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+{"text":"\\section{Introduction}\nIn recent years, significant advances have been made in learning representations of graph-structured data and predicting quantities of interest for nodes, edges or graphs themselves~\\cite{gcn,gat}. This new subfield has attracted an increasing amount of interest, leading to the development of numerous methods~\\cite{2019survey}. However, several earlier works have noted issues with existing standard benchmarks, which make it difficult to rigorously compare results and accurately distinguish between the performance of competing architectures~\\cite{oleks2018pitfalls,appnp}.\n\nOur primary focus is semi-supervised node classification: given labels of a small subset of nodes (typically 1--5\\%) and features of all nodes, as well as their connectivity information, the task is to predict all other labels. This setup is often used to assess the performance of various Graph Neural Network (GNN) architectures~\\cite{gcn,gat}. These methods are usually evaluated on three citation network datasets (Cora, CiteSeer, PubMed) introduced by Yang et al~\\yrcite{planetoid}. Unfortunately, different training splits are used across studies, which makes comparisons challenging, especially since the performance on this task is very sensitive to the choice of training split~\\cite{oleks2018pitfalls}. Furthermore, all three benchmarks are derived from the same domain, with similar structural properties. In contrast, \\textsc{Wiki-CS} has a much higher connectivity rate, hence it provides a different kind of distribution for these methods to be tested on.\n\nSimilar datasets have also been used in single-relation link prediction~\\cite{vgae,graphstar}. We further use \\textsc{Wiki-CS} to benchmark relational methods for this task, along with a non-structural SVM baseline. \n\n\\section{Related Work}\nThe most commonly used semi-supervised node classification benchmarks are the previously-described citation network graphs, proposed by Yang et al.~\\yrcite{planetoid}. Larger datasets have also been used, such as Reddit and PPI~\\cite{graphsage}. However, due to the standard split sizes proposed for these benchmarks, state-of-the-art methods have already achieved F1 scores of 0.995 and 0.97 respectively~\\cite{graphsaint}, making it difficult for further improvements to be properly gauged.\n\nDue to the issues with existing datasets, there has been significant concurrent work on establishing robust GNN benchmarks:\n\\begin{itemize}\n    \\item The Open Graph Benchmark~\\cite{open-graph-benchmark} (OGB) has recently developed a range of datasets, focusing on diversity of domains, graph sizes and types of tasks and unified evaluation methods. A Wikidata Knowledge Graph is included for a link prediction task---note that this source material is entirely different from the article hyperlink graph used for \\textsc{Wiki-CS}. OGB also proposes challenging domain-specific splits based on some aspect of the data (for exaxmple, time or molecular structure), instead of selecting this randomly.\n    \\item Dwivedi et al.~\\yrcite{dwivedi2020benchmarking} similarly proposed several datasets to rigorously distinguish the aspects of GNN architectures that significantly contribute to good performance on challenging benchmarks. To achieve this, they used largely synthetic graphs.\n\\end{itemize}  \nOur contribution complements the existing ones by providing a dataset and experimental results based on a new domain. We thus further establish the generality of GNN methods and extend the range of available benchmarks.\n\n\\section{The Dataset}\n\n\\subsection{Article Selection and Label Generation}\nWe processed Wikipedia datadumps from August 2019 to extract a subgraph where accurate class labels could be provided, based on the categories of each article. Unfortunately, these category tags were not suitable for direct use as class labels, as most of them are highly specific and inconsistently applied to a small number of pages---there were around 1.5 million different categories defined on the 6 million pages, at the time of the snapshot that we used.\n\nThis problem was mitigated by using the category sanitizer tool made available by Boldi \\& Monti~\\yrcite{category-sanitizer}, with some modifications. Their method relies on the subcategory relation to aggregate articles belonging to subcategories to their parent categories. A small set of prominent categories is selected based on harmonic centrality measures in the subcategory graph; other nodes in the subcategory graph are aggregated to one of their nearest ancestors (see Figure \\ref{fig:cs-categories} for an example subgraph). See Boldi \\& Monti~\\yrcite{category-sanitizer} for the details of the process. This avoids aggregation through many indirect steps, which would often lead to articles being mapped to categories which they have little semantic overlap with.\n\nHowever, the output still required further clean-up: some aggregated categories still contained unrelated articles. Additionally, if the subcategory graph related to some topic is very dense, the selected prominent categories and the aggregation choices can be very arbitrary.\n\n\\textsc{Wiki-CS} was created by inspecting the list of $10,000$ prominent categories selected by the sanitizer and picking a subject area with few such issues. We identified three possible candidate subjects (branches of biology, US states, branches of CS), and sampled 20 pages from every class of these candidates. Although all of these had some issues, we were able to clean up the CS data by dropping some categories and manually disabling aggregation across specific subcategories to prune bad pages from others. This resulted in a dataset with 10 classes corresponding to branches of computer science, with very high connectivity. See Appendix \\ref{app:categories} for the set of prominent categories we used for each label. Finally, we dropped the articles that would have been mapped to multiple classes.\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\centering\n\n\\includegraphics[width=0.9\\columnwidth]{cs-cat.pdf} \n\n\\caption{A subgraph of the subcategory relation graph. Nodes with dark borders are the prominent categories chosen based on centrality. The others were aggregated to the nearest marked ancestor as denoted by their colors, with ties broken arbitrarily.}\n\n\\label{fig:cs-categories}\n\\vskip -0.2in\n\\end{figure}\n\n\\subsection{Node Features}\n\nSimilarly to previous work~\\cite{planetoid}, our node features were derived from the text of the corresponding articles. However, they were calculated as the average of pre-trained GloVe word embeddings~\\cite{glove} instead of using binary bag-of-words vectors. This allowed us to encode rich features corresponding to a large vocabulary in relatively small 300-dimensional input vectors, which can be an advantage for training large models on a GPU.\n\n\\subsection{Training Splits}\n\\label{subsec:training-splits}\n\nIt has been shown that the choice of the training split can seriously affect model performance for semi-supervised node classification~\\cite{oleks2018pitfalls}. Therefore, using multiple training splits can improve the robustness of a benchmark~\\cite{appnp}. For this reason, we randomly selected 20 different training splits from the data that was not used for testing.\n\nMore specifically, we split the nodes in each class into two sets, 50\\% for the test set and 50\\% potentially visible. From the visible set, we generated 20 different splits of training, validation and early-stopping sets: 5\\% of the nodes in each class were used for training in each split, 22.5\\% were used to evaluate the early-stopping criterion, and 22.5\\% were used as the validation set for hyperparameter tuning. We stored the resulting mask vectors with the rest of the dataset, so that they can be used consistently across all future work.\n\n\\subsection{Statistics and Structural Properties}\n\n\\begin{table}[t]\n\\caption{Comparison of key dataset statistics between \\textsc{Wiki-CS} and standard citation network benchmarks. SP stands for shortest path length.}\n\\label{tab:dataset-statistics}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{l|ccc|c}\n\\toprule\n                                & \\bf{Cora} & \\bf{CiteSeer} & \\bf{PubMed} & \\bf{Wiki-CS}\\\\\n\\midrule\n         \\bf{Classes}               &   7   &       6       &       3     &     10   \\\\\n         \\bf{Nodes}                 &  2708 &       3327    &   19717     &  11701   \\\\\n         \\bf{Edges}                 &  5429 &       4732    &   44338     & 216123   \\\\\n         \\bf{Features dim.}         &  1433 &       3703    &     500     &    300   \\\\\n         \\bf{Label rate}            & 3.6\\% &       5.2\\%   &   0.3\\%     &    5\\%   \\\\\n         \\bf{Mean degree}           &  4.00 &       2.84    &   4.50      &  36.94   \\\\\n         \\bf{\\shortstack{Average SP}}  & 6.31  &       9.32    &       6.34  &    3.01    \\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nTable \\ref{tab:dataset-statistics} summarises the key statistics of the citation network and the \\textsc{Wiki-CS} datasets. Note the significantly higher rate of connectivity compared to existing benchmarks and the short average distance between any two nodes. This suggests that progress could be made on the benchmark by designing more involved computations within the neighborhood of a node, rather than focusing on long-range connections. This makes \\textsc{Wiki-CS} a useful and complementary addition to existing node classification datasets.\n\nThis connectivity also leads to more varied node neighborhoods: for each node, we calculated the proportion of neighbors that belong to the same class as the node itself, and plotted this distribution for \\textsc{Wiki-CS} as well as the existing citation network benchmarks. The results shown in Figure \\ref{fig:same-class-neighbors} show that the existing datasets have a large share of nodes in homogeneous neighborhoods, while \\textsc{Wiki-CS} is significantly more varied.\n\n\\begin{figure}\n\\vskip 0.2in\n\\centering\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/cora.png} \n\\caption{Cora}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/citeseer.png} \n\\caption{CiteSeer}\n\\end{subfigure}\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/pubmed.png} \n\\caption{PubMed}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{same-class-neighbors\/wiki.png} \n\\caption{Wiki-CS}\n\\label{fig:same-class-neighbors}\n\\end{subfigure}\n\n\\caption{Distribution of the ratio of neighbors belonging to the same class. In all three the citation network datasets, almost two-thirds of all nodes have all neighbors belonging to the same class. The distribution of \\textsc{Wiki-CS} is considerably more balanced.}\n\\label{fig:same-class-neighbors}\n\\end{figure}\n\n\nWe also visualized the structure of all four datasets using Deep Graph Mapper~\\cite{dgm}, an unsupervised GNN-based visualisation technique. The results shown in Figure \\ref{fig:dgm-vis} suggest that \\textsc{Wiki-CS} might have a more centralized, hierarchical structure than the citation networks, which seems plausible considering the different source domains.\n\\begin{figure}\n\\vskip 0.2in\n\\centering\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/cora.png} \n\\caption{Cora}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/citeseer.png} \n\\caption{CiteSeer}\n\\end{subfigure}\n\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/pubmed.png} \n\\caption{PubMed}\n\\end{subfigure}\n\\begin{subfigure}{0.49\\columnwidth}\n\\includegraphics[width=\\linewidth]{dgm\/dataset-vis\/wiki.png} \n\\caption{Wiki-CS}\n\\label{fig:dgm-vis-wiki-cs}\n\\end{subfigure}\n\n\\caption{Deep Graph Mapper (DGM) visualisation of benchmarks. Each node in the figure corresponds to a cluster of similar nodes in the original graph, with edge thickness representing the amount of connections between clusters. Colors represent the most frequent class in each cluster. The DGM unsupervised embedding process did not take labels into account, only relying on the node features and edges. The hyperparameters are described in Appendix \\ref{app:hyperparameters}.}\n\\label{fig:dgm-vis}\n\\vskip -0.2in\n\\end{figure}\n\n\\section{Experiments}\n\n\\subsection{Semi-Supervised Node Classification}\n\nAs described in Section \\ref{subsec:training-splits}, 20 different training splits were created for the node classification task, each consisting of 5\\% of nodes from each class. The same test set (50\\% of the nodes) was evaluated for all splits. In each split, a different 22.5\\% of nodes is used for early-stopping: we finish training when the loss calculated on this set has not improved for 100 epochs, and evaluate the model snapshot that produced the lowest loss.\n\nThis evaluation was performed 5 times on each of the 20 splits; we report the mean accuracy with a 95\\% confidence interval based on bootstrap resampling from these results with 1,000 samples.\n\nThree GNN models were evaluated: GCN~\\cite{gcn}, GAT~\\cite{gat} and APPNP~\\cite{appnp}. Hyperparameter tuning was performed using the same training setup and measuring validation performance on 22.5\\% of the nodes disjoint from the training and early-stopping sets. For efficiency, only the first 10 (out of 20) splits were used for hyperparameter tuning. The model configurations are described in Appendix \\ref{app:hyperparameters}.\n\nTwo non-structural baselines were also included: a multi-layer perceptron (MLP) and a support vector machine (SVM). These predicted the class for each node individually, based on the node features. Since SVMs are deterministic, we only had a single data point from each training split and report the mean accuracy.\n\nThe results are shown in Table \\ref{tab:node-classification}. The relative model performances align well with the results on citation network benchmarks, providing evidence that these are indeed good general-purpose methods. It is perhaps surprising that the attention mechanism of GAT improved very little on the GCN result despite the large neighborhoods---one reason might be that it is difficult to learn what to attend to in the semi-supervised setting, as discussed in-depth by Knyazev et al.~\\yrcite{attention}.\n\nThe model predictions were also visualised with Deep Graph Mapper, and are included in Appendix \\ref{app:dgm-preds}. This was based on training each model once, on the first of the 20 training splits. As expected, the mistakes and disagreements are largely located near boundaries of classes. This reinforces the idea that more complex neighborhood aggregation methods might be able to improve prediction accuracy. There are also some less connected clusters that seem to produce consistent incorrect predictions under all models---this might be due to not having good training samples in their proximity.\n\n\\begin{table}[ht]\n\\caption{Performance of semi-supervised node classification methods on the \\textsc{Wiki-CS} dataset. Accuracies are represented as the average over 100 runs, with 95\\% confidence intervals calculated by bootstrapping.}\n\\label{tab:node-classification}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c}\n\\toprule\n          & \\textbf{Accuracy} \\\\\n\\midrule\n         \\textbf{SVM} & 72.63\\%\\\\\n         \\textbf{MLP} & 73.17 $\\pm$ 0.19\\%\\\\\n\\midrule\n         \\textbf{GCN} & 79.07 $\\pm$ 0.10\\%\\\\\n         \\textbf{GAT} & 79.63 $\\pm$ 0.10\\%\\\\\n         \\textbf{APPNP}&79.84 $\\pm$ 0.10\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\subsection{Link Prediction}\n\nFor the link prediction benchmark, we followed the experimental setup of studies performing single-relation link prediction on the Cora, CiteSeer and PubMed datasets~\\cite{vgae, graphstar}. We split the data as follows: 85\\% of the real edges for training, 5\\% for validation and 10\\% for testing. For each group, the same number of negative examples (non-edge node pairs) was sampled uniformly at random. \n\nTwo GNN methods were benchmarked for link prediction: GraphStar~\\cite{graphstar} and VGAE~\\cite{vgae}. They were trained using the configurations reported in the original works, except for the hidden layer size of GraphStar: a maximum size of 256 would fit on the GPU. Details are included in Appendix \\ref{app:hyperparameters}. An MLP baseline was also trained using concatenated pairs of node feature vectors. \n\nThe results are shown in Table \\ref{tab:link-prediction}. Note the extremely high performance of all models, even the MLP baseline. It appears that randomly selected false edges are very easy to distinguish from true edges in this dataset, and harder negative samples would be needed for more meaningful benchmarking. The large number of edges aggravates this, but it is not the main cause: we performed an experiment where we trained the models on just $10000$ examples of each class, and found the metrics to be still comfortably above $0.9$. See Table \\ref{tab:lp-10k} for the results.we\n\n\\begin{table}[t]\n\\caption{Performance of link prediction methods on the \\textsc{Wiki-CS} dataset. Metrics are represented as the average over 50 runs of VGAE, 20 runs of the MLP and 10 runs of GraphStar, with 95\\% confidence intervals calculated by bootstrapping.}\n\\label{tab:link-prediction}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c c}\n\\toprule\n          & \\textbf{ROC-AUC} & \\textbf{AP}  \\\\\n\\midrule\n         \\textbf{MLP} & $0.9785 \\pm 0.0001$ & $0.9761 \\pm 0.0002$ \\\\\n\\midrule\n         \\textbf{VGAE} & $0.9553 \\pm 0.0008$ & $0.9608 \\pm 0.0007$ \\\\\n         \\textbf{GraphStar} & $0.9793 \\pm 0.0002$ & $0.9896 \\pm 0.0001$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\begin{table}[t]\n\\caption{Performance of link prediction methods trained on only $10,000$ examples of each class.}\n\\label{tab:lp-10k}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n    \\begin{tabular}{c|c c}\n\\toprule\n          & \\textbf{ROC-AUC} & \\textbf{AP}  \\\\\n\\midrule\n         \\textbf{MLP} & $0.9192 \\pm 0.0004$ & $0.9119 \\pm 0.0006$ \\\\\n\\midrule\n         \\textbf{VGAE} & $0.8546 \\pm 0.0024$ & $0.8427 \\pm 0.0032$ \\\\\n         \\textbf{GraphStar} & $0.9577 \\pm 0.0006$ & $0.9795 \\pm 0.0003$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\n\\section{Conclusion}\n\nWe have presented \\textsc{Wiki-CS}, a new benchmark for GNN methods. We have described how its structural properties are significantly different from commonly used datasets. Our experiments show existing GNN architectures for semi-supervised node classification and link prediction performing similarly to their results on other benchmarks, which is further evidence that they are good general-purpose methods for graph-learning tasks. Our dataset is available for further study, broadening the range of available benchmarks.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nSpacetime singularities are among the most spectacular predictions of classical general relativity.\nIn the context of cosmology, the presence of an initial singularity signals a fundamental incompleteness in our understanding of the Universe, whose origin and beginning can not be explained within a classical or semiclassical treatment.\nThe singularity theorems of general relativity require energy conditions which may be violated in the early Universe if inflation was present, but one can nevertheless show that inflationary spacetimes are past incomplete \\cite{BGV03}.\n\\footnote{See e.g.\\ \\cite{YQ18} for an analysis of conditions under which continuous or differentiable extensions of the spacetime metric beyond the past incomplete region may exist for such inflationary spacetimes.}\nIt is widely expected that a quantum theory of gravity is required to resolve the classical Big Bang singularity and give a fundamental basis to theoretical cosmology; a full quantum treatment of spacetime may indeed be needed to justify the assumption that the early Universe can be studied in terms of quantum fields on a classical background \\cite{FLT17,TFLT19}.\n\nOn the other hand, the spacetime geometry of the early Universe was presumably very simple, describable in terms of a homogeneous isotropic background with only small perturbations.\nThis simplicity must ultimately be explained more fundamentally, but it also results in significant practical simplifications: at least as a first step, one can study simple homogeneous, isotropic spacetimes in quantum gravity to learn about singularity resolution.\nOne could hope that, as in the classical theory, this does not require understanding the nonlinear and presumably complicated dynamics of full quantum gravity.\nThis philosophy, systematically applied to \\ac{lqg} as a candidate theory of quantum gravity, gave birth to the field of \\ac{lqc} where, for models of quantum gravity coupled to a massless scalar field, the classical Big Bang singularity is indeed resolved and replaced by a `big bounce' \\cite{Boj01,AS11}.\nMore recently \\ac{lqc} has made direct contact with inflation, providing a quantum-gravitational extension of the usual semiclassical framework \\cite{AAN12}.\nThe precise relation between \\ac{lqg} and \\ac{lqc} has been the focus of much work in recent years \\cite{AC15,ADLP18,DLP19}.\n\nThe origin of singularity resolution in \\ac{lqc} is the fundamental discreteness of the theory, manifest in discrete spectra for areas and volumes with a gap away from zero \\cite{Rov04,BH11}.\nThis key feature of the \\ac{lqg} kinematics is shared by its reformulation in terms of \\ac{gft} \\cite{Ori17}.\n\\ac{gft} interprets the {\\em quanta of spacetime} seen in \\ac{lqg} as excitations of a quantum field (also then a quantum field {\\em of}, not {\\em on} spacetime), while the dynamics of a \\ac{gft} are generally defined such that its perturbative expansion corresponds to a sum over {\\em spin foams}, or discrete spacetime histories of \\ac{lqg} states \\cite{RR01}.\nThe continuum limit of this sum, needed to obtain continuum  quantum geometry, is to be taken in a way similar to matrix and tensor models \\cite{Ori12}.\nGiven the very close relation of \\ac{gft} to \\ac{lqg}, it is natural to ask whether cosmological models of \\ac{gft} dynamics can resolve the classical Big Bang singularity in a way similar to what is seen in \\ac{lqc}.\n\nThe key idea that led to the derivation of cosmological models in the \\ac{gft} approach was that a spatially homogeneous quantum geometry could be understood as a type of Bose--Einstein condensate in \\ac{gft} \\cite{GOS14,GS16,Ori17a,PS19}.\nAs in other quantum field theories, such a condensate can be understood as a nonperturbative vacuum of the theory, characterised by a common quantum state for a very large number of quanta with respect to the original Fock vacuum.\nThe idea that spacetime could be a kind of Bose--Einstein condensate of geometric quanta had been formulated in other approaches before \\cite{Hu05,KS12}, but the quantum field theory framework of \\ac{gft} allows studying such a condensate with relatively conventional methods, adapted to a background-independent quantum gravity context.\nIn the simplest (mean-field) approximation, the equation of motion of the condensate mean field is the analogue of the usual Gross--Pitaevskii equation in condensed matter physics.\nFrom a solution to this equation of motion, one can compute geometric observables such as the total volume of the condensate.\nDynamics are introduced, just as in \\ac{lqc} and many other models of quantum cosmology \\cite{BI75}, by coupling a relational matter clock, given by a free massless scalar field.\nConcretely, one extracts equations for the relational volume observable $V(\\phi)$, the three-dimensional volume of a condensate given a particular value of the relational clock field, and its derivatives.\nThese are then interpreted as effective Friedmann equations derived from the \\ac{gft} condensate dynamics.\nThese steps were first fully implemented in \\cite{OSW16,OSW17} where it was shown how such effective Friedmann equations, for a wide class of \\ac{gft} models and under various simplifying assumptions, are consistent with the classical Friedmann equations at large volume while showing a bouncing behaviour at high densities very similar to the one in \\ac{lqc}.\nIn particular, these effective Friedmann equations can reproduce the preferred `improved dynamics' form of \\ac{lqc} \\cite{APS06} whose derivation from Hamiltonian formulations of \\ac{lqg} is a largely outstanding challenge \\cite{DLP19}.\n\nThese very promising results for effective cosmological dynamics from \\ac{gft} relied on assuming the emergence of a condensate regime in which the mean-field approximation is valid and \\ac{gft} interactions are subdominant with respect to the quadratic (kinetic) term.\nIncluding interactions leads to interesting modifications to effective cosmological equations, which can serve as a starting point for \\ac{gft} phenomenology \\cite{CPS16,PS17}.\nTo better understand the dependence of \\ac{gft} cosmology on a mean-field approximation, a simple toy model was then studied in \\cite{AGW18}.\nIn this model, only a single mode of a \\ac{gft} field is excited and a {\\em squeezing} Hamiltonian generates evolution in relational matter time $\\phi$, so that a squeezed state emerges dynamically even from the Fock vacuum without assuming a mean-field regime.\nThe general features of a bounce at high densities and agreement with classical cosmology at large volume could be reproduced in this simpler setting.\nThe choice of Hamiltonian was rather ad-hoc, motivated by agreement with classical cosmology and the properties of squeezed states.\nIt was then shown in \\cite{WEw19} that a Legendre transformation of the free \\ac{gft} action, taking again the matter clock $\\phi$ to define time, leads essentially to a squeezing Hamiltonian, the latter representing the dynamics of an `upside-down' harmonic oscillator with negative quadratic potential.\nThe results of \\cite{WEw19} hence explained the agreement between the effective cosmological dynamics of a squeezing Hamiltonian and those of previous results for \\ac{gft} condensates.\nA different argument explaining the close connection of \\ac{gft} cosmology to \\ac{lqc} was given in \\cite{BBC19} where it was argued that the canonical \\ac{lqc} framework and the field-theoretic (bosonic) \\ac{gft} cosmology can be seen as different realisations of the same $\\liealg{su}(1,1)$ algebra.\n\nThe aim of this paper is to extend many of these recent results to further clarify the connection between fundamental \\ac{gft} dynamics and effective cosmological equations.\nRather than using mean-field approximations, we derive general dynamical equations for operators and therefore expectation values of these operators; we work similarly to the `toy model' analysis of \\cite{AGW18} but consider a more general form of the \\ac{gft} dynamics.\nIn particular, we will go beyond quadratic Hamiltonians and add simple interactions, allowing us to connect to results such as those of \\cite{CPS16} for \\ac{gft} interactions which were obtained in a mean-field approximation.\nWe will also extend the algebraic viewpoint of \\cite{BBC19} to the case of an interacting Hamiltonian.\nAn issue in the derivation of effective dynamical equations that we will encounter is that, in the general case, these equations involve additional expectation values or higher moments that are not directly identifiable with the variables of a classical \\ac{flrw} cosmology (which is fully characterised by the scale factor or volume and the energy density in the massless scalar field).\nIn other words, dynamical equations for quantum expectation values require knowledge of additional initial conditions compared to a classical cosmology.\nThis is of course the usual situation for effective equations for quantum systems.\n(See e.g.~\\cite{BHT11} for a systematic discussion.)\nChoosing a specific class of initial conditions, for instance by focussing on a class of coherent states, then simplifies these equations.\nWe will illustrate this trade-off between obtaining effective equations that are as general as possible but include a dependence on additional variables, and more specific equations in which a particular choice of state (or family of states) allows deriving equations with more direct (semiclassical) physical interpretation.\nAs concrete examples, we will discuss Fock coherent states which have been studied in most of the existing literature on \\ac{gft} cosmology, but also coherent states based on the $\\liealg{su}(1,1)$ algebra satisfied by basic \\ac{gft} operators (in particular the well-known \\ac{pg} coherent states).\n\nWe show a key property of simple Fock coherent states, which is that relative uncertainties of quantities like volume and energy can be made arbitrarily small at late times, so that these states are as semiclassical as desired.\nThis gives further justification to their interpretation as semiclassical macroscopic geometries \\cite{GOS14}.\n\\acl{pg} states that can be thought of as elements of a \\ac{gft}-like Fock space do not admit such a semiclassical interpretation and are therefore disfavoured for \\ac{gft} cosmology.\n \n\\section{Group field theory cosmology}\n\\label{sec:gft_cosmo}\nIn this section we summarise previous work on the derivation of effective cosmological equations from group field theory, in particular the previous papers we are building on in this work.\nFor more details and background we point to \\cite{GOS14,GS16,Ori17a,PS19}.\n\n\\subsection{The group field theory approach to quantum gravity}\n\\Acfp{gft} are a nonperturbative approach to quantum gravity, aiming to extend the successes of matrix and tensor models to a theory of quantum geometry in higher (in particular four) dimensions by incorporating the kinematical and dynamical structure of loop quantum gravity and spin foams \\cite{Ori17,Ori12,Ori06,Fre05,Kra11}.\n\nConcretely, rather than being based on a matrix or tensor with a number of discrete indices of finite range, a \\ac{gft} is defined in terms of a field $\\varphi$ depending on a number of continuous variables taking values in a Lie group.\nIn this sense, the purely combinatorial structure of matrix models is enriched by additional group-valued degrees of freedom, interpreted as parallel transports akin to the fundamental variables in lattice gauge theory.\nNevertheless, the main ideas are similar; a \\ac{gft} perturbative expansion should generate a sum over quantum geometries and admit a consistent continuum limit.\n\nThe prototype for \\ac{gft} as an approach to quantum gravity is the {\\em Boulatov model} in three dimensions \\cite{Bou92}.\nOne defines a real field\n\\begin{equation}\n  \\varphi:\n  G^3 \\rightarrow \\field{R}\n  \\mathcomma\n  \\quad\n  \\varphi(g_1,g_2,g_3)\n  = \\varphi(g_2,g_3,g_1)\n  = \\varphi(g_3,g_1,g_2)\n\\end{equation}\nwith an action\n\\begin{equation}\n\\eqalign{\n  S[\\varphi]\n  =\n&\n  \\frac{1}{2}\\int\n  \\intmeasure[3]{g} \\,\n  \\varphi(g_1,g_2,g_3)\n  \\varphi(g_1,g_2,g_3)\n\\\\\n&\n  -\\frac{\\lambda}{4!}\\int\n  \\intmeasure[6]{g} \\,\n  \\varphi(g_1,g_2,g_3)\\varphi(g_1,g_4,g_5)\\varphi(g_2,g_5,g_6)\\varphi(g_3,g_6,g_4)\n  \\mathcomma\n}\n\\end{equation}\nwhere $G$ is a Lie group and $\\intmeasure{g}$ is the Haar measure on $G$.\nNotice the `nonlocal' combinatorial pairing of field arguments in the interaction term which is the generalisation of trace invariants such as ${\\rm tr}M^n$ in the case of a matrix model.\nOne can then show that, for $G=\\liegroup{SU}(2)$, the \\ac{gft} partition function admits a perturbative expansion of the form\n\\begin{equation}\n  \\label{eq:gft_boulatov_exp}\n  \\int \\mathcal{D}\\varphi\\;e^{-S[\\varphi]}=\\sum_{C}\n  \\lambda^{N_{T}(C)}\\sum_{\\{j_l\\}\\in{\\rm Irrep}}\\prod_{l\\in C} (2j_l+1)\\sum_{T\\in\n  C}\\left\\{ \\matrix{\n    j_{T_1} & j_{T_2} & j_{T_3}\n    \\cr\n    j_{T_4} & j_{T_5} & j_{T_6}\n  }\\right\\}\n\\end{equation}\nwhere the first sum is over all oriented three-dimensional simplicial complexes $C$, $N_T(C)$ is the number of tetrahedra in $C$, $j_l$ is an irreducible representation of ${\\liegroup{SU}(2)}$ assigned to each link $l\\in C$, and $\\{\\cdot\\}$ is the Wigner $6j$-symbol associated to a tetrahedron $T\\in C$ (involving its six links).\nUp to the factor $\\lambda^{N_{T}(C)}$, each complex $C$ appearing in \\eref{eq:gft_boulatov_exp} is weighted by its {\\em Ponzano--Regge state sum} \\cite{PR68}, a possible definition of discrete three-dimensional quantum gravity (see e.g.~\\cite{BN09}).\nIn this sense, the perturbative expansion of the Boulatov model generates all possible triangulations (including all topologies) each weighted by a partition function for quantum gravity on this triangulation.\nThis expansion is highly divergent without further regularisation \\cite{BN09}.