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+{"text":"\\section{Introduction}\n\nSelf-healing is one of the most fascinating properties of diffraction-free optical fields\\,\\cite{Durnin1987}. These fields have the ability to reconstruct if they are partially disturbed by an obstruction placed in their propagation path. Diffraction-free beams have found applications in fields such as imaging~\\cite{Fahrbach2010,Planchon2011,Fahrbach2012}, optical trapping~\\cite{McGloin2005,McGloin2003,ARLT2001,Volke2002}, laser material processing~\\cite{Dubey2008}, amongst many others. Arguably, the most  well-known propagation invariant (self-healing) fields are Bessel modes of light, first introduced in 1987 by J. Durin\\,\\cite{Durnin1987,Durnin1987b}. However, the self-healing property is not limited to so called non-diffracting beams, but also appears in helico-conical\\,\\cite{Hermosa2013}, caustic, or self-similar fields, namely Airy\\,\\cite{Broky2008}, Pearcey\\,\\cite{Ring2012}, Laguerre-Gaussian\\,\\cite{Bouchal2002,Mendoza2015} and even standard Gaussian beams\\,\\cite{Aiello2017}. Furthermore, within the last years, it has been shown that self-healing can also be observed at the quantum level, for example, McLaren {\\it et al.} demonstrated experimentally the self-reconstruction of quantum entanglement\\,\\cite{McLaren2014}. Importantly, self-healing is not only an attribute of scalar fields but it can also apply to beams with spatially variant polarization \\cite{Milione2015d,Wu2014,Li2017}. \\\\\nBessel beams also appear as complex vector light fields, where polarisation and spatial shape can be coupled in a non-separable way\\,\\cite{Zhan2009,Rosales2017,Otte2018}. This property has fueled a wide variety of applications, from industrial processes, such as, drilling or cutting\\,\\cite{Meier2007,Niziev1999,Dubey2008}, to optical trapping\\,\\cite{Donato2012,Zhao2005,Zhan2004,Roxworthy2010,Kozawa2010,Huang2012}, high resolution microscopy\\,\\cite{Torok2004}, quantum and classical communication\\,\\cite{Zhan2002,Ndagano2017,Ndagano2018}, amongst many others. Controversially, such non-separable states of classical light are sometimes referred to as classically or non-quantum entangled \\,\\cite{simon2010nonquantum}. This stems from the fact that the quintessential property of quantum entanglement is non-separability, which is not limited to quantum systems. Indeed, the equivalence has been shown to be more than just a mathematical construct \\cite{Ndagano2017}.  While such classical non-separable fields do not exhibit non-locality, they manifest all other properties of local entangled states.\\\\\nHere, we demonstrate that the decay in such local entanglement after an obstruction can be counteracted if the carried field is Bessel. We create higher-order vector Bessel beams that are non-separable in orbital angular momentum (the azimuthal component of the spatial mode) and polarisation, and show that self-healing also comprises the non-separability of the beams. This is at a first glance surprising since self-healing is traditionally attributed to the radial component of the spatial mode, which in our field is entirely separable. In order to demonstrate the far-reaching concept of self-healing, we unambiguously quantify the degree of non-separability in different scenarios ranging from fully to partially reconstructed fields by performing a state tomography on the classical field \\cite{McLaren2015,Ndagano2016}. We show both theoretically and experimentally that even though the non-separability reduces after the obstruction, it recovers again upon propagation, proportionally to the level of self-reconstruction. Further, we confirm our findings by a Bell-like inequality measurement\\,\\cite{bell1964js} in its most commonly used version for optics, namely, the Clauser-Horne-Shimony-Holt (CHSH) inequality \\,\\cite{clauser1969}, confirming that the non-separability of vector Bessel beams also features self-healing properties. Although our tests are exerted on purely classical fields, the results are expected to be identical for the local entanglement of internal degrees of freedom of a single photon, and may be beneficial where such entanglement preservation is needed, e.g., transporting single photons through nano-apertures for plasmonic interactions.\n\n\n\n\\begin{figure*}[ht]\n\\centering\n\\def\\svgwidth{0.9\\linewidth}\\sffamily\n\\input{Fig1_Concept1.eps_tex}  \n\\caption{(a) Formation of Bessel beams by off-axis interfering plane waves. If obstacles are included (radius $R$) a shadow region is formed. Scalar BG modes (intensity profile at $z_0$ in (c)) are realized by applying the binary Bessel function (b1) in combination with a blazed grating and Gaussian aperture (b2). (d) Concept of the realization and analysis of non-separable vBG modes (SLM: spatial light modulator; L$_{(\\mathcal{F})}$: (Fourier) lens with focal distance $f$; A: aperture; $\\lambda\/n$, $n=\\lbrace2,\\,4\\rbrace$: wave plates; q: $q$-plate; CCD: camera). (e) Propagation behavior of azimuthal vBG mode (e1) without obstruction and (e2) obstructed by on-axis absorbing object with $R=200\\,\\mbox{\\textmu m}$, indicated by white circle.\\label{fig:Concept}} \n\\end{figure*}\n\n\\section{Vector Bessel modes} \\label{sec:VectorBesselModes}\n\\subsection{Bessel-Gaussian beams}\n\\noindent Over finite distances, a valid approximation of Bessel beams is given by the so called Bessel-Gaussian (BG) modes\\,\\cite{Gori1987}. Besides other properties, these BG fields have the same ability of self-reconstruction in amplitude and phase\\,\\cite{McGloin2003,Litvin2009}. In polar coordinates $(r,\\varphi, z)$, BG modes are defined as\n\\begin{align}\nE_{\\ell}^{\\text{BG}}(r,\\varphi, z) = &\\sqrt{\\frac{2}{\\pi}}J_{\\ell}\\left(\\frac{z_R k_r r}{z_R-\\text{i}z}\\right)\\exp\\left(\\text{i}\\ell\\varphi-\\text{i}k_z z\\right) \\nonumber \\\\\n&\\cdot \\exp\\left(\\frac{\\text{i}k_r^2z w_0-2kr^2}{4(z_R-\\text{i}z)}\\right),\n\\label{eq:BGmodes}\n\\end{align}\nwhereby $\\ell$ represents the azimuthal index (topological charge), and $k_r$ and $k_z$ are the radial and longitudinal wave numbers, respectively. Further, $J_{\\ell}(\\cdot)$ defines the Bessel function, whereas the Gaussian information is encoded in the last factor with the initial beam waist $w_0$ of the Gaussian profile and the Rayleigh range $z_R = \\pi w_0^2\/\\lambda$, $\\lambda$ being the wavelength. The finite propagation distance of BG modes (``non-diffracting length\") is limited by $z_{\\text{max}}$. This distance describes the length of a rhombus-shaped region created by the superposition of plane waves with wave vectors lying on a cone described by the angle $\\alpha=k_r\/k$ (wave number $k=2\\pi\/\\lambda$)\\,\\cite{Gori1987}, as indicated in Fig. \\ref{fig:Concept}(a). The center of the rhombus-shaped region is positioned at $z_0$. For small $\\alpha$, i.e., $\\sin \\alpha \\approx \\alpha$, the non-diffracting distance is given by $z_{\\text{max}} =2\\pi w_0\/\\lambda k_r$ \\cite{McGloin2005}. If an obstruction is included within the non-diffracting distance, a shadow region is formed of length $z_{\\text{min}} \\approx R\/\\alpha \\approx \\frac{2\\pi R}{k_r \\lambda}$ \\cite{Bouchal1998} (Fig.~\\ref{fig:Concept}(a)). Here, $R$ describes the radius of the obstruction. After this distance $z_{\\text{min}}$, the beam starts to recover due to the plane waves passing the obstruction \\cite{McGloin2003,Litvin2009}. A fully reconstructed BG beam will be observed at $2z_{\\text{min}}$, as visualised in Fig.~\\ref{fig:Concept}(a).\n\n\\subsection{Realization of obstructed BG modes}\n\\noindent An established tool for the realization of complex beams are spatial light modulators (SLMs). These modulators allow for an on-demand dynamic modulation of structured beams by computer generated holograms \\cite{SPIEbook}. For the formation of BG modes, we choose a binary Bessel function as phase-only hologram, defined by the transmission function\n\\begin{equation}\nT(r,\\varphi) = \\text{sign}\\lbrace J_{\\ell}(k_r r)\\rbrace \\exp (\\text{i}\\ell\\varphi), \\label{eq:BinaryBessel}\n\\end{equation}\nwith the sign function $\\text{sign}\\lbrace\\cdot \\rbrace$ \\cite{Turunen1988,Cottrell2007}. This approach has the advantage of generating a BG beam immediately after the SLM. An example of this function is shown in Fig. \\ref{fig:Concept}(b1). Note that for encoding this hologram we use a blazed grating (see Fig.~\\ref{fig:Concept}(b2)), so that the desired beam is generated in the first diffraction order of the grating\\,\\cite{Davis1999}.\\\\\nHere, we set $k_r = 18\\,\\mbox{rad\\,mm}^{-1}$ and $\\ell =0$ for the fundamental Bessel mode. Furthermore, we multiply the hologram by a Gaussian aperture function for the realization of a Gaussian envelope with $w_0 = 0.89\\,\\mbox{mm}$ (see Eq. (\\ref{eq:BGmodes}), Fig. \\ref{fig:Concept}(b2)). These settings result in a BG beam with $z_{\\text{max}} = 49.16\\,\\mbox{cm}$ for a wavelength of $\\lambda = 633\\,\\mbox{nm}$, whose intensity profile in the $z_0$-plane is depicted in Fig.~\\ref{fig:Concept}(c).\\\\\nBeyond the generation of BG modes, the SLM can also be used for the realization of obstructions within the $z_0$-plane (see Fig. \\ref{fig:Concept}(d1)): Absorbing obstacles are created by including a circular central cut in the hologram, such that within this area no blazed grating is applied. This means, the respective information of the BG mode is deleted in the first diffraction order. Furthermore, phase obstructions can be realized by adding the chosen phase object to the hologram. Hence, this artificial generation of obstructions facilitates the realisation of any chosen kind of obstacle of defined radius $R$ in the $z_0$-plane. Moreover, the relation $z_{\\text{min}} \\approx \\frac{2\\pi R}{k_r \\lambda}$ shows that a decrease in the radius R of the circular obstruction at $z_0$ results in a decrease in the length $z_{\\text{min}}$ of the shadow region, which is equivalent to moving the detection plane in $\\pm z$-direction (see Fig. \\ref{fig:Concept}(d2)), in order to analyse the evolution from a partially to a fully reconstructed beam.\n\n\\subsection{Self-healing vector Bessel modes}\n\\noindent In order to investigate the relation between the self-healing of propagation invariant beams and the non-separability of light modes, we apply vector Bessel beams, or, more precisely, vector Bessel-Gaussian (vBG) modes. These modes are classically entangled in their spatial and polarisation degrees of freedom (DoF) as explained in the next section. As illustrated in Fig. \\ref{fig:Concept}(d1), these beams are generated by a suitable combination of an SLM (SLM$_1$), half wave plates ($\\frac{\\lambda}{2}$) and a $q$-plate (q), a device capable of correlating the polarisation and spatial DoFs\\,\\cite{Marrucci2006}. First, we create the fundamental scalar BG mode (linearly polarised in the horizontal direction) by encoding the binary Bessel hologram (Fig. \\ref{fig:Concept}(b2)) on SLM$_1$. Within a $4f$-system we filter the first diffraction order with an aperture (A). If we now position, for example, a half wave plate, whose fast axis is oriented in a $45^{\\circ}$ ($\\pi\/4$) angle with respect to the incoming horizontal polarisation, in combination with a $q$-plate ($q=1\/2$) in the beam path, an azimuthally polarised vBG mode is created in the image plane of SLM$_1$ (see Fig. \\ref{fig:Concept}(d1)). The desired mode is generated from the $q$-plate by coupling the polarisation DoF with the orbital angular momentum (OAM) via a geometric phase control, imprinting an OAM charge of $\\pm 2q$ per circular polarisation basis to the passing beam\\,\\cite{Marrucci2006}. Note that wave and $q$-plate(s) do not need to be placed within the non-diffracting distance as we work in the paraxial regime ($\\sin \\alpha \\approx \\alpha$). The plates could even be located within the Fourier plane of SLM$_1$\\,\\cite{Milione2015d}. Moreover, we are of course not limited to azimuthally polarised vBG modes. Depending on the chosen number and orientation of wave plates, different polarisation structures are accessible\\,\\cite{Cardano2012} as depicted in Fig. \\ref{fig:Polarization}. Here, we demonstrate the intensity distribution of different vBG beams in the $z_0$-plane analysed by a polariser (orientation indicated by white arrows).\\\\ \n\n\\begin{figure}[h]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{Polarization2.eps_tex}  \n\\caption{Examples of non-separable vBG modes with respective polarisation analysis. The normalised intensity distribution in the $z_0$-plane for different orientations of a polariser are shown. The respective orientation is indicated by white arrows in (a). The according polarisation distribution is highlighted by black arrows. \\label{fig:Polarization}} \n\\end{figure}\nIn Fig. \\ref{fig:Concept}(e1) we present the experimentally measured propagation invariant properties of these modes by the example of the azimuthal vBG beam (cf. Fig. \\ref{fig:Polarization}(a)). The transverse intensity profile is shown for different positions $z\\in [z_0,\\,z_{\\text{max}}]$. Consider that the outer rings of the vBG mode disappear with increasing $z$ due to the rhombus shape of the non-diffracting region.\\\\\nAs explained above, SLM$_1$ enables the inclusion of an obstruction within the holographically created scalar BG field. Following this, we are also able to apply the SLM for imparting an obstruction within the vBG mode: As SLM$_{1}$ is imaged by a $4f$-system to the $z_0$-plane of the vBG beam, an obstacle created by SLM$_1$ is also imaged to the $z_0$-plane of formed vector mode (cf. Fig.~\\ref{fig:Concept}(d1)). As an example, we investigated the propagation properties of the azimuthal vBG mode if obstructed by an absorbing object with $R=200\\,\\mbox{\\textmu m}$ created by SLM$_1$. Here, we included an additional horizontally oriented polariser to analyse the polarisation properties simultaneously. Results are shown in Fig. \\ref{fig:Concept}(e2). For the programmed obstacle we calculate a self-healing distance of $z_{\\text{min}} = 11.02\\,\\mbox{cm}$. The shown intensity distributions reveal a self-reconstruction of the beam including its polarisation properties after approximately $2z_{\\text{min}}$, as expected.\n\n\n\\section{Non-separability of self-healing beams} \n\\noindent\nWe define the non-separability of a self-healing vBG beam in the framework of quantum mechanics. This can be easily understood through the formal definition of the non-separability or entanglement of quantum systems; two systems A and B are separable if they can be written as a factorisable product of the two subsystems, i.e, $\\ket{\\Psi_{AB}}=\\ket{A}\\otimes\\ket{B}$, conversely, the systems can be entangled if ($\\ket{\\Psi_{AB}} \\neq \\ket{A}\\otimes\\ket{B}$). Here, the two subsystems are analogously replaced with the internal DoF of the photons in the vBG beam. Importantly, a non-separable vector mode has maximally entangled polarisation and spatial components.\\\\\nTo quantify the non-separability between the polarisation and spatial components of the field, we employ simple measures borrowed from quantum mechanics  \\cite{McLaren2015,Ndagano2016}. Consider an arbitrary (vBG) field with each photon described by the following state  \n\\begin{equation}\n\\ket{\\Psi}_{k_r \\ell} = \\cos(\\theta)\\ket{u_{k_r,  \\ell}} \\ket{R} + \\sin(\\theta)\\ket{u_{k_r,  -\\ell}} \\ket{L}, \\label{eq: Field}\n\\end{equation}\n\\noindent where $\\ket{R}$ and $\\ket{L}$ are the canonical right and left circular polarisation states spanning the qubit Hilbert space $\\mathcal{H}_2$. The infinite dimensional state vectors  $\\ket{u_{k_r, \\pm\\ell}} \\in \\mathcal{H_\\infty}$ represent the self-healing transverse eigenstates, namely scalar Bessel or BG modes of light (cf. Eq. (\\ref{eq:BGmodes})), characterised by the continuous radial wave number $k_r$ and the topological charge $\\ell$. The parameter $\\theta$ determines whether $\\ket{\\Psi}_{k_r \\ell}$  is purely vector (non-separable; $\\theta=(2n+1)\\pi\/4,\\, n\\in \\mathbf{Z})$, scalar (separable; $\\theta=n\\pi\/2,\\,\\, n\\in \\mathbf{Z})$, or some intermediate state.\n\nThe ``vectorness\" of a given classical state can be determined via a vector quality analysis which is equivalent to measuring the concurrence of a quantum state $C$ \\cite{Wootters2001}. Mathematically, the respective vector quality factor (VQF) has the form\n\\begin{equation}\n\\text{VQF}=\\text{Re}(C)=\\sqrt{1-s^2}=|\\sin(2\\theta)|. \\label{eq:VQFpure}\n\\end{equation}\n\\noindent Here, $s$ is the Bloch vector defined as $s=\\sum_{i}^{3}\\langle \\sigma_i\\rangle$ with $\\sigma_i$, $i=\\lbrace 1,2,3\\rbrace$, being the traceless Pauli operators spanning the so-called higher-order Poincar\\'{e}  sphere (HOPS, see Fig. \\ref{fig:HOPS})\\,\\cite{Millione2011}. The VQF takes values in the interval $[0,1]$, with 0 corresponding to a separable scalar mode and  1 representing a maximally non-separable vector mode. We employ the VQF measure as a figure of merit for determining the non-separability of partially obstructed and consequently self-healing vBG modes.\n\n\\subsection{Vector quality factor of obstructed beams}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\t\\input{HOPs2.eps_tex}\n\n\t\\caption{Higher-order Poincar\\'e spheres of self-healing BG beams spanned by the Stokes parameters (a) $S^{+1}_{1,2,3}$ for $\\mathcal{H}_{2,\\sigma,\\ell=1}$ and (b) $S^{-1}_{1,2,3}$ for $\\mathcal{H}_{2,\\sigma,\\ell=-1}$. The poles represent separable BG states with circular polarisation and $\\ell=\\pm1$, while non-separable vBG states are found on the equator (i.e. the plane intersecting the $S^{\\pm1}_{1,2}$ plane).}\t\n\t\\label{fig:HOPS}\n\\end{figure}\n\\noindent\nFrom wave optics, it is well-known that an obstructed beam can be modified in both phase and amplitude due to the interaction between the beam and the outer edges of an obstruction, resulting in diffraction. At the single photon level, this can be understood as modal coupling\\,\\cite{Sorelli2018}. That is, if some field with a transverse profile given by $\\ket{ u_{k_{r_1}, \\ell_1} }$ interacts with an obstruction, the state evolves following the mapping \n\\begin{equation}\n\\ket{ u_{k_{r_1},\\ell_1 }} \\rightarrow \\int\\sum_{\\ell} \\alpha_{\\ell}(k_r) \\ket{ u_{{k_r},\\ell}} \\text{d}k_r, \\label{eq. Diff}\n\\end{equation}\nwhere the input mode spreads over all eigenmodes $\\ket{u_{{k_r},\\ell}}$. The coefficients $|\\alpha_{\\ell}(k_r)|^2$ represent the probability of the state $\\ket{ u_{k_{r_1  },\\ell_1}}$ scattering into the eigenstates $\\ket{u_{{k_r},\\ell}}$ with the property that $\\int\\sum_{\\ell}|\\alpha_{\\ell}(k_r)|^2\\text{d}k_r=1$. Applying the mapping of Eq. (\\ref{eq. Diff}) to the scattering of the self-healing vBG mode presented in Eq. (\\ref{eq: Field}) and post selecting a particular $k_r$ and $\\ell$ values yields the state \n\\begin{eqnarray}\n\\ket{\\Phi_{k_r, \\ell}}=& a \\ket{ u_{{k_r},\\ell}}\\ket{R} + b \\ket{ u_{{k_r},-\\ell}}\\ket{R}\\nonumber\\\\ \n&+ c \\ket{ u_{{k_r},\\ell}}\\ket{L} + d \\ket{u_{k_r,\\ell}}\\ket{L}, \\label{post}\n\\end{eqnarray}\nwhere $|a|^2 +|b|^2+|c|^2+|d|^2=1$. Consequently, $\\ket{\\Phi_{k_r, \\ell}}$ restricts the measurement of the photons to the four-dimensional Hilbert space  $\\mathcal{H}_4=\\text{span}\\big(\\{\\ket{R}, \\ket{L}\\}\\otimes\\{\\ket{ u_{{k_r},\\ell}}, \\ket{ u_{{k_r},-\\ell}}\\}\\big)$ which can be written as the direct sum\n\\begin{equation}\n\t\\mathcal{H}_4= \\mathcal{H}_{2,\\sigma,\\ell} \\oplus \\mathcal{H}_{2,\\sigma,-\\ell}.\\label{eq:Hilbert}\n\\end{equation}\n\\noindent The subspaces $\\mathcal{H}_{2,\\sigma,\\pm\\ell}=\\text{span}(\\ket{u_{k_r, \\pm \\ell}}\\ket{R}, \\ket{u_{k_r,\\mp\\ell}}\\ket{L})$ are topological unit spheres for spin-orbit coupled beams belonging to the family of HOPS\\,\\cite{Millione2011}. \\\\\n\\noindent Invoking the equivalence between VQF and concurrence enables us to exploit the definition of concurrence with $C=\\sqrt{1-\\text{Tr}(\\rho_{\\psi}^2)}$ for a given density matrix $\\rho_\\psi=\\ket{\\psi}\\bra{\\psi}$. In quantum mechanics, if the state of a two qubit system written in the logical computation basis is given by the pure state\n\\begin{equation}\n \\ket{\\Psi} = c_1\\ket{0}\\ket{0}+c_2\\ket{0}\\ket{1}+c_3\\ket{1}\\ket{0}+c_4\\ket{1}\\ket{1},\n\\end{equation}\n\\noindent satisfying the normalization condition $\\sum_{j=1}^4 |c_{j}|^2=1$, the concurrence is $C=2|c_1c_4-c_3c_2|$. By replacing the two qubit logical basis with the $\\mathcal{H}_4$ basis from the two HOPSs, we can equivalently write the VQF as \n\\begin{equation}\n\\text{VQF}=2|ad-cb|, \\label{VQF_VM}\n\\end{equation}\n\\noindent following Eq. (\\ref{post}).\\\\\nAs an illustrative example, consider the azimuthally polarised vBG mode given by \n\\begin{equation}\n\\ket{\\psi}_{k_r,1} =\\frac{1}{\\sqrt{2}}\\big( \n \\ket{u_{k_r, 1}} \\ket{R} - \\ket{u_{k_r,  -1}} \\ket{L}\\big). \\label{eq:Fieldradial}\n\\end{equation}\n\n\\noindent Upon diffracting off the edges of an obstruction, one expects the mode coupling profiled in Eq. (\\ref{post}) to occur (by restricting $|\\ell|=1$). However, since the radial profile of BG modes enables the transverse structure to self-heal, $cb\\rightarrow0$ and $ad\\rightarrow-\\frac{1}{2}$, and therefore $\\text{VQF}\\rightarrow 1$ giving rise to the self-healing of the non-separability. This behavior is predicated to occur best after twice of the minimum self-healing distance $z_{\\text{min}}$.\\\\\n\n\\noindent To experimentally quantify the characteristics of vectorness, i.e. classical entanglement, in relation to the self-healing properties of vGB modes, we apply a configuration as indicated in Fig. \\ref{fig:Concept}(d2). By this configuration, consisting of a quarter wave plate ($\\frac{\\lambda}{n}$, $n=4$), a polarisation sensitive SLM (SLM$_2$), a Fourier lens (L$_{\\mathcal{F}}$) and a CCD camera, we determine the expectation values of the Pauli operators $\\langle\\sigma_{i}\\rangle$, $i = \\lbrace 1,\\,2,\\,3\\rbrace$ by 12 on-axis intensity measurements or six identical measurements for two different basis states\\,\\cite{McLaren2015,Ndagano2016}. These values are used to determine the VQF according to Eq. (\\ref{eq:VQFpure}). \\\\\nIf circular polarisation is chosen as basis ($\\ket{R},\\,\\ket{L}$), the projection measurements represent two OAM modes of topological charge $\\ell$ and $-\\ell$, namely $\\ket{u_{k_r,\\pm\\ell}} = E_{\\pm\\ell}^{\\text{BG}}$, as well as four superposition states $\\ket{u_{k_r,\\ell}} + \\exp(\\text{i}\\gamma) \\ket{u_{k_r,-\\ell}}$ with $\\gamma = \\lbrace 0, \\pi \\ell \/2, \\pi \\ell, 3\\pi\\ell\/2 \\rbrace$. As we use a $q$-plate with $q=1\/2$, our measurements are performed for $\\ell = 1$. Following Tab. \\ref{tab:Tomography}, we calculate expectation values $\\langle\\sigma_i \\rangle$ from\n\\begin{align*}\n&\\langle\\sigma_1\\rangle = I_{13}+I_{23}-(I_{15}+I_{25}),\\\\\n&\\langle\\sigma_2\\rangle = I_{14}+I_{24}-(I_{16}+I_{26}),\\\\\n&\\langle\\sigma_3\\rangle = I_{11}+I_{21}-(I_{12}+I_{22}).\n\\end{align*}\nOn-axis intensity values $I_{uv}$, $u = \\lbrace1,2\\rbrace$ , $v = \\lbrace1,2,...,6\\rbrace$, are normalised by $I_{11}+I_{12}+I_{21}+I_{22}$. The respective polarisation projections are performed by inserting a quarter wave plate, set to $\\pm 45^{\\circ}$, in combination with the polarisation selective SLM$_2$. Further, SLM$_2$ is responsible for the OAM projections. For this purpose, we encode the OAM as well as superposition states as phase-only holograms according to the binary Bessel function in Eq.~(\\ref{eq:BinaryBessel}). Finally, the on-axis intensity is measured in the focal plane of a Fourier lens by means of a CCD camera.\\\\\nCrucially, the decoding SLM$_2$ is placed at a adequately chosen distance $\\Delta z = 23\\,\\mbox{cm}$ from the $z_0$-plane so that we are able to access different levels of self-healing without the need to move the detection system. That is, by changing the radius $R$ of the digitally created obstruction, the vBG mode can fully self-heal in front of ($2z_{\\text{min}}< \\Delta z$) or behind ($2z_{\\text{min}}> \\Delta z$) this SLM$_2$, as indicated in Fig.~\\ref{fig:Concept}(d2) by the black shadow region or the green and yellow dashed lines, respectively. \n  \\begin{table}[h\n \\caption{Normalised intensity measurements $I_{uv}$ for the determination of expectation values $\\langle \\sigma_i \\rangle$. \\label{tab:Tomography}}\n \\begin{ruledtabular}\n \\begin{tabular}{c|c c c c c c c}\nBasis states & $l=1$ & $-1$ & $\\gamma =0$ & $\\pi\/2$ & $\\pi$ & $3\\pi\/2$& \\\\ \\hline \\hline\nLeft circular $\\vert L\\rangle$& $I_{11}$ & $I_{12}$ & $I_{13}$ & $I_{14}$ & $I_{15}$ & $I_{16}$ &\\\\ \nRight circular $\\vert R\\rangle$& $I_{21}$ & $I_{22}$ & $I_{23}$ & $I_{24}$ & $I_{25}$ & $I_{26}$& \\\\ \n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n \n\n\\begin{figure*}[tb]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{stateTomography.eps_tex}  \n\\caption{Vector quality analysis of self-healing vBG mode (azimuthally polarised) without obstacle (a), with absorbing obstacles of radius (b) $R = 150\\,\\mbox{\\textmu m}$, (c) $R=200\\,\\mbox{\\textmu m}$, (e) $R = 500\\,\\mbox{\\textmu m}$, and (f) $R=600\\,\\mbox{\\textmu m}$, as well as (d) phase obstacle (homogeneous phase shift of $\\pi$) of radius $R=200\\,\\mbox{\\textmu m}$. Results for beams which (do not) fully self-heal before the analysis are marked (blue) red with respective VQF within the measured normalized matrices of on-axis intensity values arranged according to Table.~\\ref{tab:Tomography}.  \\label{fig:StateTomography}} \n\\end{figure*}\n\n\\subsection{Experimental quantification of vectorness}\n\\noindent First, we prove that pure undisturbed vBG modes show the maximum degree of non-separability. This is exemplified by the analysis of an azimuthally polarised vBG beam. The experimentally measured on-axis intensity values are visualised in Fig. \\ref{fig:StateTomography}(a), arranged according to Tab. \\ref{tab:Tomography}. The measurements result in a VQF of $0.99$, verifying our expectation. In a next step, we digitally impart different obstacles and determine the respective VQF. On the one hand, we chose obstacles allowing the beam to self-reconstruct within $\\Delta z =23\\,\\mbox{cm}$, namely absorbing obstructions with $R=150\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 8.27\\,\\mbox{cm} = 16.54\\,\\mbox{cm}$) or $R=200\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 11.03\\,\\mbox{cm} = 22.06\\,\\mbox{cm}$) and a phase obstacle with $R=200\\,\\mbox{\\textmu m}$ creating a homogeneous phase shift of $\\pi$ (Fig. \\ref{fig:StateTomography}(b)-(d)). On the other hand, we program absorbing obstructions with $R = 500\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 27.57\\,\\mbox{cm} = 55.14\\,\\mbox{cm}$) and $R = 600\\,\\mbox{\\textmu m}$ ($2z_{\\text{min}} = 2\\cdot 33.03\\,\\mbox{cm} = 66.06\\,\\mbox{cm} $), for which $2z_{\\text{min}}>\\Delta z$ and even $z_{\\text{min}}>\\Delta z$ (Fig. \\ref{fig:StateTomography}(e), (f)). The respective measured VQFs are shown within each subfigure. \\\\\nObviously, the degree of non-separability, i.e. the VQF, decreases with increasing absorbing obstacle size $R$ (Fig. \\ref{fig:StateTomography}). For self-healed beams with absorbing obstacles (b)-(c), the VQF differs only minimally from the non-obstructed case (a). Note that a phase obstruction of the same radius (d) results in a slightly larger deviation. In contrast to absorbing obstacles, causing a loss of information, phase obstructions do not cut but vary information. Since in the case of absorbing obstacles the cut information is also included within the passing plane waves, the loss can be compensated within $2z_{\\text{min}}$. In the phase obstruction case, the varied information represent additional information, i.e. noise, within the non-diffracting beam, which is not eliminated when being decoded by the SLM. Hence, the beam stays disturbed and, as a consequence, the VQF decreases.\\\\\nFurther, if the beam cannot fully reconstruct before being decoded by SLM$_2$, thus, if we analyse the degree of entanglement within the self-healing distance (Fig. \\ref{fig:StateTomography}(e), (f)), the decrease in VQF is relatively large in comparison to the self-healed versions in Fig. \\ref{fig:StateTomography}(b), (c). However, note that in both cases, the self-reconstructed and non-reconstructed beams, the VQF is $\\geq 0.88$. Consequently, the beams are closer to being vector or non-separable (VQF $=1$) than scalar or separable (VQF $=0$). This is due to the fact that there is always undisturbed information reaching the SLM$_2$, and only little scattering into other modes, i.e., little modal coupling. We thus may conclude that the beam is always non-separable, but due to noise caused by the obstacle the measurement shows deviations from pure non-separability. As the noise is annihilated with propagation distance (for absorbing obstacles), the quantitative value for non-separability, namely VQF, recovers. This effect is similar to what we call ``self-healing\" in the case of amplitude, phase and polarisation of vBG modes: Obstacles add noise to the information on these degrees of freedom. Upon propagation, these perturbations vanish and the pure vBG mode information is left so that the beam and its properties seem to self-reconstruct.\\\\\nConsequently, we conclude that the lower the level of self-healing, i.e. the larger the obstacle, the smaller the VQF. This means, not only amplitude, phase and polarisation properties of the vBG mode reconstruct with distance behind an obstruction, but also the degree of non-separability. \\\\\n\n\\begin{figure*}[tb]\n\\centering\n\\def\\svgwidth{1.0\\linewidth}\\sffamily\n\\input{BellAnalysis.eps_tex}  \n\\caption{Bell-type curves for azimuthal vBG mode with four different orientations $2\\theta_A$  of the half wave plate to determine the Bell parameter $|S|$. The investigations were performed undisturbed (a) as well as for differently sized obstacles (radii $R=\\lbrace 150,\\,200,\\,500,\\,600\\rbrace\\,\\mbox{\\textmu m}$ in (b)-(f)) and a $\\pi$-phase obstacle ($R=200\\,\\mbox{\\textmu m}$) (d). Shown on-axis intensity measurements $I'(\\theta_A, \\theta_B)$ for (a)-(f) are normalised according to the maximum intensity measured without obstacle (a). Dashed curves represent $\\cos^2$-fits used to determine $S$. \\label{fig:BellAnalysis}} \n\\end{figure*}\n\n\n\\subsection{CHSH Bell-like inequality violation}\n\\noindent To confirm our results with respect to the VQF, we performed an additional investigation of the degree of entanglement or non-separability using the Bell parameter\\,\\cite{McLaren2015}. More specifically, we perform a Clauser-Horne-Shimony-Holt (CHSH) inequality measurement\\,\\cite{clauser1969}, the most commonly used Bell-like inequality for optical systems, to demonstrate the degree of entanglement between polarisation and spatial DoFs. Instead of measuring a single DoF, e.g. polarisation or OAM, non-locally, we analyse two DoF locally on the same classical light field\\,\\cite{McLaren2015}. For this purpose, we placed a half wave plate ($\\frac{\\lambda}{n}$, $n=2$) in front of SLM$_2$ (see Fig. \\ref{fig:Concept}(d2)) and measured the on-axis intensity $I(\\theta_A,\\theta_B)$ for different angles $2\\theta_A = \\lbrace 0,\\, \\pi\/8,\\,\\pi\/4,\\,3\\pi\/4\\rbrace$ of the half wave plate. Here, $\\theta_B \\in [0,\\,\\pi]$ represents the rotation angle of the hologram encoded on SLM$_2$ by $\\ket{u_{k_r,\\ell}} + \\exp(\\text{i}2\\theta_B) \\ket{u_{k_r,-\\ell}}$ ($\\ell = 1$, $k_r = 18\\,\\mbox{rad\\,mm}^{-1}$). \\\\\nWe define the CHSH-Bell parameter $S$ as\n\\begin{equation}\nS = E(\\theta_A,\\theta_B)-E(\\theta_A,\\theta_B')+E(\\theta_A',\\theta_B)+E(\\theta_A',\\theta_B'), \\label{eq:BellParameter}\n\\end{equation}\nwith $E(\\theta_A,\\theta_B)$ being calculated from measured on-axis intensity according to\n\\begin{align}\n& E(\\theta_A,\\theta_B) = \\frac{A(\\theta_A,\\theta_B)-B(\\theta_A,\\theta_B)}{A(\\theta_A,\\theta_B)+B(\\theta_A,\\theta_B)}.\\\\\n& A(\\theta_A, \\theta_B) = I(\\theta_A,\\theta_B)+I\\left(\\theta_A+\\frac{\\pi}{2},\\theta_B+\\frac{\\pi}{2}\\right),\\nonumber\\\\\n& B(\\theta_A, \\theta_B) = I\\left(\\theta_A+\\frac{\\pi}{2},\\theta_B\\right)+I\\left(\\theta_A,\\theta_B+\\frac{\\pi}{2}\\right).\\nonumber\n\\label{eq:BellParameterE}\n\\end{align}\nThe values of $S$ ranges from $|S|\\leq 2$ for separable states, up to $|S|= 2\\sqrt{2}$ for entangled or non-separable states. Our experimental results with according Bell parameters $|S|$ are presented in Fig. \\ref{fig:BellAnalysis}. As before, we performed our investigation without obstacle (a) as well as with different absorbing obstacles of radii $R = \\lbrace 150,\\, 200,\\, 500,\\, 600\\rbrace\\,\\mbox{\\textmu m}$ (b)-(f), or (d) phase-obstructing. Note that the intensity values are normalised with respect to the maximum value measured in the unobstructed case ($I'(\\theta_A, \\theta_B)$). Obviously, the ratio of measurable maximum intensity ($I'_{\\text{max}}(R)$) decreases dramatically depending on the size of the obstruction, as it can be seen on the $I'(\\theta_A, \\theta_B)$-axis (vertical) of Fig. \\ref{fig:BellAnalysis}. However, the Bell parameter does not change significantly if we measure in the fully reconstructed regime ($2z_{\\text{min}}<\\Delta z$ for (b)-(d)) compared to the undisturbed vBG mode (a). In accordance with our VQF analysis, $|S|$ reveals bigger changes if the beam is not fully self-healed when it is analysed (e), (f). In total, even if the intensity lowers to some percentage (f) of the original maximum value (a), all measurements validate a violation of the Bell inequality, matching our non-separability analysis results based on vectorness. \n\n\\subsection{The relation of self-healing and non-separability}\n \\begin{table}[h]\n \\caption{Quantification of non-separability properties of self-reconstructing vBG modes as function of the self-healing level, given by the obstacle radius $R$.  \\label{tab:Summarize}}\n \\begin{ruledtabular}\n \\begin{tabular}{c|c c c c c c c}\n$R$ in \\textmu m & $0$ & $150$ & $200$ & $200$, $\\pi$-obst. & $500$ & $600$& \\\\ \\hline \\hline\nVQF & $0.99$ & $0.98$ & $0.97$ & \\textcolor{gray}{$0.95$} & $0.94$ & $0.88$ &\\\\ \n$|S|$ & $2.81$ & $2.81$ & $2.79$ & \\textcolor{gray}{$2.79$} & $2.74$ & $2.75$& \\\\ \n$I'_{\\text{max}}$ & $1$ & $0.93$ & $0.82$ & \\textcolor{gray}{$0.53$} & $0.09$ & $0.03$ & \\\\\n \\end{tabular}\n \\end{ruledtabular}\n \\end{table}\n \n\\noindent In Table \\ref{tab:Summarize} we summarize our results emphasizing the dependence of the VQF, the maximum intensity $I'_{\\text{max}}$ as well as Bell parameter $|S|$ on the size $R$ of included obstructions, i.e. on the self-healing level. The maximum intensity $I'_{\\text{max}}(R)$ reveals an approximately Gaussian decrease with increasing obstacle size which reflects the Gaussian envelope of investigated vBG mode. Simultaneously to the intensity, the VQF as well as $|S|$ decrease as demonstrated in previous sections. However, only small changes are observed as only minor noise is disturbing vBG modes if obstructions are included. In short, both, vector quality as well as Bell analysis reveal a similar behavior of non-separability with respect to changes in the obstruction size, demonstrating the self-healing of the degree of entanglement within obstructed vBG beams. \n\n\n\\section{Conclusion and Discussion}\n\n\\noindent It is well-known that the phase and amplitude of vector Bessel beams self-heal in the presence of an obstacle that partially blocks its path. However, there are no reports about the effect of this on the coupling between phase and polarisation, known as classical entanglement. In this work, we presented for the first time to our knowledge experimental evidence that even though the coupling between these two degrees of freedom decreases after passing an obstruction, it eventually restores itself to its maximum value. For this purpose, we dynamically realized vector Bessel Gaussian (vBG) modes with digital obstructions by combining holography-based generation of structured light with a q-plate. We quantified the degree of non-separability between the spatial shape and polarisation using two different means: the vector quality factor (VQF) and a classical version of the CHSH Bell-like inequality. By a specific design of our detection system combined with digital variation of the obstruction size, different levels of self-healing were accessible, which enabled the relation between degree of non-separability and self-healing level to be analysed.