diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpudj" "b/data_all_eng_slimpj/shuffled/split2/finalzzpudj" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpudj" @@ -0,0 +1,5 @@ +{"text":"\\section*{References}}\n\n\n\n\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{framed}\n\\usepackage{amsmath,color}\n\\usepackage{mathrsfs}\n\\usepackage{graphicx}\n\\usepackage{epstopdf}\n\\usepackage{float}\n\\usepackage{caption}\n\\usepackage{subcaption}\n\\usepackage{bm}\n\\usepackage{bbm}\n\\usepackage{mathrsfs}\n\\usepackage{cleveref}\n\\usepackage{soul}\n\\usepackage{accents}\n\\usepackage{color,soul}\n\\usepackage{color}\n\\usepackage{tabu}\n\\usepackage{longtable}\n\\usepackage{nomencl}\n\\makenomenclature\n\\setlength{\\nomitemsep}{-\\parskip}\n\\biboptions{sort&compress}\n\\soulregister\\citep7\n\\soulregister\\citet7\n\\soulregister\\citealp7\n\\newcommand{\\highlight}[1]{\\colorbox{yellow}{$\\displaystyle #1$}}\n\\newsavebox{\\measurebox}\n\\newcommand{\\vect}[1]{\\boldsymbol{\\mathbf{#1}}}\n\n\n\\journal{Engineering Fracture Mechanics}\n\n\\makeatletter\n\\def\\@author#1{\\g@addto@macro\\elsauthors{\\normalsize%\n \\def\\baselinestretch{1}%\n \\upshape\\authorsep#1\\unskip\\textsuperscript{%\n \\ifx\\@fnmark\\@empty\\else\\unskip\\sep\\@fnmark\\let\\sep=,\\fi\n \\ifx\\@corref\\@empty\\else\\unskip\\sep\\@corref\\let\\sep=,\\fi\n }%\n \\def\\authorsep{\\unskip,\\space}%\n \\global\\let\\@fnmark\\@empty\n \\global\\let\\@corref\\@empty \n \\global\\let\\sep\\@empty}%\n \\@eadauthor={#1}\n}\n\\makeatother\n\n\\begin{document}\n\n\\begin{frontmatter}\n\n\n\n\\title{A cohesive zone framework for environmentally assisted fatigue}\n\n\n\\author{Susana {del Busto}\\fnref{Uniovi}}\n\n\\author{Covadonga Beteg\\'on\\fnref{Uniovi}}\n\n\\author{Emilio Mart\\'{\\i}nez-Pa\\~neda\\corref{cor1}\\fnref{DTU}}\n\\ead{mail@empaneda.com}\n\n\\address[Uniovi]{Department of Construction and Manufacturing Engineering, University of Oviedo, Gij\\'on 33203, Spain}\n\n\\address[DTU]{Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark}\n\n\\cortext[cor1]{Corresponding author. Tel: +45 45 25 42 71; fax: +45 25 19 61.}\n\n\n\\begin{abstract}\nWe present a compelling finite element framework to model hydrogen assisted fatigue by means of a hydrogen- and cycle-dependent cohesive zone formulation. The model builds upon: (i) appropriate environmental boundary conditions, (ii) a coupled mechanical and hydrogen diffusion response, driven by chemical potential gradients, (iii) a mechanical behavior characterized by finite deformation J2 plasticity, (iv) a phenomenological trapping model, (v) an irreversible cohesive zone formulation for fatigue, grounded on continuum damage mechanics, and (vi) a traction-separation law dependent on hydrogen coverage calculated from first principles. The computations show that the present scheme appropriately captures the main experimental trends; namely, the sensitivity of fatigue crack growth rates to the loading frequency and the environment. The role of yield strength, work hardening, and constraint conditions in enhancing crack growth rates as a function of the frequency is thoroughly investigated. The results reveal the need to incorporate additional sources of stress elevation, such as gradient-enhanced dislocation hardening, to attain a quantitative agreement with the experiments.\n\\end{abstract}\n\n\\begin{keyword}\n\nHydrogen embrittlement \\sep Cohesive zone models \\sep Hydrogen diffusion \\sep Finite element analysis \\sep Fatigue crack growth\n\n\n\n\\end{keyword}\n\n\\end{frontmatter}\n\n\n\n\\begin{framed}\n\\nomenclature{$a$}{crack length}\n\\nomenclature{$b, \\, b_0$}{current and initial crack opening displacement}\n\\nomenclature{$\\vect{B}$}{standard strain-displacement matrix}\n\\nomenclature{$\\vect{B_c}$}{global cohesive displacement-separation matrix}\n\\nomenclature{$C$}{total hydrogen concentration}\n\\nomenclature{$C_L, \\, C_T$}{hydrogen concentration in lattice and trapping sites}\n\\nomenclature{$c_q$}{specific heat capacity}\n\\nomenclature{$\\mathcal{C}, \\, m$}{Paris law coefficients}\n\\nomenclature{$\\mathcal{D}, \\, \\mathcal{D}_e$}{standard and effective diffusion coefficients}\n\\nomenclature{$D, \\, D_c, \\, D_m$}{damage variable: total, cyclic and monotonic}\n\\nomenclature{$E$}{Young's modulus}\n\\nomenclature{$f$}{load frequency}\n\\nomenclature{$\\vect{f_c}$}{cohesive internal force vector}\n\\nomenclature{$\\Delta g_b^0$}{Gibbs free energy difference}\n\\nomenclature{$\\vect{J}$}{hydrogen flux vector}\n\\nomenclature{$\\vect{K_c}$}{cohesive tangent stiffness matrix}\n\\nomenclature{$K_T$}{trap equilibrium constant}\n\\nomenclature{$K, \\, K_0$}{remote and reference stress intensity factor}\n\\nomenclature{$\\vect{L}$}{local displacement-separation matrix}\n\\nomenclature{$\\vect{\\mathscr{L}}$}{elastoplastic constitutive matrix}\n\\nomenclature{$\\vect{N}$}{shape functions matrix}\n\\nomenclature{$N$}{number of cycles}\n\\nomenclature{$\\mathcal{N}$}{strain hardening exponent}\n\\nomenclature{$N_A$}{Avogadro's number}\n\\nomenclature{$N_L, \\, N_T$}{number of lattice and trapping sites per unit volume}\n\\nomenclature{$q$}{heat flux per unit area}\n\\nomenclature{$\\mathcal{R}$}{universal gas constant}\n\\nomenclature{$\\vect{R}$}{rotational matrix}\n\\nomenclature{$R$}{load ratio}\n\\nomenclature{$R_0$}{reference plastic length}\n\\nomenclature{$T$}{elastic T-stress}\n\\nomenclature{$\\mathcal{T}$}{absolute temperature}\n\\nomenclature{$\\vect{T}, \\, \\tilde{\\vect{T}}$}{standard and effective cohesive traction vectors}\n\\nomenclature{$\\vect{t}$}{external traction vector}\n\\nomenclature{$T_n$}{normal cohesive traction}\n\\nomenclature{$U$}{internal energy per unit mass}\n\\nomenclature{$\\vect{U}$}{global nodal displacement vector}\n\\nomenclature{$\\vect{u}, \\, \\tilde{\\vect{u}}$}{field and local nodal displacement vectors}\n\\nomenclature{$\\bar{V}_H$}{partial molar volume of hydrogen}\n\\nomenclature{$V_M$}{molar volume of the host lattice}\n\\nomenclature{$W_B$}{trap binding energy}\n\\nomenclature{$\\alpha$}{compression penalty factor}\n\\nomenclature{$\\beta$}{number of lattice sites per solvent atom}\n\\nomenclature{$\\vect{\\Delta}, \\, \\tilde{\\vect{\\Delta}}$}{local field and nodal separation vectors}\n\\nomenclature{$\\Delta_n$}{normal cohesive separation}\n\\nomenclature{$\\delta_n$}{characteristic normal cohesive length}\n\\nomenclature{$\\delta_{\\Sigma}$}{accumulated cohesive length}\n\\nomenclature{$\\varepsilon_p$}{equivalent plastic strain}\n\\nomenclature{$\\vect{\\varepsilon}$}{Cauchy strain tensor}\n\\nomenclature{$\\theta_H$}{hydrogen coverage}\n\\nomenclature{$\\theta_L, \\, \\theta_T$}{occupancy of lattice and trapping sites}\n\\nomenclature{$\\mu_L$}{lattice chemical potential}\n\\nomenclature{$\\rho$}{density}\n\\nomenclature{$\\sigma_f$}{cohesive endurance limit}\n\\nomenclature{$\\sigma_H$}{hydrostatic stress}\n\\nomenclature{$\\sigma_Y$}{initial yield stress}\n\\nomenclature{$\\vect{\\sigma}$}{Cauchy stress tensor}\n\\nomenclature{$\\sigma_{max}, \\, \\sigma_{max,0}$}{current and original cohesive strength}\n\\nomenclature{$\\phi_n$}{normal cohesive energy}\n\\printnomenclature\n\\end{framed}\n\n\\section{Introduction}\n\\label{Sec:Introduction}\n\nMetallic materials play a predominant role in structures and industrial components because of their strength, stiffness, toughness and tolerance of high temperatures. However, hydrogen has been known for over a hundred years to severely degrade the fracture resistance of advanced alloys, with cracking being observed in modern steels at one-tenth of the expected fracture toughness. With current engineering approaches being mainly empirical and highly conservative, there is a strong need to understand the mechanisms of such hydrogen-induced degradation and to develop mechanistic-based models able to reproduce the microstructure-dependent mechanical response at scales relevant to engineering practice.\\\\\n\nModels based on the hydrogen enhanced decohesion (HEDE) mechanism have proven to capture the main experimental trends depicted by high-strength steels in aqueous solutions and hydrogen-containing gaseous environments \\cite{G03}. The use of cohesive zone formulations is particularly appealing in this regard, as they constitute a suitable tool to characterize the sensitivity of the fracture energy to hydrogen coverage. The cohesive traction separation law can be derived from first principles quantum mechanics \\cite{S04} or calibrated with experiments \\cite{S08,Y16}. The statistical distribution of relevant microstructural features has also fostered the use of weakest-link approaches \\cite{N10,A14}. Very recently, Mart\\'{\\i}nez-Pa\\~neda \\textit{et al.} \\cite{M16} integrated strain gradient plasticity simulations and electrochemical assessment of hydrogen solubility in Gerberich \\cite{G12} model. The investigation of a Ni-Cu superalloy and a modern ultra-high-strength steel revealed an encouraging quantitative agreement with experimental data for the threshold stress intensity factor and the stage II crack growth rate. However, and despite the fact that most industrial components experience periodic loading, modeling efforts have been mainly restricted to monotonic conditions. Recently, Moriconi \\textit{et al.} \\cite{M14} conducted experiments and simulations to investigate the role of hydrogen on a 15-5PH martensitic steel intended for gaseous hydrogen storage. Model predictions provided a very good agreement with experimental data for low hydrogen pressures but failed to capture the deleterious effect of hydrogen on the fatigue crack propagation under high pressures. Understanding the role of hydrogen in accelerating crack growth rates under cyclic loading could be crucial to enable the use of high-strength steels in the energy sector and to develop reliable transport and storage infrastructure for future energy systems.\\\\\n\nIn this work, we present a general numerical framework for hydrogen-assisted fatigue. The main ingredients of the model are: (i) realistic Dirichlet type conditions to account for stress-assisted diffusion at the boundaries, (ii) an extended hydrogen transport equation governed by hydrostatic stresses and plastic straining through trapping, (iii) higher order elements incorporating a coupled mechanical-diffusion response, (iv) continuum large strains elastoplasticity, (v) a hydrogen coverage dependent cohesive strength, and (vi) a Lemaitre-type damage response for an irreversible traction-separation law. The influence of diffusible hydrogen in fatigue crack growth is systematically investigated, the main experimental trends captured and valuable insight achieved.\n\n\\section{Numerical framework}\n\\label{Sec:NumModel}\n\nHydrogen transport towards the fracture process zone and subsequent cracking under cyclic loading conditions are investigated by means of a coupled mechanical-diffusion-cohesive finite element framework. Section \\ref{Sec:UMATHT} describes the mechanical-diffusion coupling that builds upon the analogy with heat transfer, Section \\ref{Sec:CZM} provides details of the cyclic and hydrogen dependent cohesive zone formulation employed and finally Section \\ref{Sec:FEM} outlines the general assemblage and implementation. \n\n\\subsection{Coupled mechanical-diffusion through the analogy with heat transfer}\n\\label{Sec:UMATHT}\n\nThe hydrogen transport model follows the pioneering work by Sofronis and McMeeking \\cite{SM89}. Hence, hydrogen transport is governed by hydrostatic stress and plastic straining through trapping. Hydrogen moves through normal interstitial lattice site diffusion and the diffusible concentration of hydrogen $C$ is defined as the sum of the hydrogen concentrations at reversible traps $C_T$ and lattice sites $C_L$. The latter is given by,\n\\begin{equation}\\label{Eq:CL}\nC_L=N_L \\theta_L\n\\end{equation}\n\n\\noindent where $N_L$ is the number of sites per unit volume and $\\theta_L$ the occupancy of lattice sites. The former can be expressed as a function of $V_M$, the molar volume of the host lattice, as:\n\\begin{equation}\\label{Eq:NLVA}\nN_L=\\frac{\\beta N_A}{V_M}\n\\end{equation} \n\n\\noindent with $N_A$ being Avogadro's number and $\\beta$ the number of interstitial lattice sites per solvent atom. On the other hand, the hydrogen concentration trapped at microstructural defects is given by,\n\\begin{equation}\\label{Eq:CT}\nC_T=N_T \\theta_T \n\\end{equation}\n\n\\noindent where $N_T$ denotes the number of traps per unit volume and $\\theta_T$ the occupancy of the trap sites. Here, focus will be placed on reversible trapping sites at microstructural defects generated by plastic straining - dislocations; a key ingredient in the mechanics of hydrogen diffusion \\cite{O14,D15}. A phenomenological relation between the trap density and the equivalent plastic strain is established based on the permeation tests by Kumnick and Johnson \\cite{KJ80},\n\\begin{equation}\\label{Eq:NTKJ}\n\\log N_T = 23.26 - 2.33 \\exp \\left( -5.5 \\varepsilon_p \\right)\n\\end{equation} \n\nOriani's equilibrium theory \\cite{O70} is adopted, resulting in a Fermi-Dirac relation between the occupancy of trap and lattice sites,\n\\begin{equation}\\label{Eq:Oriani}\n\\frac{\\theta_T}{1-\\theta_T}=\\frac{\\theta_L}{1-\\theta_L} K_T \n\\end{equation}\n\n\\noindent with $K_T$ being the trap equilibrium constant,\n\\begin{equation}\nK_T=\\textnormal{exp}\\left( \\frac{-W_B}{\\mathcal{R}\\mathcal{T}} \\right)\n\\end{equation}\n\nHere, $W_B$ is the trap binding energy, $\\mathcal{R}$ the gas constant and $\\mathcal{T}$ the absolute temperature. Under the common assumption of low occupancy conditions ($\\theta_L << 1$), the equilibrium relationship between $C_T$ and $C_L$ becomes,\n\\begin{equation}\\label{Eq:CT2}\nC_T=\\frac{N_T K_T C_L}{K_T C_L + N_L}\n\\end{equation}\n\nIn a volume, $V$, bounded by a surface, $S$, with outward normal, $\\vect{n}$, mass conservation requirements relate the rate of change of $C$ with the hydrogen flux through $S$,\n\\begin{equation}\\label{Eq:MassBal}\n\\frac{\\textnormal{d}}{\\textnormal{d}t} \\int_V C \\,\\, \\textnormal{d}V + \\int_S \\vect{J} \\cdot \\vect{n} \\,\\, \\textnormal{d} S=0\n\\end{equation}\n\nFick's law relates the hydrogen flux with the gradient of the chemical potential $\\nabla \\mu_L$,\n\\begin{equation}\\label{Eq:J1}\n\\vect{J} = - \\frac{\\mathcal{D} C_L}{\\mathcal{R}\\mathcal{T}} \\nabla \\mu_L\n\\end{equation}\n\n\\noindent with $\\mathcal{D}$ being the diffusion coefficient. The chemical potential of hydrogen in lattice sites is given by,\n\\begin{equation}\\label{Eq:muL}\n\\mu_L = \\mu_L^0 + \\mathcal{R}\\mathcal{T} \\, \\textnormal{ln} \\frac{\\theta_L}{1-\\theta_L} - \\bar{V}_H \\sigma_H\n\\end{equation}\n\nHere, $\\mu_L^0$ denotes the chemical potential in the standard state and the last term corresponds to the so-called stress-dependent part of the chemical potential $\\mu_{\\sigma}$, with $\\bar{V}_H$ being the partial molar volume of hydrogen in solid solution. Assuming a constant interstitial sites concentration and substituting (\\ref{Eq:muL}) into (\\ref{Eq:J1}), one reaches\n\\begin{equation}\\label{Eq:JL2}\n\\vect{J}=-\\mathcal{D} \\nabla C_L + \\frac{\\mathcal{D}}{\\mathcal{R}\\mathcal{T}} C_L \\bar{V}_H \\nabla \\sigma_H\n\\end{equation}\n\nReplacing $\\vect{J}$ in the mass balance equation (\\ref{Eq:MassBal}), using the divergence theorem and considering the arbitrariness of $V$ renders,\n\\begin{equation}\\label{Eq:Bal2}\n\\frac{dC_L}{dt}+\\frac{dC_T}{dt} = \\mathcal{D} \\nabla^2 C_L - \\nabla \\cdot \\left( \\frac{\\mathcal{D} C_L \\bar{V}_H}{\\mathcal{R}\\mathcal{T}} \\nabla \\sigma_H \\right)\n\\end{equation}\n\nIt is possible to phrase the left-hand side of (\\ref{Eq:Bal2}) in terms of $C_L$ by making use of Oriani's theory of equilibrium,\n\\begin{equation}\n\\frac{\\mathcal{D}}{\\mathcal{D}_e} \\frac{d C_L}{dt}= \\mathcal{D} \\nabla^2 C_L - \\nabla \\cdot \\left( \\frac{\\mathcal{D} C_L \\bar{V}_H}{\\mathcal{R}\\mathcal{T}} \\nabla \\sigma_H \\right)\n\\end{equation}\n\n\\noindent where an effective diffusion constant has been defined,\n\\begin{equation}\\label{Eq:Deff}\n\\mathcal{D}_e=\\mathcal{D} \\frac{C_L}{C_L + C_T (1 - \\theta_T)}\n\\end{equation}\n\nRegarding the boundary conditions, a constant hydrogen concentration $C_b$ is prescribed at the crack faces in the vast majority of hydrogen embrittlement studies. However, as noted by Turnbull \\cite{T15}, such scheme may oversimplify the electrochemistry-diffusion interface and the use of generalized boundary conditions is particularly recommended for materials with high hydrogen diffusivity. Here, we follow Mart\\'{\\i}nez-Pa\\~neda \\textit{et al.} \\cite{M16b} and adopt Dirichlet-type boundary conditions where the lattice hydrogen concentration at the crack faces depends on the hydrostatic stress. Hence, the lattice hydrogen concentration at the crack faces equals,\n\\begin{equation}\\label{eq:DISP}\nC_L=C_b \\exp \\left( \\frac{\\bar{V}_H \\sigma_H}{\\mathcal{R}\\mathcal{T}} \\right)\n\\end{equation}\n\n\\noindent which is equivalent to prescribing a constant chemical potential. To this end, a user subroutine DISP is employed in ABAQUS to relate the magnitude of $C_L$ to a nodal averaged value of the hydrostatic stress. Also, the domain where the boundary conditions are enforced changes with crack advance. Consequently, a multi-point constraint (MPC) subroutine is defined to update the boundary region throughout the analysis - see Section \\ref{Sec:FEM}.\\\\\n\nFinite deformation J2 plasticity theory is used to compute the two mechanical ingredients of the present hydrogen transport scheme, $\\varepsilon_p$ and $\\sigma_H$. We develop a fully coupled mass transport - continuum elastoplastic finite element framework that is solved in a monolithic way. Higher order elements are used, with nodal displacements and lattice hydrogen concentration being the primary variables. The numerical implementation is carried out in the well-known finite element package ABAQUS. To this end, a UMATHT subroutine is developed to exploit the analogy with heat transfer \\cite{B16,D16}. Thus, the energy balance for a stationary solid in the absence of heat sources is given by,\n\\begin{equation}\n\\int_V \\rho \\, \\dot{U} \\, \\textnormal{d}V - \\int_S q \\, \\textnormal{d}S = 0\n\\end{equation}\n\n\\noindent where $\\rho$ is the density, $q$ the heat flux per unit area of the solid and $\\dot{U}$ the material time rate of the internal energy, the latter being related to the temperature change through the specific heat capacity: $\\dot{U}=c_q \\dot{\\mathcal{T}}$. The similitude with (\\ref{Eq:MassBal}) is clear and an appropriate analogy can be easily established (see Table \\ref{Tab:DiffusionHeat}), enabling the use of the coupled temperature-displacement capabilities already available in ABAQUS.\n\n\\begin{table}[H]\n\\centering\n\\caption{Analogy between heat transfer and mass diffusion.}\n\\label{Tab:DiffusionHeat}\n {\\tabulinesep=1.2mm\n \\begin{tabu} {cc}\n \\hline\n Heat transfer & Mass diffusion \\\\ \\hline\n $\\rho c_p \\frac{\\partial \\mathcal{T}}{\\partial t} + \\nabla q = 0$ & $\\frac{\\partial C_i}{\\partial t} + \\nabla \\vect{J}=0$\\\\\n $\\dot{U}=c_p \\dot{\\mathcal{T}}$ & $\\frac{\\partial C}{\\partial t}=\\frac{\\partial (C_L + C_T)}{\\partial t}$ \\\\\n $\\mathcal{T}$ & $C_L$ \\\\\n $c_p$ & $\\mathcal{D}\/\\mathcal{D}_{e}$ \\\\\n $\\rho$ & 1 \\\\\\hline\n \\end{tabu}}\n\\end{table}\n\n\\subsection{Cohesive zone model}\n\\label{Sec:CZM}\n\nA cohesive zone formulation will be employed to model crack initiation and subsequent growth. Based on the pioneering works by Dugdale \\cite{D60} and Barenblatt \\cite{B62}, cohesive zone models introduce the notion of a cohesive force ahead of the crack that prevents propagation. The micromechanisms of material degradation and failure are thus embedded into the constitutive law that relates the cohesive traction with the local separation. Damage is restricted to evolve along the pre-defined cohesive interface, and consequently, the numerical implementation is generally conducted by inserting cohesive finite elements in potential crack propagation paths. Hence, in the absence of body forces, the weak form of the equilibrium equations for a body of volume $V$ and external surface $S$ renders,\n\\begin{equation}\\label{Eq:CoheWeak}\n\\int_V \\vect{\\sigma} : \\delta \\vect{\\varepsilon} \\, \\textnormal{d}V + \\int_{S_c} \\vect{T} \\cdot \\delta \\vect{\\Delta} \\, \\textnormal{d}S= \\int_S \\vect{t} \\cdot \\delta \\vect{u} \\, \\textnormal{d}S\n\\end{equation}\n\nHere, $\\vect{T}$ are the cohesive tractions and $S_c$ is the surface across which these tractions operate. The standard part of the mechanical equilibrium statement is characterized by the Cauchy stress tensor $\\vect{\\sigma}$, the work-conjugate strain tensor $\\vect{\\varepsilon}$, the external tractions $\\vect{t}$ and the displacement vector $\\vect{u}$; the latter being obtained by interpolating the global nodal displacement $\\vect{u}=\\vect{N}\\vect{U}$. The local nodal separation $\\tilde{\\vect{\\Delta}}$ is related to the local nodal displacement $\\tilde{\\vect{u}}$ by\n\\begin{equation}\n\\tilde{\\vect{\\Delta}} = \\vect{L} \\tilde{\\vect{u}}\n\\end{equation}\n\n\\noindent where $\\vect{L}$ is a local displacement-separation relation matrix. The separation along a cohesive surface element is interpolated from the nodal separation by means of standard shape functions,\n\\begin{equation}\n\\vect{\\Delta} = \\vect{N} \\tilde{\\vect{\\Delta}}\n\\end{equation}\n\n\\noindent and the global nodal displacement is related to the local nodal displacement by means of a rotational matrix:\n\\begin{equation}\n\\tilde{\\vect{u}} = \\vect{R} \\vect{U}\n\\end{equation}\n\nThe relationship between the local separation and the global nodal displacement can be then obtained by combining the previous equations,\n\\begin{equation}\n\\vect{\\Delta} = \\vect{B}_c \\vect{U}\n\\end{equation}\n\n\\noindent where $\\vect{B}_c$ is a global displacement-separation relation matrix: $\\vect{B}_c=\\vect{N} \\vect{R} \\vect{L}$. Thus, accounting for the classic finite element discretization in (\\ref{Eq:CoheWeak}) and requiring the variational statement to hold for any admissible field, it renders\n\\begin{equation}\n\\int_V \\vect{B}^T \\vect{\\mathscr{L}} \\vect{\\varepsilon} \\, \\textnormal{d}V + \\int_{S_c} \\vect{B}_c^T \\vect{T} \\, \\textnormal{d}S= \\int_S \\vect{N}^T \\vect{t} \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent where $\\vect{\\mathscr{L}}$ is the elastoplastic constitutive matrix and $\\vect{B}$ the standard strain-displacement matrix. Considering the dependence of $\\vect{\\varepsilon}$ and $\\vect{T}$ on $\\vect{U}$,\n\\begin{equation}\n\\vect{U} \\left( \\int_V \\vect{B}^T \\vect{\\mathscr{L}} \\vect{B} \\, \\textnormal{d}V + \\int_{S_c} \\vect{B}_c^T \\frac{\\partial \\vect{T}}{\\partial \\vect{\\Delta}} \\vect{B}_c \\, \\textnormal{d}S \\right) = \\int_S \\vect{N}^T \\vect{t} \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent and the components of the classic finite element global system of equations can be readily identified. The stiffness matrix of the cohesive elements is therefore given by,\n\\begin{equation}\n\\vect{K}_c = \\int_{S_c} \\vect{B}_c^T \\frac{\\partial \\vect{T}}{\\partial \\vect{\\Delta}} \\vect{B}_c \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent which corresponds to the gradient of the internal cohesive force vector,\n\\begin{equation}\n\\vect{f}_c=\\int_{S_0} \\vect{B}_c^T \\vect{T} dS\n\\end{equation} \n\nThe pivotal ingredient of cohesive zone models is the traction-separation law that governs material degradation and separation. The exponentially decaying cohesive law proposed by Xu and Needleman \\cite{XN93} is here adopted. Focus will be placed on pure mode I problems and consequently, the constitutive equations related to the tangential separation will be omitted for the sake of brevity. As depicted in Fig. \\ref{fig:CoheLaw}, for a given shape of the traction-separation curve, the cohesive response can be fully characterized by two parameters, the cohesive energy $\\phi_n$ and the critical cohesive strength $\\sigma_{max,0}$.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.8]{CoheLaw.eps}\n\\caption{Traction-separation law characterizing the cohesive zone model in the absence of cyclic loading and hydrogen degradation.}\n\\label{fig:CoheLaw}\n\\end{figure}\n\nThe cohesive response is therefore characterized by the relation between the normal tractions ($T_n$) and the corresponding displacement jump ($\\Delta_n$) as,\n\\begin{equation}\nT_n=\\frac{\\phi_n}{\\delta_n} \\left( \\frac{\\Delta_n}{\\delta_n} \\right) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right)\n\\end{equation}\n\n\\noindent with the normal work of separation $\\phi_n$ being given by,\n\\begin{equation}\n\\phi_n= \\exp(1) \\sigma_{max,0} \\delta_n\n\\end{equation}\n\n\\noindent where $\\delta_N$ is the characteristic cohesive length under normal separation. The effect of hydrogen in lowering the cohesive strength, and subsequently the fracture toughness, is captured here by employing the impurity-dependent cohesive law proposed by Serebrinsky et al. \\cite{S04}. Hence, a first-principles-based relation between the \\emph{current} cohesive strength $\\sigma_{max}$ and the original cohesive strength in the absence of hydrogen $\\sigma_{max,0}$ is defined,\n\\begin{equation}\n\\frac{\\sigma_{max}(\\theta_H)}{\\sigma_{max,0}}=1 - 1.0467 \\theta_H + 0.1687 \\theta_H^2\n\\end{equation}\n\n\\noindent where $\\theta_H$ is the hydrogen coverage, which is defined as a function of hydrogen concentration and Gibbs free energy difference between the interface and the surrounding material, as expressed in the Langmuir-McLean isotherm:\n\\begin{equation}\n\\theta_H=\\frac{C}{C+\\exp \\left( \\frac{- \\Delta g_b^0}{\\mathcal{R}\\mathcal{T}} \\right)}\n\\end{equation}\n\nA value of 30 kJ\/mol is assigned to the trapping energy $\\Delta g_b^0$ in \\citep{S04} from the spectrum of experimental data available. Thus, from first principles calculations of hydrogen atoms in bcc Fe, a quantum-mechanically informed traction-separation law can be defined as a function of the hydrogen coverage \\cite{S04} (see Fig. \\ref{fig:CoheLawH}).\\\\ \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{CoheLawH.eps}\n\\caption{Effect of hydrogen coverage $\\theta_H$ on the traction-separation law characterizing the cohesive response.}\n\\label{fig:CoheLawH}\n\\end{figure}\n\nOn the other hand, cyclic damage is incorporated by means of the irreversible cohesive zone model proposed by Roe and Siegmund \\cite{RS03}. The model incorporates (i) loading-unloading conditions, (ii) accumulation of damage during subcritical cyclic loading, and (iii) crack surface contact. A damage mechanics approach is adopted to capture the cohesive properties degradation as a function of the number of cycles. A damage variable $D$ is defined so that it represents the effective surface density of micro defects in the interface. Consequently, an effective cohesive zone traction can be formulated: $\\tilde{\\vect{T}}=\\vect{T}\/(1-D)$. Subsequently, the current or effective cohesive strength $\\sigma_{max}$ is related to the initial cohesive strength $\\sigma_{max,0}$ as,\n\\begin{equation}\n\\sigma_{max}=\\sigma_{max,0} (1 - D)\n\\end{equation}\n\nA damage evolution law is defined so that it incorporates the relevant features of continuum damage approaches, namely: (i) damage accumulation starts if a deformation measure is greater than a critical magnitude, (ii) the increment of damage is related to the increment of deformation, and (iii) an endurance limit exists, bellow which cyclic loading can proceed infinitely without failure. From these considerations, cyclic damage evolution is defined as,\n\\begin{equation}\\label{Eq:Damage}\n\\dot{D}_c= \\frac{|\\dot{\\Delta}_n|}{\\delta_{\\Sigma}} \\left[ \\frac{T_n}{\\sigma_{max}} - \\frac{\\sigma_f}{\\sigma_{max,0}} \\right] H \\left( \\bar{\\Delta}_n - \\delta_n \\right)\n\\end{equation}\n\n\\noindent with $\\bar{\\Delta}_n=\\int |\\dot{\\Delta}_n| dt$ and $H$ denoting the Heaviside function. Two new parameters have been introduced: $\\sigma_f$, the cohesive endurance limit and $\\delta_{\\Sigma}$, the accumulated cohesive length - used to scale the normalized increment of the effective material separation. The modeling framework must also incorporate damage due to monotonic loading; as a consequence, the damage state is defined as the maximum of the cyclic and monotonic contributions,\n\\begin{equation}\nD= \\int \\textnormal{max} \\left( \\dot{D}_c, \\dot{D}_m \\right) dt\n\\end{equation}\n\n\\noindent being the latter characterized as:\n\\begin{equation}\n\\dot{D}_m = \\frac{ \\left. \\textnormal{max} \\left( \\Delta_n \\right) \\right|_{t_i} - \\left. \\textnormal{max} \\left( \\Delta_n \\right) \\right|_{t_{i-1}}}{4 \\delta_n}\n\\end{equation}\n\n\\noindent and updated only when the largest stored value of $\\Delta_n$ is greater than $\\delta_N$. Here, $t_{i-1}$ denotes the previous time increment and $t_i$ the current one. In addition to damage evolution, the cohesive response must be defined for the cases of unloading\/reloading, compression, and contact between the crack faces. Unloading is defined based on the analogy with an elastic-plastic material undergoing damage. Thereby, unloading takes place with the stiffness of the cohesive zone at zero separation, such that\n\\begin{equation}\nT_n=T_{max} + \\left( \\frac{\\exp(1) \\sigma_{max}}{\\delta_n} \\right) \\left( \\Delta_n - \\Delta_{max} \\right)\n\\end{equation}\n\n\\noindent where $\\Delta_{max}$ is the maximum separation value that has been attained and $T_{max}$ its associated traction quantity. Compression behavior applies when the unloading path reaches $\\Delta_n=0$ at $T_n < 0$. In such circumstances, the cohesive response is given by,\n\\begin{align}\nT_n = & \\frac{\\phi_n}{\\delta_n} \\left( \\frac{\\Delta_n}{\\delta_n} \\right) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right) + T_{max} - \\sigma_{max} \\exp(1) \\frac{\\Delta_{max}}{\\delta_n} \\nonumber \\\\\n& + \\alpha \\sigma_{max,0} \\exp(1) \\frac{\\Delta_n}{\\delta_n} \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right)\n\\end{align}\n\n\\noindent being $\\alpha$ a penalty factor that is taken to be equal to 10, following \\cite{RS03}. Contact conditions are enforced if $\\Delta_n$ is negative and the cohesive element has failed completely ($D=1$). At this instance the cohesive law renders,\n\\begin{equation}\nT_n= \\alpha \\sigma_{max,0} \\exp(1) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right) \\frac{\\Delta_n}{\\delta_n} \n\\end{equation}\n\n\\noindent where friction effects have been neglected. Fig. \\ref{fig:CoheLawF} shows a representative response obtained by applying a stress-controlled cyclic loading $\\Delta \\sigma \/ \\sigma_{max,0}=1$ with a zero stress ratio. The accumulated separation increases with the number of loading cycles, so that it becomes larger than $\\delta_n$ and damage starts to play a role, lowering the stiffness and the cohesive strength.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{CoheLawF.eps}\n\\caption{Cohesive response under stress-controlled cyclic loading conditions.}\n\\label{fig:CoheLawF}\n\\end{figure}\n\nThis novel cyclic- and hydrogen concentration-dependent cohesive zone framework is implemented in ABAQUS by means of a user element UEL subroutine. The code can be downloaded from www.empaneda.com\/codes and is expected to be helpful to both academic researchers and industry practitioners.\\\\\n\nIn some computations, numerical convergence is facilitated by employing the viscous regularization technique proposed by Gao and Bower \\cite{GB04}. Such scheme leads to accurate results if the viscosity coefficient, $\\xi$, is sufficiently small \\cite{Y16}. A sensitivity study has been conducted in the few cases where viscous regularization was needed; values of $\\xi$ on the order of $10^{-6}$ have proven to be appropriate for the boundary value problem under consideration. Other approaches to overcome snap-back instabilities, less suitable for cyclic loading, include the use of explicit finite element solution schemes \\cite{S16} or determining the equilibrium path for a specified crack tip opening by means of control algorithms \\cite{T76,SL04,M17b}.