{"arxiv_id":"hep-ph/0202122","incoming":{"papers":131,"sentences":199,"groups":[{"kind":"method_or_tool","label":"Uses its method or tool","papers":5,"sentences":6,"quotes":[{"arxiv_id":"2202.12673","title":"$\\mathbf{{N}_{eff}}$ from Excited DM state","url":"/paper/2202.12673","section":"The effective number of neutrinos","sentence":"We neglect the influence of chemical potential and write down the temperature evolution equations from Eq.([ref]) and Eq.([ref]) by assuming that the temperature of three generation active neutrinos are the same, [equation] where $δρ_ν_e/δt$ and $δρ_ν_μ/δt$ are collision terms induced by neutrino interactions calculated in the Refs. [ref] .","intent_p":0.91,"contests_p":null},{"arxiv_id":"1812.05605","title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","url":"/paper/1812.05605","section":"Derivation of the temperature evolution equations","sentence":"Here we use the matrix elements outlined in Table 2 of [ref] and follow the integration method developed in [ref] to obtain the relevant rates[The reader is referred to the appendices of [ref] and [ref] which very recently appeared in the literature and which explain in more detail and with explicit formulae how to perform the different integration steps.].","intent_p":0.96,"contests_p":null},{"arxiv_id":"2201.00854","title":"Imprints of MeV Scale Hidden Dark Sector at Planck Data","url":"/paper/2201.00854","section":"Boltzmann equations and Numerical analysis","sentence":"Following [ref] , we have computed the collision terms for SM $ν$ decoupling, considering Fermi-Dirac (FD) distribution function for all the SM fermions and appropriate Pauli blocking factors.","intent_p":1.0,"contests_p":null},{"arxiv_id":"0810.5133","title":"Sterile Neutrinos in Light of Recent Cosmological and Oscillation Data: a Multi-Flavor Scheme Approach","url":"/paper/0810.5133","section":"Sterile Neutrino Production in the Multi-Flavor Approach","sentence":"The coherence loss in the evolution of the non-diagonal terms is given by $ρ̇ = - Γρ$ [ref] and a simplified but accurate way to describe the neutrino production and destruction is given by the anticommutator term in Eq. ([ref]) [ref] .","intent_p":0.87,"contests_p":null},{"arxiv_id":"2212.09759","title":"The domain of a cannibal dark matter","url":"/paper/2212.09759","section":"Evolution of the hidden sector","sentence":"Eqs. [ref] and [ref] are separated by the neutrino-decoupling temperature $T_νd$, which we calculate by following [ref] , i.e. by solving $T_νd^5/T_νd,SM^5 = H(T_νd)/H(T_νd, SM)$ with the neutrino-decoupling temperature $T_νd,SM ≈1.4 MeV$ [ref] in the SM.","intent_p":0.87,"contests_p":null}]},{"kind":"data_used","label":"Uses its data or values","papers":5,"sentences":5,"quotes":[{"arxiv_id":"2403.02721","title":"Light Thermal Dark Matter Beyond $p$-Wave Annihilation in Minimal Higgs Portal Model","url":"/paper/2403.02721","section":"Cosmological and astrophysical constraints","sentence":"Following Ref. [ref] , we apply a 95% upper limit $N_eff<0.17$ [ref] as a half Gaussian likelihood.","intent_p":0.94,"contests_p":null},{"arxiv_id":"2011.06519","title":"Updated BBN constraints on electromagnetic decays of MeV-scale particles","url":"/paper/2011.06519","section":"Evolution of the decaying particle $ϕ$","sentence":"[ref] over $∫^̣3 p E g_ϕ/(2π)^3$.[However, we find that for numerical calculations it is best to determine $q̇_ϕ(t)$ directly from its definition $q̇_ϕ(t) = ρ̇_ϕ(t) + 3H(t)[ ρ_ϕ(t) + _ϕ(t)]$, since the numerical evaluation of $f_ϕ(t, p) - f̅_ϕ(t, p)$ can easily lead to cancellation of significant digits.] Given that the reaction rate which keeps the neutrinos in thermal equilibrium scales as $T^5$, it is a good approximation to assume that neutrino decoupling happens instantaneously when $T^5 / H$ reaches the value it has at neutrino decoupling only considering the SM degrees of freedom, i.e. $T_ν^5 / H (T_ν) ≃1.62 ×10^-10 GeV^4$ taking $T_ν^≃1.4 MeV$ [ref] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"1812.05605","title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","url":"/paper/1812.05605","section":"Neutrino collision terms in the SM in the Maxwell-Boltzmann approximation","sentence":"By assuming Maxwell-Boltzmann statistics for all the particles involved in the process, neglecting the electron mass, neglecting statistical factors, and by considering the matrix elements reported in Table 2 of [ref] we arrive to the following annihilation collision term: [equation] where $G_F$ is the Fermi constant, $g_L^ν_e = 1/2 + s_W^2$, $g_R^ν_e = s_W^2$, $g_L^ν_μ, τ = -1/2 + s_W^2$, $g_L^ν_μ, τ = s_W^2$, and $s_W$ is the sinus of the Weinberg angle.","intent_p":0.88,"contests_p":null},{"arxiv_id":"2005.01629","title":"Dirac neutrinos and $N_{{\\rm eff}}$","url":"/paper/2005.01629","section":"Boltzmann equation","sentence":"Numerical calculations find that the distortion is about [ref]","intent_p":0.88,"contests_p":null},{"arxiv_id":"1609.07081","title":"Implication of Sterile Fermions in Particle Physics and Cosmology","url":"/paper/1609.07081","section":"Relic neutrino abundance","sentence":"A more refined analysis taking into account the flavour effects gives the decoupling temperatures [ref] T_ν_e^dec = 1.34 MeV, T_ν_μ,τ^dec = 1.5 MeV.","intent_p":0.93,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":2,"sentences":2,"quotes":[{"arxiv_id":"1607.02797","title":"The surprising influence of late charged current weak interactions on Big Bang Nucleosynthesis","url":"/paper/1607.02797","section":"The","sentence":"If we take $𝒩≃4!$, $=10.75$, and $Y_p≃1$, we calculate $≃0.9 MeV$ in line with the nominal estimate of $0.7 MeV$ for WFO [see Eq. (57) in Ref. [ref] and references therein].","intent_p":0.84,"contests_p":null},{"arxiv_id":"1812.05605","title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","url":"/paper/1812.05605","section":"Neutrino collision terms in the SM in the Maxwell-Boltzmann approximation","sentence":"The scattering rate between neutrinos and electrons can similarly be obtained to give: [equation] If we were to consider only the forward part of the annihilation and scattering processes ($ν̅ ν→e^+ e^-, νe^±→νe^±$) between electrons and neutrinos we will recover Equation 63 in [ref] .","intent_p":0.95,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites it as support","papers":2,"sentences":2,"quotes":[{"arxiv_id":"2111.12726","title":"What can CMB observations tell us about the neutrino distribution function?","url":"/paper/2111.12726","section":"BBN and model building","sentence":"Firstly, neutrinos were tightly coupled to the thermal plasma for temperatures $T≳2 MeV$ [ref] , and secondly, the successful prediction of the light element abundances require neutrinos to have a Fermi-Dirac-like distribution at the time at which the proton-to-neutron interactions freeze-out $T∼0.7 MeV$, see [ref] .","intent_p":0.96,"contests_p":null},{"arxiv_id":"1601.05265","title":"Active Dirac Neutrinos via $SU(2)_L$ Doublets in 5d","url":"/paper/1601.05265","section":"Summary and Discussion","sentence":"The sterile fields were decoupled from thermal bath at a sufficient early universe, and the chirality-flip production from the active $SU(2)_L$ doublets, $(ν_i,e_i)^(0)_L$, through the tiny Dirac mass terms are negligible as shown in Ref. [ref] .","intent_p":0.99,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":67,"sentences":77,"quotes":[{"arxiv_id":"2306.05460","title":"Neff in the Standard Model at NLO is 3.043","url":"/paper/2306.05460","section":"Introduction","sentence":"A precision calculation of $N_eff^SM$ requires one to solve the process of neutrino decoupling in the early Universe and this has been a subject of intense study for more than 40 years now, see [ref] for references devoted to this problem, see [ref] for reviews and [ref] for a book.","intent_p":1.0,"contests_p":null},{"arxiv_id":"2111.01096","title":"Cosmological Implications of a Neutrino Mass Detection","url":"/paper/2111.01096","section":"Introduction","sentence":"The prospect that precision cosmological measurements could reveal the effects of non-zero neutrino masses has drawn broad attention from both the cosmology and high energy physics communities [ref] .","intent_p":0.94,"contests_p":null},{"arxiv_id":"1909.13328","title":"Cosmological Dependence of Non-resonantly Produced Sterile Neutrinos","url":"/paper/1909.13328","section":"Non-resonant sterile neutrino production","sentence":"This is called the Dodelson-Widrow mechanism (DW) [ref] because they derived the analytic solution for the relic number density of sterile neutrinos produced in this manner[The original Dodelson-Widrow results were subsequently corrected by a factor of 2 [ref] .] (see also earlier work [ref] ).","intent_p":0.91,"contests_p":null},{"arxiv_id":"1001.4815","title":"Neutrino oscillations: Quantum mechanics vs. quantum field theory","url":"/paper/1001.4815","section":"Introduction","sentence":"In the QFT treatment [ref] , one considers neutrino production, propagation and detection as a single process, described by a tree-level Feynman diagram with the neutrino in the intermediate state (see