Andreas Brandhuber, Gabriele Travaglini, Massimiliano Vincon
We consider the two-loop four-point amplitude in N=2 super QED, and show that there exists an approximate recursive structure similar to that captured by the ABDK/BDS ansatz for MHV amplitudes in N=4 super Yang-Mills. Furthermore, we present a simple relation between the box coefficients of one-loop photon MHV amplitudes in (super) QED, and sums of box coefficients of one-loop MHV amplitudes in (super) Yang-Mills.
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We describe the origins of recurrence relations between field theory amplitudes in terms of the construction of Feynman diagrams. In application we derive recurrence relations for the amplitudes of QED which hold to all loop orders and for all combinations of external particles. These results may also be derived from the Schwinger-Dyson equations.
The amplitudes of purely photonic and photon{2-fermion processes of non- commutative QED (NCQED) are derived for different helicity configurations of photons. The basic ingredient is the NCQED counterpart of Yang-Mills recursion relations by means of Berends and Giele. The explicit solutions of recursion relations for NCQED photonic processes with special helicity configurations are presented.
This is a personal recollection of several results related to the study of the high energy limit of scattering amplitudes in gravitational theories. They would not have been possible without the encouragement and constant support from Lev Nikolaevich Lipatov.
The exact $q\bar{q}$ Bethe-Salpeter bound state amplitude is investigated in the space of relative energy $E$ for fixed value of relative position. By means of approximate analysis it is shown to possess singularities in $E$ whenever one of the quarks reaches the energy threshold for the creation of certain number of Schwinger bosons. These results are afterwards confirmed by numerical plots.
An update is given on our long-term effort to perform a three-loop check on the Affleck-Alvarez-Manton/Lebedev-Ritus exponentiation conjecture for the imaginary part of the Euler-Heisenberg Lagrangian, using 1+1 dimensional QED as a toy model. After reviewing the history and significance of the conjecture, we present trigonometric integral representations for the single electron loop contributions to the three-loop Lagrangian, and develop a symmetry-based method for the calculation of their weak-field expansion coefficients.
In recent work, Gies and Karbstein have discovered that the two-loop Euler-Heisenberg Lagrangians for scalar and spinor QED have non-vanishing reducible contributions in addition to the well-studied irreducible ones. This invalidates previous applications of those Lagrangians to the computation of the two-loop $N$-photon amplitudes in the low energy limit. Here we compute the corrections to those amplitudes due to the reducible contributions.
We derive an explicit formula for the low energy limits of the one-loop, on-shell, massive N-photon amplitudes, for arbitrary N and all helicity assignments, in scalar and spinor QED. The two-loop corrections to the same amplitudes are obtained for up to the ten point case. All photon amplitudes with an odd number of `+' helicities are shown to vanish in this limit to all loop orders.
We investigate generic properties of one-loop amplitudes in unordered gauge theories in four dimensions. For such theories the organisation of amplitudes in manifestly crossing symmetric expressions poses restrictions on their structure and results in remarkable cancellations. We show that one-loop multi-photon amplitudes in QED with at least eight external photons are given only by scalar box integral functions. This QED `no-triangle' property is true for all helicity configurations and has similarities to the `no-triangle' property found in the case of maximal N = 8 supergravity. Results are…
It is explained how first-quantized worldline path integrals can be used as an efficient alternative to Feynman diagrams in the calculation of QED amplitudes and effective actions. The examples include the one-loop photon splitting amplitude, the two-loop Euler-Heisenberg Lagrangian, and the one-loop axial vacuum polarization tensor in a general constant electromagnetic field.
