Vacuum Energy Decay
Enrique Álvarez, Roberto Vidal
通过对真空残存振幅${\mathcal A}(z, z')$的分析，研究了真空能量衰减问题。过渡振幅计算有限的时间跨度，$Z\equiv z^\prime-z$，和他们的{ \em后期}的行为进行了讨论，直到一阶的耦合常数，$ł$。
The problem of the vacuum energy decay is studied through the analysis of the vacuum survival amplitude ${\mathcal A}(z, z')$. Transition amplitudes are computed for finite time-span, $Z\equiv z^\prime-z$, and their {\em late time} behavior is discussed up to first order in the coupling constant, $ł$.
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Heretics of the False Vacuum: Gravitational Effects On and Of Vacuum Decay 2
本文重新考察了量子引力理论中的真空衰变问题。特别是它表明，衰变成稳定的单位或AdS真空，从来没有发生。相反，真空衰变发生，如果有的话，进入宇宙时空。如果后者具有负的宇宙学常数，它会经历一次大挤压，这表明整个图景是不一致的。德西特空间的衰变问题必须非常仔细地定义。
Vacuum decay into Anti de Sitter space
我们提出了一个解释的衰变到一个$AdS$区域的假真空。$AdS$区域被解释为一个双场论生活在世界尽头的膜扩展到假真空。
Finite Time Vacuum Survival Amplitude and Vacuum Energy Decay
通过对真空存活振幅的分析，研究了宇宙常数的两种符号下真空能量的衰减问题。真空存活振幅由${\mathcal A}(z,z^\prime)\equiv <\text{vac}\,z|\text{vac}\,z^\prime>$定义为{\em共形时间} $z$。过渡振幅计算有限的时间跨度，$Z\equiv z^\prime-z$，和他们的{\em后期}的行为（直接关系到推定的衰变宽度的状态），以及瞬态讨论了一阶的耦合常数，$λ$。
Brane-induced decay of the Kaluza-Klein vacuum
研究了在测试膜近似和超越测试膜近似的情况下，由于膜的存在而导致卡鲁扎-克莱因真空衰变速率的增强。还描述了卡鲁扎-克莱因真空中的自发膜物化。
Notes on false vacuum decay in quantum Ising models
本文研究量子伊辛模型的假真空衰变理论，将自旋链作为类比系统，扩展相关基础结果，探究泡壁动力学并提出二维模型衰变率的推测性猜想。
Semiclassical calculation of an induced decay of false vacuum
我们考虑一个标量场发展成亚稳真空态并与另一个标量场弱相互作用的模型。在这种情况下，我们发现了第二场粒子的存在和碰撞所激发的假真空衰变的概率。所讨论的计算是最近建议的热处理诱导半经典过程的方法的一个例证。
Gauss-Bonnet Term on Vacuum Decay
研究了高斯-邦纳项对Coleman-De Luccia形式中真空衰变过程的影响。Gauss-Bonnet项与实标量场存在指数耦合，这种耦合出现在弦论的低能有效作用量中。数值计算了描述真空衰减过程的瞬子解，得到了空泡的临界尺寸。我们发现，高斯-博内项有一个非平凡的影响假真空衰减，取决于高斯-博内系数。
Deformed Bubbles and Lorentz Invariance in Vacuum Decay
最近，有人提出了关于洛伦兹不变性在假真空衰变中的作用的问题。有人认为，无穷大可能会出现在积分洛伦兹提升的最终状态。这个建议激发了对衰变率的闵可夫斯基空间分析。我们试图阐明的振幅计算的功能，并认为，包括激发的总速率是有限的和洛伦兹不变的。
Attractor Explosions and Catalyzed Vacuum Decay
我们提出了一种机制，催化真空气泡的生产，通过结合模稳定与广义吸引子现象，其中模来源于紧凑的对象。这直接导致了一类例子，其中黑洞的霍金衰变过程揭示了与环境真空不同的真空泡，推广了霍洛维茨最近发现的霍金蒸发的新终点。带电和不带电的物体都可以产生催化真空泡，包括史瓦西黑洞，霍金过程中产生的大质量粒子可以触发真空衰变。我们简要地讨论了这个过程的应用亚稳真空的人口和稳定性。
Fermions and Kaluza-Klein vacuum decay: a toy model
我们讨论了具有扭曲周期性条件的费米子是否抑制M^4xS^1卡鲁扎-克莱因真空的半经典衰变的问题。我们考虑了一个玩具（1+1）维模型与扭曲的费米子在雪茄形的欧几里得背景几何和计算费米子行列式。我们发现，与预期相反，行列式是有限的。我们认为这是一个迹象，扭曲费米子不稳定的卡鲁扎-克莱因真空。
Particle decay in false vacuum
我们重新讨论了由粒子的存在引起的亚稳态真空的衰变问题。对于“主场”的玻色子，这个问题在任意维数下都可以用假真空的自发衰变率来解决，而对于费米子，我们找到了（1+1）维衰变率的封闭表达式。结果表明，在（1+1）维的情况下，一个红外问题的单回路校正的玻色子的衰变率是解决由于软模之间的抵消场。我们还发现玻色子衰变率在'sine-Gordon楼梯'模型的强，弱耦合的限制。
Fate of the false monopoles: induced vacuum decay
我们研究了一个规范理论模型，其中存在一个中间对称破缺到一个Meta稳真空，它使一个简单规范群破缺到一个U（1）因子。这样的模型承认亚稳定磁单极子的存在，我们称之为假单极子。在薄壁近似下证明了这些单极子的存在性。我们确定的集体坐标，对应于半径的磁墙的瞬子，我们计算半经典的隧穿率为这些单极子的衰变。因此，负真空的衰变触发了假真空的衰变。随着电子质量的增加，我们发现相对于Coleman [1]的著名均匀隧穿率，假真空的衰减率增加。
