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Inclusive P-wave Quarkonium Decay Widths from Lattice QCD and pNRQCD
Authors:
Nora Brambilla,
Viljami Leino,
Julian Mayer-Steudte,
Panayiotis Panayiotou,
Andrea Shindler,
Antonio Vairo,
Xiang-Peng Wang
Abstract:
Inclusive hadronic decay widths remain a long-standing challenge for first-principles QCD. We present a framework combining lattice QCD with strongly-coupled potential nonrelativistic QCD (pNRQCD) to compute inclusive P-wave heavy quarkonium decays to light hadrons. At leading order in the velocity expansion, all nonperturbative effects, apart from the square of the derivative of the wavefunction…
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Inclusive hadronic decay widths remain a long-standing challenge for first-principles QCD. We present a framework combining lattice QCD with strongly-coupled potential nonrelativistic QCD (pNRQCD) to compute inclusive P-wave heavy quarkonium decays to light hadrons. At leading order in the velocity expansion, all nonperturbative effects, apart from the square of the derivative of the wavefunction at the origin, are encoded in a single universal moment of the two-point chromoelectric correlator, which we determine for the first time from a quenched lattice QCD calculation matched to $\overline{\mathrm{MS}}$ via the gradient flow. Combined with perturbative short-distance coefficients and the square of the derivative of the wavefunction at the origin, our result reproduces the observed $χ_{cJ}(1P)$ widths and, at the same time, provides predictions for the unmeasured $χ_{bJ}(nP)$ widths. The framework extends naturally to inclusive decays and production of ordinary and exotic hadrons.
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Submitted 14 July, 2026;
originally announced July 2026.
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Extraction of charmonium branching fractions from $J/ψ\toγη_c$ radiative decays
Authors:
Magnus C. Schaaf,
Antonio Vairo
Abstract:
We assess the tension between theoretical predictions and the values quoted by the Particle Data Group (PDG) for the partial decay width and branching fraction associated with the radiative charmonium decay $J/ψ\toγη_c$. A profile scan over the most recent PDG data depending on the branching fraction $\mathcal{B}(J/ψ\toγη_c)$ suggests that the correlation between measured branching fractions is co…
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We assess the tension between theoretical predictions and the values quoted by the Particle Data Group (PDG) for the partial decay width and branching fraction associated with the radiative charmonium decay $J/ψ\toγη_c$. A profile scan over the most recent PDG data depending on the branching fraction $\mathcal{B}(J/ψ\toγη_c)$ suggests that the correlation between measured branching fractions is compatible with lattice QCD determinations of the partial decay widths $Γ(J/ψ\toγη_c)$ and $Γ(η_c\toγγ)$. We propose a theoretically grounded photon line shape for the radiative decay spectrum and a prescription for the extraction of (product) branching fractions involving the magnetic dipole (M1) transition $J/ψ\toγη_c$. This approach obviates the need to modify the photon energy spectrum line shape using empirical damping functions, as done in the most recent experimental extractions of $\mathcal{B}(J/ψ\toγη_c)$ from the photon line shape, thereby eliminating an inherent ambiguity in the determination of the derived observables.
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Submitted 18 June, 2026;
originally announced June 2026.
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Open-flavor threshold effects on quarkonium spectrum in the BOEFT
Authors:
Nora Brambilla,
Abhishek Mohapatra,
Tommaso Scirpa,
Antonio Vairo
Abstract:
The impact of open-flavor thresholds on the quarkonium spectrum has been a subject of study since the introduction of the Cornell potential and has been quantified through various phenomenological approaches, most notably the $^3P_0$ model. We revisit this problem using the Born--Oppenheimer effective field theory (BOEFT), an effective field theory systematically derived from QCD by exploiting hie…
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The impact of open-flavor thresholds on the quarkonium spectrum has been a subject of study since the introduction of the Cornell potential and has been quantified through various phenomenological approaches, most notably the $^3P_0$ model. We revisit this problem using the Born--Oppenheimer effective field theory (BOEFT), an effective field theory systematically derived from QCD by exploiting hierarchies of energy scales and symmetries. Within the BOEFT, open-flavor threshold effects emerge from the mixing between quarkonium and tetraquark static potentials sharing the same Born--Oppenheimer quantum numbers. The shapes of the static potentials are constrained by lattice QCD calculations. Furthermore, we account for the distinctive behavior of the BOEFT tetraquark static potentials at short and large distances: at short distances they are repulsive, reflecting the color-octet configuration of the heavy quark-antiquark pair, while at large distances they asymptotically approach heavy-light meson-antimeson thresholds. To quantify threshold effects on the quarkonium spectrum below threshold, we solve a set of coupled Schrödinger equations dictated by the BOEFT, whose only free parameter, the adjoint meson mass, is fixed to the mass of the $χ_{c1}(3872)$ state. These coupled equations are solved both in the spin-isospin averaged threshold limit and, for the first time, including the spin splittings of the physical thresholds. We validate our results by computing the same threshold effects as self-energy corrections to the quarkonium propagator. We compare our predictions with existing experimental data and previous literature. Finally, we provide a field-theoretical interpretation of the pair-creation constant $γ$ appearing in the $^3P_0$ model.
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Submitted 14 April, 2026;
originally announced April 2026.
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Strong coupling constant from the 1-loop improved static energy
Authors:
Viljami Leino,
Alexei Bazavov,
Nora Brambilla,
Georg von Hippel,
Andreas S. Kronfeld,
Julian Mayer-Steudte,
Peter Petreczky,
Sipaz Sharma,
Sebastian Steinbeißer,
Antonio Vairo,
Johannes H. Weber
Abstract:
The static energy is an excellent observable for extracting the strong coupling $α_s$ on the lattice. For short distances, the static energy can be calculated both on the lattice using Wilson line correlators, and with perturbation theory up to three loop accuracy with leading ultrasoft log resummation. Comparing the perturbative expression and lattice data allows for precise determination of…
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The static energy is an excellent observable for extracting the strong coupling $α_s$ on the lattice. For short distances, the static energy can be calculated both on the lattice using Wilson line correlators, and with perturbation theory up to three loop accuracy with leading ultrasoft log resummation. Comparing the perturbative expression and lattice data allows for precise determination of $α_s$. We present early results for 1-loop lattice perturbation theory improvement of the Wilson loop and show how it improves the $α_s$ extraction. We present a preliminary reanalysis of the TUMQCD (2+1)-flavor QCD data.
