diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzggts" "b/data_all_eng_slimpj/shuffled/split2/finalzzggts" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzggts" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n As quantitatively pioneered by those such as \n\\cite{Hoyle}~(1953), \\cite{Silk77}~(1977) and \\cite{ReesO78}~(1978) and more\nrecently noted by authors such as Frenk \\mbox{et\\,al.}\\ (1995), hierarchical galaxy\nformation models in $\\Omega_0=1$ cold dark matter (CDM) universes \ntypically combine assumptions on up to six distinct physical\nprocesses: (1) the non-linear growth \nphase of matter density peaks (known as ``haloes''),\n(2) cooling gas dynamics, (3) star formation, (4) star-to-gas\nenergy feedback, (5) stellar evolution, (6) galaxy mergers. In principle,\nif there are more free parameters describing these processes\nthan independent observational galaxy statistics, \nthen the observations should provide little constraint on\n galaxy formation ``recipes''.\nFortunately, the contrary is\npresently the case for the ``semi-analytical {\\em ab initio}'' models which \nmake various \nanalytical estimates of process (1), \ncombine semi-empirical and simple scaling \nparametrisations to represent processes (2)-(4) and (6) and\nuse evolutionary stellar population synthesis for process (5). \nSince each of these models have problems explaining at least some \nof the observations\nmeans, the models are better constrained than might\nhave been hoped for.\n\n\tThese models can be considered to be semi-analytical because rather\nthan calculating what is possibly the most important process, \nthe non-linear formation and merging history \nof collapsed objects [process (1)], via N-body simulations, various \nstatistical analytical approximations are used.\nThe models of Lacey \\mbox{et\\,al.}\\ (1993) use an approximation developed \nin \\cite{LS91}~(1991) from the BBKS peaks formalism \n(\\cite{BBKS}~1986),\nKauffmann et {al.} (\\cite{KW93}~1993; \\cite{KWG93}~1993)\nuse a probabilistic method (\\cite{Bower91}~1991) \nbased on the Press-Schecter formalism \n(e.g., \\cite{PS74}~1974; White \\& Frenk 1991) and \nexcursion set mass function calculations (\\cite{Bond91}~1991) while \n\\cite{ColeGF94}~(1994) use a spatially quantised ``block'' method\ndescribed in \\cite{ColeKais}~(1988). \n\nFurther semi-analytical developments \ninclude those adding spatial auto-correlation information \nto a Press-Schechter formalism (\\cite{MoWh96}~1996) or to the\n``block'' model (\\cite{RT96}~1996; \\cite{Naga97}~1997) and a technique \nof separately treating global, weakly non-linear and local, strongly \nnon-linear dynamics (the ``peak-patch'' formalism, \\cite{BoMy96}~1996).\n\nEach of the models which has been compared \nto observational statistics has difficulty in \nsimultaneously explaining the flatness of the\npresent-day (surface-brightness limited) galaxy luminosity function\n(e.g., Loveday \\mbox{et\\,al.}\\ 1992), the steepness of the faint galaxy counts\n(e.g., \\cite{TSei88}~1988; \\cite{Tys88}~1988), the shape of the moderately\nfaint galaxy ($B\\ltapprox23$) spectroscopic redshift distributions \n(e.g., \\cite{Coll90}~1990; \\cite{Coll93}~1993), the Tully-Fisher\nrelation and the colour distributions of present-day galaxies, \nin a CDM $\\Omega_0=1$ universe. Even though the \\cite{ColeGF94}~(1994) model\nis better than the previous models in allowing big $z=0$ galaxies to\nbe at least as red as higher $z$ galaxies, it \nshares the problem of the other models in lacking big red\nellipticals. It also shares with the \\cite{KWG93} model the problem\nthat if the large number of small haloes predicted by CDM models at \n$z\\approx0$\nfollow the IR Tully-Fisher relation (e.g., \\cite{PTully92}~1992), \nthen the slope of \nthe faint end of the general galaxy luminosity function should be\nsteeper than that estimated locally (e.g.,\nLoveday \\mbox{et\\,al.}\\ 1992). Changing the cosmology in the \n\\cite{ColeGF94} models \n(\\cite{Heyl95}~1995: low $H_0$, low $\\Omega_0$, non-zero $\\lambda_0$ and\nCHDM models) is insufficient to match the observations.\nAnother way of allowing these models to fit the\nobservations is to make a strong assumption for process (6)---to\nsuppose that\ngalaxies can merge as fast as galaxy haloes merge, or even faster---but \nsimple present-day \nconstraints on the products of the mergers \n(\\cite{Dalc93}~1993) and the relative weakness of the faint galaxy \nangular auto-correlation function (\\cite{RY93}~1993) strongly\nrestrict this possibility.\n\nIn order to avoid problems which may be due to the approximation of \nnon-linear gravitational collapse and evolution by the \nsemi-analytical techniques \nmentioned above, an alternative technique is to calculate both processes (1)\nand (2) from first principles \nin numerical N-body simulations, folding in the other physical \nprocesses as simple scaling formulae or using stellar population synthesis\nfor (5). Several authors \n(e.g., \\cite{Ev88} 1988; \\cite{NavB91}~1991; \n\\cite{CO92cdmhydro}~1992; \\cite{Ume93}~1993; \\cite{StM95}~1995) \nhave \nexperimented with these techniques, but resolution limits\non present-day computers make the results hard to interpret. For example,\n\\cite{WeinbHK97}~(1997) point out that although low resolution \ngravito-hydrodynamic simulations suggest that a UV photoionisation \nbackground can suppress galaxy formation (by heating the gas so that\nit is unable to cool and form stars), higher resolution simulations\nshow that this is a numerical artefact: the higher resolution \nsimulations show little sensitivity to either the details of\nphotoionisation or star formation.\n\n\tIn this article, rather than claiming a global ``recipe'' for galaxy\nformation, our primary purpose is to concentrate on \nprocess (1) in a way complementary to that of other techniques. \nThis is unlikely to be sufficient to solve all the observational conflicts.\nOn the contrary, this method should increase the ability of modellers \nto verify the extent to which model predictions are sensitive to the \nprecision of modelling of gravity.\n\n\tThe method presented here is \nto derive merging history trees of dark matter haloes \ndirectly from N-body simulations. \nRather than just investigating virialised haloes for\na particular dark matter model (e.g., CDM), (a) both $n=-2$ and $n=0$\ninitial perturbation spectrum simulations \n(where $n$ is the index of the power spectrum)\nare examined, and (b) since the halo-to-galaxy relationship may be\nmore complex than a simple one-to-one mapping, two significantly different\ndensity thresholds are used for halo detection. \nThis reveals the sensitivity of halo\nmerger history trees and halo statistics to these parameters. \nThe N-body simulations used are presented in \\S\\ref{s-N-body}, \nthe choice of a group-finding algorithm in \\S\\ref{s-peakalg} and \nthe defining criterion and algorithm for calculating the merging history trees\nin \\S\\ref{s-tree}.\n\n\tProperties of the haloes detected are discussed in \\S\\ref{s-halostats}.\nIn particular, the resulting merging history trees are presented in graphical\nform in \\S\\ref{s-mhtrees}, enabling patterns of halo\nmerging calculated from fully non-linear simulations to be visualised directly.\n\n\tIf processes (2)-(5) are simple enough, and if process (6), \ngalaxy merging, corresponds in a one-to-one way with halo merging,\nthen these halo merger history trees would lead directly to galaxy\nmerger history trees. We therefore examine an example application\nof the merger history trees by \nmaking minimal assumptions for processes (2)-(4), using stellar evolutionary\npopulation synthesis for process (5), and for process (6), \nassuming maximal galaxy merging (every halo merger corresponds to\na galaxy merger). \n\\S\\ref{s-bursts} presents \n(6)+(3) merger-induced star formation and \\S\\ref{s-geps} explains how\nprocess (5) is modelled.\n\nIn order for these processes to have an effect on\nthe luminosity function, an option is considered in which\neach merger induces a burst of star formation, \nscaled according to the appropriate halo and gas masses\nand the dynamical time scale. \nApart from this star formation rate option, \nwe do not explore parameter space for non-gravitational processes \nin this paper; we merely adopt simple observationally normalised \nscaling laws.\nResulting luminosity functions are presented in \\S\\ref{s-galstats}.\n\n\tApplications of N-body derived \nhalo merger trees with more complex assumptions for processes (2)-(6)\nare of course possible, and indeed to be welcomed. The galaxy formation\n``recipe'' explored here is only one simple example. \n\n\tCosmological conventions adopted for this paper are a Hubble constant \nof $H_0=50 \\,$km\\,s$^{-1}$Mpc$^{-1}$, comoving units (at $t=t_0$) and \nan $\\Omega_0=1\\.0, \\Lambda=0$ universe is assumed, \nexcept where otherwise specified.\n\n\n\\section{Halo Merger Histories (Gravity)}\n\n\\subsection{Method}\n\n\\subsubsection{N-body Models of Matter Density} \\label{s-N-body} \n\tThe non-linear gravitational evolution of matter \ndensity is modelled by \nN-body cosmological simulations run by \n\\cite{Warr92}~(1992). \nThese simulations use a 128$^3$ initial \nparticle mesh, of side length 10~Mpc. \n[The simulations analysed here \nare for power law initial perturbation spectra ($n=-2$ and $n=0$), so this is \nsimply a default choice for the scaling of units. This default scaling \nis used hereafter except where otherwise specified.]\nParticles are placed on this mesh, making a cube of $\\sim 2 {\\scriptstyle \\times} 10^{6}$\nparticles. \n\nAn initial perturbation spectrum is imposed on this cube by Fourier transforming\nthe initial complex amplitudes from the perturbation spectrum and using \nthe Zel'dovich\ngrowing mode method (\\cite{Warr92}~1992)\non this Fourier transform and the $128^3$ particle mesh.\nThe amplitude of the perturbation spectrum is chosen\nsuch that linear perturbation growth implies that \n$(\\delta M\/M) (r=0\\.5 \\mbox{$h^{-1}\\mbox{\\rm Mpc}$})=2\\.0$ at $z=0,$ where $(\\delta M\/M)(r)$ \nis the r.m.s. value of the excess mass (over uniform\ndensity) in spheres of radius $r$ (\\cite{Warr92}~1992).\nThis choice ensures that the haloes which collapse are about the same size\nfor different values of $n,$ so that the dependence on $n$ of properties \nof halo dynamics---or merging histories---can easily be studied. \nThe absolute normalisation of the spatial correlation function \nof the haloes cannot be directly interpreted in terms of observational\nquantities. The relative amount of power on different scales \n(or slope of the spatial correlation function), and the halo detection\nthreshold, are the parameters which may affect the rates and ways in\nwhich haloes merge with one another.\n\nThe initial cube of perturbed particles is trimmed to a sphere,\ni.e., particles more than $5000\\,$kpc from the centre of the \ncube are removed, resulting in a sphere of $\\sim 1\\.1 {\\scriptstyle \\times} 10^{6}$ particles.\n\nThis is then evolved forward gravitationally via a tree-code \n(e.g., see \\cite{BHut86}~1986), initially\nwith roughly logarithmic time steps up to $t=0\\.3 \\,$Gyr, after which equal \ntime steps of $0\\.03 \\,$Gyr are used. Every hundredth time\nstep is stored on disk; these are the time steps available for\nhalo analysis (hereafter ``time stages''). A vacuum boundary condition \nis used and the softening parameter is 5~kpc (proper units).\n\n\n\\subsubsection{Group-Finding Algorithm} \\label{s-peakalg} \n The simulation data are searched for density peaks at each \ntime step by an algorithm which \nuses the ``oct-tree'' method to find all overdense regions without \noverwhelming computer memory, followed by an iterative means of joining \ntogether contiguous overdense regions. \n\n\tAlternative group-finding methods which could be used\ninclude the ``friends-of-friends'' (FOF) algorithm \n(e.g., \\cite{WDEF87}~1987), the algorithm used by \\cite{Warr92}~(1992) \nor the DENMAX algorithm (\\cite{GB94}~1994).\n\nThe FOF group-finder has the advantage of low memory requirements and\nan obvious relation between the mean particle separation and the \ngroup-finding resolution, but has the disadvantages that if the link\nparameter $l$ is too low, then low density haloes---or the low density \nenvelopes of haloes---are missed, while if $l$ is higher, small but \ndistinct haloes may be erroneously joined together as single objects.\n\n\\cite{Warr92}'s (1992) \nmethod, based on the accelerations of individual particles,\nand the DENMAX algorithm, which includes a de-binding procedure to separate\nhaloes which are only temporarily close to one another, are both more\nphysically motivated than FOF. However, for a first investigation of \nthe use of N-body generated merging history trees in galaxy formation \nmodels, the use of the simple method outlined below seems prudent. Since \ntwo different density detection thresholds are used, the implications \nof having either a low or a high fixed \ndensity threshold (which are similar to the cases of high or low $l$ \nrespectively in FOF) can be seen. For further \ndevelopment, it would certainly be useful to consider\nuse of a more complex algorithm such as DENMAX.\n\n\tDetails of the method are as follows.\n\n Conceptually, a cube concentric to the sphere of particles, \nhaving as side length the diameter of the sphere, is divided into\neight equally sized subcubes.\nAny of these subcubes\ncontaining more than one particle is itself subdivided into eight\nsubcubes. By not subdividing cubes \nwith only one or zero particles, computer memory is not wasted on\nanalysing ``empty'' space.\nThe subdividing process is iterated to a depth of $n_{\\mbox{\\rm \\small levels}}$\n levels below the original cube,\nunless at some level \nall the cubes have one or zero particles in them, in which \ncase subdividing stops (this would happen at $n_{\\mbox{\\rm \\small levels}}=8$\nfor this $1\\.1 {\\scriptstyle \\times} 10^{6}$-particle model for a uniform particle\ndistribution).\nThe side length of the smallest cube is $174\\,$kpc and $20 \\,$kpc\nfor $n_{\\mbox{\\rm \\small levels}}=6$ and $n_{\\mbox{\\rm \\small levels}}=9$ \nrespectively at $z=0.$ \n\nThe ``primary'' list of density peaks is then simply the list of \neach cube at the deepest level (i.e., of size $2^{-\\nlev_tiny}$ times the\nsimulation sphere diameter) whose density is at or \nabove $r_{\\protect\\mbox{\\rm \\small thresh}}$ times the mean density. The list of particles in\neach of these peaks is recorded.\n\nThe results presented here are for $r_{\\protect\\mbox{\\rm \\small thresh}} =5$ and $r_{\\protect\\mbox{\\rm \\small thresh}} =1000.$ \nFor a flat rotation curve of the Galaxy of $220 \\mbox{\\rm\\,km\\,s$^{-1}$},$ \nthe cumulative mass to a radius $r$ is $M( 0$, we make the approximation that all mass loss results from supernovae. We do not resolve the energy-conserving, momentum-generating phase of supernova blast-wave expansion in our simulations, such that we must calculate the terminal momentum of the blast-wave explicitly to prevent over-cooling, following the prescription of~\\cite{KimmCen14}. We use the (unclustered) parametrisation of the terminal momentum injected into the gas cells $k$ neighbouring a central cell $j$, derived from the high-resolution simulations of~\\cite{Gentry17}, and given by\n \\begin{equation} \\label{Eqn::gentry17}\n \\frac{p_{{\\rm t}, k}}{{\\rm M}_\\odot {\\rm kms}^{-1}} = 4.249 \\times 10^5 N_{j, {\\rm SN}} \\Big(\\frac{n_k}{{\\rm cm}^{-3}}\\Big)^{-0.06},\n \\end{equation}\nwhere $N_{j, {\\rm SN}}$ is the cumulative number of supernovae received by a gas cell $j$ from all of the star particles for which it is the nearest neighbour. This terminal momentum is then spread into the cells surrounding the central cell, as in~\\cite{2020arXiv200403608K,2020MNRAS.498..385J,Jeffreson21a}, with an upper limit set by kinetic energy conservation as the shell sweeps up the mass in the cells surrounding the central one~\\citep[see also][for similar prescriptions]{Hopkins18,Smith2018}. A convergence test for a single supernova explosion, implemented via the above method, is presented in Appendix~\\ref{App::res-tests}.\n\nThe chemical composition of the gas in our simulations evolves according to the simplified network of hydrogen, carbon and oxygen chemistry described in~\\cite{GloverMacLow07a,GloverMacLow07b} and in~\\cite{NelsonLanger97}. For each Voronoi gas cell, fractional abundances are computed and tracked for the chemical species ${\\rm H}$, ${\\rm H}_2$, ${\\rm H}^+$, ${\\rm He}$, ${\\rm C}^+$, ${\\rm CO}$, ${\\rm O}$ and ${\\rm e}^-$. The chemistry is coupled to the heating and cooling of the interstellar medium via the atomic and molecular cooling function of~\\cite{Glover10}. The full list of heating and cooling processes is given in their Table 1. As such, the heating and cooling rates in our simulations depend not only on the gas density and temperature, but also on the strength of the interstellar radiation field, the cosmic-ray ionisation rate, the dust fraction and temperature, and on the set of chemical abundances tracked for each gas cell. We assign a value of $1.7$~Habing fields to the UV component of the ISRF according to~\\cite{Mathis83}, a value of $3 \\times 10^{-17}$~s$^{-1}$ to the cosmic ionisation rate~\\citep{2000A&A...358L..79V}, and assume the solar value for the dust-to-gas ratio.\n\n\\section{HII region feedback} \\label{Sec::HII-region-fb}\nIn this section, we derive the momentum per unit time provided by a single HII region to the surrounding interstellar medium. We develop a novel sub-grid model for injecting this momentum in simulations that do not resolve the median Str{\\\"o}mgren radius. We also describe our prescription for heating the interstellar medium within this radius. Parts of the following prescription are used in the simulations `HII heat', `HII spherical mom.', `HII beamed mom.' and `HII heat \\& beamed mom.', as listed in Table~\\ref{Tab::sims}.\n\n\\begin{table}\n\\begin{center}\n\\label{Tab::sims}\n \\caption{The five simulations run in this work and their feedback prescriptions.}\n \\begin{tabular}{@{}l m{4.5cm}@{}}\n \\hline\n Simulation name & Feedback prescription \\\\\n \\hline\n SNe (control) & Supernovae only \\\\\n HII heat & Supernovae plus thermal HII regions \\\\\n HII spherical mom. & Supernovae plus spherical HII region momentum \\\\\n HII beamed mom. & Supernovae plus beamed HII region momentum \\\\\n HII heat \\& beamed mom. & Supernovae plus thermal HII regions plus beamed HII region momentum \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\subsection{Analytic theory} \\label{Sec::Theory}\nWe consider the momentum injected into a molecular cloud by a massive star or stellar cluster that produces ionising photons of energy $\\epsilon_0 \\sim 13.6$~eV at a rate $S$. We choose a system of co-ordinates that is centred on the ionising source, and following~\\citealt{KrumholzMatzner09} (hereafter KM09), we parametrize the density profile of the surrounding gas as a power-law of the form $\\rho(r) = \\rho_0(r\/r_0)^{-k_\\rho}$. A source may be fully `embedded' within the host cloud, such that it is surrounded on all sides by dense gas, or it may be located at the edge of a molecular cloud, such that it produces a `blister-type', hemispherical HII region, for which we take $\\rho = 0$ for the outward-facing hemisphere. The photons from the source transfer their energy to the surrounding gas via two primary mechanisms. Firstly, kinetic energy is carried away by the products of ionisation (free electrons, hydrogen nuclei and helium nuclei), heating the ionised material to a temperature of $T_{\\rm II}$~\\citep{Spitzer78}. As the sound speed $c_{\\rm II}$ in the ionised gas is much higher than that in the surrounding neutral gas, the initial expansion of the HII region sweeps up a thin shell of neutral material that separates the ionised region from its surroundings. Secondly, photons may be absorbed by dust grains and hydrogen atoms, delivering a momentum kick that accelerates the particles away from the ionising source. The radiative acceleration will always be highest closest to the source (KM09), again contributing to the production of a thin shell bounding the HII region. As such, the momentum delivered to the dense gas outside the HII region is very well approximated by the momentum of this bounding shell. The momentum equation for this shell may be written in the form of~\\cite{Matzner02} as\n\\begin{equation}\n\\label{Eqn::momentumequation_orig}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} = \\frac{\\mathrm{d}}{\\mathrm{d} t}(M_{\\rm sh} \\dot{r}_{\\rm II}) = A_{\\rm sh}\\Big[P_{\\rm gas} + P_{\\rm rad}\\Big],\n\\end{equation}\nwhere $M_{\\rm sh} = (2,4)\\pi r_{\\rm II}^3 \\overline{\\rho}(r_{\\rm II})$ is the mass of neutral material swept into the shell of ionisation front radius $r_{\\rm II}$ during its initial rapid expansion, $A_{\\rm sh} = (2,4)\\pi r_{\\rm II}^2$ is its surface area, and $\\overline{\\rho}(r) = 3\/(3-k_\\rho) \\rho_0 (r\/r_0)^{-k_\\rho}$ is the mean volume density inside a radius $r$ in the initial molecular cloud. The first value in the above parentheses corresponds to the case of a blister-type HII region, in which the gas pressure at the HII-cloud interface is augmented by a thrust of equal magnitude and direction, due to the flux of gas through the opposing hemisphere. The equality of the pressure and thrust terms depends on the assumption that the ejected gas can escape freely from the HII region, such that its velocity relative to the velocity $\\dot{r}_{\\rm II}$ of the ionisation front tends towards the speed of sound within the HII region~\\citep{Kahn54}. The second value in parentheses then corresponds to the case of an embedded HII region, in which no thrust is produced. The momenta delivered by thermal heating and radiative acceleration are given in terms of a gas pressure $P_{\\rm gas}$ and a radiation pressure $P_{\\rm rad}$, respectively. The gas pressure is given by\n\\begin{equation}\n\\label{Eqn::Pgas}\nP_{\\rm gas} = (2,1) \\rho_{\\rm II} c_{\\rm II}^2,\n\\end{equation}\nwhere $\\rho_{\\rm II}$ refers to the density of the heated, ionised gas inside the swept-up shell. The radiation pressure term in Equation (\\ref{Eqn::momentumequation_orig}) can be written in the form of KM09 as\n\\begin{equation}\n\\label{Eqn::Prad}\nP_{\\rm rad} = \\frac{f_{\\rm trap} \\epsilon_0 S}{4\\pi c r_{\\rm II}^2},\n\\end{equation}\nwhere the factor $f_{\\rm trap}$ quantifies the enhancement of the radiative force via the trapping of photons and stellar winds in the expanding shell, and $c$ is the speed of light.\n\n\\begin{figure*}\n \\label{Fig::FoF-convergence-test}\n \\includegraphics[width=\\linewidth]{figs\/FoF-convergence-test.pdf}\n \\caption{Distributions of birth masses (left), ionising luminosities (centre) and momentum injection rates (right) for the star particles (thin lines) and FoF groups of star particles with overlapping ionisation front radii (bold lines) in our three `HII heat \\& beamed mom.' simulations at time $t=600$~Myr. Each line colour corresponds to a different numerical mass resolution: $10^5~{\\rm M}_\\odot$ (LOW, dark blue), $10^4~{\\rm M}_\\odot$ (MED, grey), and $10^3~{\\rm M}_\\odot$ (HI, yellow). The arrows in the left-hand panel point to the median mass resolution in each galaxy.}\n\\end{figure*}\n\n\\begin{figure}\n \\label{Fig::rst0}\n \\includegraphics[width=\\linewidth]{figs\/rst0.pdf}\n \\caption{Distributions of ionisation front radii (solid lines) and Str\\\"{o}mgren radii (dotted lines) for the star particles in our three `HII heat \\& beamed mom.' simulations at time $t=600$~Myr. Each line colour corresponds to a different numerical mass resolution: $10^5~{\\rm M}_\\odot$ (LOW, dark blue), $10^4~{\\rm M}_\\odot$ (MED, grey), and $10^3~{\\rm M}_\\odot$ (HI, yellow). The arrows point to the median gas cell radius in each galaxy.}\n\\end{figure}\n\nTo obtain the momentum injected into the cloud per unit time, we must solve the equation of motion for the expansion of the shell. We assume that the contribution of $P_{\\rm gas}$ is well-approximated by its value in a gas pressure-dominated HII region, for which $P_{\\rm gas} \\gg P_{\\rm rad}$. In the case that $P_{\\rm gas} \\ll P_{\\rm rad}$, the inaccuracy associated with this assumption will be small in comparison to the radiation pressure. Once the rate of expansion $\\dot{r}_{\\rm II}$ slows to the speed of sound in the ionised gas, the density inside the HII region equilibriates to a uniform value on the sound-crossing time, and is described by the condition of photoionisation balance~\\citep{Spitzer78}, such that\n\\begin{equation}\n\\label{Eqn::photoionisationbalance}\n\\frac{4}{3} \\pi r_{\\rm II}^3 \\alpha_{\\rm B} \\Big(\\frac{\\rho_{\\rm II} c_{\\rm II}^2}{k_{\\rm B} T_{\\rm II}}\\Big)^2 \\eta^2 = \\phi S,\n\\end{equation}\nwhere $\\alpha_{\\rm B}$ is the case-B recombination coefficient and $\\phi$ is a dimensionless constant that quantifies the effect of photon absorption by dust grains.\\footnote{This accounts for the 27~per~cent of photons at Milky Way metallicity~\\protect\\citep{McKee&Williams97} that are absorbed by dust grains and so do not contribute to the gas pressure. Following KM09, we assume that the gas and dust are well-coupled by the ambient magnetic field, so that the direct radiation pressure does not depend on the gas-to-dust ratio. For further details, see the Appendix of KM09.} The parameter $\\eta$ is given by\n\\begin{equation}\n\\label{Eqn::eta}\n\\eta = \\frac{\\mu}{\\sqrt{\\mu_e \\mu_{\\rm H^+}}},\n\\end{equation}\nwhere $\\mu_{\\rm H^+}$ is the mean mass per proton in the ionised region, $\\mu_{e}$ is the mean mass per free electron, and $\\mu$ is the mean mass per free particle. Combining Equations (\\ref{Eqn::Pgas}) and (\\ref{Eqn::photoionisationbalance}) to rewrite the gas pressure term in Equation (\\ref{Eqn::momentumequation_orig}) gives the momentum delivered to the host cloud per unit time as\n\\begin{align}\n\\label{Eqn::momentumequation_rdim}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} &= \\frac{\\mathrm{d}}{\\mathrm{d} t} (M_{\\rm sh} \\dot{r}_{\\rm II}) = \\frac{f_{\\rm trap} \\epsilon_0 S}{(2,1)c} \\Big[1 + x_{\\rm II}^{\\tfrac{1}{2}}\\Big] \\\\\n&\\approx (1.2, 2.4) \\times 10^3 \\: S_{49} \\: M_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\Big[1 + x_{\\rm II}^{\\tfrac{1}{2}}\\Big],\n\\end{align}\nwhere we define the dimensionless scale parameter as $x_{\\rm II} = r_{\\rm II}\/r_{\\rm ch}$ with\n\\begin{align}\n\\label{Eqn::r_ch}\nr_{\\rm ch} &= \\frac{\\alpha_{\\rm B}}{12(4,1)\\pi \\phi} \\Big(\\frac{\\epsilon_0 \\eta}{k_{\\rm B} T_{\\rm II}}\\Big)^2 \\Big(\\frac{f_{\\rm trap}}{c}\\Big)^2 S \\\\\n&\\approx (0.5, 1.9) \\times 10^{-2} S_{49} \\; {\\rm pc},\n\\end{align}\nwhich is the characteristic radius at which the gas and radiation pressure make equal contributions to the rate of momentum injection. To obtain the order-of-magnitude estimates for $\\mathrm{d} p\/\\mathrm{d} t$ and $r_{\\rm ch}$ in Equations (\\ref{Eqn::momentumequation_rdim}) and (\\ref{Eqn::r_ch}), we have used the same fiducial values as in KM09, setting $T_{\\rm II} = 7000$~K, $\\alpha_{\\rm B} = 3.46 \\times 10^{-13}$~cm$^3$~s$^{-1}$ and $\\phi = 0.73$, consistent with the Milky-Way dust-to-gas ratio~\\citep{McKee&Williams97}. We set $f_{\\rm trap} = 8$, consistent with observations of the pressure inside young HII regions by~\\cite{2021ApJ...908...68O}. We take $\\eta = 0.48$\\footnote{The condition for photo-ionisation balance given in Equation (\\ref{Eqn::photoionisationbalance}) differs from Equation (2) of KM09 by a factor of $\\eta^2 \\sim 0.2$, and therefore the value of $r_{\\rm ch}$ that we obtain is smaller than theirs by the same factor.}, corresponding to a ten-to-one ratio of hydrogen to helium atoms in the neutral gas, with the helium atoms singly-ionised. The ionising luminosity has been rescaled as $S_{49} = S\/10^{49} {\\rm s}^{-1}$. We assume that the volume density of the gas swept up by the HII region is approximately uniform, so that $k_\\rho = 0$. The time-evolution of $\\mathrm{d} p\/\\mathrm{d} t$ can be computed by writing Equation (\\ref{Eqn::momentumequation_rdim}) in the following non-dimensional form\n\\begin{equation}\n\\label{Eqn::momentumequation_nondim}\n\\frac{\\mathrm{d}}{\\mathrm{d} \\tau}\\Big(x_{\\rm II}^{\\rm 3-k_\\rho} \\frac{\\mathrm{d}}{\\mathrm{d} \\tau} x_{\\rm II}\\Big) = 1 + x_{\\rm II}^{1\/2},\n\\end{equation}\nwhere $\\tau = t\/t_{\\rm ch}$ for a characteristic time $t_{\\rm ch}$ at which the gas and radiation pressure are equal, given by\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::t_ch}\nt_{\\rm ch} &= \\sqrt{\\frac{4\\pi c \\overline{\\rho}(r_{\\rm st,0})}{3 f_{\\rm trap} \\epsilon_0 S} r_{\\rm ch}^4 \\Big(\\frac{r_{\\rm ch}}{r_{\\rm st,0}}\\Big)^{-k_\\rho}} \\\\\n&\\approx (45,333) \\: \\overline{n}_{\\rm H,2}^{1\/6} S_{49}^{7\/6} \\: {\\rm yr}\n\\end{split}\n\\end{equation}\nwhere $r_{\\rm st,0}$ is the initial Str{\\\"o}mgren radius and $\\overline{\\rho}(r_{\\rm st,0})$ is the mean density inside $r_{\\rm st,0}$ in the initial molecular cloud. These two quantities are related by Equation (\\ref{Eqn::photoionisationbalance}) as\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::r_st0}\nr_{\\rm st,0} &= \\Big(\\frac{3\\phi S}{4\\pi \\alpha_{\\rm B}}\\Big)^{1\/3} \\Big(\\frac{\\mu m_{\\rm H}}{\\overline{\\rho}(r_{\\rm st,0})\\eta}\\Big)^{2\/3} \\\\\n&\\approx 2.5 \\: {\\rm pc} \\: S_{49}^{1\/3} \\overline{n}_{\\rm H,2}^{-2\/3}.\n\\end{split}\n\\end{equation}\nTo obtain Equation (\\ref{Eqn::t_ch}), we have used the radial scaling of the mean volume density to write $M_{\\rm sh} = (2,4) \\pi r_{\\rm II}^3 \\overline{\\rho}(r_{\\rm st,0}) (r_{\\rm II}\/r_{\\rm st,0})^{-k_\\rho}$. The numerical value of $t_{\\rm ch}$ is obtained by writing $\\overline{\\rho}_{\\rm st,0} = 100 \\overline{\\mu} m_{\\rm H} \\overline{n}_{\\rm H,2}$ as in KM09, where $\\overline{\\mu} \\sim 1.4$ is the mean molecular weight in the ionised gas, $m_{\\rm H}$ is the proton mass, and $\\overline{n}_{\\rm H,2}$ is the number density of hydrogen atoms inside $r_{\\rm st, 0}$, in units of $100$~cm$^{-3}$.\n\nEquation (\\ref{Eqn::momentumequation_nondim}) has an approximate analytic solution that interpolates between the gas-dominated and radiation-dominated cases to an accuracy of better than $5$~per~cent~\\citep{KrumholzMatzner09}, given by\n\\begin{equation}\n\\label{Eqn::xapprox}\nx_{\\rm II, approx} = \\Big[\\frac{3}{2}\\tau^2 + \\Big(\\frac{25}{28} \\tau^2\\Big)^{6\/5}\\Big]^{1\/3}.\n\\end{equation}\nWith this solution, we may finally write the momentum equation as\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::momentumeqn_final}\n\\frac{\\mathrm{d} p}{\\mathrm{d} t} \\approx (1.2, 2.4) \\: &S_{49} \\times 10^3 M_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\times \\\\\n&\\left\\{ 1 + \\Big[\\frac{3}{2}\\frac{t^2}{t_{\\rm ch}^2} + \\Big(\\frac{25}{28} \\frac{t^2}{t_{\\rm ch}^2}\\Big)^{6\/5}\\Big]^{1\/6}\\right\\},\n\\end{split}\n\\end{equation}\nwith $t_{\\rm ch}$ given by Equation (\\ref{Eqn::t_ch}). An HII region will deposit momentum at this rate into the surrounding dense gas until its expansion stalls (see Section~\\ref{Sec::stalling}). For a Galactic giant molecular cloud with $\\overline{n}_{\\rm H,2} \\sim 1$ and a star cluster with an ionising luminosity of $S_{49} \\sim 100$, this characteristic time is around $10,000$ years, indicating that for the majority of its life-span (of order a few Myr), the momentum output from such an HII region is dominated by gas pressure. The gas pressure contributes over 90~per~cent of the final injected momentum. Only for the most luminous clusters (such as M82 L$^{\\rm a}$ in KM09, with $t_{\\rm ch} \\sim 10$~Myr) does radiation pressure dominate the momentum budget. We note that other sub-grid models for HII region feedback at similar resolutions~\\citep[e.g.][]{2011MNRAS.417..950H,2013MNRAS.434.3142A,Agertz13,2014MNRAS.445..581H,2015ApJ...804...18A} consider only the part of $\\mathrm{d} p\/\\mathrm{d} t$ due to radiation pressure (the first term in the curly brackets in Equation~\\ref{Eqn::momentumeqn_final}). In order to inject significant quantities of momentum from HII regions, they therefore inflate $f_{\\rm trap}$ above observed values. This is discussed further in Section~\\ref{Sec::SNe-fb}.\n\n\\subsection{Numerical implementation of HII region momentum} \\label{Sec::num-methods}\n\\subsubsection{Grouping of star particles} \\label{Sec::grouping}\nThe rate of momentum injection given in Equation~(\\ref{Eqn::momentumeqn_final}) does not scale linearly with the cluster luminosity $S_{49}$. This means that the total momentum injected by the star particles in a numerical simulation will not trivially converge with increasing mass resolution. As shown in the left-hand panel of Figure~\\ref{Fig::FoF-convergence-test}, the maximum stellar particle mass in {\\sc Arepo} is equal to twice the simulation mass resolution (the median gas cell mass, given by the solid vertical lines). Upon reaching the star formation threshold $\\rho_{\\rm thresh}$, larger gas cells are decremented in mass and `spawn' star particles at twice the simulation resolution, while smaller gas cells are deleted and replaced by stars of equal mass. At mass resolutions of $\\sim 900~{\\rm M}_\\odot$, the largest stellar clusters are made up of hundreds of star particles with overlapping ionisation-front radii $r_{\\rm II,*}$, each of which is given by\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::rII}\nr_{{\\rm II}, *}(t) &= r_{\\rm ch} x_{\\rm II, approx} \\\\\n&\\approx 0.5 \\times 10^{-2} S_{49,*} \\: {\\rm pc} \\: \\times \\\\\n&\\left\\{\\frac{3}{2} \\Big(\\frac{t_*}{t_{{\\rm ch},*}}\\Big)^2 + \\Big[\\frac{25}{28} \\Big(\\frac{t_*}{t_{{\\rm ch},*}}\\Big)^2\\Big]^{6\/5}\\right\\}^{1\/3},\n\\end{split}\n\\end{equation}\nwhere $t_{\\rm ch,*}$ is the characteristic time for an individual star particle, (see Equation~\\ref{Eqn::t_ch}). Physically, a group of star particles whose ionisation fronts overlap should be treated as a single HII region with a single birth density $\\overline{n}_{\\rm H,2}$ and luminosity $S_{49}$, given that the density of the ionised gas inside the bounding shell of a subsonic HII region equilibriates on its sound-crossing time and becomes uniform. We therefore substantially improve the resolution-convergence of our momentum deposition (in a physically-motivated sense) by using a Friends-of-Friends (FoF) linking algorithm between star particles, with a linking length of $r_{\\rm II, *}$.