False Vacuum Transitions - Analytical Solutions and Decay Rate Values
在这项工作中，我们显示了一类振荡配置的演化域壁在欧几里得空间。解析地得到解。相变是从相关的波动行列式，通过假真空的衰减率。
Particle production related to the tunneling in false vacuum decay
受Mersini工作的启发，本文在薄壁近似下研究了假真空衰变中与隧穿有关的粒子产生。结果表明，在这种情况下，粒子的生产是指数抑制，即使当动量是可比的曲率尺度的气泡。产生的粒子数量是紫外线有限的。
We investigate catalysis induced by a dyonic impurity in the metastable vacuum studied by Kachru, Pearson and Verlinde, which can be relevant to vacuum decay in the KKLT scenario. The impurity is a D3-brane wrapping on $ \mathbb{S}^3$ in the Klebanov-Strassler geometry. The effect of the D3-brane can be encoded in the world-volume theory of an NS5-brane as an electromagnetic field on it. As the field strength becomes large, instability of the vacuum enhances. As a result, the lifetime of the metastable vacuum becomes drastically shorter.
Reconsidering the calculation of the false vacuum decay rate at zero temperature
本文指出了零温假真空衰变率计算中的不一致性及其对Callan-Coleman解释的潜在影响，并利用shifted-bounce解和shot解重新计算了衰变率以解决该问题。
In this work we consider kink-antikink and antikink-kink collisions in a modified $ϕ^4$ model with a false vacuum characterized by a dimensionless parameter $ε$. The usual $ϕ^4$ model is recovered for $ε=0$. We investigate the $ε<<1$ regime where the kink in the presence of false vacuum can be understood as a small deformation of the standard kink for the $ϕ^4$ model. We show that the attractive interaction between the kink-antikink pair leads to a rich scattering pattern, in some cases delaying considerably the false vacuum decay.
Inflaton Decay in an Alpha Vacuum
本文通过考虑α真空中与标量场耦合的暴胀子场的衰变率来研究de Sitter空间的α真空。我们发现一个{ \em α依赖}玻色增强相对于Bunch-Davies真空，令人惊讶的是，没有不可重正化的分歧。我们还考虑了一个修改后的α依赖于时间的费曼传播处方，并表明它导致了一个α独立的结果。这一结果表明，如果我们采用适当的因果关系保持处方，在任何α真空中进行计算都是可能的。
Comments on the vacuum energy decay
Polyakov最近讨论的de Sitter空间中的{真空能量的\em不稳定性}被认为是存在外部引力场时的一个普遍特征。它与{ \em禁止}（动量守恒）衰变的存在有关，Bros，Epstein和Moschella详细推导了这些衰变。在共形不变的情况下讨论了一些计算，并提出了一个平坦的空间模型。
Decay of Kaluza-Klein Vacuum via Singular Instanton
本文研究了五维Kaluza-Klein真空衰变中的“无气泡”解，发现带有锥形奇点的反弹解具有更高衰变率，表明奇异解在紧致内部空间理论的真空衰变中起主导作用，并从热力学角度探讨了衰变率的增强。
The 'thin-wall approximation' gives a simple estimate of the decay rate of an unstable quantum field. Unfortunately, the approximation is uncontrolled. In this paper I show that there are actually two different thin-wall approximations and that they bracket the true decay rate: I prove that one is an upper bound and the other a lower bound. In the thin-wall limit, the two approximations converge. In the presence of gravity, a generalization of this lemma provides a simple sufficient condition for non-perturbative vacuum instability.
