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+{"text":"\\section{Introduction}\nThe determination of the exact string theory low energy effective action is a very difficult problem in general. In the case of type II string theory on $\\mathds{R}^{1,10-d} \\times T^{d-1}$, the lowest order non-perturbative corrections could nonetheless have been computed  \\cite{Green:1997tv,Green:1997as,Kiritsis:1997em}. Although there is no non-perturbative formulation of the theory,  the constraints following from supersymmetry and $U$-duality have permitted to determine the non-perturbative low energy effective action from perturbative computations in string theory \\cite{D'Hoker:2005jc,Green:2008uj,Gomez:2013sla,Green:2014yxa,D'Hoker:2014gfa} and in eleven-dimensional supergravity \\cite{Green:1997as,Green:1999pu,Green:2005ba,Green:2008bf,Basu:2014uba}. The four-graviton amplitude allows in particular to determine the $\\nabla^{2k}R^4$ type correction in the effective action,\n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal L}\\sim  \\frac{1}{\\kappa^2}  R +\\sum_{p,q}  \\kappa^{2 \\frac{d-3+4p+6q}{9-d}}  E_{(p,q)} \\nabla^{4p+6q} R^4 +\\dots    \\ee \nwhere the dots stand for other terms including the supersymmetric completion, $(p,q)$ labels the different invariant combinations of derivatives compatible with supersymmetry according to the notations used in \\cite{Green:2010wi}, and $E_{(p,q)}$ are automorphic functions of the scalar fields defined on $E_{d(d)}(\\mathds{Z}) \\backslash E_{d(d)}\/ K_d$. For $(p,q) = (0,0)$, $(1,0)$ and $(0,1)$, the complete effective action at this order is determined by these functions ${\\cal E}_\\gra{p}{q}$, which have been extensively studied  \\cite{Review,Obers:1999um,Basu:2007ck,Green:2010kv,Green:2011vz,Fleig:2013psa,Minimal,D4R4,Gustafsson:2014iva,Pioline:2015yea}. \n\n$E_{(0,0)}$ is an Eisenstein series associated to the minimal unitary representation \\cite{Obers:1999um,Green:2010kv}, $E_{(1,0)}$ is an (or a sum of two) Eisenstein series associated to the next to minimal unitary representation(s) \\cite{Green:2010kv}, and both are therefore relatively well understood.  They are nonetheless very complicated functions, and the explicit expansion of $E_{(1,0)}$ in Fourier modes is not yet determined \\cite{Green:2011vz,Fleig:2013psa,Gustafsson:2014iva}. $E_{(0,1)}$ is not even an Eisenstein series, and was shown in \\cite{Green:2005ba} to satisfy to an inhomogeneous Poisson equation in type IIB. A proposal for this function in eight dimensions \\cite{Basu:2007ck}, suggested a split of the function into the sum of an Eisenstein series and an inhomogeneous solution, which was subsequently generalised in seven and six dimensions \\cite{Green:2010wi,Green:2010kv}, and  recently clarified in \\cite{Pioline:2015yea}. \n\nIn this paper we extend the analysis carried out in \\cite{Minimal,D4R4} to the study of $E_{(0,1)}$. We show that this function indeed splits into the sum of two functions that are associated to two distinct supersymmetry invariants, and therefore satisfy to inequivalent tensorial differential equations. In particular, the second satisfies to a homogeneous equation, which is solved by the Eisenstein function appearing in \\cite{Green:2010wi,Basu:2007ck,Pioline:2015yea}. One can distinguish the two functions by looking at specific higher point couplings that we identify. The new class of invariants generalises to an infinite class admitting a coupling in $F^{2k} \\nabla^4 R^4$, and we identify a unique Eisenstein function solving the corresponding tensorial differential equations in all dimensions greater than four. This function turns out to be compatible with perturbative string theory, and only admits three perturbative contributions in four dimensions, at 1-loop, $(k+2)$-loop, and $2k$-loop.  However, the only amplitude that seems to unambiguously distinguish it from others is the $(k+2)$-loop four-graviton amplitude in a non-trivial Ramond--Ramond background, which makes an explicit check extremely challenging. \n\n\nWe start with the analysis of the supersymmetry invariants in four dimensions. The two $\\nabla^6 R^4$ type invariants in the linear approximation are associated to two distinct classes of chiral primary operators of $SU(2,2|8)$ discussed in \\cite{Drummond:2003ex}. We identify the corresponding representations of $E_{7(7)}$ associated to nilpotent coadjoint orbits \\cite{E7Djo} that are summarised in figure \\ref{ClosureDiag}. \n\\begin{figure}[htbp]\n\\begin{center}\n \\begin{tikzpicture}\n \\draw (1,0) node{\\textbullet};\n \\draw (1,0 - 1) node{\\textbullet};\n  \\draw (1,0 - 2)  node{\\textbullet};\n  \\draw (1,0 + 2)  node{\\textbullet};\n  \\draw (1,0 + 4)  node{\\textbullet};\n  \\draw (1 + 1,0 + 2.5)  node{\\textbullet};\n   \\draw (1 - 1,0 + 3.5)  node{\\textbullet};\n    \\draw (1 - 1 ,0 + 1) node{\\textbullet};\n   \n  \\draw (1 + 0.3,0 - 2) node{$R$};\n  \\draw (1 + 0.4,0 - 1) node{$R^4$};\n  \\draw (1 + 0.7,0) node{$ \\nabla^4 R^4$};\n  \\draw (1 - 1.9,0 + 1) node{$ {F}^{2k} \\nabla^4 R^4$};\n  \\draw (1 + 0.7 + 1,0 + 2.5) node{$ \\nabla^6 R^4$};\n  \\draw[-,draw=black,very thick](1,0) -- (1,0 + 2 );\n   \\draw[-,draw=black,very thick](1 - 1,0 + 1) -- (1 - 1,0 + 3.5);\n    \\draw[-,draw=black,very thick](1 - 1,0 + 3.5) -- (1,0 + 2);\n    \\draw[-,draw=black,very thick](1,0 + 2) -- (1 + 1,0 + 2.5);\n    \\draw[-,draw=black,very thick](1 + 1,0 + 2.5) -- (1,0 + 4);\n    \\draw[-,draw=black,very thick](1 - 1,0 + 3.5) -- (1,0 + 4);\n     \\draw[dashed,draw=black,very thick](1,0 + 4) -- (1,0 + 4.5);\n  \\draw[-,draw=black,very thick](1,0) -- (1 - 1,0 + 1);\n\\draw[-,draw=black,very thick] (1,0 - 1) -- (1,0);\n\\draw[-,draw=black,very thick] (1,0 - 2) -- (1,0 - 1);\n\\draw[<-,draw=black,thick] (1 - 3-1,0 + 5) -- (1 - 3 - 1,0 - 2);\n\\draw (1 - 3 + 0.2 - 1,0 - 2) node{$0$};\n\\draw (1 - 3- 1,0 - 2) node{-};\n\\draw (1 - 3- 1,0 - 1) node{-};\n\\draw (1 - 3- 1,0) node{-};\n\\draw (1 - 3- 1,0 + 2) node{-};\n\\draw (1 - 3- 1,0 + 1) node{-};\n\\draw (1 - 3- 1,0 + 2.5) node{-};\n\\draw (1 - 3- 1 ,0 + 3.5) node{-};\n\\draw (1 - 3 - 1,0 + 4) node{-};\n\\draw (1 - 3 + 0.3 - 1,0 - 1) node{$34$};\n\\draw (1 - 3 + 0.3 - 1,0) node{$52$};\n\\draw (1 - 3 + 0.3 - 1,0 + 1) node{$54$};\n\\draw (1 - 3 + 0.3 - 1,0 + 2) node{$64$};\n\\draw (1 - 3 + 0.3 - 1,0 + 2.5) node{$66$};\n\\draw (1 - 3 + 0.3 - 1,0 + 3.5) node{$70$};\n\\draw (1 - 3 + 0.3 - 1,0 + 4) node{$76$};\n\\draw (1 - 3 + 0.5 - 1,0 + 5) node{dim};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small Closure diagram of nilpotent orbits  of $E_{7(7)}$ of  dimension smaller than 76.}\n\\label{ClosureDiag}\n\\end{figure}\nIn the linearised approximation, the $F^2 \\nabla^4 R^4$ type invariant does not carry a $\\nabla^6 R^4$ coupling, but we explain that the structure of the linearised invariant allows for this mixing at the non-linear level, and that the latter must occur because the two classes of invariants merge in one single $E_{8(8)}$ representation in three dimensions. We conclude that the exact threshold function in four dimensions takes the form\n\\begin{equation}}\\def\\ee{\\end{equation} E_{\\gra{0}{1}} = \\hat{{\\cal E}}_{\\grad{8}{1}{1}}  + \\frac{32}{189\\pi}\\hat{{E}}_{\\mbox{\\DEVII000000{5}}}\\ ,  \\ee\nwhere $\\hat{{\\cal E}}_{\\grad{8}{1}{1}}$ is the solution to the inhomogeneous differential equation \\eqref{E811Equation} that is consistent with perturbative string theory. The explicit relation between the tensorial differential equations and the associated nilpotent orbits permits us to determine the wavefront set of the associated functions, extending the results of \\cite{Green:2011vz,Fleig:2013psa} to the $\\nabla^6 R^4$ threshold function. It appears, as can be seen in figure \\ref{ClosureDiag}, that the two functions admit distinct wavefront sets. In particular we show that although $\\hat{{\\cal E}}_{\\grad{8}{1}{1}}$ is not an Eisenstein series, it admits the same wavefront set as $\\hat{{E}}{\\mbox{\\DEVII{\\mathnormal{6}}00000{\\mathfrak{0}}}}$.\n\n\nWe then consider the uplift of our results in higher dimensions, and exhibit that this general structure extends to all dimensions lower than eight, and is in perfect agreement with the exact threshold functions proposed in \\cite{Green:2010wi,Basu:2007ck,Pioline:2015yea}. In each dimension, the supersymmetry invariants transform in irreducible representations of $E_{d(d)}$, defined by the representation of $E_{d(d)}$ on the associated function on $E_{d(d)}\/ K_d$ satisfying to the relevant differential equations implied by supersymmetry. The inequivalent invariants are summarised in figure \\ref{DimensionMultiplets}.\n\\begin{figure}[htbp]\n\\center\n \\begin{tikzpicture}\n\\draw (1 - 1,0 + 5) node{$IIA$}; \\draw (1,0 + 5) node{$IIB$};   \\draw (1 + 2,0 + 5) node{$IIA$}; \\draw (1 + 3,0 + 5) node{$IIB$};\n \\draw (1 + 5,0 + 5) node{$IIA$}; \\draw (1 + 6,0 + 5) node{$IIB$};\n \n\\draw[-,draw=black, thick](1 - 1,0 + 4) -- (1 - 1,0 + 4.7); \\draw[-,draw=black, thick](1,0 + 4) -- (1,0 + 4.7);\n\\draw[-,draw=black, thick](1 + 2,0 + 4) -- (1 + 2,0 + 4.7); \\draw[-,draw=black, thick](1 + 3,0 + 4) -- (1 + 3,0 + 4.7);\n\\draw[-,draw=black, thick](1 + 5,0 + 4) -- (1  + 5,0 + 4.7); \\draw[-,draw=black, thick](1 + 6,0 + 4) -- (1 + 6,0 + 4.7);\n\\draw[dashed,draw=black, thick](1 + 7,0 + 4) -- (1 + 7,0 + 4.5);\n\\draw[-,draw=black, thick](1,0 + 4) -- (1 - 0.9,0 + 4.7);\n\\draw[-,draw=black, thick](1 + 3,0 + 4) -- (1 + 2.1,0 + 4.7);\n\n\\draw[dashed,draw=black, thick](1 + 2.5,0 - 1) -- (1 + 2.5,0 - 1.3);\n\\draw[dashed,draw=black, thick](1 - 0.5,0 - 1) -- (1 - 0.5,0 - 1.3);\n\\draw[dashed,draw=black, thick](1 + 3 + - 1 + 7.5\/2,0 - 1) -- (1 + - 1  + 3 + 7.5\/2 ,0 - 1.3);\n\n\\draw[<-,draw=black,thick] (1 - 3,0 + 6) -- (1 - 3,0 - 2);               \n\\draw (1 - 3,0 + 5) node{-};\n\\draw (1 - 3,0 + 4) node{-};   \\draw (1 + - 1,0 + 4) node{\\color{rouge}  \\textbullet};\\draw (1 + 1 + - 1,0 + 4) node{ \\textbullet};  \\draw (1 + - 1 + 3,0 + 4) node{\\textbullet};\\draw (1 + - 1 + 4,0 + 4) node{$\\circ$};\n\\draw (1 - 3,0 + 3) node{-};                \\draw (1 + - 1 + 0.5,0 + 3) node{\\textbullet};   \\draw (1 + - 1 + 3,0 + 3) node{\\textbullet};\\draw (1 + 4 + - 1 ,0 + 3) node{$\\circ$};\n\\draw (1 - 3,0 + 2) node{-};                \\draw (1 + - 1 + 0.5,0 + 2) node{\\textbullet};   \\draw (1 + - 1 + 3,0 + 2) node{\\textbullet};\\draw (1 + 4 + - 1,0 + 2) node{\\color{rouge} \\textbullet};\n\\draw (1 - 3,0 + 1) node{-};                \\draw (1 + - 1 + 0.5,0 + 1) node{\\textbullet};   \\draw (1 + - 1 + 3.5,0 + 1) node{\\textbullet};\n\\draw (1 - 3,0) node{-};\t\t\t    \\draw (1 + - 1 + 0.5,0) node{\\textbullet};         \\draw (1 + - 1 + 3.5,0) node{\\textbullet};\n\\draw (1 - 3,0 - 1) node{-};                 \\draw (1 + - 1 + 0.5 ,0 - 1) node{\\textbullet};   \\draw (1 + - 1 + 3.5,0 - 1) node{\\textbullet};\n\\draw (1 - 3 + 0.3,0 + 5) node{$10$};\n\\draw (1 - 3 + 0.3,0 + 4) node{$8$};\n\\draw (1 - 3 + 0.3,0 + 3) node{$7$};\n\\draw (1 - 3 + 0.3,0 + 2) node{$6$};\n\\draw (1 - 3 + 0.3,0 + 1) node{$5$};\n\\draw (1 - 3 + 0.3,0) node{$4$};\n\\draw (1 - 3 + 0.3,0 - 1) node{$3$};\n\\draw (1 - 3 + 0.5,0 + 6) node{dim};\n\n\\draw (1 + - 1 + 6\/2 + 7.5\/2 ,0 - 1) node{\\textbullet}; \n\\draw (1 + - 1 + 6 ,0 ) node{\\color{rouge}  \\textbullet}; \n\\draw (1 + - 1 + 7.5 ,0 ) node{ \\textbullet}; \n\\draw (1 + - 1 + 6 ,0 +1) node{$\\circ$}; \n\\draw (1 + - 1 + 7.5 ,0 +1) node{\\textbullet}; \n\\draw (1 + - 1 + 6 ,0 +2) node{$\\circ$}; \n\\draw (1 + - 1 + 7.5 ,0 +2) node{\\color{rouge} \\textbullet}; \n\\draw (1 + - 1 + 6 ,0 +3) node{$\\circ$}; \n\\draw (1 + - 1 + 7.5 ,0 +3) node{$\\circ$}; \n\\draw (1 + - 1 + 6 ,0 +4) node{$\\circ$}; \n\\draw (1 + - 1 + 7 ,0 +4) node{$\\circ$};\n\\draw (1 + - 1 + 8 ,0 +4) node{$\\circ$}; \n\n\\draw (1 + 0.5 + - 1,0 - 1.6) node{$R^4$};\n\\draw (1 + 3.5 + - 1,0 - 1.6) node{$\\nabla^4 R^4$};\n\\draw (1 + 6\/2 + 7.5\/2 + - 1,0 - 1.6) node{$\\nabla^6 R^4$};\n\n\\draw[-,draw=black, thick](1 + 8 + - 1,0  + 4) -- (1 + 0.5 + 7 + - 1,0 + 3);\n\\draw[-,draw=black, thick](1 + 0.5 + 6.5 + - 1,0  + 4) -- (1 + 0.5 + 7 - 1,0 + 3);\n\\draw[-,draw=black, thick](1 + 0.5 + - 1,0 -1) -- (1 + 0.5 + - 1,0 + 3);\n\\draw[-,draw=black, thick](1 + 3.5 + - 1 ,0 -1) -- (1 + 3.5 + - 1,0 + 1);\n\\draw[-,draw=black, thick](1 + 3 + 7.5\/2 + - 1 ,0 -1) -- (1 + 6 + - 1,0);\n\\draw[-,draw=black, thick](1 + 3 + 7.5\/2 + - 1 ,0 -1) -- (1 + 7.5 + - 1,0);\n\\draw[-,draw=black, thick](1 + 6 + - 1,0) -- (1 + 6 + - 1,0 + 4);\n\\draw[-,draw=black, thick](1 + 7.5 + - 1,0) -- (1 + 7.5 + - 1,0 + 3);\n\n\\draw[-,draw=black, thick](1 + 3.5 + - 1,0 + 1) -- (1 + - 1 + 4,0 + 2);  \\draw[-,draw=black, thick](1 + - 1 + 3.5,0 + 1) -- (1 + - 1 + 3,0 + 2);\n\\draw[-,draw=black, thick](1 + - 1 + 0.5,0 + 3) -- (1 + - 1 ,0 + 4); \\draw[-,draw=black, thick](1 + - 1 + 0.5,0 + 3) -- (1 + - 1 + 1,0 + 4);\n\n\\draw[-,draw=black, thick](1 + - 1 + 4,0 + 4) -- (1 + - 1 + 4,0 + 2);  \\draw[-,draw=black, thick](1 + - 1 + 3,0 + 4) -- (1 + - 1 + 3,0 + 2);\n\n\n\\end{tikzpicture}\n\n\n\\caption{\\label{DimensionMultiplets}\\small Each node corresponds to an inequivalent supersymmetry invariant, white if it cannot be written in harmonic superspace in the linearised approximation, and red if the corresponding harmonic superspace is chiral. For $\\nabla^6 R^4$, the links to 10 dimensions are valid for the homogeneous solution, while all the eight-dimensional invariants uplift to type IIA for the inhomogeneous solution.}\n\\end{figure}\nThe tensorial differential equations satisfied by Eisenstein functions relevant to our analysis are reviewed in the appendices. \n\n\\section{$\\mathcal{N}=8$ supergravity in four dimensions}\nMaximal supergravity includes 70 scalar fields parametrising the symmetric space $E_{7(7)} \/ SU_{\\scriptscriptstyle \\rm c}(8)$ \\cite{Cremmer:1979up}, and can be defined in superspace by promoting these fields to superfields $\\phi^\\upmu$ \\cite{Brink:1979nt,Howe:1981gz}. One defines the Maurer--Cartan form \n\\begin{equation}}\\def\\ee{\\end{equation} d {\\cal V} \\,  {\\cal V}^{-1} = \\left( \\begin{array}{cc}\\  2 \\delta_{[i}^{[k} \\omega^{l]}{}_{j]} \\ &\\  P_{ijkl} \\ \\\\ \\ P^{ijkl}\\ & -2 \\delta^{[i}_{[k} \\omega^{j]}{}_{l]} \\ \\end{array}\\right) \\ , \\ee\nwith  \n\\begin{equation}}\\def\\ee{\\end{equation} P^{ijkl} = \\frac{1}{24} \\varepsilon^{ijklpqrs} P_{pqrs} \\ . \\label{ComplexSelfual}\\ee\nThe metric on $E_{7(7)} \/ SU_{\\scriptscriptstyle \\rm c}(8)$ is defined as\n\\begin{equation}}\\def\\ee{\\end{equation} G_{\\upmu\\upnu}(\\phi) d\\phi^\\upmu d\\phi^\\upnu = \\frac{1}{3} P_{ijkl} P^{ijkl} \\ , \\ee\nand the derivative in tangent frame is defined such that for any function \n\\begin{equation}}\\def\\ee{\\end{equation} d {\\cal E} = 3 P^{ijkl} {\\cal D}_{ijkl} {\\cal E} \\ . \\ee\nThe superfields satisfy to\n\\begin{equation}}\\def\\ee{\\end{equation} D_\\alpha^i {\\cal E} = \\frac{1}{4} \\varepsilon^{ijklpqrs} \\chi_{\\alpha jkl} \\,  {\\cal D}_{pqrs} {\\cal E} \\ , \\qquad \\bar D_{\\adt i} {\\cal E} = 6 \\bar \\chi_{\\adt}^{jkl} \\,  {\\cal D}_{ijkl} {\\cal E} \\ , \\ee\nwhere $\\chi_{\\alpha ijk}$ is the Dirac superfield in Weyl components, and $\\bar \\chi_\\adt^{ijk}$ its complex conjugate. The expansion of the scalar fields include the 28 Maxwell field strengths $F_{\\alpha\\beta ij}$, the 8 Rarita--Schwinger field strengths $\\rho_{\\alpha\\beta\\gamma i}$ and the Weyl tensor $C_{\\alpha\\beta\\gamma\\delta}$, satisfying to $\\mathcal{N}=8$ supergravity classical (two derivatives) field equations. The supervielbeins are the solutions to the Bianchi identities defined such that the Riemann tensor is valued in $\\mathfrak{sl}(2,\\mathds{C}) \\oplus \\mathfrak{su}(8)$ and the $\\mathfrak{su}(8)$ component is identified with the scalar field curvature \\cite{Brink:1979nt,Howe:1981gz},\n\\begin{equation}}\\def\\ee{\\end{equation} R^i{}_j = \\frac{1}{3} P_{jklp} \\wedge P^{iklp} \\ . \\ee\nThe covariant derivative on $E_{7(7)} \/ SU_{\\scriptscriptstyle \\rm c}(8)$ in tangent frame satisfies to \n\\begin{equation}}\\def\\ee{\\end{equation} [ {\\cal D}^{ijkl} , {\\cal D}_{pqrs} ]  {\\cal D}_{tuvw}= - 24 \\delta^{ijkl}_{qrs][t} {\\cal D}_{uvw][p} + 3 \\delta^{ijkl}_{pqrs} {\\cal D}_{tuvw} \\ ,  \\label{Comut} \\ee\nand the Laplace operator is\n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta = \\frac{1}{3} {\\cal D}^{ijkl} {\\cal D}_{ijkl} \\ . \\ee\nIn the linearised approximation, the scalar superfield $W_{ijkl}$ satisfies to the reality constraint \\eqref{ComplexSelfual} and to\n\\begin{equation}}\\def\\ee{\\end{equation} D_\\alpha^p W_{ijkl} = 2 \\delta^p_{[i} \\chi_{\\alpha jkl]} \\ , \\qquad \\bar D_{\\adt p } W_{ijkl} = \\frac{1}{12} \\varepsilon_{ijklpqrs} \\bar \\chi_\\adt^{qrs} \\ . \\ee \nIn this approximation the superfield $W^{ijkl}$ transforms in the minimal unitary representation of the superconformal group $SU(2,2|8)$ \\cite{Gunaydin:1984vz}. This property permits a complete classification of supersymmetry invariants in the linearised approximation in terms of irreducible representations of $SU(2,2|8)$ of Lorentz invariant top component \\cite{Drummond:2003ex,Drummond:2010fp}. In our analysis, we rely on the assumption of absence of supersymmetry anomaly, such that there is no algebraic obstruction to the extension of a linearised invariant to a full non-linear invariant. This implies a bijective correspondence between the set of linearised invariants and the non-linear invariants, such that one can deduce the explicit gradient expansion of the functions (or tensor functions) of the scalar fields on $E_{7(7)} \/ SU_{\\scriptscriptstyle \\rm c}(8)$ that determine the invariants.\n\\subsection{The standard $\\nabla^6 R^4$ type invariant}\\label{811D6R4}\nOne can define a $\\nabla^6 R^4$ type invariant in harmonic superspace, using the harmonic variables $u^1{}_i,\\, u^r{}_i,\\, u^8{}_i$ parametrising $SU(8)\/ S(U(1) \\times U(6)\\times U(1))$, such that $r=2$ to $7$ of $SU(6)$ \\cite{Drummond:2003ex,Drummond:2010fp,Hartwell:1994rp}. In this case the harmonic superspace integral can be defined at the non-linear level \\cite{Bossard:2011tq}, but we will only consider its linearised approximation. The superfield  in the ${\\bf 20}$ of $SU(6)$\n\\begin{equation}}\\def\\ee{\\end{equation} W_{rst} = u^i{}_8 u^j{}_r u^k{}_s u^l{}_t W_{ijkl} \\ , \\ee\nsatisfies to the G-analyticity constraints \n\\begin{equation}}\\def\\ee{\\end{equation} u^1{}_i D_\\alpha^i W_{rst} = 0   \\ , \\qquad u^i{}_8 \\bar D_{\\adt i} W_{rst} = 0 \\  . \\ee\nOne can therefore integrate any function of $W_{rst}$ on the associated analytic superspace. To understand the most general integrand, we must decompose monomials of $W_{rst}$ in irreducible representations of $SU(6)$. At quadratic order we have the representation $[0,0,2,0,0]$ and the combination \n\\begin{equation}}\\def\\ee{\\end{equation} W^{rtu} W_{stu} = \\frac{1}{6} \\varepsilon^{rtuvwx} W_{stu} W_{vwx} \\ee\nin the $[1,0,0,0,1]$. Because one obtains the $[0,0,2,0,0]$ by simply adding the Dynkin labels of $W_{rst}$, we will say that this representation is freely generated, whereas we shall consider the $[1,0,0,0,1]$ as a new generator at order two. At cubic order, we have the two elements freely generated by the ones already discussed, {\\it i.e.}\\  $[0,0,3,0,0]$ and $[1,0,1,0,1]$, and the additional combination \n\\begin{equation}}\\def\\ee{\\end{equation} W_{u[rs} W_{t]vw} W^{uvw} \\ , \\ee\nin the $[0,0,1,0,0]$. At quartic order we have the four elements freely generated by the ones already discussed, and the two additional elements \n\\begin{equation}}\\def\\ee{\\end{equation} W_{vw[r} W^{vw[t} W_{s]xy} W^{u]xy}  \\ , \\quad  W_{urs} W_{tvw} W^{uvw} W^{rst} \\ , \\ee\nthat decompose into the $[0,1,0,1,0]$ and the singlet representation. One checks that these elements freely generate the general polynomials in $W_{rst}$, such that the latter are labeled by five integers. \n\nTo integrate such a function in analytic superspace, one needs to consider these generating monomials with additional harmonic variables in order to compensate for the $S(U(1)\\times U(6)\\times U(1))$ representation, {\\it i.e.}\\  \n\\begin{eqnarray} \\label{Uintegral811}  \\int du\\,  u^8{}_i u^r{}_j u^s{}_k u^t{}_l W_{rst} &=& W_{ijkl} \\, , \\\\\n\\int du\\,  u^8{}_i u^s{}_j u_1{}^k u_r{}^l  W^{rtu} W_{stu}  &=& W_{ijpq} W^{klpq} - \\frac{1}{28} \\delta_{ij}^{kl} W_{pqrs} W^{pqrs}  \\, , \\nonumber \\\\*\n\\int du\\,  u_1{}^q u^8{}_p u^8{}_i u^r{}_j u^s{}_k u^t{}_l W_{u[rs} W_{t]vw} W^{uvw}   &=& W_{po[ij} W_{kl]mn} W^{qomn} - \\frac{|W|^2\\hspace{-1.7mm}}{108} \\scal{ \\delta_p^q W_{ijkl} - \\delta^p_{[i} W_{jkl]p}}    \\, , \\nonumber \\\\*\n\\int du u_1{}^k u_1{}^l u^8{}_i u^8{}_j  W_{urs} W_{tvw} W^{uvw} W^{rst} &=& W_{npq(i} W_{j)mp^\\prime q^\\prime} W^{np^\\prime q^\\prime(k} W^{l)pqm}  - \\delta_{(i}^{(k} \\delta_{j)}^{l)} ( \\dots ) \\ ,\\nonumber \n\\end{eqnarray}\nwhich are respectively in the $[0,0,0,1,0,0,0]$, the $[0,1,0,0,0,1,0]$, the $[1,0,0,1,0,0,1]$ and the $[2,0,0,0,0,0,2]$ irreducible representations of $SU(8)$, whereas  \n\\begin{equation}}\\def\\ee{\\end{equation} \\int du u_1{}^m u_1{}^n u^8{}_k u^8{}_l u^r{}_i u^s{}_j  u_t{}^p u_u{}^q  W_{vwr} W^{vwt} W_{sxy} W^{uxy}  = W_{i^\\prime j^\\prime [ij} W_{k]lk^\\prime l^\\prime} W^{i^\\prime j^\\prime [pq} W^{m]nk^\\prime l^\\prime} + \\dots \\label{Uintegral811B}\n\\ee\ngives rise to the fourth order monomial in the $[1,0,1,0,1,0,1]$ irreducible representation. \n\nOne obtains in this way that the harmonic superspace integral of a general monomial of order $n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime + 4$ in the $[n_2,n_4,n_1+n_3,n_4,n_2]$ of $SU(6)$ gives rise to a term in $\\nabla^6 R^4$ with a monomial of order $n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime $ in the $[n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]$ of $SU(8)$, {\\it i.e.}\\  \n\\begin{eqnarray} &&\\int du  D^{14} \\bar D^{14} \\, F(u)_{\\scriptscriptstyle [n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]}^{[ n_2,n_4,n_1+n_3,n_4,n_2]}  W^{n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime + 4}|_{[n_2,n_4,n_1+n_3,n_4,n_2]} \\nonumber \\\\*\n&&\\sim \\nabla^6 R^4  \\, W^{n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime }|_{[n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]} +\\dots\\label{811Linear} \\end{eqnarray}\nwhere the function $F(u)$ is the function of the harmonic variable defined as a product of the generating functions defined in (\\ref{Uintegral811},\\ref{Uintegral811B}). One needs at least one quartic singlet in the G-analytic superfield to get a non-vanishing integral \\cite{Drummond:2003ex}. \n\nReferring to the one to one correspondence between linearised and non-linear invariants \\cite{Drummond:2003ex}, one deduces that the non-linear invariant must admit the same gradient expansion, {\\it i.e.}\\  \n\\begin{eqnarray}&&  {\\cal L}_\\grad811[{\\cal E}_\\grad811] \\nonumber \\\\*\n&=& \\hspace{-5mm}\\sum_{n_1,n_2,n_3,n_4,n_4^\\prime} \\hspace{-5mm} {\\cal D}^{n_1 + 2 n_2 + 3 n_3 + 4 n_4 + 4 n^\\prime_4 }_{\\scriptscriptstyle [n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]} \\hspace{2mm} \n{\\cal E}_{\\grad{8}{1}{1}} \\, {\\cal L}^{\\scriptscriptstyle [n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]}_\\grad811 \\hspace{5mm} \\  \\label{811GradExpand} \\end{eqnarray}\nwhere each $ {\\cal L}^{\\scriptscriptstyle [n_3+n_4 +2n_4^\\prime,n_2,n_4,n_1+n_3,n_4,n_2,n_3+n_4 +2n_4^\\prime]}_\\grad811$ is an $E_{7(7)}$ invariant superform in the corresponding representation of $SU(8)$. Note that although the irreducible representation remains unchanged under the substitution \n\\begin{equation}}\\def\\ee{\\end{equation} (n_1,n_3,n_4^\\prime) \\rightarrow(n_1+2, n_3-2,n_4^\\prime +1)\\ee\nthe corresponding superforms and the tensor structure of the derivative are different, and are really labelled by the five integers $n_1,n_2,n_3,n_4,n_4^\\prime$ without any further identification. Of course the mass dimension implies that these integers are bounded from above, and the maximal weight terms in $\\chi^{14} \\bar \\chi^{14}$ can only be in representations like $[2,6,0,8,0,6,2]$, $[2,6,1,6,1,6,2]$, \\dots $[2,10,0,0,0,10,2]$, \\dots $[11,1,0,0,0,1,11]$. \n\n\nThis gradient expansion implies in particular that the third order derivative of ${\\cal E}_\\grad811$ in the $[0,2,0,0,0,0,0]$ and its complex conjugate must vanish, {\\it i.e.}\\  \n\\begin{eqnarray}  \\Scal{ 4 {\\cal D}_{ijpq} {\\cal D}^{pqmn} {\\cal D}_{mnkl} - {\\cal D}_{ijkl} \\scal{ \\Delta + 24} } {\\cal E}_\\grad811 &=& 0 \\ , \\label{CubicC}\\nonumber \\\\*\n  \\Scal{ 4 {\\cal D}^{ijpq} {\\cal D}_{pqmn} {\\cal D}^{mnkl} - {\\cal D}^{ijkl} \\scal{ \\Delta + 24} } {\\cal E}_\\grad811 &=& 0 \\ .  \\end{eqnarray}\nThese equations imply all the higher order constraints on the function such that  its gradient expansion is in agreement with \\eqref{811GradExpand}. \nDefining the covariant derivative in tangent frame as a Lie algebra generator in the fundamental representation of $E_{7(7)}$, this equation reads equivalently \n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf D}_{56}^{\\; 3} {\\cal E}_\\grad811  = {\\bf D}_{56} \\Scal{ 6 + \\tfrac{1}{4} \\Delta } {\\cal E}_\\grad811 \\ . \\label{Cubic56} \\ee\nThis implies in particular that all the Casimir operators are determined by the quadratic one such that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\rm tr} \\scal{ {\\bf D}_{56}^{\\; 2+2n}} \\, {\\cal E}_\\grad811= 6 \\Delta \\scal{ 6 + \\tfrac{1}{4} \\Delta}^{n} {\\cal E}_\\grad811 \\ , \\ee\nbut the quadratic Casimir is not a priori determined by equation \\eqref{CubicC} alone. We will need to consider the other invariants to finally conclude that supersymmetry moreover implies \\cite{Green:2010kv}\n\\begin{equation}}\\def\\ee{\\end{equation}  \\Delta  {\\cal E}_\\grad811 = - 60  {\\cal E}_\\grad811 - ({\\cal E}_\\grad844)^2  \\ .\\label{Laplace811} \\ee\nEquation \\eqref{Cubic56} defines a qantization of the algebraic condition ${\\bf Q}_{56}^{\\; 3}=0$ associated to the complex nilpotent orbit of $E_{7}$ of Dynkin label \\DEVII{2}00000{\\mathfrak{0}}, while the condition that the fourth order derivative does not vanish generically in the ${\\scriptstyle [2,0,0,0,0,0,2]}$ distinguishes its real form of  $SU(8)$ Dynkin label [$\\scriptstyle \\mathfrak{2}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}\\mathfrak{2}$] \\cite{E7Djo}, which defines the graded decomposition of $SU(8)$ associated to the $(8,1,1)$ harmonic superspace we consider in this section. The property that the linearised structure does not permit to determine the eigenvalue of the Laplace operator in this case, implies that the quantization of the associated nilpotent orbit is not unique, and depends on one free parameter. This property follows from the fact that a nilpotent element of this kind can be obtained as the appropriate limit of a semi-simple element satisfying to the characteristic equation ${\\bf Q}_{56}^{\\; 3}=\\frac{1}{24} {\\rm tr}({\\bf Q}_{56}^{\\; 2}) {\\bf Q}_{56} $. \n\n\n\\subsection{$F^2 \\nabla^4 R^4$ type invariant and its relation to $\\nabla^6 R^4$}\n\\label{F2D4R4} \nAlthough the $\\nabla^6 R^4$ type invariant provides the unique supesymmetric invariant preserving $SU(8)$ one can write at this order, there is another class of invariants that can be defined form the chiral harmonic superspace defined in terms of the harmonic variables $u^{\\hat{r}}{}_i,\\, u^r{}_i$ parametrising $SU(8)\/S(U(2)\\times U(6))$ \\cite{Drummond:2003ex,Hartwell:1994rp},  with $\\hat{r},\\hat{s}$ equal to  $1, 2$ of $SU(2)$, and $r ,s$ running from $3$ to $8$ of $SU(6)$. One defines the superfield \n\\begin{equation}}\\def\\ee{\\end{equation}\nW^{rs} = u^1{}_i u^2{}_j u^r{}_k u^s{}_l W^{i j k l}\n\\ee\nthat satisfies to the G-analiticity constraint \n\\begin{equation}}\\def\\ee{\\end{equation} \nu^{\\hat{r}}{}_{i} \\bar D^i_\\alpha W^{rs} = 0 \\ . \n\\ee\nSimilarly as in the preceding section, the most general function of $W^{rs}$ is freely generated by the three monomials \n\\begin{equation}}\\def\\ee{\\end{equation} W^{rs} \\ , \\qquad \\frac{1}{2} \\varepsilon_{rstuvw} W^{tu} W^{vw} \\ , \\qquad \\frac{1}{2} \\varepsilon_{rstuvw} W^{rs} W^{tu} W^{vw} \\ . \\ee\nOne must supplement them with harmonic variables to preserve $S(U(2)\\times U(6))$ invariance, using \n\\begin{eqnarray} \\int du\\, u^i{}_1 u^j{}_2 u^k{}_r u^l{}_s \\, W^{rs} &=& W^{ijkl} \\ , \\\\\n \\int du\\, u^i{}_1 u^j{}_2 u^r{}_k u^s{}_l \\,  \\frac{1}{2} \\varepsilon_{rstuvw} W^{tu} W^{vw}  &=& W^{ijpq} W_{klpq} - \\frac{1}{28} \\delta^{ij}_{kl}W^{pqrs} W_{pqrs} \\ , \\nonumber \\\\*\n  \\int du\\, u^i{}_1 u^j{}_2 u^k{}_1 u^l{}_2 \\,  \\frac{1}{2} \\varepsilon_{rstuvw} W^{rs} W^{tu} W^{vw}  &=& W^{ijpq} W_{pqrs} W^{rskl}  - \\frac{1}{12} W^{ijkl} W_{pqrs} W^{pqrs}  \\ . \\nonumber\n\\end{eqnarray}\nOne only gets a non-trivial integral if the cubic $SU(6)$ singlet in $W^{rs}$ appears at least quadratically, which can be understood from the property that the associated chiral primary operator of $SU(2,2|8)$ is otherwise in a short representation \\cite{Drummond:2003ex}. Because the $U(1)$ weight of the measure is compensated by a single factor of this cubic $SU(6)$ singlet, it appears that there is no $SU(8)$ invariant that exists in this class. \n\n For a general monomial, one gets an invariant of the form \n\\begin{eqnarray}&&  \\int du \\bar D^{16}  D^{12}  \\, F(u)^{[0,n_1,0,n_2,0]}_{[0,n_2+2n_3+2,0,n_1,0,n_2,0]} \\, W^{n_1 + 2n_2 + 3n_3 + 6} |_{[0,n_1,0,n_2,0]}  \\\\\n&\\sim&W^{n_1 + 2n_2 + 3n_3}_{\\scriptscriptstyle [0,n_2+2n_3,0,n_1,0,n_2,0]} \\bar F^{2}_{\\scriptscriptstyle[0,2,0,0,0,0,0]} \\nabla^4 R^4+\\dots + W^{n_1+2n_2+n_3-22}_{\\scriptscriptstyle [0,n_2+2n_3-8,0,n_1-8,0,n_2-4,0]}\\bar \\chi^{16}_{\\scriptscriptstyle[0,8,0,4,0,0,0]}  \\chi^{12}_{\\scriptscriptstyle[0,2,0,4,0,4,0]} \\ , \\nonumber\\end{eqnarray}\nwhere all terms are projected to the $\\scriptstyle [0,n_2+2n_3+2,0,n_1,0,n_2,0]$ irreducible representation, and the term in $\\bar F^2$ is\n\\begin{equation}}\\def\\ee{\\end{equation} \\bar F_{\\adt\\bdt}^{ij} \\bar F^{\\adt\\bdt  kl}  -   \\bar F_{\\adt\\bdt}^{[ij} \\bar F^{kl]\\adt\\bdt} \\ . \\ee\nFor a generic function ${\\cal F}[W]$ of $W_{rs}$, one obtains\n\\begin{equation}}\\def\\ee{\\end{equation} D^{16} \\bar D^{12}  {\\cal F}[W] = \\sum_{n_1,n_2,n_3} \\frac{\\partial^{n_1+2n_2+3n_3+6} {\\cal F}[W]}{\\partial W^{n_1+2n_2+3n_3+6}}{}\\Big|_{[0,n_2,0,n_1,0]} {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{[0,n_1,0,n_2,0]\\ord{n_1+2n_2+3n_3+3}} \\ , \\ee  \nwhere the densities $ {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{[0,n_1,0,n_2,0]\\ord{n_1+2n_2+3n_3+3}}$ are of order $n_1+2n_2+3n_3+6$ in the fields and only depend on the scalar fields through their space-time derivative. The number $n_1+2n_2+3n_3+3$ is the $U(1)$ weight of the density. These densities determine by construction covariant superforms in the linearised approximation \\cite{Voronov,Gates:1997kr,Gates:1997ag}, such that \n\\begin{eqnarray} d^\\ord{0}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl} &=& 0 \\ ,  \\nonumber \\\\*\n\\Scal{ d^\\ord{0}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,pqrs} + 3 P^{pqrs} \\wedge   {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl} }{}_{\\scriptscriptstyle [0,2,0,1,0,0,0]} &=& 0 \\ ,  \\nonumber \\\\*\n\\Scal{ d^\\ord{0}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,pqrs,mntu} + 3 P^{pqrs} \\wedge   {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,mntu} }{}_{\\scriptscriptstyle [0,2,0,2,0,0,0]} &=& 0 \\  , \\nonumber \\\\*\n\\Scal{ d^\\ord{0}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,pq}{}_{rs}+18 P_{rsmn} \\wedge   {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,pqmn} }{}_{\\scriptscriptstyle [0,3,0,0,0,1,0]} &=& 0 \\  ,\n\\end{eqnarray}\nwhere $d^\\ord{0}$ is the superspace exterior derivative in the linear approximation. At the next order, because \n\\begin{equation}}\\def\\ee{\\end{equation} d = \\sum_{n=0}^{\\infty} d^\\ord{n} \\ee\nsatisfies to $d^2=0$, one has \n\\begin{equation}}\\def\\ee{\\end{equation} \\{ d^\\ord{0},d^\\ord{1}\\} = 0\\ , \\ee\nand therefore \n\\begin{equation}}\\def\\ee{\\end{equation} d^\\ord{0} \\Scal{ d^\\ord{1}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl} } = 0 \\ . \\ee\nWe assume in this paper that the structure of superconformal multiplets implies the absence of supersymmetry anomaly, or equivalently that the fifth cohomology of $d^\\ord{0}$ is empty. Nevertheless, even if $d^\\ord{1}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl} $ only depends on the covariant superfields, nothing prevents its $d^\\ord{0}$ antecedent to depend explicitly on the scalar fields.  This implies in this case that\n\\begin{equation}}\\def\\ee{\\end{equation} d^\\ord{1}  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl}  =-d^\\ord{0}  {\\cal L}_{\\grad820\\, \\ord{1}}^{ij,kl} +  P_{pqrs} \\wedge  {\\cal M}^{ij,kl,pqrs} + P^{pqij} \\wedge {\\cal M}_{pq}{}^{kl} + P^{pqkl} \\wedge {\\cal M}_{pq}{}^{ij} -2 P^{i]pq[k} \\wedge {\\cal M}_{pq}{}^{l][j} \\ , \\label{d1F2R4} \\ee\nwhere $ {\\cal L}_{\\grad820\\, \\ord{1}}^{ij,kl}$ is the covariant correction to the superform, whereas ${\\cal M}^{ij,kl,pqrs}$ and ${\\cal M}^{ij}{}_{kl}$ are superforms of order six in the fields in the $[0,2,0,1,0,0,0]$ and the $[0,1,0,0,0,1,0]$, respectively, that must satisfy to\n\\begin{eqnarray}\nd^\\ord{0} {\\cal M}^{ij,kl,pqrs} &=& \\scal{ P^{pqrs} \\wedge \\mathcal{N}^{ij,kl} }{}_{\\scriptscriptstyle [0,2,0,1,0,0,0]}\\ ,  \\nonumber \\\\*\nd^\\ord{0} {\\cal M}^{ij}{}_{kl} &=& P^{ijpq} \\wedge \\mathcal{N}_{klpq} - \\frac{1}{28} \\delta^{ij}_{kl} P^{pqrs} \\wedge \\mathcal{N}_{pqrs} \\ . \\end{eqnarray}\nIn order to have such corrections that could not be reabsorbed in a covariant correction as $ {\\cal L}_{\\grad820\\, \\ord{1}}^{ij,kl}$, one must have a corresponding short multiplet associated to a linearised invariant of the same dimension. The only candidate for a superform $ {\\cal M}^{ij,kl,pqrs}$ is $  {\\cal L}_{\\grad820\\, {\\rm \\scriptscriptstyle lin}}^{ij,kl,pqrs} $, but it is of order seven in the fields, and therefore  $ {\\cal M}^{ij,kl,pqrs}=0$ at this order. However, there is a candidate for $ {\\cal M}^{ij}{}_{kl} $ which is $  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}ij}{}_{kl} $, the superform that appears in the $\\nabla^6 R^4$ type invariant discussed in the last section. Following \\eqref{811Linear}, we have \n\\begin{eqnarray} d^\\ord{0}  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}   &=&  0 \\ , \\nonumber \\\\*\n d^\\ord{0}  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{ijkl}   &=&  - 3 P^{ijkl} \\wedge   {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}  \\ , \\nonumber \\\\*\n d^\\ord{0}  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{ijkl,pqrs}     &=&  - 3 \\scal{ P^{ijkl} \\wedge {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{pqrs}  }{}_{\\scriptscriptstyle [0,0,0,1,0,0,0]}  \\ , \\nonumber \\\\*\n   d^\\ord{0}   {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}ij}{}_{kl}  &=&  - 18 \\scal{ P_{klpq} \\wedge {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{ijpq}  }{}_{\\scriptscriptstyle [0,1,0,0,0,1,0]}   \\ ,\\label{Linear811}\n\\end{eqnarray}\nand therefore \n\\begin{eqnarray} && d^\\ord{0} \\Bigl(\\Scal{ W^{ijpq} W_{pqrs} W^{rskl}  - \\tfrac{1}{12} W^{ijkl} W_{pqrs} W^{pqrs} }   {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}  \\Bigr . \\nonumber \\\\*\n&&  \\qquad + W^{ijpq} W_{pqrs}   {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{rskl} + W^{ijpq} W^{klrs}  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}{}_{pqrs}+  W^{klpq} W_{pqrs}   {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{rsij}     \\nonumber \\\\*\n&& \\hspace{10mm} \\Bigl . + 6 W^{pqij} {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}kl}{}_{pq} + 6 W^{pqkl} {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}ij}{}_{pq} -12 W^{i]pq[k} {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}l][j}{}_{pq} \\Bigr)\\nonumber \\\\*\n&=& 18 \\Scal{ P^{pqij}\\wedge  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}kl}{}_{pq} + P^{pqkl} \\wedge {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}ij}{}_{pq} -2 P^{i]pq[k}\\wedge  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}l][j}{}_{pq}} \\ , \\label{d0Cohom}\n \\end{eqnarray}\nsuch that $  {\\cal L}_{\\grad811\\, {\\rm \\scriptscriptstyle lin}}^{\\hspace{10mm}ij}{}_{kl} $ is indeed a consistent candidate. Moreover, the structure of the linearised $(8,1,1)$ invariant does not permit to have the tensor function $W^{ijpq} W_{pqrs} W^{rskl}$, such that \\eqref{d0Cohom} is not the exterior derivative of a superform that does not depend on the naked scalar fields (uncovered by a space-time derivative). It follows that such a correction, if it appeared in \\eqref{d1F2R4}, could not be reabsorbed in a redefinition of $ {\\cal L}_{\\grad820\\, \\ord{1}}^{ij,kl}$. \n\n\nIf this mixing between the $(8,2,0)$ and the $(8,1,1)$ superforms was not appearing at the non-linear level, then the action of the exterior derivative in the function of the scalar fields should not introduce lower derivative terms such that it should satisfy then to \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{ijpq} \\Scal{  4 {\\cal D}_{pqrs} {\\cal D}^{rsmn} {\\cal D}_{mnkl} - {\\cal D}_{pqkl} \\scal{ \\Delta + 24} } {\\cal E}_{\\grad{8}{2}{0}} = 0 \\ . \\label{D4Consistency} \\ee\nIf the mixing did appear, then the unicity of the linearised invariants \\eqref{Linear811} would imply that the corresponding non-linear superform should be the same as in \\eqref{811GradExpand}, such that once again the exterior derivative acting on $D^3_{\\scriptscriptstyle [0,2,0,0,0,0,0]} {\\cal E}_\\grad820$ should not generate lower derivative terms and one would conclude again that \\eqref{D4Consistency} must be satisfied. Therefore this equation must be satisfied in either cases.