diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzojfo" "b/data_all_eng_slimpj/shuffled/split2/finalzzojfo" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzojfo" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThere is an ever-growing interest in single and few photon scattering from a one-dimensional (1D) continuum because of their possible applications in quantum information processing. This subject has been studied both theoretically \\cite{c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14,c15,c16,c17,c18,c19,c20,c21,c22,c23,c24,c25,c26,c27,c28,c29,c30,c31,c32,c33,c34,c35,c36,c37} and experimentally \\cite{c38,c39,c40,c41,c42,c43,c44,c45,c46,c47,c48} from a variety of perspectives and reviewed quite elaborately in Reference \\cite{c49}. Some commonly used 1D waveguides are conducting nanowires \\cite{c38,c39}, photonic crystal waveguides \\cite{c47}, and superconducting microwave transmission lines \\cite{c43,c44}. Collective effects emerging from a periodic array of two-level atoms coupled to a waveguide can lead to Fano minima in the reflection spectrum \\cite{c28,c29,c37}, superradiant decays \\cite{c47}, rearrangement of the optical band structure \\cite{c50} and realization of Bragg mirrors \\cite{c51,c52}. The role of spatial separation between the atoms in the context of photon scattering from a 1D continuum has been manifested in recent works \\cite{c5,c31,c32,c51,c52,c53,c54,c55,c56,c57}. Very recently, multiple Fano interference channels due to the waveguide-mediated phase coupling between the atoms leading to the appearance of transparency points were demonstrated in \\cite{c37}.\n\nTheoretical analysis of photon scattering from a collection of disparately detuned atoms coupled to a waveguide is a hard problem in real space, because of the waveguide mediated phase coupling between the atoms. Here, we address the collective behavior of such a system by constraining the spatial periodicity to be an integral or half-integral multiple of the resonance wavelength. Working at this phase substantially simplifies the problem and provides insight into the various kinds of collective properties of the output radiation. Since our interest is primarily in the realizability of induced transparency, we demonstrate the emergence of new Fano minima in the reflection spectrum for a system of even number of emitters via appropriate choice of detunings. For an odd chain size, the system manifests an effective re-emergence of single-atom behavior. These are new effects that were not observed in \\cite{c37}, as the atoms were all supposed to have identical transition frequencies. Our results are strong enough to withstand dissipative effects as long as the dissipative channel is weak compared to the waveguide channel and the atomic detunings are large compared to the decay rate. In other words, much like the phenomenon of EIT \\cite{harris1991,eitreview}, while perfect transparency emerges in an ideal dissipation-free scenario, a fairly high degree of transparency can still be observed in the presence of weak dissipation. However, in the usual scenario for EIT, the system comprises of non-interacting atoms, whereas in a waveguide, an effective phase coupling between the emitters persists even at long spatial separations (of the order of a wavelength). Another important feature that surfaces at this choice of phase is the reciprocity in optical transport, i.e. with regard to both the reflected and transmitted amplitudes. More generally, the transport is insensitive to the order in which the atoms are placed in the chain. This is starkly different from the usual scenario, where the phase coupling makes the system strongly non-reciprocal. Reciprocity relations for a large class of one-dimensional systems were derived in \\cite{rec1}. A general discussion on the subject of Lorentz reciprocity for lossless systems can be found in \\cite{recrev}.\n\nThe paper is organized as follows: in Sec. \\ref{s2}, we review the transport model for a single photon through a waveguide coupled to a chain of non-identical emitters. In Sec. \\ref{s3}, we demonstrate how one can observe transparent behavior for an atomic chain consisting of two disparately detuned atoms, when $kL$ is chosen to be an integral multiple of $\\pi$. In Sec. \\ref{s4}, we write down the analytical forms for the reflection and transmission coefficients for an arbitrary chain size and generalize some of the observations to the case of an even chain size. Concurrently, we indicate, for an odd chain size, how one can effect a complete suppression of collective effects so as to recover single-atom behavior. In Sec. \\ref{s5}, we argue that the commutativity between the transfer matrices leads to reciprocal transport properties Finally, in Sec. \\ref{s6}, we probe the effect of including radiative decay into nonwaveguide modes on the spectral characteristics, with particular emphasis on the observability of transparent behavior. Section \\ref{s7} summarizes the key ideas explored.\n\n\\section{Single-photon transport in a waveguide coupled to $N$ non-identical emitters} \\label{s2}\n \nFor a periodic 1-D array of $N$ two-level emitters evanescently coupled to a 1-D continuum (Fig. \\ref{f1}), when the atomic transition frequency far exceeds the waveguide cutoff frequency, one can write down the Hamiltonian of the system in real space as:\n\\begin{align}\n\\mathcal{H}=i\\hbar v_g\\int_{-\\infty}^{\\infty}\\dd{x}\\bigg(a_L^{\\dagger}(x)\\dfrac{\\partial a_L(x)}{\\partial x}-a_R^{\\dagger}(x)\\dfrac{\\partial a_R(x)}{\\partial x}\\bigg) \\notag \\\\+\\hbar\\sum_{j=1}^{N}(\\omega_j-i\\Gamma_0)\\ket{e}_j\\bra{e}+ \\notag \\\\ \\hbar\\sum_{j=1}^N\\bigg[\\{\\mathcal{J}a_L(x_j)+\\mathcal{J}a_R(x_j)+\\text{h.c.}\\}\\ket{e}_j\\bra{g}\\bigg],\n\\end{align}\nwhere $a_L(x)$ and $a_R(x)$ describe the real-space Bosonic operators corresponding to the left and the right propagating fields, $\\omega_j$ corresponds to the transition frequency of the $j^{\\text{th}}$ atom and $x_j=(j-1)L$ to its location along the waveguide, $v_g$ is the group velocity of the propagating photon and $\\mathcal{J}$ denotes the coupling strength between the propagating field and any of the emitters. $\\Gamma_0$ denotes the rate of spontaneous emission into all modes outside of the waveguide continuum, assumed equal for all the atoms. We disregard the dipole-dipole interaction (DDI) between the atoms by assuming the interatomic separation to be comparable or larger than the resonance wavelength.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.80]{f1.pdf} \n\\caption{\\small {Atomic array coupled to a waveguide.}} \\label{f1} \n\\end{figure}\n\nThe scattering eigenstate has the form\n\\begin{align}\n\\ket{\\mathcal{E}_k}=\\int_{-\\infty}^{\\infty}\\dd{x}\\bigg[\\phi_{kL}(x)a_L^{\\dagger}(x)+\\phi_{kR}(x)a_R^{\\dagger}(x)\\bigg]\\ket{\\Psi} \\notag\\\\\n+\\sum_{j=1}^Nc_k^{(j)}\\ket{0;e_j},\n\\end{align}\nwhere $\\ket{\\Psi}$ refers to the state with the field in vacuum and all atoms in the ground state, and $\\ket{0;e_j}$ to the one where the field is still in vacuum but only the $j^{\\text{th}}$ atom has been excited. $\\phi_{kL}$ and $\\phi_{kR}$ denote the wavefunctions corresponding to left- and right-propagating photonic excitations respectively. The Schr$\\ddot{\\text{o}}$dinger equation $\\mathcal{H}\\ket{\\mathcal{E}_k}=\\hbar v_g k\\ket{\\mathcal{E}_k} $ leads to a system of ODEs for the various probability amplitudes:\n\\begin{align}\n\\bigg(-iv_g\\dfrac{\\dd}{\\dd x}-v_gk\\bigg)\\phi_{kR}(x)+\\mathcal{J}\\sum_{j=1}^{N}c_k^{(j)}\\delta(x-x_j)&=0, \\notag\\\\\n\\bigg(iv_g\\dfrac{\\dd}{\\dd x}-v_gk\\bigg)\\phi_{kL}(x)+\\mathcal{J}\\sum_{j=1}^{N}c_k^{(j)}\\delta(x-x_j)&=0, \\notag\\\\\n(\\Omega_j-v_gk)c_k^{(j)}+\\mathcal{J}\\phi_{kL}(x_j)+\\mathcal{J}\\phi_{kR}(x_j)&=0,\n\\end{align}\n\nwhere $\\Omega_j=\\omega_j-i\\Gamma_0$. For a wave incident from the left, one can solve these ODEs subject to the boundary condition that $\\phi_{kL\/kR}(x_j)=\\frac{1}{2}[\\phi_{kL\/kR}(x_j^+)+\\phi_{kL\/kR}(x_j^-)]$ and also the discontinuity imposed on the wavefunctions due to the the delta-function source, i.e. $-iv_g[\\phi_{kR}(x_j^+)-\\phi_{kR}(x_j^-)]+\\mathcal{J}c_k^{(j)}=iv_g[\\phi_{kL}(x_j^+)-\\phi_{kL}(x_j^-)]+\\mathcal{J}c_k^{(j)}=0$. The solutions take the form\n\n\\begin{align}\n\\phi_{kL}(x)&=\\begin{cases} r_1e^{-ikx}, & x<0 \\\\\nr_{j+1}e^{-ik(x-jL)}, & (j-1)L\n(N-1)L\n\\end{cases}\\hspace{1mm},\n\\end{align}\n\n\\begin{align}\n\\phi_{kR}(x)&=\\begin{cases} e^{ikx}, & x<0 \\\\\nt_{j}e^{ik(x-jL)}, & (j-1)L(N-1)L\n\\end{cases}\\hspace{1mm},\n\\end{align}\n\nwhich subsequently lead to a system of simultaneous equations involving the transmission and reflection coefficients and the atomic excitation amplitudes. Eliminating the excitation amplitudes from the system engenders in a recursive matrix relation\n\n\\begin{align}\n\\begin{bmatrix}\nr_j\\\\\nt_{j-1}\\\\\n\\end{bmatrix}=\\mathcal{L}_j\\begin{bmatrix}\nr_{j+1}\\\\\nt_j\\\\ \\end{bmatrix},\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{L}_j=\\begin{bmatrix}\ne^{ikL}(1-i{\\delta^{-1}_{k (j)}}) & -ie^{-ikL}\\delta^{-1}_{k (j)}\\\\\nie^{ikL}\\delta^{-1}_{k (j)} & e^{-ikL}(1+i\\delta^{-1}_{k (j)}) \\\\\n\\end{bmatrix}, \\label{E1}\n\\end{align}\n\nwith ${\\delta_{k (j)}}=\\dfrac{\\omega_{k}-\\Omega_j}{\\Gamma}$ and $\\Gamma=\\dfrac{\\mathcal{J}^2}{v_g}$ . Iterative use of this relation, conjoined with appropriate boundary conditions, yields the reflection and transmission coefficients:\n\\begin{align}\nr=\\dfrac{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{12}}{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{22}}, \\hspace{5mm}t=\\dfrac{e^{-ikNL}}{\\bigg(\\prod_{j=1}^N\\mathcal{L}_j\\bigg)_{22}}. \\label{E2}\n\\end{align}\n\nBecause of the presence of the phase factors $e^{\\pm ikL}$ and the differential detunings assigned to the emitters, it is analytically hard to find out compact expressions for the above. The matrix product, however, becomes simple to evaluate for $kL=n\\pi$, $n$ being a natural number. \n\n\\section{Transparency and collective behavior for a two-atom system} \\label{s3}\n\nLet us first consider the simpler scenario of two differentially detuned atoms in a waveguide and define the mean laser detuning $\\overline{\\Delta}=\\omega_k-\\frac{1}{2}(\\omega_1+\\omega_2)$ and relative atomic detuning $s=\\omega_1-\\omega_2$. It turns out that while $[\\mathcal{L}_1\\mathcal{L}_2]_{22}$ is symmetric in $s$, $[\\mathcal{L}_1\\mathcal{L}_2]_{12}$ is not, owing to the phase coupling between the emitters mediated by the waveguide. In other words, in view of Eq. \\ref{E2}, even though transmission is perfectly reciprocal, reflection is not. For this system, the reflection and transmission coefficients reduce to\n\\begin{align}\nr&=\\dfrac{-i\\Gamma[(e^{i\\alpha}+1)(\\overline{\\Delta}+i\\Gamma_0)-(e^{i\\alpha}-1){s}\/{2}]-\\Gamma^2(e^{i\\alpha}-1)}{[\\overline{\\Delta}+i(\\Gamma+\\Gamma_0)]^2+\\Gamma^2e^{i\\alpha}-(s\/2)^2}, \\notag\\\\\nt&=\\dfrac{(\\overline{\\Delta}+i\\Gamma_0+s\/2)(\\overline{\\Delta}+i\\Gamma_0-s\/2)}{[\\overline{\\Delta}+i(\\Gamma+\\Gamma_0)]^2+\\Gamma^2e^{i\\alpha}-(s\/2)^2}, \\label{E4}\n\\end{align}\n\nwhere we have taken $\\alpha=2kL$. We note, however, in the special case $kL=n\\pi$, that the above expressions turn symmetric in $s$, thereby leading to reciprocity in the transport properties. In subsequent considerations, we analyse the results pertaining to this special choice of phase.\n\nWe also assume an idealized scenario where $\\Gamma_0$ can be ignored. The effect of $\\Gamma_0$ on the transmission is studied later in Sec. \\ref{s6}. With $\\Gamma_0$ set to $0$, one observes an emergence of transparent behavior when the two atoms are equally detuned, albeit in opposite directions, with respect to the laser frequency. In other words, $t$ becomes unity when $\\overline{\\Delta}$ equals zero, or $\\omega_k-\\omega_1=\\omega_2-\\omega_k$, i.e. for a pair of antisymmetrically detuned emitters. It follows, from Eq. \\ref{E4} and the plots in Fig. \\ref{figure2}, that there is a transmission peak at $\\overline{\\Delta}=0$, while there are two roots of the profile at $\\overline{\\Delta}=\\frac{s}{2}$ (or $\\omega_k=\\omega_1$) and $\\overline{\\Delta}=-\\frac{s}{2}$ (or $\\omega_k=\\omega_2$) corresponding to perfect reflection. The peak has unit height in the absence of decay, signifiying transparent behavior. The height of this peak is strictly less than unity for any other choice of phase, as can be verified from \\ref{E4} (for instance, in the specific scenario, when $kL=\\frac{n\\pi}{2}$ with odd $n$, the height of this peak is $\\bigg[\\dfrac{s^2}{s^2+8\\Gamma^2}\\bigg]^2$ - see Fig. \\ref{f3}). For a sufficiently small yet non-zero value of $\\abs{\\omega_1-\\omega_2}$, one finds a very narrow window of size $s$ over which the system is capable of demonstrating both opacity as well as transparency. As $s\\rightarrow 0$, the two roots come progressively closer. Fig. \\ref{figure2} illustrates this scenario for various choices of $\\abs{\\omega_1-\\omega_2}$.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f2-eps-converted-to.pdf} \n\\caption{\\small {Transmission for a two-atom system without decay for a couple of values of $s=\\omega_1-\\omega_2$ and with $kL=n\\pi$. Perfect transparency is observed at $\\omega_k=\\frac{1}{2}(\\omega_1+\\omega_2)$ (zero mean detuning), unless $\\omega_1=\\omega_2$, in which case, the system is perfectly reflecting at zero detuning. The two roots of the transmission come closer as the atomic frequencies approach each other.}} \\label{figure2} \n\\end{figure}\n\nWe also note, in passing, that the poles of the transmission and reflection demonstrate features remindful of level attraction. These poles occur at \n\\begin{align}\n\\overline{\\Delta}_{\\pm}^{(\\text{p})}=-i\\Gamma\\pm\\sqrt{\\bigg(\\frac{s}{2}\\bigg)^2-\\Gamma^2}\\hspace{0.5mm}.\n\\end{align}\nLevel attraction is typically observed between the normal modes of two coupled oscillators when one of the bare modes has negative energy and the modes have comparable decay rates. When the coupling strength equals or exceeds a critical value, the level separation vanishes. As a direct analogy, we see, in our case, that the real parts of the transmission poles coincide and become $0$ in the region $\\frac{s}{2}\\leq\\Gamma$, while the imaginary parts expand and shrink respectively. The point of transition where the coupling equals this critical value is referred to as an exceptional point where the complex eigenfrequencies coincide \\cite{E1}. Realizing level attraction has been quite a challenge from an experimental perspective and consequently, there is burgeoning interest in level attraction and phenomena occuring in the vicinity of exceptional points. Recently, level attraction has been observed in a variety of systems and topological behavior around an exceptional point has also been explored \\cite{L1,L2,L3,L4,L5,L6,L7}.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f3-eps-converted-to.pdf} \n\\caption{\\small {Comparison of transmission spectra corresponding to $kL=\\pi$ and $kL=\\frac{\\pi}{2}$, with $s=1.5\\Gamma$. The transmission peak attains unit height for $kL=\\pi$ whereas it is much shorter than unity for $kL=\\frac{\\pi}{2}$.}} \\label{f3} \n\\end{figure}\n\nIn the waveguide case with two atoms, we see that the transmission has zeros at\n\\begin{align}\n\\overline{\\Delta}_{\\pm}^{(\\text{r})}=\\pm\\frac{s}{2},\n\\end{align} \nwhich simultaneously determine the peaks of the reflection spectrum. These zeros are close to the the real parts of the poles when $\\frac{s}{2}\\gg\\Gamma$. In the complementary regime, when $\\frac{s}{2}$ is comparable to $\\Gamma$, the real parts of the poles become small compared to the respective imaginary parts, as a consequence of which, the resolution between the two levels (or the two poles) becomes difficult. This problem of resolution arises fundamentally because $\\Gamma$ not only appears in the discriminant of the poles, but also acts as a natural broadening term.\\\\\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.65]{f4-eps-converted-to.pdf} \n\\caption{\\small {Transmission at $kL=n\\pi$ for $\\frac{s}{2}=\\Gamma-\\eta$ with $\\eta=0.1\\Gamma$ and $\\eta=0.25\\Gamma$.}} \\label{f4} \n\\end{figure}\n\nThe shrinking of the transmission width as $\\frac{s}{2}$ goes below $\\Gamma$ is clearly reflected in the transmission plots shown in Fig. \\ref{f4}. Letting $\\frac{s}{2}=\\Gamma-\\eta$, where $\\frac{\\eta}{\\Gamma}\\sim 10^{-1}$, we find that the pole $\\overline{\\Delta}_{+}^{(\\text{p})}$ shrinks in width, with the relevant width given by $\\bigg(1-\\sqrt{\\frac{2\\eta}{\\Gamma}}\\bigg)\\Gamma$. Thus, the transmission window becomes narrower as $\\eta$ becomes larger.\n\n\\section{Transparency in a multiatom chain} \\label{s4}\n\nWe now bring out some interesting features of the spectral behavior collectively induced by a chain of multiple emitters with non-identical detunings, corresponding to a spatial periodicity so chosen that $kL=n\\pi$. In Ref. \\cite{c37}, the analytical expressions for the spectral amplitudes were derived for a system of identical emitters for which the transfer matrices were also identical. Through a diagonalization procedure, the matrix product was calculated. However, $kL=n\\pi$ is an exceptional point of the transfer matrices, since the eigenvalues coincide and become $(-1)^n$. Hence, diagonalization fails. However, in the special case when $kL$ is an integral multiple of $\\pi$, one can derive exact analytical expressions for the relevant matrix product in \\ref{E2}:\n\\begin{align}\n\\prod_{j=1}^N\\mathcal{L}_j=(-1)^{nN}\\begin{bmatrix}\n1-i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)} & -i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)}\\\\\ni\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)} & 1+i\\Gamma\\sum_{j=1}^{N}\\delta^{-1}_{k (j)}\n\\end{bmatrix}.\n\\end{align}\n\nIt is easy to check this result for $N=2$, and the general result for arbitrary $N$ can be be verified using the procedure of mathematical induction. We then have the following transmission and reflection coefficients:\n\n\\begin{align}\nt=\\frac{1}{1+i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}\\hspace{1mm}, \\notag\\\\\nr=-\\frac{i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}{1+i\\Gamma\\sum_{j=1}^{N}(\\omega_k-\\omega_j+i\\Gamma_0)^{-1}}\\hspace{1mm}.\\label{E5}\n\\end{align}\n\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.70]{f5.pdf} \n\\caption{\\small {Even number of emitters with equal and opposite detunings assigned in pairs generates transparency. The order of the atoms is not important, so the arrangement shown here is just one of the possible permutations.}} \\label{f5} \n\\end{figure}\n\nThe collective effects due to emission from multiple periodically spaced emitters is clearly embodied in the aforementioned expressions. The spectral dependence on the detunings has a close resemblance with that in the single-emitter scenario. The key factor that modifies the spectrum is $\\sum_{j=1}^{N}(\\omega_k-\\Omega_j)^{-1}$, an additive effect of the inverse detunings pertaining to the individual emitters. The expression is, of course, not as simple for other choices of phase. As a further simplification, let us focus on the case where $\\Gamma_0$ is small enough to be dropped from consideration. This, in principle, entails in the possibility of generating perfect transmission through suitable arrangements of the individual detunings. The condition for transparency ($r=0$ and $t=1$)is given by\n\n\\begin{align}\n\\sum_{j=1}^{N}\\frac{1}{\\Delta_{k(j)}}=0.\n\\end{align}\n\nwhere $\\Delta_{k(j)}=\\omega_k-\\omega_j$ is the laser detuning relative to the transition frequency of the $j^{\\text{th}}$ atom. For a single emitter, this relation is clearly impossible to satisfy and therefore, a single atom in a waveguide does not lead to transparent behavior. One must have multiple atoms to enable the achievement of transparency. In a double-emitter scenario, where $N=2$, the condition translates to $2\\omega_k=\\omega_1+\\omega_2$, which implies an exactly antisymmetric assignment of detunings to the two emitters. This is in line with what was highlighted in the previous section dedicated to the study of a two-atom chain (see Fig. \\ref{f2}). \n\nFor $N=3$, the corresponding constraint appears as a quadratic equation\n\n\\begin{align}\n3\\omega_k^2-2(\\omega_1+\\omega_2+\\omega_3)\\omega_k+\\omega_1\\omega_2+\\omega_2\\omega_3+\\omega_3\\omega_1=0,\n\\end{align}\nwith roots given by \n$$\\frac{1}{3}\\bigg[\\omega_1+\\omega_2+\\omega_3 \\pm\\sqrt{\\omega_1^2+\\omega_2^2+\\omega_3^2-\\omega_1\\omega_2-\\omega_2\\omega_3-\\omega_3\\omega_1}\\bigg].$$ The discriminant can be re-expressed as $\\frac{1}{2}[(\\omega_1-\\omega_2)^2+(\\omega_2-\\omega_3)^2+(\\omega_3-\\omega_1)^2]$, which, being a sum of squares, is strictly non-negative, and hence, the roots are real.\n\n In general, it is easy to see that for even number of emitters in the chain, it is always possible to make the system transparent if the atomic transition frequencies can be so adjusted that for any atom detuned by a certain amount, there exists another atom in the chain detuned by the same amount but in the opposite sense (Fig. \\ref{f5}). In other words, for a chain of $2l$ atoms, an assignment of the frequency detunings $+\\Delta_1$, $-\\Delta_1$, $+\\Delta_2$, $-\\Delta_2$,... $+\\Delta_l$, $-\\Delta_l$, in no particular order, would give rise to transparency in the system. Such an asymmetric pairwise assignment of detunings lead to Fano minima in the reflection spectrum, which signifies a destructive interference between the reflected waves emanating from the emitters. Concomitantly, the transmitted waves constructively interfere, leading to perfect transmission. This extreme resonant inhibition of the reflection amplitude relative to the single-atom emission is a new phenomenon that is not observed at $kL=n\\pi$ for a system of identically detuned emitters. \n\nOn the other hand, if one has odd number of emitters in the chain, one can recover the single-atom emission spectra by assigning pairwise asymmetric detunings to any randomly chosen $(N-1)\/2$ emitter pairs, leaving out a single atom. It then follows from Eqs. \\ref{E5}, that the remaining atom completely determines the spectral characteristics. That is, if this particular atom has a transition frequency $\\omega_0$, the transmitted spectrum due to the entire atomic chain reduces simply to\n\\begin{align}\nt=\\frac{1}{1+i\\Gamma(\\omega_k-\\omega_0)^{-1}}\\hspace{1mm},\n\\end{align}\n\nwhich is identical to the transmission coefficient with just that single atom coupled to the waveguide (Fig. \\ref{f6}). Stated differently, when an even number of atoms with a pairwise asymmetric assignment of frequency detunings are added, in a periodic fashion, to a single atom coupled to a waveguide, no discernible collective effects emerge. The order of this arrangement and therefore, the location of the odd atom are not important. This makes sense from the perspective of Fano interference, since the reflected waves from the appended atoms destructively interfere, while that from the residual atom effectively goes through unperturbed. As a consequence, if the odd atom is in resonance with the laser frequency, the system resembles a perfectly reflecting mirror, regardless of the frequency detunings of the other atoms.\n\n\\begin{figure}\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.80]{f6.pdf} \n\\caption{\\small {A system of three atoms, out of which two carry equal and opposite detunings $+\\Delta$ and $-\\Delta$. The odd one out (the middle one, in this figure) with a detuning of $\\bm{\\Delta_0}$ determines the spectral behavior, and no collective effects exist. This behavior transcends to the case of any odd number of emitters in the chain with a commensurate assignment of frequency detunings. When $\\bm{\\Delta_0}$$=0$ , the system behaves as a perfectly reflecting mirror.}} \\label{f6} \n\\end{figure}\n\n\nFinally, in the event that all the atoms have identical frequencies, one can observe Dicke-type superradiant behavior due to enhancement of the reflection amplitude. If the atomic frequencies are all set equal to $\\omega_0$, the corresponding transmitted spectrum is obtained to be\n\n\\begin{align}\nt=\\frac{1}{1+iN\\Gamma(\\omega_k-\\omega_0)^{-1}}\\hspace{1mm},\n\\end{align}\n\n\nwhich pertains to a Lorentzian with a half-width of $N\\Gamma$. This superradiant behavior observed for a collection of identical emitters, at $kL=n\\pi$, was also derived in \\cite{c37}. However, the possibility of controlling collective effects by tuning the individual atomic frequencies was not explored in that work. Having this added flexibility of adjusting the atomic frequencies brings out different types of interesting radiant behavior that one can observe, in principle. We do not merely encounter the possibility of superradiant reflection, but also come across new points of transparency. In particular, if we have an even number of emitters coupled to the waveguide, a pairwise asymetric allotment of detunings generates new Fano minima. For an odd chain size, we see how a similar assignment of frequencies to any $N-1$ of the atoms can lead to a complete disappearance of collective behavior and give rise to a spectrum governed entirely by the transition property of the leftover atom.\n\n\\section{Reciprocal behavior for $kL=n\\pi$} \\label{s5}\n\nAs has been discussed previously, in the context of a two-atom system, reciprocity entails for the choice of phase $kL=n\\pi$. It follows, quite generally, from the expressions in \\ref{E5}, that $kL=n\\pi$ ensures perfect optical reciprocity for any arbitrary chain size. In fact, this choice of phase is both necessary and sufficient for reciprocity in both the reflection and the transmission. The fundamental property that brings about this reciprocal character is the commutativity between any two transfer matrices, i.e.\n\\begin{align}\n[\\mathcal{L}_j,\\mathcal{L}_k]=0\\hspace{2mm} \\forall \\hspace{1mm}j,k \\label{E3}.\n\\end{align}\n\nThe commutation relation follows from the form in \\ref{E1}. As an essential implication of this, one finds that the matrix product $\\prod_{j=1}^N\\mathcal{L}_j$ is insensitive to the order in which the individual matrices are multiplied. Consequently, no matter what the order of the atoms is, one has the same transmission and reflection coefficients. Of course, this result is valid under the assumption that each emitter couples identically to the left as well as the right-propagating fields, as far as the two coupling strengths are concerned. It is easy to verify that Eq. \\ref{E3} holds true for any arbitrary assisgnment of detunings if and only if $kL=n\\pi$.\n\n\\section{Effect of dissipation into nonwaveguide modes on transparency} \\label{s6}\n\nThe simplistic results laid out in the preceding discussions in Secs. \\ref{s3} and \\ref{s4} hold only when $\\Gamma_0$ is small enough to be ignored. However, we can look at more realistic scenarios with dissipation included ($\\Gamma_0\\neq 0$) and examine the effect of the same on those observations. For a two-atom chain, we discover that the behavior changes drastically depending on how the relative detuning between the atomic frequencies compares to this decay rate. Fig. \\ref{f7} shows the plots for $\\abs{t}^2$ vs. $\\abs{\\frac{\\overline{\\Delta}}{\\Gamma}}$ for a good-quality waveguide with a weak dissipative channel ($\\Gamma_0=0.1\\Gamma$), for varying values of $\\abs{\\omega_1-\\omega_2}$. It is observed that for sufficiently small values of the latter, the transmission peak almost disappears, whereas for large values, the height of the peak approaches unity. In other words, by adjusting the relative frequency detuning between the emitters, one can achieve either high opacity or high transparency around $\\omega_k=\\frac{1}{2}(\\omega_1+\\omega_2)$. For perfectly matched up atomic frequencies, i.e. $\\omega_1=\\omega_2$, one observes a diametrically opposite behavior as the two roots coincide - the central peak is replaced by a trough. This is a Dicke-type superradiant effect - for negligible decay, the transmission profile is a vertically inverted Lorentzian with a half-width of $2\\Gamma$.\n\n\\begin{figure}[!t]\n \\captionsetup{justification=raggedright,singlelinecheck=false}\n\\centering\n\\includegraphics[scale=0.70]{f7-eps-converted-to.pdf} \n\\caption{\\small {Effect of dissipation on the transmission of a two-atom system. If the dissipative channel is weak compared to the waveguide channel, the profile closely resembles the dissipation-free spectrum, except when the frequency mismatch between the atoms is smaller than or comparable to the rate of dissipation.}} \\label{f7} \n\\end{figure}\n\nOne can analytically understand this behavior by considering two specific regimes, (i) $s\\ll 2\\Gamma_0$ and (ii) $s\\gg 2\\Gamma_0$. At $\\overline{\\Delta}=0$, one obtains\n\n\\begin{align}\nt=\\frac{(\\frac{s}{2})^2+\\Gamma_0^2}{(\\frac{s}{2})^2+\\Gamma_0(\\Gamma_0+2\\Gamma)}\\hspace{1mm}.\n\\end{align}\n\nFor small relative detuning between the atoms, i.e. $s\\ll 2\\Gamma_0$, the approximate form is given as $\\abs{t}^2\\approx \\frac{\\Gamma_0^2}{4\\Gamma^2}$, which is vanishingly dimunitive, as long as the decay rate is much less than $\\Gamma$. In the opposite scenario when $s\\gg 2\\Gamma_0$, we get $\\abs{t}^2\\approx 1-\\mathcal{O}(\\frac{8\\Gamma\\Gamma_0}{\\delta^2})$. Thus, a fairly high degree of transparency can be achieved by specifically working with a large relative detuning $\\abs{\\omega_1-\\omega_2}$.\n\nFor a system of even number of emitters, in which half of them have detuning $+\\Delta$ whereas the other half have detuning $-\\Delta$, the transmission goes as\n\n\\begin{align}\nt=\\frac{\\Delta^2+\\Gamma_0^2}{\\Delta^2+\\Gamma_0(\\Gamma_0+N\\Gamma)}\\hspace{1mm}.\n\\end{align}\n\nFor $\\Delta\\ll\\Gamma_0$, $\\abs{t}^2\\approx (\\frac{\\Gamma_0}{N\\Gamma})^2$, which testifies to highly reflecting behavior. However, when $\\Delta\\gg \\Gamma_0$, we have $\\abs{t}^2\\approx 1-\\mathcal{O}(\\frac{4N\\Gamma\\Gamma_0}{\\Delta^2})$, implying near-transparency. The situation here is reminiscent of EIT where perfect transparency emerges in the absence of dissipative transitions \\cite{harris1991,eitreview}.\n\nSimilarly, when there are odd number of emitters, with $(N-1)\/2$ emitters each having a detuning of $+\\Delta$ and $(N-1)\/2$ other emitters each detuned by $-\\Delta$, one has, for the transmission coefficient\n\n\\begin{align}\nt=\\frac{1}{1+\\frac{(N-1)\\Gamma\\Gamma_0}{\\Delta^2+\\Gamma_0^2}+\\frac{i\\Gamma}{\\omega_k-\\omega_0+i\\Gamma_0}}\\hspace{1mm},\n\\end{align}\n\nwhere $\\omega_0$ is the frequency of the remaining atom. When $\\Delta$ is large compared to $\\Gamma_0$, one can discern a re-emergence of single-atom behavior.\n\n\\section{Conclusions} \\label{s7}\n\nTo summarize, we have thrown light on new possibilities that emerge in relation to the collective effects of a chain of atoms side-coupled to a waveguide when the interemitter separation is fixed to satisfy $kL=n\\pi$, where $n$ is an integer. For a chain of $N$ atoms, we have demonstrated the emergence of new Fano minima (transparency points) in the reflection spectrum for negligible dissipation. When $N$ is even, we have seen how transparency can be generated by assigning equal and opposite detunings to the atoms in pairs, while for odd $N$, we have highlighted the possibility of reproducing single-atom behavior through a similar assignment, so that the odd one out completely determines the emission spectrum. A system of identically detuned emitters, on the other hand, demonstrates superradiant behavior. We have also shown that the optical system demonstrates reciprocal behavior with respect to both transmission and reflection. In general, the system turns out to be insensitive to the order in which the atoms are arranged. Finally, it has been demonstrated, both analytically and graphically, that when dissipation into nonwaveguide modes cannot be neglected, one can still produce highly transparent behavior by implementing a considerable disparity in the atomic transition frequencies. For a small mismatch in the frequencies, one, however, observes predominantly opaque behavior in its place.\\\\\n\n\n\\begin{acknowledgments}\n\nD. M. is supported by the Herman F. Heep and Minnie Belle Heep Texas A\\&M University Endowed Fund held\/administered by the Texas A\\&M Foundation.\n\n\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{I.Introduction}\n\\baselineskip 0.32in\n\\indent Gravity in 1+2 dimensional spacetime has been a\npopular subject of\ndiscussion from a decade ago\\cite{s1}-\\cite{s6}. Though some\nof the models\nare toy models, the studies may shed light on the understanding\nof not\nonly quasi-(1+2) dimensional physics (e.g. QHE and high $T_{c}$),\nbut also\nthe realistic 1+3 dimensional gravity. In any gauge theory in\nodd dimensions,\nthere exists a special term, i.e. the Chern-Simons term,\nwhich can be\nincorporated into the model Lagrangian. The concept of\ngravitational anyons\nis a simple non-Abelian generalization of the U(1)\nChern-Simons theory to\nnon-compact gauge group\\cite{s1}\\cite{s2}. Using his sloution\nto the\nlinearized field equations, Deser studied the mass and spin of the\ngravitational anyons\\cite{s4}. The conclusion states that the\ngravitational\nand inertial quantities are not equal to each other in general and\nthus\nthe equivalence principle is violated. Hence, there exist much\ndifference\nbetween gravitational anyons and 1+3 Einstein gravity.\\\\\n\\indent It seems to us that in order to understand the difference,\nwe should\nhave a well-defined definition of gravitational conservative\nquantities, or\nin other words, we should have generally covariant conservation\nlaws. In\nour previous work\\cite{s7}, we have obtained a generally covariant\nconservation law of angular-momentum for gravitational anyons.\nAs suggested\nin\\cite{s7}, the present paper is to study the generally covariant\nconservation law of energy-momentum in the approach proposed\nin\\cite{s8}.\nThe paper is arranged as follows. In section II, we give a general\ndescription of the scheme for establishing generally covariant\nconservation\nlaws in general relativity. In section III, we use the general\ndisplacement\ntransform and the scheme to obtain a generally covariant\nconservation law\nof energy-momentum. In this section, we will use a first order\nLagrangian\ninstead of the original one which is in second order. In section\nIV, we\ncalculate the total energy-momentum for Deser's,\nCle$\\acute{e}$ment's\nsolutions, and a solution in the vanishing Chern-Simons coupling\nlimit.\nThe last section is devoted to some remarks and discussions.\\\\\n\\section*{II. General Scheme for Conservation Laws in General\nRelativity}\n\\indent As in 1+3 Einstein gravity, conservation laws are also\nthe consequence\nof the invariance of the action corresponding to some transforms.\nIn order\nto study the covariant energy-momentum of more complicated systems,\nit is\nbenifecial to discuss conservation laws by Noether theorem in general.\nSuppose that the spacetime is of dimension $D=1+d$ and the Lagrangian\nis in the first order formalism, i.e.\n\\begin{equation}\nI=\\int_{G}{\\cal L}(\\phi^{A}, \\partial_{\\mu}\\phi^{A})d^{D}x\n\\end{equation}\nwhere $\\phi^{A}$ denotes the generic fields. If the action is invariant\nunder the infinitesimal transforms\n\\begin{equation}\nx^{\\prime\\mu}=x^{\\mu}+\\delta x^{\\mu}\\,\\,\\,\\,\\,\\,\\,\\, \\phi^{\\prime A}\n(x^{\\prime})=\\phi^{A}(x)+\\delta\\phi^{A}(x)\n\\end{equation}\n (it is not required that $\\delta\\phi^{A}_{\\mid\\partial G}=0$), then following relation\nholds\\cite{s8}-\n\\cite{s10}.\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\delta_{0}\\phi^{A}\n)+[{\\cal L}]_{\\phi^{A}}\\delta_{0}\\phi^{A}=0\n\\end{equation}\nwhere\n\\begin{equation}\n[{\\cal L}]_{\\phi^{A}}=\\frac{\\partial{\\cal L}}{\\partial\\phi^{A}}-\\partial_{\\mu}\n\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\end{equation}\nand $\\delta_{0}\\phi^{A}$ is the Lie variation of $\\phi^{A}$\n\\begin{equation}\n\\delta_{0}\\phi^{A}=\\phi^{\\prime A}(x)-\\phi^{A}(x)=\n\\delta\\phi^{A}(x)-\\partial_{\\mu}\n\\phi^{A}\\delta x^{\\mu}\n\\end{equation}\n\\indent If ${\\cal L}$ is the total Lagrangian of the system,\nthe field equations of\n$\\phi^{A}$ is just $[{\\cal L}]_{\\phi^{A}}=0$. Hence from eq.(3), we\ncan obtain the\nconservation equation corresponding to transform eq.(2)\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}}{\\partial\\p_{\\mu}\\phi^{A}}\n\\delta_{0}\n\\phi^{A})=0\n\\end{equation}\nIt is important to recognize that if ${\\cal L}$ is not the total\nLagrangian\n, e.g. the gravitational part ${\\cal L}_{g}$, then so long as the\naction of\n${\\cal L}_{g}$ remains invariant under transform eq.(2), eq.(3) is\nstill valid\nyet eq.(6) is no longer admissible because of\n$[{\\cal L}_{g}]_{\\phi^{A}}\\not=0$.\\\\\n\\indent Suppose that $\\phi^{A}$ denotes the Riemann tensors\n$\\phi^{A}_{\\mu}$\nand Riemann scalars $\\psi^{A}$ (for gravitational anyons,\nthey are dreibein\n$e^{a}_{\\mu}$, SO(1,2) connection $\\omega^{a}_{\\mu}$, the Lagrangian\nmultiplier $\\lambda^{a}_{\\mu}$ and the matter field $\\psi^{A}$).\nEq.(3) reads\n(suppose that ${\\cal L}_{g}$ does not contain $\\psi^{A}$)\n\\begin{equation}\n\\partial_{\\mu}({\\cal L}_{g}\\delta x^{\\mu}+\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\\phi^{A}_{\\nu}}\n\\delta_{0}\n\\phi^{A}_{\\nu})+[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\delta_{0}\\phi^{A}_{\\mu}=0\n\\end{equation}\nUnder transforms eq.(2), the Lie variations are\n\\begin{equation}\n\\delta_{0}\\phi^{A}_{\\nu}=-\\delta x^{\\alpha}_{,\\nu}\\phi^{A}_{\\alpha}\n-\\phi^{A}_{\\nu,\\alpha}\\delta x^{\\alpha}\n\\end{equation}\nwhere the dot \",\" denotes partial derivative. So eq.(7) reads\n\\begin{equation}\n\\partial_{\\mu}[{\\cal L}_{g}\\delta x^{\\mu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\\phi^{A}_{\\lambda ,\\nu}\n\\delta x^{\\nu})]\n-[{\\cal L}_{g}]_{\\phi^{A}_{\\lambda}}(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\n\\phi^{A}_{\\lambda ,\\nu}\\delta x^{\\nu})=0\n\\end{equation}\nComparing the coefficients of $\\delta x^{\\nu},\n\\delta x^{\\nu}_{,\\lambda}\n$ and $\\delta x^{\\nu}_{,\\mu\\lambda}$, we may obtain an identity\n\\begin{equation}\n\\partial_{\\lambda}([{\\cal L}_{g}]_{\\phi^{A}_{\\lambda}}\\phi^{A}_{\\nu})=\n[{\\cal L}_{g}]_{\\phi^{A}\n_{\\lambda}}\\phi^{A}_{\\lambda,\\nu}\n\\end{equation}\nThen eq.(9) can be written as\n\\begin{equation}\n\\partial_{\\mu}[{\\cal L}_{g}\\delta x^{\\mu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n(\\delta x^{\\nu}_{,\\lambda}\\phi^{A}_{\\nu}+\\phi^{A}_{\\lambda ,\\nu}\n\\delta\nx^{\\nu})-[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu}\\delta x^{\\nu}]=0\n\\end{equation}\nor\n\\begin{equation}\n\\partial_{\\mu}[({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\n\\phi^{A}_{\\lambda,\\nu}-[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu})\n\\delta x^{\\nu}\n-\\frac{\\partial{\\cal L}_{g}}{\\partial\\phi^{A}_{\\lambda,\\mu}}\\phi^{A}_{\\nu}\\delta\nx^{\\nu}_{,\\lambda}]=0\n\\end{equation}\nBy definition, we introduce\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}=-({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\n\\frac{\\partial{\\cal L}_{g}}{\\partial\\p_{\\mu}\n\\phi^{A}_{\\lambda}}\\phi^{A}_{\\lambda,\\nu}-\n[{\\cal L}_{g}]_{\\phi^{A}_{\\mu}}\\phi^{A}_{\\nu})\n\\end{equation}\n\\begin{equation}\n\\tilde{Z}^{\\lambda\\mu}_{\\nu}=\\frac{\\partial{\\cal L}_{g}}{\\partial\\phi^{A}_{\\lambda,\\mu}}\n\\phi^{A}_{\\nu}\n\\end{equation}\nThen eq.(12) gives\n\\begin{equation}\n\\partial_{\\mu}(\\tilde{I}^{\\mu}_{\\nu}\\delta x^{\\nu}+\n\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\delta x^{\\nu}_{,\\lambda})=0\n\\end{equation}\nSo by comparing the coefficients of $\\delta x^{\\nu},\n\\delta x^{\\nu}_{,\\mu}$\nand $\\delta x^{\\nu}_{,\\mu\\lambda}$, we have the following from eq.(15)\n\\begin{equation}\n\\partial_{\\mu}\\tilde{I}^{\\mu}_{\\nu}=0\n\\end{equation}\n\\begin{equation}\n\\tilde{I}^{\\lambda}_{\\nu}=-\\partial_{\\mu}\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\,\\,\\,\\,\\,\\,\\,\\,\n\\tilde{Z}^{\\mu\\lambda}_{\\nu}=-\\tilde{Z}^{\\lambda\\mu}_{\\nu}\n\\end{equation}\nEq.(16)-(17) are fundamental to the establishing of conservation law of\nenergy-momentum.\\\\\n\\section*{III. Conservation Law of Energy-momentum for Gravitational\nAnyons}\n{\\it 3.1. General Displacement Transform}\\\\\n\\indent In 1+3 Einstein gravity, a generally covariant conservation law\nof energy-momentum was obtained by means of whant we usually call\nthe {\\it general displacement transform}\\cite{s8}\n\\begin{equation}\n\\delta x^{\\mu}=e^{\\mu}_{a}\\epsilon^{a}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n(\\epsilon^{a}\n=const.)\n\\end{equation}\nIt really represents an infinitesiaml displacement while\n\\begin{equation}\n\\delta x^{\\mu}=b^{\\mu}=const.\n\\end{equation}\ndoes not because $x^{\\mu}$ can be any coordinates. For instance,\nit can be\nspherical coordinates, in which case, the resulting conservation\nlaw, if\nexists, should be that of angular-momentum other than energy-momentum.\nUsing\nthe invariance of the action with respect to eq.(18) and Einstein\nequations\n, the following general covariant conservation law of energy-momentum\nis\nobtained\n\\begin{equation}\n\\nabla_{\\mu}(T^{\\mu}_{a}+t^{\\mu}_{a})=0\n\\end{equation}\nand it was shown that there exist superpotentials $V^{\\mu\\nu}_{a}$\n\\begin{equation}\nT^{\\mu}_{a}+t^{\\mu}_{a}=\\nabla_{\\nu}V^{\\mu\\nu}_{a}\n\\end{equation}\n\\begin{equation}\nV^{\\mu\\nu}_{a}=\\frac{c^{4}}{8\\pi G}[e^{\\mu}_{b}e^{\\nu}_{c}\n\\omega_{a}\\,^{bc}\n+(e^{\\mu}_{a}e^{\\nu}_{b}-e^{\\mu}_{b}e^{\\nu}_{a})\\omega^{b}]\n\\end{equation}\nThis definition of energy-momentum has the following main\npropertities:\\\\\n1). It is a covariant definition with respect to general coordinate\ntransforms. But the energy-momentum tensor is not covariant under\nlocal\nLorentz transforms, this is reasonable because of the equivalence\nprinciple.