\nThe measured VQF values showed that the degree of classical entanglement increases as function of decreasing object size. Analogously, the measured CHSH $S$ parameter values show a similar dependence on the objects size or self-healing level of the vBG beam, showing in all cases a clear violation of the CHSH inequality. This behavior can be interpreted as a self-healing in the degree of non-separability as function of the distance from an obstacle since a decrease in obstacle size is comparable to placing the detector further from the object.\\\\\nThe complex fields used in this study were separable in radial profile, defined by the $k_r$ vector of the conical waves, but non-separable in the azimuthal profile and polarisation, the latter being defined by the OAM of each polarisation component. Intriguingly, it is the radial profile that leads to the self-healing of the non-separability, despite itself being separable. Thus the local entanglement in angular momentum (spin and OAM) can be made more resilient to decay from obstructions by engineering the unused degree of freedom in a judicious manner.\n  \n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nTopological phases of matter are classified into two categories\\cite{Wen11,Wen21}: symmetry-protected topological\\,(SPT) phase and topologically ordered\\,(TO) phase. The SPT phase is distinguished by short-ranged entanglement, whereas the TO phase is distinguished by long-range entanglement. Furthermore, the SPT phase protects boundary gapless states, and it cannot be adiabatically connected to a trivial product state under perturbations preserving a certain symmetry. In contrast, in the TO phase, the global pattern of entanglement causes topological ground state degeneracy, which is robust to local perturbation regardless of symmetry. Furthermore, the topological degeneracy generates a universal subleading term in the entanglement entropy, which is known as the topological entanglement entropy (TEE)\\cite{Kitaev11, Levin11}. This entanglement entropy has been mainly used to detect the topological order\\cite{Bal}. Topological order frequently results in topological excitations with fractional quantum numbers. Entanglement entropy may also show signs of topologically ordered insulators\\cite{Haldane191}.\n\n\n\n\nRecently, it was revealed that undoped interacting disordered  graphene\\cite{Nov,Zhang,Neto} zigzag ribbons\\cite{Fujita} are a new TO Mott--Anderson insulator displaying $e^-\/2$ fractional charges,\nspin-charge separation, and two degenerate ground states\\cite{Jeong11,eeyang11}. The disorder is a singular perturbation that couples electrons on opposing zigzag edges, resulting in instantons. This effect converts zigzag ribbons from a STP to a TO phase and generates $e^-\/2$  fractional charges on the opposite zigzag edges. These fractional charges are protected\\cite{Girvin117} by an exponentially decaying soft gap\\cite{Efros,Mac1} $\\Delta_s$, as shown in Fig. \\ref{degGap}. Furthermore, numerical work\\cite{Kim11} showed that an interacting disordered zigzag nanoribbon has a finite TEE.\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Figure1.eps}\n\\caption{\\normalfont (\\textbf{a}) Schematic band structure of a disorder-free interacting zigzag graphene nanoribbon at half-filling with a hard Mott gap $\\Delta\\sim 0.1t$, where the hopping parameter $t\\sim 3$\\;eV. Soliton zigzag edge states are near $k=\\pm \\pi\/a_0$, and their charge is $e^-$ (the ribbon period is $a_0$). (\\textbf{b}) Schematic density of states (DOS) of a half-filled disordered zigzag ribbon (dashed line). It decays exponentially with an energy scale of $\\Delta_s$ (a soft gap), which decreases with increasing disorder strength\\cite{Jeong11}. The van Hove singularities of the DOS at $\\epsilon=\\pm \\Delta\/2$, originating from   the band structure displayed in a, are reduced due to the formation of the gap states. These gap states are spin--split, and many of them are soliton states with fractional charge $e^-\/2$. Note that the DOS has particle-hole symmetry after disorder averaging. The soft gap $\\Delta_s$ protects the fractional charges from quantum fluctuations.}\n\\label{degGap}\n\\end{center}\n\\end{figure}\n\n\n\n\nRecent advancements in fabrication methods have enabled the production of atomically precise graphene nanoribbons\\cite{Cai2,Kolmer}.  But it is unclear how to unequivocally measure the presence of fractional charges. We believe that doped zigzag nanoribbons are ideal for observing exotic anyons with fractional charges. The properties of doped disordered zigzag ribbons, on the other hand, are largely unknown.\nA doped ribbon is not expected to be a topologically ordered insulator because there is no hard gap (the density of states (DOS) at the Fermi is non-zero but small). However,\n  the system is still an insulator with localized edge states near the Fermi energy, displaying doubly degenerate ground states. In the dilute limit the added fractional charges will still be well defined. Let us explain this, following  Ref.\\cite{Girvin117}. These fractional charges are analogous to quasiparticles of the fractional quantum Hall effect's $1\/m$ Laughlin state ($m$ is an odd integer). In such a system's low doped regime, the added electrons divide into fractional charges.  \\textcolor{black}{Recent experimental works provide evidence for these anyons\\cite{Nakamura01,Barto1}.}   Suppose one adds $\\delta N$ electrons to such a state. In the dilute limit, each of these electrons fractionalizes into $m$ quasiparticles that are well separated from each other (the charge of a quasiparticle is $e^-\/m$). The total energy of the new system is thus $E_-=E_m+\\delta N m \\Delta_-$, where $E_m$ is the ground state energy and $\\Delta_{-}$ is the quasiparticle excitation energy. \\textcolor{black}{Despite that the quasiparticles form quasi-degenerate states, the} excitation gap $\\Delta$ and localization of quasiparticles protect fractional charges against quantum fluctuations~\\textcolor{black}{\\cite{eeyang11,Girvin117}}.\n \n \nThe role of fractional charges in low doped disordered systems is one of the fundamental questions in doped disordered zigzag ribbons. What exactly is the ground state? This concerns the applicability of mean field approaches to such a system: quantum fluctuations\\cite{Girvin117} not included in the Hartree--Fock (HF) approximation may be significant because gap states are no longer empty.\nFurthermore, in contrast to the uniform spin density of undoped ribbons, the ground state of a doped disorder-free ribbon exhibits an edge spin density wave. It is unknown how localization and charge quantization affect the nature of the ground state.\nWe use the density matrix renormalization group \\,(DMRG) approach in the matrix product states (MPS) representation to investigate the ground state of a doped ribbon and the importance of quantum fluctuations beyond the HF approach. The MPS representation is a powerful tool for solving eigenvalue problems of quantum many-body systems\\cite{White1992,Schollwock2011}.\n\n\n\n\nRibbons are in a new disordered anyon phase, according to our investigation of the low doping regime\\cite{Lei13,Wilczek03}.\nWe discover that a low doped disordered zigzag ribbon contains a large number of anyons with a fractional charge (but as doping concentration increases they disappear).\nThey cause numerous magnetic domain walls and localized magnetic moments residing on the zigzag edges.\nAlso, objects that display spin-charge separation proliferate in this phase.\nAs a result, the ground state is drastically reorganized, with highly distorted edge charge and spin modulations, as well as non-local correlations between the left and right zigzag edges.\nWe will define this new phase as a disordered anyon phase because its electron and spin densities are highly inhomogeneous.  Furthermore, we make the following new experimentally testable predictions.\n(1) The disordered anyon phase has an unusual shape of tunneling density of states (TDOS), depending on the number of extra electrons \\textcolor{black}{(for experimental measurement of a soft gap in the TDOS, see, for example, Refs. \\cite{Ashoori01,JPEisen})}.\nThe TDOS has one sharp peak at the midgap energy and two other peaks, one on each side of the sharp peak at the midgap energy, at the low doping limit. (2) However, the midgap peak disappears as the doping concentration increases. The detection of these peaks will provide strong evidence for the presence of $e^-\/2$ fractional charges. Furthermore, our findings indicate that doped zigzag ribbons may have unusual transport, magnetic, and inter-edge tunneling properties. \\textcolor{black}{Theoretical calculations\\cite{Pisa1} of disorder-free zigzag ribbons show \\textcolor{black}{that} antiferromagnetism is favored over ferromagnetism  for ribbon widths $< 100$\\AA. In the presence of disorder, the new disordered anyon phase is expected for these width values.} \n\\section*{Results}\n\\subsection*{Model}\nTo model the graphene zigzag nanoribbons, we apply the Hubbard model with the nearest neighbor hopping and a diagonal disorder $V_i$.\n\\begin{eqnarray}\n H=-t\\sum_{\\langle ij \\rangle,\\sigma} c^{\\dag}_{i,\\sigma}c_{j,\\sigma} +\\sum_{i,\\sigma} V_i c_{i,\\sigma}^{\\dag}c_{i,\\sigma}\n +U\\sum_i n_{i,\\uparrow} n_{i,\\downarrow},\n\\label{Hubbard}\n\\end{eqnarray}\nwhere $i=(x,y)$ denotes the site indices (see Fig. \\ref{dmrg-res}a), $c^{\\dagger}_{i,\\sigma}$\/$c_{i,\\sigma}$ are the creation\/destruction operators at site $i$, $t$ is the nearest neighbor hopping parameter and  $U$ is the on-site repulsion. The ratio of the numbers of impurities and carbon atoms is given by $n_{imp}=N_I\/N_s$. The values of the disorder strength $V_i$ at $N_I$ impurity sites  are  uniformly distributed in the interval \n$ [-\\Gamma,\\Gamma]$ (the sum in the second term of $H$ is only over impurity sites). The dimensionless coupling constant of the problem is the ratio of the disorder strength and on-site repulsion $g= \\frac{\\Gamma\\sqrt{n_{imp}}}{U}$. The doping concentration is defined as $\\delta N\/N_s$, where $\\delta N$ is the total number of added electrons. The mean field version of this Hamiltonian for a doped ribbon is given  in method, see Eq. (\\ref{MFhspin}).\nThe HF results of undoped zigzag ribbons show that fractionalization occurs  independent of the disorder potential range, density, and strength. Note that disorder is a singular perturbation\\cite{Jeong11,eeyang11}.\n\n\n\n\n\n\n\nIn graphene systems mean field approximations are widely used because they give accurate results\\cite{Stau}.   \\textcolor{black}{ However, there are several nearly degenerate HF ground states in graphene nanoribbons that can be generated using different HF initial states,and one does not know which of these states is close to the true ground state because quantum fluctuations are missing in the HF approximation. In this paper, we conducted the DMRG to determine which HF initial state generates the  HF state that is close to the true ground state. We will concentrate on two types of HF initial states in this section. The first, labeled AF, is generated from an antiferromagnetic initial state, while the second, labeled PM, is generated from a paramagnetic initial state with a small spin-splitting.}\nThe DMRG found that the \\textcolor{black}{undoped} ground state at clean limit exhibits the N\\'eel magnetic ordering, where spins at two zigzag edges align antiparallel to each other \\textcolor{black}{(these results agree with those results obtained using the AF initial state)}. Nonetheless, the addition of enough extra electrons results in an edge spin density wave. The corresponding DMRG results, presented in Supplementary material,    \\textcolor{black}{ agree with those results obtained using  the PM initial state, see Fig. \\ref{dmrg-res}b.}\nThe results were then tested at half-filling for an undoped disordered interacting zigzag nanoribbon. The PM initial state produces a state with fractional charges. Site spin values computed from this ground state agree well qualitatively with those of the DMRG approach, as shown in \\textcolor{black}{Figs.~\\ref{dmrg-res}c,d.} We will show below\nthat DMRG results for doped interacting disordered zigzag ribbons also support the results obtained from  the PM initial state. \\textcolor{black}{The methods of both DMRG and HF approximation are explained in detail in Supplementary material.}\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Figure2.eps}\n\\caption{\\normalfont (\\textbf{a}) Vertical and horizontal lines of carbon atoms are numbered. All lengths and widths in this paper are measured in the number of these lines. (\\textbf{b})  Site spin values $ s_{iz}$ of a disorder-free  doped zigzag ribbon. This state is generated from  $N_e=N_s+20$ ($\\delta N\/N_s=0.017$), $L_x=301$, $L_y=4$, and $U=t$. (\\textbf{c}) DMRG result of the ground state site spins $s_{iz}$ at half-filling for $U=t$, $n_{imp}=1$, and $\\Gamma=0.5t$ ($g=0.5$). We discover that other spin components $s_{ix}$ and $s_{iy}$ are very small. (\\textbf{d}) HF site spin values  at half-filling are shown.  Here, $U=t$, $n_{imp}=0.1$, and $\\Gamma=0.5t$ ($g=0.16$ is smaller compared with the value used in (\\textbf{c})).}\n\\label{dmrg-res}\n\\end{center}\n\\end{figure}\n\n\\subsection*{New Anyon Phase and TDOS of Low Doping Region}\nUsing the PM initial state, we investigated the shape of the TDOS as a function of doping concentration \\textcolor{black}{(all the HF results below are generated by using this HF initial state).}  As shown in Fig. \\ref{SolPhas}, adding a few extra electrons to the half-filled ribbon results in a sharp peak near the midgap energy $E=0$ inside an exponentially decaying small soft gap. The peak's physical origin is as follows: A tunneling electron enters into a soliton state and divides into two fractionally charged quasiparticles\n\\begin{equation}\ne^-\\rightarrow e^-\/2+e^-\/2.\n\\end{equation}\n(Ref.\\cite{Girvin} gives a good account of this process). A soliton state is described by a non-local wave function, as shown in the upper left inset of Fig. \\ref{SolPhas}.\nThe width of the central peak is $\\sim 0.02\\Delta\/2$.\n\\textcolor{black}{In the low doping limit, when an entering electron has a non-zero energy $E\\neq 0$, it has a significant chance not to split into $e^-\/2$ charges because fractionalization is only approximate at non-zero energies\\cite{eeyang11}. The lower left inset of Fig. \\ref{SolPhas} shows the highly non-linear dependence of the peak value at E=0 on doping concentration $\\delta N\/N_s$. The zero energy peak in the DOS disappears for $\\delta N\/N_s>0.005$ (the shape of this DOS will be shown below). Such non-linear behavior is unusual and provides compelling evidence for fractional charges. The shape of the TDOS at the low doping limit differs significantly from that of the half-filled undoped state} (there are also two side peaks, one on each side of the sharp peak at the midgap energy. These peaks are not shown in Fig. \\ref{SolPhas} because their energies  lie outside the energy range $|E|>0.05\\Delta\/2$). Edge site occupation numbers and site spins are displayed in Fig. \\ref{SolPhas3}. These findings show modulated ferromagnetic edges.\n\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure3.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$), $\\Gamma=0.01 t $, $L_x=101$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ $(g=0.0032)$. DOS of a slightly doped ribbon away from half-filling is shown \\textcolor{black}{in the weak disorder regime}. Sharp peak is present inside the soft gap at the midgap energy $E=0$  (the magnitude of this peak is rather {\\it small} in comparison to the peaks at $E=\\pm\\Delta\/2$ shown in Fig. \\ref{degGap}). Since there are excess electrons, the Fermi energy $E_F\/(\\Delta\/2)=0.14 $ is {\\it above} the mid gap energy. \\textcolor{black}{The DOS of $L=300$ in a larger  energy interval $E<\\Delta$  is shown in the lower right inset.}\nA charge fractionalized HF eigenstate is shown in the upper left inset. Note that energy is measured in units of $\\Delta\/2$. The number of disorder realization is $N_D\\sim 400$. A tunneling electron is fractionalized in the upper right inset. The DOS is determined by measuring the differential I-V.  \\textcolor{black}{Lower left  inset displays  the dependence of the midgap peak on doping concentration for $L=100$.}}\n\\label{SolPhas}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure4.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$), $\\Gamma=0.01 t $, $L_x=101$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ $(g=0.0032)$. Their disorder-free values are represented by dashed lines. (\\textbf{a}) A disorder realization of zigzag edge site occupation numbers $n_{i\\sigma}$ for a doped ribbon. (\\textbf{b}) Total site occupation numbers $n_i$ are shown. Some charges are transferred between the zigzag edges on the left and right. (\\textbf{c}) Site spins $s_{iz}$ are plotted. Their disorder-free values are represented by dotted lines.}\n\\label{SolPhas3}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\pagebreak\n\n Many of the added electrons have $q_A$ values $\\sim 1\/2$,  which can be seen by comparing Figs. \\ref{SolPhas1}a,b (the quantity $q_A$ gives the total probability to find an electron with energy $E$ on A-type carbon atoms. Charge fractionalization occurs when $q_A=1\/2$). As a result, the extra electrons enter soliton states with well-defined fractional charges. Thus, our results for low doped ribbons indicate that doping does not destroy anyons. The average energy cost to create\\cite{Girvin117} an $e^-\/2$ fractional charge from the undoped ground state with an exponentially small gap is $\\Delta_s\/2$, which corresponds to the midgap energy $E=0$ in the excitation spectrum. This effect is similar to the formation of polyacetylene soliton midgap states\\cite{yang1}. An undoped zigzag ribbon, unlike the chiral edges of Laughlin fractional quantum Hall states, lacks significant gapless edge excitations.\n\n\n\n\n\n\n\n Following  Ref.\\cite{Wilczek03}  let us argue that an $e^-\/2$ fractional charge of a disordered ribbon is an anyon.\nConsider two single-particle HF mixed chiral states that display $e^-\/2$ fractional charges, as shown in Fig. \\ref{semi}. If we exchange these two electrons, the total many-body wave function of $N$ electrons acquires a statistical phase of $e^{i\\pi}=-1$. This exchange is also equivalent to exchanging two $e^-\/2$ charges on the left zigzag edge and two others on the right zigzag edge. Thus, we expect that each of these exchanges generates the statistical phase of $e^{i\\pi\/2}$ to yield the final phase of $e^{i\\pi}=-1$. An anyon with the statistical phase $e^{i\\pi\/2}$ is called \\textcolor{black}{a semion\\cite{Kalme,Canri}.}\nThe presence of semions is consistent with the presence of anyons in TO phases.\nIt is also consistent with \nthe shape of the entanglement spectrum.\n It is believed that the ground state entanglement spectrum of a TO phase resembles the corresponding edge spectrum of the system\\cite{Haldane191,Van,fid}.\n The shape of the entanglement spectrum is computed and found to be similar to the DOS of the edge states (see Supplementary material).  The entanglement spectrum of undoped interacting disordered zigzag ribbons differs from that of zigzag ribbons in the disorder-free SPT phase\\cite{Kim11}.\n\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Figure5.eps}\n\\caption{\\normalfont $q_A$ values of the HF eigenstates of a ribbon are plotted for (\\textbf{a}) $N_e=N_s$  and (\\textbf{b}) $N_e=N_s+3$ ($\\delta N\/N_s=0.0037$). A gap state electron with $q_A=1\/2$ is fractionalized. Here $L_x=101$, $L_y=8$, $N_D=400$, $n_{imp}=0.1$, $U=t$, $\\Gamma=0.01t$, and $N_e=N_s+3$ $(g=0.0032)$. In case of (\\textbf{b}), the spectrum does not have particle-hole symmetry. }\n\\label{SolPhas1}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{Figure6.eps}\n\\caption{\\normalfont  The HF results for Two HF mixed chiral states are shown. Exchanges of two $e^-\/2$ charges on the left zigzag edge and two others on the right zigzag edge are displayed.}\n\\label{semi}\n\\end{center}\n\\end{figure}\n\nAs more electrons are added, the sharp peak at the midgap energy  {\\it disappears} in the DOS, but the two side peaks near $E\\sim \\pm 0.05\\Delta\/2 $ {\\it persist}, as shown in Fig. \\ref{SolPhas0}.\nSimultaneously, the edge occupation number profile becomes highly nonuniform, as shown in Fig. \\ref{SolPhas2}a.  These edge occupation numbers and site spin profiles appear to be quite different from those with fewer electrons, see Figs.~\\ref{SolPhas3}a-c.\nThe nature of the disordered ground state of the doped system is as follows. To begin, it is important to note that the ground state of doped disorder-free zigzag ribbons differs from that of undoped ribbons, which have ferromagnetic edges that are antiferromagnetically coupled.  In sufficiently doped disorder-free ribbons both the DMRG (see Supplementary material) and HF display {\\it spin density} type periodic modulations on the zigzag edges, see Fig. \\ref{dmrg-res}b (the opposite edges are still antiferromagnetically coupled). The ground state changes again in the presence of disorder, and the periodic spin density is destroyed (a disorder potential is a {\\it singular} perturbation\\cite{eeyang11}). As the vertical dashed lines in Fig. \\ref{SolPhas2}a show many HF values of $s_{iz}$ of each zigzag edge change\nsign at sites where $n_i$ abruptly changes (see Fig. \\ref{SolPhas2}b).\nSimilar behavior is also observed in the DMRG calculation, as shown in Fig. \\ref{SolPhas2}c. \nMoreover, sites $i$, where the values of $n_{i\\sigma}$ and $s_{iz}$ abruptly change, have almost identical values for the $x$-coordinate on the left and right zigzag edges. This effect is a consequence of the {\\it nonlocal} correlation between the left and right zigzag edges. This correlation between opposite zigzag edges strongly suggests that the formation of non-local soliton states is responsible for the drastic reorganization of the ground state, as well as the zigzag edge modifications.\nFurthermore, we find {\\it local magnetic moments} with non-zero values of $s_{iz}$ that is extended over several sites.\nThese objects\n proliferate in comparison to the case of undoped ribbons.  There are also objects extended over several sites with rather small values of\n$s_i=\\frac{1}{2}(n_{i\\uparrow}-n_{i\\downarrow})\\approx 0$, see Fig. \\ref{SolPhas2}b. In such an object, {\\it spin-charge separation} would take place. The following procedure is used to create these objects. An $e^-\/2$ fractional charge moves along the zigzag edges from left to right, while another fractional charge with the opposite spin moves in the opposite direction (see Ref.\\cite{eeyang11} for a detailed explanation).\n\n\n\nThe resulting ground state displays a highly distorted edge spin density.\nThis phase is characterized by localized edge magnetic moments, spin-charge separation, and correlation between the left and right zigzag edges, a {\\it disordered anyon phase} of zigzag nanoribbons.  \\textcolor{black}{Charge fractionalization is not exact in this phase since some of the nearly zero energy states are not fractionalized, see the left inset in Fig. \\ref{SolPhas0}. This is in contrast to the results of slightly doped and undoped disordered ribbons, where the fractional charge of zero energy  states is well-defined (see Fig. \\ref{SolPhas} of the current manuscript and Fig.9 in Ref.\\cite{eeyang11}, respectively).}\nWhen doping concentration is increased further ($\\delta N\/N_s\\sim 0.04$), the distorted edge spin density wave   and   charge fractionalization almost disappear. We also discover that the HF gap states are no longer localized along  the ribbon direction.   \\textcolor{black}{ These findings imply that as doping concentration increases from zero, a topological phase transition with a significant crossover region occurs.}\n\n\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Figure7.eps}\n\\caption{\\normalfont  The HF results for $N_e=N_s+20$ ($\\delta N\/N_s=0.0083$), $\\Gamma=0.06 t$, $L_x=301$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ ($g=0.019$). Two side peaks on the DOS. In the limit of large ribbon length or, equivalently, in the limit of zero doping, the profile of these two peaks becomes symmetric. Since there are excess electrons, the Fermi energy $E_F\/(\\Delta\/2)=0.46$. The number of disorder realization is $N_D\\sim 200$.}\n\\label{SolPhas0}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{Figure8.eps}\n\\caption{\\normalfont The HF results for $N_e=N_s+20$ ($\\delta N\/N_s=0.0083$), $\\Gamma=0.06 t$, $L_x=301$, $L_y=8$, $n_{imp}=0.1$, and $U=t$ ($g=0.019$):   (\\textbf{a}) Vertical lines indicate sites where $n_{i}$ or $s_{iz}$ abruptly change. On the zigzag edges, there are numerous localized magnetic moments. (\\textbf{b}) Site occupation numbers $n_{i\\sigma}$ of a disordered ribbon. The arrows point to locations where spin-charge separation occurs. (\\textbf{c}) DMRG results for $\\delta n_i \\equiv n_i - n_i^{\\rm clean}$ plotted as a function of $x$ and $y$. Here, $n_i^{\\rm clean}$ is the site occupation number for $\\Gamma=0$. Site spins $s_{iz}$ are also shown. The parameters are as follows:\n$N_e=N_s+12$ ($\\delta N\/N_s=0.025$), $\\Gamma=t$, $L_x=120$, $L_y=4$, $n_{imp}=0.2$, and $U=t$. Note that the length of this ribbon is considerably shorter than the one used in (\\textbf{a}) and (\\textbf{b}). These are the results for a more strongly disordered ribbon with $g=0.45$, and the overall magnetization is significantly reduced.  }\n\\label{SolPhas2}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\\newpage\n\\section*{Discussion}\nQuantum fluctuations beyond the HF approximation do not mitigate charge fractionalization in a ribbon at low doping concentration, according to our findings. Despite the presence of disorder, as doping increases, the edge magnetic ordering weakens and charge fractionalization disappears. Furthermore, we discovered that the low doped state is a new disordered anyon phase with highly distorted edge charge and spin modulations, as well as localized magnetic moments with non-local correlations between the left and right zigzag edges. Anyons play a key role in the formation of this new phase.\n\n As a result of spin-charge separation, our findings suggest that doped zigzag nanoribbons may exhibit new magnetic and low temperature transport properties: the conductivity may display a usual behavior while the spin susceptibility may be rather small, as was observed in polyacetylene\\cite{Chung15}.\nFurthermore, we demonstrated that the TDOS profile is significantly affected by doping concentration. The measurement of the differential  I-V curve may reveal this effect and may provide a strong test for the presence of $e^-\/2$ fractional charges. \\textcolor{black}{Ribbons with width less than $100$~\\AA\\,  are\nwell-suited for the observation of these fractional charges as the  antiferromagnetic phase  is more stable than  the ferromagnetic phase\\cite{Pisa1}.}\n\n\n\nThe following additional investigations may be interesting to pursue. A worthwhile but challenging task is to compute the anyon statistical phase using a microscopic approach. Recently such an adiabatic DMRG simulation, utilizing the quantized Hall response,  was successfully conducted\\cite{Zhu} for the non-Abelian Moore-Read state on a Haldane honeycomb lattice model.\nA similar DMRG calculation in a Mott-Anderson insulator of disordered zigzag nanoribbon with Abelian quasiparticles, where electron localization is critical, is not clear.\nAnother method is to compute the statistical Berry phase of Abelian quasiparticles using a trial wave function\\cite{Arovas}. But a good trial wave function is not yet available for disordered zigzag ribbons.\n\n\\textcolor{black}{In the limit of small doping and in weak disorder regime, ribbons  with a sharp midgap peak in the DOS have a universal value of the  TEE\\cite{Kim11}. When doping is high enough, the midgap peak disappears, as does the exact fractionalization of zero energy states. When this occurs, we can expect non-universal TEE values. It may be worthwhile }to  probe the topological phase transition as  doping concentration increases. The following issues must be addressed: \\textcolor{black}{Does} the TEE of doped disordered ribbons decay to zero as a function of doping concentration, and does the transition exhibit non-universal dependence on physical parameters\\cite{Kim11}?  This investigation  may require very high cpu resources to compute accurately small values of the TEE\\cite{Kim11}. This type of calculation could provide more information about the phase transition from modulated ferromagnetic edges at zero doping to distorted spin-wave edges at finite doping\n\nThe following experiments would also be fascinating. Investigation of tunneling between zigzag edges, as seen in fractional quantum Hall bar systems \\cite {Kang}, may be fruitful.\n Scanning tunneling microscopy can reveal the presence of fractional charges by measuring the electron density on the zigzag edges\\cite{Andrei}. Finally, it would be interesting to look into the new disordered anyon phase in other antiferromagnetic zigzag nanoribbon systems, e.g., silicene and boron nitride nanoribbons \\cite{Yao,Bar}. Chiral gauge theory can be used to describe $e^-\/2$ fractional charges\\cite{ChiralTH}.  It would be fascinating to look into the new anyon phase using random chiral gauge fields.\n\\section*{Methods}\n\\subsection*{HF Approximation}\nTo model graphene zigzag nanoribbons, the mean field Hubbard model is commonly used \\begin{eqnarray}\nH_{MF}=-t\\sum_{\\langle ij\\rangle,\\sigma} c^{\\dag}_{i,\\sigma}c_{j,\\sigma} +\\sum_{i,\\sigma} V_ic_{i,\\sigma}^{\\dag}c_{i,\\sigma}\n+U\\sum_i[ n_{i,\\uparrow}\\langle n_{i,\\downarrow}\\rangle +n_{i,\\downarrow}\\langle n_{i,\\uparrow}\\rangle-\\langle n_{i,\\downarrow}\\rangle\\langle n_{i,\\uparrow}\\rangle]\n+ \\sum_i [s_{ix} \\langle h_{ix}\\rangle+s_{iy} \\langle h_{iy}\\rangle]\n\\label{MFhspin}\n\\end{eqnarray}\nwhere $ \\langle h_{ix}\\rangle=-2U\\langle s_{ix}\\rangle$ and $ \\langle h_{iy}\\rangle=-2U\\langle s_{iy}\\rangle$ are the self-consistent ``magnetic fields.\"\nThe last term of Eq. (\\ref{MFhspin}) describes spin flips and is present but only separately from half-filling. This term mixes spin-up and spin-down. Note that the band structure no longer has particle-hole symmetry when away from half-filling.\\\\\n\n\\subsection*{DMRG}\nWe apply the DMRG\\cite{White1992, Rommer,Schollwock2011} to obtain the ground state of the model Eq.~(\\ref{Hubbard}) in the MPS representation.\nFurthermore, we illustrate the geometry of the MPS for the graphene zigzag nanoribbon of the size $(L_x \\times L_y)$ (see Supplementary material).\nFor a quasi-one-dimensional system, the complexity of the DMRG scales exponentially in the width of the system\\,($L_y$), whereas it scales polynomially in the length\\,($L_x$) of the system. Therefore, our MPS setup allows us to consider the graphene strip with long zigzag edges, and we focus on the system with $(L_x,L_y) = (120,4)$, which is far beyond the reach of exact diagonalization in the present calculations. The precision of the DMRG can be controlled by the number of basis states kept or the maximum bond dimension of the MPS ($\\chi_{\\rm max}$), and we use up to $\\chi_{\\rm max} = 1600$ to achieve the typical error of the total energy lower than $10^{-6}$ (For a short introduction to MPS, see Supplementary material). To fix a gauge redundancy of MPS specially, i.e., the canonical form, the DMRG optimizes each tensor considering the global information of the wave function, which makes the algorithm extremely stable and reliable\\cite{Schollwock2011}. Nonetheless, the DMRG can become trapped in a local minimum, particularly for models with a quasi-one-dimensional lattice. To avoid local minima, we apply the noise perturbation\\cite{White05} with the two-site algorithm\\cite{Schollwock2011} at each optimization step.\nMoreover, we exploit the $U(1)$ symmetry of the model such that the DMRG preserves the total number of electrons, e.g., $\\sum_{i,\\sigma} c^\\dag_{i,\\sigma} c_{i,\\sigma} = N_s\/2$, thereby improving greatly its convergence speed and accuracy\\cite{Schollwock2011}.\n\n\\section*{Data availability}\nOn reasonable request, the corresponding author will provide all relevant data in this paper.\n\n\\section*{Code availability}\nOn reasonable request, the corresponding author will provide all numerical codes in this paper.\n\n\n\\section{DMRG.}\n \n\n\n\\section{Matrix Product States}\n\nA large class of quantum many-body wavefunctions, $\\Psi$, can be efficiently factorized into a product of tensors as follows:\n\\begin{align}\n\t|\\psi\\rangle &= \\sum_{ \\{ s_i \\} } \\Psi_{\\cdots s_{i-1} s_i s_{i+1} \\cdots} \\,\\, \\big| \\cdots s_{i-1} s_i s_{i+1} \\cdots \\big\\rangle \\nonumber\\\\\n\t&= \\sum_{ \\{ s_i \\} } {\\rm tTr}[ \\cdots A_{i-1}^{s_{i-1}} A_i^{s_i} A_{i+1}^{s_{i+1}} \\cdots ] \\big|\\cdots s_{i-1} s_i s_{i+1} \\cdots \\big\\rangle,\n\\end{align}\nwhere $A_{i}^{s_i}$ stands for a tensor at site $i$, $\\cdots A_{i-1}^{s_{i-1}} A_i^{s_i} A_{i+1}^{s_{i+1}} \\cdots$ indicates the product of tensors forming a network, depending on how tensors are connected, and ${\\rm tTr[\\cdots]}$ stands for the tensor trace or contraction of all connected indices in the network. Here the index $s_i$ denotes a local state at site $i$, e.g., $|s_i\\rangle = |\\uparrow\\rangle, |\\downarrow\\rangle$ for a spin-half fermion system or $|s_i\\rangle = |0\\rangle, |\\uparrow\\rangle, |\\downarrow\\rangle, |\\uparrow\\downarrow\\rangle$ for a spin-full fermion system. Particularly, the  so-called MPS, which is a chain-like product of rank-3 tensors, may represent quantum states for one-dimensional and quasi-one-dimensional systems accurately. Specifically, the many-body wave function in the MPS representation is written as follows:\n\n\\begin{align}\n\t\\Psi_{\\cdots s_{i-1} s_i s_{i+1} \\cdots}\t\n \t= \\sum_{\\{l_i\\}, \\{r_i\\}} \\cdots \\,\\, \\delta_{r_{i-2} l_{i-1}} \\,\\, [A_{i-1}^{s_{i-1}}]_{l_{i-1}, r_{i-1}} \\,\\, \\delta_{r_{i-1} l_i} \\,\\, [A_i^{s_i}]_{l_i, r_i} \\,\\, \\delta_{r_i l_{i+1}} \\,\\, [A_{i+1}^{s_{i+1}}]_{l_{i+1}, r_{i+1}} \\,\\,\\delta_{r_{i+1} l_{i+2}} \\,\\, \\cdots,\n\\end{align}\n\n\nwhere $[A_i^{s_i}]_{l_i, r_i}$ is a rank-3 tensor with two virtual indices $l_i$ and $r_i$\\,(say left and right, respectively), which are traced out, $s_i$ is the physical index, and $\\delta_{ij}$ stands for the Kronecker delta or the identity matrix. Note that the Kronecker delta contracts the right and left indices of tensor at $i$ and $i+1$, respectively. Hence, the element of the wavefunction for a given set of $\\{s_i\\}$ is identical to the product of {\\it matrices} $\\{A_i^{s_i}\\}$, i.e., $\\Psi_{\\cdots s_{i-1} s_i s_{i+1} } = \\cdots A_{i-1}^{s_{i-1}} A_i^{s_{i}} A_{i+1}^{s_{i+1}} \\cdots$. It is also convenient to introduce a graphical representation for the tensor and its network. A tensor is depicted by an object with open legs denoting its indices. For instance, the tensor $A_i^{s_i}$ can be illustrated as follows:\n\\begin{align}\n\t\\includegraphics[width=0.2\\textwidth]{mps_tensor.eps},\n\\end{align}\nwhere the vertical open leg $s_i$ denotes the physics index and the horizontal ones $l_i$ and $r_i$ stand for the virtual indices. Furthermore, a contraction of two indices, particularly one left and one right horizontal indices, occurs by connecting the legs\\,(or indices) as shown below:\n\\begin{align}\n\t\\includegraphics[width=0.4\\textwidth]{contraction.eps}.\n\\end{align}\nThus, the total wavefunction in the MPS representation is illustrated as follows:\n\\begin{align}\n\t\\includegraphics[width=0.48\\textwidth]{mps.eps}.\n\\end{align}\nHere, the size of the matrix $A_i^{s_i}$, which is referred to as the bond dimension $\\chi$, determines the expressibility of the MPS. In other words, the accuracy of the MPS can be systematically enhanced by increasing the value of $\\chi$. We define the MPS on the zigzag nanoribbon in a so-called snake pattern, as illustrated in Fig.~\\ref{fig:mps_geometry}. This definition allows us to consider the graphene strip with long zigzag edges.\n\n\\begin{figure}[htpb]\n\\begin{center}\n\t\\includegraphics[width=0.4\\textwidth]{mps_geometry.eps}\n\\caption{Schematic figure of the snake pattern of the matrix product state defined on the zigzag nanoribbon lattice.