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Submitted 27 April, 2026; v1 submitted 1 April, 2026;
originally announced April 2026.
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Inclusive hadroproduction of $χ_{c1}(3872)$, $X_b$ and pentaquarks
Authors:
Nora Brambilla,
Mathias Butenschoen,
Simon Hibler,
Abhishek Mohapatra,
Antonio Vairo,
Xiangpeng Wang
Abstract:
We use the Born--Oppenheimer effective field theory factorization to compute the inclusive production cross sections of the $χ_{c1}(3872)$ and its partner in the bottomonium sector. In the same framework, we compute the production cross sections of the pentaquark states $P_{c\bar{c}}(4312)^+$, $P_{c\bar{c}}(4457)^+$, $P_{c\bar{c}}(4380)^+$ and $P_{c\bar{c}}(4440)^+$ within two possible scenarios f…
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We use the Born--Oppenheimer effective field theory factorization to compute the inclusive production cross sections of the $χ_{c1}(3872)$ and its partner in the bottomonium sector. In the same framework, we compute the production cross sections of the pentaquark states $P_{c\bar{c}}(4312)^+$, $P_{c\bar{c}}(4457)^+$, $P_{c\bar{c}}(4380)^+$ and $P_{c\bar{c}}(4440)^+$ within two possible scenarios for the Born--Oppenheimer potentials. Also for pentaquarks, we extend the results to the bottomonium sector. All our results are genuine predictions that do not involve fits to prompt hadroproduction data.
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Submitted 16 February, 2026;
originally announced February 2026.
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Born-Oppenheimer EFT: a unified description of ordinary and exotic quarkonia
Authors:
Antonio Vairo
Abstract:
We show how the Born-Oppenheimer effective field theory (BOEFT) provides a unified description of ordinary and exotic quarkonia grounded on the non-relativistic expansions of QCD and supplemented with lattice QCD inputs. We apply BOEFT to tetraquarks, pentaquarks, quarkonium hybrids and to assess threshold effects in the quarkonium spectrum.
We show how the Born-Oppenheimer effective field theory (BOEFT) provides a unified description of ordinary and exotic quarkonia grounded on the non-relativistic expansions of QCD and supplemented with lattice QCD inputs. We apply BOEFT to tetraquarks, pentaquarks, quarkonium hybrids and to assess threshold effects in the quarkonium spectrum.
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Submitted 3 December, 2025; v1 submitted 16 September, 2025;
originally announced September 2025.
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Unravelling Pentaquarks with Born--Oppenheimer effective theory
Authors:
Nora Brambilla,
Abhishek Mohapatra,
Antonio Vairo
Abstract:
The hidden-charm pentaquark states $P_{c\bar{c}}\left(4312\right)^+$, $P_{c\bar{c}}\left(4380\right)^+$, $P_{c\bar{c}}\left(4440\right)^+$, and $P_{c\bar{c}}\left(4457\right)^+$, all with isospin $I = 1/2$, were discovered by the LHCb collaboration in the decay process $Λ_b^0 \to J/ψp K^-$. Although their quantum numbers remain undetermined, these states have generated significant theoretical inte…
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The hidden-charm pentaquark states $P_{c\bar{c}}\left(4312\right)^+$, $P_{c\bar{c}}\left(4380\right)^+$, $P_{c\bar{c}}\left(4440\right)^+$, and $P_{c\bar{c}}\left(4457\right)^+$, all with isospin $I = 1/2$, were discovered by the LHCb collaboration in the decay process $Λ_b^0 \to J/ψp K^-$. Although their quantum numbers remain undetermined, these states have generated significant theoretical interest. We analyze their spectrum and decay patterns-including those of their spin partners-within the Born--Oppenheimer effective field theory (BOEFT), a framework grounded in QCD. At leading order in BOEFT, we identify these pentaquark states as bound states in BO potentials that exhibit at short-distance a repulsive octet behavior and a nonperturbative shift due to the adjoint baryons masses, while asymptotically approaching the $Σ_c\bar{D}$ threshold. We further incorporate ${\cal O}(1/m_Q)$ spin-dependent corrections to compute pentaquark multiplet spin splittings. Based on the spectrum, semi-inclusive decay widths to $J/ψ$ and $η_c$, and the decay width ratios to $Λ_c\bar{D}$ and $Λ_c\bar{D}^*$, we provide the first theoretical predictions for the adjoint baryon masses, which can be confirmed by future lattice QCD studies. Moreover, our analysis supports the quantum number assignments: $J^{P} = (1/2)^-$ for $P_{c\bar{c}}\left(4312\right)^+$, $(3/2)^-$ for $P_{c\bar{c}}\left(4380\right)^+$, $(1/2)^-$ for $P_{c\bar{c}}\left(4457\right)^+$, and $(3/2)^-$ for $P_{c\bar{c}}\left(4440\right)^+$. We also present results for the lowest bottom pentaquarks.
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Submitted 18 August, 2025;
originally announced August 2025.
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Thermalization of Bottomonium in the Quark-Gluon Plasma
Authors:
Nora Brambilla,
Tom Magorsch,
Antonio Vairo
Abstract:
We study the approach to equilibrium of bottomonium in the quark-gluon plasma within the open quantum system framework. We perform large-scale simulations of the long-time behavior in three dimensions using the quantum trajectory method to observe the emergence of steady states and determine the timescale of thermalization in position-, angular-momentum-, and color-space. We find that the thermali…
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We study the approach to equilibrium of bottomonium in the quark-gluon plasma within the open quantum system framework. We perform large-scale simulations of the long-time behavior in three dimensions using the quantum trajectory method to observe the emergence of steady states and determine the timescale of thermalization in position-, angular-momentum-, and color-space. We find that the thermalization timescale increases with decreasing temperature and decreasing coupling to the medium, which is given by transport coefficients of the medium. Additionally, we observe that the steady states exhibit small corrections to the Gibbs state due to medium interactions and show that these corrections diminish for weaker medium coupling and higher temperature. At a temperature of $450\,$MeV, quarkonium relaxes to a state that is approximately thermal, with the most significant correction being a smaller overlap of the $1S$ state with respect to the Gibbs state. We compare these findings with the master equation obtained at leading order in the expansion of the binding energy over the temperature, which we find to have a trivial steady state.
