Andrea Guerrieri, Joao Penedones, Pedro Vieira
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.
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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。
We revisit from a modern bootstrap perspective the longstanding problem of solving QCD in the large $N$ limit. We derive universal bounds on the effective field theory of massless pions by imposing the full set of positivity constraints that follow from $2 \to 2$ scattering. Some features of our exclusion plots have intriguing connections with hadronic phenomenology. The exclusion boundary exhibits a sharp kink, raising the tantalizing scenario that large $N$ QCD may sit at this kink. We critically examine this possibility, developing in the process a partial analytic understanding of the geom…
Bootstrapping Pions at Large $N$. Part II: Background Gauge Fields and the Chiral Anomaly
本文利用S矩阵自举方法研究大N禁闭规范理论中π介子与背景电磁场的有效场论，推导出2→2散射振幅的完整正定性约束，并以ρ介子质量为单位得到Wilson系数比值的新数值界限，同时通过手征反常匹配导出显含N依赖性的界限。
Bootstrapping Pion Form Factors at Large $N$
利用解析性、幺正性和Regge有界性，对大$N$ QCD中介子形状因子进行自举研究，约束低能形状因子系数，并结合微扰紫外行为给出介子衰变常数等界限。
We show that Skyrmions with massless pions in hyperbolic space provide a good approximation to Skyrmions with massive pions in Euclidean space, for a particular relationship between the pion mass and the curvature of hyperbolic space. Using this result we describe how a Skyrmion with massive pions in Euclidean space can be approximated by the holonomy along circles of a Yang-Mills instanton. This is a generalization of the approximation of Skyrmions by the holonomy along lines of an instanton, which is only applicable to massless pions.
Revisiting Schrödinger CFTs: Factorization, Massless Particles, and a Path to the Bootstrap
从现代视角重新审视薛定谔CFT，引入谐波陷阱几何并建立态-算子对应，系统分类物理谱与幺正性界限，为非微扰分解和自举方法奠定基础。
In this paper we discuss a class of 1+1 dimensional matrix field theories in the light-cone Hamiltonian approach that are obtained from a dimensional reduction of 3+1 dimensional Yang-Mills theory. We present exact massless solutions of the boundstate integral equations in the large N limit. These massless states have constant wavefunctions in momentum space and are therefore fundamental excitations of the theory.
The Gauge Theory Bootstrap: Predicting pion dynamics from QCD
通过QCD约束计算π子动力学，结合形式因子和谱密度，确定共振参数和低能有效作用系数，探讨π子气体热力学。
The EFT Bootstrap at Finite $M_{PL}$
本文研究有效场论中圈效应对正定性的影响，处理引力耦合无质量粒子的前向极限发散，并证明即使圈效应小，保持一致性也会显著改变树图层次的约束。
Gauge Theory Bootstrap: Pion amplitudes and low energy parameters
本文用规范理论自举法，仅以π介子质量、衰变常数和QCD参数为输入，计算π散射相移及低能系数，成功重现ρ(770)等共振态，为连接低能与高能强耦合物理提供新途径。
A non-perturbative method based on the Form Factor bootstrap approach is proposed for the analysis of correlation functions of 2-D massless integrable theories and applied to the massless flow between the Tricritical and the Critical Ising Models.