False vacuum decay around black holes: calculation from first principles in the (3+1)-dimensional case
该研究针对(3+1)维时空，数值计算热浴与黑洞共存时假真空衰变概率的抑制指数，为黑洞相关的假真空衰变研究提供第一性原理结果。
Decay of Vacuum Energy
本文研究了de Sitter背景下大质量粒子的相互作用。据发现，真空作为一个反向填充介质，这是能够产生受激辐射。如果没有反作用（本文未考虑），这种效应会导致爆炸。我们期望所提出的“宇宙激光”机制能耗尽曲率，并可能有助于解决宇宙学常数问题。
Topics in vacuum decay (Ph.D Thesis)
如果一个理论有不止一个经典稳定的真空，量子隧穿和热跳跃使真空之间的过渡成为可能。转变通过由新真空的气泡的成核开始的一阶相变发生。更真实的真空的向外压力使气泡膨胀，从而侵蚀掉更多的旧相。在引力存在的情况下，这种现象变得更加复杂，同时也更加有趣。它可能会产生重要的宇宙学后果。本文对这种衰变的某些方面进行了研究。与一般使用的O（4）对称性不同的对称性的解决方案进行了研究和计算他们的行动。研究了空间矢量场中的真空衰减，并提出了类似扭结畴壁的新特征。本文研究了势景中真空的稳定性问题，并指出了在大维场中可能存在的不稳定性。最后研究了Einstein-Maxwell理论的紧致化，这可以作为理解任意维数紧致化模型中衰变率的一个很好的实验。
One-loop corrections to the metastable vacuum decay
本文计算了3+1维自相互作用标量场理论中假真空衰减率的单圈前因子。我们使用的数值方法，建立了一段时间前，这是基于一个著名的定理功能决定。零模式和重整化的适当处理进行了讨论。数值结果特别表明，量子修正变得更小，远离薄壁的情况下。在薄壁极限的数值结果被发现加入到那些通过梯度展开。
Vacuum decay in multidimensional field landscapes: thin, thick and intersecting walls
我们研究了多维场空间中真空之间的隧穿。工作在严格的薄壁近似，我们发现，传统的瞬子假真空衰变发展一个新的消失的本征值在其波动行列式，所产生的装饰的成核气泡壁与额外的真空的小点。天真地，这将表明，在场空间中的额外的真空的存在下，导致成核率的大幅提高。然而，我们认为，这种潜在的增强是由物理气泡壁交叉点的有限厚度。然后，我们讨论了新的鞍点的薄壁行动，在某些制度的参数空间，有可能破坏传统的瞬子假真空衰变。
Vacuum decay and fermion total reflection by the Klein step
本文在量子场论中以渐近态视角重新分析 Klein 隧穿，通过渐近算符和相似变换揭示 Klein 能区的真空衰变，并证明入射费米子以概率 1 反射且伴随费米子—反费米子对产生。
Saddle-point approximation to the false vacuum decay at finite temperature in one-dimensional quantum mechanics
本文基于有限温度欧几里得路径积分的鞍点近似，计算了一维量子力学中的假真空衰变率，识别了有限温度反弹解与射击解，重新检验了零温结果，并通过数值计算分析了衰变率的温度依赖性。
Decay of False Vacuum via Fuzzy Monopole in String Theory
我们研究了IIB型弦论中假真空中的介电膜。介电膜由球形或管状膜内的低能量真空支撑以防止坍缩。我们认为，这样的膜可以是假真空的半经典（或量子力学）衰变的种子，这使得假真空的寿命更短。此外，我们还讨论了溶解基本弦所触发的模糊子所对应的气泡的拓扑变化。
We provide a novel, concise and self-contained evaluation of true- and false vacuum decay rates in general relativity. We insist on general covariance and choose observable boundary conditions, which yields the well known false-vacuum decay rate and a new true-vacuum decay rate that differs significantly from prior work. The rates of true- and false vacuum decays are identical in general relativity. The second variation of the action has a negative mode for all parameters. Our findings imply a new perspective on cosmological initial conditions and the ultimate fate of our universe.
