cross sections \nof GHU from those in the SM,\n$\n\\frac{d\\sigma^{\\text{GHU}}}{d\\cos\\theta}\/\n\\frac{d\\sigma^{{\\rm SM}}}{d\\cos\\theta}-1\n$,\nfor unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams\nare plotted.  \nThe left plot is for the A-model and the right plot is for the B-model. In each plot,\nthe red-solid curve represents the deviation  of the sum of all the components of the differential cross section.\nBlue-dashed, purple-dotted and green dot-dashed curves correspond to\n the deviations of $s$-, $t$-channels and interference components of differential cross sections, respectively.\n Error-bars are estimated for $L_{\\mathrm{int}} = 250 \\text{ fb}^{-1}$. \n}\\label{fig:Delta-dsigma}\n\\end{figure}\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P+8-3.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-A4-P-8+3.pdf}\\\\\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P-8+3.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{Delta-dsigma-ee-GHU-B1-P+8-3.pdf}%\n\\caption{%\nDeviations of differential cross sections\nof GHU models from the SM \nfor polarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams.\nGHU-A [(a) and (b)] and GHU-B [(c) and (d)].\n(a) and (c) are for $(P_{e^{-}},P_{e^{+}}) = (-0.8,+0.3)$.\n(b) and (d) are for$(P_{e^{-}},P_{e^{+}}) = (+0.8,-0.3)$.\nMeanings of the curves and error-bars are the same as in Figure~\\ref{fig:Delta-dsigma}.\n}\\label{fig:Delta-dsigma0}\n\\end{figure}\nIn Figures~\\ref{fig:Delta-dsigma} and \\ref{fig:Delta-dsigma0}, \nthe differences of differential cross sections of the GHU from the SM for unpolarized and polarized beams are plotted, respectively.\nIn the figures, differences of $s$-channel, $t$-channel and interference contributions are also plotted.\nIn the $s$-channel, the NP effects contribute destructively in the forward scattering.\nOn the other hand, in the $t$-channel NP effects contribute constructively.\nSince the cross section is dominated by $t$-channel, \nthe total of \n$s$-, $t$- and interference channels \nincreases due to the NP effects.\n\nIn the A-model $Z'$ bosons have larger couplings to right-handed electrons than to left-handed electrons.\nTherefore\nthe cross section of the $\\mathrm{e}_{R}^{-}\\mathrm{e}_{L}^{+}$ initial states  becomes larger than that of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$. \nOn the other hand,\nin the B-model $Z'$ bosons have larger couplings to left-handed electrons than to right-handed electrons,\nand the cross section of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$ initial states becomes larger.\n\nNP effects become smaller when $\\theta$ becomes smaller.\nThe statistical uncertainty, however, also becomes small since the cross section\nbecomes very large.\nTherefore deviations of the cross section relative to statistical uncertainties may become large.\n\nFor unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams (Figure~\\ref{fig:Delta-dsigma0}),\nthe new physics effect in  both models tends to enhance the cross section at\nforward scattering with almost the same magnitude.\nIn the B-model the suppression of NP effects due to larger $Z'$ masses \nis compensated by larger couplings of $Z'$ bosons than the couplings  in the A-model.\nThe enhancement of the differential cross section due to the NP effects at $\\cos\\theta \\sim 0.3$ is around 1\\%.\n\nFor polarized beams  deviations can be much larger.\nIn the A-model, electrons have large right-handed couplings to $Z'$ bosons and for right-handed polarized beam relative deviations of the cross-section \nbecome as much as $2\\,\\%$ [Figure~\\ref{fig:Delta-dsigma}-(b)], whereas for right-handed beams relative deviations are around $0.1\\,\\%$ [Figure~\\ref{fig:Delta-dsigma}-(a)].\nContrarily, in the B-model a left-handed electron has large couplings to $Z'$ bosons. Therefore in the B-model deviations become large for left-polarized beam [Figures~\\ref{fig:Delta-dsigma}-(c) and (d)].\nIn Figure \\ref{fig:Delta-dsigma},\nwe have also shown statistical 1 $\\sigma$ relative errors at $L_{\\mathrm{int}} =250$ fb$^{-1}$ \nfor bins\n$[\\cos\\theta_{0}-0.05,\\cos\\theta_{0}+0.05]$ ($\\cos\\theta_{0} = -0.90, -0.80,\\dots, +0.90$) as vertical bars.\nIn each bin, the observed number of events and statistical uncertainty are given by $N$ and $\\sqrt{N}$, respectively.\nTherefore relative error of the cross section is \nestimated as the inverse of the square of the number of events $N$.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.58\\textwidth]{sigdsigma.pdf}\n\\caption{Estimated significances on differential cross sections of GHU models\nwith unpolarized $\\mathrm{e}^{+} \\mathrm{e}^{-}$ beam with the integrated luminosity $L_{\\mathrm{int}} = 2\\text{ ab}^{-1}$.\nA statistical significance of 0.1\\% non-statistical error is plotted with a purple dot-dashed line.}\n\\label{fig:sigdsigma}\n\\end{figure}\n\nIn Figure~\\ref{fig:sigdsigma}, the statistical significances in the GHU models are plotted.\nAn estimated significance of the deviation of the cross section \nin a bin is given by\n\\begin{align}\n\\frac{|N_{\\text{GHU}} - N_{{\\rm SM}}|}{N_{{\\rm SM}}} \\bigg\/ \\frac{\\sqrt{N_{{\\rm SM}}}}{N_{{\\rm SM}}}\n= \\frac{|N_{\\text{GHU}} - N_{{\\rm SM}}|}{\\sqrt{N_{{\\rm SM}}}},\n\\end{align} \nwhere $N_{\\text{GHU}}$ and $N_{{\\rm SM}}$ are observed numbers of events in a bin.\nIn Figure \\ref{fig:sigdsigma},\nsignificances are larger than 5 $\\sigma$\nfor $\\cos\\theta\\gtrsim0.1$. \nSignificances are very large for forward scatterings,\nbut are very small for backward scatterings.\nIn Figure \\ref{fig:sigdsigma}, relative 0.1\\% errors are also shown. Errors due to the NP effects become very small and are around $0.1\\%$ for $\\cos\\theta \\simeq 0.9$. \nA similar analysis has been given in Ref.\\cite{Richard:2018zhl}.\n\nFor small scattering angles $\\theta$, the scattering amplitude is dominated by\n$t$-channel contributions which are constructed with the blocks $T_{LL}$, $T_{RR}$ and $T_{LR}$.\nWhen $|t| \\simeq s\\theta^{2}\/4 \\ll M_{Z}^{2}, M_{Z'}^{2}$,\nwe can approximate the SM and NP contributions to the block $T_{LL}$ in the scattering amplitude as \n\\begin{align}\nT_{LL}^{{ Nucl.\\ Phys.} } &\\equiv \n\\sum_{Z'} \\frac{\\ell_{Z'}^{2}}{t - M_{Z'}^{2} + iM_{Z'}\\Gamma_{Z'}}\n\\simeq \n\\sum_{Z'} \\frac{\\ell_{Z'}^{2} }{- M_{Z'}^{2}+ i M_{Z'}\\Gamma_{Z'}}.\n\\end{align}\nWhen $s\\theta^{2}\/4 \\ll M_{Z}^{2}$, \n$T_{LL}$ is dominated by the QED part $T_{LL}^{\\rm QED} \\simeq -4e^{2}\/(\\theta^{2}s)$ and the NP effects are estimated as\n\\begin{align}\n\\frac{T_{LL}^{{ Nucl.\\ Phys.} }}{T_{LL}^{\\rm QED}} &\\simeq \\theta^{2} \\sum_{Z'} \\frac{\n(\\ell_{Z'}^{2}\/4e^{2})s }{ M_{Z'}^{2} - i M_{Z'}\\Gamma_{Z'}}\n= \\theta^{2} \\cdot \\mathcal{O}(s\/M_{Z'}^{2}),\n\\end{align}\nand similar analysis is applied to $T_{LR}$ and $T_{RR}$.\nConsequently, this correction arises not only in amplitudes but also in\ndifferential cross sections.\nFor $\\sqrt{s} = 250\\,\\text{GeV}$ and  $\\theta \\lesssim 300\\text{ mrad}$,\nthe QED $t$-channel contribution dominates \nand corrections due to $Z'$ bosons are suppressed by a factor $\\theta^{2}s\/M_{Z'}^{2}$.\nIn Figure~\\ref{fig:farforward}, deviations of differential cross sections of GHU from the SM for $\\theta<300\\text{ mrad}$ are plotted.\nDeviations of cross sections from the SM are proportional to the square of \n$\\theta$ and become smaller than $0.1$\\% when $\\theta < 250\\text{ mrad}$.\nThe measurement of Bhabha scatterings at small scattering angle is used for the determination of the luminosity of $\\mathrm{e}^{+} \\mathrm{e}^{-}$ collision and uncertainties of the luminosity should be smaller than $0.1$\\%. In GHU models the NP effects on such uncertainty are well suppressed when $\\theta \\lesssim 100\\text{ mrad}$.\nAt ILC, the luminosity calorimeter in ILD (SiD) operates between 43 and 68 (40 and 90) mrad \\cite{ILC}, where the influence of the $Z'$ bosons is below 0.1 \\%.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{farforward.pdf}\n\\caption{Deviations of differential cross sections of GHU models from the SM in the forward-scattering region ($\\theta \\le 300$\\text{ mrad}).}\\label{fig:farforward}\n\\end{figure}\n\nWhen the initial electron and positron beams are longitudinally polarized,\nthe left-right asymmetry $A_{\\text{LR}}$ \\eqref{eq:ALR} can be measured.\nIn Figure~\\ref{fig:ALR}, the left-right asymmetries of the SM and GHU models are plotted.\nThe measured asymmetries become larger when $|P_{\\mathrm{e}^{-}} - P_{\\mathrm{e}^{+}}|$ are larger. \nAs seen in Figure~\\ref{fig:Delta-dsigma}, in the A-model cross section of $\\mathrm{e}_{R}^{-}\\mathrm{e}_{L}^{+}$ initial states becomes large whereas\nin the B-model cross section of $\\mathrm{e}_{L}^{-}\\mathrm{e}_{R}^{+}$ initial states is enhanced\ndue to the large left-handed $Z'$ couplings.\nTherefore $A_{\\text{LR}}$ of B-models are larger than the SM, whereas $A_{\\text{LR}}$ of A-models are smaller. \n Since the $A_{\\text{LR}}$ is proportional to $|P_{\\mathrm{e}^{-}} - P_{\\mathrm{e}^{+}}|$,\nthe asymmetries in Figure~\\ref{fig:ALR}-(a) are almost twice as large as in (b).\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{ALR-ee-GHU-s=250-Pmppm.pdf}\n\\includegraphics[width=0.48\\textwidth]{ALR-ee-GHU-s=250-Pmpmp.pdf}\n\\caption{Left-right asymmetries. In both plots,\nblue-dotdashed, and red-dashed and black-solid curves correspond to\n the GHU-A, GHU-B and the SM, respectively. \n Error-bars are indicated for $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ in each polarization. \n (a) Asymmetries for $(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}}) = (\\mp0.8,\\pm0.3)$.\n (b) Asymmetries for $(P_{\\mathrm{e}^{-}},P_{\\mathrm{e}^{+}}) = (\\mp0.8,\\mp0.3)$. }\\label{fig:ALR}\n\\end{figure}\nIn Figure \\ref{fig:ALR}, an asymmetry $A_{\\text{LR}}$ in a bin and the \nstatistic error $\\Delta A_{\\text{LR}}$ are also shown. \nHere\n\\begin{align}\nA_{\\text{LR}} &= \\frac{N_{L} - N_{R}}{N_{L}+N_{R}},\n&\n\\Delta A_{\\text{LR}} &= \n\\sqrt{\\frac{2 (N_{L}^{2} + N_{R}^{2})}{ (N_{L} + N_{R})^{3}}} \n\\end{align} \nwith $N_{L}$ and $N_{R}$ being observed number of events for the left-handed  ($P_{\\mathrm{e}^{-}}<0$) and right-handed ($P_{\\mathrm{e}^{-}}>0$) electron beams, respectively.\nFor small scattering angle $\\cos\\theta \\gtrsim 0.8$, both \n$A_{\\text{LR}}^{\\text{GHU}}$ and $A_{\\text{LR}}^{{\\rm SM}}$ \nbecome close to each other.\n\nTo see how NP effects against statistical uncertainty grow for small $\\theta$,\nwe plotted in Figure~\\ref{fig:sigALR} the averages and statistical significances of\nleft-right asymmetries in GHU models in each bin which are estimated as  \n\\begin{align}\n\\frac{A_{LR}^{\\text{GHU}} - A_{LR}^{{\\rm SM}} }{\\Delta A_{\\text{LR}}}.\n\\end{align}\nFor the forward scattering with $\\cos\\theta \\gtrsim 0.2$,\nthe deviations are bigger than several times of standard deviations.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.58\\textwidth]{sigALR.pdf}\n\\caption{%\nEstimated statistical significance on left-right asymmetries in GHU models.\nThe blue-dotdashed, red-dashed and\nblack-dotted lines indicate\nGHU-A, GHU-B and\n 0.1\\% non-statistical errors, respectively.}\\label{fig:sigALR}\n\\end{figure}\nIn Figure~\\ref{fig:sigALR}, \nthe significance is larger than $5\\,\\sigma$ for $\\cos\\theta\\gtrsim0.2$. \nBoth models are well distinguished from the SM.\nUsing the magnitude and sign of deviations, the A-model\nand B-model can be distinguished.\n\nIn Figure~\\ref{fig:AX},\nthe asymmetry $A_{X}$ defined in Eq.~\\eqref{eq:AX} is plotted\nfor $\\sqrt{s} = 250\\,\\text{GeV}$ and \n$\\sqrt{s} = 3\\,\\text{TeV}$.\nAt $\\sqrt{s} = 250\\,\\text{GeV}$, the NP effect on $A_{X}$ is very small.\nFor $\\sqrt{s} = 3\\,\\text{TeV}$, the asymmetry $A_{X}$ of the SM and GHU models is clearly different and may be discriminated experimentally.\nIn the present analysis of NP effects, only first KK excited states of neutral bosons are taken into account. At $\\sqrt{s} \\sim  3\\,\\text{TeV}$, effects of second KK modes on $A_{X}$ are estimated as a few percent. These effects are much smaller than the effects of the first KK modes.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{AX-ee-GHU-A4B1-s=250.pdf}\\hspace{0.2cm}\n\\includegraphics[width=0.48\\textwidth]{AX-ee-GHU-A4B1_s=3000.pdf}\n\\caption{Plot of an asymmetry $A_{X}$ in Eq.~\\eqref{eq:AX}.\nThe left plot (a) is for $\\sqrt{s} = 250\\,\\text{GeV}$ with $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ for each polarizations.\nThe right plot (b) is for $\\sqrt{s} = 3\\,\\text{TeV}$ with $L_{\\mathrm{int}} = 3\\,\\mathrm{ab}^{-1}$ for each polarizations.\nThe red-dashed, blue-dotdashed and black-solid curves correspond to the A-model, B-model and the SM, respectively.\n Error-bars are indicated for $L_{\\mathrm{int}} = 250\\,\\mathrm{fb}^{-1}$ in each polarization.}\\label{fig:AX}\n\\end{figure}\n\n\n\\section{Summary}\nIn this paper we examined the effects of $Z'$ bosons\nin the gauge-Higgs unification (GHU) models in the $\\mathrm{e}^{+}\\mathrm{e}^{-}\\to \\mathrm{e}^{+}\\mathrm{e}^{-}$ (Bhabha) scatterings.\nWe first formulated  differential cross sections in Bhabha scattering including  $Z'$ bosons.\nWe then numerically evaluated the deviations of differential cross sections in the two \n$\\mathrm{SO}(5) \\times \\mathrm{U}(1) \\times \\mathrm{SU}(3)$ GHU models (the A- and B-models) at $\\sqrt{s} = 250\\,\\text{GeV}$. \nWe found that at $L_{\\mathrm{int}} = 2\\,\\mathrm{ab}^{-1}$ with unpolarized $\\mathrm{e}^{+}\\mathrm{e}^{-}$ beams, the deviation due to $Z'$ bosons \nin the GHU models from the SM can be clearly seen.\nWe also found that for $80\\%$-longitudinally polarized electron and $30\\%$-polarized positron beams, \ndeviations of the differential cross sections from the SM \nbecome as large as a few percent for $\\cos\\theta \\sim 0.2$, and that\nthe A-model and the B-model are well distinguished\nat more than $3\\,\\sigma$ significance\nat $L_{\\mathrm{int}} =250\\,\\mathrm{fb}^{-1}$.\nWe also checked the effects of $Z'$ bosons are negligible\nfor the scattering angle smaller than $100$ mrad at $\\sqrt{s} = 250\\,\\text{GeV}$. Therefore Bhabha scatterings at very small $\\theta$ \ncan be safely used as the measurements of the luminosity in the $\\mathrm{e}^{+}\\mathrm{e}^{-}$ collisions.  \nFinally we introduced the new observable $A_{X}$. \nWe numerically evaluated it at $\\sqrt{s} = 250\\,\\text{GeV}$ and\n$3\\,\\text{TeV}$. \nEffects of the GHU models on $A_{X}$\ncan be seen at future TeV-scale $\\mathrm{e}^{+}\\mathrm{e}^{-}$ colliders.\n\nIn this paper the effects of $Z'$ bosons are calculated at the Born level. Higher-order QED effects should be taken \ninto account for more precise evaluation \\cite{Bardin:1990,Bardin:2017}.\n\n\n\\section*{Acknowledgements}\n\nThis work was supported in part \nby European Regional Development Fund-Project Engineering Applications of \nMicroworld Physics (No.\\ CZ.02.1.01\/0.0\/0.0\/16\\_019\/0000766) (Y.O.), \nby the National Natural Science Foundation of China (Grant Nos.