diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzbzov" "b/data_all_eng_slimpj/shuffled/split2/finalzzbzov" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzbzov" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\\label{sect:intro}\n\nClassical Cepheid variables are of great importance in astrophysics. They obey the famous period-luminosity relation (Leavitt 1908),\nnow also named the \"Leavitt Law\", which has made them excellent standard candles to calibrate the first rungs of the extragalactic distance scale in the\nlocal Universe (e.g. Gieren et al. 2005a, Freedman \\& Madore 2010, Riess et al. 2011, Kodric et al. 2015). Our currently most accurate approach to determine\nthe Hubble constant uses a distance scale building on classical Cepheids in tandem with Ia-type Supernovae (e.g. Riess et al. 2011). \nCepheids are also excellent tools to check on the validity, and improve stellar pulsation and stellar evolution theories (Caputo et al. 2005).\nOne of the serious and long-standing problems of these theories was an inconsistency, at a level of $\\approx 20-30\\%$, between the masses\npredicted by the evolutionary and pulsational routes\n(Stobie 1969; Cox 1980; Keller 2008; Neilson et al. 2011 and references therein). The obvious way to solve this \"mass discrepancy problem\" was to find \nCepheid variables in double-lined eclipsing binary systems which would allow to accurately determine their dynamical masses. However, it took more than 40 years until\nsuch a system (OGLE-LMC-CEP-0227) was finally found by Soszy{\\'n}ski et al. (2008) and subsequently studied\nby our group, yielding the dynamical mass of the Cepheid \nwith an exquisite accuracy of 1\\% (Pietrzynski et al. 2010; Pilecki et al. 2013). This dynamical mass determination has already led to\nimprovements in stellar evolution (Cassisi and Salaris 2011, Neilson et al. 2011, Prada Moroni et al. 2012, Neilson and Langer 2012), and stellar \npulsation theories (Marconi et al. 2013).\n\nThe detection of a Cepheid in a double-lined eclipsing system not only allows to determine its mass with excellent accuracy, but also allows \nto measure highly accurate values of other physical parameters which are impossible to determine, with a similar accuracy, for a single Cepheid,\nor a Cepheid in a non-eclipsing binary system. Our previous work has shown that radii accurate to 1-3\\% can be obtained, depending on the\nconfiguration of the system components at the primary and secondary eclipses (Pilecki et al. 2013, 2015). Such an accurate \nradius determination poses a strong constraint on the pulsation mode of the Cepheid. The orbital and photometric solutions also allow to determine the p-factor\nof the Cepheid which is needed in Baade-Wesselink (BW)-type distance determinations of Cepheids to convert their measured radial velocities\nto the pulsational velocities of the Cepheid surfaces, and currently constitutes the largest source\nof systematic uncertainty in any type of BW analyses (Storm et al. 2004; Gieren et al. 2005b; Fouqu{\\'e} et al. 2007; Storm et al. 2011). A direct\nand accurate measurement of the p-factors for a number of Cepheids spanning a range of pulsation periods will be of enormous value in the effort\nto achieve distance determinations accurate to 1-3\\% for single Cepheids with the BW method. Apart from the p-factor, the full analysis\nof a Cepheid-containing eclipsing binary, including the analysis of high-quality NIR data, is also able to provide a precise estimation of the limb darkening of the Cepheid \n(see Pilecki et al. 2013;\nhereafter P13), which cannot be determined empirically in any other way. \n\nThe object of this study, OGLE LMC562.05.9009, was discovered as an eclipsing binary with a Cepheid component from OGLE IV data by Soszy{\\'n}ski et al. (2012) \nin the OGLE South Ecliptic Pole LMC fields prepared with the aim of providing tests for the Gaia satellite mission. No orbital period\nfor the system could be derived from these data however. It is not in the list of Cepheids discovered\nby the MACHO project (Alcock et al. 2002), but is contained in the list of Cepheids published by the EROS-2 group (Kim et al. 2014). Its name in\nthe EROS-2 database is lm0240n14595, but there was no information about eclipses. Given the discovery of Soszy{\\'n}ski et al. (2012) of eclipses,\nwe initiated extensive\nhigh-resolution spectroscopy and\nfollow-up photometry of the LMC562.05.9009 system (see section 2 of this paper), which led to the spectroscopic confirmation of its genuine binary\nnature, and eventually allowed an accurate determination of its orbital period. With the photometric data, and an extensive catalog of radial velocity\nobservations of the system, we were able to precisely disentangle the pulsational and orbital radial velocity variations, and provide full and accurate\norbital and photometric solutions of the system, following the methodology which was used by P13 in the analysis\nof the OGLE-LMC-CEP-0227 system. This results in the determination of accurate physical parameters for both the Cepheid and its non-pulsating binary\ncompanion. Future near-infrared photometric coverage of the next eclipses, which will occur in about four years from now, will improve on the characterization \nof the physical parameters of the two stars. \n\nOur paper is organized as follows. In section 2, we describe the data underlying this study. In secion 3, we will present the data analysis methods, and the results emerging\nfrom our analysis of the observational data. In section 4, conclusions and an outlook on future work will be presented.\n\n\n\\section{Data}\n\\label{sect:data}\n\nIn order to reliably and accurately separate the orbital motion from the radial velocity variations due to the pulsation of the Cepheid component\nin the system, a large number of precise radial velocity measurements, providing good phase coverage of both the pulsational, and orbital radial\nvelocity curves, was necessary. This task was not made easier by the close-to-integer value of 2.988 days of the pulsation period of the Cepheid. We obtained\nhigh-resolution echelle spectra using the UVES spectrograph at the ESO-VLT on Paranal (49 epochs), the MIKE spectrograph at the 6.5-m Magellan Clay\ntelescope at Las Campanas Observatory (22 epochs), and the HARPS spectrograph at the 3.6-m telescope at ESO-La Silla (10 epochs). The UVES data were reduced\nusing a standard ESO pipeline and software obtained from the ESO webpage (http:\/\/www.eso.org\/sci\/software.html) (Freudling et al. 2013). The MIKE data were reduced\nwith the pipeline software written by Dan Kelson, following the approach outlined in Kelson (2003). The HARPS data were reduced on-site by the Online Reduction System.\n\nRadial velocities were measured using the Broadening Function method (Rucinski 1992, 1999) implemented in the RaveSpan software (Pilecki et al. 2012). Measurements \nwere made in the wavelength interval 4125 to 6800 $\\mathring{A}$ which contains numerous metallic lines. Synthetic spectra taken from the\nlibrary of Coelho et al. (2005) were used as templates. The typical formal errors of the derived velocities are $\\sim 370$ m\/s. The individual radial velocity\nmeasurements for both components of the OGLE LMC562.05.9009 system are available online at: \n\n\\centerline{ http:\/\/araucaria.astrouw.edu.pl\/p\/cep9009 }\n \nIn some cases where line profiles of both companions were blended, only the velocity of the Cepheid (number 1 in the table) was measured. By fitting the systemic\nradial velocities with the datasets from the different instruments, we found offsets of +250 m\/s for MIKE,\nand -210 m\/s for HARPS, respectively with respect to the UVES radial velocity system. These small offsets have been taken into account. Even\nif we would not have corrected for these very small velocity shifts between the different instruments, the orbital solution and the physical parameters of the \ncomponent stars derived in the following sections would not have changed in any significant way.\n\nA total of 588 photometric measurements in the $I$-band and 143 in the $V$-band were collected with the Warsaw telescope by the OGLE project (Udalski et al. 2015), \nand during observing time granted to the Araucaria project by the Chilean National Time Allocation Committee (CNTAC). The images were reduced with\nthe OGLE standard photometric pipeline based on difference image analysis, DIA (Udalski et al. 2015), and instrumental magnitudes were calibrated onto\nthe standard system using Landolt standards. The typical accuracy of the measurements was at the\n5 mmag level. We have also used instrumental $V$-band and $R$-band data \nfrom the MACHO project (ID: 71.11933.15) downloaded from the webpage http:\/\/macho.anu.edu.au and converted to the Johnson-Cousins system using equations from \nFaccioli et al.(2007). We augmented our data with $R_{EROS}$ (equivalent to Johnson-Cousins I) data from the EROS project \n(Kim et al. 2014)\\footnote{http:\/\/stardb.yonsei.ac.kr}. While the very precise OGLE data are crucial for our analysis,\nthe inclusion of the MACHO and EROS data in our study was important for an accurate determination of the orbital period of the OGLE LMC562.05.9009 system,\nand to improve the coverage of the secondary eclipse in the light curve. They also helped to considerably improve the pulsational V-band\nlight curve of the Cepheid. The\nMACHO $V$-band and EROS data were shifted in flux and magnitude to fit the OGLE data by the minimization of the difference between the out-of-eclipse light curves. \nThis way the light curves were forced to have the same average magnitude. The MACHO $R$-band light curve was left unmodified. The flux shift \nwas later modeled by adding a third light in this band which simulates a flux shift, which may appear due to any calibration error.\n\nThe individual photometric data are given on the same webpage as the radial velocity data (see above), and\nare shown in Figure 1. In Figure 2, we show the pulsational light curves of the Cepheid in the I, R and V bands, as obtained from the out-of-eclipse photometric data\nfolded with the pulsation period of the star. These data resemble light curves of very low scatter with an asymmetrical shape typical for a Cepheid pulsating in the \nfundamental mode. The Fourier decomposition parameters of the I-band light curve of the Cepheid, shown in Figure 3, clearly confirm that the star is indeed \na fundamental mode pulsator.\n\nThe pulsation period of the Cepheid is very accurately determined from the current data. From the V band data, we obtain a period of 2.9878463 (09) days,\nwhile the I band data yield a period of 2.9878466 (16) days, leading to the uncertainty of the pulsation period quoted in Table 3 which is \nconsistent with the absolute value of rate of period change\nbeing $<$0.1 s\/yr. The O-C diagrams for both the I and V band data do not display any secular systematic change, confirming that the total uncertainty\non the period as given in Table 3 is correctly estimated.\n\nFigure 4 shows the orbital light curve of the system for the OGLE IV I-band data, folded on the orbital period of 1550.4 days, and with the \npulsational variations of the Cepheid removed. It is seen that the orbit of the LMC562.05.9009 system is highly eccentric, and that both the primary and secondary eclipses \nare covered by the data.\n\nIn order to determine the effective temperatures of the component stars, we augmented our dataset with 12 epochs of J and K photometry which were obtained outside\nthe eclipses. These data were taken with the SOFI near-infrared camera attached to the ESO NTT 3.5 m telescope on La Silla. The reduction and calibration of the data\nto the UKIRT system (Hawarden et al. 2001) was done following the procedure described in detail in Pietrzy{\\'n}ski et al. (2006).\nThe accuracy of the zero points in both bands is 0.015 mag, and instrumental errors are not larger than 0.01 mag. \n\n\n\\section{Analysis and Results}\n\\label{sect:results}\n\n>From the analysis of the radial velocity curve of a binary star one can obtain the orbital parameters of the system. In the case of the studied system the procedure \nis complicated by the pulsational variability of the Cepheid superimposed on the orbital motion. Using the RaveSpan software we have fitted a model of Keplerian orbit \n(i.e. proximity effects were ignored, which is justified by the large distance of the components even at closest approach) with an additional Fourier series \nrepresenting the pulsational radial velocity curve of the Cepheid.\n\nWe simultaneously fitted the reference time $T_0$, the eccentricity $e$, the argument of periastron $\\omega$, the velocity semi-amplitudes $K_1$ and $K_2$, \nthe systemic velocity $\\gamma$, and Nth-order Fourier series. In the beginning systemic velocities of both components were fitted, but without any improvement \nin the fit and with the values equal within the errors. Eventually only one velocity was kept.\n\nThe period $P$ was initially held fixed at the estimated value of 1550.4 days. The fitting was later repeated with a fixed value of $P=1550.354$ d and $T_0$ calculated \nfrom the photometric epoch of a primary minimum $T_{I}$ as a function of eccentricity and argument of periastron:\n$$T_0 = f(e, \\omega; P=1550.354~d, T_{I}=3959.23~d)$$\nto ensure the consistency of the model.\n\nThe error of the eccentricity turned out to be 10 times higher and the error for the argument of periastron 6 times lower (see solution 3 in Table~\\ref{tab:spec}) \nthan the ones obtained from the photometry. For this reason we have tried to solve the system with $e$ and $\\omega$ fixed (solution 1), or only $e$ fixed (solution 2). \nEventually we decided to adopt solution 2 as the final one because of the low error for $e$ from the photometry and the low error of $\\omega$ from the orbital solution. \n\nIn this way we have obtained the coefficients describing the pulsational radial velocity curve and the parameters describing the orbital motion separately. \nThe orbital radial velocity curve along with the best fitting model is shown in Fig.~\\ref{fig:rvorb}.\nTo obtain the pulsational radial velocity curve of the Cepheid we then subtracted the orbital motion from the measured velocities. The resulting radial velocity curve is shown \nin Fig.~\\ref{fig:rvpuls} together with the radius variation curve calculated with the $p$-factor 1.37 obtained from the fit. The orbital solutions are presented \nin Table~\\ref{tab:spec}.\n\nThe photometric data were analyzed using a version of the JKTEBOP code (Popper \\& Etzel 1981, Southworth et al. 2004, 2007) modified to allow the inclusion of \npulsation variability. We have previously used this package in the analysis of the OGLE-LMC-CEP-0227 system (P13), and we refer the reader \nto this work for more details.\n\nWe varied the following parameters in deriving the final model: the fractional radius of the pulsating component at phase 0.0 (pulsational), $r_1$;\nthe fractional radius of the second component, $r_2$; the orbital inclination $i$; the orbital period, $P_{orb}$; the epoch of the primary minimum, $T_{I}$; \nthe component surface brightness ratios in all three photometric bands at phase 0.0 (pulsational), $j_{21}$; and the third light in the $R_C$-band $l_{3} (R_C)$. \nThe radius change \nof the Cepheid was calculated from the pulsational radial velocity curve using the $p$-factor value of 1.37 and the change of the surface brightness ratios from \nthe instantaneous radii and out-of-eclipse pulsational light curves (for details, see P13). The third light in the $R_C$-band was introduced because we were unable \nto transform it directly to the OGLE photometric system.\n\nThe search for the best model (lowest $\\chi^2$ value) was made using the Markov chain Monte Carlo (MCMC) approach (Press et al. 2007) as described in P13. \nThe best fit photometric parameters are presented in Table~\\ref{tab:photpar}. We present two photometric solutions in this Table. In the first one, the argument\nof periastron $\\omega$ is taken from the orbital solution, in the second one it is fitted. We consider Solution 1 as the final one, being consistent with\nthe above discussion of the $\\omega$ errors. In this way we take the best advantage from the photometric and orbital radial velocity data.\nUsing these parameters we generated a model for each light curve. \nIn Fig.~\\ref{fig:ieclmodel} we show a close-up of selected eclipses for each passband. The magnitude range is the same for all plots to facilitate the comparison.\n\nMost of the parameters fitted in our approach are independent and do not exhibit any significant correlation. The only significant correlation is between \nthe orbital plane inclination $i$ and the sum of the radii $r_{1}+r_{2}$ as shown in Fig.~\\ref{fig:corr_i}.\n\n\n\\subsection{Eclipses}\nIn order to better understand the configuration of the system using the derived parameters we calculated the distances between the stars at important phases. \nAt the phase of the primary eclipse the distance between the components is about $650 R_{\\odot}$, while at the phase of the secondary eclipse it is \nabout $725 R_{\\odot}$. Both eclipses occur when the stars are relatively close to each other. The minimum and maximum separations during the orbital cycle \nare 425 and 1760 $R_{\\odot}$, respectively.\nAt the primary eclipse the projected distance between the centers of the stars is $22.9 R_{\\odot}$, and at the secondary eclipse the projected distance is $25.6 R_{\\odot}$, \nwhile the sum of the radii changes between 53.4 and 56.4 $R_{\\odot}$ depending on the instantaneous radius of the Cepheid. The configuration at both phases \nis illustrated in Fig.~\\ref{fig:config}.\n\n\\subsection{Radius and projection factor}\nTo test the results of our analysis, we have calculated the expected radius of the Cepheid from period-radius (PR) relations for classical Cepheids in the literature. \nThe relation \nof Gieren et al. (1998) for fundamental mode pulsators yields an expected mean radius value of $27.0 \\pm 1.2 R_{\\odot}$ for the pulsation period of the Cepheid in our system\nwhich agrees with our determination ($28.6 \\pm 0.2 R_{\\odot}$) within the combined 1 $\\sigma$ errors. The PR relations of Sachkov (2002) for fundamental mode\nand first overtone Cepheids predict radii of $27.4 \\pm 0.9 R_{\\odot}$ and $35.6 \\pm 5.4 R_{\\odot}$, respectively, for a pulsation period of 2.988 days. The first value \nmatches our derived radius value for the OGLE LMC562.05.9009 Cepheid much better, and is in agreement with the radius prediction from the Gieren et al. (1998) PR relation. \nWe conclude that the radius value of the Cepheid clearly supports fundamental mode pulsation, in agreement with the conclusion reached\nfrom the Fourier decomposition parameters of the I-band light curve.\nThe radius value together with the other physical parameters of the Cepheid given in Table 3, particularly its mass, \nalso leave no doubt that the pulsating star in the system is a classical (and not a Type-II) Cepheid.\n\nOur models constrain the projection factor of a Cepheid in an eclipsing binary system in the way which has been discussed in detail in P13. Briefly,\nthe shape of a Cepheid light curve in a given photometric band is determined by the change of its surface temperature and its radius. The radius change is\nparticularly important if the Cepheid resides in an eclipsing binary system. The beginning and end of an eclipse may be shifted in time according to the\ninstantaneous radius of the Cepheid, and the visible area of the eclipsed stellar disk depends on the phase of the pulsating component. In our approach\nthe Cepheid variability is a part of the model, so we can trace the influence of the related parameters on the light curve. As a base we use the raw (unscaled) \nabsolute radius change obtained from the pulsational radial velocity curve. Then we scale its amplitude with the projection factor (the $p$-factor scales\nlinearly with the amplitude of the radius variation curve). A conversion from the absolute radii to the relative radii (used in the light curve analysis)\nis done by using the orbital solution. A comparison of the resulting model light curves with the data then directly constrains the $p$-factor value. From\nour best model we obtain a radius variation amplitude of 3.04 R$_\\odot$ for the Cepheid, which corresponds to $p=1.37$ (see Figs. 6 and 9).\n \nOur current determination of the projection factor of the Cepheid in the OGLE LMC562.05.9009 system is the second reliable measurement of this important quantity\nfor a Cepheid in a binary, after the first determination made by P13 for OGLE-LMC-CEP-0227. The value of $p = 1.37 \\pm 0.07$ is smaller than the predicted p-factor value\nfrom the most recent calibration of the p-factor relation of Storm et al. (2011) which yields $p = 1.46 \\pm 0.04$ for the pulsation period of the Cepheid. However, there is possible \nagreement within the combined uncertainties of the two values. This is contrary to the finding for CEP-0227 which has a pulsation period of 3.80 days \nand $p = 1.21 \\pm 0.04$ from\nour analysis in P13, whereas its expected p-factor value from the Storm et al. calibration is $p = 1.44 \\pm 0.04$, with both values clearly discrepant within their respective\nuncertainties. The large difference of the p-factor values for the two binary Cepheids for which we could determine\nthis number so far with our method is also noteworthy (the difference is 0.16, whereas the p-factor relation of Storm et al. predicts a difference of only 0.02 for a change of the period \nfrom 2.988 to 3.80 days. Other p-factor relations, such as the theoretical relations of Neilson et al. (2012), predict an even smaller change of $p$ between the two\nperiod values). Our finding hints at the possibility that the p-factor - period relation may have an intrinsic dispersion, particularly in the short pulsation period range,\nwhere the discrepancy of the p-factor values predicted by different calibrations of the relation in the literature is largest (see discussion in Storm et al. 2011\nand Gieren et al. 2013). \n\n\\subsection{Extinction and temperature}\n\nThe extinction in the direction to the target was calculated in a similar way as described in Pilecki et al.(2015). We utilized the observed \n(not extinction-corrected) period-magnitude relations for fundamental mode Cepheids in the LMC (Soszy{\\'n}ski et al. 2008) in the optical V and I bands.\nBy comparing the observed mean magnitudes of the Cepheid with the expected magnitudes for its period,\n we determined the differential color excess (with respect to the LMC mean value) as $\\Delta E(B\\!-\\!V)=-0.016$ mag, and a total color excess \n of $E(B\\!-\\!V)=0.106$ mag using the mean extinction for the LMC given by Imara \\& Blitz (2007) - see Table~\\ref{tab:magnitud}. This color excess corresponds \n to a total extinction in the K-band of $A_K=0.036$ mag.\n\nThe mean (over the pulsation cycle of the Cepheid) observed IR magnitudes of the OGLE LMC562.05.9009 system are $J=14.232\\pm 0.018$ and $K=13.873\\pm 0.018$ mag. \nThey were transformed \nonto the 2MASS system using the equations of Carpenter (2001). We calculated an expected exctinction-free K-band magnitude of the Cepheid using relations 4 and 13 from \nRipepi et al.(2012). The observed and de-reddened magnitudes of both components in the $V$, $I_C$ and $K$ bands are given in Table~\\ref{tab:magnitud}. \nThe effective temperatures of the two stars were then calculated from their intrinsic colors, using the calibrations by Worthey \\& Lee (2011). \nThe extinction-corrected magnitudes and colors of the primary and secondary components and their temperatures are given in Table 3.\n\nIt is very interesting to note that within the uncertainties both components have the same effective temperatures, luminosities and surface gravities. \nHowever, according to the very precise OGLE-IV photometry the secondary does not show any pulsations with amplitude larger than 0.01 mag.\nThis is a striking result \nbecause, assuming the same chemical composition for both components in the system, we would expect both stars to be located \nwithin the instability strip (see discussion next section). The fact that the secondary is non-pulsating and thus outside the instability strip could imply\nthat the two components of OGLE LMC562.05.9009 have significantly different abundances, which would make this system \nunique among known binary stars. \n\nA different, and probably more likely explanation is that the secondary is just a little cooler that the Cepheid ($\\sim 70$ K), as suggested by \nour photometric Solution 2 in Table 2. In that case the Cepheid would reside almost exactly on the red boundary of the instability strip, with the secondary \nlocated just beyond the red edge. If this scenario is the correct one, the present work would provide the best known observational constraint on the exact position \nof the instability strip red edge. \n\n\\subsection{Evolutionary status and age of the Cepheid and its companion}\n\nWe computed the evolutionary tracks of the two component stars of the OGLE LMC562.05.9009 system by means of the Pisa release of the FRANEC code\n(Degl'Innocenti et al. 2008; Tognelli et al. 2011) adopting the same input physics and prescriptions described in detail in Dell'Omodarme et al. (2011).\nAn important exception is the neglecting of microscopic diffusion of helium and metals, because of their negligible impact on the evolution of\nintermediate-mass stars, as we did in our previous paper on OGLE-LMC-CEP-0227 (Prada Moroni et al. 2012). During the central hydrogen burning phase,\nwe took into account an overshooting of $l_{ov}$=$\\beta_{ov} H_p$ - where $H_p$ is the pressure height-scale and $ \\beta_{ov}=0.25 $ - beyond the Schwarzschild \nclassical border of the convective core. We computed the evolutionary tracks and isochrones adopting a value of the mixing-length parameter - which\nparametrizes the efficiency of the super-adiabatic convection - $\\alpha$ = 1.74. This value results from a solar calibration with our own\nStandard Solar Model computed with the same version of the FRANEC code used to compute the evolutionary tracks in this work. For a quantitative evaluation\nof some of the main sources of uncertainty affecting the theoretical evolutionary models of He-burning stars of intermediate mass we refer to\nValle et al. (2009).\n\nThe initial metal and helium abundances adopted for the calculations are Z=0.005 and Y=0.258, respectively.\n\nIn Figure 10, the locations of the two stars on the luminosity-effective temperature diagram from the parameters derived in this study\n(see Table 3) are shown. Also plotted are the boundaries of the classical fundamental mode Cepheid instability strip, for metallicities of Z=0.004\nand Z=0.008, taken from Bono et al. (2005). It is seen that for both metallicities, not only the Cepheid, but also the stable companion star\nare located inside the instability strip. A likely explanation is that the current uncertainty on the effective temperature of the companion star \nis somewhat underestimated and that a future, more accurate determination of the temperature will move the non-pulsating star in OGLE LMC562.05.9009\nslightly beyond the Cepheid instability strip; but there is also the possibility of significant different metallicities of the two stars.\n\nAlso shown in Fig. 10 are the evolutionary tracks computed for the masses of the two stars, using the prescripts detailed above. A isochrone for an age of\n205 Myr fits the position of both stars on the diagram reasonably well within the observational uncertainties on their luminosities and temperatures.\nThe age of the Cepheid expected from the theoretical period-age relation for fundamental mode classical Cepheids of Bono et al. (2005, their Table 4) \nfor a metallicity of Z=0.004 (very slightly smaller than our assumed metallicity of Z=0.005 for the calculation of the isochrone) is $130 \\pm 35 Myr$. Our\ncurrent age determination for the classical Cepheid in the binary system is about $2\\sigma$ larger than its age as predicted from the Bono et al.\nperiod-age relation, but given the uncertainties involved the two values are marginally consistent. We will check on this more deeply once we have\nnew data which will allow us a more accurate determination of the temperatures and luminosities of the two stars, and of their metallicities, leading to a more\naccurate age determination from the isochrone method. The current\nresults do however support the conclusion that both stars in the OGLE LMC562.05.9009 system are coeval, with an age larger than, but within the errors consistent with\nthe value predicted for the Cepheid from a theoretical period-age relation.\n\n\n\\section{Conclusions}\n\\label{sect:concl}\n\nWe have confirmed from high-resolution spectra that the eclipsing binary system OGLE LMC 562.05.9009 contains a classical Cepheid pulsating\nwith a period of 2.988 days in orbit with a stable secondary component. We performed the analysis\nof our extensive spectroscopic and photometric datasets in the same way as described in our previous analysis of the OGLE-LMC-CEP-0227\nsystem by P13, and have derived very accurate masses (to 0.8\\%) and radii (0.7\\%) for both the Cepheid and its non-pulsating companion star, which has a nearly identical\nmass and radius as the Cepheid. The orbit is highly eccentric with $e=0.61$ and a very long period of 1550 days, or 4.2 years. Our solution defines the orbital\nradial velocity curves of both components, disentangled from the pulsational velocity variations of the Cepheid, extremely well, as well as the pulsational radial\nvelocity curve of the Cepheid. Our analysis yields the second precise determination of the p-factor of a Cepheid in a binary so far in the literature,\nand was used to determine the radius variation of the Cepheid over its pulsation cycle. Our model reproduces the observed light curves extremely well, particularly the\nprimary eclipse when the companion star transits in front of the Cepheid. We calculated evolutionary tracks for the two component stars in the system\nand find that a isochrone for an age of 205 Myr fits the observed positions of both stars in the luminosity-effective temperature plane, arguing for the same age\nof the Cepheid and its red giant companion.\n\nThe p-factor value for the Cepheid is marginally consistent with the prediction of the p-factor relation of Storm et al. (2011), as opposed to the p-factor we derived \nfor OGLE-LMC-CEP-0227 in P13, which is in significant disagreement with the prediction of the Storm et al. relation. Currently the situation regarding the correct p-factor\nvalues to use in Baade-Wesselink-type Cepheid distance determinations is still very confusing. The measurements from the two binary Cepheids in this paper\nand in P13 seem to support the idea that the p-factor for classical Cepheids is not only period-dependent, but might also possess an intrinsic dispersion,\nat least for short pulsation periods in the range of a few days. Clearly more work is needed to clarify this question, and one of the very few observational approaches\nwhich promise to solve the issue is the analysis of more Cepheids in eclipsing binaries whose characteristics allow the determination of their p-factors.\nThe most\nimportant parameter in this context is the radius variation amplitude of the Cepheid; the larger the amplitude, the stronger the effect of the radius variation on \nthe binary light curve,\nand the smaller the uncertainty on the p-factor derived from our model. This was the reason why we could not measure the p-factor for the first overtone Cepheid\nin the eclipsing system OGLE-LMC-CEP-2532 whose radius variation is too small to cause a significant effect on the binary light curve, given\nthe quality of the photometric data (Pilecki et al. 2015).\nSince fundamental mode Cepheids tend to have larger radius variations, precise measurements of Cepheid projection factors with our binary method will mostly be restricted\nto eclipsing systems containing fundamental mode Cepheids.\n\nIn order to analyze the OGLE LMC562.05.9009 system, and in particular its Cepheid more fully, we plan to observe more eclipses (both primary and secondary) in the\nfuture, including coverage in near-infrared bands. A high quality out-of-eclipse the pulsational K-band\nlight curve of the Cepheid in tandem with the V-band light and pulsational radial velocity curves as determined in this paper will allow us to calculate\nthe distance to the Cepheid with the BW-type Infrared Surface Brightness Technique (Fouqu{\\'e} \\& Gieren 1997; Storm et al. 2011) and compare it to the distance\nof its companion star determined from the binary analysis and a surface brightness-color relation, as described in Pietrzynski et al. (2009, 2013). Such a comparison\nwill put further constraints on the p-factor relation valid for classical Cepheid variables.\n\nOur work has now revealed and analyzed the fifth eclipsing binary system containing a classical Cepheid in orbit with a stable giant star. Previous binary Cepheids\nanalyzed by our group are OGLE-LMC-CEP-0227 (Pietrzynski et al. 2010, P13), OGLE-LMC-CEP-1812 (Pietrzynski et al. 2011), OGLE-LMC-CEP-1718 (Gieren et al. 2014), and\nOGLE-LMC-CEP-2532 (Pilecki et al. 2015). The most exotic system is OGLE-LMC-CEP-1718 which contains two classical Cepheids in a 413-day orbit. Its analysis in\nGieren et al. (2014) has been very challenging due to the multiple superimposed variations in the light- and radial velocity curves. We hope to improve\non the analysis of that exciting system in the near future with additional data and possible improvements in our analysis code. For all systems but one,\nOGLE-LMC-CEP-1812, the mass ratio is very close to, or consistent with unity. The exception in the case of OGLE-LMC-CEP-1812 is probably explained by the result \nreported by Neilson et al. (2015a) that the Cepheid in that system is actually the product of a stellar merger of two main sequence stars. From an observational \npoint of view, there is a bias which favors the finding of systems composed of a Cepheid in orbit with a giant star of similar mass and radius which leads not only to\na higher probability to observe both eclipses, but also to observe the lines of both components in the composite spectra. For this reason, we cannot argue\nthat our results to-date on Cepheids in double-lined eclipsing binary systems in the LMC contradict results regarding the binary distribution of Cepheids\nas obtained by Evans et al. (2015), or Neilson et al. (2015b).\n\nThe binary Cepheids in the LMC, with their dynamical masses determined to better than 2\\%, will be a cornerstone for\nimproving our detailed understanding of Cepheid pulsation and post-main sequence stellar evolution, and in general of our understanding of Cepheid physics. With future\nprecise distance determinations to these systems we hope to determine from the stable binary companions, these binary Cepheids will also become excellent\nabsolute calibrators of the extragalactic distance scale.\n\n\n\n\\acknowledgments\nWe gratefully acknowledge financial support for this work from the BASAL Centro de Astrof{\\'i}sica y Tecnolog{\\'i}as Afines (CATA) PFB-06\/2007, from\nthe Polish National Science Center grant MAESTRO DEC-2012\/06\/A\/ST9\/00269, and from the Polish NCN grant DEC-2011\/03\/B\/ST9\/02573.\nWG, MG, DG, DM and MC also gratefully acknowledge support for this work from the Chilean Ministry of Economy, Development and Tourism's Millennium Science Initiative through grant IC120009 \nawarded to the Millennium Institute of Astrophysics (MAS). AG acknowledges support from FONDECYT grant 3130361, and MC from FONDECYT grant 1141141.\nThe OGLE Project has received funding \nfrom the National Science Center, Poland, grant MAESTRO 2014\/14\/A\/ST9\/00121 to AU.\n\n\nThis paper utilizes public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory\n under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, \n and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.