diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhuzf" "b/data_all_eng_slimpj/shuffled/split2/finalzzhuzf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhuzf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\n\nUltracold quantum gases provide an excellent and highly controllable testbed for realizing a multitude of systems without the inherent complexity of their \ncondensed matter counterparts \\cite{Lewe}. \nKey features of ultracold atoms include the ability to manipulate their interparticle interactions by employing Feshbach resonances \\cite{Fesh1,Fesh2}, \nto tune the dimensionality of the system \\cite{Petrov,lower-D}, as well as to trap few-body ensembles possessing unique properties \\cite{Brouzos,Greene,Blume,Sowinski,Lompe}. \nTwo-dimensional (2D) systems are of particular interest due to their peculiar scattering properties, the emergent phase transitions, such as the \nBerezinskii-Kosterlitz-Thouless transition \\cite{Dalibard,Bloch,Dalibard2,Cirone,Shlyapnikov,Pricoup} and \nthe existence of long-range thermal fluctuations in the homogeneous case. \nThese thermal fluctuations in turn prohibit the development of a condensed phase, but can allow the occurence of a residual \nquasi-ordered state \\cite{Thouless}. \n\nOne among the few solvable quantum problems, is the system of two ultracold atoms confined in an isotropic harmonic oscillator. \nHere the two atoms interact via a contact pseudo-potential where only $s$-wave scattering is taken into account \\cite{Huang}, an \napproximation which is valid at ultralow temperatures where two-body interactions dominate \\cite{Tannoudji}. \nThe stationary properties of this system have been extensively studied for various dimensionalities and for arbitrary values of the coupling \nstrength \\cite{Busch,Zyl,Shea,Richard}. \nGeneralizations have also been reported including, for instance, the involvement of anisotropic traps \\cite{Calarco}, higher partial waves \\cite{Stock,Zinner} \nand very recently long-range interactions \\cite{Koscik} and hard-core interaction potentials \\cite{Diakonos}. \nRemarkably enough, exact solutions of few-body setups have also been obtained regarding the stationary properties of three harmonically trapped \nidentical atoms in all dimensions \\cite{Pethick,Drummond,Portegies,Harshman,Polls,Deck}. \n\nA quench of one of the intrinsic system's parameters is the most simple way to drive it out-of-equilibrium \\cite{Langen}. \nQuenches of $^{87}$Rb condensates confined in a 2D pancake geometry have been employed, for instance, by changing abruptly the trapping frequency \nto excite collective breathing modes \\cite{Chevy,Perrin} in line with the theoretical predictions \\cite{Pitaevskii,Pitaevskii2}. \nOn the contrary, the breathing frequency of two-dimensional Fermi gases has been recently measured experimentally \\cite{Anomaly1,Anomaly2} \nand found to deviate from theoretical predictions at strong interactions, a behavior called quantum anomaly. \nAlso, oscillations of the density fluctuations \nbeing reminiscent of the Sakharov oscillations \\cite{Sakharov} have been \nobserved by quenching the interparticle repulsion. \nFurthermore, it has been shown that the dynamics of an expanding Bose gas when switching off the external trap leads to the fast and slow equilibration of \nthe atomic sample in one- and two-spatial dimensions respectively \\cite{Demler}. \nMoreover, the collisional dynamics of two $^6$Li atoms has been experimentally monitored after quenching the frequencies of a three-dimensional harmonic trap \\cite{Jochim}.\nTurning to two harmonically trapped bosons, the existing analytical solutions have been employed in order to track the interaction quench dynamics mainly in \none- \\cite{Simos,Bolsinger,Garcia,Corson2}, but also in three-dimensional systems \\cite{Sykes}. \nFocusing on a single dimension, an analytical expression regarding the eigenstate transition amplitudes after the quench has \nbeen derived \\cite{Simos}. \nMoreover, by utilizing the Bose-Fermi mapping theorem \\cite{Tonks,Girardeau} a closed form of the time-evolved two-body wavefunction for quenches \ntowards the infinite interaction strength has been obtained \\cite{Bolsinger}, observing also a dynamical crossover from bosonic \nto fermionic properties. \n\nBesides these investigations the interaction quench dynamics of the two-boson system in two spatial dimensions employing an \nanalytical treatment has not been addressed. \nHere, the existence of a bound state for all interaction strengths might be crucial giving rise to a very different dynamics compared to its one-dimensional analogue. \nAlso, regarding the strongly interacting regime the Bose-Fermi theorem does not hold. \nTherefore it is not clear whether signatures of fermionic \nproperties can be unveiled although there are some suggestions for their existence \\cite{Mujal,Yannouleas,Mujal2}. \nAnother interesting feature is the inherent analogy between three bosons interacting via a three-body force in one-dimension and two bosons \ninteracting via a two-body force in two spatial dimensions \\cite{Valiente2,Trimers,Guijarro,Nishida,Sekino}. \nTherefore, our work can provide additional hints on the largely unexplored three-body dynamics of three bosons in one spatial dimension \\cite{Pastukhov}. \nThe present investigation will enable us to unravel the role of the different eigenstates for the dynamical response of the system and might inspire \nfuture studies examining state transfer processes \\cite{Fogarty,Reshodko} which are currently mainly restricted to one-dimensional setups. \n\nIn this work we study the interaction quench dynamics of two harmonically confined bosons \nin two spatial dimensions for arbitrary interaction strengths. \nTo set the stage, we briefly review the analytical solution of the system for an arbitrary stationary eigenstate and discuss the \ncorresponding two-body energy eigenspectrum \\cite{Busch}. \nSubsequently, the time-evolving two-body wavefunction is derived as an expansion over the postquench eigenstates of the system with \nthe expansion coefficients acquiring a closed form. \nThe quench-induced dynamical response of the system is showcased via inspecting the fidelity evolution. \nThe underlying eigenstate transitions that predominantly participate in the dynamics are identified in the fidelity spectrum \\cite{Mistakidis1,Mistakidis2,Thies}. \nIt is found that initializing the system in its ground state, characterized by finite interactions of either sign, it is driven more efficiently out-of-equilibrium \nwhen employing an interaction quench in the vicinity of the non-interacting limit. \nDue to the interaction quench the two bosons perform a breathing motion, visualized in the temporal evolution of the single-particle density and \nthe radial probability density in both real and momentum space. \nThese observables develop characteristic structures which signal the participation of the bound and energetically higher-lying excited states of the postquench system.\nThe dynamics of the short-range correlations is captured by the two-body contact, which is found to perform an oscillatory motion possessing a multitude of frequencies. \nIn all cases the predominantly involved frequency corresponds to the energy difference between the bound and the ground state. \nAdditionally, the amplitude of these oscillations is enhanced when quenching the system from weak to infinite interactions. \nMoreover, it is shown that the system's dynamical response crucially depends on the initial state and in particular starting from an energetically higher excited state, \nthe system is perturbed to a lesser extent, and a fewer amount of postquench eigenstates contribute in the dynamics \\cite{Sowinski_ent,Katsimiga_diss_flow,Pia,Katsimiga_quantum_DBs,Katsimiga_bent}. \nHowever, if the quench is performed from the bound state \nthe system is perturbed in the most efficient manner compared to \nany other initial state configuration. \nFinally, we observe that quenching the system from its ground state at zero interactions towards the infinitely strong ones the time-evolved wavefunction becomes almost \northogonal to the initial one at certain time intervals. \n\nThis work is structured as follows. \nIn Sec. \\ref{theory} we introduce our setup, provide a brief summary of its energy spectrum and most importantly derive a closed \nform of the time-evolved wavefunction discussing also basic observables. \nSubsequently, we investigate the interaction quench dynamics from attractive to repulsive interactions in Sec. \\ref{quench_att} and vice versa in Sec. \\ref{quench_rep} as well as from zero to infinitely large coupling strengths in Sec. \\ref{inf_quench}. \nWe summarize our results and provide an outlook in Sec. \\ref{conclusions}. \n\t\n\t\n\n\n\\section{Theoretical framework} \\label{theory}\n\n\\subsection{Setup and its stationary solutions} \\label{stationary_sol}\n \n \nWe consider two ultracold bosons trapped in a 2D isotropic harmonic trap. \nThe interparticle interaction is modeled by a contact $s$-wave pseudo-potential, which is an adequate approximation \nwithin the ultracold regime. \nThe Hamiltonian of the system, employing harmonic oscillator units ($\\hbar=m=\\omega=1$), reads\n\\begin{equation} \\label{hamilt}\n\\mathcal{H}= \\frac{1}{2} \\sum_{i=1}^{2} \\left[-\\nabla_i^2 +\\boldsymbol{r}_i ^{2} \\right] + 2V_{\\textrm{pp}}(\\boldsymbol{r}_1-\\boldsymbol{r}_2),\n\\end{equation}\nwhere $\\boldsymbol{r}_1$ and $\\boldsymbol{r}_2$ denote the spatial coordinates of each boson. \nNote that the prefactor 2 is used for later convenience in the calculations. \nThe contact regularized pseudo-potential can be expressed as \\cite{Olshanii} \n\\begin{equation}\nV_{\\textrm{pp}}(\\boldsymbol{r})= -\\frac{\\pi \\delta(\\boldsymbol{r})}{\\ln(Aa\\Lambda)}\\left(1-\\ln(A\\Lambda r)r\\frac{\\partial}{\\partial r} \\right),\n\\end{equation}\nwith $\\Lambda$ being an arbitrary dimensionful parameter possessing the dimension of a wavevector and $A= e^{\\gamma}\/2$ where $ \\gamma= 0.577\\ldots $ is the \nEuler-Mascheroni constant. \nWe remark that the parameter $\\Lambda$ does not affect the value of any observable or the energies and eigenstates of the system as it has been shown \nin \\cite{Pricoup,Olshanii}. \nFurthermore, the 2D $s$-wave scattering length is given by $a$. \n\nTo proceed, we perform a separation of variables in terms of the center-of-mass, $\\boldsymbol{R}=\\frac{1}{\\sqrt{2}}(\\boldsymbol{r}_1+\\boldsymbol{r}_2)$, and the relative \ncoordinates $ \\boldsymbol{\\rho}= \\frac{1}{\\sqrt{2}}(\\boldsymbol{r}_1-\\boldsymbol{r}_2)$. \nEmploying this separation, the Hamiltonian \\eqref{hamilt} acquires the form $\\mathcal{H}= \\mathcal{H}_{\\textrm{CM}}+\\mathcal{H}_{\\textrm{rel}}$ with\n\\begin{equation} \\label{separation_hamilt}\n\\mathcal{H}_{\\textrm{CM}}= -\\frac{1}{2} \\nabla_{\\boldsymbol{R}}^2+\\frac{1}{2}R^2, \n\\end{equation}\nbeing the Hamiltonian of the center-of-mass and\n\\begin{equation}\n \\mathcal{H}_{\\textrm{rel}} = -\\frac{1}{2} \n \\nabla_{\\boldsymbol{\\rho}}^2 +\\frac{1}{2}\\rho^2 +V_{\\textrm{pp}}(\\boldsymbol{\\rho}),\n\\end{equation}\nis the Hamiltonian corresponding to the motion in the relative coordinate frame.\n \nAs a result, the Schr\\\"odinger equation can be casted into the form $\\mathcal{H}\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=E\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)$. \nHere the total energy of the system has two contributions, namely $E=E_{\\textrm{CM}}+E_{\\textrm{rel}}$, and the system's wavefunction is a product of a \ncenter-of-mass and a relative coordinate part i.e. $\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=\\Psi_{\\textrm{CM}}(\\boldsymbol{R})\\Psi_{\\textrm{rel}}(\\boldsymbol{\\rho})$. \nSince the center-of-mass hamiltonian $\\mathcal{H}_{\\textrm{CM}}$ is interaction independent [see Eq. \\eqref{separation_hamilt}] its eigenstates \ncorrespond to the well-known non-interacting 2D harmonic oscillator states \\cite{Sakurai}. \nWe assume that the center-of-mass wavefunction takes the form $\\Psi_{\\textrm{CM}}(\\boldsymbol{R})=\\frac{e^{-\\boldsymbol{R}^2\/2}}{\\sqrt{\\pi}}$, namely the non-interacting \nground state of the 2D harmonic oscillator. \nSince we are interested in the interaction quench dynamics of the two interacting bosons we omit the center-of-mass wavefunction in what follows \nfor simplicity. \nFollowing the above-mentioned separation of coordinates, the problem boils down to solving the relative part of the Hamiltonian, $\\mathcal{H}_{\\textrm{rel}}$, \nwhich is interaction dependent. \nFor this purpose, we assume an ansatz for the relative wavefunction, which involves an expansion over the non-interacting energy eigenstates \nof the 2D harmonic oscillator \n\\begin{equation} \\label{states}\n\\begin{split}\n\\varphi_{n,m}(\\rho,\\theta&)=\\\\&\\sqrt{\\frac{n!}{\\pi\\Gamma(n+|m|+1)}}e^{-\\rho^2\/2}\\rho^{|m|}L_n^{(m)}(\\rho^2)e^{{i\\mkern1mu} m\\theta}.\n\\end{split}\n\\end{equation} \nIn this expression, $\\Gamma(n)$ is the gamma function while $L_n^{(m)}$ refer to the generalized Laguerre polynomials of degree $n$ and value of angular momentum $m$. \nAlso, $\\boldsymbol{\\rho}=(\\rho,\\theta)$ where \n$\\rho$ is the relative polar coordinate and $\\theta$ is the relative angle. \nThe energy of the non-interacting 2D harmonic oscillator eigenstates in harmonic oscillator units is $E_{\\textrm{rel},n,m}=2n+|m|+1$ \\cite{Sakurai}. \nWithin our relative coordinate wavefunction ansatz [see Eq. \\eqref{ansatz} below] we will employ, however, only those states that are affected by the \npseudo-potential and thus have a non-vanishing value at $\\rho=0$. \nThese are the states with bosonic symmetry $m=0$, i.e. zero angular momentum. \nThe states with odd m are fermionic, since under the exchange $\\theta \\rightarrow \\theta -\\pi$, they acquire an extra minus sign due to \nthe term $e^{{i\\mkern1mu} m\\theta}$. \nTherefore, the ansatz for the relative wavefunction reads \n\\begin{equation} \\label{ansatz}\n\\Psi_{\\textrm{rel}}(\\rho)= \\sum_{n=0}^{\\infty} c_n \\varphi_n(\\rho), \n\\end{equation} \nwhere the summation is performed over the principal quantum number $n$ and we omit the angle $\\theta$ since only the states with $m=0$ \nare taken into account. \nNote that this ansatz has already been reported previously e.g. in Refs. \\cite{Busch,Simos}.\nIn order to determine the expansion coefficients $c_n$, we plug Eq. \\eqref{ansatz} into the Schr\\\"odinger equation that $\\mathcal{H}_{\\textrm{rel}}$ satisfies \nand project the resulting equation onto the state $\\varphi_{n'}^*(\\rho)$. \nFollowing this procedure we arrive at\n\\begin{equation} \\label{exp_coeff}\n\\begin{split}\nc_{n'}&(E_{\\textrm{rel},n'}-E_{\\textrm{rel}})=\\frac{\\pi\\varphi_{n'}^*(0) }{\\ln(Aa\\Lambda)}\\\\& \\times\\left[ \\left(1-\\ln(\\sqrt{2}A\\Lambda\\rho)\\rho\\frac{\\partial}{\\partial \\rho} \n\\right)\\sum_{n=0}^{\\infty} c_n \\varphi_n(\\rho) \\right]_{\\rho\\rightarrow 0}.\n\\end{split}\n\\end{equation} \nThe right hand side of Eq. \\eqref{exp_coeff} is related to a normalization factor of the relative wavefunction $\\ket{\\Psi_{\\textrm{rel}}}$. \nIndeed it has been shown \\cite{Busch,Simos} that the coefficients take the form\n\\begin{equation}\n c_n=A_1\\frac{\\varphi_n^*(0)}{E_{\\textrm{rel},n}-E_{\\textrm{rel}}},\n\\end{equation}\n with \n$A_1=\\frac{2\\sqrt{\\pi}}{\\sqrt{\\psi^{(1)}\\left(\\frac{1-E_{\\textrm{rel}}}{2}\\right)}}$ being a normalization constant and $\\psi^{(1)}(z)$ the trigamma function. \n\nBy inserting this expression of $c_n$ into Eq. \\eqref{ansatz}, we can determine the relative wavefunction. \nThis can be achieved by making use of the generating function of the Laguerre polynomials i.e. $\\sum_{n=0}^{\\infty} t^n L_n(x)=\\frac{1}{1-t}e^{-\\frac{tx}{1-t}}$. \nThus, the relative wavefunction takes the form \\cite{Drummond}\n\\begin{equation} \\label{station}\n\\begin{split}\n\\Psi_{\\textrm{rel},\\nu_i}(\\rho)=\\frac{\\Gamma(-\\nu_i)}{\\sqrt{\\pi\\psi^{(1)}(-\\nu_i)}} e^{-\\rho^2\/2}U(-\\nu_i,1,\\rho^2),\n\\end{split}\n\\end{equation}\nwhere $U(a,b,z)$ refers to the confluent hypergeometric function of the second type (also known as Tricomi's function) and $2\\nu_i+1$ is the energy of the $i=0,1,\\dots$ interacting eigenstate \\cite{Stegun}.\nIn what follows we will drop the subscript rel and denote these relative coordinate states by $\\ket{\\Psi_{\\nu_i}}$. \nIt is important to note at this point that this relative wavefunction ansatz solves also the problem of three one-dimensional harmonically trapped bosons interacting \nvia three-body forces, see e.g. Ref. \\cite{Pastukhov} for more details.\n\nTo find the energy spectrum of $\\mathcal{H}_{\\textrm{rel}}$, we employ Eq. \\eqref{exp_coeff} along with the form of $c_{n,i}=\\frac{\\sqrt{\\pi}\\varphi_n^*(0)}{(n-\\nu_i)\\sqrt{\\psi^{(1)}(-\\nu_i)}}$. \nNote that in order to determine the right hand side of Eq. \\eqref{exp_coeff}, we make use of the behavior of the relative wavefunction \\eqref{station} \nclose to $\\rho=0$. \nIn this way, we obtain the following algebraic equation regarding the energy of the relative coordinates \\cite{Busch,Zyl}, \n$2\\nu_i+1$, \n\\begin{equation} \\label{spectrum}\n\\psi(-\\nu_i)= \\ln\\left( \\frac{1}{2a^2}\\right) +2\\ln2 -2\\gamma,\n\\end{equation}\nwhere $\\psi(x)$ is the digamma function. \nNote here that a different form of the algebraic Eq. \\eqref{spectrum} can be found in \\cite{Busch} and stems from a different definition of the \nscattering length $a$ \\cite{Zyl}. \nIt is also important to emphasize that the energy spectrum given by Eq. \\eqref{spectrum} is independent of the form of the pseudo-potential, $V_{\\textrm{pp}}(\\boldsymbol{r})$, i.e. independent of $\\Lambda$, A, or any short range potential, \nas long as its range is much smaller than the harmonic oscillator length \\cite{Zyl}. \nDenoting $a_0\\equiv \\frac{a}{2}e^{\\gamma}$, the algebraic Eq. \\eqref{spectrum} can be casted into the simpler form \n$\\psi(-\\nu_i)= \\ln\\left( \\frac{1}{2a_0^2}\\right)$. \nAlso, we define the interparticle interaction strength \\cite{Doganov,Busch,Zyl,Petrov,Shlyapnikov,comment} to be\n\\begin{equation}\ng=\\frac{1}{\\ln \\left( \\frac{1}{2a_0^2}\\right)}.\n\\end{equation} \n \n\\begin{figure*}[t!] \n\\centering\n\\includegraphics[width=\\textwidth,height=8cm]{figure1-eps-converted-to.pdf}\n\\caption{Energy spectrum of two bosons trapped in a 2D harmonic trap for varying interaction strength $g$. \nIn the spectrum for $g>-0.51$ we display the bound state, $\\nu_0$, and higher-lying eigenstates up to the fourth excited state, $\\nu_5$. \nOn the other hand, for $g<-0.51$ the spectrum contains the bound state, $\\nu_0$, as well as higher excited states up to the third excited, $\\nu_4$.\nThe black solid horizontal lines indicate the asymptotic values of the energy determined by $\\psi(-\\nu_i)=0$, in the limit of strong interactions. \nThe black solid vertical line at $g=-0.51$ marks the boundary at which the bound state for negative interaction strengths becomes the ground state for $g>-0.51$.\nThe insets show the radial probability density of the bound states $\\nu_0$ for different attractive (left panel) and repulsive (right panel) interactions, \nas well as the radial probability density of the ground state, $\\nu_1$, at $g=0.3$ (left panel).} \n\\label{fig:en_spectrum}\n\\end{figure*} \n\nThe energy $E_{\\textrm{rel}}$ of the two bosons as a function of the interparticle interaction strength is presented in Fig. \\ref{fig:en_spectrum}. \nAs it can be seen, for $g=0$ $E_{\\textrm{rel}}$ has the simple form $E_{\\textrm{rel},n}=2n+1$, and thus we recover the non-interacting energy spectrum of a 2D harmonic oscillator \nwith zero angular momentum \\cite{Sakurai,Tannoudji}. \nIn this case the energy spacing between two consecutive eigenenergies is independent of n, i.e. $\\Delta E=E_{\\textrm{rel},n+1}-E_{\\textrm{rel},n}=2$. \nFor repulsive (attractive) interactions, the energy is increased (lowered) with respect to its value at $g=0$. \nAlso and in contrast to the one-dimensional case, there are bound states $\\ket{\\Psi_{\\nu_0}}$, namely eigenstates characterized by negative energy, in both interaction regimes. \nNote that herein we shall refer to these eigenstates with negative energy as bound states ($\\nu_0$) whilst the corresponding eigenstates with positive energy in increasing energetic order \nwill be denoted e.g. as the first ($\\nu_1$), second ($\\nu_2$) etc eigenstates and called ground, first excited state etc.\nThe presence of these bound states can be attributed to the existence of the centripetal term $-\\frac{1}{4r^2}$, in the 2D radial Schr\\\"odinger equation \\cite{Sakurai}, which \nsupports a bound state even for weakly attractive potentials, in contrast to the 3D case \\cite{Cirone,Gezerlis}. \nThese energy states, $\\nu_0$, correspond to the molecular branch of two cold atoms in two dimensions. \nThis is clearly captured by the lowest energy branch of Fig. \\ref{fig:en_spectrum}, as has been demonstrated in Ref. \\cite{Drummond}. \nNote that due to a different definition of the coupling constant compared to Ref. \\cite{Drummond}, which possesses a bijective mapping to our definition of \nthe coupling strength \\cite{comment}, the molecular branch \nmaps to the bound states ($\\nu_0$) herein in both the repulsive and the attractive interaction regime. \nTo further appreciate the influence of these bound states we also provide in the insets of Fig. 1 their radial probability densities $2\\pi\\rho|\\Psi|^2$ \\cite{Cirone}\nfor various interaction strengths as well as the radial probability density of the ground state $\\ket{\\Psi_{\\nu_1}}$ at $g=0.3$. \nIn the repulsive regime of interactions (right panel) the full-width-at-half-maximum of $2\\pi\\rho|\\Psi|^2$ is smaller \nthan the one of the attractive regime (left panel). \nThis behavior is caused by the much stronger energy of the bound state at $g>0$ compared to the $g<0$ case. \nFor large interaction strengths, $|g|>8$, the widths of $2\\pi\\rho|\\Psi|^2$ tend to be the same. \nAnother interesting feature of the 2D energy spectrum is the occurrence of a boundary signifying a crossover from the bound to the ground state ($\\nu_0\\rightarrow\\nu_1$) at \n$g=-0.51$, see the corresponding vertical line in Fig. \\ref{fig:en_spectrum}. \nThis means that the negative eigenenergy of $\\ket{\\Psi_{\\nu_0}}$ crosses the zero energy axis and \nbecomes the positive eigenenergy of $\\ket{\\Psi_{\\nu_1}}$ at $g=-0.51$. \nThis crossover is captured, for instance, by $2\\pi\\rho|\\Psi|^2$ which changes from a delocalized [e.g. at $g=0.3$] to a localized [e.g. at $g=-1$] distribution. \nThe existence of this boundary affects the labeling of all the states and therefore $\\nu_i$ becomes $\\nu_{i+1}$ as it is crossed from the \nrepulsive side of interactions. \nWe note here that with $\\ket{\\Psi_{\\nu_1}}$ [$\\ket{\\Psi_{\\nu_0}}$] we label the ground [bound] state and with $\\ket{\\Psi_{\\nu_i}}, \\, i>1$, the \ncorresponding excited states. \nFor repulsive interactions the energy of the bound state diverges at $g=0$ as $-1\/a_0^2$ \\cite{Gezerlis,Zinner} or as $-2e^{1\/g}$ in terms of the \ninterparticle strength, while it approaches its asymptotic value for very strong interactions [see Fig. \\ref{fig:en_spectrum}]. \nThe two bound states share the same asymptotic value $E_{\\textrm{rel}}=-1.923264$ at $g\\to \\pm \\infty$. \nWe remark that this behavior of the bound state in the vicinity of $g=0$ is the same as the one of the so-called universal bound state of \ntwo cold atoms in two dimensions in the absence of a trap \\cite{Zinner}. \nWe also note that the states $\\ket{\\Psi_{\\nu_i}}$ with $i \\neq 0$, approach their asymptotic values faster (being close to their asymptotic value already for $g=2$) than the \nbound states. \nThe asymptotic values are determined via the algebraic equation $\\psi(-\\nu_i)=0$. \nMoreover, it can be shown that approximately the positive energy in the infinite interaction limit is given by the formula \n$E_{\\textrm{rel}} \\approx2n+1-\\frac{2}{\\ln (n) }+\\mathcal{O}\\left((\\ln n)^{-2}\\right)$ when $n \\gg 1$ \\cite{Stegun}. \n\n\n\n\\subsection{Time-evolution of basic observables} \n\n\nTo study the dynamics of the two harmonically trapped bosons, we perform an interaction quench starting from a stationary state of the system, \n$\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$, at $g^{\\textrm{in}}$ to the value $g^f$. \nLet us also remark in passing that the dynamics of two bosons in a 2D harmonic trap employing an analytical treatment has not yet been reported. \nThe time-evolution of the system's initial wavefunction reads \n\\begin{equation} \\label{quench_wave}\n\\begin{split}\n\\ket{\\Psi_{\\nu_i}(t)}&= e^{-{i\\mkern1mu} \\hat{H}t}\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}\\\\&= \\sum_{j} e^{-{i\\mkern1mu} (2\\nu_j^f+1) t}\\ket{\\Psi_{\\nu_j}^f}\\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}(0)},\n\\end{split}\n\\end{equation} \nwhere $\\ket{\\Psi_{\\nu_j}^f}$ denotes the $j$-th eigenstate of the postquench Hamiltonian $\\hat{H}$ with energy $(2\\nu_j^f+1)$. \nNote that the indices in and $f$ indicate that the corresponding quantities of interest refer to the initial (prequench) and final (postquench) state of the system respectively. \nMoreover, the overlap coefficients, $\\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$, between the initial wavefunction and a final eigenstate \n$\\ket{\\Psi_{\\nu_j}^f}$ determine the degree of participation of this postquench eigenstate in the dynamics. \nRecall also here that the center-of-mass wavefunction, $\\Psi_{\\textrm{CM}}(\\boldsymbol{R})$, is not included in Eq. \\eqref{quench_wave} since the latter \nis not affected by the quench [see also Sec. \\ref{stationary_sol}] and therefore does not play any role in the description of the dynamics. \n\nIt can be shown that initializing the system in the eigenstate $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ at $g^{\\textrm{in}}$, the probability to \noccupy the eigenstate $\\ket{\\Psi_{\\nu_j}^f}$ after the quench is given by \n\\begin{eqnarray} \nd_{\\nu_j^f,\\nu_i^{\\textrm{in}}}&\\equiv& \\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}}= \\frac{\\Gamma(-\\nu_i^{\\textrm{in}})\\Gamma(-\\nu_j^f)}{\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}} \\nonumber \\times \\\\\n& & \\times \\int_0^{\\infty}dr e^{-r}U(-\\nu_i^{\\textrm{in}},1,r)U(-\\nu_j^f,1,r) \\nonumber \\\\ & =& \\frac{\\Gamma(-\\nu_j^f)G^{32}_{33}\\left( \\begin{array}{l|lll} \n1& 0&0 &-\\nu_j^f \\\\ & 0& 0 & -1-\\nu_i^{\\textrm{in}} \n\\end{array}\\right)} {\\Gamma(-\\nu_i^{\\textrm{in}})\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}},\n\\end{eqnarray} \nwith $G^{p,q}_{m,n}\\left( \\begin{array}{l|l} \nz& a_1, \\ldots a_p \\\\ & b_1, \\ldots b_q \n\\end{array}\\right)$ being the Meijer G-function \\cite{Gradshteyn}. \nRemarkably enough, the coefficients $d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}$ can also be expressed in a much simpler form if we make use of the ansatz of Eq. \\eqref{ansatz}. \nIndeed, by employing the orthonormality properties of the non-interacting eigenstates $\\varphi_n(\\rho)$ and the explicit expression of the expansion coefficients appearing in the \nansatz \\eqref{ansatz}, the overlap coefficients between a final and the initial eigenstate reads \n\\begin{equation}\\label{overlap} \n\\begin{split}\nd_{\\nu_j^f,\\nu_i^{\\textrm{in}}} = \\frac{\\left[\\frac{1}{g^f}-\\frac{1}{g^{\\textrm{in}}} \\right]}{(\\nu_i^{\\textrm{in}}-\\nu_j^f)\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}}. \n\\end{split}\n\\end{equation} \nIt should be emphasized here that this is a closed form of the overlap coefficients and the only parameters that need to be determined are the energies, \nwhich are determined from the algebraic equation \\eqref{spectrum}. \nAs a result in order to obtain the time-evolution of $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$ we need to numerically evaluate Eq. (\\ref{quench_wave}) which is an infinite \nsummation over the postquench eigenstates denoted by $\\ket{\\Psi_{\\nu_j}^f}$. \nIn practice this infinite summation is truncated to a finite one with an upper limit which ensures that the values of all observables have been converged \nwith respect to a further adding of eigenstates. \n\nHaving determined the time-evolution of the system's wavefunction [Eq. (\\ref{quench_wave})] enables to determine any observable of interest in the course of the dynamics. \nTo inspect the dynamics of the system from a single-particle perspective we monitor its one-body density \n\\begin{widetext} \n\\begin{gather} \n\\rho^{(1)}(\\boldsymbol{r}_1,t)=\\int d\\boldsymbol{r}_2 \\tilde{\\Psi}(\\boldsymbol{r}_1,\\boldsymbol{r}_2;t)\\tilde{\\Psi}^*(\\boldsymbol{r}_1,\\boldsymbol{r_2};t) \\nonumber \\\\ \n= \\frac{e^{-(x^2+y^2)}}{\\pi^2} \\sum_{j,k} \\frac{e^{2{i\\mkern1mu}(\\nu_j^f-\\nu_k^f)t}\\Gamma(-\\nu_k^f)\\Gamma^*(-\\nu_j^f)d_{\\nu_k^f,\\nu_i^{\\textrm{in}}}d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}^*}{\\sqrt{\\psi^{(1)}(-\\nu_k^f)\\psi^{(1)*}(-\\nu_j^f)}} \\times \\nonumber \\\\ \n\\times\\int_{-\\infty}^{\\infty} dz dw e^{-z^2-w^2}U^*\\left(-\\nu_j^f,1,(x-z)^2\/2+(y-w)^2\/2 \\right) \nU\\left(-\\nu_k^f,1,(x-z)^2\/2+(y-w)^2\/2 \\right). \n\\label{density_matrix} \n\\end{gather} \n\\end{widetext}\nIn this expression, the total wavefunction of the system is denoted by \n$\\tilde{\\Psi}(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=\\Psi_{\\textrm{CM}}(R(\\boldsymbol{r}_1,\\boldsymbol{r}_2),t)\\Psi_{\\textrm{rel},\\nu_i}(\\rho(\\boldsymbol{r}_1,\\boldsymbol{r}_2),t)$ \\cite{Sakmann}. \nTo arrive at the second line of Eq. \\eqref{density_matrix} we have expressed the relative, $\\rho^2= \\frac{1}{2} (r_1^2+r_2^2-2\\boldsymbol{r}_1\\cdot\\boldsymbol{r}_2)$, and the \ncenter-of-mass coordinates, $R^2= \\frac{1}{2} (r_1^2+r_2^2+2\\boldsymbol{r}_1\\cdot\\boldsymbol{r}_2)$, in terms of the Cartesian coordinates ($\\boldsymbol{r_1}$, $\\boldsymbol{r_2}$) and integrated \nout the ones pertaining to the other particle. \nIn particular, we adopted the notation $\\boldsymbol{r}_1=(x,y)$ and $\\boldsymbol{r}_2=(z,w)$ for the coordinates that are being integrated out. \nMoreover, the integral $I_{\\nu_j^f,\\nu_k^f}$ appearing in the last line of Eq. \\eqref{density_matrix} can be further simplified by employing the replacements \n$z'=x-z$ ,$w'=y-w$ and then express the new variables in terms of polar coordinates. \nThe emergent angle integration can be readily performed and the integral with respect to the radial coordinate becomes \n\\begin{eqnarray} \n&I_{\\nu_j^f,\\nu_k^f}= 2\\pi e^{-(x^2+y^2)} \\int_0^{\\infty} dr\\,re^{-r^2} I_0\\left(2r\\sqrt{x^2+y^2} \\right)\\times \\nonumber \\\\& \\times \nU^*\\left(-\\nu_j^f,1,\\frac{r^2}{2}\\right) U\\left(-\\nu_k^f,1,\\frac{r^2}{2}\\right).\n\\end{eqnarray} \nHere, $I_0(x)$ is the zeroth order modified Bessel function of the first kind \\cite{Stegun,Gradshteyn}. \n\nAnother interesting quantity which provides information about the state of the system on the two-body level is the radial probability density of the relative \nwavefunction \n\\begin{equation} \\label{prob_dens}\n\\mathcal{B}(\\rho,t)=2\\pi\\rho|\\Psi(\\rho,t)|^2.\n\\end{equation} \nIt provides the probability density to detect two bosons for a fixed time instant $t$ at a relative distance $\\rho$. \nIt can be directly determined by employing the overlap coefficients of Eq. \\eqref{overlap}.\nMoreover, the corresponding radial probability density in momentum space reads \n\\begin{equation}\n\\mathcal{C}(k,t)=2\\pi k |\\tilde{\\Psi}(k,t)|^2. \n\\end{equation} \nHere, the relative wavefunction in momentum space is obtained from the two dimensional Fourier transform \n\\begin{equation}\n\\tilde{\\Psi}(k,t)= 2\\pi \\int_{0}^{\\infty} d\\rho\\, \\rho \\Psi(\\rho,t) J_0(2\\pi\\rho k) \\quad,\n\\label{Fourier_wave}\n\\end{equation}\nwhere $J_0(x)$ is the zeroth order Bessel function. \n \nTo estimate the system's dynamical response after the quench we resort to the fidelity evolution $F(t)$. \nIt is defined as the overlap between the time-evolved wavefunction at time $t$ and the initial one \\cite{Gorin}, namely\n\\begin{equation} \\label{f(t)}\n\\begin{split}\nF(t)=&\\bra{\\Psi(0)}e^{-{i\\mkern1mu} \\hat{H}t}\\ket{\\Psi(0)} \\\\&= \\sum_{j} e^{-{i\\mkern1mu} (2 \\nu_j^f+1 ) t}|d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}|^2.\n\\end{split}\n\\end{equation} \nEvidently, $F(t)$ is a measure of the deviation of the system from its initial state \\cite{Simos}. \nIn what follows, we will make use of the modulus of the fidelity, $\\abs{F(t)}$. \nMost importantly, the frequency spectrum of the modulus of the fidelity $F(\\omega)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} dt\\, |F(t)|e^{{i\\mkern1mu} \\omega t}$ grants \naccess to the quench-induced dynamical modes \\cite{Mistakidis1,Mistakidis2,Mistakidis3,Mistakidis4,Jannis}. \nIndeed, the emergent frequencies appearing in the spectrum correspond to the energy differences of particular postquench eigenstates of the system and therefore \nenable us to identify the states that participate in the dynamics (see also the discussion below).\n\nAnother observable of interest is the two-body contact $\\mathcal{D}$. \nThe latter is defined from the momentum distribution in the limit of very large momenta i.e. $\\mathcal{C}(k,t) \\xrightarrow{k\\rightarrow \\infty} \\frac{2\\pi\\mathcal{D}(t)}{k^3}$ \nand captures the ocurrence of short-range two-body correlations \\cite{Bellotti,Valiente,momentum_2}. \nMoreover, this quantity can be experimentally monitored \\cite{Contact_1,Contact_2} and satisfies a variety of universal relations independently \nof the quantum statistics, the number of particles or the system's dimensionality \\cite{Tan1,Tan2,Tan3,momentum_2}. \nHaving at hand the eigenstates of the system, we can expand the time evolved contact after a quench from $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ at \n$g^{\\textrm{in}}$ to an arbitrary $g^f$ in terms of the contacts of the postquench eigenstates \\cite{Colussi}. \nNamely\n\\begin{equation}\n \\mathcal{D}(t)=\\left| \\sum_j e^{-{i\\mkern1mu} (2\\nu_j^f+1)t}d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}\\sqrt{|\\mathcal{D}_j|} \\right|^2.\n\\end{equation}\nThe contacts $\\mathcal{D}_j$ of the postquench eigenstates $\\ket{\\Psi_{\\nu_j}^f}$ can be inferred by employing the behavior of the \neigenstates [Eq. \\eqref{station}] close to zero distance, $\\rho \\rightarrow 0$, between the atoms\n\\begin{equation}\n \\Psi_{\\nu_j}(\\rho) \\xrightarrow[\\rho\\rightarrow 0]{} -\\frac{2 \\ln \\rho}{\\sqrt{\\pi \\psi^{(1)}(-\\nu_j)}}.\n \\label{waves_small} \n\\end{equation}\nBy plugging Eq. \\eqref{waves_small} into Eq. \\eqref{Fourier_wave} and restricting ourselves to small $\\rho$ values we obtain the contact from \nthe leading order term ($\\sim 1\/k^2$) of the resulting expression. \nThe contact for the postquench eigenstates $\\ket{\\Psi_{\\nu_j}^f}$ reads\n\\begin{equation}\n \\mathcal{D}_j=\\frac{1}{\\pi^3\\psi^{(1)}(-\\nu_j)}.\n\\end{equation}\nNote that in order to capture the quench-induced dynamical modes that participate in the dynamics of the contact, \nwe employ its corresponding frequency spectrum i.e. $\\mathcal{D}(\\omega)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} \\mathcal{D}(t)e^{{i\\mkern1mu} \\omega t}$.\t \n\t \nHaving analyzed the exact solution of the two bosons trapped in a 2D harmonic trap both for the stationary and the time-dependent cases, \nwe subsequently explore the corresponding interaction quench dynamics. \nIn particular, we initialize the system into its ground state $ \\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ for attractive interactions and perform interaction quenches towards the repulsive \nregime (Sec. \\ref{quench_att}) and vice versa (Sec. \\ref{quench_rep}). \n\n\\section{Quench dynamics of two attractive bosons to repulsive interactions} \\label{quench_att} \n\nWe first study the interaction quench dynamics of two attractively interacting bosons confined in a 2D isotropic harmonic trap. \nMore specifically, the system is initially prepared in its corresponding ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{\\textrm{in}}}=-1$. At $t=0$ we perform an interaction quench towards the \nrepulsive interactions letting the system evolve. \nOur main objective is to analyze the dynamical response of the system and identify the underlying dominant microscopic mechanisms. \n \n \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure2-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of the two bosons following an interaction quench from $g^{\\textrm{in}}=-1$ and \n$\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ to various postquench interaction strengths. \n(b) Fidelity evolution at representative postquench interaction strengths (see legend).}\n\\label{fig:attractive_ground} \n\\end{figure} \n\n\n\\subsection{Dynamical response} \\label{response_attract_repul}\n \nTo examine the dynamical response of the system after the quench we employ the corresponding fidelity evolution $\\abs{F(t)}$ [see Eq. \\eqref{f(t)}] \\cite{Fogarty2}. \nFigure \\ref{fig:attractive_ground} (a) shows $\\abs{F(t)}$ for various postquench interaction strengths $g^f$. \nWe observe the emergence of four distinct dynamical regions where the fidelity exhibits a different behavior. \nIn region I, $-10.9$ performs small amplitude oscillations that resemble the \nones already observed within region I [Fig. \\ref{fig:attractive_ground} (b) at $g^f=7$]. \nAn important difference with respect to region I is that the oscillations of $\\abs{F(t)}$ are faster and there is more than one frequency involved, compare $\\abs{F(t)}$ at $g^f=-0.5$ and $g^f=7$ in \nFig. \\ref{fig:attractive_ground} (b).