}\n\\label{fig:mps_geometry}\n\\end{center}\n\\end{figure}\n\n\n\nSimilarly, a large class of Hamiltonians can be represented exactly as a product of matrices or as the matrix product operator\\,(MPO) as follows:\n\\begin{align}\n\t\\hat{H} &= \\sum_{\\{s_i\\},\\{s_i'\\}} H_{\\cdots s_{i-1} s_i s_{i+1} \\cdots}^{\\cdots s_{i-1}' s_i' s_{i+1}' \\cdots} \\,\\,\\big|\\cdots s_{i-1}' s_i' s_{i+1}' \\cdots \\big \\rangle \\big\\langle \\cdots s_{i-1} s_i s_{i+1} \\cdots \\big| \\nonumber \\\\\n\t& = \\sum_{\\{s_i\\},\\{s_i'\\}} {\\rm tTr}[ \\cdots W_{i\\!-\\!1,s_{i\\!-\\!1}}^{s_{i\\!-\\!1}'} W_{i,s_{i}}^{s_{i}'} W_{i\\!+\\!1,s_{i\\!+\\!1}}^{s_{i\\!+\\!1}'} \\cdots ] \\big|\\cdots s_i' \\cdots \\big \\rangle \\big\\langle \\cdots s_i \\cdots \\big|,\n\\end{align}\nwhere $W_{i,s_i}^{s_i'}$ is a rank-4 tensor with its graphical representation given as\n\\begin{align}\n\t\\includegraphics[width=0.2\\textwidth]{mpo_tensor.eps}.\n\\end{align}\nThen, the Hamiltonian is represented in the graphical representation as follows:\n\\begin{align}\n\t\\includegraphics[width=0.48\\textwidth]{mpo.eps}.\n\\end{align}\nFor example, the Hamiltonian of the one-dimensional Anderson--Hubbard model is produced with the tensor $W_{i,s}^{s'}$ as follows:\n\n\\begin{align}\n\t[W_{i,s}^{s'}]_{lr} = \t\n\t\\begin{bmatrix}\n\t\t(I_i)_{ss'}\t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\uparrow}^\\dagger)_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\downarrow}^\\dagger)_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\uparrow})_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(c_{i\\downarrow})_{ss'} \t& 0 & 0 & 0 & 0 & 0 \\\\\n\t\t(U n_{i\\uparrow} n_{i\\downarrow} + v_i (n_{i\\uparrow} + n_{i\\downarrow}))_{ss'} & t (c_{i\\uparrow})_{ss'} & t (c_{i\\downarrow})_{ss'} & -t (c_{i\\uparrow}^\\dagger)_{ss'} & -t (c_{i\\downarrow}^\\dagger)_{ss'} & (I_i)_{ss'}\n\t\\end{bmatrix}_{lr},\n\\end{align}\n\nwhere $I_i$ is the trivial operator acting on site $i$, $c_i^{(\\dagger)}$ is the annihilation\\,(creation) operator, $v_i$ is the Anderson random potential, and $U$ is the Hubbard interaction. One can easily generalize the above $W-$tensor and MPO into the proper quantities for the model of the quasi-one-dimensional honeycomb lattice.\n\nThe total energy of a given wavefunction is given as\n\\begin{align}\n\tE = \\frac{\\langle \\Psi | H | \\Psi \\rangle}{\\langle \\Psi | \\Psi \\rangle}\n\t= \\frac{\\sum_{\\{s_i\\},\\{s_i'\\}} (\\Psi_{\\cdots s_i' \\cdots})^* H_{\\cdots s_i \\cdots}^{\\cdots s_i' \\cdots} (\\Psi_{\\cdots s_i \\cdots})}\n\t{\\sum_{\\{s_i\\}} (\\Psi_{\\cdots s_i \\cdots})^* (\\Psi_{\\cdots s_i \\cdots})},\n\\end{align}\nand this total energy is graphically recast as\n\\begin{align}\n\t\\includegraphics[width=0.4\\textwidth]{energy.eps}.\n\\end{align}\nThus, to measure the total energy $E$, one should contract two different tensor networks in the denominator and numerator in the above equation. The contraction of tensor networks can be efficiently conducted by choosing the order of the contraction properly~\\cite{Schollwock2011}. With the above expression, one can apply a variational principle to optimize tensors $\\{A_i\\}$ by minimizing the energy, e.g.,\n\t$\\partial E\/\\partial A_i^* = 0$,\nThereby leading to an eigenvalue problem to update the tensor $A_i$\\,(see Ref.~\\cite{Schollwock2011} for more details).\n\n\n\n\n\\subsection{Density Matrix Renormalization Group: Clean limit}\n\n\n\\begin {figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{idmrg_scaling.eps}\n\\caption{The bond dimension scaling of entanglement entropy (a), energy density (b),  edge magnetization ($\\hbar=1$) (c),  and  correlation length (d).  Here  $U\/t=1$ is used.   The energy  unit is $t$ and the length unit is $\\sqrt{3}a$, where  $a$ is the carbon-carbon distance.}\n\\label{fig:idmrg_scaling}\n\\end{center}\n\\end{figure}\n\n\nAs discussed in the previous section, the bond dimension of the MPS determines the amount of the MPS. The number of MPS should be carefully chosen to appropriately represent the ground state. Specifically, the MPS is optimal for representing quantum states that satisfy the area law of the entanglement, i.e., its entanglement entropy scales as the size of the boundary between two subsystems. By contrast, the entanglement entropy of a gapless ground state diverges logarithmically with the subsystem size. Thus, we first check whether the ground state of our Hamiltonian is gapped or gapless. Then, we determine how large the bond dimension should be to appropriately represent the ground state. To this end, we perform the infinite-size variant of the DMRG ~\\cite{Schollwock2011} and see how the energy density, total magnetization, entanglement entropy scale and correlation length as a function of the maximum bond dimension $\\chi_{\\rm max}$. The result is presented in Fig.~\\ref{fig:idmrg_scaling}. It is certain that the entanglement entropy does not diverge but converges to a finite value as $\\chi_{\\rm max} \\rightarrow \\infty$\\,[see red dashed line in Fig.~\\ref{fig:idmrg_scaling}(a)], which implies that the ground state is gapped or satisfies the area law of the entanglement. Also, based on the scaling result of the energy density and magnetization,\\, as shown in Figs.~\\ref{fig:idmrg_scaling}(a) and \\ref{fig:idmrg_scaling}(b), we conclude that keeping $1600$ states, i.e., $\\chi_{\\rm max}=1600$, is enough to capture the essential physics of the ground state. Thus, we fix $\\chi_{\\rm max}$ to 1600 in all calculations. Although we did not present the results here, we however directly confirmed that keeping 1600 states provides convergent results in the finite system regardless of disorder.\n\nAs a benchmark result, we present in Fig.~\\ref{fig:dmrg_clean} the DMRG results of magnetization profile in the clean system. The system is at half-filling\\,($\\delta N = 0$) with size $(L_x, L_y) = (30,8)$, as shown in Fig.~\\ref{fig:dmrg_clean}(a). Expectedly, the ground state shows the N\\'eel order where spins at each zigzag edge align antiparallel to each other. In Fig.~\\ref{fig:dmrg_clean}(b), the system is slightly away from the half-filling, i.e., $\\delta N = 12$ with size $(L_x, \\, L_y) = (120,4)$. The doping introduces a spin-density wave. Hence, the magnetization oscillates along the ribbon direction.\n\n\n\\begin{figure}[htpb]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{dmrg_Mz_clean.eps}\n\\caption{Magnetization profile at (a) $(L_x,L_y, \\delta N) = (30, 8, 0)$ and (b) $(L_x, L_y, \\delta N) = (120, 4, 12)$ in the clean limit with $U\/t=1$.}\n\\label{fig:dmrg_clean}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{HF Approximation and Topological Entanglement Spectrum.}%\n\n\n\n\n\n\n\n\n\n\nThe ground state entanglement spectrum~\\cite{Haldane191} is computed at half-filling using the HF approach. The  spectrum differs from that of disorder-free SPT zigzag ribbons: degenerate eigenvalues of the SPT phase are split and distributed similarly to that of the DOS of the edge states of a disordered TO zigzag ribbon.\nThe entanglement spectrum may be obtained from the reduced density matrix.\nTo compute the bulk entanglement spectrum, we choose the region~\\cite{Bal} $A$ separate from the zigzag edges. We use the HF approximation; thus, the relevant reduced density matrix for a region $A$ can be written as~\\cite{Peschel119}\n\\begin{eqnarray}\n\\rho_A=K e^{ -\\tilde{h}}.\n\\end{eqnarray}\n\n\n\n\\begin{figure}[!hbpt]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{W72G03ESfit.eps}\n\\caption{Consider a rectangular region with length $l_x=76$ and width $l_y=36$. The rectangular region is inside a ribbon with length $L_x=150$ and width $L_y=72$. Distribution of eigenvalues $\\tilde{\\epsilon}_k $ of the reduced density matrix of this region is plotted (there are also positive values of $\\tilde{\\epsilon}_k $, but the distribution is identical). The distribution follows the exponential curve $B[e^{(\\tilde{\\epsilon}-\\tilde{\\epsilon_0})^2\/\\delta^2}-1]$ (black solid line). The parameters are $\\Gamma=0.3 t$, $n_{imp}=0.1$, and $U=0.5 t$ $(g=0.19)$. The number of disorder realization is $N_D \\sim 50$.}\n\\label{calg0g3}\n\\end{center}\n\\end{figure}\n\n\n\nWhen the operator $\\tilde{h}$ is diagonalized, we get the following Hamiltonian matrix\n\\begin{eqnarray}\n\\tilde{h}_{ij}=\\sum_{k}\\psi^*_k(i)\\psi_k(j)\\tilde{\\epsilon}_k,\n\\end{eqnarray}\nwhere $\\tilde{\\epsilon}_k$ and $\\psi_k(j)$ are eigenvalues and eigenstates of the ``Hamiltonian\" $\\tilde{h}$, respectively.\nNote that this particular density matrix describes a Fermi gas at temperature $k_B T=1$.\nThe reduced density matrix of either spin-up or spin-down electrons is equal to\n\\begin{eqnarray}\n\\rho_{ij}= Tr(\\rho c^{\\dagger}_ic_j)=\\sum_k \\psi^*_k(i)\\psi_k(j)\\frac{1}{ e^{ \\tilde{\\epsilon}_k }+1 },\n\\end{eqnarray}\nwhere $c_i=\\sum_k \\psi_k(i)a_k$, and $a_k$ is the electron destruction operator corresponding to the eigenstate $\\psi_k(i)$. The distribution of the eigenvalues $\\tilde{\\epsilon}_k $ of $\\tilde{h}$ is called the entanglement spectrum. (Note that $\\tilde{\\epsilon}_k $ are not the eigenenergies of the Hartree--Fock Hamiltonian.) The eigenvalues of a density matrix are given as\n\\begin{eqnarray}\n\\lambda_k=\\frac{1}{ e^{ \\tilde{\\epsilon}_k }+1 }.\n\\end{eqnarray}\nThe values $\\tilde{\\epsilon}_k \\approx 0$, corresponding to $\\lambda_k\\approx 1\/2$, dominate the entanglement~\\cite{KunY}.\nThe entanglement spectrum of the SPT phase of a disorder-free zigzag nanoribbon exhibits numerous nearly degenerate eigenvalues, thereby reflecting the presence of nearly degenerate edge states. Fig.~\\ref{calg0g3} shows the HF entanglement spectrum of a disordered interacting graphene zigzag ribbon at half-filling. This entanglement spectrum is different from that of the disorder-free SPT phase of zigzag ribbons.\nIn the presence of disorder, the degenerate eigenvalues of the entanglement spectrum are split and exponentially distributed in a way similar to that of the DOS of edge states in TO zigzag ribbons, as shown in Fig. 1(b) in main article. \nIn contrast, the DOS of disorder-free SPT phase of zigzag ribbons has a van Hove singularity, see  Fig. 1 in main article.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nLet $k$ be a field of characteristic different from $2$ or $3$. By an elliptic curve $E$ over $k$ we mean a smooth curve of genus one together with a specified base point $0$. These curves are described by equations   \n$$y^2z = x^3 + Axz^2 + Bz^3,$$ with $A,B \\in k^\\times. $ The Brauer group of $E$, $\\Br(E)$, is an important invariant of the elliptic curve and has applications in both arithmetic and algebraic geometry. In a famous construction, Artin and Mumford used the Brauer group of the function field of $\\mathbb{P}^2$ to give a counterexample to the L\\\"uroth problem; they constructed a complex unirational $3$-fold which is not rational \\cite{artinmumford}. Furthermore, Manin described an obstruction to the Hasse principle for varieties that lies in the Brauer group \\cite{Manin, Skorobogatov:Beyond-the-Manin-Obstruction}. There has been an ongoing effort to gain a better understanding of the Brauer group. \\\\\n\nThe goal of this paper is an explicit description of $\\Br(E)$. Note that the Brauer group of $E$ is naturally isomorphic to the unramified Brauer group of $k(E)$, the function field of $E$ \\cite{colliot-thelene-sansuc:therationalityproblem,Gabber:ANoteOnTheUnramifiedBrauerGroupAndPurity}. Furthermore, $\\Br(E)$ is a torsion abelian group and therefore it will suffice to study its prime power torsion subgroups. Fix an integer $d\\geq 2$, coprime to the characteristic of $k$, and suppose that $k$ contains a primitive $d$-th root of unity $\\rho$. Some well-understood elements in the $d$-torsion of $\\Br k(E)$ are symbol (or cyclic) algebras, a generalization of quaternion algebras. These are $k(E)$-algebras with two generators $x$ and $y$ subject to the relations\n$x^d= a, y^d = b,$ and $xy=\\rho yx$ for some $a,b \\in k(E)^\\times$.  \nThe Merkurjev-Suslin theorem \\cite{merkurjev-suslin} implies that every element in the $d$-torsion of the Brauer group of $E$, $\\tor[d][\\Br(E)]$, can be described as a tensor product of these symbol algebras over $k(E)$. We aim to describe a complete set of generators and relations of $\\tor[d][\\Br(E)]$ in terms of these products. \\\\\n\nThe initial theory for the techniques used in the case $d=2$ was developed in \\cite{GuletskiiV.Yanchevskii1998} and \\cite{Pumplun1998}. This case was investigated in \\cite{Guletskii1997} and \\cite{chernousov-guletskii:2-torsion}. In the present paper, we give generators and relations of $\\tor[d][\\Br(E)]$ in the following cases:\n\\begin{itemize} \n\t\\item Any $d$ coprime to the characteristic of $k$, assuming that $k$ contains a primitive $d$-th root of unity and that the $d$-torsion of $E(\\kbar)$ is $k$-rational (see \\cref{Prop:main result in split case}).\n\t\\item $d=q$, an odd prime coprime to the characteristic of $k$, assuming only that $k$ contains a primitive $q$-th root of unity (see \\cref{sec:Algorithm}). \n\\end{itemize} \n The abstract methods we use were developed in \\cite[Chapter 4]{skorobogatov}. We describe elements in the $d$-torsion of the Brauer group as a cup-product of certain torsors over $E$ under the $d$-torsion of $E(\\kbar)$. There is an ongoing effort to calculate \\etale cup-products explicitly (see for example \\cite{Poonen-Rains:SelCupProducts} and \\cite{Aldrovandi-Ramachandra:Cup-products-the-Heisenberg-group-and-codimension-two-algebraic-cycles}). In this paper, we overcome this difficulty by calculating the cup product at the generic point and using results from Galois cohomology. We will now discuss our methods in some detail. \\\\\n\nConsider the Hochschild-Serre spectral sequence $\\H^i\\left( k, \\H^j \\left(\\overline{E}, \\mathbb{G}_m \\right) \\right) \\Rightarrow \\H^{i+j}\\left(E, \\mathbb{G}_m \\right).$\nThe $d$-torsion of its sequence of low degree terms is\n\\begin{align}\\label{Eq:exact sequence on tor[d]Br(E)} \n\\xymatrix{0 \\ar[r]& \\tor[d][\\Br(k)] \\ar[r]^i & \\tor[d][\\Br(E)] \\ar[r]^(.4)r & \\tor[d][\\H^1\\left(k, E\\left(\\kbar\\right)\\right)]  \\ar[r] & 0 },\n\\end{align}\nwhere the first map sends the class of a central simple algebra $A$ to the class of $A \\otimes k(E)$ under the canonical map $E \\rightarrow \\Spec k(E)$. For more details on this sequence, see also \\cite{Faddeev:SimpleAlgebrasOverAFieldOfAglebraicFunctionsOfOneVariable} and \\cite{Lichtenbaum1969}. We will discuss the second map in more detail in \\cref{Mkrational}.\nDenote the zero-section of $E$ by $s: \\Spec(k) \\ra E$. Sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}) is split on the left by the induced map $s^*: \\Br(E) \\ra \\Br(k)$.\nThus the $d$-torsion of the Brauer group decomposes as $$\\tor[d][\\Br(E)] = \\tor[d][\\Br(E)] \\oplus \\tor[d][\\H^1\\left(k, E\\left(\\kbar\\right)\\right)].$$ Hence, in order to calculate $\\tor[d][\\Br(E)]$ it is sufficient to describe a split to $r$ in sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}). To that end, consider the Kummer sequence \n\\begin{align*}\n\\xymatrix{0 \\ar[r]& M \\ar[r]& E(\\kbar) \\ar[r]^{[d]}& E(\\kbar) \\ar[r] & 0 },\n\\end{align*} \nwhere $[d]$ denotes multiplication by $d$ on the elliptic curve and $M$ is the full $d$-torsion of $E$. This induces a short exact sequence on group cohomohology\n\\begin{align}\\label{eq:Kummer induces on cohomology} \n\\xymatrix{ 0 \\ar[r] & E(k)\/[d]E(k) \\ar[r]^\\delta & \\H^1(k,M) \\ar[r]^(0.45){\\lambda} & \\tor[d][\\H^1\\left(k,E(\\kbar)\\right)] \\ar[r]& 0 }.\n\\end{align} \nWe want to relate the sequences (\\ref{Eq:exact sequence on tor[d]Br(E)}) and (\\ref{eq:Kummer induces on cohomology}). \nDefine $\\epsilon: \\H^1(k,M) \\ra \\tor[d][\\Br(E)]$ as the following composition \n\\begin{equation}\\label{Eq:epsilon}\n\\mbox{\\large{$\\epsilon$ :}}\\left\\{ \n\\begin{gathered}\n\\xymatrix{\n\t\\H^1(k,M) \\ar[r]^(.35)\\sim & \\H^1_{\\text{\\etale}}\\left(\\Spec(k),M\\right) \\ar[r]^(.6){p^*}& \\H^1_{\\text{\\etale}}(E,M)}\\\\\n\\xymatrix{\\ar[r]^(.2){- \\cup [\\mathcal{T}]}& \\H^2_{\\text{\\etale}}(E,M \\otimes M) \\ar[r]^(.55)\\e& \\H^2_{\\text{\\etale}}\\left(E, \\mu_d \\right)  \\ar[r]& \\tor[d][\\Br(E)] },\n\\end{gathered}\\right.\n\\end{equation} \nwhere \n\\begin{itemize} \n\t\\item $p: E \\ra \\Spec k$ is the structure map,\n\t\\item $\\mathcal{T}$ is the torsor given by multiplication by $d$ on $E$, and \n\t\\item $e$ is the map induced by the Weil-pairing. \n\\end{itemize} In \\cite[Chapter 4]{skorobogatov}, the author proves, using abstract methods, that $\\epsilon$ induces a split of squence (\\ref{Eq:exact sequence on tor[d]Br(E)}). We reprove this result using explicit methods and compute $\\epsilon$ in terms of group cohomology. We use this explicit description to give generators and a complete set of relations of $\\tor[d][\\Br(E)]$ under the assumption that $M$ is $k$-rational. Here is our first result. \n\\begin{thm}\\label{Prop:main result in split case} \n\tSuppose that the $d$-torsion $M$ of $E$ is $k$-rational and fix two generators $P$ and $Q$ of $M$. Denote by $0$ the identity of the group law on $E$. Let $t_P,t_Q \\in k(E)$ with divisors  $\\divisor(t_P) = d(P) - d(0)$ and $\\divisor(t_Q) = d(Q)- d(0)$ so that $t_P\\circ [d], t_Q \\circ [d] \\in (k(E)^\\times)^d$. Then the $d$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[d][\\Br(E)] =\\tor[d][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\left( a, t_P \\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ with $a,b \\in k^\\times$. Such a tensor product is trivial if and only if it is similar to one of the following \n\t\\begin{itemize} \n\t\t\\item $\t\\left( t_Q(P), t_P \\right)_{d,k(E)} \\otimes \\left( \\frac{t_P(P+Q)}{t_P(Q)}, t_Q \\right)_{d,k(E)}$, \n\t\t\\item $\t\\left( \\frac{ t_Q(P+Q)}{t_Q(P)}, t_P \\right)_{d,k(E)} \\otimes \\left( t_P(Q), t_Q  \\right)_{d,k(E)}$, or \n\t\t\\item $ \\left( t_Q (R), t_P(R) \\right)_{d,k(E)} $ for some $R\\in E(k), R \\neq 0,P,Q$.\\end{itemize}  \n\\end{thm}\nIn addition to recovering the generators of $\\tor[d][\\Br(E)]$ as calculated in \\cite[Remark 6.3]{Chernousov2016}, we give a full set of relations of the group. For the proof of \\cref{Prop:main result in split case} see \\cref{Mkrational}.\n\\\\\n\nThe main result of this paper is the algorithm given in \\cref{sec:Algorithm}. Let $q$ be an odd prime coprime to the characteristic of $k$. The algorithm can be used to determine a complete set of generators and relations of the $q$-torsion of $\\Br(E)$ over an arbitrary field $k$ containing a primitive $q$-th root of unity. This algorithm is proved in \\cref{Mnotkrational}. The main idea of the proof is the following: Let $L$ be the smallest Galois extension of $k$ so that the $q$-torsion $M$ of $E$ is $L$-rational. Use \\cref{Prop:main result in split case} to describe the Brauer group of $E \\times_{\\Spec k} \\Spec L$. Note that the order of $L$ over $k$ divides the order of $SL_2\\left( \\mathbb{F}_q \\right)$, which is $q (q-1) (q+1)$. If $q$ does not divide the order of $L$ over $k$, we use the fact that restriction followed by corestriction is an isomorphism to determine generators and relations of $\\Br(E)$. If the order of $L$ over $k$ equals $q$, we apply the inflation restriction exact sequence to our problem. Finally, if $q$ divides the order of $\\Br(E)$, we combine the two previous cases. \\\\\n\nA direct consequence of the algorithm is the following corollary on the symbol length. \n\n\\begin{cor}Let $q$ be an odd prime coprime to the characteristic of $k$ and assume that $k$ contains a primitive $q$-th root of unity. For any elliptic curve $E$ over $k$, the $q$-torsion of the Brauer group decomposes as $\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$ and every element in $I$ can be written as a tensor product of at most $n_q = 2(q-1)(q+1)$ symbol algebras over $k(E)$. This means that the symbol length in $\\tor[q][\\Br(E)]\/\\tor[q][\\Br(k)]$ is at most $n_q$.\n\\end{cor} \n\nFinally, we calculate multiple examples in \\cref{ch:Examples}. For further examples, see also \\cite{dissertation}. \\\\\n\n\\noindent\\textbf{Notation.} \nThroughout this paper, $k$ denotes a base field of characteristic different from $2$ and $3$ and $E$ is an elliptic curve over $k$. We denote the addition on $E$ by $\\oplus$ and the point at infinity by $0$. Let $d\\geq 2$ be an integer coprime to the characteristic of $k$ so that $k$ contains a primitive $d$-th root of unity. $M$ denotes the $d$-torsion of $E$, and $[d]$ the multiplication by $d$-map on $E$. $P$ and $Q$ will always be generators of $M$ and $\\e(.,.)$ the Weil-pairing on $E$ with $\\e(P,Q) = \\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right]_\\rho = i $. Furthermore, identify $\\Z\/d\\Z$ with the subset $\\{0, \\ldots, d-1\\}$ of the integers and denote the image of $\\rho^i$ under the composition by $\\left[\\rho^i\\right]_\\rho^\\Z= i \\in \\Z$. When we assume that $d$ is an odd prime $q$, we write $d=q$. \\\\\n\nFor a field $F$, denote an algebraic closure by $\\overline{F}$ and its absolute Galois group by $G_F$. Furthermore, $H^i(F,A) = H^i(G_F,A)$ is the $i$-th (profinite) group cohomology group of $G_F$ with coefficients in a $G_F$-module $A$. $\\res, \\Cor,$ and $\\inflation$ denote the restriction, the corestriction, and the inflation map, respectively. \\\\\n\n\\noindent\\textbf{Acknowledgements.}\nI would like to thank my advisor Rajesh Kulkarni, for his expertise and support throughout the process of writing this paper. I also thank Igor Rapinchuk for helpful conversations and Alexei N. Skorobogatov for a useful correspondence. I thank David Saltman and Adam Chapman for comments on an earlier draft of this article. \n\n\\section{The Algorithm}\\label{sec:Algorithm}\nLet $k$ be a field of characteristic different from $2$ or $3$ and let $q$ be an odd prime. Assume that $q$ is coprime to the characteristic of $k$ and that $k$ contains a primitive $q$-th root of unity. Let $E$ be an elliptic curve over $k$. Denote by $M$ the $q$ torsion of $E(\\kbar)$. The Brauer group of $E$ decomposes as \n$$\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$$ \nand generators $\\mathcal{G}$ and relations $\\mathcal{R}$ of $I$ can be calculated using the following algorithm. \n\\begin{enumerate}\n\t\\item Determine the kernel of the natural Galois representation \n\t$$\\Psi: G_k \\rightarrow GL_2\\left(\\mathbb{F}_q\\right).$$\n\tSince $k$ contains a primitive $q$-th root of unity, the image of $\\Psi$ lies in $SL_2(\\mathbb{F}_q)$ (for more details see \\cref{Mnotkrational}). Denote by $L$ the fixed field of this kernel. \n\t\\item \n\t\\begin{enumerate} \n\t\t\\item If $q$ divides the order of $L\/k$, fix some intermediate field $L'$ so that $L\/L'$ is a Galois extension of degree $q$. Let $P$ and $Q$ be elements in $M$ so that $\\Gal(L\/L')$ is generated by $\\overline{\\sigma}$ with $\\sigma(P) = P$ and $\\sigma(Q) = P\\oplus Q$. Set \n\t\t$$\\mathcal{G}_{L'} = \\left\\{\\left( l^q, n_Q \\right)_{L'(E)}, (a,t_P)_{L'(E)}:\n\t\ta \\in L'^\\times \\right\\},$$\n\t\twhere $t_P \\in L'(E)$ with $\\divisor(t_P)= q(P)-q(0)$ and $n_Q \\in L'(E)$ with $\\divisor(n_Q) = \\sum_{i=0}^{q-1} \\sigma^i(Q) = \\sum_{i=0}^{q-1} (iP+Q).$ Furthermore, let $t_Q \\in L(E)$ with $\\divisor(t_Q) = q(Q) - q(0)$ and $n_Q^q = N_{L(E)\/k(E)}(t_Q)$. \n\t\t\\item Else fix generators $P$ and $Q$ of $M$. Set $$\\mathcal{G}_{L'} = \\left\\{ \\left( a,t_P \\right)_{L(E)}, \\left( b,t_Q\\right)_{L(E)}: a,b \\in L^\\times  \\right\\},$$\n\t\twhere $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) -q(0)$. \n\t\\end{enumerate}\n\t\\item \n\tSet \n\t\\begin{align*} \n\t\\mathcal{R}_L = &\n\t\\left\\{\n\t\\left( t_Q(P), t_P \\right)_{L(E)}  \\otimes \\left( \\frac{t_P(P\\oplus Q)}{t_P(Q)} , t_Q \\right)_{L(E)},\n\t\\left( \\frac{ t_Q(P\\oplus Q)}{t_Q(P)}, t_P \\right)_{L(E)} \\otimes \\left( t_P(Q), t_Q  \\right)_{L(E)} \\right\\} \\\\\n\t&\\cup \\left\\{\n\t \\left( t_Q (R), t_P \\right)_{L(E)} \\otimes \\left( t_P(R), t_Q \\right)_{L(E)} : R \\in E(L), R \\neq 0,P,Q\\right\\}.\n\t\\end{align*} \n\t\t\\begin{enumerate} \n\t\t\\item If $q$ divides the order of $L\/k$ let \n\t\t$$\\mathcal{R}_{L',\\res} = \\left\\{ (a,t_P)_{L'(E)} : \\res(a,t_P)_{L'(E)} \\in \\mathcal{R}_L \\right\\}.$$\n\t\tFurther, if the quotient $\\frac{E(k) \\cap [q]E(L)}{[q]E(k)}$ is not trivial let \n\t\t$$\\mathcal{R}_{L',\\inflation} = \\left\\{ \\left( l^q,n_Q \\right)_{L'(E)} \\right\\}.$$ Otherwise, let $\\mathcal{R}_{L',\\inflation} = \\emptyset$. Set $\\mathcal{R}_{L'} = \\mathcal{R}_{L', \\res} \\cup \\mathcal{R}_{L',\\inflation}.$\n\t\t\\item Else let $L=L'$ and $\\mathcal{R}_{L'} = \\mathcal{R}_L$.\n\t\\end{enumerate}\n\t\\item Set \n\t$$\\mathcal{G} = \\left\\{ \\Cor(A): A \\in \\mathcal{G}_{L'} \\right\\}$$\n\tand \n\t$$\\mathcal{R} = \\left\\{ \\Cor(A): A \\in \\mathcal{R}_{L'}\\right\\}.$$\n\\end{enumerate}\n\nNote that there are additional relations that may arise from the fact that the corestriction map is not injective. These relations require a more careful treatment. See for example \\cref{subsec:Ex degree L\/k coprime to 3}.\n\n\n\\begin{rem} \n\tWe assume that the characteristic of $k$ is different form $2$ or $3$ for simplicity of the presentation of the elliptic curves and its torsion subgroups. The general results still hold in characteristic equal to $2$ or $3$. \n\\end{rem}  \n\n\n\\section{Background}\\label{ch:background}\nLet $E$ be an elliptic curve defined over a field $k$ of characteristic different from $2$ and $3$ described by the affine equation\n$$y^2 = x^3 + Ax + B$$\nwith $A,B \\in k$. Denote the point addition on $E(\\kbar)$ by $\\oplus$, the point subtraction by $\\ominus$, and the point at infinity by $0$. Furthermore, for any natural number we denote by $[d]$ the multiplication by $d$ map on $E(\\kbar)$. \\\\\n\nWe now proceed to describe the correspondence between torsors and elements in the first cohomology group. For more details we refer the reader to \\cite{skorobogatov}. Let $A$ be an algebraic group defined over a field $F$. An $F$-torsor under $A$ is a non-empty $F$-variety $T$ equipped with a right-action of $A$ so that $T(\\overline{F})$ is a principal homogeneous space under $A(\\overline{F})$. There is a bijection  \n$$ H^1(F,A) \\leftrightarrow \\left\\{ \\begin{gathered} \\text{$F$-torsors under $A$}\\\\\\text{ up to isomorphism }\\end{gathered} \\right\\}$$\nthat is explicitly given as follows. Let $T$ be an $F$-torsor under $A$. Choose an $\\overline{F}$-point $x_0$ of $T$. By the definition of $F$-torsor, for any $\\sigma \\in G_F$, there exists a unique $a_\\sigma \\in A(\\overline{F})$ so that \n$\\sigma(x_0) = x_0. a_\\sigma$. The map $\\sigma \\mapsto a_\\sigma$ defines the cocycle in $H^1(F,A)$ corresponding to $T$. \nNow consider the more general case, where $X$ is an abelian variety over a field $k$. An $X$-torsor under an $X$-group scheme $\\mathcal{A}$ is a scheme $\\mathcal{T}$ over $X$ together with an $\\mathcal{A}$-action compatible with the projection to $X$ that is \\'etale-locally trivial. As before, there is a one-to-one correspondence between $X$-torsors under $\\mathcal{A}$ and elements of the \\etale cohomology $H^1_{\\text{\\et}}(X,\\mathcal{A})$.  \nA $d$-covering of an abelian variety $X$ is a pair $(\\mathcal{T},\\psi)$, where $\\mathcal{T}$ is a $k$-torsor under $X$ and $\\psi: \\mathcal{T} \\ra X$ is a morphism such that $\\psi(x.t) = d x + \\psi(t)$ for any $t \\in \\mathcal{T}(\\kbar), x \\in X(\\kbar).$ By \\cite[Proposition 3.3.4 (a)]{skorobogatov}, any $d$-covering is an $X$-torsor under the $d$-torsion $\\tor[d][X]$.\\\\\n\nIn the following, we describe the  correspondence between the Brauer group and the second cohomology group. Let $F$ be a field and let $d\\geq 2$ be an integer. We say that two central simple algebras $A$ and $B$ are Morita equivalent if there exist some natural numbers $n$ and $m$ so that $A \\otimes M_n(F)$ and $B \\otimes M_m(F)$ are isomorphic as $F$-algebras. Elements in the Brauer group are given by equivalence classes of central simple algebras modulo Morita equivalence and the group structure is given by the tensor product. There is a group isomorphism between $\\Br(F)$ and $H^2(F, \\overline{F}^\\times)$ that can be described as follows. Let $K\/F$ be a finite Galois extension with Galois group $G$ and let $f$ be a cocycle representing an element in $H^2(G,K^\\times)$. Consider the $F$-vector space \n$A = F \\left<x_g : g \\in G \\right>$ with multiplication $\\lambda x_g = x_g g(a)$ and $x_g x_h = f(g,h) x_{gh}$. This turns $A$ into a finite dimensional central simple algebra over $F$. \n\nFrom now on suppose that the field $F$ contains a primitive $d$-th root of unity $\\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right]_\\rho = i $. Furthermore, identify $\\Z\/d\\Z$ with the subset $\\{0, \\ldots, d-1\\}$ of the integers and denote the image of $\\rho^i$ under the composition by $\\left[\\rho^i\\right]_\\rho^\\Z= i \\in \\Z$. For $a,b \\in F^\\times$, the symbol algebra \n\t\\begin{equation}\\label{defn:symbolalgebra} \n\t\t(a,b)_{d,F}= (a,b)_{d,F,\\rho} := F \\left< x,y : x^d = a , y^d = b, xy = \\rho yx \\right>\n\t\t\\end{equation} \n\tis a central simple algebra over $F$. The element in $H^2(F,\\overline{F}^\\times)$ corresponding to $(a,b)_{d,F,\\rho}$ can be represented by the cocycle\n\t\\begin{equation}\\label{eqn:cocyclesymbolalgebra}  \n\t(\\gamma, \\tau) \\mapsto \\begin{cases} a & \\text{if }\\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\sqrt[d]{b}}\\right]_\\rho^\\Z \\geq d \\\\ 1 & \\text{else} \\end{cases}.\n\t\\end{equation} \n\tFor more details we refer the reader to \\cite[Chapter 7 \\S 29]{Reiner:Maximal-orders}. The following cocycle representing the symbol algebra $(a,b)_{d,F}$ will prove more useful for our purposes.\n\n\\begin{prop}\\label{prop:sumbolalgebra-cocycle} \n\tLet $M$ be the $d$-torsion of an elliptic curve $E$ over $k$ with generators $P$ and $Q$. Assume that the Weil-pairing satisfies $\\e(P,Q)=\\rho$. Let $F$ be a field containing a primitive $d$-th root of unity $\\rho$. The symbol algebra $(a,b)_{d,F}$ can be represented by the cocycle \n\t$$(\\gamma,\\tau) \\mapsto \\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right)^{-1}$$\n\tfor every pair $a,b \\in F^\\times$. \n\tWe will often consider the case $F = k(E)$. \n\\end{prop}\n\n\\begin{proof}\n\tConsider the map $$g: \\gamma \\rightarrow {\\sqrt[d]{a}}^{ \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }.$$\n\tThe differential of $g$ is \n\t\\begin{align*}\n\tdg(\\gamma,\\tau) &= \\gamma \\left( {\\sqrt[d]{a}}^{ \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z } \\right) \n\t{\\sqrt[d]{a}}^{ \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }\n\t{\\sqrt[d]{a}}^{ - \\left[ \\frac{\\gamma\\tau\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b}\\right)}\\right]_\\rho^\\Z }\\\\\n\n\n\n\n\t&= \\begin{cases} \n\ta \\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right) &\\text{if } \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[ \\frac{\\tau\\left( \\sqrt[d]{b} \\right)}{\\sqrt[d]{b}}\\right]_\\rho^\\Z \\geq d \\\\\n\t\\e \\left(\\frac{\\gamma\\left( \\sqrt[d]{a} \\right)}{\\left( \\sqrt[d]{a} \\right)} P, \\frac{\\gamma\\left( \\sqrt[d]{b} \\right)}{\\left( \\sqrt[d]{b} \\right)} Q \\right)& \\text{else} \t\t\t\t\t\n\t\\end{cases} \n\t\\end{align*}\n\tSubtracting this trivial cocycle from the cocycle in \\cref{eqn:cocyclesymbolalgebra} gives the desired result.\n\\end{proof}\n\n\\section{$M$ is $k$-rational}\\label{Mkrational}\n\nLet $k$ be a field of characteristic different from $2$ or $3$. Let $d\\geq 2$ be an integer coprime to the characteristic of $k$. Assume additionally that $k$ contains a primitive $d$-th root of unity $\\rho$. Fix an isomorphism $[.]_\\rho: \\mu_{d} \\rightarrow \\Z\/d\\Z$ with $\\left[\\rho^i\\right] = i $. Furthermore, for $\\rho^i \\in \\mu_d$, let $\\left[\\rho^i \\right]_\\rho^\\Z = i \\in \\left\\{ 0, \\ldots, d-1 \\right\\}\\subset \\Z$. Let $E$ be an elliptic curve over $k$ and denote its $d$-torsion by $M$. Assume throughout this section that $M$ is $k$-rational. Fix two generators $P$ and $Q$ of $M$. Denote by $\\e(.,.)$ the Weil pairing and assume that $\\e(P,Q) = \\rho$. Let $t_P, t_Q \\in \\kbar(E)$ with $\\divisor(t_P) = d(P) - d(0)$ and $\\divisor(t_Q) = d(Q) - d(0)$. We may assume that $t_P, t_Q \\in k(E)$ since their divisors are invariant under the Galois action of $G_k$. Let $\\mathcal{T}$ be the torsor given by multiplication by $d$ on $E$.\n\n\\begin{prop}\\label{Prop:torsor mult by 3 over kbar}\n\tThe pull-back  $\\eta^*\\left( \\mathcal{T}\\right)$ along the generic point $\\eta: \\Spec k(E) \\ra E$ corresponds to the element in $H^1\\left(k(E),M\\right)$ given by the cocycle\n\t$$\\gamma \\mapsto \\left[ \\frac{\\gamma\\left(\\alpha_Q\\right)}{\\alpha_Q} \\right]_\\rho P - \\left[\\frac{\\gamma\\left(\\alpha_P\\right)}{\\alpha_P} \\right]_\\rho Q,$$\n\twhere $\\alpha_P, \\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_P^d = t_P$ and $\\alpha_Q^d = t_Q$.   \n\\end{prop}\n\n\\begin{proof}\n\tFor the correspondence between torsors and elements in the first cohomology group see \\cref{ch:background}. Let $P' \\in E(\\kbar)$ so that $[d]P' = P$. Then there is some $g_P \\in \\kbar(E)$ with $$\\divisor\\left(g_P \\right)= [d]^* (P) - [d]^* (0) = \\sum_{R \\in M} \\left((P' \\oplus R) - (R) \\right).$$ Note that we may choose $g_P \\in k(E)$ since the divisor is invariant under the action of the absolute Galois group of $k$. Now $\\divisor\\left(g_P^d\\right) = \\divisor\\left( [d]^* t_P\\right)$ and thus we may assume that $g_P^d = [d]^*t_P$. Similarly we find $g_Q \\in k(E)$ with $g_Q^d = [d]^*t_Q$. \\\\\n\tNow consider the pullback of the torsor $\\mathcal{T}$ along the generic point $\\eta: \\Spec k(E) \\rightarrow \\Spec k$. Fix a $\\overline{k(E)}$-point $x_0$ of this pullback, i.e. a map of algebras so that $x_0 \\circ [d]^* = \\iota$, where $\\iota: k(E) \\rightarrow \\overline{k(E)}$ is the inclusion. This means, that the following diagramms commute\n\t$$\n\t\\xymatrix{\n\t\t& \\Spec k(E) \\ar[r]^-\\eta \\ar[d]^{[d]} & E \\ar[d]^{[d]} \\\\\n\t\t\\Spec \\overline{k(E)} \\ar[ru] \\ar[r] & \\Spec k(E) \\ar[r]^-\\eta & E}\n\t\\qquad \t\n\t\\xymatrix{ & k(E) \\ar[ld]_{x_0} \\\\\n\t\t\\overline{k(E)}& k(E)  \\ar[u]_{[d]^*} \\ar[l]^\\iota}$$\n\tAfter possibly renaming $\\alpha_P$ and $\\alpha_Q$, we may assume that $x_0\\left(g_P\\right) = \\alpha_P$ and $x_0\\left(g_Q\\right) = \\alpha_Q$. \n\tThere is a group isomorphism\n\t$$\tM \\rightarrow \\Gal\\left(k(E)\/[d]^*k(E) \\right): R\\mapsto \\tau_R^*,$$\n\twhere $\\tau_R: E \\rightarrow E$ is the translation by $R$-map; $\\tau_R: E \\rightarrow E: S \\mapsto S \\oplus R$. By the definition of the Weil-pairing $\\e(R,P) = \\frac{g_P (X\\oplus S)}{g_P(X)} = \\frac{ \\tau_S^*(X)}{g_P(X)},$ for any $R \\in M$, $X \\in E(\\kbar)$ any point so that $g_P(X)$ and $g_P(X \\oplus S)$ are defined. The analogous result holds for $g_Q$ as well. Finally, we calculate   \n\t\\begin{align*} \n\tx_0 \\circ \\tau^*_{\\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q} \\left( g_P\\right)\n\t&= x_0 \\left( \\e \\left( \\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q, P \\right) g_P \\right)\\\\\n\t&= x_0 \\left( \\e\\left(-\\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q, P\\right) g_P\\right)\\\\\n\t&= x_0 \\left({\\frac{\\gamma(\\alpha_P)}{\\alpha_P}}g_P \\right)\\\\\n\t&= \\gamma(\\alpha_P)\n\t\\end{align*}\n\tand similarly\n\t\\begin{align*} \n\tx_0 \\circ \\tau^*_{\\left[\\frac{\\gamma(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\gamma(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q} \\left( g_Q \\right)\n\t&= \\gamma(\\alpha_Q).\t\n\t\\end{align*}\n\tThe statement follows since $k(E)\/[d]^*k(E)$ is generated by $g_P$ and $g_Q$.\n\\end{proof}\n\nRecall that $\\epsilon$ is the composition \n\\begin{equation*}\n\\mbox{\\large{$\\epsilon$ :}} \\qquad \n\\begin{gathered}\n\\xymatrix{\n\tH^1(k,M) \\ar[r]^-\\sim & H^1_{\\text{\\et}}\\left(\\Spec(k),M\\right) \\ar[r]^-{p^*}& H^1_{\\text{\\et}}(E,M)}\\\\\n\\xymatrix{ \\ar[r]^-{- \\cup [\\mathcal{T}]}& H^2_{\\text{\\et}}(E,M \\otimes M) \\ar[r]^-\\e& H^2_{\\text{\\et}}\\left(E, \\mu_d\\right)  \\ar[r]& \\tor[d][\\Br(E)]}.