document: in the 1970s many - body theory flourished . 
   a topic that attracted much attention was the many - body effects on xps ( x - ray photoelectron spectroscopy ) spectra , from deep core levels in metals@xcite . 
   related effects were singularities that appeared near edges in absorption and emission spectra@xcite . 
   the cb ( conduction - band ) electrons can have many different effects on the xps peak from the core level . if the excitation could be considered adiabatic the screening of the core hole by the cb electrons would lead to a shift of the peak to higher kinetic energy ( lower binding energy of the core level ) . 
   now the excitation is not adiabatic , the excitation is swift . 
   the excitation frequency is large compared to the frequency components taking part in the screening of the core hole . 
   this has the effect that the cb electrons are shaken up and are left in a nonequilibrium state . 
   this is analogous to the following case : assume that a cork is floating in a glass of water . 
   if we remove the cork very slowly the water will be left very quiet and without any ripples . 
   if we on the other hand remove it very briskly the water will be left in an upset state with many surface waves excited . in a metal 
   the water waves correspond to plasmons , collective excitations of the cb electrons . 
   the shake - up effects will show up as plasmon replicas in the spectrum ; the main peak will be at the adiabatic position ; the first replica , a smaller copy of the main peak , will be at a shifted position to lower kinetic energy where the shift is equal to the plasma frequency ; the second replica is even smaller and shifted with two plasma frequencies ; this goes on until further peaks are absorbed by the background . 
   the main peak corresponds to the system left in quasi equilibrium(fully equilibrium would demand that the core hole were filled by one of the electrons ) , the second to a state where one plasmon is excited , the third to a state where two plasmons are excited and so on . 
   since all plasmons do not have identical energies , the plasmon curve shows dispersion , the replicas are not completely identical smaller versions of the main peak .    in a metal also 
   single - particle excitations can take place . 
   these electron - hole pair excitations form a continuum , starting from zero frequency and upwards . 
   these excitations lead to a deformation , including a low - energy tail , of both the main peak and the plasmon replicas . 
   furthermore the finite life - time of the core hole causes a lorentzian broadening of all peaks and experimental uncertainties give a gaussian broadening . 
   the low - energy tailing is a characteristic of a metallic system , i.e. a system where the chemical potential is inside an energy band and not in a band - gap , and can e.g. be used by experimentalists to find out where in a complex sample the core hole is situated . 
   several parameters were introduced by doniach and sunji@xcite to characterize the xps line shapes and are still broadly used by experimentalists in the fitting procedure for xps spectra .    in the present work we address pristine and doped graphene . in the pristine case 
   the chemical potential is neither inside an energy band nor in a band - gap . 
   the fermi surface is just two points in the brillouin zone . 
   this makes this system special . 
   as we will see there is still a low - energy tailing . in the doped case 
   the chemical potential is inside an energy band and we would expect to find , and find , a tailing . 
   however , the 2d ( two - dimensional ) character of the system means that the collective excitations are 2d plasmons with a completely different dispersion than in the ordinary 3d metallic systems . 
   the 2d plasmons give contributions that start already from zero frequency and upwards . 
   this means that they contribute to the tail and no distinct plasmon replicas are distinguishable . 
   making any quantitative interpretation of graphene core - hole spectra using doniach and sunji fitting of the spectra is not feasible . 
   it is more reasonable to calculate the spectra . 
   one purpose of this work is to provide the reader with the tools needed for such a calculation . 
   the material is organized in the following way : in sec . [ 
   the ] we show how the core - hole spectra are derived for a 2d system . 
   the results and comparison with experimental spectra are presented in sec . 
   finally , we end with a brief summary and conclusion section , sec . 
   we use a model that is based on the one used by langreth@xcite for the core - hole problem in the 1970s and here modified and extended to fit our problem . 
   we have earlier@xcite with success used another modified version for the problem of exciton annihilation in quantum wells .    in the excitation process 
   the photoelectron leaves the system and a core hole is left behind . 
   the shape of the xps spectrum depends on how fast the process is . 
   if it is very slow one may use the adiabatic approximation in which one assumes that the electrons in the system have time to relax around the core hole during the process . 
   when we derive the xps line shape we assume that the excitation process is very fast ; we use the sudden approximation in which the core - hole potential is turned on instantaneously . 
   the electrons have not time , during the process , to settle down and reach equilibrium in the potential from the core hole . 