\n\n\nUsing moreover the property that the gradient expansion of the linearised invariant is inconsistent with the presence of the third order derivative in the $[1,0,0,1,0,0,1]$ of $SU(8)$, one requires\n\\begin{equation}}\\def\\ee{\\end{equation} \\Scal{ 36 {\\cal D}_{jr[kl} {\\cal D}^{irmn} {\\cal D}_{pq]mn} - \\delta^i_j {\\cal D}_{klpq} ( \\Delta + 42)   + \\delta^i_{[k} {\\cal D}_{lpq]j} ( \\Delta-120)} {\\cal E}_{\\grad{8}{2}{0}} = 0 \\ . \\label{CubicR} \n\\ee\nUsing this equation one computes independently of \\eqref{D4Consistency} that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{ijpq} \\Scal{  4 {\\cal D}_{pqrs} {\\cal D}^{rsmn} {\\cal D}_{mnkl} - {\\cal D}_{pqkl} \\scal{ \\Delta + 24} } {\\cal E}_{\\grad{8}{2}{0}}  = \\frac{1}{12} \\Scal{ 28 {\\cal D}^{ijpq} {\\cal D}_{klpq} - 3 \\delta^{ij}_{kl} \\Delta} \\scal{ \\Delta + 60 } {\\cal E}_\\grad820 \\ee\nand we conclude that \\eqref{D4Consistency}  and \\eqref{CubicR} imply together\n\\begin{equation}}\\def\\ee{\\end{equation}  \\Delta {\\cal E}_\\grad820 = -60  {\\cal E}_\\grad820 \\ .\\label{Laplace820}  \\ee\nThis eigenvalue is such that the structure of the invariant is consistent with the mixing between the $(8,2,0)$ and the $(8,1,1)$ superforms. Only in this case can they reduce to the same invariant for a function ${\\cal E}_{\\grad{8}{2}{2}}$ satisfying to both \\eqref{CubicC} and \\eqref{CubicR}, as for the $\\nabla^4 R^4$ type invariant. \n\nWe are going to argue now that this chiral invariant must indeed include a $\\nabla^6 R^4$ coupling, because the two classes of invariants reduce to one single class in three dimensions. But before to do this, let us mention that \\eqref{CubicR} can be rewritten as\n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf D}_{133}^{\\; 3} {\\cal E}_\\grad820 =\\frac{1}{3}   {\\bf D}_{133} \\Delta {\\cal E}_\\grad820 \\ , \\ee\nwhich defines a qantization of the algebraic eqation ${\\bf Q}_{133}^{\\; 3}=0$ associated to the complex nilpotent orbit of $E_{7}$ of Dynkin label \\DEVII{0}00000{\\mathfrak{2}} with the real form defined with the $SU(8)$ Dynkin label [$\\scriptstyle \\mathfrak{0}\\mathfrak{2}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}\\mathfrak{0}$] \\cite{E7Djo}, which defines the graded decomposition of $SU(8)$ associated to the $(8,2,0)$ harmonic superspace we consider in this section. In this case the choice of real form moreover implies that the complex charge in the ${\\bf 70}$ defining the nilpotent orbit through the Kostant--Sekiguchi correspondence satisfies to \n\\begin{equation}}\\def\\ee{\\end{equation} Q^{ijpq} Q_{pqmn} Q^{mnkl} = 0 \\ , \\ee\nsuch that it admits a unique quantization, with the eigenvalue of the Laplace operator $-60$. However, we will see in the following that the constraint \\eqref{D4Consistency} can be relaxed while keeping the property that the associated representation of $E_{7(7)}$ is a highest weight representation. \n\n\n\\subsection{Dimensional reduction to three dimensions}\nIn  three dimensions, the duality group is $E_{8(8)}$, of maximal compact subgroup $Spin(16)\/\\mathds{Z}_2$. We denote $i, j$ the $SO(16)$ vector indices and $A, B$ the positive chirality Weyl spinor indices. The covariant derivative in tangent frame is a chiral Weyl spinor, {\\it i.e.}\\  in the  \\WSOXVI00000001 representation. In the linearised approximation, the covariant fields all descend from the Weyl spinor scalar field, satisfying to \\cite{Greitz:2011vh}\n\\begin{equation}}\\def\\ee{\\end{equation}\nD_{\\alpha}^{i} W^{A} = \\Gamma^{i A \\dot{A}} \\chi_{\\alpha \\dot A} \\ . \n\\ee\nBoth four-dimensional $(8,1,1)$ and $(8,2,0)$ harmonic superspaces descend to the same $(16,2)$ harmonic superspace in three dimensions, defined through the introduction of harmonic variables parametrising $SO(16)\/(U(2) \\times SO(12))$ \\cite{Howe:1994ms}. The Weyl spinor representation decomposes with respect to $U(2) \\times Spin(12)$ as\n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf 128} \\cong {\\bf 32}_+^{\\ord{-1}} \\oplus \\scal{ {\\bf 2}\\otimes {\\bf 32}_-}^\\ord{0} \\oplus {\\bf 32}_+^{\\ord{1}}\\ , \\label{128inD6} \\ee\nsuch that the grad $1$ Weyl spinor $W$ of $Spin(12)$ satisfies to a G-analyticity constraint with respect to the positive grad covariant derivative in the ${\\bf 2}$ of $U(2)$. The general polynomial in the $Spin(12)$ Weyl spinor is parametrised by four integers, just as for the rank three antisymmetric tensor of $SU(8)$ in section \\eqref{811D6R4}.\\footnote{This property follows from the fact that the classification of duality orbits of the black hole charges are the same in the $\\mathcal{N}=2$ supergravity theories of duality group $SO^*(12)$ and $SU(3,3)$ \\cite{Ferrara:1997uz}.} One computes in a similar way the general integral \n\\begin{eqnarray} && \\int du F(u)_{{\\mbox{\\WSOXVI0{n_3\\hspace{-0.5mm}\\mbox{+}n_4\\hspace{-0.5mm}\\mbox{+}2n_4^\\prime}0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} } }^{{\\mbox{\\WSOXII0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} }} D^{28}  W^{n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime + 4}|_{{\\mbox{\\WSOXII0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} }}  \\nonumber \\\\*\n&\\sim &\\nabla^{10} P^4  \\, W^{n_1 + 2n_2 +3n_3+ 4n_4+4n_4^\\prime }|_{{\\mbox{\\WSOXVI0{n_3\\hspace{-0.5mm}\\mbox{+}n_4\\hspace{-0.5mm}\\mbox{+}2n_4^\\prime}0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} } } +\\dots\\end{eqnarray}\nwhere $\\nabla^{10} P^4 $ is a $Spin(16)$ invariant quartic term in the scalar field momentum, that replaces the $\\nabla^6 R^4$ type term that vanishes modulo the equations of motion in three dimensions. In three dimensions it is not established if there is a one to one correspondence between non-linear and linear invariants defined as harmonic superspace integrals. Nevertheless, the class of invariants we discuss descends from four dimensions, and we can therefore assume they admit the same structure, {\\it i.e.}\\  \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal L}_\\gra{16}{2}[{\\cal E}_{\\gra{16}{2}}] = \\sum_{n_1,n_2,n_3,n_4,n_4^\\prime} {\\cal D}_{{\\mbox{\\WSOXVI0{n_3\\hspace{-0.5mm}\\mbox{+}n_4\\hspace{-0.5mm}\\mbox{+}2n_4^\\prime}0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} }} {\\cal E}_{\\gra{16}{2}}\\, {\\cal L}^{{\\mbox{\\WSOXVI0{n_3\\hspace{-0.5mm}\\mbox{+}n_4\\hspace{-0.5mm}\\mbox{+}2n_4^\\prime}0{n_2}0{n_4}0{n_1\\hspace{-0.5mm}\\mbox{+}n_3}} }}\\ .  \\ee\nThis expansion implies that the fourth order derivative of the function ${\\cal E}_{\\gra{16}{2}}$ restricted  to the \\WSOXVI10001000 must vanish, {\\it i.e.}\\  \\begin{equation}}\\def\\ee{\\end{equation} \\scal{ {\\cal D} \\Gamma_{i[jk}{}^r {\\cal D}} \\scal{ {\\cal D} \\Gamma_{lpq]r} {\\cal D} }{\\cal E}_{\\gra{16}{2}} = - \\delta_{i[j} \\scal{ {\\cal D} \\Gamma_{klpq]} {\\cal D}} ( \\Delta+48  ) {\\cal E}_{\\gra{16}{2}} \\ ,\\label{quarticE8}  \\ee\nwhere the Laplace operator $\\Delta$ is defined as\n\\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta = {\\cal D}_{A} {\\cal D}^{A}\\ . \n\\ee\nBy dimensional reduction of the four-dimensional equation  \\eqref{Laplace820}, one computes that   \\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta {\\cal E}_{\\gra{16}{2}} = - 198  {\\cal E}_{\\gra{16}{2}} \\ . \n\\ee\nOne can understand that the two kinds of 1\/8 BPS invariants discussed in the preceding section dimensionally reduce to this single class. If one consider the decomposition of \\eqref{128inD6} with respect to $U(6)\\subset Spin(12)$, one obtains for one embedding \n \\begin{equation}}\\def\\ee{\\end{equation} {\\bf 32}_+\\cong {\\bf  6}^\\ord{-2} \\oplus {\\bf 20}^\\ord{0} \\oplus  \\overline{\\bf 6}^\\ord{2}\\ , \\label{32inA5} \\ee\nsuch that the G-analytic superfield in the ${\\bf 32}_+$ includes the four-dimensional $(8,1,1)$ G-analytic scalar $W^{rst}$ as well as some components of the vector fields. A generic spinor of non-zero quartic invariant can be represented by $W^{rst}$. For the other embedding  $U(6)\\subset Spin(12)$, one gets\n \\begin{equation}}\\def\\ee{\\end{equation} {\\bf 32}_+\\cong  \\overline{\\bf  1}^\\ord{-3} \\oplus {\\bf 15}^\\ord{-1}  \\oplus \\overline{\\bf 15}^\\ord{1} \\oplus {\\bf 1}^\\ord{3}\\ , \\label{32inA5p} \\ee\nsuch that  the G-analytic superfield in the ${\\bf 32}_+$ includes the four-dimensional $(8,2,0)$ G-analytic scalar $W^{rs}$ as well as some components of the vector fields, and a Ehlers complex scalar parametrising the four-dimensional metric. The scalar field alone only parametrises a null spinor of $Spin(12)$ of vanishing quartic invariant, and only together with the Ehlers scalar field it can provide a representative of a generic spinor. One could have naively concluded that the function ${\\cal E}_\\grad820$ should give rise to a  function on $E_{8(8)}\/Spin_{\\scriptscriptstyle \\rm c}(16)$ satisfying moreover to \n\\begin{equation}}\\def\\ee{\\end{equation} 5\\scal{ {\\cal D} \\Gamma_{ijpq}  {\\cal D}} \\scal{ {\\cal D} \\Gamma^{klpq} {\\cal D}} {\\cal E} = - 20 \\scal{{\\cal D} \\Gamma_{ij}{}^{kl} {\\cal D}} \\scal{ \\Delta + 48 } {\\cal E}  + 28 \\delta_{ij}^{kl} \\Delta \\scal{\\Delta + 120} {\\cal E} \\ , \\ee\nbut this equation only admits solutions for functions satisfying to the Laplace equation\n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta {\\cal E} = -210\\,  {\\cal E} \\ , \\ee\nexcepted for the functions satisfying to the quadratic and cubic constraints that define the $R^4$ and $\\nabla^4 R^4$ type invariants. We see therefore that this equation is incompatible with supersymmetry.\n\n It follows that both $(8,1,1)$ and $(8,2,0)$ type invariants dimensionally reduce to three-dimensional invariants depending of functions on $E_{8(8)}\/Spin_{\\scriptscriptstyle \\rm c}(16)$ that belong to the same representation of $E_{8(8)}$. Being in the same representation, they both carry a quartic component in the linearised approximation and they must both include a $\\nabla^6 R^4$ type term in their uplift to four dimensions. This proves that the mixing between the two different linearised structures must occur such that the non-linear $\\bar F^2 \\nabla^4 R^4$ type invariant cannot exist without including a $\\nabla^6 R^4$ type term as well. \n\nBefore to end this section on the three-dimensional theory, let us discuss the modification of the supersymmetry constraint due to the completion of the $R^4$ type invariant at the next order. As it is argued in \\cite{Green:2005ba}, the appearance of a $R^4$ correction with threshold function ${\\cal E}_\\gra{16}{8}$, will modify the Laplace equation with a non-zero right-hand-side, {\\it i.e.}\\  \n \\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta {\\cal E}_{\\gra{16}{2}} = - 198  {\\cal E}_{\\gra{16}{2}} - {\\cal E}_{\\gra{16}{8}}^{\\; 2} \\ . \n\\ee\nBecause the function $ {\\cal E}_{\\gra{16}{8}}$ satisfies to \\cite{Minimal}\n\\begin{equation}}\\def\\ee{\\end{equation} \\scal{ {\\cal D} \\Gamma_{ijkl} {\\cal D}}  {\\cal E}_{\\gra{16}{8}} = 0 \\ , \\ee\nthe second derivative of its square must necessarily vanish in the  \\WSOXVI10001000, and we get accordingly a modification of \\eqref{quarticE8} to\n\\begin{equation}}\\def\\ee{\\end{equation} \\scal{ {\\cal D} \\Gamma_{i[jk}{}^r {\\cal D}} \\scal{ {\\cal D} \\Gamma_{lpq]r} {\\cal D} } {\\cal E}_{\\gra{16}{2}}= 150 \\delta_{i[j} \\scal{ {\\cal D} \\Gamma_{klpq]} {\\cal D}}  {\\cal E}_{\\gra{16}{2}} + \\delta_{i[j} \\scal{ {\\cal D} \\Gamma_{klpq]} {\\cal D}} {\\cal E}_{\\gra{16}{8}}^{\\; 2}\\ . \\ee\n\n\\subsection{$E_{7(7)}$ Eisenstein series}\nIn this section we shall discuss some properties of Einstein series that solve the differential equations we have derived for the $\\nabla^6 R^4$ type invariants.\n\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{1}}\n\\subsubsection{Fundamental representation}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{2}}\nAs discussed in \\cite{Obers:1999um,D4R4}, one can define the Eisenstein series \n\\begin{equation}}\\def\\ee{\\end{equation} E_{\\mbox{\\DEVII000000s}} = \\sum_{\\vspace{-2mm}\\begin{array}{c}\\scriptstyle \\vspace{-4mm}  \\Gamma\\in \\mathds{Z}^{56} \\vspace{2mm}\\\\ \\scriptscriptstyle I_4^{\\prime\\prime}(\\Gamma)|_{\\bf 133}=0\\end{array}} |Z(\\Gamma)_{ij} Z(\\Gamma)^{ij}|^{-s} \\   , \\label{E56s} \\ee\nas a sum over the rank one integral charge vectors $\\Gamma$ in the ${\\bf 56}$ of $E_{7(7)}$ satisfying to the constraint that the quadratic tensor  $\\Gamma\\otimes \\Gamma$ vanishes  in the adjoint representation. This formula is rather useful to identify the differential equations satisfied by the Eisenstein function, because one can simply consider the case of one charge $\\Gamma$, with  $Z(\\Gamma)_{ij} = {\\cal V}_{ij}{}^I\\Gamma_I$, such that the quadratic constraint becomes \n\\begin{equation}}\\def\\ee{\\end{equation} Z_{[ij} Z_{kl]} = \\frac{1}{24} \\varepsilon_{ijklpqrs} Z^{pq} Z^{rs} \\ , \\qquad Z_{ik} Z^{jk} = \\frac{1}{8} \\delta_i^j Z_{kl} Z^{kl} \\ ,\\ee\nand the differential operator acts on $Z_{ij} $ as an element of $\\mathfrak{e}_{7(7)}$\n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijkl} Z^{pq} = 3 \\delta^{pq}_{[ij} Z_{kl]} \\ , \\qquad {\\cal D}_{ijkl} Z_{pq} = \\frac{1}{8} \\varepsilon_{ijklpqrs} Z^{rs} \\ . \\ee\nUsing the definition $|Z|^2 = Z_{ij} Z^{ij}$, one computes that the function $|Z|^{-2s}$ satisfies to \n\\begin{eqnarray} {\\cal D}_{ijpq} {\\cal D}^{klpq} |Z|^{-2s} &=& 2s(s-2) Z_{ij} Z^{kl} |Z|^{-2s-2} + \\frac{s(s-11)}{4} \\delta_{ij}^{kl} |Z|^{-2s} \\  , \\nonumber \\\\*\n{\\cal D}_{ijpq} {\\cal D}^{pqrs} {\\cal D}_{rskl} |Z|^{-2s} &=& - 3 s(s-2)(s-4) Z_{ij} Z_{kl} |Z|^{-2s-2} + \\frac{s^2-15s + 8}{4} {\\cal D}_{ijkl} |Z|^{-2s}  \\ , \\nonumber \\\\*\n {\\cal D}_{jr[kl} {\\cal D}^{irmn} {\\cal D}_{pq]mn} |Z|^{-2s} &=&\\frac{(s-2)(s-7)}{12}  \\delta^i_j{\\cal D}_{klpq}  |Z|^{-2s} -\\frac{ s^2-9s -40}{12} \\delta^i_{[k} {\\cal D}_{pql]j} |Z|^{-2s}  \\ , \\hspace{10mm} \\label{ConstraintsZs} \n\\end{eqnarray}\nand to the Laplace equation \n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta |Z|^{-2s} = 3s(s-9) |Z|^{-2s} \\ . \\ee\nFor $s\\ne2,\\, 4$, the function admits a generic gradient expansion in the irreducible representations $[0,n_2+2n_3,0,n_1,0,n_2,0]$ and their complex conjugate. To exhibit this property, it is convenient to consider a restricted set of indices as follows \n\\begin{eqnarray} &&  \\scal{ {\\cal D}_{12ij} {\\cal D}^{ijkl} {\\cal D}_{kl12}}^{n_3} \\scal{  {\\cal D}_{12pq} {\\cal D}^{78pq}}^{n_2} \\scal{{\\cal D}_{1234}}^{n_1} |Z|^{-2s} \\\\ &=& \\tfrac{(s+n_1+n_2+n_3-1)!(s+n_2+n_3-3)!(s+n_3-5)!}{(s-1)!(s-3)!(s-5)!} \\scal{\\mbox{-}3 Z_{12}^{\\; 2}}^{n_3} \\scal{2 Z_{12} Z^{78}}^{n_2} \\scal{\\mbox{-}6 Z_{[12} Z_{34]}}^{n_1} |Z|^{-2(s+n_1+n_2+n_3)} \\, . \\nonumber \\label{GradExpandZs} \\end{eqnarray}\nOne computes moreover that for $m\\le n$ \n\\begin{eqnarray} &&  \\scal{{\\cal D}^{78ij} {\\cal D}_{ijkl} {\\cal D}^{kl78}}^{m} \\scal{{\\cal D}_{12pq} {\\cal D}^{pqrs} {\\cal D}_{rs12}}^n |Z|^{-2s} \\\\\n&=& \\tfrac{(s+n-1)!(s+n-3)!(s+n-5)!(s+n+m-1)!(s+n+m-3)!(s-n+m-5)!}{(s-1)!(s-3)!(s-5)!(s+n-1)!(s+n-3)!(s-n-5)!} \\scal{\\mbox{-}3 Z^{78\\; 2}}^{m} \\scal{\\mbox{-}3 Z_{12}^{\\; 2}}^{n}|Z|^{-2(s+n+m)} \\nonumber \\\\*\n&=&\\scal{\\mbox{-}\\tfrac{3}{2}}^{n+m} \\tfrac{(s+n-5)!(s+n+m-1)!(s+n+m-3)!(s-n+m-5)!}{(s+n-m-5)! (s+2n-1)!(s+2n-3)!(s-n-5 )! } \\scal{{\\cal D}_{12ij} {\\cal D}^{ijkl} {\\cal D}_{kl12}}^{n-m}\\scal{{\\cal D}_{12pq} {\\cal D}^{78pq}}^{n+m} |Z|^{-2s} \\nonumber  \\end{eqnarray}\nsuch that acting with a derivative operator in the conjugate representation $[0,0,0,0,0,2m,0]$ does not produce an independent tensor. One has in particular for $s$ an integer greater than $5$ \n\\begin{equation}}\\def\\ee{\\end{equation} \\scal{{\\cal D}^{78ij} {\\cal D}_{ijkl} {\\cal D}^{kl78}} \\scal{{\\cal D}_{12pq} {\\cal D}^{pqrs} {\\cal D}_{rs12}}^{s-4} |Z|^{-2s} = 0 \\ . \\ee\nThis equation is the equivalent on $E_{7(7)}\/SU_{\\scriptscriptstyle \\rm c}(8)$ of the equation on $SL(2)\/SO(2)$ \n\\begin{equation}}\\def\\ee{\\end{equation} \\bar {\\cal D} {\\cal D}^{s-1} E_{[s]} = 0 \\ ,  \\ee\nfor integral $s$, and we would like to see that the function $ E{\\mbox{\\DEVII000000s}}$ also decomposes somehow into a ``holomorphic'' part $ {\\cal F}_s$ and a ``anti-holomorphic'' part $\\bar {\\cal F}_s$, satisfying respectively to \n\\begin{equation}}\\def\\ee{\\end{equation} \\scal{{\\cal D}_{12pq} {\\cal D}^{pqrs} {\\cal D}_{rs12}}^{s-4} \\bar {\\cal F}_s = 0 \\ , \\qquad \\scal{{\\cal D}^{78ij} {\\cal D}_{ijkl} {\\cal D}^{kl78}}^{s-4} {\\cal F}_s = 0 \\  , \\ee\nsuch that \n\\begin{equation}}\\def\\ee{\\end{equation}   \\scal{{\\cal D}_{12pq} {\\cal D}^{pqrs} {\\cal D}_{rs12}}^{s-4} E_{\\mbox{\\DEVII000000s}} =  \\scal{{\\cal D}_{12pq} {\\cal D}^{pqrs} {\\cal D}_{rs12}}^{s-4} {\\cal F}_s  \\ , \\label{D3sF}  \\ee\nand respectively for the complex conjugate. By consistency, this requires for instance that acting with further derivatives on this tensor does not permit to get back lower order tensors with $n_3<s-4$ in \\eqref{GradExpandZs}. \n\n\nThrough representation theory, one obtains that \n\\begin{eqnarray}&&  {\\cal D}_{[0,0,0,1,0,0,0]} {\\cal D}^{n_1+2n_2+3n_3}_{[0,n_2+2n_3,0,n_1,0,n_2,0]} |Z|^{-2s} \\\\\n&\\sim& \\Bigl( {\\cal D}^{n_1+1+2n_2+3n_3}_{[0,n_2+2n_3,0,n_1+1,0,n_2,0]} +{\\cal D}^{n_1-1+2(n_2+1)+3n_3}_{[0,n_2+2n_3+1,0,n_1-1,0,n_2+1,0]}  +{\\cal D}^{n_1+2(n_2-1)+3(n_3+1)}_{[0,n_2+2n_3+1,0,n_1,0,n_2-1,0]}  \\Bigr . \\nonumber \\\\*\n&& \\Bigl . + {\\cal D}^{n_1-1+2n_2+3n_3}_{[0,n_2+2n_3,0,n_1-1,0,n_2,0]} +{\\cal D}^{n_1+1+2(n_2-1)+3n_3}_{[0,n_2+2n_3-1,0,n_1+1,0,n_2-1,0]}  +{\\cal D}^{n_1+2(n_2+1)+3(n_3-1)}_{[0,n_2+2n_3-1,0,n_1,0,n_2+1,0]} \\Bigr) |Z|^{-2s} \\nonumber\n\\end{eqnarray}\nfor some coefficients that are not specified. So the only way to reduce $n_3$, is to increase $n_2$ by $1$ unit. We will check this equation in the case $n_1=n_2=0$. The restriction of the derivative ${\\cal D}^{3n} |Z|^{-2s}$ to the $[0,2n,0,0,0,0,0]$ with two free indices reads \n\\begin{eqnarray} && {\\cal D}^{3n\\, [0,2n,0,0,0,0,0]}_{ij1^{2n-1}2^{2n-1}} |Z|^{-2s} \\nonumber \\\\*\n&=& \\tfrac{(s+n-1)!(s+n-3)!(s+n-5)!}{(s-1)!(s-3)!(s-5)!} \\frac{(-3)^n}{2n} \\Scal{ Z_{ij} Z_{12}^{\\; 2n-1} - (2n-1) Z_{1[i} Z_{j]2} Z_{12}^{\\; 2n-2}} |Z|^{-2(s+n)} \\ , \\end{eqnarray}\nand one computes that \n\\begin{eqnarray} && {\\cal D}^{78ij} \\frac{1}{2n}\\Scal{ Z_{ij} Z_{12}^{\\; 2n-1} - (2n-1) Z_{1[i} Z_{j]2} Z_{12}^{\\; 2n-2}} |Z|^{-2(s+n)} \\nonumber \\\\*\n&=& \\frac{(2n+5 ) ( n - s+4)}{8n} Z^{78} Z_{12}^{\\; 2n-1} |Z|^{-2s} \\ , \\end{eqnarray}\nsuch that \n\\begin{eqnarray}&&  {\\cal D}^{78ij} {\\cal D}^{3n\\, [0,2n,0,0,0,0,0]}_{ij1^{2n-1}2^{2n-1}} |Z|^{-2s} \\nonumber \\\\*\n& =& \\frac{3(s+n-5)(2n+5)(s-n-4)}{16n}   \\scal{ {\\cal D}_{12ij} {\\cal D}^{ijkl} {\\cal D}_{kl12}}^{n-1} \\scal{  {\\cal D}_{12pq} {\\cal D}^{78pq}} |Z|^{-2s} \\ . \\end{eqnarray}\nIn particular we have that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{78ij} {\\cal D}^{3(s-4)\\, [0,2(s-4),0,0,0,0,0]}_{ij1^{2s-5}2^{2s-5}} |Z|^{-2s} = 0  \\ , \\label{ConsSF2k} \\ee\nconsistently with the assumption that no lower order tensor is produced out of the tensor function \\eqref{D3sF}. We conclude therefore that the tensor \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal F} \\hspace{-0.4mm} E_{[0,2(s-4),0,0,0,0,0]} = {\\cal D}^{3(s-4)}_{[0,2(s-4),0,0,0,0,0]} E_{\\mbox{\\DEVII000000s}} \\ , \\label{FE} \\ee\nis an $E_{7(7)}(\\mathds{Z})$ modular form that is in some sense holomorphic, such that its gradient expansion is restricted to derivative of this tensor in the symmetric representations $[0,n_2+2(n_3+s-4),0,n_1,0,n_2,0]$, {\\it i.e.}\\ \n\\begin{equation}}\\def\\ee{\\end{equation}  {\\cal D}^{n_1+2n_2+3n_3}_{[0,n_2+2n_3,0,n_1,0,n_2,0]}   {\\cal F} \\hspace{-0.4mm} E_{[0,2(s-4),0,0,0,0,0]}  \\in [0,n_2+2(n_3+s-4),0,n_1,0,n_2,0] \\ . \\ee\n\n\n\nUsing Langlands functional identity  \\cite{Green:2010kv}, one computes that the only integer values of $s\\ge5$ for which the function  diverges are  \n\\begin{eqnarray}   E_{\\mbox{\\DEVII000000{5\\mbox{+}\\epsilon}}} &=& \\frac{63}{16\\pi \\, \\epsilon} E_{\\mbox{\\DEVII0000004}} + \\hat{E}_{\\mbox{\\DEVII000000{5}}} + \\mathcal{O}(\\epsilon) \\ , \\label{DivergenceF2D4R4} \\nonumber \\\\*\nE_{\\mbox{\\DEVII000000{7\\mbox{+}\\epsilon}}} &=&\\frac{1\\, 964\\, 655 \\zeta(5)}{2048 \\pi^{5} \\, \\epsilon} E_{\\mbox{\\DEVII0000002}} + \\hat{E}_{\\mbox{\\DEVII000000{7}}} + \\mathcal{O}(\\epsilon) \\ , \\nonumber \\\\*\n E_{\\mbox{\\DEVII000000{9\\mbox{+}\\epsilon}}} &=& \\frac{12\\, 642\\, 554\\, 925 \\zeta(5)\\zeta(9)}{2\\, 097\\, 152 \\pi^{9}\\, \\epsilon} +\\hat{E}_{\\mbox{\\DEVII000000{9}}}+ \\mathcal{O}(\\epsilon) \\ . \\end{eqnarray}\nHowever, according to \\eqref{ConstraintsZs}, the function $E_{\\mbox{\\DEVII000000s}}$ satisfies to \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{3}_{[0,2,0,0,0,0,0]} E_{\\mbox{\\DEVII000000s}} = 0 \\ ,\\quad  \\mbox{for}\\, s=0,\\, 2,\\, 4 \\ , \\ee\nand it follows that the tensor $ {\\cal F} \\hspace{-0.4mm} E_{[0,2(s-4),0,0,0,0,0]}$ \\eqref{FE} is finite for all $s$. However, we have argued in the preceding section that the $\\bar F^2 \\nabla^4 R^4$ type invariant must include a  $\\nabla^6 R^4$ type term, which will be multiplied by the function itself. In this case the relevant Eisenstein function diverges, and one must regularise it such that the differential equation \\eqref{Laplace820} will be modified to\n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta \\, \\hat{E}_{\\mbox{\\DEVII000000{5}}}   = - 60  \\, \\hat{E}_{\\mbox{\\DEVII000000{5}}} + \\frac{189}{16 \\pi}  E_{\\mbox{\\DEVII0000004}} \\ . \\ee\nSuch a correction is reminiscent of a 1-loop logarithm divergence of the $\\nabla^4 R^4$ type invariant form factor. \n\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{1}}\n\\subsubsection{Adjoint representation}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{2}}\n\nOne can also consider the Eisenstein series \\footnote{We assume here that all the elements of $\\mathds{Z}^{133}$ are in the $E_{7(7)}(\\mathds{Z})$ orbit of a relative integer times a normalised representative of the continuous orbit. This property does not affect our conclusions in any case, which only requires the generating character to satisfy to the differential equations we discuss.}\n\\begin{equation}}\\def\\ee{\\end{equation} E_{\\mbox{\\DEVII{\\mathnormal{s}}00000{\\mathfrak{0}}}} =  \\sum_{{\\bf Q}\\in \\mathds{Z}^{133} |{\\bf Q}^2=0} \\scal{  X(Q)_{ijkl} X(Q)^{ijkl} }^{-s} \\ ,  \\ee \nas the sum over integral charges ${\\bf Q}\\in \\mathfrak{e}_{7(7)}$ satisfying to the constraint ${\\bf Q}^2 = 0 $, and such that the adjoint action of the coset representative ${\\cal V}$ on ${\\bf Q}$ decomposes into the anti-Hermitian traceless matrix $\\Lambda^i{}_j$ and the complex-selfdual antisymmetric tensor $X_{ijkl}$ satisfying to the constraints \n\\begin{eqnarray} \\Lambda^i{}_k \\Lambda^k{}_j &=& - \\frac{1}{48} \\delta^i_j X^{klpq} X_{klpq} \\ ,\\nonumber \\\\*\n\\Lambda^{[i}{}_{[k} \\Lambda^{j]}{}_{l]} &=& - \\frac{1}{2} X^{ijpq} X_{klpq}  + \\frac{1}{48} \\delta^{ij}_{kl} X^{pqrs} X_{pqrs} \\ , \\nonumber \\\\*\n\\Lambda^{[i}{}_p X^{j]pkl} &=& \\Lambda^{[k}{}_p X^{l]pij}  \\ . \\end{eqnarray}\nThe action of the derivative on these tensors is defined as the $\\mathfrak{e}_{7(7)}$ action \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijkl} X^{pqrs}  = 12 \\delta^{[pqr}_{[ijk} \\Lambda^{s]}{}_{l]} \\ , \\qquad \n{\\cal D}_{ijkl} \\Lambda^p{}_q = 2 \\delta^p_{[i} X_{jkl]q} + \\frac{1}{4} \\delta^p_q X_{ijkl} \\ . \\ee\nOne computes for $|X|^2 = X_{ijkl} X^{ijkl} $ that\n\\begin{eqnarray} {\\cal D}_{ijkl} |X|^2 &=& -24 X_{p[ijk} \\Lambda^p{}_{l]} \\ , \\qquad \\qquad\\qquad  {\\cal D}_{ijpq} X^{klpq} = 10 \\delta_{[i}^{[k} \\Lambda^{j]}{}_{l]} \\ , \\\\\n{\\cal D}_{ijpq} {\\cal D}^{klpq} |X|^2 &=& 30 X_{ijpq} X^{klpq} + 3 \\delta_{ij}^{kl} |X|^2 \\ , \\qquad {\\cal D}_{ijpq} |X|^2 {\\cal D}^{klpq} |X|^2 = 12 X_{ijpq} X^{klpq} |X|^2 \\ , \\nonumber \\end{eqnarray}\nwhich permits to derive that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijpq} {\\cal D}^{klpq} |X|^{-2s} = 6s(2s-3) X_{ijpq} X^{klpq} |X|^{-2s-2} - 3 s\\delta_{ij}^{kl} |X|^{-2s} \\ . \\ee\nOne gets therefore a solution to the equation \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijpq} {\\cal D}^{klpq} {\\cal E}_\\grad844 = - \\frac{9}{2} \\delta_{ij}^{kl} {\\cal E}_\\grad844 \\ , \\label{R4Quadra} \\ee\nfor $s=\\frac{3}{2}$. One computes in general that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijpq} {\\cal D}^{pqrs}{\\cal D}_{rskl} |X|^{-2s} = \\scal{ s^2 - \\tfrac{17}{2} s+6} {\\cal D}_{ijkl} |X|^{-2s} \\ , \\ee\nand the function satisfies to  \\eqref{CubicC} and its complex conjugate for all $s$. The restriction of the third order derivative to the $[1,0,0,1,0,0,1]$ gives\n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{1k[12} {\\cal D}_{34]ij} {\\cal D}^{8ijk} |X|^{-2s} = - \\frac{3}{4} s(2s-3)(2s-5) \\Lambda^8{}_1 X_{1234} |X|^{-2s-2} \\ , \\ee\nshowing that the function solves the cubic equation \\eqref{CubicR} for $s=\\frac{5}{2}$. These functions satisfy the same equations as their analog Eisenstein functions defined in the fundamental representation, consistently with the property that \\footnote{We are grateful to Axel Kleinschmidt who provided the explicit coefficients.}\n\\begin{equation}}\\def\\ee{\\end{equation}  E_{\\mbox{\\DEVII{\\frac{3}{2}}00000{\\mathfrak{0}}}}  = \\frac{2}{\\pi}   E_{\\mbox{\\DEVII{0}000002}} \\ , \\qquad  E_{\\mbox{\\DEVII{\\frac{5}{2}}00000{\\mathfrak{0}}}} = \\frac{8}{15\\pi}   E_{\\mbox{\\DEVII{0}000004}} \\ . \\ee \nOne can also consider the restriction of the fourth order derivative to the $[2,0,0,0,0,0,0,2]$ to vanish, which defines a further restriction on solutions to \\eqref{CubicC}. In this case one obtains \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{8kij} {\\cal D}_{1lij} {\\cal D}_{1kpq} {\\cal D}^{8lpq}   |X|^{-2s} = - \\frac{9}{2} s(2s-3)(2s-5)(s-4) \\Lambda^8{}_1 \\Lambda^8{}_1 |X|^{-2s-2} \\ , \\ee\nand this further restriction distinguishes the value $s=4$. \nWe have in general \n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta  |X|^{-2s} = 2s(2s-17) |X|^{-2s}  \\ , \\ee\nand the function $E{\\mbox{\\DEVII{4}00000{\\mathfrak{0}}}}$ does not solve the 1\/8 BPS equation. \nUsing the same normalisation as   \\cite{Green:2010kv} with a factor of $2\\zeta(2s)$ in the definition of the Eisenstein function, one computes using Langlands formula \n\\begin{equation}}\\def\\ee{\\end{equation}  E_{\\mbox{\\DEVII{\\mathnormal{s}}00000{\\mathfrak{0}}}}   = \\pi^{\\frac{33}{2}} \\tfrac{\\Gamma(s-8) \\Gamma(s-\\stfrac{13}{2}) \\Gamma(s-\\stfrac{11}{2}) \\Gamma(2s-\\stfrac{17}{2}) \\zeta(2s-16) \\zeta(2s-13)\\zeta(2s-11) \\zeta(4s-17) }{ \\Gamma(s-\\stfrac{5}{2}) \\Gamma(s-\\stfrac{3}{2}) \\Gamma(s) \\Gamma(2s-8)\\zeta(17-2s)\\zeta(2s-5)\\zeta(2s-3)\\zeta(4s-16)} E_{\\mbox{{{\\tiny $ { \\left[ \\begin{array}{cccccc}  & & \\mathfrak{0} \\hspace{-0.7mm}&&& \\vspace{ -1.5mm} \\\\ \\stfrac{17}{2}\\mbox{-}s\\hspace{-0.7mm} &  \\mathfrak{0} \\hspace{-0.7mm}& \\mathfrak{0} \\hspace{-0.7mm} & \\mathfrak{0}\\hspace{-0.7mm}&\\mathfrak{0}\\hspace{-0.7mm}& \\mathfrak{0}  \\end{array}\\right] }$}}\n}} \\ . \\ee\n The function is singular for various values of $s$, {\\it i.e.}\\  $\\frac{9}{2}$, $\\frac{11}{2}$, $6$, $\\frac{13}{2}$, $7$, and $\\frac{17}{2}$, and in particular for $s=6$, which is the relevant value to solve equation \\eqref{CubicC} with \\eqref{Laplace811}. One should therefore consider the regularised series\n\\begin{equation}}\\def\\ee{\\end{equation} E_{\\mbox{{{\\tiny $ { \\left[ \\begin{array}{cccccc}  & & \\mathfrak{0} \\hspace{-0.7mm}&&& \\vspace{ -1.5mm} \\\\ 6\\mbox{+}\\epsilon\\hspace{-0.7mm} &  \\mathfrak{0} \\hspace{-0.7mm}& \\mathfrak{0} \\hspace{-0.7mm} & \\mathfrak{0}\\hspace{-0.7mm}&\\mathfrak{0}\\hspace{-0.7mm}& \\mathfrak{0}  \\end{array}\\right] }$}}}}= \\frac{\\pi^5}{8 \\zeta(9)\\, \\epsilon}  E_{\\mbox{{{\\tiny $ { \\left[ \\begin{array}{cccccc}  & & \\mathfrak{0} \\hspace{-0.7mm}&&& \\vspace{ -1.5mm} \\\\ \\frac{5}{2}\\hspace{-0.7mm} &  \\mathfrak{0} \\hspace{-0.7mm}& \\mathfrak{0} \\hspace{-0.7mm} & \\mathfrak{0}\\hspace{-0.7mm}&\\mathfrak{0}\\hspace{-0.7mm}& \\mathfrak{0}  \\end{array}\\right] }$}}}} +  \\hat{E}_{\\mbox{{{\\tiny $ { \\left[ \\begin{array}{cccccc}  & & \\mathfrak{0} \\hspace{-0.7mm}&&& \\vspace{ -1.5mm} \\\\ 6 \\hspace{-0.7mm} &  \\mathfrak{0} \\hspace{-0.7mm}& \\mathfrak{0} \\hspace{-0.7mm} & \\mathfrak{0}\\hspace{-0.7mm}&\\mathfrak{0}\\hspace{-0.7mm}& \\mathfrak{0}  \\end{array}\\right] }$}}}} + \\mathcal{O}(\\epsilon)\\ . \\ee\nHowever, we will see in the following that this function does not appear in string theory, similarly as the $\\nabla^6 R^4$ threshold function is not described by an Eisenstein series in type IIB supergravity. Nonetheless, some components of this function should appear, as we will argue in the following. \n\n\\subsection{$F^{2k} \\nabla^4 R^4$ type invariants}\nThe $F^2 \\nabla^4R^4$ type invariants we have discussed in section \\ref{F2D4R4} have a natural generalisation to higher order invariants. Considering the same chiral harmonic superspace defined in terms of the harmonic variables $u^{\\hat{r}}{}_i,\\, u^r{}_i$ parametrising $SU(8)\/S(U(2)\\times U(6))$ \\cite{Drummond:2003ex}, one can define the G-analytic superfields \n\\begin{equation}}\\def\\ee{\\end{equation} \\bar F_{\\adt\\bdt}^{12} = u^1{}_i u^2{}_j  F_{\\adt\\bdt}^{ij}\\  ,\\qquad  \\bar \\chi_\\adt^{12r} = u^1{}_i u^2{}_j u^r{}_k \\bar \\chi_\\adt^{ijk} \\ . \\ee\nThey do not permit to define directly chiral primary operators of $SU(2,2|8)$, because \n\\begin{equation}}\\def\\ee{\\end{equation} \\bar D_{\\adt t} W^{rs}  = \\delta_t^{[r} \\bar \\chi_\\adt^{12s]} \\ , \\qquad \\bar D_{\\adt r} \\chi_\\bdt^{12s} = \\delta_r^s \\bar F_{\\adt\\bdt}^{12} \\ . \\ee\nChiral primary operators are annihilated by the special supersymmetry generators at the origin, {\\it i.e.}\\  \n\\begin{equation}}\\def\\ee{\\end{equation} S_{\\dot{\\gamma}}^r \\bar F_{\\adt\\bdt}^{12} = \\varepsilon_{\\cdt(\\adt} \\bar \\chi_{\\bdt)}^{12r} \\ , \\qquad S_\\adt^r \\bar \\chi_\\bdt^{12s} = \\varepsilon_{\\adt\\bdt} W^{rs} \\ , \\qquad S_\\adt^t W^{rs} = 0 \\ . \\ee\nOne can enforce this property by defining a chiral primary as \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathcal{O}^\\ord{k}_{{\\cal F}} = (S)^{12} \\Scal{  ( \\bar F_{\\adt\\bdt}^{12} \\bar F^{\\adt\\bdt 12} )^{2+k} {\\cal F}[W] } \\propto \\scal{ \\varepsilon_{rstuvw} W^{rs} W^{tu} W^{vw}}^2  ( \\bar F_{\\adt\\bdt}^{12} \\bar F^{\\adt\\bdt 12} )^{k-1}  {\\cal F}[W]+\\dots \\label{ChiralPrimary}   \\ee\nfor an arbitrary function ${\\cal F}$ of the G-analytic superfield $W^{rs}$. By construction such a chiral primary operator is never short, and defines a non-trivial integrand for the $(8,2,0)$ measure. Because one can consider an arbitrary representative up to a total fermionic derivative, one can as well consider the first term in \\eqref{ChiralPrimary} as the integrand. \n\nSo similarly as in section  \\ref{F2D4R4}, we get the general class of linearised invariants for an arbitrary positive integer $k$,\n\\begin{eqnarray}&&  \\int du \\bar D^{16}  D^{12}  \\, F(u)^{[0,n_1,0,n_2,0]}_{[0,n_2+2n_3+2k,0,n_1,0,n_2,0]} \\, \\bar F^{2k-2}\\, W^{n_1 + 2n_2 + 3n_3 + 6} |_{[0,n_1,0,n_2,0]}  \\\\\n&\\sim&W^{n_1 + 2n_2 + 3n_3}_{\\scriptscriptstyle [0,n_2+2n_3,0,n_1,0,n_2,0]} \\bar F^{2k}_{\\scriptscriptstyle[0,2k,0,0,0,0,0]} \\nabla^4 R^4+\\dots \\nonumber \\\\*\n&& \\hspace{10mm} + W^{n_1+2n_2+n_3-22}_{\\scriptscriptstyle [0,n_2+2n_3-8,0,n_1-8,0,n_2-4,0]} \\bar F^{2k-2}_{\\scriptscriptstyle [0,2k-2,0,0,0,0,0]} \\bar \\chi^{16}_{\\scriptscriptstyle[0,8,0,4,0,0,0]}  \\chi^{12}_{\\scriptscriptstyle[0,2,0,4,0,4,0]} \\ . \\nonumber\\end{eqnarray}\nWe conclude that the corresponding supersymmetry invariants admit the same gradient expansion in \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal L}^\\ord{k}_\\grad820[{\\cal E}_\\grad820^\\ord{k}] =\\sum_{n_1\\ge 0,n_2\\ge0,n_3\\ge k}  {\\cal D}^{n_1+2n_2+3n_3}_{[0,n_2,0,n_1,0,n_2+2n_3,0]} {\\cal E}_\\grad820^\\ord{k} \\, {\\cal L}^{k\\, [0,n_2+2n_3,0,n_1,0,n_2,0]}_\\grad820 \\ , \\ee\nfor a function ${\\cal E}_\\grad820^\\ord{k}$ satisfying to \\eqref{CubicR}, and $k\\ge2$. The coupling at the lowest number of points is then of the kind \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{3k}_{[0,0,0,0,0,2k,0]} {\\cal E}_\\grad820^\\ord{k} \\, {\\cal L}^{k\\, [0,2k,0,0,0,0,0]}_\\grad820 = {\\cal D}^{3k}_{[0,0,0,0,0,2k,0]} {\\cal E}_\\grad820^\\ord{k} \\, \\bar F^{2k}_{[0,2k,0,0,0,0,0]} \\, \\nabla^4 R^4 + \\dots \\ee\nIn principle one could expect to have a non-trivial mixing with another class of linearised invariant at the non-linear level, just as the one of the $\\bar F^2 \\nabla^4 R^4$ type invariant with the $\\nabla^6 R^4$ type invariant described in section  \\ref{F2D4R4}. However, there is no higher order chiral primary operator that can define a non-trivial  $(8,1,1)$ harmonic superspace integral, and we did not find any linearised invariant with the right structure to define a possible cohomology class as does \\eqref{d0Cohom}. Therefore we expect these invariants to have the same structure as the associated linearised invariants, {\\it i.e.