Aditya Hebbar, Denis Karateev, Joao Penedones
We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.
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Bootstrapping Classical Spinning Compton Amplitudes with Colour-Kinematics
本文提出一种利用因式分解和色-运动学对偶自举任意自旋理论中类康普顿振幅的方法，并探讨Wilson系数的约束及其与对偶的关系。
Spinning the Large-Charge Bootstrap: Parity-Even Operators
在三维共形场论中利用特定探针研究大电荷引导，限于宇称偶数交换算符，最小假设下需特定雷杰轨迹，额外假设时该轨迹唯一，非戈德斯通轨迹仅对标量-标量通道有此阶贡献。
We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric a…
We study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)-conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed-form expressions f…
Higher-point functions of scalar operators are a rich observable in CFTs, as they contain OPE data involving multiple spinning operators. We derive the lightcone blocks for five- and six-point functions in the snowflake channel and use them to bootstrap these correlators in the lightcone limit. As a result we determine the large spin expansion of OPE coefficients involving two or three spinning operators. We verify our results by comparing to the block decomposition of higher-point functions in generalized free theory and in theories with a cubic coupling.
In this work we discuss an analytic bootstrap approach [1,2] in the context of spinning 4D conformal blocks [3,4]. As an example we study the simplest spinning case, the scalar-fermion correlator $\langleϕψϕ\barψ\rangle$. We find that to every pair of primary scalar $ϕ$ and fermion $ψ$ correspond two infinite towers of fermionic large spin primary operators. We compute their twists and products of OPE coefficients using both s-t and u-t bootstrap equations to the leading and sub-leading orders. We find that the leading order is represented by the scalar-fermion generalized free theory and the …
Bootstrapping spinning two body problem in dynamical Chern-Simons gravity using worldline QFT
本文利用自旋世界线量子场论计算动力学Chern-Simons引力中黑洞散射的eikonal相位，通过IBP约化和微分方程技术处理两圈积分，得到3PM阶线性自旋相位，并分析其自旋取向依赖。
The Gross-Neveu model defines a unitary CFT of interacting fermions in $2<d<4$ which has perturbative descriptions in the $1/N$ expansion and in the epsilon-expansion near two and four dimensions. In each of these descriptions, the CFT has an infinite tower of nearly conserved currents of all spins. We determine the structure of the non-conservation equations both at large $N$ and in the epsilon-expansion, and use it to find the leading order anomalous dimensions of the broken currents. Similarly, we use the fact that the CFT spectrum includes a nearly free fermion to fix the leading anomalous…
Three-dimensional conformal field theories (CFTs) with slightly broken higher spin symmetry provide an interesting laboratory to study general properties of CFTs and their roles in the AdS/CFT correspondence. In this work we compute the four-point functions at arbitrary 't Hooft coupling $λ$ in the CFTs with slightly broken higher spin symmetry. We use a bootstrap approach based on the approximate higher spin Ward identity. We show that the bootstrap equation is separated into two parts with opposite parity charges, and it leads to a recursion relation for the $λ$ expansions of the correlation…
Helical Inflation Correlators: Partial Mellin-Barnes and Bootstrap Equations
本文研究轴子耦合诱导的螺旋暴胀关联子，通过部分 Mellin-Barnes 表示和自举方程给出四点树图交换及三点函数的完整解析结果，并验证化学势对振荡信号的指数增强与宇宙学切割规则。
We explore the stability of a recently found class of spinning dielectric M2-branes in the 11-dimensional maximally supersymmetric plane-wave background. We find two small windows of instabilities in the dipole (j=1) and quadrupole (j = 2) sector of linear multipole perturbations.
We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients of the chiral lagrangian controlling the low-energy dynamics of massless pions thus providing a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on EFTs in more than two spacetime dimensions.
Upgrading Extremal Flows in the Space of Derivatives
本文提出一种方法，将极值流扩展到更广泛的流中，用于提升旋转模bootstrap中间隙最大化的解，并在小规模测试案例中成功验证了该方法的可行性。
Large Spin Systematics: Patterns from Reciprocity for Multiple Spinning Operators
该研究通过分析多重旋转算符的互易性模式，推导出大自旋行为的新约束，并展示了 l=0 情况下约束的简化方法。
Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a t…
We extend the cosmological bootstrap to correlators involving massless particles with spin. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields w…
Thermal Bootstrap of Large-N Matrix Models via Conic Optimization
本文通过锥优化方法改进大-N矩阵模型的热Bootstrap，实现了对长弦激发能和耦合系数的高精度估计。
We consider the 2D S-matrix bootstrap in the presence of supersymmetry, $\mathbb{Z}_2$ and $\mathbb{Z}_4$ symmetry. At the boundary of the allowed S-matrix space we encounter well known integrable models such as the supersymmetric sine-Gordon and restricted sine-Gordon models, novel elliptic deformations thereof, as well as a two parameter family of $\mathbb{Z}_4$ elliptic S-matrices previously proposed by Zamolodchikov. We highlight an intricate web of relations between these various S-matrices.
