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The $Zα^2$ correction to superallowed beta decays in effective field theory and implications for $|V_{ud}|$
Authors:
Zehua Cao,
Richard J. Hill,
Ryan Plestid,
Peter Vander Griend
Abstract:
Superallowed ($0^+\rightarrow0^+$) beta decays currently provide the most precise extraction of quark mixing in the Standard Model. Their interpretation as a measurement of $|V_{ud}|$ relies on a reliable first-principles computation of QED radiative corrections expressed as a series in $Zα$ and $α$. In this work, we provide the first model-independent result for two-loop, $O(Zα^2)$, long-distance…
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Superallowed ($0^+\rightarrow0^+$) beta decays currently provide the most precise extraction of quark mixing in the Standard Model. Their interpretation as a measurement of $|V_{ud}|$ relies on a reliable first-principles computation of QED radiative corrections expressed as a series in $Zα$ and $α$. In this work, we provide the first model-independent result for two-loop, $O(Zα^2)$, long-distance radiative corrections where the nuclei are treated as heavy point-like particles.
We use renormalization group analysis to obtain new results at $O(Zα^3)$ for the coefficient of double-logarithms in the ratio of the maximal beta energy to the inverse nuclear size, $\Em/R^{-1}$. We use the Kinoshita-Lee-Nauenberg theorem to obtain new results at $O(Z^2α^3)$ for the coefficient of logarithms in the ratio of maximal beta energy to the electron mass, $\log(2\Em/\me)$. We identify a structure-dependent, and therefore short-distance, contribution to the traditional $Zα^2$ correction that should be revisited.. We provide the first comprehensive update to the long-distance corrections in almost forty years and comment on the impact of our findings for extractions of $|V_{ud}|$. We find that shifts in the long-distance corrections are $2.5\times$ larger than past estimates of their uncertainty, $1.5\times$ larger than the statistical uncertainty from the combined fit of superallowed decays, and about $1/2$ the size of estimated systematic error, which stems dominantly from nuclear structure effects.
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Submitted 7 November, 2025;
originally announced November 2025.
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Factorization and resummation of QED radiative corrections for neutron beta decay
Authors:
Zehua Cao,
Richard J. Hill,
Ryan Plestid,
Peter Vander Griend
Abstract:
Details of the two-loop analysis of long-distance QED radiative corrections to neutron beta decay are presented. Explicit expressions are given for hard, jet, and soft functions appearing in the factorization formula that describes the small mass/large energy limit. Power corrections, cancellation of singularities in the small mass expansion, renormalization scheme dependence, and bound state effe…
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Details of the two-loop analysis of long-distance QED radiative corrections to neutron beta decay are presented. Explicit expressions are given for hard, jet, and soft functions appearing in the factorization formula that describes the small mass/large energy limit. Power corrections, cancellation of singularities in the small mass expansion, renormalization scheme dependence, and bound state effects are discussed. The results impact the determination of $|V_{ud}|$ from the measured neutron lifetime.
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Submitted 7 August, 2025;
originally announced August 2025.
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Bottomonium suppression in pNRQCD and open quantum system approach
Authors:
Ajaharul Islam,
Nora Brambilla,
Miguel Ángel Escobedo,
Michael Strickland,
Antonio Vairo,
Peter Vander Griend
Abstract:
By employing the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory within an open quantum system framework, we derive a Lindblad equation governing the evolution of the heavy-quarkonium reduced density matrix, accurate to next-to-leading order (NLO) in the ratio of the state's binding energy to the medium's temperature [1]. The derived NLO Lindblad equation provides…
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By employing the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory within an open quantum system framework, we derive a Lindblad equation governing the evolution of the heavy-quarkonium reduced density matrix, accurate to next-to-leading order (NLO) in the ratio of the state's binding energy to the medium's temperature [1]. The derived NLO Lindblad equation provides a more reliable description of heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological applications, we numerically solve this equation using the quantum trajectories algorithm. By averaging over Monte Carlo-sampled quantum jumps, we obtain solutions without truncation in the angular momentum quantum number of the considered states. Our analysis highlights the importance of quantum jumps in the nonequilibrium evolution of bottomonium states within the quark-gluon plasma [2]. Additionally, we demonstrate that the quantum regeneration of singlet states from octet configurations is essential to explain experimental observations of bottomonium suppression. The heavy-quarkonium transport coefficients used in our study align with recent lattice QCD determinations.