Primal Bootstrap for Pion Scattering at Large-N
针对大N极限下π介子散射，构造了基于Lovelace-Shapiro类振幅线性组合的树级亚纯振幅基，满足解析性、交叉对称和Regge行为，并通过部分波正定性实现自举约束。
From data to the analytic S-matrix: A Bootstrap fit of the pion scattering amplitude
本文提出结合S矩阵Bootstrap与粒子群优化、利用解析性、交叉对称性和幺正性约束拟合ππ散射实验与格点数据，成功构建紫外完备振幅并预言了约2 GeV的四夸克共振。
This is the 6th paper in the series developing the formalism to manage the effective scattering theory of strong interactions. Relying on the theoretical scheme suggested in our previous publications we concentrate here on the practical aspect and apply our technique to the elastic pion-nucleon scattering amplitude. We test numerically the pion-nucleon spectrum sum rules that follow from the tree level bootstrap constraints. We show how these constraints can be used to estimate the tensor and vector $NNρ$ coupling constants. At last, we demonstrate that the tree-level low energy expansion coef…
In massless quantum field theories the Landau equations are invariant under graph operations familiar from the theory of electrical circuits. Using a theorem on the $Y$-$Δ$ reducibility of planar circuits we prove that the set of first-type Landau singularities of an $n$-particle scattering amplitude in any massless planar theory, in any spacetime dimension $D$, at any finite loop order in perturbation theory, is a subset of those of a certain $n$-particle $\lfloor{(n{-}2)^2/4}\rfloor$-loop "ziggurat" graph. We determine this singularity locus explicitly for $D=4$ and $n=6$ and find that it co…
We use the amplitude soft bootstrap method to explore the space of effective field theories (EFT) of massless vectors and scalars. It is known that demanding vanishing soft limits fixes uniquely a special class of EFTs: non-linear sigma model, scalar Galileon and Born-Infeld theories. Based on the amplitudes analysis, we conjecture no-go theorems for higher-derivative vector theories and theories with coupled vectors and scalars. We then allow for more general soft theorems where the non-trivial part of the soft limit of the (n+1)-pt amplitude is equal to a linear combination of n-pt amplitude…
Bootstrapping string theory EFT
本文结合幺正性和单值性关系引导开弦有效场论，发现Type-I超弦是唯一解，并利用几何方法从低能系数推导出临界维度10。
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。
Bootstrap is a novel and ambitious paradigm for quantum physics. It aims to solve the target problems by exploiting theoretical constraints from general physical principles and self-consistency conditions. The bootstrap philosophy dates back to the 1960s. Its real power has been recognized only recently in, e.g., conformal field theories and relativistic scattering amplitudes. Inspired by [X. Han, S. A. Hartnoll, and J. Kruthoff, Phys. Rev. Lett. 125, 041601 (2020)], we report the first bootstrap results in low-energy nuclear physics, where deuteron, with its Hamiltonian given by pionless effe…
UV completion of 2D Ising CFT:a golden E_8 massless $S$-matrix
通过求解无质量左右S矩阵的引导方程，完整分类了RG流到二维伊辛CFT的UV完备QFT，发现四种完备化，其中一种新“金色流”的UV固定点为对角su(2)陪集CFT，且E8对称性在所有RG尺度上保持。
The recent work by Iha et al shows an upper bound on mass anomalous dimension $γ_m$ of multi-flavor massless QCD at the renormalization group fixed point from the conformal bootstrap in $SU(N_F)_V$ symmetric conformal field theories under the assumption that the fixed point is realizable with the lattice regularization based on staggered fermions. We show that the almost identical but slightly stronger bound applies to the regularization based on Wilson fermions (or domain wall fermions) by studying the conformal bootstrap in $SU(N_f)_L \times SU(N_f)_R$ symmetric conformal field theories. For…
Poincaré invariance is a well-tested symmetry of nature and sits at the core of our description of relativistic particles and gravity. At the same time, in most systems Poincaré invariance is not a symmetry of the ground state and is hence broken spontaneously. This phenomenon is ubiquitous in cosmology where Lorentz boosts are spontaneously broken by the existence of a preferred reference frame in which the universe is homogeneous and isotropic. This motivates us to study scattering amplitudes without requiring invariance of the interactions under Lorentz boosts. In particular, using on-shell…