Massless S-matrix Bootstraps and RG Flows
本文求解了A_n对称性下(1+1)维无质量粒子S矩阵自举方程，发现仅三种S矩阵能产生UV完备的RG流，并推导了连接IR与UV CFT的TBA方程和Y系统。
Bootstrapping Ground State Correlators in Matrix Theory, Part I
本文提出一种基于超荷运动方程和SO(9)对称性的自举方法，用于在零能量和无穷大$N$下推导矩阵理论中简单关联函数的界，并初步验证了该方法在强耦合区域的高效性。
Finite-$N$ Bootstrap Constraints in Matrix and Tensor Models
研究有限N下矩阵和张量模型中Bootstrap技术的约束作用，发现多迹期望值性质影响约束，张量模型中约束随N变化，提出新的双点函数约束。
Asymptotic bootstrap for unitary matrix integrals at complex coupling
本文提出一种基于渐近bootstrap方法的非微扰研究单位ary矩阵积分的高效工具，用于计算Wilson环并验证instanton效应。
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 1 + 1 dimensions due to crossing symmetry and unitarity. In this way we establish rigorous bounds on the cubic couplings of a given theory with a fixed mass spectrum. In special cases we identify interesting integrable theories saturating these bounds. Our analytic bounds match precisely with numerical bounds obtained in a companion paper where we consider massive QFT in an AdS box and study boundary correlators using the technology of the conformal bootstrap.
Primal S-matrix bootstrap with dispersion relations
本文提出了一种通过参数化部分波的虚部并利用色散关系、交叉对称性和完全单色性来构建一致散射幅空间的新方法，研究了leading couplings的约束和Regge行为，并探讨了色散关系对耦合约束灵敏度的影响。
Matrix Bootstrap Approximation without Positivity Constraint
该研究提出了一种无正性约束的单矩阵模型Bootstrap近似方法，通过最小二乘法确定自洽的特征值分布和矩，实现了对欧几里得和Minkowski型模型的高精度求解。
Spinning Mellin amplitudes
我们定义了任意自旋算符的Mellin振幅，通过离散Mellin变换处理极化矢量，建立了极点结构和因式分解性质，并给出了直至自旋4的新结果，应用于4维N=2超共形场论中的流关联函数，与AdS5×S3中的胶子散射振幅对偶。
We propose a strategy to study massive Quantum Field Theory (QFT) using conformal bootstrap methods. The idea is to consider QFT in hyperbolic space and study correlation functions of its boundary operators. We show that these are solutions of the crossing equations in one lower dimension. By sending the curvature radius of the background hyperbolic space to infinity we expect to recover flat-space physics. We explain that this regime corresponds to large scaling dimensions of the boundary operators, and discuss how to obtain the flat-space scattering amplitudes from the corresponding limit of…
The S-matrix bootstrap is extended to a 1+1d theory with $O(N)$ symmetry and a boundary in what we call the R-matrix bootstrap since the quantity of interest is the reflection matrix (R-matrix). Given a bulk S-matrix, the space of allowed R-matrices is an infinite dimensional convex space from which we plot a two dimensional section given by a convex domain on a 2d plane. In certain cases, at the boundary of the domain, we find vertices corresponding to integrable R-matrices with no free parameters. In other cases, when there is a one-parameter family of integrable R-matrices, the whole bounda…
We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering…
The S-matrix Bootstrap originated on the idea that the S-matrix might be fully constrained by global symmetries, crossing, unitarity, and analyticity without relying on an underlying dynamical theory that may or may not be a quantum field theory. Recently this approach was revived from a somewhat different point of view. Using the same constraints, one numerically maps out the (infinite-dimensional) space of allowed S-matrices that should contain all consistent quantum field theories (and quantum theories of gravity). Moreover, in the best case scenario one finds special points in the space th…
We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. In the cases where our results can be compared with the older S-matrix literature they are in excellent agreement. We also extremize a cubic coupling in 2+1 dimensions which we can directly compare to a universal bound for a QFT in AdS. This paper generalizes our previous 1+1 dimens…
The principle of boundary bootstrap plays a significant role in the algebraic study of the purely elastic boundary reflection matrix $K_a(θ)$ for integrable quantum field theory defined on a space-time with a boundary. However, general structure of that principle in the form as was originally introduced by Fring and Köberle has remained unclear. In terms of a new matrix $J_a(θ)=\sqrt{K_a(θ)/K_{\bar{a}}(iπ+θ)}$, the boundary bootstrap takes a simple form. Incidentally, a hypothesised expression of the boundary reflection matrix for simply-laced $ADE$ affine Toda field theory defined on a half l…
The S-matrix bootstrap with neural optimizers I: zero double discontinuity
本研究将神经网络参数化与基于色散表示的S矩阵自举结合，用梯度下降算法探索全同标量粒子2对2零双不连续性散射的自洽S矩阵空间，推导低能泰勒系数边界，且与传统自举方法结果完全一致。
The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toric 1-point functions in the Liouville field theory with DOZZ structure constants is proved.
