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+{"text":"\\section{Section 1}\n\n\nUnderstanding how turbulent flows develop and organize has puzzled scientists and engineers for centuries~\\cite{Lavin:2018}. The foundational characterization of turbulent flow began with Reynolds over a century ago~\\cite{Reynolds:1895} and was quickly followed by rigorous statistical interpretations of how turbulent flows develop~\\cite{Taylor:1935,Richardson:1922,Falkovich:2006}. \nIn 1937, Taylor and Green~\\cite{Taylor:1937} introduced an initial flow condition which produces a cascade of energy from large to small scales. Subsequently, Kolmogorov postulated that turbulent flows exhibit universal behaviors over many length scales. \nKolmogorov~\\cite{Kolmogorov:1941} predicted that within an inertial subrange, the energy spectrum of a turbulent flow has a universal, self-similar form wherein the energy scales as the inverse $5\/3$ power of the wavenumber  $k$.  Kolmogorov's energy cascade has been observed in a plethora of experimental systems and numerical simulations, from wind tunnels to river beds, see e.g. Fig.13 of~\\cite{Chapman:1979}.\n\nThe efficient conveyance of energy from the large scales, where it is injected, to the small scales, where it is dissipated, is at the heart of how complex, three dimensional flows are maintained. It is thus critical to understand how small-scale flow structures are formed and maintained at high Reynolds numbers. In spite of major progress in providing an effective statistical description of turbulent flows~\\cite{Pope:2000}, our understanding of the mechanisms by which interactions between eddies are mediated remains limited. In fact, the explanations of how this occurs in real-space are often abstract and ``poetic''~\\cite{Richardson:1922, Betchov:1976, Keylock:2016}.\n\nThe temporal development of the turbulent cascade remains one of the most intriguing mysteries in fluid mechanics. \nIn particular, it is not well understood what specific mechanisms lead to the development of large velocity gradients in turbulent flows.  \nThese large velocity gradients, which derive from the interactions of turbulent eddies, amplify the kinetic energy dissipation rate, $\\epsilon$, in a manner that is independent of the fluid viscosity in the high-Reynolds number limit~\\cite{Falkovich:2006, Frisch:1995}.\nThis implies the existence of an inertial mechanism by which vortices locally interact to convey energy across scales such that the statistical properties of the energy cascade develop in accordance with the scaling laws established by Kolmogorov.\nNote that the limit to how fast an initially regular flow may produce extremely large velocity gradients is also a celebrated mathematical problem~\\cite{Fefferman:2006}.\nIt is therefore of great interest to look directly for elementary flow configurations of interacting vortices that begin smooth and rapidly develop into turbulence. \nThis approach was implemented by Lundgren~\\cite{Lundgren:1982}, who analytically examined the breakdown of a single vortex under axial strain, bursting into an ensemble of helical vortex bundles. While this configuration has been observed to lead to the development of a turbulent flow, it requires the presence of a particular, large-scale strain configuration acting on an isolated vortex~\\cite{Cuypers:2003}. \nWe implement a more general flow configuration that typifies the fundamental components of the turbulent cascade: the collision of two identical vortices.\nRecent numerical and experimental works demonstrate that the breakdown of colliding vortex rings at intermediate Reynolds numbers gives rise to small-scale flow structures~\\cite{McKeown:2018}, mediated by the iterative flattening and splitting of the vortex cores to smaller and smaller filaments~\\cite{Brenner:2016,PumirSig:1987}.\n\nHere, we revisit the emergence of a turbulent burst of fine-scale flow structures that results from the violent, head-on collision of two coherent vortex rings~\\cite{Lim:1992,McKeown:2018}. \nThis classical configuration is a unique model system for probing the development of turbulence without any rigid boundaries or large-scale constraints.\nWe show that for high Reynolds numbers, the violent breakdown of the colliding vortex rings into a turbulent ``soup'' of interacting vortices is mediated by the elliptical instability. \nDuring the late-stage, nonlinear development of the elliptical instability, an ordered array of antiparallel secondary vortex filaments emerges perpendicular to the collision plane.\nLocally, these pairs of counter-rotating secondary filaments spawn another generation of tertiary vortex filaments, resulting in the expeditious formation of a hierarchy of vortices over many scales. Our numerical simulations show that at this stage of the breakdown, the interacting tangle of vortices reaches a turbulent state, such that the energy spectrum of the flow exhibits Kolmogorov scaling. We observe both experimentally and numerically how the elliptical instability precipitates the onset of turbulence, generating and maintaining the means by which the energy of the flow cascades from large to small scales. \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{fig1.png}\n\\caption{\nVortex ring collisions. (A) Schematic side-view showing the formation and collision of dyed vortex rings in experiments. Fluorescent dye (Rhodamine B) is injected into the core of the vortex via a thin gap in the orifice of the vortex cannon. The dashed horizontal line denotes the symmetry axis. (B) Vortex ring radius vs. rescaled time for collisions at various Reynolds numbers. Both cores are dyed, the core centerlines are extracted from 3D reconstructions, and the centerlines are fitted to circles with a fixed center point. The initial time begins when the vortex rings enter the scanning volume and ends when the vortex cores break down. (inset) All experimental curves shifted by $\\tilde{t}$ to collapse. The radial growth of the rings coincides with the Biot-Savart prediction.\n}\n\\label{fig:1}\n\\end{figure}\n\nThe geometry of the experimental setup is depicted schematically in Fig.~\\ref{fig:1}(A). Two identical counter-rotating vortex rings are fired head-on in a 75-gallon aquarium filled with deionized water, as shown in Movie S1. \nThe vortex rings are formed via a piston-cylinder configuration in which a slug fluid with viscosity, $\\nu$, is pushed through a cylinder of diameter, D (2.54 cm), at a constant velocity, U, with a stroke, L. The resulting flow is controlled by two dimensionless parameters: the Reynolds number, Re = UD\/$\\nu$, and the stroke ratio, SR = L\/D~\\cite{Gharib:1998}.\nFluorescent dye (Rhodamine B) is injected into the cores of the rings as they are formed. Since the collision occurs at a fixed plane in the laboratory frame, this configuration is attractive for directly observing the rapid formation of small-scale flow structures. \nThe dynamics and eventual breakdown of the dyed cores are visualized in full 3D by imaging over the collision plane with a scanning laser sheet ($\\lambda = 532$~nm), which is pulsed synchronously with a high-speed camera (Phantom V2511). The technical details of how the vortex rings are formed and visualized in 3D are described in previous work~\\cite{McKeown:2018} and in supplemental section 1. Additionally, we perform direct numerical simulations (DNS) of interacting vortices at Reynolds numbers equivalent to the experiments (see supplemental sections 1-2 for how the definitions of the Reynolds numbers in simulations and experiments compare).\n\nAs the vortex rings collide, they exert mutual strains on one another, causing them to stretch radially at a constant velocity before breaking down at a terminal radius, as shown in Fig.~\\ref{fig:1}(B). At low Reynolds numbers, Re $\\lesssim$ 5000, the dyed cores break down, ejecting a tiara of secondary vortex rings or smoky turbulent puffs~\\cite{Lim:1992,McKeown:2018} at approximately 6 times the initial radius, $R_0$. The initial vortex ring radius and core radius, $\\sigma$, were measured separately through particle image velocimetry (PIV), as described in supplemental section 2. Strikingly, for collisions at higher Reynolds numbers, Re $\\gtrsim$ 5000, the cores ``burst'' into an amorphous turbulent cloud of dye at a maximum radius of approximately 5$R_0$, indicating the onset of a different breakdown mechanism at this high Reynolds number regime. The mean radial growth of the colliding rings is well described by the Biot-Savart model~\\cite{Batchelor:1970}, as described in supplemental section 3(A-B). Additionally, the radial expansion of the rings is encapsulated by a universal functional form, as shown in the inset of Fig.~\\ref{fig:1}(B).\n\nWhile the mean radial growth of the colliding vortex rings follows the same linear evolution at any Reynolds number, the cores themselves develop different forms of perturbations, due to their mutual interaction. \nThe formation of these perturbations can arise from two different types of instabilities. The Crow instability~\\cite{Crow:1970} causes the cores to develop symmetric circumferential perturbations with long wavelengths, much larger than the core radius, $\\sigma$. This instability stems from the mutual advection of the interacting vortices~\\cite{Leweke:2016} and governs the breakdown dynamics for collisions at lower Reynolds numbers~\\cite{Lim:1992,McKeown:2018, Leweke:2016}. The nonlinear development of the Crow instability causes the rings to deflect into one another and form ``tent-like'' structures~\\cite{Hormoz:2012,Brenner:2016,McKeown:2018}, which interact locally at the collision plane. \n\nAt higher Reynolds numbers, both our experiments and simulations show that the breakdown dynamics are governed by the elliptical instability, causing the vortex cores to develop short-wavelength perturbations on the order of the core radius~\\cite{Tsai:1976,Moore:1975, Schaeffer:2010} (see supplemental section 3(C)). This instability originates from the parametric excitation of Kelvin modes in the vortex cores due to the resonant interaction of the strain field from the other vortex~\\cite{Leweke:2016, Kerswell:2002}. A hallmark of the elliptical instability, these short-wavelength perturbations grow synchronously in an antisymmetric manner, as shown for two typical  experimental and numerical examples in Fig.~\\ref{fig:2}(A-B). \n\nAs the elliptical instability grows radially along the collision plane, the mean spacing between the cores, $d$, decreases linearly. However, the separation distance between the cores saturates when the perturbations deflect out-of-plane, just prior to breaking down, as shown in Fig.~\\ref{fig:2}(C-D) and Movie S2.\nThe minimum mean spacing between the cores is approximately equal to twice the initial core radius, $ \\sigma$. For the experiment, $\\sigma = 0.14 \\pm 0.01 ~R_0$, and for the simulation, $\\sigma  = 0.1 R_0$ (see supplemental section 2).\nAfter the elliptical instability develops and the symmetry of the two cores is broken, a periodic array of satellite flow structures is shed from each core, bridging the gap between them, as shown in the inset of Fig.~\\ref{fig:2}(D) and Movie S3. Notably, in our experiments, the emergence of these secondary flow structures can only be resolved if the fluid outside of the cores is dyed.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\linewidth]{fig2.png}\n\\caption{\nAntisymmetric perturbations in vortex ring collisions. A montage of core centerline trajectories for vortex ring collisions in both (A) experiment and (B) direct numerical simulation. The top ($z>0$) cores are indicated by the red lines, and the bottom ($z<0$) cores are indicated by the blue lines. For the experimental collision, Re = 7000, SR = 2, and $R_0$ = 17.5 mm. For the DNS collision, Re$_\\Gamma$ = $\\Gamma$\/$\\nu$ = 4500 and $\\sigma = 0.1R_0$.  The cores are segmented from the 3D flow visualization in the experimental collision and from the pressure distribution in the simulation. Mean core separation distance vs. rescaled time for the same (C) experimental and (D) numerical collisions. The blue circles correspond to the trajectories in (A) and (B), and the red dashed lines correspond to the visualizations in the insets. (C, inset) 3D visualization of the dyed vortex cores in the experimental collision. (D, inset) 3D visualization of the dyed vortex rings in the simulation, showing both the dye in the cores (dark) and surrounding them (light).  \n}\n\\label{fig:2}\n\\end{figure}\n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{fig3.png}\n\\caption{\nFormation of perpendicular secondary filaments in a typical experimental vortex ring collision. 3D reconstruction of two fully-dyed vortex rings colliding head-on, viewed from overhead (top) and from the side (bottom). Re = $6000$ and SR = $2.5$. (A-C) As the rings grow, they interdigitate as the dye from the upper ring is wrapped around the lower ring and vice-versa. (D-E) The colliding rings form an array of secondary vortex filaments that are perpendicular to the vortex cores. (F) The cores and perpendicular filaments break down into a fine-scale turbulent cloud.}\n\\label{fig:3}\n\\end{figure*}\n\nIn order to better resolve the late-stage development of the elliptical instability and the resulting breakdown, we dye the full vortex rings, as shown in Fig.~\\ref{fig:3}(A-F) and Movie S4. \nObserving the fully dyed vortex rings reveals the intricate structure of the flow that develops in response to the core dynamics.\nThe antisymmetric coupling of the perturbations break the azimuthal symmetry of the flow, leading to the exchange of fluid between the two rings. \nThis periodic wrapping of dye causes the outer layers of the rings to interdigitate around one another along alternating ``tongues,''~\\cite{Leweke:1998} as shown in Fig.~\\ref{fig:3}(A-B). \nAt the boundaries of adjacent tongues, the dyed fluid curls into vortex filaments, perpendicular to the cores, as shown in Fig.~\\ref{fig:3}(C).\nThese alternating filaments are stretched by the circulating vortex cores into an array of counter-rotating secondary vortices, as shown in Fig.~\\ref{fig:3}(D-E). The secondary filaments have a fleeting lifetime of only tens of milliseconds before they break down. \nViolent interactions between the secondary filaments and primary cores result in the rapid ejection of fine-scale vortices and the formation of a turbulent cloud, as shown in Fig.~\\ref{fig:3}(F).\n\nBy performing direct numerical simulations of the colliding vortices, we additionally probe how energy is transferred through the flow via the onset of the elliptical instability.\nSince the breakdown of the vortices is localized to the area around the cores, we implement a new configuration for the simulations, which consists of two initially parallel, counter-rotating vortex tubes with circulation $\\Gamma$, initially spaced a distance, $b = 2.5\\sigma$, apart. The flow is simulated in a cubic domain of side length, $\\mathcal{L}=16.67\\sigma$, and the Reynolds number of this configuration is given by Re$_\\Gamma = \\Gamma\/\\nu$ (see supplemental section 1). From PIV measurements, we find that Re$_\\Gamma \\approx 0.678$Re as shown in supplemental section 2. \nThe dynamics of the vorticity distribution in the simulated flow are qualitatively equivalent to the experimental flow visualizations, as shown for a typical example at Re$_\\Gamma = 4500$ in  Fig.~\\ref{fig:4}(A-C) and Movie S5.\nWhen the antisymmetric perturbations resulting from the elliptical instability materialize, the tips of the perturbed cores deform into flattened vortex sheets, as illustrated in Fig.~\\ref{fig:4}(A). These sheets are stretched by strains applied by the other core and roll up along the edges into an alternating series of hairpin vortices, as shown in Fig.~\\ref{fig:4}(B) and supplemental section 4(A).\nUpon stretching across the gap to the other perturbed core, these hairpin vortices form an ordered array of secondary vortex filaments perpendicular to the initial tubes, as shown in Fig.~\\ref{fig:4}(C). Adjacent pairs of secondary filaments counter-rotate relative to one another~\\cite{Leweke:1998}, as shown in Fig.~\\ref{fig:4}(D). \nIntegrating the transverse vorticity along the symmetry plane, we find that as the secondary filaments are stretched, approximately $25\\%$ of the streamwise circulation from the initial vortex tubes is conveyed to each filament (see supplemental section 4(B)). \nAs the vorticity of the original tubes is transferred to the secondary filaments, the circulation of the flow is conserved.\n\n\\begin{figure\n\\centering\n\\includegraphics[width=\\linewidth]{fig4.png}\n\\caption{\nGeneration of perpendicular secondary filaments in DNS. (A-C) Vorticity modulus for simulated interacting tubes where Re$_\\Gamma=4500$, $\\sigma = 0.06\\mathcal{L}$, $b = 2.5\\sigma$, and $t^{*} = \\Gamma t \/ b^2$. The vorticity modulus is normalized by the maximum vorticity modulus during the simulation, $|\\omega|_{\\text{max}}$. (A) The initial antisymmetric perturbations of the cores develop as the tips of the perturbations locally flatten ($0.103 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.117$). (B) At the same time, low-vorticity perpendicular filaments form as a result of the perturbations ($0.046 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.092$). (C) Once the secondary filaments form, their vorticity amplifies ($0.076 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.114$). (D) Vorticity distribution in the $z$-direction along the center plane ($z=0$) indicated by the dashed line in (C). Adjacent secondary filaments counter-rotate. \n}\n\\label{fig:4}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.95\\linewidth]{fig5.png}\n\\caption{\nThe development of a turbulent cascade. (A-F) Vorticity modulus for simulated interacting tubes where Re$_\\Gamma = 6000$, $\\sigma = 0.06\\mathcal{L}$, $b = 2.5\\sigma$, and $t^{*} = \\Gamma t \/ b^2$. Each panel shows the front view of the full cores (left) and a close-up top view of the interacting secondary filaments indicated in the full view (right).  (A) The antisymmetric perturbations of the cores develop. (B) Perpendicular secondary filaments form between the cores. (C) Secondary filaments begin to interact with each other and break down. (D) Tertiary filaments begin to form perpendicular to the secondary filaments. (E) Tertiary filaments are fully formed. (F) The flow breaks down into a disordered tangle of vortices. The vorticity thresholds are $0.079 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.099$ for (A) and $0.110 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.211$ for (B-F), where $|\\omega|_{\\text{max}}$ is the maximum vorticity modulus for the entire simulation. (G) Normalized shell-to-shell energy transfer spectra indicate whether a mode is an energy source ($T(k)>0$) or sink ($T(k)<0$). At early times ($t^{*} = 65.56, 74.44$), the secondary filaments switch from energy sinks to sources as they are generated and then interact to form new vortices. At late times, the spectra flatten as energy is transferred more uniformly across the scales of the flow. (G, inset) Normalized kinetic energy dissipation rate as a function of time.  The energy dissipation rate increases with the development of the secondary filaments and peaks as the secondary filaments and residual cores break down into a tangle of fine-scale vortices. (H) Normalized kinetic energy spectra show the rapid development of a sustained turbulent state with Kolmogorov scaling--as indicated by the black line--around the peak dissipation rate.\n}\n\\label{fig:5}\n\\end{figure*}\n\nOnce formed, each pair of secondary filaments can be locally viewed as a replica of the initial flow configuration on a smaller scale and with a reduced circulation, hence corresponding to a smaller effective Reynolds number. \nThe resulting close-range interactions of neighboring filaments can lead to an iterative cascade by which even more generations of small-scale vortices are formed. \nFor collisions at moderately high Reynolds numbers (e.g. Re$_\\Gamma =3500$), the concentrated strains exerted by the counter-rotating secondary filaments cause one of them to flatten into an extremely thin vortex sheet and split into two smaller tertiary vortex filaments, as shown in Movies S6-S7 and supplemental section 5. This behavior is consistent with the breakdown mechanism observed experimentally in the head-on collision of vortex rings at comparatively lower Reynolds numbers mediated by the Crow instability~\\cite{McKeown:2018}.\n\nIn the high-Reynolds number limit, the secondary filaments may give rise to another generation of perpendicular tertiary vortex filaments, as shown for a typical example at Re$_\\Gamma =6000$ in Fig.~\\ref{fig:5}(A-F) and Movies S8-S9. \nThe secondary filaments are drawn into one another due to their mutual counter-rotation, as shown in the close-ups in Fig.~\\ref{fig:5}(B-C). \nThe narrow gap between these writhing vortices, which experiences intense strain, almost instantaneously becomes enveloped by several high-vorticity tertiary filaments, as shown in Fig.~\\ref{fig:5}(D-E). \nThese tertiary filaments align perpendicular to the previous generation of vortices, wrapping tightly around them. The tertiary filaments develop locally in an ordered manner while the remnants of the primary cores and secondary filaments become increasingly entangled into a disordered ``soup'' of vortices, as shown in Fig.~\\ref{fig:5}(F).\n\nThe iterative breakdown process occurs over diminutive length scales and fleeting time scales. This rapid generation of small-scale vortices leads to a dramatic increase in the energy dissipation rate, $\\epsilon$, as shown in the inset of Fig.~\\ref{fig:5}(G).\nThe initial increase in $\\epsilon$ is triggered by the onset of the elliptical instability and the formation of antisymmetric perturbations in the cores. The following precipitous rise in the dissipation rate coincides with the formation of the perpendicular secondary filaments. \n\nThe rate at which kinetic energy is transferred across scales is calculated for the simulation in three-dimensional Fourier space through the instantaneous shell-to-shell energy transfer spectrum, $T(k,t)$, as shown in Fig.~\\ref{fig:5}(G) ~\\cite{Pope:2000,Lin:1947}. At a fixed time, $T(k)$ \nis positive for a wavenumber, $k$, when energy flows toward the corresponding spatial scale $(\\sim k^{-1})$. Conversely, a negative value of $T(k)$ indicates the flow of energy away from that corresponding spatial scale to other modes (see supplemental section 6 for details). \nWhen initially formed, the secondary filaments become pronounced energy sinks, given the large positive value of $T(k)$ at the intermediate wavenumber of approximately $kb = 6.75$. This coincides with their absorption of energy from the primary vortex cores.\nNext, as the secondary filaments become fully developed and interact with each other, they change behavior and become sources of energy, as indicated by the negative value of $T(k)$. Coupled with the simultaneous increase in the dissipation rate, this change in behavior of the secondary filaments from energy sinks to energy sources indicates the existence of a cascade by which kinetic energy is conveyed to smaller scales. \n\nThrough the breakdown of the secondary filaments and residual vortex cores into a disordered tangle of vortices, the dissipation rate reaches a maximum value. \nAt this point, the flow is most vigorous and the energy transfer spectra asymptote toward a flattened profile, indicating that energy is conveyed more uniformly across the various scales of the system, as shown in Fig.~\\ref{fig:5}(G).\nThus, for this brief time, $\\epsilon$ maintains an approximately constant maximum value, as the energy is smoothly transferred to the smallest, dissipative scale, $\\eta = \\epsilon^{-\\frac{1}{4}} \\nu^{\\frac{3}{4}}$. Kolmogorov proposed that for turbulent flows under similar conditions, the kinetic energy spectrum follows a distinct scaling of $(k \\eta)^{-5\/3}$~\\cite{Kolmogorov:1941}.\nStrikingly, we find that the fully developed turbulent cloud formed by the collision of the two vortices, indeed, exhibits Kolmogorov scaling. \nThe evolution of the normalized energy spectra, $E(k)\/(\\eta^{\\frac{1}{4}} \\nu^{\\frac{5}{4}})$, demonstrates how the flow reaches a sustained turbulent state around the peak dissipation rate, as shown in Fig.~\\ref{fig:5}(H). \nThis turbulent energy spectrum scaling at the peak dissipation rate also emerges during the breakdown of interacting vortex tubes mediated by the elliptical instability at lower Reynolds numbers, as shown in supplemental section 7. \nSince the energy input of the system is finite, this turbulent state cannot be maintained indefinitely.\nAs time progresses further, the viscosity of the fluid damps out the motion of the vortices at the smallest scales. \nWhile much energy remains at the large scales of the flow, it is unable to be transmitted to smaller scales following this iterative breakdown.\nAccordingly, the energy dissipation rate decreases and the turbulent state decays.\n\nThe violent interaction between two counter-rotating vortices leads to the rapid emergence of a turbulent cascade, resulting in a flow with an energy spectrum that--for an ethereal moment--obeys Kolmogorov scaling. We find that the emergence of this turbulent cascade is initiated by the late-stage, nonlinear development of the elliptical instability, which forms an ordered array of counter-rotating secondary vortex filaments perpendicular to the primary cores. \nIn the high-Reynolds number limit, the neighboring secondary filaments may interact to form a new generation of perpendicular tertiary vortex filaments. \nThese interactions of the secondary filaments with each other and the remnants of the vortex cores lead to the rapid formation of small-scale vortices. This ensemble of vortices interacting over the full range of scales of the system provides a conduit through which energy cascades down to the dissipative scale. \n\nThe iterative cascade, which leads to the generation of vortices at decreasingly small length scales, is strongly reminiscent of the mechanism proposed by Brenner, Hormoz, and Pumir~\\cite{Brenner:2016}. One may speculate that the self-similar process suggested by this work could be modeled by assuming, in the spirit of~\\cite{Brenner:2016}, that at each iteration, the circulation is multiplied by a factor $x_\\Gamma < 1$, and the characteristic scale of the vortices by a factor $x_\\delta < 1$, resulting after $n$ steps in a generation of vortices with circulation, $\\Gamma_n = x_\\Gamma^{n-1} \\Gamma_1$, and a spatial scale, $\\delta_n = x_{\\delta}^{n-1} {\\delta}_1$. \nThe corresponding time scale over which each step evolves can be estimated as $ t_n \\sim \\delta_n^2\/\\Gamma_n \\sim (\\delta_1^2\/\\Gamma_1) \\, (x_{\\delta}^2\/x_\\Gamma)^{n-1}$. The cascade can go all the way down to vanishingly small spatial scales in a finite time provided $x_{\\delta}^2 < x_\\Gamma$. The numerical results presented here, during the first steps of the cascade, suggest that $x_\\Gamma \\sim 0.25$, and $x_{\\delta} \\sim 0.2-0.4$, and therefore that the cascade may proceed in a finite time. It would be interesting to understand whether the cascade suggested by this work proceeds faster or slower than the Kolmogorov cascade. Whereas Kolmogorov theory implies $t_n\/t_1 \\sim (\\delta_n\/\\delta_1)^{2\/3}$~\\cite{Frisch:1995}, our results imply that $t_n\/t_1 \\sim (\\delta_n\/\\delta_1)^{2 - \\ln(x_\\Gamma)\/\\ln(x_\\delta)}$. Therefore, the cascade proposed in this work proceeds faster than the Kolmogorov cascade for $x_\\delta < x_\\Gamma^{3\/4}$. Our estimates suggest that the two cascades may proceed asymptotically at a comparable rate. A more precise understanding of the development of the elliptical instability is necessary to determine accurately the scaling factors $x_\\Gamma$ and $x_\\delta$. \n\nThe essential element of the cascade process is that at each scale, discrete pairs of antiparallel vortices are able to locally interact and produce a subsequent iteration via the elliptical instability. \nVortices of similar size and circulation locally align in an antiparallel manner when they interact. This is a well established consequence of Biot-Savart dynamics~\\cite{Siggia:1985}. \nThus, the largest strains that drive the cascade will arise from the interactions of nearby vortices. we suggest that iterations of this cascade could proceed down to ever-smaller scales until viscous effects take over. \nYet, the proliferation of other small-scale vortices, clearly visible in Fig. 5, could conceivably prevent vortex pairs from forming at some stage of the process, and we do not rule out that it may influence the dynamics. \nWe remark, however, that the present work shows that only two clear iterations of the cascade are sufficient to produce a Kolmogorov spectrum. \nEven in the high-Reynolds number limit, a finite set of iterations occurring simultaneously for many independent pairs of interacting vortices might suffice to establish and sustain a turbulent cascade. The details of how this iterative process unfolds in the limit of large Reynolds number is an important question for future research.\n\nThis framework strongly agrees with recent works by Goto et al. in a fully turbulent flow regime. Namely, their numerical results demonstrate the existence of many independent pairs of antiparallel vortices interacting and locally forming smaller generations of perpendicular vortex filaments in both fully developed homogeneous isotropic turbulence and wall-bounded turbulence~\\cite{Goto:2012,Goto:2017,Motoori:2019}.\nThese discrete interactions of antiparallel vortex pairs appear simultaneously throughout Goto's simulations over four distinct scales~\\cite{Goto:2017}.\nDue to the striking similarities between the iterative mechanism we observe and the results of Goto, we propose that the elliptical instability is likely the means by which these successive generations of perpendicular filaments are formed.\nEstablishing a precise connection between our own results and Goto's observations requires a fully quantitative analysis, which is beyond the scope of the present work.\n\nOur work thus demonstrates how the elliptical instability provides a long-sought-after mechanism for the formation and perpetuation of the turbulent energy cascade through the local interactions of vortices over a hierarchy of scales. Supplied by the injection of energy at large scales, discrete iterations of this instability effectively channel the energy of a flow down to the dissipative scale through the formation of new vortices. \nFrom a quantitative point of view, the approximate estimates provided in this work suggest that the corresponding cascade proceeds in a finite time, although a precise comparison with the Kolmogorov cascade requires a better understanding of the nonlinear development of the elliptical instability.\nWhile the dynamics of turbulent flows likely involve other multi-scale vortex interactions, this fundamental mechanistic framework can begin to unravel the complexity that has long obscured our understanding of turbulence. \n\n\\section{Methods}\nTechnical details for the experiments and simulations are provided in supplemental section 1.\n\n\\section{Acknowledgements}\nThis research was funded by the National Science\nFoundation through the Harvard Materials Research Science and Engineering Center DMR-1420570 and through the\nDivision of Mathematical Sciences DMS-1411694 and\nDMS-1715477. M.P.B. is an investigator of the Simons\nFoundation. R.O.M. thanks the Core facility for Advanced Computing and Data Science (CACDS) at the University of Houston for providing computing resources.\n\n\n\n\n\\section{Methods and materials}\n\\label{sec:Methods}\n\nWe use both experiments and simulations to probe the dynamic formation of the turbulent cascade resulting from the interaction between counter-rotating vortices. Experimentally, we examine the head-on collision of vortex rings, and numerically we examine the collision of vortex rings and vortex tubes. \nIn the experiments~\\cite{McKeown:2018}, fluorescent dye (Rhodamine B) is injected into the initially formed rings to visualize the core dynamics, as shown in Fig. 1(A) in the main text. Two vortex rings are fired head-on into one another in a 75-gallon aquarium. The vortex cannons are positioned a distance of $8D$ apart, where $D$ is the vortex cannon diameter. The full, three-dimensional dynamics of the resulting collision are visualized by tomographically scanning over the collision plane with a rapidly translating pulsed laser sheet ($\\lambda = 532$ nm). The pulsing of the  laser (Spectraphysics Explorer One 532-2W) is synchronized with the exposure signal of a high speed camera (Phantom V2511), which images the illuminated plane head-on. Each image plane spans along the $xy$-plane, and the laser sheet scans in the $z$-direction. Thus, for each scan, the image slices are stacked together to form a 3D reconstruction of the collision. The spatial resolution of each volume is 145 $\\times$ 145 $\\times$ 100 ~$\\mu$m$^3$ per voxel in ($x$,$y$,$z$), and the time resolution is up to 0.5 ms per scan. The number of voxels in each scanned volume depends on the imaging window size, the camera frame rate, and the scanning rate. For example, the volume size is 512 $\\times$ 512 $\\times$ 64 voxels in $(x,y,z)$ for the dyed core collision in Movie~\\ref{mov:exp_cores} and 384 $\\times$ 288 $\\times$ 75 voxels for the fully dyed ring collision in Movie~\\ref{mov:exp_rings_dye}.  \nThe series of volumetric scans are reconstructed in full 3D with temporal evolution using Dragonfly visualization software (Object Research Systems).\nThe imaging apparatus can only detect the dyed regions of the fluid, so any flow structures that emerge during the breakdown that are undyed cannot be observed. When only the vortex cores are dyed, we probe through the volumes of each 3D scan along the azimuthal direction to locate the centroids of the cores at each cross section. This enables us to extract the vortex core centerlines, which we use to track the deformation of the cores and measure the vortex ring radius, $R(t)$, and the average spacing between the cores, $d(t)$. In order to visualize the development of secondary flow structures that emerge during the collisions, we fill the vortex cannons with fluorescent dye prior to driving the pistons to form the rings. As a result, the regions around the vortex cores are dyed, as shown in Fig. 3 in the main text.\n\nWe use  direct numerical simulations (DNS) to further probe the unstable interactions between the vortices. This allows us to directly examine the evolution of the vorticity field, relate it to experimental visualizations, and compute statistical quantities characterizing the flow. We solve the incompressible Navier-Stokes equations using an energy-conserving, second-order centered finite difference scheme with fractional time-stepping. We implement a third-order Runge-Kutta scheme for the non-linear terms and second-order Adams-Bashworth scheme for the viscous terms  \\cite{Kim:1985,Verzicco:1996}. We simulate both vortex rings and tubes, in cylindrical and Cartesian coordinates, respectively. To avoid singularities near the axis, the cylindrical solver uses $q_r = r v_r$ as a primitive variable~\\cite{Verzicco:1996}. The time-step is dynamically chosen such that the maximum Courant-Friedrich-Lewy (CFL) condition number is 1.2. Resolution adequacy is checked by three methods: monitoring the viscous dissipation and the energy balance, examining the Fourier energy spectra, and using the instantaneous Kolmogorov scale. White noise is added to all all initial conditions to trigger the most unstable modes. \n\nThe rings are initialized as two counter-rotating Gaussian (Lamb-Oseen) vortices, each with a core radius $\\sigma$ wrapped into a torus of radius $R_0$. The control parameters for this system are the circulation Reynolds number, Re$_\\Gamma = \\Gamma\/\\nu$ and the slenderness ratio of the rings, $\\Lambda = \\sigma\/R_0$. The circulation of the vortex rings, $\\Gamma$, and the initial ring radius, $R_0$, are used to non-dimensionalize parameters in the code. We simulate the collision in a closed cylindrical domain, bounded by stress-free walls at a distance far enough to not affect the collision. After testing several configurations, the bounds on the domain were placed a distance $R_0$ below and above the rings, and $5R_0$ from the ring axis in the radial direction. For the simulation presented in this paper, we selected a ring slenderness of $\\Lambda=0.1$, a circulation Reynolds number of Re$_\\Gamma=4500$, and an initial ring-to-ring distance of $d=2.5R_0$. These parameters are comparable to the experimental vortex rings, as shown by the measurements in supplemental section~\\ref{sec:PIV}. Points are clustered near the collision regions in the axial and radial directions, while uniform resolution is used in the azimuthal direction \\cite{Gharib:1998,McKeown:2018}. A rotational symmetry of order five is forced on the simulation to reduce computational costs. The vortex core centerlines are located by slicing azimuthally through the pressure field at every time step and identifying the local minima of each vortex cross section. Additionally, a simulated passive scalar is injected into the vortex rings to visualize the dynamics of the collision and compare with experiments, as shown in Movie~\\ref{mov:sim_rings_dye}. Due to computational restrictions, the diffusivity of the dye is equal to the kinematic viscosity of the fluid (i.e. the Schmidt number is unity).\n\nFor the vortex tubes, we simulate a triply periodic cubic domain of period $\\mathcal{L}$, which is discretized using a uniform grid. The two counter-rotating, parallel tubes are both initialized with a Gaussian (Lamb-Oseen) vorticity profile of radius, $\\sigma$, and circulation, $\\Gamma$, initially separated a distance, $b$, apart. The system is characterized by two dimensionless parameters: the circulation Reynolds number, Re$_\\Gamma$, and the ratio $\\sigma\/b$. Again, the circulation, $\\Gamma$, is used as a non-dimensional parameter, along with $b$. We set the core size to $\\sigma = 0.06 \\mathcal{L}$, fix $b\/\\sigma = 2.5$, and run simulations with Re$_\\Gamma$ at $2000$, $3500$, $4500$, and $6000$, with grid sizes of $256^3$, $360^3$, $540^3$, and $540^3$, respectively. As the counter-rotating tubes interact and break down, they naturally propagate through the periodic domain. For each visualization, the propagation of the tubes is subtracted so that the tubes remain in the center of the domain. Additionally, in all 3D visualizations of the vorticity modulus, $|\\omega|(t)$, the vorticity modulus at each voxel is normalized by the maximum vorticity modulus for all time, $|\\omega|(t)_{\\text{max}}$ (see supplemental section~\\ref{subsec:Vorticity_Across_Re}).\n\n\\section{PIV analysis of vortex ring geometry}\n\\label{sec:PIV}\nThe  vortex rings are characterized experimentally through 2D patricle image velocimetry (PIV). The fluid is seeded with polyamide particles with a diameter of $50 ~\\mu$m and a density of 1.03 g\/mL (Dantec Dynamics). \nA laser sheet is positioned along the central axis of the vortex cannon in order to illuminate the cross section of the ejected vortex rings. The motion of the particles in the vortex rings along this cross section is imaged with a high-speed camera (Phantom V2511) with a window size of 1280 $\\times$ 800 pixels at a maximum frame rate of 2000 fps, such that the resolution is 0.12 mm\/pixel. \nVortex rings are formed over a range of stroke ratios (SR = $L\/D$) and Reynolds numbers (Re$ = UD\/\\nu$), where $L$ is the stroke of the piston, $D= 25.4$ mm is the diameter of the vortex cannon, $U$ is the piston velocity, and $\\nu$ is the kinematic viscosity of water.\n\nThe velocity field for each generated vortex ring is calculated using MATLAB PIVsuite. The cores of the vortex rings are identified by computing the vorticity field, and each core is fitted to a two-dimensional Gaussian function, as shown for a typical example in Fig.~\\ref{fig:PIV} and Movie~\\ref{mov:PIV}.