\n\nWe would like to thank the support staffs at the ESO Paranal and La Silla and Las Campanas Observatories for their help in obtaining the\nobservations.\nWe thank the ESO OPC and the CNTAC for generous allocation of observing time for this project.\n\nThis research has made use of NASA's Astrophysics Data System Service.\n\n{\\it Facilities:} \\facility{ESO:3.6m (HARPS)}, \\facility{ESO:NTT (SOFI)}, \\facility{VLT:Kueyen (UVES)}, \\facility{Magellan:Clay (MIKE)}, \\facility{Warsaw telescope}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:Intro}\n \\emph{Swift} J164449.3+573451 (hereafter, Swift J1644+57) was first discovered by the \\emph{Swift} Burst Alert Telescope (BAT) at 12:57:45 UT on 28 March 2011 \\citep{Burrows2011,Levan2011}. Some evidences suggest that Swift J1644+57 is a tidal disruption of a star by a supermassive black hole (SMBH). This phenomenon triggered the BAT three times after the initial trigger during the first few days \\citep{Burrows2011, Levan2011}. The late-time X-ray light curve was extended to a longer period by following the expected power-law decay for the tidal disruption of a star, i.e., $t^{-5\/3}$ \\citep[e.g.,][]{Rees1988}. Finally, the source of X-ray, IR, and radio emissions were well matched up with the center of the host galaxy where a SMBH resides \\citep{Levan2011,Zauderer2011}. \n\n There have been many studies performed to understand the nature of this event, such as the characteristics of the star that was disrupted. Such a question is closely connected to the SMBH mass ($M_{\\rm BH}$). The disruption of a solar-type star is possible for all $M_{\\rm BH} < 10^8\\, M_{\\odot}$ \\citep{Rees1988,Cannizzo1990,Bloom2011}, but compact stars like a white dwarf can be disrupted only if $M_{\\rm BH} < 10^{5}\\, M_{\\odot}$ \\citep{kp2011}. If so, then this kind of event provides the interesting possibility of discovering intermediate-mass black holes.\n\n\n Unfortunately, there has been controversy concerning the mass of the SMBH. \\citet{Burrows2011} provided a rough estimate of the SMBH mass of $\\sim2\\times 10^{7}\\,M_{\\odot}$ using a black hole mass -- luminosity relation and the lower limit of $\\sim10^6\\,M_{\\odot}$ based on the X-ray variability. Similarly, \\citet{Levan2011} estimated $M_{\\rm BH}$ to be $2\\times 10^{6}$ -- $10^{7}\\,M_{\\odot}$, derived from $K$-band luminosity, but at that time, $K$-band luminosity contained a significant amount of the transient light. \\citet{mg2011} utilized a relation between the black hole mass, the radio luminosity, and the X-ray luminosity, and found $M_{\\rm BH}$ $\\sim10^{5.5}\\,M_{\\odot}$. Using a quasi-periodic oscillation resonance hypothesis, \\citet{al2012} provided an $M_{\\rm BH}$ estimate of $\\sim10^{5}\\,M_{\\odot}$. \\citet{kp2011} concluded that a white dwarf was tidally disrupted and the mass of SMBH is less than $10^{5}\\,M_{\\odot}$ in light of the short timescales of the X-ray light curve. In summary, the $M_{\\rm BH}$ estimates have centered around the two discrepant values of $10^{7}\\,M_{\\odot}$ and $10^{5}\\,M_{\\odot}$ or less. A better understanding of the host galaxy properties is needed to clear up the situation. \n\n\\begin{deluxetable*}{cccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{\\emph{HST} WFC3 Data Log} \n\\tablehead{\n\\colhead{Observation date (UT)} & \\colhead{MJD\\tablenotemark{a}} & \\colhead{Days since trigger} & \\colhead{band} & \\colhead{Exptime (s)} & \\colhead{Magnitude (AB)}\n}\n\\startdata \\\\\n2011-04-04 & 55655.147614 & 6.6 & F606W & 1260 & 22.69$\\pm$0.01\\\\\n2011-08-04 & 55777.276876 & 129 & F606W & 4160 & 22.76$\\pm$0.01\\\\\n2011-12-02 & 55897.684390 & 249 & F606W & 1113 & 22.77$\\pm$0.01\\\\\n2013-04-12 & 56394.429204 & 746 & F606W & 2600 & 22.74$\\pm$0.01\\\\\n\\\\\n\\tableline \\\\\n2011-04-04 & 55655.132654 & 6.6 & F160W & 997 & 20.68$\\pm$0.01\\\\\n2011-08-04 & 55777.257148 & 129 & F160W & 1412 & 21.09$\\pm$0.01\\\\\n2011-12-02 & 55897.702220 & 249 & F160W & 1209 & 21.22$\\pm$0.02\\\\\n2013-04-12 & 56394.295795 & 746 & F160W & 2812 & 21.55$\\pm$0.02\\\\\n\\enddata\n\\tablenotetext{a}{Exposure start time in Modified Julian Date (MJD)}\n\\label{HSTtab}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{cccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{\\emph{Spitzer} IRAC Data Log} \n\\tablehead{\n\\colhead{Observation date (UT)} & \\colhead{MJD\\tablenotemark{a}} & \\colhead{Days since trigger} & \\colhead{band} & \\colhead{Exptime (s)} & \\colhead{Magnitude (AB)}\n}\n\\startdata \\\\\n2011-04-28 & 55679.975316 & 31.4 & 3.6$\\mu$m & 1250 & 19.50$\\pm$0.02 \\\\ \n2011-10-31 & 55865.023052 & 216 & 3.6$\\mu$m & 1253 & 21.49$\\pm$0.06 \\\\ \n2012-02-24 & 55981.541644 & 333 & 3.6$\\mu$m & 1252 & 21.70$\\pm$0.07 \\\\ \n\\\\\n\\tableline \\\\\n2011-04-28 & 55679.975316 & 31.4 & 4.5$\\mu$m & 1250 & 19.30$\\pm$0.01 \\\\ \n2011-10-31 & 55865.023052 & 216 & 4.5$\\mu$m & 1253 & 21.25$\\pm$0.05 \\\\ \n2012-02-24 & 55981.541644 & 333 & 4.5$\\mu$m & 1252 & 21.57$\\pm$0.08 \\\\ \n\\enddata\n\\tablenotetext{a}{MJD in UTC at data collection event (DCE) start}\n\\label{spitzertab}\n\\end{deluxetable*}\n\n In order to more accurately estimate the SMBH mass and better constrain the properties of the host galaxy, we analyze the morphology and the surface brightness profile of the host galxaxy based on high-resolution \\emph{Hubble Space Telescope} (\\emph{HST}) images and estimate the multi-band fluxes of the host galaxy using our long-term monitoring data lasting more than 2.4 years. We fit the multi-band spectral energy distribution (SED) of the host galaxy luminosity with stellar population synthesis models, and then obtain the properties of the galaxy. Finally, we provide our best estimate of $M_{\\rm BH}$ based on the host galaxy properties.\n\n This is the second of a series of two papers. In the first paper (M. Im et al. 2015 in preparation, hereafter, Im15), we present the dataset of the long-term monitoring campaign and an analysis of the late-time light curve. \n\nThroughout the paper, we selected \\emph{H$_0=70$}km s$^{-1}$Mpc$^{-1}$, $\\Omega_{\\Lambda}=0.7$, and $\\Omega_{m}=0.3$ as cosmological parameters and adopt the AB magnitude system.\n\\\\\n\n\n\n\\section{Observations and Data}\nWe observed \\emph{Swift} J1644+57 using Wide Field Camera (WFCAM) on United Kingdom Infrared Telescope (UKIRT) for nearly 2.4 years following the burst as a part of our gamma-ray burst (GRB) and transient observation program \\citep{Lee2010}. We observed intensively in the $K$ band among $Z, Y, J ,H $, and $K$ bands of WFCAM. The number of epochs of $K$ band data used for thie analysis is $101$ and the last data were observed at $884.7$ days after the initial BAT trigger. The numbers of epochs for the $Y,J,H$-band data are $3,15,28$ and the last data were observed at $\\Delta t = 712.1$, $710.1$, and $884.7$ days, respectively, where $\\Delta t$ is the number of days since the initial BAT trigger. We have only one epoch of data for the UKIRT $Z$ band which was observed at $\\Delta t = 723.0$ days. \n\n We also observed \\emph{Swift} J1644+57 using Camera for QUasars in EArly uNiverse \\citep[CQUEAN;][]{KimE2011,Park2012,Lim2013} on the 2.1m Otto-Struve telescope of the McDonald Observatory in $g, r, i, z$, and $Y$ bands. The numbers of epochs for the $g, r, i, z,$ and $Y$ band data are $2,2,14,14$, and $2$ and the last data were observed at $\\Delta t = 25.7, 217.5, 526.6, 526.6$, and $25.8$ days respectively. The UKIRT and CQUEAN observation logs and photometry results are described in Im15. \n\n In addition, we also used data from \\citet{Burrows2011} and \\citet{Levan2011} for the earlier optical and near-infrared (NIR) data. \n\n Morphology analysis requires high-resolution images because this object is so compact that it is virtually a point source in the UKIRT and CQUEAN images. For the high-resolution images, we obtained \\emph{HST} WFC3 multi-drizzled, stacked images\\footnote{Based on observations made with the NASA\/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute. STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555.} available in the MAST database. We used F606W-, F160W-band data and the number of epochs in each two band is four. These data were observed at $\\Delta t = 6.6, 129, 249,$ and $746$ days. The \\emph{HST} WFC3 data are summarized in Table~\\ref{HSTtab}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.225,angle=00]{Swift_Mor.eps}\n\t\\caption{Images of the host galaxy in F606W, the best-fit two-dimensional models from GALFIT, and the residuals for the single S\\'{e}rsic component model and the S\\'{e}rsic bulge $+$ exponential disk model. Right panels show one-dimensional profiles of the host galaxy and each model component.\n\t\t\\label{sbfig}}\n\\end{figure*}\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.225,angle=00]{Swift_Mor2.eps}\n\t\\caption{Same as Figure~\\ref{sbfig}, but for the single exponential disk model and the double exponential profile model. \n\t\t\\label{sb2fig}}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.22,angle=00]{Swift_HST.eps}\n\t\\caption{Flux fractions of the GALFIT models as a function of time. We used the model which consists of a single S\\'{e}rsic bulge with $n=3.43$ and a point source component. The upper panel shows the results for the F606W-band images, while the lower panel shows the results for the F160W-band images. Magnitudes of the model components are also shown.\n\t\t\\label{hstfig}}\n\\end{figure}\n\n \n To supplement the NIR observation data, we used the \\emph{Spitzer} IRAC 3.6, 4.5$\\mu$m post basic calibrated data (PBCD) from the NASA\/IPAC Infrared Science Archive. These were observed at $\\Delta t = 31.4, 216.5$, and $333.0$ days. A log of the \\emph{Spitzer} IRAC 3.6, 4.5$\\mu$m data are shown in Table~\\ref{spitzertab}.\n\nThe flux measurements were performed by SExtractor software\\footnote{We used aperture magnitudes with aperture correction.} \\citep{Bertin1996} except for the \\emph{HST} images for which GALFIT \\citep{Peng2010} models were used for the flux measurements. \n\n For the X-ray data, we used \\emph{Swift}\/XRT data taken from the \\emph{Swift} archive and \\emph{XMM-Newton} data\\footnote{Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA} from the \\emph{XMM-Newton} Science Archive.\\\\\n\n\n\n\\section{Morphology of the host galaxy} \\label{sec:Mor}\n We analyzed the surface brightness profile of the host galaxy, in order to determine the bulge fraction and its nature. We used the \\emph{HST} images and GALFIT software to fit two-dimensional models to the light distribution of the host galaxy. To construct the point spread function (PSF), we selected $\\sim 5$ isolated stars with signal-to-noise ratios $\\gtrsim 300$ in the vicinity of \\emph{Swift} J1644+57, and co-added them.\n\n We used error images that are created by GALFIT for the fitting. For GALFIT to create the error image properly, we modified the unit of ADU and the image header values such that $\\mathrm{GAIN} \\times \\mathrm{ADU} \\times \\mathrm{NCOMBINE} = \\mathrm{[electrons]}$ as recommended in GALFIT website\\footnote{http:\/\/users.obs.carnegiescience.edu\/peng\/work\/galfit\/TOP10.html}.\n\n A crucial factor affecting the fitting results is background subtraction. For the background determination, we set 6 annuli with the radii logarithmically increasing between 2.5 and 9 times the radius of an ellipse for which pixel values are $1.5\\sigma$ of the background noise. We centered the annuli on the center of the host galaxy, set the minimum width of the annuli to be $\\sim1.3$ arcsec ($\\sim33$ pixel) for F606W images and $\\sim2$ arcsec ($\\sim16$ pixel) for F160W images, and augmented the widths in step with the logarithmically growing radii. We then derived the mean pixel values of each annulus. Finally, we adopted their mean value as the background value.\n\n\n\n Our surface brightness fit was carried out using a deep, stacked image of the data taken with F606W at $\\Delta t = $ 129, 249, and 749 days. It has been known that the transient component is negligible in the optical bands bluer than $i$ even at the early time \\citep{Burrows2011, Levan2011}. The use of the stacked, late-time image in the F606W band makes the transient component more negligible. On the other hand, NIR-bands, including F160W (similar to $H$ band of WFCAM), are known to contain a significant transient component which may affect the host galaxy analysis. Furthermore, the spatial resolution of the F606W images is better by a factor of three than that of F160W, which greatly helps the surface brightness fitting. The other \\emph{HST} data were also analyzed to understand the importance of the transient component, and the results for the transient component are presented later in this section.\n\n For the galaxy models, we used the S\\'{e}rsic \\citep{Sersic1968}, de Vaucouleurs \\citep{deVa1948}, and exponential disk profiles or a combination of thereof. The S\\'{e}rsic profile is described as\n\\begin{displaymath}\n \\Sigma(r)=\\Sigma_{e}\\mathrm{exp}\\Bigg[-\\kappa\\Bigg(\\bigg(\\frac{r}{r_e}\\bigg)^{1\/n}-1\\Bigg)\\Bigg],\n\\end{displaymath}\nwhere $\\Sigma_{e}$ is the surface brightness at the effective radius $r_e$, and $n$ is the S\\'{e}rsic index. $\\kappa$ is a variable parameter denpendent on $n$, where $n=4$ and $1$ correspond to the de Vaucouleurs and exponential profiles, respectively. Although $n=4$ is commonly quoted for the ellipticals and classical bulges, the S\\'{e}rsic index of ellipticals and classical bulges can assume a value in the range $2\\lesssim n \\lesssim6$, whereas pseudobulges have $n\\lesssim 2$ \\citep{Fisher2008, Fisher2010}.\n\n All of the model parameters such as ellipticity and center positions of the different components, were set free in the fitting procedure. \n\n Figures~\\ref{sbfig} and \\ref{sb2fig} show images of the host galaxy, the two-dimensional models, and the residuals (i.e., the model subtracted images), for four different models: (1) a single S\\'{e}rsic; (2) a S\\'{e}rsic bulge + exponential disk; (3) an exponential disk; and (4) a double exponential profile models. The figures also show one-dimensional profiles (along the major axis) of the host galaxy and those of each model component, which are converted through the IRAF\\footnote{IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation.} ELLIPSE task. In addition to the profiles, the differences between the data and the model profiles are shown. The results of each fit are summarized in Table~\\ref{galfittab}. Both the single S\\'{e}rsic model with $n=3.43\\pm0.05$ and the S\\'{e}rsic bulge with $n=3.39\\pm0.11$ $+$ exponential disk model fit the data well ($\\chi_{\\nu}$ $\\sim 1.2$ -- $1.3$). When the disk component is added, the bulge to total host galaxy flux ratio (B\/T) is $0.83\\pm0.03$. On the other hand, the single exponential disk model provides a poor fit to the data as shown in Figure 2 and with $\\chi_{\\nu}=6.54$. The double exponential profile model fits the data nearly as well as the single S\\'{e}rsic model and the S\\'{e}rsic bulge+disk model in terms of $\\chi_{\\nu}^{2}$. However, the analysis of the one-dimensional surface brightness profile shows that the model does not follow the outer part of the profile well, demonstrating a relatively steeper decline than that of the single S\\'{e}rsic model and the S\\'{e}rsic bulge $+$ exponential disk model. This model gives B\/T $= 0.36$, suggesting a significant bulge component. Therefore, we conclude that the host galaxy of \\emph{Swift} J1644+57 is bulge-dominant. We also conclude that the bulge is likely to have a S\\'{e}rsic index higher than $3$ regardless of the existence of the disk. This value corresponds to the range of the classical bulges \\citep{Fisher2008, Fisher2010}. We cannot completely exclude the case where the bulge is pseudobulge with $n \\sim 1$, but even in this case, the object has a significant bulge.\n\n\\begin{deluxetable*}{ccccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Surface Brightness Fitting Result} \n\\tablehead{\n\\colhead{} & \\multicolumn{3}{c}{Bulge} & \\colhead{} & \\multicolumn{2}{c}{Disk}& \\colhead{} & \\colhead{} & \\colhead{} & \\colhead{} \\\\ \n\\cline{2-4}\\cline{6-7}\\\\\n\\colhead{Galaxy model} & \\colhead{$m_{\\mathrm{b}}$ [AB]} & \\colhead{$n$} & \\colhead{$r_{\\mathrm{eff}}[\\mathrm{kpc}]$} & & \\colhead{$m_{\\mathrm{d}}$ [AB]} & \\colhead{$r_{\\mathrm{s}}[\\mathrm{kpc}]$} & \\colhead{} & \\colhead {B\/T} & \\colhead {$m_{\\mathrm{t}}$ [AB]} & \\colhead{$\\chi_{\\nu}^{2}$}\\\\ \n\\colhead{(1)} & \\colhead{(2)} & \\colhead{(3)} & \\colhead{(4)} & \\colhead{} & \\colhead{(5)} & \\colhead{(6)} & \\colhead{} & \\colhead{(7)} & \\colhead{(8)} & \\colhead{(9)}\n}\n\\startdata\nB & $22.77\\pm0.01$ & $3.43\\pm0.05$ & $1.01\\pm0.01$ & & \\nodata & \\nodata & & \\nodata & $22.77\\pm0.01$ & 1.322\\\\\nB$+$D & $23.01\\pm0.03$ & $3.39\\pm0.11$ & $0.79\\pm0.03$ & & $24.75\\pm0.14$ & $1.21\\pm0.07$ & & $0.83\\pm0.03$ & $22.81\\pm0.03$ & 1.223\\\\\n\\tableline \\\\\nB$(n=4)$ & $22.72\\pm0.00$ &4 (fixed) & $1.09\\pm0.01$ & & \\nodata & \\nodata & & \\nodata & $22.72\\pm0.00$ & 1.275\\\\\nD & \\nodata & \\nodata & \\nodata & & $23.07\\pm0.00$ & $0.47\\pm0.00$ & & \\nodata & $23.07\\pm0.00$ & 6.547\\\\\nB$(n=4)$$+$D & $23.01\\pm0.03$ & 4 (fixed) & $0.85\\pm0.02$ & & $24.64\\pm0.10$ & $0.95\\pm0.03$ & & $0.82\\pm0.03$ & $22.79\\pm0.03$ & 1.228\\\\\nS$(n=1)$$+$D & $23.99\\pm0.01$ & 1 (fixed) & $0.30\\pm0.00$ & & $23.35\\pm0.00$ & $0.92\\pm0.01$ & & $0.36\\pm0.00$ & $22.87\\pm0.00$ & 1.351\n\\enddata\n\\tablecomments{Column~1: galaxy model for the two-dimensional fitting. B: S\\'{e}rsic bulge, D: exponential disk , B$(n=4)$: de Vaucouleurs bulge. S$(n=1)$: S\\'{e}rsic profile with fixed $n=1$ (exponential profile). Column~2: AB magnitude of the bulge component. Column~3: S\\'{e}rsic index for the bulge model. Column~4: effective radius. Column~5: AB magnitude of the disk component. Column~6: scale length of the disk component. Column~7: bulge to total light ratio. Column~8: total magnitude. Column~9: reduced $\\chi^2$ for the fitting model defined as\n\\begin{displaymath}\n \\chi_{\\nu}^{2}=\\frac{1}{N_\\mathrm{DOF}} \\sum_{x=1}^{nx}\\sum_{y=1}^{ny}\\frac{(f_\\mathrm{data}(x,y)-f_\\mathrm{model}(x,y))^2}{\\sigma(x,y)^2},\n\\end{displaymath} \nwhere $f_\\mathrm{data}(x,y)$ and $f_\\mathrm{model}(x,y)$ mean the input data and the model images, respectively. $N_\\mathrm{DOF}$ is the degree of freedom. $\\sigma(x,y)$ is the error image. Here, sum is only over all $nx$ and $ny$ pixels satisfying $1.5\\sigma$ of the background noise.}\n\\label{galfittab}\n\\end{deluxetable*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.36,angle=00]{Swift_Ukirt.eps}\n\t\\caption{NIR and the X-ray light curves of the \\emph{Swift} J1644+57.\n The light curves of $Y, J ,H ,K$ bands with the early data from \\citet{Burrows2011}, \\citet{Levan2011}, and \\emph{Swift}\/XRT $0.3$ -- $10$keV are shown. All of the light curves have similar shapes except that the X-ray light curve is $\\sim$15 days ahead of the NIR light curves. After $\\sim$500 days the X-ray emission was rapidly declined as shown in with star mark for the last X-ray data from the \\emph{Swift}\/XRT.\n \\label{ukirtfig}}\n\\end{figure*}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.22,angle=00]{Swift_Xray.eps}\n\t\\caption{\\emph{Swift}\/XRT and \\emph{XMM-Newton} data as well as recent \\emph{Chandra} observation of \\emph{Swift} J1644+57. The X-ray flux abruptly declined after $\\sim$500 days since the BAT trigger. \n\t\t\\label{xfig}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_Ukirtlinear.eps}\n\t\\caption{$H$- and $K$-band light curves in linear scale. Some of the data points at very early times are cut in order to highlight the late-time light curves. The magnitudes of the latest $H, K$ bands converge to single values, suggesting that the transient component has disappeared at $\\Delta t > 500$ days.\n\t\t\\label{linfig}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.30,angle=00]{Swift_lc.eps}\n\t\\caption{Light curves of CQUEAN $i$ and $z$ bands, UKIRT $Z$-band data, and \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands. Note that the $z$-band flux decreases with time, while the $i$-band flux is almost constant with time. Considerable magnitude changes in the \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands can be seen in the right plot.\n\t\t\\label{lcfig}}\n\\end{figure*}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_change.eps}\n\t\\caption{Temporal change of the SED of \\emph{Swift} J1644+57. The fluxes in the redder bands show the substantial changes with time, whereas the bluer-band fluxes do not vary with time.\n\t\t\\label{changefig}}\n\\end{figure}\n\nAdditionally we fit the observed surface brightness profile with a single de Vaucouleurs bulge model and a de Vaucouleurs bulge $+$ exponential disk model. The results of these fits are nearly identical to that of the single S\\'{e}rsic and the S\\'{e}rsic bulge+disk models.\n\n To estimate the transient component flux, we fit all the F606W and F160W images with a model containing both the point source (transient) and the host galaxy components. Here, we adopt a single S\\'{e}rsic profile with a fixed S\\'{e}rsic index ($n=3.43$) for the host galaxy component, and a PSF profile for the transient component. The compactness of the host galaxy and the bright transient component in the F160W images create a serious degeneracy, particularly between the effective radius and the S\\'{e}rsic index, when fitting multi-component models. To alleviate the degeneracy, we fixed the S\\'{e}rsic index to be $n=3.43$, similar to that of the F606W band. The flux fractions of the models as a function of time are shown in Figure~\\ref{hstfig}. In the case of F606W, the flux fraction from the transient component is very small or nonexistent, while the transient component is very bright in the earliest F160W band, even brighter than the entire host galaxy. The result reflects a very red color to the transient well, and justifies the exclusion of the point source component in the late-time F606W images during the host galaxy analysis. The point source contribution declines rapidly as time goes on in F160W, but it contributes to the total flux of the object until around $\\Delta t = 750$ days. On the other hand, the fluxes of the host galaxy component are almost constant in both bands over the entire period. The magnitude of host galaxy in the F160W band is $\\sim21.75$ mag, and as we shall see in the next section, this is the same as for the last data point of the UKIRT $H$-band light curve, suggesting that the flux of the last data point in the NIR light curve represents the host galaxy flux. \n\\\\\n\n\n\n\\section{Light Curves} \\label{sec:LC}\n In this section, we show long-term observation results and estimate multi-band fluxes of the host galaxy of \\emph{Swift} J1644+57. Figure~\\ref{ukirtfig} shows the $Y, J, H$, and $K$ light curves. The gray data points in the background show \\emph{Swift}\/XRT $0.3$ -- $10$keV data. The $J,H,K$ light curves resemble each other. The NIR light curves rapidly decline until $\\Delta t \\simeq 10$ days, turn up again with a second peak at $\\Delta t \\simeq 30$ days, and decline again steadily. The behaviors of these NIR light curves are very similar to that of the X-ray light curve except that the X-ray light curve appears shifted ahead of the NIR light curves at a time of $\\sim$15 days. The similar shapes of these light curves indicate that the origins of the X-ray emission and NIR emission are related to each other. On the other hand, the time gap between these two emissons denotes that the X-ray source and NIR source are separated from each other as much as the time gap. \\citet{Bloom2011} suggested that X-ray source is in the close vicinity of the black hole due to the fact that the X-ray emission shows very rapid, high variability, while the IR and the radio sources are located a large distance from the black hole on account of the relatively smooth and small variability. They argued that the jet generated by black hole collides with the surrounding medium where the electrons are accelerated by the jet. These high-speed electrons emit the IR to radio photons through synchrotron radiation.\n\n\n The jet seems to be nearly turned off at $\\Delta t \\simeq 500$ days in light of the fact that there is an abrupt decrease in flux of a factor of $\\sim10$ or more, which can be seen in all the \\emph{Swift}\/XRT, \\emph{Chandra}, and \\emph{XMM-Newton} data \\citep{Levan2012, Zauderer2013}. This late-stage turn-off of X-ray flux is also shown in Figure~\\ref{xfig}. If the jet was turned off, then the transient components of the NIR fluxes must be quenched following the X-ray flux, and it is expected that the fluxes of the pure host galaxy of \\emph{Swift} J1644+57 were revealed at that time. As we can see in Figure~\\ref{linfig}, which shows the $H$-, $K$- bands light curves in linear scale, the fluxes of the $H, K$ bands converge to single values at late-time. Furthermore, the latest $H$-band magnitude is nearly the same as that of the host galaxy of the F160W-band images, shown in the results of the model fitting in \\S\\ref{sec:Mor}. This evidence indicates that it is reasonable to regard the NIR ($Y, J, H$, and $K$ band) fluxes of the last data, taken at $\\Delta t = \\sim 700$ or $884$ days, as those of the pure host galaxy. \n\n\n The left panel of Figure~\\ref{lcfig} shows the light curves of the CQUEAN $i$- and $z$-band and the UKIRT $Z$-band data. In the case of the $z$ band, the fluxes from the object slightly decrease with time. We also regard the flux of the last data, that is the UKIRT $Z$-band data, as the $Z$-band flux of the host galaxy since it is observed far beyond expected quenching time of jet. On the other hand, $i$-band fluxes are virtually constant, demonstrating that the transient components are basically non-existent in the $i$ or bluer bands \\citep[Figure~\\ref{hstfig};][]{Burrows2011, Levan2011}. We take the flux of the last data of the $i$ band as that of the host galaxy. The number of CQUEAN $g$- and $r$-band data points are scarce compared to the NIR data and the last data were observed at early time ($\\Delta t =25.7, 217.5$ days, respectively). However, there is little or no change between the very early-time magnitudes from \\citet{Levan2011} and our $g$- and $r$-band magnitudes in the same way as the fluxes of the $i$ and F606W bands. Therefore, we consider the $g$- and $r$-band fluxes of the last epoch data as those from the host galaxy. Furthermore, we also consider the magnitude of the single S\\'{e}rsic model of the stacked \\emph{HST} WFC3 F606W image as that of the host galaxy since the point source contribution to whole flux is negligible.\n\n\\begin{deluxetable}{ll}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Magnitudes of Host Galaxy} \n\\tablehead{\n\\colhead{Band} & \\colhead{Magnitude (AB)}\n}\n\\startdata\n$B$ & 24.24$\\pm$0.10 \\citep{Levan2011}\\\\\n$g$ & 23.67$\\pm$0.19\\\\\nF606W & 22.72$\\pm$0.01\\\\\n$r$ & 22.73$\\pm$0.06\\\\\n$i$ & 22.25$\\pm$0.04\\\\\n$Z$ & 22.16$\\pm$0.06\\\\\n$Y$ & 22.18$\\pm$0.07\\\\\n$J$ & 21.96$\\pm$0.08\\\\\n$H$ & 21.74$\\pm$0.12\\\\\n$K$ & 21.55$\\pm$0.10\\\\\n3.6$\\mu$m & 21.70$\\pm$0.07 (Including transient)\\\\\n4.5$\\mu$m & 21.57$\\pm$0.08 (Including transient)\n\\enddata\n\\tablecomments{Magnitudes are corrected by Galactic extinction based on \\citet{Schlafly2011}.}\n\\label{hosttab}\n\\end{deluxetable}\n\n\\begin{deluxetable*}{rl}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Input Parameters for SED Fitting} \n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value}\n}\n\\startdata\n$\\tau$ ($e$-folding time scale of stellar population) & $6.5\\leq$ $\\log$[$\\tau$\/yr] $\\leq11.0$ with a step size of 0.1 \\\\\n$t$ (age of stellar population) & $8.0\\leq$ $\\log$[$t$\/yr] $\\leq10.3$ with a step size of 0.1 \\\\\nIMF (initial mass function) & \\citet{Salpeter1955} \\\\\nZ (metallicity) & 0.004, 0.008, 0.020, 0.050 \\\\\nExtinction law & \\citet{Calzetti2000} \\\\\n$A_{V}$ ($V$-band attenuation for stellar population in magnitude) & $0.0\\leq$ $A_{V}$ $\\leq3.0$ with a step size of 0.1 \n\\enddata\n\\label{SEDtab}\n\\end{deluxetable*}\n\n\n\\begin{deluxetable}{rr}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Best-fit Parameters from SED Fitting} \n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value}\n}\n\\startdata\nStellar mass [$\\log(M_{\\star}\/M_{\\odot})$] & $9.14^{+0.13}_{-0.10}$\\\\\n\\\\\nSFR [$M_{\\odot}$\/yr] & $0.03^{+0.28}_{-0.03}$\\\\\n\\\\\nSpecific SFR [$\\log(\\rm{sSFR}\/\\rm{yr}^{-1})$] & $-10.62^{+0.90}_{-\\infty}$\\\\\n\\\\\n$t$ [Gyr] & $0.63^{+0.95}_{-0.43}$\\\\\n\\\\\n$\\tau$ [Gyr] & $0.10^{+0.24}_{-0.10}$\\\\\n\\\\\n$A_{V}$ & $0.00^{+0.97}_{-0.00}$\\\\ \n\\\\\nZ & $0.050^{+0.000}_{-0.046}$\\\\\n\\\\\n$\\chi^2$ & 1.64\n\\enddata\n\\label{SED2tab}\n\\end{deluxetable}\n\n The right panel of Figure~\\ref{lcfig} shows the light curves of the \\emph{Spitzer} IRAC 3.6$\\mu$m and 4.5$\\mu$m bands. The magnitude changes in these bands are the more significant than those for the other optical\/NIR bands. The fluxes from the transient component seem to be non-negligible even in the last epoch data ($\\Delta t = 333$ days), because the last IRAC epochs were still in the rapidly decreasing phase. Therefore, we consider the fluxes of the last epoch IRAC data to be the upper limit fluxes of the host galaxy.\n\n The multi-band magnitudes of the host galaxy of \\emph{Swift} J1644+57 are shown in Table~\\ref{hosttab}. We took the Galactic extinction of photometric data into account based on \\citet{Schlafly2011}. We added the $B$-band photometric data from \\citet{Levan2011} to expand the data points to the short wavelength band.\n\n The temporal change of the observed SED of \\emph{Swift} J1644+57 are summarized in Figure~\\ref{changefig}. The fluxes in redder bands show substantial changes with time. Meanwhile, bluer band fluxes are constant. This red feature of the transient has been suggested to be due to dust extinction. From previous studies, it is known that the hydrogen column density of the source of X-ray is large ($N_{\\mathrm{H}}\\sim10^{22}$cm$^{-2}$), meaning the line of sight to the SMBH has a very large extinction value ($A_{V}=4.5$ -- $10$) \\citep{Bloom2011,Burrows2011,Levan2011,Shao2011,Saxton2012}.\\\\\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.296,angle=00]{SED.eps}\n\t\\caption{SED fitting result of the host galaxy of \\emph{Swift} J1644+57. Plotted are the multi-band fluxes of the host galaxy in Table~\\ref{hosttab}. Two \\emph{Spitzer} data are the upper limit fluxes. The $\\chi^2$ value for the fit is 1.64.\n\t\t\\label{SEDfig}}\n\\end{figure}\n\n\n\\section{SED Fitting} \\label{sec:SED}\n\nWe performed SED model fitting of the multi-band fluxes of host galaxy of \\emph{Swift} J1644+57 in order to determine the properties of the host galaxy such as the stellar mass ($M_{\\star}$) which is an important parameter for the $M_{\\rm BH}$ estimation. We utilized the code Fitting and Assessment of Synthetic Templates \\citep[FAST;\\footnote{http:\/\/astro.berkeley.edu\/$\\sim$mariska\/FAST.html}][]{Kriek2009}, which is a public SED fitting tool for the investigation of the galaxy properties using the photometric data ranging from UV to IR. The code is based on the IDL and fits of UV to IR stellar population templates to photometric data or galaxy spectra. FAST runs with the method of $\\chi^2$ fitting and using stellar population grids to derive the best-fit model and its parameters. \n\n We used the 2003 version of the Bruzual \\& Charlot (BC03) model \\citep{BC03} for the stellar population model. There are three initial mass functions (IMFs) available in the FAST \\citep{Salpeter1955,Kroupa2001,Chabrier2003}. We chose the Salpeter IMF. To define the star formation history (SFH), we assumed an exponentially decreasing star formation rate (SFR). The stellar population was modeled with the $e$-folding time scales, $6.5\\leq$ $\\log$[$\\tau$\/yr] $\\leq11.0$ with a step size of 0.1 and ages of $8.0\\leq$ $\\log$[$t$\/yr] $\\leq10.3$ with a step size of 0.1. We used several metallicity values such as Z = 0.004, 0.008, 0.02, and 0.05. The model SEDs were attenuated by dust, for which we used attenuation curves based on \\citet{Calzetti2000}. We adopted $0.0\\leq$ $A_{V}$ $\\leq3.0$ with a step size of 0.1. All the input parameters for the SED fitting are summarized in Table~\\ref{SEDtab}. \n\n Figure~\\ref{SEDfig} shows the SED fitting result. The two \\emph{Spitzer} data were treated as the upper limit fluxes of the host galaxy. The estimated stellar mass of the host galaxy is $\\log(M_{\\star}\/M_{\\odot}) = 9.14^{+0.13}_{-0.10}$. The $e$-folding time scale is $\\tau = 0.10^{+0.24}_{-0.10}$ Gyr and the age of the stellar population is $t = 0.63^{+0.95}_{-0.43}$ Gyr. The SFR of galaxy is $0.03^{+0.28}_{-0.03} \\, M_{\\odot}$\/yr, and the specific SFR is $\\log(\\rm{sSFR}\/\\rm{yr}^{-1}) = -10.62^{+0.90}_{-\\infty}$. \\citet{Levan2011} derived SFR of $0.3$ -- $0.7 \\, M_{\\odot}$\/yr from the H$\\alpha$ and [O II] emission line luminosities. The value of $0.3 \\, M_{\\odot}$\/yr from H$\\alpha$ is consistent with our 1$\\sigma$ upper limit. The SFR from [O II] line ($0.7 \\, M_{\\odot}$\/yr) is about twice larger but the [O II] line based SFRs are known to be dependent on physical condition such as the reddening and metallicity \\citep[e.g.,][]{Kewley2004}, and less reliable than H$\\alpha$ based SFRs. Another possible cause of the discrepancy is the different timescales that are probed by different SFR indicators (emission line indicators probing recent star formation). \\citet{Levan2011} estimated $E(B-V)_{gas}=-0.01 \\pm 0.15$mag, i.e., no extinction using the intensity ratio of H$\\alpha$ and H$\\beta$ lines. This is consistent with our SED fitting result $A_{V}=0.00^{+0.97}_{-0.00}$. The $\\chi^2$ value for the fit is 1.64. \n\n\n The host galaxy of \\emph{Swift} J1644+57 is a low mass, low SFR galaxy with a low extinction. Also it seems to have experienced a rapid decline of SFR not very long ago. This fits in well with a recent suggestion by \\citet{Arcavi2014} that host galaxies of tidal disruption events are E+A galaxies with $<1$ Gyr stellar population and low or no SFRs.\n\n The output parameters are given in Table~\\ref{SED2tab}. The errors correspond to 1$\\sigma$ confidence intervals derived from 100 times Monte Carlo simulations in which the input photometric data are changed according to their errors.\n \n We also tried the Chabrier IMF instead of the Salpeter IMF for the fit. The change of the IMF influenced to the stellar mass, decreasing the stellar mass by $\\sim0.25$ dex.\n\n We also fitted the SED model with the \\citet{Maraston2005} stellar population instead of the BC03 model. We set the input parameter ranges identical to the case of the BC03 stellar populations. The results were nearly identical to the BC03 result. \n\n Our analysis of the host galaxy shows that the host galaxy is a bulge-dominated and nearly extinction free ($A_{V} \\sim 0$ mag). On the other hand, the spectral properties of the nuclear transient suggests a high extinction ($A_{V} \\sim 6$ mag). These two facts may appear contradictory, but we note that a significant amount of dust can be found easily in nuclear region of bulge-dominated galaxies when their nuclei are acitve. For example, hosts of luminous AGNs are mostly early-type, bulge-dominated galaxies \\citep[e.g.,][]{Hong2015}, and such AGNs are known to contain a significant amount of dust in nuclear region as a form of hot or warm dusty torus \\citep[e.g.,][]{Kim2015}. \n\\\\\n\n\n\\section{Discussion on Black Hole Mass} \\label{sec:BH}\n Our results on the properties of the host galaxy of \\emph{Swift} J1644+57 can be summarized as follows. It is a bulge-dominant galaxy (B\/T=$0.83\\pm0.03$). The mass of the host galaxy is somewhat low at $10^{9.14} \\,M_{\\odot}$, even though the galaxy is bulge-dominated. Now we estimate the mass of the SMBH that played the main role in the transient phenomenon.