Gavin Cullen, Maciej Koch-Janusz, Thomas Reiter
In this work, the library spinney is presented, which provides an implementation of helicity spinors and related algorithms for the symbolical manipulation program Form. The package is well suited for symbolic amplitude calculations both in traditional, Feynman diagram based approaches and unitarity-based techniques.
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Spinor Helicity Technique and String Reorganization for Multiloop Diagrams
方法适用于旋量螺旋技术和字符串重组多回路振幅使用费曼和Schwinger参数表示进行审查和扩展。
From Multiloop Spinor Helicity Technique to String Reorganization
回顾了旋量螺旋度技术在树过程中的成功应用。要将其应用于多回路图之一是自然导致的施温格适当的时间表示，其性质进行了讨论。这个表示也是场论和弦论之间的一个有用的联系。
Simplifying Algebra in Feynman Graphs, Part II: Spinor Helicity from the Spacecone
显然，洛伦兹协变费曼规则给出了一个“标量”场的每一个螺旋，大大简化了计算振幅与无质量粒子。旋量螺旋度形式被正确地识别为零复类空（而不是类光）规范，其中两个无质量外动量定义了参考系。通常这种规范只适用于外部线因素，我们扩展这种方法的顶点和传播修改的行动本身使用光锥方法。
The spinor-helicity formalism has become an invaluable tool for understanding the S-matrix of massless particles in four dimensions. In this paper we construct a spinor-helicity formalism in six dimensions, and apply it to derive compact expressions for the three, four and five point tree amplitudes of Yang-Mills theory. Using the KLT relations, it is a straightforward process to obtain amplitudes in linearized gravity from these Yang-Mills amplitudes; we demonstrate this by writing down the gravitational three and four point amplitudes. Because there is no conserved helicity in six dimensions…
We calculate analytically tree level amplitudes for Higgs production via Weak Bosons Fusion (WBF) and QCD corrections to the Standard Model prediction (SM) by using the helicity spinor formalism. We provide the explicit expression for the amplitude with two gluons emission in the final state together with massless final two quarks. The massive generalization of the Britto-Cachazo-Feng-Witten (BCFW) recursion scheme is applied successfully obtaining compact results which agree with, and overcome in simplicity, the conventional Feynman diagrams approach.
We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form $\mathrm{Tr}\, F^n$ where $F$ is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a gen…
Spinor-helicity calculation of the $g^* g^* \to q\overline{q} V^*$ amplitude at the tree level
该研究采用旋量螺旋度形式计算树图阶$g^* g^* \to q \overline q V^*$过程的振幅，得到比迹方法更简洁的解析结果，可用于高能因子化框架下Drell-Yan过程唯象研究的数值加速。
Massive Helicity-Chirality Spinor Formalism from Massless Amplitudes with On-shell Mass Insertion
本文提出大质量螺旋度-手性旋量形式，通过在壳质量插入刻画螺旋度与手性翻转，建立大质量振幅与无质量振幅（含 Higgs 插入）的紫外-红外对应，并用于解释弱衰变质量增强及构造高点多点振幅。
We present SpinorsExtras package implementing massive spinor-helicity formalism in Mathematica on top of S@M package. Package defines new objects for Mathematica - massive spinors and reference, associated and polarization vectors. Various properties of introduced symbols and relations among them are implemented together with functions to manipulate and interchange such objects. The package can be used for symbolic calculation of helicity amplitudes involving massive particles with real or complex momenta, using on- or off-shell recursive techniques or Feynman diagrams.