\n\\end{gathered}\n\\end{equation*}\n\nIn \\cite[Theorem 4.1.1 and the following example]{skorobogatov}, the author proves abstractly that $\\epsilon$ induces a split to \\cref{Eq:exact sequence on tor[d]Br(E)} using general properties of torsors and the cup-product. We will now determine $\\epsilon$ explicitly and prove directly that the map induces the desired split. \n\n\\begin{prop}\\label{Prop:epsilonk}On the level of cocycles $\\epsilon$ coincides with the map that assigns to a 1-cocycle $f: G_k \\rightarrow M$ the 2-cocycle \n\t\\begin{equation*}\n\t\\begin{gathered}  \n\t\\epsilon(f): G_{k(E)} \\times G_{k(E)} \\ra \\overline{k(E)}^\\times\\\\\n\t(\\gamma,\\tau) \\mapsto \\e\\left( f(\\gamma), \\gamma \\left( \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q \\right)  \\right).\n\t\\end{gathered} \n\t\\end{equation*} \n\\end{prop}\n\n\\begin{proof}The cup-product commutes with $\\eta^*$ by \\cite[Chapter 2, 8.2]{Bredon:Sheaf-Theory}. Consider the following diagram with commutative squares\n\t$$\n\t\\xymatrix{\n\t\tH^1(k,M) \\ar[r]^{p^*} & H^1_{\\text{\\et}}(E,M) \\ar[d]^{\\eta^*} \\ar[rr]^-{-\\cup [\\mathcal{T}]} && H^2_{\\text{\\et}}(E,M \\otimes M) \\ar[r]^-\\e_-\\sim \\ar[d]^{\\eta^*}\t& H^2_{\\text{\\et}}(E,\\mu_d)\\ar[d] \\\\ \n\t\t& H^1_{\\text{\\et}}(k(E),M) \\ar[rr]^-{-\\cup [\\eta^*\\mathcal{T}]} && H^2_{\\text{\\et}}(k(E), M \\otimes M ) \\ar[r]^-\\e_-\\sim & \\tor[d][\\Br k(E)]\n\t}$$\n\tThe statement follows from \\cref{prop:torsormult3} and by the definition of the cup-product in group cohomology \\cite[Chapter VIII, Section 3]{serrelocal}. Recall that by \\cite[Theorem 5.11]{colliot-thelene-sansuc:therationalityproblem} the map on the right is given by the injection that identifies $\\Br(E)$ with the unramified Brauer group of $k(E)$. \n\\end{proof}\n\nWe are now ready to calculate a set of generators of $\\tor[d][\\Br(E)]$. Since we assume that $M$ is $k$-rational, Kummer theory implies that there is an isomorphism \n\\begin{align}\\label{Eq:phi definition}  \\phi: k^\\times\/(k^\\times)^d \\times k^\\times\/(k^\\times)^d \\rightarrow H^1(k,M): (a,b) \\mapsto c_{a,b},\\end{align} \nwhere $c_{a,b}$ can be represented by the cocycle\n$$G_k \\rightarrow M: \\gamma \\mapsto \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P \\oplus \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q.$$ \n\n\\begin{prop}\\label{Prop:epsilon-split-case}The cocycle\n\t$\\epsilon \\circ \\phi (a,b)$ corresponds to the Brauer class of $$\\left(a,t_P\\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ for any \n\t$(a,b) \\in k^\\times\/(k^\\times)^d \\times k^\\times\/(k^\\times)^d$. \n\\end{prop}\n\n\\begin{proof}Observe that  $\t\\epsilon_k\\circ \\phi(a,b)$ can be represented by the cocycle that takes $(\\gamma,\\tau) \\in G_{k(E)} \\times G_{k(E)}$ to \n\t\\begin{align*}\n\t&\\e\\left( \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P \\oplus \\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q, \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q  \\right)\\\\\n\t=& \\e\\left( \\left[ \\frac{\\gamma\\left( \\sqrt[d]{a} \\right) }{\\sqrt[d]{a}} \\right]_\\rho P, \\left[\\frac{\\tau(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q  \\right)^{-1} \\e\\left(\\left[ \\frac{\\gamma\\left( \\sqrt[d]{b} \\right) }{\\sqrt[d]{b}} \\right]_\\rho Q, \\left[\\frac{\\tau(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P \\right).\n\t\\end{align*}\n\tThe statement follows from \\cref{prop:sumbolalgebra-cocycle}.\n\\end{proof}\nRecall that we need to prove that $\\epsilon$ induces a split to the sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}) on the right, i.e. we need to show that $r \\circ \\epsilon = \\lambda$ and $\\epsilon(\\ker (\\lambda))= 0$. We first describe $r$ explicitly. For more details see also \\cite{Faddeev:SimpleAlgebrasOverAFieldOfAglebraicFunctionsOfOneVariable} or \\cite{Lichtenbaum1969}. Let \n$\\alpha \\in \\Br(E) \\subset \\Br k(E)$. Using Tsen's theorem we view $\\alpha$ as an element in $\\H^2\\left( G_k, \\kbar(E)^\\times \\right) \\cong \\Br k(E)$. \nConsider the exact sequence \n$$0 \\rightarrow \\kbar(E)^\\times \\rightarrow \\Prin (\\overline{E}) \\rightarrow \\Div(\\overline{E}) \\rightarrow 0,$$\nwhere $\\Prin(\\overline{E})$ denotes the set of pricipal divisors on $\\overline{E}$ and $\\Div(\\overline{E})$ the set of divisors on $\\overline{E}$. \nThe sequence induced on group cohomology is \n$$\\H^2(G_k,\\kbar(E)^\\times) \\rightarrow \\H^2(G_k, \\Prin(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div(\\overline{E})),$$\nwhere the first map takes $\\alpha$ to some $\\alpha'$ in the kernel of the second map. Now consider the degree sequence \n$$0 \\rightarrow \\Div^0(\\overline{E}) \\rightarrow \\Div(\\overline{E}) \\rightarrow \\Z \\rightarrow 0, $$\nwhere $\\Div^0(\\overline{E})$ is the group of degree zero divisors. Since $\\H^1(G_k, \\Z) =0$,  \nthe map $$\\H^2(G_k, \\Div^0(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div(\\overline{E}))$$ is injective. Finally, the exact sequence\n$$0\\rightarrow \\Prin(\\overline{E}) \\rightarrow \\Div^0(\\overline{E}) \\rightarrow E(\\kbar) \\rightarrow 0$$\ninduces an exact sequence\n$$1 \\rightarrow \\H^1(G_k,E(\\kbar)) \\rightarrow \\H^2(G_k, \\Prin(\\overline{E})) \\rightarrow \\H^2(G_k, \\Div^0(\\overline{E})).$$\nThe element $\\alpha'$ is in the kernel of the second map, and therefore there exists a unique $\\alpha'' \\in \\H^1(G_k, E(\\kbar))$ with image $\\alpha'$. Set $r(\\alpha) = \\alpha''$. \n\n\\begin{prop}\\label{prop:kappcircepsilon=lambda_split}\n\t$r \\circ \\epsilon = \\lambda$. \n\\end{prop} \n\n\\begin{proof}We will only prove that $r \\circ \\epsilon\\circ \\phi(a,1) = \\lambda\\circ \\phi(a,1).$ The other cases are similar. We showed previously that $\\epsilon \\circ \\phi(a,1) = (a,t_P)_{d,k(E)}$, which corresponds to the cocycle \n\t$$(\\gamma,\\tau) \\mapsto \\begin{cases} 1 & \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z < d \\\\\n\tt_P  & \\text{else}\\end{cases}$$\n\tin $H^2(G_k, \\kbar(E)^\\times)$. This gives an element in $H^2(G_k, \\Prin(E))$ via \n\t$$(\\gamma,\\tau) \\mapsto \\begin{cases} 1 & \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z < d \\\\\n\td(P) - d(0)  & \\text{else} \\end{cases}.$$\n\tOn the other hand for any $\\gamma \\in G_k$\n\t$$\\lambda(\\phi(a,1)) (\\gamma) = \\phi(a,1)(\\gamma) =  \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z P.$$\n\tNow we follow the proof of the snake lemma to calculate the image of $\\lambda(f)$ under the connecting homomorphism \n\t$$H^1(k, E(\\kbar)) \\rightarrow H^2(k, \\Prin(E))$$\n\tinduced by the sequence \n\t$$0 \\ra \\kbar(E)^\\times \\ra \\Prin(\\overline{E}) \\ra \\Div (\\overline{E}) \\ra 0. $$\n\tFirst lift it to \n\t$$\\gamma \\mapsto \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) \\in \\Div^0(\\overline{E}).$$\n\tNow use the boundary map to get \n\t\\begin{align*} \n\t(\\gamma,\\tau) \\mapsto &\\gamma \\left( \\left[ \\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\right) -  \\left[ \\frac{\\gamma\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) +  \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\\\\n\t&= \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) - \\left[ \\frac{\\gamma\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\left(  (P) - (0) \\right) + \\left[ \\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}} \\right]_\\rho^\\Z \\left(  (P) - (0) \\right)\\\\\n\t&=  \\begin{cases} d(P) - d(0) & \\text{if } \\left[\\frac{\\gamma(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z + \\left[\\frac{\\tau(\\sqrt[d]{a})}{\\sqrt[d]{a}}\\right]_\\rho^\\Z \\geq d \\\\\n\t1 & \\text{else} \\end{cases},\n\t\\end{align*}\n\twhich coincides with what we previously calculated. The statement follows.  \n\\end{proof} \n\n\\begin{prop}\\label{prop:epsilonkerlambda=0} \n\t$\\epsilon\\left( \\ker(\\lambda) \\right) = 0.$ \n\\end{prop}\n\\begin{proof} \n\tRecall that $\\ker(\\lambda) = \\text{Im}(\\delta)$ and let $R \\in E(k)$. By the previous proposition $r\\circ\\epsilon\\circ \\delta (R) = \\lambda \\circ \\delta(R)$ is trivial. Thus the algebra $\\epsilon\\circ \\delta (R) $ is in the image of the $\\Br(k) \\rightarrow \\Br(E)$. It remains to show that the specialization of $\\epsilon \\circ \\delta(R)$ at $0$ is trivial. The cup-product commutes with specialization at a closed point \\cite[Chapter 2, 8.2]{Bredon:Sheaf-Theory}, i.e. $\\left( [\\mathcal{T}'] \\cup [\\mathcal{T}]\\right)_S = \\left( [\\mathcal{T}']\\right)_S \\cup \\left( [\\mathcal{T}]\\right)_S$ for every $S \\in E(k)$ and every $[\\mathcal{T}']\\in H^1_{\\text{\\et}} (E, M)$. By definition of $\\epsilon$ $$\\left( \\epsilon\\circ \\delta (R) \\right)_S =  \\left( \\epsilon\\circ \\delta (R) \\right)_S =   \\delta(R) \\cup \\mathcal{T}_S \\in \\Br(k)$$ for any $S \\in E(k)$. \n\tIn particular, $\\left( \\epsilon\\circ \\delta (R) \\right)_0 = \\delta(R) \\cup \\mathcal{T}_0$. The specialization of $\\mathcal{T}$ at $0$ admits a point (the point $0$) and is therefore the trivial torsor. We deduce that  $\\left( \\epsilon\\circ \\delta (R) \\right)_0 $ is trivial and thus so is $\\epsilon\\circ \\delta (R)$. \n\\end{proof} \n\nWe conclude that under the above assumptions, the $d$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[d][\\Br(E)] =\\tor[d][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\left( a, t_P \\right)_{d,k(E)} \\otimes \\left( b, t_Q \\right)_{d,k(E)}$$ with $a,b \\in k^\\times$.\n\tWe now proceed to determine the relations in $I$.  \nRecall that that an element in $I$ as before is trivial if and only if it is in the image of the composition \n$$\\xymatrix{\n\tE(k)\/[d]E(k) \\ar[r]^-\\delta & H^1(k,M) \\ar[r]^-\\epsilon & \\tor[d][\\Br(k)]\n}.$$\nWe first need to describe the image of $\\phi^{-1} \\circ \\delta$ explicitly, where $\\phi$ is the isomorphism from \\cref{Eq:phi definition}.\n\\begin{prop}Let $R \\in E(k)\/[d]E(k)$ and let $t_P, t_Q \\in k(E)$ with $\\divisor(t_P) = d(P) - d(0)$, $\\divisor (t_Q) = d(Q) -d(0)$. Assume that  $t_P\\circ [d], t_Q \\circ [d] \\in (k(E)^\\times)^d$. Then \n\t$$\\phi^{-1} \\circ \\delta (R) = \\begin{cases} \n\t(1,1) & R = 0\\\\\n\t\\left( t_Q(P), \\frac{t_P(P\\oplus Q)}{t_P(Q)} \\right) & R = P \\\\\n\t\\left( \\frac{ t_Q(P\\oplus Q)}{t_Q(P)}, t_P(Q)  \\right) & R= Q \\\\ \n\t\\left( t_Q (R), t_P(R) \\right) & \\text{else} \\end{cases} $$\n\\end{prop}\t\n\nThe proof of this proposition is inspired by a computation of the Kummer pairing in \\cite[Ch. X, Theorem 1.1]{silverman}.  \n\\begin{proof} \n\tLet $R \\in E(k)\/dE(k)$ and suppose that $R \\neq 0,P,Q$. Fix some $S \\in E(\\kbar)$ with $[d]S = R$. Let $t_P, t_Q$ as above and fix $g_P, g_Q \\in \\kbar(E)$ with $g_P^d = t_P \\circ [d]$ and $g_Q^d = t_Q \\circ [d]$. Since the divisors of $g_P$ and $g_Q$ are $G_k$-invariant, we may choose $g_P, g_Q \\in k(E)$.\n\tBy the definition of $\\phi$ we see that $\\phi(f)=(a,b)$ for some cocycle $f$ representing a class in $ H^1(k,M)$ means exactly that \n\t$\\e\\left(f(\\gamma),P\\right) = \\frac{\\gamma\\left( \\sqrt[d]{b}\\right) }{\\sqrt[d]{b}} \\text{ and } \\e\\left(f(\\gamma),Q\\right) = \\frac{\\gamma\\left( \\sqrt[d]{a}\\right) }{\\sqrt[d]{a}}.$\n\tThe Weil pairing satisfies \n\t$$\\e(\\gamma(S) \\ominus S,P ) = \\frac{ g_P(\\gamma(S) \\ominus S \\oplus S)}{g_P(S)}=  \\frac{ g_P(\\gamma(S))}{g_P(S)} =\\frac{\\gamma( g_P(S))}{g_P(S)}.$$\n\tAdditionally by definition of $g_P$, we see that $g_P(S)^d = t_P \\circ [d] (S) = t_P(R)$. A similar result holds for $Q$ as well. Therefore $\\phi^{-1}\\circ \\delta(R) = \n\t\\left( t_Q (R), t_P(R) \\right)$. The other results follow by linearity of the Weil pairing. \n\\end{proof} \n\nSummarizing these results we conclude \\cref{Prop:main result in split case}. \n\n\\section{$M$ is not $k$-rational}\\label{Mnotkrational}\n\nNow let $k$ be any field and assume that $d=q$ is an odd prime coprime to the characteristic of $k$. Assume that $k$ contains a primitive $q$-th root of unity $\\rho$. Let $E$ be an elliptic curve over $k$ and denote its $q$-torsion by $M$. We do not assume that $M$ is $k$-rational throughout this section. Consider the Galois representation \n$$\\Psi: G_k \\rightarrow GL_2(\\mathbb{F}_q)$$ given by the action of $G_k$ on $M$ and denote the fixed field of its kernel by $L$. Since $k$ contains a primitive $q$-th root of unity $\\rho$, the image of $\\Psi$ lies in $SL_2(\\mathbb{F}_q)$. For $\\sigma \\in G_k$ with $\\Psi(\\sigma) = \\begin{pmatrix} a&b\\\\c&d \\end{pmatrix}$, we have that \n$$\\rho = \\sigma(\\rho) = \\sigma \\e (P,Q) = \\e \\left(\\sigma(P), \\sigma(Q) \\right) = \\e \\left( aP + cQ, bP + dQ \\right) = \\e(P,Q)^{ad-bc} = \\rho^{ad-bc}.$$\nTherefore $ad-bc=1$ and $\\Psi(\\sigma) \\in SL_2(\\mathbb{F}_q)$. \n\nConsider the tower of field extensions \n$$\\xymatrix{\n\t& \\overline{k(E)}\\ar@{-}[d]\\\\\n\t& L(E) \\ar@{-}[ld]_M \\ar@{-}[rd]^{\\Gal(L\/k)}\\\\\n\t[q]^*L(E)\\ar@{-}[rd]^{\\Gal(L\/k)} & & k(E) \\ar@{-}[ld]\\\\\n\t& [q]^* k(E)\n}$$\nFix a set $\\tilde{G}_{L\/k} \\subset G_{k(E)}$ of coset representatives of $G_{k(E)} \/ G_{L(E)} \\cong \\Gal(L\/k)$. Note that $\\tilde{G}_{L\/k}$ is also a set of coset representatives of \n$G_{[q]^*k(E)} \/ G_{[q]^*L(E)} \\cong \\Gal(L\/k)$ and every $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ fixes $k(E)$. Let $\\gamma \\in G_{[q]^* k(E)}$, then $\\gamma$ decomposes as $\\gamma= \\gamma' \\tilde{\\sigma}$ for some $\\gamma' \\in G_{[q]^*L(E)}$ and some $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$. \\\\\n\nLet $\\mathcal{T}$ be the torsor given by multiplication by $q$ on $E$. Fix a set of coset representatives $\\tilde{G}_{L\/k}$ as before. We now describe the cocycle corresponding to the pullback of $\\mathcal{T}$ along the generic point using the correspondence in  \\cref{ch:background}.\nLet $x_1$ be a $\\overline{k(E)}$ point. We may assume, that $x_0 = \\iota_1 \\circ x_0$ for some $x_0$ as in the following commutative diagram. \n$$\n\\xymatrix{\n\tk(E) \\ar[r]^{\\iota_1}\\ar@\/^3pc\/[drr]^{x_1} & L(E) \\ar[rd]^{x_0} \\\\ \n\tk(E) \\ar[u]^{[q]^*} \\ar[r]^{\\iota_1}  &L(E) \\ar[u]^{[q]^*} \\ar[r]^\\iota & \\overline{k(E)} }.\n$$\nAny $\\gamma = \\gamma' \\tilde{\\sigma} \\in G_{k(E)}$ with $\\gamma' \\in G_{L(E)}, \\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ acts on $x_1$ by \n$$\\gamma.x_1 = \\gamma' \\circ \\tilde{\\sigma} x_0 \\circ \\iota_1 =  \\gamma'. x_1.$$\nFinally, $\\gamma'.x_1$ can be computed as in \\cref{Prop:torsor mult by 3 over kbar}.\nSummarizing this we conclude the following proposition. \n\n\\begin{prop}\\label{prop:torsormult3}\n\tThe pull-back of $\\mathcal{T}$ to the generic point corresponds to the element in $H^1(k(E),M)$ given by the cocycle \n\t$$G_{k(E)} \\ra M: \\gamma \\mapsto \\left[\\frac{\\gamma'(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[ \\frac{\\gamma'(\\alpha_P)}{\\alpha_P}\\right]_\\rho Q,$$\n\twhere $\\gamma = \\gamma' \\tilde{\\sigma}$ as above, for some $\\tilde{\\sigma} \\in \\tilde{G}_{L\/k}$ and $\\gamma' \\in G_{L(E)}$, $\\alpha_P, \\alpha_Q\\in \\overline{k(E)}$ so that $\\alpha_P^q = t_P$ and $\\alpha_Q^q = t_Q$.  \n\\end{prop}\n\nAs in the previous section, we describe the map $\\epsilon$ explicitly. \n\n\\begin{prop}On the level of cocycles $\\epsilon$ coincides with the map that assigns to a 1-cocycle $f: G_k \\rightarrow M$ the 2-cocycle \n\t\\begin{equation*}\n\t\\begin{gathered}  \n\t\\epsilon(f): G_{k(E)} \\times G_{k(E)} \\ra \\overline{k(E)}^\\times\\\\\n\t(\\gamma,\\tau) \\mapsto \\e\\left( f(\\gamma), \\gamma \\left( \\left[\\frac{\\tau'(\\alpha_Q)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{\\tau'(\\alpha_P)}{\\alpha_P}\\right]_\\rho  Q \\right)  \\right),\n\t\\end{gathered} \n\t\\end{equation*} \n\tfor $\\gamma,\\tau \\in G_{k(E)}, \\tau = \\tau' \\tilde{\\sigma}$ for some $\\sigma \\in \\tilde{G}_{L\/k}$ and $\\tau' \\in G_{\\kbar(E)}$. \n\\end{prop}\n\n\\subsection{$[L:k]$ is coprime to $q$ }\\label{sec:Lkcoprimetoq}\n\nSuppose throughout this section that $q$ does not divide the order $[L:k]$. \nLet $\\mathcal{T}$ be the torsor given by multiplication by $d$ on $E$. Consider the pull-back $\\eta_L^*(\\mathcal{T})$ of $\\mathcal{T}$ to $L(E)$ \nand the pull-back $\\eta_k^*(\\mathcal{T})$ of $\\mathcal{T}$ to $k(E)$. By \\cref{Prop:torsor mult by 3 over kbar} and \\cref{prop:torsormult3} we see immediately that $\\res\\left( \\eta^*_k (\\mathcal{T}) \\right) = \\eta^*_L (\\mathcal{T})$. By \\cite[Ch. 1, Proposition 1.5.3 (iii) and (iv)]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields} and the construction of $\\epsilon$ the following diagram commutes\n$$\\xymatrix{\n\tH^1(k,M) \\ar[r]^\\res \\ar[d]^{\\epsilon_k} & H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L} & H^1(k,M) \\ar[d]^{\\epsilon_k}\\\\\n\t\\tor[q][\\Br(E)] \\ar[r]^\\res & \\tor[q][\\Br(E \\otimes L)] \\ar[r]^\\Cor & \\tor[q][\\Br(E)]\n}.$$\nTherefore, the corestriction map $$\\Cor: \\tor[q][\\Br(E \\otimes L)] \\rightarrow \\tor[q][\\Br(E)]$$ is surjective and every element in $I$ can be written as $\\Cor(A)$ with $A \\in \\tor[q][\\Br(E)]$. We summarize this observation in the following theorem. \n\\begin{thm}\n\tLet $t_P,t_Q \\in L(E)$ with divisors  $\\divisor(t_P) = q(Q) - q(0)$ and $\\divisor(t_Q) = q(Q)- q(0)$. Then the $q$-torsion of $\\Br(E)$ decomposes as \n\t$$\\tor[q][\\Br(E)] =\\tor[q][\\Br(k)] \\oplus I$$ and every element in $I$ can be represented as a tensor product \n\t$$\\Cor\\left( a, t_P \\right)_{q,L(E)} \\otimes \\Cor\\left( b, t_Q \\right)_{q,L(E)}$$ with $a,b \\in L^\\times$. \n\\end{thm}\n\n\\begin{rem}\\label{rem:image of restriction} The corestriction map is in general not injective. To get a smaller set of generators remark that by \\cite[Ch, 1, Corollary 1.5.7]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields} the image of the restriction map $H^1(k,M) \\ra H^1(L,M)$ coincides with the image of the norm map $$N_{L\/k}: H^1(L,M) \\ra H^1(L,M).$$ Now by Kummer theory $H^1(L,M) \\cong L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ via the isomorphism $\\phi$. Let $g \\in G_k$ and $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$. Suppose that $g^{-1}(P) = c_1 P \\oplus c_2 Q$ and $g^{-1}(Q) = c_3P \\oplus c_4Q$. The action of $g$ compatible with $\\phi$ is  \n\t\\begin{align*}\n\tg.(a,b) =& \\phi^{-1} g. \\phi(a,b) \\\\\n\t=& \\left( \\left(  g^{-1}(a) \\right)^{c_1} \\left( g^{-1}(b) \\right)^{c_3}, \\left(  g^{-1}(a) \\right)^{c_2} \\left( g^{-1}(b) \\right)^{c_4}\\right) .\n\t\\end{align*}\n\tNow the image of the restriction followed by $\\phi$ coincides with the image of the norm on $L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ under the above action. \n\\end{rem} \nTo calculate the relations consider the following commutative diagram \n$$\\xymatrix{\n\tE(k)\/[q]E(k) \\ar[r]^{\\res} \\ar[d]^{\\delta_k}& E(L)\/[q]E(L) \\ar[r]^{\\Cor} \\ar[d]^{\\delta_L}& E(k)\/[q]E(k) \\ar[d]^{\\delta_k}\\\\\n\tH^1(k,M) \\ar[r]^{\\res} & H^1(L,M) \\ar[r]^{\\Cor} & H^1(k,M) \n},$$where the horizontal compositions coincide with multiplication by $[L:k]$, which is an isomorphism. Thus, the image of $\\delta_k$ is also given by the image of the composition $\\delta_k \\circ \\Cor = \\Cor \\circ \\delta_L$. Using the description of the image of $\\delta_L$ in the previous section we deduce the following result. \n\n\\begin{prop} \n\tSuppose that $[L:k]$ is not divisible by $q$. Fix two generators $P$ and $Q$ of $M$ and and let $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) - q(0)$. Assume additionally that $t_P \\circ [q], t_Q \\circ[q] \\in \\left( L(E)^\\times \\right)^d.$ An element \n\t$$\\Cor(a,t_P)_{L(E)} \\otimes \\Cor(b,t_Q)_{L(E)}$$in $I$ is trivial if it is similar to one of the following \n\t\\begin{itemize}\n\t\t\\item $\\Cor\\left( t_Q(P), t_P \\right)_{k(E)}  \\otimes \\Cor\\left( \\frac{t_P(P\\oplus Q)}{t_P(Q)} , t_Q \\right)_{k(E)}$,\n\t\t\\item $ \\Cor\\left( \\frac{ t_Q(P \\oplus Q)}{t_Q(P)}, t_P \\right)_{k(E)} \\otimes \\Cor\\left( t_P(Q), t_Q  \\right)_{k(E)},$ or  \n\t\t\\item $\\Cor\\left( t_Q (R), t_P \\right)_{k(E)} \\otimes \\Cor\\left( t_P(R), t_Q \\right)_{k(E)}$ for some $R \\in E(k)\\setminus\\{0,P,Q\\}$. \n\t\\end{itemize}\n\\end{prop} \n\nThe following observation will be useful to calculate these corestrictions explicitly. Consider the following commutative diagram\n\\begin{equation}\\label{eqn:relationscorrescorres} \n\\xymatrix{\n\tE(k) \\ar[r] \\ar[d]^{\\delta_k}& E(L) \\ar[r] \\ar[d]^{\\delta_L} & E(k) \\ar[r] \\ar[d]^{\\delta_k}& E(L) \\ar[r] \\ar[d]^{\\delta_L} & E(k) \\ar[d]^{\\delta_k}\\\\\n\tH^1(k,M) \\ar[r]^\\res \\ar[d]^{\\epsilon_k}& H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L}& H^1(k,M)\\textbf{} \\ar[r]^\\res\\ar[d]^{\\epsilon_k}  & H^1(L,M) \\ar[r]^\\Cor \\ar[d]^{\\epsilon_L}& H^1(k,M) \\ar[d]^{\\epsilon_k}\\\\\n\t\\tor[q][\\Br E] \\ar[r] & \\tor[q][\\Br E_L] \\ar[r] &  \\tor[q][\\Br E] \\ar[r] & \\tor[q][\\Br E_L] \\ar[r] &  \\tor[q][\\Br E]\n},\\end{equation} \nwhere the rows compose to multiplication by $[L:k]^2$, which is an isomorphism. Furthermore, the composition $\\res \\circ \\Cor$ coincides with the norm map \\cite[Ch. 1, Corollary 1.5.7]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields}. Therefore, an element in $I$ is trivial if it lies in the image of the composition $\\Cor\\circ \\epsilon_L \\circ N_{L\/k} \\circ \\delta_L \\circ \\res$. In \\cref{subsec:Ex degree L\/k coprime to 3}, we see how this observation can be applied to the calculation of the relations in $I$. \n\n\\subsection{$[L:k]$ equals $q$}\\label{sec:Generators-div-q} \n\nSuppose for this section that $L$ is of degree $q$ over $k$. After renaming $P$ and $Q$ we may assume without loss of generality that there is some $\\sigma \\in G_k$ such that $\\sigma(Q) = P\\oplus Q$ and $\\overline{\\sigma}$ generates $G_k\/G_L$.\nFix a coset representative $\\tilde{\\sigma}$ of $\\sigma$ in $G_{k(E)}$.\nAdditionally denote a primitive element for the extension $L\/k$ by $l$.\nConsider the diagram\n$$\\xymatrix{ 0 \\ar[r]& H^1\\left(G_k\/G_L, M\\right) \\ar[r]^-{\\inflation}& H^1(k,M)\\ar[rr]^-\\res\\ar[d]^{\\epsilon_k} && H^1(L,M)^{G_k\/G_L}\\ar[d]^{\\epsilon_L}\\\\\n\t& & \\tor[q][\\Br(E)]\\ar[rr]^-\\res && \\tor[q][\\Br(E \\times_k \\Spec L)] },$$\nwhere the first row is the inflation-restriction exact sequence. The diagram commutes by construction of $\\epsilon$ and since the restriction map and the cup-product commute \\cite[Ch. 1 Proposition 1.5.3 (iii)]{Neukirch-Schmidt-Wingberg-Cohomology-of-number-fields}. We will first describe the image of the inflation map and explore the restriction afterwards. We will apply the following technical lemma throughout. \n\n\\begin{lem}\\label{lem:technical}\n\t$\\sum_{i=0}^{q-1} \\sigma^i(R) = 0$ for every $R \\in M$.\n\\end{lem} \n\n\\begin{proof} \n\tLet $R= mP\\oplus nQ \\in M$. We calculate directly that\n\t$$\n\t\\sum_{i=0}^{q-1} \\sigma^i(mP \\oplus nQ) =  \\sum_{i=0}^{q-1} \\left( mP \\oplus inP\\oplus nQ \\right) = mqP \\oplus \\frac{q (q-1)}2 n P \\oplus n q Q = 0.\n\t$$  \n\\end{proof}\n\n\\begin{lem} The group\n\t$H^1\\left( G_k\/G_L, M \\right)$ is cyclic of rank $q$ with generator the class of the cocycle $f_L$ defined by $f_L(\\overline{\\sigma}) = Q$. \n\\end{lem} \n\n\\begin{proof} \n\t\\Cref{lem:technical} implies that  $f_L(\\overline{\\sigma}^q) = \\sum_{i=0}^{q-1} \\sigma^i f_L(\\overline{\\sigma}) = 0$ and thus $f_L$ defines a cocycle. Since $G_k\/G_L$ is cyclic with generator $\\overline{\\sigma}$, every element $f$ in $H^1(G_k,G_L,M)$ is determined by $f(\\overline{\\sigma})$. Furthermore, if $f(\\overline{\\sigma}) = mP\\oplus nQ$, then \n\t\\begin{align*}\n\tf(\\overline{\\sigma}) - \\overline{\\sigma}(mP) = mP \\oplus nQ \\ominus mP =  nQ = f_L^n (\\overline{\\sigma}).\n\t\\end{align*} \n\tThe statement follows. \n\\end{proof} \nLet $\\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_Q^q = t_Q$ and consider $n_Q = \\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right)$. Fix $\\gamma \\in G_{L(E)}$. By our previous calculations and with $x_0$ and $g_Q$ as in the proof of \\cref{prop:torsormult3} we deduce that there is some $R \\in M$ such that $\\gamma(\\alpha_Q) = R.x_0(g_Q)$. Then \nby \\cref{lem:technical} \n\\begin{align}\n\\gamma \\left( \\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right) \\right)  = \\left( \\sum_{i=0}^{p-1} \\sigma^i(R) \\right). x_0(g_Q) = \\alpha_Q.\n\\end{align} Now $\\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q \\right) $ is obviously fixed by $\\tilde{\\sigma}$ and therefore $\\prod_{i=0}^{q-1} \\tilde{\\sigma}^i \\left(\\alpha_Q\\right) \\in k(E).$ Thus $n_Q$ is defined to be the element in $k(E)$ with $n_Q^q = N_{L(E)\/k(E)} (t_Q)$. Note additionally that $\\divisor n_Q = \\sum_{i=0}^{q-1} \\left(\\sigma^i(Q) - (0)\\right)$.\n\n\\begin{prop}\\label{prop:ImageofInflation}\n\t$\\epsilon_k\\left(\\inflation\\left(f_L\\right)\\right)$ is the inverse of the Brauer class of the symbol algebra\n\t$\\left(l^q, n_Q\\right)_{q,k(E)},$ where $\\alpha_Q \\in \\overline{k(E)}$ with $\\alpha_Q^q = t_Q$.\n\\end{prop} \n\n\\begin{proof} Let $\\gamma, \\tau \\in G_{k(E)}$ and denote $\\gamma = \\gamma' \\tilde{\\sigma}^i, \\tau = \\tau' \\tilde{\\sigma}^j$ with $\\gamma',\\tau' \\in G_{L(E)}$. Then by definition of $\\epsilon$ we see that  \n\t\\begin{align*} \n\t\\epsilon_k \\left( \\inflation\\left( f_L \\right) \\right) (\\gamma,\\tau) \n\t&=\\e \\left( \\frac{(i-1) i }{2} P \\oplus iQ , \\sigma^i \\left( \\left[\\frac{ \\tau'\\left( \\alpha_Q \\right)}{\\alpha_Q}\\right]_\\rho P - \\left[\\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P}\\right]_\\rho Q \\right) \\right)  \\\\\n\n\n\n\t&= \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{- \\frac{(i-1)i}{2}} \\left(\\frac{  \\tau'(\\alpha_Q)}{\\alpha_Q} \\right)^{-i} \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{i^2}\\\\\n\t&= \\left( \\frac{ \\tau' \\left( \\alpha_P \\right)}{\\alpha_P} \\right)^{ \\frac{(i+1)i}{2}} \\left(\\frac{  \\tau'(\\alpha_Q)}{\\alpha_Q} \\right)^{-i}\\end{align*} \n\tNow consider the map \n\t\\begin{equation}\n\tg: G_{k(E)} \\rightarrow \\overline{k(E)}^\\times : \\gamma \\mapsto \\gamma' \\left( \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) \\right),\n\t\\end{equation}\n\twhere $\\gamma= \\gamma' \\tilde{\\sigma}^i$ for some $\\gamma' \\in G_{L(E)}$. The differential of $g$ is \n\t$$dg(\\gamma,\\tau) = \n\t\\begin{cases} \n\t\\frac{ \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) }{ \\tilde{\\sigma}^i \\tau' \\tilde{\\sigma}^{-i} \\left( \\prod_{n=0}^{i-1} \\tilde{\\sigma}^n(\\alpha_Q) \\right)} & \\text{ if } i + j < q\\\\\n\t\\left(  \\prod_{n=0}^{q-1} \\tilde{\\sigma}^n(\\alpha_Q)\\right) \\left( \\frac{ \\alpha_P}{\\tau'(\\alpha_P)} \\right)^{-\\frac{(i+1)i}{2}}  \\left(  \\frac{ \\alpha_Q}{\\tau'(\\alpha_Q)}\\right)^i & \\text{ else}\n\t\\end{cases}. $$\n\tThe statement follows by subtracting this trivial cocycle and applying \\cref{prop:sumbolalgebra-cocycle}.\n\\end{proof} \n\n\\begin{prop} \n\t$\\epsilon$ induces a split to sequence (\\ref{Eq:exact sequence on tor[d]Br(E)}).\n\\end{prop} \n\n\\begin{proof}\n\tIt suffices to show that $r \\circ \\epsilon \\circ \\inflation (f_L) = \\lambda \\circ \\inflation (f_L)$. \n\tBy the previous lemma, $\\epsilon(\\inflation(f_L))$ gives\n\t$$(\\gamma'\\sigma^i,\\tau'\\sigma^j) \\mapsto \\begin{cases} 1 & \\\\ i+j < q \\\\\n\t\\sum_{i=0}^{q-1} \\left(\\sigma(Q) - (0)\\right) & i+j \\geq q \\end{cases}$$\n\tin $H^2(G_k, \\Prin(E))$, where $\\gamma',\\tau' \\in G_k$. On the other hand, $r(\\inflation(f_L))$ can be presented by the cocycle \n\t$$\\gamma' {\\sigma}^i \\mapsto \\sum_{m=0}^i {\\sigma}^m(Q) - 0.$$\n\tThis lifts to the map\n\t$$\\gamma' {\\sigma}^i \\mapsto \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0) \\in \\Div^0(\\overline{E}),$$\n\tand a direct computation of the boundary map gives \n\t\\begin{align*} \n\t(\\gamma' {\\sigma}^i,\\tau'{\\sigma}^j) \\mapsto &\n\t\\begin{cases}  \\sigma^i \\left( \\sum_{m=0}^j \\left( \\sigma^m(Q)\\right)  - (0)\\right) - \\left( \\sum_{m=0}^{i+j} \\left( \\sigma^m(Q)\\right)  - (0)\\right) \\\\\n\t+ \\left( \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j < q \\\\\n\t\\sigma^i \\left( \\sum_{m=0}^j \\left( \\sigma^m(Q)\\right)  - (0)\\right) - \\left( \\sum_{m=0}^{i+j-q} \\left( \\sigma^m(Q)\\right)  - (0)\\right) \\\\\n\t+ \\left( \\sum_{m=0}^i \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j \\geq q \\end{cases} \\\\\n\t&=  \\begin{cases} 1 & i+ j < q \\\\\n\t\\left( \\sum_{m=0}^{q-1} \\left( \\sigma^m(Q)\\right)  - (0)\\right) & i+ j \\geq q \\end{cases},\n\t\\end{align*}\n\twhich coincides with the previous calculation.\n\\end{proof}\n\nWe now calculate the image of the composition $\\phi^{-1} \\circ \\res$ with $\\phi$ as in \\cref{Eq:phi definition}. By \\cite[Chapter VII, Section 5]{serrelocal} the action of $G_k$ on $L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ compatible with $\\phi$ is given by \n$$\\sigma.(a,b) = \\phi^{-1} \\sigma. \\phi(a,b) = \\left( \\frac{\\sigma^{-1}(a)}{ \\sigma^{-1}(b)} , \\sigma^{-1}(b) \\right).$$  \n\n\\begin{lem}\\label{lem:fixedSet}\n\tUnder the isomorphism $\\phi^{-1}$ the fixed set $H^1(L,M)^{G_k\/G_L}$ corresponds to \n\t$$\\left\\{ \\left( a, \\frac{a}{\\sigma(a)} \\right) : \\sigma(a)^2 \\equiv \\sigma^2(a) a \\mod (L^\\times)^q \\right\\}.$$\n\\end{lem} \n\n\\begin{proof} \n\tLet $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ be fixed by the above action. Then \n\t$a \\equiv \\frac{\\sigma^{-1}(a)}{ \\sigma^{-1}(b)}$, which implies that $b \\equiv \\frac{a}{\\sigma(a)}$. Now $b \\equiv \\sigma^{-1}(b)$ and thus \n\t$ \\frac{a}{\\sigma(a)} \\equiv \\frac{ \\sigma^{-1}(a)}{a}$ which implies that $a^2 \\equiv \\sigma(a) \\sigma^{-1}(a)$ or equivalently $\\sigma(a)^2 \\equiv \\sigma^2(a) a$. \n\\end{proof} \n\n\\begin{lem}\\label{lem:ImageRes1} \n\t$f \\in H^1\\left( L,M\\right)^{G_k\/G_L}$ is in the image of the restriction map if and only if \n\t$f(\\gamma^q) = 0$ for any $\\gamma \\in G_k$. \n\\end{lem} \n\n\\begin{proof}\n\tLet $\\gamma \\in G_k$ and suppose that $f$ is in the image of the restriction map with preimage $g$. Then we can write $\\gamma = \\gamma' \\sigma^i$ for some $\\gamma' \\in G_L$. We calculate directly using \\cref{lem:technical} that\n\t$$f(\\gamma^q) = g(\\gamma^q) = \\sum_{i=0}^{q-1} \\gamma^q g(\\gamma) = \\sum_{i=0}^{q-1} \\sigma^{iq} g(\\gamma) = \\sum_{i=0}^{q-1} \\sigma^i  g(\\gamma) = 0.$$\n\tFor the converse assume that $f$ satisfies the condition that $f(\\gamma^q) = 0$ for any $\\gamma \\in G_k$. In particular $f(\\sigma^q) = 0$. Define $g \\in H^1(k,M)$ by setting $g(\\gamma) = f(\\gamma')$, where $\\gamma = \\gamma' \\sigma^i$ for $\\gamma' \\in G_L$. This is well-defined as for any $\\gamma,\\tau\\in G_k$ with $\\gamma = \\gamma' \\sigma^i, \\tau = \\tau' \\sigma^j$ and $\\gamma',\\tau' \\in G_L$ we have that \n\t$\n\tg\\left( \\gamma\\tau\\right) = g\\left( \\gamma' \\left( \\sigma^i \\tau' \\sigma^{-i}\\right)  \\sigma^{i+j}\\right)  \n\t= g\\left( \\gamma'\\right) g\\left( \\sigma^i \\tau' \\sigma^{-i}\\right)\n\t= g\\left( \\gamma' \\right) \\sigma^i g(\\tau') \n\t= g\\left( \\gamma \\right) \\gamma g(\\tau).\n\t$\\end{proof} \n\nWe will now prove a technical lemma that will be useful to determine the image of the restriction. \n\n\\begin{lem}\\label{lem:technical-lemma-Galois-theory}\n\tLet $k$ be a field of characteristic prime to $q$ containing a primitive $q$-th root of unity. Let $k \\subset L \\subset F$ be a tower of field extensions so that each extension is Galois of degree $q$. Let $a \\in L^\\times$ such that $F = L\\left( \\sqrt[q]{a}\\right)$. Fix a representative $\\sigma \\in G_k$ that generates $\\Gal(L\/k)$. Suppose that for every $\\gamma \\in G_k$ we have that $\\gamma^q \\left( \\sqrt[q]{\\sigma^i(a)}\\right)= \\sqrt[q]{\\sigma^i(a)}$ for $0 \\leq i < q$. Then there exists some $b \\in k^\\times$ such that $a \\equiv b \\mod (L^\\times)^q$. \t\n\\end{lem} \n\n\\begin{proof}Assume that $a \\not\\in k^\\times$. Fix $\\sigma \\in G_k$ such that $\\overline{\\sigma}$ generates $\\Gal(L\/k)$. Denote the Galois closure of the extension $F$ over $k$ by $\\tilde{L}$. By Galois theory \n\t$\\tilde{L} = L\\left( \\sqrt[q]{a}, \\sqrt[q]{\\sigma(a)}, \\ldots, \\sqrt[q]{\\sigma^{q-1}(a)} \\right)$ and $\\Gal(\\tilde{L}\/L)$ is isomorphic to $\\left( \\Z\/q\\Z \\right)^{r}$ for $r=1$, or $r=q$. \n\tWe prove the lemma by contradiction. Assume that $r=q$. Let $\\tau \\in \\Gal(\\tilde{L}\/L)$ be the element with $\\tau \\left( \\sqrt[q]{a} \\right) = \\rho \\sqrt[q]{a}$ and $\\tau \\left( \\sigma^i \\sqrt[q]{a} \\right) = \\sigma^i \\sqrt[q]{a}$ for $1 \\leq i <q$. \n\tNow $\\left( \\sigma\\tau \\right) \\left( \\sqrt[q]{a} \\right)= \\left( \\rho \\sigma^q\\right) \\left( \\sqrt[q]{a} \\right) =\\rho \\left(   \\sqrt[q]{a}\\right)$, which is a contradiction to our assumptions. We conclude that $F\/k$ is Galois of degree $q^2$. We want to show that $\\Gal(F\/k) = \\Z\/q\\Z \\times \\Z\/q\\Z$.\t\n\tNext suppose that $\\Gal(F\/k) = \\Z\/q^2 \\Z$ and denote the generator by $\\tau$. Then $\\tau^q$ fixes $L$, as $\\tau^q(\\sqrt[q]{a}) =\\sqrt[q]{a}$ implies that $\\tau^q(a) = a$ and $L= k(a)$. Hence $\\tau \\in \\Gal(F\/L)$, which implies that $\\tau$ is of order $q$, a contradiction. We conclude that $\\Gal(F\/k) \\cong \\Z\/q\\Z \\times \\Z\/q\\Z$. Consider the fixed field $F^{\\left< \\sigma \\right>}$, which is a degree $q$ extension of $k$. By Kummer theory, there exists an element $b \\in k^\\times$ so that  $F^{\\left< \\sigma \\right>} = k\\left( \\sqrt[q]{b}\\right)$. Finally, $F = L \\left( \\sqrt[q]{b}\\right) = L \\left( \\sqrt[q]{a}\\right)$ and thus by Kummer theory $a \\equiv b \\mod (L^\\times)^q$. \n\\end{proof} \n\n\\begin{prop}\\label{prop:ImageOfRestrictionmap}\n\tThe image of the restriction map corresponds to the set $$k^\\times\/\\left((L^\\times)^q\\cap k^\\times\\right) \\times \\{1\\} \\subset L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$$ under the isomorphism $\\phi^{-1}$. \n\\end{prop} \n\n\\begin{proof} \n\tLet $(a,b) \\in L^\\times\/(L^\\times)^q \\times L^\\times\/(L^\\times)^q$ be in the image of the restriction map. There exists some $f \\in H^1(k,M)$ so that $\\phi^{-1}\\circ \\res(f) = (a,b)$. Then $(a,b)$ is necessarily in the preimage $\\phi^{-1} \\left( H^1(L,M)^{G_k\/G_L} \\right) $ and by \\cref{lem:fixedSet} we see that $(a,b) \\equiv \\left( a, \\frac{a}{\\sigma(a)} \\right)$ so that $\\sigma(a)^2 \\equiv \\sigma^2(a) a \\mod (L^\\times)^q$. It remains to show that we can choose $a \\in k^\\times$. \\\\\n\tBy definition of $\\phi$ and by \\cref{lem:ImageRes1} we get that $\\gamma^q\\left(\\sqrt[q]{a}\\right) = \\sqrt[q]{a}$ and $\\gamma^q\\left( \\sqrt[q]{\\frac{a}{\\sigma(a)}}\\right) = \\sqrt[q]{\\frac{a}{\\sigma(a)}}$ for any $\\gamma \\in G_k$ and for any choice of root. Using the condition that $\\sigma(a)^2 \\equiv \\sigma^2(a) a$, we deduce that $\\gamma^p \\left( \\sqrt[q]{\\sigma^i}{a} \\right) = \\sqrt[q]{\\sigma^i}{a}$ for any $i$. The statement follows from \\cref{lem:technical-lemma-Galois-theory}.\t\n\n\n\n\\end{proof} \n\nThe following theorem summarizes the results of this section. \n\n\\begin{thm}\\label{thm:Br(E) three case} \n\tSuppose that $[G_k:G_L]$ is of order $q$ and assume that there is some $\\sigma \\in G_k$ with $\\sigma(Q) = P\\oplus Q$ such that $\\overline{\\sigma}$ generates $G_k\/G_L$.  Additionally denote a primitive element for the extension $L\/k$ by $l$.  \n\t$$\\tor[q][\\Br(E)] = \\tor[q][\\Br(k)] \\oplus I$$\n\tand $I$ has generators $\\left\\{\t\\left( l^q, n_Q \\right)_{k(E)}, (a,t_P)_{k(E)}:\n\ta \\in k^\\times\n\t\\right\\},$ where $n_Q \\in k(E)$ with $n_Q^q = N_{L(E)\/k(E)}(t_Q)$.\n\\end{thm} \nFor the relations consider the commutative diagram with exact rows and columns \n$$\\xymatrix{\n\t& & 0 \\ar[d] & 0 \\ar[d] \\\\\n\t0 \\ar[r] & \\frac{E(k) \\cap [q]E(L)}{[q]E(k)} \\ar[r]^{\\inflation} \\ar[d]^{\\delta_{L\/k}} & E(k)\/[q]E(k) \\ar[r]^{\\res}\\ar[d]^{\\delta_k} & E(L)\/[q]E(L) \\ar[d]^{\\delta_L}   \\\\\n\t0 \\ar[r]& H^1(\\Gal(L\/k),M) \\ar[r]^{\\inflation} & H^1(k,M) \\ar[r]^{\\res} & H^1(L,M) \n},$$\nwhere $\\delta_{L\/k}$ is the map induced by $\\delta_k$. It is immediate that $\\delta_{L\/k}$ is injective. Recall that $H^1\\left( \\Gal(L\/k),M\\right)$ is cyclic of order $q$ with generator $f_L$. Furthermore, we saw that $\\epsilon_L\\left( \\inflation(f_L) \\right) = \\left( l^q, n_Q \\right)_{k(E)}$ (\\cref{prop:ImageofInflation}). We deduce the following results. \n\n\\begin{enumerate} \n\t\\item The Brauer class of $\\left( l^q, n_Q \\right)_{k(E)} $ is trivial, that is it is in the image of the map $\\Br(k) \\rightarrow \\Br(E)$, if and only if the quotient \n\t$\\frac{E(k) \\cap [q]E(L)}{[q]E(k)} $ is non-trivial. \n\t\\item \tThe Brauer class of $\\left( a, t_P \\right)_{k(E)}$ is trivial if and only if there is some $R \\in E(k)\/[q]E(k)$ so that $\\phi^{-1}(a,1) = \\delta_L (R)$. \n\\end{enumerate}  \n\n\n\\subsection{$q$ divides the degree $[L:k]$}\\label{sec:Generators:qdividesLk} \n\nSuppose for this section that $q$ divides $[L:k]$. Let $k \\subset L' \\subset L$ be an intermediate field so that $L\/L'$ is a Galois extension of degree $q$ and $q$ does not divide the degree $L'\/k$. After renaming $P$ and $Q$ we may assume that there is some generator $\\overline{\\sigma}$ of $\\Gal(L\/L')$ so that $\\sigma(P) = P$ and $\\sigma(Q) = P\\oplus Q$. Furthermore, let $l \\in L$ with $l^q \\in L'$ and $L = L'(l)$.