Two-loop Diagrams Research Articles

Higgs interference effects in top-quark pair production in the 1HSM

We present a next-to-leading-order (NLO) study of the process pp (→ {h1, h2}) →tt¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t} $$\\end{document} + X in the 1-Higgs-singlet extension of the Standard Model with an additional heavy Higgs boson h2 that mixes with the light Higgs boson h1. This process is subject to large interference effects between loop-induced Higgs-mediated amplitudes and the QCD continuum background which tend to overcompensate any resonance contributions. A reliable modelling of the resulting top-pair invariant mass shapes requires the inclusion of higher-order QCD corrections, which are presented here. The computation of these NLO corrections is exact in all contributions but in the class of non-factorisable two-loop diagrams which are included in an approximate way such that all infrared singular limits are preserved. We present numerical results for several benchmark points with heavy Higgs masses in the range 700–3000 GeV considering the production of stable top quarks. We find that the interference effects dominate the BSM signal yielding sharp dip structures instead of resonance peaks. The significance and excludability of the BSM effect is explored for the LHC Run 2, Run 3 and HL-LHC.

γ5 schemes and the interplay of SMEFT operators in the Higgs-gluon coupling

We calculate the four-top-quark operator contributions to Higgs production via gluon fusion in the Standard Model effective field theory. The four-top operators enter for the first time via two-loop diagrams. Owing to their chiral structure they contain γ5, so special care needs to be taken when using dimensional regularization for the loop integrals. We use two different schemes for the continuation of γ5 to D space-time dimensions in our calculations and present a mapping for the parameters in the two schemes. This generically leads to an interplay of different operators, such as four-top operators, chromomagnetic operators, or Yukawa-type operators at the loop level. We validate our results by examples of matching onto UV models. Published by the American Physical Society 2024

Binary black holes and quantum off-shell recursion

The quantum off-shell recursion provides an efficient and universal computational tool for loop-level scattering amplitudes. In this work, we present a new comprehensive computational framework based on the quantum off-shell recursion for binary black hole systems. Using the quantum perturbiner method, we derive the recursions and solve them explicitly up to two-loop order. We introduce an indicator that enables straightforward distinguishing diagrams and deriving integral families in the context of off-shell recursions. By employing the soft expansion technique, we remove irrelevant terms from the loop integrands and express them in terms of master integrals. We classify the one-loop and the two-loop classical diagrams, and their loop integrands are represented by linear combinations of the master integrals. Finally, we explicitly calculate the classical scalar 2 → 2 amplitudes in the potential region up to the 3PM order and reproduce the known results.

QCD corrections to the Golden decay channel of the Higgs boson

Future colliders aim to provide highly precise experimental measurements of the properties of the Higgs boson. In order to benefit from these precision machines, theoretical errors in the Higgs sector observables have to match at least the experimental uncertainties. The theoretical uncertainties in the Higgs sector observables can be reduced by including missing higher-order terms in their perturbative calculations. In this direction, we compute the mixed QCD-electroweak corrections at \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}\\left({\\alpha \\alpha }_{s}\\right)$$\\end{document} to the Higgs decay into four charged leptons by considering the golden decay channel, H → e+e−μ+μ−. Due to color conservation, these corrections receive contributions only from the two-loop virtual diagrams. In the complex mass scheme, we find that the mixed QCD-electroweak corrections to the partial decay width, relative to the leading order predictions, are positive and about 0.27% for αs(MZ). Relative to the next-to-leading order electroweak corrections, the mixed QCD-electroweak corrections are found to be approximately 18% for αs(MZ). With respect to the leading order, we observe a flat effect of the mixed QCD-electroweak corrections on the invariant mass distribution of the lepton pairs with fixed QCD coupling. The ϕ distribution, due to the mixed QCD-electroweak corrections, follows a (1 − cos ϕ) dependence.

