Feynman Gauge Research Articles

Automatic generation of helicity amplitudes in the Feynman-diagram gauge

We develop a method to calculate helicity amplitudes of an arbitrary tree-level process in the Feynman-diagram (FD) gauge for an arbitrary gauge model with adraph5_a@. We start from the ’t Hooft–Feynman gauge Lagrangian in eynules and generate scattering amplitudes by identifying the Goldstone boson as the 5th component of each weak boson. All the vertices of the 5-component weak bosons are then created automatically by assembling the relevant weak boson and Goldstone boson vertices in the Feynman gauge. The 5-component weak boson vertices are then connected by the 5×5 matrix propagator in the FD gauge. As a demonstration of the method we calculate the cross section for the process μ−μ+→νμν¯μtt¯H with complex top Yukawa coupling, which can be obtained by adding a gauge invariant dimension-6 operator to the Standard Model (SM) Lagrangian. The FD gauge and the unitary (U) gauge amplitudes give exactly the same cross section, and subtle gauge theory cancellation among diagrams in the U gauge at high energies is absent in the FD gauge, as has been observed for various SM processes. In addition, we find that the total cross sections at high energies are dominated by a single, or a set of nonvanishing Feynman amplitudes with the higher dimensional vertices in the FD gauge. Published by the American Physical Society 2024

One-loop QCD amplitudes in the Feynman-diagram gauge

Scattering amplitudes for the massless QCD process, qq¯→q′q¯′, are calculated in the one-loop order in the Feynman-Diagram (FD) gauge, where gluons are quantized on the light cone with opposite direction of the three-momenta. We find non-decoupling of the Faddeev-Popov ghosts and nonconventional UV singularities in dimensional regularization. The known QCD amplitudes with asymptotic freedom are reproduced only after summing propagator and vertex corrections. By quantizing gluons in the Feynman gauge on the FD gauge background, we obtain the one-loop improved FD gauge amplitudes. Published by the American Physical Society 2024

The β-function of supersymmetric theories from vacuum supergraphs: A three-loop example

In this paper, we verify a method which allows to obtain the [Formula: see text]-function of supersymmetric theories regularized by higher covariant derivatives by calculating only specially modified vacuum supergraphs. With the help of this method for a general renormalizable [Formula: see text] supersymmetric gauge theory, a part of the three-loop [Formula: see text]-function depending on the Yukawa couplings is constructed in the general [Formula: see text]-gauge. The result is written in the form of an integral of double total derivatives with respect to the loop momenta. It is demonstrated that all gauge-dependent terms cancel each other in agreement with the general statements. The result in the Feynman gauge (found earlier) coincides with the one obtained by the standard technique which requires to calculate the sum of superdiagrams with two external gauge lines. This fact confirms the correctness of the considered method by a highly nontrivial multiloop calculation.

Color-kinematics relation from the Feynman diagram perspective

Feynman diagrams for gluon tree amplitudes are studied in the Feynman gauge and in any number of spacetime dimensions. The color-kinematics combinations $\mathrm{\ensuremath{\Delta}}={n}_{s}\ensuremath{-}{n}_{t}\ensuremath{-}{n}_{u}$ of numerators are explicitly calculated for $N=4$, 5, 6 gluons to see whether the color-kinematics relation $\mathrm{\ensuremath{\Delta}}=0$ is satisfied. This is a tedious task because of the presence of four-gluon vertices, and the large number of Feynman diagrams, numerators, and $\mathrm{\ensuremath{\Delta}}$ combinations involved, especially when $N=6$. For on-shell amplitudes, it is found that $\mathrm{\ensuremath{\Delta}}=0$ for $N=4$, but $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$ for $N=5$ and $N=6$ owing to the presence of the four-gluon vertex. However, a local generalized gauge transformation can bring about $\mathrm{\ensuremath{\Delta}}=0$ for $N=5$, but not for $N=6$. This raises the question whether gluon amplitudes satisfying the color-kinematics relation contain nonlocal interactions.

Lattice QCD Calculation of the Pion Mass Splitting.