\n\n It is now generally accepted that the SMBHs ($10^6$ -- $10^{10} \\,M_{\\odot}$) reside in the bulges of all massive galaxies. Tight scaling relations have been derived between SMBH mass and several physical properties of the bulges (velocity dispersion, mass, luminosity, etc.) in many previous studies \\citep{Magorrian1998,Ferrarese2000,Gebhardt2000,McLure2002,Marconi2003,Haring2004,Aller2007,Hopkins2007,Gultekin2009,Kormendy2009,Sani2011,Kormendy2013}. Some argue that ellipticals and classical bulges follow identical relations, while the pseudobulges follow a somewhat different relation with large scatter \\citep{Hu2008,Kormendy2011,Sani2011,Kormendy2013}. We conclude from the best-fit galaxy models that the host galaxy of \\emph{Swift} J1644+57 has a classical bulge, and have also found a minor possibility of the pseudobulge with B\/T=$0.36$. For now, we consider only the best model, that is, the case of the host galaxy having a classical bulge and being bulge-dominant. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.175,angle=00]{Swift_BHrange.eps}\n\t\\caption{Results on the black hole mass from this work and previous studies. The red circles and dashed line represent our results, while the black squares and lines denote the results from previous studies. Error bars correspond to deviation of 1$\\sigma$ and the arrow indicates the upper bound value.\n\t\t\\label{BHfig}}\n\\end{figure}\n\n In order to esimate the central SMBH mass in the host galaxy of \\emph{Swift} J1644+57, we used the scaling relation between $M_{\\rm BH}$ and the stellar mass of the bulge ($M_{\\star,\\mathrm{bul}}$). We expect that a large part of the stellar mass derived in \\S\\ref{sec:SED} belongs to the bulge component. \\citet{Sani2011} present the $M_{\\mathrm{BH}}$ -- $M_{\\star,\\mathrm{bul}}$ relation, where $M_{\\star,\\mathrm{bul}}$ is directly obtained from the bulge luminosity ($L_{\\mathrm{bul}}$) of \\emph{Spitzer} 3.6$\\mu$m and the calibrated $M_{\\star,\\mathrm{bul}}$ -- $L_{\\mathrm{bul}}$ relation. They excluded pseudobulges when constructing the relation. The relation is \n\\begin{equation}\n\t\\mathrm{log}(M_\\mathrm{BH}\/M_{\\odot})=\\alpha+\\beta\\times[\\mathrm{log}(M_\\mathrm{\\star,bul}\/M_{\\odot})-11],\n\\end{equation}\nwhere $\\alpha=8.16\\pm0.06$, $\\beta=0.79\\pm0.08$, and the intrinsic scatter is $0.38\\pm0.05$ . The estimated mass of the SMBH is $10^{6.7\\pm0.4} \\,M_{\\odot}$ based on the stellar mass of the host galaxy and the above relation. If we consider the B\/T=0.83 and assume that the mass-to-light ratio is constant in the bulge and disk, the stellar mass is decreased by $\\sim0.1$ dex. It leads to a decrease in $M_{\\mathrm{BH}}$ by $\\sim0.1$ dex.\n\n The tight scaling relations between $M_{\\mathrm{BH}}$ and host galaxy properties suggest a close link between the SMBH growth and the galaxy evolution. There may be a cosmic evolution of the scaling relations, for which there have been various studies \\citep{Treu2004,McLure2006,Shields2006,Woo2006,Salviander2007,Treu2007,Woo2008,Jahnke2009,Bennert2010,Decarli2010,Merloni2010,Bennert2011}. The evolution of the scaling relations is still controversial in terms of the selection effects in the high redshift regime. Nevertheless, we can consider a case where the growth of $M_{\\mathrm{BH}}$ happened ahead of the assembly of the stellar mass as suggested by some of these studies. \\citet{Bennert2011} suggest the redshift evolution out to $z\\sim2$, in the form of $M_{\\mathrm{BH}}\/M_{\\star,\\mathrm{bul}}\\propto(1+z)^{1.96\\pm0.55}$, based on 11 X-ray-selected broadline AGNs. Considering this evolution effect and the redshift of \\emph{Swift} J1644+57 $z=0.35$, the mass of the SMBH could be larger by $\\sim0.3$ dex.\n\n\n We also estimated $M_{\\mathrm{BH}}$ through the $M_{\\mathrm{BH}}$ -- $K$ band luminosity of bulge ($L_{K,\\mathrm{bul}}$) relation in \\citet{Kormendy2013}, assuming that most of the NIR fluxes come from the bulge. Their relation is much improved compared with previous studies in view of serveral things. They excluded galaxies with black hole monsters which have over-massive SMBHs despite having relatively small bulges and ellipticals. They also omitted galaxies with black hole masses that are measured based on the kinematics of ionized gas without taking line widths into account, since this method may yield underestimated masses. Galaxies in the process of merging generally have low-mass black holes for their luminosities. Thus they excluded these galaxies from the relation. They also did not include pseudobulges. The relation of \\citet{Kormendy2013} is\n\\begin{equation}\n\t\\mathrm{log}(M_\\mathrm{BH}\/10^9\\,M_{\\odot})=-\\alpha-\\beta\\times(M_{K,\\mathrm{bul}}+24.21),\n\\end{equation}\nwhere the $\\alpha$ is $0.265\\pm0.050$, the $\\beta$ is $0.488\\pm0.033$, and intrinsic scatters is 0.30. $M_{K,\\mathrm{bul}}$ is the $K$-band absolute magnitude of the bulge based on the photometric system of 2MASS. Using the best-fit SED derived earlier, we applied K-correction and evolution correction. We find $M_{K} = -19.84$ Vega mag including the evolutionary correction of 1.38 mag, which is derived by the difference in the $K$-band magnitude between 0.63 Gyr old population as in Table~\\ref{SED2tab} and 4.5 Gyr old population, which is the age of this host galaxy in the local universe. Then, the mass of the SMBH is estimated to be $10^{6.6\\pm0.3}\\,M_{\\odot}$. If we take B\/T=0.83 into account and assume that this value is also applicable to the NIR bands, then $M_{\\mathrm{BH}}$ decreases by $\\sim0.1$ dex. If we use an $M_{K}$ value that has not been corrected for evolution, then $M_{\\mathrm{BH}}$ becomes $\\sim0.7$ dex larger.\n\n The tidal disruption of normal stars by a black hole is not possible for $M_\\mathrm{BH}>10^8\\,M_{\\odot}$, since the tidal radius where the disruption can occur is located inside the Schwarzschild radius \\citep{Rees1988,Cannizzo1990,Bloom2011}. Our results on the mass of the SMBH satisfy the condition for a tidal disruption event. \n\nAlthough we favor the model in which the host galaxy of \\emph{Swift} J1644+57 is a classical bulge, our analysis shows that it could be a galaxy with a pseudobulge with B\/T=0.36 (Table~\\ref{galfittab}). If so, it is rather difficult to obtain an $M_{\\rm BH}$ value, since the scaling relation is not well established for pseudobulges, especially in the low mass range of $M_{\\star}\\sim10^9\\,M_{\\odot}$ for the host galaxy. Several works have shown that the $M_{\\rm{BH}}$ -- host galaxy scaling relations are weak or zero with a large scatter for pseudobulges. Over the $M_\\mathrm{\\star,bul}$ range of $10^{9.3}$ to $10^{10.5}\\,M_{\\odot}$ where such a relation has been studied, $M_{\\rm BH}$ can have any value between $10^{6}$ to $10^{8}\\,M_{\\odot}$ \\citep[Figure 21 of][]{Kormendy2013}. To reach down to $M_{\\star,\\mathrm{bul}} \\sim 5 \\times 10^{8}\\,M_{\\odot}$ as implied from the pseudobulge fit of our data, currently one can barely do so by relying on results from low mass AGNs \\citep{Barth2005,Greene2008,Jiang2011,Xiao2011}. In such a case, an $M_{\\rm BH}$ value between $10^{5}$ to $10^{6.3}\\,M_{\\odot}$ is possible \\citep[Figure 32 of][]{Kormendy2013}. Overall, if the host galaxy harbors a pseudobulge, then we can only loosely constrain $M_{\\rm BH}$ to have a value between $10^{5}$ to $10^{7}\\,M_{\\odot}$ considering our current poor knowledge of the $M_{\\rm BH}$ value in pseudobulges. \n\nIt is also known that a small fraction of pseudobulges have a S\\'{e}rsic index of $n > 3$. The best example is Pox 52, for which $n\\sim3.6$ -- $4.3$, $M_{\\rm BH} \\sim2\\times10^{5} \\,M_{\\odot}$, and $M_{\\star} \\sim10^9 \\,M_{\\odot}$ \\citep{Barth2004,Thornton2008}. Therefore, even if we accept the S\\'{e}rsic index of $n=3.43$ as the best-fit result, we need to keep this kind of caveat in mind.\n\n Figure~\\ref{BHfig} shows our overall results on $M_{\\mathrm{BH}}$ and the results from the previous studies we mentioned in \\S\\ref{sec:Intro}. It shows that our favorite results are compatible with the previous rough estimates from \\citet{Burrows2011} and \\citet{Levan2011}, who also used scaling relations. However, our results are improved compared to the previous results, by revealing that the host galaxy has a significant bulge component through a two-dimensional bulge + disk decomposition of the surface brightness profile, and removing the transient component in NIR light using a long-term light curve. The $M_{\\rm BH}$ limit could be much looser (the dashed line) if the host galaxy harbors a pseudobulge. A critical test of the pseudobulge model would be to obtain a deep, high-resolution image to see how the surface brightness profile behaves at the outer region of the host galaxy. \n\\\\\n\n\n\n\n\n\n\n \n\n\n\\section{Summary}\n We investigated the host galaxy properties of tidal disruption event, \\emph{Swift} J1644+57 through morphology analysis, light curve analysis, and SED fitting. We also estimated $M_{\\mathrm{BH}}$ which played the main role of this phenomenon, through scaling relations.\n\n We decomposed the surface brightness profile of the host galaxy based on high-resolution \\emph{HST} WFC3 images. We found that the host galaxy of \\emph{Swift} J1644+57 is a bulge-dominated galaxy which is well described by a single S\\'{e}rsic model with the S\\'{e}rsic index, $n=3.43\\pm0.05$. If we add a disk component, the bulge to total host galaxy flux ratio (B\/T) is $0.83\\pm0.03$, still indicating a bulge-dominant galaxy. We conclude that the host galaxy of \\emph{Swift} J1644+57 has a classical bulge from the best-fit galaxy models, although we cannot completely exclude the possibility of this galaxy containing a pseudobulge with B\/T=$0.36$.\n\n The NIR light curves enabled us to isolate the fluxes from the host galaxy after $\\sim 500$ days following the dissipation of the X-ray flux. On the other hand, we found that there are no significant changes in the light curves of the short wavelength bands, supporting the red feature of the transient possibly being caused by severe dust extinction.\n\n We fit SEDs to the multi-band fluxes of the host galaxy which are derived in the light curve analysis. The estimated stellar mass of the host galaxy is $\\log(M_{\\star}\/M_{\\odot}) = 9.14^{+0.13}_{-0.10}$. The $e$-folding time scale $\\tau$ is $0.10^{+0.24}_{-0.10}$ Gyr and the age of stellar population is $0.63^{+0.95}_{-0.43}$ Gyr. The SFR of galaxy is $0.03^{+0.28}_{-0.03} \\,M_{\\odot}$\/yr. In terms of the surface brightness profile and the stellar mass, this galaxy resembles M32, a small companion galaxy of M31. \n\n We estimated the central $M_{\\mathrm{BH}}$ through scaling relations. The mass of the SMBH is estimated to be $10^{6.7\\pm0.4} \\,M_{\\odot}$ from $M_{\\mathrm{BH}}$ -- $M_{\\star,\\mathrm{bul}}$ and $M_{\\mathrm{BH}}$ -- $L_{\\mathrm{bul}}$ relations for the $K$ band. However, the limit on $M_{\\mathrm{BH}}$ can be much looser if the host galaxy has a pseudobulge. Future high-resolution, deep imaging should be able to unambiguosly distinguish the two possibilities.\n\\\\\n\n\n\\acknowledgments \nThis work was supported by the National Research Foundation of Korea (NRF) grant, No. 2008-0060544, funded by the Korea government (MSIP). We thank the observers who obtained the CQUEAN and UKIRT data that were used in our analysis. This paper includes the data taken at the McDonald Observatory of the University of Texas at Austin. At the time of the UKIRT observation, UKIRT was operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. We acknowledge the use of public data from the Swift data archive. CP, TS, and NG acknowledge support from the NASA research grant, NNX10AF39G. MI gratefully acknowledges the hospitality and the support from the Korea Institute of Advanced Study where part of this was carried out.\n\\\\\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\@startsection {section}{1}{\\z@}%\n\n\n\\newcommand{\\pic}[1]{\\;\\parbox[c]{45pt}{\\begin{picture}(45,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pics}[1]{\\;\\parbox[c]{30pt}{\\begin{picture}(30,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pib}[1]{\\;\\parbox[c]{36pt}{\\begin{picture}(22.5,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picb}[1]{\\;\\parbox[c]{48pt}{\\begin{picture}(45,30)(-9,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picc}[1]{\\;\\parbox[c]{45pt}{\\begin{picture}(45,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\piccb}[1]{\\;\\parbox[c]{75pt}{\\begin{picture}(75,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\pivv}[1]{\\;\\parbox[c]{36pt}{\\begin{picture}(30,15)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picl}[1]{\\;\\parbox[c]{60pt}{\\begin{picture}(60,30)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\\newcommand{\\picss}[1]{\\;\\parbox[c]{26pt}{\\begin{picture}(20,26)(0,0)\n\\SetWidth{1.0}\\SetScale{1.0} #1 \\end{picture}}\\;}\n\n\\def1{1}\n\\special{! \/Ldensity {0.25} def}\n\\def\\Agl(#1,#2)(#3,#4,#5){\\PhotonArc(#1,#2)(#3,#4,#5){1}\n{6.283 #3 mul 360 div #4 #5 sub #4 #5 sub mul sqrt mul Ldensity mul}}\n\\def\\Lgl(#1,#2)(#3,#4){\\Photon(#1,#2)(#3,#4){1}\n{#1 #3 sub #1 #3 sub mul #2 #4 sub #2 #4 sub mul add sqrt Ldensity mul}}\n\\def\\Agh(#1,#2)(#3,#4,#5){\\DashArrowArc(#1,#2)(#3,#4,#5){1}}\n\\def\\Aagh(#1,#2)(#3,#4,#5){\\DashArrowArcn(#1,#2)(#3,#5,#4){1}}\n\\def\\Lgh(#1,#2)(#3,#4){\\DashArrowLine(#1,#2)(#3,#4){1}}\n\\def\\Lagh(#1,#2)(#3,#4){\\DashArrowLine(#3,#4)(#1,#2){1}}\n\\def\\Ahh(#1,#2)(#3,#4,#5){\\DashCArc(#1,#2)(#3,#4,#5){1}}\n\\def\\Lhh(#1,#2)(#3,#4){\\DashLine(#1,#2)(#3,#4){1}}\n\\def\\Aqu(#1,#2)(#3,#4,#5){\\ArrowArc(#1,#2)(#3,#4,#5)}\n\\def\\Aaqu(#1,#2)(#3,#4,#5){\\ArrowArcn(#1,#2)(#3,#5,#4)}\n\\def\\Lqu(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\Laqu(#1,#2)(#3,#4){\\ArrowLine(#3,#4)(#1,#2)}\n\\def\\Aqq(#1,#2)(#3,#4,#5){\\CArc(#1,#2)(#3,#4,#5)}\n\\def\\Lqq(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\Asc(#1,#2)(#3,#4,#5){\\ArrowArc(#1,#2)(#3,#4,#5)}\n\\def\\Lsc(#1,#2)(#3,#4){\\ArrowLine(#1,#2)(#3,#4)}\n\\def\\DAsc(#1,#2)(#3,#4,#5){\\DashCArc(#1,#2)(#3,#4,#5){3}}\n\\def\\DLsc(#1,#2)(#3,#4){\\DashLine(#1,#2)(#3,#4){3}}\n\\def\\TAsc(#1,#2)(#3,#4,#5){\\SetWidth{2.0}\\CArc(#1,#2)(#3,#4,#5)\\SetWidth{1.0}}\n\\def\\TLsc(#1,#2)(#3,#4){\\SetWidth{2.0}\\ArrowLine(#1,#2)(#3,#4)\\SetWidth{1.0}}\n\n\n\n\\begin{document}\n\n\\preprint{ECT*-06-04, HIP-2006-18\/TH, TUW-06-02}\n\\pacs{11.10.Wx, 12.38.Mh}\n\n\\title{The pressure of deconfined QCD\nfor all temperatures and\\\\ quark chemical potentials}\n\n\\author{A. Ipp}\n\\affiliation{ECT*, Villa Tambosi, Strada delle Tabarelle 286,\\\\\nI-38050 Villazzano Trento, Italy}\n\\author{K. Kajantie}\n\\affiliation{Department of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland}\n\\author{A. Rebhan}\n\\affiliation{Institut f\\\"ur Theoretische Physik, Technische\nUniversit\\\"at Wien, \\\\Wiedner Hauptstr.~8-10,\nA-1040 Vienna, Austria }\n\\author{A. Vuorinen}\n\\affiliation{Department of Physics, University of Washington, Seattle, WA 98195, U.S.A.}\n\n\n\\begin{abstract}\n We present a new method for the evaluation of the perturbative\n expansion of the QCD pressure which is\n valid at all values of the temperature and quark chemical potentials\n in the deconfined phase and which we work out up to and including\n order $g^4$ accuracy. Our calculation is manifestly four-dimensional\n and purely diagrammatic --- and thus independent of any effective\n theory descriptions of high temperature or high density QCD.\n In various limits, we recover the\n known results of dimensional reduction and the HDL and HTL\n resummation schemes, as well as the equation of state of\n zero-temperature quark matter, thereby verifying their respective\n validity. To demonstrate the overlap of the various regimes, we\n furthermore show how the predictions of dimensional reduction and\n HDL resummed perturbation theory agree in the regime $T\\sim\n \\sqrt{g}\\mu$. At parametrically smaller temperatures $T\\sim g\\mu$,\n we find that the dimensional reduction result agrees well with those\n of the nonstatic resummations down to the remarkably low value\n $T\\approx 0.2 m_\\rmi{D}$, where $m_\\rmi{D}$ is the Debye mass at $T=0$. Beyond\n this, we see that only the latter methods connect smoothly to the\n $T=0$ result of Freedman and McLerran, to which the leading\n small-$T$ corrections are given by the so-called non-Fermi-liquid terms,\n first obtained through HDL resummations. Finally, we outline the\n extension of our method to the next order, where it would include\n terms for the low-temperature entropy and specific heats\n that are unknown at present.\n\\end{abstract}\n\\maketitle\n\n\n\n\\tableofcontents\n\n\\section{Introduction}\n\nThe most fundamental thermodynamic quantity in the theory of strong\ninteractions, the QCD pressure $p_\\rmi{QCD}(T,\\mu)$, can at large\nvalues of the temperature $T$ or the quark chemical potentials $\\mu$\nbe computed in a weak coupling expansion in the gauge coupling\nconstant $g$, defined in the ${\\overline{\\mbox{\\tiny\\rm{MS}}}}$ renormalization scheme. In\nthe region where $T$ is larger than all other relevant mass scales in\nthe problem, the expansion has been extended to include terms of order\n$g^6\\log g$ \\cite{es}-\\cite{avpres}, while at $T=0$ and $\\mu$ much\ngreater than the critical chemical potential $\\mu_c$, the pressure is\nknown up to and including terms of order $g^4$ \\cite{fmcl}. In between\nthese regimes, at $01$ these become\nmore important than the two-loop terms, as $T m_\\rmi{D}^3 \\sim g^{3+x}\\mu \\gg g^{2+2x}\\mu\n\\sim g^2\\mu^2 T^2$. As we shall see, the $T$-dependent\ncontributions from ring diagrams indeed become more important\nthan the 2-loop term $g^2\\mu^2 T^2$ here, though they are not enhanced\nby a relative factor $g^{-(x-1)}$ as suggested by dimensional\nreduction, but instead only by a logarithm.\n\nFor temperatures $T \\lesssim g\\mu$, where dimensional reduction\nis no longer applicable, it becomes important to keep the nonstatic\nparts of the gluon self energy in the ring diagrams.\nAt low momenta and frequencies of the order $g\\mu$, the leading\nterms in the gluon self-energy are given by the\nso-called hard thermal loops (HTL) approximation with the overall\n$m_\\rmi{D}^2$ factor replaced by its zero-temperature value ---\na special case occasionally referred to as hard dense loops (HDL).\nIn the longitudinal gluon propagator, one can observe\nthe usual Debye screening effect at the frequency $\\omega\\ll g\\mu$,\nbut in the transverse propagator the situation\nis more complicated. At strictly zero frequency the\nmagnetostatic HDL propagator is massless, but for\nsmall but nonvanishing frequencies\n$\\omega\\ll q\\lesssim m_\\rmi{D}$ its inverse has the form\n\\begin{eqnarray}\nq^2-\\omega^2+\\Pi^\\rmi{HDL}_\\rmi{T}(\\omega,q)\n&=&q^2-{i\\pi m_\\rmi{D}^2\\04}{\\omega\\0q}+O(\\omega^2).\n\\end{eqnarray}\nThe transverse part of the propagator thus has a pole\nat imaginary $q$ and $|q|=m_\\rmi{mag}(\\omega)$,\nintroducing a new parametrically small dynamical\nscreening mass \\cite{Weldon:1982aq,Kraemmer:2003gd}\n\\begin{eqnarray}\n\\label{mmdyn}\nm_\\rmi{mag}(\\omega)&=&\\left( {\\pi m_\\rmi{D}^2 \\omega \\over 4}\n\\right)^{1\/3},\\qquad \\omega\\ll m_\\rmi{D},\n\\end{eqnarray}\nwhich represents an in-medium version of Lenz's law.\nAs soon as the temperature is small but nonvanishing, the ring\ndiagrams obtain contributions involving the Bose-Einstein\ndistribution function which leads to sensitivity\nto this additional scale.\nIn these contributions, we effectively have\n$m_\\rmi{mag}(\\omega\\sim T) \\sim g^{(2+x)\/3}\\mu$\nfor $T\\sim g^x\\mu$ and $x>1$.\nNote that\nthis is parametrically smaller than $m_\\rmi{D}\\sim g\\mu$, but always larger than\nthe magnetic mass scale of MQCD, $m_\\rmi{mag}(\\omega\\!=\\!0)=\ng^2 T \\sim g^{2+x}\\mu$.\n\nThe resummation of the nonstatic transverse gluon self-energy\ngives rise to terms nonanalytic in the temperature which to lowest\norder in a low-temperature expansion turn out to be of the order\n$g^2 \\mu^2 T^2 \\ln\\,T$. This gives rise to\nso-called anomalous or non-Fermi-liquid behavior in the\nentropy and specific heat at low $T$, because instead of\nthe usual linear behavior in $T$ the entropy then has a $T\\ln\\,T$\nterm which is the hallmark of a breakdown of the Fermi-liquid\npicture (first discussed in the context\nof nonrelativistic QED by Norton, Holstein and Pincus \\cite{Holstein:1973}).\nIndeed, inspection of the dispersion laws of\nfermionic quasiparticles reveals that there is a logarithmic\nsingularity in the group velocity at the therefore no longer\nsharply defined Fermi surface\\footnote{A systematic\ncalculation of the group velocity beyond the leading-log approximation has only\nrecently been carried out in Ref.~\\cite{Gerhold:2005uu}.}.\n\nFor a long time, only the multiplicative coefficient of the $T\\ln\\,T$ term\nin the specific heat was known. It was only rather recently \\cite{Ipp:2003cj}\nthat also the scale under the logarithm was determined\ntogether with the next order terms in the low-temperature ($T\\ll g\\mu$)\nseries which in addition involves fractional\npowers of $T$ due to the cubic root in Eq.~(\\ref{mmdyn}).\nFor the pressure, these ``anomalous'' $T$-dependent contributions\nare contained in an expression,\nwhich was first derived in Ref.~\\cite{Gerhold:2004tb}\nand which we shall label by HDL$^+$,\n\\begin{eqnarray}\\label{PHDL}\n{1\\0N_g}\\delta p^\\rmi{HDL$^+$}\n&=&-{g^2T_F\\048\\pi^2}\\mu^2T^2\n-{1\\02\\pi^3}\\int_0^\\infty dq_0\\, n_b(q_0)\n\\int_0^\\infty dq\\,q^2\\,\\biggl[ 2\\,{\\rm Im}\\, \\ln \\left( q^{2}-q_{0}^{2}+\\Pi^\\rmi{HDL} _\\rmi{T} \\over q^{2}-q_{0}^{2} \\right )\\nonumber\\\\\n&+&{\\rm Im}\\, \\ln \\left( \\frac{q^{2}-q_{0}^{2}+\\Pi^\\rmi{HDL} _\\rmi{L}}{q^{2}-q_{0}^{2}}\\right)\n\\biggr] + O(g^2T^4) + O(g^3 \\mu T^3)+ O(g^4\\mu^2T^2),\n\\end{eqnarray}\nwith $\\delta p$ denoting the temperature-dependent part of the interaction\npressure\n\\begin{eqnarray}\\label{deltaDeltap}\n\\delta p & \\equiv & \\Delta p - \\Delta p|_{T=0}, \\nonumber \\\\\n\\Delta p & \\equiv & p - p_\\rmi{SB}.\n\\end{eqnarray}\nThe expression (\\ref{PHDL}) can be viewed as a minimal\\footnote{As\nopposed to the HTL\/HDL resummation considered in \\cite{ABS,BIR}\nwhich aims at improving the convergence of the perturbative\nseries at high temperature\nby retaining higher-order effects from HTL\/HDL physics\nbeyond what is needed from a perturbative point of\nview. The + in HDL$^+$ and HTL$^+$\nis meant as a reminder that the\ncorresponding quantities are not expressed in terms\nof HTL\/HDL quantities only, but combined with\nunresummed infrared-safe contributions.}\nresummation of HDL diagrams, where the HDL self-energies are only kept in\nthe infrared sensitive part of the ring diagrams involving the distribution function $n_b$, while\ninfrared safe two-loop contributions are treated in an unresummed form.\n\n\\subsubsection{$T$ parametrically smaller than $m_\\rmi{D}$}\n\nWith $g\\ll 1$ and $x>1$ in $T\\sim g^x\\mu$, the temperature is parametrically\nsmaller than the Debye mass $m_\\rmi{D}\\sim g\\mu$ and\nEq.~(\\ref{PHDL}) contains the leading contributions\nto the temperature-dependent parts of the interaction pressure,\nwhich ignoring logarithms are of order $g^2\\mu^2 T^2\\sim g^{2+2x}\\mu^4$,\nwhile the higher-order terms in Eq.~(\\ref{PHDL}) are at least of\norder $g^{4+2x}\\mu^4$.\nThe Freedman-McLerran result for the $T=0$ pressure, Eq.~(\\ref{pFMcL}),\nis accurate to order $g^4\\mu^2$ and its error is of order $g^6\\mu^4$\n(again ignoring logarithms of $g$).\nEq.~(\\ref{PHDL}) thus represents the leading correction to the\nFreedman-McLerran result\nas long as $x<2$ (i.e., $T\\gtrsim g^2\\mu$), whereas\nin quantities such as the entropy density $s=\\6p\/\\6T$ and the various\nspecific heats, where the $T=0$ part of the pressure drops out,\nit is in fact the leading term in the interaction part for all $x\\ge1$.\n\n\\def\\bar g } %{g_{\\rm eff}{\\bar g }\n\\def\\bar\\tau } %{\\tau_{\\rm eff}{\\bar\\tau }\nIn Eq.~(\\ref{PHDL}), $g$ appears only in the\ncombination $\\bar g } %{g_{\\rm eff}^2\\equiv g^2 T_F$, and it is therefore convenient\nto define a reduced temperature variable\n\\begin{equation}\\label{bartau}\n\\bar\\tau } %{\\tau_{\\rm eff}=\\pi T\/(\\bar g } %{g_{\\rm eff}^x \\mu).\n\\end{equation}\nFor $x>1$, the perturbative content of Eq.~(\\ref{PHDL}) is that\ngiven by the low-temperature expansion worked out in\nRefs.~\\cite{Ipp:2003cj,Gerhold:2004tb}. With the above variables, this reads\n\\begin{eqnarray}\\label{PHDLx}\n{1\\0N_g}{\\delta p^\\rmi{HDL$^+$}\\0m_\\rmi{D}^4}\n&=& {\\bar\\tau } %{\\tau_{\\rm eff}^2\\bar g } %{g_{\\rm eff}^{2(x-1)}\\over72}\\left(\\ln\\left({1\\over \\bar\\tau } %{\\tau_{\\rm eff} \\bar g } %{g_{\\rm eff}^{x-1}}\\right)\n+\\ln{4\\over \\pi}\n +\\gamma_E-{6\\over\\pi^2}\\zeta^\\prime(2)-{3\\02}\\right)\\nonumber\\\\\n &-&{2^{2\/3}\\Gamma\\left({8\\over3}\\right)\\zeta\\left({8\\over3}\\right)\\over3\\sqrt{3}\\pi^{7\/3}}\n \\bar\\tau } %{\\tau_{\\rm eff}^{8\/3}\\bar g } %{g_{\\rm eff}^{8(x-1)\/3}\n +8{2^{1\/3}\\Gamma\\left({10\\over3}\\right)\\zeta\\left({10\\over3}\\right)\n \\over9\\sqrt{3}\\pi^{11\/3}}\\bar\\tau } %{\\tau_{\\rm eff}^{10\/3}\\bar g } %{g_{\\rm eff}^{10(x-1)\/3}\\nonumber\\\\\n &+&{2048-256\\pi^2-36\\pi^4+3\\pi^6\\over2160\\pi^2}\\bar\\tau } %{\\tau_{\\rm eff}^4 \\bar g } %{g_{\\rm eff}^{4(x-1)}\n \\left[\\ln\\left({1\\over \\bar\\tau } %{\\tau_{\\rm eff}\\bar g } %{g_{\\rm eff}^{x-1}}\\right)+\\ln\\pi+\\bar c\\, \\right]\n\\nonumber\\\\\n&+&{O}(\\bar g } %{g_{\\rm eff}^{14(x-1)\/3})\n+{O}(g^{2x}),\n\\end{eqnarray}\nwhere $\\bar c\\approx 4.099348\n\\ldots$ is given by a numerical integral\ndefined in Ref.~\\cite{Gerhold:2004tb}.\nThe latter of the error terms in Eq.~(\\ref{PHDLx}) corresponds to the\nleading-order terms to be expected from three- and higher-loop contributions\\footnote{There,\n$g^2$ no longer appears exclusively in combination with $T_F$.}\nproportional to $g^4\\mu^2 T^2$ which are presumably\nenhanced by logarithms of $T$ and $g$. Depending on the value of $x>1$, a finite number of terms in the\nlow-$T$ expansion remain more important than this\n(see Fig.~\\ref{fig:orders} in Sec.~\\ref{sec:summary}).\n\nWhen $x=1$, i.e.\\ $T\\sim g\\mu$, the expansion of Eq.~(\\ref{PHDLx}) clearly breaks\ndown (unless $\\bar\\tau } %{\\tau_{\\rm eff}\\ll 1$) and the HDL-resummed expression of Eq.~(\\ref{PHDL})\ntherefore needs to be evaluated numerically as in Ref.~\\cite{Gerhold:2004tb}.\nThis expression has then the form of $g^4\\mu^4$ times\na function of $T\/(g\\mu)$, and is therefore of the same order as\nthe $g^4$ term of the $T=0$ pressure of Freedman and McLerran, to\nwhich it is to be added. As displayed in Ref.~\\cite{Gerhold:2004tb}\nfor the case of the entropy,\nand as we shall see for the pressure in the plots of Section \\ref{sec:numres}\nof the present paper,\nthe $T$-dependent terms of Eq.~(\\ref{PHDLx}) smoothly\ninterpolate between a dominant $g^2 T^2\\mu^2\\ln\\,T$ behavior at low temperature\nand the terms of order $g^2T^2 \\mu^2$, $g^3 \\mu^3 T$, and $g^4\\mu^4\\ln\\,T$\nof the dimensional reduction pressure which\nshould be the dominant terms at sufficiently high temperatures and which\nremain comparable to $g^4\\mu^4$ as long as the parametric equality $T\\sim g\\mu$ holds.\n\n\n\\subsubsection{$T$ parametrically larger than $m_\\rmi{D}$}\n\n\nWhen $x<1$ in $T\\sim g^x\\mu$, \\textit{i.e.}~$T\\gg g\\mu$,\ndimensional reduction provides the most accurate\nresult available for the QCD pressure. Up to an error of the order of\nthree-loop contributions proportional to $g^4\\mu^2 T^2\\sim g^{4+2x}\\mu^4$,\none can however reproduce its prediction\nby extending the above HDL-resummed calculation\nto include the leading thermal corrections to the gluon self-energy. In practice,\nthis means replacing the HDL approximation by the HTL one and also keeping the order $g^2 T^4$\nterms originating from infrared-safe two-loop contributions to the pressure that were\nomitted in Eq.~(\\ref{PHDL}) because they were of too high order when $x\\ge1$.\nThis possibility was mentioned in Ref.~\\cite{Gerhold:2004tb}, but\nnot considered further because that work concentrated\non the region of $T\\lesssim g\\mu$. For the purposes of the present paper, we however\nwrite down the straightforward extension of Eq.~(\\ref{PHDL}) to the HTL approximation\nin the form\n\\begin{eqnarray}\\label{PHTL}\n{1\\0N_g}\\delta p^\\rmi{HDL$^+$}&=&-{g^2T_F\\048\\pi^2}\\mu^2T^2+{g^2(2C_A-T_F)\\0288}T^4\\nonumber \\\\\n&&-{1\\02\\pi^3}\\int_0^\\infty dq_0\\, n_b(q_0)\n\\int_0^\\infty dq\\,q^2\\,\\biggl[ 2\\,{\\rm Im}\\, \\ln \\left( q^{2}-q_{0}^{2}+\\Pi^\\rmi{HTL} _\\rmi{T} \\over q^{2}-q_{0}^{2} \\right )\\nonumber\\\\&&\\qquad\n+{\\rm Im}\\, \\ln \\left( \\frac{q^{2}-q_{0}^{2}+\\Pi^\\rmi{HTL} _\\rmi{L}}{q^{2}-q_{0}^{2}}\\right)\n\\biggr] + O(g^4\\mu^2T^2).\n\\end{eqnarray}\n\nCombining the above expression with the Freedman-McLerran result of Eq.~(\\ref{pFMcL}) to obtain\n\\begin{eqnarray}\\label{PHTLtot}\n\\Delta p^\\rmi{HDL$^+$} &\\equiv & p^\\rmi{HDL$^+$} - p_\\rmi{SB} \\;\\;\\equiv\\;\\;\n\\Delta p^\\rmi{FMcL}+\\delta p^\\rmi{HDL$^+$},\n\\end{eqnarray}\nwe have an expression for the interaction pressure\nwhose error is of order $g^{{\\rm min}(4+2x,6)}$\nfor all $T\\sim g^x \\mu$. This we shall compare (and thus test) in the following\nwith our new approach which resums the complete one-loop gluon self-energy\n(i.e., not only the leading HTL\/HDL contribution) in ring diagrams.\nNote that the accuracy of (\\ref{PHTLtot}) is at least of order $g^4$\nfor all parametrically small temperatures, excluding only the case of $x=0$,\nwhere $T\\sim \\mu$.\n\n\n\\section{The new approach}\\label{sec:newappr}\n\nIn this Section, we introduce our novel and strictly four-dimensional\ncalculational scheme designed to reproduce the perturbative expansion\nof the QCD pressure up to and including order $g^4$ at all values of\n$\\mu$ and $T$.\nOur guiding principle is that when faced with the necessity to\nsum up graphs with multiple self energy insertions to circumvent\ninfrared problems, we consider the entire self energy\nand not only those parts which are identified as relevant\nin some effective field theory description,\nsuch as the Debye mass in dimensional reduction or\nthe HTL\/HDL self energy in the corresponding resummation schemes.\nBecause we (at present) limit ourselves to order $g^4$ accuracy,\nit will be sufficient to resum only one-loop self-energies in the\ninfrared sensitive graphs, while IR safe diagrams will be treated\nperturbatively, using bare propagators.\nThis will introduce gauge dependence to our results, but only at orders\nbeyond $g^4$ which we will explicitly discard by either considering values\nof $g$ low enough for the higher order terms to be negligible or by performing\nnumerical series expansions up to ${\\mathcal O}(g^4)$.\n\n\nWe begin our treatment with a general diagrammatic analysis where we identify\nthe relevant classes of Feynman graphs that need to be considered. After that,\nwe describe their evaluation and show how adding them together leads to the\nfinal result displayed in Section \\ref{subsec:result}. Many details of the\ncalculations as well as the results of several individual pieces of the result\nare left to be covered in the Appendices.\n\n\\subsection{Identification of the relevant diagrams}\n\n\n\n\\def\\Elmeri(#1,#2,#3){{\\pic{#1(15,15)(15,0,180)%\n #2(15,15)(15,180,360)%\n #3(0,15)(30,15)}}}\n\n\\def\\Petteri(#1,#2,#3,#4,#5,#6){\\pic{#3(15,15)(15,-30,90)%\n #1(15,15)(15,90,210)%\n #2(15,15)(15,210,330) #5(2,7.5)(15,15) #6(15,15)(15,30) #4(15,15)(28,7.5)}}\n\n\\def\\Jalmari(#1,#2,#3,#4,#5,#6){\\picc{#1(15,15)(15,90,270)%\n #2(30,15)(15,-90,90) #4(30,30)(15,30) #3(15,0)(30,0) #5(15,0)(15,30)%\n #6(30,30)(30,0) }}\n\n\\def\\Oskari(#1,#2,#3,#4,#5,#6,#7,#8){\\picc{#1(15,15)(15,90,270)%\n #2(30,15)(15,-90,90) #4(30,30)(15,30) #3(15,0)(30,0) #6(15,0)(15,15)%\n #5(15,15)(15,30) #8(30,30)(30,15) #7(30,15)(30,0) }}\n\n\\def\\Sakari(#1,#2,#3){\\picb{#1(15,15)(15,30,150)%\n#1(15,15)(15,210,330) #2(0,15)(7.5,-90,90) #2(0,15)(7.5,90,270) %\n#3(30,15)(7.5,-90,90) #3(30,15)(7.5,90,270) }}\n\n\\def\\Maisteri(#1,#2){\\picb{#1(15,15)(15,0,150)%\n#1(15,15)(15,210,360) #2(0,15)(7.5,-90,90) #2(0,15)(7.5,90,270) #1(37.5,15)(7.5,0,360) }}\n\n\\def\\Tohtori(#1,#2){\\picb{#1(15,15)(15,0,360)#1(45,15)(15,0,360) }}\n\n\\def\\Pietari(#1){\\pic{#1(15,15)(15,0,360) #1(15,1)(20,40,140)#1(15,29)(20,220,320)}}\n\n\\def\\Ari(#1,#2){\\pic{#1(15,15)(15,0,360) #2(0,15)(27,22) #2(0,15)(27,8)}}\n\n\\def\\Kari(#1){\\pic{#1(15,15)(15,-90,270)}}\n\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lllll}\na)~ \\Kari(\\Agl) & b)~\n\\Kari(\\Agh)\n\\;\\;\\;\\; c)~\n\\Kari(\\Asc)\n\\nonumber \\\\\n\\nonumber \\\\\nd)~ \\Elmeri(\\Agl,\\Agl,\\Lgl) & e)~\n\\Elmeri(\\Agh,\\Agh,\\Lgl)\n\\;\\;\\;\\; f)~\n\\Elmeri(\\Asc,\\Asc,\\Lgl)\n\\;\\;\\;\\; g)~\n\\!\\!\\!\\!\\Tohtori(\\Agl,\\Agl)\n\\nonumber \\\\\n\\nonumber \\\\\nh)~\\Petteri(\\Asc,\\Agl,\\Asc,\\Lsc,\\Lsc,\\Lgl)\n&i)~\\Petteri(\\Asc,\\Asc,\\Asc,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\;\nj)~\n\\Petteri(\\Agh,\\Agh,\\Agh,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\; k)~\n\\Petteri(\\Agh,\\Agl,\\Agh,\\Lgh,\\Lgh,\\Lgl)\n\\;\\;\\;\\; l)~\n\\Jalmari(\\Asc,\\Asc,\\Lsc,\\Lsc,\\Lgl,\\Lgl)\n\\nonumber \\\\\n\\nonumber \\\\\n\\!m)~\n\\!\\Ari(\\Agl,\\Lgl)\n&\\!n)~\n\\Petteri(\\Agl,\\Agl,\\Agl,\\Lgl,\\Lgl,\\Lgl)\n\\;\\;\\;\\; o)~\n\\Pietari(\\Agl)\n\\;\\;\\;\\; p)~\n\\Jalmari(\\Agh,\\Agh,\\Lgh,\\Lgh,\\Lgl,\\Lgl)\n\\nonumber \\\\\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{The one-, two- and three-loop two-gluon-irreducible (2GI) graphs of QCD. The wavy line stands for a gluon, the\ndotted line a ghost and the solid line a quark.}\n\\end{figure}\n\nTo determine the QCD pressure up to and including order $g^4$ on the entire\ndeconfined phase diagram of the theory, our first task is to identify all\nFeynman diagrams that contribute to the partition function at this order.\nThese trivially include the two-{\\em gluon}-irreducible (2GI) diagrams\nup to three-loop order, displayed in Fig.~1 which a straightforward power\ncounting as well as the explicit calculation of\nRef.~\\cite{avpres} confirms as infrared finite for all temperatures and\nchemical potentials.\n\nIn addition to these cases, there are, however, several other classes of IR\nsensitive diagrams that need to be resummed to infinite loop order, as a\npower counting reveals that the dressing of (at least some of) their gluon lines with an arbitrary\nnumber of one loop gluon polarization tensors does not increase their order beyond $g^4$. These\ndiagrams are shown in Fig.~2, where the first set corresponds to the well-known class\nof ring diagrams that leads to the known $g^3$ and $g^4\\ln\\,g$ contributions to the\npressure at high $T$ \\cite{jk,tt} and to the $g^4\\ln\\,g$ term at $T=0$ \\cite{fmcl}.\nAmong others, this class contains the set of all three-loop two particle reducible (2PR)\ngraphs of the theory which are missing from Fig.~1.\n\nAs we shall see (in contradiction to the opposite assertion in Ref.~\\cite{jk}),\nthe resummation of the ring diagrams is, however, not enough to obtain the entire\norder $g^4$ term correctly at nonzero $T$.\nAlthough without resummation starting at orders $g^6$, $g^8$ and $g^6$, respectively,\nthe classes of Fig.~2 b-d, corresponding to\nself-energy insertions in the gluonic two-loop 2GI diagrams 1d and 1g,\nhave the potential to give rise to contributions of order\n${\\mathcal O}(g^4T^2\\mu^2)$ and ${\\mathcal O}(g^4T^4)$ to the pressure.\nWhen $T$ is not parametrically larger than $m_\\rmi{D}$, it turns out that\nonly the class b gives a non-zero contribution at this order, being proportional to\n$g^2T^2m_\\rmi{D}^2$. When $T\\sim g^x\\mu$ with $x>0$, none of the three classes\ncontributes to the pressure to order $g^4\\mu^4$, but in the calculation of the\nlow-temperature entropy and specific heat they have to be taken into account already at\norder $g^4 \\mu^2 T$.