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Submitted 28 March, 2025;
originally announced March 2025.
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The Fermi function and the neutron's lifetime
Authors:
Peter Vander Griend,
Zehua Cao,
Richard Hill,
Ryan Plestid
Abstract:
The traditional Fermi function ansatz for nuclear beta decay describes enhanced perturbative effects in the limit of large nuclear charge $Z$ and/or small electron velocity $β$. We define and compute the quantum field theory object that replaces this ansatz for neutron beta decay, where neither of these limits hold. We present a new factorization formula that applies in the limit of small electron…
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The traditional Fermi function ansatz for nuclear beta decay describes enhanced perturbative effects in the limit of large nuclear charge $Z$ and/or small electron velocity $β$. We define and compute the quantum field theory object that replaces this ansatz for neutron beta decay, where neither of these limits hold. We present a new factorization formula that applies in the limit of small electron mass, analyze the components of this formula through two loop order, and resum perturbative corrections that are enhanced by large logarithms. We apply our results to the neutron lifetime, supplying the first two-loop input to the long-distance corrections. Our result can be summarized as \begin{equation*}
τ_n \times |V_{ud}|^2\big[1+3λ^2\big]\big[1+Δ_R\big]
=
\frac{5263.284(17)\,{\rm s}}
{1 + 27.04(7)\times 10^{-3} }~, \end{equation*} with $|V_{ud}|$ the up-down quark mixing parameter, $τ_n$ the neutron's lifetime, $λ$ the ratio of axial to vector charge, and $Δ_R$ the short-distance matching correction. We find a shift in the long-distance radiative corrections compared to previous work, and discuss implications for extractions of $|V_{ud}|$ and tests of the Standard Model.
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Submitted 18 August, 2025; v1 submitted 29 January, 2025;
originally announced January 2025.
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Bottomonium suppression from the three-loop QCD potential
Authors:
Nora Brambilla,
Tom Magorsch,
Michael Strickland,
Antonio Vairo,
Peter Vander Griend
Abstract:
We compute the suppression of bottomonium in the quark-gluon plasma using the three-loop QCD static potential. The potential describes the spin-averaged bottomonium spectrum below threshold with a less than 1% error. Within potential nonrelativistic quantum chromodynamics and an open quantum systems framework, we compute the evolution of the bottomonium density matrix. The values of the quarkonium…
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We compute the suppression of bottomonium in the quark-gluon plasma using the three-loop QCD static potential. The potential describes the spin-averaged bottomonium spectrum below threshold with a less than 1% error. Within potential nonrelativistic quantum chromodynamics and an open quantum systems framework, we compute the evolution of the bottomonium density matrix. The values of the quarkonium transport coefficients are obtained from lattice QCD measurements of the bottomonium in-medium width and thermal mass shift; we additionally include for the first time a vacuum contribution to the dispersive coefficient $γ$. Using the three-loop potential and the values of the heavy quarkonium transport coefficients, we find that the resulting bottomonium nuclear modification factor is consistent with experimental observations, while at the same time reproducing the lattice measurements of the in-medium width.
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Submitted 22 March, 2024;
originally announced March 2024.