\n \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure3-eps-converted-to.pdf} \n\\caption{(a) The fidelity spectrum $F(\\omega)$ after an interaction quench from $g^{\\textrm{in}}=-1$ to different final interaction strengths $g^f$. \n(b) The corresponding largest overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ (see legend). \nThe black dashed vertical line at $g^f=-0.51$ marks the boundary at which the bound state for negative interaction strengths becomes the \nground state for $g^f>-0.51$, see also Fig. \\ref{fig:en_spectrum}. The inset presents a magnification of $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ for $-1\\leq g^f \\leq-0.4$ .} \n\\label{fig:attractive_spec} \n\\end{figure}\n \nTo gain more insights onto the dynamics, we next resort to the frequency spectrum of the fidelity $F(\\omega)$, shown in Fig. \\ref{fig:attractive_spec} (a) for a varying \npostquench interaction strength. \nThis spectrum provides information about the contribution of the different postquench states that participate in the dynamics. \nIndeed, the square of the fidelity [see Eq. \\eqref{f(t)}] can be expressed as \n\\begin{equation} \\label{fidelity}\n\\begin{split}\n|F(t)|^2=& \\sum_{j} |d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^4\\\\&+2\\sum_{j \\neq k}|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2|d_{\\nu_k^f,\\nu_1^{\\textrm{in}}}|^2\\cos(\\omega_{\\nu_j^f,\\nu_k^f}t),\n\\end{split}\n\\end{equation}\nwhere $d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}$ are the overlap coefficients between the initial (prequench) $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and the final (postquench) $\\ket{\\Psi_{\\nu_j}^f}$ eigenstates. \nThe corresponding overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ for an increasing postquench interaction strength are presented in Fig. \\ref{fig:attractive_spec} (b). \nMoreover, the frequencies $\\omega_{\\nu_j^f,\\nu_k^f}$ are determined by the energy differences between two distinct eigenstates of the postquench Hamiltonian, \nnamely $\\omega_{\\nu_j^f,\\nu_k^f}=2( \\nu_j^f- \\nu_k^f)\\equiv \\omega_{\\nu_j,\\nu_k}$ with $\\quad j\\neq k$. \nNote also that the amplitudes of the frequencies [encoded in the colorbar of Fig. \\ref{fig:attractive_spec} (a)] mainly depend on the product of their respective overlap coefficients, \ni.e. $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2|d_{\\nu_k^f,\\nu_1^{\\textrm{in}}}|^2$. \nFinally, the values of the frequencies $\\omega_{\\nu_j,\\nu_k}$ along with the coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:attractive_spec} (b)] determine the \ndominantly participating postquench eigenstates \\cite{Simos,Mistakidis1,Mistakidis2,Mistakidis3}. \n\nFocusing on region I we observe that in $F(\\omega)$ there are two frequencies, hardly visible in Fig. \\ref{fig:attractive_spec} (a). The most dominant one corresponds to $\\omega_{\\nu_1,\\nu_0}$ for \n$-10.27$ [Fig. \\ref{fig:attractive_spec} (b)] \ngiving rise to the frequency branch $\\omega_{\\nu_1,\\nu_0}$ that at $g^f\\approx 0.54$ has a quite large value of approximately 14.9 and decreases rapidly as $g^f$ increases. \nOf course, this behavior stems directly from the energy gap between the bound, $\\ket{\\Psi_{\\nu_0}^f}$, and the ground, $\\ket{\\Psi_{\\nu_1}^f}$, states as it can be easily confirmed \nby inspecting the eigenspectrum [Fig. \\ref{fig:en_spectrum}]. \nIn the intersection between regions II and III, $\\omega_{\\nu_1,\\nu_0}$ becomes degenerate with the other frequency branches [see the black circles in \nFig. \\ref{fig:attractive_spec} (a)], e.g. $\\omega_{\\nu_4,\\nu_1}$ in the vicinity of $g^f=1$ and $\\omega_{\\nu_3,\\nu_1}$ close to $g^f=3$ [Fig. \\ref{fig:attractive_spec} (a)]. \nThe aforementioned frequency branches are much fainter when compared to $\\omega_{\\nu_1,\\nu_0}$, since the overlap coefficients between the relevant eigenstates are small, \ne.g. $|d_{\\nu_3^f,\\nu_1^{\\textrm{in}}}|^2<|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:attractive_spec} (b)]. \nFinally in region IV, there are mainly two dominant frequencies, namely $\\omega_{\\nu_1,\\nu_0}$ and $\\omega_{\\nu_2,\\nu_1}$, that acquire constant values as $g^f$ increases. \nIndeed, in this region $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$, $|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2$ are the most significantly populated coefficients \n[Fig. \\ref{fig:attractive_spec} (b)], which in turn yield these two frequencies. \n \n \n\\subsection{Role of the initial state}\\label{role_intial_state_attract_repul}\n \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure4-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution when performing a quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from energetically \nhigher excited states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, as well as the bound state $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$ (see legend). \nThe corresponding fidelity spectrum when initializing the system in (b) $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and (c) $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$.}\n\\label{excited_att}\n\\end{figure} \n\nTo investigate the role of the initial eigenstate in the dynamical response of the two bosons, we consider an interaction quench from \n$g^{\\textrm{in}}=-1$ to $g^f=1$ but initializing the system at energetically different excited states i.e. $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, and the bound \nstate $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$. \nIn particular, Fig. \\ref{excited_att} (a) illustrates $\\abs{F(t)}$ with a prequench eigenstate being the bound state, the first, the third, the fifth and the \nseventh excited state. \nIn all cases, $\\abs{F(t)}$ exhibits an irregular oscillatory motion as in the case of $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, see also Fig. \\ref{fig:attractive_ground} (b). \nEvidently, for an energetically higher initial eigenstate (but not the bound state) $\\abs{F(t)}$ takes larger values and therefore the system is less perturbed. \nHowever, when the two bosons are prepared in the bound state, $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$, of the system then $\\abs{F(t)}$ drops to smaller \nvalues as compared to the case of energetically higher initial states and the system becomes more perturbed. \n\nThe impact of the initial state on the oscillation amplitude of $\\abs{F(t)}$ is reflected on the values of the corresponding overlap coefficients that appear in \nthe expansion of the fidelity in Eq. (\\ref{fidelity}). \nMore precisely, when an overlap coefficient possesses a dominant population with respect to the others then $\\abs{F(t)}$ exhibits a smaller oscillation \namplitude than in the case where at least two overlap coefficients possess a non negligible population. \nFor convenience and in order to identify the states that take part in the dynamics, we provide the relevant overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, \nfor the quench $g^{\\textrm{in}}=-1\\rightarrow g^f=1$ in Table \\ref{table1} for various initial eigenstates $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nIndeed, an initial energetically higher-lying excited state results in the dominant population of one postquench state while the other states \nexhibit a very small contribution, e.g. see the last column of Table \\ref{table1}. \nFor this reason an initially energetically higher excited state leads to a smaller oscillation amplitude of $\\abs{F(t)}$. \nMoreover, the large frequency oscillations appearing in $\\abs{F(t)}$ are caused by the presence of several higher than first order eigenstate transitions as e.g. \n$\\omega_{\\nu_6,\\nu_4}$, $\\omega_{\\nu_7,\\nu_4}$, $\\omega_{\\nu_4,\\nu_0}$ in the case of starting from $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ [Fig. \\ref{excited_att} (b)]. \nThe transition mainly responsible for these large frequency oscillations of $\\abs{F(t)}$ involves the bound state $\\ket{\\Psi_{\\nu_0}^f}$.\nIndeed, by inspecting $\\abs{F(t)}$ of different initial configurations shown in Fig. \\ref{excited_att} (a) we observe that starting from energetically higher excited states \nsuch that $\\nu_j>\\nu_4$ the respective contribution of $\\ket{\\Psi_{\\nu_0}^f}$ diminishes [see also Table \\ref{table1}] leading to a decay of the amplitude of these large frequency oscillations of $\\abs{F(t)}$. \nThe aforementioned behavior becomes evident e.g. by comparing $\\abs{F(t)}$ for $\\nu_2^{\\textrm{in}}$ and $\\nu_8^{\\textrm{in}}$ in Fig \\ref{excited_att} (a). \n\nOn the other hand, in order to unveil the participating frequencies in the dynamics of $\\abs{F(t)}$ we calculate its spectrum $|F(\\omega)|$, shown in Figs. \\ref{excited_att}(b), (c). \nWe observe that starting from an energetically higher excited state several frequencies, referring to different eigenstate transitions, are triggered. \nMost of these frequencies which refer to different initial states almost coincide e.g. $\\omega_{\\nu_5,\\nu_4}$ with $\\omega_{\\nu_9,\\nu_8}$, \nsince the energy gap of the underlying eigenstates is approximately the same [see also Fig. \\ref{fig:en_spectrum}]. They possess however a distinct amplitude. \nAdditionally, there are also distinct contributing frequencies e.g. compare $\\omega_{\\nu_4,\\nu_0}$ with $\\omega_{\\nu_8,\\nu_0}$. \nThe latter are in turn responsible for the dependence of the oscillation period of $\\abs{F(t)}$ on the initial eigenstate of the system. \nFinally, let us note that if the system is quenched to other final interaction strengths (not shown here for brevity reasons), \nacross the four dynamical regions identified in Fig. \\ref{fig:attractive_ground}(a), then $\\abs{F(t)}$ follows a similar pattern as discussed \nin Fig. \\ref{excited_att} (a). \n\n\n \n\\begin{table}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|}\\hline\n& {$|d_{\\nu_j^f,\\nu_0^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_2^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_4^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_6^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_8^{\\textrm{in}}}|^2$} \\\\\n\\hline\n{$\\nu_j^f=\\nu_0$} & 0.7896 & 0.0367 & 0.0147 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_1$} & 0.1214 & 0.0198 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_2$} & 0.0351 & 0.8765 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_3$} & 0.0163 & 0.0464 & 0.0187 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_4$} & 0.0092 & 0.0092 & 0.9078 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_5$} & - & - & 0.0351 & 0.0164 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_6$} & - & - & - & 0.9249 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_7$} & - & - & - & 0.0286 & 0.0145 \\\\ \\hline\n{$\\nu_j^f=\\nu_8$} & - & -& -& - & 0.9358 \\\\ \\hline\n{$\\nu_j^f=\\nu_9$} & - & -& -& - & 0.0243 \\\\ \\hline\n\\end{tabular}\n\\caption{Overlap coefficients $|d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}|^2$ for the quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from various excited states, \nnamely $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_2}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_6}^{\\textrm{in}}}$, and $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$. \nOnly the coefficients with a value larger than 0.9\\% are presented. }\\label{table1}\n\\end{table} \n\n\n\\subsection{One-body density evolution}\n \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5 \\textwidth]{figure5-eps-converted-to.pdf}\n\\caption{(a)-(f) Time-evolution of the one-body density following an interaction quench from $g^{\\textrm{in}}=-1$ to $g^f=1$. \nThe system of two bosons is initialized in its ground state, $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, trapped in a 2D harmonic oscillator. \n(g)-(j) The corresponding one-body densities for the pre- and postquench eigenstates (see legends) whose overlap coefficients are the dominant ones \nfor the specific quench.} \n\\label{fig:1RD_-1_1} \n\\end{figure} \n \nTo monitor the dynamical spatial redistribution of the two atoms after the quench at the single-particle level, we next examine the evolution of the \none-body density $\\rho^{(1)}(x,y,t)$ [Eq. \\eqref{density_matrix}]. \nFigures \\ref{fig:1RD_-1_1} (a)-(f) depict $\\rho^{(1)}(x,y,t)$ following an interaction quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ when the system is initialized \nin its ground state configuration $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nNote that the shown time-instants of the evolution lie in the vicinity of the local minima and maxima of the fidelity [see also Fig. \\ref{fig:attractive_ground} (b)], where the \nsystem deviates strongly and weakly from its initial state respectively. \nOverall, we observe that the atoms undergo a breathing motion manifested as a contraction and expansion dynamics of $\\rho^{(1)}(x,y,t)$, see for instance the \nincrease of the density close to $x=y=0$ [Figs. \\ref{fig:1RD_-1_1} (b), (c)] and its subsequent spread [Figs. \\ref{fig:1RD_-1_1} (d), (e)]. \nTo provide further hints on the dynamical superposition \\cite{Sowinski_ent,Katsimiga_diss_flow,Katsimiga_bent} of states we show in Figs. \\ref{fig:1RD_-1_1} (g)-(j) \nthe corresponding $\\rho^{(1)}(x,y,t=0)$ of the initial state, i.e. $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, and the densities of the three most significant, in terms of the overlap coefficients, final states namely \n$\\ket{\\Psi_{\\nu_1}^f},\\ket{\\Psi_{\\nu_0}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$. \nComparing these $\\rho^{(1)}(x,y,t=0)$ with the $\\rho^{(1)}(x,y,t)$ we can deduce that during evolution the one-body density of the system is mainly in a superposition of the \n$\\ket{\\Psi_{\\nu_1}^f}$ and the $\\ket{\\Psi_{\\nu_0}^f}$. \nThe excited state $\\ket{\\Psi_{\\nu_2}^f}$ has a smaller contribution to the dynamics of $\\rho^{(1)}(x,y,t)$ [e.g. see Fig. \\ref{fig:1RD_-1_1} (e)] compared to the other \nstates.\n\n\n\n\\subsection{Evolution of the radial probability density} \n\n\nIn order to gain a better understanding of the nonequilibrium dynamics of the two bosons, we also employ the time-evolution of the radial probability density of \nthe relative wavefunction $\\mathcal{B}(\\rho,t)$ [Eq. \\eqref{prob_dens}]. \nRecall that this quantity provides the probability density of finding the two bosons at a distance $\\rho$ apart for a fixed time-instant. \nThe dynamics of $\\mathcal{B}(\\rho,t)$ after a quench from $g^{\\textrm{in}}=-1$ to $g^f=1$, starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, is \nillustrated at selected time-instants in Fig. \\ref{fig:2RD_attractive} (a). \nWe can infer that the emergent breathing motion of the two bosons is identified via the succession in time of a single [e.g. at $t=0.46 ,1.31$] and a double peak [e.g. at $t=0.84,2.63$] \nstructure in the dynamics of $\\mathcal{B}(\\rho,t)$. \nHere, the one peak is located close to $\\rho=0$ and the other close to the harmonic oscillator length (unity in our choice of units). \nMoreover, by comparing $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_attractive} (a)] with $\\rho^{(1)}(x,y,t)$ [Fig. \\ref{fig:1RD_-1_1}] suggests that a double peak \nstructure in $\\mathcal{B}(\\rho,t)$ refers to an expansion of $\\rho^{(1)}(x,y,t)$ [e.g. at $t=6.09$], while a single peaked $\\mathcal{B}(\\rho,t)$ corresponds to a contraction of $\\rho^{(1)}(x,y,t)$ [e.g. at $t=1.31$]. Indeed, for a double peak structure of $\\mathcal{B}(\\rho,t)$, its secondary maximum always occurs at slightly larger radii than the maximum of a single peak distribution of $\\mathcal{B}(\\rho,t)$, possessing also a more extended tail. This further testifies the expanding (contracting) tendency of the cloud in the former (latter) case.\nTo reveal the microscopic origin of the structures building upon $\\mathcal{B}(\\rho,t)$ we also calculate this quantity [see the inset of Fig. \\ref{fig:2RD_attractive} (a)] \nfor the states $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_0}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$ that primarily contribute to the dynamics in terms \nof the overlap coefficients [see also Fig. \\ref{fig:attractive_spec} (b)]. \nIndeed, comparing $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_attractive} (a)] with $\\mathcal{B}(\\rho)$ of the stationary eigenstates \n[inset of Fig. \\ref{fig:2RD_attractive} (a)], enables us to deduce that $\\mathcal{B}(\\rho,t)$ resides mainly in a superposition of the \nground ($\\ket{\\Psi_{\\nu_1}^f}$), the bound ($\\ket{\\Psi_{\\nu_0}^f}$) and the first excited ($\\ket{\\Psi_{\\nu_2}^f}$) eigenstates. \nAlso, it can be clearly seen that the main contribution stems from the ground state, while the other two states possess a smaller contribution. \nIn particular, the participation of the bound state can be inferred due to the existence of the peak close to $\\rho=0$, which e.g. for $t=0.84$ becomes \nprominent, whereas the presence of the excited state $\\ket{\\Psi_{\\nu_2}^f}$ is discernible from the spatial extent of the $\\mathcal{B}(\\rho,t)$ e.g. \nat $t=2.63$ [Fig. \\ref{fig:2RD_attractive} (a)]. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure6-eps-converted-to.pdf} \n\\caption{(a) Time-evolution of the radial probability density, $\\mathcal{B}(\\rho,t)$, of the two atoms at selected time-instants (see legend) for an interaction quench \nfrom $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from the ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset illustrates $\\mathcal{B}(\\rho)$ of the initial state and different postquench eigenstates (see legend). \n(b) Temporal evolution of the corresponding radial probability density in momentum space, $\\mathcal{C}(k,t)$ at specific time-instants (see legend). \nThe inset depicts $\\mathcal{C}(k)$ of the initial state and various postquench eigenstates (see legend).}\n\\label{fig:2RD_attractive}\n\\end{figure} \n\nTo showcase the motion of the two atoms in momentum space we invoke the evolution of the radial probability density in momentum space $\\mathcal{C}(k,t)$ \\cite{Selim_momentum} illustrated in Fig. \\ref{fig:2RD_attractive} (b) \nfor the quench $g^{\\textrm{in}}=-1\\rightarrow g^f=1$ starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nWe observe that in the course of the dynamics a pronounced peak close to $k=0$ and a secondary one located at values of larger $k$ appear in $\\mathcal{C}(k,t)$. \nMoreover, the breathing motion in momentum space is manifested by the lowering and raising of the zero momentum peak accompanied by a subsequent \nenhancement or reduction of the tail of $\\mathcal{C}(k,t)$, as shown e.g. at $t=0.84, 6.09$. \nNote also that the tail of $\\mathcal{C}(k,t)$ decays in a much slower manner compared to the tail of $\\mathcal{B}(\\rho,t)$. \nIndeed, the latter decays asymptotically as $\\sim e^{-\\rho^2}$ [see also Eq. (\\ref{station})] while by fitting the tail of $\\mathcal{C}(k,t)$ \nwe observe a decay law $\\sim 1\/k^3$ (not shown here for brevity reasons) \\cite{Bellotti,Valiente,momentum_1,momentum_2}. \nAdditionally, in order to unveil the corresponding superposition of states that contribute to the momentum distribution, the inset of Fig. \\ref{fig:2RD_attractive} (b) \npresents $\\mathcal{C}(k)$ of the postquench eigenstates that possess the most significantly populated overlap coefficients [see also Fig. \\ref{fig:attractive_spec} (b)]. \nAs it can be seen, the bound state ($\\ket{\\Psi_{\\nu_0}^f}$) exhibits a broad momentum distribution with a tail that extends to large values of $k$, \nwhile $\\mathcal{C}(k)$ of the ground state ($\\ket{\\Psi_{\\nu_1}^f}$) contributes the most and has a main peak around $k=0$. \nOn the other hand, the excited state ($\\ket{\\Psi_{\\nu_2}^f}$) contributes to a lesser extent, and its presence is mainly identified when the momentum \ndistribution exhibits two nodes, e.g. at $t=2.63$.\n\n\\subsection{Evolution of the contact}\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure7-eps-converted-to.pdf}\n\t\\caption{(a) Temporal evolution of the normalized contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ upon considering an interaction quench from $g^{\\textrm{in}}=-1$ \n\tto $g^f=1$. \n\t(b) The corresponding frequency spectrum.}\n\t\\label{fig:Contact_attractive}\n\\end{figure} \n\nSubsequently we examine the contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ in the course of the evolution after a quench from $g^{\\textrm{in}}=-1$ to \n$g^f=1$, see Fig. \\ref{fig:Contact_attractive} (a). \nRecall that the contact reveals the existence of short-range two-body correlations. \nEvidently $\\mathcal{D}(t)\/\\mathcal{D}(0)$ exhibits an irrregular oscillatory behavior containing a variety of different frequencies. \nIndeed, by inspecting the corresponding frequency spectrum depicted in Fig. \\ref{fig:Contact_attractive} (b), a multitude of frequencies appear. \nThe most predominant frequencies possessing the largest amplitude originate from the energy difference between the bound state, $\\ket{\\Psi_{\\nu_0}}$ and \nenergetically higher-lying states, such as $\\omega_{\\nu_1,\\nu_0}, \\omega_{\\nu_2,\\nu_0}$ and $\\omega_{\\nu_3,\\nu_0}$. \nAlso here $\\omega_{\\nu_2,\\nu_1}$ has a comparable value to $\\omega_{\\nu_3,\\nu_0}$ and thus contributes non-negligibly to the dynamics of \n$\\mathcal{D}(t)\/\\mathcal{D}(0)$. \nMoreover, there is a multitude of other contributing frequencies e.g. $\\omega_{\\nu_8,\\nu_0}$ having an amplitude smaller than $\\omega_{\\nu_3,\\nu_0}$. \nThese frequencies indicate the presence of higher-lying states in the dynamics of the contact. \nThe above-described behavior of $\\mathcal{D}(t)\/\\mathcal{D}(0)$ is expected to occur since the contact is related to short-range \ntwo-body correlations, and as such its dynamics \ninvolves a large number of postquench eigenstates, giving rise to the frequencies observed in Fig. \\ref{fig:Contact_attractive} (b). \n\n\n\\section{Quench dynamics of two repulsive bosons to attractive interactions} \\label{quench_rep} \n\nAs a next step, we shall investigate the interaction quench dynamics of two initially repulsive bosons towards the attractive \nside of interactions. \nIn particular, throughout this section we initialize the system in its ground state configuration $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ \nat $g^{\\textrm{in}}=1$ (unless it is stated otherwise) and perform an interaction quench to the attractive side of the spectrum. \n \n\\subsection{Dynamical response}\n \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure8-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of two bosons after an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to different \nfinal interaction strengths $g^f$. \n(b) Time-evolution of the fidelity for selected postquench interaction strengths (see legend).}\n\\label{fig:repulsive_ground} \n\\end{figure} \n \nIn order to study the dynamical response of the system, we invoke the fidelity evolution [Eq. \\eqref{f(t)}] \\cite{Fogarty2} shown in Fig. \\ref{fig:repulsive_ground} (a) \nwith respect to $g^f$. \nWe observe the appearance of three different dynamical regions, in a similar fashion with the response of the reverse quench scenario \ndiscussed in Section \\ref{response_attract_repul}. \nWithin region I, $0.35-0.51$, see also Fig. \\ref{fig:en_spectrum}.}\n\\label{fig:repulsive_spec}\n\\end{figure} \n\nTo identify the postquench eigenstates that participate in the nonequilibrium dynamics of the two bosons, we next calculate the fidelity spectrum $F(\\omega)$ \n[Fig. \\ref{fig:repulsive_spec} (a)] as well as the most notably populated overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:repulsive_spec} (b)] for a varying \npostquench interaction strength. \nIn region I we observe the occurrence of a predominant frequency, namely $\\omega_{\\nu_2,\\nu_1}$, in $F(\\omega)$. \nThis frequency is associated with the notable population of the coefficients $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:repulsive_spec} (b)]. \nRecall that the amplitude of the frequency peaks appearing in $F(\\omega)$ depends on the participating overlap coefficients, as it \nis explicitly displayed in Eq. \\eqref{fidelity}. \nEntering region II there is a multitude of contributing frequencies, the most prominent of them being $\\omega_{\\nu_2,\\nu_1}$. \nThe appearance of the different frequencies is related to the fact that in this regime $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$ drops significantly for more attractive interactions \naccompanied by the population of other states such as $\\ket{\\Psi_{\\nu_2}^f}$ and $\\ket{\\Psi_{\\nu_3}^f}$ [see Fig. \\ref{fig:repulsive_spec} (b)]. \nIt is important to remember here that at the vertical line $g^f=-0.51$ [see also Fig. \\ref{fig:en_spectrum}] there is a change in the labeling of the eigenstates, \nresulting in the alteration of the frequencies from $\\omega_{\\nu_j,\\nu_k}$ to $\\omega_{\\nu_{j-1},\\nu_{k-1}}$ when crossing this line towards the attractive regime. \nIn region III there are essentially two excited frequencies, namely $\\omega_{\\nu_1,\\nu_0}$ and $\\omega_{\\nu_2,\\nu_1}$. \nThe former is the most dominant since here the mainly contributing states are $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_0}^f}$ as it can be seen \nfrom Fig. \\ref{fig:repulsive_spec} (b). \nNote also that $\\omega_{\\nu_1,\\nu_0}$ increases for decreasing $g^f$, a behavior that reflects the increasing energy gap in the system's energy \nspectrum [Fig. \\ref{fig:en_spectrum}]. \nOn the other hand, the amplitude of $\\omega_{\\nu_2,\\nu_1}$ is weaker and essentially fades away for strong attractive interactions. \nThis latter behavior can be attributed to the fact that the contribution of the $\\ket{\\Psi_{\\nu_2}^f}$ state in this region decreases substantially. \n\n\n\n\n\\subsection{Role of the initial state}\\label{role_intial_state_repul_attract}\n \n \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure10-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of the two bosons when performing a quench from $g^{\\textrm{in}}=1$ to $g^f=-1$ starting from various \nexcited states (see legend). \nThe fidelity spectrum when the system is initially prepared in (b) $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and (c) $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$.}\n\\label{excited_rep}\n\\end{figure} \n\nIn order to expose the role of the initial state for the two-boson dynamics, we explore interaction quenches from $g^{\\textrm{in}}=1$ towards $g^f=-1$ but \ninitializing the system in various excited states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, or the bound state $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$. \nThe emergent dynamical response of the system as captured via $\\abs{F(t)}$ is depicted in Fig. \\ref{excited_rep} (a) starting from the bound, the first, the third, the fifth \nand the seventh excited state. \nInspecting the behavior of $\\abs{F(t)}$ we can infer that the system becomes more perturbed when it is prepared in an energetically lower excited state since the oscillation \namplitude of $\\abs{F(t)}$ increases accordingly, compare for instance $\\abs{F(t)}$ for $\\nu_2^{\\textrm{in}}$ and $\\nu_6^{\\textrm{in}}$. \nMoreover, starting from the bound state the system is significantly perturbed compared to the previous cases and $\\abs{F(t)}$ showcases an irregular oscillatory \nbehavior. \nThis pattern is maintained if the quench is performed to other values of $g^f$ which belong to the attractive regime (not shown here for brevity reasons). \nRecall that a similar behavior of $\\abs{F(t)}$ occurs for the reverse quench process, see Sec. \\ref{role_intial_state_attract_repul} and also Fig. \\ref{excited_att} (a). \n\nThe above-mentioned behavior of the fidelity evolution can be understood via employing the corresponding overlap coefficients $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, see also Eq. (\\ref{fidelity}). \nAs already discussed in Sec. \\ref{role_intial_state_attract_repul}, the fidelity remains close to its initial value in the case that one overlap coefficient dominates with respect \nto the others and deviates significantly from unity when at least two overlap coefficients possess a notable population. \nThe predominantly populated overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, are listed in Table \\ref{Table2} when starting from different initial eigenstates $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nA close inspection of this Table reveals that starting from an energetically higher excited state leads to a lesser amount of contributing overlap coefficients \nwith one among them becoming the dominant one. \nThis behavior explains the decreasing tendency of the oscillation amplitude of $\\abs{F(t)}$ for an initially energetically higher excited state, e.g. compare $\\abs{F(t)}$ of \n$\\ket{\\Psi_{\\nu_2}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_6}^{\\textrm{in}}}$ in Fig. \\ref{excited_rep} (a). \nAccordingly, an initially lower (higher) lying excited state results in a larger (smaller) amount of excitations and thus to more (less) contributing frequencies. The latter can be readily seen by resorting to the fidelity spectrum $|F(\\omega)|$ show in Figs. \\ref{excited_rep} (b) and (c) when starting from $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$ respectively.\n\n\n \n\\begin{table}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|}\\hline\n& {$|d_{\\nu_j^f,\\nu_0^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_2^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_4^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_6^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_8^{\\textrm{in}}}|^2$} \\\\\n\\hline\n{$\\nu_j^f=\\nu_0$} & 0.7896 & 0.0351 & 0.0092 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_1$} & 0.0729 & 0.0556 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_2$} & 0.0367 & 0.8765 & 0.0092 & - &- \\\\ \\hline\n{$\\nu_j^f=\\nu_3$} & 0.0221 & 0.0198 & 0.0399 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_4$} & 0.0147 & - & 0.9078 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_5$} & - & -& 0.0175 & 0.0315 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_6$} & - &- & - & 0.9248 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_7$} & - &- &- & 0.0154 & 0.0262 \\\\ \\hline\n{$\\nu_j^f=\\nu_8$} & -& -& -& - & 0.9357 \\\\ \\hline\n{$\\nu_j^f=\\nu_9$} & - & -& -&- & 0.0138 \\\\ \\hline\n\\end{tabular}\n\\caption{The most significantly populated overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, for the quench from $g^{\\textrm{in}}=1$ to $g^f=-1$ initializing the system at various \ninitial states. \nOnly the coefficients with a value larger than 0.9\\% are shown.}\\label{Table2}\n\\end{table}\n \t \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5 \\textwidth]{figure11-eps-converted-to.pdf}\n\\caption{(a)-(f) Snapshots of the one-body density evolution following an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \n(g)-(j) The corresponding one-body densities for different stationary eigenstates (see legend), that possess the largest overlap coefficients.}\n\\label{fig:1RD_1_-0.2}\n\\end{figure}\n \n \n\\subsection{One-body density evolution} \n\n\nTo visualize the nonequilibrium dynamics of the two-bosons, we next monitor the time-evolution of the one-body density [Eq. \\eqref{density_matrix}] depicted \nin Figs. \\ref{fig:1RD_1_-0.2} (a)-(f) for a quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \nNote that the time-instants portrayed in Fig. \\ref{fig:1RD_1_-0.2} refer to roughly the minima and maxima of the respective fidelity evolution [see Fig. \\ref{fig:repulsive_ground} (b)]. \nOverall, the atomic cloud performs a breathing motion during evolution, namely it expands and contracts in a periodic manner. \nMoreover, we deduce that when the fidelity is minimized [e.g. at $t=1.5,4.53,7.54$], the one-body density expands [Figs. \\ref{fig:1RD_1_-0.2} (a), (c) and (e)], while for the case of a maximum fidelity \n$\\rho^{(1)}(x,y,t)$ contracts [Figs. \\ref{fig:1RD_1_-0.2} (b), (f)]. \nTo understand which states are imprinted in $\\rho^{(1)}(x,y,t)$ we further show in Figs. \\ref{fig:1RD_1_-0.2} (g)-(j) $\\rho^{(1)}(x,y,t=0)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ \nand the three most significantly populated, according to the overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$, final states i.e. $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$ and \n$\\ket{\\Psi_{\\nu_3}^f}$ \\cite{Katsimiga_bent,Katsimiga_quantum_DBs}. \nComparing the $\\rho^{(1)}(x,y,t=0)$ of these stationary states with $\\rho^{(1)}(x,y,t)$ it becomes evident that during evolution $\\rho^{(1)}(x,y;t)$ is mainly in a superposition of the ground \nstate [Fig. \\ref{fig:1RD_1_-0.2} (i)] and the first excited state [Fig. \\ref{fig:1RD_1_-0.2} (h)]. \n \n \n \n\\subsection{Evolution of the radial probability density} \n\nAs a next step, we examine the evolution of the radial probability density $\\mathcal{B}(\\rho,t)$ [Eq. \\eqref{prob_dens}] presented \nin Fig. \\ref{fig:2RD_repulsive} (a) for a quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \nNote that the snapshots of $\\mathcal{B}(\\rho,t)$ depicted in Fig. \\ref{fig:2RD_repulsive} (a) correspond again to time-instants at which the \nfidelity evolution exhibits local minima and maxima [see also Fig. \\ref{fig:repulsive_ground} (b)]. \nWe observe that when $\\abs{F(t)}$ is minimized, e.g. at $t=1.50,4.00,7.74$, $\\mathcal{B}(\\rho,t)$ shows a double peak structure around \n$\\rho \\approx0.5$ and $\\rho \\approx 2$ respectively. \nHowever, for times that correspond to a maximum of the fidelity, e.g. at $t=3.1,6.17$, $\\mathcal{B}(\\rho,t)$ deforms to a single peak \ndistribution around $\\rho \\approx1.2$. \nTo relate this alternating behavior of $\\mathcal{B}(\\rho,t)$ with the breathing motion of the two bosons we can infer that when \n$\\mathcal{B}(\\rho,t)$ possesses a double peak distribution the cloud expands while in the case of a single peak \nstructure it contracts, see also Fig. \\ref{fig:1RD_1_-0.2}. \nIt is also worth mentioning here that for the times at which $\\mathcal{B}(\\rho,t)$ exhibits a double peak structure there \nis a quite significant probability density tail for $\\rho>1.5$. \nThis latter behavior is a signature of the participation of energetically higher-lying excited states as we shall discuss below. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure12-eps-converted-to.pdf}\n\\caption{(a) Temporal evolution of the radial probability density, $\\mathcal{B}(\\rho,t)$, upon considering a quench from $g^{\\textrm{in}}=1$ to $g^f=-0.2$ \nstarting from the ground state, $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset shows $\\mathcal{B}(\\rho)$ of the prequench state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of the postquench eigenstates $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$ \nwith the most relevant overlap coefficients. \n(b) The corresponding $\\mathcal{C}(k,t)$ of (a). \nThe inset presents $\\mathcal{C}(k)$ of the $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of the $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$. }\n\\label{fig:2RD_repulsive}\n\\end{figure} \n\nIndeed, the inset of Fig. \\ref{fig:2RD_repulsive} (a) depicts $\\mathcal{B}(\\rho)$ of the initial ($\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$) and the postquench \n($\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$) states that have the major contribution for this specific quench in terms of the overlap coefficients [see also Fig. \\ref{fig:repulsive_spec} (b)]. \nComparing $\\mathcal{B}(\\rho,t)$ with $\\mathcal{B}(\\rho)$ we can deduce that mainly the ground, $\\ket{\\Psi_{\\nu_1}^f}$, and the first excited, $\\ket{\\Psi_{\\nu_2}^f}$, states \nof the postquench system are imprinted in the dynamics of the relative density. \nMore specifically, $\\ket{\\Psi_{\\nu_2}^f}$ gives rise to the enhanced tail of $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_repulsive} (a)], while the participation of $\\ket{\\Psi_{\\nu_1}^f}$ \n(possessing also the major contribution) leads to the central peak of $\\mathcal{B}(\\rho,t)$ close to $\\rho=0$. \n\nThe radial probability density in momentum space \\cite{Selim_momentum}, $\\mathcal{C}(k,t)$, is shown in Fig. \\ref{fig:2RD_repulsive} (b) for selected time instants \nof the evolution following the quench $g^{\\textrm{in}}=1\\rightarrow g^f=-0.2$. \nWe observe that $\\mathcal{C}(k,t)$ exhibits always a two peak structure with the location and amplitude of the emergent peaks being changed \nin the course of the evolution. \nIn particular, when the atomic cloud contracts e.g. at $t=3.10,9.19$, see also Figs. \\ref{fig:1RD_1_-0.2} (b), (f), $\\mathcal{C}(k,t)$ has a large amplitude peak around $k\\approx0.1$ \nand a secondary one of small amplitude close to $k\\approx0.4$. \nHowever, for an expansion of the two bosons e.g. at $t=1.50$ [Figs. \\ref{fig:1RD_1_-0.2} (a)] the radial probability density in momentum space shows a small \nand a large amplitude peak around $k\\approx0.05$ and $k\\approx0.3$ respectively. \nMoreover, the momentum distribution during evolution is mainly in a superposition of the ground $\\ket{\\Psi_{\\nu_1}^f}$ and the first excited state $\\ket{\\Psi_{\\nu_2}^f}$, see \nin particular the inset of Fig. \\ref{fig:2RD_repulsive} (b) which illustrates $\\mathcal{C}(k)$ of these stationary states. \nAs it can be readily seen, $\\ket{\\Psi_{\\nu_2}^f}$ is responsible for the secondary peak of $\\mathcal{C}(k,t)$ at higher momenta, while the ground state contributes \nmainly to the peak close to $k=0$.\n\n\\subsection{Dynamics of the contact}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure13-eps-converted-to.pdf}\n\t\\caption{(a) Time-evolution of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ following a quench from $g^{\\textrm{in}}=1$ to $g^f=-1$. \n\t(b) The corresponding frequency spectrum.