\n\n\\subsection{Finite element implementation}\n\\label{Sec:FEM}\n\nThe aforementioned mechanical-diffusion-cohesive numerical framework is implemented in the commercial finite element package ABAQUS. Fortran modules are widely employed to transfer information between the different user subroutines. Thus, as described in Fig. \\ref{fig:Abaqus}, a user material UMAT subroutine is developed to characterize the mechanical response by means of a finite strain version of conventional von Mises plasticity. The nodal averaged value of the hydrostatic stress at the crack faces is then provided to a DISP subroutine, so as to prescribe a more realistic $\\sigma_H$-dependent lattice hydrogen concentration. The hydrostatic stress gradient is computed by means of linear shape functions and, together with the equivalent plastic strain, is afterward given as input to the UMATHT subroutine to capture the effects of chemical expansion and trapping. The UMATHT subroutine provides the cohesive elements with the diffusible concentration of hydrogen in their adjacent continuum element. The damage variable is then transferred from the user elements to the MPC subroutine to keep track of the crack extension. Multi-point constraints have been defined between the nodes ahead of the crack and a set of associated dummy nodes that are activated as the crack advances. Hydrogen diffusion is assumed to be instantaneous, such that the lattice hydrogen concentration at the boundary is immediately prescribed when a new portion of crack surface is available.\\\\\n\n\\begin{figure}[H]\n \\makebox[\\textwidth][c]{\\includegraphics[width=1.2\\textwidth]{Abaqus.eps}}%\n \\caption{Schematic overview of the relations between the different Abaqus subroutines.}\n \\label{fig:Abaqus}\n\\end{figure}\n\nHigher order elements are used in all cases: 8-node quadrilateral elements with reduced integration are employed to model the bulk response, and crack initiation and growth are captured by 6-node quadrilateral cohesive elements with 12 integration points. Results post-processing is carried out in MATLAB by making use of \\emph{Abaqus2Matlab} \\cite{P17}, a novel tool that connects the two aforementioned well-known software suites. \n\n\\section{Results}\n\\label{Sec:Results}\n\nWe investigate the pernicious effect of hydrogen in fatigue crack growth, of great relevance in both energy storage and transport. The synergistic interaction of cyclic plastic deformation and local hydrogen uptake is particularly detrimental, with catastrophic failure being observed in cases where hydrogen-assisted cracking is negligible under monotonic loading \\cite{G90}.\\\\\n\nThe boundary layer model employed by Sofronis and McMeeking \\cite{SM89} is taken as a benchmark. Hence, hydrogen transport and subsequent cracking are investigated in an iron-based material with a diffusion coefficient of $\\mathcal{D}=0.0127$ mm$^2$\/s, Young's modulus of $E=207$ GPa, Poisson's ratio of $\\nu=0.3$ and initial yield stress of $\\sigma_Y=250$ MPa. Work hardening is captured by means of the following isotropic power law,\n\\begin{equation}\n\\sigma = \\sigma_Y \\left(1 + \\frac{E \\varepsilon_p}{\\sigma_Y} \\right)^\\mathcal{N}\n\\end{equation}\n\n\\noindent with the strain hardening exponent being equal to $\\mathcal{N}=0.2$. Isotropic hardening has been adopted to reproduce the conditions of \\cite{SM89}, but one should note that other plastic flow models can be easily incorporated; the use of non-linear kinematic hardening laws is particularly convenient to appropriately capture the Bauschinger effect under low load ratios. As described in Fig. \\ref{fig:Mesh1}, the crack region is contained within a circular zone and a remote Mode I load is applied by prescribing the displacements of the nodes at the outer boundary,\n\\begin{equation}\nu \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\cos \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right)\n\\end{equation}\n\\begin{equation}\nv \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\sin \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right)\n\\end{equation}\n\n\\noindent where $u$ and $v$ are the horizontal and vertical components of the displacement boundary condition, $r$ and $\\theta$ the radial and angular coordinates of each boundary node in a polar coordinate system centered at the crack tip, and $K_I$ is the applied stress intensity factor that quantifies the remote load in small scale yielding conditions. The lattice hydrogen concentration is prescribed in the crack surface as a function of $\\sigma_H$ and the boundary concentration in the absence of hydrostatic stresses $C_b$. Following \\cite{SM89}, an initial bulk concentration equal to $C_b$ is also defined in the entire specimen at the beginning of the analysis. Only the upper half of the circular domain is modeled due to symmetry and the outer radius is chosen to be significantly larger than the initial crack tip blunting radius. As shown in Fig. \\ref{fig:Mesh1}, a very refined mesh is used, with the characteristic element size in the vicinity of the crack, $l_e$, being significantly smaller than a reference plastic length,\n\\begin{equation}\nR_0 = \\frac{1}{3 \\pi \\left( 1 - \\nu^2 \\right)} \\frac{E \\phi_n}{\\sigma_Y^2}\n\\end{equation}\n\n\\noindent ($l_e<2000R_0$). A sensitivity study is conducted to ensure that the mesh resolves the cohesive zone size - approximately 14000 quadrilateral 8-node plane strain elements are employed. The modeling framework is suitable for both low and high cycle fatigue, with computations of $10^4$ cycles (with at least 10 load increments per cycle) running overnight on a single core.\\\\\n\n\\begin{figure}[H]\n \\makebox[\\textwidth][c]{\\includegraphics[width=1.2\\textwidth]{Mesh1.eps}}%\n \\caption{General and detailed representation of the finite element mesh employed for\nthe boundary layer model. Mechanical and diffusion boundary conditions are shown superimposed.}\n \\label{fig:Mesh1}\n\\end{figure}\n\nWe first validate the coupled mechanical-diffusion implementation by computing crack tip fields under monotonic loading conditions in the absence of crack propagation. Thus, the load is increased from zero at a rate of 21.82 MPa$\\sqrt{mm} \\,$s$^{-1}$ for 130 s and held fixed afterward, when the crack opening displacement is approximately 10 times the initial blunting radius $b=5b_0=10r_0$. Fig. \\ref{fig:SH} shows the estimated hydrostatic stress distribution along with the predictions by Sofronis and McMeeking \\cite{SM89} (symbols); results are shown along the extended crack plane with the distance to the crack tip normalized by the current crack tip opening $b$. A very good agreement is observed, verifying the finite strains J2 plasticity implementation. Fig. \\ref{fig:CLCT} shows the results obtained for the lattice and trapped hydrogen concentrations for a boundary concentration of $C_b=2.08 \\cdot 10^{12}$ H atoms\/mm$^3$ at 130 s and after reaching steady-state conditions. The quantitative response described by the lattice hydrogen concentration when accounting for the dilatation of the lattice significantly differs to that obtained prescribing a constant $C_L$, as highlighted by Di Leo and Anand \\cite{DA13} in the context of their constant lattice chemical potential implementation. The results achieved by means of the present $\\sigma_H$-dependent Dirichlet scheme accurately follow the analytical steady-state solution for the distribution of the lattice hydrogen concentration ahead of the crack. On the other hand, $C_T$ shows a high peak at the crack tip and negligible sensitivity to the diffusion time (the curves for 130 s and steady state fall on top of each other); this is due to the governing role of plastic deformation as a result of the direct proportional relationship between $C_T$ and $N_T$.\\\\\n\n\\begin{figure}[H]\n \\centering\n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{SH.eps}\n \\caption{}\n \\label{fig:SH}\n \\end{subfigure}\n \n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{CLCT.eps}\n \\caption{}\n \\label{fig:CLCT}\n \\end{subfigure}\n \n \\caption{Crack tip fields for a stationary crack in an iron-based material under monotonic loading conditions, (a) normalized hydrostatic stress distribution for $K_I=2836.7$ MPa$\\sqrt{mm}$ and (b) lattice and trap sites hydrogen concentrations at steady state and after 130 s.}\\label{fig:CrackTipFields}\n\\end{figure}\n\nEnvironmentally assisted fatigue is subsequently investigated by scaling in time the external load by a sinusoidal function. The cyclic boundary conditions prescribed are characterized by the load amplitude $\\Delta K = K_{max} - K_{min}$ and the load ratio $R=K_{min}\/K_{max}$. An initial prestressing is defined, such that the mean load equals the load amplitude, and both $R$ and $\\Delta K$ remain constant through the analysis.\\\\\n\nFollowing \\cite{RS03}, the accumulated cohesive length in (\\ref{Eq:Damage}) is chosen to be $\\delta_{\\Sigma}=4 \\delta_n$ and the endurance coefficient $\\sigma_f \/ \\sigma_{max,0}=0.25$. The initial cohesive strength is assumed to be equal to $\\sigma_{max,0}=3.5 \\sigma_Y$ based on the seminal work by Tvergaard and Hutchinson \\cite{TH92}. One should, however, note that such magnitude is intrinsically associated with the stress bounds of conventional plasticity - more realistic values can be obtained if the role of the increased dislocation density associated with large gradients in plastic strain near the crack tip is accounted for \\cite{M17b,MB15}. A reference stress intensity factor,\n\\begin{equation}\nK_0 = \\sqrt{ \\frac{E \\phi_n}{(1-\\nu^2)}}\n\\end{equation}\n\n\\noindent is defined to present the results.\\\\\n\nThe capacity of the model to capture the sensitivity of fatigue crack growth rates to a hydrogenous environment is first investigated by computing the crack extension $\\Delta a$ as a function of the number of cycles for different values of $C_L$ at the boundary. Figure \\ref{fig:InfluenceH} shows the results obtained for a load ratio of $R=0.1$ and frequency of 1 Hz. The magnitude of the load ratio is appropriate for applications in the context of hydrogen-fueled vehicles, where load ratios between $R=0.1$ and $R=0.2$ accurately characterize the mechanical stresses resulting from filling cycles. Results reveal a strong influence of the environment, with crack propagation rates increasing significantly with the hydrogen content; the model appropriately captures the deleterious effect of hydrogen on crack growth resistance under cyclic loading conditions.\\\\\n\n\\begin{figure}[H]\n \\centering\n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{FigHa.eps}\n \\caption{}\n \\label{fig:InfluenceHa}\n \\end{subfigure}\n \n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{FigHb.eps}\n \\caption{}\n \\label{fig:InfluenceHb}\n \\end{subfigure}\n \n \\caption{Predicted influence of the environment on (a) crack extension versus number of cycles for $\\Delta K\/K_0=0.1$ and (b) fatigue crack growth rate versus load amplitude. Results have been obtained for an iron-based material under a load ratio of $R=0.1$ and frequency of $f=1$ Hz}\\label{fig:InfluenceH}\n\\end{figure}\n\nLattice hydrogen concentrations at the boundary range from 1 wppm ($4.68 \\cdot 10^{15}$ H atoms\/mm$^3$), which corresponds to a 3\\% NaCl aqueous solution \\cite{G86}, to 0.1 wppm. The important role of hydrogen in the fatigue crack growth behavior can be clearly observed in the crack growth versus crack amplitude curves (Fig. \\ref{fig:InfluenceHb}). By making use of the well-known Paris equation \\cite{P61}:\n\\begin{equation}\n\\frac{da}{dN}=\\mathcal{C} \\Delta K^m\n\\end{equation}\n\n\\noindent one can easily see that $\\mathcal{C}$ significantly increases with the environmental hydrogen content, in agreement with the experimental trends. On the other hand, results render a Paris exponent that shows little sensitivity to the hydrogen concentration, falling in all cases within the range of experimentally reported values for metals in inert environments ($m \\approx 4$). The cycle-dependent contribution of the environment manifests significantly, while the influence of $C_b$ as $\\Delta K$ increases is governed by a trade-off between larger levels of equivalent plastic strain (increasing $C_T$ and subsequently $C$) and shorter diffusion times due to greater crack growth rates. Thus, for a given frequency, the effect of hydrogen on the slope of the $da\/dN$ versus $\\Delta K$ curve depends heavily on the diffusion and mechanical properties of the material under consideration. The sensitivity of a steel to hydrogen embrittlement is therefore bounded between two limit cases: (a) \\emph{slow tests}, where the testing time significantly exceeds the diffusion time for hydrogen within the specimen, and (b) \\emph{fast tests}, where the testing time is much less than the diffusion time. In the former bound, hydrogen atoms accumulate in the fracture process zone and the distribution of lattice hydrogen concentration is governed by Eq. (\\ref{eq:DISP}). For sufficiently rapid tests the initial (pre-charged) hydrogen concentration and the contributions from reversible microstructural traps dominate the response. As a consequence, experiments reveal a relevant increase in $da\/dN$ with decreasing frequency until an upper bound is reached where the load-cycle duration is sufficient to allow hydrogen to diffuse and fully saturate the crack tip fracture process zone \\cite{M12}.\\\\\n\nWe subsequently investigate the influence of the loading frequency. A normalized frequency is defined as,\n\\begin{equation}\n\\bar{f}=\\frac{f R_0^2}{\\mathcal{D}}\n\\end{equation}\n\n\\noindent so as to quantify the competing influence of test and diffusion times. Fig. \\ref{fig:InfluenceFreqIronA} shows crack growth resistance curves obtained for the iron-based material under consideration in the aforementioned asymptotic limits - \\emph{slow tests} ($\\bar{f} \\to 0$) and \\emph{fast tests} ($\\bar{f} \\to \\infty$). In agreement with expectations, crack propagation is enhanced by larger testing times but results reveal very little sensitivity to the loading frequency, as opposed to experimental observations. Fig. \\ref{fig:InfluenceFreqIronB} provides the basis for the understanding of the small susceptibility of crack growth rates to the loading frequency; the $C_L$ elevation in the $\\sigma_H$-dominated case is less than 10\\% of the lattice hydrogen concentration in the \\emph{fast test}.\\\\\n\n\\begin{figure}[H]\n \\centering\n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{FigFreqIron.eps}\n \\caption{}\n \\label{fig:InfluenceFreqIronA}\n \\end{subfigure}\n \n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{CLIronFreq.eps}\n \\caption{}\n \\label{fig:InfluenceFreqIronB}\n \\end{subfigure}\n \n \\caption{Influence of the frequency in an iron-based material: (a) crack extension versus number of cycles, and (b) lattice hydrogen distribution ahead of the crack at the maximum $\\Delta K$ and for a crack extension of $\\Delta a \/R_0=0.8$. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\\label{fig:InfluenceFreqIron}\n\\end{figure}\n\nThe difference between the two limiting cases is governed by the exponential dependence of the lattice hydrogen concentration to hydrostatic stresses ahead of the crack, given the independence of the trap density to the loading frequency in Sofronis and McMeeking's \\cite{SM89} framework. Since the maximum level of $\\sigma_H$ is load-independent in finite strain J2 plasticity \\cite{M77}, we investigate the influence of yield strength, strain hardening and triaxiality conditions in providing a response closer to the experimental observations.\\\\\n\nThe role of the yield strength is first investigated by considering a high-strength steel with $\\sigma_Y=1200$ MPa and otherwise identical properties as the iron-based material assessed so far. As shown in Fig. \\ref{fig:InfluenceFreqHSA}, a considerably larger effect of the loading frequency is observed, even without the need of considering the two limiting cases. The lattice hydrogen concentration ahead of the crack tip is shown in Fig. \\ref{fig:InfluenceFreqHSB}; results reveal a much larger stress elevation when compared to the low-strength case (Fig. \\ref{fig:InfluenceFreqIronB}).\\\\ \n\n\\begin{figure}[H]\n \\centering\n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{FigFreqHS.eps}\n \\caption{}\n \\label{fig:InfluenceFreqHSA}\n \\end{subfigure}\n \n \\begin{subfigure}[h]{1\\textwidth}\n \\centering\n \\includegraphics[scale=0.8]{CLHSFreq.eps}\n \\caption{}\n \\label{fig:InfluenceFreqHSB}\n \\end{subfigure}\n \n \\caption{Influence of the frequency in a high-strength steel: (a) crack extension versus number of cycles, and (b) lattice hydrogen distribution ahead of the crack at the maximum $\\Delta K$ and for a crack extension of $\\Delta a \/R_0=0.6$. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\\label{fig:InfluenceFreqHstrength}\n\\end{figure}\n\nThe increase in fatigue crack growth rates with decreasing frequency is quantified in Fig. \\ref{fig:FreqHS}. Again, the model qualitatively captures the main experimental trends; low loading frequencies enable hydrogen transport to the fracture process zone, augmenting crack propagation rates. This $da\/dN$-dependence with frequency reaches a plateau when approaching the two limiting responses, a clear transition between the upper and lower bounds can be observed. However, crack growth rates on the lower frequency bound are less than 1.5 times the values attained when $da\/dN$ levels out at high loading frequencies; these quantitative estimations fall significantly short of reaching the experimentally reported differences. A 5-10 times crack growth rate elevation has been observed when decreasing frequency in a mid-strength martensitic SCM435 steel \\cite{T07}, and similar data have been obtained for a 2.25Cr\u20131Mo (SA542-3) pressure vessel steel \\cite{SR82} and an age-hardened 6061 aluminum alloy \\cite{K07}, among many other (see, e.g., the review by Murakami \\cite{M12}).\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{FreqHS.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in a high-strength steel. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:FreqHS}\n\\end{figure}\n\nThe gap between the maximum and minimum $da\/dN$ levels can also be affected by the strain hardening of the material under consideration. We, therefore, estimate the fatigue crack growth rates as a function of the loading frequency for three different strain hardening exponents. As shown in Fig. \\ref{fig:N}, higher values of $\\mathcal{N}$ lead to higher crack propagation rates. This comes as no surprise as larger strain hardening exponents translate into higher stresses. However, the $da\/dN$-elevation is not very sensitive to the range of loading frequencies examined. The effect of the stress elevation due to larger $\\mathcal{N}$ values could be attenuated by the intrinsic reduction of the plastic strain contribution to $N_T$. One should, however, note that, for the present choice of cohesive parameters, cracking takes place without significant plastic deformation. A different choice will probably increase the differences between the two limiting cases but highly unlikely to the level required to attain a quantitative agreement with the experiments.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{N.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in high-strength steel for different strain hardening exponents. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:N}\n\\end{figure}\n\nCrack tip constraint conditions are also expected to play a role in augmenting crack growth rates in sufficiently slow tests. Here, we make use of the elastic $T$-stress \\cite{BH91} to prescribe different triaxiality conditions by means of what is usually referred to as a modified boundary layer. Hence, the displacements at the remote boundary now read,\n\\begin{equation}\nu \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\cos \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right) + T \\frac{1-\\nu^2}{E} r \\cos \\theta\n\\end{equation}\n\\begin{equation}\nv \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\sin \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right) - T \\frac{\\nu (1+\\nu)}{E} r \\sin \\theta\n\\end{equation}\n\nFig. \\ref{fig:Tstress} shows the sensitivity of $da\/dN$ to the loading frequency under different constraint conditions. We restrict our attention to positive values of the $T$-stress, as lower triaxialities will not contribute to increasing crack growth rates in the lower frequency bound. Results reveal a substantial increase of $da\/dN$ with increasing crack tip constraint. However, the influence on the ratio between the crack propagation rates for \\emph{slow} and \\emph{fast} tests is almost negligible.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{Tstress.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in high-strength steel for different constraint conditions. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:Tstress}\n\\end{figure}\n\nResults provide a mechanistic interpretation of the reduction in fatigue crack resistance with decreasing frequency observed in the experiments. By properly incorporating the kinetics of hydrogen uptake into the fracture process zone, model predictions can be employed to identify the critical frequency above which the time per load cycle is insufficient for diffusible hydrogen to degrade the crack growth resistance. Accurate estimations are however hindered by the lack of quantitative agreement with experimental data regarding the impact of loading frequency on $da\/dN$. Tests conducted at the low-frequency bound lead to crack growth rates that are 5-10 times larger than the values attained in experiments with a duration much shorter than the diffusion time. Such differences cannot be attained in the framework of conventional J2 plasticity, where the peak $\\sigma_H$ (on the order of $5 \\sigma_Y$) is insufficient to draw in sufficient levels of hydrogen to cause a 5-fold increase in $da\/dN$. Crack growth rates at low loading frequencies increase with yield strength, material hardening and constraint conditions, but not even the most critical combination of these parameters appears to provide a quantitative agreement with a phenomenon that is observed in a wide range of metallic alloys and testing configurations. Additional sources of stress elevation are therefore needed to provide a reliable characterization of environmentally assisted fatigue for different frequencies. One possibility lies on the large gradients of plastic strain present in the vicinity of the crack, which exacerbate dislocation density and material strength. Geometrically necessary dislocations (GNDs) arise in large numbers to accommodate lattice curvature due to non-uniform plastic deformation, and act as obstacles to the motion of \\emph{conventional} statistically stored dislocations. Strain gradient plasticity theories have been developed to extend plasticity theory to the small scales by incorporating this dislocation storage phenomenon that significantly contributes to material hardening (see \\cite{M16c} and references therein). Gradient plasticity models have been consistently used to characterize behavior at the small scales involved in crack tip deformation, predicting a much higher stress level than classic plasticity formulations (see, e.g., \\cite{MN16,M17}). This stress elevation due to dislocation hardening has proven to play a fundamental role in fatigue \\cite{BS08,BS08b} and hydrogen-assisted cracking \\cite{M16,MB17}.\n\n\\section{Conclusions}\n\\label{Concluding remarks}\n\nWe propose a predictive cohesive modeling framework for corrosion fatigue. The model is grounded on the mechanism of hydrogen embrittlement, which governs fatigue crack initiation and subsequent propagation in a wide range of metallic alloys exposed to gasses and electrolytes. Mechanical loading and hydrogen transport are coupled through lattice dilatation due to hydrostatic stresses and the generation of traps by plastic straining. An irreversible cohesive zone model is employed to capture material degradation and failure due to cyclic loads. The impact of the hydrogen coverage in the cohesive traction is established from first principles quantum mechanics. Finite element analysis of a propagating crack reveals a relevant increase in crack growth rates with (i) hydrogen content in the surrounding environment and (ii) decreasing load frequency; in agreement with experimental observations. A robust and appropriate numerical model for hydrogen-assisted fatigue opens up many possibilities, enabling rapid predictions that could be key to risk quantification in industrial components. Moreover, important insight can be gained into the mechanisms at play, identifying the relevant variables and their critical magnitudes for a given material, environment and loading scenario.\\\\\n\nThe influence of the yield strength, work hardening and constraint conditions is extensively investigated aiming to quantitatively reproduce the relation between the loading frequency and the crack growth rates observed in the experiments. Results reveal the need to incorporate additional sources of stress elevation to sufficiently enhance hydrogen uptake into the fracture process zone. Future work will focus on extending the present scheme to encompass the role of geometrically necessary dislocations through strain gradient plasticity formulations. \n \n\\section{Acknowledgments}\n\\label{Acknowledge of funding}\n\nThe authors gratefully acknowledge financial support from the Ministry of Economy and Competitiveness of Spain through grant MAT2014-58738-C3. E. Mart\\'{\\i}nez-Pa\\~neda additionally acknowledges financial support from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7\/2007-2013) under REA grant agreement n$^{\\circ}$ 609405 (COFUNDPostdocDTU).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDeep inelastic production of heavy quarks at HERA is important for \nseveral reasons:\n\\begin{itemize}\n\\item At small $x$ ($x \\approx 10^{-3}$) the charm contribution \n$F_2^c$ to $F_2^{ep}$ amounts to $20 \\%$--$30 \\%$ \\cite{adloff96} \nmaking a reliable theoretical treatment of $F_2^c$ necessary for \na precision measurement of $F_2^{ep}$.\n\\item The bulk of heavy quarks is produced via the photon gluon \nfusion mechanism \\cite{adloff96,zeus97} providing the possibility to \nconstrain the gluon distribution $g(y,\\mu^2)$ in the proton \n\\cite{vogt96,laenen96,daum96}.\n\\item By measuring differential distributions, the charm production \nmechanism can be tested, and it can be studied whether and when the \ncharm quark behaves like a massless parton \\cite{laenen96,daum96}.\n\\end{itemize}\n\nIn order to estimate radiative corrections to heavy quark production, \nwe employ the well-known leading log approximation (LLA) \n\\cite{km88,kms91,jbl90,jbl95}.\nSee also ref.~\\cite{jbl91} where radiative corrections to heavy\nflavour production have been studied in the Weisz\\\"acker-Williams\napproximation (WWA) using hadronic variables.\nThe ${\\cal O}(\\alpha)$ (QED-)corrections to \nthe process $e+g\\longrightarrow e+c+\\bar{c}$ are shown in \nfig.~\\ref{Feynman} (not shown are crossed and virtual diagrams).\nThe main part of the corrections comes from bremsstrahlung from the \nelectron line (plus corresponding virtual corrections), see a) and b). \nThese two contributions are enhanced by a large collinear logarithm \n$L_e=\\ln(Q^2\/m_e^2)$. In the same way, calculating c), d), and e), \none gets logarithms $L_c=\\ln(Q^2\/m_c^2)$ which are much smaller than \n$L_e$ because of the large charm mass $m_c$, compared to the electron \nmass $m_e$. For example, for $Q^2=10$ GeV$^2$, one finds \n$L_e \\approx 17.5>>L_c\\approx 1.5$ ($m_c=1.5$ GeV).\nDiagrams d) and e) do not contribute for another reason: the photons \nare not resolved from the final state jets in general.\n(Only a light intermediary quark in diagram c) could give rise for a \ncollinear singularity, which would have to be handled the same way as \nthe corresponding collinear gluon emission, leading to a negligible \n(at least in LLA) modification of the parton distributions.)\nThus, we only take the leptonic corrections a) and b) \n(in ${\\cal O}(\\alpha)$-LLA) into consideration.\n\nThe paper is organized as follows: In Sect.~2 the formulae needed to \ncalculate radiative corrections to neutral current heavy quark \nproduction in ${\\cal O}(\\alpha)$-LLA are collected. \nIn Sect.~3 numerical results are presented.\nFinally, the main results are briefly summarized in Sect.~4.\n\\section{Formalism}\nWe calculate the ${\\cal O}(\\alpha)$ QED-corrections to deep inelastic \nproduction of heavy quarks in leading log approximation (LLA) \n\\cite{kms91, jbl90}.\nFollowing the formalism of ref.~\\cite{kms91}, the cross section for \ndeep inelastic production of charm quarks via the photon gluon fusion \n(PGF) mechanism ($e+g\\rightarrow e+c+\\bar{c}$) reads:\n\\begin{equation}\nd\\sigma(ep\\rightarrow ec\\bar{c}X)=\\int_0^1 dz_1\\int_0^1dz_2\n\\int_0^1dz_3 D_{e\/e}(z_1,Q^2)\ng(z_2,\\mu^2)\\bar{D}_{e\/e}(z_3,Q^2)\nd\\hat{\\sigma}(e+g\\rightarrow e+c+\\bar{c})\\ .\n\\label{wq}\n\\end{equation}\nThe functions $D_{e\/e}$, $\\bar{D}_{e\/e}$ contain the factorized mass \nsingularities from the electron line and are, up to ${\\cal O}(\\alpha)$, \ngiven by \\cite{kms91}\n\\begin{eqnarray}\nD_{e\/e}(x,Q^2) = \\bar{D}_{e\/e}(x,Q^2)= \\delta(1-x) + \n\\left(\\frac{\\alpha L_e}{2\\pi}\\right)P_{ee}(x)\n\\label{dee}\n\\end{eqnarray}\nwith \n\\begin{equation}\nL_e\\equiv \\ln \\frac{Q^2}{m_e^2}\n\\end{equation}\nand the splitting function\n\\begin{equation}\nP_{ee}(x) = \\left[\\frac{1+x^2}{1-x}\\right]_{+}= \n\\frac{1+x^2}{(1-x)_{+}}+\\frac{3}{2}\\delta(1-x).\n\\end{equation}\nThe $(+)$-distribution is defined by\n\\begin{equation}\n\\int_x^1\\ dz\\ P_{ee}(z)\\ f(z)=\\int_x^1\\ dz\\ \\frac{1+z^2}{1-z}\\ \n(f(z)-f(1))-\\int_0^x\\ dz\\ \\frac{1+z^2}{1-z}\\ f(1)\\ .\n\\end{equation}\n$D_{e\/e}(z_1,Q^2)$ can be interpreted as the probability of finding an \nelectron with momentum fraction $z_1$ \ninside the incoming electron, $\\bar{D}_{e\/e}(z_3,Q^2)$ describes the \nfragmentation of the scattered electron \ninto the observed outgoing electron with momentum fraction $z_3$, and \n$g(z_2,\\mu^2)$ is the gluon density inside the proton. Finally, \n$d\\hat{\\sigma}$ ist the cross section of the underlying hard scattering\nsubprocess.\n\nThe variables $x$, $y$ and $Q^2$ are reconstructed by the 4-momentum \n$p'_e$ of the observed electron\nand the 4-momenta $p_e$, $p$ of the initial state electron and proton:\n\\begin{eqnarray}\nq_{l}\\equiv p_e-{p^{\\prime}_e},\\qquad Q_{l}^{2}\\equiv -q_{l}^{2},\n\\qquad y_{l}\n=\\frac{p\\cdot q_{l}}{p\\cdot p_e},\n\\qquad x_{l}\\equiv \\frac{Q_{l}^{2}}{2p\\cdot q_{l}}\n=\\frac{Q_l^2}{S y_l}\\ ,\n\\label{outerkin}\n\\end{eqnarray}\nwhere $S\\equiv (p+p_e)^2$. In the same way, one \ndefines for the hard scattering process\n$e(\\hat{p}_e=z_1 p_e)+g(\\hat{p}=z_2 p)\\longrightarrow e(\\hat{p}^\\prime_e\n=p_{e}^{\\prime}\/z_3)+c(p_c)+X$: \n\\begin{eqnarray}\n\\label{subkin}\n\\hat{q}&=&{\\hat{p}_e}-{{\\hat{p}^\\prime_e}},\\qquad \n\\hat{Q}^{2}=-\\hat{q}^{2}=\\frac{z_1}{z_3}Q_l^2,\\qquad\n\\hat{y}=\\frac{\\hat{p}\\hat{q}}{\\hat{p}{\\hat{p}_e}}\n=\\frac{z_1z_3-1+y_l}{z_1z_3},\n\\\\\n\\hat{x}&=&\\frac{\\hat{Q}^{2}}{2\\hat{p}\\cdot \\hat{q}}\n=\\frac{\\hat{Q}^2}{\\hat{S} \\hat{y}}\n=\\frac{z_1x_ly_l}{z_2(z_1z_3-1+y_l)},\\qquad\n\\hat{S}=(\\hat{p}_e +\\hat{p})^2=z_1 z_2 S\\ .\n\\end{eqnarray}\nBecause we only use leptonic variables $x_l$, $y_l$, etc.~, the index\n$l$ will be suppressed from now on.\n\nIn the following, we calculate the radiative corrections to the photon \ngluon fusion (PGF) subprocess \n\\begin{eqnarray}\ne(\\hat{p_e})+g(\\hat{p})\\longrightarrow e(\\hat{p}_e^\\prime)+c(p_c)\n+\\bar{c}(p_{\\bar{c}}).