\nAfter pinching off, the vortex ring reaches a steady size with radius, $R_0$, as it propagates forward through the fluid, as shown in Fig.~\\ref{fig:PIV}(A). Additionally, the core radius, $\\sigma$, is calculated by averaging the standard deviations of each Gaussian fit, $\\sigma_x$ and $\\sigma_y$, as shown in Fig.~\\ref{fig:PIV}(B).\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=\\textwidth]{figS1.png}\n\\caption{\nVortex ring tracking and measurement through PIV. (A) 2D vorticity distribution and core trajectories for a vortex ring formed \nvia a piston-cylinder assembly, where  Re = $7000$ and SR = $2.5$. Each vorticity distribution results from fitting the raw vorticity data to a 2D Gaussian function. The time steps correspond to $t = 0, 0.158, 0.368, 0.578, ~$and$~ 0.788$ seconds, and the vortex cannon orifice is located at $x=0$. (B) Zoomed-in view of the final vortex core indicated by the dashed gray box. The vortex core is defined by the level curve shown by the black ellipse, with a center point, $(\\mu_x, \\mu_y)$, a rotation angle, $\\theta$, and standard deviations, $\\sigma_x$ and $\\sigma_y$.\n}\n\\label{fig:PIV}\n\\end{figure}\n\nBy fitting to the vortex cores to a Gaussian function, we evaluate the geometry of the rings over a wide range of Reynolds numbers at various stroke ratios, as shown in Fig.~\\ref{fig:Vortex_Data}. \nThis parameterization of the vortex rings and their core structures enables us to directly relate the initial state of the vortex rings used in the experimental collisions to those of the simulations, as described in the main text. \nAccordingly, all of the parameterizations for the vortex ring geometry are performed when the vortex ring propagates a distance between $2D$ and $4D$ from the orifice of the vortex cannon. This corresponds to the range of distances required for the vortex ring to pinch off and reach a steady morphology prior to reaching where the collision plane is located in the experiments at $4D$. \n\nAt every stroke ratio, the core size is larger for the vortex rings produced with lower Reynolds numbers, as shown in Fig.~\\ref{fig:Vortex_Data}(A). However, for vortex rings with a Reynolds number greater than $\\sim 10,000$, the core size remains relatively constant. \nThis is because vortex rings formed at lower Reynolds numbers are more susceptible to viscous dissipation from the ambient fluid as they propagate forward. \nThis dissipation leads to a spreading of the vorticity distribution in the cores through the diffusion of momentum via the viscosity of the fluid, thereby resulting in the thicker cores of the vortex rings at lower Reynolds numbers. \nThe vortex ring radius, $R_0$, remains roughly constant for each stroke ratio over this wide range of Reynolds numbers, as shown in Fig.~\\ref{fig:Vortex_Data}(B). Like with the core size, the vortex rings formed with a larger stroke ratio naturally have a slightly larger radius, as more fluid is injected to form vortex rings. \nHowever, by normalizing the core radius with the vortex ring radius at each Reynolds number to compute the slenderness ratio, $\\Lambda$, the data collapses, as shown in Fig.~\\ref{fig:Vortex_Data}(C). \nThis consistency of the vortex ring and core geometry across various stroke ratios informes the selection of the vortex ring parameters in the simulated collisions described in the main text. \nIn particular, for all of the simulated vortex ring collisions, a slenderness ratio of $\\Lambda = 0.1$ is used.\n\nThe circulation of the vortex cores, $\\Gamma$, is calculated by integrating the raw vorticity data for each core over an elliptical contour defined by the Gaussian fit that overcompensates the core size by a factor of 1.5 for both standard deviations. \nThe magnitude of the circulation is then normalized by the kinematic viscosity of the fluid in order to compute the circulation Reynolds number, Re$_{\\Gamma} = \\Gamma\/\\nu$. This parameter, as previously discussed in the main text and in supplemental section~\\ref{sec:Methods}, is used to define the Reynolds number in the numerical simulations of the colliding vortex rings and the interacting vortex tubes. \nThis calculation thus allows to relate the Reynolds number of the vortex rings formed experimentally with those formed numerically, as shown in Fig.~\\ref{fig:Vortex_Data}(D). \nAccordingly, we find that the experimental Reynolds number scales linearly with the circulation Reynolds number, such that Re$_{\\Gamma} \\approx 0.678$Re, which is consistent with previous experimental works ~\\cite{Gharib:1998}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\textwidth]{figS2.png}\n\\caption{\n Vortex ring and core geometry. From the Gaussian fits of the vorticity data, the following parameters are measured as a function of the Reynolds number across various stroke ratios: (A) vortex core radius, $\\sigma$, (B) vortex ring radius, $R_0$, (C) slenderness ratio, $\\Lambda$, and (D) circulation Reynolds number, Re$_\\Gamma$. The dashed line in (D) corresponds to the linear fit to the data: Re$_{\\Gamma} = 0.678$Re. All calculations are performed when the vortex rings reach a distance between $2D$ and $4D$ from the vortex cannon orifice.\n}\n\\label{fig:Vortex_Data}\n\\end{figure}\n\n\\section{Simulating vortex ring collisions using the Biot-Savart approximation}\n\\subsection{Biot-Savart model and regularization}\n\\label{sec:BS}\n\nIn this section, we establish the conditions that lead to the emergence of the elliptical instability in vortex ring collisions, focusing on the range of Reynolds numbers in which this instability is observed. Our approach consists of modeling the rings using the Biot-Savart model~\\cite{Moore:1972,Siggia:1985,PumirSig:1985} while accounting for the evolution of the core size, which is assumed to have a circular cross-section for all time. With the parameters given by this model, we determine the growth rate of the elliptical instability, as calculated in~\\cite{LeDizes:2002}. \n\nRegularization of the Biot-Savart model is required due to the logarithmic divergence of the principal integral. Here, we use the regularization used in~\\cite{Siggia:1985,PumirSig:1985}. \nDenoting $\\sigma_i(t)$ as the core radius of each vortex filament, $i$, at time, $t$, we replace the Biot-Savart integral (up to a prefactor) by:\n\n\\begin{equation}\n\\frac{\\partial \\mathbf{r}_i(\\theta,t)}{\\partial t} = \\sum_j \\frac{\\Gamma_j}{4 \\pi}\n\\int_{{\\rm filament}_j} d\\theta' \n\\frac{\\frac{\\partial \\mathbf{r}_j}{\\partial \\theta'} \\times (\\mathbf{r}_i(\\theta) - \\mathbf{r}_j(\\theta')) }{[ (\\mathbf{r}_i(\\theta) - \\mathbf{r}_i (\\theta') )^2 + \\sigma_i(\\theta)^2 + \\sigma_j(\\theta')^2 ]^{3\/2} }.\n\\label{eq:BS_motion}\n\\end{equation}\nIn this configuration, we consider two filaments, with indices $i = 1 $ and $i = 2$. The centerline positions of the filaments with circulation, $\\Gamma_i$, are given by $\\mathbf{r}_i(\\theta,t)$.\nWe start with an initially axisymmetric configuration, consisting of two counter-rotating vortex rings, perfectly aligned along the same central axis. We parametrize the centerlines of the two rings in the $(\\hat{\\mathbf{x}}, \\hat{\\mathbf{y}}, \\hat{\\mathbf{z}})$ directions by:\n\n\\begin{equation}\n\\mathbf{r}_1(\\theta,t) = \\begin{pmatrix} R(t) \\cos(\\theta) \\\\ R(t) \\sin( \\theta) \\\\-d(t)\/2 \\end{pmatrix}  ~~~ {\\rm and } ~~~\n\\mathbf{r}_2(\\theta,t) = \\begin{pmatrix} R(t) \\cos(\\theta) \\\\ R(t) \\sin( \\theta) \\\\+d(t)\/2 \\end{pmatrix},\n\\label{eq:parametr}\n\\end{equation} \nwhere $R(t)$ is the vortex ring radius, and the azimuthal angle is given by $0 \\le \\theta \\le 2 \\pi$. The perpendicular distance between the two rings, $d(t)$, is taken to be positive, and the circulations are $\\Gamma_1 = -\\Gamma_2 = \\Gamma > 0$. With these conventions, filament $1$, at $z = - d(t)\/2$, \nmoves upward, toward filament $2$ at $z = d(t)\/2$.\n\nIn the following, as was the case in~\\cite{Siggia:1985,PumirSig:1985}, we impose incompressibility, by enforcing that the total volume of the rings is conserved:\n\\begin{equation}\nR(t) \\sigma^2(t) = R_0\\sigma_0^2.\n\\label{eq:sigma}\n\\end{equation}\nThe evolution equations reduce to two simple ordinary differential equations for $R(t)$ and $d(t)$, as explained in turn. With the parameterization proposed in Eq.~\\eqref{eq:parametr}, an elementary calculation shows that the contribution of the filament $1$ to the velocity at the point $\\mathbf{r}_1(\\theta)$ reduces to a uniform velocity in the positive $z$-direction:\n\n\\begin{equation}\nv_z^{1,s} = \n\\frac{\\Gamma}{4 \\pi}\n\\int_{0}^{2\\pi} d\\theta' \n\\frac{ R(t)^2 ( 1 - \\cos( \\theta - \\theta') ) }\n{[ 2 R(t)^2 (1 - \\cos (\\theta - \\theta' ) )  + 2 \\sigma^2 ]^{3\/2} }.\n\\label{eq:vz_1s}\n\\end{equation}\nThe contribution of filament $2$ to the velocity of filament $1$ consists of a component in the radial direction:\n\n\\begin{equation}\nv_r^{1,m} = \n\\frac{\\Gamma}{4 \\pi}\n\\int_0^{2 \\pi} d \\theta' \\frac{  d(t) R(t)  \\cos( \\theta - \\theta' ) }\n{[ 2 R(t)^2 (1 - \\cos (\\theta - \\theta' ) ) + d(t)^2  + 2 \\sigma^2 ]^{3\/2} }\n\\label{eq:vr_1m}\n\\end{equation}\nand a component in the vertical direction:\n\n\\begin{equation}\nv_z^{1,m} = - \\frac{\\Gamma}{4 \\pi}\n\\int_{0}^{2\\pi} d\\theta' \n\\frac{ R(t)^2 ( 1 - \\cos( \\theta - \\theta') ) }\n{[ 2 R(t)^2 (1 - \\cos (\\theta - \\theta' ) )  + d(t)^2 +  2 \\sigma^2 ]^{3\/2} }.\n\\label{eq:vz_1m}\n\\end{equation}\nHence, the evolution equation of the vortex ring radius, $R(t)$ reduces to:\n\n\\begin{equation}\n\\dot{R}(t)  = v_r^{1,m}  = \n\\frac{\\Gamma d(t) }{2 \\pi R(t)^2} \\Psi[ (d(t)^2+ 2\\sigma^2)\/(2 R(t)^2) ],\n\\label{eq:R_dt}\n\\end{equation}\nwhere $\\Psi$ is defined as:\n\\begin{equation}\n\\Psi(X^2) = \\int_0^{2 \\pi} d \\theta' \\frac{  \\cos( \\theta - \\theta' ) }\n{[ 2 (1 - \\cos (\\theta - \\theta' ) ) + 2 X^2 ]^{3\/2} }.\n\\label{eq:Psi}\n\\end{equation}\nSimilarly, the evolution equation of the spacing between the rings, $d(t)$, is given by:\n\n\\begin{equation}\n\\dot{d}(t) = v_z^{1,s} + v_z^{1,m}\n= \\frac{\\Gamma }{ 2 \\pi R(t)} \\bigg( \n\\Phi[ ( d(t)^2 + \\sigma^2 )\/(2 R(t)^2) ] \n- \n\\Phi[\\sigma^2\/R(t)^2 ] \n\\bigg),\n\\label{eq:d_dt}\n\\end{equation}\nwhere $\\Phi$ is defined as:\n\\begin{equation}\n\\Phi(X^2) = \\int_0^{2 \\pi} d \\theta' \n\\frac{ ( 1 -  \\cos( \\theta - \\theta' ) ) }\n{[ 2 (1 - \\cos (\\theta - \\theta' ) ) + 2 X^2 ]^{3\/2} }.\n\\label{eq:Phi}\n\\end{equation}\nIt is a simple matter to compute asymptotic expressions for the functions $\\Phi$ and $\\Psi$ when $X^2 \\rightarrow 0$ or $X^2 \\rightarrow \\infty$. \nTo determine the evolution of $R(t) $ and $d(t)$, we numerically integrate Eq.~\\eqref{eq:sigma}, Eq.~\\eqref{eq:R_dt} and Eq.~\\eqref{eq:d_dt}.\n\n\\subsection{Dynamic evolution of the vortex rings}\n\\label{sec:num_results}\n\nEq.~\\eqref{eq:sigma}, Eq.~\\eqref{eq:R_dt}, and Eq.~\\eqref{eq:d_dt} are used to compare the radial growth of the colliding vortex rings from the Biot-Savart model with the experimental data in the inset of Fig. 1(B) in the main text. This model agrees well with the mean radial growth of the rings prior to breaking down, at which point the assumptions of Biot-Savart are clearly violated.\nHere, we further compare the model against direct numerical simulations which describe the head-on collision of two vortex rings with the following initial conditions: $R(0) = R_0 $, $d(0) = 2.5R_0$, and $\\sigma(0) = 0.1R_0$. This choice of parameters allows for the direct comparison of the model with the DNS, starting from the same initial configuration. The DNS collision, where Re$_{\\Gamma} = 4500$, was already presented in Fig. 2(B,D) of the main text and in Movie~\\ref{mov:sim_rings_dye}.\n\nFig.~\\ref{fig:compare_bs_ns} shows the evolution of the vortex ring radius, $R(t)$, (blue curves) and the perpendicular distance between the filaments, $d(t)$, (red curves). The solution of the Biot-Savart model is shown with dashed lines, and the full lines correspond to the same configuration evaluated numerically by solving the Navier-Stokes equations.\nAs the two rings initially approach one another, the distance between them decreases linearly. However, in both the Biot-Savart model and the DNS, the rings reach a minimum distance on the order of the core size before breaking down. \nDuring the late stage of the collision, the rings grow radially outwards; however, the growth of the ring radius predicted by the Biot-Savart model significantly exceeds that of the collision obtained from the DNS because the core dynamics--and hence the breakdown--is not captured by Biot-Savart model.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{figS3.png}\n\\caption{\nComparison between the evolution of the perpendicular spacing between the cores, $d(t)$, and the vortex ring radius, $R(t)$, obtained from DNS of the Navier-Stokes equations where Re$_\\Gamma = 4500$ (full lines) and from the Biot-Savart model (dashed lines). For both configurations, $d(0) =  2.5R_0$ and $\\sigma(0) = 0.1R_0$.\n}\n\\label{fig:compare_bs_ns}\n\\end{figure}\n\n\n\\subsection{ Elliptical instability }\n\\label{sec:instab}\n\nHere we examine the the onset of the elliptical instability when the vortex rings collide head-on. We are primarily interested in the growth of short-wavelength modes, characteristic of the elliptical instability~\\cite{Leweke:1998,Leweke:2016}. \nFor this reason, we neglect the curvature of the rings and approximate the filaments with a pair of straight antiparallel vortices, located at $z = \\pm d(t)\/2$, with circulations $\\pm \\Gamma$, and each with a core radius $\\sigma(t)$. The values of $d(t)$ and $\\sigma(t)$ are determined from the solutions of the Biot-Savart model.\n\nThe onset of the elliptical instability results in the development of perturbations with a wavelength on the order of the core size~\\cite{Leweke:2016}.\nOur analysis is based on the work of LeDizes, who examined the elliptical instability in the same flow configuration ~\\cite{LeDizes:2002,LeDizes:2002_corr}. LeDizes derived the following equation for the growth rate, $\\gamma$, of an infinitesimal perturbation of the vortex cores with a longitudinal wavenumber $k_z$:\n\n\\begin{equation}\n\\gamma = \\frac{\\Gamma}{8 \\pi d^2 } \n\\sqrt{ \\left(\\frac{3}{4}\\right)^4 K_{NL}(0)^2 - \\frac{64 d^4}{\\sigma^4} \n\\left(\\frac{1}{2} - \\cos(\\zeta^{(m)}) \\right)^2 } - \\frac{8 \\pi k_z^2 d^2 }{\\text{Re}_{\\Gamma} \\cos(\\zeta^{(m)})   },\n\\label{eq:instab_ell}\n\\end{equation}\n\nwhere $K_{NL}(0) = 1.5 + 0.038 \\times 0.16^{-9\/5} \\approx 2.52$, and\n$\\cos (\\zeta^{(m)}) = \\big( \\frac{1}{2} - \\frac{(2.26 + 1.69 m) - k_z \\sigma}{14.8 + 9m } \\big)$.\n\nIn the expression for $\\cos (\\zeta^{(m)})$, $m$ is an integer that distinguishes between several branches of solutions. We only examine the most unstable branch, corresponding to $m = 0$. To estimate the growth rate of the elliptical instability, we consider an azimuthal perturbation mode, $n$, along the rings. This corresponds to a wavenumber $q_n(t) = n\/R(t)$. The growth of the radius, $R(t)$, implies that the wavenumber $q_n(t)$ decreases as time progresses for a fixed value of $n$. Note that when neglecting the curvature of the colliding rings, $k_z = q_n(t)$.\n\nWe evaluate the growth rate of the elliptical instability, $\\gamma_n^{(m=0)}$, as a function of time for $n = 120, 180,$ and $240$, focusing on only the times when the growth rate is positive, as shown in Fig.~\\ref{fig:growth_rate}.\nThese values of $n$ were selected because they correspond to wavelengths that are on the order of the core radius, $\\sigma(t)$, when the instability begins.\nWe examine how the growth rates of these modes evolve when Re$_\\Gamma = 3500$ (dotted-dashed lines), Re$_\\Gamma = 4500$ (dashed lines) and Re$_\\Gamma = 5000$ (full lines). \nThe elliptical instability develops at each value of $n$  only over a short period of time. The magnitude of the growth rate $\\gamma_n^{(m=0)}$ increases as the Reynolds number increases.\n\n\\begin{figure}[hb!]\n\\centering\n\\includegraphics[width=\\textwidth]{figS4.png}\n\\caption{\nOnset of the elliptical instability for colliding vortex rings. The normalized growth rate, $\\gamma_{n}$, of three azimuthal perturbation modes, $n$, is evaluated for the head-on collision of two vortex rings using Eq.~\\eqref{eq:instab_ell}. (inset) zoomed-in view of the plot of the $n=120$ mode indicated by the dashed gray box.\nThe full lines, dashed lines, and dot-dashed lines correspond to Re$_\\Gamma = 5000, 4500, \\text{and} ~3500$, respectively. The values of $R(t)$ and $d(t)$ are calculated from the Biot-Savart model, shown by the dashed lines in Fig.~\\ref{fig:compare_bs_ns} above. \n}\n\\label{fig:growth_rate}\n\\end{figure}\n\nThe Biot-Savart model--Eq.~\\eqref{eq:sigma}, Eq.~\\eqref{eq:R_dt}, and Eq.~\\eqref{eq:d_dt}--provides a semi-quantitative description of the solutions of the Navier-Stokes equations when the $n=120$ mode becomes unstable, as shown in the inset of Fig.~\\ref{fig:growth_rate}. At this time, $\\Gamma t\/ R_0^2=5.9$, the unstable wavelength is $\\lambda \\approx 2 \\pi R(t)\/120 \\approx 2 \\sigma(t)$. The model therefore predicts that perturbations with a wavelength on the order of the core size are unstable due to the elliptical instability mechanism when Re$_\\Gamma = 4500$, but are stable for Re$_\\Gamma = 3500$, in qualitative agreement with our findings. The elliptical instability is still triggered for the Re$_\\Gamma = 3500$ configuration, albeit at later times.\nThe model also demonstrates that the onset of the elliptical instability for Re$_\\Gamma = 4500$ occurs when the spacing between the rings, $d(t)$, reaches the minimum threshold on the order of the initial core thickness, $2\\sigma_0 = 0.2R_0$, as shown in Fig.~\\ref{fig:compare_bs_ns}. This is consistent with the experimental and DNS results shown in Fig. 2(C-D) in the main text.\n\n\\section{Nonlinear development of the elliptical instability }\n\\label{sec:transverse_filaments}\n\n\\subsection{ Formation of secondary vortex filaments }\nFollowing the development of antisymmetric perturbations that result from the elliptical instability, an array of secondary vortex filaments spontaneously forms perpendicular to the original vortex cores. This has been directly observed experimentally and numerically for vortex ring collisions, as detailed in the main text. The same flow structures emerge via the elliptical instability during the interaction between two antiparallel vortex tubes~\\cite{Leweke:1998,Laporte:2000}. \nThis observation seems at first surprising, as it indicates the emergence of a component of circulation in the plane separating the two vortices, where the vorticity is initially zero. If the midplane separating the two vortices (i.e. the $z=0$ plane) were a plane of symmetry--as is the case in many studies examining the Crow instability~\\cite{Crow:1970} of interacting vortex tubes~\\cite{Pumir:1990,Kerr:1993}--the vorticity along this plane would remain zero at all times. \nThe antisymmetric nature of the perturbed vortex cores, shown in Fig. 2(A-B) of the main text, however, allows for the development of a non-zero circulation in the collision plane. Here, we examine how a significant component of the circulation can accumulate on the plane $z = 0$, referred to here as the plane of reflection, as shown in Fig. 4(C-D) in the main text.\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=0.92\\textwidth]{figS5.png}\n\\caption{\nFormation of a secondary vortex filament. Temporal evolution of the axial vorticity distribution, $\\omega_x(y,z)$, along a fixed cross section at $x\/\\mathcal{L} = 0.48$ for DNS of counter-rotating vortex tubes where Re$_\\Gamma = 4500$. The black arrows indicate the propagation direction, though a horizontal offset is applied at each time to keep the vortices in the center of the domain. (A) The vorticity from the two vortex tubes is initially separated along the ($z=0$) reflection plane. (B) As the perturbations develop, the lower core migrates to the leading direction and the upper core migrates to the trailing end. (C) The cores flatten into sheet-like structures, and vorticity from the upper vortex is advected down to the lower vortex. (D) The cores contract into highly curved, sheetlike structures where the vorticity is concentrated. The advected vorticity forms a secondary vortex filament across the ($z=0$) reflection plane. Note: $\\sigma = 0.06 \\mathcal{L}$, $b = 2.5 \\sigma$, and $t^* = \\Gamma t \/ b^2$.  \n}\n\\label{fig:transv_tubes}\n\\end{figure}\n\nWe examine the same direct numerical simulation presented in Fig. 4 in the main text and Movie~\\ref{mov:tubes_Re_4500}, consisting of two, initially parallel, counter-rotating vortex tubes with Re$_\\Gamma = 4500$. We examine the evolution of the normal vorticity component,  $\\omega_x$, along a fixed axial cross-section of the tubes, as shown in Fig.~\\ref{fig:transv_tubes}. \nThe upper vortex rotates in the clockwise direction, and the lower vortex rotates in the opposite direction, resulting in their propagation in the negative $y-$direction, indicated by the black arrows.\nInitially, the vorticity of the tubes largely remains on the respective sides of the plane of reflection, as shown in Fig.~\\ref{fig:transv_tubes}(A). However, once the tubes become perturbed, the centroids of the vortex cores deform such that the lower core moves forward in the propagation direction and the upper core moves backward, as shown in Fig.~\\ref{fig:transv_tubes}(B-D). This contrary motion of the cores illustrates the antisymmetric structure of the perturbations, as this particular cross section is located along an anti-node of the pair. While the amplitudes of the perturbations grow, the vorticity distributions of the cores contract and amplify into flattened, sheet-like structures, as shown in Fig.~\\ref{fig:transv_tubes}(D) and Fig. 4(A) in the main text. The curvatures of the deformed cores are of opposite sign; the leading vortex core is curved toward the propagation direction and vice-versa for the trailing core. \nAdditionally, the lower core, which is deflected toward the leading edge of the vortex pair, has a higher curvature than that of the upper core.\nThe tendency of the kinked, perturbed cores to locally flatten into vortex sheets results from the stretching field generated by each filament, as characterized by the Biot-Savart model. An elementary calculation predicts that, on the outer side of a kinked filament, the vorticity is stretched and grows, while on the inner side, the vorticity decreases (see Fig. 4 of~\\cite{Pumir:1990}).\nThe tendency of the perturbed vortex cores to flatten into sheets is therefore the result of the dynamics of kinked vortex tubes.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\textwidth]{figS6.png}\n\\caption{\nAlternating structure of secondary filaments. DNS of counter-rotating vortex tubes where Re$_{\\Gamma}$=4500 at a fixed time, $t^* = \\Gamma t \/ b^2 = 60.0$. (A-B) 3D vorticity modulus of the simulated flow and (C-E) distribution of axial vorticity, $\\omega_x(y,z)$, along cross sections of the tubes indicated by the dashed green lines. The black arrows indicate the propagation direction of the vortex pair. (A) The cores flatten into vortex sheets at the tips of each perturbation ($0.092 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.122$). (B) The edges of the each flattened perturbation roll up into pairs of secondary filaments ($0.031 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.053$). Due to the antisymmetric structure of the perturbations, the orientation of the secondary filament pairs alternates periodically. (C) At anti-nodes where the top vortex core leads, the secondary filament is stretched up from the lower vortex tube. (D) At nodes, neither vortex core leads and no secondary filaments form. (E) At anti-nodes where the bottom vortex core leads, the secondary filament is stretched down from the upper vortex tube. Note: $\\sigma = 0.06 \\mathcal{L}$ and and $b = 2.5\\sigma$.\n}\n\\label{fig:contours_vrt_b}\n\\end{figure}\n\nBecause the perturbations of the vortex cores are periodic, the relative positions of the cores revert every half-wavelength along the axial direction of the tubes, as shown in Fig.~\\ref{fig:contours_vrt_b}. The 3D visualization of the vorticity modulus, $|\\omega|$, shows how the peaks of the perturbed cores flatten into mushroom-cap structures, as shown in Fig.~\\ref{fig:contours_vrt_b}(A). By lowering the threshold of the vorticity modulus, as shown in Fig.~\\ref{fig:contours_vrt_b}(B), one can visualize how the edges of the flattened cores roll up into two secondary vortex filaments that are pulled toward the opposite vortex tube.     \nMoving along the axial direction, the leading vortex switches from the top core (Fig.~\\ref{fig:contours_vrt_b}(C)) to the bottom core (Fig.~\\ref{fig:contours_vrt_b}(E)). At the nodes of the perturbations, the cores are aligned with each other, as shown in Fig.~\\ref{fig:contours_vrt_b}(D).  \nNotably, the inherent asymmetry of the offset vortices causes the highly curved, leading core to locally advect the low-vorticity region of the trailing core around itself, as previously shown in Fig.~\\ref{fig:transv_tubes}. \nThis shedding of vorticity repeats along each anti-note peak of the perturbed cores, leading to an alternating array of perpendicular secondary filaments that traverse the plane of reflection, as shown in Fig.~\\ref{fig:contours_vrt_b}(B).\nThe alternation of pairs of secondary filaments accounts for the interdigitation of the colliding vortices visualized with dye both experimentally and numerically in the main text.\nThe counter-rotating structure of adjacent secondary filaments, as shown in Fig. 4(D) in the main text, results from two sources.\nFirst, the edges of each flattened perturbation roll up into a pair of secondary filaments that counter-rotate relative to one another. \nAdditionally, the alternating orientation of the secondary vortex pairs cause filaments formed from adjacent perturbations to counter-rotate with each other.\n\\subsection{ Transfer of circulation }\n\nThe dynamics of the thin secondary filaments, transported across the reflection plane ($z=0$), vary along the axial direction. This implies that the axial component of the circulation in the half-plane,\n\n\\begin{equation}\n\\Gamma_x(x,t) \\Big|_{z \\to -\\infty}^{z = 0} \\equiv \\int_{-\\infty}^\\infty dy \\int_{-\\infty}^0 dz \\, \\omega_x(x,y,z,t) \n\\label{eq:circ_x}\n\\end{equation} \nvaries, both as a function of time and of $x$. By integrating the vorticity distribution on the lower half plane at several times, we find that the axial component of circulation varies periodically with $x$, as shown in Fig.~\\ref{fig:circ_x}(A). This demonstrates how the formation and stretching of the secondary filaments develop a periodic transfer of vorticity from the original tubes through the reflection plane. Once the secondary filaments are fully formed after $t^* \\gtrsim 65$, however, the variations in the axial component of circulation saturate.\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{figS7.png}\n\\caption{Evolution of circulation. DNS of counter-rotating vortex tubes where Re$_{\\Gamma} = 4500$. (A) Evolution of the axial component of circulation, $\\Gamma_x(x,t)$, in the lower half plane $(y,z)$ for $z < 0$. (inset) Evolution of the axial and transverse components of circulation, $\\Gamma_x(x,t) + \\Gamma_z(x,t)$. (B) Evolution of the derivative of the transverse component of circulation, $\\partial \\Gamma_z\/\\partial x$ along the reflection plane $(z=0)$.\n}\n\\label{fig:circ_x}\n\\end{figure}\n\nConcomitantly, as the axial component of circulation is drained from the lower vortex tube, the secondary filaments traverse through the reflection plane, $z=0$. \nThe corresponding flux of vorticity in the transverse direction is quantified by computing \nthe transverse component of circulation $\\Gamma_z(x,t)$ along the reflection plane on $ [0, x]$: \n\\begin{equation}\n\\Gamma_z (x,t) \\equiv \\int_0^x dx' \\int_{-\\infty}^{+\\infty} dy' \\, \n\\omega_z(x',y',z = 0) \n\\label{eq:Gam_z}\n\\end{equation}\n\nThe derivative of $\\Gamma_z(x,t) $ with respect to $x$ reduces Eq.~\\eqref{eq:Gam_z} into the normalized sum of the transverse vorticity, $\\omega_z$, along a line located at the axial position, $x$, on the reflection plane, as shown in Fig.~\\ref{fig:circ_x}(B) at various times. \nThe sinusoidal nature of $\\partial\\Gamma_z(x,t)\/\\partial x |_{z=0}$--about a mean value of zero--along the axial direction showcases the counter-rotating structure of adjacent secondary filaments, as shown in Fig. 4(D) in the main text.\n\nAn important relation between $\\Gamma_z(x,t)$ and $\\Gamma_x(x,t)$ results from the observation that $\\nabla \\cdot \\mathbf{\\omega} = 0$; that is, the flux of vorticity a through a closed surface limiting a finite volume of fluid is conserved. \nWe define the half-plane ${\\cal P}_\\alpha$ by the conditions $x = \\alpha$ and $z \\le \\alpha$, and the band ${\\cal Q}_{\\alpha \\beta}$ for $\\alpha \\le \\beta$ by $z = 0$ and $\\alpha \\le x \\le \\beta$.\nAdditionally, we use the property that $\\nabla \\cdot \\mathbf{\\omega} = 0$ to the domain limited by ${\\cal P}_\\alpha$, ${\\cal Q}_{\\alpha \\beta}$, and ${\\cal P}_\\beta$, with $\\alpha \\le \\beta$. \nAn elementary calculation shows that the flux of $\\mathbf{\\omega}$ on this domain reduces to $ \\big(\\Gamma_x(\\beta,t) + \\Gamma_z(\\beta,t) \\big) - \\big(\\Gamma_x(\\alpha,t) + \\Gamma_z(\\alpha,t) \\big) $. \nThus, the condition that the flux of vorticity is zero imposes that $\\Gamma_x(x,t) + \\Gamma_z(x,t) $ does not dependent on $x$. \nWe find that our numerical results satisfy this conservation relation, as shown in the inset of  Fig.~\\ref{fig:circ_x}.\n\nPhysically, the relation $\\partial_x \\big( \\Gamma_x(x,t) + \\Gamma_z(x,t) \\big)=0$ imposes that the axial variations of $\\Gamma_x(x,t)$, clearly visible in Fig.~\\ref{fig:circ_x}(A), necessitate the variations of the flux $\\Gamma_z(x,t)$, which are proportional to the transverse circulation in the plane $z = 0$. \nThat is, because the circulation of the system is conserved--aside from viscous dissipation--any axial  circulation lost from the vortex tubes is redirected to the secondary filaments through the transfer of transverse circulation. \nThe fluctuations of $\\Gamma_x(x,t)$ correspond to $\\approx 0.25 \\Gamma_0$, implying that in the configuration studied here, the circulation of each transverse secondary filament is approximately $1\/4$ of the axial component of circulation in the original tubes.\n\n\\section{Interactions of secondary vortex filaments}\n\nThrough the creation of secondary filaments, the elliptical instability provides a mechanism by which smaller generations of counter-rotating vortex filaments form and interact to generate small-scale flow structures. \nDuring the evolution of the elliptical instability, the development of antisymmetric perturbations in the cores leads to the formation of an array of counter-rotating secondary filaments, as shown numerically for a typical example in Fig.~\\ref{fig:vortex_sheets}(A), Movie~\\ref{mov:tubes_Re_3500}, and Movie~\\ref{mov:sheets}, where Re$_\\Gamma = 3500$.    \nNeighboring secondary filaments interact with one another in the same manner as the initial vortex tubes. Because the secondary filaments are smaller and have a lower circulation than the original vortex tubes, they can be viewed as having a lower effective Reynolds number. \nThese adjacent secondary filaments align with one another in pairs, as shown in Fig.~\\ref{fig:vortex_sheets}(B-C). \nDue to their counter-rotation, the filaments exert large strains on each other, which causes one of the secondary filaments to flatten into a vortex sheet, as shown in Fig.~\\ref{fig:vortex_sheets}(D). Upon further stretching, this vortex sheet splits into two smaller tertiary vortex filaments, as shown in Fig.~\\ref{fig:vortex_sheets}(E-F).\nThis breakdown mechanism in which counter-rotating vortex filaments interact, flatten into vortex sheets, and split into even smaller generations of vortices has been previously observed in vortex ring collisions~\\cite{McKeown:2018} and is attributed to the late-stage development of the Crow instability~\\cite{Crow:1970}. \nThis instability dominates during the interaction of vortices at low-Reynolds numbers, like that of the secondary vortex filament pair. \n\nAs shown previously, only a fraction of the initial circulation is transferred to the secondary filaments. Even though the elliptical instability fully develops during the Re$_\\Gamma = 3500$ configuration, the effective Reynolds number of the interacting secondary filaments is not sufficient enough for the elliptical instability to develop again. Instead, the Crow instability dominates during this second iteration, and the secondary vortices flatten into vortex sheets and split into smaller tertiary filaments. \nIn order for the secondary filaments to, themselves, interact through the elliptical instability and form a tertiary generation of perpendicular filaments--as shown in Fig. 5 in the main text and in Movie~\\ref{mov:cascade}--the initial counter-rotating vortex tubes must have a higher Reynolds number.\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figS8.png}\n\\caption{Interactions of secondary vortex filaments. Vorticity modulus of a simulated vortex tube interaction where Re$_{\\Gamma} = 3500$. (A) Array of secondary filaments formed during the late-stage evolution of the elliptical instability. (B) Zoom-in view of two pairs of secondary filaments indicated by the dashed box. (C) Cross-sectional view of the secondary filaments through the plane of reflection $(z=0)$, indicated by the dashed line in (B). Neighboring filaments counter-rotate and interact with each other. (D) Interacting secondary filaments deform from the mutual strain and one of the filaments locally flattens into a vortex sheet. (E) The flattened vortex sheet splits into two smaller tertiary vortex filaments. (F) The newly-formed tertiary filaments unravel the secondary filament. For (A-F), the vorticty threshold is $0.122 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.206$, where $|\\omega|_{\\text{max}}$ is the maximum vorticity modulus for the entire simulation. Note: $\\sigma = 0.06 \\mathcal{L}$, $b = 2.5 \\sigma$, and $t^* = \\Gamma t \/ b^2$. }\n\\label{fig:vortex_sheets}\n\\end{figure}\n\n\\section{Analysis of the transfer of energy in a turbulent flow }\n\\label{sec:analysis_transfer}\n\nThis section examines the derivation and meaning of the shell-to-shell energy transfer spectrum, $T(k,t)$, introduced in the main text and plotted in Fig. 5(G).\nA typical method for characterizing a turbulent flow, which encompasses of a wide range of excited scales of motion, is to examine evolution of the the energy spectra in Fourier space. \nThis energy spectrum is designated by the term $E(k,t)$, such that $E(k,t)~dk$ is the amount of kinetic energy at time $t$, in a shell in wavenumber space between $k$ and $k + dk$. \nIn the absence of forcing, and in the simplified case of a homogeneous isotropic flow, one can derive from the Navier-Stokes equations the following energy balance~\\cite{Pope:2000,Lin:1947}:\n\n\\begin{equation}\n\\frac{\\partial E(k,t)}{\\partial t} + T(k,t) = - 2 \\nu k^2 E(k,t).\n\\label{eq:Lin}\n\\end{equation}\n\nThe terms in this equation state that for a given wavenumber, $k$, and at any time, t, the rate of change of the energy of that mode plus the rate of energy transferred to or from that wavenumber via other modes is balanced by the viscous dissipation of that mode.\nIn this equation, $T(k,t)$ originates from the nonlinear, advective term in the Navier-Stokes equations:\n\n\\begin{equation}\nT(k,t) \\, dk = \\sum_{k \\le | \\mathbf{k} | \\le k + dk } \n\\Re{ \\big( \\overline{ (\\mathbf{u} \\cdot \\nabla ) \\mathbf{u} } (\\mathbf{k} )\n\\cdot \\overline{ \\mathbf{u}} (- \\mathbf{k} ) \\bigr) }. \n\\label{eq:Pi}\n\\end{equation}\nWhile Eq.~\\eqref{eq:Lin} is often used in a context of fluid turbulence, it can be applied to any solution of the Navier-Stokes equations. \nIn our direct numerical simulations, $T(k,t)$ is calculated by applying a discrete Fourier transform to both our solved flow field, $\\mathbf{u}(\\mathbf{x},t)$ and the nonlinear term,\n$( \\mathbf{u} \\cdot \\nabla ) \\mathbf{u} (\\mathbf{x}, t)$, which are then applied to Eq.~\\eqref{eq:Pi}.\n\n\\section{Emergence of turbulence from the elliptical instability with increasing Reynolds number \\label{sec:Tubes_Across_Re}}\n\n\n\\subsection{Dissipation rate evolution and energy spectra \\label{subsec:Spectra_Across_Re}}\n\n\nDirect numerical simulations of the interacting, counter-rotating vortex tubes are performed at a range of Reynolds numbers to examine by what mechanism the onset of the elliptical instability leads to the development of turbulence. At each Reynolds number, the energy dissipation rate, $\\epsilon$, qualitatively follows the same temporal evolution, as shown in Fig.~\\ref{fig:epsilon_energy_spectra}(A).\nThe coherent vortex tubes initially interact, and the rapid increase in $\\epsilon$ is initiated by the onset of the elliptical instability at each Reynolds number, as shown in Fig. 5 in the main text. \nThe dissipation rate is maximized during the late-stage of the elliptical instability, in which the secondary filaments interact with each other and the remnants of the original primary vortex cores. As the Reynolds number is increased, the maximum dissipation rate increases. Because the viscous dissipation of energy in the flow primarily occurs on the smallest scales of the system, this behavior indicates that the high-Reynolds number configurations more effectively convey energy into small-scale flow structures.\n\nAdditionally, we examine the normalized kinetic energy spectra, $E(k)\/(\\eta^{\\frac{1}{4}} \\nu^{\\frac{5}{4}})$, when the dissipation rate is maximized for each Reynolds number, as shown in Fig.~\\ref{fig:epsilon_energy_spectra}(B). Each of the energy spectra generally follow the $\\sim(k \\eta)^{-5\/3}$ Kolmogorov scaling of turbulence, where $\\eta$ is the dissipative length scale~\\cite{Kolmogorov:1941}. \nNotably, the agreement of the simulated energy spectra with this turbulent scaling improves for simulations that are carried out at higher Reynolds numbers. This is because the inertial range of the breakdown is more developed at higher Reynolds numbers--i.e. there is a larger range of scales over which Kolmogorov's axioms for turbulence are valid~\\cite{Kolmogorov:1941}. \nThe emergence of this multi-scale turbulent behavior is encapsulated by the snapshots in Fig.~\\ref{fig:epsilon_energy_spectra}(C-F) which show the vorticity modulus of the interacting tubes at each Reynolds number when $\\epsilon$ is maximized. \n\nIn each configuration, the elliptical instability is fully developed at the peak dissipation rate, as an array of perpendicular secondary filaments bridges the gap between the original vortex tubes. \nThese stretched, counter-rotating secondary filaments interact with each other and the remnants of the original tubes through different means at each Reynolds number. For the Re$_{\\Gamma} = 2000$ configuration, the secondary filaments do not have sufficient circulation to interact with one another and break down further, as viscous dissipation sets in (see Movie~\\ref{mov:tubes_Re_2000}). When the Reynolds number is raised to $3500$, neighboring secondary filaments locally interact with one another, flatten into vortex sheets, and split into smaller tertiary filaments, as shown in Movie~\\ref{mov:tubes_Re_3500} and Movie~\\ref{mov:sheets}. The same interactions between secondary filaments occur for the Re$_{\\Gamma} = 4500$ configuration; however, the secondary filaments become more disordered as they undergo complex 3D motion and become wrapped around each other and the original tubes, breaking down into fine-scale vortex filaments (see Movie~\\ref{mov:tubes_Re_4500}). Lastly, in the Re$_{\\Gamma} = 6000$ configuration, the secondary filaments rapidly emerge, interact, and violently break down as they almost instantaneously burst into an ensemble of vortices interacting over a wide range of scales, as shown in Movie~\\ref{mov:tubes_Re_6000}. Notably, the high-circulation secondary filaments in this configuration locally interact to form new generations of perpendicular tertiary filaments, as shown in Fig. 5 in the main text and in Movie~\\ref{mov:cascade}.