\nThe \\ac{gft} programme aims to extend \\eref{eq:gft_boulatov_exp} to more complicated models, in particular candidates for quantum gravity in four dimensions, where the Ponzano--Regge state sum is replaced by a general {\\em spin foam amplitude} \\cite{Rov04}: the amplitudes of the Barrett--Crane model \\cite{BC98} can be obtained as Feynman amplitudes of a \\ac{gft} defined on the three-sphere $S^3=\\liegroup{SO}(4)\/\\liegroup{SO}(3)$ \\cite{PFKR00} and this extends to a one-to-one correspondence between general spin foam amplitudes and their realisations as the perturbative expansion of a \\ac{gft} \\cite{RR01}.\nThis correspondence extends from spin foam models for Euclidean quantum gravity to models with Lorentzian signature such as \\cite{BC00} which can be defined through a \\ac{gft} on a noncompact group such as $\\liegroup{SO}(3,1)$ (see e.g.~\\cite{LO07}).\nIn this sense one could say that a \\ac{gft} defines a completion of the spin foam programme in that it not only generates spin foam amplitudes for quantum gravity on a given discretisation, but also the weights in a sum over discretisations.\n\n\\subsection{Cosmology from group field theory}\nFor general \\ac{gft} models for quantum gravity, it is difficult to make sense of a perturbative expansion of the form \\eref{eq:gft_boulatov_exp}.\nThe number of terms quickly grows out of control as the number of building blocks is increased and there is no obvious physical meaning to truncating such an expansion to the first few terms, i.e. to discretisations with very few building blocks.\n\\Eref{eq:gft_boulatov_exp} is really an expansion around a `no-space' vacuum in which no geometry is present at all.\n\nHowever, there is more to quantum field theory than a perturbative expansion\naround vanishing field value: interacting field theories often exhibit phase transitions to a {\\em condensate} characterised by a nonvanishing field expectation value.\nWith respect to the original Fock vacuum in which the field vanishes, a condensate has a very large number of quanta all characterised by a single quantum state (the `condensate wavefunction').\nThis is a quantum state of high symmetry and quantum coherence.\nThe key idea of {\\em \\ac{gft} condensates} is that such a configuration in \\ac{gft} is a candidate for a macroscopic, nearly homogeneous Universe, and hence  a starting point for effective cosmology.\nWe refer the reader to \\cite{GOS14} for details and arguments for this geometric interpretation.\n\nWe generally define a \\ac{gft} field for a four-dimensional quantum gravity model coupled to scalar matter by\n\\begin{equation}\n  \\varphi:\n  G^4\\times \\field{R} \\rightarrow \\field{K}\n  \\mathcomma\n  \\quad\\varphi(g_1,\\ldots,g_4,\\phi)=\\varphi(g_1h,\\ldots,g_4h,\\phi)\\;\\forall h\\in G\n  \\mathperiod\n\\end{equation}\nwhere $G$ is the gauge group of gravity (often assumed to be $\\liegroup{SU}(2)$) and $\\field{K}$ is either the real or complex numbers.\nThe action takes the general form\n\\begin{equation}\n  S\n  =\n  \\int\n  \\intmeasure[4]{g}\\,\n  \\intmeasure{\\phi}\\,\n  \\bar\\varphi(g_I,\\phi)\\,\\mathcal{K}\\varphi(g_I,\\phi)\n  + \\mathcal{V}[\\varphi]\n\\end{equation}\nwhere for a real field $\\bar\\varphi=\\varphi$ (and, to obtain the usual normalisation of a kinetic term, one has to also rescale the field), $\\mathcal{K}$ is a kinetic operator which in general contains derivatives with respect to all arguments, and all terms that  are higher order than quadratic are part of $\\mathcal{V}[\\varphi]$.\nIn general, $\\mathcal{V}[\\varphi]$ is also nonlocal in a way similar to the Boulatov \\ac{gft} defined above.\nIn its Feynman expansion, such a field theory will generate graphs whose edges are labelled by $g_I$ (interpreted as parallel transports of a $G$-connection) and whose vertices are labelled by $\\phi$ interpreted as the values of a matter scalar field.\n\nIf we denote the expectation value of the field operator by\n\\begin{equation}\n  \\expval{\\op{\\varphi}(g_I, \\phi)}\n  =\n  \\sigma(g_I,\\phi)\n  \\mathcomma\n\\end{equation}\na condensate phase is then characterised by a nonvanishing $\\sigma(g_I,\\phi)$.\n\nThe {\\em mean-field approximation} which is used in most work on \\ac{gft} cosmology so far requires that the mean field $\\sigma(g_I,\\phi)$, for a \\ac{gft} with complex field, satisfies the classical \\ac{gft} equation of motion\n\\begin{equation}\n  \\label{eq:gft_cosmo_meanfield_eom}\n  \\mathcal{K}\\sigma(g_I,\\phi)\n  +\n  \\frac{\n    \\delta\\mathcal{V}[\\sigma]\n  }{\n    \\delta\\bar\\sigma(g_I,\\phi)\n  }\n  =\n  0\n  \\mathcomma\n\\end{equation}\nthe analogue of the Gross--Pitaevskii equation for the condensate wavefunction in condensed matter physics.\nFrom a solution $\\sigma(g_I,\\phi)$ to the equation of motion, one can then extract an observable corresponding to the total condensate volume as a function of the matter field `clock',\n\\begin{equation}\n  \\label{eq:gft_cosmo_gft_volumeop}\n  \\expval{\\op{V}(\\phi)}\n  \\equiv\n  \\int\n  \\intmeasure[4]{g}\\,\n  \\intmeasure[4]{g'}\\,\n  V(g_I,g'_I)\n  \\bar\\sigma(g_I,\\phi)\n  \\sigma(g'_I,\\phi)\n  \\mathcomma\n\\end{equation}\nwhere $V(g_I,g'_I)$ are matrix elements of the \\ac{gft} volume operator between `single-particle' states $\\ket{g_I}$ and $\\ket{g'_I}$; such an operator can be defined from the action of a volume operator in \\ac{lqg} on open spin networks with a single vertex and four links.\nDynamical equations satisfied by $\\expval{\\op{V}(\\phi)}$ and its derivatives with respect to $\\phi$ are then interpreted as effective cosmological (Friedmann) equations for the three-volume of (a patch of) the Universe, derived directly from a prescription for the microscopic dynamics of a \\ac{gft}.\n\nThe most concrete derivation of this type, for models of quantum gravity coupled to massless scalar matter, was given in \\cite{OSW16}.\nFirst the nonlinear, nonlocal equation of motion \\eref{eq:gft_cosmo_meanfield_eom} was simplified by making an `isotropic' ansatz\n\\begin{equation}\n  \\sigma(g_I,\\phi)\n  =\n  \\sum_{j\\in{\\rm Irrep}}\n  \\sigma_j(\\phi){\\bf D}^j(g_I)\n  \\mathcomma\n\\end{equation}\nwhere the \\ac{gft} gauge group is taken to be $\\liegroup{SU}(2)$ and ${\\bf D}^j(g_I)$ is a fixed convolution of four Wigner $D$-matrices for the irreducible representation $j$, encoding the `shape' of the \\ac{gft} building blocks.\n(${\\bf D}^j(g_I)$ requires a choice of intertwiner $j\\otimes j\\otimes j\\otimes j\\rightarrow {\\bf 0}$; in \\cite{OSW16} this is taken to be the intertwiner with maximum eigenvalue for the volume, see \\eref{eq:gft_volume_expv}.)\n\nAssuming a quintic potential as is done for many spin foam models related to \\ac{lqg}, this reduces \\eref{eq:gft_cosmo_meanfield_eom} to a decoupled form\n\\begin{equation}\n  \\label{eq:gft_cosmo_isotropic_eom}\n  A_j\n  \\partial_\\phi^2\\sigma_j(\\phi)\n  - B_j\\sigma_j(\\phi)\n  + w_j\\bar\\sigma_j(\\phi)^4\n  =\n  0\n  \\mathcomma\n\\end{equation}\nwhere $A_j$, $B_j$ and $w_j$ are determined by the couplings in the \\ac{gft} action.\nSince the volume operator is diagonal when written in terms of $\\liegroup{SU}(2)$ representations, the volume of a condensate in such a state is given by\n\\begin{equation}\n  \\expval{\n    \\op{V}(\\phi)\n  }\n  =\n  \\sum_{j\\in{\\rm Irrep}}\n  V_j\\,|\\sigma_j(\\phi)|^2\n\\label{eq:gft_volume_expv}\n\\end{equation}\nwhere $V_j$ is the volume eigenvalue assigned to the spin $j$ (which in general depends on the intertwiner used to define ${\\bf D}^j(g_I)$).\nIn a regime in which interactions can be neglected, and assuming that the ratios $B_j\/A_j$ take a positive maximum for some $j=j_0$, it is easy to see that for almost any solution to \\eref{eq:gft_cosmo_isotropic_eom}\\footnote{The only cases for which $V(\\phi)$ does not have the given asymptotics are solutions for which one only uses the exponentially growing or the exponentially decaying solution to \\eref{eq:gft_cosmo_isotropic_eom} for $j=j_0$.} the volume $V(\\phi) \\equiv \\expval{\\op{V}(\\phi)}$ satisfies\n\\begin{equation}\n  V(\\phi)\n  \\stackrel{\\phi\\rightarrow -\\infty}{\\sim}\n  c_1 \\exp\\left(-2\\sqrt{\\frac{B_{j_0}}{A_{j_0}}}\\phi\\right)\n  \\mathcomma\n  \\quad\n  V(\\phi)\n  \\stackrel{\\phi\\rightarrow +\\infty}{\\sim}\n  c_2 \\exp\\left(2\\sqrt{\\frac{B_{j_0}}{A_{j_0}}}\\phi\\right)\n\\end{equation}\nfor some constants $c_1$ and $c_2$ \\cite{Gie16}.\nMoreover, $V(\\phi)$ can only ever reach zero for very special initial conditions (although this case becomes generic if the \\ac{gft} field is taken to be real-valued \\cite{PS17}).\nWith the identification $\\frac{B_{j_0}}{A_{j_0}}=:3\\pi G$, this corresponds to a bounce solution interpolating between the expanding and contracting solutions to the {\\em classical} Friedmann equations for a flat \\ac{flrw} Universe filled with a massless scalar field, $V(\\phi)=V_0\\exp(\\pm\\sqrt{12 \\pi G}\\phi)$.\nSimilar conclusions apply if one considers condensates only formed by a single $j$ component, again denoted by $j_0$.\nIn the latter case, one can show that the volume satisfies an effective Friedmann equation \\cite{OSW16}\n\\begin{equation}\n  \\label{eq:gft_cosmo_friedmann}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  =\n  12\\pi G\n  \\left(\n    1-\\frac{\\rho(\\phi)}{\\rho_\\critical}\n  \\right)\n  + \\frac{4V_{j_0}E}{V(\\phi)}\n\\end{equation}\nwhere $\\rho=\\pi_\\phi^2\/(2V^2)$, with $\\pi_\\phi$ the conserved momentum conjugate to $\\phi$, is the energy density of the massless scalar field, and $\\rho_\\critical$ is a maximal (critical) energy density similar to the one found in \\ac{lqc} \\cite{Boj01,AS11} (and we have again set $\\frac{B_{j_0}}{A_{j_0}}=:3\\pi G$).\nThe last term, involving a conserved quantity $E$ (`\\ac{gft} energy'), represents a slight modification with respect to the usual \\ac{lqc} effective dynamics.\nAgain, clearly at large volumes or late times such effective dynamics reduce to the classical Friedmann equation $(V'\/V)^2=12\\pi G$.\n\nIn this article, we strengthen the foundations of these past results.\nWe aim to obtain effective cosmological dynamics from \\ac{gft} without several assumptions that were necessary to obtain \\eref{eq:gft_cosmo_friedmann}, namely: the validity of a mean-field regime in which one solves equations for the mean field; restriction of the effective equations to simple expectation values such as $\\expval{\\op{V}(\\phi)}$ without taking into account fluctuations around these expectation values; neglecting \\ac{gft} interactions by effectively setting $w_j=0$.\n\\footnote{The work of \\cite{CPS16} included \\ac{gft} interactions into the analysis, leading to additional terms on the right-hand side of \\eref{eq:gft_cosmo_friedmann}, while working in a mean-field regime.}\nIndirectly, the results outlined so far also assumed a given Fock space structure used to define a \\ac{gft} volume operator, which has not been derived from the canonical analysis of a \\ac{gft} action.\n\n\\subsection{Toy model for group field cosmology, and a Hamiltonian for \\ac{gft}}\nA first step towards deriving effective cosmological dynamics from \\ac{gft} outside of a mean-field regime was taken in \\cite{AGW18}.\nOne motivation for this work was to develop a toy model in which some of the successes of \\ac{gft} cosmology could be obtained in a simpler setting, but there was also a new technical assumption: the massless scalar field $\\phi$ was proposed as a (relational) time variable, with a Hamiltonian generating evolution with respect to this clock.\nThat is, the idea was to define a {\\em deparametrised} setting in which some degrees of freedom serve as coordinates parametrising the remaining ones, a strategy widely employed in canonical quantum gravity \\cite{BK95,DGKL10}.\\footnote{This strategy can be extended to a \\ac{gft} coupled to four massless scalar fields serving as relational coordinates for both time and space \\cite{Gie18}.}\nThis approach was different from previous work on \\ac{gft} cosmology in which the fundamental \\ac{gft} formalism treated all arguments of the field on the same footing.\nThe Hamiltonian itself was chosen so as to reproduce the correct cosmological dynamics at large volume.\n\nClassical \\ac{flrw} cosmology can be defined by a volume variable $V(\\phi)$ and conjugate momentum $p_V(\\phi)$ subject to a Hamiltonian\n\\begin{equation}\n  \\label{eq:gft_cosmo_dilat}\n  \\mathcal{H}\n  =\n  \\sqrt{12 \\pi G}\\,Vp_V\n  \\mathcomma\n\\end{equation}\ngenerating a dilatation as its time evolution, i.e.~the exponential solutions in $\\phi$ mentioned above.\nIn \\cite{AGW18} it was then observed that, for a Fock space generated by annihilation operators $\\op{A}^i$ and creation operators $\\hermconj{\\op{A}}_j$ (here $i,j$ run from 1 to 5) with algebra\n\\begin{equation}\n  \\commutator{\n    \\op{A}^i\n  }{\n    \\hermconj{\\op{A}}_j\n  }\n  =\n  \\delta^i_j\n  \\mathcomma\n\\end{equation}\na discrete analogue of the dilatation operator is given by a {\\em squeezing Hamiltonian}\n\\begin{equation}\n  \\label{eq:gft_cosmo_squeezing}\n  \\op{\\mathcal{H}}\n  =\n  \\frac{\\mathup{i}}{2}\n  \\lambda\n  \\left(\n    \\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j \\epsilon^{ij}\n    - \\op{A}^i \\op{A}^j \\epsilon_{ij}\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere $\\epsilon^{ij}$ is an appropriate symmetric tensor.\nIndeed, for a volume operator taken to be the multiple of the number operator\n\\begin{equation}\n  \\label{eq:gft_cosmo_volume_via_number_op}\n  \\op{V}\n  =\n  v_0 \\op{N}\n  :=\n  v_0 \\hermconj{\\op{A}}_i \\op{A}^i\n\\end{equation}\none can show that, for suitable states characterised by the eigenvalues of $\\op{V}$, $\\op{\\mathcal{H}}$ acts as\n\\begin{equation}\n  (\\op{\\mathcal{H} }\\Psi)(V)\n  \\stackrel{v_0\\rightarrow 0}{\\rightarrow}\n  - \\mathup{i}\\lambda\\left(V\\frac{\\partial}{\\partial V}\n  + \\frac{\\partial}{\\partial V}V\\right)\\Psi(V)\n  \\mathperiod\n\\end{equation}\nThus, with the identification $\\lambda:=\\sqrt{3\\pi G}$ the continuum limit of  squeezing \\eref{eq:gft_cosmo_squeezing} is compatible with the classical dilatation Hamiltonian \\eref{eq:gft_cosmo_dilat}.\nThe picture of a Fock space of `quanta of geometry' in which each quantum carries a given volume mimics the Fock space structure of \\ac{gft}, with the simplification that here each quantum comes with a fixed $v_0$ rather than a general state-dependent volume as in \\eref{eq:gft_cosmo_gft_volumeop}.\n\nGiven that the Hamiltonian is quadratic, expressions for the time evolution of observables of interest can be computed analytically.\nOne finds that\n\\begin{equation}\n  \\label{eq:gft_cosmo_toymodel_n}\n  \\expval{\\op{N}(\\phi)}\n  =\n  - \\frac{5}{2}\n  +\n  \\left(\n    N_0 + \\frac{5}{2}\n  \\right)\n  \\cosh(2\\lambda\\phi)\n  + Q \\sinh(2\\lambda\\phi)\n\\end{equation}\nwith\n\\begin{equation}\n  N_0\n  :=\n  \\left.\n    \\expval{\\op{N}}\n  \\right|_{\\phi=0}\n  \\mathcomma\n  \\quad\n  Q\n  :=\n  \\frac{1}{2}\n  \\left.\n    \\left(\n      \\epsilon^{ij} \\expval{\\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j}\n      + \\epsilon_{ij} \\expval{\\op{A}^i \\op{A}^j}\n    \\right)\n  \\right|_{\\phi=0}\n\\end{equation}\n(computed equivalently in the Schr\u00f6dinger or Heisenberg picture).\nAt late or early times $\\phi\\rightarrow\\pm\\infty$ (and with $\\lambda:=\\sqrt{3\\pi G}$), the expectation value $V(\\phi)\\equiv\\expval{\\op{V}(\\phi)}$ then again reproduces the classical solution $V(\\phi)=V_0\\exp(\\pm\\sqrt{12\\pi G}\\phi)$\\,.\nMoreover, one can show that only special initial conditions (such as choosing the Fock vacuum as initial state) lead to a solution that ever encounters a singularity where $V(\\phi_0)=0$ for some $\\phi_0$.\nGeneric solutions avoid the classical singularity and describe a bounce connecting the classical expanding and contracting branches.\n\nThe quantity $Q$ cannot be directly interpreted in terms of the volume or energy density of the corresponding classical cosmology; its presence leads to an asymmetry in the solution \\eref{eq:gft_cosmo_toymodel_n}. For simplicity, the further analysis of \\cite{AGW18} only considered the case $Q=0$, for which one obtains the effective Friedmann equation\n\\begin{equation}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  =\n  4\\lambda^2\n  \\left(\n    1\n    + \\frac{5v_0}{V(\\phi)}\n    - \\frac{N_0(N_0+5)v_0^2}{V(\\phi)^2}\n  \\right)\n  \\mathperiod\n\\end{equation}\nThe similarity to the effective Friedmann equation \\eref{eq:gft_cosmo_friedmann} of \\ac{gft} in the mean-field setting is apparent.\nIn this sense, the toy model based on a squeezing Hamiltonian \\eref{eq:gft_cosmo_squeezing} already reproduced several previous results in \\ac{gft} cosmology.\n\nOne reason for this close connection was uncovered in \\cite{WEw19,GPW19} where, taking again a deparametrised viewpoint, a Hamiltonian formalism was derived from a Legendre transformation of the full (free) \\ac{gft} action in which the `matter' argument $\\phi$ of the \\ac{gft} field is taken as a time coordinate.\nThe starting point is an action for real \\ac{gft} fields of the form\n\\begin{equation}\n  \\label{eq:gft_cosmo_gft_hamiltonian_action}\n  S\n  =\n  \\frac{1}{2}\n  \\int\n  \\intmeasure{\\phi}\n  \\sum_{\\vec{\\jmath},\\vec{m},\\iota}\n  \\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}(\\phi)\n  \\left[\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    + \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\\partial_\\phi^2\n  \\right]\n  \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\n  + \\mathcal{V}[\\varphi]\n  \\mathcomma\n\\end{equation}\nwhere the field $\\varphi(g_I,\\phi)$ has been decomposed into Peter--Weyl modes according to\n\\begin{equation}\n  \\varphi(g_I,\\phi)\n  =\n  \\sum_{j_I\\in{\\rm Irrep}}\n  \\sum_{m_I,n_I=-j_I}^{j_I}\n  \\sum_{\\iota}\\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\\,\n  \\intertwiner^{\\vec{\\jmath},\\iota}_{\\vec{n}}\n  \\prod_{I=1}^4\n  \\sqrt{2j_I+1}\\,\n  D^{j_I}_{m_I n_I}(g_I)\n\\end{equation}\nand $\\iota$ labels a basis of intertwiners $\\intertwiner$ for the representation labels $\\{j_I\\}$ (and the sum over $(\\vec{\\jmath},\\vec{m},\\iota)$ in \\eref{eq:gft_cosmo_gft_hamiltonian_action} is a shorthand for the sums appearing in this decomposition).\nFor a real field, these Peter--Weyl coefficients satisfy the reality condition\n\\begin{equation}\n  \\compconj*{\n    \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}(\\phi)\n  }\n  =\n  (-1)^{\\sum_I (j_I-m_I)}\\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}(\\phi)\n  \\mathperiod\n\\end{equation}\nThe Hamiltonian can then be written as\n\\begin{equation}\n  \\mathcal{H}\n  =\n  - \\frac{1}{2}\n  \\sum_{\\vec{\\jmath},\\vec{m},\\iota}\n  \\left[\n    \\frac{\n      \\pi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\\pi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n    }{\n      \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\n    }\n    +\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    \\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n  \\right]\n  - \\mathcal{V}[\\varphi]\n\\end{equation}\nwhose free part corresponds to either a harmonic oscillator or an `upside down' harmonic oscillator for each mode, depending on the signs of the couplings $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}$ and $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}$.\nAnnihilation and creation operators can then be defined by\n\\begin{eqnarray}\n  \\label{eq:gft_cosmo_annihilation}\n  \\op{a}_{\\vec{\\jmath},\\vec{m},\\iota}\n  & = &\n  \\frac{1}{\n    \\sqrt{\n      2\n      \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n      \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    }\n  }\n  \\left(\n    \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n    \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\op\\varphi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    +\\mathup{i} (-1)^{\\sum_I (j_I-m_I)}\\op\\pi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n  \\right)\n  \\\\\n  \\label{eq:gft_cosmo_creation}\n  \\hermconj{\\op{a}}_{\\vec{\\jmath},\\vec{m},\\iota}\n  & = &\n  \\frac{1}{\n    \\sqrt{\n      2\n      \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n      \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    }\n  }\n  \\left(\n    (-1)^{\\sum_I (j_I-m_I)}\n    \\abs{\\mathcal{K}^{(2)}_{\\vec{j}, \\vec{m}, \\iota}}\n    \\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n    \\op\\varphi^{\\vec{\\jmath},\\iota}_{-\\vec{m}}\n    -\\mathup{i} \\op\\pi^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n  \\right)\n  \\mathcomma\n\\end{eqnarray}\nwhere\n$\n\\omega^{\\vec{\\jmath},\\iota}_{\\vec{m}}\n=\n\\sqrt{\n  \\abs{\n    \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}\n    \/ \\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}\n  }\n}\n$.\nThe free Hamiltonian is then $\\op{\\mathcal{H}} = \\sum_{\\vec{\\jmath},\\vec{m},\\iota} \\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}$ written as a sum of single-mode Hamiltonians $\\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}$.\nFor a mode for which $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)}$ and $\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(2)}$ have different signs the single-mode Hamiltonian is given by\n\\begin{equation}\n  \\op{\\mathcal{H}}_{\\vec{\\jmath},\\vec{m},\\iota}\n  =\n  -\\frac{1}{2}\n  {\\rm sgn}(\\mathcal{K}_{\\vec{\\jmath},\\vec{m},\\iota}^{(0)})\n  \\omega_{\\vec{m}}^{\\vec{j}, \\iota}\n  \\left(\n    \\hermconj{\\op{a}}_{\\vec{\\jmath},\\vec{m},\\iota}\n    \\hermconj{\\op{a}}_{\\vec{\\jmath},-\\vec{m},\\iota}\n    + \\op{a}_{\\vec{\\jmath},\\vec{m},\\iota}\n    \\op{a}_{\\vec{\\jmath},-\\vec{m},\\iota}\n  \\right)\n\\end{equation}\nwhich is analogous to the squeezing operator \\eref{eq:gft_cosmo_squeezing} (after redefinition by a phase $\\op{A}^i \\rightarrow e^{\\mathup{i}\\pi\/4}\\op{A}^i$ and $\\hermconj{\\op{A}}_j \\rightarrow e^{-\\mathup{i}\\pi\/4}\\hermconj{\\op{A}}_j$, \\eref{eq:gft_cosmo_squeezing} becomes $\\op{\\mathcal{H}}=\\frac{1}{2}\\lambda(\\hermconj{\\op{A}}_i \\hermconj{\\op{A}}_j\\epsilon^{ij}+\\op{A}^i\\op{A}^j\\epsilon_{ij})$).\nIn this sense, at least for modes with magnetic indices $m_i=0$ for which there is no coupling between modes, the Hamiltonian dynamics coming from the quadratic part of the full \\ac{gft} action is exactly of squeezing type.\n \n\\section{A toy model revisited}\n\\label{sec:tm}\nIn this section, we  study the dynamics of \\ac{gft} for a single field mode, in the approximation where \\ac{gft} interactions are neglected.\nThe observation that a squeezing operator as used in \\cite{AGW18} is already the (free) \\ac{gft} Hamiltonian for a mode in which all the magnetic indices are zero motivates us to revisit the model studied in \\cite{AGW18}.\nThe restriction of this model to a single field mode can be partially motivated by results in \\cite{Gie16} that suggest \\ac{gft} dynamics are generically dominated by a single value for the spin $j$.\nIn the next section, we will add interactions and go beyond the assumption of free dynamics.\n\nWe make use of the observation of \\cite{BBC19} that the fundamental operators appearing in this model (representing the Hamiltonian and volume operators) generate an $\\liealg{su}(1, 1)$ algebra.\nWe will extend some results both of the toy model analysis \\cite{AGW18} and of the algebraic discussion in \\cite{BBC19}: we will discuss general algebraic expressions representing effective Friedmann equations, and then specify by choosing different classes of coherent states.\nImportantly, we will compute relative uncertainties for the main physical quantities and use them as a criterion for the selection of good semiclassical states.\n\nThe Hamiltonian we consider is the one-mode squeezing Hamiltonian\n\\begin{equation}\n  \\label{eq:tm_hamiltonian}\n  \\op{H}\n  =\n  - \\frac{\\omega}{2}\n  (\n    \\op{a}^2\n    + \\hermconj{\\op{a}}{}^2\n  )\n  \\mathperiod\n\\end{equation}\nAs in previous work the main observable of interest is the volume operator\n\\begin{equation}\n  \\label{eq:tm_n_v_relation}\n  \\op{V}\n  =\n  v_0\n  \\op{N}\n  :=\n  v_0\n  \\hermconj{\\op{a}}\n  \\op{a}\n  \\mathcomma\n\\end{equation}\nwhere $v_0$ would be the eigenvalue for the \\ac{gft} volume operator for the representation (and intertwiner) chosen for the model, i.e., the volume associated to a single quantum in this mode.\nWe are working in a deparametrised framework in which the Hamiltonian generates time evolution with respect to scalar field time $\\phi$.\nThe energy expectation value $\\expval{\\op{H}}$ thus physically represents the conjugate momentum $\\pi_\\phi$ of $\\phi$.\nWe can then define an effective energy density of the matter scalar field $\\phi$, at the level of expectation values, by\n\\begin{equation}\n  \\label{eq:tm_energy_density}\n  \\rho_\\phi(\\phi)\n  =\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    2 \\expval{\\op{V}(\\phi)}^2\n  }\n  \\mathcomma\n\\end{equation}\ngiven the classical expression $\\rho_\\phi=\\pi_\\phi^2\/(2V^2)$.\nThis definition extends the one used in the mean-field setting (see \\eref{eq:gft_cosmo_friedmann}) which also only included expectation values of elementary operators $\\op{H}$ and $\\op{V}$.\nOther definitions using composite operators would be possible and would differ from  \\eref{eq:tm_energy_density} by higher moments such as $\\expval{\\op{H^2}}-\\expval{\\op{H}}^2$.\nNotice that inverse operators such as $\\op{V}(\\phi)^{-2}$ are not obviously well-defined in the \\ac{gft} formalism.\n\n\\subsection{\\texorpdfstring{$\\liealg{su}(1, 1)$}{su(1, 1)} structure of the system}\nAs was first pointed out for \\ac{gft} cosmology in \\cite{BBC19}, the operators $\\op{V}$ and $\\op{H}$ generate the Lie algebra $\\liealg{su}(1, 1)$, which appears frequently in the context of quantum cosmology for a flat \\ac{flrw} Universe filled with a free scalar field, see e.g.\\ \\cite{Boj07,EM12,AL19}.\nThe three independent quadratic products of creation and annihilation operators form a realisation of the $\\liealg{su}(1,1)$ algebra.\nIn particular the identifications\n\\begin{equation}\n  \\label{eq:su11_bosonic_realisation}\n  \\op{K}_0\n  =\n  \\frac{1}{4}\n  \\left(\n    \\hermconj{\\op{a}}\n    \\op{a}\n    +\n    \\op{a}\n    \\hermconj{\\op{a}}\n  \\right)\n  = \\frac{1}{2}\\op{N}+\\frac{1}{4}\\op{I}\n  \\mathcomma\n  \\qquad\n  \\op{K}_+\n  =\n  \\frac{1}{2}\n  \\hermconj{\\op{a}}{}^2\n  \\mathcomma\n  \\qquad\n  \\op{K}_-\n  =\n  \\frac{1}{2}\n  \\op{a}^2\n\\end{equation}\n(where $\\op{I}$ denotes the identity)\ngive the $\\liealg{su}(1,1)$ relations with the usual normalisation\n\\begin{equation}\n  \\commutator{\n    \\op{K}_0\n  }{\n    \\op{K}_\\pm\n  }\n  =\n  \\pm \\op{K}_\\pm\n  \\mathcomma\n  \\qquad\n  \\commutator{\n    \\op{K}_-\n  }{\n    \\op{K}_+\n  }\n  =\n  2 \\op{K}_0\n  \\mathperiod\n\\end{equation}\nThe Casimir of $\\liealg{su}(1, 1)$ is given by\n\\begin{equation}\n  \\label{eq:su11_casimir}\n  \\op{C}\n  =\n  (\\op{K}_0)^2\n  -\n  \\frac{1}{2}\n  (\n    \\op{K}_+\n    \\op{K}_-\n    +\n    \\op{K}_-\n    \\op{K}_+\n  )\n  \\mathperiod\n\\end{equation}\nIn terms of the $\\liealg{su}(1, 1)$ generators the Hamiltonian \\eref{eq:tm_hamiltonian} reads\n\\begin{equation}\n  \\label{eq:tm_hamiltonian_su11}\n  \\op{H}\n  =\n  - \\omega\n  (\n    \\op{K}_+\n    +\n    \\op{K}_-\n  )\n  \\mathperiod\n\\end{equation}\nAs one can see from \\eref{eq:su11_bosonic_realisation} the dynamics of the operator $\\op{K}_0$ are intimately related to the dynamics of the volume operator $\\op{V} = v_0 \\op{N}$.\nWe consider here only the $\\liealg{su}(1, 1)$ representations of the discrete ascending series in which the operator $\\op{K}_0$, and hence the volume, is bounded from below.\\footnote{In general, for such representations one can only say that $\\expval{\\op{N}}>-\\frac{1}{2}$.\nBelow we mostly focus on Fock representations, for which $\\op{N}$ is always nonnegative.}\n(A more general discussion would also include other types of representations, for which there is no such lower bound. See also the comments in \\cite[Sec.~4]{BBC19}.)\n\nThese representations are labelled by a real positive number $k$, the so-called Bargmann index, and satisfy\n\\begin{eqnarray}\n  &\n  \\op{K}_-\n  \\ket{k, 0}\n  =\n  0\n  \\mathcomma\n  \\\\\n  &\n  \\op{K}_0\n  \\ket{k, m}\n  =\n  (k + m)\n  \\ket{k, m}\n  \\mathcomma\n  \\\\\n  &\n  \\op{C}\n  \\ket{k, m}\n  =\n  k (k-1)\n  \\ket{k, m}\n  \\mathcomma\n\\end{eqnarray}\nwhere the states $\\ket{k, m}$ are the normalised states proportional to\n$(\\op{K}_+)^m \\ket{k, 0}$.\nSee \\aref{app:su11} for more details.\n\nWhen one inserts the realisation of the $\\liealg{su}(1, 1)$ operators in terms of bosonic creation and annihilation operators \\eref{eq:su11_bosonic_realisation} into the Casimir \\eref{eq:su11_casimir}, one finds that the Casimir is $\\op{C} = - 3\/16 \\op{I}$ which implies a Bargmann index of either $k=1\/4$ or $k=3\/4$.\nThese two cases respectively correspond to representations spanned by the eigenstates of the number operator with even or odd eigenvalues.\nThe choice $k=1\/4$ appears more interesting physically since it contains the Fock vacuum (or cosmological `singularity') in which no quanta are present.\nSince we are interested in studying the Fock space representations of the \\ac{gft} field, we will mostly restrict the Bargmann index to these cases.\n\n\\subsection{Classes of coherent states and relative uncertainties}\n\\label{sec:tm_class_of_coh_stat_and_rel_uncert}\nThe time evolution of a system can be quite sensitive to its initial state.\nIn this section we discuss classes of coherent states and comment on their usefulness in the context of \\ac{gft} cosmology.\nThe coherent states we consider are the following:\n\\begin{itemize}\n  \\item\n    Fock coherent states,\n  \\item\n    \\acf{pg} coherent states of $\\liealg{su}(1, 1)$,\n  \\item\n    \\acf{bg} coherent states of $\\liealg{su}(1, 1)$.\n\\end{itemize}\n\nThe Fock coherent states correspond to the well-known coherent states of the harmonic oscillator, labelled by a complex number $\\sigma$.\nOne possible way to define the Fock coherent states is by acting with the displacement operator on the Fock vacuum,\n\\begin{equation}\n  \\label{eq:tm_coh_state_fock}\n  \\ket{\\sigma}\n  =\n  \\exp\\left(\n    \\sigma \\hermconj{\\op{a}}\n    - \\compconj{\\sigma} \\op{a}\n  \\right)\n  \\ket{0}\n  \\mathperiod\n\\end{equation}\n\nThe \\ac{pg} coherent states are obtained by acting on the $\\liealg{su}(1, 1)$ ground state $\\ket{k, 0}$ with a squeezing operator\n\\begin{equation}\n  \\op{S}(\\xi)\n  =\n  \\exp\\left(\n    \\xi\n    \\op{K}_+\n    -\n    \\compconj{\\xi}\n    \\op{K}_-\n  \\right)\n  \\mathperiod\n\\end{equation}\nWe will denote the \\ac{pg} coherent states by $\\ket{\\zeta, k}$ and they are obtained by the following choice of squeezing parameter\n\\begin{equation}\n  \\ket{\n    \\zeta, k\n  }\n  =\n  \\op{S}\\left(\n    \\frac{\\zeta}{\\abs{\\zeta}}\n    \\artanh{\\abs{\\zeta}}\n  \\right)\n  \\ket{k, 0}\n  \\mathcomma\n  \\qquad\n  \\abs{\\zeta} < 1\n  \\mathperiod\n\\end{equation}\n\nThe \\ac{bg} coherent states will be denoted by $\\ket{\\chi, k}$ and are defined to be the eigenstates of the lowering operator $\\op{K}_-$,\n\\begin{equation}\n  \\op{K}_-\n  \\ket{\\chi, k}\n  =\n  \\chi\n  \\ket{\\chi, k}\n  \\mathperiod\n\\end{equation}\n\nWe now turn to the question of which class of coherent states should be considered in the context of \\ac{gft} cosmology.\nIn the context of quantum cosmology a commonly studied quantity is the relative uncertainty of the volume operator.\nIt is argued that the magnitude of the relative uncertainty corresponds to a measure of `quantumness' of the system at some given time and it is therefore important that the theory allows for initial states which give a (comparatively) small value for the relative uncertainty at late times since then the system has become (semi-)classical.\nFor previous works commenting on this in the context of \\ac{lqc} see, e.g., \\cite{AG15} and in the context of \\ac{gft} see, e.g., \\cite{PS17}.\nIn addition one would also require the relative uncertainty of the energy (which is a constant of motion) to be very small.\n\nWe define the relative uncertainty of an operator $\\op{O}$ for a given state $\\ket{\\psi}$ as\n\\begin{equation}\n  r(\\op{O}, \\ket{\\psi})\n  =\n  \\frac{\n    \\bra{\\psi}\n    \\op{O}^2\n    \\ket{\\psi}\n    -\n    \\bra{\\psi}\n    \\op{O}\n    \\ket{\\psi}^2\n  }{\n    \\bra{\\psi}\n    \\op{O}\n    \\ket{\\psi}^2\n  }\n  \\mathperiod\n\\end{equation}\nIn the following we state the relative uncertainty of the Hamiltonian and the asymptotic relative uncertainty of the volume operator at large volumes (i.e., for $\\phi\\rightarrow\\pm\\infty$), for the three classes of coherent states that we are interested in.