On gravitational and thermal corrections to vacuum decay
我们重新考虑真空衰变的引力修正，确认和简化早期的结果，并通过允许希格斯粒子与引力的非最小耦合来扩展它们。我们发现，领导阶引力修正抑制真空衰变率。此外，我们计算轻微的修正，热真空衰减的SM通过添加一个循环的贡献的希格斯动力学项，两个循环的贡献的希格斯势，并允许随时间变化的反弹。
Dissipative Losses In Black Hole-Induced Vacuum Decay
研究黑洞催化虚假真空衰变的耗散效应，通过薄壁近似分析ϕ⁴和正弦-戈登模型，发现辐射损耗减缓气泡生长，防止失控真空衰变，尽管产率在部分参数空间增强，但仍呈指数抑制。
We study a false vacuum decay in a two-dimensional black hole spacetime background. The decay rate in the case that nucleation site locates at a black hole center has been calculated in the literature. We develop a method for calculating the decay rate of the false vacuum for a generic nucleation site. We find that the decay rate becomes larger when the nucleation site is close to the black hole horizon and coincides with that in Minkowski spacetime when the nucleation site goes to infinity.
Seeding decay of the false vacuum
本文提出由球形种子诱导的假真空衰变理论，考虑表面能边界，适用于零温与有限温度，并证明该理论对二维钾BEC的预测与实时数值模拟一致。
We investigate a vacuum decay around a spinning seed black hole by using the Israel junction condition and conclude that the spin of black hole would suppress a vacuum decay rate compared to that for a non-spinning case, provided that the surface of vacuum bubble has its ellipsoidal shape characterized by the Kerr geometry. We also find out that in the existence of a near-extremal black hole, a false vacuum state can be more stabilized than the case of the Coleman-de Luccia solution. A few necessary assumptions to carry the calculations are discussed.
We analyse vacuum decay in rotating BTZ black hole spacetimes with the thin wall approximation. Possible parameter regions for the vacuum decay are clarified. We find that the nucleation rate is dominated by the bounce solution with the static shell configuration. The nucleation rate of the static shell decreases with the mass of the initial black hole. For a larger/smaller value of the initial black hole, the nucleation rate can be smaller/larger than that of the Coleman De Luccia vacuum decay in the pure AdS spacetime. Through the vacuum decay, the black hole gains its mass and loses the hor…
We argue that the rate of Standard Model vacuum or thermal decay seeded by primordial black holes is negligible (because non-perturbatively suppressed by the small quartic Higgs coupling) and independent of the non-minimal coupling of the Higgs to gravity.
The tunneling potential method to calculate the action for vacuum decay is an alternative to the Euclidean bounce method that has a number of attractive features. In this paper we extend the formalism to general spacetime dimension $d>2$ and use it to give simple proofs of several results. For Minkowski or Anti de Sitter false vacua, we show that gravity or higher barriers increase vacuum lifetime and describe a very clean picture of gravitational quenching of vacuum decay. We also derive the thin-wall limit of the action, show how detailed balance for dS to dS transitions works in the new for…
On vacuum-energy decay from particle production
给出了de Sitter时空中粒子产生反反应效应的简化（但一致）描述。
The Probability of Vacuum Metastability and Artificial Vacuum Decay: Expert Survey Results
本研究通过对20名物理学专家的结构化调查，评估真空亚稳态及人工诱发真空衰变的概率与分歧，凸显超越标准模型理论的重要性。
We introduce a new picture of vacuum decay which, in contrast to existing semiclassical techniques, provides a real-time description and does not rely on classically-forbidden tunneling paths. Using lattice simulations, we observe vacuum decay via bubble formation by generating realizations of vacuum fluctuations and evolving with the classical equations of motion. The decay rate obtained from an ensemble of simulations is in excellent agreement with existing techniques. Future applications include bubble correlation functions, fast decay rates, and decay of non-vacuum states.