\nFix $t_P, t_Q \\in L(E)$ with $\\divisor(t_P) = q(P) - q(0)$ and $\\divisor(t_Q) = q(Q) - q(0)$. Assume additionally that $t_P \\circ [q], t_Q \\circ[q] \\in \\left( L(E)^\\times \\right)^d.$  Furthermore, let $\\alpha_Q\\in \\overline{k(E)}$ so that $\\alpha_Q^q = t_Q$. \\\\\n\nAlthough the field extension $L'\/k$ might not be Galois, restriction followed by corestriction coincides with multiplication by $[L':k]$, which is an isomorphism. Using the previous section we deduce the following result. \n\n\\begin{prop} \n\tUnder the above assumptions, the Brauer group decomposes as\n\t$$\\tor[q][\\Br(E)] =  \\tor[q][\\Br(k)] \\oplus I$$\n\tand every element in $I$ can be expressed using the generators \n\t$$\\left\\{\t\\Cor\\left( l^q, n_Q \\right)_{L'\n\t\t(E)}, \\Cor(a,t_P)_{L'(E)}:\n\ta \\in L'^\\times\n\t\\right\\}.$$\n\\end{prop}\n\n\nSuppose that $q$ divides $[L:k]$ and use the notation used in \\cref{sec:Generators:qdividesLk}. Recall that every element in $I$ can be written as $\\Cor(A)$ for some $A \\in \\tor[q][\\Br(E\\times \\Spec(L'))]$. Such an element is trivial if and only if it is similar to $\\epsilon_{L'} \\circ \\delta_{L'}$. \nRemark that some corestrictions of element in $\\tor[q][\\Br(E_L)]$ may coincide and we do not account for this in our description. \n\n\\section{Examples}\\label{ch:Examples} \n\nIn this section, we calculate the $q$-torsion of the Brauer group for some elliptic curves $E$, where $q$ is an odd prime. For computational reasons, we only consider the case $q=3$. As before, we will consider various cases depending on the extension $L$, that is the smallest Galois extension of $k$, so that $M$ is $L$-rational. \n\n\\subsection{$M$ is $k$-rational over a number field}\\label{sec:Ex:split}\nLet $k = \\Q(\\omega)\\subset \\mathbb{C}$, where $\\omega$ is a primitive third root of unity. In \\cite{Paladino2010} the author describes a family of elliptic curves such that $M$ is $\\mathbb{Q}(\\omega)$-rational, for example $E$ given by the affine equation $y^2 = x^3 + 16$. In this case, the three torsion of $E$ is generated by \n$P= (0,4)$ and $Q= (-4, 8 \\omega + 4)=(-4,4\\sqrt{3}i)$. Furthermore, the tangent lines at $P$ and $Q$, respectively, are given by $t_P = y-4$ and $$t_Q= y - \\frac{6}{2 \\omega + 1 } (x+4) - 8 \\omega - 4 = y - 4 \\sqrt{3}i x - 20 \\sqrt{3} i.$$ \nBy the previous discussion $\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)] \\oplus I$ and every element in $I$ can be written as a tensor product \n\\begin{equation}\\label{eq:Ex1}\n(a, y-4)_{3,k(E)} \\otimes\\left(b,  y - 4 \\sqrt{3}i x - 20 \\sqrt{3} i\\right)_{3,k(E)}\n\\end{equation} for some $a,b \\in k^\\times.$\nWe calculate with magma, that $E(k) = M$ and therefore also $E(k)\/3E(k) = M$. Using the algorithm we see that a tensor product as in \\cref{eq:Ex1} is trivial if and only if it is similar to an element in the subgroup generated by \n$$\\left( 4- 20 \\sqrt{3} i , t_P \\right)_{3,k(E)}$$\nand \n$$\\left( \\frac6{19} - \\frac{8}{19} \\sqrt{3} i, t_P\\right)_{3,k(E)} \\otimes \\left( 4\\sqrt{3} i -4,  t_Q \\right)_{3,k(E)}.$$\n\n\n\\subsection{Degree $L\/k$ coprime to $q$ for $k$ a number field}\\label{subsec:Ex degree L\/k coprime to 3} \n\nLet $k = \\mathbb{Q}(\\omega)$ and $E$ the elliptic curve given by the affine equation \n$$y^2 = x^3 + B,$$\nwhere $B \\equiv 2 \\mod (\\mathbb{Q}^\\times )^3$ and $B \\not\\equiv 1,-3 \\mod (\\mathbb{Q}^\\times)^2$. By \\cite[Theorem 3.2 and Corollary 3.3]{Bandini2012} we see that $L = k(\\sqrt{B})$. Let $\\sigma$ be given by $\\sigma(\\sqrt{B}) = - \\sqrt{B}$. The three torsion of $E$ has generators $P$ and $Q$ with $P = (0,\\sqrt{B})$ and $Q =  \\left(\\sqrt[3]{-4B},\\sqrt{-3B}\\right).$ Then $\\sigma(P) = 2P$ and $\\sigma(Q) = 2Q$. We need to calculate \n$$\\Cor_{L(E)\/k(E)}\\left((a,t_P)_{3,L(E)} \\otimes (b,t_Q)_{3,L(E)}\\right).$$ \tRecall that by \\cref{rem:image of restriction} it will be enough to consider $(a,b)$ a norm in $L^\\times\/(L^\\times)^3 \\times L^\\times\/(L^\\times)^3$. For $(a,b) \\in L^\\times\/(L^\\times)^3 \\times L^\\times\/(L^\\times)^3$ we have $N_{L\/k}(a,b) = \\left(\\frac{a}{\\sigma(a)}, \\frac{b}{\\sigma(b)}\\right)$, or equivalently we may assume that $N_{L\/k}(a) = 1$ and $N_{L\/k}(b) = 1$ and $a,b \\in L^\\times \\setminus k^\\times$. Note that $$t_P = y - \\sqrt{B}$$ and $$t_Q = y - \\frac{3 \\sqrt[3]{-4B}}{2\\sqrt{-3B}} x - 3 \\sqrt{-3B}.$$ Furthermore, $\\sqrt{-3} \\in k$ as $\\omega \\in k$ and $\\sqrt[3]{16B^2} \\in k^\\times$ since $B \\equiv 2 \\mod (\\mathbb{Q}^\\times )^3$. \\\\\n\nLet $a = a_1\\sqrt{B} + a_2$ with $a_1 \\neq 0$ and $N_{L\/k}(a) = 1$. We use the algorithm given in \\cite[Section 3]{Rosset1983} to calculate the corestriction explicitly as\n\\begin{align*} \n\\Cor(a,t_P)_{3,L(E)} &= \\left( a_2 - a_1y, a_1^2\\right)_{3,k(E)}.\n\\end{align*}\nFinally, let $b= b_1 \\sqrt{B} + b_2$ with $b_1 \\neq 0$ and $N_{L\/k}(b) = 1$. Then\n\\begin{align*} \n\\Cor(b, t_Q)_{3,L(E)} &= \\left(  \\frac{ y b_1}{ \\sqrt{3} i\\left(x+1\\right) } + b_2, 1 - b_2^2 - \\frac{b_1^2 (x^3 + B)}{3(x+1)^2} \\right)_{3,k(E)}.\n\\end{align*} \nOverall, the three torsion of the Brauer group decomposes as\n\t$$\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)]\\oplus I$$\n\tand every element in $I$ can be written as a tensor product \n\t$$\\left( a_2 - a_1y, a_1^2\\right)_{3,k(E)} \\otimes  \\left(  \\frac{ y b_1}{ \\sqrt{3} i\\left(x+1\\right) } + b_2, 1 - b_2^2 - \\frac{b_1^2 (x^3 + B)}{3(x+1)^2} \\right)_{3,k(E)}$$\n\tfor some $a_1, a_2, b_1, b_2 \\in k^\\times$ with $a_1, b_1 \\neq 0$, and $a_2^2 - B a_1^2 = b_2^2 - B b_1^2 = 1$.  \\\\\n\nTo calculate the relations we need to specify $B$. Consider the case $B= -1024$. Using magma we calculate that $E(k) = 0$. Thus there are no additional relations. Note that some elements might still become trivial due to the fact that the corestriction map is not surjective. \n\n\\subsection{Degree $L\/k= q$ for $k$ a number field}\\label{sec:Ex:degreeq}\nLet $k = \\Q(\\omega)$, $\\omega = -\\frac{1}{2} + \\frac{i\\sqrt{3}}{2}$ and let $E$ be the elliptic curve given by the affine equation \n$$y^2 = x^3+4.$$\nGenerators of the three torsion are given by $P= (0, 2)$ and $Q=(-2\\sqrt[3]{2} , 2i\\sqrt{3})$. Let $l = \\sqrt[3]{2}$. In our previous notation $L= k\\left(\\sqrt[3]{2}\\right)$ and the Galois group $\\Gal(L\/k)$ is generated by $\\overline{\\sigma}$ with $\\sigma(Q) = P+Q = (-\\omega 2 \\sqrt[3]{2}, 2i \\sqrt{3})$. It can be seen that \n\\begin{align*}\nt_P & = y - 2 \\\\\nt_Q & = y + i \\sqrt{3} \\sqrt[3]{4}x + 2 i \\sqrt{3}\\\\\nn_Q &= y - 2 \\sqrt{3}i )^3. \n\\end{align*}\nTherefore the $3$-torsion of $\\Br(E)$ is $$\\tor[3][\\Br(E)] = \\tor[3][\\Br(k)] \\oplus I$$ and every element in $I$ can be written as a tensor product of the symbol algebras \n\t$$\\left( 2, y - 2 \\sqrt{3}i \\right)_{3,k(E)} \\text{ and } (a,y-2)_{3,k(E)}$$ for some $a \\in k^\\times$. \nWe calculate that $E(k) = \\left<P\\right>$ and $E(L) \\cong \\Z\/6\\Z \\times \\Z\/6\\Z$. Therefore \n$E(k)\/3E(k) = \\left< P\\right>$ and $E(L)\/3E(L) = M$ and the quotient $\\frac{E(k) \\cap [3] E(L)}{[3]E(k)}$ is trivial. Therefore the symbol algebra \n$\\left(  2, y - 2 \\sqrt{3}i \\right)_{k(E)}$ is not trivial. Finally, $\\epsilon \\circ \\delta \\circ \\res (P) = \\left( 2 + i \\sqrt{3} , y-2 \\right)_{L(E)}$ and therefore a symbol algebra $(a, y-2)_{k(E)}$ is trivial if and only if it is similar to one of the following \n$$\\left\\{ \n(1,1)_{3,k(E)}, \\left( 2+ i \\sqrt{3}, y-2 \\right)_{3,k(E)}, \\left( -8 + 8 i \\sqrt{3}, y-2 \\right)_{3,k(E)}\n\\right\\}.\n$$\n\n\\subsection{Over a local field}\\label{sec:local} \n\nDenote by $\\Q_7$ the $7$-adic numbers. Note that since three divides $7-1$, the field $\\Q_7$ contains a primitive third root of unity $\\omega$. Let $E$ be the elliptic curve \n$$E: y^2 = x^3 + 16$$ over $k$. Consider the reduction $\\tilde{E}$ of $E$ modulo $7$. Then $\\tilde{E}$ is a non-singular curve and using magma we see that \n$$\\tilde{E}\\left(\\mathbb{F}_7\\right) = \\Z\/3\\Z \\oplus \\Z\/3\\Z = \\left\\{ \n0, (0,3), (0,4), (3,1), (3,6), (5,1),(5,6), (6,1), (6,6)\n\\right\\}. $$\nDenote by $\\hat{E}$ the formal group associated to $E$ and consider the group $\\hat{E}\\left(7\\Z_7\\right)$. By \\cite[IV Theorem 6.4]{silverman} there is an isomorphism \n$\\hat{E} \\left( 7 \\Z_7 \\right) \\rightarrow \\hat{\\mathbb{G}}_a \\left( 7 \\Z_7 \\right)$, where ${\\mathbb{G}_a}$ denotes the additive group. By \\cite[IV.3 and VII.2]{silverman} there is an exact sequence \n$$\\xymatrix{ 0 \\ar[r]&\\hat{\\mathbb{G}}_a\\left(7\\Z_7\\right) \\ar[r]& E(\\Q_7) \\ar[r]&  \\tilde{E}\\left(\\mathbb{F}_7\\right) \\ar[r]& 0}.$$\nFurthermore, by \\cite[VII.3 Proposition 3.1]{silverman} the reduction map $\\tor[3][E(\\Q_7)] \\rightarrow \\tilde{E}\\left(\\mathbb{F}_7\\right)$ is injective. Thus $E$ has $k$-rational $3$-torsion. Since $3$ is a unit in $\\Z_7$, we further deduce that $E(\\Q_7)\/[3]E(\\Q_7) = \\tilde{E}(\\mathbb{F})\/[3]\\tilde{E}(\\mathbb{F}) =M.$ Finally, \n$$\\Q_7^\\times \/ \\left( \\Q_7^\\times \\right)^3 \\cong \\left( \\Z_7^\\times \\times \\left(7\\Z_7\\right) \\right)\/ \\left( \\Z_7^\\times \\times \\left(7\\Z_7\\right) \\right)^3 \\cong \\Z\/3\\Z \\times \\Z\/3\\Z.$$\nTherefore, $H^1(k,M) \\cong \\left( \\Q_7^\\times \/ \\left( \\Q_7^\\times \\right)^3  \\right)^2 \\cong \\left( \\Z\/3\\Z \\right)^4$. \nBy the algorithm and using \\cite[Corollaire 2.3]{GroupedeBrauerIII}, the three torsion of the Brauer group decomposes as follows \n$$\\tor[3][\\Br(E)] \\cong \\tor[3][\\Br\\left(\\Q_7\\right)] \\oplus \\left( \\Z\/3\\Z  \\right)^2 = \\tor[3][\\left( \\Q\/\\Z \\right)] \\oplus \\left( \\Z\/3\\Z \\right)^2 \n= \\left( \\Z\/3\\Z \\right)^3.\n$$ \n\n\\begin{rem}\n\tThe above computations also show that $\\tor[3][\\Br\\left( \\tilde{E}\\right)] = \\tor[3][\\Br\\left( \\mathbb{F}_7 \\right)] = 0$. \t\n\\end{rem} \n\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs one of the nearest powerful radio-loud active galactic nuclei (AGN), Cygnus A provides an excellent laboratory to study the environment and activity of powerful AGN. The luminosity of the AGN in Cygnus A (which is predominantly expressed in the infrared) is high enough to classify it as quasar \\citep[e.g.,][]{Djorgovski91}. While not seen in total intensity, broad H$\\alpha$ is seen in polarized light \\citep{Ogle97}, lending support for the existence of an obscured broad-line region (BLR). The polarized broad lines were detected within the ionization cone seen by \\citet{Jackson96}, giving them a possible scattering origin in the NLR. \n\nThe host galaxy of Cygnus A is a cD elliptical near the center of a cluster which appears to be undergoing a merger with cluster of similar size \\citep{Ledlow05}. \\citet{Tadhunter03} obtained Hubble Space Telescope (HST) Space Telescope Imaging Spectrograph (STIS) spectra of the nuclear region. Stepping the slit across the nucleus, a velocity gradient indicative of rotation around the radio axis was observed. Modeling the velocity as due to the potential of a supermassive black hole (SMBH) and stellar mass distribution (measured from a $1.6\\mu$m NICMOS image) gives a SMBH mass of $2.5\\pm0.7\\times10^{9}$ $M_{\\odot}$. This measurement is consistent with black-hole-mass--host-galaxy relations \\citep[e.g.,][]{Magorrian98,Ferrarese00,Gebhardt00,Gultekin09}.\n\nThe large scale radio morphology shows prominent hotspots, lobes, and a radio core. Both a jet and counterjet are visible in Very Large Array (VLA) and Very Long Baseline Array (VLBA) observations \\citep[e.g.,][]{Sorathia96}. Due to the edge-brightened morphology it is classified as an FR II source \\citep[Fanaroff-Riley Type II;][]{Fanaroff74}.\n\nX-ray observations of Cygnus A are consistent with the presence of a hidden quasar. Chandra ACIS observations from $0.7$ to $9$ keV by \\citet{Young02} show the hard X-ray flux to be peaked at a location consistent with that of the radio core and unresolved (less than $0.\\arcsec 4$ in size, determined by comparison with a model PSF). Higher energy INTEGRAL observations show emission between $20$ and $100$ keV \\citep{Beckmann06}, although there may be some contamination from intracluster gas.  This hard X-ray emission is likely due to accretion disk emission Comptonized in the AGN corona, although the UV\/optical emission is obscured.\n\nIn addition to the AGN activity, HST imaging has also revealed star formation in the central region of Cygnus A, which began $< 1$ Gyr ago \\citep{Jackson98}. It is located in a $4$ kpc ring around the nucleus, oriented orthogonal to the radio axis.\n\nBased on adaptive optics observations showing a secondary point source near the nucleus, \\citet{Canalizo03} suggest that Cygnus A may be in the late stages of a merger event. This merger event and related accretion may be related to the current epoch of nuclear activity. \n\nA near-infrared spectrum presented by \\citet{Bellamy04} showed complicated emission line properties suggesting an infalling molecular cloud, consistent with the \\citet{Canalizo03} picture of a minor merger.  The H$_2$ lines were seen in several components, both redshifted and blueshifted relative to the systemic velocity, interpreted as emission from a rotating torus. The observed near-infrared line ratios are consistent with excitation by X-rays (from the AGN), while likely ruling out shocks as a possible excitation method.\n\n\\begin{figure}\n\\includegraphics[angle=270,width=0.5\\textwidth]{fig1.eps}\n\\caption{Spitzer IRS spectrum for Cygnus A. Extracted in a $20\\arcsec$ circular aperture from spectral mapping data.}\n\\label{fig:irs}\n\\end{figure}\n\nFor a more detailed summary of Cygnus A, including properties of the larger host galaxy and environment, see the review by \\citet{Carilli96}.\n\nWhile the bulk of the infrared emission is likely related to the presence of an AGN, the star formation can contribute to the infrared emission as well. Additionally, the bolometric luminosity of the AGN is uncertain. Estimates have been made using template spectral energy distributions (SEDs) and X-ray observations. However, a significant portion of the bolometric luminosity comes out in the infrared, via dust reprocessing of the UV\/optical continuum. An accurate determination of the bolometric luminosity then requires an understanding of the contributions to the observed infrared luminosity. \n\nIn this paper, we present modeling of the SED of Cygnus A. To accomplish this we combined new infrared measurements obtained by the Spitzer Space Telescope's Infrared Spectrograph \\citep[IRS;][]{Werner04} with existing Spitzer observations with the Imaging Array Camera \\citep[IRAC;][]{Fazio04} and the Multiband Imaging Photometer for SIRTF \\citep[MIPS;][]{Rieke04}, and measurements of the radio core from the literature to construct a radio through infrared ($\\sim2-10^{5}$ GHz; $4-10^{5}$ $\\mu$m) SED of the inner regions of Cygnus A. The resulting SED was subsequently modeled using components intended to replicate the physical processes likely to produce the observed emission. Using the results of the fitting we have been able to decompose the infrared emission and determine the bolometric luminosity of the AGN in Cygnus A. This in turn also provides an estimate of the star formation rate.\n\nThe paper is organized as follows: in Section \\ref{sec:data}, we discuss the new Spitzer IRS observations, new reductions of Spitzer IRAC and MIPS data, as well as the data from the literature. In Section \\ref{sec:modeling}, we describe the components used to model the SED, for which the results are described in Section \\ref{sec:results}. We conclude with a general discussion in Section \\ref{sec:discussion} and a summary in Section \\ref{sec:summary}.\n\n\\section{Data}\n\\label{sec:data}\n\nNew observations were combined with data compiled from multiple archives and new reductions in order to obtain a SED covering almost five orders of magnitude in frequency. Our analysis is focused on the properties of the continuum emission in this system. Below we discuss the reduction and analysis of the new Spitzer IRS observations and the re-reduction of archival IRAC and MIPS observations\n\n\\subsection{Spitzer IRS Observations}\n\nCygnus A was observed (PI: Baum) in ``mapping mode'' using the low resolution mode of IRS on board the Spitzer Space Telescope. Both the short- and long-wavelength slits were stepped across the source in increments of half the slit width. After pipeline calibration at the Spitzer Science Center, the observations were combined into a spectral data cube using the \\emph{Cube Builder for IRS Spectra Maps} \\citep[CUBISM;][]{Smith07b}. From this data cube, a spectrum was extracted in a $20\\arcsec$ aperture (Figure \\ref{fig:irs}).\n\n\\subsubsection{Removal of Mid-infrared Emission Lines}\n\nThe IRS spectrum was fit using PAHFIT \\citep{Smith07a} to measure and remove contributions from narrow emission lines. Integrated line fluxes and widths are provided in Table \\ref{table:mirlines}. While a study of the emission line properties is beyond the scope of this paper, we note the detection of multiple high ionization lines such as [O IV], [Ne V] and [Ne VI]. These are all consistent with the presence of an AGN \\citep[][and references therein]{Genzel98,Armus07}. Of particular note are [Ne V] and [Ne VI] which require the presence of ionizing photons of at least 97.1 and 125.8 eV respectively. All the measured emission lines in Table \\ref{table:mirlines} were subtracted from the IRS spectrum before fitting the SED.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Mid-infrared Emission Line Properties}\n\\tablehead{\\colhead{Line} & \\colhead{Flux} & \\colhead{FWHM}\\\\\n\\colhead{} & \\colhead{\\emph{($\\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$)}} & \\colhead{\\emph{($\\mu$m)}}}\n\\startdata\n[Ar II] 6.9\t& $1.93\\pm0.08$\t& 0.11 \\\\\n{}[Ne VI] 7.6\t& $1.37\\pm0.09$\t& 0.11 \\\\\n{}[Ar III] 8.9\t& $1.11\\pm0.04$ & 0.11 \\\\\n{}[S IV\t10.5\t& $1.55\\pm0.03$ & 0.09 \\\\\n{}[Ne II] 12.8\t& $2.67\\pm0.03$ & 0.11 \\\\\n{}[Ne V] 14.3\t& $1.51\\pm0.04$ & 0.10 \\\\\n{}[Ne III] 15.6\t& $4.13\\pm0.04$ & 0.15 \\\\\n{}[S III] 18.7\t& $2.42\\pm0.06$ & 0.13 \\\\\n{}[Ne V] 24\t& $2.51\\pm0.06$ & 0.37 \\\\\n{}[O IV] 25.9\t& $4.58\\pm0.08$ & 0.32 \\\\\n{}[S III] 33\t& $3.33\\pm0.08$ & 0.31 \\\\\n\\hline\nH$_2$ S(7)\t& $0.944\\pm0.130$ & 0.06 \\\\\nH$_2$ S(5)\t& $0.853\\pm0.112$ & 0.06 \\\\\nH$_2$ S(3)\t& $0.286\\pm0.030$ & 0.09 \\\\\nH$_2$ S(2)\t& $0.333\\pm0.025$ & 0.11 \\\\\nH$_2$ S(0)\t& $0.128\\pm0.038$ & 0.31\n\\enddata\n\\tablecomments{Only detections $>3\\sigma$ are listed. Errors quoted from PAHFIT output.}\n\\label{table:mirlines}\n\\end{deluxetable}\n\n\\subsubsection{Dust Features}\n\nIn addition to fitting the emission lines, we have also fit the mid-infrared dust features using PAHFIT. Table \\ref{table:dustfeatures} shows the integrated flux and profile FWHM based on the PAHFIT output. In contrast with the emission lines, the dust features were not removed as they are expressed in the starburst models.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Mid-infrared Dust Features}\n\\tablehead{\\colhead{$\\lambda$} & \\colhead{Flux} & \\colhead{FWHM}\\\\\n\\colhead{($\\mu$m)} & \\colhead{\\emph{($\\times 10^{-13}$ erg s$^{-1}$ cm$^{-2}$)}} & \\colhead{\\emph{($\\mu$m)}}}\n\\startdata\n$6.2$   &       $1.76\\pm0.23$ &       $0.19$ \\\\\n$6.7$   &       $5.99\\pm0.49$ &       $0.47$ \\\\\n$7.4$   &       $22.3\\pm0.8$ &       $0.94$ \\\\\n$7.8$   &       $2.26\\pm0.32$ &       $0.42$ \\\\\n$8.3$   &       $5.50\\pm0.23$ &       $0.42$ \\\\\n$8.6$   &       $4.42\\pm0.20$ &       $0.34$ \\\\\n$11.3$  &       $0.685\\pm0.101$ &       $0.36$ \\\\\n$12.0$  &       $2.71\\pm0.12$ &       $0.54$ \\\\\n$12.6$  &       $3.82\\pm0.21$ &       $0.53$ \\\\\n$13.5$  &       $2.49\\pm0.12$ &       $0.54$ \\\\\n$14.2$  &       $1.28\\pm0.11$ &       $0.36$ \\\\\n$17.0$  &       $2.52\\pm0.27$ &       $1.11$ \\\\\n$17.4$  &       $0.397\\pm0.065$ &       $0.21$ \\\\\n$17.9$  &       $0.802\\pm0.099$ &       $0.29$ \\\\\n$18.9$  &       $2.19\\pm0.14$ &       $0.36$ \\\\\n$33.1$  &       $3.54\\pm0.51$ &       $1.66$ \n\\enddata\n\\tablecomments{Only detections $>3\\sigma$ are listed. Errors quoted from PAHFIT output.}\n\\label{table:dustfeatures}\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcccc}\n\\tablecaption{Additional Infrared Flux Densities from Spitzer}\n\\tablehead{\\colhead{Instrument\/Channel} & \\colhead{$\\lambda$ ($\\mu m$)} & \\colhead{F$_{\\nu}$ (Jy)} & \\colhead{$\\sigma$ (Jy)} & \\colhead{Aperture ($\\arcsec$)}}\n\\startdata\nIRAC-2\t& $4.5$     & $0.010$ & $0.003$ & $12.2$ \\\\\nIRAC-4\t& $8$       & $0.054$ & $0.013$ & $12.2$ \\\\\nMIPS-70\t& $70$      & $2.20$ & $0.11$ & $30$ \\\\\nMIPS-160\t& $160$     & $0.668$ & $0.033$ & $48$\n\\enddata\n\\label{table:spitzer}\n\\end{deluxetable}\n\n\\citet{Spoon07} developed a diagnostic diagram using the $6.2~\\mu$m polycyclic aromatic hydrocarbon (PAH) and the strength of the $9.7~\\mu$m silicate feature (S$_{sil}$), which is seen in absorption in Cygnus A. These two spectral features can be used in tandem to classify the relative dominance of PAH emission, continuum emission, and silicate absorption. We follow their method of spline fitting to measure the depth of the silicate feature, finding S$_{sil}\\approx-0.8$, where S$_{sil}$ is the negative of the apparent optical depth. The EQW of the  $6.2~\\mu$m PAH is $0.0552~\\mu$m, placing Cygnus A on the border of region 1A and 2A in their diagnostic diagram, corresponding to objects dominated by continuum emission in the mid-infrared. This is consistent with the presence of a strong AGN.\n\n\\subsection{Archival Spitzer IRAC+MIPS Data}\n\nThe nucleus and hotspots were imaged using IRAC at $4.5$ and $8.0~\\mu$m \\citep[see ][for a presentation of the data and analysis of hotspot properties]{Stawarz07}. As fluxes for the core were not presented, the archival data were retrieved, re-reduced and calibrated according to the IRAC instrument manual. The images showed strong emission at the location of both radio hotspots and the radio core in both channels. The emission was unresolved in the core component, the measured flux densities for this component are given in Table \\ref{table:spitzer}. Extraction apertures of $12\\arcsec.2$ were used for both channels. \n\n\\citet{Shi05} presented observation of Cygnus A using the MIPS instrument on Spitzer at 24, 70, and 160 $\\mu$m. Their 24 $\\mu$m flux is consistent with our IRS observations. The slope between their 70 and 160 $\\mu$m points is steeper than that of the Rayleigh--Jeans tail. The 70 and 160 $\\mu$m data were re-reduced from the BCD products in the Spitzer Science Center archive. Our measured flux densities are given in table \\ref{table:spitzer}. The core of Cygnus A was unresolved in both the $70$ and $160~\\mu$m bands. Apertures of $30\\arcsec$ and $48\\arcsec$ were used at 70 and 160 $\\mu$m, respectively. Our 160 $\\mu$m measurement has larger error bars, but is broadly consistent with their value.\n\n\\subsection{Published Data}\n\nCygnus A's strong synchrotron emission at radio frequencies suggests that synchrotron may contribute to the infrared as well. To anchor the synchrotron spectrum we supplemented the Spitzer observations with radio measurements from the literature. Core fluxes were obtained from \\citet{Eales89,Salter89,Alexander84,Wright84}. Submillimeter nuclear fluxes were also taken from \\citet{Robson98}. Based on these archival data the unresolved radio core\\footnote{Here ``radio core'' refers to the core seen by the VLA with on kpc-scale resolution. This encompasses flux from the VLBI scale core and jet.} is flat spectrum ($\\alpha=0.18$, $F_{\\nu}\\propto \\nu^{-\\alpha}$), until $\\sim1$ THz, where thermal emission from dust begins to dominate the SED. The compiled radio through infrared SED is shown in Figure \\ref{fig:SED} and the flux densities taken from the literature are provided in Table \\ref{table:literature}. The resolution of the observations is also provided.\n\nThere is some uncertainty associated with the highest frequency sub-mm observations. The $450~\\mu$m observation suggests that the synchrotron break may be occurring in this spectral regime. It is unclear if this is a genuine break or if the measurements are affected by variability. Additional observations at these frequencies may be able to clarify this issue. \n\nA consideration of the apertures is important when assembling data across several orders of magnitude in frequency. The resolution of the data used to assembled the SED in Figure \\ref{fig:SED} varies by over a factor of 10, but in all cases the core component is unresolved. At lower frequencies the emission is due solely to synchrotron emission from the flat spectrum radio core. There is no evidence to suggest that the larger scale steep spectrum jet will contribute emission in the infrared. Accordingly we use the unresolved VLA-scale core to anchor the synchrotron emission in the nucleus, so the difference in apertures should not affect the construction of this SED for the nucleus.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.5\\textwidth]{fig2.eps}\n\\caption{Radio through mid-infrared SED with new Spitzer observations and other data from the literature. See the text for details and references.}\n\\label{fig:SED}\n\\end{figure}\n\n\\begin{deluxetable}{lcccc}\n\\tablecaption{Flux densities from the literature}\n\\tablehead{\\colhead{$\\lambda$ ($\\mu m$)} & \\colhead{F$_{\\nu}$ (Jy)} & \\colhead{$\\sigma$ (Jy)} & \\colhead{Resolution ($\\arcsec$)} & \\colhead{Ref}}\n\\startdata\n450 & 0.34 & 0.06 & $8$ & 1\\\\\n800 & 0.56 & 0.08 & $13$ & 2 \\\\\n850 & 0.53 & 0.05 & $14$ & 1\\\\\n1100 & 0.58 & 0.06 & $19$ & 2 \\\\\n1300 & 0.59 & 0.07 & $11$ & 3 \\\\\n$3.3\\times10^3$ (89 GHz) & 0.70 & 0.07 & $2$ & 4\\\\\n$19.5\\times10^3$ (15.4 GHz)& 1.22 & 0.20 & Not provided in ref & 5 \\\\\n$60.6\\times10^3$ (5 GHz)& 0.97 & 0.20 & $2.0\\times3.1$ & 5\\\\\n$109\\times10^3$ (2.7 GHz) & 1.5 & 0.4 & $3.7\\times5.8$ & 5\n\\enddata\n\\label{table:literature}\n\\tablerefs{1 - \\citet{Robson98}, 2 - \\citet{Eales89}, 3 - \\citet{Salter89}, 4 - \\citet{Wright84}, 5 - \\citet{Alexander84}.}\n\\end{deluxetable}\n\n\\section{Modeling}\n\\label{sec:modeling}\n\nWe aim to reproduce the major features in the SED: the powerlaw emission at radio wavelengths, the strong thermal emission at infrared wavelengths, and the overall character of the silicate absorption. Extrapolating the powerlaw from the radio to the infrared requires a modification of the powerlaw spectrum to avoid exceeding the observed mid-infrared flux. The infrared emission is thermal in nature, coming from dust at a variety of temperatures ranging from the cold ISM ($T\\sim20$ K) through hot dust near the AGN, up to the sublimation temperature ($T\\sim1500$ K). The power source for this dust heating is a combination of star formation and AGN activity, with an uncertain balance between the two.\n\nAs noted in the introduction previous studies of Cygnus A have found evidence for simultaneous ongoing AGN activity and star formation, both of which can contribute to the infrared emission. We model the continuum emission from $\\sim2-10^5$ GHz ($3\\times10^{6}-5$ $\\mu$m; see Figure \\ref{fig:SED}) using components to represent the AGN torus model, a starburst, and synchrotron emission. The selection of models has $15$ free parameters, and the particular choices are discussed in the following sub-sections.\n\n\\subsection{AGN Torus Model}\n\nAccording to the unified scheme for radio loud AGN, the SMBH and accretion disk can be hidden from view along some lines of sight by an obscuring torus. Along these obscured lines of sight the UV\/optical radiation is absorbed and re-radiated in the mid- and far-infrared. As noted above, Cygnus A shows evidence for a hidden BLR through observations of H$\\alpha$ in polarized light. This suggests that Cygnus A harbors a ``hidden'' accretion disk and a BLR which is obscured along our line of sight.\n\n\\citet{Nenkova08} have constructed a model for an obscuring AGN torus, where the obscuration is due to the presence of multiple clouds along the line of sight to the AGN. We select this set of models because clumpy models seem to provide better fits to Sil features than smooth dust distributions \\citep[e.g.][]{Baum10}. In order to reproduce the observed ratio of Type I\/II AGNs, the obscuring clouds collectively populate a rough toroidal structure with some opening angle. \n\nThe CLUMPY torus model is specified by multiple geometrical parameters. The outer radius of the torus ($R_o$) is $Y$ times the inner radius ($R_d$), where the inner radius is determined from a dust sublimation temperature of $T=1500$ K ($R_d=0.4\\times L_{45}^{0.5}$ pc. $L_{45}$ is the bolometric luminosity of the AGN in units of $10^{45}$ erg s$^{-1}$). The geometry of the model is shown in Figure \\ref{fig:clumpygeom}. The clumps have a Gaussian angular distribution,  with $\\sigma$ parameterizing the width of the angular distribution from the mid-plane. The radial distribution is a power law with index $q$: $r^{-q}$. The inclination of the torus symmetry axis to the line of sight is $i$ and the average number of clouds along a given line of sight is $N$.\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.5\\textwidth]{fig3.eps}\n\\caption{Geometry of the CLUMPY torus model. Figure from \\citet{Nenkova08}, used with permission. See the text for an explanation of the labels.}\n\\label{fig:clumpygeom}\n\\end{figure}\n \nIn addition to the geometrical parameters, the models also vary the optical depth of each clump ($\\tau_V$), using the \\citet{Ossenkopf92} dust composition and \\citet{Mathis77} grain size distribution. The overall scaling of the model flux is $F_{AGN}$, the bolometric flux of the AGN's accretion disk component (treating the synchrotron emission separately). \n\nThe clumpy nature of the obscuration implies that there is a finite probability of a direct line of sight to the BLR, even at high inclinations. However, as Cygnus A shows no evidence for directly observed broad lines, we have only fit models where the central regions are fully obscured along our line of sight.\n\n\\subsection{Starburst}\n\nStar formation is represented using the \\citet{Siebenmorgen07} models which assume spherical symmetry and an ISM with dust properties characteristic of the Milky Way. Emission is broken down into two components: an old stellar population uniformly distributed through the volume and hot, luminous O and B stars embedded in dusty hot spots. The density of OB stars is centrally peaked although the model output is the emission integrated over the entire starburst.\n\nThe free parameters for the model are: starburst radius $r$, total luminosity $L_{SB}$, ratio of luminosity in O and B stars to total luminosity $f_{OB}$, visual extinction from the center to the edge of the nucleus $A_V$, and dust density in hotspots around O and B stars $n$. \n\nOwing to the spherical symmetry of this starburst model and the known ring morphology of the star formation in Cygnus A, parameters such as $A_V$ and the radius $r$ do not have straight forward interpretations here. The assumption of optically thick star formation does not have strong evidence (either for or against), given the absence of high resolution far infrared observations.\n\n\\subsection{Synchrotron}\n\\label{sec:sync-model}\n\nThe strong radio emission in Cygnus A justifies the final model component. VLA core fluxes are consistent with powerlaw emission to the submillimeter where thermal emission begins to dominate. Extrapolating the powerlaw to higher frequencies suggests that the synchrotron spectrum must either be modified or subjected to attenuation. The AGN is known to sit behind a dust lane with significant extinction \\citep[$A_V=50\\pm30$;][]{Djorgovski91}. Though some attenuation of the synchrotron flux is expected at higher frequencies, this alone is not sufficient to explain why the synchrotron powerlaw does not continue through the mid-infrared. With extinction alone, the flux densities at 30 and $10~\\mu$ m would be similar for the observed spectral index of $\\alpha=0.18$. For $A_V=50$, the expected flux from the extrapolated synchrotron emission would by itself exceed the measured infrared flux at 10$\\mu m$. We conclude that there must be a break in the population of emitting electrons.\n\nOne possible explanation is a simple cutoff in the population of relativistic electrons at some energy (Case I). This would manifest itself as a cutoff in the synchrotron spectrum:\n\n\\begin{equation}\nF_{\\nu} \\propto \\nu^{-\\alpha_1} e^{-\\frac{\\nu}{\\nu_c}}e^{-\\tau_r(\\nu)}\n\\label{eq:synchrotron}\n\\end{equation}\n\nwhere $\\nu_c$ is the frequency corresponding to the cutoff in the energy distribution of the particles, $\\alpha_1$ is the spectral index in the optically thin regime, and $\\tau_r(\\nu)$ is the dust screen between the synchrotron emitting region and the observer. $\\tau_r(\\nu)$ follows \\citet{Draine84}.\n\nIn Cygnus A the spectral index $\\alpha_1$ is measured from VLA core radio fluxes, and the amplitude of the powerlaw fixed from the same observations. The only free parameters are the cutoff frequency $\\nu_c$ and the optical depth $\\tau_{r}$. These parameters art partly degenerate in that in trial runs we experienced a situation where $\\tau_r$ and $\\nu_c$ would both increase, effectively offsetting each other. To combat this we limited $\\tau_r$ such that $A_V$ does not exceed 500 towards the radio source.\n\nAn alternate model for the synchrotron emission at higher frequencies is a broken powerlaw behind a dust screen (Case II). Aging of the population of relativistic electrons results in a broken powerlaw whose spectral index increases for frequencies higher than a break frequency \\citep{Kardashev62}. The functional form adopted is:\n\n\\begin{equation}\nF_{\\nu}  \\propto e^{-\\tau_r(\\nu)} \\times\n  \\begin{cases} \n    \\nu^{-\\alpha_1}  & \\text{if } \\nu < \\nu_{break} \\\\ \n    \\nu^{-\\alpha_2}  & \\text{if } \\nu \\geq \\nu_{break}\n  \\end{cases}\n\\label{eq:brokenpw}\n\\end{equation}\n\nFor a simple aging of the electron population, the post-break spectral index in Cygnus A would be $\\alpha_2=1.24$. As with Case I, we fit a dust screen in front of the synchrotron emission ($\\tau_r$).\n\n\\subsection{Stellar Contribution to the Mid-infrared}\n\nStarlight can potentially contribute to the mid-infrared flux, especially in a large aperture. Using the flux of the stellar component from \\citet{Jackson98} and their brightness profile, we determined the contribution due to starlight in a $20\\arcsec$ aperture. The relative fluxes were consistent with an elliptical galaxy template from \\citet{Silva98}. Expected flux densities at 2.2, 5, and 10 $\\mu$m were computed using the same template spectrum. After subtracting a nuclear point source, the K-band flux density in starlight is consistent with the flux in a $20\\arcsec$ aperture as measured using Two Micron All Sky Survey (2MASS) data products \\citep{Skrutskie06}. At 5 and 10 $\\mu$m, the starlight contributes roughly 14\\% and 2\\% respectively, of the flux measured by IRS. Therefore, we do not include any contribution to the mid-infrared flux from an old stellar population in our modeling.\n\n\\subsection{Dust in the NLR}\n\nWarm dust in the NLR directly illuminated by the AGN can also contribute to the mid- and far-infrared emission \\citep[e.g.,][]{Groves06}. \\citet{Ramos09} find that a significant contribution to the mid-infrared flux can come from sources other than the AGN torus, particularly additional hot dust. In Cygnus A \\citet{Radomski02} find an extended component to the mid-infrared emission which is consistent with $T\\sim150$ K dust. Some of this extended emission is co-spatial with sites of possible star formation. Dust of this temperature is well reproduced with our choice of starburst models (with the implicit assumption that this dust is heated by star formation). \n\nHigher temperature dust is also present in the inner regions of Cygnus A. However the resolved mid-infrared images of the nuclear regions by \\citet{Radomski02} are unable to distinguish between dust in the ``torus'' and dust in the NLR heated by the AGN. Without strong observational motivation for an additional dust component we opt not to include one. As will be shown in the next section we are able to reproduce the observed emission without this additional component.