\nWe propose that these tertiary filaments form through another iteration of the elliptical instability.\n\nThese results demonstrate how the elliptical instability provides a mechanism by which counter-rotating vortex tubes at high Reynolds numbers interact and break down to develop a turbulent cascade. This iterative instability generates new vortices that  interact with each other and ``grind down'' into smaller and smaller vortex filaments before being dissipated through viscosity~\\cite{Taylor:1937}.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figS9.png}\n\\caption{Dissipation rate evolution and energy spectra for simulated vortex tube interactions at various Reynolds numbers. (A) Evolution of normalized kinetic energy dissipation rate. The markers indicate the maximum energy dissipation rate. (B) Normalized kinetic energy spectra at the peak dissipation rate, where the black line corresponds to Kolmogorov scaling. (C-F) Snapshots of the 3D vorticity modulus at the times corresponding to the maximum dissipation rate for each Reynolds number. The vorticity thresholds are $0.153 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.305$ for (C) and $0.061 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.183$ for (D-F).\nFor each simulation, $\\sigma = 0.06\\mathcal{L}$, $b = 2.5\\sigma$, Re$_{\\Gamma} = \\Gamma\/\\nu$, and $t^* = \\Gamma t \/ b^2$. \n}\n\\label{fig:epsilon_energy_spectra}\n\\end{figure}\n\n\\subsection{Vorticity evolution}\n\\label{subsec:Vorticity_Across_Re}\nThe evolution of the vorticity modulus in the simulated vortex tube interactions also indicates the onset of a turbulent state during the breakdown of the tubes, which is especially pronounced for high-Reynolds number configurations. \nFor each Reynolds number, the maximum vorticity modulus, $|\\omega|_{\\text{max}}$, remains initially constant until it increases during onset of the elliptical instability, as shown in Fig.~\\ref{fig:vorticity_evolution}(A). \nFor the Re$_{\\Gamma} = 2000$ configuration, the maximum vorticity modulus increases slightly during the formation and stretching of the secondary filaments; however, because the filaments do not interact due to the onset of viscous dissipation, the maximum vorticity modulus decreases. \nFor the higher-Reynolds number configurations, the maximum vorticity modulus increases precipitously due to the formation and stretching of the secondary filaments and remains sustained at a heightened level before decreasing due to viscous dissipation. \nThis heightened level of $|\\omega|_{\\text{max}}$ coincides with the maximization of the energy dissipation rate, $\\epsilon$, as shown in Fig.~\\ref{fig:epsilon_energy_spectra}(A). \nThe sustained amplification of the maximum vorticity modulus thus results from the new generation and local interactions of small-scale vortices during the turbulent breakdown. The higher the Reynolds number, the longer the maximum vorticity modulus remains at this elevated level before viscosity damps out the motion of the vortices.\n\nAdditionally, the mean vorticity modulus, $\\overline{|\\omega|}$, increases during the onset of the elliptical instability, reaches a peak value approximately when the dissipation rate, $\\epsilon$, is maximized, and decreases as viscosity damps out the small-scale motion of the flow, as shown in Fig.~\\ref{fig:vorticity_evolution}(B). \nThe increase in the mean vorticity modulus indicates how the initially localized and coherent flow becomes more distributed throughout the domain during the breakdown. \nThe amplification of the mean vorticity with increasing Reynolds number further demonstrates how the turbulent breakdown in the high-Reynolds number configurations is more three-dimensional and quasi-isotropic.   \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{figS10.png}\n\\caption{Vorticity evolution for simulated vortex tube interactions at various Reynolds numbers. (A) Maximum vorticity modulus and (B) mean vorticity modulus evolution. Both moduli are normalized by the initial mean vorticity modulus. For each simulation, $\\sigma = 0.06\\mathcal{L}$, $b = 2.5\\sigma$, Re$_{\\Gamma} = \\Gamma\/\\nu$, and $t^* = \\Gamma t \/ b^2$. \n}\n\\label{fig:vorticity_evolution}\n\\end{figure}\n\n\\section{Supplemental movie descriptions}\n\n\\begin{movie}{Head-on collision of vortex rings. \\\\\n \\normalfont{Underwater view of the head-on collision of two vortex rings dyed separately, where Re $ = UD\/\\nu = 6000$ and SR $= L\/D = 2.5$. The rings expand radially as they collide at the midplane before rapidly breaking down into a turbulent cloud of dye.}\n }\n\\label{mov:GoPro}\n\\end{movie}\n \n \\begin{movie}{Experimental vortex ring collision with dyed cores. \\\\\n \\normalfont{Head-on view (top) and side view (bottom) showing the core dynamics of two colliding vortex rings, where Re $= 7000$ and SR $= 2.0$. As the rings stretch radially during the collision, the cores develop antisymmetric, short wavelength perturbations, indicative of the elliptical instability. Once fully formed, the perturbations deflect out-of-plane and break down into a turbulent cloud of dye.}\n }\n \\label{mov:exp_cores}\n \\end{movie}\n \n\n \\begin{movie}{DNS of dyed vortex ring collision. \\\\\n \\normalfont{Overhead view (top) and side view (bottom) showing the simulated collision of two vortex rings dyed red and blue, respectively (Re$_{\\Gamma} = \\Gamma\/\\nu = 4500$ and $t^{*} = \\Gamma t \/ R_0^2$). The dye in the cores of the rings (dark) is differentiated from the dye surrounding the cores (light). As the rings collide, they stretch radially and develop antisymmetric, short-wavelength perturbations, indicative of the elliptical instability. As a result of these perturbations, the vortex rings interdigitate, forming alternating pairs of secondary vortex filaments, perpendicular to the cores. The rings then rapidly break down into a turbulent cloud of dye.}\n }\n \\label{mov:sim_rings_dye}\n \\end{movie}\n \n\n \\begin{movie}{Experimental fully dyed vortex ring collision. \\\\\n\\normalfont{Overhead view (top) and side view (bottom) showing the collision of two vortex rings, where Re $= 6000$ and SR $= 2.5$. As the vortex rings collide, they develop alternating ``tongues'' that interdigitate around one another. The edges of these tongues roll up into an ordered array of secondary vortex filaments, perpendicular to the vortex cores. These secondary filaments interact and rapidly break down into a turbulent cloud of dye.}\n}\n\\label{mov:exp_rings_dye}\n\\end{movie}\n\n\\begin{movie}{DNS of vortex tube interaction: Re$_{\\Gamma} = 4500 $.   \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 4500$ and $t^{*} = \\Gamma t \/ b^2$. As a result of the elliptical instability, the cores develop antisymmetric perturbations, and an array of counter-rotating secondary vortex filaments forms perpendicular to the cores. The secondary filaments interact with each other and the remains of the cores before breaking down into a ``soup'' of small-scale vortices that are dissipated by viscosity. The vorticity threshold is $0.076 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.198$, and the tubes propagate in the $-y$ direction. }\n}\n\\label{mov:tubes_Re_4500}\n\\end{movie}\n\n\\begin{movie}{DNS of vortex tube interaction: Re$_{\\Gamma} = 3500 $. \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 3500$ and $t^{*} = \\Gamma t \/ b^2$. As a result of the elliptical instability, the cores develop antisymmetric perturbations, and an array of counter-rotating secondary vortex filaments forms perpendicular to the cores. The secondary filaments interact with each other and the remains of the cores before breaking down into a ``soup'' of small-scale vortices that are dissipated by viscosity. The vorticity threshold is $0.122 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.275$, and the tubes propagate in the $-y$ direction. }\n}\n\\label{mov:tubes_Re_3500}\n\\end{movie}\n\n\\begin{movie}{Interaction and splitting of secondary vortex filaments. \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 3500$ and  $t^{*} = \\Gamma t \/ b^2$. Neighboring secondary filaments counter-rotate and interact with one another. This close-range interaction causes one of the filaments to become flattened into a vortex sheet before splitting into smaller tertiary vortex filaments. The vorticity threshold is $0.122 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.206$. }\n}\n\\label{mov:sheets}\n\\end{movie}\n\n\\begin{movie}{DNS of vortex tube interaction: Re$_{\\Gamma} = 6000$. \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 6000$ and $t^{*} = \\Gamma t \/ b^2$. As a result of the elliptical instability, the cores develop antisymmetric perturbations, and an array of counter-rotating secondary vortex filaments forms perpendicular to the cores. The secondary filaments and remaining cores interact violently and rapidly ``burst'' into a turbulent flow of vortices interacting over many scales. Viscosity damps out the motion of the remaining vortices. The vorticity threshold is $0.077 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.153$, and the tubes propagate in the $-y$ direction.}\n}\n\\label{mov:tubes_Re_6000}\n\\end{movie}\n\n\\begin{movie}{Iterative cascade of elliptical instabilities. \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 6000$ and $t^{*} = \\Gamma t \/ b^2$. Neighboring secondary filaments violently interact and form another generation of perpendicular, tertiary filaments through the elliptical instability. The vorticity threshold is $0.092 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.214$.}\n}\n\\label{mov:cascade}\n\\end{movie}\n\n\\begin{movie}{Gaussian fit to vortex core PIV data. \\\\\n\\normalfont{2D PIV measurement of the vorticity distribution of a formed vortex ring, where Re $= 7000$ and SR $= 2.5$. The raw PIV vorticity data is plotted in the top panel and the Gaussian fit of the top and bottom cores is plotted in the bottom panel.}\n}\n\\label{mov:PIV}\n\\end{movie}\n\n\\begin{movie}{DNS of vortex tube interaction: Re $_{\\Gamma} = 2000$. \\\\\n\\normalfont{Vorticity modulus for the simulated interaction of two antiparallel vortex tubes, where Re$_{\\Gamma} = 2000$ and $t^{*} = \\Gamma t \/ b^2$. As a result of the elliptical instability, the cores develop antisymmetric perturbations, and an array of counter-rotating secondary vortex filaments forms perpendicular to the cores. The secondary filaments have little circulation and quickly dissipate due to viscosity. The vorticity threshold is $0.229 \\leq |\\omega|\/|\\omega|_{\\text{max}} \\leq 0.458$, and the tubes propagate in the $-y$ direction.}\n}\n\\label{mov:tubes_Re_2000}\n\\end{movie}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{i. introduction}\n\nFor the past three decades, significant efforts were applied to study thermal and thermoelectric transport in mesoscopic and nanoscale systems  of various kinds including quantum dots and\/or molecules  attached to conducting electrodes. The latter serve as source and drain reservoirs for traveling charge carriers. Here, we concentrate on the analysis of thermoelectric properties of these systems. Below they are referred to as thermoelectric junctions.\nIn part, the research interest appeared because thermoelectric junctions are expected to be useful in building up highly efficient energy conversion devices.  Also, studies of thermoelectric properties of these systems can result in a deeper insight into the nature of general transport mechanisms and bring additional information on the electronic and vibrational excitation spectra of molecules \\cite{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}.  Presently, it is established that thermoelectric properties of quantum dots and\/or molecules may be strongly affected by Coulomb interactions between charge carriers \\cite{16,17,18,19,20,21,22}. Coulomb interactions lead to violation of the Wiedemann-Franz law in nanoscale thermoelectric junctions thus providing an enhancement of thermoelectric efficiency of these systems \\cite{21,23}. The thermal efficiency may be further increased due to the influence of quantum interference effects which may strongly affect electron transport characteristics  \\cite{9,24,25,26,27,28,29}.\n\nIn  general studies of thermoelectric transport through molecules, quantum dots and similar systems  one must imply that both atomic vibrations and charge carriers contribute to the energy transfer. Therefore, a unified description of the electrons and phonons dynamics is needed to thoroughly analyze thermoelectric properties of molecular junctions. The means for such analysis are provided by the nonequilibrium Green's functions formalism (NEGF), as described in the review \\cite{30}, and some other works (see e.g. \\cite{13,31,32,33,34,35}). However, application of this formalism to realistic models simulating thermoelectric junctions is extremely difficult. Several simplified approaches were developed and used to analyze the specifics of heat transfer and other related phenomena in quantum dots and molecules taking into account  the contribution of phonons and electron-phonon interactions \\cite{2,13,36,37,38}. Nevertheless, these studies are not completed so far. \n\nThe phonons contributing to the charge and energy transfer may be subdivided in two classes: vibrational phonons associated with molecular vibrations and thermal phonons associated with random nuclear motions in the environment. In the present work we aim at theoretical analysis of the effect  of thermal phonons on the thermoelectric characteristics of molecules and other similar systems. To carry on this analysis we combine NEGF with  the approach first suggested by Buttiker to describe quantum transport through molecules \\cite{39}. An important advantage of this approach is that it could be easily adapted to analyze  various aspects of incoherent\/inelastic transport through molecules (and some other mesoscopic systems) avoiding complicated and time-consuming methods based on more advanced formalisms.\n\n\\section{ii. Main equations}\n\nFor simplicity, in the following computations we simulate a molecule\/quantum dot by a single level with the energy $ E_0. $ We assume that this single-level bridge is coupled to the pair of dissipative reservoirs, as shown in the Fig. 1.  While on the bridge, an electron could be scattered into one of the reservoirs through the channels 3 and 4 (or 5 and 6) with a certain probability. In the reservoir, it undergoes inelastic scattering accompanied by phase breaking, and afterwards returns to the bridge with the same probability. In the present analysis, the reservoirs are treated as phonon baths representing thermal phonons associated with the left and right electrodes. Within the accepted model we imply that there is no phonon thermal conductance through the junction. This seems a reasonable assumption for experiments give low values of phonon thermal conductance in several thermoelectric molecular junctions \\cite{40,41}. This may be attributed to the fact that in many molecules the majority of vibrational transitions lie above the range determined by thermal energy when temperature takes on values of the order of or lower than the room temperature \\cite{14,29}.\nWithin the Buttiker model, the electron transport through a thermoelectric junction is considered as combination of tunnelings through potential barriers separating the electrodes from  the molecule\/quantum dot and interaction with the reservoirs coupled to the bridge site.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=5.7cm,height=8.8cm,angle=-90]{n24_1.eps}\n\\caption{(Color online) Schematics of the considered system. Semicircles  represent the left and right electrodes, square stands for the molecule\/quantum dot sandwiched in between.  Dephasing\/dissipative reservoirs are associated with the electrodes and characterized by temperatures $ T_L $ and $ T_R, $ respectively.\n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\nThe Buttiker approach was applied to describe and analyze electron transport through molecules in several works (see e.g. Refs. \\cite{42,43,44}. Following this approach, one can present particle fluxes $ J_i' $  outgoing from the system as linear combinations of incoming fluxes $ J_k $ where the indexes $ i,k $ label the channels for transport. For the adopted model $ 1 \\leq i \\leq k \\leq 6. $\n\\be \nJ_i' = \\sum_k T_{ik} J_k  \\label{1}\n\\ee\nIn these equations, the coefficients $ T_{ik} $ are related to  matrix elements of the scattering matrix $ M $ namely: $ T_{ik} = |M_{ik}|^2. $ The scattering matrix expresses outgoing wave amplitudes $ b_L',b_R',a_3',a_4',a_5',a_6' $ in terms of incident ones $ b_L,b_R,a_3,a_4,a_5,a_6. $ To provide the charge conservation in the system, zero net current should flow in the channels linking the  bridge site with the dephasing reservoirs, so we may write the following equations:\n\\begin{align}\nJ_3 + J_4 - J_3' - J_4' = 0 ,\\nn\n\\\\ \nJ_5 + J_6 - J_5' - J_6' = 0 . \\label{2}\n\\end{align} \nTo find the expressions for the matrix elements $ M_{ik} $ we first consider a subsystem including the left electrode with the associated dephasing reservoir and the bridge site. The expression for the matrix $s^{(1)} $ relating the wave amplitudes $a_1', a_2', a_3', a_4' $ to the wave amplitudes $a_1, a_2, a_3, a_4 $ has the form \\cite{39}:\n\\be \ns^{(1)} = \\left( \\ba{cccc}\n0  & \\sqrt{1 - \\epsilon_L} & \\sqrt {\\epsilon_L} &  0\n\\\\\n \\sqrt{1 - \\epsilon_L} & 0  & 0 & \\sqrt {\\epsilon_L}\n\\\\\n \\sqrt{ \\epsilon_L} & 0  & 0 & - \\sqrt {1-\\epsilon_L}\n\\\\\n0  & \\sqrt{\\epsilon_L} & -\\sqrt {1-\\epsilon_L} &  0 \n\\\\\n\\ea  \\right) \\label{3}\n\\ee\nwhere the phenomenological scattering probability $ \\epsilon_L $ corresponds to the reservoir associated with the left electrode. Also, an electron tunneling through a single potential barrier separating  this electrode from the molecule\/quantum dot is characterized by the transmission and reflection amplitudes $(t_L$ and $r_L,$ respectively). These are the matrix elements of a $ 2\\times 2 $ matrix:\n\\be\ns_L = \\left( \\ba{cc}\nt_L & r_L \\\\  r_L & t_L \\ea \\right)  . \\label{4}\n\\ee\nCombining Eqs. (\\ref{3}) and (\\ref{4}) one obtains the expression for the scattering matrix $ M^{(1)} $ relating $b_L', a_2', a_3', a_4' $ to  $b_L, a_2, a_3, a_4 $:\n\\be\nM^{(1)} = \\left(\\ba{cccc}\nr_L & \\alpha_Lt_L & \\beta_Lt_L & 0 \n\\\\\n\\alpha_Lt_L & \\alpha_L^2r_L & \\alpha_L\\beta_Lr_L & \\beta_L\n\\\\\\beta_Lt_L & \\alpha_L\\beta_Lr_L & \\beta_L^2r_L & -\\alpha_L\n\\\\\n0 & \\beta_L & -\\alpha_L & 0  \\ea \\right).  \\label{5}\n\\ee\nHere, $ \\alpha_L = \\sqrt{1 - \\epsilon_L},\\ \\beta_L = \\sqrt{\\epsilon_L}. $\n\n\nNow, we take into consideration the remaining elements of the original system. The matrix $ M^{(2)} $ which relates $a_2', b_R', a_5', a_6' $ to  $a_2, b_R, a_5, a_6 $ is \\cite{43,44}:\n\\be\nM^{(2)} = \\left( \\ba{cccc}\n\\alpha_R^2 r_R & \\alpha_R t_R & \\beta_R & \\alpha_R \\beta_R r_R \n\\\\\n\\alpha_R t_R & r_R & 0 & \\beta_R t_R\n\\\\\n\\beta_R & 0 & 0 & -\\alpha_R\n\\\\\n\\alpha_R \\beta_R r_R & \\beta_R & - \\alpha_R & \\beta_R^2 r_R\n\\ea \\right) \\label{6}\n\\ee\nwhere $ \\alpha_R = \\sqrt{1 - \\epsilon_R},\\ \\beta_R = \\sqrt{\\epsilon_R}, $ the scattering probability $ \\epsilon_R $ is associated with the right reservoir, and the transmission $(t_R) $ and reflection $(r_R) $ amplitudes characterize electron tunneling through the potential  barrier between the molecule (bridge) and the right electrode. Using Eqs. (\\ref{5}) and (\\ref{6}) and excluding the wave amplitudes $a_2,a_2' $ which correspond to the transport inside the system, we get the following expression for the scattering matrix:\n\n\\begin{widetext}\n\\be \nM = \\frac{1}{Z}  \\left \\{\\ba{cccccc}\nr_L + \\alpha_L^2\\alpha_R^2r_R  &\\alpha_L \\alpha_R t_L t_R   &\n\\beta_L t_L    &  \\alpha_L\\alpha_R^2 \\beta_L t_L r_R   &\n\\alpha_L  \\beta_R t_L   &  \\alpha_L \\alpha_R \\beta_R t_L r_R  \n\\\\ \n\\alpha_L \\alpha_R t_L t_R   &  r_R + \\alpha_L^2 \\alpha_R^2 r_L   &\n\\alpha_L \\alpha_R \\beta_L t_R r_L   &  \\alpha_R \\beta_L t_R &  \n\\alpha_L^2 \\alpha_R \\beta_R t_R r_L  &  \\beta_R t_R\n\\\\ \n\\beta_L t_L  & \\alpha_L \\alpha_R \\beta_L t_R r_L  &\n\\beta_L^2 r_R  & \\alpha_L(\\alpha_R^2 r_L r_R -1) &\n\\alpha_L \\beta_L \\beta_R r_L  &  \\alpha_L \\alpha_R \\beta_L \\beta_R r_L  r_R\n \\\\ \n\\alpha_L\\alpha_R^2 \\beta_L t_L r_R  & \\alpha_R \\beta_L t_R &\n\\alpha_L(\\alpha_R^2 r_L r_{R}-1) & \\alpha_R^2 \\beta_L^2 r_R &   \\beta_L \\beta_R   &  \\alpha_R \\beta_L \\beta_R r_R \n \\\\\n\\alpha_L \\beta_R t_L  & \\alpha_L^2 \\alpha_R \\beta_R t_R r_L &\n\\alpha_L \\beta_L \\beta_R r_L  & \\beta_L \\beta_R &\n\\alpha_L^2 \\beta_R^2 r_L  & \\alpha_R (\\alpha_L^2 r_L r_R -1) \n\\\\\n\\alpha_L \\alpha_R \\beta_R r_R t_L & \\beta_R t_R &\n\\alpha_L\\alpha_R \\beta_L \\beta_R r_L r_R & \\alpha_R \\beta_L \\beta_R r_R & \\alpha_R(\\alpha_L^2 r_L r_R-1) & \\beta_R^2 r_R \\\\\n\\ea \\right \\} .  \\label{7}\n\\ee\n\\end{widetext}\nHere, $ Z = 1 - \\alpha_L^2 \\alpha_R^2 r_L r_R. $\n\nSolving the system of linear equations (\\ref{1}), (\\ref{2}) one obtains the following expression for the electron transmission $ T(E) $ which coincides with the corresponding result reported by D'Amato and Pastawski \\cite{42}:\n\\be\nT(E) = \\frac{J_2'}{J_1} = T_{21} + \\sum_{i,j} K_i^{(2)} (W^{-1})_{ij} K_j^{(1)} .\n \\label{8}  \\ee\nWithin the considered model, $ 1 \\leq i,j \\leq 2, $\n\\begin{align}\nK_i^{(1)} = T_{2i+1,1} + T_{2i+2,1},\n\\nn \\\\ \nK_i^{(2)} = T_{2,2i+1} + T_{2,2i+2}, \\label{9}\n\\end{align}\nand $W^{-1} $ is the matrix inversed with respect to $ 2\\times 2 $ matrix $ W , $ whose matrix elements are given by:\n\\be\nW_{ij} = (2 - R_{ii}) \\delta_{ij} - \\tilde R_{ij} (1 - \\delta_{ij}). \\label{10}\n\\ee\nIn this expression, the following denotations are used:\n\\begin{align} &\nR_{ii}= T_{2i+1, 2i+1} + T_{2i+2, 2i+2} + T_{2i+2, 2i+1} + T_{2i+1, 2i+2},\n\\nn\\\\ &\n\\tilde R_{ij} = T_{2i+1, 2j+1} + T_{2i+1, 2j+2} + T_{2i+2, 2j+1} + T_{2i+2, 2j+2}.           \\label{11}\n\\end{align}\n\nAssuming that both dephasing reservoirs are detached from the  bridge $(\\epsilon_L = \\epsilon_R = 0), $ the transport through the system becomes coherent and elastic. In this case, the electron transmission given by Eqs. (\\ref{8})-(\\ref{11}) is reduced to a simple form: \n\\be\nT(E) = \\frac{t_L^2t_R^2}{(1 + r_Lr_R)^2}. \\label{12}\n\\ee \nAs known, the expression for the electron transmission in the case of coherent transport may be presented as follows:\n\\be\nT(E) \\equiv  g^2(E) = Trace \\big[(\\Gamma_{L\\sigma}(E) G^r_\\sigma(E) \\Gamma_{R\\sigma}(E) G^a_\\sigma (E)\\big]                         \\label{13}\n\\ee\nwhere $ G^{r,a}_\\sigma (E)$  are the retarded and advanced Green's functions associated with the molecule\/quantum dot bridging the electrodes, and self-energy terms $\\Gamma_{L,R;\\sigma} $ describe the coupling between the electron of a certain spin orientation on the bridge and the corresponding electrode. For a symmetrically coupled system $(\\Gamma_{L\\sigma} = \\Gamma_{R\\sigma} = \\Gamma),$ the expression for  electron transmission may be reduced to the form: \n\\be\n T(E) = \\frac{i}{2}\\Gamma(E) \\sum_\\sigma\\big[G_\\sigma^r(E) - G_\\sigma^a(E) \\big].  \\label{14}\n\\ee\nProvided that electron transport through the system is undisturbed  by electron-phonon interactions, and disregarding spin-flip processes, the retarded Green's function $ G_\\sigma^r(E) $ may be approximated as  \\cite{45}:\n\\be\nG_\\sigma^r(E) = \\frac{E - E_0 - \\Sigma_2^\\sigma - U\\big(1 - \\big<n_{-\\sigma}\\big>\\big)}{\\big(E - E_0 - \\Sigma_{0\\sigma}\\big)\\big(E - E_0 - U - \\Sigma_2^\\sigma\\big) + U\\Sigma_{1\\sigma}}. \\label{15}\n\\ee \nHere,  $ U $ is the charging energy associated with Coulomb repulsion between the electrons on the molecular bridge\/quantum dot and $ \\big<n_{\\pm\\sigma}\\big> $ are one-particle occupation numbers:\n\\be\n \\big<n_{\\sigma}\\big> = \\frac{1}{2\\pi} \\int dE \\mbox{Im} \\big[G_\\sigma^<(E) \\big] \\label{16}\n\\ee\nwhere $ G_\\sigma^<(E) $ is the lesser Green's function for electrons on the bridge. \n  Self-energy corrections $ \\Sigma_{0\\sigma},\\ \\Sigma_{1\\sigma},\\ \\Sigma_{2\\sigma} $ appear in the expression for $ G_\\sigma^r $ due to the coupling of the bridge to the electrodes. For example:\n\\be\n\\Sigma_{0\\sigma}   = \\sum_{r\\beta} \\frac{|\\tau_{r\\beta\\sigma}|^2}{E - \\epsilon_{r\\beta\\sigma} + i\\eta} \\equiv \\Sigma_{0\\sigma}^L +  \\Sigma_{0\\sigma}^R.                     \\label{17}\n\\ee\nIn this expression, $ \\epsilon_{r\\beta\\sigma} $ are  single-electron energies on the electrode $ \\beta\\ (\\beta \\in L,R), \\ \\tau_{r\\beta\\sigma} $ are coupling parameters characterizing the coupling of the electron states on the bridge to the electrodes and $ \\eta $ is an infinitesimal positive parameter. These self-energy terms are closely related to the previously introduced  coupling strengths $ \\Gamma_{\\beta\\sigma}, $ namely: $ \\Gamma_{\\beta\\sigma} (E) = - 2\\mbox{Im} \\Sigma_{0\\sigma}^\\beta. $ The expressions (\\ref{14})-(\\ref{16}) were repeatedly employed in studies of thermal transport through quantum dots (see e.g Ref. \\cite{26,46}).\n\nFor  a symmetrically coupled system, one may assume that the potential barriers separating the electrodes from the bridge are identical:$ t_L = t_R = t,\\ r_L = r_R = r. $ Then the transmission amplitude could be easily expressed in terms of the corresponding Green's functions:\n\\be\nt^2 = \\frac{2g}{1 + g}. \\label{18}\n\\ee\n\nWithin the Buttiker approach, the scattering probabilities $ \\epsilon_{L,R} $ are introduced as phenomenological parameters. However, these parameters may be given an explicit physical meaning by expressing them in terms of the relevant energies. In the considered\n system, dissipation and loss of coherency appear due to the interaction of charge carriers with thermal phonons associated with the electrodes and represented by the dephasing reservoirs. Therefore, as was suggested in an earlier work \\cite{47}, one can approximate these parameters as follows: \n\\be\n\\epsilon_\\beta = \\frac{\\Gamma_{ph}^\\beta}{2(\\Gamma_L + \\Gamma_R) + \\Gamma_{ph}^\\beta} .  \\label{19}\n\\ee\nHere, $ \\Gamma_{ph}^\\beta$ represents the self-energy term originating from electron-phonon interactions occurring in the reservoir associated with the left\/right electrode. Using NEGF and computing the relevant electron and phonon Green's functions within the self-consistent Born approximation, one can arrive at a relatively simple expression for $ \\Gamma_{ph}^\\beta $ \\cite{30}:\n\\begin{align}\n\\Gamma_{ph}^\\beta (E) = &\\, 2\\pi \\lambda^2_\\beta \\int_0^\\infty d\\omega \\rho_{ph}^\\beta (\\omega)\n\\nn\\\\  & \\times \\big\\{\nN(\\omega) \\big[\\rho_{el}(E - \\hbar\\omega) + \\rho_{el}(E + \\hbar\\omega) \\big]  \n\\nn\\\\ & +\n\\big[1 - n(E - \\hbar\\omega)\\big] \\rho_{el}(E - \\hbar\\omega)\n\\nn \\\\ & +\nn(E + \\hbar\\omega) \\rho_{el}(E + \\hbar\\omega) \\big\\}. \\label{20}\n \\end{align}\nIn this expression, $ \\rho_{el}(E) $ and $ n(E) $ are respectively the electron density of states associated with the bridge level and its steady state occupation, and $ \\rho_{ph}^\\beta(\\omega) $ is the phonon spectral function  for the corresponding reservoir. We assume that the electrodes may be kept at different temperatures $T_\\beta, $ so we introduce phonon distribution functions $ N_\\beta (\\omega) = \\big\\{\\exp\\big[\\hbar\\omega\/k_BT_\\beta\\big] - 1\\big\\}^{-1} $ where $ k_B $ is the Boltzmann constant.\nFinally, the constant $ \\lambda_\\beta $ characterizes the coupling strength for electron interactions with the thermal phonons belonging to the bath $\\beta. $ \n\nThe particular form of the phonon spectral functions $ \\rho_{ph}^\\beta (\\omega) $ may be found basing on the molecular dynamic simulations. However, to qualitatively analyze the effect of dephasing on the thermoelectric transport, one may employ the approximation \\cite{48}:\n\\be\n\\rho_{ph}^\\beta(\\omega) = \\rho_{0\\beta}\\left(\\frac{\\omega}{\\omega_{c\\beta}}\\right) \\exp\\left[-\\frac{\\omega}{\\omega_{c\\beta}}\\right] \\label{21}\n\\ee \nwhere the parameter $ \\rho_{0\\beta} $ is related to the electron-phonon coupling strength, and $ \\omega_{c\\beta} $ characterizes the relaxation time for the thermal phonons.\n\nThe electron density of states includes self-energy corrections which appear due to electron-phonon interactions. Therefore, Eq. (\\ref{20}) is an integral equation for $\\Gamma_{ph}^\\beta. $ Substituting  the approximation (\\ref{17}) into this equation, one may see that the major contribution to the integral over $ \\omega $ originates from the region where $ \\omega \\ll \\omega_c. $ Omitting the terms $ \\hbar\\omega $ in the arguments of all slowly varying terms in the integrand, we may reduce Eq. (\\ref{20}) to the form:\n\\be\n\\Gamma_{ph}^\\beta = \\rho_{el} \\big(E,\\Gamma_L,\\Gamma_R,\\Gamma_{ph}^L,\\Gamma_{ph}^R\\big) \\cdot Q(\\lambda_\\beta,\\omega_{c\\beta},T_\\beta)  \\label{22}\n\\ee\nwhere  \n\\be\nQ(\\lambda_\\beta,\\omega_c,T_\\beta ) = \\frac{4\\pi\\lambda_\\beta}{\\hbar\\omega_c}(k_B T_\\beta)^2 \\zeta \\left(2; 1 +\\frac{ k_B T_\\beta}{\\hbar\\omega_{c\\beta}} \\right)  \\label{23}\n\\ee\nand $ \\zeta(x;q) $ is the Riemann's $\\zeta $ function.\n\nThe suggested approach gives means to theoretically analyze the effect of thermal phonons  on the thermoelectric properties of thermoelectric junctions. Using the obtained results given by Eqs. (\\ref{7})-(\\ref{21}), one may compute electron transmission implying that the difference in the  temperatures $ T_L $ and $ T_R $ can take on an arbitrary value. Therefore, these results may be employed to study thermoelectric properties of the  considered systems beyond the linear regime. As known, nonlinear thermoelectric properties of molecular junctions and similar systems presently attract significant interest \\cite{33,37,46,49,50}.      \n  However, in studies of thermoelectric characteristics of such systems beyond linear regime, one inevitably encounters a nontrivial task of introducing  and defining the local temperature for the bridge which differs from temperatures $ T_{L,R} $ associated with the electrodes. The definition of local temperature and related problems are thoroughly discussed in the recent review \\cite{49}.\n\nIn the present work we avoid these difficulties by restricting further analysis with the linear temperature and bias regime. Also, we remark again that within the considered model the thermal conductivity associated with phonons is omitted for we do not include into consideration vibrational modes coupled to the bridge. Therefore, we may employ the following commonly used expressions for measurable thermoelectric characteristics:\n\\begin{align}  \nS &  = - \\frac{1}{eT} \\frac{L_1}{L_0},  \\label{24}\n\\\\ \nZT & = \\frac{S^2GT}{\\kappa} = \\frac{L_1^2}{L_0L_2 - L_1^2}.  \\label{25}\n\\end{align}\nHere, $ G $ and $ \\kappa $ are electron electrical and thermal conductances, $ S $ is the thermopower (Seebeck coefficient) and $ ZT $ is the dimensionless  thermoelectric figure of merit characterizing the efficiency of charge-driven cooling devices and\/or heat-driven current generators. In deriving these expressions, it is assumed that $ T_R = T $ and $ T_L = T + \\Delta T\\ (\\Delta T \\ll T). $ The integrals $ L_n $ included in Eqs. (\\ref{24}), (\\ref{25}) are given by:\n\\be\nL_n = \\int(E - \\mu)^n T(E) \\frac{\\partial f}{\\partial E} dE \\label{26}\n\\ee\nwhere $ f $ is the Fermi distribution function for the energy $ E, $ and the chemical potential $ \\mu $ characterizes the electrodes at zero bias.  Coulomb interactions between electrons on the molecule\/quantum dot may be accounted for by using appropriate expressions for the electron Green's functions incorporated into the expression for the coherent transmission (\\ref{13}) as well as into the expression for the electron density of states$ \\rho_{el}.$ \n\n\\section{iii. Results and discussion}\n\nSpecific thermoelectric properties of the considered systems depend on the relation of four relevant energies. These are the strength of coupling of the bridge to the electrodes $ \\Gamma, $ the electron-phonon coupling strength $ \\lambda, $ the charging energy $ U $ characterizing  Coulomb interactions of electrons on the bridge, and the thermal energy $ k_B T. $ It was established that the greater values of $ ZT $ could be achieved in weakly coupled systems where the condition $ \\Gamma \\ll k_BT $ may be satisfied at reasonably low temperatures (see e.g. Ref. \\cite{21}), so in further analysis we assume that considered system complies with this condition.\n\nAlso,  we assume that the considered quantum dot\/molecule is symmetrically coupled to the electrodes $ (\\Gamma_L = \\Gamma_R = \\Gamma) $ and two thermal baths are identical $(\\omega_{cL} = \\omega_{cR} = \\omega_c,\\ \\lambda_L = \\lambda_R = \\lambda). $ Omitting for a while Coulomb interactions, one may derive a simple Lorentzian  expression for the electron density of states:\n\\be\n\\rho_{el} = \\frac{1}{2\\pi} \\frac{\\Gamma}{(E - E_0)^2 + (\\Gamma + \\Gamma_{ph})^2} . \\label{27}\n\\ee\nwhere $ \\Gamma_{ph} = \\Gamma_{ph}^L + \\Gamma_{ph}^R. $\nSubstituting this expression into  Eq. (\\ref{22}), we may solve this equation and arrive at a reasonable asymptotic expression for $ \\Gamma_{ph}: $\n\\be\n\\Gamma_{ph} = \\frac{\\Gamma\\delta^2(1 + \\sqrt{1 + \\delta^2})}{(E - E_0)^2 +  (1 + \\sqrt{1 + \\delta^2})^2 }   \\label{28}\n\\ee\nwhere $ \\delta^2 = 2Q (\\lambda,\\omega_c,T)\/\\Gamma. $ Using this result and the expression (\\ref{19}) for the scattering probabilities, we obtain:\n\\be\n\\epsilon_L = \\epsilon_R =\n\\epsilon = \\frac{1}{2} \\frac{\\delta^2(1 + \\sqrt{1 + \\delta^2})}{\\ds \\left(\\frac{E - E_0}{\\Gamma}\\right)^2 + \\frac{1}{2}\\big(1 + \\sqrt{1 + \\delta^2}\\big)^3}. \\label{29}\n\\ee\n\nThe  parameter  $\\epsilon $ values vary between 0 and 1. When $ \\epsilon = 0, $ the bridge is detached from the reservoirs, and the electron transport is completely coherent and elastic. Within the opposite limit $ (\\epsilon = 1) $ the transport is characterized by the overall phase randomization typical for inelastic sequential hopping. Within the adopted approach, the scattering probabilities depend on tunnel energy $ E. $ As well as electron transmission function $ T(E),$ they reach their maximum values $( \\epsilon_{max} $ and $ T_{max} ,$ respectively) at $ E = E_0. $ This is shown in the left panels of the Fig. 2.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.8cm,height=4.5cm]{tr2.eps}\n\\includegraphics[width=8.8cm,height=4.5cm]{tr1.eps}\n\\caption{(Color online) Left panels: Scattering probability (top) and electron transmission (bottom) as functions of tunnel energy $ E. $ Right panels: Temperature dependencies of peak values of $ \\epsilon $ (top) and the electron transmission (bottom). All curves are plotted for a symmetrically coupled system $(\\Gamma_L = \\Gamma_R = \\Gamma) $ with identical dephasing reservoirs assuming that $ T_L = T_R = T,\\ \\Gamma = 1 meV,\\ E_0 = 0, $  $ \\lambda = 1.5 meV $ (dotted line), $ \\lambda = 3meV $ (dash-dotted line), $ \\lambda = 6meV $ (dashed line), $ \\lambda = 9meV $ (solid line).\nIn the left panels $ k_BT = 2.6meV.  $ \n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\nAs follows from Eq. (\\ref{29}), the character of the electron transport is determined by the value of the dimensionless parameter $ \\delta. $ Transport remains nearly coherent when $ \\delta \\ll 1. $ On the contrary, the strong dephasing\/dissipation occurs when $ \\delta $ takes on values significantly greater than 1. To find a suitable estimate for $ \\delta, $ one needs to approximate the Riemann's $\\zeta $ function included into expression (\\ref{23}). The approximation depends of the relation between the energies $ \\hbar\\omega_c $ and $ k_B T. $ As discussed in an earlier work \\cite{43}, the effect of the thermal bath on the electron transport is significantly more pronounced when the lifetime of thermal excitations is sufficiently long $(\\hbar\\omega_c \\ll k_B T),$ Under this condition, one may apply the estimation $ Q \\approx 4k_B T\\lambda. $ Correspondingly, $ \\delta^2 \\approx 4k_BT\\lambda\/\\Gamma^2. $ This shows that the maximum value of the scattering probabilities $ \\epsilon_{max} $  is determined with two parameters, namely, $T $ and $ \\lambda. $ \nWe remark that in the absence of electron-phonon interactions $(\\lambda = 0),\\ \\epsilon \\equiv \\epsilon_m = 0 $ regardless of the energy $ E $ value, and $ T_{max} = 1.$ In general, \n $ \\epsilon_{max} $ increases when the temperature rises, and it takes on greater values when the electron-phonon interactions are getting stronger as illustrated in the Fig. 2. The enhancement of $\\epsilon_{max} $ is accompanied by the decrease of maximum value of electron transmission $T_{max}. $ These results have an obvious physical sense because in the considered situation the phase randomization is inseparable from inelastic scattering of electrons by thermal phonons hindering  electron transport through the system.\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.8cm,height=4.5cm]{block30_31.eps}\n\\includegraphics[width=8.8cm,height=4.5cm]{block29_27.eps}\n\\caption{(Color online) Thermopower and figure of merit as functions of the bridge  level position  (left panels) and of the parameter $ \\Gamma $ characterizing the coupling between the bridge and the electrodes (right panels). The curves are plotted at $ k_BT = 2.6 meV, $   $ \\lambda = 0 $ (dash-dotted lines), $ \\lambda = 0.25 meV $ (dashed lines), and $ \\lambda = 1meV $ (solid lines) assuming $ \\Gamma = 1.25 meV $ (left panels) and $ E_0 = 10 meV $ (right panels).\n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\nIt was first shown by Sofo and Mahan \\cite{51} and then confirmed in several later works (see e.g. Ref. \\cite{21}) \n  that the figure of merit diverges when $ \\Gamma $ approaches zero provided that the system is characterized by zero phonon contribution to the thermal conductance, and the effects of Coulomb interactions between electrons on the bridge are disregarded. The results obtained in the present work agree with this conclusion. In the right bottom panel of the Fig. 3, the divergence of $ ZT $ within the limit $ \\Gamma \\to 0 $ in the absence of the electron-phonon interactions is clearly illustrated. The junction figure of merit is limited due to the effect of thermal phonons  associated with the electrodes. The stronger these interactions are, the lower in magnitude maximum values of both $ ZT $ and thermopower become. Comparing the present results with those reported in Ref. \\cite{21} one may presume that the thermal phonons take the part of phonon thermal conductance (which equals zero for the considered system) in limiting the maximum value of $ ZT $ and removing the divergence. Unlike $ ZT ,$ the thermopower remains finite at small values of $ \\Gamma $ even when the electron-phonon interactions are disregarded, as illustrated in the Fig. 3 (see right top panel). This means that the divergence of $ ZT $ originates from extremely strong violation of the Wiedemann-Franz law resulting in the divergence of the Lorentz ratio.