   this results in shake - up effects in the form of single particle ( electron - hole pair ) excitations and collective ( plasmon ) excitations . 
   the electrons contributing to the shake - up effects are the electrons in the valence and conduction bands . from here on we refer to them as the electrons . 
   we use the assumption that the core hole does not recoil in the shake - up process and that there are no excitations within the core . 
   we approximate the core - hole potential with a pure coulomb potential . 
   the following model hamiltonian for the system is used : @xmath0 where @xmath1 , @xmath2 , @xmath3 , @xmath4 , and @xmath5 are the core - hole energy , the creation operator for the core hole , the annihilation operator for the core hole , the interaction potential for the interaction between the core hole and the electrons , and the hamiltonian for the electrons , respectively . 
   the operators @xmath4 and @xmath5 contain creation and annihilation operators for the electrons and no core - hole operators . 
   the core - hole number operator is @xmath6 . 
   it commutes with the hamiltonian , i.e. , @xmath7 = 0 $ ] .    in the ground state of the system 
   there is no core hole and the electrons are in their ground state . 
   let us introduce two hamiltonians @xmath8 where the first is the hamiltonian before the core hole has been created and the second the hamiltonian after the creation . 
   let @xmath9 denote the ground state of @xmath10 . 
   it is also the ground state of @xmath11 . 
   then we have @xmath12 where @xmath13 is the ground state energy of the electron system . from now on we 
   drop the subscript @xmath14 on the core - hole operators and introduce the functions @xmath15 these functions are connected to the time ordered and retarded green s functions , according to @xmath16 and @xmath17\left| 0 \right\rangle \\ 
   \quad \quad   =   - i\theta \left ( t \right)\left [ { { g^ > } ( t ) - { g^ < } ( t ) } \right ] . 
   \end{array } \label{equ6}\ ] ]    the fourier transformed versions , @xmath18 have a direct physical meaning . 
   their sum is the spectral function , @xmath19 is the density of states for an occupied core - hole state , and @xmath20 for an unoccupied . 
   note that the creation and annihilation operators are for holes . 
   we want the density of states for a core electron . 
   thus we need to calculate @xmath20 . 
   we first determine the time dependent form . 
   we have @xmath21t/\hbar } } { c^\dag } \left| 0 \right\rangle \\   = { e^{ie_0^et/\hbar } } \left\langle 0 \right|c{e^ { - i\varepsilon nt/\hbar } } { e^ { - i\left [ { nv + { h_e } } \right]t/\hbar } } { c^\dag } \left| 0 \right\rangle \\   = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } \left\langle 0 \right|c{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } { c^\dag } \left| 0 \right\rangle \\   = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } { } _ e\left\langle 0 \right|{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } { \left| 0 \right\rangle _ e}{}_{ch}\left\langle 0 \right|1 - n{\left| 0 \right\rangle _ { ch}}\\   = { e^{i\left ( { e_0^e - \varepsilon } \right)t/\hbar } } \left\langle 0 \right|{e^ { - i\left [ { v + { h_e } } \right]t/\hbar } } \left| 0 \right\rangle , \end{array } \label{equ8}\ ] ] where we have let the operators operate to the left and right . 
   we have made use of the general relation @xmath22}}$ ] , for operators @xmath23 and @xmath24 , and that the core - hole number operator commutes with all terms of the hamiltonian . 
   it is now time to define our @xmath4 and the rest of the hamiltonian . 
   the excitation spectrum of the electrons is given by the dynamical structure factor , which is related to the dielectric function according to @xmath25 where @xmath26 is the 2d fourier transform of the coulomb potential . 
   the structure factor is nonzero in two regions ; one region is where electron - hole pairs are excited ; the other is on the plasmon dispersion curve . 
   all these excitations are bosons . 
   we assume that the bosons are independent . 
   let us for simplicity start by assuming that the bosons have a distinct dispersion curve , @xmath27 . 
   this is the case for plasmons and phonons . 
   later we will generalize this to include contributions from the electron - hole pair continuum . 
   we now get the following hamiltonian : @xmath28 where @xmath23 is the area of the 2d system and @xmath29 the coupling constant between the core hole and the boson excitation . 