\n\n\\subsection{Prior Constraints on Model Parameters}\n\nPrior to fitting the models, the available parameter space was constrained using results from previous studies of Cygnus A.  The opening angle and covering fraction of the CLUMPY torus are determined from the $\\sigma$ and $N$ parameters \\citep[see Equations (3) and (4) in ][]{Mor09}. We define the half opening angle of the torus as the angle at which the escape probability of a photon drops below $e^{-1}$:\n\n\\begin{equation}\n\\theta_{half}=90-\\sigma\\sqrt{\\ln{N_0}}\n\\label{eq:halfopening}\n\\end{equation}\n\n\\citet{Tadhunter99} measured the opening angle of the ionization cone in Cygnus A using near infrared HST data, finding $\\theta_{half}=(58\\pm4)^{\\circ}$. We use this value as the torus opening angle and limit the parameter range of $\\sigma$ and $N$ according to Equation \\ref{eq:halfopening}.\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=0.5\\textwidth]{fig4a.eps}\\includegraphics[angle=270,width=0.5\\textwidth]{fig4b.eps}\n\\caption{LEFT: Case I fit to the Cygnus A SED. The shaded region shows all acceptable model fits within the 95.4\\% confidence interval. The lines are the components for the best-fit (lowest $\\chi^2$). RIGHT: Same as left, but for Case II fit to the Cygnus A SED. Insets are a zoom of the region around the $9.7~\\mu$m Sil absorption.}\n\\label{fig:SEDfit}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[angle=270,width=0.8\\textwidth]{fig5.eps}\\\\\n\\end{center}\n\\caption{Histogram of bolometric AGN luminosity from best fit torus parameters (left) and bolometric starburst luminosity (right) for Cases I (blue, solid line) and II (red, dashed line).}\n\\label{fig:lumhist}\n\\end{figure*}\n\nThe inclination range of the torus $i$ was limited by very long baseline interferometry (VLBI) observations and modeling of the inner pc-scale jet \\citep[$50^{\\circ}\\leq i \\leq 85^{\\circ}$,][]{Sorathia96}. Additionally, the available VLA core radio fluxes were used to fix the synchrotron spectrum at radio frequencies ($\\alpha_1$, and the amplitude). \n\n\n\\section{Results}\n\\label{sec:results}\n\nTo fit the observed SED, the flux from each model was summed in each wavelength bin and compared to the observed flux within a $20\\arcsec$ aperture. Optimization was performed using Levernberg--Marquardt least-squares minimization. \n\n\nSome degeneracy between parameters was seen in the model results. Comparison model runs with fewer prior constraints on the parameters resulted in similar $\\chi^2$ values to those quoted below for regions of parameter space which are unlikely to be physically reasonable matches to Cygnus A (e.g., CLUMPY torus fits with covering fractions of unity and $\\theta_{half}=0^{\\circ}$). Prior constraints from other observations are thus important in eliminating regions of parameter space which may be statistically reasonable but physically unrealistic.\n\nAfter the models were fit, confidence intervals were determined separately for Case I and Case II through bootstrapping with replacement \\citep{Efron81}. These were used to determine the ``acceptable'' range of parameter values.\n\nThe results of the fits are shown in Figure \\ref{fig:SEDfit}. The range of parameter values for fits within at $95.4\\%$ confidence interval are shown in Figures \\ref{fig:lumhist}-\\ref{fig:synchist}. Section \\ref{ssec:sync-exp} discusses the Case I fits (exponential cutoff in the relativistic electron population) and Section \\ref{ssec:sync-break} discusses the Case II fits (broken powerlaw for the synchrotron emission).\n\n\\subsection{Case I: Synchrotron Exponential Cutoff}\n\\label{ssec:sync-exp}\n\nThe best fit combination of parameters for Case I has $\\chi^2$\/DOF = $1.35$. The range of acceptable matches for the 1641 model combinations is shown in Figure \\ref{fig:SEDfit} (left). Histograms of the best fit parameters are given in Figures \\ref{fig:lumhist} (left) and \\ref{fig:agnhist}, where $f$ is the fraction of models within the 95.4\\% confidence interval in a given bin. Parameters for the best fit model are marked with a small blue horizontal bar.\n\n\\subsubsection{AGN\/Torus Properties and Contribution}\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig6.eps}\n\\caption{Histograms of torus parameters for acceptable fits within the 95.4\\% confidence interval for Cases I (blue, solid line) and II (red, dashed line). The small horizontal bars denote the parameter value for the best-fit model. From left to right, top to bottom: viewing angle of the torus ($90$ meaning the torus is viewed edge-on), N - average number of clouds along an equatorial line of sight, A$_V$ - for an individual cloud, $\\sigma$ - angular width of the distribution of torus clouds, Y - ratio of inner to outer radii, q - power law index of the radial distribution of torus clouds, covering fraction of the torus (equivalent to the escape probability of optical\/UV photons, assuming optically thick clouds), integrated A$_V$ along an equatorial line of sight, integrated A$_V$ along our line of sight (calculated using N, $\\sigma$, the cloud A$_V$, and the inclination). Parameters in the first two rows are discrete and the histograms represent the relative number of models with those specific values. Parameters in the bottom row are derived from the parameters in the top two rows as well as the luminosity of the AGN component (shown in Figure \\ref{fig:lumhist}), and are continuous.}\n\\label{fig:agnhist}\n\\end{figure*}\n\nThe fits favor a bolometric accretion disk luminosity of $log(L_{AGN}\/L_{\\odot})\\sim 11.8-12.0$ (median of $11.82$ with an interquartile spread of $0.09$). The fits clearly favor an extended torus ($Y=200$) with low opacity clouds ($A_V\\sim10-30$). For the median bolometric luminosity, the inner radius of the torus is $R_d=0.6$ pc, giving a corresponding outer torus edge of $\\sim125$ pc (for Y$=200$). \n\nThe range of values for $i$ was limited based on previous work in the radio regime. The fits prefer an inclination for the torus on the high end of the range, $i\\approx80^{\\circ}$. $N$ has a bimodal distribution, with $25$\\% of fits having $N\\leq6$ clouds along an equatorial line of sight and $\\sim60$\\% having $N\\geq20$ clouds. Most of the fits within the 95.4\\% confidence interval have a flat radial distribution of clouds ($q=0$). \n\nThe opacity of individual clouds is anti-correlated with the radial extent of the torus; small torus sizes favor high $A_V$ values. High values of $N$ (i.e., many clouds along a given line of sight) are favored for small torus sizes, however for large torus sizes, both small and large $N$ values are acceptable.\n\nWe calculate the total extinction through the torus along both an equatorial line of sight (Figure \\ref{fig:agnhist} middle, bottom row), and along our line of sight to the torus (Figure \\ref{fig:agnhist} right, bottom row). The integrated equatorial $A_V$ spans a range between $100$ and $250$. When viewing angle is taken into consideration, the range narrows, with most fits showing the line of sight $A_V$ between $80$ and $120$. However the $A_V$ is poorly constrained beyond having a well-defined lower bound.\n\nThe torus covering fraction was computed using the method of \\citet[their Equations (3) and (4)]{Mor09}. By design, the derived covering fractions for the torus are between $50\\%$ and $70\\%$. \n\n\\subsubsection{Starburst}\n\nThe median value of the starburst luminosity for Case I is log$(L_{SB}\/L_{\\odot})\\sim10.8$, with a tail up to $\\sim11.6$ containing approximately $40\\%$ of the fits (Figure \\ref{fig:lumhist}). This covers a range of star formation rates from $10$ to $70$ M$_{\\odot}$ yr$^{-1}$, as determined by the $L_{IR}$ calibration from \\citet{Kennicutt98a} \\citep[for a review of SFR estimates, see][]{Kennicutt98b}.\n\n\\begin{equation}\n\\frac{SFR}{(M_{\\odot}~yr^{-1})} = 4.5\\times10^{-44}~\\frac{L_{FIR}}{erg~s^{-1}}=1.72\\times10^{-10}~\\frac{L_{FIR}}{L_{\\odot}}\n\\label{eq:sfr}\n\\end{equation}\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig7.eps}\n\\caption{Histograms of starburst parameters for acceptable fits within the 95.4\\% confidence interval for Cases I (blue, solid line) and II (red, dashed line). The small horizontal bars denote the parameter value for the best-fit model. From left to right, top to bottom: the fraction of luminosity in O and B stars, integrated extinction through the starburst, size of the starburst, dust density in hotspots around O and B stars. f$_{OB}$ and the size are discrete and the histograms represent the relative frequency of models with those specific values. The A$_V$ and $n$ parameters are more finely sampled than the histogram represents; the plots show the relative fraction of models with parameters in the noted ranges.}\n\\label{fig:sbhist}\n\\end{figure*}\n\nRoughly half the fits within this confidence interval have $40$\\% of the starburst luminosity in the form of OB stars. The distribution of size parameters is relatively flat, although even the largest sizes would be unresolved by our Spitzer observations. The distribution of dust density $n$ peaks around $10^3$ cm$^{-3}$. The extinction through this starburst is relatively unconstrained.\n\nThe strengths of dust features in starburst models falling within the 95.4\\% confidence interval were also measured using PAHFIT to compare with the intensities of observed dust feature. Figure \\ref{fig:dust} shows histograms of the predicted intensities from our models divided by the measured intensity from the IRS spectrum. Only dust features which are detected in the IRS spectrum have been plotted. In general the agreement is good with most models matching the measured line intensities. However for some dust features a significant number of the models within the confidence interval predict emission in excess of what is observed. While a detailed comparison of the relative strengths of dust features is beyond the scope of the paper, we suggest the models are able to generally reproduce the dust features seen in the spectrum. The dust features which show the greatest discrepancy between observed fluxes and those predicted from the modeling ($6.2$, $7.8$, and $11.3~\\mu$m) comprise 3 of the 5 lowest signal-to-noise dust feature fits in the spectrum. The discrepancy may then be due to the difficulty of fitting low equivalent width dust features.\n\nPAHFIT also attempted to fit eight other dust features in the IRS spectrum. The upper limits for three ($5.7$, $14.0$, and $15.9~\\mu$m) are consistent with expectations from the starburst models. One dust feature ($16.4~\\mu$m) has a measured limit below that expected from the best fitting starburst model, and so is discrepant. The other four features ($7.6$, $10,7$, $11.2$, and $12.7~\\mu$m) also have upper limits from PAHFIT which are lower than the expected value from the starburst models. However, these are coincident with or near other spectral features (e.g., narrow emission lines or the Sil absorption). Thus an accurate measurement of these dust features in the IRS spectrum would rely more heavily on fitting the wings, which could prove difficult in a continuum dominated source such as Cygnus A. \n\n\\subsubsection{Synchrotron Radiation}\n\nThe synchrotron emission amplitude was fixed using the non-thermal emission from the radio core, assuming it to be a point source at all frequencies observed. The synchrotron model, therefore, has only two parameters: cutoff frequency ($\\nu_c$) and extinction due to dust ($\\tau_r$).\n\nThe distribution of cutoff frequencies is somewhat broad, covering the range of $\\nu\\approx10-60$ THz ($5-30~\\mu$m) (Figure \\ref{fig:synchist}). The fits show a range of acceptable extinction for the dust screen, peaking in the range of $A_V\\approx60-80$, slightly lower than the predicted line of sight $A_V$ from the torus models. The integrated luminosity of the ``core'' synchrotron flux is log$(L_{sync}\/L_{\\odot})\\sim11.2$. \n\nThis break in the synchrotron spectrum is consistent other powerful FR II radio sources, where an extrapolation of the radio synchrotron emission significantly exceeds the observed optical flux \\citep[e.g.,][]{Schwartz00,Sambruna04,Mehta09}. In these cases, the similarity of the radio and X-ray spectral indices suggests that the X-ray may be produced by inverse Compton (IC) scattering.\n\nWith the modeled synchrotron spectrum, a magnetic field strength, and the assumption that each electron emits at a single frequency, the energy distribution of relativistic electrons can be determined:\n\n\\begin{equation}\n\\gamma(\\nu)= \\sqrt{\\frac{4\\pi m_e c \\nu}{3eB}}\n\\label{eq:e-gamma}\n\\end{equation}\n\nwhere all constants and values are in the cgs system of units.\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig8.eps}\n\\caption{Histograms of log(Predicted Flux \/ Measured Flux) for dust features detected in the IRS spectrum of Cygnus A. The predicted values were measured from the \\citet{Siebenmorgen07} models using PAHFIT. The solid blue line denotes Case I while the red dashed line denotes Case II. (see Table \\ref{table:dustfeatures} for the measured flux values).}\n\\label{fig:dust}\n\\end{figure*}\n\n\\clearpage\n\n\\begin{figure*}\n\\includegraphics[angle=270,width=\\textwidth]{fig9.eps}\n\\caption{Histograms of synchrotron parameters for acceptable fits within the 95.4\\% confidence interval for Cases I and II. For Case I $\\nu_{crit}\\equiv\\nu_{cutoff}$ and for Case II $\\nu_{crit}\\equiv\\nu_{break}$. $A_V=1.08\\tau_r (\\nu= \\text{V-Band} )$.}\n\\label{fig:synchist}\n\\end{figure*}\n\nThe magnetic field strength has been estimated from VLBI observations to be roughly $17$ mG ($100$ mG) in the jet (core) \\citep{Roland88,Kellermann81}. Using $\\nu\\approx30$ THz and Equation \\ref{eq:e-gamma} we find $\\gamma\\sim2\\times10^4$ ($8\\times10^3$) for the jet (core).\n\nIn addition to radiating energy via synchrotron emission, relativistic electrons can also lose energy via IC scattering. The ratio of the energy lost through synchrotron to the energy lost to IC is simply given by the ratio of the magnetic and radiation energy densities, with the cosmic microwave background (CMB) setting a minimum value for the radiation energy density. Using the magnetic field strength assumed above ($17$ mG), and $T_{CMB}=2.73$ K, synchrotron losses are dominant by a factor $>10^8$. A larger magnetic field would further enhance this ratio, so IC losses from scattering of CMB photons have a negligible effect on the population of relativistic particles.  \n\n\\subsection{Case II: Synchrotron Broken Power Law}\n\\label{ssec:sync-break}\n\nThe range of best fit models for Case II is shown in Figure \\ref{fig:SEDfit} (right). The best fit model had $\\chi^2$\/DOF = $1.03$. Figure \\ref{fig:SEDfit} (right) shows the range of the 1580 model fits within the 95.4\\% confidence interval. Histograms of the best fit parameters are given in Figures \\ref{fig:lumhist} (left) and \\ref{fig:agnhist}. A red horizontal bar marks the parameter for the parameters with the lowest $\\chi^2$ value.\n\n\\subsubsection{AGN\/Torus}\n\nIn Case II fits the bolometric accretion disk luminosity is roughly log$(L_{AGN}\/L_{\\odot})\\sim 11.8$ (median $11.88$ with an interquartile range of $0.08$). The distribution of luminosities is more strongly peaked than for Case I, with over 80\\% of fits occupying the log$(L\/L_{\\odot})=11.8-12.0$ bin. A large torus ($Y=200$) is exclusively preferred, corresponding to $R_{out}\\approx135$ pc (using the median value for the luminosity). Again, by design, the torus covering fraction is between 0.5 and 0.7. An inclination of $i=80^{\\circ}$ is exclusively preferred.\n\nThe distribution of the average number of clumps along a line of sight $N$ is again bimodal with either large ($<14$) or small ($<8$) numbers of clouds preferred. The number of clouds is anti-correlated with the extinction through individual clouds. The spread in equatorial $A_V$ is somewhat large ($120-260$), but the line of sight $A_V$ is more constrained ($100-160$ for 80\\% of fits). \n\n\\subsubsection{Starburst}\n\nThe starburst component in Case II fits show qualitatively similar behavior to Case I. The typical luminosity is comparable, but shows a smaller tail up to log$(L_{SB}\/L_{\\odot})=11.6$ ($\\sim30\\%$ of fits). Other parameters are similar to those in Case I fits, with the exception of the size which appears to have a very slight preference for a larger size, suggesting an overall cooler dust temperature for a given luminosity. This may indicate a degeneracy between the starburst and synchrotron models. The critical frequency for the synchrotron spectrum (see the next section) is at lower frequency for Case II, resulting in a smaller contribution to the far-infrared flux. The starburst model compensates with a larger overall size to provide additional cool dust emission (for a given starburst luminosity).  The dust feature strengths expressed in the models generally compare favorably with Case I.\n\n\\subsubsection{Synchrotron Properties and Contribution}\n\nAn alternate mechanism for limiting the influence of synchrotron emission at shorter wavelengths is for the spectrum to break at some frequency (see Section \\ref{sec:sync-model}). Case II fits used a fixed pre-break spectral index of $\\alpha_1=0.18$ and a post-break spectral index of $\\alpha_2=1.24$, consistent with aging of the relativistic population \\citep[without injection of additional particles;][]{Kardashev62}. As the flux density decreases in a slower fashion when compared to an exponential cutoff, the powerlaw must break at lower frequencies to ensure that the observed mid-infrared flux is not exceeded. \n\nThe break frequency is roughly 5 THz ($60~\\mu$m). The unobscured synchrotron luminosity is found to be log$(L_{sync}\/L_{\\odot})\\sim11.1$. Following the same arguments as Case I, the electrons emitting at the break frequency have $\\gamma\\sim7\\times10^3$ ($3\\times10^{3}$) for jet (core) magnetic field values. Again, this is consistent with the observed properties of jets in other FR II radio sources.\n\nThe extinction of the dust screen in front of the radio source is lower than Case I fits, with $A_V<60$, still within range of the extinction to the central source estimated by \\citet{Djorgovski91}, but lower than the computed line of sight $A_V$ from the torus. The larger discrepancy compared with Case I is due to the enhanced short wavelength emission of the broken power law at shorter wavelengths (when compared to the Case I exponential cutoff). The equatorial torus $A_V$ increases to provide an overall cooler temperature, and the modeled dust screen in front of the synchrotron component remains small (in order to contribute sufficient flux at shorter wavelengths).\n\n\\section{Discussion of General SED Results}\n\\label{sec:discussion}\n\n\\subsection{Luminosity and Kinetic Power in Cygnus A}\n\nOur best estimate of the bolometric AGN luminosity in Cygnus A (log$(L\/L_{\\odot})\\sim12$, including the synchrotron component and X-ray emission) is above the \\citet{Whysong04} estimate using Keck mid-infrared observations and a PG-quasar spectrum ($3.9\\times10^{11}$ $L_{\\odot}$), but is somewhat below the estimates of \\citet{Tadhunter03} who find $L_{bol}=12-55\\times10^{11}$ $L_{\\odot}$ (although consistent with the low end of their range). The bolometric AGN luminosity is the best constrained parameter, and is insensitive to the synchrotron model adopted.\n\nThe kinetic power in the expansion of the radio lobes in Cygnus A can be inferred from X-ray observations of the cluster environment \\citep{Wilson06}. From this analysis of Chandra observations Cygnus A has a kinetic power of $L_{kin} = 1.2 \\times 10^{46}$ erg s$^{-1}$ (log$(L_{kin}\/L_{\\odot})=12.5$), a factor of three larger than the bolometric AGN luminosity inferred from the modeling. Given the uncertainties in the determinations of both the kinetic and bolometric power, it is unclear if the kinetic power dominates significantly over the power emitted as radiation.\n\n\\subsection{Implications of the Modeling}\n\nThe probable torus sizes from the SED modeling give outer radii of $R_o=130$ pc ($\\approx0\\arcsec.2$ at the distance of Cygnus A). Generally, the torus parameters provide reasonable estimates of the properties of the obscuring structure. This predicted outer radius is significantly larger than what is generally assumed to be reasonable for the obscuring torus. A torus with the angular diameter of the $\\approx0.\\arcsec2$ is well within range of multiple mid and near-infrared instruments. At large radii, the distinction between torus clouds and narrow line region clouds may be somewhat arbitrary. The observations by \\citet{Canalizo03} show extended emission on scales including and larger than our fit torus sizes. The degree of contamination by emission lines is unclear, and it would be interesting to attempt to obtain a line-free continuum image to ascertain the true size and shape of the continuum emitting region.\n\nThe torus size parameters are consistent with those found from modeling of the $9.7~\\mu$m Sil feature by \\citet{Imanishi00}. Their radiative transfer modeling suggested some torus properties similar to those presented here, namely a small inner radius ($<10$ pc) and an inner-to-outer radius of $80-500$. In contrast they find a steeper radial dependence for the dust distribution $q\\sim2-2.5$, possibly due to the use of a smooth dust distribution.\n\nThe preferred parameters for the obscuring torus; low $q$, high $A_V$, and large $Y$ suggest the model is being driven toward a cooler spectrum. This could influence the starburst component, leading to a lower inferred star formation rate. In order to combat this it would be beneficial to place tighter constraints on the far-infrared SED to enable better modeling of the star formation in Cygnus A.\n\nA comparison of our IRS data with mid-infrared observations by \\citet{Radomski02} is consistent with the suggestion of our model that at $10$ and $18~\\mu$m, the emission is dominated by the torus+synchrotron component. \\citet{Tadhunter99} measure a K-band nuclear point source with $F_{\\nu}=(49\\pm10)~\\mu$Jy. This flux limit is broadly consistent with some model fits for the torus+synchrotron component.\n\nFor the estimated bolometric luminosity $10^{12}$ $L_{\\odot}$, and a black hole mass of $2.5\\times10^9$ $M_{\\odot}$, the Eddington Ratio ($L\/L_{edd}$) is $\\sim1.3\\times10^{-2}$. This is similar to, but slightly lower than previous estimates from \\citet{Tadhunter03}, likely due to the fact that our model attributes some of the IR luminosity to the heating of dust from star formation. This starburst luminosity is relatively well constrained, with consistent values across the bulk of fits. The AGN (torus plus jet) contributes $\\sim90\\%$ of the infrared luminosity and star formation produces the remaining $10\\%$.\n\n\\subsection{Future Work and Observations}\n\nThe results of our modeling can be tested and improved with the help of future observations in various wavelength regimes.\n\nAround 1 THz ($\\sim300\\mu$m) the synchrotron and starburst model components are of similar flux density, with the synchrotron contribution decreasing and the thermal contribution from star formation beginning to dominate. Unfortunately, data at this location are limited in resolution, sensitivity, and wavelength coverage. Disentangling the emission from the starburst and emission from the AGN at this frequency will be possible with the Atacama Large Millimeter Array \\citep[ALMA; e.g.,][]{Carilli05}, particularly with the availability of Band 10 and full science capabilities.\n\nThe far-infrared suffers from poor sampling of the SED. In this regime the Herschel Space Observatory \\citep{Pilbratt10} provides an opportunity to improve on the understanding of the continuum emission. SPIRE and PACS cover wavelength ranges of interest: the region of the possible jet-break and the long wavelength side of the infrared bump. Observations here can provide improved constraints for the synchrotron and starburst models. \\citet{Gonzales10} demonstrate decomposition of far-infrared emission for Mrk 231, with the Herschel SPIRE observations providing important constraints. Similar observations of Cygnus A will provide important data on this sparsely observed portion of the SED. PACS observations would contribute measurements of far-infrared fine structure lines which could be used in concert with the mid-infrared lines to provide an alternate method of determining the relative contribution of star formation and AGN activity to the infrared luminosity \\citep[e.g.,][for Mrk 231]{Fischer10}.\n\nAlthough the synchrotron spectrum breaks in our fits, it may still dominate the flux between $5$ and $10~\\mu$m. This result can be tested by future mid-infrared polarimetry or variability studies. Synchrotron emission from a compact source such as the radio core or jet knots should be subject to flaring. If the emission in this wavelength range is instead of a thermal origin (e.g., hot dust in the host galaxy), the $10~\\mu$m flux will remain relatively stable.\n\n\\section{Summary}\n\\label{sec:summary}\n\nUsing a combination of a new mid-infrared spectrum from the Spitzer Space Telescope and radio data from the literature, the nuclear emission in Cygnus A has been modeled as a combination of powerlaw emission from a synchrotron jet, reprocessed AGN emission from a dusty torus, and emission from a dusty circumnuclear starburst.\n\nThe data are well fit by a combination of these three models, and all three are necessary to reproduce the observed emission. Statistically acceptable fits were found for both an exponential cutoff in the population of relativistic electrons (Case I) as well as emission from an aging electron population (Case II), however we are unable to distinguish between the two cases. For Case I we find the cutoff frequency to be between $10$ and $50$ THz ($5-30~\\mu$m) while for Case II fits the predicted break frequency is $5$ THz ($60~\\mu$m). Degeneracy between the starburst and synchrotron components makes a more precise determination of the break or cutoff frequency difficult. Better observations on the long-wavelength side of the thermal bump will provide tighter constraints on the models.\n\nFrom this modeling, we find the following:\n\\begin{itemize}\n  \\item{The bolometric luminosity of the AGN in Cygnus A is $\\sim10^{12}$ $L_{\\odot}$}\n  \\item{The mid-infrared emission is consistent with emission from a clumpy obscuring torus with an outer size of $\\sim130$ pc.}\n  \\item{The far-infrared emission is consistent with being dominated by star formation which is occurring at a rate between $10$ and $70$ $M_{\\odot}$ yr$^{-1}$.}\n\\end{itemize}\n\nIn Cygnus A, the infrared emission is a combination of AGN and starburst heated dust, with the AGN contributing $\\sim90\\%$ of the luminosity. \n\n\n\\acknowledgements\n\nThe authors thank the anonymous referee who's comments have improved the quality of this paper. \n\nThis work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL\/Caltech. Computing resources were provided in part by the Research Computing group of the Rochester Institute of Technology. This research has made use of the NASA\/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of NASA's Astrophysics Data System. This work is supported in part by the Radcliffe Institute for Advanced Study at Harvard University.\n\nG.P. thanks D. Whelan, D. M. Whittle, and A. Evans for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur understanding of galaxy evolution has immensely progressed over the last decades thanks to deep photometric observations mainly at ultraviolet, optical and near-infrared wavelengths enabled by the {\\it Hubble Space Telescope} (HST). The {\\it Spitzer} and {\\it Herschel} space telescopes have complemented such datasets at mid- and far-infrared wavelengths, allowing us to characterize galaxy populations out to high redshifts in unprecedented detail.\nThese observations suggest that the Universe has undergone a peak of star formation activity at $1<z<3$ (e.g.~\\citealt{hopkins2006,bouwens2007,cucciati2012}). Mapping this epoch of intense galaxy assembly is thus crucial to understand galaxy evolution.\nThe new WFC3 camera on board HST has revolutionized the study of the galaxy population at this epoch. In particular, the Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey (CANDELS; \\citealt{grogin2011,koekemoer2011}) HST Multi-Cycle Treasury Survey is providing high spatial resolution images of the emission in the most extensively studied extragalactic fields (GOODS, COSMOS, EGS and UDS) for which a wealth of ancillary photometric data from the ultraviolet to the infrared have been compiled over the years. At the same time, the 3D-HST WFC3-G141 grism survey has been obtaining spatially resolved rest-frame optical spectra of $\\sim 7000$ galaxies ($H < 23$) at $z>1$ in these fields, providing redshift estimates with an accuracy of $\\sigma(z)=0.0034(1+z)$, i.e. an order of magnitude better than that of photometric redshift based on broad-band observations \\citep{brammer2012}. Crucially, these new spectroscopic data also enable detailed studies of the ionized gas and dust geometry (via rest-frame optical emission lines) and stellar population properties (via stellar absorption features) of unprecedentedly large galaxy samples at $z>1$ (e.g.~\\citealt{brammer2012,fumagalli2012,fumagalli2013,schmidt2013,whitaker2013,price2014}).\n\nA main caveat in current statistical studies of the galaxy population at $z\\ga1$ is that the way in which the physical properties of galaxies are generally derived from rich multi-wavelength data-sets does not reflect recent advances in the sophisticated modelling of galaxy spectral energy distributions (SEDs). For example, spectral analyses often rely on oversimplified colour diagnostics and\/or limited modelling of the stellar spectral continuum using simple star formation histories (SFHs), such as exponentially declining  $\\tau$-models.\\footnote{SFHs in which the star formation rate depends on time $t$ as $\\propto\\exp(-t\/\\tau)$ are often referred to as `$\\tau$-models'.} As shown for example by \\cite{maraston2010}, such models do not account well for the observed colours of $z\\approx 2$ galaxies from the GOODS-South sample, which these authors reproduce by appealing to exponentially rising SFHs \\cite[see also][]{pforr2012}.  \\cite{simha2014} propose a 4-parameter model (delayed exponential + linear ramp at some transition time) to best represent the SFHs of galaxies from smoothed particle hydrodynamics simulations. Other studies have already shown that more sophisticated SFH parametrizations provide better agreement with the data (e.g.~\\citealt{lee2010,pacifici2013,behroozi2013,lee2014}). The inclusion of nebular emission is also important to interpret observed spectral energy distributions of galaxies. While most studies including nebular emission rely on empirically calibrated emission-line template spectra (e.g.~\\citealt{fioc1997,anders2003,schaerer2009,schaerer2010}), more elaborate prescriptions have been proposed, based on combination of stellar population synthesis and photoionization codes (e.g., \\citealt{charlot2001,groves2008}).\n\n\\cite{pacifici2012} provide the sophisticated modelling framework required to interpret multi-wavelength photometric and spectroscopic galaxy observations in a physically and statistically consistent way. This approach overcomes several main limitations of galaxy spectral modelling mentioned above. Specifically, \\cite{pacifici2012} build a comprehensive library of model galaxy SEDs, which can be used to derive statistical constraints on physical parameters. This combines: (i) physically motivated star formation and chemical enrichment histories from cosmological simulations; (ii) state-of-the-art stellar population synthesis and nebular emission modelling computed consistently using the photoionization code {\\small CLOUDY} \\citep{ferland1996} and (iii) a sophisticated treatment of dust attenuation, which includes uncertainties in the spatial distribution of dust and in galaxy orientation \\citep{chevallard2013}. One of the main features of the \\cite{pacifici2012} approach is that it allows one to interpret simultaneously the stellar and nebular emission from galaxies at any spectral resolution.\n\nIn this paper, we use the wealth of multi-wavelength data provided by the 3D-HST survey to investigate, in a systematic way, how different SED modelling approaches (regarding star formation and chemical enrichment histories, dust attenuation and nebular emission lines) and the availability of spectroscopic (in addition to photometric) information affect the constraints derived on the physical parameters of high-redshift galaxies. We use the \\cite{pacifici2012} models to reproduce simultaneously the observed $0.35$ to $3.6\\mu$m photometry and emission line strengths of a sample of 1048 galaxies at $0.7<z<2.8$. We investigate the advantages provided by these sophisticated models when deriving galaxy physical parameters, with respect to standard approaches relying on widely used, oversimplified parametrizations of star formation histories (continuous, exponentially declining $\\tau$-models), fixed assumptions about metallicity and dust properties and neglecting nebular emission (e.g.~\\citealt{noll2009,kriek2009,forster2009,marchesini2009,pozzetti2010}). We also quantify the importance of accounting for nebular emission in the interpretation of broad-band galaxy observations by comparing the results obtained using the original \\cite{pacifici2012} models with those obtained using the same sophisticated prescriptions for the star formation and chemical enrichment histories and attenuation by dust, but not including nebular emission lines.\nTo further illustrate the importance of using appropriate spectral models to derive physical properties of galaxies and, more generally, to constrain galaxy evolution, we also explore how the different SED-fitting methods considered in this paper affect the interpretation of the correlation between SFR and stellar mass of galaxies (commonly referred to as the `star-formation main sequence' - \\citealt{noeske2007}) and the scatter about this relation.\n\nOur paper is organized as follows. In Section~\\ref{sec:data}, we describe the sample of 1048 galaxies in GOODS-South extracted from the 3D-HST catalogue, for which multi-wavelength photometry and HST\/WFC3 grism spectra are available. In Section~\\ref{sec:modelintro}, we introduce three model libraries of galaxy SEDs with different levels of sophistication to interpret these observations. We also assess how well each of these libraries can account for the data. Then, in Section~\\ref{sec:results}, we quantify the differences in the constraints derived on the stellar masses, SFRs and attenuation by dust for the galaxies in the sample when adopting the different spectral libraries. We investigate in Section~\\ref{sec:ms} how these differences affect the measured correlation between SFR and stellar mass of galaxies. Finally, in Section~\\ref{sec:summary}, we summarize our results. Throughout this paper we express magnitudes in the AB system and we adopt the \\cite{chabrier2003} stellar IMF and the following cosmology: $\\Omega_\\Lambda=0.7$, $\\Omega_m=0.3$, $H_0=70$ km s$^{-1}$ Mpc$^{-1}$.\n\n\\section{Data}\n\\label{sec:data}\n\nWe select a sample of galaxies from the 3D-HST Treasury Survey\\footnote{http:\/\/3dhst.research.yale.edu\/} \\citep[PI: van Dokkum]{brammer2012}, which provides a multi-wavelength photometric catalogue \\citep{skelton2014} and low-resolution WFC3 grism observations. In this Section, we present the photometric catalogue, spectroscopic observations and describe the sample selection.