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.8cm,height=4.5cm]{block32_33.eps}\n\\caption{(Color online) Maximum value of $ ZT $ as a function of temperature. The curves shown in the left panel are plotted assuming $ \\lambda = 0.25meV,\\ \\Gamma = 0.3 meV $ (solid line), $ \\Gamma = 0.7meV $ ( dashed line), $ \\Gamma =1.25 meV $ (dash-dotted line). In the right panel, the curves are plotted at $ \\Gamma = 0.7 meV, \\ \\lambda = 0 $ (dash-dotted line), $ \\lambda = 0.25meV $ (dashed line), $ \\lambda = 1meV $ (solid line). \n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\n\nIn analyzing temperature dependencies of  thermoelectric characteristics  of systems consisting of a  molecule\/quantum dot linking two electrodes, it was established that usually the figure of merit $ ZT $ is a nonmonotonous function of temperature (see e.g. Refs. \\cite{9,19,20,21,25,46}. Also, the present results show that at low temperatures $ ZT $ increases as the temperature enhances and it reaches a maximum value at certain temperature $ T_0. $ As $ T $ further  rises, the figure of merit decreases approaching zero when the temperature significantly exceeds $ T_0. $ This is illustrated in the Fig. 4. The value of the optimal temperature $ T_0 $ as well as the corresponding value of $ ZT_{max} $ is determined by the relation between the coupling energies $ \\lambda $ and $ \\Gamma. $ Assuming that $ \\lambda $ is fixed, one observes that $ ZT_{max} $ takes on greater values and the optimal temperature $ T_0 $ becomes higher as $ \\Gamma $ increases. On the contrary, enhancement of $ \\lambda $ at fixed $ \\Gamma $ leads to a significant decrease of $ ZT_{max}, $ and shifts $ T_0 $ to a lower value.  Thus the electron interactions with the thermal baths  suppress $ ZT $ values. Molecular vibrations may affect thermoelectric efficiency of the considered nanoscale systems in a similar way, as discussed in several works (see e.g. Refs. \\cite{13,21,30}).\n\nThe character  of temperature dependence of $ ZT $ displayed in the Fig. 4 indicates that while the transition from coherent and elastic tunneling to the dissipative transport significantly reduces $ZT $ values, the general character of temperature dependence of the figure of merit remains unchanged.\n At low temperatures, erosion of the sharp step in the Fermi distribution functions for the electrodes occurring at $ E  = \\mu $ creates better opportunities for the electron tunneling through the system. However, when the temperature exceeds a certain value, the same process starts to hinder electron transport. Also, at sufficiently strong electron-phonon interactions, the peak value of the electron transmission decreases   \n  bringing further reduction of the thermoelectric efficiency.\n\nAlthough considerable efforts are applied to reach understanding of combined effects of electron-electron and electron-phonon interactions on the thermoelectric transport, this subject is not fully investigated so far. Now, we reconsider the above results taking into account previously disregarded Coulomb interactions between electrons on the bridge of a thermoelectric junction.  \n Then the expression (\\ref{15}) for the electron Green's function may be employed to compute the scattering probabilities and, ultimately, the electron transmission $ T(E) $ and measurable characteristics of thermoelectric transport. In further analysis we assume that the linker (molecule\/quantum dot) is weakly coupled to the electrodes so that the charging  energy  $ U $  significantly exceeds the coupling parameter $ \\Gamma. $   The results of these  computations are displayed in the Figs. 5,6.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.8cm,height=4.5cm]{block37_38.eps}\n\\caption{(Color online) Combined effect of electron-electron and electron-phonon interactions  on the dependence of $ ZT $  of the bridge level position. The curves are plotted at $ U = 10meV.$ Left panel: $ \\Gamma = 1,25 meV,\\ \\lambda = 0.25 meV ,$  $ k_BT = 2.6 meV $ (dash-dotted line), $ k_BT = 1.3 meV $ (dashed line), $ k_BT = 0.7 meV $ (solid line). Right panel: $ \\Gamma = 1.25meV,\\ k_BT = 0.7meV,$  $ \\lambda = 0  $ (dash-dotted line), $ \\lambda = 0.25 meV $ (dashed line), $ \\lambda = 1meV $ (solid line).\n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\n\nAs shown in the Fig. 5, the dependence of $ ZT $ of  $ E_0 $ undergoes significant changes as the temperature increases. At low temperatures, $ ZT $ exhibits two pairs of peaks of unequal height situated near $ E_0 = \\mu $ and $ E_0 = \\mu - U,$ respectively. At higher temperatures two peaks making a pair cling together so that each pair is transformed to a single peak. At sufficiently high temperatures, these peaks become nearly equal in height, and their tops are shifted farther away from each other. The  curves displayed in the left panel of the Fig. 5  are plotted assuming that temperature is noticeably lower than the temperature $ T_0 $ providing the maximum value of $ ZT ,$ as shown in the Fig. 4. We cannot explicitly compare the results represented in these figures  because the curves shown in the Fig. 4 are plotted disregarding electron-electron interactions. However, we may conjecture that further increase of temperature accompanied by intensification of scattering processes will bring furthermost rise of $ ZT $ peaks as well as it happens in the case when one neglects electron-electron interactions. Also, we may expect that the increase in the peaks heights would be replaced by their reduction as the temperature would exceed a certain value. An explicit effect of electron-phonon interactions on the figure of merit is shown in the right panel of the Fig. 5. Again, one may observe the suppression of $ ZT $ originating from these interactions.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=8.8cm,height=4.5cm]{block35_36.eps}\n\\caption{(Color online) Left panel: Combined effect of electron-electron and electron-phonon interactions on the dependence of ZT of the level position at higher temperatures. Right panel: maximum value of $ ZT $ as function of charging energy. Curves are plotted assuming $ k_BT = 2.6meV,\\ \\Gamma = 1.25meV,\\ U = 10 meV $ (left panel) $ \\lambda = 0 $ (dash-dotted lines), $ \\lambda = 0.25 meV $ (dashed lines), $ \\lambda = 1meV $ (solid lines). \n}\n \\label{rateI}\n\\end{center}\\end{figure}\n\nFurther  illustration of the influence of thermal phonons on the figure of merit is presented in the Fig. 6. The curves shown in the left panel of this figure are plotted at a moderately high temperature $(k_B T = 2.6 meV) $ when the adjacent peaks are already merged, so that  $ ZT $ exhibit only two maxima. Omitting electron-phonon interactions, one observes a significant difference in the peaks heights. This difference originates from the characteristic features of electron density of states on the bridge level which are manifested in the characteristics of coherent electron transport. As the  electrons interaction with thermal phonons strengthens, the peaks heights become leveled. At fixed temperature, maximum value of $ ZT $ is determined by the relation between the charging energy $ U, $ and coupling strengths $ \\lambda $ and $ \\Gamma. $ We remark that the suppression of $ ZT $ due to electron-phonon interactions may be replaced by its promotion which occurs due to a combined effect of electron-electron and electron-phonon interactions \\cite{33}. However, this increase of thermoelectric efficiency is expected to appear when electron-phonon interactions and Coulomb repulsion between electrons are comparable in strength. These conditions are different from those considered in the present work.\n Disregarding  for a while the effect of phonons, one observes that $ ZT $ takes on greater values within the limits of low $(U \\ll k_B T)$ and high $ (U \\gg k_BT)$ values of charging energy, and it drops at intermediate values of $ U. $ This behavior was previously described and explained within the sequential hopping approximation for the electron transmission through a thermoelectric junction \\cite{21}. Electron-phonon interactions promote washing out of these features.\n\n\n\n \n\\section{iv. Conclusion} \n\nIn conclusion, we remark that thorough studies of thermoelectric properties of nanoscale systems  taking into account both electron and phonon transport as well as diverse effects arising due to electron-electron and electron-phonon interactions are not completed so far. In several earlier works this theoretical research was carried out employing single-particle scattering approach pioneered by Landauer in the context of charge transport in mesoscopic and nanoscale systems. These ideas were generalized to phonon transport through nanoscale junctions \n\\cite{30,32,42,52,53}. Within this approach, transport characteristics of a considered nanoscale system are expressed in terms of electron and phonon transmission functions. The latter were computed employing several methods, including some based on scattering matrices formalism \\cite{42,54}. Later, these methods were mostly abandoned in favor of more advanced formalisms such as NEGF and\/or various modifications of quantum rate equations. However, potential usefulness of the approaches based on scattering theory is not exhausted so far. \n\nThese approaches have an advantage of being computationally simple and less time and effort consuming than advanced formalisms. At the same time, their shortcomings  could be largely removed by incorporating some NEGF based results into a computational scheme. In the present work we suggest such approach, and we employ it to theoretically analyze some effects of electron-phonon interactions on the efficiency of nanoscale thermoelectric junctions. Presently, various manifestations of electron-phonon interactions in thermoelectric transport characteristics of nanoscale molecular junctions are already explored, and the research is still going on. However, the research efforts were and still are mostly concentrated on the effects arising from vibrational modes on the molecules linking the electrodes. Less attention was paid to the influence of thermal phonons associated with random nuclear motions in the ambience. Here, we focus on the analysis of thermal phonons on the electron transport. We show that direct interaction of electrons with thermal phonons assuming that  these phonons are assembled in two baths associated with the electrodes.\nmay significantly affect thermoelectric efficiency of molecular junctions and similar nanoscale systems. \n\nSpecifically, we show that electron-phonon interactions assist the increase of the scattering probabilities thus destroying the coherence of electron transport and promoting energy dissipation. When the electron-phonon coupling  becomes sufficiently strong, this brings a significant suppression of both thermopower and thermoelectric figure of merit thus worsening thermoelectric efficiency of a considered system. This effect is illustrated in the Figs. 3,4. We remark that $ \\lambda $ and thermal energy $ k_BT $ appear as cofactors in the expression for the scattering probability $ \\epsilon $ (see Eq. (\\ref{29})), so they affect it in a similar way. When either $ \\lambda $ or $ k_BT $ increases, this results in strengthening of dephasing in the electron transport. However, entire effects of these two parameters on the thermoelectric properties of considered systems are unidentical.\nWhile the strengthening of electron-phonon interactions always leads to reduction of $ ZT, $ the rise of temperature can promote the figure of merit increase provided that temperature does not exceed a certain value. This may be explained by the fact that besides affecting the intensity of scattering, the temperature influences distributions of electrons in the electrodes, and it may either assist or hinder their transport through the system. Also, we analyzed the combined effect of electron-electron and electron-phonon interactions on thermoelectric properties. Obtained results agree with those reported in the earlier works \\cite{21,33}. In particular, it was confirmed that $ ZT $ exhibits a minimum at a certain value of the charging energy $ U $ which becomes less distinct at stronger values of $ \\lambda. $ \n\n\nThe suggested computational scheme may be generalized to include vibrational modes.\nFor this purpose, one may add to the adopted model simulating a thermoelectric junction an extra reservoir representing vibrons. Also, one may mimic the molecular bridge by a set of energy levels thus opening the way to studies of interference effects.  For a realistic molecular junction, relevant energies may be computed  using either density functional theory or other method of electronic structure calculations. Finally, the proposed scheme may appear helpful in studies of thermoelectric transport beyond linear regime in temperature. On these grounds, we believe that presented method and results could help to reach better understanding of some important aspects of thermal transport in molecular junctions and similar nanoscale systems.\n\\vspace{2mm}\n\n {\\bf Acknowledgments:}\n This work was  supported  by  NSF-DMR-PREM 0934195 and NSF-EPS-1010094. The author  thank  G. M. Zimbovsky for help with the manuscript.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe quark model \\cite{GELLMANN1964214,Zweig:1981pd,Zweig:1964jf}  is very successful in describing the properties of \n hadrons observed in the experiment. Nevertheless, \nnot all predicted particles by the quark model are  experimentally well established: we only have the doubly charmed $\\Xi_{cc}$ state seen in the experiment among the doubly heavy baryons. The triply heavy  baryons are also missing in the experiments and efforts for their hunting are continued. More experimental and theoretical studies on these states are required. Even in the case of  $\\Xi_{cc}$ there is a  puzzle  in the experiment.\nFirst evidence for this state was reported in 2005 by the SELEX experiment   in\n $\\Xi_{cc}^{+}$ decaying into $ \\Lambda_c^+ K^- \\pi^+ $ and $pD^+K^-  $  in final states\nusing a $600\\text{MeV}\/c^2$ charged hyperon beam impinging on a fixed target.\nThe mass measured by SELEX, averaged over the two decay\nmodes, was found to be $ 3518.7\\pm 1.7~\\text{MeV}\/c^2 $.\nThe lifetime was measured to be less than $ 33~\\text{fs} $  at $ 90\\% $ confidence level.\nIt was estimated that about $  20\\% $ of $ \\Lambda_c^+ $ baryons in the SELEX experiment were produced from  $\\Xi_{cc}$  decays~\\cite{Mattson:2002vu,Ocherashvili:2004hi}.\nHowever, the FOCUS \\cite{Ratti:2003ez}, BaBar \\cite{Aubert:2006qw}, LHCb \\cite{Aaij:2013voa} and Belle \\cite{Kato:2013ynr} experiments\ndid not then confirm the SELEX results. \nIn 2017, the doubly charmed baryon $ \\Xi^{++}_{cc} $ was\nobserved by the LHCb collaboration via the decay channel $\\Xi^{++}_{cc}\\rightarrow\\Lambda_c^+ K^-\\pi^+\\pi^+$\n\\cite{Aaij:2017ueg}, \nand confirmed via measuring another decay channel $\\Xi^{++}_{cc}\\rightarrow \\Xi^+_c \\pi^+$ \\cite{Aaij:2018gfl}.\nThe weighted average of the $ \\Xi^{++}_{cc} $ mass of the two decay modes was determined to be\n$3621.24 \\pm 0.65 (\\text{stat.})\\pm 0.31 (\\text{syst.})~\\text{MeV}\/c^2 $~\\cite{Aaij:2018gfl}.\nThe lifetime of the  $ \\Xi^{++}_{cc} $ baryon was measured to be\n$0.256\\,^{+0.024}_{-0.022}(\\text{stat.})\\pm\n0.014(\\text{syst.})~\\text{ps}$~\\cite{Aaij:2018wzf}.\nRecently,\nwith a data sample corresponding to an integrated luminosity  of 1.7~\\ensuremath{\\mbox{fb}^{-1}}, \nthe $\\Xi^{++}_{cc}\\rightarrow D^+p K^- \\pi^+$ decay has been searched for by the LHCb\ncollaboration, \nbut no signal was found~\\cite{Aaij:2019dsx}. Certainly, experiments will\ncontinue to solve the unexpected difference in parameters of these states and also will search for other doubly heavy particles.\n\n The discrepancy between the results of the  SELEX and LHCb collaborations\nwas the starting point of theoretical investigations deciding to explain this difference. In Ref. \\cite{Brodsky:2017ntu},\nthe authors have shown that the SELEX and\nthe LHCb results for the production of doubly charmed baryons can both be correct\nwith application of supersymmetric algebra to hadron spectroscopy, together with the intrinsic heavy-quark QCD mechanism for the hadroproduction of heavy hadrons at large $ x_F $.\n\nOn theoretical side, studies on  doubly heavy baryons,  are needed to provide many inputs to experiments. Some aspects of doubly heavy baryons have been  discussed in Refs.  \\cite{Aliev:2012ru,Aliev:2012iv,Azizi:2014jxa,Wang:2017mqp,Wang:2017azm,\nGutsche:2017hux,Li:2017pxa,Xiao:2017udy,Sharma:2017txj,Ma:2017nik,Hu:2017dzi,Shi:2017dto,Yao:2018ifh,Zhao:2018mrg,Wang:2018lhz,\nLiu:2018euh,Xing:2018lre,Dhir:2018twm,Berezhnoy:2018bde,Jiang:2018oak,Zhang:2018llc,Gutsche:2018msz,Shi:2019fph,Hu:2019bqj,Brodsky:2011zs,Yan:2018zdt, Ozdem:2018uue}.  \nThe mechanism of production and decays of such systems  have  also been of interest to researchers for\nmany years \\cite{Baranov:1995rc,Berezhnoy:1998aa,Gunter:2001qy,Ma:2003zk,Li:2007vy,Yang:2007ep,Zhang:2011hi,Jiang:2012jt,Martynenko:2014ola,Brown:2014ena,Trunin:2016uks,Huan-Yu:2017emk,Niu:2018ycb,Yao:2018zze}.\nThe production of doubly heavy baryons can be divided into two steps: the first step is\nthe perturbative production of a heavy quark pair in the hard interaction.\nIn the second step  this pair is transformed to the baryon within the soft hadronization process. The doubly heavy baryons can participate in many interactions and processes. The fusion of two $\\Lambda_{c}$ to produce $\\Xi_{cc}^{++} n$ which results in an energy release of about $12 \\text{MeV}$, or a much higher one of about $138 \\text{MeV}$ through the fusion of two $\\Lambda_{b}$ to $\\Xi_{bb}^{0}n$ suggest related experimental setups to use the resultant huge energy. Although the very short lifetimes of $cc$ and $bb$ baryons may prevent practical applications in the present time \\cite{Karliner}.   \n\nIn this study, we\ninvestigate  the strong coupling constants among the doubly heavy spin-1\/2 baryons and light pseudoscalar mesons, $\\pi$, $K$,  $ \\eta $ and $ \\eta^\\prime $, which is the extension of our previous work \\cite{Olamaei:2020bvw}. In \\cite{Olamaei:2020bvw}, we investigated only the symmetric $  \\Xi_{QQ^\\prime }$ and calculated its coupling constant with   $\\pi $ meson. In the present study, we investigate the strong coupling constants of $  \\Xi_{QQ^\\prime } $, $  \\Xi^{\\prime}_{QQ^\\prime }$, $  \\Omega_{QQ^\\prime}$  and  $ \\Omega^{\\prime}_{QQ^\\prime}$ doubly heavy baryons with all  light pseudoscalar $\\pi$, $K$,  $ \\eta $ and $ \\eta^\\prime $ mesons with different charges. Here $ Q $ and $ Q^\\prime $\ncan both be $ b $ or $ c $ quark. \n We use the well established non-perturbative method of light cone QCD sum rule (LCSR) in the calculations. \n In the framework of LCSR, the nonperturbative dynamics of the quarks and gluons in the baryons are described by the light-cone distribution amplitudes (DAs). The LCSR approach carries out the\noperator product expansion (OPE) near the lightcone $ x^2\\approx 0 $  instead of the short distance  $ x\\approx 0 $\nand the nonperturbative matrix elements are parameterized by the light cone DAs which are classified according to their twists \\cite{Balitsky:1989ry,Khodjamirian:1997ub,Braun:1997kw}.\n\nThe rest of the paper is organized as follows: In  the next section, we describe the formalism and obtain the sum rules for the strong coupling constants under study. In Section~\\ref{NA}, the numerical analysis and results are\npresented. Section~\\ref{SC}  is reserved for summary and concluding notes.\n\n\\section{The strong coupling constants among doubly heavy baryons and pseudoscalar mesons }\\label{LH}\n\nBefore going to the details of the  calculations for the strong coupling constants, we take a look at the ground state of the doubly heavy baryons in the quark model. In the case of the doubly heavy baryons having two identical heavy quarks, i.e. $Q=Q^{\\prime}$, the two heavy quarks form a diquark system with spin-1 which is after adding the light quark spin, the whole baryon may have spin 1\/2 ($\\Xi_{QQ}$ or $\\Omega_{QQ}$) or 3\/2 ($\\Xi^*_{QQ}$ or $\\Omega^*_{QQ}$) where the latter is not the subject of this work. Here the interpolating current should be symmetric with respect to the exchange of identical heavy quarks. \nMoreover, for the different heavy quark contents ($Q\\neq Q^{\\prime}$), the diquark portion can have spin zero where together with the light quark, the total spin of the whole baryon would be 1\/2, which obviously leads to anti-symmetric interpolating current for the doubly heavy baryon due to the exchange of the two heavy quarks. They are usually denoted by primes as $\\Xi^{\\prime}_{bc}$ and $\\Omega^{\\prime}_{bc}$.\n\n\nThe main inputs in the sum rule method are interpolating currents, which are written based on the general properties of the baryons and in terms of their quark contents.   In the case of doubly heavy baryons, the symmetric and anti-symmetric interpolating fields for spin-1\/2 particles are given as\n\\begin{eqnarray}\\label{etaS}\n\\eta^{\\cal S}&=&\\frac{1}{\\sqrt{2}}\\epsilon_{abc}\\Bigg\\{(Q^{aT}Cq^b)\\gamma_{5}Q'^c+\n(Q'^{aT}Cq^b)\\gamma_{5}Q^c+t (Q^{aT}C\\gamma_{5}q^b)Q'^c \\nonumber \\\\\n&&+t(Q'^{aT}C\n\\gamma_{5}q^b)Q^c\\Bigg\\},\n\\end{eqnarray}\n\\begin{eqnarray}\\label{etaA}\n\\eta^{\\cal A}&=&\\frac{1}{\\sqrt{6}}\\epsilon_{abc}\\Bigg\\{2(Q^{aT}CQ'^b)\\gamma_{5}q^c+\n(Q^{aT}Cq^b)\\gamma_{5}Q'^c-(Q'^{aT}Cq^b)\\gamma_{5}Q^c \\nonumber\\\\\n&&+2t (Q^{aT}C\n\\gamma_{5}Q'^b)q^c\n+ t(Q^{aT}C\\gamma_{5}q^b)Q'^c-t(Q'^{aT}C\\gamma_{5}q^b)Q^c\\Bigg\\},\n\\end{eqnarray}\nwhere $C$ stands for the charge conjugation operator, $T$ denotes the transposition and $t$ is an arbitrary mixing parameter where the case $t=-1$ corresponds to the Ioffe current. $Q^{(')}$ and $q$ stand for the heavy and light quarks respectively and $a$, $b$, and $c$ are the color indices. The quark contents for different members are shown in Table \\ref{tab:baryon}.\n\nThe main goal in this section is to find the strong coupling constants among the doubly heavy baryons with spin-1\/2, \n$  \\Xi_{QQ^\\prime }$   $ \\Xi^\\prime_{QQ^\\prime} $, $ \\Omega_{QQ^\\prime}$ and  $ \\Omega^\\prime_{QQ^\\prime} $, with the light pseudoscalar mesons $ \\pi $, $ K $,  $\\eta$ and $ \\eta^\\prime $. To this end, we use the LCSR\napproach as  one of the most powerful non-perturbative methods which is based on the light-cone OPE.\nThe starting point is to write the corresponding correlation function (CF) as\n\\begin{eqnarray}\\label{equ1} \n\\Pi(p,q)= i \\int d^4x e^{ipx} \\left< {\\cal P}(q) \\vert {\\cal T} \\left\\{\n\\eta (x) \\bar{\\eta} (0) \\right\\} \\vert 0 \\right>~,\n\\end{eqnarray}\nwhere the two time-ordered interpolating currents of doubly-heavy baryons are sandwiched between the QCD vacuum and the on-shell pseudoscalar meson $ {\\cal P}(q) $.  Here,  $p$ is the external four-momentum of the outgoing doubly heavy baryon. As the theory is translationally invariant we can choose one of the interpolating currents, $\\bar{\\eta}(0)$, at the origin.\nIt is worth noting again that in the symmetric interpolating current ($ \\eta^{\\cal S} $), heavy quarks may be identical or different whereas in the anti-symmetric one ($ \\eta^{\\cal A} $) they must be different.\n\\begin{table}\n\\setlength{\\tabcolsep}{1.5em}\n\\centering\n\\begin{tabular}{cccc}\n\\hline\n\\hline\n\\textcolor{red}{Baryon}    &\\textcolor{red}{ $ q $ }& \\textcolor{red}{$ Q $ }& \\textcolor{red}{$ Q^\\prime $ }\\\\\n\\hline\n\\hline\n\\\\\n$  \\Xi_{QQ^\\prime }$ or $ \\Xi^\\prime_{QQ^\\prime} $     & $ u $ or $ d $    & $b  $   or $ c $ & $b  $   or $ c $    \\\\\n          &         &  &     \\\\\n$ \\Omega_{QQ^\\prime}$ or $ \\Omega^\\prime_{QQ^\\prime} $      & $ s $   & $b  $   or $ c $ & $b  $   or $ c $       \\\\\n    \\\\\n\\hline\n\\end{tabular}\n\\caption{The  quark contents of the doubly heavy spin-1\/2 baryons.} \\label{tab:baryon} \n\\end{table}\n\n\n\nIn the LCSR approach, the cornerstone is the CF. It can be calculated in two different ways. In the timelike region, one can insert the complete set of hadronic states with the same quantum numbers as the interpolating currents to extract and isolate the ground states. It is called the phenomenological or physical side of the CF. In the spacelike region which is free of singularities, one can calculate the CF in terms of QCD degrees of freedom using OPE. It is known as the QCD or theoretical side. These two representations, which respectively are the real and imaginary parts of the CF, can be matched via a dispersion relation to find the corresponding sum rule. The divergences coming from the dispersion integral as well as higher states and continuum are suppressed using the well-known method of Borel transformation and continuum subtraction.\n\nOn the phenomenological side, after inserting the  complete sets of\nhadronic states with the same quantum numbers as the interpolating currents and performing the Fourier integration over $x$, we get\n\\begin{eqnarray}\\label{phside} \n\\Pi^{\\text{Phys.}}(p,q)=\\frac{\\langle 0\\vert \\eta\\vert B_2(p,r)\\rangle  \\langle B_2(p,r){\\cal P}(q)\\vert B_1(p+q,s)\\rangle\\langle B_1(p+q,s) \\vert \\bar{\\eta}\\vert 0\\rangle}{(p^2-m_1^2)[(p+q)^2-m_2^2]} +\\cdots~,\n\\end{eqnarray}\nwhere the ground states are isolated and dots represent the contribution of the  higher states and continuum. $B_1(p+q,s)$ and $B_2(p,r)$ are the initial and final doubly heavy baryons with spins $s$ and $r$ respectively.\nThe matrix element $\\langle 0\\vert \\eta\\vert B_i(p,s)\\rangle$ is defined as \n\\begin{eqnarray}\\label{me1} \n\\langle 0\\vert \\eta\\vert B_i(p,s)\\rangle &=&\\lambda_{B_i}u(p,s),\n\\end{eqnarray}\nwhere $\\lambda_{B_i}$ are the residues and $u(p,s)$ is the Dirac spinor for the baryons $B_i$ with momentum $p$ and spin $s$.\nBy the Lorentz and parity consideration,  one can write the matrix element $\\langle B_2(p,r){\\cal P}(q)\\vert B_1(p+q,s)\\rangle$ in terms of the strong coupling constant and Dirac spinors as\n\\begin{eqnarray}\\label{me2} \n\\langle B_2(p,r){\\cal P}(q)\\vert B_1(p+q,s)\\rangle &=& g_{B_1 B_2{\\cal P}}\n\\bar{u}(p,r)\\gamma_5 \nu(p+q,s) ~,\n\\end{eqnarray}\nwhere $ g_{B_1 B_2{\\cal P}} $, representing the strong coupling constant for the strong decay $ B_1\\rightarrow B_2 {\\cal P}$. The final expression for the phenomenological side of the correlation function is obtained by inserting Eqs.~\\ref{me1} and \\ref{me2} into Eq.~\\ref{phside} and summing over spins:\n\\begin{eqnarray}\\label{CFPhys}\n\\Pi^{\\text{Phys.}}(p,q)=\\frac{ g_{B_1 B_2{\\cal P}}\\lambda_{B_1}\\lambda_{B_2}}{(p^2-m_{B_2}^2)[(p+q)^2-m_{B_1}^2]}[\\rlap\/q \\rlap\/p\\gamma_5 +\\cdots~] +\\cdots,\n\\end{eqnarray}\nwhere dots inside the bracket denote several $\\gamma$-matrix structures that may appear in the final expression due to the spin summation.  Here we select the structure $\\rlap\/q \\rlap\/p\\gamma_5$ to perform analyses. \n\nTo kill the higher states and continuum contributions we apply the double Borel transformation with respect to the square of the doubly heavy baryon momenta\n$ p^2_1=(p+q)^2 $ and $ p^2_2=p^2 $, which leads to \n\\begin{eqnarray}\\label{CFPhysB}\n{\\cal B}_{p_1}(M_1^2){\\cal B}_{p_2}(M_2^2)\\Pi^{\\text{Phys.}}(p,q) &\\equiv & \\Pi^{\\text{Phys.}}(M^2)\\nonumber\\\\\n&=& g_{B_1 B_2 {\\cal P}} \\lambda_{B_1} \\lambda_{B_2} e^{-m_{B_1}^2\/M_1^2} e^{-m_{B_2}^2\/M_2^2}\\rlap\/q \\rlap\/p\\gamma_5~ + \\cdots~,\n\\end{eqnarray}\nwhere  $ M^2_1 $ and $ M^2_2 $ are the Borel parameters correspond to the square momenta $p_1^2$ and $p_2^2$, respectively and $ M^2= M^2_1 M^2_2\/(M^2_1+M^2_2) $. As the masses of the initial and final state baryons are the same or to a good approximation equal, the Borel parameters are chosen to be equal and therefore $ M^2_1 = M^2_2 = 2M^2$.\n\nOn the QCD side, choosing the corresponding structure as \\ref{CFPhys} one can express the CF function as\n\\begin{equation}\\label{QCD1}\n\\Pi^{\\text{QCD}}(p,q)=\\Pi\\big(p,q\\big)  \\rlap\/q \\rlap\/p\\gamma_5,\n\\end{equation}\nwhere $\\Pi\\big(p,q\\big)$ is an invariant function of $(p+q)^2$ and $p^2$.  The main aim in this part is to determine this function in the Borel scheme. To this end,  we insert the explicit forms of the\ninterpolating currents \\ref{etaS} and \\ref{etaA} into the correlation function \\ref{equ1} \nand use the Wick theorem to contract all the heavy quark fields. The result for the symmetric interpolating current is as follows:\n\n\n\\begin{eqnarray}\\label{QCD2S}\n\\Pi^{\\text{QCD}}_{(\\cal{S})\\rho\\sigma}(p,q) &=& \\frac{i}{2}\\epsilon_{abc}\\epsilon_{a'b'c'} \\int d^4 x e^{i q.x} \\langle {\\cal P}(q) \\vert \\bar{q}^{c^\\prime}_{\\alpha}(0)q^{c}_{\\beta}(x)\\vert 0\\rangle \\Bigg\\{ \\Bigg[ \\Big(\\tilde{S}^{aa^{\\prime}}_{Q}(x) \\Big)_{\\alpha\\beta}\\Big( \\gamma_5 S^{bb^{\\prime}}_{Q^{\\prime}}(x) \\gamma_5\\Big)_{\\rho\\sigma}\\nonumber \\\\\n&+& \\Big(\\gamma_5 S_{Q^{\\prime}}^{bb^{\\prime}}(x)C\\Big)_{\\rho\\alpha}\\Big(CS^{aa^{\\prime}}_{Q}(x)\\gamma_5 \\Big)_{\\beta\\sigma} + t\\Big\\{\\Big(\\gamma_5 \\tilde{S}_{Q}^{aa^{\\prime}}(x)\\Big)_{\\alpha\\beta}\\Big(\\gamma_5 S^{bb^{\\prime}}_{Q^{\\prime}}(x) \\Big)_{\\rho\\sigma} \\nonumber\\\\\n&+& \\Big( \\tilde{S}_{Q}^{aa^{\\prime}}(x)\\gamma_5\\Big)_{\\alpha\\beta}\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma_5 \\Big)_{\\rho\\sigma} +\\Big(\\gamma_5 S_{Q^{\\prime}}^{bb^{\\prime}}(x)C\\gamma_5\\Big)_{\\rho\\alpha}\\Big(CS^{aa^{\\prime}}_{Q}(x)\\Big)_{\\beta\\sigma} \\nonumber\\\\\n&-&\\Big( S_{Q^{\\prime}}^{bb^{\\prime}}(x)C\\Big)_{\\rho\\alpha}\\Big(\\gamma_5CS^{aa^{\\prime}}_{Q}(x)\\gamma_5\\Big)_{\\beta\\sigma}\n \\Big\\}+ t^2 \\Big\\{\\Big(\\gamma_5 \\tilde{S}_{Q}^{aa^{\\prime}}(x)\\gamma_5\\Big)_{\\alpha\\beta}\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\rho\\sigma} \\nonumber\\\\\n  &-& \\Big( S_{Q^{\\prime}}^{bb^{\\prime}}(x)C\\gamma_5\\Big)_{\\rho\\alpha}\\Big(\\gamma_5CS^{aa^{\\prime}}_{Q}(x)\\Big)_{\\beta\\sigma} \\Big\\}\\Bigg]+\\Bigg(Q \\longleftrightarrow Q^{\\prime}\\Bigg)\\Bigg\\},\n\\end{eqnarray}\nwhere the $\\rho$ and $\\sigma$ are Dirac indices which run through 1 to 4, $S_{Q}^{aa^{\\prime}}(x)$ is the heavy quark propagator, $\\tilde{S}=C S^T C$ and the subscripts ${\\cal S}$ denotes the symmetric part. $\\langle {\\cal P}(q) \\vert \\bar{q}^{c^\\prime}_{\\alpha}(x)q^{c}_{\\beta}(0)\\vert 0\\rangle$ are the non-local matrix elements for the light quark contents of the doubly heavy baryons and purely non-perturbative.\nFor anti-symmetric part we have\n\\begin{eqnarray}\\label{QCD2A}\n\\Pi^{\\text{QCD}}_{(\\cal A)\\rho\\sigma}(p,q) &=& \\frac{i}{6}\\epsilon_{abc}\\epsilon_{a'b'c'} \\int d^4 x e^{i q.x} \\langle {\\cal P}(q) \\vert \\bar{q}^{c^\\prime}_{\\alpha}(0)q^{c}_{\\beta}(x)\\vert 0\\rangle \\Bigg\\{ 4 \\text{Tr}\\big[ \\tilde{S}^{aa^{\\prime}}_{Q}(x)S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\big]\\gamma^{5}_{\\alpha\\sigma}\\gamma^{5}_{\\rho\\beta}\\nonumber\\\\\n&-&2\\Big( \\tilde{S}^{aa^{\\prime}}_{Q}(x)S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\alpha\\sigma}\\gamma^{5}_{\\rho\\beta} -2\\Big(\\gamma^5  S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\tilde{S}^{aa^{\\prime}}_{Q}(x) \\Big)_{\\rho\\beta}\\gamma^5_{\\alpha\\sigma}\\nonumber\\\\\n&-&2\\Big(\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)S^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\alpha\\sigma}\\gamma^5_{\\rho\\beta}-2\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\rho\\beta}\\gamma^5_{\\alpha\\sigma} \\nonumber\\\\\n &+&\\Big(\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\Big)_{\\alpha\\beta}\\Big(\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5 \\Big)_{\\rho\\sigma} +\\Big(\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\alpha\\beta}\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\rho\\sigma}\\nonumber\\\\\n&+& \\Big(\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)C\\Big)_{\\rho\\alpha}\\Big(CS^{aa^{\\prime}}_{Q}(x)\\gamma^5 \\Big)_{\\beta\\sigma}+\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)C\\Big)_{\\rho\\alpha}\\Big(CS^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\beta\\sigma} \\nonumber\\\\\n &+&t\\Bigg[ 4\\text{Tr}\\big[\\tilde{S}^{aa^{\\prime}}_{Q}(x)S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\big]\\gamma^5_{\\rho\\beta}\\delta_{\\alpha\\sigma} +4\\text{Tr}\\big[S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5\\big]\\gamma^5_{\\alpha\\sigma}\\delta_{\\rho\\beta}\\nonumber\\\\\n &+&2\\Big(\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5S^{aa^{\\prime}}_{Q}(x)C\\gamma^5\\Big)_{\\alpha\\sigma}\\delta_{\\beta\\rho} -2\\Big(S^{aa^{\\prime}}_{Q}(x)\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\beta\\rho}\\gamma^5_{\\alpha\\sigma} \\nonumber\\\\\n &-&2\\Big(\\gamma^5  \\tilde{S}^{aa^{\\prime}}_{Q}(x)S^{bb^{\\prime}}_{Q^{\\prime}}(x) \\Big)_{\\alpha\\sigma}\\gamma^5_{\\rho\\beta} -2\\Big(\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5 \\tilde{S}^{aa^{\\prime}}_{Q}(x)\\Big)_{\\rho\\beta}\\delta_{\\sigma\\alpha} \\nonumber \\\\\n&-&2\\Big(\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)S^{aa^{\\prime}}_{Q}(x)\\Big)_{\\alpha\\sigma}\\gamma^5_{\\rho\\beta}-2\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\rho\\beta}\\delta_{\\alpha\\sigma}\\nonumber\\\\\n&-&2\\Big(\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5 \\Big)_{\\alpha\\sigma}\\delta_{\\rho\\beta} -2\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\rho\\beta}\\gamma^5_{\\alpha\\sigma}\\nonumber\\\\\n &+&\\Big(\\gamma^5\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\Big)_{\\alpha\\beta}\\Big(\\gamma^5 S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\rho\\sigma} +\\Big(\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)C\\gamma^5\\Big)_{\\rho\\alpha}\\Big(C S^{aa^{\\prime}}_{Q}(x)\\Big)_{\\beta\\sigma}\\nonumber\\\\\n &+&\\Big(\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\alpha\\beta}\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\rho\\sigma} +\\Big(\\gamma^5CS^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\beta\\sigma}\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)C\\Big)_{\\rho\\alpha} \\nonumber\\\\\n &+&\\Big(S^{aa^{\\prime}}_{Q}(x)C\\Big)_{\\rho\\alpha}\\Big(\\gamma^5CS^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\beta\\sigma} +\\Big(\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\alpha\\beta}\\Big(S^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\rho\\sigma}\n  \\nonumber \\\\\n &+&\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)C\\gamma^5\\Big)_{\\rho\\alpha}\\Big( CS^{bb^{\\prime}}_{Q^{\\prime}}(x) \\Big)_{\\beta\\sigma} +\\Big(\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\alpha\\beta}\\Big(\\gamma^5S^{aa^{\\prime}}_{Q}(x)\\Big)_{\\rho\\sigma} \\Bigg] \\nonumber\\\\\n &+& t^2\\Bigg[ 4\\text{Tr}\\big[\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\big] \\delta_{\\alpha\\sigma}\\delta_{\\beta\\rho} -2\\Big(\\gamma^5\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\alpha\\sigma}\\delta_{\\rho\\beta} \\nonumber\\\\\n &+&2\\Big(S^{aa^{\\prime}}_{Q}(x)\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\rho\\beta}\\delta_{\\alpha\\sigma} -2\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\rho\\beta}\\delta_{\\alpha\\sigma} \\nonumber \\\\ &+&2\\Big(\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5S^{aa^{\\prime}}_{Q}(x)\\Big)_{\\alpha\\sigma}\\delta_{\\beta\\rho} +\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\rho\\sigma}\\Big(\\gamma^5\\tilde{S}^{aa^{\\prime}}_{Q}(x)\\gamma^5\\Big)_{\\alpha\\beta} \\nonumber \\\\ \n &+& \\Big(S^{aa^{\\prime}}_{Q}(x)\\Big)_{\\rho\\sigma}\\Big(\\gamma^5\\tilde{S}^{bb^{\\prime}}_{Q^{\\prime}}(x)\\gamma^5\\Big)_{\\alpha\\beta} +\\Big(S^{bb^{\\prime}}_{Q^{\\prime}}(x)C\\gamma^5\\Big)_{\\sigma\\alpha}\\Big(\\gamma^5CS^{aa^{\\prime}}_{Q}(x)\\Big)_{\\beta\\rho} \\nonumber \\\\ &+&\\Big(S^{aa^{\\prime}}_{Q}(x)C\\gamma^5\\Big)_{\\rho\\alpha}\\Big(\\gamma^5CS^{bb^{\\prime}}_{Q^{\\prime}}(x)\\Big)_{\\beta\\sigma}\\Bigg]\\Bigg\\}.\n\\end{eqnarray}\nThere is also a symmetric-anti-symmetric form of the CF, $\\Pi^{\\text{QCD}}_{(\\cal{S}\\cal{A})}(p,q)$, which is the result of taking one interpolating current (say $\\eta$) to be in the symmetric and the other ($\\bar{\\eta}$) in the anti-symmetric form which represents the strong decays in which the initial baryon is anti-symmetric and the final one is symmetric. \n\nThe explicit expression for the heavy quark propagator is \n\\begin{eqnarray}\\label{HQP}\nS_Q^{aa^{\\prime}}(x) &=& {m_Q^2 \\over 4 \\pi^2} {K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2}}\\delta^{aa^{\\prime}} -\ni {m_Q^2 \\rlap\/{x} \\over 4 \\pi^2 x^2} K_2(m_Q\\sqrt{-x^2})\\delta^{aa^{\\prime}}\\nonumber \\\\& -&\nig_s \\int {d^4k \\over (2\\pi)^4} e^{-ikx} \\int_0^1\ndu \\Bigg[ {\\rlap\/k+m_Q \\over 2 (m_Q^2-k^2)^2} \\sigma^{\\mu\\nu} G_{\\mu\\nu}^{aa^{\\prime}} (ux)\n \\nonumber \\\\\n&+&\n{u \\over m_Q^2-k^2} x^\\mu  \\gamma^\\nu G_{\\mu\\nu}^{aa^{\\prime}}(ux) \\Bigg]+\\cdots.\n\\end{eqnarray}\nThe first two terms are the free part or perturbative contributions where $K_1$ and $K_2$ are the modified Bessel functions of the second kind. Terms $\\sim G_{\\mu\\nu}^{ab}$ are due to the expansion of the propagator on the light-cone and correspond to the interaction with the gluon field. Here we use the shorthand notation \n\\begin{eqnarray}\nG^{aa^{\\prime}}_{\\mu \\nu }\\equiv G^{A}_{\\mu \\nu }t^{aa^{\\prime}}_{A},\n\\end{eqnarray}\nwith $A=1,\\,2\\,\\ldots 8$ and $t_{A}=\\lambda_{A}\/2$  where $\\lambda _{A}$ are the Gell-Mann matrices.\n\nInserting the heavy quark propagator \\ref{HQP} into the CFs \\ref{QCD2S} and \\ref{QCD2A} would lead to several kinds of contributions each representing a different Feynman diagram. There are two heavy quark propagators in each term of the CFs.\nThe leading order contribution consists of a bare loop that is depicted in Fig. \\ref{fig:PertD} in which every heavy quark propagator is replaced by its perturbative terms and the non-perturbative part of this contribution comes from the non-local matrix elements of the pseudoscalar meson which are defined in terms of distribution amplitudes (DAs) of twist two and higher.\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.47\\textwidth]{11}\n\t\\caption{The leading order diagram contributing to $ \\Pi(p,q) $.}\n\t\\label{fig:PertD}\n\\end{figure}\n\nMultiplication of perturbative part of one heavy quark propagator and the gluon interaction part of another one leads to the  contributions which can be calculated using the pseudoscalar meson three particle DAs. It is responsible for the exchange of one gluon between one of the heavy quarks and the outgoing meson as shown in Fig. \\ref{fig:Npert1}. \n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.47\\textwidth]{222}\n\t\\caption{The one-gluon exchange diagrams.}\n\t\\label{fig:Npert1}\n\\end{figure}\n The higher order contributions corresponding to at least four-particle DAs, which are not available yet,  are  neglected in this work. However,  we take into account the two-gluon condensates contributions $\\sim g_s^2 \\langle GG \\rangle$. \n To proceed we use the replacement\n \\begin{equation}\\label{sumcolor1}\n \\overline{q}_{\\alpha }^{c^{\\prime }}(0)q_{\\beta }^{c}(x)\\rightarrow \\frac{1}{%\n \t3}\\delta^{cc^{\\prime }}\\overline{q}_{\\alpha }(0)q_{\\beta }(x).\n \\end{equation}\n  which applies the projector onto the color singlet product of quark fields. One can decompose $\\overline{q}_{\\alpha }(x)q_{\\beta }(0)$ into terms that have definite transformation properties under the Lorentz group and parity using the completeness relation to get the expansion \n  \\begin{equation}\n  \\overline{q}_{\\alpha }(0)q_{\\beta }(x)\\equiv \\frac{1}{4}\\Gamma_{\\beta \\alpha }^{J}\\overline{q}(0)\\Gamma_{J}q(x),  \\label{eq:Expan}\n  \\end{equation}\nwhere $\\Gamma^J$ runs over all possible $\\gamma-$matrices with definite parity and Lorentz transformation property as\n\\begin{equation}\\label{gammaexp}\n\\Gamma ^{J}=\\mathbf{1,\\ }\\gamma _{5},\\ \\gamma _{\\mu },\\ i\\gamma _{5}\\gamma\n_{\\mu },\\ \\sigma _{\\mu \\nu }\/\\sqrt{2}.\n\\end{equation}\nThis helps us to project quarks onto the corresponding distribution amplitudes.\n\nIn the following, we would like to briefly explain how the contributions of for instance the diagrams in Fig. \\ref{fig:PertD} and Fig. \\ref{fig:Npert1} are calculated.\nFor symmetric current and Fig. \\ref{fig:PertD}, we get\n\\begin{eqnarray}\\label{QCD2Spert}\n\\Pi^{\\text{QCD}(\\text{1})}_{(\\cal{S})\\rho\\sigma}(p,q) &=&   \\frac{i}{4}\\int d^4 x e^{i q.x} \\langle {\\cal P}(q) \\vert \\bar{q}(0)\\Gamma^Jq(x)\\vert 0\\rangle \\Bigg\\{\\Bigg[ \\text{Tr}\\big[ \\Gamma_J\\tilde{S}^{(\\text{pert.})}_{Q}(x) \\big]\\Big( \\gamma_5 S^{(\\text{pert.})}_{Q^{\\prime}}(x) \\gamma_5\\Big)_{\\rho\\sigma}\\nonumber \\\\\n&+& \\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\tilde{\\Gamma}_JS_{Q}^{(\\text{pert.})}(x)\\gamma_5 \\Big)_{\\rho\\sigma} + t\\Big\\{ \\text{Tr}\\big[\\Gamma_J\\gamma_5 \\tilde{S}_{Q}^{(\\text{pert.