Ruth Britto
This article reviews on-shell methods for analytic computation of loop amplitudes, emphasizing techniques based on unitarity cuts. Unitarity techniques are formulated generally but have been especially useful for calculating one-loop amplitudes in massless theories such as Yang-Mills theory, QCD, and QED.
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We introduce and discuss Monte Carlo methods in quantum field theories. Methods of independent Monte Carlo, such as random sampling and importance sampling, and methods of dependent Monte Carlo, such as Metropolis sampling and Hamiltonian Monte Carlo, are introduced. We review the underlying theoretical foundations of Markov chain Monte Carlo. We provide several examples of Monte Carlo simulations, including one-dimensional simple harmonic oscillator, unitary matrix model exhibiting Gross-Witten-Wadia transition and a supersymmetric model exhibiting dynamical supersymmetry breaking.
We raise the issue whether gauge theories, that are not renormalizable in the usual power-counting sense, are nevertheless renormalizable in the modern sense that all divergences can be cancelled by renormalization of the infinite number of terms in the bare action. We find that a theory is renormalizable in this sense if the {\em a priori} constraints that we impose on the form of the bare action correspond to the cohomology of the BRST transformations generated by the action. Recent cohomology theorems of Barnich, Brandt, and Henneaux are used to show that conventionally nonrenormalizable th…
Systematic Analytic Regularization in $φ^4$ and Yukawa Theories
本文提出系统解析正则化(SAR)方案，用于φ⁴和Yukawa理论在NLO级的自洽正则化，通过解析继续动能算子的幂次确保理论形式上有限。
For gauge theories with confinement, the analytic structure of amplitudes is explored. It is shown that the analytic properties of physical amplitudes are the same as those obtained on the basis of an effective theory involving only the composite, physical fields. The corresponding proofs of dispersion relations remain valid. Anomalous thresholds are considered. They are related to the composite structure of particles. It is shown, that there are no such thresholds in physical amplitudes which are associated with confined constituents, like quarks and gluons in QCD. Unphysical amplitudes are c…
The mechanism by which gauge and gravitational anomalies cancel in certain string theories is reviewed. The presentation is aimed at theorists who do not necessarily specialize in string theory.
We analyse two principal approaches to the quantization of physical models worked out to date. There are the Faddeev-Popov "heuristic" approach, based on fixing a gauge in the FP path integrals formalism, and the "fundamental" approach by Dirac based on the constraint-shell reduction of Hamiltonians with deleting unphysical variables. The relativistic invariant FP "heuristic" approach deals with the enough small class of problems associated with S-matrices squared taking on-shell of quantum fields. On the other hand, the "fundamental" quantization approach by Dirac involves the manifest relati…
We present a survey on the mathematical structure of zero- and single scale quantities and the associated calculation methods and function spaces in higher order perturbative calculations in relativistic renormalizable quantum field theories.
It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP^3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formul…
The high-energy Regge behavior of gauge theories is studied via the formalism of Analytic Multi-Regge Theory. Perturbative results for spontaneously-broken theories are first organised into reggeon diagrams. Unbroken gauge theories are studied via a reggeon diagram infra-red analysis of symmetry restoration. Massless fermions play a crucial role and the case of QCD involves the Super-Critical Pomeron as an essential intermediate stage. An introductory review of the build up of transverse momentum diagrams and reggeon diagrams from leading log calculations in gauge theories is presented first. …
We summarise recent work on superconformal field theories using analytic superspace. All operators of N=4 SYM can be given as unconstrained superfields on analytic superspace. We show how to write down operators as superfields on analytic superspace and how to completely solve the Ward indentities for their correlation functions. We discuss the non-renormalisation of certain operators, and of some of their correlation functions. We discuss the relationship between harmonic and analytic superspace. Finally we discuss applications of these techniques to superconformal field theory in 6 dimension…
In this work we investigate the matrix elements of the energy-momentum tensor for massless on-shell states in four-dimensional unitary, local, and Poincaré covariant quantum field theories. We demonstrate that these matrix elements can be parametrised in terms of covariant multipoles of the Lorentz generators, and that this gives rise to a form factor decomposition in which the helicity dependence of the states is factorised. Using this decomposition we go on to explore some of the consequences for conformal field theories, deriving the explicit analytic conditions imposed by conformal symmetr…
The analytic continuation from the Euclidean domain to real space of the one-particle irreducible quantum effective action is discussed in the context of generalized local equilibrium states. Discontinuous terms associated with dissipative behavior are parametrized in terms of a conveniently defined sign operator. A generalized variational principle is then formulated, which allows to obtain causal and real dissipative equations of motion from the analytically continued quantum effective action. Differential equations derived from the implications of general covariance determine the space-time…
A summary is reported on our previous publications about four-dimensional N=1 supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of a-maximization, we studied decoupling operators in the infrared and the renormalization flow of the theory. We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory. This is a review article invited to International Journal of Mo…
The evolution of ideas which has led from the first proofs of the renormalizability of non-abelian gauge theories, based on Slavnov--Taylor identities, to the modern proof based on the BRS symmetry and the master equation is recalled. This lecture has been delivered at the {\bf Symposium in the Honour of Professor C. N. Yang}, Stony-Brook, May 21-22 1999.