}\\  to only contribute to $(4+2k)$-point amplitudes and higher. \n\n\nIndependently of this assumption, the structure of these invariants requires that the action of the derivative ${\\cal D}_{ijkl}$ on ${\\cal D}^{3k} {\\cal E}_\\grad820^\\ord{k}$ does not generate lower order derivatives of the function. This condition is precisely \\eqref{ConsSF2k}, and we conclude therefore that the eigenvalue of the Laplace operator is determined in the same way as\n\\begin{equation}}\\def\\ee{\\end{equation} \\Delta {\\cal E}_\\grad820^\\ord{k} = 3 (k+4)(k-5) {\\cal E}_\\grad820^\\ord{k} \\ , \\ee\nsuch that the function satisfies to \n\\begin{equation}}\\def\\ee{\\end{equation} 12 {\\cal D}_{jr[kl} {\\cal D}^{irmn} {\\cal D}_{pq]mn} {\\cal E}_\\grad820^\\ord{k} =  (k+2)(k-3)  \\delta^i_j {\\cal D}_{klpq}  {\\cal E}_\\grad820^\\ord{k}  - ( k(k-1) -60)  \\delta^i_{[k} {\\cal D}_{lpq]j} {\\cal E}_\\grad820^\\ord{k}\\ .  \\label{CubicK} \\ee\nIt is therefore tempting to conjecture that \n\\begin{equation}}\\def\\ee{\\end{equation}  {\\cal E}_\\grad820^\\ord{k} \\propto  E_{\\mbox{\\DEVII{0}00000{4\\mbox{+}k}}} \\ , \\label{EisF2k}  \\ee\nin the string theory effective action, and we will indeed show in section \\ref{StringPerturb} that this function admits a consistent perturbative string theory limit. Moreover, we will see in section \\ref{HigherDimensions} that it also admits an appropriate decompactification limit. \n\n\n\\subsection{Wavefront set and Poisson equation source term}\nWe have seen that there are two classes of $\\nabla^6 R^4$ type invariants in four dimensions, that preserve tree-level supersymmetry modulo the classical field equations. However, considering that the effective action already includes an $R^4$ type correction, we must take into account the action of the accordingly modified supersymmetry transformation on the $R^4$ type invariant itself. This is a very difficult task to carry out in practice, but one can nonetheless show general properties on these corrections. We recall that the $R^4$ type invariant admits the following gradient expansion in derivatives of the function ${\\cal E}_\\grad844$ \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal L}_\\grad844[{\\cal E}_\\grad844] = \\sum_{n=0}^{12} {\\cal D}^n_{[0,0,0,n,0,0,0]}  {\\cal E}_\\grad844 {\\cal L}_\\grad844^{[0,0,0,n,0,0,0]} , \\ee\nwith ${\\cal E}_\\grad844$ satisfying to \\eqref{R4Quadra}. The first order modification of the supersymmetry transformations will therefore necessarily admit the same gradient expansion in the function  ${\\cal E}_\\grad844$, such that\n\\begin{equation}}\\def\\ee{\\end{equation} \\delta = \\delta^\\ord{0} + \\sum_{n=0}^{12}   {\\cal D}^n_{[0,0,0,n,0,0,0]}  {\\cal E}_\\grad844 \\delta^{\\ord{1}\\, [0,0,0,n,0,0,0]} +\\dots \\ , \\ee\nwhere the dots stand for higher order corrections. It follows that the correction at second order will admit the expansion \n\\begin{eqnarray} && \\delta \\int {\\cal L}_\\grad844[{\\cal E}_\\grad844] \\nonumber \\\\*\n&=&  \\int \\Scal{ \\sum_{n=0}^{12}   {\\cal D}^n_{[0,0,0,n,0,0,0]}  {\\cal E}_\\grad844 \\delta^{\\ord{1}\\, [0,0,0,n,0,0,0]} } \\Scal{ \\sum_{m=0}^{12} {\\cal D}^m_{[0,0,0,m,0,0,0]}  {\\cal E}_\\grad844 {\\cal L}_\\grad844^{[0,0,0,m,0,0,0]} }  \\nonumber \\\\*\n&=& \\int \\sum_{m \\, n} \\sum_R   \\scal{ {\\cal D}^n_{[0,0,0,n,0,0,0]}  {\\cal E}_\\grad844 {\\cal D}^m_{[0,0,0,m,0,0,0]}{\\cal E}_\\grad844  }_R \\Psi^R_{m,n} \\,  \\end{eqnarray}\nwhere the sum over $R$ runs over all irreducible representations of $SU(8)$ in the tensor product  $[0,0,0,n,0,0,0]\\otimes [0,0,0,m,0,0,0]$, and $ \\Psi^R_{m,n} $ are understood to be $E_{7(7)}$ invariant densities function of the fields and their covariant derivatives in the irreducible representation $R$. One checks that all the appearing irreducible representations $R$ are self-conjugate, {\\it i.e.}\\   of the type $[n_4,n_3,n_2,n_1,n_2,n_3,n_4]$, by property of the tensor product $[0,0,0,n,0,0,0]\\otimes[0,0,0,m,0,0,0] $. The $\\bar F^2 \\nabla^4 R^4$ type invariant admits a gradient expansion with non-self conjugate irreducible representations, and all its components in self-conjugate representations do in fact coincide with ones appearing in the $(8,1,1)$ $\\nabla^6 R^4$ type invariant. It follows that the analysis of the supersymmetry constraints on the $\\bar F^2 \\nabla^4 R^4$ type invariant is not modified by the presence of the $R^4$ correction, and equations (\\ref{CubicR},\\ref{Laplace820}) are the exact equations to be solved by the corresponding function ${\\cal E}_\\grad820$ in the Wilsonian action. \n\nHowever, all the irreducible representations that appear in the gradient expansion \\eqref{811GradExpand} are included in the tensor product $[0,0,0,n,0,0,0]\\otimes[0,0,0,m,0,0,0] $ for $m$ and $n$ running from 1 to twelve, and the differential equations (\\ref{CubicC},\\ref{Laplace811}) must be modified in the presence of the $R^4$ type correction. Following the analysis carried out in \\cite{Green:2005ba,Green:2010kv}, we conclude that \n\\begin{equation}}\\def\\ee{\\end{equation}  \\Delta  {\\cal E}_\\grad811 = - 60  {\\cal E}_\\grad811  - ({\\cal E}_\\grad844)^2 \\ .\\label{Laplace811R4} \\ee\nAs explained in \\cite{Minimal}, this requires then to modify \\eqref{CubicC} to \n\\begin{eqnarray} {\\cal D}_{ijpq} {\\cal D}^{pqmn} {\\cal D}_{mnkl}  {\\cal E}_\\grad811 &=& - 9 {\\cal D}_{ijkl}   {\\cal E}_\\grad811 - \\frac{1}{2} {\\cal E}_\\grad844 {\\cal D}_{ijkl} {\\cal E}_\\grad844  \\ , \\label{CubicCR4}\\nonumber \\\\*\n {\\cal D}^{ijpq} {\\cal D}_{pqmn} {\\cal D}^{mnkl} {\\cal E}_\\grad811 &=& -9  {\\cal D}^{ijkl} {\\cal E}_\\grad811 - \\frac{1}{2} {\\cal E}_\\grad844 {\\cal D}^{ijkl} {\\cal E}_\\grad844 \\ . \\end{eqnarray}\nUsing in particular the tensor product \n\\begin{equation}}\\def\\ee{\\end{equation} {\\scriptstyle [0,0,0,2,0,0,0]}\\otimes {\\scriptstyle[0,0,0,1,0,0,0]}\\cong {\\scriptstyle [0,0,0,3,0,0,0]}\\oplus {\\scriptstyle [0,1,0,1,0,1,0] } \\oplus {\\scriptstyle [1,0,0,1,0,0,1] }\\oplus  {\\scriptstyle [0,0,1,1,1,0,0]}\\oplus  {\\scriptstyle [0,0,0,1,0,0,0]} \\nonumber \\ee\none shows that \n\\begin{equation}}\\def\\ee{\\end{equation} \\Scal{ 4 {\\cal D}_{ijpq} {\\cal D}^{pqmn} {\\cal D}_{mnkl} - {\\cal D}_{ijkl} \\scal{ \\Delta + 24} }   ({\\cal E}_\\grad844)^2 = 0 \\ , \\ee\nwhereas\n\\begin{equation}}\\def\\ee{\\end{equation} \\Scal{ 36 {\\cal D}_{jr[kl} {\\cal D}^{irmn} {\\cal D}_{pq]mn} - \\delta^i_j {\\cal D}_{klpq} ( \\Delta + 42)   + \\delta^i_{[k} {\\cal D}_{lpq]j} ( \\Delta-120)}   ({\\cal E}_\\grad844)^2  \\ne 0 \\ . \\ee\nWe therefore conclude that no higher derivative correction in $ ({\\cal E}_\\grad844)^2$ can consistently modify \\eqref{CubicCR4} without contradicting \\eqref{Laplace811R4}. \n\nThese properties of the source term in the Poisson equation \\eqref{Laplace811R4} can also be understood through the structure of the Fourier modes of these functions. In the decompactification limit, the Fourier modes of a function are the coefficients,  functions on $E_{6(6)}\/Sp_{\\scriptscriptstyle \\rm c}(4)$, of $e^{2\\pi i (q,a)}$, with the axion field $a$ in the ${\\bf 27}$ of the $E_{6(6)}$ subgroup. We have shown in \\cite{Minimal} that \\eqref{R4Quadra} implies then that the associated momenta are rank one vectors, {\\it i.e.}\\  using the cubic Jordan norm $3 \\, {\\rm det\\,}(q) = {\\rm tr} \\, q (q\\times q )$,\n\\begin{equation}}\\def\\ee{\\end{equation} q\\times q = 0\\ ,\\label{qxq} \\ee\n consistently with the properties of the $R^4$ threshold function.  If we consider the square of ${\\cal E}_\\grad844$, it admits by construction Fourier modes of momenta $q_1 + q_2$ where $q_1$ and $q_2$ satisfy to \\eqref{qxq}, such that \n \\begin{equation}}\\def\\ee{\\end{equation} {\\rm det\\,} (q_1+q_2) = {\\rm det\\,}(q_1) + {\\rm tr} \\, q_1 (q_2\\times q_2 )+ {\\rm tr} \\, q_2 (q_1\\times q_1 )+ {\\rm det\\,}(q_2) = 0 \\ , \\ee\nAs one can see in \\cite{Minimal}, equation \\eqref{Laplace811R4}  implies that the Fourier modes of the function ${\\cal E}_\\grad811$ must indeed carry momenta satisfying to the rank 2 constraint ${\\rm det\\,}(q)=0$, whereas the Fourier modes of the function ${\\cal E}_\\grad820$ are generic by construction in the parabolic decomposition. The  nilpotent orbit associated to  ${\\cal E}_\\grad820$  is indeed defined from the graded decomposition \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{e}_{7(7)} \\cong \\overline{\\bf 27}^\\ord{-2}\\oplus \\scal{ \\mathfrak{gl}_1\\oplus \\mathfrak{e}_{6(6)}}^\\ord{0} \\oplus {\\bf 27}^\\ord{2}\\ , \\ee\nsuch that a representative of the nilpotent orbit is a generic element of the grad two component in the ${\\bf 27}$. \n\nConsidering instead the string theory limit, the non-abelian Fourier modes are defined over a Heisenberg algebra with $32$ momenta in the positive chirality Weyl spinor representation of $Spin(6,6)$ associated to Ramond--Ramond D-brane charge $Q$, and an additional momentum associated to the Neveu--Schwarz 5-brane charge $N_5$. The nilpotent orbit associated to ${\\cal E}_\\grad811$ is defined from the associated graded decomposition \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{e}_{7(7)} \\cong{\\bf 1}^\\ord{-4}  \\oplus {\\bf 32}^\\ord{-2}\\oplus \\scal{ \\mathfrak{gl}_1\\oplus \\mathfrak{so}(6,6)}^\\ord{0} \\oplus {\\bf 32}^\\ord{2}\\oplus {\\bf 1}^\\ord{4} \\ . \\label{D6Grad}\\ee\nA representative of the nilpotent orbit is defined as a generic Weyl spinor in the grad 2 component \\cite{SpinorOrbits}, \n\\begin{equation}}\\def\\ee{\\end{equation} Q \\in Spin(6,6)\/ SU(2,4) \\ , \\quad {\\rm or}\\quad Q \\in Spin(6,6)\/ SL(6) \\ , \\label{OrbitsRank4}  \\ee\nto which one can add an arbitrary element of the grad 4 component $N_5$. This implies in particular that equation \\eqref{CubicCR4} does not imply any constraint on the Fourier modes. Equation \\eqref{R4Quadra} implies instead that $Q$ must be a rank 1 spinor, \\cite{SpinorOrbits}\n\\begin{equation}}\\def\\ee{\\end{equation}(Q^2)|_{\\bf 66} \\ \\hat{=}\\   (Q\\Gamma^{MN} Q) = 0 \\ , \\qquad Q \\in Spin(6,6)\/ \\scal{SL(6)\\ltimes \\mathds{R}^{15}} \\ , \\ee\n as for example the grad 3 singlet in the decomposition \n\\begin{eqnarray} \\mathfrak{so}(6,6) &\\cong& \\overline{\\bf 15}^\\ord{-2}\\oplus \\scal{ \\mathfrak{gl}_1\\oplus \\mathfrak{sl}_6}^\\ord{0} \\oplus {\\bf 15}^\\ord{2}\\ , \\nonumber \\\\* \n {\\bf 32}& \\cong& {\\bf 1}^\\ord{-3} \\oplus {\\bf 15}^\\ord{-1} \\oplus \\overline{\\bf 15}^\\ord{1} \\oplus  {\\bf 1}^\\ord{3} \\ . \n \\end{eqnarray}\nA generic rank 1 charge vector can always be rotated to the grad 3 component. Considering the sum of two rank one charges, respectively in the grad -3 and the grad 3 components, one obtains a generic rank 4 spinor of stabilizer $SL(6)\\subset Spin(6,6)$. All the rank four charges defined as the sum of two rank 1 charges with a non-trivial symplectic product can be written in this form. Therefore the right-hand-side in \\eqref{CubicCR4}  indeed sources generic Fourier modes of ${\\cal E}_\\grad811$. More precisely, all the Fourier modes with a negative quartic invariant $I_4(Q)\\le 0$ (belonging to the second orbit in \\eqref{OrbitsRank4}) are sourced by the function ${\\cal E}_\\grad{8}{1}{1}^{\\; 2}$, whereas the Fourier modes with a strictly positive quartic invariant $I_4(Q)$  (belonging to the first orbit in \\eqref{OrbitsRank4}) satisfy to a homogeneous equation. \n\nOn the contrary, a representative of the nilpotent orbit associated to ${\\cal E}_\\grad820$ satisfies that its third power in the adjoint representation vanishes, which according to \\eqref{D6Grad} implies that \\cite{SpinorOrbits}\n\\begin{equation}}\\def\\ee{\\end{equation} (Q^3)|_{{\\bf 32}}  \\ \\hat{=}\\  ( Q \\Gamma^{MN} Q  ) \\Gamma_{MN} Q = 0 \\ , \\quad Q \\in Spin(6,6)\/\\scal{ SL(2)\\times Spin(3,4)\\ltimes \\mathds{R}^{2\\times 8 +1}} \\ .\\label{Q3String}  \\ee \nThe relation with the Fourier modes is not completely straightforward in the presence of a non-trivial NS5-brane charge, because in that case the nilpotent subgroup is a non-abelian Heisemberg group, such that the corresponding Killing vector \n\\begin{equation}}\\def\\ee{\\end{equation} \\kappa_\\alpha = \\frac{\\partial\\, }{\\partial a^\\alpha} - \\frac{1}{2}C_{\\alpha\\beta} a^\\beta\\frac{\\partial\\, }{\\partial b} \\ , \\qquad k_5 = \\frac{\\partial\\, }{\\partial b} \\ , \\ee\nsatisfy to\n\\begin{equation}}\\def\\ee{\\end{equation} [ \\kappa_\\alpha , \\kappa_\\beta ] = C_{\\alpha\\beta} k_5 \\ , \\ee\nwhere $C_{\\alpha\\beta}$ is the antisymmetric charge conjugation matrix of $Spin(6,6)$. For a Fourier mode of vanishing NS5-brane charge, $k_5 {\\cal E}_{Q,0} = 0 $, and one can define the spinor charge $Q$ such that $\\kappa_\\alpha   {\\cal E}_{Q,0}  = i  Q_\\alpha  {\\cal E}_{Q,0}$, and $Q$ must satisfy to the same algebraic equations as the representatives of the nilpotent orbits associated to the differential equations. For a non-zero NS5-brane charge the relevant equations are more complicated, but still involve the Killing vector $\\kappa_\\alpha$ to the third order in the same combination. \n\n\nLet us now consider the M-theory limit, for which one considers the decomposition \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{e}_{7(7)} \\cong{\\bf 7}^\\ord{-4}  \\oplus \\overline{\\bf 35}^\\ord{-2}\\oplus \\scal{ \\mathfrak{gl}_1\\oplus \\mathfrak{sl}_7}^\\ord{0} \\oplus {\\bf 35}^\\ord{2}\\oplus \\overline{\\bf 7}^\\ord{4} \\ , \\label{A6Grad}\\ee\nIn this case the nilpotent subgroup also generate a non-abelian Heisenberg type algebra \n\\begin{equation}}\\def\\ee{\\end{equation} \\kappa_{mnp} = \\frac{\\partial\\, }{\\partial a^{mnp} } - \\frac{1}{12} \\varepsilon_{mnpqrst} a^{qrs}  \\frac{\\partial\\, }{\\partial b_t } \\ , \\qquad k^m =  \\frac{\\partial\\, }{\\partial b_m }  \\ , \\ee\nsuch that \n\\begin{equation}}\\def\\ee{\\end{equation} [\\kappa_{mnp} ,\\kappa_{qrs}] = \\frac{1}{6}  \\varepsilon_{mnpqrst}  k^t \\ . \\ee\nFor a Fourier mode of vanishing M5-brane charge, $k^m  {\\cal E}_{q,0} = 0 $, and one can define the M2-brane charge $\\kappa_{mnp}  {\\cal E}_{q,0}  = i  q_{mnp}  {\\cal E}_{q,0}$. For a non-zero M5-brane charge $k^m  {\\cal E}_{q,p} = i p^m {\\cal E}_{q,p}  $ the relevant equations are more complicated, but still involve the Killing vector in a way similar as does the corresponding nilpotent orbit characteristic equation involves the algebraic charges. For a 1\/2 BPS charge satisfying to the quadratic constraint, one obtains \\cite{Lu:1997bg}\n\\begin{equation}}\\def\\ee{\\end{equation} \\varepsilon^{mnqrstu} q_{pqr} q_{stu} = 0 \\ , \\qquad q_{mnp} p^p = 0 \\ , \\ee\ngiving $17=13+4$ linearly independent solutions, with typical representative\n\\begin{equation}}\\def\\ee{\\end{equation} \\frac{1}{6} q_{mnp} dy^m \\wedge dy^n \\wedge dy^p = q_1 dy^1 \\wedge dy^2 \\wedge dy^3 \\ . \\ee\nThe cubic constraint in the adjoint representation implies \n\\begin{equation}}\\def\\ee{\\end{equation} \\varepsilon^{nrstuvw} q_{rst}q_{uv[p} q_{qm]w} = 0 \\ ,  \\qquad \\varepsilon^{mnqrstu} q_{pqr} q_{stu} p^p = 0 \\ , \\ee \nthat gives $27=21+6$ linearly independent solutions, with typical representative\n\\begin{equation}}\\def\\ee{\\end{equation}  \\frac{1}{6} q_{mnp} dy^m \\wedge dy^n \\wedge dy^p = dy^1 \\wedge \\scal{ q_1 dy^2 \\wedge dy^3 + q_2 dy^4 \\wedge dy^5 + q_3 dy^6 \\wedge dy^7}  \\ . \\label{Rank3} \\ee\n The cubic constraint in the fundamental implies instead \n\\begin{equation}}\\def\\ee{\\end{equation}  \\varepsilon^{mnrstuv} q_{mnr}q_{st(p} q_{q)uv} = 0 \\ , \\label{Rank4} \\ee\nthat gives $33=26+7$ linearly independent solutions, such that the $SL(7)$ M2-brane charge orbits are either \n\\begin{equation}}\\def\\ee{\\end{equation} q \\in SL(7,\\mathds{R})\/ (SL(3,\\mathds{C}) \\ltimes \\mathds{C}^3) \\ , \\quad {\\rm or}\\quad q\\in SL(7,\\mathds{R})\/ (SL(3,\\mathds{R}) \\ltimes \\mathds{R}^3)^{\\times 2}  \\ , \\label{SL7orbit} \\ee\nwith typical representative \n\\begin{equation}}\\def\\ee{\\end{equation}   \\frac{1}{6} q_{mnp} dy^m \\wedge dy^n \\wedge dy^p = q_1 dy^1 \\wedge  dy^2 \\wedge dy^3 + q_2 dy^1\\wedge  dy^4 \\wedge dy^5 + q_3 dy^2 \\wedge dy^6 \\wedge dy^4 + q_4 dy^3 \\wedge dy^5 \\wedge dy^6  \\ , \\ee \nand \n\\begin{equation}}\\def\\ee{\\end{equation}  \\frac{1}{2\\times 12^3}\\varepsilon^{pmnrstu}  \\varepsilon^{qm^\\prime n^\\prime r^\\prime s^\\prime t^\\prime u^\\prime } q_{mnr}q_{stm^\\prime} q_{n^\\prime r^\\prime u} q_{s^\\prime t^\\prime u^\\prime} =  q_1 q_2 q_3  q_4 \\delta^p_7 \\delta^q_7 \\ , \\ee\nsuch that the orbit \\eqref{SL7orbit} is determined by the sign of the eigenvalue of this rank one symmetric tensor, $I_4 =  q_1 q_2 q_3 q_4$. The generic sum of two rank one charges takes the form \n\\begin{equation}}\\def\\ee{\\end{equation} \\frac{1}{6} q_{mnp} dy^m \\wedge dy^n \\wedge dy^p = q_1 dy^1 \\wedge dy^2 \\wedge dy^3  + q_2dy^4 \\wedge dy^5 \\wedge dy^6 \\ ,\\ee\nand is a generic solution to \\eqref{Rank4} associated to the second orbit ({\\it i.e.}\\  $I_4<0$), and violates equation \\eqref{Rank3}. Therefore we confirm that a quadratic source in ${\\cal E}_\\grad{8}{4}{4}$ is in contradiction with the cubic equation satisfied by ${\\cal E}_\\grad{8}{2}{0}$, whereas it is consistent with the one satisfied by ${\\cal E}_\\grad{8}{1}{1}$. \n\n\n\n\n\n\n\nLet us now argue that all the invariants of the infinite series of $\\bar F^{2k} \\nabla^4 R^4$ do not get modified at the same order by lower order modifications to the supersymmetry transformations. By power counting, the next order correction to the $R^4$ type invariant and a $\\bar F^{2k} \\nabla^4 R^4$ type invariant can in principle contribute to a right-hand-side for the classical supersymmetry variation of a $\\bar F^{2k+6} \\nabla^4 R^4$ type invariant. So in principle one could expect that the function ${\\cal E}_\\grad820^\\ord{k}$ satisfies to a Poisson equation of the kind \n\\begin{multline} \\Delta {\\cal E}_\\grad820^\\ord{k} =3 (k+4)(k-5) {\\cal E}_\\grad820^\\ord{k}   - a^\\ord{k}_3 {\\cal E}_\\grad844 \\, {\\cal E}_\\grad820^\\ord{k-3} \\\\ - a^\\ord{k}_5   {\\cal E}_\\grad822 \\, {\\cal E}_\\grad820^\\ord{k-5} - a^\\ord{k}_6 ({\\cal E}_\\grad844)^2 \\, {\\cal E}_\\grad820^\\ord{k-6} - \\sum_{p=0}^{k-8} b^\\ord{k}_{p}   {\\cal E}_\\grad820^\\ord{k-8-p} {\\cal E}_\\grad820^\\ord{p} + \\dots \\end{multline}\nHowever, the solutions to the differential equation \\eqref{CubicK} admit restricted Fourier modes in the string theory limit, satisfying to \\eqref{Q3String} for a vanishing NS5-brane charge. As we have already explained, the product of two functions including non-perturbative corrections admits generic Fourier modes in the string theory limit, because the sum of two pure spinors can be a generic spinor. We see therefore that a source term modifying \\eqref{CubicK} would necessarily involve the third order differential operator such as to source these Fourier modes. Such a modification would destroy completely the structure of the equations, which would reduce then to some kind of Poisson equation. \n\n\n\\subsection{String theory perturbation theory}\n\\label{StringPerturb} \nIn order to deduce constraints on the contributions that can possibly appear in perturbative string theory, it is important to solve the differential equations satisfied by the threshold functions in the parabolic gauge with manifest T-duality symmetry \\eqref{D6Grad}. In this section we will solve these equations on an ansatz function depending only on the string theory dilaton $e^{2\\phi}$ and the scalar fields parametrising $SO(6,6)\/(SO(6)\\times SO(6))$. We have not computed explicitly the decomposition of the differential equations, but using the manifest covariance, and the known solutions for the $R^4$ and the $\\nabla^4 R^4$ threshold functions \\cite{Green:2010kv}, we can determine unambiguously all the unknown coefficients. \n\nWe define the covariant derivative ${\\cal D}_{a\\hat{b}}$ on $SO(6,6)\/(SO(6)\\times SO(6))$ in tangent frame, such that $a=1$ to $6$ of one $SO(6)$ and $\\hat{b}=1$ to $6$ of the other. It is convenient to define the covariant derivative as an $SU(4)\\times SU(4)$ tensor\n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ij\\hat{k}\\hat{l}} = \\frac{1}{4} \\gamma^a{}_{ij} \\gamma^{\\hat{b}}{}_{\\hat{k}\\hat{l}} {\\cal D}_{a\\hat{b}} \\ , \\ee\nwith $i=1,\\, 4$ of one $SU(4)$ and $\\hat{\\imath}=1,\\, 4$ of the other. \n\nCalibrating the equations on the known solutions, one obtains that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf D}_{56}^{\\;3} {\\cal E} = \\scal{ s^2 - \\tfrac{17}{2} s+6}    {\\bf D}_{56} {\\cal E} \\label{D563sString} \\ee\ndecomposes on $\\mathds{R}_+^* \\times SO(6,6)\/(SO(6)\\times SO(6))$ as\n\\begin{eqnarray}\\biggl(  \\Scal{  \\frac{1}{64} \\partial_\\phi^{\\; 3} + \\frac{17}{32} \\partial_\\phi^{\\; 2} + \\frac{3}{2} \\partial_\\phi -  {\\cal D}_{a\\hat{b}} {\\cal D}^{a\\hat{b}} } \\delta_a^b + \\Scal{ \\frac{3}{4} \\partial_\\phi + 6 } {\\cal D}_{a\\hat{c}} {\\cal D}^{b\\hat{c}} \\biggr) {\\cal E} &=&\\scal{ s^2 - \\tfrac{17}{2} s+6}   \\ \\delta_{a}^b \\,  \\frac{1}{4} \\partial_\\phi{\\cal E} \\ , \\qquad \\nonumber \\\\*\n\\biggl( {\\cal D}_{a\\hat{c}} {\\cal D}^{d\\hat{c}} {\\cal D}_{d\\hat{b}} + \\Scal{ \\frac{3}{16} \\partial_\\phi^{\\; 2} + \\frac{31}{8} \\partial_\\phi + 9 } {\\cal D}_{a\\hat{b}} \\biggr) {\\cal E} &=&\\scal{ s^2 - \\tfrac{17}{2} s+6}   {\\cal D}_{a\\hat{b}} {\\cal E}\\ ,  \\nonumber \\\\*\n\\biggl( 8 {\\cal D}_{ip}{}^{\\hat{k}\\hat{q}} {\\cal D}^{pr}{}_{\\hat{q}\\hat{s}} {\\cal D}_{jr}{}^{\\hat{l}\\hat{s}} + \\Scal{ \\frac{5}{4} \\partial_\\phi + 2} 2{\\cal D}_{ij}{}^{\\hat{k}\\hat{l}}\\biggr) {\\cal E} &=& 2\\scal{ s^2 - \\tfrac{17}{2} s+6}  {\\cal D}_{ij}{}^{\\hat{k}\\hat{l}} {\\cal E} \\ , \\ \n\\end{eqnarray}\nwhereas \n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf D}_{133}^{\\;3} {\\cal E} =s(s-9)   {\\bf D}_{133} {\\cal E} \\label{AdjointEquation}  \\ee\ngives the components in the ${\\bf 32}$ of $Spin(6,6)$\n\\begin{eqnarray} \\biggl(\\Scal{ \\frac{1}{64} \\partial_\\phi^{\\; 3} + \\frac{5}{8} \\partial_\\phi^{\\; 2} - \\frac{5}{16} \\partial_\\phi - {\\cal D}_{pq{\\hat{r}\\hat{s}}} {\\cal D}^{pq{\\hat{r}\\hat{s}}} } \\delta_i^k \\delta_{\\hat{\\jmath}}^{\\hat{l}} + 3 ( \\partial_\\phi +6)  {\\cal D}_{ip\\hat{\\jmath}\\hat{q}} {\\cal D}^{kp\\hat{l}\\hat{q}} \\biggr) {\\cal E} &=& s(s-9) \\ \\delta_i^k \\delta_{\\hat{\\jmath}}^{\\hat{l}}\\,  \\frac{1}{4} \\partial_\\phi {\\cal E}\\ , \\qquad  \\nonumber \\\\*\n\\biggl(8  {\\cal D}_{ip\\hat{\\jmath}\\hat{q}} {\\cal D}^{pr\\hat{q}\\hat{s}}{\\cal D}_{kr\\hat{l}\\hat{s}} + \\Scal{ \\frac{3}{16} \\partial_\\phi^{\\; 3} + \\frac{31}{8} \\partial_\\phi}2 {\\cal D}_{ik\\hat{\\jmath}\\hat{l}} \\biggr) {\\cal E} &=&   2 s(s-9){\\cal D}_{ik\\hat{\\jmath}\\hat{l}} {\\cal E} \\ . \\label{D1333sString}\n \\end{eqnarray}\nThe $\\nabla^4 R^4$ threshold function solves \\eqref{D563sString} for $s=\\frac{3}{2}$ and \\eqref{D1333sString} for $s=4$. One reads directly from these equations, that a solution of type $e^{a\\phi} {\\cal E}_{D_6}$  on $\\mathds{R}_+^* \\times SO(6,6)\/(SO(6)\\times SO(6))$ must be such that ${\\cal E}_{D_6}$ satisfies to the quadratic equations in all fundamental representations ({\\it i.e.}\\  the vector and Weyl spinor of positive and negative chirality), unless $a=-6$ or $a=-8$. The only other solutions are therefore such that ${\\cal E}_{D_6}$ is either a constant, or solves \\eqref{MinimalD6}. One finds the unique  solution $e^{-10\\phi}$.  For the values $a=-8$, the function ${\\cal E}_{D_6}$ satisfies to a quadratic equation in the spinor representation, and solve \\eqref{VecD6} for $s=4$ (or $1$ which is equivalent). For $a=-6$, ${\\cal E}_{D_6}$ satisfies to a quadratic equation in the vector representation, and cubic equations in the two spinor representations, and must therefore satisfy to  \\eqref{NextMinimalD6}. We find therefore that supersymmetry and T-duality alone already determine the $\\nabla^4 R^4$ type corrections in perturbative string theory, up to three free coefficients, that are given in \\cite{Green:2010kv}\n\\begin{equation}}\\def\\ee{\\end{equation} \\frac{1}{2} {E}_{\\mbox{\\DEVII{\\frac{5}{2}}00000{\\mathfrak{0}}}}  = \\zeta(5) e^{-10\\phi}  + \\frac{4}{15\\pi} e^{-8\\phi} E_{\\mbox{\\WSOXII{4}00000}}  + \\frac{2}{3} e^{-6\\phi} E_{\\mbox{\\WSOXII{0}00020}} + \\mathcal{O}(e^{-e^{-\\phi}})  \\ . \\label{E4} \\ee\nThis confirms that supersymmetry alone already prevents any perturbative correction to the $\\nabla^4 R^4$ threshold function beyond 2-loop in perturbative string theory. \n\nThe functions defining the $\\bar F^{2k} \\nabla^4 R^4$ solve equation  \\eqref{D1333sString} for $s=k+4$. \nSimilarly one obtains the general $SO(6,6,\\mathds{Z})$ invariant solution \n \\begin{equation}}\\def\\ee{\\end{equation} {\\cal E}_\\grad820^\\ord{k} = c^\\ord{k}_1 e^{-2(k+4)\\phi} {E}_{\\mbox{\\WSOXIIT{k\\mbox{+}4}000{\\mathfrak{0}}0}}   + c_{k+2}^\\ord{k} e^{-6\\phi} {E}_{\\mbox{\\WSOXIIT{\\mathfrak{0}}000{k\\mbox{+}2}0}} +c^\\ord{k}_{2k}   e^{2(k-5)\\phi} {E}_{\\mbox{\\WSOXIIT{k}000{\\mathfrak{0}}0}} +\\mathcal{O}(e^{-e^{-\\phi}})  \\ .\\ee\nIt is quite remarkable that the only solutions we get all correspond to a strictly positive number of loops in perturbative string theory. After implementing the Weyl rescaling, one obtains indeed that $c^\\ord{k}_\\ell$ is a coefficient for a $\\ell$-loop correction in string theory for the $\\bar F^{2k} \\nabla^4 R^4$ threshold function. For $k=1$ and $k=2$, equation   \\eqref{D1333sString} is exact for the Wilsonian effective action (not taking into account linear corrections associated to logarithms in the complete effective action). $U$-duality therefore implies that ${\\cal E}_\\grad820$ must be an Eisenstein function as in \\eqref{EisF2k}. Assuming that our argumentation in the preceding section is correct, and that equation \\eqref{D1333sString} is satisfied for all $k$, we arrive at the conjecture that  the $\\bar F^{2k} \\nabla^4 R^4$ threshold function is defined by the Eisenstein series $E{\\mbox{\\DEVII{0}00000{4\\mbox{+}k}}}$ for all $k$. It is rather remarkable that this coupling would only get three corrections in perturbation theory, at 1-loop, $k+2$-loop and $2k$-loop. \n\n\nThis Eisenstein function diverges precisely for $k=1$, corresponding to the $\\bar F^2 \\nabla^4 R^4$ threshold related to the $\\nabla^6 R^4$  threshold function by supersymmetry.  One must therefore consider the regularised Eisenstein series \n\\begin{multline}  \\frac{16}{63} \\hat{{E}}_{\\mbox{\\DEVII000000{5}}} = \\frac{16}{63}e^{-10\\phi}  \\hat{E}_{\\mbox{\\WSOXIIT{5}000{\\mathfrak{0}}0}}-\\frac{15\\zeta(5)}{4} \\phi\\,  e^{-10\\phi}  + \\frac{1}{\\pi} e^{-8\\phi} \\Scal{ 2 \\phi  {E}_{\\mbox{\\WSOXIIT{4}000{\\mathfrak{0}}0}} - \\partial_s {E}_{\\mbox{\\WSOXIIT{s}000{\\mathfrak{0}}0}} \\big|_{s=4} }   \\\\ + \\frac{\\pi}{9} e^{-6\\phi} \\hat{E}_{\\mbox{\\WSOXIIT{\\mathfrak{0}}000{3}0}}  +   \\mathcal{O}(e^{-e^{-\\phi}}) \\ .  \\end{multline} \nHere we have fixed all the coefficients by consistency with \\eqref{DivergenceF2D4R4} and \\eqref{E4}. The logarithm of the dilaton indicates a divergence of the $\\nabla^4 R^4$ form factor into  $\\nabla^6 R^4$ in supergravity. Note nonetheless that the 3-loop contribution in the last line violate T-duality parity in $O(6,6,\\mathds{Z})$, and the string theory effective action must include the same function with opposite chirality. Because it is a three-loop contribution, it cannot come from the completion of the $R^4$ type invariant and it must appear as a solution to equation \\eqref{D563sString} for $s=6$. \n\nConsidering the general $SO(6,6,\\mathds{Z})$ invariant solution of \\eqref{D563sString}, one finds indeed \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal E}_\\grad811  =  c_{\\mbox{-}6}e^{-4s\\phi} + c_{\\mbox{-}\\frac{1}{2}}e^{-(2s+1)\\phi}{E}_{\\mbox{\\WSOXII{0}0000{\\mathnormal{s}\\mbox{-}\\hspace{-0.2mm}\\stfrac{1}{2}}}}  +c_1 e^{2(2s-17)\\phi}  +c_2 e^{-8\\phi} {E}_{\\mbox{\\WSOXII0{\\mathnormal{s}\\mbox{-}2}0000}}+ c_3 e^{2(s-9)\\phi} {E}_{\\mbox{\\WSOXII00000{\\mathnormal{s}\\mbox{-}3}}}  \\ee\nwhere the coefficients $c_\\ell$ are constants that would correspond to $\\ell$-loop contributions for the $\\nabla^6 R^4$ threshold function in string theory.  Note that the first two terms do not make sense in perturbative string theory. The corresponding Eisenstein function ${E}{\\mbox{\\DEVII{s}00000{\\mathfrak{0}}}}$ includes generically all these terms, and therefore cannot define the string theory threshold function, consistently with the property that \\eqref{D563sString} is corrected by a source term \\eqref{CubicCR4}. However, the three-loop contribution is not affected by the source term, and one can take seriously the last contribution, which is precisely the one required to restore  $O(6,6,\\mathds{Z})$ invariance  for $s=6$.\n\nThis is indeed confirmed by the expression obtained in \\cite{Pioline:2015yea} for the $\\nabla^6 R^4$ threshold function, and using these results we conclude therefore that the exact threshold function for the $\\nabla^6 R^4$ coupling is defined as\n\\begin{equation}}\\def\\ee{\\end{equation} E_{\\gra{0}{1}} = \\hat{{\\cal E}}_{\\grad{8}{1}{1}}  + \\frac{32}{189\\pi}\\hat{{E}}_{\\mbox{\\DEVII000000{5}}}\\ ,  \\ee\nwhere the function $ {\\cal E}_{\\grad{8}{1}{1}}$ solve the differential equation \n\\begin{eqnarray}\\label{E811Equation} \\Delta \\hat{{\\cal E}}_{\\grad{8}{1}{1}} &=& - 60 \\hat{{\\cal E}}_{\\grad{8}{1}{1}} - \\Scal{ {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{3}{2}}}00000{\\mathfrak{0}}}}}^2 + \\frac{35}{\\pi} \\Scal{ \\frac{1}{2}  {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{5}{2}}}00000{\\mathfrak{0}}}}}\\ \\\\\n {\\cal D}_{ijpq} {\\cal D}^{pqmn} {\\cal D}_{mnkl}  \\hat{{\\cal E}}_\\grad811 &=& - 9 {\\cal D}_{ijkl}   \\hat{{\\cal E}}_\\grad811 - \\frac{1}{2} {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{3}{2}}}00000{\\mathfrak{0}}}} {\\cal D}_{ijkl} {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{3}{2}}}00000{\\mathfrak{0}}}}  + \\frac{35}{4\\pi}   {\\cal D}_{ijkl}  \\Scal{ \\frac{1}{2}  {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{5}{2}}}00000{\\mathfrak{0}}}}}\\ , \\nonumber \\\\*\n {\\cal D}^{ijpq} {\\cal D}_{pqmn} {\\cal D}^{mnkl} \\hat{{\\cal E}}_\\grad811 &=& -9  {\\cal D}^{ijkl} \\hat{{\\cal E}}_\\grad811 - \\frac{1}{2}{E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{3}{2}}}00000{\\mathfrak{0}}}} {\\cal D}^{ijkl} {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{3}{2}}}00000{\\mathfrak{0}}}} +\\frac{35}{4\\pi}   {\\cal D}^{ijkl}  \\Scal{ \\frac{1}{2}  {E}_{\\mbox{\\DEVII{\\mathnormal{\\tfrac{5}{2}}}00000{\\mathfrak{0}}}}}\\ . \\nonumber\n\\end{eqnarray}\nHere the anomalous right-hand-side is determined such as to coincide with the one obtained in \\cite{Pioline:2015yea} for the complete function $ E_{\\gra{0}{1}}$. These coefficients can also be directly computed from the properties of the Eisenstein functions and the structure of the differential equations \\cite{Axel}. \n\\section{Supergravity in higher dimensions}\n\\label{HigherDimensions}\nIn this section we will consider the extension of the results of the preceding section in five, six, seven and eight dimensions. We will see that the two $\\nabla^6 R^4$ type invariants both lift to higher dimensions, even if they cannot be defined as harmonic superspace integrals in the linearised approximation in general.\n\n\\subsection{$\\mathcal{N} =4$ supergravity in five dimensions}\nLet us recall in a first place some properties of maximal supergravity in five dimensions. The scalar fields parametrise the symmetric space $E_{6(6)}\/Sp(4)_{\\scriptscriptstyle \\rm c}$. We use  $i, j = 1, ..., 8$ as indices in the fundamental representation of  $Sp(4)$, and $\\O^{i j}$ defines the symplectic form with the normalisation $\\O^{i k} \\O_{j k} = \\delta^{i}_{j} $. The covariant derivative in tangent frame ${\\cal D}_{i j k l}$ is a symplectic traceless rank four antisymmetric tensor in the representation $[0,0,0,1]$ of $Sp(4)$.\n\n\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{1}}\n\\subsubsection{Linearised $\\nabla^6 R^4$ type invariants}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{2}}\nIn five dimensions there is only one kind of 1\/8 BPS harmonic superspace integral that one can define \\cite{Bossard:2009sy}. For this purpose, one considers $Sp(4)\/(U(1)\\times Sp(3))$ harmonic variables $(u^{1}{}_{i}, u^{r}{}_{i}, u^{8}{}_{i})$ with $r = 2, ... ,7$ in the fundamental of  $Sp(3)$, and the decomposition \n\\begin{eqnarray}\n\\mathfrak{sp}(4) &\\cong& {\\bf 6}^\\ord{-1} \\oplus \\scal{ \\mathfrak{u}(1)\\oplus \\mathfrak{sp}(3)}^\\ord{0} \\oplus {\\bf 6}^\\ord{1} \\nonumber \\\\*\n{\\bf 42} &\\cong&  {\\bf 14}^{\\ord{-1}}_{3} \\oplus {\\bf 14}^{\\ord{0}}_{2} \\oplus {\\bf 14}^{\\ord{1}}_{3} \\ . \n\\end{eqnarray}\nOne defines the G-analytic superfield $W^{r s t}$ in the $[0,0,1]$ of $Sp(3)$\n\\begin{equation}}\\def\\ee{\\end{equation}\nW^{r s t} = u^{1}{}_{i}  u^{r}{}_{j} u^{s}{}_{j} u^{t}{}_{k} L^{i j k l} \\ , \n\\ee\nwhich satisfies the constraint \\begin{equation}}\\def\\ee{\\end{equation}\nu^1{}_{ i} D_{\\alpha}^{i} W^{r s t} = 0 \\ . \n\\ee\nFollowing the same reasoning as in section \\ref{811D6R4}, we consider a general monomial of $W^{rst}$ in an irreducible representation of $Sp(3)$. In this case we obtain equivalently that the monomials are freely generated by $W^{rst}$ in the $[0,0,1]$,  the elements \n\\begin{equation}}\\def\\ee{\\end{equation} W^{r t p} W^{s q r} \\O_{t q} \\O_{p r} \\ , \\ee\nin the $[2,0,0]$,\n\\begin{equation}}\\def\\ee{\\end{equation} W^{r p q} W^{s u}{}_{p} W^{t}{}_{q u} \\, , \\ee\nin the $[0,0,1]$, and  \n\\begin{equation}}\\def\\ee{\\end{equation} W^{r] t u} W^{[s}{}_{t u} W^{p] v w} W^{[q}{}_{v w} - {\\rm ``symp\\ trace\"} \\ , \\qquad W^{r s t} W^{p q}{}_{r} W^{u}{}_{s p} W{}_{t q u}\\ , \\ee\nrespectively in the $[0,2,0]$ and the singlet representation. The general linearised invariant takes therefore the form \n\\begin{eqnarray}\n&&  \\int du D^{28} F(u)^{[2n_2,2n_4,n_1 + n_3]}_{[2 n_3 + 2 n_{4} + 4 n'_{4}, 2n_2, 2n_4, n_1 + n_3]} W^{4 + n_1 + 2 n_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{[2n_2,2n_4,n_1 + n_3]} \\nonumber \\\\*\n&=&   L^{n_1 + 2 n_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{[2 n_3 + 2 n_{4} + 4 n'_{4}, 2n_2, 2n_4, n_1 + n_3]} \\scal{ \\nabla^6 R^4  + \\dots } + \\dots \n\\end{eqnarray}\nThe structure of these linearised invariants suggests that the complete non-linear invariant admits the following gradient expansion \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal L}_{\\gra{4}{1}}[ {\\cal E}_{\\gra{4}{1}}] = \\sum_{n_1,n_2,n_3,n_4,n_4^\\prime} {\\cal D}^{n_1 + 2 n_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{[2 n_3 + 2 n_{4} + 4 n'_{4}, 2n_2, 2n_4, n_1 + n_3]} {\\cal E}_{\\gra{4}{1}} {\\cal L}_{\\gra{4}{1}}^{[2 n_3 + 2 n_{4} + 4 n'_{4}, 2n_2, 2n_4, n_1 + n_3]}\\ . \n\\ee\nThe consistency of this ansatz requires that the function ${\\cal E}_{\\gra{4}{1}} $ must be an eigenfunction of the Laplace operator, and that its third order derivative restricted to the $[0,2,0,0]$ is proportional to its second derivative in the same representation. This linearised analysis is consistent with the one of the $(8,1,1)$ type invariant in four dimensions, and we are going to see that the relevant equation is \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ijpq} {\\cal D}^{pqrs} {\\cal D}_{rskl}  {\\cal E}_{\\gra{4}{1}} = \\frac{1}{4} {\\cal D}_{ijkl} \\scal{ 34 + \\Delta}  {\\cal E}_{\\gra{4}{1}}  \\ .