})}(x)\\big]\\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x) \\Big)_{\\rho\\sigma} \\nonumber\\\\\n&+& \\text{Tr}\\big[\\Gamma_J \\tilde{S}_{Q}^{(\\text{pert.})}(x)\\gamma_5\\big]\\Big(S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5 \\Big)_{\\rho\\sigma} +\\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5\\tilde{\\Gamma}_JS_{Q}^{(\\text{pert.})}(x)\\Big)_{\\rho\\sigma} \\nonumber\\\\\n&-&\\Big( S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\tilde{\\Gamma}_J\\gamma_5S_{Q}^{(\\text{pert.})}(x)\\gamma_5\\Big)_{\\rho\\sigma}\n\\Big\\} + t^2 \\Big\\{ \\text{Tr}\\big[\\Gamma_J\\gamma_5 \\tilde{S}_{Q}^{(\\text{pert.})}(x)\\gamma_5\\big]\\Big(S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\Big)_{\\rho\\sigma} \\nonumber\\\\\n&-& \\Big( S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5\\tilde{\\Gamma}_J\\gamma_5S_{Q}^{(\\text{pert.})}(x)\\Big)_{\\rho\\sigma} \\Big\\}\\Bigg] + \\Bigg( Q \\leftrightarrow Q^{\\prime}\\Bigg) \\Biggr\\},\n\\end{eqnarray}\nwhere $\\tilde{\\Gamma}_J=C\\Gamma^{\\text{T}}_JC$ and\n\\begin{eqnarray}\\label{HQPpert}\nS_Q^{(\\text{pert.})}(x) &=& {m_Q^2 \\over 4 \\pi^2} {K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2}} -\ni {m_Q^2 \\rlap\/{x} \\over 4 \\pi^2 x^2} K_2(m_Q\\sqrt{-x^2}).\n\\end{eqnarray}\n Fig. \\ref{fig:Npert1} denotes contributions to  the CF due to the exchange of one gluon between heavy quark $Q$ or $Q^{\\prime}$ and the pseudoscalar meson $\\cal{P}$. To be precise, taking the emission of the gluon from $Q$, Fig. \\ref{fig:Npert1}a, one can write the corresponding CF by replacing $S^{bb^{\\prime}}_{Q^{\\prime}}(x)$ with its perturbative free part \\ref{HQPpert} and $S^{aa^{\\prime}}_{Q}(x)$ with its non-perturbative gluonic part\n\\begin{eqnarray}\\label{HQPnp}\nS^{aa^{\\prime}(non-p.)}_{Q}(x)&=&-\nig_s \\int {d^4k \\over (2\\pi)^4} e^{-ikx} \\int_0^1\ndu G^{aa^{\\prime}}_{\\mu\\nu}(ux) \\Delta^{\\mu\\nu}_{Q}(x),\n\\end{eqnarray}\nwhere $ \\Delta^{\\mu\\nu}_{Q}(x)$ is defined to be as\n\\begin{eqnarray}\\label{HQPgamma}\n\\Delta^{\\mu\\nu}_{Q}(x)&=& \\dfrac{1}{2 (m_Q^2-k^2)^2}\\Big[(\\rlap\/k+m_Q)\\sigma^{\\mu\\nu} + 2u (m_Q^2-k^2)x^\\mu \\gamma^\\nu\\Big].\n\\end{eqnarray}\nAfter summing over color indices one can find the following relation for the symmetric current and   contribution of  the exchange of the gluon between the heavy quark $Q$ and the light pseudoscalar meson ${\\cal P}$  (Fig. \\ref{fig:Npert1}a) as follows:\n\\begin{eqnarray}\\label{QCD2Snp}\n\\Pi^{\\text{QCD(2a)}}_{({\\cal S})\\rho\\sigma}(p,q) &=&- \\frac{g_s}{12}\\int d^4 x \\int {d^4k \\over (2\\pi)^4}  \\int_0^1\ndu  e^{i (q-k).x} \\langle {\\cal P}(q) \\vert \\bar{q}(x)\\Gamma^J G_{\\mu\\nu}(ux)q(0)\\vert 0\\rangle \\nonumber \\\\\n&\\times& \\Bigg\\{ \\Bigg[ \\text{Tr}\\big[ \\Gamma_J\\tilde{\\Delta}^{\\mu\\nu}_{Q}(x) \\big]\\Big( \\gamma_5 S^{(\\text{pert.})}_{Q^{\\prime}}(x) \\gamma_5\\Big)_{\\rho\\sigma}+ \\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\tilde{\\Gamma}_J\\Delta^{\\mu\\nu}_{Q}(x)\\gamma_5 \\Big)_{\\rho\\sigma} \\nonumber \\\\ &+& t\\Big\\{ \\text{Tr}\\big[\\Gamma_J\\gamma_5 \\tilde{\\Delta}^{\\mu\\nu}_{Q}(x)\\big]\\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x) \\Big)_{\\rho\\sigma} + \\text{Tr}\\big[\\Gamma_J \\tilde{\\Delta}^{\\mu\\nu}_{Q}(x)\\gamma_5\\big]\\Big(S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5 \\Big)_{\\rho\\sigma} \\nonumber \\\\ \n&+& \\Big(\\gamma_5 S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5\\tilde{\\Gamma}_J\\Delta^{\\mu\\nu}_{Q}(x)\\Big)_{\\rho\\sigma} - \\Big( S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\tilde{\\Gamma}_J\\gamma_5\\Delta^{\\mu\\nu}_{Q}(x)\\gamma_5\\Big)_{\\rho\\sigma}\n\\Big\\} \\nonumber\\\\ \n&+& t^2 \\Big\\{ \\text{Tr}\\big[\\Gamma_J\\gamma_5 \\tilde{\\Delta}^{\\mu\\nu}_{Q}(x)\\gamma_5\\big]\\Big(S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\Big)_{\\rho\\sigma} - \\Big( S_{Q^{\\prime}}^{(\\text{pert.})}(x)\\gamma_5\\tilde{\\Gamma}_J\\gamma_5\\tilde{\\Delta}^{\\mu\\nu}_{Q}(x)\\Big)_{\\rho\\sigma} \\Big\\}\\Bigg] \\nonumber\\\\ \n&+& \\Bigg( \\Delta^{\\mu\\nu}_{Q}(x) \\leftrightarrow S^{(\\text{pert.})}_{Q^{\\prime}}(x) \\Bigg) \\Bigg\\}.\n\\end{eqnarray}\nwhere $ \\tilde{\\Delta}^{\\mu\\nu}_{Q}(x) =C \\Delta^{T,\\mu\\nu}_{Q}(x) C$.\nThe other contribution, $\\Pi^{\\text{QCD(2b)}}_{({\\cal S})}$, which is responsible for the exchange of one gluon between the heavy quark $Q^{\\prime}$ and the pseudoscalar meson $\\cal P$, can simply be calculated by taking into account the perturbative part of  $Q$-propagator and one-gluon emission part of   $Q^{\\prime}$-propagator. Other contributions as well as  the anti-symmetric CF are calculated in a similar way. The results are too lengthy and not presented here.\n\nFrom Eqs. \\ref{QCD2Spert} and \\ref{QCD2Snp} it is clear that the non-perturbative nature of the interaction is represented by the non-local matrix elements \n\\begin{eqnarray}\\label{matel}\n&\\langle {\\cal P}(q) \\vert \\bar{q}(x)\\Gamma^J q(0)\\vert 0\\rangle~,&  \\nonumber\\\\\n&\\langle {\\cal P}(q) \\vert \\bar{q}(x)\\Gamma^J G_{\\mu\\nu}(ux)q(0)\\vert 0\\rangle~,&\n\\end{eqnarray}\nwhich can be expressed in terms of two- and three-particle DAs of different twists for the light pseudoscalar meson ${\\cal P}$. The expressions for the above matrix elements in terms of DAs and the explicit form of the DAs are given in the appendices \\ref{APA} and \\ref{DAs}, respectively. \n\nInserting the heavy quark propagators \\ref{HQP} and the expressions for non-local matrix elements \\ref{matel} from Appendix \\ref{APA} into the CFs \\ref{QCD2Spert} and \\ref{QCD2Snp} one obtains the version of CF that is ready for performing the Fourier and Borel transformations as well as continuum subtraction. In this stage the CF contains several kinds of configurations with a general form of\n\\begin{eqnarray}\\label{STR1}\nT_{[~~,\\alpha,\\alpha\\beta]}(p,q)&=& i \\int d^4 x \\int_{0}^{1} dv  \\int {\\cal D}\\alpha e^{ip.x} \\big(x^2 \\big)^n  [e^{i (\\alpha_{\\bar q} + v \\alpha _g) q.x} \\mathcal{G}(\\alpha_{i}) , e^{iq.x} f(u)] \\nonumber\\\\\n&\\times& [1 , x_{\\alpha} , x_{\\alpha}x_{\\beta}]  K_{\\mu}(m_Q\\sqrt{-x^2})  K_{\\nu}(m_Q\\sqrt{-x^2}).\n\\end{eqnarray} \nHere the expressions in the brackets represent different structures that come from calculations. The blank subscript bracket indicates no $x_\\alpha$ in the structure and $n$ is an integer. The two and three-particle matrix elements lead to wave functions that are denoted by $f(u)$ and $\\mathcal{G}(\\alpha_{i})$  respectively.\n${\\cal D}\\alpha$ is called measure and defined as \n\\begin{equation*}\n\\int \\mathcal{D}\\alpha =\\int_{0}^{1}d\\alpha _{q}\\int_{0}^{1}d\\alpha _{\\bar{q}%\n}\\int_{0}^{1}d\\alpha _{g}\\delta (1-\\alpha _{q}-\\alpha _{\\bar{q}}-\\alpha\n_{g}).\n\\end{equation*}\nThere are several representations for the Bessel function $K_\\nu$. Here we use the cosine representation as \n\\begin{equation}\\label{CosineRep}\nK_\\nu(m_Q\\sqrt{-x^2})=\\frac{\\Gamma(\\nu+ 1\/2)~2^\\nu}{\\sqrt{\\pi}m_Q^\\nu}\\int_0^\\infty dt~\\cos(m_Qt)\\frac{(\\sqrt{-x^2})^\\nu}{(t^2-x^2)^{\\nu+1\/2}},\n\\end{equation}\nwhich helps us to increase the radius of convergence of the CF \\cite{Azizi:2018duk}.\nTo perform the Fourier transformation we use the exponential representations of the $x-$structures as\n\\begin{eqnarray}\\label{trick1}\n(x^2)^n &=& (-1)^n \\frac{d^n}{d \\beta^n}\\big(e^{- \\beta x^2}\\big)\\arrowvert_{\\beta = 0}, \\nonumber \\\\\nx_{\\alpha} e^{i P.x} &=& (-i) \\frac{d}{d P^{\\alpha}} e^{i P.x}.\n\\end{eqnarray} \nTo be specific, one specific configuration that appears has the generic form \n\\begin{eqnarray}\\label{Z1}\n{\\cal Z}_{\\alpha\\beta}(p,q) &=& i \\int d^4 x \\int_{0}^{1} dv  \\int {\\cal D}\\alpha e^{i[p+ (\\alpha_{\\bar q} + v \\alpha _g)q].x} \\mathcal{G}(\\alpha_{i}) \\big(x^2 \\big)^n  \\nonumber\\\\\n&\\times& x_\\alpha x_\\beta  K_{\\mu}(m_Q\\sqrt{-x^2})  K_{\\nu}(m_Q\\sqrt{-x^2}).\n\\end{eqnarray}\nUsing some variable changes and performing the double Borel transformation by employing \n\\begin{equation} \\label{Borel1}\n{\\cal B}_{p_1}(M_{1}^{2}){\\cal B}_{p_2}(M_{2}^{2})e^{b (p + u q)^2}=M^2 \\delta(b+\\frac{1}{M^2})\\delta(u_0 - u) e^{\\frac{-q^2}{M_{1}^{2}+M_{2}^{2}}},\n\\end{equation}\nin which  $u_0 = M_{1}^{2}\/(M_{1}^{2}+M_{2}^{2})$ one can find the final Borel transformed result for the corresponding structure as\n\\begin{eqnarray}\\label{STR4}\n{\\cal Z}_{\\alpha\\beta}(M^2) &=& \\frac{i  \\pi^2 2^{4-\\mu-\\nu} e^{\\frac{-q^2}{M_1^2+M_2^2}}}{M^2 m_{Q_1}^{2\\mu} m_{Q_2}^{2\\nu}}\\int  \\mathcal{D}\\alpha  \\int_{0}^{1} dv \\int_{0}^{1} dz \\frac{\\partial^n }{\\partial \\beta^n} e^{-\\frac{m_1^2 \\bar{z} + m_2^2 z}{z \\bar{z}(M^2 - 4\\beta)}} z^{\\mu-1}\\bar{z}^{\\nu-1} (M^2 - 4\\beta)^{\\mu+\\nu-1} \\nonumber\\\\\n&\\times & \\delta[u_0 - (\\alpha_{q} + v \\alpha_{g})]  \\Big[ p_\\alpha p_\\beta + (v \\alpha_{g} +\\alpha_{q})(p_\\alpha q_\\beta +q_\\alpha p_\\beta ) + (v \\alpha_{g} +\\alpha_{q})^2 q_\\alpha q_\\beta \\nonumber \\\\ \n&&+ \\frac{M^2}{2}g_{\\alpha\\beta} \\Big].\n\\end{eqnarray}\nThe details of calculation can be found in \\cite{Olamaei:2020bvw}.\n\nAccording to \\ref{CFPhys}, we choose the structure $\\rlap\/q \\rlap\/p \\gamma_{5}$ in QCD side as well. Therefore, one can write \n\\begin{eqnarray}\n\\Pi^{\\text{QCD}}_{{ B}_1 { B}_2 {\\cal P}}(M^2) = \\Pi_{{ B}_1 { B}_2 {\\cal P}}(M^2) \\rlap\/q \\rlap\/p \\gamma_{5}, \n\\end{eqnarray}\nwhere ${ B}_1$, ${B}_2$ and ${\\cal P}$ represent the initial baryon,  final baryon and the pseudoscalar meson, respectively.\n The coefficient of the structure $\\rlap\/q \\rlap\/p \\gamma_{5}$, i.e. $\\Pi_{{ B}_1 {B}_2 {\\cal P}}(M^2)$, is obtained considering all the contributions discussed above.\nAs an example,  we present the expression for the invariant function for the specific channel $\\Omega_{bb}\\rightarrow\\Xi_{bb}\\bar{K}^0$ after the Borel transformation, which reads\n\\begin{eqnarray}\n\\Pi_{\\Omega_{bb}\\Xi_{bb}\\bar{K}^0}(M^2) &=& \\dfrac{e^{-\\frac{m_{\\bar{K}^0}^2}{4M^2}}}{6912\\pi^2M^6m_b}\\int_{0}^{1}dz\\dfrac{e^{-\\frac{m_b^2}{M^2 z \\bar{z}}}}{z^2 \\bar{z}^2}\\nonumber \\\\ \n&\\times&\\Bigg\\{ 72 m_b M^6 z \\bar{z}^2 \\Bigg( 3f_{\\bar{K}^0}m_{\\bar{K}^0}^2m_b(t^2-1)\\big(m_b^2+2M^2z\\bar{z}\\big){\\mathbb A}(u_0) \\nonumber \\\\\n&+&2M^2z\\Big[ -6f_{\\bar{K}^0}m_bM^2(t^2-1)\\bar{z}\\varphi_{\\bar{K}^0}(u_0) \\nonumber \\\\\n&+&\\mu_{\\bar{K}^0}(\\tilde \\mu_{\\bar{K}^0}^2 -1)\\Big( 2m_b^2(1+t^2)+3M^2(t-1)^2 z\\bar{z}\\Big)\\varphi_{\\sigma}(u_0)\\Big] \\Bigg)\\nonumber\\\\\n&+&432m_b M^8 z\\bar{z}^2\\int_{0}^{1}dv\\int \\mathcal{D}\\alpha \\Bigg( f_{\\bar{K}^0}m_{\\bar{K}^0}^2m_b(t^2-1)\\delta[u_0-(\\alpha_{q}+v\\alpha_{g})] \\nonumber\\\\\n&\\times& \\Big( (2v-1)z{\\cal A}_\\parallel (\\alpha_i)+(2z-3){\\cal V}_\\parallel(\\alpha_i) +2\\bar{z}{\\cal V}_\\perp(\\alpha_i)\\Big) \\\\\n&-&\\mu_{\\bar{K}^0}M^2(t-1)^2z\\bar{z}\\delta^{\\prime}[u_0-(\\alpha_{q}+v\\alpha_{g})] {\\cal T}(\\alpha_i)\\Bigg) \\nonumber\\\\\n&+& g_s^2\\langle GG \\rangle\\Bigg[ -3f_{\\bar{K}^0}m_{\\bar{K}^0}^2(t^2-1)\\Big( 2m_b^6-3m_b^4 M^2\\bar{z}^2 -6m_b^2M^4z \\bar{z}^3-6M^6z^2\\bar{z}^4{\\mathbb A}(u_0) \\Big)\\nonumber\\\\\n&+&4M^2\\bar{z}\\Big[-3f_{\\bar{K}^0}(t^2-1)z\\Big(-2m_b^4-m_b^2M^2(5z-3)\\bar{z}+6M^4z\\bar{z}^3 \\Big)\\varphi_{\\bar{K}^0}(u_0)\\nonumber\\\\\n&-&\\mu_{\\bar{K}^0}(\\tilde \\mu_{\\bar{K}^0}^2 -1)m_b\\Big( 2m_b^4(1+t^2)+m_b^2M^2\\big[(1+t^2)(1-4z)-6t \\big]z \\nonumber\\\\\n&+&M^4\\big[ (1+t^2)(1-4z)-6t\\big]z^2\\bar{z}\\varphi_{\\sigma}(u_0)\\Big) \\Big]\\nonumber\\\\\n&+&6\\int_{0}^{1}dv\\int\\mathcal{D}\\alpha \\Bigg(-f_{\\bar{K}^0}m_{\\bar{K}^0}^2M^2(t^2-1)\\bar{z}\\delta[u_0-(\\alpha_{q}+v\\alpha_{g})]\\nonumber\\\\\n&\\times&\\Big\\{ (2v-1)\\Big[m_b^4-m_b^2M^2z(1+2z)-M^4z^2(1+2z)\\bar{z}\\Big]{\\cal A}_\\parallel (\\alpha_i) \\nonumber\\\\\n&+&\\Big[ -4m_b^2+m_b^2M^2z(1+2z)+M^4z^2(1+z-2z^2) \\Big]{\\cal V}_\\parallel(\\alpha_i)+2m_b^4{\\cal V}_\\perp(\\alpha_i) \\Big\\}\\nonumber\\\\\n&+&m_b M^4\\mu_{\\bar{K}^0}(t-1)2(2v-1)z\\bar{z}(m_b^2+M^2z\\bar{z})\\delta^{\\prime}[u_0-(\\alpha_{q}+v\\alpha_{g})] {\\cal T}(\\alpha_i) \\Bigg) \\Bigg]\\Bigg\\}.\\nonumber\n\\end{eqnarray}\nThe invariant functions for other channels in QCD side are calculated from the same manner but they are  not presented here because of their very lengthy expressions. \n\nThe next step is the  continuum subtraction. To this end,  we set the argument in  $ e^{-\\frac{m_1^2 \\bar{z} + m_2^2 z}{M^2 z \\bar{z}}} $ (in the case of different heavy quarks) or  $ e^{-\\frac{m_1^2}{M^2 z \\bar{z}}} $ (in the case of identical heavy quarks) equal to  $s_0$, with  $s_0$ being the continuum threshold for higher states and continuum.  Here, $ m_1 $ and $ m_2 $ stand for the heavy quarks' masses. As a result,   the limits of $z$ are changed  as\n\\begin{eqnarray}\\label{zsubtraction}\n\\int_{0}^{1}dz \\rightarrow \\int_{z_{\\text{min}}}^{z_{\\text{max}}}dz,\n\\end{eqnarray}\t\nwhere for two different heavy quarks we get,\n\\begin{eqnarray}\\label{zlimits}\nz_{\\text{max}(\\text{min})}=\\frac{1}{2s_0}\\Big[(s_0+m_1^2-m_2^2)+(-)\\sqrt{(s0+m_1^2-m_2^2)^2-4m_1^2s_0}\\Big].\n\\end{eqnarray}\nIn the case of the same heavy quarks, we need to just put $ m_1=m_2 $ in this result. \n\n\nAfter performing continuum subtraction, the invariant function  becomes $s_0$-dependent. We had also another auxiliary parameter $ t $ in the currents. Hence, in terms of three auxiliary parameters, that we are going to numerically analyze the results with respect to their variations, and after equating the coefficients of the selected structures from both the physical and QCD sides, we get the strong coupling constants as\n\\begin{eqnarray}\\label{SR}\ng_{B_1 B_2  {\\cal P}}(M^2,s_0,t)=\\frac{1}{\\lambda_{B_1}\\lambda_{B_2}} e^{\\frac{m_{B_1}^2+m_{B_2}^2}{2M^2}}\\Pi_{B_1 B_2  {\\cal P}}(M^2,s_0,t).\n\\end{eqnarray}\nWe are going to numerically analyze these sum rules in the next section.\n\n\\section{Numerical results}\\label{NA}\nThere are two sets of input parameters needed to perform the numerical analysis. One corresponds to the mass and decay constants of the  light pseudoscalar meson as well as the non-perturbative parameters coming from the light-cone DAs of different twists calculated at the renormalization scale $\\mu=1\\text{GeV}$. These parameters are collected in Tables \\ref{tabmeson} and \\ref{tabDAs}, respectively. The other corresponds to the doubly heavy baryons masses and residues which are  taken from \\cite{Aliev:2012ru} and are presented in Table \\ref{tabBaryon}.\n\\begin{table}[t]\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\addtolength{\\arraycolsep}{1pt}\n\t$$\n\t\\rowcolors{2}{cyan!10}{white}\n\t\\begin{tabular}{|c|c|c|c|}\n\t\\rowcolor{cyan!30}\n\t\\hline \\hline\n\t\\mbox{Parameters}         &  \\mbox{Values  $[\\text{MeV}]$ }  \n\t\\\\\n\t\\hline\\hline\n\t$ m_{c} $    &  $ 1.275^{+0.025}_{-0.035}~\\mbox{GeV} $      \\\\\n\t$ m_b $    & $  4.18^{+0.04}_{-0.03}~\\mbox{GeV} $      \\\\\n\t$ m_{\\eta } $    &   $547.862\\pm0.018 $  \\\\\n\t$ m_{\\eta^{\\prime} } $    &   $957.78\\pm0.06 $  \\\\\n\t$ m_{K^0}  $  &   $497.648\\pm0.022$    \\\\\n\t$ m_{K^{\\pm}}  $  &   $493.677\\pm0.013$    \\\\\n\t$  f_\\pi $    & $  131  $     \\\\\n\t$  f_\\eta $    & $  130  $     \\\\\n\t$  f_{\\eta^{\\prime}} $    & $  136  $     \\\\\n\t$  f_{K} $    & $  160  $     \\\\\n\t\\hline \\hline\n\t\\end{tabular}\n\t$$\n\t\\caption{The meson masses and leptonic decay constants along with the heavy quark masses \\cite{Tanabashi:2018oca,Ball:2005vx,Ball:2004ye,Ball:1998je}.} \\label{tabmeson} \n\t\\renewcommand{\\arraystretch}{1}\n\t\\addtolength{\\arraycolsep}{-1.0pt}\n\\end{table} \n\n\\begin{table}[t]\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\addtolength{\\arraycolsep}{1pt}\n\t$$\n\t\t\t\\rowcolors{2}{cyan!10}{white}\n\t\\begin{tabular}{|c|c|c|c|c|c|}\n\t\t\t\t\\rowcolor{cyan!30}\n\t\\hline \\hline\n\t\\text{meson}  &  $a_2$ & $\\eta_3$ & $w_3$ & $\\eta_4$ & $w_4$  \n\t\\\\\n\t\\hline\\hline\n\t $\\pi$  & 0.44 & $0.015$ & -3 & 10 & 0.2     \\\\\n\n\t$ $K$ $ & 0.16  & $0.015$ & -3 & 0.6 & 0.2     \\\\\n\t$\\eta $ & 0.2 & 0.013 & -3 & 0.5 & 0.2     \\\\\n\t\\hline \\hline\n\t\\end{tabular}\n\t$$\n\t\\caption{Input parameters for twist 2, 3 and 4 DAs at the renormalization scale $\\mu=1\\text{GeV}$ \\cite{Ball:2005vx,Ball:2004ye}.}  \\label{tabDAs} \n\t\\renewcommand{\\arraystretch}{1}\n\t\\addtolength{\\arraycolsep}{-1.0pt}\n\\end{table} \n\n\\begin{table}[t]\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\addtolength{\\arraycolsep}{1pt}\n\t$$\n\t\\rowcolors{2}{cyan!10}{white}\n\t\\begin{tabular}{|c|c|c|c|}\n\t\\rowcolor{cyan!30}\n\t\\hline \\hline\n\t\\mbox{Baryon}         &  \\mbox{Mass} $[\\text{GeV}]$  & \\mbox{Residue $[\\text{GeV}^3]$}\n\t\\\\\n\t\\hline\\hline\n\t\n\t$ \\Xi_{cc}  $    &  $ 3621.4\\pm0.8~\\mbox{MeV} $ \\cite{Tanabashi:2018oca}  & $0.16\\pm0.03$ \\\\\n\t$ \\Xi_{bc}  $    &  $ 6.72\\pm0.20 $ & $0.28\\pm0.05$  \\\\\n\t$ \\Xi^\\prime_{bc}  $    &  $ 6.79\\pm0.20$ & $0.3\\pm0.05$  \\\\\n\t$ \\Xi_{bb} $    &  $ 9.96\\pm0.90  $  & $0.44\\pm0.08$  \\\\\n\t$ \\Omega_{cc}  $    &  $ 3.73\\pm0.20 $  & $0.18\\pm0.04$ \\\\\n\t$ \\Omega_{bc}  $    &  $ 6.75\\pm0.30 $  & $0.29\\pm0.05$ \\\\\n\t$ \\Omega^\\prime_{bc}  $    &  $ 6.80\\pm0.30 $ & $0.31\\pm0.06$  \\\\\n\t$ \\Omega_{bb} $    &  $ 9.97\\pm0.90  $  & $0.45\\pm0.08$ \\\\\n\t\n\t\\hline \\hline\n\\end{tabular}\n$$\n\\caption{The baryons' masses and residues   \\cite{Aliev:2012ru}.}  \\label{tabBaryon} \n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-1.0pt}\n\\end{table} \n \n\n\nThe sum rules for the strong coupling constants depend also on three auxiliary parameters $M^2$,  $s_0$ and $t$  that should be fixed. To find the working intervals of these parameters the standard prescriptions of the method are used: the variations of the results with respect to the changes in these parameters should be minimal. \nFirst of all we would like to fix the working region of $t$. Considering $t=\\tan\\theta$,  we plot the strong coupling constants with respect to $\\cos\\theta$. We choose $\\cos\\theta$ and vary it in the interval $ [-1,1] $ in order to explore $ t $  at all regions.  Our analyses show that the variations of the  results are minimal in the intervals $0.5 \\leq \\cos\\theta \\leq 0.7$ and $-0.7 \\leq \\cos\\theta \\leq -0.5$   for all the strong decays under study.\n\n\n\n\n\n\nThe continuum threshold depends on the energy of the first excited state at each channel. Unfortunately, we have no experimental information on the first excited doubly heavy baryons. We choose it in the interval  $(m_B+0.3)^2\\leq s_0\\leq(m_B+0.7)^2 \\text{GeV}^2$, which dependence of the results on $s_0$ are  weak. As examples, we show the dependence of $g_{\\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\eta (\\eta^\\prime)}  $  on continuum threshold  in Fig~\\ref{fig:s0}. From this figure, we see that the strong coupling constants   depend on the variation of the $s_0$ very weakly in the selected intervals.\n\\begin{figure}[h!]\n\\includegraphics[width=1.0\\textwidth]{s0}\n\\caption{The strong couplings $g_{\\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\eta (\\eta^\\prime)}  $ \n\t\tas  functions of continuum threshold $s_0$ at average value of $\\cos\\theta$.}\n\\label{fig:s0}\n\\end{figure} \n\n\\begin{figure}[h!]\n\t\\includegraphics[width=1.0\\textwidth]{m2}\n\t\\caption{The strong couplings $g_{\\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\eta (\\eta^\\prime)}  $ \n\t\tas  functions of $M^2$ at average value of $\\cos\\theta$.}\n\t\\label{fig:msq}\n\\end{figure}\nFor $M^2$, the lower and higher limits are determined as follows. The lower bound, $M^2_{\\text{min}}$, is found demanding the OPE convergence. The higher bound, $M^2_{\\text{max}}$, is determined by the requirement of pole dominance, i.e.,\n\\begin{eqnarray}\nR=\\dfrac{\\int_{(m_Q+m_{Q^{\\prime}})^2}^{s_0}ds \\rho(s) e^{-s\/M^2}}{\\int_{(m_Q+m_{Q^{\\prime}})^2}^{\\infty}ds \\rho(s) e^{-s\/M^2}}\\geq \\frac{1}{2}.\n\\end{eqnarray} \n\\begin{table}[h!]\n\t\\begin{center}\n\t\t\\rowcolors{2}{cyan!10}{white}\n\t\t\\begin{tabular}{ c c c c }\n\t\t\t\\rowcolor{cyan!30}\n\t\t\n\t\t\n\t\t\tChannel & $ M^2 $(GeV$ ^2 $) & $ s_0 $ (GeV$ ^2 $) & strong coupling constant\\\\\n\t\t\t\t\t\t\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{c}{  \\textcolor{cyan}{Decays to $ \\pi $}} \\\\\n\t\t\t\\hline\n$ \\Xi_{bb}\\rightarrow \\Xi_{bb} \\pi^0  $&$ 14\\leq M^2\\leq18 $ &$ 105.3\\leq s_0\\leq113.6 $ &$  17.63^{\\:0.38}_{\\:0.24}  $\\\\\n$ \\Xi_{bb}\\rightarrow \\Xi_{bb} \\pi^\\pm  $&$ 14\\leq M^2\\leq18 $ &$ 105.3\\leq s_0\\leq113.6 $ &$  24.93^{\\:0.53}_{\\:0.33}  $\\\\\n\\hline \n$ \\Xi_{bc}\\rightarrow \\Xi_{bc} \\pi^0  $&$ 7\\leq M^2\\leq10 $ &$ 49.3\\leq s_0\\leq55$ &$3.76^{\\:0.17}_{\\:0.10}   $\\\\\n$ \\Xi_{bc}\\rightarrow \\Xi_{bc} \\pi^\\pm  $&$ 7\\leq M^2\\leq10 $ &$ 49.3\\leq s_0\\leq55$ &$5.32^{\\:0.24}_{\\:0.14}   $ \\\\\n\\hline\n$ \\Xi_{cc}\\rightarrow \\Xi_{cc} \\pi^0  $ & $ 3\\leq M^2\\leq6 $ &$ 15.4\\leq s_0\\leq18.7 $ & $  5.27^{\\:0.97}_{\\:0.70} $ \\\\\n$ \\Xi_{cc}\\rightarrow \\Xi_{cc} \\pi^\\pm  $ & $ 3\\leq M^2\\leq6 $ &$ 15.4\\leq s_0\\leq18.7 $ & $  7.45^{\\:1.37}_{\\:0.98} $\\\\\n\\hline\n$ \\Xi^\\prime_{bc}\\rightarrow \\Xi^\\prime_{bc} \\pi^0  $&$ 7\\leq M^2\\leq10 $ &$ 50.3\\leq s_0\\leq56.1 $ &$  7.84^{\\:0.24}_{\\:0.24}$   \\\\\n$ \\Xi^\\prime_{bc}\\rightarrow \\Xi^\\prime_{bc} \\pi^\\pm $&$ 7\\leq M^2\\leq10 $ &$ 50.3\\leq s_0\\leq56.1 $ &$  11.08^{\\:0.33}_{\\:0.34}$ \\\\\n\\hline\n$ \\Xi^\\prime_{bc}\\rightarrow \\Xi_{bc} \\pi^0  $&$ 7\\leq M^2\\leq10 $ &$ 50.3\\leq s_0\\leq56.1 $ &$  0.62^{\\:0.14}_{\\:0.13}$   \\\\\n$ \\Xi^\\prime_{bc}\\rightarrow \\Xi_{bc} \\pi^\\pm $&$ 7\\leq M^2\\leq10 $ &$ 50.3\\leq s_0\\leq56.1 $ &$  0.89^{\\:0.20}_{\\:0.19}$ \\\\\n\\hline\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{c}{ \\textcolor{cyan}{Decays to $K$}} \\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{bb}\\rightarrow \\Xi_{bb} \\bar{K}^0  $&$ 14\\leq M^2\\leq18 $ &$ 105.3\\leq s_0\\leq113.6 $ &$ 22.36^{\\:1.30}_{\\:0.91}  $\\\\\n\t\t\t$ \\Omega_{bb}\\rightarrow \\Xi_{bb} K^-  $& $ 14\\leq M^2\\leq18 $ &$105.5\\leq s_0\\leq113.8 $ &$ 22.90^{\\:1.31}_{\\:0.91} $\\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\t$ \\Omega_{bc}\\rightarrow \\Xi_{bc} \\bar{K}^0  $&$ 7\\leq M^2\\leq10 $ &$ 49.7\\leq s_0\\leq55.5 $ &$ 4.04^{\\:0.42}_{\\:0.25}  $\\\\\n\t\t\t$ \\Omega_{bc}\\rightarrow \\Xi_{bc} K^-  $& $ 7\\leq M^2\\leq10 $ &$49.7\\leq s_0\\leq55.5 $ &$  4.05^{\\:0.42}_{\\:0.25} $\\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t$ \\Omega_{cc}\\rightarrow \\Xi_{cc} \\bar{K}^0  $&$ 3\\leq M^2\\leq6 $ &$ 16.2\\leq s_0\\leq19.6 $ &$  5.76^{\\:1.40}_{\\:0.80} $\\\\\n\t\t\t$ \\Omega_{cc}\\rightarrow \\Xi_{cc} K^-  $& $ 3\\leq M^2\\leq6 $ &$ 16.2\\leq s_0\\leq19.6 $ &$  5.78^{\\:1.42}_{\\:0.84}  $\\\\\n\t\t\t\\hline\n\t\t\t\n\t\t\t$ \\Omega^\\prime_{bc}\\rightarrow \\Xi^\\prime_{bc} \\bar{K}^0  $&$ 7\\leq M^2\\leq10$ &$ 50.4\\leq s_0\\leq56.2 $ &$  11.11^{\\:1.20}_{\\:0.76}  $\\\\\n\t\t\t$ \\Omega^\\prime_{bc}\\rightarrow \\Xi^\\prime_{bc} K^-  $& $ 7\\leq M^2\\leq10 $ &$ 50.4\\leq s_0\\leq56.2 $ &$  11.14^{\\:1.19}_{\\:0.75} $\\\\\n\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{c}{  \\textcolor{cyan}{Decays to $\\eta$}} \\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{bb}\\rightarrow \\Omega_{bb} \\eta  $&$ 14\\leq M^2\\leq18 $ &$ 105.3\\leq s_0\\leq113.6 $ &$ 17.20^{\\:0.75}_{\\:0.75}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{bc}\\rightarrow \\Omega_{bc} \\eta $& $ 7\\leq M^2\\leq10 $ &$49.7\\leq s_0\\leq55.5 $ &$ 3.36^{\\:0.25}_{\\:0.15}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{cc}\\rightarrow \\Omega_{cc} \\eta $&$ 3\\leq M^2\\leq6 $ &$ 16.2\\leq s_0\\leq19.6 $ &$ 4.14^{\\:0.90}_{\\:0.48}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega^\\prime_{bc}\\rightarrow \\Omega^\\prime_{bc} \\eta$& $ 7\\leq M^2\\leq10 $ &$ 50.4\\leq s_0\\leq56.2 $ &$ 8.38^{\\:0.64}_{\\:0.42}  $\\\\\n\t\t\t\\hline\n\t\t\t\\hline\n\t\t\t\\multicolumn{4}{c}{  \\textcolor{cyan}{Decays to $\\eta^{\\prime}$}} \\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{bb}\\rightarrow \\Omega_{bb} \\eta^\\prime  $&$ 14\\leq M^2\\leq18 $ &$ 105.3\\leq s_0\\leq113.6 $ &$ 9.54^{\\:0.90}_{\\:0.77}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{bc}\\rightarrow \\Omega_{bc} \\eta^\\prime $& $ 7\\leq M^2\\leq10 $ &$49.7\\leq s_0\\leq55.5 $ &$ 1.78^{\\:0.24}_{\\:0.18}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega_{cc}\\rightarrow \\Omega_{cc} \\eta^\\prime  $&$ 3\\leq M^2\\leq6 $ &$ 16.2\\leq s_0\\leq19.6 $ &$ 1.80^{\\:0.80}_{\\:0.80}  $\\\\\n\t\t\t\\hline\n\t\t\t$ \\Omega^\\prime_{bc}\\rightarrow \\Omega^\\prime_{bc} \\eta^\\prime $& $ 7\\leq M^2\\leq10 $ &$ 50.4\\leq s_0\\leq56.2 $ &$ 4.43^{\\:0.76}_{\\:0.62}  $\\\\\n\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{Working regions of the Borel mass $ M^2 $ and   continuum threshold $ s_0 $  as well as the numerical values for different strong coupling constants extracted from the analyses.} \\label{tab:g} \n\\end{table}\n\n\n Fig. \\ref{fig:msq}  displays  dependence of  the $g_{\\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\Omega^{(\\prime)}_{QQ^{(\\prime)}} \\eta (\\eta^\\prime)}  $ on $M^2$ at its working window and at average values of other auxiliary parameters. From this figure we see mild variations of the results with respect to the changes in the Borel parameter $M^2$. Extracted from the analyses, the working intervals for all auxiliary parameters at all strong decay channels are depicted in Table \\ref{tab:g}.\n \n \n\nThe final results for the strong coupling constants under study are also  presented in Table \\ref{tab:g}. The errors in the presented results are due to the uncertainties in determinations of the working intervals for the auxiliary parameters, the errors in the masses and residues of the doubly heavy baryons and the uncertainties coming from the DAs parameters as well as other inputs. As we previously said, in Ref. \\cite{Olamaei:2020bvw} we investigated the symmetric $ \\Xi_ {QQ^{(\\prime)}}$ couplings to $ \\pi $ meson, in which the continuum subtraction procedure is different than that of the present study. We extracted those coupling constants in the present study, as well. Comparing the results on $ g_{\\Xi_{bb}\\Xi_{bb}\\pi^0} $, $ g_{\\Xi_{bc}\\Xi_{bc}\\pi^0} $ and $ g_{\\Xi_{cc}\\Xi_{cc}\\pi^0} $ from the present study and Ref. \\cite{Olamaei:2020bvw}, we see that the extracted values are close to each other within the presented errors. The small differences are due to the fact that, the values in   Ref. \\cite{Olamaei:2020bvw} were extracted in a single value for $ \\cos\\theta $ inside its working window, while in the present study we take the average of many values obtained at different values of $ \\cos\\theta $  in its working interval. We should remind that the errors are small in the present study compared to the results of  Ref. \\cite{Olamaei:2020bvw}. From Table \\ref{tab:g}, it is clear that, overall, the couplings for each symmetric\/anti-symmetric case and  pseudoscalar meson in $ bb $ channels are greater than those in $ cc $ channels and the later are greater than the couplings in $ bc $ channels. In extracting the results at $ \\eta $ and  $ \\eta^\\prime $, we have ignored from the mixing between these two states. Our results may be checked via different phenomenological approaches.\n\n\n\n\\section{Summary and concluding notes}\\label{SC}\nMotivated by the LHCb observation of  $\\Xi_{cc}^{++} (ccu)$ state, we investigated the strong vertices of the doubly heavy baryons of various quark contents with the light pseudoscalar  $ \\pi $, $ K $,  $\\eta$ and $ \\eta^\\prime $ mesons. We extracted the strong coupling constants at $ q^2=m^2_{\\cal P} $ from the strong coupling form factors. In the calculations, we used the general forms of the interpolating currents in their symmetric and anti-symmetric forms. We also used the light cone DAs of the pseudoscalar mesons entering the calculations. The strong coupling constants are fundamental objects that their investigation can help us get useful knowledge on the nature of the strong interactions among the participated particles. Such objects can also help us in our understanding of QCD as the theory of strong interactions. One of the main problems on the strong interactions between hadrons is to determine their interaction potential. The obtained results may be used in the  construction of such strong potentials.  \nThe obtained results may also help experimental groups in the analyses of the results obtained at hadron colliders. Investigation of doubly heavy baryons may help experiments in the course of search for the doubly heavy baryons. Note again that, we have only one state,  $\\Xi_{cc}$, detected in the experiment and listed in PDG. However, even for this particle there is a tension between the SELEX and LHCb results on the mass and width of this particle. More theoretical and experimental studies  on the properties of doubly heavy baryons and their various interactions with other hadrons are needed.\n\n\n\n\\section*{Acknowledgment}\n\n K. Azizi and S.~Rostami are thankful  to  Iran Science Elites Federation (Saramadan) for the financial support provided under the grant number ISEF\/M\/99171.\n\n\n\\section*{Note added:}\nAfter completing this study and at final stages of the proof readings we noticed that the paper Ref. \\cite{Alrebdi:2020rev} appeared in arXiv, where the authors calculate the strong coupling constants among some of  the doubly heavy baryons and  $\\pi$ and $K$ mesons. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe details of how the star formation rate (SFR) is regulated in galaxies in not well understood. Evidently, star formation is fueled by cold gas in and around galaxies \\citep{Kennicutt+12, Genzel+15, Saintonge+17,Tumlinson+17,Catinella+18}. External mechanisms that are governed by dark matter halos or the environment and\/or internal processes such as blackhole activity may regulate the amount, the in and out flows, and\/or the thermodynamic state of the cold gas in galaxies or their surrounding halos, thereby impacting SFR \\citep{Schaye+15, Henriques+17,Pillepich+18, Matthee+19}. Consequently, some of the observable properties of galaxies correlate with the properties of halos and the environment while others reflect the baryonic physics within galaxies. Most observable galaxy properties depend strongly on stellar mass ($M_\\star$), which is likely the primary parameter of galaxies characterizing the internal mechanisms. For example, star-forming galaxies (SFGs) follow a tight relationship (scatter of $\\sim 0.3-0.4$\\,dex) between $M_\\star$ and SFR \\citep[e.g.,][]{Brinchmann+04, Elbaz+07, Noeske+07,Speagle+14}. The residuals of this relation, $\\Delta\\,\\mathrm{SSFR}$, for SFGs depend weakly on the environment. In contrast, the fraction of quenched galaxies depends strongly on both $M_\\star$ and environment \\citep{Baldry+06, Bamford+09, Peng+10,Darvish+16, Kawinwanichakij+17,WangHuiyuan+18}. The environmental quenching mechanisms are effective in low-mass galaxies ($M_\\star \\lesssim10^{10}\\,M_\\odot$); high-mass galaxies quench by internal processes such as blackhole feedback \\citep[e.g.,][]{Croton+06, Hopkins+08,Fabian+12, Kormendy+13}. The majority of low-mass galaxies are quenched as satellites of more massive galaxies by interacting with the hot intracluster media (ICM) of their host halos, being starved of cold gas, or merging\/interacting with other satellites \\citep[for reviews see][]{Boselli+06,Cortese+21}.  \n\nNumerous studies have explored trends of galaxy properties with the environment to establish its importance in shaping galaxy mergers\/interactions, morphology, star formation, black hole activity, and cold gas \\citep[e.g.,][]{Dressler+80, Gomez+03, Balogh+04, Kauffmann+04, LinLihwai+10, Ellison+10, Wetzel+12, Kampczyk+13, vandeVoort+17, Donnari+21}. The following points list some noteworthy observational trends with the environment: (1) galaxy mergers\/interactions happen in all environments. Those that induce SFR enhancements, however, are favored in galaxies in low-density environments, likely because of their high gas fractions \\citep{Ellison+10}. (2) The morphology-environment relation is partly driven by $M_\\star$, in the sense that higher mass galaxies live in denser environments \\citep{Kauffmann+04}. (3) The impact of the environment on SFR is stronger than on morphology \\citep{Kauffmann+04, Ball+08, Bamford+09,Skibba+09}; at fixed SFR, the dependence of morphology on the environment is very weak. (4) Both mass quenching and environmental quenching are associated with morphological transformation of galaxies \\citep{Carollo+16}. (5) The effects of the environment on SFR and morphology are relatively local, within a single halo, $\\sim 1-2$\\,Mpc \\citep{Kauffmann+04, Blanton+07, Park+07,Blanton+09}. The observed environmental dependencies can be understood in terms of the host halo mass, and the galaxy's position within that halo \\citep{WooJoanna+13}. The large-scale environment probably has little influence on galaxy properties. (6) The properties of dark matter haloes such as their growth rates, concentrations, and interaction histories correlate with the environment in scale-dependent ways \\citep[][]{Wechsler+18,Behroozi+22}. These correlations are imprinted in observables such as the two-point correlation functions and $k$-nearest neighbor distances of galaxies, more prominently at $\\lesssim 2$\\,Mpc scale \\citep{Behroozi+22}.\n\nIt is not straightforward to attribute the quenching of typical quiescent galaxies (QGs) to their current dense environments, because they might have been quenched by internal processes or in a different environment at much earlier time. Either way, the correlation with their current environments may not be causal. Factors such as the star formation history, the assembly history of the dense environment\/cluster, and the pre-processing that happens before a small group falls onto the cluster need to be accounted for \\citep{Dressler+13, Mahajan13, Donnari+21}. Alternatively, studying environments and structures of galaxies that transition rapidly between SFGs and QGs may provide valuable insights on the causal relationship between environment and star formation quenching. These galaxies, namely, starbursts and their descendants, are the focus of this study. Because of their short evolution span, we expect these galaxies to have similar $M_\\star$, structures, and environments. This work examines this expectation in detail.\n\nQuenched post-starburst galaxies \\citep[QPSBs, also known as k+a galaxies;][]{Dressler+83, Quintero+04, Goto05, French+21} are a rare class of galaxies in transition from the star-forming to quiescent phase. Their spectra reveal very little ongoing star formation (as evidenced by weak emission lines due to lack of ionizing O \\& B stars) but a substantial recent star formation within the last $\\lesssim $1\\,Gyr (as indicated by strong Balmer absorption lines due to abundance of A stars). Because of this unusual combination of two spectral features, QPSBs are relatively easy to identify in large spectroscopic surveys of nearby galaxies. Starbursts are also relatively easy to identify because they have unusually high ongoing SFRs. Note that the progenitor-descendant relationship between these two classes is inferred from a stellar population analysis indirectly. Do all starbursts become QPSBs? Can some QPSBs form without prior bursts? Do they collectively pass through the AGN phase? These are the kinds of questions we want to answer by using additional information about their environments and structures. Unfortunately, identifying a complete and pure sample of transition PSBs with ongoing star formation or\/and AGN activity is extremely challenging \\citep{Wild+10,Yesuf+14, Alatalo+16,Baron+22}. Some uncertainty remains on how AGNs with strong Balmer absorption lines relate to the starburst and QPSB populations. This work provides some clarity on these AGNs. \n\n\\subsection{The Scope and Contributions of This Paper}\n\nThe main aim of this work is to do thorough consistency checks of the evolution from SFGs $\\rightarrow$ starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs $\\rightarrow$ QGs at $z < 0.16$ using multiple environmental indicators and structural properties based on the Sloan Digital Sky Survey (SDSS). The new contributions are: (1) examining this evolution as a whole using the same data and measurements in four narrow $M_\\star$ ranges. As later discussion will elucidate, previous studies rarely examined the environments and structures of starbursts and QPSBs together, let alone keeping their $M_\\star$ fixed. (2) Unlike many previous studies of starbursts or QPSBs, this study uses several multiscale environmental indicators to study their evolution. This is crucial because various environmental indicators have different information content and meaning, and they often lead to different results \\citep{Wilman+10, Haas+12, Muldrew+12,WooJoanna+13}. Therefore, this paper explores fixed aperture environmental density indicators ranging from scales of $0.5-8\\,h^{-1}$Mpc, $k$-nearest neighbor distances\/densities ($k=1, 3, 5$), the tidal parameter, two estimates of halo masses \\citep{Lim+17,Tinker21}, satellite\/central classification, and group\/cluster membership \\citep{Tempel+14}. Compared to related past studies, this work uses entirely new measurements or similar measurements with updated data. (3) To put this study in the broader context of galaxy evolution, the environments of starbursts and QPSBs are meticulously compared with those of galaxies in the upper SFMS, SFMS, lower SFMS, green valley, and those of QGs as well as AGNs. Due to the improvements or approaches described above, this study demonstrates that all QPSBs can be linked to some SFGs, starbursts, AGNs, and QGs that have similar $M_\\star$, structures, and multiscale environments. However, not all starbursts can be linked to QPSBs of similar properties, primarily because these starbursts are disc-dominated. In addition, this paper also shows that some QGs plausibly originated from recent quenching of starbursts, but most QGs live in utterly different environments today and are unlikely to be descendants of recently quenched galaxies.\n\n\\subsection{Inconsistencies of Previous Studies of Environments of Starbursts and QPSBs}\n\nAs summarized in this section and also discussed in section~\\ref{sec:disc}, the environments of starbursts and QSBs have been investigated by many previous studies. However, the results from these studies were not completely consistent because of differences in methodology (e.g., quantifying environment) and sample selection and characteristics (e.g., sample size, redshift, and $M_\\star$). Nonetheless, the majority of the studies at $z \\lesssim 0.3$ found that most QPSBs reside in the low-density\/field environments similar to those of SFGs \\citep{Zabludoff+96, Blake+04, Hogg+06, Goto05, Nolan+07, Yan+09,Wilkinson+17, Pawlik+18}. The existence of a large population of QPSBs in the low-density environments indicates that environmental mechanisms are not the dominant route by which QPSBs formed recently. On the other hand, there is evidence that at least some low-$z$ QPSBs have quenched due to environmental mechanisms \\citep{Owers+19, Paccagnella+17, Paccagnella+19, Vulcani+20}. For example, \\citet{Paccagnella+19} found that the QPSB fraction increases from the outskirts toward the cluster center and from the least to the most massive halos, suggesting that clusters are more efficient at producing QPSBs. As clusters are rare and most galaxies do not reside in clusters, it is still unclear how important the cluster environment is for QPSB formation.