\n\nFor the Fock coherent states one obtains for the relative uncertainty of the Hamiltonian and the asymptotic relative uncertainty of the volume operator\n\\begin{eqnarray}\n  \\label{eq:tm_rel_uncertainty_h_fock}\n  r(\n    \\op{H},\n    \\ket{\\sigma}\n  )\n  =\n  \\frac{\n    2 (1 + 2 \\abs{\\sigma}^2)\n  }{\n    (\\sigma^2 + \\compconj{\\sigma}^2)^2\n  }\n  \\mathcomma\n  \\\\\n  \\label{eq:tm_rel_uncertainty_v_fock}\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\sigma}\n  )\n  =\n  \\frac{\n    2\n    \\left(\n      1 \\mp 2 \\mathup{i} ( \\sigma \\pm \\mathup{i} \\compconj{\\sigma} )^2\n    \\right)\n  }{\n    \\left(\n      1 \\mp \\mathup{i} ( \\sigma \\pm \\mathup{i} \\compconj{\\sigma} )^2\n    \\right)^2\n  }\n  \\mathperiod\n\\end{eqnarray}\nIn principle the value of the parameter $\\sigma$ is arbitrary and therefore for suitable choices of $\\sigma$ the asymptotic relative uncertainty in both energy and volume becomes arbitrarily small.\nThese states can hence be interpreted as becoming semiclassical, consistent with arguments from \\ac{gft} that suggest that such `condensates' are good candidates for effective semiclassical macroscopic geometries \\cite{GOS14,GS16,Ori17a,PS19}.\n\nFor the \\ac{pg} coherent states the relative uncertainties of interest are\n\\begin{eqnarray}\n  r(\n    \\op{H},\n    \\ket{\\zeta, k}\n  )\n  =\n  \\frac{1}{2k}\n  \\frac{\n    (1 + \\zeta^2)\n    (1 + \\compconj{\\zeta}^2)\n  }{\n    (\\zeta + \\compconj{\\zeta})^2\n  }\n  \\mathcomma\n  \\\\\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\zeta, k}\n  )\n  =\n  \\frac{1}{2k}\n  \\mathperiod\n\\end{eqnarray}\nThe asymptotic relative uncertainty of the volume operator is independent of the parameter labelling the different \\ac{pg} coherent states.\nThis suggests that for \\ac{pg} coherent states the classical limit is reached for $k \\rightarrow \\infty$.\nHowever, we saw before that if we want to consider coherent states living in a bosonic Fock representation (rather than a more general $\\liealg{su}(1,1)$ representation), this restricts the values of the Bargmann index to either $k = 1\/4$ or $k = 3\/4$.\nThus we conclude that in the context of \\ac{gft} cosmology the class of \\ac{pg} coherent states do not `classicalise' at late times and hence, even though these states are naturally suggested by the  $\\liealg{su}(1,1)$ structure, they do not appear to be good candidate states for effective macroscopic cosmologies in \\ac{gft}.\n\nFor completeness we also state the relative uncertainties for the \\ac{bg} coherent states\n\\begin{eqnarray}\n  &r(\n    \\op{H},\n    \\ket{\\chi, k}\n  )\n  =\n  \\frac{2}{\n    (\\chi + \\compconj{\\chi})^2\n  }\n  \\left[\n    k\n    +\n    \\abs{\\chi}\n    \\frac{\n      I_{2k}(2 \\abs{\\chi})\n    }{\n      I_{2k-1}(2 \\abs{\\chi})\n    }\n  \\right]\n  \\mathcomma\n  \\\\\n&\n\\eqalign{\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\chi, k}\n  )\n  =\n  &\n  2\n  \\Big[\n    - 2 \\abs{\\chi}^2\n    I_{2k}(2 \\abs{\\chi})^2\n    +\n    (3 - 4k)\n    \\abs{\\chi}\n    I_{2k}(2\\abs{\\chi})\n    I_{2k-1}(2\\abs{\\chi})\n    \\\\\n  &\n    \\qquad\n    +\n    (k \\mp \\mathup{i}(\\chi - \\compconj{\\chi}) + 2 \\abs{\\chi}^2)\n    I_{2k-1}(2\\abs{\\chi})^2\n  \\Big]\n  \\\\\n  &\n  \\times\n  \\Big[\n    2 \\abs{\\chi}\n    I_{2k}(2 \\abs{\\chi})\n    +\n    (2 k \\mp \\mathup{i}(\\chi - \\compconj{\\chi}))\n    I_{2k-1}(2 \\abs{\\chi})\n  \\Big]^{-2}\n  \\mathcomma\n}\n\\end{eqnarray}\nwhere $I_\\alpha(x)$ is the modified Bessel function of the first kind.\nWe can use an asymptotic expansion of the modified Bessel functions to get the relative uncertainties for large values of $\\abs{\\chi}$,\n\\begin{eqnarray}\n  &r(\n    \\op{H},\n    \\ket{\\chi, k}\n  )\n  \\stackrel{\\abs{\\chi} \\rightarrow \\infty}{\\sim}\n  \\frac{\n    2\n    \\abs{\\chi}\n  }{\n    (\\chi + \\compconj{\\chi})^2\n  }\n  \\mathcomma\n\\\\\n&\n  r(\n    \\op{V}(\\pm \\infty),\n    \\ket{\\chi, k}\n  )\n  \\stackrel{\\abs{\\chi} \\rightarrow \\infty}{\\sim}\n  \\frac{\n    2\n  }{\n    2 \\abs{\\chi} \\mp \\mathup{i} (\\chi - \\compconj{\\chi})\n  }\n  \\mathcomma\n\\end{eqnarray}\nwhich shows that the asymptotic relative uncertainties for the \\ac{bg} coherent states are also arbitrarily small for large values of $\\abs{\\chi}$.\nHence these states would also be suitable states for \\ac{gft} cosmology.\nHowever, the ubiquitous appearance of the modified Bessel functions makes calculations with the \\ac{bg} states quite cumbersome.\nBelow we mostly focus on Fock coherent states which are easier to calculate with.\n\nIn \\fref{fig:tm_rel_uncertainty_comp} we provide an overview of the time dependence of the relative uncertainty of the various states discussed here.\nOne notable aspect is that the uncertainties are asymmetric with respect to time.\nWhile the uncertainty can be asymptotically small in the future, it might have been asymptotically large in the past.\nIn particular, one could in general not conclude from the emergence of a semiclassical regime at late times, in which the relative uncertainties remain small, that the same was true at early times in the collapsing pre-bounce phase.\nIn order to quantify this asymmetry of the asymptotic relative uncertainty, we define an `asymptotic asymmetry parameter'\n\\begin{equation}\n  \\label{eq:tm_asymmetry_parameter}\n  \\eta(\\ket{\\psi})\n  =\n  1\n  -\n  \\min\n  \\left\\{\n    \\frac{\n      r(\\op{V}(+\\infty), \\ket{\\psi})\n    }{\n      r(\\op{V}(-\\infty), \\ket{\\psi})\n    }\n    ,\n    \\frac{\n      r(\\op{V}(-\\infty), \\ket{\\psi})\n    }{\n      r(\\op{V}(+\\infty), \\ket{\\psi})\n    }\n  \\right\\}\n  \\mathperiod\n\\end{equation}\nThe values this parameter can take lie between zero and one, where small values signify that the relative uncertainty is the same in the asymptotic past and future whereas values close to one correspond to a large past-future asymmetry.\nIn \\fref{fig:tm_asymmetry_parameter} the asymptotic asymmetry parameter is shown for Fock and \\ac{bg} coherent states as a function of the argument $\\theta$ of the complex parameters characterising the state, i.e.\\ $\\chi = \\abs{\\chi} \\exp(\\mathup{i} \\theta)$ and $\\sigma = \\abs{\\sigma} \\exp(\\mathup{i} \\theta)$.\n    Note that even though the plot is done for some specific value of the absolute value of the coherent state parameters, the situation is generic.\n    Only for very small absolute values ($\\abs{\\sigma} \\ll 1, \\abs{\\chi} \\ll 1$) is the asymmetry parameter close to one for all values of the argument $\\theta$.\nFrom this it becomes apparent that the past-future asymmetry is rather generic.\nSimilar questions have been discussed previously in the context of \\ac{lqc} \\cite{Boj07a,CP08,Boj08}; our analysis here extends them from the mean-field calculations in \\cite{PS17} to broader classes of states of interest for \\ac{gft} cosmology.\n\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-tm_rel_uncertainty_comp.pdf}\n  \\caption{The relative uncertainty of the volume operator as a function of $\\omega \\phi$ for $k=1\/4$.\n    The complex parameter $x$ given in the figure is related to the parameters of the coherent states in the following manner.\n    \\ac{pg}:\n    $\\ket{\\zeta, k} \\equiv \\ket{(x\/\\abs{x}) \\tanh(\\abs{x}),1\/4}$\n    \\ac{bg}:\n    $\\ket{\\chi, k} \\equiv \\ket{x, 1\/4}$,\n    Fock:\n    $\\ket{\\sigma} \\equiv \\ket{x}$.\n    For the bottom-left plot, $|x|\\ll 1$ and these states have very small volume around $\\phi=0$, which leads to the large relative uncertainties.\n  }\\label{fig:tm_rel_uncertainty_comp}\n\\end{figure}\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-tm_asymmetry_parameter.pdf}\n  \\caption{The asymptotic asymmetry parameter \\eref{eq:tm_asymmetry_parameter} as a function of the argument of the complex coherent state parameters for $k=1\/4$.\n    The argument $\\theta$ is related to the parameters of the coherent states in the following way.\n    \\ac{bg}:\n    $\\ket{\\chi, k} \\equiv \\ket{100 \\exp(\\mathup{i} \\theta), 1\/4}$,\n    Fock:\n    $\\ket{\\sigma} \\equiv \\ket{100 \\exp(\\mathup{i} \\theta)}$.\n  }\\label{fig:tm_asymmetry_parameter}\n\\end{figure}\n\n\\subsection{Effective Friedmann equations}\nIn order to derive the cosmological implications of the model, we derive in this section an effective Friedmann equation\n\\begin{equation}\n  \\label{eq:tm_eff_fried_formal}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  f[V(\\phi)]\n  \\mathcomma\n\\end{equation}\nwhere we introduced the compact notation $V(\\phi) \\equiv \\expval{\\op{V}(\\phi)}$ and $f[V(\\phi)]$ is some functional to be specified later.\nThe method we will be employing to solve this problem is an algebraic approach introduced in \\cite{BBC19} which we extend to noncommuting variables and connect to the Fock representation underlying the kinematics of \\ac{gft}.\n\nWe work in the Heisenberg picture and assume that the Schr\u00f6dinger and Heisenberg picture coincide at $\\phi = 0$.\nOperators without argument denote the Schr\u00f6dinger picture operators.\nThe equations of motion for $\\op{K}_0$ and $\\op{K}_+ - \\op{K}_-$ are given by ($\\op{K}_+ + \\op{K}_-$ is proportional to the Hamiltonian and therefore constant under time evolution)\n\\begin{eqnarray}\n  &\n  \\label{eq:tm_k0_eom}\n  \\op{K}_0'\n  (\\phi)\n  =\n  \\mathup{i} \\omega\n  (\n    \\op{K}_+\n    -\n    \\op{K}_-\n  )\n  (\\phi)\n  \\mathcomma\n  \\\\\n  &\n  (\\op{K}_+ - \\op{K}_-)'(\\phi)\n  =\n  - 4 \\mathup{i} \\omega \\op{K}_0\n\\end{eqnarray}\nwhich are solved by\n\\begin{eqnarray}\n  \\label{eq:tm_k0_heisenberg}\n  &\n  \\op{K}_0(\\phi)\n  =\n  \\op{K}_0\n  \\cosh(2 \\omega \\phi)\n  +\n  \\frac{\\mathup{i}}{2}\n  (\\op{K}_+ - \\op{K}_-)\n  \\sinh(2 \\omega \\phi)\n  \\mathcomma\n  \\\\\n  &\n  (\\op{K}_+ - \\op{K}_-)(\\phi)\n  =\n  (\\op{K}_+ - \\op{K}_-)\n  \\cosh(2 \\omega \\phi)\n  -\n  2 \\mathup{i} \\op{K}_0\n  \\sinh(2 \\omega \\phi)\n  \\mathperiod\n\\end{eqnarray}\nFrom this one gets for the time dependence of the number operator\n\\begin{equation}\n  \\label{eq:tm_n_heisenberg}\n  \\op{N}(\\phi)\n  =\n  -\\frac{1}{2}\n  +\n  \\left(\n    \\op{N}\n    + \\frac{1}{2} \\op{I}\n  \\right)\n  \\cosh(2 \\omega \\phi)\n  +\n  \\mathup{i}\n  (\\op{K}_+ - \\op{K}_-)\n  \\sinh(2 \\omega \\phi)\n  \\mathperiod\n\\end{equation}\nThe expectation value of this is nonnegative for all Fock states and grows exponentially at early or late times ($|\\omega\\phi|\\gg 1$).\nA nonvanishing expectation value $\\expval{\\mathup{i}(\\op{K}_+ - \\op{K}_-)}$ implies a time asymmetry in the resulting effective cosmological history, i.e., different pre- and post-bounce phases.\nFor generic states $\\expval{\\op{N}(\\phi)}$ is positive for all $\\phi$; the only cases for which it becomes zero at some point during the evolution is for states which satisfy $ \\abs{\\expval{\\op{K}_+ - \\op{K}_-}}^2 \\geq \\expval{\\op{N}} (\\expval{\\op{N}} + 1)$.\nFor Fock coherent states this is only the case for the Fock vacuum for which both sides are zero.\nFor both \\ac{pg} and \\ac{bg} coherent states it depends on the value of the Bargmann index $k$.\nFor $k > 1\/4$ this inequality never holds, for $k < 1\/4$, however, there are states for which the inequality is satisfied and for $k=1\/4$ it is only the ground state (analogue to the Fock vacuum) for which the inequality holds.\n\\Eref{eq:tm_n_heisenberg} reproduces the result obtained in the `toy model' context of \\cite{AGW18} (cf.~\\eref{eq:gft_cosmo_toymodel_n} and below), the only difference being that a factor 5 is replaced by 1 since we consider a single field mode, not five modes as in the toy model.\n\nOne can derive an effective Friedmann equation directly by taking the expectation value of the explicit expression \\eref{eq:tm_n_heisenberg}.\nOne then finds\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_from_heisenberg}\n  \\fl\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4\\omega^2\n  \\left(\n    1\n    +\n    \\frac{v_0}{V(\\phi)}\n    -\n    \\frac{v_0^2}{V(\\phi)^2}\n    \\left[\n      \\expval{\\op{N}}^2 + \\expval{\\op{N}}\n      -\n      \\expval{\\mathup{i}(\\op{K}_+ - \\op{K}_-)}^2\n    \\right]\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere $V(\\phi) \\equiv v_0 \\expval{\\op{N}(\\phi)}$ (cf.~\\eref{eq:tm_n_v_relation}).\n\nOne can, however, also obtain this effective Friedmann equation without knowing the time dependence of the number operator explicitly by using the algebraic structure of the system.\nThis was shown in \\cite{BBC19} for the corresponding classical system, where the variables commute.\nHere we extend this algebraic approach to the noncommutative case.\n\nStarting from \\eref{eq:tm_k0_eom} and the definition of the Casimir \\eref{eq:su11_casimir} one arrives at\n\\begin{equation}\n    \\op{K}_0' (\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    \\op{K}_0(\\phi)^2\n    -\n    \\left(\n      \\frac{\\op{H}^2}{4 \\omega^2}\n      +\n      \\op{C}\n    \\right)\n  \\right]\n\\end{equation}\nor written in terms of the number operator\n\\begin{equation}\n  \\label{eq:gft_alg_friedmann}\n    \\op{N}' (\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    \\op{N}(\\phi)^2\n    +\n    \\op{N}(\\phi)\n    -\n    \\left(\n      \\frac{\\op{H}^2}{\\omega^2}\n      +\n      4 \\op{C}\n      -\n      \\frac{1}{4}\n      \\op{I}\n    \\right)\n  \\right]\n\\mathperiod\n\\end{equation}\n\nIn order to get the effective Friedmann equation one has to take the expectation value of \\eref{eq:gft_alg_friedmann}.\nHowever, it is crucial to note that in \\eref{eq:tm_eff_fried_formal} the expectation value of the volume operator enters, rather than the expectation value of the volume operator squared.\nThe difference between the two is related to the variance of the volume, which is in general state-dependent.\nIndeed, rearranging the expectation value of \\eref{eq:gft_alg_friedmann} gives\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_k0}\n  N'(\\phi)^2\n  =\n  4 \\omega^2\n  \\left[\n    N(\\phi)^2\n    +\n    N(\\phi)\n    +\n    X\n  \\right]\n\\end{equation}\nwith $N(\\phi) \\equiv \\expval{\\op{N}(\\phi)}$ and $X$ being given by\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n  -\n  \\frac{1}{4\\omega^2}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}'\n    (\\phi)\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}^2}\n  }{\n    \\omega^2\n  }\n  -\n  4 \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathcomma\n\\end{equation}\nwhere the covariance $\\covariance{\\op{A}}{\\op{B}}$ is defined as\n\\begin{equation}\n  \\covariance{\n    \\op{A}\n  }{\n    \\op{B}\n  }\n  =\n  \\frac{1}{2}\n  \\expval{\n    \\op{A}\n    \\op{B}\n    +\n    \\op{B}\n    \\op{A}\n  }\n  -\n  \\expval{\\op{A}}\n  \\expval{\\op{B}}\n\\end{equation}\n(and for the case $\\op{A}=\\op{B}$ this would be called the variance of $\\op{A}$).\n\nWe will now show that the quantity $X$ is indeed time-independent as suggested by the notation.\nNoting that the variance of $\\op{N}'(\\phi)$ can be written as\n\\begin{equation}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}'\n    (\\phi)\n  }\n  =\n  - 4 \\covariance{\\op{H}}{\\op{H}}\n  + 16 \\omega^2\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  \\mathcomma\n\\end{equation}\none can write $X$ as\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n  -\n  4\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    \\omega^2\n  }\n  -\n  4 \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nA quick calculation shows that\n\\begin{equation}\n  \\dfrac{\\phi}\n  \\covariance{\n    \\op{K}_+\n    (\\phi)\n  }{\n    \\op{K}_-\n    (\\phi)\n  }\n  =\n  \\frac{1}{2}\n  \\covariance{\n    \\op{N}'\n    (\\phi)\n  }{\n    \\op{N}\n    (\\phi)\n  }\n\\end{equation}\nwhich shows that $X$ is time-independent, since both $\\op{H}$ and $\\op{C}$ are constants\nof motion.\nIt follows that $X$ can be written as\n\\begin{equation}\n  X\n  =\n  \\covariance{\n    \\op{N}\n  }{\n    \\op{N}\n  }\n  -\n  4\n  \\covariance{\n    \\op{K}_+\n  }{\n    \\op{K}_-\n  }\n  -\n  \\frac{\n    \\expval{\\op{H}}^2\n  }{\n    \\omega^2\n  }\n  -\n  4\n  \\expval{\\op{C}}\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\n\nUsing the definition of the Casimir \\eref{eq:su11_casimir} one can show that $X$ can be written in the alternative form\n\\begin{equation}\n  X\n  =\n  - \\expval{\n    \\op{N}\n  }^2\n  -\n  \\expval{\n    \\op{N}\n  }\n  +\n  4\n  \\expval{\n    \\op{K}_+\n  }\n  \\expval{\n    \\op{K}_-\n  }\n  -\n  \\frac{\n    \\expval{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  \\mathperiod\n\\end{equation}\nYet another form can be obtained by inserting the Hamiltonian \\eref{eq:tm_hamiltonian_su11} to get\n\\begin{equation}\n  X\n  =\n  -\n  \\expval{\n    \\op{N}\n    +\n    \\frac{1}{2}\n    \\op{I}\n  }^2\n  +\n  \\expval{\n    \\mathup{i}\n    (\n      \\op{K}_+\n      -\n      \\op{K}_-\n    )\n  }^2\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nThe operators appearing in this last expression for $X$ are exactly those appearing in the formula for the operator $\\op{K}_0(\\phi)$ in the Heisenberg picture \\eref{eq:tm_n_heisenberg} and one recovers the form of the effective Friedmann equation given in \\eref{eq:tm_eff_friedmann_from_heisenberg}.\nFrom the exact solution \\eref{eq:tm_n_heisenberg}, one can see that $X\\le 0$ in all Fock states, since $X>0$ would be equivalent to the number operator $N(\\phi)$ taking a negative expectation value somewhere.\nThere are Fock states with $X=0$; these states encounter a singularity in their geometric interpretation, in the sense that the expectation value of the volume reaches zero somewhere and hence the effective energy density defined according to \\eref{eq:tm_energy_density} diverges, even though the quantum evolution is completely regular even for these states.\n\nRecalling that the volume operator is the rescaled number operator, i.e.\\ $\\op{V} = v_0 \\op{N}$, one arrives at the effective Friedmann equation\n\\begin{equation}\n  \\label{eq:tm_eff_friedmann_full}\n  \\fl\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval{\n        \\op{N}\n      }\n      (\n        \\expval{\n          \\op{N}\n        }\n        +\n        1\n      )\n    }{\n      V(\\phi)^2\n    }\n    +\n    \\frac{\n      4\n      v_0^2\n      \\expval{\n        \\op{K}_+\n      }\n      \\expval{\n        \\op{K}_-\n      }\n    }{\n      V(\\phi)^2\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n  \\right)\n\\end{equation}\nTaking the late time limit (corresponding to large volumes) suggests that one should identify $12 \\pi G := 4 \\omega^2$ in order for the leading term to be compatible with the classical Friedmann dynamics.\nThis identification of fundamental couplings with an `emergent' Newton's constant is common in \\ac{gft} cosmology \\cite{OSW16,AGW18}.\nFurthermore, identifying an energy density as in \\eref{eq:tm_energy_density} and defining a critical energy density\n\\begin{equation}\n  \\rho_\\critical\n  =\n  \\frac{\n    \\omega^2\n  }{\n    2 v_0^2\n  }\n  =\n  \\frac{\n    3 \\pi G\n  }{\n    2 v_0^2\n  }\n  =\n  \\frac{3\\pi}{2}\n  \\rho_\\planck\n  \\left(\n    \\frac{v_\\planck}{v_0}\n  \\right)^2\n  \\mathcomma\n\\end{equation}\nwhere $\\rho_\\planck$ is the Planck mass density and $v_\\planck$ is the Planck volume, the last term inside the parentheses in \\eref{eq:tm_eff_friedmann_full} takes the form $-\\rho\/\\rho_\\critical$ familiar from \\ac{lqc}.\nThe value for $\\rho_\\critical$ appearing here agrees with the critical density found in \\cite{OSW16,OSW17}.\nIn close analogy to the results obtained in \\ac{lqc} we then write for the effective Friedmann equation (see also \\eref{eq:gft_cosmo_friedmann})\n\\begin{equation}\n\\label{eq:tm_friedman_with_rho_eff}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    -\n    \\frac{\n      \\rho_\\effective(\\phi)\n    }{\n      \\rho_\\critical\n    }\n  \\right)\n  +\n  4 \\omega^2\n  \\frac{\n    v_0\n  }{\n    V(\\phi)\n  }\n  \\mathcomma\n\\end{equation}\nwhere the effective energy density $\\rho_\\effective(\\phi)$ is defined as\n\\begin{equation}\n  \\rho_\\effective(\\phi)\n  =\n  \\rho_\\phi(\\phi)\n  +\n  \\frac{\n    \\omega^2\n    \\expval{\n      \\op{N}\n    }\n    (\n      \\expval{\n        \\op{N}\n      }\n      +\n      1\n    )\n  }{\n    2V(\\phi)^2\n  }\n  -\n  \\frac{\n    2\n    \\omega^2\n    \\expval{\n      \\op{K}_+\n    }\n    \\expval{\n      \\op{K}_-\n    }\n  }{\n    V(\\phi)^2\n  }\n  \\mathperiod\n\\end{equation}\nThe first contribution to this effective energy density is given by the energy density $\\rho_\\phi(\\phi)=\\expval{\\op{H}}^2\/(2V(\\phi)^2)$ associated to a massless scalar field (as defined in  \\eref{eq:tm_energy_density}), but there are two additional contributions depending on the expectation values $\\expval{\\op{N}},\\,\\expval{\\op{K}_+}$ and\n    $\\expval{\n      \\op{K}_-\n    }$\nin the initial state.\nAs these additional contributions to $\\rho_\\effective$ also scale as $V(\\phi)^{-2}$, their effect is equivalent to a shift in the scalar field momentum compared to its classical value $\\expval{\\op{H}}$.\nThe last term in \\eref{eq:tm_friedman_with_rho_eff}, scaling as $1\/V(\\phi)$, is similar to a correction found in mean-field calculations and takes the form of an effective matter contribution for matter with equation of state $p=2\\rho$ (cf.~\\cite{CPS16}).\n\nDepending on the initial state, the quantity $X$ (or, alternatively, the additional contributions to the effective energy density) can take different forms.\nFor the \\ac{pg} coherent states one finds the following form\n\\begin{equation}\n  \\label{eq:gft_su11_x_pg}\n  X_{\\mathsubscript{PG}}\n  =\n  - 4 k^2\n  \\frac{\n    (1 + \\zeta^2)\n    (1 + \\compconj{\\zeta}^2)\n  }{\n    (1 - \\abs{\\zeta}^2)^2\n  }\n  +\n  \\frac{1}{4}\n  =\n  -\n  4 k^2\n  -\n  \\frac{\n    \\expval[\\mathsubscript{PG}]{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\nWe note that for $k = 1\/4$ this reduces to\n$\n  X_{\\mathsubscript{PG}}\n  =\n  -\n  \\expval[\\mathsubscript{PG}]{\n    \\op{H}\n  }^2\n  \/\n  \\omega^2\n$, and the effective energy density and classical energy density exactly coincide.\nThe effective Friedmann equation for \\ac{pg} coherent states therefore reads\n(for general $k$)\n\\begin{equation}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{PG}]{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n    - \\frac{\n      v_0^2\\left(16 k^2\n    - 1\\right)\n    }{\n     4V(\\phi)^2\n    }\n  \\right)\n  \\mathperiod\n\\end{equation}\n\nFor Fock coherent states one gets for $X$\n\\begin{equation}\n  X_{\\mathsubscript{F}}\n  =\n  - \\frac{1}{4} (\\sigma^2 + \\compconj{\\sigma}^2)^2\n  - \\abs{\\sigma}^2\n  =\n  -\n  \\expval[\\mathsubscript{F}]{\n    \\op{N}\n  }\n  -\n  \\frac{\n    \\expval[\\mathsubscript{F}]{\n      \\op{H}\n    }^2\n  }{\n    \\omega^2\n  }\n  \\mathperiod\n\\end{equation}\nTherefore the Friedmann equation for Fock coherent states is given by\n\\begin{equation}\n  \\label{eq:tm_eff_fried_fock}\n  \\left(\n    \\frac{\n      V'(\\phi)\n    }{\n      V(\\phi)\n    }\n  \\right)^2\n  =\n  4 \\omega^2\n  \\left(\n    1\n    +\n    \\frac{\n      v_0\n    }{\n      V(\\phi)\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{F}]{\n        \\op{N}\n      }\n    }{\n      V(\\phi)^2\n    }\n    -\n    \\frac{\n      v_0^2\n      \\expval[\\mathsubscript{F}]{\n        \\op{H}\n      }^2\n    }{\n      \\omega^2\n      V(\\phi)^2\n    }\n  \\right)\n  \\mathperiod\n\\end{equation}\n\nFor completeness we also state the value of $X$ one gets for \\ac{bg} coherent states,\n\\begin{equation}\n  X_{\\mathsubscript{BG}}\n  =\n  -\n  (\\chi - \\compconj{\\chi})^2\n  -\n  4\n  \\left(\n    k\n    +\n    \\abs{\\chi}\n    \\frac{\n      I_{2k}(2 \\abs{\\chi})\n    }{\n      I_{2k - 1}(2 \\abs{\\chi})\n    }\n  \\right)^2\n  +\n  \\frac{1}{4}\n  \\mathperiod\n\\end{equation}\n\nThe Friedmann equations derived here are compatible with previous results in \\ac{gft} cosmology \\cite{OSW16,OSW17,AGW18,WEw19} where either a mean-field approach was used or a simplifying assumption was imposed on the initial conditions.\nWe emphasise that no approximations were used which resulted in the appearance of extra terms.\nIn particular, we were able to identify one of those extra terms with the energy density of the real scalar field acting as a relational clock variable.\nCorrections to the classical Friedmann dynamics of the \\ac{lqc}-like form $-\\rho\/\\rho_\\critical$, which lead to effective repulsive behaviour and a bounce at high energies, were found for all coherent states considered.\nWe also found that in general the `effective' energy density appearing in the Friedmann equation \\eref{eq:tm_friedman_with_rho_eff} is not equal to the classical energy density $\\rho_\\phi=\\pi_\\phi^2\/(2V^2)$ of a massless scalar field, but contains additional terms depending on the initial conditions chosen.\n \n\\section{Interacting toy model}\nIn this section we extend the toy model discussed in \\sref{sec:tm} by adding an interaction term to the Hamiltonian.\nThe resulting interacting model still represents a simplification of the dynamics of full \\ac{gft}, since we continue to assume that only one mode is relevant.\nWhile the general expectation is that the dynamics should depend on the coupling of different modes, studying this simpler model can provide insights on how \\ac{gft} interactions can change the interpretation of the dynamics in terms of effective cosmology.\nA similar model, which included polynomial interactions for a single \\ac{gft} field mode, was previously studied in \\cite{CPS16} in a mean-field approximation (see also \\cite{PS17}), leading to corrections to the effective Friedmann equations coming from these interactions.\nThese corrections become more significant at late times as the impact of \\ac{gft} interactions grows with the number of quanta.\nHere we will be able to contrast these mean-field results with effective modified Friedmann equations obtained in a more general setting.\n\nWe now consider a Hamiltonian given in terms of the $\\liealg{su}(1, 1)$ variables by\n\\begin{equation}\n  \\label{eq:int_hamiltonian}\n  \\op{H}\n  =\n  -\\omega\n  (\n    \\op{K}_+\n    + \\op{K}_-\n  )\n  +\n  \\lambda \\omega\n  (\n    \\op{K}_+\n    + \\op{K}_-\n    + 2 \\op{K}_0\n  )^2\n  \\mathperiod\n\\end{equation}\nIn terms of the bosonic realisation of the $\\liealg{su}(1, 1)$ algebra \\eref{eq:su11_bosonic_realisation} the Hamiltonian reads\n\\begin{equation}\n  \\label{eq:int_hamiltonian_fock_vars}\n  \\op{H}\n  =\n  - \\frac{\\omega}{2}\n  (\n    \\op{a}^2\n    + \\hermconj{\\op{a}}{}^2\n  )\n  +\n  \\frac{\n    \\lambda \\omega\n  }{\n    4\n  }\n  (\n    \\op{a}\n    + \\hermconj{\\op{a}}\n  )^4\n  \\mathperiod\n\\end{equation}\nRecalling the definitions of the creation and annihilation operators in terms of the \\ac{gft} field and its conjugate momentum \\eref{eq:gft_cosmo_annihilation}, \\eref{eq:gft_cosmo_creation} one can rewrite the Hamiltonian as\n\\begin{equation}\n  \\label{eq:int_hamiltonian_gft_vars}\n  \\op{H}\n  =\n  \\frac{1}{2 \\abs{\\mathcal{K}^{(2)}}}\n  \\op{\\pi}^2\n  -\n  \\frac{1}{2}\n  \\abs{\\mathcal{K}^{(0)}}\n  \\op{\\varphi}^2\n  +\n  \\lambda\n  \\abs{\\mathcal{K}^{(0)}}^{3\/2}\n  \\abs{\\mathcal{K}^{(2)}}^{1\/2}\n  \\op{\\varphi}^4\n  \\mathcomma\n\\end{equation}\nwhere we suppressed the Peter--Weyl representation labels and we also assume the mode to be of the type discussed at the end of \\sref{sec:gft_cosmo} with magnetic indices $m_i = 0$.\nFrom this expression one sees that the interaction term would correspond to a $\\varphi^4$ interaction term in an appropriately defined \\ac{gft} action.\n\nThe dynamics of this system crucially depend on the sign of $\\lambda$.\nIndeed, for positive $\\lambda$ this Hamiltonian will be bounded from below, whereas for the case that $\\lambda$ is negative the Hamiltonian is unbounded as it is in the free case.\nInterpreting \\eref{eq:int_hamiltonian_gft_vars} as a mechanical system with kinetic and potential terms, one sees that for positive $\\lambda$ one gets a `Mexican hat' type potential, whereas in the case of negative $\\lambda$ the potential is an `upside-down' anharmonic oscillator.\nIn the cosmological context one expects from this that for negative $\\lambda$ the Universe will undergo an enhanced exponential expansion and for positive $\\lambda$ the Universe will recollapse after some time leading to a cyclic cosmology.\nSuch a cyclic cosmology was indeed found in \\cite{CPS16}, where $\\lambda > 0$ was assumed.\nIn \\sref{sec:int_alg_poisson} we argue that this expectation is correct if \\eref{eq:int_hamiltonian} is seen as the Hamiltonian of a classical system.\n\nWhen one tries to use the algebraic approach detailed in \\sref{sec:tm} to derive an effective Friedmann equation for the interacting model \\eref{eq:int_hamiltonian} one faces several challenges.\nFirstly, the noncommutativity does not allow the reduction to a small set of `basis operators'.\nSecondly, the expressions involved feature products of three operators and it is technically challenging to relate them to the expectation values of `simple' operators such as the Hamiltonian and number operator.\n\nTo begin with, we restrict ourselves to the classical case, where the variables commute and one can employ the algebraic approach to derive an effective Friedmann equation.\nWe find an exact Friedmann equation whose limits at early and late times are given.\nAfter that we turn to the general (quantum) case, where the operators do not commute.\nIn that case we resort to a perturbative treatment which is valid at early times.\nFurthermore, we perform a numerical analysis for Fock coherent states.\nWe find that a linear (perturbative) correction to the effective Friedmann equation can capture the effect of the interaction term for a short time, after which the dynamics become nonperturbative.\n\n\\subsection{Algebraic approach for classical analogue system}\n\\label{sec:int_alg_poisson}\nIn this section we study a classical dynamical system with time evolution generated by the Hamiltonian \\eref{eq:int_hamiltonian}.\nIn this approach the $\\liealg{su}(1,1)$ variables $K_0$, $K_+$ and $K_-$ are not viewed as quantum operators but as coordinates on a Poisson manifold, subject to an $\\liealg{su}(1,1)$ Poisson algebra which then defines the Hamiltonian dynamics.\nIn contrast to the full quantum case, these variables commute and we interpret the variables themselves as the observables of interest, i.e.\\ one does not have to take expectation values.\nIn this sense, this approximation neglects all quantum corrections coming from operator orderings and uncertainties in a quantum state.\nWe will identify the variable $K_0$ with the particle number $N$ such that $N \\equiv 2 K_0$.\nAs above, we assume the total volume to be proportional to the particle number, $V = v_0 N$, and switch between $N$ and $V$ freely.\n\nThe equation of motion of the variable $N$ is given by\n\\begin{equation}\n  N'(\\phi)\n  =\n  2 \\mathup{i} \\omega\n  (K_+(\\phi) - K_-(\\phi))\n  (\n    1 - 2 \\lambda (K_+(\\phi) + K_-(\\phi) + N(\\phi))\n  )\n  \\mathperiod\n\\end{equation}\nAfter squaring this equation one can use the Casimir \\eref{eq:su11_casimir} to replace the combination $(K_+ - K_-)$.\nOne is then left with an expression where only the combination $(K_+ + K_-)$ appears,\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_intermediate}\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    &\n    4 \\omega^2\n    \\left(\n      N(\\phi)^2\n      - 4C\n      - (K_+(\\phi) + K_-(\\phi))^2\n    \\right)\n    \\\\\n    &\n    \\qquad\n    \\times\n    \\left(\n      1\n      - 2 \\lambda (K_+(\\phi) + K_-(\\phi) + N(\\phi))\n    \\right)^2\n    \\mathperiod\n  }\n\\end{equation}\nFrom the Hamiltonian one can derive an explicit formula for $(K_+ + K_-)$\n\\begin{equation}\n  \\label{eq:int_kP_plus_kM}\n  K_+(\\phi)\n  +\n  K_-(\\phi)\n  =\n  \\frac{1}{2 \\lambda}\n  \\left(\n    1\n    - 2 \\lambda N(\\phi)\n    -\n    \\sqrt{\n      1\n      + 4 \\lambda \\left( \\frac{H}{\\omega} - N(\\phi) \\right)\n    }\n  \\right)\n  \\mathcomma\n\\end{equation}\nwhere we chose the solution connected to the free theory.\nInserting this into \\eref{eq:int_classical_eff_friedmann_intermediate} one gets the nonperturbative effective Friedmann equation\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_nonpert}\n  \\fl\n\\eqalign{\n  N'(\\phi)^2\n  =\n  &\n  -\n  \\frac{\n    2\n    \\omega^2\n  }{\n    \\lambda^2\n  }\n  \\left(\n    1 + 4 \\lambda\n    \\left(\n      \\frac{H}{\\omega} - N(\\phi)\n    \\right)\n  \\right)\n    \\times\n    \\Bigg[\n      1 - 4 \\lambda N(\\phi)\n  \\\\\n  &\n  \\qquad\n  \\qquad\n      - (1 - 2 \\lambda N(\\phi))\n      \\sqrt{\n        1 + 4 \\lambda\n        \\left(\n          \\frac{H}{\\omega} - N(\\phi)\n        \\right)\n      }\n      + 2 \\lambda\n      \\left(\n        4 \\lambda C\n        + \\frac{H}{\\omega}\n      \\right)\n    \\Bigg]\n    \\mathperiod\n}\n\\end{equation}\nWe already see that $\\lambda > 0$ implies an upper bound on the value of $N$ and hence the volume, since the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} has to be real and positive.