The full analytic trans-series in integrable field theories
该研究针对含外场的可积量子场论广义能量密度问题，用Wiener-Hopf技术求解线性热力学贝特拟设方程，推导完整解析跨级数，揭示其复苏结构并验证中位数重求和可得物理值。
Quenched reduction is revisited from the modern viewpoint of field-orbifolding. Fermions are included and it is shown how the old problem of preserving anomalies and field topology after reduction is solved with the help of the overlap construction.
Conformal theory correlators are characterized by the spectrum and three- point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by…
We apply the method proposed by Fröhlich, Morchio, and Strocchi to analyze the bound state spectrum of various gauge theories with a Brout-Englert-Higgs mechanism. These serve as building blocks for theories beyond the standard model but also stress the exceptional role of the standard model weak group. We will show how the Fröhlich-Morchio-Strocchi mechanism relates gauge-invariant bound state operators to invariant objects of the remaining unbroken gauge group after gauge fixing. In particular, this provides a strict gauge-invariant formulation of the latter in terms of the original gauge sy…
Einstein- Schroedinger (ES) non-symmetric theory has been extended to accommodate the Abelian and non-Abelian gauge theories of dyons in terms of the quaternion-octonion metric realization. Corresponding covariant derivatives for complex, quaternion and octonion spaces in internal gauge groups are shown to describe the consistent field equations and generalized Dirac equation of dyons. It is also shown that quaternion and octonion representations extend the so-called unified theory of gravitation and electromagnetism to the Yang-Mill's fields leading to two SU(2) gauge theories of internal spa…
Conformal field theories play a central role in theoretical physics with many applications ranging from condensed matter to string theory. The conformal bootstrap studies conformal field theories using mathematical consistency conditions and has seen great progress over the last decade. In this thesis we present an implementation of analytic bootstrap methods for perturbative conformal field theories in dimensions greater than two, which we achieve by combining large spin perturbation theory with the Lorentzian inversion formula. In the presence of a small expansion parameter, not necessarily …
We give a modern geometric viewpoint on anomalies in quantum field theory and illustrate it in a 1-dimensional theory: supersymmetric quantum mechanics. This is background for the resolution of worldsheet anomalies in orientifold superstring theory.
We study next-to-leading corrections to the integral kernel of the BFKL equation for high energy cross-sections in QCD and in supersymmetric gauge theories. The eigenvalue of the BFKL kernel is calculated in an analytic form as a function of the anomalous dimension γof the local gauge-invariant operators and their conformal spin n. For the case of an extended N=4 SUSY the kernel is significantly simplified. In particular, the terms non-analytic in n are canceled. We discuss the relation between the DGLAP and BFKL equations in the N=4 model.
We explain some applications of bicategories in both classical and quantum field theory. This includes a modern perspective on some pioneering work of Max Kreuzer and Bert Schellekens on rational conformal field theory.