\\ee\nHowever, the $(8,2,0)$ type invariants cannot be defined in the linearised approximation through a harmonic superspace integral, and we shall instead consider the uplift of the general invariant to five dimensions. \n\n\\subsubsection{Decompactification limit from four to five dimensions}\nWe are therefore going to solve the differential equations  \\eqref{CubicR} and \\eqref{E811Equation}  for a function depending only of the Levi subgroup $\\mathds{R}_*^+\\times  E_{6(6)}$ of the parabolic subgroup associated to the decompactification limit, such that \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{e}_{7(7)}\\cong \\overline{\\bf 27}^\\ord{-2} \\oplus \\scal{ \\mathfrak{gl}_1 \\oplus \\mathfrak{e}_{6(6)} }^\\ord{0} \\oplus {\\bf 27}^\\ord{2} \\ , \\qquad \n{\\bf 56} \\cong {\\bf 1}^\\ord{-3} \\oplus {\\bf 27}^\\ord{-1} \\oplus \\overline{\\bf 27}^\\ord{1} \\oplus {\\bf 1}^\\ord{3} \\ .\\label{56E6}  \\ee\nFor this purpose we use the same conventions as in \\cite{Minimal}, such that the coset representative in $E_{7(7)}\/SU(8)_{\\scriptscriptstyle \\rm c}$ is defined as\n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal V} = \\left( \\begin{array}{cccc} \\ e^{3\\phi} \\ & \\ 0 \\ & \\ 0 \\ & \\ 0 \\\\\n0 &\\ e^{\\phi} V_{ij}{}^I \\ &0&0\\\\\n0&0& \\ e^{-\\phi} V^{ \\scriptscriptstyle \\rm{\\mbox{\\tiny-1}}}{}_I{}^{ij} \\ &0\\\\\n0&0&0& e^{-3\\phi} \\end{array}\\right) \\left( \\begin{array}{cccc} \\ 1 \\ & \\ a^J \\ & \\ \\tfrac{1}{2} t_{JKL} a^K a^L  \\ & \\ \\tfrac{1}{3} t_{KLP} a^K a^L a^P \\\\\n0 &\\ \\delta_I^J \\ &t_{IJK} a^K&\\tfrac{1}{2} t_{IKL} a^K a^L\\\\\n0&0& \\delta_J^I \\ &a^I \\\\\n0&0&0& 1 \\end{array}\\right)  \\ , \\label{corset_elem} \\ee\nwhere $V_{i j}{}^{I}$ is the coset representative in $E_{6(6)}$ with the $Sp(4)$ pair $ij$ being antisymmetric symplectic traceless and the index $I$ in the fundamental of $E_{6(6)}$. $t_{I J K}$ is the $E_{6(6)}$ invariant symmetric tensor normalised as in \\cite{Minimal}. We have already computed the decomposition of the cubic equation \\eqref{Cubic56} in \\cite{Minimal}, which is\n\\begin{eqnarray} \\Bigl( \\frac{1}{64} \\partial_\\phi^{\\; 3} + \\frac{21}{32} \\partial_\\phi^{\\; 2} +\\frac{9}{2} \\partial_\\phi - \\frac{3}{4} \\Delta \\Bigr) {\\cal E}_\\grad{8}{1}{1}   &=& -  \\frac{1}{4} \\partial_\\phi\\Scal{9 {\\cal E}_\\grad{8}{1}{1}  + \\frac{1}{4}   {\\cal E}_\\grad{8}{4}{4}^{\\; 2}}  \\\\\n \\biggl(  {\\cal D}_{ijpq} {\\cal D}^{pq}{}_{rs} {\\cal D}^{rskl}  + {\\cal D}_{ij}{}^{kl} \\Bigl( \\frac{1}{48} \\partial_\\phi^{\\; 2} + \\frac{27}{24} \\partial_\\phi + \\frac{7}{2} \\Bigr)  + {\\cal D}_{ijpq} {\\cal D}^{klpq} \\Bigl( \\frac{1}{4} \\partial_\\phi+3 \\Bigr)\\hspace{-32mm} && \\nonumber \\\\* +\\delta_{ij}^{kl} \\Bigl( \\frac{1}{12^3} \\partial_\\phi^{\\; 3} + \\frac{5}{96} \\partial_\\phi^{\\; 2} +\\frac{1}{6} \\partial_\\phi - \\frac{1}{4} \\Delta  \\Bigr) \n \\biggr) {\\cal E}_\\grad{8}{1}{1} &=& -  \\Bigl( \\frac{1}{12} \\delta_{ij}^{kl} \\partial_\\phi + {\\cal D}_{ij}{}^{kl} \\Bigr)  \\Scal{ 9 {\\cal E}_\\grad{8}{1}{1}  + \\frac{1}{4}   {\\cal E}_\\grad{8}{4}{4}^{\\; 2}} \\nonumber \\ ,\n \\end{eqnarray}\n where \n \\begin{equation}}\\def\\ee{\\end{equation}\n\\delta_{ij}^{kl} = \\delta_{[i}^{[k} \\delta_{j]}^{l]} - \\frac{1}{8} \\Omega_{ij} \\Omega^{k l} \\ , \\qquad \\Delta = \\frac{1}{3} {\\cal D}_{ijkl} {\\cal D}^{ijkl} \\ ,\n\\ee\nand indices are raised and lowered with the symplectic matrix $\\Omega_{ij}$.  Because of the Weyl rescaling required to stay in Einstein frame, the relevant radius power in the decompactification limit for a $\\nabla^{2n} R^4$ threshold function is such that  ${\\cal E}_{E_{7}} = e^{-( 6 + 2 n)\\phi} {\\cal E}_{E_6}$, and because we are interested in the constraint on the $\\nabla^{6} R^4$ threshold function, we use the ansatz \n \\begin{equation}}\\def\\ee{\\end{equation}\n {\\cal E}_{\\grad{8}{1}{1}} = e^{- 12 \\phi} {\\cal E}_{\\gra{8}{1}} \\ , \\qquad  {\\cal E}_\\grad844 = e^{-6 \\phi}  {\\cal E}_{\\gra{8}{4}} \\label{LaplaceE6}  \\ ,\n \\ee\nwhere $ {\\cal E}_{\\gra{8}{1}}$ and $ {\\cal E}_{\\gra{8}{4}}$ are functions on $E_{6(6)}\/Sp(4)_{\\scriptscriptstyle \\rm c}$. Using this ansatz, one derives \n \\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta {\\cal E}_{\\gra{8}{1}} = - 18  {\\cal E}_{\\gra{8}{1}}  - {\\cal E}^{\\; 2}_{\\gra{8}{4}} \\ , \n \\ee\nand\n\\begin{equation}}\\def\\ee{\\end{equation}\n\\bigl( {\\cal D}_{ijpq} {\\cal D}^{pqrs} {\\cal D}_{rskl}  + 2  {\\cal D}_{i jkl} \\bigr) {\\cal E}_{\\gra{8}{1}} = - \\frac{1}{4}  {\\cal D}_{i jkl} {\\cal E}^{\\; 2}_{\\gra{8}{4}} \\ . \\label{D6R4E6} \n\\ee\nThese equations are satisfied by the 1\/8 BPS threshold functions in the Wilsonian effective action, but the U-duality invariant function appearing in the 1PI effective action satisfied to anomalous equations with additional terms linear in ${\\cal E}_\\gra{8}{4}$ in the right-hand-side \\cite{Pioline:2015yea}. \n\nWe shall now consider the uplift of the  $F^{2 k} \\nabla^4 R^4$ type invariants, but for this purpose it will be more convenient to consider directly the decompactification limit of the Eisenstein function  ${E}{\\mbox{\\DEVII000000{k\\mbox{+}4}}}$. We shall only consider the term with the correct power of the compactification radius $r$ to lift to a diffeomorphism invariant in five dimensions, as computed in appendix \\ref{E66Eis},\n\\begin{equation}}\\def\\ee{\\end{equation} \\int d^4 x \\, \\sqrt{-g} \\  {E}_{\\mbox{\\DEVII000000{k\\mbox{+}4}}}\\,  \\nabla^{4+2k} R^4 \\rightarrow  \\frac{\\pi^{\\frac{1}{2}} \\Gamma(k + \\frac{7}{2})}{\\Gamma(k+4)}   \\int d^5x \\, \\sqrt{-g}\\  E_{\\mbox{\\DEVI00000{k\\mbox{+}\\frac{7}{2}}}}   \\nabla^{4+2k} R^4 \\ . \\ee\nWe conclude in this way that the threshold function  ${\\cal E}_{\\frac{1}{8}}^\\ord{k}$ defining the $F^{2k} \\nabla^{4} R^4$ type invariants satisfies to \n\\begin{eqnarray}\n\\Scal{ {\\cal D}_{ijpq} {\\cal D}^{pq}{}_{rs} {\\cal D}^{rskl}  - \\frac{2k(k+1)-10}{3} {\\cal D}_{ij}{}^{kl}} {\\cal E}_{\\frac{1}{8}}^\\ord{k} &=&\n\\frac{k+2}{2} \\biggl({\\cal D}_{i j p q} {\\cal D}^{k lp q}  - \\frac{2(2 k-5) (2 k+7)}{27}  \\delta_{ij}^{kl} \\biggr) {\\cal E}_{\\frac{1}{8}}^\\ord{k} \\nonumber \\\\*\n\\Delta {\\cal E}_{\\frac{1}{8}}^\\ord{k} = \\frac{2}{3} (2 k-5) (2 k+7) {\\cal E}_{\\frac{1}{8}}^\\ord{k} \\ , && \\qquad {\\cal D}^{3}_{[2,0,0,1]} {\\cal E}_{\\frac{1}{8}}^\\ord{k} = 0 \\ . \\label{F2kD4R4E6}\n\\end{eqnarray}\nIt follows from representation theory that such equations are indeed implied by \\eqref{AdjointEquation}, and this explicit example permits to determine them uniquely. \n\nWe therefore obtain that the threshold function is the regularised Eisenstein series \n\\begin{equation}}\\def\\ee{\\end{equation} \\hat{{\\cal E}}_{\\frac{1}{8}}^\\ord{1} =   \\frac{5}{108} \\hat{E}_{\\mbox{\\DEVI00000{\\mbox{$\\frac{9}{2}$}}}} \\ , \\ee\nsuch that the exact $\\nabla^6 R^4$ threshold function $E_{(0,1)}$ is \n\\begin{equation}}\\def\\ee{\\end{equation} E_{(0,1)} =  \\hat{{\\cal E}}_{\\gra{8}{1}} +  \\frac{5}{108}  \\hat{E}_{\\mbox{\\DEVI00000{\\mbox{$\\frac{9}{2}$}}}}  \\ .\\ee\nThe series $E{\\mbox{\\DEVI00000{s}}}$ admits a pole at $s=\\frac{9}{2}$ proportional to the series $E{\\mbox{\\DEVI{\\mathnormal{\\mbox{$\\frac{3}{2}$}}}0000{\\mathfrak{0}}}}$ defining the $R^4$ threshold, exhibiting that the $R^4$ invariant form factor diverges at two loop into the $\\nabla^6 R^4$ form factor associated to the same function. This is in agreement with \\cite{Pioline:2015yea}, where the explicit coefficient is computed. \n\n\n\n\\subsection{$\\mathcal{N} = (2,2)$ supergravity in six dimensions}\nWe shall now discuss these invariants in $\\mathcal{N}=(2,2)$ supergravity in six dimensions. We recall that the scalar fields parametrise in this case  the symmetric space $SO(5,5) \/ ( SO(5) \\times SO(5) )$. \n\\subsubsection{Linearised invariant}\nIn the linearised approximation, the theory is defined from the scalar superfield $L^{ij \\hi \\hj}$ in the  $[0,1] \\times [0,1]$ of $Sp(2) \\times Sp(2)$, where $i,j $ and $\\hat{\\imath},\\hat{\\jmath}$ run from $1$ to $4$ in the fundamental of the two respective $Sp(2)$. One can define a $\\nabla^6 R^4$ type invariant by considering harmonic variables $u^{1}{}_{i}, u^{r}{}_{i} ,u^{4}{}_{i}$ parametrising $Sp(2)\/(U(1)\\times Sp(1))$ associated to one $Sp(2)$ factor, with $r=2,3$ of $Sp(1)$, such that \\begin{equation}}\\def\\ee{\\end{equation}\n\\mathfrak{sp}(2)  \\cong {\\bf 2}^\\ord{-1}  \\oplus \\scal{  \\mathfrak{u}(1)  \\oplus \\mathfrak{sp}(1) }^\\ord{0}\\oplus {\\bf 2}^\\ord{1}   \\ , \\qquad \n{\\bf 4}  \\cong{\\bf  1}^{\\ord{-1}} \\oplus {\\bf 2}^{\\ord{0}} \\oplus {\\bf 1}^{\\ord{1}} \\ . \\ee\nOne can in this way introduce the G-analytic superfield \\cite{Bossard:2009sy}\n\\begin{equation}}\\def\\ee{\\end{equation}\nW^{r \\hi \\hj} = u^{1}{}_{i} u^{r}{}_{j} L^{ij \\hi \\hj} \\ , \n\\ee\nthat transforms in the fundamental of $Sp(1)$ and as a vector of $SO(5) \\cong Sp(2)\/\\mathds{Z}_2$, and satisfies to the $1\/8$ BPS G-analyticity  constraint \n\\begin{equation}}\\def\\ee{\\end{equation}\nu^1{}_i D^{i}_{\\alpha} W^{r, \\hi \\hj} = 0 \\  . \n\\ee\nA general polynomial in $W^{r, \\hi \\hj}$ decomposes into irreducible representations of $Sp(1)\\times Sp(2)$. Similarly as in lower dimensions, one shows that the latter are freely generated by $W^{r, \\hi \\hj}$ itself in the $[1]\\times[0,1]$ of $SU(2)\\times Sp(2)$, the two quadratic monomials \\begin{equation}}\\def\\ee{\\end{equation} \nW^{r \\hi \\hj} W^{s}{}_{\\hi \\hj} \\ , \\qquad  W^{r \\hi \\hk} W_{r}{}^{\\hj}{}_{\\hk} \\ , \\ee\nin the $[2]$ and the $[2,0]$, respectively, the cubic monomial \n\\begin{equation}}\\def\\ee{\\end{equation}\n W^{s \\hi \\hj }W^{r \\hk \\hl} W_{s \\hk \\hl} \\ , \\ee\nin the $[1]\\times[0,1]$, and the two quartic monomials \n\\begin{equation}}\\def\\ee{\\end{equation}\nW^{s \\hi \\hat{p}}\\def\\hq{\\hat{q}} W_{s}{}^{\\hj}{}_{\\hat{p}}\\def\\hq{\\hat{q}}   W^{p}{}_{\\hk}{}^{\\hq} W_{p \\hl \\hq} - \\frac{1}{6} \\delta^{\\hi \\hj}_{\\hk \\hl} W^{s \\hi \\hat{p}}\\def\\hq{\\hat{q}} W_{s}{}^{\\hj}{}_{\\hat{p}}\\def\\hq{\\hat{q}}   W^{p}{}_{\\hi}{}^{\\hq} W_{p \\hj \\hq} \n\\ , \\quad  W^{s \\hi \\hat{p}}\\def\\hq{\\hat{q}} W_{s}{}^{\\hj}{}_{\\hat{p}}\\def\\hq{\\hat{q}}   W^{p}{}_{\\hi}{}^{\\hq} W_{p \\hj \\hq}  \\ , \\ee\nin the $[0,2]$ and the singlet representation, respectively. One concludes that the most general monomial is labeled by 6 integers, such that \n\\begin{eqnarray}\n && \\int du D^{12} \\bar{D}^{16} F(u)^{[n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]}_{[ 2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]} \nW^{4 + n_1 + 2 n_2 + 2 n'_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{[n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]} \\nonumber \\\\*\n&=&  L^{n_1 + 2 n_2 + 2 n'_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{[ 2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]}\\nabla^{6} R^{4} + \\dots \n\\end{eqnarray}\nThe linear analysis therefore suggests the form of the nonlinear invariant \n\\begin{eqnarray}\n&& {\\cal L}_{\\grad410}[ {\\cal E}_{\\grad410}] \\\\\n&=& \\sum_{\\substack{ n_1,n_2,n^\\prime_2\\\\ n_3 ,n_4,n_4^\\prime}} {\\cal D}^{n_1 + 2 n_2 + 2 n'_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{\\scriptscriptstyle [ 2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]} {\\cal E}_{\\grad410} \n{\\cal L}_{\\grad410}^{\\scriptscriptstyle [ 2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] [2 n'_2, n_1 + n_3 + 2 n_4]}\\nonumber \\\\*\n&& +  \\sum_{\\substack{ n_1,n_2,n^\\prime_2\\\\ n_3 ,n_4,n_4^\\prime}} {\\cal D}^{n_1 + 2 n_2 + 2 n'_2 + 3 n_3 + 4 n_4 + 4 n'_4}_{\\scriptscriptstyle[2 n'_2, n_1 + n_3 + 2 n_4] [ 2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] } {\\cal E}_{\\grad410} \n{\\cal L}_{\\grad410}^{\\scriptscriptstyle [2 n'_2, n_1 + n_3 + 2 n_4][2 n'_2 + 2 n_3  + 4 n_4 + 4 n'_4 ,n_1 + 2 n_2 + n_3] }\\nonumber\n\\end{eqnarray}\nwhere we consider the possibility of a mixing between the invariant ${\\cal L}_{\\grad410}$ with its conjugate obtained by exchanging the two $Sp(2)$ factors, according to the observation in \\cite{D4R4} for the $\\nabla^4 R^4$ type invariant in eight dimensions. \n\nFrom this structure one deduces that supersymmetry requires the function ${\\cal E}_{\\grad410}$ to be an eigenfunction of the Laplace operator, and to satisfy equations of the form \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^3_{[2,0],[2,0]} {\\cal E}_{\\grad410} \\propto  {\\cal D}^{\\; 2}_{[2,0],[2,0]} {\\cal E}_{\\grad410} \\ , \\qquad  {\\cal D}^3_{[0,1],[0,1]} {\\cal E}_{\\grad410} \\propto  {\\cal D}_{[0,1],[0,1]} {\\cal E}_{\\grad410} \\ , \\ee\nas well has a highest weight constraint \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}^{2k}_{[0,0],[0,2k]}  {\\cal E}_{\\grad410} = 0 \\ , \\label{HighestWeight} \\ee\nfor some integer $k$. We will see in the next section that the standard $\\nabla^6 R^4$ type invariant threshold function indeed satisfies to these equations for $k=2$. \n\\subsubsection{Decompactification limit from five to six dimensions}\nWe are now going to solve the differential equations  \\eqref{D6R4E6} and \\eqref{F2kD4R4E6}  for a function depending only of the Levi subgroup $\\mathds{R}_*^+\\times  SO(5,5)$ of the parabolic subgroup associated to the decompactification limit, such that \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{e}_{6(6)}\\cong {\\bf 16}^\\ord{-3} \\oplus \\scal{ \\mathfrak{gl}_1 \\oplus \\mathfrak{so}(5,5) }^\\ord{0} \\oplus \\overline{\\bf 16}^\\ord{3} \\ , \\qquad \n{\\bf 27} \\cong {\\bf 10}^\\ord{-2} \\oplus {\\bf 16}^\\ord{1} \\oplus {\\bf 1}^\\ord{4} \\ .\\label{27D5}  \\ee\nThe covariant derivative on $E_{6(6)}\/Sp_{\\scriptscriptstyle \\rm c}(4)$ acting on such a function takes the block diagonal form \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\bf D}_{27} = \\text{diag} \\biggl(\\tfrac{1}{6} \\partial_{\\phi}, {\\bf D}_{16}+ \\tfrac{1}{24} {\\bf\\mathds{1}}_{16} \\partial_{\\phi}, {\\bf D}_{10} - \\tfrac{1}{12} {\\bf\\mathds{1}}_{10} \\partial_{\\phi}  \\biggr) \\ . \n\\ee\nTo check the differential equations \\eqref{D6R4E6} and \\eqref{F2kD4R4E6} we need to compute the block diagonal decomposition of the higher order differential operators. In order to do this computation we consider a general ansatz and determine all the free coefficients by consistency with the various differential equations displayed in appendix \\ref{TensorEquationsEd}. We obtain in this way \n\\begin{multline}\n{\\bf D}^{\\; 2}_{27} = \\text{diag} \\biggl(\\tfrac{1}{6^2} \\partial^{\\; 2}_{\\phi} + \\tfrac{1}{2} \\partial_{\\phi}\\,  ,\\   {\\bf D}^{\\; 2}_{16} + \\tfrac{1}{2}  {\\bf D}_{16} \\bigl(\\tfrac{1}{6} \\partial_{\\phi}+1\\bigr)  + \\tfrac{1}{16}  {\\bf \\mathds{1}}_{16}  \\bigl( \\tfrac{1}{6^2}  \\partial^{\\; 2}_{\\phi} + 3 \\partial_{\\phi} \\bigr) \\, , \\\\\n {\\bf D}^{\\; 2}_{10} -  {\\bf D}_{10} \\bigl(  \\tfrac{1}{6} \\partial_{\\phi} +1 \\bigr) + \\tfrac{1}{4} {\\bf\\mathds{1}}_{10} \\bigl(  \\tfrac{1}{6^2} \\partial^{\\; 2}_{\\phi} + \\partial_{\\phi} \\bigr) \\biggr)\n\\end{multline}\nand\n\\begin{multline}\n{\\bf D}^{\\; 3}_{27} = \\text{diag} \\biggl(\\tfrac{1}{6^3} \\partial^{\\; 3}_{\\phi} + \\tfrac{3}{16} \\partial^{\\; 2}_{\\phi} + \\tfrac{5}{4} \\partial_{\\phi} - \\tfrac{1}{2} \\Delta \\,,\\\\\n {\\bf D}^{\\; 3}_{16}\n+\\tfrac{3}{4}  {\\bf D}^{\\; 2}_{16} \\bigl(\\tfrac{1}{6} \\partial_{\\phi} +2 \\bigr)+  {\\bf D}_{16} \\bigl( \\tfrac{3}{16}\\scal{ \\tfrac{1}{6^2} \\partial^{\\; 2}_{\\phi} +2  \\partial_{\\phi}}+1 \\bigr)+ \\tfrac{1}{64} {\\bf\\mathds{1}}_{16} \\bigl( \\tfrac{1}{6^3}  \\partial^{\\; 3}_{\\phi} + \\tfrac{1}{2} \\partial^{\\; 2}_{\\phi} - 8 \\Delta \\bigr) \\, , \\\\  {\\bf D}^{\\; 3}_{10} - \\tfrac{3}{2}  {\\bf D}^{\\; 2}_{10} \\bigl( \\tfrac{1}{6} \\partial_{\\phi}  +2 \\bigr) \n + {\\bf D}_{10} \\bigl(\\tfrac{3}{4}  \\scal{ \\tfrac{1}{6^2} \\partial^{\\; 2}_{\\phi} + \\partial_{\\phi} }+\\tfrac{5}{2} \\bigr)- \\tfrac{1}{8} \n {\\bf\\mathds{1}}_{10} \\bigl( \\tfrac{1}{6^3} \\partial^{\\; 3}_{\\phi} +\\tfrac{1}{4} \\partial^{\\; 2}_{\\phi} +\\partial_{\\phi}-2\\Delta \\bigr)  \\biggr) \\ ,\n\\end{multline}\nwhere $\\Delta \\equiv \\text{Tr}\\,  {\\bf D}_{10}^{\\; 2}$. In order to determine the constraints on the threshold function is six dimensions, we consider an ansatz with the appropriate power of the radius modulus $e^{-3\\phi}$ such as to compensate for the Weyl rescaling to Einstein frame, {\\it i.e.}\\  \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal E}_{\\gra84} = e^{-6 \\phi}{\\cal E}_{\\grad422}\\ ,  \\qquad {\\cal E}_{\\gra81} = e^{-12 \\phi} {\\cal E}_{\\grad410} \\ . \n\\ee\nThe singlet component of \\eqref{D6R4E6} gives directly the Poisson equation \n\\begin{equation}}\\def\\ee{\\end{equation} \\label{E6D_laplace}\n\\Delta {\\cal E}_{\\grad410}  = - {\\cal E}^{\\; 2}_{\\grad422} \\ , \n\\ee\nwhich is indeed consistent with \\eqref{LaplaceE6}. Working out spinor and the vector equations, using the Poisson equation \\eqref{E6D_laplace}, one obtains similarly \n\\begin{equation}}\\def\\ee{\\end{equation} \\label{E6D}\n\\biggl( {\\bf D}^{\\; 3}_{16} - \\frac{3}{4} {\\bf D}_{16} \\biggr) {\\cal E}_{\\grad410}  = - \\frac{1}{4} {\\bf D}_{16} {\\cal E}^{\\; 2}_{\\grad422} \\ , \\qquad \n\\biggl( {\\bf D}^{\\; 3}_{10} -\\frac{3}{2} {\\bf D}_{10} \\biggr) {\\cal E}_{\\grad410}   = - \\frac{1}{4}  {\\bf D}_{10} {\\cal E}^{\\; 2}_{\\grad422} \\ . \n\\ee\nThe only Eisenstein function that solves this homogenous equation (for ${\\cal E}_{\\grad422} =0$) is the Eisenstein series $\\hat{E}{\\text{\\DSOX0{\\mbox{$\\frac{7}{2}$}}000}}$, but its  expansion in the string theory limit is inconsistent with perturbation theory. As is computed in appendix \\ref{DnAdj}, one has moreover\n\\begin{equation}}\\def\\ee{\\end{equation}  D^d{}_{\\hat{a}}\\scal{  {\\cal D}_{(a}{}^{\\hat{b}} {\\cal D}_{b|\\hat{b}} {\\cal D}_{c}{}^{\\hat{c}} {\\cal D}_{d)\\hat{c}} }|_{[0,4]} \\hat{E}_{\\text{\\DSOX0{\\mbox{$\\frac{7}{2}$}}000}} = 0 \\ , \\ee\nwhich defines an integrability condition for the function to decompose into the sum of two functions satisfying to \\eqref{HighestWeight} and its conjugate obtained by exchange of the two $Sp(2)$ for $k=2$. \n\n\nLet us now consider the differential equation \\eqref{F2kD4R4E6} for the function ${\\cal E}^{(k)}_{\\frac{1}{8}\\, E_6}$ defining the  $F^{2k} \\nabla^{4} R^4$ threshold function in five dimensions. Diffeomorphism invariance in six dimensions requires an ansatz of the form \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal E}^{(k)}_{\\frac{1}{8}\\, E_{6}} = e^{- 2(k+5)\\phi} {\\cal E}^{(k)}_{\\frac{1}{8}\\, D_5} \\ , \n\\ee\nand using this ansatz, one obtains from the singlet component of \\eqref{F2kD4R4E6} the Laplace equation \n\\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta {\\cal E}^{(k)}_{\\frac{1}{8}} = \\frac{5}{2}  (k+3)(k-1) {\\cal E}^{(k)}_{\\frac{1}{8}} \\ , \n\\ee\nwhere we removed the $D_5$ label for simplicity. The spinor and the vector equations give then\n\\begin{eqnarray}\n\\biggl( {\\bf D}^{\\; 3}_{16} -\\frac{13  (k+3) (k-1)+24}{16} {\\bf D}_{16}  \\biggr) {\\cal E}^{(k)}_{\\frac{1}{8}}   &=& - \\frac{3(k+1)}{4}  \\Scal{ {\\bf D}^{\\; 2}_{16} -\\frac{5 (k+3) (k-1)}{16}   \\mathds{1}_{16} } {\\cal E}^{(k)}_{\\frac{1}{8}} \\ , \\nonumber \\\\*\n {\\bf D}^{\\; 2}_{10}  \\, {\\cal E}^{(k)}_{\\frac{1}{8}} &=&  \\frac{ (k+3) (k-1)  }{4} \\mathds{1}_{10} \\, {\\cal E}^{(k)}_{\\frac{1}{8}}   \\ ,\n\\end{eqnarray}\nwhere we used that the even and odd powers of ${\\bf D}_{10}$ lie in different irreducible representations of $Sp(2)\\times Sp(2)$, and must therefore vanish separately. The unique Eisenstein function satisfying to this equation is \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal E}^{(k)}_{\\frac{1}{8}} \\propto E_{\\mbox{\\DSOX{0}000{\\mbox{$k$+$3$}}}}\\, , \n\\ee\nconsistently with the decompactification limit of the five-dimensional function \n\\begin{equation}}\\def\\ee{\\end{equation}   E_{\\mbox{\\DEVI00000{k\\mbox{+}\\frac{7}{2}}}}    =2 \\zeta(2k+7)  e^{-4(2k+7) \\phi }   + \\frac{\\pi^\\frac{1}{2} \\Gamma(k+3)}{\\Gamma(k + \\frac{7}{2})} e^{- (2k+10)\\phi} E_{\\mbox{\\DSOX{0}000{\\mbox{$k$+$3$}}}} + \\dots \\ee\nWe conclude therefore that the exact $\\nabla^6 R^4$ function is defined as\n\\begin{equation}}\\def\\ee{\\end{equation} E_{(0,1)}= \\hat{{\\cal E}}_\\grad{4}{1}{0}+ \\hat{{\\cal E}}_\\grad{4}{0}{1} + \\frac{8}{189} \\hat{E}_{\\mbox{\\DSOX{0}000{\\mbox{$4$}}}}\\ , \\ee\nwhere $\\hat{{\\cal E}}_\\grad{4}{1}{0}$ satisfies to \\eqref{E6D}, and to an anomalous Poisson equation with an additional constant source term. This function is consistent with \\cite{Pioline:2015yea}, where the second Eisenstein function appears with this normalisation, and the 2-loop five dimensional threshold function must indeed solve \\eqref{E6D}, because the equation is parity invariant with respect to $O(5,5)$. \n\n\n\n\n\\subsection{$\\mathcal{N} = 2$ supergravity in seven dimensions}\nNone of the $\\nabla^6 R^4$ type invariants can be defined in the linearised approximation as harmonic superspace integrals in seven dimensions. We will therefore consider the uplift of the four-dimensional invariants in the decompactification limit. In seven dimensions the scalar fields parametrize the symmetric space $SL(5)\/SO(5)$, and the covariant derivative ${\\cal D}_{ij}$ transforms as a symmetric traceless tensor of $SO(5)$, with  $i, j = 1,\\dots,5$ of $SO(5)$. We consider therefore the parabolic subgroup of $E_{7(7)}$ of semi-simple Levi subgroup $SL(5) \\times SL(3)$ associated to the decomposition \n\\begin{eqnarray} \\label{133Graded}\n\\mathfrak{e}_{7(7)} &\\cong&  \\bar{{\\bf 5} }^\\ord{-6} \\oplus  ( {\\bf 3} \\otimes {\\bf 5})^\\ord{-4}  \\oplus   ( \\bar{ \\bf 3} \\otimes {\\bf \\overline{10}})^\\ord{-2}  \\oplus \\scal{  \\mathfrak{gl}_1 \\oplus \\mathfrak{sl}_3 \\oplus \\mathfrak{sl}_5 }^\\ord{0} \\oplus ( { \\bf 3} \\otimes { \\bf 10})^\\ord{2}\n\\oplus (\\bar{ \\bf 3} \\otimes \\bar{ \\bf 5})^\\ord{4} \\oplus { \\bf 5}^\\ord{6} \\nonumber \\\\*\n{\\bf 56} &\\cong&\\overline{\\bf 3}^\\ord{-5} \\oplus {\\bf 10}^\\ord{-3} \\oplus ({\\bf 3}\\otimes  \\overline{\\bf  5})^\\ord{-1} \\oplus (\\overline{\\bf 3}\\otimes {\\bf 5})^\\ord{1} \\oplus \\overline{\\bf 10}^\\ord{3} \\oplus {\\bf 3}^\\ord{5}\\ . \n\\end{eqnarray}\nWe will use the same conventions as in \\cite{D4R4}, where the decompactification limit of equations  \\eqref{CubicR} and \\eqref{E811Equation}  is already discussed in details. We consider the ansatz \n \\begin{equation}}\\def\\ee{\\end{equation}\n {\\cal E}_{\\grad{8}{1}{1}} = e^{- 36 \\phi} {\\cal E}_{\\frac{1}{8}} \\ , \\qquad    {\\cal E}_{\\grad{8}{4}{4}} = e^{- 18 \\phi} {\\cal E}_{\\gra{4}{2}} \\ , \n \\ee\nwith $ {\\cal E}_{\\frac{1}{8}}$ and ${\\cal E}_{\\gra{4}{2}} $ defined on $SL(5)\/SO(5)$, and the appropriate power of the volume modulus $e^{-3\\phi}$ required by diffeomorphism invariance in seven dimensions. The ${\\bf 3}^\\ord{3}$ component of the equation reduces to the Poisson equation \n\\begin{equation}}\\def\\ee{\\end{equation}\\label{Laplace7D} \n\\Delta {\\cal E}_{\\frac{1}{8}} \\equiv  2 {\\cal D}^{ij} {\\cal D}_{ij} {\\cal E}_{\\frac{1}{8}}  = \\frac{42}{5} {\\cal E}_{\\frac{1}{8}} - {\\cal E}^{\\; 2}_{\\gra{4}{2}} \\ . \n\\ee\nUsing this equation in the $ (\\overline{\\bf 3}\\otimes {\\bf 5})^\\ord{1} $ component of \\eqref{E811Equation}, one obtains \n\\begin{equation}}\\def\\ee{\\end{equation}\\label{Vec7D} \n\\biggl( {\\cal D}_{i}{}^{k} {\\cal D}_{k}{}^{l}  {\\cal D}_{l}{}^{j} + \\frac{1}{5} {\\cal D}_{i}{}^{k} {\\cal D}_{k}{}^{j} - \\frac{1053}{400} {\\cal D}_{i}{}^{j} +\\frac{177}{500} \\delta_{i}^{j} \\biggr) {\\cal E}_{\\frac{1}{8}} =\n\\frac{1}{4} \\biggl(\\frac{1}{10} \\delta_{i}^{j} - {\\cal D}_{i}{}^{j} \\biggr) {\\cal E}^{\\; 2}_{\\gra{4}{2}}  \\ . \n\\ee\nUsing moreover these equations to simplify the $ \\overline{\\bf 10}^\\ord{3}$ components, one obtains \n\\begin{equation}}\\def\\ee{\\end{equation} \\label{Tens7D}   \\Scal{ 6{\\cal D}^{[i}{}_{[k} {\\cal D}^{j]}{}_{p} {\\cal D}^{p}{}_{l]}  + \\frac{3}{10} {\\cal D}^{[i}{}_{[k} {\\cal D}^{j]}{}_{l]}} {\\cal E}_{\\frac{1}{8}} = \\frac{3}{5}  \\delta^{[i}_{[k} \\biggl(  {\\cal D}^{j]}{}_p {\\cal D}^p{}_{l]}  -\\frac{71}{20}{\\cal D}^{j]}{}_{l]} + \\frac{9}{25}  \\delta_{l]}^{j]} \\biggr) {\\cal E}_{\\frac{1}{8}}  \\ . \\ee\nThe solution to the homogenous equation (with $ {\\cal E}_{\\gra{4}{2}} =0$) can be written as the Eisenstein function $E_{\\scriptscriptstyle [3, 0,0,\\frac{5}{2}]}$, using the formulae of Appendix \\ref{SL5Review}. \n\nWe consider now the $F^{2k} \\nabla^4 R^4$ threshold function, with the ansatz \n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal E}_{\\grad820}^\\ord{k} = e^{-6(k +5) \\phi}{\\cal E}_{\\frac{1}{8}}^\\ord{k} \\ . \n\\ee\nAppropriate linear combinations of the grad 6 and 4 components of \\eqref{CubicR} in \\eqref{133Graded} give the two equations \n\\begin{eqnarray} \\biggl( {\\cal D}_{i}{}^{k} {\\cal D}_{k}{}^{l}  {\\cal D}_{l}{}^{j}  + \\frac{1}{2}  {\\cal D}_{i}{}^{k} {\\cal D}_{k}{}^{j}  \\biggr) {\\cal E}_{\\frac{1}{8}}^\\ord{k}  &=&\\biggl( \\frac{16 k(7k+20) + 75}{400} {\\cal D}_{i}{}^{j} +\\frac{3 k(k+5)(2k+5)}{125} \\delta_{i}^{j}- \\frac{1}{40}  \\delta_{i}^{j} \\Delta  \\biggr) {\\cal E}_{\\frac{1}{8}}^\\ord{k}   \\nonumber \\\\*\nk \\,   {\\cal D}_{i}{}^{k} {\\cal D}_{k}{}^{j}  {\\cal E}_{\\frac{1}{8}}^\\ord{k}  &=& \\biggl(k \\frac{4k+5}{20} {\\cal D}_{i}{}^{j} +\\frac{3k(k+5)(2k+5)}{25} \\delta_{i}^{j} - \\frac{1}{2}  \\delta_{i}^{j} \\Delta \\biggr)     {\\cal E}_{\\frac{1}{8}}^\\ord{k}   \\ .  \\end{eqnarray}\nFor $k\\ge 1$ we conclude that the function must satisfy to the quadratic equation\n\\begin{equation}}\\def\\ee{\\end{equation}  \\label{fundamental_1}\n{\\cal D}_{i}{}^{p} {\\cal D}_{p}{}^{j} {\\cal E}_{\\frac{1}{8}}^\\ord{k} = \\frac{4 k + 5}{20} {\\cal D}_{i}{}^{j} {\\cal E}_{\\frac{1}{8}}^\\ord{k} + \\frac{3 k (2k + 5)}{25} \\delta_{i}^{j} {\\cal E}_{\\frac{1}{8}}^\\ord{k} \\ , \n\\ee\nsuch that \n\\begin{equation}}\\def\\ee{\\end{equation}\n\\Delta {\\cal E}_{\\frac{1}{8}}^\\ord{k} = \\frac{6 k (2k + 5) }{5} {\\cal E}_{\\frac{1}{8}}^\\ord{k} \\ . \n\\ee\nOne can then check that all the other equations implied by  \\eqref{CubicR} are indeed satisfied provided that \\eqref{fundamental_1} is.  This equation is satisfied by the Eisenstein function $E_{\\scriptscriptstyle [0,0,k + \\frac{5}{2},0]}$, which appears in the decompactification limit of the corresponding Eisenstein function on $E_{7(7)}\/ SU_{\\scriptscriptstyle \\rm c}(8)$, {\\it i.e.}\\  \n\\begin{equation}}\\def\\ee{\\end{equation}\nE_{\\mbox{\\DEVII000000{k\\mbox{+}4}}} =e^{- 10(k+4) \\phi} E_{\\scriptscriptstyle [k+4,0]}  +  \\frac{\\pi^{\\frac{3}{2}}\\Gamma(k+\\frac{5}{2})}{\\Gamma(k+4)}  e^{- 6(k+5) \\phi} E_{\\scriptscriptstyle [0,0,k + \\frac{5}{2},0]} + \\dots \n\\ee\nwhere the Eisenstein series is normalised with an extra  $2\\zeta(2s)$ factor with respect to the Langlands normalisation. We conclude therefore that the exact $\\nabla^6 R^4$ threshold function is \n\\begin{equation}}\\def\\ee{\\end{equation}  E_{(0,1)}  = {\\cal E}_{\\frac{1}{8}}  + \\frac{5 \\pi}{378} E_{\\scriptscriptstyle [0,0,\\frac{7}{2},0]}  \\ ,\\ee\nwhere ${\\cal E}_{\\frac{1}{8}}$ is a solution to (\\ref{Laplace7D},\\ref{Vec7D},\\ref{Tens7D}) in agreement with \\cite{Green:2010wi}.\\footnote{Note that the normalisation of the Eisenstein function $E_{\\scriptscriptstyle [0,0,s,0]} $ does not include the additional factor of $\\zeta(2s-1)$ as in  \\cite{Green:2010wi}.}\n\n\n\n\\subsection{$\\mathcal{N} = 2$ supergravity in eight dimensions}\nWe shall now consider the oxidation of the  seven-dimensional  $\\nabla^6 R^4$ and $F^{2n} \\nabla^4 R^4$ type invariants to eight dimensions. Because there is a 1-loop divergence in eight dimensions, the exact $R^4$ threshold function differs from the Wilsonian effective action function. In the dimensional reduction, the divergence appears to be absorbed into the infinite sum of Kaluza--Klein states over the circle such that the function is finite in seven dimensions, but involves a logarithm of the radius modulus in the decompactification limit \\cite{Green:2010sp}. In order to consider the non-analytic terms in eight dimensions, we will take these logarithms into account in the decompactification limit. \n\n\n We shall use the same conventions as in \\cite{D4R4}, {\\it i.e.}\\  the complex scalar field $\\tau$ parametrises the coset representative $v_\\alpha{}^j \\in SL(2) \/ SO(2)$, with $\\alpha, \\beta = 1,2$ of $SO(2)$ and $i,j = 1,2$ of $SL(2)$, whereas the five real scalar fields $t$ parametrise the coset representative $V^{a}{}_J \\in SL(3)\/SO(3)$, with $a, b = 1,2,3$ of $SO(3)$ and $I, J = 1,2,3$ of $SL(3)$. The corresponding covariant derivative in tangent frame are then traceless symmetric tensors ${\\cal D}_{\\alpha \\beta}$ and  ${\\cal D}_{a b}$, respectively. In the decompactification limit, one writes the  $SL(5)\/SO(5)$ coset element in the parabolic gauge \n \\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal V} = \\left( \\begin{array}{cc} \\ e^{-3\\phi} v^{ \\scriptscriptstyle \\rm{\\mbox{\\tiny-1}}}_j{}^\\alpha  \\ &  \\ 0 \\ \\\\ \n\\  e^{2\\phi} V^a{}_K a_j^K & \\ e^{2\\phi} V^{a}{}_J \\ \\end{array} \\right) \\ , \n\\ee\nassociated to the graded decomposition \n\\begin{equation}}\\def\\ee{\\end{equation} \\mathfrak{sl}_5 \\cong ( {\\bf 2}\\otimes \\overline{\\bf 3})^\\ord{-5} \\oplus \\scal{ \\mathfrak{gl}_1 \\oplus \\mathfrak{sl}_2 \\oplus \\mathfrak{sl}_3}^\\ord{0} \\oplus  ( {\\bf 2}\\otimes {\\bf 3})^\\ord{5} \\ . \\ee\nIn this way one computes that the covariant derivative over $SL(5)\/SO(5)$ in tangent frame acts on a function of $\\phi, \\tau,t$ as \\cite{D4R4}\n\\begin{equation}}\\def\\ee{\\end{equation} {\\bf D}_{5}  =\\text{diag} \\left( -  \\tfrac{1}{20} \\partial_\\phi  \\delta_\\alpha^\\beta - {{\\cal D}}_\\alpha{}^\\beta,  \\tfrac{1}{30} \\partial_\\phi \\delta_a^b +{\\cal D}_a{}^b \\right) \\  .  \\ee\nOne computes that the higher order derivative operators obtained as products of ${\\bf D}_{5} $ in the ${\\bf 5}$ of $SL(5)$ decompose similarly as\n\\begin{multline} {\\bf D}^{\\; 2}_{5} = \\text{diag} \\biggl(  {{\\cal D}}_\\alpha{}^\\beta \\scal{\\tfrac{1}{10} \\partial_\\phi+ \\tfrac{3}{4} }+ \\scal{  \\tfrac{1}{400} \\partial_\\phi^{\\; 2} + \\tfrac{1}{16} \\partial_\\phi + \\tfrac{1}{2} {\\cal D}_{\\gamma\\delta} {\\cal D}^{\\gamma\\delta}  }   \\delta_\\alpha^\\beta , \\hspace{3 cm} \\\\ \n  {\\cal D}_a{}^c {\\cal D}_c{}^b + {\\cal D}_a{}^b \\scal{ \\tfrac{1}{15} \\partial_\\phi+\\tfrac{1}{2}  } +\\scal{ \\tfrac{1}{900}  \\partial_\\phi^{\\; 2} + \\tfrac{1}{24} \\partial_\\phi  }  \\delta_a^b \\biggr) \\ , \\end{multline}\nand\n\\begin{multline} {\\bf D}^{\\;3}_{5} = \\text{diag} \\biggl( - \\tfrac{1}{4}  {{\\cal D}}_\\alpha{}^\\beta \\scal{ \\tfrac{3}{10^2} \\partial_{\\phi}^{\\; 2}+ \\tfrac{7}{10} \\partial_\\phi  +\\tfrac{9}{4} +2  {\\cal D}_{\\gamma \\delta} {\\cal D}^{\\gamma \\delta}} \\\\\n\\hspace{10mm} - \\tfrac{1}{8}  \\scal{ \\tfrac{1}{10^3} \\partial_\\phi^{\\; 3} +\\tfrac{1}{30} \\partial_\\phi^{\\; 2}+\\tfrac{1}{8} \\partial_\\phi + 6  {\\cal D}_{\\gamma \\delta} {\\cal D}^{\\gamma \\delta} \\scal{ \\tfrac{1}{10}  \\partial_\\phi +1} -2 {\\cal D}_{a b} {\\cal D}^{a b}}   \\delta_\\alpha^\\beta\\ ,  \\\\ \n  {\\cal D}_a{}^c {\\cal D}_c{}^d  {\\cal D}_d{}^b +{\\cal D}_a{}^c {\\cal D}_c{}^b \\scal{\\tfrac{1}{10} \\partial_{\\phi} + 1} + \\tfrac{1}{3} {\\cal D}_a{}^b \\scal{ \\tfrac{1}{10^2} \\partial_\\phi^{\\; 2} +  \\tfrac{11}{40}\\partial_\\phi + \\tfrac{3}{4} }\\\\ + \\tfrac{1}{8} \\scal{ \\tfrac{1}{15^3}  \\partial_\\phi^{\\; 3} + \\tfrac{1}{180} \\partial_\\phi^{\\; 2} -\\tfrac{1}{12} \\partial_\\phi -2 {\\cal D}_{\\alpha \\beta} {\\cal D}^{\\alpha \\beta} }  \\delta_a^b \\biggr) \\ . \\end{multline}\nWe can now solve equation \\eqref{Vec7D} in the docompactification limit, with $ {\\cal E}_{\\frac{1}{8}}(\\phi,\\tau,t)$ and \\cite{Green:2010wi}\n\\begin{equation}}\\def\\ee{\\end{equation}  {\\cal E}_{\\gra{4}{2}}  = e^{-6\\phi} \\Scal{ 2 \\hat{E}_{[1]}(\\tau) + \\hat{E}_{[\\stfrac{3}{2},0]}(t) - 20 \\pi ( \\phi - \\phi_0)} \\ , \\ee\nwhere $\\phi_0$ is a constant that depends on the renormalisation scheme. Using the property that \n\\begin{equation}}\\def\\ee{\\end{equation} {\\cal D}_{ac} {\\cal D}^{bc}   \\hat{E}_{[\\stfrac{3}{2},0]}(t)  = - \\frac{1}{4} {\\cal D}_a{}^b   \\hat{E}_{[\\stfrac{3}{2},0]}(t)  + \\frac{2\\pi}{3} \\ , \\qquad 2 {\\cal D}_{\\alpha\\beta} {\\cal D}^{\\alpha\\beta}  \\hat{E}_{[1]}(\\tau) = \\pi \\ , \\ee\none shows that the general solution to \\eqref{Vec7D}, as a function of $\\phi, \\tau$ and $t$ takes the general form \n\\begin{multline}  {\\cal E}_{\\frac{1}{8}}(\\phi,\\tau,t) = e^{-12\\phi} \\biggl(\\hat{{\\cal F}}_{[4]}(\\tau)  +\\hat{{\\cal F}}_{[4,-2 ]}(t) + \\frac{1}{3} \\hat{E}_{[1]}(\\tau) \\hat{E}_{[\\stfrac{3}{2},0]}(t) + \\frac{\\pi}{18} \\Scal { \\hat{E}_{[1]}(\\tau)  + \\frac{19\\pi}{12}} \\biggr . \\\\ \\biggl .  - \\frac{10\\pi}{3} ( \\phi - \\phi_0 ) \\Scal{2 \\hat{E}_{[1]}(\\tau) + \\hat{E}_{[\\stfrac{3}{2},0]}(t)  + \\frac{\\pi}{3}} + \\frac{100\\pi^2}{3} ( \\phi- \\phi_0)^2\\biggr)  \\ ,  \\end{multline} \nwhere ${\\cal F}_{[4]}(\\tau)$ and ${\\cal F}_{[4,-2 ]}(t)$ are solutions to \n\\begin{eqnarray} \\Delta\\hat{{\\cal F}}_{[4]}(\\tau) &=& 12\\hat{{\\cal F}}_{[4]}(\\tau)  - ( \\hat{E}_{[1]}(\\tau))^2 \\ , \\qquad  \\Delta\\hat{{\\cal F}}_{[4,-2 ]}(t)  = 12\\hat{{\\cal F}}_{[4,-2 ]}(t) -(  \\hat{E}_{[\\stfrac{3}{2},0]}(t) )^2 \\ , \\nonumber \\\\*\n{\\cal D}_{ac} {\\cal D}^{cd} {\\cal D}_{db} \\hat{{\\cal F}}_{[4,-2 ]}(t)  &=& \\frac{49}{16} {\\cal D}_{ab}\\hat{{\\cal F}}_{[4,-2 ]}(t) - \\frac{3}{2} \\delta_{ab}\\hat{{\\cal F}}_{[4,-2 ]}(t) - \\frac{1}{2}   \\hat{E}_{[\\stfrac{3}{2},0]}(t)  {\\cal D}_{ab}   \\hat{E}_{[\\stfrac{3}{2},0]}(t)  \\nonumber \\\\*\n&& \\hspace{30mm} + \\frac{1}{2} \\delta_{ab} \\Scal{ \\frac{1}{4} (  \\hat{E}_{[\\stfrac{3}{2},0]}(t) )^2 + \\frac{\\pi}{9}  \\hat{E}_{[\\stfrac{3}{2},0]}(t)  + \\frac{\\pi^2}{27} } \\ . \\end{eqnarray}\nHere the notation is used to emphasize that these solutions are defined modulo the homogeneous solutions $E_{[4]}(\\tau)$ and $E_{[4, -2]}(t)$, respectively, as one can see using the formulae of Appendix \\ref{SL3Equa}. Note nonetheless that these homogeneous solutions are inconsistent with the string  theory perturbation expansion, and the exact threshold function is uniquely determined by these equations and consistency with string theory \\cite{Basu:2007ck}. \n\nThe structure of the threshold function exhibits that there is a 1-loop divergence of the $R^4$ type invariant form factor proportional to the $\\nabla^6 R^4$ type invariant. This implies the presence of an addition renormalisation scheme ambiguity in the definition of the analytic part of the effective action. It appears that the renormalisation scheme used in \\cite{Green:2010wi,Basu:2007ck}, cannot be obtained by simply neglecting the terms in $\\phi-\\phi_0$, but one finds nonetheless that the threshold function only differs by terms proportional to the linear and the quadratic term in  $\\phi-\\phi_0$, {\\it i.e.}\\  \n\\begin{multline}  \\hat{ {\\cal E}}_{\\frac{1}{8}}(\\tau,t) = \\hat{{\\cal F}}_{[4]}(\\tau)  +\\hat{{\\cal F}}_{[4,-2 ]}(t) + \\frac{1}{3} \\hat{E}_{[1]}(\\tau) \\hat{E}_{[\\stfrac{3}{2},0]}(t) + \\frac{\\pi}{18} \\Scal { \\hat{E}_{[1]}(\\tau)  + \\frac{19\\pi}{12}}  \\\\ +\\frac{\\pi}{36}  \\Scal{2 \\hat{E}_{[1]}(\\tau) + \\hat{E}_{[\\stfrac{3}{2},0]}(t)  - \\frac{5\\pi}{2}} \\ . \\end{multline} \n\n\n\nLet us now consider the oxidation of the $F^{2k} \\nabla^4 R^4$ type invariants, {\\it i.e.