\\footnote{Note that this method cannot produce perfect resolution convergence because the ionisation front radii also depend weakly on both the stellar luminosity $S_{49}$ and the stellar birth density $\\overline{n}_{\\rm H,2}$.} A sample of the FoF groups produced by this algorithm in the low-resolution run is shown in Figure~\\ref{Fig::FoF-schematic}.\n\nIn practice, the total momentum injected by an FoF-grouped HII region during a numerical time-step $\\Delta t_{\\rm HII}$ is then given by\n\\begin{equation}\n\\Delta p_{\\rm HII} = {\\Big(\\frac{\\mathrm{d} p}{\\mathrm{d} t}\\Big)_{\\rm FoF} \\Delta t_{\\rm HII}},\n\\end{equation}\nwith a momentum injection rate of\n\\begin{equation}\n\\begin{split}\n\\label{Eqn::momentumequation_fof}\n\\Big( \\frac{\\mathrm{d} p}{\\mathrm{d} t} &\\Big)_{\\rm FoF} = \\; (1.2, 2.4) \\times 10^3 \\times \\\\ &\\sum_{*=1}^N{S_{49,*}} \\: {\\rm M}_\\odot \\: {\\rm km} \\: {\\rm s}^{-1} {\\rm Myr}^{-1} \\times \\\\\n &\\left\\{1 + \\Big[\\frac{3}{2} \\Big(\\frac{\\langle t \\rangle_S}{t_{\\rm ch, FoF}}\\Big)^2 + \\Big(\\frac{25}{28} \\frac{\\langle t \\rangle_S}{t_{\\rm ch,FoF}}\\Big)^{6\/5}\\Big]^{1\/6} \\right\\},\n\\end{split}\n\\end{equation}\nand a characteristic time of\n\\begin{equation}\n\\label{Eqn::t_ch_fof}\nt_{\\rm ch,FoF} = \\sqrt{(0.6, 33) \\Big(\\sum^N_{*=1}{S_{49,*}}\\Big)^{7\/3} \\langle \\overline{n}_{\\rm H,2} \\rangle_S^{1\/3} \\: {\\rm pc} \\: {\\rm s}^2}\n\\end{equation}\nfor each FoF group. The angled brackets $\\langle ... \\rangle_S$ denote ionising luminosity-weighted averages over the star particles $*=1 ... N$ in the group, such that $\\langle t \\rangle_S$ is the luminosity-averaged age of the star particles. The momentum is injected at the luminosity-weighted centre of the group, given by\n\\begin{equation}\n\\label{Eqn::centre_fof}\n\\langle \\bm{x} \\rangle_S = \\frac{\\sum_{*=1}^N{S_{49,*}\\bm{x}_*}}{\\sum_{*=1}^N{S_{49,*}}}.\n\\end{equation}\nTo ensure that all star particles in a single group have their ionising luminosities, ionisation front radii and ages updated on the same time-step, we set $\\Delta t_{\\rm HII}$ to be the global time-step for the simulation. This means that we inject HII region feedback on global time-steps only, which have a maximum value of $0.1$~Myr for the simulations presented in this work.\\footnote{Due to the hierarchical time-stepping procedure used in {\\sc Arepo}~\\citep[see][for details]{Springel10}, the ionising properties of different star particles from the same FoF group would otherwise be updated at different time intervals. Our global time-step has a maximum value of $0.1$~Myr, and the peak ionisation-front expansion rate for the most massive star particles in our high-resolution simulation ($\\sim 2 \\times 10^3$~M$_\\odot$, see Figure~\\ref{Fig::FoF-convergence-test}) is $\\sim 20$~pc Myr$^{-1}$, so in the very worst case, our FoF groups may be affected by an error of order $\\sim 2$~pc.}\n\n\\begin{figure}\n \\label{Fig::FoF-schematic}\n \\includegraphics[width=\\linewidth]{figs\/FoF-schematic.pdf}\n \\caption{FoF groups of HII regions with overlapping ionisation front radii in a $(950~{\\rm pc})^3$ box within the LOW-resolution simulation with spherical HII region feedback (`HII spherical mom.') at $\\sim 600~{\\rm Myr}$ (the galaxy is centred at the origin). The coloured circles represent the ionisation front radii of individual HII regions, while the small black stars mark the positions of unlinked HII regions (FoF group size of one) and the coloured stars mark the luminosity-weighted centres of the linked HII regions (FoF group size greater than one). The colour bar gives the z-position of the HII regions.}\n\\end{figure}\n\n\\begin{figure}\n \\label{Fig::2HII-convergence}\n \\includegraphics[width=\\linewidth]{figs\/2HII-convergence.pdf}\n \\caption{Energy (upper panel) and radial momentum (lower panel) of the gas cells in a box of size $(950~{\\rm pc})^3$ and density $100~{\\rm cm}^3$ containing two star particles of mass $1 \\times 10^4~{\\rm M}_\\odot$, separated by a distance of $5$~pc, as a function of time. The star particles both inject spherical momentum from HII regions (no thermal energy injection) according to the prescription outlined in Section~\\ref{Sec::num-methods}. In the lower panel, the combined momentum from the individual star particles (dashed lines) is compared to the momentum injected by the FoF-grouped pair (solid lines). The resolution varies from a lowest value of $10^7~{\\rm M}_\\odot$ per gas cell (dark blue) up to a highest resolution of $10^3~{\\rm M}_\\odot$ per gas cell (yellow).}\n\\end{figure}\n\n\\begin{figure*}\n \\label{Fig::blister-projections}\n \\includegraphics[width=\\linewidth]{figs\/blister-projections.pdf}\n \\vspace{-0.5cm} \\caption{Column density projections of the gas cells surrounding an HII region of stellar mass $10^4$~M$_\\odot$ in a $(950~{\\rm pc})^3$ box containing $128^3$ gas cells at a volume density of $100~{\\rm cm}^{-3}$. The top row shows a spherical\/embedded HII region, while the bottom row shows a beamed\/blister-type HII region with momentum injected accordering to Equation (\\ref{Eqn::jet-profile}), using an opening angle of $\\Theta = \\pi\/12$ and axis aligned along $(x=1\/\\sqrt{2}, y=1\/\\sqrt{2})$. All projections are computed parallel to the simulation $z$-axis.}\n\\end{figure*}\n\nIn Figure~\\ref{Fig::FoF-convergence-test} we show the effect of grouping on the HII region masses (left-hand panel), the ionising luminosities (centre panel), and the momentum injection rate (right-hand panel) in the low-resolution (dark blue lines), medium-resolution (grey lines) and high-resolution (yellow lines) Agora disc simulations. Comparison of the bold lines (FoF groups) and thin lines (star particles) demonstrates that the distributions of stellar birth masses $m_{\\rm birth}$, luminosities $S_{49}$, and momentum injection rates $\\mathrm{d} p\/\\mathrm{d} t$, are brought closer to convergence by the FoF grouping. The ionisation front radii $r_{\\rm II}$ used to compute the groups are displayed for each simulation in Figure~\\ref{Fig::rst0}. We might also consider using the Str{\\\"o}mgren radius $r_{\\rm st,0}$ (dotted lines, left-hand panel) as the FoF linking-length, however $r_{\\rm st,0}$ is more heavily-dependent on the stellar birth density than is $r_{\\rm II}$ (see Equations~\\ref{Eqn::r_st0} and~\\ref{Eqn::rII}), and so is more heavily-dependent on the simulation resolution, making it a less favourable choice.\n\nWhile our FoF grouping corrects for the non-linear dependence of Equation~(\\ref{Eqn::momentumeqn_final}) on the ionising luminosity, and reduces the spurious cancellation of the momentum injected between adjacent star particles, it \\textit{does not} address resolution-dependent variations in the spatial distribution of stellar mass in our simulations, caused partly by the variation in star particle mass, and partly by the suppressed clustering of star particles at lower resolutions. These effects change the spatial distribution of the energy injected by stellar feedback. As discussed in~\\cite{2020arXiv200911309S,2020arXiv200403608K}, an increase in the clustering of supernovae leads to burstier feedback with larger outflows perpendicular to the galactic mid-plane. We discuss these effects further in Section~\\ref{Sec::discussion}.\n\n\\begin{figure}\n \\label{Fig::heat-convergence}\n \\includegraphics[width=\\linewidth]{figs\/heat-convergence.pdf}\n \\caption{Ionising luminosity emitted by star particles ($S_{49, *}$, solid lines) and absorbed by surrounding gas cells ($S_{49, {\\rm g}}$, thin lines) at a density of $100~{\\rm cm}^{-3}$ in a box of size $(950~{\\rm pc})^3$ containing 100 star particles of mass $1 \\times 10^4~{\\rm M}_\\odot$, as a function of time. The star particles inject HII region heating feedback according to the prescription outlined in Section~\\ref{Sec::num-methods}. The resolution varies from a lowest resolution of $10^7~{\\rm M}_\\odot$ per gas cell (dark blue) up to a highest resolution of $10^3~{\\rm M}_\\odot$ per gas cell (yellow). Note that lines for the $10^3~{\\rm M}_\\odot$ and $10^4~{\\rm M}_\\odot$ overlap.}\n\\end{figure}\n\n\\begin{figure*}\n \\label{Fig::morphology}\n \\includegraphics[width=\\linewidth]{figs\/morphology.pdf}\n \\caption{Column density maps of the gas at each resolution (LOW = $10^5~{\\rm M}_\\odot$ per cell, MED = $10^4~{\\rm M}_\\odot$, HI = $10^3~{\\rm M}_\\odot$) and with each feedback prescription, at $t=600$~Myr. Momentum injection from HII regions changes the phase structure of the interstellar medium in the MED- and HI-resolution cases only. It also improves the numerical convergence of the gas-disc scale-height. Heating from HII regions makes no difference to the disc morphology at any resolution.}\n\\end{figure*}\n\n\\begin{figure}\n \\label{Fig::tuning-forks}\n \\includegraphics[width=\\linewidth]{figs\/tuning-forks.pdf}\n \\caption{The gas-to-SFR flux ratio relative to the galactic average value as a function of aperture size, for each of our high-resolution galaxies at $t = 600$~Myr. The upper branch represents apertures focussed on molecular gas peaks, while the lower branch represents apertures focussed on the peaks of surface density of `young stars' (ages $0$-$5$~Myr). The error bars on each data point represent the 1$\\sigma$ uncertainty on the value of the gas-to-SFR flux ratio. The shaded areas on each data point indicate the effective $1\\sigma$ uncertainty range that accounts for the covariance between the data points. The grey-shaded region shows the result of applying the same analysis to observations of the Milky Way-like galaxy NGC 628~\\protect\\citep{Chevance20}. The dashed coloured lines represent the best-fit model of~\\protect\\cite{Kruijssen18a}.}\n\\end{figure}\n\n\\begin{figure*}\n \\label{Fig::outflows}\n \\includegraphics[width=\\linewidth]{figs\/outflow-rate.pdf}\n \\caption{Global galactic star formation rate (top row), gas outflow rate (see Section~\\protect\\ref{Sec::outflows}) from the galactic mid-plane (middle row) and mass-loading of outflows (bottom row) as a function of simulation time for the simulation with both thermal and beamed momentum from HII regions (HII thermal \\& beamed mom.), at low-resolution (LOW, dark blue), at medium-resolution (MED, grey), and high-resolution (HI, yellow).}\n\\end{figure*}\n\n\\begin{figure*}\n \\label{Fig::vss}\n \\includegraphics[width=\\linewidth]{figs\/vss.pdf}\n \\caption{Azimuthally- and temporally-averaged cold-gas ($\\leq 10^4~{\\rm K}$) gas-disc scale-height (top row), molecular gas (dashed lines) and total gas (solid lines) surface density (second row), thermal (dashed lines) and total (thermal plus turbulent, solid lines) velocity dispersion (third row), and the Toomre $Q_{\\rm g}$ parameter of the total gas distribution (bottom row) as a function of the galactocentric radius for each simulated galaxy at each resolution, across the time interval $t=300$-$600$~Myr.}\n\\end{figure*}\n\n\\subsubsection{Injection of momentum from HII regions} \\label{Sec::injection}\nWe inject the radial momentum $\\Delta p_{{\\rm HII}, *}$ from each star particle at the luminosity-weighted centre $\\langle \\bm{x} \\rangle_S$ of its FoF group via the same procedure as used for supernovae, described in~\\cite{2020arXiv200403608K,2020MNRAS.498..385J}. Briefly, the algorithm proceeds as follows.\n\\begin{enumerate}\n \\item For each FoF group, find the nearest-neighbour gas particle $j$ to the luminosity-weighted centre of mass.\n \\item Increment the total radial momentum $\\Delta p_{j, {\\rm HII}}$ received by cell $j$ from all of the FoF groups it hosts, such that\n\\begin{equation}\n\\Delta p_{j, {\\rm HII}} = \\sum_{{\\rm FoF}=1}^N {\\Big(\\frac{\\mathrm{d} p}{\\mathrm{d} t}\\Big)_{\\rm FoF}} \\Delta t_{\\rm HII}.\n\\end{equation}\n \\item For each gas cell $j$ that has received HII-region momentum, find the set of neighbouring gas cells $k$ with which it shares a Voronoi face. Compute the fraction of the radial momentum received by each facing cell according to\n\\begin{equation}\n\\label{Eqn::j-to-k-inj}\n\\Delta \\bm{p}_{k, {\\rm HII}} = w_k \\hat{\\bm{r}}_{j \\rightarrow k} \\Delta p_{j, {\\rm HII}},\n\\end{equation}\nwhere $\\hat{\\bm{r}}_{j \\rightarrow k}$ is the unit vector from the centre of the host cell to the centre of the cell receiving the feedback, and the weight factor $w_k$ is the fractional Voronoi face area $A_{j \\rightarrow k}$ shared between these cells, such that\n\\begin{equation}\n\\label{Eqn::weight-fn}\nw_k = \\frac{A_{j \\rightarrow k}}{\\sum_k{A_{j \\rightarrow k}}}.\n\\end{equation}\nEquation (\\ref{Eqn::j-to-k-inj}) ensures that the momentum injection is perfectly isotropic, regardless of the distribution over the volumes of the cells $k$.\n \\item Ensure conservation of linear momentum by subtracting the sum of the injected momenta $\\sum_k{\\Delta p_{k, {\\rm HII}}}$ from the momentum of the central cell $j$.\\footnote{We note that we do not account for a full tensor renormalisation of the injected momentum, as in~\\cite{Hopkins18b,Smith2018,2020arXiv200911309S}, for example.}\n\\end{enumerate}\n\nIn Figure~\\ref{Fig::2HII-convergence}, we check the numerical convergence of the momentum and energy injected according to the algorithm described above, at mass resolutions varying between $10^7~{\\rm M}_\\odot$ and $10^3~{\\rm M}_\\odot$ per gas cell. We take a box of side-length $950~{\\rm pc}$ and uniform gas density $100~{\\rm cm}^{-3}$, containing a single pair of star particles of mass $10^4~{\\rm M}_\\odot$ each. We record the radial momentum of the gas cells in the box as a function of time when the stars inject momentum from their individual HII regions (dashed lines, lower panel), and when the stars are grouped via the FoF procedure described in Section~\\ref{Sec::grouping} (solid lines, lower panel). We also record the kinetic and total energies of the gas cells in the FoF-grouped case, represented by the thin and bold lines, respectively, in the top panel of Figure~\\ref{Fig::2HII-convergence}. The bottom panel demonstrates that the radial momentum injected is converging to within 1.1 dex in momentum per 3 dex in mass resolution in both the FoF-grouped (solid lines) and ungrouped (dashed lines) cases, for the mass resolutions between $10^3~{\\rm M}_\\odot$ and $10^5~{\\rm M}_\\odot$ spanned by our isolated disc galaxies. At lower resolutions the injected momentum does not persist, but rather begins to drop steeply after about $10$~Myr of evolution. This is because the ionisation front bounding the HII region is never resolved at mass resolutions of $>10^5~{\\rm M}_\\odot$, and so the neighbouring gas cells $k$ have a combined mass much larger than that of the swept-up shell. This greatly reduces their final velocities\/kinetic energies (shown in the top panel of Figure~\\ref{Fig::2HII-convergence}), and so the injected momentum is quickly lost. This behaviour is not inaccurate, as entirely-unresolved feedback processes should not have any impact on the simulated interstellar medium.\n\n\\subsubsection{Directional injection for blister-type HII regions}\nThe weight factor $w_k$ in Section~\\ref{Sec::injection} results in isotropic momentum injection, appropriate for embedded HII regions. To mimic the directional outflow from a blister-type HII region along an axis $\\hat{\\bm{z}}_{\\rm FoF}$, we instead weight the momenta $\\Delta p_{k, {\\rm HII}}$ by the following axisymmetric factor,\n\\begin{equation}\n\\label{Eqn::jet-profile}\n\\begin{split}\nw(\\theta_k, A_k) &= \\frac{A_{j \\rightarrow k} f(\\theta_k)}{\\sum_k{A_{j \\rightarrow k} f(\\theta_k)}} \\\\\nf(\\theta_k) &= \\Big[\\log{\\Big(\\frac{2}{\\Theta}\\Big)(1+\\Theta^2-\\cos^2{\\theta_k})}\\Big]^{-1}\n\\end{split}\n\\end{equation}\nwhere $\\Theta$ controls the width of the beam and $\\theta_k$ is the angle between the beam-axis and the unit vector $\\hat{\\bm{r}}_{j \\rightarrow k}$ connecting cells $j$ and $k$, defined by\n\\begin{equation}\n\\cos{\\theta_k} = \\frac{\\hat{\\bm{r}}_{j \\rightarrow k} \\cdot \\hat{\\bm{z}}_{\\rm FoF}}{|\\hat{\\bm{z}}_{\\rm FoF}|}.\n\\end{equation}\nThe opening angle is set to $\\Theta = \\pi\/12$ in our simulations, and the beam-axis vector $\\hat{\\bm{z}}_*$ for each star particle is drawn randomly from a uniform distribution over the spherical polar angles about the star's position, $\\phi_*$ and $\\theta_*$. This value is fixed throughout the star particle's lifetime, and the beam-axis $\\hat{\\bm{z}}_{\\rm FoF}$ of each FoF group is calculated as a luminosity-weighted average of $\\hat{\\bm{z}}_*$ across the constituent star particles. In Figure~\\ref{Fig::blister-projections} we compare the density profiles for spherical- (top row) and blister-type (bottom row) momentum injection, at simulation times $1$~Myr, $10$~Myr and $30$~Myr after the birth of the stellar cluster in a uniform medium of density $100~{\\rm cm}^{-3}$. Qualitatively, the blister-type momentum injection results in a faster and wider ejection of gas away from the cluster centre than does the spherical momentum injection, despite the fact that the ionisation front radius (solid white lines) is only marginally larger. We note that in this uniform-density box, the number of Voronoi cells surrounding the star particle is relatively small, resulting in a deviation from perfect spherical symmetry when the feedback is injected isotropically (top row of Figure~\\ref{Fig::blister-projections}, the momentum propagates along rays joining the star particle to the centroids of the neighbouring cells). This effect will be less marked in the highly-overdense star-forming regions of isolated disc galaxies.\n\n\\subsection{Stalling of HII regions} \\label{Sec::stalling}\nIn computing the FoF groups via the method presented in Section~\\ref{Sec::grouping}, we must be careful to exclude star particles whose ionisation fronts have stalled, and which are no longer depositing significant quantities of momentum into the surrounding gas. Stalling occurs when the rate of HII region expansion becomes comparable to the velocity dispersion of the host cloud, at which point the ionised and neutral gas are able to intermingle and the swept-up shell loses its coherence~\\citep{Matzner02}. After this transition, it no longer makes sense to include the stalled HII region in an FoF group of expanding HII regions, as its radius and internal density are no longer well-defined. In particular, we want to avoid the case where such an HII region links together two active HII regions, spuriously shifting the origin of their momentum ejection to a position halfway between the two particles. Before the FoF groups are calculated, we therefore compute the rate of HII region expansion $\\dot{r}_{{\\rm II},*}$ for each star particle, and if this is found to be smaller than the velocity dispersion of the surrounding gas at the same scale, we flag the particle as `stalled'. Star particles with stalled ionisation fronts are not allowed to be FoF group members, but are still allowed to contribute to HII region feedback with what little remains of their ionising luminosity. Following KM09, we approximate the ambient velocity dispersion by considering a blister-type HII region centred at the origin of a cloud with an average density of $\\overline{\\rho}(r) = 3\/(3-k_\\rho) \\rho_0 (r\/r_0)^{-k_\\rho}$ and a virial parameter $\\alpha_{\\rm vir}$ as measured on the scale of the HII region. This gives a cloud velocity dispersion of\n\\begin{equation}\n\\label{Eqn::sigma}\n\\begin{split}\n\\sigma_{\\rm cl}(r_{\\rm II}) &= \\sqrt{\\frac{\\alpha_{\\rm vir} G M( S_{j, {\\rm cons}}$, ionise cell $j$ and compute the `residual' ionisation rate $S_{j, {\\rm res}}$ to be spread to the facing cells $k$, such that $S_{j, {\\rm res}} = S_{j, {\\rm in}} - S_{j, {\\rm cons}}$. Each ionised cell is heated to a temperature of $7000$~K.\n\\end{enumerate}\nIn the case that $S_{j, {\\rm in}} < S_{j, {\\rm cons}}$, the algorithm ends here. Otherwise we continue as follows.\n\\begin{enumerate}\n\t\\setcounter{enumi}{6}\n\t\\item For each gas cell $j$ with $S_{j, {\\rm res}} > 0$, find the set of neighbouring cells $k$ with which it shares a Voronoi face. Compute the fraction of photons it receives according to\n\\begin{equation}\nS_{k, {\\rm in}} = w_k S_{j, {\\rm res}},\n\\end{equation}\nwhere $w_k = A_{j \\rightarrow k}\/\\sum_k{A_{j \\rightarrow k}}$, as for the injection of HII region momentum in Section~\\ref{Sec::injection}.\n\t\\item Ionise each facing cell $k$ with a probability of $S_{k, {\\rm in}}\/S_{k, {\\rm cons}}$. Summed over the set of facing cells for many HII regions, this ensures that the expectation value for the rate of ionisation converges to $S_{j, {\\rm res}}$.\n\\end{enumerate}\nSubsequent to the above procedure for thermal energy injection, the chemistry and cooling for each gas cell is computed using {\\sc SGChem}, as described in Section~\\ref{Sec::SNe-only}. During this computation, we impose a temperature floor of $7000$~K, which is enforced until the next HII-region update. We rely on the chemical network to collisionally-ionise the gas cells in a manner that is self-consistent with their temperatures. This will only produce an ionisation fraction of $10^{-5}$ when cold gas is heated to $7000$~K, but in the non-equilibrium case, whereby gas cools from much higher temperatures to a floor of $7000$~K, much higher ionisation fractions can be achieved. After the chemistry computation, the ionised cells are unflagged and are ready to absorb more photons.\n\nIn Figure~\\ref{Fig::heat-convergence}, we check that at mass resolutions between $10^3$ and $10^7~{\\rm M}_\\odot$, the above method ensures convergence of the quantity of photoionised gas. We consider a box of side-length $950~{\\rm pc}$ containing a gas of uniform density $100~{\\rm cm}^{-3}$, along with $100$ star particles of mass $10^4~{\\rm M}_\\odot$ each. We record the total cumulative value of $S_{49, *}$ emitted by these particles as a function of time (solid lines), as well as the total cumulative $S_{49, {\\rm g}}$ absorbed by the surrounding gas cells (dashed lines). Cooling and chemistry are switched on. We see that the bold and solid lines match at all resolutions, indicating that none of the emitted photons are `wasted' by our restriction of photon injection to the set of facing cells $k$ surrounding each star particle. The offset for the lowest-resolution ($10^7~{\\rm M}_\\odot$ per gas cell) case is due to the stochastic procedure for choosing the gas cells to ionise: in the limit of a very large number of HII regions ($\\gg 100$), we would expect this offset to approach zero. The star particle mass used in this test is in the 99th percentile for FoF grops in the highest-resolution isolated disc simulation used in this work ($10^3~{\\rm M}_\\odot$ per gas cell), and the gas density is ten times lower than the birth density of these star particles. If all photons are absorbed in this case, then the algorithm described above is valid in its modelling of the heating due to the vast majority of our marginally-resolved HII regions.\n\n\\section{Results} \\label{Sec::results}\nIn this section, we analyse the properties of the four simulated disc galaxies with thermal HII region feedback (`HII heat'), spherically-injected HII region momentum (`HII spherical mom.'), blister-type HII region momentum with $\\Theta = \\pi\/12$ (`HII beamed mom.'), and a combination of blister-type momentum and thermal energy (`HII heat \\& beamed mom.'), relative to our control simulation with supernova feedback on its own (`SNe only'). The simulations are summarised in Table~\\ref{Tab::sims}. We consider the morphology, stability, global star formation rate and phase structure of the interstellar medium (Section~\\ref{Sec::disc-props}), and the distribution of the lifetimes, masses, star formation rate densities, and velocity dispersions of its molecular clouds (Section~\\ref{Sec::molecular-props}).\n\nIn this section, whenever we compare to observed quantities involving molecular hydrogen, we use synthetic $^{12}{\\rm CO}(J=1\\rightarrow 0)$ maps obtained by post-processing the simulations using the {\\sc Despotic} code~\\citep{Krumholz14}, rather than using the ${\\rm CO}$ or ${\\rm H}_2$ abundances determined from the {\\sc SGChem} during run-time. We convert these ${\\rm CO}$ maps back to synthetic ${\\rm H}_2$ maps using a constant ${\\rm H}_2$-to-${\\rm CO}$ conversion factor $\\alpha_{\\rm CO} = 4.3~{\\rm M}_\\odot ({\\rm K}~{\\rm kms}^{-1}{\\rm pc}^{-2})^{-1}$, mimicking the procedures used in observations~\\citep{Bolatto13}. This allows a direct comparison of our results in Section~\\ref{Sec::molecular-props} to observed molecular cloud populations. Our motivation for this method is that, while {\\sc SGChem} produces fully time-dependent chemical abundances, it does not calculate ${\\rm CO}$ excitation or line emission, whereas {\\sc Despotic} includes a full treatment of the ${\\rm CO}$ emission, out of local thermal equilibrium. This allows us to capture the effects of local variations in the ${\\rm CO}$ luminosity per unit ${\\rm H}_2$ mass, which may be important for comparing to observations. Full details of the post-processing procedure are provided in Appendix~\\ref{App::postproc}.\n\n\\subsection{Galactic-scale properties of the interstellar medium} \\label{Sec::disc-props}\n\\subsubsection{Disc morphology} \\label{Sec::morphology}\nThe face-on and edge-on gas column densities across all simulation resolutions and feedback prescriptions are displayed in Figure~\\ref{Fig::morphology}. In the medium- (centre row) and high-resolution (top row) cases, the addition of momentum from HII regions visibly reduces the sizes of the largest voids in the gas of the interstellar medium, blown by supernova feedback. This corresponds to a qualitative reduction in the amount of outflowing gas from the galactic mid-plane, as seen in the edge-on view, and so to a visible reduction in the gas disc scale-height. The introduction of thermal energy from HII regions without momentum (`HII heat') has no effect on the interstellar medium. In the low-resolution case (bottom row), the difference between the simulations with and without HII region momentum is eradicated. This can likely be attributed to the reduction in supernova clustering with decreasing resolution, as discussed in Section~\\ref{Sec::discussion}.\n\nFigure~\\ref{Fig::tuning-forks} quantifies the structure of the multi-scale molecular gas distribution in our simulations, relative to the distribution of young stars. This is the result of measuring the gas-to-stellar flux ratio enclosed in apertures centred on ${\\rm H}_2$ peaks (top branch) and SFR peaks using `young stars' with ages in the range $0$ to $5$~Myr (bottom branch), and for aperture sizes ranging between $50$~pc and $4000$~pc, following~\\cite{Kruijssen2014} and~\\cite{Kruijssen18a}. The deviation of the lower branch from the top branch, which sets in at around the gas-disc scale-height~\\citep[see also][]{2019Natur.569..519K,Jeffreson21a}, indicates how effectively (on average) molecular gas is removed from around young star clusters in each simulation. If the regions surrounding young stars are effectively cleared of dense gas, then the lower branch drops significantly below the galactic average gas-to-stellar flux ratio at small scales. By contrast, if the young stars remain embedded for long periods of time, then the lower branch remains close to the galactic average value. This is seen in the simulations of~\\cite{Fujimoto19}, who find a duration of $23 \\pm 1$~Myr, nearly an order of magnitude longer than observed~\\citep{2014ApJ...795..156W,2015MNRAS.449.1106H,2018MNRAS.481.1016G,2019MNRAS.483.4707G,2019MNRAS.490.4648H,2019Natur.569..519K,Chevance20,2020arXiv201107772K,2021ApJ...909..121M}. In our simulations, this time-scale ranges from $4.4$~Myr (HII region momentum runs) up to $>5$~Myr (runs without HII region momentum; representing a lower limit, because the duration of co-existence cannot exceed the adopted duration of the young stellar phase, which is $5$~Myr). All of the above numbers are comparable to those obtained for the galaxies with the highest gas surface densities (appropriate for the Agora initial conditions) in the observational sample of~\\cite{Chevance20}, who used the same diagnostic to infer time-scales. This provides a qualitative indication that our feedback implementation broadly matches observed feedback-driven dispersal rates of molecular clouds. Indeed, we see that our HII region momentum feedback moves the morphology of the molecular gas and stellar distribution towards that observed in NGC 628. The qualitative result that the top branch is flatter than the bottom branch indicates a cloud lifetime that is longer than the lifetimes of the young stellar groups (here chosen to be 5 Myr). In Section~\\ref{Sec::GMC-lifetimes}, we further discuss the influence of our feedback prescription on molecular cloud lifetimes and cloud properties.\n\n\\subsubsection{Galactic outflows} \\label{Sec::outflows}\nThe top row of Figure~\\ref{Fig::outflows} shows the total galactic star formation rate as a function of the simulation time $t$ at each simulation resolution and for each feedback prescription. At the beginning of the simulation, the disc collapses vertically and a burst of star formation is produced, after which the interstellar medium settles into a state of dynamical equilibrium. In our simulations, equilibrium is achieved after around $200$~Myr. In the medium- and high-resolution cases, the introduction of HII region momentum suppresses the initial starburst at earlier times and so decreases its magnitude. No such effect is seen for the thermal HII regions (`HII heat'), or in any of the low-resolution simulations, mirroring the qualitative results presented in Section~\\ref{Sec::morphology}. At $\\sim 600$~Myr the star formation rate is consistent with current observed values in the Milky Way~\\citep{Murray&Rahman10,Robitaille&Whitney10,Chomiuk&Povich,Licquia&Newman15}. The feedback prescription does not have a perceivable effect on the global star formation rate after the galaxies have equilibriated.\n\nIn the centre row of Figure~\\ref{Fig::outflows}, we show the rate of gas outflow from each galaxy. The outflow rates are calculated as the total momentum of the gas moving away from the disc, summed over two planar slabs of thickness $500~{\\rm pc}$, located at $\\pm 5$~kpc above and below the galactic disc. This is the same definition used in~\\cite{2014MNRAS.442.3013K,2020arXiv200403608K}. In the medium- and high-resolution simulations, the outflow rate is decreased by around an order of magnitude upon the introduction of HII region momentum feedback. This is again consistent with a reduced level of supernova clustering, which decreases the effectiveness of supernova feedback in driving outflows~\\citep{2020arXiv200911309S,2020arXiv200403608K}. The mass-loading $\\eta$ of the stellar feedback in our model (bottom row of Figure~\\ref{Fig::outflows}) divides the outflow rate by the star formation rate. We note that there is a clear resolution-dependence of the feedback-induced outflow rates and mass-loadings for all feedback prescriptions, likely due to the increased clustering of supernovae at higher resolutions. This is discussed further in Section~\\ref{Sec::SNe-fb}.\n\n\\subsubsection{Resolved disc stability} \\label{Sec::vss}\nThe presence of momentum feedback from HII regions makes a significant difference to the velocity dispersion $\\sigma_{\\rm g}$ and gravitational stability $Q_{\\rm g}$ of the cold gas ($T \\leq 10^4$~K) in our high- and medium-resolution simulations (left and centre columns in Figure~\\ref{Fig::vss}, respectively), as well as to the scale-height $h_{\\rm g}$ of the total gas distribution. We calculate the line-of-sight turbulent velocity dispersion as\n\\begin{equation}\n\\sigma_{\\rm los, g}^2 = \\frac{\\langle |\\bm{v}_i - \\langle \\bm{v}_i \\rangle|^2 \\rangle}{3},\n\\end{equation}\nwhere $\\{\\bm{v}_i\\}$ are the velocity vectors of the gas cells in each radial bin, and angled brackets denote mass-weighted averages over these cells. The~\\cite{Toomre64} $Q$ parameter of the cold gas is then defined as\n\\begin{equation}\nQ_{\\rm g} = \\frac{\\kappa \\sigma_{\\rm g}}{\\pi G \\Sigma_{\\rm g}},\n\\end{equation}\nwith $\\kappa$ the epicyclic frequency of the galactic rotation curve and $\\sigma_{\\rm g} = \\sqrt{c_s^2 + \\sigma_{\\rm los, g}^2}$ for gas sound speed $c_s$. In the top row of Figure~\\ref{Fig::vss}, we quantitatively show the result for the disc scale-height that was demonstrated qualitatively in Figure~\\ref{Fig::morphology}: the reduction in the violence of feedback-induced outflows perpendicular to the galactic mid-plane leads to a smaller disc scale-height when momentum from HII regions is incorporated. In the second row, we demonstrate that for galactocentric radii $R < 8$~kpc in the high-resolution simulation, the amount of cold gas is increased by up to 50~per~cent when HII region momentum is included (solid lines) and that the amount of molecular gas is almost doubled (dashed lines). This is due to two effects: (1) the overall mass of the interstellar medium is larger in the simulations with HII region momentum, due to the suppression of the initial `starburst' (see Section~\\ref{Sec::outflows}), and (2) the fraction of the interstellar medium in the cold and molecular phases is increased (see Section~\\ref{Sec::phases}). We also find that the cold gas has a lower velocity dispersion by $\\sim 5~{\\rm kms}^{-1}$ at all galactocentric radii. Accordingly, the Toomre $Q$ factor ($Q_{\\rm g}$, bottom row of panels) is suppressed by a factor of $\\sim 2$ out to $R \\sim 8~{\\rm kpc}$. The HII region momentum causes the interstellar medium to become clumpier and less gravitationally-stable, leading to the formation of more molecular clouds, as will be discussed in Section~\\ref{Sec::molecular-props}. This is again consistent with the idea that the HII region feedback reduces the momentum injected by supernova feedback, likely by reducing its clustering. In the low-resolution case, none of the observables associated with galactic disc stability are altered by the addition of HII region momentum, consistent with the results presented in Sections~\\ref{Sec::morphology} and~\\ref{Sec::outflows}.