We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large $N$ limit, developing and improving the recent proposal of H.Lin. It gives rigorous inequalities on the single trace moments of the matrices up to a given "cutoff" order (length) of the moments. The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be e…
Exploring Inelasticity in the S-Matrix Bootstrap
本文通过引入非弹性作为额外输入，在S-矩阵自举中探索非弹性效应，并在一般维度及二维情形下建立相应方法，得到比标准设置更强的低能界限。
We formulate and close the boundary state bootstrap for factorizing K-matrices in AdS/CFT. We found that there are no boundary degrees of freedom in the boundary bound states, merely the boundary parameters are shifted. We use this family of boundary bound states to describe the D3-D5 system for higher dimensional matrix product states and provide their asymptotic overlap formulas. In doing so we generalize the nesting for overlaps of matrix product states and Bethe states.
We explore the space of consistent three-particle couplings in $\mathbb Z_2$-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [arXiv:1607.06110] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [arXiv:1607.06109]. The space of allowed couplings that we carve out is rich in features, som…
Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interact…
The constraints of conformal bootstrap are applied to investigate a set of conformal field theories in various dimensions. The prescriptions can be applied to both unitary and non unitary theories allowing for the study of the spectrum of low-lying primary operators of the theory. We evaluate the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity for $2 \le D \le 6$. Likewise we obtain the scaling dimensions of six scalars and four spinning operators for the 3d critical Ising model. Our findings are in agreement with existing results to a per mill pr…
We investigate the boundary bootstrap programme for finding exact reflection matrices of integrable boundary quantum field theories with N=1 boundary supersymmetry. The bulk S-matrix and the reflection matrix are assumed to take the form S=S_1S_0, R=R_1R_0, where S_0 and R_0 are the S-matrix and reflection matrix of some integrable non-supersymmetric boundary theory that is assumed to be known, and S_1 and R_1 describe the mixing of supersymmetric indices. Under the assumption that the bulk particles transform in the kink and boson/fermion representations and the ground state is a singlet we p…
Hidden simplicity in AdS spinning Mellin amplitudes via scaffolding
通过scaffolding方法，研究AdS中旋进算子的Mellin幅度，发现其形式与平坦空间相似，且顶点比例系数简洁，体现正确极限。
We construct nonrelativistic spinning string solutions corresponding to $ SU(1,2|3) $ Spin-Matrix theory (SMT) limit of strings in $ AdS_5 \times S^5 $. Considering various nonrelativistic spinning string configurations both in $ AdS_5 $ as well as $ S^5 $ we obtain corresponding dispersion relations in the strong coupling regime of SMT where the strong coupling ($ \sim \sqrt{\mathfrak{g}} $) corrections near the BPS bound have been estimated in the slow spinning limit of strings in $ AdS_5 $. We generalize our results explicitly by constructing three spin folded string configurations that has…
We work out constraints imposed by channel duality and analyticity on tree-level amplitudes of four identical real scalars, with the assumptions of a linear spectrum of exchanged particles and Regge asymptotic behaviour. We reduce the requirement of channel duality to a countably infinite set of equations in the general case. We show that channel duality uniquely fixes the soft Regge behaviour of the amplitudes to that found in String theory, $(-s)^{2t}$. Specialising to the case of tachyonic external particles, we use channel duality to show that the amplitude can be any one in an infinite-di…
Spinning Partial Waves for Scattering Amplitudes in $d$ Dimensions
本研究提出任意维自旋粒子2→2散射分波的计算方法，通过调和分析构建权重移位算子代数生成自旋分波，配套Mathematica工具，为自旋S矩阵自举等研究奠定基础。
The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the $2\rightarrow 2$ scattering matrix $S_{2\rightarrow 2}$ such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider $3+1$ dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical an…
We use the conformal bootstrap equations to study the non-perturbative gravitational scattering between infalling and outgoing particles in the vicinity of a black hole horizon in AdS. We focus on irrational 2D CFTs with large $c$ and only Virasoro symmetry. The scattering process is described by the matrix element of two light operators (particles) between two heavy states (BTZ black holes). We find that the operator algebra in this regime is (i) universal and identical to that of Liouville CFT, and (ii) takes the form of an exchange algebra, specified by an R-matrix that exactly matches with…
An asymptotic bootstrap method and its applications to Hermitian matrix models
