We present a method to simulate toponium formation events at the LHC using the Green’s function of non-relativistic QCD in the Coulomb gauge, which governs the momentum distribution of top quarks in the presence of the QCD potential. This Green’s function can be employed to re-weight any matrix elements relevant for $$t{\bar{t}}$$ production and decay processes where a colour-singlet top–antitop pair is produced in the -wave at threshold. As an example, we study the formation of $$\eta _t$$ toponium states in the gluon fusion channel at the LHC, combining the re-weighted matrix elements with parton showering.

France

Benjamin Fuks

Japan

Kaoru Hagiwara

China

Kai Ma

Ya-Juan Zheng

Simulating toponium formation signals at the LHC

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.

Junichi Kanzaki

Belgium

Olivier Mattelaer

Kentarou Mawatari

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

We study the top-Higgs coupling with a CP violating phase ξ at a future multi-TeV muon collider. We focus on processes that are directly sensitive to the top quark Yukawa coupling:   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>h</mi> </math>  $$ t\overline{t}h $$ , tbhμν, and   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>hν</mi> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ t\overline{t} h\nu \overline{\nu} $$  with   <math> <mi>h</mi> <mo>→</mo> <mi>b</mi> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ h\to b\overline{b} $$  and semileptonic top decays. At different energies, different processes dominate the cross section, providing complementary information. At and above an energy of   <math> <mi>O</mi> <mfenced> <mn>10</mn> </mfenced> </math>  $$ \mathcal{O}(10) $$  TeV, vector boson fusion processes dominate. As we show, in the Standard Model there is destructive interference in the vector boson fusion processes   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>hν</mi> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ t\overline{t} h\nu \overline{\nu} $$  and tbhμν between the top quark Yukawa and Higgs-gauge boson couplings. A CP-violating phase changes this interference, and the cross section measurement is very sensitive to the size of the CP-violating angle. Although we find that the cross sections are measured to   <math> <mi>O</mi> <mfenced> <mrow> <mn>50</mn> <mo>%</mo> </mrow> </mfenced> </math>  $$ \mathcal{O}\left(50\%\right) $$  statistical uncertainty at 1σ, a 10 and 30 TeV muon collider can bound the CP-violating angle |ξ| ≲ 9.0° and |ξ| ≲ 5.4°, respectively. However, cross section measurements are insensitive to the sign of the CP-violating angle. To determine that the coupling is truly CP violating, observables sensitive to CP-violation must be measured. We find in the   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>h</mi> </math>  $$ t\overline{t}h $$  process the azimuthal angle between the   <math> <mi>t</mi> <mo>+</mo> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ t+\overline{t} $$  plane and the initial state muon+Higgs plane shows good discrimination for ξ = ±0.1π. For the tbhμν and   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>hν</mi> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ t\overline{t} h\nu \overline{\nu} $$  processes, the operator proportional to   <math> <mfenced> <mrow> <msub> <mover> <mi>p</mi> <mo>→</mo> </mover> <mi>μ</mi> </msub> <mo>×</mo> <msub> <mover> <mi>p</mi> <mo>→</mo> </mover> <mi>h</mi> </msub> </mrow> </mfenced> </math>  $$ \left({\overrightarrow{p}}_{\mu}\times {\overrightarrow{p}}_h\right) $$  ·   <math> <msub> <mover> <mi>p</mi> <mo>→</mo> </mover> <mi>t</mi> </msub> </math>  $$ {\overrightarrow{p}}_t $$  is sensitive to the sign of CP phase ξ. From these observables, we construct asymmetry parameters that show good distinction between different values and signs of the CP violating angle.

United States

Morgan Cassidy

Zhongtian Dong

Kyoungchul Kong

Ian Lewis

Yanzhe Zhang

Probing the CP structure of the top quark Yukawa at the future muon collider

The total cross section of the process <math><msup><mrow><mi>μ</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>→</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><msub><mrow><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>μ</mi></mrow></msub><mi>t</mi><mover><mrow><mi>t</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>H</mi></math> has strong dependence on the CP phase ξ of the top Yukawa coupling, where the ratio of <math><mi>ξ</mi><mo>=</mo><mi>π</mi></math> and <math><mi>ξ</mi><mo>=</mo><mn>0</mn></math> (SM) grows to 670 at <math><msqrt><mrow><mi>s</mi></mrow></msqrt><mo>=</mo><mtext>30</mtext></math> TeV, 3400 at 100 TeV. We study the cause of the strong energy dependence and identify its origin as the <math><msup><mrow><mo>(</mo><mi>E</mi><mo>/</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>W</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> growth of the weak boson fusion sub-amplitudes, <math><msubsup><mrow><mi>W</mi></mrow><mrow><mi>L</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msubsup><mrow><mi>W</mi></mrow><mrow><mi>L</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mi>t</mi><mover><mrow><mi>t</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>H</mi></math>, with the two W's are longitudinally polarized. We repeat the study in the SMEFT framework where EW gauge invariance is manifest and find that the highest energy cross section is reduced to a quarter of the complex top Yukawa model result, with the same energy power. By applying the Goldstone boson (GB) equivalence theorem, we identify the origin of this strong energy growth of the SMEFT amplitudes as associated with the dimension-6 <math><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>t</mi><mi>t</mi><mi>H</mi></math> vertex, where <math><msup><mrow><mi>π</mi></mrow><mrow><mo>±</mo></mrow></msup></math> denotes the GB of <math><msup><mrow><mi>W</mi></mrow><mrow><mo>±</mo></mrow></msup></math>. We obtain the unitarity bound on the coefficient of the SMEFT operator by studying all <math><mn>2</mn><mo>→</mo><mn>2</mn></math> and <math><mn>2</mn><mo>→</mo><mn>3</mn></math> cross sections in the <math><mi>J</mi><mo>=</mo><mn>0</mn></math> channel.