fig.","intent_p":0.83,"contests_p":null},{"arxiv_id":"1812.05605","title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","url":"/paper/1812.05605","section":"Introduction","sentence":"Given the usefulness of the number of effective neutrino species as a tool to test the SM and its extensions, the neutrino decoupling process in the early Universe has been a subject of intense study [ref] .","intent_p":0.98,"contests_p":null}]}],"unlabelled_sentences":107},"outgoing":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":{"arxiv_id":"hep-ph/0202122","kind":"model","paper_version":2,"prompt_version":"held-up-v3.5","generated_at":"2026-10-27T15:30:58+00:00","loaded_at":"2026-10-28T01:02:25.451517+00:00","evidence":"substantial","account":[],"groups":[{"kind":"data_used","label":"Uses its data or values","papers":2,"entries":[{"p":0.88,"quote":null,"citing":[{"self":false,"year":2018,"title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","arxiv_id":"1812.05605"}],"verified":true,"paraphrase":"Used the matrix elements reported in Table 2 of this paper to arrive at its annihilation collision term."},{"p":0.92,"quote":null,"citing":[{"self":false,"year":2017,"title":"BBN for the LHC: constraints on lifetimes of the Higgs portal scalars","arxiv_id":"1706.01920"}],"verified":true,"paraphrase":"Said that thermal refinements in the interaction rates and phase space tend to yield a lower value of the neutrino decoupling temperature, 1.4, citing this paper."}],"papers_all":7,"self_papers":0,"self_entries":[]},{"kind":"comparison","label":"Compares with it","papers":2,"entries":[{"p":0.95,"quote":null,"citing":[{"self":false,"year":2018,"title":"Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise $N_{\\rm eff}$ evaluation","arxiv_id":"1812.05605"}],"verified":true,"paraphrase":"Compared its scattering rate obtained by considering only the forward part of the annihilation and scattering processes with this paper, recovering Equation 63 in this paper."},{"p":0.84,"quote":null,"citing":[{"self":false,"year":2016,"title":"The surprising influence of late charged current weak interactions on Big Bang Nucleosynthesis","arxiv_id":"1607.02797"}],"verified":true,"paraphrase":"Compared its calculated value of about 0.9 MeV with the nominal estimate of 0.7 MeV for WFO given in Eq. (57) of this paper."}],"papers_all":2,"self_papers":0,"self_entries":[]}],"counts":{"by_kind":{"data_used":7,"comparison":2,"method_or_tool":5,"supports_this_paper":2},"dropped":{"relied_on":0,"unknown_ids":0,"account_lines":0,"contest_vetoed":0,"below_threshold":0,"contest_drawback":0,"contest_without_basis":0,"contest_not_one_sentence":0,"contest_on_group_citation":0},"papers_read":18,"self_papers":2,"citing_papers":143,"sentences_read":20,"citing_sentences":216,"unambiguous_read":20,"typed_independent":14,"independent_papers":141,"by_kind_independent":{"data_used":7,"comparison":2,"method_or_tool":5,"supports_this_paper":2}},"input_sha256":"014a536f02612fb256ffe372e7498505883c038dc3622b69606fa76554f25621","method":"For a paper with five or more citing arXiv papers, Pith selected up to 40 citing sentences its citation classifier typed, read those whose citation markers point at this paper alone, and wrote one paraphrase for each later paper in each group, each resting on one sentence. A later paper is listed under a label only when the classifier gives one of its sentences that label at intent probability 0.8 or more (0.95 for support), Pith's reading of the sentence finds that the citing paper itself did it, and a second check of the paraphrase against the sentence passes. The section lists uses only: building on the paper, using its method, tool, data or values, comparing with it and citing it as support. It does not list criticism or disagreement, so their absence says nothing about how the paper has been received. Code counted the papers, wrote the summary from the listed entries, set the evidence level and linked each entry to its citing paper. Quotes are copied from the citing sentence and shown only when the citing paper's licence is CC BY, CC BY-SA or CC0; otherwise the paraphrase is shown. Papers that share an author with this one are listed apart. A paper with fewer citing papers gets a line counted by code, which names uses only: a later paper whose only label is a contest is counted as not described.","line":null,"summary":"3 later papers by other authors are listed below: 2 used its data or values and 2 compared their own results with it. 1 of them appears under more than one label."},"_links":{"html":"https://pith.science/paper/hep-ph/0202122"}}