SpinorHelicity4D: a Mathematica toolbox for the four-dimensional spinor-helicity formalism
本文介绍了 Mathematica 软件包 SpinorHelicity4D，用于四维无质量和有质量旋量-螺旋度形式的解析与数值计算，提供表达式简化、Dirac 迹计算和复动理学生成等功能，兼顾易用性与高级定制需求。
$\texttt{SMaSH}$ : Simplify Massive Spinor Helicity
提出SMaSH，一个Mathematica包，用于四维时空中的旋量螺旋度计算，支持有质量粒子、显式小群指标、离壳变量，并集成高能极限、离散对称性检查、接触项和规范不变性验证。
S@M, a Mathematica Implementation of the Spinor-Helicity Formalism
在本文中，我们提出的软件包S@M（旋量@Mathematica），实现旋量螺旋形式主义在Mathematica中。该软件包允许使用复数旋量代数以及Mathematica的多用途功能。该包定义了旋量对象及其基本属性以及操作它们的函数。它还提供了在每一个计算步骤中数值评估旋量对象的可能性。因此，该软件包非常适合在壳上技术的背景下使用，特别是用于树和环路级的散射振幅的评估。
We use the spinor helicity formalism to calculate the cross section for production of three partons of a given polarization in Deep Inelastic Scattering (DIS) off proton and nucleus targets at small Bjorken x. The target proton or nucleus is treated as a classical color field (shock wave) from which the produced partons scatter multiple times. We reported our result for the final expression for the production cross section and studied the azimuthal angular correlations of the produced partons in [1]. Here we provide the full details of the calculation of the production cross section using the …
Compton amplitude and Contact term(s) in the Spinor Helicity formalism
该研究利用规范不变可观测量与参考矢量选择无关的原理，确定电磁 Compton 振幅的接触项，并证明所得树级结果无虚假极点。
A New Method for Computing Helicity Amplitudes
本文提出了一种用于微扰计算的螺旋度形式。它基于在旋量线中正式插入由非物理旋量建立的一整套状态。当存在大量旋量时，这是特别方便的。简要讨论了它在$e^+\,e^-\,\rightarrow b\,\bar b\,W^+W^-$中的应用。
In this article, we extend the %Weyl-van der Waerden spinor technique for calculating helicity amplitudes to general massive fields of half-integer spins. We find that the little group generators can be represented as first-order differential operators in the spinor formalism. We use the spinor forms of the generators to get the explicit form of the massive fields of any spin and any helicity. We also deal with the three-particle S-matrix by these spinor form generators, and find that we are able to extend the explicit form of the three-particle vertex obtained by Benincasa and Cachazo to the …
Two-Loop Five-Parton Leading-Colour Finite Remainders in the Spinor-Helicity Formalism
本文利用自旋螺旋度变量重构全部两圈五部分子领头色有限余项，得到更紧凑、稳定且高效的表达式，并以少一个数量级的样本提供有限域重构的替代方案。
Massive On-shell Splitting Functions in Spinor-Helicity Formalism
研究大量粒子壳上分裂函数，基于Soper-Weinberg共线旋量提出构造形式，推导标准模型粒子分裂函数集，建立匹配字典，通过递归关系获更高点函数，为对撞机物理和部分子喷注发展提供计算工具。
The spinor-helicity formalism has proven to be very efficient in the calculation of scattering amplitudes in quantum field theory, while the loop tree duality (LTD) representation of multi-loop integrals exhibits appealing and interesting advantages with respect to other approaches. In view of the most recent developments in LTD, we exploit the synergies with the spinor-helicity formalism to analyse illustrative one- and two-loop scattering processes. We focus our discussion on the local UV renormalisation of IR and UV finite amplitudes and present a fully automated numerical implementation th…
We show that a natural spinor-helicity formalism that can describe massive scattering amplitudes exists in $D=6$ dimensions. This is arranged by having helicity spinors carry an index in the Dirac spinor {\bf 4} of the massive little group, $SO(5) \sim Sp(4)$. In the high energy limit, two separate kinds of massless helicity spinors emerge as required for consistency with arXiv:0902.0981, with indices in the two $SU(2)$'s of the massless little group $SO(4)$. The tensors of ${\bf 4}$ lead to particles with arbitrary spin, and using these and demanding consistent factorization, we can fix $3-$ …
Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity techniques
本文用旋量-螺旋度方法计算轻标量衰变至两中微子与两轻子的全部螺旋度振幅，证明在正确反称化与求和下，狄拉克-马约拉纳混淆定理成立。
Comments on Exploring Quantum Statistics for Dirac and Majorana Neutrinos using Spinor-Helicity technique (arXiv:2507.07180 [hep-ph])
本文评论了先前工作中关于利用量子统计区分狄拉克与马约拉纳中微子的思路，指出[1]中对狄拉克情况幅平方的任意对称化无物理依据，导致标准模型中狄拉克中微子轻子数违反。
A New Method for Helicity Calculations
我们提出了一个新的螺旋度形式主义的基础上正式插入旋量线的一套完整的国家建立非物理旋量。该方法是为无质量和大质量的费米子，它被证明是特别快。给出了所有有关公式。
Inspired by the flow description of su(N) colour calculations, we recently showed how to simplify the spinor-helicity formalism (at the algebra level two copies of complexified su(2)) by treating each Weyl spinor as part of a flow line with definite chirality and momentum. This formalism, dubbed the chirality-flow formalism, eliminates all non-trivial algebra from tree-level spinor-helicity calculations, thus allowing the shortest possible route from Feynman diagrams to complex numbers (spinor inner products). In this presentation, we briefly introduce the main features of this method and show…
Helicity amplitude for the process $q\Bar{q} \rightarrow q\Bar{q} g$
本文利用旋量-螺旋度形式计算由光子和胶子介导的过程 $q\Bar{q}\rightarrow q\Bar{q}g$ 的 $S$ 矩阵元，并证明忽略 $Z$ 玻色子交换后计算可被大幅简化。
Optimization of symbolic evaluation of helicity amplitudes
我们提出了一种符号化的费曼振幅的评估方法。构造了旋量粒子的特殊极化基，得到了基旋量张量积的紧致表达式。
Methods of spinor calculus in particle physics are described in this series of four chapters written in french. A special emphasis is made on the well-known helicity-coupling scheme which is the guiding thread of all these methods. Numerous explicit formulas are given for projectors onto two-particles helicity states as well as for bilinears of Dirac spinors, for the s and t channels of a given two-body process involving spin 1/2 particles. In the last chapter, applications to various elementary processes are given.