\\\\\n2). For closed system, the total energy-momentum does not depend\non the\nchoice of Riemann coordinates and transforms in the covariant way\n\\begin{equation}\nP_{a}^{\\prime}=L_{a}\\,^{b}P_{b}\n\\end{equation}\nunder local Lorentz transform $\\Lambda^{a}\\,_{b}$ which is constant\n$L^{a}\\,\n_{b}$ at spatial infinity.\\\\\n3). For a closed system with static mass center, the total\nenergy-momentum is\n$P_{a}=(Mc,0,0,0)$.\\\\\n4). For a rather concentrated matter system, the gravitational\nenergy\nradiation is\\cite{s11}\n\\begin{equation}\n-\\frac{\\partial E}{\\partial t}=\\frac{G}{45c^{5}}(\\stackrel{\\cdots}{D})^{2}\n\\end{equation}\n5). For Bondi's plane wave, the energy current is\\cite{s11}\n\\begin{equation}\nt^{\\mu}_{0}=(\\frac{1}{4\\pi}\\beta^{\\prime 2},\n\\frac{1}{4\\pi}\\beta^{\\prime 2}, 0,0)\n\\end{equation}\n6). For the solution of gravitational solitons, we can obtain\nfinite energy\nwhile the Landau-Lifshitz definition leads to infinite energy\n\\cite{s12}.\\\\\n7). In Ashtekar's complex formalism of general relativity,\nthe energy-momentum and angular-momentum constitute a 3-Poincare\nalgebra\nand the energy coincides with the ADM energy\\cite{s10}.\\\\\nWith these foundations, we next use 1+2 dimensional transform\neq.(18) to\nobtain conservative energy-momentum for gravitational anyons.\\\\\n{\\it 3.2 The Energy-momentum for Gravitational Anyons}\\\\\n\\indent We take the Lagrangian for gravitational anyons to be\nin the\nfirst order ($\\omega^{a}_{\\mu}=\\frac{1}{2}\\epsilon^{abc}\n\\omega_{\\mu bc},\n\\omega_{\\mu bc}$ is the SO(1,2) connection.)\n\\begin{equation}\n{\\cal L}={\\cal L}_{g}+{\\cal L}_{m}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\cal L}_{g}=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\omega^{a}_{\\nu}\\partial_{\\mu}\ne_{\\alpha a}+\\frac{1}{2\\kappa}\\epsilon^{\\mu\\nu\\alpha}e_{\\alpha a}\n\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\nu c}\n+{\\cal L}_{c-s}+\\epsilon^{\\mu\\nu\\alpha}\\lambda^{a}_{\\mu}(\\partial_{\\nu}\ne_{\\alpha a}\n+\\epsilon_{abc}\\omega^{b}_{\\nu}e^{c}_{\\alpha})\n\\end{equation}\n\\begin{equation}\n{\\cal L}_{c-s}=\\frac{1}{2\\kappa\\mu}\\epsilon^{\\mu\\nu\\alpha}(\\omega_{\\mu a}\n\\partial_{\\nu}\\omega^{a}_{\\alpha}+\\frac{1}{3}\\epsilon_{abc}\\omega^{a}_{\\mu}\n\\omega^{b}_{\\nu}\\omega^{c}_{\\alpha})\n\\end{equation}\nand ${\\cal L}_{m}$ denotes the matter part. The field equations for\n$e^{a}_{\\mu},\n\\omega^{a}_{\\mu}$ and $\\lambda^{a}_{\\mu}$ are $[{\\cal L}]_{e^{a}_{\\mu}}=0,\n[{\\cal L}]_{\\omega^{a}_{\\mu}}=0 $ and $[{\\cal L}]_{\\lambda^{a}_{\\mu}}=0$, i.e.\n\\begin{equation}\n\\frac{1}{2\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\epsilon^{abc}\\omega_{\\mu b}\n\\omega_{\\nu c}+\\epsilon^{\\mu\\nu\\alpha}\\lambda^{b}_{\\mu}\\epsilon_{abc}\n\\omega^{c}_{\\nu}+\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\\partial_{\\mu}\n\\omega^{a}_{\\nu}+\\epsilon^{\\mu\\nu\\alpha}\\partial_{\\mu}\\lambda_{\\nu}^{a}=\n-[{\\cal L}_{m}]_{e^{a}_{\\alpha}}\n\\end{equation}\n\\begin{equation}\n\\epsilon^{\\mu\\nu\\alpha}(\\partial_{\\nu}e_{\\alpha a}+\\epsilon_{abc}\n\\omega^{b}_{\\nu}\ne^{c}_{\\alpha})=0\n\\end{equation}\n$$\n\\frac{1}{\\kappa}\\epsilon^{\\nu\\mu\\alpha}(\\partial_{\\mu}e^{a}_{\\alpha}+\n\\epsilon^{a}\\,\n_{bc}\\omega^{b}_{\\mu}e^{c}_{\\alpha})+\\frac{1}{\\kappa\\mu}\n\\epsilon^{\\nu\\mu\\alpha}\n(\\partial_{\\mu}\\omega^{a}_{\\alpha}$$\n\\begin{equation}\n+\\frac{1}{2}\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\alpha c})+\n\\epsilon^{\\nu\\mu\n\\alpha}\\lambda_{b \\mu}\\epsilon^{abc}e_{\\alpha c}\n=-[{\\cal L}_{m}]_{\\omega^{a}_{\\nu}}\n\\end{equation}\nUsing eq.(30), eq.(31) can be rewritten as\n\\begin{equation}\n\\frac{1}{\\kappa\\mu}\\epsilon^{\\nu\\mu\\alpha}\n(\\partial_{\\mu}\\omega^{a}_{\\alpha}\n+\\frac{1}{2}\\epsilon^{abc}\\omega_{\\mu b}\\omega_{\\alpha c})+\n\\epsilon^{\\nu\\mu\n\\alpha}\\lambda_{b \\mu}\\epsilon^{abc}e_{\\alpha c}\n=-[{\\cal L}_{m}]_{\\omega^{a}_{\\nu}}\n\\end{equation}\nThese equations are the same as those given in\\cite{s13}.\\\\\n\\indent From eq.(14)\n$$\\tilde{Z}^{\\lambda\\mu}_{\\nu}=\\frac{\\partial{\\cal L}_{g}}{\\partial e^{a}\n_{\\lambda,\\mu}}e^{a}_{\\nu}\n+\\frac{\\partial{\\cal L}_{g}}{\\partial\\omega^{a}_{\\lambda,\\mu}}\\omega^{a}_{\\nu}\n+\\frac{\\partial{\\cal L}_{g}}{\\partial\\lambda^{a}_{\\lambda,\\mu}}\\lambda^{a}_{\\nu}$$\n\\begin{equation}\n=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\alpha\\lambda}\\omega_{\\alpha a}\ne^{a}_{\\nu}\n+\\frac{1}{2\\kappa\\mu}\\epsilon^{\\alpha\\mu\\lambda}\\omega_{\\alpha a}\n\\omega^{a}_{\\nu}+\\epsilon^{\\alpha\\mu\\lambda}\\lambda_{\\alpha a}\ne^{a}_{\\nu}\n\\end{equation}\nFor transform eq.(18), eq.(15) implies\n\\begin{equation}\n\\partial_{\\mu}(\\tilde{I}^{\\mu}_{\\nu}e^{\\nu}_{a}+\\tilde{Z}^{\\lambda\\mu}_{\\nu}\ne^{\\nu}_{a,\\lambda})=0\n\\end{equation}\nDefine\n\\begin{equation}\n\\tilde{I}^{\\mu}_{a}=-\\partial_{\\lambda}\\tilde{Z}^{\\lambda\\mu}_{a},\\,\\,\\,\\,\n\\,\\,\\,\\,\\,\n\\tilde{Z}^{\\lambda\\mu}_{a}=\\tilde{Z}^{\\lambda\\mu}_{\\nu}e^{\\nu}_{a}\n\\end{equation}\nwe then have\n\\begin{equation}\n\\partial_{\\mu}\\tilde{I}^{\\mu}_{a}=0\n\\end{equation}\nSince $[{\\cal L}_{g}]_{e^{a}_{\\mu}}=-[{\\cal L}_{m}]_{e^{a}_{\\mu}}$, and\n$T^{\\mu}_{a}\n=-\\frac{1}{e}[{\\cal L}_{m}]_{e^{a}_{\\mu}}$, we have from eq.(13)\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}=-({\\cal L}_{g}\\delta^{\\mu}_{\\nu}-\n\\frac{\\partial{\\cal L}_{g}}{\\partial e^{a}_{\\lambda,\n\\mu}}e^{a}_{\\lambda,\\nu}-\\frac{\\partial{\\cal L}_{g}}{\\partial \\omega^{a}_{\\lambda,\n\\mu}}\\omega^{a}_{\\lambda,\\nu}\n-[{\\cal L}_{g}]_{\\omega^{a}_{\\mu}}\\omega^{a}_{\\nu}-\n[{\\cal L}_{g}]_{\\lambda^{a}_{\\mu}}\n\\lambda^{a}_{\\nu})+eT^{\\mu}_{a}e^{a}_{\\nu}\n\\end{equation}\nDefine $t^{\\mu}_{a}$ by\n\\begin{equation}\n\\tilde{I}^{\\mu}_{\\nu}e^{\\nu}_{a}+\\tilde{Z}^{\\lambda\\mu}_{\\nu}\ne^{\\nu}_{a,\\lambda}=e(T^{\\mu}_{a}+t^{\\mu}_{a})\n\\end{equation}\nThen we have\n\\begin{equation}\ne(T^{\\mu}_{a}+t^{\\mu}_{a})=e\\nabla_{\\lambda}Z^{\\lambda\\mu}_{a}\n\\end{equation}\nwhere $\\tilde{Z}^{\\lambda\\mu}_{a}=eZ^{\\lambda\\mu}_{a}$. Eq.(39) is the\ndesired general covariant conservation law of energy-momentum for\ngravitational anyons. The total energy-momentum is\n\\begin{equation}\nP_{a}=\\int e(T^{0}_{a}+t^{0}_{a}) d^{2}x=\\int \\partial_{i}\\tilde{Z}^{i0}_{a}\nd^{2}x\n\\end{equation}\n{\\it 3.3 The iso(1,2) Algebra}\\\\\n\\indent The pure Einstein case is restored by setting\n$\\mu\\rightarrow\\infty$\nand ${\\cal L}_{m}=0$. In this limit, we have $\\lambda^{a}_{\\mu}=0$\nand the\nsuperpotential is simply\n\\begin{equation}\n\\tilde{Z}^{\\mu\\nu}_{a}=-\\frac{1}{\\kappa}\\epsilon^{\\mu\\nu\\alpha}\n\\omega_{\\alpha\na}\n\\end{equation} and the total energy-momentum is\n\\begin{equation}\nP_{a}=\\frac{1}{\\kappa}\\oint_{\\partial\\Sigma}\\omega_{a}\n\\end{equation}\nwhere $\\partial\\Sigma$ is the spatial infinity which is 1 dimensional.\nFrom the\nangular-momentum\\cite{s7}\n\\begin{equation}\nJ_{a}=-\\frac{1}{\\kappa}\\oint_{\\partial\\Sigma}e_{a}=\\frac{1}{\\kappa}\\int\n\\epsilon^{ij}\\epsilon_{abc}\\omega^{b}_{i}e^{c}_{j} d^{2}x\n\\end{equation} and the Poisson brackets given in \\cite{s5}\n$$\\{\\omega^{a}_{i}(x),e^{b}_{j}(y)\\}=\\kappa\\epsilon_{ij}\\eta^{ab}\n\\delta^{2}(x-y)\n$$\n\\begin{equation}\n\\{\\omega^{a}_{i}(x),\\omega^{b}_{j}(y)\\}=\\{e^{a}_{i}(x),e^{b}_{j}\n(y)\\}=0\n\\end{equation}\nwe have\n\\begin{equation}\n\\{J_{a},J_{b}\\}=-\\frac{1}{\\kappa^2}\\{\\oint e_{a}(x), \\int\n\\epsilon^{ij}\n\\epsilon_{bcd}\\omega^{c}_{i}e^{d}_{j} d^{2}y\\}=\\epsilon_{abc}J^{c}\n\\end{equation}\n\\begin{equation}\n\\{P_{a}, P_{b}\\}=0\n\\end{equation}\n\\begin{equation}\n\\{J_{a}, P_{b}\\}=\\{\\frac{1}{\\kappa}\\int \\epsilon^{ij}\\epsilon_{acd}\n\\omega^{c}_{i}e^{d}_{j}d^{2}x, \\frac{1}{\\kappa}\\oint\\omega_{b}(y)\\}\n=\\epsilon_{abc}P^{c}\n\\end{equation}\nThus the ${\\it iso}(1,2)$ algebra can be restored.\n\\section*{IV. Examples}\n\\indent We now consider the special case that $[{\\cal L}_{m}]_{\\omega^{a}\n_{\\mu}}=0$. Using the identities in 3-dim Riemann geometry\n$$\nR_{\\alpha\\beta\\gamma\\delta}=g_{\\alpha\\gamma}\\tilde{R}_{\\beta\\delta}+\ng_{\\beta\\delta}\\tilde{R}_{\\alpha\\gamma}-g_{\\alpha\\delta}\\tilde{R}\n_{\\beta\\gamma}-g_{\\beta\\gamma}\\tilde{R}_{\\alpha\\delta}$$\n$$\\tilde{R}_{\\mu\\nu}=R_{\\mu\\nu}-\\frac{1}{4}g_{\\mu\\nu}R\n\\,\\,\\,\\,\\,\\,\\,\\, R^{\\alpha\\beta}\\,_{\\gamma\\delta}=\n-\\epsilon^{\\alpha\\beta\\mu}\n\\epsilon_{\\gamma\\delta\\nu}G^{\\nu}_{\\mu}\\,\\,\\,\\,\\,\\,\\, G^{\\mu\\nu}=\nR^{\\mu\\nu}\n-\\frac{1}{2}g^{\\mu\\nu}R$$\nwe have\n\\begin{equation}\n-\\frac{1}{\\kappa\\mu}\\epsilon^{\\mu\\alpha\\beta}(\\partial_{\\alpha}\n\\omega_{\\beta a}\n+\\frac{1}{2}\\epsilon_{abc}\\omega^{b}_{\\alpha}\\omega^{c}_{\\beta})=\n\\frac{1}{\\kappa\\mu}eG^{\\mu}_{a}\n\\end{equation}\nSo\n\\begin{equation}\n\\lambda^{a}_{\\mu}=-\\frac{1}{\\kappa\\mu}\\tilde{R}^{a}_{\\mu}\n\\end{equation}\nSubstitute into eq.(29), we have\n\\begin{equation}\nG^{\\mu}_{a}+\\frac{1}{\\mu}C^{\\mu}_{a}=-\\kappa T^{\\mu}_{a}\n\\end{equation}\nwhere $C^{\\mu}_{a}$ is the Cotton tensor. For Deser's solution\n\\cite{s4}.\n$$e^{0}_{0}\\simeq 1,\\enspace\ne^{1}_{1}=e^{2}_{2}\\simeq \\sqrt{\\frac{m\\kappa^{2}}{\\pi}}\nln^{\\frac{1}{2}}r$$\n $$e^{0}_{2}\\simeq\n-\\frac{\\kappa^{2}}{\\mu}\\frac{(m+\\mu\\sigma)}{2\\pi}\\frac{x}\n {r^{2}},\\enspace e^{0}_{1}\\simeq\n\\frac{\\kappa^{2}(m+\\mu\\sigma)}{2\\pi\\mu}\n\\frac{y}{r^{2}}$$\n\\begin{equation}\ne^{1}_{2}\\simeq\n-\\frac{\\kappa^{4}(m+\\mu\\sigma)^{2}}{4\\pi^{2}\\mu^{2}\\sqrt{\\frac\n{m\\kappa^{2}}{\\pi}}}\\frac{xy}{r^{4}ln^{1\/2}r}\n\\end{equation}\n we can obtain the asymptotical behaviour of the\nspin connection \\begin{equation}\n\\omega_{\\mu ab}\\simeq \\frac{1}{rf(r)}\n\\end{equation}\n where$f(r)$ represents some monototically\nincreasing functions of $r$. Thus we have\n\\begin{equation}\n\\oint_{\\partial\\Sigma}\\omega_{\\mu ab}dx^{\\mu}=0\n\\end{equation}\nThus the total energy-momentum vanishes.\nFor Cl$\\acute{e}$ment's \\cite{s14}\n self-dual exact solution\n\\begin{equation}\n ds^{2}=A^{-1}[dt-(\\omega_{0}+A)d\\theta]^{2}-dr^{2\n}-Ad\\theta^{2} \\end{equation}\n$$A=a+ce^{-\\mu r}$$\n where $a,c$ are constants and $\\mu$ should be\npositive since $r\\in (0,+\\infty)\n$. In terms of rectangular coordinates, it takes\nthe following form\n $$ds^{2}=A^{-1}dt^{2}-2(\\omega_{0}A^{-1}+1)(\\frac\n{x}{r^{2}}dtdy-\\frac{y} {r^{2}}dtdx)$$\n $$+\\{[A^{-1}(\\omega_{0}+A)^{2}-A]\\frac{x^{2}}{r^{\n4}}-1\\}dy^{2}+\\{[A^{-1}\n(\\omega_{0}+A)^{2}-A]\\frac{y^{2}}{r^{4}}-1\\}dx^{2}$$\n\\begin{equation}\n-[A^{-1}(\\omega_{0}+A)^{2}-A]\\frac{2xy}{r^{4}}dydx\n\\end{equation}\n We obtain the following\n asymptotical dreibein \\cite{s7}\n $$e^{0}_{0}=\\frac{1}{\\sqrt{a}}, \\enspace\ne^{0}_{1}=\\sqrt{a}(1+\\frac{\\omega\n_{0}}{a})\\frac{y}{r^{2}}$$\n\\begin{equation}\n e^{0}_{2}=-\\sqrt{a}(1+\\frac{\\omega_{0}}{a})\\frac{\nx}{r^{2}}, \\enspace e^{1}_{1}\n=1+\\frac{ay^{2}}{2r^{4}}\n\\end{equation}\n $$ e^{1}_{2}=\\frac{axy}{r^{4}}, \\enspace\ne^{2}_{2}=1+\\frac{ax^{2}}{2r^{4}}$$\nHence it can be shown that\n$$\\lim_{r\\rightarrow \\infty}r\\omega_{\\mu ab}=0$$\nThus\n$$\\int_{\\partial\\Sigma}\\omega_{\\mu ab}dx^{\\mu}=0$$\nSo the total energy vanishes also. In the limit,\n$\\mu\\rightarrow\\infty$, eq.(50) has a solution with\ndreibein and spin connection\\cite{s15}\n$$e^{0}=dt+\\frac{\\kappa J}{2\\pi r^{2}}{\\bf r}\\times d{\\bf r}\n\\,\\,\\,\\,\\,\\,\\,\n{\\bf e}=(1-\\frac{\\kappa m}{2\\pi})d{\\bf r}+\\frac{\\kappa m}\n{2\\pi r^{2}}{\\bf r}\n({\\bf r}\\cdot d{\\bf r})$$\n\\begin{equation}\n\\omega^{0}=\\frac{\\kappa m}{2\\pi r^{2}}{\\bf r}\\times d{\\bf r}\n\\,\\,\\,\\,\\,\\,\\,\n\\omega^{i}=0\n\\end{equation}\nwe have\n\\begin{equation}\nP_{a}=(m,0,0)\n\\end{equation}\nwhich is the same as in\\cite{s16}.\n\\section*{V. Discussions}\n\\indent As an end, we make some discussions. First, general\ncovariance is a\nfundamental demand for conservation laws in general relativity.\nour definition\neq.(39) (40) of energy-momentum is coordinate independent. As\nthe definition\nof angular-momentum\\cite{s7}, under local SO(1,2) transform\n$e^{a}\\rightarrow\n\\Lambda^{a}\\,_{b}(x)e^{b}$, where $\\Lambda^{a}\\,_{b\\mid\\partial\\Sigma}\n=L^{a}\\,_{b}\n=const.$, we have $P^{a}\\rightarrow L^{a}\\,_{b}P^{b}$. Second,\nit is worth\nwhile noting that for Deser's solution, the source stress-energy\ntensor\nof which is given {\\it a priori}(the energy-momentum vanishes\nwhile Deser's\ngravitational mass vanishes only when $m+\\sigma\\mu=0$). This is\nquite\ndifferent from the solution eq.(57) in the limit\n$\\mu\\rightarrow\\infty$.\nThe reason is that, though the form of eq.(51) agrees with eq.(57),\nthe\nfall-off is substantially different. Remember that in Deser's\nsolution,\nthe metric is linearized $g_{\\mu\\nu}=\\eta_{\\mu\\nu}+h_{\\mu\\nu}$,\nwhich\nis a good approximation on condition that $\\mid h_{\\mu\\nu}\\mid\\ll 1$.\nYet in Deser's\nsolution, $h_{ij}=\\phi\\delta_{ij}$ and $\\phi\\sim \\ln r$, so\n$h_{\\mu\\nu}$ does\nnot satisfy the confition. We expect a solution with the same\n$T^{\\mu\\nu}$ as Deser's while without the difficulty.\n\n\\vskip 0.3in\n\\underline{\\bf Acknowledgement} S.S. Feng is indebted to Prof. S.\nRandjbar-Daemi for his invitation for working at ICTP for three\nmonths.This work is supported by the National Science\nFoundation of China under Grant No. 19805004 and the Funds for Young Teachers\nof Shanghai Education Council.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\subsection{Overview and Related Work}\n\nThe problem of reaching consensus on the ordering of acts by participants in a distributed system has been investigated for four decades~\\cite{shostak1982byzantine}, with efforts in the last decade falling into two categories: \\emph{Permissioned}, where the set of participants is determined by some authority, and \\emph{permissionless}, where anyone may join and participate provided that they pass some `sybil-proof' test, notably proof-of-work~\\cite{bitcoin} or proof-of-stake~\\cite{kiayias2017ouroboros}. Two leaders-of-the-pack in the permissioned category are the State-Machine-Replication protocol (SMR, consensus on an ordering of proposals) for the eventual-synchrony model -- Hotstuff~\\cite{yin2019hotstuff} and its extensions and variations~\\cite{cohen2021aware}, and the Byzantine Atomic Broadcast protocol (BAB, consensus on an ordering of all proposals made by correct participants) for the asynchronous model -- DAG-Rider~\\cite{keidar2021need} and its extensions and variations~\\cite{giridharan2022bullshark}. Since the emergence of Bitcoin~\\cite{bitcoin}, followed by Ethereum with its support for smart contracts~\\cite{buterin2014next}, permissionless consensus protocols have received the spotlight.\n\nRecent conceptual and computational advances, notably stake-based sampling, have allowed permissioned consensus protocols to join the cryptocurrency fray (e.g. Cardano~\\cite{kiayias2017ouroboros} and Algorand~\\cite{gilad2017algorand}), offering much greater efficiency and throughput compared to proof-of-work protocols. According to this approach, in every epoch (which could be measured in minutes or weeks) a new set of \\emph{miners} \nis chosen in a random auction, where the probability of being an auction winner is correlated with the stake bid by the miner.\nMechanism design ensures that miners benefit from performing the protocol well, benefit less if they perform the protocol less well, and lose their stake if they subvert the protocol. \n\n\n\n\\begin{table}[t] \\smaller\n\\makebox[\\linewidth]{\n\\begin{tabular}{|c|cccc|c|} \n\\hline\n\\multirow{3}{*}{\\textbf{Protocol}} & \\multicolumn{4}{c|}{\\textbf{Latency} (number of communication rounds)} & \\multirow{3}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Message \\\\ complexity\\end{tabular}}} \\\\ \\cline{2-5}\n & \\multicolumn{2}{c|}{Eventual synchrony} & \\multicolumn{2}{c|}{Asynchrony} & \\\\ \\cline{2-5}\n & \\multicolumn{1}{c|}{Good case} & \\multicolumn{1}{c|}{Expected case} & \\multicolumn{1}{c|}{Good-case} & Expected case & \\\\ \\hline\n\\multicolumn{1}{|l|}{Cordial Miners} & \\multicolumn{1}{c|}{3} & \\multicolumn{1}{c|}{4.5} & \\multicolumn{1}{c|}{6} & $9$ & amortized $O(n)$ \\\\ \\hline\n\\multicolumn{1}{|l|}{\\begin{tabular}[c]{@{}l@{}}Bullshark~\\cite{giridharan2022bullshark} (same \\\\ protocol as DAG-Rider~\\cite{keidar2021need} \\\\ in asynchrony)\\end{tabular}} & \\multicolumn{1}{c|}{4} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Fallback \\\\ to asynchrony\\end{tabular}} & \\multicolumn{1}{c|}{8} & 12 & amortized $O(n)$ \\\\ \\hline\n\\end{tabular}\n}\n\t\\caption{\\textbf{Performance summary.} Latency is measured in communication rounds.\n\tThe Cordial Miners protocols for both models have better latency than Bullshark in both the good-case and the expected case. All protocols have the same concrete and asymptotic message complexity. Note that each round of Bullshark and DAG-Rider uses reliable broadcast, meaning two rounds of communication.\n\t}\n\t\\vspace{-2em}\n\t\\label{table:performance}\n\\end{table}\n\nWith this in mind, the expectation is that miners will do their best, not their worst, to execute the protocol, and hence the focus of analyses of permissioned consensus protocols has shifted from worst-case complexity to \\emph{good-case complexity}~\\cite{abraham2021good,giridharan2022bullshark}, where miners are generally expected to behave as well as they can, given compute and network limitations, as opposed to as bad as they can. Still, standard protections against a malicious adversary are needed, for example to prevent a double-spending, a hostile takeover, or a meltdown of the cryptocurrency supported by the consensus protocol.\n\nThe use of a DAG-like structure to solve consensus has been introduced in previous works, especially in asynchronous networks~\\cite{moser1999byzantine}.\nHashgraph~\\cite{RN284} builds an unstructured DAG, with each block containing two references to previous blocks, and on top of the DAG the miners run an inefficient binary agreement protocol.\nThis leads to expected exponential time complexity.\nAleph~~\\cite{gkagol2018aleph} builds a structured round-based DAG, where miners proceed to the next round once they receive $2f+1$ DAG nodes from other miners in the same round.\nOn top of the DAG protocols run a binary agreement protocol to decide on the order of vertices to commit. Nodes in the DAG are reliably broadcast.\nBlockmania~\\cite{danezis2018blockmania} uses a variant of PBFT~\\cite{RN581} in the eventual synchrony model.\n\n\n\nDAG-Rider~\\cite{keidar2021need} is a BAB protocol for the asynchronous model. It assumes an adaptive adversary that eventually delivers messages between any two correct miners.\nIn DAG-Rider the miners jointly build a DAG of blocks, with blocks as vertices and pointers to previously-created blocks as edges, divided into strong and weak edges.\nStrong edges are used for the commit rule, and weak edges are used to ensure fairness.\nThe protocol employs an underlying reliable broadcast protocol of choice, which ensures that eventually the local DAGs of all correct miners converge and equivocation is excluded.\nEach miner independently converts its local DAG to an ordered sequence of blocks, with the use of threshold signatures to implement a global coin that retrospectively chooses one of the miners as the leader for each round.\nThe decision rule for delivering a block is if the vertex created by the leader is observed by at least $2f+1$ miners three rounds after it is created.\nThe DAG is divided to waves, each consisting of the nodes of four rounds. When a wave ends, miners locally check whether a decision rule is met, similar to our protocol.\nDAG-Rider has an expected amortized linear message complexity, and expected constant latency.\nTusk~\\cite{danezis2021narwhal} is an implementation based on DAG-Rider.\nBullshark~\\cite{giridharan2022bullshark} is the current state-of-the-art dual consensus protocol based on DAG-Rider that offers a fast-track to commit nodes every two rounds in case the network is synchronous.\nOther DAG-based consensus protocols include~\\cite{chockler1998adaptive,dolev1993early,sompolinsky2015secure,RN349}.\n\n\n\nHotStuff~\\cite{yin2019hotstuff} is an SMR protocol designed for the eventual synchrony model. \nThe protocol employs all-to-leader, leader-to-all communication: In each round, a deterministically-chosen designated leader proposes a block to all and collects from all signatures on the block. Once the leader has $2f+1$ signatures, it can combine them into a threshold signature~\\cite{boneh2001short} which it sends back to all.\nThe decision rule for delivering a block is three consecutive correct leaders.\nThis leads to a linear message complexity and constant latency in the good case. The protocol delivers a block if there are three correct leaders in a row, which is guaranteed to happen after GST.\nHotStuff is based on Tendermint~\\cite{buchman2016tendermint} and is also the core of several other consensus protocols~\\cite{kamvar2019celo,cypherium,flow,thunder}.\nIn this model, there are number of leader-based protocols such as DLS~\\cite{dwork1988consensus}, PBFT~\\cite{RN581}, Zyzzyva~\\cite{kotla2007zyzzyva}, and SBFT~\\cite{gueta2019sbft}.\n\n\nIt is within this context that we introduce \\textbf{Cordial Miners} -- a family of simple, efficient, self-contained Byzantine Atomic Broadcast~\\cite{cachin2001secure} protocols, and present two of its instances: Retrospective random leader selection for the asynchronous model and deterministic leader selection for the eventual synchrony model (See Table \\ref{table:performance} for their performance).\n\nWe believe that the simplicity-cum-efficiency of the Cordial Miners protocols stems from the use of the blocklace data structure and its analysis for all key algorithmic tasks (the following refers to correct miners):\n\\begin{enumerate}\n \\item \\textbf{The Blocklace}~\\cite{shapiro2021multiagent} is a partially-ordered generalization of the totally-ordered blockchain (Def. \\ref{definition:block}), that consists of cryptographically-signed blocks, each containing a payload and a finite number of cryptographic hash pointers to previous blocks. The blocklace induces a DAG, as cryptographic hash pointers cannot form cycles by a compute-bound adversary. The DAG induces a partial order $\\succ$ (Def. \\ref{definition:acknowledge}) on the blocks that includes Lamport's `happened-before' causality relation~\\cite{lamport1978time} among correct miners.\n \n The globally-shared blocklace is constructed incrementally and cooperatively by all miners, who cordially disseminate it to each other.\n \n \\item \\textbf{Ordering}: The ordering algorithm (Algorithm \\ref{alg:CMO}) is used locally by each miner to topologically-sort its locally-known part of the blocklace into a totally-ordered output sequence of blocks, excluding equivocation along the way. This conversion is monotonic (Prop. \\ref{proposition:tau-finality}) -- the output sequence is extended as the miner learns of or produces ever-larger portions of the global blocklace, and in this sense every output block of each miner is final. We say that two sequences are \\emph{consistent} if one is a prefix of the other (Def. \\ref{definition:prefix-consistent}), a notion stronger than the common prefix property of Ouroboros~\\cite{kiayias2017ouroboros}. We\n assume that less than one-third of the miners are faulty,\n and prove that the following holds for the remaining correct miners of the Cordial Miners protocols:\n \\begin{itemize}\n \\item \\textbf{Safety}: Outputs of different miners are consistent (Prop. \\ref{proposition:safety}). \n \\item \\textbf{Liveness:} A block created by one is eventually output by everyone (Proposition \\ref{proposition:CMAbd-liveness}).\n \n \n \\end{itemize}\n\n \\item \\textbf{Dissemination:} Any new block created by a miner $p$ acknowledges blocks known to $p$ by including pointers to the tips (DAG sources) of $p$'s local blocklace. Correspondingly, a miner $p$ will buffer, rather than include in its blocklace, any received block with dangling pointers -- pointers to blocks not known to $p$.\n Hence, a block $b$ by $p$ informs any recipient $q$ of blocks not known to $p$ at the time of $b$'s creation. Thus $q$, being cordial, when sending to $p$ a new $q$-block, will include with it blocks $q$ knows but, to the best of $q$'s knowledge, are not yet known to $p$ and have not already been sent by $q$ to $p$, thus ensuring block dissemination (Prop. \\ref{proposition:CMAbd-dissemination}). \n \n \\item \\textbf{Equivocation exclusion}: An \\emph{equivocation} (Def. \\ref{definition:da-faulty-consistent}) is a pair of blocks by the same miner that are not causally-related -- have no path of pointers from one to the other; a miner that creates an equivocation is considered faulty and is referred to as an \\emph{equivocator}. The shared blocklace will eventually include any equivocating block known to a correct miner, and hence eventually known to all correct miners. The question is: What should miners do with this knowledge? \n \n A block $b$ \\emph{acknowledges} block $b'$ if there is a (possibly empty) path from $b$ to $b'$, namely $b \\succeq b'$ (Def. \\ref{definition:acknowledge}). Let $[b]$ denote the set of blocks acknowledged by $b$, also referred to as the \\emph{closure} of $b$ (Def. \\ref{definition:closure}). A block $b$ \\emph{approves} block $b'$ if it acknowledges $b'$ and does not acknowledge any block $b''$ equivocating with $b'$ (Def. \\ref{definition:approval}, see Fig. \\ref{figure:approve-sr}.A). A key observation is that a miner cannot approve both blocks of an equivocation without being itself an equivocator (Ob. \\ref{observation:approve-da}). Hence, if less than one-third of the miners are equivocators, then no equivocation will ever receive an approval from blocks created by a \\emph{supermajority} (at least two-thirds) of the miners. This is the basis of equivocation-exclusion by the blocklace: A miner finalizes a block $b$ once its local blocklace includes blocks that approve $b$ by a supermajority (Algorithm \\ref{alg:CMO}).\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=14cm]{Figs\/approve-sr.jpeg}\n\\caption{\nAcknowledgement, Approval, Equivocation, Ratification, Super-Ratification: (A) Observing an Equivocation: The `drop shape' depicts a blocklace with a block at the tip of the drop, where all the blocks that the tip block acknowledges being the entire volume of the drop. Initial blocks of the blocklace are at the bottom, and drop inclusion implies $\\succ$. Assume $b_1, b_2$ are an equivocation (Def. \\ref{definition:da-faulty-consistent}) by the red miner. According to the figure, $b''$ approves $b_2$ (Def. \\ref{definition:approval}) since it acknowledges $b_2$ (Def. \\ref{definition:acknowledge}, namely $b'' \\succ b_2$) and does not acknowledge any conflicting red block, in particular it does not acknowledge $b_1$. However, since $b'$ acknowledges $b''$ it acknowledges both $b_1$ and $b_2$ and hence does not approve the equivocating $b_1$ (nor $b_2$). (B) \\emph{Ratified}: The block (black dot) at round $r$ is ratified by another block (gray dot), if the black block is approved by a supermajority (green horizontal line) that is acknowledged by the gray block. (C) \\emph{Super-Ratified}: A block (blue dot) at round $r$ is super-ratified if there is a supermajority (light green horizontal line) at round $r+\\alpha$, each member of which ratifies the blue block at round $r$ by acknowledging a supermajority (dark green horizontal line) that approves the blue block. In the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round (black dot). In asynchrony, the leader is only elected in the next round (grey dot). The parameters are $\\alpha=1, \\beta =2$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model. Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 5 for asynchrony.}\n\n\n\\label{figure:approve-sr}\n\\end{figure}\n \n \\item \\textbf{Cordial Miners}: The \\emph{depth}, or \\emph{round}, of a block $b$ is the maximal length of any path emanating from $b$ (Def. \\ref{definition:block}). A \\emph{round} is a set of blocks of the same depth. Miners are cordial in two respects. First, as explained above, in informing other miners of blocks they believe the other miner lacks. Second, awaiting a supermajority of round $d$ before producing a block of round $d+1$ (Def. \\ref{definition:cordial}).\n \n \\item \\textbf{Leader Selection}: The Cordial Miners protocol for the eventual synchrony model employs deterministic leader selection (e.g. via a shared pseudorandom function). In the asynchronous model, the adversary has complete control over the order of message delivery, indefinitely. The panacea to such an adversary, employed for example by DAG-Rider~\\cite{keidar2021need}, is to use a shared random coin~\\cite{cachin2005random} and elect the leader retroactively. \n \n \n \n \\item \\textbf{Ratified and Super-Ratified Leaders}: A block $b$ of round $r$ is \\emph{ratified} by block $b'$ if $[b']$ includes a supermajority of round $r+\\alpha$ that approves $b$. A leader block $b$ of round $r$ is \\emph{super-ratified} if there is a supermajority of round $r+\\beta$ that include the leader, each member of which acknowledges a supermajority of round $r+\\alpha$ that approves $b$. In the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round. In asynchrony, the leader is only elected in the next round. The parameters are $\\alpha=1, \\beta=1$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model. Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 6 for asynchrony. (Def. \\ref{definition:ratified-leaders}, see Fig. \\ref{figure:approve-sr}).\n \n \\item \\textbf{Common Core}: Since miners are cordial, the notion of \\emph{common core}~\\cite{canetti1996studies} can be applied to prove a lower bound on the probability of a leader being super-ratified despite such a powerful adversary. This approach is applied in the Cordial Miners protocol for asynchrony, at the cost of delaying leader selection and hence expected finality to 4 rounds. As the approach uses threshold signatures, a secure method for key distribution is needed~\\cite{damgaard2001practical}.\n \n The common core property implies that there is a set $V$ of at least $2f+1$ blocks in round $r$ such that every correct miner that issues block $b$ in round $r+2$ has $ b \\succeq V$. Hence, if a miner issues a block at round $r+3$ then it has a set $U$ of at least $2f+1$ blocks in round $r+3$ such that for $u \\in U$ and $v \\in V$ $u\\succ v$. \n\n \n \\item \\textbf{Ordering by Leaders}: We assume a topological sort procedure that takes a blocklace as an input and produces a sequence of its blocks while respecting their causal partial order $\\succ$. With it, a recursive \\emph{ordering function} $\\tau$, applied to a leader block $b$, is defined as follows (Def. \\ref{definition:tau}): Let $b'$ be the highest-depth leader block in $[b]$ ratified by $b$.\n If there is no such $b'$, output the topological sort of $[b]$ and terminate. Else \n recursively call the ordering function with $b'$ and output its result following by the topological sort of $[b]\\setminus [b']$.\n \n \\item \\textbf{Finality by Super-Ratified Leaders}: The key insight to finality is this (Proposition \\ref{proposition:finality-of-dr}): A super-ratified leader will be ratified by any subsequent leader (see Figure \\ref{figure:b-hat}). Hence, the ordering algorithm (Algorithm \\ref{alg:CMO}) used by any miner is as follows: When identifying a new super-ratified leader $b$ in its blocklace, apply the ordering function $\\tau$ to $b$ and output the newly added suffix since the previous super-ratified leader.\n As the ordering function is guaranteed to be called with the previous super-ratified leader in one of its recursive calls, it can be optimized to return if it does, rather than recompute the prefix it has already delivered.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=10cm]{Figs\/finality.jpeg}\n\\caption{Finality of a Super-Ratified Leader (Definition \\ref{definition:leaders}): Assume\nthat the blue leader block of round $r$ (blue dot) is super-ratified. We show that it is ratified by any subsequent cordial leader.\n(A) If the cordial leader is at round $r+\\beta$ (black dot, eventual synchrony instance) then it is part of the ratifying supermajority (light green) by definition. Else if the cordial red leader is at round $r+\\beta+1$ then it acknowledges a supermajority (red horizontal line) of \nthe previous round $r+\\beta$, which must have a correct agent (purple) in common with the ratifying supermajority at round $r+\\beta$ (light green horizontal line), which acknowledges the approving supermajority at round $r+\\alpha$, and via which the red leader at $r+\\beta+1$ ratifies the black leader at $r$.\n(B) Otherwise, there are two purple blocks $b_1$, $b_2$ by the same correct purple agent in the red $\\hat{B}$ and light green $B'$ supermajorities, which are connected via a path since the purple agent is correct, connecting the red leader $\\hat{b}$ to the blue leader $b$ via the approving supermajority $B$ (dark green horizontal line) at round $r+\\alpha$. Namely, the red leader $\\hat{b}$ ratifies the blue leader $b$.}\n\n\\label{figure:b-hat}\n\\end{figure*}\n\n\n \\item \\textbf{Identification and exclusion of faulty miners}: Any faulty $p$-block known to some correct miner will eventually be known to all, resulting in correct miners suspending any further communication with $p$. For example, any miner can easily verify whether another miner $p$ is cordial in the second sense by examining the blocklace. In addition, an equivocation by $p$, with each block of the pair known to a different correct miner, will eventually be known to all and result in the exclusion of $p$.\n \n \\item \\textbf{Exclusion of nonresponsive miners}: A miner $p$ need not be cordial to a miner $q$ as long as $q$ has not acknowledged a previous block $b$ sent to $q$ by $p$. If $q$ fail-stopped, then $p$ should definitely not waste resources on $q$; if $q$ is only suspended or delayed, then eventually it will send to $p$ a block acknowledging $b$, following which $p$---being cordial---will send to $q$ all the backlog $p$ has previously refrained from sending it, and is not acknowledged by the new block received from $q$.\n \n \n \n\\end{enumerate}\nMiners accomplish all the above by simple and efficient analyses of their local blocklace.\n\nThe protocols are presented as algorithmic components written in pseudo-code:\nAlgorithm \\ref{alg:blocklace} contains a description of a block in a blocklace and blocklace utilities. Algorithm \\ref{alg:CMO} describes the local conversion each miner executes with the function $\\tau$ converting its local copy of the partially-ordered blocklace into a totally-ordered sequence of blocks. Algorithm \\ref{alg:CMAbd} incorporates Algorithms \\ref{alg:blocklace} and \\ref{alg:CMO}, \ndescribes how a miner creates a block and sends it to all correct miners and\nincludes any backlog needed for dissemination. It has two instances that specify the conditions to create a block, one for asynchrony and one for eventual synchrony.\n\n\n\n\\subsection{Models, Problem, Safety and Liveness Proof Outline}\nWe assume $n\\ge 3$ miners $\\Pi$, of which $f < n\/3$ may be faulty (act arbitrarily, be \\emph{Byzantine}), and the rest are \\emph{correct}, and that a message sent from one correct miner to another will eventually arrive. We assume that every miner has a single and unique key-pair (PKI) and can cryptographically sign messages.\nEach miner $p \\in \\Pi$ has an input call $\\textit{payload}()$ that returns a payload (e.g., a proposal from a user or a mempool), and an output call $\\textit{deliver}(b)$ where $b$ is a block.\n\n\\begin{definition}\\label{definition:models}\nWe consider two models of communication with an adversary:\n\\begin{itemize}\n \\item \\emph{Asynchrony}, where an adversary controls the finite message delay of each message. \n \\item \\emph{Eventual Synchrony}~\\cite{dwork1988consensus}, where an adversary can delay arbitrarily an unknown but finite number of messages, beyond which messages arrive possibly under adversarial control but within a bounded delay.\n\\end{itemize}\n\\end{definition}\n\n\nIn each model the protocol addresses Byzantine Atomic Broadcast~\\cite{cachin2001secure}. \n\n\n\\begin{definition}[Prefix, $\\preceq$, Consistent Sequences]\\label{definition:prefix-consistent}\nA sequence $x$ is a \\emph{prefix} of a sequence $x'$, $x \\preceq x'$ if $x'$ can be obtained from $x$ by appending to it zero or more elements. Two sequences $x, x'$ are \\emph{consistent} if $x \\preceq x'$ or $x'\\preceq x$.\n\\end{definition}\n\n\\begin{definition}[Safety and Liveness]\\label{definition:safety-liveness}\nThese are the requirements of a blocklace-based Byzantine Atomic Broadcast protocol: \n\\begin{itemize}\n \\item \\emph{Safety}: Outputs of correct miners are consistent.\n \\item \\emph{Liveness:} A block created by a correct miner is eventually output by every correct miner with probability 1.\n\\end{itemize}\n\\end{definition}\nWe note that these safety and liveness requirements, combined with the uniqueness of a block in a blocklace, imply the standard Byzantine Atomic Broadcast guarantees: Agreement, Integrity, Validity, and Total Order~\\cite{bracha1987asynchronous, keidar2021need}.\n\nThe proofs of safety and liveness proceed as follows, assuming less than one-third of the miners are faulty, and referring to the remaining correct miners.\\\\\n\\textbf{Safety}: \n \\begin{enumerate}\n \\item Prove that the function $\\tau$ that converts a blocklace to a sequence of blocks is monotonic with respect to the superset relation (Prop. \\ref{proposition:tau-finality}).\n \\item Observe that if two sequences are each a prefix of a third sequence, then they are consistent (Ob.\\ref{observation:consistent-triplet}). Given any two local blocklaces $B, B'$ of miners $p, p'$, then due to the monotonicity of $\\tau$, both $\\tau(B)$ and $\\tau(B')$ are prefixes of $\\tau(B\\cup B')$. Therefore they are consistent (Corollary \\ref{corollary:consistency}).\n \\item Argue that Algorithm \\ref{alg:CMO} correctly implements $\\tau$ and hence is safe (Prop. \\ref{proposition:safety}).\n \\end{enumerate}\n\\textbf{Liveness:} \n \\begin{enumerate}\n \\item Observe that the conversion function $\\tau$, applied to a super-ratified leader block $b$, includes in the output sequence any block known to that leader, namely any block in $[b]$ (Ob. \\ref{observation:fairness}).\n \\item Given a block $b$ known to a miner at some point $t$ in the computation, argue that $b$ will eventually be known to every miner at some later point $t'$ in the computation (Prop. \\ref{proposition:CMAbd-dissemination}).\n \\item Argue that eventually some leader block $b'$ of a miner will be ratified at a point later than $t'$ with probability 1.\n \\item By construction, $b \\in [b']$, hence $b$ is included in the output of $\\tau$ applied to $b'$ (Prop. \\ref{proposition:CMAbd-liveness}).\n \\end{enumerate}\n\nFor asynchrony, we employ a shared random global coin.\n\\begin{definition} [Shared random coin] \\label{def:sharedCoin}\nWe use a \\emph{global perfect coin}, which is unpredictable by the\nadversary. \nIn round $r$, $r \\in \\mathbb{N}$, of the coin is invoked by miner $p_i \\in \\Pi$ by calling $\\textit{leader}_i(r)$.\nThis call returns a miner $p_j \\in \\Pi$, which is the chosen leader for round $r$.\nLet $X_r$ be the random variable that represents the probability that the coin returns miner $p_j$ as the return value of the call $\\textit{leader}_i(r)$.\nThe global perfect coin has the following guarantees:\n\\begin{description}\n \\item[Agreement] If two correct miners call $\\textit{leader}_i(r)$ and $\\textit{leader}_j(r)$ with respective return values $p_1$ and $p_2$, then $p_1=p_2$.\n \\item [Termination] If at least $f+1$ miners call $\\textit{leader}(r)$, then every $\\textit{leader}(r)$ call eventually returns.\n \\item[Unpredictability] As long as less than $f+1$ miners call $\\textit{leader}(r)$, the return value is indistinguishable from a random value except with negligible probability $\\epsilon$.\n Namely, the probability $pr$ that the adversary can guess the returned\n miner $p_j$ of the call $\\textit{leader}(r)$ is $pr \\leq\n \\Pr [X_r = p_j] + \\epsilon$. \n \\item[Fairness] The coin is fair, i.e., $\\forall r \\in \\mathbb{N}, \\forall p_j \\in \\Pi \\colon \\Pr[X_r = p_j] = 1\/n$.\n\\end{description}\n\\end{definition}\nExamples of such a coin implementation using a PKI and threshold signatures~\\cite{boneh2001short,libert2016born,shoup2000practical} are in~\\cite{cachin2005random,keidar2021need}.\nSee DAG-Rider~\\cite{keidar2021need} on details on how to implement such a coin as part of a distributed blocklace-like structure.\n\n\\section{The Blocklace: A Partially-Ordered Generalization of the Totally-Ordered Blockchain}\n\nThe blocklace was introduced in reference~\\cite{shapiro2021multiagent}. For completeness we include here needed definitions and results.\n\n\\begin{definition}[Block]\\label{definition:block}\nGiven a set of \\emph{miners} $\\Pi$ and a set of \\emph{payloads} $\\mathcal{A}$, a \\emph{block} is a triple $b=(p,a,H)$, referred to as a \\emph{$p$-block}, $p \\in \\Pi$, with $a \\in \\mathcal{A}$ being the \\emph{payload} of $b$, and $H$ is a (possibly empty) finite set of hash pointers to blocks, namely for each $h \\in H$, $h=\\textit{hash}(b')$ for some block $b'$. Such a hash pointer $h$ is a \\emph{$q$-pointer} if $b'$ is a $q$-block, and \\emph{same-miner} if $h$ is a $p$-pointer, in which case $b'$ is the \\emph{immediate predecessor} of $b$.