\n\n\\begin{figure}\n \\label{Fig::phases}\n \\includegraphics[width=\\linewidth]{figs\/phases.pdf}\n \\caption{{\\it Upper panel:} Density-temperature phase diagram for the HI-resolution simulation with both thermal and beamed momentum from HII regions (HII thermal + beamed mom.), at $t=600$~Myr. Dashed lines delineate the regions of phase space corresponding to the warm neutral medium (WNM), the thermally-unstable phase (Unstable), the cold neutral median (CNM), gas heated by HII regions (HII), and gas heated by supernovae (SN). {\\it Lower panel:} Partitioning of the gas mass in each HI-resolution simulation into five ISM phases from warmest to coolest, as a fraction of the total gas mass in the simulation: hot gas that has received thermal energy from stellar feedback ($M_{\\rm SN+HII}$), the warm neutral medium ($M_{\\rm WNM}$), the unstable phase ($M_{\\rm unstable}$), the cold neutral medium ($M_{\\rm CNM}$), and the star-forming gas in the molecular phase ($M_{\\rm H_2}$, as computed using {\\sc Despotic}, see Appendix~\\protect\\ref{App::postproc}).}\n\\end{figure}\n\n\\begin{figure*}\n \\label{Fig::KS-rln}\n \\includegraphics[width=\\linewidth]{figs\/KS-rln.pdf}\n \\caption{Pixel density as a function of $\\Sigma_{\\rm H_2}$ and $\\Sigma_{\\rm SFR}$ each disc, at a spatial resolution of $750$~pc, corresponding to the resolved molecular star-formation relation of~\\protect\\cite{Kennicutt98}. Gas depletion times of $10^8$, $10^9$ and $10^{10}$~Myr are given by the black solid, dashed and dotted lines respectively. The contours encircle 90~per~cent (dotted), 50~per~cent (dashed) and 10~per~cent (solid) of the observational data for nearby galaxies from~\\protect\\cite{Bigiel08}. All maps are computed at $t=600$~Myr.}\n\\end{figure*}\n\n\\begin{figure}\n \\label{Fig::cloud-lifetimes}\n \\includegraphics[width=\\linewidth]{figs\/cloud-lifetimes.pdf}\n \\caption{\\textit{Top:} Cumulative distribution of trajectory lifetimes $t_{\\rm life}$ in each of the high-resolution simulations. The characteristic cloud lifetimes, obtained from the exponential distributions by fitting a function $\\exp{(-t\/\\tau_{\\rm life})}$ according to Equation~(\\ref{Eqn::lifetime-dstbn}), are annotated according to the legend colours. \\textit{Bottom:} Characteristic cloud lifetime for each simulation (transparent solid lines) as a function of the cloud mass, with the cloud mass PDFs below. The mean values with and without HII region momentum feedback are given by the solid and dashed black lines, respectively. The error-bars correspond to the errors associated with the exponential fits to the distributions $D(t_{\\rm life}>t)$ in each mass bin. Three regimes are annotated: $(i)$ for clouds destroyed preferentially by HII region feedback, $(ii)$ for clouds destroyed preferentially by supernovae, and $(iii)$ for clouds dominated by interactions.}\n\\end{figure}\n\n\\begin{figure}\n \\label{Fig::node-connectivity}\n \\includegraphics[width=\\linewidth]{figs\/node-connectivity.pdf}\n \\caption{Fraction of nodes in the cloud evolution network (molecular clouds observed at an instant in time) that split into two or more children ($\\theta_{\\rm child}>1$) or are the result of a merger between two or more parents ($\\theta_{\\rm par}>1$), as a function of cloud mass. We see that the connectivity of the network increases exponentially from $\\sim 10$~per~cent of multiply-connected nodes for cloud masses $M \\sim 5\\times 10^5 M_\\odot$ up to $\\sim 70$~per~cent at $M \\sim 3\\times 10^7 M_\\odot$. This is the same mass range over which cloud lifetimes decrease with mass in the lower panel of Figure~\\protect\\ref{Fig::cloud-lifetimes}, and cease to depend on the feedback prescription used. The three mass regimes are annotated as in Figure~\\protect\\ref{Fig::cloud-lifetimes}.}\n\\end{figure}\n\n\\subsubsection{ISM phase structure} \\label{Sec::phases}\nIn the top panel of Figure~\\ref{Fig::phases} we display the mass-weighted distribution of gas temperature as a function of the gas volume density (the phase diagram) for the high-resolution simulation including both thermal and beamed HII region momentum (`HII heat \\& beamed mom'). The gas cells cluster around a state of thermal equilibrium in which the rate of cooling (dominated in our simulations by line emission from ${\\rm C}^+$, ${\\rm O}$ and ${\\rm Si}^+$) balances the rate of heating due to photoelectric emission from PAHs and dust grains. The thin horizontal line of particles at high volume densities and $T \\sim 7000$~K contains the particles that are heated by the thermal feedback from HII regions. The dashed black lines delineate the partitioning of the interstellar medium into the feedback-heated phases (SN and HII) the warm neutral medium (WNM), the unstable phase, the cold neutral medium (CNM) and the set of gas cells that are predominantly molecular (${\\rm H}_2$). We have chosen the partitioning of the WNM and CNM gas by eye, according to the major regions of gas accumulation along the thermal equilibrium curve in the phase diagram. The region bridging the WNM and CNM is then classified as `unstable' following~\\cite{Goldbaum16}, and material that is lifted above the equilibrium curve is attributed to feedback-related heating. In the lower panel of Figure~\\ref{Fig::phases} we show the fraction of the total gas mass in each of these phases for the five high-resolution simulations. The mass of molecular hydrogen we use is that which would be inferred by an observer from the CO luminosity, as computed by {\\sc Despotic} (see Appendix~\\ref{App::postproc}). The addition of thermal feedback from HII regions does nothing to the phase structure of the interstellar medium, relative to the case of supernovae only. By contrast, explicit injection of momentum from HII regions leads to almost double the mass of molecular gas and $\\sim 50$~per~cent more cold gas overall ($T \\leq 10^4$~K). The masses of warm and hot, feedback-heated gas are correspondingly reduced. We also note that the overall gas mass remaining in the galaxy at $t=600$~Myr is larger by around $0.8 \\times 10^9~{\\rm M}_\\odot$. This is because the initial `bursts' of star formation, as the galaxy settles into equilibrium, are smaller in the case of effective pre-supernova feedback, as discussed in Section~\\ref{Sec::outflows}.\n\n\\subsubsection{Star formation in molecular gas} \\label{Sec::molecular-SF}\nAlthough the global star formation rate in our simulations appears insensitive to the feedback prescription applied (top row of Figure~\\ref{Fig::outflows}), a slightly greater level of variation is revealed when we look explicitly at the star formation rate surface density $\\Sigma_{\\rm SFR}$ as a function of the molecular gas surface density $\\Sigma_{\\rm H_2}$ (the molecular Kennicutt-Schmidt relation) in Figure~\\ref{Fig::KS-rln}. We find that with the addition of momentum feedback from HII regions, the gradient of the slope in the $\\Sigma_{\\rm H_2}$-$\\Sigma_{\\rm SFR}$ plane is flattened slightly and the molecular gas surface densities are increased by a factor of around two. This means that they fall closer to the observed values delineated by the closed black contours. However this fact should not be over-interpreted, given that the size of the shift in surface density is smaller than the uncertainty in the ${\\rm H}_2$-to-CO conversion factor used to compute the molecular gas abundances (see Appendix~\\ref{App::postproc}). Again, the addition of thermal HII region feedback on its own has no effect.\n\n\\subsection{Properties of molecular clouds} \\label{Sec::molecular-props}\nIn this section, we analyse the molecular clouds identified at the native spatial resolution ($6$~pc) of the column-density projections for our high-resolution simulations. These clouds span a size range from $18$~pc up to $200$~pc and a mass range from $100~{\\rm M}_\\odot$ up to $10^8~{\\rm M}_\\odot$. We identify clouds by taking a threshold of $\\log{(\\Sigma_{\\rm H_2}\/{\\rm M}_\\odot~{\\rm pc}^{-2})} = -3.5$ on the molecular gas column density, as calculated using {\\sc Despotic} (see the beginning of Section~\\ref{Sec::results} and Appendix~\\ref{App::postproc}). The clouds themselves are identified using the {\\sc Astrodendro} package for Python. This procedure is described in detail in Appendix~\\ref{App::postproc}, and is discussed at length in Section 2.9 of~\\cite{2020MNRAS.498..385J}, where we also show that the molecular clouds identified by this method have properties in agreement with observations of clouds in Milky Way-like galaxies, including their masses, sizes, velocity dispersions, surface densities, pressures and star formation rate surface densities. Similarly to~\\cite{2020MNRAS.498..385J}, we discard clouds spanning fewer than 9 pixels ($324~{\\rm pc}^2$), or containing fewer than 20 Voronoi gas cells.\n\nOnce the molecular clouds in our simulations have been identified at every simulation time-step, we follow their evolution as a function of time according to the procedure described in Section 3.2 of~\\cite{Jeffreson21a}. Briefly, we take the two-dimensional pixel masks associated with the sets of molecular clouds in two consecutive snapshots at times $t=t_1$ and $t=t_2$. We project the mask positions of the clouds at $t=t_1$ using the positions and velocities of the gas cells that they span, such that $(x_1, y_1) \\rightarrow (x_1 + v_x \\Delta t, y_1 + v_y \\Delta t)$. We then compare the projected masks to the true pixel masks of the clouds at $t=t_2$. If the projected and true pixel maps overlap by one pixel, then the clouds are indistinguishable at the spatial resolution ($6$~pc) and temporal resolution ($1$~Myr) of the snapshots, and so each cloud at $t=t_1$ is assigned as a parent of the children at $t=t_2$. A given cloud can spawn multiple children (\\textit{cloud splits}) or have multiple parents (\\textit{cloud mergers}). We store the network of parents and children using the {\\sc NetworkX} package for python~\\citep{NetworkX}, and `prune away' unphysical nodes produced by regions of faint CO background emission in our astrochemical post-processing, which do not contain sufficient quantities of CO-luminous gas. We find that these nodes can be removed by taking a cut of $\\sigma \\sim 0.03~{\\rm kms}^{-1}$ on the cloud velocity dispersion, as described in~\\cite{Jeffreson21a}.\\footnote{The mass cut applied in~\\cite{Jeffreson21a} is not required here, as we discard clouds containing fewer than 20 Voronoi cells.}\n\n\\subsubsection{Molecular cloud lifetimes} \\label{Sec::GMC-lifetimes}\nUsing the \\textit{cloud evolution network} described above, we calculate the lifetime $t_{\\rm life}$ of each distinct molecular cloud identified at a given time in our simulations, by performing a Monte Carlo (MC) walk through the network. At the beginning of each MC iteration, walkers are initialised at every \\textit{formation node} in the network (nodes corresponding to a net increase in cloud number). The walkers step along time-directed edges of length $1$~Myr between consecutive nodes, until an \\textit{interaction node} is reached. An interaction may be a merger, a split, or a transient meeting. A random number from a uniform distribution is used to choose between the possible subsequent trajectories for each walker, including the possibility of cloud destruction, if it exists at that node. If the cloud is destroyed, the final lifetime $t_{\\rm life}$ is returned. This algorithm satisfies the requirements of:\n\\begin{enumerate}\n\t\\item \\textit{Cloud uniqueness:} Edges between nodes in the network represent time-steps in the evolution of a single cloud, so must not be double-counted.\n\t\\item \\textit{Cloud number conservation:} The number of cloud lifetimes retrieved from the network must be equal to the number of cloud formation events and cloud destruction events, as each cloud can be formed and destroyed just once.\n\\end{enumerate}\nSeventy MC iterations are performed to reach convergence of the characteristic molecular cloud lifetime $\\tau_{\\rm life}$ for the cloud population of the entire galaxy.\n\nIn the top panel of Figure~\\ref{Fig::cloud-lifetimes}, we show the cumulative distributions $D(t_{\\rm life} > t)$ of lifetimes $t_{\\rm life}$ for the molecular clouds in our high-resolution simulations. These distributions have an exponential form, as expected if the formation and destruction of clouds has reached a steady state. The simulations with HII region momentum feedback do not appear significantly different to those without. We have annotated the \\textit{characteristic cloud lifetime} $\\tau_{\\rm life}$ for each simulation by fitting an exponential profile to each distribution, and assuming the steady-state proportionality\n\\begin{equation} \\label{Eqn::lifetime-dstbn}\n\\ln{D(t_{\\rm life})} \\propto -\\frac{t}{\\tau_{\\rm life}},\n\\end{equation}\nas in~\\cite{Jeffreson21a}. We find only a marginal increase of $4$~Myr in the overall value of $\\tau_{\\rm life}$ upon the introduction of HII region momentum feedback (an average of $37 \\pm 2$~Myr with HII region momentum vs. $33 \\pm 2$~Myr without). However, in the lower panel of Figure~\\ref{Fig::cloud-lifetimes}, we see that the the influence of the feedback prescription is dependent on the cloud mass. Its influence can be divided into three regimes as follows:\n\\begin{enumerate}\n\t\\item $M\/{\\rm M}_\\odot \\la 5.6 \\times 10^4$: HII region momentum depresses the cloud lifetime by $\\sim 10$~Myr.\n\t\\item $5.6 \\times 10^4 \\la M\/{\\rm M}_\\odot \\la 5 \\times 10^5$: HII region momentum increases the cloud lifetime by $\\sim 7$~Myr.\n\t\\item $5 \\times 10^5 \\la M\/{\\rm M}_\\odot$: HII region momentum has no effect on the cloud lifetime.\n\\end{enumerate}\nThis result is consistent with the following scenario: the least massive molecular clouds in $(i)$ are less likely to contain the massive stellar clusters required for the fastest and most efficient injection of supernova energy. This results in an uptick of the characteristic cloud lifetime for the simulations without HII region momentum feedback (blue and purple lines in Figure~\\ref{Fig::cloud-lifetimes}) at small masses. However, the least-massive clouds are also the easiest to disperse, and so the relatively-small amount of momentum injected by HII regions can truncate the cloud lifetime in the absence of efficient supernova feedback. At larger cloud masses $(ii)$, the HII region momentum is too puny to cause disruption, so its main influence is to reduce supernova clustering and thus decrease the efficacy of the supernova feedback, consistent with its effect on the large-scale properties of the interstellar medium, presented in Section~\\ref{Sec::disc-props}. This increases the characteristic cloud lifetime. Finally, the most massive molecular clouds in $(iii)$ are often unresolved cloud complexes, and undergo increasingly more mergers and splits as the cloud mass is increased from $10^{5.8}$ through $10^{7.5}~{\\rm M}_\\odot$, as shown in Figure~\\ref{Fig::node-connectivity}. Across this mass regime, the fraction of multiply-connected nodes increases from 10~per~cent up to 70~per~cent, elevating the number of short MC trajectories containing high-mass nodes. The trajectory lifetimes returned by the MC walk are therefore likely to be determined by the level of graph connectedness, rather than by the feedback-induced destruction of the molecular clouds. This also explains the drop in the cloud lifetime for the most massive clouds. In order to determine the effects of stellar feedback on these high-mass clouds, we will need to develop the algorithm put forward in~\\cite{Jeffreson21a}, to distinguish between cloud mergers of varying mass ratio. Overall, the cloud lifetimes across masses span the range from $10$-$40$~Myr, similar to observations~\\citep{Engargiola03,Blitz2007,Kawamura09,Murray11,Meidt15,Corbelli17,Chevance20}.\n\nFinally, we note that the number of molecular clouds generated per unit mass of the interstellar medium in our simulations is increased by the presence of HII region momentum. In the `SNe only' and `HII heat' simulations, the average number of clouds identified is $3.7$ per $10^7~{\\rm M}_\\odot$; this increases to $6.2$ per $10^7~{\\rm M}_\\odot$ for the `HII spherical mom.', `HII beamed mom.' and `HII heat \\& beamed mom.' simulations. Combined with the reduced number of high-mass clouds, this result indicates that the molecular interstellar medium is slightly more fragmented in the case of the HII region feedback.\n\n\\begin{figure*}\n \\label{Fig::cloud-veldisp-surfdens}\n \\includegraphics[width=\\linewidth]{figs\/HII_veldisp-surfdens.pdf}\n \\caption{Distributions of molecular cloud surface densities $\\Sigma$ and velocity dispersions $\\sigma$ in each of the high-resolution simulations. Each value is a median over a trajectory in the cloud evolution network. \\textit{Left\/Centre:} Cumulative distribution of the surface density\/velocity dispersion, with medians indicated by the vertical dashed lines. \\textit{Right:} Scaling relation of the surface density with the velocity dispersion. The contours enclose 90~per~cent of the identified molecular clouds. The grey-shaded histogram contains the full cloud distribution for the `HII heat \\& beamed mom.' simulation. Virial parameters of $\\alpha_{\\rm vir}=0.5$ and $1$ for spherical clouds of size $6$~pc are given by the dashed and dot-dashed lines, respectively. We see that the introduction of HII region momentum predominantly (but hardly) affects the cloud velocity dispersion.}\n\\end{figure*}\n\n\\subsubsection{Cloud velocity dispersions and surface densities} \\label{Sec::GMC-veldisp-surfdens}\nWe now turn to the physical properties of the molecular clouds in our simulations: first to the scaling relation between the cloud surface density $\\Sigma$ and velocity dispersion $\\sigma$. Each value corresponds to an average (median) taken over the cloud lifetime $t_{\\rm life}$ (i.e.~along a unique trajectory in the cloud evolution network). The right-hand panel of Figure~\\ref{Fig::cloud-veldisp-surfdens} shows the scaling relation itself, for which the clouds fall along a line of constant virial parameter, as observed in nearby Milky Way-like galaxies~\\citep[e.g.][]{Sun18}. Lines representing virial parameters of $1$ and $2$ for spherical clouds at a fixed size of $6$~pc (our native resolution) are given by the dashed and dot-dashed lines, respectively. The coloured contours enclose 90~per~cent of the clouds for each high-resolution simulation, while the grey-shaded histogram displays the entire cloud population for the `HII heat \\& beamed mom.' simulation. In the left and central panels of Figure~\\ref{Fig::cloud-veldisp-surfdens} we show the cumulative distributions of the cloud surface density and velocity dispersion separately. We see that the introduction of HII region momentum makes little difference to the distribution of surface densities, and reduces the median cloud velocity dispersion by only $0.5~{\\rm kms}^{-1}$. This reduction is consistent with the drop in the bulk velocity dispersion of the cold gas in our simulations, presented in Figure~\\ref{Fig::vss}.\n\n\\subsubsection{Cloud masses and star formation rates} \\label{Sec::GMC-mass-SFR}\nThe influence of the stellar feedback prescription on the masses and star formation rate surface densities of our molecular clouds is shown in Figure~\\ref{Fig::cloud-mass-SFR}. In the right-hand panel, the number of clouds $N(>M)$ with mass greater than $M$ is compared to the power-law form $\\mathrm{d} N\/\\mathrm{d} M \\propto M^{-\\beta}$ observed for clouds in the Milky Way over the mass range of $\\log{M} \\in [4.8, 6.5]$, with $\\beta \\in [1.6, 1.8]$~\\citep{Solomon87,Williams&McKee97,Heyer+09,Roman-Duval+10,Miville-Deschenes17,Colombo+19}. When we fit corresponding powerlaws to the PDF of each mass spectrum (via simple linear regression in the mass range $\\log{(M\/M_\\odot)} \\in [4.8, 6.5]$), we find a slope of $\\beta = 1.75 \\pm 0.09$ for the `SNe only' simulation and a slope of $\\beta = 1.80 \\pm 0.12$ for the `HII heat \\& beamed mom.' simulation. That is, the number of the most massive clouds is reduced slightly by the presence of HII region feedback.\n\nWe note that this result (a steeper mass function with HII region momentum) is the opposite of that expected if the characteristic rates $\\xi_{\\rm form}$ and $1\/\\tau_{\\rm life}$ for cloud formation and destruction in each galaxy are independent of the cloud mass, as assumed in~\\cite{2017ApJ...836..175K}. These authors use an analytic rate equation for the number of clouds $N$, explicitly accounting for the process of cloud coagulation, to derive a mass function slope of $\\mathrm{d} N\/\\mathrm{d} M \\propto -(\\xi_{\\rm form} \\tau_{\\rm life})^{-1}$. In the steady-state approximation of Equation (\\ref{Eqn::lifetime-dstbn}), the number of clouds present in the galaxy at a given time approaches $N \\rightarrow \\tau_{\\rm life} \\xi_{\\rm form}$, so the predicted slope goes as $\\mathrm{d} N\/\\mathrm{d} M \\propto -1\/N$. We find $N$ to be higher in the simulations with HII region momentum, but $\\mathrm{d} N\/\\mathrm{d} M$ to be steeper, in contradiction with this work. We attribute this to the fact that $\\tau_{\\rm life}$ is manifestly dependent on the cloud mass (see Figure~\\ref{Fig::cloud-lifetimes}) and that the mass-dependence of $\\xi_{\\rm form}$ is not studied here, but likely non-negligible.\n\nIn the central panel of Figure~\\ref{Fig::cloud-mass-SFR}, we show the star formation rate $\\Sigma_{\\rm SFR}$ per unit area of the molecular clouds in each simulation. The introduction of HII region momentum causes a three-fold drop in the value of $\\Sigma_{\\rm SFR}$. In the right-hand panel, we show that this drop in the star formation rate occurs across the whole range of cloud masses. This result agrees broadly with the results from high-resolution simulations of resolved HII regions~\\citep[e.g.][]{2016ApJ...829..130R,2018MNRAS.478.4799H,2018MNRAS.475.3511G,2019MNRAS.489.1880H,2020MNRAS.497.3830F,2020MNRAS.499..668G,2020MNRAS.492..915G,2020arXiv201107772K}, which show that HII region feedback can efficiently suppress the overall star formation efficiency within individual molecular clouds.\n\n\\begin{figure*}\n \\label{Fig::cloud-mass-SFR}\n \\includegraphics[width=\\linewidth]{figs\/HII_CDFs_mass-SFR.pdf}\n \\caption{Distributions of molecular cloud masses $M$ and star formation rates per unit area of the galactic mid-plane $\\Sigma_{\\rm SFR}$ in each of the high-resolution simulations. \\textit{Left:} Normalised number of clouds $N(>M)$ with mass larger than or equal to $M$. The solid black and dashed black lines denote the range of power-law slopes for the observed cloud mass distribution in the Milky Way, given by $\\mathrm{d} N\/\\mathrm{d} M \\propto M^{-\\beta}$ with $\\beta \\in [1.6, 1.8]$. \\textit{Centre:} Cumulative distribution of the star formation rate surface density, with medians indicated by the vertical dashed lines. \\textit{Right:} Scaling relation of the star formation rate surface density with the cloud mass. The contours enclose 90~per~cent of the identified molecular clouds. The grey-shaded histogram contains the full cloud distribution for the `HII heat \\& beamed mom.' simulation. We see that the introduction of HII region momentum reduces the star formation rates of all clouds, and slightly decreases the number of massive clouds.}\n\\end{figure*}\n\n\\subsection{Beamed vs.~spherical HII region momentum} \\label{Sec::beamed-vs-spherical}\nAside from the finding that thermal feedback from marginally-resolved HII regions is ineffective in transferring energy to the surrounding interstellar medium, a recurring theme in the preceding sub-sections is that there is no discernible difference between our simulations with spherical HII region feedback and beamed HII region feedback. The morphology, phase structure and stability of the interstellar medium are identical in these cases, and the properties of the molecular clouds are unaffected. This might be surprising, considering the qualitative difference in the appearance of HII regions in the blistered and spherical cases (see Figure~\\ref{Fig::blister-projections}) and the difference in their ionisation-front and Str\\\"{o}mgren radii. We find that it is only the quantity of momentum injected in our simulations that matters (this is roughly equivalent in the spherical and beamed cases), and not the direction in which it is injected. However, we might expect that if the direction of momentum injection were not chosen randomly for each FoF group and star particle, but rather preserved over the evolution of each molecular cloud, the blistered HII region feedback might be more effective in removing the gas from around star particles.\n\n\\section{Discussion} \\label{Sec::discussion}\nWe have shown in Section~\\ref{Sec::results} that the injection of momentum from HII regions, according to a novel numerical model based on the analytic framework of KM09, reduces the mass-loading of outflows perpendicular to the mid-plane of isolated disc galaxies, and increases the fraction of cold gas within these discs, while decreasing its velocity dispersion and scale-height. The resolved molecular clouds formed from this cold gas reservoir suffer alterations in their lifetimes, masses, star formation rates and velocity dispersions. We find that these results apply across a mass resolution range of $10^3$-$10^4~{\\rm M}_\\odot$ in the moving-mesh code {\\sc Arepo.}\n\nIt is important to note that all of the above results depend not just on our modelling of HII regions, but on a number of other assumptions made during the construction of our stellar population and its feedback, including its supernova feedback. In Section~\\ref{Sec::caveats}, we outline the key caveats of our model, their possible effects, and how these could in the future be disentangled from the relative roles of HII region and supernova feedback in isolated galaxy simulations. In Section~\\ref{Sec::literature-sims} we compare our results to studies of HII region feedback in the literature.\n\n\\subsection{Caveats of our model} \\label{Sec::caveats}\n\\subsubsection{Photon trapping and escape} \\label{Sec::comp-highres}\nWithin the model for HII region feedback, we have used a value of\n\\begin{equation}\nf_{\\rm trap} = 1 + f_{\\rm trap, w} + f_{\\rm trap, IR} + f_{{\\rm trap, Ly}\\alpha} \\sim 8\n\\end{equation}\nto account for the enhancement of pressure inside the ionisation front, due to the trapping of energy from stellar winds ($f_{\\rm trap, w}$), infrared photons ($f_{\\rm trap, IR}$) and Lyman-$\\alpha$ ($f_{{\\rm trap, Ly}\\alpha}$) photons. The value of $f_{\\rm trap}$ is constrained by~\\cite{2021ApJ...908...68O} using infrared observations to infer the pressures inside young HII regions in the Milky Way. By using $f_{\\rm trap} \\sim 8$, we therefore implicitly assume that $f_{{\\rm trap, Ly}\\alpha} \\approx 0$ and $f_{\\rm trap, w} \\approx 0$, because an estimation of the effects of Lyman-$\\alpha$ photons and winds would require observations of the dust and diffuse gas surrounding the sources, in the optical and the X-ray, respectively. Our model therefore does not account for the absorption of Lyman-$\\alpha$ photons, or for the trapping of stellar winds. In addition, the interaction of stellar winds with radiation pressure is a complex problem: numerous high-resolution, radiative-transfer numerical studies of HII regions inside individual clouds~\\citep[e.g.][]{2017MNRAS.467.1067D,2017ApJ...850..112R,2018ApJ...859...68K,2019ApJ...883..102K} have shown that stellar winds (along with inhomogeneities in the gas surrounding HII regions) can lead to the escape of radiation through holes in the shell bounding the ionised gas, and so to a reduction in the overall radiation pressure by factors of $\\sim 5$-$10$~\\citep{2019ApJ...883..102K}. The HII regions in our model are assumed perfectly spherical or hemi-spherical, and we have not accounted for stellar winds. It is therefore possible that we have under-estimated the strength of radiation pressure by ignoring Lyman-$\\alpha$ photon and wind trapping, or have over-estimated it by ignoring photon escape. However, this is unlikely to have a large effect on the total amount of momentum injected by our HII regions, because for the ionising luminosities between $S_{49} \\sim 1$ and $100$ spanned by the FoF groups in our simulations, the momentum contribution made by the gas pressure is around ten times that made by the radiation pressure, according to our Equation~(\\ref{Eqn::momentumeqn_final}).\n\n\\subsubsection{Resolution-dependence of supernova feedback} \\label{Sec::SNe-fb}\nAs noted throughout Section~\\ref{Sec::results}, the differences between the simulations with and without HII region momentum do not persist down to resolutions of $10^5~{\\rm M}_\\odot$ per gas cell. In addition, when supernova feedback is used on its own, the outflow rates and their mass-loadings, as well as the gravitational stabilities of the gas in the galactic discs, are substantially different for the low-, medium- and high-resolution simulations. This may be due to a decrease in the effectiveness of supernova clustering at resolutions of $<10^5~{\\rm M_\\odot}$ (i.e. the resolution is too low for clustering to be resolved). As discussed by~\\cite{2020arXiv200911309S}, early feedback from HII regions and stellar winds affects the interstellar medium by reducing the degree of supernova clustering and so the violence of the resulting explosions, decreasing the sizes of galactic outflows and the mid-plane gas velocity dispersion. Therefore, if clustering is not resolved, the effect of our HII region feedback on the large-scale properties of the interstellar medium may be spuriously-weakened at low resolutions.~\\cite{2020arXiv201010533S} also discuss the non-convergence of various stellar feedback prescriptions due to an under-sampling the IMF at high mass resolutions. However, this is not a problem in our simulations, due to the use of the Poisson sampling procedure from~\\cite{Krumholz15}. By this procedure, the number of stars assigned to a given star particle\/cluster depends on the star particle mass, but the form of the resulting distribution of stellar masses is not affected.\n\n\\subsection{Comparison to the literature} \\label{Sec::literature-sims}\n\\subsubsection{High-resolution simulations of molecular clouds}\nThe molecular cloud sample in our simulations has yielded two key results: (1) the lifetimes of the least-massive clouds are truncated by HII region feedback (while those of intermediate-mass clouds are extended), and (2) HII region feedback suppresses the star formation rate within molecular clouds by a factor of three. These findings can be qualitatively compared to high-resolution simulations of resolved HII regions in individual molecular clouds. In particular,~\\cite{2012MNRAS.424..377D,2013MNRAS.430..234D,2017ApJ...851...93K} find that only the least-massive molecular clouds are prone to dispersal by HII region feedback, and that this dispersal occurs on time-scales of $<10$~Myr, as we have found in Section~\\ref{Sec::GMC-lifetimes}. Larger clouds can only be disrupted by supernovae. Across molecular clouds of all masses,~\\cite{2016ApJ...829..130R,2018MNRAS.478.4799H,2018MNRAS.475.3511G,2019MNRAS.489.1880H,2020MNRAS.497.3830F,2020MNRAS.499..668G,2020MNRAS.492..915G,2020arXiv201107772K} show that the star formation efficiency per free-fall time is suppressed by the presence of HII region feedback, as we have discussed in Section~\\ref{Sec::GMC-mass-SFR}. Although it will be important to check the convergence of our sub-grid model with high-resolution simulations such as these, it is encouraging to note that the main results for our molecular cloud sample echo existing results in single-cloud studies.\n\n\\subsubsection{Isolated disc simulations}\nWe may also compare our results with other implementations of radiation\/thermal pressure from HII regions in isolated disc galaxies at similar mass resolutions.~\\cite{2020arXiv200911309S} investigate the role of pre-supernova feedback in suppressing supernova clustering in dwarf galaxies in {\\sc Arepo}, reaching mass resolutions of $20~{\\rm M}_\\odot$. The Str\\\"{o}mgren radii of the HII regions in their simulations are well-resolved, allowing for the explicit ionisation and heating of gas cells to be converted to momentum. Using this prescription, the authors find that supernova clustering is decreased by the presence of HII region feedback. This leads to a significant suppression of outflows and their mass-loadings, as well as a reduction in the sizes of supernova-blown voids within the interstellar medium, in agreement with our results.~\\cite{Fujimoto19} investigate molecular clouds in an isolated disc galaxy at a comparable resolution to ours, but with only thermal HII region feedback~\\citep[see also][]{Goldbaum16}. These authors find that both the pre-supernova and supernova feedback in their simulations are inefficient at disrupting the parent molecular clouds around young stars, resulting in a flat scale-dependence of the gas-to-stellar flux ratio when apertures are centred on young stellar peaks (by comparison to the diverging branch we find in our Figure~\\ref{Fig::tuning-forks}). This leads to a much longer duration of the embedded phase of star formation, as derived via the method of~\\cite{Kruijssen18a}: $23$~Myr in~\\cite{Fujimoto19} vs. $4.4$~Myr in our simulations.