Leptonic ALP portal to the dark sector

We discuss the leptonic axionlike particle (ALP) portal as a simple scenario that connects observed discrepancies in anomalous magnetic moments to the dark matter relic abundance. In this framework an axionlike particle in the multi-MeV range couples to SM leptons and a dark matter (DM) fermion, with mass above the ALP mass but below 1 GeV. The ALP contributes to (g−2)μ and (g−2)e dominantly through two-loop Barr-Zee diagrams, while the DM abundance is generated by p-wave annihilation to ALP pairs. Constraints from beam-dump experiments, colliders, and cosmic microwave background probes are very stringent, and restrict the viable parameter space to a rather narrow region that will be tested in the near future. Published by the American Physical Society 2024

Two-loop vertices with vacuum polarization insertion

We present the analytic evaluation of the second-order corrections to the massive form factors, due to two-loop vertex diagrams with a vacuum polarization insertion, with exact dependence on the external and internal fermion masses, and on the squared momentum transfer. We consider vector, axial-vector, scalar and pseudoscalar interactions between the external fermion and the external field. After renormalization, the finite expressions of the form factors are expressed in terms of polylogarithms up to weight three.

Nucleon Self-Energy Including Two-Loop Contributions

The nucleon self-energy is calculated in SU(2) covariant chiral perturbation theory to study the pion mass dependence of the nucleon mass up to chiral order O(q6), i.e., including two-loop diagrams. The contributions of the diagrams are expressed by a small set of (scalar) master integrals. The extended-on-mass-shell (EOMS) renormalization scheme is applied by systematically subtracting infrared regular parts of the integrals (besides the divergences) to ensure that the renormalized results are consistent with the power counting. The master integrals are evaluated in two ways: Firstly, they are calculated by means of the chiral expansion in d dimensions, using the strategy of regions to differentiate between the infrared singular and regular part. This yields the physical nucleon mass in a 1/m-expansion and is in agreement with the infrared-renormalized result. Secondly, the master integrals are solved numerically without relying on the 1/m-expansion using the sector decomposition method.

Lepton g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g-2$$\\end{document} and W-boson mass anomalies in the DFSZ axion model

With regard to the leptonic magnetic dipole moment anomaly as well as the W-boson mass excess, we study the DFSZ axion model. Considering theoretical and experimental constraints, we show that the muon and electron g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g-2$$\\end{document} anomalies can be explained within the parameter space of the model for extra Higgs bosons with mass spectra around the electroweak scale and for an almost equivalent contribution of one-and two-loop diagrams. A negative electron g-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g-2$$\\end{document} could be achieved by introducing heavy neutrinos. Furthermore, the W-boson mass excess can be consistently addressed within the mass range of the matter content testable at collider experiments.

Vacuum free energy, quark condensate shifts and magnetization in three-flavor chiral perturbation theory to [formula omitted] in a uniform magnetic field

We study three-flavor QCD in a uniform magnetic field using chiral perturbation theory (χPT). We construct the vacuum free energy density, quark condensate shifts induced by the magnetic field and the renormalized magnetization to O(p6) in the chiral expansion. We find that the calculation of the free energy is greatly simplified by cancellations among two-loop diagrams involving charged mesons. In comparing our results with recent 2+1-flavor lattice QCD data, we find that the light quark condensate shift at O(p6) is in better agreement than the shift at O(p4). We also find that the renormalized magnetization, due to its small-ness, possesses large uncertainties at O(p6) due to the uncertainties in the low-energy constants.

The Casimir effect in a dilute Bose gas in broken symmetry phase within the improve Hartree–Fock approximation

The Casimir effect in a weakly interacting Bose gas confined between two parallel plates is studied at finite temperature, which is smaller than the phase transition temperature, within improved Hartree–Fock (IHF) approximation, in which the number of the Goldstone bosons are conserved. When condensate and non-condensate phases co-exist, considering the contribution of the two-loop diagrams, the order-parameter and effective mass are not constant. The Casimir energy and resulting Casimir force are separated into two parts: the one corresponds to the corresponding one-loop approximation, and the other is associated with the contribution of the two-loop diagrams. Our result can be reduced to the one for weakly interacting gas at zero temperature, or for the ideal Bose gas.

Evaluating master integrals in non-factorizable corrections to t-channel single-top production at NNLO QCD

We studied the two-loop non-factorizable Feynman diagrams for the t-channel single-top production process in quantum chromodynamics. We present a systematic computation of master integrals of the two-loop Feynman diagrams with one internal massive propagator in which a complete uniform transcendental basis can be built. The master integrals are derived by means of canonical differential equations and uniform transcendental integrals. The results are expressed in the form of Goncharov polylogarithm functions, whose variables are the scalar products of external momenta, as well as the masses of the top quark and the W boson. We also gave a discussion on the diagrams with potential elliptic sectors.

Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles

We develop an operator approach to the evaluation of multiple integrals for multiloop Feynman massless diagrams. A commutative family of graph building operators Hα for ladder diagrams is constructed and investigated. The complete set of eigenfunctions and the corresponding eigenvalues for the operators Hα are found. This enables us to explicitly express a wide class of four-point ladder diagrams and a general two-loop propagator-type master diagram (with arbitrary indices on the lines) as Mellin-Barnes-type integrals. Special cases of these integrals are explicitly evaluated. A certain class of zig-zag four-point and two-point planar Feynman diagrams (relevant to the bi-scalar D-dimensional “fishnet” field theory and to the calculation of the β-function in ϕ4-theory) is considered. The graph building operators and convenient integral representations for these Feynman diagrams are obtained. The explicit form of the eigenfunctions for the graph building operators of the zig-zag diagrams is fixed by conformal symmetry and these eigenfunctions coincide with the 3-point correlation functions in D-dimensional conformal field theories. By means of this approach, we exactly evaluate the diagrams of the zig-zag series in special cases. In particular, we find a fairly simple derivation of the values for the zig-zag multi-loop two-point diagrams for D = 4. The role of conformal symmetry in this approach, especially a connection of the considered graph building operators with conformal invariant solutions of the Yang-Baxter equation is investigated in detail.

Locally finite two-loop QCD amplitudes from IR universality for electroweak production

We describe the implementation of infrared subtractions for two-loop QCD corrections to quark-antiquark annihilation to electroweak final states. The subtractions are given as form-factor integrands whose integrals are known. The resulting subtracted amplitudes are amenable to efficient numerical integration. Our procedure is based on the universality of infrared singularities and requires a relatively limited set of subtractions, whose number grows as the number of two-loop diagrams, rather than with the number of singular regions of integration.

Free energy on the sphere for non-abelian gauge theories

We compute the Sd partition function of the fixed point of non-abelian gauge theories in continuous d, using the ϵ-expansion around d = 4. We illustrate in detail the technical aspects of the calculation, including all the factors arising from the gauge-fixing procedure, and the method to deal with the zero-modes of the ghosts. We obtain the result up to NLO, i.e. including two-loop vacuum diagrams. Depending on the sign of the one-loop beta function, there is a fixed point with real gauge coupling in d > 4 or d < 4. In the first case we extrapolate to d = 5 to test a recently proposed construction of the UV fixed point of 5d SU(2) Yang-Mills via a susy-breaking deformation of the E1 SCFT. We find that the F theorem allows the proposed RG flow. In the second case we extrapolate to d = 3 to test whether QCD3 with gauge group SU(nc) and nf fundamental matter fields flows to a CFT or to a symmetry-breaking case. We find that within the regime with a real gauge coupling near d = 4 the CFT phase is always favored. For lower values of nf we compare the average of F between the two complex fixed points with its value at the symmetry-breaking phase to give an upper bound of the critical value {n}_f^{ast } below which the symmetry-breaking phase takes over.

Feynman integrals of Grassmannians

We embed Feynman integrals in the subvarieties of Grassmannians through homogenization of the integrands in projective space, then obtain Gel'fand-Kapranov-Zelevinsky systems satisfied by those scalar integrals. The Feynman integral can be written as linear combinations of the hypergeometric functions of a fundamental solution system in neighborhoods of regular singularities of the Gel'fand-Kapranov-Zelevinsky system, whose linear combination coefficients are determined by the integral on an ordinary point or some regular singularities. Taking some Feynman diagrams as examples, we elucidate in detail how to obtain the fundamental solution systems of Feynman integrals in neighborhoods of regular singularities. Furthermore we also present the parametric representations of Feynman integrals of the two-loop self-energy diagrams that are convenient to embed in the subvarieties of Grassmannians.