~11775092, \n11675061, 11521064, 11435003 and 11947213) (S.F.), \nby the International Postdoctoral Exchange Fellowship Program (IPEFP) (S.F.), \nand by Japan Society for the Promotion of Science, \nGrants-in-Aid  for Scientific Research, Nos.\nJP19K03873 (Y.H.) and JP18H05543 (N.Y.).\n\n\n\\vskip 1.cm\n\n\\def\\jnl#1#2#3#4{{#1}{\\bf #2},  #3 (#4)}\n\n\\def{ Z.\\ Phys.} {{ Z.\\ Phys.} }\n\\def{ J.\\ Solid State Chem.\\ }{{ J.\\ Solid State Chem.\\ }}\n\\def{ J.\\ Phys.\\ Soc.\\ Japan }{{ J.\\ Phys.\\ Soc.\\ Japan }}\n\\def{ Prog.\\ Theoret.\\ Phys.\\ Suppl.\\ }{{ Prog.\\ Theoret.\\ Phys.\\ Suppl.\\ }}\n\\def{ Prog.\\ Theoret.\\ Phys.\\  }{{ Prog.\\ Theoret.\\ Phys.\\  }}\n\\def{ Prog.\\ Theoret.\\ Exp.\\  Phys.\\  }{{ Prog.\\ Theoret.\\ Exp.\\  Phys.\\  }}\n\\def{ J. Math.\\ Phys.} {{ J. Math.\\ Phys.} }\n\\def Nucl.\\ Phys. \\textbf{B}{ Nucl.\\ Phys. \\textbf{B}}\n\\def{ Nucl.\\ Phys.} {{ Nucl.\\ Phys.} }\n\\def{ Phys.\\ Lett.} \\textbf{B}{{ Phys.\\ Lett.} \\textbf{B}}\n\\def{ Phys.\\ Lett.} {{ Phys.\\ Lett.} }\n\\defPhys.\\ Rev.\\ Lett. {Phys.\\ Rev.\\ Lett. }\n\\defPhys.\\ Rev. \\textbf{B}{Phys.\\ Rev. \\textbf{B}}\n\\defPhys.\\ Rev. \\textbf{D}{Phys.\\ Rev. \\textbf{D}}\n\\defPhys.\\ Rep. {Phys.\\ Rep. }\n\\def{Ann.\\ Phys.\\ (N.Y.)}{{Ann.\\ Phys.\\ (N.Y.)}}\n\\defRev.\\ Mod.\\ Phys. {Rev.\\ Mod.\\ Phys. }\n\\defZ.\\ Phys. \\textbf{C}{Z.\\ Phys. \\textbf{C}}\n\\defScience{Science}\n\\defComm.\\ Math.\\ Phys. {Comm.\\ Math.\\ Phys. }\n\\defMod.\\ Phys.\\ Lett. \\textbf{A}{Mod.\\ Phys.\\ Lett. \\textbf{A}}\n\\defEur.\\ Phys.\\ J. \\textbf{C}{Eur.\\ Phys.\\ J. \\textbf{C}}\n\\defJHEP {JHEP }\n\\def{\\it ibid.} {{\\it ibid.} }\n\n\n\\renewenvironment{thebibliography}[1]\n         {\\begin{list}{[$\\,$\\arabic{enumi}$\\,$]} \n         {\\usecounter{enumi}\\setlength{\\parsep}{0pt}\n          \\setlength{\\itemsep}{0pt}  \\renewcommand{\\baselinestretch}{1.2}\n          \\settowidth\n         {\\labelwidth}{#1 ~ ~}\\sloppy}}{\\end{list}}\n\n\n\n\\leftline{\\Large \\bf References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\vskip -0.25cm\nCompressed sensing \\cite{Donoho2006,Candes2008} is a mathematical framework that defines the conditions and tools for the recovery of a signal from a small number of its linear projections (i.e. measurements). In the CS framework, the measurement device acquires the signal in the linear projections domain, and the full signal is reconstructed by convex optimization techniques. CS has diverse applications including image acquisition \\cite{Romberg2008}, radar imaging \\cite{5420035}, Magnetic Resonance Imaging (MRI) \\cite{4472246, 6153065}, spectrum sensing \\cite{6179814}, indoor positioning \\cite{6042868}, bio-signals acquisition \\cite{6184345}, and sensor networks \\cite{6159081}. In this paper we address the problem of block-based CS (BCS) \\cite{Fowler-NOW}, which employs CS on distinct low-dimensional segments of a high-dimensional signal. BCS is mostly suitable for processing very high-dimensional images and video, where it operates on distinct local patches. Our approach is based on a deep neural network \\cite{Bengio-2009}, which simultaneously learns the linear sensing matrix and the non-linear reconstruction operator.\\\\\nThe contributions of this paper are two-fold: (1) It presents for the first time, to the best knowledge of the authors, the utilization of a fully-connected deep neural network for the task of BCS; and (2) The proposed network performs both the linear sensing and non-linear reconstruction operators, and during training these operators are \\emph{jointly} optimized, leading to a significant advantage compared to state-of-the-art.\\\\\nThis paper is organized as follows: section \\ref{Problem Formulation} introduces CS concepts, and motivates the utilization of BCS for very high-dimensional images and video. Section \\ref{The Proposed Approach} presents the deep neural network approach, and discusses structure and training aspects. Section \\ref{Results} evaluates the performance of the proposed approach for compressively sensing and reconstructing natural images, and compares it with state-of-the-art BCS methods and full-image Total Variation-based CS. Section \\ref{Conclusions} concludes the paper and discusses future research directions. \\vskip -0.25cm\n\\section{Compressed Sensing Overview}\n\\vskip -0.25cm\n\\label{Problem Formulation}\n\\subsection{Full-Signal Compressed Sensing}\nGiven a signal $\\signal \\in \\mathbf{R}^N$, an $M \\times N$ sensing matrix $\\Phi$ (such that $M \\ll N$) and a measurements vector $\\measurement = \\Phi \\signal$, the goal of CS is to recover the signal from its measurements. The sensing rate is defined by $R=M\/N$, and since $R \\ll 1$ the recovery of $\\signal$ is not possible in the general case. According to CS theory \\cite{Donoho2006,Candes2008}, signals that have a sparse representation in the domain of some linear transform can be exactly recovered with high probability from their measurements: let $\\signal = \\Psi \\reps $, where $\\Psi$ is the inverse transform, and $\\reps$ is a sparse coefficients vector with only $S \\ll N$ non-zeros entries, then the recovered signal is synthesized by $ \\hat{\\signal} = \\Psi \\repsHat$, and $\\repsHat$ is obtained by solving the following convex optimization program:\n\\begin{equation}\n \\repsHat  = \\argmin_{\\reps{'}} \\left\\|\\reps{'}\\right\\|_1 \\text{ subject to } \\measurement = \\Phi \\Psi \\reps{'},\n\\end{equation}\n\\vskip -0.25cm\n\\noindent where $\\left\\|\\alpha\\right\\|_1$ is the $l_1$-norm, which is a convex relaxation of the $l_0$ pseudo-norm that counts the number of non-zero entries of $\\alpha$. The exact recovery of $\\signal$ is guaranteed with high probability if $\\reps$ is sufficiently sparse and if certain conditions are met by the sensing matrix and the transform.\n\n\\begin{table*}[]\n  \\caption{Average reconstruction PSNR [dB] and SSIM vs. sensing rate (R=M\/N): for each method and sensing rate, the result is displayed as PSNR | SSIM  (each result is the average over the 10 test images).}\n  \\label{Reconstruction-Quality}\n  \\centering\n\\resizebox{\\textwidth}{!}{\n  \\begin{tabular}{lccccc}\n   \\hline\n    Method                        & R = 0.1  & R = 0.15 & R = 0.2 & R = 0.25 &  R = 0.3\\\\\n   \\hline\n   Proposed (block-size = 16$\\times$16)     & \\textbf{28.21} | \\textbf{0.916} & \\textbf{29.73} | \\textbf{0.948} & \\textbf{31.03} | \\textbf{0.965}  & \\textbf{32.15} | \\textbf{0.976} & \\textbf{33.11} | \\textbf{0.983}\\\\\n   BCS-SPL-DDWT (16$\\times$16)\\cite{Fowler2009}  & 24.92 | 0.789 & 26.12 | 0.834 & 27.17 | 0.873 & 28.16 | 0.898 & 29.02 | 0.917 \\\\\n   BCS-SPL-DDWT (32$\\times$32)\\cite{Fowler2009}   & 24.99 | 0.781 & 26.40 | 0.833 & 27.46 | 0.868 & 28.43 | 0.894 & 29.29 | 0.914 \\\\\n   MH-BCS-SPL (16$\\times$16)  \\cite{MH-Fowler}    & 26.01 | 0.827 & 27.92 | 0.888 & 29.46 | 0.919 & 30.69 | 0.939 & 31.69 | 0.952 \\\\\n   MH-BCS-SPL (32$\\times$32)  \\cite{MH-Fowler}    & 26.79 | 0.845 & 28.51 | 0.895 & 29.81 | 0.923 & 30.77 | 0.938 & 31.73 | 0.950 \\\\\n   MS-BCS-SPL \\cite{MS-Fowler}                      & 27.32 | 0.883 & 28.77 | 0.909 & 30.04 | 0.934 & 31.15 | 0.956 & 32.05 | 0.974 \\\\\n   MH-MS-BCS-SPL \\cite{MH-Fowler}                     & 27.74 | 0.889 & 29.10 | 0.919 & 30.78 | 0.947 & 31.38 | 0.960 & 32.82 | 0.979 \\\\\n   TV (Full Image) \\cite{Romberg2008}               & 27.41 | 0.867 & 28.57 | 0.890 & 29.62 | 0.909 & 30.63 | 0.926 & 31.59 | 0.939 \\\\\n   \\hline\n  \\end{tabular}}\n\\end{table*}\n\\subsection{Block-based Compressed Sensing}\n\\label{Block-based Compressed Sensing}\nConsider applying CS to an image of $L \\times L$ pixels: the technique described above can be employed by column-stacking (or row-stacking) the image to a vector $\\signal \\in \\mathbb{R}^{L^2}$, and the dimensions of the measurement matrix $\\Phi$ and the inverse transform $\\Psi$ are $M \\times {L^2}$ and $ {L^2} \\times {L^2}$, respectively. For modern high-resolution cameras, a typical value of $L$ is in the range of $2000$ to $4000$, leading to overwhelming memory requirements for storing $\\Phi$ and $\\Psi$: for example, with $L=2000$ and a sensing rate $R=0.1$ the dimensions of $\\Phi$ are $400,000 \\times 4,000,000$ and of $\\Psi$ are $4,000,000 \\times 4,000,000$. In addition, the computational load required to solve the CS reconstruction problem becomes prohibitively high. Following this line of arguments, a BCS framework was proposed in \\cite{Gan2007}, in which the image is decomposed into non-overlapping blocks (i.e. patches) of $B \\times B$ pixels, and each block is compressively sensed independently. The full-size image is obtained by placing each reconstructed block in its location within the reconstructed image canvas, followed by full-image smoothing. The dimensions of the block sensing matrix $\\Phi_B$ are ${B^2}R\\times{B^2}$, and the measurement vector of the \\emph{i}-th block is given by:\n\\begin{equation}\n\\label{block-based sensing}\n\\measurement_i = \\Phi_B \\signal_i,\n\\end{equation}\n\\noindent where $\\signal_i \\in \\mathbb{R}^{B^2}$ is the column-stacked block, and $\\Phi_B$ was chosen in \\cite{Gan2007} as an orthonormalized i.i.d Gaussian matrix. Following a per-block minimum mean squared error reconstruction stage, a full-image iterative hard-thresholding algorithm is employed for improving full-image quality. An improvement to the performance of this approach was proposed by \\cite{Fowler2009}, which employed the same BCS approach as \\cite{Gan2007} and evaluated the incorporation of directional transforms such as the Contourlet Transform (CT) and the Dual-tree Discrete Wavelet Transform (DDWT) in conjunction with a Smooth Projected Landweber (SPL) \\cite{bertero1998introduction} reconstruction of the full image. The conclusion of the experiments conducted in \\cite{Fowler2009} was that in most cases the DDWT transform offered the best performance, and we term this method as BCS-SPL-DDWT. A multi-scale approach was proposed by \\cite{MS-Fowler}, termed MS-BCS-SPL, which improved the performance of BCS-SPL-DDWT by applying the block-based sensing and reconstruction stages in multiple scales and sub-bands of a discrete wavelet transform. A different block dimension was employed for each scale and with a 3-level transform, dimensions of $B=64,32,16$ were set for the high, medium and low resolution scales, respectively. A multi-hypothesis approach was proposed in \\cite{MH-Fowler} for images and videos, which is suitable for either spatial domain BCS (termed MH-BCS-SPL) or multi-scale BCS (termed MH-MS-BCS-SPL). In this approach, multiple predictions of a block are computed from neighboring blocks in an initial reconstruction of the full image, and the final prediction of the block is obtained by an optimal linear combination of the multiple predictions. For video frames, previously reconstructed adjacent frames provide the sources for multiple predictions of a block. The multi-scale version of this approach provides the best performance among all above mentioned BCS methods. A detailed survey of BCS theory and performance is provided in \\cite{Fowler-NOW}, which describes additional applications such as BCS of multi-view images and video, and motion-compensated BCS of video.\n\\section{The Proposed Approach}\n\\vskip -0.25cm\n\\label{The Proposed Approach}\nIn this paper we propose to employ a deep neural network that performs BCS by processing each block independently\\footnote{In this paper we treat only block-based processing, and a full-image post-processing stage is not performed.} as described in section \\ref{Block-based Compressed Sensing}. Our choice is motivated by the outstanding success of deep neural networks for the task of full-image denoising \\cite{Burger} in which a 4-layer neural network achieved state-of-the-art performance by block-based processing. In our approach, the first hidden layer performs the linear block-based sensing stage (\\ref{block-based sensing}) and the following hidden layers perform the non-linear reconstruction stage. The advantage and novelty of this approach is that during training, the sensing matrix and the non-linear reconstruction operator are \\emph{jointly} optimized, leading to better performance than state-of-the-art at a fraction of the computation time. \\\\ The proposed fully-connected network includes the following layers: (1) an input layer with $B^2$ nodes; (2) a compressed sensing layer with $B^{2}R$ nodes, $R\\ll1$ (its weights form the sensing matrix); (3) $K\\ge1$ reconstruction layers with $B^{2}T$ nodes, each followed by a ReLU \\cite{icml2010_NairH10} activation unit, where $T>1$ is the redundancy factor; and (4) an output layer with $B^2$ nodes. Note that the performance of the network depends on the block-size $B$, the number of reconstruction layers $K$, and their redundancy $T$. We have evaluated\\footnote{The network was implemented using Torch7 \\cite{Collobert_NIPSWORKSHOP_2011} scripting language, and trained on NVIDIA Titan X GPU card.} these parameters by a set of experiments that compared the average reconstruction PSNR of 10 test images, depicted in Figure \\ref{test_images}, and by training the network with 5,000,000 distinct patches, randomly extracted from 50,000 images in the LabelMe dataset \\cite{LabelMe}. The chosen optimization algorithm was AdaGrad \\cite{AdaGrad} with a learning rate of $0.005$ (100 epochs), and batch size of 16. Our study revealed that best\\footnote{Note that by increasing significantly the training set, slightly different values of $B$, $K$, and $T$ may provide better results, as discussed in \\cite{Burger}.} performance were achieved with a block size $B\\times B = 16\\times16$, $K=2$ reconstruction layers and redundancy $T=8$, leading to a total of 4,780,569 parameters ($R=0.1$). Table \\ref{Reconstruction-Quality-vs-Size} provides a comparison for varying the block size between $8\\times8$ to $20\\times20$ (with 2 reconstruction layers and redundancy of 8), and indicates that block size of $16\\times16$ provides the best results. Table \\ref{Reconstruction-Quality-vs-Redundancy} provides a comparison for varying the redundancy between 2 to 12 (with 2 reconstruction layers and block size of $16\\times 16$), and indicates that a redundancy of 8 provides the best results. Table \\ref{Reconstruction-Quality-vs-Layers} provides a comparison for varying the number of hidden reconstruction layers between 1 to 8 (with redundancy of 8 and block size of $16\\times 16$), and indicates that two reconstruction layers provided the best performance.