\n\\label{pgf}\n\\end{eqnarray}\nand compare the results with the radiative corrections to the charm \nexcitation (CE) subprocess\n\\begin{eqnarray}\ne(\\hat{p_e})+c(\\hat{p})\\longrightarrow e(\\hat{p}_e^\\prime)+c(p_c) \n\\label{eqeq}\n\\end{eqnarray}\nwhere, in contrast to the PGF, the charm quark is treated as an\nintrinsic (massless) parton of the proton.\nThe latter process has been used by the H1 Collab.~\\cite{glazov} in a \nrecent analysis of leptoproduction of D-mesons \\cite{adloff96},\nemploying some charm density $c(x,\\mu^2)$. \n\\subsection{Inclusive case}\n\\subsubsection{Photon gluon fusion (PGF)}\nThe hard cross section for the process (\\ref{pgf}) can be written as:\n\\begin{equation}\n\\frac{d^2\\hat{\\sigma}}{d\\hat{x}d\\hat{y}}=\n\\frac{4\\pi\\alpha^{2} \\hat{S}}{{\\hat{Q}}^4}\n\\left[(1-{\\hat{y}})f_2\\left({\\hat{x}},{\\hat{Q}}^{2}\\right)\n+{\\hat{x}}{\\hat{y}}^{2}f_1\\left({\\hat{x}},{\\hat{Q}}^{2}\\right)\\right]\n\\label{partonwq}\n\\end{equation}\nwith the partonic structure functions\n\\cite{witten76,gr79,leveille79,lrsn93} \n\\begin{eqnarray}\nf_k({\\hat{x}},{\\hat{Q}}^2,m_c^2,\\mu^2) & = & \n\\frac{\\alpha_s(\\mu^2)}{\\pi}\\ \\frac{\\hat{\\xi}}{4\\pi}\n\\ e_c^2\\ c_{k,g}^{(0)}({\\hat{x}},{\\hat{Q}}^2)\\ \n\\theta(\\hat{W}^2-4m_c^2),\\ \\: k=2,L \\: ,\n\\\\\nf_1({\\hat{x}},{\\hat{Q}}^2,m_c^2,\\mu^2) & = & \\frac{1}{2{\\hat{x}}}\n(f_2-f_L)\\: ,\n\\\\\n\\frac{\\hat{\\xi}}{4\\pi}c_{2,g}^{(0)} & = & \n\\left[\\frac{{\\hat{x}}}{2} -{\\hat{x}}^2(1-{\\hat{x}})+\n2\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^2(1-3{\\hat{x}})\n-4\\frac{m_c^4}{{\\hat{Q}}^4}{\\hat{x}}^3\\right]\n\\ln \\frac{1+\\hat{\\beta}}{1-\\hat{\\beta}}\\nonumber \\\\\n& & +\\hat{\\beta}\\left[4{\\hat{x}}^2(1-{\\hat{x}})\n-\\frac{{\\hat{x}}}{2}-2\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^2\n(1-{\\hat{x}})\\right],\n\\\\\n\\frac{\\hat{\\xi}}{4\\pi}c_{L,g}^{(0)} & = & -4\n\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^3\n\\ln \\frac{1+\\hat{\\beta}}{1-\\hat{\\beta}}+2\\hat{\\beta} {\\hat{x}}^2\n(1-{\\hat{x}}), \n\\label{HQProd}\n\\end{eqnarray}\nwhere $e_c$, $m_c$ are the charm quark charge and mass, respectively, \n$\\displaystyle \\hat{\\xi}\\equiv \\xi(\\hat{Q}^2) = \n\\frac{{\\hat{Q}}^2}{m_c^2}$\\,,\\,$\\displaystyle \n\\hat{W}^2\\equiv W^2(\\hat{x},\\hat{Q}^2)=\\frac{\\hat{Q}^2\n(1-\\hat{x})}{\\hat{x}}$, and\n$\\hat{\\beta}^2\\equiv \\beta^2(\\hat{x},\\hat{Q}^2)=1-4m_c^2\/\\hat{W}^2$.\n\nBy insertion of eq.~(\\ref{dee}), (\\ref{partonwq}) into eq.~(\\ref{wq}),\none obtains the cross section \n$d\\sigma=d\\sigma ^0+d\\sigma^{i}+ d\\sigma^{f}$.\nThe Born cross section $d\\sigma^0$ is given by\n\\begin{equation}\n\\frac{d^{2} \\sigma^0}{dx dy}=\\frac{4\\pi\\alpha^{2} S}{Q^4}\n\\left[(1-y)F_2\\left(x,Q^{2}\\right)+xy^{2}F_1\\left(x,Q^{2}\\right)\\right]\n\\label{wqem}\n\\end{equation}\nwith the leading order structure functions\n\\begin{eqnarray}\nF_1(x,Q^2,\\mu^2,m_c^2)=\\int_{ax}^1\\ \\frac{dz_2}{z_2}\\ g(z_2,\\mu^2)\n\\ f_1(x\/z_2,Q^2,m_c^2,\\mu^2),\n\\\\\nF_2(x,Q^2,\\mu^2,m_c^2)=\\int_{ax}^1\\ \\frac{dz_2}{z_2} z_2\\ g(z_2,\\mu^2)\n\\ f_2(x\/z_2,Q^2,m_c^2,\\mu^2) ,\n\\end{eqnarray} \nwhere $a=1+4m_c^2\/Q^2$.\n$d\\sigma^{i,f}$ contain the contribution due to radiation of a collinear\nphoton from the incoming (initial state radiation, ISR) and outgoing \nelectron (final state radiation, FSR),\nrespectively:\n\\begin{eqnarray}\n\\frac{d^2\\sigma^i}{dxdy}& = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{1}^{min}}dz_1\\left[\\frac{1+z_1^2}{1-z_1}\n(\\sigma_0(z_1,1)-\\sigma_0(1,1))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi} H(z_{1}^{min})\\sigma_0(1,1),\n\\label{initial}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{d^2\\sigma^f}{dxdy} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{3}^{min}}dz_3\\left[\\frac{1+z_3^2}{1-z_3}\n(\\sigma_0(1,z_3)-\\sigma_0(1,1))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi} H(z_{3}^{min})\\sigma_0(1,1),\n\\label{final}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\label{pgfs0}\n\\sigma_0(z_1,z_3) & = &\\int^1_{z_{2}^{min}}dz_2\\ g\\left(z_2,\\mu ^2\\right)\n\\ \\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n\\ \\frac{d^2\\hat{\\sigma}}{d\\hat{x}d\\hat{y}}, \n\\\\\nH(z) & = & -\\int_0^z dx \\frac{1+x^2}{1-x}=2\\ln(1-z)+z+\\frac{1}{2}z^2,\n\\end{eqnarray}\nand the Jacobian\n\\begin{equation}\n \\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n=\\frac{y}{z_2z_3(z_1z_3-1+y)}.\n\\label{jacobi1}\n\\end{equation}\nThe electromagnetic coupling is taken to be \n$\\alpha\\equiv \\alpha(Q^2=m_e^2)=1\/137.036$\\footnote{\nUsing a running coupling $\\alpha(Q^2)$ according to \neq.~(14) in \\protect\\cite{kms91}, \ntaking effective quark masses \\protect\\cite{hector}\n$m_u=m_d=0.041$, $m_s=.15$, $m_c=1.5$, and $m_b=4.5\\ {\\rm GeV}$, \ninstead of a constant $\\alpha$, leads to negligible differences.}.\nThe integration bounds\n\\begin{eqnarray}\n\\label{bounds}\nz_{1}^{min} & = & \\frac{1-y}{1-xy}+\\frac{4m_c^2}{S(1-xy)},\n\\nonumber\\\\\nz_{3}^{min} & = & \\frac{1-y+xy}{1-4m_c^2\/S},\n\\\\\nz_{2}^{min}(z_1,z_3)&=&\\left(1+\\frac{4m_c^2}{Q^2}\\frac{z_3}{z_1}\\right)\nz_1xy\\frac{1}{z_1z_3+y-1}\n\\nonumber\n\\end{eqnarray}\nfollow from the conditions $\\hat{W}^2 \\ge 4 m_c^2\\ , \n\\ 0 \\le z_{2}^{min}(z_1,z_3) \\le 1$. \nIf no photon is radiated ($z_1=1$, $z_3=1$), one finds the well-known\nexpression $z_{2}^{min}(1,1)=(1+4m_c^2\/Q^2)x\\equiv a x$.\nIn the kinematical region of small $x$, the bounds read approximately: \n$z_{1}^{min}= z_{3}^{min}= 1-y+{\\cal O}(x)+{\\cal O}(m_c^2\/S)$.\nIt should be remarked that $\\sigma_0$ is related to the Born \ncross section by $d^2\\sigma^0\/dxdy=\\sigma_0(1,1)$.\n\\subsubsection{Charm excitation (CE)}\nIn the case of electron quark scattering, the radiative corrections \nin LLA are known up to\n${\\cal O}(\\alpha^2)$ \\cite{km88,kms91,jbl90,jbl95,hector}.\nFor comparison with eq.~(\\ref{initial}), (\\ref{final}), we use \nequations (22)--(29) in \\cite{kms91},\nwith the charm quark as the only massless parton (MP) in the \ninitial state. \nFurthermore, contributions due to $Z$-exchange can be neglected \nin the realm of $Q^2 \\le 50$ GeV$^2$, i.~e.~, we make the\nreplacements $A^c\\to e_c^2$, $B^c\\to 0$ \n(in eq.~(22)--(29) in \\cite{kms91}).\nFor the partonic structure functions, the simple relations\n\\begin{eqnarray}\nf_2(\\hat{x},\\hat{Q}^2) & = & e_c^2\\ \\delta(1-\\hat{x})\\ ,\n \\\\\nf_1(\\hat{x},\\hat{Q}^2) & = & \\frac{1}{2\\hat{x}}\\ f_2(\\hat{x},\\hat{Q}^2)\n\\label{mpstrukt}\n\\end{eqnarray}\nhold, and one easily finds \\cite{kms91}\n\\begin{equation}\n\\sigma_0(z_1,z_3)=\\frac{2\\pi\\alpha^2 y}{z_1 z_3^2 \\hat{y}^3 \\hat{S}}\n[1+(1-\\hat{y})^2]\ne_c^2\\left(c(\\bar{z_2},\\hat{Q}^2)+\\bar{c}(\\bar{z_2},\\hat{Q}^2)\\right)\n\\label{mps0}\n\\end{equation}\nwith\n\\begin{equation}\n\\bar{z}_2=\\frac{z_1 x y}{z_1 z_3 + y -1}\\ , \\ z_{1}^{min}\n=\\frac{1-y}{1-xy}\\ , \\ z_{3}^{min}=1-y(1-x)\\ .\n\\label{mpbounds}\n\\end{equation}\n(Of course, eq.~(\\ref{mps0}) and (\\ref{mpbounds}) have to be \ninserted into eq.~(\\ref{initial}) and (\\ref{final}).)\n\nFinally, we briefly discuss the Compton contribution $d\\sigma^C$ \nto the radiative corrections\nwhich can be obtained from eq.~(21) ($Q_0=200\\ {\\rm MeV}$) \nand (29) in \\cite{kms91}, only \nallowing for the charm (or anti-charm) quark in the initial state.\nFig.~\\ref{compton} displays the Compton \ncontribution $\\displaystyle \\delta^C=\\frac{d^2\\sigma^C}{dxdy}\n\/\\frac{d^2\\sigma^0}{dxdy}$ for $x=10^{-2}$ (dashed line) and $x=10^{-3}$\n(full line) using the CTEQ4L parton distributions \\cite{cteq4}.\nFor $y\\lesssim 0.7$, $\\delta^C$ is small and can be safely neglected for \nexperimentally relevant values of $y$ \\cite{adloff96}.\n\\subsection{$z$-differential case}\nIn order to calculate radiative corrections to the $z$-differential \ncross section \n$\\displaystyle d^3\\sigma^D\/dx dy dz$, where $z=p\\cdot p_D\/p\\cdot q$, \nthe fragmentation of the charm quark into \nthe observed $D$-meson must be taken into consideration.\nThus, it is necessary to extend eq.~(\\ref{wq}):\n\\begin{equation}\nd\\sigma(ep\\rightarrow eDX)=\\int_0^1 dz_1\\int_0^1dz_2\n\\int_0^1dz_3 D_{e\/e}(z_1,Q^2)\ng(z_2,\\mu^2)\\bar{D}_{e\/e}(z_3,Q^2)\n\\int_0^1dz_4 D_c(z_4)d\\hat{\\sigma} .\n\\label{fragds}\n\\end{equation}\nThe hadronization of the outgoing charm quark with momentum $p_c$ \ninto the observed $D$-meson \nwith momentum $p_D=z_4p_c$ is modeled by the fragmentation \nfunction $D_c(z_4)$.\nWriting the partonic cross section differential in \n$\\hat{z}_c=\\hat{p}\\cdot p_c\/\\hat{p}\\cdot \\hat{q}$, the relations in\neq.~(\\ref{subkin}) have to be accomplished with\n\\begin{equation}\n\\hat{z}_c=\\frac{z y z_3}{z_4(z_1z_3+y-1)}\\equiv \\frac{z}{z_4}r,\n\\label{subkin2}\n\\end{equation}\nwith $r$ defined by \n\\begin{equation}\nr\\equiv \\frac{y\\ z_3}{z_1z_3+y-1},\n\\label{r}\n\\end{equation}\nwhich can be derived from the definitions of $\\hat{z}_c$, $z$ \nand $\\hat{q}$, using $p_D=z_4 p_c$.\n\nIn analogy to the above discussion,\nwe compare the ``massive'' (PGF) with the ``massless'' (CE) \ncorrections, using a massless charm parton (MP).\n\\subsubsection{Photon gluon fusion}\nThe partonic cross section reads\n\\begin{equation}\n\\frac{d^3\\hat{\\sigma}}{d\\hat{x}d\\hat{y}d\\hat{z}_c}=\n\\frac{4\\pi\\alpha^{2} \\hat{S}}{{\\hat{Q}}^4}\n\\left[(1-{\\hat{y}})f_2\\left({\\hat{x}},{\\hat{Q}}^{2};\\hat{z}_c\\right)\n+{\\hat{x}}{\\hat{y}}^{2}f_1\\left({\\hat{x}},{\\hat{Q}}^{2};\n\\hat{z}_c\\right)\\right],\n\\end{equation}\nwhere the structure functions are given by\n\\footnote{The corresponding expressions for\ngeneral boson gluon fusion can be found in \\protect\\cite{ks97}.}\n\\begin{eqnarray}\nf_2(x,Q^2;{\\zeta}) &=& \\frac{\\alpha_s}{\\pi}e_{c}^2x\n\\left[\\frac{1}{4}{\\rm BG}\\left(x,Q^2,{\\zeta},{m_c}\\right)\n+\\frac{3}{4}{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right)\\right],\n\\label{dpgfdz1}\\\\\nf_L(x,Q^2;{\\zeta})&=&\\frac{\\alpha_s}{\\pi}e_{c}^2x\\frac{1}{2}\n{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right),\n\\label{dpgfdz2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right)&=&\n4x\\left[1-x-x\\frac{{m_c}^2}{Q^2}\\frac{1}{{\\zeta}(1-{\\zeta})}\\right],\n\\nonumber\\\\\n{\\rm BG}\\left(x,Q^2,{\\zeta},{m_c}\\right)&=&-2+\n\\left[1-2x+2x^2+4\\frac{{m_c}^2}{Q^2}x(1-x)\\right]\n\\frac{1}{{\\zeta}(1-{\\zeta})}\n\\nonumber\\\\\n&&+2x^2\\frac{{m_c}^2}{Q^2} \\left(1-2\\frac{{m_c}^2}{Q^2}\\right)\n\\frac{1}{(1-{\\zeta})^2{\\zeta}^2},\n\\nonumber\n\\end{eqnarray}\n\\cite{lrsn93,schuler88} and\n$\\displaystyle f_1(\\hat{x},\\hat{Q}^2;\\hat{z}_c)= \\frac{1}{2\\hat{x}}\n\\left[f_2(\\hat{x},\\hat{Q}^2;\\hat{z}_c)-\nf_L(\\hat{x},\\hat{Q}^2;\\hat{z}_c)\\right]$.\n\nThe cross sections for ISR and FSR have the same structure\nas in eq.~(\\ref{initial}) and (\\ref{final}) with a $z$ dependent \nfunction $\\sigma_0$:\n\\begin{eqnarray}\n\\frac{d^3\\sigma^i}{dxdydz} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{1}^{min}}dz_1\\left[\\frac{1+z_1^2}{1-z_1}\n(\\sigma_0(z_1,1;z)-\\sigma_0(1,1;z))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi}H(z_{1}^{min})\\sigma_0(1,1;z),\n\\label{fragini}\\\\\n\\frac{d^3\\sigma^f}{dxdydz} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{3}^{min}}dz_3\\left[\\frac{1+z_3^2}{1-z_3}\n(\\sigma_0(1,z_3;z)-\\sigma_0(1,1;z))\\right] \\nonumber\n\\\\\n&& +\\frac{\\alpha L_{e}}{2\\pi}H(z_{3}^{min})\\sigma_0(1,1;z),\n\\label{fragfin}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\sigma_0(z_1,z_3;z) & = &\\int^1_{z_{2}^{min}}dz_2 g\n\\left(z_2,\\mu ^2\\right)\\int_{z_{4}^{min}}^{z_{4}^{max}}dz_4\n\\frac{\\partial{(\\hat{x},\\hat{y},\\hat{z_c})}}{\\partial{(x,y,z)}}\n\\frac{d^3\\hat{\\sigma}}{d\\hat{x}d\\hat{y}d\\hat{z_c}}D_c(z_4)\\ .\n\\label{frags0}\n\\end{eqnarray}\nThe Jacobian in eq.~(\\ref{frags0}) can easily be calculated as\n\\begin{eqnarray}\n\\frac{\\partial{(\\hat{x},\\hat{y},\\hat{z_c})}}{\\partial{(x,y,z)}}=\n\\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n\\frac{\\partial \\hat{z}_c}{\\partial z}=\n\\frac{y^2}{z_2z_4(z_1z_3-1+y)^2}.\n\\end{eqnarray}\nThe integration bounds can be derived from the conditions\n$\\displaystyle \\hat{W}^2=\\frac{\\hat{Q}^2(1-\\hat{x})}{\\hat{x}} \n\\ge 4m_c^2$, leading to eq.~(\\ref{bounds}), and \n$\\displaystyle \\hat{z}_{c}^{min}\\le \\hat{z}_c \\le \n\\hat{z}_{c}^{max}$ with\n$\\displaystyle \\hat{z}_{c}^{max \\atop min}=\n\\frac{1\\pm \\beta(\\hat{x},\\hat{Q}^2)}{2}$ \\cite{schuler88}, where\n$\\beta^2(\\hat{x},\\hat{Q}^2)=1-4m_c^2\/{\\hat{W}}^2$, implying\n\\begin{eqnarray}\nz_{4}^{max}&=&\\min \\left[1,\\frac{z r}{\\hat{z}_{c}^{min}}\\right],\n\\\\\nz_{4}^{min}&=&\\min \\left[z_{4}^{max},\\frac{z r}{\\hat{z}_{c}^{max}}\n\\right]\n\\end{eqnarray}\nwith $r$ from eq.~(\\ref{r}).\nFor $D_c(z)$, we use a Peterson et al.\\ fragmentation function \n\\cite{peterson83}\n\\begin{equation}\nD_c(z) = N \\left\\{ z \\left[ 1-z^{-1}-\\varepsilon_c\/(1-z)\n\\right]^2\\right\\}^{-1}\\ ,\n\\label{peterson}\n\\end{equation}\nnormalized to $\\int_0^1 dz D_c(z) = 1$, i.~e.~,\n\\begin{equation}\nN^{-1} =\n\\frac{({\\varepsilon_c}^2-6{\\varepsilon_c}+4)}{(4-{\\varepsilon_c}) \n\\sqrt{4{\\varepsilon_c}-{\\varepsilon_c}^2}}\n\\left\\{\n\\arctan\\frac{{\\varepsilon_c}}{\\sqrt{4{\\varepsilon_c}\n-{\\varepsilon_c}^2}}\n+ \\arctan\\frac{2-{\\varepsilon_c}}{\\sqrt{4{\\varepsilon_c}\n-{\\varepsilon_c}^2}} \\right\\}\n+ \\frac{1}{2} \\ln {\\varepsilon_c} + \\frac{1}{4-{\\varepsilon_c}}\\ .\n\\end{equation} \n\n\\subsubsection{Flavor excitation}\nIn the subprocess $e+c\\longrightarrow e+c$ we have\n$\\displaystyle \\hat{z}_c=\\hat{p}\\cdot p_c\/\\hat{p}\\cdot \\hat{q}=1$, \nfollowing from $p_c=\\hat{q}+\\hat{p}$.\nThus, one obtains the $\\hat{z}_c$-differential cross section \nby inserting the structure functions\n\\begin{eqnarray}\nf_2(\\hat{x},\\hat{Q}^2,\\hat{z}_c) & = & \ne_c^2\\ \\delta(1-\\hat{x})\\delta(1-\\hat{z}_c)\n \\\\\nf_1(\\hat{x},\\hat{Q}^2,\\hat{z}_c) & = & \n\\frac{1}{2\\hat{x}}\\ f_2(\\hat{x},\\hat{Q}^2,\\hat{z}_c)\n\\label{mpstruktdz}\n\\end{eqnarray}\ninto eq.~(\\ref{partonwq}).\nBecause of $\\displaystyle \\delta(1-\\hat{z}_c)=z_4\\delta(z_4-r z)$ and\n$\\displaystyle \\partial\\hat{z}_c\/\\partial z=r\/z_4$ one finds\n\\begin{equation}\n\\sigma_0(z_1,z_3;z)=\\sigma_0(z_1,z_3)\\ r D_c(rz)\n\\label{mps0dz}\n\\end{equation}\nwith $\\sigma_0(z_1,z_3)$ from eq.~(\\ref{mps0}) and $r$ \nfrom eq.~(\\ref{r}).\n\nFinally the $z$-differential corrections are given by \neq.~(\\ref{fragini}) and (\\ref{fragfin}) \nwith $\\sigma_0$ from eq.~(\\ref{mps0dz}) and the integration bounds\n\\begin{eqnarray}\nz_{1}^{min}&=&\\max\\left[\\frac{1-y}{1-xy},1-y(1-z)\\right],\n\\\\\nz_{3}^{min}&=&\\max\\left[1-y(1-x),\\frac{1-y}{1-yz}\\right] .\n\\label{mpboundsdz}\n\\end{eqnarray}\nThe second boundary in $\\max[ \\ldots , \\ldots ]$ can be deduced \nfrom $0 \\le rz \\le 1$.\n\nEq.~(\\ref{fragini}), (\\ref{fragfin}) can easily be tested:\n\\begin{equation}\n\\int_0^1 dz\\frac{d^3\\sigma^{i,f}}{dxdydz}\\stackrel{!}{=}\n\\frac{d^2\\sigma^{i,f}}{dxdy}\\ ,\n\\end{equation}\nwith $d^2\\sigma^{i,f}\/dxdy$ from (\\ref{initial}), (\\ref{final}).\nThe Compton contribution will be neglected for the same reasons as \nin the inclusive case.\n\\section{Numerical results}\nAs usual, the radiative corrections will be shown in form of a \ncorrection factor $\\delta$ defined by \n$d\\sigma=d\\sigma^0(1+\\delta)$, i.~e., \n$\\delta=\\delta^i+\\delta^f+\\ldots$\nwith \n\\begin{displaymath}\n\\delta^{i,f}(x,y)=\\frac{d^2 \\sigma^{i,f}}{dx dy}\/\n\\frac{d^2 \\sigma^0}{dx dy}\n\\end{displaymath}\nor, in the $z$-differential case, \n\\begin{displaymath}\n\\delta^{i,f}(x,y,z)=\\frac{d^3 \\sigma^{i,f}}{dx dy dz}\/\n\\frac{d^3 \\sigma^0}{dx dy dz}.\n\\end{displaymath}\nIn all figures we use HERA centre-of-mass energies \n$S=4\\cdot27.5\\cdot 820$ GeV$^2$.\n\nFig.~\\ref{rcpgf} shows the radiative corrections to heavy quark \nproduction (PGF) for experimentally relevant values\\footnote{For \na clean extraction of the \ngluon density it is necessary to extend the $x$-range \nto $x\\lesssim 5 \\cdot 10^{-4}$ \\cite{daum96}.} \nof Bjorken-$x$ \\cite{adloff96} as a function of $y$ according to\neq.~(\\ref{initial}) and (\\ref{final}) using \nthe GRV94(LO) parton distributions \\protect\\cite{grv94}.\nThe factorization scale has been chosen to be \n$\\mu^2=Q^2+4m_c^2$ \\cite{vogt96} \nwith $m_c=1.5\\ {\\rm GeV}$.\nFor $x= 10^{-2}$, the typical shape of radiative corrections \nin leptonic variables can be seen, with large corrections \nfor $y\\to 1$, whereas for smaller $x\\le 10^{-3}$ the curves \nbecome more and more flat for $y \\to 1$.\n\nThe theoretical uncertainties due to different choices of \nparton distributions, factorization scales, and charm masses turn \nout to be small, as can be seen in fig.~\\ref{pdfcomp} \nand fig.~\\ref{compare}.\nOne finds $\\delta({\\rm CTEQ})-\\delta({\\rm GRV})< 0.02$,\n$\\delta(m_c=1.3)-\\delta(m_c=1.7)< 0.03$ for relevant $y\\le 0.7$, and\n$\\delta(\\mu^2=4m_c^2)-\\delta(\\mu^2=Q^2+4m_c^2)< 0.03$, where the \nscale $\\mu^2=4m_c^2$ has been favored in \\cite{grs94}.\nThis could be expected because variations of $\\mu$, $m_c$, or \nthe parton distributions lead to rather similar changes in \n$d\\sigma^0$ and $d\\sigma^{i,f}$, so that the quotient $\\delta$\ndoes not change too much.\n\nIn the recent analysis of deep inelastic charm production by the \nH1 Collab.~\\cite{adloff96}, the radiative corrections have been \ncalculated in ${\\cal O}(\\alpha)$-LLA with the \nHECTOR package \\cite{hector} using the charm excitation \nsubprocess ($F_L^c=0$) and the GRV92 parton distributions \\cite{grv92}.\nFurthermore, only initial state radiation has been taken into \nconsideration, assuming the collinear final state photon not to be \nseparated from the outgoing electron \\cite{glazov},\nleading to small corrections. Because PGF has been measured to be \nthe dominant \n($>95 \\%$) charm production mechanism in the small\n$x$ (and $Q^2$) range \\cite{adloff96}, it is necessary to check \nif the ``massless'' corrections ($\\delta^{MP}$) agree with the \n``massive'' ones ($\\delta^{PGF}$).\nIn fig.~\\ref{mpcpgfgrvi} we compare the massive \n($\\mu^2=Q^2+4m_c^2$; $m_c=1.5$ GeV) \nwith the massless corrections due to initial state radiation.\nThe experimentally relevant values of $Q^2$ \\cite{adloff96} are\nindicated by dotted vertical lines. \nThe conversion of ``masslessly corrected'' ($\\delta^{MP}$) data to \n``massively corrected'' ($\\delta^{PGF}$) data\ncan be performed by applying the factor \n$R\\equiv (1+\\delta^{MP})\/(1+\\delta^{PGF})$ because of\n\\begin{eqnarray*}\nd\\sigma({\\rm PGF})&=&d\\sigma({\\rm exp})\\frac{1}{1+\\delta^{\\rm PGF}}\n\\\\\n&=&d\\sigma({\\rm MP})\\frac{1+\\delta^{\\rm MP}}{1+\\delta^{\\rm PGF}}\n\\equiv d\\sigma({\\rm MP})R\\ .\n\\end{eqnarray*}\nFor the $(x,Q^2)$ data points one finds $|R-1|=|\\delta^{MP}-\n\\delta^{PGF}|\/(1+\\delta^{PGF})< 3\\%$,i.~e.~, the differences \nbetween massless and massive radiative corrections lead to a small\nincrease ($\\delta^{MP}>\\delta^{PGF}$) of the data.\nHowever, this difference is small enough\nto use the simpler charm excitation subprocess for calculating \nthe radiative corrections in the inclusive case.\n\nOf course, heavy quark production processes are exclusive in\nthe heavy quark momentum and on this more differential level the\nphoton gluon fusion and the charm excitation processes are not \ncompatible which can be seen, e.g., from the different shapes\nof the $x_D=|\\vec{p}_D^*|\/|\\vec{p}_p^{\\, *}|=2|\\vec{p}_D^{\\, *}|\/W$ \ndistributions shown in the experimental analyses\n\\cite{adloff96,zeus97} (fig.~6, fig.~1 resp.).\nWe prefer to employ the lorentz-invariant variable\n$z=p\\cdot p_D\/p\\cdot q$ which approximately transforms\ninto $x_D$ in the $\\gamma^{\\, *}$-$p$ centre-of-mass \nsystem, following the argumentation in \\cite{adloff96}.\nThis can be easily seen by calculating $z$\nin the $\\gamma^{\\, *}$-$p$-CMS. One finds $z=x_D \\sin^2 \\theta^{\\, *}\/2$\nwith\\footnote{$\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{p}^{\\, *}=0$, \n$\\vec{p}_{g,c}^{\\, *}=z_2\\vec{p}_{p}^{\\, *} \\Rightarrow \n\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{g,c}^{\\, *}=-(1-z_2)\\vec{p}_{p}^{\\, *}$.\nThis means for the\nCE-subprocess: $\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{c}^{\\, *}=\\vec{p'}_{c}^{\\, *}\n=z_4 \\vec{p}_D^{\\, *} \\Rightarrow \\theta^{\\, *}=\\pi$\nand for the PGF: $\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{g}^{\\, *}=\\vec{p}_{c}^{\\, *}\n+\\vec{p}_{\\bar{c}}^{\\, *}\n\\Rightarrow \\theta^{\\, *}\\approx \\pi$ \n(the collinear configuration is dominant). Note that an \nangle $\\theta^{\\, *}=160^{\\circ}$ still yields \n$\\sin^2 \\theta^{\\, *}\/2=0.97$.} \n $\\theta^{\\, *}=\\angle (\\vec{p}_p^{\\, *},\\vec{p}_D^{\\, *})\n\\approx \\pi$. \n\nIn fig.~\\ref{mpcpgffrag} we show the $z$-differential radiative \ncorrections according to eq.~(\\ref{fragini}), (\\ref{fragfin}), \n(\\ref{frags0}) (PGF, full line) and\n(\\ref{mps0dz}) (MP, dashed line), where it has been integrated \nover the kinematical range \n$10\\ {\\rm GeV}^2\\le Q^2\\le 100\\ {\\rm GeV}^2,\\ 0.01 \\le y\\le 0.7$.\nThe left hand side shows the correction factor \n$\\delta^{i}=d\\sigma^{i}\/d\\sigma^0$ as well as\nthe sum of initial and final state radiation $\\delta^i+\\delta^f$. \nIn all cases, we employed the GRV92(LO) (massless) \n\\protect\\cite{grv92} and GRV94(LO) (massive) \\protect\\cite{grv94} \nparton distributions, in the massive case\nwith $m_c=1.5$ GeV and $\\mu^2=Q^2+4m_c^2$. For $D_c$ \na Peterson et al.\\ fragmentation function \\protect\\cite{peterson83} \nwith $\\varepsilon=0.15$ has been taken.\nTo study the dependence of $\\delta$ on $\\varepsilon$, we \ncompare the radiative corrections for two\ndifferent choices of $\\varepsilon$ in fig.~\\ref{epscomp}.\nIn the massive case the $\\varepsilon$-dependence is obviously small, \nwhereas for massless corrections\none finds big differences in the steep region $z\\lesssim 0.4$.\nFig.~\\ref{mpcpgffrag} reveals a rather big difference between \nmassive and massless corrections for $z\\lesssim 0.5$.\nFor $z\\to 1$ only soft photon radiation is allowed, so that the \ncorrections factorize and become independent of the \nunderlying subprocess.\nOn this more differential level it is necessary to\ncalculate the radiative corrections using the photon gluon fusion \nsubprocess because of the big deviations of the massless\nfrom the massive corrections for $z\\lesssim 0.5$.\nOn the right side of fig.~\\ref{mpcpgffrag} the $z$-dependence of the \ncross sections $d\\sigma^{0,i,f}$ is displayed.\nIntegration over $0\\le z\\le 1$ leads to the observed small \ndifferences between massless and massive corrections because \npositive and negative contributions compensate each \nother. To get an impression of the effect of $z$-cuts, \nwe show in fig.~\\ref{zcut1} \nthe correction factor $\\delta^{i,f}(x,y;z^{min},z^{max})=\n\\int_{z^{min}}^{z^{max}}dz \\frac{d\\sigma^{i,f}}{dx dy dz}\/\n\\int_{z^{min}}^{z^{max}}dz \\frac{d\\sigma^{0}}{dx dy dz}$ \nfor three $z$-integration ranges, from top to bottom \n$z^{min}=0 \\le z\\le z^{max}=1$, $0.2 \\le z\\le 0.8$, and\n$0.3 \\le z \\le 0.9$ for $x=10^{-3}$.\nFor completeness, we used the GRV94\/92(LO) parton distributions \n\\cite{grv94}, \\cite{grv92} for the massive ($m_c=1.5$ GeV, \n$\\mu^2=Q^2+4m_c^2$) and massless corrections\nand a Peterson et al.\\ fragmentation function \\cite{peterson83} with \n$\\varepsilon=0.15$.\nAs can be seen from fig.~\\ref{mpcpgffrag}, a cut $z\\gtrsim 0.2$\nleads to the exclusion of positive contributions of $d\\sigma^{i,f}$, \ndiminishing the radiative corrections.\n\\section{Summary}\nThe ${\\cal O}(\\alpha)$-QED corrections to inclusive and \n$z$-differential deep inelastic electroproduction of heavy\nquarks have been calculated in the leading log approximation,\nusing electron variables.\nThe results have been compared to the radiative corrections in the \nMP scheme, where the charm quark is assumed to be a massless parton\nin the proton.\nIn the inclusive case, the differences between these two approaches \nturned out to be negligible.\nHowever, the measurement of heavy quark production is of course\ndifferential in the momentum of the (observed) heavy quark, \nrecommending to perform the radiative corrections on the same\ndifferential level.\nThus, we have considered the semi-inclusive $z$-differential \ncase, in which the massive \ncorrections have to be applied, i.e., using the photon gluon fusion \nsubprocess, because for $0.2 \\lesssim z \\lesssim 0.5$ the \nmassless corrections differ from the massive ones by about\n$\\approx 40 \\%$--$10 \\%$.\nFurthermore, we studied the effect of cuts on the $z$-integration range. \nA cut $z \\gtrsim 0.2$, e.g., \nexcludes positive contributions of $d\\sigma^{i,f}$, so that\nthe ($z^{min}\\le z\\le z^{max}$)-integrated corrections are smaller as \nin the fully inclusive, i.e., ($0\\le z\\le 1$)-integrated case.\n\\section*{Acknowledgements}\nWe thank E.\\ Reya and M.\\ Gl\\\"{u}ck for advice and useful discussions.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{Introduction}\n\nSince Macdonald \\cite{Mac88} defined Macdonald polynomials $\\widetilde{H}_\\mu({\\mathbf {x}};q,t)$ and conjectured the Schur positivity of them, the Macdonald polynomial has been one of the central objects in algebraic combinatorics. Even though a combinatorial formula for Macdonald polynomials is given \\cite{HHL04} and the Schur positivity of Macdonald polynomials is proved \\cite{Hai01}, not much is known about an explicit combinatorial formula for the $(q,t)$-Kostka polynomials, which are the Schur coefficients of the Macdonald polynomial. In this paper, we discuss some enumerative results involving the $(q,t)$-Kostka polynomials in the words of cyclic sieving phenomena. This will provide a series of identities between the number of matrices with certain cyclic symmetries and evaluation of $(q,t)$-Kostka polynomials at a root of unity, uncovering a part of the mystery of the $(q,t)$-Kostka polynomials.\n\nTo begin, we define the cyclic seiving phenomena. Let $X$ be a set with an action of a cyclic group $C$. Fix a generator $c$ of $C$ and let $\\zeta$ be a root of unity having the same multiplicative order as $c$. Let $X(q)\\in{\\mathbb {Z}}[q]$ be a polynomial in $q$. We say that the triple $(X, C, X(q))$ exhibits the \\emph{cyclic sieving phenomenon (CSP)} \\cite{RSW04} if for any integer $r$ the number of fixed points of $c^r$ in $X$ is equal to the evalation of $X(q)$ at $q=\\zeta^r$, i.e.\n\\begin{equation*}\n\\left|X^{c^r}\\right| = \\left|\\{ x \\in X \\,:\\, c^r \\cdot x = x \\}\\right| = X(\\zeta^r).\n\\end{equation*}\n\nMore generally, let $X$ be a set with an action of a direct product of $k$ cyclic groups $C_1\\times C_2\\times \\cdots \\times C_k$. For each $i=1,2,\\dots,k$, fix a generator $c_i$ for $C_i$. Let $X(q_1,q_2,\\dots,q_k)\\in{\\mathbb {Z}}[q_1,q_2,\\dots,q_k]$ be a polynomial in $k$ variables. Following \\cite{BRS08}, we say the triple $\\left(X, C_1\\times C_2\\times \\cdots \\times C_k, X(q_1,q_2,\\dots,q_k)\\right)$ exhibits the \\emph{k-ary-cyclic sieving phenomenon (k-ari-CSP)} if for any integers $r_1, r_2, \\dots, r_k$ the number of fixed points of $(c_1^{r_1},c_2^{r_2},\\dots, c_k^{r_k})$ in $X$ is equal to the evalation of $X(q_1,q_2,\\dots,q_k)$ at $(q_1,q_2,\\dots,q_k)=(\\zeta_1^{r_1}, \\zeta_2^{r_2},\\dots, \\zeta_k^{r_k})$, i.e.\n\\begin{equation*}\n\\left|X^{(c_1^{r_1},c_2^{r_2},\\dots,c_k^{r_k})}\\right| = \\left|\\{ x \\in X \\,:\\, (c_1^{r_1},c_2^{r_2},\\dots,c_k^{r_k}) \\cdot x = x \\}\\right| = X(\\zeta_1^{r_1}, \\zeta_2^{r_2}, \\dots, \\zeta_k^{r_k}),\n\\end{equation*}\nwhere $\\zeta_i$ is a root of unity having the same multiplicative order as $c_i$. In this paper, we provide instances of tricyclic sieving and quadracyclic sieving phenomena, i.e. $k$-ary-CSP for $k=3$ and $k=4$.\n\nThe main tool we used to prove our results is the theory of orbit harmonics. The orbit harmonics is a tool in combinatorial representation theory that promotes an (ungraded) action of $\\mathfrak{S}_n$ on a finite set $X$ to a graded action of $\\mathfrak{S}_n$ on a polynomial ring quotient, by viewing $X$ as an $\\mathfrak{S}_n$-stable point locus in $\\mathbb{C}^n$. The idea goes as follows. Let $X\\subseteq\\mathbb{C}^n$ be a finite set which is closed under the action of $\\mathfrak{S}_n\\times C$, where\n\\begin{itemize}\n \\item a symmetric group $\\mathfrak{S}_n$ acts on $\\mathbb{C}^n$ by permuting coordinates, and\n \\item $C$ is a finite cyclic group acting on $\\mathbb{C}^n$ by scaling a root of unity.\n\\end{itemize}\nLet $\\mathbf{I}(X)$ be the ideal of polynomials in $\\mathbb{C}[{\\mathbf {x}}_n]:=\\mathbb{C}[x_1,\\dots,x_n]$ which vanish on $X$. Then we have an isomorphism\n\\begin{equation}\\label{isomorphism I-ideal}\n \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{I}(X),\n\\end{equation}\nwhere $\\mathbb{C}[X]$ is the algebra of all functions $X\\rightarrow \\mathbb{C}$. We further define a homogeneous ideal\n\\begin{equation*}\n \\mathbf{T}(X):=\\langle\\tau(f):f\\in\\mathbf{I}(X)\\setminus\\{0\\}\\rangle\\subseteq\\mathbb{C}[{\\mathbf {x}}_n],\n\\end{equation*}\nwhere $\\tau(f)$ denote the top degree homogeneous part of $f$. Then the isomorphism~\\eqref{isomorphism I-ideal} extends to an isomorphism \n\\begin{equation}\\label{isomorphism T-ideal}\n \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{I}(X)\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{T}(X).\n\\end{equation}\nNote that $\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{T}(X)$ admits an additional structure of a graded $\\mathfrak{S}_n$-module.\n\nUsing the isomorphism~\\eqref{isomorphism T-ideal}, the author and Rhoades provided a `generating theorem' for sieving results \\cite[Theorem 3.4]{OR20}. By varying the choice of combinatorial locus $X$, one can obtain various sieving results concerning $X$. The associated polynomial is given by a variant of the graded Frobenius image ${\\mathrm {grFrob}}({\\mathbb {C}}[{\\mathbf {x}}_n]\/\\mathbf{T}(X);q)$.\n\nIn this paper, we adopt orbit harmonics to the diagonal orbit harmonics to obtain a `generating theorem' (Theorem~\\ref{sieving-generator}) for sieving results of the combinatorial locus $X\\subseteq\\mathbb{C}^{2n}$ with a diagonal action of $\\mathfrak{S}_n$ on $X$. A precise explanation of the diagonal orbit harmonics is given in Section~\\ref{subsection Orbit harmonics and cyclic sieving}. \n\nTo prove the Macdonald positivity conjecture, Garsia and Haiman \\cite{GH93} suggested a bigraded ${\\mathfrak{S}}_n$-module ${\\mathbf {H}}_\\mu$ (called the Garsia--Haiman module) for a partition $\\mu$ whose Frobenius image is the Macdonald polynomial of $\\mu$. This module is defined as the $\\mathbb{C}$-span of a variant of the Vandermonde determinant and its partial derivatives (See Section~\\ref{modules of garsia--Haiman} for detail). Garsia and Haiman defined another module ${\\mathbf {R}}_\\mu$ via orbit harmonics which is later shown to be isomorphic to the original Garsia--Haiman module ${\\mathbf {H}}_\\mu$. Therefore by taking $\\mu=(m^n)$, we can apply Theorem~\\ref{sieving-generator} to this module ${\\mathbf {R}}_\\mu$ to obtain triCSP for $n\\times m$ matrices of content $(1^{mn})$ (each of $1,2,\\dots,mn$ appears once) and biCSP for $n\\times m$ matrices of given content $\\nu$ ($i$ appears $\\nu_i$ times). This relates roots of unity specializations of $(q,t)$-Kostka polynomials with enumerations of matrices invariant under certain rotation of row index and column index and translation of entries.\n\n\\begin{theorem}\\label{Main theorem 1}\nLet $X_{(m^n)}$ be the set of $n\\times m$ matrices of content $(1^{mn})$. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{mn}$ acts on $X_{(m^n)}$ by row rotation, column rotation, and adding 1 modulo $mn$ to each entry. In addition for a composition $\\nu\\models mn$, let $X_{(m^n),\\nu}$ be the set of $n\\times m$ matrices of content $\\nu$ where a product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m$ acts on $X_{(m^n),\\nu}$ by row and column rotation. Then we have the followings.