   the operators @xmath30 and @xmath31 are boson creation and annihilation operators , respectively , obeying the commutation relation @xmath32 = { \delta _ { { \bf{q}},{\bf{q}}'}}$ ] and @xmath33 are the density operators , where @xmath34 denotes the position of the core hole . we treat the core hole as a classical particle , i.e. , the core - hole operators are c - numbers . 
   this is why the density operators do not contain any creation and/or annihilation operators . 
   let us here take the opportunity to derive the results using the adiabatic approximation . 
   since there is no kinetic energy terms for the core hole the hamiltonian can be diagonalized . 
   this is achieved by using the following unitary transformation : @xmath35 , \label{equ12}\ ] ] where @xmath36 letting @xmath37 we find that @xmath38 = { f^\dag } ( - { \bf{q } } ) = f({\bf{q}})$ ] and @xmath39 = { f^\dag } ( { \bf{q}})$ ] . 
   this gives @xmath40    the transformation of the hamiltonian gives @xmath41 } \\ 
   - \frac{1}{{{a^{{1 \mathord{\left/   { \vphantom { 1 2 } } \right . 
   \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } \left ( { { c_{\bf{q } } } + f({\bf{q } } ) + c _ { - { \bf{q}}}^\dag   + { f^\dag } ( - { \bf{q } } ) } \right)\\   = \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag { c_{\bf{q } } } + { 1 \mathord{\left/   { \vphantom { 1 2 } } \right .   \kern-\nulldelimiterspace } 2 } } \right ) }   - \frac{1}{{{a^{{1 \mathord{\left/   { \vphantom { 1 2 } } \right .   \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } \left ( { { c_{\bf{q } } } + c _ { - { \bf{q}}}^\dag } \right)\\   + \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag f({\bf{q } } ) + { c_{\bf{q}}}{f^\dag } ( { \bf{q } } ) + { f^\dag } ( { \bf{q}})f({\bf{q } } ) } \right ) } \\   - \frac{2}{{{a^{{1 \mathord{\left/   { \vphantom { 1 2 } } \right . 
   \kern-\nulldelimiterspace } 2}}}}}\sum\limits_{\bf{q } } { g({\bf{q}})\rho _ { ch}^\dag ( { \bf{q } } ) } f({\bf{q } } ) . 
   \end{array } \label{equ16}\ ] ] substituting for the expression of @xmath42 we find @xmath43    thus @xmath44 where @xmath45 we see that the transformation of the hamiltonian in eq.([equ10 ] ) has the effect that the interaction term is changed into a constant energy term . 
   the interactions with the core hole produce an energy shift of the ground state . 
   this is the relaxation energy ; the gain in energy when the electrons relax around the core hole . 
   this is the shift of the xps peak ( towards higher kinetic energy ) one would get in the adiabatic approximation . 
   we now return to the sudden approximation and our green s function @xmath46{t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle $ ] . 
   let us insert the identity on both sides of the exponential inside the ground state matrix element . 
   @xmath47{t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } u\left| 0 \right\rangle .   \end{array } \label{equ20}\ ] ] now , since there is no boson excited in the ground state we may write @xmath48{t \mathord{\left/   { \vphantom { t \hbar } } \right .   \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle   = { e^{i\left ( { \sum\limits_{\bf{q } } { { \textstyle{1 \over 2}}\hbar { \omega _ { \bf{q } } } } } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle \\ 
   = { e^{ie_0^h{t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle . 
   \end{array } \label{equ21}\ ] ] thus we have @xmath49{t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } u{e^{i\left [ { \sum\limits_{\bf{q } } { \hbar { \omega _ { \bf{q}}}\left ( { c_{\bf{q}}^\dag { c_{\bf{q } } } + { 1 \mathord{\left/   { \vphantom { 1 2 } } \right .   \kern-\nulldelimiterspace } 2 } } \right ) } } \right]{t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } \left| 0 \right\rangle \\ 
   = { e^ { - i\left ( { \varepsilon   + \delta \varepsilon } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } \left\langle 0 \right|{u^\dag } \left ( 0 \right)u\left ( t \right)\left| 0 \right\rangle , \end{array } \label{equ22}\ ] ] where @xmath50 and we see that @xmath51 is also a unitary transformation . now , from eqs.([equ12 ] ) and ( [ equ23 ] ) we find @xmath52\;\\   = \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } ( { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } - c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } ) } \right]\\   = \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } ( - c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } + { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } ) } \right ] . \end{array } \label{equ24}\ ] ]    if we now once again make use of the relation @xmath22}}$ ] we find @xmath53\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad   \times \exp \left [ { { \textstyle{1 \over 2}}\sum\limits_{{\bf{k}},{\bf{l } } } { { f^\dag } ( { \bf{l } } ) } { f^\dag } ( { \bf{k}}){e^ { - i{\omega _ { \bf{l}}}t}}{e^{i{\omega _ { \bf{k}}}t}}\underbrace { \left [ { c _ { - { \bf{l}}}^\dag , { c_{\bf{k } } } } \right ] } _ { - { \delta _ { { \bf{k } } , - { \bf{l } } } } } } \right]\\   = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad   \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { f^\dag } ( - { \bf{q } } ) } \right]\\   = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad   \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } f({\bf{q } } ) } \right]\\   = \exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad   \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right ] . 