\nIndeed, at exactly that upper limit the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} becomes zero leading to a recollapse.\nIn the case $\\lambda < 0$, for the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert} to be real and positive $N$ has to be greater than some minimal value (which can be zero).\nThe right-hand side remains real and positive for all values of $N$ greater than that minimal value, implying that the Universe expands indefinitely.\nFor clarity, we recall that $H$ here and throughout the paper denotes the Hamiltonian or energy (interpreted as the canonical momentum conjugate to the scalar field $\\phi$), not a Hubble rate in cosmology.\n\nWe would like to interpret \\eref{eq:int_classical_eff_friedmann_nonpert} in terms of a cosmological model given by one or several matter components which contribute to the matter energy density on the right-hand side.\nFor a general such model in usual classical cosmology, with $n$ matter components labelled by an index $i$ viewed as perfect fluids with each having an equation of state $p_i=w_i \\rho_i$, the Friedmann equation for the volume as a function of relational time $\\phi$ would be of the form\n\\begin{equation}\n  \\label{eq:int_friedmann_eos}\n  \\left(\n    \\frac{V'(\\phi)}{V(\\phi)}\n  \\right)^2\n  = \\sum_{i=1}^n\n  A_i V(\\phi)^{1-w_i}\n\\end{equation}\nwhere $A_i$ are constants of motion.\nWhile \\eref{eq:int_classical_eff_friedmann_nonpert} is valid at all times, its interpretation in terms of cosmological models of the form \\eref{eq:int_friedmann_eos} is not clear due to the appearance of a square root on the right-hand side.\nMoreover, additional matter components as in \\eref{eq:int_friedmann_eos} would come with new conserved quantities $A_i$, whose values can be varied independently (or, in other words, are determined by the initial conditions).\nIn our model, the presence of \\ac{gft} interactions does not introduce new parameters set by initial conditions, only a new coupling constant $\\lambda$; these interactions therefore modify the dynamics of gravity rather than matter.\nHere we follow the convention in which quantum gravity corrections are written as modifying the right-hand side rather than the left-hand side of Friedmann equations (as is usually done in \\ac{lqc}) in order to give intuition for the effective dynamics.\\footnote{An alternative possible interpretation of effective Friedmann equations obtained from quantum gravity models is to view them as equivalent to Friedmann equations of a modified theory of gravity.\n  A general reconstruction method of this type for mimetic gravity theories was developed in \\cite{Ces19}.\n}\n\nOne might be interested in the Friedmann equation valid at relatively small volumes where interactions have become relevant but not dominant, so that one can employ perturbation theory.\nExpanding \\eref{eq:int_classical_eff_friedmann_nonpert} as a series around $\\lambda=0$ one gets\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_series}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4\\omega^2\n    \\Bigg\\{\n  &\n      N(\\phi)^2\n      - \\frac{H^2}{\\omega^2}\n      - 4C\n      \\left(\n        1\n        -\n        4 \\lambda\n        \\left(\n          N(\\phi) - \\frac{H}{\\omega}\n        \\right)\n      \\right)\n  \\\\\n  &\n    \\eqalign{\n      -\n      \\sum_{n=1}^\\infty\n      \\bigg[\n    &\n      12 \\lambda^n\n      \\frac{\n        (2n - 2)!\n      }{\n        (n - 1)!\n        (n + 2)!\n      }\n      \\left(\n        (2n - 1) \\frac{H}{\\omega}\n        + (3 - n) N(\\phi)\n      \\right)\n    \\\\\n    &\n      \\times\n      \\left(\n        N(\\phi)\n        - \\frac{H}{\\omega}\n      \\right)^{n + 1}\n      \\bigg]\n      \\Bigg\\}\n      \\mathperiod\n    }\n  }\n\\end{equation}\nFrom this form one can see that it is the product $\\lambda N(\\phi)$ that must be small for the perturbative expansion to make sense.\nComparing with \\eref{eq:int_friedmann_eos}, one could interpret the leading (linear) correction coming from the interaction term as an effective matter component with an equation of state parameter $w=0$, i.e., a dust component.\nThis result would then agree with the results of a mean-field calculation in \\cite{CPS16} where adding a $\\varphi^4$ interaction to the \\ac{gft} Lagrangian led to such a dust-like contribution in the effective cosmology.\nHowever, the full expansion given in \\eref{eq:int_classical_eff_friedmann_series} shows that such an interpretation would only be valid in an intermediate regime in which the product $\\lambda N(\\phi)$ is no longer negligible, but also not yet large enough for higher orders to contribute.\nIndeed, as the volume grows further, $\\lambda N(\\phi)$ would soon be $\\bigO{1}$ and the perturbative expansion receives contributions from \\emph{all} orders.\nIn particular, there is never a regime in which the effective Friedmann equation is dominated by the `dust-like' component, as it would be in the mean-field form obtained in \\cite{CPS16}.\nOne possible interpretation of this discrepancy is that mean-field methods are strictly only valid in the free theory, since they assume the absence of correlations between quanta.\nHence one would expect them to become inaccurate as the contribution to the effective dynamics coming from \\ac{gft} interactions becomes strong.\nCare has to be taken when interpreting \\eref{eq:int_classical_eff_friedmann_series} up to some order.\nFor instance, if one only considers terms up to order $\\lambda^2$ in \\eref{eq:int_classical_eff_friedmann_series}, one would conclude that there is always a recollapse (even for negative $\\lambda$), which is not true for the full solution.\n\nTo understand the late-time behaviour of the system, recall that $N(\\phi)$ grows without bound while the other dynamical quantities on the right-hand side of \\eref{eq:int_classical_eff_friedmann_nonpert}, $C$ and $H$, are constants of motion.\nIn the limit of large $N(\\phi)$ (corresponding to late times) the leading order contribution of \\eref{eq:int_classical_eff_friedmann_nonpert} is given by\n\\begin{equation}\n  \\label{eq:int_classical_eff_friedmann_asymptotic}\n  \\left(\n    \\frac{N'(\\phi)}{N(\\phi)}\n  \\right)^2\n  =\n  32 \\omega^2\n  \\sqrt{-\\lambda N(\\phi)}\n  +\n  \\bigO{1}\n  \\mathperiod\n\\end{equation}\nInterpreting the right-hand side of \\eref{eq:int_classical_eff_friedmann_asymptotic} as an energy density of matter, one finds from \\eref{eq:int_friedmann_eos} that it corresponds to matter with an equation of state parameter $w = 1\/2$.\nThe solutions to this asymptotic form of the effective Friedmann equation behave as $N(\\phi)\\sim|\\phi-\\phi_0|^{-4}$, diverging at some $\\phi=\\phi_0$.\nWe would hence expect the evolution of our system to terminate at some finite value of $\\phi$, depending on initial conditions.\nNotice that in this interpretation the energy density of this new `matter' is fixed by the \\ac{gft} coupling $\\lambda$ and hence, as we mentioned before, the effect of adding \\ac{gft} interactions should rather be seen as modifying the dynamics of gravity on large scales.\nThe scale at which such a modification would become relevant depends on the chosen value of $\\lambda$.\n\nTo characterise the dynamics of the effective cosmology at arbitrary times, from \\eref{eq:int_friedmann_eos} we define an `effective equation of state parameter'\n\\begin{equation}\n  \\label{eq:int_eos_eff}\n  w_{\\effective}(\\phi)\n  =\n  1\n  -\n  \\frac{\n    \\mathrm{d} \\log \\left( N'(\\phi)\/N(\\phi) \\right)^2\n  }{\n    \\mathrm{d} \\log N(\\phi)\n  }\n\\end{equation}\nwhich is plotted in \\fref{fig:int_eos_poisson} as a function of $N(\\phi)$ for various truncations of \\eref{eq:int_classical_eff_friedmann_series}.\nIn the plot one sees that for $\\abs{\\lambda N(\\phi)} \\lesssim 10^{-2}$ the interactions become relevant and that for $\\abs{\\lambda N(\\phi)} \\lesssim 1$ the difference between the exact result and the first order truncation becomes large as expected.\nAnother observation is that while for small values of $\\abs{\\lambda N(\\phi)}$ the second order truncation is in better agreement with the nonperturbative results, this truncation quickly diverges for $\\abs{\\lambda N(\\phi)} \\gtrsim 1$.\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-int_poisson.pdf}\n  \\caption{The relative relational expansion rate squared \\eref{eq:int_classical_eff_friedmann_nonpert} and the `effective equation of state parameter' \\eref{eq:int_eos_eff} as functions of $N$.\n    The solid lines correspond to a truncation at zeroth order in $\\lambda$.\n    The dashed lines correspond to a truncation at first order in $\\lambda$.\n    The dotted lines correspond to a truncation at second order in $\\lambda$.\n    The dash-dotted lines correspond to the full nonperturbative case.\n    The parameters are:\n    $\\omega = 1$,\n    $\\lambda = - 10^{-7}$,\n    $H = -10040$,\n    $C = - 3\/16$.\n    (The choice of $H$ corresponds to $\\bra{\\sigma} \\op{H} \\ket{\\sigma}$ for $\\sigma = 100$.)\n  }\n  \\label{fig:int_eos_poisson}\n\\end{figure}\n\n\\subsection{Quantum calculation}\nWe are not able to derive an exact solution for $\\hat{N}(\\phi)$ in the interacting quantum mechanical case, where operators do not commute and higher moments and simple expectation values are independent.\nTo give approximate solutions, we first give a perturbative analytical method which we then contrast with numerical results.\nWe expand the number operator as a series with expansion parameter $\\lambda$, i.e.\\\n\\begin{equation}\n  \\label{eq:int_n_pert_series}\n  \\op{N}(\\phi)\n  =\n  \\sum_{n=0}^\\infty\n  \\lambda^n\n  \\op{N}_n(\\phi)\n  \\mathperiod\n\\end{equation}\nSplitting the Hamiltonian $\\op{H}$ into a $\\lambda$-independent part, $\\op{H}_0$, and a $\\lambda$-dependent part, $\\op{H}_1$, one can split the time evolution operator in a way similar to what is done in the interaction picture as $\\op{U}(\\phi) = \\op{U}_0(\\phi) \\op{U}_\\interaction(\\phi)$, where $\\op{U}_0(\\phi) = \\exp(-\\mathup{i} \\op{H}_0 \\phi)$ is the time evolution operator of the free system and the interaction time evolution operator is defined by the time-ordered exponential\n\\begin{equation}\n  \\op{U}_\\interaction(\\phi)\n  =\n  \\timeordering\n  \\exp\n  \\left(\n    -\\mathup{i}\n    \\int_0^\\phi\n    \\intmeasure{\\phi'}\\,\n    \\op{U}_0^{-1}(\\phi')\n    \\op{H}_1\n    \\op{U}_0(\\phi')\n  \\right)\n  \\mathperiod\n\\end{equation}\nNote that the inverse interaction time evolution operator requires anti-time-ordering.\nExpanding the interaction time evolution operator as a series in $\\lambda$,\n\\begin{equation}\n  \\op{U}_\\interaction(\\phi)\n  =\n  \\sum_{n=0}^\\infty\n  \\lambda^n\n  (\\op{U}_\\interaction)_n(\\phi)\n  \\mathcomma\n\\end{equation}\nand inserting into \\eref{eq:int_n_pert_series} gives for the $\\op{N}_n(\\phi)$\n\\begin{equation}\n  \\op{N}_n(\\phi)\n  =\n  \\sum_{m=0}^n\n  (\\op{U}_\\interaction^{-1})_m(\\phi)\n  \\op{U}_0^{-1}(\\phi)\n  \\op{N}\n  \\op{U}_0(\\phi)\n  (\\op{U}_\\interaction)_{n-m}(\\phi)\n  \\mathperiod\n\\end{equation}\nThe strategy would be to find the exact form of these $\\op{N}_n(\\phi)$ up to some order and then derive an effective Friedmann equation from these.\nWe will only do this to first order.\n\nThe leading term $\\op{N}_0(\\phi)$ is of course the same as in the free theory \\eref{eq:tm_n_heisenberg}.\nThe first order term $\\op{N}_1(\\phi)$ is given by\n\\begin{equation}\n\\eqalign{\n  \\fl\n  \\op{N}_1(\\phi)\n  =\n  \\frac{\\lambda}{2}\n  \\Big\\{\n    &\n    \\mkern-12mu (\\op{K}_+)^2\n    \\left[\n      3\n      - 2 (2 + 3\\mathup{i}\\omega\\phi) \\cosh(2\\omega\\phi)\n      + \\cosh(4\\omega\\phi)\n    \\right]\n    + \\hermconjtext\n    \\\\\n    &\n    +\n    \\op{K}_+\n    (2\\op{N} + 3)\n    \\left[\n      (\\mathup{i} - 3\\omega\\phi) \\sinh(2\\omega\\phi)\n      - \\mathup{i} \\sinh(4\\omega\\phi)\n    \\right]\n    + \\hermconjtext\n    \\\\\n    &\n    -\n    \\frac{1}{2}\n    \\sinh^2(2\\omega\\phi)\n    [\n      3 + 4 \\op{N}^2 + 8 \\op{N} + 8 \\op{K}_+ \\op{K}_-\n    ]\n  \\Big\\}\n  \\mathperiod\n}\n\\end{equation}\n\nAs outlined above, what one would like to do is take the expectation values of the perturbative expansion \\eref{eq:int_n_pert_series} and derive an effective Friedmann equation for arbitrary states, valid up to some order in $\\lambda$.\nHowever, already the first order expression for $\\op{N}(\\phi)$ is quite complicated and we were not able to derive a corresponding effective Friedmann equation for general states.\nTo ameliorate this we resort to taking the expectation values for some specific classes of coherent states.\n\nFirstly, we turn to the Fock coherent states.\nWriting for the parameter of the Fock coherent states \\eref{eq:tm_coh_state_fock} $\\sigma = \\sigma_1 + \\mathup{i} \\sigma_2$, one finds for the case $\\sigma_2 = 0$, i.e.\\ for real $\\sigma$,\n\\begin{equation}\n  \\label{eq:int_eff_fried_fock_real_order1}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4 \\omega^2\n    \\bigg(\n    &\n    N(\\phi)^2\n    + N(\\phi)\n    - \\sigma_1^2 (1 + \\sigma_1^2)\n    \\\\\n    &\n    \\eqalign{\n      +\n      \\frac{\n        \\lambda\n      }{\n        (1 + 2 \\sigma_1^2)^2\n      }\n      &\n      \\Big[\n        -4 N(\\phi)^3 (3 + 12 \\sigma_1^2 + 4 \\sigma_1^4)\n        \\\\\n        &\n        - 6 N(\\phi)^2 (3 + 15 \\sigma_1^2 + 12 \\sigma_1^4 + 4 \\sigma_1^6)\n        \\\\\n        &\n        + 6 N(\\phi) (-1 - 5 \\sigma_1^2 + 4 \\sigma_1^6)\n        \\\\\n        &\n        +  2 \\sigma_1^2 (3 + 24 \\sigma_1^2 + 51 \\sigma_1^4 + 48 \\sigma_1^6 +\n        20 \\sigma_1^8)\n      \\Big]\n      + \\bigO{\\lambda^2}\n      \\bigg)\n      \\mathperiod\n    }\n  }\n\\end{equation}\nFor the case $\\sigma_1 = 0$, i.e.\\ imaginary $\\sigma$, one finds the similar expression\n\\begin{equation}\n  \\fl\n  \\eqalign{\n    N'(\\phi)^2\n    =\n    4 \\omega^2\n    \\bigg(\n    &\n    N(\\phi)^2\n    + N(\\phi)\n    - \\sigma_2^2 (1 + \\sigma_2^2)\n    \\\\\n    &\n    \\eqalign{\n      +\n      \\frac{\n        \\lambda\n      }{\n        (1 + 2 \\sigma_2^2)^2\n      }\n      &\n      \\Big[\n        -4 N(\\phi)^3 (3 + 12 \\sigma_2^2 + 4 \\sigma_2^4)\n        \\\\\n        &\n        - 6 N(\\phi)^2 (3 + 9 \\sigma_2^2 - 4 \\sigma_2^4 - 4 \\sigma_2^6)\n        \\\\\n        &\n        + 6 N(\\phi) (-1 + \\sigma_2^2 + 16 \\sigma_2^4 + 12 \\sigma_2^6)\n        \\\\\n        &\n        +  2 \\sigma_2^2 (3 + 6 \\sigma_2^2 - 15 \\sigma_2^4 - 24 \\sigma_2^6\n          - 4 \\sigma_2^8)\n      \\Big]\n      + \\bigO{\\lambda^2}\n      \\bigg)\n      \\mathperiod\n    }\n  }\n\\end{equation}\nThe expression for the general case, where $\\sigma$ can be any complex number, is quite involved and we do not state it here.\n\nSecondly, we turn to the \\ac{pg} coherent states.\nFor the these states one finds the following general expression\n\\begin{equation}\n  \\fl\n  \\eqalign{\n  N'(\\phi)^2\n  =\n  4 \\omega^2\n  \\Bigg\\{\n  &\n    \\left(\n      N(\\phi)\n      + \\frac{1}{2}\n    \\right)^2\n    - 4 k^2\n    - \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}^2}{\\omega^2}\n  \\\\\n  &\n    \\eqalign{\n    +\n    \\lambda\n    \\frac{2k + 1}{4k}\n    \\Big[\n    &\n      - 8 N(\\phi)^3\n      + 12 N(\\phi)^2\n      \\left(\n        \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega} - 1\n      \\right)\n    \\\\\n    &\n      + 2 N(\\phi)\n      \\left(\n        6 \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        + 16 k^2\n        - 3\n      \\right)\n    \\\\\n    &\n      -\n      \\left(\n        \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        - 1\n      \\right)\n      \\left(\n        2 \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}^2}{\\omega^2}\n        + \\frac{\\expval[\\mathsubscript{PG}]{\\op{H}}}{\\omega}\n        + 16 k^2\n        - 1\n      \\right)\n    \\Big]\n    }\n  \\\\\n  &\n    +\n    \\bigO{\\lambda^2}\n    \\Bigg\\}\n    \\mathperiod\n  }\n\\end{equation}\nIt is remarkable that it is possible to write the right-hand side only in terms of $N(\\phi)$ and $\\expval[\\mathsubscript{PG}]{H}$.\nAs before, we would expect all higher order corrections to become relevant as soon as a regime is reached in which $\\abs{\\lambda N(\\phi)} \\lesssim 1$.\n\n\\begin{figure}[htpb]\n  \\centering\n  \\includegraphics{fig-int_quantum.pdf}\n  \\caption{\n    The relative relational expansion rate squared and the `effective equation of state parameter' \\eref{eq:int_eos_eff} as functions of $N$ for Fock coherent states with real parameter $\\sigma$.\n    The solid lines correspond to a truncation at zeroth order in $\\lambda$.\n    The dashed lines correspond to a truncation at first order in $\\lambda$.\n    The dotted lines correspond to a truncation at second order in $\\lambda$.\n    The dash-dotted lines correspond to the full nonperturbative case.\n    The parameters are:\n    $\\omega = 1$,\n    $\\lambda = - 10^{-3}$,\n    $\\sigma = 10$\n  }\\label{fig:int_quantum}\n\\end{figure}\n\nIn \\fref{fig:int_quantum} the relative relational expansion rate squared and the effective equation of state parameter \\eref{eq:int_eos_eff} are plotted as a function of $N$ for different truncations of the perturbative expansion and for the result of a numerical calculation.\\footnote{The numerical results were obtained by solving the time-dependent Schr\u00f6dinger equation of the Hamiltonian \\eref{eq:int_hamiltonian_fock_vars} in the position representation.}\nThe zeroth order truncation corresponds to the free case and corresponds to \\eref{eq:tm_eff_fried_fock} and the first order truncation is given in \\eref{eq:int_eff_fried_fock_real_order1}.\nNote that we do not state the second order truncation explicitly, since the expression is rather convoluted.\nThe state considered in this plot is a Fock coherent state with real parameter $\\sigma$.\nThe numerical results are in good agreement with the second order truncation.\nHowever, the second order truncation diverges quickly for values $\\abs{\\lambda N(\\phi)} \\gtrsim 1$, whereas the first order truncation does not.\nNote that the parameters chosen do not accommodate a regime in which $\\abs{\\lambda N(\\phi)} \\ll 1$, explaining the mismatch with the linearised theory.\nDue to numerical limitations we were not able to enter the asymptotic regime and determine the corresponding effective equation of state parameter for late times.\n\nWe conclude the present discussion of the quantum behaviour of the interacting toy model by revisiting the relative uncertainties of the volume as a function of relational time presented in \\sref{sec:tm_class_of_coh_stat_and_rel_uncert} and  illustrated in \\fref{fig:tm_rel_uncertainty_comp}.\nHere we restrict ourselves to the class of Fock coherent states.\nThe results of numerical calculations comparing the free and interacting cases for a range of parameters are given in \\fref{fig:int_rel_uncertainty_comp}.\nAn immediate effect of the interactions is that the expectation values diverge at finite relational time, resulting in different ranges of the relational time coordinate.\nThese divergences can already be anticipated from the discussion below \\eref{eq:int_classical_eff_friedmann_asymptotic}.\nThe key observation is that when interactions are present the relative uncertainties are not asymptotically constant but start growing as soon as the interactions become dominant.\nThe general statement that Fock coherent states become semiclassical at late times can therefore not be extended to this interacting case in an obvious way.\nAgain, this is also consistent with the expectation that mean-field methods break down in the interacting case when interactions begin to dominate over the quadratic Hamiltonian.\n\n\\begin{figure}[htbp]\n  \\centering\n  \\includegraphics{fig-int_rel_uncertainty_comp.pdf}\n  \\caption{The relative uncertainty of the volume operator as a function of $\\omega \\phi$ for Fock coherent states.\n    The solid line corresponds to the free case, $\\lambda = 0$.\n    The dashed line corresponds to the interacting case with $\\lambda = -10^{-3}$.\n  }\n  \\label{fig:int_rel_uncertainty_comp}\n\\end{figure}\n \n\\section{Conclusions}\nOur aim was to present a general perspective on the derivation of reliable effective Friedmann equations from given quantum dynamics of a \\ac{gft} model, building on various recent developments in the derivation of effective cosmological dynamics from \\ac{gft}.\nWithin this general perspective, the most important assumption was to restrict the \\ac{gft} dynamics to those of a single field mode, i.e., fixed values of the representation labels in the Peter--Weyl decomposition.\nThis simplification of the full dynamics in which all modes would be present can be seen as the most important limitation of our work.\nWhile there are arguments suggesting the dynamical emergence of a regime dominated by a single field mode in \\ac{gft}, showing such an emergence in models of interest for four-dimensional quantum gravity remains an outstanding challenge.\nOn the other hand, we were able to derive effective cosmological dynamics without relying on a mean-field approximation, and in general no assumptions needed to be made on the initial state.\n\nWe first focussed on the case of dynamics defined by a free (quadratic) Hamiltonian.\nSuch a Hamiltonian can be of harmonic form or of `upside-down' harmonic form; the latter case in which the Hamiltonian is unbounded from below is most relevant to \\ac{gft} cosmology, since it admits solutions expanding to infinity \\cite{WEw19,GPW19}.\nFor this case, we recovered and extended results of \\cite{AGW18} and \\cite{BBC19} for the resulting effective Friedmann equations.\nGeneric solutions exhibit singularity resolution in the sense of a minimal non-zero value for the volume, and interpolate between a collapsing and an expanding branch which both at large volumes approach classical \\ac{flrw} solutions.\nThese solutions depend on a parameter determined by the initial conditions (with no obvious classical analogue) which generates an asymmetry between the solution before and after the minimum for the volume.\nThe main new result in this part was a discussion of relative uncertainties in the two main physical observables---volume and energy---in three different classes of coherent states: Fock coherent states which have been used previously in \\ac{gft} cosmology, and \\acf{pg} and \\acf{bg} coherent states of $\\liealg{su}(1, 1)$.\nWe found that Fock coherent states approach a semiclassical regime at large volume, where relative uncertainties can be arbitrarily small, while PG states of interest here never reach such a regime.\nThe difference in our treatment compared with works such as \\cite{EM12} was that we assumed the Fock space structure of \\ac{gft} and only considered PG states that live in bosonic Fock representations.\nIn \\ac{gft}, Fock coherent states are a good choice for initial conditions that become semiclassical at low curvature.\nThis part of our analysis could be broadened by going to more general types of coherent states for which only a Casimir condition and the saturation of uncertainty relations are assumed, as was done for $\\liealg{su}(1, 1)$ in \\cite{BT14}.\n\nWe then added a quartic interaction term to the Hamiltonian to extend the derivation of effective Friedmann equations for \\ac{gft} models with polynomial interactions given in \\cite{CPS16} to situations where no mean-field approximation is assumed.\nAs for the free case, one has a choice between a quartic term for which the Hamiltonian is bounded from below, which leads to a recollapse and cyclic cosmology \\cite{CPS16}, or an interaction that admits solutions escaping to infinity, corresponding to a Universe expanding forever.\nWe focussed on the second case, the more common one in usual cosmology.\nTo understand the dynamics for this system, we first used a classical approximation in which one considers the basic dynamical variables as commuting phase space functions, with commutators replaced by Poisson brackets.\nIn this case we could derive an exact nonperturbative Friedmann equation; its unusual feature is the appearance of a square root involving the volume and energy on the right-hand side, preventing its straightforward interpretation in terms of effective perfect fluids.\nLinearising this equation in the interaction constant leads to a Friedmann equation with an effective dust term, which would reproduce the mean-field result of \\cite{CPS16}.\nThis linear correction only describes an intermediate regime in the expansion history, after which all orders become relevant.\nWe interpret this as signifying a failure of mean-field methods as soon as interactions become strong.\nThe asymptotic form of the effective Friedmann equation at late times would correspond to a matter component with equation of state $p=\\frac{1}{2}\\rho$ (instead of dust with $p=0$), or a modification of gravity on large scales in this model.\nWe then turned to the full quantum case in which one has to resort to a perturbative or numerical treatment.\nFor Fock coherent states, we find qualitatively similar results to the classical case: we derived a linearised correction to the effective Friedmann equation, and the full numerical solution quickly deviates from this regime as interactions become stronger.\nWe found numerical evidence that relative uncertainties in the volume start growing for Fock coherent states when the interactions become relevant, spoiling the property of these states to become semiclassical at late times that we observed for a quadratic Hamiltonian.\n\nA main direction for future work will be lifting the assumption that only a single field mode contributes to the \\ac{gft} dynamics.\nSince the quadratic part of the full \\ac{gft} Hamiltonian only couples pairs of modes, in the free case it would not be difficult to include additional modes into the analysis.\nFor this case one question would be whether some modes would always dominate asymptotically, as in the mean-field analysis of \\cite{Gie16}.\nWhen adding interaction terms however, we would expect the interaction of different modes to lead to a substantial modification of the effective cosmological dynamics away from the effectively free regime described by an \\ac{lqc}-like bounce.\nIn particular this could apply to the recent proposal of \\cite{GO18} for the generation of cosmological perturbations through quantum fluctuations in \\ac{gft} cosmology.\nIn the long term, we would then also aim to bring the interacting \\ac{gft} `toy' models studied in cosmological applications closer to candidate theories for full quantum gravity.\n\nAn entirely different but conceptually important direction would be to contrast the deparametrised framework used here, in which the scalar field $\\phi$ is used as a clock from the beginning, with a covariant setting in which one is free to choose different clocks, following e.g. the ideas of \\cite{Van18,Hoe18}.\nWe plan to investigate this in models which include multiple candidate matter clocks.\n\n \n\\section*{Acknowledgments}\nWe thank Martin Bojowald, Marco de Cesare, Daniele Oriti, Andreas Pithis and Edward Wilson-Ewing for helpful comments on an earlier version of the manuscript.\nWe also thank the referees for helpful suggestions for improvement.\nThe work of SG was funded by the Royal Society through a University Research Fellowship (UF160622) and a Research Grant for Research Fellows (RGF\\textbackslash R1\\textbackslash 180030).\nAP was supported by the same Research Grant for Research Fellows awarded to SG.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn \\cite{Lo2}, J.-L. Loday introduced a non-antisymmetric version of\nLie algebras, whose bracket satisfies the Leibniz relation (see\n(2.5)), therefore  called {\\it Leibniz algebra}. The Leibniz\nrelation, combined with antisymmetry, is a variation of the Jacobi\nidentity, hence Lie algebras are anti-symmetric Leibniz algebras. In\n\\cite{Lo4}, Loday also introduced an 'associative' version of\nLeibniz algebras, called {\\it associative dialgebras}, equipped with\ntwo binary operations, $\\vdash$ and $\\dashv$, which satisfy the five\nrelations (see the axiom (Ass) in section 2). These identities are\nall variations of the associative law, so associative algebras are\ndialgebras for which the two products coincide. The peculiar point\nis that the bracket $[a, b]=:a\\dashv b-b\\vdash a$ defines a Leibniz\nalgebra which is not antisymmetric, unless the left and right\nproducts coincide. Hence dialgebras yield a commutative diagram of\ncategories and functors\n\\begin{eqnarray*}\n{\\bf Dias}&\\stackrel{-}{\\to}& {\\bf Leib}\\\\\n\\downarrow&&\\downarrow\\\\\n{\\bf Assoc}&\\stackrel{-}{\\to}& {\\bf Lie}\n\\end{eqnarray*}\n\n\nSteinberg Lie algebras come from Steinberg groups, which are closely\nconnected with K-theory, and play a key role in the study of Lie\nalgebras graded by finite root systems of type $A$. By definition,\nthe {\\it Steinberg Lie algebra} $\\frak{st}\\,(n, A)$ over a $K$-algebra $A$\nis a Lie algebra generated by symbols $v_{ij}(a)$, $1\\le i\\ne j\\le\nn$, $a\\in A$, subject to the relations\n\\begin{eqnarray}\n&&v_{ij}(k_1a+k_2b)=k_1v_{ij}(a)+k_2v_{ij}(b),\\  \\hbox{ for } \\ a, b\\in D, \\ k_1,\nk_2\\in K;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=v_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k.\n\\end{eqnarray}\nIt is clear that the relation (3) makes sense only if $n\\ge3$.\n\n From \\cite{Fa} we see that the map $\\eta: a\\to v_{ij}(a)$ is one-to-one if and\n only if $A$ is an associative algebra for $n\\ge 4$ and  $A$ is an alternative\n algebra for $n=3$.\n\n\nIn 1992, S. Berman and R.V. Moody (\\cite{BM}) studied Lie algebras\ngraded by finite root systems of $A_l\\,(l\\ge 2)$, $D_l\\, (l\\ge 4)$,\n$E_l\\, (l=6, 7, 8)$ and obtained the structure of a Lie algebra over\n$K$ graded by the root system $\\Delta$ of type $X_l\\, ( l\\ge2)\\, (\nX_l=A_l, D_l, E_l)$.\n\n\nThe universal central extensions of Lie algebras graded by finite\nroot systems were studied in several papers (\\cite{B}, \\cite{Gar},\n\\cite{KL}, \\cite{Gao1}, \\cite{ABG1}, etc.).\n\n In this paper we shall consider Leibniz algebras graded by finite root systems of\n types $A, D$ and $E$. We also prove that\n\\begin{theo}{\\bf(Recognition Theorem).}\n Let $L$ be a Leibniz algebra over $K$ graded by the root system $\\Delta$ of\n type $X_l(l\\ge2)\\, (X_l=A_l, D_l, E_l)$.\n\n(1) If $X_l=A_l, l\\ge 3$, then there exists a unital associative\n$K$-dialgebra $R$ such that $L$ is centrally isogenous with\n$\\frak{sl}\\,(l+1, R)$;\n\n(2)  If $X_l=A_l, l=2$, then there exists a unital alternative\n$K$-dialgebra $R$ such that $L$ is centrally isogenous with\n$\\frak{stl}\\,(l+1, R)$, where $\\frak{stl}\\,(n, R)$ is defined in Section 2.4;\n\n(3)  If $X_l=D_l\\, (l\\ge 4), E_l\\, (l=6, 7, 8)$, then there exists a\nunital associative commutative $K$-dialgebra $R$ such that $L$ is\ncentrally isogenous with $\\dot{\\mg}\\otimes R$.\n\n\\end{theo}\n{\\bf Remark.} Two perfect Lie algebras $L_1$ and $L_2$ are called\n{\\it centrally isogenous} if they have the same universal central\nextension (up to isomorphism).\n\n\nThe paper is organized as follows. In Section 2, we recall some\nnotions of Leibniz algebras and dialgebras. In Section 3, we give\nthe definition of Leibniz algebras graded by finite root systems. In\nSections 4 and 5, we mainly prove the Recognition Theorem (Theorem\n1.1). Throughout this paper, $K$ denotes a field of characteristic\n0, $R$ a unital dialgebra over $K$.\n\n\n\n\n\\section{Dialgebras and Leibniz algebras}\n\n\nWe recall the notions of associative dialgebras, alternative\ndialgebras, Leibniz algebras and their (co)homology as defined in\n\\cite{Lo1}---\\cite{Lo4} and \\cite{L}.\n\n\n\\subsection{Dialgebras.}\n\n\\begin{defi}\\cite{Lo4}  A {\\it dialgebra} $D$ over $K$ is a $K$-vector space\n$D$ with two operations $\\dashv, \\vdash:D\\otimes D\\to D$, called left\nand right products.\n\\end{defi}\n\nA dialgebra is called unital if it is given a specified bar-unit: an\nelement $1\\in D$ which is a unit for the left and right products\nonly on the bar-side, that is, $1\\vdash a=a=a\\dashv 1$, for any\n$a\\in D$. A morphism of dialgebras is a $K$-linear map $f:D\\to D'$\nwhich preserves the products, i.e. $f(a\\star b)=f(a)\\star f(b)$,\nwhere $\\star$ denotes either the product $\\dashv$ or the product\n$\\vdash$.\n\n\n\\begin{defi} \\cite{Lo4} A dialgebra $D$ over $K$ is called {\\it associative}\nif the two operators $\\dashv$ and $\\vdash$ satisfy the following\nfive axioms:\n$$\\cases{a\\dashv(b\\dashv c)=(a\\dashv b)\\dashv c=a\\dashv(b\\vdash c),\\cr\n        (a\\vdash b)\\dashv c=a\\vdash(b\\dashv c),\\cr\n         (a\\vdash b)\\vdash c=a\\vdash (b\\vdash c)=(a\\dashv b)\\vdash c.  }\\leqno(Ass)$$\n\\end{defi}\n\n\nDenote by {\\bf Dias, Assoc} the categories of associative dialgebras\nand associative algebras over $K$ respectively. Then the category\n{\\bf Assoc} is a full subcategory of {\\bf Dias}.\n\nObviously, an associative dialgebra is an associative algebra if and\nonly if $a\\dashv b=a\\vdash b=ab$.\n\nThe concept of alternative dialgebras was introduced in \\cite{L} for\nthe study of the Steinberg Leibniz algebras.\n\n\\begin{defi}\\cite {L} A dialgebra $D$ over $K$ is called {\\it alternative}\nif the two operators $\\dashv$ and $\\vdash$ satisfying the following five axioms:\n$$\\cases{J_{\\dashv}(a, b, c)=-J_{\\vdash}(c, b, a), \\quad J_{\\dashv}(a, b, c)=J_{\\vdash}(b, c, a),\\cr\n        J_{\\times}(a, b, c)=-J_{\\vdash}(a, c, b),\\cr\n         (a\\vdash b)\\vdash c=(a\\dashv b)\\vdash c,\\quad  a\\dashv (b\\vdash c)\n         =a\\dashv (b\\dashv c), }\\leqno(Alt)$$\nwhere $J_{\\dashv}(a, b, c)=(a\\dashv b)\\dashv c-a\\dashv(b\\dashv c),\\  J_{\\vdash}(a, b, c)\n=(a\\vdash b)\\vdash c-a\\vdash(b\\vdash c)$ and $J_{\\times}(a, b, c)\n=(a\\vdash b)\\dashv c-a\\vdash(b\\dashv c)$.\n\\end{defi}\n\nObviously, an alternative dialgebra is an alternative algebra if\n$a\\dashv b=a\\vdash b=ab$. Moreover, the following formulae are clear\nfor an alternative dialgebras according to the definition.\n$$J_{\\dashv}(a, b, c)=-J_{\\dashv}(a, c, b),\\eqno(2.1)$$\n$$J_{\\vdash}(a, b, c)=-J_{\\vdash}(b, a, c),\\eqno(2.2)$$\n$$J_{\\times}(a, b, c)=-J_{\\times}(c, b, a).\\eqno(2.3)$$\nSo we also have $$J_{\\dashv}(a, b, b)=0, \\  J_{\\vdash}(a, a, b)=0, \\\nJ_{\\times}(a, b, a)=0.\\eqno(2.4)$$ {\\bf Examples.}\n\n1. Obviously, an associative (alternative) dialgebra is an\nassociative (alternative) algebra if and only if $a\\dashv b=a\\vdash\nb=ab$.\n\n2. {\\it Differential associative (alternative) dialgebra.} Let $(A, d)$ be\na differential associative (alternative) algebra. So by hypothesis,\n$d(ab)=(da)b+adb$ and $d^2=0$. Define left and right products on\n$A$ by the formulas\n$$x\\dashv y=xdy, \\quad x\\vdash y=(dx)y.$$\nThen $A$ equipped with these two products is an associative (alternative) dialgebra.\n\n3. Tensor product. Let $D$ and $D'$ be two associative dialgebras,\nthen $D\\otimes D'$ with multiplication $(a\\otimes a')\\star (b\\otimes b')=(a\\star\nb)\\otimes (a'\\star b')$, $\\star=\\dashv, \\vdash$, is also an associative\ndialgebra. Especially, if $D$ is a unital associative dialgebra,\nthen $M_n(D)=M_n(K)\\otimes D$ is also a unital associative dialgebra.\n\n4. Let $D$ be an associative (alternative) algebra.  On the module of $n$-space $D=A^{\\otimes n}$ one puts\n$$(x\\dashv y)_i=x_i(\\sum_{j=1}^ny_j), \\  i=1, \\cdots, n\\quad \\hbox {and}$$\n$$(x\\vdash y )_i=(\\sum_{j=1}^nx_j)y_i, \\  i=1, \\cdots, n. $$\nThen $(D, \\dashv, \\vdash)$ is an associative (alternative) dialgebra. For $n=1$, this is example 1.\n\n\n\n\\subsection{Leibniz algebras.} A {\\it Leibniz algebra} \\cite {Lo2} $L$\nis a vector space over a field $K$ equipped with a $K$-bilinear\nmap\n$$[-,-]: L\\times L\\to L$$\nsatisfying the Leibniz identity\n$$[x, [y, z]]= [[x, y], z]-[[x, z], y], \\quad \\forall \\;x, \\,y, \\,z\\in L.\\eqno(2.5)$$\n\nObviously, a Lie algebra is a Leibniz algebra. A Leibniz algebra is a Lie algebra if\nand only if\n$[x, x]=0$ for all $x\\in L$.\n\n\nSuppose that $L$ is a Leibniz algebra over $K$. For any $z\\in L$,\nwe define $\\hbox{\\rm ad}\\, z\\in \\hbox{End}_kL$ by\n$$\\hbox{\\rm ad}\\, z(x)=-[x, z], \\quad\\forall x\\in L.$$\nIt follows (2.5) that\n$$\\hbox{\\rm ad}\\, z([x, y])=[\\hbox{\\rm ad}\\, z(x), y]+[x, \\hbox{\\rm ad}\\, z(y)]$$\nfor all $x, y\\in L$. This says that $\\hbox{\\rm ad}\\, z$ is a derivation of\n$L$. We also call it an inner derivation of $L$.\n\n\nSimilarly, we also have the definition of general derivation of a\nLeibniz algebra and we denote by $\\hbox{\\rm Inn}\\,(L)$, $\\hbox{Der}\\,(L)$ the set of all\ninner derivations, derivations of $L$ respectively. They are also\nLeibniz algebras.\n\n\nLet $L$ be a Leibniz algebra over $K$. Consider the boundary map: $\\delta_n:L^{\\otimes n}\\to L^{\\otimes (n-1)}$ defined by\n$$\\delta_n(x_1\\otimes\\cdots\\otimes x_n)=\\sum_{1\\le i<j\\le n}(-1)^{j+1}x_1\\otimes \\cdots \\otimes x_{i-1}\\otimes[x_i, x_j]\\otimes x_{i+1}\\otimes\\cdots\\otimes\\hat x_j\\otimes\\cdots\\otimes x_n,$$\nwhere $\\hat x_j$ indicates that the term $x_j$ is omitted. One can\nshow that $\\delta^2=0$ (see \\cite{LP1}) and the complex $(L^{\\otimes\nn},\\, \\delta)\\ (L^0=K,\\, \\delta_1=0)$ gives the Leibniz homology\n$HL_*(L)$ of the Leibniz algebra $L$.\n\nLet $L$ be a Leibniz algebra over $K$. It is called perfect if\n$[L, L]=L$. A central extension of $L$ is a pair $(\\hat L, \\pi)$\nwhere $\\hat L$ is a Leibniz algebra and $\\pi: \\hat L\\to L$ is a\nsurjective homomorphism such that $\\hbox{Ker}\\,\\pi$ lies in the center of\n$\\hat L$ and the exact sequence $0\\to \\hbox{Ker}\\,\\pi\\to \\hat L\\to L\\to 0$\nsplits as $K$-module. The pair $(\\hat L, \\pi)$ is a universal\ncentral extension of $L$ if for every central extension $(\\tilde\nL, \\tau)$ of $L$ there is a unique homomorphism $\\psi:\\hat L\\to\n\\tilde L$ for which $\\tau\\circ\\psi=\\pi$. So the universal central\nextension is unique, up to isomorphism. A Leibniz algebra $L$ has\na universal central extension if and only if $L$ is perfect. If\n$(\\hat L, \\pi)$ is the universal central extension of $L$, then\n$$HL_2(L)\\cong \\hbox{Ker}\\,\\pi.\\eqno(2.6)$$\n\n\n\\noindent{\\bf Remark.} In \\cite{LP1}, \\cite{Gao1}, \\cite{Gao2} and\n\\cite{LH1}, the universal central extensions of  many infinite\ndimensional Lie algebras in the category of Leibniz algebras are\ndetermined.\n\nWe also denote by {\\bf Leib} and {\\bf Lie} the categories of Leibniz\nalgebras and Lie algebras over $K$ respectively.\n\n\nFor any associative dialgebra $D$, define $$[x, y]=x\\dashv y-y\\vdash x,$$\nthen $D$ equipped with this bracket is a Leibniz algebra. We denote it by $D_L$.\n The canonical map $D\\mapsto D_{L}$ induces a functor $(-):$ {\\bf Dias}$\\to${\\bf Leib}.\n\n\nFor a Leibniz algebra $L$, let $L_{Lie}$ be the quotient of $L$ by\nthe ideal generated by the elements $[x, y]+[y, x]$, for all $x,\ny\\in L$. It is clear that $L_{Lie}$ is a Lie algebra. The canonical\nprojection $L\\to L_{Lie}$ is universal among the maps from $L$ to\nLie algebras. In other words, the functor $(-)_{Lie}:$ {\\bf\nLeib}$\\to${\\bf Lie} is left adjoint to $inc:$ {\\bf Lie}$\\to${\\bf\nLeib}.\n\nMoreover, we have the following commutative diagram of categories\nand functors\n\\begin{eqnarray*}\n{\\bf Dias}&\\stackrel{-}{\\to}& {\\bf Leib}\\\\\n\\downarrow&&\\downarrow\\\\\n{\\bf Assoc}&\\stackrel{-}{\\to}& {\\bf Lie}\n\\end{eqnarray*}\n\nAs in the Lie algebra case, the universal enveloping associative\ndialgebra of a Leibniz algebra $L$ is defined as\n$$Ud(L):=(T(L)\\otimes L\\otimes T(L))\/\\{[x, y]-x\\dashv y+y\\vdash x|x, y\\in L\\},$$\nwhere elements $x,y$ of $L$ are regarded as elements in $K\\otimes\nL\\otimes K$.\n\n\\begin{prop}$\\cite{Lo4}$\n\n The functor $Ud:{\\bf Leib}\\to {\\bf Dias}$ is left adjoint to the functor $-:{\\bf Dias}\\to {\\bf Leib}$.\n\\end{prop}\n\n\n\nLet $L$ be a Leibniz algebra, then $M$ is said to be a right $L$-module if $M$ is a $K$-vector space  equipped with the action of $L$:\n$$[-,-]: M\\times L\\to M$$\nsatisfying\n\n$$[m,[x,y]]=[[m,x],y]-[[m,y],x],\\ \\forall x,\\, y\\in L, m\\in M.$$\n\n\n\nSo for a Lie algebra ${\\bf \\frak g}$, any right ${\\bf \\frak g}$-module in the Leibniz algebra case is just the right ${\\bf \\frak g}$-module in the Lie algebra case.\n\n\n\n\\subsection{Lie algebras graded by finite root system.}\n\nFirst we introduce the Steinberg Lie algebra. Steinberg Lie algebras\ncome from Steinberg groups, which are closely connected with\nK-theory.\n\nIf $R$ is a (nonassociative) ring with 1, and $n\\ge3$, the {\\it\nSteinberg group} (see \\cite{Fa}) St$_n(R)$ is the group generated by\nthe symbols $x_{ij}(a), 1\\le i\\ne j\\le n, a\\in R$, subject to the\nrelations\n\\begin{eqnarray*}\n&&x_{ij}(a+b)=x_{ij}(a)x_{ij}(b),\\  \\hbox{ for } \\ a, b\\in R, \\ k_1, k_2\\in K;\\\\\n&&[x_{ij}(a), x_{kl}(b)]=1\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[x_{ij}(a), x_{kl}(b)]=x_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k,\n\\end{eqnarray*} where $[x, y]=xyx^{-1}y^{-1}$ is the group commutator.\n\nBy definition, the {\\it Steinberg Lie algebra} (see \\cite{Fa})\n$\\frak{st}\\,(n, A)$ $(n\\ge 3)$ over a $K$-algebra $A$ is a Lie algebra\ngenerated by symbols $u_{ij}(a)$, $1\\le i\\ne j\\le n$, $a\\in A$,\nsubject to the relations\n\\begin{eqnarray*}\n&&u_{ij}(k_1a+k_2b)=k_1u_{ij}(a)+k_2u_{ij}(b),\\  \\hbox{ for } \\ a, b\\in A, \\ k_1, k_2\\in K;\\\\\n&&[u_{ij}(a), u_{kl}(b)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[u_{ij}(a), u_{kl}(b)]=u_{il}(ab)\\  \\hbox{ if } \\ i\\ne l\\\n\\hbox{and } \\  j= k.\n\\end{eqnarray*}\n\nDefine $i_n(A)=\\{a\\in A\\mid u_{ij}(a)=0 \\}$. Clearly, it is an ideal\nof $A$ and does not depend on the choice of $i\\ne j$.\n\n From \\cite{Fa}, we see that $i_n(A)=0$ if and only if $A$ is an associative algebra\n for $n\\ge 4$ and  $A$ is an alternative algebra for $n=3$.\n\nBy definition, a $K$-algebra $A$ is called {\\it alternative} if $(a, b,\nc)=-(c, b, a)=(b, c, a)$, where $(a, b, c)=(ab)c-a(bc)$.\n\n\nA Lie algebra $L$ over a field $K$ of characteristic 0 is {\\it\ngraded by the (reduced) root system} (see \\cite{BM}) $\\Delta$ or\nis {\\it $\\Delta$-graded} if\n\n(1) L contains as a subalgebra a finite-dimensional simple Lie\nalgebra $\\dot{\\mg}=H\\oplus\\bigoplus_{\\alpha\\in\\Delta}\\dot{\\mg}_{\\alpha}$ whose root\nsystem is $\\Delta$ relative to a split Cartan subalgebra $H=\\dot{\\mg}_0$;\n\n(2) $L=\\bigoplus_{\\alpha\\in\\Delta\\cup\\{0\\}}L_{\\alpha}$, where\n$L_{\\alpha}=\\{x\\in L\\mid [h, x]=\\alpha(h)x, \\forall h\\in H\\}$ for $\\alpha\\in \\Delta\\cup\\{0\\}$; and\n\n(3) $L_0=\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}]$.\n\n\n\n\n\\begin{theo} \\cite{BM}\n Let $L$ be a Lie algebra over $K$ graded by the root system $\\Delta$ of\n type $X_l\\, ( l\\ge2)\\, ( X_l=A_l, D_l, E_l)$ .\n\n(1) If $X_l=A_l,\\, l\\ge 3$, then there exists a unital associative\n$K$-algebra $A$ such that $L$ is centrally isogenous with\n$\\frak{sl}\\,(l+1, A)$.\n\n(2)  If $X_l=A_l,\\, l=2$, then there exists a unital alternative\n$K$-algebra $A$ such that $L$ is centrally isogenous with\n$\\frak{st}(l+1, A)$.\n\n(3)  If $X_l=D_l\\,(l\\ge 4), \\, E_l\\,(l=6, 7, 8)$, then there\nexists a unital associative commutative $K$-algebra $A$ such that\n$L$ is centrally isogenous with $\\dot{\\mg}\\otimes A$.\n\\end{theo}\n\n\n\n\n\n\n\n\\subsection{Steinberg Leibniz algebras }\n\n\nThe matrix Leibniz algebra\n$\\frak{gl}(n, D)$ is generated by all $n\\times n$ matrices with coefficients from a unital\nassociative dialgebra $D$, and $n\\ge 3$ with the bracket\n$$[E_{ij}(a), E_{kl}(b)]=\\delta_{jk}E_{il}(a\\dashv b)-\\delta_{il}E_{kj}(b\\vdash a),$$ for all $a, b\\in D$, where $E_{ij}(a)$ is the $n\\times n$ matrix with coefficient $a$ on $(i, j)$-th position and 0 in all others.\n\nClearly, $\\frak{gl}(n, D)$ is a Leibniz algebra. If $D$ is an associative algebra, then  $\\frak{gl}(n, D)$ becomes a Lie algebra.\n\n\nNow we consider the subalgebra $\\frak{sl}\\,(n, D):=[\\,\\frak{gl}(n, D), \\frak{gl}(n, D)\\,]$, which is called the {\\it special linear Leibniz algebra} with coefficients in $D$, of $\\frak{gl}(n, D)$.\n\n\nBy definition, the {\\it special linear Leibniz algebra} $\\frak{sl}\\,(n, D)$\nhas generators $E_{ij}(a),\\, 1\\le i\\ne j\\le n, a\\in D$, which\nsatisfy the following relations:\n\\begin{eqnarray*}\n&&[E_{ij}(a), E_{kl}(b)]=0\\  \\hbox{if}\\ i\\ne l\\ \\hbox{and}\\  j\\ne k;\\\\\n&&[E_{ij}(a), E_{kl}(b)]=E_{il}(a\\dashv b)\\  \\hbox{if}\\ i\\ne l\\ \\hbox{and}\\  j= k;\\\\\n&&[E_{ij}(a), E_{kl}(b)]=-E_{kj}(b\\vdash a)\\ \\hbox{if}\\ i= l\\\n\\hbox{and}\\  j\\ne k.\n\\end{eqnarray*}\n\n\nThe Steinberg Leibniz algebra was first introduced in \\cite{LP1} for\nassociative algebras and in \\cite{L} for associative dialgebras. By\ndefinition, the {\\it Steinberg Leibniz algebra} $\\frak{stl}\\,(n, D)$ is a\nLeibniz algebra generated by symbols $v_{ij}(a)$, $1\\le i\\ne j\\le\nn$, $a\\in D$, subject to the relations\n\\begin{eqnarray}\n&&v_{ij}(k_1a+k_2b)=k_1v_{ij}(a)+k_2v_{ij}(b),\\  \\hbox{ for } \\ a, b\\in D, \\ k_1, k_2\\in K;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=0,\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=v_{il}(a\\dashv b)\\  \\hbox{ if } \\ i\\ne l\\ \\hbox{and } \\  j= k;\\\\\n&&[v_{ij}(a), v_{kl}(b)]=-v_{kj}(b\\vdash a)\\  \\hbox{ if } \\ i= l\\\n\\hbox{and} \\  j\\ne k.\n\\end{eqnarray}\nIt is clear that the relations (6)---(7) make sense only if\n$n\\ge3$.\n\n\n\nLet $H_{ij}(a, b):=[v_{ij}(a), v_{ji}(b)]$ for $1\\le i\\ne j\\le n, a,\nb\\in D$, and $H$ the submodule of $\\frak{stl}\\,(n, D)$ generated by\n$H_{ij}(a, b), i\\ne j, a, b\\in D$. Define $i_n(D)=\\{a\\in D\\mid\nv_{ij}(a)=0 \\}$. Clearly, it is an ideal of $D$ and does not depend\non the choice of $i\\ne j$. The same consideration as in Steinberg\nLie algebras, we have\n\\begin{prop}\\cite{L}\n For a unital dialgebra $D$, $i_n(D)=0$ in $\\frak{stl}\\,(n, D)$ if and only if $D$ is\n associative for $n\\ge 4$ and  $D$ is alternative for $n=3$.\n\\end{prop}\n\nThe Steinberg Leibniz algebra $\\frak{stl}\\,(n, D)$ with $n\\ge3$ is perfect.\n\nThe homomorphism $\\psi$ of Leibniz algebras\n$$\\psi: \\quad \\frak{stl}\\,(n, D)\\to \\frak{sl}\\,(n, D)$$ by the rule\n$\\psi(v_{ij}(a))=E_{ij}(a)$ is a surjective homomorphism.\n\n\\begin{theo}\\cite{L}\nIf $n\\ge3$, then $(\\frak{stl}\\,(n, D), \\psi)$ is the universal central\nextension of the Leibniz algebra $\\frak{sl}\\,(n, D)$ with kernel $HHS_1(D)$\nfor a unital associative dialgebra $D$, where $HHS_1(D)$ is the\nfirst homology group of chain complex $(CS_*(D), d)$ defined in\n\\cite{Fa} (or see \\cite{L}) .\n\\end{theo}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{ Leibniz algebras graded by finite root systems}\n\n\\begin{defi}\nA Leibniz algebra $L$ over a field $K$ of characteristic 0 is\ngraded by the $($reduced$)$ root system $\\Delta$ or is\n$\\Delta$-graded if\n\n$(1)$ L contains as a subalgebra a finite-dimensional simple Lie\nalgebra $\\dot{\\mg}=H\\oplus\\bigoplus_{\\alpha\\in\\Delta}\\dot{\\mg}_{\\alpha}$ whose root\nsystem is $\\Delta$ relative to a split Cartan subalgebra $H=\\dot{\\mg}_0$;\n\n$(2)$ $L=\\bigoplus_{\\alpha\\in\\Delta\\cup\\{0\\}}L_{\\alpha}$, where\n$L_{\\alpha}=\\{x\\in L\\mid \\hbox{\\rm ad}\\, h(x)=-[x, h]=\\alpha(h)x, \\forall h\\in H\\}$\nfor $\\alpha\\in \\Delta\\cup\\{0\\}$; and\n\n$(3)$ $L_0=\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}]$.\n\\end{defi}\n\n{\\bf Remarks.}\n\n1. The conditions for being a $\\Delta$-graded Leibniz algebra imply\nthat $L$ is a direct sum of finite-dimensional irreducible right\n$\\dot{\\mg}$-modules whose highest weights are roots, hence either the\nhighest long root or short root or $0$.\n\n2. If $L$ is $\\Delta$-graded, then $L$ is perfect. Indeed,  the\nresult follows from $L_\\alpha=[L_\\alpha, H]$ for all $\\alpha\\in\\Delta$ and (3)\nas above.\n\n3. The Steinberg Leibniz algebra $\\frak{stl}\\,(n, D)$ is graded by the\nroot system of type $A_{n-1}$. Let $D$ be a commutative dialgebra,\nthen the Leibniz algebra $\\dot{\\mg}\\otimes D$ and its central extensions\nare graded by the root system of type $\\dot{\\mg}$ (see \\cite{LL}).\n\nNow we shall prove the Recognition Theorem (Theorem 1.1). So from\nnow on, we always set $\\Delta$ to be the root system of type $A_l\\,\n(l\\ge2)$, $D_l\\, (l\\ge 4)$ or $E_l \\,(l=6, 7, 8)$.\n\nSuch as that in \\cite{BM}, we also have the following results.\n\\begin{defi}\nAn ordered pair $(\\beta, \\gamma)\\in \\Delta\\times\\Delta$ is an $A_2$-pair if\n$(\\beta, \\gamma)=-1$. Thus $(\\beta, \\gamma)$ is an $A_2$-pair if and only if it is\na base for an $A_2$ subroot system\n of $\\Delta$.\n Two $A_2$-pairs $(\\beta, \\gamma)$ and $(\\beta', \\gamma')$ are equivalent,\n written  $(\\beta, \\gamma)\\sim(\\beta', \\gamma')$,\n if there is an element $w$ of the Weyl group $W$ of $\\Delta$ such that $\\beta'=w\\beta$\n and $\\gamma'=w\\gamma$.\nThe equivalent class of $(\\beta, \\gamma)$ is denoted by $[(\\beta, \\gamma)]$.\n\\end{defi}\n\n\\begin{lemm}\\cite{BM}\n$(1)$ If $\\Delta$ is of type $D$ or $E$, then there is only one\nequivalent class of $A_2$-pairs.\n\n$(2)$ If $\\Delta$ is of type $A$, then there are exactly two\nequivalent classes of $A_2$-pairs and furthermore, if $(\\beta, \\gamma)$ is\nan $A_2$-pair, then\n$$(\\beta, \\gamma)\\sim(-\\gamma, -\\beta), \\  (\\beta, \\gamma)\\not\\sim(\\gamma, \\beta).$$\n\n$(3)$ In all cases if $(\\beta, \\gamma)$ and $(\\gamma, \\delta)$ are $A_2$-pairs\nwith $(\\beta\\mid\\delta)=0$, then\n$$(\\beta, \\gamma)\\sim(\\gamma, \\delta)\\sim(\\beta, \\gamma+\\delta)\\sim(\\beta+\\gamma, \\delta).\\quad  \\rule[-.23ex]{1.0ex}{2.0ex}$$\n\\end{lemm}\n\n\n{\\bf Remark.} In type $A_l$, for definiteness, we distinguish the\ntwo classes as follows. We choose a base $\\Pi=\\{\\alpha_1, \\cdots,\n\\alpha_l\\}$ for $\\Delta$ once and for all with Coxeter-Dynkin diagram\n$\\alpha_1\\circ$----$\\circ$----$\\cdots$----$\\circ\\alpha_l$. Then $[(\\alpha_1,\n\\alpha_2)]$ is the positive class and an $A_2$-pair $(\\alpha, \\beta)\\in [\\alpha_1,\n\\alpha_2]$ is called positive pair, $[(\\alpha_2, \\alpha_1)]$ is the negative\nclass and an $A_2$-pair $(\\alpha, \\beta)\\in [\\alpha_2, \\alpha_1]$ is called\nnegative pair. For convenience, any $A_2$-pair in types $D_l$ and\n$E_l$ is either positive and negative.\n\nLet $L$ be a Leibniz algebra graded by $\\Delta$ and $\\dot{\\mg}\\subset L$\nthe split simple Lie algebra of Definition 3.1. Let $\\{e_\\alpha, H_i\\mid\n\\alpha\\in\\Delta, i=1, \\cdots, l\\}$ be a Chevalley basis of $\\dot{\\mg}$ so\nthat $e_\\alpha\\in L_\\alpha$ for any $\\alpha\\in\\Delta$, and $\\{H_i, i=1, \\cdots,\nl\\}$ a basis of $H$. Let $G=\\langle \\exp te_\\alpha\\mid t\\in K\\rangle$ be\nthe corresponding simply connected Chevalley group. For each\n$\\alpha\\in\\Delta$, $\\{e_\\alpha,\\, \\alpha^{\\vee}=[e_\\alpha, e_{-\\alpha}],\\, e_{-\\alpha}\\}$ is\nan $\\frak{sl}\\,_2$-triplet. Let\n$$n_\\alpha(t)=\\exp te_\\alpha\\exp(-t^{-1}e_{-\\alpha})\\exp te_\\alpha$$ and\nset $$N=\\langle n_\\alpha(t)\\mid \\alpha\\in\\Delta,\\, t\\in K^{\\times}\\rangle,$$\nwhere $K^{\\times}$ is the set of non-zero elements of $K$.\n\nLet $h_\\alpha(t)=n_\\alpha(t)n_\\alpha(1)^{-1}$ and $T=\\langle h_\\alpha(t)\\mid\n\\alpha\\in\\Delta,\\, t\\in K^{\\times}\\rangle$. Then $$T\\lhd N,\\, N\/T\\cong\nW.$$\n\nClearly, from the $\\Delta$-grading $\\hbox{\\rm ad}\\, e_\\alpha$ and  $\\hbox{\\rm ad}\\, e_{-\\alpha}$ are\nnilpotent on $L$, so we can define a homomorphism\n$$\\hbox{\\rm Ad}\\,: G\\to \\hbox{Aut}\\,(L)\\quad \\hbox{by}\\quad \\exp te_\\alpha\\to \\exp\\hbox{\\rm ad}\\, te_\\alpha, \\  \\alpha\\in \\Delta,\\, t\\in K.$$\n\nNext, recall that if $M$ is any integrable (right) $\\dot{\\mg}$-module\n(one on which $e_\\alpha$ acts locally nilpotent for all $\\alpha\\in\\Delta$)\nwith weight space decomposition $M=\\oplus M_\\lambda$ relative to $\\dot{\\mg}$\nthen, letting $r_\\alpha$ denote the reflection $r_\\alpha\\lambda=\\lambda-\\langle \\lambda,\n\\alpha^{\\vee}\\rangle\\alpha$, we have\n$$n_\\alpha(t)M_\\lambda=M_{r_\\alpha\\lambda} \\quad \\hbox{for all weights}\\ \\lambda,\\ \\hbox{and} \\eqno(3.1)$$\n$$\\hbox{the action of}\\  h_\\alpha(t)\\ \\hbox{restricted to}\\  M_\\lambda\\  \\hbox{is a\nscalar-multiplication by}\\  t^{\\langle \\lambda, \\alpha^{\\vee}\\rangle}. \\eqno(3.2)$$\n\nIn particular, the adjoint representation restricted to $\\dot{\\mg}$ acts\non $L$ in this way.\n\nFix any $\\alpha\\in\\Delta$. Let $W^\\alpha$ denote the stabilizer of $\\alpha$ in\n$W$ and let $N^\\alpha:=W^{\\alpha}T$. Then\n$$W^\\alpha=\\langle r_\\beta\\mid \\beta\\in\\Delta, (\\beta|\\alpha)=0\\rangle, \\eqno(3.3)$$\n$$N^\\alpha=\\langle n_\\beta(1)\\mid \\beta\\in\\Delta, (\\beta|\\alpha)=0\\rangle\\cdot T.\\eqno(3.4)$$\n\nFix any $\\beta\\in \\Delta$ and choose $w\\in W$ with $w\\alpha=\\beta$. Choose any\n$n\\in N$ with $nT\\leftrightarrow w$ in the isomorphism $N\/T\\cong W$.\nThen $\\hbox{\\rm Ad}\\,(n)L_\\alpha=L_\\beta$ by (3.1). Let the restriction of $\\hbox{\\rm Ad}\\,(n)$ to\n$L_\\alpha$ be denoted by $\\tilde\\lambda_{\\beta, \\alpha}$. If also $w'\\alpha=\\beta$ and\n$n'T\\leftrightarrow w'$, then $w^{-1}w'\\in W^\\alpha$ and $n^{-1}n'\\in\nN^\\alpha$. Thus $n'\\in nN^\\alpha=n\\langle n_\\gamma(1)\\mid \\gamma\\in\\Delta,\\,\n(\\beta|\\alpha)=0\\rangle T$. Elements of $T$ acts as scalar multiplications\non $L_\\alpha$. Furthermore,\n$$(\\gamma|\\alpha)=0\\Rightarrow (\\alpha\\pm\\gamma, \\alpha\\pm\\gamma)=4\\Rightarrow \\alpha\\pm\\gamma\\not\\in\\Delta\\cup\\{0\\}$$\n$$\\Rightarrow \\hbox{\\rm Ad}\\,\\exp(te_{\\pm\\gamma})\\ \\hbox{acts as the identity on}\\ L_\\alpha, \\forall\\, t\\in K^{\\times}\\Rightarrow n_\\gamma(1)\\ \\hbox{acts as the identity on}\\ L_\\alpha.$$\nThis establishes that $\\tilde\\lambda_{\\beta, \\alpha}$ is determined, up to a\nnonzero scalar multiple, by $\\alpha$ and $\\beta$ and does not otherwise\ndepend on our choice of $w$ or $n$. Since also $n\\dot{\\mg}_\\alpha=\\dot{\\mg}_\\beta, \\\nne_\\alpha=\\varepsilon e_\\beta$ for some $\\varepsilon\\in K^{\\times}$ (actually $\\varepsilon=\\pm\n1$).\n\n\\begin{defi}  $\\lambda_{\\beta, \\alpha}: L_\\alpha\\to L_\\beta$ is the $K$-linear map\n$\\varepsilon^{-1}\\tilde\\lambda_{\\beta, \\alpha}.$\n\\end{defi}\n\nClearly, $\\lambda_{\\beta, \\alpha}$ depends only on $\\alpha, \\beta$ and the choice of\nChevally basis. We see that\n$$\\lambda_{\\alpha, \\beta}=\\lambda_{\\beta, \\alpha}^{-1}, \\eqno(3.5)$$\n$$\\lambda_{\\alpha, \\beta}\\lambda_{\\beta, \\gamma}=\\lambda_{\\alpha, \\gamma}, \\eqno(3.6)$$\n$$\\lambda_{\\alpha, \\alpha}=1. \\eqno(3.7)$$\n\nWith $\\alpha$ fixed as above, define $R=L_\\alpha$ (as a $K$-space).\nEventually, $R$ will have a life of its own, independent of $\\alpha$.\nFor this reason, given $r\\in R$, we shall write it as $e_\\alpha(r)$ when\nwe think of it as being in $L_\\alpha$. We identify $K$ as a subspace of\n$L_\\alpha$ by $ae_\\alpha=e_\\alpha(a)$ for some $a\\in K$.\n\nNow if $\\beta\\in \\Delta $ is arbitrary, then the $K$-linear map\n$\\lambda_{\\beta, \\alpha}:L_\\alpha\\to L_\\beta$ is defined and we define\n$$e_\\beta(r)=\\lambda_{\\beta, \\alpha}e_\\alpha(r).\\eqno(3.8)$$\n From (3.7), we see that this definition is consistent when $\\beta=\\alpha$.\nFrom (3.5) and (3.6), we see that (3.8) holds for all pairs of roots\n $\\alpha, \\beta\\in\\Delta.$\n\nWe note that for all $a, b\\in K,\\, r, s\\in R$, we have\n$e_\\beta(ar+bs)=ae_\\beta(r)+be_\\beta(s)$. This allows us to write $x_\\beta(r),\\,\nr\\in R$, for any $x_\\beta\\in \\dot{\\mg}_\\beta$ unambiguously. Clearly,\n$$x_\\beta(r)=x_\\beta(s)\\Longleftrightarrow r=s.$$\n\n\\begin{lemm}\nFor all $n\\in N,\\, \\beta\\in\\Delta,\\, r\\in R$, we have\n$$(\\hbox{\\rm Ad}\\, n)(e_\\beta(r))=((\\hbox{\\rm Ad}\\, n)e_\\beta)(r).$$\n\\end{lemm}\n{\\bf Proof.} Let $nT\\leftrightarrow w\\in W$ and $\\gamma=w\\beta$, so both\nsides of the asserted equality lie in $L_\\gamma$. Since $\\lambda_{\\gamma, \\beta}$\nequals, up to a scalar, $\\hbox{\\rm Ad}\\,(n)|_{L_\\beta}$, $b\\lambda_{\\gamma, \\alpha}=\\hbox{\\rm Ad}\\,\nn(\\lambda_{\\beta, \\alpha})$, for some $b\\in K^{\\times}$. Then\n$$be_\\gamma=b\\lambda_{\\gamma, \\alpha}e_\\alpha=(\\hbox{\\rm Ad}\\, n)\\lambda_{\\beta, \\alpha}e_\\alpha=(\\hbox{\\rm Ad}\\, n)e_\\beta$$ and hence\n$$(\\hbox{\\rm Ad}\\, n)e_\\beta(r)=(\\hbox{\\rm Ad}\\, n)\\lambda_{\\beta, \\alpha}e_\\alpha(r)=b\\lambda_{\\gamma, \\alpha}e_\\alpha(r)=be_\\gamma(r)\n=((\\hbox{\\rm Ad}\\, n)e_\\beta)(r).\\quad \\rule[-.23ex]{1.0ex}{2.0ex}$$\n\nNow we set $$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](m_{(\\beta, \\gamma)}(r,\ns)).\\eqno(3.9)$$\n\n\\begin{lemm}\nIf $(\\beta', \\gamma')\\in [(\\beta, \\gamma)]$, then $m_{(\\beta', \\gamma')}=m_{(\\beta, \\gamma)}$.\n\\end{lemm}\n{\\bf Proof.}  Let $m'=m_{(\\beta', \\gamma')}, m=m_{(\\beta, \\gamma)}$. Choose $w\\in\nW$ with $\\beta'=w\\beta$ and $\\gamma'=w\\gamma$. Let $a, b, c, \\varepsilon, \\varepsilon'\\in\nK^{\\times}$ be chosen such that\n$$a\\hbox{\\rm Ad}\\,(n)e_\\beta=e_{\\beta'}, \\  b\\hbox{\\rm Ad}\\,(n)e_\\gamma=e_{\\gamma'},\\  c\\hbox{\\rm Ad}\\,(n)e_{\\beta+\\gamma}=e_{\\beta'+\\gamma'},$$\n$$[e_\\beta, e_\\gamma]=\\varepsilon e_{\\beta+\\gamma}, \\  [e_{\\beta'}, e_{\\gamma'}]=\\varepsilon' e_{\\beta'+\\gamma'}.$$\nSince $\\hbox{\\rm Ad}\\,(n)$ is an automorphism, $\\varepsilon'=abc^{-1}\\varepsilon$. Now for\n$r, s\\in R$,\n\\begin{eqnarray*}\n&&[e_{\\beta'}, e_{\\gamma'}]m'(r, s)=[e_{\\beta'}(r), e_{\\gamma'}(s)]\\\\\n&=&[\\lambda_{\\beta', \\beta}e_{\\beta}(r), \\lambda_{\\gamma', \\gamma}e_{\\gamma}(s)]\\\\\n&=&[a\\hbox{\\rm Ad}\\,(n)e_\\beta(r), b\\hbox{\\rm Ad}\\,(n)e_\\gamma(s)]=ab\\hbox{\\rm Ad}\\,(n)[e_\\beta(r), e_\\gamma(s)]\\\\\n&=&ab\\hbox{\\rm Ad}\\,(n)[e_\\beta, e_\\gamma](m(r, s))\\\\\n&=&ab\\varepsilon\\hbox{\\rm Ad}\\,(n)e_{\\beta+\\gamma}(m(r, s))=ab\\varepsilon c^{-1} e_{\\beta'+\\gamma'}(m(r, s))\\\\\n&=&\\varepsilon'e_{\\beta'+\\gamma'}(m(r, s))=[e_{\\beta'}, e_{\\gamma'}](m(r, s)).\n\\end{eqnarray*}\nThus  $m_{(\\beta', \\gamma')}=m_{(\\beta, \\gamma)}$  and $m=m'$.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n From the above Lemma, we can define two multiplications on $R$:\n\n(1) For a positive $A_2$-pair $(\\beta, \\gamma)$ (see the Remark after Lemma\n3.3), we define $\\dashv: R\\times R\\to R$ given by\n$$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](r\\dashv s).$$\n\n(2) For a negative $A_2$-pair $(\\beta, \\gamma)$, we define $\\vdash: R\\times\nR\\to R$ given by\n$$[e_\\beta(r), e_\\gamma(s)]=[e_\\beta, e_\\gamma](s\\vdash r).$$\n\nClearly from Lemmas 3.5 and 3.6, we know that the above definition\nis well-defined on $D$. Moreover, if $\\Delta$ is  of type $D_l\\;\n(l\\ge4)$ or $E_l\\; (l=6, 7, 8)$, then $r\\dashv s=s\\dashv r$, i.e.,\n$R$ is a commutative dialgebra since $m_{\\beta, \\gamma}(r, s)=m_{\\gamma, \\beta}(r,\ns)$.\n\n\nLet $(\\alpha, \\beta)$ be a $A_2$-pair. Then $r_\\beta\\alpha=\\alpha+\\beta$. Suppose that\n$e_\\alpha, e_\\beta$ have already been chosen as our Chevalley basis. Let\n$n_\\beta=\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})\\exp\\hbox{\\rm ad}\\, e_\\beta$. Then by\ndefinition, we get that\n\\begin{eqnarray*}\nn_\\beta e_\\alpha(r)=n_\\beta(e_\\alpha(r))&=&\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})\\exp\\hbox{\\rm ad}\\, e_\\beta(e_\\alpha(r))\\\\\n&=&\\exp\\hbox{\\rm ad}\\, e_\\beta\\exp\\hbox{\\rm ad}\\, (-e_{-\\beta})(e_\\alpha(r)-[e_\\alpha(r), e_\\beta])\\\\\n&=&\\exp\\hbox{\\rm ad}\\, e_\\beta(e_\\alpha(r)-[e_\\alpha(r), e_\\beta]-[e_\\alpha(r)+[e_\\alpha(r), e_\\beta], e_{-\\beta}])\\\\\n&=&-[e_\\alpha(r), e_\\beta].\n\\end{eqnarray*}\nThus $$(n_\\beta e_\\alpha)(r)=-[e_\\alpha(r), e_\\beta].$$ In particular, $$n_\\beta\ne_\\alpha=-[e_\\alpha, e_\\beta].$$\n\nThe element $e_\\alpha\\in L_\\alpha$ defines an element of $R$ that has been\nidentified with $1\\in K: e_\\alpha(1)=1e_\\alpha$. For any $\\beta\\in \\Delta$ we\nhave $e_\\beta(1)=\\lambda_{\\beta, \\alpha}(e_\\alpha(1))=e_\\beta$, so every basis element\n$e_\\beta$ defines the same element of $R$.\n\nNote that $1$ is bar-unit of $R$. For a positive $A_2$-pair $(\\alpha,\n\\beta)$, we have $[e_\\alpha(r), e_\\beta(1)]=[e_\\alpha, e_\\beta](r\\dashv 1)$. But\n\\begin{eqnarray*}\n[e_\\alpha(r), e_\\beta(1)]&=&[e_\\alpha(r), e_\\beta]=-n_\\beta(e_\\alpha(r))\\\\\n&=&-(n_\\beta e_\\alpha)(r)=[e_\\alpha, e_\\beta](r).\n\\end{eqnarray*}\nThus $r\\dashv 1=r$. Similarly, for a negative $A_2$-pair $(\\alpha, \\beta)$,\nwe can prove that $1\\vdash r=r.$ So $R$ is a unital dialgebra.\n\nNext, we investigate the associativity of the multiplication on $R$.\n\nNow we assume that $l\\ge 3$. Let $(\\beta, \\gamma)$ and $(\\gamma, \\delta)$ be two\npositive $A_2$-pairs with $(\\beta, \\delta)=0$ (such pairs exist if\n$l\\ge3$). By Lemma 3.3 (3), $m_{(\\beta, \\gamma)}=m_{(\\gamma, \\delta)}$. By the\nJacobi identity and $[L_\\beta, L_\\delta]=0$, we obtain that\n$$[[e_\\beta(r), e_\\gamma(s)], e_\\delta(t)]=[e_\\beta(r), [e_\\gamma(s),  e_\\delta(t)]],$$\n$$[e_\\beta(r), [e_\\delta(t), e_\\gamma(s)]]=-[[e_\\beta(r), e_\\gamma(s)],  e_\\delta(t)],$$\n$$[[e_\\gamma(s), e_\\beta(r)], e_\\delta(t)]=-[[e_\\gamma(r), e_\\delta(t)],  e_\\gamma(s)].$$\nIt follows that $(r\\dashv s)\\dashv t=r\\dashv (s\\dashv\nt)=r\\dashv(s\\vdash t)$ and $(r\\vdash s)\\dashv t=r\\vdash (s\\dashv\nt)$. Similarly, we can prove the left identities in the axiom(Ass).\nThe $l=2$ case will be handled in \\S5.2.\n\n\nFrom the above, we have proved the following theorem.\n\\begin{theo}\nLet $L$ be a $\\Delta$-graded Leibniz algebra over a field $K$ of\ncharacteristic $0$, where $\\Delta$ is a simply-laced finite\nindecomposable root system of rank $l\\ge 2$. Let $\\dot{\\mg}$ be the\nassociated split simple Lie subalgebra with root system $\\Delta$.\nFor any root $\\alpha\\in\\Delta$ and let $R=L_\\alpha$ as a $K$-vector space.\nRelative to a Chevally basis $\\{e_\\beta\\mid \\beta\\in\\Delta\\}\\cup\\{H_i\\mid\ni=1, \\cdots, l\\}$, define the map $\\lambda_{\\beta, \\alpha}:L_\\alpha\\to L_\\beta$ of\nDefinition 3.4 and the element $e_\\beta(r), r\\in R$ of (3.8). Then $R$\nis a unital $K$-dialgebra. Moreover, $R$ is associative if the rank\n$l\\ge 3$. If $\\Delta$ is of type $D$ or $E$, then $R$ is\ncommutative.\n\\end{theo}\n\n\n\n\n\n\\section{Centrally isogenous of $\\Delta$-graded Leibniz algebra}\n\nLet $L$ and $L'$ be $\\Delta$-graded Leibniz algebras over $K$ for\nthe same finite root system $\\Delta$. Then their associated split\nsimple subalgebras $\\dot{\\mg}$ and $\\dot{\\mg}'$ are isomorphic. Simply denoted\nthem by $\\dot{\\mg}$, so $\\dot{\\mg}$ is a subalgebra both of $L$ and $L'$.\n\n\\begin{defi}\nA homomorphism $\\varphi: L\\to L'$ is $\\Delta$-homomorphic if\n$\\varphi|_{\\dot{\\mg}}=id_{\\dot{\\mg}}$.\n\\end{defi}\n\nLet $\\varphi: L\\to L'$ be a $\\Delta$-homomorphism. From the definition,\nit follows at once that $\\varphi(L_\\alpha)\\subset L_\\alpha'$ for all $\\alpha\\in\n\\Delta\\cup\\{0\\}$. Let $R$ and $R'$ be the $K$-dialgebras associated\nto $L$ and $L'$ respectively defined by a certain choice of\nChevalley basis for $\\dot{\\mg}$. For each $\\alpha\\in \\Delta$, we have\n$R=L_\\alpha\\stackrel{\\varphi}{\\to}L_\\alpha'=R'$, so $\\varphi$ determines a map\n$$\\bar\\varphi_\\alpha: R\\to R'.$$\n\\begin{prop}\n$(1)$ $\\bar\\varphi_\\alpha: R\\to R'$ is independent of the choice of\n$\\alpha\\in\\Delta$ (so we can denote it by $\\bar\\varphi$) and is a\nhomomorphism of dialgebras. Furthermore, $\\bar\\varphi$ is injective\n(resp. surjective, bijective) if $\\varphi$ is injective (resp.\nsurjective, bijective).\n\n$(2)$ If $\\bar\\varphi$ is an isomorphism, then the $\\Delta$-homomorphism\n$\\varphi$ is central and $L$ and $L'$ are centrally isogenous.\n\\end{prop}\n\n{\\bf Proof.} (1) Let $\\{e_\\beta\\}_{\\beta\\in\\Delta}\\cup \\{H_i\\}_{i=1}^l$ be\nthe Chevalley basis of $\\dot{\\mg}$. We have $L_\\alpha=\\{e_\\alpha(r)\\mid r\\in R\\}$\nand $L_\\alpha'=\\{e_\\alpha(r)\\mid r\\in R'\\}$. Furthermore, $ke_\\alpha=e_\\alpha(k)$\nfor all $k\\in K$ and $e_\\alpha(k_1r+k_2s)=k_1e_\\alpha(r)+k_2e_\\alpha(s)$ for all\n$r, s\\in R, R'$, $ k_1, k_2\\in K$. By definition,\n$\\varphi(e_\\alpha(r))=e_\\alpha(\\bar\\varphi_\\alpha(r))$ for all $r\\in R$, and it follows\nthat $\\bar\\varphi_\\alpha$ is $K$-linear and maps $1\\in R$ to $1\\in R'$.\n\nNow suppose that $(\\alpha, \\beta)$ is a positive $A_2$-pair. Then\n\\begin{eqnarray*}\n[e_\\alpha, e_\\beta](\\bar\\varphi_{\\alpha+\\beta}(r\\dashv s))&=&\\varphi([e_\\alpha, e_\\beta](r\\dashv s))\n=\\varphi([e_\\alpha(r), e_\\beta(s)])\\\\\n&=&[e_\\alpha(\\bar\\varphi_\\alpha(r)), e_\\beta(\\bar\\varphi_\\beta(s))]=[e_\\alpha,\ne_\\beta](\\bar\\varphi_\\alpha(r)\\dashv\\bar\\varphi_\\beta(s)).\n\\end{eqnarray*}\nThus  $$\\bar\\varphi_{\\alpha+\\beta}(r\\dashv\ns)=\\bar\\varphi_\\alpha(r)\\dashv\\bar\\varphi_\\beta(s).\\eqno(4.1)$$ Similarly, we have\n$$\\bar\\varphi_{\\alpha+\\beta}(r\\vdash\ns)=\\bar\\varphi_\\alpha(r)\\vdash\\bar\\varphi_\\beta(s).\\eqno(4.2)$$\n\n\nWith $s=1$, we get $\\bar\\varphi_{\\alpha+\\beta}(r)=\\bar\\varphi_\\alpha(r)$. From this, it\nis easy to see that $\\bar\\varphi$ is independent of $\\alpha$. Thus, $(4.1)$\nand (4.2) show the homomorphism property of $\\bar\\varphi$. The remaining\nparts of (1) are obvious.\n\n(2) If $\\bar\\varphi$ is an isomorphism, then  $\\bar\\varphi_\\alpha$ is an\nisomorphism for each $\\alpha\\in\\Delta$, so $\\hbox{Ker}\\,\\varphi\\subset L_0$. Since\n$[L_\\alpha, \\hbox{Ker}\\,\\varphi]\\subset L_\\alpha\\cap\\hbox{Ker}\\,\\varphi=\\{0\\}$, we see that\n$\\hbox{Ker}\\,\\varphi$ lies in the center of $L$ from Definition 3.1. Since $L$\nand $L'$ are perfect and $L'\\cong L\/Z$ for some central ideal $Z$,\nthey have the same universal central extension.