}\n\t\\label{fig:Contact_repulsive}\n\\end{figure}\n\nTo unravel the emergence of short-range two-body correlations we next track the time-evolution of the rescaled contact \n$\\mathcal{D}(t)\/\\mathcal{D}(0)$ after an interaction quench from $g^{\\textrm{in}}=1$ to $g^f=-1$, see Fig. \\ref{fig:Contact_repulsive} (a). \nAs it can be seen, the rescaled contact exhibits an irregular multifrequency oscillatory pattern in time. \nIt is also worth mentioning that here the involved frequencies in the dynamics of $\\mathcal{D}(t)\/\\mathcal{D}(0)$ are smaller \nwhen compared to the ones excited in the reverse quench scenario, see in particular Fig. \\ref{fig:Contact_repulsive} (b) and \nFig. \\ref{fig:Contact_attractive} (b). \nBy inspecting the corresponding frequency spectrum presented in Fig. \\ref{fig:Contact_repulsive} (b), we can deduce that the most prominent \nfrequency $\\omega_{\\nu_1,\\nu_0}\\approx2.5$ corresponds to the energy difference between the bound and the ground state. \nMoreover this predominant frequency is smaller than the corresponding dominant frequency $\\omega_{\\nu_1,\\nu_0}\\approx7.5$ occuring at the reverse \nquench process [Fig. \\ref{fig:Contact_attractive} (b)]. \nThere is also a variety of other contributing frequencies which signal the participation of higher-lying states in the evolution of the contact, \nsuch as $\\omega_{\\nu_7,\\nu_0}$, $\\omega_{\\nu_2,\\nu_1}$, $\\omega_{\\nu_3,\\nu_1}$ and $\\omega_{\\nu_2,\\nu_0}$, \nexhibiting however a much smaller amplitude as compared to $\\omega_{\\nu_1,\\nu_0}$. \nThese frequencies are essentially responsible for the observed irregular motion of $\\mathcal{D}(t)\/\\mathcal{D}(0)$. \n\n\n\n\\section{Quench from zero to Infinite interactions} \\label{inf_quench} \n\nUp to now we have discussed in detail the interaction quench dynamics of two bosons trapped in a 2D harmonic trap for weak, intermediate and strong coupling in both the \nattractive and the repulsive regime. \nNext, we aim at briefly analyzing the corresponding interaction quench dynamics from $g^{\\textrm{in}}=0$ to $g^f=\\infty$. \nWe remark here that when the system is initialized at $g^{\\textrm{in}}=0$ the formula of Eq. (\\ref{overlap}) is no longer valid and the overlap coefficients between the \neigenstates $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_j}^{f}}$ are given by \n\\begin{eqnarray}\n d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}&=&\\frac{2\\Gamma(-\\nu_j^f)}{\\sqrt{\\psi^{(1)}(-\\nu_j^f)}} \\int_0^{\\infty} dr \\,r e^{-r^2} U(-\\nu_j^f,1,r^2) L_{\\nu_i^{\\textrm{in}}} (r^2) \\nonumber \\\\ &= &\\frac{1}{(\\nu_i^{\\textrm{in}}-\\nu_j^f)\\sqrt{\\psi^{(1)}(-\\nu_j^f)}}.\n\\end{eqnarray}\n\n \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure14-eps-converted-to.pdf}\n\\caption{Fidelity evolution when applying an interaction quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$. \nThe system is initialized in different eigenstates (see legend).}\n\\label{inf_fid}\n\\end{figure} \n\nThe dynamical response of the system after such a quench [$g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$] as captured by the fidelity evolution [Eq. \\eqref{f(t)}] is illustrated \nin Fig. \\ref{inf_fid} when considering different initial states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nEvidently, when the system is initialized in its ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, $\\abs{F(t)}$ performs large amplitude oscillations. \nThe latter implies that the time-evolved wavefunction becomes almost orthogonal to the initial one at certain time intervals and as a consequence the system \nis significantly perturbed. \nAlso, it can directly be deduced by the fidelity evolution that when the system is prepared in an energetically higher excited state it is less perturbed since the \noscillation amplitude of $\\abs{F(t)}$ is smaller, e.g. compare $\\abs{F(t)}$ for $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_5}^{\\textrm{in}}}$. \nThis tendency which has already been discussed in Secs. \\ref{role_intial_state_attract_repul} and \\ref{role_intial_state_repul_attract} can be explained \nin terms of the distribution of the amplitude of the overlap coefficients, see also Eq. (\\ref{fidelity}). \nIndeed, if there is a single dominant overlap coefficient then $|F(t)|\\approx1$, while if more than one overlap coefficients possess large values $\\abs{F(t)}$ deviates \nappreciably from unity. \nHere, for instance, the first two most dominant overlap coefficients when starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_5}^{\\textrm{in}}}$ \nare $|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2=0.4837$, $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2=0.4402$ and $|d_{\\nu_4^f,\\nu_5^{\\textrm{in}}}|^2=0.6453$, $|d_{\\nu_5^f,\\nu_5^{\\textrm{in}}}|^2=0.1894$ respectively. \n \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure15-eps-converted-to.pdf}\n\\caption{(a) Radial probability, $\\mathcal{B}(\\rho,t)$, at specific time-instants of the evolution following an interaction quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$. \nThe system is prepared in its ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset illustrates $\\mathcal{B}(\\rho)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and some of the postquench eigenstates $\\ket{\\Psi_{\\nu_0}^f}$, $\\ket{\\Psi_{\\nu_1}^f}$ \nand $\\ket{\\Psi_{\\nu_2}^f}$. \n(b) Time-evolution of the corresponding radial probability density in momentum space, $\\mathcal{C}(k,t)$. \nThe inset shows $\\mathcal{C}(k)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of certain postquench eigenstates, namely $\\ket{\\Psi_{\\nu_0}^f}$, \n$\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$.}\n\\label{inf_relwave}\n\\end{figure} \n\nTo further unravel the motion of the two bosons we next employ the time-evolution of their radial probability density, $\\mathcal{B}(\\rho,t)$, in real space [see also Eq. \\eqref{prob_dens}]. \nFigure \\ref{inf_relwave} (a) shows snapshots of $\\mathcal{B}(\\rho,t)$ after an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=0$ to\n$g^f=\\infty$. \nAs it can be seen for the time intervals that $\\abs{F(t)}$ is minimized [Fig. \\ref{inf_fid}], e.g. at $t=0.78,2.42,5.61$, $\\mathcal{B}(\\rho,t)$ exhibits a pronounced peak close to $\\rho=0$ \nand a secondary one at a larger radii $\\rho\\approx 1.5$. \nHowever, when $|F(t)|\\approx1$ ($t=1.62, 3.13, 8.04$) $\\mathcal{B}(\\rho,t)$ shows a more delocalized distribution. \nTo explain this behavior of $\\mathcal{B}(\\rho,t)$ we next calculate $\\mathcal{B}(\\rho)$ of the initial state (i.e. $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$) and of the postquench eigenstates \nthat possess the most dominant overlap coefficients, namely $\\ket{\\Psi_{\\nu_0}^f}$, $\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$, following the above-described \nquench scenario [see the inset of Fig. \\ref{inf_relwave} (a)]. \nComparing $\\mathcal{B}(\\rho,t)$ with $\\mathcal{B}(\\rho)$ we observe that the bound state, $\\ket{\\Psi_{\\nu_0}^f}$, gives rise to the prominent peak close \nto $\\rho=0$ [see Fig. \\ref{inf_relwave} (a)]. \nMoreover, the states $\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$ are responsible for the emergent spatial delocalization of $\\mathcal{B}(\\rho,t)$. \nOf course, the ground state ($\\ket{\\Psi_{\\nu_1}^f}$) plays a more important role here than the first excited state ($\\ket{\\Psi_{\\nu_2}^f}$), since \n$|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2=0.4402$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2=0.0406$ respectively [see the inset of Fig. \\ref{inf_relwave} (a)]. \n\nTurning to the dynamics in momentum space, Fig. \\ref{inf_relwave} (b) presents $\\mathcal{C}(k,t)$ at specific time-instants for the quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$ \nstarting from the ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nWe observe that when the system deviates notably from its initial state (i.e. $t=0.78,2.42,5.61$) meaning also that $|F(t)|\\ll1$, then $\\mathcal{C}(k,t)$ shows a two peak structure \nwith the first peak located close to $k=0$ and the second one at $k\\approx0.4$. \nNotice also here that the tail of $\\mathcal{C}(k,t)$ has an oscillatory behavior. \nOn the other hand, if $\\abs{F(t)}$ is close to unity (e.g. at $t=1.62, 3.13, 8.04$) where also $\\mathcal{B}(\\rho,t)$ is spread out [Fig. \\ref{inf_relwave} (a)], \nthe corresponding $\\mathcal{C}(k,t)$ has a narrow momentum peak close to zero and a fastly decaying tail at large $k$. \n\nThe inset of Fig. \\ref{inf_relwave} (b) illustrates $\\mathcal{C}(k)$ of the initial eigenstate and some specific postquench ones which possess the \nlargest contributions for the considered quench according to the overlap coefficients. \nIt becomes evident that both the bound state, $\\ket{\\Psi_{\\nu_0}^f}$, and the ground state, $\\ket{\\Psi_{\\nu_1}^f}$, of the postquench system are mainly \nimprinted in $\\mathcal{C}(k,t)$. \nIndeed, the bound state has a broad momentum distribution whereas the ground state possesses a main peak close to $k=0$. \nOn the other hand, the first excited state ($\\ket{\\Psi_{\\nu_2}^f}$) has a smaller contribution compared to the previous ones and its presence can be discerned in Fig. \\ref{inf_relwave} (b) \nfrom the oscillatory tails of $\\mathcal{C}(k,t)$ at large momenta. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure16-eps-converted-to.pdf}\n\t\\caption{(a) Time-evolution of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ for the interaction quench from $g^{\\textrm{in}}=0.2$ to $g^f=\\infty$. \n\t(b) The respective frequency spectrum $D(\\omega)$.}\n\t\\label{fig:Contact_inf}\n\\end{figure}\n\nFinally, we examine the dynamics of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ illustrated in Fig. \\ref{fig:Contact_inf} (a) \nfollowing a quench from $g^{\\textrm{in}}=0.2$ to $g^f=\\infty$. \nNote here that we choose $g^{\\textrm{in}}=0.2$, and not exactly $g^{\\textrm{in}}=0$, since the contact is well-defined only for interacting eigenstates \\cite{Tan1}. \nEvidently $\\mathcal{D}(t)\/\\mathcal{D}(0)$ undergoes a large amplitude multifrequency oscillatory motion. \nThe large amplitude of these oscillations stems from the fact that the system is quenched to unitarity and therefore the built up of short-range two-body \ncorrelations is substantial especially when compared to the correlations occuring for finite interactions as e.g. the ones displayed in \nFig. \\ref{fig:Contact_attractive} (a) and Fig. \\ref{fig:Contact_repulsive} (a). \nWe remark that similar large amplitude oscillations of the contact, at the frequency of the two-body bound state, have already been observed in Ref. \\cite{Corson} \nduring the interaction quench dynamics of a three dimensional homogeneous BEC from zero to very large interactions.\nRegarding the participating frequencies identified in the spectrum of the contact shown in Fig. \\ref{fig:Contact_inf} (b), we can clearly infer \nthat the dominant frequencies refer to the energy differences between the bound state, $\\ket{\\Psi_{\\nu_0}}$ and higher-lying states \ne.g. $\\omega_{\\nu_1,\\nu_0}$, $\\omega_{\\nu_2,\\nu_0}$. \nThe existence of other contributing frequencies in the spectrum, such as $\\omega_{\\nu_2,\\nu_1}$ and $\\omega_{\\nu_3,\\nu_0}$, has also an impact on the \ndynamics of the contact and signal the involvement of higher-lying states.\n\n\n\n\\section{Conclusions}\\label{conclusions} \n\n\nWe have explored the quantum dynamics of two bosons trapped in an isotropic two-dimensional harmonic trap, and interacting via a contact \n$s$-wave pseudo-potential. \nAs a first step, we have presented the analytical solution of the interacting two-body wavefunction for an arbitrary stationary eigenstate. \nWe also briefly discuss the corresponding two-body energy eigenspectrum covering both the attractive and \nrepulsive interaction regimes, showcasing the importance of the existing bound state. \n\nTo trigger the dynamics we consider an interaction quench from repulsive to attractive interactions and vice versa as well as a quench from zero to \ninfinite interactions. \nHaving the knowledge of the stationary properties of the system the form of the time-evolving two-body wavefunction is provided. \nMost importantly, we showcase that the expansion coefficients can be derived in a closed form and therefore the dynamics of the two-body wavefunction can be \nobtained by numerically determining its expansion with respect to the eigenstates of the postquench system. \nIn all cases, the dynamical response of the system has been analyzed in detail and the underlying eigenstate transitions that mainly contribute to the dynamics have \nbeen identified in the fidelity spectrum together with the system's eigenspectrum. \n\nWe have shown that initializing the system in its ground state, characterized by either repulsive or attractive interactions, it is driven more efficiently out-of-equilibrium, \nas captured by the fidelity evolution, when performing an interaction quench towards the vicinity of zero interactions. \nHowever, if we follow a quench towards the intermediate or strong coupling regimes of either sign, then the system remains close to its initial state. \nAs a consequence of the interaction quench the two bosons undergo a breathing motion which has been visualized by monitoring the temporal evolution \nof the single-particle density and the radial probability density, in both real and momentum space. \nThe characteristic structures building upon the above-mentioned quantities enable us also to infer about the participation of energetically higher-lying \nexcited states of the postquench system. \n\nTo inspect the dependence of the system's dynamical response we have examined also quenches for a variety of different initial states such as the bound state \nor an energetically higher excited state in both the repulsive and attractive interaction regimes. \nIt has been found that starting from energetically higher excited states, the system is perturbed to a lesser extent, and a fewer amount of postquench eigenstates \ncontribute in the emergent dynamics. \nA crucial role here is played by the bound state of the postquench system, both in the attractive and the repulsive regime, whose contribution is essentially \ndiminished as we initialize the two bosons at higher excited states. \nOn the other hand, when the quench is performed from the bound state, independently of the interaction strength, the system is driven out-of-equilibrium \nin the most efficient manner than any other initial state configuration. \n\nAdditionally, upon quenching the system from zero to infinite interactions starting from its ground state the time-evolved wavefunction becomes even orthogonal to the initial \none at certain time intervals. \nAgain here, if the two bosons are prepared in an energetically higher excited state then the system becomes more unperturbed. \nInspecting the evolution of the radial probability density we have identified that it mainly resides in a superposition of the bound and the ground state alternating from a \ntwo peaked structure to a more spread distribution.\n\nTo unveil the emergence of short-range two-body correlations we have examined the dynamics of the Tan's contact in all of the above-mentioned quench scenaria. \nIn particular, we have found that the contact performs a multifrequency oscillatory motion in time. \nThe predominant frequency of these oscillations refers to the energy difference between the bound and the ground states. \nThe participation of other frequencies possessing a comparable smaller amplitude signals the contribution of higher-lying states in the dynamics of the contact. \nMoreover, upon quenching the system from weak to infinite interactions, the oscillation amplitude of the contact is substantially enhanced indicating the significant development \nof short-range two-body correlations as compared to the correlations occuring at finite postquench interactions. \n\nThere is a variety of fruitful directions to follow in future works. \nAn interesting one would be to consider two bosons confined in an anisotropic two-dimensional harmonic trap and examine the stationary properties of this system \nin the dimensional crossover from two- to one-dimensions. \nHaving at hand such an analytical solution would allow us to study the corresponding dynamics of the system upon changing its dimensionality e.g. by considering \na quench of the trap frequency in one of the spatial directions which enable us to excite higher than the monopole mode. \nAlso one could utilize the spectra with respect to the different anisotropy in order to achieve controllable state transfer processes \\cite{Fogarty,Reshodko}. \nBesides the dimensionality crossover, it would be interesting to study the effect of the presence of the temperature in the interaction quench dynamics examined herein. \nFinally, the dynamics of three two-dimensional trapped bosons requires further investigation. \nEven though the Efimov effect is absent in that case \\cite{Nielsen}, the energy spectrum is rich possessing dimer and trimer states \\cite{Drummond} and the corresponding \ndynamics might reveal intriguing dynamical features when quenching from one to another configuration. \n\n\n\\begin{acknowledgements} \n\nG. B. kindly acknowledges financial support by the State Graduate Funding Program Scholarships (HmbNFG). S. I. M and P. S gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB \n925 ``Light induced dynamics and control of correlated quantum systems\". The authors thank G.M. Koutentakis for fruitful discussions. \n\n\\end{acknowledgements}\n \n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Conformal generators on the circle}\n\n\\subsection{Virasoro generators}\n\nA basis of generators of conformal symmetry on the circle $x \\in [0,L)$ is given by the Virasoro generators $L_n$ and $\\bar{L}_{n}$ \\cite{CFT2}, \n\\begin{eqnarray}\nL_n &\\equiv& \\frac{L}{(2\\pi)^2} \\int_{0}^{L} dx~e^{i\\frac{2\\pi}{L}nx } T(x) + \\frac{c}{24}\\delta_{n,0},\\\\\n\\bar{L}_n &\\equiv& \\frac{L}{(2\\pi)^2} \\int_{0}^{L} dx~e^{-i\\frac{2\\pi}{L}nx } \\bar{T}(x) + \\frac{c}{24}\\delta_{n,0},\n\\end{eqnarray}\nwhich are the Fourier modes of the holomorphic and antiholomorphic components of the stress tensor\n\\begin{eqnarray}\nT(x) &\\equiv& 2\\pi\\frac{h(x) + p(x)}{2},\\\\\n\\bar{T}(x) &\\equiv& 2\\pi\\frac{h(x) - p(x)}{2}.\n\\end{eqnarray}\nHere $h(x)$ and $p(x)$ are the CFT hamiltonian and momentum densities, which we can also use to write the CFT hamiltonian and momentum operators on the circle,\n\\begin{eqnarray}\nH^{\\mbox{\\tiny CFT}} &\\equiv& \\int_0^{L} dx~h(x) = \\frac{2\\pi}{L} \\left(L_0 +\\bar{L}_0 - \\frac{c}{12}\\right),\\\\\nP^{\\mbox{\\tiny CFT}} &\\equiv& \\int_0^{L} dx~p(x) = \\frac{2\\pi}{L} \\left(L_0 -\\bar{L}_0\\right).\n\\end{eqnarray} \n\n\n\nThe Virasoro generators $L_n, \\bar{L}_n$ can be seen to close two copies of the Virasoro algebra,\n\\begin{eqnarray} \\label{eq:Virasoro}\n\\big[L_n, L_m\\big] &=& (n-m)L_{n+m} + \\frac{c}{12}n(n^2-1)\\delta_{n+m,0},~~~~~ \\\\\n\\big[ L_n, \\bar{L}_m \\big] &=& 0, ~~~\\\\\n\\big[\\bar{L}_n, \\bar{L}_m\\big] &=& (n-m)\\bar{L}_{n+m} + \\frac{c}{12}n(n^2-1)\\delta_{n+m,0},~~~~~\n\\end{eqnarray} \nwhere $c$ is the central charge of the CFT.\n\n\n\n\\subsection{Generators of non-uniform euclidean time evolution}\n\nLet us now consider the Fourier mode expansion of the Hamiltonian density $h(x)$,\n\\begin{eqnarray}\nh(x) &\\equiv& \\left(\\frac{2\\pi}{L}\\right) \\frac{1}{L} \\sum_n H_n ~e^{-i\\frac{2\\pi}{L}nx},\\\\\nH_n &\\equiv& \\left(\\frac{L}{2\\pi}\\right) \\int_{0}^{L}dx~ e^{i\\frac{2\\pi}{L}nx} ~h(x),\n\\end{eqnarray}\nwhere we use the unusual factor $L\/2\\pi$ so that \n\\begin{equation}\nH_n \\equiv L_{n} + \\bar{L}_{-n} -\\frac{c}{12}\\delta_{n,0}.\n\\end{equation} \nIn particular the Hamiltonian reads $H^{\\mbox{\\tiny CFT}} = (2\\pi\/L) H_0$.\nSimilarly, let $a(x)$ be a real function on the circle $x \\in [0,L)$, with Fourier expansion\n\\begin{eqnarray}\na(x) &\\equiv& \\sum_n a_n ~e^{i\\frac{2\\pi}{L}nx},\\\\\na_n &\\equiv& \\frac{1}{L} \\int_{0}^{L} dx~ e^{-i\\frac{2\\pi}{L}nx}~a(x).\n\\end{eqnarray}\nThen we have that the generator $Q_0$ of a non-uniform euclidean time evolution with profile function $a(x)$ reads\n\\begin{eqnarray}\nQ_0 &\\equiv& \\int_0^{L} dx~ a(x) h(x) \\\\\n&=& \\frac{2\\pi}{L}\\frac{1}{L}\\int_0^L dx~ \\sum_n a_n ~e^{i\\frac{2\\pi}{L}nx} \\sum_m H_m ~e^{-i\\frac{2\\pi}{L}mx}~~~~\\\\\n&=& \\frac{2\\pi}{L}~ \\sum_n a_n ~ \\sum_m H_m \\left(\\frac{1}{L}\\int_0^L dx~e^{i\\frac{2\\pi}{L}(n-m)x}\\right)~~~~\\\\\n&=& \\frac{2\\pi}{L} \\sum_{n} a_nH_n,\n\\end{eqnarray}\nwhere we used that $(1\/L)\\int_0^L dx ~e^{i\\frac{2\\pi}{L}(n-m)x} = \\delta_{m,n}$.\n\n\\subsection{Generators of non-uniform rescaling}\n\nLet us now consider the Fourier mode expansion of the momentum density $p(x)$,\n\\begin{eqnarray}\np(x) &\\equiv& \\left(\\frac{2\\pi}{L}\\right) \\frac{1}{L} \\sum_n P_n ~e^{-i\\frac{2\\pi}{L}nx},\\\\\nP_n &\\equiv& \\left(\\frac{L}{2\\pi}\\right) \\int_{0}^{L}dx~ e^{i\\frac{2\\pi}{L}nx} ~p(x),\n\\end{eqnarray}\nwhere we again use the unusual factor $L\/2\\pi$ so that \n\\begin{equation}\nP_n \\equiv L_{n} - \\bar{L}_{-n},\n\\end{equation}\nwith the momentum operator reading $P^{\\mbox{\\tiny CFT}} = (2\\pi\/L)P_0$. Let $b(x)$ be a real function on the circle $x \\in [0,L)$, with Fourier expansion\n\\begin{eqnarray}\nb(x) &\\equiv& \\sum_n b_n ~e^{i\\frac{2\\pi}{L}nx},\\\\\nb_n &\\equiv& \\frac{1}{L} \\int_{0}^{L} dx~e^{-i\\frac{2\\pi}{L}nx}~b(x).\n\\end{eqnarray}\nThen the generator $Q_1$ of a non-uniform rescaling with profile function $b(x)$ reads\n\\begin{eqnarray}\nQ_1 \\equiv \\int_0^{L} dx~ b(x) p(x) = \\frac{2\\pi}{L} \\sum_{n} b_nP_n.\n\\end{eqnarray}\n\n\\section{Low energy matrix elements}\n \nConsider a CFT on a circle of perimeter $L$ and a conformal transformation $V^{\\mbox{\\tiny CFT}} \\equiv e^{- Q}$, with generator $Q$, acting on its Hilbert space. Our goal is to compute the matrix elements\n\\begin{eqnarray}\nV^{\\mbox{\\tiny CFT}}_{\\alpha \\beta} &\\equiv& \\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} V^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}\\\\\n&=& \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{n!}\\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} Q^n \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}},\n\\end{eqnarray}\nbetween simultaneous eigenstates of the hamiltonian and momentum operators $H^{\\mbox{\\tiny CFT}}$ and $P^{\\mbox{\\tiny CFT}}$,\n\\begin{eqnarray}\nH^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} &=& E^{\\mbox{\\tiny CFT}}_{\\alpha} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}, \\\\\nP^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} &=& P^{\\mbox{\\tiny CFT}}_{\\alpha} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}},\n\\end{eqnarray}\nwhere, by the CFT operator-state correspondence \\cite{CFT2}, \n\\begin{eqnarray}\nE^{\\mbox{\\tiny CFT}}_{\\alpha} &=& \\frac{2\\pi}{L}\\left( \\Delta_{\\alpha} -\\frac{c}{12}\\right),\\\\\nP^{\\mbox{\\tiny CFT}}_{\\alpha} &=& \\frac{2\\pi}{L}S_{\\alpha}.\n\\end{eqnarray}\nHere $\\Delta_{\\alpha} \\equiv h_{\\alpha}+\\bar{h}_{\\alpha}$ and $S_{\\alpha}\\equiv h_{\\alpha} -\\bar{h}_{\\alpha}$ are the scaling dimension and conformal spin of the scaling operator $\\phi_{\\alpha}$ corresponding to state $\\ket{\\phi_{\\alpha}}$, with $h_{\\alpha}$ and $\\bar{h}_{\\alpha}$ its holomorphic and antiholomorphic conformal dimensions \\cite{CFT2}. \n\nThe resulting $\\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} V^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}$ for specific choices of profile functions $a(x)$ and $b(x)$ (described below), were compared in Figs. \\ref{fig:time}-\\ref{fig:scale} of the main text with matrix elements $\\bra{\\phi_\\beta} V \\ket{\\phi_{\\alpha}}$ numerically obtained on the spin chain, where $\\ket{\\phi_{\\alpha}}$ and $V$ denote an energy\/momentum eigenstate of the spin chain and a linear map implemented by a tensor network, respectively.\n\nThe generator $Q = Q_0 + iQ_1$ can be expanded in terms of the Fourier modes $H_n = L_n +\\bar{L}_{-n}$ and $P_n=L_n -\\bar{L}_{-n}$ of the hamiltonian and momentum densities $h(x)$ and $p(x)$, namely\n\\begin{equation}\nQ = \\frac{2\\pi}{L}\\sum_{n\\in \\mathbb{Z}} \\left(a_n H_n + ib_n P_n\\right)\n\\end{equation} \nThus, we would like to compute the matrix elements $\\bra{\\phi_\\beta} Q^n \\ket{\\phi_{\\alpha}}$, were $Q^n$ is a sum of products of powers of $H_n$ and $P_n$. To do this computation, we express $Q^n$ as a sum of products of Virasoro generators such as $X \\equiv \\left(L_{n_1} \\dots L_{n_r}\\right) \\left(\\bar{L}_{\\bar{n}_1} \\dots \\bar{L}_{\\bar{n}_s} \\right)$ for some integers $r$ and $s$ and apply the Virasoro algebra. As a first step, we express the states $|\\phi_\\alpha^{\\mbox{\\tiny CFT}}\\rangle$ in terms of Virasoro generators acting on primary states.\n\n\\subsection{Descendant states in terms of primary states}\n\nTo keep the notation relatively simple, given a primary state $\\ket{h,\\bar{h}}$ we consider a descendant state $\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}$ that is obtained by applying a sequence $L_{n_1}L_{n_2} \\cdots L_{n_q}$ of holomorphic Virasoro generators $L_n$,\n\\begin{equation} \n\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} = \\frac{1}{N_{\\alpha}} L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}},\n\\end{equation}\nwhere $n_i \\leq -1$ and $N_{\\alpha}$ is a normalization factor such that $\\braket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}=1$. $N_{\\alpha}$ can be expanded as\n\\begin{eqnarray}\nN_{\\alpha}^2 &=& \\bra{h,\\bar{h}}\\left(L_{n_1}L_{n_2} \\cdots L_{n_q}\\right)^{\\dagger} L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}} ~~~\\\\\n&=& \\bra{h,\\bar{h}} \\left(L_{n_q}^{\\dagger} \\cdots L_{n_2}^{\\dagger} L_{n_1}^{\\dagger} \\right) L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}}~~~\\\\\n&=& \\bra{h,\\bar{h}} \\left(L_{-n_q} \\cdots L_{-n_2} L_{-n_1} \\right) L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}},~~~~~\n\\end{eqnarray}\nwhere we used that $L_n^{\\dagger} = L_{-n}$. A more general descendant state is descended from a primary state via both $L_n$ and $\\overline{L}_{\\bar{n}}$ generators:\n\\begin{equation}\\label{eq:desc_state}\n\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} = \\frac{1}{N_{\\alpha}} \\left(L_{n_1}L_{n_2} \\cdots L_{n_q}\\right) \\left(\\bar{L}_{\\bar{n}_1}\\bar{L}_{\\bar{n}_2} \\cdots \\bar{L}_{\\bar{n}_q} \\right)\\ket{h,\\bar{h}},\n\\end{equation}\nwith $n_i, \\bar{n}_i \\leq -1$, and its norm $N_{\\alpha}$ can be expanded similarly.\n\n\\subsection{Matrix elements of virasoro generators}\n\nGiven a product of Virasoro generators $X$, we wish to compute\n\\begin{equation}\n X_{\\alpha\\beta} \\equiv \\langle \\phi_\\beta^{\\mbox{\\tiny CFT}} | X | \\phi_\\alpha^{\\mbox{\\tiny CFT}} \\rangle.\n\\end{equation}\nWe first expand the states $|\\phi_\\alpha^{\\mbox{\\tiny CFT}}\\rangle$ and $|\\phi_\\beta^{\\mbox{\\tiny CFT}}\\rangle$ in terms of their primary states and products of Virasoro generators as in the previous section. If the two primary states are different, thus belonging to different representations of the Virasoro algebra, $X_{\\alpha\\beta}=0$. If the primary states are the same, we have\n\\begin{equation}\n X_{\\alpha\\beta} \\equiv \\frac{1}{N_\\alpha N_\\beta}\\langle h,\\bar{h} | \\left(L_{n_1} \\dots L_{n_r}\\right) \\left(\\bar{L}_{\\bar{n}_1} \\dots \\bar{L}_{\\bar{n}_s} \\right) | h,\\bar{h} \\rangle,\n\\end{equation}\nwhere we have also expanded $X$, resulting in an expectation value on the state $|h,\\bar{h}\\rangle$ of a long sequence of generators. We have also used the commutativity of $L_n$ with $\\bar{L}_m$. To evaluate the expectation value, we apply the Virasoro commutators of Eq.~(\\ref{eq:Virasoro}) to produce ``normal ordered'' terms, moving all generators with $n>0$ ($\\bar{n}>0$) to the right and all generators with $n<0$ ($\\bar{n}<0$) to the left. Since $L_n\\bar{L}_{m}|h,\\bar{h}\\rangle = 0$ for $n>0$ or $m>0$, and $L_n^\\dagger = L_{-n}$, $\\bar{L}_n^\\dagger = \\bar{L}_{-n}$, these terms are zero and we are left with terms proportional to\n\\begin{equation}\n \\langle h,\\bar{h}|(L_0)^a(\\bar{L}_0)^b|h,\\bar{h}\\rangle = (h)^a (\\bar{h})^b,\n\\end{equation}\nfor integers $a,b \\ge 0$, where we have used $L_0|h,\\bar{h}\\rangle=h|h,\\bar{h}\\rangle$ and $\\bar{L}_0|h,\\bar{h}\\rangle=\\bar{h}|h,\\bar{h}\\rangle$. We sum these terms to compute the final result.\n\nThe normalization factors $N_\\alpha$ can be computed in the same way. For example, the norm-squared of the state $|\\phi^{\\mbox{\\tiny CFT}}\\rangle = L_{-2}|h,\\bar{h}\\rangle$ is\n\\begin{align}\n N_\\phi^2 &= \\langle h,\\bar{h} | L_2 L_{-2} | h,\\bar{h} \\rangle \\\\\n &= \\langle h,\\bar{h} | 4L_0 + \\frac{c}{2} | h,\\bar{h} \\rangle\\\\\n &= 4h+\\frac{c}{2},\n\\end{align}\nwhere we applied the Virasoro commutator once. In particular, the norm of the state $|T^{\\mbox{\\tiny CFT}}\\rangle = L_{-2}|\\mathbb{I}^{\\mbox{\\tiny CFT}}\\rangle$ is $N_T = \\sqrt{c\/2}$, since for the identity primary $h=\\bar{h}=0$. An exemplary off-diagonal matrix element involving this state is\n\\begin{align}\n& \\langle T^{\\mbox{\\tiny CFT}} | \\bar{L}_3 L_{-2}\\bar{L}_{-3} |\\mathbb{I}^{\\mbox{\\tiny CFT}}\\rangle \\\\\n=~ &\\frac{1}{N_T}\\langle \\mathbb{I}^{\\mbox{\\tiny CFT}} | \\bar{L}_3 \\bar{L}_{-3} L_2 L_{-2} | \\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n=~ & N_T \\langle \\mathbb{I}^{\\mbox{\\tiny CFT}} | \\bar{L}_3 \\bar{L}_{-3} | \\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n= ~& N_T \\langle \\mathbb{I}^{\\mbox{\\tiny CFT}}| 6L_0 + 2c |\\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n= ~& \\sqrt{2}c^{3\/2}.\n\\end{align}\n\n\n\\subsection{Finite non-uniform euclidean evolution}\n\nWe wish to compute matrix elements of a conformal transformation equivalent to evolving non-uniformly in euclidean time. In other words, the generator should have the form from Eq.~(\\ref{eq:Q0})\n\\begin{equation} \nQ_0 \\equiv \\int_0^{L} dx~a(x)~h(x), \n\\end{equation}\nwith $a(x)$ a real-valued periodic function of $x$ so that $a(0) = a(L)$. Since we want to compare the action of the finite transformation $V^{\\mbox{\\tiny CFT}}_0 \\equiv e^{-Q_0}$ on eigenstates of the CFT Hamiltonian to the action of the tensor network of Fig.~\\ref{fig:time}(b) on corresponding low-energy eigenstates of a critical spin chain, we guess a form for $a(x)$ based on the structure of the tensor network. We first take each lattice site to represent an interval of the continuum one lattice-spacing in length and centered on that site. Then we interpret each euclideon as evolving its interval by one unit of euclidean time, which is consistent with the action of the full transfer matrix $\\mathcal{T}$. We then guess that a smoother evolves its interval by, on average, \\emph{half} a unit of time, with the amount of evolution changing linearly from zero at one end of the interval associated with the smoother, to one unit at the other end. This leads to the profile function shown in Fig.~\\ref{fig:time}(a), where the way the smoother is ``cut'' from the euclideon determines the direction of its sloping profile: See section \\ref{sec:time_smoothers}.\n\nThe profile function, defined as a continuous function of $x \\in [0,L)$, is\n\\begin{equation}\n a(x) = \\begin{cases}\n x & 0 < x \\le 1 \\\\\n 1 & 1 < x \\le L\/4 \\\\\n -x + L\/4 & L\/4 < x \\le L\/4+1 \\\\\n 0 & L\/4+1 < x \\le L\/2 \\\\\n x - L\/2 & L\/2 < x \\le L\/2+1 \\\\\n 1 & L\/2+1 < x \\le 3L\/4 \\\\\n -x + 3L\/4 & 3L\/4 < x \\le 3L\/4+1 \\\\\n 0 & 3L\/4+1 < x \\le L\n \\end{cases},\n\\end{equation}\nand has the Fourier coefficients\n\\begin{equation} \\label{eq:Fourier_boxy}\n a_n = e^{in S \\frac{2\\pi}{N}} \\frac{1}{4} \\;\\mathrm{sinc}\\left(\\frac{n}{4}\\right) \\;\\mathrm{sinc}\\left(\\frac{n}{N}\\right) \\left(1 + e^{in \\pi}\\right),\n\\end{equation}\nwhere the shift $S$ reflects the freedom of choosing the origin along the x-axis. The corresponding conformal generator is\n\\begin{equation} \\label{eq:Q_boxy}\n Q_0 = \\frac{2\\pi}{N} \\sum_{n=-N\/2}^{N\/2} a_n H_n,\n\\end{equation}\nwhere we have summed over only those modes that can be distinguished on a lattice of $N$ sites ($N$ is assumed to be even). For the plot in Fig.~\\ref{fig:time}(b), where $N=24$, we expand the exponential $V^{\\mbox{\\tiny CFT}}_0 \\equiv e^{-Q_0}$ to 4th order in $Q_0$ so that all the matrix elements shown are reasonably well converged.\n\nNote that there are some reasons to expect mismatches between $V^{\\mbox{\\tiny CFT}}_0$ and the euclideon tensor network of Fig.~\\ref{fig:time}(a), even with an appropriate choice of $a(x)$. Perhaps most significantly, due to the smoothers, the tensor network is not Hermitian, unlike $V^{\\mbox{\\tiny CFT}}_0$. This is the cause of the slight differences in magnitude between matrix elements of the tensor network and their Hermitian conjugate counterparts in the plot of Fig.~\\ref{fig:time}(b). Furthermore, we neglect the effect of finite-size corrections to the eigenstates of the critical spin chain due to irrelevant terms present in the lattice model, as well as the effects of any relevant or irrelevant operator contributions to the smoothers, which are indeed present, as demonstrated by the observed matrix elements connecting different conformal towers.\n\n\\subsection{Finite non-uniform scale transformation}\n\nAn appropriate conformal transformation for comparison with the local scaling tensor network of Fig.~\\ref{fig:scale}(b) is generated by a position-dependent translation\n\\begin{equation} \nQ_1 \\equiv \\int_0^{L} dx~b(x)~p(x), \n\\end{equation}\nas in Eq.~(\\ref{eq:Q1}), where $b(x)$ is a real-valued, periodic function of $x \\in [0,L)$. Unlike in the euclidean time case, however, the function $b(x)$ cannot simply be scaled to produce the position-dependent translation function associated with a \\emph{finite} transformation. Indeed, $b(x)$ represents a position-dependent \\emph{velocity} and a point $x(s_0)$ on the x-axis is translated by an infinitesimal transformation $e^{i\\epsilon Q_1}$ as\n\\begin{equation}\n x(s_0+\\epsilon) = x(s_0) + \\epsilon b(x(s_0)).\n\\end{equation}\nTo find the final location of a point $x(s_0)$ under a finite transformation $e^{isQ_1}$, this equation must be integrated, for example numerically. By doing this for every starting point of interest $x_j$, the translation\n\\begin{equation}\n B(x_j) \\equiv x_j(s-s_0) - x_j(s_0)\n\\end{equation}\nthese points experience under the finite transformation can be computed.\n\nBy inspecting the tensor network of Fig.~\\ref{fig:scale}(b) we can guess at the appropriate form for $B(x)$. We consider the action of the network on a state at the top, with each lattice site representing an unit interval, centered on the site, of continuous $x$-axis. The network first performs fine-graining on sites $1$ and $8$. We thus assign a constant scale-factor of $2$ to the interval $(8,1]$, \\emph{between} sites $1$ and $8$. Since we do not know exactly how the smoothers contribute, we leave the behavior in the intervals $(7.5,8]$ and $(1,1.5]$ undetermined. The network then performs a coarse-graining of sites $4$ and $5$, hence we assign the scale factor $1\/2$ to the interval $(3.5,5.5]$. We further assume there is no scaling in the neighborhood of points $2$ and $7$, since these each mark the midpoint of a pair of smoothers and, by symmetry, the scale factor should pass through zero here.\n\nTo derive shifts, we must further pay attention to how the outgoing legs at the bottom of the network are arranged relative to the ingoing legs at the bottom. Although the unscaled sites $2$ and $7$ pass through the smoothers without being translated, translations are needed at the bottom to match up the outgoing legs with lattice sites. Hence site $2$ is translated by $1$ unit (to the right) and site $7$ is translated by $-1$ (to the left). By assuming, based on the symmetry of the network, that the points $4.5$ and $8.5$ are not translated, we can then derive the translation of points in the intervals $(8,1]$ and $(3.5,5.5]$ from the above scale factors (a constant scale factor corresponds to a constant derivative of the translation function). We use only the values in the stated regions and at the stated points in arriving at a possible generator, reflecting our lack of knowledge concerning the precise action of the smoothers. The resulting profile\n\\begin{align}\n B(x) = \\begin{cases}\n 1 & x = 2 \\\\\n -x\/2 + 7\/4 & 3.5 < x \\le 5.5 \\\\\n -1 & x = 7 \\\\\n 2x - 16 & 8 < x \\le 9 \\;(\\equiv 1)\n \\end{cases}\n\\end{align}\nis shown in Fig.~\\ref{fig:scale}(a).\n\nTo constrain the set of generating functions $b(x)$, we restrict to those described by just the first two odd Fourier modes, which is enough to reproduce the regions of constant scaling quite accurately. We may restrict to odd functions since we know there must be nodes at $x=4.5$ and $x=8.5$. While it is possible to fit these parameters to $B(x)$, we choose Fourier coefficients that are very close to the optimal ones, but which better match the numerical matrix elements of Fig.~\\ref{fig:scale}(b):\n\\begin{align} \\label{eq:Fourier_cosy}\n b_1 = e^{i S \\frac{2\\pi}{N}} \\; 0.5, \\quad b_3 = e^{i 2 S \\frac{2\\pi}{N}} \\; 0.0275,\n\\end{align}\nwhere $S$ is a shift determined by the location of the origin along the x-axis. The generator of the corresponding scale transformation is\n\\begin{equation}\n Q_1 = \\frac{2\\pi}{N}\\left( b_1 P_1 + b_1^* P_{-1} + b_3 P_3 + b_3^* P_{-3} \\right).\n\\end{equation}\nFor the plot in Fig.~\\ref{fig:scale}(b), where $N=8$, we expand the exponential $V^{\\mbox{\\tiny CFT}}_1 \\equiv e^{-iQ_1}$ to 7th order so that the plotted matrix elements are reasonably well converged.\n \n\\subsection{Final remark on $V_0^{\\mbox{\\tiny CFT}}$ and $V_1^{\\mbox{\\tiny CFT}}$ } \n \nAbove we have carefully specified two profiles $a(x)$ and $b(x)$ that led to generators $Q_{0}$ and $Q_1$ for the specific conformal transformations $V_0^{\\mbox{\\tiny CFT}}$ and $V_1^{\\mbox{\\tiny CFT}}$ acting on the the CFT that we used in Figs.~\\ref{fig:time} and \\ref{fig:scale} for comparison with the linear maps implemented with tensor networks. The match, both qualitative and quantitative, between matrix elements of the linear maps $V$ and $V^{\\mbox{\\tiny CFT}}$ obtained on the lattice with tensor networks and on the CFT with conformal transformations is remarkable, confirming that the proposed tensor networks indeed implement lattice versions of non-uniform euclidean time evolution and rescaling.\n\nWe remark that although the profiles $a(x)$ and $b(x)$ were obtained above through a somewhat convoluted derivation that required making ad hoc decisions, slightly different choices of profile $a(x)$ and $b(x)$ were also tested and seen to lead to very similar $V^{\\mbox{\\tiny CFT}}$ whose matrix elements continued to accurately match the matrix elements of the proposed lattice linear map $V$. \n \n\\section{Euclideons, disentanglers, isometries, and smoothers} \n\nIn this section we briefly review the well-established construction of the tensors called eucliodeons $e$, disentanglers $u$, and isometries $w$, and sketch how to build smoothers $e_{L}, e_{R}, u_{L}, u_{R}$. \n\n\\subsection{Euclideons}\n\nAn \\textit{euclideon}, denoted $e$, is a tensor that implements euclidean time evolution. In Refs.~\\cite{TNR,TNRMERA,TNRscale} the tensor $e$ was instead denoted as $A$. Two possible ways of building an euclideon are (i) from a quantum spin chain Hamiltonian $H$ (see e.g.\\ Supplemental Material in \\cite{TNRMERA}) and (ii) from the Boltzmann weights of the statistical partition function of a classical two-dimensional lattice system (see e.g.\\ Ref.~\\cite{TNR}). We briefly review those constructions for completeness. \n\nGiven a local quantum spin chain Hamiltonian $H$, the defining property of an euclideon $e$ for $H$ is that a periodic row $\\mathcal{T}$ of $N$ euclideons should implement the euclidean time evolution $\\exp(-\\tau H)$ for one unit of euclidean time $\\tau=1$. If the spin chain is described at low energies by a relativistic quantum field theory (QFT), then we normalize $H$ such that the ground state energy is zero in the thermodynamic limit ($N\\rightarrow \\infty$) and the speed of light is $1$ \\cite{Ash}. When the relativistic QFT is in addition a conformal field theory (CFT), the above normalization of $H$ implies that the euclideons are (up to lattice effects) isotropic in the two-dimensional euclidean space-time. \nAfter normalizing the Hamiltonian $H$ one first builds a matrix product operator (MPO) for $\\exp(-\\delta \\tau H)$ for small $\\delta \\tau \\ll 1$, e.g.\\ using a Suzuki-Trotter decomposition, and then multiplies together $1\/\\delta \\tau$ copies of the resulting MPO while appropriately truncating the bond indices of the resulting product $\\left(\\exp (-\\delta \\tau H) \\right)^{1\/\\delta \\tau} = \\exp(- \\tau H)$. This results in a new MPO made of (roughly) isotropic euclideons $e$, see e.g.\\ part A of the Supplemental Material of Ref.~\\cite{TNRMERA} for further details.\n\nThe construction of euclideons is even simpler if the quantum spin chain relates to a two-dimensional statistical model, since in this case we can express an euclideon $e$ directly in terms of Boltzmann weights. For instance, for the critical quantum Ising spin chain $H = -\\sum_l \\sigma^{x}_l\\sigma^{x}_{l+1} - \\sigma_l^{z}$, we can build euclideons using the Boltzmann weights $e^{\\sigma_i\\sigma_j\/T}$ of the statistical partition function of the (isotropic) critical 2d classical Ising model, namely\n\\begin{equation} \\label{eq:A}\ne_{ijkl} \\equiv e^{\\left(\\sigma_i\\sigma_j + \\sigma_j\\sigma_k + \\sigma_k\\sigma_l + \\sigma_l\\sigma_i \\right)\/T},\n\\end{equation}\nwhere $\\sigma_i = \\pm 1$ labels the two possible values of a classical Ising spin on site $i$ of a two-dimensional lattice, see e.g.\\ Ref.~\\cite{TNR} for further details.\n\n\n\\subsection{Euclidean time smoothers}\n\\label{sec:time_smoothers}\n\nWe can use a truncated row of euclideons $e$ on a region of a quantum spin chain in order to apply an euclidean time evolution that only evolves the spins on that region. However, at both ends of the truncated row of euclideons we must place special tensors that we call \\textit{smoothers} $e_L$ and $e_R$, see Fig.~\\ref{fig:timesmoother0}. The purpose of each smoother is to smooth out lattice effects that occur at the ends of the truncated row of euclideons. Without smoothers there would be an open bond index at each end. Thus a first role of the smoothers, which only have one bond index (as opposed to the two bond indices of regular euclideons) is to eliminate open bond indices. However, not any tensor with a single bond index will do. Indeed, a generic choice of coefficients inside the smoother will result in a transformation that is not the intended conformal transformation (non-uniform euclidean time evolution). As a matter of fact, the resulting transformation is generically not even approximately diagonal in the conformal towers and therefore does not correspond to any conformal transformation.\n\n\n\\begin{figure}\n\\includegraphics[width=7cm]{timesmoother0_v2.pdf}\n\\caption{\n(a) A truncated row of euclideons $e$ has a dangling bond index at each end. As a result, it does not define a linear map in the Hilbert space of the spin chain.\n(b) Left and right smoother $e_L$ and $e_R$ are special tensors placed at the two ends of the truncated row of euclideons in order to eliminate the dangling bond indexes, producing a linear map in the spin chain. \n\\label{fig:timesmoother0} \n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=7cm]{timesmoother1_v2.pdf}\n\\caption{\nComputation of smoothers $e_L$ and $e_R$ starting from an euclideon $e$.\n(a) In order to create a smoother $e_L$, first we temporarily regard an euclideon $e$ as a matrix $M$ (by joining its indexes into two pairs) and then compute its singular value decomposition $M= USV^{\\dagger}$, where $U$ and $V$ are unitary matrices and $S$ is a diagonal matrix with the singular values of $M$ in its diagonal. Then we create tensors $a_L = \\sqrt{S}V^{\\dagger}$ and $\\tilde{a}_L \\equiv U\\sqrt{S}$, where $\\sqrt{S}$ is a diagonal matrix with the square root of the singular values in the diagonal. Tensor $a_L$, which is a precursor of the smoother $e_L$, has two indices (the ones which originally belonged to the euclideon $e$, depicted in black) of dimension $d$ and one index (the one coming from the singular value decomposition, depicted in red) of dimension $d^2$. \n(b) The smoother $e_L$ is then obtained by multiplying $a_L$ by an isometry $v$ of size $d\\times d^2$ that maps the $d^2$-dimensional (red) index into a $d$-dimensional (black) index, thus implementing a truncation of the former. The variational parameters in $v$ correspond to a choice of truncation basis and are determined as indicated in Fig.~\\ref{fig:timesmoother2}. \n(c) Notice that the product of $\\tilde{e}_L$ (obtained analogously to $e_L$) and $e_L$ amounts to the original euclideon $e$, up to effects due to the truncation implemented through the isometry $v$.\n(d)-(f) The smoother $e_R$ is created analogously. \n\\label{fig:timesmoother1} \n}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{timesmoother2_v2.pdf}\n\\caption{\nIn order to determine the variational parameters in the smoother $e_L$, which are contained in the isometry $v$ of Fig.~\\ref{fig:timesmoother1}(b), we require that the dominant eigenvector of the transfer matrix $\\mathcal{T}$ (a), corresponding to the ground state on the circle, has maximal overlap with the dominant eigenvector of other transfer matrices, such as those in (b) and (c), which can be seen to become close to the ground state on the circle. The variational parameters in the smoother $e_R$ are determined analogously using the dominant eigenvectors of transfer matrices $(d)$ and $(e)$.\n\\label{fig:timesmoother2} \n}\n\\end{figure}\n\nIn practice we build the smoothers through a two-step procedure, see Fig.~\\ref{fig:timesmoother1}. First we split an euclideon diagonally using a singular value decomposition (SVD). Then we multiply it by an isometry, that we determine by demanding that it optimally connects to the physical indices of the original euclideon $e$, in the following sense. We build a \\textit{diagonal transfer matrix} $M'$ made of euclideons and smoothers and demand that its dominant eigenvector has maximal overlap with the dominant eigenvector of the horizontal transfer matrix $M$ made of euclideons. The resulting smoothers, when placed at the ends of a truncated row of euclideons, are then seen to indeed produce linear maps that not only act (approximately) diagonally in the conformal towers (as any conformal transformation does) but whose matrix elements between low energy states accurately correspond to the intended non-uniform euclidean time evolution, as seen in the example of Fig.~\\ref{fig:time} in the main text.\n\n\n\\subsection{Optimized disentanglers and isometries} \n \nDisentanglers $u$ and isometries $w$, the tensors in the multi-scale entanglement renormalization ansatz (MERA), are in general full of variational parameters constrained in such a way that the tensors are unitary\/isometric. In the context of this work, however, by a disentangler $u$ and an isometry $w$ we refer exclusively to such tensors after the variational parameters have been chosen so that the MERA represents the ground state of the quantum spin Hamiltonian $H$. \n\nThis optimization can be carried out variationally using iterative energy minimization algorithms \\cite{siMERA}, as we did here. Alternatively, one can extract optimized disentanglers and isometries from the tensor network renormalization (TNR) algorithm \\cite{TNRMERA}, which manipulates a network of euclideons.\n\n\\subsection{Scale smoothers}\n\nWe can use a truncated double layer of optimized disentanglers and isometries on a region of a quantum spin chain in order to apply a non-uniform scale transformation that only rescales the spins in that region. Again, at both ends of the truncated layer of disentanglers and isometries we must place special tensors called smoothers $u_L$ and $u_R$. One first reason for including these smoothers is that otherwise there is a mismatch in index connectivity, since the lower indices of an isometry $w$ must connect with the upper indices of a disentangler $u$ (by design of the MERA). However, a generic choice of coefficients inside the scale smoothers $u_L$ and $u_R$ (even when compatible with the unitary\/isometric character of these tensors) will produce a linear map that does not correspond to a conformal transformation. \n\nIn practice, we build the smoothers $u_L$ and $u_R$ by demanding that the equality in Fig.~\\ref{fig:scalesmoother1} be (approximately) fulfilled. These equalities ensure that the scale smoothers properly connect upper and lower indices. The resulting smoothers $u_L$ and $u_R$, when placed at the ends of a truncated double layer of optimized disentanglers and isometries, indeed produce a linear map that not only acts (approximately) diagonally in the conformal towers but whose matrix elements between low energy states accurately correspond to the intended non-uniform scale transformation.\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{scalesmoother1_v2.pdf}\n\\caption{\n(a) Isometry $w$, disentangler $u$, and smoothers $u_L$ and $u_R$, with two types of indices: in red, the top index of an isometry $w$, which connects with the bottom indices of a disentangler $u$; in black, the bottom indices of an isometry $w$, which connect with the top indices of a disentangler $u$. Smoothers map a pair of red and black top indices into a pair of bottom red indices.\n(b) At each end of a truncated double layer of MERA tensors $\\mathcal{W}$ there is a dangling black index that cannot be directly connected to a red index. Scale smoothers $u_L$ and $u_R$ can map these dangling black indexes, together with an adjacent red index, into a pair of red indexes. \n(c) The left scale smoother $u_L$ is determined variationally by demanding that this tensor network equality be fulfilled (approximately, but as accurately as possible).\n(d) Similarly, the right scale smoother $u_R$ is determined variationally by demanding that this other tensor network equality be approximately fulfilled.\n\\label{fig:scalesmoother1} \n}\n\\end{figure}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\nReasoning about actions and planning concern the representation of a dynamic\nsystem. This representation consists of a description of the interaction\nbetween an agent and its environment and aims at enabling reasoning and\ndeliberation on the possible course of action for the agent \\cite{Reit01}.\nPlanning in fully observable nondeterministic domains (FOND), say in Planning Domain Definition Language (PDDL),\n\\cite{GhNT04,GeBo13} exemplifies the standard methodology for expressing\ndynamic systems: it represents the world using finitely many \\emph{fluents}\nunder the control of the \\emph{environment} and a finitely many \\emph{actions}\nunder the control of the \\emph{agent}. Using these two elements a model of the\ndynamics of world is given. Agent goals, e.g., reachability objectives, or,\nsay, temporally extended objectives written in \\text{{LTL}}\\xspace\n\\cite{BacchusK00,CTMBM17,DR-IJCAI18}, are expressed over such models in terms of\nsuch fluents and actions.\n\nAn important observation is that, in devising plans, the agent takes\nadvantage of such a representation of the world. Such a representation\ncorresponds to knowledge that the agent has of the world. In other\nwords, the agent \\emph{assumes} that the world works in a certain\nway, and \\emph{exploits such an assumption in devising its plans}.\nA question immediately comes to mind: \n \\begin{quote}\n \\emph{Which kinds of environment assumptions can the agent make?}\n \\end{quote}\nObviously the planning domain itself (including the initial state) with its\npreconditions and effects is such an assumption. That is, as long as the agent \nsticks to its preconditions, the environment acts as described by the domain. \nSo, the agent can exploit the effect of its actions in order to reach a certain goal \n{(state of affairs)}.\n{Another common assumption is to assume the domain is \\emph{fair}, i.e.,\nso-called \\emph{fair FOND}~\\cite{DaTV99,PistoreT01,Cimatti03,CTMBM17,DIppolitoRS18}. In this case\nthe agent can exploit not only the effects, but also the guarantee that by\ncontinuing to execute an action {from a given state} the environment will\neventually respond with all its possible nondeterministic effects.}\\footnote{There are two notions of fairness in planning. One stems from the fact that nondeterminism is resolved stochastically. The other is a logical notion analogous to that used in the formal-methods literature. These two notions coincide in the context reachability goals~\\cite{DIppolitoRS18}, but diverge with more general LTL goals~\\cite{Pnueli:STOC83,Pnueli:IC93}. In this paper, we focus on the logical notion.}\nMore recently \\cite{DBLP:conf\/ijcai\/BonetG15,DBLP:conf\/ijcai\/BonetGGR17} \ntrajectory constraints over the domain, expressed in \\text{{LTL}}\\xspace, have been proposed\nto model general restrictions on the possible environment behavior.\nBut is any kind of \\text{{LTL}}\\xspace formula on the fluents and actions of the domain a\npossible trajectory constraint for the environment?\nThe answer is obviously not! To see this, consider a formula expressing that\neventually a certain possible action must actually be performed (the agent may\ndecide not to do it). But then \n\\begin{quote}\n\\emph{Which trajectory constraints are suitable as assumptions in a given\ndomain?} \n\\end{quote}\nFocusing on \\text{{LTL}}\\xspace, the question can be rephrased as: \n\\begin{quote}\n\\emph{Can any {linear-time} specification be used as an assumption for the environment?}\n\\end{quote}\nWe can summarize these questions, ultimately, by asking:\n\\begin{quote}\n\\emph{What is an environment assumption?}\n\\end{quote}\n\n\nThis is what we investigate in this paper. We take the view that environment\nassumptions are ways to talk about \\emph{the set of strategies the environment can enact}. Moreover, \nthe plan for the goal, i.e., the agent strategy for fulfilling\nthe goal, need only fulfill the goal against the strategies of the\nenvironment {from the given set of environment strategies}.\nWe formalize this insight and {\\emph{define}} synthesis\/planning under assumptions and\nthe relationship between the two in a general linear-time setting. {In particular, our definitions\nonly allow linear-time properties to be assumptions if the environment can enforce them.}\nIn doing this \\emph{we answer the above questions}.\n\nWe also concretize the study and express goals and assumptions in \\text{{LTL}}\\xspace, automata over infinite words (deterministic parity word automata)~\\cite{ALG02}, as well as formalisms over finite traces, i.e., \\text{{LTLf}}\\xspace\/\\text{{LDLf}}\\xspace~\\cite{DegVa13,DR-IJCAI18} and finite word automata. \nThis allows us to study \\emph{how to solve} synthesis\/planning under assumptions problems. One may\nthink that the natural way to solve such synthesis problems is to have \nthe agent synthesize a strategy for the implication \n\\[ Assumption \\supset Goal\\] \nwhere both $Assumption$ and $Goals$ are expressed, say, in \\text{{LTL}}\\xspace. \nA first problem with such an implication is that the agent should not devise\nstrategies that make $Assumption$ false, because in this case the agent would\nlose its model of the world without \\emph{necessarily} fulfilling its $Goal$. \n{This undesirable situation is avoided by our very notion of environment assumption.}\n{A second issue is this:}\n\\begin{quote}\n\\emph{Does synthesis\/planning under assumptions amount to\nsynthesizing for the above implication?} \n\\end{quote}\n{We show that this is not the case. \nNote that an agent that synthesizes for the implication is too pessimistic: the agent, having chosen a candidate agent strategy, considers as possible all environment strategies that satisfy $Assumption$ against the specific candidate strategy it is analyzing. But, in this way the agent gives too much power to the environment, since, in fact, the environment does not know the agent's chosen strategy. On the other hand, surprisingly, we show that if there is an agent strategy fulfilling $Goal$ under $Assumption$, then also there exists one that indeed enforces the implication.\n}\nThus, even if the implication \n\\emph{cannot be used for characterizing the problem of\nsynthesis\/planning under assumptions}, it \\emph{can be used to solve it}.\nExploiting this result, we give techniques to solve synthesis\/planning under\nassumptions, and study the worst case complexity of the problems when goals and assumptions are\nexpressed {in the logics and automata mentioned above.}\n\n\n\n\n\\section{Synthesis and Linear-time specifications} \\label{sec:prelims}\n\n\\emph{Synthesis} is the problem of producing a module that satisfies a given property no matter how the environment behaves ~\\cite{PnueliR89}.\nSynthesis can be thought of in the terminology of games. Let ${\\sf{Var}}$ be a finite set of Boolean variables (also called atoms), and assume it is partitioned into two sets: $A$, those controllable by the agent, and $E$, those controllable by the environment. \nLet $\\modecal{A}} \\newcommand{\\B}{\\modecal{B} = 2^A$ be the set of \\emph{actions} and $\\modecal{E}} \\newcommand{\\F}{\\modecal{F} = 2^E$ the set of \\emph{environment states} (note the symmetry: we \nthink of $\\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ as a set of actions that are compactly\nrepresented as assignments of the variables in $A$). The game consists of infinitely many phases. \nIn each phase of the game, both players assign values to their variables, with the environment going first. These assignments \nare given by \\emph{strategies}: an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace:\\modecal{E}} \\newcommand{\\F}{\\modecal{F}^+ \\to \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ and an environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$. The resulting infinite sequence of assignments is denoted $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$.\\footnote{Formally, \nwe say that $\\pi = \\pi_0 \\pi_1 \\cdots$ complies with $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ if $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace((\\pi_0 \\cap E) \\cdots (\\pi_k \\cap E)) = \\pi_k \\cap A$ for all $k$; and we say that $\\pi$ complies with $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ if $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace((\\pi_0 \\cap A) \\cdots (\\pi_k \\cap A)) = \\pi_{k+1} \\cap E$ for all $k$. Then $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is defined to be the unique infinite trace that complies with both strategies.} \n\n{In classic synthesis the agent is trying to ensure that the produced sequence satisfies a given linear-time property. In what follows \\emph{we write \\SF to denote a generic formalism for defining linear-time properties.} Thus, the reader may substitute their favorite formalism for \\SF, e.g., one can take \\SF to be linear temporal logic, or deterministic parity automata. We use logical notation throughout. For instance, when $\\phi$ refers to a logical formula, then $\\phi_1 \\wedge \\phi_2$ refers to conjunction of formulas, but when $\\phi$ refers to an automaton then $\\phi_1 \\wedge \\phi_2$ refers to intersection of automata. If $\\phi \\in \\SF$ write $[[\\phi]] \\in (2^{\\sf{Var}})^\\omega$ for the set it defines. For instance, if $\\phi \\in \\text{{LTL}}\\xspace$ then $[[\\phi]]$ is the set of infinite sequences that satisfy $\\phi$, but when $\\phi$ is an automaton operating on infinite sequences, then $[[\\phi]]$ is the set of infinite sequences accepted by the automaton. Moreover, in both cases we say that the sequence \\emph{satisfies} $\\phi$.\n}\n\nWe say that \\emph{$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]$, i.e., if no matter which strategy the environment uses, the resulting sequence satisfies $\\phi$. Similarly, we say that \\emph{$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]$. We write $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\phi)$ (resp. $Str_\\ensuremath{\\mathsf {ag}}\\xspace(\\phi)$) for the set of environment (resp. agent) strategies that realize $\\phi$, and in case this set is non-empty we say that \\emph{$\\phi$ is environment (resp. agent) realizable}. We write $Str_\\ensuremath{\\mathsf {env}}\\xspace$ (resp. $Str_\\ensuremath{\\mathsf {ag}}\\xspace$) for the set of all environment (resp. agent) strategies.\n\n\\emph{Solving $\\SF$ environment- (resp. agent-) synthesis} asks, given $\\phi \\in \\SF$ to decide if $\\phi$ is environment- (resp. agent-) \nrealizable, and to return such a finite-state strategy (if one exists). \n{In other words, realizability is the {recognition problem} associated to synthesis.}\nWe now recall two concrete specification formalisms $\\SF$, namely \\text{{LTL}}\\xspace (linear temporal logic) and \\text{{DPW}}\\xspace (deterministic parity word automata), and then state that results about solving \\text{{LTL}}\\xspace\/\\text{{DPW}}\\xspace synthesis.\n\n\\subsubsection{Linear temporal Logic (LTL)} \\label{sec:prelims:LTL}\n\n\t \\emph{Formulas of $\\text{{LTL}}\\xspace({\\sf{Var}})$}, or simply \\emph{$\\text{{LTL}}\\xspace$}, are generated by the following grammar:\n\t\\[\\varphi \\!::=\\! p \\!\\mid\\! \\varphi \\vee \\varphi \\!\\mid\\! \\neg \\varphi \\!\\mid\\! \\nextX \\! \\varphi \\!\\mid \\! \\varphi \\until \\varphi\\]\n\twhere $p \\in {\\sf{Var}}$. The \\emph{size $|\\varphi|$} of a formula $\\varphi$ is the number of symbols in it.\n\t\\text{{LTL}}\\xspace formulas are interpreted over infinite sequences $\\pi \\in (2^{{\\sf{Var}}})^\\omega$. Define the satisfaction relation\n\t$\\models$ as follows:\n\t\\begin{enumerate*}\n\t\\item[] $(\\pi,n) \\models p$ iff $p \\in \\pi_n$;\n\t\\item[] $(\\pi,n) \\models \\varphi_1 \\vee \\varphi_2$ iff $(\\pi,n) \\models \\varphi_i$ for some $i \\in \\{1,2\\}$;\n\t\\item[] \t$(\\pi,n) \\models \\neg \\varphi$ iff it is not the case that $(\\pi,n) \\models \\varphi$;\n\t\\item[] $(\\pi,n) \\models \\nextX \\varphi$ iff $(\\pi,n+1) \\models \\varphi$;\n\t\\item[] $(\\pi,n) \\models \\varphi_1 \\until \\varphi_2$ iff there exists $i \\geq n$ such that $(\\pi,i) \\models \\varphi_2$ and for all $i \\leq j < n$, $(\\pi,j) \\models \\varphi_1$.\n\t\\end{enumerate*}\n\tWrite $\\pi \\models \\varphi$ if $(\\pi,0) \\models \\varphi$ and say that $\\pi$ \\emph{satisfies} $\\varphi$ and $\\pi$ is a \\emph{model} of $\\varphi$. \n\tAn \\text{{LTL}}\\xspace formula $\\varphi$ \\emph{defines} the set $[[\\pi]] \\doteq \\{\\pi \\in (2^{\\sf{Var}})^\\omega: \\pi \\models \\varphi\\}$.\n\tWe use the usual abbreviations, $\\varphi \\supset \\varphi' \\doteq \\neg \\varphi \\vee \\varphi'$, $\\mathsf{true} := p \\vee \\neg p$, $\\mathsf{false} \\doteq \\neg \\mathsf{true}$, \n\t$\\eventually \\varphi \\doteq \\mathsf{true} \\until \\varphi$,\n\t$\\always \\varphi \\doteq \\neg \\eventually \\neg \\varphi$. \n\tWrite $Bool({\\sf{Var}})$ for the set of Boolean formulas over ${\\sf{Var}}$. We remark that every result in this paper that mentions \\text{{LTL}}\\xspace also holds for \\text{{LDL}}\\xspace (linear dynamic logic)~\\cite{Var11,Eisner:2006tu}. \n\t\n\\subsubsection{Deterministic Parity Word Automata (DPW)}\n\nA \\emph{\\text{{DPW}}\\xspace} over ${\\sf{Var}}$ is a tuple $M = (Q,q_{in},T,col)$ where $Q$ is a finite set of \\emph{states}, \n$q_{in} \\in Q$ is an \\emph{initial state}, $T:Q \\times 2^{{\\sf{Var}}} \\to Q$ is the \\emph{transition function}, and $col:Q \\to \\mathbb{Z}$ \nis the \\emph{coloring}. The \\emph{run} $\\rho$ of $M$ on the \\emph{input} word $x_0 x_1 x_2 \\cdots \\in (2^{{\\sf{Var}}})^\\omega$ \nis the infinite sequence of transitions $(q_0,x_0,q_1) (q_1,x_1,q_2) (q_2,x_2,q_3) \\cdots$ such that $q_0 = q_{in}$. \nA run is \\emph{successful} if the largest color occurring infinitely often is even. \nIn this case, we say that the input word is \\emph{accepted}. The \\text{{DPW}}\\xspace $M$ \n\\emph{defines} the set $[[M]]$ consisting of all input words it accepts. The \\emph{size of $M$}, written $|M|$, is the cardinality of $Q$.\nThe \\emph{number of colors} of $M$ is the cardinality of $col(Q)$. \n\n{\\text{{DPW}}\\xspace{s} are effectively closed under Boolean operations, see e.g., \\cite{ALG02}:\n\\begin{lemma} \\label{lem:DPW Boolean combinations} \nLet $M_i$ be \\text{{DPW}}\\xspace with $n_i$ states and $c_i$ colors, respectively.\n\\begin{enumerate} \n \\item One can effectively form a \\text{{DPW}}\\xspace with $n_1$ states and $c_1$ colors for the complement of $M_1$.\n \\item One can effectively form a \\text{{DPW}}\\xspace with with $O(n_1 n_2 d^2d!)$ many states and $O(d)$ many colors, where $d = c_1 + c_2$, for the disjunction $M_1 \\lor M_2$.\n\\end{enumerate}\n\\end{lemma}\nThus, e.g., from \\text{{DPW}}\\xspace $M_1,M_2$ one can build a DPW for $M_1 \\supset M_2$ whose number of states is $O(n_1 n_2 d^2d!)$ and whose number of colors is $O(d)$.\n \n\nEvery \\text{{LTL}}\\xspace formula $\\varphi$ can be translated into an equivalent \\text{{DPW}}\\xspace $M$, i.e., $[[\\varphi]] = [[M]]$, see e.g. \\cite{DBLP:conf\/banff\/Vardi95,DBLP:journals\/lmcs\/Piterman07}.\nMoreover, the cost of this translation and the size of $M$ are at most doubly exponential in the size of $\\varphi$, \nand the number of colors of $M$ is at most singly exponential in the size of $\\varphi$.\n\nHere is a summary of the complexity of solving synthesis:\n\\begin{theorem}[Solving Synthesis] \\label{fact:synthesis} \\hspace{0cm}\n\\begin{enumerate} \n\\item Solving \\text{{LTL}}\\xspace environment (resp. agent) synthesis is $2$\\textsc{exptime}\\xspace-complete~\\cite{PnueliR89}.\n\\item Solving \\text{{DPW}}\\xspace environment (resp. agent) synthesis is \\textsc{ptime}\\xspace \nin the size of the automaton and $\\textsc{exptime}\\xspace$ in the number of its colors~\\cite{PnueliR89,finkbeiner2016synthesis}. \n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Synthesis under Assumptions}\n\nIn this section we give core definitions of environment assumptions and synthesis under such assumptions.\nIntuitively, the assumptions are used to select the environment strategies that the agent considers possible, i.e., although the agent does not know \nthe particular environment strategy it will encounter, it knows that it comes from such a set.\nWe begin in the abstract, and then move to declarative specifications. Unless explicitly specified, we assume fixed sets $E$ and $A$ of environment and agent atoms.\n\n{Here are the main definitions of this paper:}\n\\begin{definition}[Environment Assumptions -- abstract] \\label{def:abstract:assumptions}\nWe call any {non-empty} set $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ of environment strategies an \\emph{environment assumption}.\n\\end{definition}\n\nInformally, the set $\\Omega$ represents the set of environment strategies that the agent considers possible.\n\\begin{definition}[Agent Goals -- abstract] \nWe call any set $\\Gamma$ of traces an \\emph{agent goal}. \n\\end{definition}\n\n\\begin{definition}[Synthesis under assumptions -- abstract] \\label{def:abstract:sua}\nLet $\\Omega$ be an environment assumption and $\\Gamma$ an agent goal. \nWe say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\Gamma$ assuming $\\Omega$} if \n\\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\in \\Omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in \\Gamma\\] \n\\end{definition}\n\n\\begin{remark}[On the non-emptiness of $\\Omega$]\nNote that the requirement that $\\Omega$ be non-empty is a consistency requirement; if it were empty then there would be no $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ to test for membership in $\\Gamma$ and so synthesis under assumptions would trivialize and all agent strategies would realize all goals. \n\\end{remark}\n\nFor the rest of this paper we will specify agent goals and environment assumptions as linear-time properties. \\textbf{In particular, we assume that $\\SF$ is a formalism for specifying linear-time properties over ${\\sf{Var}}$, e.g., $\\SF = \\text{{LTL}}\\xspace$ or $\\SF = \\text{{DPW}}\\xspace$. }\n\nHow should $\\omega \\in \\SF$ determine an assumption $\\Omega$? In general, $\\omega$ talks about the interaction between the agent and the environment.\nHowever, we want that the agent can be guaranteed that whatever it does the resulting play satisfies $\\omega$. Thus, a given $\\omega$ induces the set $\\Omega$ consisting of {all} environment strategies $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ such that for all agent strategies $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ the resulting trace satisfies $\\omega$. In particular, for $\\Omega$ to be non-empty (as required for it to be an environment assumption) we must have that $\\omega$ is environment realizable. This justifies the following definitions.\n\n\n\\begin{definition}[Synthesis under Assumptions -- linear-time] \\label{dfn:LT:assumptions} \\hspace{0cm}\n\\begin{enumerate}\n \\item We call $\\omega \\in \\SF$ an \\emph{environment assumption} if it is environment realizable.\n \\item We call any $\\gamma \\in \\SF$ an \\emph{agent goal}.\n \\item An \\emph{$\\SF$ synthesis under assumptions problem} is a tuple $P = (E,A,\\omega,\\gamma)$ where $\\omega \\in \\SF$ is an environment assumption and $\\gamma \\in \\SF$ is an agent goal.\n \\item We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\gamma$ assuming $\\omega$}, or that it \\emph{solves} $P$, if \n $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma$. \n \\item The corresponding decision problem is to decide, given $P$, if there is an agent strategy solving $P$.\n\\end{enumerate}\n\\end{definition}\nFor instance, \\emph{solving \\text{{LTL}}\\xspace synthesis under assumptions} means, given $P = (E,A,\\omega,\\gamma)$ with environment assumption \n$\\omega \\in \\text{{LTL}}\\xspace(E \\cup A)$ and agent goal $\\gamma \\in \\text{{LTL}}\\xspace(E \\cup A)$, to decide if there is an agent strategy solving $P$, and to return such a finite-state strategy (if one exists).\nWe remark that solving \\text{{LTL}}\\xspace synthesis under assumptions is not immediate; we will provide algorithms in the next section. For now, we point out that \n\\emph{deciding whether $\\omega$ is an environment assumption amounts to checking if $\\omega$ is environment realizable}, itself a problem that can be solved by {known results (i.e., Theorem~\\ref{fact:synthesis}).}\n\\begin{theorem}\n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTL}}\\xspace$ formula is an environment assumption is $2$\\textsc{exptime}\\xspace-complete. \n \\item Deciding if a $\\text{{DPW}}\\xspace$ is an environment assumption is in \\textsc{ptime}\\xspace in the size of the \\text{{DPW}}\\xspace and exponential in its number of colors.\n\\end{enumerate}\n\\end{theorem}\n\nWe illustrate such notions with some examples.\n\\begin{example} \\label{ex:asmp}\n\\begin{enumerate}\n \\item The set $\\Omega = Str_\\ensuremath{\\mathsf {env}}\\xspace$, definable in \\text{{LTL}}\\xspace by the formula $\\omega \\doteq \\mathsf{true}$, is an environment assumption. It captures the situation that the agent assumes that the environment will use any of the strategies in $Str_\\ensuremath{\\mathsf {env}}\\xspace$.\n \n \\item In robot-action planning problems, typical environment assumptions encode the physical space, e.g., ``if robot is in Room 1 and does action $Move$ then in the next step it can only be in Rooms 1 or 4''. \n The set $\\Omega$ of environment strategies that realize these properties is an environment assumption, \n definable in \\text{{LTL}}\\xspace by a conjunction of formulas of the form $\\always((R_1 \\wedge Move) \\supset \\nextX (R_1 \\vee R_4))$. \n We will generalize this example by showing that the set of environment strategies in a planning domain $D$ can be viewed as an environment assumption definable in \\text{{LTL}}\\xspace. \n \\end{enumerate}\n\\end{example}\n\n\n\n\n\\section{Solving Synthesis under Assumptions}\n\nIn this section we show how to solve synthesis under assumptions when the environment assumptions and agent goals are given in \\text{{LTL}}\\xspace or by \\text{{DPW}}\\xspace.\n{The general idea is to reduce synthesis under assumptions to ordinary synthesis, i.e., synthesis of the implication $\\omega \\supset \\gamma$. \nAlthough correct, understanding why it is correct is not immediate.}\n\n\\begin{lemma}\nLet $\\omega \\in \\SF$ be an environment assumption and $\\gamma \\in \\SF$ an agent goal. Then, \nevery agent strategy that realizes $\\omega \\supset \\gamma$ also realizes $\\gamma$ assuming $\\omega$. \n\\end{lemma}\n\\begin{proof}\nLet $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ be an agent strategy realizing $\\omega \\supset \\gamma$ (a). To show that $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega$ let \n$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ be an environment strategy realizing $\\omega$ (b). Now consider the trace $\\pi = \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$. We must show that $\\pi$ satisfies $\\gamma$. By (a) $\\pi$ satisfies $\\omega \\supset \\gamma$ and by (b) $\\pi$ satisfies $\\omega$. \n\\end{proof}\n\nWe now observe that the converse is not true. Consider $A \\doteq \\{x\\}$ and $E \\doteq \\{y\\}$, and let $\\omega \\doteq y \\supset x$ and $\\gamma \\doteq y \\supset \\neg x$. First note that $\\omega$ is an environment assumption formula (indeed, the environment can realize $\\omega$ by playing $\\neg y$ at the first step). Moreover, every environment strategy realizing $\\omega$ begins by playing $\\neg y$ (since otherwise the agent could play $\\neg x$ on its first turn and falsify $\\omega$). Thus, every agent strategy realizes $\\gamma$ assuming $\\omega$ (since the environment's first move is to play $\\neg y$ which makes $\\gamma$ true no matter what the agent does). On the other hand, not every agent strategy realizes $\\omega \\supset \\gamma$ \n(indeed, the strategy which plays $x$ on its first turn fails to satisfy the implication on the trace in which the environment plays $y$ on its first turn).\nIn spite of the failure of the converse, the realizability problems are inter-reducible:{\\footnote{For all reasonable expressions $\\omega$, e.g., that define Borel sets~\\cite{Mar75}.}}\n\\begin{theorem} \\label{thm:det}\nSuppose $\\omega \\in \\SF$ is an environment assumption.\n The following are equivalent:\n \\begin{enumerate}\n \\item There is an agent strategy realizing $\\omega \\supset \\gamma$.\n \\item There is an agent strategy realizing $\\gamma$ assuming $\\omega$.\n \\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe previous lemma gives us $1 \\rightarrow 2$. For the converse, suppose $1$ does not hold, i.e., $\\omega \\supset \\gamma$ is not agent-realizable. \nNow, an immediate consequence of Martin's Borel Determinacy Theorem~\\cite{Mar75} is that for every $\\phi$ in any reasonable specification formalism (including all the ones mentioned in this paper), $\\phi$ is not agent realizable iff $\\neg \\phi$ is environment realizable. \nThus, $\\neg (\\omega \\supset \\gamma)$ is environment-realizable, i.e., \n$\\exists \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\omega \\wedge \\neg \\gamma$. Note in particular that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\omega$, i.e., $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega$. Now, suppose for a contradiction that $2$ holds, and take $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizing $\\gamma$ assuming $\\omega$. Then by definition of realizability under assumptions and using the fact that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega$ we have that $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma$. On the other hand, we have already seen that $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\neg \\gamma$, a contradiction.