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Submitted 15 August, 2025;
originally announced August 2025.
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Lattice study of correlators for quarkonium decay
Authors:
Saumen Datta,
Debasish Banerjee,
Nora Brambilla,
Marc Janer,
Viljami Leino,
Julian Mayer-Steudte,
Peter Petreczky,
Balbeer Singh,
Antonio Vairo
Abstract:
While there has been a lot of progress in developing a
formalism for the study of quarkonia in QGP, a nonperturbative
study is still difficult. For bottomonia, where the system size
is much less than the inverse temperature, the interaction of the
system with the medium can be approximated by a dipole interaction with
the color electric field. The decay of the quarkonia can be connected…
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While there has been a lot of progress in developing a
formalism for the study of quarkonia in QGP, a nonperturbative
study is still difficult. For bottomonia, where the system size
is much less than the inverse temperature, the interaction of the
system with the medium can be approximated by a dipole interaction with
the color electric field. The decay of the quarkonia can be connected to
a correlation function of the color electric field.
We present preliminary results from a lattice study of the relevant color
electric field correlator. The structure of the correlator, and its
difference from the corresponding correlator studied for heavy quark
diffusion, is discussed.
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Submitted 27 June, 2025;
originally announced June 2025.
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The chromoelectric adjoint correlators in Euclidean space at next-to-leading order
Authors:
Nora Brambilla,
Panayiotis Panayiotou,
Saga Säppi,
Antonio Vairo
Abstract:
The physics of quarkonium created in heavy-ion collisions is intrinsically connected to the correlation functions of adjoint chromoelectric fields in quantum chromodynamics. We study such correlation functions in a weak-coupling expansion in a thermal medium. We identify three distinct gauge-invariant correlators, and evaluate them to next-to-leading order. Two of the resulting correlators turn ou…
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The physics of quarkonium created in heavy-ion collisions is intrinsically connected to the correlation functions of adjoint chromoelectric fields in quantum chromodynamics. We study such correlation functions in a weak-coupling expansion in a thermal medium. We identify three distinct gauge-invariant correlators, and evaluate them to next-to-leading order. Two of the resulting correlators turn out to be asymmetric. We pinpoint the source of this asymmetry to Matsubara zero modes associated with Wilson lines. The results are shown to agree well with recent lattice calculations at high temperatures.
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Submitted 8 August, 2025; v1 submitted 22 May, 2025;
originally announced May 2025.
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Lattice study of correlators of chromoelectric fields for heavy quarkonium dynamics in the quark-gluon plasma
Authors:
Nora Brambilla,
Saumen Datta,
Marc Janer,
Viljami Leino,
Julian Mayer-Steudte,
Peter Petreczky,
Antonio Vairo
Abstract:
We perform a lattice calculation of the correlators of two chromoelectric fields in the adjoint representation connected by adjoint Wilson lines at non-zero temperature. These correlators arise in the study of quarkonium dynamics and of adjoint heavy quark diffusion in deconfined matter. We work in SU(3) gauge theory using either gradient flow or multi-level algorithms for noise reduction, and dis…
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We perform a lattice calculation of the correlators of two chromoelectric fields in the adjoint representation connected by adjoint Wilson lines at non-zero temperature. These correlators arise in the study of quarkonium dynamics and of adjoint heavy quark diffusion in deconfined matter. We work in SU(3) gauge theory using either gradient flow or multi-level algorithms for noise reduction, and discuss the renormalization of the correlators on the lattice. We find that a Casimir factor rescaling relates the adjoint correlators corresponding to the diffusion of an adjoint heavy quark and the octet-octet quarkonium transitions to the chromoelectric correlator in the fundamental representation describing the diffusion of a heavy quark.
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Submitted 22 May, 2025;
originally announced May 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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Strong coupling in (2+1+1)-flavor QCD
Authors:
Viljami Leino,
Alexei Bazavov,
Nora Brambilla,
Andreas Kronfeld,
Julian Mayer-Steudte,
Peter Petreczky,
Sebastian Steinbeißer,
Antonio Vairo,
Johannes Heinrich Weber
Abstract:
The strong coupling $α_\mathrm{s}$ can be obtained from the static energy as shown in previous lattices studies. For short distances, the static energy can be calculated both on the lattice with the use of Wilson line correlators, and with the perturbation theory up to three loop accuracy with leading ultrasoft log resummation. Comparing the perturbative expression and lattice data allows for prec…
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The strong coupling $α_\mathrm{s}$ can be obtained from the static energy as shown in previous lattices studies. For short distances, the static energy can be calculated both on the lattice with the use of Wilson line correlators, and with the perturbation theory up to three loop accuracy with leading ultrasoft log resummation. Comparing the perturbative expression and lattice data allows for precise determination of $α_\mathrm{s}(m_Z)$. We will present preliminary results for the determination of $α_\mathrm{s} {(M_Z)}$ in (2+1+1)-flavor QCD using the configurations made availableby the MILC-collaboration with smallest lattice spacing reaching 0.0321fm.
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Submitted 3 February, 2025;
originally announced February 2025.