On helicity and spin on the light cone
从一个单体前形式方程与Lepage-Brodsky旋量，我们展示了，与相当数量的新技术，如何在标准的动量空间与Bjorken-Drell旋量的积分方程可以得到。积分方程对单重态和三重态进行解耦。
Two-loop self-dual Euler-Heisenberg Lagrangians (I): Real part and helicity amplitudes
我们表明，无论是标量和旋量QED，两个循环的欧拉-海森堡有效拉格朗日常数欧几里德自对偶背景有一个非常简单的双伽玛函数的封闭形式的表达。此外，标量和旋量QED有效拉格朗日量是非常相似的。与其他背景的结果相比，这些结果是戏剧性的简化。我们应用它们来计算的低能量限制的双圈大规模N-光子的“所有+”螺旋度振幅。我们的结果的简单性可以与自对偶性，螺旋性和超对称性之间的联系。
Two-Loop Five-Point One-Mass Amplitudes in the Spinor-Helicity Formalism
本文首次在旋量-螺旋度形式下表示两圈五点单质量振幅，基于先前解析结果重算部分子过程，使表达式规模减小三个数量级，以促进唯象研究和阐明振幅数学性质。
We establish a simple formula for the minimal dimension of operators leading to any helicity amplitude. It eases the systematic enumeration of independent operators from the construction of massless non-factorizable on-shell amplitudes. Little-group constraints can then be solved algorithmically for each helicity configuration to extract a complete set of spinor structures with lowest dimension. Occasionally, further reduction using momentum conservation, on-shell conditions and Schouten identities is required. A systematic procedure to account for the latter is presented. Dressing spinor stru…
The entanglement between $SU(2) \otimes SU(2)$ internal degrees of freedom of parity and helicity for reflected and transmitted waves of Dirac-like particles scattered by a potential step along an arbitrary direction on the $x-y$ plane is quantified. Diffusion ($E \geq V$) and Klein zone ($V \geq E$) energy regimes are considered. It has been shown that, for $SU(2) \otimes SU(2)$ polarized structures of helicity eigenstates impinging the barrier, the local interaction with a {\em step} potential destroys the {\em parity-spin} separability. The framework presented here can be straightforwardly …
We study representations of the Poincaré group that have a privileged transformation law along a p-dimensional hyperplane, and uncover their associated spinor helicity variables in D spacetime dimensions. Our novel representations generalize the recently introduced celestial states and transform as conformal primaries of SO(p,1), the symmetry group of the p-hyperplane. We will refer to our generalized states as ``partially celestial." Following Wigner's method, we find the induced representations, including spin degrees of freedom. Defining generalized spinor helicity variables for every D and…
Multiphoton Production at High Energies in the Standard Model II
我们研究多光子生产的电弱部门的标准模型在高能量极限使用的等效定理结合旋量螺旋技术。我们利用由辐射$n$光子的带电标量、旋量或矢量线组成的电流。在这些电流中，带电线只有一端离壳，这是已知的类螺旋光子和一个不类螺旋光子的情况。我们得到了各种各样的螺旋度振幅的过程，涉及两对带电粒子通过考虑四个电流的组合。我们研究的情况下，相对于电流的两端的带电线的壳外，并提出解决方案的情况下，螺旋光子。这些新的电流可能与两个原始电流结合，产生额外的振幅，涉及希格斯，纵向$Z$或中微子对。
On Higgs+jet production at next-to-leading power accuracy
本文计算了胶子聚变道希格斯粒子+单喷注产生的次领先幂次修正，通过旋量平移方法简化次软辐射计算，并给出不同螺旋度组态的次领先幂次领先对数解析表达式。
Multiphoton Production at High Energies in the Standard Model I
我们研究多光子生产的电弱部门的标准模型在高能量极限使用的等效定理结合旋量螺旋技术。我们得到的电流组成的带电标量，旋量，或矢量线辐射$n$光子的递归关系。封闭形式的解决方案，这些递归关系的任意$n$的情况下，类似的螺旋和一个unlike-helicity光子生产。我们应用电流单独和成对获得振幅的过程涉及生产的$n$光子与两个不同的螺旋从一对带电粒子。本文还讨论了用横向极化的$ $Z$ -玻色子代替一个或多个光子的问题.