\n\\begin{table}[]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. block size ($B\\times B$)}\n  \\label{Reconstruction-Quality-vs-Size}\n  \\centering\n  \\begin{tabular}{cllll}\n    \\hline\n   \n   \n    Training Examples & $B=8$        & $B=12$     & $B=16$      & $B=20$ \\\\\n    \\hline\n   \n    $5 \\times 10^6$   &     27.21  &  27.66   & \\textbf{28.21}   &  27.73 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. network redundancy}\n  \\label{Reconstruction-Quality-vs-Redundancy}\n  \\centering\n  \\begin{tabular}{cllll}\n    \\hline\n   \n   \n    Training Examples & $T=2$ & $T=4$ & $T=8$ & $T=12$ \\\\\n    \\hline\n     $5 \\times 10^6$   & 27.99  &  28.11 & \\textbf{28.21}   &  28.15 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[htb]\n \\vskip -0.25cm\n  \\caption{Reconstruction PSNR [dB] vs. no. of reconstruction layers}\n  \\label{Reconstruction-Quality-vs-Layers}\n  \\centering\n  \\begin{tabular}{ccccc}\n    \\hline\n   \n   \n    Training Examples & $K=1$ & $K=2$ & $K=4$ & $K=8$ \\\\\n    \\hline\n     $5 \\times 10^6$   & 27.98  &  \\textbf{28.21} & 28.18   &  27.07 \\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\begin{table}[t]\n  \\caption{Computation time at R=0.25 ($512 \\times 512$ images):}\n  \\label{Comp-Time}\n   \\centering\n  \\begin{tabular}{lc}\n  \\hline\n    Method    &  Time [seconds]\\\\\n     \\hline\n   \n    Proposed  & 0.80\\\\\n    BCS-SPL-DDWT ($16 \\times 16$) \\cite{Fowler2009}   & 13.57\\\\\n    BCS-SPL-DDWT ($32 \\times 32$) \\cite{Fowler2009}   & 13.10\\\\\n    MH-BCS-SPL ($16 \\times 16$) \\cite{MH-Fowler}   &  144.61\\\\\n    MH-BCS-SPL ($32 \\times 32$) \\cite{MH-Fowler}   &  69.73\\\\\n    MS-BCS-SPL \\cite{MS-Fowler}   & 6.39\\\\\n    MH-MS-BCS-SPL \\cite{MH-Fowler} & 207.32\\\\\n    TV (Full Image) \\cite{Romberg2008} & 1675.09\\\\\n    \\hline\n  \\end{tabular}\n\\end{table}\n\\section{Performance Evaluation}\n\\vskip -0.25cm\n\\label{Results}\nThis section provides performance evaluation results of the proposed approach\\footnote{A MATLAB package implementing the proposed approach is available at: \\url{http:\/\/www.cs.technion.ac.il\/~adleram\/BCS_DNN_2016.zip}} vs. the leading BCS approaches: BCS-SPL-DDWT \\cite{Fowler2009}, MS-BCS-SPL \\cite{MS-Fowler}, MH-BCS-SPL \\cite{MH-Fowler} and MH-MS-BCS-SPL \\cite{MH-Fowler}, using the original code provided by their authors. The proposed approach was employed with block size $16 \\times 16$, BCS-SPL-DDWT with block sizes of $16 \\times 16$ and $32 \\times 32$ (the optimal size for this method), MH-BCS-SPL with block sizes of $16 \\times 16$ and $32 \\times 32$ (the optimal size for this method). MS-BCS-SPL and MH-MS-BCS-SPL utilized a 3-level discrete wavelet transform with block sizes as indicated in section \\ref{Block-based Compressed Sensing} (their optimal settings). In addition, we also compared to the classical full-image Total Variation (TV) CS approach of \\cite{Romberg2008} that utilizes a sensing matrix with elements from a discrete cosine transform and Noiselet vectors. Reconstruction performance was evaluated for sensing rates in the range of $R=0.1$ to $R=0.3$, using the average PSNR and SSIM \\cite{SSIM} over the collection of 10 test images ($512 \\times 512$ pixels), depicted in Figure \\ref{test_images}. Reconstruction results are summarized in Table \\ref{Reconstruction-Quality}, and reveal a consistent advantage of the proposed approach vs. all BCS methods as well as the full-image TV approach. Visual quality comparisons (best viewed in the electronic version of this paper) are provided in Figures \\ref{results_comp_01}-\\ref{results_comp_025_2}, and demonstrate the high visual quality of the proposed approach. Computation time comparison at a sensing rate $R=0.25$, with a MATLAB implementation of all tested methods (without GPU), is provided in Table \\ref{Comp-Time} and demonstrates that the proposed approach is over 200-times faster than state-of-the-art (MH-MS-BCS-SPL), and over 1600-times faster than full-image TV CS.\n\\begin{figure*}[]\n\\begin{minipage}{\\linewidth}\n\\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm, scale=0.55]{Test_Images.eps}}\n\\vskip -1cm\n \\caption{Test images ($512 \\times 512$): 'lena', 'bridge', 'barbara', 'peppers', 'mandril', 'houses', 'woman', 'boats', 'cameraman' and 'couple'.}\n\\label{test_images}\n \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_01_couple.eps}}\n\\vskip -1cm\n  \\caption{Reconstruction of 'couple' at R = 0.1 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (27.1691 | 0.8812); (c) MS-BCS-SPL (26.8429 | 0.8756); (d) MH-MS-BCS-SPL (27.1804 | 0.8877); and (e) Proposed (28.5902 | 0.9414).}\n \\label{results_comp_01}\n \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_02_houses.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'houses' at R = 0.2 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (31.0490 | 0.9304); (c) MS-BCS-SPL (31.4317 | 0.9497); (d) MH-MS-BCS-SPL (31.7030 | 0.9544); and (e) Proposed (32.9328 | 0.9766).}\n \\label{results_comp_02}\n  \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_025.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'lena' at R = 0.25 (PSNR [dB] | SSIM): (a) Original; (b) Full image TV (35.4202 | 0.9718); (c) MS-BCS-SPL (36.5555 | 0.9861); (d) MH-MS-BCS-SPL (35.7346 | 0.9825; and (e) Proposed (36.3734 | 0.9910).}\n \\label{results_comp_025_1}\n  \\end{minipage}\n\\end{figure*}\n \\begin{figure*}[]\n \\begin{minipage}{\\linewidth}\n \\vskip -1cm\n \\makebox[\\linewidth]{\n\\centering\n\\includegraphics[width=200mm,scale=0.5]{compare_reconst_rate_03_boats.eps}}\n\\vskip -1cm\n \\caption{Reconstruction of 'boats' at R = 0.3 (PSNR[dB] | SSIM): (a) Original; (b) Full image TV (32.5986 | 0.9598); (c) MS-BCS-SPL (32.5063 | 0.9835); (d) MH-MS-BCS-SPL (32.7697 | 0.9847); and (e) Proposed (34.0065 | 0.9914).}\n \\label{results_comp_025_2}\n  \\end{minipage}\n\\end{figure*}\n\\section{Conclusions}\n\\vskip -0.25cm\n\\label{Conclusions}\nThis paper presents a deep neural network approach to BCS, in which the sensing matrix and the non-linear reconstruction operator are jointly optimized during the training phase. The proposed approach outperforms state-of-the-art both in terms of reconstruction quality and computation time, which is two orders of magnitude faster than the best available BCS method. Our approach can be further improved by extending it to compressively sense blocks in a multi-scale representation of the sensed image, either by utilizing standard transforms or by deep learning of new transforms using convolutional neural networks.\n\\vskip -0.25cm\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe scalar sectors of the supersymmetric field theories especially\nthe supergravity theories which are the theories that govern the\nmassless sector low energy background coupling of the relative\nsuperstring theories \\cite{kiritsis} can be formulated as\nprincipal sigma models or non-linear sigma models whose target\nspaces are group manifolds or coset spaces. In particular a great\nmajority of the supergravity scalar sectors are constructed as\nsymmetric space sigma models\n\\cite{sm1,sm2,sm3,nej1,nej2,nej3,sssm1}. When the Abelian\n(Maxwell) vector multiplets are coupled to the graviton multiplets\nin these theories the scalar sector which has the non-linear sigma\nmodel interaction with in itself is coupled to the Abelian gauge\nfields through a kinetic term in the Lagrangian\n\\cite{julia1,julia2,ker1,ker2}.\n\nIn this work, we will focus on the gauge field equations of the\nprincipal sigma model and the Abelian gauge field couplings\nmentioned above. Bearing in mind that the non-linear sigma model\ncan be obtained from the principal sigma model by imposing extra\nrestrictions for the sake of generality we will consider the\ngeneric form of the principal sigma model as the non-linear\ninteraction of the scalars. We will simply show that the gauge\nfield equations can be locally integrated so that they can be\nexpressed as first-order equations containing arbitrary locally\nexact differential forms. The first-order form of the gauge field\nequations will be used to show that there exists a one-sided\ndecoupling between the scalars and the gauge fields in a sense\nthat the scalar field equations do not contain the gauge fields in\nthem whereas the scalars enter as sources in the gauge field\nequations. Thus the scalar solution space of the coupled theory\ncoincides with the general solution space of the pure sigma model.\nFurthermore we will also discuss that this hidden on-shell\nscalar-matter decoupling results in a number of coupled Maxwell\ntheories with sources whose currents contain the general solutions\nof the principal sigma model which is completely decoupled from\nthe Maxwell sector. Therefore we will show that when the general\nsolutions of the pure principal sigma model are obtained and when\none fixes the sector of the solution space of the coupled theory\nby fixing the field dependence or the independence of the locally\nexact differential forms appearing in the first-order gauge field\nequations one may determine the currents of the coupled Maxwell\nfields. In this respect one may solve the gauge fields from the\nMaxwell sector field equations. Consequently the solution space of\nthe scalar-matter coupling can be entirely generated by the\ngeneral solutions of the pure principal sigma model and the\narbitrary choice of the locally exact differential forms. This\nfact is a consequence of the solution methodology which is based\non the implicit on-shell decoupling between the matter fields and\nthe scalar sector which provides current sources to the former.\n\\section{Hidden Decoupling Structure of the Gauge Fields and Their Sources}\nIn a $D$-dimensional spacetime $M$ the Lagrangian which gives the\ninhomogeneous Maxwell equations can be given as\n\\begin{equation}\\label{de1}\n {\\mathcal{L}}=-\\frac{1}{2}dA\\wedge\n \\ast dA-A\\wedge\\ast J,\n\\end{equation}\nwhere $A$ is the $U(1)$ electromagnetic gauge potential one-form\nand $F=dA$ is the field strength of it. Also in the units where\nthe speed of light is unity the current one-form in a local\ncoordinate basis $\\{dt,dx^{a}\\}$ is\n\\begin{equation}\\label{de2}\n J=-\\rho dt+\\mathbf{J}_{a}dx^{a},\n\\end{equation}\nwhere in the temporal component $\\rho$ is the charge density and\nthe spatial components $\\mathbf{J}_{a}$ are the current densities.\nThe Lagrangian \\eqref{de1} defines a media in which the charge\ndensity and the currents are not influenced by the electromagnetic\nfield. The current one-form is predetermined and static that is to\nsay although it acts as a source for the electromagnetic field it\ndoes not interact with it dynamically. From \\eqref{de1} the\ninhomogeneous Maxwell equations read\n\\begin{equation}\\label{de2.5}\n d\\ast F=-\\ast J.\n\\end{equation}\nIn this section we will consider the coupling of $N$ $U(1)$ gauge\nfield one-forms $A^{i}$ to the principal sigma model whose target\nspace is a group manifold $G$. The sigma model Lagrangian can be\ngiven as\n\\begin{equation}\\label{de3}\n {\\mathcal{L}}=\\frac{1}{2}\\, tr(\\ast dg^{-1}\\wedge\n dg).\n\\end{equation}\nHere we take a differentiable map\n\\begin{equation}\\label{de4}\n h: M\\longrightarrow G,\n\\end{equation}\nwe also consider a representation $f$ of $G$ in $Gl(N,\\Bbb{R})$\n\\begin{equation}\\label{de5}\n f: G\\longrightarrow Gl(N,\\Bbb{R}),\n\\end{equation}\nwhich may be taken as a differentiable homomorphism. Then the map\n$g$ can be given as\n\\begin{equation}\\label{de6}\n g=f\\circ h: M\\longrightarrow Gl(N,\\Bbb{R}),\n\\end{equation}\nwhich is a matrix-valued function on $M$\n\\begin{equation}\\label{de7}\ng(p)=\\left(\\begin{array}{ccc}\n  \\varphi^{11}(p) & \\varphi^{12}(p) & \\cdots \\\\\n  \\varphi^{21}(p) & \\varphi^{22}(p) & \\cdots \\\\\n  \\vdots & \\vdots & \\vdots \\\\\n\\end{array}\\right),\n\\end{equation}\n$\\forall p\\in M$. Due to the presence of the scalar fields\n$\\{\\varphi^{ij}\\}$ the theory can be considered to be a scalar\nfield theory. In \\eqref{de3} we have a matrix multiplication with\nthe wedge product used between the components and the trace is\nover the representation chosen in \\eqref{de5}. Depending on the\nrestrictions on the sigma model and the nature of $G$ the scalars\nin \\eqref{de7} can be all independent or not. This model also\ncovers the non-linear (coset) sigma models \\cite{westsugra,tanii}\nand in particular the symmetric space sigma models\n\\cite{sm1,sm2,sm3} which shape the scalar sectors of the\nsupergravities thus the low energy effective string theories. The\nconstruction of the Lagrangian of the symmetric space sigma model\nin which an internal metric substitutes the map $g$ can be found\nin \\cite{nej1,nej2,nej3}.\n\nBy generalizing the supersymmetric coupling of the supergravity\nmatter multiplets with the graviton multiplets\n\\cite{julia1,julia2,ker1,ker2,sssugradivdim} we can write down the\ncoupling of $N$ $U(1)$ gauge field one-forms $A^{i}$ to the\nprincipal sigma model whose target space is a group manifold as\n\\begin{equation}\\label{de8}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}\\ast F^{T}g\\wedge\n F,\n\\end{equation}\nwhere we define the column vector $F$ whose components are $F^{i}$\nthus the coupling term can be explicitly written as\n\\begin{equation}\\label{de9}\n-\\frac{1}{2}\\ast F^{T}g\\wedge\n F =-\\frac{1}{2}g^{i}_{\\:\\:\\:j} \\ast F_{i}\\wedge\n F^{j}.\n\\end{equation}\n Now if we\nvary the Lagrangian \\eqref{de8} with respect to $A^{i}$ we find\nthe corresponding field equations as\n\\begin{equation}\\label{de10}\nd(\\mathcal{T}^{i}_{\\:\\:\\:k}\\ast F_{i})=0,\n\\end{equation}\nwhere\n\\begin{equation}\\label{de11}\n\\mathcal{T}^{i}_{\\:\\:\\:k}=g^{i}_{\\:\\:\\:k}+(g^{T})^{i}_{\\:\\:\\: k}.\n\\end{equation}\nThe expression \\eqref{de10} defines a closed form. Since locally\nany closed form is equal to an exact form we can integrate\n\\eqref{de10} to write\n\\begin{equation}\\label{de12}\n\\mathcal{T}^{i}_{\\:\\:\\:k}\\ast F_{i}=dC_{k},\n\\end{equation}\nwhere $\\{C_{k}\\}$ are arbitrary $N$ $(D-3)$-forms on $M$. They may\nbe chosen to depend on the fields $\\{\\varphi^{ij},A^{i}\\}$ or they\nmay be fixed. The former case is the most general one. The general\nsolutions of the field equations of the Lagrangian \\eqref{de8}\nmust satisfy \\eqref{de12} on-shell in which one can freely change\nthe set of $(D-3)$-forms $\\{C_{k}\\}$ which may be functions of\n$\\{\\varphi^{ij},A^{i}\\}$ or not. To generate the entire solution\nspace one can first choose a set of field-dependent or\nfield-independent $\\{C_{k}\\}$ and solve the system of field\nequations of \\eqref{de8} to find the corresponding solutions then\none can repeat this procedure for a different set of $\\{C_{k}\\}$.\nIn this respect the field-independent or the fixed $(D-3)$-forms\n$\\{C_{k}\\}$ can be considered as the integration constants coming\nfrom the reduction of the degree of the gauge field equations\nwhich are second-order differential equations. Now we will follow\nthe methodology described above to show that there exists an\non-shell decoupling between the scalars and the gauge fields. Let\nus first take a look at the more restricted case of\nfield-independent $\\{C_{k}\\}$ and let us fix a set of\nfield-independent $\\{C_{k}\\}$. In this case if we take the partial\nderivative of both sides of \\eqref{de12} with respect to the\nscalar fields $\\{\\varphi^{ml}\\}$ we immediately see that\n\\begin{equation}\\label{de13}\n\\frac{\\partial\\mathcal{T}^{i}_{\\:\\:\\:k}}{\\partial\\varphi^{ml}}\\ast\nF_{i}=0,\n\\end{equation}\nsince the right hand side of \\eqref{de12} is a fixed $(D-2)$-form\nand it does not depend on the scalar fields $\\{\\varphi^{ml}\\}$.