We use the infinite volume reconstruction method to calculate the charged and neutral pion mass difference. The hadronic tensor is calculated using lattice QCD and then combined with an analytic photon propagator, and the mass splitting is calculated with exponentially suppressed finite-volume errors. The calculation is performed using six gauge ensembles generated with 2+1-flavor domain wall fermions, and five ensembles are at the physical pion mass. Both Feynman and Coulomb gauges are adopted in the calculation and agree well when the lattice spacing approaches zero. After performing the continuum extrapolation and examining the residual finite-volume effects, we obtain the pion mass splitting Δm_{π}=4.534(42)_{stat}(43)_{sys} MeV, which agrees well with experimental measurements.

Parton distribution function for topological charge at one loop

We present results for the gg-part of the one-loop corrections to the recently introduced “topological charge” GPD overset{sim }{F} (x, q2). In particular, we give expression for its evolution kernel. To enforce strict compliance with the gauge invariance requirements, we have used on-shell states for external gluons, and have obtained identical results both in Feynman and light-cone gauges. No “zero mode” δ(x) terms were found for the twist-4 gluon GPD overset{sim }{F} (x, q2).

Using Covariant Polarisation Sums in QCD

Covariant gauges lead to spurious, non-physical polarisation states of gauge bosons. In QED, the use of the Feynman gauge, sum _{lambda } varepsilon _mu ^{(lambda )}varepsilon _nu ^{(lambda )*} = -eta _{mu nu }, is justified by the Ward identity which ensures that the contributions of non-physical polarisation states cancel in physical observables. In contrast, the same replacement can be applied only to a single external gauge boson in squared amplitudes of non-abelian gauge theories like QCD. In general, the use of this replacement requires to include external Faddeev–Popov ghosts. We present a pedagogical derivation of these ghost contributions applying the optical theorem and the Cutkosky cutting rules. We find that the resulting cross terms mathcal {A}(c_1,bar{c}_1;ldots )mathcal {A}(bar{c}_1,c_1;ldots )^* between ghost amplitudes cannot be transformed into (-1)^{n/2}|mathcal {A}(c_1,bar{c}_1;ldots )|^2 in the case of more than two ghosts. Thus the Feynman rule stated in the literature holds only for two external ghosts, while it is in general incorrect.

One-loop structure of parton distribution for the gluon condensate and \u201czero modes\u201d

We present results for one-loop corrections to the recently introduced “gluon condensate” PDF F(x). In particular, we give expression for the gg-part of its evolution kernel. To enforce strict compliance with the gauge invariance requirements, we have used on-shell states for external gluons, and have obtained identical results both in Feynman and light-cone gauges. No “zero mode” δ(x) terms were found for the twist-4 gluon PDF F(x). However a q2δ(x) term was found for the ξ = 0 GPD F(x, q2) at nonzero momentum transfer q. Overall, our results do not agree with the original attempt of one-loop calculations of F(x) for gluon states, which sets alarm warning for calculations that use matrix elements with virtual external gluons and for lattice renormalization procedures based on their results.

Heavy particle non-decoupling in flavor-changing gravitational interactions

Abstract The flavor-changing gravitational process, d → s + graviton, is evaluated at the one-loop level in the standard electroweak theory with on-shell renormalization. The results that we present in the ’t Hooft–Feynman gauge are valid for on- and off-shell quarks and for all external and internal quark masses. We show that there exist non-decoupling effects of the internal heavy top quark in interactions with gravity. A naive argument taking account of the quark Yukawa coupling suggests that the amplitude of the process d → s + graviton in the large top quark mass limit would possibly acquire an enhancement factor $m_{t}^{2}/M_{W}^{2}$, where mt and MW are the top quark and the W-boson masses, respectively. In practice this leading enhancement is absent in the renormalized amplitude due to cancellation. Thus the non-decoupling of the internal top quark takes place at the ${\cal O}(1)$ level. The flavor-changing two- and three-point functions are shown to satisfy the Ward–Takahashi identity, which is used as a consistency check for the aforementioned cancellation of the ${\cal O}(m_{t}^{2}/M_{W}^{2})$ terms. Among the ${\cal O}(1)$ non-decoupling terms, we sort out those that can be regarded as due to the effective Lagrangian in which quark bilinear forms are coupled to the scalar curvature.