提出一种渐近自举方法，用于计算厄米矩阵模型中的矩积分，即使耦合常数为复数也能指数快速收敛，并用于研究正交多项式及相结构。
In this work we apply the S-matrix bootstrap maximization program to the 2d bosonic O(N) integrable model which has N species of scalar particles of mass m and no bound states. Since in previous studies theories were defined by maximizing the coupling between particles and their bound states, the main problem appears to be to find what other functional can be used to define this model. Instead, we argue that the defining property of this integrable model is that it resides at a vertex of the convex space determined by the unitarity and crossing constraints. Thus, the integrable model can be fo…
The purpose of the "bootstrap program" is to construct integrable quantum field theories in 1+1 dimensions in terms of their Wightman functions explicitly. As an input the integrability and general assumptions of local quantum field theories are used. The object is to be achieved in tree steps: 1) The S-matrix is obtained using a qualitative knowledge of the particle spectrum and the Yang-Baxter equations. 2) Matrix elements of local operators are calculated by means of the "form factor program" using the S-matrix as an input. 3) The Wightman functions are calculated by taking sums over interm…
S-matrices in the holomorphic modular bootstrap approach
本文提出了一种基于全纯模 Bootstrap 方法和 MLDE 的方法，通过 S 矩阵的整数性质确定其精确公式，并通过示例展示了该方法的实用性。
We consider 2-2 scattering in four spacetime dimensions in Celestial variables. Using the crossing symmetric dispersion relation (CSDR), we recast the Celestial amplitudes in terms of crossing symmetric partial waves. These partial waves have spurious singularities in the complex Celestial variable, which need to be removed in local theories. The locality constraints (null constraints) admit closed form expressions, which lead to novel bounds on partial wave moments. These bounds allow us to quantify the degree of low spin dominance(LSD) for scalar theories. We study a new kind of positivity t…
The main objective of this work is to isolate Effective Field Theory scattering amplitudes in the space of non-perturbative two-to-two amplitudes, using the S-matrix Bootstrap. We do so by introducing the notion of Effective Field Theory cutoff in the S-matrix Bootstrap approach. We introduce a number of novel numerical techniques and improvements both for the primal and the linearized dual approach. We perform a detailed comparison of the full unitarity bounds with those obtained using positivity and linearized unitarity. Moreover, we discuss the notion of Spin-Zero and UV dominance along the…
Corners and Islands in the S-matrix Bootstrap of the Open Superstring
本文通过自底向上的 S 矩阵自举方法约束 10 维最大超对称 Yang-Mills 有效场论，证明最低大质量态必为标量，并在不同质量谱假设下发现 Veneziano 振幅对应允许区域中的尖锐角点或参数空间孤岛，从而从底层原理隔离出开超弦振幅。
S-matrix bootstrap and positivity bounds are usually viewed as constraints on low-energy theories imposed by the requirement of a standard UV completion. By considering graviton--photon scattering in the Standard Model, we argue that the low-energy theory can be used to put constraints on the UV behaviour of the gravitational scattering amplitudes.
Thermal Bootstrap of Matrix Quantum Mechanics
本文提出一种结合稳态条件、热不等式和矩阵对数半定松弛的自举方法，用于未规范化的单矩阵量子力学，确定有限温度可观测量，并给出热相变界限及双矩阵初步结果。
We study the $2\rightarrow2$ $S$-matrix element of a generic, gapped and Lorentz invariant QFT in $d=1+1$ space time dimensions. We derive an analytical bound on the coupling of the asymptotic states to unstable particles (a.k.a. resonances) and its physical implications. This is achieved by exploiting the connection between the S-matrix phase-shift and the roots of the S-matrix in the physical sheet. We also develop a numerical framework to recover the analytical bound as a solution to a numerical optimization problem. This later approach can be generalized to $d=3+1$ spacetime dimensions.
We compute, to the first non-trivial order in the $ε$-expansion of a perturbed scalar field theory, the anomalous dimensions of an infinite class of primary operators with arbitrary spin $\ell=0,1,..$, including as a particular case the weakly broken higher-spin currents, using only constraints from conformal symmetry. Following the bootstrap philosophy, no reference is made to any Lagrangian, equations of motion or coupling constants. Even the space dimensions d are left free. The interaction is implicitly turned on through the local operators by letting them acquire anomalous dimensions. Whe…
Discrete Basis Parameterization for the Gauge Theory Bootstrap
本研究针对规范理论自举框架提出离散基参数化方案，大幅降低计算耗时，提升收敛性分析精度，还成功提取出远离实轴的σ介子极点。
The O(N) Monolith reloaded: Sum rules and Form Factor Bootstrap
本研究拓展二维O(N)对称有隙场论的S矩阵自举框架，纳入应力张量与守恒流的形状因子，通过求和规则约束紫外CFT中心荷，发现边界大部分区域源自无穷大中心荷的CFT，并分析了振幅行为的物理意义。
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