\n\\end{proof}\n\nMoreover, we see that one can actually extract a strategy solving \nsynthesis by assumptions simply by extracting a strategy for solving\nthe implication $\\omega \\supset \\gamma$, which itself can be done by\nknown results, i.e., for \\text{{LTL}}\\xspace use Theorem~\\ref{fact:synthesis} (part 1), and for \\text{{DPW}}\\xspace use Lemma~\\ref{lem:DPW Boolean combinations} and Theorem~\\ref{fact:synthesis} (part 2).\n\\begin{theorem}\\label{thm:solving:SUA}\n\\begin{enumerate} \n\\item Solving \\text{{LTL}}\\xspace synthesis under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\item Solving \\text{{DPW}}\\xspace synthesis under assumptions is in \\textsc{ptime}\\xspace in the size of the automata and in \\textsc{exptime}\\xspace in the number of colors of the automata.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Planning under Assumptions} \\label{sec:Planning Under Assumptions}\nIn this section we define planning under assumptions,\n{that is synthesis wrt a domain\\footnote{Domains can be thought of as compact representations of the arenas in games on graphs~\\cite{ALG02}. The player chooses actions, also represented compactly, and the environment resolves the nondeterminism. In addition, not every action needs to be available in every vertex of the arena.}.}\nWe begin with a representation of fully-observable non-deterministic (FOND) domains~\\cite{GhNT04,GeBo13}. Our representation considers actions symmetrically to fluents, i.e., as assignments to certain variables.\n\nA \\emph{domain} $D = (E,A,I,Pre,\\Delta)$ consists of:\n\\begin{itemize} \n \\item a non-empty set $E$ of \\emph{environment} Boolean variables, also called \\emph{fluents}; \n the elements of $\\modecal{E}} \\newcommand{\\F}{\\modecal{F} = 2^E$ are called \\emph{environment states},\n \\item a non-empty set $A$ (disjoint from $E$) of \\emph{action} Boolean variables; the elements of $\\modecal{A}} \\newcommand{\\B}{\\modecal{B} = 2^A$ are called \\emph{actions}, \n \\item a non-empty set $I \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ of \\emph{initial environment states},\n \\item a relation $Pre \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ of \\emph{available actions} such that for every $s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ there is an $a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ with $(s,a) \\in Pre$ (we say that $a$ is \\emph{available} in $s$), and \n \\item a relation $\\Delta \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B} \\times \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ such that $(s,a,t) \\in \\Delta$ implies that $(s,a) \\in Pre$.\n\\end{itemize}\n\nAs is customary in planning and reasoning about actions, we assume domains are represented compactly by tuples $(E,A,init,pre,\\delta)$ where \n$init \\in Bool(E)$, \n$pre \\in Bool(E \\cup A)$, \nand $\\delta \\in Bool(E \\cup A \\cup E')$ (here $E' \\doteq \\{e' : e \\in E\\}$). This data induces the domain $(E,A,I,Pre,\\Delta)$ where \n\\begin{enumerate}\n \\item $s \\in I$ iff $s \\models init$,\n \\item $(s,a) \\in Pre$ iff $s \\cup a \\models pre$, \n \\item $(s,a,t) \\in \\Delta$ iff $s \\cup a \\cup \\{e' : e \\in t\\} \\models \\delta$.\n\\end{enumerate}\n\nWe emphasize that when measuring the size of $D$ we use this compact representation:\n\\begin{definition} \nThe \\emph{size of $D$}, written $|D|$, is $|E| + |A| + |init| + |pre| + |\\delta|$. \n\\end{definition}\n\nWe remark that in PDDL action preconditions are declared using $\\texttt{:precondition}$, conditional effects using the $\\texttt{when}$ operator, and nondeterministic outcomes using the $\\texttt{oneof}$ operator (note that we code actions with action variables).\n\n\\begin{example}[Universal Domain]\nGiven $E$ and $A$ define the \\emph{universal} domain $U = (E,A,I,Pre,\\Delta)$ where $I \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$, $Pre \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ and $\\Delta \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B} \\times \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$. \n\\end{example}\n\nWe now define the set of environment strategies induced by a domain. We do this by describing a property $\\omega_D$, that itself can be \nrepresented in \\text{{LTL}}\\xspace and \\text{{DPW}}\\xspace, as shown below.\n\n\\begin{definition} \\label{dfn:domain-env} \nFix a domain $D$.\nDefine a property $\\omega_D$ (over atoms $E \\cup A$) as consisting of all traces $\\pi = \\pi_0 \\pi_1 \\ldots$ such that \n\\begin{enumerate} \n\\item $\\pi_0 \\in I$ and \n\\item for all $n \\geq 1$, if $\\pi_i \\cap A$ is available in $\\pi_i \\cap E$ for every $i \\in [0,n-1]$ then $(\\pi_{n-1} \\cap E, \\pi_{n-1} \\cap A, \\pi_{n} \\cap E) \\in \\Delta$.\n\\end{enumerate}\n\n\\end{definition}\nObserve that $\\omega_D$ is an environment assumption since, by the definition of domain, whenever an action is available in a state there is at least one possible successor state. \nIntuitively, an environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ is in $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D)$ if i) its first move is to pick an initial environment state, and ii) thereafter, if the current action $a$ is available in the current environment state $x$ (and the same holds in all earlier steps) then the next environment state $y \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ is constrained so that $(x,a,y) \\in \\Delta$. Notice that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ is unconstrained the moment $a$ is not available in $x$, e.g., in PDDL these would be actions for which the preconditions are not satisfied. Intuitively, this means that it is in the interest of the agent to play available actions because otherwise the agent can't rely {on the fact that the trace comes from the domain.}\n\n\\begin{remark}\nThe reader may be wondering why the above definition does not say i') $\\pi_0 \\in I$ and ii') for all $n \\geq 1$, $(\\pi_{n-1} \\cap E, \\pi_{n-1} \\cap A, \\pi_{n} \\cap E) \\in \\Delta$. Consider the linear-time property $\\omega'_D$ consisting of traces $\\pi$ satisfying i' and ii'. Observe that, in general, $\\omega'_D$ is not environment realizable. Indeed, condition ii' implies that $\\pi_n \\cap A$ is available in $\\pi_n \\cap E$. However, no environment strategy can force the agent to play an available action.\n\\end{remark}\n\n\nWe now observe that one can express $\\omega_D$ in \\text{{LTL}}\\xspace.\n\n\\begin{lemma} \\label{lem:omegaD:LTL}\nFor every domain $D$ there is an \\text{{LTL}}\\xspace formula equivalent to $\\omega_D$. \nFurthermore, the size of the \\text{{LTL}}\\xspace formula is linear in the size of $D$. \n\\end{lemma}\n\nTo see this, say domain $D = (E,A,I,Pre,\\Delta)$ is represented compactly by $(E,A,init,pre,\\delta)$. For the \\text{{LTL}}\\xspace formula, let $\\delta'$ be the $\\text{{LTL}}\\xspace(E \\cup A)$ formula formed from the formula $\\delta \\in Bool(E \\cup A \\cup E')$ by replacing every term of the form $e'$ by $\\nextX e$. Note that $(\\pi,n) \\models \\delta'$ iff $(\\pi_n \\cap E, \\pi_n \\cap A, \\pi_{n+1} \\cap E) \\in \\Delta$. \nThe promised $\\text{{LTL}}\\xspace(E \\cup A)$ formula is \n\\[ init \\wedge (\\always \\delta' \\vee \\delta' \\until \\neg pre).\\]\n\n{One can also express $\\omega_D$ directly by a \\text{{DPW}}\\xspace.}\n\\begin{lemma} \\label{lem:omegaD:DPW}\nFor every domain $D$ there is a \\text{{DPW}}\\xspace $M_D$ equivalent to $\\omega_D$. \nFurthermore, the size of the \\text{{DPW}}\\xspace is at most exponential in the size of $D$ and has two colors.\n\\end{lemma}\n\nTo do this we define the \\text{{DPW}}\\xspace directly rather than translate the \\text{{LTL}}\\xspace formula (which would give a double exponential bound). \nDefine the \\text{{DPW}}\\xspace $M_D \\doteq (Q,q_{in},T,col)$ over $E \\cup A$ as follows. \nIntroduce fresh symbols $q_{in}, q_{+},q_{-}$. Let $q_{in}$ be the initial state. Define $Q \\doteq \\{q_{in},q_{+},q_{-}\\} \\cup (\\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B})$.\nDefine $col(q_{-}) = 1$, and $col(q) = 0$ for all $q \\neq q_{-}$.\nFor all $e,e' \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}, a,a' \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ the transitions are given in Table~\\ref{tab:trans}. Intuitively, on reading the input $e' \\cup a'$ the \\text{{DPW}}\\xspace goes to the rejecting sink $q_{-}$ if $\\Delta$ (resp. $I$) is not respected, \nit goes to the accepting sink $q_{+}$ if $\\Delta$ (resp. $I$) is respected but $Pre$ is not, \nand otherwise it continues (and accepts).\n \\begin{table}[h!]\n \\begin{tabular}{llll}\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $q_{-}$ & if $e' \\not \\in I$\\\\\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $(e',a')$ & if $e' \\in I$ and $(e',a') \\in Pre$ \\\\\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $q_{+}$ & if $e' \\in I$ and $(e',a') \\not \\in Pre$\\\\\n$(e,a)$ & $\\xrightarrow{e' \\cup a'}$& $q_{-}$ & if $(e,a,e') \\not \\in \\Delta$\\\\\n$(e,a)$ & $\\xrightarrow{e' \\cup a'}$& $(e',a')$ & if $(e,a,e') \\in \\Delta$ and $(e',a') \\in Pre$\\\\\n$(e,a)$ & $\\xrightarrow{e' \\cup a'}$& $q_{+}$ & if $(e,a,e') \\in \\Delta$ and $(e',a') \\not \\in Pre$\\\\\n$q_{-}$ & $\\xrightarrow{e' \\cup a'}$& $ q_{-}$ &\\\\\n$q_{+}$ & $\\xrightarrow{e' \\cup a'}$& $ q_{+}$ &\\\\\n\\end{tabular}\n \\caption{Transitions for \\text{{DPW}}\\xspace for $\\omega_D$}\n \\label{tab:trans}\n\\end{table}\n\n\n\\begin{definition}\\label{dfn:planning:assumption}\nLet $D$ be a domain. \n\\begin{itemize} \\item A set $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ is an \\emph{environment assumption for the domain $D$} if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap \\Omega$ is non-empty. \n \\item $\\omega \\in \\SF$ is an \\emph{environment assumption for the domain $D$} if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega)$ is non-empty, i.e., if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega)$ is an environment assumption for the domain $D$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{We illustrate the notion with some examples.}\n\\begin{example} \\label{ex} \\hspace{0cm}\n\\begin{enumerate} \n \n \\item $\\omega \\doteq \\mathsf{true}$ is an environment assumption for $D$ since $\\omega_D \\wedge \\omega \\equiv \\omega_D$ is environment realizable.\n \n \\item Let $\\omega_{D,fair}$ denote the following property: $\\pi \\in \\omega_{D,fair}$ iff for all $(s,a) \\in Pre$, if there are infinitely many $n$ such that $s = \\pi_n \\cap E$ and $a = \\pi_n \\cap A$, then for every $t \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ with $(s,a,t) \\in \\Delta$ there are infinitely many $n$ such that $s = \\pi_n \\cap E, a = \\pi_n \\cap A$ and $t = \\pi_{n+1} \\cap E$. In words, this says that if a state-action pair occurs infinitely often, then infinitely often this is followed by every possible effect.\n \n Note that $\\omega_{D,fair}$ is an environment assumption for domain $D$ since, e.g., the strategy that\n resolves the effects in a round-robin way realizes $\\omega_D \\wedge \\omega_{D,fair}$. Note that $\\omega_{D,fair}$ is definable in \\text{{LTL}}\\xspace by a formula of size exponential in $D$: \n \\[\n\\bigwedge_{s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}} \\bigwedge_{a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}} (\\always \\eventually (s \\wedge a) \\supset \\bigwedge_{s': (s,a,s') \\in \\Delta} \\always \\eventually (s \\wedge a \\wedge \\nextX s')). \n\\]\n\n \n\n\\item In planning, trajectory constraints, e.g., expressed in LTL, have been introduced for expressing temporally extended goals ~\\cite{BacchusK00,DBLP:journals\/ai\/GereviniHLSD09}.\nMore recently, especially in the context of generalized planning, they have been used to describe restrictions on the environment as well \\cite{DBLP:conf\/ijcai\/BonetG15,DeGiacomoMRS16,DBLP:conf\/ijcai\/BonetGGR17}. \nHowever, not all trajectory constraints $\\omega$ can be used as assumptions. In fact, Definition~\\ref{dfn:planning:assumption}, which says that a formula $\\omega$ is an environment assumption for the domain $D$ if $\\omega_D \\wedge \\omega$ is environment realizable, characterizes those formulas that can serve as trajectory constraints.\n\\end{enumerate}\n\\end{example}\n\n\n\nWe can check if $\\omega \\in \\text{{LTL}}\\xspace$ is an environment assumption for $D$ by converting it to a DPW $M_\\omega$, converting $D$ into the \\text{{DPW}}\\xspace $M_D$ (as above), and then checking if the \\text{{DPW}}\\xspace $M_D \\wedge M_\\omega$ is environment realizable. Hence we have:\n\\begin{theorem} \n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTL}}\\xspace$ formula $\\omega$ is an environment assumption for the domain $D$ is $2$\\textsc{exptime}\\xspace-complete. Moreover, it can be solved in \n \\textsc{exptime}\\xspace in the size of $D$ and $2$\\textsc{exptime}\\xspace in the size of $\\omega$. \n \\item Deciding if a \\text{{DPW}}\\xspace $\\omega$ is an environment assumption for the domain $D$ is in \\textsc{exptime}\\xspace. Moreover, it can be solved in \\textsc{exptime}\\xspace in the size \n of $D$ and \\textsc{ptime}\\xspace in the size of $\\omega$ and \\textsc{exptime}\\xspace in the number of colors of $\\omega$.\n\\end{enumerate}\n\\end{theorem}\nFor the lower bound take $D \\doteq U$ to be the universal domain and apply the lower bound from Theorem~\\ref{fact:synthesis}.\n\n\n{Now we turn to planning under assumptions.}\n\\begin{definition}[Planning under Assumptions -- abstract] \\label{def:PUA}\\hspace{0cm}\n\\begin{enumerate}\n\\item A \\emph{planning under assumptions problem $P$} is a tuple $((D,\\Omega),\\Gamma)$ where \n \\begin{itemize} \n \\item $D$ is a domain,\n \\item $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ is an environment assumption for $D$, and \n \\item $\\Gamma$ is an agent goal.\n \\end{itemize}\n \\item We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{solves} $P$ if \n \\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\in Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap \\Omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in \\Gamma\\] \n\\end{enumerate}\n\\end{definition}\n\n\n\nWe can instantiate this definition to environment assumptions and agent goals definable in $\\SF$.\n\\begin{definition}[Planning under Assumptions -- linear-time] \\hspace{0em}\n\\begin{enumerate}\n \\item An \\emph{$\\SF$ planning under assumptions problem} is a tuple $P = ((D,\\omega),\\gamma)$ where $\\omega \\in \\SF$ is an environment assumption for $D$ and $\\gamma \\in \\SF$ is an agent goal.\n \\item We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\gamma$ assuming $\\omega$}, or that it \\emph{solves} $P$, if \n \\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright (\\omega_D \\wedge \\omega). \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma\\] \n\\end{enumerate}\n\\end{definition}\n\nThe corresponding decision problem asks, given an \\SF planning under assumptions problem $P$ to decide whether there is an agent strategy that solves $P$. For instance, \\emph{\\text{{LTL}}\\xspace planning under assumptions} asks, given $P = ((D,\\omega),\\gamma)$ with $\\omega,\\gamma \\in \\text{{LTL}}\\xspace$, \nto decide if there is an agent strategy that solves $P$, and to return such a finite-state strategy (if one exists). Similar definitions apply to \\text{{DPW}}\\xspace planning under assumptions, etc.\n\n \n\n \n{It turns out that {virtually all forms of planning {(with linear-time temporally extended goals)}} in the literature are special cases of planning under \\text{{LTL}}\\xspace assumptions, i.e., {the set of strategies that solve a given planning problem are exactly the set of strategies that solve the corresponding planning under assumptions problem.} \nIn the following, $\\mathit{Goal} \\in Bool(E \\cup A)$, and \n$\\mathit{Exec}$ is the \\text{{LTL}}\\xspace formula $\\always \\bigwedge_{a \\in A} (a \\supset \\mathit{pre_a})$ expressing that if an action is done then its precondition holds.}\n\n\n\\begin{example}\n\n \\begin{enumerate}\n\\item FOND planning with reachability goals~\\cite{Rintanen:ICAPS04} corresponds to \\text{{LTL}}\\xspace planning under assumptions with $\\omega \\doteq \\mathsf{true}$ and {$\\gamma \\doteq \\mathit{Exec} \\land \\eventually Goal$.}\n\n\\item FOND planning with \\text{{LTL}}\\xspace (temporally extended) goals $\\gamma$ \n\\cite{BacchusK00,PistoreT01,CTMBM17}.\ncorresponds to \\text{{LTL}}\\xspace planning under assumptions with $\\omega \\doteq \\mathsf{true}$ {and goal $\\mathit{Exec} \\land \\gamma$.}\n\n\\item FOND planning with \\text{{LTL}}\\xspace trajectory constraints $\\omega$ and \\text{{LTL}}\\xspace (temporally extended) goals $\\gamma$ \\cite{DBLP:conf\/ijcai\/BonetG15,DeGiacomoMRS16,DBLP:conf\/ijcai\/BonetGGR17} corresponds to \\text{{LTL}}\\xspace planning under assumptions {with assumptions $\\omega$ and goal $\\mathit{Exec} \\land \\gamma$.}\n\n\\item {Fair FOND planning} with reachability goals~\\cite{DaTV99,GeBo13,DIppolitoRS18} corresponds to planning under assumptions with \n$\\omega \\doteq \\omega_{D,fair}$ and $\\gamma \\doteq \\textit{Exec} \\land \\eventually Goal$.\n\n\\item {Fair FOND planning} with (temporally extended) goals $\\gamma$ as defined in~\\cite{DBLP:conf\/ijcai\/PatriziLG13,CTMBM17} corresponds to planning under assumptions with $\\omega \\doteq \\omega_{D,fair}$ and goal $\\mathit{Exec} \\land \\gamma$.\n\n\\item Obviously adding \\text{{LTL}}\\xspace trajectory constraints $\\omega_{tc}$ to {fair FOND planning} with (temporally extended) goals corresponds to planning under assumptions with $\\omega \\doteq \\omega_{D,fair} \\wedge \\omega_{tc}$ {and goal $\\mathit{Exec} \\land \\gamma$.} \n\\end{enumerate}\n\n\\end{example}\n\n\n{We also observe that the Fair FOND planning problems just mentioned can be captured by \\text{{LTL}}\\xspace planning under assumptions since $\\omega_{D,fair}$ can be written in \\text{{LTL}}\\xspace (see Example~\\ref{ex}).\n\n\n\n\n\n\\section{Translating between planning and synthesis}\nIn this section we ask the question if there is a {fundamental} difference between synthesis and planning in our setting (i.e., assumptions and goals given as linear-time properties). We answer by observing that there are translations between them. The next two results follow immediately from the definitions:\n\n\\begin{theorem}[Synthesis to Planning] \\label{prop:synthesis to planning}\nLet $(E,A,\\omega,\\gamma)$ be a synthesis under Assumptions problem, and let \n$P = ((U,\\omega),\\gamma)$ be the corresponding \nPlanning under Assumptions problem where $U$ is the universal domain. Then, for every agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ we have that \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ solves $P$ iff $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega$.\n\\end{theorem}\n\n\\begin{theorem}[Planning to Synthesis] \\label{prop:planning to synthesis}\nLet $D = (E,A,I,Pre,\\Delta)$ be a domain and let $P = ((D,\\omega),\\gamma)$ be a Planning under Assumptions problem. \nLet $(E,A,\\omega_D \\wedge \\omega,\\gamma)$ be the corresponding Synthesis under Assumptions problem. Then, for every agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ we have that \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ solves $P$ iff $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega_D \\wedge \\omega$.\n\\end{theorem}\n\nThus, we can solve \\text{{LTL}}\\xspace planning under assumptions by reducing to \\text{{LTL}}\\xspace synthesis under assumptions, which itself can be solved by known results (i.e., Theorem~\\ref{fact:synthesis}):\n\\begin{corollary}\\label{thm:solving:PUA:LTL:combined}\n Solving \\text{{LTL}}\\xspace planning under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\end{corollary}\nHowever, this does not distinguish the complexity measured in the size of the domain from that in the size of the assumption and goal formulas. \n{We take this up next.}\n\n\n\n\\section{Solving Planning under Assumptions}\n\nIn this section we show how to solve Planning under Assumptions for concrete specification languages \\SF, i.e., \\SF = \\text{{LTL}}\\xspace and \\SF = \\text{{DPW}}\\xspace. We measure the complexity in two different ways: we fix the domain $D$ and measure the complexity with respect to the size of the formulas or automata for the environment assumption and the agent goal, this is called \\emph{goal\/assumption complexity}; \nand we fix the formulas\/automata and measure the complexity with respect to the size of the domain, this is called the \\emph{domain complexity}. \\footnote{Formally, if $C$ is a complexity class, we say that \\emph{goal\/assumption complexity is in $C$} if for every domain $D_0$ the complexity of deciding if there is an agent strategy solving $P = ((D_0,\\omega),\\gamma)$, is in $C$. A similar definition holds for domain complexity. Also, we say that the \\emph{goal\/assumption complexity is $C$-hard} if there exists a domain $D_0$ such that the problem of deciding if there is an agent strategy solving $P = ((D_0,\\omega),\\gamma)$, is $C$-hard.} \n\n\nWe begin with \\SF = \\text{{DPW}}\\xspace and consider the following algorithm:\nGiven $P = ((D,\\omega),\\gamma)$ in which $\\omega$ is represented by a \\text{{DPW}}\\xspace $M_\\omega$ and $\\gamma$ is represented by a \\text{{DPW}}\\xspace $M_\\gamma$, perform the following steps:\n\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 1. Solving \\text{{DPW}}\\xspace planning under assumptions}\\\\\nGiven domain $D$, assumption $M_\\omega$, goal $M_\\gamma$.\\\\\n1:\\>Form \\text{{DPW}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form \\text{{DPW}}\\xspace $M$ for $(M_D \\wedge M_\\omega) \\supset M_\\gamma$.\\\\\n3:\\>Solve the parity game on $M$.\n\\end{tabbing}\n\nThe first step results in a \\text{{DPW}}\\xspace whose size is exponential in the size of $D$ and with a constant number of colors (Lemma~\\ref{lem:omegaD:DPW}). \nThe second step results in a \\text{{DPW}}\\xspace whose size is polynomial in the number of states of the \\text{{DPW}}\\xspace{s} involved (i.e., $M_D,M_\\omega$ and $M_\\gamma$), and exponential in the number of their colors (Lemma~\\ref{lem:DPW Boolean combinations}).\nFor the third step, the think of the \\text{{DPW}}\\xspace $M$ as a parity game: play starts in the initial state, and at each step, if $q$ is the current state of $M$, first the environment picks $s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ and then the agent picks an action $a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$, i.e., an evaluation of the action variables. The subsequent step starts in the state of $M$ resulting from taking the unique transition from $q$ labeled $s \\cup a$. This produces a run of the \\text{{DPW}}\\xspace which the agent is trying to ensure is successful (i.e., the largest color occurring infinitely often is even). \n\nFormally, we say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ is \\emph{winning} if \nfor every environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$, the unique run of the \\text{{DPW}}\\xspace on input word $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is successful. \nDeciding if the a player has a winning strategy, and returning a finite-state strategy (it one exists), is called \\emph{solving} the game. \nParity games can be solved in \ntime polynomial in the size of $M$ and exponential in the number of colors of $M$~\\cite{ALG02}.\\footnote{Better algorithms are known, e.g.~\\cite{CaludeJKL017}, but are not helpful for this paper.} \n\n{The analysis of the above algorithm shows the following.}\n\\begin{theorem} \\label{thm:solving:PUA:DPW} \\hspace{0em}\n\\begin{enumerate} \\item The domain complexity of solving \\text{{DPW}}\\xspace planning under assumptions is in \\textsc{exptime}\\xspace. \n\\item The goal\/assumption complexity of solving \\text{{DPW}}\\xspace planning under assumptions is in \\textsc{ptime}\\xspace in their sizes and \\textsc{exptime}\\xspace in the number of their colors.\n\\end{enumerate}\n\\end{theorem}\n\n{Moreover,} by converting \\text{{LTL}}\\xspace formulas to \\text{{DPW}}\\xspace with exponentially many colors and double-exponential many states \\cite{DBLP:conf\/banff\/Vardi95,DBLP:journals\/lmcs\/Piterman07}, we get the upper bounds in the following:\n\\begin{theorem} \\label{thm:solving:PUA:LTL}\n\\begin{enumerate}\n\\item The domain complexity of solving \\text{{LTL}}\\xspace planning under assumptions is \\textsc{exptime}\\xspace-complete.\n\\item The goal\/assumption complexity of solving \\text{{LTL}}\\xspace planning under assumptions is $2$\\textsc{exptime}\\xspace-complete. \n\\end{enumerate}\n\\end{theorem}\n{For the matching lower-bounds, we have that}\nthe domain complexity is \\textsc{exptime}\\xspace-hard follows from the fact that planning with reachability goals and no assumptions is \\textsc{exptime}\\xspace-hard~\\cite{Rintanen:ICAPS04}; to see that the goal\/assumption complexity is $2$\\textsc{exptime}\\xspace-hard note that \\text{{LTL}}\\xspace synthesis, known to be $2$\\textsc{exptime}\\xspace-hard~\\cite{PnueliR89,rosner1992modular}, is a special case (take $\\omega \\doteq \\mathsf{true}$ and $D$ to be the universal domain). \n\n{Similarly, one can apply this technique to solving Fair \\text{{LTL}}\\xspace planning under assumptions. The exact complexity, however, is open. See the conclusion for a discussion.}\n\n\n\n\n\n\\section{Focusing on finite traces}\nIn this section we revisit the definitions and results in case that assumptions and goals are expressed as linear-time properties over \\emph{finite} traces. There are two reasons to do this. \nFirst, in AI and CS applications executions of interest are often finite~\\cite{DegVa13}.\nSecond, the algorithms presented for the infinite-sequence case involve complex constructions on automata\/games that are notoriously hard to optimize~\\cite{DFogartyKVW13}. Thus, we will not simply reduce the finite-trace case to the infinite-trace case~\\cite{GMM14}. We begin by carefully defining the setting.\n\n\\subsubsection{Synthesis and linear-time specifications over finite traces}\nWe define synthesis over finite traces in a similar way to the infinite-trace case, {cf.~\\cite{DegVa15,Camacho:KR18}.}\nThe main difference is that agent strategies \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace:\\modecal{E}} \\newcommand{\\F}{\\modecal{F}^+ \\to \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ can be partial. This represents the situation that the agent stops the play. Environment strategies $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ are total (as before). Thus, the resulting play $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ may be finite, if the agent chooses to stop, as well as infinite.\\footnote{Formally, $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is redefined to be the longest trace (it may be finite or infinite) that complies with both strategies.} Objectives may be expressed in general specification formalisms \\SFf for finite traces, e.g., \\SFf = \\text{{LTLf}}\\xspace (\\text{{LTL}}\\xspace over finite traces\\footnote{All our results for \\text{{LTLf}}\\xspace also hold for linear-dynamic logic over finite traces (\\text{{LDLf}}\\xspace)~\\cite{DegVa13}. \n}), \\SFf = \\text{{DFA}}\\xspace (deterministic finite word automata). For $\\phi \\in \\SFf$, we overload notation and write $[[\\phi]]$ for the set of finite traces $\\phi$ defines.\n\nWe now define realizability in the finite-trace case:\n\\begin{definition}\nLet $\\phi \\in \\SFf$.\n\\begin{enumerate} \n\\item We say that \\emph{$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace. \\left(\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite and } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]\\right)$. \n\\item We say that \\emph{$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\left(\\text{if } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite, then } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}\\in [[\\phi]]\\right)$. \n\\end{enumerate}\n\\end{definition}\nThe asymmetry in the definition results from the fact that stopping is controlled by the agent.\n\n\nDuality still holds, and is easier to prove since it amounts to determinacy of reachability games~\\cite{ALG02}:\n\\begin{lemma}[Duality] \\label{lem:duality:finite}\nFor every $\\phi \\in \\SFf$ we have that $\\phi$ is not agent realizable iff $\\neg \\phi$ is environment realizable.\n\\end{lemma}\n\n\\subsubsection{Linear temporal logic on finite traces (\\text{{LTLf}}\\xspace)} The logic $\\text{{LTLf}}\\xspace$ has the same syntax as $\\text{{LTL}}\\xspace$ but is interpreted on finite traces $\\pi \\in (2^{\\sf{Var}})^+$. Formally, for $n \\leq len(\\pi)$ (the length of $\\pi$) we only reinterpret the temporal operators:\n\\begin{itemize}\n\t\\item $(\\pi,n) \\models \\nextX \\varphi$ iff $n < len(\\pi)$ and $(\\pi,n+1) \\models \\varphi$;\n\t\\item $(\\pi,n) \\models \\varphi_1 \\until \\varphi_2$ iff \n\tthere exists $i$ with $n \\leq i \\leq len(\\pi)$ such that $(\\pi,i) \\models \\varphi_2$ and for all $i \\leq j < n$, $(\\pi,j) \\models \\varphi_1$.\n\\end{itemize}\nLet $\\tilde{\\nextX}$ denote the dual of $\\nextX$, i.e., $\\tilde{\\nextX} \\doteq \\neg \\nextX \\neg \\varphi$. Semantically we have that\n\\begin{itemize}\n\t\\item $(\\pi,n) \\models \\tilde{\\nextX} \\varphi$ iff $n < len(\\pi)$ implies $(\\pi,n+1) \\models \\varphi$.\n\\end{itemize}\n\n\n\\subsubsection{Deterministic finite automata (\\text{{DFA}}\\xspace)} \n\nA \\text{{DFA}}\\xspace over ${\\sf{Var}}$ is a tuple $M = (Q,q_{in},T,F)$ which is like a \\text{{DPW}}\\xspace except that $col$ is replaced by a set $F \\subseteq Q$ of final states. The run on a finite input trace $\\pi \\in (2^{\\sf{Var}})^*$ is successful if it ends in a final state. We recall that \\text{{DFA}}\\xspace are closed under Boolean operations using classic algorithms (e.g., see \\cite{DBLP:conf\/banff\/Vardi95}).\nAlso, \\text{{LTLf}}\\xspace formulas $\\varphi$ (and also \\text{{LDLf}}\\xspace formulas) can be effectively translated into \\text{{DFA}}\\xspace. This is done in three classic simple steps that highlight the power of the automata-theoretic approach: convert $\\varphi$ to an alternating automaton (poly), then into a nondeterministic finite automaton (exp), and then into a \\text{{DFA}}\\xspace (exp). These steps are outlined in detail in, e.g., \\cite{DegVa13}. \n\n\n\\subsubsection{Solving Synthesis over finite traces} \\SFf agent synthesis is the problem, given $\\phi \\in \\SFf$, of deciding if the agent can realize $\\phi$. Now, solving \\text{{DFA}}\\xspace agent synthesis is \\textsc{ptime}\\xspace-complete: it amounts to solving a reachability game on the given \\text{{DFA}}\\xspace $M$, which can be done with an algorithm that captures how close the agent is to a final state, i.e., a least-fixpoint of the operation. Finally, to solve \\text{{LTLf}}\\xspace agent synthesis first translate the \\text{{LTLf}}\\xspace formula to a \\text{{DFA}}\\xspace and then run the fixpoint algorithm (also, \\text{{LTLf}}\\xspace agent synthesis is $2$\\textsc{exptime}\\xspace-complete)~\\cite{DegVa15}.\n\n\n\nNote that, by Duality, solving \\text{{LTLf}}\\xspace environment realizability and solving \\text{{LTLf}}\\xspace agent realizability are inter-reducible (and thus the former is also $2$\\textsc{exptime}\\xspace-complete). Thus, to decide if $\\phi$ is environment realizable we simply negate the answer to whether $\\neg \\phi$ is agent realizable. However, to extract an environment strategy, one solves the dual safety game.\n\n\n\n\\subsubsection{Synthesis under assumptions}\nWe say that $\\omega \\in \\SFf$ is an \\emph{environment assumption} if $\\omega$ is environment realizable. Solving \n\\SFf synthesis under assumptions means to decide if there is an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ such that\n\\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega. \\left(\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite and } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma\\right).\\]\n\nWe now consider the case that $\\SFf = \\text{{LTLf}}\\xspace$. Checking if $\\omega \\in \\text{{LTLf}}\\xspace$ is an environment assumption is, by definition, \nthe problem of deciding if $\\omega$ is environment realizable, as just discussed. Hence we can state the following:\n\\begin{theorem} \\hspace{0em}\n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTLf}}\\xspace$ formula $\\omega$ is an environment assumption is $2$\\textsc{exptime}\\xspace-complete. \n \\item Deciding if a $\\text{{DFA}}\\xspace$ $\\omega$ is an environment assumption is \\textsc{ptime}\\xspace-compete (cf.~\\cite{ALG02}).\n\\end{enumerate}\n\\end{theorem}\n\nTurning to \\text{{LTLf}}\\xspace synthesis under assumptions we have \nthat synthesis under assumptions and synthesis of the implication are equivalent. Indeed, as before, the key point is the duality which we have in \nLemma~\\ref{lem:duality:finite}:\n\n\n\n\n\\begin{theorem} \\label{thm:det:finite}\nSuppose $\\omega \\in \\SFf$ is an environment assumption.\n The following are equivalent:\n \\begin{enumerate}\n \\item There is an agent strategy realizing $\\omega \\supset \\gamma$.\n \\item There is an agent strategy realizing $\\gamma$ assuming $\\omega$.\n \\end{enumerate}\n\\end{theorem}\n\nHence to solve synthesis under assumptions we simply solve agent synthesis for the implication. Hence we have:\n\\begin{theorem}\\label{thm:solving:SUA:finite} \\hspace{0em}\n\\begin{enumerate} \n\\item Solving \\text{{LTLf}}\\xspace synthesis under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\item Solving \\text{{DFA}}\\xspace synthesis under assumptions is \\textsc{ptime}\\xspace-complete.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsubsection{Planning under assumptions}\nPlanning and fair planning have recently been studied for \\text{{LTLf}}\\xspace goals~\\cite{DR-IJCAI18,Camacho:KR18,Camacho:ICAPS18}. Here we define and study how to add environment assumptions.\n\nRecall that we represent a planning domain $D$ by the linear-time property $\\omega_D$ (Definition~\\ref{dfn:domain-env}) which itself was defined as those infinite traces satisfying two conditions. The exact same conditions determine a set of finite traces, also denoted $\\omega_D$. Moreover, \nthis $\\omega_D$ is equivalent to an \\text{{LTLf}}\\xspace formula of size linear in $D$ and a \\text{{DFA}}\\xspace of size at most exponential in $D$. \nTo see this, replace $\\nextX$ by $\\tilde{\\nextX}$ in the \\text{{LTL}}\\xspace formula from Lemma~\\ref{lem:omegaD:LTL}. That is, \nlet $\\delta''$ be the $\\text{{LTLf}}\\xspace$ formula formed from $\\delta$ by replacing every term of the form $e'$ by $\\tilde{\\nextX} e$. Note that if \n$n < len(\\pi)$ then $(\\pi,n) \\models \\delta''$ iff $(\\pi_n \\cap E, \\pi_n \\cap A, \\pi_{n+1} \\cap E) \\in \\Delta$, and if $n = len(\\pi)$ then $(\\pi,n) \\models \\delta''$ iff $(\\pi_n \\cap E, \\pi_n \\cap A) \\in Pre$. \nThe promised $\\text{{LTLf}}\\xspace(E \\cup A)$ formula is \n$init \\wedge (\\always \\delta'' \\vee \\delta'' \\until \\neg pre)$. \nAlso, similar to the \\text{{DPW}}\\xspace before there is a \\text{{DFA}}\\xspace of size at most exponential in the size of $D$ equivalent to $\\omega_D$. \nTo see this, take the \\text{{DPW}}\\xspace $M_D \\doteq (Q,q_{in},T,col)$ from Lemma~\\ref{lem:omegaD:DPW} and instead of $col$ define the set \nof final states to be the set $col^{-1}(0)$.\n\nAs before, say that $\\omega \\in \\SFf$ is an \\emph{environment assumption for the domain $D$} if $\\omega_D \\wedge \\omega$ is environment realizable. \nDefine an \\emph{\\SFf planning under assumptions problem} to be a tuple $P = ((D,\\omega),\\gamma)$ with $\\omega,\\gamma \\in \\SFf$ \nsuch that $\\omega$ is an environment assumption for $D$. To decide if $\\omega \\in \\text{{LTLf}}\\xspace\/\\text{{DFA}}\\xspace$ is an environment assumption for $D$ \nwe use the next algorithm:\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 2. Deciding if $\\omega$ is an environment assumption for $D$}\\\\\nGiven domain $D$, and \\text{{DFA}}\\xspace $M_\\omega$.\\\\\n1:\\>Convert $D$ into a \\text{{DFA}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form the \\text{{DFA}}\\xspace $M$ for $(M_D \\wedge M_\\omega)$.\\\\\n3:\\>Decide if $M$ is environment realizable.\n\\end{tabbing}\nFurther, if $\\omega$ is given as an \\text{{LTLf}}\\xspace formula, first convert it to a \\text{{DFA}}\\xspace $M_\\omega$ and then run the algorithm. We then have:\n\\begin{theorem} \\hspace{0em}\n\\begin{enumerate}\n \\item Deciding if $\\text{{LTLf}}\\xspace$ formula $\\omega$ is an environment assumption for the domain $D$ is $2$\\textsc{exptime}\\xspace-complete. Moreover, it can be solved in \n \\textsc{exptime}\\xspace in the size of $D$ and $2$\\textsc{exptime}\\xspace in the size of $\\omega$. \n \\item Deciding if $\\text{{DFA}}\\xspace$ $\\omega$ is an environment assumption for the domain $D$ is in \\textsc{exptime}\\xspace. Moreover, it can be solved in \\textsc{exptime}\\xspace in the size \n of $D$ and \\textsc{ptime}\\xspace in the size of $\\omega$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsubsection{Solving Planning under Assumptions}\nAs before, there are simple translations between \\SFf planning under assumptions and \\SFf synthesis under assumptions. And again, solving \\text{{LTLf}}\\xspace planning under assumptions via such a translation is not fine enough to analyze the complexity in the domain vs the goal\/assumption. \nTo solve \\text{{DFA}}\\xspace\/\\text{{LTLf}}\\xspace planning under assumptions use the following simple algorithm:\n\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 3. Solving \\text{{DFA}}\\xspace planning under assumptions}\\\\\nGiven domain $D$, assumption $M_\\omega$, goal $M_\\gamma$.