\n\nFor any other classes of diagrams apart from those shown in Figs.~1 and 2, it is very straightforward to\nsee that the contributions will be beyond order $g^4$. In particular, if we were to add an additional\ngluon line with some number of self energy insertions into the graphs of Fig.~2 b-d (\\textit{i.e.~}dressing\nthe three-loop 2GI diagrams with self energies), we would notice that\nthe two extra insertions of the coupling constant due to the new vertices (vertex) ensure that these graphs only contribute\nto the pressure at order $g^6$. Similarly, one can see that the inclusion of the two-loop self energy into the ring\ndiagrams only has an effect on the pressure starting at ${\\mathcal O}(g^5)$.\n\n\n\\subsection{The 2GI diagrams}\n\nIn Feynman gauge,\nthe sum of the 2GI diagrams in Fig.~1 at arbitrary $T$ and $\\mu$ can be directly\nextracted from from Ref.~\\cite{avpres} with the result\n\\begin{eqnarray}\np_\\rmi{2GI}\n&=&\\pi^2 d_A T^4 \\Bigg(\\fr{1}{45}\\bigg\\{1+\\fr{d_F}{d_A}\\left(\\fr{7}{4}+30\\bar{\\mu}^2+60\\bar{\\mu}^4\\right)\\bigg\\}\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n&-&\\fr{g^2}{9(4\\pi)^2}\\bigg\\{C_A + \\fr{T_F}{2}(1+12\\bar{\\mu}^2)(5+12\\bar{\\mu}^2)\\bigg\\}\\nonumber \\\\\n&-&\\fr{g^4}{54(4\\pi)^4}\\Bigg\\{\\fr{23C_A^2-C_AT_F(29+360\\bar{\\mu}^2+720\\bar{\\mu}^4) +\n4T_F^2(5+72\\bar{\\mu}^2+144\\bar{\\mu}^4)}{\\epsilon}\\nonumber \\\\\n&+&C_A^2\\(182\\,\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+247+272\\frac{\\zeta'(-1)}{\\zeta(-1)}-90\\frac{\\zeta'(-3)}{\\zeta(-3)}\\)\\nonumber \\\\\n&+&C_AT_F\\bigg(-16\\(5+36\\bar{\\mu}^2+72\\bar{\\mu}^4\\)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}-\n\\fr{217}{5}-56\\frac{\\zeta'(-1)}{\\zeta(-1)}+\\fr{72}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)} \\nonumber \\\\\n&+&24\\(9+4\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2+432\\bar{\\mu}^4+144(1+4\\bar{\\mu}^2)\\aleph(1,z)+3456\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&4T_F^2\\bigg((1+12\\bar{\\mu}^2)\\(4(5+12\\bar{\\mu}^2)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+15+8\\frac{\\zeta'(-1)}{\\zeta(-1)}+36\\bar{\\mu}^2\\)\n\\label{2pi}\\nonumber \\\\\n&+&144(1+4\\bar{\\mu}^2)\\aleph(1,z)\\bigg)\n-9C_FT_F\\bigg(\\fr{35}{2}-16\\frac{\\zeta'(-1)}{\\zeta(-1)}+4\\(59+16\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2\\nonumber \\\\\n&+&664\\bar{\\mu}^4+96\\(i\\bar{\\mu}(1+4\\bar{\\mu}^2)\\aleph(0,z)+2(1+8\\bar{\\mu}^2)\\aleph(1,z)-12i\\bar{\\mu}\\aleph(2,z)\\)\n\\bigg)\\Bigg\\}\\Bigg),\n\\end{eqnarray}\nwhere $\\bar{\\mu}\\equiv\n\\mu\/(2\\pi T)$ and where we have\nrenormalized the gauge coupling using the usual zero-temperature renormalization constant\n$Z_g$. The sum, however, still contains uncanceled UV $1\/\\epsilon$ divergences and depends on the choice of gauge, so that it\nhas no separate physical significance.\n\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lll}\n\\nonumber \\\\\\nn\na)~\\sum_{n=2}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(0,15){2.8}{0.0} \\GCirc(7,27){2.8}{0.0} \\GCirc(7,3){2.8}{0.0}\\GCirc(23,27){2.8}{0.0} \\GCirc(23,3){2.8}{0.0}\n\\GCirc(30,15){2.8}{0.0}}\n\\;\\;\\;\\;b)~\\sum_{n_1,n_2=1}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(4,25){2.8}{0.0} \\GCirc(4,5){2.8}{0.0}\\GCirc(15,30){2.8}{0.0} \\GCirc(15,0){2.8}{0.0}\n\\GCirc(26,25){2.8}{0.0} \\GCirc(26,5){2.8}{0.0}\\Lgl(0,15)(30,15)}\n\\;\\;\\;\\;c)~\\sum_{n_1,n_2,n_3=1}^{\\infty}\\!\\!\\picb{\\Agl(15,15)(15,0,360)\\GCirc(4,25){2.8}{0.0} \\GCirc(4,5){2.8}{0.0}\\GCirc(15,30){2.8}{0.0} \\GCirc(15,0){2.8}{0.0}\n\\GCirc(26,25){2.8}{0.0} \\GCirc(26,5){2.8}{0.0}\\GCirc(9,15){2.8}{0.0}\\GCirc(21,15){2.8}{0.0}\\Lgl(0,15)(30,15)}\\nonumber \\\\\\nn\nd)~\\sum_{n_1,n_2=1}^{\\infty} \\;\\, \\piccb{\\Agl(15,15)(15,0,360)\\GCirc(0,15){2.8}{0.0} \\GCirc(7,27){2.8}{0.0} \\GCirc(7,3){2.8}{0.0}\\GCirc(23,27){2.8}{0.0}\n\\GCirc(23,3){2.8}{0.0}\\GCirc(60,15){2.8}{0.0} \\GCirc(53,27){2.8}{0.0} \\GCirc(53,3){2.8}{0.0}\\GCirc(37,27){2.8}{0.0} \\GCirc(37,3){2.8}{0.0}\\Agl(45,15)(15,0,360)}\n\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{Classes of IR sensitive vacuum graphs contributing to the QCD pressure at order $g^4$.\nThe black dots represent the one-loop gluon polarization tensor given in Fig.~3a\nand the indices $n_i$ stand for the numbers of loop insertions on the respective lines.}\n\\end{figure}\n\n\\subsection{The ring sum}\n\nTo order $g^4$,\nthe ring sum of Fig.~2a can be separated into three pieces\n$p_\\rmi{VV}$, $p_\\rmi{VM}$ and $p_\\rmi{ring}$\naccording to Fig.~3 by decomposing the one-loop gluon polarization tensor\n(see Appendix \\ref{app:Pi}) into its vacuum ($T=\\mu =0$)\nand matter parts. Note that only the matter part has to be resummed, as the vacuum\nparts contribute to order $g^4$ only through the two three-loop diagrams\nin Figs.~3b and c.%\n\\footnote{Take any graph $G$ belonging to the ring sum and having four or more loops\nand at least one vacuum tensor insertion,\nand consider it in the Feynman gauge. Applying Eq.~(\\ref{polarvac}) to it and contracting\nthe Lorentz indices of the vacuum tensor with one of its neighboring gluon\npropagators, we see that $G$ is proportional to $g^2$ times a similar graph with the\nvacuum insertion removed. But this graph is nothing but one of those diagrams\nthat appeared in the original sum which implies that $G$ has to be proportional to\nat least the fifth power of the coupling. \\label{footnoteG}}\nThe evaluation of $p_\\rmi{VV}$ and $p_\\rmi{VM}$ is relatively\nstraightforward, and\nfully analytic expressions for them\nare given in Appendix \\ref{sec:anlI}.\n\n\nTo evaluate the remaining matter ring sum $p_\\rmi{ring}$ we define the standard\nlongitudinal and transverse parts of the vacuum-subtracted polarization tensor at $d=4-2\\epsilon$ by\n\\begin{eqnarray}\n\\Pi_\\rmi{L}(P)\\delta^{ab}&=&\\fr{P^2}{p^2}\\(\\Pi_{00}^{ab}(P)-\\Pi_{00}^{ab}(P)\\mid_{\\rmi{vac}}\\), \\label{pil1}\\\\\n\\Pi_\\rmi{T}(P)\\delta^{ab}&=&\\fr{1}{d-2}\\(\\Pi_{\\mu\\mu}^{ab}(P)-\\Pi_{\\mu\\mu}^{ab}(P)\\mid_{\\rmi{vac}} -\n\\fr{P^2}{p^2}\\(\\Pi_{00}^{ab}(P)-\\Pi_{00}^{ab}(P)\\mid_{\\rmi{vac}}\\)\\),\\quad\\label{pit1}\n\\end{eqnarray}\nwhere we have used the fact \\cite{Heinz:1986kz} that the one-loop\ngluon polarization tensor is transverse\nwith respect to the four-momentum $P$ in the Feynman gauge.\nIn terms of $\\Pi_\\rmi{T}$ and $\\Pi_\\rmi{L}$,\nthe sum of the ring diagrams is then readily performed with the result\n\\begin{eqnarray}\np_\\rmi{ring}&=&-\\fr{d_A}{2}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\ln\\Big[1+\\Pi_\\rmi{L}(P)\/P^2\\Big]-\\Pi_\\rmi{L}(P)\/P^2\\nonumber \\\\\n&+&(d-2)\\(\\,\\ln\\Big[1+\\Pi_\\rmi{T}(P)\/P^2\\Big]-\\,\\Pi_\\rmi{T}(P)\/P^2\\)\\bigg\\}, \\label{logsi}\n\\end{eqnarray}\nwhich is now explicitly IR safe.\n\\begin{figure}[t]\n\\centering\n\\begin{eqnarray} \\nonumber\n\\begin{array}{lll}\na)\\pic{\\Lgl(0,15)(10,15)%\n \\Asc(20,15)(10,0,180) \\Asc(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,15)(10,15)%\n \\Agh(20,15)(10,0,180) \\Agh(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,15)(10,15)%\n \\Agl(20,15)(10,0,180) \\Agl(20,15)(10,180,360) \\Lgl(30,15)(40,15)}\\!+\\pic{\\Lgl(0,10)(20,10)%\n \\Agl(20,20)(10,0,360) \\Lgl(20,10)(40,10)}\\equiv\\, \\pic{\\Lgl(0,15)(10,15)%\n\\Lgl(30,15)(40,15)\\GCirc(20,15){10}{0.8} \\Text(20,15)[c]{V}} \\!\\!+\n\\pic{\\Lgl(0,15)(10,15)\\GCirc(20,15){10}{0.8}\\Text(20,15)[c]{M}%\n \\Lgl(30,15)(40,15)}\\nonumber \\\\\\nn\n b)~ p_\\rmi{VV}\\;\\equiv\\;\\picb{\\Agl(15,15)(15,30,150)%\n\\Agl(15,15)(15,210,330) \\GCirc(0,15){7.5}{0.8}\\Text(0,15)[c]{V} %\n\\GCirc(30,15){7.5}{0.8}\\Text(30,15)[c]{V}}\n\\;\\;\\;\\;\\;\\;\nc)~ p_\\rmi{VM}\\;\\equiv\\;\\picb{\\Agl(15,15)(15,30,150)%\n\\Agl(15,15)(15,210,330) \\GCirc(0,15){7.5}{0.8}\\Text(0,15)[c]{V} %\n\\GCirc(30,15){7.5}{0.8}\\Text(30.2,15)[c]{M}}\n\\;\\;\\;\\;\\;\\;\nd)~ p_\\rmi{ring}\\;\\equiv\\;\\sum_{n=2}^{\\infty}\\picb{\\Agl(15,15)(15,110,160)%\n\\Agl(15,15)(15,200,250) %\n\\GCirc(0,15){5}{0.8} \\GCirc(15,30){5}{0.8} \\GCirc(15,0){5}{0.8} \\Agl(15,15)(15,40,70)\\Agl(15,15)(15,290,320)\n\\DAsc(15,15)(15,-40,40)\\Text(15,0)[c]{{${\\mbox{\\scriptsize{M}}}$}}\n\\Text(15,30)[c]{{${\\mbox{\\scriptsize{M}}}$}}\\Text(0,15)[c]{{${\\mbox{\\scriptsize{M}}}$}} }\n\\end{array}\n\\end{eqnarray}\n\\caption[a]{a) The one-loop gluon polarization tensor $\\Pi_{\\mu\\nu}(P)$ divided\ninto its vacuum ($T=\\mu=0$) and matter (vacuum-subtracted) parts. \\\\\nb) The IR-safe Vac-Vac diagram contributing to the pressure at $\\mathcal{O}(g^4)$. \\\\\nd) The IR-safe Vac-Mat diagram contributing to the pressure at $\\mathcal{O}(g^4)$. \\\\\nd) The remaining 'matter' ring sum.}\n\\end{figure}\n\nAs the functions $\\Pi_\\rmi{L}(P)$ and $\\Pi_\\rmi{T}(P)$ behave at large $P^2$ like\n(see Sec.~B.1.2 of Ref.~\\cite{avthesis})\n\\begin{eqnarray}\n\\Pi_\\rmi{L\/T}(P)&\\xrightarrow[P^2\\rightarrow\\infty]{}&-2(1+\\epsilon)C_Ag^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_Q \\fr{1}{Q^2}\n+\\mathcal{O}(1\/P^2)\n\\;\\;\\,\\equiv\\;\\;\\,\\Pi_\\rmi{UV}+\\mathcal{O}(1\/P^2),\n\\end{eqnarray}\nit is, however, immediately obvious that the sum-integral of Eq.~(\\ref{logsi}) is still\nlogarithmically divergent in the ultraviolet at $T\\neq 0$. To regulate the divergence, we add\nand subtract a term of the form $(1+d-2)(\\Pi_\\rmi{UV})^2\/(2(P^2+m^2)^2)$ from the integrand, with\n$m$ being an arbitrary mass parameter shielding it from IR divergences. By further adding and subtracting\nthe corresponding massless term from the counterterm, we obtain three separate contributions to $p_\\rmi{ring}$:\nan UV and IR finite (at least to order $g^4$ --- see below), $m$-dependent ring sum\n$p_\\rmi{ring}^\\rmi{finite}$, an UV finite, but IR divergent and $m$-dependent $p_\\rmi{ring}^\\rmi{IR}$ and an\nUV and IR divergent and massless $p_\\rmi{ring}^\\rmi{UV}$\n\\begin{eqnarray}\np_\\rmi{ring}&=&p_\\rmi{ring}^\\rmi{finite}+p_\\rmi{ring}^\\rmi{IR}+p_\\rmi{ring}^\\rmi{UV},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\np_\\rmi{ring}^\\rmi{finite}&=&-\\fr{d_A}{2}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\ln\\Big[1+\\Pi_\\rmi{L}(P)\/P^2\\Big]-\n\\Pi_\\rmi{L}(P)\/P^2+C_A^2g^4T^4\/(72(P^2+m^2)^2)\\label{prf}\\nonumber \\\\\n&+&2\\(\\,\\ln\\Big[1+\\Pi_\\rmi{T}(P)\/P^2\\Big]-\\,\\Pi_\\rmi{T}(P)\/P^2+C_A^2g^4T^4\/(72(P^2+m^2)^2)\\)\\bigg\\},\n\\label{eq:pringfinite}\\\\\np_\\rmi{ring}^\\rmi{IR}&\\equiv&\\fr{d_AC_A^2g^4T^4}{48}\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P\\bigg\\{\\fr{1}{(P^2+m^2)^2}-\n\\fr{1}{P^4}\\bigg\\}\\label{pring37}\\\\\np_\\rmi{ring}^\\rmi{UV}&=&\\fr{1}{4}(d-1)d_A\\(\\Pi_\\rmi{UV}\\)^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_P{1\\over P^4}.\n\\label{pUV}\n\\end{eqnarray}\nThe two first terms can be\nevaluated numerically at $\\epsilon=0$ while the third one needs to be regulated with finite $\\epsilon$. It is noteworthy that\none can set $\\epsilon = 0$ even in the formally divergent $p_\\rmi{ring}^\\rmi{IR}$ due to the fact\nthat its IR divergence originates solely from the zeroth Matsubara mode of its second\nterm which vanishes identically in dimensional regularization. The explicit values of $p_\\rmi{ring}^\\rmi{IR}$ and $p_\\rmi{ring}^\\rmi{UV}$\nare given in Appendix \\ref{app:ringsum}, while the numerical evaluation of\n$p_\\rmi{ring}^\\rmi{finite}$ is the subject of Appendix \\ref{app:num}.\n\n\n\n\\subsection{The double and triple sums}\n\nIf the sums in Figs.~2b--d were to start from $n=0$, these multiple resummations would clearly correspond to the dressing of the propagators in three two-loop diagrams\nwith the one-loop gluon polarization tensor. In the present case, we instead define a four-dimensionally transverse\\footnote{Thus decomposable into three-dimensionally\ntransverse and longitudinal parts.} tensor $\\Delta G_{\\mu\\nu}(P)$ by the equations\n\\begin{eqnarray}\n\\Delta G_\\rmi{L}(P)&=&\n\\fr{1}{P^2+\\Pi_\\rmi{L}(P)}-\\fr{1}{P^2}=\n-\\fr{\\Pi_\\rmi{L}(P)}{P^2(P^2+\\Pi_\\rmi{L}(P))},\\\\\n\\Delta G_\\rmi{T}(P)&=&\n\\fr{1}{P^2+\\Pi_\\rmi{T}(P)}-\\fr{1}{P^2}=\n-\\fr{\\Pi_\\rmi{T}(P)}{P^2(P^2+\\Pi_\\rmi{T}(P))},\n\\end{eqnarray}\ncorresponding to the difference of a dressed (with the vacuum-subtracted self energy) and a bare gluon propagator in the Feynman gauge. It is a straightforward\nexercise in combinatorics to show that the symmetry factors of all graphs in Figs.~2b--d equal $1\/4$ independently of $n$ --- a result particularly obvious in\n2PI formalism. To order $g^4$, these three classes of diagrams, denoted here by $p_\\rmi{b}$, $p_\\rmi{c}$ and $p_\\rmi{d}$, can then be written in the forms\n\\begin{eqnarray}\np_\\rmi{b}&=&\\fr{d_AC_A}{4}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\fr{\\Delta G_{\\mu\\mu '}(P)\\Delta G_{\\rho\\rho '}(Q)}{(P+Q)^2}\\nonumber \\\\\n&\\times&\\(g^{\\mu\\nu}(2P+Q)^{\\rho}-g^{\\nu\\rho}(2Q+P)^{\\mu}+g^{\\rho\\mu}(Q-P)^{\\nu}\\)\\nonumber \\\\\n&\\times&\\(g^{\\mu '\\nu}(2P+Q)^{\\rho '}-g^{\\nu\\rho '}(2Q+P)^{\\mu '}+g^{\\rho '\\mu '}(Q-P)^{\\nu}\\) + {\\mathcal O}(g^6),\\label{pb}\\\\\np_\\rmi{c}&=&\\fr{d_AC_A}{12}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\Delta G_{\\mu\\mu '}(P)\\Delta G_{\\rho\\rho '}(Q)\\Delta G_{\\nu\\nu '}(P+Q)\\nonumber \\\\\n&\\times&\\(g^{\\mu\\nu}(2P+Q)^{\\rho}-g^{\\nu\\rho}(2Q+P)^{\\mu}+g^{\\rho\\mu}(Q-P)^{\\nu}\\)\\nonumber \\\\\n&\\times&\\(g^{\\mu '\\nu '}(2P+Q)^{\\rho '}-g^{\\nu '\\rho '}(2Q+P)^{\\mu '}+g^{\\rho '\\mu '}(Q-P)^{\\nu '}\\)+ {\\mathcal O}(g^6),\\\\\np_\\rmi{d}&=&-\\fr{d_A C_A}{2}g^2\\hbox{$\\sum$}\\!\\!\\!\\!\\!\\!\\!\\int_{PQ}\\(\\Delta G_{\\mu\\mu}(P)\\Delta G_{\\nu\\nu}(Q)-\\Delta G_{\\mu\\nu}(P)\\Delta G_{\\mu\\nu}(Q)\\)+ {\\mathcal O}(g^6).\\label{pd}\n\\end{eqnarray}\nAll contributions involving the vacuum piece of the polarization tensor have been discarded as being of order $g^6$, following a reasoning similar to that in\nFootnote~\\ref{footnoteG}.\n\nIt is worthwhile to first perform a power counting analysis to determine at which order the above sum-integrals start to contribute to the pressure.\nIn the regime of dimensional reduction, where $T\\sim g^x \\mu$ with $x<1$, one merely needs to\nconsider the contributions of the zeroth Matsubara modes, as for the others the temperature acts as an infrared cutoff, leading to their values being proportional\nto at least $g^5T\\mu^3\\sim g^{5+x}\\mu^4$. In the region $T\\sim g\\mu$, the Debye mass is, however, of the same order as the temperature,\nimplying that all Matsubara modes give contributions to the pressure parametrically similar in magnitude. Scaling the three momenta in the integrals of\nEqs.~(\\ref{pb})--(\\ref{pd}) by $g\\mu$, one quickly sees that the results for the sum-integrals in this regime can up to ${\\mathcal O}(g^4)$ be written in the form\n$g^4T^2\\mu^2f(T\/(g\\mu))$, where the contributions of the non-static modes to the function $f$ vanish as the parameter $T\/(g\\mu)$\napproaches infinity, while in the opposite limit $T\/(g\\mu)\\rightarrow 0$ the function approaches a constant. As long as we are interested in the value of the pressure\nonly to order $g^4\\mu^4$, these graphs can clearly be altogether\nignored. They will become relevant in the determination of the ${\\mathcal O}(g^4\\mu^2 T)$ contributions to the specific heats, but this is outside the scope of the\npresent work.\n\nFor now, we can concentrate our attention to the regime of dimensional reduction and therefore to the zero Matsubara mode parts of the above sum-integrals.\nHere, we encounter an important simplification which results from the fact that only the longitudinal part of the static gluon polarization tensor has a\nnon-zero zero momentum limit at one-loop order. As the finite momentum corrections to the functions $\\Pi_\\rmi{L}(P)$ and $\\Pi_\\rmi{T}(P)$ clearly correspond\nto higher perturbative orders, we can simply replace\n\\begin{eqnarray}\n\\Delta G_{\\mu\\nu}(P)&\\rightarrow & -\\fr{m_\\rmi{D}^2}{p^2(p^2+m_\\rmi{D}^2)}\\delta_{\\mu 0}\\delta_{\\nu 0} \\label{glimit}\n\\end{eqnarray}\nin the integrals, leading to a dramatic reduction: both $p_\\rmi{c}$ and $p_\\rmi{d}$ then vanish identically. This can, however, be easily understood from the point of\nview of the three-dimensional effective theory EQCD as a demonstration of the\nfact that the $A_0^3$ and $A_0^4$ operators in its Lagrangian\nare not accompanied by couplings of order $g$ and $g^2$, respectively,\nbut only $g^3$ (at nonzero $\\mu$) and $g^4$.\n\nIn contrast to the above, for $p_\\rmi{b}$ one does obtain a non-zero value which has a direct parallel in EQCD in\nthe form of an ${\\mathcal O}(g)$ coupling between one massless $A_i$ and two massive $A_0$ fields and a corresponding two-loop diagram with one $A_i$ and two $A_0$\nlines.\nApplying the limit of Eq.~(\\ref{glimit}) to the sum-integral of Eq.~(\\ref{pb}), it is easy to see that we can reduce the expression of $p_\\rmi{b}$ (to order $g^4$) to the\nsimple form\n\\begin{eqnarray}\np_\\rmi{b}\\;\\,=\\;\\,\\fr{d_AC_A}{4}T^2m_\\rmi{D}^4g^2\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\!\\int\\!\\fr{d^3q}{(2\\pi)^3}\n\\fr{(\\mathbf{p}-\\mathbf{q})^2}{\\mathbf{p}^2(\\mathbf{p}^2+m_\\rmi{D}^2)\\mathbf{q}^2(\\mathbf{q}^2+m_\\rmi{D}^2)(\\mathbf{p}+\\mathbf{q})^2}\n\\end{eqnarray}\nwhich can be solved straightforwardly by introducing three Feynman parameters and using standard formulae for one-loop integrals in three dimensions.\nAfter some work, we get\n\\begin{eqnarray}\np_\\rmi{b}&=&\\fr{d_AC_A}{4}T^2m_\\rmi{D}^4g^2\\bigg\\{\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\fr{1}{\\mathbf{p}^2(\\mathbf{p}^2+m_\\rmi{D}^2)}\\int_0^1\\!\\!\\! dx\\!\\!\\int\\!\\fr{d^3q}{(2\\pi)^3}\n\\fr{1}{\\mathbf{q}^2+2x\\mathbf{q}\\cdot \\mathbf{p}+x\\mathbf{p}^2+ (1-x)m_\\rmi{D}^2}\\nonumber \\\\\n&-&\\(\\int_0^1\\!\\!\\! dx\\!\\!\\int\\!\\fr{d^3p}{(2\\pi)^3}\\fr{1}{(\\mathbf{p}^2+xm_\\rmi{D}^2)^2}\\)^2\\bigg\\}\\nonumber \\\\\n&=&\\fr{d_AC_A}{4}\\fr{T^2m_\\rmi{D}^4g^2}{(4\\pi m_\\rmi{D})^2}\\bigg\\{\\fr{1}{\\pi}\\int_0^1\\!\\!\\! dx\\fr{1}{\\sqrt{x(1-x)}}\\!\\!\\int_0^1\\!\\!\\! dy\\fr{1}{\\sqrt{y}}\\fr{1}{1-y+y\/x}\n\\!\\!\\int_0^1\\!\\!\\! dz\\fr{1}{\\sqrt{z}}-1\\bigg\\}\\nonumber \\\\\n&=&-\\fr{d_AC_A}{4}T^2m_\\rmi{D}^2\\fr{g^2}{(4\\pi)^2}(1-4\\,\\ln\\,2)\\label{pbres}\n\\end{eqnarray}\nwhich we identify as the entire contribution of the classes b-d of Fig.~2 to the QCD pressure up to order $g^4$.\n\n\n\n\n\\subsection{The result}\\label{subsec:result}\n\nWe are now ready to write down our final result for the pressure,\nvalid on the entire deconfined phase of QCD\nand accurate up to and including order $g^4$.\nAssembling all the various pieces, this function reads\n\\begin{eqnarray}\np&=&(p_\\rmi{2GI}+p_\\rmi{VV}+p_\\rmi{VM}+p_\\rmi{ring}^\\rmi{UV}+ p_\\rmi{b}) + (p_\\rmi{ring}^\\rmi{IR} + p_\\rmi{ring}^\\rmi{finite})\n+{\\mathcal O}(g^5T\\mu^3) +\n{\\mathcal O}(g^6\\mu^4)\\label{res1}\\\\\n&\\equiv &p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}+{\\mathcal O}(g^5T\\mu^3) +\n{\\mathcal O}(g^6\\mu^4),\n\\end{eqnarray}\nwhere $p_\\rmi{anl}$ stands for the sum of the first five terms in Eq.~(\\ref{res1}) and\n\\begin{eqnarray}\np_\\rmi{ring}^\\rmi{safe}&\\equiv& p_\\rmi{ring}^\\rmi{finite}+p_\\rmi{ring}^\\rmi{IR}\n\\end{eqnarray}\nis to be evaluated numerically. One should note that in this notation all $m$-dependence\nin contained in the two pieces of $p_\\rmi{ring}^\\rmi{safe}$, naturally canceling\nin their sum. In addition, it is worthwhile to point out that the inclusion of the term $p_b$ in Eq.~(\\ref{res1})\nis inconsistent in the region of $T\\sim g^x\\mu$, $x\\geq 1$ where we have neglected several\ncontributions of the same magnitude. As this term, however, is of order $g^{4+2x}\\mu^4$, \\textit{i.e.~}at least\nof order $g^6\\mu^4$ in the region in question, the inconsistency is in any case beyond the order to which our result is\nindicated to be valid and can therefore be ignored.\n\nCollecting the expressions for all of its parts from above and from\nAppendix \\ref{app:anl}, the function $p_\\rmi{anl}$ reads\n\\begin{eqnarray}\np_\\rmi{anl}&=&\\pi^2d_AT^4\\Bigg(\\fr{1}{45}\\bigg\\{1+\\fr{d_F}{d_A}\\(\\fr{7}{4}+30\\bar{\\mu}^2+60\\bar{\\mu}^4\\)\\bigg\\}\\nonumber \\\\\n&-&\\fr{g^2}{9(4\\pi)^2}\\bigg\\{C_A + \\fr{T_F}{2}(1+12\\bar{\\mu}^2)(5+12\\bar{\\mu}^2)\\bigg\\}\\nonumber \\\\\n&+&\\fr{g^4}{27(4\\pi)^4}\\bigg\\{-C_A^2\\bigg(22\\,\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+63-18\\gamma\n+110\\frac{\\zeta'(-1)}{\\zeta(-1)}-70\\frac{\\zeta'(-3)}{\\zeta(-3)}\n\\bigg)\\nonumber \\\\\n&-&C_AT_F\\bigg(\\(47+792\\bar{\\mu}^2+1584\\bar{\\mu}^4\\)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+\\fr{2391}{20}+4\\frac{\\zeta'(-1)}{\\zeta(-1)}+\\fr{116}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)} \\nonumber \\\\\n&+&6\\(257+88\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)\\bar{\\mu}^2+2220\\bar{\\mu}^4+792(1+4\\bar{\\mu}^2)\\aleph(1,z)+3168\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&T_F^2\\bigg((1+12\\bar{\\mu}^2)\\(4(5+12\\bar{\\mu}^2)\\ln\\fr{\\bar{\\Lambda}}{4\\pi T}+16\\frac{\\zeta'(-1)}{\\zeta(-1)}\\)+\\fr{99}{5} + \\fr{16}{5}\\frac{\\zeta'(-3)}{\\zeta(-3)}\\nonumber \\\\\n&+&312\\bar{\\mu}^2+624\\bar{\\mu}^4+288(1+4\\bar{\\mu}^2)\\aleph(1,z)+1152\\aleph(3,z)\\bigg)\\nonumber \\\\\n&+&\\fr{9}{4}C_FT_F\\bigg(35-32(1-4\\bar{\\mu}^2)\\frac{\\zeta'(-1)}{\\zeta(-1)}+472\\bar{\\mu}^2+1328\\bar{\\mu}^4\\nonumber \\\\\n&+&192\\(i\\bar{\\mu}(1+4\\bar{\\mu}^2)\\aleph(0,z)+2(1+8\\bar{\\mu}^2)\\aleph(1,z)-12i\\bar{\\mu}\\aleph(2,z)\\)\\bigg)\\bigg\\}\\Bigg). \\label{panl}\n\\end{eqnarray}\nNot only have all the UV divergences canceled between the\ndifferent parts of this result,\nonce the renormalization of the gauge coupling $g$ has been taken care of,\nbut this expression actually contains all the (explicit)\nrenormalization scale dependence\nof the pressure up to the present order in perfect agreement with Ref.~\\cite{avpres},\nleaving $p_\\rmi{ring}^\\rmi{safe}$ entirely independent of the parameter $\\bar{\\Lambda}$.\nEq.~(\\ref{panl}) is also valid for all values of $T$ and $\\mu$; the limit for $\\mu\\to0$ is given in\nEq.~\\nr{mutozero} and the limit $T\\rightarrow0$ in Eq.~\\nr{ttozero}. All terms non-analytic in $g^2$\nare contained in the piece $p_\\rmi{ring}^\\rmi{safe}$ awaiting numerical evaluation.\n\n\nIn the following we shall denote our final result for the pressure\n--- which is accurate to order $g^4$ for all values of $T$ and $\\mu$\n(while also containing some incomplete contributions of higher order, to be discarded later) --- by\n\\begin{equation}\np_\\rmi{IV}=p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}.\n\\end{equation}\n\n\n\\subsection{Numerical infrared issues}\\label{sec:numiss}\n\nBefore moving on to examining our result by numerically evaluating the function $p_\\rmi{ring}^\\rmi{finite}$ in Eq.~(\\ref{eq:pringfinite}),\nthere is one more practical issue related to the magnetic\nmass problem \\cite{Linde:1980ts,Kalashnikov:1980tk} that needs to be dealt with. To wit, in the limit $P\\rightarrow 0$, the argument of\n$\\ln(1+\\Pi_\\rmi{T}(P)\/P^2)$ becomes negative, resulting in an unwanted imaginary\ncontribution to the integral which actually renders $p_\\rmi{ring}^\\rmi{finite}$\ninfrared singular beyond order $g^4$. This problem depends on the choice of gauge, but is present in all\ncovariant gauges (as well as the Coulomb gauge).\n\nThe origin of the problem can be traced back to the fact that when dressed with\nthe full one-loop self-energy,\nthe transverse part of the gluon propagator develops a space-like pole.\nFor $p_0=0$ this pole is determined by the equation \\cite{Kalashnikov:1980tk}\n\\begin{eqnarray}\np^2+\\Pi_\\rmi{T}(p_0=0,p)&=&p^2-g^2N_cT{8+(\\xi+1)^2\\064}p\n\\end{eqnarray}\nwhere $\\xi$ is the gauge parameter of covariant gauges.\\footnote{Replacing the ordinary one-loop gluon self-energy by one that includes\nresummation of the Debye mass does not cure the problem, but only produces a different\ngauge-dependent spacelike pole \\cite{Kalashnikov:1982sc,Kraemmer:2003gd}.}\nIt is evidently unphysical and appears only at the non-perturbative magnetic mass scale $g^2T$,\nwhich contributes to the pressure starting at order $g^6T^4$. This suggests that\nwe can in fact eliminate the entire problem by adding by hand\na magnetic mass term\nto the transverse self-energy in Eq.~(\\ref{eq:pringfinite})\n\\begin{eqnarray}\n\\Pi_\\rmi{T}(P) & \\rightarrow & \\Pi_\\rmi{T}(P) + m_\\rmi{mag}^2\n\\end{eqnarray}\nwith (for $\\xi=1$)\n\\begin{equation}\\label{mmagf}\nm_\\rmi{mag}=c_{f}\\frac{3}{32}g^{2}C_A T\n\\end{equation}\nand $c_f\\ge1$,\nwhich only has an effect on the pressure beyond $O(g^4)$.\nIndeed, comparing with the effective magnetic mass for nonzero frequencies,\nEq.~(\\ref{mmdyn}), we find that the magnetic screening behaviour\nis modified only for frequencies $p_0 \\lesssim g^4 T$ when $\\mu\\sim T$\nand even $p_0 \\lesssim g^4 T(T^2\/\\mu^2)$ when $T\\ll \\mu$.\nNote, however, that the introduction of this magnetic mass for the transverse self-energy\nalters the UV behavior of $p_\\rmi{ring}^\\rmi{finite}$, implying that both\n$p_\\rmi{ring}^\\rmi{finite}$ and $p_\\rmi{ring}^\\rmi{IR}$ have to be modified to account\nfor this reorganization. In\n$p_\\rmi{ring}^\\rmi{finite}$, this change is crucial because it renders the result finite,\nbut for the already finite $p_\\rmi{ring}^\\rmi{IR}$ the effects are beyond the order of interest\n(see App.~\\ref{app:num}).\n\nThe numerical evaluation of $p_\\rmi{ring}^\\rmi{finite}$ is performed along the lines of\nRefs.~\\cite{moore,ippreb}, with the sum over Matsubara frequencies being converted to an integration\nin the usual way (see \\textit{e.g.~}Ref.~\\cite{kap}). Contributions containing the bosonic\ndistribution function $n_b$ are best evaluated in Minkowski space, as UV problems are cut off\nby $n_b$, while the other contributions are evaluated in Euclidean space in order to\nnumerically exploit the Euclidean invariance of UV contributions.\nBy varying the parameter $c_f$ in Eq.~(\\ref{mmagf}), we can verify that the effects of this\ninfrared regulator are indeed beyond the order $g^4$ we are aiming at.\nThe remaining part,\nhowever, gives rise to yet another type of unphysical pole,\nwhich (at least in the long-wavelength limit) has been well-known since the earliest\nperturbative calculations in finite-temperature QCD \\cite{Kalashnikov:1979cy}:\nin covariant gauges, the one-loop gluon self-energy, evaluated at the location of the\npoles corresponding to time-like propagating plasmon modes, gives rise\nto a (gauge-dependent) damping constant $\\propto g^2 T$ with negative sign\n(for all gauge parameters $\\xi$, though not in Coulomb or axial gauges\n\\cite{Kajantie:1982xx,Heinz:1986kz}). A consistent systematic calculation of the\nplasmon damping constant to order $g^2T$ requires the use of a HTL-resummed gluon\nself-energy which finally leads to a positive and gauge-independent result\n\\cite{Braaten:1990it,Kraemmer:2003gd}. The corresponding pole is then on the unphysical\nsheet where it would cause no problem for the evaluation of $p_\\rmi{ring}^\\rmi{finite}$.\nWith the bare one-loop gluon self-energy appearing in our integrand we, however, have poles\non the physical sheet, connected to the light-cone by a branch cut, and we need to avoid\nthem by deforming the contour of the numerical integration in\nMinkowski space as sketched in Fig.~\\ref{fig:analyticstructure}.\nThe details of this procedure and the entire numerical calculation are described further in Appendix \\ref{app:num}.\n\n\n\n\n\n\n\\section{Numerical results}\\label{sec:numres}\n\nHaving the result of Eq.~(\\ref{res1}) for the QCD pressure now finally at hand, we move on to examine\nit numerically by evaluating the function\n$p_\\rmi{ring}^\\rmi{safe}$ using methods reviewed in Appendix \\ref{app:num} and adding to it the analytic\npart of Eq.~(\\ref{panl}).\nThe sum total we call $p_\\rmi{IV}$ as a reminder that its\naccuracy is of order $g^4$ for all $T$ and $\\mu$, while it also includes\nincomplete and gauge dependent\nhigher-order contributions. For the most part of the following analysis, we shall\nexplicitly eliminate the latter effects by either considering sufficiently small values of $g$\nor performing numerical expansions of our results in powers of $g$.\n\nWe begin by inspecting\nthe region where the temperature is parametrically larger than the Debye scale and the\nresults of dimensional reduction should be applicable,\nthen continue towards making contact with HDL results\non non-Fermi liquid behavior at $T \\lesssim m_\\rmi{D}$, and\nfinally the Freedman-McLerran result for $T\\to0$.\nIn all plots of the present section we use the values\n$N_\\rmi{c}=3$, $N_\\rmi{f}=2$. Because of the latter, we conveniently\nhave $T_F=1$ and therefore $\\tau=\\bar\\tau$ for the reduced\ntemperature variables introduced in Eqs.~(\\ref{tau}) and (\\ref{bartau}), respectively.\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{perturbative_g4_02a.eps}\n\\caption{Comparison of the $g^4\\log g$ and $g^4$ terms of the numerical computation\nand the analytic DR result, for various values of $\\mu\/T$.\nThe perturbative terms are subtracted up to order $g^3$.\n \\label{fig:X}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{$T$ parametrically larger than $m_\\rmi{D}$}\n\nThe first non-trivial check on our result --- and that of dimensional reduction ---\nis to verify that their predictions for the pressure agree to\norder $g^4$ for all temperatures and chemical potentials that are of\nequal parametric order in $g$.\nThis is particularly important in order\nto clarify that the (entirely correct) statement in the literature about\ndimensional reduction being valid as long as $\\pi T$ is the\nlargest dynamical energy scale does not imply a condition\n$\\pi T > \\mu$, but rather $\\pi T \\gg m_\\rmi{D}$ (or even $\\pi T \\gtrsim m_{\\rmi D}$, as\nwe shall find to be sufficient below).\nTo this end, we start from the most\nwidely studied region of $\\mu=0$ by comparing our numerical result\nto that of the analytic one of dimensional reduction, and then increase the\n${\\mathcal O}(g^0)$ value of $\\mu\/T$ up to $\\mu \\gg \\pi T$\nwhile still having $\\pi T \\gg m_\\rmi{D}\\sim g\\mu$.\n\nThe results of this comparison are shown in Fig.~\\ref{fig:X}, where we plot the order $g^4\\ln\\,g$ and $g^4$\ncontributions of the ring sum of Eq.~(\\ref{prf}) to the pressure together with the same quantity\nextracted from the result of dimensional reduction (obtained by subtracting the analytic part of our result\nfrom the DR one). The agreement is perfect up to the numerical accuracy of our result, and only at larger values of $g$\ncan one see that the agreement is getting slightly worse with increasing $\\mu\/T$. This was, however, to be expected,\nsince there $\\mu\/T$ is coming closer to the value $g^{-1}$, making $m_\\rmi{D}\/T$ of order one which is parametrically\nthe limit of applicability of dimensional reduction. Our conclusion is that the result of dimensional reduction\nis valid at in principle arbitrarily large ${\\mathcal O}(g^0)$ values of $\\mu\/T$, though the expansion in $g$ only\nmakes sense at smaller and smaller values of $g$ as this parameter is increased. This statement will be made more\nconcrete in the following sections.\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{sqrtg_tau02_g9o2_02a.eps}\n\\caption{Comparison of the HTL+ pressure and our numerical result $p_\\rmi{IV}$ in the region of $T=\\tau \\sqrt{g} \\mu$, $\\tau=0.2$, with the known\nperturbative terms from dimensional reduction\n subtracted and the entire quantities divided by $g^{9\/2}$.\nThis plot shows that both the HTL+ result and our numerical one are accurate\nat least up to order $g^{9\/2}$. The renormalization scale has\nbeen varied between $\\mu$ and $4\\mu$. While $p_\\rmi{IV}-p_\\rmi{DR}$\nis scale independent, $p_\\rmi{HTL+}-p_\\rmi{DR}$ has a scale dependence\nat order $g^4\\mu^2 T^2\\sim g^5\\mu^4$.\n \\label{fig:ninehalf}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{sqrtg_tau02_g5_02a.eps}\n\\caption{Same as Fig.~\\ref{fig:ninehalf}, but normalized to $g^5$.\nWhile the HTL+ result is no longer accurate to this order and diverges logarithmically,\nour numerical result still correctly reproduces the\ndimensional reduction result for the pressure at order $g^5\\mu^4$.\n \\label{fig:ninehalfb}}\n\\end{center}\n\\end{figure}\n\nThe logical next step is to test the validity of dimensional reduction at temperatures larger\nthan but now parametrically closer to the Debye scale.\nFor concreteness, we specialize to the case of $T\\sim \\sqrt{g}\\mu$, for which the prediction\nof dimensional reduction is given in Eq.~(\\ref{pDRsqrtg}).\nIn this region, the error in our result is of order $g^{11\/2}\\mu^4$ and that of the minimal\nHTL resummation $g^5 \\mu^4$,\nso that the first one should be able to reproduce the first seven and the latter the\nfirst six terms of the series (\\ref{pDRsqrtg}). And indeed, a numerical\nevaluation of both Eqs.~(\\ref{res1}) and (\\ref{PHTL}) and the subtraction of\nthe first terms of Eq.~(\\ref{pDRsqrtg}) shows the expected results: as\ndisplayed in Fig.~\\ref{fig:ninehalf}, we find perfect agreement in comparing the\ndimensional reduction result with the HTL one (\\ref{PHTLtot}) and\nwith that of our new approach up to order $g^{9\/2}$.\nIn Fig.~\\ref{fig:ninehalfb}, we see that our numerical evaluation\nof $p_\\rmi{IV}$ is accurate enough to even\nverify the $g^5\\mu^4$ term in the dimensional reduction result, while the HTL result starts deviating from the\nDR one at this order.\n\n\n\\subsection{$T$ comparable to $m_\\rmi{D}$}\n\n\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltap_g01_03b.eps}\n\\caption{\nThermal contribution to the interaction pressure $\\delta p$\nas a function of $T\/m_\\rmi{D}^{T=0}$ for fixed chemical potential $\\mu$\nand coupling $g=0.1$.\nFor this value of the coupling, the results of the numerical evaluation\nof $p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}$ and $\\rm{HTL}^{+}$\ncoincide within plot resolution.\nThe result is compared to the dimensional reduction pressure\nat orders $g^2$, $g^3$, $g^4$, and $g^5$ (where the latter\nis included only for completeness, as neither\n$p_\\rmi{IV}$ nor $p_{\\rmi{HTL}^{+}}$\ncontain contributions of order $g^5$). The effect of varying the\nrenormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu \\,...\\, 4\\mu$ is not visible\nfor this value of the coupling. \\label{fig:p01}}\n\\end{center}\n\\end{figure}\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltap_g05_06a.eps}\n\\caption{\nSame as Fig.~\\ref{fig:p01}, but for $g=0.5$.\nThe results of the numerical evaluation\nof $p_\\rmi{anl}+p_\\rmi{ring}^\\rmi{safe}$ and $\\rmi{HTL}^{+}$\ncan now be distinguished due to their different content of higher-order terms.\nWhen two lines of the same type run close to each other, they differ by\nchanging the renormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu \\,...\\, 4\\mu$.\n\\label{fig:p05}}\n\\end{center}\n\\end{figure}\n\n\nIn Figs.~\\ref{fig:p01}--\\ref{fig:p05parts}, we plot the the temperature-dependent contributions to the interaction pressure $\\delta p$\n(see Eq.~(\\ref{deltaDeltap})) for $T\\sim m_\\rmi{D}$ as extracted from our numerical\ncalculation of $p_\\rmi{IV}$ but with no expansions in powers of $g$. We compare this with $p_\\rmi{HTL+}$ as well\nas with the dimensional reduction result expanded to orders $g^2$, $g^3$, $g^4$ and $g^5$ which refer\nto the counting in powers of $g$ when $T\\sim\\mu$.\nFor $T\\sim g\\mu$, however,\nthe terms $g^2 \\mu^2 T^2$, $g^3 \\mu^3 T$, and $g^4\\mu^4\\ln\\,T$ all\nbecome of the same order of magnitude and together constitute the leading temperature-dependent\ncontribution to the interaction pressure $p-p_\\rmi{SB}$ which is contained in the result marked by\nthe dashed line ``$g^4$''. For completeness, we also\ninclude the complete dimensional reduction result to\n(explicit) order $g^5$, but it should be remembered that\nthe term $g^5 T\\mu^3$ is already of the same magnitude as the unknown $g^6\\mu^4$ piece\nwhen $T\\sim g\\mu$, and is therefore both incomplete and beyond our scope which also explains why the $g^4$ curve seems to produce better agreement\nwith our results than the $g^5$ one.