Quadratic gravity and restricted Weyl symmetry

In this paper, we investigate the relationship between quadratic gravity and a restricted Weyl symmetry where a gauge parameter [Formula: see text] of Weyl transformation satisfies a constraint [Formula: see text] in a curved spacetime. First, we briefly review a model with a restricted gauge symmetry on the basis of QED, where a [Formula: see text] gauge parameter [Formula: see text] obeys a similar constraint [Formula: see text] in a flat Minkowski spacetime, and explain that the restricted gauge symmetry removes one on-shell mode of gauge field, which together with the Feynman gauge leaves only two transverse polarizations as physical states. Next, it is shown that the restricted Weyl symmetry also eliminates one component of a dipole field in quadratic gravity around a flat Minkowski background, leaving only a single scalar state. Finally, we show that the restricted Weyl symmetry cannot remove any dynamical degrees of freedom in static background metrics by using the zero-energy theorem of quadratic gravity. This fact also holds for the Euclidean background metrics without imposing the static condition.

Renormalization in a Landau-to-Coulomb interpolating gauge in Yang–Mills theory

In a previous paper (Andrasi and Taylor (2020)), we studied the renormalization of a gauge interpolating between the Feynman and Coulomb gauges. The present paper extends the analysis to a gauge interpolating between the Landau and Coulomb gauges. This interpolating gauge is characterized by a parameter θ and the Coulomb gauge is obtained in the limit θ→0. We study the renormalization for all values of θ, and note some special features of it. The purpose of these interpolating gauges is to make the Coulomb gauge well-defined, in spite of the energy-divergences which occur in individual Feynman integrals for θ=0.

CP violation in neutral lepton transition dipole moment

The CP violation in the neutrino transition electromagnetic dipole moment is discussed in the context of the Standard Model with an arbitrary number of right-handed singlet neutrinos. A full one-loop calculation of the neutrino electromagnetic form factors is performed in the Feynman gauge. A non-zero CP asymmetry is generated by a required threshold condition for the neutrino masses along with non-vanishing CP violating phases in the lepton flavour mixing matrix. We follow the paradiagm of CP violation in neutrino oscillations to parametrise the flavour mixing contribution into a series of Jarlskog-like parameters. This formalism is then applied to a minimal seesaw model with two heavy right-handed neutrinos denoted N1 and N2. We observe that the CP asymmetries for decays into light neutrinos N → νγ are extremely suppressed, maximally around 10−17. However the CP asymmetry for N2→ N1γ can reach of order unity. Even if the Dirac CP phase δ is the only source of CP violation, a large CP asymmetry around 10−5–10−3 is comfortably achieved.

The renormalization structure of [formula omitted] supersymmetric higher-derivative gauge theory

We consider the harmonic superspace formulation of higher-derivative 6D,N=(1,0) supersymmetric gauge theory and its minimal coupling to a hypermultiplet. In components, the kinetic term for the gauge field in such a theory involves four space-time derivatives. The theory is quantized in the framework of the superfield background method ensuring manifest 6D,N=(1,0) supersymmetry and the classical gauge invariance of the quantum effective action. We evaluate the superficial degree of divergence and prove it to be independent of the number of loops. Using the regularization by dimensional reduction, we find possible counterterms and show that they can be removed by the coupling constant renormalization for any number of loops, while the divergences in the hypermultiplet sector are absent at all. Assuming that the deviation of the gauge-fixing term from that in the Feynman gauge is small, we explicitly calculate the divergent part of the one-loop effective action in the lowest order in this deviation. In the approximation considered, the result is independent of the gauge-fixing parameter and agrees with the earlier calculation for the theory without a hypermultiplet.

Four-loop singularities of the massless fermion propagator in quenched three-dimensional QED

We calculate the three- and four-loop corrections to the massless fermion propagator in three-dimensional quenched Quantum Electrodynamics with four-component fermions. The three-loop correction is finite and gauge invariant but the four-loop one has singularities except in the Feynman gauge where it is also finite. Our results explicitly show that, up to four loops, gauge-dependent terms are completely determined by lower order ones in agreement with the Landau-Khalatnikov-Fradkin transformation.