The properties of the high energy behavior of the scattering amplitude of massive, neutral and spinless particles in higher dimensional field theories are investigated. The axiomatic formulation of Lehmann, Symanzik and Zimmermann is adopted. The analyticity properties of the causal, the retarded and the advanced functions associated with the four point elastic amplitudes are studied. The analog of the Lehmann-Jost-Dyson representation is obtained in higher dimensional field theories. The generalized J-L-D representation is utilized to derive the t-plane analyticity property of the amplitude. …
Quantum fluctuations generate in three-dimensional gauge theories not only radiative corrections to the Chern-Simons coupling but also non-analytic terms in the effective action. We review the role of those terms in gauge theories with massless fermions and Chern-Simons theories. The explicit form of non-analytic terms turns out to be dependent on the regularization scheme and in consequence the very existence of phenomena like parity and framing anomalies becomes regularization dependent. In particular we find regularization regimes where both anomalies are absent. Due to the presence of non-…
The Quantum Theory Of Gravitation, Effective Field Theories, and Strings: Yesterday And Today
本文从引力量子理论视角分析了有效场论，回顾了1930年代至1994年量子引力探索如何推动粒子物理和广义相对论的观念转变，并探讨了弦论与有效场论在现代物理学中的关系及研究焦点变化。
From a path integral point of view (e.g. \cite{Q98}) physicists have shown how {\it duality} in antisymmetric quantum field theories on a closed space-time manifold $M$ relies in a fundamental way on Fourier Transformations of formal infinite-dimensional volume measures. We first review these facts from a measure theoretical point of view, setting the importance of the Hodge decomposition theorem in the underlying geometric picture, ignoring the local symmetry which lead to degeneracies of the action. To handle these degeneracies we then apply Schwarz's Ansatz showing how duality leads to a fa…
We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and the Seiberg-Witten invariants, for a three-manifold with arbitrary first Betti number. We also present the Hamiltonian quantization of our theories, find the modular transformations of states, and various properties of loop operators. Our results for the U(1|1) theory are in genera…
Static regular black holes in Horndeski theories: analytic no-go and nonanalytic obstructions
研究霍恩德斯基理论中静态正则黑洞，通过分析标量动能项\(X\)在不同分支情况，利用解析性得出非微扰无毛定理等，还分类边缘非解析偏差，确定标量 - 高斯 - 博内链为完备化方式，揭示相关黑洞特性。
This thesis is about conceptual aspects of gauge theories. Gauge theories lie at the heart of modern physics: in particular, they constitute the standard model of particle physics. At its simplest, the idea of gauge is that nature is best described using a descriptively redundant language; the different descriptions are said to be related by a gauge symmetry. The over-arching question the thesis aims to answer is: how can descriptive redundancy be fruitful for physics?
We study the nonperturbative dynamics of nonsupersymmetric asymptotically free gauge theories with fermionic matter in distinct representations of the SO(N) and Sp(2N) gauge groups. We use different analytic methods to unveil the associated conformal windows for the relevant matter representations. We propose a direct test for confronting and establishing the validity of the analytic methods used to constrain the conformal windows. By comparing the resulting windows for SU, Sp and SO a pleasing universal picture emerges.
An attempt to shed light on the various belief/idea systems in high Tc superconductivity that are at present popular. This text is in first instance intended to serve both string theorists and junior condensed matter physicists who want to enter this field. It departs from the premise that the often confusing, mutually contradicting portfolio of theories can be best appreciated by viewing it from a historical perspective. The histories of the following subjects are chronicled: the spin fluctuation superglue, Mottness, Resonating Valence Bonds and the gauge theories, pseudo-gap and competing or…
One of the perspectives in modern quantum field and string theory is related with the attempts to go beyond the perturbation theory. It turns out that a key principle in the formulation of all known non-perturbative results is {\it integrability}, i.e. arising of the structures of completely integrable systems. I discuss several important steps in this direction and speculate on its further possible development.
This series of lectures is planned as a generalization of author's large (more than fifteen years) experience of work in the theoretical physics. The modern theoretical physics is based on the group-theoretical approach which generates the formalism of the principal fibre bundles and the instanton approach. The latter is based on the Pontrjagin's degree of map theorem and this theorem is the original ``bridge'' between homology and cohomology theories. The author plans to devote his two first lectures to fibre bundle theory: this is the foundation on which the modern physics rests -- the theor…
We point out that modern brane theories suffer from a severe vacuum energy problem. To be specific, the Casimir energy associated with the matter fields confined to the brane, is stemming from the one and the same localization mechanism which forms the brane itself, and is thus generically unavoidable. Possible practical solutions are discussed, including in particular spontaneously broken supersymmetry, and quantum mechanically induced brane tension.
We investigate a dynamical mass generation mechanism for the off-diagonal gluons and ghosts in SU(N) Yang-Mills theories, quantized in the maximal Abelian gauge. Such a mass can be seen as evidence for the Abelian dominance in that gauge. It originates from the condensation of a mixed gluon-ghost operator of mass dimension two, which lowers the vacuum energy. We construct an effective potential for this operator by a combined use of the local composite operators technique with the algebraic renormalization and we discuss the gauge parameter independence of the results. We also show that it is …
Bootstrapping 3D Conformal Field Theories with Product Analytic Functionals
该研究提出基于乘积解析泛函的三维共形场论自举方法，其在单关联函数能隙最大化问题中表现更优，可提升边界、加速收敛，为相关共形理论研究提供新工具。
This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common mathematical pattern on which the paper concludes.