\n\nLikewise, the environments of starbursts are not well established. The handful of previous studies on environments of starbursts at $z < 0.3$, with the exception of \\citet{Owers+07}, are limited to a special class of galaxies known as ultraluminous infrared galaxies (ULIRGs) and LIRGs \\citep[][]{Koulouridis+06, HwangHS+10,Tekola+12, Tekola+14}. \\citet{Owers+07} found that starbursts are less clustered at $\\sim 1-15$\\,Mpc scales than the overall 2dFGRS\\footnote{2dF Galaxy Redshift Survey} galaxy population, and they do not preferentially live in clusters. \\citet{HwangHS+10} found that, at fixed $M_\\star$, the fraction of ULIRGs and LIRGs and their infrared luminosities show weak or no dependance on the environmental density, which was estimated using the projected distance to the 5th nearest neighbor galaxy ($r_{p,5}$). However, they found that the probability of being (U)LIRG and its IR luminosity both increase with the decreasing distance to the nearest late-type galaxy. Other similar studies found LIRGs live preferentially in low-density ($\\delta_{5}$) environments \\citep[][]{Goto05b,Ellison+13, Burton+13}. In contrast, \\citet{Tekola+12} found that LIRGs live in denser environments than those of non-LIRGs. These authors measured the overdensity around LIRGs by counting their numbers in a cylinder of $2\\,h^{-1}$\\,Mpc radius and $10\\,h^{-1}$\\,Mpc length ($|\\Delta v| < 1000\\,\\mathrm{km\\,s^{-1}}$) and comparing their distribution with a random catalog. They found that LIRGs show a strong correlation between this environmental density and the infrared luminosity, while non-LIRGs do not show such correlation. In short, galaxies with high infrared luminosities preferentially reside in low-mass ($\\lesssim 10^{13}\\,M_\\odot$) haloes \\citep{Tekola+14} or low-density environments on large scales. But the small-scale ($ \\lesssim 2 \\,h^{-1}$\\,Mpc) environments of LIRGs may be relatively higher than those of normal SFGs \\citep{Koulouridis+06, Tekola+12}. The environmental studies of local LIRGs and ULIRGs are limited by small sample sizes; therefore, their conclusions are tentative. This study complements them by selecting $\\sim 8,600$ starbursts using SFR based on UV-optical-mid IR spectral energy density (SED) fitting \\citep{Salim+16,Salim+18}. \n\nSimilarly, the environments of starbursts and QPSBs at $z \\gtrsim 0.3$ are also not well constrained. Some studies found that the fractions of QPSBs\/k+a are higher in high-density environments than in low-density environments \\citep[e.g.,][]{Tran+04, Poggianti+09, Yan+09, Vergani+10, Muzzin+12, Dressler+13, Socolovsky+18, McNab+21} and others did not \\citep{Balogh+99, Yan+09, Lemaux+17}. \\citet{Yan+09} found that the distribution of $k$-nearest neighbor densities of QPSBs at $z \\sim 0.8$ is similar to that of QGs, whereas \\citet{Lemaux+17} found that it is similar to that of SFGs. Discrepancies between these studies are caused primarily by differences in their sample selection and\/or methodology \\citep[for a detailed discussion, see][]{Yan+09, Lemaux+17}. The typical environment of distant starbursts is also unknown. There is a long-standing debate about the SFR-density relation (the sign of the correlation or lack thereof) of distant galaxies \\citep{Elbaz+07,Cooper+08, Patel+11,Sobral+11, Ziparo+14}.\n\n\\subsection{The Merger Origin and Structures of Starbursts and PSBs}\n\n A merger\/interaction-driven galaxy evolution is one of the hypotheses that have been proposed to explain the observed link between structure and SFR. Major galaxy mergers in simulations lead to a central mass concentration and black hole growth \\citep{Sanders+88, Mihos+96,Hopkins+08}. During galaxy mergers, the gas is perturbed and funneled to the center to power a nuclear starburst and an AGN activity. The gas is then depleted quickly by the starburst or removed or heated by stellar and\/or AGN feedback, eventually forming a quiescent, early-type galaxy. For example, QPSBs in Magneticum cosmological simulations were shut down rapidly by a merger-triggered AGN feedback, which redistributed and heated gas in the PSBs. About 89\\% of the simulated QPSBs at $z \\approx 0$ had at least one merger within the last 2.5\\,Gyr and about 65\\% had major ($M_\\star$ ratio greater than 1:3) mergers \\citep{Lotz+21}. In EAGLE simulations, around 14\\% of the simulated QPSBs have experienced mergers with $M_\\star$ ratios greater than 1:10 and $\\sim 3\\%$ with ratios greater than 1:3 within 0.5\\,Gyr of the onset of the PSB episode \\citep{Davis+19}. While AGNs are important in removing star-forming gas from some galaxies, they are not the dominant cause of gas removal in PSBs in EAGLE; only $\\sim 10\\%$ of the QPSBs had enhanced AGN accretion\/outburst. \n\nConsistent with the prevalence of mergers in simulated PSBs, a significant fraction of observed QPSBs show prominent signs of recent mergers or interactions \\citep{Yamauchi+08, YangYujin+08, Pracy+09, Pawlik+16, Sazonova+21}. Most QPSBs are also more centrally concentrated than SFGs \\citep{Nolan+07,YangYujin+08,Pracy+09, Sazonova+21}. Likewise, starbursts (including LIRGs and ULIRGs) are highly disturbed merger remnants \\citep{Sanders+96, Veilleux+02, Luo+14, Larson+16, Cibinel+19, Shangguan+19,Yesuf+21}. Alternatively, minor mergers may produce a significant fraction of rejuvenated PSBs by triggering new cycles of starbursts in passive, bulge-dominated galaxies \\citep{Dressler+13, Rowlands+18, Davis+19, Pawlik+18}. Although rare at $z \\approx 0$, the compaction process, triggered by an intense gas inflow episode, involving mergers, counter-rotating streams, or recycled gas may also produce PSBs \\citep{Dekel+14,Zolotov+15, Tacchella+16}. Several works have shown that PSBs at $z \\sim 1-2$ are compact \\citep{Yano+16, Almaini+17, Maltby+18, WuPo-Feng+20, Suess+21, Setton+22}.\n\nMerger signatures are hard to identify in observations; they fade with time \\citep{Lotz+08}, are hard to accurately quantify, and require deep imaging. Thus, disturbance parameters such as asymmetry cannot be measured reliably for typical starbursts and QPSBs using shallow SDSS images \\citep{Pawlik+16,Pawlik+18}. Alternatively, we will demonstrate that easily measurable structural parameters such as $C$ and $\\sigma_\\star$ and multiscale environmental parameters provide valuable information for large samples of starbursts and their descendants. In fact, complementary to the asymmetry parameter, $C$ and $\\sigma_\\star$ are good predictors of $\\Delta\\,\\mathrm{SSFR}$ for central galaxies \\citep[][]{Yesuf+21}.\n\nThe rest of the paper is structured as follows: Section~\\ref{sec:data} presents the stellar, structural, and environmental data used in this study. Section~\\ref{sec:res} presents the main results. To help interpret these results, Section~\\ref{sec:disc} presents in-depth discussion and comparisons with previous studies of starbursts and QPSBs. The summary and conclusions of this work are given in Section~\\ref{sec:conc}. We adopt a cosmology with $\\Omega_\\Lambda = 0.7$, $\\Omega_m =  0.3$ and $h = 0.7$.\n\n\\section{Data and Methodology}\\label{sec:data}\n\n\n\\subsection{Measurements of Galaxy Properties}\n\nOur sample is selected from the Sloan Digital Sky Survey \\citep[SDSS;][]{Aihara+11}. We use the publicly available Catalog Archive Server to retrieve some of the measurements used in this work such as the stellar velocity dispersion $\\sigma_\\star$, Petrosian radii, the spectral index H$\\delta_A$, and emission-line fluxes \\citep{Brinchmann+04}. The $M_\\star$ and SFR measurements are taken from version 2 of the GALEX-SDSS-WISE Legacy Catalog \\citep[GSWLC-2][]{Salim+16, Salim+18}. They are derived by spectral energy distribution (SED) fitting of UV-optical photometry with additional IR luminosity constraints using the Code Investigating GALaxy Emission \\citep[CIGALE]{Noll+09}. \\citet{Salim+18} estimated the total infrared luminosities from WISE 22\\,$\\mu$m or 12\\,$\\mu$m photometry using luminosity-dependent infrared templates of \\citet{Chary+01} and calibrations derived from a subset of galaxies that have Herschel far-infrared data. For narrow-line AGNs, their IR luminosities are corrected using \\ion{O}{3} 5007\\,{\\AA} emission-line equivalent width before using them in the SED fitting. Broad-line AGNs are excluded from the current sample because their SFRs estimates are not reliable. The SED fitting used template superposition of two exponential star formation histories (SFHs) of a younger population (100\\,Myr to 5\\,Gyr) and an old stellar population (formed 10\\,Gyr ago), with the young mass fraction varying between zero and 50\\%. The stellar population models were calculated for four stellar metallicities ($0.2-2.5\\,Z_\\odot$) using \\citet{Bruzual+03} models and assuming a \\citet{Chabrier03} stellar initial mass function.\n\n\\subsection{Sample Definitions}\n\nFrom the SDSS main galaxy sample, we select $387, 258$ galaxies at $z = 0.02-0.16$, with $M_\\star = 3\\times 10^9 - 3\\times 10^{11}\\,M_\\odot$,  with good SFR measurements ($\\mathrm{flag\\_SED=0}$), and which are at least 4\\,$h^{-1}$Mpc away from the survey edge. We define the difference from the ridge line of the SFMS, $\\Delta\\,\\mathrm{SSFR}$, by fitting a simple linear relation of the form $\\log\\,(\\mathrm{SSFR\/Gyr^{-1}}) = (a - 1)[\\log(M_\\star \/M_\\odot) - 10.5] + b$ galaxies to the galaxies with SSFR $> 0.01\\,\\mathrm{Gyr^{-1}}$. We fix $\\alpha=0.48$ based on the estimate of \\citet{Speagle+14} and derive $\\beta = -1.24$ from the median of the residuals. Based on their $\\Delta\\,\\mathrm{SSFR}$ we group galaxies into starbursts ($\\Delta\\,\\mathrm{SSFR}  > 0.6$\\,dex), upper SFMS ($\\Delta\\,\\mathrm{SSFR}  = 0.2-0.6$\\,dex), SFMS ($\\Delta\\,\\mathrm{SSFR} = -0.2-0.2$\\,dex), lower SFMS ($\\Delta \\,\\mathrm{SSFR}  = -(0.2-0.5)$\\,dex) , and green valley ($\\Delta\\,\\mathrm{SSFR} =  -(0.5-1)$\\,dex), and quiescent ($\\Delta\\,\\mathrm{SSFR}  < -1$\\,dex) galaxies. The total number of starbursts in our sample is 8,\\,614.\n\nWe define QPSBs as galaxies with the equivalent width (EW) of H$\\alpha< 3$\\,{\\AA} in emission and H$\\delta_A > 4${\\AA} in absorption \\citep{Yesuf+14}. We only select QPSBs with reliable spectral measurements. Namely, we exclude galaxies whose median signal-to-noise ratio per pixel (S\/N)  of the entire spectra are $\\mathrm{S\/N} < 10$ or have gaps around their H$\\alpha$ continua. To automatically identify objects with bad H$\\alpha$ measurements, we require the H$\\alpha$ continuum fluxes to be non-zero and the EWs of H$\\beta$ to be $< 3\\,${\\AA}. The total number of selected QPSBs is 752, which is about 10 times smaller than that of starbursts.\n\nAs mentioned before, selecting a complete and unbiased sample of the PSBs is not easy. Some attempts have been made to improve the definition of PSBs to include galaxies with ongoing star formation and\/or AGN activity \\citep{Wild+10,Yesuf+14, Alatalo+16}. We simply select a subsample of AGNs with H$\\delta_A > 4$\\,{\\AA} and show that these galaxies have similar structural and environmental properties as QPSBs, whereas AGNs with H$\\delta_A < 3$\\,{\\AA} are different from QPSBs. We identify ``pure\" AGNs using the emission-line ratios \\ion{O}{3}\/H$\\beta$ and \\ion{N}{2}\/H$\\alpha$ \\citep{Kewley+06} and H$\\alpha > 3$\\,{\\AA}. We require $\\mathrm{S\/N} > 3$ for the emission line fluxes and $\\mathrm{S\/N} > 10$ for the entire continuum spectra. The total number of AGNs with H$\\delta_A > 4$\\,{\\AA} (H$\\delta_A < 3$\\,{\\AA}) is 1,\\,006 (9,\\,544).\n\n\\subsection{Measurements of Environmental Properties}\n\nThis paper uses several environmental indicators measured by its author or others \\citep{Tempel+14, Lim+17, Tinker21}. The author calculates the mass density of galaxies within radii of 0.5, 1, 2, 4, and 8$\\,h^{-1}\\,\\mathrm{Mpc}$ centered around each galaxy and the projected distances to its $k = {1, 3, 5}$ nearest neighbors ($r_{p,k}$). In these calculations only galaxies that have $|\\Delta v| < 1000$\\,km\\,s$^{-1}$ relative to the primary galaxies are considered, thereby excluding unrelated foreground and background galaxies along the line of sight. The main results do not change if $|\\Delta v| < 500$\\,km\\,s$^{-1}$ or $|\\Delta v| < 1500$\\,km\\,s$^{-1}$ is used instead. The projected $k$-nearest neighbor surface density is defined as $\\Sigma_{k} = k\/(\\pi r_{p,k}^2)$. All densities are normalized by the median densities of all galaxies in a given mass range and in a redshift bin of $\\Delta z = 0.02$. This converts the densities into overdensities relative to the median density (e.g., $\\delta_{k} \\equiv \\Sigma_{k}\/\\tilde{{\\Sigma_{k}}}-1$) and effectively accounts for redshift variations in the selection rate of SDSS spectroscopy \\citep{Cooper+09}. Subscripts of $\\delta$ with or without units differentiate fixed radius densities from nearest neighbor densities. For example, $1+\\delta_5$ denotes overdensity within the 5th nearest neighbor, and $1+\\delta_\\mathrm{8Mpc}$ denotes normalized mass density within 8$\\,h^{-1}\\,\\mathrm{Mpc}$.\n\nThe environmental density indicators are significantly correlated with each other. For example, $\\delta_\\mathrm{0.5Mpc}$ is strongly correlated with $\\delta_\\mathrm{1Mpc}$ ($\\rho = 0.8$) and $\\delta_\\mathrm{2Mpc}$ ($\\rho=0.6$) and it is moderately correlated with $\\delta_\\mathrm{4Mpc}$ ($\\rho =0.5$) and $\\delta_\\mathrm{8Mpc}$ ($\\rho \\approx 0.4$). The correlation between $\\delta_\\mathrm{8Mpc}$ and $\\delta_\\mathrm{4Mpc}$ is also strong ($\\rho \\approx 0.8$). Likewise, $\\delta_5$ is strongly correlated ($\\rho \\approx 0.8$) with $\\delta_\\mathrm{1Mpc}$, $\\delta_\\mathrm{2Mpc}$, and $\\delta_\\mathrm{4Mpc}$.\n\nFurthermore, we quantify the tidal strength exerted by the five nearest neighbor galaxies using the tidal parameter \\citep{Verley+07,Argudo-Fernandez+13} defined as follows:\n\n\\begin{equation}\n\\log \\,Q_k = \\log \\left(\\sum^k_{i=1} \\frac{M_{\\star, i}}{M_\\star}\\left(\\frac{D}{r_{p,i}}\\right)^3\\right)\n\\end{equation}\n\n\\noindent where $M_\\star$ and $D$ are the stellar mass and diameter of the main galaxy of interest, $M_{\\star,i}$ is  the stellar mass and $r_{p,i}$ is the projected distance to the $i$th nearest neighbor. Following \\citet{Argudo-Fernandez+13}, we set $D = 2\\alpha\\,R_{90}$, where $R_{90}$ is the $r$-band Petrosian radius containing 90\\% of the total flux of the main galaxy and it is scaled by a factor $\\alpha = 1.43$ to recover the  isophotal diameter $D_{25}$ at the surface brightness of 25 mag\/arcsec$^2$. In our measurements of $Q_{k}$ and mass overdensities, we use redshift and $M_\\star$ data from the SDSS DR17. The $M_\\star$ estimates used here were derived using the methodology of \\citet{Chen+12} and they agree well ($\\rho = 0.97$ and $\\sigma \\approx 0.1$\\,dex) with the estimates from \\citet{Salim+18}. When $M_\\star$ estimates of nearby neighbors are not available, we estimate them using their $i$-band luminosities and the mean relationship between $g-i$ color and the mass-to-light ratio ($\\log\\, M_\\star\/L_i = -0.5 +0.66 \\times (g-i)$). If a neighbor of a galaxy is classified as QSO, we set its $\\log\\,M_\\star = 0$. Our main conclusions do not change if remove galaxies with QSO neighbors. Figure~\\ref{fig:vis_sp} visualizes the information content of some of the environmental indicators measured by the author. They are conspicuously correlated but are also discrepant.\n\n\\begin{figure*}[hbt!]\n\\gridline{\\fig{vis_spz08_all_dist.pdf}{0.40\\textwidth}{}\n          \\fig{vis_spz08_del5_dist.pdf}{0.45\\textwidth}{}}\n\\gridline{\\fig{vis_spz08_denM1_dist.pdf}{0.45\\textwidth}{}\n         \\fig{vis_spz08_denM4_dist.pdf}{0.45\\textwidth}{}}\n\\gridline{\\fig{vis_spz08_Qp_dist.pdf}{0.45\\textwidth}{}\n         \\fig{vis_spz08_rp1_dist.pdf}{0.45\\textwidth}{}}\n\\caption{Visualizing the various environmental indicators. Panel (a): the SDSS galaxy distribution at $z= 0.075-0.08$; panel (b): the 5th nearest neighbor overdensity; panel (c): the mass overdensity within $1\\,h^{-1}$\\,Mpc;\npanel (d): the mass overdensity within $4\\,h^{-1}$\\,Mpc; panel (e): the tidal parameter of the five nearest neighbors; panel (f): the nearest neighbor distance. The color bars are truncated outside $5\\%-95\\%$ range of the data. \\label{fig:vis_sp}}\n\\end{figure*}\n\n\\citet{Tempel+14} utilized a modified friends-of-friends (FoF) algorithm with a variable linking length ($\\sim 0.3-0.7$\\,Mpc) in the transverse and radial directions to identify groups of galaxies within certain neighborhood radius. Their flux-limited catalogue (based on SDSS DR10) includes 82,458 galaxy groups with two or more members. The flux-limited and volume-limited catalogs agree well, especially for large groups\/clusters. About $80\\%-90\\%$ of groups are identified at the same locations in both catalogs, and $\\sim 60\\%-70\\%$ of the matched groups have similar mass estimates within a factor of two. We use the flux-limited catalog, which includes measurements of several group characteristic (richness, size, radial velocity dispersion, etc) and normalized environmental luminosity densities estimated using different smoothing radii (1, 2, 4, and 8$\\,h^{-1}$Mpc). Following \\citet{Poggianti+09}, groups with more six members are classified as rich groups if the have group radial velocity $\\sigma_g < 400$\\,km\\,s$^{-1}$ or as clusters if $\\sigma_g > 400$\\,km\\,s$^{-1}$.\n\n\\citet{Tempel+14}'s measurements indicate that starbursts and upper SFMS galaxies have high 1$\\,h^{-1}$Mpc luminosity densities but low 8$\\,h^{-1}$Mpc luminosity overdensities relative to other galaxies (see Appendix~\\ref{sec:appA}). Suspecting this could be due to a mass-to-light ratio bias, the author of this paper did the mass overdensity measurements. Starbursts and upper SFMS galaxies have relatively low mass overdensities at all scales. \\citet{Tempel+14}'s luminosity overdensities are used only in the Appendix. Note that in all overdensity calculations, the mass or luminosity of the primary galaxy is included. Galaxies within 4$\\,h^{-1}$Mpc from the survey edge are excluded using distance measurements in \\citet{Tempel+14}'s catalog.\n\nIn \\citet{Lim+17} catalog (based on SDSS DR13), the groups were identified with an updated version of the iterative halo-based group finder of \\citet{YangX+05,YangX+07}. This version of the group finder uses $M_h$ proxies that scale with $M_\\star$ of central galaxies and the the ratio of $M_\\star$ between the central galaxy and the $k$th (typically $k=4$) most massive satellite to estimate preliminary $M_h$ for every galaxy. If the galaxy is isolated, the mean relation between the halo mass and stellar mass of isolated galaxies based on EAGLE simulations is used. The group finder then assigns galaxies into groups using halo properties such as halo radius and velocity dispersion. NFW density profile \\citep{Navarro+97} and a Gaussian distribution for the redshifts of galaxies within a halo ($P(z-z_\\mathrm{group})$) is assumed. An abundance matching between the mass function of the preliminary groups and a theoretical halo-mass function is used to update $M_h$ at each iteration. The update continues until group memberships converge. Mock galaxy samples constructed from EAGLE simulations were used to test and calibrate the group finder. The tests  showed that the group finder can find $\\sim$ 95\\% of the true member galaxies for $\\sim 85$\\% of the groups in the SDSS. The typical uncertainty of $M_h$ is $\\sim 0.2$\\,dex. The group finder also classifies galaxies into centrals and satellites.\n\nSimilarly, in the self-calibrated model of \\citet{Tinker21}, which also attempts to improve the halo-based algorithm of \\citet{YangX+05}, groups are rank-ordered by their weighted total group luminosity; the multiplicative weight factors are calibrated by comparing the predictions of the group catalog based $N$-body simulations to measurements of galaxy clustering and the total $r$-band luminosity of satellites, $L_\\mathrm{sat}$, around spectroscopic central galaxies. The latter quantity is based on deep-imaging data from the DECALS Legacy Imaging Survey. Besides probing $M_h$, $L_\\mathrm{sat}$ is also sensitive to the formation histories of halo, with younger haloes having more substructure, and thus more satellite galaxies. To break this degeneracy, the concentration of the central galaxy is used in the calibration. The concentration index of a galaxy is defined as the ratio between the radius that contains 90\\% of the galaxy light to the half-light radius, $C = R_{90}\/R_{50}$. \\citet{Tinker21} gave several reasons for using $C$ in the halo mass estimate: (1) $C$ for central galaxies does not correlate with large-scale environment (proxy for halo age) at fixed $M_\\star$.  (2) The correlation of $C$ with $L_\\mathrm{sat}$ is roughly independent of $M_\\star$. (3) $C$ varies minimally with galaxy luminosity\/mass. Furthermore, \\citet{Tinker21} showed that his halo masses estimates are in good agreement with weak-lensing estimates for star-forming and quiescent central galaxies. However, the results inferred from his group catalog also differ in several ways from previous catalogs. He found significantly different fractions of satellite in SFGs and QGs than other methods. He claimed that his method is more sensitive to low-mass halos, $M_h < 10^{12}\\,M_\\odot$.\n\n\\subsection{Statistical Methods}\n\nTo estimate the 95\\% simultaneous confidence intervals of the fractions of galaxies of a given class (say starbursts), that are isolated, pairs, in groups, or in cluster, we assume a multinomial distribution and compute the confidence intervals of the multinomial proportions using the Goodman method\\footnote{For a multinomial distribution with $k$ categories and a total sample size $N_\\mathrm{tot}$, the Goodman method approximates the true proportions $p_1,p_2 \\cdots p_k$ with the observed sample proportion for $i$th category $\\hat{p}_i$ as  $(p_i - \\hat{p}_i) \\le \\pm \\chi(\\alpha\/k, 1) \\sqrt{\\hat{p}_i(1-\\hat{p}_i)\/N_\\mathrm{tot}}$ for $i= 1, 2, \\cdots k$, where $\\alpha=0.05$ and $\\chi^2(\\alpha\/k, 1)$ is the ($1-\\alpha\/k) \\times 100$th percentile of the chi-square distribution with 1 degree of freedom.}\\citep{Goodman65} as implemented in \\texttt{statsmodels} python package. The Multinomial distribution is an extension of the binomial distribution for $k > 2$ categories.\n\nFor \\citet{Lim+17}'s central\/satellite classification, the mean central fraction in a given $M_\\star$ range is estimated as $\\hat{f}_\\mathrm{cen}= N_\\mathrm{cen}\/N_\\mathrm{tot}$, where $N_\\mathrm{cen}$ and $N_\\mathrm{tot}$ are number of centrals and the total number of galaxies in a given sample, respectively. The standard error of $\\hat{f}_\\mathrm{cen}$ is estimated by the Normal approximation of the Binomial distribution as $\\sigma_{\\hat{f}_\\mathrm{cen}} = \\sqrt{\\frac{\\hat{f}_\\mathrm{cen} (1-\\hat{f}_\\mathrm{cen})}{N_\\mathrm{tot}}}$. Since \\citet{Tinker21} provides the probability of satellite ($p_\\mathrm{sat} = 1-p_\\mathrm{cen}$) for each galaxy, in his case, assuming the Poisson-binomial distribution for the number of centrals in a given sample $\\hat{f}_\\mathrm{cen} = \\sum \\limits _{{i=1}}^{N_\\mathrm{tot}}p_{\\mathrm{cen},i}\/N_\\mathrm{tot}$ and $\\sigma_{\\hat{f}_\\mathrm{cen}}= \\sqrt{\\sum \\limits _{{i=1}}^{N_\\mathrm{tot}}p_{\\mathrm{cen},i}(1-p_{\\mathrm{cen},i})}\/N_\\mathrm{tot}$. The Binomial distribution is a special case of the Poisson-binomial distribution, when all probabilities are equal to each other.\n\nThe Anderson-Darling (AD) test is used to test the null hypothesis that two samples come from the same but unspecified distribution. This non-parametric test is more powerful than the Kolmogorov-Smirnov (KS) test; it is especially sensitive to the tail of a distribution and requires a small sample size. The KS statistics compares two empirical cumulative distribution functions (ECDFs) by looking only at their maximum absolute difference. In contrast, the AD statistics compares two ECDFs by looking at the weighted sum of all their squared differences, which are calculated at each point in the joint sample. The weights are determined by the inverse-variance of the joint ECDF at each point; points in the tail of the distribution receive more weights than points close to the median. The AD test is done using the \\texttt{scipy.stats} python package.\n\nTo match two samples in a multi-dimensional space, the function \\texttt{NearestNeighbors} in \\texttt{sklearn} python package is used with a setting of euclidian distance and ball tree algorithm ($\\mathrm{leaf\\,size=30)}$;  each input measurement is transformed to the logarithmic scale and is standardized by subtracting its mean and dividing by its standard deviation.\n\n\\section{Results}\\label{sec:res}\n\nThis section first compares the environments of galaxies classified by their $\\Delta\\,\\mathrm{SSFR}$, ranging from starbursts to QGs. It demonstrates the existence of a general trend with $\\Delta\\,\\mathrm{SSFR}$, in the sense that starbursts occupy the lowest density environments among SFGs and QGs occupy the highest densities. The section then compares the environments of QPSBs and (post-starburst) AGNs with those of starbursts and normal galaxies. It ends by assessing the consistency of the evolution from starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs $\\rightarrow$ QGs in terms of their structures and multiscale environments.\n\n\\subsection{The Environments of Starbursts and Comparison Samples and their Trends with $\\Delta\\,\\mathrm{SSFR}$}\n\n\\begin{figure*}[hbt!]\n\\gridline{\\fig{M10105_delM05Mpc_dist_R1.pdf}{0.47\\textwidth}{}\n          \\fig{M10511_delM05Mpc_dist_R1.pdf}{0.47\\textwidth}{}}\n\\gridline{\\fig{M10105_delM8Mpc_dist_R1.pdf}{0.47\\textwidth}{}\n          \\fig{M10511_delM8Mpc_dist_R1.pdf}{0.47\\textwidth}{}}\n\\caption{The cumulative distributions of the normalized environmental mass densities within $0.5\\,h^{-1}$Mpc ($\\delta_\\mathrm{0.5Mpc}$) or $8\\,h^{-1}$Mpc ($\\delta_\\mathrm{8Mpc}$) for galaxies grouped into two $M_\\star$ ranges and $\\Delta\\,\\mathrm{SSFR}$. The $\\delta_\\mathrm{0.5Mpc}$ distributions of starbursts are different from those of normal SFGs, and they are narrow and peak at lower densities compared to those of QGs. In contrast, the $\\delta_\\mathrm{8Mpc}$ distributions of starbursts are slightly lower than those of normal SFGs, but they are much lower than those of green-valley galaxies and QGs.\\label{fig:del18_sb}}\n\\end{figure*}\n\nStarbursts inhabit lower density environments than galaxies with lower $\\Delta\\,\\mathrm{SSFR}$. For example, Figure~\\ref{fig:del18_sb} shows the distributions of stellar mass overdensities within $0.5\\,h^{-1}$\\,Mpc and $8\\,h^{-1}$\\,Mpc in two narrow $M_\\star$ ranges: $\\log\\,M_\\star\/M_\\odot = 10-10.5$ and $\\log\\,M_\\star\/M_\\odot = 10.5-11$, and for subsets of galaxies ranging from starbursts to QGs. Notably, $\\Delta\\,\\mathrm{SSFR}$ is significantly anticorrelated with $\\delta_\\mathrm{0.5Mpc}$ ($\\rho = -0.39$ and $p < .001$) and $\\delta_\\mathrm{8Mpc}$ ($\\rho = -0.15$ and $p < .001$); the $\\delta_\\mathrm{0.5Mpc}$ and  $\\delta_\\mathrm{8Mpc}$ distributions of the different samples generally shift toward higher overdensity as $\\Delta\\,\\mathrm{SSFR}$ decreases, as the galaxies move toward quiescence. Table~\\ref{tab:del18_sb} presents the summary statistics (median, 15\\%, and 85\\%) for the distributions of overdensities measured within apertures $r_{ap} \\in \\{0.5, 1, 2, 4, 8\\}\\,h^{-1}$Mpc for all $M_\\star$ ranges. Furthermore, Figure~\\ref{fig:massdel18} compares the summary statistics of the overdensity distributions of starbursts, SFMS galaxies, and QGs for all $M_\\star$ ranges. Starbursts have lower overdensities than QGs at all scales and for all $M_\\star$ ranges. The difference between the overdensties of starbursts and SFMS galaxies is more evident at the scales of $\\sim 0.5-2h^{-1}$\\,Mpc. The environmental difference of starbursts\/SFGs and QGs is not sensitive to the adopted mass bins of 0.5\\,dex. The results still hold if the $M_\\star$ bins are made 0.25\\,dex.\n\n\n\\begin{figure*}[hbt!]\n\\fig{SFGroup_deltaM_Mass_YH.pdf}{0.9\\textwidth}{}\n\\caption{Comparing the $0.5-4\\,h^{-1}$\\,Mpc mass overdensities of starbursts with those of SFMS galaxies and QGs at different $M_\\star$ ranges. The names of the samples indicate the starting points of the $\\log\\, M_\\star$ ranges, which span 0.5\\,dex. For example, M11 denotes $\\log M_\\star\/M_\\odot = 11-11.5$. The box marks 15\\%, 50\\%, and 85\\% of the mass overdensity distribution at a given scale and the error bar extends to show the rest of the distribution, ignoring outliers. The dashed line shows the median overdensity for M105: $\\log M_\\star\/M_\\odot = 10.5-11$ at a given scale. The small-scale mass overdensities strongly depend on $M_\\star$. At all $M_\\star$ ranges, the mass overdensity distributions of starbursts are different from those of SFMS galaxies and QGs at $\\sim 0.5-2\\,h^{-1}$\\,Mpc scales. At $4\\,h^{-1}$\\,Mpc scale, they are more similar to those SFMS galaxies than they are to the distributions of QGs. \\label{fig:massdel18}}\n\\end{figure*}\n\n\\begin{rotatetable*}\n\\movetableright=0.1in\n\\begin{deluxetable*}{llcccccc}\n\\tabletypesize{\\footnotesize}\n\\tablecaption{The Multiscale Environments of Starbursts, QPSBs, and Comparison Samples \\label{tab:del18_sb}}\n\\tablewidth{0pt}\n\\tablenum{1}\n\\tablehead{\n\\colhead{Sample} & \\colhead{Measurement} & \\colhead{Starburst} & \\colhead{QPSB}  & \\colhead{AGN H$\\delta_A > 4$\\,{\\AA}} & \\colhead{AGN H$\\delta_A < 3$\\,{\\AA}} & \\colhead{SFMS} & \\colhead{QG}\n}\n\\decimalcolnumbers\n\\startdata\n & $\\log\\,(1+\\delta_\\mathrm{0.5Mpc})$ & $-0.21\\,(-0.42, 0.09)$ & $-0.31\\,(-0.47, 0.03)$ & $-0.23\\,(-0.41, 0.13)$ & $-0.02\\,(-0.27, 0.32)$ & $-0.12\\,(-0.33, 0.22)$ & $0.07\\,(-0.19,  0.43)$\\\\\n  & $\\log\\,(1+\\delta_\\mathrm{1Mpc})$ & $-0.23\\,(-0.53, 0.15)$ & $-0.33\\,(-0.59,  0.27)$ & $-0.27\\,(-0.53, 0.23)$ & $-0.03\\,(-0.36, 0.37)$ & $-0.13\\,(-0.44,  0.28)$ & $0.07\\,(-0.27, 0.49)$\\\\\n& $\\log\\,(1+ \\delta_\\mathrm{2Mpc})$  & $-0.18\\,(-0.62, 0.21)$ &  $-0.20\\,(-0.61, 0.23)$ & $-0.28\\,(-0.72, 0.21) $ &  $-0.03\\,(-0.44,  0.38) $ & $-0.11\\,(-0.54, 0.31)$ & $0.05\\, (-0.35, 0.46) $\\\\\nM11 & $\\log\\,(1+\\delta_\\mathrm{4Mpc})$  & $-0.14\\,(-0.57, 0.24)$ & $-0.15\\,(-0.47, 0.28)$ & $-0.19\\,(-0.72, 0.25)$ & $-0.03\\,(-0.43, 0.34)$ & $-0.08\\,(-0.51,  0.30)$ & $0.04\\,(-0.36,  0.40)$\\\\\n& $\\log\\,(1+\\delta_\\mathrm{8Mpc})$  & $-0.11\\,(-0.51, 0.24)$ & $-0.11\\,(-0.41, 0.25)$ & $-0.13\\,(-0.53, 0.29)$ & $-0.02\\,(-0.38, 0.29)$ & $-0.06\\,(-0.43, 0.27)$ & $0.03\\,(-0.32, 0.33)$\\\\  \n& $\\log\\,(1+\\delta_5)$  & $-0.19\\,(-0.69, 0.39)$ & $-0.26\\,(-0.64, 0.42)$ & $-0.24\\,(-0.77, 0.48)$ & $-0.05\\,(-0.59, 0.67)$ & $-0.13\\,(-0.64, 0.49)$ & $0.08\\,(-0.48, 0.86)$\\\\  \n& $\\log\\,Q_5$  & $-4.05\\,(-5.30, -2.31)$ & $-4.24\\,(-5.33, -2.78)$ & $-4.04\\,(-5.39,-2.19)$ & $-3.59\\,(-4.97, -1.83)$ & $-3.85\\,(-5.09, -2.24)$ & $-3.41\\,(-4.97, -1.77)$\\\\  \n\\hline\n & $\\log\\,(1+\\delta_\\mathrm{0.5Mpc})$ & $-0.22\\,(-0.52, 0.26)$ & $-0.26\\,(-0.54, 0.23)$ & $-0.16\\,(-0.40, 0.30)$ & $-0.02\\,(-0.31, 0.51)$ & $-0.11\\,(-0.39, 0.39)$ & $0.11\\,(-0.20,  0.72)$\\\\\n  & $\\log\\,(1+\\delta_\\mathrm{1Mpc})$ & $-0.23\\,(-0.67, 0.36)$ & $-0.26\\,(-0.72,  0.33)$ & $-0.21\\,(-0.56, 0.35)$ & $-0.04\\,(-0.48, 0.51)$ & $-0.13\\,(-0.57,  0.44)$ & $0.13\\,(-0.37, 0.71)$\\\\\n& $\\log\\,(1+ \\delta_\\mathrm{2Mpc})$  & $-0.18\\,(-0.83, 0.32)$ &  $-0.21\\,(-0.81, 0.31)$ & $-0.17\\,(-0.75, 0.32) $ &  $-0.03\\,(-0.58,  0.45) $ & $-0.10\\,(-0.67, 0.40)$ & $0.10\\, (-0.46, 0.59)$\\\\\nM105 & $\\log\\,(1+\\delta_\\mathrm{4Mpc})$  & $-0.13\\,(-0.68, 0.29)$ & $-0.15\\,(-0.63, 0.28)$ & $-0.11\\,(-0.70, 0.30)$ & $-0.03\\,(-0.50, 0.38)$ & $-0.07\\,(-0.56, 0.34)$ & $0.07\\,(-0.41, 0.47)$\\\\\n& $\\log\\,(1+\\delta_\\mathrm{8Mpc})$  & $-0.08\\,(-0.50, 0.26)$ & $-0.10\\,(-0.52, 0.25)$ & $-0.06\\,(-0.52, 0.26)$ & $-0.02\\,(-0.40, 0.31)$ & $-0.04\\,(-0.43, 0.29)$ & $0.04\\,(-0.34,  0.37)$\\\\  \n& $\\log\\,(1+\\delta_5)$  & $-0.15\\,(-0.65, 0.42)$ & $-0.18\\,(-0.64, 0.45)$ & $-0.14\\,(-0.70,  0.46)$ & $-0.05\\,(-0.60,  0.67)$ & $-0.11\\,(-0.63, 0.53)$ & $0.12\\,(-0.50, 0.96)$\\\\  \n& $\\log\\,Q_5$  & $-4.14\\,(-5.39, 2.40)$ & $-4.44\\,(-5.57,-2.85)$ & $-4.25\\,(-5.44, -2.61)$ & $-3.78\\,(-5.17, -2.10)$ & $-3.86\\,(-5.10, -2.28)$ & $-3.63\\,(-5.18, -1.99)$\\\\  \n\\hline\n & $\\log\\,(1+\\delta_\\mathrm{0.5Mpc})$ & $-0.18\\,(-0.59, 0.56)$ & $-0.05\\,(-0.51, 0.66)$ & $-0.07\\,(-0.46, 0.62)$ & $0.03\\,(-0.44, 0.80)$ & $-0.11\\,(-0.53, 0.66)$ & $0.39\\,(-0.28, 1.15)$\\\\\n & $\\log\\,(1+\\delta_\\mathrm{1Mpc})$ & $-0.20\\,(-0.92, 0.55)$ & $-0.11\\,(-0.86,  0.55)$ & $-0.11\\,(-0.73, 0.59)$ & $-0.02\\,(-0.76, 0.61)$ & $-0.12\\,(-0.84,  0.53)$ & $0.29\\,(-0.58, 0.93)$\\\\\n& $\\log\\,(1+ \\delta_\\mathrm{2Mpc})$  & $-0.13\\,(-0.95, 0.42)$ &  $-0.10\\,(-0.80, 0.38)$ & $-0.03\\,(-0.84, 0.50) $ & $-0.02\\,(-0.64, 0.53) $ & $-0.09\\,(-0.76, 0.45)$ & $0.20\\, (-0.48, 0.73)$\\\\\nM10 & $\\log\\,(1+\\delta_\\mathrm{4Mpc})$  & $-0.09\\,(-0.68, 0.33)$ & $-0.07\\,(-0.62, 0.34)$ & $-0.05\\,(-0.55, 0.37)$ & $-0.03\\,(-0.54, 0.43)$ & $-0.06\\,(-0.57, 0.37)$ & $0.13\\,(-0.39, 0.55)$\\\\\n& $\\log\\,(1+\\delta_\\mathrm{8Mpc})$  & $-0.06\\,(-0.50, 0.28)$ & $-0.02\\,(-0.50, 0.30)$ & $0.00\\,(-0.39, 0.31)$ & $-0.02\\,(-0.39, 0.35)$ & $-0.04\\,(-0.44, 0.30)$ & $0.08\\,(-0.31, 0.42)$\\\\  \n& $\\log\\,(1+\\delta_5)$  & $-0.11\\,(-0.63, 0.47)$ & $-0.12\\,(-0.66, 0.61)$ & $-0.05\\,(-0.59, 0.68)$ & $0.00\\,(-0.61, 0.78)$ & $-0.10\\,(-0.64, 0.57)$ & $0.29\\,(-0.47, 1.20)$\\\\  \n& $\\log\\,Q_5$  & $-3.98\\,(-5.18, -2.10)$ & $-4.21\\,(-5.68, -2.46)$ & $-4.06\\,(-5.39, -2.54)$ & $-3.85\\,(-5.29, -2.04)$ & $-3.64\\,(-4.92, -2.05)$ & $-3.40\\,(-5.17, -1.77)$\\\\  \n\\hline\n & $\\log\\,(1+\\delta_\\mathrm{0.5Mpc})$ & $-0.19\\,(-0.68, 0.91)$ & $0.20\\,(-0.54, 1.27)$ & $0.22\\,(-0.62, 0.74)$ & $0.01\\,(-0.73, 0.80)$ & $-0.14\\,(-0.76, 0.90)$ & $0.77\\,(-0.38, 1.43)$\\\\\n & $\\log\\,(1+\\delta_\\mathrm{1Mpc})$ & $-0.20\\,(-1.29, 0.58)$ & $0.07\\,(-0.83,  0.84)$ & $-0.14\\,(-1.03, 0.45)$ & $-0.11\\,(-1.02, 0.60)$ & $-0.10\\,(-1.13,  0.62)$ & $0.53\\,(-0.33, 1.13)$\\\\\n& $\\log\\,(1+ \\delta_\\mathrm{2Mpc})$  & $-0.18\\,,(-0.92, 0.44)$ & $0.06\\,(-0.39, 0.61)$ & $0.02\\,(-0.63, 0.45) $ & $-0.09\\,(-0.78, 0.44) $ & $-0.06\\,(-0.79, 0.49)$ & $0.36\\, (-0.27, 0.87)$\\\\\nM95 & $\\log\\,(1+\\delta_\\mathrm{4Mpc})$  & $-0.13\\,(-0.65, 0.35)$ & $0.04\\,(-0.41, 0.42)$ & $0.01\\,(-0.48, 0.32)$ & $-0.01\\,(-0.60, 0.37)$ & $-0.04\\,(-0.56, 0.39)$ & $0.23\\,(-0.24, 0.64)$\\\\\n& $\\log\\,(1+\\delta_\\mathrm{8Mpc})$  & $-0.10\\,(-0.48, 0.27)$ & $0.02\\,(-0.28, 0.35)$ & $0.02\\,(-0.32, 0.35)$ & $-0.02\\,(-0.36, 0.27)$ & $-0.02\\,(-0.42, 0.31)$ & $0.15\\,(-0.20, 0.48)$\\\\  \n& $\\log\\,(1+\\delta_5)$  & $-0.15\\,(-0.65, 0.45)$ & $0.09\\,(-0.47, 0.94)$ & $-0.06\\,(-0.52  0.69)$ & $-0.02\\,(-0.65, 0.78)$ & $-0.07\\,(-0.64, 0.61)$ & $0.68\\,(-0.26, 1.44)$\\\\  \n& $\\log\\,Q_5$  & $-3.78\\,(-5.12, -2.00)$ & $-3.44\\,(-4.76, -1.70)$ & $-3.38\\,(-4.74, -1.77)$ & $-3.21\\,(-4.72, -1.61)$ & $-2.67\\,(-4.57, -1.30)$ & $-2.64\\,(-4.78, -1.18)$\\\\  \n\\enddata\n\\tablecomments{Column (1) : the $M_\\star$ ranges of the samples binned by 0.5\\,dex. The names of the samples indicate the minima of the $\\log\\, M_\\star$ ranges. For example, M11 denotes $\\log M_\\star\/M_\\odot = 11-11.5$. Column (2) : gives the measurements of mass overdensities with radii of \\{0.5,1, 2, 4, 8\\}\\,$h^{-1}$Mpc, the 5th nearest neighbor overdensity, and the tidal parameter due to the five nearest neighbors. We use the notation $X\\,(Y, Z)$ to denote $X$ = median (50\\%), $Y=$15\\%, and $Z=85$\\% of a distribution. The results for additional comparison samples are given in Appendix~\\ref{sec:appB}.}\n\\end{deluxetable*}\n\\end{rotatetable*}\n\nAs expected from the clear trends in Figure~\\ref{fig:del18_sb} and Figure~\\ref{fig:massdel18}, in almost all cases the AD test rejects the null hypothesis that the multiscale overdensities of starbursts for each $M_\\star$ range are the same as those of the other samples ($p < .001$). The exceptions being that the distributions of $\\delta_\\mathrm{4Mpc}$ and $\\delta_\\mathrm{8Mpc}$ of starbursts and upper SFMS galaxies in $\\log M_\\star\/M_\\odot = 11-11.5$ are similar ($p=.01$ for $\\delta_\\mathrm{8Mpc}$ and $p=.1$ for $\\delta_\\mathrm{4Mpc}$). Furthermore, the AD test indicates that the mass overdensity distributions of upper and lower SFMS galaxies are different at all scales and for all $M_\\star$ ranges ($p < .001$). Therefore, SFGs do not scatter randomly above and below SFMS.\n\nThe overdensity of galaxy counts within the fifth nearest neighbor distances ($\\delta_5$) of the samples divided by $\\Delta\\,\\mathrm{SSFR}$ give similar results as fixed aperture mass overdensities (Figure~\\ref{fig:del5_sb} and Table~\\ref{tab:del18_sb}). The median and the dispersion of $\\delta_5$ increase as $\\Delta\\,\\mathrm{SSFR}$ decreases. Starbursts have the lowest $\\delta_5$, while QGs have the highest $\\delta_5$. The AD test confirms that the differences in $\\delta_5$ distributions shown in Figure~\\ref{fig:del5_sb} are significant ($p < .001$). In general, the $\\delta_5$ distributions of starbursts are significantly different from the distributions of the SFMS galaxies at all $M_\\star$ ranges ($p \\lesssim .01$). Likewise, the $\\delta_5$ distributions of upper SFMS and lower SFMS galaxies are different for all $M_\\star$ ranges ($p < .001$). The trends presented in Figure~\\ref{fig:del5_sb} are similar if $\\delta_3$ is used instead.\n\n\\begin{figure}[hbt]\n\\fig{M10105_del5_dist_R1.pdf}{0.5\\textwidth}{}\n\\fig{M10511_del5_dist_R1.pdf}{0.5\\textwidth}{}\n\\caption{Similar to Figure~\\ref{fig:del18_sb}, but here the cumulative distributions of the normalized 5th nearest neighbor number densities ($\\delta_5$) are shown. The $\\delta_5$ of starbursts are slightly lower than those of normal SFGs, but they are much lower than those of green-valley galaxies and QGs.\\label{fig:del5_sb}}\n\\end{figure}\n\nFurthermore, Figure~\\ref{fig:Q5_sb} compares the ECDFs of the tidal parameter $Q_5$ and the nearest neighbor distance $r_{p,1}$ of starbursts and the other samples grouped by $\\Delta\\,\\mathrm{SSFR}$ for two $M_\\star$ ranges: $\\log\\,M_\\star\/M_\\odot = 10-10.5$ and $\\log\\,M_\\star\/M_\\odot = 10.5-11$. The two environmental indicators are strongly anticorrelated ($\\rho=-0.9$); the first nearest neighbors contributes about $55\\%-60\\%$ to the total $Q_5$ values. Again, starbursts have the lower $Q_5$ or the higher $r_{p,1}$ than those of the other samples (see also Table~\\ref{tab:del18_sb}). The distributions of $r_{p,1}$  and $Q_5$ of starbursts are significantly different (AD test $p <.001$) from all of the other samples at all $M_\\star$ ranges. In most cases, the ECDFs of $Q_5$ and $r_{p,1}$ of various samples are significantly different from each other. In other words, $\\Delta\\,\\mathrm{SSFR}$ and $r_{p,1}$ show a weak but significant trend ($\\rho \\approx 0.2$), in the sense that the median nearest neighbor distances decrease from starbursts to SFMS galaxies to QGs. Only $15$\\% (8\\%) of the starbursts with $\\log\\,M_\\star\/M_\\odot = 10-10.5$ have nearby neighbors within $r_{p,1} < 0.2$\\,Mpc ($< 0.1$\\,Mpc). The difference in percentage between the nearby neighbors of starbursts and SFMS galaxies is only $\\sim 2\\%-3$\\%. Therefore, we do not find compelling evidence that most starbursts are triggered by interactions with their nearest neighbors. \n\n\\begin{figure*}[hbt!]\n\\gridline{\\fig{M10105_Q5_dist_R1.pdf}{0.49\\textwidth}{}\n          \\fig{M10511_Q5_dist_R1.pdf}{0.49\\textwidth}{}}\n\\gridline{\\fig{M10105_rp1_dist_R1.pdf}{0.49\\textwidth}{}\n          \\fig{M10511_rp1_dist_R1.pdf}{0.49\\textwidth}{}}\n\\caption{Similar to Figure~\\ref{fig:del5_sb}, but here the cumulative distributions of the tidal parameter of the five nearest neighbors ($Q_5$) and the projected distance to the nearest neighbor ($r_{p,1}$) are shown. The $Q_5$ and $r_{p,1}$ of starbursts are slightly lower than to those of SFMS galaxies, and are much lower than those of green-valley galaxies and QGs. Note that $\\sim 10\\%-15\\%$ of the starbursts have $Q_5 > -2$ or nearby neighbors within $r_{p,1} < 0.1-0.2$\\,Mpc, with only $\\sim 2\\%-3\\%$ enhancement compared to SFMS galaxies. About $55\\%-60\\%$ of $Q_5$ is contributed by the nearest neighbor. \\label{fig:Q5_sb}}\n\\end{figure*}\n\nFigure~\\ref{fig:Mh_sb} compares the ECDFs of $M_h$ estimated by \\citet{Lim+17} and \\citet{Tinker21}, again dividing the samples by $M_\\star$ and $\\Delta\\,\\mathrm{SSFR}$. Both estimates broadly agree that starbursts have lower $M_h$ than QGs. However, below few times $10^{12}\\, M_\\odot$, \\citet{Tinker21}'s estimates indicate that starbursts have high $M_h$ among SFGs and that $M_h$ increases with $\\Delta\\,\\mathrm{SSFR}$, while \\citet{Lim+17}'s estimates indicate the opposite. For the purpose of this work, the difference between the two $M_h$ estimates is not so important. It suggests that starbursts are useful in discriminating between different methods of connecting observed galaxies with simulated dark matter haloes. Simulators should also consider comparing their predictions with less model-dependent environmental indicators such as $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$.