\nThe set $H$ may have at most one $q$-pointer for any miner $q \\in \\Pi$, and if $H$ has no same-miner pointer then $b$ is called \\emph{initial}. The \\emph{depth} of $b$, $\\textit{depth}(b)$, is the maximal length of any path emanating from $b$.\n\\end{definition}\n\nNote that $\\textit{hash}$ being cryptographic implies that a cycle cannot be effectively computed.\n\n\\begin{definition}[Dangling Pointer, Closed]\nA hash pointer $h=\\textit{hash}(b)$ for some block $b$ is \\emph{dangling} in $B$ if $b \\notin B$. \nA set of blocks $B$ is \\emph{closed} if no block $b \\in B$ has a pointer dangling in B. \n\\end{definition}\n\nThe non-dangling pointers of a set of blocks $B$ induce finite-degree directed graph $(B, E)$, $E \\subset B \\times B$, with blocks $B$ as vertices and directed edges $(b,b') \\in E$ if $b, b' \\in B$ and $b$ includes a hash pointer to $b'$. We overload $B$ to also mean its induced graph $(B,E)$. \n\nIn the following we assume a given set of payloads $\\mathcal{A}$.\n\n\\begin{definition}[Blocklace]\nLet $\\mathcal{B}$ be the maximal set of blocks over $\\mathcal{A}$ and $\\textit{hash}$ for which the induced directed graph $(\\mathcal{B},\\mathcal{E})$ is acyclic. A \\emph{blocklace} over $\\mathcal{A}$ is a set of blocks $B \\subset \\mathcal{B}$.\n\\end{definition}\n\n\n\n\n\nThe two key blocklace notions used in our protocols are \\emph{acknowledgement} and \\emph{approval}.\n\\begin{definition}[$\\succ$, Acknowledge]\\label{definition:acknowledge}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, the strict partial order $\\succ_B$ is defined by $b'\\succ_B b$ if $B$ has a non-empty path of directed edges from $b'$ to $b$ ($B$ is omitted if $B = \\mathcal{B}$). Given a blocklace $B$, $b'$ \\emph{acknowledges $b$ in} $B$ if $b' \\succ_B b$. Miner $p$ \\emph{acknowledges $b$ via} $B$ if there is a $p$-block $b' \\in B$ that acknowledges $b$, and a group of miners $Q \\subseteq \\Pi$ \\emph{acknowledge $b$ via} $B$ if for every miner $p \\in Q$ there is a $p$-block $b' \\in B$ that acknowledges $b$.\n\\end{definition}\n\n\\begin{definition}[Closure, Tip]\\label{definition:closure}\nThe \\emph{closure of $b \\in \\mathcal{B}$ wrt $\\succ$} is the set $[b] := \\{b'\\in \\mathcal{B} : b \\succeq b' \\}$. The \\emph{closure of $B\\subset \\mathcal{B}$ wrt $\\succ$} is the set $[B] := \\bigcup_{b \\in B} [b]$.\nA block $b \\in \\mathcal{B}$ is a \\emph{tip} of $B$ if $[b] =[B] \\cup \\{b\\}$.\n\\end{definition}\nNote that a set of blocks is closed iff it includes its closure (and thus is identical to it):\n\\begin{observation}\n$B\\subset \\mathcal{B}$ is closed iff $[B] \\subseteq B$. \n\\end{observation}\n\n\n\n\nWith this, we can define the basic notion of \\emph{equivocation} (aka \\emph{double-spending} when payloads are conflicting financial transactions).\n\\begin{definition}[Equivocation, Equivocator]\\label{definition:da-faulty-consistent}\nA pair of $p$-blocks $b\\ne b'\\in \\mathcal{B}$, $p \\in \\Pi$, form an \\emph{equivocation} of $p$ if they are not consistent wrt $\\succ$, namely $b' \\not\\succ b$ and $b \\not\\succ b'$. A miner $p$ is an \\emph{equivocator} in $B$ if $[b]$ has an equivocation of $p$. \n\\end{definition}\nNamely, a pair of $p$-blocks form an equivocation of $p$ if they do not acknowledge each other in $\\mathcal{B}$. In particular, two initial $p$-blocks constitute an equivocation by $p$. As $p$-blocks are cryptographically signed by $p$, an equivocation of $p$ is a volitional fault of $p$. \n\n\n\\begin{definition}[Approval]\\label{definition:approval}\nGiven a blocklace $B$, a block $b$ \\emph{approves $b'$ in $B$} if $b$ acknowledges $b'$ in $B$ and does not acknowledge any block $b''$ in $B$ that together with $b'$ forms an equivocation. An miner $p$ \\emph{approves $b'$ in $B$} if there is a $p$-block $b$ that approves $b'$ in $B$, in which case we also say that \\emph{$p$ approves $b'$ in $B$ via $b$}. A set of miners $Q \\subseteq \\Pi$ \\emph{approve $b'$ via $B'$ in $B$} if every miner $p \\in Q$ approves $b'$ in $B$ via some $p$-block $b \\in B'$, $B'\\subseteq B$.\n\\end{definition}\n\n\\begin{observation}\nApproval is monotonic wrt $\\supset$.\n\\end{observation}\nNamely, if $b$ or $p$ approve $b'$ in $B$ they also approve $b'$ in $B' \\supset B$.\n\nA key observation is that a miner cannot approve an equivocation of another miner without being an equivocator itself (Fig. \\ref{figure:approve-sr}.A):\n\n\\begin{observation}\\label{observation:approve-da}[Approving an Equivocation]\nIf miner $p \\in \\Pi$ approves an equivocation $b_1, b_2$ in a blocklace $B \\subseteq \\mathcal{B}$, then $p$ is an equivocator in $B$.\n\\end{observation}\n\n\n\nAs equivocation is a fault, at most $f$ miners may equivocate. \n\\begin{definition}[Supermajority]\nA set of miners $P \\subset \\Pi$ is a \\emph{supermajority} if $|P| \\ge 2f+1$. A set of\nblocks $B$ is a \\emph{supermajority} if the set $\\{p\\in \\Pi : b \\in B \\text{ is a $p$-block}\\}$ is a supermajority.\n\\end{definition}\n\n\n\\begin{lemma}[No Supermajority Approval for Equivocation]\\label{lemma:no-double-majority}\nIf there are at most $f$ equivocators in a blocklace $B\\subset \\mathcal{B}$ with an equivocation $b, b' \\in B$, then not both $b, b'$ have supermajority approval in $B$.\n\\end{lemma}\n\nBlocklace utilities that realize these definition are presented in Algorithm \\ref{alg:blocklace}.\n\n\\begin{algorithm*}[t!] \n \\caption{\\textbf{Cordial Miners\\xspace: Blocklace Utilities} \n \\\\ pseudocode for miner $p \\in \\Pi$}\n \n \\label{alg:blocklace}\n \\small\n \\begin{algorithmic}[1] \\smaller\n \n \\Statex \\textbf{Local variables:}\n \\StateX struct $\\textit{block } b$: \\Comment{The struct of the most recent block $b$ created by miner $p$} \n \n \\StateXX $b.\\textit{creator}$ -- the miner that created $b$ \n \\StateXX $b.\\textit{payload}$ -- a set of transactions\n \n \\StateXX $b.\\textit{pointers}$ -- a possibly-empty set of hash pointers to other blocks\n \\StateX \\textit{blocks} $\\gets \\{\\}$ \n \n \n\t\t\n\t\t\\vspace{0.5em}\n\t\t\\Procedure{$\\textit{create\\_block}$}{\\textit{blocks'}} \\Comment create into $b$ a new block pointing to the sources of \\textit{blocks}$'$\n\t\t\\State $b.\\textit{payload} \\gets \\textit{payload}()$ \\Comment e.g. dequeue a payload from a queue of proposals (aka mempool)\n\t\t\\State $b.\\textit{creator} \\gets p$\n\t\t\\State $b.\\textit{pointers} \\gets \\{hash(b') ~:~ b' \\in \\textit{blocks}', \\text{ $b'$ has no incoming pointers in \\textit{blocks}}' \\}$ \n\t \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\n\t \\State $\\textit{deliver\\_blocks}()$ \\label{alg:CMA:addedOutgoingBlock}\n\t \\State \\Return $b$\n\t\t\\EndProcedure\n\t\t\n\t\t\n \\vspace{0.5em}\n\t\t\\Procedure{\\textit{hash}}{$b$}\n\t \\Return collision-free cryptographic hash pointer(s) to block(s) $b$ \\Comment{Returns a set if applied to a set}\n \\EndProcedure\n \n\t\t\n \\vspace{0.5em}\n \\Procedure{\\textit{path}}{$b,b'$} \n \\Return $\\exists b_1,b_2,\\ldots,b_k \\in \\textit{blocks}$, $k\\ge 1$, s.t.\\\n $b_1 = b $, $b_k = b'$ and $\\forall i \\in [k-1] \\colon b_{i+1} \\in b_i.\\textit{pointers}$\n \\EndProcedure\n \n \\vspace{0.5em}\n \\Procedure{\\textit{depth}}{$b$} \n \\Return max~ $\\{k : \\exists b' \\in$ \\textit{blocks} and \\textit{path$(b,b')$} of length $k$\\}. \\Comment{Depth of block $b$}\n \n \n \\EndProcedure\n \n \\vspace{0.5em}\n \\Procedure{\\textit{depth}}{\\textit{blocks}} \n \\Return $\\max~ \\{\\textit{depth}(b) : b \\in \\textit{blocks}\\}$ \\Comment{Depth of set of blocks}\n \\EndProcedure\n \n\t \n \\vspace{0.5em}\n \\Procedure{\\textit{blocks\\_prefix}}{$d$}\n \\Return $\\{b \\in \\textit{blocks} : \\textit{depth}(b) \\le d\\}$\n \\EndProcedure\n \n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{closure}}{$b$} \\label{alg:SMR:closure}\n \\Return \\{$b' \\in \\textit{blocks} : \\textit{path}(b,b')$\\} \\Comment{also referred to as $[b]$. $b$ could also be a set of blocks}\n \n \\EndProcedure\n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{leader}}{$d$} \n \\Return \\Comment{Returns a leader at depth $d$ if there is one: Predetermined for every even $d$ for eventual synchrony; random\/unpredictable using a shared coin for every $d$ divisible by 5 for asynchrony (Def.~\\ref{def:sharedCoin}).} \n \n \n \\EndProcedure\n \n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{leaders}}{\\textit{blocks}} \n \\Return $\\{b \\in \\textit{blocks} : b.\\textit{creator} = \\textit{leader}(\\textit{depth}(b))\\}$\n \\EndProcedure\n \n \n \n\t\t\\vspace{0.5em} \n\t \\Procedure{\\textit{equivocation}}{$b_1,b_2$} \\label{alg:SMR:da}\n \\Return \n $b_1.\\textit{creator} = b_2.\\textit{creator} \\wedge\n b_1 \\notin [b_2] \\wedge\n b_2 \\notin [b_1]\n $ \\Comment{See Figure \\ref{figure:approve-sr}.A} \n \n \n \\EndProcedure\n\t\t\n\t\t\n\t\t\\vspace{0.5em} \n\t \\Procedure{\\textit{equivocator}}{$q$} \\label{alg:SMR:dactor}\n \\Return \n $\\exists b_1,b_2 \\in \\textit{blocks} \\wedge\n b_1.\\textit{creator} = b_2.\\textit{creator} = q \\wedge\n \\textit{equivocation}(b_1,b_2)\n $ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n \\EndProcedure\n \n \n \t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{approved}}{$b_1,b$} \\label{alg:SMR:approved}\n \\Return $b_1 \\in [b] \\wedge \\forall b_2 \\in [b] : \\lnot$\\textit{equivocation}$(b_1,b_2)$ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n \\EndProcedure\n \n \n\t\t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{ratified}}{$b_1,b_2$} \\label{alg:SMR:sm_approved} \n \\Return\n $|\n \\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge \n b \\in [b_2] \\wedge\n \\textit{approved}(b_1,b) \\}\n | \\ge 2f+1$ \\Comment{See Figure \\ref{figure:approve-sr}.B}\n \\EndProcedure\n \n \n\t \n \\vspace{0.5em}\n \\Procedure{\\textit{cordial\\_block}}{$b$}\n \\Return \n $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}| \\geq 2f+1$\n \\EndProcedure\n \n\t \n\t\t\\vspace{0.5em} \n\t \\Procedure{\\textit{faulty}}{$q$} \\label{alg:faulty}\n \\Return \n $\\textit{equivocator}(q) \\vee\n (\\exists b \\in \\textit{blocks} \\wedge\n b.\\textit{creator} = q \\wedge\n \\lnot\\textit{cordial\\_block}(b))$ \\Comment{Can be expanded}\n \\EndProcedure\n \n\t\n \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{cordial\\_round}()$}{} \\label{alg:CMO:cr} \\Comment{Returns recent new cordial round, singleton or the empty set, Def. \\ref{definition:cordial}}\n \\State \\Return $\\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$ where $R =$\n \\label{alg:CMA:computeCordial}\n \\State $\\{ r \\in \\textit{depth}(\\textit{blocks}) : \n |\\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge b.\\textit{depth} = r \\wedge \\lnot\\textit{equivocator}(b.\\textit{creator})\\}| \\ge 2f+1 \\wedge\n \\not\\exists b \\in \\textit{blocks} : (b.\\textit{creator} = p \\wedge \\textit{depth}(b) \\ge r)\\}$ \n \\EndProcedure \n \n \\alglinenoNew{counter}\n \\alglinenoPush{counter}\n\n \\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\\section{Converting a Blocklace into a Sequence of Blocks}\n\nHere we present a deterministic function $\\tau$ that incrementally converts a blocklace into a sequence of some of its blocks, respecting $\\succ$, and show that it is monotonic wrt to the subset relation, provided no more than $f$ miners equivocate. Each miner in the Cordial Miners protocol employs $\\tau$ to locally compute the final output sequence of blocks from its local copy of the blocklace as input, as realized by Algorithm \\ref{alg:CMO}. The monotonicity of $\\tau$ ensures finality, as it implies that the output sequence will only extend while the input local blocklace increases over time. It also ensures the safety of a protocol that uses $\\tau$ (Proposition \\ref{proposition:tau-finality} below). \nTo ensure liveness, one has to argue that every block in the input blocklace will eventually be in the output of $\\tau$; this argument is made in Proposition \\ref{proposition:CMAbd-liveness}.\n\n\\begin{definition}[Cordial]\\label{definition:cordial}\nA block $b\\in \\mathcal{B}$ is \\emph{cordial} if the set $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}|$ is a supermajority.\nMiner $p$ is \\emph{cordial} in blocklace $B \\subseteq \\mathcal{B}$ if every $p$-block $b \\in B$ is cordial.\n\\end{definition}\n\n\\begin{definition}[Round, Leader, Leader Block]\\label{definition:leaders}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, then a \\emph{round} $r \\ge 1$ in $B$ is the set of blocks \n$\\{b\\in B : \\textit{depth}(b)=r\\}$. We assume a leader selection function $\\textit{leader}: \\mathbb{N} \\mapsto \\Pi \\cup \\{\\bot\\}$ that is defined only for certain depths. If $\\textit{leader}(r)=p$ then $p$ is the \\emph{leader} of round $r$, and if, in addition, $b\\in B$ is a $p$-block of depth $r$, then $b$ is a \\emph{leader block} of round $r$ in $B$.\n\\end{definition}\n\n\nSee Figure \\ref{figure:approve-sr}.B for the following definition. We employ parameters \n$\\alpha$ and $\\beta$, where $\\alpha=1, \\beta=1$ for deterministic leader selection in the eventual synchrony model and $\\alpha=2, \\beta=5$ for retrospective random leader selection via a shared coin in the asynchronous model.\n\\begin{definition}[Ratified and Super-Ratified Blocks]\\label{definition:ratified-leaders}\nA block $b \\in \\mathcal{B}$ is \\emph{ratified} if there is a supermajority of blocks $B$ of depth $d(b)+\\alpha$ (light green in Figure \\ref{figure:approve-sr}.B) that approves $b$; $b$ is \\emph{super-ratified} if, in addition, there is a supermajority of blocks $B'$ of depth $d(b)+\\beta$ (dark green in Figure \\ref{figure:approve-sr}.B) such that each member of $B'$ acknowledges $B$. \nIn the case of eventual synchrony, we also require the acknowledging supermajority of round $r+\\beta$ to include the leader of that round. \n\\end{definition}\nIn asynchrony, the leader is only elected in the next round. Hence the \\emph{wave length} (gap between leaders) is 3 for eventual synchrony and 6 for asynchrony.\n\nThe following proposition ensures that a super-ratified block is ratified by any subsequent cordial block (Fig. \\ref{figure:b-hat}).\n\n\\begin{proposition}[Finality of Super-Ratification]\\label{proposition:finality-of-dr}\nIf a leader block $b$ is super-ratified in $B$, then it is ratified by any subsequent cordial leader block $b'$. \n\\end{proposition}\n\\begin{proof}\nAssume a leader block $b$ super-ratified via supermajorities $B'$ and $B$ (Fig. \\ref{figure:approve-sr}.B), and assume that $\\hat{b}$ is a subsequent cordial leader block.\nBeing cordial, $\\hat{b}$ acknowledges a supermajority of the preceding round $\\hat{B}$. \nWe consider three cases (Figure \\ref{figure:b-hat}): \n\\begin{enumerate}\n \\item In the eventual synchrony instance, $d(b') = d(b) + \\beta$, (Figure \\ref{figure:b-hat}.A), $\\hat{b}$ (black dot) is part of the supermajority that acknowledges $B$.\n \\item Consider the asynchrony instance. If $d(b') = d(b) + \\beta+1$ (Figure \\ref{figure:b-hat}.A), then $\\hat{B}$ and $B'$ are supermajorities of the same depth and hence must have a block (purple dot Figure \\ref{figure:b-hat}.A) in in their intersection, via which $b'$ acknowledges $B$. \n \\item Else, $d(b') > d(b) + \\beta + 1$ (Figure \\ref{figure:b-hat}.B), then, by counting, at least one block $b_1 \\in B'$ and one block $b_2 \\in \\hat{B}$ are both $p$-blocks by the same correct (purple) miner $p \\in \\Pi$. Since $p$ is not an equivocator $b_2 \\succ b_1$. Hence $\\hat{b} \\succ b_2 \\succ b_1$ and $b_1$ acknowledges $B$, thus $\\hat{b}$ acknowledges $B$ and hence ratifies $b$.\n\\end{enumerate}\n\n\\end{proof}\n\nProposition \\ref{proposition:finality-of-dr} ensures that given a blocklace $B$, a super-ratified leader $b$ in $B$ will be ratified by any subsequent cordial leader, hence included in the sequence of final leaders. Hence, the final sequence up to a super-ratified leader $b$ is fully-determined by $b$ itself independently of the (continuously changing) identity of the last super-ratified leader. Hence the final sequence up to a super-ratified leader $b$ can be `cached' and will not change as the blocklace increases.\n\n\nSuper-ratified leaders are the anchors of finality in a growing chain, each `writes history' backwards till the preceding ratified leader. We use the term `Okazaki fragments'~\\cite{okazaki2017days} for the sequences computed backwards from each super-ratified leaders to its predecessor, acknowledging the analogy with the way one of the DNA strands of a replicated DNA molecule is elongated via the stitching of backwards-synthesized Okazaki-fragments.\n\n\nThe following recursive ordering function $\\tau$ maps a blocklace into a sequence of blocks. Formally, the entire sequence is computed backwards from the last super-ratified leader, afresh by each application of $\\tau$. Practically, a sequence up to a super-ratified leader is final (Prop. \\ref{proposition:tau-finality}) and hence can be cached, allowing $\\tau$ to be computed one `Okazaki-fragment' at a time, from the new super-ratified leader backwards to the previous super-ratified leader. \n\n\n\n\n\n\\begin{definition}[$\\tau$]\\label{definition:tau}\nWe assume an fixed topological sort function (e.g. lexicographic) $s$ that maps a blocklace to a sequence of blocks consistent with $\\succ$. The function $\\tau: 2^{\\mathcal{B}} \\xrightarrow{} \\mathcal{B}^*$ is defined for a blocklace $B\\subset \\mathcal{B}$ backwards, from the last output element to the first, as follows:\nIf $B$ has no super-ratified leaders then $\\tau(B) :=\\Lambda$.\nElse let $b$ be the last super-ratified leader in $B$. Then $\\tau(B) := \\tau'(b)$, where $\\tau'$ is defined recursively:\n$$\n\\tau'(b) := \n\\begin{cases}\ns([b]) \\text{\\ \\ if $[b]$ has no leader ratified by $b$, else }\\\\\n\\tau'(b') \\cdot s([b]\\setminus [b']) \\text{\\ \\ if $b'$ is the last leader ratified by $b$ in $[b]$}\n\\end{cases}\n$$\n\\end{definition}\nNote that $\\tau'$ uses the notion of a leader ratified by another leader, not a super-ratified leader.\n\n\nProposition \\ref{proposition:finality-of-dr} shows that if there are less than $f$ equivocators, a super-ratified leader is final, in that it will be ratified by any subsequent leader, super-ratified or not. Hence the following:\n\n\\begin{proposition}[Monotonicity and Finality of $\\tau$]\\label{proposition:tau-finality}\nIf there are at most $f$ equivocators, the function $\\tau$ is monotonic wrt $\\subseteq$, namely $B \\subseteq B'$ for two blocklaces with at most $f$ equivocators implies that $\\tau(B) \\preceq \\tau(B')$, and in this sense membership in $\\tau$ is final. \n\\end{proposition}\n\\begin{proof\nAssume given blocklaces $B \\subset B' \\subseteq \\mathcal{B}_{21}$.\nIf $\\tau(B)=\\Lambda$ the claim holds vacuously.\nSo let $b$ be the last super-ratified leader in $B$. If $B'$ has no super-ratified leader not in $B$, then $\\tau(B) = \\tau(B')$ and the claim holds vacuously. So let $b_1,\\ldots, b_k$, $k\\ge 1$, be \nthe sequence of final leaders in $B' \\setminus B$.\nThen by the definition of $\\tau$, $\\tau(B') = \\tau(B) \\cdot s([b_1]\\setminus [b]) \\cdot \\ldots \\cdot s([b_k]\\setminus [b_{k-1}])$, namely $\\tau(B) \\preceq \\tau(B')$.\n\\end{proof}\n\\begin{observation}[Consistent triplet]\\label{observation:consistent-triplet}\nGiven three sequences $x, x', x''$, if $x' \\preceq x$ and $x'' \\preceq x$ then $x'$ and $x''$ are consistent.\n\\end{observation}\n\nThe following Corollary ensures that the output sequences computed by miners based on their local blocklaces would be consistent as long as there are at most $f$ equivocators.\n\n\\begin{corollary}[Consistency of Output Sequences]\\label{corollary:consistency}\nIf $B, B'\\subset \\mathcal{B}$ are closed with at most $f$ equivocators in $B := B' \\cup B''$ then $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{corollary}\n\\begin{proof\nBy monotonicity of $\\tau$ with at most $f$ equivocators (Prop. \\ref{proposition:tau-finality}), $\\tau(B') \\preceq \\tau(B)$ and $\\tau(B'') \\preceq \\tau(B)$. By Observation \\ref{observation:consistent-triplet}, $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{proof}\n\nWhile $\\tau$ does not output all the blocks in its input, as blocks not acknowledged by the\nlast super-ratified leader in its input are not delivered, the following observation reminds us of the half-full glass:\n\\begin{observation}[Fairness of $\\tau$]\\label{observation:fairness}\nIf a block $b \\in B$ is acknowledged by the last super-ratified leader in $B$, then it is included in $\\tau(B)$.\n\\end{observation}\nIf every block known to a correct miner will be known to all correct miners, \nand if every super-ratified leader has a successor, then\nevery block will be eventually acknowledged by some super-ratified leader\nand therefore be delivered by $\\tau$.\n\n\n\n\\begin{algorithm*}[t] \n\t\\caption{\\textbf{Cordial Miners\\xspace: Conversion of Blocklace to Sequence of Blocks with $\\tau$} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMO}\n\t\\small\n\t\\begin{algorithmic}[1] \n\t\\alglinenoPop{counter} \n\n\t\t\\vspace{0.5em}\n\t\t\\Statex \\textbf{Local Variable:}\n\t\t\\StateX $\\textit{deliveredBlocks} \\gets \\{\\}$\n\t\t\\StateX $\\textit{currentLeader} \\gets \\{\\}$\n\t\t\\StateX $\\alpha$ \\Comment{A constant; $\\alpha\\gets 1$ for eventual synchrony and $\\alpha\\gets 2$ for asynchrony}\n\t\t\\StateX $\\beta$ \\Comment{A constant; $\\beta\\gets 2$ for eventual synchrony and $\\beta\\gets 5$ for asynchrony}\n\t\t\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\\Procedure{\\textit{deliver\\_blocks}()}{} \n\t\t\\label{alg:CMO:waveCompletion}\n\t\t\\If{$\\textit{super\\_ratified\\_leader}() \\ne \\textit{currentLeader}$} \n\t\t $\\textit{tau}(\\textit{super\\_ratified\\_leader}())$\n\t\t \\EndIf\n\t \\EndProcedure\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\n\t\n\t\t\\label{alg:CMO:commitrule}\n\t\t\n\t\t \\Procedure{\\textit{tau}}{$b_1$} \\label{alg:CMO:da}\n\t\t \t\\If{$b_1 \\in \\textit{deliveredBlocks} \\vee b_1 = \\emptyset$}\n\t\t \\Return\n\t\t \\EndIf\n\t\t \\State $b_2 \\gets \\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$\n \\textit{ where } $R = \\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\textit{ratified}(b,b_1)\\}$ \n \\Comment{Previous ratified leader or $\\emptyset$, Fig. \\ref{figure:b-hat}}\n \\label{alg:CMO:previous_ratified}\n \\State \\textit{tau}$(b_2)$ \\Comment{Recursive call to $tau$}\n\t\t \\For{$\\textbf{every} ~b \\in [b_1] \\setminus [b_2]$ s.t. $\\textit{approved}(b,b_1)$, ordered by topological sort} \\Comment{Deliver a new equivocation-free `Okazaki-fragment'}\n\t \t\t \\State \\textbf{deliver} $(b)$ \n\t \t\t \\label{alg:CMO:Okazaki}\n\t\t \\State $\\textit{deliveredBlocks} \\gets \\textit{deliveredBlocks} \\cup \\{b\\}$\n\t\t \\EndFor\n\t\n\t\t \\EndProcedure\n\t\t\n \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{super\\_ratified\\_leader}()$}{} \\label{alg:CMO:dr} \\Comment{Returns last super-ratified leader, if any, Figure \\ref{figure:approve-sr}.B}\n \\State \\Return $\\textit{arg}_{u \\in U} \\max~ \\textit{depth}(u)$ where $U =$\n \\State $\\{ b \\in \\textit{leaders}(\\textit{blocks}) : \n B = \\{b' : \\textit{depth}(b) +\\alpha = \\textit{depth}(b') \\wedge\n \\textit{approved}(b,b')\\}\n \\wedge |B| \\ge 2f +1 \\wedge$ \n $|\\{b' : \\textit{depth}(b) +\\beta = \\textit{depth}(b') \\wedge\n B \\subseteq [b']\\}| \\ge 2f +1 \n \\wedge (\\alpha = 1 \\xrightarrow{} B \\subseteq [\\textit{leader\\_block}(r+\\beta)])\n \\}$ \n \n \n \\EndProcedure\n \n\t\n\t\t\n\t\\alglinenoPush{counter}\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\n\n\nAlgorithm \\ref{alg:CMO} is a fairly literal implementation of the mathematics described above:\nIt maintained \\emph{deliveredBlocks} that includes the prefix of the output of $\\tau$ that it has already computed. Upon adding a new block to its blocklace (\\cref{alg:CMO:waveCompletion}), it computes the most-recent super-ratified leader $b_1$ according to Definition \\ref{definition:ratified-leaders}, and applies \\emph{tau} to it, which is intended to be a literal realization of the mathematical definition of $\\tau$ (Def. \\ref{definition:tau}), with the optimization, discussed above, that a recursive call with a delivered block is returned. Hence the following proposition:\n\n\\begin{proposition}[Tau implements $\\tau$]\\label{proposition:tau-tau}\nThe procedure \\emph{tau} in Algorithm \\ref{alg:CMO} correctly implements $\\tau$ in Definition \\ref{definition:tau}.\n\\end{proposition}\n\nAnd based on it, we conclude the following, which ensures the safety of a protocol that employs Algorithm \\ref{alg:CMO} for ordering a blocklace.\n\\begin{proposition}\\label{proposition:safety}\nAlgorithm \\ref{alg:CMO} satisfied the safety requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof\nProposition \\ref{proposition:tau-tau} establishes that\nAlgorithm \\ref{alg:CMO} implements $\\tau$ correctly. Together with Corollary \\ref{corollary:consistency}, we conclude that the outputs of all miners using Algorithm \\ref{alg:CMO} are consistent, satisfying the safety requirement. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Cordial Miners Protocols}\n\n\n \n\nThe Cordial Miners protocols employ the blocklace for dissemination. To do so, each miner maintains a \\emph{history} array that records its communication history with the other miners, and updates it upon receiving a block (\\cref{alg:CMAbd:receive}) and upon sending one (\\cref{alg:CMS:historySend}).\nA miner $p$ sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to each non-faulty miner $q$, and if $q$ is responsive, it includes in the package all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]]$ . This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all blocks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: The Generic Protocol and Instances for Asynchrony and Eventual Synchrony}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMAbd}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$ \\Comment{Communication history of $p$ with others} \n\t\\StateX \\textit{timer} \\Comment{A timer that can be \\textbf{reset} and tested}\n \\StateX $\\Delta$ \\Comment{A constant; set $\\Delta$ to $0$ for asynchrony and to the expected network round time after GST for eventual synchrony}\n \n\t\\vspace{0.5em}\n\t\\State \\textbf{reset} \\textit{timer}\n\t\\Upon{\\textbf{receipt} of $b$} \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$ \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t } \\Comment{Accept cordial block with no dangling pointers}\n\t \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMAbd:receive}\n \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMAbd:historyReceive}\n\n \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t \n\t \\If{ $\\textit{can\\_proceed}()$} \\label{alg:CMA:isCordial} \\Comment{Defined at \\cref{alg:CMO:cr}}\n\t \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$ \\Comment{Create a cordial block} \n\t\t\t\\For{$q \\in \\Pi \\wedge q \\ne p \\wedge \\lnot\\textit{faulty}(q)$} \n\t\t\t \\If{$\\textit{responsive}(q)$} \n\t \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$\n\t\t\\label{alg:CMS:CreatePackage}\n\t\n\t\t\\Else{ $\\textit{package} \\gets b$} \t\\label{alg:CMS:CreateBlockPackage}\n\t\t \\EndIf\n\t\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySend}\n\t\t \\State \\textbf{send} \\textit{package} to $q$\n\t\t \\State \\textbf{reset} \\textit{timer}\n\t\t \\EndFor\n\t\t \\EndIf\n\t \\EndWhile\n\t \n\t \t\\vspace{0.5em}\n \n \\Procedure{\\textit{responsive}}{$q$} \\label{alg:CMS:responsive}\n \\Return \n $\\forall b \\exists b'\\in \\textit{history}[q] : b.\\textit{creator} = p \\wedge b'.\\textit{creator} = q \\wedge b \\in [b'] $\n \\EndProcedure\n\n \n \\vspace{0.5em}\n \\Procedure{$\\textit{can\\_proceed}()$}{}\n \\Return $\\textit{cordial\\_round}()$ \\Comment{Instance for asynchrony}\n\t\t\\EndProcedure\\\n\t\t\n \\vspace{0.5em}\n \\Procedure{$\\textit{can\\_proceed}()$}{}\n \\Return $ r \\in \\textit{cordial\\_round}() \n \\wedge (timer \\ge \\Delta \\vee \\textit{leader}(r))$\n \\Comment{Instance for eventual synchrony}\n \n\t\t\\EndProcedure\n\t\t\n \n\t\t\n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\nBased on this, we argue the following:\n\\begin{proposition}[Algorithm \\ref{alg:CMAbd} Dissemination]\\label{proposition:CMAbd-dissemination}\nIn any run of Algorithm \\ref{alg:CMAbd}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$. If $q$ is correct then eventually it will send a block to $p$, and consider a subsequent round $r$. If at round $r$ the communication history of $p$ with $q$ shows that $q$ knows $b$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history. Then $q$ will eventually receive the package. The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\n\\begin{proposition}[Liveness of the Cordial Miners Protocol Instances for Asynchrony and Eventual Synchrony]\\label{proposition:CMAbd-liveness}\nThe instances of Algorithm \\ref{alg:CMAbd} for asynchrony and eventual synchrony satisfy the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMAbd}. According to Proposition \\ref{proposition:CMAbd-dissemination}, every block $b$ known to a correct miner will be known eventually to every correct miner. The probability of each leader block in this suffix being super-ratified is greater than some fixed $\\epsilon$ (specifically, $\\epsilon = \\frac{2}{3}$ for asynchrony and for eventual asynchrony after GST). Hence the probability measure of an infinite computation in which no correct leader block being super-ratified is zero. \nHence, for any block $b$ and point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be super-ratified at a point later than $t'$ with probability 1. Thus for any block $b$ known to a correct miner, there will be a subsequent super-ratified leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\\section{Performance analysis}\n\\textbf{Latency} (See Table \\ref{table:performance}).\nIn the asynchronous instance of the protocol, each wave (rounds till leader finality) consists of six rounds.\nTo calculate the probability that the decision rule is met in a wave, we use the common-core~\\cite{canetti1996studies} abstraction that is also used (and proved) in DAG-Rider and Bullshark.\n\\begin{claim} [Blocklace common-core]\nEventually, for every miner that completes round $r+5$ there exists a set $V$ of blocks in round $r+5$ and a set $U$ of blocks in round $r+2$ such that $|V| \\geq 2f+1, |U| \\geq 2f+1$ and for any $u \\in U, v \\in V$ there exists a path from $u$ to $v$.\n\\end{claim}\n\nTherefore, for every wave $w$, any miner that completes round $r+5$ has a supermajority (the blocks in $U$) that acknowledge a supermajority in round $r+2$ (the blocks in $V$).\nThus, the probability that the decision rule is met in wave $w$ is the probability that the blocks in $V$ acknowledge the retrospective elected leader block in round $r$.\nSince each block in $b \\in V$ acknowledges at least $2f+1$ blocks in round $r+1$, which in turn acknowledge $2f+1$ blocks in round $r$, then the probability of such a block $b$ to acknowledge a supermajority that approves a correct leader of round $r$ is $(1 - (\\frac{f}{3f+1})^{2f+1})^{2f+1}$, which is $1 - \\epsilon$ where $\\epsilon$ is a function of $f$ that converges exponentially to $0$ as $f$ increases. Hence, the probability that each of the blocks in $V$ acknowledges the elected leader block is close to 1 if the elected leader is a correct process, and $\\frac{2}{3}-\\varepsilon$ overall.\n\nTherefore, in the good-case, where the decision rule is met in wave $w$, the latency is $6$ rounds of communication (the length of a single wave in asynchrony).\nIn the expected case, the decision rule is met on average every $1.5$ waves, and therefore the expected latency is $9$ rounds of communication.\n\nNote that the adversary can equivocate up to $f$ times, but after each Byzantine process $p$ equivocates, all correct processes eventually detect the equivocation and do not consider $p$'s blocks as part of their cordial rounds when building the blocklace.\nThus, in an infinite run, equivocations do not affect the overall expected latency.\n\nIn the eventual synchrony case, the probability that the decision rule is met in each wave is the probability that the elected leader is a correct miner, therefore, it is at least $2\/3$.\nThus, in the good case, the latency is $3$ rounds of communication (the length of a single wave in eventual synchrony), and in the expected case it is $1.5$ waves, i.e., $4.5$ rounds of communication.\n\n\n\\textbf{Bit complexity.}\nEach block in the blocklace is linear in size, since it has a linear number of hash pointers to previous blocks.\nIn the worst-case, each block is sent to all miners by all the other miners, i.e., in the worst-case the bit complexity is $O(n^3)$ per block.\nBut, if we batch per block a linear number of transactions, when the decision rule is met, a quadratic number of transactions is committed each time.\nThus, the amortized bit complexity per decision is $O(n)$.\nThis is on par with the amortized bit complexity of DAG-Rider and Bullshark.\n\nThe concrete message complexity of the protocols depends on the security parameter of the hash function and the implementation of the threshold signatures in the shared random coin for the asynchronous protocol.\nIn any case, using the same security parameters as DAG-Rider and Bullshark achieves the same concrete amortized bit complexity as those protocols as well, and not just the same asymptotic bit complexity.\n\n\n\n\n\\section{Optimizations and Future Instances of the Cordial Miners Protocol Family}\n\nSeveral optimizations are possible to the protocol instances presented.\nFirst, as faulty miners are uncovered, they are excommunicated and therefore need not be counted as parties to the agreement, which means that the number of remaining faulty miners, initially bounded by $f$, decreases. As a result, the supermajority needed for finality is not $\\frac{n+f}{2n}$ (namely $2f+1$ votes in case $n = 3f+1$), but $\\frac{n+f-2f'}{2(n-f')}$, where $f'$ is the number of uncovered faulty miners, which converges to simple majority ($\\frac{1}{2}$) among the correct miners as more faulty miners are uncovered and $f'$ tends to $f$.\n\nSecondly, once faulty miners are uncovered and excommunicated, their slots as leaders could be taken by correct miners, improving good-case and expected complexity.\n\nThirdly, a hybrid protocol in the spirit of Bullshark~\\cite{giridharan2022bullshark} can be explored. Such a protocol would employ two leaders per round -- deterministic and random, try to achieve quick finality with the deterministic leader, and fall back to the randomly-selected leader if this attempt fails.\n\nWe also intend to explore the use of reliable broadcast~\\cite{bracha1987asynchronous} in the first round of each wave as it may improve the good case and expected case latency of our protocol in the asynchronous model.\n\n\\section{Conclusions}\n\nThe Cordial Miners protocols are simple and efficient. We believe simplicity has many ramifications when practical applications are considered: Simpler algorithms are easier to debug, to optimize, to make robust, and to extend.\n\nAn important next step towards making Cordial Miners a useful foundation for cryptocurrencies is to design a mechanism that will encourage miners to cooperate---as needed by these protocols---as opposed to compete, which is the current standard in mainstream cryptocurrencies.\n\n\\newpage\n\n\\section{Introduction}\n\\subsection{Overview and Related Work}\n\nhttps:\/\/arxiv.org\/pdf\/1809.01620.pdf\n\nThe problem of reaching consensus on the ordering of acts by participants in a distributed system has been investigated for four decades~\\cite{shostak1982byzantine}, with efforts in the last decade falling into two categories: \\emph{Permissioned}, where the set of participants is predetermined by an outside authority, and \\emph{permissionless}, where anyone may join and participate provided that they pass some `sybil-proof' test, notably proof-of-work~\\cite{bitcoin} or proof-of-stake~\\cite{kiayias2017ouroboros}. Two leaders-of-the-pack in the permissioned category are the State-Machine-Replication protocol (SMR, consensus on an ordering of proposals) for the eventual-synchrony model -- Hotstuff~\\cite{yin2019hotstuff} and its extensions and variations~\\cite{cohen2021aware}, and the Byzantine Atomic Broadcast protocol (BAB, consensus on an ordering of all proposals made by correct participants) for the asynchronous model -- DAG-Rider~\\cite{keidar2021need} and its extensions and variations~\\cite{giridharan2022bullshark}. Since the emergence of Bitcoin~\\cite{bitcoin}, followed by Ethereum with its support for smart contracts~\\cite{buterin2014next}, permissionless consensus protocols have received the spotlight.\n\nRecent conceptual and computational advances, notably stake-based sampling, have allowed permissioned consensus protocols to join the cryptocurrency fray (e.g. Cardano~\\cite{kiayias2017ouroboros} and Algorand~\\cite{gilad2017algorand}), offering much greater efficiency and throughput compared to proof-of-work protocols. According to this approach, in every epoch (which could be measured in minutes or weeks) a new set of \\emph{miners} \nis chosen in a random auction, where the probability of being a winner is correlated with the stake bid by the miner.\nMechanism design ensures that miners benefit from performing the protocol well, benefit less if they perform the protocol less well, and lose their stake if they subvert the protocol. \n\nWith this in mind, the expectation is that miners will do their best, not their worst, to execute the protocol, and hence the focus of analyses of permissioned consensus protocols has shifted from worst-case complexity to \\emph{good-case complexity}~\\cite{abraham2021good,giridharan2022bullshark}, where miners are generally expected to behave as well as they can, given compute and network limitations, as opposed to as bad as they can. Still, standard protections against a malicious adversary are needed, for example to prevent a double-spending, a hostile takeover, or a meltdown of the cryptocurrency supported by the consensus protocol.\n\nIt is within this context that we present \\textbf{Cordial Miners} -- a family of Byzantine Atomic Broadcast~\\cite{cachin2001secure} protocols that includes simpler, yet as efficient, counterparts to both DAG-Rider and Hotstuff.\n\nWe believe that the simplicity-cum-efficiency of the Cordial Miners protocols stems from the data structure and associated ordering algorithm that they share: \n\\begin{enumerate}\n \\item \\textbf{The Blocklace}~\\cite{shapiro2021multiagent}: The shared data structure is a partially-ordered generalization of the totally-ordered blockchain, termed \\emph{blocklace} (Def. \\ref{definition:block}), that consists of cryptographically-signed blocks, each containing a payload and a finite number of cryptographic hash pointers to previous blocks. The blocklace induces a DAG, as cryptographic pointers cannot form cycles even by a compute-bound adversary. The DAG induces a partial order $\\succ$ (Def. \\ref{definition:acknowledge}) on the blocks that includes Lamport's `happened-before' causality relation~\\cite{lamport1978time} among correct miners.\n \n The globally-shared blocklace is constructed incrementally and cooperatively by all miners, who also disseminate it to each other.\n \n \\item \\textbf{Ordering}: The ordering algorithm (Algorithm \\ref{alg:CMO}) is used locally by each miner to topologically-sort its locally-known part of the blocklace into a totally-ordered output sequence of blocks, excluding equivocation along the way. This conversion is monotonic (Prop. \\ref{proposition:tau-finality}) -- the output sequence is extended as the miner learns of or produces ever-larger portions of the global blocklace, and in this sense every output block of each miner is final. We say that two sequences are \\emph{consistent} if one is a prefix of the other (Def. \\ref{definition:prefix-consistent}), a notion stronger than the common prefix property of Ouroboros~\\cite{kiayias2017ouroboros}. We\n assume that less than one third of the miners are faulty,\n and prove that the following holds for the remaining correct miners of the Cordial Miners protocols:\n \\begin{itemize}\n \\item \\textbf{Safety}: Outputs of different miners are consistent (Prop. \\ref{proposition:safety}). \n \\item \\textbf{Liveness:} A block created by one is eventually output by everyone (Props. \\ref{proposition:aCM-liveness}, \\ref{proposition:sCM-liveness}, \\ref{proposition:CMAbd-liveness}).\n \n \n \\end{itemize}\n\\end{enumerate}\n\nThe simplicity of the protocols in the Cordial Miners family stems from their use of the blocklace and its analysis for all key algorithmic tasks (the following refers to correct miners):\n\\begin{enumerate}\n \\item \\textbf{Dissemination:} Any new block created by a miner $p$ acknowledges blocks known to $p$ by including pointers to the tips (DAG sources) of $p$'s local blocklace, as well as a pointer to $p$'s previous block. Correspondingly, a miner $p$ will buffer, rather than include in its blocklace, any received block with dangling pointers -- pointers to blocks not known to $p$.