\nThese conditions must be satisfied on-shell by a sector of the\nsolution space of $\\{\\varphi^{ml},A^{i}\\}$ which is restricted to\nthe condition of choosing fixed $\\{C_{k}\\}$. After fixing the set\nof $(D-3)$-forms $\\{C_{k}\\}$ if we use \\eqref{de12} in \\eqref{de8}\nwe can obtain the on-shell Lagrangian as\n\\begin{equation}\\label{de14}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}dC_{j}\\wedge\n F^{j}.\n\\end{equation}\nSince we have made use of the gauge field equations varying this\non-shell Lagrangian with respect to $A^{i}$ yields an identity. On\nthe other hand by varying the above Lagrangian with respect to the\nscalars to find the solutions which satisfy \\eqref{de12} for a\nchosen fixed set of $\\{C_{k}\\}$ we find that the scalar field\nequations are the same ones which can be obtained directly from\nthe pure sigma model Lagrangian \\eqref{de3} since the coupling\npart in \\eqref{de14} does not depend on the scalars. This result\ncan also be obtained by directly varying \\eqref{de8} and by using\nthe conditions \\eqref{de13} which are satisfied by the sector of\nthe general solutions of the theory which we have restricted\nourselves in by fixing $\\{C_{k}\\}$. The same result would be\nobtained if one chooses another field-independent set of\n$\\{C_{k}\\}$. Thus we observe that the scalar solutions of a\nsub-sector of the scalar-gauge field theory defined by the\ncoupling Lagrangian \\eqref{de8} are the same with the general\nsolution space of the pure principal sigma model. This is a\nconsequence of the local integration given in \\eqref{de12} and\nfixing the arbitrary $\\{C_{k}\\}$.\n\nOn the other hand a similar result with a minor difference can\nalso be derived for the rest of the solution space of the theory.\nIf we consider the most general case of field-dependent\ndifferential forms $\\{C_{k}(\\varphi^{ml},A^{i})\\}$ and use\n\\eqref{de12} in the Lagrangian \\eqref{de8} we get\n\\begin{equation}\\label{de14.5}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n -\\frac{1}{2}dC_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}.\n\\end{equation}\nThis on-shell Lagrangian gives us the same decoupling conditions\ndiscussed above for the restricted sub-sector case. To see this we\nshould realize that the second term in the Lagrangian\n\\eqref{de14.5} which is written in an on-shell form is a closed\ndifferential form. Locally any closed differential form is an\nexact one. Thus the second term in the Lagrangian \\eqref{de14.5}\ncan be written as an exact differential form as\n\\begin{equation}\\label{de14.6}\n -\\frac{1}{2}dC_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}=dB(\\varphi^{ml},A^{i}).\n\\end{equation}\nIn fact we may simply calculate $B(\\varphi^{ml},A^{i})$ as\n\\begin{equation}\\label{de14.65}\n B(\\varphi^{ml},A^{i})=-\\frac{1}{2}C_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}.\n\\end{equation}\n Thus the Lagrangian \\eqref{de14.5} becomes\n\\begin{equation}\\label{de14.7}\n \\mathcal{L}_{tot}=\\frac{1}{2}tr( \\ast dg^{-1}\\wedge dg)\n +d(-\\frac{1}{2}C_{j}(\\varphi^{ml},A^{i})\\wedge\n F^{j}).\n\\end{equation}\nIf one varies the above Lagrangian one immediately sees that the\nsecond term does not contribute to the field equations as by using\nthe Stoke's theorem we have\n\\begin{equation}\\label{de14.75}\n\\int\\limits_{M}d\\delta B(\\varphi^{ml},A^{i})=\\int\\limits_{\\partial\nM}\\delta B(\\varphi^{ml},A^{i}).\n\\end{equation}\nIf $M$ does not posses a boundary the right hand side of the above\nequation is automatically zero whereas if it has a boundary then\nthe usual variation principles demand that the variation of the\nfields on the boundary are chosen to be zero in which case again\nthe right hand side of \\eqref{de14.75} becomes zero. Therefore we\nconclude that the on-shell Lagrangian \\eqref{de14.5} which is\nresponsible for the most general structure of the solution space\ngives us the scalar field equations which are the same with the\npure sigma model field equations since the coupling part in\n\\eqref{de14.5} does not contribute to the scalar field equations\nat all as we have proven as an on-shell condition above. Also like\nwe have already encountered for the non-field-dependent\n$\\{C_{k}\\}$ sub-sector case varying \\eqref{de14.5} does not give\nany information (which may also mean an identity) about the gauge\nfields $A^{i}$ since we have already made use of their field\nequations in writing it.\n\nIn summary, we have shown that the solutions of the theory must\nobey \\eqref{de12} for arbitrary right hand sides. If one\ndetermines the right hand sides in \\eqref{de12} one restricts him\nor herself to a layer of solutions. In this case \\eqref{de12}\nwhich are the descendants of the gauge field equations become\non-shell conditions. We have proven that if we use these\nfirst-order field equations which are on-shell conditions back in\nthe general Lagrangian \\eqref{de8} then the scalars of this\nparticular layer of solutions do not have gauge fields in their\nfield equations. Therefore the scalars of this particular layer\nbelong to the general solutions of the pure principal sigma model.\nBy running the right hand sides in \\eqref{de12} over the entire\nlocal field-dependent exact $(D-2)$-forms one can generalize this\nresult to the whole solution space. Thus in this way we have\nproven that the entire scalar solutions of the coupled theory\ncoincide with the pure sigma model solution space. Now from\n\\eqref{de12} we can write down the gauge field strengths as\n\\begin{equation}\\label{de15}\nF_{l}=(-1)^{s}(\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\ast\ndC_{k}(\\varphi^{mn},A^{i}),\n\\end{equation}\nwhere $s$ is the signature of the spacetime. We observe that if we\nconsider the sector of the solution space generated by the\nfield-independent $\\{C_{k}\\}$ then after obtaining the general\nsolutions of the pure principal sigma model and after choosing a\nfixed set $\\{C_{k}\\}$ one can use these in \\eqref{de15} to find\nthe corresponding $U(1)$ gauge field strengths. Also for the more\ngeneral field-dependent case of $\\{C_{k}(\\varphi^{ml},A^{i})\\}$\none again obtains the general solutions of the pure principal\nsigma model then one inserts these solutions in \\eqref{de15} to\nsolve for the gauge fields. We can say that in general we have a\npartial decoupling between the scalars and the gauge fields. The\nscalars are not affected by the presence of the gauge fields on\nthe other hand as we will show next they act as sources for the\ngauge fields. Now after multiplying by the Hodge star operator if\nwe take the exterior derivative of both sides of \\eqref{de15} we\nobtain\n\\begin{equation}\\label{de16}\nd\\ast F_{l}=(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\wedge\ndC_{k}(\\varphi^{mn},A^{i}).\n\\end{equation}\nWhen we compare this result with the inhomogeneous Maxwell\nequations \\eqref{de2.5} we observe that the current one-forms\nbecome\n\\begin{equation}\\label{de17}\nJ_{l}=(-1)^{(D+s)}\\ast((d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}\\wedge\ndC_{k}(\\varphi^{mn},A^{i})).\n\\end{equation}\nOne can furthermore verify that the currents in \\eqref{de17} obey\nthe current conservation law\n\\begin{equation}\\label{de18}\nd\\ast J_{l}=0,\n\\end{equation}\nwhich guarantees that the equations \\eqref{de16} have solutions.\nIn a local moving co-frame field $\\{e^{\\alpha}\\}$ on the spacetime\n$M$ if we introduce the components of the one-forms\n$(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}$, the $(D-2)$-forms\n$dC_{k}(\\varphi^{ml},A^{i})$, and the one-forms $J_{l}$ as\n\\begin{subequations}\\label{de19}\n\\begin{align}\n(d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}&=(\\mathcal{T}^{-1k}_{l})_{\\alpha}e^{\\alpha},\\notag\\\\\n\\notag\\\\\ndC_{k}(\\varphi^{ml},A^{i})&=\\frac{1}{(D-2)!}(\\mathcal{C}_{k})_{\\alpha_{1}\\cdots\\alpha_{(D-2)}}e^{\\alpha_{1}\\cdots\n\\alpha_{(D-2)}},\\notag\\\\\n \\notag\\\\\nJ_{l}&=\\mathcal{J}_{l\\beta}e^{\\beta},\\tag{\\ref{de19}}\n\\end{align}\n\\end{subequations}\nfrom \\eqref{de17} we can calculate the components of the current\none-forms as\n\\begin{equation}\\label{de20}\n\\mathcal{J}_{l\\beta}=\\frac{(-1)^{s}\\sqrt{|detH|}}{(D-2)!}(\\mathcal{T}^{-1k}_{l})_{\\alpha}\n(\\mathcal{C}_{k})_{\\alpha_{1}\\cdots\\alpha_{(D-2)}}H^{\\alpha_{1}\\beta_{1}}\\cdots\nH^{\\alpha\\beta_{(D-1)}}\\varepsilon_{\\beta_{1}\\cdots\\beta_{(D-1)}\\beta},\n\\end{equation}\nwhere we have introduced the metric $H$ on $M$ and the Levi-Civita\nsymbol $\\varepsilon$. In \\eqref{de20} $\n\\alpha,\\beta,\\alpha_{i},\\beta_{j}=1,\\cdots,D$. Also for\n$\\alpha_{i}$ $i=1,\\cdots,D-2$ and for $\\beta_{j}$\n$j=1,\\cdots,D-1$.\n\nIf one cancels an exterior derivative (performs integration) on\nboth sides of \\eqref{de16} one finds\n\\begin{equation}\\label{de21}\n\\ast\nF_{l}=(\\mathcal{T}^{-1})^{k}_{\\:\\:\\:l}dC_{k}(\\varphi^{mn},A^{i})+dC^{\\prime}_{l}(\\varphi^{mn},A^{i}).\n\\end{equation}\nNow if we compare this with the first-order equations \\eqref{de12}\noriginating from \\eqref{de8} we see that our model is a sub-sector\nof the one defined in \\eqref{de21} with the choice of\n$dC^{\\prime}_{l}(\\varphi^{mn},A^{i})=0$. One may also inspect\nwhich scalar-gauge field coupling kinetic term would result in a\nfirst-order formulation of gauge fields in the form \\eqref{de21}.\n\nWe should finally state that by using the local first-order\nformulation \\eqref{de12} of the gauge field equations of\n\\eqref{de8} we have shown that there exists a decoupling between\nthe $U(1)$ gauge fields and the scalar fields of the theory. The\nscalars are proven to be the general solutions of the pure\nprincipal sigma model and they generate current sources for the\ngauge fields as can be explicitly seen in \\eqref{de16}. For the\nsub-sector of the solution space which is generated by the\nfield-independent $\\{C_{k}\\}$ we end up with a decoupled set of\n$N$ non-interacting Maxwell theories with prescribed and known\ncurrents whose sources are predetermined by the principal sigma\nmodel scalar fields. These currents interact with each other via\nthe sigma model which is completely decoupled from the Maxwell\nsector and they define a media which does not interact with the\nMaxwell fields. For this sub-sector the $N$ decoupled and\nnon-interacting Maxwell theories can alternatively be formulated\nby the Lagrangian\n\\begin{equation}\\label{de22}\n {\\mathcal{L}}^{\\prime}=\\sum\\limits_{i=1}^{N}\\bigg(-\\frac{1}{2}dA^{i}\\wedge\n \\ast dA^{i}+A^{i}\\wedge (d\\mathcal{T}^{-1})^{k}_{\\:\\:\\:i}\\wedge\n dC_{k}\\bigg).\n\\end{equation}\nWe see that for a single scalar field the above Lagrangian drops\nto be the ordinary Maxwell Lagrangian with a known current\none-form. We also realize that the Maxwell fields in this\nrestricted case are coupled to each other only by means of\nintegration constants. Each of these decoupled Maxwell theories is\nan embedding into the ordinary Maxwell theory with known sources.\nWe should remark that \\eqref{de22} must only be used to\nderive\\footnote{By integrating the field equations to first-order\nand by choosing $dC^{\\prime}_{l}=0$.} the field equations of the\ngauge fields which belong to a restricted sector of the solution\nspace generated by fixing $\\{C_{k}\\}$ and in this case the field\nequations of the scalars must again be derived from \\eqref{de3}.\nOn the other hand for the most general solution space elements\ngenerated by choosing field-dependent\n$\\{C_{k}(\\varphi^{ml},A^{i})\\}$ we can again adopt the general\nsolutions of the pure sigma model as the general scalar solutions\nof our theory however in this case we have $N$ coupled Maxwell\ntheories whose potentials may enter in the currents. For the\nsub-sector generated by $\\{C_{k}(\\varphi^{ml})\\}$ which are not\ndependent on the gauge fields we have a similar situation of\ndecoupled Maxwell theories with prescribed currents discussed\nabove.\n\nFinally before concluding we should summarize the local general\nsolution methodology of the principal sigma model and $U(1)$ gauge\nfields coupling defined in \\eqref{de8}. The method has three steps\nfirst find the general scalar field solutions of the pure\nprincipal sigma model, secondly define field-dependent or\nfield-independent $\\{C_{k}\\}$ and use the general scalar solutions\nin them then insert these in \\eqref{de15} to solve the\ncorresponding gauge field strengths. By combining the general pure\nprincipal sigma model solutions with a different set of\n$\\{C_{k}\\}$ each time one can generate the entire set of\nsolutions. Such a solution methodology of the general solution\nspace of \\eqref{de8} is a consequence of the first-order\nformulation of the gauge field equations in \\eqref{de12} and the\non-shell decoupling between the scalars and the gauge fields which\nwe have described in detail for both the restricted sector of the\nsolution space and for the entire solution space in its most\ngeneral generation process. From the scalars point of view there\nis a complete decoupling so that the scalars of the coupled theory\ncoincide with the pure sigma model solution space. However the\ngauge fields are not decoupled from the scalars since we have\nshown that the scalars act in the current part of the gauge field\nequations.\n\\section{Conclusion}\nBy integrating the gauge field equations of the principal sigma\nmodel and Abelian gauge field coupling Lagrangian we have\nexpressed these equations in a first-order form. Then we have\nshown that when these first-order and on-shell expressions which\ncontain local $(D-3)$-forms in them are used in the scalar-matter\nLagrangian one reveals an implicit decoupling between the scalars\nand the matter gauge fields. We have proven that this decoupling\nstructure exists for the entire solution space. Therefore it is\nshown that the scalar solutions of the coupled theory are the\ngeneral solutions of the pure principal sigma model. We should\nstate here that a similar result is derived in \\cite{consist2} for\nthe heterotic string. However in that work the decoupling occurs\ndue to the existence of a dilatonic field and one derives the\ndecoupling structure of the coset scalars from the field equation\nof the dilaton. On the other hand in the present work we prove\nthat such a decoupling exists for a generic principal sigma model\nwith an arbitrary number of Abelian gauge field couplings.\n\nWe have also mentioned that the on-shell hidden scalar-matter\ndecoupling mentioned above generates a theory composed of the pure\nsigma model and a number of coupled Maxwell theories with sources\ninduced by the scalars. In particular we have discriminated a\nsub-sector of the general theory generated by the\nfield-independent integration constants. This sub-sector contains\na number of separate and dynamically non-interacting Maxwell\ntheories whose current sources are drawn from the general scalar\nsolutions of the pure principal sigma model and the integration\nconstants of the first-order formulation.