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Effective field theories for dark matter pairs in the early universe: Debye mass effects
Authors:
Simone Biondini,
Nora Brambilla,
Andrii Dashko,
Gramos Qerimi,
Antonio Vairo
Abstract:
In some scenarios for the early universe, non-relativistic thermal dark matter chemically decouples from the thermal environment once the temperature drops well below the dark matter mass. The value at which the energy density freezes out depends on the underlying model. In a simple setting, we provide a comprehensive study of heavy fermionic dark matter interacting with the light degrees of freed…
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In some scenarios for the early universe, non-relativistic thermal dark matter chemically decouples from the thermal environment once the temperature drops well below the dark matter mass. The value at which the energy density freezes out depends on the underlying model. In a simple setting, we provide a comprehensive study of heavy fermionic dark matter interacting with the light degrees of freedom of a dark thermal sector whose temperature $T$ decreases from an initial value close to the freeze-out temperature. Different temperatures imply different hierarchies of energy scales. By exploiting the methods of non-relativistic effective field theories at finite $T$, we systematically determine the thermal and in-vacuum interaction rates. In particular, we address the impact of the Debye mass on the observables and ultimately on the dark matter relic abundance. We numerically compare the corrections to the present energy density originating from the resummation of Debye mass effects with the corrections coming from a next-to-leading order treatment of the bath-particle interactions. We observe that the fixed-order calculation of the inelastic heavy-light scattering at high temperatures provides a larger dark matter depletion, and hence an undersized yield for given benchmark points in the parameter space, with respect to the calculation where Debye mass effects are resummed.
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Submitted 8 January, 2025; v1 submitted 6 January, 2025;
originally announced January 2025.
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The nature of $χ_{c1}\left(3872\right)$ and $T_{cc}^+\left(3875\right)$
Authors:
Nora Brambilla,
Abhishek Mohapatra,
Tommaso Scirpa,
Antonio Vairo
Abstract:
Two decades ago the $χ_{c1}\left(3872\right)$ was discovered in the hadron spectrum with two heavy quarks. The discovery fueled a surge in experimental research, uncovering dozens of so called XYZ exotics states lying outside the conventional quark model, as well as theoretical investigations into new forms of matter, such as quark-gluon hybrids, tetraquarks, pentaquarks, with the potential of dis…
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Two decades ago the $χ_{c1}\left(3872\right)$ was discovered in the hadron spectrum with two heavy quarks. The discovery fueled a surge in experimental research, uncovering dozens of so called XYZ exotics states lying outside the conventional quark model, as well as theoretical investigations into new forms of matter, such as quark-gluon hybrids, tetraquarks, pentaquarks, with the potential of disclosing new information about the fundamental strong force. Among the XYZs, the $χ_{c1}\left(3872\right)$ and $T_{cc}^+\left(3875\right)$ stand out for their striking characteristics and unlashed many discussions about their nature. Here, we address this question using the Born--Oppenheimer Effective Field Theory (BOEFT) and show how QCD settles the issue of their composition. Not only we describe well the main features of the $χ_{c1}\left(3872\right)$ and $T_{cc}^+\left(3875\right)$ but obtain also predictions in the bottomonium sector. This opens the way to systematic applications of BOEFT to all XYZs.
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Submitted 26 September, 2025; v1 submitted 21 November, 2024;
originally announced November 2024.
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One Born$-$Oppenheimer Effective Theory to rule them all: hybrids, tetraquarks, pentaquarks, doubly heavy baryons and quarkonium
Authors:
Matthias Berwein,
Nora Brambilla,
Abhishek Mohapatra,
Antonio Vairo
Abstract:
The discovery of XYZ exotic states in the hadronic sector with two heavy quarks, represents a significant challenge in particle theory. Understanding and predicting their nature remains an open problem. In this work, we demonstrate how the Born$-$Oppenheimer (BO) effective field theory (BOEFT), derived from Quantum Chromodynamics (QCD) on the basis of scale separation and symmetries, can address X…
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The discovery of XYZ exotic states in the hadronic sector with two heavy quarks, represents a significant challenge in particle theory. Understanding and predicting their nature remains an open problem. In this work, we demonstrate how the Born$-$Oppenheimer (BO) effective field theory (BOEFT), derived from Quantum Chromodynamics (QCD) on the basis of scale separation and symmetries, can address XYZ exotics of any composition. We derive the Schrödinger coupled equations that describe hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia at leading order, incorporating nonadiabatic terms, and present the predicted multiples. We define the static potentials in terms of the QCD static energies for all relevant cases. We provide the precise form of the nonperturbative low-energy gauge-invariant correlators required for the BOEFT: static energies, generalized Wilson loops, gluelumps, and adjoint mesons. These are to be calculated on the lattice and we calculate here their short-distance behavior. Furthermore, we outline how spin-dependent corrections and mixing terms can be incorporated using matching computations. Lastly, we discuss how static energies with the same BO quantum numbers mix at large distances leading to the phenomenon of avoided level crossing. This effect is crucial to understand the emergence of exotics with molecular characteristics, such as the $χ_{c1}(3872)$. With BOEFT both the tetraquark and the molecular picture appear as part of the same description.
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Submitted 22 November, 2024; v1 submitted 8 August, 2024;
originally announced August 2024.
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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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Center-of-mass recoil effects on the annihilation and formation of dark matter bound pairs
Authors:
Simone Biondini,
Nora Brambilla,
Gramos Qerimi,
Antonio Vairo
Abstract:
For a quantitative investigation on the time evolution of heavy thermal dark matter at and after thermal freeze-out, near-threshold processes need to be taken into account which have a large impact on the observed dark matter relic abundance. In this conference paper, we study the recoil effect of heavy dark matter pairs in a thermal bath and compute the annihilation cross section and the decay wi…
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For a quantitative investigation on the time evolution of heavy thermal dark matter at and after thermal freeze-out, near-threshold processes need to be taken into account which have a large impact on the observed dark matter relic abundance. In this conference paper, we study the recoil effect of heavy dark matter pairs in a thermal bath and compute the annihilation cross section and the decay width as well as the bound-state formation cross section of dark matter fermion-antifermion pairs in the laboratory frame within the framework of potential non-relativistic effective field theories at finite temperature. For the considered hierarchy of energy scales, we highlight the effect of the recoil corrections to the thermal rates.
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Submitted 21 February, 2024;
originally announced February 2024.
