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 madgraph5_amc@nlo. We start from the ’t Hooft–Feynman gauge Lagrangian in feynrules 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 <math><mn>5</mn><mo>×</mo><mn>5</mn></math> matrix propagator in the FD gauge. As a demonstration of the method we calculate the cross section for the process <math><msup><mi>μ</mi><mo>−</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><mo>→</mo><msub><mi>ν</mi><mi>μ</mi></msub><msub><mover><mi>ν</mi><mo>¯</mo></mover><mi>μ</mi></msub><mi>t</mi><mover><mi>t</mi><mo>¯</mo></mover><mi>H</mi></math> 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.

Japan

Kaoru Hagiwara

Junichi Kanzaki

Belgium

Olivier Mattelaer

Kentarou Mawatari

Ya-Juan Zheng

Automatic generation of helicity amplitudes in the Feynman-diagram gauge

Scattering amplitudes for the massless QCD process, <math><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover><mo>→</mo><msup><mi>q</mi><mo>′</mo></msup><msup><mover><mi>q</mi><mo>¯</mo></mover><mo>′</mo></msup></math>, 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.

Youichi Yamada

One-loop QCD amplitudes in the Feynman-diagram gauge

We calculate helicity amplitudes for <math><msup><mi>e</mi><mo>−</mo></msup><msup><mi>e</mi><mo>+</mo></msup><mo>→</mo><msup><mi>W</mi><mo>−</mo></msup><msup><mi>W</mi><mo>+</mo></msup></math> analytically in the Feynman-diagram (FD) gauge. We show that, unlike in the unitary gauge, there is no energy growth of the individual Feynman amplitudes for the longitudinally polarized W bosons, and the contributions from the associated Goldstone bosons are manifest even without taking the high-energy limit. We also find that the physical distributions can be interpreted by the individual amplitudes in the FD gauge.

Hiroyuki Furusato

Yutaro Suzuki

<math><mi>W</mi></math>-boson pair production at lepton colliders in the Feynman-diagram gauge

China

Junmou Chen

Erratum: Helicity amplitudes without gauge cancellation for electroweak processes

In the 5-component representation of weak bosons, the first four components make a Lorentz four vector, representing the transverse and longitudinal polarizations excluding the scalar component of the weak bosons, whereas its fifth component corresponds to the Goldstone boson. We obtain the  $$5\times 5$$  <math> <mrow> <mn>5</mn> <mo>×</mo> <mn>5</mn> </mrow> </math>   component propagators of off-shell weak bosons, proposed previously and named after the Goldstone boson equivalence theorem, by starting from the unitary-gauge representation of the tree-level scattering amplitudes, and by applying the BRST (Becchi–Rouet–Stora–Tyutin) identities to the two sub-amplitudes connected by each off-shell weak-boson line. By replacing all weak boson vertices with those among the off-shell 5-component wavefunctions, we arrive at the expression of the electroweak scattering amplitudes, where the magnitude of each Feynman amplitude has the correct on-shell limits for all internal propagators, and hence with no artificial gauge cancellation among diagrams. Although our derivation is limited to the tree-level only, it allows us to study the properties of each Feynman amplitude separately, and then learn how they interfere in the full amplitudes. We implement the 5-component weak boson propagators and their vertices in the numerical helicity amplitude calculation code HELAS (Helicity Amplitude Subroutines), so that an automatic amplitude generation program such as MadGraph can generate the scattering amplitudes without gauge cancellation. We present results for several high-energy scattering processes where subtle gauge-theory cancellation among diagrams takes place in all the other known approaches.

Helicity amplitudes without gauge cancellation for electroweak processes

We introduce a methodology and investigate the feasibility of measuring quantum properties of tau lepton pairs in the <math><mi>H</mi><mo>→</mo><msup><mi>τ</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>−</mo></msup></math> decay at future lepton colliders. In particular, observation of entanglement, steerability, and violation of Bell inequalities are examined for the International Linear Collider (ILC) and Future Circular <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> Collider (FCC-ee). We find that detecting quantum correlation crucially relies on precise reconstruction of the tau lepton rest frame, and a simple kinematics reconstruction does not suffice due to the finite energy resolution of the colliding beams and detectors. To correct for energy mismeasurements, a log-likelihood method is developed that incorporates the information of impact parameters of tau lepton decays. We demonstrate that an accurate measurement of quantum properties is possible with this method. As a by-product, we show that a novel model-independent test of <math><mi>C</mi><mi>P</mi></math> violation can be performed and the <math><mi>C</mi><mi>P</mi></math> phase of <math><mi>H</mi><mi>τ</mi><mi>τ</mi></math> interaction can be constrained with an accuracy comparable to dedicated analyses, i.e., up to 7.9° and 5.4° at ILC and FCC-ee, respectively.