\n\nIn this work, we prove that the scalars of the coupled theory are\nnot affected by the presence of the $U(1)$ gauge fields however\nthey take role in the currents of the gauge fields. Thus one may\ncall such a coupling between the scalars and the gauge fields a\none-sided or a one and a half coupling. We also see that for the\nrestricted sub-sector of the theory which is obtained by fixing\nthe integration constants the currents of the electromagnetic\nfields interact with in each other via the sigma model but they do\nnot interact with the corresponding gauge fields. Thus for this\nsub-sector the currents form up a subset of the general currents\nof the prescribed current-Maxwell theory. In this sub-sector the\nscalar-gauge field decoupling induces another decoupling scheme\nalso among the gauge fields and one obtains $N$ non-interacting\nMaxwell theories whose sources come from the pure principal sigma\nmodel. For each $U(1)$ gauge field we have an embedded sector of\nthe general predetermined current-Maxwell theory which has the\nmost general form of currents. In our case these currents are\nrestricted to the solutions of the pure principal sigma model. The\n$N$ $U(1)$ gauge fields in this case probe each other only by the\npresence of the integration constants however this does not\ncorrespond to a dynamic coupling. The coupling among the gauge\nfields occur only at the level of determining the integration\nconstants from the boundary conditions. Since we have a static and\nunchanged current structure which emerges from the pure principal\nsigma model where the currents interact with each other and since\nthese currents are not affected by the presence of the gauge\nfields this sub-sector of the general theory defines a theory of a\nstatic media with $N$ non-dynamically interacting Maxwell fields.\nOne can inspect the other sectors of the general theory based on\nvarious choice of field dependent $\\{C_{k}\\}$ which define dynamic\nmedia with current-gauge field interactions. For example one can\nstudy the sectors that result in gauge field equations that define\ndynamic media with specially defined conducting properties\n\\cite{thring}. The analysis of this work can furthermore be\nextended on another scalar-gauge field coupling which might have\nbroader dynamic media sub-sectors namely on the gauged sigma\nmodel.\n\nIn conclusion, we may state that the decoupling structure studied\nin this work is an important remark on the solutions of the\nscalar-gauge field interactions which form up a basic sector of\nthe Bosonic dynamics of the supergravities and the effective\nstring theories. Therefore the scalar-gauge field decoupling\nrevealed here points out a simplification in seeking solutions to\nthese theories.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe wrote a small class library to render\nwith computer graphics images of\nthe highly mathematical structures\ncreated by the artist Elias Wakan \\cite{wakan06}.\nThe \\hbox{{\\tt C++}}\\ classes we wrote will be extended\nto output the type of description\nrequired by a rendering package such as POV-Ray \\cite{povRay06}.\nWe were surprised that this thoroughly practical enterprise\nled us to two fascinating fundamental issues:\nthe limits of computability\nand the abstract nature of axiomatic geometry.\nHence our title ``Computational Euclid''.\n\n\\vspace{0.05in}\nTo accommodate the limits of computability,\nwe use interval arithmetic.\nWe use it in such a way that when we pose the question\nwhether two straight lines intersect, \nthe answer ``no'' has the force of a mathematical proof they do not.\nThe other possible answer is a box in 3-space, typically very small.\nThis answer means that the intersection is in this box,\n\\emph{if there is an intersection}.\nThis proviso is probably essential\nbecause we sense that a decision procedure\nfor the intersection of two lines can be used\nto implement a decision procedure\nfor the equality of any two real numbers,\nwhich has been shown to be impossible by Turing \\cite{trng37,aberth98}.\nHowever, we have not pursued the details of such a problem reduction.\n\nThe other fundamental issue\nis the abstract nature of an axiomatic approach to geometry.\n\nEuclid is widely credited with inaugurating the axiomatic method in which\naxioms contain references to undefined concepts of which the meaning is\nonly constrained by the axioms. However, we should not look to the Elements\nfor a literal embodiment of the axiomatic method.\nIt fell to Hilbert in 1902 \\cite{hlbrt02} to cast these in a form that\nis recognized today as an axiomatic treatment of the subject.\n\nLack of space prevents us here to go into a detailed analysis of the concepts\nof Euclid's geometry.\nSuffice it to say that Hilbert's formulation contains as undefined concepts,\namong others, the following that we found useful in our work:\n\\emph{point},\n\\emph{line},\n\\emph{segment},\n\\emph{plane},\n\\emph{angle}.\n\nIn the modern conception of the axiomatic method these are undefined.\nTheir meaning is only constrained by the relations between them as\nasserted by the axioms. In logic this is formalized by the axioms being\na theory, which, if consistent, can have a variety of models.\nIt is only the model that says what a point \\emph{is}.\nFor example, in one type of model,\npoints, lines, and planes are solution spaces\nof sets of linear equations in three variables.\n\nWhat we find surprising is how well the way in which the axiomatic method,\nas realized in formal logic, combines with the most widely accepted principles\nof object-oriented software design \\cite{wbww90}.\nAccording to it, one looks for the \\emph{nouns} in an informal specification\nof the software to be written.\nThese are candidates for the \\emph{classes} of an object-oriented program.\n\nWe used Hilbert's axioms as specification.\nThe recipe of \\cite{wbww90} has, of course, to be taken with a grain of salt:\nonly the \\emph{important} nouns are candidates for classes.\nUsually, informal specifications contain a majority of not-so-important nouns.\nTo our delight,\nwe found that Hilbert's axioms\ncontain an unusually small number\nof not-so-important nouns.\n\n\\section{The structure of our class library}\n\nOf the nouns occurring in Hilbert's axioms,\n{\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are a special subset.\nThey are special in the sense\nthat any unordered pair of these determine\nan object in this trinity,\nunless a specific condition prevails.\nIn the case of a point and a line determining a  plane,\nthe condition is ``unless the point is on the line''.\nNote that these conditions are called \\emph{predicates}\nby some authors~\\cite{effpred02,itar98}.\nThe table in Figure~\\ref{pointsLinesPlanes}\nsummarizes the operations for all unordered pairs,\neach with the attendant disabling condition.\n\\begin{figure*}\n\\begin{center}\n\\begin{tabular}{l|l}\n\\hline\n Construction &  Disabling Condition   \\\\\n\\hline\n{\\tt Point}\\ $\\times$ {\\tt Point}\\ $\\rightarrow$ {\\tt Line}\\ & equal                          \\\\\n{\\tt Point}\\ $\\times$ {\\tt Line}\\ $\\rightarrow$ {\\tt Plane}\\ & on                             \\\\\n{\\tt Point}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Line}\\ & (none)                         \\\\\n{\\tt Line}\\ $\\times$ {\\tt Line}\\ $\\rightarrow$ {\\tt Line}\\ & parallel or intersect          \\\\\n{\\tt Line}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Point}\\ & in                             \\\\\n{\\tt Plane}\\ $\\times$ {\\tt Plane}\\ $\\rightarrow$ {\\tt Line}\\ & parallel                       \\\\\n \\hline\n\\end{tabular}\n\\caption{Operations for all unordered pairs formed from {\\tt Point}, {\\tt Line}\\ and {\\tt Plane}.\nThe operations cannot be performed if the condition listed holds between\nthe input arguments of the construction.\n}\n\\label{pointsLinesPlanes}\n\\end{center}\n\\end{figure*}\n\nTwo of the constructions involve perpendiculars.\nThe line determined by the point\nand the plane is the perpendicular\nto the plane through the point.\nThe line determined by two lines in general position\nlikewise is a perpendicular:\nthe unique one that is perpendicular to both given lines.\n\nIn this way, an object-oriented reading\nof Hilbert's axioms determines that the class {\\tt Line}\\\ncontains constructors with parameters\n({\\tt Point}, {\\tt Point}), with ({\\tt Point}, {\\tt Plane}), and with ({\\tt Plane}, {\\tt Plane}).\nThe class {\\tt Point}\\ contains a constructor\nwith parameters ({\\tt Line}, {\\tt Plane}).\nThe class {\\tt Plane}\\ contains a constructor\nwith arguments ({\\tt Point}, {\\tt Line}).\n\nThe constructors cannot be invoked\nwhen the conditions noted in Figure~\\ref{pointsLinesPlanes} hold\nbetween the arguments.\nFor example, if a {\\tt Point}\\ is on a {\\tt Line}, then these do not determine a plane.\nThese conditions are semi-decidable: they either determine that the\ncondition does not hold, or that the condition \\emph{may} hold.\nHowever, in rare cases it can be determined that, say, two\ninstances of {\\tt Point}\\ are equal.\nThe conditions therefore return the truth values of a 3-valued logic.\n\n\nThe abstract nature of an axiomatic approach to geometry\nrequires that the {\\tt Point}, {\\tt Line}\\ and {\\tt Plane}\\ \nare left undefined.\nThis abstraction is not only essential in \nthe axiomatic treatment of mathematical theories,\nbut it is also the essence of object-oriented design.\n\nIn object-oriented design one may distinguish two forms of abstraction.\nThe weaker form is achieved by any class in which the variables\nare private.\nOne can then modify the representation of the objects without\nconsequences for the code using the class.\nThere is also a stronger form of abstraction in which\npolymorphism makes it possible\nto use more than one implementation of the same abstraction simultaneously.\nThe concept is then represented by an abstract class for the concept in\nwhich the representation-dependent methods are virtual.\nFor each representation there is a separate derived class of\nwhich the methods are dispatched at run time.\nWe have found this stronger form of abstraction\nadvantageous in our suite of \\hbox{{\\tt C++}} classes.\n\nA UML class diagram summarizing the classes \nand the conditions above is shown in Figure~\\ref{geomuml}. \nThe purpose of the extra classes in the diagram is explained later.\n\\begin{figure}[!htbp]\n\\begin{center}\n\\epsfxsize=3.5in\n\\leavevmode\n\\epsfbox{Figures\/euclclassdiagram.eps}\n\\caption{\n\\label{geomuml}\nUML class diagram for our system.}\n\\end{center}\n\\end{figure}\n\n\\section{Consequences of computability limitations}\n\nWhatever computer representation is chosen,\nthere will only be finitely many points, lines, and planes\nthat can be represented.\nThe conventional method of mapping\nthe infinity of abstract objects\nto the finitely many representable ones\nis to choose a representation in terms of reals\nand then to map each real to a nearby floating-point number.\nWhen this method is followed,\nit has so far not been found possible\nto give precise meaning to the outcomes\nof tests such as whether a point is on a line.\nThe outcomes have to be interpreted\nas ``probably not'' and ``possibly'',\ndepending on whether the computed distance\n(subject to an unknown error)\nis greater than a certain tolerance.\n\nIt may seem that this degree of uncertainty\nis inherent in the limitation\nto a finite number of representations.\nThis is not case.\nEven when restricted to floating-point numbers,\nit is possible to represent the point $p$\nby a set $P$ of points containing $p$;\nlikewise, the line $l$ can be represented by a set $L$.\nThese sets are specified in terms of floating-point numbers,\nso there are only finitely many of these.\nBecause of this finiteness it is decidable\nwhether the set of points contains any\nthat is on any in the set of lines.\nIt may seem computationally formidable\nto make such a determination.\nActually, the techniques of interval constraints\nmake this perfectly feasible \\cite{hcqvn99},\nand this is what we use.\n\nIf it is determined that no point in $P$ is on any line in $L$,\nthen it is clear that $p$ is not on $l$.\nIf, on the other hand, some point in $P$ is on some line in $L$,\nthis says nothing about  whether $p$ is on $l$.\nHowever, if $P$ and $L$ are, in a suitable sense, small,\nthen it follows that $p$ is close to $l$.\nIt is this asymmetry that is a consequence of the fact that the test for a point\non a line can at best be a \nsemi-decision algorithm.\nSimilarly, the other tests\nin Figure~\\ref{pointsLinesPlanes} are semi-decision algorithms.\n\nIt is worth mentioning that to cope with the computability limitations \nin the area of computational geometry, the \\emph{exact geometric computing} \nparadigm was proposed \\cite{yap95}. This paradigm encompasses all techniques \nfor which the outcomes are correct. As shown in \\cite{itar98}, \ninterval arithmetic can be used to do exact geometric computing.\nThis paper is also classified under this paradigm.\n\n\n\n\\section{Our implementation}\nIn the previous section we explained the need for interval methods\nto ensure that in most cases where a test should have a negative\noutcome, this is indeed proved numerically.\nInterval methods can do this in several ways.\nIn \\cite{itar98}, Br{\\\"o}nnimann {\\it et al.} used \ninterval arithmetic to dynamically bound \narithmetic errors when computing tests (i.e. to compute dynamic filters).\nIn our case, we use interval constraints\nnot only to compute tests but also to implement\ngeometrical constructions. \nThis means that the representations\nof {\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are in the form of constraint\nsatisfaction problems.\nFor example, a plane is represented by the constraint\n$ax+by+cz+d=0$, where\n$a$, $b$, $c$ and $d$ are real-valued constants and\n$x$, $y$, and $z$ are real-valued variables.\nDue to computability limitations\ndiscussed in the previous section,\nthe coefficients $a$, $b$, $c$ and $d$\nare implemented as floating-point intervals.\nFor each point with coordinates\nin these intervals, the constraint has a different\nplane as solution. In this way our concrete representation is\na set of planes in the abstract sense.\nThe reader may refer to the following papers \n\\cite{rthpra01}, \\cite{bhlgfg99}, \\cite{hckmdw98}, \nand \\cite{hckvnmdn01} for more information\non constraints, propagation algorithms, interval constraints,\ncorrectness and implementation of interval constraints.\n\nAs shown in Figure~\\ref{geomuml},\nthe abstract classes {\\tt Point}, {\\tt Line}, and {\\tt Plane}\\ are extended \nand modelled using intervals and constraints.\nThe abstract class {\\tt Constraint}\\ represents the constraint class,\nwhich can be extended to implement primitive constraints\nsuch as {\\tt Sum} and {\\tt Prod}. \nEach of these primitive constraints has \na \\emph{domain reduction operator} (DRO), \nrepresented by {\\tt shrink}() method,\nwhich removes inconsistent values from \nthe domains of the variables in the constraint.