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\n\\begin{prop}\nLet $L$ be a Leibniz algebra graded by $\\Delta$ and $(\\frak u, \\varphi)$\nthe universal central extension of $L$. Then $\\frak u$ is graded by\n$\\Delta$ and has the same associated dialgebra as that of $L$.\nFurthermore, $\\varphi$ is a $\\Delta$-homomorphism and $\\varphi: \\frak\nu_\\alpha\\to L_\\alpha$ is a homomorphism for all $\\alpha\\in\\Delta$. In\nparticular, $\\hbox{Ker}\\,\\varphi\\subset \\frak u_0$.\n\\end{prop}\n{\\bf Proof.} It is well known that $\\dot{\\mg}$ is centrally closed in the\ncategory of Leibniz algebras (\\cite{Gao2}). Thus the central\nextension $\\varphi: \\varphi^{-1}(\\dot{\\mg})\\to \\dot{\\mg}$ splits and we may view\n$\\dot{\\mg}$ as a subalgebra of $\\frak u$. In particular, $H$ is a\nsubalgebra of $\\frak u$. We define\n\\begin{eqnarray*}\n&&\\tilde\\frak u_\\alpha:=\\varphi^{-1}(L_\\alpha), \\quad \\alpha\\in \\Delta\\cup\\{0\\},\\\\\n&&\\frak u_\\alpha:=\\cases{[\\tilde\\frak u_\\alpha, H], \\quad \\alpha\\in\\Delta,\\cr\n\\tilde\\frak u_0, \\quad \\alpha=0.}\n\\end{eqnarray*}\nFor all $h_1, h_2\\in H, x\\in \\tilde\\frak u_\\alpha$, we have\n$$\\hbox{\\rm ad}\\, h_1([x, h_2])=-[[x, h_2], h_1]=-[[x, h_1], h_2]=[\\alpha(h_1)x+z, h_2]=\\alpha(h_1)[x, h_2],$$\nwhere $z\\in\\hbox{Ker}\\,\\varphi$. This proves that\n$$\\frak u_\\alpha \\ \\hbox{is an  $\\alpha$-weight space for}\\  \\hbox{\\rm ad}\\,_{\\frak u}H,\\, \\alpha\\in\\Delta.\\eqno(4.3)$$\n It follows that for $\\alpha\\in\\Delta$, $\\frak u_\\alpha\\cap \\hbox{Ker}\\,\\varphi=\\{0\\}$ and hence\n$$\\varphi|_{\\frak u_\\alpha}:\\frak u_\\alpha\\to L_\\alpha\\eqno(4.4)$$ is an isomorphism of vector spaces.\n\nLet $x\\in\\frak u$ and write $x=\\sum_{\\alpha\\in\\Delta\\cup\\{0\\}}\\tilde x_\\alpha$,\nwhere $\\tilde x_\\alpha\\in\\tilde\\frak u_\\alpha$. Fix $h\\in H$ so that for all\n$\\alpha\\in\\Delta, \\alpha(h)\\ne 0$. Then\n$$\\tilde x_\\alpha-\\alpha(h)^{-1}[\\tilde x_\\alpha, x]\\in\\hbox{Ker}\\,\\varphi\\subset \\tilde \\frak u_0=\\frak u_0.\\eqno(4.5)$$\nThus we may rewrite $x$ as $\\sum_{\\alpha\\in\\Delta\\cup\\{0\\}} x_\\alpha$, where\n$x_\\alpha\\in\\frak u_\\alpha$ and $x_0\\in \\frak u_0$. It follows that\n$$\\frak u=\\frak u_0+\\sum_{\\alpha\\in\\Delta}\\frak u_\\alpha.$$\n\nNow\n$$\\frak u_0=\\tilde\\frak u_0=\\varphi^{-1}(\\sum_{\\alpha\\in\\Delta}[L_\\alpha, L_{-\\alpha}])\n=\\sum_{\\alpha\\in\\Delta}[\\tilde \\frak u_\\alpha, \\tilde \\frak u_{-\\alpha}]+\\hbox{Ker}\\,\\varphi.$$\nSince $\\tilde \\frak u_\\alpha=\\frak u_\\alpha+\\hbox{Ker}\\,\\varphi$ from (4.3), we have\n$$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak u_{-\\alpha}]+\\hbox{Ker}\\,\\varphi.$$\nThis proves that $\\frak u_0$ is a $0$-eigenspace for $H$. Now we see\nthat $\\frak u_\\alpha$ is exactly the $\\alpha$-eigenspace for $H$ for all\n$\\alpha\\in\\Delta\\cup\\{0\\}$ and $[\\frak u_\\alpha, \\frak u_\\beta]\\subset \\frak\nu_{\\alpha+\\beta}$, whenever $\\alpha+\\beta\\in\\Delta\\cup\\{0\\}$. Also since $\\frak\nu=[\\frak u, \\frak u]$, we see that\n$$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak u_{-\\alpha}]+[\\frak u_0, \\frak u_0].$$\nBut\n$$[\\frak u_0, \\frak u_0] =\\sum_{\\alpha, \\beta\\in\\Delta}[[\\frak u_\\alpha, \\frak u_{-\\alpha}],\n[\\frak u_\\beta, \\frak u_{-\\beta}]]\\subset \\sum_{\\gamma\\in\\Delta} [\\frak u_\\gamma,\n\\frak u_{-\\gamma}],$$\n so $$\\frak u_0=\\sum_{\\alpha\\in\\Delta}[\\frak u_\\alpha, \\frak\nu_{-\\alpha}].$$ This proves that $\\frak u$ is graded by $\\Delta$. From\n(4.4) and the construction of $\\varphi$, we see that $\\varphi$ is a\n$\\Delta$-homomorphism and $\\bar\\varphi$ is an isomorphism. Thus the\ndialgebra associated to $\\frak u$ is the same as that associated to\n$L$. \\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\\begin{prop}\nLet $L$ be a Leibniz algebra graded by $\\Delta$, $Z$ the center of\n$L$ and $Z'$ any subspapce of $Z$. Then\n\n$(1)$  $Z\\subset L_0$.\n\n$(2)$  $L\/Z'$ has a unique structure as a Leibniz algebra graded\nby $\\Delta$ that makes the natural map $\\pi: L\\to L\/Z'$ a\n$\\Delta$-homomorphism.\n\n$(3)$  For all $\\alpha\\in\\Delta$, $L_\\alpha\\stackrel{\\pi}{\\cong}(L\/Z')_\\alpha$ as\nvector spaces.\n\n$(4)$  The dialgebras associated to $L$ and $L\/Z'$ are isomorphic\nby the map $\\bar\\pi$ induced by $\\pi$ as in Proposition 4.2.\n\\end{prop}\n{\\bf Proof.} Since for all $h\\in H$, $\\hbox{\\rm ad}\\, h|_{L_\\alpha}$ is a scalar by\n$\\langle \\alpha, h \\rangle$, we see that $Z\\subset L_0$. The remaining\nresults are now obvious. \\rule[-.23ex]{1.0ex}{2.0ex}\n\n\\begin{prop}\nLet $L$ and $L'$ be centrally isogenous Leibniz algebras and suppose\nthat $L$ is graded by a finite root system $\\Delta$. Then $L'$ is\nalso graded by $\\Delta$ and in such a way that the associated\ndialgebras are isomorphic.\n\\end{prop}\n{\\bf Proof.} Let $\\frak u$ be a universal central extension of $L$.\nBy Proposition 4.3, $\\frak u$ is graded by $\\Delta$. By assumption,\n$L'\\cong \\frak u\/Z_1$ for some subspace of the center of $\\frak u$.\nBy Proposition 4.4, $L'$ is graded by $\\Delta$. The associated\ndialgebras are isomorphic by Propositions 4.3 and 4.4.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\nIn short, all Leibniz algebras in a given isogeny class are\n$\\Delta$-graded if one of them is, and all have isomorphic root spaces\nfor all $\\alpha\\in\\Delta$. They differ only by central elements in the\n$0$-root space.\n\n\\section{Proof of Recognition Theorem}\n\nIn this section we shall complete the proof of the Recognition\nTheorem. We use the results and notation of $\\S 3$ and $\\S4$ freely.\n\n\\subsection{The Recognition Theorem for types $D$ and $E$}\n\nNow we assume that $L$ is a Leibniz algerba graded by $\\Delta$, where\n$\\Delta$ is a finite root system of type $D_l$, $l\\ge 4$, or $E_6, E_7,\nE_8$.\n\n\\begin{lemm} Let $\\alpha, \\beta\\in\\Delta, r, s \\in R $ and set $\\rho=:\\hbox{\\rm ad}\\,([e_\\alpha(r), e_{-\\alpha}(s)])$. Then\n$$\\rho(e_\\beta(t))=\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv s).$$\n\\end{lemm}\n{\\bf Proof.} Case 1. $(\\alpha|\\beta)=-1$:  Then $\\beta-\\alpha\\not\\in\\Delta$ and we\nhave\n\\begin{eqnarray*}\n\\rho(e_\\beta(t))&=&-[e_\\beta(t), [e_\\alpha(r), e_{-\\alpha}(s)]]\\\\\n             &=&-[[e_\\beta(t), e_\\alpha(r)], e_{-\\alpha}(s)]\\\\\n             &=&-[[e_\\beta, e_\\alpha](t\\dashv r), e_{-\\alpha}(s)]\\\\\n             &=&-[[e_\\beta, e_\\alpha], e_{-\\alpha}]((t\\dashv r)\\dashv s)\\\\\n             &=&-[[e_\\beta, [e_\\alpha, e_{-\\alpha}]]((t\\dashv r)\\dashv s)\\\\\n             &=&\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv\n             s).\n\\end{eqnarray*}\n\nCase 2. $(\\alpha|\\beta)=1$: Then $(-\\alpha|\\beta)=-1$ and we may use Case 1.\n\nCase 3. $(\\alpha|\\beta)=0$: Then neither $\\beta+\\alpha$ nor $\\beta-\\alpha$ is a root\nand both sides of our equation are 0.\n\nCase 4. $\\alpha=\\beta$:  It is easy to find a pair of roots $\\gamma,\n\\varepsilon\\in\\Delta$ with $\\gamma+\\varepsilon=\\alpha$, $\\gamma, \\varepsilon\\not\\in\\{\\alpha, -\\alpha\\}$. Then\n$e_\\beta(t)=e_\\alpha(t)=[e_\\gamma, e_\\varepsilon](t')$ for some $t'\\in R$ which is\nsome $K$-multiple of $t$. Applying $\\rho$ and using the previous\ncases,\n\\begin{eqnarray*}\n\\rho(e_\\beta(t))&=&\\rho([e_\\gamma, e_\\varepsilon](t'))=\\rho([e_\\gamma(t'), e_\\varepsilon(1)])\\\\\n             &=&\\langle \\gamma, \\alpha^{\\vee}\\rangle [e_\\gamma((t'\\dashv r)\\dashv s), e_\\varepsilon(1)]+\\langle \\varepsilon, \\alpha^{\\vee}\\rangle [e_\\gamma(t'), e_\\varepsilon(r\\dashv s)]\\\\\n             &=&\\langle \\gamma+\\varepsilon, \\alpha^{\\vee}\\rangle [e_\\gamma, e_\\varepsilon](t'\\dashv(r\\dashv s))\\\\\n             &=&\\langle \\alpha, \\alpha^{\\vee}\\rangle e_\\beta(t\\dashv (r\\dashv s))\\\\\n             &=&\\langle \\alpha, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv\n             s).\n\\end{eqnarray*}\n\nCase 5. $-\\alpha=\\beta$: This is similar to Case 4.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\n\nWe wish to define a homomorphism\n$$\\phi:L\\to \\frak t(R, \\Delta)=\\dot{\\mg}\\otimes R\\eqno(5.1)$$\nby defining\n$$\\phi(e_\\alpha(r))=e_\\alpha\\otimes r, \\quad \\hbox{for all}\\ \\alpha\\in\\Delta, r\\in R.\\eqno(5.2)$$\n\nSince $L$ is generated by the subspace $L_\\alpha, \\alpha\\in\\Delta$, it is clear\nthat (5.2) uniquely defines $\\phi$.\n\nSuppose that $\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}[e_\\alpha(r(\\alpha, i)),\ne_{-\\alpha}(s(\\alpha, i))]=0$ for some $r(\\alpha, i), s(\\alpha, i)\\in R$. Then using\nLemma 5.1, we have for all $\\beta\\in\\Delta$\n$$\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta(r(\\alpha, i)\\dashv\ns(\\alpha, i))=0$$\nand hence\n$$\\sum_{\\alpha\\in\\Delta}\\langle \\beta, \\alpha^{\\vee}\\rangle\\sum_{i=1}^{n_\\alpha} e_\\beta(r(\\alpha, i)\n\\dashv s(\\alpha, i))=0.$$ Since $e_\\beta(r)=e_\\beta(s)\\Leftrightarrow r=s$. It\nfollows that in $H\\otimes R$,\n$\\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\alpha^{\\vee}\\otimes(r(\\alpha, i)\\dashv s(\\alpha,\ni))=0$ (it is killed by all the functionals $1\\otimes \\beta$). Thus we may\nextend $\\phi$ to $L_0$ by\n$$\\phi: \\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}[e_\\alpha(r(\\alpha, i)), e_{-\\alpha}(s(\\alpha, i))]\\mapsto \\sum_{\\alpha\\in\\Delta}\\sum_{i=1}^{n_\\alpha}\\alpha^{\\vee}\\otimes (r(\\alpha, i)\\dashv s(\\alpha, i)) .$$\n\nWe check that $\\phi$ is a homomorphism: If $(\\alpha, \\beta)$ is an\n$A_2$-pair, then\n\\begin{eqnarray*}\n\\phi([e_\\alpha(r), e_\\beta(s)])&=&\\phi([e_\\alpha, e_\\beta](r\\dashv s))\\\\\n                        &=&[e_\\alpha, e_\\beta]\\otimes(r\\dashv s)\\\\\n                        &=&[\\phi(e_\\alpha(r)), \\phi(e_\\beta(s))].\n\\end{eqnarray*}\n\nIf $(\\alpha, \\beta)\\ge 0$, then $\\phi([e_\\alpha(r),\ne_\\beta(s)])=0=[\\phi(e_\\alpha(r)), \\phi(e_\\beta(s))]$.\n\nIf $\\beta=-\\alpha$ then  $\\phi([e_\\alpha(r),\ne_{-\\alpha}(s)])=\\alpha^{\\vee}\\otimes(r\\dashv s)=[e_\\alpha\\otimes r, e_\\beta\\otimes s]$. Now\nlet $h=[e_\\alpha(r), e_{-\\alpha}(s)]$. We have\n\\begin{eqnarray*}\n\\phi([e_\\beta(t), h])&=&-\\phi(\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta((t\\dashv r)\\dashv s))\\\\\n                  &=&-\\langle \\beta, \\alpha^{\\vee}\\rangle e_\\beta\\otimes ((t\\dashv r)\\dashv s)\\\\\n                  &=&[e_\\beta\\otimes t, \\alpha^{\\vee}\\otimes (r\\dashv s)]\\\\\n                  &=&[\\phi(e_\\beta(t)), \\phi(h)],\n\\end{eqnarray*} and\n\\begin{eqnarray*}\n&&\\phi([[e_\\beta(t), e_{-\\beta}(u)], h])\\\\\n&=&-\\phi(\\langle \\beta, \\alpha^{\\vee}\\rangle [e_{\\beta}(t), e_{-\\beta}((u\\dashv r)\\dashv s)]+\\langle \\beta, \\alpha^{\\vee}\\rangle [e_\\beta((t\\dashv r)\\dashv s), e_{-\\beta}(u)])\\\\\n                  &=&-\\langle \\beta, \\alpha^{\\vee}\\rangle \\Big(\\beta^{\\vee}\\otimes (t\\dashv ((u\\dashv r)\\dashv s))-\\beta^{\\vee}\\otimes (((t\\dashv r)\\dashv s)\\dashv u)\\Big)\\\\\n                  &=&0=[\\phi([e_\\beta(t), e_{-\\beta}(u)]), \\phi(h)]\n\\end{eqnarray*}\nsince $ ((t\\dashv r)\\dashv s)\\dashv u=u\\vdash((t\\dashv r)\\dashv\ns)=(u\\vdash(t\\dashv r))\\dashv s=((u\\vdash t)\\dashv r)\\dashv\ns=((t\\dashv u)\\dashv r)\\dashv s=t\\dashv ((u\\dashv r)\\dashv s)$,\nwhere we have used the commutativity and associativity of $R$.\n\n\\begin{prop}\nThe homomorphism $\\phi$ define by (5.1) and (5.2) is a surjective\n$\\Delta$-homomorphism and $\\hbox{Ker}\\,\\varphi$ is contained in the center of\n$L$.\n\\end{prop}\n{\\bf Proof.} Since $L_\\alpha=e_\\alpha(R)\\stackrel{\\phi}{\\to}e_\\alpha\\otimes R$, it\nis clear that $\\phi$ is bijective on the root spaces of $L_\\alpha,\n\\alpha\\in\\Delta$ and hence $\\phi$ is surjective and $\\hbox{Ker}\\,\\varphi\\subset L_0$.\nThus $[\\hbox{Ker}\\,\\phi, L_\\alpha]\\subset L_\\alpha\\cap (\\hbox{Ker}\\,\\phi)=(0)$ and hence\n$\\hbox{Ker}\\,\\phi$ is central by (2) of Definition 3.1.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\nWe have proved that $L$ is a central extension of $\\dot{\\mg}\\otimes R$, thus\nproving the third part of the Recognition Theorem 1.1.  The\nuniversal central extension of $\\dot{\\mg}\\otimes R$ is given in \\cite{LL} for\na unital commutative associative dialgebra $R$.\n\n\n\\subsection{The Recognition Theorem for type $A$}\n\nNow we assume that $L$ is a Leibniz algerba graded by $\\Delta$, where\n$\\Delta$ is a finite root system of type $A_l$, $l\\ge 2$.\n\nLet $n=l+1$ and $\\varepsilon_1, \\cdots, \\varepsilon_n$ be an orthonormal basis\nfor $\\mathbb R^n$. Identify $\\Delta$ with $\\{\\varepsilon_i-\\varepsilon_j\\mid\ni\\ne j\\}$ and define a base $\\{\\alpha_1, \\cdots, \\alpha_l\\}$ of $\\Delta$ by\n$\\alpha_i=\\varepsilon_i-\\varepsilon_{i+1}$. The positive class of $A_2$-pair in\n$\\Delta$ will be taken as $[(\\alpha_1, \\alpha_2)]$.\n\nFor $\\alpha=\\varepsilon_i-\\varepsilon_j\\in\\Delta$ we let $L_{ij}=L_\\alpha$. The simple Lie\nalgebra $\\dot{\\mg}$ over $K$ of type $A_l$ may be identified with\n$\\frak{sl}\\,(n, K)$. We choose as our Chevalley basis the matrix units\n$E_{ij}, i\\ne j$ and the elements $h_i=[E_{i, i+1}, E_{i+1, i}],\ni=1, \\cdots, l$. To bring the notation in line with $\\S2.4$, we\nwrite $e_{ij}$ for $E_{ij}$. Let $R$ be the $K$-dialgebra derived\nfrom $L$ with this choice of positive $A_2$-pairs and the given\nChevalley basis. Then we have\n$$[(\\alpha_1, \\alpha_2)]=\\{\\varepsilon_i-\\varepsilon_j, \\varepsilon_j-\\varepsilon_k\\mid i, j, k\\ \\hbox{distinct}\\}$$ and\n$$[(\\alpha_2, \\alpha_1)]=\\{\\varepsilon_i-\\varepsilon_j, \\varepsilon_k-\\varepsilon_i\\mid i, j, k\\ \\hbox{distinct}\\}.$$\n\nThus by results in $\\S4$, we have\n\\begin{eqnarray}\n&(i)&  L_{ij}=K\\{e_{ij}(r)\\mid r\\in R\\}, \\,i\\ne j;\\\\\n&(ii)& e_{ij}(k_1r+k_2s)=k_1e_{ij}(r)+k_2e_{ij}(s) ;\\\\\n&(iii)& [e_{ij}(r), e_{kl}(s)]=0\\  \\hbox{ if }\\ i\\ne l\\ \\hbox{ and } \\  j\\ne k;\\\\\n&(iv)& [e_{ij}(r), e_{kl}(s)]=e_{il}(r\\dashv s)\\  \\hbox{ if } \\ i\\ne l\\ \\hbox{and } \\  j= k;\\\\\n&(v)& [e_{ij}(r), e_{kl}(s)]=-e_{kj}(s\\vdash r)\\  \\hbox{ if } \\ i=l\\\n\\hbox{and} \\  j\\ne k,\n\\end{eqnarray}\nfor all  $r, s\\in R, \\ k_1, k_2\\in K$.\n\nNow whether $R$ is associative or not, $L$ is homomorphic to the\nimage of $\\frak{stl}\\,(n, R)$, under the map\n$$\\phi: v_{ij}(r)\\to e_{ij}(r), \\quad r\\in R,\\, i\\ne j.\\eqno(5.6)$$\n\nIn Section 3, we show that $R$ is associative if $l\\ge3$. Similar to\nthe proof of Proposition 3.1 in \\cite{L}, we can show that $R$ is\nalternative if $l=2$.\n\n\n\\begin{prop}\nThe homomorphism of $(5.6)$ is central.\n\\end{prop}\n{\\bf Proof.} It is clear that $\\frak{stl}\\,(n, R)$ is graded by $\\Delta$ and\nits associated dialgebra is $R$. The homomorphism (5.6) sends\n$v_{ij}(R)\\to e_{ij}(R)=L_{ij}$ isomorphically and hence the\n$\\hbox{Ker}\\,(\\phi) \\subset \\frak{stl}\\,(n, R)_0$.  The same argument as used in\nProposition 5.2 gives that $\\phi$ is central.\n\\rule[-.23ex]{1.0ex}{2.0ex}\n\n\nThis completes (1) and (2) of the Recognition Theorem 1.1.\n\n\n\\noindent{\\bf Remark.} 1. The structures of Leibniz algebras graded\nby finite root systems of other types are determined in \\cite{LH2}\nby using the methods in \\cite{BZ} and \\cite{ABG2}.\n\n2. By Theorem 1.3 in \\cite{MZ} and Theorem 1.1, we can easily obtain\nthe following theorem.\n\\begin{theo}\n A $Q(n)$-graded Leibniz superalgebra over $K$ is centrally isogenous to $\\frak{stl}\\,(n+1, D)$,\n where $D=D_{\\bar0}+D_{\\bar1}$ is an associative or an alternative (if $n=2$)\n unital dialgebra such that there exists $\\nu\\in D_{\\bar1},\n \\nu^2=1$. (The definition of $Q(n)$-graded Leibniz superalgebra immediately follows\n from Definition 1.3 in \\cite{MZ} and\n Definition 3.1 in section 3, also see \\cite{LH3}).\n\\end{theo}\n\n\n\n\n\n\n\\vskip45pt \\centerline{\\bf ACKNOWLEDGMENTS}\n\n\\vskip15pt    Part of the work was done during the first author's\nvisit in the Department of Mathematics at University of Bielefeld\nand study in the Department of Mathematics at Shanghai Jiaotong\nUniversity for his Post-doctor study. He expresses his gratitude to\nthe `AsiaLink Project' for financial support. He is also indebted to\nC.M. Ringel and C.P. Jiang for the hospitality and encouragement.\nAuthors give their thanks to Y. Gao for his useful comments. Liu is\nsupported by the NNSF (Grant 10671027, 10701019 and 10571119), the\nZJNSF(Grant Y607136), and Qianjiang Excellence Project of Zhejiang\nProvince (No. 2007R10031).  Hu is supported in part by the NNSF\n(Grants 10431040, 10728102), the PCSIRT, the Priority Academic\nDiscipline from the MOE of China,  the Shanghai Priority Academic\nDiscipline from the SMEC. Authors are grateful to the referee for\ncorrection in some errors and invaluable suggestions.\n\n\n\n\n\\vskip30pt\n\\def\\cen{\\normalsize\\bf REFERENCES}{\\centerline{\\normalsize\\bf REFERENCES}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Cherenkov Telescope Array (CTA) project design study \n\\cite{cta-url} is under way to\ndesign and optimise the next generation of stereoscopic Imaging\nAtmospheric Cherenkov Telescope (IACT) arrays. CTA aims to achieve\nan order of magnitude improvement in point-source sensitivity \nover current instruments and a\nwider energy coverage in order to study astrophysical processes of\nmore sources in more detail.\n\nWhile extrapolation from current generation instruments may be a first\nguide to what is needed to achieve the CTA goals, only detailed\nsimulations can give definite answers. Physical processes in air\nshowers may ultimately limit the gamma{}-hadron separation power of\nthe instruments, as well as their angular and energy resolution.\nDepending on area coverage and instrumental capabilities, any\naffordable solution will typically not reach such ultimate limits.\nThe approach in this paper is the evaluation of realistic -- or conservative --\nperformance parameters by detailed simulations of air showers and\nthe potential instruments, followed by analysis methods well tested\non current instruments.\n\nThe air{}-shower simulation is based on the CORSIKA \\cite{CORSIKA} code with its\nchoice of different interaction models. With the IACT option of CORSIKA\nwe can simulate the Cherenkov light hitting arbitrary arrays of\ntelescopes, each defined separately by its position and a fiducial radius. \nSince the CORSIKA stage requires the largest amount of\nCPU time, each shower is re-used at different random displacements.\nDue to the excellent gamma-hadron separation of all the\nconfigurations envisaged, a huge number of\nhadron showers has to be simulated in order to estimate the\nremaining backgrounds -- in most cases billions of events.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{pixel_size_gray}\n\\caption{Different pixel sizes can have an\nimpact on the image obtained from the same shower (here\n0.07{\\textdegree} to 0.28{\\textdegree} pixels on a 23 m telescope, only\npart of f.o.v. shown). CORSIKA output can be piped directly into\nmultiple telescope simulations running in parallel, allowing for\nefficient simulation of different configurations. \n(See Figure \\ref{fig:pixel-size-results} for results.)\n\\label{fig:pixel-size-image}}\n\\end{figure}\n\nIn a second processing stage, the atmospheric extinction\nand the details of the detector response are simulated by the\n{\\tt sim\\_telarray} program \\cite{simtelarray}. \nThis program is very flexible and\ndifferent detectors with different read{}-out, different triggering\nschemes, or just different reflectors are treated by the same code,\nby just specifying different configuration files. Simulations include\noptical ray{}-tracing, night{}-sky background, all relevant electronic\npulse shapes, switching behaviour of comparators or discriminators, as\nwell as many other details. Telescope configurations are tested\nfor random night-sky trigger rates before being used for shower\nsimulations.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{arrays}\n\\caption{Selection of different array\nconfigurations simulated for altitudes of 1800 to 2000 m.\nSee text for more details.\n\\label{fig:arrays}}\n\\end{figure}\n\nFor efficiency reasons, the CORSIKA output data is usually piped into the\ntelescope simulation, without intermediate on-disk storage.\nBy placing a utility program termed {\\tt multipipe\\_corsika} into\nthis pipe, the CORSIKA data can be piped into multiple telescope\nsimulations at the same time, e.g. for different pixel sizes (see\nFigure \\ref{fig:pixel-size-image}),\ndifferent offset angles etc.\n\n\n\\section{A variety of candidate configurations}\n\nSince the performance is a complex function of array layout (which may\nconsist of multiple types of telescopes), and of a variety of\ntelescope parameters, detailed simulations can be performed only\nfor a limited {--} and therefore rather complementary {--} subset of layout and\ntelescope parameters. These candidate configurations \n(see Figure \\ref{fig:arrays}) include:\n\\begin{itemize}\n\\item A 9{}-telescope array aiming at low energies (420 m{\\texttwosuperior}\nmirror area each, with 0.1{\\textdegree} pixels covering a\n5{\\textdegree} field{}-of{}-view (f.o.v.). This configuration was\nsimulated at different altitudes (2000 m, 3500 m, and 5000 m\na.s.l.). The 2000 m simulations were carried out with a variety of\npixel sizes (from 0.07{\\textdegree} up to 0.28{\\textdegree}, see \nFigure~\\ref{fig:pixel-size-image}). It was also cross-checked with\nan different telescope simulation code \\cite{ICRC2007paper}.\n\\item \nAn array of 41 H.E.S.S.{}-I type telescopes (106 m{\\texttwosuperior}),\nto see how gamma{}-hadron rejection can be improved by better coverage\nwith current{}-generation instruments. It also allows to test\narrays of 4, 5, 9, or 16 telescopes with different inter-telescope\nseparations by selecting subset of the 41 telescopes in the\nanalysis.\n\\item \nAn array consisting of telescopes of two different sizes (600\nm{\\texttwosuperior} and 106 m{\\texttwosuperior} mirror area,\n5{\\textdegree} and 7{\\textdegree} f.o.v.) aiming at a rather wide\nenergy coverage. For the large telescopes, three sets of four\ntelescopes each, with different separations, were included in\nthe simulation -- usually selecting only one set of four large\nplus the 85 small telescopes in the analysis.\nAll telescopes were assumed to have a 50\\% higher quantum efficiency\nthan H.E.S.S.\n\\item \nDifferent arrays of rather small (37 m{\\texttwosuperior}) telescopes\nwith large separations and wide{}-f.o.v. cameras (up to 8{\\textdegree}\nwith 0.3{\\textdegree} pixels), aiming at high energies. \n\\end{itemize}\n\nSimulations include gammas, protons, and electrons as\nprimaries {--} nuclei, being generally easy to distinguish from\ngammas, were not considered. \nAll simulations were carried out at the Max Planck Institute in\nHeidelberg, except for the 3500 m 9{}-telescope configuration which was\nprocessed at SLAC.\n\n\n\\section{Analysis}\n\nThe analysis of simulation data is based on methods similar to the\nH.E.S.S. Hillas{}-parameter based standard analysis, with some\nextensions like using the height of shower maximum and how well the\nlateral distribution conforms with expectations for gamma showers.\nImage shape cuts include the mean reduced scaled width and length\n\\cite{HESS-Crab-paper}. Since, after geometrical shower reconstruction,\nthe image in each telescope can be used for an energy estimate,\nthe consistency of the individual estimates for the same event \nis used as another cut. Showers where the expected accuracy of\nthe final (weighted mean) energy estimate is poor for the\ngiven energy, e.g. showers outside the array, are also discarded.\nAt the lowest energies, a few tens of GeV, the image shape cuts\ntypically have little gamma-hadron discrimination power. \nThe most important parameter in this energy range turned out to\nbe the height of shower maximum.\nSelection cuts were manually optimised as simple functions of\nreconstructed energy or telescope multiplicity (e.g. of the\nform $a+b\\;\\log E$, with additional lower and upper bounds).\nFurther improvements\nby more sophisticated shower reconstruction and advanced cut\noptimisations may be feasible. In that sense, the following\nperformance results are very conservative.\n\nThe set of cuts from the H.E.S.S. standard analysis was also\nused on H.E.S.S.-I simulations as well as corresponding\nfour-telescope subsets of the 41-telescope array, with \ncorresponding sensitivities matching the actual H.E.S.S.\nsensitivity \\cite{HESS-Crab-paper}.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{sens_var3}\n\\caption{Differential sensitivity limit\n(with 5 sigma, 10 events in each energy bin) in HEGRA Crab units \n\\cite{HEGRA-Crab} for\nthe 9{}-telescope array at 2000 m altitude, at a zenith\nangle of 20{\\textdegree}. Each curve is for one of\nthe pixel sizes in Figure \\ref{fig:pixel-size-image}. Since the\nlarger pixels will see more night-sky background noise, it is\nimportant to adapt image cleaning thresholds to this noise.\nOnly at very low energies any improvement\ncould be achieved by small pixels.\n\\label{fig:pixel-size-results}}\n\\end{figure}\n\n\n\\section {Performance results}\n\nThe performance of the tested CTA candidate configurations was\nevaluated for point sources, usually taking advantage of the angular\nresolution improving with increasing number of telescopes with usable\nimages. Since the rejection of hadronic background improves with\nincreasing energy, most selection cuts have to be rather strict at\nlow energies and get looser towards higher energies. Both proton and\nelectron background was included, with the electron flux\nextrapolated ${\\propto}{E}^{-3.3}$ beyond available measurements. \nIn contrast to na\\\"ive expectations, electrons never dominated\nthe background at the lowest energies, due to the energy-dependent\nhadron rejection efficiency.\nIn the evaluation of flux limits, the\nLi\\&Ma formula was used to find the minimum number of gammas to achieve\na 5{}-sigma significance. In addition, a minimum of 10 gammas was required. At\nvery low energies, another limitation had to be taken into account: the\nsystematics in the subtraction of remaining background, here\nassumed at a level of 1\\% being equivalent to a 1{}-sigma fluctuation.\n\nIt turns out that with the arrays of nine 420~m{\\texttwosuperior} telescopes\nthe energy threshold can in fact be reduced to some 20~GeV. Rather strict\nselection cuts are needed at the lowest energies to achieve\nsufficient gamma-hadron separation to avoid running into the\nlimitation by background systematics. \n\nThe 41-telescope array of \n106~m{\\texttwosuperior} telescopes would not substantially improve the \nenergy threshold compared to the four-telescope H.E.S.S. phase I array.\nThanks to the much better sampling of the showers it would however\nimprove gamma-hadron separation, angular resolution, and effective\narea compared to H.E.S.S.-I, resulting in a sensitivity improvement\nmuch faster than the na\\\"ive $1\/\\sqrt{N}$ expectation ($N$ being\nthe number of telescopes).\n\nThe 97-telescope array (of which usually 85 small plus 4 large\ntelescopes were used in the analysis) combines low energy threshold\nwith much improved gamma-hadron rejection. In fact, the presence\nof the smaller telescopes helps to improve the gamma-hadron\nrejection even in the energy domain normally only accessible\nto the large telescopes -- by rejecting hadron showers where\na sub-shower may look like a low-energy gamma-ray shower.\nAt intermediate energies, this configuration fully meets\ninitial goals for CTA (`` milli-Crab sensitivity'').\n\nAt the highest energies envisaged for CTA, even the square kilometre\narea of the 97-telescope configuration may not be sufficient to\ndetect enough events. It could be supplemented with additional\nsmall telescopes instrumented with much coarser pixels (up to \n0.3{\\textdegree}) and operated at larger inter-telescope separations.\nThis is demonstrated by the assumed 39-telescope array.\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{cta-he-sens-prelim}\n\\caption{Point source sensitivity limits\nF({\\textgreater}E) of the four arrays from Figure \\ref{fig:arrays} \nat 20{\\textdegree} zenith angle, \ncompared with current instruments \n\\cite{HESS-Crab-paper,MAGIC-II-sens,VERITAS-proposal,GLAST-sens}. \nOne C.U. (Crab unit) in this case is\n$F_{\\rm{cu}}(>E)=1.78\\cdot10^{-7}(E\/\\mathrm{TeV})^{-1.57}\\mathrm{photons}\/(\\mathrm{m}^{2}\\;\\mathrm{s})$\n\\cite{HEGRA-Crab}.\n\\label{fig:sensitivity-comparison}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\hsize]{differential2rn}\n\\caption{Differential sensitivity limit\n(with 5 sigma, 10 events in each energy bin of 0.2 dex) for the 97{}-tel.\nconfiguration, with different exposure times, from 3 minutes to 50 hours. \nA 1 hour exposure would\nbe enough for a high{}-quality spectrum of a 0.1 Crab source over two\norders of magnitude in energy. 50 hours would be needed for the\nspectrum of a milli{}-Crab source.\nOne C.U. (Crab unit) is assumed here as \n$dF_{\\rm{cu}}\/dE=2.79\\cdot10^{-7}(E\/\\mathrm{TeV})^{-2.57}\\mathrm{photons}\/(\\mathrm{m}^{2}\\;\\mathrm{s}\\;\\mathrm{TeV})$\n\\cite{HEGRA-Crab}.\n\\label{fig:sensitivity-time}}\n\\end{figure}\n\n\n\\section {Conclusions and outlook}\n\nThe combination of the CORSIKA and sim\\_telarray programs is very well\nsuited for simulations of large telescope arrays, even with hundreds\nof telescopes. It is also very\nflexible and can be adapted to arbitrary array configurations and\ntelescope setups just by means of run{}-time configuration. The\nsubsequent analysis with its Hillas{}-parameter based shower\nreconstruction results in rather conservative performance\nestimates. Arrays consisting of multiple telescope types are easily\nhandled in all stages. First simulations included a range of\ndifferent test configurations, with emphasis on different energy\nranges, and demonstrated that the CTA sensitivity goal can be\nachieved \\cite{ICRC2007paper} {--} at least for energies above 50 to 100 GeV. Future\nsimulations will include more realistic CTA configurations (within\nbudget constraints) as well as some corner cases needed to improve the\nCTA design optimisation scheme.\n\n\n\n\\begin{theacknowledgments}\nI would like to thank Emiliano Carmona, Jim Hinton, and many others\nfor useful discussions on the CTA configurations. Stefan Funk has\ncomplemented my 9-telescope simulations for 2000 and 5000~m altitude\nwith corresonding ones for 3500~m altitude, carried out at the\nSLAC computing center.\n\\end{theacknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\nThe large hadron collider (LHC) has discovered the  Higgs boson and measured its properties consistent with the precision measurements of the Standard Model (SM). The discovery of the last particle of the standard model strengthened the foundation of the SM. However, overwhelming evidence and hints require physics beyond the Standard Model (BSM). While the intensive searches for new particles at the weak scale or heavier have been performed experimentally, we have not found any convincing evidence of such new particles yet. In these circumstances, there is growing attention to feebly interacting light particles from both theoretical and experimental points of view. \n\n\nNew light particles with mass less than tens of GeV are compelling addition to the SM as they can directly resolve the central issues of the SM, such as the strong CP problem and the finite neutrino masses,  explain  many flavor physics anomalies, and be a portal to the dark sector accommodating the dark matter and the baryon asymmetry  of the universe. On the experimental grounds, the light particles, if they exist, need to be feebly interacting with the SM sector due to various experimental constraints. However, one should be aware that the bounds are strongly model-dependent because the signature depends on the nature of the light degrees of freedom. This situation is contrasted with heavy new particle exploration with mass beyond the reach of current collider energies, which can be probed through  Effective Field Theories~(EFT). Therefore, there is growing interest in various scenarios based on the current experimental data and the potential for future high-intensity experiments. The status of recent progress is well-summarized in the report of {\\it Physics Beyond Collider}~\\cite{Beacham:2019nyx}. \n\n\nIn this paper, we study the physics potential of the dump facilities of the International Linear Collider (ILC) to hunt new light particles. The ILC is a proposed future $e^+e^-$ linear collider with beam-energy of 125~GeV, which can be upgraded to 500~GeV. The main goal of the ILC is to study events of $e^+e^-$ collision to perform high precision measurements of the Higgs bosons and search for new particles produced by the electroweak interaction.   