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Effective field theories for dark matter pairs in the early universe: center-of-mass recoil effects
Authors:
Simone Biondini,
Nora Brambilla,
Gramos Qerimi,
Antonio Vairo
Abstract:
For non-relativistic thermal dark matter, close-to-threshold effects largely dominate the evolution of the number density for most of the times after thermal freeze-out, and hence affect the cosmological relic density. A precise evaluation of the relevant interaction rates in a thermal medium representing the early universe includes accounting for the relative motion of the dark matter particles a…
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For non-relativistic thermal dark matter, close-to-threshold effects largely dominate the evolution of the number density for most of the times after thermal freeze-out, and hence affect the cosmological relic density. A precise evaluation of the relevant interaction rates in a thermal medium representing the early universe includes accounting for the relative motion of the dark matter particles and the thermal medium. We consider a model of dark fermions interacting with a plasma of dark gauge bosons, which is equivalent to thermal QED. The temperature is taken to be smaller than the dark fermion mass and the inverse of the typical size of the dark fermion-antifermion bound states, which allows for the use of non-relativistic effective field theories. For the annihilation cross section, bound-state formation cross section, bound-state dissociation width and bound-state transition width of dark matter fermion-antifermion pairs, we compute the leading recoil effects in the reference frame of both the plasma and the center-of-mass of the fermion-antifermion pair. We explicitly verify the Lorentz transformations among these quantities. We evaluate the impact of the recoil corrections on the dark matter energy density. Our results can be directly applied to account for the relative motion of quarkonia in the quark-gluon plasma formed in heavy-ion collisions. They may be also used to precisely assess thermal effects in atomic clocks based on atomic transitions; the present work provides a first field theory derivation of time dilation for these processes in vacuum and in a medium.
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Submitted 27 June, 2024; v1 submitted 20 February, 2024;
originally announced February 2024.
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Static force from generalized Wilson loops on the lattice using the gradient flow
Authors:
Nora Brambilla,
Viljami Leino,
Julian Mayer-Steudte,
Antonio Vairo
Abstract:
The static QCD force from the lattice can be used to extract $Λ_{\overline{\textrm{MS}}}$, which determines the running of the strong coupling. Usually, this is done with a numerical derivative of the static potential. However, this introduces additional systematic uncertainties; thus, we use another observable to measure the static force directly. This observable consists of a Wilson loop with a…
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The static QCD force from the lattice can be used to extract $Λ_{\overline{\textrm{MS}}}$, which determines the running of the strong coupling. Usually, this is done with a numerical derivative of the static potential. However, this introduces additional systematic uncertainties; thus, we use another observable to measure the static force directly. This observable consists of a Wilson loop with a chromoelectric field insertion. We work in the pure SU(3) gauge theory. We use gradient flow to improve the signal-to-noise ratio and to address the field insertion. We extract $Λ_{\overline{\textrm{MS}}}^{n_f=0}$ from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value $\sqrt{8t_0} Λ_{\overline{\textrm{MS}}}^{n_f=0} =0.629^{+22}_{-26}$, where $t_0$ is a flow-time reference scale. We also obtain precise determinations of several scales: $r_0/r_1$, $\sqrt{8 t_0}/r_0$, $\sqrt{8 t_0}/r_1$ and we compare these to the literature. The gradient flow appears to be a promising method for calculations of Wilson loops with chromoelectric and chromomagnetic insertions in quenched and unquenched configurations
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Submitted 5 September, 2024; v1 submitted 28 December, 2023;
originally announced December 2023.
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Effective Field Theories for Dark Matter Pairs in the Early Universe
Authors:
Simone Biondini,
Nora Brambilla,
Gramos Qerimi,
Antonio Vairo
Abstract:
In this conference paper, we consider effective field theories of non-relativistic dark matter particles interacting with a light force mediator in the early expanding universe. We present a general framework, where to account in a systematic way for the relevant processes that may affect the dynamics during thermal freeze-out. In the temperature regime where near-threshold effects, most notably t…
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In this conference paper, we consider effective field theories of non-relativistic dark matter particles interacting with a light force mediator in the early expanding universe. We present a general framework, where to account in a systematic way for the relevant processes that may affect the dynamics during thermal freeze-out. In the temperature regime where near-threshold effects, most notably the formation of bound states and Sommerfeld enhancement, have a large impact on the dark matter relic density, we scrutinize possible contributions from higher excited states and radiative corrections in the annihilations and decays of dark-matter pairs.
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Submitted 14 April, 2023;
originally announced April 2023.
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Effective field theories for dark matter pairs in the early universe: cross sections and widths
Authors:
Simone Biondini,
Nora Brambilla,
Gramos Qerimi,
Antonio Vairo
Abstract:
In order to predict the cosmological abundance of dark matter, an estimation of particle rates in an expanding thermal environment is needed. For thermal dark matter, the non-relativistic regime sets the stage for the freeze-out of the dark matter energy density. We compute transition widths and annihilation, bound-state formation, and dissociation cross sections of dark matter fermion pairs in th…
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In order to predict the cosmological abundance of dark matter, an estimation of particle rates in an expanding thermal environment is needed. For thermal dark matter, the non-relativistic regime sets the stage for the freeze-out of the dark matter energy density. We compute transition widths and annihilation, bound-state formation, and dissociation cross sections of dark matter fermion pairs in the unifying framework of non-relativistic effective field theories at finite temperature, with the thermal bath modeling the thermodynamical behaviour of the early universe. We reproduce and extend some known results for the paradigmatic case of a dark fermion species coupled to dark gauge bosons. The effective field theory framework allows to highlight their range of validity and consistency, and to identify some possible improvements.
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Submitted 20 February, 2024; v1 submitted 31 March, 2023;
originally announced April 2023.
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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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Implications of gradient flow on the static force
Authors:
Nora Brambilla,
Viljami Leino,
Julian Mayer-Steudte,
Antonio Vairo
Abstract:
We use gradient flow to compute the static force based on a Wilson loop with a chromoelectric field insertion. The result can be compared on one hand to the static force from the numerical derivative of the lattice static energy, and on the other hand to the perturbative calculation, allowing a precise extraction of the $Λ_0$ parameter. This study may open the way to gradient flow calculations of…
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We use gradient flow to compute the static force based on a Wilson loop with a chromoelectric field insertion. The result can be compared on one hand to the static force from the numerical derivative of the lattice static energy, and on the other hand to the perturbative calculation, allowing a precise extraction of the $Λ_0$ parameter. This study may open the way to gradient flow calculations of correlators of chromoelectric and chromomagnetic fields, which typically arise in the nonrelativistic effective field theory factorization.