\\\\\n1:\\>Convert $D$ into a \\text{{DFA}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form the \\text{{DFA}}\\xspace $M$ for $(M_D \\wedge M_\\omega) \\supset M_\\gamma$.\\\\\n3:\\>Solve the reachability game on \\text{{DFA}}\\xspace $M$.\n\\end{tabbing}\n\nFurther, if $\\omega$ is given as an \\text{{LTLf}}\\xspace formula, first convert it to a \\text{{DFA}}\\xspace $M_\\omega$ and then run the algorithm. This gives the upper bounds in the following:\n\\begin{theorem} \\label{thm:solving:PUA:DFW} \\hspace{0em}\n\\begin{enumerate} \n\\item The domain complexity of solving \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) planning under assumptions is \\textsc{exptime}\\xspace-complete.\n\\item The goal\/assumption complexity of solving \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) planning under assumptions is \\textsc{ptime}\\xspace-complete (resp. $2$\\textsc{exptime}\\xspace-complete). \n\\end{enumerate}\n\\end{theorem}\nFor the lower bounds, setting $\\omega \\doteq \\mathsf{true}$ results in FOND with reachability goals, known to be \\textsc{exptime}\\xspace-hard~\\cite{Rintanen:ICAPS04}; and additionally taking the domain $D$ to be the universal domain results in \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) synthesis, known to be \\textsc{ptime}\\xspace-hard~\\cite{ALG02} (resp. $2$\\textsc{exptime}\\xspace-hard~\\cite{DegVa15}).\n\nFinally, if $P = ((D,\\omega),\\gamma)$ is an \\text{{LTLf}}\\xspace planning under assumptions problem, say that $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{fairly solves} $P$ if for every $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega_D \\wedge \\omega$ we have that if $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\omega_{D,fair}]]$ then $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is finite and satisfies $\\gamma$ (here $\\omega_{D,fair}$ from Example~\\ref{ex} is defined so that it now also includes all finite traces). We remark that Alg $2$ applies unchanged. \n{However, to solve the fair \\text{{LTLf}}\\xspace planning problem, we do not know a better way, in general, than translating the problem into one over infinite traces and applying the techniques from the previous section.}\n\n\n\n\n\\section{Conclusion and Outlook}\n \n\nWhile we illustrate synthesis and planning under assumptions expressed in\nlinear-time specifications, our definitions immediately apply to assumptions\nexpressed in branching-time specifications, e.g., $\\ensuremath{\\mathsf{CTL}^*}\\xspace$, $\\mu$-calculus, and\ntree automata. As future work, it is of great interest to study synthesis\nunder assumptions in the branching time setting so as to devise restrictions on\n\\emph{possible agent behaviors} with certain guarantees, e.g., remain in an\narea from where the agent can enforce the ability to reach the recharging doc,\nwhenever it needs to, in the spirit of~\\cite{LagoPT02}.\n\nAlthough our work is in the context of reasoning about actions and planning, we\nexpect it can also provide insights to verification and to multi-agent systems.\nIn particular, the undesirable drawback of the agent being able to falsify an\nassumption when synthesizing $Assumption \\supset Goal$ is well known, and it has\nbeen observed that it can be overcome when the $Assumption$ is environment\nrealizable~\\cite{DBLP:journals\/tosem\/DIppolitoBPU13,DBLP:journals\/acta\/BrenguierRS17}.\nOur Theorem~\\ref{thm:det} provides the principle for such a solution.\nInterestingly, various degrees of cooperation to fulfill assumptions among\nadversarial agents has been considered, e.g.,\n\\cite{CH07,BEK15,DBLP:journals\/acta\/BrenguierRS17} and we believe that a work\nlike present one is needed to establish similar principled foundations.\n\nTurning to the multi-agent setting, there, agents in a common environment\ninteract with each other and may have their own objectives. Thus, it makes\nsense to model agents not as hostile to each other, but as rational, i.e.,\nagents that act to achieve their own objectives. \\emph{Rational\nsynthesis}~\\cite{KPV14} (as compared to classic synthesis) further requires\nthat the strategy profile chosen by the agents is in equilibrium (various\nnotions of equilibrium may be used). It would be interesting to investigate\nrational synthesis under environment assumptions, in the sense that all agents\nalso make use of their own assumptions about their common environment. We\nbelieve that considering assumptions as sets of strategies rather than sets of\ntraces will serve as a clarifying framework also for the multi-agent setting.\n\n{Finally, there are a number of open questions regarding the computational complexity \nof solving synthesis\/planning under assumptions, i.e., what is the exact complexity of {Fair \\text{{LTL}}\\xspace\/\\text{{LTLf}}\\xspace planning under assumptions}? what is the assumption complexity of \\text{{LTL}}\\xspace\/\\text{{LTLf}}\\xspace synthesis under assumptions? Here, the \\emph{assumption} complexity is the complexity of the problem assuming the domain and goal are fixed, and the only input to the problem is the assumption formula\/automaton.}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sectionintroduction}\n\nThe use of techniques and concepts from effective field theories\n(EFT's) has lead to an encouraging record of achievements for the\ndescription of nonrelativistic fermion-antifermion systems. The\nformulation of a nonrelativistic EFT for such systems has gone through\na number of important conceptual states~\\cite{Reviews}. Bodwin,\nBraaten and \nLepage~\\cite{Caswell,BBL} initially formulated a method for separating \nfluctuations at short distances of order the heavy particle mass $m$\nfrom those at long distances related to the nonrelativistic momentum\nand energy, $mv$ and $mv^2$ ($v$ being the relative velocity) and the\nhadronic scale $\\Lambda_{\\rm QCD}$. Subsequent work helped to clarify\nthe power counting in $v$, and the relevant degrees of freedom needed\nto construct an EFT. \n\nFor very heavy quarkonium systems, characterized by the scale\nhierarchy $m\\gg m v\\gg mv^2 > \\Lambda_{\\rm QCD}$ the formulation has\nreached a mature state. In Ref.~\\cite{Labelle} it was realized that it\nis necessary to distinguish soft ($\\sim mv$) and ultrasoft ($\\sim\nmv^2$) fluctuations and in Ref.~\\cite{LM} that the original NRQCD\naction~\\cite{BBL} is problematic concerning the simultaneous power\ncounting of soft and ultrasoft terms. In particular, the Lagrangian\nneeds to be multipole-expanded in the presence of ultrasoft\ngluons~\\cite{Labelle,GR}. Technically this can be formulated by\nintroducing soft as well as ultrasoft gluons in the theory~\\cite{LS}.\n\nIn Refs.~\\cite{PS,PS2,PS3}, based on the hierarchy $m\\gg m v\\gg\nmv^2$, it was suggested to employ a series of EFT's, one for the\nscales $m > \\mu \\mathrel{\\raise.3ex\\hbox{$>$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}} mv$ and one for $mv > \\mu \\mathrel{\\raise.3ex\\hbox{$>$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}} mv^2$, called pNRQCD\n(\"potential\" NRQCD). The pNRQCD is derived in a two-step matching\nprocedure where the intermediate soft matching scale and the pNRQCD\nrenormalization scale are independent at the field theoretic level. \nThis is appropriate e.g. for static quarks where the momentum is fixed\nby the quark separation $r\\sim 1\/mv$ and not correlated with the\ndynamical energy fluctuations $E\\sim mv^2$. The potential interactions\nare generated from integrating out soft fluctuations at the soft\nmatching scale. \n\nIn Ref.~\\cite{LMR} is was pointed out that for dynamical heavy\nquarkonium systems the dispersive correlation between ultrasoft energy\nand soft momentum scales, $E\\sim \\mathbf p^2\/m$, can be implemented\nsystematically at the field theoretic level by matching to the proper\nEFT directly at the hard scale $\\mu\\sim m$ (one-step matching). The EFT,\ncalled vNRQCD \n(\"velocity\" NRQCD) has a strict power counting in $v$.\nIt contains soft and ultrasoft degrees of freedom as well as soft and\nultrasoft renormalization scales, $\\mu_S$ and $\\mu_U$. Through the\ndifferent velocity counting of soft and ultrasoft fields they are\ncorrelated as $\\mu_U\\propto \\mu_S^2\/m$, where usually the constant of\nproportionality is set to $1$, i.e. $\\mu_U=\\mu_S^2\/m=m\\nu^2$. The\nrenormalization group running of the theory is then conveniently\nexpressed in terms of the dimensionless parameter $\\nu$. The matching\nis carried out at the hard scale $\\nu=1$ and the theory is evolved to\n$\\nu\\sim v\\sim \\alpha_s$ of order of the relative velocity of the\ntwo-body system for computations of matrix elements. The running\nproperly sums logarithms of the momentum and the energy scale at the\nsame time~\\cite{LMR,amis4,mss1,HoangStewartultra} \nand is referred to as the velocity renormalization group\n(VRG)~\\cite{LMR}. Within dimensional regularization the powers of\n$\\mu_S^\\epsilon$ and $\\mu_U^\\epsilon$ multiplying the \noperators of the Lagrangian are uniquely determined by the $v$\ncounting and the dimension of the operators in $d=4-2\\epsilon$\ndimensions. \n\nIn Ref.~\\cite{Hoang3loop}, where a three-loop renormalization of\nquark-antiquark vertex diagrams was carried out for fermion-antifermion\nS-wave states, is was demonstrated that the one-step matching principle, upon\nwhich vNRQCD is based, is consistent under renormalization at the subleading\norder level, when subdivergences need to be subtracted and when UV\ndivergences from heavy quark-antiquark loops and from soft or\nultrasoft gluons appear simultaneously in single diagrams. Up to now there is \nno analogous demonstration for the two-step matching principle. \n\nWith the VRG, the running of potentials relevant for the\nnext-to-next-to-leading logarithmic (NNLL) description of the\nnonrelativistic dynamics of pairs of heavy quarks and colored scalars\nhas been determined in Refs.~\\cite{amis,amis2,amis3,HoangStewartultra}\nand \\cite{HoangRuizSvNRQCD}. The NLL running of leading (in the $v$\nexpansion) S-wave currents for heavy quark-antiquark production and of\nthe leading S- and P-wave currents for pairs of colored scalars were\ndetermined in Refs.~\\cite{amis3,HoangStewartultra} and\n\\cite{HoangRuizSvNRQCD}. Corresponding work in the pNRQCD formalism\nfor heavy quarks was carried out in\nRefs.~\\cite{PSstat,Pineda1,Pineda2,Pineda3}. In these computations,\ndiagrams which simultaneously contain both UV divergences from heavy\nquark-antiquark loops and from soft or ultrasoft gluon loops do not arise. The\nfinal results based on the VRG and on pNRQCD agree, after the\nscale correlations from the one-step matching procedure of vNRQCD are\nimposed in pNRQCD. Concerning the NNLL running of \nthe production currents, only the contributions from three-loop vertex\ndiagrams for the leading currents describing quark-antiquark pair\nproduction in an S-wave configuration~\\cite{Hoang3loop} are \nknown at this time in vNRQCD. For the NNLL contributions that arise from the \nsubleading order evolution of the coefficients that appears in the NLL\nanomalous dimension of the current at present only the results for the\nCoulomb potential~\\cite{PSstat,HoangStewartultra}, the spin-dependent\n$1\/m^2$ potentials~\\cite{Peninspin1,Peninspin2} and from\nultrasoft quark loops~\\cite{Stahlhofen1} have been determined. \n\nThe main phenomenological application of the nonrelativistic EFT for\nthe situation $m\\gg mv\\gg mv^2>\\Lambda_{\\rm QCD}$ is top quark pair\nproduction close to threshold at a future Linear Collider~\\cite{LC}. \nHere the summation of logarithms of $v$ is crucial to control the\nlarge normalization uncertainties~\\cite{hmst,hmst1} that arise in \nfixed-order nonrelativistic computations that only account for the\nsummation of Coulomb potential insertions~\\cite{HT1,Hoangsynopsis}. One\ncan expect that the summation of logarithms of the velocity is also\nimportant for the description of squark pair production at\nthreshold~\\cite{HoangRuizSvNRQCD}. \nRecently, it was also found that it is crucial to apply\nthe EFT for predictions of the $e^+e^-\\to t\\bar t H$ cross section at \n$\\sqrt{s}=500$~GeV~\\cite{Farrell1,Farrell2} since here the $t\\bar t$\nfinal state interactions are dominated by nonrelativistic dynamics.\n\nIn this work we address the form and construction of non-relativistic\ninterpolating currents for production and annihilation of a color\nsinglet heavy particle-antiparticle pair with general quantum\nnumbers ${}^{2S+1}L_J$ in $n=d-1=3-2\\epsilon$ dimensions for quarks\nand colored scalars within NRQCD. In general, the interpolating\ncurrents for a specific ${}^{2S+1}L_J$ state are associated to\nirreducible tensor representations of the SO(n) rotation group that\ncan be built from \nthe particle-antiparticle relative momentum and the bilinear\ncovariants of the particle-antiparticle field operators. \nWhile in three dimensions the basis of interpolating currents for the \ndifferent ${}^{2S+1}L_J$ states is, by construction, unique, in\ngeneral $n\\neq 3$ dimensions for fermions there exist evanescent\noperator structures that make the choice of basis for the\ninterpolating currents ambiguous. This is because the structure of\nirreducible SO(n) representations for $n\\neq 3$ is inherently\ndifferent and in general richer than in three dimensions. Technically\nthe ambiguity is related to the number of sigma \nmatrices that generate the SO(n) rotations for the spinor wave\nfunctions. For explicit computations a specific choice of basis has\nto be made, which is associated to a specific choice of a\nrenormalization scheme. A very similar problem has been treated\nsystematically already some time ago in the framework of relativistic\nQCD and the effective weak\nHamiltonian~\\cite{DuganGrin,HerrlichNierste,ChetyrkinMisiak}, and much\nof the statements that apply for the relativistic case can be directly\ntransferred to the nonrelativistic theory. However, due to the \nnonrelativistic power counting some even more specific statements,\ncan be made. In particular, \nthe NLL order matching conditions and anomalous dimensions of the\n${}^{2S+1}L_J$ interpolating currents that are leading in the\nnonrelativistic expansion are scheme independent.\n\nUsing the explicit form of the interpolating currents obtained in this\nwork we also determine the NLL anomalous dimensions of the leading (in\nthe $v$ expansion) currents describing color singlet heavy \nquark-antiquark and squark-antisquark pair production for arbitrary\nspin and angular momentum ${}^{2S+1}L_J$ configurations. The results\nfor low angular \nmomentum states (S,P,D) are e.g. relevant for angular distributions at\nthe threshold or for production and decay rates of squark-antisquark\nresonances in certain supersymmetric scenarios where squarks can\nhave a very long lifetime. The results also shed some light on the\nimportance of summing logarithms of \n$v$ for the production and annihilation rates of high angular momentum\nstates. In this respect our results help to complete analogous higher\norder considerations in previous literature for the energy\nlevels~\\cite{hms1} and the\nwave-functions~\\cite{Pineda2,Melnikov:1998ug}.\n\nFor the presentations in this work we employ the\nnotations from vNRQCD based on a label formalism for soft fields and\nthe heavy quarks (or scalars)~\\cite{LMR}. We note, however, that \nmost of the results obtained in this work are applicable to NRQCD in\ngeneral. In particular, the NLL order anomalous dimensions obtained\nhere are also valid in pNRQCD once the vNRQCD scale correlations from\nthe one-step matching are imposed. \n\nThe outline of the paper is as follows: In Sec.~\\ref{sectionsinglet}\nwe discuss the form of the spherical harmonics in $n$ dimensions and\npresent the form of currents describing states with arbitrary angular\nmomentum $L$ and total spin zero. We also present the $n$-dimensional\nform of the Legendre polynomials and demonstrate in an example why the\n$n$-dimensional form of the spherical harmonics is needed in\ndimensional regularization. \nIn Sec.~\\ref{sectiontriplet} we discuss the properties of\n$\\sigma$-matrices for $n\\neq 3$ and the importance of evanescent\noperator structures that can be built from the $\\sigma$-matrices.\nWe present a simple basis of interpolating currents describing\nfermion-antifermion pairs in a spin triplet state and having arbitrary\n$L$ and total angular momentum $J$. \nIn Sec.~\\ref{sectionNLLsinglet} we determine the anomalous dimension\nof the current for spin singlets, and in \nSec.~\\ref{sectionNLLtriplet} those of the current for spin triplets.\nSection~\\ref{sectionNLLtriplet} also contains a discussion on the\nscheme-dependence of the spin-dependent potentials.\nIn Sec.~\\ref{sectiondiscussion} we determine and solve the resulting\nanomalous dimensions for heavy quarks and colored scalars. The results\nare analysed numerically for a few cases.\nIn Sec.~\\ref{sectioncomments} we comment on the scheme-dependence of\nresults that can be found in previous literature. \nThe conclusions are given in Sec.~\\ref{sectionconclusion}.\nWe have added a few appendices: Appendices~\\ref{appendix2} and\n\\ref{appendix3} contain the derivation of the tree level NRQCD\nmatching conditions for the well known processes $q\\bar q\\to 2\\gamma$\nand $3\\gamma$ accounting properly for the existence of evenescent \noperator structures. In App.~\\ref{appendix4} some\nmore details are given on the determination of the spin triplet\ncurrents shown in Sec.~\\ref{sectionNLLtriplet}. We also derive an\nalternative set of currents that is equivalent for $n=3$ but \ninequivalent for $n\\neq3$. Finally in App.~\\ref{appendix5} we give \nresults for the UV-divergences for the elementary integrals\nthat are needed for the determination of the anomalous dimensions\nof the currents.\n\n\n\\section{Spin Singlet Currents} \n\\label{sectionsinglet}\n\nIn this section we discuss the form of the interpolating currents\ndescribing the production of a particle-antiparticle pair with total\nspin zero for arbitrary relative angular momentum $L$\n(${}^{2S+1}L_J={}^1L_L$). They are relevant for pairs of scalars or for\nquark-antiquark pairs in a spin singlet state. The currents naturally\nhave their simplest form in the c.m.\\,frame where relevant angular\ndependence can only arise from the c.m.\\,spatial momenta of the\npair. Thus the generic structure of the production currents is\n\\begin{eqnarray}\n \\psi_{\\mathbf p}^\\dagger(x)\\,\\Gamma(\\mathbf p)\\,\\tilde \\chi_{-\\mathbf p}^*(x)\n\\,,\n\\end{eqnarray}\nwhere $\\Gamma(\\mathbf p)$ represents an arbitrary tensor depending on the\nc.m.\\,momentum label $\\mathbf p$. For the case of scalars $\\psi_{\\mathbf p}$ and\n$\\tilde\\chi_{\\mathbf p}$ are scalar fields with\n$\\psi_{\\mathbf p}^\\dagger=\\psi_{\\mathbf p}^*$, while for fermions $\\psi_{\\mathbf p}$ and\n$\\chi_{\\mathbf p}$ are Pauli spinor fields with $\\tilde\\chi_{-\\mathbf p}^*\\equiv\n(i\\sigma_2)\\chi_{-\\mathbf p}^*$. (Note that we adopt the \nstandard nonrelativistic spinor convention of antiparticles with {\\it\npositive} energies. In this convention fermions and antifermions have\nthe same spin operators.) The corresponding\nannihilation current can be obtained by hermitian conjugation. \nThe interpolating currents associated to a definite angular momentum\nstate $L$ are related to irreducible representations of the tensor\n$\\Gamma$ with respect to the rotation group SO(n). Since the transformation\nproperties of the fields are not relevant in this respect for the spin\nzero case, it is sufficient to identify the irreducible tensors that\ncan be built from the spatial momentum vector $p^i$, $i=1,\\ldots,n$,\nwhere $n=d-1$. The irreducible tensors associated to the\nangular momentum $L$ are up to normalization just the spherical\nharmonics of degree $L$. In the following we give a general discussion\nin $n$ dimensions. All results reduce to the well known results for $n=3$.\n\n\\subsection*{Spherical Harmonics}\n\nThe spherical harmonics of degree $L$ are the\npolynomials $u(\\mathbf x)$ in $\\mathbb{R}^n$ which are homogeneous of degree $L$\n(i.e. $u(r \\mathbf x)=r^L u(\\mathbf x)$), harmonic (i.e. they satisfy the Laplace equation \n$\\Delta_{\\mathbb{R}^n}u(\\mathbf x)\\equiv\\nabla^2 u(\\mathbf x)=0$) and restricted to the unit\nsphere $S^{n-1}$. The \nspherical harmonics ${Y}_{LM}(n,\\bar\\mathbf x)$ of degree $L$, with\n$M=1,\\dots,n_L$ and $\\bar\\mathbf x=\\mathbf x\/|\\mathbf x|$, form an orthogonal basis of a\n$n_L$-dimensional vector space with\n\\begin{equation}\nn_L\\,=\\,\n\\left(\\begin{array}{c}n+L-1\\\\ L\\end{array}\\right)\n-\\left(\\begin{array}{c}n+L-3\\\\ L-2\\end{array}\\right)\n\\, = \\,\n(2L+n-2)\\,\\frac{\\Gamma(n+L-2)}{\\Gamma(n-1)\\Gamma(L+1)}\\,.\n\\label{nL}\n\\end{equation}\nAny differentiable function on the unit sphere $S^{n-1}$ in\n$\\mathbb{R}^n$ can be expanded in terms of a convergent series of\nspherical harmonics. \nA representation in terms of cartesian coordinates~\\footnote{\nAlthough it is possible to write down results with spherical\ncoordinates for integer $n$ (see e.g. Ref.~\\cite{Normand:1981dz}), it\nis more convenient in practice and for actual \ncomputations to use cartesian coordinates. \n} \nof the spherical\nharmonics of degree $L$ is given by the totally symmetric and traceless\ntensors with $L$ indices \n$T^{i_1\\dots i_L}(\\mathbf x)$, where the indices $i_1,\\ldots,i_L$ are cartesian\ncoordinates~\\cite{grouptheory}. The restriction to the unit sphere is not\nessential regarding the transformation properties under SO(n) rotations and\ncan be dropped for our purposes. An explicit expression for \n$T^{i_1\\dots i_L}(x)$ reads: \n\\begin{eqnarray}\nT^{i_1\\dots i_L}(\\mathbf x) \n &=& x^{i_1}\\ldots x^{i_L}\n -\\frac{\\mathbf x^2 \\, \\Phi_1^{i_1\\dots i_L}(\\mathbf x)}{(2L+n-4)}\n +\\frac{(\\mathbf x^2)^2 \\, \\Phi_2^{i_1\\dots i_L}(\\mathbf x)}{(2L+n-4)(2L+n-6)}\n - \\dots\n\\nonumber\\\\ & & \n\\dots + (-1)^{[L\/2]}\n \\frac{(\\mathbf x^2)^{[L\/2]} \\, \\Phi_{[L\/2]}^{i_1\\dots i_L}(\\mathbf x)}\n {(2L+n-4)\\dots(2L+n-2-2[L\/2])}\n\\nonumber\\\\[3mm] \n &=& \n \\sum_{k=0}^{[L\/2]}\\, C_k^L \\, \\mathbf x^{2k}\\, \\Phi_k^{i_1\\dots i_L}(x) \\;,\n\\nonumber\\\\[3mm] \nC_k^L &=& (-1)^k \\, 2^{-k} \\frac{\\Gamma(\\frac{n}{2}+L-k-1)}{\\Gamma(\\frac{n}{2}+L-1)} \\;,\n\\nonumber\\\\[3mm] \n\\Phi_k^{i_1\\dots i_L}(\\mathbf x) & = & \n \\sum_{\\mathrm{inequ. \\,\\,permut.}\\atop(p_1\\dots p_L)\\,\\mathrm{of}\\, (1\\dots L) }\n \\underbrace{ \\,\\delta^{i_{p_1}i_{p_2}}\\dots\n \\delta^{i_{p_{(2k-1)}}i_{p_{2k}}} }_{k \\;\\delta'\\mathrm{s}} \n \\, x^{i_{p_{(2k+1)}}}\\dots x^{i_{p_L}} \n\\nonumber\\\\ & = &\n \\delta^{i_1i_2}\\dots\\delta^{i_{2k-1}i_{2k}}\\,x^{i_{2k+1}}\\dots x^{i_L}\n \\, + \\, \\mbox{(inequ. permut.'s)}\n\\,,\n\\label{Tdef}\n\\end{eqnarray}\nwhere the symbol $[\\ldots]$ denotes the Gauss bracket. \nThe sum in the $\\Phi_k$ tensors extends over all sets of $k$ pairs of\nindices that can be constructed \nfrom $L$ indices ($L\\ge 2k$); the number of terms in the sum is thus \n${L\\choose 2k}(2k-1)!!$. Note that we call two\npermutations equivalent if they lead to the same term in the sum.\nAlthough we believe that the expression for $T^{i_1\\dots i_L}(x)$ has been shown somewhere in the\nliterature before, we were not able to locate a suitable reference. \nA few simple examples are\n\\begin{eqnarray} &&\nT^i(\\mathbf x) \\, = \\, x^i\\,,\\qquad\nT^{ij}(\\mathbf x) \\, = \\, x^ix^j-\\frac{\\mathbf x^2}{n}\\delta^{ij}\\,,\n\\nonumber\\\\ & &\nT^{ijk}(\\mathbf x) \\, = \\, x^ix^jx^k-\\frac{\\mathbf x^2}{n+2}\n \\left(x^i\\delta^{jk}+x^j\\delta^{ik}+x^k\\delta^{ij}\\right)\\,.\n\\end{eqnarray}\nFrom the Laplacian \n\\begin{eqnarray}\n\\Delta_{\\mathbb{R}^n} & \\equiv & \n\\frac{\\partial^2}{\\partial x^{i\\,2}} \\, = \\,\n\\left(\\frac{\\partial^2}{\\partial r^2} +\n \\frac{n-1}{r}\\frac{\\partial}{\\partial r}\\right) -\n\\frac{1}{r^2}\\,\\mathbf L^2\n\\,,\n\\nonumber\\\\[2mm]\n-\\mathbf L^2 & = &\n\\frac{\\partial}{\\partial x^j}x^k\\frac{\\partial}{\\partial x^j} x^k-\n\\frac{\\partial}{\\partial x^j} x^k \\frac{\\partial}{\\partial x^k} x^j\n\\, = \\,\n\\mathbf x^2\\nabla^2 - \nx^k x^j \\frac{\\partial}{\\partial x^k}\\frac{\\partial}{\\partial x^j} -\n(n-1) x^k \\frac{\\partial}{\\partial x^k}\n\\,,\n\\end{eqnarray}\none can find an explicit form for the squared angular momentum\noperator and, using the homogeneity relation \n$\\Delta_{\\mathbb{R}^n}T^{i_1\\dots i_L}=0$, the eigenvalue equation\n\\begin{eqnarray}\n\\mathbf L^2\\,\\, T^{i_1\\dots i_L}(\\mathbf x) & = & L(L+n-2)\\,\\, T^{i_1\\dots i_L}(\\mathbf x)\n\\,.\n\\end{eqnarray}\nWe define the interpolating currents that describe production of a\nspin singlet and angular momentum $L$ state (${}^{2S+1}L_J={}^1L_L$)\nas\n\\begin{eqnarray}\n(j_L^{S=0})^{i_1\\ldots i_L} & \\equiv & \n\\psi_{\\mathbf p}^\\dagger(x)\\, T^{i_1\\dots i_L}(\\mathbf p)\\,\\tilde\n\\chi_{-\\mathbf p}^*(x)\n\\,.\n\\label{singletcurrent1}\n\\end{eqnarray}\nThe current in Eq.~(\\ref{singletcurrent1}) is the dominant ${}^1L_L$\ncurrent in the $v$-expansion. Higher order currents are obtained by\nincluding additional powers of the scalar term $\\mathbf p^2$. \n\nAs an example, let us consider the currents with angular momenta $S,\\,P$ and $D$.\nThey are relevant in the electromagnetic production of\nheavy charged scalars from $e^+e^-$ and $\\gamma\\gamma$ collisions. \nThe $e^+e^-$ annihilation at lowest \norder in the expansion of the electromagnetic coupling produces $P$-wave states:\n\\begin{eqnarray}\ni{\\cal M}_{e^+e^-} =\n-i\\,\\frac{8\\pi\\alpha}{s}\\,Q_q \\,\\bar{v}(k^\\prime)\\,\\gamma^i\\, u(k)\n\\,T^i(\\mathbf p)\n\\,,\n\\label{pwave}\n\\end{eqnarray}\nwith $k,\\,k^\\prime$ the momenta of the incoming $e^+e^-$ in the c.m.\\,frame \nand $Q_q$ denoting the electromagnetic\ncharge of the scalars. Note that potential color indices are always implied.\nPairs of heavy scalars in $S$- and $D$-waves are first\nproduced in $\\gamma\\gamma$ collisions. The first terms of the amplitude in the\nexpansion in the c.m.\\,three-momentum of the outgoing particles read:\n\\begin{eqnarray}\ni{\\cal M}_{\\gamma\\gamma} &=&\n32 i\\,\\pi\\alpha\\,Q_q^2 \\,\\epsilon_1^i\\epsilon_2^j\\,\n\\left[\\,-\\frac{1}{4}\\,\\delta^{ij}+\\frac{p^i p^j}{2m^2} \n+ \\dots \\,\\right]\n\\nonumber\\\\[3mm]\n&=& -8i\\,\\pi\\alpha\\,Q_q^2 \\,\\epsilon_1^i\\epsilon_2^j\\left(\n1-\\frac{2\\,\\mathbf p^2}{n \\,m^2}\n\\right) +\n\\frac{16i\\,\\pi\\alpha}{m^2}\\,Q_q^2 \\,\n\\epsilon_1^i\\epsilon_2^j \\, T^{ij}(\\mathbf p)\n+ \\dots\\,.\n\\label{s-d-waves}\n\\end{eqnarray}\nThe first term in the last equality is the $S$-wave\ncontribution ($T=1$) whereas the second is the $D$-wave contribution described\nby the spherical harmonic of degree 2, $T^{i_1i_2}(\\mathbf p)$.\n\nSome useful relations for the $T$ tensors read:\n\\begin{eqnarray}\nT^{\\,i_1\\ldots i_L}(\\mathbf x)\\, x^{i_L}\n &=& \\frac{L+n-3}{2L+n-4}\\,\\mathbf x^2\\,T^{\\,i_1\\ldots i_{L-1}}(\\mathbf x) \\;,\n \\nonumber\\\\[3mm] \nT^{\\,i_1\\ldots i_L}(\\mathbf x)\\, x^{i_{L+1}} \n &=& T^{\\,i_1\\ldots i_{L+1}}(\\mathbf x) \n - \\frac{2 \\,\\mathbf x^2}{(2L+n-2)(2L+n-4)}\\, \\sum_{j< k \\atop j=1}^L \\, \\delta^{i_j i_k} \\, \n T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots i_{L+1} } (\\mathbf x)\n \\nonumber\\\\[1mm] \n && +\\;\\frac{\\mathbf x^2}{2L+n-2} \\, \\sum_{j=1}^L \\, \\delta^{i_j i_{L+1}} \\, \n T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_{L} } (\\mathbf x) \\;,\n \\nonumber\\\\[3mm] \n\\frac{\\partial}{\\partial x^\\ell}\\,\\Phi_k^{i_1\\dots i_L}(\\mathbf x) &=&\n \\sum_{j=1}^L \\, \\delta^{i_j\\ell}\\,\\Phi_{k}^{i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_L}(\\mathbf x)\n\\, = \\,\n \\Phi_{k+1}^{i_1\\dots i_L \\ell}(\\mathbf x) - x^\\ell\\,\\Phi_{k+1}^{i_1\\dots i_L}(\\mathbf x) \\;,\n \\nonumber\\\\[3mm]\nx^\\ell\\,\\Phi_{k}^{i_1\\dots i_L}(\\mathbf x) &= &\n \\Phi_{k}^{i_1\\dots i_L \\ell}(\\mathbf x) - \n \\sum_{j=1}^L\\,\\delta^{i_j\\ell}\\,\\Phi_{k-1}^{i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_L}(\\mathbf x) \\;,\n\\nonumber\\\\[3mm]\nx^{[\\ell}\\,\\partial^{\\,k]}_x\\,T^{\\,i_1\\ldots i_L}(\\mathbf x) &=& \n \\frac{1}{2}\\sum_{j=1}^{L} \\, \\delta^{i_j k} \\,\n T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_{L} \\ell} (\\mathbf x) - \\{ \\ell \\leftrightarrow k \\}\\;, \n\\label{Trelations}\n\\end{eqnarray}\nwhere we use the notation that hatted indices are dropped, and \n$A^{[ij]}\\equiv\\frac{1}{2}(A^{ij}-A^{ji})$.\n\n\\subsection*{Legendre Polynomials}\n\nThe spherical harmonics of degree $L$ satisfy the addition theorem\n\\begin{equation}\n\\sum_{M=1}^{n_L} {Y}_{LM}(n,\\bar \\mathbf x) \\, {Y}_{LM}^*(n,\\bar\n\\mathbf y)=\\frac{n_L}{\\sigma_n}\\,P_L(n,\\bar \\mathbf x.\\bar\\mathbf y) \n\\,,\n\\label{addtheorem1}\n\\end{equation}\nwhere $P_L(n,t)$, $t\\in[-1,1]$, is the Legendre polynomial\nof degree $L$ generalized to $n$ dimensions, with $P_L(n,1)=1$ and\n$\\sigma_n=\\frac{2\\pi^{n\/2} }{\\Gamma(n\/2)}$ is the area of the unit sphere $S^{n-1}$. The\nLegendre polynomial is related to the Gegenbauer polynomial\n$C_L^{\\lambda}(t)$.\\footnote{\n The index $\\lambda$ is\n chosen such that the $C_L^{\\lambda}$ are orthogonal on the\n $n$-dimensional sphere\n} \nOne has\n$P_L(n,t)=\\frac{\\Gamma(n-2)\\Gamma(L+1)}{\\Gamma(L+n-2)}C_L^{\\lambda}(t)$,\nwhere $\\lambda=\\frac{n-2}{2}$.\nThe explicit form of the generalized Legendre polynomial\nreads~\\cite{Gradstheyn}, \n\\begin{equation}\nP_L(n,t)=\\frac{\\Gamma(n-2)\\,\\Gamma(L+1)}{\\Gamma(\\frac{n-2}{2})\\, \n\\Gamma(n+L-2)}\\,\\sum_{k=0}^{\\left[\\frac{L}{2}\\right]}\n\\,\\frac{(-1)^k\\,\\Gamma(\\frac{n-2}{2}+L-k)}{\\Gamma(k+1)\\,\\Gamma(L-2k+1)}\n\\;(2t)^{L-2k} \n\\;.\n\\label{Legendredef}\n\\end{equation}\nFrom Eq.~(\\ref{addtheorem1}) follows an addition theorem for the $T$ tensors\nupon total contraction of their indices:\n\\begin{equation}\nT_{L,n}(\\mathbf x,\\mathbf y)\n\\, \\equiv \\,\nT^{\\,i_1\\dots i_{L}}(\\mathbf x)\\, T^{\\,i_1\\dots i_{L}}(\\mathbf y) \n\\, = \\, \n|\\mathbf x|^L\\,|\\mathbf y|^L\\,N_L \\, P_L(n,t) \\;,\n\\qquad t=\\frac{\\mathbf x.\\mathbf y}{|\\mathbf x||\\mathbf y|}\\;,\n\\label{Tcontract}\n\\end{equation}\nwhere\n\\begin{equation}\nN_L \\, \\equiv \\,\n\\frac{\\Gamma(L+n-2)\\,\\Gamma(\\frac{n-2}{2})}{2^{L}\\,\\Gamma(L+\\frac{n}{2}-1)\\,\\Gamma(n-2)}\\,.\n\\end{equation}\nAnother very useful contraction is:\n\\begin{eqnarray}\n\\lefteqn{\nT^{\\,i_1\\dots i_{L-1}j}(\\mathbf x)\\,T^{\\,i_1\\dots\n i_{L-1}\\ell}(\\mathbf y) \\,\\left(y^j x^{\\ell}-x^j y^\\ell \\right) \n\\,=\\,\n}\n\\nonumber\\\\ & &\n|\\mathbf x|^{L+1}\\,|\\mathbf y|^{L+1}\\,\nN_L \\,\n\\frac{L+n-2}{2L+n-2}\\,\\left[ P_{L+1}(n,t) -P_{L-1}(n,t) \\right]\n\\;.\n\\label{TyTx}\n\\end{eqnarray}\n\n\n\\subsection*{Consistency Consideration}\n\nThe use of the generalized currents in\nEqs.~(\\ref{Tdef}) and~(\\ref{singletcurrent1}) is mandatory to obtain\nconsistent results in dimensional regularization in accordance with\nSO(n) rotational invariance. As an example let us consider the\nnonrelativistic three-loop vacuum polarization diagram shown in\nFig.~\\ref{fig:3loopCoul} with two insertions of the Coulomb potential.\nInserting the currents that produce and annihilate the\nparticle-antiparticle pair with angular momentum $L$ and fully\ncontracting the indices, the quantity we want to compute is, up to a\nglobal factor \n($D^n\\mathbf p\\equiv \\tilde{\\mu}_s^{2\\epsilon} d^n\\mathbf p\/(2\\pi)^{n},\\;\n\\tilde{\\mu}_s^{2\\epsilon}=\\mu_s^{2\\epsilon} (4\\pi)^{-\\epsilon} e^{\\epsilon\\gamma}$):\n\\begin{figure}[t] \n\\begin{center}\n \\leavevmode\n \\epsfxsize=6cm\n \\leavevmode\n \\epsffile{figs\/3loopCoul.eps}\n \\vskip 0.0cm\n \\caption{Three-loop diagram with two insertions of the Coulomb\n potential (black dots). \n \\label{fig:3loopCoul} }\n\\end{center}\n\\end{figure}\n\\begin{eqnarray}\n\\frac{1}{N_L}\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n\\frac{T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)\\, T^{\\,i_1\\dots i_{L}}(\\mathbf q_3) }\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,, \n\\end{eqnarray}\nwith $\\delta=-mE-i\\epsilon$. We can now proceed in two different ways\nto do the computation. From Eq.~(\\ref{Tcontract}), the contraction of\nthe $T$'s at the ends generates a Legendre polynomial \ndepending on the angle between the loop momenta on the sides:\n\\begin{eqnarray}\n{\\cal A}_1^{(L)} = \\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\\frac{|\\mathbf q_1|^L |\\mathbf q_3|^L \\,\nP_L(n,\\bar \\mathbf q_1 . \\bar \\mathbf q_3) }\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,.\n\\label{A1}\n\\end{eqnarray}\nOn the other hand, we can first shift the loop momenta dependence of\nthe current $T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)$ by using the \nrelation\n\\begin{eqnarray}\n &&\\int d^n \\mathbf q_1 \\, T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)\\,f( \\mathbf q_1, \\mathbf q_2)= A\\,\n T^{\\,i_1\\dots i_{L}}(\\mathbf q_2)\\,,\n\\nonumber\\\\[2mm]\n&&\nA= \\int d^n \\mathbf q_1 \\,|\\mathbf q_1|^L |\\mathbf q_2|^{-L} \\,\nP_L(n,\\bar \\mathbf q_1 . \\bar \\mathbf q_2) \\,f( \\mathbf q_1, \\mathbf q_2) \\,,\n\\label{shift}\n\\end{eqnarray}\nwhich is a consequence of rotational invariance if \n$f( \\mathbf q_1, \\mathbf q_2)$ is a scalar function depending on\n$\\mathbf q_1^2,\\,\\mathbf q_2^2,\\,\\mathbf q_1\\cdot\\mathbf q_2$. \nThis gives an alternative expression\nwith two Legendre polynomials:\n\\begin{eqnarray}\n{\\cal A}_2 ^{(L)} = \n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n\\frac{|\\mathbf q_1|^L |\\mathbf q_3|^L \\,\nP_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_2)\\,P_L(n,\\bar\\mathbf q_2.\\bar\\mathbf q_3)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,. \n\\label{A2}\n\\end{eqnarray}\nHad we worked in $n=3$ from the beginning, we would have written down\nEqs.~(\\ref{A1}) and~(\\ref{A2}) with \nthe generalized Legendre polynomials replaced by their $n=3$\nexpressions. Let us call the corresponding \nexpressions as $\\tilde{{\\cal A}}_1^{(L)},\\;\\tilde{{\\cal A}}_2^{(L)}$. \nSince the integrals are \npower divergent, we can compute them using dimensional regularization in $n=3-2\\epsilon$\ndimensions and check whether they give the same result. \nThe first non-trivial case is $L=2$, because one has $P_{0,1}(3,x)=P_{0,1}(n,x)$. \nThe results for $\\tilde{{\\cal A}}_1^{(2)},\\;\\tilde{{\\cal A}}_2^{(2)}$ then read\n\\begin{eqnarray}\n\\tilde{{\\cal A}}_1^{(2)} &=& \\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{15+4\\pi^2+\\epsilon\\left[32\\pi^2-120\\,\\zeta(3)\n +231\\right]+{\\cal O}(\\epsilon^2) \\Big\\}\\,,\n\\nonumber\\\\[3mm]\n\\tilde{{\\cal A}}_2^{(2)} &=& \\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{\n18+4\\pi^2+\\epsilon\\left[ 32\\pi^2-120\\,\\zeta(3)\n +264-4\\log 8 \\right]+{\\cal O}(\\epsilon^2)\\Big\\}\\,.\n\\nonumber\\\\\n\\end{eqnarray}\nA difference arises even in the first term in the $\\epsilon$\nexpansion, which reflects the fact that \n$\\tilde{{\\cal A}}_1^{(2)},\\;\\tilde{{\\cal A}}_2^{(2)}$ are not the same\nquantity. The reason is that the shift (\\ref{shift}) performed in\n$n=3$ dimensions and the following evaluation using dimensional\nregularization do not commute if the integrals are divergent. \nRotational invariance in $n$ dimensions\nmust be satisfied at every step if dimensional regularization is\nused. This is automatically achieved by using the SO(n) tensors $T$ as\nshown in Eqs.~(\\ref{A1}) and~(\\ref{A2}). The computation \nwith generalized Legendre polynomials corresponding to ${\\cal\n A}_1^{(L)}$ and ${\\cal A}_2^{(L)}$ in Eqs.~(\\ref{A1},\\ref{A2}) can\nbe shown to produce the correct outcome. For $L=2$ the correct result\nreads \n\\begin{eqnarray}\n{\\cal A}_1^{(2)} = {\\cal A}_2^{(2)} =\\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{\n15+4\\pi^2+\\epsilon\\left[ 32\\pi^2-120\\,\\zeta(3)\n +236\\right]+{\\cal O}(\\epsilon^2) \\Big\\}\\,,\n \\nonumber\n\\end{eqnarray}\nwhich disagrees with $\\tilde{{\\cal A}}_1^{(2)}$ in terms proportional\nto $\\epsilon^m$ for $m>0$, and with $\\tilde{{\\cal A}}_2^{(2)}$ for all\nterms in the $\\epsilon$ expansion. \n\n\\section{Spin Triplet Currents} \n\\label{sectiontriplet}\n\nIn this section we discuss the form of the interpolating currents describing\nthe production of a fermion-antifermion pair in a spin triplet $S=1$ state for\narbitrary relative angular momentum $L$ (${}^{2S+1}L_J={}^3L_J$). As in\nSec.~\\ref{sectionsinglet} the currents are defined in the c.m.\\,frame.\n\n\\subsection*{Pauli Matrices}\n\nSince the treatment of Pauli $\\sigma$-matrices in $n$ dimensions involves a\nnumber of subtleties, we briefly review some of their properties relevant for\nthe formulation of the currents. Many of the properties can be directly\nobtained from the corresponding properties of the $\\gamma$-matrices in $d=n+1$\ndimensions. The \n$\\sigma$-matrices $\\sigma^i$ ($i=1,\\ldots,n$) are the generators of SO(n)\nrotations for spin $1\/2$. They are traceless, hermitian, satisfy the Euclidean\nClifford algebra $\\{\\sigma^i,\\sigma^j\\}=2\\delta^{ij}$, and can be defined\nsuch that $\\mathbf \\mbox{\\boldmath $\\sigma$}^T=\\mathbf \\mbox{\\boldmath $\\sigma$}^*=-\\sigma_2\\mathbf \\mbox{\\boldmath $\\sigma$}\\sigma_2$. While traces of\na product of an even number of $\\sigma$-matrices in $n$ dimensions can be\nexpressed as a \nmultiple of $\\mbox{Tr}[\\mathbf 1]$ using the anticommutator, traces of a product\nof an odd number are identically zero and for the case of three\n$\\sigma$-matrices can require additional\nrules to yield results that are consistent with the relations known from $n=3$. \nIn this respect the product of three different $\\sigma$-matrices can be\nsomewhat considered the three-dimensional\nanalog of $\\gamma_5$ in four dimensions.