   \end{array } \label{equ25}\ ] ]    we also need @xmath54\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \quad \quad   \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right ] . 
   \end{array } \label{equ26}\ ] ] substituting the results of eqs.([equ25 ] ) and ( [ equ26 ] ) in eq.([equ22 ] ) gives @xmath55\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\ \quad \quad   \times \exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\\ \quad \quad   \times \exp \left [ { \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } { c_{\bf{q}}}{e^{i{\omega _ { \bf{q}}}t } } } \right]\exp \left [ { - { \textstyle{1 \over 2}}\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\left| 0 \right\rangle . \end{array } \label{equ27}\ ] ] the factors that do not contain boson operators can be taken outside . 
   then we can let the left most factor operate to the left . 
   it produces a factor of unity . 
   the rightmost factor may operate to the right and it also produces a factor of unity . 
   we are left with @xmath56 here we make use of the relation @xmath57}}$ ] and find @xmath58\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\   \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right]\\   \times \exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\   \times \exp \left\ { { \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } , - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right ] } \right\}\left| 0 \right\rangle \\   = \exp \left [ { - i\left ( { \varepsilon   + \delta \varepsilon } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right .   \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\   \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\   \times \exp \left\ { { \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } , - \sum\limits_{\bf{q } } { { f^\dag } ( { \bf{q } } ) } c _ { - { \bf{q}}}^\dag { e^ { - i{\omega _ { \bf{q}}}t } } } \right ] } \right\}\left| 0 \right\rangle \\ 
   = \exp \left [ { - i\left ( { \varepsilon   + \delta \varepsilon } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right .   \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\   \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\\   \times \exp \left\ { { \sum\limits_{{\bf{k}},{\bf{l } } } { f({\bf{k}}){f^\dag } ( { \bf{l}}){e^ { - i{\omega _ { \bf{l}}}t } } } \underbrace { \left [ { { c _ { - { \bf{k}}}},c _ { - { \bf{l}}}^\dag } \right]}_{{\delta _ { { \bf{k}},{\bf{l } } } } } } \right\}\left| 0 \right\rangle \\   = \exp \left [ { - i\left ( { \varepsilon   + \delta \varepsilon } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right .   \kern-\nulldelimiterspace } \hbar } } \right]\exp \left [ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } \right]\\   \times \exp \left [ { \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2}{e^ { - i{\omega _ { \bf{q}}}t } } } } \right]\\   \times \left\langle 0 \right|\exp \left [ { - \sum\limits_{\bf{q } } { f({\bf{q } } ) } { c _ { - { \bf{q } } } } } \right]\left| 0 \right\rangle \\   = { e^ { - i\left ( { \varepsilon   + \delta \varepsilon } \right){t \mathord{\left/   { \vphantom { t \hbar } } \right . 
   \kern-\nulldelimiterspace } \hbar } } } { e^ { - \sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2 } } } } { e^{\sum\limits_{\bf{q } } { { { \left| { f({\bf{q } } ) } \right|}^2}{e^ { - i{\omega _ { \bf{q}}}t } } } } } . 
   \end{array } \label{equ29}\ ] ]    thus we have @xmath59    we want the fourier transformed version . 
   it is @xmath60 where the so - called satellite generator is @xmath61    $ ] for doped graphene . 
   we see the typical 2d plasmon dispersion . 
   the plot is for the doping density @xmath62 but the result is very similar for any finite doping density . , width=226 ]    in the unperturbed case , i.e. , when there is no interaction with the core hole we have @xmath63 so we can write @xmath64 where @xmath65 now we generalize this result to include all excitation processes in our system . 
   we first rewrite eq . 