France

Poland

Mohammad Mahdi Altakach

Priyanka Lamba

Italy

Fabio Maltoni

Kazuki Sakurai

Quantum information and <math><mi>C</mi><mi>P</mi></math> measurement in <math><mi>H</mi><mo>→</mo><msup><mi>τ</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>−</mo></msup></math> at future lepton colliders

We present the cross section for  $$e^{+}e^{-}\rightarrow hZ$$  <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> <mo>→</mo> <mi>h</mi> <mi>Z</mi> </mrow> </math>   with arbitrary sets of electron and Z boson polarizations at the full next-to-leading order in various extended Higgs models, such as the Higgs singlet model (HSM), the inert doublet model (IDM) and the two Higgs doublet model (2HDM). We systematically perform complete one-loop calculations to the helicity amplitudes in the on-shell renormalization scheme, and present the full analytic results as well as numerical evaluations. The deviation  $$\Delta R^{hZ}$$  <math> <mrow> <mi>Δ</mi> <msup> <mi>R</mi> <mrow> <mi>hZ</mi> </mrow> </msup> </mrow> </math>   in the total cross section from its standard model (SM) prediction is comprehensively analyzed, and the differences among these models are discussed in details. We find that new physics effects appearing in the renormalized hZZ vertex almost govern the behavior of  $$\Delta R^{hZ}$$  <math> <mrow> <mi>Δ</mi> <msup> <mi>R</mi> <mrow> <mi>hZ</mi> </mrow> </msup> </mrow> </math>  , and it takes a negative value in most cases. The possible size of  $$\Delta R^{hZ}$$  <math> <mrow> <mi>Δ</mi> <msup> <mi>R</mi> <mrow> <mi>hZ</mi> </mrow> </msup> </mrow> </math>   reaches several percent under the theoretical and experimental bounds. We also analyze the deviation  $$\Delta R^{hZ}_{XY}$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>XY</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>   in the total cross section times decay branching ratios of the discovered Higgs boson by utilizing the H-COUP program. It is found that the four types of 2HDMs can be discriminated by analyzing the correlation between  $$\Delta R^{hZ}_{\tau \tau }$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>τ</mi> <mi>τ</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>   and  $$\Delta R^{hZ}_{bb}$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>bb</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>   and those between  $$\Delta R^{hZ}_{\tau \tau }$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>τ</mi> <mi>τ</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>   and  $$\Delta R^{hZ}_{cc}$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>cc</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>  . Furthermore, the HSM and the IDM can be discriminated from the 2HDMs by measuring  $$\Delta R^{hZ}_{WW}$$  <math> <mrow> <mi>Δ</mi> <msubsup> <mi>R</mi> <mrow> <mi>WW</mi> </mrow> <mrow> <mi>hZ</mi> </mrow> </msubsup> </mrow> </math>  . These signatures can be tested by precision measurements at future Higgs factories such as the International Linear Collider.

Masashi Aiko

Shinya Kanemura

Next-to-leading-order corrections to the Higgs strahlung process from electron–positron collisions in extended Higgs models

We study a pseudoscalar dark-matter model arising from a complex singlet extension of the standard model (SM), and show that the dark-matter nucleon scattering is suppressed when two <math><mrow><mi>C</mi><mi>P</mi></mrow></math>-even scalars are degenerate. In such a degenerate-scalar scenario we explore the model parameter space which satisfies constraints from the direct detection experiments and the relic density of dark matter. In addition, we discuss a possibility to verify such a scenario by using the recoil-mass technique at the International Linear Collider. We find that a pair of states separated by 0.2 GeV can be distinguished from the single SM-like Higgs state at <math><mrow><mn>5</mn><mi>σ</mi></mrow></math> with an integrated luminosity of <math><mrow><mn>2</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>ab</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>.

Sachiho Abe

Gi-Chol Cho

Probing a degenerate-scalar scenario in a pseudoscalar dark-matter model

We discuss a possibility that the parameter space of the two Higgs doublet model is significantly narrowed down by considering the synergy between direct searches for additional Higgs bosons at the LHC and its luminosity upgraded operation and precision measurements of the Higgs boson properties at future electron-positron colliders such as the International Linear Collider. We show that, in the case where the coupling constants of the discovered Higgs boson are slightly different from the predicted values in the standard model, most of the parameter space is explored by the direct searches of extra Higgs bosons, in particular for the decays of the extra Higgs bosons into the discovered Higgs boson, and also by the theoretical arguments such as perturbative unitarity and vacuum stability. This can be done because there appears an upper limit on the mass of the extra Higgs bosons as long as the deviation exists in the Higgs boson coupling. We also show that in the alignment limit where all the Higgs boson couplings take the standard model like values most of the parameter space cannot be excluded because most of the Higgs to Higgs decays are suppressed and also there is no upper limit on the masses from the theoretical arguments.

Mariko Kikuchi

Germany

Kodai Sakurai

Kei Yagyu

Probing extended Higgs sectors by the synergy between direct searches at the LHC and precision tests at future lepton colliders

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.

QED and QCD helicity amplitudes in parton-shower gauge