\\footnote{ \nThe vanishing of traces of a product of an odd number of $\\sigma$-matrices\nin $n$ dimensions\nis related to the simultaneous use of the cyclicity property of the trace\noperators and the Euclidean Clifford algebra relations for all \n$\\sigma$-matrices. Products of an odd number of $\\sigma$-matrices,\nhowever, do not arise from insertions of\npotentials which we consider in this work. Such traces can, however, arise\ne.g.\\,when the spin-dependent radiation of ultrasoft gluons or annihilation\npotentials need to be considered. Thus, in general, the cyclicity\nproperty of the trace operation should only be used after\nrenormalization is completed - if the Euclidean Clifford algebra is applied\nfor all $\\sigma$-matrices. \n}\nAs for the case of the $\\gamma$-matrices~\\cite{DuganGrin} products of\n$\\sigma$-matrices in arbitrary number of dimensions cannot be reduced to a\nfinite basis, but represent an infinite set of independent structures. In\nanalogy to Ref.\\,\\cite{DuganGrin} it is convenient to define \n\\begin{eqnarray}\n\\sigma^{i_1\\cdots i_m} & \\equiv &\n\\sigma^{[i_1}\\sigma^{i_2}\\cdots \\sigma^{i_m]}\n\\,,\\quad (m=1,2,\\ldots)\n\\end{eqnarray}\nfor the averaged antisymmetrized product of $\\sigma$-matrices, so e.g.\\,\n$\\sigma^{ij}=1\/2(\\sigma^i\\sigma^j-\\sigma^j\\sigma^i)$. It is straightforward to\nderive the following useful relations:\n\\begin{eqnarray}\n\\sigma^{i_1\\cdots i_m}\\sigma^\\ell & = &\n(-1)^m\\,\\Big(\\, \\sigma^{\\ell i_1\\cdots i_m} \\, + \\,\n\\sum_{j=1}^{m}(-1)^j\\,\\delta^{\\ell i_j}\\,\\sigma^{i_1\\cdots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\cdots i_m}\\Big)\n\\,,\n\\label{sigma1}\n\\\\\n\\sigma^\\ell\\sigma^{i_1\\cdots i_m} & = &\n\\sigma^{\\ell i_1\\cdots i_m} \\, + \\,\n\\sum_{j=1}^{m}(-1)^{j+1}\\,\\delta^{\\ell i_j}\\,\\sigma^{i_1\\cdots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\cdots i_m}\n\\,,\n\\label{sigma2}\n\\\\\n\\left[ [\\sigma^j,\\sigma^\\ell],\\sigma^{i_1\\ldots i_m} \\right]\n & = & 4(-1)^m\\, \\sum_{k=1}^{m}\\, (-1)^{k+1}\n \\left( \\delta^{j i_k}\\sigma^{i_1\\ldots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots i_m \\ell} \n - \\delta^{\\ell i_k}\\sigma^{i_1\\ldots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots i_m j} \\right)\n\\,,\n\\label{sigma3}\n\\\\\n\\frac{1}{2} \\left( \\sigma^{j}\\sigma^{i_1\\ldots i_m}\\sigma^{\\ell}\n+ \\sigma^{\\ell}\\sigma^{i_1\\ldots i_m}\\sigma^{j} \\right)\n& = &\n(-1)^m\\,\\delta^{j\\ell}\\,\\sigma^{i_1\\dots i_m} \n\\nonumber \\\\ \n&& + \\sum_{k=1}^{m}\\, (-1)^{k+1}\\left( \\delta^{j i_k}\\sigma^{i_1\\ldots\n \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots i_m \\ell} \n + \\delta^{\\ell i_k}\\sigma^{i_1\\dots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots i_m j} \\right)\n\\,,\n\\label{sigma4}\n\\\\\n\\sum_k\\,\\sigma^k\\sigma^{i_1\\dots i_m}\\sigma^k & = &\n(-1)^m\\,(n-2m)\\,\\sigma^{i_1\\dots i_m}\n\\,,\n\\label{sigma5}\n\\\\\n\\sum_{i_1\\dots i_k}\\,\\sigma^{i_1\\dots i_k}\\,\\sigma^{i_1\\dots i_k}\n& = &\n(-1)^{\\frac{k(k-1)}{2}}\\,n\\dots (n-k+1)\n\\, = \\,\n(-1)^{\\frac{k(k-1)}{2}}\\,\\frac{\\Gamma(n+1)}{\\Gamma(n-k+1)}\\,,\\quad\n\\nonumber\\\\\n\\label{sigma6}\n\\\\\n\\mbox{Tr}(\\sigma^{i_1}\\dots \\sigma^{i_{2m}}) &=& \\mbox{Tr}\\,[\\mathbf 1]\n \\!\\!\\! \\!\\!\\!\\sum_{\\mathrm{inequ. \\,\\,permut.}\\atop (p_1\\dots p_{2m})\\,\\mathrm{of}\\, (1\\dots 2m) }\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\delta^{i_{p_1}i_{p_2}}\\dots\\delta^{i_{p_{2m-1}}i_{p_{2m}}}\\,\\delta_P\\,.\n\\label{sigma7}\n\\end{eqnarray}\nIn the last relation, $\\delta_P$ is the signature (sign) of the respective\npermutation \n${1\\;\\,2 \\,\\ldots\\, \\;2m \\choose p_1\\,p_2\\ldots \\,p_{2m}}$, and for \neach $\\delta^{i_m i_n}$ only $m3$. Thus the latter are evanescent for $n\\neq 3$. For $m\\leq 3$ the\n$\\sigma^{i_1\\cdots i_m}$ are related to physical spin operators.\n\n\n\\subsection*{S-Wave Currents}\n\nThe general interpolating current describing the production of a\nfermion-antifermion pair in an S-wave state and in an arbitrary spin state in\n$n$ dimensions has the form\n\\begin{eqnarray}\n(j_{L=0})^{[i_1\\cdots i_m]} & = &\n\\psi_{\\mathbf p}^\\dagger(x)\\, \\sigma^{i_1\\cdots i_m}\\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*(x)\n\\,.\n\\label{Swavecurrent1}\n\\end{eqnarray}\nA SO(n) rotation by an angle $\\theta$ around the axis $\\mathbf n$ leads to the\ntransformed current \n$D_\\theta(\\psi_{\\mathbf p}^\\dagger\\, \\sigma^{i_1\\cdots i_m}\\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*)=\\psi_{\\mathbf p}^\\dagger S(\\theta)^\\dagger \\sigma^{i_1\\cdots i_m}\\,\nS(\\theta)(i\\sigma_2)\\chi_{-\\mathbf p}^*$, where \n$S(\\theta)=\\exp(-i \\theta \\mathbf n.\\mathbf \\mbox{\\boldmath $\\sigma$}\/2)$. Since \n$S(\\theta)^\\dagger \\sigma^i S(\\theta)=D^{ij}(\\theta)\\sigma^j$, where \n$D^{ij}(\\theta)$ is the rotation matrix for $n$-vectors, the currents in\nEq.\\,(\\ref{Swavecurrent1}) are tensors with respect to SO(n) rotations. The\ntensors are irreducible due to the antisymmetry of the \n$\\sigma^{i_1\\cdots i_m}$~\\cite{grouptheory} and each have \n$(^n_m)$ independent free components. Their eigenvalues with respect to the\nsquare of the total spin operator \n$\\mathbf S^2=[\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\otimes\\mathbf 1_{ap}+\\mathbf 1_{p}\\otimes\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}]^2\/4=\n(n[\\mathbf 1_a\\otimes\\mathbf 1_{ap}]+[\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\otimes\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}])\/2$, where\nthe indices \nof the $\\sigma$-matrices are summed over, read\n\\begin{eqnarray}\n\\mathbf S^2\\,(\nj_{L=0})^{[i_1\\cdots i_m]}\n & = &\n\\frac{1}{2}\n\\psi_{\\mathbf p}^\\dagger\\,\\Big( \nn\\,\\sigma^{i_1\\cdots i_m}(i\\sigma_2) + \n\\mathbf \\mbox{\\boldmath $\\sigma$}\\sigma^{i_1\\cdots i_m}(i\\sigma_2)\\mathbf \\mbox{\\boldmath $\\sigma$}^T \\Big)\\,\n\\chi_{-\\mathbf p}^*\n\\nonumber\\\\\n& = &\n\\frac{1}{2}\n\\left(n+(-1)^m\\,(2m-n)\n\\right)\\,\n(j_{L=0})^{[i_1\\cdots i_m]}\n\\,.\\qquad\n\\label{S2op}\n\\end{eqnarray}\nFor the physical currents with $m=(0,1,2,3)$ one thus finds the spin\neigenvalues $(0,n-1,2,n-3)$. Note that the spin eigenvalue for the unphysical\nS-wave currents $\\sigma^{i_1\\cdots i_m}$ for $m>3$ are non-zero.\nWhile for $n=3$ the spin singlet currents for\n$m=0,3$ are equivalent, and likewise the triplet currents for $m=1,2$, each of\nthe currents represents a different irreducible representation of SO(n) for $n\\neq\n3$. It is a very instructive fact that the action of $\\mathbf S^2$ onto the\nsinglet current $j_{L=0}^{[i_1i_2i_3]}$ does not give zero for $n\\neq\n3$. Thus to achieve that spin-dependent potentials do not contribute for\nphysical predictions involving spin singlet currents, in general additional finite\nrenormalizations are required in analogy to Ref.~\\cite{DuganGrin}, unless a\nscheme for potentials or currents is chosen such that they vanish\nautomatically. However, even if such a scheme is adopted, matrix elements,\nmatching conditions and anomalous dimensions can depend on the spin-dependent\npotentials at nontrivial subleading order\\footnote{\nTo be more specific,\nwe refer to orders of perturbation theory where subdivergences need to be\nrenormalized.\n}.\n\nAll of the physical currents can arise in important processes. In\nTab.~\\ref{tab1} the leading order matching for a number of\ndifferent currents is displayed. In general there are several\nrelativistic currents that can lead to the same nonrelativistic\ncurrent~\\cite{Fadin:1991zw}. Note that the respective production\nand annihilation currents are related either by hermitian or\nantihermitian conjugation. Except for $\\gamma_5$, which is\ntreated as fully anticommuting ($\\gamma_5=(^{0\\,\\mathbf I}_{\\mathbf I\\,0})$),\nthree dimensional relations have not been used.\nThe full theory \ncurrent with the $\\gamma$-structure $\\gamma^{ijk}$ for example arises for\nfermion pair production in $\\gamma\\gamma$ collisions, as shown in \nAppendix~\\ref{appendix2}.\nHowever, also evanescent currents naturally occur in standard processes, such as\nthe current $j_{L=0}^{[i_1\\cdots i_5]}$ that arises in fermion-antifermion\npair annihilation into three photons. The explicit computation can be found in\nAppendix~\\ref{appendix3}. Also note that the differences between\nthe two different singlet and triplet currents correspond to evanescent\noperators as well. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||cc|}\\hline Full theory $\\Gamma$ & &\n$\\Gamma(\\mathbf p,\\bm{\\sigma})$ \\\\ \\hline \\hline \n $\\gamma^{i_1\\ldots i_{2k}}$ && $(-1)^{k+1} \\frac{p^\\ell}{m}\\,\\sigma^{\\ell\n i_1\\ldots i_{2k}}$ \\\\ \\hline \n $\\gamma^{i_1\\ldots i_{2k+1}}$ && $(-1)^{k} \\,\\sigma^{ i_1\\ldots i_{2k+1}}$\n \\\\ \\hline \n $\\gamma^0\\gamma^{i_1\\ldots i_{2k}}$ & $(\\star)$ & $(-1)^{k} \\sum_{j=1}^{2k}\n (-1)^{j+1}\\frac{p^{i_j}}{m} \\sigma^{ i_1\\ldots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\ldots i_{2k}}$ \\\\\n \\hline \n $\\gamma^0\\gamma^{i_1\\ldots i_{2k+1}}$ & $(\\star)$ & $(-1)^{k} \\,\\sigma^{ i_1\\ldots\n i_{2k+1}}$ \\\\ \\hline \n $\\gamma^{i_1\\ldots i_{2k}}\\gamma^5$ & $(\\star)$ & $(-1)^{k} \\,\\sigma^{ i_1\\ldots i_{2k}}$\n \\\\ \\hline \n$\\gamma^{i_1\\ldots i_{2k+1}}\\gamma^5$ & $(\\star)$ & $(-1)^{k}\\frac{p^{\\ell}}{m}\n \\sigma^{\\ell i_1\\dots i_{2k+1}}$ \\\\\n \\hline \n$\\gamma^0\\gamma^{i_1\\ldots i_{2k}}\\gamma^5$ && $(-1)^{k} \\,\\sigma^{ i_1\\ldots\n i_{2k}}$ \\\\ \\hline \n$\\gamma^0\\gamma^{i_1\\ldots i_{2k+1}}\\gamma^5$ && $(-1)^{k} \\sum_{j=1}^{2k+1}\n (-1)^{j}\\frac{p^{i_j}}{m} \\sigma^{ i_1\\ldots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\ldots i_{2k+1}}$ \\\\\n \\hline \n \\end{tabular} \n\\end{center}\n\\caption{Nonrelativistic production currents \n$\\psi_{\\mathbf p}^\\dagger\\,\\Gamma(\\mathbf p,\\bm{\\sigma}) \\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*$ arising from the leading order matching with the full\ntheory currents $\\bar{\\psi}\\,\\Gamma \\, \\psi$. The nonrelativistic\nnormalization of the full theory spinors has been used (see\nEq.~(\\ref{spinors})). The notation $\\gamma^{i_1\\ldots i_{m}}$ stands for the\naveraged antisymmetrized product of $\\gamma$-matrices for indices other than\nzero. For the corresponding annihilation currents \n$\\chi_{-\\mathbf p}^T(-i\\sigma_2)\\,\\Gamma^\\prime(\\mathbf p,\\bm{\\sigma})\\,\\psi_{\\mathbf p}$ \none has $\\Gamma^\\prime=-\\Gamma$ for all entries \nwith the $(\\star)$ symbol and $\\Gamma^\\prime=\\Gamma$ for the others.\n} \n\\label{tab1} \n\\end{table}\nIt is well known from subleading order computations based on the effective\nweak Hamiltonian that one needs to consistently account for the evanescent\noperator structures that arise in matrix elements of physical operators when\nbeing dressed with gluons. In Ref.\\,\\cite{DuganGrin} it was shown that a\nrenormalization scheme can be adopted such that a mixing of evanescent\noperators into physical ones does not arise. Moreover it is also\nknown~\\cite{HerrlichNierste,ChetyrkinMisiak} that modifications of the\nevanescent operator basis (e.g.\\,by adding physical operators multiplied by\nfunctions of $\\epsilon$ that vanish for $\\epsilon\\to 0$) and similar\nmodifications to the physical operator basis correspond to a change of the\nrenormalization scheme. While this does not affect physical predictions, it\ndoes affect matrix elements, matching conditions and anomalous dimensions at\nnontrivial subleading order.\nThus precise definitions of the schemes being employed have to be given to\nrender such intermediate results useful. \n\nIn the framework of the nonrelativistic EFT these properties still apply.\nHowever, the velocity power counting in the EFT allows for even\nmore specific statements. Concerning interactions through potentials,\ntransitions between S-wave currents in Eq.\\,(\\ref{Swavecurrent1}) that are\ninequivalent cannot occur because the potentials are \nSO(n) scalars and the currents are (due to the different symmetry\npatterns of their indices~\\cite{grouptheory}) inequivalent irreducible\nrepresentations of SO(n). Even for currents with $L\\neq 0$ and for the\nspin-dependent spin-orbit and tensor potentials (which is all we need\nto consider at NNLL order) one can show with\nEqs.\\,(\\ref{sigma3}) and (\\ref{sigma4}) that transitions between currents\ncontaining $\\sigma^{i_1\\cdots i_m}$ with a different number of indices\ncannot occur (see Sec.\\,\\ref{sectionNLLtriplet}.1). Transitions can,\nhowever, arise between currents with \ndifferent angular momentum, such as for the tensor potential that can\ngenerate transitions between $L$ and $L^\\prime=L\\pm 2$ (see\nSec.\\,\\ref{sectionNLLtriplet}). The same arguments apply to the\nexchange of soft gluons (in the framework of vNRQCD) since they cannot appear\nas external particles and furthermore need to be exchanged in pairs due to\nenergy conservation. In this respect the effects from soft gluon exchange\neffectively represent modifications to the potential interactions (see\ne.g. Ref.\\,\\cite{LMR}). Concerning the \nexchange of ultrasoft gluons, transitions between currents\ncontaining $\\sigma^{i_1\\cdots i_m}$ with a different number of indices\ncan arise, but only if the interaction is spin-dependent. The dominant\namong these interactions corresponds to the operator \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ij} k^j \\psi_{\\mathbf p} A^i$, where $A^i$ is\nan ultrasoft gluon. This operator can only\ncontribute at N${}^4$LL order, which is beyond the present need and\ntechnical capabilities. Thus in practice, transitions between\ncurrents containing $\\sigma^{i_1\\cdots i_m}$ with a different number\nof indices do not need to be considered. \n\nSo for the S-wave currents in $n$ dimensions one can employ either one \nof the two spin singlet ($k=0,3$) or triplet currents ($k=1,2$) in the\nEFT and the difference corresponds to a change in the renormalization\nscheme. This means in particular that as long as the renormalization\nprocess is restricted e.g.\\,to time-ordered products of the currents, one\ncan (but does not have to) freely use three-dimensional relations to reduce\nthe basis of the physical currents. However, once the basis of the\nphysical currents is fixed (where each current is irreducible with\nrespect to SO(n)), one has to consistently apply the computational\nrules in $n$ dimensions discussed in the previous sections. As an\nexample, instead of using the current \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ijk}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ that arises\nin $\\gamma\\gamma$ collision, one can employ the current \n$\\psi_{\\mathbf p}^\\dagger(i\\sigma_2)\\chi_{-\\mathbf p}^*$, defined in $n$ dimensions,\ntimes the $\\epsilon$-tensor $i\\epsilon^{ijk}$ defined in three dimensions,\nwhich means that the $\\epsilon$-tensor is zero if any of its indices takes a\nvalue different from $1,2,3$. Concerning the\n$\\epsilon$-tensor, this works because the $\\epsilon$-tensor does not\nplay any role during \nthe renormalization procedure as long as we only consider time-ordered\nproducts of the currents.\\footnote{This procedure cannot be applied in\nthis simple form, if the initial state that is involved in the\nquark-antiquark production process is also involved in the\nrenormalization procedure, as can be the case for QED corrections.} \nThis justifies the approach in Sec.\\,\\ref{sectionsinglet} where we\nhave only considered spin singlet currents for fermions involving the\ncurrent $\\psi_{\\mathbf p}^\\dagger(i\\sigma_2)\\chi_{-\\mathbf p}^*$. This scheme is\nalso advantageous practically because for the current\n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ijk}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ the\nmatrix elements of the spin-dependent potentials that vanish for $n=3$\ncan, as discussed above, give evanescent contributions for $n\\ne 3$. \nMoreover, from these considerations we can also conclude that currents\ncontaining evanescent $\\sigma^{i_1\\cdots i_m}$ matrices ($m>3$) can\n(but do not have to) be dropped from the very beginning in the EFT as\nlong as one does not need to account for spin-dependent ultrasoft gluon\ninteractions. \n\nAn important lesson to learn from this discussion is that partial results at\nnontrivial subleading order obtained from the threshold \nexpansion~\\cite{Beneke} of full theory diagrams such as e.g. the \ncontributions from the hard regions obtained in\nRefs.~\\cite{Czarnecki1,Beneke4,firstc0,2loop_hard}, \nare equal to the EFT matching conditions only in schemes for the effective \ntheory currents and potentials that are compatible with the nonrelativistic\nreduction of the $\\gamma$ matrices that has been used during the threshold \nexpansion. In general, there are additional contributions to EFT matching\nconditions to account for the scheme choices made in the EFT.\nNote that in the threshold expansion different scheme choices can also \nbe possible, in particular for the treatment of $\\gamma_5$.\nIn Sec.~\\ref{sectioncomments} we comment on scheme dependences\nof a number of partial results that can be found in the literature.\n\nAt this point one might also ask whether $\\gamma_5$ needs special\ntreatment in the nonrelativistic EFT in $n$ dimensions. Chirality\nand the flavor symmetries are not relevant in the EFT for a single heavy\nparticle-antiparticle system and their potential\neffects are contained in the matching contributions of the EFT to \nthe full theory. For the\nmatching relations shown in Tab.~\\ref{tab1} we have used a totally\nanticommuting $\\gamma_5$. Since the resulting effective theory\ncurrents have well defined SO(n) transformation properties and reduce\nto the proper nonrelativistic currents for $n\\to 3$, this represents a\nconsistent scheme choice from the point of view of the effective theory\ncomputations. A different ansatz for $\\gamma_5$ such as\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ (which is the fully consistent\none in the full theory) just corresponds to a different choice of scheme for \nthe currents in the nonrelativistic EFT, and both schemes can be used\nconsistently. Note that from the results in Tab.~\\ref{tab1} without an\nexplicit $\\gamma_5$ one can also \nderive the form of the EFT currents for the ansatz\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ (see also\nSec.~\\ref{sectioncomments}). \n\n \n\\subsection*{Arbitrary Angular Momentum}\n\nBased on the results obtained in the previous sections it is\nstraightforward to construct spin-triplet currents with arbitrary\nangular momentum $L$ (${}^{2S+1}L_J={}^3L_J$)). They can be obtained\nby determining irreducible SO(n) representations from products of the\ntensors $T^{i_1\\dots i_L}(\\mathbf p)$ describing angular momentum $L$ and\nthe spin-triplet $S=1$ currents discussed in the previous\nparagraphs. Due to the different symmetry patterns of the symmetric\n$T^{i_1\\dots i_L}(\\mathbf p)$ tensors and the antisymmetric \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{i_1\\cdots i_k}(i\\sigma_2)\\chi_{-\\mathbf p}^*$\ncurrents one needs to apply the construction principles for general\ntensors~\\cite{grouptheory} which state that tensors are irreducible\nwith respect to SO(n) if and only if they are traceless and if their\nindices have a symmetry pattern according to a standard Young tableau. \nAs for the case of the S-wave currents the physical basis for\narbitrary spatial angular momentum is not unique due to the existence\nof evanescent operator structures. \n\nHere we construct currents with fully symmetric indices because of\ntheir comparatively simple form and because their number of indices is\nequal to the total angular momentum $J$ quantum number. For the\ncase $J=L\\pm 1$, these currents are contained in the reduction of the \nreducible tensor $A^{i_1\\cdots i_{L+1}} = \\psi_{\\mathbf p}^\\dagger \n\\,T^{i_1\\dots i_L}(\\mathbf p)\\sigma^{i_{L+1}}(i\\sigma_2)\\chi_{-\\mathbf p}^*$. Upon\nsymmetrization and removal of all traces one obtains the \n${}^{2S+1}L_J={}^3L_{L+1}$ current\n\\begin{eqnarray}\n \\left(j_{J=L+1}^{S=1}\\right)^{i_1\\dots i_{L+1}} \n & \\equiv & \n\\psi_{\\mathbf p}^\\dagger\n\\,\\Big[\\Gamma^{S=1}_{L+1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L+1}} \\Big]\n(i\\sigma_2)\\chi_{-\\mathbf p}^* \n\\,, \\nonumber\\\\\n \\Gamma^{S=1}_{L+1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L+1}} \n&\\equiv& \n\\sum_{k=1}^{L+1}\\,T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots i_{L+1}}(\\mathbf p)\\,\\sigma^{i_k}\n - \\frac{2}{2L+n-2}\\,\\sum_{k$5\\arcsec) visual companions to our sample members. \nHowever, the PSC does not always distinguish multiple point sources in close \nproximity ($\\la$5\\arcsec), instead reporting only the brightest source. This \nsuggests that wide neighbors to our sample members should be identified in the \nPSC, but most close neighbors are probably absent.\n\nWe address this incompleteness by working directly with the processed survey \nimages to identify close ($\\la$5\\arcsec) companions via PSF-fitting photometry. \nFrom the 2MASS website, we extracted postage-stamp (60x60$\\arcsec$) and \nwide-field (510x1024$\\arcsec$) images for each of the association members \ndescribed in Section 2. The wide-field images were used to create reference PSFs \nfor each science target, while the postage-stamp images have been used to \nidentify close visual companions. The width of the wide-field images \n(510$\\arcsec$) corresponds to the width of each 2MASS survey tile; any image \nwith larger width would include data taken at different epochs, and therefore \nwith different seeing conditions. The height was chosen to allow for $\\ga$10 PSF \nreference stars brighter than $K\\sim$11 in all fields. The size of the overlap \nregion between adjacent tiles was 60\\arcsec\\, in right ascension and \n8.5\\arcmin\\, in declination, so each science target appeared to be $>$30\\arcsec \naway from the edge in at least one tile.\n\nThe 2MASS survey images were produced by coadding multiple exposures taken in \nsequence, each offset by $\\sim$85\\arcsec\\, in declination, so drawing PSF \nreference stars from several arcminutes away could lead to nonuniform images. \nOnly sources $\\la$40$\\arcsec$\\, north or south of a science target were observed \nin all six exposures that the science target was observed, and sources \n$\\ga$500$\\arcsec$\\, north or south do not share any simultaneous scans. However, \nall of the scans which contribute to a wide-field image were observed within \n$\\sim$30 seconds. We do not expect the seeing-based PSF to change on this short \ntimescale, and we have found that the PSF is usually constant over each entire \nwide-field image ($\\sigma$$_{FWHM}$$\\sim$0.1$\\arcsec$).\n\n\\subsection{Data Reduction and Source Identification}\n\nWe identified candidate companions and measured their fluxes from the \npostage-stamp image of each sample member using the \nIRAF\\footnote{IRAF is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association of Universities \nfor Research in Astronomy, Inc., under cooperative agreement with the National \nScience Foundation.} package DAOPHOT (Stetson 1987), specifically with the \nPSF-fitting photometry routine ALLSTAR. The template PSFs for each \npostage-stamp image were created \nusing the PSTSELECT and PSF tasks. We selected template stars for each source \nfrom the corresponding wide-field image; each PSF was based on the eight \nbrightest, unsaturated stars which appeared to be isolated under visual \ninspection. The appropriate photometric zero-point was extracted from the image \nheaders. We compared PSF-fitting magnitudes for single stars to the \ncorresponding PSC values in order to test our results; there is no systematic \noffset, and the standard deviation of the random scatter in $m_{PSF}-m_{PSC}$ is \n$\\sim$0.03 magnitudes.\n\nAs we have discussed in previous publications (Kraus et al. 2005, 2006), one \nlimitation of ALLSTAR-based PSF photometry is that binaries with very close \n($\\la$$\\theta$$_{FWHM}$) separations are often not identified, even when their \ncombined PSF deviates significantly from that of a true point source. This \nlimitation can be overcome for known or suspected binaries by manually adding a \nsecond point source in approximately the correct location and letting ALLSTAR \nrecenter it to optimize the fit. However, this method requires objective \ncriteria for identifying suspected binaries; subjective selection methods like \nvisual inspection would not allow us to rigorously choose and characterize a \nstatistically complete sample. We have found that ALLSTAR's $\\chi$$^2$ \nstatistic, which reports the goodness-of-fit between a source and the template \nPSF, is an excellent diagnostic for this purpose. Since there are images in \nthree bandpasses, we use a single diagnostic value, denoted $\\chi$$_3$, which is \nthe sum of the three $\\chi$$^2$ values obtained for each association member when \nfit with a single point source. We list the value of $\\chi$$_3$ for each \nassociation member in Table 2.\n\nIn Figure 1, we plot the values of $\\chi$$_3$ as a function of K-band\nmagnitude for a subset of sample members with no known companions between\n0.5$\\arcsec$ and 15$\\arcsec$ (according to the surveys of Leinert et al.\n1993; Ghez et al. 1993; Simon et al. 1995; Duchene 1999; Kohler et al.\n2000; Kraus et al. 2005, 2006; White et al. 2006). The goodness of fit\ndegrades rapidly for saturated stars ($K\\la8$), so our technique does not\ndiscriminate betweeen single stars and candidate binaries in this regime.\nHowever, since there are few stars brighter than the saturation limit, we\ndecided not to reject them until we were certain we could not identify any\nbinary systems via other methods. The distribution of $\\chi$$_3$ values for\nunambiguously unsaturated stars ($K>8.5$) is not normally distributed, but\n95\\% of these stars produce fits with $\\chi$$_3$$<$2.5, so we have selected all \nsources with $\\chi$$_3$$\\ge$2.5 as candidate binary systems.\n\nThe mean value of $\\chi$$_3$ for single stars should be $\\sim$3 since it \nrepresents the sum of 3 variables which follow a $\\chi$$^2$ distribution. However, \nwe find that the mean value reported by ALLSTAR for unsaturated single sources is \n$\\sim$1.75. This disagreement is caused by an overestimate of the photometric \nerrors in each observation by ALLSTAR. The coadding and subsampling process used \nin the 2MASS image processing pipeline results in correlated noise between \nadjacent pixels of the final survey images, so the true uncertainties are lower \nthan those estimated solely by Poisson statistics (Skrutskie et al. 2006).\n\nWe identified the candidate binaries in our sample based on this empirically \nmotivated $\\chi$$_3$ selection criterion, and then we attempted to fit each with \na pair of point sources separated initially by the PSF FWHM (3$\\arcsec$) and \nwith position angle corresponding to the angle of maximum elongation of the \nsystem PSF. The ALLSTAR routine optimized the components' separation, position \nangle, and magnitudes to produce the optimal fit; as we further summarize in \nSection 3.3, known binaries were typically fit with consistent positions and \nflux ratios in all three bandpasses while contaminants (such as sources with an \nerroneous template PSF in one filter) did not produce consistent fits in \nmultiple images. We adopt the criterion that any candidate binary with component \npositions within 1$\\arcsec$ (3$\\sigma$ for astrometry of very close, faint \ncompanions; Section 3.4) in all three filters is a bona-fide visual binary. We \nfound that saturated stars produced fits for erroneous companions at separations \nof 1.0-1.5\\arcsec, so we have rejected all candidate companions to saturated \ntargets ($K_{tot}<8$) with separations of $<$2\\arcsec. Known binaries with wider \nseparations produced consistent fits even in the saturated regime for systems \nfainter than $K\\sim6$, so we adopted this as a maximum brightness limit for our \nsample.\n\nFinally, we compared the location of each candidate companion with the online \ncatalog of 2MASS image artifacts. We found that a candidate companion to \nMHO-Tau-4 was coincident with a persistence artifact flag. Furthermore, a \nprevious high-resolution imaging survey with HST (Kraus et al. 2006) found no \noptical counterpart to a limit of $z'$$\\sim$24, so we removed this candidate \ncompanion from further consideration and treat MHO-Tau-4 as a single star.\n\n\\begin{figure*}\n\\plotone{f1.eps}\n\\caption{A plot of the goodness-of-fit as a function of K-band magnitude for 203 \nobjects with no wide companions (0.5-15$\\arcsec$). The sharp increase in \n$\\chi$$_3$ at $K\\sim$8 is due to the onset of image saturation; the stars in \nthis brightness range are typically late K or early M, so saturation begins \nsimultaneously in all three bands. The solid line at $\\chi$$_3$$=2.5$ denotes \nthe 95\\% confidence interval for nominally single stars; we have selected all \nsample members above this limit as candidate close binaries. We found that our \nfitting algorithm for identifying companions is effective for mildly saturated \nstars, so we include association members up to $K=6$.\n} \n\\end{figure*}\n\n\\subsection{Sensitivity Limits}\n\nWe determined companion detection limits as a function of distance from the \nprimary stars via a Monte Carlo simulation similar to that of Metchev et al. \n(2003). We used the IRAF task DAOPHOT\/ADDSTAR to add artificial stars at a \nrange of radial separations and magnitudes to the images of FO Tau, MHO-Tau-5, \nKPNO-Tau-8, and KPNO-Tau-9. These four sources have been shown to be single to \nthe limits of high-resolution imaging (Ghez et al. 1993; Kraus et al. 2006) \nand span the full range of brightness in this sample. We then attempted to \nidentify the artificial companions via PSF-fitting photometry. Our photometric \nroutines attempt simultaneous source identification in all three filters in \norder to separate erroneous detections from genuine companions, so we created \nthe same synthetic source in all three filters using colors from the 2 Myr \nBaraffe isochrones (Baraffe et al. 1998).\n\nIn Figure 2, we show our survey's 50\\% detection limits as a function of \nseparation for identifying candidate companions using the same PSF-fitting \nalgorithm as our actual search program. The minimum separation at which we can \ndetect equal-flux companions is $\\sim$1\\arcsec\\, for bright, unsaturated sources \nand $\\sim$1.6\\arcsec\\, for sources just above our adopted $K$ band magnitude \nlimit ($K=14.3$). The 10\\% and 90\\% detection limits are typically $\\sim$0.5 \nmagnitudes below and above the 50\\% limit. The sensitivity of PSF-fitting \nphotometry falls at separations $\\ga$5\\arcsec\\, since objects become cleanly \nresolved and most companion flux falls outside the fitting radius for the \nprimary. However, the PSC is complete to at least $K=14.3$ at larger separations, \nso wider companions will be recovered by our search of the catalog.\n\nWe also show the separation and flux ratio for known binary systems which \nhave been detected in K-band surveys (Kohler et al. 2000; White et al. \n2006) and whether these systems were unambiguously recovered (via either \nPSF-fitting photometry or the PSC), identified as candidate systems based on \nthe $\\chi$$_3$ criterion, or not recovered. The limits between detected and \nnondetected systems are roughly consistent with our empirically determined \nmagnitude limits, but there are few known systems which fall near these \nlimits. There are only two known wide systems among the faintest members of \nour sample ($K>11$), so we can not significantly test the detection limits \nof our search method in this brightness range. However, we identified four \nadditional candidate companions to sources in this brightness range, plus \nnumerous likely background stars, so our survey appears to be sensitive to \ncompanions in this regime.\n\n\\begin{figure} \n\\plotone{f2.eps} \n\\caption{Detection frequencies as a function of separation for \nartificially-introduced companions to four known single objects spanning the \nsurvey sample's brightness range: FO Tau ($K=8.12$), MHO-Tau-5 ($K=10.06$), \nKPNO-Tau-8 ($K=11.99$) and KPNO-Tau-9 ($K=14.19$). The solid lines denote the \n50\\% detection limit for our PSF-fitting photometry. The symbols represent \nknown binary companions from high-resolution K-band multiplicity surveys in \nUpper Scorpius (Kohler et al. 2000) and Taurus (White et al. 2006 and \nreferences therein). Filled circles denote companions which we recovered, open \ncircles denote companions which passed our $\\chi$$^2$ criterion but did not \nproduce significant fits, and crosses denote companions which were not \nrecovered. The dotted line shows the minimum separation at which the PSC will \nidentify all companions bright enough to be considered in our search \n($K<14.3$).\n} \n\\end{figure}\n\n\\begin{figure} \n\\plotone{f3.eps} \n\\caption{\nThe uncertainty in the measured binary companion brightness as a function of \nseparation for simulated binary images spanning the range of primary and \nsecondary brightnesses. The flux ratios shown are $\\Delta$$K=$0, 1, 2, 4, and 6 \n(solid, dotted, short-dashed, long-dashed, dash-dotted lines, respectively). The \nphotometric uncertainties increase sharply at separations of $\\la$3\\arcsec, \nsuggesting that observed photometric colors will not be accurate in \nthis separation range.\n} \n\\end{figure}\n\n\\begin{figure} \n\\plotone{f4.eps} \n\\caption{\nAs in Figure 3, showing uncertainties in binary secondary positions as a \nfunction of separation. \n} \n\\end{figure}\n\n\\subsection{Uncertainties in Binary Properties}\n\nMany of our candidate binaries have separations of $\\la$$\\theta$$_{FWHM}$, \nso our measurements could be subject to significant uncertainties. We \ntested these uncertainties by using a Monte Carlo routine to produce \nsynthetic images for binaries spanning a range of primary brightnesses, \nflux ratios, and separations. Specifically, we used ADDSTAR to construct \nsimulated $JHK$ images, and then we measured the binary fluxes and \nseparations for each set of simulated images using ALLSTAR. For each \ncombination of parameters, we produced 100 sets of synthetic images with \nrandomly distributed position angles. The $J-K$ and $H-K$ colors for the \nsecondaries were drawn from the 2 Myr isochrone of Baraffe et al. (1998) \nin order to determine realistic values for $\\Delta$$K$, $\\Delta$$H$, and \n$\\Delta$$J$.\n\nIn Figure 3, we show the standard deviation in the measured brightness for our \nsimulated binary companions as a function of separation. These simulations \npredict that photometric uncertainties increase significantly at separations of \n$\\la$3\\arcsec, so measured colors may not be reliable at small separations. As \nwe describe in Section 3.5, these colors are necessary at large separations \n($>$5\\arcsec) to distinguish candidate companions from background stars. \nHowever, contamination from background sources should be low at small \nseparations ($\\la$3\\arcsec) due to their low surface density, so we can neglect \nthese selection criteria with only a minor increase in the number of erroneous \nbinary identifications.\n\nIn Figure 4, we show a similar plot of the RMS scatter in the measured position \nof the secondary. The typical standard deviations are $\\la$0.3\\arcsec\\, for all \nbut the faintest companions, so the uncertainties in our measured separations \nshould have similar precision. Given these positional uncertainties, the \ncorresponding uncertainties in position angles range from 1 to 10 degrees, \ndepending on the binary separation. The standard deviations in secondary \nposition for our simulated images are consistent with the scatter between the \nthree filters for each observed binary, so we adopt the results from these \nsimulations as our estimated uncertainties.\n\nWe also conducted Monte Carlo tests to determine the probability of mistakenly\nidentifying a true single star as a binary. We constructed a series of simulated\nimages (100 each for four objects spanning our sample's range of brightness),\nand then tried to fit each object with two point sources. We found that this\nnever produced consistent fits in 3 filters, though faint peaks due to noise\nwere occasionally identified in one of the 3 images. This suggest that the\nprobability of an erroneous binary identification due to statistical errors is\nlow ($<1\\%$). This agrees with our results for known single stars; as we note in\nSection 3.2, 5\\% of known single stars fall above our $\\chi$$_3$ criterion for\nidentifying candidate binaries. However, none of these yielded fits for multiple \npoint sources in all 3 filters.\n\n\\subsection{Field Star Contamination}\n\nThe identification of binary companions based solely on proximity is\ncomplicated by contamination from foreground dwarfs, background giants, and\nreddened early-type background dwarfs. We have not conducted followup\nspectroscopic or astrometric observations to confirm association membership,\nso we must limit the survey to a total area in which the contamination from\nbackground stars is small compared to the number of candidate binary\ncompanions. We estimate the surface density of contaminants for each\nassociation based on the total number of objects within an annulus of\n30-90\\arcsec\\, from all of the association members in our sample. Field\nsurveys (e.g. Duquennoy \\& Mayor 1991; Reid \\& Gizis 1997) have identified few\nbinaries with projected separations of $\\ga$500 AU ($\\ga$30\\arcsec at the\ndistance of our sample members), so this method will also address the\nprobability of chance alignment with other association members.