Feynman Amplitudes Research Articles

Motivic Galois coaction and one-loop Feynman graphs

Following the work of Brown, we can canonically associate a family of motivic periods -- called the motivic Feynman amplitude -- to any convergent Feynman integral, viewed as a function of the kinematic variables. The motivic Galois theory of motivic Feynman amplitudes provides an organizing principle, as well as strong constraints, on the space of amplitudes in general, via Brown's small graphs principle. This serves as motivation for explicitly computing the motivic Galois action, or, dually, the coaction of the Hopf algebra of functions on the motivic Galois group. In this paper, we study the motivic Galois coaction on the motivic Feynman amplitudes associated to one-loop Feynman graphs. We study the associated variations of mixed Hodge structures, and provide an explicit formula for the coaction on the four-edge cycle graph -- the box graph -- with non-vanishing generic kinematics, which leads to a formula for all one-loop graphs with non-vanishing generic kinematics in four-dimensional space-time. We also show how one computes the coaction in some degenerate configurations -- when defining the motive of the graph requires blowing up the underlying family of varieties -- on the example of the three-edge cycle graph.

Renormalization of Feynman amplitudes on manifolds by spectral zeta regularization and blow-ups

Our goal in this paper is to present a generalization of the spectral zeta regularization for general Feynman amplitudes on Riemannian manifolds. Our method uses complex powers of elliptic operators but involves several complex parameters in the spirit of analytic renormalization by Speer, to build mathematical foundations for the renormalization of perturbative interacting quantum field theories. Our main result shows that spectrally regularized Feynman amplitudes admit analytic continuation as meromorphic germs with linear poles in the sense of the works of Guo–Paycha and the second author. We also give an explicit determination of the affine hyperplanes supporting the poles. Our proof relies on suitable resolution of singularities of products of heat kernels to make them smooth. As an application of the analytic continuation result, we use a universal projection from meromorphic germs with linear poles on holomorphic germs to construct renormalization maps which subtract singularities of Feynman amplitudes of Euclidean fields. Our renormalization maps are shown to satisfy consistency conditions previously introduced in the work of Nikolov–Todorov–Stora in the case of flat space-times.

Long-range multi-scalar models at three loops

We compute the three-loop beta functions of long-range multi-scalar models with general quartic interactions. The long-range nature of the models is encoded in a kinetic term with a Laplacian to the power 0 < ζ < 1, rendering the computation of Feynman diagrams much harder than in the usual short-range case (ζ = 1). As a consequence, previous results stopped at two loops, while seven-loop results are available for short-range models. We push the renormalization group analysis to three loops, in an ϵ = 4ζ − d expansion at fixed dimension d < 4, extensively using the Mellin–Barnes representation of Feynman amplitudes in the Schwinger parametrization. We then specialize the beta functions to various models with different symmetry groups: O(N), , and O(N) × O(M). For such models, we compute the fixed points and critical exponents.

The two-photon production of W-bosons pair at colliders at the leading order and the one-loop correction

In this paper the process of two-photon production of W-bosons pair was studied at hadron and electron-positron colliders in the leading and the next-to-leading order approximation. The process can be used for investigation precision parameters of Standard Model (tri-linear and quartic couplings) and search effects of «new physics» (for example anomalous couplings). This process of production W-bosons has a small background therefore it can be studied at colliders with high precision. The program LoopTools was used to numerical calculate one-loop integrals, the program Mathematica and the packages FeynArts and FeynCalc were used to obtain the Feynman diagrams and amplitudes, the square of module of matrix element. The total and differential cross sections for hadron and electron-positron colliders were calculated using the approximation of equivalent of photons (Weizsäcker – Williams approximation). In this approximation, the scattering particles are detected in the forward-detectors at small angles and the production particles are detected in the main detector. The Monte-Carlo generator TwoPhotonGen written in C++ was used for simulation. It is shown, the total and differential cross section are increasing with increasing of the total energy of colliding particles.