\n\\begin{itemize}\n \\item $\\left(X_{(m^n)},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{mn},X_{(m^n)}(q,t,z)\\right)$ exhibits triCSP, where $$X_{(m^n)}(q,t,z)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)f^{\\lambda}(z).$$\n \\item $\\left(X_{(m^n),\\nu},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m,X_{(m^n),\\nu}(q,t)\\right)$ exhibits biCSP, where $$X_{(m^n),\\nu}(q,t)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)K_{\\lambda,\\nu}.$$\n\\end{itemize}\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ ($K_{\\lambda,\\mu}$, respectively) denotes the modified $(q,t)$-Kostka polynomial (Kostka number, respectively) and $f^{\\lambda}(z):=\\sum_{T\\in{{\\mathrm {SYT}}}(\\lambda)}z^{\\operatorname{maj}(T)}$ is the fake degree polynomial, where ${{\\mathrm {SYT}}}(\\lambda)$ is the set of standard tableaux of shape $\\lambda$ and $\\operatorname{maj}$ is the major index.\n\\end{theorem}\n\n\nWe generalize Theorem~\\ref{Main theorem 1} in two directions. First direction is to generalize biCSP in the second bullet point of Theorem~\\ref{Main theorem 1} to triCSP. We say a composition $\\nu$ has a cyclic symmetry of order $a$ if $\\nu_i=\\nu_{i+a}$ always, where the subscripts are interpreted modulo the length $l(\\nu)$ of $\\nu$. In the second bullet point of the above theorem, if $\\nu$ has a cyclic symmetry of order $a$ dividing $l(\\nu)$, the set $X_{(m^n),\\nu}$ possesses an additional action of a cyclic group ${\\mathbb {Z}}_{l(\\nu)\/a}$ by adding $a$ modulo $l(\\nu)$ to each entry. Then one might ask if there is natural $z$-analogue of $X_{(m^n),\\nu}(q,t)$ to give triCSP for $X_{(m^n),\\nu}$. We give an answer of this question in the following theorem.\n\n\\begin{theorem}\\label{Main theorem 2}\nLet $\\nu\\models mn$ be a composition with a cyclic symmetry of order $a$ dividing $l(\\nu)$. Let $X_{(m^n),\\nu}$ be the set of $n\\times m$ matrices of content $\\nu$. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a}$ acts on $X_{(m^n),\\nu}$ by row rotation, column rotation, and adding $a$ modulo $l(\\nu)$ to each entry. Then the triple $\\left(X_{(m^n),\\nu},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a},X_{(m^n),\\nu}(q,t,z)\\right)$ exhibits the triCSP, where\n$$X_{(m^n),\\nu}(q,t,z)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,\\nu}(z).$$\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ ($\\widetilde{K}_{\\lambda,\\mu}(z)$, respectively) denotes the modified $(q,t)$-Kostka polynomial ($z$-Kostka polynomial, respectively).\n\\end{theorem}\n\nThe second generalization starts from rephrasing Theorem~\\ref{Main theorem 1}. For a $mn \\times mn$ permutation matrix $M$, we associate a $n \\times m$ matrix $\\phi(M)$ of content $(1^{mn})$ by letting $k$ be the $(i,j)$-entry if the $(n(i-1)+j, k)$-entry of $M$ is 1. For example, if we set $m=2, n=2$, then the following permutation matrix $M$ in the left corresponds to a $2 \\times 2$ matrix $\\phi(M)$ in the right.\n\\begin{figure}[h]\n $M=\\begin{pmatrix}\n0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1\n \\end{pmatrix}$\n \\qquad$\\phi(M)=\\begin{pmatrix}\n 3 & 1\\\\\n 2 & 4\n \\end{pmatrix}$\n \\end{figure}\\\\\nUnder this correspondence, the row rotation, column rotation, and adding 1 modulo $mn$ to each entry of $\\phi(M)$ corresponds to `external' row rotation (of order $m$), `internal' row rotation (of order $n$), and column rotation of $M$. Here, by external row rotation, we mean sending the $k$th $n$ rows (from $n(k-1)+1$-th to $nk$-th row) to the next $n$ rows (from $nk+1$-th to $n(k+1)$-th row). Here the row numbers are interpreted modulo $mn$. By internal row rotation, we mean sending each row to the next row except for the $nk$-th row for $k=1,2,\\dots,m$. For the $nk$-th row we send this row to the $n(k-1)+1$-th row. In our running example, if we apply a row rotation and a column rotation to $\\phi(M)$, we get $\\begin{pmatrix} 2&4 \\\\ 3&1\\end{pmatrix}$ and $\\begin{pmatrix} 1&3 \\\\ 4&2\\end{pmatrix}$. Each of these corresponds to the matrix \n\\[\n\\begin{pmatrix}\n0&1&0&0\\\\0&0&0&1 \\\\ 0&0&1&0\\\\ 1&0&0&0 \\end{pmatrix} \n\\text{ and } \n\\begin{pmatrix}\n1&0&0&0\\\\0&0&1&0 \\\\ 0&0&0&1\\\\ 0&1&0&0 \\end{pmatrix}.\n\\] \nThe first matrix can also be obtained from $M$ by sending the first two rows to the third and the fourth row and sending the third and the fourth row to the first and the second row which is an external row rotation (of order 2). Similarly, the second matrix can be obtained from $M$ by applying an internal row rotation (of order 2). Now we can understand Theorem~\\ref{Main theorem 1} as a tricyclic sieving result concerning $mn \\times mn$ matrices under external row rotation, internal row rotation, and column rotation. One might ask if we could get a quadracyclic sieving result concerning external and internal rotation to both columns and rows. The following theorem gives a positive answer.\n\n\\begin{theorem}\\label{Main theorem 3}\nLet $l=mn=ab$ be a positive integer with two factorizations. Let $\\mathfrak{S}_{l}$ be the set of $l\\times l$ permutation matrices. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}$ acts on $\\mathfrak{S}_l$ by external row rotation, internal row rotation, external column rotation, and internal column rotation. Then the triple $\\left(\\mathfrak{S}_l, {\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}, \\mathfrak{S}_l(q,t,z,w)\\right)$ exhibits the quadraCSP, where\n\n$$\\mathfrak{S}_l(q,t,z,w)=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).$$\n\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ denotes the modified $(q,t)$-Kostka polynomial.\n\\end{theorem}\n\n\nThere have been similar results discovered which relate root of unity specializations of $q$-Kostka polynomials or Macdonald polynomials and fixed point enumerations of matrices or fillings of tableaux (see \\cite{Rho10, AU19, BRS08} for example). We remark some results which are especially close to our results. First of all, the set $X_{(m^n),\\nu}$ in Theorem~\\ref{Main theorem 2} bijects with the set of $0,1$-matrices with column content $\\mu$ and row content $(1^{mn})$. If we specialize $n=1$, then Theorem~\\ref{Main theorem 2} recovers \\cite[Theorem 1.2]{Rho10}. In addition, Barcelo, Reiner and Stanton considered biCSP concerning row and column rotation of permutation matrices (\\cite[Theorem 1.4]{BRS08}, in the case $W=\\mathfrak{S}_n$). Theorem~\\ref{Main theorem 3} specializes to their result if we take $n=1, b=1$.\n\nIn \\cite[Theorem 1.3]{Rho10}, using Hall--Littlewood polynomial, Rhoades showed that ${\\mathbb{N}}$-matrices with fixed column content $\\mu$ and row content $\\nu$ exhibits biCSP. It should be mentioned that we modify the argument of Rhoades to prove Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3} in Section~\\ref{subsection: proof of the main theorem 2}. \n\nThe remainder of this paper is organized as follows. In Section~\\ref{Preliminaries} we give background on combinatorics, symmetric functions, the representation theory of ${\\mathfrak{S}}_n$, and the Garsia--Haiman modules. In Section~\\ref{Section: Sieving generating theorem and orbit harmonics}, we illustrate the diagonal orbit harmonics and how to obtain sieving results from orbit harmonics. We then present a combinatorial locus that gives a graded module isomorphic to Garsia--Haiman module via orbit harmonics. In Section~\\ref{Section: proofs}, we provide proofs of Theorem~\\ref{Main theorem 1}, Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3}. We conclude this paper with some remarks in Section~\\ref{concluding remarks}.\n\n\\section{Preliminaries}\n\\label{Preliminaries}\n\n\\subsection{Combinatorics}\\label{subsection Combinatorics}\nA \\emph{weak composition} of $n$ is a sequence of non-negative integers which sum to $n$. A \\emph{composition} is a weak composition which consists of positive integers. A \\emph{partition} of $n$ is a composition of $n$ which is weakly decreasing. We denote $\\mu\\models n$ and $\\lambda\\vdash n$ for a composition $\\mu$ and a partition $\\lambda$ of $n$. For a composition $\\mu=(\\mu_1,\\dots,\\mu_k)$, the \\emph{length} $l(\\mu)$ of $\\mu$ is $k$.\n\nFor a partition $\\lambda\\vdash n$ we abbuse our notation so that a partition $\\lambda$ also denotes for its \\emph{Young diagram}. We draw Young diagrams in French style:\n$$\\lambda=\\{(i,j)\\in{\\mathbb {Z}}_{\\ge0}\\times{\\mathbb {Z}}_{\\ge0}:i<\\lambda_{j+1}\\}.$$ The elements of Young diagram are called cells. For example,\n\\begin{align*}\n \\lambda &= (4,3,1)\\\\\n &=\\{(0,0),(1,0),(2,0),(3,0),(0,1),(1,1),(2,1),(0,2)\\}\n\\end{align*}\n\n\\quad\\qquad\\qquad\\qquad\\qquad = \\quad\\begin{young}\n \\cr\n & & \\cr\n & & & \\cr\n\\end{young}\\\\\nWe define the \\emph{conjugate} $\\lambda'$ to be the partition obtained by reflecting $\\lambda$ with respect to the diagonal line $x=y$ in the plane. \n\nA \\emph{tableau} of a partition $\\lambda$ is a filling $T:\\lambda\\rightarrow{\\mathbb {Z}}_{>0}$. In this case, we call the \\emph{shape} of $T$ is $\\lambda$. The \\emph{content} of a tableau $T$ of $\\lambda$ is a weak composition $(T_1,T_2,\\dots)$ of $n$, where $T_i$ is the number of $i$'s appearing in $T$. A tableau $T$ is called \\emph{semistandard} if the entries in each row are weakly increasing (left to right) and the entries in each column are strictly increasing (bottom to top). The \\emph{Kostka number} $K_{\\lambda,\\mu}$ is the number of semistandard tableaux of shape $\\lambda$ and content $\\mu$. A semistandard tableau is called \\emph{standard} if its content is $(1,1,\\dots)$. The set of standard tableaux of shape $\\lambda$ is denoted by ${\\mathrm {SYT}}(\\lambda)$. Examples of semistandard tableau and standard tableau of shape $(4,3,1)$ are shown in the left and the right below, respectively.\n\n\\begin{center}\n\\begin{young}\n 4 \\cr\n 2& 2& 4\\cr\n 1& 1& 2& 3\\cr\n\\end{young}\\qquad \\qquad\n\\begin{young}\n 5 \\cr\n 2 & $\\mathbf{4}$ & 7\\cr\n $\\mathbf{1}$ & $\\mathbf{3}$ & $\\mathbf{6}$ & 8\\cr\n\\end{young}\n\n\\end{center}\n\nFor a standard tableau $T$, a \\emph{descent} is an index $i$ such that $i+1$ is in the upper row than $i$. The \\emph{major index} ${\\mathrm {maj}}(T)$ of $T$ is defined as the sum of all descents in $T$. For the standard tableau given in the right above, $1, 3, 4$ and $6$ are descents (descents of the tableau are written in bold), so the major index of the tableau is $1+3+4+6=14$. The \\emph{fake degree polynomial} of a partition $\\lambda$ is defined by the major index generating function for the standard tableaux of shape $\\lambda$, i.e.,\n\\begin{equation*}\n f^\\lambda(q):=\\sum_{T\\in {\\mathrm {SYT}}(\\lambda)}q^{{\\mathrm {maj}}(T)}.\n\\end{equation*}\n\\subsection{Symmetric functions}\n\\label{subsection Symmetric functions}\nLet $\\Lambda=\\bigoplus_{d\\ge0}\\Lambda_d$ be the ring of symmetric functions in an infinite number of variables ${\\mathbf {x}}=(x_1,x_2,\\dots)$ over $\\mathbb{C}(q,t)$. Here $\\Lambda_d$ denotes the subspace of $\\Lambda$ consisting of symmetric functions of homogeneous degree $d$. \n\nBases of $\\Lambda_n$ are indexed by partitions $\\lambda\\vdash n$. Among various bases of $\\Lambda_n$, one of the most important basis is given by Schur functions. The \\emph{Schur function} $s_\\lambda$ of a partition $\\lambda$ is defined by\n\\begin{equation*}\n s_\\lambda({\\mathbf {x}}):=\\sum_{T}{\\mathbf {x}}^T,\n\\end{equation*}\nwhere the sum is over all semistandard tableaux of shape $\\lambda$ and ${\\mathbf {x}}^T=x_1^{T_1}x_2^{T_2}\\cdots$.\n\nWe write $\\langle \\cdot, \\cdot\\rangle$ for the Hall inner product \n\\begin{equation*}\n \\langle s_\\lambda, s_\\mu \\rangle =\\delta_{\\lambda,\\mu},\n\\end{equation*}\nwhere $\\delta_{\\lambda,\\mu}$ is the Kronecker delta. The (modified) \\emph{Macdonald polynomials} $\\widetilde{H}_\\lambda({\\mathbf {x}};q,t)$ indexed by partitions $\\lambda\\vdash n$ form another basis of $\\Lambda_n$. They are defined by the unique family satisfying the following \\emph{triangulation} and \\emph{normalization} axioms \\cite{HHL04},\n\\begin{itemize}\n \\item $\\widetilde{H}_\\lambda[{\\mathbf {x}}(1-q);q,t]=\\sum_{\\lambda\\ge\\mu} a_{\\lambda,\\mu}(q,t)s_\\lambda$,\n \\item $\\widetilde{H}_\\lambda[{\\mathbf {x}}(1-t);q,t]=\\sum_{\\lambda\\ge\\mu'} b_{\\lambda,\\mu}(q,t)s_\\lambda$, \n \\item $\\langle \\widetilde{H}_\\mu, s_{(n)}\\rangle=1$,\n\\end{itemize}\nfor suitable coefficients $a_{\\lambda,\\mu}, b_{\\lambda,\\mu'}\\in{\\mathbb {Q}}(q,t)$. Here, a partial order $\\le$ called \\emph{dominance order} of partitions of $n$ is defined by \n\\begin{equation*}\n \\lambda\\le\\mu \\text{ if } \\lambda_1+\\cdots+\\lambda_k\\le\\mu_1+\\cdots\\mu_k \\text{ for all } k,\n\\end{equation*} and $[\\cdot]$ denotes the plethystic substitution. These axioms are equivalent with Macdonald's triangularity and orthogonality axioms.\n\nExpanding the Macdonald polynomial with Schur functions, we may write \n\\begin{equation*}\n \\widetilde{H}_\\mu({\\mathbf {x}};q,t)=\\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)s_{\\lambda}({\\mathbf {x}}),\n\\end{equation*}\nwhere the sum is over partitions $\\lambda$ of $n$. The coefficients $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ are called the (modified) $(q,t)$-\\emph{Kostka polynomials}. A combinatorial description of general $(q,t)$-Kostka polynomials is unknown, and it is one of the most important open problems in algebraic combinatorics. Since $\\widetilde{K}_{\\lambda,\\mu}(1,1)=|{\\mathrm {SYT}}(\\lambda)|$, the most desirable form of the combinatorial formula would be a generating function for the standard tabelaux.\n\nThe (modified) \\emph{Hall--Littlewood polynomial} $\\widetilde{Q}_\\mu(X;q)$ and $q$-\\emph{Kostka polynomial} $\\widetilde{K}_{\\lambda,\\mu}(q)$ can be obtained by specializing $t=0$ to the Macdonald polynomial $\\widetilde{H}_\\mu(X;q,t)$ and the $(q,t)$-Kostka polynomial $\\widetilde{K}_{\\lambda,\\mu}(q,t)$. The $q$-Kostka polynomial $\\widetilde{K}_{\\lambda,\\mu}(q)$ can also be defined as the generating function of the cocharge statistics for the semistandard tableaux of shape $\\lambda$ and content $\\mu$ (see \\cite{Rho10} for a definition of cocharge statistics). \n\n\\subsection{Representation theory of $\\mathfrak{S}_n$}\\label{subsection rep of Sn}\n\nIrreducible representations of the symmetric group $\\mathfrak{S}_n$ are in one to one correspondence with partitions $\\lambda$ of $n$. We let $S^\\lambda$ be the corresponding irreducible representation. If $V$ is a finite dimensional $\\mathfrak{S}_n$-module, there is a unique way of decomposing $V$ into irreducibles as $V=\\bigoplus_{\\lambda\\vdash n} c_\\lambda S^\\lambda$. The \\emph{Frobenius image} of $V$ is the symmetric function defined by \n\\begin{equation*}\n {\\mathrm {Frob}}(V):=\\sum_{\\lambda\\vdash n} c_\\lambda s_\\lambda.\n\\end{equation*}\nIf $V$ is graded (or bigraded) $\\mathfrak{S}_n$-module as $V=\\bigoplus_{d\\ge0} V_d$ (or $V=\\bigoplus_{d,e\\ge0} V_{d,e}$) the graded Frobenius image is the symmetric function over ${\\mathbb {C}}(q)$ (or ${\\mathbb {C}}(q,t)$) given by\n\n\\begin{equation*}\n{\\mathrm {grFrob}}(V;q) :=\\sum_{d\\ge0}{\\mathrm {Frob}}(V_d)q^d\n\\end{equation*}\n\\begin{equation*}\n (\\text{or }{\\mathrm {grFrob}}(V;q,t):=\\sum_{d,e\\ge0} {\\mathrm {Frob}}(V_{d,e})q^d t^e).\n\\end{equation*}\nIf $V = \\bigoplus_{d \\geq 0} V_d$ (or $V=\\bigoplus_{d,e\\ge 0} V_{d,e}$ is any graded (or bigraded) vector space, its {\\em Hilbert series} is\n \\begin{equation*}\n {\\mathrm {Hilb}}(V;q) = \\sum_{d \\geq 0} \\dim (V_d) q^d\n \\end{equation*}\n \\begin{equation*}\n (\\text{or }{\\mathrm {Hilb}}(V;q,t):=\\sum_{d,e\\ge0} \\dim(V_{d,e})q^d t^e).\n\\end{equation*}\n\n\nWe recall two facts that we use in the proof of the main results. The first one is the theorem of Springer. Note that the original result of Springer deals with the action of a regular element of an arbitrary complex reflection group. For simplicity, we focus only on the action of a long cycle in a symmetric group ${\\mathfrak{S}}_n$.\n\n\\begin{theorem}(\\cite{Spr74})\n\\label{springer-theorem}\nLet $c=(1,2,\\dots,n)$ be a long cycle of ${\\mathfrak{S}}_n$ and $\\chi^\\lambda:{\\mathfrak{S}}_n \\rightarrow {\\mathbb {C}}$ be the irreducible character associated to the $\\mathfrak{S}_n$-representation $S^\\lambda$. Then we have\n\\begin{equation*}\n\\chi^\\lambda(c^r) = f^{\\lambda}(\\zeta^r),\n\\end{equation*}\nwhere $f^\\lambda(q) \\in {\\mathbb {C}}[q]$ is the fake degree polynomial and $\\zeta$ is a (primitive) $n$-th root of unity.\n\\end{theorem}\n\nThe second fact we need is about the Kronecker coefficients. For ${\\mathfrak{S}}_n$-modulues $V$ and $W$, define the \\emph{inner tensor product} $V\\otimes W$ to be the the usual tensor product of vector spaces with $\\mathfrak{S}_n$-module structure given by $$\\sigma\\cdot(v\\otimes w)=(\\sigma \\cdot v)\\otimes (\\sigma \\cdot w).$$ For given partitions $\\lambda, \\mu$ and $\\nu$ of $n$, the \\emph{Kronecker coefficients} $g^\\lambda_{\\mu,\\nu}$ is the multiplicity of $S^\\lambda$ in the inner tensor product $S^\\mu\\otimes S^\\nu$, i.e. $$S^\\mu\\otimes S^\\nu\\cong\\bigoplus g^\\lambda_{\\mu,\\nu}S^\\lambda.$$ Although giving an explicit combinatorial description of general Kronecker coefficients is difficult in general (it is one of the major open problems in algebraic combinatorics), we have the following identity for the special case when $\\lambda$ is a single row.\n\n\\begin{proposition}\\label{prop:Kronecker} For partitions $\\mu,\\nu$ of $n$, the Kronecker coefficient\n$g^{(n)}_{\\mu,\\nu}=\\delta_{\\mu,\\nu},$\nwhere $\\delta_{\\mu,\\nu}$ is the Kronecker delta.\n\\end{proposition}\n\n\n\n\n\\subsection{Modules of Garsia and Haiman}\\label{modules of garsia--Haiman}\n\nTo each partition $\\mu$ of $n$, let $(a_1, b_1), \\dots, (a_n, b_n)$ be the cells of $\\mu$, taken in some arbitrary order. Then we define\n\\[\n\\Delta_\\mu:=\\mathrm{det}(x_i^{a_j} y_i^{b_j})_{1\\le i,j\\le n}.\n\\]\nThe ${\\mathfrak{S}}_n$-module $\\mathbf{H}_\\mu$ is the smallest vector space over ${\\mathbb {C}}$ that contains the determinant $\\Delta_\\mu$ and its partial derivatives with respect to any of the variables $x_i$'s and $y_i$'s for $1\\le i \\le n$. For example, for a partition $\\mu=(3,2)$, the cells of $\\mu$ are $(0,0), (1,0), (2,0), (0,1)$ and $(1,1)$ and the corresponding determinant is given by \n\\begin{equation*}\n\\Delta_\\mu:=\\mathrm{det}\\left[\\begin{array}{ccccc}\n1 & x_1 & x_{1}^{2} & y_1 & x_{1} y_1 \\\\\n1 & x_2 & x_{2}^{2} & y_2 & x_{3} y_3 \\\\\n1 & x_3 & x_{3}^{2} & y_3 & x_{3}y_3 \\\\\n1 & x_4 & x_4^2 & y_4 & x_{4}y_4 \\\\\n1 & x_5 & x_5^{2} & y_5 & x_{5}y_5\n\\end{array}\\right].\n\\end{equation*}\nThen the module $\\mathbf{H}_\\mu$ is given by the $\\mathbb{C}$-span \n\\begin{equation*}\n\\mathbb{C}\\{\\partial_{{\\mathbf {x}}_I}\\partial_{{\\mathbf {y}}_J}\\Delta_\\mu\\}_{I,J}\n\\end{equation*}\nwhere $\\partial_{{\\mathbf {x}}_I}:=\\partial_{x_{i_1}}\\dots\\partial_{x_{i_k}}$ for a multiset $I=\\{i_1,\\cdots,i_k\\}$ and $\\partial_{{\\mathbf {y}}_J}$ is defined similarly. The symmetric group ${\\mathfrak{S}}_n$ acts on ${\\mathbf {H}}_\\mu$ diagonally i.e. permuting $x$ and $y$ coordinates simultaneously. Since the action of ${\\mathfrak{S}}_n$ preserves the degree of ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$, the module ${\\mathbf {H}}_\\mu$ is bigraded ${\\mathfrak{S}}_n$-module, where the bigrade is given by degree of ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$. This is called the \\emph{Garsia--Haiman module}. Haiman \\cite{Hai01} proved the \\emph{$n!$ conjecture} which asserts that this module is of dimension $n!$ regardless of $\\mu$, and moreover, the graded Frobenius image of ${\\mathbf {H}}_\\mu$ is the Macdonald polynomial of $\\mu$:\n\\begin{equation}\\label{graded Frob=Macdonald}\n {\\mathrm {grFrob}}({\\mathbf {H}}_\\mu;q,t)=\\widetilde{H}_\\mu({\\mathbf {x}};q,t)=\\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)s_\\lambda({\\mathbf {x}}),\n\\end{equation}\nwhich proves the Schur positivity of the Macdonald polynomials.\n\n\\section{Sieving generating theorem and diagonal orbit harmonics}\n\\label{Section: Sieving generating theorem and orbit harmonics}\n\n\n\\subsection{Orbit harmonics and cyclic sieving}\n\\label{subsection Orbit harmonics and cyclic sieving}\nIn this section, we introduce a systematic way to generate sieving results using orbit harmonics. The author and Rhoades provided a `generating theorem' for sieving results \\cite[Theorem 3.4]{OR20} by exploiting orbit harmonics applied to a locus $X\\subseteq \\mathbb{C}^n$ with ${\\mathfrak{S}}_n$ acting on $X$ by permuting coordinates. To modify this idea for our purpose, we first explain the \\emph{diagonal orbit harmonics} (see \\cite{GH96} for more details). Consider a finite set $X\\subseteq {\\mathbb {C}}^{2n}$ which is closed under $\\mathfrak{S}_n\\times C_1\\times C_2$-action where \n\\begin{itemize}\n \\item a symmetric group $\\mathfrak{S}_n$ acts on $\\mathbb{C}^{2n}$ by permuting coordinates diagonally, i.e. $$\\sigma(x_1,\\dots,x_n,y_1,\\dots,y_n)=(x_{\\sigma(1)},\\dots,x_{\\sigma(n)},y_{\\sigma(1)},\\dots,y_{\\sigma(n)}),$$\n \\item a finite cyclic group $C_1$ is acts on $\\mathbb{C}^n$ by scaling $x$-coordinates by a root of unity, and\n \\item a finite cyclic group $C_2$ is acts on $\\mathbb{C}^n$ by scaling $y$-coordinates by a root of unity.\n\\end{itemize}\nThen the method of orbit harmonics gives us an isomorphism of $\\mathfrak{S}_n\\times C_1 \\times C_2$-modules:\n\\begin{equation}\\label{isomorphism T-ideal xy}\n \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{I}(X).\n\\end{equation}\nWe further define homogeneous ideal\n\\begin{equation*}\n {\\mathbf {T}}(X):=\\langle \\tau_x\\circ\\tau_y(f): f\\in {\\mathbf {I}}(X)\\setminus \\{0\\}\\rangle \\subseteq {\\mathbb {C}}[{\\mathbf {x}}_n],\n\\end{equation*}\nwhere $\\tau_x$ and $\\tau_y$ is the map taking top degree homogeneous part with respect to ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$, respectively. Then the isomorphism~\\eqref{isomorphism T-ideal xy} extends to an isomorphism\n\\begin{equation*}\n \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{I}(X)\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X),\n\\end{equation*}\nwhere the last item $\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X)$ has an additional structure of graded $\\mathfrak{S}_n\\times C_1\\times C_2$-module on which $C_1$ and $C_2$ acts by scaling in each fixed (bi)degree. Thanks to this isomorphism, we can provide a generating theorem for sieving results in diagonal orbit harmonics whose proof is analogous to the proof of Theorem 3.4 in \\cite{OR20}.\n\n\\begin{theorem}\n\\label{sieving-generator}\nLet $C$ be the subgroup of $\\mathfrak{S}_n$ generated by a long cycle $c=(1,2,\\dots,n)$. Fix positive integers $k_1$ and $k_2$. For $j=1,2$, let\n$\\zeta_j := \\exp(2 \\pi i \/ k_j) \\in {\\mathbb {C}}^{\\times}$ and $C_j = \\langle c_j \\rangle \\cong {\\mathbb {Z}}_{k_j}$ be a cyclic group of order $k_j$. Consider the action of $\\mathfrak{S}_n \\times C_1\\times C_2$ on ${\\mathbb {C}}^{2n}$ where $c_1$ scales $x$-coordinates by $\\zeta_1$, $c_2$ scales $y$-coordinates by $\\zeta_2$ and $\\mathfrak{S}_n$ acts by permuting coordinates diagonally.\n\nLet $X \\subseteq {\\mathbb {C}}^{2n}$ be a finite point set which is closed under the action of $\\mathfrak{S}_n\\times C_1\\times C_2$.\n\\begin{enumerate}\n\\item Suppose that for $d,e \\geq 0$, the isomorphism type of the degree $(d,e)$-piece of \n${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ is given by\n\\begin{equation*}\n( {\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X) )_{d,e} \\cong \\bigoplus_{\\lambda\\vdash n} c_{\\lambda,d,e}S^\\lambda.\n\\end{equation*}\nThe triple $(X, C_1\\times C_2 \\times C, X(q,t,z))$ exhibits the tricyclic sieving phenomenon where \n\\begin{equation*}\nX(q,t,z) = \\sum_{\\lambda\\vdash n} c_{\\lambda}(q,t) f^{\\lambda}(z).\n\\end{equation*}\nwhere $c_{\\lambda}(q,t) := \\sum_{d,e \\geq 0} c_{\\lambda,d,e} q^d t^e$.\n\\item Let $G \\subseteq \\mathfrak{S}_n$ be a subgroup. The set $X\/G$ of $G$-orbits in $X$ carries a natural\n$C_1\\times C_2$-action and the triple $(X\/G, C_1\\times C_2, X\/G(q,t))$ exhibits the bicyclic sieving phenomenon where\n\\begin{equation*}\nX\/G(q,t) = {\\mathrm {Hilb}}( ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t).\n\\end{equation*}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} Applying orbit harmonics to the action of ${\\mathfrak{S}}_n \\times C_1 \\times C_2$ on $X$ \nyields an isomorphism of ungraded ${\\mathfrak{S}}_n \\times C_1 \\times C_2$-modules\n\\begin{equation}\n\\label{ungraded-isomorphism}\n{\\mathbb {C}}[X] \\cong {\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X).\n\\end{equation}\nLet $\\zeta := \\exp(2 \\pi i \/ n)$.\nTo prove (1), apply Theorem~\\ref{springer-theorem} to obtain that \nfor any integers $r, s, k$, the trace of $(c_1^r, c_2^s, c^k) \\in C_1\\times C_2 \\times C'$ acting on ${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ is given by $$\\sum_{\\lambda\\vdash n} c_\\lambda(\\zeta_1^r, \\zeta_2^s) f^{\\lambda}(\\zeta^k) = X(\\zeta^r, \\zeta_2^s, \\zeta^k).$$\nBy the isomorphism \\eqref{ungraded-isomorphism}, this coincides with the trace of $(c_1^r, c_2^s, c^k)$ which is the number of fixed\npoints of $(c_1^r, c_2^s, c^k)$ acting on $X$, completing the proof of (1).\n\nFor (2), we take $G$-invariants of both sides of the isomorphism \\eqref{ungraded-isomorphism}\nto get an isomorphism of $C_1\\times C_2$-modules\n\\begin{equation}\n\\label{g-invariants}\n{\\mathbb {C}}[X\/G] \\cong ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G.\n\\end{equation}\nSince $C_1\\times C_2$ acts on the graded vector space $({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G$ by a root of unity scaling for each $x$ and $y$ variables, the trace of $(c_1^r, c_2^s)$ on the right hand side of the isomorphism \\eqref{g-invariants} is given by\n$$[{\\mathrm {Hilb}}( ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t)]_{q = \\zeta_1^r, t=\\zeta_2^s} = X\/G(\\zeta_1^r,\\zeta_2^s).$$ \nThe trace of $(c_1^r, c_2^s)$ on the left hand side of the isomorphism \\eqref{g-invariants} coincides with the number \nof $G$-orbits in $X\/G$ fixed by $(c_1^r, c_2^s)$.\n\\end{proof}\n\n\\begin{remark}\\label{remark sieving generator}\nIn order to obtain a sieving result involving a combinatorial set $X$ with a cyclic group action using Theorem~\\ref{sieving-generator}, we must \n\\begin{itemize}\n \\item realize $X$ (or its quotient $X\/G$) and the relevant action on it as a point locus in ${\\mathbb {C}}^{2n}$ and the compatible action,\n \\item calculate the graded Frobenius image of ${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ or the Hilbert series of the quotient ${\\mathrm {Hilb}}\\left(({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t\\right)$.\n\\end{itemize}\n\n\\end{remark}\n\n\\subsection{Orbit harmonics and Garsia--Haiman module}\\label{Modules of Garsia--Haiman and Garsia--Procesi}\nThere is a way to understand the Garsia--Haiman module ${\\mathbf {H}}_\\mu$ via orbit harmonics. Let $\\mu$ be a partition of $n$ with $l(\\mu)=l$ and $l(\\mu')=l'$. Let $\\{\\alpha_0,\\dots,\\alpha_{l-1}\\}$ and $\\{\\beta_0,\\dots,\\beta_{l'-1}\\}$ be two sets of distinct complex numbers. Recall that a \\emph{injective tableau} $T$ of shape $\\mu\\vdash n$ is a filling of cells of $\\mu$ by integers $1,2,\\dots,n$ without repetition. The collection of such tableaux will be denoted by $\\mathbf{IT}(\\mu)$. For each $T\\in\\mathbf{IT}(\\mu)$, we assign a point $p_T\\in{\\mathbb {C}}^{2n}$ by letting the $i$-th and the $(n+i)$-th coordinates of $p_T$ record the position of $i$ in $T$:\n\\begin{equation*}\n p_T=(\\alpha_{y_T(1)},\\dots,\\alpha_{y_T(n)},\\beta_{x_T(1)},\\dots,\\beta_{x_T(n)}),\n\\end{equation*}\nwhere $x_T(i)$ and $y_T(i)$ are $x$ and $y$ coordinates of the cell which contains $i$ in $T$. For example, for a partition $\\mu=(2,1)$ and an injective tableau $T=$\\begin{young}\n 2 \\cr\n 3& 1\\cr\n\\end{young} of shape $\\mu$, the point assigned for $T$ is $p_T=(\\alpha_0, \\alpha_1, \\alpha_0, \\beta_1, \\beta_0, \\beta_0)$. Let us denote the collection of points associated to the injective tableaux by\n\\begin{equation*}\n X_\\mu=\\{p_T\\in{\\mathbb {C}}^{2n}:T\\in\\mathbf{IT}(\\mu)\\}.\n\\end{equation*}\nNote that there are exactly $n!$ points in $X_\\mu$. The point locus $X_\\mu$ possesses a natural diagonal action of ${\\mathfrak{S}}_n$: For $\\sigma\\in{\\mathfrak{S}}_n$,\n\\begin{equation*}\n \\sigma (x_1,\\dots,x_n,y_1,\\dots,y_n)=(x_{\\sigma(1)},\\dots,x_{\\sigma(n)},y_{\\sigma(1)},\\dots,y_{\\sigma(n)}).\n\\end{equation*}\nUsing orbit harmonics, one can promote the ungraded ${\\mathfrak{S}}_n$-module ${\\mathbb {C}}[X_\\mu]$ to the bigraded ${\\mathfrak{S}}_n$-module. As usual, let $\\mathbf{I}(X_\\mu)$ be the ideal of polynomials in $\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]$ which vanish on $X$ and define a homogeneous ideal \n\\begin{equation*}\n \\mathbf{T}(X_\\mu):=\\langle\\tau_x\\circ\\tau_y(f):f\\in\\mathbf{I}(X_\\mu)\\setminus\\{0\\}\\rangle\\subseteq\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n].\n\\end{equation*}\nThen the module ${\\mathbf {R}}_\\mu:=\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X_\\mu)$ has an additional structure of (bi)graded $\\mathfrak{S}_n$-module.\n\nGarsia and Haiman \\cite{GH96} proved that the Garsia--Haiman module ${\\mathbf {H}}_\\mu$ embedds into this graded module ${\\mathbf {R}}_\\mu$. Thanks to the $n!$-conjecture, we can conclude the following isomorphism between ${\\mathbf {R}}_\\mu$ and ${\\mathbf {H}}_\\mu$.\n\\begin{theorem}\n\\label{theorem:Garsia--Haiman module} We have an isomorphism as bigraded ${\\mathfrak{S}}_n$-modules:\n$${\\mathbf {R}}_\\mu\\cong{\\mathbb {C}}\\left[X_\\mu\\right]\\cong {\\mathbf {H}}_\\mu.$$\n\\end{theorem}\n\n\\section{Proofs of main theorems}\n\\label{Section: proofs}\n\\subsection{A proof of Theorem~\\ref{Main theorem 1}}\n\\label{subsection: proof of the main theorem}\n\nWe first construct a point locus $X_{(m^n)}$ associated to a rectangular partition $\\mu=(m^n)$. Following Section~\\ref{Modules of Garsia--Haiman and Garsia--Procesi}, to consider $X_{(m^n)}$ as a point locus in $\\mathbb{C}^{2n}$, we choose two sets of distinct complex numbers $\\{\\alpha_0,\\dots,\\alpha_{n-1}\\}$ and $\\{\\beta_0,\\dots,\\beta_{m-1}\\}$. For our purpose, let $\\zeta_1=\\exp(\\frac{2\\pi i}{n})$ and $\\zeta_2=\\exp(\\frac{2\\pi i}{m})$, then set $\\alpha_j=\\zeta_1^{j}$ for $0\\le j \\le n-1$ and $\\beta_k=\\zeta_2^{k}$ for $0\\le k \\le m-1$. Then the corresponding locus $X_\\mu$ possesses\n\\begin{itemize}\n \\item diagonal action of ${\\mathfrak{S}}_n$,\n \\item action of a cyclic group $C_1$ of order $n$ acting by scaling a root of unity $\\zeta_1$ to each $x$-coordinates, and \n \\item action of a cyclic group $C_2$ of order $m$ acting by scaling a root of unity $\\zeta_2$ to each $y$-coordinates.\n\\end{itemize}\n\nNow we can present a proof of Theorem~\\ref{Main theorem 1}. By the construction above, $X_{(m^n)}$ has ${\\mathfrak{S}}_n\\times C_1\\times C_2$ action which corresponds to permutation of letters, row rotation, and column rotation on $\\mathbf{IT}(\\mu)$, respectively. Combining an isomorphism between ${\\mathbf {R}}_\\mu$ and ${\\mathbf {H}}_\\mu$ (Theorem~\\ref{theorem:Garsia--Haiman module}), the fraded Frobenius image of ${\\mathbf {H}}_\\mu$ (Equation~\\eqref{graded Frob=Macdonald}) and the sieving generating theorem (Theorem~\\ref{sieving-generator}), the first bullet point of Theorem~\\ref{Main theorem 1} immediately follows.