Analytic Properties of Infrared-Finite Amplitudes in Theories with Long-Range Forces
本文以带电标量粒子在固定库仑背景下的严格可解模型为研究对象，通过正则量子化等方法，揭示长程力理论中散射振幅的新颖解析性质，提出一种消除红外发散的新途径。
The multiplets that occur in four dimensional rigidly supersymmetric theories can be described either by chiral superfields in Minkowski superspace or analytic superfields in harmonic superspace. The superconformal Ward identities for Green's functions of gauge invariant operators of these types are derived. It is shown that there are no chiral superconformal invariants. It is further shown that the Green's functions of analytic operators are severely restricted by the superconformal Ward when analyticity is taken into account.
A survey is given on the present status of analytic calculation methods and the mathematical structures of zero- and single scale Feynman amplitudes which emerge in higher order perturbative calculations in the Standard Model of elementary particles, its extensions and associated model field theories, including effective field theories of different kind.
We review the concept of anomaly field theory, namely the fact that the anomalies of a $d$-dimensional field theory can be encoded in a $d+1$-dimensional field theory functor. We give numerous examples of anomaly field theories, explain how classical facts about anomalies are recovered from the anomaly field theory, and review recent work on global anomaly cancellation in 6d supergravity where this concept was instrumental. We also sketch the status of global anomaly cancellation checks in string theory. This paper is based on a talk given at the Durham Symposium `Higher Structures in M-theory…
We consider a massive, neutral, scalar field theory of mass $m_0$ in a five dimensional flat spacetime. Subsequently, one spatial dimension is compactified on a circle, $S^1$, ofradius $R$. The resulting theory is defined in the manifold, $R^{3,1}\otimes S^1$. The mass spectrum is a state of lowest mass, $m_0$, and a tower of massive Kaluza-Klein states. The analyticity property of the elastic scattering amplitude is investigated in the Lehmann-Symanzik-Zimmermann (LSZ) formulation of this theory. In the context of nonrelativistic potential scattering, for the $R^3\otimes S^1$ spatial geometry…
We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogs of the field theories proposed recently through the notion of `braided $L_\infty$-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy $2$-sph…
We study the Janus interface, a domain wall characterized by spatially varying couplings, in two-dimensional N=(2,2) supersymmetric gauge theories on the two-sphere. When the variations of the couplings are small enough, SUSY localization in the Janus background gives an analytic continuation of the sphere partition function. This directly demonstrates that the interface entropy is proportional to the quantity known as Calabi's diastasis, as originally shown by Bachas et al. When the variations are not small, we propose that an analytic continuation of the sphere partition function coincides w…
We investigate spectral functions of matter-gauge theories that are asymptotically free in the ultraviolet and display a Banks-Zaks conformal fixed point in the infrared. Using perturbation theory, Callan-Symanzik resummations, and UV-IR connecting renormalisation group trajectories, we analytically determine the gluon, quark, and ghost propagators in the entire complex momentum plane. At weak coupling, we find that a Källén-Lehmann spectral representation of propagators is achieved for all fields, and determine suitable ranges for gauge-fixing parameters. At strong coupling, a proliferation o…
We study Ω-deformation of B-twisted gauge theories in two dimensions. As an application, we construct an Ω-deformed, topologically twisted five-dimensional maximally supersymmetric Yang-Mills theory on the product of a Riemann surface $Σ$ and a three-manifold $M$, and show that when $Σ$ is a disk, this theory is equivalent to analytically continued Chern-Simons theory on $M$. Based on these results, we establish a correspondence between three-dimensional $\mathcal{N} = 2$ superconformal theories and analytically continued Chern-Simons theory. Furthermore, we argue that there is a mirror symmet…
We propose a distinction between the physical and the mathematical parts of gauge field theories. The main problem we face is to uphold a strong and meaningful criterion of what is physical. We like to call it "Field's dilemma", referring to Hartry Field's nominalist proposal which we consider to be inadaequate. The resolution to the dilemma, we believe, is implicitly provided by the so-called fiber bundle formalism. We shall demonstrate, in detail, that the bundle structure underlying modern quantum and gravitational gauge field theories allows for a genuine distinction between the physically…
The IR/UV mixing and the violation of unitarity are two of the most intriguing aspects of noncommutative quantum field theories. In this paper the relation between these two phenomena is explained and established. We start out by showing that the S-matrix of noncommutative field theories is hermitian analytic. As a consequence, a noncommutative field theory is unitary if the discontinuities of its Feynman diagram amplitudes agree with the expressions calculated using the Cutkosky formulae. These unitarity constraints relate the discontinuities of amplitudes with physical intermediate states; a…
In this work we explicitly show that the perturbative unitarity of analytic infinite derivative (AID) scalar field theories can be achieved using a modified prescription for computing scattering amplitudes. The crux of the new prescription is the analytic continuation of a result obtained in the Euclidean signature to the Minkowski external momenta. We intensively elaborate an example of a non-local $ϕ^4$ model for various infinite derivative operators. General UV properties of amplitudes in non-local theories are discussed.