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Submitted 23 December, 2022;
originally announced December 2022.
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50 Years of Quantum Chromodynamics
Authors:
Franz Gross,
Eberhard Klempt,
Stanley J. Brodsky,
Andrzej J. Buras,
Volker D. Burkert,
Gudrun Heinrich,
Karl Jakobs,
Curtis A. Meyer,
Kostas Orginos,
Michael Strickland,
Johanna Stachel,
Giulia Zanderighi,
Nora Brambilla,
Peter Braun-Munzinger,
Daniel Britzger,
Simon Capstick,
Tom Cohen,
Volker Crede,
Martha Constantinou,
Christine Davies,
Luigi Del Debbio,
Achim Denig,
Carleton DeTar,
Alexandre Deur,
Yuri Dokshitzer
, et al. (70 additional authors not shown)
Abstract:
This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD,…
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This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed.
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Submitted 26 December, 2022; v1 submitted 21 December, 2022;
originally announced December 2022.
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Heavy hybrid decays to quarkonia
Authors:
Nora Brambilla,
Wai Kin Lai,
Abhishek Mohapatra,
Antonio Vairo
Abstract:
The decay rates of the XYZ exotics discovered in the heavy quarkonium sector are crucial observables for identifying the nature of these states. Based on the framework of nonrelativistic effective field theories, we calculate the rates of semi-inclusive decays of heavy quarkonium hybrids into conventional heavy quarkonia. We compute them at leading and subleading power in the inverse of the heavy-…
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The decay rates of the XYZ exotics discovered in the heavy quarkonium sector are crucial observables for identifying the nature of these states. Based on the framework of nonrelativistic effective field theories, we calculate the rates of semi-inclusive decays of heavy quarkonium hybrids into conventional heavy quarkonia. We compute them at leading and subleading power in the inverse of the heavy-quark mass, extending and updating previous results. We compare our predictions with experimental data of inclusive decay rates for candidates of heavy quarkonium hybrids.
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Submitted 18 December, 2022;
originally announced December 2022.
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Charm mass effects in the static energy computed in 2+1+1 flavor lattice QCD
Authors:
Johannes Heinrich Weber,
Nora Brambilla,
Rafael L. Delgado,
Andreas Kronfeld,
Viljami Leino,
Peter Petreczky,
Sebastian Steinbeißer,
Antonio Vairo
Abstract:
We report our analysis for the static energy in (2+1+1)-flavor QCD over a wide range of lattice spacings and several quark masses. We obtain results for the static energy out to distances of nearly 1 fm, allowing us to perform a simultaneous determination of the lattice scales $r_2$, $r_1$ and $r_0$ as well as the string tension, $σ$. While our results for ${r_0}/{r_1}$ and $r_0$ $\sqrtσ$ agree wi…
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We report our analysis for the static energy in (2+1+1)-flavor QCD over a wide range of lattice spacings and several quark masses. We obtain results for the static energy out to distances of nearly 1 fm, allowing us to perform a simultaneous determination of the lattice scales $r_2$, $r_1$ and $r_0$ as well as the string tension, $σ$. While our results for ${r_0}/{r_1}$ and $r_0$ $\sqrtσ$ agree with published (2+1)-flavor results, our result for ${r_1}/{r_2}$ differs significantly from the value obtained in the (2+1)-flavor case, likely due to the effect of the charm quark. We study in detail the effect of the charm quark on the static energy by comparing our results on the finest lattices with the previously published (2+1)-flavor QCD results at similar lattice spacing. The lattice results agree well with the two-loop perturbative expression of the static energy incorporating finite charm mass effects.
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Submitted 15 December, 2022;
originally announced December 2022.
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Inclusive production of $J/ψ$, $ψ(2S)$, and $Υ$ states in pNRQCD
Authors:
Nora Brambilla,
Hee Sok Chung,
Antonio Vairo,
Xiang-Peng Wang
Abstract:
Under some assumptions on the hierarchy of relevant energy scales, we compute the nonrelativistic QCD (NRQCD) long-distance matrix elements (LDMEs) for inclusive production of $J/ψ$, $ψ(2S)$, and $Υ$ states based on the potential NRQCD (pNRQCD) effective field theory. Based on the pNRQCD formalism, we obtain expressions for the LDMEs in terms of the quarkonium wavefunctions at the origin and unive…
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Under some assumptions on the hierarchy of relevant energy scales, we compute the nonrelativistic QCD (NRQCD) long-distance matrix elements (LDMEs) for inclusive production of $J/ψ$, $ψ(2S)$, and $Υ$ states based on the potential NRQCD (pNRQCD) effective field theory. Based on the pNRQCD formalism, we obtain expressions for the LDMEs in terms of the quarkonium wavefunctions at the origin and universal gluonic correlators, which do not depend on the heavy quark flavor or the radial excitation. This greatly reduces the number of nonperturbative unknowns and substantially enhances the predictive power of the nonrelativistic effective field theory formalism. We obtain improved determinations of the LDMEs for $J/ψ$, $ψ(2S)$, and $Υ$ states thanks to the universality of the gluonic correlators, and obtain phenomenological results for cross sections and polarizations at large transverse momentum that agree well with measurements at the LHC.
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Submitted 1 April, 2023; v1 submitted 31 October, 2022;
originally announced October 2022.