\nThe DROs of primitive constraints are computed based on \ninterval arithmetic. As an example, \nthe {\\tt Sum} constraint defined \nby $x+y=z$ has the\nfollowing DRO\n\\begin{eqnarray}\n&&X^{new}=X^{old} \\cap (Z^{old} - Y^{old})\\nonumber \\\\\n&&Y^{new}=Y^{old} \\cap (Z^{old} - X^{old})\\nonumber \\\\\n&&Z^{new}=Z^{old} \\cap (X^{old} + Y^{old})\\nonumber \n\\end{eqnarray}\nwhere the intervals $X^{old}$, $Y^{old}$, \nand $Z^{old}$ are the domains of $x$, $y$, and $z$\nrespectively before applying the DRO,\nand $X^{new}$, $Y^{new}$ and $Z^{new}$ are the\ndomains of $x$, $y$, and $z$ respectively after applying the DRO.\nFor a non-primitive constraint, such as {\\tt Line}\\ and {\\tt Plane}, \nwe first decompose it into primitive constraints and then\nuse the propagation algorithm to implement the {\\tt shrink}() method.\nA simple version of this algorithm is shown in Figure~\\ref{gpa}.\n\\begin{center}\n\\begin{figure}\n\\begin{center}\n\\fbox{\n\\parbox{1 cm}{\n\\small{\n\\begin{tabbing}\nmake $A$ the set of primitive constraints;\\\\\nwhile \\=( $A \\neq \\emptyset$) $\\{$\\\\  \n         \\>choose a constraint $C$ from $A$ and apply its DRO;\\\\\n         \\> if one of the domains becomes empty, then stop;\\\\\n         \\>add \\=to $A$ all constraints involving variables whose\\\\\n         \\>\\> domains have changed, if any;\\\\\n         \\>remove $C$ from $A$; $\\}$\n\\end{tabbing}\n}\n}\n}\n\\end{center}\n\\caption{\n\\label{gpa}\nPropagation algorithm.}\n\\end{figure}\n\\end{center}\nIn what follows, we present some examples\nin two dimensions illustrating the use of our implementation.\nWe ran the examples on a Pentium II machine with a CPU rate of 400 MHz, \nand with 128 MB of memory. \n\\paragraph{Are two points in the same side of a line?}\nLet $L$ be a line represented by $[2.0,2.5]*x - [0.5,1.0]*y=[1.0,1.05]$.\nLet $P$ and $Q$ be the two points represented respectively by\n$([0.0,0.0],[0.0,0.0])$ and $([0.5,0.5],[0.5,0.5])$.\nThe question we are interested in is to\ndetermine whether $P$ and $Q$ are on the same side of $L$.\nUsing the function \\emph{sameSide(Point, Point)},\nwhich checks whether two points are in the same side of a line,\nour system outputs the following results:\n\\begin{verbatim}\nDuration (musec): 179\nTrue: the points are in the same side\n\\end{verbatim}\nThis means that our system was able to prove that\n$P$ and $Q$ are in the same side of $L$.\n\nNow suppose that $Q$ is represented by $([1 , 1],[0.5 , 0.5])$.\nIn this case, our system returned the following output:\n\\begin{verbatim}\nDuration (musec): 133\nFalse: the points are not in the same side\n\\end{verbatim}\nIf, somehow, the point $Q$ is only known to be represented by\n$([0.75 , 1],[0.25 , 0.5])$ (note that the intervals are not singletons),\nthen our system was not able to prove that\nthe points $P$ and $Q$ are in the same side.\nThe output in this case is:\n\\begin{verbatim}\nDuration (musec): 265\nUndetermined\n\\end{verbatim}\n\\paragraph{Circumcenter of a triangle}\nGiven three points $P$, $Q$ and $R$ represented respectively by\n$([0.0,0.0],[0.0,0.0])$, $([1.0,1.0],[0.5,0.5])$ and $([0.5,0.5],[1.0,1.0])$\nwe wish to find the center of the circle passing through $P$, $Q$ and $R$.\nThis example is taken from \\cite{fnpr02}.\nSince this center is given by the intersection of $L_1$ and $L_2$,\nwhere $L_1$ is the line that passes through the middle of the segment $PQ$ and is\nperpendicular to the line passing through $P$ and $Q$, and $L_2$\nis the line that passes through the middle of the segment $QR$ and\nis perpendicular to the line passing through $Q$ and $R$.\nUsing the \\emph{intersect (Line)} function that\nchecks whether a line intersects with another\nline, our system returned the following output:\n\\begin{verbatim}\nDuration (msec):  6\nTrue: the lines intersect at\nx = [0.41666666666666663 , \n     0.41666666666666669]\ny = [0.41666666666666663 , \n     0.41666666666666669]\n\\end{verbatim}\nWe then checked whether the point\n$(x,y)$\nis on the line $L_3$\nthat passes through the middle of the segment $PR$\nand is perpendicular to the line passing through $P$ and $R$.\nThe output of our system indicates it is possible:\n\\begin{verbatim}\nDuration (musec): 112\nUndetermined\n\\end{verbatim}\n\n\\section*{Acknowledgements}\nThis research was supported by the University of Victoria and by\nthe Natural Science and Engineering Research Council of Canada.\n\n\\section*{For further information please contact us:}\n\\begin{flushleft}\n\\begin{tabular}{l@{\\quad}l@{\\hspace{3mm}}l@{\\quad}l}\n$\\bullet$&\\multicolumn{3}{@{}l}{\\bfseries Springer-\\kern-2pt Verlag\nHeidelberg}\\\\[1mm]\n&\\multicolumn{3}{@{}l}{Department New Technologies\/Product\nDevelopment}\\\\\n&\\multicolumn{3}{@{}l}{Springer-Verlag, Postfach 105280, D-69042\nHeidelberg\n1, FRG}\\\\[0.5mm]\n & Telefax:   & (0\\,62\\,21)487688\\\\\n &            & (0\\,62\\,21)487366\\\\\n & Internet:  & \\tt lncs@springer.de    & for editorial questions\\\\\n &            & \\tt texhelp@springer.de & for \\TeX{} problems\n\\end{tabular}\n\\end{flushleft}\n\\rule{\\textwidth}{1pt}\n\\section*{Acceptable formats of your disk\/magnetic tape and output:}\nThe following formats are acceptable: 5.25$^{\\prime\\prime}$ diskette\nMS-DOS, 5.25$^{\\prime\\prime}$ CP\/M, 3.5$^{\\prime\\prime}$ diskette\nMS-DOS, 3.5$^{\\prime\\prime}$ diskette Apple MacIntosh, 9-track 1600\nbpi magnetic tape VAX\/VMS, 9-track 1600 bpi magnetic tape ANSI with\nlabel, SUN-Streamer Tape.\n\nOnce you have completed your work using this macro package,\nplease submit your own printout of the {\\em final\nversion together with the disk or magnetic tape}, containing your\n\\LaTeX{} input (source) file und the final DVI-file and make sure\nthat the text is {\\em identical in both cases.}\n\n\\bigskip\nThis macro package, as well as all other macro packages, style\nfiles, and document classes that Springer distributes, are also\navailable through our mailserver (for people with only e-mail access).\n\n{\\tt svserv@vax.ntp.springer.de}\\hfil first try the \\verb|help|\ncommand.\n\n\\noindent We are also reachable through the world wide web:\n\\begin{flushleft}\n\\tt\n\\begin{tabular}{@{}l@{\\quad}r@{\\tt:}l}\n\\rmfamily URLs are &   http&\/\/www.springer.de       \\\\\n             & gopher&\/\/ftp.springer.de \\\\\n             &    ftp&\/\/ftp.springer.de\n\\end{tabular}\n\\end{flushleft}\n\n\n\n\n\\newpage\n\\tableofcontents\n\\newpage\n\\section{Introduction}\nAuthors wishing to code their contribution\nwith \\LaTeX{}, as well as those who have already coded with \\LaTeX{},\nwill be provided with a document class that will give the text the\ndesired layout. Authors are requested to\nadhere strictly to these instructions; {\\em the class\nfile must not be changed}.\n\nThe text output area is automatically set within an area of\n12.2\\,cm horizontally  and 19.3\\,cm vertically.\n\nIf you are already familiar with \\LaTeX{}, then the\nLLNCS class should not give you any major difficulties.\nIt will change the layout to the required LLNCS style\n(it will for instance define the layout of \\verb|\\section|).\nWe had to invent some extra commands,\nwhich are not provided by \\LaTeX{} (e.g.\\\n\\verb|\\institute|, see also Sect.\\,\\ref{contbegin})\n\nFor the main body of the paper (the text) you\nshould use the commands of the standard \\LaTeX{} ``article'' class.\nEven if you are familiar with those commands, we urge you to read\nthis entire documentation thoroughly. It contains many suggestions on\nhow to use our commands properly; thus your paper\nwill be formatted exactly to LLNCS standard.\nFor the input of the references at the end of your contribution,\nplease follow our instructions given in Sect.\\,\\ref{refer} References.\n\nThe majority of these hints are not specific for LLNCS; they may improve\nyour use of \\LaTeX{} in general.\nFurthermore, the documentation provides suggestions about the proper\nediting and use\nof the input files (capitalization, abbreviation etc.) (see\nSect.\\,\\ref{refedit} How to Edit Your Input File).\n\\section{How to Proceed}\nThe package consists of the following files:\n\\begin{flushleft}\n\\begin{tabular}{@{}p{2.5cm}l}\n{\\tt history.txt}& the version history of the package\\\\[2pt]\n{\\tt llncs.cls}  & class file for \\LaTeX{}\\\\[2pt]\n{\\tt llncs.dem}  & an example showing how to code the text\\\\[2pt]\n{\\tt llncs.doc}  & general instructions (source of this document),\\\\\n        & {\\tt llncs.doc} means {\\itshape l\\\/}atex {\\itshape doc\\\/}umentation for\\\\\n        & {\\itshape L\\\/}ecture {\\itshape N}otes in {\\itshape C\\\/}omputer {\\itshape S\\\/}cience\\\\\n{\\tt llncsdoc.sty}  & class modifications to help for the instructions\\\\\n{\\tt llncs.ind} & an external (faked) author index file\\\\\n{\\tt subjidx.ind} & subject index demo from the Springer book package\\\\\n{\\tt llncs.dvi}    & the resultig DVI file (remember to use binary transfer!)\\\\[2pt]\n{\\tt sprmindx.sty} & supplementary style file for MakeIndex\\\\\n                   & (usage: {\\tt makeindex -s sprmindx.sty <yourfile.idx>})\n\\end{tabular}\n\\end{flushleft}\n\\subsection{How to Invoke the LLNCS Document Class}\nThe LLNCS class is an extension of the standard \\LaTeX{} ``article''\ndocument class. Therefore you may use all ``article'' commands for the\nbody of your contribution to prepare your manuscript.\nLLNCS class is invoked by replacing ``article'' by ``llncs'' in the\nfirst line of your document:\n\\begin{verbatim}\n\\documentclass{llncs}\n\\begin{document}\n  <Your contribution>\n\\end{document}\n\\end{verbatim}\n\\subsection{Contributions Already Coded with \\protect\\LaTeX{} without\nthe LLNCS document class}\nIf your file is already coded with \\LaTeX{} you can easily\nadapt it a posteriori to the LLNCS document class.\n\nPlease refrain from using any \\LaTeX{} or \\TeX{} commands\nthat affect the layout or formatting of your document (i.e. commands\nlike \\verb|\\textheight|, \\verb|\\vspace|, \\verb|\\headsep| etc.).\nThere may nevertheless be exceptional occasions on which to\nuse some of them.\n\nThe LLNCS document class has been carefully designed to produce the\nright layout from your \\LaTeX{} input. If there is anything specific you\nwould like to do and for which the style file does not provide a\ncommand, {\\em please contact us}. Same holds for any error and bug you\ndiscover (there is however no reward for this -- sorry).\n\\section{General Rules for Coding Formulas}\nWith mathematical formulas you may proceed as described\nin Sect.\\,3.3 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport (2nd~ed. 1994), Addison-Wesley Publishing\nCompany, Inc.\n\nEquations are automatically numbered sequentially throughout your\ncontribution using arabic numerals in parentheses on the right-hand\nside.\n\nWhen you are working in math mode everything is typeset in italics.\nSometimes you need to insert non-mathematical elements (e.g.\\\nwords or phrases). Such insertions should be coded in roman\n(with \\verb|\\mbox|) as illustrated in the following example:\n\\begin{flushleft}\n{\\itshape Sample Input}\n\\end{flushleft}\n\\begin{verbatim}\n\\begin{equation}\n  \\left(\\frac{a^{2} + b^{2}}{c^{3}} \\right) = 1 \\quad\n  \\mbox{ if } c\\neq 0 \\mbox{ and if } a,b,c\\in \\bbbr \\enspace .\n\\end{equation}\n\\end{verbatim}\n{\\itshape Sample Output}\n\\begin{equation}\n  \\left(\\frac{a^{2} + b^{2}}{c^{3}} \\right) = 1 \\quad\n  \\mbox{ if } c\\neq 0 \\mbox{ and if } a,b,c\\in \\bbbr \\enspace .\n\\end{equation}\n\nIf you wish to start a new paragraph immediately after a displayed\nequation, insert a blank line so as to produce the required\nindentation. If there is no new paragraph either do not insert\na blank line or code \\verb|\\noindent| immediately before\ncontinuing the text.\n\nPlease punctuate a displayed equation in the same way as other\nordinary text but with an \\verb|\\enspace| before end punctuation.\n\nNote that the sizes of the parentheses or other delimiter\nsymbols used in equations should ideally match the height of the\nformulas being enclosed. This is automatically taken care of by\nthe following \\LaTeX{} commands:\\\\[2mm]\n\\verb|\\left(| or \\verb|\\left[| and\n\\verb|\\right)| or \\verb|\\right]|.\n\\subsection{Italic and Roman Type in Math Mode}\n\\begin{alpherate}\n\\item\nIn math mode \\LaTeX{} treats all letters as though they\nwere mathematical or physical variables, hence they are typeset as\ncharacters of their own in\nitalics. However, for certain components of formulas, like short texts,\nthis would be incorrect and therefore coding in roman is required.\nRoman should also be used for\nsubscripts and superscripts {\\em in formulas\\\/} where these are\nmerely labels and not in themselves variables,\ne.g. $T_{\\mathrm{eff}}$ \\emph{not} $T_{eff}$,\n$T_{\\mathrm K}$ \\emph{not} $T_K$ (K = Kelvin),\n$m_{\\mathrm e}$ \\emph{not} $m_e$ (e = electron).\nHowever, do not code for roman\nif the sub\/superscripts represent variables,\ne.g.\\ $\\sum_{i=1}^{n} a_{i}$.\n\\item\nPlease ensure that {\\em physical units\\\/} (e.g.\\ pc, erg s$^{-1}$\nK, cm$^{-3}$, W m$^{-2}$ Hz$^{-1}$, m kg s$^{-2}$ A$^{-2}$) and\n{\\em abbreviations\\\/} such as Ord, Var, GL, SL, sgn, const.\\\nare always set in roman type. To ensure\nthis use the \\verb|\\mathrm| command: \\verb|\\mathrm{Hz}|.\nOn p.\\ 44 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport you will find the names of\ncommon mathe\\-matical functions, such as log, sin, exp, max and sup.\nThese should be coded as \\verb|\\log|,\n\\verb|\\sin|, \\verb|\\exp|, \\verb|\\max|, \\verb|\\sup|\nand will appear in roman automatically.\n\\item\nChemical symbols and formulas should be coded for roman,\ne.g.\\ Fe not $Fe$, H$_2$O not {\\em H$_2$O}.\n\\item\nFamiliar foreign words and phrases, e.g.\\ et al.,\na priori, in situ, brems\\-strah\\-lung, eigenvalues should not be\nitalicized.\n\\end{alpherate}\n\\section{How to Edit Your Input (Source) File}\n\\label{refedit}\n\\subsection{Headings}\\label{headings}\nAll words in headings should be capitalized except for conjunctions,\nprepositions (e.g.\\ on, of, by, and, or, but, from, with, without,\nunder) and definite and indefinite articles (the, a, an) unless they\nappear at the beginning. Formula letters must be typeset as in the text.\n\\subsection{Capitalization and Non-capitalization}\n\\begin{alpherate}\n\\item\nThe following should always be capitalized:\n\\begin{itemize}\n\\item\nHeadings (see preceding Sect.\\,\\ref{headings})\n\\item\nAbbreviations and expressions\nin the text such as  Fig(s)., Table(s), Sect(s)., Chap(s).,\nTheorem, Corollary, Definition etc. when used with numbers, e.g.\\\nFig.\\,3, Table\\,1, Theorem 2.\n\\end{itemize}\nPlease follow the special rules in Sect.\\,\\ref{abbrev} for referring to\nequations.\n\\item\nThe following should {\\em not\\\/} be capitalized:\n\\begin{itemize}\n\\item\nThe words figure(s), table(s), equation(s), theorem(s) in the text when\nused without an accompanying number.\n\\item\nFigure legends and table captions except for names and abbreviations.