\n\nTo proceed to the second bullet point, consider a composition $\\nu$ of $mn$. For the Young subgroup $G={\\mathfrak{S}}_\\nu={\\mathfrak{S}}_{\\nu_1}\\times{\\mathfrak{S}}_{\\nu_2}\\cdots$ of $\\nu$, the $G$-orbits of $X_{(m^n)}$ are in one-to-one correspondence with the set of $n\\times m$ matrices with content equal to $\\nu$. \n\nTo obtain a sieving result for $X_{(m^n)}\/G$, we must calculate the Hilbert series of $G$-fixed subspace of ${\\mathbf {R}}_\\mu$. Let ${\\bf 1}$ be the trivial representation of ${\\mathfrak{S}}_\\nu$. It is a standard fact that the induction of ${\\bf 1}$ from ${\\mathfrak{S}}_\\nu$ to ${\\mathfrak{S}}_n$\ncan be written as\n\\begin{equation*}\n{\\bf 1} \\uparrow_{{\\mathfrak{S}}_\\nu}^{{\\mathfrak{S}}_n} \\cong \\bigoplus_\\lambda K_{\\lambda,\\nu} S^\\lambda,\n\\end{equation*}\nwhere $K_{\\lambda,\\nu}$ denotes a Kostka number. Applying Frobenius reciprocity, it follows that the dimension of the ${\\mathfrak{S}}_\\nu$-fixed subspace of \nthe ${\\mathfrak{S}}_n$-irreducible $S^{\\lambda}$ is given by the character inner product:\n\\begin{equation*}\n\\dim (S^{\\lambda})^{{\\mathfrak{S}}_\\nu} = \\langle {\\bf 1}, S^{\\lambda} \\downarrow^{{\\mathfrak{S}}_n}_{{\\mathfrak{S}}_\\nu} \\rangle_{{\\mathfrak{S}}_\\nu} = \n\\langle {\\bf 1} \\uparrow_{{\\mathfrak{S}}_\\nu}^{{\\mathfrak{S}}_n}, S^{\\lambda} \\rangle_{{\\mathfrak{S}}_n} = K_{\\lambda,\\nu}.\n\\end{equation*}\nCorrespondingly, if $V$ is any bigraded ${\\mathfrak{S}}_n$-module with Frobenius image\n\\begin{equation*}\n {\\mathrm {grFrob}}(V;q,t)=\\sum_{\\lambda\\vdash n}c_{\\lambda}(q,t)S^\\lambda,\n\\end{equation*}\nthe Hilbert series of ${\\mathfrak{S}}_\\nu$ fixed subspace will be\n\\begin{equation*}\n {\\mathrm {Hilb}}(V^{{\\mathfrak{S}}_\\nu};q,t)=\\sum_{\\lambda\\vdash n} c_{\\lambda}(q,t)K_{\\lambda,\\nu}.\n\\end{equation*}\nBy (2) of Theorem~\\ref{sieving-generator}, this concludes the second bullet point.\n\n\\subsection{A proof of Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3}}\n\\label{subsection: proof of the main theorem 2}\n\nIn previous section, we proved that for a composition $\\nu$ of $mn$, the triple $\\left(X_{(m^n),\\nu}, {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m, \\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)K_{\\lambda,\\nu}\\right)$ exhibits biCSP. Suppose, furthermore, $\\nu$ has a cyclic symmetry of order $a$. Then the set $X_{(m^n),\\nu}$ possesses another cyclic group action by adding $a$ modulo $l(\\nu)$ to each entry. Therefore, it is natural to seek for a sieving result that reflects this additional cyclic group action. \n\n\nBefore we begin, we recall the Tanisaki locus. For a composition $\\nu\\models d$, let $W_\\nu$ be the set of length $d$ words $w=(w_1,\\dots,w_d)$ of content $\\nu$ ($i$ appears $\\nu_i$ times). Let $\\zeta=\\exp\\left(\\frac{2\\pi i}{l(\\nu)}\\right)$. We assign a point $p_w$ in ${\\mathbb {C}}^d$ so that we can realize $W_\\nu$ as a point locus $Y_\\nu$ (called the Tanisaki locus) in ${\\mathbb {C}}^d$ as follows:\n\\begin{equation*}\n p_w=(\\zeta^{w_1},\\dots,\\zeta^{w_d}).\n\\end{equation*}\nGarsia and Procesi \\cite{GP92} proved that the T-ideal corresponding to the Tanisaki locus is given by the ideal generated by elementary symmetric polynomials with extra conditions (for precise definition of this `Tanisaki ideal', we refer \\cite{GP92}). By orbit harmonics, there is an isomorphism\n\\begin{equation}\\label{Garsia--Procesi module orbit harmonics}\n {\\mathbb {C}}[Y_\\nu]\\cong {\\mathbf {L}}_\\nu:={\\mathbb {C}}[{\\mathbf {x}}_d]\/{\\mathbf {T}}(Y_\\nu).\n\\end{equation}\nMoreover, they showed that the graded Frobenius image coincides with the Hall--Littlewood symmetric function, $${\\mathrm {grFrob}}({\\mathbf {L}}_\\nu;q)=\\widetilde{Q}_\\nu({\\mathbf {x}};q).$$\nFurthermore, if a composition $\\nu$ has a cyclic symmetry of order $a$, the set $W_\\nu$ has additional cyclic group action given by adding $a$ modulo $l(\\nu)$ to each letter. This action corresponds with the action of scaling a root of unity $\\zeta^a$ in each coordinates in $Y_\\nu$. In this setting, the isomorphism~\\eqref{Garsia--Procesi module orbit harmonics} extends to an isomorphism as graded ${\\mathfrak{S}}_d\\times C$-modules, where $C$ is a cyclic group of order $l(\\nu)\/a$.\n\nNow let $\\mu=(m^n)$ be a rectangular partition and $\\nu$ be a composition of $mn$ with a cyclic symmetry of order $a$. Then the product $X_\\mu\\times Y_\\nu$ carries an $\\mathfrak{S}_{mn}\\times {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m \\times {\\mathbb {Z}}_{l(\\nu)\/a}$-action, where $\\mathfrak{S}_{mn}$ acts diagonally on $X_\\mu$ and $Y_\\nu$ and the cyclic groups ${\\mathbb {Z}}_n$, ${\\mathbb {Z}}_m$ and ${\\mathbb {Z}}_{l(\\nu)\/a}$ acts by row rotation on $X_\\mu$, column rotation on $X_\\mu$ and translation on the entries on $Y_\\nu$, respectively. By Theorem~\\ref{theorem:Garsia--Haiman module} and the isomorphism~\\eqref{Garsia--Procesi module orbit harmonics}, we have an isomorphism\n\\begin{equation}\\label{equation: isomorphism between product}\n{\\mathbb {C}}[X_\\mu\\times Y_\\nu]\\cong {\\mathbf {R}}_\\mu \\otimes {\\mathbf {L}}_\\nu\n\\end{equation}\nas $\\mathfrak{S}_{mn}\\times{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a}$-modules. Since the graded Frobenius image of the module ${\\mathbf {R}}_\\mu$ is given by the Macdonald polynomial and the graded Frobenius image of the Garsia--Procesi module ${\\mathbf {L}}_\\nu$ is given by the Hall--Littlewood polynomial, the Frobenius image is given by\n\\begin{equation*}\n\\label{Frob Xmu times Ynu}\n {\\mathrm {grFrob}}\\left({\\mathbb {C}}[X_\\mu\\times Y_\\nu];q,t,z\\right)=\\sum_{\\rho,\\lambda,\\lambda'\\vdash mn} \\widetilde{K}_{\\lambda,\\mu}(q,t)\\widetilde{K}_{\\lambda',\\nu}(z)g^{\\rho}_{\\lambda,\\lambda'}s_\\rho,\n\\end{equation*}\nwhere $g^\\rho_{\\lambda,\\lambda'}$ denotes a Kronecker coefficient.\nBy taking isotypic components for a trivial representation $S^{(mn)}$ of $\\mathfrak{S}_{mn}$ on both sides of equation~\\eqref{equation: isomorphism between product}, we have\n\\begin{equation}\\label{equation: isomorphism isotypic components}\n{\\mathbb {C}}[X_\\mu\\times Y_\\nu]^{{\\mathfrak{S}}_{mn}}\\cong [{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu]^{{\\mathfrak{S}}_{mn}}.\n\\end{equation}\n\nThere is a natural basis of ${\\mathbb {C}}[X_\\mu\\times Y_\\nu]^{\\mathfrak{S}_{mn}}$ indexed by $\\mathfrak{S}_{mn}$-orbits of $X_\\mu\\times Y_\\nu$, given by the sum of elements in each orbit. Note that each of these orbits corresponds to a $n\\times m$ matrix with content equal to $\\nu$. It is clear that the cyclic groups ${\\mathbb {Z}}_n$, ${\\mathbb {Z}}_{m}$ and ${\\mathbb {Z}}_{l(\\nu)\/a}$ act on these matrices by row rotation, column rotation, and translation of the entries. For an element $g\\in {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m \\times {\\mathbb {Z}}_{l(\\nu)\/a}$, we can count the number of fixed points of $g$ in $(X_\\mu\\times Y_\\nu)\/{\\mathfrak{S}_{mn}}$ is given by the trace of $g$ acting on the left hand side of the isomorphism~\\eqref{equation: isomorphism isotypic components}. On the other hand, this can be calculated by trigraded Hilbert series \n\\begin{equation}\\label{Hilbert poly of S_mn invariant of R_mu times L_nu}\n {\\mathrm {Hilb}}\\left(\\left[{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu\\right]^{{\\mathfrak{S}}_{mn}};q,t,z\\right)=\\sum_{\\lambda, \\lambda'\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda',\\nu}(z)g^{(mn)}_{\\lambda,\\lambda'},\n\\end{equation}\nof $[{\\mathbf {R}}_\\mu\\otimes_{{\\mathbb {C}}}{\\mathbf {L}}_\\nu]^{\\mathfrak{S}_{mn}}$ at roots of unity. By Proposition~\\ref{prop:Kronecker}, taking the coefficient of the Schur function $s_{(mn)}$ in Equation~\\eqref{Hilbert poly of S_mn invariant of R_mu times L_nu}, we have the following polynomial\n$$ {\\mathrm {Hilb}}\\left([{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu]^{{\\mathfrak{S}}_{mn}};q,t,z\\right)=X_{\\mu,\\nu}(q,t,z):=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,\\nu}(z)$$\nfor sieving result. This proves Theorem~\\ref{Main theorem 2}.\n\\begin{example}\nTake $\\mu=(2,2)$ and $\\nu=(2,2)$. We have six $2\\times2$ matrices with content equal to $\\nu$ listed in the following.\n\n\\begin{figure}[h]\n $\n \\begin{pmatrix}\n 1 & 1 \\\\\n 2 & 2\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 1 & 2\\\\\n 1 & 2\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 1 & 2\\\\\n 2 & 1\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 2 & 1\\\\\n 1 & 2\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 2 & 1\\\\\n 2 & 1\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 2 & 2\\\\\n 1 & 1\n \\end{pmatrix}$\n\n \\end{figure}\n\nFixed points of $(0,1,1)\\in{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2$ correspond to the following four matrices and there is no fixed point of $(0,0,1)\\in{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2$\n\n\\begin{figure}[h]\n $\n \\begin{pmatrix}\n 1 & 2\\\\\n 1 & 2\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 1 & 2\\\\\n 2 & 1\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 2 & 1\\\\\n 1 & 2\n \\end{pmatrix}$\n \\qquad$\n \\begin{pmatrix}\n 2 & 1\\\\\n 2 & 1\n \\end{pmatrix}$\n \\end{figure}\n\nThe polynomial $X(q,t,z)$ is given by\n\\begin{align*}\nX(q,t,z)\n &=\\sum_{\\lambda\\vdash 4} \\widetilde{K}_{\\lambda,(2,2)}(q,t)\\widetilde{K}_{\\lambda,(2,2)}(z)\\\\\n &=\\widetilde{K}_{(4),(2,2)}(q,t)\\widetilde{K}_{(4),(2,2)}(z)+\\widetilde{K}_{(3,1),(2,2)}(q,t)\\widetilde{K}_{(3,1),(2,2)}(z)+\\widetilde{K}_{(2,2),(2,2)}(q,t)\\widetilde{K}_{(2,2),(2,2)}(z)\\\\\n &\\equiv 3+qz+tz+qtz \\qquad \\operatorname{mod} \\quad(q^2-1, t^2-1, z^2 -1).\n\\end{align*}\nNote that $X(-1,1,-1)=4$ and $X(1,1,-1)=0$, which agrees with Theorem~\\ref{Main theorem 2}.\n\\end{example}\n\nTo keep this paper concise, instead of providing a precise proof of Theorem~\\ref{Main theorem 3}, we rather sketch a proof that is very similar to the proof of Theorem~\\ref{Main theorem 2}. Let $l=mn=ab$ be a positive integer with two factorizations. Following the argument in the proof of Theorem~\\ref{Main theorem 2}, we consider point locus $X_{(m^n)}$ to the $l\\times l$ permutation matrices by the map $\\phi$ defined in Section~\\ref{Introduction}. We also consider $X_{(a^b)}$ as the set of $l\\times l$ permutation matrices in the same way.\n\nA product of groups ${\\mathfrak{S}}_l\\times \\mathbb{Z}_n\\times \\mathbb{Z}_m$ acts on $X_{(m^n)}$ by permuting columns, external row rotation and internal row rotation, while a product of groups ${\\mathfrak{S}}_l \\times\\mathbb{Z}_b\\times \\mathbb{Z}_a$ acts on $X_{(a^b)}$ by permuting columns, external row rotation and internal column rotation. Each $\\mathfrak{S}_l$-orbits of $X_{(m^n)}\\times X_{(a^b)}$ corresponds to a $l\\times l$ permutation matrix. One can see that there is a natural action of $\\mathbb{Z}_n\\times \\mathbb{Z}_m\\times\\mathbb{Z}_b\\times \\mathbb{Z}_a$ on orbits in $[X_{(m^n)}\\times X_{(a^b)}]\/\\mathfrak{S}_l$ by external and internal row rotation, and external and internal column rotation.\n\nOn the other hand, let ${\\mathbf {R}}_{(m^n)}$ and ${\\mathbf {R}}_{(a^b)}$ be two bigraded ring obtained from point loci $X_{(m^n)}$ and $X_{(a^b)}$ by orbit harmonics. Since their graded Frobenius images are Macdonald polynomials, by Proposition~\\ref{prop:Kronecker}, we have\n\\begin{align*}\\label{Hilbert poly of S_mn invariant of R_mu times R_nu}\n {\\mathrm {Hilb}}\\left(\\left[{\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{(a^b)}\\right]^{{\\mathfrak{S}}_{l}};q,t,z,w\\right)&=\\sum_{\\lambda, \\lambda'\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda',(a^b)}(z,w)g^{(mn)}_{\\lambda,\\lambda'}\\\\\n &=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).\n\\end{align*}\nBy applying orbit harmonics, we have an isomorphism\n\\[\n\\mathbb{C}\\left[X_{(m^n)}\\times X_{(a^b)}\\right]^{\\mathfrak{S}_l}\\cong \\left[{\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{(a^b)}\\right]^{{\\mathfrak{S}}_{l}}.\n\\]\nTherefore, we can conclude that the triple $\\left(\\mathfrak{S}_l, {\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}, \\mathfrak{S}_l(q,t,z,w)\\right)$ exhibits the quadracyclic sieving phenomenon, where\n$$\\mathfrak{S}_l(q,t,z,w)=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).$$\n\n\\section{Concluding remarks}\\label{concluding remarks}\n\n\\subsection{Other combinatorial loci}\nRecall that we obtained Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3} by taking the tensor product of modules of Garsia--Haiman and Garsia--Procesi, and then taking ${\\mathfrak{S}}_{mn}$-invariant part. We could replace one of those modules to obtain various sieving results. One way to do this is replacing one of the modules with the module ${\\mathbf {R}}_{n,k}$ defined in \\cite{HRS18}. They defined this module to construct a graded ${\\mathfrak{S}}_n$-module for the Delta conjecture at $t=0$. The module ${\\mathbf {R}}_{n,k}$ can also be obtained by applying orbit harmonics to the locus corresponding to the set of surjective functions from $[k]$ to $[n]$. For $mn\\le k$ by taking ${\\mathfrak{S}}_{mn}$ invariant part of ${\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{mn,k}$, we could obtain a triCSP for $n$ times $m$ matrices (or fillings of a rectangular partition) with entries given by nonempty set partitions of $[k]$ into $[mn]$ parts (which may be called the surjective tableaux).\n\nThis process can be applied to a broad class of modules obtained via orbit harmonics. One of the interesting modules which we did not consider in this paper is the module ${\\mathbf {R}}_{\\mu,k}$ defined by Griffin \\cite{Gri21}. This module is a common generalization of the Garsia--Procesi module and the module ${\\mathbf {R}}_{n,k}$ of Haglund--Rhoades--Shimozono and it is possible to obtain this module via orbit harmonics. \n\n\\subsection{Combinatorial proof of main theorems}\nRhoades used two facts to prove cyclic sieving results involving $q$-Kostka polynomials \\cite{Rho10}. The first one is about the evaluation of the Hall--Littlewood polynomial at a root of unity due to Lascoux, Leclerc and Thibbon \\cite{LLT94, LLT97}. The second one is the rim hook correspondence of Stanton and White \\cite{SW85}.\n\nFor the evaluation of the Macdonald polynomials of a rectangular partition at a root of unity, Descouens, and Morita gave a formula \\cite{DM08}. If one can provide a counterpart for rim hook correspondence of Stanton--White in the setting of Theorem~\\ref{Main theorem 1}, Theorem~\\ref{Main theorem 2} or Theorem~\\ref{Main theorem 3}, it would give more combinatorial proof of those.\n\n\\section{Acknowledgements}\n\nThe author is grateful to Brendon Rhoades for helpful conversations about orbit harmonics. The author also thanks anonymous referees for their careful reading and valuable comments. In particular, the author thanks a referee's suggestion to clearly spell out the connections and differences between the results in this paper and known CSP.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\subsection{Mondrian Processes \\& Mondrian Forest}\n\\textbf{Mondrian Processes}\nare families of (potentially infinite) hierarchical binary\npartitions of a subdomain ${\\mathcal{D}} \\subseteq \\mathbb{R}^D$;\nthey can be thought of as a family of $k$d trees with height $h$, which\nsequentially refine the partition of ${\\mathcal{D}}$ as $h$ increases \n\\cite{roy2008mondrian}.\nA Mondrian Tree can be defined as a restriction of the underlying \nMondrian Process to an observed set of data points \n\\cite{lakshminarayanan2014mondrian}.\nUnlike the Mondrian Process, it allows for the online sampling of the \nstored tree as more data is observed.\nSpecifically, a \\textbf{Mondrian Tree} $T$ can be represented by the tuple\n$(\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ for a decision tree\n$(\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi} )$ whose cut dimensions ${\\boldsymbol \\delta}$ are \nchosen with probability proportional to the feature lengths of data \nstored in a node\nand ${\\boldsymbol \\tau}$ is a \nsequence of cut times\n${\\boldsymbol \\tau} = (\\tau_j)_{j \\in \\mathsf{T}}$\nwhich begin from 0 at the root ($\\tau_{\\epsilon}$) while monotonically increasing up to a \\textbf{lifetime budget} $\\lambda > 0$. \nFor any node $j$, the \\textbf{time} or \\textbf{weighted depth} is the value\n$\\tau_j$, whereas the \\textbf{(absolute) depth} is the length of the\n(unweighted) path from the root to $j$.\nGiven \\emph{observations} ${\\mathbf{X}}$, the generative process for sampling Mondrian\nTrees is denoted $\\MTree{{\\mathbf{X}}}{\\lambda}$.\nFor every node $j \\in \\mathsf{T}$, the indices of the data stored at $j$ is \ndenoted $N(j)$ (so we clearly have $N(\\epsilon) = \\{1,\\ldots, n\\}$) and the \nregions of space a every node $B_j$ are the \\emph{minimal axis-aligned box}\ncontaining the data ${\\mathbf{X}}_{N(j)}$.\nAdditionally, the dimension-wise minima and maxima over \n${\\mathbf{X}}_{N(j)}$ are stored in the vectors ${\\mathbf{l}}_j^{{\\mathbf{X}}}$ and \n${\\mathbf{u}}_j^{{\\mathbf{X}}}$.\nAn example implementation is given in \\Cref{alg: mondrian-forest}.\n\nMondrian Trees are attractive models as they can be sampled \\emph{online}\nas new data is observed. The key principle for this is \\textbf{projectivity}, meaning that if \n$T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ and ${\\mathbf{X}}'$ is a subset of the data from ${\\mathbf{X}}$,\nthen the tree restricted to the datapoints ${\\mathbf{X}}'$ is drawn from \n$\\MTree{{\\mathbf{X}}'}{\\lambda}$ \\cite{lakshminarayanan2014mondrian}.\nCrucially, this enables the sequential building of Mondrian Trees:\n\n\\begin{lemma}[Projectivity] \\label{lem:projectivity}\nLet ${\\mathbf{X}} = \\{{\\mathbf{x}}_i\\}_{i=1}^n, {\\mathbf{X}}' = {\\mathbf{X}} \\cup {\\mathbf{x}}_{n+1}$.\nSuppose $\\textsf{MTx}({\\mathbf{X}}',\\lambda)$ is a random\nfunction to extend the tree $T$.\n If $T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ and \n \n $T' | T, {\\mathbf{X}}' \\sim \\textsf{MTx}({\\mathbf{X}}',\\lambda)$ then \n $T' \\sim \\MTree{{\\mathbf{X}}'}{\\lambda}$.\n\\end{lemma}\n\nHence, Mondrian \\emph{Trees} are essentially, finite, truncated versions of \nMondrian Processes in the regions of $\\mathbb{R}^D$ where data is observed.\nAn ensemble of trees each independently sampled from $\\MTree{{\\mathbf{X}}}{\\lambda}$ is referred\nto as a \\textbf{Mondrian Forest}.\n\n\n\n\n\n\\subsection{\\Polya{} Tree} \\label{sec: ptree-intro}\nThe \\Polya{} Tree{} is a nonparametric model for estimating the density function \nover a nested binary partition of a bounded input domain, ${\\mathcal{D}}$.\nWe require the \\Polya{} Tree{} to decide how to distribute mass about the space\nrepresented according to a random binary partition that we will sample.\nFirst we will introduce the infinite version of the \\Polya{} Tree{} and then\ndemonstrate a restricted, finite \\Polya{} Tree{}\n(further details can be found in \\cite{muller2013polya}).\n\n\nSuppose $\\Pi_m = \\{ A_{b(j)}: j = 1,\\dots,2^m\\} $ is the depth $m$ partition of ${\\mathcal{D}}$\ninto $2^m$ disjoint subsets $j$, indexed by the binary\nstring $b(j) = e_0 e_1 \\dots e_{m-1}$.\nIf we refine $\\Pi_{m}$ to $\\Pi_{m+1}$ by splitting every \n$A_{b(j)} = A_{b(j)0} \\cup A_{b(j)1} $ with \n$A_{b(j)0} \\cap A_{b(j)1} = \\emptyset$ to generate \nthen it remains to understand how mass is allocated to all subsets in \n$\\Pi_{m+1}$.\nThe \\Polya{} Tree{} treats probability mass as a random variable which is \ndistributed throughout $\\Pi_m$ through split \nprobabilities $\\pi_{b(j)} \\sim \\BetaDist{\\alpha_{b(j)0}}{\\alpha_{b(j)1}}$,\neach $\\pi_{b(j)}$ being sampled independently across all levels of \nrefinement, $m$.\nThe probability $\\pi_{b(j)}$ is the probability of reaching the \n``right-hand side'' of the split: that is, choosing a point that is \nin $A_{b(j)1}$ given that the point is in $A_{b(j)}$.\nOverall, the \\Polya{} Tree{} has two sets of parameters: the nested partition\n$\\Pi = \\{ \\Pi_m : m \\ge 0 \\}$ and the Beta distribution parameters\n${\\mathcal{A}} = \\{ (\\alpha_{b(j)0}, \\alpha_{b(j)1} ) : j = 1,\\dots, 2^m \\}$.\n\n\nA \\textbf{\\Polya{} Tree{} over infinite depth partition} allows \n$m \\rightarrow \\infty$ and \nis capable of modelling absolutely continuous functions if the\n$\\alpha_{b(j)} = \\Theta(m^2)$ or discrete functions if \n$\\alpha_{b(j)} = \\Theta(2^{-m})$.\nRather than let $m \\rightarrow \\infty$ \na \\textbf{\\Polya{} Tree{} over a finite depth partition} assumes the partition\nis truncated at some fixed $m$.\nProbability mass is then assumed to be distributed uniformly \nwithin the final $2^m$ bins.\nAn implementation of the \\Polya{} Tree{} is given in \\Cref{alg: polya-tree-sampling} when the \npartition $\\Pi_m$ is defined by a binary tree of height $m$,\nas opposed to the online setting.\nThe predicitve distribution for density estimation over a finite \npartition is the product of expectations of the Beta distributions along the \nleaf-to-root path \\cite{muller2013polya}.\n\n\\subsection{Streaming Mondrian \\Polya{} Tree} \\label{sec: mpt-intro}\n\\input{MondrianPolyaTreeExample}\nThe standard Mondrian Tree \\emph{only} considers (sub-)regions \nwhere data is observed: information about space \nwithout observations is discarded.\nWe decouple the splitting method used to generate Mondrian Trees into\na two-step procedure which will allow a tree sampled\nin the Mondrian Tree $T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ to implicitly \nrepresent the entirety of a given input domain so we can appeal to the \n\\Polya{} Tree{} model.\n\n\nOur modified Mondrian Tree is the \n\\textbf{streaming Mondrian P{\\'o}lya{} Tree}\n(\\acrshort{mpt}) and draws from this structure are \ndenoted $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$.\nWe generate an \\acrshort{mpt} by drawing $T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$\nand introducing `pseudosplits' in $T$ to generate a new, implicitly \ndefined $T_s$.\nThese pseudosplits distinguish between the regions of space where data is\nobserved, and those which do not contain any data so the only extra\nspace cost we incur is that of storing the parameters for the Beta \ndistributions necessary for the \\Polya{} Tree.\nAn example is illustrated in \\Cref{fig:mpt-example-plot} and is \nimplemented in \\Cref{alg: mpt}.\n\n\n\nLet $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau}) \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$\nbe a Mondrian Tree.\nWe define two functions over nodes $j \\in \\mathsf{T}$ to decouple the \ncutting of space from the restriction to bounding boxes.\nFirst recall that every node $j$ has a minimal axis-aligned bounding box $B_j$:\n(i) $\\Cut{j}$ samples a cut dimension $\\delta_j$ and cut location $\\xi_j$ splitting\n$j$ into disjoint sets $R_{j0}, R_{j1}$ with $R_{j0} \\cup R_{j1} = B_j$, and \n$R_{j0}, R_{j1}$ representing the \\emph{region} of space less than cut $\\xi_j$ and greater \nthan $\\xi_j$, respectively.\nFor $k=0,1$, the set $R_{j k}$ is a bounded region of space which is \\emph{not \nrestricted} to the observations located at the child nodes ${\\mathbf{X}}_{N(\\Child{j})}$.\nThis motivates the subsequent `split' to maintain the Mondrian Tree \nstructure of only performing $\\Cut{\\cdot}$ on bounding boxes of \nobserved data; \n(ii) $\\Restrict{j}$ acts on the pair $R_{j0}, R_{j1}$, returning \n$\\left\\{ \\left( B_{\\lChild{j}}, B_{\\lChild{j}}^C \\right), \n\\left( B_{\\rChild{j}}, B_{\\rChild{j}}^C \\right) \\right\\}$ such that \n$B_{\\lChild{j}} \\cup B_{\\lChild{j}}^C = R_{j0}$ and are pairwise disjoint.\nThe same property holds for $R_{j1}$ with the right-hand child nodes.\nWe refer to $B_{\\Child{j}}$ as the \\textbf{observed region} and \n$B_{\\Child{j}}^C$ as the \\textbf{complementary region} where $\\Child{j}$ can be \neither $\\lChild{j}$ or $\\rChild{j}$.\nWe term this as a `pseudosplit' because all of the information \nrequired to perform $\\Restrict{\\Cut{j}}$ is already defined in the generation \nof the Mondrian Tree.\n\n\n\\textbf{Combining \\acrshort{mpt} with Finite \\Polya{} Tree{}.}\nDecoupling the `cut-then-restrict' allows the Mondrian Tree $T$ \nto encode a valid hierarchical partition over the entirety of the input domain ${\\mathcal{D}}$.\nAdditionally, we only ever store the $T$ and the extra Beta parameters\nfrom the \\Polya{} Tree{} structure as $T$\nimplicitly defines the \\acrshort{mpt} $T_s$.\nThese Beta parameters are \\textbf{cut parameters}: $\\chi_{j0}, \\chi_{j1}$, index 0 for less than $\\xi_j$, 1 otherwise; \n\\textbf{restriction parameters} \n$\\rho_{j0 \\in}, \\rho_{j0 \\neg}$ indexed by $j0\\in$ for the observed region\n$B_{\\lChild{j}}$ and $j0\\neg$ for the complementary region $B_{\\lChild{j}}^C$\n(and similarly for $\\rho_{j1 \\in}, \\rho_{j1 \\neg}$) at node $j$.\n\n\nThe following distinctions are necessary to ensure all volume\n comparisons for the \\Polya{} Tree{} construction are on $D$-dimensional \n hypervolumes while the final distinction is necessary to account for \n the mass associated to regions with no observations.\n \\begin {enumerate*} [label=(\\roman*)]\n \\item{\\textbf{Observation leaves (Type I)}:\n Any leaf $l$ for which the bounding box \n at $l$, $B_l = \\bbox{X_{N(l)}}$ has at least one of the dimensions with zero\n length.\n Note that this includes the case when only one datapoint is stored\n in $l$; \n }\n\n \\item{\\textbf{Observation leaves (Type II)}:\n Any leaf $l$ formed from a cut at node $j$ which contains two or more datapoints and $B_l = \\bbox{X_{N(l)}}$ has all $D$ lengths positive; \n }\n\n \\item{\\textbf{Complementary leaves}: Leaves formed in a region where no\n observations are made.\n }\n \\end{enumerate*}\n\nPseudosplits in the Mondrian Tree $T$ are used to generate the \n\\acrshort{mpt}, so it is necessary to revise the indexing scheme of the\nnested partition over domain ${\\mathcal{D}}$.\nFor every node $j \\in \\mathsf{T}$ of (absolute) depth $k$ in\n$T$, we generate a set of encodings for the spaces represented in $T_s$:\n$b(j) = c_0 r_0 \\dots c_k r_k$.\nThe length of $b(j)$ is at most twice the maximum absolute depth \nof $T$ and indexes all nodes in the (implicitly defined) $T_s$.\nThe symbol $c_i \\in \\{ 0,1 \\}$ indicates ``less than'' or ``greater than'' the cut at level\n$i$, and $r_i \\in \\{ \\in, \\neg, \\emptyset \\}$ indicates whether the node represents\nthe observed region ($\\in$), complementary region ($\\neg$), or can be $\\emptyset$\nif the leaf is a Type I observed leaf as no restriction is performed.\n\n\\subsubsection{Model Parameters for the \\acrshort{mpt}} \\label{sec: mpt-params}\nFor a Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ we now\nshow how to set the parameters for the \\Polya{} Tree{} over the induced \n\\acrshort{mpt}, $T_s$.\n\\begin{itemize}\n\n\\item{Let $P_j$ denote the probability mass associated with node $j$.\nWe define the probability mass associated with the root $\\epsilon$ to\nbe $P_{\\epsilon} = 1$.}\n\n\\item{\\textbf{Setting the prior.}\nAt every internal node $j \\in \\mathsf{T}$, the probability of a point\nbeing greater than the cut $\\xi_j$ is given by a Bernoulli \nwith parameter $\\kappa_j$ whose prior is \n$\\BetaDist{\\chi_{j0}}{\\chi_{j1}}$.\nLikewise, the probability of a point being in the observed \nregion after cut $\\xi_j$ follows a\\footnote{Note that we choose this\nordering so the expected\nvalue of the Beta distribution is associated with being in \nthe observed region after every cut.\nSee example in \\Cref{fig:mpt-example-tree}.} $\\text{Bernoulli}(1-\\theta_j)$ whose prior is \n$\\BetaDist{\\rho_{j k \\in}}{\\rho_{j k \\neg}}$\nfor $k=0,1$, depending on which side \nof the cut the point lies.\nTo set the \\Polya{} Tree{} parameters, we need to evaluate the volumes of \nvarious parts of the space, this will be denoted $V_X = \\vol{X}$ \nfor $X \\in \\{R_{\\cdot}, B_{\\cdot}, B_{\\cdot}^C\\}$.\nThe \\acrshort{mpt} $T_s$ is defined from a two-stage split so we correct\nthe `depth' of nodes in $T_s$ from the usual \\Polya{} Tree{} construction by\na simple translation: if $j \\in \\mathsf{T}$ has depth $d_j$ then\n$\\PolyaTreeDepth{j} = (2d_j, 2d_j+1)$.\nThe prior strength is controlled by hyperparameter $\\gamma > 0$.\nParameters for cut ($\\chi_{\\cdot}$) and \nrestriction ($\\rho_{\\cdot}$) are then:\n\\begin{equation} \\label{eq: mpt-prior}\n \\begin{split}\n \\chi_{j0} &= \\gamma (2d_j + 1)^2 \n \\nicefrac{ V_{R_{j0}}}{ ( V_{R_{j0}} + V_{R_{j1}} ) }\\\\\n \\chi_{j1} &= \\gamma (2d_j + 1)^2 \n \\nicefrac{ V_{R_{j1}}}{ ( V_{R_{j0}} + V_{R_{j1}} ) }\\\\\n \\end{split}\n\\quad\n \\begin{split}\n \\rho_{j k \\in} &= \\gamma (2d_j + 2)^2 \n \\nicefrac{ V_{B_{jk\\in}}}{ ( V_{B_{jk\\in}} + V_{B_{jk\\neg}} ) }\\\\\n \\rho_{j k \\neg} &= \\gamma (2d_j + 2)^2 \n \\nicefrac{ V_{B_{jk\\neg}}}{ ( V_{B_{jk\\in}} + V_{B_{jk\\neg}} ) }\\\\\n \\end{split}\n\\end{equation}\n\n}\n\n\\item{\\textbf{Distributing Mass.}\nThe predictive distribution of the \\Polya{} Tree{} over a finite \ndepth partition is the product of expected value of Beta distributions on the leaf-root path.\nThere can be maintained exactly over all nodes for both cutting \\& restricting.\nWe allocate a \n$\\mu_{\\chi_j} = \\mathbb{E}(\\BetaDist{\\chi_{j0}}{\\chi_{j1}})$ fraction of $j$'s mass to $R_{j0}$ so $P_{j0} = P_j \\mu_{\\chi_j}$ \\& \n$P_{j1} = P_j (1 - \\mu_{\\chi_j})$.\nNext, we repeat for the restriction step which allots \n$\\mu_{\\rho_{j0}} = \\mathbb{E}(\\BetaDist{\\rho_{j0\\in}}{\\rho_{j0\\neg}})$ to $B_{j0}$ so $P_{j0\\in} = P_{j0} \\mu_{\\rho_{j0}}$ and \n$P_{j0\\neg} = P_{j0} (1 - \\mu_{\\rho_{j0}})$; likewise for \nabove the cut $\\xi_j$.\n\n}\n\n\\item{\\textbf{The Posterior Distribution.}\nBy Beta-Binomial conjugacy, on inserting data, the parameters of any given\nBeta distribution can be updated by the number of datapoints observed at a node.\nIn the Mondrian Tree, $n_0$ points are passed from $j$ to $\\lChild{j}$\n and $n_1$ points to $\\rChild{j}$.\n Hence, all of the $n_0$ points in $\\lChild{j}$ are both at most the \n cut value $\\xi_j$ and present in the bounding box $B_{\\lChild{j}}$ \n while the opposite is true for $n_1$ and $\\rChild{j}$.\n Therefore, we obtain the simple posterior update procedure:\n\n\\begin{equation} \\label{eq: mpt-posterior}\n \\chi_{jk}^* = \\chi_{jk} + n_k, \\quad\n \\rho_{j k \\in}^* = \\rho_{j k \\in} + n_k, \\quad\n \\rho_{j k \\neg}^* = \\rho_{j k \\neg}, \\quad \\mbox{for }k=0,1\n\\end{equation}\n}\n\\item{Mass in the leaves of a finite \\Polya{} Tree{} is assumed to be \ndistributed uniformly; if a point \nfalls into a leaf $j$, then the mass associated to that point is \nsimply the product of the expected Beta distributions on the path from \n$\\epsilon$ to $j$, and its density is the mass divided by the volume.}\n\n\\end{itemize}\nIt is necessary to retain the volumes of both observed and complementary regions for the restriction parameters.\nHowever, this is straightforward given the cut and volume at node $j$ (see \\Cref{sec: mpt-appendix}). \n\n\n\n\\textbf{Complexity: Instantiating the \\acrshort{mpt}.}\nThe complexity of combining the \\Polya{} Tree{} with the Mondrian Tree incurs only mild overhead.\nLet $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ denote the stored Mondrian Tree which \ngenerates the \\acrshort{mpt}.\nThe extra space necessary to use \\acrshort{mpt} for density estimation is $ 7 |\\mathsf{T}|$ due to the extra counters needed for every Beta distribution (i.e. $\\chi_{j0},\\chi_{j1},\n\\rho_{jk\\in},\\rho_{jk\\neg}, k=0,1$) and the probability mass float $P_j$. \nAt every node we must compute the volume at a cost\nof $O(D)$ which is $O(D |\\mathsf{T}|)$ over the entire tree but this can be done on-the-fly as the tree is constructed.\n\n\\textbf{\\acrshort{mpt}: Insertions and Deletions.}\nFor a \\acrshort{mpt}, we provide efficient algorithms to insert and \ndelete points over the data stream.\nA full treatment is given in \\Cref{sec: mpt-appendix}: the pertinent points\nbeing that we retain \\emph{projectivity} due to the underlying Mondrian \nTree which generates the \\acrshort{mpt}.\nDeletions require a little more work as removal points could lie on a \nbounding box, so it is necessary to check how this interacts with the\nlifetime of the stored tree.\n\n\n\n\n\n\\textbf{Example.}\nIn \\Cref{fig:mpt-example-tree} we present an instantiation of the \\acrshort{mpt}.