\n\nAt mass resolutions of $10^3$-$10^4$ solar masses per gas cell, ~\\cite{2011MNRAS.417..950H,2013MNRAS.434.3142A,2014MNRAS.445..581H,Agertz13,2015ApJ...804...18A} inject HII region momentum via a similar prescription to ours, but in the analytic form of a `direct radiation pressure' during the radiation-dominated phase of HII region expansion. As discussed in KM09 and in our Section~\\ref{Sec::Theory}, radiation pressure dominates the expansion of only the largest HII regions, while those with ionising luminosities in the range of $1 < S_{49} < 100$ (as for the FoF groups in our high-resolution simulation) suffer a factor of ten or more reduction in the momentum injected, if the gas-pressure term in Equation~(\\ref{Eqn::momentumeqn_final}) is ignored. Despite this, the above works find that their radiation pressure prescriptions are necessary to achieve a realistic interstellar medium. This can be attributed to their use of an $f_{\\rm trap}$ factor far exceeding that found in observations~\\citep[e.g.][]{2021ApJ...908...68O}. In~\\cite{2013MNRAS.434.3142A,Agertz13,2015ApJ...804...18A} a value of $f_{\\rm trap} \\sim 25$ is used, and in~\\cite{2011MNRAS.417..950H,2014MNRAS.445..581H} this value is further increased within the range $f_{\\rm trap} \\sim 10$-$100$. By contrast, later works of~\\cite{Hopkins18,2019MNRAS.489.4233M} reduce the value of $f_{\\rm trap}$ back to order one, and the authors find that in this case, the direct radiation pressure has a negligible effect~\\citep[see Figure 36 of][]{Hopkins18}. In summary, the above works agree with our results in the sense that a pre-supernova momentum injection of $\\sim 10$-$100 \\times L\/c$ has a substantial influence on the intermediate- and large-scale properties of the interstellar medium. This momentum injection is needed to achieve an interstellar medium consistent with observations. However, according to the calculations presented in KM09 and in our Section~\\ref{Sec::Theory}, the vast majority of this momentum comes from the gas pressure inside the HII region, and not from the radiation pressure.\n\nFinally, we note that an identical feedback prescription (but using $f_{\\rm trap} = 2$ rather than $f_{\\rm trap} = 8$) was adopted in~\\cite{2020MNRAS.498..385J} and in~\\cite{Jeffreson21a} to investigate molecular cloud properties in three isolated disc galaxies with external, analytic galactic potentials. The molecular cloud population at the native resolution in these studies was on average less massive (maximum mass of $\\sim 10^7~{\\rm M}_\\odot$ vs. $\\sim 10^8~{\\rm M}_\\odot$ here) and had a shorter median cloud lifetime ($\\sim 20$~Myr vs. $\\sim 35$~Myr here). This can be attributed to the fact that the mid-plane turbulent pressure in the Agora disc used here is approximately eight times that of the discs introduced in~\\cite{2020MNRAS.498..385J}, i.e.~the mid-plane gas surface density and velocity dispersion are both doubled. This may be due to the use of a live dark matter and stellar potential, which allows for a greater degree of baryon clustering. The star formation efficiency per free-fall time of $\\epsilon_{\\rm ff} = 10$~per~cent used in this work is also ten times the value of $1$~per~cent used in~\\cite{2020MNRAS.498..385J}, because we have found that the lower star formation efficiency results in unphysically-bursty star formation, and an unphysically-high turbulent velocity dispersion of the cold gas on kpc-scales.\n\n\\section{Conclusions} \\label{Sec::conclusion}\nIn this work, we have developed a novel model for the momentum imparted by marginally-resolved HII regions in simulations with mass resolutions between $10^3$ and $10^5~{\\rm M}_\\odot$ per gas cell. The model can be applied in a spherical or a beamed configuration, where the latter corresponds to the directed momentum injected from blister-type HII regions on the edges of molecular clouds. We have compared simulations with only supernova feedback to simulations with supernova and thermal HII region feedback, spherical HII region momentum, beamed HII region momentum, and a combination of beamed momentum and thermal injection, across the mass resolution range $10^3$-$10^5~{\\rm M}_\\odot$. In general, we find that:\n\\begin{enumerate}\n\t\\item Thermal feedback from marginally-resolved HII regions has no influence on the interstellar medium, at any scale or resolution.\n\t\\item The geometry of momentum injection (spherical or beamed) from HII regions similarly has very little effect.\n\\end{enumerate}\nWhen HII region momentum is introduced at mass resolutions between $10^3$ and $10^4~{\\rm M}_\\odot$, the large-scale interstellar medium responds in the following ways:\n\\begin{enumerate}\n\t\\item The mass-loading and magnitude of galactic outflows are reduced by an order of magnitude.\n\t\\item The gas-disc scale-height is reduced by $0.5$~dex for galactocentric radii $>5$~kpc.\n\t\\item The velocity dispersion of the cold gas is supressed by $\\sim 5~{\\rm kms}^{-1}$, and the gravitational stability of the gas disc is correspondingly decreased by a factor of around two in the~\\cite{Toomre64} $Q$ parameter.\\footnote{We recall that the factors of decrease in the velocity dispersion and Toomre $Q$ parameter do not match exactly because of the difference in gas density at the simulation time of comparison.}\n \\item The mass fraction of cold gas is increased by $\\sim 50$~per~cent and the mass fraction of cold molecular gas is approximately doubled.\n\\end{enumerate}\nThe above results are consistent with the idea that HII region feedback (and pre-supernova feedback in general) reduces the clustering of supernovae and therefore their efficacy in depositing large quantities of momentum into the interstellar medium, as studied by~\\cite{2020arXiv200911309S}. The results do not persist down to resolutions of $10^5~{\\rm M}_\\odot$, although the HII regions are still marginally-resolved. This non-convergence is likely due to a decrease in the effectiveness of supernova clustering.\n\nFor molecular clouds specifically, we find the following results:\n\\begin{enumerate}\n\t\\item The lifetimes of the least massive molecular clouds ($M\/{\\rm M}_\\odot \\la 5.6 \\times 10^4$) are reduced from $\\sim 18$~Myr to $<10$~Myr. That is, HII region momentum is able to disperse low-mass clouds that do not contain supernovae from massive stellar clusters.\n\t\\item The lifetimes of intermediate-mass ($5.6 \\times 10^4 \\la M\/{\\rm M}_\\odot \\la 5 \\times 10^5$) clouds are increased by $\\sim 7$~Myr. That is, HII region momentum decreases the efficiency of supernovae in dispersing intermediate-mass clouds.\n\t\\item The molecular cloud star formation rate surface density is suppressed by a factor of three.\n\t\\item The molecular cloud velocity dispersion is reduced by $\\sim 0.5~{\\rm kms}^{-1}.$\n\\end{enumerate}\n\nIn summary, we find that the large- and intermediate-scale properties of the simulated interstellar medium, as well as the properties of its molecular clouds, are significantly altered by the introduction of momentum from HII regions at numerical mass resolutions from $10^3~{\\rm M}_\\odot$ to $10^5~{\\rm M}_\\odot$ per gas cell. More than 90~per~cent of the injected momentum is due to the thermal expansion of the HII regions, rather than radiation pressure. The injection of thermal energy without momentum has no discernible effect on the simulated galaxies.\n\n\\section*{Acknowledgements}\nWe thank an anonymous referee for a constructive report, which improved the paper. We thank Volker Springel for providing us access to Arepo. SMRJ is supported by Harvard University through the ITC. We gratefully acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through an Emmy Noether Research Group (SMRJ, MC, JMDK; grant number KR4801\/1-1) and the DFG Sachbeihilfe (MC, JMDK; grant number KR4801\/2-1), as well as from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme via the ERC Starting Grant MUSTANG (SMRJ, BWK, JMDK; grant agreement number 714907). SMRJ, MRK, YF, MC and JMDK acknowledge support from a UA-DAAD grant. BWK acknowledges funding in the form of a Postdoctoral Research Fellowship from the Alexander von Humboldt Stiftung. MRK acknowledges support from the Australian Research Council through Future Fellowship FT80100375 and Discovery Projects award DP190101258. The work of LA was partly supported by the Simons Foundation under grant no.~510940. The work was undertaken with the assistance of resources and services from the National Computational Infrastructure (NCI; award jh2), which is supported by the Australian Government. We are grateful to Jeong-Gyu Kim, Eve Ostriker and Vadim Semenov for helpful discussions.\n\n\\section*{Data Availability Statement}\nThe data underlying this article are available in the article and in its online supplementary material.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n Scattering amplitudes as well as further quantities in Quantum Field Theory contain a rich mathematical structure, whose understanding has frequently expanded our calculational reach -- benefiting both phenomenological tests of the Standard Model of Particle Physics at the LHC as well as more formal studies.\n \n At one-loop order, and in certain cases also at higher loop orders, the functions that occur in Feynman integrals and thus in Quantum Field Theory are multiple polylogarithms (MPLs) \\cite{Chen:1977oja,G91b,Goncharov:1998kja,Remiddi:1999ew,Borwein:1999js,Moch:2001zr}, which are by now very well understood.\n Increasing the loop order, the next class of functions we encounter are elliptic multiple polylogarithms (eMPLs), on which there has been much recent progress \\cite{Laporta:2004rb,MullerStach:2012az,brown2011multiple,Bloch:2013tra,Adams:2013nia,Adams:2014vja,Adams:2015gva,Adams:2015ydq,Adams:2015ydq,Adams:2016xah,Adams:2017ejb,Adams:2017tga,Bogner:2017vim,Broedel:2017kkb,Broedel:2017siw,Remiddi:2017har,Chen:2017soz,Bourjaily:2017bsb,Adams:2018yfj,Broedel:2018iwv,Broedel:2018qkq,Honemann:2018mrb,Bogner:2019lfa,Broedel:2019hyg,Duhr:2019rrs,Walden:2020odh,Weinzierl:2020fyx,Kristensson:2021ani,Frellesvig:2021hkr},\n in particular by studying the two-loop massive sunrise integral in two dimensions \\cite{SABRY1962401,Broadhurst:1993mw,Laporta:2004rb,Muller-Stach:2011qkg,Adams:2013nia,Bloch:2013tra,Remiddi:2013joa,Bloch:2014qca,Adams:2014vja,Adams:2015gva,Adams:2015ydq,Adams:2015ydq,Bloch:2016izu,Remiddi:2016gno,Adams:2017ejb,Broedel:2017siw} (see figure \\ref{subfig:sunrise}). More recently, also more complicated Feynman integrals are starting to be understood,\n in particular the \n two-loop ten-point double-box integral with massless internal propagators in four dimensions \\cite{Bourjaily:2017bsb,Kristensson:2021ani} (see figure \\ref{subfig:doublebox}).\n Beyond eMPLs, also integrals over more complicated geometries than elliptic curves occur \\cite{Brown:2009ta,Brown:2010bw,Bourjaily:2018yfy,Bourjaily:2018ycu,Festi:2018qip,Broedel:2019kmn,Besier:2019hqd,Bourjaily:2019hmc,Vergu:2020uur,mirrors_and_sunsets,\nBroadhurst:1987ei,Adams:2018kez,Adams:2018bsn,Huang:2013kh,Klemm:2019dbm,Bonisch:2020qmm,Bonisch:2021yfw,Muller:2022gec,Chaubey:2022hlr}; an understanding of the corresponding spaces of functions is still in its infancy.\n For a recent review on functions in scattering amplitudes beyond MPLs, see \\cite{Bourjaily:2022bwx}.\n \nOur good understanding of MPLs is to a large extend due to the Hopf algebra structure underlying these functions \\cite{Gonch2,Goncharov:2010jf,Brown:2011ik,Duhr:2011zq,Duhr:2012fh}, and in particular the symbol \\cite{Goncharov:2010jf}.\nThe symbol map associates to each MPL $f$ a simple tensor product, $\\mathcal{S}(f)=\\sum \\log(\\phi_{i_1})\\otimes\\dots\\otimes \\log(\\phi_{i_k})$. \nThe entries in this tensor product, called symbol letters, are logarithms of rational or algebraic functions $\\phi_i$ of the kinematic invariants.\nSince a tensor product is easy to manipulate, and the identities of the symbol letters, $\\log(a)+\\log(b)=\\log(ab)$, are well understood, the symbol provides a powerful way of finding identities between MPLs and for simplifying expressions. \nThe symbol moreover manifests physical properties of the corresponding function. For example, the first entry of the symbol describes the discontinuities, which are heavily restricted in particular in massless theories resulting in so-called first-entry conditions \\cite{Gaiotto:2011dt}.\nMoreover, discontinuities in overlapping channels are forbidden by the so-called Steinmann conditions \\cite{Steinmann,Steinmann2}, restricting which symbol letter in the second entry can follow a particular letter in the first entry.\nThe symbol has made enormous progress possible for quantities consisting of MPLs, both in relation to phenomenology and more formal studies, including in particular powerful bootstrap techniques \\cite{Dixon:2011pw,Dixon:2011nj,Dixon:2013eka,Dixon:2014voa,Dixon:2014iba,Drummond:2014ffa,Dixon:2015iva,Caron-Huot:2016owq,Dixon:2016apl,Dixon:2016nkn,Drummond:2018caf,Caron-Huot:2019vjl,Li:2016ctv,Almelid:2017qju,Dixon:2020bbt,Dixon:2022rse,Abreu:2020jxa,Chicherin:2020umh,Dixon:2012yy,Chestnov:2020ifg,He:2021fwf,Drummond:2017ssj,Chicherin:2017dob,Caron-Huot:2018dsv,Henn:2018cdp,He:2021non,He:2021eec,Heller:2019gkq,Heller:2021gun,Duhr:2021fhk}.\n\nWhile the symbol for eMPLs was defined in \\cite{BrownNotes,Broedel:2018iwv}, it has so far not been put to much use, and is still much less understood than its analog for MPLs.\nOne reason is that the symbol letters $\\Omega^{(i)}$ for eMPLs are themselves elliptic functions of the kinematic invariants. The simplest letters $\\Omega^{(0)}$ satisfy simple identities as the consequence of the group law on the elliptic curve.\nIn \\cite{Kristensson:2021ani}, some identities for the elliptic letters $\\Omega^{(i)}$ with $i=1,2$ were observed numerically in the study of the symbol of the ten-point elliptic double-box integral. \nUsing these identities, it was found that the elliptic letters in the first two entries combine to logarithms, manifesting the same first-entry condition as for polylogarithmic amplitudes as well as the Steinmann conditions.\nMoreover, the last entries were found to be given by simple elliptic integrals $\\Omega^{(0)}$, with $\\Omega^{(2)}$ only occurring in the third entry preceding the modular parameter $\\tau$ in the last entry.\n\nIn this paper, we show how the identities observed in \\cite{Kristensson:2021ani} for $\\Omega^{(1)}$ are a consequence of Abel's theorem \\cite{abel1841}.\nMoreover, we demonstrate that the identities observed in \\cite{Kristensson:2021ani} for $\\Omega^{(2)}$ are a consequence of the elliptic Bloch relation \\cite{Zagier2000,bloch2011higher} for the elliptic dilogarithm, which generalizes the Bloch relation for the classic dilogarithm and which have also been studied in the context of finding identities between elliptic multiple polylogarithms \\cite{Broedel:2019tlz,Bolbachan:2019dsu}.\nWhile the identities for $\\Omega^{(1)}$ can be reduced to three-term identities similar to $\\log(a)+\\log(b)=\\log(ab)$ in the case of the logarithm, the elliptic Bloch relation, and thus the identities for $\\Omega^{(2)}$, are five-term identities similar to the Bloch relation for the classical dilogarithm, which are made manifest only by the symbol.\nThus, we introduce a symbol prime $\\mathcal{S}'$ for the symbol letters $\\Omega^{(2)}$\n(and similarly for $\\Omega^{(n>2)}$)\nin analogy to the symbol for MPLs and eMPLs, which makes the identities due to the elliptic Bloch relation manifest. \n\nIn general, eMPLs transform under modular transformations of $\\tau$ in a complicated way \\cite{Duhr:2019rrs,Weinzierl:2020fyx}, and results given in terms of eMPLs are not manifestly double periodic.\nHowever, in the examples we studied, we find that the symbol prime makes both double periodicity as well as a simple behaviour under modular transformations manifest.\nFinally, it makes also part of the integrability conditions manifest, which result from the requirement that partial derivatives commute.\n\nTo illustrate the use of the symbol for eMPLs, the application of the identities of elliptic letters as well as the symbol prime, we study two concrete examples.\nThe first example is the two-loop sunrise integral in two dimensions with all internal masses being unequal (see figure \\ref{subfig:sunrise}). \nThe second example is the ten-point two-loop double-box integral in four dimensions with massless internal propagators (see figure \\ref{subfig:doublebox}).\nIn addition to the aforementioned properties and techniques, we also demonstrate how the (elliptic) symbol reduces to polylogarithmic symbol in kinematic limits where the elliptic curve degenerates. \n\n\n\n\\begin{figure}\n \\begin{subfigure}{.5\\textwidth}\n \\centering\n \\begin{tikzpicture}[scale=1.5,label distance=-1mm]\n\\clip (0,14.6) rectangle (3,16.4);\n\t\t\\node (0) at (0, 15.5) {};\n\t\t\\node (4) at (3.0, 15.5) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (1.5,15.5) circle (0.75);\n\\end{tikzpicture} \n \\caption{\\textcolor{white}{.}}\n \\label{subfig:sunrise}\n\\end{subfigure}\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\begin{tikzpicture}[scale=1.5,label distance=-1mm]\n\\clip (0,14) rectangle (3,16);\n\t\t\\node (0) at (0, 15.5) {};\n\t\t\\node (1) at (1.5, 16) {};\n\t\t\\node (2) at (0.5, 16) {};\n\t\t\\node (3) at (2.5, 16) {};\n\t\t\\node (4) at (3.0, 15.5) {};\n\t\t\\node (5) at (0.5, 14) {};\n\t\t\\node (6) at (0, 14.5) {};\n\t\t\\node (7) at (1.5, 14) {};\n\t\t\\node (8) at (3, 14.5) {};\n\t\t\\node (9) at (2.5, 14) {};\n \\node (11) at (5, 15) {};\n\t\t\\node (12) at (6, 15) {};\n\t\t\\draw[thick] (0.center) to (4.center);\n\t\t\\draw[thick] (6.center) to (8.center);\n\t\t\\draw[thick] (2.center) to (5.center);\n\t\t\\draw[thick] (1.center) to (7.center);\n\t\t\\draw[thick] (3.center) to (9.center);\n\\end{tikzpicture}\n \\caption{\\textcolor{white}{.}}\n \\label{subfig:doublebox}\n\\end{subfigure}\n\\caption{The sunrise integral in two dimensions with unequal internal masses (\\subref{subfig:sunrise}) as well as the ten-point double-box integral in four dimensions with massless internal propagators (\\subref{subfig:doublebox}).}\n\\label{fig: diagrams intro} \n\\end{figure}\n\n\n\nThe remainder of this paper is organized as follows. We review \n elliptic multiple polylogarithms in section \\ref{sec:2}. In section \\ref{sec:3}, we derive identities for elliptic symbol letters -- based on Abel's addition theorem for $\\Omega^{(1)}$'s and by introducing the symbol prime map for $\\Omega^{(2)}$'s. We illustrate the use of these techniques for the unequal-mass sunrise integral in section \\ref{sec: example 1} and for the ten-point double-box integral in section \\ref{sec: example 2}.\n In particular, we provide analytic results for the non-elliptic nine-point double-box integral and its symbol, which result from taking the soft limit of the ten-point double-box integral.\n We conclude with a summary and an outlook on open questions in section \\ref{sec:5}.\nIn appendix \\ref{app:sunrise}, we review the calculation of the unequal-mass sunrise integral via Feynman parameters using our conventions and notation. \nThe details of simplifying the symbols for the sunrise integral, as well as the expressions of the functions and \nsymbols for the ten-point elliptic integral and its soft limit, are included as ancillary files (\\texttt{sunrise\\_symbol.nb}, \\texttt{doublebox\\_omega2} and \\texttt{doublebox\\_soft}).%\n\\footnote{In this article, \nwe only provide the expression for the ten-point double box in the normalization by the period $-\\omega_2$ since the corresponding expressions in the normalization by the period $\\omega_1$ can be found in the ancillary files of \\cite{Kristensson:2021ani}.} \n\n\n\n\\section{Review of Elliptic Multiple Polylogarithms} \\label{sec:2}\n\n\nLet us first review several elementary facts about elliptic multiple polylogarithms; see \\cite{Broedel:2017kkb,Broedel:2017siw,Broedel:2018iwv,Broedel:2018qkq} for further details. We follow the notations and conventions of \\cite{Kristensson:2021ani}, which differ slightly from those of \\cite{Broedel:2017kkb,Broedel:2017siw,Broedel:2018iwv,Broedel:2018qkq}.\n\n\\subsection{Elliptic multiple polylogarithms on the elliptic curve}\n\nIn this paper, the elliptic curves $\\mathcal{C}$ are described by monic quartic equations:\n\\begin{equation}\n y^{2} =P_{4}(x)= x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0} \\:. \\label{quarticEcurve}\n\\end{equation}\nSuch elliptic curves can be converted to the standard Weierstrass form \n\\begin{equation}\n Y^{2}=4X^{3}-g_{2}X-g_{3} \\end{equation}\nusing the rational point at $(x,y)=(+\\infty,+\\infty)$, where $(X,Y)$ are related to $(x,y)$ by \n\\begin{equation} \\label{XYtoxy}\n \\begin{aligned}\n X &= \\frac{1}{12}\\bigl( a_{2} + 3a_{3}x + 6x^{2}+6y\\bigr) \\:, \\\\\n Y &= \\frac{1}{4}\\bigl( a_{1} + 2a_{2}x + 3a_{3}x^{2} + 4x^{3} +a_{3}y+4xy\\bigr) \\:.\n \\end{aligned} \n\\end{equation}\n\n\nOn the curve $\\mathcal{C}$, we can introduce elliptic multiple polylogarithms $\\mathrm{E}_{4}$, which are recursively defined as \\cite{Broedel:2017kkb}%\n\\footnote{The subscript ``$4$'' indicates that the elliptic curve is given by a quartic polynomial. Analogous functions for a cubic polynomial were also defined in \\cite{Broedel:2017kkb}.}\n\\begin{equation}\n\\label{Eiterateddefinition}\n \\Ef{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}=\n \\int_{0}^{x}\\dif x'\\,\\psi_{n_{1}}(c_{1},x')\\Ef{n_{2} & \\ldots & n_k}{c_{2} & \\ldots& c_k}{x'}\n\\end{equation}\nwith $\\mathrm{E}_{4}(;x)=1$, where\n\\begin{equation} \\label{psibasis}\n \\begin{split}\n &\\psi_{0}(0,x)=\\frac{1}{y} \\,,\\qquad \\:\\:\\psi_{-1}(\\infty,x)=\\frac{x}{y}\\,, \\\\\n &\\psi_{1}(c,x)=\\frac{1}{x-c}\\,,\\quad \n \\psi_{-1}(c,x)=\\frac{y_{c}}{y(x-c)}\\,,\n \\end{split} \n\\end{equation}\nwith $y_{c}=y\\vert_{x=c}$. The definitions of $\\psi_{n}(c,x)$ for $n=\\pm2,\\pm3,\\dots$ can be found in \\cite{Broedel:2017kkb}; the kernels \\eqref{psibasis} are sufficient for the purpose of this paper, though. \n\nThe class of elliptic multiple polylogarithms $\\Ef{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}$ contains in particular all non-elliptic (Goncharov) multiple polylogarithms, defined by \n\\begin{equation}\n G(c_1,\\dots,c_n;x)=\\int_0^x\\frac{\\dif x'}{x'-c_1}G(c_2,\\dots,c_n;x')\n\\end{equation}\nwith $G(;x)=1$,\nsince by definition $\\Ef{1 & \\ldots & 1}{c_1 & \\ldots& c_k}{x}\\equiv G(c_1,\\dots,c_k;x)$.\n\n\nIn general, any integral of the form\n\\begin{equation}\n \\int\\frac{\\dif x}{y} \\:\\mathcal{G}(x,y)\\,, \\label{1dIntofPolylog}\n\\end{equation}\nwhere $\\mathcal{G}$ is a polylogarithm whose letters are rational functions of $x$ and $y$, can be converted to $\\mathrm{E}_{4}$ functions with only the four kinds of integration kernels defined in \\eqref{psibasis}.%\n\\footnote{Since $\\mathcal{G}$ is a polylogarithm, the integration kernels have only simple poles, in addition to being rational in $x$ and $y$. While all integration kernels $\\psi_n$ have only simple poles, only $\\psi_{-1,0,+1}$ are rational functions of $x,y$.}\n In particular, this is the case for the (unequal-mass) sunrise integral and the ten-point double-box integral, which we will study as examples in sections \\ref{sec: example 1}--\\ref{sec: example 2}.\n \n \\subsection{From the elliptic curve to the torus}\n\nThe functions $\\mathrm{E}_{4}$ are closely related to Feynman integrals since they are defined in an algebraic way. However, the purity of some elliptic Feynman integrals, such as integrals of the form \\eqref{1dIntofPolylog}, is hidden in terms of $\\mathrm{E}_{4}$ functions since taking the total derivative of a $\\mathrm{E}_{4}$ function does not necessarily decrease its length.%\n\\footnote{This can be seen concretely by how the integration kernels $\\psi_{-1}(\\infty,x)\\dif x$ and $\\psi_{-1}(c,x) \\dif x$ are related to the kernels of pure functions given below in \\eqref{psitog1}.\n}\nOn the other hand, iterated integrals defined on the torus, such as the $\\tilde{\\Gamma}$ functions we will review below, are manifestly pure and hence allow a symbol map defined via the total derivative. \n\nTo connect both sides, we first need a bijection between the elliptic curve $\\mathcal{C}$ and the torus $\\mathbb{C}\/\\Lambda$, where $\\Lambda$ is the lattice generated by the periods $\\omega_{1}$ and $\\omega_{2}$ of the elliptic curve. For an elliptic curve of the form \\eqref{quarticEcurve}, one can find such a map through the birationally \nequivalent curve in the Weierstrass normal form: first solve $(x,y)$ in terms of $(X,Y)$ from \\eqref{XYtoxy}, then replace $X$ and $Y$ with the Weierstrass elliptic function $\\wp(z)$ and its derivative $\\wp'(z)$, respectively. \nThis gives \\begin{equation}\n z\\mapsto (x,y)=(\\kappa(z),\\kappa'(z))\\,,\n \\end{equation}\nwhere\n \\begin{equation}\n \\kappa(z)=\\frac{6a_{1}-a_{2}a_{3}+12 a_{3}\\wp(z)-24\\wp'(z)}{3a_{3}^{2}-8a_{2}-48\\wp(z)}.\n \\end{equation} \nIt is easy to see that $\\kappa(0)=\\infty$, and hence all lattice points are mapped to the infinity point in the $(x,y)$ space. Furthermore, each point $c$ in the $x$-space corresponds to two points $(c,\\pm y_{c})$ on the elliptic curve $\\mathcal{C}$ and hence to two images on the torus $\\mathbb{C}\/\\Lambda$, which we denote by $z_{c}^{\\pm}$;\n these two images satisfy \n \\begin{equation} \\label{wpwmrelation}\n z_{c}^{+}+z_{c}^{-}= z^{-}_{\\infty} + z^{+}_{\\infty}\\equiv z^{-}_{\\infty} \\: \\operatorname{mod} \\Lambda \\:,\n \\end{equation}\n since the corresponding points $(X_{c}^{\\pm},Y_{c}^{\\pm})$ and\n \\begin{equation}\n (X_{\\infty}^{-},Y_{\\infty}^{-}) = \\bigl(\\tfrac{1}{48}(3a_{3}^{2}-8a_{2}),\\tfrac{1}{32}(-8a_{1}+4a_{2}a_{3}-a_{3}^{3}) \\bigr) \n \\end{equation}\n are on the same line. \n \n\n\n\\tikzset{cross\/.style={cross out, thick, draw=black, fill=none, minimum size=2*(#1-\\pgflinewidth), inner sep=0pt, outer sep=0pt}, cross\/.default={2pt}}\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}\n \\draw[->, thick] (0,0) to (0,3);\n \\draw[->, thick] (-3,1.5) to (1,1.5);\n \\draw[<-,line width=0.23mm,red] (-0.8,0.0) to (-0.8,3.0);\n \\draw[->,line width=0.23mm,blue] (-3,1.4) .. controls (-1,1.4) .. (-1.0,0.0) ; \n \\node[cross,label=above:$r_{1}$] at (-1.5, 2.0) {};\n \\node[cross,label=below:$r_{4}$] at (-1.5, 1.0) {};\n \\node[cross,label=above:$r_{2}$] at (-0.5, 0.8) {};\n \\node[cross,label=below:$r_{3}$] at (-0.5, 2.2) {};\n \\node at (-0.5,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n \\node at (-1.3,0.1) {\\textcolor{blue}{$\\gamma_{-}$}};\n \\node at (0.4,3) {$\\Im x$};\n \\node at (1.3,1.5) {$\\Re x$};\n \\end{tikzpicture}\n \\caption{The distribution of the four roots of $y^{2}(x)$ and the two integration contours $\\gamma_1$ and $\\gamma_-$ defining the period $\\omega_1$ and $z_\\infty^-$. The contour $\\gamma_2$ which defines the period $\\omega_2$ runs along the real axis.\n }\n \\label{fig: contours}\n \\end{figure}\n\nThe inverse map from the torus to the elliptic curve is simply given by the Abel-Jacobi map. We assume that the four roots of $y^{2}(x)$ come in complex conjugate pairs as shown in figure \\ref{fig: contours}.%\n\\footnote{For a discussion of other possible distributions of roots, see \\cite{Broedel:2017kkb}.}\nThen, the torus image $z_{c}^{+}$ for any real $c$ is given by\n\\begin{equation}\n z_{c}^{+}=\\int_{-\\infty}^{c} \\frac{\\dif x}{y}\\,.\n\\end{equation} \nHence, $z_{\\infty}^{+}$ is one period of the torus, and we choose it to be $\\omega_{2}$. The image $z_{c}^{-}$ can be obtained by \\eqref{wpwmrelation} together with $z_{\\infty}^{-}=\\int_{\\gamma_{-}} \\dif x\/y$, and the other period is $\\omega_{1}=\\int_{\\gamma_{1}} \\dif x\/y$; see figure \\ref{fig: contours} for the definitions of the integration contours $\\gamma_{-}$ and $\\gamma_{1}$.\nDue to the distribution of roots, $\\omega_{2}$ and $\\mi\\omega_{1}$ are \\emph{positive} reals.\n\n\n\\subsection{Elliptic multiple polylogarithms on the torus}\n\n\nDue to the equivalence between the elliptic curve and the torus, \nanother way to define elliptic multiple polylogarithms is via iterated integrals on a torus. Such iterated integrals can be formulated in several ways. \nIn this paper, we use the so-called $\\tilde{\\Gamma}$ functions \\cite{Broedel:2017kkb,Broedel:2018iwv}, which are defined as\\footnote{Another variant of such iterated integrals extensively used in one-loop string amplitudes are the so-called $\\Gamma$ functions, whose integration kernels $f^{(n)}$ are double periodic but \\emph{not} meromorphic; see e.g.\\ \\cite{Broedel:2014vla}.} \n\\begin{equation}\n \\gamt{n_1 & \\ldots & n_k}{w_1 & \\ldots& w_k}{w|\\tau}=\n \\int_{0}^{w} \\dif w'\\,g^{(n_{1})}(w'{-}w_{1},\\tau)\\gamt{n_{2} & \\ldots & n_k}{w_{2} & \\ldots& w_k}{w'|\\tau}\n\\end{equation}\nwith $\\tilde{\\Gamma}(;w|\\tau)=1$; we will frequently suppress the dependence on $\\tau$ for ease of notation. Such an iterated integral is said to have length $k$ and weight $\\sum_kn_k$, and in contrast to the case of MPLs both quantities are not necessarily equal. \nThe integration kernels $g^{(n)}(w,\\tau)$ are generated by the \\emph{Eisenstein-Kronecker series}\n\\begin{equation}\n \\frac{\\partial_{w}\\theta_{1}(0|\\tau)\\theta_{1}(w+\\alpha|\\tau)}{\\theta_{1}(w|\\tau)\\theta_{1}(\\alpha|\\tau)} = \\sum_{n\\geq 0}\\alpha^{n-1}g^{(n)}(w,\\tau)\\:,\n\\end{equation}\nwhere $\\theta_{1}(w|\\tau)$ is the odd Jacobi theta function. \nAll the integration kernels $g^{(n)}$ except $g^{(0)}=1$ are quasi double periodic,\n\\begin{equation}\n g^{(n)}(w+1)=g^{(n)}(w)\\:, \\qquad g^{(n)}(w+\\tau)=\\sum_{j=0}^{n}\\frac{(-2\\pi \\mi)^{j}}{j!}g^{(j)}(w)\\:,\n\\end{equation}\nbut meromorphic with only logarithmic poles at most \\cite{Broedel:2017kkb,Broedel:2018iwv}. \n\nNote that the functions $\\tilde\\Gamma$ and the integration kernels $g^{(n)}$ are defined on the normalized torus, and that the torus with periods $(\\omega_{1},\\omega_{2})$ has the two normalizations $[1:\\tau=\\omega_{2}\/\\omega_{1}]$ and $[1:\\tau'=-\\omega_{1}\/\\omega_{2}]$, which are related by the modular $S$-transformation $\\tau\\to -\\tau^{-1}$. We denote the images of $c$ on $[1:\\tau=\\omega_{2}\/\\omega_{1}]$ as $w^{\\pm}_{c}=z^{\\pm}_{c}\/\\omega_1$ and the images on $[1:\\tau'=-\\omega_{1}\/\\omega_{2}]$ as $\\xi_{c}^{\\pm}=z^{\\pm}_{c}\/(-\\omega_2)$. The two are related by $\\xi_c^\\pm=\\tau' w_c^\\pm$. In what follows, most of the results are written in terms of $w$-coordinates, but it should be understood that the analogous results also hold in terms of $\\xi$-coordinates unless otherwise indicated.\n\n\nThe integration kernels $\\psi_{n}$ can be identified as combinations of $g^{(j)}$'s by matching poles on both sides. On the torus $[1:\\tau=\\omega_{2}\/\\omega_{1}]$, one can easily find the following relations between $g^{(j)}$'s and $\\psi_{n}$'s,\n\\begin{subequations} \\label{psitog1}\n \\begin{align}\n \\psi_{1}(c,x)\\dif x &=\\Bigl(g^{(1)}(w-w_{c}^{+})+g^{(1)}(w-w_{c}^{-}) \\nonumber \\\\ \n &\\qquad \\qquad \\qquad \\qquad -g^{(1)}(w-w_{\\infty}^{+}) -g^{(1)}(w-w_{\\infty}^{-})\\Bigr)\\dif w \\,, \n \\label{psitog1a}\\\\\n \\psi_{-1}(c,x)\\dif x &= \n \\Bigl( g^{(1)}(w-w_{c}^{+})-g^{(1)}(w-w_{c}^{-}) + g^{(1)}(w_{c}^{+}) -g^{(1)}(w_{c}^{-}) \\Bigr) \\dif w \\,, \\label{psitog1b}\\\\\n \\psi_{-1}(\\infty,x)\\dif x &= \\Bigl(g^{(1)}(w-w_{\\infty}^{-})-g^{(1)}(w) + g^{(1)}(w_{\\infty}^{-})-\\omega_{1}a_{3}\/4 \\Bigr)\\dif w\\,, \\\\\n \\psi_{0}(0,x)\\dif x&= \\omega_{1}\\dif w \\,.\n \\end{align} \n\\end{subequations}\nOn the torus $[1:\\tau']$, the corresponding relations can be obtained by replacing $w\\to\\xi$ and $\\omega_1\\to -\\omega_2$ in \\eqref{psitog1}.\n\nSometimes, it is more convenient to combine $\\tilde{\\Gamma}$ functions into the so-called $\\mathcal{E}_{4}$ functions \\cite{Broedel:2018qkq}, especially if the $\\tilde{\\Gamma}$ functions originally arose from an expression of $\\mathrm{E}_{4}$ functions. The elliptic multiple polylogarithms $\\mathcal{E}_{4}$ are defined in complete analogy to \\eqref{Eiterateddefinition}:\n\\begin{equation}\n\\label{curlyEiterateddefinition}\n \\cEf{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}=\n \\int_{0}^{x}\\dif x'\\,\\Psi_{n_{1}}(c_{1},x')\\cEf{n_{2} & \\ldots & n_k}{c_{2} & \\ldots& c_k}{x'}\n\\end{equation}\nwith $\\mathcal{E}_{4}(;x)=1$,\n\\begin{align}\n \\Psi_{\\pm (n>0)}(c,x) \\dif x=\\Bigl(&g^{(n)}(w-w_{c}^{+})\\pm g^{(n)}(w-w_{c}^{-}) \n\\nonumber \\\\\n &\n - \\delta_{\\pm n,1}\\bigl( g^{(1)}(w-w_{\\infty}^{+})+g^{(1)}(w-w_{\\infty}^{-} )\\bigr) \\Bigr) \\dif w \n\\label{Psikernels} \n\\end{align}\nand $\\Psi_0(x)\\dif x=\\dif w$, as well as analogous expressions in terms of $\\xi$.\nThe weight of a function $\\cEf{n_1 & \\ldots & n_k}{c_1 & \\ldots& c_k}{x}$ is defined as $\\sum_i |n_i|$.\n\n\\subsection{Symbol}\n\nBy construction,\n the total derivative of $\\tilde{\\Gamma}$ admits a recursive structure \\cite{Broedel:2018iwv},\n\\begin{align} \\nonumber\\label{devoftG}\n &\\quad \\dif\\tilde{\\Gamma}(A_{1},\\ldots,A_{k};w) \\\\\n &= \\sum_{p=1}^{k-1}(-1)^{n_{p+1}}\\tilde{\\Gamma}(A_{1},\\ldots,A_{p-1},\\vec{0},A_{p+2},\\ldots,A_{k};w) \n \\times\\omega^{(n_{p}+n_{p+1})}(w_{p+1,p})\n \\\\\n &\\quad+ \\sum_{p=1}^{k}\\sum_{r=0}^{n_{p}+1} \\Biggl[ \n \\binom{n_{p-1}{+}r{-}1}{n_{p-1}{-}1}\\tilde{\\Gamma}(A_{1},\\ldots, A_{p-1}^{[r]},A_{p+1},\\ldots,A_{k};w) \n \\times\\omega^{(n_{p}-r)}(w_{p-1,p}) \\nonumber \\\\\n &\\qquad \\qquad - \\binom{n_{p+1}{+}r{-}1}{n_{p+1}{-}1} \\tilde{\\Gamma}(A_{1},\\ldots, A_{p-1},A_{p+1}^{[r]},\\ldots,A_{k};w) \n \\times \\omega^{(n_{p}-r)}(w_{p+1,p})\n \\Biggr],\\nonumber\n\\end{align}\nwhere $\\vec{0}\\equiv\\bigl(\\begin{smallmatrix}\n 0 \\\\\n 0 \n \\end{smallmatrix}\\bigr)$, $w_{i,j}\\equiv w_{i}-w_{j}$, as well as \n\\begin{equation}\n A_{i}^{[r]}\\equiv\\bigl(\\begin{smallmatrix}\n n_{i}+r \\\\\n w_{i} \n \\end{smallmatrix}\\bigr)\\:, \\qquad A_{i}^{[0]}\\equiv A_{i}\\:.\n\\end{equation}\nThe forms $\\omega^{(j)}(w)$ are exact, and we can thus write them as \n\\begin{equation}\n \\omega^{(j)}(w,\\tau)=(2\\pi i)^{j-1}\\dif\\Omega^{(j)}(w,\\tau)\\,,\n\\end{equation}\nwith\n\\begin{align} \\label{wdef}\n \\Omega^{(-1)}(w,\\tau) &= -2\\pi\\mi\\tau\\:, \\quad \\Omega^{(0)}(w,\\tau) =2\\pi\\mi w\\:, \\quad \\Omega^{(1)}(w,\\tau)=\\log \\frac{\\theta_{1}(w|\\tau)}{\\eta(\\tau)} \\:, \\nonumber \\\\\n \\Omega^{(\\text{odd } j>1)}(w,\\tau)&=- \\frac{2j\\zeta_{j+1}\\tau}{(2\\pi \\mi)^{j}} +\n \\frac{1}{(j{-}1)!}\\sum_{n=1}^{\\infty}\n n^{j-1}\\log\\bigl((1-\\me^{2\\pi \\mi (n\\tau-w)})(1-\\me^{2\\pi \\mi (n\\tau+w)})\\bigr), \\nonumber \\\\ \n \\Omega^{(\\text{even } j)}(w,\\tau)&=-\\frac{2\\zeta_{j}w}{(2\\pi i)^{j-1}} \n + \\frac{1}{(j{-}1)!}\\sum_{n=1}^{\\infty} n^{j-1}\\log\\frac{1-\\me^{2\\pi \\mi (n\\tau+w)}}{1-\\me^{2\\pi \\mi (n\\tau-w)}} ,\n\\end{align}\nwhere $\\eta(\\tau)$ is the Dedekind eta function and $\\zeta_{j}=\\sum_{n\\in\\mathbb{Z}_{+}} n^{-j}$ are the Riemann zeta values.