\n Hence, a block $b$ by $p$ informs any recipient $q$ of blocks not known to $p$ at the time of $b$'s creation. Thus $q$, being cordial, when sending to $p$ a new $q$-block, will include with it blocks $q$ knows but, to the best of $q$'s knowledge, are not yet known to $p$ and have not already been sent to $p$, thus ensuring block dissemination (Prop. \\ref{proposition:sCM-dissemination}). \n \n \\item \\textbf{Equivocation exclusion}: An \\emph{equivocation} (Def. \\ref{definition:da-faulty-consistent}) is a pair of blocks by the same miner that are not causally-related -- have no path of pointers from one to the other; such blocks are \\emph{conflicting} and a miner that creates them is considered faulty and is referred to as an \\emph{equivocator}. The shared blocklace will eventually include any conflicting block known to a correct miner, and hence eventually known to all correct miners. The question is: What should miners do with this knowledge? \n \n A block $b$ \\emph{acknowledges} block $b'$ if there is a (possibly empty) path from $b$ to $b'$, namely $b \\succeq b'$ (Def. \\ref{definition:acknowledge}). Let $[b]$ denote the set of blocks acknowledged by $b$, also referred to as the \\emph{closure} of $b$ (Def. \\ref{definition:closure}). A block $b$ \\emph{approves} block $b'$ if it acknowledges $b'$ and does not acknowledge any block $b''$ conflicting with $b'$ (Def. \\ref{definition:approval}, see Fig. \\ref{figure:approve-sr}.A). A key observation is that a miner cannot approve both blocks of an equivocation without being itself an equivocator (Ob. \\ref{observation:approve-da}). Hence, if less than one-third of the miners are equivocators, then no equivocation will ever receive an approval from blocks created by a \\emph{supermajority} (at least two-thirds) of the miners. This is the basis of equivocation-exclusion by the blocklace: A miner finalizes a block $b$ once its local blocklace includes blocks that approve $b$ by a supermajority (Algorithm \\ref{alg:CMO}).\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=15cm]{Figs\/approve-sr.jpeg}\n\\caption{\\oded{A few reviewers said that image A isn't clear. Maybe remove it?} Acknowledgement, Approval, Equivocation, Ratification, Super-Ratification: (A) Observing an Equivocation: Initial blocks are at the bottom, inclusion implies $\\succ$. Assume $b_1, b_2$ are an equivocation (Def. \\ref{definition:da-faulty-consistent}) by the red miner. According to the figure, $b''$ approves $b_2$ (Def. \\ref{definition:approval}) since it acknowledges $b_2$ (Def. \\ref{definition:acknowledge}, namely $b'' \\succ b_2$) and does not acknowledge any conflicting red block, in particular it does not acknowledge $b_1$. However, since $b'$ acknowledges $b''$ it acknowledges both $b_1$ and $b_2$ and hence does not approve the equivocating $b_1$ (nor $b_2$). (B) \\emph{Ratified}: The block (black dot) at round $r$ is ratified by the block (gray dot) at round $\\ge r+2$, as the black block is approved by a supermajority at round $r+1$ (green horizontal line) that is acknowledged by the gray block. (C) \\emph{Finalized}: A leader block (blue dot) at round $r$ is finalized if there is a supermajority (light green horizontal line) that includes a leader block (red dot) at round $r+2$, each member of which ratifies the blue leader at round $r$, namely each acknowledges a supermajority (dark green horizontal line) at round $r+1$ that approves the blue leader.}\n\n\n\\label{figure:approve-sr}\n\\end{figure}\n \n \\item \\textbf{Cordial Miners}: The \\emph{depth}, or \\emph{round}, of a block $b$ is the maximal length of any path emanating from $b$ (Def. \\ref{definition:block}). A \\emph{round} is a set of blocks of the same depth. Miners are cordial in two respects. First, as explained above, in informing other miners of blocks they believe the other miner lacks. Second, in waiting for a supermajority of round $d$ before producing a block of round $d+1$ (Def. \\ref{definition:cordial}).\n \n \n \\item \\textbf{Leader Selection}: Our protocols assume a fair \\emph{leader selection} function, that chooses a miner $p$ as \\emph{leader} for every round $d$, in which case a $p$-block of depth $d$ is a \\emph{leader block}. The leader selection function can be round-robin, pseudo-random, or random, selecting the leader retroactively using a shared coin (as in DAG-Rider~\\cite{keidar2021need}). The communication model (asynchrony, eventual synchrony) and the leader selection function determine the strength of the adversary (bounded, unbounded) the protocol can withstand. In particular, to withstand an unbounded adversary in the asynchronous model, the leader needs to be selected randomly and retroactively via a shared coin.\n \n \\item \\textbf{Ratified and Finalized Leaders}: A block $b$ of round $r$ is \\emph{ratified} by block $b'$ if $[b']$ includes a supermajority of round $r+1$ that approves $b$. A leader block $b$ of round $r$ is \\emph{final} if there is a supermajority of round $r+2$ that include a leader, each member of which acknowledges a supermajority of round $r+1$ that approves $b$. (Def. \\ref{definition:ratified-leaders}, see Fig. \\ref{figure:approve-sr}).\n\n \n \\item \\textbf{Ordering by Leaders}: We assume a given topological sort procedure that takes a blocklace as an input and produces a sequence of its blocks while respecting their causal partial order $\\succ$. With it, a recursive \\emph{ordering function} $\\tau$, applied to a leader block $b$, is defined as follows (Def. \\ref{definition:tau}): Let $b'$ be the highest-depth leader block in $[b]$ ratified by $b$.\n If there is no such $b'$, output the topological sort of $[b]$ and terminate. Else \n recursively call the ordering function with $b'$ and output its result following by the topological sort of $[b]\\setminus [b']$.\n \n \\item \\textbf{Finality by Finalized Leaders}: The key insight to finality is this (Proposition \\ref{proposition:finality-of-dr}): A finalized leader will be ratified by any subsequent leader (see Figure \\ref{figure:b-hat}). Hence, the ordering algorithm (Algorithm \\ref{alg:CMO}) used by any miner is as follows: When identifying a new finalized leader $b$ in its blocklace, apply the ordering function $\\tau$ to $b$ and output the newly added suffix since the previous finalized leader.\n As the ordering function is guaranteed to select the previous finalized leader in one of its recursive calls, it can be optimized to return if it does so, rather than recompute the prefix it has already delivered.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=14cm]{Figs\/finality.jpeg}\n\\caption{Finality of a Finalized Leader (\\ref{definition:leaders}): If a blue leader block of round $r$ (blue dot) is finalized then it is ratified by any subsequent leader leader (red dot).\n(A) If the red leader is at round $r+2$ then by definition of super-ratification it acknowledges a supermajority that approves the blue leader. (B+C) If the red leader is at a subsequent round $r+k$, $k>2$, then, being cordial, it acknowledges a supermajority (red horizontal line) of the previous round $r+k-1$, which must have a correct agent (purple) in common with the super-ratification supermajority at round $r+2$ (light green horizontal line).\n(B) If these supermajorities are of the same round $k=2$, then there is a single purple block that connects the red leader to the blue leader via the approving supermajority (dark green horizontal line) at round $r+1$. (C) Otherwise $k>2$, there are two purple blocks $b_1$, $b_2$ by the purple agent in the red $\\hat{B}$ and light green $B'$ supermajorities, which are connected via a path since the purple agent is correct, connecting the red leader $\\hat{b}$ to the blue leader $b$ via the approving supermajority $B$ (dark green horizontal line) at round $r+1$. In all cases, the red leader $\\hat{b}$ ratified the blue leader $b$.}\n\n\\label{figure:b-hat}\n\\end{figure*}\n\n\n \\item \\textbf{Identification and exclusion of faulty miners}: Any faulty $p$-block known to some correct miner will eventually be known to all, resulting in correct miners suspending any further communication with $p$. For example, any miner can easily verify whether another miner $p$ is cordial in the second sense by examining the blocklace. In addition, an equivocation by $p$, with each block of the pair known to a different correct miner, will eventually be known to all and result in the exclusion of $p$.\n \n \\item \\textbf{Exclusion of nonresponsive miners}: Being cordial endows legitimacy to ignore nonresponsive miners. A miner $p$ need not send new blocks to a miner $q$ as long as $q$ has not acknowledged a previous block $b$ sent to $q$ by $p$. If $q$ fail-stopped, then $p$ should definitely not waste resources on $q$; if $q$ is only suspended or delayed, then eventually it will send to $p$ a block acknowledging $b$, following which $p$---being cordial---will send to $q$ all the backlog $p$ has previously refrained from sending it, and is not acknowledged by the new block received from $q$.\n \n \n \\item \\textbf{Retroactive Leader Election via Shared Coin}\n In the model of asynchrony the adversary has complete control over the order of message delivery, indefinitely. The panacea to such an adversary, employed for example by DAG-Rider~\\cite{keidar2021need}, is to use a shared random coin~\\cite{cachin2005random} and elect the leader retroactively, and to allow more rounds for a finalized leader to emerge. Since miners are cordial, the notion of \\emph{common core}~\\cite{canetti1996studies} can be applied to prove a lower bound on the probability of a leader being finalized despite such a powerful adversary. The same approach can be applied here for the Cordial Miners protocols, at the cost of delaying expected finality from 2 rounds to 4 rounds, as shown in Figure \\ref{figure:coin-core}. As the approach uses threshold signatures, a secure method for key distribution is needed~\\cite{damgaard2001practical}.\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=12cm]{Figs\/coin-core.png}\n\\caption{Finality with a Shared Coin: }\n\n\\label{figure:coin-core}\n\\end{figure*}\n\n \n\\end{enumerate}\nMiners accomplish all the above by simple and efficient analyses of their local blocklace.\n\nWe present three simple instances of the Cordial Miners protocol family (Figure \\ref{figure:family}): A counterpart to DAG-Rider~\\cite{keidar2021need}, a BAB protocol in the asynchronous model; a counterpart to HotStuff~\\cite{yin2019hotstuff}, an SMR protocol in the eventual synchrony model; and a hybrid protocol that takes the best of both.\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=15cm]{Figs\/cm-family.png}\n\\caption{\\oded{Add the new algorithm 2 variant for the asynchronous setting} The Cordial Miners Protocols and their Algorithmic Components. Algorithms 1 \\& 2 are shared. Algorithm 1 includes basic blocklace utilities, Algorithm 2 realizes blocklace ordering. Algorithm 3 incorporates Algorithms 1 \\& 2 with a modular asynchronous data dissemination protocol to realize Byzantine Atomic Broadcast in the model of eventual asynchrony\/asynchrony, depending on the leader selection function employed. Algorithm 4 incorporates Algorithms 1 \\& 2 with cordial all-to-leader-to-all blocklace dissemination to realize Byzantine Atomic Broadcast in the eventual synchrony model. Algorithm 5 integrates ideas from the first two: It incorporates Algorithms 1 \\& 2 with cordial all-to-all block communication and cordial leader-to-all blocklace dissemination to realize Byzantine Atomic Broadcast in the model of eventual asynchrony\/asynchrony, depending on the leader selection function employed. Length in lines of pseudocode is noted for each. The asynchronous data-dissemination protocol employed by the Algorithm 3 is not shown.}\n\\label{figure:family}\n\\end{figure*}\nThe protocols are presented as algorithmic components written in pseudo-code, with some components shared between them:\n\\begin{enumerate}\n \\item The \\emph{Asynchronous Cordial Miners protocol with modular Asynchronous Data Dissemination} consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMA}.\n \\item The \\emph{Synchronous Cordial Miners protocol with Blocklace Dissemination} consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMS}.\n \\item The \\emph{Asynchronous Cordial Miners protocol with Blocklace Dissemination} consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO}, and \\ref{alg:CMAbd}.\n \\item Optimizations for all, which exclude equivocating and nonresponsive miners, are described in Algorithm \\ref{alg:CMOpt}.\n\\end{enumerate}\nFigure \\ref{figure:family} presents the shared (gray) and distinct (green\/blue\/red) components of three protocols in the family. Table \\ref{table:intro} compares their various properties with DAG-Rider and HotStuff.\n\\begin{table*}[]\n\\centering\n\n \\begin{tabular}{ | m{9em} | m{6em}| m{6em}| m{6em} | m{6em} |m{6em} |} \n\n \\hline\n & \\textbf{\\textsc{DAG-Rider}} & \\textbf{\\textsc{Asynch. Cordial Miners + ADD}}& \\textbf{\\textsc{HotStuff}} & \\textbf{\\textsc{Synch. Cordial Miners + BD}}& \\textbf{\\textsc{Asynch. Cordial Miners + BD}} \\\\\n \\hline\\hline\n \\textbf{Problem} & BAB & BAB & SMR & BAB & BAB \\\\\n \\hline\n \\textbf{Model} & Asynchrony & Asynchrony\/ Eventual Asynchrony & Eventual Synchrony & Eventual Synchrony & Asynchrony\/ Eventual Asynchrony\\\\ \n \n \n \\hline\n \\textbf{Cryptography} & Threshold signatures & Threshold signatures\/PKI & Threshold signatures & PKI & Threshold signatures\/PKI \\\\\n \\hline\n \\textbf{Data Types} & DAG of blocks, weak and strong edges & Blocklace & Blocks & Blocklace & Blocklace \n \\\\\n \n \\hline\n \\textbf{Message Types} & Block, Threshold signature & Signed Block, Threshold signature\n & Block, Signature, Threshold signature & Blocklace & Blocklace, w\/Threshold signatures\n \\\\\n \\hline\n \\textbf{Communication} & All-to-All & All-to-All & Leader-based & Leader-based & All-to-All + Leader-based backlog\n \\\\\n \\hline\n \\textbf{Underlying Protocol} & Reliable Broadcast & Asynchronous Data Dissemination & Reliable Links & Reliable Links & Reliable Links \n \\\\\n \\hline\n \\textbf{Leader Election} & Random & Random\/ Deterministic & Deterministic & Deterministic & Random\/ Deterministic \n \\\\\n \\hline\n \\textbf{Decision Rule} & Supermajority of $4^{th}$ round observing $1^{st}$ round leader & Super-ratified leader & Three consecutive correct leaders & Super-ratified leader & Super-ratified leader\n \\\\\n \n \\hline\n \\textbf{Expected Latency} & $\\frac{1}{(2\/3)*8}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1) & $\\frac{1}{(2\/3)^3}$ O(1)& $\\frac{1}{(2\/3)^3}$ O(1)\\textbf{}\\\\\n \\hline\n \\textbf{Bit complexity} & Amortized expected linear & Amortized expected linear & Good case linear & Amortized linear & Amortized linear\\\\\n \\hline\n \n \n \n\n\\hline\n \n \n \n \n\n\n\n\\end{tabular}\n\\caption{A comparison between DAG-Rider, Asynchronous Cordial Miners with Modular Dissemination, Hotstuff, Synchronous Cordial Miners, and Asynchronous Cordial Miners with blocklace Dissemination \\oded{Like we talked, need to think if removing the table is better than leaving it.}}\n\\label{table:intro}\n\\end{table*}\n\n\nAlgorithm \\ref{alg:blocklace} contains a description of a block in a blocklace and blocklace utilities. Algorithm \\ref{alg:CMO} describes the local conversion each miner executes with the function $\\tau$ converting its local copy of the partially-ordered blocklace into a totally-ordered sequence of blocks. Then there are two algorithms: Algorithm \\ref{alg:CMA}, an all-to-all communication protocol that is analogue to DAG-Rider~\\cite{keidar2021need}, and uses asynchronous data dissemination (e.g. ADD~\\cite{Das2021ADD}) as the underlying communication protocol; and Algorithm \\ref{alg:CMS}, a leader-based communication protocol that is analogue to HotStuff~\\cite{yin2019hotstuff}. Algorithm \\ref{alg:CMAbd} combines the best of both Algorithm \\ref{alg:CMA} and Algorithm \\ref{alg:CMS}: It uses all-to-all communication as in Algorithm \\ref{alg:CMA}, but eschews an external dissemination protocol by the leader of each round sending any backlog needed for dissemination as in Algorithm \\ref{alg:CMS}.\nAlgorithm \\ref{alg:CMOpt} presents several optimizations that exclude faulty and nonresponsive leaders in all protocols. \n\n\n\n\\subsection{Models, Problem, Safety and Liveness Proof Outline}\nWe assume $n\\ge 3$ miners $\\Pi$, of which $f < n\/3$ may be faulty (act arbitrarily, be \\emph{Byzantine}), and the rest are \\emph{correct}, and that a message sent from one correct miner to another will eventually arrive. We assume that every miner has a single and unique key-pair (PKI) and can cryptographically sign messages.\nEach miner $p \\in \\Pi$ has an input call $\\textit{payload}()$ that returns a payload (e.g., a proposal from a user or a mempool), and an output call $\\textit{deliver}(b)$ where $b$ is a block.\n\n\\begin{definition}\\label{definition:models}\nWe consider three models of communication with an adversary:\n\\begin{itemize}\n \\item \\emph{Asynchrony}, where an adversary controls the finite message delay of each message. \n \\item \\emph{Eventual Synchrony}~\\cite{dwork1988consensus}, where an adversary controls message delay for an unknown but finite number of messages, beyond which messages arrive within bounded delay.\n \\item \\emph{Eventual Random Asynchrony}, where an adversary controls message delay for an unknown but finite number of messages, beyond which messages arrive with random delay of finite expectation.\n \\item \\change{\\emph{Asynchrony}, where the adversary controls the delivery times of the messages.}\n\\end{itemize}\n\\end{definition}\n\nEventual asynchrony is a natural counterpart to eventual synchrony, yet novel to the best of our knowledge.\nThe Synchronous Cordial Miners protocol is devised for eventual synchrony and the Asynchronous Cordial Miners protocols are can be applied for eventual asynchrony with pseudorandom leader selection and to asynchrony with random retroactive shared-coin leader selection \\oded{Maybe cite some works about a common coin}.\nAll protocols address Byzantine Atomic Broadcast~\\cite{cachin2001secure}. \n\n\n\\begin{definition}[Prefix, $\\preceq$, Consistent Sequences]\\label{definition:prefix-consistent}\nA sequence $x$ is a \\emph{prefix} of a sequence $x'$, $x \\preceq x'$ if $x'$ can be obtained from $x$ by appending to it zero or more elements. Two sequences $x, x'$ are \\emph{consistent} if $x \\preceq x'$ or $x'\\preceq x$.\n\\end{definition}\n\n\\begin{definition}[Safety and Liveness]\\label{definition:safety-liveness}\nThese are the requirements of correct miners of a blocklace-based Byzantine Atomic Broadcast protocol: \n\\begin{itemize}\n \\item \\emph{Safety}: Outputs of miners are consistent.\n \\item \\emph{Liveness:} A block created \n \n by a correct miner is eventually output by every miner with probability 1.\n\\end{itemize}\n\\end{definition}\nWe note that these safety and liveness requirements, combined with the uniqueness of a block in a blocklace, imply the standard Byzantine Atomic Broadcast guarantees: Agreement, Integrity, Validity, and Total Order~\\cite{bracha1987asynchronous, keidar2021need}.\n\nProofs of safety and liveness of the two protocols proceed as follows, assuming less than one third of the miners are faulty, and referring to the remaining correct miners.\\\\\n\\textbf{Safety}: \n \\begin{enumerate}\n \\item Prove that the function $\\tau$ that converts a blocklace to a sequence of blocks is monotonic with respect to the superset relation (Prop. \\ref{proposition:tau-finality}).\n \\item Observe that if two sequences are each a prefix of a third sequence, then they are consistent (Ob.\\ref{observation:consistent-triplet}). Given any two local blocklaces $B, B'$ of miners $p, p'$, then due to the monotonicity of $\\tau$, both $\\tau(B)$ and $\\tau(B')$ are prefixes of $\\tau(B\\cup B')$. Therefore they are consistent (Corollary \\ref{corollary:consistency}).\n \\item Argue that Algorithm \\ref{alg:CMO} correctly implements $\\tau$ and hence is safe (Prop. \\ref{proposition:safety}).\n \\end{enumerate}\n\\textbf{Liveness:} \n The proofs of the protocols are different but have a common structure.\n \\begin{enumerate}\n \\item Observe that the conversion function $\\tau$, applied to a finalized leader block $b$, includes in the output sequence any block known to that leader, namely any block in $[b]$ (Ob. \\ref{observation:fairness}).\n \\item Given a block $b$ known to a miner at some point $t$ in the computation, argue that $b$ will eventually be known to every miner at some later point $t'$ in the computation (follows from the correctness of the underlying asynchronous data dissemination protocol used by Algorithm \\ref{alg:CMA}; claimed by Props. \\ref{proposition:sCM-dissemination},\\ref{proposition:CMAbd-dissemination} for Algs. \\ref{alg:CMS}, \\ref{alg:CMAbd}).\n \\item Argue that eventually some leader block $b'$ of a miner will be ratified at a point later than $t'$ with probability 1.\n \\item By construction, $b \\in [b']$, and hence $b$ is included in the output of $\\tau$ applied to $b'$ (Props. \\ref{proposition:aCM-liveness}, \\ref{proposition:sCM-liveness}, \\ref{proposition:CMAbd-liveness}).\n \\end{enumerate}\n\n\n\\subsection{Additional Related Work}\nThe use of a DAG-like structure to solve consensus has been introduced in previous works, especially in asynchronous networks~\\cite{moser1999byzantine}.\nHashgraph~\\cite{RN284} builds an unstructured DAG, with each block containing two references to previous blocks, and on top of the DAG the miners run an inefficient binary agreement protocol.\nThis leads to expected exponential time complexity.\nAleph~~\\cite{gkagol2018aleph} builds a structured round-based DAG, where miners proceed to the next round once they receive $2f+1$ DAG nodes from other miners in the same round.\nOn top of the DAG protocols run a binary agreement protocol to decide on the order of vertices to commit. Nodes in the DAG are reliably broadcast.\n\nDAG-Rider~\\cite{keidar2021need} is also a structured round-based DAG protocol that proceeds in rounds.\nNodes are reliably broadcast.\nThe DAG is divided to waves, each consisting of the nodes of four rounds. When a wave ends, miners locally check whether a decision rule is met, similar to our protocol.\nTusk~\\cite{danezis2021narwhal} is an implementation based on DAG-Rider.\nBullshark is a dual consensus protocol based on DAG-Rider that offers a fast-track to commit nodes every two rounds in case the network is synchronous.\nOther DAG-based consensus protocols include~\\cite{chockler1998adaptive,dolev1993early,sompolinsky2015secure,RN349}.\n\nHotStuff~\\cite{yin2019hotstuff} has a similar commit rule to our protocol of a finalized leader.\nThe commit rule is met when there are three consecutive correct leaders in a row.\nHotStuff is based on Tendermint~\\cite{buchman2016tendermint} and is also the core of several other consensus protocols~\\cite{kamvar2019celo,cypherium,flow,thunder}.\nUnlike HotStuff, Cordial miners guarantees fairness, i.e., each block proposed by a miner is eventually guaranteed to be included in the blockchain.\nHotStuff works in eventually synchronous networks and is a leader-based consensus protocol.\nIn this model, there are number of leader-based protocols such as DLS~\\cite{dwork1988consensus}, PBFT~\\cite{RN581}, Zyzzyva~\\cite{kotla2007zyzzyva}, SBFT~\\cite{gueta2019sbft}, and more.\n\n\\section{The Blocklace: A Partially-Ordered Generalization of the Totally-Ordered Blockchain}\n\nThe blocklace was introduced in reference~\\cite{shapiro2021multiagent}. For completeness we include the needed definitions and results, and for space considerations relegate them to Appendix \\ref{appendix:section:blocklace}. Blocklace utilities that realize these definition are presented in Algorithm \\ref{alg:blocklace}.\n\n\n\\begin{algorithm*}[t]\n \\caption{\\textbf{Cordial Miners\\xspace: Blocklace Utilities} \n \\\\ pseudocode for miner $p \\in \\Pi$}\n \n \\label{alg:blocklace}\n \\small\n \\begin{algorithmic}[1]\n \\Statex \\textbf{Local variables:}\n \\StateX struct $\\textit{block } b$: \\Comment{The struct of the most recent block $b$ created by miner $p$} \n \n \\StateXX $b.\\textit{creator}$ -- the miner that created $b$ \n \\StateXX $b.\\textit{payload}$ -- a set of transactions\n \\StateXX $b.\\textit{share}$ -- a share of the random coin, if needed \n \\StateXX $b.\\textit{pointers}$ -- a possibly-empty set of hash pointers to other blocks\n \\StateX \\textit{blocks} $\\gets \\{\\}$ \n \n \n\t\t\n\t\t\\vspace{0.5em}\n\t\t\\Procedure{$\\textit{create\\_block}$}{\\textit{blocks'}} \\Comment create into $b$ a new block pointing to the sources of \\textit{blocks}$'$\n\t\t\\State $b.\\textit{payload} \\gets \\textit{payload}()$ \\Comment e.g. dequeue a payload from a queue of proposals (aka mempool)\n\t\t\\State $b.\\textit{creator} \\gets p$\n\t\t\\State $b.\\textit{pointers} \\gets \\{hash(b') ~:~ b' \\in \\textit{blocks}', \\text{ $b'$ has no incoming pointers in \\textit{blocks}}' \\}$ \n\t\t\\State $b.\\textit{share} \\gets \\textit{choose\\_leader}_p(\\textit{depth}(b)+3)$ \\Comment If a shared coin is employed\n\t \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMA:addedOutgoingBlock}\n\t \\State $\\textit{deliver\\_blocks}()$\n\t \\State \\Return $b$\n\t\t\\EndProcedure\n\t\t\n\t\t\n \\vspace{0.5em}\n\t\t\\Procedure{\\textit{hash}}{$b$}\n\t \\Return collision-free cryptographic hash pointer(s) to block(s) $b$ \\Comment{Returns a set if applied to a set}\n \\EndProcedure\n \n\t\t\n \\vspace{0.5em}\n \\Procedure{\\textit{path}}{$b,b'$} \n \\Return $\\exists b_1,b_2,\\ldots,b_k \\in \\textit{blocks}$, $k\\ge 1$, s.t.\\\n $b_1 = b $, $b_k = b'$ and $\\forall i \\in [k-1] \\colon b_{i+1} \\in b_i.\\textit{pointers}$\n \\EndProcedure\n \n \\vspace{0.5em}\n \\Procedure{\\textit{depth}}{$b$} \n \\Return the max~ $\\{k : \\exists b' \\in$ \\textit{blocks} and \\textit{path$(b,b')$} of length $k$\\}. \\Comment{Depth of block $b$}\n \n \n \\EndProcedure\n \n \\vspace{0.5em}\n \\Procedure{\\textit{depth}}{\\textit{blocks}} \n \\Return $\\max~ \\{\\textit{depth}(b) : b \\in \\textit{blocks}\\}$ \\Comment{Depth of set of blocks}\n \\EndProcedure\n \n\t \n \\vspace{0.5em}\n \\Procedure{\\textit{blocks\\_prefix}}{$d$}\n \\Return $\\{b \\in \\textit{blocks} : \\textit{depth}(b) \\le d\\}$\n \\EndProcedure\n \n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{closure}}{$b$} \\label{alg:SMR:closure}\n \\Return \\{$b' \\in \\textit{blocks} : \\textit{path}(b,b')$\\} \\Comment{also referred to as $[b]$. $b$ could also be a set of blocks}\n \n \\EndProcedure\n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{leader}}{$d$} \n \\Return an index in $[n]$ or $\\bot$. \n \\oded{Otherewise return $\\bot$? Why not just use round robin? odd is not defined, better to use $d \\bmod 2 = 1$ or ``if d is odd''}\n \\udi{Let's discuss how to define it parametric for all scenarios including shared coin.}\n \n \n \\EndProcedure\n \n \n \n \\vspace{0.5em}\n \\Procedure{\\textit{leaders}}{\\textit{blocks}} \n \\Return $\\{b \\in \\textit{blocks} : b.\\textit{creator} = \\textit{leader}(\\textit{depth}(b))\\}$\n \\EndProcedure\n \n \n \n\t\t\\vspace{0.5em} \n\t \\Procedure{\\textit{equivocation}}{$b_1,b_2$} \\label{alg:SMR:da}\n \\Return \n $b_1.\\textit{creator} = b_2.\\textit{creator} \\wedge\n b_1 \\notin [b_2] \\wedge\n b_2 \\notin [b_1]\n $ \\Comment{See Figure \\ref{figure:approve-sr}.A} \n \n \n \\EndProcedure\n\t\t\n\t\t\n \t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{approved}}{$b_1,b$} \\label{alg:SMR:approved}\n \\Return $b_1 \\in [b] \\wedge \\forall b_2 \\in [b] : \\lnot$\\textit{equivocation}$(b_1,b_2)$ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n \\EndProcedure\n \n \n\t\t \\vspace{0.5em}\t\n\t\t\\Procedure{\\textit{ratified}}{$b_1,b_2$} \\label{alg:SMR:sm_approved} \n \\Return\n $|\n \\{b.\\textit{creator} : b \\in \\textit{blocks} \\wedge \n b \\in [b_2] \\wedge\n \\textit{approved}(b_1,b) \\}\n | \\ge 2f+1$ \\Comment{See Figure \\ref{figure:approve-sr}.B}\n \\EndProcedure\n \n \n\t \n \\vspace{0.5em}\n \\Procedure{\\textit{cordial\\_block}}{$b$}\n \\Return \n $|\\{b'\\in [b] : \\textit{depth}(b') = \\textit{depth}(b)-1\\}| \\geq 2f+1$\n \\EndProcedure\n \n\t \n \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{cordial\\_round}()$}{} \\label{alg:CMO:cr} \\Comment{Returns recent new cordial round, singleton or the empty set, Def. \\ref{definition:cordial}}\n \\State \\Return $\\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$ where $R =$\n \\label{alg:CMA:computeCordial}\n \\State $\\{ r \\in \\textit{depth}(\\textit{blocks}) : \n |\\{b.\\textit{create} : b \\in \\textit{blocks} \\wedge b.\\textit{depth} = r\\}| \\ge 2f+1 \\wedge\n \\not\\exists b \\in \\textit{blocks} : (b.\\textit{creator} = p \\wedge \\textit{depth}(b) \\ge \\textit{depth}(r))\\}$ \n \\EndProcedure \n \n \n \\vspace{0.5em}\n\t\t\\change{\\Procedure{$\\textit{choose\\_leader}$}{$r$} \\Comment{A random shared coin}\n\t\t \\State \\Return a random leader process drawn by the coin \n\t\t \\State \\oded{Need to add somewhere in the text the properties of the coin: doesn't return until at least f+1 processes call it, the adversary cannot guess the outcome of the coin in advance, the probability of each process to be chosen is equal, the coin returns the same results for each $r$.}}\n\t\t \n \n \\alglinenoNew{counter}\n \\alglinenoPush{counter}\n\n \\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\\section{Converting a Blocklace into a Sequence of Blocks}\n\nHere we present a deterministic function $\\tau$ that incrementally converts a blocklace into a sequence of some of its blocks, and show that it is monotonic wrt to the subset relation, provided no more than $f$ miners equivocate. The intention is that each miner in a blockchain consensus protocol employs $\\tau$ to locally compute the final output sequence of blocks from its local copy of the blocklace as input, as realized by Algorithm \\ref{alg:CMO}. Monotonicity ensures finality, as it implies that the output sequence will only extend while the input local blocklace increases over time. Monotonicity is a sufficient condition for the safety of each protocol that uses $\\tau$ (Proposition \\ref{proposition:tau-finality} below). To ensure liveness, one has to argue that every block in the input blocklace will eventually be in the output of $\\tau$; this argument is protocol-specific and argued separately for the asynchronous protocol (Prop. \\ref{proposition:aCM-liveness}) and the synchronous protocol (Prop. \\ref{proposition:sCM-liveness}).\n\n\\begin{definition}[Cordial]\\label{definition:cordial}\nLet $b', b\\in \\mathcal{B}$ be two consecutive $p$-blocks.\nThen block $b$ is \\emph{cordial} if $[b] \\setminus ( [b'] \\cup \\{b\\})$ is a supermajority.\nMiner $p$ is \\emph{cordial} in blocklace $B \\subseteq \\mathcal{B}$ if every $p$-block $b \\in B$ is cordial.\n\\end{definition}\n\n\\begin{definition}[Round, Leader, Leader Block]\\label{definition:leaders}\nGiven a blocklace $B \\subseteq \\mathcal{B}$, then a \\emph{round} $r \\ge 1$ in $B$ is the set of blocks \n$\\{b\\in B : \\textit{depth}(b)=r\\}$. We assume a leader selection function $\\textit{leader}: \\mathbb{N} \\mapsto \\Pi \\cup \\{\\bot\\}$ that is defined only for depths. If $\\textit{leader}(r)=p$ then $p$ is the \\emph{leader} of round $r$, and if, in addition, $b\\in B$ is a $p$-block of depth $r$, then $b$ is a \\emph{leader block} of round $r$ in $B$.\n\\end{definition}\n\n \n \nSee Figure \\ref{figure:approve-sr}.B for the following definition.\n\\begin{definition}[Ratified and Finalized Leader Blocks]\\label{definition:ratified-leaders}\nA block $b \\in \\mathcal{B}$ is \\emph{ratified} if there is a supermajority of depth $d(b)+1$ that approves $b$. A leader block $b$ of depth $r$ is \\emph{final} if there is a supermajority $B$ of depth $d(b)+2$ that includes a leader block such that each member of $B$ ratifies $b$.\n\\end{definition}\n\nThe following proposition ensures that a finalized block is ratified by any subsequent cordial block (Fig. \\ref{figure:b-hat}).\n\n\\begin{proposition}[Finality of Super-Ratification]\\label{proposition:finality-of-dr}\nIf $b$ is finalized leader in $B$, then it is ratified by any subsequent leader \n$b' \\succ b$ in $B$. \n\\end{proposition}\n\\begin{proof}\nAssume a leader block $b$ finalized via supermajorities $B'$ and $B$ (Fig. \\ref{figure:approve-sr}.B), and assume that $\\hat{b}$ is some subsequent leader block that, being cordial, acknowledges a supermajority of the preceding round $\\hat{B}$. By counting, at least one block $b_1 \\in B'$ and one block $b_2 \\in \\hat{B}$ are both $p$-blocks by the same correct miner $p \\in \\Pi$ (Fig. \\ref{figure:b-hat}). Since $p$ is not an equivocator $b_2 \\succ b_1$. Hence $\\hat{b} \\succ b_2 \\succ b_1$ and $b_1$ ratifies $b$, thus $\\hat{b}$ ratifies $b$.\n\\end{proof}\n\nProposition \\ref{proposition:finality-of-dr} ensures that given a blocklace $B$, a finalized leader $b$ in $B$ will be ratified by any subsequent finalized leader, hence included in the sequence of final leaders. Hence, the final sequence up to a finalized leader $b$ is fully-determined by $b$ itself independently of the (continuously changing) identity of the last finalized leader. Hence the final sequence up to a finalized leader $b$ can be `cached' and will not change as the blocklace increases.\n\n\nSuper-ratified leaders are the anchors of finality in a growing chain, each `writes history' backwards till the preceding ratified leader. We use the term `Okazaki fragments'~\\cite{okazaki2017days} for the sequences computed backwards from each finalized leaders to its predecessor, acknowledging the analogy with the way one of the DNA strands of a replicated DNA molecule is elongated via the stitching of backwards-synthesized Okazaki-fragments.\n\n\nThe following recursive ordering function $\\tau$ maps a blocklace into a sequence of blocks. Formally, the entire sequence is computed backwards from the last finalized leader, afresh by each application of $\\tau$. Practically, a sequence up to a finalized leader is final (Prop. \\ref{proposition:tau-finality}) and hence can be cached, allowing $\\tau$ to be computed in `Okazaki-fragment' increments from the new finalized leader backwards to the previous finalized leader. \n\n\n\n\n\n\\begin{definition}[$\\tau$]\\label{definition:tau}\nWe assume an fixed ordering function (e.g. lexicographic) $s$ that maps a blocklace to a sequence of blocks consistent with $\\succ$. The function $\\tau: 2^{\\mathcal{B}} \\xrightarrow{} \\mathcal{B}^*$ is defined for a blocklace $B\\subset \\mathcal{B}$ backwards, from the last output element to the first, as follows:\nIf $B$ has no finalized leaders then $\\tau(B) :=\\Lambda$.\nElse let $b$ be the last finalized leader in $B$. Then $\\tau(B) := \\tau'(b)$, where $\\tau'$ is defined recursively:\n$$\n\\tau'(b) := \n\\begin{cases}\ns([b]) \\text{\\ \\ if $[b]$ has no leader ratified by $b$, else }\\\\\n\\tau'(b') \\cdot s([b]\\setminus [b']) \\text{\\ \\ if $b'$ is the last leader ratified by $b$ in $[b]$}\n\\end{cases}\n$$\n\\end{definition}\nNote that $\\tau'$ uses the notion of a leader ratified by another leader, not a finalized leader.\n\n\nProposition \\ref{proposition:finality-of-dr} shows that if there are less than $f$ equivocators, a finalized leader is final, in that it will be ratified by any subsequent leader, finalized or not. Hence the following:\n\n\\begin{proposition}[Monotonicity and Finality of $\\tau$]\\label{proposition:tau-finality}\nIf there are at most $f$ equivocators, the function $\\tau$ is monotonic wrt $\\subseteq$, namely $B \\subseteq B'$ for two blocklaces with at most $f$ equivocators implies that $\\tau(B) \\preceq \\tau(B')$, and in this sense membership in $\\tau$ is final. \n\\end{proposition}\n\\begin{proof}[Proof of Proposition \\ref{proposition:tau-finality} ]\nAssume given blocklaces $B \\subset B' \\subseteq \\mathcal{B}_{21}$.\nIf $\\tau(B)=\\Lambda$ the claim holds vacuously.\nSo let $b$ be the last finalized leader in $B$. If $B'$ has no finalized leader not in $B$, then $\\tau(B) = \\tau(B')$ and the claim holds vacuously. So let $b_1,\\ldots, b_k$, $k\\ge 1$, be \nthe sequence of final leaders in $B' \\setminus B$.\nThen by the definition of $\\tau$, $\\tau(B') = \\tau(B) \\cdot s([b_1]\\setminus [b]) \\cdot \\ldots \\cdot s([b_k]\\setminus [b_{k-1}])$, namely $\\tau(B) \\preceq \\tau(B')$.\n\\end{proof}\n\n\n\\begin{observation}[Consistent triplet]\\label{observation:consistent-triplet}\nGiven three sequences $x, x', x''$, if $x' \\preceq x$ and $x'' \\preceq x$ then $x'$ and $x''$ are consistent.\n\\end{observation}\n\nThe following Corollary ensures that the output sequences computed by miners based on their local blocklaces would be consistent as long as there are there are at most $f$ equivocators.\n\n\\begin{corollary}[Consistency of Output Sequences]\\label{corollary:consistency}\nIf $B, B'\\subset \\mathcal{B}$ are grounded with at most $f$ equivocators in $B := B' \\cup B''$ then $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{corollary}\n\\begin{proof}\nBy monotonicity of $\\tau$ with at most $f$ equivocators (Prop. \\ref{proposition:tau-finality}), $\\tau(B') \\preceq \\tau(B)$ and $\\tau(B'') \\preceq \\tau(B)$. By Observation \\ref{observation:consistent-triplet}, $\\tau(B')$ and $\\tau(B'')$ are consistent.\n\\end{proof}\n\nWhile $\\tau$ does not output all the blocks in its input, as blocks not acknowledged by the\nlast finalized leader in its input are not delivered, the following observation reminds us of the half-full glass:\n\\begin{observation}[Fairness of $\\tau$]\\label{observation:fairness}\nIf a block $b \\in B$ is acknowledged by the last finalized leader in $B$, then it is included in $\\tau(B)$.\n\\end{observation}\nIf every block known to a correct miner will be known to all correct miners, \nand if every finalized leader has a successor, then\nevery block will be eventually acknowledged by some finalized leader\nand therefore be delivered by $\\tau$.\n\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Conversion of Blocklace to Sequence of Blocks with $\\tau$} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMO}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\n\t\t\\vspace{0.5em}\n\t\t\\Statex \\textbf{Local Variable:}\n\t\t\\StateX $\\textit{deliveredBlocks} \\gets \\{\\}$\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\\Upon{ $\\textit{deliver\\_blocks}()$} \\Comment{Signal that a new block was added to \\textit{blocks} (\\cref{alg:CMA:addedOutgoingBlock}, \\cref{alg:CMA:receive}}, \\cref{alg:CML:addedIncomingBlock})\n\t\t\\label{alg:CMO:waveCompletion} \n\t\t\n\t\t \n\t\t\\EndUpon\n\t\t\\vspace{0.5em}\n\t\t\n\t\t\n\t\n\t\t\\label{alg:CMO:commitrule}\n\t\t\n\t\t \\Procedure{\\textit{tau}}{$b_1$} \\label{alg:CMO:da}\n\t\t \t\\If{$b_1 \\in \\textit{deliveredBlocks} \\vee b_1 = \\emptyset$}\n\t\t \\Return\n\t\t \\EndIf\n\t\t \\State $b_2 \\gets \\textit{arg}_{r \\in R} \\max~ \\textit{depth}(r)$\n \\textit{ where } $R = \\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\textit{ratified}(b,b_1)\\}$ \n \\Comment{Previous ratified leader or $\\emptyset$, Fig. \\ref{figure:b-hat}}\n \\label{alg:CMO:previous_ratified}\n \\State \\textit{tau}$(b_2)$ \\Comment{Recursive call to $tau$}\n\t\t \\For{$\\textbf{every} ~b \\in [b_1] \\setminus [b_2]$, ordered by topological sort} \\Comment{Deliver a new `Okazaki-fragment'}\n\t \t\t \\State \\textbf{deliver} $(b)$ \n\t \t\t \\label{alg:CMO:Okazaki}\n\t\t \\State $\\textit{deliveredVertices} \\gets \\textit{deliveredVertices} \\cup \\{b\\}$\n\t\t \\EndFor\n\t\n\t\t \\EndProcedure\n\t\t\n \\vspace{0.5em}\n\t\t\\Procedure{$\\textit{super\\_ratified\\_leader}()$}{} \\label{alg:CMO:dr} \\Comment{Returns last finalized leader, singleton or the empty set, Figure \\ref{figure:approve-sr}.B}\n \\State \\Return $\\textit{arg}_{u \\in U} \\max~ \\textit{depth}(u)$ where $U =$\n \\State $\\{ b \\in \\textit{leaders}(\\textit{blocks}) : \\exists b', b'' \\in \\textit{leaders}(\\textit{blocks}) \\wedge$ \n $\\textit{depth}(b) +4 = \\textit{depth}(b')+2 = \\textit{depth}(b'') \\wedge$ \n $\\textit{ratified}(b,b') \\wedge \\textit{ratified}(b',b'')\\}$ \n \n \n \\EndProcedure\n \n \n\t\t \n\t\\alglinenoPush{counter}\t\n\t\\end{algorithmic}\n\\end{algorithm*}\n\n\n\n\n\n\nAlgorithm \\ref{alg:CMO} is a fairly literal implementation of the mathematics described above:\nIt maintained \\emph{deliveredBlocks} that includes the prefix of the output of $\\tau$ that it has already computed. Upon adding a new block to its blocklace (\\cref{alg:CMO:waveCompletion}), it computes the most-recent finalized leader $b_1$ according to Definition \\ref{definition:ratified-leaders}, and applies \\emph{tau} to it, which is intended to be a literal realization of the mathematical definition of $\\tau$ (Def. \\ref{definition:tau}), with the optimization, discussed above, that a recursive call with a delivered block is returned. Hence the following proposition:\n\n\\begin{proposition}[Tau implements $\\tau$]\\label{proposition:tau-tau}\nThe procedure \\emph{tau} in Algorithm \\ref{alg:CMO} correctly implements $\\tau$ in Definition \\ref{definition:tau}.\n\\end{proposition}\n\nAnd based on it, we conclude the following, which ensures the safety of every protocol that employs Algorithm \\ref{alg:CMO} for ordering a blocklace; in particular, the safety of the asynchronous and synchronous Cordial Miners protocols.\n\\begin{proposition}\\label{proposition:safety}\nAlgorithm \\ref{alg:CMO} satisfied the safety requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nProposition \\ref{proposition:tau-tau} establishes that\nAlgorithm \\ref{alg:CMO} implements $\\tau$ correctly. Together with Corollary \\ref{corollary:consistency}, we conclude that the outputs of all miners using Algorithm \\ref{alg:CMO} are consistent, satisfying the safety requirement. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Asynchronous Cordial Miners Protocol}\n\nDAG-Rider~\\cite{keidar2021need} is a BAB protocol designed for the asynchronous model. It assumes an adaptive adversary that eventually delivers messages between any two correct miners.