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Static Energy in ($2+1+1$)-Flavor Lattice QCD: Scale Setting and Charm Effects
Authors:
TUMQCD Collaboration,
Nora Brambilla,
Rafael L. Delgado,
Andreas S. Kronfeld,
Viljami Leino,
Peter Petreczky,
Sebastian Steinbeißer,
Antonio Vairo,
Johannes H. Weber
Abstract:
We present results for the static energy in ($2+1+1$)-flavor QCD over a wide range of lattice spacings and several quark masses, including the physical quark mass, with ensembles of lattice-gauge-field configurations made available by the MILC Collaboration. We obtain results for the static energy out to distances of nearly $1$~fm, allowing us to perform a simultaneous determination of the scales…
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We present results for the static energy in ($2+1+1$)-flavor QCD over a wide range of lattice spacings and several quark masses, including the physical quark mass, with ensembles of lattice-gauge-field configurations made available by the MILC Collaboration. We obtain results for the static energy out to distances of nearly $1$~fm, allowing us to perform a simultaneous determination of the scales $r_{1}$ and $r_{0}$, as well as the string tension $σ$. For the smallest three lattice spacings we also determine the scale $r_{2}$. Our results for $r_{0}/r_{1}$ and $r_{0}\sqrtσ$ agree with published ($2+1$)-flavor results. However, our result for $r_{1}/r_{2}$ differs significantly from the value obtained in the ($2+1$)-flavor case, which is most likely due to the effect of the charm quark. We also report results for $r_{0}$, $r_{1}$, and $r_{2}$ in~fm, with the former two being slightly lower than published ($2+1$)-flavor results. We study in detail the effect of the charm quark on the static energy by comparing our results on the finest two lattices with the previously published ($2+1$)-flavor QCD results at similar lattice spacing. We find that for $r > 0.2$~fm our results on the static energy agree with the ($2+1$)-flavor result, implying the decoupling of the charm quark for these distances. For smaller distances, on the other hand, we find that the effect of the dynamical charm quark is noticeable. The lattice results agree well with the two-loop perturbative expression of the static energy incorporating finite charm mass effects. This is the first time that the decoupling of the charm quark is observed and quantitatively analyzed on lattice data of the static energy.
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Submitted 12 April, 2023; v1 submitted 7 June, 2022;
originally announced June 2022.
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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 24 November, 2025; v1 submitted 20 May, 2022;
originally announced May 2022.
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The strong coupling constant: State of the art and the decade ahead
Authors:
D. d'Enterria,
S. Kluth,
G. Zanderighi,
C. Ayala,
M. A. Benitez-Rathgeb,
J. Bluemlein,
D. Boito,
N. Brambilla,
D. Britzger,
S. Camarda,
A. M. Cooper-Sarkar,
T. Cridge,
G. Cvetic,
M. Dalla Brida,
A. Deur,
F. Giuli,
M. Golterman,
A. H. Hoang,
J. Huston,
M. Jamin,
A. V. Kotikov,
V. G. Krivokhizhin,
A. S. Kronfeld,
V. Leino,
K. Lipka
, et al. (33 additional authors not shown)
Abstract:
Theoretical predictions for particle production cross sections and decays at colliders rely heavily on perturbative Quantum Chromodynamics (QCD) calculations, expressed as an expansion in powers of the strong coupling constant $α_s$. The current $\mathcal{O}(1\%)$ uncertainty of the QCD coupling evaluated at the reference Z boson mass, $α_s(m_Z) = 0.1179 \pm 0.0009$, is one of the limiting factors…
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Theoretical predictions for particle production cross sections and decays at colliders rely heavily on perturbative Quantum Chromodynamics (QCD) calculations, expressed as an expansion in powers of the strong coupling constant $α_s$. The current $\mathcal{O}(1\%)$ uncertainty of the QCD coupling evaluated at the reference Z boson mass, $α_s(m_Z) = 0.1179 \pm 0.0009$, is one of the limiting factors to more precisely describe multiple processes at current and future colliders. A reduction of this uncertainty is thus a prerequisite to perform precision tests of the Standard Model as well as searches for new physics. This report provides a comprehensive summary of the state-of-the-art, challenges, and prospects in the experimental and theoretical study of the strong coupling. The current $α_s(m_Z)$ world average is derived from a combination of seven categories of observables: (i) lattice QCD, (ii) hadronic $τ$ decays, (iii) deep-inelastic scattering and parton distribution functions fits, (iv) electroweak boson decays, hadronic final-states in (v) $e^+e^-$, (vi) e-p, and (vii) p-p collisions, and (viii) quarkonia decays and masses. We review the current status of each of these seven $α_s(m_Z)$ extraction methods, discuss novel $α_s$ determinations, and examine the averaging method used to obtain the world-average value. Each of the methods discussed provides a ``wish list'' of experimental and theoretical developments required in order to achieve the goal of a per-mille precision on $α_s(m_Z)$ within the next decade.
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Submitted 29 November, 2024; v1 submitted 15 March, 2022;
originally announced March 2022.
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Production and polarization of $S$-wave quarkonia in potential nonrelativistic QCD
Authors:
Nora Brambilla,
Hee Sok Chung,
Antonio Vairo,
Xiang-Peng Wang
Abstract:
Based on the potential nonrelativistic QCD formalism, we compute the nonrelativistic QCD long-distance matrix elements (LDMEs) for inclusive production of $S$-wave heavy quarkonia. This greatly reduces the number of nonperturbative unknowns and brings in a substantial enhancement in the predictive power of the nonrelativistic QCD factorization formalism. We obtain improved determinations of the LD…
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Based on the potential nonrelativistic QCD formalism, we compute the nonrelativistic QCD long-distance matrix elements (LDMEs) for inclusive production of $S$-wave heavy quarkonia. This greatly reduces the number of nonperturbative unknowns and brings in a substantial enhancement in the predictive power of the nonrelativistic QCD factorization formalism. We obtain improved determinations of the LDMEs and find cross sections and polarizations of $J/ψ$, $ψ(2S)$, and excited $Υ$ states that agree well with LHC data. Our results may have important implications in pinning down the heavy quarkonium production mechanism.
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Submitted 27 June, 2022; v1 submitted 15 March, 2022;
originally announced March 2022.
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The static force from generalized Wilson loops using gradient flow
Authors:
Viljami Leino,
Nora Brambilla,
Julian Mayer-Steudte,
Antonio Vairo
Abstract:
We explore a novel approach to compute the force between a static quark-antiquark pair with the gradient flow algorithm on the lattice. The approach is based on inserting a chromoelectric field in a Wilson loop. The renormalization issues, associated with the finite size of the chromoelectric field on the lattice, can be solved with the use of gradient flow. We compare numerical results for the fl…
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We explore a novel approach to compute the force between a static quark-antiquark pair with the gradient flow algorithm on the lattice. The approach is based on inserting a chromoelectric field in a Wilson loop. The renormalization issues, associated with the finite size of the chromoelectric field on the lattice, can be solved with the use of gradient flow. We compare numerical results for the flowed static potential to our previous measurement of the same observable without a gradient flow.