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Comparative Study of Quarkonium Transport in Hot QCD Matter
Authors:
A. Andronic,
P. B. Gossiaux,
P. Petreczky,
R. Rapp,
M. Strickland,
J. P. Blaizot,
N. Brambilla,
P. Braun-Munzinger,
B. Chen,
S. Delorme,
X. Du,
M. A. Escobedo,
E. G. Ferreiro,
A. Jaiswal,
A. Rothkopf,
T. Song,
J. Stachel,
P. Vander Griend,
R. Vogt,
B. Wu,
J. Zhao,
X. Yao
Abstract:
This document summarizes the efforts of the EMMI Rapid Reaction Task Force on "Suppression and (re)generation of quarkonium in heavy-ion collisions at the LHC", centered around their 2019 and 2022 meetings. It provides a review of existing experimental results and theoretical approaches, including lattice QCD calculations and semiclassical and quantum approaches for the dynamical evolution of quar…
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This document summarizes the efforts of the EMMI Rapid Reaction Task Force on "Suppression and (re)generation of quarkonium in heavy-ion collisions at the LHC", centered around their 2019 and 2022 meetings. It provides a review of existing experimental results and theoretical approaches, including lattice QCD calculations and semiclassical and quantum approaches for the dynamical evolution of quarkonia in the quark-gluon plasma as probed in high-energy heavy-ion collisions. The key ingredients of the transport models are itemized to facilitate comparisons of calculated quantities such as reaction rates, binding energies, and nuclear modification factors. A diagnostic assessment of the various results is attempted and coupled with an outlook for the future.
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Submitted 6 February, 2024;
originally announced February 2024.
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Regeneration of bottomonia in an open quantum systems approach
Authors:
Nora Brambilla,
Miguel Ángel Escobedo,
Ajaharul Islam,
Michael Strickland,
Anurag Tiwari,
Antonio Vairo,
Peter Vander Griend
Abstract:
We demonstrate the importance of quantum jumps in the nonequilibrium evolution of bottomonium states in the quark-gluon plasma. Based on nonrelativistic effective field theory and the open quantum system framework, we evolve the density matrix of color singlet and octet pairs. We show that quantum regeneration of singlet states from octet configurations is necessary to understand experimental resu…
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We demonstrate the importance of quantum jumps in the nonequilibrium evolution of bottomonium states in the quark-gluon plasma. Based on nonrelativistic effective field theory and the open quantum system framework, we evolve the density matrix of color singlet and octet pairs. We show that quantum regeneration of singlet states from octet configurations is necessary to understand experimental results for the suppression of both bottomonium ground and excited states. The values of the heavy-quarkonium transport coefficients used are consistent with recent lattice QCD determinations.
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Submitted 8 August, 2023; v1 submitted 23 February, 2023;
originally announced February 2023.
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Heavy quarkonium dynamics at next-to-leading order in the binding energy over temperature
Authors:
Nora Brambilla,
Miguel Ángel Escobedo,
Ajaharul Islam,
Michael Strickland,
Anurag Tiwari,
Antonio Vairo,
Peter Vander Griend
Abstract:
Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-qu…
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Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological application, we numerically solve the resulting NLO Lindblad equation using the quantum trajectories algorithm. To achieve this, we map the solution of the three-dimensional Lindblad equation to the solution of an ensemble of one-dimensional Schrödinger evolutions with Monte-Carlo sampled quantum jumps. Averaging over the Monte-Carlo sampled quantum jumps, we obtain the solution to the NLO Lindblad equation without truncation in the angular momentum quantum number of the states considered. We also consider the evolution of the system using only the complex effective Hamiltonian without stochastic jumps and find that this provides a reliable approximation for the ground state survival probability at LO and NLO. Finally, we make comparisons with our prior leading-order pNRQCD results and experimental data available from the ATLAS, ALICE, and CMS collaborations.
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Submitted 20 May, 2022;
originally announced May 2022.
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Bottomonium observables in an open quantum system using the quantum trajectories method
Authors:
Peter Vander Griend
Abstract:
We solve the Lindblad equation describing the Brownian motion of a Coulombic heavy quark-antiquark pair in a strongly coupled quark gluon plasma using the Monte Carlo wave function method. The Lindblad equation has been derived in the framework of pNRQCD and fully accounts for the quantum and non-Abelian nature of the system. The hydrodynamics of the plasma is realistically implemented through a 3…
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We solve the Lindblad equation describing the Brownian motion of a Coulombic heavy quark-antiquark pair in a strongly coupled quark gluon plasma using the Monte Carlo wave function method. The Lindblad equation has been derived in the framework of pNRQCD and fully accounts for the quantum and non-Abelian nature of the system. The hydrodynamics of the plasma is realistically implemented through a 3+1D dissipative hydrodynamics code. We compute the bottomonium nuclear modification factor and elliptic flow and compare with the most recent LHC data. The computation does not rely on any free parameter, as it depends on two transport coefficients that have been evaluated independently in lattice QCD. Our final results, which include late-time feed down of excited states, agree well with the available data from LHC 5.02 TeV PbPb collisions.
