The electroweak phase transition serves as a crucial portal to explore physics beyond the Standard Model, with profound implications for gravitational waves, baryogenesis, dark matter, and vacuum stability. We review the computational workflow for analyzing cosmological phase transitions, which includes constructing the finite-temperature effective potential, identifying possible phases, tracing transition history, calculating transition rates, milestone temperatures, and thermal parameters, as well as the numerical tools developed for each step. We compare the functionalities, strategies, and applicable scopes of these tools, aiming to provide a practical guide that helps researchers select the most appropriate computational resources for their studies.
Aviv Padawer-Blatt, Zhiwei Shao, Renier T. Hough
et al.
We employ the <span style="font-variant: small-caps;">simba-c</span> cosmological simulation to study the impact of its upgraded chemical enrichment model (Chem5) on the distribution of metals in the intragroup medium (IGrM). We investigate the projected X-ray emission-weighted abundance profiles of key elements over two decades in halo mass (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mn>13</mn></msup><mo>≤</mo><msub><mi>M</mi><mn>500</mn></msub><mo>/</mo><msub><mi mathvariant="normal">M</mi><mo>⊙</mo></msub><mo>≤</mo><msup><mn>10</mn><mn>15</mn></msup></mrow></semantics></math></inline-formula>). Typically, <span style="font-variant: small-caps;">simba-c</span> generates lower-amplitude abundance profiles than <span style="font-variant: small-caps;">simba</span> with flatter cores, in better agreement with observations. For low-mass groups, both simulations over-enrich the IGrM with Si, S, Ca, and Fe compared to observations, a trend likely related to inadequate modeling of metal dispersal and mixing. We analyze the 3D mass-weighted abundance profiles, concluding that the lower <span style="font-variant: small-caps;">simba-c</span> IGrM abundances are primarily a consequence of fewer metals in the IGrM, driven by reduced metal yields in Chem5, and the removal of the instantaneous recycling of metals approximation employed by <span style="font-variant: small-caps;">simba</span>. Additionally, an increased IGrM mass in low-mass <span style="font-variant: small-caps;">simba-c</span> groups is likely triggered by changes to the AGN and stellar feedback models. Our study suggests that a more realistic chemical enrichment model broadly improves agreement with observations, but physically motivated sub-grid models for other key processes, like AGN and stellar feedback and turbulent diffusion, are required to realistically reproduce observed group environments.
In comparing the two alternative explosion mechanisms of core-collapse supernovae (CCSNe), I examine recent three-dimensional (3D) hydrodynamical simulations of CCSNe in the frame of the delayed neutrino explosion mechanism (neutrino mechanism) and argue that these valuable simulations show that neutrino heating can supply a non-negligible fraction of the explosion energy but not the observed energies, and hence cannot be the primary explosion mechanism. In addition to the energy crisis, the neutrino mechanism predicts many failed supernovae that are not observed. The most challenging issue of the neutrino mechanism is that it cannot account for point-symmetric morphologies of CCSN remnants, many of which were identified in 2024. These contradictions with observations imply that the neutrino mechanism cannot be the primary explosion mechanism of CCSNe. The alternative jittering jets explosion mechanism (JJEM) seems to be the primary explosion mechanism of CCSNe; neutrino heating boosts the energy of the jittering jets. Even if some simulations show explosions of stellar models (but usually with energies below that observed), it does not mean that the neutrino mechanism is the explosion mechanism. Jittering jets, which simulations do not include, can explode the core before the neutrino heating process does. Morphological signatures of jets in many CCSN remnants suggest that jittering jets are the primary driving mechanism, as expected by the JJEM.