}\\  solve the differential equation \\eqref{fundamental_1} for a function of the form $e^{- (10 + 2k) \\phi}{\\cal E}_{\\frac{1}{8}}^\\ord{k}(\\tau, t)$, as required by diffeomorphism invariance in eight dimensions. One obtains straightforwardly \n\\begin{eqnarray}\n  2 {\\cal D}_{\\alpha \\beta} {\\cal D}^{\\alpha \\beta} {\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau,t)  &=& (1 + k) (2 + k){\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau,t)   \\ ,  \\nonumber \\\\*\n{\\cal D}_{a}{}^{c} {\\cal D}_{c}{}^{b}{\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau,t)  &=& \\frac{(5 + 4 k)}{12} {\\cal D}_{a}{}^{b}  {\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau,t)+ \\frac{(2 + k)(1 + 2 k)}{9} \\delta_{a}^{b} {\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau,t) \\ . \n\\end{eqnarray}\nUsing the results of Appendix \\ref{SL3Equa} one obtain that the solution can be written in terms of Eisenstein functions as\n\\begin{equation}}\\def\\ee{\\end{equation}\n{\\cal E}^\\ord{k}_{\\frac{1}{8}}(\\tau, t) \\propto  E_{[k + 2]}(\\tau) E_{[0, k+2]}(t) \\ , \n\\ee\nconsistently with the decompactification limit of the $SL(5)\/SO(5)$ Eisenstein function \\cite{D4R4},\n \\begin{multline}  E_{[0,0,k + \\stfrac{5}{2},0]} =2 \\zeta( 2k+5)\\zeta(2k+4)  e^{-6(2k+5) \\phi} + \\frac{2\\pi^2 \\zeta(2k+2)}{(2k+3)(k+1)}  e^{8k \\phi} E_{[k+\\stfrac{3}{2},0]}(t) \\\\  +\\frac{\\sqrt{\\pi} \\Gamma(k+2)}{2\\Gamma(k + \\stfrac{5}{2})}  e^{-2(k+5) \\phi} E_{[k+2]}(\\tau) E_{[0,k+2]}(t)  + \\mathcal{O}(e^{- e^{-5 \\phi}}) \\ . \\end{multline}\nThe sum of the two functions reproduces correctly the threshold function obtained in \\cite{Green:2010wi,Basu:2007ck}. \n\n\n\n\\section*{Acknowledgment}\nWe would like to thank Axel Kleinschmidt and Boris Pioline for useful discussions, and are particularly grateful to Axel Kleinschmidt for providing some explicit relations between Eisenstein series. \n\n        \n\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecent changes of the law in China and Turkey mean that their current leaders (President Xi Jinping and President Recep Tayyip Erdo\\^{g}an respectively) may hold onto their positions indefinitely. Conversely, in other countries the leader may change relatively often; for instance the UK has had five different Prime Ministers between the years 2007-2019. Among smaller groups some, such as private companies, have a permanent leader while other groups, such as university departments or social societies, have more transient and flexible leadership structures. \n\nA notable feature of many human groups is that they often do not explicitly coerce their members to join a hierarchy. Instead, soft power and prestige play a strong role \\cite{sapolsky_social_2004,van_vugt_evolutionary_2015}, with status being an abstraction of more tangible material resources such as land, food, weapons or other commodities \\cite{lin_social_2002}. Status is then voluntarily conferred upon leaders by those they are allied with \\cite{hawley_ontogenesis_1999,henrich_evolution_2001,lin_social_2002,pratto_social_2006,castells_network_2011,fiske_virtuous_2014,henrich_secret_2015,van_vugt_evolutionary_2015}, with many of these alliances being asymmetric \\cite{hall_asymmetric_1985,henrich_evolution_2001,henrich_secret_2015,van_vugt_evolutionary_2015,holekamp_aggression_2016}. The member with the highest status usually deemed the leader \\cite{hawley_ontogenesis_1999,pratto_social_2006,fiske_virtuous_2014,van_vugt_evolutionary_2015}, creating hierarchical societies. What is not clear, however, is what factors determine why some groups have no leader, others have transient leaders and yet others have relatively permanent leaders.\n\nA notable example of a shift between different forms of leadership dynamics is found from evidence from the Neolithic Era. Before this era, human societies consisted of egalitarian hunter-gatherer groups where material resources such as food were shared relatively equally \\cite{jaeggi_reciprocity_2013}\nand leadership roles were facultative and of a temporary duration. There followed a transition to sedentary groups where high-status individuals had more resources but leaders still changed relatively regularly \\cite{bar-yosef_sedentary_2001}. Finally, hereditary leadership became institutionalised, where the role of a chief was passed down a paternal line, which monopolised most of the resources \\cite{earle_how_1997}. Many scholars argue that these shifts were due to social and technological developments, which meant that interactions between individuals became increasingly asymmetric. These asymmetries were likely due to control of agricultural surpluses \\cite{carneiro_theory_1970,testart_significance_1982,boone_competition_1992,kaplan_hillard_s._evolutionary_2009}, land \\cite{carneiro_theory_1970}, ideologies \\cite{cauvin_birth_2001}, or military units and weapons \\cite{earle_how_1997}.  In light of this evidence, the model we present here investigates how asymmetry in status interaction can generate the different classes of leadership dynamics observed during the Neolithic Era.\n\nNetwork analysis has proved to be a useful approach for studying the interactions of members of a population \\cite{lin_social_2002,latour_reassembling_2007,castells_network_2011}. Quantitative study of hierarchical networks studied is usually static in nature and they are presented as snapshots in time \\cite{padgett_robust_1993,albert_statistical_2002}. However, members of a population will often change their properties and who they interact with over time. Recently, a number of studies have concentrated on the dynamical properties of networks \\cite{gross_adaptive_2008,bryden_stability_2011}. To look at the factors behind different types of leadership dynamics, we build on this work to model a dynamic coevolutionary network which incorporates the status of individuals as properties of the nodes and links between individuals representing alliances.\n\n\\section{Model}\n\nThe model consists of a dynamic network of $n$ individuals and is based on the following five assumptions: 1. individuals are equivalent to one another in every way and are unable to coerce one another to form alliances; 2. each individual has a status level which depends on their associations with others, meaning that status is adjusted according to the status of those who they are linked to; 3. individuals share a proportion of their status amongst those they are linked with; 4. status can be asymmetrically passed from one individual to another; 5. individuals can change who they associate with according to the marginal utility of the associations.\n\nEach individual $i$ maintains a status $s_i$ which translates to how much influence they have within the group. Status acts as a multivariate aggregate of an individual's level of money, prestige (titles, jobs, etc), and ownership (land, valuable resources, etc). We assume that, to participate in society, an individual must maintain relationships where they choose a fixed number (constant for all individuals) of associations with other individuals. Consequently, in our model, each individual is given $\\lambda$ unidirectional `outgoing' links which are assigned to others in a network. Individuals can have any number of `incoming' links from others in the network. \n\nThe statuses of the individuals are updated according to their links in the status update stage, and individuals are then given an opportunity to rewire their links in the rewiring stage. The model is then run forward in time to observe the distribution of status and changes of that distribution amongst the individuals. We especially looked for `leader' individuals with high status and checked to see whether they were superceded by other leaders. We ran models until patterns of leadership dynamics had stabilised, or for a substantially long time (up to 5 million time steps) to check that new leaders were extremely unlikely to rise to high status.\n\n{\\bf Status update stage:} In the model, a proportion of $r$ of each individual's status is shared amongst each of its links (including both incoming and outgoing links). This formalisation of sharing status amongst links is based on Katz's prestige measure \\cite{katz_new_1953}. For each link, we calculate that link's status by adding the status contribution from the outgoing individual (for whom the link is outgoing) to the status contribution from the incoming individual. To model unequal alliances we introduce the \\emph{inequality parameter} $q$ which reassigns the link's status back to those linked to unequally. A proportion $q$ of a link's status is assigned to the incoming individual, with a proportion $(1-q)$ remaining with the outgoing individual. In this formulation, the total amount of status in the model is constant, and because every status is initialised at $s_i =1.0$, the sum always equals $n$, i.e., $\\sum_i s_i=n$.\n\n{\\bf Rewiring phase: } In order to maximise status, each individual determines the outgoing link which is worth the least value to itself.  This link is rewired randomly, with probability $w$, to an individual with whom they are not currently linked. The assumption of random rewiring reflects the fact that an individual is unlikely to be able to tell the value of an association with another until they have formed the link.\n\n\n\\section {Results}\n\nIn the following, we will present our analysis of the dynamics that result from the interplay between the processes outlined above. Simulations were run of the model choosing parameter values over a complete range. Depending on the parameters, we either observe relatively equal statuses among the population, or a relatively high status level for one or a few individuals. An example of a typical network with a single dominant individual can be seen in Fig. 1. \n\n\\begin {figure}\n  \\includegraphics[width=0.5\\textwidth]{figure_1.png}\n  \\caption{A plot of the network showing a single individual with a high level of status compared to the others. Individuals with higher status are lighter coloured and have larger circles. }\n\\end {figure}\n\nA key parameter in the model is the inequality parameter ($q$), which models an unequal transfer of status from a link originator to a link receiver. This parameter can represent how social and technological development in the Neolithic Era created inequalities in interactions. As we increase $q$, theory predicts different phases of dynamics in the model, which are shown in \\figreftext{2}. We dub the individual with the highest level of status as the leader. We find three different types of leadership dynamics in the model: No leader, transient leader(s), and permanent leader(s).\n\n\\begin {figure}\n  \\includegraphics[width=0.5\\textwidth]{figure_2.png}\n  \\caption{Increasing the inequality takes the model through five different phases. First there is effectively no leader at all (panel A). Then we see that a single individual can rise to a high leadership status, but this is transient and leaders are replaced by other individuals (panel B). The length of time that individuals stay as leader then increases as we increase $q$ until the leader is effectively permanently in charge (panel C). In the next phase, a second individual can rise to a high status alongside the first leader, but these individuals' leadership position is transient (panel D). Finally, two individuals share leadership status and remain so permanently (panel E). \nThe value of $q$ is shown, other parameters are $r=0.2$, $n=50$, $\\lambda=3$, $w=0.5$. }\n\\end {figure}\n\n\nThe dynamics shown in Fig. 2 panel B indicate that as one leader is removed, this can create a power vacuum which leads to a new leader quickly gaining status. We have produced a video animation of the model of this phase of the model which is available at www.youtube.com\/watch?v=LrQWDhjwsYA. To investigate how the model transitions in and out of this phase, we looked in more detail at how the numbers of leaders and the lengths of time that they were leaders varied over different levels of the inequality parameter ($q$). We calculated the distributions of the different numbers of leaders over a threshold status level ($s_i > 3.0$) and the median length of time new leaders stay over the threshold after they have become the highest status individual. These are plotted in \\figreftext{3} with varying numbers of links originating from each individual ($\\lambda$). The plot demonstrates there are ranges of the inequality parameter ($q$) where leader turnover is relatively high, but the number of leaders is relatively constant. This shows that there is a power-vacuum effect in our model.\n\n\\begin {figure}\n  \\includegraphics[width=0.5\\textwidth]{figure_3.png}\n  \\caption{During transient periods, as one leader loses status, another quickly replaces it. Increasing the inequality parameter increases the time that leaders stay above the status level. The lines in panel {\\bf A} show the mean numbers of leaders over a threshold status level throughout a complete run of the model. For each value of $q$, vertical boxes indicate the relative proportion of timesteps with each number of leaders, for instance the width of $\\approx 0.98$ at $q=0.532$ and $\\lambda=3$ indicates that there was generally only one leader above the threshold throughout the simulation. Panel {\\bf B} shows the mean time that an individual stays above the threshold after they have reached highest status.Leader turn over was quite high at $q=0.532$ with the average leader lasting 7000 time steps with simulations run over 2 million time steps. Parameters as in \\figreftext{2} unless shown.\n  }\n\\end{figure}\n\n\nTo understand the dynamics created by the model in more detail, we compare them to a simpler model of a branching process. Branching processes specify the rates at which an individual with $x_i$ links acquires or loses links. In such processes, we observe a power-law distribution of the number of links when the link gain rate is close to the loss rate. We observe such a power law distribution when this is the case in our model (see Fig. 4, panel A). The formation of a hump in the distribution to the right is also consistent with branching processes with a reflecting boundary. However, the results suggest our model is not simply a superposition of branching processes for each individual. A superposition would not explain why we find a power vacuum effect, where there is always a leader which changes identity over time (see, e.g., \\figreftext{3}, $q=0.532$, $\\lambda=3$). The reason the power vacuum happens is because the presence of a leader suppresses the others (see \\figreftext{4} panel {\\bf B}), which is due to the relatively high number of links to the leader.\n\n\n\\begin {figure}\n  \\includegraphics[width=0.5\\textwidth]{figure_4.png}\n  \\caption{Frequency distributions of the numbers of links to individuals demonstrate how individuals show critical behaviour of a branching process over a range of values of $q$ (panel {\\bf A}. There is a truncated power law distribution with exponent $\\approx -8.0$ on panel {\\bf A} at a relatively low level of $q=0.525$. At higher levels of $q$  we can see how dominant individuals will suppress others to subcritical behaviour (panel {\\bf B}). We can see how the rewiring of links to an extra leader between $q=0.54$ and $q=0.55$ (see \\figreftext{3} panel {\\bf A}) suppresses the frequencies of individuals with mid-range number of links as $q$ is increased. Parameters are the same as in \\figreftext{2}, $q$ as shown.  Simulations were run over 2 million time steps.}\n\\end{figure}\n\n\nIn order to check whether our model is consistent with evidence from the Neolithic Era, we looked to see if the increase in leadership time-length is consistent over a wide range of parameters and that our results can be found in large groups. We ran simulations over a wide range of parameters to investigate the role of each parameter on the dynamics. Over each simulation run, we recorded the number of times there was a change of leader. We counted new leaders when they hadn't been one of the previous $l-1$ leaders. Our results confirm that as we increased the inequality parameter, we recorded fewer leaders as individuals spend longer time periods as leader. The full range of parameters are presented in the Supplementary Material (see Figures S1-9).\n\n\\section{Discussion}\n\nThe model presented here demonstrates a rich set of leadership structure dynamics amongst individuals in a non-coercive environment. The model reveals an interesting phase where competition to be leader is suppressed by the temporary presence of one leader, meaning that when a leader loses status it will be quickly replaced. We show how the dynamics can depend on the level of inequality of alliances between individuals, and on the numbers of alliances an individual can form. This suggests that technology and social norms can modulate such a system and implies that self-organisation in a society can play a role in keeping a system near to an equilibrium point where leadership changes relatively frequently.\n\nThis work pushes forward our understanding of hierarchies in human networks. This is currently largely based on static networks which are formed by preferential-attachment where nodes are more likely to connect to other nodes which are already of high status \\cite{albert_statistical_2002}. Our model presents an alternative where the hierarchy is dynamic: nodes have high numbers of connections (alliances) at some points and then other nodes take over. \n\nThis work also contributes new insights into the Neolithic transitions in human societies from relatively flat power structures, through a period where leaders changed over time, to dominant institutionalised leaders \\cite{bar-yosef_sedentary_2001}.  Our model presents a potential explanation for this, given that status would be closely linked to control of food or other monopolisable resources. Contemporary to the political transitions were innovations in agriculture, which enabled a high status individual to control a large food surplus. These high status individuals were able to feed a large number of supporters at relatively little cost to themselves, for example funding a military, enabling them to maintain and eventually institutionalise their power.  Monopolisable resources could also be less tangible, such as religious authority \\cite{cauvin_birth_2001} however the evidence suggests these changes followed technological advances \\cite{whitehouse_complex_2019}. In either case, the growth of population size and the transition to a sedentary, agricultural, lifestyle would have made it more difficult for followers to leave their group and hence easier for a dominant individual to monopolise \\cite{carneiro_theory_1970,powers_evolutionary_2014}. These factors, especially the ability to monopolise resources, relate to a high level of the inequality parameter ($q$) in our model as the leader individual is able to form alliances where they exchange a small proportion of these resources to gain loyalty from their supporters.\n\nThe three phases of human leadership dynamics correspond to three phases identified in the organisational psychology literature. Lewin has identified three modes of leadership: Laissez Faire, Democratic and Autocratic \\cite{lewin_patterns_1939}. In this analysis, the Laissez Faire mode was found to be the case where there is no central resource to coordinate and corresponds to the no-leader phase. When there is a central resource, our model predicts that those individuals with higher status are able to control this central resource, and thus not lose out when more individuals join their group. This is because a controlled surplus of this central resource enables a leader to pay off many individuals and maintain their leadership \\cite{bueno_de_mesquita_logic_2005}. The ability to control a central resource means that such groups will switch from the Laissez Faire mode to more Democratic and Autocratic modes. \n\nIn this paper we focussed primarily on applying this model to develop insights regarding the Neolithic transitions from flat power structures to hierarchical societies.  Future work can build upon these foundations to examine whether this model can be applied to other changes in societal structure, such as the movements from monarchy toward parliamentary democracies in 18th-century Europe, or the transitions of Roman civilization betwween monarchy, through annually electing two concurrent consuls in the Roman Republic, to a single Imperator Caesar in the Roman Empire.  Other work might investigate the impact of relaxing some of our assumptions. For instance, exploring different rewiring rules where individuals have different numbers of links, or rewire to others based on a similar or higher levels of status or numbers of links. The model also can be extended in various ways to better represent the real-world contexts in which leadership dynamics operate; these could include representations of technological innovations, changes in social norms, or power struggles between potential leaders.  These extensions would enable us to develop the model further into a powerful exploratory tool for human leadership dynamics. \n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\nAn accurate description of excited nucleons and their interaction with probes such as photons at GeV energies has remained elusive for decades.   \nThe standard model~\\citep{Gross:1973id,Politzer:1973fx} underpins the structure of the \nnucleons and their excitations, but in the low-energy non-perturbative regime, competing semi-phenomenological models of specific\nreaction dynamics are all that are available.  Present-day  \nlattice QCD calculations~\\citep{Edwards:2011jj, Edwards:2012fx} and quark models~\\citep{Capstick:2000qj, Capstick:1998uh, Capstick:1986bm,Loring:2001kx,Glozman:1997ag, Giannini:2001kb} predict a richer baryon spectrum than experimentally observed~\\citep{Patrignani:2016xqp, Klempt:2009pi,Koniuk:1979vw}: the so-called {\\it missing resonance problem}.  There are theoretical approaches to the nucleon resonance spectrum that predict that some quark-model states do not exist, including models with quasi-stable diquarks~\\citep{Anselmino:1992vg},\nAdS\/QCD string-based models~\\citep{Brodsky:2006uq}, and ``molecular'' models in which some baryon resonances are dynamically generated from\nthe unitarized interaction among ground-state baryons and mesons~\\citep{Kolomeitsev:2003kt}.\nBut finding such missing states may in part be an experimental problem:  high-mass nucleon resonances may couple weakly to $\\pi N$ and may thus have escaped detection in the analysis of $\\pi N$ elastic scattering experiments.\nFurther, they are wide and overlapping,  and partial-wave analysis (PWA) of reaction data for specific final states remains difficult due to channel-coupling effects and insufficient experimental constraints.\nThe experimental results discussed here represent one step in the direction of adding constraints to the hyperon photoproduction database, which ultimately impacts models for nucleon excitations.  \n\nCross-section measurements alone are not enough to constrain PWA models of meson production amplitudes.  Polarization observables related to the spins of the beam photons, target, and recoiling baryons are also needed.\nPhotoproduction of pseudoscalar mesons is governed by four complex amplitudes that lead to an interaction cross sections and 15 spin observables~\\citep{Chew:1957tf, Barker:1975bp,Fasano:1992es, Chiang:1996em, Keaton:1996pe, Sandorfi:2010uv, Nys:2016uel}.  \nA \\textit{mathematically complete} experiment would require data, with negligible uncertainties, on a minimum of eight well-chosen observables at each center-of mass (c.m.) energy, $W$,  and meson polar angle, $\\cos \\theta_{c.m.}$. In practice, with realistically achievable uncertainties, measurements of many more are needed to select between competing partial wave solutions, and even knowledge of the sign of an asymmetry can provide valuable discrimination~\\cite{Sandorfi:2010uv}. Furthermore, avoiding ambiguities in PWA solutions requires measurements of observables from each spin configuration of the three combinations of beam-target, target-recoil, and beam-recoil polarization~\\cite{Sandorfi:2010uv, Nys:2016uel}.\n\nFurthermore, while isospin $I=3\/2$ transitions ($\\Delta^*$ excitations) can be studied with proton target data alone, both proton- and neutron-target\nobservables are necessary to study $I=1\/2$ transitions and isolate the separate $\\gamma p N^*$ proton and $\\gamma n N^*$ neutron photo-couplings~\\citep{Sandorfi:2013cya}.\nInformation from neutron targets is comparatively scarce~\\citep{ Anisovich:2017afs},  particularly in the hyperon channels~\\citep{Compton_PhysRevC.96.065201,  AnefalosPereira:2009zw},\nwhich is why the present measurement is of value.   Furthermore, the hyperon photoproduction channels   $\\gamma N\\rightarrow K \\Lambda (\\Sigma^{0})$\nare  attractive for analysis for two reasons.  First,  the threshold for two-body hyperon final states is at $W \\simeq 1.6$~GeV, above which lie numerous poorly known resonances.   Two-body strange decay modes, rather than cascading non-strange many-body decays, may be easier to interpret.\nSecond, the hyperon channels give easy access to recoil polarization observables on account of their self-analyzing weak decays.  While the present work  does not involve measurement of hyperon polarizations, previous work has shown the benefit of using such information to extract properties of higher-mass nucleon resonances~\\citep{Paterson:2016vmc,Bradford_CxCz, Bradford_xsec, McNabb, Anisovich:2007bq, Paterson:2016vmc, McCracken, Dey, Anisovich:2017ygb}.  Thus, \npursuing ``complete\" amplitude information in the hyperon photoproduction channels can be complimentary to the analogous quest in, say, pion photoproduction.\n\nIn this article, we present first-time measurements of the beam-target observable $E$ on a longitudinally polarized neutron bound in deuterium in the quasi-free reaction $\\gamma n(p) \\to K^0Y(p)$.\nThe helicity asymmetry $E$  is formally defined as the normalized difference in photoproduction yield between  antiparallel ($\\sigma^{A}$) and parallel ($\\sigma^{P}$) configurations, {\\it i.e.},  settings where the incident photon beam polarization is antialigned or aligned, respectively, with the longitudinal polarization of the target.  Following Ref.~\\cite{Barker:1975bp} and Ref.~\\cite{Sandorfi:2010uv}  write \n\\begin{equation}\nE=\\frac{\\sigma^{A}-\\sigma^{P}}{\\sigma^{A}+\\sigma^{P}}.\n\\end{equation}\nThis helicity asymmetry, $E$, is related to the  cross section by\n\\begin{equation}\n\\left(\\frac{d\\sigma}{d\\Omega}\\right)=\\left(\\frac{d\\sigma}{d\\Omega}\\right)_{0}\\left(1-P_{T}P_{\\odot}E\\right),\n\\label{equation2}\n\\end{equation}\n %\nwhere $\\left(d\\sigma \/ d\\Omega\\right)_{0}$ is the differential cross section averaged over initial spin states and summed over the final states, and $P_{T}$\nand $P_{\\odot}$ are the target longitudinal and beam circular polarizations, respectively. \n\nThe asymmetry results obtained are compared with several model predictions.  The first is a single-channel effective Lagrangian approach,  \nKaonMAID~\\citep{Mart:1999ed,Lee:1999kd}, with parameter constraints largely imposed from SU(6).  Without experimental constraints on the $N^* \\Lambda K^0$ and  $\\gamma n N^*$ vertices, the reaction of interest is difficult to model accurately.   \nThe second model giving predictions for the present results is the data description given by  SAID~\\citep{SAID, Adelseck:1986fb}.   In general, SAID is more up to date than KaonMAID;  for the present reaction channels the SAID predictions are a polynomial fit to all available data  before  2008, assuming final state interactions for these polarization observables can be neglected~\\cite{strakovsky}.\nThe third  comparison is made to the multichannel K-matrix formalism of the  Bonn-Gatchina~\\citep{Anisovich:2012ct} group, which is the most up to date, being constrained by recent first-time measurements~\\citep{Compton_PhysRevC.96.065201}  of the differential cross section for the reaction $\\gamma n(p) \\to K^0\\Lambda (p)$  [with $(p)$ as the spectator proton].\n\n\n\\section{Experimental Procedures\n\\label{sec:Section-II}} \n\nThe experiment was performed at the Thomas Jefferson National Accelerator Facility (JLab) using the CEBAF Large Acceptance\nSpectrometer (CLAS)~\\citep{CLAS-NIM}.  \nThis setup has been used for several studies of $K^+$ photoproduction of  \nhyperonic final states on a proton target~\\citep{Bradford_CxCz, Bradford_xsec, McNabb, Moriya:2014kpv, Moriya:2013hwg, Moriya:2013eb, McCracken, Dey} \nand on an effective neutron (deuteron) target~\\citep{Compton_PhysRevC.96.065201,AnefalosPereira:2009zw}.\nThe present results stem from the so-called ``\\textit{g14}\" run period between December 2011 and May 2012, from which non-strange \nresults have been previously reported~\\citep{Ho:2017kca}.   \nThe CEBAF accelerator provided longitudinally polarized electron beams with energies of \n$E_{e}=2.281$, \n$2.257$, \nand $2.541$ GeV, \nand an \\textit{average} electron beam polarization for the present study of $P_{e}=0.82\\pm0.04$, which was measured routinely\nby the Hall-B M\\\"oller polarimeter~\\cite{Moller2}. The electron beam helicity was pseudorandomly flipped between +1 and $-1$ with a 960 Hz flip\nrate. The electron beam was incident on the thin gold radiator of the Hall-B Tagger system~\\citep{Sober} and produced circularly polarized\ntagged photons. The polarization of the photons was determined using the Maximon and Olsen formula~\\citep{Olsen:1959zz}\n\\begin{equation}\nP_{\\odot}=P_{e}\\frac{4k-k^{2}}{4-4k+3k^{2}},\n\\end{equation}\nwhere $P_{\\odot}$ and $P_{e}$ are the photon and electron polarizations, respectively, and $k=E_{\\gamma} \/ E_e$ is the ratio between\nthe photon energy and the electron beam energy. \n\nA 5-cm-long solid target of hydrogen deuteride (HD)  was used in the experiment~\\citep{Lowry:2016uwa,Bass:2013noa}.\nIt  achieved vector polarizations of   25\\%-30\\% for deuterons, i.e., for \n{\\it bound} neutrons in the deuteron with relaxation times of about a year.  \nThe polarized target was held at the center of CLAS using an in-beam cryostat that produced a 0.9~T holding field and operated at 50 mK.  The target polarization was monitored using nuclear magnetic resonance measurements~\\citep{Lowry:2016uwa}.   The orientation of the target longitudinal polarization direction was inverted between periods of data taking, either parallel or antiparallel to the direction of the incoming photon beam.   Background events from the unpolarizable target wall material and aluminum cooling wires~\\citep{Bass:2013noa} were removed using empty-target data, as discussed in Secs.~\\ref{sec:Section-IIIa} and~\\ref{sec:Section-IIIb}.\n\nThe specific reaction channel for this discussion came from events of the type $\\gamma d \\to \\pi^+ \\pi^- \\pi^- p (X)$ using a readout trigger requiring a minimum of two charged particles in different CLAS sectors.  After particle identification  we required  the ``spectator,\"  $X$, to be an undetected  low-momentum proton and possibly a photon, via the missing mass technique, as explained in the next section. In order to determine the $E$ asymmetry experimentally, the event yields in a given kinematic bin of $W$ and kaon center-of-mass angle were obtained by counting events with total c.m. helicity $h=$3\/2 (laboratory-frame antiparallel configuration), called $N_{A}$, and events with $h=$1\/2 (laboratory frame parallel configuration) called $N_{P}$, respectively. The $E$ observable \nwas then computed as \n\\begin{equation}\nE=\\frac{1}{\\overline{P_{T}}\\cdot\\overline{P_{\\odot}}}\\left(\\frac{N_{A}-N_{P}}{N_{A}+N_{P}}\\right),\n\\label{Eq_Eval}\n\\end{equation}\nwhere $\\overline{P_{T}}$ and $\\overline{P_{\\odot}}$ are the run-averaged\ntarget and beam polarizations, respectively. \n\n\n\n\\section{Data Analysis \n\\label{sec:Section-III}}\n\nThe performance of the system was extensively studied for a reaction with much higher count rates than the present one.   The nonstrange reaction  $\\gamma d \\to \\pi^- p (X) $ was investigated using many of the same analysis steps and methods discussed in this article to extract the $E$ observable for $\\gamma n \\to \\pi^- p$~\\citep{Ho:2017kca}.    The analysis steps outlined below were all tested on that reaction.   In particular, the boosted decision tree (BDT) selection procedure~\\cite{DruckerCortes, ROE2005577} used below was validated against alternative ``cut-based\" and kinematic fit  methods, with the result that the BDT procedure resulted in $\\sim30\\%$ larger yields of  signal events and therefore gave better statistical precision on the final $E$ asymmetry.\n\n\\subsection{Particle identification}\n\\label{sec:Section-IIIa}\n\nFor this particular analysis, we required that every selected event consists of at least two positive tracks and two negative tracks with associated photon tagger hits~\\cite{Sober}.\nThe CLAS detector system determined the path length, the charge type,\nthe momentum and the flight time for each track~\\citep{Sharabian,Mestayer,Smith}.\nFor each track of momentum $\\overrightarrow{p}$, we compared the measured time of flight, $TOF_{m}$, to a hadron's expected time of\nflight, $TOF_{h}$, for a pion and proton of identical momentum and path length. \nCLAS-standard cuts were placed on the difference between the measured and the expected time of flight, $\\triangle TOF=TOF_{m}-TOF_{h}$. We selected events for which the two positively charged particles were the proton and $\\pi^{+}$, and the two negatively charged were both  $\\pi^{-}$. Well-established CLAS fiducial cuts were applied to select events with good spatial reconstruction. \n\nEvents originating from unpolarized target material\\textit{\\footnotesize{}\\textemdash }aluminum cooling wires and polychlorotrifluoroethylene (pCTFE)\\textit{\\footnotesize{}\\textemdash } dilute $E$ and must be taken into account. A  period of data taking was dedicated to an \\textit{empty} target cell in\nwhich the HD material was not present. This set of data was used to study and remove the bulk of the target material background on the basis of a loose missing mass cut.   \nFigure \\ref{Z-vertex} shows the resulting reconstructed reaction vertex for four-track data along the beam line both for a full target and for an empty target scaled to match the counts in several downstream target foils. \nThe full-to-empty ratio of about 3.3:1 in the target region was important in selecting the optimal BDT cut discussed below.\n\n\\begin{figure}\n\\vspace{-0.5cm}\n\\includegraphics[width=0.5\\textwidth]{Z_vertex_RS2}\\protect\n\\vspace{-0.5cm}\n\\caption{The open histogram shows the vertex  distribution of events along the beam\nline for a full target is the open histogram. Dashed red lines show the nominal target boundaries.  The peaks at $z>0$ are from target-independent foils in the cryostat;  the positions of two are  highlighted with dotted blue lines~\\cite{Lowry:2016uwa}. The filled histogram shows the scaled target-empty  background  distribution.   \n\\label{Z-vertex} }\n\\end{figure}\n\nFigure \\ref{MMass} shows the resulting target-full  missing mass distribution for spectator $X$ in $\\gamma d\\rightarrow\\pi^{-}\\pi^{+}\\pi^{-}p(X)$,\nafter these cuts.  A clear peak corresponding to the spectator proton is seen at point 1 for events that produced a $\\Lambda$ particle.  A loose cut was applied to reject events with missing mass larger than 1.4 GeV\/c$^{2}$ at point 4 because of the presence of $\\Sigma^{0}\\rightarrow\\pi^{-}p(\\gamma)$ events. These have a 73-MeV photon in the final state in addition to the proton, and the distribution peaks at point 2 and has a kinematic tail to about point 3.