\nObserve that the \\emph{Mondrian Tree} which is used to generate the partition\nin \\Cref{fig:mpt-example-plot} splits the entire input domain into \ndisjoint subsets of Type I\/II observed leaves and complementary regions.\nIt also implicitly encodes the associated \\acrshort{mpt} as given in \n\\Cref{fig:mpt-binary-tree}.\nThe calculations to obtain density estimates over this tree are given in \\Cref{sec:mpt-example-calcs}.\n\n \n \\subsection{Mondrian P{\\'o}lya{} Forest for Density Estimation\n \\& Anomaly Detection}\n \\label{sec: mpf-anomaly}\n Recall that an independently sampled ensemble of \n batch\/streaming Mondrian \\Polya{} Tree{s} is \n referred to as a batch\/streaming Mondrian P{\\'o}lya{} Forest\n (\\acrshort{bmpf}), (\\acrshort{smpf}),\n $F = \\cup_i T_i$.\n Each $T_i$ defines a function over its leaves $p_i({\\mathbf{x}})$\n which is a noisy estimate of the true underlying \n density function $p({\\mathbf{x}})$.\n \n \\begin{definition}[Density Estimation]\nLet $p({\\mathbf{x}})$ be a density function and suppose \n $F = \\cup_i T_i$ is a \\acrshort{bmpf} or \\acrshort{smpf}.\n Let $l$ denote the leaf in $T_i$ which contains ${\\mathbf{x}}$\n and whose mass is $P^{(i)}_l$.\nThe \\emph{density estimate} of ${\\mathbf{x}}$ in $T_i$ is \n$p_i({\\mathbf{x}}) = P_l \/ \\vol{l}$ while the density estimate over\nthe forest is $\\hat{p}({\\mathbf{x}}) = \\frac1n \\sum_{i=1}^{|F|}p_i({\\mathbf{x}})$.\n \\end{definition}\nRather than using density estimates, we adopt the following simple approach to declare anomalies while \nremaining in probability space; using simply\nthe $P^{(i)}_l$ rather than $p_i({\\mathbf{x}})$.\nThis alteration is to prevent a small number of trees from \ncorrupting the `score' if they are not good trees.\n\nThe simplicity of this approach is one of the strengths of \nour work.\nWhile previous works add an extra scoring mechanism over \nthe forest, ours is an inherent property of the underlying\nprobabilistic framework.\nWe can threshold exactly in probability space which makes \nthese `scores' more interpretable than prior work.\nSynthetic density estimation \\& anomaly detection examples\nare in \\Cref{sec: synthetic-examples}.\n\n\n\\begin{definition}[${\\varepsilon}$-anomaly \\& $({\\varepsilon},\\phi)$-anomaly]\n\\label{def: anomaly}\nLet $F = \\cup_i T_i$ be a \\acrshort{bmpf} or \\acrshort{smpf} and\n${\\varepsilon}, \\phi \\in [0,1]$.\nA point ${\\mathbf{x}} \\in \\mathbb{R}^d$ is an \\emph{${\\varepsilon}$-anomaly} in tree $T$ \nif the probability mass of the leaf in which ${\\mathbf{x}}$ is stored is at most\n${\\varepsilon}$.\nA point ${\\mathbf{x}} \\in \\mathbb{R}^d$ is an \\emph{$({\\varepsilon},\\phi)$-anomaly}\nif ${\\mathbf{x}}$ is and ${\\varepsilon}$-anomaly in at least $\\phi |F|$\ntrees from $F$.\n\\end{definition}\n\n\n\n\\section{Sampling Mondrian Trees, \\Polya{} Tree{s} and Mondrian \\Polya{} Tree{s}}\n\n\\input{tab_lay_summary}\n\nFor clarity we describe the structures necessary to introduce our Mondrian \\Polya{} Tree{s} which \nare summarised in \\Cref{tab:lay-summary}.\\footnote{\nPlease note that between paper submission and supplementary submission\nwe added \\Cref{tab:lay-summary} so the table indexing has been incremented\nby 1 from the paper originally submitted.}\nWe will begin with the \\textbf{Mondrian Process} which can be succinctly described as:\ngiven an input domain ${\\mathcal{D}}$ and a lifetime $\\lambda > 0$, choose a direction (feature)\nto cut with probability proportional to length.\nNext, choose a cut location uniformly at random on the selected feature and \nsplit into two sets less than and greater than the cut location.\nThis cut procedure has a random cost associated to the ``linear dimension''\n(sum of the lengths) of the region at a given node and the process is \nrepeated until the lifetime is exhausted by accumulating the random costs.\nAn implementation is given in \\cite{roy2008mondrian}.\n\nThe \\textbf{Mondrian Tree} builds on the Mondrian Process by building the trees in a more\ndata-aware fashion.\nAt a high-level this process is similar to the Mondrian Process except every cut\ntakes place on a restriction of space to the bounding box on which observations \nare made.\nThe advantage of this is that cuts are guaranteed to pass through observations\nwhich in high dimensions could result in substantially shortened trees.\nMondrian Trees can also be sampled online which makes them highly efficient.\nHowever, the price to pay for these efficiency gains is that behaviour outside of the \nbounding boxes cannot be modelled.\n\nWhile the previous two methods are useful for partitioning the data into clusters,\nthey make no statements about the underlying density of the dataset.\nTo accommodate this we introduce the \\textbf{\\Polya{} Tree{}} which is a Bayesian nonparametric\nmodel for estimating the underlying density function generating the data.\nThe \\Polya{} Tree{} model takes as input a binary nested partition of an input space ${\\mathcal{D}}$,\n(represented by a binary tree) and assigns probability to each of the bins (nodes in \nthe tree).\nGiven a point in a bin indexed $A_{b(j)}$, the presence of a point in the bins \n$A_{b(j)1}$ is modelled by a Bernoulli distribution with parameter $p$.\nLet $d_j$ denote the depth of $A_{b(j)}$ and \n$V_0, V_1$ denote the volumes of the the bins $A_{b(j)0}, A_{b(j)1}$, \nrespectively.\nThe prior distribution for $p$ is a Beta distribution which has parameters:\n\\begin{align}\n \\alpha_{j0} &= \\gamma \\left(d_j + 1\\right)^2 \\frac{V_0}{V_0 + V_1} \\\\\n \\alpha_{j1} &= \\gamma \\left(d_j + 1\\right)^2 \\frac{V_1}{V_0 + V_1}\n\\end{align}\nfor a hyperparameter $\\gamma > 0$ denoting the strength of the prior distribution.\nThe posterior parameters for the $\\alpha_{jk}$ are then incremented by the \nnumber of points observed in the $A_{b(j)k}$ bin for $k=0,1$.\nAn implementation is given in \\Cref{alg: polya-tree-sampling} which takes \nas input the partition of space ${\\mathcal{D}}$, thus requiring an extra pass through the tree.\nHowever, for our applications as defined in \\Cref{sec: mpf-anomaly}, \nwe will be able to implement this in an online fashion.\n\nOur \\textbf{Mondrian \\Polya{} Tree{}} can be implemented in either a batch or streaming fashion.\nFor a batch computation, we can adapt the Mondrian Process and easily combine this \nwith the \\Polya{} Tree{}.\nHowever, for streaming computation, the `empty space' caused by restricting to \nbounding boxes in the Mondrian Tree procedure is highly problematic and this \nmotivated our revised construction, the \\acrshort{mpt} as described in \n\\Cref{sec: mpt-intro}.\nWe describe this revision in \\Cref{alg: mpt} while the parameter \nupdate algorithms are presented in \\Cref{alg: mpt-parameters}.\n\n\\textbf{Generating Mondrian \\Polya{} Tree{s}: Computational Complexity.}\nCombining the \\Polya{} Tree{} with either the Mondrian Process or Mondrian Tree \nincurs only a mild overhead in both time and space as all that needs to be stored is an\nextra set of parameters.\nFor the \\textbf{batch Mondrian \\Polya{} Tree{}} (\\Cref{sec: mondrian-polya-forest}) this is simply\n3 counters per node ($\\alpha_{j0},\\alpha_{j1}, P_j$).\nIn \\Cref{sec: mpt-intro} we showed that a two-stage split was necessary for the \n\\textbf{streaming Mondrian \\Polya{} Tree{}} and this \nslightly increases the number of parameters to at most 7 per node (see \\ref{sec: mpt-params})\nwhich come from the 2 cut parameters, at most 4 restriction parameters, and the mass \nfloat $P_j$.\nOverall, both methods need $O(|\\mathsf{T}|)$ extra space which, nevertheless, is only a\nconstant factor more space than is required to build the partitioning tree.\n\nThe time cost to evaluate these parameters is $O(d | \\mathsf{T} |)$ as computing the volume \nof every node costs $O(d)$.\nSince we make the distinction between \\textbf{type I\/II observation \\& complementary\nleaves}, volume comparisons are made over nonzero $D$-dimensional hypervolumes.\nThis permits the following distinctions\nat every node to avoid incurring complex volume computations of the complementary regions.\n\n\\textbf{Volume Computation for \\acrshort{mpt}.}\nRecall that for a node $j$ we sample a cut dimension $\\delta_j$ and in that dimension \na cut location $\\xi_j$.\nThe node $j$ contains the restriction to bounding box $B_j$ which is split into two \nregions $R_{j0}$ and $R_{j1}$ either side of $\\xi_j$.\nNode $j$ has volume $V_{B_j} = \\vol{j}$\nand let $h_j$ denote the length of the sampled dimension $\\delta_j$;\nthe volumes associated \nwith $R_{j0}$ and $R_{j1}$ are:\n\\begin{align}\n V_{R_{j0}} &= \\frac{V_{B_j}}{h_j} |\\min_{{\\mathbf{x}} \\in j} {\\mathbf{x}}_{\\delta_j} - \\xi_j | \\\\\n V_{R_{j1}} &= \\frac{V_{B_j}}{h_j}|\\max_{{\\mathbf{x}} \\in j} {\\mathbf{x}}_{\\delta_j} - \\xi_j |.\n\\end{align}\n\n\n\n\n\nWe obtain the volume of the observed region when \n computing $\\Restrict{j}$ for the restriction to bounding boxes either side of \n the cut $\\xi_j$ at $j$.\n Recall that \n $V_{\\lChild{j}} = \\vol{ B_{\\lChild{j}} }$, so the subtraction \n $V_{R_{j0}} - V_{\\lChild{j}} = V_{B_{j0}^C}$ \n yields the complementary volume necessary for setting the restriction\n P{\\'o}lya{} parameters $\\rho_{\\cdot \\in}, \\rho_{\\cdot \\neg}$.\nAll volumes being supported on $D$-dimensional boxes ensures that none of these\nquantities trivially collapse to zero.\nIf one of the feature lengths is zero then we simply treat such a node as a Type I\n observation leaf.\n \n\n\n\n\n\n\n\n\n\\begin{algorithm}[htb]\n\\KwIn{Training data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$, lifetime $\\lambda > 0$ }\n\\SetKwFunction{FMain}{SampleMondrianTree}\n\\SetKwFunction{FSub}{SampleMondrianBlock}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{${\\mathbf{X}}, \\lambda$}}{\nInitialise $\\mathsf{T} = \\emptyset, \\leaves{\\mathsf{T}} = \\emptyset,\n{\\boldsymbol \\delta} = \\emptyset, {\\boldsymbol \\xi} = \\emptyset, {\\boldsymbol \\tau} = \\emptyset,\nN(\\epsilon) = \\{1,2,\\dots,n\\}.$\\\\\n \\FSub{$\\epsilon, {\\mathbf{X}}_{N(\\epsilon)}, \\lambda$} \\\\\n}\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FSub{$j, {\\mathbf{X}}_{N(j)}, \\lambda$}}{\n $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\} $ \\\\\n For all $d \\in [D]$ set\n ${\\mathbf{l}}_{jd}^{{\\mathbf{X}}} = \\min_d {\\mathbf{X}}_{N(j)},{\\mathbf{u}}_{jd}^{{\\mathbf{X}}} = \\max_d {\\mathbf{X}}_{N(j)}$\n to be the dimension-wise minima and maxima of the observations in $j$ \\\\\n Let $L = \\sum_d (u_{jd}^{{\\mathbf{X}}} - l_{jd}^{{\\mathbf{X}}})$ denote the\n \\emph{linear dimension} of the data in $j$ \\\\\n Sample $E \\sim \\ExpDist{L}$ \\\\\n \\uIf{$\\tau_{\\parent{j}} + E < \\lambda$}{\n Set $\\tau_j = \\tau_{\\parent{j}} + E$ \\\\\n Sample cut dimension $\\delta_j$ with probability proportional to\n $u_{jd}^{{\\mathbf{X}}} - l_{jd}^{{\\mathbf{X}}}$ \\\\\n Sample cut location uniformly on the interval\n $[ l_{j \\delta_j }^{{\\mathbf{X}}}, u_{j \\delta_j}^{{\\mathbf{X}}}]$ \\\\\n\n Set $N(\\lChild{j}) = \\{ n \\in N(j) : X_{n \\delta_j} \\le \\xi_j \\}$ and\n $N(\\rChild{j}) = \\{ n \\in N(j) : X_{n \\delta_j} > \\xi_j \\}$ \\\\\n\n \\FSub{$\\lChild{j}, {\\mathbf{X}}_{N(\\lChild{j})}, \\lambda$}\\\\\n \\FSub{$\\rChild{j}, {\\mathbf{X}}_{N(\\rChild{j})}, \\lambda$}\n }\n \\Else{\n $\\tau_j \\leftarrow \\lambda$ and\n $\\leaves{\\mathsf{T}} \\leftarrow \\leaves{\\mathsf{T}} \\cup \\{j\\}$\n }\n}\n \\caption{Mondrian Forest Sampling \\cite{lakshminarayanan2014mondrian}}\n \\label{alg: mondrian-forest}\n\\end{algorithm}\n\n\\begin{algorithm}[htb]\n\\KwIn{Input domain ${\\mathcal{D}} \\subset \\mathbb{R}^{D}$,\n a decision tree $T$ which partitions ${\\mathcal{D}}$,\n hyperparameter $\\gamma > 0$}\n\\KwOut{Probability distribution ${\\mathcal{P}} = (P_l)_{l \\in \\leaves{T}}$}\n\\SetKwFunction{FMain}{SampleP{\\'o}lya{}Tree}\n\\SetKwFunction{FSub}{UpdateP{\\'o}lya{}Parameters}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{${\\mathcal{D}}, T, \\gamma$}}{\n$\\epsilon = \\text{root}\\left( T \\right)$ \\\\\n$P_{\\epsilon} = 1$ \\Comment{Assume all mass is located in the region ${\\mathcal{D}}$}\\\\\n \\FSub{$\\epsilon, \\gamma$} \\\\\n }\n\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$j, \\gamma$}}{\n \\uIf{$j \\in \\leaves{T}$}{\n $\\text{ProbDensity}(j) = P_j \/ \\vol{j}$\n \\Comment{$P_j$ was defined at the preceeding level.}\n }\n \\uElse{\n $V_j = \\vol{j}$ \\\\\n Let $R_j = [l_1,u_1] \\times \\dots \\times [l_d,u_d]$ define the region in\n as a product of intervals from the minimum in dimension $i$, $l_i$, to\n the maximum in dimension $i$, $u_i$. \\\\\n $L_j = \\sum_{j=1}^d (u_j - l_j)$ is the linear dimension of the region.\\\\\n $V_0 = \\frac{V_j}{u_{\\delta_j} - l_{\\delta_j}} \\cdot | l_{\\delta_j} - \\xi_j |$\n \\Comment{Volume less than cut $\\xi_j$}\\\\\n $V_1 = \\frac{V_j}{u_{\\delta_j} - l_{\\delta_j}} \\cdot | u_{\\delta_j} - \\xi_j |$\n \\Comment{Volume greater than cut $\\xi_j$}\\\\\n $n_0 = \\text{NumberOfPoints}(\\lChild{j}),\n n_1 = \\text{NumberOfPoints}(\\rChild{j})$\n \\Comment{Number of points in children nodes}\\\\\n $\\alpha_0 = \\gamma (d+1)^2 \\frac{V_0}{V_0 + V_1} + n_0,\n \\alpha_1 = \\gamma (d+1)^2 \\frac{V_1}{V_0 + V_1} + n_1$\n \\Comment{Set prior parameters using \\Polya{} Tree~ and then increment using\n Beta-Binomial conjugacy}\\\\\n $\\mu_j = \\frac{\\alpha_0}{\\alpha_0 + \\alpha_1}$\n \\Comment{$\\mathbb{E}(\\BetaDist{\\alpha_0}{\\alpha_1})$}\\\\\n $P_{\\lChild{j}} = \\mu_j P_j, P_{\\rChild{j}} = (1 - \\mu_j) P_j$\\\\\n \\FSub{$\\lChild{j}, \\gamma$}, \\FSub{$\\rChild{j}, \\gamma$}\n }\n}\n \\caption{\\Polya{} Tree{} Sampling. Sets probability (density) for all nodes in the\n given random partition $T$.}\n \\label{alg: polya-tree-sampling}\n\\end{algorithm}\n\n\\begin{algorithm}[p]\n \\KwIn{Training data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$, at least one of lifetime\n $\\lambda > 0$ or maximum tree height $m$, \\Polya{} Tree{} hyperparameter $\\gamma > 0$}\n \\KwOut{A classical Mondrian Tree data structure $T$;\n Partition $\\Pi$ over $T$ such that\n ${\\mathcal{P}} = (P_l)_{l \\in \\leaves{T}}$\n is a probability distribution over the leaves of $T$ induced by \\Polya{} Tree{} prior}\n \\SetKwFunction{FMain}{SampleMondrianP{\\'o}lya{}Tree}\n \\SetKwFunction{FSub}{SampleMondrianP{\\'o}lya{}Block}\n \\SetKwFunction{FSubCut}{SetCutParameters}\n \\SetKwFunction{FSubObs}{SetRestrictionParameters}\n\\SetKwFunction{FRestrict}{Restrict}\n\\SetKwFunction{FCut}{Cut}\n \\SetKwProg{Fn}{Function}{:}{}\n \\Fn{\\FMain{${\\mathbf{X}}, \\lambda$}}{\n Initialise $\\mathsf{T} = \\emptyset, \\leaves{\\mathsf{T}} = \\emptyset,\n {\\boldsymbol \\delta} = \\emptyset, {\\boldsymbol \\xi} = \\emptyset, {\\boldsymbol \\tau} = \\emptyset,\n N(\\epsilon) = \\{1,2,\\dots,n\\}.$\\\\\n $\\epsilon.\\text{ObservedVolume} = \\vol{\\bbox{{\\mathbf{X}}_{N(\\epsilon)}}}$ \\\\\n \\FSub{$\\epsilon, {\\mathbf{X}}_{N(\\epsilon)}, \\lambda$} \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$j, {\\mathbf{X}}_{N(j)}, \\lambda$}}{\n $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\} $ \\\\\n $B_j \\leftarrow \\bbox{ {\\mathbf{X}}_{N(j)}}, L = \\LinDim{B_j}$\n \\Comment{\\emph{linear dimension} of the bounding box for $j$ }\\\\\n \n \n Sample $E \\sim \\ExpDist{L}$ \\\\\n \\uIf{$\\tau_{\\parent{j}} + E < \\lambda \\text{~and all feature lengths are positive}$}{\n Set $\\tau_j = \\tau_{\\parent{j}} + E$ \\\\\n \\FCut{$j, {\\mathbf{X}}_{N(j)}, B_j$} \\\\\n\n Set $N(\\lChild{j}) = \\{ n \\in N(j) : {\\mathbf{X}}_{n \\delta_j} \\le \\xi_j \\}$ and\n $N(\\rChild{j}) = \\{ n \\in N(j) : {\\mathbf{X}}_{n \\delta_j} > \\xi_j \\}$ \\\\\n\n \\FRestrict{$j, N(\\lChild{j})$} \\\\\n \\FRestrict{$j, N(\\rChild{j})$}\n\n \\FSub{$\\lChild{j}, {\\mathbf{X}}_{N(\\lChild{j})}, \\lambda$}\\\\\n \\FSub{$\\rChild{j}, {\\mathbf{X}}_{N(\\rChild{j})}, \\lambda$}\n }\n \\Else{\n \\uIf{any feature length is 0}{\n $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\}$\n \\Comment{Bounding box supported on $ < d$ dimensions: Type I Observed leaf}\n }\n \\Else{\n \\FRestrict{$j, N(j)$}\n \\Comment{Restrict once more to generate a \\emph{complementary leaf}} \\\\\n }\n $\\tau_j \\leftarrow \\lambda$ and\n $\\leaves{\\mathsf{T}} \\leftarrow \\leaves{\\mathsf{T}} \\cup \\{j\\}$\n \\Comment{Type II Observed leaf (see \\Cref{sec: mpt-intro})}\n }\n }\n\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FCut{$j, {\\mathbf{X}}_{N(j)}, B_j$}}{\n Sample cut dimension $\\delta_j$ with probability proportional to\n $u_{jd}^{{\\mathbf{X}}} - l_{jd}^{{\\mathbf{X}}}$ \\\\\n Sample cut location $\\xi_j$ uniformly on the interval\n $[ l_{j \\delta_j }^{{\\mathbf{X}}}, u_{j \\delta_j}^{{\\mathbf{X}}}]$ \\\\\n ${R_{\\textsf{left}}} = \\{{\\mathbf{z}} \\in B_j : {\\mathbf{z}}_{\\delta_j} \\le \\xi_j\\}$ \\\\\n ${R_{\\textsf{right}}} = \\{{\\mathbf{z}} \\in B_j : {\\mathbf{z}}_{\\delta_j} > \\xi_j\\}$ \\\\\n $n_{\\textsf{left}} = |{\\mathbf{X}}_{N(j)} \\cap {R_{\\textsf{left}}}|$ \\\\\n $n_{\\textsf{right}} = |{\\mathbf{X}}_{N(j)} \\cap {R_{\\textsf{right}}}|$ \\\\\n $V_{\\textsf{left}} = \\vol{{R_{\\textsf{left}}}},\n V_{\\textsf{right}} = \\vol{{R_{\\textsf{right}}}}$ \\\\\n $d_j = \\TreeDepth{j}$ \\Comment{Absolute depth in Mondrian Tree} \\\\\n \\FSubCut{$2d_j, n_{\\textsf{left}}, n_{\\textsf{right}},V_{\\textsf{left}}, V_{\\textsf{right}}$}\n}\n\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FRestrict{$j, {\\mathbf{X}}_{N(j)}$}}{\n $d = j.\\text{depth}$\n \\Comment{Get absolute depth in Mondrian tree} \\\\\n $n_{\\textsf{obs}} = |N(j)|$ \\Comment{Num. points in node} \\\\\n $V_p = \\parent{j}.\\text{ObservedVolume}$\n \\Comment{Parent volume} \\\\\n $V_o = \\vol{B({\\mathbf{X}}_{N(j)})}$\n \\Comment{Observed volume} \\\\\n $V_c = V_p - V_o$\n \\Comment{Complementary volume} \\\\\n $\\rho_0^*, \\rho_1^*$ = \\FSubObs{$2d+1, n_{\\textsf{obs}}, V_o, V_c$}\n \\Comment{Set Beta parameters.}\\\\\n \\Return{$\\rho_0^*, \\rho_1^*$}\n}\n\n \\caption{Mondrian \\Polya{} Tree{} Sampling. Subroutines:\n \\Cref{alg: mpt-parameters}}\n \\label{alg: mpt}\n \\end{algorithm}\n \n \\begin{algorithm}[htb]\n \\SetKwFunction{FSubCut}{SetCutParameters}\n \\SetKwFunction{FSubObs}{SetRestrictionParameters}\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSubCut{$\\text{depth}, n_{\\textsf{left}}, n_{\\textsf{right}},\n V_{\\textsf{left}}, V_{\\textsf{right}}$}}{\n $d = \\text{depth}$ \\\\\n $j.\\chi_0^* = \\gamma (d + 1)^2\n \\frac{ V_{\\textsf{left}} }{ V_{\\textsf{left}} + V_{\\textsf{right}}} + n_{\\textsf{left}},\n j.\\chi_1^* = \\gamma (d + 1)^2\n \\frac{ V_{\\textsf{right}} }{ V_{\\textsf{left}} + V_{\\textsf{right}}} + n_{\\textsf{right}}$ \\\\\n }\n %\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSubObs{$\\text{depth},\\text{NodeSize}, \\text{ObservedVolume}, \\text{ComplemetaryVolume}$}}{\n $d = \\text{depth}; \\qquad$ \n $n = \\text{NodeSize}$ \\Comment{Number of points in observed bounding box} \\\\\n $V_{obs} = \\text{ObservedVolume}, V_{comp} = \\text{ComplemetaryVolume}$ \\\\\n $\\rho_0^* = \\gamma (d+1)^2 \\frac{V_{obs}}{V_{obs} + V_{comp}} + n$,\n $\\rho_1^* = \\gamma (d+1)^2 \\frac{V_{comp}}{V_{obs} + V_{comp}}$ \\\\\n \\Return{$\\rho_0^*, \\rho_1^*$}\n }\n \\caption{Subroutines for setting Beta Distribution parameters for the\n Mondrian \\Polya{} Tree.\n Note that the depth parameters in these subroutines refer to depth in \\Polya{} Tree, not absolute\n depth in Mondrian Tree!}\n \\label{alg: mpt-parameters}\n \\end{algorithm}\n\\newpage\n\\section{\\acrshort{mpt}: Insertions and Deletions} \n\\label{sec: mpt-appendix}\n\nA substantial benefit of the Mondrian Tree construction is that it can \nbe built online as new data is seen.\nThe key idea underpinning this is\n\\emph{projectivity} (\\Cref{lem:projectivity} \\Cref{sec: preliminary}), \nwhich asserts that if a Mondrian tree \n$T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ is sampled and a new point ${\\mathbf{z}}$ is \nobserved, then inserting ${\\mathbf{z}}$ into $T$ to generate $T'$ yields\n$T' \\sim \\MTree{{\\mathbf{X}} \\cup {\\mathbf{z}}}{\\lambda}$ \\cite{lakshminarayanan2014mondrian}; \nmoreover, this process is efficient.\nThis is where the restriction of a cut $\\xi_j$ to the \nbounding box $B_j$ is critical, because it permits the sequential\naddition of ${\\mathbf{z}}$ into $T$ while preserving the distribution over which \n$T$ was sampled had ${\\mathbf{z}}$ been seen prior to sampling $T$!\nWe adapt the online update procedures from Mondrian Trees\nto streaming Mondrian \\Polya{} Tree{s} by invoking projectivity and then recognising that \nthe necessary parameters can be easily incremented as the \ndata is observed.\nInserting a point ${\\mathbf{z}}$ into tree $T$ is denoted \n$T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$.\n\n\n\nHowever, for data streams we also need the capability to delete from the tree;\nthis is where the link with the RRCF{} work becomes \nnecessary, as we can adapt their deletion mechanism for the \nMondrian \\Polya{} Tree{} setting.\nOur alteration is necessary for the Mondrian Tree setting as\nnodes have an associated time which cannot\nexceed the lifetime budget $\\lambda$ and deleting a point on the bounding box\ncan affect the times of all nodes in the subtree rooted at that node.\nIn this setting, the point to delete, ${\\mathbf{z}}$, is chosen\nahead of time, hence the algorithm is deterministic which is why we will\nwrite $T' = \\MPTreeDelete{T}{{\\mathbf{z}}}$ (in contrast to \n$T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$) for deleting ${\\mathbf{z}}$ from $T$.\nThe following lemmas summarise the insertion and deletion \nprocedures from \\cite{lakshminarayanan2014mondrian} and \\cite{guha2016robust}\nto account for the additional \\Polya{} Tree{} parameters that we need\nwhen using the Mondrian \\Polya{} Tree.\nThe insertions procedure is described in \\Cref{alg: mpt-insertion}, \nand \\Cref{alg: mpt-deletion} illustrates the deletion mechanism.\n\n\n\n\n\n\n\\begin{lemma}[Insertions] \\label{lem:mpt-insertion}\nLet $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$ be a Mondrian \\Polya{} Tree sampled over \ndata ${\\mathbf{X}}$ with lifetime $\\lambda > 0$.\nIf ${\\mathbf{z}}$ is a new observation and $T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$ then \n$T' \\sim \\MondPolyaTree{{\\mathbf{X}} \\cup {\\mathbf{z}}}{\\lambda}$.\n\\end{lemma}\n\n\\begin{proof} \nThe tree that we sample and store is exactly a Mondrian Tree, hence we \ninvoke projectivity so that $T'$ is a valid Mondrian Tree over \n${\\mathbf{X}} \\cup {\\mathbf{z}}$.\nSince the Mondrian Tree $T$ implicitly but uniquely defines a Mondrian \\Polya{} Tree{}\nwhich partitions the input space, projectivity also applies to the \nMondrian \\Polya{} Tree{} structure as a random partition.\nAdditionally, we need to alter the (cut and restrict) Beta parameters for\nevery node which are affected by the insertion of ${\\mathbf{z}}$ in tree $T$. \nHowever, this amounts to simply incrementing counts over the subtree: \nupdating the parameters is sufficient as we only need the expected value\nof every Beta distribution.\n\n\n\\end{proof}\n\n\\begin{lemma}[Deletions] \\label{lem:mpt-deletion}\nLet $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$ and \nlet ${\\mathbf{z}}$ be the point to be removed from ${\\mathbf{X}}$ and $T$.\nIf $T' = \\MPTreeDelete{T}{{\\mathbf{z}}}$ then \n$T' \\sim \\MondPolyaTree{{\\mathbf{X}} \\setminus {\\mathbf{z}}}{\\lambda}$.\n\\end{lemma}\n\n\\begin{proof}\n\nFirst, locate the deepest node $j$ containing ${\\mathbf{z}}$, there are two cases:\n(i) ${\\mathbf{z}}$ is \\emph{internal} to the box $B_j$\n(ii) ${\\mathbf{z}}$ is a \\emph{boundary point} defining part of the bounding box\n$B_j$ (i.e. it is maximal or minimal at $j$ in one dimension).\nIf ${\\mathbf{z}}$ is internal to $B_j$ then we are free to simply remove it from \n$j$ and decrement the necessary counts.\nOtherwise, deleting ${\\mathbf{z}}$ causes a change to the bounding box: let \n$B_j'$ denote the new bounding box for $j$ under the removal of ${\\mathbf{z}}$.\nNow, it must be the case that $L_j' = \\LinDim{B_j'}$ is \\emph{at most}\n$L_j = \\LinDim{B_j}$.\nHowever, if there is a ${\\mathbf{u}} \\ne {\\mathbf{z}}$ \nin $j$ but is equal to ${\\mathbf{z}}$ in \\emph{all} dimensions on which ${\\mathbf{z}}$ lies \non the boundary, then we could treat ${\\mathbf{z}}$ as an internal point and \nremove then decrement.\nSo assume ${\\mathbf{z}}$ uniquely defines $B_j$ in the required dimensions, hence\n$L_j' < L_j$ so the exponential distribution used to generate the \nnode time $\\tau_j$ is different under the absence of ${\\mathbf{z}}$.\nLet $F(t) = \\cdf{\\ExpDist{L_j}}(t)$ and $G(t) = \\cdf{\\ExpDist{L_j'}}(t)$ be the\nCDF functions of the exponential distributions $\\ExpDist{L_j}$ and \n$\\ExpDist{L_j'}$, respectively as functions of time $t$.\nThe mass associated to time $\\tau_j$ is $\\psi = F(\\tau_j)$ hence, the time \nwith the same mass in $G(t)$ is\n$\\tau_j' = G^{-1}(\\psi)$ (these are straightforward for the exponential\ndistribution since $\\cdf{\\ExpDist{\\zeta}}(t) = 1 - \\exp(-\\zeta t)$).\nFinally, since $L' < L$, we must have $\\tau_j' > \\tau_j$ so the time \nhas increased, meaning we must check whether $\\tau_j' < \\lambda$.\nIf so, then keep $j$, else contract $j$ and its descendants into $\\parent{j}$.\nThis approach must be done for every node on the path from $\\epsilon$ to \n $j$ which contains ${\\mathbf{z}}$ so in the worst case is \n $O(d \\cdot \\TreeDepth{T})$.\n Finally, it remains to decrement all necessary counts which were affected \n by the presence of ${\\mathbf{z}}$ on the path from $\\epsilon$ to $j$ \n (or the contracted ancestor of $j$).\n\\end{proof}\n\n \\textbf{Complexity: Insertions \\& Deletions}\n Both procedures are efficient and are dominated by the time it takes to locate\nthe locate the node which stores query point and requires checking inclusion in a\nbounding box at $O(D)$ cost a maximum of $\\TreeDepth{T}$ times, hence $O(D \\TreeDepth{T})$ overall.\n Note that this is the absolute depth measured in the Mondrian\n Tree sense, not the adjusted depth to account for the \\Polya{} Tree{}\n construction as defined prior to \\Cref{eq: mpt-prior},\n nor the lifetime $\\lambda$ which could potentially be large.\n Since we only store the Mondrian Tree which generates the Mondrian \\Polya{} Tree{}\n which, in expectation, should be balanced and hence \n $\\TreeDepth{T} = \\Theta(\\log n)$.\n In the random forest literature (\\cite{liu2008isolation}, \n \\cite{guha2016robust}, \\cite{gopalan2019pidforest}), the depth is \n typically a parameter of small magnitude relative to \n the size of input data, usually 10.\n Hence, the presence of the maximum tree depth term in the above\n time complexity bounds is not problematic.\n \n\n\n\n\n\\begin{algorithm}[h]\n \\KwIn{Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$}\n \\KwOut{Mondrian Tree $T$ sampled over ${\\mathbf{X}} \\cup {\\mathbf{z}}$}\n\\SetKwFunction{FMain}{$\\textsf{MT}_+$}\n\\SetKwFunction{FSub}{MTx}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}$}}{\n$\\epsilon = \\textsf{root}(T)$ \\\\ \n \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\epsilon$} \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, j$}}{\n ${\\mathbf{e}}^l = \\max(l_j^{{\\mathbf{X}}} - {\\mathbf{z}},0), {\\mathbf{e}}^u = \\max({\\mathbf{z}} - u_j^{{\\mathbf{X}}},0)$ \n \\Comment{${\\mathbf{e}}^l, {\\mathbf{e}}^u = {\\mathbf{0}}_d$ iff $z \\in B_j$}\\\\\n Increment the \\emph{observed restriction} parameter $\\rho_0$ by 1 \\\\ \n Sample $E \\sim \\ExpDist{ \\sum_{i=1}^d \\left({\\mathbf{e}}^u + {\\mathbf{e}}^l\\right)_i}$ \\\\ \n \\uIf{$\\tau_{\\parent{j}} + E< \\tau_j$}{\n Sample $\\delta$ with probability proportional to ${\\mathbf{e}}^u + {\\mathbf{e}}^l$ \\\\\n Sample a cut $\\chi \\sim \\textsf{Uniform}\\left(a,b\\right)$ with \n $a = u_{j\\delta}^{{\\mathbf{X}}}, b = {\\mathbf{z}}_{\\delta}$ if ${\\mathbf{z}}_{\\delta} > u_{j\\delta}$,\n else $a = {\\mathbf{z}}_{\\delta}, b = l_{j\\delta}^{{\\mathbf{X}}}$ \\\\\n Insert $j'$ ($j$ but below $\\parent{j}$) to $j$ with: $N(j') = N(j) \\cup \\{{\\mathbf{z}}\\}$,\n $\\delta_{j'} = \\delta, \\xi_{j'} = \\chi, \\tau_{j'} = \\tau_{\\parent{j}} + E,\n l_{j'}^{{\\mathbf{X}}} = \\min (l_j^{{\\mathbf{X}}}, {\\mathbf{z}}),\n u_{j'}^{{\\mathbf{X}}} = \\max (u_j^{{\\mathbf{X}}}, {\\mathbf{z}})$ \\\\\n Insert sibling $\\sib{j}$ containing ${\\mathbf{z}}$ such that \n $\\lChild{j'} = j, \\rChild{j'} = \\sib{j}$ if ${\\mathbf{z}}_{\\delta_j} > \\xi_j$ or \n $\\rChild{j'} = j, \\lChild{j'} = \\sib{j}$, otherwise \\\\ \n Set the Beta parameters according to number of points either side of $\\xi_j$\n }\n \\uElse{\n Update $l_{j}^{{\\mathbf{X}}} \\leftarrow \\min (l_j^{{\\mathbf{X}}}, {\\mathbf{z}}),\n u_{j'}^{{\\mathbf{X}}} \\leftarrow \\max (u_j^{{\\mathbf{X}}}, {\\mathbf{z}})$ \\\\ \n \\uIf{$j \\in \\leaves{\\mathsf{T}}$}{\n \\Return{}\n }\n \\uElse{\n \\uIf{${\\mathbf{z}}_{\\delta_j} < \\xi_j$}{\n $\\Child{j} = \\lChild{j}$\n }\n \\uElse{\n $\\Child{j} = \\rChild{j}$\n }\n Increment $\\chi_{\\Child{j}}$ by 1 \\\\ \n \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\Child{j}$}\n }\n }\n }\n \\caption{ Mondrian \\Polya{} Tree{} Insertion: $\\MPTreeInsert{T}{{\\mathbf{z}}}$ }\n \\label{alg: mpt-insertion}\n\\end{algorithm}\n \n\\begin{algorithm}[!htb]\n \\KwIn{Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$}\n \\KwOut{Mondrian Tree $T$ sampled over ${\\mathbf{X}} \\setminus {\\mathbf{z}}$}\n\\SetKwFunction{FMain}{$\\textsf{MT}_-$}\n\\SetKwFunction{FSub}{MTd}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}$}}{\n$\\epsilon = \\textsf{root}(T), \\textsf{path} = \\{ \\epsilon \\} $\\\\\n \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\epsilon$} \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, j, \\textsf{path}$}}{\n Find the deepest node $j$ containing ${\\mathbf{z}}$ \\\\ \n Let $\\textsf{path} = \\{\\epsilon, u_1, u_2, \\dots, u_k\\}$ be the set of nodes from $\\epsilon$ to $j$ \\\\ \n \\For{$j \\in \\textsf{path}$}{\n Check if ${\\mathbf{z}}$ is \\emph{internal} to the bounding box, or a point which defines\n the bounding box in one of dimensions $i \\in \\{1,2,\\dots,d\\}$ \\\\ \n \\uIf{${\\mathbf{z}}$ is \\emph{internal}}{\n $\\Child{j} = \\lChild{j}$ iff ${\\mathbf{z}}_{\\delta_j} \\le \\xi_{j}$\n Decrement the cut and restriction counter corresponding to ${\\mathbf{z}}$ at $v$ by 1 \\\\\n Decrement all counts by 1 at the subtree rooted at $\\Child{j}$ \n \\Comment{This only decrements on the side of the cut that ${\\mathbf{z}}$ should go.}\\\\\n \\Return\n }\n \\uElse{\n Let $B_j'$ be the bounding box at $j$ with ${\\mathbf{z}}$ ignored which has linear \n dimension $L' = \\LinDim{B_j'}$ \\\\\n Evaluate new node time $\\tau_j'$ through inverse CDF \\\\\n \\uIf{$\\tau_j' \\ge \\lambda$}{\n Contract the entire subtree rooted at $j$ into $j$\\\\\n Set $\\tau_j' = \\lambda$ \\\\ \n \\Return\n }\n }\n }\n \\Return{$T$}\n }\n \\caption{Inplace deletions for the Mondrian \\Polya{} Tree{}, $\\MPTreeDelete{T}{{\\mathbf{z}}}$.}\n \\label{alg: mpt-deletion}\n\\end{algorithm}\n\n\\newpage\n\\newpage\n\n\\section{Calculations for \\Cref{fig:mpt-example-tree}} \\label{sec:mpt-example-calcs}\n\nLet us consider the generative process for sampling a \n\\emph{streaming Mondrian \\Polya{} Tree} (\\acrshort{mpt}) to clarify the interplay between the underlying\nMondrian and \\Polya{} Tree{s}.\nLet ${\\mathbf{X}} = \\{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3,{\\mathbf{x}}_4\\}$ with ${\\mathbf{x}}_1 = (0,0), {\\mathbf{x}}_2=(1\/4,1\/4),\n{\\mathbf{x}}_3=(2\/5,4\/5),{\\mathbf{x}}_4=(1,1)$.\nSet ${\\mathcal{D}} = \\bbox{{\\mathbf{X}}}$ to be the bounding box of the region containing ${\\mathbf{X}}$ and denote \nthe two directions which span $\\mathbb{R}^2$ be $x$ and $y$.\nWe show how sampling a depth 2 \\emph{Mondrian Tree}\nencodes a depth 4 \\acrshort{mpt} which can be used to estimate the \ndensity over ${\\mathcal{D}}$.\nNote that the only tree we store is the Mondrian Tree (with\nlifetime $\\lambda = \\infty$), albeit with the extra parameters necessary\nfor \\Polya{} Tree{} density estimation.\nRecall that the root of the tree is the node $\\epsilon$ which has an\nempty index bitstring $b(\\epsilon) = \\emptyset$ so it can be ignored\nfrom node\/parameter index strings.