   ( [ equ35 ] ) @xmath66 where @xmath67 is the boson propagator . 
   now the boson propagator is replaced by @xmath68 times the dynamical structure factor of eq.([equ9 ] ) and the coupling constant by @xmath69 . 
   thus we get @xmath70    $ ] for doped graphene . 
   the plot is for the doping density @xmath62 but the result is very similar for any finite doping density.,width=207 ]    the xps spectrum reflects the density of states of the core electron and becomes @xmath71}}{e^ { - a(t ) } } } \\ 
   = \frac{1}{\pi } \int\limits_0^\infty   { dt\cos \left [ { \left ( { \omega   - d } \right)t + b\left ( t \right ) } \right]{e^ { - a(t ) } } } ,   \end{array } \label{equ38}\ ] ] where @xmath72,\\ b\left ( t \right ) =   - \frac{{2{e^2}}}{{\pi \hbar } } \int\limits_0^\infty   { \int\limits_0^\infty   { dqd\omega } } \frac{1}{{{\omega ^2}}}{\mathop{\rm im}\nolimits } \frac { { - 1}}{{\varepsilon \left ( { q,\omega } \right)}}\sin \left ( { \omega t } \right),\\ d = \frac{{2{e^2}}}{{\pi \hbar } } \int\limits_0^\infty   { \int\limits_0^\infty   { dqd\omega } } \frac{1}{\omega } { \mathop{\rm im}\nolimits } \frac { { - 1}}{{\varepsilon \left ( { q,\omega } \right)}}. \end{array } \label{equ39}\ ] ] we give the energy relative to the core - level position when the interaction with the electrons are neglected . taking the finite life time of the core hole into account we arrive at the central theoretical result of this work , the xps spectrum . 
   the xps spectrum can be written as @xmath73dt } ,   \label{equ40}\ ] ] where @xmath74\left [ { 1 - \cos \left ( { wt } \right ) } \right]dq } dw } , \end{array } \label{equ41}\ ] ] @xmath75\sin \left ( { wt } \right)dq } dw } , \label{equ42}\ ] ] and @xmath76dq } dw}. \label{equ43}\ ] ]    $ ] for pristine graphene . 
   the scaling of @xmath77 and @xmath78 is the same as in fig.[figu2].,width=207 ]    all variables have been scaled according to @xmath79 and are now dimensionless . 
   the two last relations will be used below . 
   the quantity @xmath80 is the energy shift of the adiabatic peak . 
   we have taken the finite life - time of the core hole into account by introducing the first factor of the integrand of eq.([equ40 ] ) , where @xmath81 is the fwhm ( full width at half maximum ) of the lorentz broadened peak . 
   we have here assumed that the core - hole potential can be represented by a pure coulomb potential . 
   the results are valid for a general 2d system . 
   the particular system enters the problem through @xmath82 $ ] . for pristine graphene the dielectric function is@xcite @xmath83 where we have used the values@xcite @xmath84 m 
   / s for the fermi velocity @xmath85 and @xmath86 for @xmath87 , respectively . 
   the background dielectric constant @xmath87 is the result of high frequency electronic excitations to higher lying empty energy bands and from lower lying filled energy bands . 
   when the graphene sheet is doped the dielectric function becomes much more complicated . 
   however it has been derived by several groups@xcite . the dielectric function in a general point in the complex frequency plane , @xmath77 , away from the real axis is@xcite @xmath88 } \right\};}\\ { f\left ( { q , w } \right ) = { \rm{asin}}\left ( { \frac{{1 - w}}{q } } \right ) + { \rm{asin}}\left ( { \frac{{1 + w}}{q } } \right)}\\ { \qquad \qquad   - \frac{{w - 1}}{q}\sqrt { 1 - { { \left ( { \frac{{w - 1}}{q } } \right)}^2 } }   + \frac{{w + 1}}{q}\sqrt { 1 - { { \left ( { \frac{{w + 1}}{q } } \right)}^2 } } , } \end{array } \label{equ46}\ ] ] where @xmath89 is the density of states at the fermi level and @xmath90 is the doping concentration . 
   in both eqs . 