Di-Higgs production in SUSY models at the LHC

We study the modification to di-Higgs production via gluon fusion within the context of the minimal supersymmetric standard model and the next-to-minimal supersymmetric standard model in the parameter space allowed by current experimental and theoretical constraints, and also relevant to the Large Hadron Collider (LHC) experiments in the near future. The calculation is based on the analytical expression of the leading-order Feynman amplitudes (which includes both quark and squark loops). We separate the di-Higgs production cross section into resonant, non-resonant, and interference parts, in order to better understand the mechanisms that are responsible for the modification to di-Higgs production rate in different regions of the allowed parameter space. We also investigate the sensitivity of high-luminosity LHC (HL-LHC) to the di-Higgs production in these low-energy supersymmetry models. Furthermore, we examine the complementarity between di-Higgs searches and direct searches for BSM particles and precision Higgs couplings measurements at the HL-LHC. We found that the di-Higgs production cross section can be enhanced significantly through resonant production. In the region where the resonant production cross section is small, di-Higgs production receives a moderate enhancement due to the modifications in the Higgs couplings. In addition, there is a strong correlation between di-Higgs production and single Higgs production, and di-Higgs production is not as sensitive as single Higgs production at the HL-LHC.

Feynman amplitudes on moduli spaces of graphs

This article introduces moduli spaces of coloured graphs on which Feynman amplitudes can be viewed as “discrete” volume densities. The basic idea behind this construction is that these moduli spaces decompose into disjoint unions of open cells on which parametric Feynman integrals are defined in a naturalway. Renormalisation of an amplitude translates then into the task of assigning to every cell a finite volume such that boundary relations between neighboring cells are respected. It is shown that this can be organized systematically using a type of Borel–Serre compactification of these moduli spaces. The key point is that in each compactified cell the newly added boundary components have a combinatorial description that resembles the forest structure of subdivergences of the corresponding Feynman diagram.

Testing binary dynamics in gravity at the sixth post-Newtonian level

We calculate the motion of binary mass systems in gravity up to the sixth post–Newtonian order to the GN3 terms ab initio using momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates. For these contributions we construct a canonical transformation to isotropic and to EOB coordinates at 5PN and agree with the results in the literature [1,2] by Bern et al. and Damour. At 6PN we compare to the Hamiltonians in isotropic coordinates either given in [1] or resulting from the scattering angle. We find a canonical transformation from our Hamiltonian in harmonic coordinates to [1], but not to [2]. This implies that we also agree on all observables with [1] to the sixth post–Newtonian order to GN3.

QED and QCD helicity amplitudes in parton-shower gauge

We introduce photon and gluon propagators in which the scalar polarization component is subtracted systematically by making use of the BRST invariance of the off-shell vector boson created from physical on-shell states. The propagator has the light-cone gauge form, where the spacial component of the gauge vector points along the negative of the off-shell vector boson momentum. We call the gauge as parton-shower gauge, since in collinear configurations the absolute value squared of each Feynman amplitude reproduces all the singular behaviors of the corresponding parton shower in this gauge. We introduce new HELAS codes that can be used to calculate the tree-level helicity amplitudes of arbitrary QED and QCD processes by using MadGraph. The absence of subtle gauge cancellation among Feynman amplitudes allows numerical codes to evaluate singular behaviors accurately, and helps us gaining physical insights on interference patterns.

Goldstone equivalence and high energy electroweak physics

The transition between the broken and unbroken phases of massive gauge theories, namely the rearrangement of longitudinal and Goldstone degrees of freedom that occurs at high energy, is not manifestly smooth in the standard formalism. The lack of smoothness concretely shows up as an anomalous growth with energy of the longitudinal polarization vectors, as they emerge in Feynman rules both for real on-shell external particles and for virtual particles from the decomposition of the gauge field propagator. This makes the characterization of Feynman amplitudes in the high-energy limit quite cumbersome, which in turn poses peculiar challenges in the study of Electroweak processes at energies much above the Electroweak scale. We develop a Lorentz-covariant formalism where polarization vectors are well-behaved and, consequently, energy power-counting is manifest at the level of individual Feynman diagrams. This allows us to prove the validity of the Effective $W$ Approximation and, more generally, the factorization of collinear emissions and to compute the corresponding splitting functions at the tree-level order. Our formalism applies at all orders in perturbation theory, for arbitrary gauge groups and generic linear gauge-fixing functionals. It can be used to simplify Standard Model loop calculations by performing the high-energy expansion directly on the Feynman diagrams. This is illustrated by computing the radiative corrections to the decay of the top quark.