\n\n\\begin{figure}[htbp]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.5\\textwidth,height=0.35\\textwidth]{MMass_RS}\\protect\n\\vspace{-0.5cm}\n\\caption{The missing mass distribution, $\\gamma d\\rightarrow\\pi^{-}\\pi^{+}\\pi^{-}pX$\nafter PID cuts showing the dominant spectator proton peak at ``1.''   The magenta line at ``4'' indicates a loose event rejection for $m_X >  1.4$~GeV\/c$^2$. This rejects unambiguous\nbackground but keeps $\\Sigma^{0}\\rightarrow\\pi^{-}p(\\gamma)$ events in which both a proton and a photon are missing between ``2'' and ``3.''  (See text.)\n\\label{MMass} }\n\\end{figure}\n\n\n\\subsection{$K^{0}Y$ event selection using BDT analysis}\n\\label{sec:Section-IIIb}\n\nBecause of the small reaction cross section in this experiment, a method was needed to optimally isolate the events of interest with minimal statistics loss.\nThe multivariate analysis tool called the boosted decision tree (BDT)  approach was used to select the exclusive events of interest in this study.   Three steps were needed to achieve this result.  The first BDT was created to select events from both the \n$\\gamma d\\rightarrow\\pi^{-}\\pi^{+}\\pi^{-}p(p_{S})$ and the $\\gamma d\\rightarrow\\pi^{-}\\pi^{+}\\pi^{-}p(p_{S}\\gamma)$ final states, \nconsistent with quasi-free production from a deuteron.  This was to reject target-material background and events with a high missing momentum of the undetected spectator nucleon, $p_S$.  The second BDT was created to remove the nonstrange pionic background with the same final states, that is, to pick out events with $\\Lambda$ and  $\\Sigma^0$ intermediate-state particles.    The third BDT was to separate the $K^{0}\\Lambda$ and $K^{0}\\varSigma^{0}$ events. \n\nThis BDT algorithm is more efficient than a simple ``cut'' method in both rejecting background and keeping signal events~\\citep{ROE2005577,Ho-thesis}.  The method  builds a ``forest'' of \\textit{distinct} \\textit{decision trees} that are linked together by a \\textit{boosting} mechanism. Each decision tree constitutes a \\textit{disjunction} of logical conjunctions (i.e., a graphical representation of a set of \\textit{if-then-else} rules). Thus, the entire reaction phase-space is considered by every decision tree.  Before employing the BDT for signal and background classification, the BDT algorithm needs to be constructed (or trained) with \\textit{training} data\\textit{\\textemdash }wherein  the category of every event is definitively known. We used the ROOT implementation of the BDT algorithm ~\\citep{Hocker:2007zz}. \nEvery event processed by the constructed BDT algorithm is assigned a value of between $-1$ and +1 that\nquantifies how likely the processed event is a background event (closer to $-1$) or a signal event (closer to $+1$). An optimal cut on the\nBDT output is chosen to maximize the\\textit{ }$S\/\\sqrt{S+B}$ ratio, where $S$ and $B$ are the estimations, based on training data, of the initial number of signal\nand background events, respectively.\n\nThe initial assignment of the $\\pi^-$ particles to either $K^0$ or $\\Lambda$ decay was studied with Monte Carlo simulation, and a loose selection based on invariant masses was made.  Specific details of these cuts are given in Ref.~\\citep{Ho-thesis}. \n\nThe first BDT was trained using real empty-target data for the background training.   A signal Monte Carlo simulating quasifree hyperon production on the neutron was used for signal training data.  The momentum distribution of the spectator proton, $p_{s}$, followed the Hulth\\`en potential~\\citep{Cladis_PhysRev.87.425,Lamia:2012zz} for the deuteron. Based on this training, an optimal BDT cut  that maximized the estimated initial\\textit{ }$S\/\\sqrt{S+B}$ ratio was selected. Figure \\ref{Z_vertex_BDT} shows the total (blue histogram) and rejected (black histogram) events by the first BDT cut. In comparing  Figs.~\\ref{Z-vertex} and \\ref{Z_vertex_BDT}, two items should be noted. First, the BDT was trained to remove target-material background events with missing momentum not consistent with a Hulth\\`en distribution. Second, the BDT background-rejection efficiency was not perfect, leaving some target-material background events that were removed in a subsequent step (Sec.~\\ref{sec:Section-IIIc}). We then rejected events with $z>-2$~cm on the reaction vertex to remove remaining unambiguous background events due to various cryostat foils. \n\n\\begin{figure}[htbp]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.5\\textwidth]{Z_vertex_BDT1_RS}\n\\vspace{-1.cm}\n\\protect\\caption{The reconstructed distribution of the reaction vertex along the beam\nline showing target-full events in the top histogram (blue) after loose $K^0Y^0$ selection  and the missing mass cut shown in Fig.~\\ref{MMass}.\nEvents selected by the first BDT are shown in the middle histogram (red), and rejected events in the bottom histogram  (black).\nThe magenta line indicates a loose cut to reject unambiguous target-material background. \n\\label{Z_vertex_BDT} }\n\\end{figure}\n\nThe second-step BDT was trained using a four-body phase-space $\\gamma d\\rightarrow\\pi^{-}\\pi^{+}\\pi^{-}p(p_{S})$\nsimulation as background training data and the $\\gamma d\\rightarrow K^{0}\\Lambda(p_{S})$ simulation as signal training data. There were two negative pions in each event:  one from the decay of the $K^0$ and one from the decay of the hyperon.   The goal of the BDT analysis was to use the available correlations among all particles to sort the pions correctly and to select events with decaying strange particles.   The main training variables at this stage of the analysis included the 3-momenta of all the particles and the detached decay vertices of the $K^0$s and the hyperons.  \nAfter the optimized BDT cut was placed, Fig.~\\ref{lambda K0 IM}  shows the total (red histogram) and\nrejected (black histogram) events after this second BDT analysis step.  The efficiency of the second BDT was less than 100\\%, thus, there are remaining\ntarget background events in the selected data sample.  The dips near the signal maxima in the background spectra show that the background is slightly undersubtracted. This issue is discussed and corrected below.  A fit with  a Breit-Wigner line shape and a polynomial was used to estimate  that the \nstrange-to-non-strange ratio of events in the data set at this stage was about 2.3:1 in the peak regions.\n\n\\begin{figure}[htpb]\n\\includegraphics[width=0.48\\textwidth]{IM_ppim_g2}\n\\includegraphics[width=0.48\\textwidth]{IM_pippim_g2}\n\n\\protect\\caption{Invariant $\\pi_{\\Lambda}^{-}p$ mass (top) and invariant $\\pi_{K^{0}}^{-}\\pi^{+}$\nmass (bottom) after target material background rejection by the first\nBDT cut. Black histograms show events rejected  by the second BDT cut.  Fits of the sum (red curve) of a Breit-Wigner line-shape (blue curve) and a third order polynomial (black curve) are shown.  The fits aid the discussion in the text but were not used in the subsequent analysis. \\label{lambda K0 IM} }\n\\end{figure}\n\nFor the final task, separating the  $K^{0}\\Lambda$ and $K^{0}\\varSigma^{0}$ channels, the third BDT was trained using $\\gamma d\\rightarrow K^{0}\\Sigma^{0}(p_{S})$ simulation as ``background'' training data and $\\gamma d\\rightarrow K^{0}\\Lambda(p_{S})$ simulation as ``signal'' training data. Note that the term background used here is just for semantic convenience, since both channels were retained after applying the third optimized BDT cut.\nFigure~\\ref{MMass_off_K0_simulation}  shows in the left [right] histogram the classification success of the third BDT on $\\gamma d\\rightarrow K^{0}\\Lambda(p_{S})$  [$\\gamma d\\rightarrow K^{0}\\Sigma^{0}(p_{S})$] simulation data. The histograms reveal that a small number of $K^{0}\\Lambda$ events  would be misclassified as $K^{0}\\varSigma^{0}$ events, and vice versa. In the next section, the correction for the contamination on both final\ndata sets is discussed. Figure \\ref{MMass_off_K0} shows the separation result from the third BDT on real data. \n\n\\begin{figure*}[t]\n\\includegraphics[width=0.5\\textwidth]{MM_pippim_K0Lambda_MC}\\includegraphics[width=0.5\\textwidth]{MM_pippim_K0Sigma0_MC}\\protect\n\\caption{Distributions of missing mass from the reconstructed $K^{0}$, $\\gamma n\\rightarrow\\pi_{K^{0}}^{-}\\pi^{+}X$\nfor simulation data, assuming that the target is an at-rest neutron.\nLeft: the magenta histogram represents events with correct $K^{0}\\Lambda$\nclassification, while the cyan histogram represents events with the wrong\n$K^{0}\\Sigma^{0}$classification. Right: the cyan histogram represents\nevents with the correct $K^{0}\\Sigma^{0}$~classification, while the magenta\nhistogram represents events with the wrong $K^{0}\\Lambda$ classification.\n\\label{MMass_off_K0_simulation} }\n\\end{figure*}\n\n\n\\begin{figure}[t]\n\\includegraphics[width=0.5\\textwidth]{MM_pippim}\\protect\\caption{Distribution of missing mass \nfrom the reconstructed $K^{0}$, $\\gamma n\\rightarrow\\pi_{K^{0}}^{-}\\pi^{+}X$\nfor real data, assuming that the target is an at-rest neutron,\nafter rejecting non-hyperon background by the second BDT cut.\nThe magenta (cyan) histogram was classified as $K^{0}\\Lambda$ ($K^{0}\\Sigma^{0}$)\nusing the third BDT selection step. \\label{MMass_off_K0} }\n\\end{figure}\n\n\\subsection{Corrections for remaining backgrounds and asymmetry calculation}\n\\label{sec:Section-IIIc}\n\nThe $E$ asymmetry values  for both target-material and non-strange background events were statistically\nconsistent with 0~\\citep{Ho-thesis}; therefore, we implemented an  approximation procedure\nto correct for the dilution effect from the remaining background.\nWe estimated two ratios: one for the remaining fraction of target background (TGT), $R^{TGT}$,  \nand one for the fraction of remaining nonstrange (NS) final-state events mixed with the hyperon events, $R^{NS}$.\nWe write \n$R^{TGT}= {N^{remain}} \/ {N^{HD}}$,\nand \n$R^{NS}={Y^{remain}}\/{Y^{K^{0}Y}}$.   \n$N^{remain}$ and $N^{HD}$ are the estimated number of remaining target-material background events\nand true deuteron events after the first BDT and $z=-2$~cm vertex cuts, respectively.\n$Y^{remain}$ and  $Y^{K^{0}Y}$ are the estimated number of remaining nonstrange  and true $K^{0}Y$ events after the second\nBDT cut, respectively. Next, let $Y_{BDT}$  be the number of events that passed the $z$-vertex cut and the first two BDT selections; then $Y_{BDT}$\ncan be partitioned into \n\\begin{align}\nY_{BDT}&=\\left(1+R^{NS}\\right)Y^{K^{0}Y} \\nonumber \\\\\n&=\\left(1+R^{NS}\\right)\\left[Y_{HD}^{K^{0}Y}+Y_{TGT}^{K^{0}Y}\\right],\\;\\label{eq:wideeq-2-3}\n\\end{align}\nsince $Y^{K^{0}Y}$ also comprises events from the remaining target-material\nbackground and the bound signal events.  If we further allow  \n$Y_{TGT}^{K^0 Y} \/ Y_{HD}^{K^0 Y} = N^{remain} \/ N^{HD} = R^{TGT}$,\nthen $Y_{BDT}$ can finally be expressed as\n\\begin{equation}\nY_{BDT}=\\left(1+R^{NS}\\right)\\left(1+R^{TGT}\\right)Y_{HD}^{K^{0}Y},\\;\\label{eq:wideeq-2}\n\\end{equation}\nor\n\\begin{equation}\nY_{HD}^{K^{0}Y}=\\left(1+R^{NS}\\right)^{-1}\\left(1+R^{TGT}\\right)^{-1}Y_{BDT}.\n\\label{eq:wideeq-2-1}\n\\end{equation}\nThese relations should remain valid for both $Y_{BDT}^{K^{0}\\Lambda}$ and $Y_{BDT}^{K^{0}\\Sigma^{0}}$, \nwhich are the $K^{0}\\Lambda$ and $K^{0}\\Sigma^{0}$ signal events from bound neutrons, respectively.  \nThe backgrounds that leak through the BDT filters will be helicity independent and will be subtracted in the numerator of Eq.~(\\ref{Eq_Eval}).  \nUsing Eq.~(\\ref{eq:wideeq-2-1}) to correct the summed yields in the denominator gives the corrected asymmetry as\n\\begin{equation}\nE_{corrected}^{K^{0}Y}=\\left(1+R^{NS}\\right)\n\\times\\left(1+R^{TGT}\\right)E_{BDT}^{K^{0}Y},\\;\\label{eq:wideeq-2-2}\n\\end{equation}\nwhere $E_{BDT}^{K^{0}Y}$ is obtained from $Y_{BDT}^{K^{0}Y}$ (or,\nmore exactly, $Y_{BDT}^{P}$ and $Y_{BDT}^{A}$\nof the $K^{0}Y$ parallel and antiparallel subsets). \nFrom the simulations we found average values of $R^{TGT}$ and $R^{NS}$ of 0.09  and 0.17, respectively, with some dependence on\nthe specific run period.\n\nNext we discuss a correction for the third BDT classification result.\nRecall that the third BDT selection separates the true signal $K^{0}Y$ events into two subsets: one is mostly $K^{0}\\Lambda$ events,\nand the other is mostly $K^{0}\\varSigma^{0}$. If we denote  $N_{\\Lambda}^{BDT}$ and $N_{\\Sigma^{0}}^{BDT}$ as the\nnumber of events the third BDT identified as $K^{0}\\Lambda$ and $K^{0}\\varSigma^{0}$ events, respectively, then we have the expressions\n\n\\begin{equation}\nN_{\\Lambda}^{BDT}=\\omega_{\\Lambda}N_{\\Lambda}^{true}+(1-\\omega_{\\Sigma^{0}})N_{\\Sigma^{0}}^{true},\\;\\label{eq:wideeq-3}\n\\end{equation}\n\\begin{equation}\nN_{\\Sigma^{0}}^{BDT}=(1-\\omega_{\\Lambda})N_{\\Lambda}^{true}+\\omega_{\\Sigma^{0}}N_{\\Sigma^{0}}^{true},\\;\\label{eq:wideeq-3-1}\n\\end{equation}\nwhere $\\omega_{\\Lambda}$ and $\\omega_{\\Sigma^{0}}$ are the fractions of events correctly identified: these values were\nestimated based on simulation data. After rearrangement, we arrive at the expressions \n\\begin{align}\nN_{\\Lambda}^{true}&=\\left[\\omega_{\\Lambda}-\\frac{(1-\\omega_{\\Sigma^{0}})}{\\omega_{\\Sigma^{0}}}(1-\\omega_{\\Lambda})\\right]^{-1} \\nonumber \\\\\n&\\times \\left[N_{\\Lambda}^{BDT}-\\frac{(1-\\omega_{\\Sigma^{0}})}{\\omega_{\\Sigma^{0}}}N_{\\Sigma^{0}}^{BDT}\\right],\\;\n\\label{eq:wideeq}\n\\end{align}\n\\begin{align}\nN_{\\Sigma^{0}}^{true}&=\\left[\\omega_{\\Sigma^{0}}-\\frac{(1-\\omega_{\\Lambda})}{\\omega_{\\Lambda}}(1-\\omega_{\\Sigma^{0}})\\right]^{-1} \\nonumber \\\\ \n&\\times \\left[N_{\\Sigma^{0}}^{BDT}-\\frac{(1-\\omega_{\\Lambda})}{\\omega_{\\Lambda}}N_{\\Lambda}^{BDT}\\right]\\;\n\\label{eq:wideeq-1}.\n\\end{align}\n\nThe \\textit{corrected} $E$ asymmetry was obtained using the derived $N_{\\Lambda}^{true}$ and $N_{\\Sigma^{0}}^{true}$ by using\nEq.~(\\ref{Eq_Eval}).  From the simulations we found average values of $\\omega_Y$ of 0.87 and 0.91 for $\\Lambda$ and $\\Sigma^0$ events, respectively.\n\nThe neutron polarization in the deuteron is smaller than the deuteron polarization because the deuteron wavefunction has, in addition to an $S$-wave component, a $D$-wave component in which the spin of the neutron need not be aligned with the deuteron spin.   This was studied using data for the $\\gamma n \\to \\pi^- p $ reaction and reported in our previous publication~\\citep{Ho:2017kca}.  It was found that for spectator recoil momenta of less than 100~MeV\/$c$ the correction was negligible.  Had we cut on the recoil momentum at 200~MeV\/$c$ rather than 100~MeV\/$c$, a measured dilution factor of $(8.6\\pm0.1)$\\%  would have been necessary for the nonstrange channel.   But different reaction channels  may exhibit different sensitivities to recoil momentum.  For the reaction under discussion here we could not afford the statistical loss by cutting on recoil momentum, and we elected to make a conservative correction based on the general considerations in \\citep{Ramachandran:1979ck}.  The neutron polarization can\nbe estimated as $P_{n}=P_{d}(1-\\frac{3}{2}P_{D})$, where $P_{n}$ and $P_{d}$ are neutron and deuteron polarizations, respectively, and $P_{D}$ denotes the deuteron $D$-state probability.  The latter is not strictly an observable and needs only to be treated consistently within a given $NN$ potential.\nFollowing  Ref.~\\citep{Ramachandran:1979ck}, we take the $D$-state contribution averaged over a range of $NN$ potentials as about  5\\%, which implies that the neutron polarization is  92.5\\% of the deuteron polarization, or a 7.5\\% dilution factor. \n\n\\subsection{Systematic Uncertainties}\n\nWe implemented four systematic studies to quantify the robustness of the trained BDT algorithms and the sensitivity of our results on\nthe  correction procedures introduced in the previous section.  \nTwo  tests studied the effect of loosening the first and the second BDT cuts, respectively.  One test focused on the sensitivity of the $E$ results on the third correction\\textit{\\textemdash }the correction procedure that was implemented to ``purify'' the final selected $K^{0}\\Sigma^{0}$($K^{0}\\Lambda$) sample.  Finally, we reduced the beam and target polarizations by one standard deviation of their respective total uncertainties (statistical and systematic) to study the changes in the $E$ results.\n\nFinally, we note a complication that could occur when summing $\\Lambda$ yields to create the $E$ asymmetries. The relative angular distribution between the $\\pi^-$ and the $p$ that are used to reconstruct a $\\Lambda$ carries information on the recoil polarization of the latter. When summed over azimuthal angles, this information is lost. However, limitations in detector acceptance could result in  incomplete integration, which in principle could introduce into Eq.~\\ref{equation2}  a dependence on six additional observables~\\cite{Sandorfi:2010uv}. The gaps in CLAS acceptance are modest, and due to the lower than expected production cross sections, the data below are presented in broad kinematic bins, which tends to dilute such effects. On the scale of our statistical uncertainties, such corrections are expected to be negligible and we have not attempted to correct for them.\n\n\n\n\\section{Results \\label{sec:Section-IV}}\n\nWe present here the results for the $E$ asymmetry in two $W$ energy bins. The lower bin is from 1.70 to 2.02 GeV and denoted $W_{1}$, while the higher bin is from 2.02 to 2.34 GeV and referred to  $W_{2}$.  Due to small cross sections for $K^0Y$ photoproduction, and to detector inefficiencies that are amplified by the  required identification of four charged particles, our statistics are  sufficient for only three bins in the $K^0$ center-of-mass production angle.  The measurements for the $\\gamma n \\to K^0 \\Lambda$ reaction  are plotted together with predictions from the KaonMAID, SAID, and Bonn-Gatchina (BnGa)  models in Fig.~\\ref{Ecostheta_K0L}.   The data show that the $K^0\\Lambda$ asymmetry is largely positive below 2~GeV and mostly negative above 2~GeV, without more discernible trends.   Values of $E$ must approach $+1$ at $\\cos \\theta^{c.m.}_{K^0}\\to \\pm 1$ to conserve angular momentum.   Thus, the values for $E$  in bin $W_2$ must change rather rapidly near the extreme angles.\n\n\nFor comparison, the PWA combine results from many experiments at different energies, and this results in varying degrees of sensitivity to energy and angle. This is illustrated in Fig.~\\ref{Ecostheta_K0L} by the SAID and BnGa PWA predictions at the limits of the energy bins. None of the models were tuned to these results;  that is, the models are all predictions based on  fits to previously published data on other observables.\nFirst, one observes that the data are not statistically strong enough to strongly discriminate among the models.  In the lower $W$ bin all three models\ncan be said to agree with the data.   In the higher $W$ bin the SAID model may be  slightly favored by the data among the three.\n\n\\begin{figure}[htbp]\n\\vspace{-0.5cm}\n\\includegraphics[angle=-90 , width=0.55\\textwidth, trim=0 3.5cm 0 0, clip]{e-k0-lambda-1.pdf}\n\\vspace{-10mm} \n\\caption{Helicity asymmetry  $E$ for the  ${K^{0}\\Lambda}$ final state (with combined statistical and systematic uncertainties) vs. $\\cos\\theta_{K^{0}}$   The asymmetries are shown with the neutron-target theoretical models KaonMaid ~\\citep{Mart:1999ed} (dashed red curve) and SAID~\\citep{SAID} (dot-dashed blue curve) and Bonn-Gatchina~\\citep{Anisovich:2007bq,Anisovich:2012ct} (solid black curve).  Because of the 0.32-GeV-wide $W$ bins, each model is represented by two curves, computed at the bin endpoint $W$ values, as labeled. \n\\label{Ecostheta_K0L} }\n\\end{figure}\n\nThe results for the $\\gamma n \\to K^0 \\Sigma^0$ channel are plotted in Fig.~\\ref{Ecostheta_K0S}, together with model predictions from SAID and Kaon-MAID.   In contrast to the $K^0 \\Lambda$  channel at lower $W$, here the data hint at less positive values for $E$.   \nIn the bin for  $W$ above 2 GeV, the data are also consistent with 0 for $K^0\\Sigma^0$, whereas the $K^0\\Lambda$ data tended to be negative.  In fact, the $K^0\\Sigma^0$ asymmetry is consistent with 0 in all available bins.   \nThe model comparisons show that the KaonMAID prediction for the $K^0\\Sigma^0$ channel in the higher $W$ bin are probably not consistent with the data, while the SAID result is consistent with the data.   For the $K^0\\Sigma^0$  case we do not have predictions from the Bonn-Gatchina model  because the unpolarized differential cross section has not been measured yet, and without it the model  does not have a prediction available. \n\n\\begin{figure}[htbp]\n\\vspace{-0.5cm}\n\\includegraphics[angle=-90 ,  width=0.55\\textwidth, trim=0 3.5cm 0 0, clip]{e-k0-sigma0-1.pdf}\n\\vspace{-10mm} \n\\caption{Helicity asymmetry  $E$ for the  ${K^{0}\\Sigma^0}$ final state (with combined statistical and systematic uncertainties) vs. $\\cos\\theta_{K^{0}}$ for two 0.32-GeV-wide  energy bands in $W$, as labeled. Model curves are as in Fig.~\\ref{Ecostheta_K0L}.\n\\label{Ecostheta_K0S} }\n\\end{figure}\nIn order to show  one other comparison between data and theory, we plot some of the present results for a neutron target together with the model predictions for the $K^+ \\Lambda$ reaction on a {\\textit {proton}} target in Fig.~\\ref{Ecostheta_K0L2}.  This is intended to show the difference in  the model predictions on protons versus neutrons.  One sees how different the three model predictions are for protons versus neutrons.   One notes that the predictions for the proton target calculations all tend to be closer to the new data we are presenting for a neutron target.   This suggests that calculations of the $E$ observable for a neutron target can be improved.   \nThus, we may expect these present results to have some impact on the further development of these models.\n\n\\begin{figure}[htbp]\n\\vspace{-0.5cm}\n\\includegraphics[angle=-90 ,  width=0.55\\textwidth, trim=0 3.5cm 0 0, clip]{e-k0-lambda-2.pdf}\n\\vspace{-10mm} \n\\caption{\nHelicity asymmetry  $E$ for the  ${K \\Lambda}$ final state vs. $\\cos\\theta_{K^{0}}$ for energy band  $W_2$.   Left: Data from Fig.~\\ref{Ecostheta_K0L} together with model  predictions for a neutron target. Right: Model calculations  for the $K^+ \\Lambda$ reaction on a proton target, as computed using KaonMaid~\\citep{Mart:1999ed} (dashed red curve),  SAID~\\citep{SAID} (dot-dashed blue curve) and Bonn-Gatchina~\\citep{Anisovich:2007bq,Anisovich:2012ct} (solid and dashed black curves).   Curves on the right are closer to the (reaction mismatched) data shown on the left.\n\\label{Ecostheta_K0L2} }\n\\end{figure}\n\nSo far unpublished CLAS results for the corresponding reaction $\\gamma p \\to K^+ \\Lambda$ have higher statistics and finer energy bins than the present results (since the identification of this final state requires the detection of fewer particles).   The present  $K^0 \\Lambda$ results are, within our uncertainties, similar to the $K^+\\Lambda$ asymmetries in Ref.~\\cite{LiamCasey}. The numerical values of the measured  $K^0 \\Lambda$  and $K^0 \\Sigma^0$ $E$ asymmetries, together with their statistical and systematic uncertainties, are reported in Table \\ref{Tab:E_sys_stat}.\n\n\\begin{table*}[htbp]\n\\hfill{}%\n\\begin{tabular}{ccccc}\n\\hline\\hline \n\\multicolumn{1}{c}{   } &  & \\multicolumn{3}{c}{$\\cos\\theta_{K^{0}}$}\\tabularnewline\n\\cline{3-5} \n\\multicolumn{1}{c}{} &  & $-$0.6 & 0.0 & $+$0.6\\tabularnewline\n\\hline \n\\multirow{2}{*}{$K^{0}\\Lambda$ } & $W_{1}$ & 0.834$\\pm$0.499$\\pm$0.287 & $-$0.144$\\pm$0.436$\\pm$0.098 & 1.066$\\pm$0.419$\\pm$0.231\\tabularnewline\n\\cline{2-5} \n & $W_{2}$ & $-$0.533$\\pm$0.752$\\pm$0.345 & $-$0.263$\\pm$0.618$\\pm$0.101 & $-$0.648$\\pm$0.464$\\pm$0.136\\tabularnewline\n\\hline \n\\multirow{2}{*}{$K^{0}\\Sigma^{0}$} & $W_{1}$ & $-$0.110$\\pm$0.723$\\pm$0.406 & 0.581$\\pm$0.539$\\pm$0.144 & $-$0.319$\\pm$0.541$\\pm$0.460\\tabularnewline\n\\cline{2-5} \n & $W_{2}$ & $-$0.471$\\pm$0.446$\\pm$0.391 & 0.0002$\\pm$0.317$\\pm$0.150 & 0.054$\\pm$0.281$\\pm$0.065\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\hfill{}\n\n\\protect\\caption{Numerical values of the $E$ asymmetry measurements for the $K^{0}\\Lambda$\/$K^{0}\\Sigma^{0}$\nchannels. The uncertainties are statistical and systematic, respectively.  Center-of-mass energy ranges are $1.70 < W_1 < 2.02$~GeV and $2.02 < W_2 < 2.34$~GeV.\n\\label{Tab:E_sys_stat}}\n\\end{table*}\n\n\\section{Conclusions \\label{sec:Section-V}}\n\nWe have reported the first set of $E$ asymmetry measurements for the reaction $\\gamma d\\rightarrow K^{0}Y(p_{s})$ for 1.70~GeV$\\leq W \\leq$ 2.34~GeV. In particular, we have described the three-step BDT-based analysis method developed to select a clean sample of $p\\pi^{+}\\pi^{-}\\pi^{-}$ with intermediate hyperons. We have plotted the $E$ asymmetry as a function of $\\cos \\theta_{K^{0}}^{CM}$.\nSeveral systematic uncertainty tests led to the conclusion that statistical uncertainties dominated the final results.  The numerical values of the measured $E$ asymmetries and their statistical and systematic uncertainties are reported in Table \\ref{Tab:E_sys_stat}.\n\nEvidently,  this analysis is limited by the small cross sections of the channels of interest, leading to large uncertainties in the measurements of the $E$ asymmetry.  At present, comparison with several models makes no decisive selections among the model approaches.  \nOverall, the BnGa predictions are of a quality similar to that of the SAID predictions. The Kaon-MAID predictions for both channels seem less successful.  Among all three model comparisons, the distinction between proton- and neutron-target predictions are  differentiated by the data:  The proton-target predictions compare better than the neutron-target predictions with the experimental results.  In principle, this information is valuable since it hints at the necessary isospin decomposition of the hyperon photoproduction mechanism.   \n\nAt present, multipole analyses for $K^0Y$ channels are severely limited by the available data.   Higher statistical data on these channels for a number of other polarization  observables, from a much longer (unpolarized) target, have been collected during the $g13$ running period with CLAS and are under analysis.   A greater number of different polarization observables is generally more effective than precision at determining the  photoproduction amplitude~\\cite{Sandorfi:2010uv}.   When these $g13$ results become available, the present data on the beam-target $E$ asymmetry are likely to have a larger impact.   \n\n\\begin{acknowledgments}\nWe acknowledge the outstanding efforts of the staff of the Accelerator\nand Physics Divisions at Jefferson Lab who made this experiment\npossible. The work of the Medium Energy Physics group at Carnegie\nMellon University was supported by DOE Grant No. DE-FG02-87ER40315.  The\nSoutheastern Universities Research Association (SURA) operated the\nThomas Jefferson National Accelerator Facility for the United States\nDepartment of Energy under Contract No. DE-AC05-84ER40150.  Further\nsupport was provided by \nthe National Science Foundation, \nthe Chilean Comisi\\'on Nacional de Investigaci\\'on Cient\\'ifica y Tecnol\\'ogica (CONICYT),\nthe French Centre National de la Recherche Scientifique,\nthe French Commissariat \\`{a} l'Energie Atomique,\nthe Italian Istituto Nazionale di Fisica Nucleare,\nthe National Research Foundation of Korea,\nthe Scottish Universities Physics Alliance (SUPA),\nand the United Kingdom's Science and Technology Facilities Council.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn a previous communication$^1$ it was suggested that a typical elementary\nparticle, the electron can be considered to be what was termed a Quantum\nMechanical Black Hole (or QMBH), made up of a relativistic fluid of subconstituents,\ndescribed by the Kerr-Newman metric giving both its gravitational and\nelectromagnetic fields$^2$. It was pointed out that alternatively the QMBH could\nbe described as a relativistic vortex in the hydrodynamical formulation. It\nwas pointed out that the QMBH or vortex could also be thought of as a\nrelativisitc rotating shell.\\\\\nIn Section 2 we examine this model which explains several observed facts,\nwhile in Section 3 we try to explore the mechanism which triggers off the\nformation of these QMBH particles. In Section 4 we examine the cosmological\nimplications of the model and again discover that a surprisingly large\nnumber of observed facts are neatly explained. Finally in Section 5 we\nmake some comments and observations.\n\\section{Quantum Mechanical Black Holes}\nIf we adhoc treat an electron as a charged and spinning black hole, described\nby the Kerr-Newman metric, the pleasing fact which emerges is that this\nmetric describes the gravitational and electromagnetic field of an electron\nincluding the anomalous gyromagnetic ratio$^2$, $g=2.$\\\\\nHowever the horizon of the Kerr-Newman Black Hole becomes in this case\ncomplex$^3$,\n\\begin{equation}\nr_+ = \\frac{GM}{c^2} + \\imath b,b \\equiv (\\frac{G^2Q^2}{c^8} + a^2 -\n\\frac{G^2M^2}{c^4})^{1\/2}\\label{e1}\n\\end{equation}\nwhere $G$ is the gravitational constant, $M$ the mass and $a \\equiv L\/Mc,L$\nbeing the angular momentum. That is, we have a naked singularity apparently\ncontradicting the cosmic censorship conjecture. However, in the Quantum\nMechanical domain, (\\ref{e1}) can be seen to be meaningful.\\\\\nInfact, the position coordinate for a Dirac particle is\ngiven by$^4$\n\\begin{equation}\nx_\\imath = (c^2p_\\imath H^{-1} t + a_\\imath)+\\frac{\\imath}{2}\nc\\hbar (\\alpha_\\imath - cp_\\imath H^{-1})H^{-1},\n\\label{e2}\n\\end{equation}\nwhere $a_\\imath$ is an arbitrary constant and $c\\alpha_\\imath$ is the velocity operator\nwith eigen values $\\pm c$. The real part in (\\ref{e2}) is the usual position\nwhile the imaginary part arises from Zitterbewegung. Interestingly, in both\n(\\ref{e1}) and (\\ref{e2}), the imaginary part is of the order of $\\frac{\\hbar}\n{mc}$, the Compton wavelength, and leads to an immediate identification of\nthese two equations. We must remember that our physical measurements are\ngross - they are really measurements averaged over a width\nof the order $\\frac{\\hbar}{mc}$. Similarly, time measurements are imprecise\nto the tune $\\sim \\frac{\\hbar}{mc^2}$. Very precise measurements if possible,\nwould imply that all Dirac particles would have the velocity of light, or\nin the Quantum Field Theory atleast of Fermions, would lead to divergences.\n(This is closely related to the non-Hermiticity of position operators in\nrelativistic theory as can be seen from equation (\\ref{e2}) itself$^5$.\nPhysics begins after an averaging over the above unphysical\nspace-time intervals. In the process as is known (cf.ref.5), the imaginary\nor non-Hermitian part of the position operator in (\\ref{e2}) disappears.\nThat is in the case of the QMBH (Quantum Mechanical Black Hole), obtained by\nidentifying (\\ref{e1}) and (\\ref{e2}), the naked singularity is shielded\nby a Quantum Mechanical censor.\\\\\nTo examine this situation more closely we reverse the arguments after\nequation (\\ref{e2}), and consider instead the complex displacement,\n\\begin{equation}\nx^\\mu \\to x^\\mu + \\imath a^\\mu\\label{e3}\n\\end{equation}\nwhere $a^o \\approx \\frac{\\hbar}{2mc^2}, \\mbox{and} a^\\mu \\approx \\frac{\\hbar}{mc}$\nas before. That is, we probe into the QMBH or the Zitterbewegung region\ninside the Compton wavelength as suggested by (\\ref{e1}) and (\\ref{e2}).\nRemembering that $|a^\\mu| < < 1$, we have, for the wave function,\n\\begin{equation}\n\\psi (x^\\mu) \\to \\psi (x^\\mu + \\imath a^\\mu) = \\frac{a^\\mu}{\\hbar}\n[\\imath \\hbar \\frac{\\partial}{\\partial x^\\mu} + \\frac{\\hbar}{a^\\mu}]\n\\psi (x^\\mu)\\label{e4}\n\\end{equation}\nWe can identify from (\\ref{e4}), by comparison with the well known electromagnetism-\nmomentum coupling, the usual electrostatic charge as,\n\\begin{equation}\n\\Phi e = \\frac{\\hbar}{a^o} = mc^2\\label{e5}\n\\end{equation}\nIn the case of the electron, we can verify that the equality (\\ref{e5})\nis satisfied. Infact it was shown that from here we can get a rationale\nfor the value of the fine structure constant (cf.ref.1).\\\\\nWe next consider the spatial part of (\\ref{e3}), viz.,\n$$\\vec x \\to \\vec x + \\imath \\vec a, \\mbox{where} |\\vec a| = \\frac{\\hbar}{2mc},$$\ngiven the fact that the particle is now seen to have the charge $e$ (and mass\n$m$). As is well known this leads in General Relativity from the static Kerr\nmetric to the Kerr-Newman metric where the gravitational and electromagnetic\nfield of the particle is given correctly, including the anomalous factor\n$g = 2$. In General Relativity, the complex transformation (\\ref{e3}) and\nthe subsequent emergence of the Kerr-Newman metric has no clear explanation. Nor\nthe fact that, as noted by Newman$^6$ spin is the orbital angular momentum\nwith an imaginary shift of origin. But in the above context we can see the\nrationale: the origin of (\\ref{e3}) lies in the QMBH and Zitterbewegung\nprocesses inside the Compton wavelength.\\\\\nHowever the following question has to be clarified: How can an electron\ndescribed by the Quantum Dirac spinor $(\\theta_\\chi)$, where $\\theta$\ndenotes the positive energy two spinor and $\\chi$ the negative energy\ntwo spinor, be identified with the geometrodynamic Kerr-Newman Black\nHole characterised by the curved space time (without any doublevaluedness,\ncf.ref.2).\\\\\nWe observe that as is well known,$^7$ at and within the Compton wavelength\nit is the negative energy $\\chi$ that dominates. Further, under reflection,\nwhile $\\theta \\to \\theta, \\chi$ behaves like a psuedo-spinor,\\\\\n$$\\chi \\to -\\chi$$\nHence the operator $\\frac{\\partial}{\\partial x^\\mu}$ acting on $\\chi$, a\ndensity of weight $N = 1,$ has the following behaviour$^8$,\n\\begin{equation}\n\\frac{\\partial \\chi}{\\partial x^\\mu} \\to \\frac{1}{\\hbar} [\\hbar \\frac{\\partial}\n{\\partial x^\\mu} - NA^\\mu]\\chi\\label{e6}\n\\end{equation}\nwhere,\n\\begin{equation}\nA^\\mu = \\hbar \\Gamma_\\sigma^{\\mu \\sigma} = \\hbar \\frac{\\partial}{\\partial x^\\mu}\nlog (\\sqrt{|g|})\\label{e7}\n\\end{equation}\nAs before we can identify $NA^\\mu$ in (\\ref{e6}) with the electromagnetic\nfour potential. That $N = 1$, explains the fact that charge is discrete.\\\\\nIn this formulation, electromagnetism arises from the covariant derivative\nwhich is the result of the Quantum Mechanical behaviour of the negative\nenergy components of the Dirac spinor at the Compton wavelength scale. We can see\nat once how an electron can be associated with curvature and how the double\nconnectivity of spin half surfaces in the geometrodynamical formulation.\n(\\ref{e7}) strongly resembles Weyl's formulation for the unification\nof electromagnetism and gravitation$^9$. However it must be noted that the\noriginal Christofell symbol of Weyl contained two independent entities viz.\nthe metric tensor \\underline{and} the electromagnetic potential, so that there was\nreally no unification. In our formulation we have used only the Quantum\nMechanical psuedo spinor property.