\n\nThe stellar mass overdensities significantly correlate with \\citet{Lim+17}'s $M_h$; the smaller the scale the overdensities probe, the stronger the correlation (see Appendix~\\ref{sec:appC}). For example, the correlation between $\\delta_\\mathrm{0.5Mpc}$ and $M_h$ for the whole sample has $\\rho \\approx 0.8$ and that of $\\delta_\\mathrm{8Mpc}$ and $M_h$ has $\\rho \\approx 0.3$. Furthermore, the nature of the relationship between $\\delta_\\mathrm{0.5Mpc}$ and $M_h$ (i.e., its shape, scatter, and $\\rho$) also depend subtly on satellite\/central classification, $\\Delta\\,\\mathrm{SSFR}$, and $M_\\star$ (Figure~\\ref{fig:Sig05Mh_sat} \\&~\\ref{fig:Sig05Mh}). Generally, $\\rho$ decreases as $M_\\star$ or $\\Delta\\,\\mathrm{SSFR}$ increases. For a given $M_\\star$ range, the correlation for QGs has $\\rho \\approx 0.7$, while that of the SFGs has $\\rho \\approx 0.5-0.7$. In particular, starbursts with with $M_\\star < 3 \\times 10^{10}\\,M_\\odot$ have the lowest $\\rho \\approx 0.5$ among SFG subsamples. The bimodal distributions of galaxies in Figure~\\ref{fig:Sig05Mh} change with $M_\\star$ due to changing satellite fractions; the plume of points in the upper corner of the panels (for $M_\\star < 10^{11}\\,M_\\odot$) are mainly satellites. Centrals occupy the almost horizontal cluster of points at low $M_h$, which marches downward and leftward in Figure~\\ref{fig:Sig05Mh} as $M_\\star$ decreases, as expected from the $M_\\star-M_h$ relation. In short, $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ are decent proxies for $M_h$.\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10105_Mh_Lim_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_Mh_Lim_dist_R1.pdf}{0.48\\textwidth}{}}\n\\gridline{\\fig{M10105_Mh_Tinker_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_Mh_Tinker_dist_R1.pdf}{0.48\\textwidth}{}}\n\\caption{The cumulative distributions of halo mass ($M_h$) estimates from \\citet[][panels (a) and (b)]{Lim+17} and \\citet[][panels (c) and (d)]{Tinker21} for galaxies grouped by $M_\\star$ and $\\Delta\\,\\mathrm{SSFR}$. Both $M_h$ estimates indicate that $\\sim 90\\%$ of starbursts have $M_h < 10^{13}\\,h^{-1}M_\\odot$, which is a significantly higher percentage than that of QGs below this halo mass. The two estimates however disagree on the trends of $M_h$ with $\\Delta\\,\\mathrm{SSFR}$ below $M_h \\lesssim 3 \\times 10^{12}\\,h^{-1}M_\\odot$. \\label{fig:Mh_sb}}\n\\end{figure*}\n\n\n\\begin{deluxetable*}{ccccccc}\n\\tablenum{2}\n\\tablecaption{The Mean Central Fractions and Median (15\\%, 85\\%) Halo Masses of Starbursts, QPSBs, and Comparison Samples.  \\label{tab:fcentMh_sb}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Sample} & \\colhead{Measurement} & \\colhead{Starburst} & \\colhead{QPSB} & \\colhead{AGN H$\\delta_A > 4$\\,{\\AA}} & \\colhead{AGN H$\\delta_A < 3$\\,{\\AA}} & \\colhead{QG}\n}\n\\decimalcolnumbers\n\\startdata    \n& $f$ Central Tinker & $0.839 \\pm 0.005$ & $0.983 \\pm 0.007$ & $0.902 \\pm 0.012$ & $0.788 \\pm 0.002$ & $0.777 \\pm 0.001$\\\\\n& $f$ Central Lim et al. & $0.832 \\pm 0.016$ & $0.907 \\pm 0.030 $ & $0.845 \\pm 0.034$ & $0.785 \\pm 0.007$ & $0.767 \\pm 0.002$ \\\\\nM11 & $\\log M_h$ Tinker & $12.3\\,(11.8, 13.2)$ & $12.4\\,(12.2, 12.6)$ &$12.4\\,(12.2, 12.7)$  & $12.4\\,(11.7, 13.6)$ & $12.9\\,(11.8,13.8)$\\\\\n& $\\log M_h$ Lim et al. & $12.5\\,(12.3, 12.8)$ & $12.7\\,(12.6, 13.0)$ &$12.6\\,(12.4, 13.1)$  & $12.9\\,(12.6, 13.4)$ & $13.1\\,(12.7,13.6)$\\\\\n\\hline\n & $f$ Central Tinker & $0.801 \\pm 0.002$ & $0.880 \\pm 0.008$ & $0.874 \\pm 0.007$ & $0.770 \\pm 0.003$ & $0.708 \\pm 0.001$ \\\\\n& $f$ Central Lim et al. & $0.799 \\pm 0.006$ & $0.833 \\pm 0.021$ & $0.786 \\pm 0.017$ & $0.723 \\pm 0.007$ & $0.647 \\pm 0.002$ \\\\\nM105 & $\\log M_h$  Tinker & $12.4\\,(11.7, 13.4)$ & $12.5\\,(12.2, 12.7)$ & $12.4\\,(12.1, 12.7)$ & $12.3\\,(11.7, 13.4)$ & $12.5\\,(11.7,13.8)$ \\\\\n& $\\log M_h$  Lim et al. & $12.1\\,(11.9, 12.5)$ & $12.3\\,(12.1, 12.6)$ & $12.2\\,(12.0, 12.6)$ & $12.3\\,(12.1, 13.0)$ & $12.5\\,(12.2,13.4)$\\\\\n\\hline\n  & $f$ Central Tinker & $0.816 \\pm 0.003$ & $0.798 \\pm 0.012$ & $0.828 \\pm 0.013$ & $0.692 \\pm 0.006$ & $0.606 \\pm 0.001$ \\\\\n  & $f$ Central Lim et al. & $0.787 \\pm 0.008$ & $0.728 \\pm 0.027$ & $0.733 \\pm 0.027$ & $0.666 \\pm 0.015$ & $0.526 \\pm 0.003$ \\\\\nM10 & $\\log M_h$  Tinker & $12.1\\,(11.8, 12.9)$ & $12.2\\,(11.9, 12.6)$ &$12.2\\,(11.8, 12.6)$ & $12.0\\,(11.4, 13.1)$ & $12.4\\,(11.5,13.9)$\\\\\n& $\\log M_h$ Lim et al. & $11.8\\,(11.6, 12.3)$ & $12.0\\,(11.8, 12.6)$ & $11.9\\,(11.7, 12.5)$ & $11.9\\,(11.8, 12.9)$ & $12.0\\,(11.8,13.7)$\\\\\n\\hline\n & $f$ Central Tinker & $0.791 \\pm 0.007$ & $0.664 \\pm 0.024$ & $0.755 \\pm 0.040$ & $0.718 \\pm 0.017$ & $0.276 \\pm 0.0016$\\\\\n& $f$ Central Lim et al. & $0.766 \\pm 0.016$ & $0.563 \\pm 0.055$ & $ 0.642 \\pm 0.066$ & $0.647 \\pm 0.047$ & $0.362 \\pm 0.009$\\\\\nM95 & $\\log M_h$  Tinker & $12.0\\,(11.7, 12.4)$ & $12.0\\,(11.5, 12.9)$ & $11.9\\,(11.4, 12.3)$ & $11.6\\,(11.2, 12.3)$ & $13.2\\,(11.3,14.3)$ \\\\\n& $\\log M_h$  Lim et al. & $11.6\\,(11.4, 12.2)$ & $11.7\\,(11.5, 13.1)$ & $11.7\\,(11.5, 12.6)$ & $11.6\\,(11.5, 12.6)$  & $12.7\\,(11.6,14.1)$\\\\\n\\enddata\n\\tablecomments{The halo mass are in units of $h^{-1}M_\\odot$. The results for other comparison samples can be found in Appendix~\\ref{sec:appB}.}\n\\end{deluxetable*}\n\n\n\\begin{figure*}[hbt!]\n\\includegraphics[width=0.5\\textwidth]{M95115_Mh_denM05_all_YHLi.pdf}\n\\includegraphics[width=0.5\\textwidth]{M95115_Mh_denM05_all_YHTJ_R1.pdf}\n\\caption{The correlations between the stellar mass overdensity $\\delta_\\mathrm{0.5Mpc}$ and halo mass $M_h$ for satellites and centrals. Panel (a) uses \\citet{Lim+17}'s measurements of $M_h$ and central\/satellite classification, whereas panel (b) uses \\citet{Tinker21}'s measurements. The $\\rho$ values are the Spearman correlation coefficients. \\label{fig:Sig05Mh_sat}}\n\\end{figure*}\n\n\\begin{figure*}[hbt!]\n\\gridline{\\fig{M11115_Mh_denM1_all_YHLi.pdf}{0.48\\textwidth}{}\n\\fig{M10511_Mh_denM1_all_YHLi.pdf}{0.48\\textwidth}{}}\n\\gridline{\\fig{M10105_Mh_denM1_all_YHLi.pdf}{0.48\\textwidth}{}\n\\fig{M9510_Mh_denM1_all_YHLi.pdf}{0.48\\textwidth}{}}\n\\caption{The correlations between the stellar mass overdensity $\\delta_\\mathrm{0.5Mpc}$ and halo mass $M_h$ \\citep{Lim+17} as a function of $M_\\star$ and $\\Delta\\,\\mathrm{SSFR}$. The $\\rho$ value in each panel is the Spearman correlation coefficient for the indicated $M_\\star$ range and $\\Delta\\,\\mathrm{SSFR}$ class. For guidance, the dashed lines mark the median values of $\\delta_\\mathrm{0.5Mpc}$ and $M_h$ of starbursts; they are same for all panels in a given $M_\\star$ range. The orange point denotes the median values of $\\delta_\\mathrm{0.5Mpc}$ and $M_h$ for a given panel. Generally, $\\delta_\\mathrm{0.5Mpc}$ shows a moderate ($\\rho \\approx 0.4-0.7$) correlation with $M_h$. The strength of the correlation decreases as $M_\\star$ increases or as $\\Delta\\,\\mathrm{SSFR}$ increases from QGs to starbursts. The contours represent number density of galaxies. The plume of points on the upper right corner of the panels are mainly satellites. Centrals occupy the horizontal contours at low $M_h$. Their distribution shifts leftward and downward as $M_\\star$ decreases, as expected from $M_\\star-M_h$ relation. \\label{fig:Sig05Mh}}\n\\end{figure*}\n\nWe have checked that the group masses (with three or more members) estimated by \\citet{Tempel+14} assuming NFW profile are also significantly correlated with our mass overdensity measurements. The strength of the correlations between the group mass and mass overdensities are similar to the corresponding results using \\citet{Lim+17}'s $M_h$ estimates. In addition, the mass overdensities also significantly correlate with \\citet{Tinker21}'s $M_h$ estimates. His estimates, however, give lower $\\rho$ values. \n\nFurthermore, $\\sim 80\\%-85\\%$ starbursts are isolated or pairs. The fraction of starbursts in rich groups or clusters is significantly smaller than those SFGs and QGs (Figure~\\ref{fig:grpercent_sb}, Table~\\ref{tab:richness2_sb}, and Appendix~\\ref{sec:appB}). Noting that the number of galaxies in groups increases with decreasing $M_\\star$, let us discuss the results for $\\log\\,M_\\star\/M_\\odot = 10.5-11$ sample as an example. In this $M_\\star$ range, $65\\%$ of starbursts are isolated (have no neighbors within the FoF linking length), $19\\%$ are pairs, $3\\%$ are in rich groups, and $1\\%$ are in clusters. In contrast, $43\\%$ QGs of are isolated, $17\\%$ are pairs, $13\\%$ are in rich groups, and $7\\%$ are in clusters. Likewise, $60\\%$ of SFMS galaxies are isolated, $17\\%$ are pairs, $6\\%$ are in rich groups, and $2\\%$ are in clusters. In addition, the fraction of starbursts that are satellites is only $\\sim 15\\%-20\\%$, and it is significantly lower than those of QGs of similar $M_\\star$ (Table ~\\ref{tab:fcentMh_sb}).\n\nHaving shown that starbursts live in distinctly low density environments, by all measures, and are unlike most non-bursty SFGs and QGs, next we check if the environments of PSBs conform with those of starbursts. \n\n\\subsection{The Environments of QPSBs and Strong-H$\\delta_A$ AGNs }\n\nLike starbursts, most QPSBs and AGNs with H$\\delta_A> 4$\\,{\\AA} are isolated or centrals residing in dark matter halos of $M_h  < 10^{13}\\, M_\\odot$. The fractions of QPSBs and strong-H$\\delta_A$ AGNs that are isolated, pairs, in groups, or in clusters are similar to those of starbursts (Figure~\\ref{fig:grpercent_psb} and Table~\\ref{tab:richness2_sb}). So are their fractions of centrals\/satellites (Table~\\ref{tab:fcentMh_sb}).\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10105_groupfrac.pdf}{0.48\\textwidth}{}\n\\fig{M10511_groupfrac.pdf}{0.48\\textwidth}{}}\n\\caption{Comparing the mean percentages of starbursts that are isolated, in pairs, in groups with $m=3-6$ members, in rich groups ($m > 6$ and $\\sigma < 400\\,$km\\,s$^{-1}$), or in clusters ($m > 6$ and $\\sigma > 400$\\,km\\,s$^{-1}$) with their counterparts in SFMS galaxies and QGs. The error bars denote the 95\\% multinomial confidence intervals.\\label{fig:grpercent_sb}}\n\\end{figure*}\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10105_groupfrac_psb_R1.pdf}{0.48\\textwidth}{}\n\\fig{M10511_groupfrac_psb_R1.pdf}{0.48\\textwidth}{}}\n\\caption{Similar to Figure~\\ref{fig:grpercent_sb}, here the fractions of QPSBs and H$\\delta_A > 4$\\,{\\AA} AGNs that are isolated, pairs, in groups or clusters are shown to be similar to the corresponding fractions of starbursts. \\label{fig:grpercent_psb}}\n\\end{figure*}\n\n\n\\begin{rotatetable*}\n\\movetableright=0.1in\n\\begin{deluxetable*}{cccccccc}\n\\tablenum{3}\n\\tablecaption{The Mean Fractions (and 95\\% Confidence Intervals) of Galaxies that are Isolated, Pairs, in Groups, or in Clusters. \\label{tab:richness2_sb}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{Sample} & \\colhead{Measurement} &\\colhead{Starburst} & \\colhead{QPSB}  & \\colhead{AGNs H$\\delta_A > 4$\\,{\\AA}} & \\colhead{AGN H$\\delta_A < 3$\\,{\\AA}} &\\colhead{SFMS} & \\colhead{QG}}\n\\decimalcolnumbers\n\\startdata\nM11 & Isolated &$0.632\\,(0.576, 0.684)$ & $0.659\\,(0.519, 0.776)$ & $0.612\\,(0.490,  0.722)$  & $0.498\\,(0.476, 0.52 )$ & $0.605\\,(0.593, 0.616)$  & $0.447\\,(0.441, 0.453) $ \\\\\n      & Pairs & $0.200\\,(0.158, 0.248)$  & $0.205\\,(0.115, 0.338)$ & $0.190\\,(0.112, 0.302)$ & $0.194\\,(0.177, 0.212)$ & $0.184\\,(0.175, 0.194)$ & $0.188\\,(0.183, 0.193)$ \\\\\n      & $2-3$ neighbors & $0.108\\,(0.078, 0.148)$ & $0.080\\,(0.031, 0.190)$  & $0.112\\,(0.056, 0.212)$ & $0.139\\,(0.124, 0.154)$ & $ 0.116\\,(0.108, 0.124)$ & $0.156\\,(0.152, 0.161)$\\\\ \n      & $4-5$ neighbors & $0.033\\,(0.018, 0.060)$ & $0.023\\,(0.004, 0.111)$  & $0.034\\,(0.010,  0.112)$ & $0.057\\,(0.048, 0.068)$ & $0.035\\,(0.031, 0.040)$ & $0.066\\,(0.062, 0.069)$ \\\\\n      & Rich group & $0.015\\,(0.006, 0.036)$  & $0.023\\,(0.004, 0.111)$ & $0.052\\,(0.019, 0.136])$ & $0.083\\,(0.072, 0.960)$ & $0.045\\,(0.040, 0.050)$  & $0.100\\,(0.096, 0.104)$ \\\\\n      & Cluster & $0.013\\,(0.005, 0.033)$ & $0.011\\,(0.001, 0.093)$ & $0.000\\,(0.000, 0.057)$ & $0.029\\,(0.023, 0.038)$ & $0.015\\,(0.012, 0.018)$ & $0.043\\,(0.041, 0.046)$ \\\\\n\\hline\nM105 & Isolated & $0.654\\,(0.636, 0.673)$  & $0.667\\,(0.595, 0.731)$ & $0.626\\,(0.572, 0.678)$  & $0.501\\,(0.482, 0.521)$ & $0.597\\,(0.590, 0.603)$ & $0.431\\,(0.426, 0.436) $  \\\\\n      & Pairs & $0.191\\,(0.176, 0.207)$  &  $0.162\\,(0.115, 0.223)$  & $0.172\\,(0.134, 0.218)$ &  $0.175\\,(0.161, 0.191)$ & $0.169\\,(0.164, 0.175)$ & $0.167\\,(0.163, 0.171) $ \\\\\n      & $2-3$ neighbors & $0.092\\,(0.081, 0.103)$  & $0.107\\,(0.070, 0.161)$  & $0.111\\,(0.080, 0.150)$ & $0.134\\,(0.122, 0.148)$ & $ 0.117\\,(0.112, 0.121)$ &  $0.139\\,(0.136, 0.143)$ \\\\ \n      & $4-5$ neighbors & $0.028\\,(0.022, 0.035)$  & $0.028\\,(0.012, 0.063)$  & $0.032\\,(0.017, 0.057)$ & $0.058\\,(0.049, 0.067)$ & $0.040\\,(0.038, 0.043)$ &  $0.064\\,(0.061, 0.066)$ \\\\\n      & Rich group & $0.026\\,(0.021, 0.033)$  & $0.018\\,(0.007, 0.050)$ & $0.051\\,(0.032, 0.081)$ & $0.102\\,(0.091, 0.115)$ & $0.060\\,(0.057, 0.064)$ &  $0.131\\,(0.128, 0.134)$ \\\\\n      & Cluster & $0.008\\,(0.006, 0.013)$ & $0.018\\,(0.007, 0.050)$ & $0.009\\,(0.003, 0.027)$ & $0.029\\,(0.023, 0.036)$ & $0.017\\,(0.016, 0.019)$  & $0.068\\,(0.065, 0.070)$ \\\\\n\\hline\nM10 & Isolated & $0.653\\,(0.629, 0.676)$ &  $0.551\\,(0.472, 0.628)$ & $0.555\\,(0.475, 0.632)$  & $0.447\\,(0.407, 0.488)$  & $0.549\\,(0.541, 0.556)$  & $0.332\\,(0.323, 0.340) $ \\\\\n      & Pairs & $0.176\\,(0.158, 0.196)$  & $0.186\\,(0.132, 0.256)$  & $0.164\\,(0.114, 0.231)$ & $0.175\\,(0.146, 0.208)$  & $0.171\\,(0.165, 0.177)$ & $0.136\\,(0.130, 0.142)$ \\\\\n      & $2-3$ neighbors & $0.096\\,(0.082, 0.112)$ & $0.117\\,(0.075, 0.178)$  & $0.135\\,( 0.090, 0.199)$ & $0.136\\,(0.111, 0.167)$ & $0.123\\,(0.118, 0.128)$ & $0.123\\,(0.118, 0.130)$  \\\\ \n      & $4-5$ neighbors & $0.026\\,(0.020, 0.036)$  & $0.055\\,(0.028, 0.103)$  & $0.036\\,(0.016, 0.080)$ & $0.057\\,(0.041, 0.079)$ & $0.045\\,(0.042, 0.048)$ & $0.065\\,(0.061, 0.070)$ \\\\\n      & Rich group & $0.037\\,(0.028, 0.047)$ & $0.062\\,(0.033, 0.112)$ & $0.095\\,(0.058, 0.152)$ & $0.139\\,(0.113, 0.170)$ & $0.089\\,(0.085, 0.093)$ & $0.216\\,(0.208, 0.223)$ \\\\\n      & Cluster &$0.012\\,(0.008, 0.019)$ & $0.029\\,(0.012, 0.070)$ & $0.015\\,(0.004, 0.049)$ & $0.046\\,(0.032, 0.066)$ &  $0.024\\,(0.021, 0.026)$  & $0.128\\,(0.122, 0.134)$ \\\\\n\\hline\nM95 & Isolated & $0.623\\,(0.576, 0.668)$  & $0.439\\,(0.305, 0.583)$ & $0.500\\,(0.331, 0.669)$  & $0.471\\,(0.348, 0.598)$ &  $0.529\\,(0.519, 0.539)$ & $0.216\\,(0.196, 0.238)$  \\\\\n      & Pairs & $0.179\\,(0.145, 0.219)$  & $0.085\\,(0.033, 0.202)$  & $0.148\\,(0.062, 0.315)$ & $0.135\\,(0.069, 0.246)$ & $0.175\\,(0.167, 0.183)$ & $0.090\\,(0.077, 0.106)$\\\\\\\n      & $2-3$ neighbors &  $0.103\\,(0.077, 0.135)$ & $0.232\\,(0.133, 0.373)$  & $0.111\\,(0.040, 0.271)$ & $0.144\\,(0.076, 0.257)$ & $0.123\\,(0.116, 0.129)$ & $0.091\\,(0.077, 0.107)$  \\\\ \n      & $4-5$ neighbors &$0.027\\,(0.016, 0.047)$  & $0.012\\,(0.001, 0.099)$  & $0.019\\,(0.002, 0.145)$ & $0.038\\,(0.011, 0.124)$ & $0.050\\,(0.046, 0.055)$ & $0.063\\,(0.052, 0.077)$  \\\\\n      & Rich group & $0.045\\,(0.029, 0.070)$  & $0.146\\,(0.071, 0.277)$ & $0.185\\,(0.085, 0.357)$ & $0.183\\,(0.104, 0.301)$ & $0.099\\,(0.094, 0.106)$ &  $0.336\\,(0.312, 0.361)$ \\\\\n      & Cluster & $0.022\\,(0.012, 0.041)$ & $0.085\\,(0.033, 0.202)$ & $0.037\\,(0.007, 0.173)$ & $0.024\\,(0.007, 0.110)$ &$0.024\\,(0.021, 0.027)$ & $0.203\\,(0.184, 0.225)$\\\\\n\\enddata\n\\tablecomments{The results for the upper SFMS, lower SFMS, and green-valley galaxies are given in the Appendix~\\ref{sec:appB}.}\n\\end{deluxetable*}\n\\end{rotatetable*}\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10105_delM05Mpc_psb_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_delM05Mpc_psb_dist_R1.pdf}{0.48\\textwidth}{}}\n\\gridline{\\fig{M10105_delM1Mpc_psb_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_delM1Mpc_psb_dist_R1.pdf}{0.48\\textwidth}{}}                   \n\\caption{Comparing the cumulative distributions of mass densities $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ of QPSBs to those of starbursts, SFMS galaxies, and QGs. \nDepending on $M_\\star$, the $0.5-1\\,h^{-1}$Mpc scale environments of QPSBs similar to or slightly lower than those of starbursts, but they are quite different from the environments of QGs. As shown in panel (a), QPSBs and starbursts do not necessarily have similar $\\delta_\\mathrm{0.5Mpc}$ distributions. This can be understood if only a subset of starbursts become QPSBs; the number of starbursts is much higher than QPSBs. The  environments of AGNs with H$\\delta_A > 4$\\,{\\AA}  are similar to upper SFMS galaxies. \\label{fig:del18_psb}}\n\\end{figure*}\n\nThe multiscale environments of QPSBs are generally similar to those of starbursts, especially at the scale of $>1h^{-1}$\\,Mpc. The AD test, however, sometimes indicates that the $\\delta_\\mathrm{0.5Mpc}$ or $\\delta_\\mathrm{1Mpc}$ distributions of the two populations are significantly different. Figure~\\ref{fig:del18_psb} shows the distributions of $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ in two $M_\\star$ ranges for QPSBs and for the comparison samples of starbursts, AGNs with H$\\delta_A > 4$\\,{\\AA}, SFGs, and QGs. Due to the small sample size, the Dvoretzky-Kiefer-Wolfowitz confidence interval\\footnote{The true CDF, $F(\\delta)$, is bounded by the CDF estimated from a sample of size $n$ as {$\\displaystyle F_{n}(\\delta)-\\varepsilon \\leq F(\\delta)\\leq F_{n}(\\delta)+\\varepsilon \\;{\\text{, where}}\\;\\varepsilon ={\\sqrt {\\frac {\\ln {\\frac {2}{\\alpha }}}{2n}}}\\; \\text{and $\\alpha$ is the significance level}$}. The CDFs of QPSBs in a given mass range have $\\epsilon \\approx 0.1$, using $\\alpha = 0.05$. The CDFs of starbursts have $\\epsilon \\approx 0.02$ and those of the other samples have $\\epsilon < 0.01$.} of the ECDFs of QPSBs and AGNs with H$\\delta_A > 4$\\,{\\AA} are quite large. Nevertheless, $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ of distributions of QPSBs and AGNs with H$\\delta_A > 4$\\,{\\AA} are clearly different from those of QGs. For $\\log\\,M_\\star\/M_\\odot = 10.5-11$, the $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ of QPSBs are similar those of starbursts ($p = .2$ for $\\delta_\\mathrm{0.5Mpc}$ and $p > .2$ for $\\delta_\\mathrm{1Mpc}$) but are dissimilar to those of SFMS and upper SFMS galaxies ($p < .001$). In contrast, for $\\log\\,M_\\star\/M_\\odot = 10-10.5$, $\\delta_\\mathrm{0.5Mpc}$ of QPSBs are more similar ($p > .2$) to upper SFMS and SFMS galaxies than they are to starbursts ($p=.001$). The multiscale environments of AGNs with H$\\delta_A > 4$\\,{\\AA} are generally similar those of the upper SFMS galaxies. Except for galaxies with $M_\\star > 10^{11}\\,M_\\odot$, the AD test does not indicate significant differences between the overdensity distributions of the two populations at all scales. In some cases, the distributions of $\\delta_\\mathrm{0.5Mpc}$ and $\\delta_\\mathrm{1Mpc}$ of AGNs with H$\\delta_A > 4$\\,{\\AA} are significantly different from those of starbursts and QPSBs (as also shown in Figure~\\ref{fig:del18_psb}).\n\nWe see later that not all starbursts are equally likely to become QPSBs and the subtle discrepancies of the small-scale overdensities can be understood with a selective draw of QPSBs and strong-H$\\delta_A$ AGNs from the parent samples of starbursts and\/or upper SFMS galaxies. Starbursts and upper SFMS galaxies outnumber QPSBs and strong-H$\\delta_A$ AGNs by more than ten times. Thus, the fact that their environments are different sometimes, does not necessarily imply that they are not evolutionary connected. Because the selection from the parent sample is not random, it is not necessary to match the overdensity distributions of the parent samples; there is no inconsistency as long as there are enough galaxies in the parent samples that have similar environments (and structures) as QPSBs and strong-H$\\delta_A$ AGNs. \n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10105_del5_psb_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_del5_psb_dist_R1.pdf}{0.48\\textwidth}{}}\n          \\gridline{\\fig{M10105_Q5_psb_dist_R1.pdf}{0.48\\textwidth}{}\n          \\fig{M10511_Q5_psb_dist_R1.pdf}{0.48\\textwidth}{}}\n\\caption{The ECDFs of the 5th nearest neighbor overdensity ($\\delta_5$) and the tidal parameter($Q_5$) of QPSBs and AGNs with H$\\delta_A > 4$\\,{\\AA} are compared with starbursts, other SFGs, and QGs. \nThe distributions of $\\delta_5$ and $Q_5$ of QPSBs and AGNs with H$\\delta_A > 4$\\,{\\AA} are more similar to those of SFGs than to those of QGs. \\label{fig:del5_psb}}\n\\end{figure*}\n\nThe $\\delta_5$, $r_{p,1}$, and $Q_5$ distributions of QPSBs and strong-H$\\delta_A$ AGNs are also clearly different from those of QGs (Figure~\\ref{fig:del5_psb}). The $\\delta_5$ distributions of QPSBs and strong-H$\\delta_A$ AGNs are generally similar to the $\\delta_5$ distributions starbursts and\/or upper SFMS galaxies. The $r_{p,1}$ and $Q_5$ distributions of QPSBs are significantly different from starbursts and upper SFMS galaxies ($p < .001$). The $\\delta_5$, $r_{p,1}$, and $Q_5$ distributions of QPSBs and strong-H$\\delta_A$ AGNs are not inconsistent.\n\nLastly, the environments of AGNs with H$\\delta_A > 4$\\,{\\AA} and of those with H$\\delta_A  < 3$\\,{\\AA} are very different. Figure~\\ref{fig:del18_agn} compares, for example, the distributions of $\\delta_\\mathrm{0.5Mpc}$, $\\delta_\\mathrm{1Mpc}$, $\\delta_5$, and $r_{p,1}$ of AGNs with SFMS galaxies and QGs in $\\log\\,M_\\star\/M_\\odot = 10.5-11$. Strong-H$\\delta_A$ AGNs live in environments that are slightly sparser than those SFMS galaxies while weak-H$\\delta_A$ AGNs live in environments denser than those SFMS galaxies (more consistent with those of lower SFMS galaxies). Both weak and strong-H$\\delta_A$ AGNs reside in distinctly lower density environments than QGs do.\n\n\\begin{figure*\n\\gridline{\\fig{M10511_delM05Mpc_AGN_dist_R1.pdf}{0.47\\textwidth}{}\n\\fig{M10511_delM1Mpc_agn_dist_R1.pdf}{0.47\\textwidth}{}}\n\\gridline{\\fig{M10511_del5_agn_dist_R1.pdf}{0.47\\textwidth}{}\n        \\fig{M10511_rp1_agn_dist_R1.pdf}{0.47\\textwidth}{}}          \n          \\caption{Comparing the environments of AGNs with those of the other samples. The environments of AGNs have lower densities than those of QGs. The overdensities (nearest neighbor distances) of AGNs with H$\\delta_A > 4$\\,{\\AA} are significantly lower (longer) than those of SFMS galaxies while the opposite is true for AGNs with H$\\delta_A < 3$\\,{\\AA} .\\label{fig:del18_agn}}\n\\end{figure*}\n\n\\subsection{Structural and Environmental Consistency of the Evolution from Starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs $\\rightarrow$ QGs}\n\nBecause the starburst to QPSB evolution is rapid ($\\lesssim 1$\\,Gyr), galaxies that are on this evolution track are expected to have similar $M_\\star$, environment, and structure. We saw in the previous subsection that starbursts, strong-H$\\delta_A$ AGNs, and QPSBs live in broadly similar environments, albeit exhibiting some subtle differences. Comparing their structure together with their environments (Figure~\\ref{fig:morphSB}) shows that (1) starbursts have a wide range of $\\sigma_\\star$ and $C$ and the typical starburst is less centrally concentrated than QPSBs. (2) The distribution of $\\sigma_\\star$ and $C$ of QPSBs and strong-H$\\delta_A$ AGNs are more similar to those of QGs than to those of SFMS galaxies, although their environments are more similar to the latter. (3) Most AGNs with H$\\delta_A > 4$\\,{\\AA} have similar environments and structures as QPSBs, while only few are structurally similar to starbursts. (4) The correlation between environments and structures of these three populations is weak (e.g., $\\rho \\approx 0.2-0.3$ for the $\\delta_\\mathrm{0.5Mpc}$ vs. $\\sigma_\\star$). The implications of the trends in Figure~\\ref{fig:morphSB} is that only some starbursts become QPSBs, and the mechanism that produces QPSBs is associated with structural transformations and AGN activities of galaxies in low-density environments. \n\nNext, we match each QPSB with its closet starburst, upper SFMS galaxy, strong-H$\\delta_A$ AGN, and QG simultaneously in multi-dimensional space of $M_\\star$, $\\sigma_\\star$, $C$, and environments (Figure~\\ref{fig:matched_psb}), thereby demonstrating that subsamples of SFGs, AGNs, and QGs that are similar to QPSBs in many aspects exist. In other words, we confirm using structure and multiscale environments  that QPSBs are descendants of some starbursts, as their spectral indices suggest, and in turn QPSBs are progenitors of some QGs. Moreover, most AGNs with H$\\delta_A > 4$\\,{\\AA} are associated with the evolution from starbursts to QPSBs. They too are PSBs. All AGNs with H$\\delta_A > 4$\\,{\\AA} can also be matched to some starbursts and upper SFMS galaxies of similar structures and environments. A significant fraction ($\\sim 20\\%-30\\%$) of starbursts cannot be matched to QPSBs or QGs of similar $M_\\star$, structures, and environments. Note that we only compare face-on galaxies with axis ratio $b\/a > 0.3$ in this subsection. The $M_\\star$ distributions before matching are given in Appendix~\\ref{sec:appD}.\n\nThe matching procedure does not impose a limit on the maximum acceptable distance to remove cases where the first nearest neighbor may be too distant. This is not consequential in our case (Figure~\\ref{fig:matched_psb}) as the samples are well-matched. Moreover, the nearest neighbor may be a match to one or more QPSBs. Excluding duplicate matches does not change the matched distributions of the different samples shown in Figure~\\ref{fig:matched_psb}. Majority ($\\sim 85\\%$) of QPSBs have at least one match within a distance of $\\sim 1 - 1.5$ in each matched sample; the 85\\% (or 50\\%) of the absolute difference between the variables used to match the samples is $\\lesssim 0.2$\\,dex (or $\\lesssim 0.1$\\,dex). Matching to the closest $k=3$ neighbors also gives similar results.\n\nGiven the considerable uncertainty of estimating the total $M_\\star$ and the burst mass fraction in starbursts and their descendants, it is reasonable to match by the stellar mass at the time of observation and ignore the evolution of $M_\\star$. The amount of stellar mass formed in a recent burst in massive local galaxies may be $5\\%-30\\%$. The molecular gas fractions of starbursts are $\\sim 15\\%-30\\%$ \\citep[e.g.,][]{Saintonge+17}. A significant fraction of PSBs may still have molecular gas fractions of $\\sim 5\\%-10\\%$ \\citep[e.g.,][]{French+15,Yesuf+17b}. Thus, the stellar mass added by a recent burst is likely below 30\\% for most PSBs. The median (or 85\\%) difference $\\Delta \\log\\,M_\\star$ between PSBs and the matched sample is $\\sim 0.1$\\,dex (or $\\sim 0.2$\\,dex). \n\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{M10511_logVdisp_C.pdf}{0.48\\textwidth}{}\n\\fig{M10511_delM05Mpc_Vdisp.pdf}{0.48\\textwidth}{}}\n\\gridline{\\fig{M10511_logCz_delM05.pdf}{0.48\\textwidth}{}\n\\fig{M10511_delM05Mpc_MhLim.pdf}{0.48\\textwidth}{}}\n\\caption{Comparing the stellar velocity dispersions ($\\sigma_\\star$), $z$-band concentration indices ($C$), $\\delta_\\mathrm{0.5Mpc}$, and $M_h$ from \\citet{Lim+17} of starbursts, strong-H$\\delta_A$ AGNs, QPSBs, SFMS galaxies, and QGs. Although the first three samples overlap in the plotted parameters, starbursts have similar $\\sigma_\\star$ and $C$ as SFMS galaxies (some are disc-dominated\/less centrally concentrated), while QPSBs and strong-H$\\delta_A$ AGNs have similar structures as QGs (i.e., most are bulge-dominated). Starbursts also have lower $M_h$ and $\\delta_\\mathrm{0.5Mpc}$ than QGs. Their structures correlate weakly ($\\rho \\approx 0.2$) with $M_h$ and $\\delta_\\mathrm{0.5Mpc}$. \\label{fig:morphSB}}\n\\end{figure*}\n\n\\begin{figure*}[ht!]\n\\gridline{\\fig{Mall_delM05Mpc_matched_dist_R1.pdf}{0.32\\textwidth}{}\n        \\fig{Mall_delM1Mpc_matched_dist_R1.pdf}{0.32\\textwidth}{}\n        \\fig{Mall_delM4Mpc_matched_dist_R1.pdf}{0.32\\textwidth}{}}\n\\gridline{\\fig{Mall_delM8Mpc_matched_dist_R1.pdf}{0.32\\textwidth}{}\n        \\fig{Mall_delk5_matched_dist_R1.pdf}{0.32\\textwidth}{} \n         \\fig{Mall_Q5_matched_dist_R1.pdf}{0.32\\textwidth}{}}      \n\\gridline{\\fig{Mall_Mstar_matched_dist_R1.pdf}{0.32\\textwidth}{}\n        \\fig{Mall_veldisp_matched_dist_R1.pdf}{0.32\\textwidth}{} \n         \\fig{Mall_C_matched_dist_R1.pdf}{0.32\\textwidth}{}}                       \n\\caption{Simultaneously matching all QPSBs to starbursts, upper SFMS galaxies, strong-H$\\delta_A$ AGNs, and QGs of similar multiscale ($0.5-8\\,h^{-1}$Mpc) environmental indicators (panels (a) to (f)), $M_\\star$, $\\sigma_\\star$, and $C$ (panels (g) to (i), respectively). The figure demonstrates that some starbursts, upper SFMS galaxies, and AGNs are progenitors of QPSBs, which in turn are progenitors of some QGs.\\label{fig:matched_psb}}\n\\end{figure*}\n\n\\section{Discussion}\\label{sec:disc}\n\nEnvironment, galaxy mergers\/interactions, rapid gas consumption during the starburst phase, stellar and AGN feedback have all been invoked to explain the origin of starbursts and\/or QPSBs \\citep{Mihos+96, Hopkins+08, Snyder+11,Davis+19, Lotz+21}. In this section, we discuss how the results presented in this paper may help discriminate among competing mechanisms. We also compare our results with those of previous studies. Future analyses of the multiscale environments and structures of starbursts and their descendent in cosmological simulations may provide more insightful interpretations.\n\n\\subsection{Starbursts and QPSBs are Rare in High-density Environments}\n\nMajorities of starbursts and QPSBs are isolated\/centrals and do not live in massive groups and clusters (Table~\\ref{tab:fcentMh_sb} and Table~\\ref{tab:richness2_sb}). Environmental effects such as ram pressure stripping cannot explain the origin of typical starbursts and QPSBs. In particular, for galaxies with $M_\\star > 3 \\times 10^{10} \\, M_\\odot$, $\\sim 80\\%-85\\%$ starbursts and $\\sim 85\\%-95\\%$ strong-H$\\delta_A$ AGNs and QPSBs are centrals. Half of them have $M_h \\approx 3 \\times 10^{12} \\, M_\\odot$, $\\sim 85\\%-90$\\% have $M_h \\lesssim 10^{13} \\,M_\\odot$, and only $\\sim 2\\%-4\\%$ have $M_h > 10^{14} \\,M_\\odot$. In contrast, $\\sim 50\\%-55\\%$ of QGs have $M_h \\lesssim 10^{13} \\,M_\\odot$ and  $\\sim 8\\%-14\\%$ of QGs have $M_h > 10^{14} \\, M_\\odot$. These results broadly agree with expectations from cosmological simulations \\citep{Davis+19,Lotz+21,Wilkinson+18}.\n\n\\citet{Davis+19} studied the cold interstellar medium of 1244 simulated QPSBs selected from the EAGLE cosmological simulations \\citep{Schaye+15}. The simulated QPSBs have $M_\\star > 3 \\times 10^{9}\\,M_\\odot$, only 19\\% of them are satellites or become satellites within $\\pm 0.5$\\,Gyr of the time of burst, and only $ \\sim 10$\\% show gas streams consistent with ram pressure stripping (compared to 5\\% for the control sample). The satellite fraction of QPSBs in EAGLE agrees reasonably with our results. Satellite-related environmental effects are not the dominant formation channel of QPSBs in EAGLE simulations.\n \nSimilarly, \\citet{Lotz+21} studied 647 QPSBs with $M_\\star \\ge 5 \\times 10^{10}\\,M_\\odot$ in the Magneticum Pathfinder cosmological hydrodynamical simulations. The authors found that their simulated QPSBs at $z \\approx 0$ live in low-mass halos, similar to stellar mass-matched SFGs, but unlike mass-matched QGs. More quantitatively, 89\\% of QPSBs, 85\\% SFGs, and 79\\% of QGs in the simulation have $M_h < 10^{13}\\,M_\\odot$. In contrast, only 2\\% of QPSBs, 3\\% of SFGs, and 7\\% QGs have $M_h > 10^{14}\\,M_\\odot$. \n\n\\citet{Wilkinson+18} selected 196 starbursts at $z = 0.15$ with $M_\\star > 10^{9}\\,M_\\odot$ in Illustris simulations using similar definition ($\\Delta\\,\\mathrm{SSFR} > 0.6$\\,dex) to ours. After dividing their sample based on the merger history within 2\\, Gyr prior to the starburst, the authors found that the mean values of $M_h$ for the merger starbursts and non-merger starbursts are $M_h = 3.4 \\pm 2.0 \\times 10^{12} \\,M_\\odot$ and $M_h = 6.1 \\pm 2.1 \\times 10^{12} \\,M_\\odot$, respectively. These values are not significantly discrepant with the observational estimates. Specifically, noting that most of the starbursts in Illustris have $M_\\star <  10^{10}\\,M_\\odot$, we find, for the same $M_\\star$ threshold, a median (15\\%, 85\\%) $M_h$ value of $\\log M_h \\approx  1\\,(0.6, 3) \\times 10^{12} M_\\odot$ based on \\citet{Tinker21}'s catalog. Similarly, the median values for \\citet{Lim+17} is catalog is $M_h \\approx  0.5\\,(0.4, 2) \\times 10^{12} M_\\odot$. In addition, \\citet{Wilkinson+18} found that their starbursts are located in the lower-density environments (under density within 0.5\\,Mpc or 1\\,Mpc as shown in their Figure 4) compared to random samples from Illustris, consistent with our results. Given our comprehensive results on multiscale environments of starbursts, a future study can definitely do better comparisons, preferably by also classifying the observed starbursts into mergers and non-mergers to confirm or refute the predicted trend.\n\nUsing EAGLE cosmological simulations, \\citet{Matthee+19} found that the SFMS residuals at $z = 0.1$ arise from fluctuations on short ($<0.2-2$\\,Gyr) and long ($\\sim10$\\,Gyr) timescales. The long-timescale fluctuations are related to $M_h$ and halo formation time. At $M_\\star < 10^{10}\\,M_\\odot$, EAGLE predicted a clear trend that halos that form later have galaxies with higher SSFRs in them. The simulated starbursts (SSFR $\\gtrsim 0.1$\\,Gyr) at $M_\\star < 10^{10}\\,M_\\odot$ have lower halo formation time and higher $M_h$ than SFMS galaxies at similar $M_\\star$. At least qualitatively, this agrees with our result that the starbursts with $M_\\star < 10^{10}\\,M_\\odot$ have lower environmental densities than SFMS galaxies do. It is expected that haloes that assembled more recently are substantially less clustered than those that assembled earlier \\citep[][]{Gao+05,Wechsler+18}. \\citet{Berti+21} also found that galaxies above the SFMS are less clustered than those below, at fixed $M_\\star$. Our results are consistent with short and long-term changes in $\\Delta\\,\\mathrm{SSFR}$ as indicated by the existence of galaxies evolving from starbursts to QPSBs and by the clear difference in environments of upper SFMS and lower SFMS galaxies, quantified in several ways.\n\n\\subsection{Mergers and Interactions may Trigger Some Starbursts}\n\nAs discussed in the introduction, the merger hypothesis is a viable pathway for some galaxies to evolve from SFGs $\\rightarrow$ starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs $\\rightarrow$ QGs. Furthermore, it is a common view that classical bulges and ellipticals form mainly by dissipative, gas-rich mergers, while pseudobulges form by disc-related, secular processes \\citep{Kormendy+04,Brooks+16}. Gas-rich mergers lead to central mass concentration and may destroy discs \\citep{Bournaud+05, Martin+18}. The starbursts, QPSBs, and AGNs with high $C$ and $\\sigma_\\star$ (Figure~\\ref{fig:morphSB}) are classical bulges or ellipticals \\citep{Yesuf+20a}. Thus, demonstrating the existence of a population of galaxies along the aforementioned evolutionary sequence of similar $M_\\star$, environments, $C$, and $\\sigma_\\star$ provides indirect support for the merger hypothesis. On the other hand, a large fraction of nearby starbursts have low $C$ and $\\sigma_\\star$, unlike most QPSBs and QGs. These pseudobulge starbursts likely were not mergers. Likewise, \\citet{Dressler+13} and \\citet{Abramson+13} found that most starbursts in their sample $z \\approx 0.4$ are not centrally concentrated, just like the SFGs in their sample. These starbursts were identified by their anomalously strong H$\\delta$ absorption or \\ion{O}{2} emission. Most of them, however, do not lie significantly above SFMS\\footnote{The SFRs in \\citet{Dressler+13} and \\citet{Abramson+13} were estimated using \\emph{Spitzer}\/MIPS 24\\,$\\mu$m and\/or emission lines (\\ion{O}{2}, H$\\beta$ or\/and H$\\alpha$).}. Their structures were quantified using S\\'{e}rsic indices in $J$- and $K$-bands. Using $C$ and S\\'{e}rsic index based on $K$-band images from the GAMA survey \\citep{Kelvin+12}, we have confirmed that the structures of $z < 0.2$ starbursts (and upper SFMS galaxies) are very different from QGs. Therefore, the low-concentration starbursts in our sample do not simply fade and become QPSBs and QGs in a short period of time -- a major structural rearrangement is required. Unlike \\citet{Dressler+13}'s work, this study does not find that QPSBs live in similar environments as QGs at $z < 0.2$.\n\nWell-separated galaxy interactions do not account for most starbursts, although they may trigger some starbursts. As shown in Figure~\\ref{fig:Q5_sb},  less than 15\\% (8\\%) of massive galaxies above $10^{10}\\,M_\\odot$ have first nearest neighbors within $r_{p,1} < 200$\\,kpc (100\\,kpc). The fraction of starbursts with $r_{p,1} < 200$\\,kpc is marginally different ($\\lesssim 2\\%-3\\%$) from that of SFMS galaxies of similar $M_\\star$. Likewise, \\citet{Yamauchi+08} found that only $\\sim 8\\%$ of QPSBs have nearby companion within 50\\,kpc compared to 5\\% for normal galaxies. Therefore, tidal interactions are not the dominant mechanism for triggering starbursts although they may trigger some starbursts \\citep[e.g., ][]{ Li+08,Patton+20}. \n\nIn summary, mergers may drive the evolution of some galaxies from SFGs $\\rightarrow$ starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs. The existence of a significant number of isolated and diffuse\/pseudobulge starbursts indicates that not all starbursts are triggered by mergers\/interactions. What else can be a triggering mechanism? Next, we discuss the possible role of gas accretion modulated by environment and feedback in triggering starbursts. This hypothesis will be tested rigorously in the future.\n\n\\subsection{Relationship Between Environment and Gas Accretion or AGN Activity}\n\nObservational estimates indicate that gas accretion rates from minor mergers are not enough ($\\sim 0.1-0.3\\,M_\\odot\\,\\mathrm{yr}^{-1}$) to sustain the observed SFR in local galaxies \\citep{Sancisi+08, DiTeodoro+14}. Moreover, halo gas estimates made using background quasars of 17 low-$z$ starbursts or PSBs indicates a total gas mass of $\\sim 3-6 \\times 10^{9}\\,M_\\odot$ in their the circum-galactic medium \\citep{Heckman+17}. These galaxies are mostly normal late-type galaxies. ``A few appear to be interacting with companions, but few (if any) appear to be recent or on-going mergers.