A generating functional for infrared-safe QED amplitudes
该研究受AFS构造启发，引入Faddeev-Kulish修饰构建有质量QED的红外有限散射振幅生成泛函，树阶下可约化为带修饰边界条件的AFS路径积分。
Helicity amplitudes in massless QED to higher orders in the dimensional regulator
本文通过四维张量分解解析计算无质量QED中单圈和双圈螺旋度振幅，提出高效算法组织积分族，并研究其奇异结构，结果以广义多对数表示至权重六。
Strong background fields require a non-perturbative treatment, which is afforded in QED by the Furry expansion of scattering amplitudes. It has been conjectured that this expansion breaks down for sufficiently strong fields, based on the asymptotic growth of loop corrections with increasing "quantum nonlinearity", essentially the product of field strength and particle energy. However, calculations to date have assumed that the background is constant. We show here, using general plane waves of finite duration, that observables at high quantum nonlinearity scale differently depending on whether …
Recently it has been shown that the vacuum state in QED is infinitely degenerate. Moreover a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computations of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construc…
We apply the worldline formalism to amplitudes in scalar quantum electrodynamics (QED) involving open scalar lines, with an emphasis on their non-perturbative gauge dependence. At the tree-level, we study the scalar propagator interacting with any number of photons in configuration space as well as in momentum space. At one-loop we rederive, in an efficient way, the off-shell vertex in an arbitrary dimension and any covariant gauge. Generalizing the Landau-Khalatnikov-Fradkin transformation (LKFT) for the non-perturbative propagator, we find simple non-perturbative transformation rules for arb…
Light-by-light scattering at three loops in massless QCD and QED: amplitudes and cross sections
本文通过洛伦兹张量分解计算了三重环无质量QCD和QED的轻子-轻子散射振幅，结果简洁且可高效数值计算，并用于预测双光子系统的横截面，与ATLAS实验数据一致。
The finding of an explicit effective action describing amplitude of photon triple splitting is investigated through Krein space quantization and quantum metric fluctuation. Using the derivative expansion method, the one-loop radiative corrections from Lorentz- and CPT- violating extended Krein QED shows that it is not possible to find a nonzero effective action.
Scattering amplitudes in dimensionless quadratic gravity coupled to QED
研究了在无标度二次量子引力与QED耦合下的超普朗克能2→2散射，推导了多种树级过程的非极化平方矩阵元表达式，并验证了结果与引力规范固定参数无关。
In this talk, I will summarize the present state of a long-term effort to obtain information on the high-order asymptotic behaviour of the QED perturbation series through the effective action. Starting with the constant-field case, I will discuss the Euler-Heisenberg Lagrangian in various dimensions, and up to the three-loop level. This Lagrangian holds the information on the N-photon amplitudes in the low-energy limit, and combining it with spinor helicity methods explicit all-N results can be obtained at the one-loop and, for the "all + " amplitudes, also at the two-loop level. For the imagi…
Three-loop helicity amplitudes of four-lepton scattering in QED
本文研究了QED中四轻子散射的三环极性振幅，通过优化策略计算并表达为广义对数函数，推导了三种散射过程的振幅。
We construct the de Sitter QED in Coulomb gauge assuming that the quantum modes are prepared by a global apparatus which is able to determine a stable and invariant vacuum state, independent on the local coordinates. Then we proceed in traditional manner postulating the appropriate equal-time commutators and anti-commutators of the interacting fields and deriving the perturbation expansion of the scattering operator. In this approach the $in -out$ transitions amplitudes, measured by the same global apparatus, can be calculated exactly by using the reduction formalism and the perturbation proce…
Instability in the scalar channel of the fermion-antifermion scattering amplitude in massless QED_3 for number of flavours less than the critical value 128/3π^2 is demonstrated. The anomalous dimensions of gauge-invariant composite operators are determined to O(1/N). Exponentiation of the O(1/N) infrared logarithm is explicitly demonstrated by evaluating the contribution of the ladder diagrams.
Subleading soft dressings for QED scattering states
研究QED散射过程中软发射的次主导软修饰，明确展示软因子对硬粒子动量和总角动量的依赖，证明软修饰消除红外发散，使硬幅度等效于对应的Fock基幅度的有限部分。
Two-loop massive QCD and QED helicity amplitudes for light-by-light scattering
本文提出了含质量费米子的光子-光子散射过程的双圈QCD和QED螺旋度振幅解析表达式，通过优化主积分表示，实现了快速稳定的数值计算，适用于唯象应用。
We study the application of BCFW recursion relations to the QED processes $0\to e^- e^+ n γ$. Based on 6-point amplitudes (both MHVA and NMHVA) computed from Feynman diagrams in the Berends-Giele gauge, we conduct a comprehensive study on all different shifts. Then we propose a new shift (LLYZ shift) which can lead to the full amplitudes for these processes and can have some realistic computation advantages. We compare the number of terms and the independent amplitudes of this novel shift with a few typical shifts.