   ( [ equ45 ] ) and ( [ equ46 ] ) we let @xmath77 be real valued but add a small imaginary part to get the retarded forms of the dielectric functions .    in figs.[figu1 ] and [ figu2 ] 
   we show the contour plot and surface plot , respectively , of @xmath91 $ ] for doped graphene . 
   the integrands in eqs.([equ41 ] ) , ( [ equ42 ] ) , and ( [ equ43 ] ) all have an extra factor @xmath92 for small @xmath77 which means that small energy transfers are important in the shake up structure of the spectra . in fig.[figu3 ] we give the corresponding surface plot for pristine graphene . here 
   the excitations are of electron - hole pair type . 
   we have taken the lorentzian broadening into account but how should we include the gaussian instrumental broadening ? here we use a trick . 
   if all shake up processes were involving only one discrete frequency , @xmath93 , the @xmath94- , @xmath95- , and @xmath96-functions would become @xmath97,\\ { b_g}\left ( t \right ) =   - \eta \sin \left ( { { w_0}t } \right),\\ { d_g } = \eta { w_0},\\ \eta   = \int { \frac{{{d^2}q}}{{{{\left ( { 2\pi } \right)}^2}}}\frac{{{{\left| { g\left ( { \bf{q } } \right ) } \right|}^2}}}{{{{\left ( { \hbar { \omega _ 0 } } \right)}^2 } } } } ,   \end{array } \label{equ47}\ ] ]    -xps ) c 1s core - level spectrum of a suspended graphene sheet obtained in ref.@xcite using the photon energy 480 ev , red circles . 
   the solid curve is the theoretical result for an undoped free - standing graphene sheet . 
   a linear background was subtracted from the experimental data . 
   we used lorentzian and gaussian broadening of .4 ev and .666 ev , respectively , width=302 ]    where @xmath98 is a strength constant and the spectrum would consist of a series of delta functions , @xmath99     but now for a gold supported graphene sheet . 
   the theoretical curve was obtained assuming a doping density of @xmath100 . 
   we used the same lorentzian and gaussian broadening as in fig.[figu5].,width=302 ]    the amplitudes of the delta functions form a poisson distribution . 
   why is it a poisson distribution ? 
   the average number of bosons surrounding the core hole is @xmath98 . 
   the bosons do not interact and the core hole does not recoil when a boson is excited . 
   thus the probability that a boson is excited in a given instant does not depend on how many bosons are already excited . 
   then , according to probability theory the probability for having exactly @xmath90 bosons excited at a certain time is given by the poisson distribution . 
   when the strength parameter @xmath98 is large this turns into a gaussian distribution . 
   it is close to a gaussian already at @xmath101 . 
   the gaussian fwhm is given by @xmath102 . 
   the trick is now to choose a small enough value for @xmath103 so that @xmath98 becomes large enough . 
   then we add the resulting @xmath94- , @xmath95- , and @xmath96-functions to the original functions in eq.([equ40 ] ) . 
   thus we get the final spectrum with both lorentzian and gaussian broadened structures from @xmath104 } } } \right . } \\ 
   \quad \quad \quad 
   \quad   \times \cos \left\ { { \left [ { w - \left ( { d + { d_g } } \right ) } \right]t - \left [ { b\left ( t \right ) + { b_g}\left ( t \right ) } \right ] } \right\ } } \right ] . \end{array } \label{equ49}\ ] ] this is the relation we have used in finding the xps spectra presented in figs.[figu4]-[figu7 ] . 
   in fig.[figu4 ] we compare our results to an experimental xps spectrum@xcite represented by red circles . 
   the experiment was performed on a single quasi - free - standing epitaxial graphene layer on sic obtained by hydrogen intercalation@xcite . 
   this leads to a virtually undoped graphene sheet . 
   the photon energy used was 750 ev . 
   the leftmost peak is from graphene c1s and the rightmost from the c1s core level in the sic substrate . 
   the dashed curve is the result one would get if there were no interaction between the electrons and the core hole . 
   another way to express this is to say that the electron wave functions are frozen . 
   the energies are given relative to the position of this peak . 
   the peak is symmetric and has the full broadening but no structure from shake - up effects . 
   the thick solid curve is the adiabatic peak . 
   this is what the spectrum would look like if the excitation were adiabatic , i.e. , the excitation were so slow that the electrons were left in a quasiequilibrium state ; the core - hole potential was fully screened . 
   this peak is just the dashed peak shifted by the energy @xmath80 . 