\\footnote{Recall that $\\zeta_{2n}=\\frac{(-1)^{n+1}B_{2n}(2\\pi)^{2n}}{2(2n)!}$ with $B_{2n}$ being the $(2n)^{\\rm th}$ Bernoulli number, such that the first terms in \\eqref{wdef} can equivalently be written in terms of Bernoulli numbers.}\nThe functions $\\Omega^{(j)}$ satisfy \n\\begin{equation}\n g^{(j)}(w,\\tau)=(2\\pi\\mi)^{j-1}\\partial_{w}\\Omega^{(j)}(w,\\tau)=\\frac{(2\\pi\\mi)^{j-1}}{j-1}\\partial_{\\tau}\\Omega^{(j-1)}(w,\\tau).\n\\end{equation}\nThe sum representation \\eqref{wdef} can be derived using the sum representation of the $g^{(n)}$ functions given in \\cite{Broedel:2018iwv}.\nIn particular, \n\\begin{equation}\n (2\\pi i)^{1-n}\\gamt{n}{0}{w}=\\Omega^{(n)}(w)-\\Omega^{(n)}(0)\\,,\n\\end{equation}\nwhere $\\Omega^{(n)}(0)$ vanishes for even $n$ and is the primitive of the Eisenstein series for odd $n$; see \\cite{Broedel:2018iwv}.\nWe will see below that the functions $\\Omega^{(j)}$ play the role of elliptic symbol letters.\nAs can be seen from \\eqref{wdef}, $\\Omega^{(1)}$ has a logarithmic singularity at all lattice points, while \n$\\Omega^{(j>1)}$ has a logarithmic singularity at all lattice points except for the origin \\cite{Broedel:2018iwv}.\n\n\nFor a function $\\widetilde{\\Gamma}_{k}^{(n)}$ of weight $n$ and length $k$, we can define \n$ \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}=(2\\pi i)^{k-n}\\widetilde{\\Gamma}_{k}^{(n)}.$\nSchematically, the differential of $\\widetilde{\\underline{\\Gamma}}_{k}^{(n)}$ then takes the form \n\\begin{equation}\n \\dif \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}=\\sum_i \\widetilde{\\underline{\\Gamma}}^{(n-j_{i})}_{k-1} \\dif \\Omega^{(j_i)}(w_i) \\,,\n\\end{equation}\nThus, it is natural to define the symbol of $\\tilde{\\underline{\\Gamma}}_{k}^{(n)}$ as \n\\begin{equation}\n\\label{eq: elliptic symbol}\n \\mathcal{S}(\\widetilde{\\underline{\\Gamma}}_{k}^{(n)})=\\sum_i \\mathcal{S}(\\tilde{\\underline{\\Gamma}}_{k-1}^{(n-j_{i})})\\otimes \\Omega^{(j_i)}(w_i) \\,.\n\\end{equation}\n\n\n\nNote that in contrast to \\cite{Broedel:2018iwv} we have included additional factors of $(2\\pi\\mi)$ in the definition of the elliptic letters $\\Omega^{(n)}$ and consider the symbol of $ \\widetilde{\\underline{\\Gamma}}_{k}^{(n)}$ rather than $\\widetilde{\\Gamma}_{k}^{(n)}$.\\footnote{In \\cite{Broedel:2018iwv}, there is also a projection operator $\\pi_{k}$ in the definition of the symbol for $\\tilde{\\Gamma}$ functions due to the fact that some eMPLs of weight $0$ evaluate to rational numbers, such as $\\gamt{0}{0}{1}=1$. Here we exclude it by introducing these $2\\pi \\mi$ factors.} This is such that the elliptic letters and symbols degenerate to logarithms and polylogarithmic symbols without additional factors of $(2\\pi i)$ in the limit where the elliptic curve degenerates, see sections \\ref{sec: example 1}--\\ref{sec: example 2}.\n\n\n\\subsection{Shuffle regularization}\n\nLet us close this section by remarking on shuffle regularization. One can easily see that $\\gamt{1}{0}{z}=\\Omega^{(1)}(z)-\\Omega^{(1)}(0)$ is divergent since $\\Omega^{(1)}(0)$ is singular according to \\eqref{wdef}. The shuffle regularization used in \\cite{brown2011multiple,Broedel:2018iwv} takes $\\Omega^{(1)}(0)\\equiv2\\log\\eta(\\tau)$.\nHowever, that regularization leads to an issue if we start with an integral of the form \\eqref{1dIntofPolylog} since it is inconsistent with the usual shuffle regularization for polylogarithms, $G(0;x)\\equiv\\log x$. To see this, we apply \\eqref{psitog1} to $G(0;1)=\\log 1=0$ and find\n\\begin{equation}\n 0 \\stackrel{?}{=} \\gamt{1}{0}{w_{1}^{+}-w_{0}^{+}} + \\gamt{1}{w_{0}^{-}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}} - \\gamt{1}{w_{\\infty}^{+}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}} - \\gamt{1}{w_{\\infty}^{-}-w_{0}^{+}}{w_{1}^{+}-w_{0}^{+}} \\:,\n\\end{equation}\nwhich is in general not true if we use $\\Omega^{(1)}(0)\\equiv2\\log\\eta(\\tau)$.\nTo reconcile both sides, we expand $\\gamt{1}{w'}{w}=\\Omega^{(1)}(w-w')-\\Omega^{(1)}(-w')$ to arrive at the \nfollowing regularization for elliptic multiple polylogarithms:\n\\begin{align}\n \\Omega^{(1)}(0)&\\equiv\\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{-})+\\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{+}) -\\Omega^{(1)}(w_{0}^{+}-w_{0}^{-}) \\nonumber \\\\\n &\\quad +\\Omega^{(1)}(w_{1}^{+}-w_{0}^{-}) +\\Omega^{(1)}(w_{1}^{+}-w_{0}^{+}) -\\Omega^{(1)}(w_{1}^{+}-w_{\\infty}^{-}) - \\Omega^{(1)}(w_{1}^{+}-w_{\\infty}^{+}) \\nonumber \\\\\n &=2\\log \\eta(\\tau)+\\log\\frac{2\\pi\\mi}{\\omega_{1}}-\\log y_{0} = \\frac{1}{12}\\log \\Delta -\\log y_{0}\\:, \\label{shuffreg2}\n\\end{align} \nwhere $\\Delta= g_{2}^{3}-27g_{3}^{2}$ is the discriminant of the elliptic curve. The second equality will be explained in the next section and the third equality shows that this regularization is actually independent of the normalization of the torus. \n\n\n\n\n\n\\section{Identities of Elliptic Symbol Letters and the Symbol Prime} \n\\label{sec:3}\n\nWe have briefly reviewed several elementary facts about elliptic multiple polylogarithms, and we saw that the symbol letters of elliptic multiple polylogarithms are the functions $\\Omega^{(n)}(w,\\tau)$. These functions stand in the way of analyzing the elliptic symbols. For one thing, the relations among $\\Omega^{(n)}$'s are much more complicated than the manipulation rules $\\log a+\\log b=\\log ab$ for the symbol letters of multiple polylogarithms. For another, they depend on the kinematics in a rather indirect way -- their arguments $w$ and $\\tau$ are (ratios) of elliptic integrals involving kinematics. \n\n\nIn this section, we investigate the identities of the elliptic letter $\\Omega^{(n)}(w,\\tau)$. \nThe most trivial identities these letters satisfy are the following: \n\\begin{align}\n \\text{Parity :}&\\quad \\Omega^{(n)}(-w)=(-1)^{n+1} \\Omega^{(n)}(w) \\:, \n \\label{parity}\\\\\n \\text{Quasi periodicity :} &\\quad \\Omega^{(n)}(w+\\tau) = \\sum_{j=0}^{n+1}\\frac{(-1)^{j}}{j!}\\Omega^{(n-j)}(w) \\:. \\label{quasiperiodicity}\n\\end{align}\nThey immediately follow from \\eqref{wdef}.\nOur investigation of more non-trivial identities will be focussed on the cases $n=0,1,2$ since -- for the two examples considered in this paper, namely the sunrise integral and the double-box integral -- the identities among $\\Omega^{(n\\leq 2)}$ are sufficient to simplify the symbols after using \\eqref{parity} and \\eqref{quasiperiodicity}. We comment on a generalization to identities among $\\Omega^{(n>2)}$ at the end of this section.\n\nLet us start with the slightly trivial identity\n\\begin{align} \\label{identity1}\n \\quad \\log\\frac{c-b}{c-a} &= \\sum_{\\sigma\\in \\pm} \\Bigl(\\Omega^{(1)}(w_{c}^{\\sigma}-w_{b}^{+})-\\Omega^{(1)}(w_{c}^{\\sigma}-w_{a}^{+}) \n \\nonumber \n \\\\\n &\\qquad \\qquad \\qquad \n -\\Omega^{(1)}(w_{\\infty}^{\\sigma}-w_{b}^{+})+\\Omega^{(1)}(w_{\\infty}^{\\sigma}-w_{a}^{+})\\Bigr)\\,,\n\\end{align}\nwhich is a simple consequence of applying \\eqref{psitog1a} to $\\int_{a}^{b}\\psi_{1}(c,x)\\dif x = \\log\\frac{c-b}{c-a} $. \nThe identity \\eqref{identity1} has two important special cases. One is obtained by taking $a\\to \\infty$, giving\n\\begin{align} \\label{identity2}\n \\sum_{\\sigma\\in\\pm} \\Omega^{(1)}(w_{c}^{\\sigma}-w_{b}^{+}) \n &=\\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+})+\\sum_{\\sigma\\in\\pm}\\left[\\Omega^{(1)}(w_{c}^{\\sigma}-w_{\\infty}^{+})+\\Omega^{(1)}(w_{b}^{\\sigma}-w_{\\infty}^{+}) \\right] \\nonumber \\\\ \n &\\quad - \\log\\biggl(\\frac{2\\pi\\mi}{\\omega_{1}}\\biggr) -2\\log\\eta(\\tau)+\\log(c-b)\\,;\n\\end{align}\nthe other one is obtained by further taking $b\\to c$, which yields\n\\begin{align} \\label{identity3}\n \\Omega^{(1)}(w_{c}^{-}{-}w_{c}^{+}) &= 2\\biggl(\\Omega^{(1)}(w_{c}^{-}-w_{\\infty}^{+})+\\Omega^{(1)}(w_{c}^{+}-w_{\\infty}^{+})\n -2\\log\\eta(\\tau) -\\log\\frac{2\\pi \\mi}{\\omega_{1}} \\biggr) \\nonumber \\\\ \n &\\quad - \\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+}) - \\log y_{c} \\:.\n\\end{align}\nNow one can easily see that the second equality in \\eqref{shuffreg2} is the consequence of applying \\eqref{identity2} and \\eqref{identity3}. The two special cases are particularly useful since the letters $\\Omega^{(1)}$ on their right-hand sides always involve $w^{\\pm}_{\\infty}$ and hence can serve as a basis.\n\n\\subsection{Abel's addition theorem} \\label{sec:3.1}\n\nSurprisingly, a very classical and powerful theorem, Abel's addition theorem \\cite{abel1841}, \nyields other identities for $\\Omega^{(0)}$ and $\\Omega^{(1)}$.\\footnote{See e.g.\\ \\cite{GriffithsHarris} for a textbook treatment of Abel's addition theorem and \\cite{Tarasov:2017yyd,Tarasov:slides} for previous applications of it to Feynman integrals.}\n\nLet us first spell out this theorem: Let $\\mathcal{C}$ and $\\mathcal{C}'$ be curves given by two polynomial equations\n\\begin{align}\n \\mathcal{C}:& \\quad F(x,y)=0 \\:, \\\\\n \\mathcal{C}':&\\quad Q(x,y)=0\\:,\n\\end{align}\nwhere $\\mathcal{C}$ is viewed as a \\emph{fixed} curve and $\\mathcal{C}'$ as a \\emph{variable} curve with coefficients collectively denoted as $\\{b_{i}\\}$. \nSuppose that these two curves intersect at $n$ points $(x_{1},y_{1})$, ..., $(x_{n},y_{n})$. Let $R(x,y)$ be a rational function defined on $\\mathcal{C}$. Then the following holds.\n\\begin{theorem}[Abel]\n The integral \n \\begin{equation}\n I(\\{b_{i}\\}) =\\sum_{i=1}^{n} \\int_{x_{\\ast}}^{x_{i}}R(x,y)\\:\\dif x\\:,\n \\end{equation}\n where $x_{\\ast}$ is an arbitrary base point, contains at most rational functions and logarithms of $\\{b_{i}\\}$. \n\\end{theorem}\n\\noindent This theorem can be proven by showing that $\\partial_{b_{\\nu}}I$ is always a rational function of $\\{b_{i}\\}$ for all $b_{\\nu}$.\n\n\nIf a symbol letter $\\phi(u)$ can be expressed as $\\int^{u} R(x,y)\\dif x$, one can try to find the composition rule of $\\phi(u)$ through Abel's addition theorem.\nOf all applications of this theorem, we are most interested in the cases that $\\mathcal{C}'$ has only \\emph{two} degrees of freedom and intersects $\\mathcal{C}$ at \\emph{three} points. \nFor this case, Abel's addition theorem gives \n\\begin{equation}\n \\phi(u)+\\phi(v)=\\phi\\bigl(T(u,v)\\bigr)+\\cdots,\n\\end{equation}\nwhere $T(u,v)$ is an algebraic function of $u$ and $v$ and `$\\cdots$' denotes simpler objects, like logarithms. \n\nAn example is the composition rule $\\log(x_1)+\\log(x_2)=\\log(x_1x_2)$ for logarithms,\\footnote{This example can e.g.\\ be found in \\cite{Tarasov:slides}.} which is given by choosing \n\\begin{align}\n \\mathcal{C}:&\\quad y= \\frac{1}{x} \\:, \\\\\n \\mathcal{C}': &\\quad y=x^{2}+b_{1}x+b_{2} \\:.\n\\end{align}\nThese two curves intersect at the three points $x_1,x_2,x_3$ that solve \n\\begin{equation}\n x^3+b_1x^2+b_2x-1=0,\n\\end{equation}\nand hence satisfy $x_1 x_2 x_3=1$.\nNow consider\n \\begin{equation}\n I=\\sum_{i=1}^3\\int_1^{x_i}\\frac{\\dif x}{x},\n \\end{equation}\n which satisfies\n\\begin{equation}\n \\partial_{b_j}I=\\sum_{i=1}^3\\frac{1}{x_i}\\frac{\\partial x_i}{\\partial b_j}=\\frac{1}{x_1 x_2 x_3}\\frac{\\partial}{\\partial b_j}x_1x_2x_3=0 ,\n\\end{equation}\nsince $x_1 x_2 x_3=1$. Thus, $I$ is a constant. To fix this constant, we can pick $b_1=-3,b_2=3$, such that $x_1=x_2=x_3=1$, yielding $I=0$.\nAgain using $x_1 x_2 x_3=1$, we thus have \n\\begin{equation}\n 0=I=\\log(x_1)+\\log(x_2)+\\log(x_3)=\\log(x_1)+\\log(x_2)-\\log(x_1 x_2),\n\\end{equation}\nas claimed.\n\n\n\nFor the case we are most interested in, the fixed curve $\\mathcal{C}$ is given by \\eqref{quarticEcurve}, and we find that a convenient choice for $\\mathcal{C}'$ is \n\\begin{equation}\n y=-x^{2}+b_{1}x+b_{2} \\:.\n\\end{equation} \nOne can easily check that these two curves intersects at three point at most.\nSuppose that two intersection points are $(x_{1},y_{1}=\\sqrt{P_{4}(x_{1})})$ and $(x_{2},y_{2}=\\sqrt{P_{4}(x_{2})})$, then\n\\begin{align}\n b_{1}&=\\frac{y_{1}-y_{2}}{x_{1}-x_{2}}+x_{1}+x_{2} \\:, &&\n b_{2}= \\frac{x_{1}y_{2}-x_{2}y_{1}}{x_{1}-x_{2}}-x_{1}x_{2} \\:, && \\text{for }x_{1}\\neq x_{2} \\:, \\\\\n b_{1} &= \\frac{P_{4}'(x_{1})}{2 y_{1}}+2x_{1} \\:, && b_{2}=y_{1}+x_{1}^{2}-b_{1}x_{1} \\:, && \\text{for }x_{1}= x_{2} \\:,\\intertext {and} \n x_{3} &= \\frac{b_{1}^{2}-2b_{2}-a_{2}}{2b_{1}+a_{3}} - x_{1}-x_{2} \\:, && \n y_{3} = -\\sqrt{P_{4}(x_{3})} \\:.&&\n\\end{align}\nSince $z_{c}^{+}=\\int_{-\\infty}^{c} \\dif x\/y$, Abel's addition theorem tells us \n\\begin{subequations} \\label{zIdentity}\n \\begin{align}\n z_{x_{1}}^{+}+ z^{+}_{x_{2}} &\\equiv z^{+}_{x_{3}} \\operatorname{mod} \\Lambda \\:, && \\text{for }b_{1}\\neq -a_{3}\/2 \\:, \\\\ \n z_{x_{1}}^{+}+ z^{+}_{x_{2}} &\\equiv 0 \\operatorname{mod} \\Lambda \\:, && \\text{for }b_{1}= -a_{3}\/2 \\:,\n \\end{align} \n\\end{subequations}\nwhich is the well-known group law on the elliptic curve.\nFurthermore, if we take $b_{2}=(a_{3}^{2}-4 a_{2})\/8$ aside $b_{1}=-a_{3}\/2$, then $\\mathcal{C}$ and $\\mathcal{C}'$ only intersect at one point,\n\\begin{equation}\n \\chi=\\frac{a_{3}^{4}-8a_{2}a_{3}^{2}+16a_{2}^{2}-64a_{0}}{8\\bigl(a_{3}^{3}-4a_{2}a_{3}+8a_{1}\\bigr)} \\:.\n\\end{equation}\nTogether with a little divisor theory, \nthis gives\\footnote{For any meromorphic function $F$ on a torus, by using $\\oint \\dif \\log F(z)=0$ and $\\oint z \\:\\dif \\log F(z)=0$, one can conclude that the number and the sum of its poles are the same as its zeros, where poles and zeros of order $n$ are counted $n$ times. Now consider the function \n\\[ \n F=-\\kappa'(z)-\\kappa(z)^{2}-a_{3}\\kappa(z)\/2+(a_{3}^{2}-4a_{2})\/8\\:,\n\\]\nwhich has poles at lattice points but vanishes only at $z_{\\chi}^{-}$ and $z_{\\infty}^{-}$, two intersection points of the curve $ y=-x^{2}-a_{3}x\/2+(a_{3}^{2}-4a_{2})\/8$ and the elliptic curve. We then obtain \\eqref{minfid} by using \\eqref{wpwmrelation}.}\n\\begin{equation}\n 2z_{\\infty}^{-} \\equiv \\omega_{1}+z_{\\chi}^{+} \\operatorname{mod} \\omega_{2} \\:. \\label{minfid}\n\\end{equation}\n\nSimilarly, for the integral $\\int \\psi_{-1}(c,x)\\dif x$, the same procedure gives \n\\begin{align} \\label{eq1}\n \\int_{\\ast}^{x_{1}}\\frac{y_{c}\\,\\dif x}{y(x-c)}+\\int_{\\ast}^{x_{2}}\\frac{y_{c}\\,\\dif x}{y(x-c)}-\\int_{\\ast}^{x_{3}}\\frac{y_{c}\\,\\dif x}{y(x-c)} &= \\log\\frac{c^{2}-b_{1} c-b_{2}+y_{c}}{c^{2}-b_{1}c-b_{2}-y_{c}} + \\text{const.} \n\\end{align}\nIf $z_{x_{1}}^\\pm+z_{x_{2}}^\\pm=z_{x_{3}}^\\pm$, applying \\eqref{psitog1b} to \\eqref{eq1} gives%\n\\begin{align}\n &\\quad \\sum_{i=1}^{2}\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{-}) \\label{Abel1}\\\\\n &=\\Omega^{(1)}(w_{c}^{+}{-}w_{x_{3}}^{+})-\\Omega^{(1)}(w_{c}^{+}{-}w_{x_{3}}^{-}) \n +\\Omega^{(1)}(w_{c}^{+})-\\Omega^{(1)}(w_{c}^{-})+\\log\\frac{c^{2}-b_{1}c-b_{2}+y_{c}}{c^{2}-b_{1}c-b_{2}-y_{c}}. \\nonumber \n\\end{align}\nIf $z_{x_{1}}^\\pm+z_{x_{2}}^\\pm\\equiv z_{x_{3}}^\\pm \\mod \\Lambda$, a corresponding identity can be found from \\eqref{Abel1} using the quasi double periodicity of $\\Omega^{(1)}$ \\eqref{quasiperiodicity}.\n\nThree boundary cases of \\eqref{Abel1} require special care: \\\\\n(i) taking $c\\to\\infty$ gives\n\\begin{align}\n \\sum_{i=1}^{2}\\Omega^{(1)}(w_{x_{i}}^{+})-\\Omega^{(1)}(w_{x_{i}}^{-})\n &=\\Omega^{(1)}(w_{x_{3}}^{+})-\\Omega^{(1)}(w_{x_{3}}^{-}) \\nonumber \\\\\n &\\quad-\\Omega^{(1)}(\\omega_{\\infty}^{-}-\\omega_{\\infty}^{+})-\\log\\frac{2b_{1}+a_{3}}{4}+\\frac{1}{12}\\log\\Delta \\:, \\label{Abel2} \n\\end{align}\n(ii) taking $x_{3}\\to \\infty$ gives \n\\begin{align}\n & \\quad \\sum_{i=1}^{2}\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{c}^{+}-w_{x_{i}}^{-}) \\label{Abel3} \\\\\n &=\\Omega^{(1)}(w_{c}^{+}{-}w_{\\infty}^{+})-\\Omega^{(1)}(w_{c}^{+}{-}w_{\\infty}^{-}) \n +\\Omega^{(1)}(w_{c}^{+})-\\Omega^{(1)}(w_{c}^{-})+\\log\\frac{c^{2}+a_{3}c\/2-b_{2}+y_{c}}{c^{2}+a_{3}c\/2-b_{2}-y_{c}} \\:. \\nonumber \n\\end{align} \n(iii) taking $c\\to\\infty$ and $x_{3}\\to \\infty$ gives \n\\begin{align}\n \\sum_{i=1}^{2}\\left[\\Omega^{(1)}(w_{\\infty}^{+}-w_{x_{i}}^{+})-\\Omega^{(1)}(w_{\\infty}^{+}-w_{x_{i}}^{-})\n \\right] &=-2\\Omega^{(1)}(w_{\\infty}^{+}-w_{\\infty}^{-})+\\Omega^{(0)}(w_{\\infty}^{-}-w_{\\infty}^{+}) \\nonumber \\\\\n &\\quad+\\frac{1}{6}\\log\\Delta-\\log\\frac{4a_{2}-a_{3}^{2}+8b_{2}}{16} \\:. \\label{Abel4}\n\\end{align} \n\nEqs.\\ \\eqref{zIdentity}, \\eqref{identity1}--\\eqref{identity3} as well as \\eqref{Abel1}--\\eqref{Abel4} explain the subset of the identities numerically found in \\cite{Kristensson:2021ani} which only involve $\\Omega^{(0)}$'s and $\\Omega^{(1)}$'s.\n\nNote that the identities we presented in this subsection can be equivalently formulated in terms of divisor theory, see e.g.\\ \\cite{Broedel:2019tlz}.\n\n\\subsection{Elliptic Bloch relation and the symbol prime}\n\\label{subsec: symbol prime}\n\nIn \\cite{Kristensson:2021ani}, also five identities involving $\\Omega^{(2)}$'s were observed which are much lengthier than the other identities; each of these five identities contains at least 100 terms in the form that they were found. It turns out all these identities are consequences of the so-called elliptic Bloch relation \\cite{bloch2011higher,Zagier2000}, \nan elliptic generalization of the five-term identity for dilogarithms,\n\\begin{equation}\n\\label{eq: five term identity}\n D(x)+D(y)+D\\biggl(\\frac{1-x}{1-xy}\\biggr)+D(1-xy)+D\\biggl(\\frac{1-y}{1-xy}\\biggr)=0 \\:,\n\\end{equation}\nwhere $D(z)=\\Im(\\Li_{2}(z))+\\arg(1-z)\\log |z|$ is the Bloch-Wigner function.%\n\\footnote{To show this concretely, one would need to do the divisor-theory analog of finding a curve that intersects the elliptic curve at precisely the points given by the more than 100 terms in the identities.\nAn algorithm for doing this is given in \\cite{Bolbachan:2019dsu}.}\n\nIn practice, it is difficult to simplify even expressions containing dilogarithms by using the above five-term identity directly. Instead, we introduce the symbol map \\cite{Goncharov:2010jf} for polylogarithms as an assistance; we associate to each polylogarithm a tensor product -- the so-called symbol -- whose entries satisfy simpler identities. We then exploit that the symbol of a combination of polylogarithms vanishes if that combination of polylogarithms vanishes. \n\n\n\nA similar strategy can be used for the elliptic letters $\\Omega^{(2)}(w)=(2\\pi \\mi)^{-1}\\gamt{2}{0}{w}$, although they already serve as entries of the symbol for elliptic multiple polylogarithms. Inspired by the proof of the elliptic Bloch relation for $\\gamt{2}{0}{w}$ in \\cite{Broedel:2019tlz}, we associate to $\\Omega^{(2)}(w)$ a rank-two tensor through the \\emph{symbol prime} map,\n\\begin{equation} \\label{symbolp1}\n \\mathcal{S}'\\bigl(\\Omega^{(2)}(w)\\bigr) = \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w).\n\\end{equation} \nThis map has a property similar to that of the symbol map:\n\\begin{equation}\n\\label{eq: symbol prime property}\n \\sum_{j}c_j \\Omega^{(2)}(w_{j})=0 \\quad \n \\text{``$\\Rightarrow$''}\n \\quad\n \\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))\\equiv\n \\sum_{j}c_j\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w_{j})=0\n\\end{equation}\nfor some rational coefficients $c_j$.\nTo show this, consider the sum $\\sum_{j}c_j\\gamt{1&0}{0&0}{w_{j}}$.\nAccording to \\eqref{devoftG}, \n\\begin{align}\n \\label{dev_tG2}\n\\mathcal{S}(2\\pi i \\gamt{1&0}{0&0}{w})= \\Omega^{(0)}(w) \\otimes \\Omega^{(1)}(w)- \\Omega^{(2)}(w) \\otimes (2\\pi i\\tau)\\:,\n\\end{align}\nwhere we used that $\\gamt{0}{0}{w}=w$, $\\gamt{2}{0}{w}=2\\pi \\mi\\Omega^{(2)}(w)$ and $\\Omega^{(-1)}=-2\\pi i \\tau$.\nIf the arguments $w_j$ and coefficients $c_j$ are such that $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ due to an elliptic Bloch relation, $\\sum_{j}c_j\\gamt{1&0}{0&0}{w_{j}}=0$ according to an analogous elliptic Bloch relation \\cite{Broedel:2019tlz}, which in turn implies that the second term in \\eqref{dev_tG2} drops out in the sum, i.e.\\ $\\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))$.\nIn this sense, the symbol prime makes the elliptic Bloch relations manifest.\n\nNote that we have assumed that $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ vanishes \\emph{due to an elliptic Bloch relation} here in order to show that $\\sum_{j}c_j\\mathcal{S}'(\\Omega^{(2)}(w_{j}))=0$. We have indicated this in \\eqref{eq: symbol prime property} as ``$\\Rightarrow$''.\nHowever, we currently have no way of proving that all identities $\\sum_{j}c_j\\Omega^{(2)}(w_{j})=0$ are due to an elliptic Bloch relation \\cite{bloch2011higher,Zagier2000}.\nThis is similar to the case of dilogarithms, where \\eqref{eq: five term identity} is only conjectured but not proven to generate all functional identities among dilogarithms.%\n\nThe symbol map itself has a kernel, and the same is true for the symbol prime.\nIf $\\mathcal{S}'(\\sum_j \\Omega^{(2)}(w_{j}))=0$, the first term in \\eqref{dev_tG2} drops out in the sum. This implies that $\\sum_{j}\\gamt{1&0}{0&0}{w_{j}}$ and thus $\\sum_j \\Omega^{(2)}(w_{j})$ is a function \\emph{only} of $\\tau$. However, not all functions only of $\\tau$ are in the kernel of the symbol prime; for example, $\\Omega^{(2)}(\\tau\/n)$ with some positive integer $n$ only depend on $\\tau$ but has a non-vanishing symbol prime.%\n\\footnote{However, the appearance of such a letter means that the kinematic image $\\kappa(\\omega_1\\tau\/n)$ relates to the physical problem and hence is algebraic in general. This is not the case for the examples of the unequal-mass sunrise integral and the ten-point double-box integral studied in sections \\ref{sec: example 1}--\\ref{sec: example 2}, but it is the case for the \\emph{equal}-mass sunrise integral.}\n\n\nOne can find the action of the symbol prime map on the letters $\\Omega^{(n<2)}$ by expressing them in terms of $\\Omega^{(2)}$ using the quasi periodicity \\eqref{quasiperiodicity} of $\\Omega^{(n)}$:\n\\begin{align}\n \\Omega^{(1)}(w) &= \\tfrac{1}{6}\\Omega^{(2)}(w+2\\tau)-\\Omega^{(2)}(w+\\tau)+\\tfrac{1}{2}\\Omega^{(2)}(w)\n +\\tfrac{1}{3}\\Omega^{(2)}(w-\\tau) \\:, \\\\\n \\Omega^{(0)}(w)&=\\Omega^{(2)}(w+\\tau)+\\Omega^{(2)}(w-\\tau)-2\\Omega^{(2)}(w) \\:, \\\\\n \\Omega^{(-1)}&=-\\Omega^{(2)}(w+2\\tau)+3\\Omega^{(2)}(w+\\tau)-3\\Omega^{(2)}(w)+\\Omega^{(2)}(w-\\tau)\n \\:.\n\\end{align}\nThis yields \n\\begin{subequations} \\label{symbolp2}\n \\begin{align} \n \\mathcal{S}'\\bigl(\\Omega^{(1)}(w)\\bigr)&= \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(0)}(w)\n +\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(1)}(w) \\:, \\\\\n \\mathcal{S}'\\bigl(\\Omega^{(0)}(w)\\bigr) &= \\Omega^{(0)}(w)\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(-1)}\n + 2 \\Omega^{(-1)} \\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(0)}(w) \\:, \\\\\n \\mathcal{S}'\\bigl(\\Omega^{(-1)}\\bigr) &= 3\\Omega^{(-1)}\\otimes'}%{\\otimes_{\\mathrm{p}} \\Omega^{(-1)} \\:,\n \\end{align} \n\\end{subequations}\nwhere we have moreover used quasi periodicity to simplify the entries of the symbol prime.\nIn particular, by expressing a logarithm in terms of $\\Omega^{(1)}$'s and $\\Omega^{(0)}$'s either through \\eqref{identity1} or \\eqref{Abel1}, one finds\n\\begin{equation} \\label{symbolp3}\n \\mathcal{S}'\\bigl(\\log c\\bigr) = \\Omega^{(-1)} \\otimes'}%{\\otimes_{\\mathrm{p}} \\log c\\:.\n\\end{equation}\nThus, for a combination of $\\Omega^{(n\\leq 2)}$'s and logarithms, one can compute its symbol prime. \nIt involves only $\\Omega^{(n\\leq1)}$ and can thus be simplified using the techniques discussed in subsection \\ref{sec:3.1}. If the symbol prime is not zero, one may search for a simpler combination of $\\Omega^{(n\\leq 2)}$'s and logarithms with the same symbol prime according to \\eqref{symbolp1}, \\eqref{symbolp2} and \\eqref{symbolp3}.%\n\\footnote{In particular, if the first entry of the symbol prime is only $\\tau$, then the function is the sum of logarithms and a function of $\\tau$.} The difference of these two combinations has to be a function of $\\tau$ only, which can be fixed by sending the independent $w$-variables to any values, say $0$. In this way, we have proven the five identities involving $\\Omega^{(2)}$'s found in \\cite{Kristensson:2021ani}.\n\n\nThe current definitions for $\\Omega^{(n\\leq 2)}$ are sufficient for the two examples treated in this paper. \nFor $n>2$, one might similarly define the symbol prime for $\\Omega^{(n)}$'s as \n\\begin{equation}\n \\mathcal{S}^{(n-1)}\\bigl(\\Omega^{(n)}(w)\\bigr) = \\frac{1}{n-1}\\Omega^{(0)}(w)\\otimes^{(n-1)} \\Omega^{(n-1)}(w),\n\\end{equation}\ndue to the fact that,\n\\begin{equation}\n\\mathcal{S}\\bigl((2\\pi i)^{2-n} \\gamt{n-1&0}{0&0}{w}\\bigr)=\\Omega^{(0)}(w) \\otimes \\Omega^{(n-1)}(w)- (n-1)\\bigl( \\Omega^{(n)}(w)-\\Omega^{(n)}(0)\\bigr)\\otimes (2\\pi i\\tau)\\:,\n\\end{equation}\nwhere $\\Omega^{(n)}(0)$ is either zero or a function only depending on $\\tau$ for even or odd $n$, respectively.\nWith the knowledge of the identities among $\\Omega^{(n-1)}$, we can then find identities among $\\Omega^{(n)}$ recursively. \nWe leave the exploration of the symbol prime for $\\Omega^{(n>2)}$ to future work.\n\n\\section{Example I: Unequal-Mass Sunrise Integral}\n\\label{sec: example 1}\n\nTwo particularly interesting cases of elliptic Feynman integrals are the unequal-mass sunrise integral in two dimensions and the double-box integral in four dimensions. We will investigate these two integrals through the tools developed so far. The main focus will be on the sunrise integral treated in this section, since this integral is simple enough such that the main results can be written within a couple of lines. After applying the symbol prime map, we will see that several properties, such as double periodic invariance, modular invariance (covariance) and part of integrability are manifest. \n\n\n\nWe calculate the unequal-mass sunrise integral in terms of elliptic multiple polylogarithms $\\mathcal{E}_{4}$ in appendix \\ref{app:sunrise}.\nThis integral was originally calculated in terms of iterated integrals on the moduli space $\\overline{\\mathcal{M}}_{1,3}$ in \\cite{Bogner:2019lfa}. \nWe closely follow the Feynman-parameter approach of \\cite{Broedel:2017siw} for the equal-mass case.\n\nThe resulting expression when normalizing the torus by the period $\\omega_1$ is \n\\begin{equation}\n\\label{eq: sunrise general}\n I_{\\sr} = \\frac{\\omega_{1}}{2\\pi\\mi m_{1}^{2}} (2\\pi \\mi T^{(1)}_{\\sr})\n \\,,\n\\end{equation}\nwhere the periods were defined in figure \\ref{fig: contours} and $T_{\\sr}^{(1)}$ is a pure combination of elliptic multiple polylogarithms of weight one and length two,\n\\begin{align}\n T_{\\sr}^{(1)} &= \\cEf{0&-1}{0&-1}{\\infty|\\tau}-\\cEf{0&-1}{0&0}{\\infty|\\tau} + \\cEf{0&-1}{0&r}{\\infty|\\tau} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau} \\nonumber \\\\\n &\\quad +4\\pi \\mi\\cEf{0&0}{0&0}{\\infty|\\tau} -\\cEf{0}{0}{\\infty|\\tau} \\log\\frac{t^{2}_{2}}{t_{3}^{2}} \\:, \\label{eq:sunrise normalization 1}\n \\end{align}\nwhere we introduced $t_{i}^{2}=m_{i}^{2}\/p^{2}$ and $r=-t_{3}^{2}\/t^{2}_{1}$. \nNote that we have included seemingly redundant factors of $(2\\pi i)$ in the numerator and denominator of \\eqref{eq: sunrise general} that ensure that the prefactor degenerates to an algebraic function in the limit where the elliptic curve degenerates, and the term in parentheses degenerates to a pure logarithm of transcendental weight two; see subsection \\ref{sec:4.1}. \n\nHowever, we can also normalize the torus by the period $\\omega_2$, finding\n\\begin{equation}\n\\label{eq: sunrise general 2}\n I_{\\sr} = \\frac{-\\omega_{2}}{2\\pi\\mi m_{1}^{2}} (2\\pi \\mi T^{(2)}_{\\sr})\\,,\n\\end{equation}\nwith\n\\begin{align}\n T_{\\sr}^{(2)} &= \\cEf{0&-1}{0&-1}{\\infty|\\tau'}-\\cEf{0&-1}{0&0}{\\infty|\\tau'} + \\cEf{0&-1}{0&r}{\\infty|\\tau'} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau'} \\nonumber \\\\ \n &\\quad -\\cEf{0}{0}{\\infty|\\tau'} \\log\\frac{t^{2}_{2}}{t_{3}^{2}} \\:,\\label{eq:sunrise normalization 2}\n\\end{align}\n(Recall that $\\tau'=-\\omega_1\/\\omega_2$.)\n\nAccording to \\eqref{eq: sunrise general} and \\eqref{eq: sunrise general 2}, the values of $T_{\\sr}^{(1)}$ and $T_{\\sr}^{(2)}$ are related by $T_{\\sr}^{(1)}= -\\tau T_{\\sr}^{(2)}$, but this relation is not obvious from their expressions in terms of eMPLs. \nIn general, eMPLs transform non-trivially under the modular $S$-transformation $\\tau\\to \\tau'=-1\/\\tau$; for example,\n\\begin{equation}\n\\cEf{-1}{c}{x|\\tau}= \\cEf{-1}{c}{x|\\tau'}-\\frac{2\\pi \\mi(\\xi_{c}^{+}-\\xi^{-}_{c})}{\\tau'} \\cEf{0}{0}{x|\\tau'} \\:.\n\\end{equation}\nSee \\cite{Duhr:2019rrs} for the cases of iterated integrals of modular forms. The same is true for the symbol, as we will see soon. \nHowever, we will see that the application of the symbol prime map makes the behavior under the modular $S$-transformation manifest.\n\n\\subsection{Symbol of the sunrise integral} \\label{sec:4.1}\n\nThe symbol of $ T_{\\sr}^{(1,2)}$ can be calculated by first rewriting $\\mathcal{E}_4$'s in terms of $\\tilde\\Gamma$'s via \\eqref{Psikernels} and then applying \\eqref{devoftG}--\\eqref{eq: elliptic symbol}.\nFor example,\n\\begin{equation}\n \\cEf{0&-1}{0&c}{x}= \\gamt{0&1}{0&w_{c}^{+}-w_{0}^{+}}{w_{x}^{+}-w_{0}^{+}}- \\gamt{0&1}{0&w_{c}^{-}-w_{0}^{+}}{w_{x}^{+}-w_{0}^{+}}\n\\end{equation}\nand \n\\begin{align}\n \\mathcal{S}\\bigl(2\\pi i\\gamt{0&1}{0&w_{1}}{w_{2}}\\bigr)&= \\Bigl(\\Omega^{(2)}(-w_{1})-\\Omega^{(2)}(w_{2}-w_{1})\\Bigr)\\otimes \\Omega^{(-1)}\n -\\Omega^{(0)}(w_{2})\\otimes \\Omega^{(1)}(w_{1}) \\nonumber\\\\ \n &\\quad + \\Bigl(\\Omega^{(1)}(w_{2}-w_{1})-\\Omega^{(1)}(-w_{1})\\Bigr)\\otimes \\Omega^{(0)}(w_{2}-w_{1}) \\,.\n\\end{align}\n\n\nThe simplification of the symbols in this case \nis slightly non-trivial: it involves some non-trivial relations of $\\Omega^{(1)}$'s, $\\Omega^{(0)}$'s and logarithms as described in \nsection \\ref{sec:3.1}; for instance, \n\\begin{align}\n \\log \\frac{t_{1}}{t_{3}} &= \\Omega^{(1)}(w_{-1}^{+}-w_{0}^{+})-\\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{+}) \n +\\Omega^{(1)}(w_{-1}^{+}-w_{0}^{-})-\\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{-}) \\:, \\\\\n \\log \\frac{t_{2}}{t_{3}} &= \\Omega^{(1)}(w_{0}^{+}-w_{\\infty}^{+}) - \\Omega^{(1)}(w_{-1}^{+}-w_{\\infty}^{+}) \n +\\Omega^{(1)}(w_{-1}^{+}-w_{0}^{-})-\\Omega^{(1)}(w_{\\infty}^{-}-w_{\\infty}^{+}) \\:.\n\\end{align}\n(Recall that $w_{c}^{+}=\\omega_{1}^{-1}\\int_{-\\infty}^{c} \\dif x\/y$). \nAll the relations involving $\\Omega^{(2)}$ in this case are relatively trivial; they are the consequences of \\eqref{quasiperiodicity}. \nWe present the full simplification in the attached file \\texttt{sunrise\\_symbol.nb}.\n\nThe final result is\n\\begin{align}\n \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(1)} \\bigr) &= \\log \\frac{t_{2}^{2}}{t_{1}^{2}} \\otimes \\Omega^{(0)}(w_{0}^{+})\n + \\log \\frac{t_{1}^{2}}{t_{3}^{2}}\\otimes \\Omega^{(0)}(w_{-1}^{+}) \\nonumber \\\\\n &\\quad+\\biggl[\\frac{1}{2\\pi\\mi} \\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr)+\\log\\frac{t_{3}^{2}}{t_{2}^{2}} \\biggr]\\otimes (2\\pi\\mi\\tau)\\:. \\label{symbolw1} \n\\end{align}\nwhere we have moreover used \n\\begin{align}\n\\label{eq:Ecal-gammat-relation}\n \\frac{\\cEf{-n}{c}{\\infty}}{(2\\pi\\mi)^{n-1}}=\\Omega^{(n)}(w_{\\infty}^{+}-w_{c}^{+})-\\Omega^{(n)}(w_{0}^{+}-w_{c}^{+})\n -\\Omega^{(n)}(w_{\\infty}^{+}-w_{c}^{-})+\\Omega^{(n)}(w_{0}^{+}-w_{c}^{-}) \\:.\n\\end{align}\nSimilarly, \n \\begin{align}\n \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(2)} \\bigr) &= \\log \\frac{t_{2}^{2}}{t_{1}^{2}} \\otimes \\Omega^{(0)}(\\xi_{0}^{+})\n + \\log \\frac{t_{1}^{2}}{t_{3}^{2}}\\otimes \\Omega^{(0)}(\\xi_{-1}^{+}) \\nonumber \\\\\n &\\quad+ \\biggl[\\frac{1}{2\\pi \\mi}\\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr)\\nonumber \\\\\n &\\qquad \\qquad +\\log\\frac{t_{1}}{t_{3}} +\n \\Omega^{(1)}(\\xi_{\\infty}^{+}-\\xi_{-1}^{+})-\\Omega^{(1)}(\\xi_{-1}^{+}-\\xi_{0}^{+})\\biggr] \\otimes (2\\pi\\mi\\tau^{\\prime}) \\:, \\label{symbolw2}\n\\end{align}\nwhere we used \\eqref{eq:Ecal-gammat-relation} in terms of $\\xi$-coordinates. \n\nAt this point, the symbols of the sunrise integral partially show some desired properties; for example, the first entries of the first two terms in \\eqref{symbolw1} and \\eqref{symbolw2} indicate the physical first-entry conditions known from the massless case, and their last entries are related by simple $S$-transformations $w\\to\\xi$.\nHowever, the first two terms on their own are neither double periodic nor integrable.\n\nThe first entries of the last terms in \\eqref{symbolw1} and \\eqref{symbolw2}, i.e., $\\partial_\\tau T_{\\sr}^{(1)}$ and $\\partial_{\\tau'} T_{\\sr}^{(2)}$,\nare relatively complicated and the main obstacles to understanding the entire symbols, since it is hard to see how they render the whole symbol double periodic and integrable. In this respect, it is instructive to consider the symbol primes of these entries:\n\\begin{align}\n \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &= \\Omega^{(0)}(w_{0}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}+\\Omega^{(0)}(w_{-1}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}} \\:, \\label{spofsr1}\\\\\n \\mathcal{S'}\\bigl(\\partial_{\\tau'} T_{\\sr}^{(2)} \\bigr) &=\\Omega^{(0)}(\\xi_{0}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}+\\Omega^{(0)}(\\xi_{-1}^{+})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}} \\:.\\label{spofsr2}\n\\end{align}\nThey have the following advantageous properties:\n\\begin{enumerate}\n \\item It is obvious that $\\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)} \\bigr)$ differs from $\\mathcal{S'}\\bigl(\\partial_{\\tau'} T_{\\sr}^{(2)} \\bigr) $ only by a modular $S$-transformation $w\\to\\xi$.\n \\item If we shift $w_{-1}^{+}$ or $w_{0}^{+}$ by $\\tau$, then $\\partial_\\tau T_{\\sr}^{(1)}$ changes by $\\log\\frac{t_{3}^2}{t_{1}^{2}} $ or $\\log\\frac{t_{1}^2}{t_{2}^{2}}$, respectively, (and similarly for $\\partial_{\\tau'} T_{\\sr}^{(2)}$) since\n \\begin{align}\n \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)}|_{w_{-1}^{+}\\to w_{-1}^{+}+\\tau}-\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &= (2\\pi\\mi\\tau)\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{1}^2}{t_{3}^{2}}=\\mathcal{S'}\\biggl(-\\log\\frac{t_{1}^2}{t_{3}^{2}}\\biggr) \\:, \\\\\n \\mathcal{S'}\\bigl(\\partial_\\tau T_{\\sr}^{(1)}|_{w_{0}^{+}\\to w_{0}^{+}+\\tau}-\\partial_\\tau T_{\\sr}^{(1)} \\bigr) &= (2\\pi\\mi\\tau)\\otimes'}%{\\otimes_{\\mathrm{p}} \\log\\frac{t_{2}^2}{t_{1}^{2}}=\\mathcal{S'}\\biggl(-\\log\\frac{t_{2}^2}{t_{1}^{2}}\\biggr) \\:.