Scattering Amplitudes and Resonant Processes in QED with Chiral Chemical Potential and Chiral Magnetic Conductivity
分析了恒定手征化学势和手征磁导率介质中的QED散射振幅，发现1→2、2→2和2→3过程中的共振行为，计算了决定准稳态费米子和光子态宽度的1→2过程速率，阐明了共振起源、出现条件及正则化原理。
Lepton-Pair Scattering With an Off-Shell and an On-Shell Photon at Two Loops in Massless QED
本研究计算了无质量QED中含一个离壳和一个在壳光子的轻子对散射两圈振幅，采用多尺度QCD振幅计算的现代特殊函数基与有限域数值重构方法，完善了MUonE实验所需的电子-μ子散射N3LO预测振幅要素。
In recent years, the Euler-Heisenberg Lagrangian has been shown to be a useful tool for the analysis of the asymptotic growth of the N-photon amplitudes at large N. Moreover, certain results and conjectures on its imaginary part allow one, using Borel analysis, to make predictions for those amplitudes at large loop orders. Extending work by G.V. Dunne and one of the authors to the three-loop level, but in the simpler context of 1+1 dimensional QED, we calculate the corresponding Euler-Heisenberg Lagrangian, analyse its weak field expansion, and study the congruence with predictions obtained fr…
We obtain information on the QED photon amplitudes at high orders in perturbation theory starting from known results on the QED effective Lagrangian in a constant electric field. A closed-form all-order result for the weak field limit of the imaginary part of this Lagrangian has been given years ago by Affleck, Alvarez and Manton (for scalar QED) and by Lebedev and Ritus (for spinor QED). We discuss the evidence for its correctness, and conjecture an analogous formula for the case of a self-dual field. From this extension we then obtain, using Borel analysis, the leading asymptotic growth for …
The Constructive Method for Massive Particles in QED
针对QED中三点振幅与含质量外粒子胶合涉及无质量内光子的构造性方法难题，采用旧式微扰论的在壳方法，证明正确识别留数即可推导费曼振幅，为高效计算振幅提供新路径。
Massive two-flavor QED_2 is known to have many similarities to the two-flavor QCD_4. Here we compare the π-πscattering amplitudes (actually an analog process in QED_2) of the two theories. The QED_2 amplitude is computed from the bosonized version of the model while the QCD_4 amplitude is computed from an effective low energy chiral Lagrangian. A number of interesting features are noted. For example, the contribution of the two-dimensional Wess-Zumino-Witten (WZW) term in QED_2 is structurally identical to the vector meson exchange contribution in QCD_4. Also it is shown that the QED_2 amplitu…
In this note, we present a new method to investigate the positronium states in ${\rm{QED_3}}$. According to the $\rm{K\ddot{a}ll\acute{e}n-Lehmann}$ spectral representation, the energy eigenvalues of bound states are poles of the correlation function. Using the chain approximation, we obtain the energy eigenvalues of the vector positronium states by taking into account the single-valued branches of multi-valued function. Using the same method, we also find the electron physical mass at some single-valued branch of multi-value function. Our results are agreement with the known ones.
The QED four-photon amplitudes off-shell: part 2
本文使用worldline formalism首次完整计算标量和自旋QED四光子离壳振幅，在低能极限下积分掉两个光子腿，构建两圈真空极化张量并应用于Delbrück散射。
It is shown that the ground state or vacuum to the lattice Euclidean quantum gravity is significantly different from the ground states to the well-known vacua in QED, QCD, et cetera. In the case of the lattice Euclidean quantum gravity, the long-wavelength scale vacuum structure is similar to that in QED, moreover the quantum fluctuations to gravity are very reduced in comparison with the situation in QED. But the small scale (of the order of the lattice scale) vacuum structure to gravity is significantly different from that to the long-wavelength scales: the fluctuation values of geometrical …
We construct Faddeev-Kulish states in QED and perturbative quantum gravity to subleading order in the soft momentum expansion and to first order in the coupling constant, using the charge conservation formula of asymptotic symmetries associated with the tree-level subleading soft theorems. We demonstrate that the emission and absorption of soft photons/gravitons in dressed amplitudes vanish. The fact that no additional soft radiation may be added to a dressed amplitude supports the claim that, in the dressed state formalism, the soft and hard sectors of scattering processes are correlated. We …
A survey is given of applications of the ``string-inspired'' worldline formalism to the computation of amplitudes and effective actions in QED.
The leading long-distance 1-loop quantum corrections to the Coulomb potential are derived for scalar QED and their gauge-independence is explicitly checked. The potential is obtained from the direct calculation of the 2-particle scattering amplitude, taking into account all relevant 1-loop diagrams. Our investigation should be regarded as a first step towards the same programme for effective Quantum Gravity. In particular, with our calculation in the framework of scalar QED, we are able to demonstrate the incompleteness of some previous studies concerning the Quantum Gravity counterpart.