\\\\\nSo we could treat the Quantum Mechanical Black Hole as a relativistic fluid\nof subconstituents (or Ganeshas). In a linearized theory (cf.ref.2) we\nhave\n\\begin{equation}\ng_{\\mu v} = \\eta_{\\mu v} + h_{\\mu v}, h_{\\mu v} = \\int \\frac{4T_{\\mu v}(t -\n|\\vec x - \\vec x'|, \\vec x')}{|\\vec x - \\vec x'|}d^3 x'\\label{e8}\n\\end{equation}\nIt was then shown, (cf.ref.1), that not only do we recover the Quantum\nMechanical spin but using equation (\\ref{e7}) and (\\ref{e8}) that for\n$r = |\\vec x| >> |\\vec x'|$ we get\n\\begin{equation}\n\\frac{e'e}{r} = A_o \\sim \\frac{\\hbar c^3}{r} \\int \\rho \\omega d^3 x'\n\\sim (Gmc^3)\\frac{mc^2}{r}\\label{e9}\n\\end{equation}\nwhere $e' = 1 \\mbox{esu}$ corresponds to the charge $N = 1$ and $e$ is the\ntest charge. (\\ref{e9}) is correct and infact leads to the well known empirical result,\n\\begin{equation}\n\\frac{e^2}{Gm^2} \\sim 10^{40},\\label{e10}\n\\end{equation}\nThe above model gives a rationale for the left handedness of the neutrino,\nwhich can be treated as an electron with vanishing mass so that the\nCompton wavelength becomes arbitrarily large. For such a particle, we\nencounter in effect the region within the Compton wavelength with the pseudo\nspinorial property discussed above, that is left handedness.\\\\\nFinally it may be remarked that the electron, the positron and its special case the neutrino\nare the fundamental elementary particles which could be used to generate\nthe mass spectrum of elementary particles$^{10}$.\\\\\nWe now briefly examine why the Compton wavelength emerges as a fundamental\nlength. Our starting point could be the Dirac or Klein-Gordon equations. For\nsimplicity we consider the Klein-Gordon equation. It is well known that the\nposition operator is given by$^{5}$\n\\begin{equation}\n\\vec X_{op} = \\vec x_{op} - \\frac{\\imath \\hbar c^2}{2} \\frac{\\vec p}{E^2}\\label{e11}\n\\end{equation}\n(The Dirac equation also has a similar case).\\\\\nWe saw in (ref.1) that the imaginary part in equation (\\ref{e11}) which\nmakes $\\vec X_{op}$ non-Hermitian, and for the Dirac particle gives Zitterbewegung\ndisappears on averaging over intervals $\\Delta t \\sim \\frac{\\hbar}{mc^2} (\\mbox{and}\n\\Delta r \\sim \\frac{\\hbar}{mc})$ so that $\\vec X_{op}$ becomes Hermitian (this is\nalso the content of the Foldy-Wothuysen transformation). Our physics, as\npointed out begins after such an average or Hermitization. Our measurement\nin other words are necessarily gross to this extent - we will see this more\nclearly. From equation (\\ref{e11})\nwe now get\n\\begin{equation}\n\\hat X^2_{op} \\equiv \\frac{2m^3c^4}{\\hbar^2} X^2_{op} = \\frac{2m^3c^6}{\\hbar^2}\nx^2 + \\frac{p^2}{2m}\\label{e12}\n\\end{equation}\nMathematically equation (\\ref{e12}) shows that $\\hat X^2_{op}$ gives a problem\nidentical to the harmonic oscillator with quantized levels: Infact the quantized\n\"space-levels\" for $\\vec X^2_{op}$ turn out to be multiples of $(\\hbar\/mc)^2$!\nFrom here, we get $\\Delta t = \\frac{\\Delta x}{c} = \\frac{\\hbar}{mc^2}$.\n\\section{The formation of QMBH particles}\nWe now investigate how such QMBH can be formed. For this we digress\ntemporarily to vaccuum fluctuations. It is well known that there is a zero\npoint field (ZPF). According to QFT this arises due to the virtual quantum\neffects of the electromagnetic field already present. Whereas according to\nwhat has now come to be called Stochastic Electrodynamics (SED), it is\nthese ZPF that are primary and result in Quantum Mechanical primary effects\n$^{11}$. Many Quantum Mechanical effects can indeed be explained this way.\nWithout entering into the debate about the ZPF fluctuations for the moment, we observe that\nthe energy of the fluctuations of the magnetic field in a region of length\n$\\lambda$ is given by$^{2}$ $(\\vec E$ and $\\vec B$ are electromagnetic field\nstrengths)\n\\begin{equation}\nB^2 \\sim \\frac{\\hbar c}{\\lambda^4}\\label{e13}\n\\end{equation}\nIf $\\lambda$ as in the QMBH is taken to be the Compton wavelength,\n$\\frac{\\hbar}{mc}$ (\\ref{e13}) gives us for the energy in this volume\nof the order $\\lambda^3$,\\\\\n$$\\mbox{Total\\quad energy\\quad of\\quad QMBH\\quad}\\sim \\frac{\\hbar c}{\\lambda} =\nmc^2,$$\nexactly as required. In other words the entire energy of the QMBH\nof mass $m$ can be thought to have been generated by the fluctuations\nalone. Further the fluctuation in curvature over the length $l$ is given\nby$^{2}$,\n\\begin{equation}\n\\Delta R \\sim \\frac{L^*}{l^3},\\label{e14}\n\\end{equation}\nwhere $L^*$ is the Planck length of the order $10^{-33}cms$.\\\\\nFor the electron which we consider, $l$ is of the order of the Compton\nwavelength, that is $10^{-11}cms$. Substitution in (\\ref{e14}) therefore\ngives\n$$\\Delta R \\sim 1$$\nIn other words the entire curvature of the QMBH is also generated by these\nfluctuations. That is the QMBH can be thought to have been created by\nthese fluctuations alone.\\\\\nWithin the framework of QED, we can come to this conclusion in another\nway$^{12}$. It is known that the vaccuum energy of the electron field with\na cut off $k_{max}$ is given by,\n\\begin{equation}\n\\frac{\\mbox{Energy}}{\\mbox{Volume}} \\sim \\hbar c k^4_{max}\\label{e15}\n\\end{equation}\nThis is the same as equation (\\ref{e13}) encountered earlier. Also the\ninfinite energy of the vaccuum is avoided by the assumption of the cut\noff normally taken to be of the order of a typical Compton wavelength on\nthe ground that we do not know that the laws of electromagnetism are\nvalid beyond these high frequencies, that is within these length scales.\\\\\nBut the preceding discussion shows that it is natural to take $k_{max} =\n\\frac{mc}{\\hbar},$ the inverse Compton wavelength of the electron. The energy\nof the electron from equation (\\ref{e15}) then comes out to be\n$$E \\sim mc^2,$$\nas before. So we are led to the important conclusion that the\ninfinity of QED is avoided by the fact that QMBH are formed, rather than by\nthe arbitrary prescription of a cut off. Infact there is a further bonus\nand justification for the above interpretation. Let us use in (\\ref{e15})\nthe pion Compton wavelength as the cut off. The reason we choose the pion\nis that it is considered to be a typical elementary particle in the sense\nthat it plays a role in the strong interactions, and further it could\nbe used as a building block for developing a mass spectrum, and finally\nas seen in (ref.1) can be considered to be made up of an electron and\na positron. Then from (\\ref{e15}) we can recover the pion mass,\n$m_\\pi$ and moreover,\n\\begin{equation}\nNm_\\pi = M,\\label{e16}\n\\end{equation}\nwhere $N$ is the number of elementary\nparticles, typically pions, $N \\sim 10^{80}$ and $M$ is the mass\nof the universe,viz. $10^{56}gms$.\\\\\nIn other words, in our interpretation we have not only avoided the QED\ninfinity but have actually recovered the mass of the universe. We will return\nto this point shortly.\\\\\nWe now consider the same scenario from a third point of view, viz. from the\nstandpoint of Quantum Statistical Mechanics. Here also the spirit is that\nof randomness$^{13}$. A state can be written as\n\\begin{equation}\n\\psi = \\sum_{n} c_n \\phi_n,\\label{e17}\n\\end{equation}\nin terms of basic states $\\phi_n$ which\ncould be eigen states of energy for example, with eigen values $E_n$. It\nis known that (\\ref{e17}) can be written as\n\\begin{equation}\n\\psi = \\sum_{n} b_n \\phi_n\\label{e18}\n\\end{equation}\nwhere $|b_n|^2 = 1$ if $E<E_n<E+\\Delta$, and $= 0$ otherwise under the assumption\n\\begin{equation}\n\\overline{(c_n,c_m)} = 0, n \\ne m\\label{e19}\n\\end{equation}\n(Infact $n$ could stand for not a single state but for a set of states\n$n_\\imath$ and so also $m$). Here the bar denotes a time average over a\nsuitable interval. This is the well known Random Phase Axiom and arises\ndue to the total randomness amongst the phases $c_n$. Also the expectation\nvalue of any operator $O$ is given by\n\\begin{equation}\n<O> = \\sum_{n} |b_n|^2 (\\phi_n, O \\phi_n)\/ \\sum_{n} |b_n|^2\\label{e20}\n\\end{equation}\n(\\ref{e18}) and (\\ref{e20}) show that effectively we have incoherent states\n$\\phi_1, \\phi_2,....$ once averages over time intervals for the phases\n$c_n$ in (\\ref{e19})vanish owing to their relative randomness.\\\\\nIn the light of the preceding discussion of random fluctuations in the context of QMBH\nin SED and QED, we can interpret the above meaningfully: We can identify\n$\\phi_n$ with the ZPF. The time averages are the Zitterbewegung averages\nover intervals $\\sim \\frac{\\hbar}{mc^2}$. We then get disconnected or\nincoherent particles or QMBH from a single background of vaccuum\nfluctuations exactly as before. The incoherence arises because of the well\nknown random phase relation (\\ref{e19}) that is after averaging over the\nsuitable interval.\\\\\nBut in all of the above considerations, and in present day theory the question\nthat comes up is: How can we reconcile the fact that the various particles\nin the universe are not infact incoherent but rather occupy a single\ncoherent space-time. The answer which can now be seen to emerge in the\nlight of the above discussion is that all these particles are linked by\ninteraction. These interactions as pointed out in (ref.1) arise within\nthe Compton wavelength or Zitterbewegung region, that is in phenomena\nwithin the time scale $\\frac{\\hbar}{mc^2}$. It will be observed in the\nabove discussion that at these time scales the equation (\\ref{e19}) is\nno longer valid and we have to contend with equation (\\ref{e17}) rather\nthan equation (\\ref{e18}). So interactions arising within the\nCompton wavelength link or make coherent\nthe various particles.\\\\\nInfact all this is perfectly in tune with the QFT picture wherein the\ninteractions are caused by virtual particles with life time less than\n$\\frac{\\hbar}{mc^2}$. It may also be observed that in Wheeler's Geometrodynamical\nmodel$^{14}$, the various particles are linked by exactly such\nwormholes linking distant regions.\\\\\nIn the above formulation we could take $\\phi_n$ to be the particlets or Ganeshas instead of\nenergy eigen states, that is to be position eigen states and consider sets\n$$\\overline{(c_{n_\\imath}, c_{m_j})} = 0,$$\nexactly as before (cf.(\\ref{e19})). Each set $\\phi_{n_\\imath}$ defines a particle\n$P_n$ consisting of $n_\\imath$ Ganeshas or particlets. It is the link\nat $\\Delta t < \\hbar\/mc^2$ between $P_n$ and $P_m$ which puts otherwise incoherent particles into a\nsingle space-time, that is allows interactions.\\\\\nIn other words, a set of particles can be said to be in the same space-time\nif every particle interacts with atleast one other member of the set.\\\\\nFor completeness we mention that the above bunching could be carried out\nin principle for two\nor more universes. Thus a set of particles constitutes universe $U_1$ while\nanother set of particles constitute an incoherent universe $U_2$. Again the\nincoherence can be broken at a suitable time scale (cf.ref.15 for a pictorial\nmodel in terms of wormholes).\\\\\nThere is another way of looking at all this.\nWe first note that the space-time symmetry of relativity has acquired\na larger than life image. Infact our perception of the universe is\nessentially one of all space (or as much of it as possible) at one instant\nof time (cf.also ref.2). Further, time is essentially an ordering or sequencing\nof events. To understand time we must know on what basis this ordering is\ndone so that causality and other laws of physics hold or in other words we\nhave the universe of the physical hyperboloid.\\\\\nWe now approach this problem by trying to liberate the sequence of\nevents in time from any ordering at all. At first sight it would appear that\nthis approach would lead to a chaotic universe without physics that is\ncausality, interaction and so on. We will actually try to attempt to explain\nthe emergence of physics from such a, what may be called pre-space-time scenario. It must be\nnoted that both Special and General Relativity work in a deterministic\nspace-time. Even relativistic Quantum Mechanics and Quantum Field Theory\nassume the space-time of Special Relativity. Quantum Gravity on the other\nhand which has not yet proved to be a completely successful theory questions\nthis concept of space-time$^{16}$.\\\\\nWhile a random time sequence is ruled out at what may be called the macro\nlevel, in our case above the Compton wavelength, within the framework of\nQMBH and as seen above this is certainly possible below the Compton\nwavelength scale. Infact this is the content of non locality and non Hermiticity\nof the Zitterbewegung in the region of QMBH.\\\\\nSo we start with truly instantaneous point particles or particlets (or Ganeshas)\nwhich are therefore indistinguishable, (cf. ref.1) (and could be denoted by\n$\\phi_n$ of (\\ref{e17})). We then take a random sequence of such\nparticlets$^{13}$. Such a sequence for the interval $\\Delta t \\sim \\frac\n{\\hbar}{mc^2}$ in time collectively constitutes a particle that has come\ninto existence and is spread over a space interval of the order of the\nCompton wavelength. In other words we have made a transition from pre-\nspace-time to a particle in space-time. This is exactly the averaging over random phases in\nequation (\\ref{e19}). Hermiticity of position operators has now been\nrestored and we are back with the states $\\phi_n$ in equation\n(\\ref{e18}). All this is in the spirit that our usual time is such that,\nwith respect to it vaccuum fluctuations are perfectly random as pointed\nby Macrea$^{17}$. So the subconstituents of the relativistic fluid given\nin (\\ref{e8}) (or the Quantized Vortex in the hydrodynamical formulation\n(cf. ref.1)), are precisely these particlets.\\\\\nTo visualize the above consideration in greater detail we first consider strictly\npoint particles obtained by taking the random sequences over time\nintervals $\\sim \\frac{\\hbar}{mc^2}$. We consider an assembly of such truly\npoint particles which as yet we cannot treat either with Fermi-Dirac or\nBose-Einstein statistics but rather as a Maxwell-Boltzman distribution. If there\nare $N$ such particles in a volume $V$, it is known that$^{13}$, the volume\nper particle is of the order of,\n$$(\\frac{V}{N})^{1\/3} \\sim \\lambda_{thermal} \\approx \\frac{\\hbar}{\\sqrt{m^2c^2}} =\n\\frac{\\hbar}{mc},$$\nwhere we take the average velocity of each particle to be equal to $c$. Infact,\nthis is exactly what happens, as Dirac pointed out (cf. ref.4), for a truly\nhypothetical point electron, in the form of Zitterbewegung within the\nCompton wavelength.\\\\\nSo the Compton wavelength arises out of the (classical) statistical inability to\ncharacterise a point particle precisely: It is not that the particle has an\nextension per se. In this sense the Compton wavelength has a very\nCopenhagen character, except that it has been deduced on the basis of an\nassembly of particles rather than an isolated particle.\n\\section{The Universe of Fluctuations}\nThe question that arises is, what are the cosmological implications of the\nabove scenario, that is, if we treat the entire universe as arising from\nfluctuations, is this picture consistent with the observed universe? It\nturns out that not only is there no inconsistency, but on the contrary\na surprising number of correspondences emerge.\\\\\nThe first of these is what we have encountered a little earlier viz.\nthe fact that we recover the mass of the universe as in equation (\\ref{e16}).\\\\\nWe can next deduce another correspondence. The ZPF gives the correct\nspectral density viz.\n$$\\rho (\\omega) \\alpha \\omega^3$$\nand infact the Planck spectrum$^{18}$. We then get the total intensity\nof radiation from the fluctuating field due to a single star as$^{11}$,\n$$I(r) \\alpha \\frac{1}{r^2}$$\nIt then follows that given the observed isotropy and homogeneity of the\nuniverse at large, as is well known,\n\\begin{equation}\nM \\alpha R\\label{e21},\n\\end{equation}\nwhere $R$ is the radius of the universe.\\\\\nEquation (\\ref{e21}) is quite correct and infact poses a puzzle, as is\nwell known and it is to resolve this dependence that dark matter has\nbeen postulated$^{19}$ whereas in our formulation the correct mass radius\ndependence has emerged quite naturally without any other adhoc postulates.\\\\\nAs we have seen above the Compton wavelength of a typical particle, the pion viz $l_\\pi$ can be\ngiven in terms of the volume of uncertainity. However in actual observation\nthere is an apparent paradox. If the universe is $n$ dimensional then we\nshould have,\n$$Nl^n_\\pi \\sim R^n$$\nfor the universe itself. This relation is satisfied with $n = 2$ in which\ncase we get a relation that has been known emperically viz.,\n$$l_\\pi \\sim \\frac{R}{\\sqrt{N}}$$\n(Even Eddington had used this relation).\\\\\nSo in conjunction with (\\ref{e16}) we have an apparent paradox where the actual universe appears to be\ntwo dimensional. This will be resolved shortly and it will be seen that\nthere is no contradiction.\\\\\nAnother interesting consequence is as follows: According to our formulation\nthe gravitational potential energy of a pion in a three dimensional isotropic\nsphere of pions is given by\n$$\\frac{Gm_\\pi M}{R}$$\nThis should be equated with the energy of the pion viz. $m_\\pi c^2$. We then\nget,\n\\begin{equation}\n\\frac{GM}{c^2} = R,\\label{e22}\n\\end{equation}\na well known and observationally correct relation. In our formulation we\nget $m_\\pi$ from the ZPF and given $N$ we know $M$ so that from equation\n(\\ref{e22}) we can deduce the correct radius $R$ of the universe.\\\\\nProceeding further we observe that the fluctuations in the particle number $N$ itself is\nof the order $\\sqrt{N}^{13,20}$. Also $\\Delta t$ above is the typical\nfluctuating time. So we get,\n$$\\frac{dN}{dt} = \\frac{\\sqrt{N}}{\\Delta t} = \\frac{m_\\pi c^2}{\\hbar} \\sqrt{N}$$\nwhence as $t = 0, N = 0,$\n\\begin{equation}\n\\sqrt{N} = \\frac{2m_\\pi c^2}{\\hbar} .T\\label{e23}\n\\end{equation}\nwhere $T$ is the age of the universe $\\approx 10^{17}secs$. It is remarkable\nthat equation (\\ref{e23}) is indeed correct. One way of looking at this is\nthat not only the radius but also the age of the universe is correctly\ndetermined. As we saw before,\n$$R = \\frac{GM}{c^2} = \\frac{GNm_\\pi}{c^2}$$\nso that\n\\begin{equation}\n\\frac{dR}{dt} = \\frac{Gm_\\pi}{c^2} \\frac{dN}{dt} = \\frac{Gm^2_\\pi}{\\hbar} \\sqrt{N} = HR\\label{e24}\n\\end{equation}\nwhere\n\\begin{equation}\nH = \\frac{Gm_\\pi^3 c}{\\hbar^2}\\label{e25}\n\\end{equation}\nOne can easily verify that (\\ref{e25}) is satisfied for the Hubble constant so\nthat (\\ref{e24}) infact gives the Hubble's velocity distance relation.\\\\\nFurthermore from (\\ref{e25}) we deduce that,\n\\begin{equation}\nm_\\pi = (\\frac{\\hbar^2 H}{Gc})^{1\/3}\\label{e26}\n\\end{equation}\nIt is remarkable that equation (\\ref{e26}) is known to be true from a\npurely empirical standpoint$^{21}$. However we have actually deduced it in\nour formalism. Another way of interpreting equation (\\ref{e26}) is that\ngiven $m_\\pi$ (and $\\hbar, G$ and $c$) we can actually deduce the value of\n$H$ in our formalism.\\\\\nFrom Equation (\\ref{e24}), we deduce that,\n\\begin{equation}\n\\frac{d^2R}{dt^2} = H^2R\\label{e27}\n\\end{equation}\nThat is, effectively there is a cosmic repulsion. Infact, from (\\ref{e27}) we\ncan identify the cosmological constant as\n$$\\Lambda \\sim H^2$$\nwhich is not only consistent but agrees exactly with the limit on this constant\n(cf.ref.2).\\\\\nThe final correspondence is to do with an explanation for the microwave\ncosmic background radiation within the above framework of fluctuations. It\nis well known that the fluctuations of the Boltzmann $H$ function for interstellar\nspace is of the order of $10^{-11}secs^{13}$. These fluctuations can be\nimmediately related to the ZPF exactly as in the case of the Lamb shift\n(cf.ref.2). So $\\frac{\\hbar}{mc^2} = 10^{-11}$ or the associated wavelength\nviz.,\n$$\\frac{\\hbar}{mc} \\sim 0.3 cms,$$\nwhich corresponds to the cosmic background\nradiation$^{3}$. The same conclusion can be drawn from a statistical\ntreatment of interstellar Hydrogen$^{22}$.\n\\section{Comments}\n1. We could arrive at equation (\\ref{e13}) by a slightly different route\n(cf.ref.2). We could start with a single oscillator in the ground state\ndescribed by the wave function\n\\begin{equation}\n\\psi(x) = \\mbox{const} \\quad exp[-(m\\omega\/2 \\hbar)x^2]\\label{e28}\n\\end{equation}\nwhich would fluctuate with a space uncertainity of\n$$\\Delta x \\sim (\\hbar\/m\\omega)^{\\frac{1}{2}} = \\frac{\\hbar}{mc}$$\nThe electromagnetic ZPF could be treated as an infinite collection of\nindependent oscillators and we could recover equation (\\ref{e13}).\\\\\n2.Earlier we skirted the issue whether the ZPF is primary or secondary.\nWe now start either with the ZPF or with the pre-space-time background\nfield of the instantaneous particles (or Ganeshas). We could assign a\nprobability $p$ for them to appear in space-time and the probability\n$1-p = q$ for this not to happen. From here we get the probability for\n$N$ of them to appear as\n\\begin{equation}\n\\mbox{Probability}\\quad \\alpha \\quad exp \\quad [-\\mu^2 N^2]\\label{e29}\n\\end{equation}\nThis immediately ties up with the considerations following from equation\n(\\ref{e12}) (cf.ref.2), if we identify $N$ with $x$. The justification\nfor this can be seen by a comparison with $|\\psi (x)|^2$ from (\\ref{e28}):\nFrom (\\ref{e29}), the probability is non-negligible if\n$$\\Delta N \\sim \\frac{1}{\\mu},$$\nwhich turns out to be, from (\\ref{e28}),\n$$\\Delta N \\sim \\frac{1}{\\mu} \\approx \\frac{\\hbar}{mc},$$\nthe Compton wavelength. Thus once again we conclude from (\\ref{e29})\nthat a probabilistic fluctuational collection of instantaneous particlets\nfrom a pre-space-time background shows up as a particle in space-time.\\\\\nWheeler considers the algebra of propositions as providing the link\nbetween what he terms pre-geometry and geometrodynamics. In our formulation\nprobabilistic fluctuations lead to space-time and physics from\npre-space-time.\\\\\n3. It was pointed out that the equation\n\\begin{equation}\nl_\\pi \\sim \\frac{R}{\\sqrt{N}}\\label{e30}\n\\end{equation}\nsuggests that the universe is apparently two dimensional. This paradoxical\nresult is consistent with astrophysical data (cf.ref.19). We could resolve\nthe paradox as follows:\\\\\nWe start with the fact that the universe on the average is neutral. Further\nthe fluctuation in the number of electrons is $\\sim \\sqrt{N}$. So an extra\nelectrostatic potential energy is created which is balanced by (or in\nour formulation manifests itself as) the energy of the electron itself\n(cf.ref.20):\n$$\\frac{e^2\\sqrt{N}}{R} = mc^2$$\nwhich leads to the above relation.\\\\\nSo in the conventional theory, that is in the language of a fixed particle\nnumber universe, we would say that the universe is apparently two dimensional.\nBut once we recognise the fluctuations, the universe is really three\ndimensional. Infact the fundamental equation (\\ref{e10}) which was\nderived purely from the point of view of an isolated particle can also be\nderived on the basis of a \"two dimensional\" universe$^{23}$.\\\\\n4. The considerations of the previous section show that there exists, what\nmay be called a micro-macro nexus: Fundamental constants of Quantum Theory\nare tied up with constants from macro physics and cosmology. So the universe\nis holistic. It has a slightly different connotation from the Machian formulation,\nbecause the latter deals with a deterministic universe with rigid physical\nlaws.\\\\\nInfact from (\\ref{e30}), (\\ref{e22}) and (\\ref{e23}), we can deduce that,\n\\begin{equation}\n\\frac{2Gm_\\pi^3c}{\\hbar^2} = \\frac{1}{T}\\label{e31}\n\\end{equation}\nwhich is a variant of equation (\\ref{e10}), if we replace its right side\nby $\\sqrt{N}$. This may be interpreted as giving $e,G,c$ or $\\hbar$ in\nterms of $m_\\pi\\quad \\mbox{and}\\quad N$. More interestingly, (\\ref{e31})\ngives the variation of $G$, or more generally, the left side, with the\nage of the universe (cf.ref.2 for Dirac's conjecture in this connection).\\\\\n5. The quantization formula for space following from equation (\\ref{e12}),\nreflects an empirical formula deduced by Chacko$^{24}$ which can be used to\ngenerate a mass spectrum. It also vindicates a close connection between\nenergy and space-time: As pointed out (cf.ref.1) inertial mass arises from\nthe non local Zitterbewegung processes within the Compton wavelength$^{25}$.\\\\\n6. Finally we observe that inspite of similarities, the above scenario of\nfluctuations differs from steady-state cosmology and the $C$ field\nformulation$^{26}$.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA citation network is an important source of analysis in science.\nCitations serve multiple purposes such as crediting an idea, signaling\nknowledge of the literature, or critiquing others' work \\citep{martyn1975citation}.\nWhen citations are thought of as impact, they inform tenure, promotion,\nand hiring decisions \\citep{meho2007impact}. Furthermore, scientists\nthemselves make decisions based on citations, such as which papers\nto read and which articles to cite. Citation practices and infrastructures\nare well-developed for journal articles and conference proceedings.\nHowever, there is much less development for dataset citation. This\ngap affects the increasingly important role that datasets play in\nscientific reproducibility \\citep{task2013out,belter2014measuring,robinson2016analyzing,park2018informal},\nwhere studies use them to confirm or extend the results of other research\n\\citep{sieber1995not,darby2012enabling}. One historical cause of\nthis gap is the difficulty in archiving datasets. While less problematic\ntoday, the citation practices for datasets take time to develop. Better\nalgorithmic approaches to track dataset usage could improve this state.\nIn this work, we hypothesize that a network flow algorithm could track\nusage more effectively if it propagates publication and dataset citations\ndifferently. With the implementation of this algorithm, then, it will\nbe possible to correct differences in citation behavior between these\ntwo types of artifacts, increasing the importance of datasets as first\nclass citizens of science.\n\nDifferent researchers use citation networks to evaluate the importance\nof authors \\citep{ding2009pagerank,ding2011applying,west2013author},\npapers \\citep{chen2007finding,ma2008bringing}, journals \\citep{bollen2006journal,bergstrom2007eigenfactor},\ninstitutions \\citep{fiala2013suborganizations} and even countries\n\\citep{fiala2012bibliometric}. The \\textsc{PageRank} algorithm \\citep{page1999pagerank}\nhas served as a base for much of these citation network-based evaluations.\nFor example, \\citet{bollen2006journal} proposed a weighted \\textsc{PageRank\n}to assess the prestige of journals, while \\citet{ding2009pagerank}\nand \\citet{ding2011applying} proposed a weighted \\textsc{PageRank\n}to measure the prestige of authors. \\citet{fiala2012time} defined\na time-aware \\textsc{PageRank }method for accurately ranking the most\nprominent computer scientists. \\citet{franceschet2017timerank} introduced\nan approach called \\textsc{TimeRank} for rating scholars at different\ntime points. \\textsc{TimeRank} updates the rating of scholars based\non the relative rating of the citing and cited scholars at the time\nof the citation. Citation networks are thus an important source of\ninformation for ranking homogenous types of nodes.\n\nHistorically, ranking datasets using citation networks is significantly\nmore challenging. These challenges have technical and social issues\nalike. First, datasets cost time and labor to prepare and to share,\nresulting in some articles failing to provide datasets \\citep{alsheikh2011public}.\nSecond, archiving and searching massive datasets is prohibitively\nexpensive and difficult. Third, scholars are not used to citing datasets.\nSurvey research shows that scholars value citing dataset \\citep{kratz2015researcher}\nyet they tend to cite the \\emph{article} rather than the dataset or\nthey merely mention the dataset without explicit reference \\citep{force2014encouraging}.\nTherefore, these reasons have prevented the proper assignment of credit\nto dataset usage.\n\nSeveral initiatives attempt to improve citation practices for datasets.\nIn 2014, the Joint Declaration Of Data Citation Principles was officially\nreleased. These principles, however, mainly focus on normalizing dataset\nreferences rather than normalizing storage and some other technical\nissues \\citep{altman2015introduction,callaghan2014joint,mooney2012anatomy}.\nFor instance, some researchers have suggested assigning specific DOIs\nto datasets to mitigate differences between datasets and articles\n\\citep{callaghan2012making}. Others have proposed to automatically\nidentify uncited or unreferenced datasets used in articles \\citep{boland2012identifying,kafkas2013database,ghavimi2016identifying}.\nAll these solutions try to make citation dataset behavior more standard\nor attempt to fix the citation network by estimating which data nodes\nare missing. Therefore, these solutions necessarily modify the source\nthat algorithms use to estimate impact.\n\nIn this article, we develop a method for assigning credit to datasets\nfrom citation networks of publications, assuming that dataset citations\nhave biases. Importantly, our method does not modify the source data\nfor the algorithms. The method does not rely on scientists explicitly\nciting datasets but infers their usage. We adapt the network flow\nalgorithm of \\citet{walker2007ranking} by including two types of\nnodes: datasets and publications. Our method simulates a random walker\nthat takes into account the differences between obsolescence rates\nof publications and datasets, and estimates the score of each dataset---the\n\\textsc{DataRank}. We use the metadata from the National Center for\nBioinformatics (NCBI) GenBank nucleic acid sequence and Figshare datasets\nto validate our method. We estimate the relative rank of the datasets\nwith the \\textsc{DataRank} algorithm and cross-validate it by predicting\nthe actual usage of them---number of visits to the NCBI dataset web\npages and downloads of Figshare datasets. We show that our method\nis better at predicting both types of usages compared to citations\nand has other qualitative advantages compared to alternatives. We\ndiscuss interpretations of our results and implications for data citation\ninfrastructures and future work.\n\n\\section{Why measure dataset impact?}\n\nScientists may be incentivized to adopt better and broader data sharing\nbehaviors if they, their peers, and institutions are able to measure\nthe impact of datasets (e.g., see \\citet{silvello2018theory} and\n\\citet{kidwell2016badges}). In this context, we review impact assessment\nconceptual frameworks and studies of usage statistics and crediting\nof scientific works more specifically. These areas of study aim to\ndevelop methods for scientific indicators of the usage and impact\nof scholarly outputs. Impact assessment research also derives empirical\ninsights from research products by assessing the dynamics and structures\nof connections between the outputs. These connections can inform better\npolicy-making for research data management, cyberinfrastructure implementation,\nand funding allocation.\n\nMethods for measuring usage and impact include a variety of different\ndimensions of impact, from social media to code use and institutional\nmetrics. Several of these approaches recognize the artificial distinction\nbetween the scientific process and product \\citep{priem2013scholarship}.\nFor example, altmetrics is one way to measure engagement with diverse\nresearch products and to estimate the impact of non-traditional outputs\n\\citep{priem2014altmetrics}. Researchers predict that it will soon\nbecome a part of the citation infrastructure to routinely track and\nvalue \\textquotedblleft citations to an online lab notebook, contributions\nto a software library, bookmarks to datasets from content-sharing\nsites such as Pinterest and Delicious\\textquotedblright{} \\citep[from ][]{priem2014altmetrics}.\nIn short, if science has made a difference, it will show up in a multiplicity\nof places. As such, a correspondingly wider range of metrics are needed\nto attribute credit to the many locations where research works reflect\ntheir value. For example, datasets contribute to thousands of papers\nin NCBI\\textquoteright s Gene Expression Omnibus and these attributions\nwill continue to accumulate, just like paper accumulate citations,\nfor a number of years after the datasets are publicly released \\citep{piwowar2011data,piwowar2013altmetrics}.\nEfforts to track these other sources of impact include ImpactStory,\nstatistics from FigShare, and Altmetric.com \\citep{robinson2017datacite}.\n\nCredit attribution efforts include those by federal agencies to expand\nthe definition of scientific works that are not publications. For\nexample, in 2013 the National Science Foundation (NSF) recognized\nthe importance of measuring scientific artifacts other than publications\nby asking researchers for \\textquotedblleft products\\textquotedblright{}\nrather than just publications . This represents a significant change\nin how scientists are evaluated \\citep{piwowar2013altmetrics}. Datasets,\nsoftware, and other non-traditional scientific works are now considered\nby the NSF as legitimate contributions to the publication record.\nFurthermore, real-time science is presented in several online mediums;\nalgorithms filter, rank, and disseminate scholarship as it happens.\nIn sum, the traditional journal article is increasingly being complemented\nby other scientific products \\citep{priem2013scholarship}.\n\nYet despite the crucial role of data in scientific discovery and innovation,\ndatasets do not get enough credit \\citep{silvello2018theory}. If\ncredit was properly accrued, researchers and funding agencies would\nuse this credit to track and justify work and funding to support datasets---consider\nthe recent Rich Context Competition which aimed at to filling this\ngap by detecting dataset mentions in full-text papers \\citep{zengnyu}.\nBecause these dataset mentions are not tracked by current citation\nnetworks, this leads to biases in dataset citations \\citep{robinson2016analyzing}.\nThe FAIR (findable, accessible, interoperable, reproducible) principles\nof open research data are one major initiative that is spearheading\nbetter practices with tracking digital assets such as datasets \\citep{wilkinson2016fair}.\nHowever, the initiative is theoretical, and lacks technical implementation\nfor data usage and impact assessment. There remains a need to establish\nmethods to better estimate dataset usage.\n\n\\section{Materials and methods}\n\n\\subsection{Datasets}\n\n\\subsubsection{OpenCitations Index (COCI)}\n\nThe OpenCitations index (COCI) is an index of Crossref's open DOI-to-DOI\ncitation data. We obtained a snapshot of COCI (November 2018 version),\nwhich contains approximately 450 million DOI-to-DOI citations. Specifically,\nCOCI contains information including citing DOI, cited DOI, the publication\ndate of citing DOI. The algorithm we proposed in the paper requires\nthe publication year. However, not all the DOIs in COCI have a publication\ndate. We will introduce Microsoft Academic Graph to fill this gap.\n\n\\subsubsection{Microsoft Academic Graph (MAG)}\n\nThe Microsoft Academic Graph is a large heterogeneous graph consisting\nof six types of entities: paper, author, institution, venue, event,\nand field of study \\citep{sinha2015overview}. Concretely, the description\nof a paper consists of DOI, title, abstract, published year, among\nother fields. We downloaded a copy of MAG in November 2019, which\ncontains 208,915,369 papers. As a supplement to COCI, we extract the\nDOI and published year from MAG to extend those DOIs in COCI without\na publication date.