\nThe following example is illustrated in \\Cref{fig:mpt-example-tree}.\n\nSuppose the prior strength hyperparameter is $\\gamma=1$\nso it can be ignored from the \\Polya{} Tree{} calculations.\nThe first cut, $\\xi_{\\epsilon}$ occurs at $x = 0.5$ and traverses the\nentire bounding box ${\\mathcal{D}}$ in direction $x$.\nThis splits ${\\mathcal{D}}$ into two regions \n$R_{ 0 } \\supset \\{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3\\}$\nand \n$R_{ 1} \\supset {\\mathbf{x}}_4$,\neach with volume $V_0,V_1 = 0.5$.\nHence, the posterior cut parameters are $\\chi_0^* = 7\/2$ \\& $\\chi_1^* = 3\/2$\nwhich results in $\\mu_{\\chi_{\\epsilon}} = 7\/10$.\n\nNext, call $\\Restrict{\\epsilon}$ which\ncomputes $\\Restrict{R_{0}}$\n\\& $\\Restrict{R_{ 1}}$ (see \\Cref{alg: mpt}).\nSince ${\\mathbf{x}}_4$ is isolated in $R_{ 1}$, the bounding box containing ${\\mathbf{x}}_4$ is supported on only 1 dimension;\n$\\Restrict{R_{ 1}} = (R_{1}, \\emptyset)$ and the node \nstoring ${\\mathbf{x}}_4$ is a \\emph{Type I observation leaf} with volume $1\/2$.\nHence, the \\emph{mass} associated to this leaf is \n$P_{ 1} = 3\/10$ and dividing\nout the volume of the leaf yields the \\emph{density} as $6\/10$.\n\nLet $B_{ 0} = \\bbox{R_{ 0}}$ be the bounding box\ncontaining the points ${\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3$ from the region $R_{ 0}$.\nThen, $\\Restrict{R_{ 0}} = (B_{ 0}, B_{ 0}^C)$ which have volumes: \n$V_{B_{0}}= 32\/100$ and $ V_{B_{ 0}^C} = 18\/100$. \nThe \\text{P{\\'o}lya} depth of this node is now 1 \n(use restriction parameters \\eqref{eq: mpt-prior} with $d_{\\epsilon} = 0$)\nso the\n(posterior) parameters for the split at this level are, for inclusion in $B_{0}$\n(encoded with a $\\in$) and exclusion (encoded with a $\\neg$), respectively:\n\\begin{equation}\n \\rho_{0\\in}^* = 2^2 \\cdot \\nicefrac{32\/100}{50\/100} + 3 \n \\qquad\n \\text{and}\n \\qquad\n \\rho_{0\\neg}^* = 2^2 \\cdot \\nicefrac{18\/100}{50\/100}.\n\\end{equation}\nAccordingly, we obtain $\\mu_{\\rho_{0}} = 36\/175$, `generate' an\n\\emph{internal node} with bitstring $0\\in$ and a \n\\emph{complementary leaf} with bitstring $0 \\neg$.\nNote that neither of these nodes is ever materialised as they \nare wholly defined by the node with index $b(j) = 0$ in the \nMondrian Tree.\nThe masses allocated are $\\mu_{\\chi_0} \\mu_{\\rho_{0}}$ \nfor the node $0 \\in$ \\& $\\mu_{\\chi_0} (1 - \\mu_{\\rho_{0}})$\nfor the node $0 \\neg$.\n\nSince we have fixed a maximum depth of 2 for the Mondrian \nTree, we perform one subsequent cut, $\\xi_{ 0}$, to \nseparate $\\{{\\mathbf{x}}_1,{\\mathbf{x}}_2\\}$ from $\\{{\\mathbf{x}}_3\\}$, and perform a final \nrestriction procedure.\nHence, we have used the Mondrian Tree to correctly define\na \\Polya{} Tree{} over the partition of ${\\mathcal{D}}$.\nFurther details and calculations\ncan be found in \\Cref{sec:mpt-example-calcs}.\n\nNext, we deal with the points ${\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3$ and the region $R_0$\ngenerated left of the cut $\\xi_{\\epsilon}$ by performing the first $\\Restrict{\\epsilon}$ step.\nNote that $\\Restrict{\\epsilon}$ separately computes the restriction to \nbounding boxes either side of $\\xi_{\\epsilon}$ noting that \n$\\Restrict{R_0} = (B_0, B_0^C)$ and $\\Restrict{R_1} = (R_1, \\emptyset$).\nRecall that $R_1$ contains a bounding box supported only on one dimension\nso we set this to be a Type I observation leaf so the restriction simply\nreturns the set $R_1$.\nOn the other hand, consider $R_{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3}$ which has a\nvolume of 1\/2 and is decomposed into the subregions $B_0$ and\n$R_{0} \\setminus B_0 = B_0^C$ (here the $B_0^C$\nnotation denotes the set complement in the universe $R_0$).\nWe thus obtain $V_{0\\in} = \\vol{B_0} = 32\/100$ and \n$V_{0\\neg} = 18\/100$.\nThe \\text{P{\\'o}lya} depth of this node is now 1 so the\n(posterior) parameters for the split at this level are, for inclusion in $B_0$\n(encoded with a $\\in$) and exclusion (encoded with a $\\neg$), respectively:\n\\begin{align*}\n \\rho_{0\\in}^* &= (1+1)^2 \\cdot \\frac{32\/100}{50\/100} + 3 \\\\\n \\rho_{0\\neg}^* &= (1+1)^2 \\cdot \\frac{18\/100}{50\/100}.\n\\end{align*}\nOverall, this results in $\\mu_{\\rho_{0}} = 36\/175$, the internal node \nwhose bitstring is $0\\in$ and a \\emph{complementary leaf} encoded by \n$0 \\neg$.\nThe mass assigned to each of these nodes is \n$\\mu_{\\chi_0} \\mu_{\\rho_{0}}$ and \n$\\mu_{\\chi_0} (1 - \\mu_{\\rho_{0}})$, respectively.\n\nFollowing this restriction, we complete one more cut: $\\xi_{0\\in}$\nat $y=0.4$ defined only on the box $B_0$ and generate\nthe two regions $R_{00}, R_{01}$.\nSince $|R_{01} \\cap {\\mathbf{X}}| = 1$, we terminate the process and treat this leaf as an observed leaf of type I with mass \n$\\mu_{\\chi_0} \\mu_{\\rho_{0}} (1 - \\mu_{\\chi_{0\\in}})$.\nOn the other hand, $|R_{00} \\cap {\\mathbf{X}}| = 2$ so we again perform\n$\\Restrict{R_{0\\in0}} = (B_{00}, B_{00}^C)$\nwhich returns the bounding box $B_{00} = \\bbox{ {{\\mathbf{x}}_1, {\\mathbf{x}}_2} }$\nin an observation leaf of type II, along with its complementary region\nwhich is added to the set of complementary leaves.\nAt this point we terminate the process, so there are 5 leaves generated which partition the entire input domain as defined by the input data.\n\n\\paragraph{Numerics.}\nGiven data ${\\mathbf{X}}$ and the cuts $\\xi_{\\epsilon}, \\xi_{0}$ the following \nquantities are used to evaluate the density in each of the 5 leaves:\n\\begin{itemize}\n \\item{Cut at root node P{\\'o}lya{} depth = 0: \n $\\chi_0^* = \\frac{1\/2}{1\/2}+3$ and \n $\\chi_0^* = \\frac{1\/2}{1\/2}+1$ so that \n $\\mu_{\\chi_{\\epsilon}} = 7\/10$ and nodes $0,1$ are created.\n They are internal and observation leaf Type I, respectively.}\n \n \\item{Restrict at node $0$, P{\\'o}lya{} depth = 1:\n $ \\rho_{0\\in}^* = (1+1)^2 \\cdot \\frac{32\/100}{50\/100} + 3$,\n $ \\rho_{0\\neg}^* = (1+1)^2 \\cdot \\frac{18\/100}{50\/100}$ so that \n $ \\mu_{\\rho_{0}} = 36\/175$.\n Internal node $0\\in$ and complementary leaf $0\\neg$ are created.}\n \n \\item{Cut at node $0\\in$, P{\\'o}lya{} depth = 2:\n $\\chi_{0\\in0}^* = 9 \\frac{16}{32} + 2 = 13\/2$,\n $\\chi_{0\\in1}^* = 9 \\frac{16}{32} + 1 = 11\/2$, so that \n $\\mu_{\\chi_{0 \\in}} = 13 \/ 24$.\n Internal node $0\\in 0 $ and $0\\in 1$ are created, however, $0\\in 1$\n has exactly one datapoint in so is a Type I observation leaf.\n }\n \n \\item{Restrict at node $0\\in$ with P{\\'o}lya{} depth=3:\n $\\rho_{0\\in0\\in} = 4^2 \\frac{1\/16}{16\/100} + 2$,\n $\\rho_{0\\in0\\neg} = 4^2 \\frac{16\/100 - 1\/16}{16\/100}$ so that \n $\\mu_{\\rho_{0 \\in}} = 11\/24$ to get the Type II observation leaf\n containing $B({\\mathbf{x}}_1, {\\mathbf{x}}_2)$ and the complementary leaf.}\n\\end{itemize}\n\n\n\\section{Further Details: \\Cref{sec: anomaly-detection}} \n\\label{sec: anom-appendix}\n\nThe dataset details are given in \\Cref{table: data-details}. \nWe then present the numeric results corresponding to \\Cref{tab:mean_ranks} in \n\\Cref{tab:pidforest-baseline} and a discussion in the subsequent section.\nWe also briefly present some results on the running time as well as an\ninitial statistical analysis.\n\n\\input{tab_data_summary}\n\n\n\\subsection{Experimental Results}\nThe experimental setup is as in Section \\ref{sec: anomaly-detection} and the \n\\acrshort{auc} is recorded for each dataset.\nWe separately test the batch (iForest{}, PIDForest{} and \\acrshort{bmpf}) and \nstreaming methods (\\acrshort{smpf}, RRCF{}).\nThe results are given in \\Cref{tab:pidforest-baseline}: 5 independent trials are \nperformed for each dataset with the mean and standard deviation being reported.\nWe boldface the winner for every dataset and this is used to evaluate the \nmean rank and number of wins from \\Cref{tab:mean_ranks}. \nNote that \\Cref{tab:mean_ranks} is evaluated for every trial over all datasets, \nwhereas \\Cref{tab:pidforest-baseline} simply records the winner for the best \nreported mean \\acrshort{auc}.\nThe general behaviour is that both of the Mondrian P{\\'o}lya{} Forests behave comparably\nprior state-of-the-art methods.\n\n\\input{tab_results}\n\n\n\n\n\n\\subsection{Classical Batch Methods}\n\nAnomaly detection is a classification problem with\nimbalanced classes consisting of a (large) `normal' subset of data, and\na small subset containing anomalies.\nOne could adapt supervised learning techniques (e.g a One-Class \nSupport Vector Machines (\\acrshort{1csvm}) \n\\cite{scholkopf2001estimating}) but\nlabelling anomalies is time-consuming \\& expensive so\nsupervised learning is incompatible with the large-scale streaming model.\nFor instance, \ntraining a \\acrshort{1csvm} takes time between\n$O(n^2)$ and $O(n^3)$ depending on\nthe sizes of $n$ and $D$ \\cite{bottou2007support}.\nUnsupervised methods have also been proposed which rely on some\nnotion of local or global clustering.\nFor example, Local Outlier Factor\n(\\acrshort{lof}) \\cite{breunig2000lof};\n$k$-Nearest Neighbours (\\acrshort{knn}) (\\cite{ramaswamy2000efficient}, \\cite{angiulli2002fast});\nor Principal Components Analysis (\\acrshort{pca}),\n(\\cite{shyu2003novel}, \\cite{aggarwal2015outlier}).\nHowever, the time complexity of these methods can scale\nquadratically with $n$ or $D$ so are unsuitable in the \nlarge-scale or high-dimensional setting.\n\n\nWe are interested in \\emph{unsupervised} methods: typically, these \napproaches rely on some notion of local or global clustering, for example Local Outlier Factor \n(\\acrshort{lof}) \\cite{breunig2000lof}, $k$-Nearest Neighbours (\\acrshort{knn}) (\\cite{ramaswamy2000efficient}, \\cite{angiulli2002fast}),\nor Principal Components Analysis (\\acrshort{pca}),\n(\\cite{shyu2003novel}, \\cite{aggarwal2015outlier}).\nThese solutions do not scale for large-scale and high-dimensional datasets in the offline \nsetting, let alone when we are constrained to the data stream model;\nconsider input data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$,\n\\acrshort{lof} requires time at least $\\Omega(n)$, \nbut for high dimensions requires $\\Theta(n^2)$ \ntime \\cite{breunig2000lof}.\nAdditionally, \\acrshort{pca} requires \na singular value decomposition (\\acrshort{svd}) which takes\ntime $O(nD^2)$.\nUsing these datasets in the large-scale batch setting is problematic because of \nthe overhead incurred, let alone when we are further constrained to the \nstreaming environment.\nDue to the scalability of the batch offline methods, we only present the \nresults on a small subset of the datasets tested: these are given in \n\\Cref{tab:non-rf-baseline}.\n\n\\input{tab_offline_methods}\n\n \n\n \n \\subsection{Running Time}\n \\input{tab_runtime}\nAlthough not the focus of this investigation, we present an interesting \ncontrast between our method and PIDForest~ in terms of running time.\nThese results are summarised in \\Cref{table:runtime-comparison} in which the wall \nclock time necessary to perform the forest sampling from the previous\nexperiment (\\Cref{tab:pidforest-baseline}) is recorded.\nWe compare only \\acrshort{smpf} and PIDForest~ as both RRCF{} and iForest{}\nare heavily optimised and the other methods are not suitable for \nstreaming data.\nRecall that our algorithm uses all datapoints in ${\\mathbf{X}}$ to \n(i) cut the data at random,\n(ii) update model parameters for probability mass estimation.\nWhile the cutting is cheap, it is likely that the cuts may not be \ninformative which is why the second corrective step is required.\n\nPIDForest{} takes a complementary approach by optimising for the \ncut at every level rather than cutting at random, using only a small \nsubset of the data to build the tree.\nOur findings suggest that it is more efficient to make random cuts and \nupdate the parameters of the density model than solving the optimisation\nproblem for PIDForest{}.\nThis is borne out in \\Cref{table:runtime-comparison}, \\Cref{sec: anom-appendix} where our streaming \nimplementation of \\acrshort{mpf} is at least a (small) constant factor\nquicker than\nPIDForest{}, but can reach almost 50x (approximate) speedup over the time it \ntakes to fit a PIDForest{}.\nOf further interest is the fact that we use \\emph{all} datapoints per \ntree, whereas PIDForest{} uses only 100 points per tree meaning that,\nin aggregate, our method is substantially faster.\nWhile both implementations of \\acrshort{smpf} and PIDForest{} are \nproof-of-concept, the similarity of our proposed \\acrshort{bmpf} and \n\\acrshort{smpf} to the iForest{} and RRCF{} suggests that it should be\nsubstantial room for improvement, achieving runtime comparable to the best\nimplementations of each.\n\n\n\n\n\n\n\n\n\\subsection{Statistical Analysis}\n\n\nWe use repeated measures ANOVA as an omnibus test to determine if there are any significant differences between the mean values of the populations, shown in \\Cref{tab:anova}. We reject the null hypothesis ($F=104.844$, $p<0.001$) of the repeated measures ANOVA that there is a difference between the mean values of the for the independent variable of algorithm (the dataset and interaction were also significant). Therefore, we assume that there is a statistically significant difference between the mean values of the populations. Given that the results of the ANOVA test are significant, for post-hoc testing we use the paired two-way t-tests to infer which differences are significant.\nThe results are shown in \\Cref{tab:post_hoc}. The results at the $p<0.01$ level that show that bMPF and iForest both significantly outperform sMPF, all methods significantly outperform RRCF. All other comparisons failed to reach significance, indicating that based on these experiments, these methods cannot be separated from one another.\n\n\\input{tab_statistics}\n\n\\subsection{NAB Datasets}\nThe result in \\Cref{tab:pidforest-baseline} often suggest that the \\acrshort{auc} for the NAB datasets can be relatively low.\nAdditionally, sometimes our streaming method appears to lose out to the \nRRCF{} approach.\nWe suggest that part of the reason here for the slightly diminished \n\\acrshort{auc} performance could be to do with the labelling of the \nNAB datasets.\nThe anomalies are not labelled as specific datapoints, but rather windows\nor intervals which contain an anomaly.\nThis can clearly hurt the performance of a detector as not detecting an\nanomaly at the start of a window (which may well be \\emph{normal}\nbehaviour) would be recorded as incorrect predictions in the NAB labelling\nscheme.\nLikewise, the same applies if a detector quickly returns to normal \nbehaviour after the anomaly despite the labelling suggesting that the data\nindex still lies in an anomalous window.\nBoth of these behaviours are observed in Figures \\ref{fig:trace-rogue} and\n\\ref{fig:trace-adexch}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{figures\/trace_rogue_.pdf}\n \\caption{``Rogue\\_hold'' trace denoted by the blue curve in each panel.\n The top panel illustrates the ground truth anomalies with their \n associated window in green.\n Flagged anomalies are in the grey shading and the red dashed line\n is the threshold which achieves the optimum \\acrshort{auc}.\n }\n \\label{fig:trace-rogue}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{figures\/trace_ad_exch.pdf}\n \\caption{``Ad\\_exchange'' trace.\n Plots as described in Figure \\ref{fig:trace-rogue}}\n \\label{fig:trace-adexch}\n\\end{figure}\n\n\\newpage\n\\section{Illustrative Examples}\n\\label{sec: synthetic-examples}\n\n\n\\subsection{2d Toy Datasets}\nWe provide a simple comparison of the methods on all of the baseline\nsynthetic examples taken from the scikit-learn outlier detection \npage \\cite{outlier-sklearn} which contains unimodal and bimodal data.\nThe data is of size $n=500$ which is split between \n$n_{\\text{inliers}} = 425$ inlier points and the remaining \n$n_{\\text{outliers}} =75$ being planted outliers chosen uniformly over \nthe input domain.\nFor visual comparison, we plot the resulting classification induced by \neach of the random forest methods at the optimum threshold.\nThe results are illustrated in Figure \\ref{fig:sklearn-outlier}.\nWe additionally record the area under the ROC curve (AUC) and \narea under the\nprecision-recall-gain (PRG) curve in Table \\ref{table:synthetic-auprg}\n\\cite{flach2015precision}.\nArea under a precision-recall curve is not justified, instead use area under the PRG curve.\nWe use \\cite{prg} to evaluate the Precision-Recall-Gain and observe that\nagain our methods perform well compared to other random forests.\nThese results are presented in \\Cref{table:synthetic-auprg} but a more\nin-depth study is deferred for future work.\n\\input{appendix-synthetic-figures}\n\n\n\\section{Density Estimation}\nWe provide 4 synthetic examples to illustrate the use of our proposed models.\nA more significant experimental study will be necessary to evaluate the \nefficacy of the models in this context.\nThe synthetic datasets given below are used to generate an initial sample of \n5000 points, after which a grid is placed over the domain to estimate the \ndensity.\nFurther investigation is necessary to understand the efficicacy of both \nMondrian P{\\'o}lya{} Forests as density estimators along with a comparison to \npopular methods.\n\n\\begin{enumerate}\n \\item{Standard Normal: Figure \\ref{fig:univariate-density} (left)}\n \\item{Univariate Gaussian Mixture: \n Figure \\ref{fig:univariate-density} (right)\n taken from \n \\url{https:\/\/scikit-learn.org\/stable\/auto_examples\/neighbors\/plot_kde_1d.html}}\n \\item{Standard Bivariate Normal: Figure \\ref{fig:bivariate-density}\n (left)}\n \n \\item{Bivariate Bimodal Mixture: \n Figure \\ref{fig:bivariate-density} (right).\n As in the univariate case, except the \n covariances are adjusted to alter the shape of the clusters.\n Also the dataset used in \\Cref{fig:all-mondrians}.} \n\\end{enumerate}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{figures\/univariate_density_estimation.pdf}\n \\caption{Density estimation on univariate Gaussians}\n \\label{fig:univariate-density}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{figures\/bivariate_density_estimation.pdf}\n \\caption{Density estimation on bivariate Gaussians.\n Left-right: True density function, \\acrshort{bmpf}, \\acrshort{smpf}.}\n \\label{fig:bivariate-density}\n\\end{figure}\n\n\n\n\n\n\\section{Introduction} \\label{sec: intro}\n\n\\input{01_n_intro}\n\n\\section{Preliminaries} \\label{sec: preliminary}\n\n\\input{02_preliminaries}\n\n\\section{Mondrian P{\\'o}lya{} Forest{}} \\label{sec: mondrian-polya-forest}\n\\input{03_mondrian-polya-forest}\n\n\\section{Related Work} \\label{sec: related-work}\n\\input{04_related-work}\n\n\n\n\\section{Anomaly Detection Experiments} \n\\label{sec: anomaly-detection}\n\\input{05_anomaly-detection}\n\n\\paragraph{Conclusion.}\nWe have introduced the random forest consisting of Mondrian \\Polya{} Tree{}s.\nThese trees have natural interpretations as density estimators of the underlying\ndistribution of data.\nOur approach relates open questions concerning anomaly detection in \\cite{lakshminarayanan2016decision} through the lens of density\nestimation, thus resolving the open question in \\cite{balog2015mondrian}.\nOur method enables interpretable anomaly detection as we can threshold in the probability\ndomain and use masses rather than densities.\n\nIn addition, our random forest can be maintained on a dynamic data stream with\ninsertions and deletions, thus allowing the scalability required for large-data.\nIn future work, we plan a more in-depth analysis of the performance on data\nstreams and a rigorous study of the Mondrian \\Polya{} Tree{} as a density estimator and change-point\ndetector, rather than simply an anomaly detector.\n\nFinally, there are several directions in which this work could be\nextended to allow scalability to higher dimensions by applying random\nrotations and\/or projections after cuts.\nThis has the effect of introducing oblique cuts into the space as opposed to\naxis-aligned cuts, and could be of further benefit.\nAnother area for investigation would be to study the effect of \\emph{approximate\ncounting} for the \\Polya{} Tree{} parameters using sketches such as,\nfor example, the \\textsf{CountMin}{} sketch.\n\n\n\n\\newpage\n\n\\begin{ack}\n CD is supported by European Research Council grant ERC-2014-CoG 647557.\n We thank Shuai Tang for helpful discussions concerning the experiments and \n maunscript preparation.\n\\end{ack}\n\n\\section*{Broader Impact}\n\n\nImportant applications of anomaly detection include cybersecurity intrusion detection, operational metrics monitoring, IoT signals (such as detecting broken sensors), and fraud detection. \nThus, our contribution can have impact across all these domains. \nWhile there are many applications of varying ethical value that use anomaly detection, such as the possibility for misuse by a surveillance state to \"detect anomalous citizen behavior\", we believe that by focusing on the addition of interpretability to this solution helps to mitigate the misuses possible and allow better auditing of systems that do make use of anomaly detection \\cite{Borradaile2020}.\n\nOne application in this domain where there are fairness concerns is regarding the rate of \"anomalies\" triggered by certain subgroups in fraud detection. \nA poorly calibrated or heuristic measure of anomalous behavior in this setting has the potential to discriminate against subgroups, where the data may be more sparse and thus more likely to appear anomalous.\nIn this case, additional interpretability of how the model chooses anomalies is extremely important, as it allows the system operator to properly calibrate, using existing probabilistic fairness techniques, to remove or otherwise mitigate discrimination \\cite{Davidson2019AFF}.\n\nWe present a method that enhances the state of the art for streaming anomaly detection by casting the problem as one of probabilistic density estimation.\nModeling the problem in this way brings the immediate benefit of interpretability in the anomaly space:\ntypical approaches such as thresholding at say 3 standard deviations away from the mean or median is a standard way of declaring outliers\nin applications\nbut may not be suitable in settings when arbitrary scoring metrics are\nproposed\nImportantly however, the reframing of this into probability space allows future work to integrate other important socio-technical properties such as privacy and fairness into the same solution, for which there is much research in the field.\n\n\nDeveloping accurate, efficient methods for dealing with or summarizing streaming data has the potential to reduce environmental impact significantly, as summarized data is less expensive to send and dealing with data in a localized manner (i.e. on device) removes the need to send data into the cloud for further computation.\nThis enhancement of downstream analytics also inherently allows for more privacy, by aggregating less raw data together in the cloud. Additional research into how streaming summary methods can be applied in such cases is an exciting area in the preservation of user privacy. Privacy, differential privacy in particular, in the regime of anomaly detection involves a trade-off between knowing enough about a particular data point to determine its anomaly status and the plausible deniability of that data-point. \nImproving the capabilities of private, useful models for anomaly \ndetection could be an important area for future work;\nfor example, integrating existing differential privacy models for kd-trees \\cite{cormode2012differentially} with the interpretable\nanomaly detectors we have proposed.\n\n\\newpage\n\n\\bibliographystyle{plain}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nIn 2009, the CDF collaboration observed a narrow structure\n($Y(4140)$) near the $J\/\\psi\\phi$ threshold with statistical\nsignificance in excess of $3.8 \\sigma$ in exclusive\n$B^+\\to J\/\\psi\\phi K^+$ decays produced in $\\bar{p} p $ collisions\nat $\\sqrt{s}=1.96 \\,\\rm{TeV}$ \\cite{CDF0903}. The measured mass and width are $\\left(4143.0\\pm2.9\\pm1.2\\right)\\,\\rm{ MeV}$ and\n$\\left(11.7^{+8.3}_{-5.0}\\pm3.7\\right)\\, \\rm{MeV}$, respectively \\cite{CDF0903}. There have been several assignments, such as the molecular state \\cite{Y4140-molecule-1,Y4140-molecule-2,Y4140-molecule-3,Y4140-molecule-4,Y4140-molecule-5,Y4140-molecule-6,Y4140-molecule-7,Wang2014-4140}, charmonium hybrid \\cite{Y4140-hybrid}, rescattering effect \\cite{Y4140-rescattering}, tetraquark state \\cite{Y4140-tetraquark}, etc.\n\n\nLater, the Belle collaboration measured the process $\\gamma \\gamma \\to\n\\phi J\/\\psi$ for the $\\phi J\/\\psi$ invariant mass distributions\nbetween the threshold and $5\\,\\rm{GeV}$, and observed no signal for\n the decay $Y(4140)\\to \\phi J\/\\psi$, however, they observed a\nnarrow peak ($X(4350)$) of $8.8^{+4.2}_{-3.2}$ events with an\nsignificance of $3.2\\,\\sigma$ \\cite{Belle4350}. The measured mass\nand width are $(4350.6^{+4.6}_{-5.1}\\pm 0.7)\\,\\rm{MeV}$ and\n$(13.3^{+17.9}_{-9.1}\\pm 4.1)\\,\\rm{MeV}$, respectively\n\\cite{Belle4350}. There also have been several assignments, such as the molecular state \\cite{X4350-molecule-1,X4350-molecule-2,X4350-molecule-3,X4350-molecule-4}, conventional charmonium \\cite{X4350-charmonium-1,X4350-charmonium-2}, charmonium-molecule mixing state \\cite{X4350-mixing}, etc.\n\n\n In 2011, the CDF collaboration confirmed the\n$Y(4140)$ in the $B^\\pm\\rightarrow J\/\\psi\\,\\phi K^\\pm$ decays with\na statistical significance greater than $5\\sigma$, the measured mass and width are $\\left(4143.4^{+2.9}_{-3.0} \\pm0.6\n\\right)\\, \\rm{MeV}$ and\n$\\left(15.3^{+10.4}_{-6.1}\\pm2.5\\right)\\,\\rm{MeV}$, respectively\n\\cite{CDF1101}. Furthermore, the CDF\ncollaboration observed an evidence for a second structure ($Y(4274)$) with approximate significance of $3.1\\,\\sigma$. The\nmeasured mass and width\n are $\\left(4274.4^{+8.4}_{-6.7}\\pm1.9\\right)\\,\\rm{MeV}$ and\n$\\left(32.3^{+21.9}_{-15.3}\\pm7.6\\right)\\,\\rm{MeV}$, respectively\n\\cite{CDF1101}. The $Y(4274)$ maybe (or maybe not) a molecular state \\cite{Y4274-molecule-1,Y4274-molecule-2,Y4274-molecule-3} or a $0^{-+}$ tetraquark state \\cite{Y4274cscs}.\n\n\nIn 2013, the CMS collaboration confirmed the $Y(4140)$ in the $J\/\\psi\\phi$ mass spectrum in the $B^\\pm \\to J\/\\psi \\phi K^\\pm$ decays produced in $pp$ collisions at $\\sqrt{s} = 7\\,\\rm{ TeV}$ collected with the CMS detector at the Large Hadron Collider, and fitted the structure to a $S$-wave relativistic Breit-Wigner line-shape with the statistical significance exceeding $5 \\sigma$ \\cite{CMS1309}.\nAlso in 2013, the D0 collaboration confirmed the $Y(4140)$ in the $B^+ \\to J\/\\psi \\phi K^+$ decays in $p\\bar{p}$ collisions at $\\sqrt{s} = 1.96\\,\\rm{ TeV}$ collected by the D0 experiment at the Fermilab Tevatron collider with the statistical significance of $3.1\\sigma$ \\cite{D0-1309}.\nThe $X(4350)$ and $Y(4274)$ have not been confirmed yet.\n For detailed discussions on this subject, one can consult Ref.\\cite{Review-Jpsiphi}.\n\nThe S-wave $J\/\\psi\\phi$ systems have the quantum numbers $J^{PC}=0^{++}$, $1^{++}$, $2^{++}$, while the P-wave $ J\/\\psi\\phi$ systems have the quantum numbers $0^{-+}$, $1^{-+}$, $2^{-+}$, $3^{-+}$. The $X(4350)$ is observed in the $\\gamma\\gamma$ fusion, the $J^{PC}=1^{++}$, $1^{-+}$, $3^{-+}$ assignments are excluded due to Yang's Theorem \\cite{Review-Jpsiphi}. The possible assignments are $J^{PC}=0^{++}$, $0^{-+}$, $2^{++}$, $2^{-+}$. In the scenario of tetraquark states, the masses of the $0^{-+}$ and $2^{-+}$ states are much larger than that of the $0^{++}$ and $2^{++}$ states \\cite{EFG-2008}. The $Y(4140)$, $X(4350)$ and $Y(4274)$ are observed in the $J\/\\psi\\phi$ invariant mass distribution, if they are tetraquark states, their quark constituents must be $cs\\bar{c}\\bar{s}$. So in this article, we study the masses of the $0^{++}$ and $2^{++}$ $cs\\bar{c}\\bar{s}$ tetraquark states with the QCD sum rules, and try to identify the $Y(4140)$, $X(4350)$ and $Y(4274)$.\n\n\n\nThe article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the scalar and tensor tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.\n\n\n\\section{QCD sum rules for the scalar and tensor tetraquark states }\nIn the following, we write down the two-point correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ in the QCD sum rules,\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta}(p)&=&i\\int d^4x e^{ip \\cdot x} \\langle0|T\\left\\{J_{\\mu\\nu}(x)J_{\\alpha\\beta}^{\\dagger}(0)\\right\\}|0\\rangle \\, , \\\\\n\\Pi(p)&=&i\\int d^4x e^{ip \\cdot x} \\langle0|T\\left\\{J(x)J^{\\dagger}(0)\\right\\}|0\\rangle \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n J_{\\mu\\nu}(x)&=&\\frac{\\epsilon^{ijk}\\epsilon^{imn}}{\\sqrt{2}}\\left\\{s^j(x)C\\gamma_\\mu c^k(x) \\bar{s}^m(x)\\gamma_\\nu C \\bar{c}^n(x)+s^j(x)C\\gamma_\\nu c^k(x)\\bar{s}^m(x)\\gamma_\\mu C \\bar{c}^n(x) \\right\\} \\, , \\\\\n J(x)&=&\\epsilon^{ijk}\\epsilon^{imn}s^j(x)C\\gamma_\\mu c^k(x) \\bar{s}^m(x)\\gamma^\\mu C \\bar{c}^n(x) \\, ,\n\\end{eqnarray}\nthe $i$, $j$, $k$, $m$, $n$ are color indexes, the $C$ is the charge conjugation matrix. The\n currents $J_{\\mu\\nu}(x)$ and $J(x)$ have positive parity and charge conjugation. We take the currents $J(x)$ and $J_{\\mu\\nu}(x)$ to interpolate the scalar and tensor tetraquark states, respectively.\n\n\n\nAt the hadronic side, we can insert a complete set of intermediate hadronic states with\nthe same quantum numbers as the current operators $J_{\\mu\\nu}(x)$ and $J(x)$ into the\ncorrelation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ to obtain the hadronic representation\n\\cite{SVZ79,Reinders85}. After isolating the ground state\ncontributions of the scalar and tensor tetraquark states (denoted by $X$, $Y$ and $Z$), we get the following results,\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta} (p) &=&\\frac{\\lambda_{X\/Y\/ Z}^2}{M_{X\/Y\/Z}^2-p^2}\\left( \\frac{\\widetilde{g}_{\\mu\\alpha}\\widetilde{g}_{\\nu\\beta}+\\widetilde{g}_{\\mu\\beta}\\widetilde{g}_{\\nu\\alpha}}{2}-\\frac{\\widetilde{g}_{\\mu\\nu}\\widetilde{g}_{\\alpha\\beta}}{3}\\right) +\\cdots \\, \\, , \\\\\n\\Pi (p) &=&\\frac{\\lambda_{X\/Y\/ Z}^2}{M_{X\/Y\/Z}^2-p^2} +\\cdots \\, \\, ,\n\\end{eqnarray}\nwhere $\\widetilde{g}_{\\mu\\nu}=g_{\\mu\\nu}-\\frac{p_{\\mu}p_{\\nu}}{p^2}$, the pole residues $\\lambda_{X\/Y\/Z}$ are defined by\n\\begin{eqnarray}\n \\langle 0|J_{\\mu\\nu}(0)|X\/Y\/Z (p)\\rangle &=& \\lambda_{X\/Y\/Z} \\, \\varepsilon_{\\mu\\nu} \\, , \\nonumber\\\\\n \\langle 0|J (0)|X\/Y\/Z (p)\\rangle &=& \\lambda_{X\/Y\/Z} \\, ,\n\\end{eqnarray}\nthe summation of the polarization vector $\\varepsilon_{\\mu\\nu}$\n results in the following formula,\n \\begin{eqnarray}\n \\sum_{\\lambda}\\varepsilon^*_{\\alpha\\beta}(\\lambda,p)\\varepsilon_{\\mu\\nu}(\\lambda,p)\n &=&\\frac{\\widetilde{g}_{\\alpha\\mu}\\widetilde{g}_{\\beta\\nu}+\\widetilde{g}_{\\alpha\\nu}\\widetilde{g}_{\\beta\\mu}}{2}-\\frac{\\widetilde{g}_{\\alpha\\beta}\\widetilde{g}_{\\mu\\nu}}{3}\\,.\n \\end{eqnarray}\n\n\n\n In the following, we briefly outline the operator product expansion for the correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ in perturbative QCD. We contract the $s$ and $c$ quark fields in the correlation functions\n$\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ with Wick theorem, and obtain the results:\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta}(p)&=&\\frac{i\\epsilon^{ijk}\\epsilon^{imn}\\epsilon^{i^{\\prime}j^{\\prime}k^{\\prime}}\\epsilon^{i^{\\prime}m^{\\prime}n^{\\prime}}}{2}\\int d^4x e^{ip \\cdot x} \\nonumber\\\\\n&&\\left\\{{\\rm Tr}\\left[ \\gamma_{\\mu}C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\beta} C^{n^{\\prime}n}(-x)\\gamma_{\\nu} C S^{m^{\\prime}mT}(-x)C\\right] \\right. \\nonumber\\\\\n&&+{\\rm Tr}\\left[ \\gamma_{\\nu} C^{kk^{\\prime}}(x)\\gamma_{\\beta} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\alpha} C^{n^{\\prime}n}(-x)\\gamma_{\\mu} C S^{m^{\\prime}mT}(-x)C\\right] \\nonumber\\\\\n&&+{\\rm Tr}\\left[ \\gamma_{\\mu} C^{kk^{\\prime}}(x) \\gamma_{\\beta} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\alpha} C^{n^{\\prime}n}(-x) \\gamma_{\\nu}C S^{m^{\\prime}mT}(-x)C\\right] \\nonumber\\\\\n &&\\left.