\n\\end{itemize}\n\\end{alpherate}\n\\subsection{Abbreviation of Words}\\label{abbrev}\n\\begin{alpherate}\n\\item\nThe following {\\em should} be abbreviated when they appear in running\ntext {\\em unless\\\/} they come at the beginning of a sentence: Chap.,\nSect., Fig.; e.g.\\ The results are depicted in Fig.\\,5. Figure 9 reveals\nthat \\dots .\\\\\n{\\em Please note\\\/}: Equations should usually be referred to solely by\ntheir number in parentheses: e.g.\\ (14). However, when the reference\ncomes at the beginning of a sentence, the unabbreviated word\n``Equation'' should be used: e.g.\\ Equation (14) is very important.\nHowever, (15) makes it clear that \\dots .\n\\item\nIf abbreviations of names or concepts are used\nthroughout the text, they should be defined at first occurrence,\ne.g.\\ Plurisubharmonic (PSH) Functions, Strong Optimization (SOPT)\nProblem.\n\\end{alpherate}\n\\section{How to Code the Beginning of Your Contribution}\n\\label{contbegin}\nThe title of a single contribution (it is mandatory) should be coded as\nfollows:\n\\begin{verbatim}\n\\title{<Your contribution title>}\n\\end{verbatim}\nAll words in titles should be capitalized except for conjunctions,\nprepositions (e.g.\\ on, of, by, and, or, but, from, with, without,\nunder) and definite and indefinite articles (the, a, an) unless they\nappear at the beginning. Formula letters must be typeset as in the text.\nTitles have no end punctuation.\n\nIf a long \\verb|\\title| must be divided please use the code \\verb|\\\\|\n(for new line).\n\nIf you are to produce running heads for a specific volume the standard\n(of no such running heads) is overwritten with the \\verb|[runningheads]|\noption in the \\verb|\\documentclass| line. For long titles that do not\nfit in the single line of the running head a warning is generated.\nYou can specify an abbreviated title for the running head on odd pages\nwith the command\n\\begin{verbatim}\n\\titlerunning{<Your abbreviated contribution title>}\n\\end{verbatim}\n\nThere is also a possibility to change the text of the title that goes\ninto the table of contents (that's for volume editors only -- there is\nno table of contents for a single contribution). For this use the\ncommand\n\\begin{verbatim}\n\\toctitle{<Your changed title for the table of contents>}\n\\end{verbatim}\n\nAn optional subtitle may follow then:\n\\begin{verbatim}\n\\subtitle{<subtitle of your contribution>}\n\\end{verbatim}\n\nNow the name(s) of the author(s) must be given:\n\\begin{verbatim}\n\\author{<author(s) name(s)>}\n\\end{verbatim}\nNumbers referring to different addresses or affiliations are\nto be attached to each author with the \\verb|\\inst{<no>}| command.\nIf there is more than one author, the order is up to you;\nthe \\verb|\\and| command provides for the separation.\n\nIf you have done this correctly, this entry now reads, for example:\n\\begin{verbatim}\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\\end{verbatim}\nThe first name\\footnote{Other initials are optional\nand may be inserted if this is the usual\nway of writing your name, e.g.\\ Alfred J.~Holmes, E.~Henry Green.}\nis followed by the surname.\n\nAs for the title there exist two additional commands (again for volume\neditors only) for a different author list. One for the running head\n(on odd pages) -- if there is any:\n\\begin{verbatim}\n\\authorrunning{<abbreviated author list>}\n\\end{verbatim}\nAnd one for the table of contents where the\naffiliation of each author is simply added in braces.\n\\begin{verbatim}\n\\tocauthor{<enhanced author list for the table of contents>}\n\\end{verbatim}\n\nNext the address(es) of institute(s), company etc. is (are) required.\nIf there is more than one address, the entries are numbered\nautomatically with \\verb|\\and|, in the order in which you type them.\nPlease make sure that the numbers match those placed next to\nto the authors' names to reflect the affiliation.\n\\begin{verbatim}\n\\institute{<name of an institute>\n\\and <name of the next institute>\n\\and <name of the next institute>}\n\\end{verbatim}\n\nIn addition, you can use\n\\begin{verbatim}\n\\email{<email address>}\n\\end{verbatim}\nto provide your email address within \\verb|\\institute|. If you need to\ntypeset the tilde character -- e.g. for your web page in your unix\nsystem's home directory -- the \\verb|\\homedir| command will happily do\nthis.\n\n\\medskip\nIf footnote like things are needed anywhere in the contribution heading\nplease code\n(immediately after the word where the footnote indicator should be\nplaced):\n\\begin{verbatim}\n\\thanks{<text>}\n\\end{verbatim}\n\\verb|\\thanks| may only appear in \\verb|\\title|, \\verb|\\author|\nand \\verb|\\institute| to footnote anything. If there are two or more\nfootnotes or affiliation marks to a specific item separate them with\n\\verb|\\fnmsep| (i.e. {\\itshape f}oot\\emph note \\emph mark\n\\emph{sep}arator).\n\n\\medskip\\noindent\nThe command\n\\begin{verbatim}\n\\maketitle\n\\end{verbatim}\nthen formats the complete heading of your article. If you leave\nit out the work done so far will produce \\emph{no} text.\n\nThen the abstract should follow. Simply code\n\\begin{verbatim}\n\\begin{abstract}\n<Text of the summary of your article>\n\\end{abstract}\n\\end{verbatim}\nor refer to the demonstration file {\\tt llncs.dem} for an example or\nto the {\\em Sample Input\\\/} on p.~\\pageref{samppage}.\n\n\\subsubsection{Remark to Running Heads and the Table of Contents}\n\\leavevmode\\\\[\\medskipamount]\nIf you are the author of a single contribution you normally have no\nrunning heads and no table of contents. Both are done only by the editor\nof the volume or at the printers.\n\\section{Special Commands for the Volume Editor}\nThe volume editor can produce a complete camera ready output including\nrunning heads, a table of contents, preliminary text (frontmatter), and\nindex or glossary. For activating the running heads there is the class\noption \\verb|[runningheads]|.\n\nThe table of contents of the volume is printed wherever\n\\verb|\\tableofcontents| is placed. A simple compilation of all\ncontributions (fields \\verb|\\title| and \\verb|\\author|) is done. If you\nwish to change this automatically produced list use the commands\n\\begin{verbatim}\n\\titlerunning  \\toctitle\n\\authorrunning \\tocauthor\n\\end{verbatim}\nto enhance the information in the specific contributions. See the\ndemonstration file \\verb|llncs.dem| for examples.\n\nAn additional structure can be added to the table of contents with the\n\\verb|\\addtocmark{<text>}| command. It has an optional numerical\nargument, a digit from 1 through 3. 3 (the default) makes an unnumbered\nchapter like entry in the table of contents. If you code\n\\verb|\\addtocmark[2]{text}| the corresponding page number is listed\nalso, \\verb|\\addtocmark[1]{text}| even introduces a chapter number\nbeyond it.\n\\section{How to Code Your Text}\nThe contribution title and all headings should be capitalized\nexcept for conjunctions, prepositions (e.g.\\ on, of, by, and, or, but,\nfrom, with, without, under) and definite and indefinite articles (the,\na, an) unless they appear at the beginning. Formula letters must be\ntypeset as in the text.\n\nHeadings will be automatically numbered by the following codes.\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\section{This is a First-Order Title}\n\\subsection{This is a Second-Order Title}\n\\subsubsection{This is a Third-Order Title.}\n\\paragraph{This is a Fourth-Order Title.}\n\\end{verbatim}\n\\verb|\\section| and \\verb|\\subsection| have no end punctuation.\\\\\n\\verb|\\subsubsection| and \\verb|\\paragraph|\nneed to be punctuated at the end.\n\nIn addition to the above-mentioned headings your text may be structured\nby subsections indicated by run-in headings (theorem-like environments).\nAll the theorem-like environments are numbered automatically\nthroughout the sections of your document -- each with its own counter.\nIf you want the theorem-like environments to use the same counter\njust specify the documentclass option \\verb|envcountsame|:\n\\begin{verbatim}\n\\documentclass[envcountsame]{llncs}\n\\end{verbatim}\nIf your first call for a theorem-like environment then is e.g.\n\\verb|\\begin{lemma}|, it will be numbered 1; if corollary follows,\nthis will be numbered 2; if you then call lemma again, this will be\nnumbered 3.\n\nBut in case you want to reset such counters to 1 in each section,\nplease specify the documentclass option \\verb|envcountreset|:\n\\begin{verbatim}\n\\documentclass[envcountreset]{llncs}\n\\end{verbatim}\n\nEven a numbering on section level (including the section counter) is\npossible with the documentclass option \\verb|envcountsect|.\n\n\\section{Predefined Theorem like Environments}\\label{builtintheo}\nThe following variety of run-in headings are at your disposal:\n\\begin{alpherate}\n\\item\n{\\bfseries Bold} run-in headings with italicized text\nas built-in environments:\n\\begin{verbatim}\n\\begin{corollary} <text> \\end{corollary}\n\\begin{lemma} <text> \\end{lemma}\n\\begin{proposition} <text> \\end{proposition}\n\\begin{theorem} <text> \\end{theorem}\n\\end{verbatim}\n\\item\nThe following generally appears as {\\itshape italic} run-in heading:\n\\begin{verbatim}\n\\begin{proof} <text>    \\qed    \\end{proof}\n\\end{verbatim}\nIt is unnumbered and may contain an eye catching square (call for that\nwith \\verb|\\qed|) before the environment ends.\n\\item\nFurther {\\itshape italic} or {\\bfseries bold} run-in headings with roman\nenvironment body may also occur:\n\\begin{verbatim}\n\\begin{definition} <text> \\end{definition}\n\\begin{example} <text> \\end{example}\n\\begin{exercise} <text> \\end{exercise}\n\\begin{note} <text> \\end{note}\n\\begin{problem} <text> \\end{problem}\n\\begin{question} <text> \\end{question}\n\\begin{remark} <text> \\end{remark}\n\\begin{solution} <text> \\end{solution}\n\\end{verbatim}\n\\end{alpherate}\n\n\\section{Defining your Own Theorem like Environments}\nWe have enhanced the standard \\verb|\\newtheorem| command and slightly\nchanged its syntax to get two new commands \\verb|\\spnewtheorem| and\n\\verb|\\spnewtheorem*| that now can be used to define additional\nenvironments. They require two additional arguments namely the type\nstyle in which the keyword of the environment appears and second the\nstyle for the text of your new environment.\n\n\\verb|\\spnewtheorem| can be used in two ways.\n\\subsection{Method 1 {\\itshape (preferred)}}\nYou may want to create an environment that shares its counter\nwith another environment, say {\\em main theorem\\\/} to be numbered like\nthe predefined {\\em theorem\\\/}. In this case, use the syntax\n\\begin{verbatim}\n\\spnewtheorem{<env_nam>}[<num_like>]{<caption>}\n{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\noindent\nHere the environment with which the new environment should share its\ncounter is specified with the optional argument \\verb|[<num_like>]|.\n\n\\paragraph{Sample Input}\n\\begin{verbatim}\n\\spnewtheorem{mainth}[theorem]{Main Theorem}{\\bfseries}{\\itshape}\n\\begin{theorem} The early bird gets the worm. \\end{theorem}\n\\begin{mainth} The early worm gets eaten. \\end{mainth}\n\\end{verbatim}\n\\medskip\\noindent\n{\\em Sample Output}\n\n\\medskip\\noindent\n{\\bfseries Theorem 3.}\\enspace {\\em The early bird gets the worm.}\n\n\\medskip\\noindent\n{\\bfseries Main Theorem 4.} The early worm gets eaten.\n\n\\bigskip\nThe sharing of the default counter (\\verb|[theorem]|) is desired. If you\nomit the optional second argument of \\verb|\\spnewtheorem| a separate\ncounter for your new environment is used throughout your document.\n\n\\subsection[Method 2]{Method 2 {\\itshape (assumes {\\tt[envcountsect]}\ndocumentstyle option)}}\n\\begin{verbatim}\n\\spnewtheorem{<env_nam>}{<caption>}[<within>]\n{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\noindent\nThis defines a new environment \\verb|<env_nam>| which prints the caption\n\\verb|<caption>| in the font \\verb|<cap_font>| and the text itself in\nthe font \\verb|<body_font>|. The environment is numbered beginning anew\nwith every new sectioning element you specify with the optional\nparameter \\verb|<within>|.\n\n\\medskip\\noindent\n\\paragraph{Example} \\leavevmode\n\n\\medskip\\noindent\n\\verb|\\spnewtheorem{joke}{Joke}[subsection]{\\bfseries}{\\rmfamily}|\n\n\\medskip\n\\noindent defines a new environment called \\verb|joke| which prints the\ncaption {\\bfseries Joke} in boldface and the text in roman. The jokes are\nnumbered starting from 1 at the beginning of every subsection with the\nnumber of the subsection preceding the number of the joke e.g. 7.2.1 for\nthe first joke in subsection 7.2.\n\n\\subsection{Unnumbered Environments}\nIf you wish to have an unnumbered environment, please\nuse the syntax\n\\begin{verbatim}\n\\spnewtheorem*{<env_nam>}{<caption>}{<cap_font>}{<body_font>}\n\\end{verbatim}\n\n\\section{Program Codes}\nIn case you want to show pieces of program code, just use the\n\\verb|verbatim| environment or the \\verb|verbatim| package of \\LaTeX.\n(There also exist various pretty printers for some programming\nlanguages.)\n\\noindent\n\\subsection*{Sample Input {\\rmfamily(of a simple\ncontribution)}}\\label{samppage}\n\\begin{verbatim}\n\\title{Hamiltonian Mechanics}\n\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\n\\institute{Princeton University, Princeton NJ 08544, USA\n\\and\nUniversit\\'{e} de Paris-Sud,\nLaboratoire d'Analyse Num\\'{e}rique, B\\^{a}timent 425,\\\\\nF-91405 Orsay Cedex, France}\n\n\\maketitle\n\\begin{abstract}\nThis paragraph shall summarize the contents of the paper\nin short terms.\n\\end{abstract}\n\\section{Fixed-Period Problems: The Sublinear Case}\nWith this chapter, the preliminaries are over, and we begin the\nsearch for periodic solutions \\dots\n\\subsection{Autonomous Systems}\nIn this section we will consider the case when the Hamiltonian\n$H(x)$ \\dots\n\\subsubsection*{The General Case: Nontriviality.}\nWe assume that $H$ is\n$\\left(A_{\\infty}, B_{\\infty}\\right)$-subqua\\-dra\\-tic\nat infinity, for some constant \\dots\n\\paragraph{Notes and Comments.}\nThe first results on subharmonics were \\dots\n\\begin{proposition}\nAssume $H'(0)=0$ and $ H(0)=0$. Set \\dots\n\\end{proposition}\n\\begin{proof}[of proposition]\nCondition (8) means that, for every $\\delta'>\\delta$, there is\nsome $\\varepsilon>0$ such that \\dots \\qed\n\\end{proof}\n\\begin{example}[\\rmfamily (External forcing)]\nConsider the system \\dots\n\\end{example}\n\\begin{corollary}\nAssume $H$ is $C^{2}$ and\n$\\left(a_{\\infty}, b_{\\infty}\\right)$-subquadratic\nat infinity. Let \\dots\n\\end{corollary}\n\\begin{lemma}\nAssume that $H$ is $C^{2}$ on $\\bbbr^{2n}\\backslash \\{0\\}$\nand that $H''(x)$ is \\dots\n\\end{lemma}\n\\begin{theorem}[(Ghoussoub-Preiss)]\nLet $X$ be a Banach Space and $\\Phi:X\\to\\bbbr$ \\dots\n\\end{theorem}\n\\begin{definition}\nWe shall say that a $C^{1}$ function $\\Phi:X\\to\\bbbr$\nsatisfies \\dots\n\\end{definition}\n\\end{verbatim}\n{\\itshape Sample Output\\\/} (follows on the next page together with\nexamples of the above run-in headings)\n\\newcounter{save}\\setcounter{save}{\\value{section}}\n{\\def\\addtocontents#1#2{}%\n\\def\\addcontentsline#1#2#3{}%\n\\def\\markboth#1#2{}%\n\\title{Hamiltonian Mechanics}\n\n\\author{Ivar Ekeland\\inst{1} \\and Roger Temam\\inst{2}}\n\n\\institute{Princeton University, Princeton NJ 08544, USA\n\\and\nUniversit\\'{e} de Paris-Sud,\nLaboratoire d'Analyse Num\\'{e}rique, B\\^{a}timent 425,\\\\\nF-91405 Orsay Cedex, France}\n\n\\maketitle\n\\begin{abstract}\nThis paragraph shall summarize the contents of the paper\nin short terms.