Two enigmatic gamma-ray features in the galactic central region, known as Fermi Bubbles (FBs), were found from Fermi-LAT data. An energy release, (e.g., by tidal disruption events in the Galactic Center, GC), generates a cavity with a shock that expands into the local ambient medium of the galactic halo. A decade or so ago, a phenomenological model of the FBs was suggested as a result of routine star disruptions by the supermassive black hole in the GC which might provide enough energy for large-scale structures, like the FBs. In 2020, analytical and numerical models of the FBs as a process of routine tidal disruption of stars near the GC were developed; these disruption events can provide enough cumulative energy to form and maintain large-scale structures like the FBs. The disruption events are expected to be <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup><mo>∼</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup><msup><mi>yr</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>, providing an average power of energy release from the GC into the halo of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi mathvariant="script">E</mi><mo>˙</mo></mover><mo>∼</mo><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>41</mn></msup></mrow></semantics></math></inline-formula> erg <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="normal">s</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>, which is needed to support the FBs. Analysis of the evolution of superbubbles in exponentially stratified disks concluded that the FB envelope would be destroyed by the Rayleigh–Taylor (RT) instabilities at late stages. The shell is composed of swept-up gas of the bubble, whose thickness is much thinner in comparison to the size of the envelope. We assume that hydrodynamic turbulence is excited in the FB envelope by the RT instability. In this case, the universal energy spectrum of turbulence may be developed in the inertial range of wavenumbers of fluctuations (the Kolmogorov–Obukhov spectrum). From our model we suppose the power of the FBs is transformed partly into the energy of hydrodynamic turbulence in the envelope. If so, hydrodynamic turbulence may generate MHD fluctuations, which accelerate cosmic rays there and generate gamma-ray and radio emission from the FBs. We hope that this model may interpret the observed nonthermal emission from the bubbles.
Abstract In a special run of the LHC with $$\beta ^{\star } = 2.5$$ β ⋆ = 2.5 km, proton–proton elastic-scattering events were recorded at $$\sqrt{s} = 13$$ s = 13 TeV with an integrated luminosity of $$340~\upmu {\text {b}}^{-1}$$ 340 μ b - 1 using the ALFA subdetector of ATLAS in 2016. The elastic cross section was measured differentially in the Mandelstam t variable in the range from $$-t = 2.5 \cdot 10^{-4}$$ - t = 2.5 · 10 - 4 GeV $$^{2}$$ 2 to $$-t = 0.46$$ - t = 0.46 GeV $$^{2}$$ 2 using 6.9 million elastic-scattering candidates. This paper presents measurements of the total cross section $$\sigma _{\text {tot}}$$ σ tot , parameters of the nuclear slope, and the $$\rho $$ ρ -parameter defined as the ratio of the real part to the imaginary part of the elastic-scattering amplitude in the limit $$t \rightarrow 0$$ t → 0 . These parameters are determined from a fit to the differential elastic cross section using the optical theorem and different parameterizations of the t-dependence. The results for $$\sigma _{\text {tot}}$$ σ tot and $$\rho $$ ρ are $$\begin{aligned} \sigma _{\text {tot}}(pp\rightarrow X) = 104.7 \pm 1.1 \; \text{ mb },\quad \rho = 0.098 \pm 0.011 . \end{aligned}$$ σ tot ( p p → X ) = 104.7 ± 1.1 mb , ρ = 0.098 ± 0.011 . The uncertainty in $$\sigma _{\text {tot}}$$ σ tot is dominated by the luminosity measurement, and in $$\rho $$ ρ by imperfect knowledge of the detector alignment and by modelling of the nuclear amplitude.
Astrophysics, Nuclear and particle physics. Atomic energy. Radioactivity
The Hubble tension has now grown to a level of significance which can no longer be ignored and calls for a solution which, despite a huge number of attempts, has so far eluded us. Significant efforts in the literature have focused on early-time modifications of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>CDM, introducing new physics operating prior to recombination and reducing the sound horizon. In this opinion paper I argue that early-time new physics <i>alone</i> will always fall short of fully solving the Hubble tension. I base my arguments on seven independent hints, related to (1) the ages of the oldest astrophysical objects, (2) considerations on the sound horizon-Hubble constant degeneracy directions in cosmological data, (3) the important role of cosmic chronometers, (4) a number of “descending trends” observed in a wide variety of low-redshift datasets, (5) the early integrated Sachs-Wolfe effect as an early-time consistency test of <inline-formula><math display="inline"><semantics><mi mathvariant="sans-serif">Λ</mi></semantics></math></inline-formula>CDM, (6) early-Universe physics insensitive and uncalibrated cosmic standard constraints on the matter density, and finally (7) equality wavenumber-based constraints on the Hubble constant from galaxy power spectrum measurements. I argue that a promising way forward should ultimately involve a combination of early- and late-time (but non-local—in a cosmological sense, i.e., at high redshift) new physics, as well as local (i.e., at <inline-formula><math display="inline"><semantics><mrow><mi>z</mi><mo>∼</mo><mn>0</mn></mrow></semantics></math></inline-formula>) new physics, and I conclude by providing reflections with regards to potentially interesting models which may also help with the <inline-formula><math display="inline"><semantics><msub><mi>S</mi><mn>8</mn></msub></semantics></math></inline-formula> tension.