Fourth post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach

We calculate the motion of binary mass systems in gravity up to the fourth post–Newtonian order. We use momentum expansions within an effective field theory approach based on Feynman amplitudes in harmonic coordinates by applying dimensional regularization. We construct the canonical transformations to ADM coordinates and to effective one body theory (EOB) to compare with other approaches. We show that intermediate poles in the dimensional regularization parameter ε vanish in the observables and the classical theory is not renormalized. The results are illustrated for a series of observables for which we agree with the literature.

From little string free energies towards modular graph functions

We study the structure of the non-perturbative free energy of a one-parameter class of little string theories (LSTs) of A-type in the so-called unrefined limit. These theories are engineered by N M5-branes probing a transverse flat space. By analysing a number of examples, we observe a pattern which suggests to write the free energy in a fashion that resembles a decomposition into higher-point functions which can be presented in a graphical way reminiscent of sums of (effective) Feynman diagrams: to leading order in the instanton parameter of the LST, the N external states are given either by the fundamental building blocks of the theory with N = 1, or the function that governs the counting of BPS states of a single M5-brane coupling to one M2-brane on either side. These states are connected via an effective coupling function which encodes the details of the gauge algebra of the LST and which in its simplest (non-trivial) form is captured by the scalar Greens function on the torus. More complicated incarnations of this function show certain similarities with so-called modular graph functions, which have appeared in the study of Feynman amplitudes in string- and field theory. Finally, similar structures continue to exist at higher instanton orders, which, however, also contain contributions that can be understood as the action of (Hecke) operators on the leading instanton result.

P-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field

We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematic parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters. This result allows us to regularize the Ghoshal-Kawano amplitudes. The regularized amplitudes do not have ultraviolet divergencies. Due to the need for a certain symmetry, the theory works only for prime numbers which are congruent to 3 modulo 4. We also discuss the limit p→1 in the noncommutative effective field theory and in the Ghoshal-Kawano amplitudes. We show that in the case of four points, the limit p→1 of the regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p→1 of the noncommutative Gerasimov-Shatashvili Lagrangian.

Strong and electromagnetic amplitudes of the J/ψ decays into baryons and their relative phase

The Feynman amplitude for the decay of the J/ψ meson into baryon-antibaryon can be written as a sum of three sub-amplitudes: a purely strong, a purely electromagnetic and a mixed strong-electromagnetic. Assuming that the strong and mixed strong-electromagnetic sub-amplitudes have the same phase, the branching ratio of the decay contains an interference term that depends on the relative phase φ between strong and electromagnetic sub-amplitudes. In this work we calculate this phase, by using an effective strong Lagrangian density and considering, as final states, pairs of baryons, BB‾, belonging to the spin-1/2 SU(3) octet. Moreover, we obtain the purely strong, purely electromagnetic and mixed strong-electromagnetic contributions to the total branching ratio and hence the moduli of the corresponding sub-amplitudes, for each pair of baryons. Of particular interest is the mixed strong-electromagnetic contribution that, not only is determined for the first time, but it is proven to be crucial, in the framework of our model, for the correct description of the decay mechanism. Finally we use the purely electromagnetic branching ratio to calculate the Born non-resonant cross section of the annihilation processes e+e−→BB‾ at the J/ψ mass. By taking advantage from all available data, we obtain the relative phase between strong and electromagnetic sub-amplitudes: φ=(73±8)∘.

Structure functions at small x from worldlines: Unpolarized distributions

The world-line representation of quantum field theory is a powerful framework for the computation of perturbative multi-leg Feynman amplitudes. In particular, in gauge theories, it provides an efficient way, via point particle Grassmann functional integrals, to compute spinor and color traces in these amplitudes. Further, semi-classical approximations to quantum mechanical world-line trajectories provide useful intuition in a wide range of dynamical problems. We develop here the world-line approach to compute deeply inelastic structure functions in the small x Regge limit of QCD. In particular, in a shockwave approximation valid in this limit, we show how one recovers the well-known dipole model for unpolarized structure functions. In a follow-up work, we will discuss the world-line computation of polarized structure functions at small x.