\n\nThe different results are normalized to the leading term of the $T$-dependent part of the interaction pressure\nin the dimensional reduction result\n(\\ref{dimredpr5}),\n\\begin{equation}\\label{dpDR2}\n\\delta p^{(2)}_\\rmi{DR} = -g^2\nd_A \\biggl\\{ {T_F\\016\\pi^2} \\mu^2 T^2 + {5T_F+2C_A\\0244} T^4 \\biggr\\}.\n\\end{equation}\nTo understand the structure of these figures, note that the $g^3$ curve goes like $-1+(4\/3\\pi) m_\\rmi{D}\/T$\nfor small $T$ and like $-1+1.07g$ for large $T$. At $T\\ll m_\\rmi{D}$ it, of course, deviates from the\nexact result which is instead dominated by the\nleading $\\fr{2}{9} \\ln \\, T^{-1}$ behavior of the low-temperature series of Eq.~(\\ref{PHDLx}) when\nnormalized by the absolute value of Eq.~\\nr{dpDR2}.\n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=10cm]{deltapparts_g05_04b.eps}\n\\caption{\nSame as Fig.~\\ref{fig:p05}, but with $p_\\rmi{IV}$ separated into $p_\\rmi{anl}$ and $p_\\rmi{ring}^\\rmi{safe}$.\nAs the $g^4$ contribution in $\\delta p_\\rmi{anl}$\nonly amounts to a small correction\n(of effective order $g^6$),\nthe shape of the full pressure curve as a function of $T$\n(beyond the rather trivial $g^2$ contribution)\nis mainly determined by $p_\\rmi{ring}^\\rmi{safe}$.\nThe renormalization scale dependence\n${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}=\\mu\\, ...\\, 4\\mu$ is entirely due to $p_\\rmi{anl}$.\n\\label{fig:p05parts}}\n\\end{center}\n\\end{figure}\n\nFor small values of $g\\sim 0.1$, Fig.~\\ref{fig:p01} shows that the numerical evaluation\nof Eq.~(\\ref{res1}) perfectly agrees with the result of the\nHTL resummation (the two curves lie virtually on top of each other).\nAt this value of $g$, also the complete dimensional reduction result to\n(explicit) order $g^5$ is virtually indistinguishable from the\norder $g^4$ result. The dimensional reduction result\nreproduces the numerical results remarkably well down to temperatures of about\n$0.2\\,m_\\rmi{D}^{T=0}$, but at even lower $T$ severely overestimates the\nlogarithmic growth of\n$\\delta p\/T^2$ as $T\\to 0$.\nThis is to be expected, since, in the limit $T\\to0$,\nthe plasmon term of order\n$g^3\\mu^3 T$ in the pressure is clearly unphysical, as it would\nlead to a nonvanishing entropy at $T=0$; the $g^4 \\mu^4 \\ln\\,T$ term\nof the dimensional reduction result,\nwhile evidently crucial for good agreement down to $T\\approx 0.2 m_\\rmi{D}$,\nwould even lead to a diverging entropy as $T\\to0$.\nThe point at which the dimensional reduction result ceases to be a\ngood approximation for both $p_\\rmi{HTL+}$ and $p_\\rmi{IV}$ seems\nto agree rather well with the value of $T\/m_\\rmi{D}$ where $\\delta p$\nswitches sign.\n\nIn Fig.~\\ref{fig:p05} we consider a larger coupling $g=0.5$, for which we begin to see effects from\nvarying the renormalization scale ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ in our result by a factor of 2 around\nthe central value $2\\mu$,\nexcept in the HTL+ result, where ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ appears only\nin the $T=0$ (Freedman-McLerran) part of the result.\\footnote{\nIn Fig.~\\ref{fig:p05}, the value $g=0.5$ is kept fixed for all ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$,\nwhich means that the $x$-axis does not correspond to a\nrenormalization-group invariant variable. The (explicit)\ndependence of the results on ${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ is here shown\nonly to assess the theoretical error in the numerical comparison between the\ndifferent approaches. Taking into account the implicit\n${\\Lambda_{\\overline {\\mbox{\\tiny MS}} }}$ dependence of $g$, the scale dependence of\nall the results we are comparing is of the order of their error,\nwhich is $O(g^6\\mu^4)$ at $T\\sim g\\mu$.}\nFor small $T\/m_\\rmi{D}^{T=0}$, we find good agreement between the HTL+ result\nand $p_\\rmi{IV}$,\nwith the dimensional reduction result to order $g^4$ lying in between the two\nin the range $T\/m_\\rmi{D}^{T=0}\\approx 0.1\\ldots 10$, but deviating again\nabruptly for $T\/m_\\rmi{D}^{T=0} < 0.2$,\nwhich is where $\\delta p$ changes sign.\nAt this value of the coupling, the complete order $g^5$ result\nof dimensional reduction is still reasonably close to the order $g^4$\nresult. While it is certainly unreliable when $\\delta p>0$, the\norder $g^5$ result suggests that taking into account the next\nhigher orders in $g$ may move the onset of non-Fermi-liquid behavior\nto slightly larger $T\/m_\\rmi{D}$.\n\nFig.~\\ref{fig:p05parts} shows how the final result $\\delta p_\\rmi{IV}$\nis composed of the infrared-safe piece $p_\\rmi{anl}$ and the ring sum\n$p_\\rmi{ring}^\\rmi{safe}$. At parametrically small $T\\sim g\\mu$,\nthe $T$-dependent terms in the interaction part of $p_\\rmi{anl}$\nwhich are of effective order $g^4 \\mu^4$ come just from the terms $g^2 T^2 \\mu^2$,\nso that the shape of $\\delta p$ in the above figures is mainly\ndetermined by $\\delta p_\\rmi{ring}^\\rmi{safe}$ which is seen to\ncoincide with $\\delta p_\\rmi{HTL+}-\\delta p^{(2)}_\\rmi{DR}$ up\nto terms beyond $g^4$ accuracy.\n\nFinally, in Fig.~\\ref{fig:p1} we consider $g=1$ which is roughly the value\nof the QCD coupling at 100 GeV. Here the result for $p_\\rmi{IV}$\nstill follows $p_\\rmi{HTL+}$ for $T0$. As long as $x<1$, $T$ is parametrically\nlarger than the Debye mass $\\sim g\\mu$, and so dimensional reduction\nshould still be applicable.\nHowever, each coefficient of the original series at $x=0$ now\nhas to be expanded in powers of $T\/\\mu\\sim g^x$. The 2-loop\npressure contribution for example yields three different terms for $x>0$: one\nis proportional to $\\mu^4$ and thus is always of order $g^2$,\nanother --- proportional to $\\mu^2 T^2$ --- gives the line $y=2+2x$\nand the third term proportional to $T^4$ produces the line $y=2+4x$.\nStarting with the plasmon term which is of order $g^3$ at $x=0$,\nwe obtain an infinite series of higher-order terms for $x>0$.\nThese arise from the expansion of the third power of the Debye mass parameter in powers of $T\/\\mu$, and, for\nsubsequent terms in the dimensional reduction result, also from the expansion of the special\nfunctions $\\aleph(n,z)$. Because both the Debye mass\nand the $\\aleph$ functions can be expanded\nin even powers of $T\/\\mu$, the lines emanating from\ntheir starting points at $x=0$ come with slopes differing\nby two units. The terms proportional to $g^3$ and $g^5$\nat $x=0$ involve a single overall power of $T$, so the lines\nemanating from these have slopes 1, 3, 5, \\ldots, whereas\nthe term proportional to $g^4$ (or $g^4 \\ln\\,g$) has\nalso $T$-independent parts and thus gives rise\nto lines with slopes 0, 2, 4, \\ldots.\nIn Eq.~(\\ref{pDRsqrtg}) we have seen how this gives rise to a new series\nin $g$ at $x=1\/2$, and Fig.~\\ref{fig:orders} illustrates\nhow the individual terms of order 2, 3, $7\\over2$, 4, $9\\over2$, \\ldots\nare produced from the various coefficients of the expansion at $x=0$.\n\nMoving on to the border of applicability of the dimensional reduction\nresults, $x=1$, we see that all lines converge to points\ncorresponding to an expansion in even powers of $g$ (and\nalso involving $\\ln\\,g$). As noted before, for $x\\ge 1$ the relevant effective\ntheory is the one\ngiven by non-static hard dense loops. Their resummation\nis necessary to obtain the classic Freedman-McLerran (FMcL) result\nto order $g^4$ (again accompanied by a logarithmic term) at $T=0$\nas well as the leading thermal corrections to the interaction\npressure. In a low-$T$ expansion\nthese $T$-dependent terms start with a contribution of order\n$g^2 T^2 \\mu^2 \\ln(T\/g\\mu)$ and then involve fractional powers\n$T^{8\/3}$, $T^{10\/3}$, $T^4\\,\\ln\\, T$, $T^{14\/3}$, \\ldots$\\;$such that the corresponding lines in\nFig.~\\ref{fig:orders} (labeled by the exponent of $T$)\nmeet at $x=1$ and effective order $g^4$.\nAt this point, the leading $T$-dependent contributions are of the\nsame order as the three-loop $T=0$ (FMcL) pressure contribution\nand remain more important than the undetermined four-loop $T=0$ term even for\nparametrically lower temperatures as long as $x<2$ (i.e.\\ $T\\gg g^2 \\mu$).\nFor the entropy and specific heat, for which the zero-temperature contribution to\nthe pressure drops out, these $T$-dependent terms represent\nthe leading interaction contributions down to arbitrarily low temperatures.\nThe $T\\,\\ln\\, T$ behavior of the entropy (as well as of the specific heat) is\ncharacterized by ``anomalous'' non-Fermi-liquid behavior, caused by the only\nweakly (dynamically) screened quasi-static magnetic interactions\nwith an effective frequency-dependent screening mass, displayed in Eq.~(\\ref{mmdyn}).\n\nAs suggested by Fig.~\\ref{fig:orders} and\nshown in detail in the previous section, the HDL-resummed\nthermal pressure contributions responsible for the\nnon-Fermi-liquid behavior at $T\\ll g\\mu$ match smoothly\nto the perturbative effects at $T\\gg g\\mu$ described by\nEQCD. As the temperature is increased, electrostatic\nscreening replaces dynamical magnetic screening\nas the dominant collective phenomenon also in the $T$-dependent\ncontributions.\nFor $T$ parametrically larger than $g\\mu$ (\\textit{i.e.~}$x<1$)\nthe resummation of HDL self energies needs to be trivially extended\nto HTL self energies to avoid accuracy loss.\nWhen added to the zero temperature ${\\mathcal O}(g^4)$ result,\nthis gives an expression that gives the pressure\nfor all temperatures and chemical potentials up to\nan error of order $g^{{\\rm min}(4+2x,6)}$ (or $g^{4+x}$ throughout\nin the case of the entropy, for which the unknown four-loop T=0 pressure\ndrops out).\n\nFrom the ``flow'' of the various perturbative\ncontributions as a function of $x$ in Fig.~\\ref{fig:orders}, one notices that a single expression aiming to be\nvalid both for $x>1$ and $x<1$ needs to keep track of\ncontributions which are perhaps higher-order and irrelevant\nin some region but essential in another.\nThe novel approach we have presented here does so\nby resumming the complete one-loop gluon self-energy in all\nIR sensitive graphs while treating the infrared-safe\n2GI diagrams perturbatively. To the extent that we have worked\nit out, this procedure covers both $x>1$ and $x<1$ with an error of order\n$g^{{\\rm min}(5+x,6)}$ which improves over the HTL\/HDL result in the\nregion $x<1$ by including the contributions of all relevant three-loop\ngraphs. A drawback compared to the HTL\/HDL resummation schemes is however\nthat the resummation of the complete gluon self-energy\nleads to gauge-dependent higher-order contributions whose unphysical\nnature is highlighted by the appearance of spacelike poles in the\nlogarithmic resummation integrand with momenta $\\sim g^2T$ and also\nof an unphysical damping constant (with an incorrect sign) $\\propto g^2 T$.\nFor our expression for the pressure, the effect of these problems is, however,\nonly of the order of the nonperturbative MQCD contributions, \\textit{i.e.}~$g^6$,\nso it has not hindered us from confirming and thus validating the\nresults obtained through dimensional reduction or the HTL\/HDL approach.\n\n\n\n\\section{Conclusions and outlook}\\label{sec:concl}\n\nIn this paper, we have constructed a novel resummation scheme designed to reproduce the weak coupling expansion of the QCD pressure up to order $g^4$\non the entire $\\mu$-$T$ plane. We have used it to provide an independent check of practically all existing perturbative results. In particular,\nwe have performed\nthe first explicit test on the validity of dimensional reduction for values of $\\mu\/T$ far beyond the capability of present-day lattice\ntechniques, thus verifying that dimensionally reduced effective theories provide a solid description of the perturbative physics up to in principle arbitrarily large values\nof $\\mu\/T$ as long as $\\pi T>m_\\rmi{D}$. At temperatures parametrically smaller than the chemical potential, we have on the other hand reproduced numerically all the results of the\nHTL\/HDL resummation schemes, verifying their validity and highlighting the smooth transition taking place in the perturbative expansion of the pressure\nas one moves from the region of dimensional reduction towards the zero-temperature limit.\n\nBased on our numerical results from Section \\ref{sec:numres},\nthe dimensional reduction result for the QCD pressure appears to be provide a remarkably good\napproximation for this quantity down to the point where\nthe $T$-dependent contribution to the interaction pressure, $\\delta p$,\nceases to be negative (cf.\\ Figs.\\ \\ref{fig:p01}ff)\nwhich happens at $T\\approx 0.2 m_\\rmi{D}$. Since the\ndimensional reduction result to order $g^6\\ln\\,g$ combined\nwith optimized choices of the renormalization scale\nhas turned out to agree rather well with\nlattice results, both at zero chemical potential\n\\cite{klry,Blaizot:2003iq} and\nfor $\\mu\\sim T$ \\cite{avpres,Ipp:2003yz}, our present findings in fact suggest a remarkably wide\npractical range of applicability for the dimensional reduction method and results.\n\nProgressing down on the temperature axis to $T\\lesssim 0.2 m_\\rmi{D}$, one eventually has to switch to the nonstatic resummation schemes provided either by\nour new approach or by the calculationally much simpler HTL resummation of Eq.~(\\ref{PHTLtot}).\nAt such low temperatures, the pressure can --- up to but not including order $g^6$ --- be approximated by the Freedman-McLerran result plus positive contributions from the\nStefan-Boltzmann terms as well as the interaction pressure $\\delta p$. The latter of these is the source of the non-Fermi-liquid behavior of the entropy and specific heat.\n\nWhile we believe to have thoroughly clarified the nature of perturbative\nexpansions of the pressure in different regimes of the $\\mu$-$T$ plane,\nour new approach is, as of today, yet to produce results for the pressure beyond what has already been achieved through either dimensional reduction at\n$x<1$, the HTL\/HDL resummation schemes at $x\\geq 1$ or the Freedman-McLerran result at $T=0$. Its present relative error of order\n$g^{{\\rm min}(5+x,6)}$ can in principle be reduced through the inclusion of the two-loop gluon polarization tensor into the resummation of the\nring diagrams and in addition by taking the contributions of non-static modes into account in the multiple sums of Fig.~2.b-d. For the pressure,\nthis would bring the accuracy of our new approach up to the one currently achieved by dimensional reduction calculations (excluding the already known\n${\\mathcal O}(g^6\\ln\\,g \\,T^2(T^2+\\mu^2))$ term), so that the error, up to logarithms, would be uniformly (for all values of $x$) of order $g^6 \\ln\\,g$,\ncorresponding to the line marked ``4-loop $T=0$ pressure'' in Fig.~\\ref{fig:orders}.\nThis would then unify all existing perturbative results for the pressure of QCD, while for the entropy it would moreover lead to genuinely\nnew results. Apart from increasing \\textit{e.g.}~the accuracy of the entropy result at $x=1\/2$ to order $g^{13\/2}$\n(green open dots in Fig.~\\ref{fig:orders})\\footnote{The highest purely perturbatively calculable order at $x=1\/2$\nis $g^{15\/2}$ which would require a calculation of the contributions of order $g^6\\mu^2 T^2$ and $g^7\\mu^3T$ for the pressure.},\nit would push the error in the $T$-dependent part of the pressure up to the line denoted in Fig.~\\ref{fig:orders} by ``4-loop T contribution''\nand thus, for $x>1$, include the so far unknown order $g^4\\mu^2 T$ corrections to the non-Fermi-liquid terms in the entropy and the specific heat.\nConsidering the difficulties caused by the gauge-dependent parts of the gluon self-energy, it seems\nthat such an extension should probably aim at keeping only gauge-independent contributions such as HTL self energies\nin the ring diagrams and treating corrections to those self-energies in a perturbative manner.\nWork towards this goal is currently in progress.\n\n\n\\acknowledgments\n\nWe are grateful to Mikko Laine, York Schr\\\"oder,\nand Larry Yaffe for their helpful\ncomments and suggestions and to Dirk Rischke for discussions\non color superconductivity. This\nwork has been partially supported by the Austrian Science Foundation\nFWF, project no.\\ P16387-N08 and the Academy of Finland, project no. 109720.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Acknowledgments}\n\\noindent \nWe would especially like to thank Jenny List for many detailed\ndiscussions regarding the \\textsc{Ilc} analysis. The work has been supported\nby the Helmholtz Alliance ``Physics at the Terascale'', the \\textsc{Dfg\nSfb\/Tr9} ``Computational Particle Physics'', and the \\textsc{Dfb Sfb\/Tr33} (`The\nDark Universe'). H.K.D. would like to thank the Aspen Center for\nPhysics where part of this work was completed.\n\n\\onecolumngrid\n\\pagebreak\n\\twocolumngrid\n\n\\pagebreak\n\n\\section{Differential Cross Section for $ \\Ppositron \\Pelectron \\rightarrow \\boldsymbol{\n \\chi \\chi} \\Pphoton$}\n\n\n\\subsection{Abbreviations}\n\\label{app:xsectterms}\nWe use the following abbreviations for the final cross section list in\nTable~\\ref{tbl:2t3crosssections}:\n\\begin{align}\n\\intertext{Polarisation prefactors:}\nC_S \\equiv 1+P^+ P^-&, \\quad C_V \\equiv 1-P^+ P^-, \\\\\nC_L \\equiv (1- P^-) ( 1+P^+)&,\\quad C_R \\equiv (1+ P^-) ( 1-P^+). \\nonumber \\\\\n\\intertext{Terms with combined couplings:}\nG_{X \\pm Y} \\equiv g_{X}^2 \\pm g_{Y}^2&,\\quad G_{XY} \\equiv g_{X} g_{Y}. \\\\\n\\intertext{Relativistic velocities:}\n\\beta \\equiv \\sqrt{\\displaystyle 1-\\frac{4 M_\\chi^2}{s}}&,\\quad \\hat{\\beta} \\equiv\n\\sqrt{\\displaystyle 1-\\frac{4 M_\\chi^2}{s(1-x)}}\\;.\n\\end{align}\nKinematical functions:\n\\begin{align}\nF_{x \\theta} &\\equiv \\frac{\\alpha}{\\pi} \\frac{(x-1)^2+1}{x \\sin^2 \\theta}\\;, \\\\\nV_{x \\theta} &\\equiv \\frac{x^2\\cos(2 \\theta) + (3x -8)x + 8 }{4 \\left((x-1)^2 +\n 1\\right)}\\;.\n\\end{align}\n\nWe show terms that arise in the analytical evaluation of the differential photon cross\nsection in $\\Ppositron \\Pelectron \\rightarrow \\chi \\chi \\Pphoton$ but not in\nthe Weizs\\\"acker--Williams approximation in (\\ref{eq:analyticfirst})-(\\ref{eq:analyticlast}). They all vanish in the soft--photon\nlimit $x \\rightarrow 0$.\n\\begin{widetext}\n\\begin{align}\nA_{SF} &= \\frac{ \\left(1 - V_{x \\theta} \\right)}{4 M^2_\\Omega}\n\\frac{\\hat{s}}{1-x} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right] \\label{eq:analyticfirst}\\\\\nA_{SFr} &= \\frac{\\alpha}{8 \\pi}\n\\frac{\\hat{s}}{M_\\Omega^2} \\frac{x}{1-x} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right] \\\\\nA_{FtS} &= \\frac{(1-V_{x \\theta})}{4} \\left[ C_S (\\hat{s} - 4 M_\\chi^2) + \\frac{1}{1 - x} C_S (2 M_\\chi^2 + \\hat{s}) \\right] \\\\\nA_{VF} & = 20 G^2_{lr} C_S (1-V_{x \\theta}) \\frac{x }{1-x} (\\hat{s}^2 + 4\nM_\\chi^2 \\hat{s}-8 M_\\chi^4 )+ \\frac{(g_l^4 C_L + g_t^4 C_R)}{M_\\Omega^2 }\n\\Big[ -\\frac{1}{32}\\frac{x^4 \\sin^2(2 \\theta) \\hat{s} ( 3 \\hat{s}^2 + 26 M_\\chi^2 \\hat{s} - 32 M_\\chi^4)}{(x-1)^2\n ((x-1)^2+1)} \\nonumber \\\\\n& \\ + 6 \\frac{x}{((x-1)^2+1)} \\hat{s} ( \\hat{s}^2 + 7 M_\\chi^2 \\hat{s} - 24\nM_\\chi^4 ) - \\frac{1}{4} (1-V_{x \\theta}) (21 \\hat{s}^3 + 282 M_\\chi^2\n\\hat{s}^2 - 1144 M_\\chi^4 \\hat{s} + 160 M_\\chi^6) \\nonumber \\\\\n& \\ +\\frac{3}{2}\\frac{(1-V_{x \\theta})}{(1-x)} \\hat{s} (\\hat{s}^2-28 M_\\chi^2\n\\hat{s}+ 16 M_\\chi^4) +\\frac{1}{4} \\frac{(1-V_{x \\theta})}{(1-x)^2} \\hat{s} (7\n\\hat{s}^2-126 M_\\chi^2 \\hat{s}+ 32 M_\\chi^4) \\left. +\\frac{(1-V_{x\n \\theta})}{(1-x)^3} \\hat{s} (\\hat{s}^2 +2 M_\\chi^2 \\hat{s}+6 M_\\chi^4)\n\\right] \\\\ \nA_{VFr} & = \\frac{(g_l^4 C_L + g_r^4 C_R)}{M_\\Omega^2 } \\Big[ - \\frac{1}{32} \\frac{x^4 \\sin^2(2 \\theta)}{(x-1)^2\n ((x-1)^2+1)} \\hat{s}( \\hat{s}^2 + 32 M_\\chi^2 \\hat{s} -24 M_\\chi^4 ) + 2 \\frac{x}{((x-1)^2+1)} \\hat{s} ( \\hat{s}^2 + 12 M_\\chi^2 \\hat{s} + 56 M_\\chi^4 ) \\nonumber \\\\\n&\\ - \\frac{1}{4}(1-V_{x \\theta}) (7\\hat{s}^3 + 144 M_\\chi^2 \\hat{s}^2 -168\nM_\\chi^4 \\hat{s}+1280 M_\\chi^6) +\\frac{1}{2}\\frac{(1-V_{x \\theta})}{(1-x)} \\hat{s} (\\hat{s}^2 -48 M_\\chi^2 \\hat{s}+ 56 M_\\chi^4)\\nonumber \\\\\n&\\ +\\frac{1}{4}\\frac{(1-V_{x \\theta})}{(1-x)^2} \\hat{s} (9 \\hat{s}^2 -272\nM_\\chi^2 \\hat{s}+104 M_\\chi^4) \\left. +2 \\frac{(1-V_{x \\theta})}{(1-x)^3}\n \\hat{s} (\\hat{s}^2+2 M_\\chi^2 \\hat{s}+ 6 M_\\chi^4) \\right] \\,.\\label{eq:analyticlast}\n\\end{align}\n\\end{widetext}\n\n\\section{Astrophysical constraints}\n\\label{sec:astro}\n\nAny model which aims to describe dark matter, for example through a\n\\textsc{Wimp}, has to agree with present data. It has to give the correct relic\nabundance, and must be consistent with the bounds from direct and\nindirect detection\nsearches \\cite{Komatsu:2010fb,Aprile:2012nq,Adriani:2008zr}.\n\\allowdisplaybreaks\n\\subsection{The Relic Abundance}\nWe first consider the best measurement of the relic abundance from\n\\textsc{Wmap}-7 \\cite{Komatsu:2010fb},\n\\begin{equation}\n\\Omega^{\\text{DM}} h^2 = 0.1099 \\pm 0.0056\\,.\n\\end{equation}\nWe employ the solution of the model dependent Boltzmann equation obtained\nin \\cite{Beltran:2008xg},\n\\begin{subequations}\n\\begin{align}\n\\Omega^\\text{DM}_0 h^2 \\approx 1.04\\cdot10^9 \\,\\GeV^{-1} \\frac{x_f}\n{m_\\text{Pl} \\sqrt{g_*(x_f)} (a + 3b\/x_f)}\\;. \\label{eqn:Omega0}\n\\end{align}\nHere $m_{\\mathrm{Pl}}$ is the Planck mass. $x_f=M_\\chi\/T_f$ is the\ninverse freeze--out temperature, $T_f$, rescaled by the \\textsc{Wimp} mass,\n$M_\\chi$. It is implicitly given by the equation,\n\\begin{align}\nx_f = \\text{ln} \\left[ c (c+2) \\sqrt{\\frac{45}{8}} \\frac{1}{2 \\pi^3} \n\\frac{g\\ m_\\text{pl} M_\\chi (a + 6b\/x_f)}{\\sqrt{x_f} \\sqrt{g_*(x_f)}} \n\\right]. \\label{eqn:x0}\n\\end{align}\n\\end{subequations}\n$g_*(x_f)$ denotes the relativistic degrees of freedom in equilibrium\nat freeze-out and is given in Ref.~\\cite{Coleman:2003hs}. $a$ and $b$\nare the first two coefficients of the non-relativistic\nexpansion of the thermally averaged annihilation cross section,\n\\begin{equation}\n \\langle\\sigma v \\rangle \\approx a + b v^{2} + O(v^{4}),\n\\end{equation}\nwhere $v$ is the relative velocity of the colliding particles. Here the center-of-mass energy squared is approximated by \\cite{Zheng:2010js,Yu:2011by},\n\\begin{equation}\n s \\approx 4 M_{\\chi}^2+ M_{\\chi}^2 v^2 +3\/4 ~ M_{\\chi}^2 v^4.\n\\end{equation}\n$g$ are the internal degrees of freedom of the \\textsc{Wimp}. $c$ is an order unity parameter which is determined numerically in the solution of the Boltzmann equation and we set this parameter to 0.5.\n\nInstead of testing all the models presented in\nTable~\\ref{tbl:allmodels}, we shall focus on a few exemplary cases.\nFirst, the relic density depends on the possible Standard Model\nparticles, $f$, the \\textsc{Wimp}s can annihilate into $\\chi \\overline{\\chi}\n\\rightarrow f \\overline{f}$. We shall consider two cases for the set of particles $f$:\n(i) all leptons, (ii) all SM fermions.\nSecond, two variants of couplings are tested. In one scenario all SM\nparticles couple via the mediator to the \\textsc{Wimp} with the same strength;\nthis is called \\textit{universal coupling}. In the other they have a\ncoupling proportional to their mass, which we call \\textit{Yukawa-like\ncoupling}. In the cases where we have the same effective operator our\nresults agree with Refs.~\\cite{Zheng:2010js,Yu:2011by}, up to the\nnormalisation (see Appendix \\ref{sec:sigmarelic}).\n\n\nIn order to set constraints, we must determine the total relic\ndensity, which is the sum of the relic density of the particle and the\nanti-particle (if the latter exists). This means the relic density for\na complex particle-pair is two times the density of a real particle.\nIf we consider the \\textsc{Wmap} result as an upper bound on the relic density,\ni.e.\\ allowing for other dark matter, then this corresponds to\na lower bound on the effective coupling of the \\textsc{Wimp} to the SM\nparticles. If we require our \\textsc{Wimp} to be the only dark matter, we shall also obtain an\nupper bound on the effective coupling.\n\nThe strict interpretation that our model only contains a heavy\nmediator and a single \\textsc{Wimp} ensures that there are no\nresonances or co-annihilations. However we also note that in many\nfull theories that contain dark matter, a `co-annihilation' regime can\nexist that can significantly alter the relic density in the\nuniverse. Whilst the co-annihilation mechanism cannot be incorporated\ninto the strict definition of our model, it may actually have no\nobservable effect on the collider based phenomenology. An example of\nsuch a feature could be stau co-annihilation in \\textsc{Susy} that would not\nchange the \\textsc{Ilc} production process of the lightest supersymmetric\nparticle. Another example is that a more complicated model may contain\nresonant annihilations. Both of these examples can significantly\nweaken the relic abundance bounds.\n\n\n\\subsection{Direct Detection}\nWe shall also impose bounds on our operators from the direct detection\nsearches for \\textsc{Wimp} dark matter. The experiments are designed to measure\nthe recoil energy from the scattering between a (dark matter halo)\n\\textsc{Wimp} and the target nucleus. The interactions are difficult to detect\nsince the energy deposited is quite small, 1 to \\unit{100}{\\keV},\n\\cite{Bertone:2004pz}. These experiments give an upper limit for the \ncross section between the dark matter and the nucleus of the\ntarget. One drawback is that in the cases where the \\textsc{Wimp} does not\ncouple to quarks, the coupling can only occur through loop diagrams.\n\nThe direct detection experiments give a much stronger bound on spin\nindependent (SI) interactions than on spin dependent (SD). The reason\nis that in the SI case the interaction with all nucleons add\ncoherently which enhances the corresponding cross section by the\natomic number squared. However, the spins of the nucleons cancel if\nthey are paired. Thus SD interactions are only enhanced for very\nspecial nuclei.\n\nThe SI interactions are scalar or vector interactions in the\n$s$-channel, the axialvector and tensor interactions in the\n$s$-channel give a SD interaction. Note that due to the low\nkinetic energy of the \\textsc{Wimp}s the cross section should be\ncomputed in the non-relativistic limit. In that case the pseudoscalar\ninteraction, $\\overline{\\psi} \\gamma ^5 \\psi$, vanishes.\n\nThe $t\/u$--channel diagrams are cast into a sum of $s$--channel diagrams\nvia the Fierz identities. From this only the SI parts are employed, since any SD\ncontribution is negligibly small.\nTensor interactions occur only via the Fierz identities, since we do\nnot consider fundamental tensor interactions. However, since Fierz\n identities will always give at least one SI contribution, tensor\n terms can be dropped. \n\nFor the SI interactions we shall consider the limits set by the \\textsc{Xenon100} experiment \\cite{Aprile:2012nq}. These are the most recent and set\nthe strictest limits over a broad parameter range. For the SD\ninteractions we consider the \\textsc{Xenon}10 data \\cite{Angle:2008we} since\n\\textsc{Xenon}100 gives no statement on SD interactions. The smaller data \nset along with the physical reasons mentioned above lead to a bound \nthat is $\\sim 10^6$ times weaker than for the SI interactions. The \ncalculations for the \\textsc{Wimp}--nucleus cross sections follow \nRef.~\\cite{Agrawal:2010fh} and for identical models\nwe find the same results. See Appendix \\ref{sec:directdetect} for the\ncomplete list of cross sections.\n\n\n\\subsection{Indirect Detection}\n\nWe also consider the indirect detection searches for dark\nmatter. These are much more model dependent, as the dark matter is\nseen via an agent, for example neutrinos, which could also be produced\nvia other means. Specifically we shall consider the \\textsc{Pamela} experiment\n\\cite{Adriani:2008zr} which measured an excess of positrons. These\ncould potentially originate from dark matter annihilation. To\nimplement this we need to compute the propagation of the produced\npositrons and electrons from the source to the earth. This is\ndescribed by the diffusion--loss equation \\cite{Baltz:1998xv},\n\\begin{equation}\n\\frac{\\partial \\psi}{\\partial t} - \\nabla [K(\\textbf{x},E) \\nabla \n\\psi ] - \\frac{\\partial}{\\partial E} [b(E) \\psi] = q(\\textbf{x},E).\n\\label{diff_loss}\n\\end{equation}\nHere $\\psi(x,E) = \\mathrm{d} n_{e+}\/ \\mathrm{d}E$ is the positron\ndensity per energy. $K(x,E)$ is the diffusion coefficient which\ndescribes the interaction with the galactic magnetic field. $b(E)$\ndenotes the energy loss due to synchrotron emission and inverse\nCompton scattering. $q(x,E)$ is the source term due to dark matter\nannihilation. We note that convection and re--acceleration terms are\nignored as these do not apply to positrons \\cite{Delahaye:2008ua}.\n\nWe use the conventional formalism\n\\cite{Delahaye:2007fr,Perelstein:2010fq} to derive a solution of\nEq.~(\\ref{diff_loss}). It is also possible to use the so-called\nextended formalism that takes the corrections from sources in the free\npropagation zone into account as well as those from the diffusion\nzone. However, this increases the runtime of the calculation\nconsiderably while only giving a small correction that is less than\nthe measurement error. To perform the numerical comparison we use the\ncored isothermal dark matter density profile \\cite{Bahcall:1980fb} and\nthe galactic propagation model M2 \\cite{Delahaye:2007fr}.\n\n\nThe above choices result in the following positron flux,\n\\begin{align}\n\\Phi_{e^+}(E)&=\\frac{\\beta_{e+}}{4 \\pi} \\psi(r_{\\odot},z_{\\odot},E), \\\\\n\\psi(r,z,E) &=\\frac{\\tau_{E}}{\\epsilon^2} \\int^{\\epsilon_{max}}_{\\epsilon} d \\epsilon_S f(\\epsilon_S) I(r,z,\\epsilon,\\epsilon_S), \\\\\nI(r,z,\\epsilon,\\epsilon_S) &= \\sum_i \\sum_n J_0(\\frac{\\alpha_i r}{R}) \\sin{\\frac{n \\pi (z+L)}{2L}} \\nonumber \\\\\n&\\hspace{1.8cm}\\times \\exp{(-\\omega_{i,n} (t-t_S))}\t R_{i,n}, \\\\\n\\omega_{i,n}&= K_0 [ (\\frac{\\alpha_i}{R})^2+ (\\frac{n \\pi}{2L})^2].\n\\end{align}\n\nHere $\\tau_E,~R,~K_0,~L$ are parameters which describe the\nM2 propagation model. They are set to the standard choices \\cite{Delahaye:2007fr,Perelstein:2010fq}\n$\\tau_E = \\unit{\\power{10}{16}}{\\second}$, $R=\\unit{20}{\\kilo \\text{pc}}$ as well as to the M2 propagation model\n$L=\\unit{1}{\\kilo \\text{pc}}$, $K_0 = \\unit{0.00595}{\\kilo \\text{pc$^2$\/Myr}}$, $\\delta=0.55$. $f(\\epsilon)$ is\nthe energy distribution of the positrons from the annihilation and is\ngenerated with \\textsc{Pythia}8 \\cite{Sjostrand:2007gs}.\n$R_{i,n}$ are the coefficients of the Bessel-Fourier expansion of $R(r,z)$,\n\\begin{align}\nR(r,z)&\\equiv \\eta \\langle\\sigma v \\rangle \\left[\\frac{\\rho(r,z)}{M_{\\chi}}\\right]^2, \\\\\n\\rho(r,z) &= \\rho_{\\odot}(\\frac{r_{\\odot}}{r})^{\\gamma}\\left[\\frac{1+(r_{\\odot}\/r_s)^{\\alpha}}{1+(r_{\\odot}\/r)^{\\alpha}}\\right] ^{(\\beta-\\gamma)\/\\alpha}.\n\\end{align}\nHere $\\langle \\sigma v \\rangle$ is the thermally averaged annihilation\ncross section. We include all possible final states, not just those\nresulting in positrons. Furthermore $\\eta= 1\/2$ for real\nparticles and 1\/4 for complex particles. $ r_{\\odot}=\\unit{8.5}{\\kilo \\text{pc}}$ is the\ndistance of the solar system from the galactic center. $\\rho_{\\odot}=\n\\unit{0.3}{\\GeV\\per\\centi\\meter\\cubed}$ is the local dark matter density and $\\alpha=\\beta=2,\n~\\gamma=0, ~ r_S =\\unit{5}{\\kilo \\text{pc}}$ are chosen according to the cored isothermal\ndark matter density distribution \\cite{Delahaye:2007fr,Perelstein:2010fq}.\n\n\\textsc{Pamela} measures the ratio $\\Phi_{e^+}\/(\\Phi_{e^-}+\\Phi_{e^+})$,\nwhere the fluxes, $\\Phi_{e^\\pm}$, contain the flux from dark matter\nannihilation and from any astrophysical background. The background we\ntake is \\cite{Baltz:1998xv},\n\\begin{subequations}\n\\begin{align}\n\\dfrac{d \\Phi_{e^- bg}}{dE} &=\\left(\\frac{0.16 \\epsilon^{-1.1}}{1+11 \\epsilon^{0.9}+3.2 \\epsilon^{2.15}}\\right. \\nonumber \\\\\n&\\hspace{-1.1cm}+\\left.\\frac{0.7 \\epsilon^{0.7}}{1+110 \\epsilon^{1.5}+600 \\epsilon^{2.9}+580 \\epsilon^{4.2}}\\right)\n\\mathrm{GeV}^{-1} \\mathrm{cm}^{-2} \\mathrm{s}^{-1} \\mathrm{sr}^{-1}, \\\\\n\\dfrac{d \\Phi_{e^+ bg}}{dE} &=\\frac{4.5 \\epsilon^{0.7}}{1+650 \\epsilon^{2.3}+1500 \\epsilon^{4.2}} \n\\mathrm{\\GeV}^{-1} \\mathrm{cm}^{-2} \\mathrm{s}^{-1} \\mathrm{sr}^{-1}, \\\\[3mm]\n\\epsilon &\\equiv E\/ \\mathrm{GeV}. \\nonumber\n\\end{align}\n\\end{subequations}\nThe quantity we compare to \\textsc{Pamela} is,\n\\begin{equation}\n\\frac{\\Phi_{e^+}}{\\Phi_{e^+}+\\Phi_{e^-}} = \\frac{\\Phi_{e^+ \\chi} +\\Phi_{e^+ bg}}{\\Phi_{e^+ bg} + \\Phi_{e^+ \\chi} + \\Phi_{e^- \\chi} +\\Phi_{e^- bg}},\n\\end{equation} \nand we note that $\\Phi_{e^+ \\chi }= \\Phi_{e^- \\chi}$. \n\nWe find an upper bound on the annihilation cross section by assuming that all of the excess comes from dark matter. However, it is possible that other background sources contribute and thus we also allow models that produce a flux smaller than the one seen.\n\nWe also note that for dark matter masses above $\\sim$\\unit{1}{\\TeV}, the \\textsc{Fermi--Lat} \\cite{Atwood:2009ez} experiment may provide competitive bounds from inverse Compton scattering \\cite{Cirelli:2009vg,Bernal:2010ip}. However, since we are only interested in models that can be probed at the \\textsc{Ilc} we ignore them here.\n\nThe \\textsc{IceCube} collaboration also sets limits on heavier dark matter masses via annihilations into neutrino final states \\cite{Abbasi:2012ws,IceCube:2011aj}. In addition these bounds may be competitive for spin dependent interactions but we do not consider the limits in this study.\n\n\n\n\n\\section{Cross Sections for Annihilation} \\label{sec:sigmarelic}\n\\allowdisplaybreaks We give the full cross sections for annihilation\nof a pair of dark matter particles with mass $M_\\chi$ into a\npair of Standard Model fermions with mass $m_f$. To find the\nexpansion coefficients in $\\sigma v \\approx a + b v^2$, we perform the\nnon--relativistic approximation $s \\approx 4 M_{\\chi}^2+ M_{\\chi}^2\nv^2 + \\frac{3}{4} M_{\\chi}^2 v^4$ \\cite{Beltran:2008xg}. Note that in order to\nfind the correct result for the $v^2$ term in $\\sigma v$, it is\nnecessary to expand up to order $v^4$ because of the appearance of\n$\\sqrt{s}$ in the cross section formul{\\ae}.\n\nThe total cross section is then given as the sum of the cross\n sections over all allowed final state fermions. This set is\nrestricted both by kinematics ($m_f \\leq M_\\chi$) and by the assumed\nmodel. The latter also determines whether the coupling $G_f$ is\nuniversal or particle--dependent.\n\nWe define the mass ratio $\\xi\\equiv m_{f}\/M_{\\chi}$ and the velocities of both\nparticles $\\beta_X \\equiv \\sqrt{1-4m_X^2\/s}$ to compactify the following expressions.\n\nSome of our effective operators have been analysed before, for example \\cite{Zheng:2010js,\n Yu:2011by}, and we agree with the respective results for the annihilation\n cross sections.\n\\subsection{Scalar \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{SS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{8 \\pi s} \\frac{\\beta_f}{\\beta_\\chi}(s-4 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2}) + \\dfrac{v^{2}}{8}(5 \\xi^{2}-2)\\Big].&\\\\\n\\sigma^{\\text{SS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{8 \\pi } \\frac{\\beta_f}{\\beta_\\chi}, &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\Big[ \\sqrt{1-\\xi^{2}} + \\dfrac{v^{2}}{8}\\dfrac{3 \\xi^2 -2}{\\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{SV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{12 \\pi} \\beta_f \\beta_\\chi (s+2 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi} \\Big[ M_{\\chi}^{2} v^{2}\\sqrt{1-\\xi^{2}}(\\xi^{2}+2)\\Big].