In this review, we discuss and extend the study of the inclusive production of vector quarkonia, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Υ</mi></semantics></math></inline-formula>, emitted with large transverse momenta and rapidities at the LHC. We adopt the novel ZCW19<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> determination of fragmentation functions to depict the quarkonium production mechanism at the next-to-leading level of perturbative QCD. This approach is based on the nonrelativistic QCD formalism well adapted to describe the formation of a quarkonium state from the collinear fragmentation of a gluon or a constituent heavy quark at the lowest energy scale. We rely upon the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>NLL</mi><mo>/</mo><msup><mi>NLO</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> hybrid high-energy and collinear factorization for differential cross-sections, where the collinear formalism is enhanced by the BFKL resummation of next-to-leading energy logarithms arising in the <i>t</i>-channel. We employ the method to analyze the behavior of the rapidity distributions for double-inclusive vector quarkonium and inclusive vector quarkonium plus jet emissions. We discover that the natural stability of the high-energy series, previously seen in observables sensitive to the emission of hadrons with heavy flavor detected in the rapidity acceptance of LHC barrel calorimeters, becomes even more manifest when these particles are tagged in forward regions covered by endcaps. Our findings present the important message that vector quarkonia at the LHC via hybrid factorization offer a unique chance to perform precision studies of high-energy QCD, as well as an intriguing opportunity to shed light on the quarkonium production puzzle.
Spacecraft flybys provide access to the chemical composition of the gaseous envelope of the planetary object. Typical relative encounter velocities range from km/s to tens of km/s in flybys. For speeds exceeding about 5 km/s, modern mass spectrometers analyzing the rapidly encountering gas suffer from intrinsic hypervelocity impact-induced fragmentation processes causing ambiguous results when analyzing complex molecules. In this case, instruments use an antechamber, inside which the incoming species collide many times with the chamber wall. These collisions cause the desired deceleration and thermalization of the gas molecules. However, these collisions also dissociate molecular bonds, thus fragmenting the molecules, and possibly forming new ones precluding scientists from inferring the actual chemical composition of the sampled gas. We developed a novel time-of-flight mass spectrometer that handles relative encounter velocities of up to 20 km/s omitting an antechamber and its related fragmentation. It analyzes the complete mass range of <i>m/z</i> 1 to 1000 at an instance. This innovation leads to unambiguous analysis of complex (organic) molecules. Applied to Enceladus, Europa or Io, it will provide reliable chemical composition datasets for exploration of the Solar System to determine its status, origin and evolution.
Alexandre M. R. Almeida, Júlio C. Fabris, Mahamadou Hamani Daouda
et al.
We propose a unimodular version of the Brans–Dicke theory designed with a constrained Lagrangian formulation. The resulting field equations are traceless. The vacuum solutions in the cosmological background reproduce the corresponding solutions of the usual Brans–Dicke theory but with a cosmological constant term. A perturbative analysis of the scalar modes is performed and stable and unstable configurations appear, in contrast with the Brans–Dicke case for which only stable configurations occur. On the other hand, tensorial modes in this theory remain the same as in the traditional Brans–Dicke theory.
The exact coronal origin of the slow-speed solar wind has been under debate for decades in the Heliophysics community. Besides the solar wind speed, the heavy ion composition, including the elemental abundances and charge state ratios, are widely used as diagnostic tool to investigate the coronal origins of the slow wind. In this study, we recognize a subset of slow speed solar wind that is located on the upper boundary of the data distribution in the O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>7</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>/O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>6</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula> versus C<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>6</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>/C<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>5</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula> plot (O-C plot). In addition, in this wind the elemental abundances relative to protons, such as N/P, O/P, Ne/P, Mg/P, Si/P, S/P, Fe/P, He/P, and C/P are systemically depleted. We compare these winds (“upper depleted wind” or UDW hereafter) with the slow winds that are located in the main stream of the O-C plot and possess comparable Carbon abundance range as the depletion wind (“normal-depletion-wind”, or NDW hereafter). We find that the proton density in the UDW is about 27.5% lower than in the NDW. Charge state ratios of O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>7</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>/O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>6</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>, O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>7</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>/O, and O<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mrow><mn>8</mn><mo>+</mo></mrow></msup></semantics></math></inline-formula>/O are decreased by 64.4%, 54.5%, and 52.1%, respectively. The occurrence rate of these UDW is anti-correlated with solar cycle. By tracing the wind along PFSS field lines back to the Sun, we find that the coronal origins of the UDW are more likely associated with quiet Sun regions, while the NDW are mainly associated with active regions and HCS-streamer.