Vacuum birefringence is governed by the amplitude for a photon to flip helicity or polarisation state in an external field. Here we calculate the flip and non-flip amplitudes in arbitrary plane wave backgrounds, along with the induced spacetime-dependent refractive indices of the vacuum. We compare the behaviour of the amplitudes in the low energy and high energy regimes, and analyse the impact of pulse shape and energy. We also provide the first lightfront-QED derivation of the coefficients in the Heisenberg-Euler effective action.
We compare and contrast the implications of gauge invariance for the structure of scattering amplitudes in QED and QCD. We derive the most general analogue for QCD of the famous QED rule that the scattering amplitude must be invariant if the polarization vectors $ε^{μ_j}(k_j)$ of any number of photons undergo the replacement $ ε^{μ_j}(k_j) \rightarrow ε^{μ_j}(k_j) + c \, k_j^{μ_j} $, where $c$ is an arbitrary constant.
The QED four -- photon amplitudes off-shell: part 1
本文使用世界线形式化计算QED四光子离壳振幅，统一标量与自旋圈计算，避免Feynman图分解，实现显式横向性和UV有限性，并自然投影到张量基上。
Significant progress has been made in the past year in developing new `MHV' techniques for calculating multiparticle scattering amplitudes in Yang-Mills gauge theories. Most of the work so far has focussed on applications to Quantum Chromodynamics, both at tree and one-loop level. We show how such techniques can also be applied to abelian theories such as QED, by studying the simplest tree-level multiparticle process, e^+e^- to n γ. We compare explicit results for up to n=5 photons using both the Cachazo, Svrcek and Witten `MHV rules' and the related Britto-Cachazo-Feng `recursion relation' ap…
Among 12672 Feynman diagrams contributing to the electron anomalous magnetic moment at the tenth order, 6354 are the diagrams having no lepton loops, i.e., those of quenched type. Because the renormalization structure of these diagrams is very complicated, some automation scheme is inevitable to calculate them. We developed an algorithm to write down FORTRAN programs for numerical evaluation of these diagrams, where the necessary counterterms to subtract out ultraviolet subdivergence are generated according to Zimmermann's forest formula. Thus far we have evaluated crudely integrals of 2232 te…
Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED
本研究计算无质量QED中含离壳、在壳光子的轻子对散射两圈螺旋度振幅，用多重对数基表示主积分并通过有限域数值重构解析形式，补全MUonE实验所需的eμ散射N³LO预言的振幅组分。
We generalize the Mellin representation for a generic co-dimension flat defect CFT. We study the analytic structure of the Mellin amplitudes. We also compute Witten diagrams for a generic co-dimension flat defect CFT.
In a recent paper, Witten proposed a surprising connection between perturbative gauge theory and a certain topological model in twistor space. In particular, he showed that gluon amplitudes are localized on holomorphic curves. In this note we present some preliminary considerations on the possibility of having a similar localization for gravity amplitudes.
Unitarization of electron scattering with an external potential at NLO in QED
本文基于Faddeev-Kulish形式计算了QED中电子被库仑势散射的无红外发散单圈振幅，推导出无红外发散的分波振幅，并用三种方法进行幺正化以寻找束缚态极点。
In this short note, we propose an algorithm based on the expansions of amplitudes, the dimensional reduction technic and the differential operators, to calculate the tree level scalar-graviton amplitudes with two massive scalars, as well as the tree level scalar-photon-graviton amplitudes with two massive scalars and one photon. While applying the unitarity method, these amplitudes are necessary inputs for the calculation of post-Newtonian and post-Minkowskian expansions in general relativity for two massive charged objects interact with gravity and electromagnetic field.
We investigate the possibility of Lorentz and CPT violations in the photon sector, of the Chern-Simons form, be induced by radiative corrections arising from the Lorentz and CPT non-invariant fermionic sector of an extended version of QED. By analyzing the modified vacuum polarization tensor, three contributions are considered: two of them can be identified with well known amplitudes; the (identical) QED vacuum polarization tensor and the (closely related) $AVV$ triangular amplitude. These amplitudes are evaluated in their most general form (to include in our discussion automatically the quest…
Two new formulas which express n-graviton MHV tree amplitudes in terms of sums of squares of n-gluon amplitudes are discussed. The first formula is derived from recursion relations. The second formula, simpler because it involves fewer permutations, is obtained from the variant of the Berends, Giele, Kuijf formula given in Arxiv:0707.1035.