&\\\\\n\\sigma^{\\text{SV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{12 \\pi } \\beta_f \\beta_\\chi (s-4 m_{f}^{2}), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{6 \\pi} \\Big[ M_{\\chi}^{2}\nv^{2}(1-\\xi^{2})^{3\/2}\\Big]. &\\\\\n\\sigma^{\\text{SV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi } \\beta_f\\beta_\\chi(s-m_f^2), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}M_{\\chi}^2}{48 \\pi} v^2\\sqrt{1-\\xi^{2}} (4-\\xi^2). &\\\\\n\\sigma^{\\text{SF}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{48 \\pi s}\n\\frac{ \\beta_f}{ \\beta_\\chi} \\Big[ 2 s(4 m_{f}^{2}-2\nM_{\\chi}^{2}+3 M_{\\Omega}^{2} \\mp 6 m_{f} M_{\\Omega}) \\nonumber &\\\\ &\\qquad -8\nm_{f}^{2} \\Big(3( M_{\\Omega} \\mp m_{f})^{2}+M_{\\chi}^{2}\\Big) +s^{2} \\Big], &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} \\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2})(\\xi M_{\\chi} \\mp M_{\\Omega})^{2}. \\nonumber &\\\\*\n&\\qquad + \\dfrac{v^{2}}{24} \\Big((15 \\xi^{2}\n-6) M_{\\Omega}^{2} \\mp 6 \\xi(5 \\xi^{2}-2) M_{\\chi} M_{\\Omega}\\nonumber &\\\\\n&\\qquad +(15 \\xi^{4} -4 \\xi^{2}+4) M_{\\chi}^{2}\\Big)\\Big],&\\\\\n\\sigma^{\\text{SFr}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{2\\pi s} \\frac{ \\beta_f^3}{ \\beta_\\chi} (m_{f} \\mp M_{\\Omega})^{2} &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2}}{\\pi} \\sqrt{1-\\xi^{2}}^3 (\\xi M_{\\chi} \\mp\nM_{\\Omega})^{2} \\nonumber &\\\\ & \\qquad \\times \\Big[ 1\n+ \\dfrac{v^{2}}{8} (5 \\xi^{2}-2)\\Big].\n\\end{flalign}\n\\subsection{Fermion \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{FS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{16 \\pi } \\beta_f \\beta_\\chi(s-4 m_{f}^{2}), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{8 \\pi} v^{2} M_{\\chi}^{2} (1-\\xi^{2})^{3\/2}.&\\\\\n\\sigma^{\\text{FS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{16 \\pi } \\frac{ \\beta_f}{ \\beta_\\chi}s, &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{2 \\pi} \\Big[ \\sqrt{1-\\xi^{2}} + \\dfrac{v^{2}}{8}\\dfrac{\\xi^2 }{\\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{ \\beta_f}{ \\beta_\\chi}(s+2 M_{\\chi}^{2}) (s+2 m_{f}^{2}), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{2 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}}(2+\\xi^{2}) \\nonumber &\\\\ &\\qquad +v^{2}\\dfrac{-4+2 \\xi^{2}+11 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big[s\\Big(s-4(m_{f}^{2}+M_{\\chi}^{2})\\Big) \\nonumber &\\\\\n& \\qquad \\qquad +28 m_{f}^{2} M_{\\chi}^{2}\\Big], &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{2 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}}\\xi^{2} \\nonumber &\\\\\n& \\qquad \\qquad + v^{2}\\dfrac{8-28 \\xi^{2}+23 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(s-m_f^2+M_{\\chi}^2) \\nonumber &\\\\ & \\qquad \\qquad +4 m_f^2 M_{\\chi}^2\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{8 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\\n& \\qquad \\qquad + v^{2}\\dfrac{(2 -\\xi^2 +2 \\xi^4)}{24\n \\sqrt{1-\\xi^{2}}}\\Big]. &\\\\\n\\sigma^{\\text{FVr}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\frac{ \\beta_f}{\n \\beta_\\chi}\\Big((s-4 M_{\\chi}^{2})(s-m_f^2) \\nonumber &\\\\ & \\qquad + 6 m_f^2 M_{\\chi}^2 \\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{4\\pi} \\Big[ \\xi^2\n\\sqrt{1-\\xi^2} \\nonumber &\\\\ & \\qquad + v^{2} \\frac{16-32 \\xi^2 +19 \\xi^4}{24 \\sqrt{1-\\xi^{2}}} \\Big]. &\\\\\n\\sigma^{\\text{FtS}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(s-M_{\\chi}^{2}) \\mp 6 m_{f} M_{\\chi}s \\nonumber &\\\\ &\n\\qquad \\qquad + m_{f}^{2}(16 M_{\\chi}^{2}-s)\\Big), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{8 \\pi} (1 \\mp \\xi)^{2}\n\\Big[ \\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{2 \\pm 16 \\xi+17\n \\xi^{2}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FtSr}}_{\\text{Sc\/Ps}} &= \\frac{G_{f}^{2}}{96 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(5 s^{2}+80 m_{f}^{2} M_{\\chi}^{2} \\nonumber &\\\\ &\n\\qquad -2s(7 m_{f}^{2}+7 M_{\\chi}^{2}\\mp 6 m_{f} M_{\\chi})\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{8 \\pi}(1 \\pm \\xi)^{2} \\Big[\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\ &\\qquad + v^{2} \\dfrac{14\\mp 40 \\xi+29\n \\xi^{2}}{24 \\sqrt{1-\\xi^{2}}}\\Big].&\\\\\n\\sigma^{\\text{FtV}}_{\\text{Vec\/Ax}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(s(4s-7M_{\\chi}^{2}) \\pm 6 m_{f} M_{\\chi}s \\nonumber &\\\\\n& \\qquad \\qquad - m_{f}^{2}(7s-40 M_{\\chi}^{2})\\Big), &\\\\*\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{4 \\pi} \\Big[(3 \\pm 2\n\\xi+\\xi^{2}) \\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{14 \\mp 12\n \\xi -31 \\xi^{2} \\pm 18 \\xi^3+29 \\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtVr}}_{\\text{Vec\/Ax}} &= \\frac{G_{f}^{2}}{12 \\pi s} \\dfrac{\\beta_f}{\\beta_\\chi}\\Big(7 s^{2}+76 m_{f}^{2} M_{\\chi}^{2} \\nonumber &\\\\\n& \\qquad -4s(4 m_{f}^{2}+4 M_{\\chi}^{2} \\pm 3 m_{f} M_{\\chi})\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2}}{2 \\pi} \\Big[(2 \\mp \\xi)^{2}\n\\sqrt{1-\\xi^{2}} \\nonumber &\\\\ & \\qquad + v^{2} \\dfrac{32 \\pm 24 \\xi-64 \\xi^{2} \\mp 36 \\xi^3 +47\\xi^{4}}{24 \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtV}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{48 \\pi s} \\dfrac{\n \\beta_f}{ \\beta_\\chi}\\Big(4 m_f^2 M_{\\chi}^2 + s(s-m_f^2-M_\\chi^2)\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{8 \\pi} \\Big[\n\\sqrt{1-\\xi^{2}} + v^{2}\\dfrac{(2 -\\xi^2 +2 \\xi^4)}{24\n \\sqrt{1-\\xi^{2}}}\\Big]. \\hspace{-2cm} &\\\\\n\\sigma^{\\text{FtVr}}_{\\text{Ch}} &= \\frac{G_{f}^{2}}{24 \\pi s} \\frac{\\beta_f}{\\beta_\\chi}\\Big((s-4 M_{\\chi}^{2})(s-m_f^2) \\nonumber &\\\\ & \\qquad \\qquad + 6 m_f^2 M_{\\chi}^2\\Big), &\\\\\n\\sigma v &\\approx \\frac{G_{f}^{2} M_{\\chi}^{2} }{4\\pi} \\Big[ \\xi^2\n\\sqrt{1-\\xi^2} \\nonumber &\\\\ & \\qquad \\qquad +v^{2} \\frac{16-32 \\xi^2 +19 \\xi^4}{24 \\sqrt{1-\\xi^{2}}}\n\\Big] .\n\\end{flalign}\n\\subsection{Vector \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^{\\text{VS}}_{\\text{Sc}} &= \\frac{G_{f}^{2}}{288 M_{\\chi}^{4} \\pi s}\\frac{ \\beta_f}{ \\beta_\\chi}(s-4 m_{f}^{2}) \\nonumber \\\\\n& \\qquad \\times (12 M_{\\chi}^{4} +s^{2}-4 M_{\\chi}^{2} s), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi}\\sqrt{1-\\xi^{2}} \\Big[ (1-\\xi^{2})\n\\nonumber \\\\ & \\qquad \\qquad + \\dfrac{v^{2}}{24}(2+7 \\xi^2)\\Big].\\\\\n\\sigma^{\\text{VS}}_{\\text{Ps}} &= \\frac{G_{f}^{2}}{288 M_{\\chi}^{4} \\pi } \\frac{ \\beta_f}{ \\beta_\\chi}(12 M_{\\chi}^{4} +s^{2}-4 M_{\\chi}^{2} s), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{12 \\pi}\\sqrt{1-\\xi^{2}} \\Big[ 1 + \\dfrac{v^{2}}{24}\\dfrac{2+\\xi^{2}}{1-\\xi^{2}}\\Big].\\\\\n\\sigma^{\\text{VV}}_{\\text{Vec}} &= \\frac{G_{f}^{2}}{432 \\pi M_{\\chi}^{4} }\n\\beta_f \\beta_\\chi (s+2 m_{f}^{2}) \\nonumber \\\\ & \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{4 \\pi} M_{\\chi}^{2} v^{2}\\sqrt{1-\\xi^{2}}(\\xi^{2}+2).\\\\\n\\sigma^{\\text{VV}}_{\\text{Ax}} &= \\frac{G_{f}^{2}}{432 \\pi M_{\\chi}^{4} }\n\\beta_f \\beta_\\chi (s-4 m_{f}^{2}) \\nonumber \\\\ \n& \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{2 \\pi} M_{\\chi}^{2} v^{2} (1-\\xi^{2})^{3\/2}.\\\\\n\\sigma^{\\text{VV Ch}} &= \\frac{G_{f}^{2}}{864 \\pi M_{\\chi}^{4}} \\beta_f\n \\beta_\\chi (s-m_{f}^{2})\\nonumber \\\\ & \\qquad \\times (s^{2}+20 M_{\\chi}^{2} s +12 M_{\\chi}^{4}), \\\\\n\\sigma v &\\approx \\frac{G_{f}^{2}}{16 \\pi} M_{\\chi}^{2}\nv^{2}\\sqrt{1-\\xi^{2}}(4-\\xi^4). \\\\\n\\sigma^{\\text{VF}}_{\\text{Vec\/Ax}} &= \\frac{G^2}{4320 \\pi M_{\\chi}^4 s } \\frac{\n \\beta_f}{\\beta_\\chi} \\Big[ 8 m_{f}^{4}(-174 M_{\\chi}^{4}+2M_{\\chi}^{2}s+s^{2}) \\nonumber \\\\\n& \\qquad+4 M_{\\chi}^{2}s (s-20 M_{\\Omega}^{2})+s^{2}(10 M_{\\Omega}^{2} + 7s)\\Big) \\nonumber \\\\ \n&\\qquad -2 m_{f}^{2}\\Big(680 M_{\\chi}^{6}+152 M_{\\chi}^{4} (5 M_{\\Omega}^{2}+s) \\nonumber \\\\ \n& \\qquad+s\\Big(40 M_{\\chi}^{6}+2 M_{\\chi}^{4}(70 M_{\\Omega}^{2}-31s) \\nonumber \\\\\n& \\qquad\\pm 240 m_{f}^{3} M_{\\Omega} M_{\\chi}^2 (10 M_{\\chi}^{2}- s^{2}) \\nonumber \\\\\n&\\qquad \\pm 120 m_{f} M_{\\chi}^2 M_{\\Omega} s (M_{\\chi}^{2}-s) \\nonumber \\\\ \n&\\qquad +M_{\\chi}^{2}(76 s^{2}-40 M_{\\Omega}^{2} s )\\nonumber \\\\ & \\qquad+s^{2}(20 M_{\\Omega}^{2}+3s)\\Big) \\Big], \\\\\n\\sigma v &\\approx \\frac{G^2}{36 \\pi }\\sqrt{1-\\xi^2} \\Big[ (1-\\xi^2)\\Big((5\n\\xi^2+4) M_{\\chi}^2 \\nonumber \\\\ & \\qquad \\mp 6 \\xi M_{\\chi} M_{\\Omega}+5 M_{\\Omega}^2\\Big) \\nonumber \\\\\n& \\qquad +\\frac{v^2}{24} \\Big(\\mp 6 \\xi (19 \\xi^2+2) M_{\\chi} M_{\\Omega}\\nonumber\n\\\\ & \\qquad +3 (25 \\xi^2+6) M_{\\Omega}^2\\nonumber \\\\ & \\qquad+(83 \\xi^4+ 136 \\xi^2+156) M_{\\chi}^2\\Big) \\Big].\\\\\n\\sigma^{\\text{VFr}}_{\\text{Vec\/Ax}} &= \\frac{ G_{f}^{2}}{2160 \\pi M_{\\chi}^{4} s} \\frac{ \\beta_f}{ \\beta_\\chi} \\Big[s^4 + 22 m_f^2 M_{\\chi}^2 + 13 M_{\\chi}^4 \\nonumber \\\\ \n& \\qquad -8 s^2\\Big(8 m_f^4 +15 M_{\\Omega}^2 (m_f^2+M_{\\chi}^2) \\nonumber \\\\ \n& \\qquad \\pm 5 m_f M_{\\Omega} (4 m_f^2 + 5 M_{\\chi}^2)\\Big)\\nonumber \\\\\n&\\qquad-32 m_f^2 M_{\\chi}^4 (37 m_f^2 \\pm 50 m_f M_{\\Omega}\\nonumber \\\\\n&\\qquad +70 M_{\\chi}^2+45 M_{\\Omega}^2) \\nonumber \\\\ \n&\\qquad+ 2 s^3 (6 m_f^2 \\pm 20 m_f M_{\\Omega}+16 M_{\\chi}^2+15 M_{\\Omega}^2) \\nonumber \\\\* \n&\\qquad+8 M_{\\chi}^2 s \\Big(24 m_f^4 +15 M_{\\Omega}^2 (4 m_f^2+3M_{\\chi}^2)\n\\nonumber \\\\\n&\\qquad \\pm 50 m_f M_{\\Omega} (2 m_f^2 + M_{\\chi}^2)\\nonumber \\\\\n&\\qquad+119 m_f^2 M_{\\chi}^2+40 M_{\\chi}^4\\Big), \\\\ \n\\sigma v &\\approx \\frac{ G_{f}^{2}}{ 9 \\pi} \\sqrt{1-\\xi^{2}} \\Big[\n(1-\\xi^{2})\\Big(3 M_{\\Omega}^{2} \\pm 2 \\xi M_{\\chi} M_{\\Omega} + \\nonumber \\\\\n&\\qquad (3 \\xi^{2}+4) M_{\\chi}^{2}\\Big) + \\dfrac{v^{2}}{24} \\Big(3(2+7 \\xi^{2}) M_{\\Omega}^{2} \\nonumber \\\\*\n& \\qquad \\pm 6 \\xi (2+\\xi^{2}) M_{\\chi} M_{\\Omega} \\nonumber\\\\\n&\\qquad +(16+30 \\xi^{2}+29 \\xi^{4}) M_{\\chi}^{2}\\Big)\\Big].\n\\end{flalign}\n\\section{Cross Sections for Direct Detection} \\label{sec:directdetect}\nWe now give results for the dark matter--nucleon scattering cross\nsection at zero momentum transfer, $\\sigma^0$, for all defined\nbenchmark models. In a universal scenario, the effective coupling is\nindependent of the quark ($G_q = G$), whereas it grows proportionally\nto the quark mass in a Yukawa-like model ($G_q = G\\ m_q \/ m_e$). We\nuse the following definitions:\n\\begin{align}\n\\frac{f_{p}}{M_P} &\\equiv \\sum_{q=u,d,s} \\hspace{-0.2cm}f_{q}^{p}\\frac{G_q}{m_{q}} + \\frac{2}{27}(1-\\sum_{q=u,d,s}\\hspace{-0.2cm}f_{q}^{p})\\sum_{q=c,b,t}\\hspace{-0.05cm}\\frac{G_{q}}{m_{q}}, \\\\\nd_p &\\equiv \\sum_{q=u,d,s} G_q \\Delta_q^p, \\\\\nb_{p} &\\equiv 2 G_{u} +G_{d}, \\\\\n\\tilde{b}_p &\\equiv b_p M_\\chi + 2 G_u m_u + G_d m_d. \\\\\n\\intertext{with the numerical values for $f_q^p$ and $\\Delta_q^p$ listed in \\cite{fnumbers, deltanumbers}:}\nf_{u}^{p} &= 0.020 \\pm 0.004, \\\\\nf_{d}^{p} &= 0.026 \\pm 0.005, \\\\\nf_{s}^{p} &= 0.118 \\pm 0.062, \\\\ \n\\Delta_{u}^{p} &= -0.427 \\pm 0.013, \\\\\n\\Delta_{d}^{p} &= 0.842\\pm 0.012, \\\\\n\\Delta_{s}^{p} &= -0.085 \\pm 0.018. \\\\\n\\intertext{Furthermore we define the reduced mass of the \\textsc{Wimp} proton system,}\n\\mu &\\equiv \\frac{M_{\\chi} M_{p}}{M_{\\chi}+M_{p}}.\n\\end{align}\nThe cross sections can be evaluated in a nonrelativistic approximation for the\n\\textsc{Wimp} and by using the quark proton form factors listed above. See\ne.g.\\ \\cite{Agrawal:2010fh}. If a model is not listed, its\nscattering cross section\nequals zero, e.g.\\ for pseudoscalar interactions that always vanish in a\nnonrelativistic model. Again, we agree with the respective results in \\cite{Zheng:2010js,\n Yu:2011by} for comparable operators. \n\nCross sections for real final state particles can\neasily be derived from the following list by setting the vector form factors\n$b_p$ and $\\tilde{b}_p$ to zero and rescaling $f_p$ and $d_p$ by a\nfactor of 2.\n\\subsection{Scalar \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{SS Sc.}} &= \\frac{\\mu^{2}}{ 4 \\pi M_{\\chi}^{2}}f_{p}^{2}, &\\\\\n\\sigma^0_{\\text{SV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{SF Sc.}} &= \\frac{\\mu^{2}}{4 \\pi}(+f_{p} + \\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{SF Ps.}} &= \\frac{\\mu^{2}}{4 \\pi}(-f_{p} + \\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{SV Chi.}} &= \\frac{\\mu^{2}}{ 4 \\pi}b_{p}^{2}. &\n\\end{flalign}\n\\subsection{Fermion \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{FS Sc.}} &= \\frac{\\mu^{2}}{\\pi}f_{p}^{2},&\\\\ \n\\sigma^0_{\\text{FV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}, &\\\\ \n\\sigma^0_{\\text{FV Ax.}} &= 3 \\frac{\\mu^{2}}{ \\pi} d_p^{2},&\\\\ \n\\sigma^0_{\\text{FV Chi.}} &= \\frac{\\mu^{2}}{ 16 \\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{FVr Chi.}} &= 3 \\frac{\\mu^{2}}{\\pi} d_p^{2}, &\\\\\n\\sigma^0_{\\text{FtS Sc.}} &= \\frac{\\mu^{2}}{16 \\pi}(b_{p} + f_{p} )^{2},&\\\\ \n\\sigma^0_{\\text{FtS Ps.}} &= \\frac{\\mu^{2}}{16 \\pi}(b_{p} - f_{p} )^{2},&\\\\\n\\sigma^0_{\\text{FtV Vec.}} &= \\frac{\\mu^{2}}{\\pi}(1\/2 \\cdot b_{p} -f_{p})^{2},&\\\\ \n\\sigma^0_{\\text{FtV Ax.}} &= \\frac{\\mu^{2}}{\\pi}(1\/2 \\cdot b_{p} + f_{p})^{2},&\\\\ \n\\sigma^0_{\\text{FtV Chi.}} &= \\frac{\\mu^{2}}{ 16 \\pi}b_{p}^{2}. &\n\\end{flalign}\n\\subsection{Vector \\textsc{Wimp}}\n\\vspace{-1cm}\n\\begin{flalign}\n\\sigma^0_{\\text{VS Sc.}} &= \\frac{\\mu^{2}}{4 \\pi M_{\\chi}^{2}}f_{p}^{2},&\\\\ \n\\sigma^0_{\\text{VF Vec.}} &= \\frac{\\mu^{2}}{4 \\pi}(-f_{p}+\\frac{\\tilde{b}_{p}}{M_\\Omega})^{2},&\\\\ \n\\sigma^0_{\\text{VF Ax.}} &= \\frac{\\mu^{2}}{4 \\pi}(+f_{p}+\\frac{\\tilde{b}_{p}}{M_\\Omega})^{2}, &\\\\\n\\sigma^0_{\\text{VF Chi.}} &= \\frac{\\mu^{2}}{4 \\pi}b_{p}^{2}, &\\\\\n\\sigma^0_{\\text{VV Vec.}} &= \\frac{\\mu^{2}}{\\pi}b_{p}^{2}. & \n\\end{flalign}\n\n\\subsection{Photon Loop} \n\\label{app:PhotonLoop} \n\nIf the \\textsc{Wimp} only couples to leptons, the \\textsc{Wimp}--proton\ninteraction can only happen at the loop level. In that case, a low\nenergy photon that couples to a virtual lepton pair interacts with the\nwhole proton. This only happens for models with s--channel vector\nbilinears $\\bar{\\psi} \\gamma^\\mu \\psi$, i.e.\\ models which\ninclude either a, $b_p$, or a, $\\tilde{b}_p$, term in the low energy\ntree level cross section. Results can therefore be derived as follows,\n\\begin{align}\n\\sigma_0^\\text{Loop} &= \\frac{\\alpha^2_{\\text{em}}}{81 \\pi^2} F^2(q^2)\n\\left. \\sigma_0^\\text{Tree} \\right|_{\\text{reduced}}, \\\\\n\\intertext{where the reduced cross section has to be understood as the tree level\n cross section given above after setting $b_p,\\tilde{b}_p = 1$ and $f_p, d_p\n = 0$. This ensures\n that we only take the vector interaction parts. If the tree level cross section includes\n a $b_p$ term, the loop factor is given as,}\nF(q^2) &\\equiv \\sum_lG_l\\ f(q^2, m_l). \\\\\n\\intertext{For $\\tilde{b}_p$ terms, it reads,}\nF(q^2) &\\equiv \\sum_l \\left(m_l + M_\\chi\\right) G_l\\ f(q^2, m_l). \\\\\n\\intertext{In both cases, the loop function can be evaluated as,}\nf(q^{2},m)&\\equiv\\frac{1}{q^{2}} \\left[ 5 q^{2}+12 m^{2}-6(q^{2}+2\nm^{2})\\beta_q \\text{arcoth}~ \\beta_q \\right. \\nonumber \\\\\n& \\qquad \\qquad \\left. - 3 q^2 \\ln m^2 \/ \\Lambda^2 \\right], \\\\\n\\beta_q &\\equiv \\sqrt{1-4 m^{2} \/ q^{2}}.\n\\end{align}\nWe follow the conservative assumption of a maximum scattering angle to find $q^2 = - 4\n\\mu^2 v^2$ with $\\mu$ describing the reduced mass of the \\textsc{Wimp} nucleus system\nand $v = \\unit{500}{\\kilo\\meter\\per\\second}$ being the typical\nescape velocity of a \\textsc{Wimp} in a dark matter halo. Because of the new\n$q$--dependence of the cross section and the fact that the photon only couples\nto the protons inside the nucleus, the official \\textsc{Xenon} results have to\nbe rescaled according to,\n\\begin{align}\n\\sigma^\\text{Loop} = \\sigma^\\text{Tree} \\left[ \\frac{F(\\tilde{q}^2)}{F(q^2)} \\cdot \\frac{A}{Z}\\right]^2,\n\\end{align}\nwhere $\\tilde{q} = q(M_N = M_P)$ uses the reduced mass $\\mu$ of the \\textsc{Wimp} proton\nsystem instead. This weakens the cross section limits by about a factor\n of 10.\n\n\n\\section{Results}\n\\label{sec:results}\nWe begin by presenting the reach at the \\textsc{Ilc} in terms of the\neffective coupling constant in Sec.~\\ref{sec:ilc_bounds}. We then\ncompare these potential bounds with the couplings\npredicted by the cosmological relic density and the bounds coming from\ndirect and indirect detection experiments. Of course we would also like to discover a dark matter at the \\textsc{Ilc} and the bounds provide an estimate of the potential sensitivity of the collider. \n\n\\subsection{ILC Bounds}\n\\label{sec:ilc_bounds}\nWe determine the \\unit{90}{\\%} exclusion bound for the effective\ncoupling constant in each benchmark model for the best case\nscenario. The integrated luminosity is set to \\unit{500}{\\femto\\reciprocal\\barn} and\nthe systematic polarisation error to $\\Delta P\/P =\n\\unit{0.1}{\\%}$. For each benchmark model we choose the polarisation\nsetting that leads to the best signal to background ratio for the\ncorresponding polarisation behaviour according to\nTables~\\ref{tbl:sigoverbkgestimate} and\n\\ref{tbl:sigoverbkgestimate_1tev}. Results for different polarisation\nsettings can be found by rescaling the bound on the coupling according\nto $G^\\prime = G \\sqrt{r^\\prime \/ r}$ with $r$ denoting the ratio\n$N_\\text{S} \/ \\Delta N_\\text{B}$ given in\nTable~\\ref{tbl:sigoverbkgestimate_1tev}. We choose to present all of\nthe results for an \\textsc{Ilc} with a center of mass energy of\n\\unit{1}{\\TeV} due to the increased range of dark matter masses that\nthis option can probe. In addition, smaller effective couplings can be\nprobed, mainly due to the falling Bhabha background.\n\nIn Fig.~\\ref{img:ilcbounds} we show the derived bounds on the coupling\nconstants for an \\textsc{Ilc} center of mass energy of\n\\unit{1}{\\TeV}. The hashed area denotes the region that either\nviolates the tree level approach with a too large dimensionless\ncoupling constant $g^2 > 4 \\pi$, or by having a too small mediator\nmass $M_\\Omega < \\unit{1}{\\TeV}$, for the effective approach to be\nvalid. Note that the leading order in models with fermionic mediators\nhas a different mass dimension and therefore gives a different\ndefinition for the effective coupling constant $G_\\text{eff}$. If a\nmodel has no separate `pseudoscalar' or `axialvector' results, it is\nidentical to the corresponding `scalar'\/ `vector' line due to\nidentical cross section formulas. For masses away from the threshold,\nthe \\textsc{Ilc} is able to exclude coupling constants down to the order of \\unit{\\power{10}{-7}}{\\GeV \\rpsquared}\nor \\unit{\\power{10}{-4}}{\\reciprocal\\GeV}, depending on the mass\ndimension. This corresponds to a total cross section (for the given\nphase space criteria) of about \\unit{0.3}{\\femto\\barn}. Exceptions\nhowever arise for models with vector dark matter that tend to have\nvery strong exclusion limits for small masses. This is caused by the\n$1\/M_\\chi^4$ dependence in the photon cross section, which leads to\ndivergences for very small vector boson masses. It has been shown\n\\cite{Cornwall:1974km} that only spontaneously broken gauge theories\ncan lead to models with massive vector particles that are not\ndivergent. Therefore, our initial fundamental model cannot be the full\ntheory for all energies. In our effective approach, we restrict the\nenergy to a maximum and in that case one can still receive\nperturbative valid results for mass ranges that do not violate unitary\nbounds. However, the perturbatively allowed mass range cannot be given\nin this model independent approach, since such an analysis needs more\ninformation about the size of the individual couplings and the\nrelation between the mass of the mediator and the dark matter mass\nitself. In summary, a more detailed fundamental theory is needed to\nevaluate the breakdown of perturbation theory in this scenario.\n\n\nWe note that in models with fermionic operators, the sub-leading\norder has a negligible effect, as can be seen from the nearly identical lines\nfor fermionic mediators with different masses. \n\n\\subsection{Combined Results}\nThe combined maximum\nexclusion limits for spin independent DM--proton interaction at \\textsc{Pamela}, \\textsc{Wmap}\nand the \\textsc{Ilc} are shown in\nFigs.~\\ref{img:totalbounds1}-\\ref{img:totalbounds3}. We choose a subset of\nmodels that couple to all Standard Model fermions and give an overview\nof the bounds that we can expect. Other models behave similarly and are\ntherefore not shown again separately. We can give the following statements about\nthe comparison of the \\textsc{Ilc} exclusion bound with the current \\textsc{Xenon} limits:\n\\begin{itemize}\n\\item We have sensitivity to spin independent proton cross sections for, as an example, the\n FV Vector model down to \\unit{\\power{10}{-42}}{\\cm\\rpsquared} or\n equivalently \\unit{\\power{10}{-3}}{\\femto\\barn}, which\n is an improvement of about four orders of magnitude compared to\n current \\textsc{Lep} \\cite{Fox:2011fx} and two orders of magnitude\n compared to current Tevatron \\cite{Bai:2010hh} and \\textsc{Cms}\n \\cite{Chatrchyan:2012pa} results.\n\\item An increased center of mass energy can lead to stronger bounds\n by up to one order of magnitude. It also allows a larger dark matter\n mass range to be probed. \n\\item \\textsc{Ilc} bounds get significantly weakened if the interaction is\n Yukawa--like. At the \\textsc{Ilc} the mediator must couple to\n electrons, which have a suppressed Yukawa coupling. The production cross section\nis thus small, leading to weaker bounds.\n\\item Models with scalar mediators give weaker bounds than models with\n vector interactions. For fermionic dark matter we observe a\n difference of about two orders of magnitude, which is in agreement\n with previously mentioned results from e.g.\\\n \\textsc{Lep}. For scalar and vector dark matter the difference is\n mass--dependent and can increase to up to six orders of magnitude,\n which is due to the different mass dimension of the\n couplings.\n\\item The \\textsc{Wmap} bounds are for many effective models very\n constraining, Figs.\\ref{img:totalbounds1}--\\ref{img:totalbounds4}. However, we would like to point out that these can be\n highly dependent on the full theory whilst not affecting the\n \\textsc{Ilc} or direct detection phenomenology. For example,\n annihilation can occur via some resonance or as in some \\textsc{Susy} models,\n co-annihilation with staus or stops.\n\\end{itemize}\n\nIn Fig.~\\ref{img:totalbounds4} we show some models which allow for\nlepton couplings only. In that case, dark matter can only interact\nwith protons via photons through a fermion loop, \\textit{cf.}\nAppendix.~\\ref{app:PhotonLoop}. The loop factor significantly lowers\nthe cross section and therefore increases the bound in the case of\nvector coupled models. Other models allow quark couplings only at the\ntwo--loop level or theoretically completely forbid them\n\\cite{Fox:2011fx}. In all cases, the \\textsc{Ilc} would give the\nstrongest exclusion bounds for dark matter lepton couplings.\nFor models with fermionic mediators there is an extra subtlety when\ncomparing the bounds. In particular the exclusion limit at the\n\\textsc{Ilc} is mainly given by the leading term in the operator\nexpansion, which is scalar like. Loop couplings can only happen for\nvector currents, which in the case of a fermionic mediator is only\ngiven by the sub-leading order and has an additional factor of\n$1\/M_\\Omega^2$. In that case, when translating any exclusion limits\ninto bounds on the {\\textsc{Wimp}--proton cross section, we need to know\nthe exact mass of the mediator. We show this in\nFig.~\\ref{img:totalbounds4} for the two different chosen suppression\nscales `Low' ($M_{\\Omega} =$\\unit{1}{\\TeV}) and `High' ($M_{\\Omega} =$\\unit{10}{\\TeV}), Table~\\ref{tbl:constraints}.\n\nIn Fig.~\\ref{img:totalbounds5} we show the exclusion limits for\nthe spin--dependent interaction. In our\ncase, only the model with fermionic dark matter, a vector mediator and\nan axial--vector coupling leads to such an interaction. In that case,\nwe compare with data from the previous \\textsc{Xenon} experiment\n(\\textsc{Xenon}10), since no results for the \\textsc{Xenon}100 phase\nwere available when this study was completed. Since in this scenario\ndark matter only couples to a single nucleon on average because of the\nnatural spin anti--alignment in nuclei, the \\textsc{Xenon} bounds are\nnot coherently enhanced by the atomic number and therefore strongly\nlose sensitivity. The \\textsc{Ilc} would also give strongest exclusion\nbounds over the whole accessible mass range here.\n\\onecolumngrid\n\\twocolumngrid\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_SSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SVVectorplotter_protonxsect_yukawa}\n\\caption{Combined \\unit{90}{\\%} exclusion limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of scalar dark matter models.}\n\\label{img:totalbounds1}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_FSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_FtVRightplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FtVRightplotter_protonxsect_yukawa}\n\\caption{Combined \\unit{90}{\\%} exclusion limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of fermionic dark matter models.}\n\\label{img:totalbounds2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_VSScalarplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VSScalarplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_VFVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VFVectorplotter_protonxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_VVVectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_VVVectorplotter_protonxsect_yukawa}\n\\caption{Combined limits on the spin independent dark matter proton cross\n section from \\textsc{Ilc}, \\textsc{Pamela} and \\textsc{Wmap} for a selection of vector dark matter models.}\n\\label{img:totalbounds3}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_chargedleptons_loopxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVVectorplotter_chargedleptons_loopxsect_yukawa}\\\\\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFScalarplotter_loopxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_SFLScalarplotter_loopxsect_yukawa}\n\\vspace{-0.5cm}\n\\caption{Combined limits for a selection of models\n with loop--coupling to leptons only. `Low' corresponds to $M_{\\Omega} =$\\unit{1}{\\TeV} and `High' to $M_{\\Omega} =$\\unit{10}{\\TeV}, Table~\\ref{tbl:constraints}}\n\\label{img:totalbounds4}\n\\vspace{0.5cm}\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVAxialvectorplotter_protonxsect_universal} \\hfill\n\\includegraphics[width=0.45\\textwidth]{withpamela_FVAxialvectorplotter_protonxsect_yukawa}\n\\vspace{-0.5cm}\n\\caption{Combined limits on the spin dependent dark matter proton cross\n section.}\n\\label{img:totalbounds5}\n\\end{figure*}\n\n\\clearpage\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this paper we considered a broad range of effective models for dark\nmatter and investigated the possibility that these models could be\nexplored at the \\textsc{Ilc}. The models considered the possibility\nthat dark matter was a new scalar, fermion or vector particle and\nwould be produced at the \\textsc{Ilc} via a new, heavy intermediate\nstate, the mediator particle. For the mediator we also\n considered spins 0, 1\/2 and 1. We obtained the corresponding\n effective theories by integrating out the mediator field.\n\n To be able to compare the reach of the \\textsc{Ilc} with the other\n experimental searches, certain assumptions have to be made on how the\n mediator and dark matter couples to the Standard Model particles. We\n assume in all models that interactions only occur with the Standard\n Model fermions but the relative strength to different particles is\n varied. In the simplest variant we choose that the coupling is equal\n between all the Standard Model states. Another choice is that the\n interaction scales with the mass of the interacting Standard Model\n fermion, a `Yukawa-like' interaction. The last choice we make is the\n most optimistic for \\textsc{Ilc} phenomenology with only the Standard Model\n leptons interacting with the heavy mediator.\n\\balance\n Since the produced dark matter particles will be invisible to the\n \\textsc{Ilc} detectors, we require a radiated photon to be emitted\n from the initial state that will recoil against missing\n momentum. This topology provides a distinctive signal with which to\n discover dark matter. For the \\textsc{Ilc} study, we included the\n dominant backgrounds and most important detector effects. In addition\n we considered the possibility of using polarised initial states to\n reduce backgrounds and improve the signal strength.\n\n The effective theories that we consider provide an efficient way to\n compare the reach of the \\textsc{Ilc} with other methods to discover\n dark matter. Firstly, we consider the dark matter annihilation cross\n section required for the relic density observed by \\textsc{Wmap}. We\n also look at the direct detection bounds at \\textsc{Xenon} by\n calculating the dark matter-nucleon scattering cross section. In\n addition, we include bounds from dark matter annihilation to\n positrons from the \\textsc{Pamela} experiment.\n\nIn terms of the effective dark matter model, we found that the \\textsc{Ilc}\nshould be able to probe couplings \\unit{\\power{10}{-7}}{\\GeV \\rpsquared}, or\n\\unit{\\power{10}{-4}}{\\reciprocal\\GeV} depending on the mass dimension of the theory. In models that contain vector dark matter, the \\textsc{Ilc} may be able to probe even weaker couplings in the case of low dark matter mass.\n\nTo compare with astrophysical bounds, we found that the\n\\textsc{Ilc} reach is strongly dependent on the exact dark matter\nmodel. If we assume that dark matter is relatively heavy ($>$\n\\unit{100}{\\GeV}) and interacts with a Standard Model particle in\nproportion to its mass, then the \\textsc{Ilc} is\nuncompetitive. However, in the case that dark matter is relatively\nlight ($<$ \\unit{10}{\\GeV}) then the bounds from the \\textsc{Ilc} are\ncompetitive with astrophysical bounds in many models. In addition, if\ndark matter happens to only interact with the Standard Model\nleptons then the \\textsc{Ilc} offers a unique possibility to discover\ndark matter. For this reason, an \\textsc{Ilc} search is complementary\nto those done at the \\textsc{Lhc} thanks to the different initial\nstate.\n\n\n\\section{Dark matter search at the \\textsc{Ilc}}\n\\label{sec:ilc}\n\\subsection{Radiative Production of Dark Matter}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\columnwidth]{chichigamma_1.pdf} \\hfill\n\\includegraphics[width=0.49\\columnwidth]{chichigamma_2.pdf}\n\\caption{Diagrams for radiative pair production of dark\n matter. Terms in which the heavy mediator can emit a photon are neglected.}\n\\label{fig:signalprocess_feynmandiagrams}\n\\end{figure}\n\nFor the \\textsc{Ilc} search, we look at the process $\\Ppositron\n\\Pelectron \\rightarrow \\chi \\chi\\Pphoton$ with a hard photon being the\nonly detected particle in the final state, Fig.~\\ref{fig:signalprocess_feynmandiagrams}. We determine the polarized\ndifferential cross section for this process with respect to the\nrelative photon energy $x \\equiv 2 E_\\gamma \/ \\sqrt{s}$ and its polar\nangle $\\theta$ by integrating over the full phase space of the\nfinal state dark matter particles. The results for this\ncalculation are given in Table~\\ref{tbl:2t3crosssections}, with\nfurther explanation of the abbreviations used given in\nAppendix~\\ref{app:xsectterms}. Previous \\textsc{Ilc} studies,\ne.g.\\ \\cite{Bartels:2012ex, Birkedal:2004xn}, have used the\nWeizs\\\"acker--Williams approximation for soft photons. This formula\nrelates the differential photon cross section to the total pair\nproduction cross section $\\Ppositron\\Pelectron \\rightarrow \\chi \\chi$\nwith a reduced center of mass energy $s \\rightarrow \\hat{s} \\equiv\ns(1-x)$ and multiplied by the kinematical function $F_{x \\theta}$,\n\\begin{align}\n\\frac{\\mathrm{d} \\sigma \\left[\\Ppositron \\Pelectron \\rightarrow \\bar{\\chi} \\chi\n \\gamma \\right] }{\\mathrm{d} x \\ \\mathrm{d}\n \\cos \\theta_\\gamma} \\approx F_{x \\theta}\\ \n \\hat{\\sigma} \\left[\\Ppositron \\Pelectron \\rightarrow \\bar{\\chi} \\chi\n \\right].\n\\label{weiz-will}\n\\end{align}\n\\begin{table*}\n\\begin{tabular}{r@{\\quad}l}\n\\hline\nModel & $ \\displaystyle \\frac{ \\mathrm{d}\\sigma}{ \\mathrm{d} x \\\n \\mathrm{d}\\cos \\theta} $\\\\ [1.5ex]\n\\hline\n\\hline\nSS & $ \\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{32 \\pi M_\\Omega^4}\nG_{s+p} g_\\chi^2 C_s $\\\\[1.5ex]\nSF & $\\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}}{32 \\pi M_\\Omega^2} \n\\left[ G_{s-p}^2 C_s + \\frac{\\hat{\\beta}^2 \\hat{s}}{12 M_\\Omega^2} \n \\boldsymbol{ V_{x \\theta}} \\left[(g_s+g_p)^4 C_R + (g_s-g_p)^4 C_L \\right]\n + \\boldsymbol{A_{\\text{SF}}} \\right] $ \\\\[1.5ex]\nSFr & $\\displaystyle \\frac{ \\hat{\\beta}}{16 \\pi M_\\Omega^2} \\left[ F_{x \\theta}\nG_{s-p}^2 C_s + \\boldsymbol{A_{\\text{SFr}}}\\right]$ \\\\[1.5ex]\nSV & $ \\displaystyle \\frac{\\hat{s} \\hat{\\beta}^3 F_{x \\theta}}{96 \\pi M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \n \\left[ g_l^2C_L + g_r^2C_R \\right] g_\\chi^2 $ \\\\[1.5ex]\n\\hline\n\\hline\nFS & $ \\displaystyle \\frac{\\hat{s} \\hat{\\beta} F_{x \\theta}}{16 \\pi M_\\Omega^4} \nG_{s+p} C_s \\left[ g_{s}^2 \\hat{\\beta^2} + g_{p}^2 \\right] $ \\\\[1.5ex]\nFV & $ \\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta} }{48 \\pi\n M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \\left[\nG_{l+r} \\hat{s} \\hat{\\beta}^2 + 3 \\left(g_l + g_r \\right)^2 M_\\chi^2 \\right] \\left[ g_l^2C_L + g_r^2C_R \\right] \n$ \\\\[1.5ex]\nFVr & $ \\displaystyle \\frac{ \\hat{s} \\hat{\\beta^3} F_{x \\theta} }{48 \\pi\n M_\\Omega^4} \\boldsymbol{V_{x \\theta}} \\left(g_l - g_r \\right)^2 \\left[ g_l^2C_L + g_r^2C_R \\right] $\\\\[1.5ex]\nFtS & $\\displaystyle \\frac{ F_{x \\theta} \\hat{\\beta}}{48 \\pi M_\\Omega^4} G_{s+p}^2\n\\left[\\boldsymbol{V_{x \\theta}}(\\hat{s}-M_\\chi^2) + \\boldsymbol{A_{\\text{FtS}}} \\right]$ \\\\[1.5ex]\nFtSr & $\\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}}{192 \\pi M_\\Omega^4}\nG_{s+p}^2 \\left[ 3 (\\hat{s}-2 M_\\chi^2) C_P + \\boldsymbol{V_{x \\theta}} 2 (\\hat{s}\n - 4 M_\\chi^2) C_V \\right] $\\\\[1.5ex]\nFtV & $\\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{48 \\pi M_\\Omega^4} \\left[ \n 6 G_{lr}^2 C_s\n (\\hat{s} - 2 M_\\chi^2 ) + (\\hat{s} - M_\\chi^2) \\boldsymbol{V_{x \\theta}} (g_l^4 C_L + g_r^4 C_R) \\right] $ \\\\[1.5ex]\nFtVr & $\\displaystyle \\frac{\\hat{\\beta} F_{x \\theta}}{48 \\pi M_\\Omega^4} \\left[ \n 12 G_{lr}^2 C_s (\\hat{s} - 2 M_\\chi^2 ) + (\\hat{s} - 4 M_\\chi^2) \\boldsymbol{V_{x \\theta}} (g_l^4 C_L + g_r^4 C_R) \\right] $\\\\[1.5ex]\n\\hline\n\\hline\nVS & $ \\displaystyle \\frac{ \\hat{\\beta} F_{x \\theta}} {128 \\pi M_\\chi^4 M_\\Omega^4} \nG_{s+p} g_\\chi^2 C_s (12 M_\\chi^4-4M_\\chi^2\\hat{s}+\\hat{s}^2) $ \\\\[1.5ex]\nVF & $\\displaystyle \\frac{\\hat{\\beta}F_{x \\theta}}{3840 \\pi M_\\chi^4 M_\\Omega^2} \n\\Big[ 40 G_{lr}^2 C_s (7 M_\\chi^4 - 2 M_\\chi^2 \\hat{s} + \\hat{s}^2) +\\frac{1}{M_\\Omega^2}\\left(g_l^4C_L+g_r^4C_R\\right)\n (40 M_\\chi^6-22M_\\chi^4\\hat{s}+56 M_\\chi^2 \\hat{s}^2 + 3 \\hat{s}^3) + \\boldsymbol{A_{\\text{VF}}}\\Big]$\\\\[1.5ex]\nVFr & $\\displaystyle \\frac{\\hat{\\beta}F_{x \\theta}}{3840 \\pi M_\\chi^4 M_\\Omega^2} \n\\Big[ 60 G_{lr}^2 C_s (12 M_\\chi^4 - 4 M_\\chi^2 \\hat{s} + \\hat{s}^2) + \\frac{1}{M_\\Omega^2}\\left(g_l^4C_L+g_r^4C_R\\right)\n (320 M_\\chi^6-104^4\\hat{s}+32 M_\\chi^2 \\hat{s}^2 + \\hat{s}^3) + \\boldsymbol{A_{\\text{VFr}}}\\Big]$\\\\\nVV & $ \\displaystyle \\frac{ \\hat{s} \\hat{\\beta}^3 F_{x \\theta} \\boldsymbol{V_{x \\theta}}}{3840 \\pi M_\\chi^4 M_\\Omega^4}\\left[ g_l^2C_L + g_r^2C_R \\right] g_\\chi^2 \n( M_\\chi^4 + 20 M_\\chi^2 \\hat{s} + \\hat{s}^2)$ \\\\[1.5ex]\n\\hline\n\\end{tabular}\n\\caption{Analytical differential cross sections for the process $\\Ppositron\n \\Pelectron \\rightarrow \\chi \\chi \\gamma$ in the various effective\n models. Terms in bold do not appear in the Weizs\\\"acker--Williams\n approach and are given in Appendix~\\ref{app:xsectterms} where we also\n define all used abbreviations. Models with a suffix `r' correspond to the case of real particles. \n Cross sections for SSr, FSr and VSr are twice as large as\n in the complex case while SV and VV vanish completely for real particles.