\nIn DAG-Rider the miners jointly build a DAG of blocks, with blocks as vertices and pointers to previously-created blocks as edges, divided into strong and weak edges.\nStrong edges are used for the commit rule, and weak edges are used to ensure fairness.\nThe protocol employs an underlying reliable broadcast protocol of choice, which ensures that eventually the local DAGs of all correct miners converge and equivocation is excluded.\nEach miner independently converts its local DAG to an ordered sequence of blocks, with the use of threshold signatures to implement a global coin that retrospectively chooses one of the miners as the leader for each round.\nThe decision rule for delivering a block is if the vertex created by the leader is observed by at least $2f+1$ miners three rounds after it is created.\nDAG-Rider has an expected amortized linear message complexity, and expected constant latency.\n\nThe Asynchronous Cordial Miners protocol is our DAG-Rider counterpart. It assumes an underlying data dissemination protocol (e.g. \\cite{Das2021ADD}), consists of Algorithms \\ref{alg:blocklace}, \\ref{alg:CMO} \\& \\ref{alg:CMA}.\nAlgorithm \\ref{alg:CMA} assume an asynchronous data dissemination protocol as a sub-protocol. For example, it could use ADD~\\cite{Das2021ADD}. To do so, when a miner executing the Cordial Miners protocol calls \\textbf{disseminate}$(b)$, it sends the block $b$ to all other miners. This ensures that $2f+1 > f+1$ correct miners receive $b$, satisfying condition for ADD to operate correctly. A miner that receives a proper block $b$ (signed, cordial, with no dangling pointers) for the first time initiates the ADD protocol with $b$. Every shard of $b$ created by the ADD protocol is tagged with the hash value of $b$, so that multiple instances of the ADD protocol on multiple blocks can proceed concurrently, namely asynchronously and in parallel, without confusing between the blocks of the shards. When the ADD protocol reconstructs a block $b'$, the \\textbf{receipt} condition of the Cordial Miners protocol is fulfilled, with block $b'$. The correctness of ADD ensures that every block disseminated by a correct miner will eventually be reconstructed and received by every correct miner.\n\n\n\nThe Asynchronous Cordial Miners protocol is designed for the model of eventual asynchrony (Def. \\ref{definition:models}).\nThe dissemination component is called with $\\textit{disseminate}(m)$ for some block $b$ and can output $\\textit{receipt}(b)$. We assume that the component guarantees that if a correct miner calls $\\textit{disseminate}(b)$ or $\\textit{receipt}(b)$ then eventually all correct miners perform $\\textit{receipt}(b)$.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Asynchronous Protocol with Modular Block Dissemination}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMA}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\n\t\n\t\\vspace{0.5em}\n\t\n\t\\Upon{$\\textbf{receipt}(b)$} \\Comment{\\textbf{disseminate} and \\textbf{receipt} are calls to the Asynchronous Data Dissemination (ADD) protocol~\\cite{Das2021ADD}} \\oded{The ADD guarantees in [13] are different from the ones needed here} \\udi{ Please explain\/let's discuss}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$ \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t } \\Comment{Accept cordial block with no dangling pointers}\n \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMA:receive}\n \t\\State $\\textit{buffer} \\gets \\textit{buffer} \\setminus \\{ b \\}$\n \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t \n\t \\If{ $\\textit{cordial\\_round}()$} \\label{alg:CMA:isCordial}\n\t \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$ \n\t \\State $\\textbf{disseminate}(b)$ \n\t \\EndIf\n\t \\EndWhile\n\t \n \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n \nAlgorithm \\ref{alg:CMA} is short and sweet. All it does is buffer incoming blocks (\\cref{alg:CMA:buffer}), receive them when they have no dangling pointers (\\cref{alg:CMA:receive}), and if it observes a new cordial round (\\cref{alg:CMA:isCordial}), it creates a block for the most recent cordial round in which it has not participated (\\cref{alg:CMA:computeCordial}) and disseminates it. That's it.\n\nThe dissemination protocol ensures that all correct miners eventually the same blocks. The only concern is that the adversary, being familiar with the `celebrated FLP theorem'~\\cite{fischer1985impossibility}, will order message delivery so that in every round the supermajority first observed by all correct miners will not include the leader of that round. This way no ratified leader, let alone finalized one, will emerge. Fortunately, in the eventual asynchrony model our adversary will wane after controlling delivery times for some finite number of messages, and random network delays will rule thereafter. Hence, from any point in the computation, a finalized leader will eventually emerge and all blocks acknowledged by it and not yet delivered will be delivered. Hence the following:\n\n\n\\begin{proposition}[Liveness of Asynchronous Cordial Miners protocol with Modular Asynchronous Data Dissemination]\\label{proposition:aCM-liveness}\nAlgorithm \\ref{alg:CMA} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMA} beyond the control of the adversary. The underlying asynchronous data dissemination protocol ensures that every block $b$ known to a miner will be known eventually to every miner. The probability of each leader block in this suffix being ratified by its successor is greater than some fixed $\\epsilon$, hence the probability of a leader block being finalized is greater than some smaller but fixed $\\epsilon'$. Hence the probability measure of an infinite computation in which no correct leader block being finalized is zero. \nHence, for any block $b$ and point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be finalized at a point later than $t'$ with probability 1. Thus for any block $b$ known to a correct miner, there will be a subsequent finalized leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\n \n\\section{Synchronous Cordial Miners Protocol}\n\nHotStuff~\\cite{yin2019hotstuff} is an SMR protocol designed for the eventual synchrony model. \nThe protocol employs all-to-leader, leader-to-all communication: In each round, a deterministically-chosen designated leader proposes a block to all and collects from all signatures on the block. Once the leader has $2f+1$ signatures, it can combine them into a threshold signature~\\cite{boneh2001short} which it sends back to all.\nThe decision rule for delivering a block is three consecutive correct leaders.\nThis leads to a linear message complexity and constant latency in the good case. The protocol delivers a block if there are three correct leaders in a row, which is guaranteed to happen after GST.\n\n\nThe Synchronous Cordial Miners protocol is our Hotstuff counterpart, s designed for the model of eventual synchrony (Def. \\ref{definition:models}). It consists of Algorithms \n\\ref{alg:blocklace}, \\ref{alg:CMO} \\& \\ref{alg:CMS}).\n\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Synchronous Protocol with Blocklace Dissemination} \\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMS}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[1..n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$\n\t\\Comment{Communication history of $p$ with other miners} \n\t\\StateX $\\textit{buffer} \\gets \\{ \\}$\n \n\n\n\t\n\t\\vspace{0.5em}\n\t\n\t\\Upon{\\textbf{receipt} of $b$} \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link} \n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{b \\}$ \\Comment{New blocks are buffered}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t \\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t \\wedge \\textit{cordial\\_block}(b)$\n\t } \\Comment{Accept cordial block with no dangling pointers}\n\t \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMS:historyReceive}\n\t \\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \n\t \\label{alg:CML:addedIncomingBlock}\n\t \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n\t \\State $\\textit{deliver\\_blocks}()$\n\t \\EndFor\n\t \n\t \\If{$p= \\textit{leader}(\\textit{cordial\\_round}())$} \\Comment{Cordial leader sends packaged new block to all responsive miners} \\label{alg:CMS:codrialLeader} \n\t \n\t \n\t \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$ \n\t\t\t \\For{$q \\in \\Pi \\wedge q \\ne p \\wedge \\textit{responsive}(q) $}\n\t\t\t \\label{alg:CMAbd:isCordial}\n\t \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$ \n\t \n\t \n\t\t \\label{alg:CMS:CreatePackage}\n\t\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySendPackage}\n\t\t \\State \\textbf{send} \\textit{package} to $q$\n\t\t \\EndFor\n\t\t \\EndIf\n \\If{$p \\neq \\textit{leader}(\\textit{depth}(\\textit{blocks})+1)~ \\vee$ \\textit{timeout}()} \\Comment{Non-leader sends new block to leader} \n \\udi{It seems that this can be taken erroneously due to a Byzantine block}\n \n \\State $b \\gets \\textit{create\\_block}(\\textit{blocks}))$ \n\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\{b\\}$ \\label{alg:CMS:historySendBlock}\n\t\t \\State \\textbf{send} $b$ to \\textit{leader}(\\textit{depth}$(b)$)\n \\EndIf \n \n\t\\EndWhile\t \n \n \n\n \n \n \n \n \n\n \n \n \n \n \n \n \n \\Procedure{\\textit{responsive}}{$q$} \\label{alg:CMS:responsive}\n \\Return \n $\\forall b \\exists b'\\in \\textit{history}[q] : b.\\textit{creator} = p \\wedge b'.\\textit{creator} = q \\wedge b \\in [b'] $\n \\EndProcedure\n \n \n \n \n \n \n \n \n \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\nUnlike the Asynchronous Cordial Miners protocol (Algorithm \\ref{alg:CMA}), which relies on another protocol to perform dissemination, the Synchronous Cordial Miners protocol in Algorithm \\ref{alg:CMS} employs the the blocklace structure to perform dissemination. To achieve that, each miner maintains a \\emph{history} array that records its communication history with the other miners, and updates it upon receiving a block (\\cref{alg:CMS:historyReceive}) and upon sending one (\\cref{alg:CMS:historySend}).\nAn ordinary miner $p$ simply sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to the leader (\\cref{alg:CMS:CreateBlockPackage}). A leader $p$, on the other hand, sends to each responsive miner $q$ not only the block $b$, but also all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]$ . This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all blocks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\nWhen sending a block to miner $q$, the miner $p$ uses their communication history to create a package of all blocks $p$ knows, which $q$ may not know (\\cref{alg:CMS:SendPackage}) and sends the whole package. Based on this, we argue the following:\n\n\n\n\n\n\\begin{proposition}[Algorithm \\ref{alg:CMS} Dissemination]\\label{proposition:sCM-dissemination}\nIn any run of Algorithm \\ref{alg:CMS}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof}\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$. If $q$ is correct then eventually $p$ will send a block to $q$, with one of them being a leader. If at that time the communication history of $p$ with $q$ shows that $q$ knows $p$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history. Then $q$ will eventually receive the package. The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\nUnlike Algorithm \\ref{alg:CMA}, in which block communication is all-to-all, here in Algorithm \\ref{alg:CMS} block communication is from all-to-leader-to-all. This requires less synchronization, as only leaders need to be cordial, not all miners as in Algorithm \\ref{alg:CMA}, and in the good case reduces message complexity (but not bit complexity), since the leader of each round serves as a relay for all miners for that round. As a result, the argument for the liveness of the protocol is a bit different.\n\n\\begin{proposition}[Liveness of Synchronous Cordial Miners protocol]\\label{proposition:sCM-liveness}\nAlgorithm \\ref{alg:CMS} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nProposition \\ref{proposition:sCM-dissemination} ensures that every block known to a correct miner will be known to every correct miner.\nConsider a suffix of an infinite computation beyond the control of the adversary. Every miner will get there despite the adversary thanks to the timeout (\\cref{alg:CMS:timeout}).\nIn this synchronous suffix, the probability of a leader block being ratified by its successor is $(\\frac{2}{3})^2$, and the probability of a leader block being finalized is $(\\frac{2}{3})^3$. \nHence the probability measure of an infinite computation in which no correct leader block being finalized is zero. \n\nHence, for any block $b$ and any point $t$ in the computation, some leader block $b'$ that acknowledges $b$ will be finalized at a point later than $t'$ with probability 1. Thus for any block $b$ known to a correct miner, there will be a subsequent finalized leader block $b'$ that acknowledges $b$ and hence delivers $b$, satisfying the liveness requirement.\n\\end{proof}\n\n\n\n\\section{Asynchronous Cordial Miners with Blocklace Dissemination}\n\nAlgorithm \\ref{alg:CMAbd} is a hybrid of Algorithm \\ref{alg:CMA} and Algorithm \\ref{alg:CMS}.\nIt employs all-to-all block communication as Algorithm \\ref{alg:CMA}. But rather than employing an underlying asynchronous data dissemination protocol, it\nperforms dissemination \\emph{en passant}, similarly to Algorithm \\ref{alg:CMS}\nAn ordinary miner $p$ simply sends the block $b$ it has created, which includes pointers to the blocks $p$ knows, to all miners (\\cref{alg:CMS:CreateBlockPackage}). A leader $p$, on the other hand, operates similarly to a leader in Algorithm \\ref{alg:CMA}: It sends to each responsive miner $q$ not only the block $b$, but also all the blocks that $p$ knows but, to the best of $p$'s knowledge, $q$ does not, namely $[b] \\setminus [\\textit{history}[q]$ . This package is constructed in \\cref{alg:CMS:CreatePackage}. Excluded from the closure of $b$ it is the closure of all $q$-blocks known to $p$ and all bloks $p$ has sent to $q$, both recorded in $history[q]$ of miner $p$.\nA miner $q$ is considered \\emph{responsive} (\\cref{alg:CMS:responsive}) by $p$ if $q$ has responded to the last block $p$ has sent it.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Asynchronous Protocol with Blocklace Dissemination}\\\\ pseudocode for miner $p \\in \\Pi$, including Algorithms \\ref{alg:blocklace} \\& \\ref{alg:CMO}}\n\t\\label{alg:CMAbd}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\t\n\n\t\\Statex \\textbf{Local variables:}\n\t\\StateX array $\\textit{history}[n]$, initially $\\forall k \\in [n] \\colon \\textit{history}[k] \\gets \\{ \\}$ \\Comment{Communication history of $p$ with others} \n\n\t\\vspace{0.5em}\n\t\\Upon{\\textbf{receipt} of $b$} \\Comment{\\textbf{send} and \\textbf{receipt} are simple messaging on a reliable link}\n\t\\State $\\textit{buffer} \\gets \\textit{buffer} \\cup \\{ b \\}$ \\label{alg:CMA:buffer}\n\t\\EndUpon\n\t\n\t\\While{True}\n\t\\For{$b \\in \\textit{buffer} \\colon \\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks})\n\t\\wedge \\textit{cordial\\_block}(b)$\n\t } \\Comment{Accept cordial block with no dangling pointers}\n\t \\State $\\textit{buffer} \\gets \\textit{buffer}~ \\setminus \\{ b \\}$\n \t\\State $\\textit{blocks} \\gets \\textit{blocks}~ \\cup \\{ b\\}$ \t\\label{alg:CMAbd:receive}\n \\State $\\textit{history}[b.\\textit{creator}] \\gets \\textit{history}[b.\\textit{creator}] \\cup \\{ b \\}$ \\label{alg:CMAbd:historyReceive}\n\n \t\\State $\\textit{deliver\\_blocks}()$\n\t\\EndFor\n\t \n\t \\If{ $\\textit{cordial\\_round}()$} \\label{alg:CMA:isCordial} \\Comment{Defined at \\cref{alg:CMO:cr}}\n\t \\State $b \\gets \\textit{create\\_block}(\\textit{blocks\\_prefix}(\\textit{cordial\\_round}()))$ \\Comment{Create a cordial block} \n\t\t\t\\For{$q \\in \\Pi \\wedge q \\ne p $} \n\t\t\t \\If{$p = \\textit{leader}(\\textit{depth}(b)) \\wedge \\textit{responsive}(q)$} \\Comment{Defined at \\cref{alg:CMS:responsive}} \\label{alg:CMAbd:isCordial}\n\t \\State $\\textit{package} \\gets [b] \\setminus [\\textit{history}[q]]$\n\t\t\\label{alg:CMS:CreatePackage}\n\t\n\t\t\\Else{ $\\textit{package} \\gets b$} \t\\label{alg:CMS:CreateBlockPackage}\n\t\t \\EndIf\n\t\t \\State $\\textit{history}[q] \\gets \\textit{history}[q] \\cup \\textit{package}$ \\label{alg:CMS:historySend}\n\t\t \\State \\textbf{send} \\textit{package} to $q$\n\t\t \\EndFor\n\t\t \\EndIf\n\t \\EndWhile\n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\n\\begin{proposition}[Algorithm \\ref{alg:CMAbd} Dissemination]\\label{proposition:CMAbd-dissemination}\nIn any run of Algorithm \\ref{alg:CMAbd}, if a correct miner knows a block $b$, then eventually every correct miner will know $b$.\n\\end{proposition}\n\\begin{proof}\nConsider a correct miner $p$ with block $b \\in \\emph{blocks}$, and miner $q$. If $q$ is correct then eventually it will send a block to $p$, and consider a subsequent round $r$ in which $p$ is a cordial leader. If at round $r$ communication history of $p$ with $q$ shows that $q$ knows $p$, we are done. Else, $p$ will include $b$ in its package to $q$, together with all blocks in $[b]$ for which $p$ has no evidence that $q$ knows, based on their communication history. Then $q$ will eventually receive the package. The package has no blocks with dangling pointers since $p$ includes in the package everything $q$ might miss, hence $q$ can receive it and include it together with $b$ in its \\emph{blocks}.\n\\end{proof}\n\n\\begin{proposition}[Liveness of Asynchronous Cordial Miners protocol with Blocklace Dissemination]\\label{proposition:CMAbd-liveness}\nAlgorithm \\ref{alg:CMAbd} satisfies the liveness requirement of Definition \\ref{definition:safety-liveness}.\n\\end{proposition}\n\\begin{proof}\nConsider a suffix of an infinite computation of Algorithm \\ref{alg:CMAbd} beyond the control of the adversary. According to Proposition \\ref{proposition:CMAbd-dissemination}, every block $b$ known to a miner will be known eventually to every miner. \nThe rest of the proof is identical to the proof of Proposition \\ref{proposition:aCM-liveness}.\n\\end{proof}\n\n\n\n\\section{Optimizations}\n\nThe optimizations in Algorithm \\ref{alg:CMOpt} apply to all Cordial Miners protocols. They exclude miners that are faulty, either by equivocation or by not being cordial. These can be easily taken further in various directions. A simple example is to improve leader utilization by bypassing immediately (without timeout) rounds with a faulty leader.\n\n\\begin{algorithm*}[t]\n\t\\caption{\\textbf{Cordial Miners\\xspace: Optimizations} \\\\ pseudocode for miner $p \\in \\Pi$}\n\t\\label{alg:CMOpt}\n\t\\small\n\t\\begin{algorithmic}[1]\n\t\\alglinenoPop{counter} \n\t\n\t\t \n \\vspace{0.5em}\n \\Procedure{\\textit{proper}}{$b$}\n \\Return \n $\\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks}) \\wedge \n (\\textit{leader}(b) \\rightarrow \\textit{cordial\\_block}(b)) \\wedge\n \\lnot \\textit{equivocator}(b.\\textit{creator})$\n \\EndProcedure\n \n\t\t\\vspace{0.5em}\n\t\t\\Procedure{\\textit{send\\_blocks}}{$q$} \n\t\t\\If{$\\textit{equivocator}(q) \\vee \\textit{nonresponsive}(q) $} \\Return \\Comment{Don't bother sending to Byzantine or nonresponsive miners}\n\t\t\\EndIf \n\t \\State $\\ldots$\n\t\t\\EndProcedure\n\t\t\n\t\n \\vspace{0.5em}\n \\Procedure{\\textit{proper}}{$b$}\n \\Return \n $\\textit{b.pointers} \\subseteq \\textit{hash}(\\textit{blocks}) \\wedge \n (\\textit{leader}(b) \\rightarrow \\textit{cordial\\_block}(b)) \\wedge\n \\lnot \\textit{equivocator}(b.\\textit{creator})$\n \\EndProcedure\n \n \n \n\t\n\t\t\\vspace{0.5em} \n\t \\Procedure{\\textit{equivocator}}{$q$} \\label{alg:SMR:dactor}\n \\Return \n $\\exists b_1,b_2 \\in \\textit{blocks} \\wedge\n b_1.\\textit{creator} = b_2.\\textit{creator} = q \\wedge\n \\textit{equivocation}(b_1,b_2)\n $ \\Comment{See Figure \\ref{figure:approve-sr}.A}\n \\EndProcedure\n \n \\vspace{0.5em}\n\n \n\t\t\\alglinenoPush{counter}\n\t\\end{algorithmic}\n\t\n\\end{algorithm*}\n\n\n\n\n\n\\section{Performance analysis}\nFor the Synchronous Cordial Miners protocol, every block in the blocklace is sent from the miner that creates it to the leader.\nIn the worst-case, there is linear a number of Byzantine leaders in a row, meaning at most that each block is sent $O(n^2)$ times.\nThe block size is linear, due to its hash pointers, meaning a bit complexity of $O(n^3)$ per block.\nThe leader sends to all miners the blocks it receives, i.e., it sends a linear number of blocks, each linear in size, resulting in a bit complexity of $O(n^3)$ as well.\nBut, if we batch per block a linear number of transactions, when the commit rule is met a quadratic number of transactions is committed each time.\nThus, the amortized bit complexity per decision is $O(n)$.\nHotStuff also achieves this this complexity in the good-case, i.e., when all miners are synchronized to the same round.\n\nFor the Asynchronous Cordial Miners protocol, each block is disseminated using an $O(n^2)$ message complexity dissemination protocol.\nSince each block is linear-sized, the bit complexity of the dissemination each node is $O(n^3)$.\nIn a similar way to the synchronous protocol, each block can also be batched with a linear number of transactions.\nThus, when the commit rule is met, a quadratic number of transactions is committed, leading to an amortized $O(n)$ bit complexity per decision as well.\nThis is the same amortized bit complexity as DAG-Rider.\n\n\\section{Conclusions}\n\nThe Cordial Miners protocol family is simple and diverse. We believe simplicity has many ramifications when practical applications are considered: Simpler algorithms are easier to debug, to optimize, to make robust, and to extend.\n\nAn interesting next step towards making Cordial Miners a useful foundation for cryptocurrencies is to design a mechanism that will encourage miners to cooperate---as needed by these protocols---as opposed to compete, which is the current standard in mainstream cryptocurrencies.\n\n\n\n\n\n\\newpage\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOngoing and upcoming large galaxy surveys~\\cite{Aghamousa:2016zmz,Laureijs:2011gra,Hill:2008mv,Adams:2010un,Dawson:2015wdb,Ellis:2012rn,Spergel:2013tha} provide us a wealth of information about the initial condition of density perturbations. However, accurately modeling the low redshift matter distribution is a challenge in modern cosmology, as the history of structure formation suffers from the nonlinear evolution of density fields. Although various improvement has been made for perturbation theories~\\cite{10.1093\/mnras\/203.2.345,Fry:1983cj,Suto:1990wf,Makino:1991rp,Jain:1993jh,Scoccimarro:1996se,Bernardeau:2001qr,Crocce:2005xy,McDonald:2006hf,McDonald:2006hf,Crocce:2007dt,Taruya:2007xy,Matarrese:2007aj,Matarrese:2007wc,Takahashi:2008yk,Bernardeau:2008fa,Carlson:2009it,Shoji:2009gg,Matsubara:2007wj,Pietroni:2011iz,Tassev:2011ac,Taruya:2012ut,Sugiyama:2013mpa}, small scale matter power spectrum is still difficult to model without additional fitting parameters~\\cite{Baumann:2010tm,Carrasco:2012cv,Pajer:2013jj,Manzotti:2014loa,Carroll:2013oxa,Porto:2013qua}.\n$N$-body simulations that numerically solve the evolution of all representative matter particles in the Universe can be a solution.\nHowever, this is computationally expensive, making it challenging to browse all possible parameter spaces to find the best fit cosmological parameters to the observed data.\nAlternatively, a method to reduce the nonlinear effect in the observed galaxy distribution can potentially ease modeling the late time field and extracting cosmological information from large galaxy surveys. \n\n\nBaryon acoustic oscillation~(BAO) reconstruction has been developed in this context~\\cite{Eisenstein:2006nk}, and a series of both numerical and analytic investigations have been conducted to understand and utilize the method~\\cite{Padmanabhan:2008dd,Noh:2009bb,Tassev:2012hu,White:2015eaa,Schmittfull:2015mja,Seo:2015eyw,Hikage:2017tmm,Schmittfull:2017uhh,Chen:2019lpf,Hada:2018fde,Hada:2018ziy}.\nThe standard BAO reconstruction pioneered by Ref.~\\cite{Eisenstein:2006nk} showed that shifting the observed particles along the gradient of the smoothed density efficiently recovers the linear BAO signals.\nThe standard reconstruction works because we reduce the degradation effect due to the free streaming by minimizing displacement that the particles traveled by moving them back.\nThe reconstructed field shows an augmented correlation with the initial field, but it is not quite the same as the initial linear density field because it cannot fully reverse all the nonlinearities~\\cite{Padmanabhan:2008dd,Schmittfull:2015mja}. \nIn other words, the BAO reconstruction partially brings the cosmological information of higher-order statistics back to the two-point statistics, making the reconstructed field closer to be Gaussian~\\cite{Schmittfull:2015mja}.\nThe resulting discrepancy returns the broadband spectrum and redshift space distortion~(RSD) that deviates from the linear prediction and the prediction of the nonlinear density field. \nThis procedure introduces a challenge from the modeling perspective,\nconsequently making it difficult to combine the BAO analysis with other large-scale structure analyses. Recently, there have been a few promising studies that address the challenge and construct perturbation theory models of postreconstruction full clustering~\\cite{White:2015eaa,Hikage:2017tmm,Chen:2019lpf}. In parallel, there have been many studies on initial density field reconstruction by forward modeling~\\cite{Lavaux:2019fjr,Schmidt:2020viy,Nguyen:2020yuc,Seljak:2017rmr,Feng:2018for,Modi:2021acq}. However, we should also note that, in principle, we cannot recover the initial field without solving the equation of motion~(EoM) backward, which is not reasonably possible due to the shell crossing. \nThen a natural question arises: can we find a more consistently defined late time field that is more correlated with the initial condition and less degraded than the density perturbations and try to model the field rather than reconstruct such a patchy initial density field?\n\n\nIn this work, we argue that the Lagrangian displacement field can be an alternative to reconstructing matter density perturbations. As our $N$-body simulations show in the top panel of Fig.~\\ref{fig00}, the late time displacement divergence and initial linear density field are indeed 99\\% correlated for $k{\\rm Mpc}\/h\\lesssim 0.2$ and 95\\% for $k{\\rm Mpc}\/h \\lesssim 0.5$ at $z=0.6$~\\cite{Baldauf:2015tla}~(see Sec.~\\ref{simsec} and Sec.~\\ref{sec5} for the details of our simulations.). \nMoreover, the top and middle panels imply that the power spectrum is amplified rather than damped. \nThus, both the nonlinear instability and degradation effect are reduced for the displacement field compared to the density field, and we can potentially extract a wealth of information from the displacement field if they can be precisely measured and accurately modeled.\nIn practice, measuring the displacement fields for actual surveys is nontrivial because we do not know the initial galaxy positions a priori.\nHence, we are interested in reconstructing the displacement field from the observed galaxy distributions, which was the motivation of Ref.~\\cite{Schmittfull:2017uhh}, while their primary focus was the BAO feature. \nIf we could adequately reconstruct the displacement field, the broadband power spectrum would also be consistently reconstructed. In this paper, we construct a model for the broadband power spectrum for the reconstructed displacement field.\n\nWhile there have been a few comparable approaches developed to iteratively reconstruct the nonlinear displacement field~\\cite{Zhu:2016xyy,Zhu:2016sjc,Hada:2018fde}, the ``iterative reconstruction'' introduced by Ref.~\\cite{Schmittfull:2017uhh} iteratively moves the particles along the density gradient by progressively reducing the smoothing radius step by step until we get the almost zero density field.\nThe final position is an estimated Lagrangian position, and the displacement from the original, i.e., the observed Eulerian position, is the reconstructed displacement field.\nRef.~\\cite{Schmittfull:2017uhh} simulated the postreconstruction displacement field and showed that it is 95$\\%$ correlated with the initial linear density up to $k{\\rm Mpc}\/h=$0.35 at $z$=0.\n\nAlthough the top panel in Fig.~\\ref{fig00} shows the linearity of both the displacement field and post-iterative reconstruction field, it does not necessarily mean that we can model those fields within the linear theory.\nThe displacement field power spectrum has about an 8\\% discrepancy from the linear matter field at $k{\\rm Mpc}\/h=0.2$ due to the shift term, i.e., the term correlated directly with the linear density field as shown in the middle panel of Fig.~\\ref{fig00}.\nIn addition, the post-iterative reconstruction estimator does not perfectly recover the true displacement field even for the BAO scale.\nOn the other hand, the bottom panel shows that the power spectrum of the errors from the linear matter field is similar to the nonlinear\/reconstructed displacement field despite the discrepancy in the power spectrum in the middle panel.\nThus, the relation among those fields is not simple.\n\nThe authors of Ref.~\\cite{Schmittfull:2017uhh} called the displacement field reconstruction ``$\\mathcal O(1)$'' reconstruction, and they further reconstruct the initial density field from the reconstructed displacement field perturbatively, calling it ``$\\mathcal O(2)$''. \nThis paper takes a rigorous approach to directly model the ``$\\mathcal O(1)$'' estimator using perturbation theory.\nWe present a theoretical modeling of the reconstruction estimator in Lagrangian perturbation theory~(LPT) up to third order for dark matter without redshift space distortions just for simplicity.\nRef.~\\cite{Schmittfull:2017uhh} presented a motivation for perturbation theory modeling for their iterative reconstruction.\nHowever, they did not consider modeling the iterative steps but provided a way to find the Zel'dovich displacement from the measured density.\nThis paper will model the iterative steps explicitly and find $n$-th step displacement field up to 1-loop order perturbations.\n\n\n\nThis paper is organized as follows.\nSec.~\\ref{simsec} provides the details of our two simulations and their measurements.\nIn Sec.~\\ref{sec2}, we explain the difficulty of perturbation theory for the displacement field towards an application to the iterative reconstruction.\nSec.~\\ref{moderec} provides the reconstruction algorithm in the LPT perspective.\nIn, Sec.~\\ref{sec4}, we present our LPT modeling of the iterative reconstruction and give a comparison with the simulations in Sec.~\\ref{sec5}.\nThen we give a summary and conclusions in the last section.\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{nl-disp-lin_00}\n \\caption{\\textit{Top}: The cross correlation coefficient $r_X\\equiv P_{X\\delta_{\\rm L}}\/\\sqrt{P_X P_{{\\rm L}}}$ for the nonlinear density field $\\delta_{\\rm NL}$, nonlinear displacement field potential $\\phi$, standard reconstruction estimator $\\delta_{\\rm rec}$ and 9th step iterative reconstruction estimator $\\phi_{\\rm rec}^{(9)}$.\n The nonlinear displacement field is 95\\% correlated with the linear field for $k{\\rm Mpc}\/h\\lesssim 0.5$, which is better than that for the standard density field reconstruction for $R=20$Mpc\/$h$.\n \\textit{Middle}: power spectrum of the nonlinear fields normalized by the linear matter power spectrum $P_{\\rm L}$. The shift term is not reconstructed well in the iterative reconstruction.\n \\textit{Bottom:} $E(X,\\delta_{\\rm L})$ is the power spectrum of the error field, which is the difference between the nonlinear\/reconstructed displacement field and $\\delta_{\\rm L}$ normalized by $P_{\\rm L}$~(see Sec.~\\ref{simsec} and \\ref{sec5} for the details of simulations.).}\n \\label{fig00}\n\\end{figure}\n\n\n\n\\section{Measurement of the simulated displacements}\\label{simsec}\n\n\n\\begin{table*}\n\\caption{\\label{table1}Simulation and sampling parameters. \nBoth simulations assume a flat $\\Lambda$CDM cosmology based on Ref.~\\cite{Ade:2015xua} with $\\Omega_{\\rm m} = 0.3075$, $\\Omega_{\\rm b}h^2=0.0223$, $h=0.6774$, and $\\sigma_8=0.8159$.}\n\\begin{ruledtabular}\n\\begin{tabular}{lccccc}\nName & subsampling \\% & \\# of meshes & Box size [Mpc$\/h]^3$ & \\# of particles &\\# of simulations \\\\\n\\hline\nL500 &4 & $512^3$ & $500^3$ & $1536^3$ & 5\\\\\nsubL500 &0.15 & $512^3$ & $500^3$ & $1536^3$ & 5\\\\\nL1500 &4 \t & $1024^3$ & $1500^3$ & $1536^3$ &1 \\\\\nfullL1500 &100 & $1536^3$ & $1500^3$ & $1536^3$ &1 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\nTo compare our theoretical model with realistic nonlinear\/reconstructed displacement fields, we need to minimize nonphysical numerical artifacts in measuring the simulated displacement fields. \nIn this section, we explain the setup of our numerical simulations.\nReaders who are only interested in the theoretical implementation can skip this section.\n\n\n\n\\medskip\nMeasuring the displacement field or velocity field is challenging due to these vector fields' discrete and nonuniform sampling. \nMeasuring the displacement using mass tracers gives the mass-weighted displacement field, giving no measurement in the locations without mass, while perturbation theories derive a volume-weighted quantity. \nThe Delaunay tessellation method (e.g., Ref.~\\cite{Pueblas:2008uv}) works for a volume-weighted measurement, which is more robust to the discreteness effect. \nIn this paper, however, we adopt the mass-weighted measurement, following the convention of Ref.~\\cite{Schmittfull:2017uhh}, for both calculating the Lagrangian displacement field and the reconstructed displacement field.\nAs a caveat, one could instead directly model the mass-weighted quantity in perturbation theory~\\cite{Vlah:2012ni}.\nWe test the sanity of the mass-weighted measurement using a convergence test as a function of various levels of discreteness effect against a reference simulation without the discreteness effect. \nThe mass-weighted displacement can be written as\n\\def \\mathbf x{\\mathbf x}\n\\begin{align}\n\t\\mathbf \\Psi^{\\rm obs.}_p \n\t&= \\frac{\\sum_i W_{\\rm CIC}({\\mathbf x_p,\\mathbf x_i}) \\mathbf \\Psi^{\\rm obs.}(\\mathbf x_i) }{\\sum_i W_{\\rm CIC}({\\mathbf x_p,\\mathbf x_i}) }\n\t,\\label{mweight}\n\\end{align}\nwhere $\\mathbf \\Psi^{\\rm obs.}(\\mathbf x_i)$ is the observed displacement field of $i$-th particle, $W_{\\rm CIC}$ is the pixel window function indicating that we are using the Cloud-in-Cell assignment, and $\\mathbf \\Psi^{\\rm obs.}_p$ is the mass-weighted displacement field for a pixel centered at $\\mathbf x_p$. For pixels with no particles found, we incorrectly set $\\mathbf \\Psi^{\\rm obs.}_p=0$. \nThis operation biases the result; the overall amplitude is reduced by the fraction of the zeroed pixels, which we correct with a simple multiplicative rescaling. However, this prescription also introduces a spurious small-scale power correlated with the pixel window function and the nonuniform particle spacing, which we cannot straightforwardly correct. Note that $\\mathbf \\Psi$ is evaluated at the initial Lagrangian position, which is fixed at $\\mathbf x_{p}$, and that this is fundamentally different from estimating the velocity divergence field, which we evaluate in the Eulerian position.\n\nWe show the subsampling parameters~(i.e., particle spacing) and pixel window function of our two numerical simulations in Tab.~\\ref{table1}. Simulations assume a flat $\\Lambda$\\rm{CDM}~cosmology based on Ref.~\\cite{Ade:2015xua} with $\\Omega_{\\rm m} = 0.3075$, $\\Omega_{\\rm b}h^2=0.0223$, $h=0.6774$, and $\\sigma_8=0.8159$. \nThis is the same cosmology used in Ref.~\\cite{Schmittfull:2017uhh}. \nFull $N$-body simulations were produced using the MP-Gadget~\\cite{Springel:2000yr,Springel:2005mi,2018zndo...1451799F} with the box size of $500{\\rm Mpc}\/h$ and $1500{\\rm Mpc}\/h$. For $500{\\rm Mpc}\/h$, we use the average of five simulations. Both simulations use a particle resolution of $1536^3$. The simulation evolves $1536^3$ particles from $z=99$ by computing forces in a grid of $1536^3$, and we subsample 4\\% of the output particles at $z=0.6$ to make L1500 (for a box of $1500{\\rm Mpc}\/h$) and L500 (for a box of $500{\\rm Mpc}\/h$). We use a grid of $512^3$ to Fourier-transform, reconstruct this nonlinear field for L500 and subL500, and use a grid of $1024^3$ for fullL1500 and L1500.\n\n\nHere, fullL1500 is the simulation of $1500{\\rm Mpc}\/h$ without subsampling, therefore uniformly being distributed in the Lagrangian position. The mass-weighted displacement for this simulation should be equivalent to the volume-weighted measurement, and we expect no discreteness and no pixel window function effect for this set. We do not use this complete catalog for reconstruction, as, first, it is computationally expensive for iterative steps, and second, the reconstructed density field will suffer the discreteness and the pixel window function effect even if we used this complete set, as we could not perfectly recover the Lagrangian, uniform position even after reconstruction.\nWe use this complete set as our reference to estimate the convergence of the simulation L500, which is our main set to test the broadband shape after reconstruction.\nWe find that the displacement field of the two simulations is convergent up to $k{\\rm Mpc}\/h\\lesssim 0.2$ for those mesh resolutions within 1.1\\%. We find that further subsampling and\/or increasing the mesh size breaks this convergence.\nGiven the mesh resolution and the particle resolution of these two simulations in Tab.~\\ref{table1}, we consider 1 ${\\rm Mpc}\/h$ as the limiting resolution and account for it in our LPT model.\nThe convergence of the displacement field does not straightforwardly imply the convergence of the post-iterative reconstruction displacement field. \nHowever, in this paper, we shelve the plans for 100\\% sampling for the iterative reconstruction mainly because of our numerical resources.\nTo see the numerical stability of the iterative reconstruction, we also introduced subL500, which is a 0.15\\% sampling of the $512^3$ simulation and the rest of the parameters are the same as those for L500.\n\n\nFor the BAO feature comparison, L500 is not optimal due to its small box size, i.e., $500{\\rm Mpc}\/h$.\nFor this reason, we apply the iterative reconstruction on a pair of L1500, the one generated with an initial condition with BAO and the one generated without BAO with the same white-noise fields~\\cite{Wojtak:2016sxr,Schmittfull:2017uhh,Ding:2017gad}, which will be shown in Fig.~\\ref{fig2}. By pairing and dividing, we cancel out the cosmic variance and the spurious effect due to the discreteness aforementioned.\n\nAs a caveat, note that the observed field from galaxy surveys will be subject to many orders of magnitude more severe discreteness effects than our default simulation L500. Our companion paper~\\cite{Seo:2021} discusses a surrogate reconstruction method to avoid the discreteness in reconstruction. \nIn this paper, however, we focus on modeling the reconstructed displacement in the shot-noiseless limit without redshift space distortion to test the model with minimal complications.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The UV mistake of 3LPT displacement}\\label{sec2}\n\n\nThis paper aims at modeling the iterative reconstruction up to 1-loop order, which can also be referred to as ``3LPT'' as we expand the equation of motion up to third order in the linear density field.\nWe also call the Zel'dovich approximation, i.e., linear perturbations, ``1LPT'' in this paper.\n\nOne might wonder if the perturbation theory for the displacement field is more convergent and more manageable than that for the density field, since we mentioned that the simulated nonlinear displacement field is more correlated with the linear field than the density field.\nHowever, it does not work as we expected because of the ``UV-mistake'', which we explain in this section.\nThe UV mistake, first pointed out by Ref.~\\cite{Baldauf:2015tla}, is an issue about the cut-off dependence of a loop calculation for the displacement field.\n\n\\medskip\nLet $\\mathbf x$ and $\\mathbf q$ be the Eulerian and Lagrangian coordinates.\nThe relation between these two coordinates is written as \n\\begin{align}\n\t\\mathbf x = \\mathbf q + \\mathbf \\Psi(\\mathbf q),\\label{psidef}\n\\end{align}\nand $\\mathbf \\Psi$ is called the displacement field.\nThe Eulerian density perturbation is exponentiated by the Lagrangian displacement field as~\\cite{Matsubara:2007wj}\n\t\\begin{align}\n \\delta_{\\rm NL}(\\mathbf x)\n &= \\int \\frac{d^3kd^3 q}{(2\\pi)^3} e^{i\\mathbf k\\cdot (\\mathbf x-\\mathbf q)} \\left [ e^{ -i\\mathbf k\\cdot \\mathbf \\Psi(\\mathbf q)} \n -\n 1\\right].\\label{14}\n\\end{align}\n\n\n\n\n\\medskip\nWe introduce the displacement field potential $\\phi$ that satisfies $i\\mathbf k\\phi = \\mathbf \\Psi$ in Fourier space.\nThe 1LPT spectrum of $\\phi$ is straightforwardly written by the linear matter power spectrum $P_{\\rm L}$:\n\\begin{align}\n P^{\\rm 1LPT}_{\\phi}=& \\frac{P_{\\rm L}}{k^4},\n\\end{align}\nThen, the 3LPT power spectrum is given as\n\\begin{align}\n P^{\\rm 3LPT}_{\\phi}=& P^{\\rm 1LPT}_{\\phi}+ P_{\\phi13}+ P_{\\phi22} \\label{LPT:disp},\n\\end{align}\nwhere we defined~\\cite{Baldauf:2015tla}\n\\begin{align}\nP_{\\phi22}=& \\frac{9}{98}\\int dxd\\mu \\frac{P_{\\rm L}(kx)P_{\\rm L}(ky)}{4\\pi^2k}\n\t\t \\frac{x^2(1-\\mu^2)^2}{y^4},\\label{P22}\\\\\nP_{\\phi13}=&\n\\frac{10}{21}P_{\\rm L}(k)\\int dxd\\mu \\frac{P_{\\rm L}(k x)}{4\\pi^2k} \n\t\t \\frac{x^2(1-\\mu^2)^2}{y^2 },\n\t\t\\label{P13}\n\\end{align}\nwith $y\\equiv(1 - 2x \\mu+x^2)^{1\/2}$.