\" Galaxy simulations also indicate that gas infall rates onto dark matter halos are dominated by the diffuse component over the merger contribution, at least for galaxies with $M_h < 10^{13}\\,M_\\odot$ \\citep{Fakhouri+10,Wright+21}. \n\nGas accretions onto halos or galaxies are regulated by the environment and feedback \\citep[e.g.,][]{vandeVoort+17, Brennan+18, Correa+18}. Based on predictions of cosmological simulations, plenty of cold gas is expected in the halos of starbursts like ours. Most starbursts in our sample have $M_h \\approx \\mathrm{few} \\times 10^{12}\\,M_\\odot$ and live in the low-density environments, which are conducive for gas accretion onto their halos. As discussed in the introduction, starbursts are also overwhelmingly asymmetric\/disturbed. Diffuse gas accretion rates of a few $M_\\odot\\,\\mathrm{yr^{-1}}$ may plausibly explain the ubiquity of asymmetries in strongly star-forming galaxies \\citep{Keres+05,Bournaud+05,Jog+09, Yesuf+21}. According to cosmological simulations, gas is accreted onto halos in hot and cold modes. The contribution of the hot mode increases with $M_h$ as haloes become massive enough to efficiently shock heat the accreting gas \\citep{Keres+05,Dekel+06}. The median hot fraction increases sharply around $M_h \\approx 10^{12}\\, M_\\odot$ and almost all the gas is accreted in the hot mode for halos above $10^{13}\\,M_\\odot$ \\citep{Wright+21}. Starbursts and upper SFMS galaxies in EAGLE simulations have higher gas accretion rates than galaxies below SFMS. Furthermore, gas accretion rates in massive halos and in dense environments ($\\delta_{10}$) in EAGLE are strongly suppressed, especially for satellite galaxies at smaller halo-centric distances \\citep{vandeVoort+17}. However, the accretion rates of simulated, massive, central galaxies (in similar $M_\\star$ range as ours) are not appreciably impacted by their environments. They have gas accretion rates comparable to their SFRs. EAGLE and other simulations also predict that stellar and AGN feedback can curtail gas accretion rates onto galaxies significantly \\citep[e.g.,][]{Brennan+18, Correa+18}. We have shown that strong-H$\\delta_A$ AGNs have similar large and small-scale environments, $C$, and $\\sigma_\\star$ (black hole mass) to those of starbursts and QPSBs of similar $M_\\star$. This is expected if AGN feedback acts as a maintenance mode, preventing cooling flows onto these galaxies. \n\nApart from the maintenance role, however, AGNs may not primarily impact the evolution from starbursts to QPSBs; it is rather driven by gas consumption. First, a significant time delay is observed between starbursts and AGNs that show PSB spectral signatures \\citep{Wild+10, Yesuf+14}. Second, PSBs are a mixture of gas-poor and gas-rich systems, and about half of QPSBs have large amounts of cold gas and dust \\citep{Zwaan+13, French+15, Rowlands+15, Alatalo+16, Yesuf+17b, LiZhihui+19, YesufHo20b,Bezanson+22}, indicating that AGN feedback did not completely remove or destroy them. Strong AGNs in normal SFGs are also not necessarily gas-deficient \\citep{Ho+08, Husemann+17, Rosario+18, Shangguan+20, YesufHo20a}. Furthermore, no consensus has been reached on the effects of AGN-driven outflows in galaxies in general or in PSBs in particular \\citep[][and references therein]{Yesuf+17a, Baron+22}. Some simulations also found that AGNs do not clear out gas in PSBs \\citep[][but see \\citet{Lotz+21}]{Snyder+11,Davis+19}, and not all simulated PSBs are gas-deficient.\n\nBulge-dominated central\/isolated galaxies of similar $M_\\star$ and structures can be evolutionarily linked through a life cycle wherein the molecular gas amount already inside galaxies regulates their SFRs and levels of AGN activity \\citep{YesufHo20a}. These galaxies first consume their gas mostly through bursty star formation, then pass through a transition phase of intermediate gas richness in which star formation and AGNs coexist, before  retiring as gas-poor QGs. Strong AGNs are gas-rich; neither their gas reservoirs nor their abilities to form stars seem to be impacted instantaneously by AGN feedback. The molecular gas masses of these galaxies were estimated using an empirical relation among molecular gas, dust absorption, and metallicity \\citep{YesufHo19}. Incidentally, galaxies in low-density environments show more dust absorption than those at high-density environments \\citep{Kauffmann+04}.  \\citet{YesufHo20a}'s sample includes starbursts and PSBs, although most galaxies in their sample are ordinary. In addition, using the same empirical calibration, \\citet{YesufHo20b} found median (15\\%, 85\\%) molecular gas mass and fraction of $\\log\\,M_\\mathrm{H_2}\/M_\\odot = 9.3\\,(9.1, 9.6)$ and $f_{H2} = 4\\,(2.5, 8)$\\%, respectively, for strong-H$\\delta_A$ Seyferts with $\\log\\,M_\\star\/M_\\odot = 10.5-11$. Therefore, strong AGNs are gas-rich regardless of their SFHs. Simulations also indicate that gas content of galaxies do not change within $\\sim 1\\,$Gyr due to halo gas accretion \\citep{Scholz-Diaz+21}. Gas amount already in the galaxies determines the level of star formation and AGN activity. Environment and feedback regulate its long term replenishments and the cooling rate of halo gas.\n\n\\citet{Sabater+15} inferred from SDSS observations that the large-scale environments ($\\delta_{10}$) and galaxy interactions only affect AGN activity indirectly by influencing the central gas supply in the nuclear region. After controlling for the effects of $M_\\star$ and central SFR (which is correlated with cold gas), the authors found that the effects of the environment on AGN fraction and AGN luminosity are minimal.\n\nLastly, previous estimates of $M_h$ based on AGN clustering, weak lensing, or galaxy groups found that optically selected AGNs reside preferentially in halos of $M_h \\approx \\mathrm{few} \\times 10^{12}\\,h^{-1} M_\\odot$ \\citep{Mandelbaum+09,ZhangZiwen+21}, consistent with our finding. Obviously, the discussion above does not do justice to the rich literature on AGN environments, with a plethora of differing results \\citep[see references in][]{Sabater+15,Man+19,ZhangZiwen+21}. We add a new result that AGNs with $H\\delta_A  > 4${\\AA} live in lower-density environments than AGNs with $H\\delta_A  < 3${\\AA} do. The strong-$H\\delta_A$ AGNs are only $\\sim 10\\%$ of narrow-line AGNs.\n\n\\subsection{Comparison with Previous Studies of Starbursts and QPSBs at $z < 0.3$}\n\nThe advent of large redshift surveys such as the SDSS led to the discovery that QPSBs are not particularly prevalent in clusters or high-density environments, but rather live in a wide range of environments \\citep{Zabludoff+96,Quintero+04, Blake+04, Goto05b, Hogg+06, Helmboldt+08, Yan+09,Paccagnella+19}. With few exceptions, most these studies reached a similar conclusion as ours. Namely, QPSBs live mainly in lower density environments than QGs. Although the environments of QPSBs are widely studied, only few studies \\citep{Owers+07, Pawlik+18} attempted to directly compare their environments with those of starbursts and\/or AGNs using the same datasets and methodology. Next, we give an overview of these studies and compare our results.\n\n\\citet{Quintero+04} studied the environments of 1143 k+a galaxies in SDSS at $z=0.05-0.2$, which were identified as outliers in the diagram of H$\\alpha$ emission EW and the A\/K stellar population fraction. The authors found that k+a galaxies in their sample do not primarily live in the high-density environments or clusters typical of bulge-dominated galaxies. In particular, \\citet{Quintero+04} found that the mean overdensity of galaxies around k+a galaxies, measured within $1\\,h^{-1}$ Mpc and $8\\,h^{-1}$ scales are similar to the mean overdensities of disk-dominated galaxies (S\\'{e}rsic $n < 2$). However, the comparison samples were not mass-matched, and the average values were taken over a wide range in $M_\\star$, potentially washing out the mass trends (Figure~\\ref{fig:massdel18} and Figure~\\ref{fig:del18_psb}). \n\nIn the follow-up study of \\citet{Quintero+04}'s k+a sample, \\citet{Hogg+06} studied the fractional abundance of the k+a galaxies and comparison samples as function of environmental density. The authors quantified environment by measuring: (1) number densities of galaxies in 8\\,$h^{-1}$Mpc radius, (2) transverse distances to nearest Virgo-like galaxy clusters, and (3) the distance to the nearest neighbor galaxy, $r_{p,1}$. They found that the fractions of k+a galaxies depend on the environmental indicators above in similar way to those of SFGs. \\citet{Hogg+06} concluded that: (1) starbursts\/QPSBs are not triggered by external tidal impulses from close passages of massive galaxies (because they did not find strong dependance of k+a fraction with $r_{p,1}$). (2) A small fraction of QPSBs are created by ICM interactions on infall into clusters (because the k+a fraction does not vary strongly with the distance from the cluster center, particularly outside of the virial radius). Although \\citet{Hogg+06} did not study starbursts directly, we reach similar conclusions. Similarly, in 521 SDSS clusters at $z < 0.1$ studied by \\citet{vonderLinden+10}, the fraction of galaxies that have strong Balmer absorption lines is independent of the distance from the cluster center; it is only marginally significant within $0.2R_{200}$ and for $M _\\star > 3 \\times 10^{10}M_\\odot$.\n\n\\citet{Owers+07} studied the environment and spatial clustering of 418 starbursts at $z < 0.16$ selected from the 2dFGRS and compared them with random control samples matched in redshift and $r$-band apparent magnitude. The environmental indicators they used include $r_{p,1}$, the tidal parameter, and $\\delta_{5}$. The authors found that the distribution the tidal parameter and $r_{p,1}$ of starbursts differ significantly ($p=.0017$ and $p=.015$, respectively) from those of the random control samples; a higher proportion of starbursts have $\\log\\,Q_1 > -1$ than the control sample and 38\\% of starbursts have bright neighbors within $\\sim 20$\\,kpc. The authors concluded that galaxy-galaxy mergers\/interactions are important in triggering $\\sim 30\\%-40\\%$ starbursts in their samples, and that the rest of starbursts, in which environmental influences are not obvious, must be internally driven. On the other hand, the authors found that starbursts are less clustered on $\\sim 1-15\\,h^{-1}$\\,Mpc scales compared to control samples, in agreement with our results. Their starbursts were selected using H$\\alpha$ EW. After showing that the 56 QPSBs in 2dFGRS studied by \\citet{Blake+04} have broadly similar environmental properties as starbursts, \\citet{Owers+07} surmised that only a small fraction of their starbursts evolve through the k+a phase. In comparison, our analysis is done using samples of starbursts and QPSBs that are more than ten times larger than those of the two studies. It also controls for effects of $M_\\star$ and structure better. We find only marginal excess of neighbors around starbursts in our data (Figure~\\ref{fig:Q5_sb}). In the future, we plan to improve our analysis by combining spectroscopic and photometric redshifts of nearby neighbors of starbursts. From cursory visual inspections our sample, we do not think that 38\\% of our starbursts have close but well-separated neighbors within $\\sim 20$\\,kpc. \n\n\\citet{Helmboldt+08} showed that $M_\\star$, $\\sigma_\\star$, and $r_{p,4}$ of early-type SFGs are similar to those of QPSBs ($N= 435$, $z = 0.05-0.08$, and H$\\delta_A > 2\\,${\\AA}). To visually identify E\/S0 morphology, the authors only selected bright ($m_r < 16$) SFGs that are nearby enough that their spiral arms or prominent discs would be apparent in their $g$-band images. The authors found the fractions of early-type SFGs and QPSBs are similar, and they increase in parallel with $r_{n,4}$ (i.e., with decreasing environmental density). The environments of these galaxies are more similar to those of typical SFGs than to those of most early-type QGs, in agreement with our results. Unlike \\citet{Helmboldt+08}, we do not restrict the comparison to bright SFGs and do not exclude AGNs from our analysis. Our selection is not based on morphology, but we agree that QPSBs have E\/S0 morphology as indicated by their $C$ and $\\sigma_\\star$.\n\nFurthermore, \\citet{Pawlik+18} studied 189 strong Balmer absorption-line galaxies, including those that are QPSBs or AGNs. This sample is taken from the SDSS and has $z < 0.05$. About 80\\% of galaxies in this sample have $M_\\odot < 3 \\times 10^{10}\\, M_\\odot$. The authors studied the SFHs, structures, and environments (as quantified by $\\delta_{5}$) of different classes of galaxies with strong Balmer absorption lines to ascertain whether or not they are evolutionarily connected. The goals of the authors were similar to ours except that the authors were also interested in measuring faint merger signatures (asymmetries), which necessitated restricting their sample to $z < 0.05$. Consequently, their sample is much smaller and different from ours. Nevertheless, the authors showed that low-mass ($9.5 < \\log\\,(M\/M_\\odot) < 10.5$), QPSBs have similar distribution of $\\delta_{5}$ as SFGs. At high-mass ($\\log\\,(M\/M_\\odot) > 10.5$), the authors tentatively found QPSBs are preferentially found in lower density environments than the control sample of SFGs. We conclusively confirm these results with a larger sample. More specifically, having split our sample into two $M_\\star$ ranges as in \\citet{Pawlik+18} and also combining upper SFMS and lower SFMS galaxies together, we find that low-mass QPSBs have a similar $\\delta_{5}$ distribution to that of low-mass SFGs of (AD test $p > .25$), whereas high-mass QPSBs have a different $\\delta_{5}$ distribution from that of high-mass SFGs (AD test $p = .005$). A similar result is also shown in Figure~\\ref{fig:del5_psb}. \n\n\\citet{Pawlik+18} also found that QPSBs and strong Balmer-line AGNs have similar $r$-band Sersic indices, SFHs, and $\\delta_{5}$. This also agrees with our more compressive results on the structures ($\\sigma_\\star$ and $C$) and environments of the two populations quantified several ways, not only using $\\delta_{5}$. Therefore, \\citet{Pawlik+18}'s and our results confirm that these two populations have the same physical origin and are the same class of galaxies, i.e., PSBs. Our work goes further to demonstrate that these two populations can be evolutionary linked to a subset of starbursts, (upper) SFMS galaxies, and QGs that have similar structures and multiscale environments. \\citet{Pawlik+18} did not study starbursts; their randomly selected SFG control sample did not make distinctions between SFGs of different $\\Delta\\,\\mathrm{SSFR}$ as this study does. \n\nFurthermore, \\citet{Pawlik+18} proposed $M_\\star$ dependent evolutionary scenarios that lead to QPSBs (see their Figure 12 and \\cite{Dressler+13}). Our results do not fully support these scenarios but let us explain what they are. In the low-mass range, (a) after a violent triggering event (e.g., galaxy mergers), a SFG experiences a starburst and a morphological transformation, which are likely followed by a strong AGN activity, and then the galaxy becomes a bulge-dominated QG; (b) a less violent mechanism leads to a starburst that fades more gradually through a star-forming PSB phase, and it may or may not lead to a morphological transformation. Depending on the remaining gas reservoir, the galaxy either returns to the SFMS or becomes a QPSB or PSB AGN and subsequently joins the red sequence. In high-mass range, the aforementioned scenario (a) also happens. Besides, (c) an already quiescent galaxy experiences a relatively weak burst due to a minor merger, after which it passes through a brief PSB phase only to return back to the red sequence. In other words, to explain some PSBs, the authors proposed a cycle of rejuvenation starting from QGs but not passing through SFMS at high $M_\\star$, and a cycle of minor burst originating from SFMS but not passing through the red sequence at low $M_\\star$. Our analysis is not inconsistent with the proposed cycle of burst at low-mass; some low-mass starbursts have low environmental density and low central concentration, unlike QPSBs. However, the cycle of burst without morphological transformation may also happen in the high-mass range; some high-mass starbursts also do not evolve to QPSBs because they have low central concentration, although the environments of the two populations are similar at the high-mass range. Moreover, the our analysis does not support the proposed cycle of rejuvenation. First, this paper shows that QPSBs in all $M_\\star$ ranges can be matched to starbursts, SFMS galaxies, strong-H$\\delta_A$ AGNs, and QGs. Second, it shows that the environments of QGs, as a whole, are very different from those starbursts and PSBs. Therefore, a rejuvenation of QGs is not a dominant mechanism in originating QPSBs.\n\n\\subsection{Caveats}\n\nThe SFR used in this work is based on UV-optical-mid IR SED fitting, and it is sensitive to changes on $\\sim 100$ Myr timescale. Although H$\\alpha$ emission is more sensitive, this study does not use it to define starbursts. \\citet{Baron+22} recently showed that H$\\alpha$-based SFR can be orders of magnitude smaller than the FIR-based SFR, suggesting that some highly obscured starbursts can be misclassified as transition PSBs if H$\\alpha$ is used \\citep[see also][]{Poggianti+00}. In addition, spatial gradients in star formation could be such that the H$\\alpha$ measured in SDSS fiber may not account for the star formation in outskirts of galaxies. Although the SED-based SFRs are sufficient for our purpose, future work may investigate their sensitivity to burst amplitude, mass fraction, and burst duration. It is already shown that these SFRs may overestimate the instantaneous SFRs in PSBs \\citep{Hayward+14, Salim+16, YesufHo20b}. Differences in star formation timescales can cause H$\\alpha$ to be absent during a PSB phase, while UV and mid-IR emission still persist after the O stars have died off. Thus, our QPSBs and strong-H$\\delta_A$ AGNs are selected independent of their SFR estimates from their SEDs. The upper SFMS galaxies that are matched to QPSBs are consistent with being either low-amplitude starbursts or fading starbursts. We do not distinguish between the two cases at the moment.\n\nIt should be noted that the SDSS fiber spectroscopy only cover $\\sim$ 3\\,kpc radius at $z = 0.1$. Our work and the aforementioned studies could only select QPSBs based on their central stellar populations. Using MaNGA integral field survey data, \\citet{Chen+19} found 31 galaxies with central post-starburst (CPSBs) regions, 37 galaxies with off-centre ring-like post-starburst regions (RPSB), and 292 galaxies with irregular PSBs regions. The authors found that CPSBs have suppressed star formation throughout their bulge and disc, while RPSBs have recently suppressed star formation in their outer regions and active nuclear star formation. Most RPSBs are located in lower SFMS (median $D_n(4000)$ = 1.4), whereas CPSBs are mainly located in the green valley. RPSBs have similar kinematics ($v_\\mathrm{rot}\/\\sigma_\\star$) as a control sample of non-PSBs matched in $M_\\star$ and global $D_n(4000)$ index. Most of them have S\\'{e}rsic index $n < 2$ (median $n=1.6$). \\citet{Chen+19} provided two evidence that RPSBs and CPSBs are produced by different physical mechanisms. First, the two populations have very different SFHs (mass-weighted ages and  spectral indices) at all radii. Second, the CPSBs have lower $v_\\mathrm{rot}\/\\sigma_\\star$ at all radii. We checked that the environments of RPSBs are consistent with those of starbursts and QPSBs ($\\log\\,(1+\\delta_\\mathrm{0.5Mpc}) = 0.04\\,(-0.7, 0.4)$, $\\log\\,(1+\\delta_\\mathrm{8Mpc}) = -0.1\\,(-0.4, 0.2)$, $\\log\\, M_h\/M_\\odot = 12.0\\,(11.3, 12.3)$, 75\\% are centrals, and $\\log\\, M_\\star\/M_\\odot = 10.1\\,(9.5, 10.6)$).\n\nThe SDSS spectroscopic sample is incomplete because it is flux-limited and because of the fiber collision -- the 55$^{\\prime\\prime}$ minimum separation requirement between spectroscopic fibers. The former affects galaxies in our sample with $M_\\star \\lesssim 10^{10}\\, M_\\odot$ at $z \\gtrsim 0.1$, while the latter affects galaxies that reside in groups or are pairs. Restricting the redshifts of galaxies in our two lowest mass ranges to $z < 0.1$ does not change the main conclusions. The fraction of missing pairs in \\citet{Tempel+14}'s catalog is about 8\\%. \\citet{Lim+17}'s group catalog incorporates redshifts for the missing galaxies from different sources to achieve about 98\\% completeness. \\citet{Tinker21} assigned the spectroscopic redshift and other properties of the nearest-neighbor galaxy when they are not measured. The environmental indicators measured by this author use the SDSS DR17, which has improved redshift completeness. Furthermore, there are good agreements between the measurements based on SDSS and similar measurements based on the Galaxy And Mass Assembly (GAMA) survey \\citep{Brough+13}; the GAMA survey is more complete and deeper than SDSS. Although it is important to ameliorate the effects of survey incompleteness in the future (e.g., using photometric redshifts and deeper redshift surveys), they are not large enough to change the conclusions of this study \\citep[see also][]{Hogg+06, Luo+14}. Lastly, halo mass estimates in the group catalogs clearly need improvements and their discrepancies need a resolution.\n\n\\section{Summary and Conclusions}\\label{sec:conc}\n\nUsing SDSS data, we study how star formation and black hole activity depend on environments in galaxies at $z=0.02-0.16$ and with $M_\\star = 3\\times 10^9 - 3\\times 10^{11}\\,M_\\odot$. In particular, we check the consistency of the evolution from SFGs $\\rightarrow$ starbursts $\\rightarrow$ AGNs $\\rightarrow$ QPSBs $\\rightarrow$ QGs using multiple environmental indicators, $C$, and $\\sigma_\\star$ for galaxies in four $M_\\star$ bins of 0.5\\,dex. The following are our conclusions: \n\n\\begin{itemize}\n\n\\item All QPSBs can be matched to some SFGs, starbursts, AGNs, and QGs that have similar $M_\\star$, $C$, $\\sigma_\\star$, and multiscale environments. The environments of QPSBs are quite different from those of QGs. But, they are broadly similar to those SFGs at the scale of $\\gtrsim 2\\,$Mpc.\n\n\\item The environments of starbursts are broadly similar to or slightly lower density than those of SFGs. Starbursts clearly do not reside in high density environments populated by most QGs. Their distributions of small-scale environments ($r_{p,\\,1}$, $\\delta_\\mathrm{0.5Mpc}$, and $M_h$) are also significantly different from those of non-bursty SFGs; larger fractions of starbursts have fewer (massive) neighbors within $\\lesssim 1\\,h^{-1}$Mpc than normal SFGs.\n\n\\item The environments of AGNs with H$\\delta_A > 4$\\,{\\AA} are similar to QPSBs. Moreover, their distributions of $C$ and $\\sigma_\\star$ are also similar. Therefore, AGNs with H$\\delta_A > 4$\\,{\\AA} are also PSBs. They all can be matched to some starbursts that have similar $M_\\star$, $C$, $\\sigma_\\star$, and multiscale environments. AGNs with H$\\delta_A > 4$\\,{\\AA} live in lower density environments than those of AGNs with H$\\delta_A < 3$\\,{\\AA}. The latter live in environments similar to those of lower SFMS galaxies.\n\n\\item Depending on the $M_\\star$ range, $\\sim 70\\%-90$\\% of starbursts, QPSBs, and AGNs with H$\\delta_A > 4$\\,{\\AA} are isolated or central galaxies (are not satellites), and $\\sim 85\\%$ of them have $M_h < 10^{13}\\,M_\\odot$ and only $\\sim 2\\%-4\\%$ have $M_h < 10^{14}\\,M_\\odot$. The distributions $M_h$ and central fractions of starbursts, QPSBs, and AGNs with H$\\delta_A > 4$\\,{\\AA} are significantly different from those of QGs.\n\n\\item Unlike QGs, only a small fraction starbursts, QPSBs, and AGNs with H$\\delta_A > 4$\\,{\\AA} are found in clusters and rich groups. For example, in $\\log\\,M_\\star\/M_\\odot = 10.5-11$ range, $65\\%$ of starbursts are isolated, $19\\%$ are pairs, $9\\%$ have 2 or 3 neighbors, $3\\%$ are in rich groups, and  $\\sim 1\\%$ in clusters.\n\n\\item The distributions of environmental indicators of upper SFMS and lower SFMS are significantly different; consistent with previous finding  that the scatter of SFMS is not random \\citep{Berti+21}. Furthermore, some upper SFMS and lower SFMS galaxies may be progenitors of QPSBs because they have similar structures and environments. These SFGs are either fading\/weak starbursts or some QPSBs may have originated from a truncation of normal SFGs.\n\n\\item A significant fraction ($\\sim 20\\%-30\\%$) of starbursts cannot be matched to QPSBs or QGs of similar $M_\\star$, structures, and environments. Some starbursts have low $C$ and $\\sigma_\\star$, similar to late-type SFGs and\/or their environments can be inconsistently lower density than those of QPSBs. Thus, a significant fraction of starbursts may not quench rapidly.\n\n\\item Most starbursts are not triggered by tidal interactions. About $80\\%-90\\%$ of starbursts do not have nearby neighbors within 200\\,kpc and their tidal parameters are in fact slightly lower than SFMS galaxies. \n\n\\item The mass overdensity within 0.5\\,$h^{-1}$\\,Mpc, $\\delta_\\mathrm{0.5Mpc}$, is significantly correlated with halo mass, $M_h$. It may be provide useful information when $M_h$ is hard to measure accurately.\n\n\\end{itemize}\n\nOur work implies that the evolution from SFGs $\\rightarrow$ starbursts $\\rightarrow$ AGNs $\\rightarrow$QPSBs $\\rightarrow$ QGs or rejuvenation of QGs in the opposite direction is not a common path taken by typical galaxies at $z < 0.2$. This is mainly because the environments of typical QGs are very different from those of QPSB and starbursts. The importance of rapid quenching of starbursts to the build-up of red sequence need to be reassessed in the future taking their structures and environments into account, in addition to their inferred SFHs. Although we tried to be thorough in our analysis of multiscale environments, further checks should be done using upcoming and future data to confirm the intriguing dependance of $\\Delta\\,\\mathrm{SSFR}$ of SFGs on the small-scale environmental indicators. It would also be useful to improve our analysis with a selection of more complete samples of weak starbursts or\/and PSBs with ongoing star formation or\/and AGN activity. The merger hypothesis for the origin of PSBs is obviously incomplete. Future investigations need to quantify the prevalence of both major and minor mergers in starbursts and constrain alternative explanations. For example, a future study on the role of gas accretion in triggering starbursts in low-density environments will be useful since mergers cannot fully account for prevalence of disturbances\/asymmetries and low-$C$ discs in starbursts \\citep[e.g.,][]{Yesuf+21}. Likewise, a multivariate comparison of observed properties of starbursts and PSBs with similar measurements based on mock observations of cosmological simulations will significantly improve our understanding of the rapid quenching of star formation. \n\n\n\\emph{Software}: astropy \\citep{AstropyI,AstropyII}, scipy \\citep{Scipy}, scikit-learn \\citep{scikit-learn}, and statmodels \\citep{Statsmodels}\n\n\\begin{acknowledgments}\n\nWe thank the anonymous referee and John Silverman very much for their suggestions and comments that significantly improved the paper. H. Yesuf was supported by The Research Fund for International Young Scientists of NSFC (11950410492). Kavli IPMU is supported by World Premier International Research Center Initiative (WPI), MEXT, Japan \n\nFunding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http:\/\/www.sdss3.org\/.\nSDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State\/Notre Dame\/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nMany materials have been proposed theoretically as conventional phonon-mediated \nsuperconductors having a high superconducting transition temperature (T$_c$).\nBased on the BCS theory\\cite{bardeen}, materials with light masses and strong bonds\nare promising candidates for high-T$_c$ superconductors\\cite{moussa,cohen}\nbecause T$_c$ is scaled by the inverse square root of the atomic mass.\nTherefore, theoretical studies have been intensively performed focusing on the\ncompounds consisting of the lightest hydrogen atom. In experiments, on the other hand,\nachieving a high-T$_c$ in hydrogen compounds has not been reported yet.\nRecently, it is experimentally reported that, under extreme high pressures of \n100--200 GPa, sulfur hydride transforms to a metallic state and\nshows extremely high-T$_c$ up to $\\sim$200K\\cite{drozdov,troyan}.\n\nTo find out the crystal structure of the high-T$_c$ sulfur hydride,\nmany {\\it ab-initio} studies have been done and most of these studies \nhave concluded that cubic H$_3$S will form with a H-rich decomposition environment \nunder high pressure\\cite{duan,wang,errea,bernstein,duan2,li,errea2}.\nFurthermore, from electron-phonon coupling (EPC) calculations\n\\cite{duan,wang,errea,bernstein,duan2,li,errea2,papa,heil,sano},\nit is revealed that strong coupling happens between high-frequency phonon modes\nand electrons and these strong coupling induces high-T$_c$ \nin the body-centered cubic H$_3$S. \n\nHere we study two types of hydrides, H$_3$S and H$_3$P.\nFollowing the discovery of high-T$_c$ conventional superconductivity in sulfur hydride,\na hydride phosphine (H$_3$P) was also reported to be a possible high-T$_c$ (T$_c>100~$K \nat pressure P$>200~$GPa) superconductor via four-probe electrical measurements \\cite{drozdov2}.\nHence we compare the normal and superconducting properties of these two materials. \nFor the crystal structures of high-T$_c$ hydrides, X-ray diffraction experiments\\cite{errea2,einaga} \nconfirm that the sulfur atoms of H$_3$S form a body-centered cubic structure \nas shown in Fig.~\\ref{fig:atom}. Up to now, no available experimental data for \nthe crystal structure of H$_3$P exists. Hence for comparison purposes, \nwe assume in this study that both materials have the same crystal structure \nand analyze the effect of element change on material properties.\n\n\n\\section{Methods}\n\n\nThe following methods are used to perform the calculations of the electronic structures, \nphonon properties, and superconducting properties. For the electronic structures,\nour calculations are based on {\\it ab-initio} norm-conserving\npseudopotentials and the Perdew-Burke-Ernzerhof\\cite{perdew}\nfunctional as implemented in the SIESTA\\cite{sanchez} and \nQuantum-ESPRESSO\\cite{giannozzi} codes.\nPhonon frequencies are computed using density-functional perturbation theory\\cite{baroni}\nimplemented in Quantum-ESPRESSO\\cite{giannozzi} package.\nFinally, EPC and Eliashberg spectral functions are\nobtained via the Wannier90\\cite{mostofi} and EPW\\cite{noffsinger} packages.\n\nFor the calculation using SIESTA, electronic wavefunctions are expanded \nwith pseudoatomic orbitals (double-$\\zeta$ polarization) and a charge density \ncutoff of 800 Ry is used. We sample the Brillouin zone on a uniform \n16$\\times$16$\\times$16 k-point mesh. For the calculation with Quantum-ESPRESSO,\na plane-wave basis up to 160 Ry and a 32$\\times$32$\\times$32 k mesh size are employed.\n\nPhonon frequencies $\\omega_{{\\bf q}\\nu}$ and EPC parameters $\\lambda_{{\\bf q}\\nu}$\nare computed on a coarse mesh (8$\\times$8$\\times$8) of reciprocal space.\nNext, interpolation techniques\\cite{giustino} based on maximally localized Wannier \nfunctions\\cite{giustino,marzari,souza} are used to interpolate EPC \nparameters on a fine grid (36$\\times$36$\\times$36).\n\nThe Eliashberg spectral function $\\alpha^2 F(\\omega)$ is computed by \nintegrating the interpolated phonon frequencies $\\omega_{{\\bf q}\\nu}$ \nand the EPC $\\lambda_{{\\bf q}\\nu}$ over the Brillouin zone,\n\\begin{equation}\n \\alpha^2 F(\\omega)=\\frac{1}{2} \\sum_{{\\bf q}\\nu} {\\mathrm w}_{\\bf q} \\omega_{{\\bf q}\\nu} \n \\lambda_{{\\bf q}\\nu} \\delta(\\omega-\\omega_{{\\bf q}\\nu}).\n \\label{eq:a2f}\n\\end{equation}\nHere the ${\\mathrm w}_{\\bf q}$ is the Brillouin zone weight \nassociated with the phonon wavevectors ${\\bf q}$.\nThe total EPC $\\lambda$ is calculated as the Brillouin zone \naverage of the mode-resolved coupling strengths $\\lambda_{{\\bf q}\\nu}$:\n\\begin{equation}\n \\lambda=\\sum_{{\\bf q}\\nu} {\\mathrm w}_{\\bf q} \\lambda_{{\\bf q}\\nu}\n =2 \\int_{0}^{\\infty} d\\omega~\\alpha^2 F(\\omega) \/ \\omega.\n\\end{equation}\n\n\\begin{figure}\n\\epsfig{file=Fig_1.eps,width=5cm,clip=}\n\\caption{The {\\it Im$\\overline{3}$m} crystal structure assumed for H$_3$S and H$_3$P.\nThe large sphere (orange) is S or P, and the small sphere (white) is H.\n\\label{fig:atom}\n}\n\\end{figure}\n\n\n\\section{Electronic structure}\n\n\nHere we discuss the electronic structure of H$_3$S and H$_3$P. \nIn all of our calculations we set the conventional lattice parameter as\n3 \\AA. With this lattice parameter, the calculated pressures of \nboth materials are 220 GPa. \n\nThe overall shapes of the band structures are similar for both materials\n[Figs.~\\ref{fig:band}(a) and (c)]. Because phosphorus has one less valence \nelectron than sulfur, the Fermi level ($E_F$) is shifted down in H$_3$P. \nWith the shift, $E_F$ of H$_3$P is placed near a different peak position in the\ndensity of states (DOS). For H$_3$S, the DOS at $E_F$ is calculated to be\n0.45~states~eV$^{-1}$~f.u.$^{-1}$. A similar value\n(0.50~states~eV$^{-1}$~f.u.$^{-1}$) for the DOS is found in the case of H$_3$P.\n\nFigure~\\ref{fig:band} compares the orbital contributions to the band structure and\nDOS in H$_3$S and H$_3$P. In both materials, the DOS at $E_F$ comes\ndominantly from $3p$ orbitals of sulfur or phosphorus. The portion of $3p$ orbitals \nis twice as large as the portion of hydrogen orbitals.\nThe Fermi surfaces originated from hydrogen orbitals are almost same in\nboth case, forming small hole pockets centered at ${\\it \\Gamma}$-point.\n\n\n\\begin{figure}\n\\epsfig{file=Fig_2.eps,width=8.5cm,clip=}\n\\caption{Electronic band structures and density of states (DOS) \nper three-hydrogen formula unit (f.u.) of (a), (b) H$_3$S and (c), (d) H$_3$P. \nDominant orbital characters are represented in blue (H orbitals), \nred (S or P $s$ orbitals), and green (S or P $p$ orbitals) color. \n\\label{fig:band}\n}\n\\end{figure}\n\n\n\\section{Phonon properties}\n\n\n\\begin{figure}\n\\epsfig{file=Fig_3.eps,width=8.5cm,angle=0,clip=} \n\\caption{\nPhonon spectrum and phonon density of states (PHDOS) of \n(a) H$_3$S and (b) H$_3$P. The radius of the red circle is proportional \nto $\\omega_{{\\bf q}\\nu}\\lambda_{{\\bf q}\\nu}$.\n\\label{fig:phonon}\n}\n\\end{figure}\n\n\nIn this section we discuss the differences of the phonon properties \nbetween H$_3$S and H$_3$P. When the sulfur is changed to phosphorus, \nthe characteristics of the phonon spectra differ significantly along \n${\\it \\Gamma}$--{\\it H} and {\\it H}--{\\it N} high-symmetry lines [Fig.~\\ref{fig:phonon}]. \nThe hydrogen--phosphorus bond-bending modes become softer \nand three unstable phonon modes appear at the {\\it H} high-symmetry point. \nTherefore we expect that in the doubled unit cell these unstable modes \nwould be stabilized. We exclude these negative phonon modes \nwhen calculating $\\alpha^2 F$ so that we can make an reliable \ncomparison with H$_3$S. The structural instability of body-centered cubic \nH$_3$P is also reported by previous theoretical structural studies\\cite{shamp,liu}.\n\nNext we discuss the strength of the EPC for the two cases. In H$_3$S, \nphonon modes of 150$\\sim$200 meV frequencies (which are H--S bond-stretching \nmodes) are strongly coupled to electrons at the Fermi surface. \nIn H$_3$P, however, low-frequency modes ($<$~50~meV) are more relevant. \nThese modes originate from softened H--P bond-bending motion.\n\nTo give a more quantitative discussion about the relevant energy scales\nof the phonons, we calculate the EPC-weighted average of the phonon frequencies,\n\\begin{equation}\n \\omega_{\\mathrm{ln}}=\\mathrm{exp} \\left\\{ \\frac{2}{\\lambda} \\int_{0}^{\\infty} \n d\\omega~\\frac{\\alpha^2 F(\\omega)}{\\omega}~\\mathrm{ln}~\\omega \\right\\}.\n\\end{equation}\nThe value of $\\omega_{\\mathrm{ln}}$ is 1580~K (136~meV) for the H$_3$S \nand 610~K (53~meV) for the H$_3$P. Therefore $\\omega_{\\mathrm{ln}}$ is \nmore than twice as large in H$_3$S relative to H$_3$P.\n\n\n\\section{Superconducting properties}\n\n\nThe total EPC $\\lambda$ equals 1.38 in H$_3$S, whereas it reaches \n1.66 in H$_3$P. The Eliashberg phonon spectral functions of H$_3$S and H$_3$P \nare quite different. The EPC in H$_3$S is dominated by the phonon modes \nat the zone center ${\\it \\Gamma}$ point. In H$_3$P, however, \nwe observed an overall contribution of different modes to $\\lambda$ along \n${\\it \\Gamma}$--{\\it H}--{\\it N} directions as shown in Fig.~\\ref{fig:phonon}.\n\nHere we discuss why there is a large difference in the EPC \nbetween H$_3$S and H$_3$P. First, we consider the difference in DOS. \nSince $\\lambda$ is roughly proportional to the DOS at $E_F$, the EPC \ncould be enhanced by the large DOS. However, in our case, there is \nno sufficient change in DOS to reproduce the large enhancement in EPC for H$_3$P. \nAnother point is the coupling strength between the electrons and \nthe low-frequency hydrogen vibration. There is no significant enhancement\nin the electron-phonon matrix elements which is proportional to \n $\\omega_{{\\bf q}\\nu}\\lambda_{{\\bf q}\\nu}$ [Fig.~\\ref{fig:phonon}].\n But, the dominant modes to EPC appear at low frequencies in H$_3$P [Fig.~\\ref{fig:a2f}].\n This change causes the enhancement of the EPC $\\lambda$ value.  \n\n\n\\begin{figure}\n\\epsfig{file=Fig_4.eps,width=8.5cm,angle=0,clip=} \n\\caption{\nEliashberg spectral function $\\alpha^2F$ (red) and cumulative contribution \nto the electron-phonon coupling strength $\\lambda$ (blue) of \n(a) H$_3$S and (b) H$_3$P. The cumulative EPC is calculated as\n$ \\lambda(\\omega)=2 \\int_{0}^{\\omega} d\\omega'~\\alpha^2 F(\\omega') \/ \\omega'$.\n\\label{fig:a2f}\n}\n\\end{figure}\n\n\nFinally, we estimate the superconducting transition temperature T$_c$\nusing the McMillan equation\\cite{allen}\n\\begin{equation}\n T_c = \\frac{\\omega_{\\mathrm{ln}}}{1.20}~\\mathrm{exp} \\left\\{ - \\frac\n {1.04~(1+\\lambda)}{\\lambda-\\mu^*(1+0.62~\\lambda)} \\right\\}.\n\\end{equation}\nHere $\\mu^*$ is the Coulomb repulsion parameter.  For commonly used\n$\\mu^*=0.1$ we estimate T$_c=166$~K for H$_3$S and 76~K for H$_3$P.\nThe exact value of $\\mu^*$ here is not that important since even with\n$\\mu^*=0$ we get very similar T$_c$ (219 and 96~K).  \n\nThe value of $\\lambda$ we obtained for H$_3$S and H$_3$P is near the limit\nof applicability of the McMillan equation.  However, we find that\nthe Kresin--Barbee--Cohen model\\cite{kresin,bourne}, which is\napplicable for large $\\lambda$, gives similar estimates for T$_c$.\n\nAlthough H$_3$P has a higher $\\lambda$ value than H$_3$S,\nthe estimated T$_c$ is about half of that in H$_3$S. \nThis agrees well with the experimentally obtained T$_c$ of \n$\\sim200$~K in H$_3$S and $\\sim100$~K in H$_3$P.\nWe expect that the deviation here from experiment might occur \nbecause we ignored unstable phonon modes in our calculation, \nso softening might be overestimated for H$_3$P in the low-frequency regime.\n\n\n\\section{Conclusion}\n\n\nWith the assumption of the same body-centered cubic structure \nand lattice parameter, we compare the electronic, phonon, and \nsuperconducting properties of H$_3$S and H$_3$P. The results of \nelectronic structures show no significant difference, except for a slight \nchange in the Fermi level due to the different number of valence electrons. \nHowever, there are notable changes in phonon spectrum and \nelectron-phonon coupling properties. First, there exists phonon softening \nin low-frequency bond-bending modes, and the coupling of these modes \nto electrons near the Fermi surface is enhanced. \nAs the dominant frequency regime changes from high to low frequency, \nthe superconducting transition temperature is reduced \nfrom $\\sim$166~K in H$_3$S to $\\sim$76~K in H$_3$P.\n\n\n\\section*{Acknowledgements}\n\nThis work was supported by National Science Foundation Grant\nNo. DMR15-1508412 (electronic structure calculation) and by the\nDirector, Office of Science, Office of Basic Energy Sciences,\nMaterials Sciences and Engineering Division, U.S. Department of Energy\nunder Contract No. DE-AC02-05CH11231, within the Theory Program \n(phonon and superconducting properties calculations). \nComputational resources have been provided by\nthe DOE at Lawrence Berkeley National Laboratory's NERSC facility.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}