We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement …
We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension $d\le5$, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with $d\le3$. The results are valid for non-integer $d$ as well. We further propose an analytic continuation from $d=3$ to $d=4$ that gives the perturbative partition function for an $\mathcal{N}=1$ gauge theory. The results are consistent with the free multiplets and the one-loop $β$-functions for general $\mathcal{N}=1$ gauge theories. We also co…
Analytic Bootstrap for $O(N)$ Boundary Conformal Field Theories with Interacting Boundaries
利用解析自举和微扰重整化群分析，研究 $d=4-\varepsilon$ 和 $d=3-\varepsilon$ 中具有边界相互作用的 $O(N)$ 边界共形场论，发现新的边界固定点并首次提取三临界 $O(N)$ 模型的边界共形数据。
We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to th…
Symbolic algebra relevant to the renormalization of gauge theories can be efficiently performed by machine using modern packages. We devise a scheme for representing and manipulating the objects involved in perturbative calculations of gauge theories. This scheme is readily implemented using the general purpose package, Mathematica. The techniques discussed are used to calculate renormalization group functions for a non-abelian $SU(m)$ gauge theory with massless fermions in a representation R, in the two-loop approximation, and to simplify some expressions arising in electroweak calculations a…
Using the principles of the modern scattering amplitudes programme, we develop a formalism for constructing the amplitudes of three-dimensional topologically massive gauge theories and gravity. Inspired by recent developments in four dimensions, we construct the three-dimensional equivalent of $x$-variables for conserved matter currents coupled to topologically massive gauge bosons or gravitons. Using these, we bootstrap various matter-coupled gauge-theory and gravitational scattering amplitudes, and conjecture that topologically massive gauge theory and topologically massive gravity are relat…
The analyticity properties of the scattering amplitude for a massive scalar field is reviewed in this article where the spacetime geometry is $R^{3,1}\otimes S^1$ i.e. one spatial dimension is compact. Khuri investigated the analyticity of scattering amplitude in a nonrelativitstic potential model in three dimensions with an additional compact dimension. He showed that, under certain circumstances, the forward amplitude is nonanalytic. He argued that in high energy scattering if such a behaviour persists it would be in conflicts with the established results of quantum field theory and LHC migh…
Exact solutions to the low-energy effective action (LEEA) of the four-dimensional N=2 supersymmetric gauge field theories are known to be obtained either by quantum field theory methods from S-duality in the Seiberg-Witten approach, or by the Type-IIA superstring/M-Theory methods of brane technology. After a brief review of the standard field-theoretical results for the N=2 gauge (Seiberg-Witten) LEEA, we consider a field-theoretical derivation of the exact hypermultiplet LEEA by using the N=2 harmonic superspace methods. We illustrate our techniques on a number of explicit examples. Our main …
We derive positivity bounds on low energy effective field theories which admit gapped, analytic, unitary, Lorentz invariant, and possibly non-local UV completions, by considering 2 to 2 scatterings of Jaffe fields whose Lehmann-Källén spectral density can grow exponentially. Several properties of S-matrix, such as analyticity properties, are assumed in our derivation. Interestingly, we find that some of the positivity bounds obtained in the literature, such as sub-leading order forward-limit bounds, must be satisfied even when UV completions fall into non-localizable theories in Jaffe's langua…
We are used to thinking of an operator acting once, twice, and so on. However, an operator acting integer times can be consistently analytic continued to an operator acting complex times. Applications: (s,r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytic continued matrix representati…
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