\nThe integral variables $x$ and $\\mu$ run from $0$ to $\\infty$ and $-1$ to 1 respectively.\nWe show $P^{\\rm 3LPT}_{\\phi}$, $P_{\\phi13}$ and $P_{\\phi22}$ and their standard perturbation theory~(SPT) counterpart for the density field in Fig.~\\ref{fig0}~(see e.g., Ref.~\\cite{Bernardeau:2001qr} for the SPT 1-loop term).\nThe 3LPT correction is dominated by $P_{\\phi13}$, which is about 10\\% correction to the linear power spectrum even at the BAO scale.\nWe explain why $P_{\\phi13}\\gg P_{\\phi22}$ as follows.\nIn the large loop momentum $x\\to \\infty$ limit, we get~\\cite{Baldauf:2015tla}\n\\begin{align}\n\tk^4P_{\\phi22}\\to & \\frac{24}{245}\\frac{k^3}{4\\pi^2}\\int \\frac{dx}{x^2} P_{\\rm L}(kx)^2\\, ,\\\\\nk^4P_{\\phi13} \\to &\n\t\t \t\t \\frac{32}{63}P_{\\rm L}(k)\\frac{k^3}{4\\pi^2}\\int dx P_{\\rm L}(k x) \\label{UVmistake}\\, ,\n\\end{align}\nand the integral in Eq.~\\eqref{UVmistake} is sensitive to the large $x$ contributions.\nAs we see the gray curve in Fig.~\\ref{fig0}, $P_{\\phi}^{\\rm 1LPT}$ overestimates the displacement field at high $k$, and thus the loop integral contains the incorrect UV modes.\nAs a result, $P_{\\phi13}$, i.e., the shift term is overestimated.\nThis is why the authors in Ref.~\\cite{Baldauf:2015tla} called this problem ``UV-mistake''.\nThe UV mistake around the BAO scale is specific to the displacement field.\nWhile we have the integral like Eq.~\\eqref{UVmistake} in the 3SPT corrections, they are canceled, and the net correction is suppressed.\nOn the other hand, the fact that $|P_{\\phi 13}|\\gg |P_{\\delta13}|$, while $P_{\\phi 13}$ is positive and $P_{\\delta13}$ is negative, and $P_{\\phi 22}\\ll P_{\\delta22}$ is the reason that the displacement field is correlated with the linear field very well and that the phase shift in the BAO is negligible for the displacement field.\n\n\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{lptspt.pdf}\n \\caption{\\textit{Top}: Comparison of the simulation displacement~(gray) and density field~(black), the 1-loop displacement power sepctrum~(blue) and 1-loop density power spectrum~(orange). We evaluated at $z$=0.6. The second order LPT is shown in pink.\n The EFT~(green) is defined in Eq.~\\eqref{EFT1}.\n All power spectra are normalized by the linear matter power spectrum to better illustrate the difference.\n \\textit{Bottom}: 1-loop corrections for the density~(SPT) and displacement fields~(LPT) are shown. While $|P_{\\delta13}| \\sim P_{\\delta22} \\gg k^4P_{\\phi13}\\gg k^4P_{\\phi22}$, we have $P_{\\rm L}\\sim P^{\\rm 3SPT}_{\\delta} 0.2$ in the perturbation theory, and so is the loop integral.\n\n\n\\section{Comparison with simulations}\n\\label{sec5}\n\nIn this section, we compare our LPT modeling with simulations.\nLet us first present the simulation results and see if the iterative procedures are successfully modeled.\n\n\\subsection{Simulation of post reconstruction field}\\label{simulation:sec}\n\nWe choose the initial ($n=0$) smoothing scale of $20 {\\rm Mpc}\/h$ to reconstruct L500 and subL500, and we reduce the smoothing scale incrementally by $\\epsilon=1\/\\sqrt{2}$ until we reach $n=9$.\nIn Fig.~\\ref{fig2}, we reproduced Fig.~5 in Ref.~\\cite{Schmittfull:2017uhh}, adding displacement fields.\nWe plot the nonlinear fields and postreconstruction fields normalized by the corresponding no-wiggle spectra. The no-wiggle spectra are defined for each simulation, calculated from the paired no-wiggle simulation in Sec.\\ref{simsec}, and thus the nonlinear and numerical effect on the broadband shape is removed. Hence, we can single out the degradation effect in the BAO.\nAs we see in the plot, the initial density, nonlinear displacement, and the iterative reconstruction estimators agree well below $k{\\rm Mpc}\/h<0.3$. We overplot the standard reconstruction $\\delta_{\\rm rec}$ with the smoothing scale of $20 {\\rm Mpc}\/h$ (blue lines); its reconstructed BAO feature is less distinct than $n>7$ mainly because the smoothing scale is smaller than the effective smoothing scale of the iterative reconstructions~\\cite{Seo:2021}.\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{wg_nwg}\n \\caption{power spectrum divided by its matched no-wiggle one. The data set is L1500. The no-wiggle spectra are defined for each simulation and thus any nonlinear and numerical effect on the broadband shape are removed. $\\delta_{\\rm rec}$ is the standard reconstruction with $R=20$Mpc\/$h$. }\n \\label{fig2}\n\\end{figure}\n\n\nTo see the rest of the nonlinear effects, we plotted the nonlinear power spectrum normalized by the linear one in Fig.~\\ref{fig22}.\nOur $n=9$ visually agrees with Fig.14 in Ref.~\\cite{Schmittfull:2017uhh}. The higher noise in the $\\delta_{\\rm NL}$ power spectrum in the figure, compared to the power spectra of displacement field tracers, reflects its poor cross-correlation with the initial field. The iterative reconstruction appears to converge to the displacement field rather than to the linear power spectrum as $n$ approaches seven and then approaches closer to the linear shape as $n$ increases.\nTo better quantify how close the reconstructed field is to the nonlinear displacement field or to the linear field, \nwe introduce the error power spectrum of $X$ and $Y$ as\n\\begin{align}\n E(X,Y) = \\frac{P_{X-Y}}{P_Y},\\label{erdef}\n\\end{align}\nwhere $P_{X-Y}$ and $P_Y$ are the auto-power spectrum of $X-Y$ and $Y$, respectively. \nThen we show $E(k^2\\phi^{(n)}_{\\rm rec},k^2\\phi)$ and $E(k^2\\phi^{(n)}_{\\rm rec},\\delta_{\\rm L})$ in Fig.~\\ref{fig222}.\nWe can see $E(k^2\\phi^{(n)}_{\\rm rec},k^2\\phi)7$.\n\n\n\n Figs \\ref{1-loopres1}, \\ref{1-loopres1.1} and \\ref{1-loopres3.1} show the power spectrum, the propagator, and the cross-correlation coefficient in each step of iteration. At $n=6$ and 7, we indeed see a slight positive shift term, similar to that of the displacement (grey), but such feature is very weak and disappears after $n>7$. In terms of the cross-correlation coefficient, again, the reconstructed field converges to the displacement field up to $k {\\rm Mpc}\/h \\sim 0.2$ for $n>5$. \n \n In summary, the iterative reconstruction converges to the displacement field in mode coupling terms, but it deviates in terms of the shift term. For the shift term, the reconstructed field appears to converge to the linear field ($k {\\rm Mpc}\/h \\sim 0.2$). \n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{nl-disp-lin}\n \\caption{Power spectra normalized by the linear matter power spectrum for L500. One can see the effect of reconstruction on the full shape.}\n \\label{fig22}\n\\end{figure}\n\n\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{nl-disp-itrec}\n \\caption{The error power spectrum $E(k^2\\phi^{(n)}_{\\rm rec},Y)$ for $Y=k^2\\phi$~(solid) and $Y=\\delta_{\\rm L}$ (dotted). A lower error power spectrum implies a better agreement. The postreconstruction field is ``closer'' to the displacement field than the linear field. Iterative reconstruction is convergent around $n=6$ and the error is at most 3\\% for $k$Mpc\/$h<$0.3.}\n \\label{fig222}\n\\end{figure}\n\n\\subsection{LPT modeling and simulations}\n\nIn this section, we compare our LPT model derived in Sec.~\\ref{sec5} with the simulation result and check if the discrepancy between the iterative reconstruction and the displacement field can be explained. \nUsing Eqs.~\\eqref{1-loop:power} and \\eqref{1-loop:prop} we obtained the solid orange curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}.\nFor $n< 4$, except for the first step, the 1LPT (dashed orange curves) and 3LPT (solid orange curves) solutions are very similar, and they reproduce the simulation result (solid blue lines) in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and~\\ref{1-loopres3.1}. That is, the model explains the discrepancy between the postreconstruction field and the displacement field (gray lines) properly.\nFor higher iterations, LPT predictions begin to deviate, and only the 3LPT can predict the excess above the linear power spectrum, which is reasonable as we are trying to reconstruct smaller scale information that requires a higher-order description.\nNevertheless, we overall observe that the LPT models do not describe the numerical results very well. Looking at the propagator (Fig.~\\ref{1-loopres1.1}) and the cross-correlation coefficient (Fig.~\\ref{1-loopres3.1}), we find that the main deviation happens in the propagator, while the prediction of the cross-correlation coefficient is in excellent agreement for $n < 6$ and within less than 1\\% at $k{\\rm Mpc}\/h \\sim 0.2$ even at $n=9$. \n\nOn the other hand, we find that the 3LPT up to $n = 6$ is converging to the true displacement obtained in the simulation. The 3LPT model for $n>6$ then deviates from the true displacement field to the 3LPT model of the displacement field. This shows that the perturbation theory predicts that the iterative process we adopted should recover the true displacement field up to $k{\\rm Mpc}\/h\\lesssim 0.2$.\n The relatively weak convergence of the simulated post-iterative reconstruction field to the true nonlinear displacement field after $n=4$ implies the iterative reconstruction simulation includes some unknown numerical or technical effect that is not described in Sec.~\\ref{stdbao} and \\ref{itrbao}, particularly in the propagator.\nGiven that the offset between the model and the simulations is similarly parametrized as Eq.~\\eqref{tobias_eft}, we conduct a quick test of an EFT model by simply adding a counter term\n\\begin{align}\n\tP^{{\\rm 3EFT}}_{\\phi^{(n)}_{\\rm rec}} = P^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}} + 2\\alpha k^2 P^{{\\rm 1LPT}}_{\\phi^{(n)}_{\\rm rec}},\n\t\\label{3EFT}\n\\end{align}\nand show the green curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}. $\\alpha$ is the same as for the displacement field. This test indicates that the offset between the model and the reconstructed field in the propagator is less likely mitigated by a simple counter term, and again the perturbation theory model expects that the iterative reconstruction converges to the displacement field in the end.\n\nAnother likely source of the discrepancy is the aforementioned numerical noise\/artifact. Earlier, we justified subsampling particles (i.e., L500) by inspecting the convergence of the displacement field measurement on the L1500 (Sec.~\\ref{simsec}).\nHowever, the convergence in the measurements does not necessarily mean convergence in the process of reconstruction. \nThe UV sensitivity of the loop integral implies that the displacement field in a simulation is sensitive to the particle resolution\/density, and the effect may have been more complicated\/more significant in the backward evolution of reconstructing the field based on the subsampled tracers in the iterative reconstruction. Our numerical resource is limited conducting the iterative reconstruction on fullL1500, but we present the case of 0.15\\% subsampling (subL500) in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1}, as dotted lines to show the greater sensitivity to the particle subsampling in the process of reconstruction compared to the displacement field. Including a shot noise effect in the theoretical model and implementing a pixel window function de-convolution in the simulated field will be crucial for further comparison, which we plan to investigate in a future paper.\n\nTo account for the expected breakdown due to various missing small-scale physics of our subsampled catalog, we manually impose a minimum smoothing scale in the reconstruction in an attempt to control such an effect before reaching this breakdown. \nFig.~\\ref{fig222} showed that the reconstruction changes the convergence behavior around $n=6$, which is suggestive of the effective minimum smoothing scale in the simulation and the theory to be the smoothing scale around this step, which is $3.5.{\\rm Mpc}\/h$.\nTherefore, instead of Eq.~\\eqref{Rdef}, we set\n\\begin{align}\nR_n = &{\\rm max}(\\epsilon^{n} R,R_{\\rm min}),\n\\end{align}\nwith $R_{\\rm min}=3.5.{\\rm Mpc}\/h$ and show the results in Figs.~\\ref{1-loopres2}, \\ref{1-loopres2.1} and \\ref{1-loopres3.2}.\nIntroducing the minimum smoothing scale derives a convergence at $n=7$, as we cannot extract the information from smaller scales than the smoothing scale. However, the discrepancy between the theory and the simulation remains. \n\n\nBased on the lack of UV modes that we cannot explain, we introduce a phenomenological ad hoc model that we call 3LPT*.\nIn this model, we subtract all UV sensitive integrals from the 3LPT power spectrum and define \n\\begin{align}\n\t&P^{{\\rm 3LPT*}}_{\\phi^{(n)}_{\\rm rec}}\\equiv \t \n\t\\bar A^{(n)2}\\left [ \n\tP^{\\rm 1LPT}_{\\phi}\n\t+\n\tP_{\\phi22}\n\t\\right]\n\t+\n\tP^{{\\rm 3LPT}}_{\\phi^{(n)}_{\\rm rec}22},\\label{1-loop:power22}\n\\end{align}\t\nand we show the red curves in Figs.~\\ref{1-loopres1},~\\ref{1-loopres1.1}, and \\ref{1-loopres3.1} and Figs.~\\ref{1-loopres2}, \\ref{1-loopres2.1} and \\ref{1-loopres3.2}. This model takes the propagator of 1LPT, which better describes the observed propagator of the reconstructed field than 3LPT. \nAs a caveat, there is no reasonable explanation about this ad hoc truncation, and this is an inconsistent treatment as either loop expansion or perturbative expansion.\nHowever, we found the 3LPT* fits the simulated iterative reconstruction at 1\\% accuracy for $n>6$ for both cases with and without the minimum smoothing scale.\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n \\includegraphics[width=\\linewidth]{1_pow_lpt_rec_kmax_inf}\n \\caption{The comparison between the simulated postreconstruction power spectra~(blue), simulated true displacement~(gray) and the 3LPT~\\eqref{1-loop:power}, 1LPT~\\eqref{rec:zel:pow}, 3EFT~\\eqref{3EFT} and 3LPT*~\\eqref{1-loop:power22} models.\n Solid and doted curves for the simulation are 4\\% and 0.15\\% subsampling, respectively.\n Our 3LPT prediction agrees with the true displacement field rather than the simulated postreconstruction displacement up to $n=6$.\n 3LPT* is comparable to the simulated post reconstruction displacement field within 1\\% accuracy.\n The EFT slowly converges to the true displacement field. \n }\n \\label{1-loopres1}\n \\includegraphics[width=\\linewidth]{1_pow_lpt_rec_kmax_inf35}\n \\caption{The power spectrum in Fig.~\\ref{1-loopres1} for the minimum smoothing scale $R_{\\rm min}=3.5{\\rm Mpc}\/h$ Plots are the same with Fig.~\\ref{1-loopres1} up to $n=6$ and are almost fixed at $n=7$. \n }\n \\label{1-loopres2}\n \\end{figure*}\n\n\n\\begin{figure*}\n \\includegraphics[width=\\linewidth]{1_proc_lpt_rec_kmax_inf}\n \\caption{The propagator counterparts of Fig.~\\ref{1-loopres1}. The simulated displacement field, 3LPT, 3EFT show the shift term, but the effect is very small for the simulated postreconstruction.}\n \\label{1-loopres1.1}\n\n \\includegraphics[width=\\linewidth]{1_proc_lpt_rec_kmax_inf35}\n \\caption{The propagator in Fig.~\\ref{1-loopres1.1} with the minimum smoothing scale $R_{\\rm min}\\sim 3.5$Mpc\/$h$.\n Plots are the same with Fig.~\\ref{1-loopres1.1} up to $n=6$.\n }\n \\label{1-loopres2.1}\n\\end{figure*}\n\n\n\n\n\\begin{figure*}\n \\includegraphics[width=\\linewidth]{1_coef_lpt_rec_kmax_inf}\n \\caption{The cross-correlation coefficient counterparts of Fig.~\\ref{1-loopres1}.\n 3LPT, 3EFT, the true displacement and the simulation post reconstruction displacement agree well, while $r=1$ for 1LPT, i.e., the linear perturbations.\n While the cross-correlation coefficient should be always bounded by 1, the EFT is divergent for high $k$ since we introduced $-k^2$ term. \n }\n \\label{1-loopres3.1}\n\n \\includegraphics[width=\\linewidth]{1_coef_lpt_rec_kmax_inf35}\n \\caption{The cross correlation coefficient in Fig.~\\ref{1-loopres1.1} with the minimum smoothing scale $R_{\\rm min}\\sim 3.5$Mpc\/$h$. Plots are the same with Fig.~\\ref{1-loopres3.1} up to $n=6$.}\n \\label{1-loopres3.2}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{secdisc}\nAs a recent study~\\cite{Baldauf:2015tla} has pointed out, the nonlinear displacement at low redshift maintains a high cross-correlation with the initial field, contrary to the corresponding nonlinear density field. \nThus, if one could measure the nonlinear displacement field directly in the late time Universe, the measurement would contain the BAO feature almost without the nonlinear degradation, and it could also allow us to consistently use the information beyond the BAO, i.e., the full clustering shape, based on nonlinear perturbation theory. \n\nRef. \\cite{Schmittfull:2017uhh} then developed a non-standard extension to the density reconstruction technique, which was designed to reproduce the displacement field from the observed nonlinear density fluctuations by iteratively moving mass tracers along the gradient of the progressively smoothed density field to the Lagrangian positions. \nThis paper investigated how closely the method returned the nonlinear displacement field, focusing on the BAO and the full shape. Then, we derived an LPT-based model for the reconstructed displacement field and tested the model against the simulated reconstructed displacement field. We summarize our main results below.\n\nFirst, we confirmed that the displacement field is highly correlated with the linear field; the degradation effect and the phase shift on the BAO are negligible, and the nonlinear instability is milder than the nonlinear density field. However, the full shape of the nonlinear displacement field is different from the linear matter power spectrum on the BAO scale by $\\sim 8\\%$, and modeling with a higher-order perturbation theory with a correction term for the UV sensitive loops, such as the EFT model, is required for precision cosmology.\n\n\nWe found that the postreconstructed displacement field approaches closer to the nonlinear displacement field with increasing iterations but does not perfectly recover the nonlinear displacement field in the end. There is an 8\\% discrepancy between the true displacement field and the simulated postreconstruction displacement field at $k{\\rm Mpc}\/h\\sim 0.2$ due to unknown effects specific to the iterative process in the simulation. The discrepancy is mainly in the propagator (i.e., the shift term), while the cross-correlation coefficient (i.e., mode-coupling term) agrees well.\nAt the final iteration, the postreconstruction field tends to converge closer to the linear power spectrum. \n\nWe built perturbation theory models for the reconstructed field to understand the difference.\nIn this process, we realized that the UV mistake should largely cancel for the reconstructed displacement for moderate iterations in the process of estimating the displacement field, allowing us to avoid introducing the EFT counter term up to $k{\\rm Mpc}\/h\\lesssim 0.2$ at $z=0.6$. We, therefore, adopt the 1-loop LPT approach.\n\nWe modeled the iterative reconstruction as follows: firstly, we put forward the $n$th reconstruction step displacement field ansatz in the most general form, expanding the equations up to third order in the 0th step, i.e., the nonlinear displacement field corresponding to the observed nonlinear density. Then we derived the $(n+1)$th step displacement field and obtained the recurrence relation for the expansion kernels.\nThis allows us to systematically compute an arbitrary step's postreconstruction 1-loop spectrum using the recurrence relation.\n\nOur model predicts that the postreconstruction displacement field should converge to the true displacement field, contrary to what we found in our simulation. The model provides a good description of the cross-correlation coefficient, while the difference in the propagator is mainly responsible for the discrepancy between the model and the simulation. \n\n\n\nWhile we could not identify the reason for the discrepancy between the model and the reconstructed field in the present paper, we suspect this is potentially related to the subsampling of the tracers. Including such an effect in the theoretical model and implementing a pixel window function correction will help identify the discrepancies. We leave such an extension for future papers. \n\nOur tentative solution to the discrepancy is to drop all UV-sensitive integrals from the 3LPT spectrum, and the ad hoc model fits the postreconstruction displacement field with 1\\% accuracy for $n>6$.\n\n\n\n\n\nThis paper presents the proof of concept on how we can model the iterative reconstruction steps using the real-space mass distribution with little shot noise effect.\nImproving the agreement between the model and the simulation in future papers, we expect it would be straightforward to generalize our modeling to the redshift space. \nOn the other hand, an extension to biased tracers would be nontrivial, based on our investigation in the companion paper~\\cite{Seo:2021}. While the model we built appears quite strenuous, we note that all iteration kernels are cosmology-independent and could be precomputed and interpolated. A short come is that we find that the speed of 1-loop calculation is still slow. We confirmed that the prereconstruction 3LPT calculation could be optimized with FFTlog, using the technique presented in Ref.~\\cite{Schmittfull:2016jsw}. However, we found that the same algorithm does not apply to the postreconstruction spectrum because of the complex structure of the iteration kernels.\nThus, the iterative reconstruction requires 2D standard quadrature integration. Speeding up those calculations would be critical for actual data analysis. Additionally, it would be essential to extract a more compressed and practical form of the model that can be more efficiently utilized for data analysis, which we plan to do to implement the redshift space distortions and galaxy bias. \n\n\n\nFinally, we note that our work is still in an early stage compared to the modeling of the standard reconstruction density field using the 1-loop SPT in Ref.~\\cite{Hikage:2017tmm} and the Zeldovich approximation in Ref.~\\cite{Chen:2019lpf}, etc. Moreover, Ref.~\\cite{Seo:2021} implies that the iterative displacement reconstruction needs to be better optimized in the presence of high shot noise to take full advantage of it. We nevertheless believe that reconstructing the nonlinear displacement field could be pretty helpful in a deficient shot noise regime that can be available in future galaxy surveys due to its high correlation with the initial condition and, therefore, worth further investigation. \n\n\\begin{acknowledgments}\n\nThe authors would like to thank Marcel Schmittfull for providing simulations and valuable discussions.\nThe authors also would like to thank Stephen Chen for their helpful discussions.\nAO would like to thank Masaru Hongo for valuable discussions. AO and H.-J.S. are supported by the U.S.~Department of Energy, Office of Science, Office of High Energy Physics under DE-SC0019091. \nSS was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. \nThis project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 853291). FB is a University Research Fellow.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\par\nThe study of rare kaon decays \\cite{kdecay-rare} has played an important role \nin the establishment of the standard model (SM) in particle physics. \nIt has also been crucial for understanding the phenomenon of CP violation \\cite{kdecay-CP}. \nThe rare decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ \\cite{litt89,isidori,buras} is a direct CP violation process caused by\na flavor-changing neutral current (FCNC) with transition from a strange to down quark.\n\\par\nThe unique characteristic of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay is that the branching ratio can be \ncalculated with very small theoretical uncertainties. In the SM\nbased on the Cabibbo-Kobayashi-Maskawa (CKM) matrix \\cite{ckm} for quark flavor mixing, \nthe $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio is predicted to be $(2.49\\pm 0.39) \\times 10^{-11}$ \\cite{mescia}. \nThe uncertainty of the prediction is dominated by the\nallowed range of the imaginary part of a CKM matrix element $V_{td}$, \nwhich is determined by other measurements,\nand the intrinsic theoretical uncertainty is only 1-2\\% \\cite{buras}.\n\\par\nThe determination of CKM matrix elements has been greatly improved\nin the past ten years\nby measuring various B-decay properties \\cite{bdecay}; \nall measurements were consistent with each other within the standard model parameterization. \nBy measuring the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay precisely, we can check the consistency against the \ncurrently predicted value in the SM.\nA deviation indicates new physics beyond the SM \nbecause of the very small uncertainty in deriving the imaginary part of $V_{td}$\nfrom the branching ratio of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay.\n\\par\nThe decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ is one of the processes expected to have a significant impact on \nnew physics searches because it is an FCNC process that can proceed through loop diagrams, \nincluding the interactions at short distance and large mass scales.\nGrossman and Nir \\cite{grossman} pointed out the importance of studying new physics by using \nboth the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$ decays, where the decay $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$ is another rare kaon decay with \nsmall theoretical uncertainties \\cite{buras,mescia,brod}. \nVarious new physics models have been developed and used to predict \nthe $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and $K^+\\rightarrow\\pi^+\\nu\\bar{\\nu}\\;$ branching ratios \\cite{buras, isidori,mesciaweb}.\n\\par \nThe best upper limit obtained in previous experiments was BR($K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$) $<$\n5.9 $\\times$ 10$^{-7}$ (90\\% CL) \\cite{ktev}. \nThe ultimate goal of our experimental study is to determine the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio \nwith an accuracy less than 10\\% of the value predicted in the SM.\nThe goal can be achieved by performing a series of experiments with improved and refined \ndetection methods. \n\\par\nThe E391a experiment, which was carried out at the KEK 12-GeV proton synchrotron \n(KEK-PS), is the first step of this approach. The main objectives of E391a were not only \nto investigate the decay with a dedicated detector, but also to test and confirm our basic \nexperimental methods for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ at the highest possible sensitivity. \nData collection of the E391a experiment started in February 2004 and continued \nuntil November 2005, which was one month before the shutdown of the KEK-PS. \nThe total running time was about twelve months and it was divided into three periods \n(Run-1, Run-2, and Run-3). \nEarly results from the first and second periods were reported in \nRef.~\\cite{run1-oneweek} and Ref.~\\cite{run2}, respectively. In this article, we report \nthe final results on the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay as obtained from E391a.\n\n\n\\section{Experimental method}\n\\subsection{Basic method of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ detection}\nThe key signature of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay is detection of exactly two photons and \nnothing else. \nThe \\pizs decays dominantly into two photons ($\\gamma\\gamma$), \nand the neutrinos are undetectable. \nBecause detection of the incident $K_L^0\\;$ is difficult, measurements can only be made of \nthe energy and position of the two photons in a calorimeter located downstream of the $K_L^0\\;$ decay region, \nwithout having direct information of the incident particle.\nKinematic constraints are weak for definitive identification of the decay. \nInstead, the decay must be isolated by eliminating all possible backgrounds. \n\\par\nBecause the signal mode is identified as the final state of two photons and nothing else,\nprocesses that make two or more photons can cause background events.\nThe $K_L^0\\;$ decays such as $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ become backgrounds when\nextra photons or charged particles escape detection.\nThe $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay can be a background source because it has only two photons in the final state,\nalthough it is well suppressed by kinematical constraints.\nAnother background process is hadronic interactions of beam neutrons \nwith the residual gas in the beamline or in detectors near the beam.\nAny \\pizs or $\\eta$ produced in these interactions decay into two photons and\nproduce backgrounds. \nHyperons produced at the target can cause backgrounds through processes \nsuch as $\\Lambda \\to \\pi^0 n$;\nthese hyperon events are strongly suppressed because most of them decay in a 10 m long neutral beamline.\n\\par\nThe most important tool for reducing the background is a hermetic detector\nsystem to detect and veto extra particles. \nBecause all of the other $K_L^0\\;$ decays, except for $K_L^0\\rightarrow\\gamma\\gamma\\;$, are accompanied with \nat least two additional photons or charged particles, \nthe detector should be highly sensitive to photons and charged particles. \n\\par\nThe signature of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ can be provided by using a small-diameter $K_L^0\\;$ beam \n(called a ``pencil beam\") and measuring precisely the energy and position of the two photons.\nAlthough $K_L^0\\;$ flux is reduced with a pencil beam, it has several advantages. \nFirst, the beam hole at the center of the calorimeter, which compromises hermeticity, can be minimized. \nSecondly, the $K_L^0\\;$ decay vertex position ($Z_{\\textrm{VTX}}$) can be assumed to be on the beam axis.\nThe vertex position, which is same as the \\pizs decay position due to the short lifetime of $\\pi^0$,\nis obtained from the kinematics of the two photons from its decay (See Sec.\\ III.B).\nThe transverse momentum of the \\pizs ($P_T$) with respect to the $K_L^0\\;$ \nbeam axis is also obtained. \n\\par\nThe signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay can be defined with $Z_{\\textrm{VTX}}\\;$ and $P_T$.\nRequiring a sufficiently large missing transverse momentum \neliminates contamination from the $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay and also reduces the contamination \nof other $K_L^0\\;$ decays. Most of the $K_L^0$'s decay into multiple particles that have \nlow momenta in the $K_L^0\\;$ rest frame, and hence low $P_T\\;$ in the laboratory frame. \nThe $Z_{\\textrm{VTX}}\\;$ should be in the region away from beam counters \nand $P_T\\;$ should be in the range from 120 to 240 MeV\/$c$, \nwhere the maximum momentum of $\\pi^0$'s in the $K_L^0$-rest frame is 231 MeV\/$c$ for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$. \n\\par\nProduction of $\\pi^0$'s through beam-gas interactions in the decay fiducial region are \nreduced by having a high vacuum. Hit-rates of the beam halo (mostly neutrons) \nwith the surrounding detectors are minimized by a sharp collimation of the beam. \nIn addition, as discussed in Sec.\\ III.A and in the subsequent analysis of the\ndata, mis-combinations of two photons from background processes and\/or \nmis-measurements of energies and positions were found to be major sources \nof backgrounds in the E391a experiment. \nThe photons were often produced from interactions of the beam halo and can lead to incorrect determinations of $Z_{\\textrm{VTX}}$. \nSome backgrounds from beam interactions can be reduced by detecting low-energy deposits from recoil \nparticles emitted after the interactions. More detailed descriptions of the \ndetection method have been reported in the original proposal of E391a \\cite{design}. \n\\par\nInefficiencies of photon and charged particle detection provide backgrounds. \nThe decay $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ is the most serious background source arising from an inefficiency \nbecause it has only two extra photons in the final state. \nThe inefficiency of photon detection was investigated in a series of experiments \n\\cite{phineff}, which showed that the inefficiency monotonically decreased with the \nincident photon energy. \nIt was also found that photon detection with a very low energy threshold is necessary \nto achieve a small overall inefficiency, even for high-energy photons. If an extremely \nlow energy threshold around 1 MeV was set, the backgrounds from other $K_L^0\\;$ decays were \nreduced to a negligible level within the sensitivity of the experiment. \n\\par\nThe following sections describe the application of the basic methods in the\nE391a experiment.\n\n\n\\subsection{Beamline}\nThe production target of the $K_L^0\\;$ beam was a platinum rod with a length of 60 mm and \na diameter of 8 mm. The profile of the primary beam was $\\sigma$ = 3.3 mm and \n$\\sigma$ = 1.1 mm along the $X$ (horizontal) and $Y$ (vertical) axes.\nThe neutral beam was extracted at an angle of 4$^{\\circ}$ with respect to the primary \nproton beam. The target rod and beamline elements were aligned along a straight line \nin the direction of the neutral beam. The total length of the beamline was 10 m. \nThe neutral beamline consisted of two dipole magnets (D1, D2) to sweep charged particles \nout of the beam, with field strengths of 4 T$\\cdot$m and 3 T$\\cdot$m, respectively, and six collimators \n(C1 - C6) to collimate the beam, as shown in Fig.~\\ref{fig_beamline}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.48\\textwidth]{.\/fig1.eps}\n\\end{center}\n\\caption{Schematic view of the neutral beamline: (top) the arrangements of the components and \n(bottom) the collimation scheme. C1-C3, C5, and C6 were tungsten collimators, assembled as a \nstack of cylindrical blocks 5-cm thick with circular holes of different diameters.\nEach collimator approximated the cone-shaped aperture as indicated by the A-E lines \nin the bottom figure. The $Z = 0$ coordinate is at the center of the production target.}\n\\label{fig_beamline}\n\\end{figure}\n\n\\par\nFive collimators (C1-C3, C5, and C6) were made of tungsten. \nThe C2 and C3 collimators in the upstream end were used to define the beam with the designed \napertures, which were arranged to form a half cone angle of 2 mrad from the target center. \nThe last two collimators, C5 and C6, were used to trim the beam halo.\nThe most upstream collimator C1 reduced the size of the beam immediately after the target \nwithout producing a large penumbra. The total thickness of these five collimators \nwas approximately 6 m. A thermal-neutron absorber, which was made of polyethylene terephthalate \n(PET) sheets containing Gd$_2$O$_3$ 40\\% in weight, was used for C4. \nThe aperture of C4 was set to be larger than that of the other collimators.\n\\par\nMovable absorbers, made of lead (Pb) and beryllium (Be), were placed between C1 and C2 \nto reduce the number of photons and neutrons relative to the $K_L^0$'s. The absorbers were \n10-mm-diameter rods with the lengths of 5 cm and 30 cm for Pb and Be, respectively. \nThe downstream region, starting with a stainless steel window 100 $\\mu$m thick\nat the upstream end of C4, was evacuated to approximately 1 Pa.\n\\par\nThe primary beam on the target was monitored by a secondary-emission chamber (SEC) placed \nupstream of the target, and a target monitor (TM) that was a counter telescope\nwhich viewed the target center at 90 degrees. \nThe primary beam position at the target was adjusted with steering magnets \nby monitoring the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay events.\nIn the $3\\pi^0$ events, six photons were detected by a CsI calorimeter\nwhich will be described in Sec.\\ II.C.\nThe center of energy was defined to be $\\bm{r}_c = \\sum E_i \\, \\bm{r}_i$, where $E_i$ are the photon \nenergies and $\\bm{r}_i$ are the photon hit positions at the CsI calorimeter. \nBecause the six-photon events were mostly $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays, the peak position \nof the center of energy should be on the axis of the beamline.\nThe position of the neutral beam was maintained to be within 0.2 mm \nfrom the center throughout the entire running period.\nThe beam size at the CsI calorimeter, was also monitored by the distribution of the center of energy.\nIts diameter was $\\sigma = 40$ mm, which was consistent with the beam divergence.\n\\par\nThe peak momentum of the $K_L^0\\;$ as determined from beamline simulations was 2 GeV\/$c$ at the exit of C6, \nas shown in Fig.~\\ref{fig_spectrum}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig2.eps}\n\\end{center}\n\\caption{\nMomentum spectrum of $K_L^0\\;$ at the exit of C6, obtained from beamline simulations.}\n\\label{fig_spectrum}\n\\end{figure}\n\nThe initial $P_T\\;$ spread due to the beam divergence of 2 mrad was approximately 4 MeV\/$c$. \nThe neutron-to-$K_L^0\\;$ ratio was 60 and the halo-to-core ratio was approximately 10$^{-5}$ \nfor both neutrons and photons with energies above 1 MeV, as shown in Fig.~\\ref{fig_profile}. \nBy inserting a Be absorber, the neutron-to-$K_L^0\\;$ ratio was reduced to 40, \nwith roughly a 45\\% loss in $K_L^0\\;$ flux.\nProfiles of the $K_L^0\\;$ beam are shown in Fig.~\\ref{fig_klprofile}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig3.eps}\n\\end{center}\n\\caption{\nBeam profiles of neutrons and photons above 1 MeV at the exit of C6 collimator, \nobtained from beamline simulations.}\n\\label{fig_profile}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig4.eps}\n\\end{center}\n\\caption{\nBeam profile of $K_L^0\\;$ at the exit of C6 collimator, obtained from beamline simulations.\nFilled and open circles show the case with and without the Be absorber, respectively.\n}\n\\label{fig_klprofile}\n\\end{figure}\n\n\nPunch-through muons were emitted in the direction parallel to the beam axis. \nTheir position distribution was almost flat and the flux density was larger than the \ncosmic ray flux by roughly one order of magnitude.\nDetails of the beamline has been reported elsewhere \\cite{beamline}.\n\n\\subsection{Detectors}\nThe E391a detection system was located at the end of the beamline.\nThe detector subsystems were cylindrically arranged around the beam axis, and most of \nthem were placed inside a large vacuum vessel, as shown in Fig.~\\ref{fig_apparatus}.\nFrom here on, the origin of the coordinate system is defined to be at the upstream end of the \nE391a detector, as shown in Fig.~\\ref{fig_apparatus}.\nThis position was approximately 12 m from the production target.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\rotatebox{-90}{\\includegraphics[width=.3\\textwidth]{.\/fig5.eps}}\n\\end{center}\n\\caption{Detection system.}\n\\label{fig_apparatus}\n\\end{figure*}\n\n\n\\subsubsection{CsI calorimeter}\nThe energy and hit position of photons, such as the two photons\nthat would come from the \\pizs decay in the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ signal, were measured by \nusing a calorimeter placed at the downstream end of the decay region. \nAs shown in Fig.~\\ref{fig_csi}, the calorimeter was made of 496 undoped CsI crystals\neach with a dimension of 7 $\\times$ 7 $\\times$ 30 cm$^3$ (Main CsI)~\\cite{e162-csi} and \nassembled in a cylindrical shape with an outer diameter of 1.9 m.\nThere was a 12 cm $\\times$ 12 cm beam hole at the center of the calorimeter.\nThe beam hole was surrounded by a tungsten-scintillator collar counter (CC03) and\n24 CsI modules with dimensions of 5 $\\times$ 5 $\\times$ 50 cm$^3$ (KTeV CsI)~\\cite{ktev-csi}.\nThe gaps at the periphery of the cylinder were filled with \n56 CsI modules with trapezoidal shape and 24 modules \nof lead-scintillator sandwich counters (Lead\/Scintillator Sandwich).\nThese modules were tightly stacked such that the gaps between them were less than 0.1 mm. \nAlthough the CC03 and the sandwich counters were used only in the veto system, \nthe other parts, which consisted of CsI crystals, were used for photon measurements. \nThe average light yield of the main CsI modules was 16 pe\/MeV, where pe\/MeV is the unit \nof the number of photo-electrons emitted from the photomultiplier tube (PMT) cathode \nfor an energy deposit of 1 MeV in the detector. Details of the CsI calorimeter have \nbeen reported in Ref.~\\cite{csi}.\n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig6.eps}\n\\end{center}\n\\caption{Assembly of detector subsystems in the downstream end-cap.}\n\\label{fig_csi}\n\\end{figure}\n\n\n\\subsubsection{Charged veto counter}\nA set of plastic scintillation counters (named CV) were placed in front of the CsI \ncalorimeter to identify events that included the emission of charged particles,\nsuch as $K_L^0 \\rightarrow \\pi^-e^+\\nu\\;$ decay~\\cite{chineff}.\nA total of 32 sector-shaped modules (outer CV) were placed at a distance of 50 cm\nfrom the front face of the CsI calorimeter. \nThe outer CV modules were arranged to have overlaps between adjacent modules.\nThe beam region from the outer CV to the \nCsI was covered by the inner CV, which was a square pipe formed with four plastic scintillator plates. \nThe inner and outer CVs were closely connected with aluminum fixtures.\nIn order to eliminate gaps between the outer and inner modules, \nthe inner modules were extended to cover the edge of the outer modules.\nThe edges of the outer CV and the inner CV were located \nclose to the beam, and neutrons in the beam halo frequently interacted with them. \n\n\n\\subsubsection{Barrel counters}\nThe decay region was surrounded with two large lead-scintillator sandwich counters: \nthe main barrel (MB) and the front barrel (FB), composed of 32 and 16 modules, \nrespectively. The modules were tightly assembled with small gaps \\cite{kumitate}, \nas shown in Fig.~\\ref{fig_barrels}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.35\\textwidth]{.