}\n\\label{tbl:2t3crosssections}\n\\end{table*}\nDue to the soft collinear approximation used, we expect that the above equation will perform poorly for large angle and high $p_T$ photons. We compare the analytical result to this approximation to test\nthe reliability. In Table~\\ref{tbl:2t3crosssections} we put terms in\nbold, which are purely caused by our analytical\ntreatment. The corrections are either of the form of an additional\nkinematical factor $V_{x \\theta}$, mostly appearing in models with\nvector mediators, or completely new terms that typically appear in\nt--channel interactions. Since $\\lim_{x \\rightarrow 0} V_{x \\theta} =\n1$ and $\\lim_{x \\rightarrow 0} (A_i) = 0$, the WW--approximation is in\nagreement with our full result for small energies. In\nFig.~\\ref{img:comparison} we show the respective photon energy\ndistributions for different models in both the WW--approximation and\nthe full analytical treatment. The curves behave quite congruently\nwith differences visible in the high energy sector. Since most of the\nsignal events lie in the low energy part, the approximation gives\naccurate results for counting experiments. A shape dependent analysis\nwould need to use the analytical result to estimate the correct\nthreshold behaviour for high energies. Our subsequent analysis\nis performed with the full analytical cross section.\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{FtS_1tev.pdf}\n\\put(-45,180){(a)} \n\\hfill\n\\includegraphics[width=0.49\\textwidth]{VF_1tev.pdf} \n\\put(-45,180){(b)} \n\\caption{Comparison of tree level photon energy distributions in the\n WW--approximation and the analytical solution for $M_{\\chi} = \\unit{50}{\\GeV}$,\n $|\\cos \\theta_\\gamma|_\\text{max} = 0.98$ and\n $\\sqrt{s} = \\unit{1}{\\TeV}$. (a) SV, (b) FtS.}\n\\label{img:comparison}\n\\end{figure*}\n\n\nWhen we restrict the various couplings in our model according to the benchmark scenarios, Table~\\ref{tbl:constraints}, most of the cross sections simplify and have only one polarisation dependent term $C_i$. To\ndetermine the polarisation leading to the best signal to\nbackground ratio, we only need to consider cases with different $C_i$.\nWe therefore classify our models as follows:\n\\begin{align}\n\\text{Scalar--like}: \\sigma_{\\text{pol}} &= C_S\n\\sigma_{\\text{unpol}}, \\label{eqn:polclasses} \\\\\n\\text{Vector--like}: \\sigma_{\\text{pol}} &= C_V \\sigma_{\\text{unpol}}, \\nonumber\n\\\\\n\\text{Right--like}: \\sigma_{\\text{pol}} &= C_R \\sigma_{\\text{unpol}}, \\nonumber \\\\\n\\text{Left--like}: \\sigma_{\\text{pol}} &= C_L \\sigma_{\\text{unpol}}. \\nonumber\n\\end{align}\n\nModels with t--channel mediators usually have multiple terms with\ndifferent polarisation behaviour and do not fall into one of the basic\npolarisation classes given in Eq.~(\\ref{eqn:polclasses}). We choose\nthe following polarisation settings for those:\n\\begin{itemize}\n\\item Models with fermionic mediators are classified according to their\n leading term, which is always scalar--like. \n\\item All other models have both scalar--like and vector--like parts of about\n the same size. We analyse them in a vector--like scenario that naturally leads to a\n better background suppression.\n\\end{itemize}\n\n\\subsection{Standard Model Background for Monophotons}\nWe consider the two leading dominant Standard Model background contributions\nafter selection, determined with a full \\textsc{Ild} (International Linear Detector concept) detector simulation\n\\cite{BartelsThesis, Bartels:2012ex}. All numbers here and in the following\nparagraphs refer to the nominal \\textsc{Ilc} center of mass energy of\n\\unit{500}{\\GeV} \\cite{Phinney:2007gp}. We also consider the case of an\nincreased energy of \\unit{1}{\\text{TeV}} and mention the differences later.\n\\begin{itemize}\n\\item Neutrinos from $\\Ppositron \\Pelectron \\rightarrow\n \\Pnu \\APnu \\Pphoton (\\Pphoton)$ form a polarisation dependent background. The leading\n contribution comes from t--channel $\\PW$--exchange, which only couples to\n left--chiral leptons. Additional smaller contributions come from s--channel\n $\\PZ$--diagrams with both left-- and right--chiral couplings. We also\n consider the case of one additional undetected photon, which contributes\n with a size of roughly $\\unit{10}{\\%}$. \n\\item Bhabha scattering of leptons with an additional hard photon, $\\Ppositron \\Pelectron \\rightarrow\n \\Ppositron \\Pelectron \\Pphoton$ has a large cross section but a very small\n selection efficiency, since both final state leptons must be undetected. It has been determined to give a contribution of the same order\n of magnitude as the neutrino background, after application of all selection\n criteria. It is mostly polarisation independent \\cite{BartelsThesis, Bartels:2012ex}.\n\\end{itemize}\n Other background sources contribute with less\nthan \\unit{1}{\\%} compared to the neutrino background and are therefore omitted. \n\n\\subsection{Data Modeling}\nTo evade the use of a full detector simulation, we build on the results of\nRefs.~\\cite{BartelsThesis, Bartels:2012ex}. For the signal and monophoton neutrino\nbackground, we generate the events by ourselves with the given phase space criteria. \nWe then apply the \\textsc{Ild} estimates for the\nenergy resolution as well as the reconstruction and selection\nefficiencies\\footnote{From here on, the expression `efficiency' abbreviates\n `reconstruction and selection efficiencies'.} and compare the final energy\ndistributions. For the diphoton neutrino and Bhabha background, we model the\nfinal distributions directly from the given results performed with a full detector simulation \\cite{BartelsThesis, Bartels:2012ex}. \n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{energy_neutrino_color.pdf} \n\\put(-45,180){(a)}\n\\hfill\n\\includegraphics[width=0.49\\textwidth]{energy_FS_color.pdf}\n\\put(-45,180){(b)}\n\\caption{Photon energy distribution before and after application of beam effects\n (\\textsc{Isr} + beamstrahlung) and detector effects (resolution +\n efficiency) for a) unpolarised neutrino background and b) unpolarised FS\n scalar signal with $M_\\chi = \\unit{150}{\\GeV}$. Distributions are normalised to\n $10^6$ tree level events.}\n\\label{img:photonenergy}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\vspace{-0.55cm}\n\\includegraphics[width=0.49\\textwidth]{backgrounds_new.pdf}\n\\vspace{-0.8cm}\n\\caption{Photon energy distributions of the most dominant background\n contributions (stacked) compared to an example signal (FS Scalar, $M_\\chi =\n \\unit{150}{\\GeV}$) with a total cross section of \\unit{100}{\\femto\\barn}. All spectra are taken after selection for an unpolarised\n initial state.}\n\\label{img:backgroundcontributions}\n\\vspace{0.775cm}\n\\begin{tabular}{r@{\\qquad}crr c@{\\quad} rr c@{\\quad} rr }\n\\hline\n$P^-\/P^+$ && \\multicolumn{2}{c}{$\\Pnu \\Pnu \\Pphoton$} && \\multicolumn{2}{c}{$\\Pnu \\Pnu \\Pphoton \\Pphoton$} &&\\multicolumn{2}{c}{$\\Ppositron \\Pelectron \\Pphoton$} \\\\\n\\hline\n\\hline\n$0\/0$ && 2257 & (2240) && 226 & (228) && 1218 & (1229) \\\\\n$+0.8\/-0.3$ && 493 & (438) && 49 & (43) && 1218 & (1204) \\\\\n$-0.8\/+0.3$ && 5104 & (5116) && 510 & (523) && 1218 & (1227) \\\\\n\\hline\n\\end{tabular}\n\\caption{Total number of events in the different background\n sources after application of all selection criteria. The numbers are\n given for an integrated luminosity of \\unit{1}{\\femto\\reciprocal\\barn} in different\n polarisation settings. Numbers in brackets are taken from\n Ref.~\\cite{BartelsThesis} which employed a proper detector\n simulation.}\n\\label{tbl:neventsperscenario}\n\\end{figure}\n\n\nFor the generation of signal and monophoton neutrino events we use\n\\textsc{C}alc\\textsc{hep} \\cite{Pukhov:1999gg}. We produce signal events for all benchmark\nscenarios with dark matter masses ranging from \\unit{1}{\\GeV} to\n\\unit{240}{\\GeV}. To avoid collinear and infrared divergences,\nwe limit phase space in the event generation to $E_\\gamma \\in \\left[\\unit{8}{\\GeV}, \\unit{250}{\\GeV}\n \\right]$ and $\\cos \\theta_\\gamma \\in \\left[ -0.995, 0.995\n \\right]$. Initial State Radiation (\\textsc{Isr})\n and beamstrahlung significantly change the width and position of the neutrino\n \\PZzero--resonance, Fig.~\\ref{img:photonenergy}a), and are taken into account. We\n set the accessible parameters in \\textsc{C}alc\\textsc{hep} according to the \\textsc{Ild}\n Letter of Intent \\cite{Abe:2010aa} to \\unit{645.7}{\\nm} for the bunch size, \\unit{0.3}{\\mm}\n for the bunch length and a total number of particles per bunch of $2 \\cdot\n 10^{10}$. \n\n\nThe finite resolution of the detector components and the use of \nselection criteria to reduce beam--induced background are taken into account by\napplying the following steps to both signal and background data. First we shift the photon\nenergy, given in \\GeV, according to a Gaussian distribution by taking into account the estimated\n resolution of the \\textsc{Ild} detector components \\cite{Abe:2010aa},\n\\begin{align}\n\\frac{\\Delta E_\\gamma}{E_\\gamma} =\n\\frac{\\unit{16.6}{\\%}}{\\sqrt{E_\\gamma \\text{ in \\GeV}}} \\oplus \\unit{1.1}{\\%}.\n\\end{align}\nAfterwards we further limit the phase space to reduce background processes in\n the \\PZzero resonance peak at \\unit{242}{\\GeV} and additional collinear photons from \\textsc{Isr},\n\\begin{align}\nE_\\gamma &\\in \\left[ \\unit{10}{\\GeV}, \\unit{220}{\\GeV} \\right], \\\\\n\\cos \\theta_\\gamma &\\in \\left[ -0.98, 0.98 \\right]. \\nonumber\n\\end{align}\nThe additional angular cut ensures a good photon reconstruction within\nthe detector. Finally a random elimination of events is used\nto simulate the efficiency factor for reconstruction and selection\ndetermined in Ref.~\\cite{BartelsThesis}. The efficiency consists of an\nenergy dependent part $\\epsilon_1$ and a constant part $\\epsilon_2$\nthat are applied successively,\n\\begin{align}\n\\epsilon_1 &= \\unit{97.22}{\\%} - (E_\\gamma \\text{ in \\GeV}) \\cdot\\unit{0.1336}{\\%}, \\label{eqn:efficiency1}\\\\\n\\epsilon_2 &= \\unit{96.8}{\\%}. \\nonumber\n\\end{align}\nFig.~\\ref{img:photonenergy} shows how these settings affect the signal and\nbackground spectrum and Fig.~\\ref{img:backgroundcontributions} shows a stacked histogram of the dominant background processes along with a example dark matter signal.\n\n\n\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{r@{\\quad}r@{\\qquad}c rr@{\\qquad}c rr@{\\qquad}c rrrr}\n\\hline \n&&&&&&&&&&&& \\\\ [-2.ex] \n$P^-\/P^+$ & $N_{\\text{B}}$ && $\\Delta^\\text{stat}_{50}$ & $\\displaystyle\\tilde{\\Delta}^\\text{stat}_{500}$ && $\\Delta^\\text{sys}_{P}$\n& $\\displaystyle\\Delta^\\text{sys}_{\\tilde{P}}$ && $\\displaystyle \\Delta^\\text{tot}_{50 P}$ &\n$\\Delta^\\text{tot}_{50 \\tilde{P}}$ & $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 P}$\n& $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 \\tilde{P}}$ \\\\ [1ex]\n\\hline\\hline\n0\/0 & 184998 && & && & && & & & \\\\\n\\hline\n$+0.8$\/$+0.3$ & 97568 && 312 & 99 && 312 & 125 && 441 & 336 & 327 & 159 \\\\\n$+0.8$\/$+0.6$ & 102365 && 320 & 101 && 385 & 154 && 500 & 355 & 398 & 184 \\\\\n\\hline\n$+0.8$\/$-0.3$ & 87974 && 297 & 94 && 169 & 68 && 341 & 304 & 193 & 116 \\\\\n$+0.8$\/$-0.6$ & 83177 && 288 & 91 && 104 & 42 && 307 & 291 & 138 & 100 \\\\\n\\hline\n$-0.8$\/$+0.3$ & 341597 && 584 & 185 && 351 & 140 && 682 & 601 & 396 & 232 \\\\\n$-0.8$\/$+0.6$ & 404970 && 637 & 201 && 501 & 200 && 811 & 668 & 546 & 284 \\\\\n\\hline\n$-0.8$\/$-0.3$ & 212851 && 461 & 156 && 233 & 93 && 517 & 471 & 275 & 173 \\\\\n$-0.8$\/$-0.6$ & 148478 && 385 & 122 && 337 & 135 && 512 & 408 & 359 & 182 \\\\\n\\hline\n\\end{tabular}\n\\caption{Total amount of background events, $N_{\\text{B}}$, with statistical\n error, $\\Delta^\\text{stat}$, systematic error, $\\Delta^\\text{sys}$, and the total error,\n $\\Delta^\\text{tot}$. The subscripts 50 and 500 denote the integrated\n luminosity in inverse femtobarn. In case of a ten times larger luminosity,\n one will get ten times as many events in all channels; to better compare to the error\n of the low luminosity case,\n we give $\\tilde{\\Delta}^\\text{stat}_{500} \\equiv \\Delta^\\text{stat}_{50}\/\\sqrt{10}$. The polarisation uncertainties are set to\n \\unit{0.25}{\\%} ($P$) and \\unit{0.1}{\\%}\n ($\\tilde{P}$).}\n\\label{tbl:bkgevts}\n\\end{table*}\n\n\\subsection{Analysis}\n\nWe are interested in determining the lower bound on the effective coupling constants that the\n\\textsc{Ilc} can find for each individual model under the assumption that no signal\nevents are measured. We perform a counting experiment by using the\n\\textsc{TRolke} \\cite{Rolke:2004mj} statistical test. We determine the total\nnumber of background events along with its statistical and systematic fluctuation\n$\\Delta N_{\\text{B}}$ and exclude coupling constants which would lead to a\nlarger number of signal events than the \\unit{90}{\\%} confidence interval of the background--only\nassumption. \n\n\n\\subsection{Systematic Uncertainties}\n\nSystematic uncertainties play an important role in determining the\ntotal error on the background, $\\Delta N_B$, and for estimating the\nbounds on the effective couplings. There are two dominant\ncontributions, motivated in Ref.~\\cite{BartelsThesis} which we now discuss.\n\nThe experimental efficiency given in Eq.~(\\ref{eqn:efficiency1}) will\nbe determined at the real experiment by measuring the\n$\\PZzero$--resonance peak, which is theoretically known to a very good\naccuracy. Systematic uncertainties on that value are given by the\nfinite statistics of this measurement and further broadening of the\npeak by unknown beam effects. These errors can be extrapolated down to\nthe dark matter signal region at small photon energies and, since the\nsame efficiency factor is used for signal and background, is highly\ncorrelated between those two. This global uncertainty will therefore\napproximately cancel in the determination of the maximum coupling\n$G_\\text{eff}$.\n\nCancellation will not take place for model dependent effects\nhowever. This is due to the fact that the signal energy distribution \ndepends on the unknown mass of the dark matter particle and the\nunderlying interaction model. Therefore, the correct function\n$\\epsilon(E_\\gamma)$ for the signal will be different from the used\nneutrino background efficiency given in\nEq.~(\\ref{eqn:efficiency1}). Since we do not know the model a priori,\nwe use the same value for both and introduce an error on the\ndetermination of the signal events, $N_S$. Compared to Ref.~\\cite{BartelsThesis}, we use a\nconservative value of $\\Delta \\epsilon = \\unit{2}{\\%}$.\n\nSince the neutrino spectrum depends on the incoming lepton's\npolarisation $P^\\pm$, any fluctuation within those parameters will\ngive additional systematic uncertainties on the number of expected\nbackground events. One can not use the information from measuring the\n$\\PZzero$--resonance in this case to infer information in the low\nenergy signal range because of the polarisation dependence of the\nshape itself. Given the assumed accuracy of at least $\\Delta P\/P =\n\\unit{0.25}{\\%}$ \\cite{Abe:2010aa} with a possible improvement to\n\\unit{0.1}{\\%} at the \\textsc{Ilc}, we can derive the corresponding\nerror on the polarised number of background events. As an example we\nshow the left handed background,\n\\begin{align}\nN_{\\text{pol}} &= (1+P^+)(1-P^-)N_{\\text{unpol}}, \\nonumber \\\\\n\\Delta N_{\\text{pol}} &= \\sqrt{\\left[P^- (1+P^+) \\right]^2 + \\left[P^+ (1-P^-)\n \\right]^2} \\ \\frac{\\Delta P}{P} \\ N_{\\text{unpol}}. \\label{eqn:deltap}\n\\end{align}\nFrom the numbers in Table~\\ref{tbl:neventsperscenario}, we assume an identical\npolarisation dependence for $\\Pnu \\Pnu \\Pphoton$ and $\\Pnu \\Pnu \\Pphoton\n\\Pphoton$ events and no dependence for the Bhabha background. \n\n\\begin{table}[b]\n\\centering\n\\begin{tabular}{rr@{\\quad}rrrrrr}\n\\hline\nIA type & $P^-\/P^+$ & $N_{\\text{S}}$ && $r_{50 P}$ & $r_{50 \\tilde{P}}$ & $r_{500 P}$ &\n$r_{500 \\tilde{P}}$ \\\\\n\\hline\nScalar &$+0.8\/+0.3$ & 620 && 1.41 & 1.85 & \\textbf{1.90} & 3.90 \\\\\n &$+0.8\/+0.6$ & 740 && \\textbf{1.48} & \\textbf{2.08} & 1.86 & \\textbf{4.02} \\\\\n\\hline\nVector &$+0.8\/-0.3$ & 620 && 1.82 & 2.04 & 3.21 & 5.34 \\\\\n& $+0.8\/-0.6$ & 740 && \\textbf{2.41} & \\textbf{2.54} & \\textbf{5.36}\n& \\textbf{7.40} \\\\\n\\hline\nLeft & $-0.8\/+0.3$ & 1170 && 1.72 & 1.95 & \\textbf{2.95} & 5.04 \\\\\n& $-0.8\/+0.6$ & 1440 && \\textbf{1.78} & \\textbf{2.16} & 2.64 & \\textbf{5.07} \\\\\n\\hline\nRight &$+0.8\/-0.3$ & 1170 && 3.43 & 3.85 & 6.06 & 10.09 \\\\\n& $+0.8\/-0.6$ & 1440 && \\textbf{4.69} & \\textbf{4.95} & \\textbf{10.43} & \\textbf{14.4} \\\\\n\\hline\n\\end{tabular}\n\\caption{Determination of the best ratio $r \\equiv\n {N_{\\text{S}}}\/{\\Delta N_{\\text{B}}}$ with $\\Delta N_{\\text{B}}$ given by the different\n total errors determined in Table~\\ref{tbl:bkgevts}. $N_{\\text{S}}$ describes the\nnumber of polarised signal events for the different classes described in Sec.~\\ref{sec:models}\nwith a common reference value of 500 unpolarised events for an integrated\nluminosity of $\\unit{50}{\\femto\\reciprocal\\barn}$. We only show the polarisation signs with the largest ratios. We mark the\nnumbers which lead to the best signal to background ratio in bold.}\n\\label{tbl:sigoverbkgestimate}\n\\end{table}\n\n\n\\begin{table*}\n\\begin{tabular}{l@{\\qquad}r@{\\quad}rr@{\\quad}rr@{\\quad}rrrr}\n\\hline\n &&&&&&&&& \\\\ [-2.ex] \n$P^-\/P^+$ & $N_{\\text{B}}$ & $\\Delta^\\text{S}_{50}$ & $\\displaystyle\\tilde{\\Delta}^\\text{S}_{500}$ & $\\delta^\\text{P}_{P}$\n& $\\delta^\\text{P}_{\\tilde{P}}$ & $\\Delta^\\text{tot}_{50 P}$ & $\\Delta^\\text{tot}_{50 \\tilde{P}}$ & $\\displaystyle\\tilde{\\Delta}^\\text{tot}_{500 P}$\n& $\\displaystyle \\tilde{\\Delta}^\\text{tot}_{500 \\tilde{P}}$ \\\\ [1ex]\n\\hline\n\\hline\n$0\/0$ & 162437 & & & & & & & & \\\\\n\\hline\n$+0.8$\/$+0.3$ & 54649 & 234 & 74 & 380 & 152 & 446 & 279 & 387 & 169 \\\\\n$+0.8$\/$+0.6$ & 62791 & 251 & 79 & 469 & 188 & 531 & 314 & 476 & 203 \\\\\n\\hline\n$+0.8$\/$-0.3$ & 38365 & 196 & 62 & 201 & 82 & 281 & 212 & 210 & 102 \\\\\n$+0.8$\/$-0.6$ & 30223 & 174 & 55 & 125 & 50 & 214 & 181 & 137 & 74 \\\\\n\\hline\n$-0.8$\/$+0.3$ & 357173 & 598 & 189 & 428 & 171 & 735 & 622 & 468 & 255 \\\\\n$-0.8$\/$+0.6$ & 435979 & 660 & 209 & 612 & 245 & 900 & 704 & 647 & 322 \\\\\n\\hline\n$-0.8$\/$-0.3$ & 199561 & 447 & 141 & 284 & 114 & 530 & 461 & 317 & 181 \\\\\n$-0.8$\/$-0.6$ & 120755 & 348 & 110 & 411 & 165 & 538 & 385 & 425 & 198 \\\\\n\\hline\n\\end{tabular}\n\\caption{Total amount of background events ($N_{\\text{B}}$) and different\n error sources (see Table~\\ref{tbl:bkgevts}) for $\\sqrt{s} = \\unit{1}{\\TeV}$.}\n\\label{tbl:bkgevts_1tev}\n\\end{table*}\n\n\n\\begin{table}[b]\n\\begin{tabular}{r@{\\qquad}r@{\\quad}r@{\\quad}r@{\\quad}}\n\\hline\n$P^-\/P^+$ & $\\Pnu \\Pnu \\Pphoton$ & $\\Pnu \\Pnu \\Pphoton \\Pphoton$ & $\\Ppositron \\Pelectron$ \\\\\n\\hline\n\\hline\n$0\/0$ & 2677 & 268 & 304 \\\\\n$+0.8\/-0.3$ & 421 & 42 & 304 \\\\\n$-0.8\/+0.3$ & 6217 & 622 & 304 \\\\\n\\hline\n\\end{tabular}\n\\caption{Simulated and modeled number of events in the different background\n sources after application of all selection criteria for $\\sqrt{s} = \\unit{1}{\\TeV}$. The numbers are\n calculated for an integrated luminosity of \\unit{1}{\\femto\\reciprocal\\barn} in different\n polarisation settings.}\n\\label{tbl:neventsperscenario_1tev}\n\\end{table}\n\n\\begin{table}[b]\n\\begin{tabular}{r@{\\qquad}rr@{\\quad}rrrr}\n\\hline\nModel & $P^-\/P^+$ & $N_{\\text{S}}$ & $r_{50 P}$ & $r_{50 \\tilde{P}}$ & $r_{500 P}$ &\n$r_{500 \\tilde{P}}$ \\\\\n\\hline\n\\hline\nScalar &$+0.8$\/$+0.3$ & 620 & 1.39 & 2.22 & \\textbf{1.60} & \\textbf{3.7} \\\\\n &$+0.8$\/$+0.6$ & 740 & \\textbf{1.39} & \\textbf{2.36} & 1.55 & 3.65 \\\\\n\\hline\nVector &$+0.8$\/$-0.3$ & 620 & 2.21 & 2.92 & 2.95 & 6.08 \\\\\n& $+0.8$\/$-0.6$ & 740 & \\textbf{3.46} & \\textbf{4.09} & \\textbf{5.40}\n& \\textbf{10.00} \\\\\n\\hline\nLeft & $-0.8$\/$+0.3$ & 1170 & 1.59 & 1.88 & 2.50 & \\textbf{4.59} \\\\\n& $-0.8$\/$+0.6$ & 1440 & \\textbf{1.60} & \\textbf{2.05} & 2.23 & 4.47 \\\\\n\\hline\nRight &$+0.8$\/$-0.3$ & 1170 & 4.16 & 5.52 & 5.57 & 11.47 \\\\\n&$+0.8$\/$-0.6$ & 1440 & \\textbf{6.73} & \\textbf{7.96} & \\textbf{10.51} & \\textbf{19.46} \\\\\n\\hline\n\\end{tabular}\n\\caption{Determination of the best ratio $r \\equiv\n {N_{\\text{S}}}\/{\\Delta N_{\\text{B}}}$ (see\n Table~\\ref{tbl:sigoverbkgestimate}) for $\\sqrt{s} =\n \\unit{1}{\\TeV}$. }\n\\label{tbl:sigoverbkgestimate_1tev}\n\\end{table}\n\\subsection{Polarisation Settings}\nPolarisation can be used to significantly increase the number of\nsignal events according to Eq.~(\\ref{eqn:polclasses}) but also\nincreases the systematical contribution to the total background error,\n$\\Delta N_\\text{B}$ via Eq.~(\\ref{eqn:deltap}). We are interested in\nthe settings for each individual model that leads to the largest\n$N_\\text{S}\/\\Delta N_\\text{B}$ ratio allowing for the strictest bounds\non $G_\\text{eff}$. In Table~\\ref{tbl:bkgevts} we give the total number\nof background events in different polarisation settings $P^-$ = $\\pm\n0.8$ and $P^+$ = $\\pm 0.3$\/$\\pm 0.6$ that are feasible at the\n\\textsc{Ilc} \\cite{Phinney:2007gp}. We give the statistical\nfluctuation for integrated luminosities of\n\\unit{50}{\\femto\\reciprocal\\barn} as well as for\n\\unit{500}{\\femto\\reciprocal\\barn}. Since the latter will give ten times as much events in\nall channels, we reduce the statistical error accordingly to give a value\ncomparable to the small luminosity case.\nWe also give the systematic error that is dominated by the polarisation uncertainty for two estimates of the\npolarisation error $\\delta P\/P = \\unit{0.25}{\\%} $ and $\n\\unit{0.1}{\\%}$ \\cite{Helebrant:2008qz}. Finally we give the total errors adding all combinations of\nindividual errors in quadrature.\n\nOn the signal side, we look at the different classes derived in Sec.~\\ref{sec:models} with respect to their polarisation dependence. For comparison,\nwe use a common reference value of 500 unpolarised events for an integrated\nluminosity of $\\unit{50}{\\femto\\reciprocal\\barn}$ and derive the corresponding number of\nevents for polarised input.\n\nWe look for the maximum ratio $r \\equiv N_{\\text{S}} \/ \\Delta N_{\\text{B}}$ and the results for the best settings are displayed in Table~\\ref{tbl:sigoverbkgestimate}. In most cases the largest possible polarisation for the incoming leptons enhances the result. For high statistics and a non--reduced polarisation error, the systematic uncertainty\nfrom increased polarisation may outweigh the gain in the number of signal events though. In those\ncases, which appear only in scalar-- and left--coupling models, less polarised\nbeams lead to better results.\n\n\n\\subsection{Increasing \\boldsymbol{$\\sqrt{s}$} to \\unit{1}{\\text{TeV}}}\n\n\nWe also consider the possibility of a doubled center of mass\nenergy. This changes the previous analysis as follows:\n\\begin{itemize}\n\\item We generate events in a larger photon energy range $E_\\gamma \\in\n \\left[\\unit{8}{\\GeV}, \\unit{500}{\\GeV}\\right]$ and reduce it to the interval $\\left[\n \\unit{10}{\\GeV}, \\unit{450}{\\GeV} \\right]$ after performing the energy resolution\n shift $\\Delta E \/ E$. This again reduces background events from the\n $\\PZzero$--resonance, which now is positioned at \\unit{496}{\\GeV}.\n\\item Dark matter signal processes can now be produced with masses up to \\unit{490}{\\GeV}.\n\\item We use our previously modeled distribution for the Bhabha background\n and rescale it by a factor of 1\/4, taking into account that the full cross\n section for that process is approximately proportional to $1\/s$. \n\\item We use, as a rough approximation, the same \\textsc{Isr}-- and beamstrahlung parameters in\n\\textsc{C}alc\\textsc{hep}, efficiency factors and systematic error estimates. \n\\end{itemize}\n\nTables~\\ref{tbl:bkgevts_1tev}-\\ref{tbl:sigoverbkgestimate_1tev}\nsummarise again the number of background events per background\nscenario, the individual error sources and the determination of the\nbest polarisation setting for the increased center of mass energy. In\ncontrast to the Bhabha cross section that falls mainly according to\n$\\sigma \\propto 1\/s$, the neutrino background gets significant\ncontributions from t--channel $\\PW^{\\pm}$s, which give $s \/ m_W^4$\n--terms in the evaluation of the total cross section. The left--handed\nneutrino contribution therefore gets enhanced whereas the Bhabha\nbackground becomes less dominant in some polarisation channels. This\nleads to a larger relative polarisation error and therefore a larger\nimpact on the size of the background fluctuation. In the end, vector--\nand right--coupling models receive stronger enhancement for polarised\ninput than in the $\\sqrt{s} = \\unit{500}{\\GeV}$ case, whereas the\nother models suffer from the larger impact of polarisation on the\ntotal error and prefer smaller polarisation.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{finalplot_scalars_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_all12_couplingBounds.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{finalplot_fermions_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_fermions2_couplingBounds.pdf} \\\\\n\\includegraphics[width=0.45\\textwidth]{finalplot_all1_couplingBounds.pdf} \\hfill\n\\includegraphics[width=0.45\\textwidth]{finalplot_all2_couplingBounds.pdf} \\\\\n\\caption{\\unit{90}{\\%} exclusion limits on the effective couplings accessible at the \\textsc{Ilc} with $\\sqrt{s} =\n \\unit{1}{\\TeV}$. We only give effectively allowed regions for models with\n dimensionless fundamental couplings $g$. }\n\\label{img:ilcbounds}\n\\end{figure*}\n\n\\section{Introduction}\n\n\nWeakly interacting massive particles (\\textsc{Wimp}s) are one of the leading candidates to solve the dark matter puzzle \\cite{Bertone:2004pz}. Primarily this is due to the fact that a neutral particle that interacts with roughly the strength of the weak force, naturally gives the correct relic abundance. In addition many theoretical models predict that the masses of these \nstates should exist around the scale of electroweak symmetry breaking, e.g.\\ Supersymmetry (\\textsc{Susy}) \\cite{Martin:1997ns,Drees:2004jm}, Universal Extra Dimensions (\\textsc{Ued}) \\cite{Appelquist:2000nn}, Little Higgs \\cite{ArkaniHamed:2001nc} etc.\n\nCurrently, this \\textsc{Wimp} paradigm is being actively explored in a number of different ways. Perhaps the most well known \nare the direct detection searches that aim to observe interactions between the dark matter and an atomic nucleus \\cite{Goodman:1984dc}. As these are extremely low rate experiments, the detectors are typically placed deep underground to reduce \nbackground. The annihilation of dark matter into Standard Model particles in high density regions of our universe offers another potential method to see a signal e.g.\\ \\cite{Bouquet:1989sr}.\n\nIn particle colliders here on Earth the same interactions may be probed in the production of dark matter. Unfortunately, the fact that \\textsc{Wimp}s are neutral and only weakly interacting means that they cannot be detected directly in these experiments. Therefore collider based searches must rely on particles produced in combination with the dark matter candidates. If dark matter is produced directly, one possibility is to use initial state radiation (\\textsc{Isr}), such as gluon jets, or photons, that will recoil against the \\textsc{Wimp}s.\n\nThis idea was first explored in a model independent approach for the International Linear Collider (\\textsc{Ilc}) using mono-photons in a non-relativistic approximation \\cite{Birkedal:2004xn,Konar:2009ae}. Later, detailed detector studies have been performed to understand the full capabilities of the \\textsc{Ilc} for such a signature \\cite{Bartels:2007cv,Bartels:2009fa,Bartels:2010qv,Bartels:2012ex,Bernal:2008zk}. Furthermore the same signature has been considered in the case of \\textsc{Susy} \\cite{Dreiner:2006sb,Dreiner:2007vm}. At the \\textsc{Lhc} (Large Hadron Collider) and Tevatron similar signals have also been studied but with a mono-jet signal \\cite{Cao:2009uw,Bai:2010hh,Fox:2011pm,Goodman:2010ku,Goodman:2010yf,Beltran:2010ww,Rajaraman:2011wf,Bai:2012he,Cheung:2012gi}. All of these papers used the idea of parameterising the dark matter interactions in the form of effective operators. This has the advantage that the bounds can be compared with those coming from direct detection and also that a non-relativistic approximation is not required to compare with the relic density measurement. These methods have now been used by the \\textsc{Lhc} experiments to set bounds on different effective operators that are competitive with other methods \\cite{Chatrchyan:2012pa,ATLAS-CONF-2012-084}. In addition, \\textsc{Lep} (Large Electron-Positron Collider) data has been re-interpreted to determine corresponding constraints \\cite{Fox:2011fx}.\n\nIn this paper we take the effective field theory approach to dark matter and apply this to an \\textsc{Ilc} search \n\\cite{Kurylov:2003ra,Beltran:2008xg,Agrawal:2010fh}. To apply the effective field theory in a consistent way we \nassume that the dark matter particles can only interact with the Standard Model fields via a heavy mediator. \nThe mediator is always assumed to be too heavy to be produced directly at the \\textsc{Ilc} and thus can be \nintegrated out. For our model choices we consider the possibility that the dark matter candidate could be a scalar, \na Dirac (or Majorana) fermion or a vector particle. The same choices are taken for the heavy mediator \nand all combinations are considered. The collider phenomenology can vary significantly, depending on \nwhether the mediator is exchanged in the $s$- or $t$-channel and consequently we examine both. In addition, we also study the different ways in which the mediator can couple to both the dark matter and Standard Model particles. We note that using the effective field theory approach allows us to move away from the non-relativistic approximation that had previously been used in \\textsc{Ilc} studies. This can be especially important if the dark matter candidate happens to be light.\\footnote{The mass determination of a light neutralino dark matter candidate at the \\textsc{Ilc} has been discussed in Ref.~\\cite{Conley:2010jk}.}\n\nFor all models we compare the reach of the \\textsc{Ilc} with the bounds derived from direct and indirect detection. We also calculate the couplings expected to lead to the correct relic density and see whether the \\textsc{Ilc} can probe these regions of parameter space. We also note that an \\textsc{Ilc} search is complementary to that at the \\textsc{Lhc} thanks to the different initial state.\n\nThe paper is laid out as follows. We begin in Sec.~\\ref{sec:models} by explaining how we derive the effective field theories for the dark matter interactions and we explicitly give the Lagrangian for both the full and effective theory. We also describe the benchmark models that we use throughout the study. In Sec.~\\ref{sec:astro} we describe the various astrophysical constraints on our effective theories. We begin with the calculation of the relic density abundance before moving on to explain the bounds from direct and indirect detection.\n\nSection~\\ref{sec:ilc} describes in detail the potential search for dark matter at the \\textsc{Ilc}. Here we explain the calculation of the signal rate and the dominant backgrounds that were considered. In addition we detail how the \\textsc{Ilc} detectors are modeled to account for relevant experimental effects. We find that the polarisation of incoming beams is particularly important for many models of dark matter to discriminate the signal and background. We also investigate the advantage of a doubling of the \\textsc{Ilc} energy to $\\sqrt{s}=1$~TeV. \n\nIn Sec.~\\ref{sec:results} we present the results of the paper. We begin by examining the potential bounds of the \\textsc{Ilc} on the effective coupling of the dark matter model at the collider. Afterwards, we combine these results with those from direct and indirect detection to understand for which models and mass ranges the \\textsc{Ilc} presents a unique opportunity to discover dark matter. Finally in Sec.~\\ref{sec:conclusions} we conclude and summarise the main results of our work.\n\n\n\\section{Models}\n\\label{sec:models}\n\n\n\\subsection{General Motivation}\n\\noindent\nThe idea of parametrising the interaction of a dark matter particle\nwith Standard Model particles by using effective operators is not new,\nsee for example Refs.~\\cite{Agrawal:2010fh, Beltran:2008xg,\n Zheng:2010js, Yu:2011by, Fox:2011fx, Bai:2010hh}. Many authors\nconstruct a list of effective 4--particle-interactions with Lorentz--invariant\ncombinations of $\\gamma^\\mu$, $\\partial_\\mu$ and\nspinor--\/vector--indices up to mass dimension 5 or 6. In many cases\nthere is no explanation how those operators may arise in an underlying\nfundamental theory. That makes it difficult to judge how exhaustive\nthe lists of operators are, whether interference between different\noperators should be taken into account and how the effective model is\nconnected to realistic fundamental theories and their couplings.\n\nWe follow the effective approach introduced in \\cite{Agrawal:2010fh} by\nstarting from different fundamental theories with given renormalisable\ninteractions between Standard Model fermions and the\n hypothesized dark matter particles that are mediated by a very massive\nparticle. From these theories we deduce effective 4--particle--vertices for\nenergies significantly smaller than the mass of the mediator. Working\nwith these effective operators, one can deduce information about the\neffective coupling and propagate this information to the parameters of\nthe corresponding underlying fundamental theory. The effective \napproach allows us to reduce the dimensionality of the parameter space\nand more easily compare the different experimental searches.\n\n\n\\subsection{Deriving Effective Lagrangians}\nWe start with a list of fundamental Lagrangians taken from\n\\cite{Agrawal:2010fh}. However we do not perform a non--relativistic\napproximation, since we are interested in the phenomenology of\n this Lagrangian at a high energy\nexperiment and therefore the results for our effective operators\ndiffer. We also use a different method to evaluate the effective\nvertices, motivated in Ref.~\\cite{Haba:2011vi}, which uses the path\nintegral formalism.