We present the analytic evaluation of the two-loop corrections to the amplitude for the scattering of four fermions in Quantum Electrodynamics, $f^- + f^+ + F^- + F^+ \to 0$, with $f$ and $F$ representing a massless and a massive lepton, respectively. Dimensional regularization is employed to evaluate the loop integrals. Ultraviolet divergences are removed by renormalizing the coupling constant in the ${\overline{\text{MS}}}$-scheme, and the lepton mass as well as the external fields in the on-shell scheme. The analytic result for the renormalized amplitude is expressed as Laurent series aroun…
On (Scalar QED) Gravitational Positivity Bounds
本研究探讨引力存在下的正定性界，回顾树级结果并估计负贡献大小，提出单圈Regge振幅形式，证明其正定性界有限且不导致截断尺度参数化小界，与求和规则研究一致。
We show how the maximally helicity violating (MHV) scattering amplitudes for gravitons can be related to current correlators and vertex operators in twistor space. This is similar to what happens in Yang-Mills theory and raises the possibility of a direct twistor-string-like construction for ${\cal N}=8$ supergravity.
We compute the simplest one-loop planar amplitudes in Higgsed ABJM theory at a generic point of the moduli space. We explicitly check that they can be expressed in terms of integrals which are invariant under dual conformal symmetry involving masses, in a similar fashion as in N=4 SYM.
Note on hidden zeros and expansions of tree-level amplitudes
本研究利用树图振幅的普适展开推导并解释了杨-米尔斯等多种理论树图振幅的隐藏零点，分析了引力等无序振幅零点引发的传播子发散问题并展示了消除机制。
There exist certain intrinsic relations between the ultraviolet divergent graphs and the convergent ones at the same loop order in renormalizable quantum field theories. Whereupon we may establish a new method, the intrinsic regularization method, to regulate those divergent graphs. In this note, we present a proposal, the inserter proposal, to the method. The $ϕ^4$ theory and QED at the one loop order are dealt with in some detail. Inserters in the standard model are given. Some applications to SUSY-models are also made at the one loop order.
A new class of twistor string theories were recently constructed by Abou-Zeid, Hull and Mason. The most interesting one of these theories has the particle content of N=8 supergravity. Arguments are given for the vanishing of the (+ - -)-helicity amplitude for gravitons which suggest that this theory describes a chiral N=8 supergravity rather than Einstein supergravity.
Low-energy limit of N-photon amplitudes in a constant field
该研究利用世界线形式体系，将低能极限下N光子振幅的已有结果推广到任意恒定场情形，得到了标量和旋量QED中该类振幅的紧凑解析表示。
We extend a previously developed approach to relate thermal currents in the high temperature regime and classical limits of amplitudes. We consider the bi-adjoint scalar theory, which has the basic structure of a cubic theory and which is related to QCD and gravity through the double copy. In addition, we consider a generalization of scalar QED to model classical spin, where massive scalars are complex higher-spin fields. We derive Vlasov-type kinetic equations for bi-adjoint scalars and study their iterative solutions, while for QED we use well-known kinetic equations. In both cases we find c…
Master formulas for photon amplitudes in a combined constant and plane-wave background field
本文利用世界线形式，推导了组合恒定与平面波场“平行”特例背景下标量与旋量QED $N$光子振幅的紧凑主公式，进一步推进了已有研究。
For scattering amplitudes in strong background fields, it is -- at least in principle -- possible to perturbatively expand the background to obtain higher-point vacuum amplitudes. In the case of self-dual plane wave backgrounds we consider this expansion for two-point, one-loop amplitudes in pure Yang-Mills, QED and QCD. This enables us to obtain multicollinear limits of 1-loop vacuum amplitudes; the resulting helicity configurations are surprisingly restricted, with only the all-plus helicity amplitude surviving. These results are shown to be consistent with well-known vacuum amplitudes. We a…
In this paper, we extend our earlier one loop analysis to two loops and give a simple diagrammatic description for the retarded Greens functions at finite temperature, in terms of forward scattering amplitudes of on-shell thermal particles. We present a simple discussion, which can be easily generalized to any field theory, of the temperature dependent parts of the retarded two and three point functions in scalar field theory and QED. As an application of our result at two loops, we show how the infrared singularities in the thermal part of the retarded photon self-energy, cancel in QED_2 in t…
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