\/fig7.eps}\n\\end{center}\n\\caption{Cross-sectional end view of the MB (outer ring) and FB (inner ring).}\n\\label{fig_barrels}\n\\end{figure}\n\nThe lamination was parallel to the beam axis, and the lengths of the MB and FB modules \nwere 5.5 m and 2.75 m, respectively. We developed a new type of plastic scintillator\nmade of a resin mechanically strengthened with a mixture of styrene and methacrylate \\cite{ci-scinti}. \nThe scintillator counters were fabricated by extrusion.\nThe thickness of the MB and FB modules were 13.5$X_0$ and 17.2$X_0$, respectively. \nThe emitted light was transmitted through wavelength-shifting (WLS) fibers, \nwhich were glued to each scintillator plate with a pitch of 10 mm. \nThe fibers had a double-cladded structure with Y11 as a dopant. \nThe green light emitted from Y11 was read by a newly developed photomultiplier tube \nwith a high quantum efficiency for green light (EGP-PMT)~\\cite{pmt}. \nFor readout, each module was divided into an outer and an inner parts. \nThe MB module was viewed from both ends, and the FB module was viewed from the upstream end. \nFigure~\\ref{fig_mblight} shows the light yields for 4 readout channels of one MB module, \nobtained from a cosmic-ray test. \nThe light yield and attenuation of the FB module were similar to those shown in \nFig.~\\ref{fig_mblight} for the MB module. Details of the barrel counters has been \nreported in Ref.~\\cite{barrel}.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig8.eps}\n\\end{center}\n\\caption{Light yield and attenuation for a MB module.}\n\\label{fig_mblight}\n\\end{figure}\n\n\nA layer of plastic scintillator was placed inside of the MB modules to identify charged particles. \nIt consisted of 32 modules and was called the barrel charged veto (BCV).\nA module was made of a 10-mm-thick plastic scintillator, \nand the light signal was read from both ends by an EGP-PMT through WLS fibers glued \nto the scintillator with a 5-mm pitch. \n\n\n\\subsubsection{Counters close to the beam}\nMultiple collar-shaped counters, CC02 -- CC07, were placed along the beam axis. \nCounter CC02 at the entrance of the decay region was a lead-scintillator \nsandwich of the Shashlik type, with WLS fibers piercing the lead and \nscintillator layers. Counter CC03 at the end of the decay region was a \ntungsten-scintillator sandwich. Counters CC04 -- CC07 covered the solid-angle in \nthe downstream direction. Counters CC04 and CC05 were lead-scintillator\nsandwiches with WLS fibers glued on each scintillator plate at a pitch of 10 mm, \nwhich was the same as for the MB and FB. \nCounters CC06 and CC07 were made of SF5 lead-glass. \nThe direction of lamination was perpendicular to the beam for \nCC02, CC04, and CC05, and was parallel to the beam for CC03. The signals \nfrom CC02 -- CC05 were read by EGP-PMTs. \n\\par\nBeginning with Run-2, another collar counter called \nCC00 was installed in front of the FB and outside the vacuum vessel to reduce\nthe effects of halo neutrons. It consisted of 11 layers of a 5-mm-thick plastic \nscintillator interleaved with 10 layers of 20-mm-thick tungsten. \nHowever, CC00 did not significantly reduce the neutron halo because it was \nnot placed as close to the beam as the other collar counters. \n\\par\nA beam-plug counter, back-anti (BA), was placed at the end of detector system along \nthe beam axis. In the first two data-taking periods, Run-1 and Run-2, the BA consisted \nof six superlayers, where each superlayer had six plastic-scintillator layers\ninterleaved with lead sheets, and a single layer of quartz. In Run-3, the lead \nand plastic layers were replaced with PWO crystals as shown in Fig~\\ref{fig_ba},\nwith the intention to separate electromagnetic showers from the neutron hits.\n\\par\nA thin layer of plastic scintillator, the beam hole charged veto (BHCV), \nwas placed in front of the BA. It was effective in removing the \n$K_L^0\\rightarrow\\pi^0\\pi^+\\pi^-\\;$ decay. \nThe thickness of the BHCV was 1~mm for Run-1, and 3~mm for Run-2 and Run-3.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.32\\textwidth]{.\/fig9a.eps} \\\\\n\\includegraphics[width=.32\\textwidth]{.\/fig9b.eps}\n\\end{center}\n\\caption{BA used in Run-2 (top) and Run-3 (bottom).}\n\\label{fig_ba}\n\\end{figure}\n\n\\par\nThe upstream counters, CC02 -- CC04, were placed inside the vacuum vessel, \nand the downstream CC05 -- BA were placed outside the vessel. A vacuum duct, \nwhich was directly connected to the vacuum vessel, penetrated CC05 -- CC07 \nand the vacuum region was extended up to the front face of BHCV.\n\\par \nThe size and basic parameters of CC02 -- CC07, BHCV, and BA are summarized in Table~\\ref{tab:table1}.\n\n\\begin{table*}\n\\caption{%\n\\label{tab:table1}\nSize and basic parameters of detectors along the beam.\nThe origin of the $z$ position corresponds to the start of the E391a detector.\nIn the sizes of cross section and hole, dia. and sq. represents diameter and square, respectively.\nVisible fraction ($R_{vis}$) is defined as the ratio of the energy deposit in \nthe active material to that in the whole volume. \nThe thickness is expressed in units of radiation length.\nThe recorded number of photo-electrons per 1-MeV visible energy deposit \nwas 10 or more for sandwich counters (CC02-05), and \nat least 0.5 for lead glass detectors (CC06-07), respectively.}\n\\begin{ruledtabular}\n\\begin{tabular}{lccclccc}\ndetector & z position (cm) & outer dimension (cm) & inner dimension (cm) \n& configuration & $R_{vis}$ & thickness ($X_0$) \\\\\n\\hline\nCC02 & 239.1 & 62.0~dia. & 15.8~dia. \n& lead\/scint. & 0.32 & 15.7 \\\\\nCC03 & 609.8 & 25.0~sq. & 12.0~sq. \n& tungsten\/scint & 0.23 & 7.6\\footnotemark[1] \\\\\nCC04 & 710.3 & 50.0~sq. & 12.6~sq. \n& lead\/scint. & 0.28 & 11.8 \\\\\nCC05 & 874.1 & 50.0~sq. & 12.6~sq. \n& lead\/scint. & 0.28 & 11.8 \\\\\nCC06 & 925.6 & 30.0~sq. & 15.0~sq. \n& lead glass & 1.0 & 6.3 \\\\\nCC07 & 1000.6 & 30.0~sq. & 15.0~sq. \n& lead glass & 1.0 & 6.3 \\\\\nBHCV & 1029.3 & 23.0~sq. & no\n& plastic scint. & 1.0 & 0.007 \\\\\nBA(Run-2) & 1059.3 & 24.5~sq. & no \n & quartz & 1.0 & 1.5 \\\\\n& & & & lead\/scint. & 0.31 & 13.3 \\\\\nBA(Run-3) & 1059.3 & 24.5~sq. & no\n & quartz & 1.0 & 1.2 \\\\\n& & & & PWO & 1.0 & 16.8 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{The lamination of CC03 was parallel to the beam.}\n\\end{table*}\n\n\\subsubsection{Vacuum}\nThe vacuum pressure in the decay region had to be maintained below 10$^{-5}$ Pa in order \nto reduce the \\pizs backgrounds produced by beam-gas interactions to a negligible level \nat the sensitivity corresponding to the SM prediction of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nThe amount of dead material along the path of particles from the decay vertex to the detector had to be \nminimized in order to achieve highly efficient detection even for low-energy particles. \nA differential pumping method was adopted for this purpose. \n\\par\nThe entire detection system was placed in a large vacuum vessel, as shown in\nFig.~\\ref{fig_apparatus}. The pressure in the outer part, where the detectors were placed, \nwas around 1 Pa, and the pressure in the inner part, through which the beam passed, was \n$1 \\times 10^{-5}$ Pa. \nThe two regions were separated by a laminated membrane sheet \nwith a thickness of 20 mg\/cm$^2$. \nAs shown in Fig.~\\ref{fig_membrane},\nthe sheet was a lamination of four films. \nThe EVAL film had low transmission for oxygen gas (mostly air) and the nylon film strengthened the sheet. \nThe polyethylene layers on both sides were used to make a tight connection by using a heat iron press.\nThe bag-shaped membrane covered the inner surface of the CsI detectors by using a \nskeleton structure of thin aluminum pipe, similar to a camping tent. \n\\par\nTwo sets of rotary and root pump systems were connected through a manifold\nand eight ports to the outer-vacuum part. The pumping speed of each system was \n1200 m$^3$\/hour. Four turbo molecular pumps, each having the pumping speed of 800 l\/sec, \nwere connected between the inner vacuum part and the manifold. They produced \nthe necessary pressure difference of an additional five orders of magnitude.\n\\par \nFor the PMTs installed in vacuum, the pressure had to be less than 10 Pa to prevent \nhigh voltage discharges. The PMT operation in vacuum additionally caused a cooling \nproblem due to the absence of convection. We modified the configuration of the\nresistor chains of the PMTs, and cooled them with a water circulation system \nas described in Ref.~\\cite{csi}. The temperature was stable within $\\pm 0.1^{\\circ}$ \nfor the CsI calorimeter.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.3\\textwidth]{.\/fig10.eps}\n\\end{center}\n\\caption{\nMembrane used for the vacuum separation.\n}\n\\label{fig_membrane}\n\\end{figure}\n\n\n\\subsubsection{Electronics}\nAbout 1000 PMTs were used as readout devices for all detectors. \nSignals from the detectors in the vacuum vessel were extracted through individual \nfeed-through connectors. A connector was made of a coaxial cable simply molded to a metal \nflange with a resin. \n\\par\nAll detector signals were fed to Amplifier-Discriminator (AD) modules, \nwhich were newly developed for the experiment. An AD module accepted 16 PMT signals \nand generated 16 analog signals, 16 discriminated signals, and two 8-channel linear sums. \nBecause early discrimination prevents time deterioration due to distortion of the signal \nshape in the cable, the AD modules were placed near the vacuum vessel to shorten the cables.\n\\par\nThe analog output signal was derived from each input signal with a throughput of 95$\\%$. \nIt was sent to a charge-sensitive analog-to-digital converter (ADC) in the counting hut \nthrough a 90-m coaxial cable, while the cable length of the other outputs was 30 m. \nThe discriminator output was generated with a very low threshold of 1 mV, which corresponded \nto an energy deposit of 1 MeV for the CsI calorimeter and below 1 MeV for the other detectors.\nThe discriminator output was sent to a time-to-digital converter (TDC) in the counting hut \nthrough 30 m of twisted-pair cable after passing through a fixed delay of 300 ns in the AD \nmodule. The signal summed over 8 inputs was sent to the counting hut through 30 m of \ncoaxial cable and used to form trigger signals for data acquisition. \n\\par\nThe ADC module, LeCroy FASTBUS 1885F, had 96 input channels, \neach channel having the equivalent dynamic range of a 15-bit ADC in its 12-bit data\nby using a bi-linear technique.\nThe typical resolution at a low energy range was\n0.13 MeV\/bit for the CsI calorimeter, and 0.035 MeV\/bit for photon veto detectors.\nThe gate width for the CsI calorimeter was 200 ns. \nThe TDC module in TKO (Tristan KEK Online-system), which was an electronics platform developed in KEK~\\cite{tko},\nwas operated at a \nfull range of 200 ns and a resolution of 50 ps. The analog and discriminated signals were \nalready delayed by 300 ns (60-m cable with a propagation velocity of 20 cm\/ns) \ncompared to the timing of linear-sum signal at the entrance of the counting hut. \nAll cables to the counting hut were placed inside trays covered with copper-plated iron sheets \nto minimize ground noise caused by alternating magnetic fields. \nThe pedestal widths of almost all the ADC channels were less than 1 bit. \n\\par\nAny failures in the PMTs lead to serious problems in the experiment. Almost all \nPMTs were operated at voltages below 60\\% of the rated voltage. Such a large margin was \nmade possible by using very sensitive ADCs and very low thresholds for the TDCs. \nAll PMTs operated properly during the entire running time.\n\\par\nMulti-hit TDCs (MTDC) were used for the BHCV and BA to cope with high counting rates. \nWe did not benefit from using MTDCs in Run-1 because the input pulse width was set \nto 100 ns. For Run-2 and Run-3, the pulse width was shortened to 25 ns.\n\n\n\\subsection{Trigger and data acquisition}\nThe signals from eight adjacent CsI blocks (a segment) were collected into a summed\nsignal; there were 72 such segments as shown in Fig.~\\ref{fig_hwclustering}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig11.eps}\n\\end{center}\n\\caption{Segments of CsI blocks.}\n\\label{fig_hwclustering}\n\\end{figure}\n\nThe segments were used to form a trigger signal.\nA threshold, which corresponded to an energy deposit of 80 MeV, was applied to\neach summed signal, and the number of segments whose summed signal was above the threshold, \nN$_{\\rm HC}$, was counted. \nWe required N$_{\\rm HC} \\ge 2$ for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ trigger.\nWe also required an anti-coincidence of several veto counters, CV, MB, FB, and CC02 -- CC05 \nwith a rather high threshold. \nLater, in the offline analysis, these conditions were set tighter: \na higher threshold for photon detection and lower thresholds for the vetos. \n\n\\par\nIn addition to the trigger for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$, we prepared triggers for light pulsars: \nxenon lamps for CsI and LEDs for the other detectors. They were used to monitor\nshort-term drifts in the PMT gains. Moreover, they were useful for studying the\neffects of beam loading by flashing them within and outside the beam spills. \nTriggers for cosmic ray muons and punch-through muons were also used for \ndetector calibration. \n\\par\nWe prepared two types of triggers to record accidental hits in the detector system \nby using an electronic pulse generator and the TM, which was a counter telescope near the target. \nThe accidental activities observed in the two types of triggers were consistent with each other\nfor all detectors except the BA and BHCV. \nWhile the pulsar trigger had no correlation with the event time and was randomized \nwith respect to the asynchronous timing of events, \nthe TM trigger reflected intensity variations of the primary beam. \nThe TM trigger data were normally used because we observed \nthe micro time-structure of the beam extraction.\n\\par\nAccidental losses were estimated by applying the analysis cuts used for \n$K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ to the TM trigger data. \nThe estimated value was compared with the value estimated from Monte Carlo simulations \nwith imposition of the TM trigger data, and consistency was confirmed. \n\\par\nData were collected through multiple parallel systems of the VME crates\noperated with a CPU to control them. \nEnvironmental data such as temperatures at 100 locations, \nvacuum pressure at several places, and single counting rates of \nsampled channels were accumulated by using PCs.\nThe CPU and PCs were distributed on the network.\nThe basic software used in the E391a experiment was MIDAS \\cite{midas}. \n\\par\nThe dead time of the DAQ system was around 600 $\\mu$s\/event. \nIn a typical run, in which the proton intensity was $2.5\\times 10^{12}$ POT (protons on target) \nin a 2-s spill every 4 s, the trigger rate was 300 per spill (150 Hz), \nand the live time was 91\\%. The data size was 3 Mbytes\/spill and the typical \ndata size collected per day was 60 GB.\n \n\n\\subsection{Calibration}\nGood linearity between the energy deposit and the ADC output was observed in all \ndetector subsystems. \nTheir gains were calibrated through a constant in units of MeV\/bit.\nThe gain constants of all detectors were basically calibrated in situ, \nafter assembling and in vacuum, by using cosmic-ray muons and\/or punch-through muons \ncoming from the upstream region of the primary beamline. While the cosmic-ray muons \nprimarily traveled in the downward direction, the punch-through muons were parallel \nto the beam. In the case of sandwich detectors, we selected the muons so that their \nprimary direction was perpendicular to the lamination.\n\\par\nThe CsI modules were initially calibrated by using cosmic-ray tracks \nsuch as the example shown in Fig.~\\ref{fig_cosmic}. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.43\\textwidth]{.\/fig12.eps}\n\\end{center}\n\\caption{Cosmic ray track used for calibration. The outer ring shows the MB.}\n\\label{fig_cosmic}\n\\end{figure}\n\nThe punch-through muons were used to cross-check the cosmic-ray calibration because their \ndirections were perpendicular to each other and their penetration lengths were different \n(7 cm for cosmic-ray muons and 30 cm for punch-through muons). The gain constants were \nrefined by using the two photons from a \\pizs produced in a special run in which an aluminum \nplate was installed on the beam axis. \nFinally, the gain constants were refined by an iteration process based on the\nkinematic constraints of $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay.\nThe short-term variation of the gain was corrected by using xenon-lamp light pulses. \nThe reconstructed $K_L^0\\;$ mass and the width were well stabilized, as shown in Fig.~\\ref{fig_gainstab}. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{.\/fig13a.eps} \\\\\n\\includegraphics[width=.45\\textwidth]{.\/fig13b.eps}\n\\end{center}\n\\caption{\nStability of the kinematic variables over the entire period of Run-3.\nThe upper and lower graphs show the peak and width of the effective mass distribution \nof 3\\pizs from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay, respectively.}\n\\label{fig_gainstab}\n\\end{figure}\n\n\\par\nIn the Monte Carlo simulations, we smeared the energy deposit of each\nphoton by the function $a\/\\sqrt{E(\\rm{GeV})}+b$. The first term was given as \n$0.008\/\\sqrt{E(\\rm{GeV})}$, which was consistent with the statistical fluctuation \nof photoelectron yields of 16 pe\/MeV. The parameter $b$ was determined to be \n0.004 by tuning it to reproduce the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ invariant mass distribution.\nBecause this smearing was applied to the deposited energy instead of the incident \nphoton energy,\nthe small value indicates good understanding of the calibration.\n \\par\nThe beam loading effect was examined by flashing the light pulse during and between \nthe beam spills. \nIt was negligibly small in the current experiment for all detectors, \nexcept for the BHCV and BA in which the PMT gain was shifted by 10\\%.\n\\par\nThe timings were determined relative to one of the photon clusters in the CsI calorimeter. \nFirst, TDC constants (ns\/count) were measured for all TDC channels by using a fixed \ndelay, and the time-zero value was calibrated from the data. Cosmic-ray and\/or punch-through \nmuons were also used for the time-zero calibration. They were determined step-by-step by \nutilizing the overlapping parts of different detectors with respect to the muon tracks.\n\\par\nThe time-zero values among CsI modules were determined by using cosmic ray tracks.\nFinally the calibration was refined by using six \nphotons from $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays. The time difference between two photons was determined \nwith a standard deviation of 0.3 ns, as shown in Fig.~\\ref{fig_csitiming}, where the \ntiming of a photon cluster was estimated from the timing of the central block that had \nthe local maximum energy deposit.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig14.eps}\n\\end{center}\n\\caption{Time difference between two photons from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay,\nobtained by subtracting the timing of lower energy photon from that of higher energy photon.\n}\n\\label{fig_csitiming}\n\\end{figure}\n\n\n\n\\section{Data analysis}\n\\subsection{Summary of three runs and outline of present analysis}\nDuring the course of the experiment, there were $(1 - 2.5 ) \\times 10^{12}$ \nPOT per spill, with a total yield of $4.6 \\times 10^{18}$ POT for physics runs.\nA summary of three physics runs is listed in Table~\\ref{table_runs}.\nThe experimental setup was modified during intervals between the runs, \nand the running parameters were refined on the basis of the results of a previous run.\n\n\\begin{table}[htbp]\n\\caption{Summary of three physics runs.}\n\\label{table_runs}\n\\begin{tabular}{ccccccc}\n\\hline \\hline\nRun & \\hspace{0.15cm}& Run period & \\hspace{0.15cm} & POT & \\hspace{0.15cm} & Remarks \\\\\n\\hline\nRun-1 & & Feb.-Jun. 2004 & & $2.1 \\times 10^{18}$ & & Membrane problem \\\\\nRun-2 & & Feb.-Apr. 2005 & & $1.4 \\times 10^{18}$ & & Be absorber \\\\\nRun-3 & & Oct.-Dec. 2005 & & $1.1 \\times 10^{18}$ & & New BA \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\\par\nThe data quality of Run-1 was severely affected because the \nmembrane for vacuum separation drooped into the beam core, causing many beam interactions. \nThe results obtained from a 10\\% data sample of Run-1 have been published \nelsewhere \\cite{run1-oneweek}. The single-event sensitivity (S.E.S.) was \n$(9.11\\pm 0.20_{\\textrm{stat}}\\pm 0.64_{\\textrm{syst.}})\\times10^{-8}$ \nwith $1.9\\pm 1.0$ background events expected in the signal box ($N_{\\textrm{bg}}$). \nThe complete data sample obtained in Run-1 was analyzed with a blind analysis. \nIn the blind analysis, the signal candidate events were not examined \nuntil after the final selection criteria were set, and the criteria were not changed later. \nThe results of the complete data sample of Run-1 were found to be essentially \nconsistent with those obtained from the 10\\% data sample. \nThe S.E.S. was $(5.14\\pm0.25)\\times10^{-8}$ with \nN$_{\\textrm{bg}}=2.10$ \\cite{run1-full}. \nThe number of observed events in the signal box (N$_{\\rm{ob}}$) \nwas 0 for the 10\\% sample and 1 for the complete data sample. \nThe corresponding upper limits for the branching \nratio of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode at the 90\\% confidence level (C.L.) were found to be \n$2.1\\times10^{-7}$ and $1.26\\times10^{-7}$, respectively. \nBecause the sensitivity of the Run-1 data sample was not as high as those of \nRun-2 and Run-3, and the running condition of Run-1 was considerably \ndifferent from the later runs, the results of Run-1 are not included \nin the results in this article.\n\\par\nThe membrane problem was rectified before Run-2. \nAs described in Sec.\\ II.C.4,\nthe plastic scintillator of the BHCV was replaced with a thicker one, \nand the discriminated pulse width for the BHCV and BA was reduced. \nAn additional collar counter, CC00, was installed in front of the FB outside \nthe vacuum vessel. In Run-2 and Run-3, a neutron absorber made of beryllium \nwas inserted into the beam line. The results for the complete Run-2 data \nsample, S.E.S. = $(2.91\\pm 0.31)\\times10^{-8}$ with N$_{\\textrm{bg}}=0.42\\pm 0.14$ \nand N$_{\\rm{ob}}=0$, and the upper limit of BR($K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$) $= 6.7\\times10^{-8}$ \nat the 90\\% CL, have been published in a letter \\cite{run2}.\n\\par\nFrom the previous Run-2 analysis, the dominant source of the background was found to be \nthe interaction of the beam halo (mostly neutrons) with the detectors near the beam \n(``halo neutron background\"). The halo neutron background was from three sources:\n$\\pi^0$'s produced by the interaction with CC02, and $\\pi^0$'s and $\\eta$'s \nproduced by the interaction with the CV. \nIn the case of CC02-\\pizs background, the computed $Z_{\\textrm{VTX}}\\;$ could be shifted downstream due to shower \nleakage or photo-nuclear interactions in the CsI calorimeter. \nIn the case of CV-\\pizs background, $Z_{\\textrm{VTX}}\\;$ could be shifted upstream due to the fusion \nof multiple photons or the overlap of other hits in the calorimeter. \nIn the case of CV-$\\eta$ background, \n$Z_{\\textrm{VTX}}\\;$ could be shifted upstream by applying the \\pizs assumption in the reconstruction of the two photons.\n\\par\nIn the previous Run-2 analysis, the background level N$_{\\textrm{bg}}$ was estimated \nwith a different method for each background source. A special run with an Al plate was used \nto estimate the background level for CC02-$\\pi^0$. \nA bifurcation method with a pair of cut sets was used to estimate the CV-\\pizs background.\nAn MC calculation was used to estimate the CV-$\\eta$ background.\nThe difference in estimating these backgrounds made it difficult \nto carry out optimizations with the objective of obtaining the best signal-to-noise ratio, S\/N. \n\\par\nTo improve the analysis of Run2-3 data as described in this paper, \nwe developed a method to generate a large number of beam halo interactions \nby using the hadron-interaction code FLUKA~\\cite{fluka}, whose reliability was confirmed \nby the data from the beam survey \\cite{beamline}. \nThe simulation generated beam-halo interactions with approximately 8 times \nthe total number of real data obtained in Run-3. \nFor the CV-$\\eta$ background case, we recycled the event seeds \nto boost the $\\eta$ production and obtained approximately 80 times larger statistics \nas compared to the Run-3 data.\nAlthough the simulation was carried out for Run-3, the events were\nused for Run-2 with minor modifications. The running conditions were almost \nthe same in Run-2 and Run-3, except that the super-layers of the lead scintillator \nsandwich of the BA were replaced with PWO hodoscopes in Run-3.\n\\par\nIn the analysis presented in this article, we first carefully checked the MC events \nagainst the data from the aluminum plate run to confirm the hadronic interaction model. \nNext, we optimized the selection criteria by monitoring the S\/N ratio of MC data. \nHere, S is the number of events inside the signal region \n(340 $<$ $Z_{\\textrm{VTX}}\\;$ $<$ 500 cm and 120 $<$ $P_T\\;$ $<$ 240 MeV\/$c$) \nfor the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ MC events, and N is the MC background events caused \nby beam-halo interactions~\\cite{jiasen}. \nThe optimization based on the MC is also effective to avoid potential bias \nin reanalyzing the Run-2 data.\nFor the data processing, we first masked the events in the signal region \nfor both Run-2 and Run-3. \nOnce we fixed all the selection criteria by the optimization described above, \nwe compared the data and MC for the events outside the signal region. \nAfter confirming the consistency, we finally examined the events inside the signal region.\n\n\n\n\\subsection{Event reconstruction}\n\\subsubsection{Photon clustering}\nThe energy of each photon was measured by forming a cluster of CsI blocks with finite energy deposits. \nFirst, a search was made for a block with the locally maximum energy, \n{\\em i.e.} the maximum energy among the five blocks geometrically sharing a side. \nAn envelope surrounding the local-maximum block was developed \nby gathering blocks that shared one side and had lower energies. \nAn envelope with a fixed boundary was required\nto reject extra particles that hit the outsides of the two photon clusters. \nIt was also necessary to determine that there was only one local maximum within \nthe envelope, to discriminate clusters that arose from the fusion of \nmore than one photon.\nFigure~\\ref{fig_ncluster} shows the distribution of the number of crystals contained in one photon cluster\nby comparing the six photon events of data and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC.\nThe number of crystals was obtained by counting the crystals having the energy deposit greater than 5 MeV.\nThe size of the cluster was well reproduced by the simulation.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig15.eps}\n\\end{center}\n\\caption{\nDistribution of the number of crystals having the energy deposit greater than 5 MeV in a photon cluster, \ncomparing the data and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC.\nThe top portion shows a comparison between the number of events in the data and MC, \nand the bottom portion shows the ratio between them.\nAll analysis cuts for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode, as will be described in Sec.\\ V.B., are imposed.\n}\n\\label{fig_ncluster}\n\\end{figure}\n\n\n\\subsubsection{Energy and position correction}\nThe hit position of a photon at the front of the CsI calorimeter was first estimated \nby determining the center of energy of hit blocks in each photon cluster.\nThe $Z_{\\textrm{VTX}}\\;$ coordinate was calculated by using two photon clusters and assuming the \\pizs mass. \nThe energy, position, and injection angle were obtained for each photon. \n\\par\nIn this experiment, the energy and position of each photon \nhad to be corrected for the injection angle to obtain better resolution.\nThe thickness of the CsI crystal (16.2X$_0$) was not sufficient to prevent a \nsmall leakage of energy from its downstream end, and the energy leakage \ndepended on the injection angle. \nWe accumulated a large sample of MC data for a single photon with various energies,\npositions, and angles, \nand compiled look-up tables to correct energy and position of each photon.\nThe correction was iteratively carried out by using data from the tables. \nFigure~\\ref{fig_hslee_corr} shows a result of the correction. \n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig16.eps}\n\\end{center}\n\\caption{\nDifference of the reconstructed $x$-position of the photons on the calorimeter\nfrom the true value ($\\Delta x$), obtained by $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ MC simulation. \nThe $x$ coordinate denotes the horizontal position at the face of the CsI.\nThe resolution was improved after the energy and position corrections.\n}\n\\label{fig_hslee_corr}\n\\end{figure}\n\n\n\\subsubsection{Sorting of events}\nThe events were sorted into several samples according to the number of \nreconstructed photon clusters.\nThe two-photon sample was used to search for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay and for monitoring \nthe $K_L^0\\rightarrow\\gamma\\gamma\\;$ decay. The four-photon and six-photon samples were used for monitoring \nthe $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays, respectively. \n\n\\subsubsection{\\pizs reconstruction}\nBy using the position and energy of two photon clusters, \nthe decay vertex ($Z_{\\textrm{VTX}}$) and the momentum of the \\pizs were reconstructed\nwith assumptions that the invariant mass of two photons was equal to the \\pizs mass \nand that the decay vertex was on the beam axis.\nThe opening angle of two photons ($\\theta$) was \ncalculated from the equation\n\\begin{equation}\nM_{\\pi^0}^2 = 2E_1 E_2 (1 - \\cos{\\theta}) \\:,\n\\end{equation}\nwhere $E_1$ and $E_2$ are the energies of the two photons. \nThe momentum and transverse momentum ($P_T$) of the \\pizs were calculated \nfrom the decay vertex.\n\n\n\\subsection{Monte Carlo simulations}\nWe generated Monte Carlo events that had the same structure as the recorded events, \nand analyzed them with the same code as for the real events. \nBecause a full shower simulation would require extensive computation time, \nthe calculations were separated into several stages and streams, as shown \nin Fig.~\\ref{fig_mcscheme}. The first stage was the beamline simulation in \nwhich the target, collimators, magnetic field, and absorbers were introduced. \nThe GEANT3 package~\\cite{geant3} with the GFLUKA plug-in code for hadronic \ninteractions was used for this simulation. The momentum, position in the \ntransverse directions, and angle distributions at the exit of the last \ncollimator C6 were obtained for various beam particles.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig17.eps}\n\\end{center}\n\\caption{\nSchematic diagram of simulation.\n}\n\\label{fig_mcscheme}\n\\end{figure}\n\n\\par\nFor $K_L^0$, functions for the momentum and position \ndistributions at the C6 exit were obtained by fitting the Monte Carlo data.\nThe $K_L^0\\;$ beam beyond C6 was subsequently \ngenerated with respect to the momentum spectrum and targeting angle at C6.\nThe generated $K_L^0$'s were used for \nthe study of various $K_L^0\\;$ decays through detector simulations with the GEANT3 \npackage. \nIn the generation of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ events, we assumed V-A interactions.\nIn the detector simulations, particles were traced until their \nenergy decreased below the cutoff value (e.g. 0.05 MeV for photons and electrons).\nTypical results of $K_L^0\\;$ decay simulations: reconstructed mass, vertex position,\nmomentum, and transverse momentum for the six-photon data samples are shown in Fig.~\\ref{fig_kpithree}.\nSmall corrections to the momentum and radial position of the $K_L^0\\;$ were included.\nEach distribution is reproduced by the simulation of the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays.\nThe method used to reconstruct the six-photon events and \nthe event selections are described in a later section.\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=.44\\textwidth]{.\/fig18a.eps}\n\\includegraphics[width=.44\\textwidth]{.\/fig18b.eps} \\\\\n\\includegraphics[width=.44\\textwidth]{.\/fig18c.eps}\n\\includegraphics[width=.44\\textwidth]{.\/fig18d.eps}\n\\end{center}\n\\caption{\nDistributions of reconstructed mass (top left), vertex position (top right), momentum (bottom left), \nand transverse momentum (bottom right) obtained from the \nsix-photon data samples, compared to the data and the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ MC. \nIn each plot, the top portion shows a comparison between the number of \nevents in the data and MC, and the bottom portion shows the ratio between them.\nAll the veto cuts and kinematic selections, except those for the \nrespective abscissa variable for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode, are applied in these plots. \nDetails of the cuts are described in Sec.\\ V.B.}\n\\label{fig_kpithree}\n\\end{figure*}\n\n\n\\par\nAnother analysis stream from the MC particles at C6 was \nthe generation of halo neutrons. \nFirst, the core neutrons were removed from the neutron data sample. \nThe remaining halo neutrons were used multiple times as seeds. \nDuring the generation, we introduced Gaussian fluctuations\nto the momentum, position, and angle of \neach new event to prevent duplications of the same event. \nWe used the hadron code FLUKA to simulate the interaction between halo neutrons\nand the detector.\nSecondary particles generated by FLUKA were fed into the GEANT3 detector simulation.\n\nThe detector response was simulated by using the GEANT3 package.\nThe energy deposit and the timing in each detector subsystem was stored. \nTrigger conditions were simulated according to the energy deposits of the detector elements,\nincluding the segments of the CsI shown in Fig.~\\ref{fig_hwclustering}.\n\n\n\n\\subsection{Reproducibility of the MC simulations}\nIn addition to the distributions of kinematic variables for the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decay mode, \nthe events with four photons reconstructed in the calorimeter were analyzed\nin order to verify the detection inefficiency of photon counters in the simulation.\nFigure~\\ref{fig_4gmass} shows the reconstructed mass distribution for four-cluster events.\nIn addition to the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events at the $K_L^0\\;$ mass, there was a tail in the lower \nmass region due to contamination from $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, two out of six photons of which \nescaped detection. The number of events in the tail was reproduced by our\nsimulations.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig19.eps}\n\\end{center}\n\\caption{Reconstructed invariant-mass distribution of events with four photons in \nthe calorimeter. The points show the data and the histograms indicate the \ncontribution of $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays (and their sum), as expected from \nthe simulation, normalized by the number of events in the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ peak.\nAll veto and kinematic cuts \nexcept for the cut on the 4$\\gamma\\;$ invariant mass \nfor the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode are applied.\nIt includes the shower-leakage correction, but does not include the correction for photo-nuclear interactions.}\n\\label{fig_4gmass}\n\\end{figure}\n\nA simulation with the FLUKA hadronic-interaction package to a dedicated run \n(``Al plate run\") was carried out in order to confirm the reproducibility of hadronic interactions. \nIn the Al plate run, a 0.5-cm-thick aluminum plate was inserted \ninto the beam at 6.5 cm downstream of the rear end of CC02, as shown in \nFig.~\\ref{fig_pi0run_setup}. Because the position of the Al plate is known, \nthe invariant mass of two photons can be reconstructed from the energy and \nposition of two photons. Figure~\\ref{fig_alrun} demonstrates the simulation \nin which the invariant mass distribution (from \\pizs mass to $\\eta$ mass) \nof the events from the Al plate run was reproduced with two photons in the calorimeter.\n\nNext, by assuming the \\pizs mass and reconstructing the vertex position in the Al plate run,\nthe effect of shower leakage and photo-nuclear interactions in the CsI was examined.\nFigure~\\ref{fig_alrun_vtx} shows the distribution of $Z_{\\textrm{VTX}}\\;$ obtained from the Al plate run\nby comparing the data and MC.\nIf the photon energy is incorrectly measured as being low due to these processes, \nthe \\pizs vertex position is reconstructed downstream of the true position.\nBecause the true $Z_{\\textrm{VTX}}\\;$ is known in the Al plate run, we can estimate the effect \nof these processes by using the tail in the distribution of the reconstructed $Z_{\\textrm{VTX}}$.\n\nPhoto-nuclear interactions are not implemented in GEANT3, and therefore the\neffects were estimated from GEANT4 simulations~\\cite{geant4}. \nWe estimated the probability of photo-nuclear interactions by taking the difference \nbetween MC samples with and without photo-nuclear interactions.\nThe energies of two photons were smeared according to the probability of the interaction.\nAs shown in Fig.~\\ref{fig_alrun_vtx},\na tail on the downstream side of the peak is reproduced\nby the simulations with photo-nuclear interactions.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.38\\textwidth]{.\/fig20.eps}\n\\end{center}\n\\caption{\nSchematic layout of the Al plate run.\n}\n\\label{fig_pi0run_setup}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig21.eps}\n\\end{center}\n\\caption{Reconstructed invariant mass distribution of the two-photon event \nin the Al plate run with all the veto cuts imposed. \nThe points represent data and the solid line is from FLUKA simulations. \nThe peaks at 0.14 GeV\/c$^{2}$ 0.55 GeV\/c${^2}$ correspond \nto $\\pi^0$ and $\\eta$ particles, respectively.}\n\\label{fig_alrun}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig22.eps}\n\\end{center}\n\\caption{Reconstructed $Z_{\\textrm{VTX}}\\;$ distribution of two photon events in the Al plate run.\nThe points represent the real data and the solid (dashed) line is obtained from \nthe FLUKA MC simulaiton with (without) the effects of photo-nuclear interaction.\nIn both cases, the shower-leakage correction was applied.\n}\n\\label{fig_alrun_vtx}\n\\end{figure}\n\n\n\\section{Event Selection and background estimation}\n\n\\subsection{Event selection}\nEvent selection consisted of veto cuts and kinematic selections.\nVetoing extra activities was the primary method used to isolate the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ signal events\nfrom the background events. Kinematic selections were applied to obtain further \nrejection of background events.\n\n\\subsubsection{Veto cuts}\nParticle veto with a hermetic detector system is the primary method to reject \npossible background sources. For example, events made by the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ or \n$K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ mode can be rejected by veto cuts because these modes have \nat least two extra activities. \nThe method also works for halo neutron backgrounds because the hadronic \ninteractions of halo neutrons are often accompanied by extra particles \nsuch as protons or pions.\n\nEnergy thresholds for veto cuts were set at two levels. \nFor loose veto thresholds (typically, 10 MeV), no timing cut was made \nto reduce inefficiencies caused by accidental hits before the event time. \nFor tighter veto thresholds (typically, 1-2 MeV), events with a TDC value \nwithin roughly $\\pm5\\sigma$ from the on-time peak were rejected.\nCombining these conditions, we balanced the efficiencies of detecting extra \nparticles and the suppression of the acceptance loss caused by accidental hits.\n\n\n\\textbf{CsI veto cut:}\nIn addition to its main role of measuring the energy and position of photons, \nthe CsI also served as a veto detector for extra photons.\nExtra activities that were not reconstructed as photon clusters, {\\it i.e.} \n``single crystal hits,\" were rejected with this cut.\nHowever, a photon occasionally creates a single crystal hit near its genuine \ncluster due to fluctuations in the electromagnetic processes.\nThus, applying a tight energy threshold for a single crystal hit near the \nphoton cluster can cause acceptance loss for signal candidates. \nTo recover this loss, the energy threshold for a single crystal hit was \ndetermined as a function of the distance $d$ to the closest cluster \nas shown in Fig.~\\ref{fig_csiveto}:\n\\begin{itemize}\n \\item $E_{\\textrm{thres.}} = 10$ MeV \\;\\;\\; for $d < 17 $ cm, \n \\item $E_{\\textrm{thres.}} = 5 - (3\/8)(d-17)$ MeV for $17 < d < 25$ cm,\n \\item $E_{\\textrm{thres.}} = 2$ MeV \\;\\;\\; for $d>25$ cm,\n\\end{itemize}\nwhere $E_{\\textrm{thres.}}$ is the energy threshold for a single crystal hit.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.4\\textwidth]{.\/fig23a.eps}\\\\\n\\includegraphics[width=.4\\textwidth]{.\/fig23b.eps}\n\\end{center}\n\\caption{\nEnergy deposition in a single crystal hit versus the distance\nfrom the nearest photon cluster \nof the two-photon events from the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ (top) and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ (bottom) simulations.