Naturally light Dirac and pseudo-Dirac neutrinos from left-right symmetry

We develop a class of left-right symmetric theories based on the gauge group SU(3)c × SU(2)L × SU(2)R × U(1) with a generalized seesaw mechanism for generating the charged fermion masses. Neutrinos are naturally Dirac particles in this setup with their small masses arising from two-loop quantum corrections. We evaluate these two-loop diagrams exactly and analyze the flavor structure of the lepton sector. We find excellent fits to neutrino oscillation data, independent of the right-handed gauge symmetry breaking scale. We also explore the possibility that neutrinos are pseudo-Dirac particles in this framework, with the tiny mass splittings between active and sterile neutrinos arising from Planck-induced corrections and find possible realizations. These models can be tested in the near future with precision cosmological measurements of ∆Neff in CMB which is predicted to be ≃ 0.14. This class of models allows for a solution to the strong CP problem via parity symmetry without the need for an axion.

Two-loop tensor integral coefficients in OpenLoops

We present a new and fully general algorithm for the automated construction of the integrands of two-loop scattering amplitudes. This is achieved through a generalisation of the open-loops method to two loops. The core of the algorithm consists of a numerical recursion, where the various building blocks of two-loop diagrams are connected to each other through process-independent operations that depend only on the Feynman rules of the model at hand. This recursion is implemented in terms of tensor coefficients that encode the polynomial dependence of loop numerators on the two independent loop momenta. The resulting coefficients are ready to be combined with corresponding tensor integrals to form scattering probability densities at two loops. To optimise CPU efficiency we have compared several algorithmic options identifying one that outperforms naive solutions by two orders of magnitude. This new algorithm is implemented in the OpenLoops framework in a fully automated way for two-loop QED and QCD corrections to any Standard Model process. The technical performance is discussed in detail for several 2 → 2 and 2 → 3 processes with up to order 105 two-loop diagrams. We find that the CPU cost scales linearly with the number of two-loop diagrams and is comparable to the cost of corresponding real-virtual ingredients in a NNLO calculation. This new algorithm constitutes a key building block for the construction of an automated generator of scattering amplitudes at two loops.

Study muon g − 2 at two-loop level in the U(1)XSSM

The new experiment data of muon g − 2 is reported by the workers at Fermilab National Accelerator Laboratory (FNAL). Combined with the previous Brookhaven National Laboratory (BNL) E821 result, the departure from the standard model prediction is about 4.2σ. It strengthens our faith in the new physics. U(1)XSSM is the U(1) extension of the minimal supersymmetric standard model, where we study the electroweak corrections to the anomalous magnetic dipole moment of muon from the one-loop diagrams and some two-loop diagrams possessing important contributions. These two-loop diagrams include Barr-Zee type, rainbow type and diamond type. The virtual supersymmetric particles in these two-loop diagrams are chargino, scalar neutrino, neutralino, scalar lepton, which are supposed not very heavy to make relatively large corrections. We obtain the Wilson coefficients of the dimension 6 operators inducing the anomalous magnetic dipole moment of muon. The numerical results can reach 25 × 10−10 and even larger.

Charged lepton EDM with extra Yukawa couplings

In a two Higgs doublet model with extra Yukawa couplings, we assess the new physics contributions to charged lepton electric dipole moment. We focus especially on muon (and tau) EDM where in the coming decade several experiments — Muon g-2, J-PARC and PSI (and Belle II) — will push sensitivities down by several orders of magnitude. With the working assumption that extra Yukawa couplings are analogous to SM ones in strength and taking exotic scalar masses in sub-TeV range, we find that μ and τ EDM can be enhanced to values larger than new physics scenarios that scale with lepton mass. The main effect comes from the flavor-conserving extra top coupling ρtt via well-known two-loop diagrams. Deviating from our working assumption, if the muon g − 2 anomaly arises from the one-loop diagram, driven by singly enhanced lepton flavor violating ρτμ coupling, it can also induce rather large muon EDM, accessible at upcoming experiments.

Integral representation for three-loop banana graph

It has recently been shown that two-loop kite-type diagrams can be computed analytically in terms of iterated integrals with algebraic kernels. This result was obtained using a new integral representation for two-loop sunset subgraphs. In this paper, we have developed a similar representation for a three-loop banana integral in $d = 2-2\varepsilon$ dimensions. This answer can be generalized up to any given order in the $\varepsilon$-expansion and can be calculated numerically both below and above the threshold. We also demonstrate how this result can be used to compute more complex three-loop integrals containing the three-loop banana as a subgraph.