\n \\end{align}\n The first two terms in the symbol change by corresponding terms with opposite sign that cancel these. Thus, $\\mathcal{S}(2\\pi i T_{\\sr}^{(1,2)})$ are \\emph{double periodic}.\n \\item Moreover, the symbol prime also makes integrability with respect to $\\tau$ manifest. This is slightly trivial in the case of the length-two sunrise integral, and will thus be discussed in full generality for the case of the double-box integral in section \\ref{sec:5.1}.\n\\end{enumerate}\n\nFinally, note that the equal-mass case can be obtained smoothly by taking $t_1=t_2=t_3$; we will briefly comment on this case in section \\ref{sec:5}. \n\n\n\\subsection{Degeneration at \\texorpdfstring{$p^{2}=0$}{p**2=0} and pseudo-thresholds}\n\nNext, let us see how the symbol of the unequal-mass sunrise integral behaves in kinematic limits where the elliptic curve degenerates.\n\nThe kinematic configurations where the elliptic curve degenerates can be easily read off from the discriminant\n\\begin{equation} \\label{Discriminant}\n \\Delta_{\\sr}=\\frac{t_{2}^{4}t_{3}^{4}}{t_{1}^{20}}\\Bigl((t_{1}+t_{2}+t_{3})^{2}-1\\Bigr)\\Bigl((t_{1}+t_{2}-t_{3})^{2}-1\\Bigr)\\Bigl((t_{1}-t_{2}+t_{3})^{2}-1\\Bigr)\\Bigl((-t_{1}+t_{2}+t_{3})^{2}-1\\Bigr) \\:,\n\\end{equation}\nwhere $t_i^2=m_i^2\/p^2$ as before. \nIn particular, the sunrise integral remains finite at $p^{2}=0$, at the pseudo-thresholds $p^{2}=\\{(m_{1}+m_{2}-m_{3})^{2},(m_{1}+m_{3}-m_{2})^{2},(m_{2}+m_{3}-m_{1})^{2}\\}$ and at the threshold $p^{2}=(m_{1}+m_{2}+m_{3})^{2}$, while it diverges for $m_i=0$.\nThe values at $p^{2}=0$ and the pseudo-thresholds were given in terms of MPLs in \\cite{Bloch:2013tra}. \nIn what follows, we will show how the symbols $\\mathcal{S}(2\\pi iT_{\\sr}^{(1,2)})$ reproduce the corresponding symbols in these two limits.%\n\\footnote{The threshold can be treated in a similar way.}\n\n\n\n\n\\begin{figure} \n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\begin{tikzpicture}[scale=1.5]\n \n \\draw[->, thick] (0,0) to (0,3);\n \\draw[->, thick] (-3,1.5) to (1,1.5);\n \\draw[<-,line width=0.23mm,red] (-1.0,0.2) to (-1.0,2.8);\n \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n \\draw[->] (-1.65,2.0) to (-1.1,2.1);\n \\draw[->] (-1.65,1.0) to (-1.1,0.9);\n \\draw[->] (-0.35,0.8) to (-0.9,0.87);\n \\draw[->] (-0.35,2.2) to (-0.9,2.13);\n \\node at (-1.1,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n \\node at (0.4,3) {$\\Im x$};\n \\node at (1.3,1.5) {$\\Re x$};\n \\end{tikzpicture}\n \\caption{\\phantom{.}}\n \\label{Fig:nullmomentumlimit}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.45\\textwidth}\n \\centering\n \\begin{tikzpicture}[scale=1.5]\n \n \\draw[->, thick] (0,0) to (0,3);\n \\draw[->, thick] (-3,1.5) to (1,1.5);\n \\draw[<-,line width=0.23mm,red] (-1.0,0.2) to (-1.0,2.8);\n \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n \\draw[->] (-1.7,1.95) to (-1.7,1.6);\n \\draw[->] (-1.7,1.05) to (-1.7,1.4);\n \\node at (-1.1,0.1) {\\textcolor{red}{$\\gamma_{1}$}};\n \\node at (0.4,3) {$\\Im x$};\n \\node at (1.3,1.5) {$\\Re x$};\n \\node[cross] at (-1.7, 1.5) {};\n \\node[cross,label=above:$-1$] at (-1.3, 1.5) {};\n \\end{tikzpicture}\n \\caption{\\phantom{.}}\n \\label{Fig:pseudothreshold}\n \\end{subfigure}\n \\caption{The roots of $y^{2}(x)$ for the sunrise integral coincide in the null-momentum limit (\\subref{Fig:nullmomentumlimit}) and pseudo-thresholds (\\subref{Fig:pseudothreshold}). In the latter case, the position of the coinciding roots relative to $-1$ is shown for the case $t_3>t_1$.}\n \\label{fig: sr_degeneration}\n\\end{figure}\n\n\\paragraph{Null-momentum limit} As $p^{2}\\to 0$, $t_{1}$, $t_{2}$ and $t_{3}$ approach infinity while their ratios remain finite. The elliptic curve degenerates in a way that $r_{1} \\to r_{3}$ and $r_{2}\\to r_{4}$; cf.\\ figure \\ref{Fig:nullmomentumlimit}.\nIn this case, $\\omega_{1}\\to \\infty$ since the roots pinch the corresponding integration contour $\\gamma_1$, while \n \\begin{equation}\n \\omega_{2} \\to \\int_{-\\infty}^{\\infty} \\frac{\\dif x}{x^{2}+\\bigl(1+(t_{3}\/t_{1})^{2}-(t_{2}\/t_{1})^{2}\\bigr)x+(t_{3}\/t_{1})^{2}}\n = \\frac{2\\pi \\mi}{\\sqrt{\\Bigl(1-\\frac{t_{2}^{2}}{t_{1}^{2}}-\\frac{t_{3}^{2}}{t_{1}^{2}}\\Bigr)^{2}-4\\frac{t_{2}^{2}t_{3}^{2}}{t_{1}^{4}}}} \\:.\n \\end{equation}\n Then $-(2\\pi \\mi)^{-1}m_{1}^{-2}\\omega_{2}$ reproduces the same normalization factor as in \\cite{Bloch:2013tra} (up to a sign).\n Thus, we should expect that $\\mathcal{S}(2\\pi iT_{\\sr}^{(2)})$ reduces to the corresponding symbol.\n In this limit, $q=\\exp(2\\pi \\mi \\tau')$ vanishes, and hence all $\\cEf{-2}{c}{x}$ in \\eqref{symbolw1} vanish; cf.\\ \\eqref{eq:Ecal-gammat-relation} and \\eqref{wdef}. Furthermore, \n \\begin{align}\n &\\Omega^{(0)}(\\xi_{c}^{+} )\\to \\log \\frac{2c+1-u+v+\\sqrt{(1-u-v)^{2}-4uv}}{2c+1-u+v-\\sqrt{(1-u-v)^{2}-4uv}} \\:, \\\\\n &\\log\\frac{t_{1}}{t_{3}} +\n \\Omega^{(1)}(\\xi_{\\infty}^{+}-\\xi_{-1}^{+})-\\Omega^{(1)}(\\xi_{-1}^{+}-\\xi_{0}^{+}) \n \\to 0\\,,\n \\end{align}\n where we have introduced $u = (t_{2}\/t_{1})^{2}=z\\bar{z}$ and $v=(t_{3}\/t_{1})^{2}=(1-z)(1-\\bar{z})$. \n Then,\n\\begin{align}\n \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(2)} \\bigr) \\to \\log u \\otimes \\log\\frac{1-\\bar{z}}{1-z}-\\log v \\otimes \\log \\frac{\\bar{z}}{z} \\:.\n\\end{align}\nwhich is the symbol of $-4i$ times the Bloch-Wigner dilogarithm $D(z)$, in perfect agreement with \\cite{Bloch:2013tra}.\n\n\n \\paragraph{Pseudo-thresholds} Without loss of generality, we consider the pseudo-threshold $p^{2}=(m_{1}+m_{2}-m_{3})^{2}$. In terms of $t_{i}$, this pseudo-threshold is equal to the condition $(t_{1}+t_{2}-t_{3})^{2}=1$. We only consider the solution $t_{3}=t_{1}+t_{2}-1$ since the treatment for the other solution is similar. At $t_{3}=t_{1}+t_{2}-1$, the roots $r_{1}$ and $r_{4}$ pinch the real axis; cf.\\ figure \\ref{Fig:pseudothreshold}. Thus $\\omega_2$ diverges and we should consider the normalization of the torus by $\\omega_1$, $T_{\\sr}^{(1)}$. We can then close the contour $\\gamma_{1}$ with a large semi-circle in the left half-plane and evaluate the integral via residues:\n \\begin{align}\n \\int_{\\gamma_{1}}\\frac{t_{1}\\dif x}{\\Bigl(x+\\frac{t_{3}}{t_{1}}\\Bigr)\\sqrt{\\Bigl(t_{1}^{2}x^{2}+\\bigl(t_{1}^{2}+t_{3}^{2}-(t_{2}+1)^{2}\\bigr)x+t_{3}^{2}\\Bigr)}} =-2\\pi \\mi\\sqrt{\\frac{t_{1}^{3}}{4t_{2}t_{3}}} \\:.\n \\end{align}\n Again $(2\\pi\\mi)^{-1}m_{1}^{-2}\\omega_{1}$ reproduces the same normalization factor as in \\cite{Bloch:2013tra} at $p^{2}=(m_{1}+m_{2}-m_{3})^{2}$. Thus, we should expect that $\\mathcal{S}(2\\pi i T_{\\sr}^{(1)})$ reduces to the corresponding symbol. \n At the pseudo-threshold, $q=\\exp(2\\pi \\mi \\tau)$ vanishes. Furthermore, we assume $t_{3}>t_{1}$;\\footnote{The case $t_{1}, thick] (0,0) to (0,3);\n \\draw[->, thick] (-3,1.5) to (1,1.5);\n \\draw[->,line width=0.23mm,blue] (-3,1.4) .. controls (-1,1.4) .. (-1.0,0.0) ; \n \\node[cross,label=above:$r_{1}$] at (-1.7, 2.0) {};\n \\node[cross,label=below:$r_{4}$] at (-1.7, 1.0) {};\n \\node[cross,label=below:$r_{2}$] at (-0.3, 0.8) {};\n \\node[cross,label=above:$r_{3}$] at (-0.3, 2.2) {};\n \\draw[->] (-1.7,1.95) to (-1.7,1.7);\n \\draw[->] (-1.7,1.05) to (-1.7,1.3);\n \\draw[->] (-0.3,0.85) to (-0.3,1.45);\n \\draw[->] (-0.3,2.15) to (-0.3,1.55);\n \\node at (-1.3,0.1) {\\textcolor{blue}{$\\gamma_{-}$}};\n \\node at (0.4,3) {$\\Im x$};\n \\node at (1.3,1.5) {$\\Re x$};\n \\node[cross,label=above:${-z_{2}}$] at (-2.5, 1.5) {};\n \\node[cross,label=above:${\\bar{z}_{1}-1}$] at (-1.3, 1.5) {};\n \\node[cross,label=above:${-\\bar{z}_{2}}$] at (-0.7, 1.5) {};\n \\node[cross,label=above:${z_{1}-1}$] at (0.5, 1.5) {};\n \\end{tikzpicture}\n \\caption{In the soft limit $p_{10}\\to0$, the roots of $y^{2}(x)$ pairwise pinch the integration contours for $w^+_{\\bar{z}_{1}-1}, w^+_{-\\bar{z}_{2}}$ and $w^+_{z_{1}-1}$, which run along the real axis.\n By subtracting $w^-_{\\infty}$ and $w^+_{\\infty}$, respectively, we obtain integration contours that can be deformed such they are not pinched, thus resulting in finite integrals in the soft limit. \n }\n \\label{fig: degeneration}\n \\end{figure}\n To cancel the resulting singularities, we reorganize $\\mathcal{S}(2\\pi i T_{\\db}^{(1)})$ as \n\\begin{align}\n \\mathcal{S}(2\\pi i T_{\\db}^{(1)}) &= \\mathcal{S}(F_{-z_{2}})\\otimes \\Omega^{(0)}(w_{-z_{2}}^{+})+\n \\mathcal{S}(F_{z_{1}-1})\\otimes \\Omega^{(0)}(w_{z_{1}-1}^{+}-w_{\\infty}^{+}) \\nonumber \\\\\n &\\quad + \\mathcal{S}(F_{\\bar{z}_{1}-1})\\otimes \\Omega^{(0)}(w_{\\bar{z}_{1}-1}^{+}-w_{\\infty}^{-}) \n + \\mathcal{S}(F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(w_{-\\bar{z}_{2}}^{+}-w_{\\infty}^{-}) \\nonumber \\\\\n &\\quad \n +\\mathcal{S}(F_{\\tau}+F_{z_{1}-1}) \\otimes (2\\pi i\\tau) +\\mathcal{S}(F_{-}+F_{\\bar{z}_{1}-1}+F_{-\\bar{z}_{2}})\\otimes \\Omega^{(0)}(w_{\\infty}^{-}) \\nonumber \\\\\n &\\quad+\\mathcal{S}(I_{\\text{hex}}) \\otimes \\Omega^{(0)}(w_{c_{25}}^{+}) \\:,\n\\end{align}\ncf.\\ figure \\ref{fig: degeneration}.\nOne can easily check that not only the last term but the last three terms do not contribute in the soft limit since the three weight-three symbols making up their first three entries vanish in the soft limit. The first four terms yield the correct polylogarithmic symbol in the soft limit with last entries\n\\begin{equation}\n\\begin{aligned}\n \\Omega^{(0)}(w_{-z_{2}}^{+}) &\\to (\\rho-\\bar{\\rho}) \\int_{-\\infty}^{-z_{2}}\\frac{\\dif x}{(x-\\rho)(x-\\bar{\\rho})}\\:= \n \\log \\frac{\\rho+z_{2}}{\\bar{\\rho}+z_{2}} \\:, \\\\\n \\Omega^{(0)}(w_{z_{1}-1}^{+}-w_{\\infty}^{+})& \\to (\\rho-\\bar{\\rho}) \\int_{+\\infty}^{z_{1}-1}\\frac{\\dif x}{(x-\\rho)(x-\\bar{\\rho})} = \\log\\frac{1+\\rho-z_{1}}{1+\\bar{\\rho}-z_{1}} \\:, \\\\\n \\Omega^{(0)}(w_{\\bar{z}_{1}-1}^{+}-w_{\\infty}^{-}) & \\to (\\rho-\\bar{\\rho}) \\int_{-i\\infty}^{\\bar{z}_{1}-1} \n \\frac{-\\dif x}{(x-\\rho)(x-\\bar{\\rho})}= \\log\\frac{1+\\bar{\\rho}-\\bar{z}_{1}}{1+\\rho-\\bar{z}_{1}} \\:, \\\\\n \\Omega^{(0)}(w_{-\\bar{z}_{2}}^{+}-w_{\\infty}^{-})&\\to (\\rho-\\bar{\\rho}) \\int_{-i\\infty}^{-\\bar{z}_{2}}\\frac{-\\dif x}{(x-\\rho)(x-\\bar{\\rho})} \\: = \\log \\frac{\\bar{\\rho}+\\bar{z}_{2}}{\\rho+\\bar{z}_{2}} \\:.\n\\end{aligned} \n\\end{equation}\nNote that in the soft limit $z_{1}\\equiv z_{1,3,5,8}$ and $z_{2}\\equiv z_{3,6,8,1}$ and the reflection symmetry $R_{1}$ is broken while $R_{2}$ survives; \nthus, the symbol for this nine-point double-box integrals can be expressed as\n\\begin{align}\n \\mathcal{S}\\Bigl((\\rho-\\bar{\\rho})I_{\\softdb}\\Bigr) &= \\mathcal{S}(F_{-z_{2}}\\vert_{p_{10}\\to 0})\\otimes \\log \\frac{\\rho+z_{2}}{\\bar{\\rho}+z_{2}} +\\mathcal{S}(F_{-\\bar{z}_{2}}\\vert_{p_{10}\\to 0})\\otimes \\log \\frac{\\bar{\\rho}+\\bar{z}_{2}}{\\rho+\\bar{z}_{2}} \\nonumber \\\\\n & \\quad \n + (\\text{images under }R_2\n ) \\:,\n\\end{align}\nwhere $R_2$ acts on the last entries via $\\rho\\leftrightarrow -(1{+}\\bar{\\rho})$, $z_{1}\\leftrightarrow z_{2}$ and $\\bar{z}_{1}\\leftrightarrow \\bar{z}_{2}$.\nFurthermore, $F_{-z_{2}}\\vert _{p_{10}\\to 0}$ and $F_{-\\bar{z}_{2}}\\vert _{p_{10}\\to 0}$ are related by exchanging $z_{2}$ and $\\bar{z}_{2}$, same as the corresponding last entries.\nThe reason is that $z$ and $\\bar{z}$ occur symmetrically in their definition $\\{z\\bar{z}=u, (1-z)(1-\\bar{z})=v\\}$ , and thus have to occur symmetrically in the symbol as well.\n\nThe symbol alphabet of the nine-point double-box integral consists of 10 rational letters and 11 algebraic letters:\n\\begin{enumerate}\n \\item Rational letters:\n\\begin{equation}\n \\begin{gathered}\n u_{1}\\,,\\:\\: u_{2} \\,,\\:\\: v_{1} \\,,\\:\\: v_{2} \\,,\\:\\: u_{1}-v_{2} \\,,\\:\\: v_{1}-u_{2} \\,,\\:\\: \n u_{1}u_{2}-v_{1}v_{2} \\,,\\:\\: \\Delta_{1} \\,,\\:\\: \\Delta_{2}\\:, \\\\\n \\frac{\\langle 5(91)(23)(78)\\rangle \\langle \\bar{5}(91)(23)(78)\\rangle\\langle 1239\\rangle \\langle 1789\\rangle}{ \\langle 1459\\rangle^{2}\\langle 1569\\rangle^{2}\\langle 2378\\rangle^{3}} \\:,\n \\end{gathered} \n \\end{equation}\n where we introduced the following notations:\\footnote{Here we use $(\\bar{a})\\equiv Z_{a-1}{\\wedge}Z_{a}{\\wedge} Z_{a+1}$ to denote the dual plane of $Z_{a}$. Then a vanishing $\\langle \\bar{a} (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle$ means that the three intersection points $(i\\,i{+1}){\\cap} (\\bar{a})$, $(j\\,j{+1}){\\cap} (\\bar{a})$ and $(k\\,k{+1}){\\cap} (\\bar{a})$ are on the same line, which is the dual picture of the vanishing of $\\langle a (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle$. We are grateful to Cristian Vergu for pointing this out.}\n \\begin{align}\n \\langle a(bc)(de)(fg)\\rangle &=\\langle abde\\rangle \\langle acfg \\rangle-\\langle acde\\rangle \\langle abfg \\rangle , \\\\\n \\langle \\bar{a} (i\\,i{+}1)(j\\,j{+}1)(k\\,k{+}1) \\rangle &= \\langle (i\\,i{+}1)\\cap (\\bar{a}) \\,j\\,j{+}1\\,(k,k{+}1)\\cap (\\bar{a})\\rangle \n \\end{align}\nand \n\\begin{align}\n \\Delta_{i}=(1-u_{i}-v_{i})^{2}-4u_{i}v_{i}=(z_{i}-\\bar{z}_{i})^{2} \\:, \\qquad i=1,2\\,.\n\\end{align}\n \\item Algebraic letters:\n \\begin{itemize}\n \\item $\\displaystyle \\frac{z_{1}}{\\bar{z}_{1}}$, $\\displaystyle \\frac{1-z_{1}}{1-\\bar{z}_{1}}$, $\\displaystyle \\frac{1+\\bar{\\rho}-z_{1}}{1+\\rho-z_{1}}$, $\\displaystyle \\frac{1+\\bar{\\rho}-\\bar{z}_{1}}{1+\\rho-\\bar{z}_{1}}$, \n $\\displaystyle \\frac{\\frac{\\langle 5(23)(46)(78)\\rangle \\langle 1239\\rangle }{\\langle 5(19)(23)(78)\\rangle \\langle 2378\\rangle}-z_{1}}{\\frac{\\langle 5(23)(46)(78)\\rangle \\langle 1239\\rangle }{\\langle 5(19)(23)(78)\\rangle \\langle 2378\\rangle}-\\bar{z}_{1}}$, \\\\\n and five others generated by the reflection $R_2$. \n \\item $\\displaystyle \\frac{(z_{1}-1+\\bar{z}_{2})(\\bar{z}_{1}-1+z_{2})}{(z_{1}-1+z_{2})(\\bar{z}_{1}-1+\\bar{z}_{2})}$.\n \\end{itemize}\n\\end{enumerate}\nWe find that there are three different square roots in this alphabet; two of them are of four-mass-box type and the other, that is the square root in $\\rho$ and $\\bar{\\rho}$, arises from the leading singularity of the whole Feynman diagram. Furthermore, the new type of square root \\emph{only} appears in the last entries.\nThe symbol alphabet is organized such that the symbol is manifestly invariant (up to a sign) under the reflection $R_{2}$, as well as under each of the three transformations $z_{1} \\leftrightarrow \\bar{z}_{1}$, \n$z_{2} \\leftrightarrow \\bar{z}_{2}$, and $\\rho \\leftrightarrow \\bar{\\rho}$.\nFor an analysis of these letters through Schubert problems, see \\cite{Yang:2022gko}.\n\n\\section{Conclusion and Outlook} \\label{sec:5}\n\nIn this paper, we have investigated various techniques for manipulating and simplifying the symbol of Feynman integrals that evaluate to elliptic multiple polylogarithms. In particular, we study identities between the elliptic symbol letters $\\Omega^{(i)}$.\n\nIn contrast to ordinary multiple polylogarithms, the length of an elliptic multiple polylogarithm is not necessarily equal to its weight. A symbol letter $\\Omega^{(i)}$, whose length is by definition one, can have weight $i\\neq 1$.\nIdentities for $\\Omega^{(0)}$ follow from the well-known group law on the elliptic curve.\nMoreover, we found that various identities for $\\Omega^{(1)}$ can be derived from Abel's theorem, which generalize the identity $\\log(a)+\\log(b)=\\log(ab)$ in the polylogarithmic case.\nThe higher-weight letters $\\Omega^{(2)}$ satisfy significantly more intricate identities, closer to those of $\\Li_{2}(a)$ than those of $\\log(a)$, which are harder to exploit in a direct fashion.\nWe thus introduce the \\emph{symbol prime} $\\mathcal{S}'$ for elliptic symbol letters $\\Omega^{(2)}$, which plays the same role the symbol $\\mathcal{S}$ plays for $\\Li_{2}(a)$. We also introduced a symbol prime for $\\Omega^{(i>2)}$ but leave its exploration for future work.\n\nWe studied two concrete examples at two-loop order, namely the sunrise integral in two dimensions and the ten-point double-box integral in four dimensions. In particular, we provided proofs for the identities between symbol letters numerically found in \\cite{Kristensson:2021ani}.\n\nIn addition to identities between symbol letters, we also studied how the symbol behaves under kinematic limits in which the elliptic curve degenerates. We recover the known symbols of the sunrise integral in the null-momentum limit $p^{2}\\to 0$ and the pseudo threshold $p^2\\to(m_{1}+m_{2}-m_{3})^{2}$ \\cite{Bloch:2013tra}, as well as the nine-point limit of the double box, which has not previously appeared in the literature.\n\nThe numeric values of elliptic Feynman integrals are of course independent of whether we normalize the torus by the period $\\omega_1$ or $-\\omega_2$; the corresponding modular parameters $\\tau=\\omega_{2}\/\\omega_{1}$ and $\\tau'=-1\/\\tau$ are related by a modular $S$-transformation. In particular, for an elliptic integral of the form $\\int \\mathcal{G}(x,y)\\, \\dif x\/y$ as in \\eqref{1dIntofPolylog}, its two normalizations $T^{(1)}$ and $T^{(2)}$, which are obtained by dividing by $\\omega_{1}$ and $-\\omega_{2}$ respectively, are simply related by $T^{(2)} = \\tau' T^{(1)}$. However, this property is not manifest when expressed in terms of elliptic multiple polylogarithms or their symbols. \nInstead, we find that the application of the symbol prime to the two examples in this paper yields symbols of the form\n\\begin{align}\n & \\sum_{ij} \\mathcal{S}(f_{i})\\otimes \\Bigl( \\log a_{ij} \\otimes \\Omega^{(0)}(w_{j}) + \\bm{\\Omega}_{i}\\otimes (2 \\pi i\\tau) \\Bigr), \n \\intertext{with}\n &\\qquad\\mathcal{S}'(\\bm{\\Omega}_{i})=\\Omega^{(0)}(w_{j})\\otimes'}%{\\otimes_{\\mathrm{p}} \\log a_{ij}\\,.\n\\end{align}\nNot only the modular covariance is manifest in this form, but also the double-periodic invariance and the integrability conditions involving $\\tau$. However, one could \\emph{not} expect that the application of the symbol prime to the one-fold integral of general polylogarithms, of the form $\\int \\mathcal{G}(x,y) \\,\\dif x\/y$, yields such a structure.\nAs a simple counter-example, consider the symbol of the integral $\\int_{0}^{c} \\log(x+a)\\,\\dif x\/y$, with arbitrary values of $a$ and $c$ as well as the elliptic curve given by \\eqref{ellcurve}, which does \\emph{not} follow the above structure after applying the symbol prime map.\nIt would be very interesting to investigate why the two elliptic Feynman integrals we considered in this paper turn out to exhibit such a structure after applying the symbol prime map, and to study whether this property extends to further Feynman integrals.\n\n\n\nAlready the polylogarithmic symbol has a kernel, which is given by $i\\pi$, multiple zeta values (MZVs) and their products with MPLs.\nSimilarly, also the symbol prime has a kernel. \nAs discussed in section \\ref{subsec: symbol prime}, all functions in the kernel necessarily depend only on the modular parameter $\\tau$, but not all functions that depend only on $\\tau$ are in the kernel. \nTo see that the kernel can be non-trivial, \nconsider the symbol of the \\emph{equal-mass} sunrise integral, which is an iterated integral of modular forms and \\emph{only} depends on $\\tau$:\n\\begin{align}\n \\mathcal{S}\\bigl(2\\pi\\mi T_{\\sr}^{(1)} \\bigr) &= \\biggl[\\frac{1}{2\\pi\\mi} \\bigl(2\\cEf{-2}{-1}{\\infty}-\\cEf{-2}{0}{\\infty} - \\cEf{-2}{\\infty}{\\infty}\\bigr) \\biggr]\\otimes (2\\pi\\mi\\tau)\\:. \\label{eq:symbol equal mass sunrise} \n\\end{align}\nwhere the $\\mathcal{E}_4$ are specific combinations of $\\Omega^{(2)}$'s given in \\eqref{eq:Ecal-gammat-relation}.\nThe application of the symbol prime map to the first entry in \\eqref{eq:symbol equal mass sunrise} yields $0$.\nWe leave a \ncomprehensive treatment of the kernel of the symbol prime map to a future study.\n\nAnother interesting problem is to lift simplified symbols to simplified functions for elliptic multiple polylogarithms. \nAs a primary example, let us consider how to lift the simplified symbol \\eqref{symbolw1} to a simplified function for the sunrise integral. By rewriting the logarithms in \\eqref{symbolw1} in terms of $\\Omega^{(1)}$'s, we find a (slightly) simpler expression for $T^{(1)}_{\\sr}$,\n\\begin{equation}\n\\begin{aligned}\n T^{(1)}_{\\sr} &= 2\\cEf{0&-1}{0&-1}{\\infty|\\tau}-\\cEf{0&-1}{0&0}{\\infty|\\tau} -\\cEf{0&-1}{0&\\infty}{\\infty|\\tau} \\\\ \n &\\quad - \\biggl(2\\log\\frac{t_{2}}{t_{3}}+\\cEf{-1}{-1}{\\infty}\\biggr)\\cEf{0}{0}{\\infty|\\tau} \\:,\n\\end{aligned}\n\\end{equation}\n(and a similar expression for $T^{(2)}_{\\sr}$), in which the $\\mathcal{E}_4$ functions only contain $c=0,-1,\\infty$ but not the fourth argument $r$ that occurs in \\eqref{eq:sunrise normalization 1}. \nHowever, the general prescription of uplifting more complicated elliptic symbols to functions is still underexplored.%\n\\footnote{In particular, there are symbols that can be written in terms of only logarithms as letters that can nevertheless not be lifted to polylogarithmic functions, only to elliptic ones \\cite{Duhr:2020gdd}.\n}\n\nAlthough the simplified symbols of the elliptic Feynman integrals manifest some desired properties, such as double periodicity and modular covariance, after applying the symbol prime map, the singularity structures are not completely manifest. For example, the sunrise integral becomes singular at $m_{i}=0$ as well as at the threshold $p^{2}=(m_{1}+m_{2}+m_{3})^{2}$, as can be seen through a Landau analysis~\\cite{OKUN1960261}. One can see the branch cut at $m_{i}=0$ explicitly from eq.\\ \\eqref{symbolw1} or \\eqref{symbolw2}; however, the branch cut at the threshold is not manifest from the symbol. \nIn general, the logarithmic letter $\\log(a)$ has a logarithmic singularity if $a=0$ or $a=\\infty$.\nIn contrast, the elliptic letter $\\Omega^{(1)}(w)$ has a logarithmic singularity at all lattice points, while $\\Omega^{(j\\geq2)}(w)$ has a logarithmic singularity at all lattice points except for the origin.\nHowever, $w$ is a function of the kinematics; typically, $w=w_c^+=1\/\\omega_1 \\int_{-\\infty}^c\\dif x\/y$, where $c$ is an algebraic function of the kinematics. If the configuration of roots in $y$ does not change as we vary $c$, $w_c^+=0$ if $c=-\\infty$ and $w_c^+=\\tau$ if $c=+\\infty$. However, the configuration of roots may also vary as we vary $c$; \nwe leave a comprehensive analysis to a future study. \n\n\n\nIt would be interesting to apply the techniques used in this paper to bootstrap the symbol of scattering amplitudes or Feynman integrals that can be expressed in terms of elliptic multiple polylogarithms, such as the twelve-point elliptic double box. \nOn top of the integrability condition for the final entry $\\tau$, which is made manifest by the symbol prime, this requires understanding the integrability condition for the other last entries \\cite{integrability_in_progress}.\nMoreover, it requires an educated guess for the alphabet of symbol letters that occur in them.\nFor six- and seven-point amplitudes in $\\mathcal{N}=4$ sYM theory, the symbol alphabet was shown to be given by cluster algebras \\cite{Golden:2013xva,Golden:2014pua,Drummond:2017ssj,Drummond:2018dfd,Drummond:2018caf}, and similar techniques have recently been extended to Feynman integrals and amplitudes~\\cite{Zhang:2019vnm,He:2020vob,He:2020lcu,Li:2021bwg} containing symbol letters that are given by logarithms of algebraic functions of the kinematics \\cite{Chicherin:2020umh,He:2021esx,He:2021non,Drummond:2019cxm,Arkani-Hamed:2019rds,Henke:2019hve,Herderschee:2021dez,Henke:2021ity,Ren:2021ztg}.\nIt would be interesting to extend these techniques also to the elliptic case.\n\n\n\n\n\n\n\\acknowledgments\n\n\nWe thank Johannes Br\u00f6del, Simon Caron-Huot, Claude Duhr, Zhenjie Li, Robin Marzucca, Andrew McLeod, Cristian Vergu, Matthias Volk, Matt von Hippel and Stefan Weinzierl for interesting discussions as well as Andrew McLeod, Mark Spradlin and Stefan Weinzierl for comments on the manuscript.\nMW thanks the organizers of the conference ``Elliptics '21'', where part of this work was presented.\nThis work was supported by the research grant 00025445 from Villum Fonden and the ERC starting grant 757978.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe study of Interplanetary Shocks associated with major solar eruptions is very important not only from the theoretical point of view, but also because of potential impacts on human technologies. First because shocks are, as well as solar flares, optimal locations for the acceleration of Solar Energetic Particles (SEPs; i.e. electrons, protons and He ions with energies from a few KeV to some GeV) that constitute an important hazard for satellites and astronauts, and may affect the ionosphere around polar caps. Moreover, as the shocks reach the Earth, significant southward components of the interplanetary magnetic field associated with them can magnetically reconnect with the magnetosphere, thus disturbing the system and producing severe geomagnetic storms \\citep[see e.g. review by][]{schwenn2006}. Hence, understanding the origin, propagation and physical properties of interplanetary shocks is also crucial for future developments of our capabilities of forecasting possible Space Weather effects of solar activity. For these reasons, over the last decades huge efforts have been devoted in order to improve our knowledge of these phenomena and of the associated Coronal Mass Ejections (CMEs), by using different instrumentation taking remote sensing as well as in situ data. In particular, over the last few years, the most recent space based missions, such as the twin STEREO satellites, the Hinode and SDO observatories, provided significant new insights, thus allowing to investigate shocks from the early phases of their formation at the base of the corona out to their propagation into the interplanetary space.\n\nA clear signature of the formation and propagation of interplanetary shocks associated with CME expansion and\/or flare explosions is the detection of type-II radio bursts \\citep[see][for a review of the problem of type-II sources]{vrsnak2008}. Combination of radio data with images acquired at different wavelengths is able to provide unique new information on these phenomena. Recently, combined analysis of EUV images and radio dynamic spectra were used to demonstrate \\citep{cho2013,chen2014} that type-II bursts may be excited in the lower corona through interaction between CMEs and nearby dense structures such as streamers \\citep[see also][]{classenaurass2002, reiner2003, ma04}. A similar result was also obtained with the use of a new radio triangulation technique exploiting radio data acquired by different spacecraft \\citep{magdalenic2014}. Hence, type-II radio bursts are likely to be excited during the early propagation phase of the shocks (that is, at heliocentric distances $r < 1.5$~$R_\\odot$), around the expected location of the local minimum of $v_\\text{A}(r)$ profile \\citep{gopalswamy2012a, gopalswamy2013}. Thanks to the high cadence, good sensitivity and spatial resolution now available in EUV with SDO\/AIA, it has been shown \\citep{kouloumvakos2014} also that the sole analysis of EUV images can provide by itself an estimate of the density compression ratio $X$ (an important shock parameter given by the ratio between the downstream and the upstream plasma densities, $X = n_\\text{d} \/ n_\\text{u}$) and that this estimate is in agreement with the one derived from radio data in sheat regions. The above results clearly have important implications for the identification of SEP source regions.\n\nOver the last decade it also became clear that a significant number of information on interplanetary shocks can be derived from White Light (WL) coronagraphs data alone, as first shown by \\citet{vourlidas03}. Analysis of these data allowed to verify that shocks form when their propagation velocity $v_\\text{sh}$ (measured in a reference system at rest with the solar wind, moving at velocity $v_\\text{sw}$) is larger than the local Alfv\\'en velocity $v_\\text{A}$ ($|\\mathbf{v_\\text{sh}} - \\mathbf{v_\\text{sw}}| > v_\\text{A} = B\/ \\sqrt{4 \\pi \\rho}$). Hence, the lower is the velocity of the driver, the larger are the distances where shock front forms \\citep{eselevich2011}. Moreover, combination of EUV and WL data shows that the shock thickness $\\delta$ is of the same order as the proton mean free path $\\lambda_p$ only for heliocentric distances $r < 6$~$R_\\odot$\\, while higher up in the corona $\\delta << \\lambda_p$. Hence, during its propagation, the shock regime changes from collisional to collisionless \\citep{eselevich2012}. These information are crucial for our understanding of the physics at the base of the shock. Also, at larger heliocentric distances, the analysis of WL data provided by heliospheric imagers have demonstrated that the driver (CME) and the shock undergo different magnetic drag deceleration during their interplanetary expansion, with the shock propagating faster than the ejecta, thus leading to possible CME-shock decouplings \\citep[][]{hesszhang2014}. Statistically, the coupling has been found to be stronger for faster CMEs \\citep{mujiber2013}. Studies of interplanetary propagation of shocks have tremendous implications for Space Weather prediction capabilities as well.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{f1.pdf}\n\\caption{Top: sequence of SDO\/AIA 304 and SOHO\/LASCO-C2 images acquired on June 7, 2011 during the eruptive event analyzed here. The LASCO-C2 images are shown in inverted color scale (brighter features are darker and vice-versa) and after the application of a filter to enhance the visibility of CME structures (images created with JHelioviewer). Bottom: sequence of LASCO-C2 and -C3 images showing the CME propagation at higher altitudes; again the images are shown in inverted color scale and after the application of a filter to enhance the visibility of CME structures (images created with JHelioviewer).}\\label{fig:event}\n\\end{figure*}\n\nSignificant advances were also made from comparisons between observations and numerical simulations. At heliocentric distances $r > 2$~$R_\\odot$\\, coronal protons and electrons are no more coupled by Coulomb collisions. This leads to different temperatures for these two species, with slightly larger proton than electron temperatures (by a factor depending on the relevant altitude and coronal structure) as demonstrated by coronal UV spectra acquired by the UV coronagraph Spectrometer \\citep[UVCS; see reviews by][]{antonucci2006,kohl06}. Protons, however, being much heavier than electrons, have much smaller microscopic velocities (by a factor of 42.85). CME-driven shocks are thus supersonic only with respect to the proton thermal speed, implying that only protons are expected to be significantly heated by the transit of the shock. This was recently confirmed from both observations and simulations: in particular, \\citet{manchester2012, jin2013} demonstrate that the WL appearances of CME-driven shocks are better reproduced by 2-temperature (2T) MHD simulations with respect to 1-temperature (1T) simulations, where 2T plasma protons are heated up to $\\sim 90$~MK, and 2T shocks have larger Alfv\\'enic Mach numbers $M_\\text{A}$ (by a factor $\\sim 1.25$--1.4) with respect to the 1T plasma case. Very similar results were recently obtained by the combined analysis of UV and WL observations of a CME driven shock performed by \\citet{bem14}.\n\nThe latter work was the result of a sequence of previous researches performed on CME-driven shocks and based on the combined analyses of UV spectra acquired by UVCS and WL images acquired by the LASCO coronagraph. As first demonstrated by \\citet{bem10}, this unique combination allows to measure not only the plasma compression ratio $X$, but also the pre- and post-shock plasma temperatures. Moreover, once these informations are combined with the Rankine-Hugoniot equations written for the general case of oblique shocks, and by measuring geometrical (inclination) and kinematical (velocity) properties of the shock from WL data, it is even possible to determine both the pre- and post-shock magnetic and velocity field vectors projected on the plane of the sky. This technique allowed \\citet{bem11,bem13} to conclude that, for a few specific events, radio-loud (radio-quiet) CMEs are more likely associated with super- (sub-) critical shocks, and that only a small region around the shock center is super-critical in the early evolution phases, while higher up (i.e. later on) the whole shock becomes sub-critical. Moreover, the same technique applied to different points located along the same shock front allowed \\citet{bem14} to demonstrate that the transit of shock leads to a significant deflection of the magnetic field close to the shock nose, and a smaller deflection at the flanks, implying a draping of field lines around the expanding CME, in nice agreement with the post-shock magnetic field rotations obtained by \\citet{liu2011} with 3D MHD numerical simulations.\n\nIn this paper the above results are further extended: in particular we demonstrate here that, under some specific hypotheses, the analysis of WL coronagraphic data alone not only can provide the density compression ratios at different times and locations along the shock front, but also the $M_\\text{A}$ numbers and the pre-shock coronal magnetic fields, allowing us to derive a 2D map of magnetic field strength covering an heliocentric distance interval by $\\sim 10$~$R_\\odot$\\ and a latitude interval by $\\sim 110^\\circ$. Moreover, the combined analysis of WL and radio data allows us to derive the possible location of the source for the type-II radio burst. The paper is organized as follows: after a general description of the event being analyzed here (Section \\ref{sec:obs}), we describe the analysis of data (Section \\ref{sec:datanal}), focusing in particular on LASCO\/C2 and C3 WL coronagraphic images (Section \\ref{sec:wldata}) and WAVES\/RAD1-RAD2 radio dynamic spectra (Section \\ref{sec:radiodata}). Then, the obtained results are summarized and discussed (Section \\ref{sec:concl}).