In the linear collider the beam has to go into the beam dump after the collision, and it provides an excellent opportunity for a high-intensity experiment to produce feebly interacting light particles. The number of electrons on the beam dump is enormous, $N_{\\rm EOT}=4\\times 10^{21}\/\\rm$~per~year, and the electromagnetic (EM) shower also leads to a high-intensity photon carrying $\\cal O$(10\\%) of the beam energy. The setup is equivalent to the other fixed target experiments if a detector is placed downstream of the beam dump. Hereafter we refer to such setup as {\\it the ILC beam dump experiment}. One can search for light particles produced in the dump, fly beyond the muon shield, then decay to the SM particles or scatter the detector material. The potential of the ILC beam dump project has been investigated in Refs~\\cite{Izaguirre:2013uxa,Kanemura:2015cxa,Sakaki:2020mqb,Asai:2021ehn,Asai:2021xtg,Moroi:2022qwz}. \n\n\nIn the previous works of the electron beam dump experiments, the searches for light particles  dominantly interacting with electron or photon were mainly studied since they are produced by  the primary electron and positron or the secondary shower particles. In the ILC beam dump experiment, the initial beam energy is much higher than those of the past  electron beam dump experiments. In addition to the EM showers, heavy mesons and $\\tau$ lepton can be produced by the shower photon  hitting nuclei. The decay of the produced SM particles is another promising source of light particles. This paper examines the production yield and spectrum of the mesons and $\\tau$ lepton at the ILC beam dump setup for the first time. \n\n \nAs an application of this study, we investigate a projected sensitivity of heavy neutral leptons  (HNLs, called sterile neutrinos in some literature). The HNLs mix with the SM neutrinos with an angle of $U_\\ell \\ (\\ell=e,\\mu, \\tau)$, resulting in a suppressed weak interaction to the SM. Therefore it is very natural to consider the HNL production from the meson and $\\tau$ lepton decays where  the weak interaction dominates. The current experimental constraints are mostly for mixing with $\\nu_e$ and $\\nu_\\mu$, and the high intensity of mesons would further reach the under-explored parameter space. Also the sensitivity to the $\\tau$ neutrino mixing angle $U_\\tau$ can be significantly improved because $\\tau$ lepton is accessible. \n\n\nThe phenomenology of the HNLs is well-reviewed in Refs~\\cite{Alekhin:2015byh, Dasgupta:2021ies}, and see references therein. We stress that feebly interacting HNLs in a GeV mass range are a motivated and well-defined physics target. The seesaw mechanism can explain the neutrino masses observed by the  neutrino oscillations with at least two HNLs.  If the two HNLs are almost degenerate,  the sum of the mixing angles have the lower bound, approximately $U^2\\equiv \\sum_l |U_l|^2\\gtrsim m_{\\rm atm}\/m_N \\sim 10^{-11} (m_N\/{\\rm 1~GeV})^{-1}$~\\cite{Shaposhnikov:2006nn,Alekhin:2015byh}. Furthermore the degenerate HNLs in the early universe can produce the baryon asymmetry via the HNL oscillations~\\cite{Akhmedov:1998qx, Asaka:2005pn}. The feeble interactions are necessary for the departure of the thermal equilibrium which is one of Sakharov's conditions for generating the baryon asymmetry. Together with the neutrino mass constraints, the interesting parameter space is in a range of $10^{-11}\\lesssim U^2\\lesssim10^{-6}$.~\\footnote{The benchmark values depend on the number of HNLs involved in the seesaw mechanism. Three or more HNLs will allow a smaller value of $|U_l|^2$. However, it is important to examine the parameter space of the minimal scenarios.}\n\n\nThis paper is organized as follows. In Sec.~\\ref{sec:dumpsetup}, we briefly review the ILC beam dump experiment, and in the following section, we evaluate the spectra of mesons and $\\tau$ lepton. In Sec.~\\ref{sec:HNL}, we study the HNLs at the ILC beam dump experiment. In addition to  the HNL production from the SM particle decays, we  consider an HNL  production via deep inelastic scattering and another production from $Z$ decays at the interaction point.  We finish with the discussion in Sec.~\\ref{sec:discussion}. \n\n\n\\section{ILC Beam Dump Experiment}\\label{sec:dumpsetup}\nThe ILC  main beam dump has to absorb 2.6~MW ($125~{\\rm GeV}\\times 21~{\\rm \\mu A}$) of the $e^\\pm$ beam energy for 125~GeV in the initial stage and 13.6~MW ($500~{\\rm GeV}\\times 27.3~{\\rm \\mu A}$) for 500~GeV in an upgrade stage. Following a water dump designed with the length of $l_{\\rm dump}=11~{\\rm m}$ and the diameter of 1.8~m, a muon shield of length $l_{\\rm sh}=70~{\\rm m}$ would be placed as proposed in \\cite{Sakaki:2020mqb,Asai:2021ehn}, to remove the secondary muon background. The cylindrical decay volume of length $l_{\\rm dec}=50~{\\rm m}$ and radius $r_{\\rm det}=3$~m would lie between the muon shield and the downstream detector. The schematic view is seen in Fig.~\\ref{fig:exp}, and the similar design of the setup can be found in \\cite{Sakaki:2020mqb,Asai:2021ehn}. Here, we additionally assume a multi-layer tracker in the decay volume. We consider a time frame of 10-year run  for both ILC-250 and ILC-1000 where the  beam energy is 125~GeV and 500~GeV, respectively.  \n\n\nThe EM shower (photons, electrons, and positions) starts in the beam dump. The ILC beam dump experiment is unique compared to the past electron beam dump experiments with regards to the higher beam energy and the high intensity, i.e., the number of electrons on the beam dump  $N_{\\rm EOT}$ is about $4\\times 10^{21}$ per year\\footnote{In ILC-1000, we assume to be $N_{\\rm EOT}=4\\times 10^{21}$ per year in numerical calculations.}.\nThe energetic beam  creates photons at $\\cal O$(1-10)~GeV energy scale (Fig.~9 of \\cite{Asai:2021ehn}), and the secondary interaction between the photon and the nucleus can  produce light mesons ($\\pi$ and $K$), heavy mesons ($D$, $B$, and even $B_c$) and $\\tau$ lepton.\nThey loose the energy in the beam dump and finally decay, and some of them produce the HNLs.  The yield and energy spectrum of the SM particles at the decay is therefore important to study the sensitivity of the ILC beam dump experiment.\nWe show the evaluation in Sec.~\\ref{sec:spectrum}. \n\n\nThe secondary muons are stopped in the muon shield at ILC-250, but their penetration behind the shield cannot be neglected for $E_{\\rm beam}=500$~GeV at ILC-1000. In this case, an additional active muon shield behind the muon shield would be necessary, and we assume that the muon shield consists of the lead shield ($l^{\\rm lead}_{\\rm sh}=10$~m) and the active shield ($l^{\\rm active}_{\\rm sh}=70~{\\rm m}-l^{\\rm lead}_{\\rm sh}$). The HNL with a dominant mixing with the muon neutrino can be produced inside the muon shield by  scattering of the shower muon, and we approximate that the HNLs produced behind the lead shield do not contribute to the signal events. In Appendix~\\ref{app:muonshield}, we study how  different depth of the muon shield affects the sensitivity for the HNL at ILC-1000. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{figs\/exp.pdf}\n\\caption{A setup for ILC beam dump experiments. It consists of the main beam dump, a muon shield, and a decay volume. We assume a multi-layer tracker is placed in the decay volume so that the charged tracks are measured.}\n\\label{fig:exp}\n\\end{figure}\n\n\n\\section{Meson and $\\tau$ lepton Spectra}\\label{sec:spectrum}\n\n\nThis section presents the meson and $\\tau$ lepton spectrum obtained by Monte-Carlo simulation at the ILC beam dump. We use PHITS~3.25~\\cite{Sato:2018imy} for production and transport of particles other than heavy mesons. PHITS (Particle and Heavy Ion Transport code System) is a general-purpose Monte Carlo particle transport simulation code developed under collaboration between JAEA, RIST, KEK, and several other institutes. PHITS can transport most particle species for a given geometry of the materials, and it is tested thoroughly by benchmarks studies~\\cite{iwamoto2017benchmark,matsuda2011benchmarking}. For heavy mesons production, we implement their differential production cross sections obtained by PYTHIA8.3~\\cite{Bierlich:2022pfr} into PHITS. More details are given in the following. \n\n\n\\subsection{Light Mesons}\n\n\n\\begin{figure*}\n\\centering \n\\includegraphics[width=0.49\\textwidth]{figs\/decay_125GeV.pdf}~~\n\\includegraphics[width=0.49\\textwidth]{figs\/decay_500GeV.pdf}\n\\caption{The kinetic energy distribution of light mesons when they decay, where $\\pi^\\pm$ and $K^\\pm$ indicates the sum of charged pions and kaons.\nWe consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively.}\n\\label{fig:decay}\n\\end{figure*}\n\n\nThe light mesons are mainly produced by the interaction of real photons in the electromagnetic shower with the nucleons in the beam dump. If the decay length of the produced light mesons is in the same order of magnitude of or greater than the mean free path in the material, the particles reduce their energy or change into different flavors by  (in-)elastic scattering or  multiple scattering. We use the following codes and models which are available in PHITS to simulate the electromagnetic shower and the production and transport of the light mesons. For the electromagnetic shower, the simulation is performed by EGS5~\\cite{Hirayama:2005zm}. For the light meson production and transport, the JAM~\\cite{Nara:1999dz} and JQMD~\\cite{PhysRevC.52.2620} models modified for photoproduction (photonuclear interaction) are used. In addition to these models, INCL4.6~\\cite{PhysRevC.87.014606} is also employed to calculate the interaction of the mesons with nuclei during transport. The energy loss of the charged particles due to multiple scattering is evaluated by ATIMA~\\cite{ATIMA}.\n\n\nFig.~\\ref{fig:decay} shows the kinetic energy distribution of light mesons when they decay. The decay energy distribution is more important than the production energy distribution because the kinematics of new particles is determined by the parent particle distribution at the decay. For reference, the production energy distribution of light mesons is shown in Fig.~\\ref{fig:production}.\n\n\n\n\n\n\\subsection{Heavy Mesons}\n\n\nWe use PYTHIA8 to calculate the differential cross sections of the  $\\gamma p(n)\\to B(D)+X$ process for the heavy mesons.\\footnote{We thank the Pythia team, especially to Ilkka Helenius for helping us understand the latest photoproduction feature of PYTHIA~8.3.}\nWe have checked that the sum of the direct and non-diffractve cross section of the $D$ meson production agrees very well with the photoproduction data~\\cite{SLACHybridFacilityPhoton:1983yfx,TaggedPhotonSpectrometer:1989bpi}, see Appendix~\\ref{app:Dmeson}. Therefore we regard the sum of the two cross sections as the total cross section, \n$\\sigma_{\\rm total}(\\gamma p(n))$\n$=\\sigma_\\text{non-diff}(\\gamma p(n))$\n$+ \\sigma_{\\rm direct}(\\gamma p(n))$. The total cross section of atomic neucleus is obtained by taking into account the shadowing effect \\cite{Caldwell:1978ik,Kopeliovich:2012kw}, \n\\begin{equation}\n\\sigma_{\\rm total}(\\gamma A)=  A^{l} \\left(\\frac{Z}{A}\\sigma_{\\rm total}(\\gamma p)+\\frac{A-Z}{A}\\sigma_{\\rm total}(\\gamma n)\\right) ,\n\\label{eq:atomic}\n\\end{equation}\nwhere $l=0.92$~\\cite{Kopeliovich:2012kw}. \n\n\nThe differential production cross sections for heavy meson photoproduction are obtained by PYTHIA8~\\cite{Bierlich:2022pfr} and implemented in PHITS. Since the decay length of the heavy mesons are much shorter than the mean free path in the material, the spectra at their production and decay are similar. In Fig.~\\ref{fig:production}, we show the production rate per electron injection in the beam dump for mesons and $\\tau$ lepton with respect to the kinetic energy at production, where the energy is normalized by the beam energy. The results for $\\pi^{\\pm}(\\pi^+, \\pi^-)$, $K(K^+, K^-, K_S^0, K_L^0)$, $D(D^+, D^-, D^0, \\overline{D}^0)$, $D_s(D_s^+, D_s^-)$, $B(B^+, B^-, B^0, \\overline{B}^0)$, $B_s(B_s^0, \\overline{B}_s^0)$, $B_c(B_c^+, B_c^-)$, and $\\tau(\\tau^+, \\tau^-)$ produced in the beam dump are shown, which represent the sum of the particles in the parenthesis. We consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively. The overall yield of the heavy meson production increases as the beam energy gets higher, and $B_c$ becomes accessible at ILC-1000. For the sake of comparison, we also include $\\pi$, $K$ distributions at production. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/production_125GeV.pdf}\n\\includegraphics[width=0.45\\textwidth]{figs\/production_500GeV.pdf}\n\\caption{The production rate per electron injection in the beam dump for mesons and $\\tau$ lepton with respect to the kinetic energy at production, where the energy is normalized by the beam energy.\nThe results for $\\pi^{\\pm}(\\pi^+, \\pi^-)$, $K(K^+, K^-, K_S^0, K_L^0)$, $D(D^+, D^-, D^0, \\overline{D}^0)$, $D_s(D_s^+, D_s^-)$, $B(B^+, B^-, B^0, \\overline{B}^0)$, $B_s(B_s^0, \\overline{B}_s^0)$, $B_c(B_c^+, B_c^-)$, and $\\tau(\\tau^+, \\tau^-)$ produced in the beam dump are shown, which represent the sum of the particles in the parenthesis.\nWe consider two beam energies, $E_{\\rm beam}=125,~500$~GeV at ILC-250 and ILC-1000, respectively.}\n\\label{fig:production}\n\\end{figure*}\n\n\n\\subsection{$\\tau$ lepton}\nAs discussed in the previous literature~\\cite{2018}, the primary source of $\\tau$ lepton is the $D_s$ decay with approximately 5\\% branching ratio. PHITS simulates $D_s$ meson propagation and decay, which accounts for the $\\tau$ lepton production. The sub-dominant source of $\\tau$ lepton is the $\\tau$ lepton pair production, $\\gamma + {\\rm nucleus\/nucleon} \\to \\tau^+ + \\tau^- + X$. We implement a complete differential cross section calculated with the Born approximation in QED in the PHITS code and generate events for the process~\\cite{Tsai:1973py,Sakaki:2020cux}. The form factors for coherent (nucleus elastic), quasi-elastic and inelastic interactions are included. The spectrum of $\\tau$ lepton is shown in Fig.~\\ref{fig:production}.\n\n\nWe find that the number of $\\tau$ leptons from the pair production is about 20 times smaller than those from the decay of $D_s$. However, the pair production process becomes dominant in the high-energy region where the kinetic energy of $\\tau$ lepton is above 65\\% of the beam energy. So the pair production process will be necessary when considering the physics of high-energy $\\tau$ leptons or $\\tau$ neutrinos at the ILC beam dump.\n\n\n\n\n\\section{Heavy Neutral Leptons}\\label{sec:HNL}\n\nIf gauge singlet fermions $N$ exist in the BSM sector, a renormalizable interaction with the SM sector is possible\n\\begin{align}\n{\\cal L} = -\\lambda_{\\ell I} (\\bar L_\\ell \\tilde{H}) N_I -\\frac{1}{2}M_{I} {\\bar{N}_I^c} N_I +\\rm h.c. \\label{eq:HNL}, \n\\end{align}\nwhere $L_\\ell$ is the SM lepton doublet of flavor $\\ell=e,\\mu,\\tau$, and $I$ is the index of $N$. \nIf $M_I\\gg \\lambda_{iI} v$, the standard seesaw mechanism   make the SM neutrino mass light, $m_{\\nu, \\ell\\ell'}\\sim  \\sum_I (\\lambda_{\\ell I}\\lambda_{\\ell'I}) v^2\/M_I$. For $\\mathcal{O}(1)$  Yukawa coupling, the singlet fermion $N$ has to be extremely heavy to satisfy the bounds on the neutrino mass, $m_{\\nu} \\gtrsim 0.05~{\\rm eV}$~\\cite{ParticleDataGroup:2020ssz}. On the other hand,  if the size of Yukawa coupling $\\lambda$ is small, $M$ can be in MeV-GeV mass scale to satisfy the same condition. Such particles can be searched directly in laboratories, and they are often called  heavy neutral leptons (HNLs) or sterile neutrinos. The active neutrinos $\\nu$ from $L$ doublet and the HNLs $N$ are almost in the mass eigenstates up to a small admixture between $\\nu_i$ and $N_I$ characterized by the mixing angle\n\\begin{align}\nU_{\\ell I} = \\frac{v \\lambda_{\\ell I}}{M_I}.  \\label{eq:mixing}\n\\end{align}\nSince the mixing and mass determine the HNL interactions with the SM particles, we use the mixing parameter $U_{\\ell I}$ instead of the Yukawa couplings to discuss the HNL phenomenology. \n\n\n\\begin{figure*}[tt]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/benchmarkplot.pdf}\n\\caption{\nThe target parameter space in a scenario of two degenerate HNLs with respect to the HNL mass and the sum of the mixing squared defined in Eq.\\eqref{eq:mixingsum}. The region between dashed (dotted) green lines is favored as the HNL can generate the baryon asymmetry of the universe~\\cite{Akhmedov:1998qx, Asaka:2005pn}, and the lines are adopted from Fig.~4.17 of \\cite{Alekhin:2015byh}. \nThe bottom shaded region cannot explain the neutrino oscillation data in Type-I seesaw mechanism with the two degenerate HNLs. Another shaded region with the BBN label is excluded by that the long-lived HNLs affect the successful big bang nucleosynthesis. The bound is obtained with respect to $U_e$, $U_\\mu$, or $U_\\tau$ in \\cite{Boyarsky:2020dzc}, and we take $U^2<\\min_{\\ell=e,\\mu,\\tau}[|U_{\\ell}^{(\\rm BBN)}|^{2}]$ for this plot. \n}\n\\label{fig:bench}\n\\end{figure*}\n\n\nThe HNLs in the GeV mass range have another exciting aspect other than explaining the mass of active neutrinos and testability. They can be responsible for the baryon asymmetry by leptogenesis via HNL oscillation~\\cite{Akhmedov:1998qx, Asaka:2005pn}. The effective leptogenesis occurs for fast $N_I$ oscillation, and therefore two degenerate HNLs are an excellent benchmark model to investigate. \n\n\nIn the minimalistic scenario with two quasi-degenerate HNLs, the target parameter space is well-defined by the baryon asymmetry and the neutrino mass. We schematically show it in Fig.~\\ref{fig:bench}. Note that the vertical axis is the sum of the mixing angles  over both the active neutrinos and the HNLs, \n\\begin{align}\nU^2\\equiv \\sum_{i=e,\\mu,\\tau \\ I=1,2} |U_{\\ell I}|^2 \n\\label{eq:mixingsum}\n\\end{align}\nWe also include a bound on  $U^2$ from Big Bang Nucleosynthesis (BBN). The small mixing angle is disfavored as the HNLs are sufficiently long-lived to decay during or slightly before BBN and  affect the ratio of the neutron and proton number densities~\\cite{Boyarsky:2020dzc}. \n\n\nIt is essential to probe all flavor mixings to cover the target parameter space for baryon asymmetry characterized by $U^2$. The sensitivity to $U_{e I}$ and $U_{\\mu I}$ of the current and proposed experiments is high, but the $\\tau$ neutrino mixing is poorly constrained. In the ILC beam dump experiment, the relevant region of $U_{\\tau I}$ can be probed because  $\\tau$ leptons can be copiously produced from $D_s$ meson decay thanks to the higher beam energy.\n\n\nIn many phenomenological studies of HNLs, one assumes a single HNL (say $N_1$) in the low energy  because having two HNLs will have little impact on  the search sensitivity as long as the two HNLs are degenerate. In the following we deal with one HNL for simplicity, and thus the index $I$ is omitted. Furthermore we turn on only one $U_\\ell$, a mixing with one of the active neutrinos in the flavor eigenstate ($\\nu_e, \\nu_\\mu, \\nu_\\tau$), at a time, which helps us to understand which underlying process matters. Under these assumptions, the phenomenology is well-described by the HNL mass and the single mixing in each benchmark model. Also, these benchmarks are used commonly in the literature, which allows us to compare our results with the previous works.  \n\n\n\\subsection{Sensitivities at ILC beam dump}\n\n\nIn this subsection we evaluate the sensitivity of the ILC beam dump experiment to the HNLs. We consider the following two production mechanisms of HNL:\n\\begin{enumerate}[(i)]\n \\item Productions from meson and $\\tau$ lepton decays; \n \\item Direct productions from electrons and muons in EM shower interacting with nucleons.\n\\end{enumerate}\nIn both cases, we consider the HNLs decaying inside the decay volume as the signal. We adopt the decay widths of HNL to the SM particles based on Ref.~\\cite{2018}, and most decay patterns of the HNL would leave multiple tracks, which is distinguishable from the background. Then, we assume zero background and account for all the HNL decays inside the decay volume as signal except for the invisible decay mode, $\\rm HNL\\to \\nu\\nu\\bar\\nu$. One of the possible backgrounds is hardons produced by high energy neutrinos, such as $K_{S,L}$ decaying to charged pions. This type of background was estimated with GIANT~4 in the SHiP setup~ \\cite{Bonivento:2013jag}. They found that   the relevant hadrons are always produced at the edge of the muon shield and can be significantly reduced by requiring the charged tracks consistent with the HNL kinematics. Note that PHITS does not take into account the high energy neutrinos interactions, which is, therefore, not included in our results. Although cosmic muons may produce beam-unrelated backgrounds, the deep underground location of the detector (100 m from the ground surface), direction of tracks, and coincidence time window based on the pulsed electron beam significantly reduce the backgrounds. The detector setup and event selections to reduce the background are beyond the scope of this paper.\n\n\nIn the HNL production process (i), evaluating the position and 4-momentum of the HNL is not straightforward because it involves various intermediate particles with possible higher multiplicity. Therefore, it is suitable to  estimate the sensitivity using Monte-Carlo simulation. We include the SM particle decays to an HNL  which are summarized in Appendix~\\ref{sec:Br} and Ref.~\\cite{2018}\n\n\nFor the other production (ii), associated physical processes are  tractable   without simulation. We therefore evaluate its sensitivity by the numerical integration and provide relevant formulae for it. To help the quantitative understanding, we also provide an approximate formula of the signal rate of the process (i) that can be integrated numerically. The signal rate of the numerical integration will be compared with the one obtained by the Monte-Carlo simulation. \n\n\nIn the following, we describe the detail of each method. \n\n\n\\subsubsection{Monte-Carlo method} \nWe simulate particle production and transport by PHITS with the help of PYTHIA8 for electron injection as described in Sec.~\\ref{sec:spectrum}. We also modify the decay tables of mesons and $\\tau$ lepton of PHITS  to include decay modes to an HNL as discussed in Appendix~\\ref{sec:Br}. It is essential to perform the Monte-Carlo simulation to track how the system evolves from the incident electron since the intermediate steps are very involved in the production (i). The result of the Monte-Carlo simulation also becomes the guiding post for the course-grained integration method which is described in Sec.~\\ref{sec:nummethod}.\n\n\nHowever, a naive Monte-Carlo simulation of the long-lived particle would suffer from technical challenges in obtaining sufficient statistics, in particular in the following cases:\n\\begin{enumerate}[(a)]\n \\item Small cross section of new physics process or photoproduction of mesons.\n \\item Small decay branching ratio from a SM particle to the new particle.\n \\item The decay length of the new particle much shorter than the shield length.\n \\item The decay length of the new particle much longer than the length of the experimental setup.\n\\end{enumerate}\n\n\nIn cases (a) and (b), the processes resulting in the signal process are so rare that it is difficult to obtain the sufficient number of events. The issue can be solved in PHITS-based simulation using the {\\it biasing technique}. In this technique, PHITS provides a biasing parameter for photoproduction. In this technique, the biased process occurs more frequently according to the biasing parameter, and an appropriate weight of the produced particle lower than that of the incident particle is assigned. We obtain the correct expected value of physical quantities by adding up the weights, rather than adding up the number of particles produced. \n\n\nIn case (c), the new particles generated by the simulation predominantly decay inside the shield, so  sufficient signal statistics are not easily obtained. This issue can be avoided by using the {\\it importance sampling technique} which PHITS supports. This technique allows us to assign an {\\it importance} to different regions of the simulation geometry. When a particle passes through the boundary between different regions with increasing importance, several copies of the particle are created in an event, and their weights are reduced depending on the importance ratio between the regions. We divide the region of the shield in the direction of the beam axis and increase their importance value exponentially, so that a short lifetime particle pass through the shield more efficiently without exponential loss. \n\n\nThe importance technique is also useful in the opposite case (d) as we can set a large importance value in the decay volume to increase the event sampling in the relevant region. In addition, {\\it forced-decay technique} is more useful when the decay length of the long-lived particle $X$ is extremely longer than the length of the shield and decay volume. In this technique, we introduce a {\\it maximum decay length}, $l_{\\rm max}$. When the decay length of $X$, $l_X$, is sufficiently longer than the typical length of experiment, $l_{\\rm exp} (\\sim l_{\\rm dump}+l_{\\rm sh}+l_{\\rm dec})$, the differential decay rate of $X$ with respect to the flight distance $z$ is\n\\begin{align}\n\\left. \\frac{dP}{dz}\\right\\vert_{l_X}\n= \\frac{1}{l_X} {\\rm e}^{-\\frac{z}{l_X}}\n\\simeq\n\\frac{1}{l_X},~~(z<l_{\\rm exp}).\n\\end{align}\nThis is a very small value, so it's hard to get enough Monte-Carlo statistics for the decay of $X$ in the decay volume.  To deal with this problem, we multiply the decay probability by $b = l_X \/ l_{\\rm max} (\\,>1)$ and the weight of $X$ by $1\/b$. The rescaled probability is\n\\begin{align}\n\\left. b \\frac{dP}{dz}\\right\\vert_{l_X}\n\\simeq\n\\frac{b}{l_X}\n\\simeq\n\\left. \\frac{dP}{dz}\\right\\vert_{l_{\\rm max}}.\n\\end{align}\nThis corresponds to the decay rate when the lifetime of $X$ is multiplied by $1\/b$. In summary, when $l_{\\rm max}<l_X$, multiplying the lifetime and weight of $X$ by $1\/b$ forces a higher probability of decay within the decay volume and reduces the computational cost by a factor $b$. This approach is an approximation, and the higher order correction to weight is $\\mathcal{O}(l_X^2\/l_{\\rm max}^2)$.  We typically use $l_{\\rm max}\\sim 10^{3}$~m, so that we can safely ignore the error. We count the number of the HNL decays to the visible SM particles in the decay region counting up all the weight factors appropriately.\n\n\n\n\n\n\\subsubsection{Coarse-grained integration method}\n\\label{sec:nummethod}\nIn this subsection, we describe the approximate estimation for the HNL productions (i) and (ii). Here, we apply suitable approximations to obtain simplified expressions involving multiple integrals for the signal yield, and then perform the integrals numerically. We refer this calculation scheme to as the course-grained integration (CGI) method. The same scheme was used in Ref.~\\cite{Sakaki:2020mqb,Asai:2021ehn} in which the ALPs and dark photon productions by the EM shower were studied.\n\n\n\nFor the production mechanism (i), we consider processes where a photon from the EM shower interacts with a nucleus in the beam dump and produces the SM particle $k$. \nThe number of signal events is schematically given by\n\\begin{align}\nN_{\\rm signal}^{\\rm (i)}&=N_{\\rm EOT}\\times\\sum_{k=\\pi, K, D, B,\\tau} \\left( l_{\\gamma}\\times n_{\\rm nucleus}\\times  \\sigma_{\\gamma~{\\rm nucleus}\\to k X}\\right)\n\\notag\n\\\\ \n& \\times{\\rm Br}(k \\to {\\rm HNL}~X) \\times {\\rm Br}_{\\rm vis}\\times {\\rm Acc}^{\\rm (i)}({\\rm HNL}), \\label{eq:sig}\n\\end{align}\nwhere  $l_{\\gamma}$ is  the photon track length in the beam dump, $n_{\\rm nucleus}$ is the number density of beam dump nucleus and the production cross section of particle $k$ is denoted by $\\sigma_{\\gamma~{\\rm nucleus}\\to k X}$. The branching ratio ${\\rm Br}(k \\to {\\rm HNL}~X)$ and ${\\rm Br}_{\\rm vis}={\\rm Br}({\\rm HNL} \\to {\\rm visible~SM})$ are summarized in Ref.~\\cite{2018} and also Appendix~\\ref{sec:Br}. ${\\rm Acc}^{\\rm (i,ii)}({\\rm HNL})$ denotes the detector acceptance depending on the production process. The distribution of photon track length  $l_{\\gamma}$ in the beam dump can be  obtained by the Monte-Carlo simulation, and the result is given in Ref.~\\cite{Asai:2021ehn} and also Appendix~\\ref{app:track}.\n\n\nIn the production mechanism (ii), an incoming lepton $\\ell=e,\\mu$ from the EM shower interacts with a nucleon and would produce an HNL. The number of signal events is schematically expressed as\n\\begin{align}\nN_{\\rm signal}^{\\rm (ii)}&=N_{\\rm EOT}\\times l_{{\\ell}^{\\pm}}\\times n_{N}^{\\rm dump\/shield}\\times \\sigma_{{\\ell}^{\\pm}~{\\rm nucleon}\\to {\\rm HNL}~ X}\\times{\\rm Br}_{\\rm vis}\\times {\\rm Acc}^{\\rm (ii)}({\\rm HNL}),\\label{eq:sig2}\n\\end{align}\nwhere the track length $l_{\\ell^{\\pm}}$ of the lepton in the beam dump is provided in Ref.~\\cite{Asai:2021ehn} and Appendix~\\ref{app:track}, and $n_{N}^{\\rm dump\/shield}$ is the number density of nucleon in the beam dump or muon shild. For the incoming muon, we ignore the HNL production in the beam dump. The production cross section of  HNL is denoted by $\\sigma_{{\\ell}^{\\pm}~{\\rm nucleon}\\to {\\rm HNL}~X}$, which is summarized in Appendix~\\ref{app:DIS}.\n\n\nWe elaborate the schematic formulae Eqs.~\\eqref{eq:sig} and \\eqref{eq:sig2} in the following. Given that the differential distribution of  mesons and $\\tau$ lepton were obtained by PHITS and PYTHIA8, as the results are showed in Sec.~\\ref{sec:spectrum}, we fit the differential distribution of the number of SM particle $N_k$ as the function of $E_k$. This quantity is expressed by the more fundamental quantity as follows,\n\\begin{align}\n\\frac{d N_k}{dE_k}=\\int dE_{\\gamma}\\frac{d l_{\\gamma}}{dE_{\\gamma}}\\times n_{\\rm nucleus}\\times \\int d \\Pi _X  \\frac{d\\sigma_{\\gamma~{\\rm nucleus}\\to k X}}\n{dE_k d \\Pi_X}.\n\\end{align}\nThis corresponds to the schematical terms in the parentheses of Eq.~\\eqref{eq:sig}. Note that ${d l_{\\gamma}}\/{dE_{\\gamma}}$ and $\\sigma_{\\gamma~{\\rm nucleus}\\to k X}$ is absorbed in ${d N_k}\/{dE_k}$.\n\n\nBy using $dN_k\/dE_k$ the production rate approximately expressed as follows, \n\\begin{align}\n    N^{\\rm (i)}_{\\rm signal}&\\sim N_{\\rm EOT}\\sum_{k=\\pi, K, D, B,\\tau}\\int d E_k \\frac{d N_k}{dE_k}\\cdot {\\rm Br}(k \\to {\\rm HNL}~X)\\cdot {\\rm Br}_{\\rm vis}\\cdot{\\rm Acc}_{\\rm dump}^{\\rm (i)}({\\rm HNL}), \\label{eq:N1approx}\n\\end{align}\nwhere we assume momentum of the SM particle $k$ is aligned to the electron beam direction. The HNL production via the SM particle decay is the leading production channel at the ILC beam dump experiment for the most accessible parameter space of mass and mixing angle. As we will see later, this production mode even contributes to the sensitivity of $U_\\tau$.\n\n\nAdditionally, the direct production initiated by the charged leptons of the EM shower takes an essential role to  explore the higher mass region of the HNL in the presence of $U_e$ or $U_\\mu$. The direct production from the electron and positron occurs inside the beam dump, while the one from the muon occurs dominantly in the muon shield. Several direct production processes contribute to the HNL productions~\\cite{Formaggio:2012cpf} such as quasi-elastic scattering, resonance production, and deep inelastic scattering~(DIS). Among those, DIS would be a significant contribution to the HNL productions around $E_{e^{\\pm},\\mu^{\\pm}}\\geq 10~{\\rm GeV}$ which is favored by the acceptance of the HNL production angle. Therefore, we focus on the DIS process, and the formulae of the number of signal events are expressed as follows, \n\\begin{align}\n    N^{{\\rm (ii)},e^{\\pm}}_{\\rm signal}&=N_{\\rm EOT}\\sum_{i=e^-,e^+}\\int d E_{i}\\frac{1}{2}\\frac{dl_{i}}{d E_{i}}\n      \\sum_{N=n,p}n_{N}^{\\rm dump}\n\\int dx \\int dy \\frac{d^2\\sigma (i N\\to {\\rm HNL}~X)}{dx dy}\n\\nonumber\\\\ &\\times\n{\\rm Br}_{\\rm vis}\\cdot {\\rm Acc}_{\\rm dump}^{{\\rm (ii)},e^{\\pm}}({\\rm HNL}),\\label{eq:DISel}\n    \\\\\n    N^{{\\rm (ii)},\\mu^{\\pm}}_{\\rm signal}&=N_{\\rm EOT}\\sum_{i=\\mu^-,\\mu^+}\\int d E_{i 0}\\frac{1}{2}\\frac{d Y_{i 0}}{dE_{i 0}}\\int dE_{i}\\frac{1}{2}\\frac{dl_{i}}{d E_{i}} \\int d\\theta_{\\rm MCS} \\frac{d P_{\\rm MCS}}{d\\theta_{\\rm MCS}} \\notag\n    \\\\\n&\\times \\sum_{N=n,p} n_{N}^{\\rm shield}\\int dx \\int dy \\frac{d^2\\sigma (i N\\to {\\rm HNL}~X)}{dx dy}\\cdot {\\rm Br}_{\\rm vis}\\cdot{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},\\mu^{\\pm}}({\\rm HNL}).