\n\n\\subsection{Photometry}\n\\label{sec:phot}\n\nWe use version 4.1 of the 3D-HST Survey photometric catalogue for the GOODS-South field covering an area of 171 arcmin$^2$ \\citep{skelton2014}. The catalogue contains broad- and narrow-band photometry in 38 filters sampling ultraviolet (UV) to mid-infrared (MIR) wavelengths, from both ground- and space-based observatories, for a total of 50,507 objects detected in the HST\/WFC3-F160W band. To make the present investigation relevant to a wide range of studies of high-redshift galaxies, we focus on the following commonly used nine broad bands: $U$, ACS-F435W, ACS-F606W, ACS-F775W, ACS-F850lp, WFC3-F125W, WFC3-F140W, WFC3-F160W and {\\em Spitzer}\/IRAC 3.6$\\mu$m. These bands allow us to reasonably sample the rest-frame UV, optical and near-infrared emission of galaxies at all redshifts between 0.7 and 2.8, optimally probing the emission by stellar populations of different ages and the effects of dust attenuation. We exclude {\\em Spitzer}\/IRAC data at wavelengths longer than $3.6\\mu$m because the spectral modelling (see Section~\\ref{sec:modelintro}) does not include dust emission.\\footnote{Adding dust emission increases the complexity of the analysis and is beyond the scope of the current paper; we leave a detailed analysis combining the \\citealt{pacifici2012} model with the models by \\citealt{dacunha2008} to a future study.}\n\nThe photometry in all bands is extracted consistently across all wavelengths as described in \\cite{skelton2014}. To summarize, the 3D-HST photometric catalogue is based on WFC3 selection using a combination of the three WFC3 bands (F125W, F140W and F160W) for detection. In order to measure consistent colours across different HST bands, each of the HST images is convolved with the same point spread function (PSF) and aperture photometry is performed on the PSF-matched images. For the lower resolution ground-based optical and {\\em Spitzer}\/IRAC bands, the photometry is extracted using the method described in \\cite{labbe2006}, \\cite{wuyts2008} and \\cite{whitaker2011}, which uses the HST images as a high-resolution prior to correct for the larger PSF and confusion caused by neighboring sources.\n\n\\subsection{Grism spectroscopy}\n\\label{sec:spec}\n\nThe 248 orbits dedicated to the 3D-HST Survey are distributed among 124 individual pointings, each observed for two orbits. 34 of these pointings cover the GOODS-South field. For all 50,507 galaxies in this field, WFC3-G141 grism spectra are available, while a spectroscopic redshift ($z_\\mathrm{grism}$) has been estimated for a subset of 2699 galaxies to $H<23$ \\citep[here the $H$ band is HST\/WFC3-F160W;][]{brammer2012}.\nThe WFC3 grism spectra cover observer-range wavelengths between 1.1 and 1.6 $\\mu$m. The spectral scale is $\\approx 23$ {\\AA}\/pixel and the effective resolution depends on the spatial extent of the galaxy (see \\citealt{brammer2012}, Sec. 2.1 and 4.2). For example, the observed full width at half maximum (FWHM) of the {\\hbox{H$\\alpha$}} line (which in these spectra is blended with $\\hbox{[N\\,{\\sc ii}]}\\lambda 6548$ and $\\hbox{[N\\,{\\sc ii}]}\\lambda 6584$) can span a range as wide as 75--200 \\AA. The emission-line sensitivity is $\\sim 5 \\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ at $5\\sigma$. \nGrism redshifts are derived from a combination of the 1D spectra and broad- and medium-band photometry from the catalogue described above (see \\citealt{brammer2012} and references therein for details). The precision of grism redshifts is $\\sigma(z)=0.0034(1+z)$, i.e. an order of magnitude better than what is typically achieved with high-quality broad-band photometry alone.\n\nTo characterize the emission features in the observed spectra, we measure rest-frame equivalent widths (EWs) of the most prominent lines. Because of the low resolution, some emission lines are blended together, thus we extract the EWs of: ${\\hbox{[O\\,{\\sc ii}]}}\\lambda\\lambda3726,\\,3729$; {\\hbox{H$\\beta$}}; ${\\hbox{[O\\,{\\sc iii}]}}\\lambda\\lambda4959,\\,5007$; $\\hbox{[N\\,{\\sc ii}]}\\lambda6548+\\hbox{H$\\alpha$}+\\hbox{[N\\,{\\sc ii}]}\\lambda6584$; and ${\\hbox{[S\\,{\\sc ii}]}}\\lambda\\lambda6717,\\,6731$. In the following, we refer to these (blends of) lines simply as {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}}. To measure the continuum underneath the emission lines, we first smooth the spectra over 17 pixels (i.e., about 200 \\AA\\ in rest-frame at $z=1$). Then, we mask $\\approx350${\\AA}-wide regions centred on the 5 (groups of) emission lines, interpolate the spectra in the masked regions and finally smooth again over 17 pixels to remove possibly persisting features. Using the resulting continuum, we can calculate the emission-line EWs of {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} as defined above. The uncertainties on the EWs are measured through the propagation of the errors on the flux.\\footnote{Since the continuum is obtained by smoothing the flux, we scale the error on the continuum with the fraction between the standard deviations of the flux and the continuum in a wavelength window free of line contamination.} The signal-to-noise ratio (S\/N) of {\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}} can reach up to $\\sim30$, while that of the other lines ({\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}} and {\\hbox{[S\\,{\\sc ii}]}}) rarely exceeds $\\sim10$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{.\/figures\/fig_01.png}\n\\caption{Redshift distributions of the 3D-HST photometric sample (gray shaded histogram; 1048 galaxies) and emission-line sub-sample (black open histogram; 364 galaxies) are shown in the top panel. Coloured boxes indicate the redshift windows in which selected emission lines can be detected in the WFC3-G141 grism spectra. In the bottom panel, we show the observer-frame grism spectrum of one of the highest S\/N objects in the sample, detected at $z=1.316$. The black solid line and the grey shaded region represent the observed flux and the flux error, respectively. Dashed lines indicate the position of the four detected (blends of) emission lines.}\n\\label{fig:zz}\n\\end{center}\n\\end{figure}\n\n\\subsection{Sample selection}\n\\label{sec:sample}\n\n\\begin{table}\n\\begin{center}\n\\caption{Typical magnitudes and signal-to-noise ratios of the 3D-HST photometric sample in the nine bands considered in this paper.}\n\\label{tab:phot}\n\\begin{tabular}{l r r r}\n\\hline\n\\hline\nFilter & Effective $\\lambda$\/{\\AA} & Median magnitude & Median S\/N \\\\\n\\hline\n$U$ & 3446 & 25.1 & 52\\\\\nACS-F435W & 4302 & 24.9 & 23\\\\\nACS-F606W & 5925 & 24.3 & 52\\\\\nACS-F775W & 7705 & 23.6 & 62\\\\\nACS-F850lp & 9059 & 23.2 & 72\\\\\nWFC3-F125W & 12,516 & 22.5 & 91\\\\\nWFC3-F140W & 13,971 & 22.4 & 61\\\\\nWFC3-F160W & 15,392 & 22.2 & 105\\\\\nIRAC 3.6$\\mu$m & 35,573 & 21.5 & 106\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe first select a \\textit{photometric} sample by extracting from the 3D-HST v4.1 photometric catalogue all galaxies with a measured grism redshift in the range $0.7<z_\\mathrm{grism}<2.8$ and use\\_flag $=1$ (see Section 3.8 in \\citealt{skelton2014}). This sample includes 1140 galaxies down to $H=23$. From this sample, we select an \\textit{emission-line} sub-sample including only galaxies with at least one emission line detected in the WFC3-G141 grism spectrum with equivalent-width S\/N greater than 5. We visually inspect all selected grism spectra to check the accuracy in redshift and to make sure that contiguous emission lines ({\\hbox{H$\\beta$}} with {\\hbox{[O\\,{\\sc iii}]}} and {\\hbox{H$\\alpha$}} with {\\hbox{[S\\,{\\sc ii}]}}) are well separated. In the end, we retain 392 emission-line galaxies. We then reject all sources detected by \\textit{Chandra} (\\citealt{xue2011}, $\\log(L_\\mathrm{X})>42$) as potential AGNs, which leaves us with 1048 galaxies in the photometric sample and 364 in the emission-line sample. In Table~\\ref{tab:phot}, we summarize the median magnitudes and S\/N in the nine filters of interest to us for the photometric sample. We note that the typical S\/N on the photometry is very high for most bands. We discuss the influence of such small uncertainties on the derivation of statistical constraints on physical parameters in Section~\\ref{sec:fit} and \\ref{sec:results}.\n\nThe redshift distributions of the photometric and emission-line samples are shown in Fig.~\\ref{fig:zz} (top panel) as shaded and open histograms, respectively. The peak at $z\\approx0.7$ is associated to an over-density in the field (\\citealt{salimbeni2009}). Most of the galaxies in the emission-line sample lie at $z\\sim1$, which corresponds to where {\\hbox{H$\\alpha$}} falls in the G141 spectral range. The most `data-rich' window is the redshift range $1.2 \\lesssim z \\lesssim 1.5$, where four emission lines can be detected simultaneously; 17 per cent of the galaxies in the emission-line sample are selected from this redshift window. As an example, we show, in the bottom panel of Fig.~\\ref{fig:zz}, the observer-frame grism spectrum of one of the highest S\/N objects in the sample, detected at $z=1.316$. This galaxy shows two emission lines with S\/N~$>20$ ({\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}}), one with S\/N~$\\approx10$ ({\\hbox{H$\\beta$}}) and one with S\/N~$\\approx5$ ({\\hbox{[S\\,{\\sc ii}]}}).\n\n\\section{Comparison between different spectral modelling approaches}\n\\label{sec:modelintro}\n\nTo derive galaxy physical parameters (such as stellar mass, SFR and optical depth of the dust) from the multi-wavelength observations described above, we must appeal to spectral modelling techniques. A main focus of the present paper is to establish the appropriateness of different spectral modelling approaches to interpret photometric and spectroscopic observations of distant galaxies. Specifically, we consider three modelling approaches relying on different assumptions about several main ingredients of spectral interpretation techniques: the explored (prior) ranges of star formation and chemical enrichment histories; attenuation by dust; and nebular emission. In the next paragraphs, we describe these competing approaches in order of increasing complexity. We also quantify the extent to which the different models can account for the data of Section~\\ref{sec:data} by comparing prior libraries of predicted colours with these multi-wavelength data with a bayesian approach. We refer to Appendix A for a specific comparison between physical parameters extracted using our most sophisticated spectral library (P12, see below) and a tool widely used in the 3D-HST collaboration \\citep[FAST]{kriek2009}.\n\n\\subsection{Different model spectral libraries}\n\\label{sec:model}\n\n\\subsubsection{The `classical' spectral library (CLSC)}\n\\label{sec:basic}\n\nWe first assemble a `classical' model spectral library, corresponding to standard simplistic descriptions of the stellar and interstellar content of galaxies, which are the most widely used to derive galaxy physical properties from fits of ultraviolet to near-infrared SEDs (e.g.~\\citealt{kriek2009,pozzetti2010}). Galaxy SFHs in this spectral library are parametrized as exponentially declining functions of the form $\\psi(t)\\propto \\exp(\\gamma \\, t)$. Here $\\gamma$ is the inverse star formation timescale, drawn randomly in the range $0<\\gamma\/\\mathrm{Gyr}^{-1}<3$,\\footnote{We note that allowing for larger values of $\\gamma$ would increase the number of low-specific-SFR galaxies in the prior at fixed evolutionary stage. We have checked that, on the one hand, this would bias the SFR estimates low \\citep{wuyts2011,price2014}. On the other hand, very young ages (and thus low masses) would be required in order to match the photometry of the galaxies in the sample. In Appendix A, we show the physical parameters extracted using the code FAST \\citep{kriek2009}, in which very short $e$-folding star-formation timescales are allowed.} and $t$ is the lookback time. In this CLSC spectral library, all stars in a given galaxy are assumed to have the same metallicity, drawn randomly from the logarithmic range $-1.6<\\log(Z\/Z_{\\odot})<0.4$ (we adopt the solar metallicity $Z_{\\odot}=0.017$). By analogy with \\cite{pacifici2012}, we generate a sample of 4 million galaxy SFHs, selecting randomly the redshift of observation in the range $0.6<z<3.0$ (appropriate for the 3D-HST sample described in Section~\\ref{sec:data}) and the evolutionary stage between 0.5 Gyr and the age of the Universe at the redshift of observation (see Sections 2.1 and 3.1.2 of \\citealt{pacifici2012}).\\footnote{In \\cite{pacifici2012}, the evolutionary stage is drawn randomly from a linear range in redshift. Here, we choose to draw the evolutionary stage randomly from a linear range in lookback time to avoid having in the spectral library too many galaxies which started forming stars less than 1 Gyr before the redshift of observation.} We combine this set of exponentially declining star formation histories with the latest version of the the \\cite{bruzual2003} stellar population synthesis models, which allows one to compute the spectral evolution of stellar populations for given SFH, metallicity and IMF (see Appendix B for a comparison with the original version of the \\citealt{bruzual2003} stellar models). We model interstellar dust attenuation using a two-component model, where the $V$-band attenuation optical depth seen by stars older than 10Myr in the diffuse interstellar medium (ISM) is a fraction ($\\mu$) of the optical depth $\\hbox{$\\hat{\\tau}_{V}$}$ seen by younger stars still embedded in their birth clouds \\citep{charlot2000}. We adopt a fixed slope of the attenuation curve in both the birth clouds and the diffuse ISM ($\\hbox{$\\hat{\\tau}_\\lambda$} \\propto \\lambda^n$, with $n=-0.7$). In this spectral library, we draw randomly the total attenuation optical depth $\\hbox{$\\hat{\\tau}_{V}$}$ in the range between 0 and 4 and fix $\\mu=0.3$. The overall (i.e. galaxy-wide) attenuation law produced by the Charlot \\& Fall (2000; see their fig. 5) model depends on the relative contributions by young and old stars to the emission of a galaxy and for young starbursts, can resemble the \\cite{calzetti2000} attenuation law (as used in several studies, e.g.~\\citealt{kriek2009,forster2009,marchesini2009}).\n\n\\subsubsection{The `P12 without emission lines' spectral library (P12nEL)}\n\\label{sec:p12}\n\nThe second model spectral library we consider is based directly on the approach of \\cite{pacifici2012}. It includes physically motivated star formation and chemical enrichment histories derived from a post-treatment of the Millennium cosmological simulation \\citep{springel2005} using the semi-analytic models of \\cite{delucia2007}. This approach allows us to explore a wide range of (non-parametric) SFHs, including declining, rising, roughly constant, bursty and smooth evolutionary shapes, in addition to realistic chemical enrichment histories. As in the case of the CLSC spectral library, we first select randomly the redshift of observation and the evolutionary stage of each model galaxy. Then, following \\cite{pacifici2012}, to further widen the range of physical properties probed by this library, and we re-sample the `current' (i.e. averaged over a period of 10Myr before a galaxy is looked at) star formation and chemical properties of each model galaxy by redrawing the specific SFR (the SFR divided by the stellar mass; $\\psi_S = $~SFR~$\/M_{\\ast}$) in the range $-2<\\log(\\psi_S\/\\mathrm{Gyr}^{-1})<+2$ and the gas-phase oxygen abundance in the range $7<\\hbox{$12+\\log\\textrm{(O\/H)}$}<9.4$. For each star formation and chemical enrichment history in the original library we draw 10 different realizations of these current properties and assemble in this way a library of 4 million star formation and chemical enrichment histories. We then generate a library of 4 million model spectra by combining this set of star formation and chemical enrichment histories with the same stellar population synthesis models as used to build the CLSC spectral library in Section~\\ref{sec:basic} above. We include dust attenuation using a more sophisticated implementation of the \\cite{charlot2000} two-component dust model than in the CLSC spectral library. In this prescription, the slope of the attenuation curve is fixed in the birth clouds ($n=-1.3$), but is drawn randomly in the range $-1.1<n<-0.4$ in the diffuse ISM to reflect uncertainties about the spatial distribution of dust and the orientation of a galaxy (see \\citealt{pacifici2012,chevallard2013}). As in the case of the CLSC spectral library, we take $\\hbox{$\\hat{\\tau}_{V}$}$ to be randomly distributed between 0 and 4. In summary, the P12nEL spectral library differs from the CLSC spectral library of Section~\\ref{sec:basic} in that it includes far more realistic star formation and chemical enrichment histories and a more sophisticated prescription for attenuation by dust.\n\n\\subsubsection{The `P12 with emission lines' spectral library (P12)}\n\\label{sec:p12el}\n\nFinally, the third model spectral library we consider is the original one of \\cite{pacifici2012}. This is identical to the P12nEL library described in the previous section, except that it also includes a component of nebular emission. \\cite{pacifici2012} used the standard photoionization code {\\small CLOUDY} \\citep{ferland1996} to compute the emission lines consistently with the emission by stellar populations of different ages (neglecting the contribution by stars older than 10 Myr, which produce negligible ionizing radiation). The nebular emission in a model spectrum is parametrized in terms of the gas-phase metallicity, $Z$, the zero-age ionization parameter, $U_0$ and the dust-to-metal (mass) ratio, $\\xi_d$ (which characterizes the depletion of metals onto dust grains) of the photoionized gas (see section 2.2.2 of \\citealt{pacifici2012} for more details). These parameters are randomly drawn while re-sampling of the current chemical properties of a galaxy (Section~\\ref{sec:p12}). We build in this way a P12 library of 4 million model galaxy spectra.\n\n\\subsection{Confronting spectral models with the observations}\n\\label{sec:mockobs}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_02.png}\n\\caption{Optical-NIR colour-colour diagrams comparing the two 3D-HST samples with the three model libraries. The photometric sample and the emission-line sub-sample (including only galaxies with at least one emission line detected with equivalent-width S\/N~$>5$) are represented by gray and black symbols, respectively; open circles mark objects for which the error in at least one of the two colours is larger than 0.2 magnitudes. Contours show the colour-colour space covered by the three spectral libraries: CLSC (left-hand column, orange), built using exponentially declining SFHs, fixed metallicity, simple dust attenuation and no nebular emission (Section~\\ref{sec:basic}); P12nEL (middle column, green), built using physically motivated star formation and chemical enrichment histories and a sophisticated treatment for dust attenuation (Section~\\ref{sec:p12}); P12 (right-hand column, blue), same as previous, including a component of nebular emission (Section~\\ref{sec:p12el}). In each panel, the three contours mark 50, 16 and 2 per cent of the maximum density. While the CLSC spectral library leaves few observed galaxies with no model counterpart, the P12 spectral library allows us to cover reasonably well the entire observed colour-colour space.}\n\\label{fig:col3lib}\n\\end{center}\n\\end{figure*}\n\nHence, we have built three model spectral libraries relying on different prescriptions to describe galaxy star formation and chemical enrichment histories, attenuation by dust and nebular emission. Each spectral library contains 4 million galaxy SEDs covering the rest-frame wavelength range $912\\,\\mathrm{\\AA}<\\lambda<5\\,\\mu \\mathrm{m}$.\\footnote{We do not include here reradiation by dust grains, which is expected to dominate the emission at wavelengths greater than $\\sim 5\\,\\mu$m (e.g., \\citealt{dacunha2008}).} To compare these model SEDs with the observations of galaxies in the 3D-HST photometric and emission-line samples, we convolve the observer-frame model SEDs with the response functions of the filters presented in Section~\\ref{sec:phot}. For P12, we also compute the EW of the emission lines in each model spectrum in the same way as we compute the observed EWs from the WFC3 grism spectra. In brief, we first reproduce the average resolution of the observations ($100$ {\\AA} FWHM and 22.5{\\AA} bin width) and then calculate the EWs of the {\\hbox{[O\\,{\\sc ii}]}}, {\\hbox{H$\\beta$}}, {\\hbox{[O\\,{\\sc iii}]}}, {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} emission lines (as defined in Section~\\ref{sec:spec}).\n\nIn Fig.~\\ref{fig:col3lib}, we compare the observations of both the photometric and emission-line 3D-HST samples described in Section~\\ref{sec:sample}, in two ACS and two WFC3 bands, with the predictions of the three model spectral libraries.  We plot the ACS (F435W $-$ F775W) vs WFC3 (F125W $-$ F160W) observer-frame colours in different redshift ranges (rows) for the three different spectral libraries (columns). Contours show the colour-colour space covered by the different spectral libraries. The photometric sample (grey symbols) and the emission-line sample (black symbols) are shown on top. Since the emission-line sample can be analysed only with the P12 spectral library, we plot it only in the right hand-side column. This figure shows that the CLSC spectral library leaves few observed galaxies with no model counterpart. Thus, SED fits for these galaxies will be biased towards the models that lie the closest to the observations, at the very edge of the spectral library. The P12nEL spectral library, which does not include nebular emission, can cover reasonably well the bulk of the observations at all redshifts. This shows the importance of accounting for more realistic ranges of star formation (and chemical enrichment) histories and dust properties than included in the widely used CLSC spectral library. Few observed galaxies fall outside the contours of the P12nEL model spectral library, presumably because of the contamination of the WFC3-F160W flux by strong {\\hbox{H$\\alpha$}} emission. In fact, the P12 spectral library, which includes contamination of broad-band fluxes by nebular emission, spans a much larger range of the colour-colour space than the CLSC and P12nEL libraries in Fig.~\\ref{fig:col3lib}. This spectral library allows us to cover reasonably well the entire observed colour-colour space, with the exception of some fairly red galaxies with large photometric errors (empty circles).\n\nWe note that, the emission-line 3D-HST sample in Fig.~\\ref{fig:col3lib} (right-hand panel), although including only galaxies with at least one well-detected emission line, is not biased towards the strongest starburst galaxies (characterized by blue ACS colours), but populates a similar colour-colour space as the photometric sample. This is because the selection in equivalent-width S\/N also gathers those massive galaxies with good continuum S\/N and relatively faint emission lines. The global properties of the photometric and emission-line 3D-HST samples in Fig.~\\ref{fig:col3lib} are thus quite similar.\n\n\\subsection{Spectral fits}\n\\label{sec:fit}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_03.png}\n\\vspace{-0.3cm}\n\\caption{An example of spectral fit of a galaxy at redshift $z=1.261$ using the three model spectral libraries. (a) Observed broad-band magnitudes (black crosses) and best-fitting observer-frame model spectra in full resolution computed with the CLSC (orange), P12nEL (green) and full P12 (blue) spectral libraries. (b, c, d) Residuals between the observed magnitudes and the magnitudes of the best-fitting models. The error bars are roughly the size of the symbols or smaller. Probability density functions (PDFs) of (e) stellar mass, (f) SFR, (g) gas-phase oxygen abundance and (h) dust attenuation optical depth derived with the three libraries using the same colour code as above. (i, j, k) SFHs of the best-fitting models (black solid lines), likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, shades) from the three fits. The residuals show that all three fits are reasonably good, but the PDFs reveal large differences in the extracted parameters. The PDFs derived using the CLSC spectral library in (g) and (h) look unrealistically narrow and hit the edge of the prior (see text for details). The original P12 library yields narrower PDFs than the P12nEL library. This illustrates how accounting for the contamination of broadband fluxes by emission lines can help constrain the parameters better (see Section~\\ref{sec:resphot}). The SFHs estimated using the P12nEL (j) and P12 (k) spectral libraries rise as a function of time, in contrast to those derived using the CLSC spectral library (i).}\n\\label{fig:fitp}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{.\/figures\/fig_04.png}\n\\vspace{-0.25cm}\n\\caption{Distribution of fit residuals between the observed flux $F_\\mathrm{obs}$ and best-fitting model flux $F_\\mathrm{best-fitting}$, in units of the observational error $\\sigma_\\mathrm{obs}$, for all 1048 galaxies in the 3D-HST photometric sample. Each panel refers to a different photometric band (indicated in the top-right corner). The different histograms refer to the CLSC (yellow shaded histogram), the P12nEL (green hatched histogram) and the original P12 (blue solid histogram) model spectral libraries. We note that the residuals extend to large values because the quote photometric uncertainties on the observed fluxes are very small, of the order of 2 percent of the total flux (as discussed in Section~\\ref{sec:sample}). The three different spectral libraries provide reasonable fits to the data, although the best results (i.e. narrowest histograms most centred on zero) are obtained with the P12 library.}\n\\label{fig:resid}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.47\\textwidth]{.\/figures\/fig_05a.png}\n\\includegraphics[width=0.47\\textwidth]{.\/figures\/fig_05b.png}\n\\vspace{-0.25cm}\n\\caption{Best-fitting models and parameter PDFs for an example galaxy in the 3D-HST emission-line sample fitting the photometry alone (\\textit{left-hand panel}) and fitting the photometry and the observed emission-line EWs simultaneously (\\textit{right-hand panel}) with the P12 spectral library. (a, h) Observed SED (black crosses) and best-fitting models (blue solid lines). (b, i) Observed grism spectrum (black solid lines) and best-fitting model spectra at similar resolution (blue solid lines). PDFs of stellar mass (c, j), SFR (d, k), gas-phase oxygen abundance (e, l) and optical depth of the dust (f, m). 50th (solid line), 16th and 84th (dashed lines) percentiles of the PDFs are marked on top. (g, n) SFHs of the best-fitting models (black solid lines), likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, blue shades) in the two cases. The photometric fits look reasonable in both top panels, but the emission-line fluxes predicted when fitting the photometry alone do not match the observed ones. When including the emission-line EWs in the fit, the best-fitting model spectrum reproduces them satisfactorily and the constraints on the physical parameters are significantly tighter.}\n\\label{fig:fit}\n\\end{center}\n\\end{figure*}\n\nWe can quantify further the appropriateness of the three spectral modelling approaches investigated here to interpret photometric and spectroscopic observations of distant galaxies by comparing the constraints derived in each case on physical parameters, such as stellar mass, SFR and dust attenuation optical depth. To do so, we use the bayesian approach described in \\cite{pacifici2012} (see also \\citealt{kauffmann2003a,gallazzi2005,dacunha2008}). In brief, we compare the observational constraints (photometry$+$spectroscopy or photometry alone) on each 3D-HST galaxy to the same observable quantities predicted for each model in a given spectral library, such that $z_{\\mathrm{model}}=z_{\\mathrm{grism}} \\pm 0.02$ (in practice, this implies that each observed galaxy is compared to about 65,000 models). We then use the likelihood of each model to build probability density functions (PDFs) of selected physical parameters for that galaxy: stellar mass ($M_{\\ast}$); SFR ($\\psi$); gas-phase oxygen abundance [$\\hbox{$12+\\log\\textrm{(O\/H)}$}$]; and attenuation optical depth of the dust ($\\hbox{$\\hat{\\tau}_{V}$}$).\n\nFor galaxies in the 3D-HST photometric sample, we fit the fluxes in all nine broad bands from 0.35 to 3.6$\\mu$m.\\footnote{Since the formal photometric errors can be extremely small (Section~\\ref{sec:phot}), we adopt a minimum photometric error of 5 per cent to enlarge the number of models contributing to each fit.} In Fig.~\\ref{fig:fitp}, we show an example of spectral fit of a galaxy at redshift $z=1.261$ using the three model spectral libraries described in Section~\\ref{sec:model}. In Fig.~\\ref{fig:fitp}a, we plot the observed magnitudes (black crosses) and the best-fitting model spectra from the CLSC (orange), P12nEL (green) and P12 (blue) spectral libraries. The residuals in Fig.~\\ref{fig:fitp}bcd show that all three fits are reasonably good, but the PDFs plotted in Fig.~\\ref{fig:fitp}efgh reveal large differences in the extracted physical parameters. Since all fits are performed in the same way, these discrepancies arise purely from differences in the model spectral libraries. Not only the median values, but also the widths of the PDFs are different in the three fits. In particular, the fit obtained using the CLSC spectral library shows narrow histograms of the gas-phase oxygen abundance (g) and the dust attenuation optical depth (h). If a narrow PDF can in some occasions reflect a good fit, this must be interpreted in the context of the number of fitted data points, the errors in the data, the distribution of the priors, the number of models in the spectral library and the number of models which actually contribute to the fit. In Fig.~\\ref{fig:fitp}, differences in the derived PDFs of $M_{\\ast}$ (e), $\\psi$ (f), $\\hbox{$12+\\log\\textrm{(O\/H)}$}$ (g) and $\\hbox{$\\hat{\\tau}_{V}$}$ (h) using different libraries can arise only from differences in the prior distributions and the number of models contributing to each fit. In particular, the narrowness of the PDFs obtained using the CLSC library arises from the fact that only few models in this library are as blue as the observed galaxy (with both ACS and WFC3 colours close to zero in Fig.~\\ref{fig:col3lib}), driving the fit to the most actively star-forming, most metal-poor and least attenuated models. The observed colours are better sampled by the other libraries, the original P12 library yielding narrower PDFs than the P12nEL library. This illustrates how accounting for the contamination of broadband fluxes by emission lines can help constrain these parameters better, as we discuss in Section~\\ref{sec:resphot}. This point was emphasized by \\cite{pacifici2012}, whose models were specifically designed to allow one to maximize the constraining power of photometric observations in the absence of direct spectroscopic information on emission lines. In Fig.~\\ref{fig:fitp}ijk, we show the SFHs of the best-fitting models (black solid lines), the likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, shades) derived with the three spectral libraries. We note that the SFHs derived using the P12nEL (j) and P12 (k) spectral libraries rise as a function of time, in contrast to those derived using the CLSC spectral library.\n\nTo quantify in a more global way the ability of the different spectral libraries to account for the photometric properties of 3D-HST galaxies, we plot in Fig.~\\ref{fig:resid} the distribution of the residuals between best-fitting model and observed fluxes, in each photometric band, for all 1048 galaxies in the sample. In each panel, the different histograms refer to the CLSC (yellow shaded histogram), the P12nEL (green hatched histogram) and the original P12 (blue solid histogram) model spectral libraries. Fig.~\\ref{fig:resid} shows that, overall, the three different spectral libraries provide reasonable fits to the data, although as expected, the best results (i.e. narrowest histograms most centred on zero) are obtained with the P12 library. The CLSC spectral library shows slight systematic offsets in the $U$ band (where the best-fitting flux overestimates the observed one) and in the F606W and F775W bands (where the best-fitting flux underestimates the observed one). These discrepancies imply that the corresponding models do not reproduce the rest-frame ultraviolet-optical slopes of observed galaxies as well as those in the P12 spectral library, which we attribute to the oversimplified dust attenuation prescription and lack of stochasticity in the star formation histories in the CLSC spectral library. The most extended residual distributions in Fig.~\\ref{fig:resid} pertain to the F125W band, in which the best-fitting fluxes in both the CLSC and P12nEL spectral libraries can severely underestimate the observed fluxes. This is because both models fail to reproduce the contamination of the F125W band flux by the {\\hbox{H$\\alpha$}} emission line. This happens at $z\\lesssim1.2$, thus for more than half the galaxies in the photometric sample. Remarkably, when emission lines are included in the model spectra, as is the case for the P12 spectral library, the fit residuals improve significantly. We further discuss and quantify the contamination of observed broadband fluxes by emission lines in Section~\\ref{sec:resspec}.\n\nIn the case of the 3D-HST emission-line sample, we can add to the constraints on the nine broad-band fluxes those on up to four emission-line EWs in the rest-frame optical wavelength range. Fig.~\\ref{fig:fit} shows an example of fit (performed with the P12 spectral library) of such a galaxy at $z=1.781$ with available measurements of the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines. The left-hand panels show the results obtained when fitting the photometry alone and the right-hand panels those obtained when including also the constraints on the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines. In Fig.~\\ref{fig:fit}ah, we show the observed SED (black crosses) together with the best-fitting models (blue solid line) and in Fig.~\\ref{fig:fit}bi, the observed grism spectrum (black solid line) together with the best-fitting model spectra at similar resolution (blue solid line). The photometric fits performed with the P12 spectral library look reasonable in both top panels, but the emission-line fluxes predicted when fitting the photometry alone do not match the observed ones (b). When including the emission-line EWs in the fit, the best-fitting model spectrum reproduces them satisfactorily (i). In Fig.~\\ref{fig:fit}, we show the constraints derived in both cases on the same physical parameters as in Fig.~\\ref{fig:fitp}, i.e. stellar mass (c, j), SFR (d, k), gas-phase oxygen abundance (e, l) and attenuation optical depth of the dust (f, m). Except for the stellar mass (which is usually well constrained by multi-band photometry alone), the constraints obtained when including information on the {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} emission lines (k, l, m) are significantly tighter than those derived from photometry alone (d, e, f). In particular for the SFR, the uncertainty decreases from 0.7 to 0.35 dex. This is expected, since the nebular emission lines arise from the obscured {\\mbox{H\\,{\\sc ii}}} regions ionized by young massive stars; thus measuring these lines allows one not only to probe current, massive star formation, but also the metallicity and dust content in those regions \\citep{charlot2001,pacifici2012}. In Fig.~\\ref{fig:fit}gn, we show the SFHs of the best-fitting models (black solid lines), the likelihood-weighted average SFHs (black dashed lines) and associated confidence ranges (likelihood-weighted standard deviation, blue shades) in the two cases. Including the emission-line equivalent widths in the fit improves the constraints on the \\textit{current} (last 10 Myr) values of physical parameters, but generally does not affect the constraints on the full SFH.\n\nIn the next Section, we apply the above fitting procedure to all galaxies in the photometric and emission-line 3D-HST samples and inter-compare the results obtained with the different spectral libraries in terms of the accuracy and uncertainty of statistical constraints on physical parameters (stellar mass, SFR and attenuation optical depth of the dust) in a more global way.