\n\nOur estimate of the contamination could be influenced by variations in\nbackground source counts due to the large angular extent of these associations\nor by variations in galactic latitude or extinction. The result would be a\nsystematic overestimation of the association probability for candidate\ncompanions at points of high contamination and a corresponding underestimation\nat points of low contamination. However, any local deviation from the mean\ncontamination rate should not affect the binary statistics for the association\nas a whole since the ensemble background at 30-90\\arcsec\\, will match the\nensemble background at $<$30\\arcsec. Our subsequent cuts against color, mass\nratio, and separation will also help to homogenize the sample by preferentially\nremoving background stars.\n\nMost previous multiplicity surveys were based on observations in a single \noptical or near-infrared bandpass (e.g. Kohler et al. 2000); in the absence \nof color information, these surveys can only estimate physical association \nprobabilities for candidate companions based on the surface density of \nbackground stars of similar brightness. Since 2MASS includes images in 3 \nfilters, we can reject most background stars by requiring colors consistent \nwith regional membership (Section 4.1). Specifically, we have plotted \n($K,J-K$) and ($K,H-K$) color-magnitude diagrams for each region and we \nrequire prospective binary companions to fall above a smoothed field main \nsequence (Bessell \\& Brett 1988; Leggett et al. 2001) for the regional \ndistance in both CMDs. We have chosen to use $K$ as a proxy for luminosity \ninstead of $J$ in order to minimize the effect of extinction for background \nstars. This choice will cause disk-bearing association members to sit \npreferentially higher in our color-magnitude diagrams, but this moves them \nfurther from our selection cutoff, so our results should be robust.\n\nAs a test of these color criteria, we have plotted color-magnitude diagrams for \nthe members of our primary sample. We find that $\\sim$97\\% of the primaries have \ncolors consistent with our definition of association membership, so any \nincompleteness in the selection of binary companions should be negligible. Most \nunselected primaries fall just below our color cuts; the only sample members \nwhich fall well below the association sequences are GSC 06191-00552 and \nUSco-160803.6-181237. Both of these objects are claimed to be \nspectroscopically-confirmed members of USco-A, but the spectra are not available \nin the literature. We have not detected any binary companions to these objects, \nso their erroneous inclusion in our sample would not significantly change our \nresults. However, it might be prudent to reconsider their membership status with \nadditional spectroscopic observations in the future.\n\n\\begin{deluxetable*}{lllllllllllll}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Association Star Counts\\label{tbl2}}\n\\tablehead{\\colhead{} & \n\\multicolumn{3}{c}{Chamaeleon I ($N=147$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{Taurus-Auriga ($N=235$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{USco-A ($N=356$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{USco-B ($N=45$\\tablenotemark{a})}\n\\\\\n\\colhead{Sep} \n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n\\\\\n\\colhead{(\\arcsec)} \n& \\colhead{Count\\tablenotemark{b}} & \\colhead{Valid\\tablenotemark{b}} \n&\\colhead{Stars\\tablenotemark{b}}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n}\n\\startdata\n0-3\\tablenotemark{c}&7&-&0.9&9&-&0.9&15&-&2.0&8&-&0.3\\\\\n3-5&5&5&1.0&6&5&0.7&8&4&0.3&1&0&0.1\\\\\n5-10&8&6&4.8&10&5&3.1&12&5&1.4&6&4&0.4\\\\\n10-15&19&12&8.0&22&11&5.2&32&8&2.4&3&0&0.6\\\\\n15-20&20&13&11.2&23&13&7.2&36&6&3.4&4&0&0.8\\\\\n20-25&34&18&14.4&21&12&9.3&44&6&4.3&5&1&1.1\\\\\n25-30&39&28&17.6&33&16&11.4&60&5&5.3&9&4&1.3\\\\\n30-90&766&461&-&733&298&-&1566&138&-&215&34&-\\\\\n\\enddata\n\\tablenotetext{a}{The total sample size for each region, as \nsummarized in Table 1.}\n\\tablenotetext{b}{The number of unassociated contaminants was estimated \nfrom the surface density of sources which meet our color selection \ncriteria in the 30-90\\arcsec\\, separation range; most of these sources \nshould be foreground stars, background stars, or unbound association \nmembers.}\n\\tablenotetext{c}{We cannot use color criteria at separations of \n$<3$$\\arcsec$\\, due to the poor photometric precision (Section 3.4), so \nthe surface density of unassociated contaminants is higher.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lccccccccccl} \n\\tabletypesize{\\tiny}\n\\tablewidth{0pt} \n\\tablecaption{Candidate Wide Binary Systems\\label{tbl3_02}} \n\\tablehead{\\colhead{Name} & \\multicolumn{3}{c}{Primary} & \n\\multicolumn{3}{c}{Secondary} & \\colhead{Projected} & \n\\colhead{Position} & \\colhead{$\\mu_{\\alpha}$,$\\mu_{\\delta}$\\tablenotemark{a}}\n& \\colhead{Ident} & \\colhead{References}\n\\\\\n\\colhead{} & \\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} & \n\\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} &\n\\colhead{Sep(\\arcsec)} & \\colhead{Angle(deg)} & \\colhead{(mas yr$^{-1}$)} & \n\\colhead{Method}\n} \n\\startdata \n2M11103-7722&2.00&0.68&10.03&2.21&0.77&13.85&9.30&108.8&-&PSC&-\\\\\nC7-1&1.78&0.62&10.55&1.67&0.43&13.32&5.73&214.9&0,0&PSC&-\\\\\nCHSM1715&2.05&0.85&10.90&1.42&0.43&13.94&9.07&30.3&-58,42&PSC&-\\\\\nCHXR26&2.02&0.46&9.92&2.68&1.07&9.98&1.41&215.2&-&PSF&Luhman (2004b)\\\\\nCHXR28&1.17&0.32&8.23&1.53&0.39&8.83&1.78&121.6&-&PSF&Brandner et al. (1996)\\\\\n\\enddata \n\\tablecomments{The full table of sample members can be found in Table 9 at the \nend of this manuscript.}\n\\tablenotetext{a}{An entry of 0,0 denotes a source which was detected by \nthe USNO-B survey, but did not show a significant proper motion. An entry of \"-\" \ndenotes a source which was not detected by the USNO-B survey.}\n\\tablenotetext{b}{ScoPMS052 B is also known as GSC06209-01312; Martin et al. (1998) \nidentified it as a WTTS.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lcccccccccl} \n\\tabletypesize{\\tiny}\n\\tablewidth{0pt} \n\\tablecaption{Ultrawide Visual Companions\\label{tbl3_03}} \n\\tablehead{\\colhead{Name} & \\multicolumn{3}{c}{Primary} & \n\\multicolumn{3}{c}{Secondary} & \\colhead{Projected} & \n\\colhead{Position} & \\colhead{$\\mu_{\\alpha}$,$\\mu_{\\delta}$} &\n\\colhead{References}\n\\\\\n\\colhead{} & \\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} & \n\\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} &\n\\colhead{Sep(\\arcsec)} & \\colhead{Angle(deg)} & \\colhead{(mas yr$^{-1}$)}\n} \n\\startdata\nC1-6&3.92&1.68&8.67&1.93&0.80&14.10&27.58&156.0&&OTS12(candidate; Oasa et al. 1999)\\\\\nC1-6&3.92&1.68&8.67&2.26&0.70&13.75&24.51&123.8&-&OTS14(candidate; Oasa et al. 1999)\\\\\nCam2-19&2.40&0.74&10.25&2.36&0.72&13.45&23.13&107.6&-&-\\\\\nCam2-42&2.44&0.73&9.16&2.09&0.50&13.51&27.64&261.7&-&-\\\\\nCam2-42&2.44&0.73&9.16&2.11&0.65&14.14&28.18&180.4&-&-\\\\\n\\enddata \n\\tablecomments{The full table of sample members can be found in Table 10 at the \nend of this manuscript.}\n\\tablenotetext{a}{The source [CCE98] 2-26 is an ultrawide neighbor of both ChaHa7 and CHXR76; its \nphysical association, if any, is uncertain.}\n\\tablenotetext{b}{Haro 6-5 B is a known member of Taurus (Mundt et al. 1984), but was not included as \npart of our statistical sample because its spectral type is uncertain.}\n\\tablenotetext{c}{USco-160428.4-190441 B is also known as GSC06208-00611; Preibisch et al. (1998) \nidentified it as a field star.}\n\\tablenotetext{d}{SCH16075850-20394890 B is also known as T64-2; The (1964) identified it as a strong \nH$\\alpha$ emitter.}\n\n\\end{deluxetable*}\n\n\\begin{figure*} \n\\plotone{f5.eps} \n\\caption{$K,J-K$ and $K,H-K$ color-magnitude diagrams for the four regions in our\nsurvey. The top panels show the confirmed association members in our\nsurvey, the middle panels show all objects within 5-15\\arcsec\\, of known\nassociation members, and the bottom panels show all objects within the\nbackground annuli (30-90\\arcsec). The solid line shows the main sequence at\nthe association distance and the dashed line shows the isochrone for the\nadopted association age (Table 2). In the top panels, association members are \nshown with filled circles. In all other panels, sources which lie above a \nsmoothed main sequence in both CMDs are shown with open circles and other \nsources are shown with small dots.\n} \n\\end{figure*}\n\n\\section{Results}\n\n\\subsection{Candidate Binary Companions}\n\nWe identified a total of 451 well-resolved visual companions brighter than \n$K=14.3$ within 30\\arcsec\\, of our sample members in the 2MASS PSC (Section 3.1), \nas well as 48 close ($\\la$5\\arcsec) candidate companions based on our PSF-fitting \nphotometry of 2MASS image data (Section 3.2). We have chosen 30\\arcsec\\, \n($\\sim$5000 AU) as an absolute upper limit for for identifying candidate \ncompanions since it corresponds to the maximum separation seen for field binaries \nat the distances of these association members. We also found 3280 visual \ncompanions within 30-90\\arcsec\\, of our sample members. Since the ratio of \nsources at 0-30\\arcsec\\, and 30-90\\arcsec\\, is roughly equal to the ratio of \nareas (1\/8), we expect that most of the sources within 30\\arcsec\\, of our sample \nstars are foreground or background stars having colors inconsistent with \nassociation membership.\n\nIn Figure 5, we present ($K$,$J-K$) and ($K$,$H-K$) color-magnitude diagrams \nfor the four regions showing all confirmed association members in our \nsample and all companions in two separation ranges (5-15\\arcsec\\, and \n30-90\\arcsec) corresponding to likely companions and likely background stars. \nWe summarize the number of objects which pass or fail the color selection \ncriteria (Section 3.5) as a function of separation in Table 4. We also \nestimate the number of contaminants which are expected to pass both selection \ncriteria in each separation range, assuming that the source density at \n30-90\\arcsec\\, represents the contaminant source density.\n\nWe showed in Section 3.4 that the uncertainties in our PSF-fitting photometry \nbecome significant at small separations, so we cannot use color criteria to \nidentify candidate companions inside $\\sim$3\\arcsec. However, given the low \nsurface density of background sources and the faintness of most nonmembers, we \nexpect only a small level of contamination in this separation range. Each of the \n39 candidate companions at separations $<$3\\arcsec\\, has a sufficiently high \nprobability of physical association ($\\ga$80\\%) to merit inclusion in our sample \nwithout using color cuts.\n\nWe have defined the maximum separation at which we identify candidate binary \ncompanions by requiring that the number of sources which pass our color \nselection requirement in each separation bin be $\\ga$2 times the number of \nexpected background companions. The corresponding probability that any \nindividual source inside that separation limit is a background star will be \n$\\la$50\\%. Based on the expected contamination rates and visual companion \ncounts in Table 4, these separations are 10\\arcsec\\, for ChamI, 15\\arcsec\\, for \nTaurus, 20\\arcsec\\, for USco-A, and 30\\arcsec\\, for USco-B. The separation limit \nis lower for regions with higher extinction since a higher fraction of \nbackground stars are reddened into our selection range. We adopt these \nseparation limits as our criteria for identifying candidate binary companions. \nWe note that sources at higher separations still have a nonnegligible \nprobability of association, but the probability that any individual source is a \nbinary companion will be low.\n\nUsing the color and separation cuts described above, we have identified (of 451 \nsources identified in the PSC and 48 sources identified with PSF-fitting \nphotometry) a total of 18 candidate binary companions in ChamI, 32 in Taurus, 40 \nin USco-A, and 17 in USco-B. Of these candidates, 4, 7, 23, and 5, respectively, \nhave not been previously reported in the literature. We summarize the binary \nproperties of these candidate systems in Table 5. Some of the very wide and very \nfaint companions are likely to be unassociated foreground or background stars, so \nwe will consider a restricted range of separations and mass ratios in our \nsubsequent statistical analysis. In Table 6, we list the other visual companions \nwith separations $<$30\\arcsec\\, (but wider than the association's companion \nidentification limit) which have colors consistent with association membership and \nseparations greater than the limits given above. Many of these sources are \nexpected to be background stars, but additional information (such as optical \nphotometry or kinematic data) could be used in the future to remove additional \ncontaminants and more securely identify any ultrawide binary companions.\n\n\\subsection{Previous Observations}\n\nMany of our candidate companions have been identified previously in the \nliterature, but as we note in Tables 5 and 6, several of our candidates have also \nbeen rejected as association members based on the absence of spectroscopic \nsignatures of youth. Some of the candidates we list have probably been considered \nand rejected in previous work, but most surveys do not publish their catalogue of \nconfirmed field stars, so we cannot assess this number. \n\nWe also find that five members of our sample (USco-160700.1-203309, \nSCH16151115-24201556, and USco80 in USco-A; 2MASSJ04080782+2807280 and \n2MASSJ04414489+2301513 in Taurus) have candidate companions which are \nsignificantly brighter, and thus are likely to be the system primary \n(making the known association member a binary secondary). This result is \nnot surprising for the three Upper Sco members. Upper Sco is thought to \ncontain several thousand low-mass members, and photometric surveys have \nidentified many more candidates than could be confirmed via spectroscopy, \nso there are many more association members awaiting discovery. The two \nTaurus members are located on the edges of the association and were \ndiscovered by the only survey which considered these areas (Luhman 2006). \nOur newly-identified candidate companions are both brighter than the upper \nbrightness limit for this survey ($H=10.75$), so there were no previous \nopportunities for them to have been discovered.\n\nFinally, we find that 5 candidate companions identified in previous surveys have \n2MASS colors inconsistent with association membership: UX Tau B, V819 Tau B, \nHBC355 (HBC354 B), RXJ1524.2-3030B B, and RXJ1559.8-2556 B. Since $\\sim$3\\% of \nthe spectroscopically confirmed association members in our primary star sample \ndid not meet both color cuts, we expect (adopting the same percentage for the \nsecondaries) that only $\\sim$1-2 bona fide binary companions would not be \nselected. However, close pairs of stars have larger photometric errors, which \nincreases the probability that some companions might fall outside our selection \ncuts. Of these five companions, three fall just below the color cuts (UX Tau B, \nHBC 355, and RXJ1524.2-3030B B) in our CMDs and the other two fall significantly \nbelow the color cuts, so we suggest that the first three are erroneous \nrejections, and therefore we keep these objects, while we consider the other two \nto be valid rejections.\\footnote{V819 Tau B has also been classified as a \nbackground star by Woitas et al. (2001) due to its position on a (J,J-K) CMD and \nby Koenig et al. (2001) due to an absence of x-ray emission. UX Tau B and HBC355 \nare spectroscopically confirmed cluster members, and no membership assessments \nare available for the other two sources.}\n\n\\subsection{Inferred Stellar Properties}\n\n\\begin{deluxetable*}{lccccccccl} \n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt} \n\\tablecaption{Inferred Binary Properties\\label{tbl3_04}} \n\\tablehead{\\colhead{Name} & \\multicolumn{2}{c}{Primary}\n& \\multicolumn{2}{c}{Secondary} & \\colhead{Projected} \n& \\colhead{Mass}\n\\\\\n\\colhead{} & \\colhead{SpT} & \\colhead{Mass ($M_{\\sun}$)}\n& \\colhead{SpT\\tablenotemark{b} \\tablenotemark{b}} \n& \\colhead{Mass\\tablenotemark{a} \\tablenotemark{b} ($M_{\\sun}$)} & \n\\colhead{Separation(AU)\\tablenotemark{b}} & \n\\colhead{Ratio\\tablenotemark{b} ($q$)}\n} \n\\startdata \n2M11103&M4&0.27&(M8.5)&(0.02)&1535&0.08\\\\\n2M11103(\/ISO250)&M4&0.27&M4.75(M5.5)&0.20(0.15)&1569&0.56\\\\\nC7-1&M5&0.18&(M8)&(0.03)&945&0.18\\\\\nCHSM1715&M4.25&0.25&(M7)&(0.05)&1497&0.18\\\\\nCHXR26&M3.5&0.33&(M5)&(0.19)&233&0.57\\\\\n\\enddata \n\\tablenotetext{a}{Values in parentheses are estimated from the system flux \nratio $\\Delta$$J$ and the spectroscopically determined properties of the \nprimary.}\n\\tablenotetext{b}{Estimated statistical uncertainties are $\\sim$10\\% for \nmass ratios, $\\sim$20\\% for secondary masses, $\\sim$2-3 subclasses for \nspectral types, and $\\sim$10\\% for projected separations.}\n\\tablecomments{The full table of sample members can be found in Table 11 at the \nend of this manuscript.}\n\\end{deluxetable*}\n\nIn Table 2, we list the inferred spectral types and masses for all of the \nassociation and cluster members in our sample. Spectral types are taken from the \nprimary reference and were typically determined via low- or \nintermediate-resolution spectroscopy. We assume that the spectroscopically \ndetermined spectral type and mass for previously-unresolved binary systems \ncorresponds to the primary mass and spectral type. Equal-mass binary components \nshould have similar spectral types and the flux from inequal-mass systems should \nbe dominated by the primary; in either case, spectroscopic observations of the \nunresolved system should have been affected only marginally by the flux from the \nsecondary.\n\nWe estimated the masses of sample members by combining mass-temperature and \ntemperature-SpT relations from the literature. No single set of relations spans \nthe entire spectral type range of our sample, so we have chosen the M dwarf \ntemperature scale of Luhman et al. (2003b), the early-type ($\\le$M0) temperature \nscale of Schmidt-Kaler (1982), the high-mass stellar models of D'Antona \\& \nMazzitelli (1997; DM97), and the low-mass stellar models of Baraffe et al. (1998; \nNextGen). We apply the DM97 mass-temperature models for masses of $\\ga$1 $M_\\sun$ \nand the NextGen models for masses of $\\la$0.5 $M_\\sun$; in the 0.5-1.0 $M_\\sun$ \nregime, we have adopted an average sequence. For each association, we adopt the \nmodels corresponding to the mean age listed in Table 1; this will introduce some \nuncertainty given the unknown age spread for each association. Large systematic \nerrors may be present in these and all pre-main sequence models (e.g. Baraffe et \nal. 2002; Hillenbrand \\& White 2004; Close et al. 2005; Reiners et al. 2005), so \nthey are best used for relative comparison only.\n\nMuch of the uncertainty in theoretical mass-temperature relations can be assessed \nin terms of a zero-point shift in the mass; preliminary observational \ncalibrations by the above authors suggest that theoretical models overestimate \nmasses by 10-20\\% over most of our sample mass range. This suggests that \ntheoretical predictions of relative properties (e.g. mass ratios, \n$q=m_{s}\/m_{p}$) might be more accurate than absolute properties (e.g. individual \ncomponent masses) since the systematic mass overestimates will cancel. Relative \nquantities are also largely independent of age and extinction, which are expected \nto be similar for binary components. We have combined our adopted \nmass-luminosity-SpT relations with the near-infrared colors of Bessell \\& Brett \n(1988) and the K-band bolometric corrections of Leggett et al. (1998, 2000, 2002) \nand Masana et al. (2006) to predict values for $q$ as a function of primary \nbrightness $m$ and flux ratio $\\Delta$$m$ in all three 2MASS filters. Some of our \nsample members could possess K-band excesses due to hot inner disks, so we have \nadopted the $q$ values predicted by the J-band fluxes; this will not eliminate \nthe effect, but should minimize it. We have also combined our derived $q$ values \nwith the estimated primary masses to predict secondary masses, and we use our \nmass-SpT relations to predict the corresponding secondary spectral types.\n\nWe list the derived values for each binary system in Table 7. Some wide \nbinaries have independent SpT determinations for both components, so we \nreport derived quantities with parentheses and measured quantities \nwithout. The typical uncertainties in $q$ are $\\sim$10\\% and represent the \nuncertainties in the photometry and the assigned spectral types, though \nsome systematic effects (e.g. unresolved multiplicity or different levels \nof extinction) could produce far larger values. This can be seen in the \ndiscrepancies for some systems (e.g. GG Tau AB, MHO-2\/1) which are known \nto be hierarchical multiple systems. We can not quantify the unknown \nuncertainties in the theoretical models, but they should be considered \nwhen interpreting these results. The typical uncertainty in physical \nseparation is $\\sim$10\\% and reflects the uncertainty in angular \nseparation and the unknown depth of each system in its association; we \nassume each association has a total depth equal to its extent on the sky \n($\\sim$40 pc for Taurus and Upper Sco, $\\sim$20 pc for ChamI). The \nuncertainty in the mean association distance ($\\sim$5 pc) introduces a \nsystematic uncertainty of $\\pm$3\\%, but this is generally negligible.\n\n\\subsection{Binary Statistics}\n\nMultiplicity surveys typically consider the frequency of binary systems for \nrestricted ranges of parameter space (observed separations and mass ratios) \ncorresponding to the survey completeness limits. For our analysis, we select a \nrange of projected separations (330-1650 AU, set by the inner and outer detection \nlimits of ChamI since those limits are most restrictive) and flux ratios \n($\\Delta$$K<2$, corresponding to $q\\ga$0.25) that should be complete for all but \nthe lowest-mass brown dwarfs in our sample. The inner separation limit and mass \nratio limit are set by the resolution limit for low-mass sample members \n($K\\sim$12.3) in ChamI, while the outer separation limit is set by the background \ncontamination in ChamI, where our mass ratio cut allows us to choose a 90\\% pure \nsample for separations $<$10\\arcsec.\n\nIn Figure 6, we present plots of the wide binary frequency as a function of \nprimary mass for each region in our sample. The binary fractions plotted \ncorrespond to our designated completeness regime: mass ratios $q>0.25$ and \nprojected separations of 330-1650 AU. In the bottom panel, we show the field \nbinary frequency in the same range of mass ratios and projected separations for \nsolar-type stars (Duquennoy \\& Mayor 1991), early-mid M dwarfs (M0-M6; Reid \\& \nGizis 1997), and brown dwarfs (Bouy et al. 2003; Burgasser et al. 2003). We also \nshow the corresponding frequencies for early-type stars in USco-A and USco-B \n(Kouwenhoven et al. 2005). The bin sizes were chosen to evenly sample the mass \nrange of our survey (0.025-2.50 $M_{\\sun}$ for which the primary targets were \nbrighter than our brightness cutoff ($K=14.3$). For each region in our survey, \nwe also show the expected frequency for foreground and background sources which \npass our color selection criteria and have $\\Delta$$K<2$, assuming a background \nsource count function N(K) matching that shown in Figure 2; in all cases, the \nexpected contamination rate is negligible. USco-A, ChamI, and Taurus all show a \ndecline in the binary frequency with mass, consistent with the results shown for \nfield multiplicity surveys. USco-B does not show a decline, but the \nuncertainties are not small enough to strongly constrain the slope of any \nmass dependence.\n\nThis binary search may not be complete for objects in the lowest-mass bin where \nsome binary companions could have been fainter than the survey detection limits \n($K>14.3$), so the true upper limits may be marginally higher. However, it has \nbeen observationally determined that most very low mass binaries in the field \nhave mass ratios near unity ($q>0.7$) and much smaller separations ($\\la$20 AU), \nso we are unlikely to have missed any wider or lower-mass ratio companions \n(Close et al. 2003; Burgasser et al. 2003; Bouy et al. 2003).\n\nAnother interesting distribution to consider would be the mass ratio\ndistribution for wide binaries as a function of mass and environment.\nUnfortunately, extending our binary results along another axis of parameter\nspace exceeds the statistical limits of our sample, leaving most bins with\nonly 0-1 detections. The best solution for this is to combine all regions into\na single population. In Figure 7, we plot the mass ratio distribution in our\nsurvey separation range (330-1650 AU) for the three highest-mass bins. We also\nshow the best-fit distribution for solar-type stars in the field (Duquennoy \\&\nMayor 1991).\n\nThis result should be treated with caution since it represents an admixture of \nformation environments which likely does not match the composition of the field. \nAs we show in Figure 6, the binary frequency appears to be fundamentally \ndifferent in the dark cloud complexes (Taurus and Chamaeleon) than in USco-A. \nThis distinction suggests that binary formation processes can vary significantly \nbetween different environments, and therefore that analysis of other binary \nproperties should take the environment into account when possible.\n\n\\begin{figure*} \n\\epsscale{0.90} \n\\plotone{f6.eps} \n\\caption{\nThe wide (330-1650 AU) binary frequency as a function of mass for each region and \nas determined from field multiplicity surveys. The higher-mass histogram bins are \nequally sized in $log M$, but the three lowest-mass bins have been combined to \nillustrate the absence of any companions. The error bars are calculated assuming \nbinomial statistics. The highest-mass datapoints for USco-A and USco-B denote the \nresults of Kouwenhoven et al. (2005). The dashed lines show the expected \nfrequency for each bin solely from foreground and background sources and \nunbound association members; they are not distinguishable from zero in most \nbins. Most upper limits for the lowest-mass bins are also very close to zero.\n} \n\\end{figure*}\n\n\\begin{figure} \n\\epsscale{1.00} \n\\plotone{f7.eps} \n\\caption{\nThe mass ratio distribution for wide binaries in the three highest-mass \nbins of our survey, calculated as a frequency among all sample members. \nThe mass ratio distribution function found by Duquennoy \\& Mayor (1991) \nfor field solar-type stars is denoted with a dashed line. These results \nrepresent the sum over all associations in our sample; the binary \nfrequency varies between environments (Figure 6) and our sample represents \na different admixture of formation environments than the field sample, so \nthe sample and field frequencies should be compared with caution.\n} \n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{The Role of Mass and Environment in Multiplicity}\n\nField multiplicity surveys have established several apparent trends for the mass \ndependence of binary properties. Solar-mass binaries occur at high frequency \n($\\ga$60\\%) and possess high mean and maximum separations (30 AU and 10$^4$ AU) \nand a mass ratio distribution biased toward low-mass companions ($q<0.5$) (e.g. \nDuquennoy \\& Mayor 1991). By contrast, binaries near and below the substellar \nboundary occur at low frequency ($\\sim$10-20\\%) and possess small mean and \nmaximum separations (4 AU and 20 AU), and a mass ratio distribution biased toward \nunity ($q$$\\ga$0.7) (Close et al. 2003; Bouy et al. 2003; Burgasser et al. 2003). \nObservations of intermediate-mass M dwarfs (e.g. Fischer \\& Marcy 1992; Reid \\& \nGizis 1997) suggest that their binary properties are transitional, with an \nintermediate binary frequency and possibly an intermediate separation range, plus \na mass ratio distribution that is nearly flat for $q>0.1$.\n\nThese results have been supported by recent surveys of young open clusters and\nassociations (e.g. Kohler et al. 2000; Patience et al. 2002; Luhman et al. 2005;\nKraus et al. 2005, 2006; White et al. 2006). High-mass stars in these regions\ntypically have higher binary frequencies and wider binary separations than\nlower-mass stars. There is emerging evidence that high-density regions might have\nlower binary frequencies or preferentially smaller binary separations (e.g.\nKohler et al. 2006), but it has not yet been conclusively determined whether this\nis a primordial feature or the result of early dynamical evolution.\n\n\\subsubsection{The Frequency of Wide Binary Formation}\n\nOur results appear to be broadly consistent with the established paradigm of \nmass-dependent multiplicity. Wide (330-1650 AU) binaries are very common among \nstars of $\\ga$1 $M_{\\sun}$ and the frequency appears to decline smoothly with \nmass (Figure 6). We found few wide binaries with primaries less massive than \n$\\sim$0.25 $M_{\\sun}$. Wide binary systems also appear to be common in the \nlow-density T associations (Taurus and Cham I), but comparatively rare in the \nUSco-A OB association. This suggests that the trend against wide binaries in \ndense bound clusters might extend to unbound associations, and therefore may be \nthe result of another initial condition besides stellar density.\n\nThe high frequency of wide binary systems in USco-B also suggests that binary\nformation is not a pure function of stellar density. This population is\nkinematically associated with the Sco-Cen complex and its proper motions most\nclosely match the Upper Centaurus-Lupus OB association, but the wide binary\nfrequency for solar-type stars in USco-B is more consistent with the T\nAssociations in our sample. As we discuss further in Appendix C, this\ncould also be explained if the stars in USco-B represent an evolved low-density\nassociation analogous to the $\\rho$ Oph or Lupus complexes rather than a \nsubgroup of an OB association.\n\n\\subsubsection{The Separation Distribution of Binary Systems}\n\nThe wide binary systems discovered by our survey only represent the outer tail\nof the separation distribution function. The measurement of its functional\nform will require large high-resolution imaging surveys sensitive to the core\nof the separation distribution ($\\sim$10-100 AU for solar-type stars,\ndeclining to $\\sim$1-10 AU for brown dwarfs). The uncertainties in results\nfrom the literature do not allow for strong constraints in this separation\nrange, but our results are consistent with some of the proposed environmental\ntrends. Wide binary systems appear to be significantly less common in USco-A\nthan in USco-B, a fact which was noted by Kohler et al. (2000). Their\nhigh-resolution speckle interferometry survey found many binaries in USco-A\nwith projected separations of 20-300 AU, but most of the binaries they\ndiscovered in USco-B had significantly higher separations. This led them to\nconclude that the binary separation distribution is biased toward tighter \nsystems in USco-A than in USco-B. Numerous multiplicity surveys in Taurus and \nCham I (e.g. Ghez et al. 1993, 1997) have also found a wider separation \ndistribution than in USco-A, which is also consistent with our results.\n\nField multiplicity surveys have shown a probable mass dependence in the\nmaximum binary separation. A census of previous surveys (Reid et al. 2001)\nfound that the maximum field binary separation can be described empirically\nwith an exponential function of the total system mass,\n$a_{max}$$\\propto$$10^{3.3M}$; an extension of this study to the substellar\nregime by Burgasser et al. (2003) found a corresponding quadratic function,\n$a_{max}$$\\propto$$M^2$. Burgasser et al. demonstrated, using the formalism\nof Weinberg et al. (1987), that this is not due to interactions with field\nstars or giant molecular clouds, but instead must be a feature of the\nformation process or a result of early dynamical evolution in the formation\nenvironment.\n\nThese empirical relations predict maximum separations of 330 AU and 1650 AU for \ntotal system masses of $\\sim$0.4 and 0.6 $M_{\\sun}$, respectively. This \nprediction is consistent with the general minimum primary mass of $\\sim$0.25 \n$M_{\\sun}$ that we have identified among the wide binaries in our sample. The \nimplication is that this limit is indeed set by the T Tauri stage, either as a \nresult of the formation process or during dynamical evolution while these systems \nare still embedded in their natal gas cloud.\n\nHowever, we have identified one candidate system, USco-160611.9-193532, with an \napparent low-mass primary (0.13 $M_{\\sun}$; SpT M5) and a very wide projected \nseparation (10.8\\arcsec; 1550 AU). The USNO-B proper motion for the secondary \n($\\mu_{\\alpha}$,$\\mu_{\\delta}$=-8,-18 mas yr$^{-1}$) suggests that it is a \ngenuine USco member and not a background star. As we will report in a future \npublication, subsequent observations with Laser Guide Star Adaptive Optics on the \nKeck-II telescope also find that the primary is itself a close ($\\sim$0.1\\arcsec) \nequal-flux pair. If the wide visual companion is gravitationally bound, then this \ntriple system ($M_{tot}\\sim$0.4 $M_{\\sun}$) does not follow the empirical \nmass-maximum separation relations. There are several other candidate wide binary \nsystems which could potentially violate these relations, but the probability of \nbackground star contamination is high enough in these cases that association \nmembership should be confirmed via spectroscopy before any conclusions are drawn.\n\nFinally, a census of several star-forming regions by Simon (1997) found that\npre-main sequence stars tend to cluster on two length scales, with two-point\ncorrelation functions described by separate power laws. He concluded that\nsmall-scale clustering is a result of binary formation, while clustering on\nlarger scales is a result of the condensation of multiple cores from single\nmolecular clouds. This could potentially explain the excess of wide companions\nin Taurus, where the stars are younger and have not dispersed as far from their\nformation point. However, Simon found that the transition occurred at\nseparations of $\\sim$$10^4$ AU in Taurus, and our survey truncates at $\\sim$1500\nAU. This suggests that unless his initial estimate was significantly higher than\nthe true transition point, all of our candidate companions fall within the\nbinary regime. \n\n\\subsubsection{The Mass Ratio Distribution of Binary Systems}\n\nField multiplicity surveys have found that the mass ratio distribution\nvaries significantly with primary star mass. Most solar-mass primaries\npossess binary companions with low mass ratios (Duquennoy \\& Mayor 1991),\nearly M dwarf primaries possess companions with a uniform mass ratio\ndistribution (Fischer \\& Marcy 1992), and late-M dwarf and brown dwarf\nprimaries possess companions with mass ratios near unity (e.g. Close et al.\n2003; Siegler et al. 2005).\n\nOur results can not support any strong statistical claims, but they are largely\nconsistent with this pattern. The only exception is that our results for the\nhighest-mass stars (1.16-2.50 $M_{\\sun}$) suggest the presence of a possible\nexcess of wide similar-mass binaries. The excess is not consistent with\nbackground star contamination since the primaries are all very bright and most\nbackground stars should be significantly fainter; many of these binary\ncompanions have been confirmed independently as association members. It is also\nunlikely that we missed a significant number of companions with $0.250.25$) is a function of\nboth mass and environment, with significantly higher frequencies among high-mass\nstars than lower-mass stars and in the T associations than in the OB\nassociation. We discuss the implications for wide binary formation and conclude\nthat the environmental dependence is not a direct result of stellar density or\ntotal association mass, but instead might depend on another parameter like the\ngas temperature of the formation environment.\n\nWe also analyze the mass ratio distribution as a function\nof mass and find that it largely agrees with the\ndistribution seen for field stars. There appears to be a moderate excess\nof similar-mass ($q>0.5$) wide binaries among the highest-mass (1.16-2.50\n$M_{\\sun}$) stars in our sample, but the number statistics do not support\nany other strong conclusions. The binary populations in these associations\ngenerally follow the empirical mass-maximum separation relation observed\nfor field binaries, but we have found one candidate low-mass system\n(USco-160611.9-193532; $M_{tot}$$\\sim$0.4 $M_{\\sun}$) which has a projected\nseparation (10.8\\arcsec; 1550 AU) much larger than the limit for its mass.\n\nFinally, we find that the binary frequency in the USco-B subgroup is\nsignificantly higher than the USco-A subgroup and is consistent with the measured\nvalues in Taurus and Cham I. This discrepancy, the absence of high-mass stars in\nUSco-B, and its marginally distinct kinematics suggest that it might not be\ndirectly associated with either USco-A or Upper Centaurus-Lupus, but instead\nrepresent an older analogue of the $\\rho$ Oph or Lupus associations.\n\n\n\\acknowledgements\n\nThe authors thank R. White and C. Slesnick for helpful feedback on the manuscript \nand on various ideas presented within.\n\nThis work makes use of data products from the Two Micron All-Sky Survey, which is a\njoint project of the University of Massachusetts and the Infrared Processing and\nAnalysis Center\/California Institute of Technology, funded by the National\nAeronautics and Space Administration and the National Science Foundation. This\nresearch has also made extensive use of the SIMBAD database, operated at CDS,\nStrasbourg, France, and of the USNOFS Image and Catalogue Archive operated by \nthe United States naval Observatory, Flagstaff Station \n(http:\/\/www.nofs.navy.mil\/data\/fchpix\/).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}