+{\\rm Tr}\\left[ \\gamma_{\\nu} C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\beta} C^{n^{\\prime}n}(-x)\\gamma_{\\mu} C S^{m^{\\prime}mT}(-x)C\\right] \\right\\} \\, , \\nonumber\\\\\n \\Pi(p)&=&i\\epsilon^{ijk}\\epsilon^{imn}\\epsilon^{i^{\\prime}j^{\\prime}k^{\\prime}}\\epsilon^{i^{\\prime}m^{\\prime}n^{\\prime}}\\int d^4x e^{ip \\cdot x} \\nonumber\\\\\n&&{\\rm Tr}\\left[ \\gamma_{\\mu}C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma^{\\alpha} C^{n^{\\prime}n}(-x)\\gamma^{\\mu} C S^{m^{\\prime}mT}(-x)C\\right] \\, ,\n\\end{eqnarray}\nwhere the $S_{ij}(x)$ and $C_{ij}(x)$ are the full $s$ and $c$ quark propagators respectively,\n \\begin{eqnarray}\nS_{ij}(x)&=& \\frac{i\\delta_{ij}\\!\\not\\!{x}}{ 2\\pi^2x^4}\n-\\frac{\\delta_{ij}m_s}{4\\pi^2x^2}-\\frac{\\delta_{ij}\\langle\n\\bar{s}s\\rangle}{12} +\\frac{i\\delta_{ij}\\!\\not\\!{x}m_s\n\\langle\\bar{s}s\\rangle}{48}-\\frac{\\delta_{ij}x^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{192}+\\frac{i\\delta_{ij}x^2\\!\\not\\!{x} m_s\\langle \\bar{s}g_s\\sigma\n Gs\\rangle }{1152}\\nonumber\\\\\n&& -\\frac{ig_s G^{a}_{\\alpha\\beta}t^a_{ij}(\\!\\not\\!{x}\n\\sigma^{\\alpha\\beta}+\\sigma^{\\alpha\\beta} \\!\\not\\!{x})}{32\\pi^2x^2} -\\frac{i\\delta_{ij}x^2\\!\\not\\!{x}g_s^2\\langle \\bar{s} s\\rangle^2}{7776} -\\frac{\\delta_{ij}x^4\\langle \\bar{s}s \\rangle\\langle g_s^2 GG\\rangle}{27648}-\\frac{1}{8}\\langle\\bar{s}_j\\sigma^{\\mu\\nu}s_i \\rangle \\sigma_{\\mu\\nu} \\nonumber\\\\\n&& -\\frac{1}{4}\\langle\\bar{s}_j\\gamma^{\\mu}s_i\\rangle \\gamma_{\\mu }+\\cdots \\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\nC_{ij}(x)&=&\\frac{i}{(2\\pi)^4}\\int d^4k e^{-ik \\cdot x} \\left\\{\n\\frac{\\delta_{ij}}{\\!\\not\\!{k}-m_c}\n-\\frac{g_sG^n_{\\alpha\\beta}t^n_{ij}}{4}\\frac{\\sigma^{\\alpha\\beta}(\\!\\not\\!{k}+m_c)+(\\!\\not\\!{k}+m_c)\n\\sigma^{\\alpha\\beta}}{(k^2-m_c^2)^2}\\right.\\nonumber\\\\\n&&\\left. +\\frac{g_s D_\\alpha G^n_{\\beta\\lambda}t^n_{ij}(f^{\\lambda\\beta\\alpha}+f^{\\lambda\\alpha\\beta}) }{3(k^2-m_c^2)^4}-\\frac{g_s^2 (t^at^b)_{ij} G^a_{\\alpha\\beta}G^b_{\\mu\\nu}(f^{\\alpha\\beta\\mu\\nu}+f^{\\alpha\\mu\\beta\\nu}+f^{\\alpha\\mu\\nu\\beta}) }{4(k^2-m_c^2)^5}+\\cdots\\right\\} \\, ,\\nonumber\\\\\nf^{\\lambda\\alpha\\beta}&=&(\\!\\not\\!{k}+m_c)\\gamma^\\lambda(\\!\\not\\!{k}+m_c)\\gamma^\\alpha(\\!\\not\\!{k}+m_c)\\gamma^\\beta(\\!\\not\\!{k}+m_c)\\, ,\\nonumber\\\\\nf^{\\alpha\\beta\\mu\\nu}&=&(\\!\\not\\!{k}+m_c)\\gamma^\\alpha(\\!\\not\\!{k}+m_c)\\gamma^\\beta(\\!\\not\\!{k}+m_c)\\gamma^\\mu(\\!\\not\\!{k}+m_c)\\gamma^\\nu(\\!\\not\\!{k}+m_c)\\, ,\n\\end{eqnarray}\nand $t^n=\\frac{\\lambda^n}{2}$, the $\\lambda^n$ is the Gell-Mann matrix, $D_\\alpha=\\partial_\\alpha-ig_sG^n_\\alpha t^n$ \\cite{Reinders85}. Then we compute the integrals both in the coordinate and momentum spaces to obtain the correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ therefore the QCD spectral densities.\nIn Eq.(10), we retain the terms $\\langle\\bar{s}_j\\sigma_{\\mu\\nu}s_i \\rangle$ and $\\langle\\bar{s}_j\\gamma_{\\mu}s_i\\rangle$ originate from the Fierz re-arrangement of the $\\langle s_i \\bar{s}_j\\rangle$ to absorb the gluons emitted from the heavy quark lines to extract the mixed condensate $\\langle\\bar{s}g_s\\sigma G s\\rangle$ and four-quark condensates $g_s^2\\langle\\bar{s}s\\rangle^2$, respectively.\n\n Finally we can take the\nquark-hadron duality below the continuum thresholds $s_0$ and perform Borel transform with respect to\nthe variable $P^2=-p^2$ to obtain the QCD sum rules:\n\\begin{eqnarray}\n\\lambda^2_{X\/Y\/Z}\\, \\exp\\left(-\\frac{M^2_{X\/Y\/Z}}{T^2}\\right)= \\int_{4m_c^2}^{s_0} ds\\, \\rho(s) \\, \\exp\\left(-\\frac{s}{T^2}\\right) \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\rho(s)&=&\\rho_{0}(s)+\\rho_{3}(s) +\\rho_{4}(s)+\\rho_{5}(s)+\\rho_{6}(s)+\\rho_{7}(s) +\\rho_{8}(s)+\\rho_{10}(s)\\, ,\n\\end{eqnarray}\nthe explicit expressions of the $\\rho_i(s)$ are given in the appendix.\n\n\n\n\n\n\n We differentiate Eq.(12) with respect to $\\frac{1}{T^2}$, then eliminate the\n pole residues $\\lambda_{X\/Y\/Z}$, and obtain the QCD sum rules for\n the masses of the scalar and tensor tetraquark states,\n \\begin{eqnarray}\n M^2_{X\/Y\/Z}= \\frac{\\int_{4m_c^2}^{s_0} ds\\frac{d}{d \\left(-1\/T^2\\right)}\\rho(s)\\exp\\left(-\\frac{s}{T^2}\\right)}{\\int_{4m_c^2}^{s_0} ds \\rho(s)\\exp\\left(-\\frac{s}{T^2}\\right)}\\, .\n\\end{eqnarray}\n\n\n\n\n\n\\section{Numerical results and discussions}\nThe vacuum condensates are taken to be the standard values\n$\\langle\\bar{q}q \\rangle=-(0.24\\pm 0.01\\, \\rm{GeV})^3$, $\\langle\\bar{s}s \\rangle=(0.8\\pm0.1)\\langle\\bar{q}q \\rangle$,\n$\\langle\\bar{s}g_s\\sigma G s \\rangle=m_0^2\\langle \\bar{s}s \\rangle$,\n$m_0^2=(0.8 \\pm 0.1)\\,\\rm{GeV}^2$, $\\langle \\frac{\\alpha_s\nGG}{\\pi}\\rangle=(0.33\\,\\rm{GeV})^4 $ at the energy scale $\\mu=1\\, \\rm{GeV}$\n\\cite{SVZ79,Reinders85,Ioffe2005-1,Ioffe2005-2}.\nThe quark condensates and mixed quark condensates evolve with the renormalization group equation,\n$\\langle\\bar{q}q \\rangle(\\mu)=\\langle\\bar{q}q \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{4}{9}}$,\n$\\langle\\bar{s}s \\rangle(\\mu)=\\langle\\bar{s}s \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{4}{9}}$,\n $\\langle\\bar{s}g_s \\sigma Gs \\rangle(\\mu)=\\langle\\bar{s}g_s \\sigma Gs \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{2}{27}}$, we take into account the energy scale dependence.\n\nIn the article, we take the $\\overline{MS}$ masses $m_{c}(m_c)=(1.275\\pm0.025)\\,\\rm{GeV}$ and $m_s(\\mu=2\\,\\rm{GeV})=(0.095\\pm0.005)\\,\\rm{GeV}$\n from the Particle Data Group \\cite{PDG}, and take into account\nthe energy-scale dependence of the $\\overline{MS}$ masses from the renormalization group equation,\n\\begin{eqnarray}\nm_s(\\mu)&=&m_s({\\rm 2GeV} )\\left[\\frac{\\alpha_{s}(\\mu)}{\\alpha_{s}({\\rm 2GeV})}\\right]^{\\frac{4}{9}} \\, ,\\nonumber\\\\\nm_c(\\mu)&=&m_c(m_c)\\left[\\frac{\\alpha_{s}(\\mu)}{\\alpha_{s}(m_c)}\\right]^{\\frac{12}{25}} \\, ,\\nonumber\\\\\n\\alpha_s(\\mu)&=&\\frac{1}{b_0t}\\left[1-\\frac{b_1}{b_0^2}\\frac{\\log t}{t} +\\frac{b_1^2(\\log^2{t}-\\log{t}-1)+b_0b_2}{b_0^4t^2}\\right]\\, ,\n\\end{eqnarray}\n where $t=\\log \\frac{\\mu^2}{\\Lambda^2}$, $b_0=\\frac{33-2n_f}{12\\pi}$, $b_1=\\frac{153-19n_f}{24\\pi^2}$, $b_2=\\frac{2857-\\frac{5033}{9}n_f+\\frac{325}{27}n_f^2}{128\\pi^3}$, $\\Lambda=213\\,\\rm{MeV}$, $296\\,\\rm{MeV}$ and $339\\,\\rm{MeV}$ for the flavors $n_f=5$, $4$ and $3$, respectively \\cite{PDG}.\n\nIn Refs.\\cite{Wang2014-4140,WangHuangTao-1,WangHuangTao-2,WangHuangTao-3,Wang4430,Wang-Cu-Cu,WangHuang-molecule}, we study the acceptable energy scales of the QCD spectral densities for the hidden charmed (bottom) tetraquark states and molecular states in the QCD sum rules in details for the first time, and suggest a formula,\n\\begin{eqnarray}\n\\mu&=&\\sqrt{M^2_{X\/Y\/Z}-(2{\\mathbb{M}}_Q)^2} \\, ,\n \\end{eqnarray}\nwith the effective $Q$-quark masses ${\\mathbb{M}}_Q$ to determine the energy scales of the QCD spectral densities.\nIn Refs.\\cite{WangHuangTao-1,WangHuangTao-2,WangHuangTao-3,Wang4430,Wang-Cu-Cu}, we focus on the scenario of tetraquark states, study the diquark-antidiquark type scalar, vector, axial-vector, tensor hidden charmed tetraquark states and\naxial-vector hidden bottom tetraquark states systematically with the QCD sum rules,\nand try to make possible assignments of the $X(3872)$,\n$Z_c(3900)$, $Z_c(3885)$, $Z_c(4020)$, $Z_c(4025)$, $Z(4050)$, $Z(4250)$, $Y(4360)$, $Z(4430)$, $Y(4630)$, $Y(4660)$, $Z_b(10610)$ and $Z_b(10650)$.\nIn the operator product expansion, we calculate the vacuum condensates up to dimension-10, just like in the present case;\n the energy scale formula works very well.\n\n\nIn the conventional QCD sum rules \\cite{SVZ79,Reinders85}, we\n usually take the energy gap between the ground\nstates and the first radial excited states to be $(0.4-0.6)\\,\\rm{GeV}$. Such relation survives in the tetraquark sector,\n for example,\nthe $Z(4430)$ is tentatively assigned to be the first radial excitation of the $Z_c(3900)$ according to the\nanalogous decays,\n$Z_c(3900)^\\pm\\to J\/\\psi\\pi^\\pm$, $Z(4430)^\\pm\\to\\psi^\\prime\\pi^\\pm$,\nand the mass differences $M_{Z(4430)}-M_{Z_c(3900)}=576\\,\\rm{MeV}$, $M_{\\psi^\\prime}-M_{J\/\\psi}=589\\,\\rm{MeV}$ \\cite{Wang4430,Maiani-2014,Nielsen-1401}.\n\nFirstly, we take the $Y(4140)$, $Y(4274)$ and $X(4350)$ as the scalar and tensor $cs\\bar{c}\\bar{s}$ tetraquark states, respectively, and choose the continuum threshold parameters\nas $s^0_{Y(4140)}=(4.70\\,\\rm{GeV})^2$, $s^0_{Y(4274)}=(4.80\\,\\rm{GeV})^2$ and $s^0_{X(4350)}=(4.85\\,\\rm{GeV})^2$.\nIn Fig.1,\n the masses of the scalar and tensor tetraquark states are plotted with variations of the Borel parameters $T^2$ and energy scales $\\mu$. From the figure, we can see that the masses decrease monotonously with increase of the energy scales, and we can also obtain the allowed energy scales to reproduce the experimental values of the masses.\n\n In Table 1, we denote the allowed energy scales which can reproduce the experimental values of the masses as $\\mu_A$, and denote the resulting energy scales from the energy scale formula as $\\mu_T$.\n From the table, we can see that the $\\mu_A$ and $\\mu_T$ are compatible only in the case of the $Y(4140)$ with the assignment $J^{PC}=2^{++}$.\n\nNow, we assume the $Y(4140)$ to be the tensor tetraquark state, take the continuum threshold parameter as $s^0_{Y(4140)}=(4.7\\pm 0.1)^2\\,\\rm{GeV}^2$ and the energy scale as $\\mu=2.0\\,\\rm{GeV}$ to search for\nthe Borel parameter $T^2$ to satisfy the\ntwo criteria (pole dominance and convergence of the operator product\nexpansion) of the QCD sum rules. Furthermore, we study the scalar tetraquark state in the same way, i.e.\nwe search for the optimal Borel parameter $T^2$ and threshold\nparameter $s_0$ to satisfy the\ntwo criteria of the QCD sum rules and the energy scale formula of the QCD spectral densities.\nThe resulting Borel parameters, continuum threshold parameters and the pole contributions are shown explicitly in Table 2.\n\nIn Fig.2, we plot the contributions of different terms in the\noperator product expansion with variations of the Borel parameters $T^2$ for the threshold parameters $s^0_{J=2}=(4.7\\,\\rm{GeV})^2$ and $s^0_{J=0}=(4.5\\,\\rm{GeV})^2$, respectively. In the Borel windows, the $D_0$, $D_3$ and $D_5$\nplay an important role, the $D_6$ and $D_{8}$ play a minor important role, while the $D_4$, $D_7$ and $D_{10}$ are tiny, where the $D_i$ denote the contributions of the vacuum condensates of dimensions $D=i$. The operator product expansion is well convergent. It is obvious that the two criteria of the QCD sum rules are fully satisfied, so we expect to make reasonable predictions.\n\n\nWe take into account all uncertainties of the input parameters,\nand obtain the values of the masses and pole residues of\n the scalar and tensor tetraquark states, which are shown explicitly in Figs.3-4 and Table 2.\nThe prediction $M_{J=2} =\\left(4.13^{+0.08}_{-0.08}\\right)\\,\\rm{GeV}$ is consistent with the experimental value $M_{Y(4140)}=(4143.0\\pm 2.9\\pm1.2)\\,\\rm{MeV}$ \\cite{PDG}. The present predictions favor assigning the $Y(4140)$ to be the $J^{PC}=2^{++}$ diquark-antidiquark type tetraquark states, and disfavor assigning the $Y(4274)$ and $X(4350)$ to be the $J^{PC}=0^{++}$ or $2^{++}$ diquark-antidiquark type tetraquark states.\nAt the present time, there is no experimental candidate for the scalar $cs\\bar{c}\\bar{s}$ tetraquark state,\n we can search for the scalar tetraquark state at the BESIII, LHCb and Belle-II in the futures.\n\n\n Recently, Mo et al study the $X(4350)$ as a $ c s\\bar{c}\\bar{s}$ tetraquark state with the assignment $J^{PC}=1^{-+}$ using the QCD sum rules, and obtain the mass $M_{J=1}=(4.82\\pm 0.19)\\,\\rm{GeV}$, which is not compatible with the $X(4350)$ as a $1^{-+}$ tetraquark state \\cite{X4350-Huang}. So the $X(4350)$ is unlikely to be a $ c s\\bar{c}\\bar{s}$ tetraquark state. Furthermore, the $X(4350)$ and $Y(4274)$ are still need confirmation.\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{Y4140-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4140-J2-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4274-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4274-J2-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4350-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4350-J2-miu.EPS}\n \\caption{ The masses of the $Y(4140)$, $Y(4274)$ and $X(4350)$ with the assignments $J^{PC}=0^{++}$ and $2^{++}$ respectively vary with the\n Borel parameters $T^2$ and the energy scales $\\mu$, where the horizontal lines denote the experimental values of the masses of the $Y(4140)$, $Y(4274)$ and $X(4350)$, respectively. }\n\\end{figure}\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\\hline\n & $J^{PC}$ & $\\sqrt{s_0} (\\rm{GeV})$ & $\\mu_A(\\rm{GeV})$ & $\\mu_{T}(\\rm{GeV})$ & \\\\ \\hline\n $Y(4140)$ & $0^{++}$ & 4.70 & $1.4-1.7$ & 2.0 & $\\times$ \\\\ \\hline\n $Y(4140)$ & $2^{++}$ & 4.70 & $1.8-2.1$ & 2.0 & $\\surd$ \\\\ \\hline $Y(4274)$ & $0^{++}$ & 4.80 & $1.2-1.4$ & 2.3 & $\\times$ \\\\ \\hline\n $Y(4274)$ & $2^{++}$ & 4.80 & $1.4-1.6$ & 2.3 & $\\times$ \\\\ \\hline\n $X(4350)$ & $0^{++}$ & 4.85 & $1.1-1.2$ & 2.4 & $\\times$ \\\\ \\hline\n $X(4350)$ & $2^{++}$ & 4.85 & $1.2-1.3$ & 2.4 & $\\times$ \\\\ \\hline\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The continuum threshold parameters $s_0$, allowed energy scales $\\mu_A$, theoretical energy scales $\\mu_T$ for the $Y(4140)$, $Y(4274)$ and $X(4350)$\nwith the possible assignments $J^{PC}$, where the $\\times$ and $\\surd$ denote the compatibility between the $\\mu_A$ and $\\mu_T$. }\n\\end{table}\n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n$J^{PC}$ &$T^2(\\rm{GeV}^2)$ &$\\sqrt{s_0} (\\rm{GeV})$ &$\\mu(\\rm{GeV})$ &pole &$M_{X\/Y\/Z}(\\rm{GeV})$ &$\\lambda_{X\/Y\/Z}$ \\\\ \\hline\n$2^{++}$ &$3.0-3.4$ &$4.7\\pm0.1$ &2.0 &$(49-69)\\%$ &$4.13^{+0.08}_{-0.08}$ &$5.34^{+0.76}_{-0.68}\\times10^{-2}\\rm{GeV}^5$\\\\ \\hline\n$0^{++}$ &$2.5-2.9$ &$4.5\\pm0.1$ &1.7 &$(46-70)\\%$ &$3.98^{+0.08}_{-0.08}$ &$4.87^{+0.81}_{-0.68}\\times10^{-2}\\rm{GeV}^5$ \\\\ \\hline\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions, masses and pole residues of the scalar and tensor tetraquark states. }\n\\end{table}\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{fraction-0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{fraction-2.EPS}\n \\caption{ The contributions of different terms in the operator product expansion for the $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states with variations of the\n Borel parameters $T^2$, where the 0, 3, 4, 5, 6, 7, 8, 10 denote the dimensions of the vacuum condensates. }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{mass-J0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{mass-J2.EPS}\n \\caption{ The masses of the $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states with variations of the\n Borel parameters $T^2$, where the horizontal lines denote the experimental value of the mass of the $Y(4140)$. }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{residue-J0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{residue-J2.EPS}\n \\caption{ The pole residues of the $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states with variations of the\n Borel parameters $T^2$. }\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this article, we tentatively assign the $Y(4140)$, $Y(4274)$ and $X(4350)$ to be the scalar and tensor $cs\\bar{c}\\bar{s}$ tetraquark states, respectively, and study them with the QCD sum rules. In the operator product expansion, we calculate the contributions of the vacuum condensates up to dimension-10. Furthermore, we use the formula $\\mu=\\sqrt{M^2_{X\/Y\/Z}-(2{\\mathbb{M}}_c)^2}$ to determine the energy scales of the QCD spectral densities. The numerical results of the masses $M_{X\/Y\/Z}$ favor assigning the $Y(4140)$ to be the $J^{PC}=2^{++}$ $cs\\bar{c}\\bar{s}$ tetraquark state, and disfavor assigning the $Y(4274)$ and $X(4350)$ to be the $0^{++}$ or $2^{++}$ tetraquark states. There is no candidate for the scalar $cs\\bar{c}\\bar{s}$ tetraquark state, we can search for it at the BESIII, LHCb and Belle-II in the futures.\n\n\\section*{Appendix}\nThe spectral densities $\\rho_i(s)$ with $i=0$, 3, 4, 5, 6, 7, 8, 10 at the level of the quark-gluon degrees of\nfreedom,\n\n\n\\begin{eqnarray}\n\\rho^{2}_{0}(s)&=&\\frac{1}{15360\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\, (1-y-z)^3\\left(s-\\overline{m}_c^2\\right)^2\\left(293s^2-190s\\overline{m}_c^2+17\\overline{m}_c^4 \\right) \\nonumber\\\\\n&&+\\frac{1}{5120\\pi^6} \\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\,(1-y-z)^2\\left(s-\\overline{m}_c^2\\right)^4 \\nonumber\\\\\n&&+\\frac{m_sm_c}{128\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)\\, (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right)^2\\left(4s-\\overline{m}_c^2 \\right) \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{3}^{2}(s)&=&-\\frac{m_c\\langle \\bar{s}s\\rangle}{16\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)(1-y-z)\\left(s-\\overline{m}_c^2\\right)\\left(3s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{160\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz(1-y-z)\\left(115s^2-112s\\overline{m}_c^2+17\\overline{m}_c^4 \\right) \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{160\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s - \\overline{m}_c^2 \\right)^2 \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}s\\rangle}{4\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( s - \\overline{m}_c^2 \\right) \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{4}^{2}(s)&=&-\\frac{m_c^2}{11520\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( \\frac{z}{y^2}+\\frac{y}{z^2}\\right)(1-y-z)^3 \\nonumber\\\\\n&&\\left\\{ 56s-17\\overline{m}_c^2+10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right\\} \\nonumber\\\\\n&&-\\frac{m_c^2}{3840\\pi^4}\\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right) (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&-\\frac{1}{15360\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 185s^2-208s\\overline{m}_c^2+43\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{7680\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z) \\left( s-\\overline{m}_c^2\\right)^2 \\nonumber\\\\\n&&-\\frac{1}{2304\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 15s^2-16s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&-\\frac{1}{13824\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^3 \\left( 25s^2-24s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&-\\frac{1}{6912\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\,(1-y-z) \\left( 25s^2-24s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&-\\frac{1}{4608\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^2 \\left( s-\\overline{m}_c^2\\right)^2 \\nonumber\\\\\n&&-\\frac{1}{6912\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s-\\overline{m}_c^2\\right)\\left( 13s-5\\overline{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho^{2}_{5}(s)&=&\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{32\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z) \\left(2s-\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&+\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{144\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z) \\left(2s-\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{480\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left\\{56s - 17\\overline{m}_c^2 +10\\overline{m}_c^4 \\delta(s-\\overline{m}_c^2 )\\right\\}\\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{480\\pi^4}\\int_{y_i}^{y_f}dy \\, y(1-y) \\left(s - \\widetilde{m}_c^2 \\right)+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{16\\pi^4}\\int_{y_i}^{y_f}dy \\nonumber\\\\\n&&+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{288\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{1}{y}+\\frac{1}{z} \\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho_{6}^{2}(s)&=&\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{6\\pi^2}\\int_{y_i}^{y_f}dy +\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{3240\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\, yz \\left\\{56s-17\\overline{m}_c^2 +10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2 \\right)\\right\\}\\nonumber\\\\\n&&+\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{3240\\pi^4}\\int_{y_i}^{y_f}dy \\,y(1-y)\\left(s-\\widetilde{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{9720\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 45\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(2s-\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right.\\nonumber\\\\\n&&\\left.m_c^2\\left[ 19+20\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2 \\right)\\right]+(y+z)\\left[18\\left(3s-\\overline{m}_c^2\\right)+10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right) \\right] \\right\\} \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{9720\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 15\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(2s-\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right. \\nonumber\\\\\n&&\\left.m_c^2\\left[ 6+5\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right)\\right]+(y+z)\\left[56s-17\\overline{m}_c^2 +10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right] \\right\\}\\nonumber\\\\\n&&-\\frac{m_sm_c \\langle\\bar{s}s\\rangle^2}{12\\pi^2}\\int_{y_i}^{y_f}dy \\left\\{ 1+\\widetilde{m}_c^2\\delta(s-\\widetilde{m}_c^2)\\right\\}\\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho_7^{2}(s)&=&\\frac{m_c^3\\langle\\bar{s}s\\rangle}{144\\pi^2 T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^3}+\\frac{z}{y^3}+\\frac{1}{y^2}+\\frac{1}{z^2}\\right)(1-y-z)\\, \\overline{m}_c^2 \\, \\delta\\left(s-\\overline{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^2}+\\frac{z}{y^2}\\right)(1-y-z) \\left\\{1+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left\\{1+\\frac{\\overline{m}_c^2}{3}\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\} \\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{432\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left(\\frac{1-y}{y}+\\frac{1-z}{z}\\right)\n\\left\\{1+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber \\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{288\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\left\\{1+ \\widetilde{m}_c^2 \\, \\delta \\left(s-\\widetilde{m}_c^2\\right) \\right\\}\\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_8^{2}(s)&=&-\\frac{m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{12\\pi^2}\\int_0^1 dy \\left(1+\\frac{\\widetilde{m}_c^2}{T^2} \\right)\\delta\\left(s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&-\\frac{ m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{216\\pi^2}\\int_{0}^{1} dy \\frac{1}{y(1-y)}\\delta\\left(s-\\widetilde{m}_c^2\\right)\n \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{10}^{2}(s)&=&\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{96\\pi^2T^6}\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right)\n\\nonumber \\\\\n&&-\\frac{m_c^4\\langle\\bar{s}s\\rangle^2}{108T^4}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\left\\{ \\frac{1}{y^3}+\\frac{1}{(1-y)^3}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{36T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\left\\{ \\frac{1}{y^2}+\\frac{1}{(1-y)^2}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{324T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{864 \\pi^2T^4} \\int_0^1 dy \\frac{1}{y(1-y)} \\widetilde{m}_c^2 \\, \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{576 \\pi^2T^2} \\int_0^1 dy \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s} s\\rangle^2}{108 T^6}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho^{0}_{0}(s)&=&\\frac{1}{256\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\, (1-y-z)^3\\left(s-\\overline{m}_c^2\\right)^2\\left(7s^2-6s\\overline{m}_c^2+\\overline{m}_c^4 \\right) \\nonumber\\\\\n&&+\\frac{1}{256\\pi^6} \\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\,(1-y-z)^2\\left(s-\\overline{m}_c^2\\right)^3 \\left(3s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{m_sm_c}{128\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)\\, (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right)^2\\left(5s-2\\overline{m}_c^2 \\right) \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{3}^{0}(s)&=&-\\frac{m_c\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)(1-y-z)\\left(s-\\overline{m}_c^2\\right)\\left(2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz(1-y-z)\\left(10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4 \\right) \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\left(s-\\overline{m}_c^2\\right)\\left(2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}s\\rangle}{2\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(s-\\overline{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{4}^{0}(s)&=&-\\frac{m_c^2}{192\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( \\frac{z}{y^2}+\\frac{y}{z^2}\\right)(1-y-z)^3 \\nonumber\\\\\n&&\\left\\{ 2s-\\overline{m}_c^2+\\frac{\\overline{m}_c^4}{6}\\delta\\left(s-\\overline{m}_c^2\\right)\\right\\} \\nonumber\\\\\n&&-\\frac{m_c^2}{384\\pi^4}\\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right) (1-y-z)^2 \\left(3s-2\\overline{m}_c^2\\right) \\nonumber\\\\\n&&-\\frac{1}{768\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{384\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z) \\left( s-\\overline{m}_c^2\\right)\\left( 2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{1}{384\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{3456\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^3 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{576\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\,(1-y-z) \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{576\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^2 \\left( s-\\overline{m}_c^2\\right)\n\\left( 2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{1}{288\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s-\\overline{m}_c^2\\right)\\left( 2s-\\overline{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho^{0}_{5}(s)&=&\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{32\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z) \\left(3s-2\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z) \\left(3s-2\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\left\\{2s-\\overline{m}_c^2+\\frac{\\overline{m}_c^2}{6} \\delta\\left(s-\\overline{m}_c^2 \\right) \\right\\} \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy \\, y(1-y)\\left(3s-2\\widetilde{m}_c^2 \\right) \\nonumber\\\\\n&&+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{1}{y}+\\frac{1}{z} \\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho_{6}^{0}(s)&=&\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{3\\pi^2}\\int_{y_i}^{y_f}dy +\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{54\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\, yz \\left\\{2s-\\overline{m}_c^2 +\\frac{\\overline{m}_c^4}{6}\\delta\\left(s-\\overline{m}_c^2 \\right)\\right\\}\\nonumber\\\\\n&&+\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{324\\pi^4}\\int_{y_i}^{y_f}dy \\,y(1-y)\\left(3s-2\\widetilde{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{648\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 3\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(3s-2\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right.\\nonumber\\\\\n&&\\left.m_c^2\\left[ 2+ \\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2 \\right)\\right]+(y+z)\\left[12\\left(2s-\\overline{m}_c^2\\right)+2\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right) \\right] \\right\\} \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{1944\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 15\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(3s-2\\overline{m}_c^2 \\right)+7\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right. \\nonumber\\\\\n&&\\left.m_c^2\\left[ 2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right)\\right]+(y+z)\\left[12\\left(2s-\\overline{m}_c^2\\right) +2\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right] \\right\\} \\nonumber\\\\\n&&-\\frac{m_sm_c \\langle\\bar{s}s\\rangle^2}{12\\pi^2}\\int_{y_i}^{y_f}dy \\left\\{ 2+\\widetilde{m}_c^2\\delta(s-\\widetilde{m}_c^2)\\right\\}\\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho_7^{0}(s)&=&\\frac{m_c^3\\langle\\bar{s}s\\rangle}{144\\pi^2 }\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^3}+\\frac{z}{y^3}+\\frac{1}{y^2}+\\frac{1}{z^2}\\right)(1-y-z)\\nonumber\\\\\n&&\\left(1+\\frac{ \\overline{m}_c^2}{T^2}\\right) \\delta\\left(s-\\overline{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^2}+\\frac{z}{y^2}\\right)(1-y-z) \\left\\{2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left\\{2+ \\overline{m}_c^2 \\delta\\left(s-\\overline{m}_c^2\\right) \\right\\} \\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{144\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left(\\frac{1-y}{y}+\\frac{1-z}{z}\\right)\n\\left\\{2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber \\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{288\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\left\\{2+ \\widetilde{m}_c^2 \\, \\delta \\left(s-\\widetilde{m}_c^2\\right) \\right\\}\\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_8^{0}(s)&=&-\\frac{m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{6\\pi^2}\\int_0^1 dy \\left(1+\\frac{\\widetilde{m}_c^2}{T^2} \\right)\\delta\\left(s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{ m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{36\\pi^2}\\int_{0}^{1} dy \\frac{1}{y(1-y)}\\delta\\left(s-\\widetilde{m}_c^2\\right)\n \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{10}^{0}(s)&=&\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{48\\pi^2T^6}\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right)\n\\nonumber \\\\\n&&-\\frac{m_c^4\\langle\\bar{s}s\\rangle^2}{54T^4}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\left\\{ \\frac{1}{y^3}+\\frac{1}{(1-y)^3}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{18T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\left\\{ \\frac{1}{y^2}+\\frac{1}{(1-y)^2}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{54T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&-\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{144 \\pi^2T^4} \\int_0^1 dy \\frac{1}{y(1-y)} \\widetilde{m}_c^2 \\, \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{32 \\pi^2T^2} \\int_0^1 dy \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s} s\\rangle^2}{54 T^6}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n the subscripts $0$, $3$, $4$, $5$, $6$, $7$, $8$ and $10$ denote the dimensions of the vacuum condensates, the superscripts $0$ and $2$ denote the spin the tetraquark states, the $T^2$ denotes the Borel parameter; $y_{f}=\\frac{1+\\sqrt{1-4m_c^2\/s}}{2}$,\n$y_{i}=\\frac{1-\\sqrt{1-4m_c^2\/s}}{2}$, $z_{i}=\\frac{y\nm_c^2}{y s -m_c^2}$, $\\overline{m}_c^2=\\frac{(y+z)m_c^2}{yz}$,\n$ \\widetilde{m}_c^2=\\frac{m_c^2}{y(1-y)}$, $\\int_{y_i}^{y_f}dy \\to \\int_{0}^{1}dy$, $\\int_{z_i}^{1-y}dz \\to \\int_{0}^{1-y}dz$,\n when the $\\delta$ functions $\\delta\\left(s-\\overline{m}_c^2\\right)$ and $\\delta\\left(s-\\widetilde{m}_c^2\\right)$ appear.\n\n\n\n\\section*{Acknowledgements}\nThis work is supported by National Natural Science Foundation,\nGrant Numbers 11375063, and Natural Science Foundation of Hebei province, Grant Number A2014502017.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}