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Submitted 26 November, 2021;
originally announced November 2021.
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Bottomonium production in heavy-ion collisions using quantum trajectories: Differential observables and momentum anisotropy
Authors:
Nora Brambilla,
Miguel Ángel Escobedo,
Michael Strickland,
Antonio Vairo,
Peter Vander Griend,
Johannes Heinrich Weber
Abstract:
We report predictions for the suppression and elliptic flow of the $Υ(1S)$, $Υ(2S)$, and $Υ(3S)$ as a function of centrality and transverse momentum in ultra-relativistic heavy-ion collisions. We obtain our predictions by numerically solving a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix derived using potential nonrelativistic QCD and the formalism of open qua…
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We report predictions for the suppression and elliptic flow of the $Υ(1S)$, $Υ(2S)$, and $Υ(3S)$ as a function of centrality and transverse momentum in ultra-relativistic heavy-ion collisions. We obtain our predictions by numerically solving a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix derived using potential nonrelativistic QCD and the formalism of open quantum systems. To numerically solve the Lindblad equation, we make use of a stochastic unraveling called the quantum trajectories algorithm. This unraveling allows us to solve the Lindblad evolution equation efficiently on large lattices with no angular momentum cutoff. The resulting evolution describes the full 3D quantum and non-abelian evolution of the reduced density matrix for bottomonium states. We expand upon our previous work by treating differential observables and elliptic flow; this is made possible by a newly implemented Monte-Carlo sampling of physical trajectories. Our final results are compared to experimental data collected in $\sqrt{s_{NN}} = 5.02$ TeV Pb-Pb collisions by the ALICE, ATLAS, and CMS collaborations.
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Submitted 13 July, 2021;
originally announced July 2021.
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QTRAJ 1.0: A Lindblad equation solver for heavy-quarkonium dynamics
Authors:
Hisham Ba Omar,
Miguel Ángel Escobedo,
Ajaharul Islam,
Michael Strickland,
Sabin Thapa,
Peter Vander Griend,
Johannes Heinrich Weber
Abstract:
We introduce an open-source package called QTraj that solves the Lindblad equation for heavy-quarkonium dynamics using the quantum trajectories algorithm. The package allows users to simulate the suppression of heavy-quarkonium states using externally-supplied input from 3+1D hydrodynamics simulations. The code uses a split-step pseudo-spectral method for updating the wave-function between jumps,…
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We introduce an open-source package called QTraj that solves the Lindblad equation for heavy-quarkonium dynamics using the quantum trajectories algorithm. The package allows users to simulate the suppression of heavy-quarkonium states using externally-supplied input from 3+1D hydrodynamics simulations. The code uses a split-step pseudo-spectral method for updating the wave-function between jumps, which is implemented using the open-source multi-threaded FFTW3 package. This allows one to have manifestly unitary evolution when using real-valued potentials. In this paper, we provide detailed documentation of QTraj 1.0, installation instructions, and present various tests and benchmarks of the code.
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Submitted 7 January, 2022; v1 submitted 13 July, 2021;
originally announced July 2021.