We study the decay rate of a false vacuum in gauge theory at the one-loop level. We pay particular attention to the case where the bounce consists of an arbitrary number of scalar fields. With a multi-field bounce, which has a curved trajectory in the field space, the mixing among the gauge fields and the scalar fields evolves along the path of the bounce in the field space and the one-loop calculation of the vacuum decay rate becomes complicated. We consider the one-loop contribution to the decay rate with an arbitrary choice of the gauge parameter, and obtain a gauge invariant expression of …
Vacuum Decay in CFT and the Riemann-Hilbert problem
本文研究了具有外背景场的1+1维共形场论中的真空稳定性。我们证明了真空衰变率是由一个非局部的两种形式。这两种形式是必须添加到有效输入/输出拉格朗日量的边界项。这两种形式用背景规范场的黎曼-希尔伯特分解表示，在引力场的情况下则用其新颖的“泛函”形式表示。
We propose to test the concept of seeded vacuum decay in cosmology using an analogue gravity Bose-Einstein condensate system. The role of the nucleation seed is played by a vortex within the condensate. We present two complementary theoretical analyses that demonstrate seeded decay is the dominant decay mechanism of the false vacuum. First, we adapt the standard instanton methods to the Gross-Pitaevskii equation. Second, we use the truncated Wigner method to study vacuum decay.
We propose an ultracold atom analogue of early universe vacuum decay using all three states of a spin-1 Bose gas. We consider a one-dimensional system with both radio frequency and optical Raman coupling between internal states. An advantage of our proposal is the lack of a time-modulated coupling, which can lead to instabilities. Within the elaborate phase structure of the system we identify an effective Klein-Gordon field and use Gross-Pitaevskii simulations within the truncated Wigner approximation to model the decay of its false vacuum. We examine the dependence of the rate of vacuum decay…
We analyze the instability of the non-rotating fermion vacuum in Kerr spacetimes. We describe how the co-rotating Fermi sea is formed as a result of a spontaneous vacuum decay. Most significantly, and drawing upon intuition gained from analogous electrodynamic processes in supercritical fields, we show that this decay process is encoded entirely in a subset of quasi-normal fermion modes.
False Vacuum Decay in Gauge Theory
在规范理论中研究假真空的衰变率，特别注意它的规范不变性。虽然根据Nielsen恒等式，衰变率不应该依赖于规范参数$ξ$，但微扰计算结果的规范不变性还没有得到清楚的证明。我们给出了一个处方，执行一个循环计算的衰变率，与该衰变率的一个明显的规范不变的表达式获得。我们还讨论了使结果有限所需的重整化，并表明即使在重整化之后，衰变率也与规范参数无关。
We consider the Skyrme model modified by the addition of mass terms which explicitly break chiral symmetry and pick out a specific point on the model's target space as the unique true vacuum. However, they also allow the possibility of false vacua, local minima of the potential energy. These false vacuum configurations admit metastable skyrmions, which we call false skyrmions. False skyrmions can decay due to quantum tunnelling, consequently causing the decay of the false vacuum. We compute the rate of decay of the false vacuum due to the existence of false skyrmions.
Semiclassical Approach to Heterogeneous Vacuum Decay
在薄壁近似下，导出了含杂质的量子场论不稳定相的衰减率。这个推导是基于杂质如何改变（平坦时空）几何相对于纯假真空的情况。给出了两个例子，显示如何估计一些额外的参数，进入这个异质衰变率。这种形式主义，然后应用到希格斯真空的标准模型（SM），重子物质作为杂质的电弱希格斯真空。我们发现，非均匀真空衰变发生的概率被抑制相对于均匀的情况下。也就是说，从均匀情况下得出的结论并没有因为在计算中包括重子物质而被修改。另一方面，我们表明，超越标准模型的物理特性的规模相比，管理均匀的衰变率在SM的规模，原则上可以导致增强的衰变率。
Using a new approach to the analysis of false vacuum decay based on the so-called tunneling potential, we develop a general method to find scalar potentials with a false vacuum with exactly solvable decay at the semi-classical level, including gravitational corrections. We examine in particular the decays of de Sitter vacua providing concrete examples that allow to explore analytically the transition between the Coleman-De Luccia and Hawking-Moss regimes.