\n\n\\subsubsection{PMC Open Access Subset (PMC-OAS)}\n\nThe PubMed Central Open Access Subset is a collection of full-text\nopen access journal articles in biomedical and life sciences. We obtained\na snapshot of PMC-OAS in August 2019. It consists of about 2.5 million\nfull-text articles organized in well-structured XML files. The articles\nare identified by a unique id called PMID. We also obtained a mapping\nbetween PMIDs and DOIs from NCBI, which enabled us to integrate PMC-OAS\ninto the citation network.\n\n\\subsubsection{GenBank}\n\nGenBank is a genetic sequence database that contains an annotated\ncollection of all publicly available nucleotide sequences for almost\n420,000 formally described species \\citep{Sayers2019}. The information\nabout a nucleotide sequence in GenBank is organized as a record consisting\nof many data elements (fields) and stored in a flat-file (see Fig.\n\\ref{fig:sample-data-of}). The ACCESSION field contains the unique\nidentifier of the dataset. The last citation in the REFERENCE field\ncontains the information about the submitter, including the author\nlist and the date in which the dataset was introduced (e.g., ``16-JAN-1992''\nin Fig. \\ref{fig:sample-data-of}). We obtained a snapshot of the\nGenBank database (version 230) with 212,260,377 gene sequences. We\nremove those sequences without submission date. This left us with\n77,149,105 sequences.\n\nThe National Institutes of Health (NIH) provided us with number of\nvisits to a sequence's landing page for the top 1000 Nucleotide sequences\nduring the month of September 2012. We use these visits as a measure\nof real \\emph{usage.}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics{genbank_record}\n\\par\\end{centering}\n\\caption{\\label{fig:sample-data-of} Sample record of a sequence submission\nfrom GenBank. The ACCESSION field is the unique identifier of a dataset.}\n\\end{figure}\n\n\n\\subsubsection{Figshare}\n\nFigshare is a multidisciplinary, open access research repository where\nresearchers can deposit and share their research output. Figshare\nallows users to upload various formats of research output, including\nfigures, datasets, and other media \\citep{thelwall2016figshare}.\nIn order to encourage data sharing, all research data made publicly\navailable has unlimited storage space and is allocated a DOI. This\nDOI is used by scientists to cite Figshare resources using traditional\ncitation methods. Figshare makes the research data publicly and permanently\navailable which mitigates the resource decay problem and improves\nresearch reproducibility \\citep{zeng2019deadscience}. A Figshare\nDOI contains the string 'figshare' (e.g., '10.6084\/m9.figshare.6741260').\nWe can leverage this feature to determine whether a publication cites\nFigshare resources by detecting Figshare-generated DOIs.\n\nFigshare also keeps track of dataset accesses, such as page views\nand dataset downloads. We get the downloads statistics of Figshare\nDOI from the Figshare Stats API\\footnote{https:\/\/docs.figshare.com\/\\#stats}\nand use it as a measure of real \\textit{usage}.\n\n\\subsection{Construction of citation network}\n\nThe citation networks in this paper consist of two types of nodes\nand two types of edges. The node is represented by the paper and the\ndataset and the edge is represented by the citation between two nodes.\nConcretely, papers cite each other to form paper-paper edges as datasets\ncan only be cited by papers which are represented by the paper-dataset\nedges. As shown in the construction workflow (Fig. \\ref{fig:Network-construction-workflow}),\nwe build the paper-paper citation network using COCI and MAG and build\ntwo separate paper-dataset edge sets using GenBank and Figshare. Then\nwe integrate the paper-dataset edge sets into the paper-paper citation\nnetwork to form two complete citation networks. The construction workflow\nis illustrated in Figure \\ref{fig:Network-construction-workflow}.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1\\columnwidth]{network_construction}\n\\par\\end{centering}\n\\caption{\\label{fig:Network-construction-workflow}Citation network construction\nworkflow}\n\\end{figure}\n\n\n\\subsubsection{Paper-paper citation network}\n\nAs our proposed method takes the publication year into account, we\nneed to remove those edges without publication year. The DOI-DOI citation\nin COCI dataset provides us a skeleton of the citation network. Even\nthough COCI comes with a field named timespan which originally refers\nto the publication time span between the citing DOI and cited DOI,\nthis timespan leads to different results depending on the time granularity\n(i.e., year-only vs year+month). As explained above, we have to complement\nthe COCI dataset with MAG to complete the year of publication of cited\nand citing articles. By joining the COCI and MAG, we build a large\npaper-paper citation network consisting of 45,180,243 nodes and 443,055,788\nedges. As the usage data of GenBank only covers 2012 (see Genbank\nabove), we pruned the network to remove papers and datasets published\nafter 2012. For Figshare, we use all the nodes and edges. The statistics\nof the networks described above are listed in Table \\ref{tab:Statistics-of-network}.\n\n\\begin{table}\n\\caption{\\label{tab:Statistics-of-network}Statistics of article citation networks}\n\n\\centering{}%\n\\begin{tabular}{cccc}\n\\hline \nSources & Filter & Number of Nodes & Number of Edges\\tabularnewline\n\\hline \nCOCI (original) & N\/A & 46,534,424 & 449,840,585\\tabularnewline\nCOCI (filtered) & contains publication year & 21,689,394 & 203,884,791\\tabularnewline\nCOCI \\& MAG (for Figshare) & contains publication year & 45,180,243 & 443,055,788\\tabularnewline\nCOCI \\& MAG (for GenBank) & publication year \\ensuremath{\\le} 2012 & 30,304,869 & 212,410,743\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\n\\subsubsection{Paper-dataset citation network}\n\\begin{enumerate}\n\\item \\textbf{Constructing paper-dataset citations for GenBank. }As previously\ndescribed by \\citet{Sayers2019}, authors should use GenBank accession\nnumber with the version suffix as identifier to cite a GenBank data.\nThe accession number is usually mentioned in the body of the manuscript.\nThis practice enables us to extract GenBank dataset mention from the\nPMC-OAS dataset to build the paper-dataset citation network. \n\\begin{enumerate}\n\\item \\emph{Parsing XML file to extract full-text}\\textbf{. }We first parse\nthe XML files using XPath expressions to extract the PMID and the\nfull-text. We get 2,174,782 articles with PMID and full-text from\nPMC-OAS dataset.\n\\item \\emph{Matching the accession number to build paper-dataset citations}.\nAccording to the GenBank Accession Prefix Format, an accession number\nis composed of a fixed-number of letters plus a fixed-number of numerals\n(e.g., 1 letter + 5 numerals, 2 letters + 6 numerals). Based on the\nformat, we composed a regular expression to match individual mentions\nof accession numbers in the full-text. There are two kinds of accession\nnumber mentions: accession number only (e.g., U00096) and range of\naccession number (e.g., KK037225-KK037232). For the second kind, we\nexpanded the range to recover all the omitted accession numbers.\n\\end{enumerate}\n\\end{enumerate}\n\\begin{enumerate}[resume]\n\\item \\textbf{Extracting paper-dataset citation for Figshare. }As there\nis a string 'figshare' in a Figshare DOI, we can use a regular expression\nto search for them in COCI. \n\\begin{enumerate}\n\\item \\textit{Identifying Figshare DOI in the set of cited DOI. }In this\nstep, we extracted 918 Figshare DOIs as dataset candidates.\n\\item \\textit{Filtering by resource type. }Not all the Figshare DOIs are\ndatasets because Figshare supports many kinds of resources. We use\nthe Figshare Article API\\footnote{https:\/\/docs.figshare.com\/\\#public\\_article}\nto get the meta-data of a DOI. In the meta-data, there is a field\nindicating the resource type (type code 3 is dataset). After filtering\nthe candidates by resource type, we get 355 datasets.\n\\end{enumerate}\n\\end{enumerate}\n\n\\subsubsection{Visualization of citation network}\n\nFor the purpose of exploring this network, we sample about one thousand\nnodes and 1.5 thousand edges from the network. We observe four patterns\nof papers citing datasets (a dataset cannot cite anything): one-to-one,\none-to-many, many-to-one and many-to-many (Fig. \\ref{fig:network-of-sample}).\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1\\textwidth]{F1A_sampleData_colored}\n\\par\\end{centering}\n\\caption{\\label{fig:network-of-sample} Visualization of publication--dataset\ncitation network. Papers appear as red nodes and Datasets are green.\nThe lighter the color, the older the resource.}\n\\end{figure}\n\n\n\\subsection{Network models for scientific artifacts}\n\n\\subsubsection{\\textsc{NetworkFlow}}\n\nWe adapt the network model proposed in \\citet{walker2007ranking}.\nThis method, which we call \\textsc{NetworkFlow} here, is inspired\nby \\textsc{PageRank} and addresses the issue that citation networks\nare always directed back in time. In this model, each vertex in the\ngraph is a publication. The goal of this method is to propagate a\nvertex's impact through its citations.\n\nThis method simulates a set of researchers traversing publications\nthrough the citation network. It ranks publications by estimating\nthe average path length taken by researchers who traverse the network\nconsidering the age of resources and a stopping criterion. Mathematically,\nit defines a traffic function $T_{i}(\\tau_{\\text{pub}},\\alpha)$ for\neach paper. A starting paper is selected randomly with a probability\nthat exponentially decays in time with a \\emph{decay} parameter $\\tau_{\\text{paper}}$.\nEach occasion the researcher moves, it can stop with a probability\n$\\alpha$ or continue the search through the citation network with\na probability $1-\\alpha$. The predicted traffic $T_{i}$ is proportional\nto the rate at which the paper is accessed. The concrete functional\nform is as follows. The probability of starting at the $i$th node\nis \n\\begin{equation}\n\\rho_{i}\\propto\\exp\\left(-\\frac{\\text{age}_{i}}{\\tau_{\\text{pub}}}\\right)\\label{eq:starting-probability}\n\\end{equation}\n\nThen,\\textsc{ }the method defines a transition matrix from the citation\nnetwork as follows \n\\begin{equation}\nW_{ij}=\\begin{cases}\n\\frac{1}{k_{j}^{\\text{out}}} & \\text{if \\ensuremath{j} cites \\ensuremath{i}}\\\\\n0 & \\text{o.w.}\n\\end{cases}\\label{eq:transition-probability}\n\\end{equation}\nwhere $k_{j}^{\\text{out}}$ is the out-degree of the $j$th paper.\n\nThe average path length to all papers in the network starting from\na paper sampled from the distribution $\\rho$ is defined as \n\\begin{equation}\nT=I\\cdot\\rho+(1-\\alpha)W\\cdot\\rho+(1-\\alpha)^{2}W^{2}\\rho+\\cdots\\label{eq:average-path-length}\n\\end{equation}\n\nThe parameters $\\tau_{\\text{paper}}$ and $\\alpha$ are found by cross\nvalidation by predicting real traffic. In practice, this series can\nbe solved iteratively by computing the difference between $T_{t+1}$\nand $T_{t}$. For this and all algorithms used in this work, we stop\nthe iterations when the total absolute difference in rank between\ntwo consecutive iterations is less than $10^{-2}$ which typically\nrequired around 30 iterations.\n\n\\subsubsection{\\textsc{DataRank}}\n\nIn this article, we extend\\textsc{ NetworkFlow} to accommodate different\nkinds of nodes. The extension considers that the probability of starting\nat any single node should depend on whether the node is a publication\nor a dataset. This is, publications and datasets may vary in their\nrelevance period. We call this new algorithm \\textsc{DataRank}. Mathematically,\nwe redefine the starting probability in Eq. \\ref{eq:starting-probability}\nof the $i$th node as \n\\begin{equation}\n\\rho_{i}^{\\text{DataRank}}=\\begin{cases}\n\\exp\\left(-\\frac{\\text{age}_{i}}{\\tau_{\\text{pub}}}\\right) & \\text{if \\ensuremath{i} is a publication}\\\\\n\\exp\\left(-\\frac{\\text{age}_{i}}{\\tau_{\\text{dataset}}}\\right) & \\text{\\text{if \\ensuremath{i} is a dataset}}\n\\end{cases}\\label{eq:data-rank-starting}\n\\end{equation}\n\nThis initialization process is depicted in Figure \\ref{fig:Initialize-the-value}.\nHere, datasets have a smaller decay than papers. The size of the bubble\nindicates the initial flow as defined by Eq. \\ref{eq:data-rank-starting}.\nAfter initializing, we can easily find that nodes of the same type\nand same age have the same value, and that the younger a node is,\nthe bigger its value.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.95\\textwidth]{F2_data_paper_arrow}\n\\par\\end{centering}\n\\caption{\\label{fig:Initialize-the-value}Nodes after initialization. The size\nof the node represents its value, different color means different\nnode type: red nodes are papers and green nodes are datasets.}\n\\end{figure}\n\nNow, we estimate the traffic in a similar fashion but now the traffic\nof a node $T_{i}^{\\text{DataRank}}(\\tau_{\\text{pub}},\\tau_{\\text{dataset}},\\alpha)$\ndepends on three parameters that should be found by cross-validation\nwith the rest of the components of the method remaining the same (e.g.,\n\\eqref{transition-probability} and \\eqref{average-path-length}).\n\n\\subsubsection{\\textsc{DataRank-FB}}\n\nIn \\textsc{DataRank}, each time the walker moves, there are two options:\nto stop with a probability $\\alpha$ or to continue the search through\nthe reference list with a probability $1-\\alpha$. However, there\nmay exist a third choice: to continue the search through papers who\ncite the current paper. In other words, the researcher can move in\ntwo directions: forwards and backwards. We call this modified method\n\\textsc{DataRank-FB. }In this method, one may stop with a probability\n$\\alpha-\\beta$, continue the search forward with a probability $1-\\alpha$,\nand backward with a probability $\\beta$. To keep them within the\nunity simplex, the parameters must satisfy $\\alpha>0$, $\\beta>0$,\nand $\\alpha>\\beta$,\n\nThen\\textbf{\\textsc{, }}we define another transition matrix from the\ncitation network as follows \n\\begin{equation}\nM_{ij}=\\begin{cases}\n\\frac{1}{k_{j}^{\\text{in}}} & \\text{if \\ensuremath{j} cited by \\ensuremath{i}}\\\\\n0 & \\text{o.w.}\n\\end{cases}\\label{eq:transition-probability-1}\n\\end{equation}\nwhere $k_{j}^{\\text{in}}$ is the number of papers that cite $j$.\nWe update the average path length to all papers in the network starting\nfrom $\\rho$ as\n\n\\begin{equation}\nT=I\\cdot\\rho+(1-\\alpha)W\\cdot\\rho+\\beta M\\cdot\\rho+(1-\\alpha)^{2}W^{2}\\rho+\\beta^{2}M^{2}\\rho+\\cdots\\label{eq:average-path-length-1}\n\\end{equation}\n\n\n\\subsection{Other network models}\n\n\\subsubsection{\\textsc{PageRank}}\n\n\\textsc{PageRank} is a well-known and widely used webpage ranking\nalgorithm proposed by Google \\citep{page1999pagerank}. It uses the\ntopological structure of the web to determine the importance of a\nwebpage, independently of its content \\citep{bianchini2005inside}.\nFirst, it builds a network of the web through the link between webpages.\nSecond, each webpage is assigned a random value, which is then updated\nbased on the link relationships---an iterative process that will\neventually converge to a stationary value.\n\nThe mathematical formulation of \\textsc{PageRank} is\n\n\\begin{equation}\nPR(p_{i})=\\frac{1-d}{N}+d\\underset{p_{j}\\in M(p_{i})}{\\sum\\frac{PR(p_{j})}{L(p_{j})}},\\label{eq:pagerank-algorithm}\n\\end{equation}\nwhere $p_{1},p_{2},\\cdots,p_{N}$ are the webpages whose importance\nneed to be calculated, $M(p_{i})$ is the set of webpages that has\na link to page $p_{i}$, $L(p_{j})$ is the number of outbound links\non webpage $p_{j}$, and $N$ is the total number of webpages. The\nparameter $d$ is a dampening factor which ranges from 0 to 1 and\nis usually set to 0.85 \\citep{brin1998anatomy,page1999pagerank}.\n\n\\subsubsection{Modified \\textsc{PageRank}}\n\nConsidering that we have two types of resources in the network, we\nmodify the standard \\textsc{PageRank} to allow a different damping\nfactor for publication and for dataset. This amounts to modifying\nthe update equation to\n\n\\begin{equation}\nPR(p_{i})=\\frac{1-d_{data}}{N_{data}}+\\frac{1-d_{pub}}{N_{pub}}+d_{data}\\underset{p_{j}\\in M^{data}(p_{i})}{\\sum\\frac{PR(p_{j})}{L(p_{j})}}+d_{pub}\\underset{p_{k}\\in M^{pub}(p_{i})}{\\sum\\frac{PR(p_{k})}{L(p_{k})}},\\label{eq:modified-pagerank}\n\\end{equation}\nwhere $p_{1},p_{2},\\cdots,p_{N}$ are still the nodes whose importance\nneed to be calculated, $M^{data}(p_{i})$ is the set of datasets that\nhave a link to page $p_{i}$, $M^{pub}(p_{i})$ is the set of papers\nthat has a link to page $p_{i}$, $L(p_{j})$ is the number of outbound\nlinks on webpage $p_{j}$, and $N_{data},N_{pub}$ are the total size\nof datasets and paper collections, respectively. The parameter $d_{data}$\nis a damping factor for datasets and $d_{pub}$ is a damping factor\nfor paper sets.\n\n\\section{Results}\n\nWe aim at finding whether the estimation of the rank of a dataset\nbased on citation data is related to a real-world measure of relevance\nsuch as page views or downloads. We propose a method for estimating\nrankings that we call \\textbf{\\textsc{DataRank,}} which considers\ndifferences in citation dynamics for publications and datasets. We\nalso propose some variants to this method, and compare all of them\nto standard ranking algorithms. We use the data of visits of GenBank\ndatasets and downloads of Figshare datasets as measure of real usage.\nThus, we will investigate which of the methods work best for ranking\nthem.\n\n\\subsection{Properties of the citation networks}\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{Citation_network_publication_age_prob_dist}\n\\par\\end{centering}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{Citation_network_received_citation_prob_dist}\n\\par\\end{centering}\n\\caption{\\label{fig:paper_network_age_citation_dist}Probability density of\nage and citations. Power laws describing the growth of both aspects\nof the network have parameters $k_{\\text{age}}\\approx-4.1$ and $k_{\\text{citations}}\\approx-2.16$.}\n\\end{figure}\n\nWe first examined some statistics of the paper--paper citation network\nand the paper--dataset citation network. Specifically, we examined\ndataset age because it determines the initial rank of a node and we\nexamine the citations because they control how ranks diffuse through\nthe network. We modeled publication age with respect to 2019 as a\npower law distribution $p(a)\\propto a^{-k}$ and we found the best\nparameter to be $k\\approx-4.1$ (SE=$0.07$, $p<0.001$, Fig. \\ref{fig:paper_network_age_citation_dist}\ntop panel). Similarly, the citation count can be modeled by a power\nlaw distribution $p(d)\\propto d^{-k}$ with parameter $k\\approx-2.16062$\n(SE=$0.012$, $p<0.001$, Fig. \\ref{fig:paper_network_age_citation_dist}\nbottom panel). While both distributions can be well described by power\nlaws, the age distribution had some out-of-trend dynamics for small\nage values because the number of publications is not growing as fast\nas the power law would predict. The networks thus is expanding fast\nin nodes (e.g., age) and highly skewed for citations.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{genebank_usage}\n\\par\\end{centering}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{figureshare_usage}\n\\par\\end{centering}\n\\caption{\\label{fig:pdf-of-figshare}Probability density function of usage\n(website visits for Genbank and downloads for Figshare)}\n\\end{figure}\n\nWe then wanted to examine differences in usage between Genbank and\nFigshare. We plotted the estimated probability density function of\nthis usage for both datasets as a log-log plot (Fig. \\ref{fig:pdf-of-figshare}).\nWhile the scale of usage is significantly different (i.e., overall\nGenBank is more used than Figshare), there seems to be a long-tail\npower law relationship in usage. The GeneBank dataset had a larger\nbut not significantly different scale parameter than Figshare ($k\\approx-1.416$,\nSE=0.2699, in GenBank and $k\\approx-1.125$, SE=0.0907, in Figshare,\nfor $p(u)\\propto u^{-k}$, two-sample t-test $t(220)=-1.05$, $p=0.29$).\nThere is a significant bias for downloads below a certain threshold,\nwhere smaller number of downloads are less frequent than expected\nby the power laws. However, both datasets show similar patterns suggesting\na common mechanism driving the dataset usage behavior.\n\n\\subsection{Prediction of real usage}\n\nOne of the real tests of whether the methods work is to predict how\nthey are related to real usage data. For each algorithm and set of\nparameters, we estimated the rank of networks' nodes. We then correlated\nthese ranks with real usage (i.e., web visits for GenBank and downloads\nfor Figshare). We now describe the best performance after these parameter\nsearch.\n\nWe performed a grid search with publication decay year $\\tau_{\\text{pub}}\\in\\left\\{ 1,5,10,20,30,50,70,100\\right\\} $,\nthe dataset decay year $\\tau_{\\text{dataset}}\\in\\left\\{ 1,5,10,20,30,50,70,100\\right\\} $,\nand alpha in $\\alpha\\in\\left\\{ 0.05,0.15\\right\\} $. The best model\nof \\textsc{DataRank} achieved a correlation of 0.3336 in Genbank which\nis slightly better than \\textsc{DataRank-FB} 0.3335, \\textsc{NetworkFlow}\n0.3327, \\textsc{PageRank} 0.3324, and Modified \\textsc{PageRank} 0.3327\n(Fig. \\ref{fig:result-correlation}). The best model of \\textsc{DataRank}\nachieves a correlation of 0.1078 in Figshare, which is substantially\nbetter than \\textsc{DataRank-FB} with 0.0723. The best absolute correlations\nby all the other methods only achieved negative correlations (\\textsc{NetworkFlow}\\textbf{\\textsc{=}}$-0.072$,\n\\textsc{PageRank}=$-0.073$, and Modified \\textsc{PageRank}=$-0.073$)---we\nselected models based on absolute correlation because it would produce\nthe best predictive performance. Taken together, \\textsc{DataRank}\nhas both better absolute correlation performance and highest correlation,\nsuggesting a superior ability to predict real usage.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=1\\textwidth]{correlation}\n\\par\\end{centering}\n\\caption{\\label{fig:result-correlation}Model comparison in correlation coefficient.\n\\textsc{DataRank} is able to predict better the real usage of datasets\ncompared to other variants.}\n\\end{figure}\n\n\n\\subsection{Interpretation of best-fit model}\n\nWe wanted to explore how \\textsc{DataRank} changes in performance\nacross the different parameters. This exploration allowed us to understand\nhow the parameters tell us something about the underlying characteristics\nof the citation networks.\n\nThe results across all parameters are shown in Figure \\ref{fig:Parameter-search}\nand we now describe general trends. For experiments on GenBank dataset,\nwe found the \\textsc{DataRank} model reaches its best with $\\tau_{\\text{pub}}=100$\nand $\\tau_{\\text{dataset}}=30$ (Fig. \\ref{fig:result-correlation}\ntop panel). However, for dataset decay, the performance reaches a\nplateau after around 20 years and it reaches its peak at 30 years.\nAfter 30 years, performance goes down slowly. In terms of publication\ndecay, the performance increases significantly before 20 years. After\n20 years, the performance enters into a steady but marginal increase.\nIn all the cases, the alpha at 0.05 is better than 0.15. For Figshare\ndataset, dataset decay has divergent patterns: for small publication\ndecays, dataset decay increases the performance. For large publication\ndecays, dataset decay decreases the performance. For publication decay,\nmore specifically, we observed an opposite trend compared to GenBank:\nthe performance goes down rapidly as the publication decay increases\n(Fig. \\ref{fig:result-correlation} bottom panel). However, similar\nto GenBank, it losses momentum after 20 years but still decreases\nsteadily.\n\nOne explanation for the differences between the best parameters for\nGenbank and Figshare is that dataset age and citation distribution\nhave different patterns. We perform an analysis to confirm this hypothesis.\nIndeed, we found that Figshare datasets are significantly younger\nthan Genbank datasets (bootstrapped difference between average age\n= $-4.80$ years, SE = 0.24, $p<0.001$). We also found that Figshare\ndataset citations are more uniform than Genbank dataset citations\nalthough not significantly different (bootstrapped difference in dataset\ncitation kurtosis = -44.34, SE = 48.74, $p=0.25$). The \\textsc{DataRank}\nalgorithm uses large values of dataset decay to propagate publication\ncitations over a longer time and small values of data decay to steer\nthose citations towards a concentrated set of top cited datasets.\nThis is the case of GenBank dataset dynamics. Figshare, comparatively,\nobeys opposite age and citation dynamics which \\textsc{DataRank} attempts\nto accommodate by using small values of publication decay and large\nvalues of dataset decays. Concretely, the best publication decay for\nGenbank is 100 and for Figshare is 1, and the best dataset decay for\nGenbank is 30 and for Figshare is 100. Taken together, \\textsc{DataRank}'s\nparameters offer inferential interpretation of the citation network\ndynamics, which can help us understand how they are distributed in\ntime and credit spaces.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{genebank_parameter_search}\n\\par\\end{centering}\n\\begin{centering}\n\\includegraphics[width=0.8\\textwidth]{figshare_parameter_search}\n\\par\\end{centering}\n\\caption{\\label{fig:Parameter-search}Model performance during grid search\nwith publication decay time (years) $\\tau_{\\text{pub}}\\in\\left\\{ 1,5,10,20,30,50,70,100\\right\\} $,\nthe dataset decay time (years) $\\tau_{\\text{dataset}}\\in\\left\\{ 1,5,10,20,30,50,70,100\\right\\} $,\nand alpha in $\\alpha\\in\\left\\{ 0.05,0.15\\right\\} $}\n\n\\end{figure}\n\n\n\\section{Discussion}\n\nThe goal of this article is to better evaluate the importance of datasets\nthrough article citation network analysis. Compared with the mature\ncitation mechanisms of articles, referencing datasets is still in\nits infancy. Acknowledging the long time the practice of citing datasets\nwill take to be adopted, our research aims at recovering the true\nimportance of datasets even if their citations are biased compared\nto publications.\n\nScholars disagree on how to give credit to research outcomes. Regardless\nof how disputed citations are as a measure of credit \\citep[see][]{moed1985use,diamond1986citation,martin1996use,seglen1997impact,wallin2005bibliometric},\nthey complement other measures that are harder to quantify such as\npeer review assessment \\citep{meho2007impact,piwowar2007sharing}\nor real usage such as downloads \\citep{belter2014measuring}. Citations,\nhowever, are rarely used for datasets \\citep{altman2013evolution},\ngiving these important research outcomes less credit that they might\ndeserve. Our proposal aims at solving some of these issues by constructing\na network flow that is able to successfully predict real usage better\nthan other methods. While citations are not a perfect measure of credit,\nhaving a method that can at least attempt to predict usage is advantageous.\n\nPrevious research has examined ways of normalizing citations by time,\nfield, and quality. This relates to our \\textsc{DataRank} algorithm\nin that we are trying to normalize the citations by year and artifact\ntype. Similar work has been done in patent citations: \\citet{hall2001nber}\nattempt to eliminate the effects caused by year, field and year-field\ninteraction through dividing the number of citation received by a\ncertain patent by the corresponding year-field average number of citation\nof patents of each cohort in each field. \\citet{yang2015using} used\npatent citation networks to propagate citations and evaluate patent\nvalue. Other researchers have gone beyond time and field by also attempting\nto control by quality of the resource. Clarivate Analytics editors\nconsider other normalizing factors such as publishing standards, editorial\ncontent and citation information when creating the Data Citation Index\n(DCI) database \\citep{reuters2012repository}. However, a control\nlike this requires significantly more manual curation.\n\nWe found that the practice of following who cites an artifact (i.e.,\nbackward propagation) seemed to be not as important compared to following\na cited artifact (i.e., forward propagation). This is, the more specialized\n\\textsc{DataRank-FB} model (see Methods) did not produce higher predictability.\nThis lack of improvement could suggest that when scientists traverse\nthe network of citations, they seem to only follow one direction of\nthe graph (i.e., the reference list). This could be perhaps a limitation\nof the tools available to scientists to explore the citation network\nor because backward propagation changes constantly year after year.\nInitiatives that force the creation of identifiers (DOI) for datasets,\nsuch as the NERC Science Information Strategy Data Citation and Publication\nproject \\citep{callaghan2012making}, might change this pattern.\n\nWe find it useful to interpret the effect of different decay times\non the performance of \\textsc{DataRank}. The best-fitting parameters\nshow that \\textsc{DataRank} attempts to model the network temporal\nand topological dynamics differently for Genbank and Figshare. Because\nGenbank tends to have older datasets that have more concentrated citations\ncompared to Figshare, the large value in publication decay time and\nsmall value in dataset decay time produces rank distributions that\nmatch the underlying dynamics of Genbank (Fig. \\ref{fig:Parameter-search}).\nSimilarly, the alpha parameter, which controls the probability of\nthe random walk to stop, is larger for Figshare, intuitively suggesting\nthat due to the smaller network size and shorter temporal paths, a\nhigher probability of stopping should be in place (Fig. \\ref{fig:Parameter-search}).\nThus, \\textsc{DataRank} can help to interpret dataset citation behavior.\n\nThere are some shortcomings in our study. For GenBank and Fighsare,\nwe do not have a great deal of information on actual usage. There\nare 1000 records but we only located 693 in the publication network\nof Genbank and 355 datasets from Figshare. Also, there are significant\ndifferences in the information, with Figshare dataset being cited\nless often than GenBank. In the future, we will explore other data\nrepositories such as PANGEA, Animal QTLdb, and UK Data Archive, and\nwe will request updates about web visits to other GenBank sequences.\n\nOpen research datasets have little means for systematic attribution\neven in well-resourced disciplines, such as the biomedical community,\nand our proposal attempts to systematize this attribution more broadly\nwithout requiring changes in behavior. The lack of systematic benchmarking\ntools prevents good policy-making. Furthermore, it contributes to\nthe invisibility of digital objects and labor, an issue of serious\nconcern \\citep{scroggins2019labor}. When digital objects are not\nwell described, they tend to disappear from view. They fall to the\nbottom of classic search engine results when there are no mechanisms\nto assign credit to datasets. This lack of indexing creates an asymmetrical\nrepresentation of some kinds of objects of science and can be an obstacle\nto quality evaluation and data reuse. Thus \\textsc{DataRank} can be\nused to correct these asymmetrical discrepancies between articles\nand datasets.\n\nOur approach does not directly estimate true impact but predicts one\npossible measure of usage. However, this prediction could be a foundational\nstep toward developing technical and theoretical models of impact.\nAlso, in the literature, \\textquotedblleft impact\\textquotedblright{}\nis defined differently depending on the goal \\citep{piwowar2013altmetrics}.\nMetrics estimating impact can be defined through \\textquotedblleft use\\textquotedblright ,\n\\textquotedblleft reuse\\textquotedblright , or \\textquotedblleft engagement\\textquotedblright ,\nand have a range of proxies. For example, a download might measure\nuse while in another context it may only indicate viewing. Comparatively,\nthe edifice of citation standards in the realm of journal articles\nis relatively well-established, with refinements to the interpretation\nof citation behavior such as the relative value of citations \\citep{stuart2017data},\nthe disciplinary context \\citep{borgman1989bibliometrics}, and the\nin-text location \\citep{teplitskiy2018almost}. Nonetheless, usage\nserves as a proximal indicator of influence, and a first-order approximation\nof the impact of the scientific work. Recent work in developing impact\nassessment tools continue development and refinement \\citep{silvello2018theory}.\nHowever, existing evaluation mechanisms often fail where there are\nno direct or indirect measure for usage, which can be an important\nindicator of scientific impact. Clearly, our approach serves not only\nas a concrete tool to measure impact---albeit imperfectly---but\nalso as a vehicle to discuss why and how to measure impact on datasets.\n\n\\section{Conclusion}\n\nUnderstanding how datasets are used is an important topic in science.\nDatasets are becoming crucial for the reproduction of results and\nthe acceleration of scientific discoveries. Scientists, however, tend\nto dismiss citing datasets and therefore there is no proper measurement\nof how impactful datasets are. Our method uses the publication-publication\ncitation network to propagate the impact to the publication--dataset\ncitation network. Using two databases of real dataset networks, we\ndemonstrate how our method is able to predict actual usage more accurately\nthan other methods. Our results suggest that datasets have different\ncitation dynamics to those of publications. In sum, our study provides\na prescriptive model to understand how citations interact in publication\nand dataset citation networks and gives a concrete method for producing\nranks that are predictive of actual usage.\n\nOur study advances an investigation of datasets and publications with\nnovel ideas from network analysis. Our work puts together concepts\nfrom other popular network flow estimation algorithms such as \\textsc{PageRank}\n\\citep{brin1998anatomy} and \\textsc{NetworkFlow} \\citep{walker2007ranking}.\nWhile scientists might potentially take a great amount of time to\nchange their citation behavior, we could use techniques such as the\none put forth in this article to accelerate the credit assignment\nfor datasets. Ultimately, the need for tracking datasets will only\nbecome more pressing and therefore we must adapt or miss the opportunity\nto make datasets first-class citizens of science.\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank the Dr. Kim Pruitt from National Center\nfor Biotechnology Information, NLM, NIH, DHHS. Tong Zeng was funded\nby the China Scholarship Council \\#201706190067. Sarah Bratt was partially\nfunded by National Science Foundation award \\#1561348. Daniel E. Acuna\nwas partially funded by the National Science Foundation awards \\#1800956.\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}