\n\n\n\\section{Observations} \\label{sec:obs}\n\nOn June 7th 2011, a GOES M2.6 class flare from AR 11226 (located in the southwest quadrant at 22$^\\circ$ S and 66$^\\circ$ W) occurred between 06:16 and 06:59 UT, peaking around 06:16 UT. This soft X-ray flare was associated with significant HXR emission and even $\\gamma-$ray emission lasting for about 2 hours \\citep{ackermann2014}. The impressive eruption associated with this flare has been extensively studied by many previous authors who focused on different physical phenomena related with the event. They focused on several aspects of this event, such as the early evolution of the released CME bubble and compression front \\citep{cheng2012}, the propagating EUV wave \\citep{li2012}, the magnetic reconnections driven by the CME expansion \\citep{vandriel2014}, the flare emission \\citep{inglisgilbert2013}, and the associated type-II radio burst \\citep{dorovskyy2013, dorovskyy2015}. Moreover, this spectacular eruption was followed by the ejection of huge radial columns of chromospheric plasma, reaching the field of view of LASCO and COR1 coronagraphs, and then falling back to the sun. Thus, other authors focused also on the dynamics and plasma properties of returning plasma blobs \\citep{innes2012, williams2013, carlyle2014, dolei2014}, as well as on the energy release from falling material impact on the sun \\citep{gilbert2013, reale2013, reale2014}.\n\nIn this work we study the evolution of the shock wave associated with this eruption as observed by white light coronagraphic images. As reported by \\citet{cheng2012}, immediately after the flare onset (around 06:26 UT) a circular plasma CME bubble was observed in the SDO\/AIA images expanding at $\\sim 960$~km~s$^{-1}$; in the early phases, due to the small standoff distance, the compression front and the front of the driver (i.e. the CME bubble) cannot be discerned. The two fronts started to separate only later on, when a deceleration of the CME bubble is observed; at the same time, a type II radio burst started (as well as a type-III burst), suggesting that the compression wave had just turned itself into a shock wave. Later on, the CME enters in the field of view of the SOHO\/LASCO-C2 coronagraph starting from the frame acquired at 06:49 UT (Figure 1, top row), and then enters in the field of view of the LASCO C3 coronagraph starting form the frame acquired at 07:11 UT (Figure 1, bottom row). The LASCO C2 frames clearly show the propagation of the shock wave associated with the event, as well as the CME front and the circular flux rope, while this latter part becomes hardly discernible in the LASCO C3 frames (see \\ref{fig:event}).\n\nIn what follows we describe how the sequence of white light images acquired by LASCO C2 and C3 has been analyzed to derive the pre-CME coronal density and the different physical parameters of the shock wave.\n\n\n\\section{Data analysis} \\label{sec:datanal}\n\n\\subsection{WL coronagraphic images} \\label{sec:wldata}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{f2.pdf}\n\\caption{Appearance of the white light corona as observed on June 4, 02:48 UT (left) before the acquisition of the pB image used for the coronal density determination, and on June 7, 06:04 UT (right) before the occurrence of the eruption.}\\label{fig:corona}\n\\end{figure*}\n\n\\subsubsection{Pre-CME coronal densities}\nFor the density calculation we use SOHO\/LASCO C2 polarized brightness (pB) images. It is well known that the K-corona brightness originates from Thomson scattering of photospheric light by free electrons in the solar corona \\citep[e.g.,][]{bil66}. Because the emission is optically thin, the observer sees a contribution from electrons located all along the line of sight. In addition to the K-corona, observations will contain a component due to scattering of photospheric light from interplanetary dust (the so-called F-corona). This component must be eliminated from the data to derive the coronal electron density; however, in the case of pB observations at small altitudes ($\\lesssim 5$~$R_\\odot$), the F corona can be assumed unpolarized and thus does not contribute to the pB \\citep[][]{hay01}.\n\nThe intensity of the scattered light depends on the number of scattering electrons and several geometric factors, as was first outlined by \\citet[][]{min30}. In the absence of F corona, the polarized brightness observed on the plane of the sky is given by the following equation:\n\\begin{equation}\\label{eq:vandehulst}\n\\text{pB}(\\varrho)=C\\int_\\varrho^\\infty n_e(r)\\left[A(r)-B(r)\\right]\\frac{\\varrho^2\\,dr}{r\\sqrt{r^2-\\varrho^2}},\n\\end{equation}\nwhere $C$ is a unit conversion factor, $n_e$ is the electron density, $A$ and $B$ are geometric factors \\citep[][]{vdh50,bil66}, $\\varrho$ is the projected heliocentric distance of the point (impact distance), and $r$ is the actual heliocentric distance from Sun center. The integration is performed along the line of sight through the considered point. \\citet[][]{vdh50} developed a well known method for estimating the electron density by the inversion of Equation (\\ref{eq:vandehulst}) under the assumptions that: (1) the observed polarized brightness along a single radial can be expressed in the polynomial form $\\text{pB}(r)=\\sum_k \\alpha_k r^{-k}$ and (2) that the coronal electron density is axisymmetric. We apply this method to the latest LASCO C2 pB image acquired before the June 7th CME, in order to determine the pre-CME electron density distribution in the corona.\n\nThe pB image considered here is obtained from the polarization sequence of observations recorded on June~4th 2011, starting at 02:54~UT, i.e. about three days before the occurrence of the June 7 CME. During this three-day time lag, three other much smaller CMEs occurred having a central propagation direction in the same latitudinal sector crossed by the June 7th CME ($70^\\circ$S--$40^\\circ$N), as reported in the SOHO\/LASCO CME catalog: on June~4th, at 06:48~UT and 22:05~UT, and on June~6th, at 07:30~UT. Nevertheless, despite these smaller scale events and coronal evolution, a direct comparison between the LASCO C2 white-light images acquired on June~4th at 02:48~UT and on June~7th immediately before the eruption at 06:04~UT shows that the overall density structure of the corona above the west limb of the Sun is quite similar even after more than three days (Figure \\ref{fig:corona}), hence the electron density estimated from the inversion of the June~4th pB data can be considered at least a first order approximation of the real pre-CME coronal density configuration.\n\n\\begin{figure*}\n\\centering\n\\subfigure[]{\\includegraphics[width=0.27\\textwidth]{f3a.pdf}}\n\\subfigure[]{\\includegraphics[width=0.72\\textwidth]{f3b.pdf}}\n\\caption{LASCO C2 polarized-brightness image of the solar corona above the west limb, acquired on June 4th 2011 at 02:57~UT (a) and the corresponding 2D electron density map derived from the inversion of the pB data (b).}\\label{fig:density}\n\\end{figure*}\n\nThe electron density radial profiles obtained at different latitudes from the pB image (Fig.~\\ref{fig:density}a) are combined into a 2D map in polar coordinates, shown in Fig.~\\ref{fig:density}b. The map shows the density distribution in the latitudinal region being crossed later on by the shock, for heliocentric distances ranging between 2 and 12~$R_\\odot$; electron densities at distances from the Sun larger than 6~$R_\\odot$\\ (the outer limit of the LASCO C2 field of view) are obtained through a power-law extrapolation of the density profiles assuming a radial dependence proportional to $r^{-2}$. The presence of the coronal streamer centered around $50^\\circ$S, that is persistent till June~7, is very clear as it is associated with a local electron density maximum. Notice that in general coronal features are much less evident in the pB image and in the density map (Figure \\ref{fig:density}) with respect to the regular LASCO frames (Figure \\ref{fig:corona}) because the latter are obtained after subtraction of a monthly minimum background average to enhance the visibility of fainter structures.\n\n\\subsubsection{Shock position and kinematics}\n\nWhite-light coronagraphic images can be used to identify the shock front location at different times and to distinguish between the shock-compressed plasma and the CME material, as extensively demonstrated by several works \\citep[e.g.,][]{vourlidas03,ont09,bem10,bem11}. The CME-driven shock front can be identified as a weak brightness increase located above the expanding CME front, that is generally interpreted as the visible signature of the downstream plasma compression and density enhancement caused by the transit of the shock; for this reason, the shock front becomes visible only when the intensity scale of WL images is adjusted to bring out the fainter structures.\n\nIn this work, we determine the location of the shock front in both LASCO C2 and C3 total brightness images using a common procedure that consists of three steps: (1) we compute excess-mass (or base-difference) images by subtracting from each calibrated LASCO frame an average pre-event image that is representative of the quiescent background corona \\citep[see][]{vou00,ont09}; (2) we apply a Normalizing Radial Graded Filter (NRGF), as described by Morgan et al. (2006), in order to reveal faint emission features at high heliocentric distances in the corona (this is particularly useful for the identification of the shock front in LASCO C3 images); (3) we measure the projected altitude of the shock by locating the intensity jump at the front in the radial direction. With this technique the location of the shock can be identified with an estimated uncertainty of $\\pm 3$ pixels on average and $\\pm 5$ pixels for LASCO C2 and C3 images, respectively. Larger uncertainties could be related with the applied procedure of background subtraction, in the possible locations where the pre-eruption corona significantly changed during the event.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f4.pdf}\n\\caption{Cartesian plot showing the locations of the shock front identified at different times in LASCO C2 and C3 white-light images.}\\label{fig:positions}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f5.pdf}\n\\caption{Base-difference LASCO C3 image showing the location of the shock front (solid white line) at 07:39~UT and a schematic representation of selected vectors normal to the shock surface (white arrows) and corresponding radial directions in the same points (red arrows).}\\label{fig:lasco_norm}\n\\end{figure}\n\nWe apply this procedure to seven consecutive images where we could identify signatures of the shock: two from LASCO C2, acquired at 06:47 and 07:01~UT, and five from LASCO C3, acquired at 07:09, 07:24, 07:39, 07:54, and 08:09~UT, respectively (see Figure \\ref{fig:event}). Later on, we were not able to locate the shock front with a significant accuracy in LASCO C3 images. The curves giving the position the shock fronts identified in the considered WL images are plotted in Figure~\\ref{fig:positions}. The shock appears to propagate almost symmetrically and to exhibit only a moderate latitudinal displacement, since the center of the shock (i.e., the highest point along the front) has a latitudinal location which is always in the range 21--25$^\\circ$S. We notice here that around a latitude of about 12$^\\circ$S the identified location of the shock surface shows a clear discontinuity, which is likely due to the Northward displacement of a the pre-event coronal streamer, leading to an overestimate (underestimate) of the shock projected altitude Northward (Southward) of the streamer itself.\n\nThese curves can be easily employed to derive, all along each shock front, the angle $\\theta_\\text{sh}$ between the normal to the shock front and the radial direction, as well as the latitudinal distribution of the average shock speed, $v_\\text{sh}$. These quantities are essential for the determination of the Alfv\\'enic Mach number and the upstream plasma velocity distribution, as discussed in the following section. As an example, Figure~\\ref{fig:lasco_norm} shows the relative orientation of vectors parallel with the radial direction and those normal to the shock surface at different positions along the front as we identified in the LASCO C3 image acquired at 07:39~UT. It is evident from this Figure that $\\theta_\\text{sh}$ angles are in general larger at the flanks of the shock, and smaller near the shock center (or ``nose''). This result confirms what we already found in recent works \\citep[see, e.g.,][]{bem14} and suggests that we may expect the prevalence of quasi-perpendicular shock conditions at the flanks and quasi-parallel shock conditions at the center of the shock.\n\n\\begin{figure*}\n\\centering\n\\vspace{-4cm}\n\\includegraphics[width=\\textwidth]{f6.pdf}\n\\caption{Compression ratios $X \\equiv \\rho_\\text{d}\/\\rho_\\text{u}$ as measured along the shock fronts identified in LASCO observations and reported in Fig.~\\ref{fig:positions}. Each profile is shown as a thick shaded area representing the uncertainty in the derived $X$ values.}\\label{fig:ratios}\n\\end{figure*}\n\nThe radial component of the average shock speed is obtained at each latitude simply as $v_\\text{r}=\\Delta\\varrho\/\\Delta t$, where $\\Delta\\varrho$ is the variation of the projected heliocentric distance of the shock measured in the radial direction between two consecutive shock curves. The true shock velocity can be then derived simply as $v_\\text{sh}=v_\\text{r}\\cdot\\cos\\theta_\\text{sh}$. Note that, as in \\citet[][]{bem14}, this corresponds to assume isotropic self-similar expansion of the front in the range of common latitudes between consecutive curves, but taking into account the correction for the latitudinal shock propagation. A 2D polar map of radial velocity distribution $v_\\text{r}$ in the region where the shock propagates is obtained by interpolating with polynomial fitting the heliocentric distance values at each latitude and altitude along the shock fronts, and is shown in Figure~\\ref{fig:results1} (top-left panel). The resulting radial shock speed is (as expected) larger at the center of the shock at all altitudes, then it decreases toward the shock flanks; at a heliocentric distance of $2.5$~$R_\\odot$\\ it reaches a value as high as $\\sim 1200$~km~s$^{-1}$ near the center and $\\sim 800-900$~km~s$^{-1}$ $\\sim 20^\\circ$ away from it. The shock also appears to decelerate during its propagation, since the velocity at higher altitudes is progressively smaller: for instance, at 12~$R_\\odot$\\ $v_\\text{sh}\\simeq 1000$~km~s$^{-1}$ at the shock center. This means that the shock is losing its energy as it expands; this is also supported by the results we obtain for the compression ratio and the Alfv\\'enic Mach number, as discussed in the following section.\n\n\\subsubsection{Compression ratio, Alfv\\'enic Mach number, and Alfv\\'en speed}\n\nThe shock compression ratio $X$, defined as the ratio between the downstream (i.e., post-shock) and the upstream (i.e., pre-shock) plasma densities, $X\\equiv \\rho_\\text{d}\/\\rho_\\text{u}$, is determined here as described in \\citet[][]{bem11}. For each pixel along an identified shock front, we measure the total white-light brightness of the compressed downstream plasma, tB$_\\text{d}$, from the corresponding LASCO C2 or C3 image, and, at the same locations in the corona, the upstream brightness tB$_\\text{u}$ from the last image acquired before the arrival of the shock. This provides us with the observed ratio $(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{obs}$.\n\nOn the other hand, the upstream total brightness tB$_\\text{u}(\\varrho)$ expected at a projected altitude $\\varrho$ in the corona can be evaluated through the line-of-sight integration of the upstream electron density profile, $n_e(r)$, multiplied by a geometrical factor $K$ that includes all the geometrical parameters for Thomson scattering:\n\\begin{equation}\n\\text{tB}_\\text{u}(\\varrho)=\\int_{\\varrho}^{\\infty}{K(r,\\varrho)\\cdot n_e(r)\\,dr},\n\\end{equation}\nwhere $r$ is the heliocentric distance of the scattering point along the line of sight. The expected downstream total brightness tB$_\\text{d}$ is similarly given by the sum of two integrals: one performed over the unshocked corona (with density $n_e$) and the other over a length $L$ across the shocked plasma with density $X\\cdot n_e$ ($X \\geq 1$):\n\\begin{eqnarray}\n&&\\text{tB}_\\text{d}(\\varrho)=\\int_{\\varrho}^{\\infty}{K(r,\\varrho)\\cdot n_e(r)\\,dr} +\\\\ \\nonumber \n&&\\int_{\\varrho}^{r_\\text{sh}}{K(r,\\varrho)\\cdot (X-1)\\cdot n_e(r)\\,dr},\n\\end{eqnarray}\nwhere $r_\\text{sh}=\\sqrt{\\varrho^2+L^2}$ and $X$ is precisely the unknown compression ratio. The shock depth $L$ is estimated as in \\citet[][]{bem10}, i.e., by assuming that the shock surface has the three-dimensional shape of an hemispherical shell with thickness equal to the 2D projected thickness $d$ of the white-light intensity jump across the shock, corrected for the shock motion during the LASCO C2 or C3 exposure time. For each frame we estimated an average value of the shock depth $L$, and applied the same value to the whole shock front. Given $L$ and by adopting the radial density profiles derived from the analysis of the LASCO C2 pB, the shock compression ratio $X$ can be inferred directly from the comparison between the observed and the expected total brightness ratios: $(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{obs}=(\\text{tB}_\\text{d}\/\\text{tB}_\\text{u})_\\text{exp}$.\n\nThe corresponding curves for the compression ratio $X$ measured along the shock fronts with different LASCO C2 and C3 frames are reported in Figure~\\ref{fig:ratios}. The uncertainties in $X$ values shown in this Figure are due to the uncertainty in the identification of the exact location of the shock in C2 and C3 images (see above). The compression ratio reaches the maximum value of $\\sim 2.1$ at 06:47~UT in a point that is very close to center of the shock front at that time located around a latitude of -20$^\\circ$S; this $X$ value is quite lower than the upper limit adiabatic compression of 4 expected for a monoatomic gas. In all cases, the latitudinal dependence is similar: $X$ has a maximum around the center of the shock front, progressively but not monotonically decreasing toward the flanks. As the shock expands, the $X$ values decrease on average all along the shock fronts: for instance, at 08:09~UT the maximum value is of $\\sim 1.5$; as already pointed out in the previous section, this indicates that the shock is dissipating its energy while propagating in the corona. These results are in agreement with those reported by \\citet[][]{bem11} in their analysis of a different CME-driven shock. We notice here that, as explained above, the $X$ values have been not derived after background subtraction, but from the ratio between the total brightnesses observed at the shock location and those observed at the same pixels in the frame acquired just before the arrival of the shock. This method allows to remove in the ratio any possible uncertainty due to the instrumental calibration; moreover, because the shock is the faster feature propagating outward, no significant changes occurred in the corona aligned along the LOS between the two frames other than the compression due to the shock.\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7a.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7b.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7c.pdf}}\n\\subfigure[]{\\includegraphics[width=0.49\\textwidth]{f7d.pdf}}\n\\caption{2D maps showing the distribution of the radial shock velocity $v_\\text{r}$ (a), the Alfv\\'enic Mach number $M_\\text{A}$ (b), the Alfv\\'en speed $v_\\text{A}$ (c), and as a reference the pre-shock coronal densities $n_e$ (d). The $M_\\text{A}$ and $v_\\text{A}$ values are derived by assuming a negligible solar wind speed, as described in the text. In each panel real measurements were obtained only in the region between the two dotted lines, while values shown out of these region have been extrapolated at higher and lower altitudes.}\\label{fig:results1}\n\\end{figure*}\n\nThe Alfv\\'enic Mach number is defined as the ratio between the upstream plasma velocity $v_\\text{u}$ (i.e., the velocity of the plasma flowing toward the shock surface in the reference frame at rest with the shock itself) and the Alfv\\'en speed $v_\\text{A}$, $M_\\text{A} \\equiv v_\\text{u}\/v_\\text{A}$. $M_\\text{A}$ can be estimated from the compression ratio $X$ and the angle $\\theta_\\text{sh}$ under two assumptions: (1) the plasma $\\beta \\ll 1$ ($\\beta$ is the ratio between the thermal and magnetic plasma pressures) and (2) the upstream magnetic field is radially directed, so that the angle between the shock normal and the magnetic field vector can be assumed to be equal to $\\theta_\\text{sh}$ on the plane of the sky. These are not strong assumptions, as discussed in \\citet[][]{bem11}, and can be considered fairly verified also in our case. Under these hypotheses, as we verified observationally in \\citet[][]{bem14} and theoretically in \\citet[][]{bacchini2015}, the Alfv\\'enic Mach number is well approximated in the general case of oblique shock by the following semi-empirical formula:\n\\begin{equation}\\label{eq:mach}\nM_{\\text{A}\\angle}=\\sqrt{M_{\\text{A}\\parallel}^2\\cos^2\\theta_\\text{sh}+M_{\\text{A}\\perp}^2\\sin^2\\theta_\\text{sh}},\n\\end{equation}\nwhere $M_{\\text{A}\\parallel}=\\sqrt{X}$ and $M_{\\text{A}\\perp}=\\sqrt{\\frac{1}{2}X(X+5)\/(4-X)}$ are the expected Mach numbers for parallel and perpendicular shocks, respectively, for a $\\beta \\ll 1$ plasma. The validity of Eq.~(\\ref{eq:mach}) has been confirmed by the analysis of \\citet[][]{bem14} which takes advantage of both white-light and ultraviolet data from the \\emph{Ultra-Violet Coronagraph Spectrometer} (UVCS) on board SOHO (see discussion therein) and has been recently tested with MHD numerical simulations by \\citet[][]{bacchini2015}. This equation allowed us to derive, from different values of $X$ and $\\theta_\\text{sh}$ parameters, 2D polar maps of $M_{\\text{A}\\angle}$ values, as shown in Figure~\\ref{fig:results1} (top right panel). This map clearly shows that in the early phases the shock was super-Alfv\\'enic at all latitudes (with larger $M_\\text{A}$ values at the shock nose), while later on (i.e. higher up) keeps super-Alfv\\'enic numbers only at the nose.\n\nThe Alfv\\'en speed can be derived, in turn, from $M_\\text{A}$ values once the upstream plasma velocity is known or estimated. The upstream velocity is given by $v_\\text{u}=|\\mathbf{v_{sw}}-\\mathbf{v_{sh}}|$, where $\\mathbf{v_{sw}}$ is the outflow solar wind speed, assumed to be radial, and $\\mathbf{v_{sh}}$ is the shock speed. In our case, we have no direct measurements of the wind flows in the corona, hence we must adopt a model for the solar wind expansion in order to infer the Alfv\\'en speed from the Alfv\\'enic Mach number. To this end, a first-order approximation can be obtained by assuming $\\mathbf{v_{sw}}=\\mathbf{0}$ in the previous equation, i.e., by neglecting the solar wind at all. This is not a realistic assumption, but it is rather reasonable, considering that at low altitudes in the corona ($\\lesssim 5$~$R_\\odot$) and in the early phase of propagation, the shock speed may be up to one order of magnitude larger than typical wind velocities measured outside coronal holes \\citep[$\\approx 100$--300~km~s$^{-1}$; see, e.g.,][]{sus08}. Under this hypothesis, the estimated Alfv\\'en speed can be considered as an upper limit to the real values. Possible consequences of this assumption will be discussed in the last Section.\n\n\\begin{figure*}\n\\centering\n\\vspace{-11.5cm}\n\\includegraphics[width=0.9\\textwidth]{f8.pdf}\n\\vspace{-2cm}\n\\caption{Lower panel: Dynamic spectrum of the Wind\/WAVES radio data in the frequency range between 20 KHz and 13.8 MHz from 6 to 14 UT on 2011 June 7, showing, showing the decametric to kilometric type II radio emissions associated with the CME. The upper panel at the left shows details of the radio emission associated with the emission excited earlier at the southern flank of the shock. The curves on this plot are also explained in the text.}\\label{fig:radio}\n\\end{figure*}\n\n2D polar maps of the Alfv\\'en speed are shown in Figure~\\ref{fig:results1} (bottom-left panels); these maps have been obtained again with polynomial (third-order) interpolation of the Alfv\\'en speeds measured at different locations (i.e. latitudes and altitudes) of the shock front at different times (Figure~\\ref{fig:positions}). Results plotted in Figure ~\\ref{fig:results1} clearly show that the Alfv\\'en speed has not only radial, but also significant latitudinal modulations. The Alfv\\'en speed reaches the highest value ($\\sim 1000$~km~s$^{-1}$) at the lowest altitudes in the equatorial belt. The latitudinal dependence is rather complex, with an alternation of local minima and maxima ranging between $\\sim 600$ and $\\sim 1000$~km~s$^{-1}$. At increasing altitudes, $v_\\text{A}$ generally decreases, with values that never exceed 800~km~s$^{-1}$ at 12~$R_\\odot$. Interestingly, the regions characterized by the slowest decrease in electron density (around $\\sim 50^\\circ$S and around $\\sim 10^\\circ$N; see Fig.~\\ref{fig:density}) are also those where the Alfv\\'en speed decreases more steeply, reaching values below $\\sim 500$~km~s$^{-1}$ already at 5~$R_\\odot$. As a consequence, in the early propagation phase (i.e., at low altitudes) the shock is significantly super-Alfv\\'enic not only at the nose but also in several regions distributed in the flanks of the shock surface. These high-density and high-Mach number regions are very probable candidates as sources of particle acceleration and type-II radio bursts; we discuss in the next section possible correlations with the sources of radio emission identified from radio dynamic spectra, while the determination of the magnetic field strength is discussed in the last Section.\n\n\\subsection{Radio dynamic specrum} \\label{sec:radiodata}\n\nAs it is well known, shock waves are able to accelerate electron beams to suprathermal energies, which in turn can produce Langmuir waves that are converted by means of nonlinear wave-wave interactions into electromagnetic waves near the fundamental and\/or harmonic of the local electron plasma frequency $f_{pe}$. Since the coronal density $n_e$ decreases with increasing heliocentric distance and $f_{pe} \\propto {n_e^{1\/2}}$, the expanding shock surface produces type-II radio emissions at decreasing frequencies as it propagates through space and the measured frequency drift rate at a given time is directly related to the shock speed. The observed frequency drift rate provides therefore information on the shock dynamics through the corona, while its onset depends on the local magnetosonic speed.\n\nThe dynamic spectrum in the lower panel of Figure \\ref{fig:radio} shows the intensity of the radio data from 06:00 to 14:00~UT on 2011 June 7 in the frequency range between 20~KHz and 13.8~MHz measured by the RAD1 and RAD2 radio receivers of the WAVES experiment on the Wind spacecraft. A very intense complex type-III-like radio emissions was observed beginning at 6:24~UT. This fast-drifting radio emission can be interpreted as the first radio signature indicating the lift-off of the CME on the Sun \\citep[e.g.,][]{reinerkaiser1999} and is probably originated by the reconfiguration of the magnetic field in the lower corona that allows the energetic electrons produced by the flare to escape into the interplanetary medium \\citep{reiner2000}. Two slowly-drifting episodes of strong type-II emission were also observed in the decametric range around 07:00 UT (clearly visible in the expanded upper left panel of Figure \\ref{fig:radio}) and after 09:00 UT, abruptly intensifying between 13:00 and 14:00~UT (lower panel of Figure \\ref{fig:radio}). We interpret these bands of emissions, as usually assumed when only one band is visible, as second harmonics. The origin of the second harmonic emission in type-II bursts is well understood as a result of coalescence of two plasma waves into a transverse one at twice the plasma frequency. Less intense, additional slow-drifting, type-II-like radio emissions at different times and frequencies are also visible, probably originating from different portions of the super-Alfv\\'enically expanding shock surface.\n\n\\begin{figure*}\n\\centering\n\\vspace{-2.5cm}\n\\includegraphics[width=0.9\\textwidth]{f9.pdf}\n\\caption{Comparison between the 2D maps of coronal magnetic field strengths derived by assuming negligible wind speed (top left, upper limit for the field values) and fast wind speed at all latitudes (bottom left, lower limit for the field values). The right panel shows a comparison between the latitudinal average of magnetic fields obtained under the assumption of negligible wind speed (blue line) and assuming fast wind speed at all latitudes (red line).}\\label{fig:fastslowwind}\n\\end{figure*}\n\nIn order to model the observed complex type-II radio emissions displayed in Figure \\ref{fig:radio}, we need to know the coronal electron density profile at the time of the CME event. In fact, the density profile allows to convert the height measurements related to the shock surface dynamics to corresponding values of the coronal density as the frequencies $f$ are simply obtained as $f \\approx f_{pe} \\approx 9 \\sqrt{n_e[\\rm cm^{-3} ]}$~KHz. Instead of relying on a generic coronal electron density model, as usually done in the literature, we used the coronal electron density at different heliocentric distances and latitudes provided by the LASCO pB measurements discussed in the previous section. These density estimates, obtained for heliocentric distances greater than about 2~$R_\\odot$, correspond to radio frequencies below about 14~MHz, i.e., the range of radio emissions observed in the Wind\/WAVES dynamic spectrum. By assuming, as usual, second harmonic type-II emission and using the coronal density distribution inferred from the available LASCO pB observations to relate the type-II frequencies to their heliocentric heights, we identified, knowing the shock's surface height from the previous analysis, a set of synthetic type-II profiles that were superimposed (as dashed lines in Figure \\ref{fig:radio}) to the radio dynamic spectrum for comparison with the actual type-II emissions. This comparison allowed to characterize all observed type-II features and, in particular, two distinct regions (assuming radial propagation) along the shock's surface where the brightest radio emissions were most likely generated. An accurate estimate of the model radio profiles could only be obtained considering the coronal parameters outward from the flare longitude of 66$^\\circ$ W and not from 90$^\\circ$ W (plane of the sky). Unfortunately, at the time when the CME occurred, the STEREO-A and -B spacecraft were located at 94.9$^\\circ$ and 93.0$^\\circ$ from the Sun-Earth line, respectively. Hence, coronagraphic images acquired by the STEREO coronagraphs would not provide any useful information about the corona lying on the meridional plane at 66$^\\circ$ W. Said that, although we assume that no significant temporal and longitudinal variations are present between the density profile we inferred on the plane of the sky and the density really met by the shock, this assumption is undoubtedly much more realistic with respect to the one that involves the adoption of a generic power-law density profile, as usually done in the literature for this kind of studies \\citep[see e.g.][]{ra99, pojo2006, liu2009, kong2015, dorovskyy2015}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f10.pdf}\n\\vspace*{-6cm}\n\\caption{Comparison between radial magnetic field profiles derived in this work at different latitudes (solid black lines), other magnetic field radial profiles provided in the literature (in particular: \\citet{dulkmclean1978} - solid blue line, \\citet{patzold1987} - dash-dotted dark green line, \\citet{vrsnak2004} - dash-dotted orange line, \\citet{gopalyashi2011} - dashed green line, and \\citet{mancusogarzelli2013a} - solid red line) together with a compilation of other measurements (in particular: \\citet{sakurai1994} - blue boxes, \\citet{spangler2005} - red boxes, \\citet{ingleby2007} - green boxes, \\citet{feng2011} - orange boxes, \\citet{you2012} - cyan boxes, and \\citet{bem10} - brown boxes).}\\label{fig:bmag_literature}\n\\end{figure}\n\nWith the above caveat in mind, we show that the two strong type-II bursts in this event are probably generated by two different portions of the shock (see upper right panel of Figure \\ref{fig:radio}), one driven near the CME front and the other one at the southern flank region of the CME. We point out that the angular ranges specified in Figure \\ref{fig:radio} are not intended to designate the accuracy of our results, but that they are simply meant to illustrate the angular location of the models that better fit the observed type II features. This result supports the scenario of type-II shock generation typically\narising at the CME flank due to interaction with a nearby streamer \\citep[e.g.][]{ma04,ch08}. In this case, the type-II-emitting shock front may be quasi-perpendicular and thus apt to accelerate electrons by the shock drift acceleration mechanism \\citep{ho83}.