\n\\end{abstract}\n\\section{Fixed-Period Problems: The Sublinear Case}\nWith this chapter, the preliminaries are over, and we begin the search\nfor periodic solutions \\dots\n\\subsection{Autonomous Systems}\nIn this section we will consider the case when the Hamiltonian\n$H(x)$ \\dots\n\\subsubsection{The General Case: Nontriviality.}\nWe assume that $H$ is\n$\\left(A_{\\infty}, B_{\\infty}\\right)$-subqua\\-dra\\-tic at\ninfinity, for some constant \\dots\n\\paragraph{Notes and Comments.}\nThe first results on subharmonics were \\dots\n\\begin{proposition}\nAssume $H'(0)=0$ and $ H(0)=0$. Set \\dots\n\\end{proposition}\n\\begin{proof}[of proposition]\nCondition (8) means that, for every $\\delta'>\\delta$, there is\nsome $\\varepsilon>0$ such that \\dots \\qed\n\\end{proof}\n\\begin{example}[{{\\rmfamily External forcing}}]\nConsider the system \\dots\n\\end{example}\n\\begin{corollary}\nAssume $H$ is $C^{2}$ and\n$\\left(a_{\\infty}, b_{\\infty}\\right)$-subquadratic\nat infinity. Let \\dots\n\\end{corollary}\n\\begin{lemma}\nAssume that $H$ is $C^{2}$ on $\\bbbr^{2n}\\backslash \\{0\\}$\nand that $H''(x)$ is \\dots\n\\end{lemma}\n\\begin{theorem}[Ghoussoub-Preiss]\nLet $X$ be a Banach Space and $\\Phi:X\\to\\bbbr$ \\dots\n\\end{theorem}\n\\begin{definition}\nWe shall say that a $C^{1}$ function $\\Phi:X\\to\\bbbr$ satisfies \\dots\n\\end{definition}\n}\\setcounter{section}{\\value{save}}\n\\section{Fine Tuning of the Text}\nThe following should be used to improve the readability of the text:\n\\begin{flushleft}\n\\begin{tabular}{@{}p{.19\\textwidth}p{.79\\textwidth}}\n\\verb|\\,|   & a thin space, e.g.\\ between numbers or between units\n              and num\\-bers; a line division will not be made\n              following this space\\\\\n\\verb|--|   & en dash; two strokes, without a space at either end\\\\\n\\verb*| -- |& en dash; two strokes, with  a space at either end\\\\\n\\verb|-|    & hyphen; one stroke, no space at either end\\\\\n\\verb|$-$|  & minus, in the text {\\em only} \\\\[8mm]\n{\\em Input} & \\verb|21\\,$^{\\circ}$C etc.,|\\\\\n            &  \\verb|Dr h.\\,c.\\,Rockefellar-Smith \\dots|\\\\\n            & \\verb|20,000\\,km and  Prof.\\,Dr Mallory \\dots|\\\\\n            & \\verb|1950--1985 \\dots|\\\\\n            & \\verb|this -- written on a computer -- is now printed|\\\\\n            & \\verb|$-30$\\,K \\dots|\\\\[3mm]\n{\\em Output}& 21\\,$^{\\circ}$C etc., Dr h.\\,c.\\,Rockefellar-Smith \\dots\\\\\n            & 20,000\\,km and  Prof.\\,Dr Mallory \\dots\\\\\n            & 1950--1985 \\dots\\\\\n            & this -- written on a computer -- is now printed\\\\\n            & $-30$\\,K \\dots\n\\end{tabular}\n\\end{flushleft}\n\\section {Special Typefaces}\nNormal type (roman text) need not be coded. {\\itshape Italic}\n(\\verb|{\\em <text>}| better still \\verb|\\emph{<text>}|) or, if\nnecessary, {\\bfseries boldface} should be used for emphasis.\\\\[6pt]\n\\begin{minipage}[t]{\\textwidth}\n\\begin{flushleft}\n\\begin{tabular}{@{}p{.25\\textwidth}@{\\hskip6pt}p{.73\\textwidth}@{}}\n\\verb|{\\itshape Text}|   & {\\itshape Italicized Text}\\\\[2pt]\n\\verb|{\\em Text}|   & {\\em Emphasized Text --\n   if you would like to emphasize a {\\em definition} within an\n   italicized text (e.g.\\ of a {\\em theorem)} you should code the\n   expression to be emphasized by} \\verb|\\em|.\\\\[2pt]\n\\verb|{\\bfseries Text}|& {\\bfseries Important Text}\\\\[2pt]\n\\verb|\\vec{Symbol}| & Vectors may only appear in math mode. The default\n   \\LaTeX{} vector symbol has been adapted\\footnotemark\\\n to LLNCS conventions.\\\\[2pt]\n & \\verb|$\\vec{A \\times B\\cdot C}| yields $\\vec{A\\times B\\cdot C}$\\\\\n & \\verb|$\\vec{A}^{T} \\otimes \\vec{B} \\otimes|\\\\\n & \\verb|\\vec{\\hat{D}}$|yields $\\vec{A}^{T} \\otimes \\vec{B} \\otimes\n\\vec{\\hat{D}}$\n\\end{tabular}\n\\end{flushleft}\n\\end{minipage}\n\n\\footnotetext{If you absolutely must revive the original \\LaTeX{}\ndesign of the vector symbol (as an arrow accent), please specify the\noption \\texttt{[orivec]} in the \\texttt{documentclass} line.}\n\\newpage\n\\section {Footnotes}\nFootnotes within the text should be coded:\n\\begin{verbatim}\n\\footnote{Text}\n\\end{verbatim}\n{\\itshape Sample Input}\n\\begin{flushleft}\nText with a footnote\\verb|\\footnote{The |{\\tt footnote is automatically\nnumbered.}\\verb|}| and text continues \\dots\n\\end{flushleft}\n{\\itshape Sample Output}\n\\begin{flushleft}\nText with a footnote\\footnote{The footnote is automatically numbered.}\nand text continues \\dots\n\\end{flushleft}\n\\section {Lists}\nPlease code lists as described below:\\\\[2mm]\n{\\itshape Sample  Input}\n\\begin{verbatim}\n\\begin{enumerate}\n  \\item First item\n  \\item Second item\n  \\begin{enumerate}\n    \\item First nested item\n    \\item Second nested item\n  \\end{enumerate}\n  \\item Third item\n\\end{enumerate}\n\\end{verbatim}\n{\\itshape Sample Output}\n \\begin{enumerate}\n\\item First item\n\\item Second item\n  \\begin{enumerate}\n    \\item First nested item\n    \\item Second nested item\n  \\end{enumerate}\n\\item Third item\n\\end{enumerate}\n\\section {Figures}\nFigure environments should be inserted after (not in)\nthe  paragraph in which the figure is first mentioned.\nThey will be numbered automatically.\n\nPreferably the images should be enclosed as PostScript files -- best as\nEPS data using the epsfig package.\n\nIf you cannot include them into your output this way and use other\ntechniques for a separate production,\nthe figures (line drawings and those containing halftone inserts\nas well as halftone figures) {\\em should not be pasted into your\nlaserprinter output}. They should be enclosed separately in camera-ready\nform (original artwork, glossy prints, photographs and\/or slides). The\nlettering should be suitable for reproduction, and after a\nprobably necessary reduction the height of capital letters should be at\nleast 1.8\\,mm and not more than 2.5\\,mm.\nCheck that lines and other details are uniformly black and\nthat the lettering on figures is clearly legible.\n\nTo leave the desired amount of space for the height of\nyour figures, please use the coding described below.\nAs can be seen in the output, we will automatically\nprovide 1\\,cm space above and below the figure,\nso that you should only leave the space equivalent to the size of the\nfigure itself. Please note that ``\\verb|x|'' in the following\ncoding stands for the actual height of the figure:\n\\begin{verbatim}\n\\begin{figure}\n\\vspace{x cm}\n\\caption[ ]{...text of caption...}          (Do type [ ])\n\\end{figure}\n\\end{verbatim}\n\\begin{flushleft}\n{\\itshape Sample Input}\n\\end{flushleft}\n\\begin{verbatim}\n\\begin{figure}\n\\vspace{2.5cm}\n\\caption{This is the caption of the figure displaying a white\neagle and a white horse on a snow field}\n\\end{figure}\n\\end{verbatim}\n\\begin{flushleft}\n{\\itshape Sample Output}\n\\end{flushleft}\n\\begin{figure}\n\\vspace{2.5cm}\n\\caption{This is the caption of the figure displaying a white eagle and\na white horse on a snow field}\n\\end{figure}\n\\section{Tables}\nTable captions should be treated\nin the same way as figure legends, except that\nthe table captions appear {\\itshape above} the tables. The tables\nwill be numbered automatically.\n\\subsection{Tables Coded with \\protect\\LaTeX{}}\nPlease use the following coding:\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\begin{table}\n\\caption{Critical $N$ values}\n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n${\\mathrm M}_\\odot$ & $\\beta_{0}$ & $T_{\\mathrm c6}$ & $\\gamma$\n  & $N_{\\mathrm{crit}}^{\\mathrm L}$\n  & $N_{\\mathrm{crit}}^{\\mathrm{Te}}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\\\\n 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\\\\n120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\medskip\\noindent{\\itshape Sample Output}\n\\begin{table}\n\\caption{Critical $N$ values}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.4}\n\\setlength\\tabcolsep{3pt}\n\\begin{tabular}{llllll}\n\\hline\\noalign{\\smallskip}\n${\\mathrm M}_\\odot$ & $\\beta_{0}$ & $T_{\\mathrm c6}$ & $\\gamma$\n  & $N_{\\mathrm{crit}}^{\\mathrm L}$\n  & $N_{\\mathrm{crit}}^{\\mathrm{Te}}$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n 30 & 0.82 & 38.4 & 35.7 & 154 & 320 \\\\\n 60 & 0.67 & 42.1 & 34.7 & 138 & 340 \\\\\n120 & 0.52 & 45.1 & 34.0 & 124 & 370 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nBefore continuing your text you need an empty line. \\dots\n\n\\vspace{3mm}\nFor further information you will find a complete description of\nthe tabular environment\non p.~62~ff. and p.~204 of the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport.\n\\subsection{Tables Not Coded with \\protect\\LaTeX{}}\nIf you do not wish to code your table using \\LaTeX{}\nbut prefer to have it reproduced separately,\nproceed as for figures and use the following coding:\\\\[2mm]\n{\\itshape Sample Input}\n\\begin{verbatim}\n\\begin{table}\n\\caption{text of your caption}\n\\vspace{x cm}    \n\\end{table}\n\\end{verbatim}\n\\subsection{Signs and Characters}\n\\subsubsection*{Special Signs.}\nYou may need to use special signs.  The available ones are listed in the\n{\\em \\LaTeX{} User's Guide \\& Reference Manual\\\/} by Leslie Lamport,\npp.~41\\,ff.\nWe have created further symbols for math mode (enclosed in \\$):\n\\begin{center}\n\\begin{tabular}{l@{\\hspace{1em}yields\\hspace{1em}}\nc@{\\hspace{3em}}l@{\\hspace{1em}yields\\hspace{1em}}c}\n\\verb|\\grole| & $\\grole$ & \\verb|\\getsto| & $\\getsto$\\\\\n\\verb|\\lid|   & $\\lid$   & \\verb|\\gid|    & $\\gid$\n\\end{tabular}\n\\end{center}\n\\subsubsection*{Gothic (Fraktur).}\nIf gothic letters are {\\itshape necessary}, please use those of the\nrelevant \\AmSTeX{} alphabet which are available using the amstex\npackage of the American Mathematical Society.\n\nIn \\LaTeX{} only the following gothic letters are available:\n\\verb|$\\Re$| yields $\\Re$ and \\verb|$\\Im$| yields $\\Im$. These should\n{\\itshape not\\\/} be used when you need gothic letters for your contribution.\nUse \\AmSTeX{} gothic as explained above. For the real and the imaginary\nparts of a complex number within math mode you should use instead:\n\\verb|$\\mathrm{Re}$| (which yields Re) or \\verb|$\\mathrm{Im}$| (which\nyields Im).\n\\subsubsection*{Script.}\nFor script capitals use the coding\n\\begin{center}\n\\begin{tabular}{l@{\\hspace{1em}which yields\\hspace{1em}}c}\n\\verb|$\\mathcal{AB}$| & $\\mathcal{AB}$\n\\end{tabular}\n\\end{center}\n(see p.~42 of  the \\LaTeX{} book).\n\\subsubsection*{Special Roman.}\nIf you need other symbols than those below, you could use\nthe blackboard bold characters of \\AmSTeX{},  but there might arise\ncapacity problems\nin loading additional \\AmSTeX{} fonts. Therefore  we created\nthe blackboard bold characters listed below.\nSome of them are not esthetically\nsatisfactory. This need not deter you from using them:\nin the final printed form they will be\nreplaced by the well-designed MT (monotype) characters of\nthe phototypesetting machine.\n\\begin{flushleft}\n\\begin{tabular}{@{}ll@{ yields }\nc@{\\hspace{1.em}}ll@{ yields }c}\n\\verb|\\bbbc| & (complex numbers)   & $\\bbbc$\n  & \\verb|\\bbbf| & (blackboard bold F) & $\\bbbf$\\\\\n\\verb|\\bbbh| & (blackboard bold H) & $\\bbbh$\n  & \\verb|\\bbbk| & (blackboard bold K) & $\\bbbk$\\\\\n\\verb|\\bbbm| & (blackboard bold M) & $\\bbbm$\n  & \\verb|\\bbbn| & (natural numbers N) & $\\bbbn$\\\\\n\\verb|\\bbbp| & (blackboard bold P) & $\\bbbp$\n  & \\verb|\\bbbq| & (rational numbers)  & $\\bbbq$\\\\\n\\verb|\\bbbr| & (real numbers)      & $\\bbbr$\n  & \\verb|\\bbbs| & (blackboard bold S) & $\\bbbs$\\\\\n\\verb|\\bbbt| & (blackboard bold T) & $\\bbbt$\n  & \\verb|\\bbbz| & (whole numbers)     & $\\bbbz$\\\\\n\\verb|\\bbbone| & (symbol one)      & $\\bbbone$\n\\end{tabular}\n\\end{flushleft}\n\\begin{displaymath}\n\\begin{array}{c}\n\\bbbc^{\\bbbc^{\\bbbc}} \\otimes\n\\bbbf_{\\bbbf_{\\bbbf}} \\otimes\n\\bbbh_{\\bbbh_{\\bbbh}} \\otimes\n\\bbbk_{\\bbbk_{\\bbbk}} \\otimes\n\\bbbm^{\\bbbm^{\\bbbm}} \\otimes\n\\bbbn_{\\bbbn_{\\bbbn}} \\otimes\n\\bbbp^{\\bbbp^{\\bbbp}}\\\\[2mm]\n\\otimes\n\\bbbq_{\\bbbq_{\\bbbq}} \\otimes\n\\bbbr^{\\bbbr^{\\bbbr}} \\otimes\n\\bbbs^{\\bbbs_{\\bbbs}} \\otimes\n\\bbbt^{\\bbbt^{\\bbbt}} \\otimes\n\\bbbz \\otimes\n\\bbbone^{\\bbbone_{\\bbbone}}\n\\end{array}\n\\end{displaymath}\n\\section{References}\n\\label{refer}\nThere are three reference systems available; only one, of course,\nshould be used for your contribution. With each system (by\nnumber only, by letter-number or by author-year) a reference list\ncontaining all citations in the\ntext, should be included at the end of your contribution placing the\n\\LaTeX{} environment \\verb|thebibliography| there.\nFor an overall information on that environment\nsee the {\\em \\LaTeX{} User's Guide \\& Reference\nManual\\\/} by Leslie Lamport, p.~71.\n\nThere is a special {\\sc Bib}\\TeX{} style for LLNCS that works along\nwith the class: \\verb|splncs.bst|\n-- call for it with a line \\verb|\\bibliographystyle{splncs}|.\nIf you plan to use another {\\sc Bib}\\TeX{} style you are customed to,\nplease specify the option \\verb|[oribibl]| in the\n\\verb|documentclass| line, like:\n\\begin{verbatim}\n\\documentclass[oribibl]{llncs}\n\\end{verbatim}\nThis will retain the original \\LaTeX{} code for the bibliographic\nenvironment and the \\verb|\\cite| mechanism that many {\\sc Bib}\\TeX{}\napplications rely on.\n\\subsection{References by Letter-Number or by Number Only}\nReferences are cited in the text -- using the \\verb|\\cite|\ncommand of \\LaTeX{} -- by number or by letter-number in square\nbrackets, e.g.\\ [1] or [E1, S2], [P1], according to your use of the\n\\verb|\\bibitem| command in the \\verb|thebibliography| environment. The\ncoding is as follows: if you choose your own label for the sources by\ngiving an optional argument to the \\verb|\\bibitem| command the citations\nin the text are marked with the label you supplied. Otherwise a simple\nnumbering is done, which is preferred.\n\\begin{verbatim}\nThe results in this section are a refined version\nof \\cite{clar:eke}; the minimality result of Proposition~14\nwas the first of its kind.\n\\end{verbatim}\nThe above input produces the citation: ``\\dots\\ refined version of\n[CE1]; the min\\-i\\-mality\\dots''. Then the \\verb|\\bibitem| entry of\nthe \\verb|thebibliography| environment should read:\n\\begin{verbatim}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}