Renormalization in an interpolating gauge in Yang–Mills theory

The Coulomb gauge in QCD is the only explicitly unitary gauge. But it suffers from energy-divergences which means that it is not rigorously well-defined. One way to define it unambiguously is as the limit of a gauge interpolating between the Feynman gauge and the Coulomb gauge. This interpolating gauge is characterized by a parameter θ and the Coulomb gauge is obtained in the limit θ→0. We study the renormalization of this θ-gauge for all values of θ. Novel features include field mixing as well as scaling, the renormalization of the θ parameter itself, and the appearance of new counter-term structures at two-loop order.

Electroweak fermion triangle loop contributions to the muon anomalous magnetic moment revisited

Abstract The contribution to the muon anomalous magnetic moment from the fermion triangle loop diagrams connected to the muon line by a photon and a $Z$ boson is re-analyzed in both the unitary gauge and the ’t Hooft–Feynman gauge. With use of the anomalous axial-vector Ward identity, it is shown that the calculation in the unitary gauge exactly coincides with the one in the ’t Hooft–Feynman gauge. The part which arises from the ordinary axial-vector Ward identity corresponds to the contribution of the neutral Goldstone boson. For the top-quark contribution, the one-parameter integral form is obtained up to the order of $m_\mu^2/m_Z^2$. The results are compared with those obtained by the asymptotic expansion method.

Mass spectra of heavy pseudoscalars using instantaneous Bethe\u2013Salpeter equation with different kernels

We solved the instantaneous Bethe–Salpeter equation for heavy pseudoscalars in different kernels, where the kernels are obtained using linear scalar potential plus one gluon exchange vector potentials in Feynman gauge, Landau gauge, Coulomb gauge and time-component Coulomb gauge. Since we cannot give a complete QCD-based calculation, the results are gauge dependent. We compared the obtained mass spectra of heavy pseudoscalars between different kernels, found that using the same parameters we obtain the smallest mass splitting in time-component Coulomb gauge, the similar largest mass splitting in Feynman and Coulomb gauges, middle size splitting in Landau gauge.

All two-loop scalar self-energies and tadpoles in general renormalisable field theories

We calculate the complete tadpoles and self-energies at the two-loop order for scalars in general renormalisable theories, a crucial component for calculating two-loop electroweak corrections to Higgs-boson masses or for any scalar beyond the Standard Model. We renormalise the amplitudes using mass-independent renormalisation schemes, based on both dimensional regularisation and dimensional reduction. The results are presented here in Feynman gauge, with expressions for all 121 self-energy and 25 tadpole diagrams given in terms of scalar and tensor integrals with the complete set of rules to reduce them to a minimal basis of scalar integrals for any physical kinematic configuration. In addition, we simplify the results to a set of only 16 tadpole and 58 self-energy topologies using relations in order to substitute the ghost and Goldstone-boson couplings that we derive. To facilitate their application, we also provide our results in electronic form as a new code TLDR. We test our results by applying them to the Standard Model and compare with analytic expressions in the literature.

Off-shell Ward identities for N-gluon amplitudes

Off-shell Ward identities in non-abelian gauge theory continue to be a subject of active research, since they are, in general, inhomogeneous and their form depends on the chosen gauge-fixing procedure. For the three-gluon and four-gluon vertices, it is known that a relatively simple form of the Ward identity can be achieved using the pinch technique or, equivalently, the background-field method with quantum Feynman gauge. The latter is also the gauge-fixing underlying the string-inspired formalism, and here we use this formalism to prove the corresponding form of the Ward identity for the one-loop N-gluon amplitudes.

Unpolarized and spin-dependent DIS structure functions in Double-Logarithmic Approximation

We demonstrate how to calculate perturbative components of the structure functions F1 (for unpolarized DIS) and gi (spin-dependent DIS) in Double-Logarithmic Approximation, studying separately the cases of fixed and running QCD coupling. We show that as long as only ladder graphs are accounted for (throughout the talk we use the Feynman gauge for virtual gluons) there is no difference at all between F1 and g1. However, accounting for contributions of non-ladder graphs brings an essential difference between them. Applying Saddle-Point method to expressions for F1 and g1, we obtain their small-x asymptotics. The both asymptotics are of the Regge kind but with different intercepts. The intercept of F1 proved to be greater than unity, so it is a new contribution to Pomeron. We show the reason for the g1 intercept should be less than the one of F1 and thereby argue against using model Pomerons in the spin-dependent processes. Finally, we discuss the applicability region of Regge asymptotics.