\n\n\n\\section{Discussion and Conclusions} \\label{sec:concl}\n\nThe actual limitations in our understanding of many physical phenomena occuring in the solar corona is due in first place to our limited knowledge of the coronal magnetic fields. Knowledge of its strength and orientation is primarily based on extrapolations from observations of magnetic fields in the photosphere, where the magnetic field is strong and the Zeeman effect produces a detectable splitting of atomic levels and a subsequent polarization of the emitted light. Nevertheless, extrapolations from photospheric fields are model-dependent, static (no eruptive events) and fail to reproduce accurately complex coronal topologies. For these reasons, many different techniques have been developed to measure magnetic fields in the extended corona using radio observations and taking advantage of Faraday rotation \\citep[e.g.,][]{maspan1999, mancusogarzelli2013a, mancusogarzelli2013b} and circular polarization in radio bursts \\citep[e.g.,][]{hariharan2014}, or in the lower corona with EUV images using coronal seismology \\citep[e.g.,][]{west2011} and field extrapolations bounded to 3D reconstructions \\citep[e.g.,][]{aschwanden2014}. The recent development of spectro-polarimetric measurements of magnetic field strength and orientation is now providing very promising results \\citep[e.g.,]{tomczyk2007, dove2011}, even if (due to the required polarimetric sensitivities) these techniques can be applied only in the lower corona ($h < 0.4$~$R_\\odot$).\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\includegraphics[width=0.7\\textwidth]{f11.pdf}\n\\caption{Comparison between the pre-shock coronal white light structures observed by LASCO C2 coronagraph (left) and the magnetic field strengths derived in this work in the LASCO C2 field of view (right). The dashed lines show the location where latitudinal profiles of the WL intensity and field strength have been extracted to be plotted in Figure \\ref{fig:intensitycut}. }\\label{fig:nicefig}\n\\end{figure*}\n\nRecently, an interesting technique to measure coronal fields with CME-driven shocks was proposed by \\citet{gopalyashi2011}. This technique takes advantage of the relationship derived by \\citet{russel2002} between the standoff distance of an interplanetary shock and the radius of curvature of its driver, and is applied to derive the strength of coronal fields just above the shock nose during its propagation. This technique has been applied to images obtained from white light coronagraphic observations and, recently, to CME-driven shocks observed with EUV disk imagers \\citep{gopalswamy2012b} and white light heliospheric images \\citep{poomvises2012} allowing for the first time the derivation of magnetic field strengths up to an heliocentric distance of $\\sim 200$~$R_\\odot$. Notwithstanding the above, this technique has some limitations, in particular: 1) it can be only applied to shocks driven by CMEs, and 2) it is able to provide magnetic field measurements only along the radial located at the position of the shock nose.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{f12.pdf}\n\\caption{Comparison between the normalized pre-shock coronal white light structures observed by LASCO C2 coronagraph (dashed line) and the magnetic field strengths (solid line) at the constant altitude of 2.75 R$_\\odot$.}\\label{fig:intensitycut}\n\\end{figure}\n\nOn the other hand, the technique we developed here and in our previous works is able to provide measurements of the pre-shock coronal magnetic field strengths from white light observations of shock waves over all altitudes and latitudes crossed by the shock, independently of any hypothesis on the nature of the shock driver. In fact, once a 2D map for the Alfv\\'en speed and for the electron density $n_e$ are derived, the determination of the 2D coronal magnetic field strength is straightforward and is given by $B = v_\\text{A} \\sqrt{4 \\pi n_e m_p}$. The resulting 2D map of the magnetic field strength is shown in Figure \\ref{fig:fastslowwind} (top left panel) under the assumption that the solar wind speed is negligible with respect to the shock speed. Nevertheless, because the shock speed is decreasing with altitude ($v_\\text{sh}\\simeq 1200$~km~s$^{-1}$ at 2.5~$R_\\odot$\\ and $v_\\text{sh}\\simeq 1000$~km~s$^{-1}$ at 12~$R_\\odot$\\ as we measured at the shock center), while the wind speed is increasing, higher up in the corona the field will be more and more overestimated, leading to larger uncertainties. In order to quantify these uncertainties, lower limit estimates for the Alfv\\'en speed, and thus for the magnetic field, have been derived by assuming that the whole corona is pervaded at all latitudes by fast solar wind; in particular, here we assumed the fast solar wind radial profile provided by \\citet{hu1997}. The resulting 2D map for the lower limit estimate of the magnetic field strength is shown in Figure \\ref{fig:fastslowwind} (bottom left panel). Comparison between the two maps clearly shows that no significant differences are present in the lower corona, while larger differences may exist higher up. In particular, by averaging all the magnetic field radial profiles obtained at different latitudes, we conclude that the maximum difference between the upper and the lower limit estimates is on the order of a factor $\\sim 2.7$ at 12~$R_\\odot$, and smaller factors at lower altitudes (see Figure \\ref{fig:fastslowwind}, right panel).\n\nThe magnetic field values we derived here are in very good agreement with previous measurements provided in the literature at different altitudes and latitudes and obtained with many different techniques, as shown in Figure \\ref{fig:bmag_literature}. Hence, not only the radial variation of the field strength is comparable to other estimates obtained with completely different techniques, but the latitudinal modulation we derived in this work is reliable as well. We remind that the technique applied in this work for the determination of field strengths was only based on the analysis of white light coronagraphic images, which have been analyzed to derive 2D maps (projected on the plane of the sky) of the pre-shock coronal densities, shock compression ratios, shock velocities and inclination of the shock surface with respect to the radial. Then, some assumptions were needed in order to derive the magnetic field strengths: first, we assumed that above the lower boundary of the LASCO C2 occulter ($\\sim 2$~$R_\\odot$) the coronal field is radial, so that the shock inclination with respect to the radial also provides its inclination with respect to the upstream magnetic field. This is not a strong assumption, because it is well known that coronal structures (outlining the magnetic field orientation) are nearly radial above heliocentric distances of $\\sim 2$~$R_\\odot$. Second, we assumed an empirical formula for the determination of the Alfv\\'enic Mach number for the general case of an oblique shock starting from the measured shock compression ratios and shock inclination angles. The validity of this formula has been verified in a previous work \\citep{bem14} where the Alfv\\'enic Mach number was derived independently also form the analysis of white light and UV data; the verification of the same formula with MHD numerical simulations has been also recently provided by another work \\citep{bacchini2015}. Third, in order to convert the derived Alfv\\'enic Mach numbers in estimates for the Alfv\\'en speed, we assumed that the solar wind speed ahead of the shock is negligible with respect to the shock speed; as discussed above, this leads to an overestimate of the magnetic field by a factor no more than $\\sim 2.7$ at 12~$R_\\odot$, decreasing with altitude. For comparison with the white light coronal structures, the magnetic field values derived in this work are shown again in Figure \\ref{fig:nicefig}, plotted in the field of view of the LASCO C2 coronagraph (right panel), together with the original pre-CME coronal white light intensity (left panel). We also notice that the latitudinal distribution of coronal field strength is, in first approximation, anti-correlated with the white light intensity. This result is also better shown in Figure \\ref{fig:intensitycut} providing the latitudinal distribution of the normalized WL intensity and the magnetic field strength at a constant altitude of 2.75 R$_\\odot$. The observed anti-correlation is in nice agreement with what we could expect around the vertical axis of each coronal streamer, where the neutral current sheet corresponds to a region of minimum magnetic field strength.\n\n\\begin{figure*}\n\\centering\n\\vspace{-3cm}\n\\includegraphics[width=0.85\\textwidth]{f13.pdf}\n\\vspace{-0.5cm}\n\\caption{Left: LASCO C2 base difference image acquired on June 7, 2011 at 07:01 UT and with the contrast of faint features enhanced using the filter provided by the JHelioviewer software. The overplot shows the location of the shock (solid white line), the center of the CME flux rope (plus symbol), and the CME propagation direction (dashed black line). Right: same frame shown in the left panel, where the overplot provides again the location of the shock (solid white line), the center of the CME flux rope (plus symbol), and the location of the shock driver (black dotted line) as derived by assuming that the relationship between the Mach number at the shock nose and the $\\Delta\/R$ ratio holds also at different latitudes away from the shock nose (see text).}\\label{fig:shock_driver}\n\\end{figure*}\n\nIn order to further support the correctness of our measurements of coronal magnetic fields, we also applied the same technique proposed by \\citet{gopalyashi2011} and based on the measurement of the shock standoff distance. In order to perform the comparison between the two techniques, we selected the LASCO C2 frame where the circular shape of the CME flux rope is better visible, shown in Figure~\\ref{fig:shock_driver}. For this frame we determined the position of the center of the flux rope (plus symbol in the left plot) and (looking at previous and subsequent frames) the CME propagation direction (dashed line in the left plot). This provides us with the identification of the shock nose, as well as a measurement of the sum between the shock standoff distance $\\Delta$ and the radius $R$ of the flux rope, which turns out to be $\\Delta + R = 1.48$ R$_\\odot$. We thus used the value of the Mach number derived as decribed above at the shock nose ($M_A = 1.50$) and derived the expected $\\Delta \/ R$ ratio, which turns out to be \\citep[see][]{gopalyashi2011}\n\\begin{equation}\n\\frac{\\Delta}{R} = K \\frac{\\left( \\gamma -1 \\right)\\, M_A^2 +2}{\\left( \\gamma +1 \\right)\\, M_A^2} \\simeq 0.45,\n\\end{equation}\nwhere $K = 0.78$ for a circular shape of the shock driver, and $\\gamma = 5\/3$. With the above numbers it turns out that $\\Delta = 0.46$ R$_\\odot$ and $R = 1.02$ R$_\\odot$. The corresponding circumference (plotted in the left panel of Figure~\\ref{fig:shock_driver}) shows a quite nice agreement with the location of the CME flux rope, thus demonstrating that our results are in good agreement with those that could be derived for the same event with the technique described by \\citet{gopalyashi2011}. Moreover, since in this work we derived measurements of the shock Mach number $M_A$ not only at the shock nose, but also at different latitudes, it is interesting to test what happens by assuming that the above relationship relating $M_A$ and the $\\Delta \/ R$ ratio holds also away from the shock nose. In particular, the right plot of Figure~\\ref{fig:shock_driver} shows the locations of the shock driver (black dotted line) as inferred by assuming different values of $M_A$ away from the shock nose along each radial starting from the same position of the center of the flux rope (plus symbol). The resulting curve shows a surprisingly nice agreement with some white light features visible between the CME flux rope and shock. This may suggest that at this time a decoupling between the flux rope and the shock is already occurring away from the shock nose, or alternatively that the side parts of shock are driven at some latitudes by the expansion of other loop-like plasma features surrounding the CME flux rope and embedded within the same CME.\n\nThe analysis performed here provides not only a new technique to derive coronal field strengths with unprecedent radial and latitudinal extension, but also very important insights into the physical relation between the type-II emitting regions and the shock front. In fact, the difference between the 2D maps we derived for the shock and the Alv\\'en speed clearly show that in the early phases (2--4~$R_\\odot$) the whole shock surface is super-Alfv\\'enic, while later on (i.e. higher up) becomes super-Alfvenic only at the nose. For a better understanding of the acceleration regions of SEP, this result has also to be considered together with our previous finding that in the early propagation phases shocks are super-critical only at the nose and becomes sub-critical later on \\citep[e.g.][]{bem11}. At the same time, we demonstrate here with analysis of radio dynamic spectra that the emission near the front was generated later than the one produced by the flanks, in agreement with the conclusion we derived from the analysis of white light data. This suggests that the acceleration of SEP leading to gradual events could also involve at different times coronal regions located not only at different altitudes, but also at different latitudes and\/or longitudes along the shock front, as recently simulated for instance by \\citet{rodriguez2014}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum error correcting codes have been introduced as an alternative to classical\ncodes for use in quantum communication channels.\nSince the landmark papers~\\cite{shor} and~\\cite{steane96}, this field of research has grown rapidly.\nClassical codes have been used to construct good quantum codes~\\cite{calderbank96}.\nRecently, Lisonek and Singh~\\cite{singh} gave a variant of Construction X that\nproduces binary stabilizer quantum codes from arbitrary linear codes.\nIn their construction, the requirement on the\nduality of the linear codes was relaxed.\nIn this paper, we extend their work on construction X to obtain quantum error-correcting codes over finite fields of order $p^2$\nwhere $p$ is a prime number.\nWe apply our results to the dual of Hermitian repeated root cyclic codes to generate new quantum\nerror-correcting codes.\n\nThe remainder of the paper is organized as follows.\nIn Section 2, we present our main result on the extension of the quantum construction X.\nSection 3 characterizes the generator polynomial of the Hermitian dual of a repeated root cyclic code.\nWe also give the structure of cyclic codes of length $3p^s$ over\n$\\ensuremath{\\mathbb{F}_{p^2}}$ as well as the structure of the dual codes.\nOur interest in this class of codes comes from the importance of relaxing the\ncondition $(n,p)=1$, which allows us to consider codes other than the simple root codes.\n\\section{Extending Construction X for $\\f{p}$}\n\nLet $\\f{p}$ denote the finite field with $p$ elements and\n$\\fzero{p} = \\f{p} \\backslash \\{ 0\\}$.\nFor $x \\in \\f{p^2}$ we denote the \\emph{conjugate} of $x$ by $\\conj{x} = x^p$.\nLet $\\hip{x}{y} = \\sum^n_{i=1} x_i \\conj{y_i} $ be the Hermitian inner product.\nThen the \\emph{norm} of $x$ is defined as $\\norm{x} = \\hip{x}{x} = \\sum_{i=1}^n x^{p+1}$,\nand the \\emph{trace} of $x$ as $\\trace(x) = x + \\conj{x}$.\nBoth the trace and norm are mappings from $\\ensuremath{\\mathbb{F}_{p^2}}$ to $\\f{p}$.\n\nThe following lemmas will be used later.\n\\begin{lemma} \\label{l:existenceZ}\nLet $S$ be a subspace of $\\f{p^2}^n$ such that there exist $x,y$ with $\\hip{x}{y} \\neq 0$.\nThen for all $k \\in \\f{p}$, there exists $z\\in S$ with $\\norm{z} = k$.\n\\end{lemma}\n\\begin{proof}\nThis is a non-constructive proof of the existence of the required element $z$.\nWith the assumption on $x$ and $y$, let $g(c) = \\norm{cx+y} =\n(cx+y)^{p+1}$ be a polynomial of degree $p+1$ in $c$.\nWe claim that as $c$ ranges over the elements of $\\ensuremath{\\mathbb{F}_{p^2}}$, the rhs\nwill range over all elements of $\\f{p}$.\n\nAssume now that there exists some $k\\in \\ensuremath{\\mathbb{F}_{p^2}}$ such that $\\forall c\\in \\ensuremath{\\mathbb{F}_{p^2}}, g(c) \\neq k$.\nFor each $i \\in \\f{p} \\backslash k$, let $S_{i} = \\{c \\in \\ensuremath{\\mathbb{F}_{p^2}};\\, g(c)=i \\}$.\nSince the polynomial $g$ has degree $p+1$, $g$ can have at most $p+1$ roots in any field.\nThen $|S_i| \\leq p+1$, as the polynomial $g(c)-i$ can have at\nmost $p+1$ roots, and the $S_i$ partition the set $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThen $|\\ensuremath{\\mathbb{F}_{p^2}}| = p^2 \\leq \\sum_{i \\in \\f{p} \\backslash {k}} |S_i| \\leq (p+1)(p-1) = p^2-1$,\nwhich is a contradiction.\nHence the result follows.\n\\end{proof}\n\n\\begin{lemma}\\label{l:existenceB}\nLet $D$ be a subspace of $\\f{p^2}^n$ and assume that $M$ is a basis\nfor $D \\ensuremath{\\cap} D^\\ensuremath{{\\perp_h}}$. Then there exists an orthonormal set $B$\nsuch that $M \\ensuremath{\\cup} B$ is a basis for $D$.\n\\end{lemma}\n\\begin{proof}\nThe proof given here is a generalization of the proof for the\nanalogous case presented in \\citep[Theorem~2]{singh}.\nLet $W$ be a subspace of $\\f{p^2}^n$ such that\n\\begin{equation}\n\\label{e:3.1}\nD = (D \\ensuremath{\\cap} D^\\ensuremath{{\\perp_h}} ) \\oplus W,\n\\end{equation}\nand let $l = \\dim(W)$.\nFor each $0 \\leq i \\leq l$, we can construct an\northonormal set $S_i$ that is a basis for an $i$-dimensional\nsubspace $T_i$ of $W$ such that\n\\begin{equation}\n\\label{e:3.2Ti}\nW = T_i \\oplus (T_i^\\ensuremath{{\\perp_h}}\\ensuremath{\\cap} W).\n\\end{equation}\nThe process is iterative.\nDefine $S_0 := \\phi$ and suppose that for some $0 \\leq i < l$, the set $S_i$ is an orthonormal basis for $T_i$\nsuch that $dim(T_i) = i$ and (\\ref{e:3.2Ti}) holds.\nLet $x$ be a non-zero vector in $T^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$.\nThen there exists $y \\in T^\\ensuremath{{\\perp_h}}\\ensuremath{\\cap} W$ such that $\\hip{x}{y} \\neq 0$.\nIf no such $y$ exists, then $x\\in D^\\ensuremath{{\\perp_h}}$, which would\ncontradict (\\ref{e:3.1}) because the intersection of $D$ and $D^\\ensuremath{{\\perp_h}}$ is $\\{0\\}$.\nHence by Lemma \\ref{l:existenceZ}, there must exist a $z \\in T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$ such that $\\norm{z} = 1$.\nSet $S_{i+1}=S_i\\ensuremath{\\cup} \\{z\\}$.\nClearly all the elements in $S_{i+1}$ are orthogonal to each other.\nIn addition, $\\norm{s} = 1$ for all $s\\in S_{i+1}$.\n\nLet $T_{i+1}$ be the subspace spanned by $S_{i+1}$.\nAs $z \\not \\in T_{i}$ we have that $\\dim(T_{i+1}) = i+1$.\nTo show that\n\\begin{equation}\n\\label{3.1Ti+1} W = T_{i+1} \\oplus (T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W),\n\\end{equation}\nwe must first show that $T_{i+1}\\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W = {0}$.\nLet $v \\in T_{i+1} \\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$.\nAs $v \\in T_{i+1}$, we have $v = u + cz$ where $u \\in T_i$ and $c \\in \\f{p^2}$.\nSince $v \\in T_{i+1}^\\ensuremath{{\\perp_h}}$, we have for each $w \\in T_i$\nand each $d \\in \\f{p^2}$ that\n\\begin{equation*}\n0 = \\hip{u + cz}{w + dz} = \\hip{u}{w} + \\conj{d}\\hip{u}{z} +\nc\\hip{z}{w} + c\\conj{d}\\norm{z} = \\hip{u}{w} + c\\conj{d}.\n\\end{equation*}\nWe must have $c = 0$ or else $\\hip{u}{w} + cd$ would not remain\nconstant as $d$ runs over the elements of $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThus $\\hip{u}{w}=0$ for all $w \\in T_{i}$, and hence $u \\in T_{i}^\\ensuremath{{\\perp_h}}$. As $u \\in T_i$ and $T_i\n\\ensuremath{\\cap} T_{i}^\\ensuremath{{\\perp_h}} = {0}$, we obtain that $u=0$.\nHence $v$ is also $0$ and $T_{i+1}\\ensuremath{\\cap} T_{i+1}^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W = {0}$.\n\nNext we show that $W = T_{i+1} + (T_{i+1} \\ensuremath{\\cap} W)$.\nLet $w \\in W$.\nBy assumption $W = T_i + (T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W)$, so\nthere exist vectors $x \\in T_i$ and $y \\in T_i^\\ensuremath{{\\perp_h}} \\ensuremath{\\cap} W$\nsuch that $w = x + y$.\nNow it is shown that $W=T_{i+1}+(T_{i+1}^{\\ensuremath{{\\perp_h}}}\\ensuremath{\\cap} W)$.\nBy assumption $W=T_{i}+(T_{i}^{\\ensuremath{{\\perp_h}}}\\bigcap W)$, so there\nexist vectors $x\\in T_{i}$ and $y\\in T_{i}\\ensuremath{\\cap} W$.\nClearly $x\\in T_{i+1}$ and for any $u+dz\\in T_{i+1}$ (where $u\\in\nT_{i}$ and $d\\in\\ensuremath{\\mathbb{F}_{p^2}}$), we have\n\\begin{eqnarray}\n\\hip{y-\\hip{y}{z}z}{u+dz} &=& \\hip{y}{u}+\\conj{d}\\hip{y}{z}-\\hip{y}{z}\\hip{z}{u}-\\conj{d}\\hip{y}{z}\\norm{z} \\nonumber \\\\\n&=&\\conj{d}\\hip{y}{z}-\\conj{d}\\hip{y}{z} \\nonumber \\\\\n&=&0.\n\\end{eqnarray}\nThus $y\\in T_{i+1}\\ensuremath{\\cap} W$, and hence $W=T_{i+1}+(T_{i+1}\\ensuremath{\\cap} W)$.\nThis completes the proof that (\\ref{e:3.2Ti}) implies\n(\\ref{3.1Ti+1}) assuming that the vector $z$ is chosen as described above.\n\\end{proof}\n\n\\begin{theorem}\n\\label{th:main}\nFor an $[n,k]_{p^2}$ linear code $C$, let $e = n-k-\\dim(C \\cap C^\\ensuremath{{\\perp_h}})$.\nThen there exists a quantum code with parameters\n$\\qecc{n+e}{2k-n}{d}{p}$ with $d \\geq \\min(\\wt(C), \\wt(C + C^\\ensuremath{{\\perp_h}})+1)$.\n\\end{theorem}\n\\begin{proof}\nWe start with the observation that the equation $x^2+1=0$ always has\na solution in $\\ensuremath{\\mathbb{F}_{p^2}}$. This can be proven using the fact that\n$\\ensuremath{\\mathbb{F}_{p^2}}^\\star$ is a cyclic group. Let $\\beta $ be a generator of $\\ensuremath{\\mathbb{F}_{p^2}}^*$.\nThen $\\beta^k = -1$ for some $k$, and it is also known that $-1^2 = 1$.\nHence $\\beta^{2k}=1$ and $p^2-1 | 2k$, so that $k$ is even.\nThus, $\\beta^\\frac{k}{2}$ is the required solution.\n\nAs defined previously\n\\[\ne=\\dim(C^{\\ensuremath{{\\perp_h}}})-\\dim(C\\text{\\ensuremath{\\cap}}C^{\\ensuremath{{\\perp_h}}})=\\dim(C+C^{\\ensuremath{{\\perp_h}}})-\\dim(C).\n\\]\nLet $s=\\dim(C\\cap C^{\\ensuremath{{\\perp_h}}})$, and $G$ be the matrix\n\\begin{equation}\n\\label{m:generatorG}\nG=\\begin{pmatrix}M_{s\\times n} & 0_{s\\times e}\\\\\nA_{(n-e-2s)\\times n} & 0_{(n-e-2s)\\times e}\\\\\nB_{e\\times n} & \\beta^{k\/2}I_{e\\times e}\n\\end{pmatrix},\n\\end{equation}\nwhere the size of the matrix is indicated by the subscripts, and\n$0$ and $I$ denote the zero matrix and identity matrix, respectively.\n\nFor a matrix $P$, let $r(P)$ denote the set of rows of $P$.\nThe matrix $G$ is constructed such that $r(M)$ is a basis for $C\\cap\nC^{\\ensuremath{{\\perp_h}}}$, $r(M)\\cup r(A)$ is a basis for $C$, $r(M)\\cup r(B)$ is a\nbasis for $C$, and $r(B)$ is an orthonormal set.\nThe existence of such a matrix $B$ follows from Lemma \\ref{l:existenceB}.\nNote that $r(M)\\cup r(A)\\cup r(B)$ is a basis for $C+C^{\\ensuremath{{\\perp_h}}}$ .\n\nLet $E$ be the linear code for which $G$ is a generator matrix.\nFurther, let $S$ denote the union of the first $s$ rows of $G$ and the last $e$\nrows of $G$, i.e., $S$ is the set of rows of the matrix\n\\begin{equation}\n\\label{m:generatorS}\nS=\\begin{pmatrix}M_{s\\times n} & 0_{s\\times e}\\\\\nB_{e\\times n} & \\beta^{k\/2}I_{e\\times e}\n\\end{pmatrix}.\n\\end{equation}\nWe observe that each row of $S$ is orthogonal to each row of $G$\nbecause any row from the first $s$ rows of $S$ represents a vector in\n$C\\cap C^{\\ensuremath{{\\perp_h}}}$, and hence is orthogonal with all codewords in $C+C^{\\ensuremath{{\\perp_h}}}$,\nthe code represented by $G$.\n\nConsider a row from the last $e$ rows in $S$. This row is orthogonal\nto the first $n-e-s$ rows of $G$ because they represent the code $C$\nwhile the matrix $B$ represents codewords from $C^{\\ensuremath{{\\perp_h}}}$. The rows\nof the matrix are orthogonal. Because in the case they are different\nrows in the matrix, then they are orthogonal and the $\\beta^{k\/2}I$\nmatrix part will contribute a $0$. Any row $z$ is self-orthogonal\nsince from the construction $\\norm{z}=1$ and the identity matrix\nwill contribute $-1$, giving an inner product of $0$. This completes\nthe proof of the observation. Thus, each vector from $S$ belongs to\n$E^{\\ensuremath{{\\perp_h}}}$, and the vectors in $S$ are linearly independent because\n\\[\n\\dim(E^{\\ensuremath{{\\perp_h}}})=n+e-(n-s)=e+s=|S|.\n\\]\nHence $S$ is a basis for $E^{\\ensuremath{{\\perp_h}}}$.\nSince $S$ is a subset of $G$ by construction, it follows that $E^{\\ensuremath{{\\perp_h}}}\\subseteq E$.\n\nLet $x$ be a non-zero vector in $E$ and due to the\nvertical block structure of $G$, we can write $x=(x^{1}|x^{2})$\nwhere $x^{1}\\in\\ensuremath{\\mathbb{F}_{p^2}}^{n}$ and $x^{2}\\in\\ensuremath{\\mathbb{F}_{p^2}}^{e}$.\nThus $x$ is a linear combination of rows of $G$.\nIf none of the last $e$ rows of $G$ are contained in this linear combination with a non-zero coefficient,\nthen $x^{1}\\in C\\backslash{0}$, and so $\\wt(x)=\\wt(x^{1})\\ge\\wt(C)$.\nIf some of the last $e$ rows of $G$ are in this linear combination with\na non-zero coefficient, then $x^{1}\\in C+C^{\\ensuremath{{\\perp_h}}}$ and\n$\\wt(x)=\\wt(x^{1})+\\wt(x^{2})\\ge\\wt(C+C^\\ensuremath{{\\perp_h}})+1$.\nThus $E$ is an $[n+e,k+e,d]_{p^{2}}$ code with\n$d\\ge\\min(\\wt(C),\\wt(C+C^{\\ensuremath{{\\perp_h}}})+1)$ and $E^{\\ensuremath{{\\perp_h}}}\\subseteq E$.\nThe code $E$ satisfies the required conditions, and thus the proof is complete.\n\\end{proof}\n\nMany constructions of quantum codes use self-orthogonal\ncodes~\\cite{G2010,G-G2013}, which corresponds to the case when $e=0$.\nThe results of the next section are required to construct the quantum codes in subsequent sections.\nNote that many of the results in the next section can easily be generalized to constacyclic codes.\n\n\\section{The Hermitian Dual of Repeated Roots Cyclic Codes}\nLet $p$ is a prime number and $C$ be a cyclic code of length $n$ over the finite field $\\ensuremath{\\mathbb{F}_{p^2}}$.\nThen $C$ is given by the principal ideal $g(x)$ in $\\dfrac{\\ensuremath{\\mathbb{F}_{p^2}}[x]}{\\langle x^{n}-1 \\rangle}$,\nand so $g(x)$ is called the generator polynomial for $C$.\nWhen the length $n$ divides $p$, $C$ is called a repeated root cyclic code.\n\nIn this section, we obtain the generator polynomial of the Hermitian\ndual of a repeated root cyclic code.\nWe also give the structure of the cyclic codes of length $3p^s$ over $\\ensuremath{\\mathbb{F}_{p^2}}$ as well as the\nstructure of the dual code.\nOur interest in this class of codes comes from the importance of relaxing the condition $(n,p)=1$,\nwhich allows us to consider codes other than simple root codes.\n\nLet $f(x)=a_0+a_1 x+\\ldots + a_rx^r$ be a polynomial in\n$\\mathbb{F}_{q^2}[x]$, and $\\conj{f(x)} = \\conj{a_0} + \\conj{a_1}x + \\ldots + \\conj{a_r}x^r$.\nThe polynomial inverse of $f$ is denoted by $f^\\star(x) = x^rf(x^{-1}) = a_r+a_{r-1} x+\\ldots + a_0x^r$,\nso then $f^{\\bot}(x) = \\overline{a_r} + \\overline{a_{r-1}}x + \\ldots + \\overline{a_0}x^r$.\n\nThe following properties can easily be verified.\n\n\\begin{lemma}\n\\label{l:propConjInv}\nLet $f(x)$ and $g(x)$ be polynomials over $\\f{p^m}$.\nThen\n\\begin{enumerate}\n\\item conjugation is additive: $\\conj{f(x) + g(x)} =\\conj{f(x)} + \\conj{g(x)}$;\n\\item conjugation is multiplicative: $\\conj{f(x)g(x)} =\\conj{f(x)}\\, \\conj{g(x)}$;\n\\item polynomial inversion is additive if the polynomials have the same degree:\\\\\n${(f(x) + g(x))}^\\star ={f(x)}^\\star + {g(x)}^\\star $;\n\\item polynomial inversion is multiplicative: ${(f(x)g(x))}^\\star ={f(x)}^\\star \\,{g(x)}^\\star$;\n\\item inversion and conjugation commute with each other: $\\conj{(f(x)^\\star)} = (\\conj{f(x)})^\\star$; and\n\\item both operations are self-inverses: $(f(x)^\\star)^\\star = f(x)$ and $\\conj{\\conj{f(x)}} = f(x)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{lemma}\n\\label{l:herDualCondition1} Let $a(x) = a_0 + a_1x + \\ldots + a_{n-1}x^{n-1}$\nand\n$b(x) = b_0 + b_1x + \\ldots + b_{n-1}x^{n-1}$ be polynomials in $\\dfrac{F_{p^2}[x]}{x^n-1}$.\nThen $a(x)\\conj{b(x)} = 0$ in $\\dfrac{F_{p^2}[x]}{x^n-1}$\nif and only if $(a_0, a_1, \\ldots, a_{n-1})$ is orthogonal to\n$(\\conj{b_{n-1}}, \\conj{b_{n-2}}, \\ldots, \\conj{b_0})$ and all its cyclic shifts.\nThat is $\\hip{a}{\\conj{b^\\star}}=0 \\iff a(x)b(x)^\\bot =0$.\n\\end{lemma}\n\\begin{proof}\nIt well known (see for example~\\cite{huffman03}), that if $a(x) = a_0 + a_1x + \\ldots + a_{n-1}x^{n-1}$\nand $b(x) = b_0 + b_1x + \\ldots + b_{n-1}x^{n-1}$ are\npolynomials in $\\dfrac{F_{p^2}[x]}{x^n-1}$, then $a(x)b(x) = 0$ in\n$\\dfrac{F[x]}{x^n-1}$ if and only if $(a_0, a_1, \\ldots, a_{n-1})$\nis orthogonal to $(b_{n-1}, b_{n-2},\\ldots, b_0)$ and all its cyclic shifts.\nHence by applying this fact to $a(x)$ and\n$\\conj{b(x)}$ and noting that $\\conj{\\conj{b(x)}} = b(x)$, the result follows.\n\\end{proof}\n\nWe now use Lemma~\\ref{l:herDualCondition1} to derive an expression for the Hermitian dual\nof a cyclic code.\nLet $S\\subseteq R$ and let the annihilator be $\\ann(S) = \\{ g\\in R | fg=0, \\; \\forall f\\in S\\}$.\nThen $\\ann(S)$ is also an ideal of the ring and hence is generated by a polynomial.\n\\begin{lemma}\n\\label{l:hdToAnnihilator}\nIf $g(x)$ generates the code $C$, then $C^\\ensuremath{{\\perp_h}} = \\ann(\\conj{g(x)}^\\star)$.\n\\end{lemma}\n\\begin{proof}\nAssume that $g(x)$ generates the code $C$.\nThen each codeword in $C$ has the form $a(x) = g(x)c(x)$.\nLet a codeword $b(x)$ lie in the Hermitian dual $C^\\ensuremath{{\\perp_h}}$.\nThen by Lemma \\ref{l:herDualCondition1} we have that\n\\[\n a(x)b^\\bot(x)=0,\n\\]\nand by Lemma \\ref{l:propConjInv}, this is equivalent to\n\\begin{equation}\n\\label{l:annihilatorDerivation}\n b(x) (\\conj{g(x)}^\\star) =0.\n\\end{equation}\nThen by (\\ref{l:annihilatorDerivation}), we have that for a\ncodeword $b(x)$, $b(x)\\in C^\\ensuremath{{\\perp_h}} \\iff b(x)\\in \\ann(\\conj{g(x)}^\\star)$,\nwhich completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{l:annihilatorToGeneratorPoly}\nAssume that $C=\\codegenerated{g(x)} $ is a cyclic code of length $n$ over\n$\\mathbb{F}_{p^2}$ with generator polynomial $g(x)$.\nDefine $h(x)=\\frac{x^n-1}{g(x)}$.\nThen we have that $C^{\\ensuremath{{\\perp_h}}}= \\codegenerated{h^{\\bot}(x)}$.\n\\end{lemma}\n\\begin{proof}\nFrom Lemma \\ref{l:hdToAnnihilator} it is known that $C^\\ensuremath{{\\perp_h}} = \\ann(g(x)^\\bot)$.\nThus, we must show that $\\ann(g^\\bot(x)) = \\codegenerated{h^\\bot(x)}$.\nOne way containment is easy since $\\codegenerated{h^\\bot(x)} \\subseteq \\ann(g^\\bot(x))$,\nwhich is true because $h^\\bot(x)g^\\bot(x) = (h(x)g(x))^\\bot = (x^n-1)^\\bot = 0$ by Lemma \\ref{l:propConjInv}.\nFor containment the other way, we observe that since\n$\\ann(g^\\bot(x))$ is an ideal of the polynomial ring\n$\\dfrac{\\ensuremath{\\mathbb{F}_{p^2}}[x]}{x^n-1}$, it is generated by a polynomial, say\n$b^\\bot(x)$. Then $b^\\bot(x)g^\\bot(x) = x^n -1 =\n\\lambda(x^n-1)^\\bot$ (because $b(x)$ is the smallest polynomial, so it is an equality).\nHence $b(x)g(x) = x^n -1$, so it must be that\n$b(x)= h(x)$ since both are unitary polynomials.\nThis completes the proof.\n\\end{proof}\n\n\\begin{theorem}\nLet $p > 3$ be a prime.\nThen\n\\begin{enumerate}\n\\item there exists $\\omega \\in \\ensuremath{\\mathbb{F}_{p^2}}$ such that $\\omega^3=1$ and the factorization of $x^{3p^s}-1$ into irreducible factors over $\\ensuremath{\\mathbb{F}_{p^2}}[x]$ is\n\\[\nx^{3p^s}-1 = (x-1)^{p^s}(x-\\omega)^{p^s}(x-\\omega^2)^{p^s};\n\\]\n\\item the cyclic codes of length $3p^s$ are always of the form\n\\[ \\codegenerated{(x-1)^{i}(x-\\omega)^{j}(x-\\omega^2)^{k}},\\] where $0\n\\leq i,j,k \\leq p^s$, and the code has $p^{2(3p^s-i-j-k)}$ codewords; and\n\\item the Hermitian dual of the codes have the form\n\\begin{equation} \\label{e:hdGeneratorForm}\nC^\\ensuremath{{\\perp_h}} =\n \\begin{cases}\n \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}} & \\text{ if } p\\equiv 1\\mod 3, \\\\\n \\codegenerated{(x-1)^{p^s-i}(x-\\omega^2)^{p^s-j}(x-\\omega)^{p^s-k}} & \\text{ if } p\\equiv 2 \\mod 3.\n \\end{cases}\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n \\begin{enumerate}\n \\item Since $p$ is a prime number, $p \\neq 0 \\mod 3$, and $p^2-1 = (p+1)(p-1)$, so either\n $p+1 = 0 \\mod 3$ or $p-1 = 0 \\mod 3$. Therefore an element of order 3 exists in $\\ensuremath{\\mathbb{F}_{p^2}}$.\n Let this element be $\\omega$, so then $(x-1)(x-\\omega)(x-\\omega^2) = x^3-1$.\n In a field of characteristic $p$, it is known that $x^n-1 = (x^m-1)^p$ if $n=mp$.\n Therefore we have that $x^{3p^s}-1 = (x^3-1)^{p^s} = ((x-1)(x-\\omega)(x-\\omega^2))^{p^s}.$\n \\item\n From the previous part we know that the irreducible factors are $(x-1)$, $(x-\\omega)$ and $(x-\\omega^2)$,\n each of multiplicity $p^s$.\n As the generator polynomial divides $x^{3p^s}-1$, the statement follows.\n \\item\n We know from Lemma \\ref{l:annihilatorToGeneratorPoly} that\n \\[C^\\ensuremath{{\\perp_h}} = \\codegenerated{h^\\bot(x)}, \\]\nhence\n\\begin{align}\nC^\\ensuremath{{\\perp_h}} &= \\conj{ \\codegenerated{\\dfrac{(x-1)^{p^s}(x-\\omega)^{p^s}(x-\\omega^2)^{p^s}}{(x-1)^{i}(x-\\omega)^{j}(x-\\omega^2)^{k}}}^\\star } \\nonumber \\\\\n &= \\conj{ \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}}^\\star } \\nonumber \\\\\n &= \\conj{\\codegenerated{[(x-1)^{p^s-i}]^\\star[(x-\\omega)^{p^s-j}]^\\star [(x-\\omega^2)^{p^s-k}]^\\star} } \\nonumber \\\\\n &= \\conj{\\codegenerated{[-(x-1)^{p^s-i}][- \\omega(x-\\omega^{-1})^{p^s-j}]^\\star [- \\omega^2 (x-\\omega^{-2})^{p^s-k}]^\\star} } \\nonumber \\\\\n & \\text{Since, } (x-1)^\\star = -x + 1 = -(x-1) , (x-\\omega)^\\star = -\\omega x+1 = -\\omega (x-\\omega^2) \\nonumber \\\\\n &= \\conj{\\codegenerated{[(x-1)^{p^s-i}][(x-\\omega^2)^{p^s-j}][(x-\\omega)^{p^s-k}]} } \\nonumber \\\\\n &= \\codegenerated{[(\\conj{x-1})^{p^s-i}][(\\conj{x-\\omega^2})^{p^s-j}][(\\conj{x-\\omega})^{p^s-k}]} \\nonumber \\\\\n &= \\codegenerated{[({x-1})^{p^s-i}][({x-\\omega^{2p}})^{p^s-j}][({x-\\omega^p})^{p^s-k}]} \\nonumber \\\\\n &= \\begin{cases}\n \\codegenerated{(x-1)^{p^s-i}(x-\\omega^2)^{p^s-j}(x-\\omega)^{p^s-k}} & \\text{ if } p\\equiv 1\\mod 3, \\\\\n \\codegenerated{(x-1)^{p^s-i}(x-\\omega)^{p^s-j}(x-\\omega^2)^{p^s-k}} & \\text{ if } p\\equiv 2 \\mod 3.\n \\end{cases} \\\\\n & \\text{Since } \\omega^{p} = \\omega \\text{ if } p \\equiv 1 \\mod 3 \\text{, and } \\omega^{p} = \\omega^2 \\text{ if } p \\equiv 2 \\mod 3.\n\\end{align}\n \\end{enumerate}\nThis completes the proof.\n\\end{proof}\n\n\\section{Extension to Simple Root Cyclic Codes}\nThis section considers cyclic codes of length $n$ over $\\ensuremath{\\mathbb{F}_{p^2}}$ such that $(p,n)=1$.\nIn this case, a cyclic code can be represented by its defining set $Z$.\nIf $m$ has order $p^{2}$ modulo $n$, then $\\f{p^{2m}}$ is the splitting field of\n$x^n-1$ containing a primitive $n$th root of unity.\nConsider a primitive root $\\beta$.\nThen $\\{ k|g(\\beta^{k})=0,\\; 0\\leq k