Two-loop QCD Amplitudes from the Chiral Algebra Bootstrap
本文利用手征代数自举法，结合超对称Ward恒等式，计算了无质量QCD中两圈全正螺旋度胶子振幅，并证明全色两圈振幅可由扭量理论形状因子及低圈振幅得到，从而简化计算。
We argue that the $125 GeV$ "Higgs" particle is unlikely to arise as a fermion- antifermion composite if the underlying dynamics is a vectorial gauge theory. The reason is that the lightest scalar in such theories is heavier than the lightest pseudo-scalar with the mass difference being fixed by the scale of the theory. LHC searches suggest that the scale of any new physics, including that of a putative new theory dynamically generating the 125 GeV "Higgs" particle, is relatively high $\sim{(1/2TeV-1TeV)}$. Also the LHC analysis suggests that it is {\it scalar} namely $J^P = 0^+$ rather than p…
A study of integrable form factors in massless relativistic $AdS_2$
本文针对可积$AdS_2$超弦无质量散射的形状因子展开研究，借助Fendley $S$矩阵的极限性质，构建满足假设的相对论修饰因子积分表示，证明交叉与幺正性关系，给出两粒子形状因子约束的解。
New RG flows between non-Unitary CFTs from exact massless scattering theories
该研究构造了无质量S矩阵自举方程的完整解，得到从高融合水平极小模型到非幺正Virasoro极小模型的无穷族可积RG流，通过共形微扰论与TBA方程小体积展开的匹配提供证据，还发现非可逆Verlinde缺陷线在流中保持不变。
Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover
本文研究了连接三临界与临界Ising场论的无质量重整化群流中的Rényi纠缠熵，通过推导bootstrap方程及利用sinh-Gordon理论的roaming极限解析计算了twist场form factor，并揭示了纠缠熵展开的手性结构及$\mathbb{Z}_2$对称性分解。
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…
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 bootstrap the S-matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such S-matrices is strongly constrained by the existence of a UV completion. In the context of flux tube physics, this allows us to constrain several terms in the S-matrix low energy expansion or -- equivalently -- on Wilson coefficients of several irrelevant operators showing up in the flux tube effective action. These bounds have direct implications for other physical quantities; for instance, they allow us to further bound the ground state…
Bootstrapping mesons at large $N$: Regge trajectory from spin-two maximization
本文通过自举方法研究大 N QCD 中的介子，假设存在自旋二共振并最大化其耦合，得到包含完整 Regge 轨迹的极值解，其低能态与真实介子质量高度吻合。
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。
Analytic bootstrap bounds on masses and spins in gravitational and non-gravitational scalar theories
该研究在S矩阵自举标准假设下推导了含无质量标量理论的弱耦合能谱解析约束，推广了相关振幅结果，提出序贯自旋与质量约束，且超弦能谱可饱和该约束。
We give an overview of the implementation of the soft-bootstrap method applied to the landscape of theories where the Special Galileon couples to a massless vector particle. We also describe the corresponding traditional Lagrangian approach for this model, which takes into account the formal geometrical interpretation of the Special Galileon as fluctuations of a $D$-dimensional brane embedded in a $2D$-dimensional flat space.
We consider entanglement measures in 2-2 scattering in quantum field theories, focusing on relative entropy which distinguishes two different density matrices. Relative entropy is investigated in several cases which include $ϕ^4$ theory, chiral perturbation theory ($χPT$) describing pion scattering and dilaton scattering in type II superstring theory. We derive a high energy bound on the relative entropy using known bounds on the elastic differential cross-sections in massive QFTs. In $χPT$, relative entropy close to threshold has simple expressions in terms of ratios of scattering lengths. De…
Both gravitons and pions are described by non-linear and non-renormalizable actions at low energies. These are most usefully treated by effective field theory, which is a full quantum field theoretic approach that relies only on the low energy degrees of freedom and their interactions. The gravitational case is particularly clean because of the masslessness of the graviton and the wide separation of scales. This essay provides an overview of this approach.