I provide a basic introduction to modern helicity amplitude methods, including color organization, the spinor helicity formalism, and factorization properties. I also describe the BCFW (on-shell) recursion relation at tree level, and explain how similar ideas - unitarity and on-shell methods - work at the loop level. These notes are based on lectures delivered at the 2012 CERN Summer School and at TASI 2013.
We introduce the Majorana spinors in the momentum representation. They obey the Dirac-like equation with eight components, which has been first introduced by Markov. Thus, the Fock space for corresponding quantum fields is doubled (as shown by Ziino). The particular attention has been paid to the questions of chirality and helicity (two concepts which frequently are confused in the literature) for Dirac and Majorana states.
Covariant BCFW for All Masses
该研究提出适用于所有质量粒子的协变BCFW新形式体系，利用有质量旋量螺旋度形式体系推导通用BCFW移位，助力四维有质量粒子振幅的壳上递推实现。
Recursion for Wilson-line Form Factors
本文利用单线复动量平移推导递推关系，以低点形状因子与在壳振幅为输入高效计算树图阶螺旋度形状因子，并给出 QCD 中 $1\ o 2$ 和 $1\ o 3$ 劈裂函数的新紧凑表达式。
Two-Loop Self-Dual QED
我们提出了显式的封闭形式的表达的两圈欧拉-海森堡拉格朗日在一个恒定的自对偶领域，旋量和标量QED。这些表示的简单性使我们能够详细研究这些拉格朗日量的渐近性质，并使用Borel色散关系构建其虚部。特别是，对于这种自对偶的情况下，我们得到显式形式的Lebedev-Ritus函数出现在Schwinger表示的虚部在两个循环。利用自对偶性和螺旋性之间的联系，我们还得到了明确的公式的低能量极限的“所有+螺旋性”N光子振幅，在标量和旋量QED在一个和两个循环。
Polarization structure and spin covariance of massive vector-boson amplitudes in QCD
本文展示了矢量玻色子衰变为无质量轻子的振幅通过自旋协变性包含所有极化状态信息，包括纵向极化。
The instant form and the front form of relativistic dynamics proposed by Dirac in 1949 can be linked by an interpolation angle parameter $δ$ spanning between the instant form dynamics (IFD) at $δ=0$ and the front form dynamics which is now known as the light-front dynamics (LFD) at $δ=π/4$. We present the formal derivation of the interpolating quantum electrodynamics (QED) in the canonical field theory approach and discuss the constraint fermion degree of freedom which appears uniquely in the LFD. The constraint component of the fermion degrees of freedom in LFD results in the instantaneous co…
In this thesis I study applications of effective field theories to understand aspects of QCD jets and their substructure at the Large Hadron Collider. In particular, I introduce an observable, $D_2$, which can be used for distinguishing boosted $W/Z/H$ bosons from the QCD background using information about the radiation pattern within the jet, and perform a precision calculation of this observable. To simplify calculations in the soft collinear effective theory, I also develop a helicity operator basis, which facilitates matching calculations to fixed order computations performed using spinor-…
Recently we introduced the chirality-flow formalism, a method which builds on the spinor-helicity formalism and is inspired by the color-flow idea in QCD. With this formalism, Feynman rules and diagrams are simplified to the extent that it is often possible to immediately, by hand, write down a helicity amplitude given a Feynman diagram. In this paper we show that the method can also speed up numerical evaluation of scattering amplitudes by considering $e^+ e^-$ going to $n$ photons in a MadGraph-based tree-level implementation. We find that the computation time is reduced by roughly a factor …
We calculate the spinor helicity amplitudes of anomalous $H\to ZZ \to 4\ell$ decay. After embedding these analytic formulas into the $\texttt{MCFM}$ package, we study the interference effects between the anomalous $gg\to H\to ZZ \to 4\ell$ process and the SM processes, which are indispensable in the Higgs off-shell region. Subsequently, the constraints on the anomalous couplings are estimated using LHC experimental data.
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