Vernon Barger

CP-violating top-Higgs coupling in SMEFT

We study reported deviations between observations and theoretical predictions associated with the production of a pair of dileptonically decaying top quarks at the LHC, and we examine the possibility that they reflect a signal of toponium formation. We investigate the production by gluon fusion of a color-singlet spin-0 toponium <math><msub><mi>η</mi><mi>t</mi></msub></math> bound state of a top and antitop quark, that then decays dileptonically (<math><mi>g</mi><mi>g</mi><mo>→</mo><msub><mi>η</mi><mi>t</mi></msub><mo>→</mo><mover><mo>ℓ</mo><mo>¯</mo></mover><mo>ℓ</mo><mi>b</mi><mover><mi>b</mi><mo>¯</mo></mover><mi>ν</mi><mover><mi>ν</mi><mo>¯</mo></mover></math>). We find strong correlations favoring the production of dilepton systems featuring a small angular separation in azimuth and a small invariant mass. Although toponium production only contributes to 0.8% of the total top-quark pair-production cross section at the 13 TeV LHC, there is a possibility that it can account for observed excesses in the narrow edges of phase space. We propose a method to discover toponium formation by “reconstructing” both its top and antitop quark constituents in the dilepton channel.

Signatures of toponium formation in LHC run 2 data

We study Higgs boson production associated with single top or anti-top via t-channel weak boson exchange at the LHC. The process is an ideal probe of the top quark Yukawa coupling, because we can measure the relative phase of htt and hWW couplings, thanks to the significant interference between the two amplitudes. By choosing the emitted W momentum along the polar axis in the th (  <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>h</mi> </math>  $$ \overline{t}h $$ ) rest frame, we obtain the helicity amplitudes for all the contributing subprocesses analytically, with possible CP phase of the Yukawa coupling. We study the azimuthal asymmetry between the W emission and the Wb (  <math> <mover> <mi>b</mi> <mo>¯</mo> </mover> </math>  $$ \overline{b} $$ ) → t(  <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$ ) h scattering planes, as well as several t and   <math> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ \overline{t} $$  polarization asymmetries as a signal of CP violating phase in the htt coupling. Both the azimuthal asymmetry and the polarization perpendicular to the scattering plane are found to have the opposite sign between the top and anti-top events. We identify the origin of the sign of asymmetries, and propose the possibility of direct CP violation test in pp collisions by comparing the top and anti-top polarization at the LHC.

Probing the top Yukawa coupling at the LHC via associated production of single top and Higgs

We study the quasi-two-body decays 
					 by employing the perturbative QCD (PQCD) factorization approach, where the charmonia 
					 represents 
					 and 
					. The corresponding decay channels are studied by constructing the kaon-pion distribution amplitude (DA) 
					, which comprises important final state interactions between the kaon and pion in the resonant region. Relativistic Breit-Wigner formulas are adopted to parameterize the time-like form factor 
					 appearing in the kaon-pion DAs. The SU(3) flavor symmetry breaking effect resulting from the mass difference between the kaon and pion is taken into account, which makes significant contributions to the longitudinal polarizations. The observed branching ratios and the polarization fractions of 
					 are accommodated by tuning hadronic parameters for the kaon-pion DAs. The PQCD predictions for 
					 modes from the same set of parameters can be tested by precise data obtained in the future from LHCb and Belle II experiments.
				

Ya Li

Zhou Rui

Zhen-Jun Xiao

P-wave contributions to B(s)→ψKπ decays in perturbative QCD approach Supported by the National Natural Science Foundation of China (11947013, 11605060, 11775117, 11547020); Ya Li is also Supported by the Natural Science Foundation of Jiangsu Province (BK20190508) and the Research Start-up Funding of Nanjing Agricultural University; Zhou Rui is Supported in part by the Natural Science Foundation of Hebei Province (A2019209449)

The constituent quark model is used to compute the ground and excited state masses of QQQbaryons containing either c or bquarks. The quark model parameters previously used to describe the properties of charmonium and bottomonium states were used in this analysis. The non-relativistic three-body bound state problem is solved by means of the Gaussian expansion method which provides sufficient accuracy and simplifies the subsequent evaluation of the matrix elements. Several low-lying states with quantum numbers 
					 and 
					 are reported. We compare the results with those obtained by the other theoretical formalisms. There is a general agreement for the mass of the ground state in each sector of triply heavy baryons. However, the situation is more puzzling for the excited states, and appropriate comments about the most relevant features of our comparison are given.
				

Gang Yang

Jialun Ping

Spain

Pablo G. Ortega

Jorge Segovia

Triply heavy baryons in the constituent quark model Supported by China Postdoctoral Science Foundation (2019M650617), National Natural Science Foundation of China (11535005, 11775118), and by Spanish Ministerio de Economía, Industria y Competitividad under contractsno. FPA2017-86380-P and FPA2016-77177-C2-2-P. P.G.O. acknowledges the financial support from Spanish MINECO's Juan de la Cierva-Incorporación programme, Grant Agreement No. IJCI-2016-28525.