We investigate the noncommutative analogue of the spontaneously broken linear sigma model at the one-loop quantum level. In the commutative case, renormalization of a theory with a spontaneously broken continuous global symmetry depends on cancellations that enable the limited set of counterterms consistent with that symmetry to remove the divergences even after its spontaneous breaking, while preserving the masslessness of the associated Goldstone modes. In the noncommutative case, we find that these cancellations are violated, and the renormalized one-loop correction to the inverse pion prop…
Bootstrap for Lattice Yang-Mills theory
本文用数值bootstrap方法研究SU(∞)格点杨-米尔斯理论，结合环方程与正性条件，高效获得plaquette平均值的严格界，并验证了其在强耦合与弱耦合相的有效性，为蒙特卡洛方法提供了替代方案。
We examine the space of allowed S-matrices on the Adler zeros' plane using the recently resurrected (numerical) S-matrix bootstrap program for pion scattering. Two physical quantities, an averaged total scattering cross-section, and an averaged entanglement power for the boundary S-matrices, are studied. Emerging linearity in the leading Regge trajectory is correlated with a reduction in both these quantities. We identify two potentially viable regions where the S-matrices give decent agreement with low energy S- and P-wave scattering lengths and have leading Regge trajectory compatible with e…
We study the changes in the short-distance quark structure of the Nambu-Goldstone boson when the long-distance symmetry-breaking scales are depleted controllably. We achieve this by studying the valence Parton Distribution Function (PDF) of pion in 2+1 dimensional two-color QCD, with the number $N$ of massless quarks as the tunable parameter that slides the theory from being strongly broken for $N=0$ to the conformal window for $N>4$, where the theory is gapped by the fixed finite volume. We perform our study non-perturbatively using lattice simulations with $N=0,2,4,8$ flavors of nearly massl…
This article is an introduction to two currently very active research programs, the Conformal Bootstrap and Scattering Amplitudes. Rather than attempting full surveys, the emphasis is on common ideas and methods shared by these two seemingly very different programs. In both fields, mathematical and physical constraints are placed directly on the physical observables in order to explore the landscape of possible consistent quantum field theories. We give explicit examples from both programs: the reader can expect to encounter boiling water, ferromagnets, pion scattering, and emergent symmetries…
Classical Spin Gravitational Compton Scattering
本文提出一种新的自举方法，在重质量有效场理论中推导出涉及任意自旋无限质量粒子的经典康普顿散射振幅，并给出胶子四点和五点过程的无虚假极点显式结果。
We develop the method of calculation of the leading chiral (infrared) logarithms to an arbitrary loop order for various form factors of Nambu-Goldstone bosons. The method is illustrated on example of scalar and vector form factors in massless 4D O(N+1)/O(N) sigma-model. The analytical properties of the form factors are derived. The leading chiral (infrared) logarithms are summed up in the large N limit.
In this paper we numerically demonstrate that massless two-dimensional QCD is not integrable. To this aim, we explicitly solve the 't Hooft integral equation for bound states by an adaptive spline procedure, and compute the decay amplitudes. These amplitudes significantly differ from zero except in all cases in which the decay also produces a pion.
Scattering off of Twistorial Line Defects
本文将手征代数引导法扩展到含带电源的散射场景，以扭量理论中的自对偶双子为例，通过推广Koszul对偶构造，重现了胶子在自对偶双子上的n点MHV散射公式。
CFTs with $U(m)\times U(n)$ Global Symmetry in 3D and the Chiral Phase Transition of QCD
本研究针对具有$U(m)\times U(n)$对称性的三维共形场论，结合微扰展开与数值共形自举方法开展分析，得到新的微扰与非微扰结果，并为两味、三味无质量QCD的手征相变可能为二级相变提供了自举证据。
This work is driven by our curiosity towards some basic questions in QCD: how is the energy of a moving hadron partitioned among different gauge-invariant sectors in QCD Hamiltonian? How is the energy of an massless pion separated between quark and gluon sectors, particularly in the soft pion limit? Is it possible to decompose the celebrated Gell-Mann-Oakes-Renner (GOR) relation? To what extent can we justify the quark potential model from the field-theoretical mass decomposition for heavy quarkonium? Due to limitation of contemporary nonperturbative tools, we do not yet know answers to these …
On gauge amplitudes first appearing at two loops
研究无质量非阿贝尔规范理论中全正螺旋度散射振幅，验证特定费米子表示下单圈振幅为零并推广至两圈，修正维度正则化结果并利用天体手征代数自举法计算双迹项。
Bounds on scattering amplitudes of non-identical scalar particles in 4d
本文用原始S矩阵自举研究四维中两种不同标量粒子的散射振幅，发现不等质量时因伪物理割线导致部分可观测量无界或单侧有界，等质量或补充信息后恢复双侧界，为π-K等过程提供起点。
We use the strong coupling and hopping parameter expansions to calculate the pion and rho meson masses for lattice Yang-Mills gauge theories with fermions in irreducible two-index representations, namely the adjoint, symmetric and antisymmetric. The results are found to be consistent with orientifold planar equivalence, and leading order 1/N corrections are calculated in the lattice phase. An estimate of the critical bare mass, for which the pion is massless, is obtained as a function of the bare coupling. A comparison to data from the two-flavour SU(2) theory with adjoint fermions gives evide…
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