   the thin solid curve is the full result from eq.([equ49 ] ) . 
   when we used the dielectric function from pristine graphene the curve emerged a little above the experimental tail . 
   when we added a very small doping level , @xmath105 , the agreement with experiment was perfect . 
   the reason for that a small amount of doping pushes down the tail a little is that there are some extra contributions at the top of the peak and the normalization then leads to a reduction away from the center of the peak .    in figs.[figu5 ] and 
   [ figu6 ] we compare our results for pristine and doped free - standing graphene to the experimental results in ref.@xcite . 
   the experiments were performed using a photon energy of 480 ev on a suspended single graphene sheet and on a gold supported single graphene sheet . in the first sample 
   the graphene is more or less undoped ; in the second gold provides p - doping . 
   we subtracted a linear background from both experimental sets of data and used the same broadening parameters for both spectra . 
   the main peak in the doped sample is broader than in the undoped but this extra broadening comes from different shake - up effects in the two samples . 
   the theoretical spectrum was obtained assuming a doping density of @xmath100 . 
   this number @xmath106 should not be taken too seriously . 
   we varied the density in equidistant steps on a logarithmic scale and tried @xmath107 , @xmath108 and @xmath109 . 
   the one in the middle gave the best fit with experiments . 
   , from all spectra . 
   the symmetric peak is the result from the fully adiabatic approximation and the result when there is no interaction between the core hole and the electron - hole system . 
   the red dashed curve is the result from the undoped graphene sheet , the blue short - dashed from a sheet with doping concentration @xmath110 , and the green dotted curve from a sheet with doping concentration @xmath111 , respectively . 
   the curve for doping concentration @xmath112 has been omitted since it is very close to the curve for pristine graphene . , 
   width=302 ]    finally , in fig.[figu7 ] we show the theoretical results for different doping levels . in order to more clearly see the effect doping has on the peak shape 
   we have here removed all different shifts , @xmath80 , for the curves . 
   the solid black curve is the noninteracting curve or the adiabatic curve ( it is the same for all doping levels now when the shift has been removed ) . 
   the red dashed curve is the pristine result . 
   the blue short - dashed curve is for the doping density @xmath110 , and the green dotted curve from a sheet with doping concentration @xmath111 . 
   important doping effects show up first at doping concentrations exceeding @xmath112 . 
   the reason is that some electron - hole - pair excitation - channels are blocked with doping due to the pauli exclusion principle ; this is compensated by new electron - hole - pair excitation - channels and the new plasmon - excitation channel .    in a recent work@xcite one came to the same conclusion as we regarding the need to make a full calculation of the core - hole spectra in graphene instead of using the doniach and sunji fitting of the peaks . 
   they treated the band - structure of graphene in a different way compared to our and relied on density functional theory . 
   they seem to have assumed a constant 2d plasmon frequency and thereby obtained sharp plasmon structures in the tail region of the peaks . 
   such structures are not observed in the experimental spectra . 
   to summarize , we have presented a detailed derivation of the core - level spectra of 2d systems and presented numerical results for pristine and doped free - standing graphene . 
   although , pristine graphene is not a metal its core - level spectrum shows a peak tailing , characteristic of metallic systems . the tailing increases with doping for doping concentrations exceeding @xmath112 . 
   the peak shape changes with further increase of the doping concentration . 
   this opens up for a complementary way to estimate the degree of doping of a sample . 
   we have compared our results to three different experimental spectra from two experimental groups . 
   the agreement is quite good which is very encouraging . 
   we have furthermore introduced a convenient way to introduce the effect of gaussian instrumental broadening in the formalism . 
   we thank professor leif i. johansson and professor chung - lin wu for providing us with their experimental data . 
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summary: we calculate core - level spectra for pristine and doped free - standing graphene sheets . 
   instructions for how to perform the calculations are given in detail . 
   although pristine graphene is not metallic the core - level spectrum presents low - energy tailing which is characteristic of metallic systems . 
   the peak shapes vary with doping level in a characteristic way . 
   the spectra are compared to experiments and show good agreement . 
   we compare to two different pristine samples and to one doped sample . 
   the pristine samples are one with quasi - free - standing epitaxial graphene on sic obtained by hydrogen intercalation and one with a suspended graphene sheet . 
   the doped sample is a gold supported graphene sheet . 
   the gold substrate acts as an acceptor so the graphene sheet gets p - doped .