Viktor D. Stasenko, Alexander A. Kirillov, Konstantin M. Belotsky
The PBH clusters can be sources of gravitational waves, and the merger rate depends on the spatial distribution of PBHs in the cluster which changes over time. It is well known that gravitational collisional systems experience the core collapse that leads to significant increase of the central density and shrinking of the core. After core collapse, the cluster expands almost self-similarly (i.e., density profile extends in size without changing its shape). These dynamic processes affect the merger rate of PBHs. In this paper, the dynamics of the PBH cluster is considered using the Fokker–Planck equation. We calculate the merger rate of PBHs on cosmic time scales and show that its time dependence has a unique signature. Namely, it grows by about an order of magnitude at the moment of core collapse which depends on the characteristics of the cluster, and then decreases according to the dependence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">R</mi><mo>∝</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>1.48</mn></mrow></msup></mrow></semantics></math></inline-formula>. It was obtained for monochromatic and power-law PBH mass distributions with some fixed parameters. Obtained results can be used to test the model of the PBH clusters via observation of gravitational waves at high redshift.
Abstract Using a data sample of $$\sqrt{s}=13\,\text {TeV}$$ s = 13 TeV proton-proton collisions collected by the CMS experiment at the LHC in 2017 and 2018 with an integrated luminosity of $$103\text {~fb}^{-1}$$ 103 fb - 1 , the $$\text {B}^{0}_{\mathrm{s}} \rightarrow \uppsi (\text {2S})\text {K}_\mathrm{S}^{0}$$ B s 0 → ψ ( 2S ) K S 0 and $$\text {B}^{0} \rightarrow \uppsi (\text {2S})\text {K}_\mathrm{S}^{0} \uppi ^+\uppi ^-$$ B 0 → ψ ( 2S ) K S 0 π + π - decays are observed with significances exceeding 5 standard deviations. The resulting branching fraction ratios, measured for the first time, correspond to $${\mathcal {B}}(\text {B}^{0}_{\mathrm{s}} \rightarrow \uppsi (\text {2S})K_\mathrm{S}^{0})/{\mathcal {B}}(\text {B}^{0}\rightarrow \uppsi (\text {2S})K_\mathrm{S}^{0}) = (3.33 \pm 0.69 (\text {stat})\, \pm 0.11\,(\text {syst}) \pm 0.34\,(f_{\mathrm{s}}/f_{\mathrm{d}})) \times 10^{-2}$$ B ( B s 0 → ψ ( 2S ) K S 0 ) / B ( B 0 → ψ ( 2S ) K S 0 ) = ( 3.33 ± 0.69 ( stat ) ± 0.11 ( syst ) ± 0.34 ( f s / f d ) ) × 10 - 2 and $${\mathcal {B}}(\text {B}^{0} \rightarrow \uppsi (\text {2S})\text {K}_\mathrm{S}^{0} \uppi ^{+} \uppi ^{-})/ {\mathcal {B}}(\text {B}^{0} \rightarrow \uppsi (\text {2S})\text {K}^{0}_{\mathrm{S}}) = 0.480 \pm 0.013\,(\text {stat}) \pm 0.032\,(\text {syst})$$ B ( B 0 → ψ ( 2S ) K S 0 π + π - ) / B ( B 0 → ψ ( 2S ) K S 0 ) = 0.480 ± 0.013 ( stat ) ± 0.032 ( syst ) , where the last uncertainty in the first ratio is related to the uncertainty in the ratio of production cross sections of $$\hbox {B}^{0}_{\mathrm{s}}$$ B s 0 and $$\hbox {B}^{0}$$ B 0 mesons, $$f_{\mathrm{s}}/f_{\mathrm{d}}$$ f s / f d .