Off-shell Yang-Mills amplitude in the Cachazo-He-Yuan formalism

M\"obius invariance is used to construct gluon tree amplitudes in the Cachazo, He, and Yuan (CHY) formalism. If it is equally effective in steering the construction of off-shell tree amplitudes, then the S-matrix CHY theory can be used to replace the Lagrangian Yang-Mills theory. In the process of investigating this possibility, we find that the CHY formula can indeed be modified to obtain a M\"obius invariant off-shell amplitude, but unfortunately this modified amplitude $M_P$ is not the Yang-Mills amplitude because it lacks gauge invariance. A complementary amplitude $M_Q$ must be added to restore gauge invariance, but its construction relies on the Lagrangian and not M\"obius invariance. Although neither $M_P$ nor $M_Q$ is fully gauge invariant, both are partially gauge invariant in a sense to be explained. This partial gauge invariance turns out to be very useful for checking calculations. A Feynman amplitude so split into the sum of $M_P$ and $M_Q$ also contains fewer terms.

Removing infrared divergences from two-loop integrals

Feynman amplitudes at higher orders in perturbation theory generically have complex singular structures. Notwithstanding the emergence of many powerful new methods, the presence of infrared divergences poses significant challenges for their evaluation. In this article, we develop a systematic method for the removal of the infrared singularities, by adding appropriate counterterms that approximate and cancel divergent limits point-by-point at the level of the integrand. We provide a proof of concept for our method by applying it to master-integrals that are found in scattering amplitudes for representative 2→2 scattering processes of massless particles. We demonstrate that, after the introduction of counterterms, the remainder is finite in four dimensions. In addition, we find in these cases that the complete singular dependence of the integrals can be obtained simply by analytically integrating the counterterms. Finally, we observe that our subtraction method can be also useful in order to extract in a simple way the asymptotic behavior of Feynman amplitudes in the limit of small mass parameters.

The exchange graph and variations of the ratio of the two Symanzik polynomials

Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively. In a previous paper with Bloch, Burgos and Fresán, we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory. Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a complete description of the connected components of this graph, and use this to prove our result on boundedness of the variations.

On p-adic string amplitudes in the limit p approaches to one

In this article we discuss the limit p approaches to one of tree-level p-adic open string amplitudes and its connections with the topological zeta functions. There is empirical evidence that p-adic strings are related to the ordinary strings in the p → 1 limit. Previously, we established that p-adic Koba-Nielsen string amplitudes are finite sums of multivariate Igusa’s local zeta functions, consequently, they are convergent integrals that admit meromorphic continuations as rational functions. The meromorphic continuation of local zeta functions has been used for several authors to regularize parametric Feynman amplitudes in field and string theories. Denef and Loeser established that the limit p → 1 of a Igusa’s local zeta function gives rise to an object called topological zeta function. By using Denef-Loeser’s theory of topological zeta functions, we show that limit p → 1 of tree-level p-adic string amplitudes give rise to certain amplitudes, that we have named Denef-Loeser string amplitudes. Gerasimov and Shatashvili showed that in limit p → 1 the well-known non-local effective Lagrangian (reproducing the tree-level p-adic string amplitudes) gives rise to a simple Lagrangian with a logarithmic potential. We show that the Feynman amplitudes of this last Lagrangian are precisely the amplitudes introduced here. Finally, the amplitudes for four and five points are computed explicitly.

Supercurrent anomaly and gauge invariance in the N=1 supersymmetric Yang-Mills theory

We analyse Feynman diagram calculational issues related to the quantum breaking of supercurrent conservation in a supersymmetric non-abelian Yang-Mills theory. For the sake of simplicity, we take a zero mass gauge field multiplet interacting with a massless Majorana spin-$1/2$ field in the adjoint representation of $SU(2)$. We shed light on a long-standing controversy regarding the perturbative evaluation of the supercurrent anomaly in connection with gauge and superconformal symmetry in different frameworks. We find that only superconformal symmetry is unambiguously broken using an invariant four dimensional regularization and compare with the triangle AVV anomaly. Subtleties related to momentum routing invariance in the loops of diagrams and Clifford algebra evaluation inside divergent integrals are also discussed in connection with finite and undetermined quantities in Feynman amplitudes.

Square root of gerbe holonomy and invariants of time-reversal-symmetric topological insulators

The Feynman amplitudes with the two-dimensional Wess-Zumino action functional have a geometric interpretation as bundle gerbe holonomy. We present details of the construction of a distinguished square root of such holonomy and of a related 3d-index and briefly recall the application of those to the building of topological invariants for time-reversal-symmetric two- and three-dimensional crystals, both static and periodically forced.