\n\n\\section{Statistical constraints on galaxy physical parameters}\n\\label{sec:results}\n\nThe accuracy and uncertainty of the constraints on galaxy physical parameters derived from statistical fits of observations depend both on the type of observations considered (photometry, spectroscopy) and on the model spectral library used to interpret these. In this Section, we first explore the effects of using the three different model spectral libraries of Section~\\ref{sec:model}, which describe with different levels of sophistication the stellar and interstellar components of galaxies, to interpret the same set of observations (photometric 3D-HST sample). Then, we quantify the improvement introduced by the ability with the P12 spectral library to fit a combination of photometric and spectroscopic data (available for the 3D-HST emission-line sample) compared to fitting broad-band photometry alone. \n\n\\subsection{Fits to the photometry: classical vs realistic models}\n\\label{sec:resphot}\n\n\\begin{figure*}\n\\begin{center}\n\\vspace{-0.9cm}\n\\includegraphics[width=0.95\\textwidth]{.\/figures\/fig_06.png}\n\\caption{Comparison between constraints of physical parameters derived for all 1048 galaxies in the 3D-HST photometric sample using the CLSC (top panels, in orange) and P12nEL (bottom panels, in green) model spectral libraries to those obtained using the more sophisticated P12 library. In each box, we show the comparison between the medians of the PDFs. The shades, from dark to light, mark 75, 50, 30, 10 and 1 per cent of the maximum density. The black contours mark 50 per cent of the maximum density. In the top part of each panel, we show also the average uncertainty in three bins. Each column represents a different physical parameter: stellar mass, SFR and optical depth of the dust, from left to right. The use of simple exponentially declining SFHs (CLSC spectral library) can cause strong biases on both the stellar mass and the SFR. The uncertainties on physical parameters estimated using this library may appear to be artificially narrow, hiding the fact that the models span a too narrow prior range. Not including the emission lines in the broad-band fluxes (P12nEL spectral library) does not strongly affect the estimates of stellar mass, but can induce a slight overestimation of the SFR.}\n\\label{fig:paramp}\n\\end{center}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{16, 50 and 84 percentiles of the distributions of the differences between best estimates and uncertainties of stellar mass, SFR and optical depth of the dust when comparing constraints with different libraries as shown in Fig.~\\ref{fig:paramp}: CLSC vs P12 and P12nEL vs P12.}\n\\label{tab:diff}\n\\begin{tabular}{l c c c c c c | c c c c c c}\n\\hline\n\\hline\n& \\multicolumn{6}{c}{CLSC $-$ P12} & \\multicolumn{6}{c}{P12nEL $-$ P12} \\\\\n& \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty} & \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty} \\\\\n& 16th & 50th & 84th &  16th & 50th & 84th & 16th & 50th & 84th & 16th & 50th & 84th  \\\\\n\\hline\n$\\log(M_{\\ast}\/M_{\\sun})$ \t\t\t\t&{0.27}\t& {0.08}\t&{ -0.03}\t&{0.02}\t&{-0.02}\t& {-0.07}\t& 0.06\t& 0.00\t& -0.13\t& 0.08\t& 0.01\t& -0.02 \\\\\n$\\log[\\psi\/(M_{\\sun} \\mathrm{yr}^{-1})]$ \t&{ -0.14}\t& {-0.63}\t& {-2.23}\t&{0.03}\t&{ -0.16}\t& {-0.61}\t& {0.62}\t&{0.12}\t& -0.15\t&{0.35}\t& {0.06}\t& {-0.09} \\\\\n$\\hat{\\tau}_V$ \t\t\t\t\t\t&{ 0.28}\t& {-0.18}\t& {-1.62}\t&{0.26}\t& {-0.03}\t& {-0.41}\t& {0.42}\t& 0.07\t& {-0.12}\t& {0.17}\t& 0.01\t& {-0.16} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nIn Fig.~\\ref{fig:paramp}, we compare the constraints on stellar mass, SFR and optical depth of the dust derived for all 1048 galaxies in the 3D-HST photometric sample using the CLSC (top panels, in orange) and P12nEL (bottom panels, in green) model spectral libraries to those obtained using the more sophisticated P12 library. In all cases, the constraints are derived from fits of the 9-band photometry at observer-frame wavelengths between 0.35 and 3.6 $\\mu$m, as described in Section~\\ref{sec:fit}.\n\n\\paragraph*{Stellar mass.} The left-hand panels of Fig.~\\ref{fig:paramp} show the differences in the constraints derived on galaxy stellar mass. For each set of spectral libraries, each panel shows the comparison between the median likelihood estimates of stellar mass (colour-coded according to the number of galaxies falling into each bin of the diagram). The results show that stellar-mass estimates derived using the CLSC spectral library are systematically $\\sim0.08$ dex greater than those derived using the P12 library. This is because in the CLSC spectral library, simple exponentially declining SFHs tend to produce larger mass-to-light ratios and smaller specific SFRs than SFHs including late bursts of star formation. Such a trend can cause a bias in the mass estimate towards large values (although this depends on the allowed age range of the models; see below). Fig.~\\ref{fig:paramp} also does not reveal any systematic bias in the stellar masses obtained using the P12nEL rather than the original P12 spectral library, only a large scatter. This suggests that stellar-mass estimates are not strongly affected by the contamination of broad-band fluxes by emission lines, at least for masses $M_{\\ast}>10^9~\\hbox{$M_\\odot$}$.\n\n\\paragraph*{SFR.} The middle panels of Fig.~\\ref{fig:paramp} show the analog to the left-hand panels for the SFR. In this case, the constraints derived using the CLSC spectral library produce SFR estimates significantly smaller (by $\\sim 0.63$ dex) than those derived from the more sophisticated P12 library, again because the SFH prior favours low specific-SFR values. When comparing the constraints derived from the P12nEL and P12 spectral libraries, we see that neglecting nebular emission causes a slight bias towards larger SFRs ($\\sim 0.12$ dex). This is likely because the total (stellar$+$nebular) observed emission is interpreted as a larger amount of young stars when emission lines are not included in the broad band fluxes predicted by the models. The uncertainties are also slightly larger when the emission lines are not included in the models.\n\n\\paragraph*{Dust optical depth.} The right-hand panels of Fig.~\\ref{fig:paramp} pertain to the attenuation optical depth of the dust. In the CLSC spectral library, dust attenuation is computed using a two-component model with a fixed slope of the attenuation curve ($n=-0.7$) both in stellar birth clouds and in the diffuse ISM (as recalled in Section~\\ref{sec:basic}, the galaxy-wide attenuation curve then depends on the SFH). The P12nEL and P12 spectral libraries rely on the same two-component model, but with a steeper slope of the attenuation curve in stellar birth clouds ($n=-1.3$) and a random slope of the attenuation curve in the diffuse ISM ($n$ between $-0.4$ and $-1.1$). On the one hand, this difference would cause a bias in the estimate of the optical depth of the dust, giving a larger value when applying a shallow attenuation curve. On the other hand, the large mass-to-light ratios and small SFRs favoured by the CLSC spectral library tend to produce redder intrinsic SEDs and favour low dust attenuation. These two effects conspire to produce a large scatter ($\\pm$ 1 dex), and an offset ($\\sim -$0.18 dex) between the results derived using the CLSC and P12 spectral libraries in the upper right panel of Fig.~\\ref{fig:paramp}. The lower-right panel further shows that neglecting nebular emission in the P12 spectral library introduces a slight bias upward ($\\sim$ 0.07) in the derived attenuation optical depth with a large scatter, showing the strong degeneracy of this parameter with age, metallicity and SFR. We note that, for some dusty galaxies ($\\hbox{$\\hat{\\tau}_{V}$}>2.5$), the measurements of $\\hbox{$\\hat{\\tau}_{V}$}$ derived with the P12nEL and P12 spectral libraries are in good agreement. This is because nebular emission lines are strongly attenuated at high dust optical depths and therefore have a negligible influence on the broad-band fluxes. As a consequence, in the lower-right panel of Fig.~\\ref{fig:paramp}, the density along the identity relation is high when $\\hbox{$\\hat{\\tau}_{V}$}$ is large.\n\n\\paragraph*{} We conclude from the above analysis that the widely used, CLSC spectral library is not appropriate to extract reliable stellar masses and SFRs from photometric observations of distant galaxies such as those in the 3D-HST survey, for two main reasons: firstly, because the limited parameter range probed by this library cannot account for all observations (as discussed in Sections~\\ref{sec:mockobs} and \\ref{sec:fit}); and secondly, because this discrepancy implies important systematic biases in derived galaxy physical parameters, even if the resulting PDFs can appear artificially narrow (see Section~\\ref{sec:fit}). We note that the results presented here for the CLSC spectral library are consistent with the claim by \\citet[see also \\citealt{pforr2012}]{maraston2010} that models with exponentially declining SFHs overestimate the stellar mass and underestimate the SFR if galaxies are assumed to be at least $\\sim1\\,$Gyr old (see their section~3.2). The constraints on stellar mass, SFR and optical depth of the dust derived from 9-band photometry using the P12nEL and P12 spectral libraries are roughly similar, except for a slight bias in SFR toward large values when nebular emission is neglected. For reference, we list in Table~\\ref{tab:diff} the 16th, 50th and 84th percentiles of the distributions of differences between the median-likelihood estimates of stellar mass, SFR and optical depth of the dust (and associated uncertainties) derived using the CLSC and P12nEL spectral libraries with respect to those derived using the more sophisticated P12 library.\n\n\\subsection{Including emission-line constraints in the spectral fits}\n\\label{sec:resspec}\n\n\\begin{figure*}\n\\begin{center}\n\\vspace{-0.9cm}\n\\includegraphics[width=0.95\\textwidth]{.\/figures\/fig_07.png}\n\\vspace{-0.25cm}\n\\caption{Comparison between the constraints derived on stellar mass, SFR and optical depth of the dust for all 364 galaxies in the 3D-HST emission-line sample when fitting the photometry alone and when fitting both the photometry and emission-line EWs, using the P12 model spectral library. The format is the same as in Fig.~\\ref{fig:paramp}. Including the emission-line EWs in the fit does not affect strongly the best estimates, but improves considerably the uncertainties on all physical parameters.}\n\\label{fig:params}\n\\end{center}\n\\end{figure*}\n\n\\begin{table*}\n\\begin{center}\n\\caption{16, 50 and 84 percentiles of the distributions of the differences between best estimates and uncertainties of the stellar mass, SFR and attenuation optical depth of the dust as shown in Fig.~\\ref{fig:params}, comparing simultaneous fits to the photometry and emission-line EWs vs. fits to the photometry alone (using the P12 model spectral library in both cases).}\n\\label{tab:diffew}\n\\begin{tabular}{l c c c c c c}\n\\hline\n\\hline\n&  \\multicolumn{6}{c}{P12 (phot) $-$ P12 (phot $+$ EW)}\\\\\n&  \\multicolumn{3}{c}{bias} & \\multicolumn{3}{c}{$\\Delta$ uncertainty}\\\\\n&16th & 50th & 84th & 16th & 50th & 84th  \\\\\n\\hline\n$\\log(M_{\\ast}\/M_{\\sun})$ \t\t\t\t& 0.09\t& 0.01\t& -0.07\t& 0.07\t& 0.03\t& -0.01\\\\\n$\\log[\\psi\/(M_{\\sun} \\mathrm{yr}^{-1})]$ \t& 0.50\t& {0.13}\t& {-0.21}\t& {0.48}\t& {0.14}\t& 0.02\\\\\n$\\hat{\\tau}_V$ \t\t\t\t\t\t& {0.45}\t& {0.03}\t& {-0.23}\t& {0.35}\t& {0.06}\t& {-0.09}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_08.png}\n\\vspace{-0.5cm}\n\\caption{(a): comparison between the rest-frame emission EW of {\\hbox{H$\\alpha$}}  measured from grism spectroscopy and the EW predicted from fits of the 9-band photometry using the P12 model spectral library, for the 314 galaxies with \\hbox{H$\\alpha$}\\ measurements in the 3D-HST emission-line sample. For reference, the black solid line indicates the identity relation and the black dashed lines mark deviations by a factor of 2 ($\\pm0.3\\,$dex) between the measured and predicted EWs. (b): distribution of the difference between predicted and measured \\hbox{H$\\alpha$}\\ rest-frame EW. (c) and (d):  same as (a) and (b), but for the EW of \\hbox{[O\\,{\\sc iii}]}\\ for the 99 galaxies with \\hbox{[O\\,{\\sc iii}]}\\ measurements in the 3D-HST emission-line sample.}\n\\label{fig:ewmvsbfit}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_09a.png}\n\\includegraphics[width=0.475\\textwidth]{.\/figures\/fig_09b.png}\n\\vspace{-0.5cm}\n\\caption{Contribution by nebular emission to the WFC3-F140W magnitude of the best-fitting (P12) model for the galaxies in the 3D-HST emission-line sample as a function of the observer-frame line EWs directly measured in the grism spectra (left-hand panel) and as a function of stellar mass (right-hand panel). Crosses represent galaxies with measured {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} (235 galaxies at $0.8<z<1.4$), while circles represent galaxies with measured {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} (47 galaxies at $1.5<z<2.2$). The points are colour-coded according to star formation activity from low (red) to high (black) specific SFR. In the left-hand panel, the grey lines show the same relation considering the best-fitting model EWs instead of the observed ones. In the right-hand panel, grey symbols represent median and 16-to-84 percentiles of the contamination in bins ($\\approx 0.5$ dex wide) of stellar mass for {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} (squares) and for {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} (filled circles). The contamination of the emission lines in the broad-band WFC3-F140W filter increases from $\\approx 0.01$ to $\\approx 0.5\\,$mag as the stellar mass decreases.}\n\\label{fig:ewband}\n\\end{center}\n\\end{figure*}\n\nThe 3D-HST emission-line sample includes only those galaxies from the photometric sample with well-detected (S\/N~$>5$) emission lines, which allows us to test the effect of adding emission-line information when deriving constraints on physical parameters. In Fig.~\\ref{fig:params}, we compare (in a similar way to Fig.~\\ref{fig:paramp}) the constraints derived on stellar mass, SFR and optical depth of the dust for all 364 galaxies in this sample when fitting the photometry alone and when fitting both the photometry and emission-line EWs, using the P12 model spectral library. For the stellar mass (left-hand panel), there is no significant difference between the constraints derived in both ways and the uncertainties remain similar. The SFR (middle panel) appears to be slightly overestimated when the constraints on the emission-line EWs are not included in the fit (by $\\sim0.13$~dex). This is because emission-line EWs help break the degeneracy between star formation activity (i.e. specific SFR) and attenuation by dust (right-hand panel in Fig.~\\ref{fig:params}) at fixed SED shape. The fact that the SFR is biased high when fitting only the photometry can be considered as a consequence of the selection criteria of the emission line sample, because, for an emission line to be well detected in a low-resolution 3D-HST spectrum, attenuation by dust must be low. In the absence of emission-line information, $\\hbox{$\\hat{\\tau}_{V}$}$ is less well constrained and the corresponding PDF broadens to larger values (implying larger SFR at fixed observed colours). The introduction of line-EW constraints breaks the SFR-dust degeneracy in these lightly obscured galaxies therefore leads on average to slightly lower SFR estimates. Adding constraints on emission-line EWs reduces the uncertainties in both SFR and $\\hbox{$\\hat{\\tau}_{V}$}$ by $\\sim0.14$\\,dex and 0.06, respectively (Table~\\ref{tab:diffew}).\n\nIt is important to note that, for galaxies in the 3D-HST emission-line sample, the emission-line strengths predicted by the P12 model providing the best fit to 9-band photometry alone are in fair agreement with direct EW measurements from grism spectroscopy. This is illustrated by Fig.~\\ref{fig:ewmvsbfit}, in which we compare the rest-frame EWs of {\\hbox{H$\\alpha$}} (Fig.~\\ref{fig:ewmvsbfit}a) and {\\hbox{[O\\,{\\sc iii}]}} (Fig.~\\ref{fig:ewmvsbfit}c) emission lines as measured from the grism spectra with the predictions obtained when fitting the photometry alone. The distributions of the differences between predicted and measured EWs are plotted in Figs~\\ref{fig:ewmvsbfit}b and \\ref{fig:ewmvsbfit}d for {\\hbox{H$\\alpha$}} and {\\hbox{[O\\,{\\sc iii}]}}, respectively. Fig.~\\ref{fig:ewmvsbfit} shows that, for 52 per cent (20 per cent) of the galaxies with spectroscopic {\\hbox{H$\\alpha$}} ({\\hbox{[O\\,{\\sc iii}]}}) emission-line measurements, the EWs inferred from fits of 9-band photometry agree to within a factor of 2 with those measurements. Considering only galaxies with EW({\\hbox{H$\\alpha$}}) larger than 100\\,{\\AA} makes this fraction rise to 73 per cent. This is not surprising, as the P12 model spectral library accounts for contamination of broad-band fluxes by nebular emission, which increases and hence is more easily identifiable in the most actively star-forming galaxies. We note that, despite this good agreement, 9-band photometric fits tend to systematically overestimate EW(\\hbox{H$\\alpha$}) in Fig.~\\ref{fig:ewmvsbfit}a relative to direct measurements. This is again a consequence of the SFR-dust degeneracy described in the previous paragraph. There is no systematic bias in the predicted EW(\\hbox{[O\\,{\\sc iii}]}) but a larger scatter, because the strength of this line is strongly affected by the metallicity of the gas and is not easily constrained by purely photometric fits.\n\nTo further quantify how emission lines contaminate observed broad-band fluxes, we record, for each galaxy in the 3D-HST emission-line sample, the contribution by nebular emission to the WFC3-F140W magnitude of the best-fitting P12 model (as derived when including the constraints from both 9-band photometry and EW measurements). This is shown in Fig.~\\ref{fig:ewband} (left-hand panel) against the line EWs directly measured in the grism spectra. For galaxies at redshifts $0.8 < z < 1.4$, both {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} fall in the WFC3-F140W filter. The combined observer-frame EW of these lines is plotted in Fig.~\\ref{fig:ewband} (crosses). In the same way, we plot the combined observer-frame EW of {\\hbox{H$\\beta$}} and {\\hbox{[O\\,{\\sc iii}]}} for galaxies in the range $1.5 < z < 2.2$ (empty circles). Each galaxy is colour-coded according to star formation activity, from low (red) to high (black) specific SFR. As expected, the contamination of broad-band fluxes by emission lines increases with the level of star formation activity. The bulk of the sample shows observed emission-line EWs between 200 and 700~\\AA, which corresponds roughly to a contamination of $0.1\\,$magnitudes in the F140W broad-band magnitude. We also show for reference the relation obtained when considering the EWs of the best-fitting model instead of the observed ones (grey lines). In the right-hand panel of Fig.~\\ref{fig:ewband}, we plot the emission-line contamination of the WFC3-F140W broad-band magnitude as a function of stellar mass. For galaxies at $0.8 < z < 1.4$, where {\\hbox{H$\\alpha$}} and {\\hbox{[S\\,{\\sc ii}]}} are both sampled in the band, the contamination decreases from $\\approx 0.1$ to $\\approx 0.02\\,$magnitudes as the stellar mass increases from $\\approx 10^{9.5}$ to $\\approx 10^{11}$  M$_{\\odot}$.  At higher redshift, where {\\hbox{[O\\,{\\sc iii}]}} and {\\hbox{H$\\beta$}} are sampled in the band, the contamination is slightly larger because the SFR is on average larger at higher than at lower redshifts \\citep{noeske2007}.\n\n\\section{Implications for the scatter in the star-formation main sequence}\n\\label{sec:ms}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_10.png}\n\\vspace{-0.5cm}\n\\caption{Constraints on stellar mass and SFR (star-formation main sequence) derived in Section~\\ref{sec:resphot} from fits of the 9-band 3D-HST photometry using the CLSC (left-hand panels), P12nEL (middle panels) and P12 (right-hand panels) model spectral libraries, in different redshift ranges (rows). In each panel, we show the median and 16-to-84 percentile range of the SFR in bins of stellar mass (coloured squares) and linear least-squares fit to all points (black solid line; red solid line in the right-hand panels). For each redshift bin, we report in the 2 left-most panels the fit obtained in the right-hand panel using the P12 spectral library (red dashed line). The use of the CLSC spectral library, which relies on oversimplified prescriptions of SFHs and dust attenuation, cause the main sequence to lie, at all redshifts, significantly below the relation obtained using the more realistic P12 spectral library. The relative zero-point of the main sequence derived using the P12nEL spectral library is shifted towards higher SFRs compared to that obtained using the original P12 spectral library. The scatter about the derived star formation main sequence is reduced including the emission lines as contaminants of the broad-band fluxes. In the right-hand panel, we also report for comparisons the fits to the main-sequence by \\protect\\citet[black long-dashed line]{karim2011} and \\protect\\citet[black dot-dashed line]{whitaker2012} in the mass ranges allowed by their samples.}\n\\label{fig:msall}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{.\/figures\/fig_11.png}\n\\vspace{-0.5cm}\n\\caption{The main sequence derived from the 3D-HST emission-line sample when fitting the photometry alone (left-hand panels; analog of the right-hand panels of Fig.~\\ref{fig:msall}) and when conbining the photometry with the emission-line EWs detected in the grism spectra (right-hand panels). The format is the same as in Fig.~\\ref{fig:msall}. Estimates are extracted using the P12 spectral library. For this sample, the estimates of SFR are in general less uncertain than for the photometric sample given the strength of the emission lines. This contributes to reducing the scatter. Including the information from the emission-line EWs further reduces the scatter at all redshifts.}\n\\label{fig:ms}\n\\end{center}\n\\end{figure*}\n\nWe have shown in the previous sections that interpreting the same 3D-HST observations of distant galaxies using model spectral libraries of different levels of sophistication can lead to different constraints on galaxy stellar masses and SFRs. We also showed that accounting for nebular emission is important to interpret (even purely photometric) observations of distant star-forming galaxies. Here we explore how the biases introduced by the use of oversimplified model spectral libraries might affect the interpretation of the correlation between stellar mass and SFR observed for star-forming galaxies, commonly referred to as the star-formation main sequence. It is now established that this `main sequence' is in place both in the local Universe and at high redshift (up to at least $z\\approx3$; \\citealt{brinchmann2004,noeske2007,elbaz2007,daddi2007,karim2011,reddy2012,whitaker2012}), but the normalization, slope and scatter of the relation and the dependence of these on redshift, are still under debate. The normalization of the main sequence appears to increase with time (from redshift 0 to roughly 3) in such a way that, at fixed stellar mass, galaxies at higher redshift form stars at higher rates. The slope also appears to change with redshift, as shown by \\cite{karim2011} and \\cite{whitaker2012}. The intrinsic scatter should contain important information about the physical processes that drive the formation of galaxies \\citep{dutton2010}, but it is always contaminated by observational uncertainties and thus is hard to quantify \\citep{guo2013}.\n\nIn this Section, we assess how the use of different model spectral libraries and the availability or not of emission-line measurements can affect the determinations of the shape and scatter of the main sequence at $0.7<z<1.8.$\\footnote{Because of the lack of statistics, we limit the analysis of the star-formation main sequence to $z=1.8$. We note also that the samples we use reach a magnitude of $H=23$ (WFC3-F160W) and thus are complete only to a stellar mass of $\\approx10^{10}\\hbox{$M_\\odot$}$ at $z=1$ ($\\approx10^{10.7}\\hbox{$M_\\odot$}$ at $z=2$). Such completeness limits do not allow us to draw reliable conclusions about the absolute slope or the zero point of the main sequence. It is thus beyond the scope of this paper to assess the evolution of the main sequence in redshift.} We use the estimates of stellar mass and SFR derived in Section~\\ref{sec:results} to draw the star-formation main sequence in four redshift bins for both the 3D-HST photometric and the emission-line samples.\\footnote{We include only galaxies with specific SFR larger than 0.01 Gyr$^{-1}$.} Fig.~\\ref{fig:msall} shows the results obtained when adopting the constraints on stellar mass and SFR derived in Section~\\ref{sec:resphot} from fits of the 9-band 3D-HST photometry using the CLSC (left-hand panels), P12nEL (middle panels) and P12 (right-hand panels) model spectral libraries. In each panel, on top of the single points pertaining to individual galaxies, we plot the median and 16-to-84 percentile range of the SFR in bins of stellar mass (coloured squares) and a linear least-squares fit to all points (black solid line in the 2 left-most panels; red solid line in the right-hand panel). For the purpose of comparison, for each redshift bin, we report in the 2 left-most panels the fit obtained in the right-hand panel using the P12 spectral library (as a red dashed line). Fig.~\\ref{fig:msall} confirms our finding in Section~\\ref{sec:resphot} above that the use of the CLSC spectral library, which relies on oversimplified prescriptions of SFHs and dust attenuation, can severely bias estimates of galaxy stellar masses and SFRs from photometric data. Such biases cause the main sequence to lie, at all redshifts, significantly below the relation obtained using the more realistic P12 spectral library. The primary cause for this can be traced to the prior distribution of the specific SFR, which in the CLSC spectral library is relatively narrow with a peak around $\\log(\\psi_{S}\/\\mathrm{Gyr}^{-1})\\sim -0.75$.\n\nThe middle panels of Fig.~\\ref{fig:msall} show the results obtained when including more sophisticated treatments of star formation and chemical enrichment histories and attenuation by dust, but without accounting for nebular emission (P12nEL spectral library). This produces a stronger correlation between stellar mass and SFR than obtained with the CLSC library, but with a very large scatter spanning almost the entire range of the prior distribution of the specific SFR ($-2<\\log(\\psi_{S}\/\\mathrm{Gyr}^{-1})<2$). This is caused by the strong degeneracy between SFR and dust attenuation, which affects the uncertainties in SFR estimates at all redshifts (Section~\\ref{sec:resphot}). We also find that the relative zero-point of the main sequence derived using the P12nEL spectral library is shifted towards higher SFRs compared to that obtained using the original P12 spectral library. This is in agreement with the result of Fig.~\\ref{fig:paramp} that the SFR is slightly overestimated when the models do not account for the contamination of broadband fluxes by emission lines.\n\nAppealing to the original P12 library to include the contamination of broadband fluxes by nebular emission reduces the scatter about the derived star formation main sequence ($\\approx0.7$~dex), especially at those redshifts where the {\\hbox{H$\\alpha$}} line falls in one of the observer-frame near-infrared bands ($0.7<z<1.5$); at $z>1.5$, the scatter remains very large ($\\approx1.2$~dex) because the parameters are not as tightly constrained.\n\nThe star-formation main sequence has been studied in the past using different approaches tailored to specific redshift ranges and datasets. For comparison, we plot in the right-hand panels of Fig.~\\ref{fig:msall} the main-sequence fits derived by \\citet[black long-dashed line]{karim2011} and \\citet [black dot-dashed line]{whitaker2012} over a redshift range similar to that sampled by the 3D-HST survey. In both studies, stellar masses are derived using a classical approach (exponentially declining $\\tau$-models, possibly allowing for very young ages to widen the specific-SFR and mass-to-light-ratio priors), while SFRs are derived in different ways: \\cite{karim2011} estimate galaxy SFRs from stacked 1.4 GHz data using the prescription by \\cite{bell2003}; and \\cite{whitaker2012} from ultraviolet and infrared data using the prescription by \\cite{kennicutt1998}. The main sequences obtained in these two studies are both in good agreement with that derived from 3D-HST data using the P12 library in Fig.~\\ref{fig:msall}. An advantage of the approach presented here is the ability to derive, for each galaxy, simultaneous constraints on the stellar mass and SFR from a library of star formation and chemical enrichment histories, which allows one to also constrain other physical parameters (age, dust attenuation, metallicity).\n\nWe can now turn to the 3D-HST emission-line sample and assess whether the addition of spectroscopic emission-line measurements can improve determinations of the star formation main sequence with the P12 model spectral library. Fig.~\\ref{fig:ms} shows the main sequences derived for this sample in the same redshift bins as for the photometric sample in Fig.~\\ref{fig:msall}, both from fits of the photometry alone (left-hand panels; this is the analog of the right-hand panels of Fig.~\\ref{fig:msall}, but for the sub-sample of 3D-HST galaxies with emission-line measurements) and from combined fits of the photometry and emission-line EWs (right-hand panels). Even when fitting the photometry alone, the scatter in Fig.~\\ref{fig:ms} is, to some extent, reduced ($\\approx 0.5$~dex) compared to that in the right-hand panels of Fig.~\\ref{fig:msall}. This is because the average uncertainty in the SFR derived from 9-band photometry  in the emission-line sample (which includes only galaxies with well-detected emission lines) is slightly smaller than that for the entire photometric sample and so is the scatter in the inferred relation between stellar mass and SFR.  When we add the constraints on the emission-line EWs, the uncertainty on the SFR is further reduced (Section~\\ref{sec:resspec}) and the main sequence becomes tighter at all redshifts (with a scatter $\\approx 0.4$ dex; right-hand panels in Fig.~\\ref{fig:ms}). The slight bias towards lower SFRs at large stellar masses when including the emission-line EWs in the fits is associated to the selection of the sample as discussed in Section~\\ref{sec:resspec}.\n\n\\section{Summary and conclusions}\n\\label{sec:summary}\n\nInterpreting ultraviolet-to-infrared observations of distant galaxies in terms of constraints on physical parameters -- such as stellar mass, SFR and attenuation by dust -- requires spectral synthesis modelling. In this paper, we have investigated how increasing the level of sophistication of standard simplifying assumptions of such models can improve estimates of galaxy physical parameters. To achieve this, we have compiled a sample of 1048 galaxies at $0.7<z<2.8$ with accurate photometry at rest-frame ultraviolet to near-infrared wavelengths from the 3D-HST survey, 364 of which have strong enough optical-line emission to be measured with good S\/N from grism spectroscopy. Such spectroscopic information is important to allow us to quantify, from a pure observational point of view, the contamination of broadband photometric fluxes by emission lines. We have compared the SEDs of the galaxies in this sample with those from different model spectral libraries (of different levels of sophistication) to derive bayesian estimates of stellar mass, star formation rate and optical depth of the dust. We have found the following.\n\\begin{enumerate}\n\\item Classical spectral libraries, in which galaxy SFHs are assumed to be exponentially declining functions of time ($\\tau$-models), cannot reproduce the observed SEDs of all galaxies in the sample, because such libraries do not account for the colours of galaxies with old populations undergoing a new starburst. \\cite{maraston2010} and \\cite{pforr2012} introduced exponentially {\\em rising} SFHs to overcome this limitation, but such parametrization cannot reproduce the potential rise and fall of galaxy SFHs. The more realistic SFHs implemented in the P12 library \\citep{pacifici2012}, which rely on the versatile post-treatment of cosmological simulations, allow us to adequately reproduce the observed colours of all galaxies in the 3D-HST sample.\n\\item As a result, stellar masses derived using classical spectral libraries (generally associated with an oversimplified prescription for dust attenuation and the neglect of nebular emission) tend to be systematically overestimated (by a median $\\sim 0.1$ dex), and SFRs systematically underestimated (by a median $\\sim 0.6\\,$dex), relative to the values derived adopting the more realistic P12 spectral library (which includes a sophisticated and versatile prescription for dust attenuation as well as a careful account for nebular emission). Moreover, the uncertainties derived using classical spectral libraries can be misleadingly small because of the inappropriate sampling of the physical parameter space (Section~\\ref{sec:fit}).\n\\item Neglecting the contamination of broadband fluxes by emission lines, while using the same SFHs and dust prescription as in the P12 spectral library, would lead to systematic SFR overestimates (by a median $\\sim 0.1\\,$dex), leaving stellar-mass estimates nearly unaffected. We note that galaxies in the 3D-HST sample have masses $M \\ga10^9\\hbox{$M_\\odot$}$. In lower-mass galaxies with high specific SFR, we expect SFR and stellar-mass estimates obtained when neglecting nebular emission to be more severely biased than found here \\citep[e.g.][]{atek2011,stark2013}.\n\\item The simultaneous fit of photometric broad-band fluxes and emission-line EWs, as enabled by the \\cite{pacifici2012} approach, helps break a fundamental degeneracy between SFR and dust attenuation, which reduces considerably the uncertainties in these derived parameters (by $\\sim0.1$--0.5\\,dex; the uncertainty in stellar-mass estimates dropping by only $\\sim0.05\\,$dex).\n\\end{enumerate}\n\nThe results obtained in this paper reveal the importance of choosing appropriate spectral models to interpret deep galaxy observations. In particular, the biases introduced by the use of classical spectral libraries to derive estimates of SFR, stellar mass and attenuation by dust in distant galaxies can significantly affect the interpretation of standard diagnostics diagrams of galaxy evolution. These include, for example, the galaxy stellar-mass function \\citep[see also][]{marchesini2009,muzzin2013}; the main sequence of star-forming galaxies and its intrinsic scatter; and the relations between stellar mass, SFR and other galaxy physical parameters, such as stellar age and metallicity \\citep[e.g.][]{mannucci2010}. In this context, the spectral library developed by \\cite{pacifici2012} offers the possibility to interpret these and other fundamental diagnostics on the basis of more realistic, and at the same time more versatile models. This is all the more valuable in that the approach can be straightforwardly tailored to the analysis of any combination of photometric and spectroscopic observations of galaxies at any redshift.\n\n\n\\section*{acknowledgements}\n\nWe thank the Referee for the very useful report that helped improving the Paper. This work was funded in part by the Marie Curie Initial Training Network ELIXIR of the European Commission under contract PITN-GA-2008-214227 and in part by the KASI-Yonsei Joint Research Program for the Frontiers of Astronomy and Space Science funded by the Korea Astronomy and Space Science Institute. E.d.C. acknowledges funding through the ERC grant `Cosmic Dawn'. S.C. acknowledges support from the European Research Council via an Advanced Grant under grant agreement no. 321323-NEOGAL. S.K.Y was supported by the National Research Foundation of Korea (Doyak 2014003730).\n\n\n\\defAJ{AJ}\n\\defARA\\&A{ARA\\&A}\n\\defApJ{ApJ}\n\\defApJ{ApJ}\n\\defApJS{ApJS}\n\\defAp\\&SS{Ap\\&SS}\n\\defA\\&A{A\\&A}\n\\defA\\&A~Rev.{A\\&A~Rev.}\n\\defA\\&AS{A\\&AS}\n\\defMNRAS{MNRAS}\n\\defPASP{PASP}\n\\defPASJ{PASJ}\n\\defQJRAS{QJRAS}\n\\defNature{Nature}\n\n\\defAstrophys.~Lett.{Astrophys.~Lett.}\n\\defAAS{AAS}\n\\let\\astap=A\\&A\n\\let\\apjlett=ApJ\n\\let\\apjsupp=ApJS\n\\let\\applopt=\\ao\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}