Astrophysics, Nuclear and particle physics. Atomic energy. Radioactivity
Due to the accelerated expansion of the universe, the possibilities for the formation of singularities has changed from the classical Big Bang and Big Crunch singularities to include a number of new scenarios. In recent papers it has been shown that such singularities may appear in inflationary cosmological models with a fractional power scalar field potential. In this paper we enlarge the analysis of singularities in scalar field cosmological models by the use of generalised power expansions of their Hubble scalars and their scalar fields in order to describe all possible models leading to a singularity, finding other possible cases. Unless a negative scalar field potential is considered, all singularities are weak and of type IV.
We construct global phase portraits of inflationary dynamics in teleparallel gravity models with a scalar field nonminimally coupled to torsion scalar. The adopted set of variables can clearly distinguish between different asymptotic states as fixed points, including the kinetic and inflationary regimes. The key role in the description of inflation is played by the heteroclinic orbits that run from the asymptotic saddle points to the late time attractor point and are approximated by nonminimal slow roll conditions. To seek the asymptotic fixed points, we outline a heuristic method in terms of the “effective potential” and “effective mass”, which can be applied for any nonminimally coupled theories. As particular examples, we study positive quadratic nonminimal couplings with quadratic and quartic potentials and note how the portraits differ qualitatively from the known scalar-curvature counterparts. For quadratic models, inflation can only occur at small nonminimal coupling to torsion, as for larger coupling, the asymptotic de Sitter saddle point disappears from the physical phase space. Teleparallel models with quartic potentials are not viable for inflation at all, since for small nonminimal coupling, the asymptotic saddle point exhibits weaker than exponential expansion, and for larger coupling, it also disappears.
We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that, in addition to the tetrad or metric, they employ a flat connection as additional field variable, but dthey iffer by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanishing curvature, torsion, and nonmetricity, either by imposing them a priori and correspondingly restricting the variation of the action when the field equations are derived, or by using Lagrange multipliers. Special care must be taken, since these conditions form non-holonomic constraints. Here, we explicitly show that all of the aforementioned approaches are equivalent, and that the same set of field equations is obtained, independently of the choice of the geometric formulation and variation procedure. We further discuss the consequences arising from the diffeomorphism invariance of the gravitational action, and show how they establish relations between the gravitational field equations.
The role that is assigned to elementary particle physics in high school education differs all over the world. Even within a single country like Germany there are major differences in the depth of discussion. To help teachers to address the fundamental concepts of elementary particles and their interactions and to reduce the subject didactically with respect to the given general conditions, Netzwerk Teilchenwelt has developed teaching material with an innovative didactic spiral approach that points out connections to common contents of physics curricula and is focused on the fundamental interactions of the Standard Model of particle physics and the charges that generate them. This concept is being presented and discussed during in-service trainings for teachers all over Germany. Since 2017, roughly 290 teachers have been trained in at least two-day courses and enabled to teach this approach.
Nuclear–nuclear collisions at energies attainable at the large accelerators RHIC and the LHC are an ideal environment to study nuclear matter under extreme conditions of high temperature and energy density. One of the most important probes of such nuclear matter is the study of production of jets. In this article, several jet shape observables in Au+Au collisions at the center of mass energy per nucleon–nucleon pair of <inline-formula> <math display="inline"> <semantics> <msqrt> <msub> <mi>s</mi> <mrow> <mi>N</mi> <mi>N</mi> </mrow> </msub> </msqrt> </semantics> </math> </inline-formula> = 200 GeV simulated in the Monte Carlo generator JEWEL are presented. Jets were reconstructed using the anti-<inline-formula> <math display="inline"> <semantics> <msub> <mi>k</mi> <mi>T</mi> </msub> </semantics> </math> </inline-formula> algorithm and their shapes were studied as a function of the jet-resolution parameter <i>R</i>, transverse momentum <inline-formula> <math display="inline"> <semantics> <msub> <mi>p</mi> <mi>T</mi> </msub> </semantics> </math> </inline-formula> and collision centrality.
A measurement is presented of the associated production of a single top quark and a Z boson. The study uses data from proton–proton collisions at s=13TeV recorded by the CMS experiment, corresponding to an integrated luminosity of 35.9 fb−1. Using final states with three leptons (electrons or muons), the tZq production cross section is measured to be σ(pp→tZq→Wbℓ+ℓ−q)=123−31+33(stat)−23+29(syst)fb, where ℓ stands for electrons, muons, or τ leptons, with observed and expected significances of 3.7 and 3.1 standard deviations, respectively. Keywords: SM, Single top, Cross section, tZq