We discuss particle production associated with vacuum decay, which changes the mass of a scalar field coupled to a background field which induces the decay. By utilizing the Stokes phenomenon, we can optimally track the time-evolution of mode function and hence calculate particle production properly. In particular, we use real time formalisms for vacuum decay in Minkowski and de Sitter spacetime together with the Stokes phenomenon method. For each case, we consider the flyover vacuum decay model and stochastic inflation, respectively. Within the real time formalism, the particle production can…
Empirical constraints on vacuum decay in the stringy landscape
在弦论的图景中，我们真空的寿命不可能比宇宙的当前年龄短得多，这一点通常被认为是不言自明的。在这里，我将说明为什么这个下限是无效的。某种类型的“平行宇宙”是弦景观动力学的必然结果，很可能使我们能够“生存”真空衰变。因此，我们的粘性真空的寿命是不受经验约束的，可能很短。基于这种反直觉的见解，我提出了一种新型的实验室实验，寻找一个明显违反量子力学玻恩规则的引力效应对真空衰变。如果我们的真空寿命短于6 × 10^{-13}秒，这样的实验就足够灵敏，可以用最先进的设备来确定它的值。
We study the decay of the false vacuum in the scaling Ising and tricritical Ising field theories using the Truncated Conformal Space Approach and compare the numerical results to theoretical predictions in the thin wall limit. In the Ising case, the results are consistent with previous studies on the quantum spin chain and the $φ^4$ quantum field theory; in particular we confirm that while the theoretical predictions get the dependence of the bubble nucleation rate on the latent heat right, they are off by a model dependent overall coefficient. The tricritical Ising model allows us on the othe…
A Unified View of Vacuum Decay Channels
本研究采用隧穿势形式体系，为真空衰变提供统一视角，指出Coleman反弹属于包含Hawking-Moss瞬子、伪反弹等的连续解族。
Constrained potential method for false vacuum decays
本文给出了一个多标量场模型中假真空跃迁的数值分析方法。它是Konstandin和Huber方法的改进版本。这种改变使得有可能解决一类问题，这是困难的或无法解决的原始方法，即那些遥远的或不存在的真正真空。给出了一个自下而上无边界方向的例子。
Recently it was discovered that the Standard Model vacuum may suffer an essential metastability near the Planck scale. In this regard it makes sense to study in more detail the decay processes of the metastable vacuum and to develop methods for their more accurate analysis. In this article we develop a technique to calculate two loop radiative corrections to false vacuum decay rate in thin-wall approximation and apply it to false vacuum decay in scalar quantum field theory with cubic and quartic interactions. The results obtained use dimensional regularization and given in two different renorm…
Barnacles -- A Novel Channel for Vacuum Decay
我表明，在薄壁近似的宇宙气泡壁是不稳定的创建“藤壶”-不同的壁张力的轨迹附近的区域充满了第三个真空。藤壶的形成导致了与广泛研究的气泡碰撞场景相同的观测结果，但发生的次数更频繁。该过程由薄壁作用的鞍点描述，具有两个负模态。
Radiative corrections to false vacuum decay in quantum mechanics
本文在量子力学的框架内，考虑了形式为1/2 M q ^2（q-A）（q-B）的广义势对假真空衰变的辐射修正，其中M、A和B是任意参数。对于这类势，我们给出了相应反弹解背景下格林函数的解析结果以及假真空衰减率的单圈表达式。接下来，我们讨论了假真空衰减率的高阶修正的计算，并提供了两个和三个回路贡献的数值表达式。
Vacuum Decay Rate in D-dimensional Electroweak theories
本文提出了一种计算D维电弱理论中真空衰变率的方法，通过结合WKB展开和维度正则化，实现了对不同场的统一处理，并在不同维度下验证了其有效性。
Metastable vacuum decay in center-stabilized Yang-Mills theory at large N
在${\mathbb R}^3 \times {\mathbb S}^1$空间中，利用Ünsal和Yaffe提出的双迹形变，用$SU(N)$规范理论计算了亚稳真空态的衰变率.导出的衰减率的指数因子的解析表达式，包括依赖于参数$θ$，给出了确切的行为，在领先的权力$N$在大$N$的限制，并提供了一个更好的近似比最近发现的文献数值结果。
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