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Bottomonium suppression in an open quantum system using the quantum trajectories method
Authors:
Nora Brambilla,
Miguel Ángel Escobedo,
Michael Strickland,
Antonio Vairo,
Peter Vander Griend,
Johannes Heinrich Weber
Abstract:
We solve the Lindblad equation describing the Brownian motion of a Coulombic heavy quark-antiquark pair in a strongly coupled quark-gluon plasma using the highly efficient Monte Carlo wave-function method. The Lindblad equation has been derived in the framework of pNRQCD and fully accounts for the quantum and non-Abelian nature of the system. The hydrodynamics of the plasma is realistically implem…
▽ More
We solve the Lindblad equation describing the Brownian motion of a Coulombic heavy quark-antiquark pair in a strongly coupled quark-gluon plasma using the highly efficient Monte Carlo wave-function method. The Lindblad equation has been derived in the framework of pNRQCD and fully accounts for the quantum and non-Abelian nature of the system. The hydrodynamics of the plasma is realistically implemented through a 3+1D dissipative hydrodynamics code. We compute the bottomonium nuclear modification factor and compare with the most recent LHC data. The computation does not rely on any free parameter, as it depends on two transport coefficients that have been evaluated independently in lattice QCD. Our final results, which include late-time feed down of excited states, agree well with the available data from LHC 5.02 TeV PbPb collisions.
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Submitted 28 April, 2021; v1 submitted 2 December, 2020;
originally announced December 2020.
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Transport coefficients from in medium quarkonium dynamics
Authors:
Nora Brambilla,
Miguel A. Escobedo,
Antonio Vairo,
Peter Vander Griend
Abstract:
The in medium dynamics of heavy particles are governed by transport coefficients. The heavy quark momentum diffusion coefficient, $κ$, is an object of special interest in the literature, but one which has proven notoriously difficult to estimate, despite the fact that it has been computed by weak-coupling methods at next-to-leading order accuracy, and by lattice simulations of the pure SU(3) gauge…
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The in medium dynamics of heavy particles are governed by transport coefficients. The heavy quark momentum diffusion coefficient, $κ$, is an object of special interest in the literature, but one which has proven notoriously difficult to estimate, despite the fact that it has been computed by weak-coupling methods at next-to-leading order accuracy, and by lattice simulations of the pure SU(3) gauge theory. Another coefficient, $γ$, has been recently identified. It can be understood as the dispersive counterpart of $κ$. Little is known about $γ$. Both $κ$ and $γ$ are, however, of foremost importance in heavy quarkonium physics as they entirely determine the in and out of equilibrium dynamics of quarkonium in a medium, if the evolution of the density matrix is Markovian, and the motion, quantum Brownian; the medium could be a strongly or weakly coupled plasma. In this paper, using the relation between $κ$, $γ$ and the quarkonium in medium width and mass shift respectively, we evaluate the two coefficients from existing 2+1 flavor lattice QCD data. The resulting range for $κ$ is consistent with earlier determinations, the one for $γ$ is the first non-perturbative determination of this quantity.
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Submitted 18 September, 2019; v1 submitted 19 March, 2019;
originally announced March 2019.
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Asymptotic safety in the dark
Authors:
Astrid Eichhorn,
Aaron Held,
Peter Vander Griend
Abstract:
We explore the Renormalization Group flow of massive uncharged fermions -- a candidate for dark matter -- coupled to a scalar field through a Higgs portal. We find that fermionic fluctuations can lower the bound on the scalar mass that arises from vacuum stability. Further, we discuss that despite the perturbative nonrenormalizability of the model, it could be ultraviolet complete at an asymptotic…
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We explore the Renormalization Group flow of massive uncharged fermions -- a candidate for dark matter -- coupled to a scalar field through a Higgs portal. We find that fermionic fluctuations can lower the bound on the scalar mass that arises from vacuum stability. Further, we discuss that despite the perturbative nonrenormalizability of the model, it could be ultraviolet complete at an asymptotically safe fixed point. In our approximation, this simple model exhibits two mechanisms for asymptotic safety: a balance of fermionic and bosonic fluctuations generates a fixed point in the scalar self-interaction; asymptotic safety in the portal coupling is triggered through a balance of canonical scaling and quantum fluctuations. As a consequence of asymptotic safety in the dark sector, the low-energy value of the portal coupling could become a function of the dark fermion mass and the scalar mass, thereby reducing the viable parameter space of the model.
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Submitted 23 February, 2018;
originally announced February 2018.