\nEvents above the solid line were rejected.\n}\n\\label{fig_csiveto}\n\\end{figure}\n\n\n\\textbf{Other photon veto cuts:}\nPhoton veto detectors consisted of barrel counters (MB and FB), beam collar counters \n(CC00, CC02 - CC07), and a beam hole counter (BA).\n\nBecause the MB was read out at both the upstream and downstream ends, \nthe energy deposit was determined as the geometrical mean of the visible energies in both sides\nin order to cancel the position dependence of the light yield. \nThe timing requirement for the MB was loosened when a tighter energy threshold \nwas used because the timing was broadly distributed in the halo neutron backgrounds.\nWhen a photon hit the CsI calorimeter, low-energy photons and electrons that were created \nin an electromagnetic shower occasionally went backward and hit the MB. \nThis process caused a larger acceptance loss in the MB as compared to the loss\nin other veto detectors.\n\nFor the collar counters in the downstream region (CC06 and CC07), \nboth tighter and looser energy thresholds were\nset higher than for other detectors,\nin order to reduce accidental losses due to beam-induced activities.\nThe BA located inside the beamline rejected photons escaping into the beam hole. \nThe energy thresholds for these counters are summarized in Table~\\ref{table_veto}.\n\n\\textbf{Charged particle veto cuts:}\nBackground events produced by charged $K_L^0\\;$ decay modes were mostly rejected \nby cuts on the charged particle veto detectors, mainly CV, BCV, and BHCV.\nAs in the case of MB, an energy deposit in the BCV was determined as \nthe geometrical mean of the hits in the upstream and downstream ends, and the timing \nrequirement was loosened.\nThe energy thresholds for these \ncounters are summarized in Table~\\ref{table_veto}.\n\n\\begin{table}[htbp]\n \\caption{Summary of tighter veto energy threshold for each detector.}\n \\label{table_veto}\n \\begin{tabular}{lc}\n\t\\hline \\hline\t\n\tDetector & Threshold \\\\\n\t\\hline \\hline\n\tCC00 & 2 MeV\\\\\n\t\\hline\n\tFB & 1 MeV \\footnotemark[1] \\\\\n\t\\hline\n\tCC02 & 1 MeV \\\\\n\t\\hline\n\tBCV & 0.75 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tMB Inner & 1 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tMB Outer & 1 MeV \\footnotemark[2] \\\\\n\t\\hline\n\tCV Outer & 0.3 MeV \\\\\n\t\\hline\n\tCV Inner & 0.7 MeV \\\\\n\t\\hline\t\n\tCC03 & 2 MeV \\\\\n\t\\hline\n\tCsI & depends on $d$ \\footnotemark[3] \\\\\n\t\\hline\n\tSandwich & 2 MeV \\\\\n\t\\hline\n\tCC04 Scintillator & 0.7 MeV \\\\\n\t\\hline\n\tCC04 Calorimeter & 2 MeV \\\\\n\t\\hline\n\tCC05 Scintillator & 0.7 MeV \\\\\n\t\\hline\n\tCC05 Calorimeter & 3 MeV \\\\\n\t\\hline\n\tCC06 & 10 MeV \\\\\n\t\\hline\n\tCC07 & 10 MeV \\\\\n\t\\hline\n\tBHCV & 0.1 MeV \\\\\n\t\\hline\n\tBA Scintillator (Run-2) & 20 MeV \\footnotemark[4] \\\\\n\tBA PWO (Run-3) & 50 MeV \\footnotemark[4] \\\\\n\tBA Quartz & 0.5 MIPs \\footnotemark[5] \\\\\n\t\\hline \\hline\n \\end{tabular}\n \\footnotetext[1]{Sum of inner and outer layers.}\n \\footnotetext[2]{Geometrical mean of upstream and downstream ends.}\n \\footnotetext[3]{Detailed in the text.}\n \\footnotetext[4]{Summed over layers.}\n \\footnotetext[5]{Determined by AND logic of scintillator\/PWO and quartz.}\n\\end{table}\n\n\n\\subsubsection{Kinematic selections}\nKinematic selections were applied to discriminate the signal from background events \nby using the information of two photons in the CsI calorimeter. \nThese selections can be categorized into three types: photon-cluster quality \nselections, \nselection on photons, and \\pizs selections.\n\n\\textbf{Photon-cluster quality selections:}\nPhoton-cluster quality selections were developed from the information of each photon cluster.\nThe energy was required to be \n$E_H > 250$ MeV and $E_L > 150$ MeV, where $E_H$ and $E_L$ are\nthe energies of the higher- and lower-energy photons, respectively.\nWe also made selections on the incident position, size, profile of the energy distribution,\nand timing dispersion of each photon cluster.\nThese cuts were applied to ensure that two photon clusters were from the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay\nand were distinguished from fake clusters created by hadronic showers or photons produced by\nhalo neutron interactions.\n\nIn addition, two neural network selections were developed pertaining to the shape \nof a photon cluster. The first neural network was used to reduce events with \nclusters that overlapped those from other photons or associated particles and reconstructed \nas a single cluster (``fusion cluster\"). In the case of $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ background, a fusion cluster \nresults in an inefficiency to extra photons; in the case of CV-\\pizs background, \nit results in a larger photon energy such that the event vertex \ncomes into the signal region. This neural network was trained by single \nand fusion clusters obtained from the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ MC samples. \nThe second neural network selection was used to reject the CV-$\\eta$ background.\nBecause $\\eta$ particles were produced near the front end of CsI and had a larger invariant \nmass than a $\\pi^0$, the two photons produced by $\\eta$ decay tended to \nhave large incident angles to the CsI calorimeter.\nThis neural network was trained by the MC samples of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ and CV-$\\eta$ background. \nThe rejection power of these two neural network selections is\nshown in Fig.~\\ref{fig_nncuts}, with the distribution of the output values from \nthe neural network for the signal and background modes.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig24a.eps}\\\\\n\\includegraphics[width=.47\\textwidth]{.\/fig24b.eps}\n\\end{center}\n\\caption{\nDistributions of the output values obtained by two neural network selections \nfor the signal mode (hollow) and the background events (hatched).\nThe top figure shows the fusion neural network selection, in which the background \nis CV-$\\pi^0$. The bottom figure shows the CV-$\\eta$ neural network selection, \nin which the background is CV-$\\eta$. In both plots, all analysis cuts except \nfor the respective neural network cut are imposed for the signal mode. \nSeveral cuts are loosened for the background case in order to enhance the events.\n}\n\\label{fig_nncuts}\n\\end{figure}\n\n\\textbf{Selection on photons:}\nWe required the distance between two photons to be greater \nthan 15 cm, and the timing difference between them to be \n$-9.6 < T(E_H) - T(E_L) < 18.4$ ns, where $T(E_H)$ and $T(E_L)$ are the \ntiming of the higher and lower energy photons, respectively.\nIn addition, the energy balance of two photons defined as \n$(E_{H}-E_{L})\/(E_{H}+E_{L})$ , was required to be less than 0.75.\nThese selections were needed mainly to suppress accidental activities.\n\n\\textbf{\\pizs selections:}\nBy using the information for reconstructed $\\pi^0$'s, several selection criteria were imposed. \nWe required the kinetic energy of the reconstructed $\\pi^0$'s to be less \nthan 2 GeV to suppress neutron backgrounds with high energy.\nIn addition, the reconstructed $\\pi^0$'s had to be kinematically consistent \nwith a $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay within the proper $K_L^0\\;$ momentum range.\nA neural-network selection was applied to the discrepancy between two angles.\nOne angle was calculated by connecting the incident position of photons and the position of reconstructed $\\pi^0$'s,\nand the other angle was estimated by the neural network based on the cluster shape.\nThis selection suppressed CV-\\pizs and CV-$\\eta$ backgrounds \nbecause the discrepancy of the angle increased in these processes.\nTo reduce the $K_L^0\\rightarrow\\gamma\\gamma\\;$ background, we calculated the opening angle between \nthe two photon directions projected onto the CsI calorimeter plane, \nand required it to be less than 135 degrees (``acoplanarity angle cut\").\n\n\n\\subsubsection{Signal region}\nWe set the signal region in the $Z_{\\textrm{VTX}}$-$P_T\\;$ plane to be \n$340 \\leq Z_{VTX} \\leq 500 $ cm and $0.12 \\leq P_T \\leq 0.24$ GeV\/$c$.\nThe requirement on $Z_{\\textrm{VTX}}\\;$ was determined to avoid background $\\pi^0$'s \ncoming from the CC02 (upstream) and CV (downstream).\nThe lower limit for $P_T\\;$ was set to reduce the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and CV-$\\eta$ \nbackgrounds; these events cluster in the low $P_T\\;$ region due to kinematics \nand a large halo neutron flux near the beam center, respectively. \nThe upper bound for $P_T\\;$ was determined by the kinematic limit of the \n$K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay, whose maximum value is 0.231 GeV\/$c$.\n\n\n\\subsection{Background estimation}\n\n\\subsubsection{Halo neutron background}\nThe halo neutron background was estimated from combined MC simulations \nwith FLUKA and GEANT3.\n\nFigure~\\ref{fig_bgestim} shows the distribution of background events in the \n$Z_{\\textrm{VTX}}$-$P_T\\;$ plane as estimated from the MC samples.\nThe outside of the signal box was divided into four regions, \nas shown in Fig.~\\ref{fig_bgestim}. \nThe number of events in each region was dominated by the halo neutron backgrounds.\nThe numbers of halo neutron backgrounds were normalized \nto the number of events in the upstream region (shown as Region-(1)). \nThe numbers for events outside the signal box \nwere compared between the data and the MC, as listed in Table~\\ref{table_outside}.\nThe numbers of events in each region are consistent within the statistical uncertainties\nbetween the data and the MC.\nPossible impacts of discrepancies outside the signal box on the estimation inside the signal box\nare included in the systematic uncertainties below, and will be discussed in detail \nin a later section (Sec.\\ VI.B.).\n\nFor estimating the CC02-\\pizs background, the effects of shower leakage and photo-nuclear\ninteractions described in Sec.\\ III.D were taken into account.\nAfter applying all event selections, we estimated $0.66 \\pm 0.33_{\\textrm{stat}} \n\\pm 0.20_{\\textrm{syst}}$ background events in the signal box for the combined \ndata sample of Run-2 and Run-3. \n\nFor CV-\\pizs events, a fusion cluster or a cluster made by a neutron produces \nbackground. After applying all event selections, no events inside \nthe signal box were obtained with 3 times larger statistics as compared to \nthe combined data sample of Run-2 and Run-3. An upper limit of 0.36 events \nat the $1 \\sigma$ level was set for this process.\n\nCV-$\\eta$ events were strongly suppressed by the dedicated neural-network cut \non the cluster shape. A value of $0.19 \\pm 0.08_{\\textrm{stat}} \\pm \n0.10_{\\textrm{syst}}$ events was estimated for this process.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig25.eps}\n\\end{center}\n\\caption{\nScatter plot of $P_{T}$ vs.\\ the reconstructed $Z$ position after applying \nall cuts to the MC samples for background simulation, including halo neutron and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ \nbackgrounds. The region bounded by the solid line shows the signal box.\nEvents around $Z_{VTX} = 275$ cm are halo neutron events reconstructed at \nthe position of CC02, and those around $Z_{VTX} = 560$ cm are events \nreconstructed at the CV. The numbers of events in each region are listed \nin Table~\\ref{table_outside}.\n}\n\\label{fig_bgestim}\n\\end{figure}\n\n\\begin{table}[htbp]\n\\caption{\nEstimated numbers of events outside the signal region, compared with the\ncombined data samples of Run-2 and Run-3.\nNote that the numbers of events estimated by the MC simulation were normalized in Region-(1).\nErrors in the MC estimates indicate statistical uncertainties.\n}\n\\label{table_outside}\n\\begin{tabular}{rlcc}\n\\hline \\hline\n\\multicolumn{2}{c}{Region} & Data & MC estimation \\\\\n\\hline\nRegion-(1) & (CC02) & 360 & $360.0 \\pm 15.6$ \\\\\nRegion-(2) & (CV) & 101 & $77.2 \\pm 5.6$ \\\\\nRegion-(3) & (upstream) & 8 & $5.9 \\pm 1.1$ \\\\\nRegion-(4) & (low-$P_T$) & 8 & $2.9 \\pm 0.9$ \\\\\n\\multicolumn{2}{c}{Signal box} & (masked) & $0.87 \\pm 0.34$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsubsection{$K_L^0\\;$ background}\nThe $K_L^0\\;$ background was estimated by using GEANT3-based MC simulations \nthat were performed separately for Run-2 and Run-3 conditions.\nAmong the backgrounds from $K_L^0\\;$ decay, the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode made the \nlargest contribution. \nThe amount of Monte Carlo statistics for the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode was roughly 70 (60) times \nthat for the real data of the Run-2 (Run-3) sample.\nIn Run-2, two events remained after applying all event selections in \nthe simulations, and in Run-3, no events remained. \nThe two remaining events both corresponded to the case that \ntwo photons in the CsI came from the same \\pizs.\nOne extra photon with high energy ($\\sim$ 1 GeV) went through the CsI (``punch-through\"), \nand the other extra photon with low energy ($\\sim$ 10 MeV) hit the MB and failed \nto be detected by the fluctuation in an electromagnetic shower.\nThe estimated number of background events from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ decay was \nestimated as $(2.4 \\pm 1.8_{\\textrm{stat}} \\pm 0.2_{\\textrm{syst}}) \n\\times 10^{-2}$ for the combined data sample of Run-2 and Run-3.\n\nFor the MC $K_L^0\\rightarrow\\gamma\\gamma\\;$ samples whose statistics corresponded \nto 3.8 and 8.7 times the Run-2 and Run-3 data, respectively,\nall of the event selection cuts except for the acoplanarity angle cut were \nimposed and no events remained in the signal region.\nFurthermore, the acoplanarity angle cut strongly suppressed the $K_L^0\\rightarrow\\gamma\\gamma\\;$ events, \nwith a typical rejection of $1\\times 10^5$.\nWe concluded that the background due to $K_L^0\\rightarrow\\gamma\\gamma\\;$ was negligible in this analysis.\n\nFor charged modes, the branching ratios of the $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0\\;$ and \n$K_L^0\\rightarrow\\pi l\\nu\\;$ $(l=e, \\mu)$ modes are too large to generate sufficient statistics \nin the simulations.\nThe background from these modes \nwere estimated by assuming that the inefficiency of charged-particle vetos was \n$1.0\\times10^{-4}$ for the CV, and $1.0\\times10^{-3}$ for BCV and BHCV \\cite{chineff}. \nBy applying the event weight calculated from the inefficiency, the numbers of \nbackground events from these modes were estimated to be \n$4.2 \\times 10^{-4}$ for $K_L^0\\rightarrow\\pi^- e^+\\nu\\;$ and less than $1.0 \\times 10^{-4}$ for $K_L^0\\rightarrow\\pi^+\\pi^-\\pi^0$.\nHere, the $K_L^0\\rightarrow\\pi^- e^+\\nu\\;$ mode had the largest contribution to the background events among \n$K_L^0\\rightarrow\\pi l\\nu\\;$ modes because a charge-exchange interaction of a $\\pi^-\\;$ ($\\pi^- + p \n\\rightarrow \\pi^0 + n$) and annihilation of $e^+$ prevent the detection of \ncharged particles.\n\n\n\\subsubsection{Other background sources}\nIn addition to the backgrounds from halo neutron and $K_L^0\\;$ decay, two other \npossible background sources were considered. The first one is the \n{\\it backward-going \\pizs background}. \nWhen the halo neutron interacts in the end cap of the vacuum vessel located \n2 m downstream of the CsI, $\\pi^0$'s are occasionally produced in the upstream direction. \nBecause we were unable to discriminate whether photons came from the front or the back, \nthese events were reconstructed as coming from the front and ended up inside the \nsignal box.\nWe estimated this background by using FLUKA simulations with 20 times the amount \nof statistics as compared to the combined data sample of Run-2 and Run-3.\nNo events remained after applying all cuts, and the number of background events \nwas determined to be less than 0.05 events.\n\nThe second additional background source is the {\\it residual gas background} \nthat occurs from interactions of beam neutrons with the residual gas.\nThis process is well suppressed by a high-vacuum system that provided \na very low pressure of $10^{-5}$ Pa.\nTo estimate the background, we carried out a dedicated run at atmospheric \npressure, accumulating statistics that were roughly equivalent to 0.6\\% of the \ncombined data sample of Run-2 and Run-3. We obtained 6867 candidate events \nwith loose event selections. By considering a reduction factor of $10^{-10}$ \ndue to the air pressure, this background was concluded to be negligible.\n\nWe investigated possible contributions from unknown background sources by\nloosening the selection criteria for the data and MC. \nThe numbers of events in the data sample with loosened cuts were consistent \nwith the prediction with the known background sources.\n\nIn total, the estimated number of background events was $0.87 \\pm 0.41$, \nwhere the CV-\\pizs and the backward-going \\pizs backgrounds were not included.\nTable~\\ref{table_bgestim} summarizes estimates for the numbers of background \nevents.\n\n\n\\begin{table}\n\\caption{\nEstimated number of background events.\n}\n\\label{table_bgestim}\n\\begin{tabular}{ccc}\n\\hline \\hline\n\\multicolumn{2}{c}{Background source} & Estimated number of BG\\\\\n\\hline\nhalo neutron BG & CC02-$\\pi^0$ & $0.66 \\pm 0.39$ \\\\\n & CV-$\\pi^0$ & $<0.36$ \\\\\n & CV-$\\eta$ & $0.19 \\pm 0.13$ \\\\\n\\hline\n$K_L^0\\;$ decay BG & $K_L^0\\rightarrow\\pi^0\\pi^0$ & $(2.4 \\pm 1.8)\\times 10^{-2}$ \\\\\n & $K_L^0\\rightarrow\\gamma\\gamma\\;$ & negligible \\\\\n & charged modes & negligible (${\\cal O}(10^{-4})$) \\\\\n\\hline\nother BG & backward $\\pi^0$ & $< 0.05$ \\\\\n & residual gas & negligible (${\\cal O}(10^{-4})$)\\\\ \n\\hline\n\\multicolumn{2}{c}{total} & $0.87 \\pm 0.41$ \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Sensitivity and results}\n\n\n\\subsection{Principle of normalization}\n\nThe single event sensitivity (S.E.S.) is represented as \n\\begin{equation}\n\\textrm{S.E.S.} (K_L^0\\rightarrow \\pi^0\\nu\\bar{\\nu}) = \n\\frac{1}{N(K_L^0\\:\\textrm{decays}) \\times A_{\\textrm{signal}}} \\:,\n\\label{eq_SES}\n\\end{equation}\nwhere $N(K_{L}^{0}\\:\\textrm{decays})$ denotes the number of $K_L^0$'s that decayed \nin the fiducial region and $A_{\\textrm{signal}}$ denotes the acceptance for the signal mode.\nThe value of $N(K_{L}^{0}\\:\\textrm{decays})$ is determined from the normalization mode as\n\\begin{equation}\n N(K_L^0\\:\\textrm{decays}) = \\frac{N_{\\textrm{norm.}}^{\\textrm{data}}}{A_{\\textrm{norm.}}\n\\times BR_{\\textrm{norm.}}} \\;,\n\\label{eq_nkl}\n\\end{equation}\nwhere $N_{\\textrm{norm.}}^{\\textrm{data}}$ is the number of observed events from \nthe normalization mode and $A_{\\text{norm.}}$ and $BR_{\\textrm{norm.}}$ represent \nthe acceptance and branching ratio of the mode, respectively.\nSubstituting Eq.~\\ref{eq_nkl} into Eq.~\\ref{eq_SES}, the S.E.S. is obtained \nfrom the number of remaining events in the normalization mode and the ratio of\nacceptances between the signal and the normalization modes.\nBy taking this ratio, uncertainties arising from variations in beam condition, \n{\\em etc.}, can be canceled.\n\nWe examined three decay modes, namely, $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma\\;$, to obtain \nthe number of $K_L^0\\;$ decays, because they were fully reconstructed and clearly identified.\nBecause these three decay modes have different numbers of photons in the final states, \nwe could cross-check the reliability of the clustering method. \nThe mean value of the accepted $K_L^0\\;$ momentum was also different, which provided a \ncheck in the full range of the accepted $K_L^0\\;$ momentum for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode.\n\n\n\\subsection{Analysis of normalization modes}\n\\subsubsection{$K_L^0\\;$ reconstruction}\nFor the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ modes, $K_L^0$'s were reconstructed from \nthe six and four photons in the CsI calorimeter, respectively.\nIn the reconstruction, the number of possibile combinations of pairs of \nthe two photons was $(_6C_4 \\times _4C_2 \\times _2C_2)\/3! = 15$ for \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $(_4C_2 \\times _2C_2)\/2! = 3$ for $K_L^0\\rightarrow\\pi^0\\pi^0$. \nThe decay vertex location can be calculated from the energy and position \nof the two photons of each pair by assuming the \\pizs mass.\nTo find the best combination of the photons, the variance in the \nreconstructed vertex points was calculated, named ``pairing $\\chi_Z^2$,\" \nfor all the possible combinations.\nThe pairing $\\chi_Z^2$ was defined as\n\\begin{equation}\n\\chi_Z^2 = \\sum_{i=1}^n \\frac{ ( Z_i - \\bar{Z} )^2}{ \\sigma_i^2} \\:,\n\\label{eq_pairing_chi}\n\\end{equation}\nwhere $i$ runs over the two-photon pairs reconstructing $\\pi^0$, \n({\\it e.g.}, $n=3$ for $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $n=2$ for $K_L^0\\rightarrow\\pi^0\\pi^0$.), \n$Z_i$ is the vertex point of $i$-th two-photon pair, $\\sigma_i$ is the \nresolution in reconstructing $Z_i$ calculated from the energy and position \nresolutions of the two photons, and \n\\begin{equation}\n\\bar{Z} = \\frac{ \\sum_{i=1}^n Z_i\/\\sigma_i^2}{ \\sum_{i=1}^n 1\/\\sigma_i^2} \\:.\n\\label{eq_barZ}\n\\end{equation}\nThe decay vertex of the $K_L^0\\;$ was determined as $\\bar{Z}$ for the combination \nwith minimum $\\chi_Z^2$.\nWe required the minimum pairing $\\chi_Z^2$ to be less than 3.0 and the difference \nbetween the next-to-minimum one to be greater than 4.0, in order to reduce \nincorrect paring.\n\nFor the $K_L^0\\rightarrow\\gamma\\gamma\\;$ mode, $K_L^0$'s were reconstructed from the two photons in the \nCsI calorimeter by assuming the $K_L^0\\;$ mass.\n\nSeveral analysis cuts were imposed on the events in each decay mode to remove \ncontaminations of other decay modes. \nPhoton veto cuts were important to detect the modes with extra photons,\nsuch as $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ to the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events, and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ to the $K_L^0\\rightarrow\\gamma\\gamma\\;$ events.\nThe energy threshold for the veto in each subsystem was the same as that \nused in the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ analysis (see Table~\\ref{table_veto}).\nThe reconstructed mass of six photons in the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ events and four photons \nin the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ events had to be consistent with the $K_L^0\\;$ mass (from 0.481 to \n0.513 GeV\/$c^2$). \nThe decay vertex point of $K_L^0\\;$ also had to be located in the fiducial region \n(from 340 to 500 cm), as in the case of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nSome additional cuts, such as the neural network fusion cluster selection, were also\nimposed on these decay modes.\n\n\n\\subsubsection{Number of $K_L^0\\;$ decays}\nThe acceptance of each mode was estimated from Monte-Carlo simulations.\nBecause, in the simulations, $K_L^0$'s were generated at the exit of the \nlast collimator (C6), the probability of $K_L^0\\;$ decay in the fiducial region, \n$(2.14 \\pm 0.02)$\\%, was calculated separately\nand was taken into account in calculating the number of $K_L^0\\;$ decays.\nLosses due to accidental activities in the detector \nwere also included in the acceptance.\n\nTable~\\ref{table_flux} shows a summary of the estimated numbers of $K_L^0\\;$ decays \nfrom the three decay modes $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$.\nThe difference among the three modes was within the systematic uncertainties,\nand considered to come from the CsI veto, because it has the largest systematic uncertainties\nas well as a dependence on the number of photons.\nWe adopted the number obtained from $K_L^0\\rightarrow\\pi^0\\pi^0$,\n$(8.70 \\pm 0.17_{\\textrm{stat}} \\pm 0.59_{\\textrm{syst}}) \\times 10^9$ for the combined \ndata sample of Run-2 and Run-3, as the normalization of this analysis \nbecause the energy distribution of photons in the CsI calorimeter from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$\nwas similar to that expected to $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.\nEstimates of the systematic uncertainties are described later.\n\n\n\\begin{table*}[htbp]\n\\caption{\nEstimated numbers of $K_L^0\\;$ decays calculated from the three decay modes in the \ncombined sample of Run-2 and Run-3 data.\nUncertainties in the acceptances are statistical ones due to the amount of the MC samples.\n$N_{\\textrm{norm}}^{\\textrm{data}}$ is the number of events obtained from the \nreal data after imposing all the analysis cuts. \nIn the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode, $N_{norm}^{data}$ is obtained by subtracting the \ncontamination from the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ mode.\nStatistical uncertainties in the number of $K_L^0\\;$ decays include an ambiguity \nin $N_{\\textrm{norm}}^{\\textrm{data}}$, and systematic ones include an \nambiguity from the reproducibility of the MC (described later) and\nstatistical uncertainties of the MC in the acceptance estimate.\n}\n\\label{table_flux}\n\\begin{tabular}{cccc}\n\\hline\nmode & acceptance & $N_{norm}^{data}$ & $N(K_L^0 \\mathrm{decays})$ \\\\\n\\hline\n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ $\\;$& $\\;$ $ (7.21 \\pm 0.06) \\times 10^{-5} $ $\\;$ & $\\;$ 118334 $\\;$ \n & $\\;$ $ (8.41 \\pm 0.03_{\\textrm{stat}} \\pm 0.53_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n$K_L^0\\rightarrow\\pi^0\\pi^0\\;$ $\\;$ & $\\;$ $ (3.42 \\pm 0.03) \\times 10^{-4} $ $\\;$ & $\\;$ 2573.9 $\\;$ \n & $\\;$ $ (8.70 \\pm 0.17_{\\textrm{stat}} \\pm 0.59_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n$K_L^0\\rightarrow\\gamma\\gamma\\;$ $\\;$ & $\\;$ $ (7.18 \\pm 0.03) \\times 10^{-3} $ $\\;$ & $\\;$ 35367 $\\;$ \n & $\\;$ $ (9.02 \\pm 0.05_{\\textrm{stat}} \\pm 0.51_{\\textrm{syst}}) \\times 10^{9} $ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Acceptance and single event sensitivity}\n\\subsubsection{Signal acceptance}\nThe acceptance for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay was estimated from Monte-Carlo simulations.\nThe raw acceptance was calculated by dividing the number of remaining events\nafter imposing all the analysis cuts by the number of $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decays generated in \nthe simulation.\nIt was 1.40\\% for the Run-2 and 1.39\\% for the Run-3 data, respectively.\n\nIn the raw acceptance, the losses caused by geometrical acceptance, veto cuts, \nkinematic selections, and selection on fiducial region were included. \nThe geometrical acceptance to detect two photons in the CsI was approximately 20\\%.\nThe loss by veto cuts, or self-vetoing, was about 50\\%, which was dominated by the CsI and MB vetoes.\nThe loss in the CsI veto was caused by hits accompanied by the genuine photon cluster, as shown in \nFig.~\\ref{fig_csiveto}, and the loss in the MB was caused by photons escaping from the\nfront face of the CsI and hitting the MB.\nThe acceptance of the kinematic and signal region selections was approximately 15\\%.\nThese values were estimated through MC simulations of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode.\nThe evaluation of the acceptance loss was supported by the fact that\nthe acceptance losses in the normalization modes ($K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$)\nwere reproduced by the simulation.\n\nThe acceptance loss due to accidental activities in the detector \nwas estimated from real data taken with the TM trigger. \nThe accidental loss was estimated to be 20.6\\% for the Run-3 data,\nin which the losses in MB (7.4\\%) and BA (6.4\\%) were major contributions.\nFor Run-2, the accidental loss was estimated to be 17.4\\%; the difference \nbetween Run-2 and Run-3 was due to the difference in the BA counters used \nin the data taking. \nThe acceptance loss caused by \nthe selections on the timing dispersion of each photon cluster and \non the timing difference between two photons\nwas estimated separately by using real data and was obtained to be 8.9\\%. \nThus, the total acceptance for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ was $(1.06\\pm0.08)$\\% for Run-2 and\n$(1.00\\pm0.06)$\\% for Run-3 case, where the errors are dominated by the \nsystematic uncertainties that are discussed later. Figure~\\ref{fig_ptz_signal} \nshows the distribution of the MC $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ events in the scatter plot of \n$P_T$-$Z_{\\textrm{VTX}}\\;$ after imposing all of the other cuts.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig26.eps}\n\\end{center}\n\\caption{\nDensity plot of $P_T\\;$ vs. the reconstructed $Z$ position for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ Monte Carlo events\nafter imposing all of the analysis cuts.\nThe box indicates the signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.}\n\\label{fig_ptz_signal}\n\\end{figure}\n\n\n\\subsubsection{Single event sensitivity}\nBy using the number of $K_L^0\\;$ decays and the total acceptance, \nthe single event sensitivity for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ was\n$(1.84 \\pm 0.05_{\\textrm{stat}} \\pm 0.19_{\\textrm{syst}} )\\times 10^{-8}$ for Run-2,\n$(2.80 \\pm 0.09_{\\textrm{stat}} \\pm 0.23_{\\textrm{syst}} )\\times 10^{-8}$ for Run-3,\nand\n$(1.11 \\pm 0.02_{\\textrm{stat}} \\pm 0.10_{\\textrm{syst}} )\\times 10^{-8}$ in total.\n\n\n\\subsection{Results}\nAfter finalizing all of the event selection cuts, the candidate events inside the \nsignal region were examined. No events were observed in the signal region, as shown \nin Fig.~\\ref{fig_final}. \nAn upper limit for the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ branching ratio was set to be $2.6\\times10^{-8}$ at \nthe 90\\% confidence level, based on Poisson statistics. \nThe result improves the limit previously published~\\cite{run2}\nby a factor of 2.6.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=.47\\textwidth]{.\/fig27.eps}\n\\end{center}\n\\caption{Scatter plot of $P_T\\;$ vs.\\ the reconstructed $Z$ position for the events \nwith all of the selection cuts imposed. The box indicates the signal region for $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$.}\n\\label{fig_final}\n\\end{figure}\n\n\n\n\\section{Systematic uncertainties}\nAlthough the systematic uncertainties were not taken into account \nin setting the current upper limit on the branching ratio,\nwe will describe our treatment for them \nto provide a thorough understanding of the experiment. \nIn particular, systematic uncertainties of the single event sensitivity and \nbackground estimates due to halo neutrons are discussed in order.\n\n\\subsection{Uncertainty of the single event sensitivity}\nThe systematic uncertainty of the single event sensitivity was evaluated \nby summing the uncertainties of the number of $K_L^0\\;$ decays \nand the acceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. \nBecause the calculation of the former also includes \nthe acceptance of the normalization modes, \nthe acceptance of both the normalization mode and the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode are relevant to the acceptance evaluation\nby the Monte Carlo simulations.\nTo estimate the uncertainties in the acceptance calculation,\nwe utilized the fractional difference between data and the simulation in each selection criterion,\ndefined by the equation\n\\begin{equation}\nF^i = \\frac{A_{\\textrm{data}}^i - A_{\\textrm{MC}}^i}{A_{\\textrm{data}}^i} \n\\end{equation}\nwhere $A_{\\textrm{data}}^i$ and $A_{\\textrm{MC}^i}$ denote the acceptance values of the $i$-th cut, \ncalculated as the ratio of numbers of events with and without the cut,\nfor the data and MC simulations, respectively. \nIn $F^i$, the acceptance was calculated with all the other cuts imposed.\nThe systematic uncertainty of the acceptance was evaluated\nby summing all the fractional differences in quadrature, \nweighted by the effectiveness of each cut, as \n\\begin{equation}\n\\sigma_{\\textrm{syst.}}^2 = \\frac{\\sum_{i=\\textrm{cuts}} \n(F^i\/A_{\\textrm{data}}^i)^2}{\\sum_{i=\\textrm{cuts}}(1\/A_{\\textrm{data}}^i)^2} \\:.\n\\end{equation}\n\nFor the three decay modes used in the normalization, \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0$, $K_L^0\\rightarrow\\pi^0\\pi^0$, and $K_L^0\\rightarrow\\gamma\\gamma$, \nthe calculated uncertainties were 5.2\\%, 5.7\\%, and 3.6\\% (in Run-3), respectively.\nThe acceptances of the CsI veto cut had the largest uncertainties in all decay modes.\nThe number of $K_L^0\\;$ decays was obtained by using the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode, \nand thus its uncertainty is quoted as the same value for the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode.\n\nFor the acceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ mode, the same systematic uncertainties as the \n$K_L^0\\rightarrow\\pi^0\\pi^0\\;$ mode were adopted because there were no signal candidates in the data \nto be compared with the MC simulations.\n\nThe systematic error of the single event sensitivity was evaluated to be a\nquadratic sum of the uncertainties of the number of $K_L^0\\;$ decays and the \nacceptance of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. It was 10.3\\% in Run-2 and 8.2\\% in Run-3, \nrespectively.\n\n\n\\subsection{Uncertainty of the halo neutron backgrounds}\nThe systematic errors of halo neutron backgrounds were also estimated \nby utilizing fractional differences between data and the simulations.\nThere were two kinds of cuts in our analysis, \nveto cuts and kinematic selections.\nFor the former, the method was unable to be used directly because \nveto cuts had been applied in the early stages of the MC simulations \nto save the computing time. \nThus, for the systematic uncertainty due to veto cuts, \nthe same value obtained in the $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ analysis was assigned.\nFor the kinematic selections, the same method as described in \nthe previous section was used.\nThe acceptance of each cut was calculated \nwith all veto cuts imposed except kinematic selections.\nIn total, the systematic uncertainties due to fractional differences \nwere calculated to be 31\\%, 32\\%, and 44\\% \nfor CC02-$\\pi^0$, CV-$\\pi^0$, and CV-$\\eta$ backgrounds, respectively.\n\nIn addition, the uncertainties in the normalizations of the MC \nsimulations were taken into account. \nBecause the normalization was determined by \nusing the number of events in the CC02 region (Region-(1)),\nthere can be ambiguity in estimating the CV-related backgrounds. \nAs shown in Table~\\ref{table_outside}, \nthere was a 24\\% difference between data and MC simulation \nin the downstream region (Region-(2)).\nEven though they were statistically consistent, we assigned the \ndifference as an additional systematic uncertainty of CV-$\\pi^0$.\nFor the CV-$\\eta$ case, further ambiguity due to the reproducibility \nof $\\eta$ production should be considered.\nIt was estimated to be 24\\% from the difference between data and MC\nsimulations in the numbers of $\\eta$ events in the Al plate run, \nas shown in Fig.~\\ref{fig_alrun}.\n\nThe total systematic uncertainties were calculated by summing up\nthe contributions quadratically, \nto be 31\\%, 40\\%, and 55\\% for CC02-$\\pi^0$, CV-$\\pi^0$, and CV-$\\eta$\nbackgrounds, respectively.\n\n\n\\section{Conclusion and discussion}\nThe E391a experiment at the KEK-PS was the first dedicated experiment \nfor the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. Combining the periods of Run-2 and Run-3, \nthe single event sensitivity reached $1.11\\times10^{-8}$.\nNo events were observed inside the signal region and \nthe new upper limit on the branching ratio of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay was set \nto be BR $< 2.6 \\times 10^{-8}$ at the 90\\% confidence level.\nThe result improves the previous published limit given by the Run-2 \nanalysis by a factor of 2.6, \nand the E391a experiment as a whole has improved the limit \nfrom previous experiments by a factor of 20. \n\nThe E391a experiment was also the first step of our step-by-step approach\ntoward the accurate measurement of the $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ decay. The experiment was planned \nto confirm our experimental approach, and this purpose was well achieved. \nSeveral points need to be noted.\n\nFirst, we found solutions to several technical issues, \nsuch as the pencil beamline, differential pumping for ultra-high vacuum, \nlow-threshold particle detection with a hermetic configuration, in situ calibration, {\\it etc.},\nwhich can be successively used in the next step. \nWe encountered some technical problems that exist in the current apparatus, \nsuch as insufficient thickness and segmentation of the CsI calorimeter, \nthe structure of the CV, the limitation of the BA in an environment with higher counting rate, {\\it etc.}.\nThey will be improved in the next experiment.\n \nSecondly, we were able to control the systematic uncertainties to be small in the estimate \nof the single event sensitivity, as described in the previous section. \nA small systematic error is essential for an accurate determination of the \nbranching ratio. We are confident that the branching ratio of the decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}\\;$ \ncan be measured accurately with this method. \n\nA third point concerns the understanding and estimation of backgrounds. \nIn the experiment for the decay $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$, elimination of all possible backgrounds \nis the only effective way to identify the decay, \nand it can be achieved by a profound understanding of backgrounds.\n\nThe dominant source of backgrounds was the result of beam interactions. \nAlthough the background from this source was considered to be as \nserious as those from $K_L^0\\;$ decays in the experiment, \nits understanding and estimation were very difficult to assess before the work \nreported here. The background mechanisms were clearly understood and divided \ninto three different sources, CC02, CV-$\\pi^0$, and CV-$\\eta$. \nMethods were developed to estimate them with rather small systematic errors. \nBased on this experience, we clearly know the direction of upgrades to minimize \nthe beam backgrounds in the next step. \n\nThe backgrounds from other $K_L^0\\;$ decays were reduced to be negligibly small in \nthe current experiment, mainly due to the success of applying tight vetoes with \na hermetic configuration. One of the important results is the invariant mass \ndistribution of the events with four photons, shown in Fig.~\\ref{fig_4gmass}. \nThe distribution was well reproduced by the simulations. In particular, the\nlow-mass region was well described by $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ decays with a small contamination \nby mis-combination events of four photons from $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ decays. The relation between \n$K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ is similar to the relation between $K_L^0\\rightarrow\\pi^0\\pi^0\\;$ and $K_L^0\\rightarrow\\pi^0\\nu\\bar{\\nu}$. \nBy taking their branching ratios into account, it was found that the low-mass \nregion could to be reproduced after reducing the $K_L^0\\rightarrow\\pi^0\\pi^0\\pi^0\\;$ yield by several orders \nof magnitude. Reproduction could not be achieved without good simulations of \nsmall signals from the detector, to which a tight veto was applied. \nThis achievement indicates that a similar direction will be promising in the next step. \n\nThe E391a experiment was also able to study other decay modes\nincluding $\\pi^0$'s in the final state, such as \n$K_L^0 \\rightarrow \\pi^0 \\pi^0 \\nu \\bar{\\nu}$ and \n$K_L^0 \\rightarrow \\pi^0 \\pi^0 X$ $(X\\rightarrow \\gamma \\gamma)$.\nThe \\pizs reconstruction method and the hermetic veto system are also valid \nin the analysis of these modes.\nResults based on portions of the data have already been published\n\\cite{ppnn,ppgg}, and further studies are in progress with the entire data set.\n\nThe next step, the KOTO experiment \\cite{koto} at the new J-PARC accelerator \\cite{jparc}, \nis now in preparation. \nMost of the improvements pointed out above are being implemented. \nThe new beamline has been constructed; runs to evaluate properties \nof the beamline started in October, 2009.\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe are grateful to the crew of the KEK 12-GeV proton synchrotron\nfor successful beam operation during the experiment. We express our sincere thanks to KEK staff, \nin particular \nthe directors: Professors H. Sugawara, Y. Totsuka, A. Suzuki, S. Yamada, M. Kobayashi, \nF. Takasaki, K. Nishikawa, K. Nakai, and K. Nakamura\nfor their continuous encouragement and support. \nWe also express our sincere thanks \nto the many colleagues in universities, institutes, and the kaon-physics\ncommunity for their continuous encouragement and support.\n\nThis work was partly supported by \nGrant-in-Aids from MEXT and JSPS in Japan, \ngrants from NSF and DOE of US, \ngrants from NSC in Taiwan,\ngrants from KRF in Korea,\nand the ISTC project from CIS countries.\n\\\\\n\\end{acknowledgments}\n\n\n\n\\noindent \n\n\n\\bibliographystyle{plain}\n\\noindent\n$^a$Present address: Laboratory of Nuclear Problem, Joint Institute for Nuclear Research, \nDubna, Moscow Region, 141980 Russia \\\\\n$^b$Present address: KEK, Tsukuba, Ibaraki, 305-0801 Japan. \\\\\n$^c$Present address: CERN, CH-1211 Geneva 23, Switzerland. \\\\\n$^d$Deceased \\\\\n$^e$Present address: SPring-8, Japan. \\\\\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}