\n\nWe give one explicit example for the derivation of the effective\noperators and only mention specific peculiarities for the other cases,\nwhich are apart from that calculated similarly. Let $\\psi$ be a\nStandard Model fermion and $\\chi$ a complex scalar field representing\nthe dark matter candidate. For our example, we assume the mediator to\nbe a real scalar field, $\\phi$, with mass $M_\\Omega$ (we will keep\nthis notation for the mediator mass throughout). The relevant terms in\nthe UV completed Lagrangian are then given by,\n\\begin{align}\n\\mathscr{L_{\\text{UV}}} &= \\frac{1}{2} \\left[\\partial_\\mu \\phi(x) \\right]^2 -\n\\frac{1}{2} M_{\\Omega}^2 \\phi^2(x) - g_\\chi \\chi^\\dagger(x) \\chi(x) \\phi(x) \\nonumber \\\\\n& \\quad - \\bar{\\psi}(x) \\left( g_s +\n i g_p \\gamma^5 \\right) \\psi(x) \\phi(x)\\;, \\\\\n&\\equiv - \\frac{1}{2} \\phi(x) \\square_x \\phi(x) - \\frac{1}{2}\nM_{\\Omega}^2 \\phi^2(x) - F(x) \\phi(x)\\;. \n\\end{align}\nwhere the function $F(x)$ is given by,\n\\begin{align}\nF(x) & \\equiv g_\\chi \\chi^\\dagger(x) \\chi(x) + \\bar{\\psi}(x) \\left( g_s + i\n g_p \\gamma^5 \\right) \\psi(x) \\,.\n\\end{align}\nWe have not included the kinetic terms for $\\chi,\\,\\psi$, as they are\nnot relevant for the computation of the effective Lagrangian. In this\nparticular example, $g_s$, $g_p$ are dimensionless couplings and\n$g_\\chi$ is a dimension one parameter but these definitions can change\ndepending upon the precise model studied and we shall use this notation throughout. \nWe have included the kinetic term\nfor $\\phi$, the heavy mediator field. After integrating out $\\phi$, we\nobtain the effective Lagrangian,\n\\begin{align}\n\\mathscr{L}_\\text{eff} &= \\frac{1}{2 M_\\Omega^2} F^2 \\supset \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\dagger\n \\chi \\bar{\\psi} \\left(g_s + i g_p \\gamma^5 \\right) \\psi \\;.\n\\end{align}\n\\begin{table*}\n\\centering\n\\renewcommand{\\arraystretch}{1.525}\n\\begin{tabular}{c c c l}\n\\hline\n\\multirow{2}{*}{DM} & \\multirow{2}{*}{Med.} & \\multirow{2}{*}{Diagram} &\n$-\\mathscr{L}_{\\text{UV}}$ \\\\\n& & & $-\\mathscr{L}_{\\text{eff}}$ \\\\\n\\hline \\hline\n\\multirow{2}{*}{S} & \\multirow{2}{*}{S} & \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SS_2.pdf}}} & $g_\\chi \\chi^\\dagger \\chi \\phi + \\bar{\\psi} (g_s + i g_p \\gamma^5) \\psi \\phi$ \\\\\n& & & $\\displaystyle \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\dagger \\chi \\bar{\\psi} (g_s + i g_p \\gamma^5) \\psi$ \\\\\n\\hline\n\\multirow{2}{*}{S} & \\multirow{2}{*}{F} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SF_2.pdf}}}&\n $ \\bar{\\eta} (g_s + g_p \\gamma^5 ) \\psi \\chi + \\bar{\\psi} (g_s - g_p\n \\gamma^5 ) \\eta \\chi^\\dagger$ \\\\\n& & & $ \\displaystyle \\frac{1}{M_\\Omega} \\left[ (g_s^2 - g_p^2) \\bar{\\psi}\n \\psi \\chi^\\dagger \\chi + \\frac{i}{M_\\Omega} \\chi^\\dagger \\bar{\\psi} \\left(g_s^2 + g_p^2 - 2 g_s g_p \\gamma^5 \\right)\\gamma^\\mu \\partial_\\mu \\left( \\psi \\chi \\right) \\right]$ \\\\\n\\hline \n \\multirow{2}{*}{S} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{SV_2.pdf}}}&$\n g_\\chi (\\chi^\\dagger \\partial_\\mu \\chi - \\chi \\partial_\\mu\n \\chi^\\dagger) Z^\\mu + \\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\psi\n Z_\\mu$ \\\\\n& & & $\\displaystyle \\frac{g_\\chi}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r P_R \\right) \\psi \\left( \\phi^\\dagger \\partial_\\mu \\phi - \\phi \\partial_\\mu \\phi^\\dagger \\right)$\\\\\n\\hline \\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{S} &\n\\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FS_2.pdf}}}&\n$\\bar{\\chi} \\left(g_s + g_p \\gamma^5 \\right) \\chi \\phi + \\bar{\\psi} \\left( g_s\n + g_p \\gamma^5 \\right) \\psi \\phi$ \\\\\n& & & $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\chi} \\left(g_s + i g_p \\gamma^5 \\right) \\chi \\bar{\\psi} \\left( g_s + i g_p \\gamma^5 \\right)\\psi$\\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FV_2.pdf}}}&\n $\\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\psi Z_\\mu + \\bar{\\chi}\n \\gamma^\\mu (g_l P_L + g_r P_R) \\chi Z_\\mu $ \\\\\n& & & $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r P_R \\right) \\psi \\ \\bar{\\chi} \\gamma_\\mu \\left(g_l P_L + g_r P_R \\right) \\chi$\\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{tS} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FtS_2.pdf}}}&$\\bar{\\chi}\n \\left(g_s + g_p \\gamma^5 \\right) \\psi \\phi + \\bar{\\psi} \\left( g_s +\n g_p \\gamma^5 \\right) \\chi \\phi$ \\\\\n&&& $\\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi} \\left(g_s - g_p \\gamma^5 \\right) \\chi \\bar{\\chi} \\left( g_s + g_p \\gamma^5 \\right)\\psi $ \\\\\n\\hline\n\\multirow{2}{*}{F} & \\multirow{2}{*}{tV} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{FtV_2.pdf}}}&$\\bar{\\psi}\n \\gamma^\\mu (g_l P_L + g_r P_R) \\chi Z_\\mu + \\bar{\\chi} \\gamma^\\mu (g_l\n P_L + g_r P_R) \\psi Z_\\mu $ \\\\\n&&& $ \\displaystyle \\frac{1}{M_\\Omega^2} \\bar{\\psi}\n \\gamma^\\mu \\left(g_l P_L + g_r P_R \\right) \\chi \\bar{\\chi} \\gamma_\\mu \\left(g_l P_L + g_r P_R \\right) \\psi $ \\\\\n\\hline \\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{S} &\n\\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VS_2.pdf}}}\n& $-\\chi^\\mu \\chi_\\mu \\phi + \\bar{\\psi} (g_s + i g_p \\gamma^5)\\psi \\phi$ \\\\\n&&& $\\displaystyle - \\frac{g_\\chi}{M_\\Omega^2} \\chi^\\mu \\chi_\\mu \\bar{\\psi} \\left( g_s + i g_p \\gamma^5 \\right)\\psi$\\\\\n\\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{F} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VF_2.pdf}}}&\n $ - \\bar{\\eta} \\gamma^\\mu (g_l P_L + g_r P_R ) \\chi_\\mu\n + \\bar{\\psi} \\gamma^\\mu (g_l P_L + g_r P_R) \\eta \\chi_\\mu^\\dagger$ \\\\\n&&& $\\displaystyle \\frac{1 }{M_\\Omega} \\left[g_l g_r \\bar{\\psi} \\gamma^\\nu\n \\gamma^\\rho \\psi \\ \\chi^\\dagger_\\nu\\chi_\\rho + \\frac{i}{M_\\Omega} \\chi^\\dagger_\\nu \\bar{\\psi}\n \\gamma^\\nu\\gamma^\\mu \\gamma^\\rho \\left(g_l^2 P_L + g_r^2P_R\n \\right) \\partial_\\mu \\left( \\psi \\chi_\\rho \\right) +\\right]$\\\\\n\\hline\n\\multirow{2}{*}{V} & \\multirow{2}{*}{V} &\n \\multirow{2}{*}{\\raisebox{-0\\height}{\\includegraphics[width=0.1\\textwidth]{VV_2.pdf}}}&\n $i g_\\chi \\left[ Z_\\mu \\chi^\\dagger_\\nu {\\partial \\chi}^{\\mu \\nu} + Z_\\mu\n \\chi_\\nu \\partial \\chi^{\\mu \\nu} + \\chi^\\dagger_\\mu\n \\chi_\\nu \\partial Z^{\\mu \\nu} \\right] + \\bar{\\psi} \\gamma_\\mu (g_l P_L + g_r P_R ) \\psi$ \\\\\n&&& $\\displaystyle \\frac{i g_\\chi}{M_\\Omega^2} \\bar{\\psi} \\gamma^\\mu \\left( g_l P_L + g_r\n P_R \\right) \\psi \\left[\\chi^\\nu \\partial \\chi^\\dagger_{\\mu \\nu} - \\chi^{\\dagger, \\nu} \\partial \\chi_{\\mu \\nu} + \\partial^\\nu \\left(\\chi^\\dagger_\\nu\\chi_\\mu - \\chi^\\dagger_\\mu \\chi_\\nu \\right) \\right]$\\\\\n\\hline\n\\end{tabular}\n\\caption{List of interaction vertices for S(calar), F(ermion) and \nV(ector) dark matter, $\\chi$, before and after integrating out the \nheavy mediator scalar field $\\phi$, spinor field $\\eta$ or vector field\n$Z^\\mu$ with mass $M_{\\Omega}$. $\\psi$ denotes the Standard Model fermion.\n$\\partial X^{\\mu \\nu} \\equiv \\partial^\\mu X^\\nu - \\partial^\\nu X^\\mu$.\ntS and tV denote cases where the mediator is exchanged in the $t$-channel.}\n\\label{tbl:allmodels}\n\\end{table*}\nCases with different spin for the dark matter or the mediator particle\nare evaluated similarly. We only want to give some special remarks:\n\\begin{itemize}\n\\item For spin--$1\/2$ mediators, the Dirac propagator\nhas only one power of $M_\\Omega$ in the denominator,\n\\begin{align}\n\\frac{1}{\\slashed{p} - M_\\Omega} \\approx -\\frac{1}{M_\\Omega} - \\frac{\\slashed{p}}{M_\\Omega^2}. \n\\end{align}\nWe therefore get two effective vertices after expanding the Lagrangian up to order $1\/M_\\Omega^2$.\n\n\\item Some effective operators give derivatives on the Standard Model fermion\n fields. These are not negligible, since they only vanish if the\n Dirac equation $i \\slashed{\\partial} \\psi = m \\psi$ can be used and\n the fermion mass $m$ is small. This is not the case for e.g.\\ heavy\n quark contributions in the annihilation sector and processes with\n off--shell fermions.\n\n\\item We use the same list of effective operators for the cases of real scalar\n ($\\chi = \\chi^\\dagger$), real vector ($\\chi_\\mu = \\chi^\\dagger_\\mu$)\n or Majorana fermion \\cite{Denner:1992vza} dark matter\n fields. However, we would like to mention that for consistency we do\n not introduce additional factors of $\\nicefrac{1}{2}$ in the\n couplings as is often done in the case of real fields.\n\n\\end{itemize}\n\nThe full list of models with their respective fundamental and effective\nLagrangians is given in Table~\\ref{tbl:allmodels}. Note that all Lagrangians are\nhermitian by construction.\n\n\\subsection{Benchmark Models}\n\n\nThe effective operators described above have multiple\nindependent parameters to describe the effective coupling, for example $g_\\chi, g_l, g_r$ and $M_\n\\Omega$ in the scalar dark matter, vector mediator (SV) case or\n$g_s, g_p$ and $M_\\Omega$ in the fermion dark matter, scalar mediator (FS) case in Table~\\ref{tbl:allmodels}. \n\\begin{table}\n\\centering\n\\begin{tabular}{l@{\\quad}l@{\\quad}l}\n\\hline\nOperators & Definition & Name \\\\\n\\hline \\hline\nSS, VS, FS, & $g_p = 0$ & scalar \\\\\n FtS, FtSr: & $g_s = 0$ & pseudoscalar \\\\\n\\hline\nSF, SFr: & $g_p = 0, M_\\Omega = \\unit{1}{\\TeV}$ & scalar\\_low \\\\\n & $g_p = 0, M_\\Omega = \\unit{10}{\\TeV}$ & scalar\\_high \\\\\n & $g_s = 0, M_\\Omega = \\unit{1}{\\TeV}$ & pseudoscalar\\_low \\\\\n & $g_s = 0, M_\\Omega = \\unit{10}{\\TeV}$ & pseudoscalar\\_high \\\\\n\\hline\nSV, FV, FtV, & $g_l = g_r$ & vector \\\\\n FtVr, VV: & $g_l = -g_r$ & axialvector \\\\\n & $g_l = 0$ & right--handed \\\\\n\\hline\nVF, VFr: & $g_l = g_r, M_\\Omega = \\unit{1}{\\TeV}$ & vector\\_low \\\\\n & $g_l = -g_r, M_\\Omega = \\unit{10}{\\TeV}$ & vector\\_high \\\\\n & $g_l = g_r, M_\\Omega = \\unit{1}{\\TeV}$ & axialvector\\_low \\\\\n & $g_l = -g_r, M_\\Omega = \\unit{10}{\\TeV}$ & axialvector\\_high \\\\\n\\hline\nFVr : & $g_l = 0$ & right--handed \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{Benchmark models with specific values for the coupling constants shown in Table \\ref{tbl:allmodels}.}\n\\label{tbl:constraints}\n\\end{table}\nConsidering the full range of parameters would lead to a\nplethora of scenarios, well beyond the scope of this paper. Thus we restrict our \nanalysis to specific benchmark models (see\nTable~\\ref{tbl:constraints}) with constraints on the individual\ncouplings such that only one overall multiplicative factor\nremains. The effective coupling constant $G$ for each model is then\ndefined as $G \\equiv g_ig_j\/ M_\\Omega^2$.\nFor models with fermionic\nmediators, the leading term has only a $1\/M_\\Omega$ dependence, which is\nwhy we define $G \\equiv g_ig_j\/ M_\\Omega$ for these. We also choose two\npossible values for $M_\\Omega$ to represent different\nsuppression scales of the respective second order terms. Models with\nreal fields that are trivially connected to the corresponding complex\ncases by multiplicative prefactors are not taken into account\nseparately. We also omit models with left--handed couplings that are\nrelated to the respective right--coupled cases. Information on these\ncan easily be extracted from the related models by rescaling the\ncorresponding result accordingly.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nRecently, the H1~\\cite{H1} and ZEUS~\\cite{zeus} collaborations at HERA\nannounced an anomaly at high-$Q^2$ in the $e^+p\\to eX$ neutral current\n(NC) channel. Using a combined accumulated luminosity of\n$34.3\\invpb$ in $e^+p\\to eX$ mode at $\\sqrt{s}=300\\gev$, the two\nexperiments have\nobserved 24 events with $Q^2>15000\\gev^2$ against a Standard Model (SM)\nexpectation of $13.4\\pm1.0$, and 6 events with $Q^2>25000\\gev^2$ against\nan expectation of only $1.52\\pm0.18$. Furthermore, the\nhigh-$Q^2$ events are clustered at Bjorken-$x$ values near 0.4 to 0.5.\n\nA number of authors have presented proposals for new physics which\nmight explain the NC anomaly, including: \nnew contact interactions~\\cite{bkmrw,altarelli,contact},\n$s$-channel leptoquark production~\\cite{bkmrw,altarelli,leptoquark},\nand R-parity violating supersymmetry~\\cite{altarelli,rparity}, and\nrelated proposals~\\cite{others}.\nThere exist a variety of criticisms for each of the proposed ideas, from\ntheir highly speculative nature, to very concrete flavor problems which\nthey all share~\\cite{bkmrw}. As such, another somewhat less\nspeculative idea has been proposed~\\cite{tung}\\ in which the anomaly in the\nHERA data is simply further evidence of our inability to calculate, and often \neven reliably fit, the parton probability distribution\nfunctions (PDFs) resulting from the non-perturbative dynamics\ninside the proton. That is, Kuhlmann, \\etal~\\cite{tung},\nhave suggested a way by which the behavior of the PDFs at large-$x$ can be \nmodified to explain part, or all, of the HERA NC data without disrupting\nthe fits of the old low-$Q^2$, low-to-moderate-$x$ data. \n\nIt is actually a simple exercise to show that increasing the parton\ndensities at $Q^2\\sim20,000\\gev^2$ and $x\\sim0.4$ to 0.5 can in fact \nfit the NC data.\nIt is a far more complicated question whether these changes are consistent\nwith all other world data.\nWe will not consider here the validity of the claim that such a fit can be\ndone which is consistent with the low-$Q^2$ data. Instead we will show that\nthere can be dramatic consequences for the HERA data itself in the charged\ncurrent mode at high-$Q^2$. We will find that the HERA data already\nrules out a number of theoretically acceptable scenarios for modifying the\nPDFs to agree with the NC data.\n\nHERA is capable of running in two modes: $e^-p$ and $e^+p$. In the former mode\nH1 and ZEUS have accumulated $1.53\\invpb$ of data but have observed\nno statistically significant deviations from the SM. Further, the experiments\ndifferentiate between final state $eX$ and $\\nu X$, where the neutrino is\nidentified through its missing $p_T$. H1 has also announced its\nfindings in the $e^+p\\to\\nu X$ charged current (CC) channel. They find\n3 events at $Q^2>20000\\gev^2$ with an expectation of $0.74\\pm0.39$, but\nno events with $Q^2>25000\\gev^2$. ZEUS has not announced its CC data as of\nthis date.\nAlthough compared to the NC data, the present CC data is much sparser,\none can still conclude from it that there can be no deviations from the SM\nby more than a factor of 2 or 3 in that channel. Whether there is in fact\nany deviation at all is still too uncertain to say. (Explanations of the\npossible CC excess involving non-SM physics have been considered\nby~\\cite{altbkmrCC}.) However, given the \nsizes of the effects which we are going to find, more data in $e^+p$, and \nespecially $e^-p$, mode will provide very strong constraints on the\nviability of this suggestion.\n\n\n\\section{The proposal}\n\nThe class of proposals considered by Kuhlmann, \\etal~\\cite{tung},\nfor explaining the HERA anomaly can be thought of as\nvariations on the so-called ``intrinsic charm'' scenario\\cite{charm}\nin which it is posited that there is some non-perturbative, valence\ncontribution to the charm quark distributions at $Q^2\\to0$.\n(Current fits assume that there is no valence\ncharm component in the proton and generate non-zero\ncontributions at higher $Q^2$ only through perturbative renormalization\ngroup flow, \\ie,\ngluon splitting. In this scenario one can think of the valence\nstructure of the proton being $uudc\\bar c$ versus the usual $uud$.)\n\nThough the underlying dynamics (and motivation) may be subtle, \nthe proposal itself is straightforward: increase by hand\nthe parton densities inside the proton at $x\\sim0.5$ and large $Q^2$ to\nthe point that the NC cross-section of the Standard Model\nmatches that observed at HERA. Such an effect can arise naturally from \nnon-perturbative dynamics at low $Q^2$ if the dynamics produce a narrow\n``bump'' in the PDFs at low $Q^2$ and very large $x\\sim1$. This bump would\nmigrate down to lower $x$ as one flows (through the renormalization group) \nup to higher $Q^2$. Such a structure is difficult to rule out; the data\nat low $Q^2$ and very high $x$ is limited and extractions of the \nstructure functions are problematic due to non-perturbative and higher-twist\neffects which can be important at large $x$. Fits available now typically\nuse only data with $x\\lsim0.8$.\nThe proposal in \\cite{tung}\\ emphasized enhancements to the\n$u$-quark density rather than that of the $c$-quark, but the difference is\nirrelevant from the point of view of this paper because electroweak\nphysics does not distinguish among the generations.\nEnhancement of any or all of the parton densities\ncould in principle explain the HERA data.\n\nThis proposal has strong advantages\n{\\sl and} disadvantages. In its favor, it requires no new physics beyond the\nStandard Model and thus automatically solves the flavor problem associated\nwith many of the new physics interpretations~\\cite{bkmrw}. \nTo its detriment, it invokes\nnon-perturbative QCD effects which are not calculable and cannot be\npredicted {\\sl ab initio}, and its consistency with the moderate\n$Q^2$ data taken at both HERA and the Tevatron has not been fully studied. \n\n\\section{High-$Q^2$ tests at HERA}\n\nPutting aside any questions of how well such a proposal can really do \nat explaining the HERA NC data while\nremaining consistent with all other world data, one can ask:\nwhat are the consequences of such a suggestion\nfor the NC process in $e^-$ mode, and for CC processes in either mode?\n\nThe NC result is simple: in this scenario, the NC cross-sections in both\n$e^-p$ and $e^+p$ modes scale by approximately the same amount. The $Z$\ncouplings introduce a small helicity dependence that keeps the two modes\nfrom being exactly the same. However, the differences between the two are\nless than 10\\% because the photon contribution typically dominates the \ntotal cross-section.\nThus this proposal predicts that once HERA has\naccumulated enough data in $e^-p$ mode, they will observe a NC anomaly there\nas well, and it should be roughly the same size as that observed in $e^+p$.\n\nIn discussing the CC predictions, a number of complications arise,\nmost stemming from the fact that the $x$ and $Q^2$ dependences of the \nNC and CC cross-sections are somewhat different. Nonetheless,\nwe will present fairly precise heuristic arguments about the sizes of CC \neffects, which we will then check in a full numerical calculation.\n\nOne key simplification is this: in the SM, even at high-$Q^2$,\nthe NC scattering is largely dominated by virtual photon exchange\nin the $t$-channel. This need not have been so, since at $Q^2>>m_Z$ there\nis no additional kinematic suppression of the $Z$ contributions; however,\nthe $Z$ coupling to quarks is generally smaller than that of the\nphoton. For the arguments that follow, we will ignore the $Z$ contributions\nand reintroduce them only when going to the full calculation in the next\nsection. \n\nFor photon exchange alone, the NC cross-section at high-$Q^2$ behaves as:\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma_{NC}}{dx}\\propto u(x)+\\bar u(x)\n+\\frac{1}{4}\\left\\{d(x)+\\bar d(x)\\right\\}+\\cdots\n\\eeq\nwhere $u(x)$ is the $u$-quark parton probability distribution function (PDF)\ninside a proton,\n$\\bar u(x)$ is the $u$-antiquark\nPDF, {\\sl etc.}, and the ellipses represent heavier ($s$, $c$, $b$, $t$) \nquarks. The factor of $1\/4$ is the relative charge-squared of \n$u$- and $d$-type quarks.\n\nThere are four orthogonal classes of changes to the PDF's that can be\nconsidered, each corresponding to enhancing the densities of either\n$u$, $\\bar u$, $d$ or $\\bar d$ individually. \nThat we do not have to consider changes to\nthe charm and strange densities is clear, since the physics in question cannot\ndistinguish $u$ from $c$, or $d$ from $s$. (This is not a general statement\nfor all experiments at all $Q^2$. For example, it does not hold even at HERA\nif H1 and\/or ZEUS could tag prompt charm production.)\nEach of these four cases leads to a distinctive CC signature.\n\nWe will parameterize the effect of an ``intrinsic quark'' component on\nthe PDFs by:\n\\beq\nq(x)=q_0(x)+q_{\\rm int}(x)\\equiv q_0(x)\\epsilon_q(x)\n\\eeq\nwhere $q(x)$ is the total parton distribution,\n$q_0(x)$ is the usual fit distribution (we will use the CTEQ3 set whenever\nwe have to make an explicit choice~\\cite{cteq}, similar results would\nfollow from the MRS sets~\\cite{mrs}), \n$q_{\\rm int}(x)$ is the intrinsic component and\n$\\epsilon_q(x)\\geq1$ parameterizes the effects of the enhanced\ncomponent. (The $Q^2$ dependence of $q(x)$ and $\\epsilon(x)$\nis implicit and of little relevance to what follows since we will only \nconsider scattering within a small range of $Q^2$ values; therefore\nthe renormalization group running with $Q^2$ can be ignored.)\nFor simplicity, suppose that $\\epsilon_q(x)$\nis exactly 1 everywhere except in a small range of $x$ centered on $x=x_0$\nfor which it is much greater and roughly constant (\\ie, it is roughly a\ntop hat distribution):\n\\beq\n\\eps_q(x)=1+\\eta_q\\theta(x-x_0+\\delta)\\theta(x_0+\\delta-x)\n\\label{epsq}\n\\eeq\nwhere $\\eta_q>0$, $\\theta(x)$ is the usual step function, and $\\delta$ is\nthe half-width of the top hat.\nThough the numerical arguments do not rely on making this\nassumption, it does make the pedagogy simpler.\n\nTo begin, suppose that the correct fit to the HERA NC data is achieved\nby changing only $u(x)$ such that $x_0$, $\\delta$ and $\\eps_u(x_0)$\nhave certain fit values. This will be the canonical scenario to which we\ncompare all others. For example, suppose that instead of changing the $u(x)$\nPDFs, we would like to do a fit for which only $d(x)$ is changed.\nThen to produce the same NC cross-section that $\\eps_u(x_0)$ provided in\nthe $u$-case, $\\eps_d(x_0)$ must shift by a larger amount, though at the same\n$x=x_0$. That the shift must be larger is clear, because $d$-quark effects\non the NC cross-section are suppressed both by electric charge (the 1\/4)\nand by the smaller $d$-quark content in the proton, $d_0(x_0)\/u_0(x_0)$.\nIn terms of an equation, setting the shift due\nto the $\\eps_u$ in the one case equal to the new shift by $\\eps_d$:\n\\beq\n\\eps_u(x_0)u_0(x_0)+\\frac{1}{4}d_0(x_0)=u_0(x_0)+\\frac{1}{4}\\eps_d(x_0)\nd_0(x_0),\n\\eeq\ngiving\n$\\eps_d=1+4(\\eps_u-1)(u_0\/d_0)$, with all functions evaluated at $x=x_0$.\nTo get the same effect by enhancing $\\bar u(x)$ alone requires $\\eps_{\\bar u}\n=1+(\\eps_u-1)(u_0\/\\bar u_0)$ at $x=x_0$. Similarly, changing $\\bar d(x)$ alone\nis identical to the case of $d(x)$, but with $d\\to\\bar d$ in the expression\nabove.\n\nWhat effects do these changes have on the charged current? \nIn terms of the PDFs, the CC cross-sections scale as:\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma^+_{CC}}{dx\\,dy}\\propto d(x)(1-y)^2+\\bar u(x)\n+\\cdots,\n\\eeq\n\\beq\n\\frac{1}{x}\\,\\frac{d\\sigma^-_{CC}}{dx\\,dy}\\propto u(x) + \\bar d(x)(1-y)^2\n+\\cdots\n\\eeq\nwhere the $+[-]$ superscript denotes scattering in $e^+p[e^-p]$ mode, and\n$y$ has its usual definition in deep inelastic scattering. In the\ncenter-of-mass frame, $y=\\sin^2(\\vartheta\/2)$ where $\\vartheta$ is the $e^\\pm$\nscattering angle; its presence in the expressions above follows trivially\nfrom angular momentum conservation in the $(V-A)$ \nscattering process. Note also that\nfor the usual PDFs at large $x$, to a very good approximation\n$\\sigma^+_{CC}\\propto d(x)(1-y)^2$ and $\\sigma^-_{CC}\\propto u(x)$.\n\nFor the moment, in order to examine the effects of enhancing the PDFs\non the CC differential cross-section, we will restrict ourselves\nto the region around $x=x_0$ of half-width $\\delta$. We will\ndenote the differential cross-section in this small region\n$d\\sigma^\\pm_{CC}(x_0)$ as a shorthand. In the first case discussed above,\nin which only $u(x_0)$ is changed in response to the NC data,\nthe CC signal in $e^+$ mode is unchanged, while\n$d\\sigma^-_{CC}(x_0)$ increases by $\\eps_u(x_0)$. Thus the same relative\nchange in the NC data will also occur in the $e^-p$ CC data, at least in the\nneighborhood of $x=x_0$.\nHowever, because the kinematic dependence on $x$ (at high-$Q^2$)\nis roughly the same in $e^+u\\to e^+u$ as in $e^-u\\to\\nu d$, the relative\nscaling of the NC to CC in the $x_0$-region will hold for all $x$.\n(The preceding statement is exact in the limit $Q^2\\gg m_W^2$ and when the\n$Z$ contribution to NC can be ignored. Since these two conditions are not\nsimultaneously satisfied in general, there will be important corrections\nto the results of these heuristic arguments. Nonetheless, the results derived\nhere do not differ greatly from the exact results, as we will see.)\nThus, to have a concrete example, if $\\eps_u(x_0)$ is chosen to double the\nhigh $Q^2$ NC cross-section, it will also (approximately)\ndouble the high-$Q^2$ CC\ncross-section in $e^-$ mode, while having no effect in $e^+$ mode.\n\nThe effect on the CC is more marked in the case of using $d(x)$ to explain the\nNC anomaly. As we said above, to have the same effect on the NC data as\n$\\eps_u$, the $\\eps_d$ must be roughly $4[u_0(x_0)\/d_0(x_0)]$ times bigger\nthan the corresponding $\\eps_u$ would have been. And since to a good\napproximation\n\\beq\nd\\sigma^+_{CC}(x_0)\\propto\\eps_d(x_0)\\simeq4\\frac{u_0(x_0)}{d_0(x_0)}\n\\eps_u(x_0),\n\\eeq\nit will scale by the same amount. Consider again our\nexample, where we demanded that $\\eps_u(x_0)$ double the high-$Q^2$ NC data.\nFor $x_0\\simeq0.5$ one finds $u_0(x_0)\\simeq4d_0(x_0)$. Then to fit the\nsame NC data, $\\eps_d(x_0)$ must be $4\\cdot4=16$ times larger than\n$\\eps_u(x_0)$. Moving this enhancement to the CC one finds that\n$d\\sigma^+_{CC}(x_0)$ increases by a factor of $16\\eps_u(x_0)$,\nwhile $\\sigma^-_{CC}$ is unchanged. Again the explicit kinematic\ndependences on $x$ are (approximately) \nthe same in the CC and NC, so the overall $\\sigma^+_{CC}$ will increase \nby roughly a factor of 32 in the high-$Q^2$ region.\n\nThe same arguments go through for $\\bar u$ and $\\bar d$. For $\\bar u$,\n$d\\sigma^+_{CC}(x_0)$ scales by a factor\n\\beq \nd\\sigma^+_{CC}(x_0)\\propto 1+\\frac{\\eps_u-1}{(y-1)^2}\\,\\frac{u_0}{d_0}\n\\simeq\\frac{\\eps_u}{(1-y)^2}\\,\\frac{u_0}{d_0},\n\\eeq\nwhile $\\sigma^-_{CC}$ is unchanged. For explaining the HERA anomaly we \nwould be interested in $Q^2\\simeq2\\times10^4\\gev^2$ and $x\\simeq\n0.5$, and thus $y=Q^2\/(xs)\\simeq0.5$.\nThen in our recurring example wherein the NC signal is doubled,\n$d\\sigma^+_{CC}(x_0)$ scales by a factor of $16\\eps_u$, leading to an\noverall scaling of $\\sigma^+_{CC}$ again by 32.\n\nIn the final case, with enhanced\n$\\bar d(x)$, $d\\sigma^-_{CC}(x_0)$ scales by a factor \n\\beq\nd\\sigma^-_{CC}(x_0)\\propto 1+4(\\eps_u-1)(1-y)^2\n\\eeq\nwhile $\\sigma^+_{CC}$ is unchanged. In our recurring example,\nwith $y\\simeq0.5$, we find $d\\sigma^-_{CC}(x_0)$ scales by approximately\n$\\eps_u$, so that the full $\\sigma^-_{CC}$ at high-$Q^2$ is expected to be\ndouble the usual SM prediction.\n\nThe various scalings of the CC mode are summarized in Table~\\ref{tungtable}.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c|c|c|} \\hline\nParton & NC Factor & CC${}^+$ Factor & CC${}^-$ Factor \\\\ \\hline & & & \\\\\n$u$ & $\\eps_u$ & 1 & $\\eps_u$ \\\\ & & & \\\\\n$d$ & $1+4(\\eps_u-1)\\fract{u_0}{d_0}$ & $1+4(\\eps_u-1)\n\\fract{u_0}{d_0}$ & 1 \\\\ & & & \\\\\n$\\bar u$ & $1+(\\eps_u-1)\\fract{u_0}{\\bar u_0}$ & $1+\\frac{\\eps_u-1}{(1-y)^2}\n\\,\\fract{u_0}{d_0}$\n& 1 \\\\ & & & \\\\\n$\\bar d$ & $1+4(\\eps_u-1)\\fract{u_0}{\\bar d_0}$ & 1 &\n$1+4(\\eps_u-1)(1-y)^2$ \\\\ & & & \\\\ \\hline\n\\end{tabular}\n\\label{tungtable}\n\\caption{For each parton is shown the relative sizes of contributions needed\nto explain the HERA NC data, as well as the predictions for the enhancement\nof the CC signal, $d\\sigma^\\pm_{CC}(x_0)$. All quantities are implicitly\nfunctions of $x$ and are to be evaluated at $x=x_0$. These results are derived\nin the heuristic scenario discussed in the text. Note that for $\\eps_u\\gg1$\nthe scaling factors are independent of $\\eps_u$.}\n\\end{table}\nFor each parton, if we assume that it alone must have its density scaled to\nexplain the HERA NC data, the size of the enhancement needed (relative to\nthe enhancement $\\eps_u$ for the $u$-quark) is given in the second column.\nIn the third and fourth column are then shown the relative enhancements\nof $d\\sigma^\\pm_{CC}(x_0)$ given a fit to the NC.\n\nSome profound results can already be extracted from these simple \nconsiderations. If the PDF's of $(u, d, \\bar u, \\bar d)$ are each \nchanged respectively so as \nto double the high-$Q^2$ NC data, then the $e^+$ CC data will increase by \nfactors of about\n(1, 32, 32, 1) while the $e^-$ CC data will increase by (2, 1, 1, 2).\nSince doubling the NC cross-section at high-$Q^2$ is roughly \nconsistent with the\nHERA data, we can conclude that we either expect extremely large enhancements\nin the $e^+$ CC signal, or none at all. The data from H1 is not consistent\nwith an extremely large enhancement (such as a factor of 32), so we can\nconclude that: (1) changing the PDF's of a $d$-type quark or a $\\bar u$-type\nquark to explain the HERA NC data is inconsistent with the HERA CC data;\n(2) any other explanation\ninvolving modifications to the $u$-type or $\\bar d$-type PDFs will not\nlead to a CC signal in $e^+$ mode, but will have one in $e^-$ mode;\n(3) if HERA sees an enhancement in the $e^+p$ CC channel\ncomparable to the one in the NC channel, more than one\nPDF must be modified. Note that\n(2) above does not preclude a small CC enhancement in $e^+$ mode; but\nany such enhancement will not be enough to also explain the NC data.\n\nUp until now, our arguments have ignored the complications introduced by\nkeeping the $Z$ contributions in the NC, and by the non-zero $W$ mass in the\nCC. To show how these results are modified in a complete calculation, we have\nnumerically evaluated the CC signals which would be induced by fitting to the\nNC anomaly. In Figure~\\ref{figure},\nwe have considered each of the $u$, $d$, $\\bar u$ and $\\bar d$ cases\nseparately as a solution to the HERA NC anomaly. (We are only using the H1\nNC and CC data since as of this writing ZEUS has not announced their CC \nresults.) We have\nchosen in each case for $\\eps_q(x)$ to have a top-hat form as in \nEq.~(\\ref{epsq}) with $x_0=0.45$ and $\\delta=0.05$. This region in $x$\nenvelopes the bulk of the H1 high $Q^2$ events. The overall normalization\n$\\eta_q$ is chosen to provide the best possible $\\chi^2$ fit to the H1 data.\n(The best fit values of $\\eta_q$ for $q=u,d,\\bar u,\\bar d$ were found to be\n7.5, 95, 640, 450 using the CTEQ3M PDFs. $\\eta_q$ for the sea quarks shows\na strong dependence on the PDFs used, because of the uncertainty in the sea\nquark distributions at moderate $x$ and large $Q^2$. However the overall \nenhancement of the CC signal is independent of the choice.)\nThen we have plotted the resulting \n$d\\sigma^\\pm_{CC}\/dQ^2$ for each possibility.\n\\begin{figure}\n\\centering\n\\epsfxsize=5in\n\\hspace*{0in}\n\\epsffile{tung.eps}\n\\caption{Differential cross-section for CC $e^+p$ and $e^-p$ scattering\nas a function of $Q^2$. Dotted lines are the SM prediction with unmodified\nPDFs; solid lines are with modified PDFs.}\n\\label{figure}\n\\end{figure}\n\nThe Figure is divided into two frames for the CC processes $e^+p\\to\\bar\\nu X$\nand $e^-p\\to\\nu X$ separately. For the $e^+p$ mode, the enhancement of\nthe CC cross-sections is marked, consistent with our heuristic derivation of\nan enhancement factor of $\\sim32$. For the $e^-p$ mode, the\nenhancements are much smaller, once again consistent with our expectation\nof factors of $\\sim2$ only. \n\nNotice from the figure that the peak enhancements as a function\nof $Q^2$ can be much larger than those of the total integrated cross-sections,\nwhich we give in Table~\\ref{fittable}. There\nwe have done the $Q^2$ integration in the Figure,\nfor $Q^2>10,000\\gev^2$ and $Q^2>20,000\\gev^2$.\n\\begin{table}\n\\centering\n\\begin{tabular}{|c|c||c|c|} \\hline\nPDF & Mode & $Q^2>10,000\\gev^2$ & $Q^2>20,000\\gev^2$ \\\\ \\hline\n$u$ & $e^-p$ & 1.9 & 2.9 \\\\\n$d$ & $e^+p$ & 10 & 21 \\\\\n$\\bar u$ & $e^+p$ & 5.5 & 25 \\\\\n$\\bar d$ & $e^-p$ & 1.3 & 1.2 \\\\ \\hline\n\\end{tabular}\n\\label{fittable}\n\\caption{Multiplicative enhancements of the CC cross-sections for $Q^2$ above\nthe indicated value, in the mode indicated. The PDFs in the first column\nhave been changed to give a best fit to the H1 NC data.}\n\\end{table}\nThe most important H1 cuts have been included in the calculation:\n$y<0.9$ and $p_{T,{\\rm miss}}>50\\gev$. Our pedagogical derivation of the\nenhancements has been shown to work reasonably well, though not exactly.\nThe two PDFs which affect the $e^+p$ mode both induce very large corrections\nin the CC if they are invoked to explain the NC data. However the two\nPDFs which affect the $e^-p$ mode induce small, but observable, corrections.\nGiven high statistics, HERA should be able to probe even the hardest case,\nthat of changing $\\bar d(x)$ to explain the NC data. With the current data,\nit already appears that altering $d(x)$ or $\\bar u(x)$ cannot be invoked to\nexplain the NC anomaly.\n\nThe possibility exists of probing more complicated combinations of \nmodifications because\nthe cross-sections at HERA are linear in the PDFs. Therefore in a scenario in\nwhich several of the PDFs are modified so that the total modification is\na sum of individual parton modifications, weighted by some $\\alpha_i$, then\nthe resulting CC signals are also linear combinations of those shown, again\nweighted by the $\\alpha_i$. For example, if enhancements to the $u$ and $d$ are\nsuch that each one contributes half of the NC excess, then the CC signal\nwill be an average of the individual signals for $u$ and $d$. In particular,\nthe intrinsic charm scenario, for which $\\eps_c=\n\\eps_{\\bar c}$ is responsible for explaining the HERA data, the\n$d\\sigma^+_{CC}(x_0)$ is scaled by $1+(\\eps_u-1)(u_0\/2d_0)\/(1-y)^2$ and\n$d\\sigma^-_{CC}(x_0)$ is scaled by\n$(1+\\eps_u)\/2$ where $\\eps_u=1+2(\\eps_c-1)(c_0\/u_0)$.\nThis linear behavior provides\na powerful, and simple, way to disentangle the form of the modifications\ngiven the size of the excesses over SM in the two CC modes. Unfortunately,\nwith only 2 data points for 4 unknowns, one cannot deduce the full answer\nusing CC data alone. However, it is already enough to rule out the\nintrinsic charm scenario as an explanation of the NC data, given the\nnumbers in Table~\\ref{fittable}.\n\n\n\\section{Conclusions}\n\nModifying the PDFs to explain the HERA $e^+p$ NC data implies\nan immediate modification (by about the same relative size) of\nthe $e^-p$ NC data,\nand also implies striking patterns of modification in the CC data.\nSuch modifications should be easy to observe given the current size of\nthe NC anomaly. Further, they already rule out the intrinsic charm scenario\n(with equal modifications to $c(x)$ and $\\bar c(x)$),\nor any other attempt to use modifications of $d$-type or $\\bar u$-type\ndistribution functions to account for the anomaly in the HERA NC data.\nOur results for the CC channel are summarized in Table~\\ref{fittable}\nand Figure~\\ref{figure}.\n\nIf an excess in the CC signal is detected at HERA in the $e^+p$ data\nof the size currently suggested by H1, then one must go to a scenario in\nwhich more than one PDF are modified, but such that one dominates the \ncontribution to the $e^+p$ NC excess. This will lead to an interesting \nsignal at HERA in the $e^-p$ CC mode. In particular,\nonce significant luminosity has been collected in $e^-p$ mode at HERA, one \nshould also be able to probe solutions to the NC anomaly which involve\nonly changing the distributions of $u$- and $\\bar d$-type quarks.\n\n\n\n\\section*{Acknowledgments}\n\nWe wish to thank J.~Ellis, E.~Weinberg, F.~Wilczek and C.P.~Yuan\nfor useful conversations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}