\\label{eq:DISmu}\n\\end{align}\nThe energy spectrum of the muon yield per incident electron behind the beam dump $dY_{\\mu 0}\/d E_{\\mu 0}$ where $E_{\\mu^{\\pm}0}\\geq E_{\\mu^{\\pm}}$ is evaluated in Ref.~\\cite{Sakaki:2020cux}, {also see Eq.(A.3) of Ref.~\\cite{Sakaki:2020mqb}} and Appendix~\\ref{app:track}, and the track length of muons $l_{\\mu^{\\pm}}$ in the muon shield is provided in Ref.~\\cite{Sakaki:2020mqb} and Appendix~\\ref{app:track}. The angular distribution stemming from the multiple Coulomb scattering (MCS)  of the muon $dP_{\\rm MCS}\/d\\theta_{\\rm MCS}$ is given in Ref.~\\cite{Lynch:1990sq,ParticleDataGroup:2020ssz}. The integral variable $x\\equiv$ $Q^2\/[2M_N(E_{i}-E_{\\rm HNL})]$ denotes the Bjorken scaling, where $E_{i}$ is the incoming lepton energy at the HNL production, and $E_{\\rm HNL}$ is the HNL energy in the nucleon rest frame. The other integral variable $y\\equiv(E_{i}-E_{\\rm HNL})\/E_{i}$ is the fraction of the lepton energy loss in the rest frame. \n\n\nFinally, we give the formula we used to estimate the acceptance ${\\rm Acc}_{\\rm dump,shield}^{(\\rm i,ii)}({\\rm HNL})$ as follows,\n\\begin{align}\n{\\rm Acc}_{\\rm dump}^{{\\rm (i)}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm dump}}{dz}\\cdot \\int_0^{r_{\\rm det}\/(l_{\\rm dump}+l_{\\rm sh})}d\\theta_{\\rm HNL}^{\\rm dec} \\frac{dP_{\\rm ang}}{d\\theta_{\\rm HNL}^{\\rm dec}}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm dump}}^{{\\rm (i)}}\\right),\\label{eq:acci}\n\\\\\n{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},e^{\\pm}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm shield}}{dz}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm shield}}^{{\\rm (ii)},e^{\\pm}}\\right),\\label{eq:acciie}\n\\\\\n{\\rm Acc}_{\\rm shield}^{{\\rm (ii)},\\mu^{\\pm}}({\\rm HNL}) &=\\int_0^{l_{\\rm dec}}dz \\frac{dP_{\\rm dec}^{\\rm shield}}{dz}\\cdot \\Theta \\left(r_{\\rm det}-r_{\\perp,{\\rm shield}}^{{\\rm (ii)},\\mu^{\\pm}}\\right)\\times\n\\begin{cases}\n\\Theta \\left(l_{\\rm sh}-\\delta_{\\mu}\\right)~~~{\\rm for}~{\\rm ILC\\text{-}250}\n\\\\\n\\Theta \\left(10~{\\rm m}-\\delta_{\\mu}\\right)~~~{\\rm for}~{\\rm ILC\\text{-}1000}\n\\end{cases},\\label{eq:accmuon}\n\\end{align}\nwhere, as in the Monte-Carlo simulation, all the HNL decays inside the decay volume are assumed to be the visible signal except for the ones to entirely invisible particles ${\\rm HNL}\\to \\nu\\nu\\bar{\\nu}$.\nHere $z$ is the decay position of HNL in the decay volume, and $P_{\\rm dec}$ denotes the decay probability of HNL given by\n\\begin{align}\n\\frac{d P_{\\rm dec}^{\\rm dump}}{dz}&=\\frac{1}{l^{({\\rm lab})}_{\\rm HNL}}{\\rm exp}\\left(-\\frac{l_{\\rm dump}+l_{\\rm sh}+z}{l^{({\\rm lab})}_{\\rm HNL}}\\right),\\label{eq:decP}\n\\\\\n\\frac{d P_{\\rm dec}^{\\rm shield}}{dz}&=\\frac{1}{l^{({\\rm lab})}_{\\rm HNL}}{\\rm exp}\\left(-\\frac{l_{\\rm sh}-\\delta_{\\mu}+z}{l^{({\\rm lab})}_{\\rm HNL}}\\right), \n\\end{align} \nwhere $l^{({\\rm lab})}_{\\rm HNL}$ denotes the lab-frame decay length of HNL given by\n\\begin{align}\nl^{(\\rm lab)}_{\\rm HNL}=\\frac{p_{\\rm HNL}}{m_{\\rm HNL}}\\frac{1}{\\Gamma_{\\rm HNL}}\n\\end{align}\nwith $p_{\\rm HNL}$ being the momentum, $m_{\\rm HNL}$ the mass, and $\\Gamma_{\\rm HNL}$ the total decay width of HNL evaluated in \\cite{2018}.\nThe angular distribution of the HNLs is denoted as $dP_{\\rm ang}\/d \\theta^{\\rm dec}_{\\rm HNL}$, where $\\theta^{\\rm dec}_{\\rm HNL}$ is the decay angle of HNL from the particle $k$ with respect to the direction of $k$, and we use an approximated formula as follows,\n\\begin{align}\n    \\frac{dP_{\\rm ang}}{d\\theta_{\\rm HNL}^{\\rm dec}}=\\sin \\theta_{\\rm HNL}^{\\rm dec} \\frac{1}{2}\\left(\\frac{m_k}{(E_k+m_k)-p_k\\cos\\theta_{\\rm HNL}^{\\rm dec}}\\right)^2,\n\\end{align}\nwhere $m_k$ is the mass of the SM particle $k$, and $p_k$ denotes the momentum of $k$ in the lab-frame.\nThen, the energy of the HNL is approximately given by\n\\begin{align}\n    E_{\\rm HNL}=\\frac{1}{2}\\frac{m_k^2}{(E_k+m_k)-p_k\\cos\\theta_{\\rm HNL}^{\\rm dec}}.\n\\end{align}\nThe distance that a muon passes through the muon shield before emitting the HNL is the function of $E_{\\mu0}$ and $E_{\\mu}$~\\cite{Sakaki:2020mqb}, \n\\begin{align}\n    \\delta_{\\mu}=\\frac{E_{\\mu 0}-E_{\\mu}}{{\\langle dE\/dx\\rangle}_{\\rm Lead}},\n\\end{align}\nwith the energy independent stopping power: ${\\langle dE\/dx\\rangle}_{\\rm Lead}=0.02~{\\rm GeV}\/{\\rm cm}$. \nFor $E_{\\rm beam}=500$~GeV, the additional active shield behind the muon shield may be necessary due to their strong penetrating power, and the muon's trajectory becomes non-trivial after the active shield. Then we account for only the HNLs produced before the active shield, which leads to  the Heaviside step function in Eq.~\\eqref{eq:accmuon}. Regarding the impact of varying the active shield length, we study how  different depth of the muon shield affects the sensitivity for the HNL at ILC-1000 in Appendix~\\ref{app:muonshield}. The angular acceptance is expressed by the Heaviside step function associated with the transverse radius $r_\\perp (z)$ in Eqs.~\\eqref{eq:acci}, \\eqref{eq:acciie}, and \\eqref{eq:accmuon}. The typical deviation of the visible SM particles emitted from HNL from the beam axis is estimated as\n\\begin{align}\nr_{\\perp,{\\rm dump}}^{\\rm (i)}&=|\\theta_{\\rm HNL}^{\\rm dec}|\\cdot (l_{\\rm dump}+l_{\\rm sh}+z),\\label{eq:ri}\n\\\\\nr_{\\perp,{\\rm dump}}^{{\\rm (ii)},e^{\\pm}}&=\\sqrt{(\\theta_{e}^{\\rm shower})^2\\cdot (l_{\\rm dump}+l_{\\rm sh}+z)^2+(\\theta_{\\rm HNL}^{\\rm prod})^2\\cdot (l_{\\rm dump}+l_{\\rm sh}+z)^2},\n\\\\\nr_{\\perp,{\\rm dump}}^{{\\rm (ii)},\\mu^{\\pm}}&=\\sqrt{(\\theta_{\\mu}^{\\rm shower})^2\\cdot (l_{\\rm sh}-\\delta_{\\mu}+z)^2+(\\theta_{\\rm HNL}^{\\rm prod})^2\\cdot (l_{\\rm sh}-\\delta_{\\mu}+z)^2},\n\\end{align}\nwhere $\\theta_e^{\\rm shower}$ ($\\theta_{\\mu}^{\\rm shower}$) is the angle of shower electrons and positrons (muons) with respect to the beam axis, and $\\theta_{\\rm HNL}^{\\rm prod}$ is the production angle of HNL.\nFor the DIS process, the production angle is expressed as\n\\begin{align}\n    \\cos\\theta_{\\rm HNL}^{\\rm prod}=\\frac{1}{|p_{e,\\mu}|\\cdot|p_{\\rm HNL}|}\\left(E_{e,\\mu}E_{\\rm HNL}-\\frac{1}{2}\\left(Q^2+m_{e,\\mu}^2+m_{\\rm HNL}^2\\right) \\right),\n\\end{align}\nwhere $p_{e,\\mu}$ is the momentum of incoming electron and muon, and $Q^2=2 M_N E_{e,\\mu} xy$ with the mass of nucleus. \nAs provided in Refs.~\\cite{Sakaki:2020mqb} and \\cite{Asai:2021ehn}, $\\theta_{e}^{\\rm shower}$ is estimated by the Monte-Carlo simulation, and we use the mean value $\\theta_{e}^{\\rm shower}=16~{\\rm mrad}\\cdot {\\rm GeV}\/E_{e^{\\pm}}$.\nFor the incoming muons, we use $\\theta_{\\mu}^{\\rm shower}=\\theta_{\\rm MCS}$. Although for the production mechanism (i), the angle of shower photon and the production angle of the particle $k$ are precisely calculated in the Monte-Carlo simulation, we omit them in the coarse-grained integration method. This is because the heavy mesons are produced by high-energy photons and the mean value of the angle of shower photons $\\theta^{\\rm shower}_{\\gamma}=8~{\\rm mrad}\\cdot {\\rm GeV}\/E_{\\gamma}$~\\cite{Sakaki:2020mqb,Asai:2021ehn} is suppressed.\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/Ue.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the electron neutrino in the mass and mixing plane, assuming 10 year run at ILC-250 (black solid) and ILC-1000 (red solid).  It requires more than three signal events in the detector assuming no background, corresponding to $95\\%$ C.L. sensitivity. The current exclusion bounds are shown in the gray region, see Sec.~\\ref{sec:bounds}. The darker grey region is from the laboratory bounds, and the lighter gray region is the BBN bound for the HNL, which is roughly $\\tau_{\\rm HNL}>0.02$~s~\\cite{Boyarsky:2020dzc}. Sensitivity reach through $10^ 9$ $Z$-decays  at ILC is shown as a blue solid line. See Sec.~\\ref{sec:ZatILC}. For a comparison, dashed lines show a sensitivity reach of the DUNE experiment~\\cite{Coloma:2020lgy} (brown), the FASER2 experiment~\\cite{Kling:2018wct} (purple), the NA62 experiment~\\cite{Beacham:2019nyx} (orange), and the SHiP experiment~\\cite{Alekhin:2015byh} (magenta), the  MATHUSLA experiment\\cite{Curtin:2018mvb} (green), and $10^{12}$ $Z$-decays that could be realized at the FCC-ee experiment~(cyan). \\label{fig:sensitivityUe}\n}\n\\end{figure*}\n\n\n\n\\subsubsection{Projected sensitivities}\nWe give the prospects of the ILC beam dump experiment in ILC-1000 and ILC-250 for 10 year run. We assume the parameter region with more than three signal events can be probed, corresponding to the expected $95\\%$ C.L. exclusion sensitivity. After independently evaluating the signal events by the productions (i) and (ii), we combine the plots of the sensitivity of ILC.\n\n\n \n\n\\subsection*{$U_e$ dominance}\n\nFig.~\\ref{fig:sensitivityUe} shows the sensitivity of ILC for the HNL whose mixing to active neutrinos is dominated by the $U_e$ mixing. The region above the red (black) curves is the region {expected $95\\%$ C.L. exclusion sensitivity} that can be searched by ILC-1000 (ILC-250) with 10-year statistics. The gray-shaded regions are constrained from past experiments which is discussed in Sec.~\\ref{sec:bounds}.\n\n\nLet us explore the results in Fig.~\\ref{fig:sensitivityUe}, highlighting each of the HNL production processes.\n\\begin{enumerate}[(a)]\n\\item Meson and $\\tau$ lepton decays\n\nIn the small $|U_e|^2$ regions where the lifetime of HNL becomes longer, the decay probability in Eq.~\\eqref{eq:decP} is approximately $d P_{\\rm dec}^{\\rm dump}\/dz \\simeq 1\/l^{\\rm (lab)}_{X}$. Depending on the mass of HNL, the HNLs are produced by the decay of mesons and $\\tau$ lepton, see Figs.~\\ref{fig:brUeli}, \\ref{fig:brUeD}, \\ref{fig:brUeB}, and \\ref{fig:brUetau} in Appendix~\\ref{sec:Br}.\nBelow the kaon mass $m_{K}\\simeq 0.5~{\\rm GeV}$, the HNLs are mainly produced by the kaon decay. For $m_{\\rm HNL}\\leq m_{D,\\tau}-m_e\\sim 2~{\\rm GeV}$, the $D(D_s)$ meson decay dominates the HNL productions. In the region of $1~{\\rm GeV}\\lesssim m_{\\rm HNL}\\lesssim 2~{\\rm GeV}$ there are many thresholds of production mode, such as $D\\to K+e+{\\rm HNL}$, $D_s\\to \\eta+e+{\\rm HNL}$, $D\\to \\pi+e+{\\rm HNL}$ and $\\tau\\to \\nu_{\\tau}+e+{\\rm HNL}$, see Fig.~\\ref{fig:brUeD}.   \n\n\nThe limit obtained by the Monte-Carlo simulation can be checked using the approximate formula Eq.~\\eqref{eq:N1approx}. As explained in Appendix \\ref{app:Com}, the approximate formula agrees well with the results of the Monte-Carlo simulation. For the electron beam energy $E_{\\rm beam}=500~{\\rm GeV}$, the number of events by the leptonic $D$ meson decays $D^\\pm\\to e^\\pm +{\\rm HNL}$ is given by \n\\begin{align}\nN_{\\rm signal}^{\\rm (i)}&\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)  \\left(\\frac{l_{\\rm dec}}{50~{\\rm m}}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2\\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2\\left(\\frac{|U_e|^2}{ 10^{-10}}\\right)^2  \\left(\\frac{m_{\\rm HNL}}{1~{\\rm GeV}}\\right)^4,\n\\end{align}\nwhere we used follwing approximations:\n\\begin{align}\n {\\rm Br}(D^{\\pm}\\to e^{\\pm}+{\\rm HNL})\\propto |U_e|^2 , \\quad   \n {\\rm Br}_{\\rm vis}\\simeq 1, \\quad \n (l^{\\rm lab}_X)^{-1}\\propto |U_e|^2  m_{\\rm HNL}^4\/E^{\\rm lab}_{\\rm HNL} \\ . \n\\end{align}\nNear the kinematic threshold, the number of events is rapidly decreased by a suppression of the branching ratio ${\\rm Br}(D^{\\pm}\\to e^{\\pm}+{\\rm HNL})$. For $m_D-m_e \\lesssim m_{\\rm HNL} \\lesssim m_B-m_e$, the $D$ meson decay channels are closed, and the $B$ meson decay channels become dominant in the HNL productions up to around $m_{\\rm HNL}\\sim 3$ GeV.\n\n\n\n\n\\item Direct production\n\nAbove $m_{\\rm HNL}\\sim 3$ GeV, the direct production channel with the incoming electrons and positrons become significant. \nIn particular, the production with the high-energy primary electron is essential to extend the mass reach of the $U_e$ dominant scenario.\nIn the small $|U_e|^2$ regions, the decay probability in Eq.~\\eqref{eq:decP} is approximately $d P_{\\rm dec}^{\\rm dump}\/dz \\simeq 1\/l^{\\rm (lab)}_{X}$, and the energy of the produced HNL is approximately $E^{\\rm lab}_{\\rm HNL}\\simeq E_{e^{\\pm}}$. Then, the number of events for $E_{\\rm beam}=500~{\\rm GeV}$ is given by\n\\begin{align}\nN_{\\rm signal}^{\\rm (ii),e^{\\pm}}\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)\\left(\\frac{l_{\\rm dec}}{50~{\\rm m}}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2\\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2\\left(\\frac{|U_e|^2}{10^{-10}}\\right)^2 \\left(\\frac{m_{\\rm HNL}}{10~{\\rm GeV}}\\right)^6,\n\\end{align}\nwhere we use, \n\\begin{align}\n    &dl_{e^{\\pm}}\/d E_{e^{\\pm}}\\simeq \\left(dl_{e^{\\pm}}\/d E_{e^{\\pm}}\\right)_{\\rm primary}\\propto 1\/E_{e^{\\pm}},\\quad\n    d^2\\sigma(e^{\\pm} N\\to {\\rm HNL})\/dx dy \\propto |U_e|^2 E_{e^{\\pm}},\n    \\nonumber\\\\\n    &(l^{(\\rm lab)}_X)^{-1}\\propto |U_e|^2 m_{\\rm HNL}^6\/E^{\\rm lab}_{\\rm HNL},\\quad\n    {\\rm Br}_{\\rm vis}\\simeq 1,\\quad\n    y\\lesssim x^{-1}E_{e^{\\pm}} r^2_{\\rm det}(l_{\\rm dump}+l_{\\rm sh})^{-2}.\n\\end{align}\n\n\nIn the larger $|U_e|^2$ regions, where $l^{\\rm (lab)}_{\\rm HNL}\\ll l_{\\rm dump}+l_{\\rm sh}$, the shape of the upper contour lines in Fig.~\\ref{fig:sensitivityUe} is determined by the probability to decay inside the decay volume given in Eq.~\\eqref{eq:decP}. \nThe contour is characterized by the exponent of the decay probability\n\\begin{align}\n\\frac{m_{\\rm HNL}\\Gamma_{\\rm HNL}}{E^{\\rm lab}_{\\rm HNL}}(l_{\\rm dump}+l_{\\rm sh})\\sim {\\rm const}.\\label{eq:diupp}\n\\end{align} \nCombining $E^{\\rm lab}_{\\rm HNL}\\simeq E_{\\rm beam}$ and $\\Gamma_{\\rm HNL}\\propto |U_e|^2\\cdot m_{\\rm HNL}^5$, Eq.~\\eqref{eq:diupp} becomes $|U_e|^2\\propto m_{\\rm HNL}^{-6}$ , which is consistent with the parameter dependence of Fig.~\\ref{fig:sensitivityUe}. \n\n\\end{enumerate}\n\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{figs\/Umu.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the mu-neutrino in the mass and mixing\nplane. For the description of this plot, see the caption of Fig.~\\ref{fig:sensitivityUe}.  \\label{fig:sensitivityUmu}\n}\n\\vspace{20pt}\n\\includegraphics[width=0.6\\textwidth]{figs\/Utau.pdf}\n\\caption{\nSensitivity reach of ILC beam dump experiment to HNLs mixing with the tau-neutrino in the mass and mixing\nplane. For the description of this plot, see the caption of Fig.~\\ref{fig:sensitivityUe}. \n\\label{fig:sensitivityUtau}\n}\n\\end{figure*}\n\n\n\\subsection*{$U_\\mu$ dominance}\nFig.~\\ref{fig:sensitivityUmu} summarizes the prospects of the ILC beam dump experiment in ILC-1000 and ILC-250 for the HNL mixing dominantly with $\\nu_{\\mu}$.\nThe red (black) curves show the expected sensitivity for $95\\%$ C.L. exclusion of ILC-1000 (ILC-250) with 10-year statistics given by the meson and $\\tau$ lepton decays, and the DIS process for incoming muons from the EM shower. \nWe provide approximated formulae of the number of signal events for each of the HNL production mode for ease of understanding.\n\n\n\\begin{enumerate}[(a)]\n\\item Meson and $\\tau$ lepton decays\n\nThe production process from meson and $\\tau$ lepton decays of the HNL that mix dominantly with $\\nu_{\\mu}$ can be calculated in parallel with $U_e$ dominant case except for minor threshold difference.\nThe mass dependence of the dominant HNL production process is the same as that of $U_e$ dominant case with $e$ replaced with $\\mu$.\n\n\nAs shown in Figs.~\\ref{fig:brUmuD} and \\ref{fig:brUmuB}, for $m_D-m_{\\mu} \\lesssim m_{\\rm HNL} \\lesssim m_B-m_{\\mu}$, \nthe $B$ meson decay gives a significant contribution to the HNL production.\nThe zigzag curves near $m_{\\rm HNL}\\sim 3~{\\rm GeV}$ correspond to the threshold of $B\\to D+{\\mu}+{\\rm HNL}$.\nAbove $m_{\\rm HNL}\\simeq 3~{\\rm GeV}$, the leptonic $B$ meson decay such as $B^{\\pm}\\to \\mu^{\\pm} +{\\rm HNL}$ dominates the HNL productions.\n\n\n\\item Direct production\n\nAs shown in Fig.~\\ref{fig:exp}, incident real photons produce muon pairs in the beam dump through the electromagnetic interaction with the nucleus or nucleon~\\cite{Sakaki:2020cux}, and the HNL can be generated by the DIS process in the muon shield.\nThe muon pair production cross section is about $(m_{\\mu}\/m_e)^2\\simeq 10^{5}$ times smaller than that of the electron pair production. \n\n\nIn the small $|U_{\\mu}|^2$ regions, the projected sensitivity scales with the ratio of numbers of muon and electron in the \nshower, and the projected sensitivity via the DIS process  is less significant compared with the \n$U_e$ dominant case.\nOn the other hand in the large $|U_{\\mu}|^2$ regions, the number of signal events  is mostly determined by the muon shield length and  less sensitive to the number of muon pairs, because the lifetime is shorter.\nThe shape of the upper side of contours in Fig.~\\ref{fig:sensitivityUmu} is determined by the exponential factor in Eq.~\\eqref{eq:decP}, which is characrized by \n\\begin{align}\n    \\frac{m_{\\rm HNL} \\Gamma_{\\rm HNL}}{E^{\\rm lab}_{\\rm HNL}}(l_{\\rm sh}-\\delta_{\\mu})\\sim {\\rm const}.\\label{eq:larUmu}\n\\end{align}\nCombining $E^{\\rm lab}_{\\rm HNL}\\simeq E_{\\rm beam}$ and $\\Gamma_{\\rm HNL}\\propto |U_{\\mu}|^2\\cdot m^5_{\\rm HNL}$, Eq.~\\eqref{eq:larUmu} becomes $|U_{\\mu}|^2\\propto m^{-6}_{\\rm HNL}$. \nSince the DIS process with a muon happens in the muon shield,  the shorter distance from the HNL production point to the decay volume enhances the acceptance compared the DIS process in the beam dump.\n  \n\\end{enumerate}\n\n\n\n\\subsection*{$U_\\tau$ dominance}\n\nThe projected sensitivities in the $U_\\tau$ dominant scenario are shown in Fig.~\\ref{fig:sensitivityUtau} with a similar notation as in the $U_{e,\\mu}$ dominance.\nThe red (black) curves show the expected $95\\%$ C.L. exclusion sensitivity with 10-year statistics in ILC-1000 (ILC-250) by $B$ and $D_s$ mesons and $\\tau$ lepton decay production. The direct (DIS) production is absent in this case due to very limited flux of $\\tau$ leptons.\n\n\nAs shown in Figs.~\\ref{fig:production}, \\ref{fig:brUtauD}, \\ref{fig:brUtauB}, and \\ref{fig:brUtautau}, below the threshold of $m_{D_s}-m_{\\tau}\\simeq 0.2~{\\rm GeV}$,\nthe main HNL production channel is the $D_s\\to \\tau+{\\rm HNL}$ decay.\nFor $0.2~{\\rm GeV}\\lesssim m_{\\rm HNL}\\leq m_{\\tau}-m_e\\simeq 1.8~{\\rm GeV}$, the production of HNL is dominated by $\\tau$ lepton decay.\nMost of the $\\tau$ leptons are produced by $D_s$ meson decays, and the $\\tau$ pair production in the electromagnetic showers are subdominant, which therefore is dropped in the HNL study. \nThe branching ration of $D_s\\to \\tau +\\nu_{\\tau}$ is $5.48\\%$, and an energy dependence of $\\tau$ lepton spectra is determined by that of $D_s$ meson spectra in Fig.~\\ref{fig:production}.\nThe main decay channels of $\\tau$ are $\\tau\\to \\rho +{\\rm HNL}$, $\\tau\\to \\mu+\\nu+{\\rm HNL}$, $\\tau\\to e+\\nu+{\\rm HNL}$, $\\tau\\to \\pi+{\\rm HNL}$, $\\tau\\to K^{\\ast}+{\\rm HNL}$, and $\\tau\\to K+{\\rm HNL}$, see Fig~\\ref{fig:brUtautau}.\n\nIn the small $|U_{\\tau}|$ regions with the longer lifetime of HNL, the decay probability in Eq.~\\eqref{eq:decP} is approximated by $d P_{\\rm dec}^{\\rm dump} \/dz\\simeq 1\/l^{\\rm (lab)}_X$.\nFor the electron beam energy $E_{\\rm beam}=500~{\\rm GeV}$, the number of events by the $\\tau^{\\pm}\\to \\rho^{\\pm} +{\\rm HNL}$ is approximately given by\n\\begin{align}\n    N_{\\rm signal}^{\\rm (i)}&\\sim \\left(\\frac{N_{\\rm EOT}}{4\\times 10^{22}}\\right)\\left(\\frac{l_{\\rm dec}}{50~{\\rm }}\\right)\\left(\\frac{r_{\\rm det}}{3~{\\rm m}}\\right)^2 \\left(\\frac{81~{\\rm m}}{l_{\\rm dump}+l_{\\rm sh}}\\right)^2 \\left(\\frac{|U_{\\tau}|^2}{5\\times 10^{-9}}\\right)^2 \\left(\\frac{m_{\\rm HNL}}{0.5~{\\rm GeV}}\\right)^4, \n\\end{align}\nwhere we use\n\\begin{align}\n    {\\rm Br}(\\tau^{\\pm}\\to \\rho^{\\pm}+{\\rm HNL})\\propto |U_{\\tau}|^2,\\quad\n    {\\rm Br}_{\\rm vis}\\simeq 1,\\quad\n    (l^{\\rm lab}_X)^{-1}\\propto |U_{\\tau}|^2 \\cdot m^4_{\\rm HNL} \/E^{\\rm lab}_{\\rm HNL}.\n\\end{align}\nFor $1.8~{\\rm GeV}\\lesssim m_{\\rm HNL}$, the $B$ meson decays dominate the HNL productions, see Figs.~\\ref{fig:brUtauD}, \\ref{fig:brUtauB}, and \\ref{fig:brUtautau}.\n\n\n\n\\subsection{Sensitivities from $Z$~decays at ILC and FCC-ee}\\label{sec:ZatILC}\n\nThe future $e^+ e^-$ colliders can be seen as a $Z$ boson factory, such as Giga-$Z$ program of the ILC and Tera-$Z$ program of the CERN Future $e^+ e^-$ Circular Collider, dubbed FCC-ee. The HNLs can leave a clear signal with displaced tracks at $e^+ e^-$ colliders once  they are produced via $Z\\to N \\bar\\nu, \\bar N \\nu$. This type of signal was examined at DELPHI detector of the Large Electron-Positron collider (LEP) \\cite{DELPHI:1996qcc}. We briefly study the future sensitivities of the HNL search using the displaced tracks from the $Z$ decay, which is complementary to the sensitivities of the ILC beam dump experiment. \n\n\nThe proposed ILC detector \\cite{Behnke:2013lya} has three layers of vertex detector (VTX) starting from 1.5~cm to 19.5~cm surrounding the beam,  Time Projection Chambers (TPC) as the main tracking detector spreading from 33~cm to 170~cm, and  two layers of silicon strip detectors (SIT) arranged between TPC and VTX. We assume the ILC detector is sensitive to the HNL signal with negligible background if the HNL decays between 3~cm and 170~cm from the collision point. We adopt  the radius-dependent track-detection efficiency $\\epsilon_{\\rm trk}$ linearly decreasing from $r = 3$~cm ($\\epsilon_{\\rm trk}=100\\%$) to $r = 170$~cm ($\\epsilon_{\\rm trk}=0\\%$)~\\cite{Bertholet:2021hjl}. Then, we assume $10^9$ $Z$ decays and  zero background after requiring the displaced tracks and consider all the HNL decays except for the fully invisible mode, $N\\to \\nu\\nu\\bar\\nu$. The $95\\%$~CL future sensitivity corresponds to three HNL decays with the multi-displaced tracks in the fiducial volume, and the result is included in Figs.~\\ref{fig:sensitivityUe},~\\ref{fig:sensitivityUmu},~and~\\ref{fig:sensitivityUtau}.\n\n  \nSimilarly, we study the future projection at the FCC-ee. For the simplicity, we take the same detector setup of the ILC and rescale the statistics assuming $10^{12}$ $Z$ decays. This is also shown in Figs.~\\ref{fig:sensitivityUe},~\\ref{fig:sensitivityUmu},~and~\\ref{fig:sensitivityUtau}. Ref.~\\cite{Blondel:2014bra} studied also the projected sensitivities at the FCC-ee, but our estimate is different with respect to the detector coverage and the signal efficiency. \n\n\n\\subsection{Existing constraints on HNL and projected sensitivities at other experiments}\\label{sec:bounds}\n\nThe presence of HNL can have drastic consequences on early-universe observables and has been explored extensively in the literature. The HNL is thermally produced in the early universe, and its late decay can disrupt the standard  Big Bang Nucleosynthesis. The typical constraint is for $\\tau_{\\rm HNL}\\lesssim 1\\rm s$, see \\cite{Abazajian:2012ys} and references therein. In addition, Ref ~\\cite{Boyarsky:2020dzc}  studies that the pions from the HNL hadronic decay alter the decoupling of neutrons from the prediction of the standard BBN, which affects the ratio of neutrons to protons. The upper limit on the HNL lifetime is obtained from the $^4$He abundance as $\\tau_{\\rm HNL}\\lesssim 0.02$~s (see also other recent studies \\cite{Alonso-Alvarez:2022uxp, Bondarenko:2021cpc}).\n\n\nThe long-lived HNLs are also extensively searched in the accelerator-based experiments. The HNLs may be produced directly either in the beam-target collisions, or in the decays of secondary particles such as $B$, $D$ mesons and $\\tau$ leptons. The searches are carried out either by detecting the displaced decay of the HLN or the significant missing momentum of the events. In the following, we briefly list the existing constraints included in Fig.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau}. \n\n\n\\begin{itemize}\n\\item CHARM  proton beam dump experiment --- \nSearches for HNLs produced at a proton beam dump operating at the 400 GeV CERN SPS are  sensitive to all the scenarios we consider~\\cite{CHARM:1983ayi, CHARM:1985nku}. Recently, the result of the $U_\\tau$ dominant scenario was reanalyzed, and the bound was improved for  $ 0.3~{\\rm GeV}\\lesssim m_{\\rm HNL}\\lesssim 1.5~{\\rm GeV}$~\\cite{Boiarska:2021yho}. \n\n\n\\item Other beam damp\/neutrino beam experiments --- \nIn addition to the CHARM experiment, the HNL searches were conducted at a variety of fixed target and neutrino experiments, such as NuTeV~\\cite{NuTeV:1999kej}, BEBC~\\cite{WA66:1985mfx}, and PS191~\\cite{Bernardi:1987ek}. They are mostly sensitive to the $U_\\mu$ mixing, while PS191 placed a strong bound on the $U_e$ dominant scenario as well.\n \n\n\\item Long-baseline neutrino experiment --- \nAt the T2K experiment, the near detector ND280 was utilized to search for the HNL~\\cite{T2K:2019jwa}. The detection principle is similar to the one at  PS191. \nWe include the bounds of ``single-channel'' analysis where only one of  $U_\\ell$ is non-zero at a time. In the future, this type of search can be performed at the near detector of the DUNE experiment~\\cite{Coloma:2020lgy}.\n\n\n\\item Super-Kamiokande --- \nThe HNLs can be copiously produced from kaon and pion decays in atmospheric showers and decay in the Super-Kamiokande detector. In~\\cite{Coloma:2019htx}, the authors analyzed the Super-Kamiokande data~\\cite{Super-Kamiokande:2017yvm}, and obtained bounds on the HNL in all the scenarios. The bound on $U_{\\tau}$ is as strong as the T2K one. \n \n\n\\item Pion decay --- \nPrecision measurements of charge pions can test $\\pi^+ \\to \\ell^+ +\\rm HNL$  with no subsequent HNL decay which is sensitive to  the $U_e$ and $U_{\\mu}$ mixings \\cite{PIENU:2017wbj, PIENU:2019usb}. It gives an unique bound in the low mass region $m_{\\rm HNL}\\sim 0.1~{\\rm GeV}$ of the $U_{e}$ dominant scenario. \n\n\n\\item  Kaon decay ---  \nSimilar to the search with the charged pion, the HNL searches have been conducted by $K^+\\to \\ell^+ +\\rm HNL(invisible)$ at NA62~\\cite{NA62:2020mcv, NA62:2021bji}, E949~\\cite{E949:2014gsn}, and KEK\\cite{Yamazaki:1984sj}.  Typically they give the best limit near the threshold of $m_{\\rm HNL}\\lesssim m_{K^+}-m_{\\ell^+}$. \n\n\n\\item $B$ meson and $\\tau$ lepton decay --- \nThe heavier HNL can be looked for from the $B$ meson decays. The HNL production by $B\\to X \\ell +{\\rm HNL}\\ (\\ell=e,\\mu)$  with a subsequent displaced decay of ${\\rm HNL}\\to \\ell \\pi$ was examined using the Belle data in~\\cite{Belle:2013ytx}.  The $B$-factories also produce $\\tau$ leptons  copiously, and  the search of long-lived HNL from $\\tau$ lepton decays at Belle and Babar placed a bound on $U_{\\tau}.$ \\cite{Dib:2019tuj}\n\n\n\\item $Z$ boson decay --- \nThe search of $Z$ boson decaying into HNL using the DELPHI data at the LEP collider constraints all the scenarios. It sets strongest limit  in mass range around $m_{\\rm HNL}\\sim {\\cal O}(10)~\\rm GeV$ \\cite{DELPHI:1996qcc}. See Sec.~\\ref{sec:ZatILC} for more detail and  the  projection at the future $e^+e^-$ colliders.    \n\n\n\\item Large Hadron Collider --- \nAt the ATLAS and CMS detectors of LHC, the $W$ boson mediated process efficiently produces the HNL. Depending on the final states and decay length, different analyses were performed at ATLAS~\\cite{ATLAS:2019kpx, ATLAS:2022atq} and CMS~\\cite{CMS:2018jxx, CMS:2018iaf, CMS:2022fut}. In particular, the search for the displaced decays of HNL~\\cite{CMS:2022fut, ATLAS:2022atq} excludes the large parameter space. \n \n\n\\end{itemize}\n\n\nAmong studies of the future HNL searches, we include projected sensitivities at FASER2~\\cite{FASER:2018eoc}, NA62~\\cite{Beacham:2019nyx}, DUNE~\\cite{Coloma:2020lgy}, SHiP~\\cite{Alekhin:2015byh}, and MATHUSLA~\\cite{Curtin:2018mvb}  in Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau} to compare  to the sensitivities at the ILC beam dump experiment. Other experiments including LHCb~\\cite{Antusch:2017hhu, Cvetic:2019shl}, Codex-b~\\cite{Aielli:2019ivi}, Belle~II~\\cite{Dib:2019tuj}, Dark-Quest~\\cite{Batell:2020vqn}, IceCube~\\cite{Coloma:2017ppo},  ATLAS and CMS experiments~\\cite{ATLAS:2022atq, CMS:2022fut} at HL-LHC can also search for the HNLs beyond the current experimental constrains. \n\nIn Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau}, the estimated sensitivity contours of ILC is better than the other proposed searches. We remind here  that the contour is calculated  assuming zero background events, but it is very encouraging to explore the possibility further. The ILC sensitivity is very close to the one of SHiP in the low mass region ($m_{\\rm HNL}<2$~GeV). Above 2~GeV, the sensitivity of the ILC beam dump experiment is better than SHiP because the initial electron energy is high enough to produce $B$ mesons at a higher rate. Thanks to the higher HNL mass, the HNL decay products are more clearly separated from the background so that the neutrino background would be less critical in the region. \n\n\n\n\n\\section{Discussions}\\label{sec:discussion}\n\nThe ILC beam dump experiment is a seamless extension of the ILC program, which provides a unique opportunity to test feebly interacting light particles. The electron beam energy of ILC is much higher than the ones at the current and past electron beam dump experiments, and therefore, not only the light mesons but also the heavier SM particles, such as heavy flavor mesons and $\\tau$ lepton, can be produced in the beam dump. Moreover, a large number of electrons on target are expected thanks to the high-intensity beam. In many BSM scenarios, new particles can be efficiently produced by decays of the SM particles. Therefore, it is essential to estimate the production rate of the SM particles at the ILC beam dump experiment. In this paper, we evaluate the light and heavy mesons and $\\tau$ lepton spectra at the decay for the first time using  PHITS and PYTHIA8. PHITS is responsible for producing and transporting light SM particles, and we incorporate the heavy meson productions calculated by PYTHIA8 into PHITS. The main results are in Fig.~\\ref{fig:decay} for the light mesons and in Fig.~\\ref{fig:production} for the heavy mesons and $\\tau$ lepton. \n\n\nThe spectra of SM particles can be used to estimate the  yield of BSM particles from SM particle decay. As a demonstration, we studied the projected sensitivity of the heavy neutral leptons at the ILC beam dump experiment. Figs.~\\ref{fig:sensitivityUe}, \\ref{fig:sensitivityUmu}, and \\ref{fig:sensitivityUtau} show that  ILC would explore the HNLs with the heavier mass and small mixing and cover the large parameter space motivated by the baryon asymmetry of the Universe. We use the Monte-Carlo simulation to evaluate the HNL signal from the SM particle decay, and the results are well reproduced by the the course-grained integration method convolving the SM particle spectra given in Figs.~\\ref{fig:decay} and \\ref{fig:production}. \n\n\nApart from the SM particle decays, the HNLs can be produced through various processes at ILC. The EM shower can directly create an HNL via DIS, and we account for this production by the course-grained integration method. Moreover, copious $Z$ decays expected in the primary detector of ILC would realize  a different search method for the mildly long-lived HNLs.\n\n\nOther than the HNLs, many motivated long-lived particles can be predominantly produced by the SM particle decay. For example, dark scalars such as the QCD axion and the Higgs portal scalar can be efficiently produced via flavor-changing decays of meson (especially $B\\to K X$ and $K\\to \\pi X$ where $X$ denotes a long-lived particle).  The heavy mesons produced from the high energy electron beam allow us to probe long-lived particles heavier than several GeV. Another advantage of the abundant heavy mesons is that the experiment is sensitive to a new particle that preferably couples to the third generation fermions ($b$ quark or $\\tau$ lepton). One can estimate the sensitivity of these particles at the ILC beam dump experiment based on our results in Figs.~\\ref{fig:decay}, \\ref{fig:production} and the approximate formula like Eqs.~\\eqref{eq:sig}. \n\n\n\n\n\\section*{Note added}\\label{sec:note}\nWhile completing this work, we became aware of \\cite{Giffin:2022}, which considers related topics.\n\n\\section*{Acknowledgement}\\label{sec:ackn}\nMN is supported by \t Grant-in-Aid for Scientific Research on Innovative Areas(16H06492) JSPS KAKENHI 22K03629, YS is supported by JSPS KAKENHI JP21H05466.  KT is supported by in part the US Department of Energy grant DE-SC0010102 and JSPS KAKENHI 21H01086. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}