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Submitted 19 November, 2021;
originally announced November 2021.
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The static force from generalized Wilson loops
Authors:
Viljami Leino,
Nora Brambilla,
Owe Philipsen,
Christian Reisinger,
Antonio Vairo,
Marc Wagner
Abstract:
Recently a method to compute the static force with lattice gauge theory using an insertion of a chromoelectric field into a Wilson loop was proposed. We explore this method using the multilevel algorithm and discuss the renormalization of the chromoelectric field on the lattice.
Recently a method to compute the static force with lattice gauge theory using an insertion of a chromoelectric field into a Wilson loop was proposed. We explore this method using the multilevel algorithm and discuss the renormalization of the chromoelectric field on the lattice.
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Submitted 15 November, 2021;
originally announced November 2021.
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QCD Static Force in Gradient Flow
Authors:
Nora Brambilla,
Hee Sok Chung,
Antonio Vairo,
Xiang-Peng Wang
Abstract:
We compute the QCD static force and potential using gradient flow at next-to-leading order in the strong coupling. The static force is the spatial derivative of the static potential: it encodes the QCD interaction at both short and long distances. While on the one side the static force has the advantage of being free of the $O(Λ_{\rm QCD})$ renormalon affecting the static potential when computed i…
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We compute the QCD static force and potential using gradient flow at next-to-leading order in the strong coupling. The static force is the spatial derivative of the static potential: it encodes the QCD interaction at both short and long distances. While on the one side the static force has the advantage of being free of the $O(Λ_{\rm QCD})$ renormalon affecting the static potential when computed in perturbation theory, on the other side its direct lattice QCD computation suffers from poor convergence. The convergence can be improved by using gradient flow, where the gauge fields in the operator definition of a given quantity are replaced by flowed fields at flow time $t$, which effectively smear the gauge fields over a distance of order $\sqrt{t}$, while they reduce to the QCD fields in the limit $t \to 0$. Based on our next-to-leading order calculation, we explore the properties of the static force for arbitrary values of $t$, as well as in the $t \to 0$ limit, which may be useful for lattice QCD studies.
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Submitted 1 February, 2022; v1 submitted 15 November, 2021;
originally announced November 2021.
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The static energy in 2+1+1-flavor QCD
Authors:
Sebastian Steinbeißer,
Nora Brambilla,
Rafael L. Delgado,
Andreas S. Kronfeld,
Viljami Leino,
Peter Petreczky,
Antonio Vairo,
Johannes Heinrich Weber
Abstract:
We report on the status of the analysis of the static energy in $2+1+1$-flavor QCD. The static energy is obtained by measuring Wilson line correlators in Coulomb gauge using the HISQ action, yielding the scales $r_{0}/a$, $r_{1}/a$, $r_{2}/a$, their ratios, and the string tension $σr_{i}^{2}$. We put emphasis on the possible effects due to the dynamical charm-quark by comparing the lattice results…
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We report on the status of the analysis of the static energy in $2+1+1$-flavor QCD. The static energy is obtained by measuring Wilson line correlators in Coulomb gauge using the HISQ action, yielding the scales $r_{0}/a$, $r_{1}/a$, $r_{2}/a$, their ratios, and the string tension $σr_{i}^{2}$. We put emphasis on the possible effects due to the dynamical charm-quark by comparing the lattice results to continuum results of the static energy with and without a massive flavor at two-loop accuracy. We employ gauge-field ensembles from the HotQCD and MILC Collaborations.
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Submitted 3 November, 2021;
originally announced November 2021.
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Quarkonium production in pNRQCD: the $P$-wave case
Authors:
Antonio Vairo
Abstract:
We review our recent understanding of quarkonium production at $e^+e^-$ and hadronic colliders from the point of view of potential non-relativistic QCD, and apply it to $P$-wave charmonium and bottomonium inclusive production.
We review our recent understanding of quarkonium production at $e^+e^-$ and hadronic colliders from the point of view of potential non-relativistic QCD, and apply it to $P$-wave charmonium and bottomonium inclusive production.
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Submitted 8 October, 2021;
originally announced October 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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Inclusive Production of Heavy Quarkonia in pNRQCD
Authors:
Nora Brambilla,
Hee Sok Chung,
Antonio Vairo
Abstract:
We develop a formalism for computing inclusive production cross sections of heavy quarkonia based on the nonrelativistic QCD and the potential nonrelativistic QCD effective field theories. Our formalism applies to strongly coupled quarkonia, which include excited charmonium and bottomonium states. Analogously to heavy quarkonium decay processes, we express nonrelativistic QCD long-distance matrix…
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We develop a formalism for computing inclusive production cross sections of heavy quarkonia based on the nonrelativistic QCD and the potential nonrelativistic QCD effective field theories. Our formalism applies to strongly coupled quarkonia, which include excited charmonium and bottomonium states. Analogously to heavy quarkonium decay processes, we express nonrelativistic QCD long-distance matrix elements in terms of quarkonium wavefunctions at the origin and universal gluonic correlators. Our expressions for the long-distance matrix elements are valid up to corrections of order $1/N_c^2$. These expressions enhance the predictive power of the nonrelativistic effective field theory approach to inclusive production processes by reducing the number of nonperturbative unknowns, and make possible first-principle determinations of long-distance matrix elements once the gluonic correlators are known. Based on this formalism, we compute the production cross sections of $P$-wave charmonia and bottomonia at the LHC, and find good agreement with measurements.
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Submitted 8 September, 2021; v1 submitted 17 June, 2021;
originally announced June 2021.
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Lattice gauge theory computation of the static force
Authors:
Nora Brambilla,
Viljami Leino,
Owe Philipsen,
Christian Reisinger,
Antonio Vairo,
Marc Wagner
Abstract:
We explore a novel approach to compute the force between a static quark and a static antiquark with lattice gauge theory directly. The approach is based on expectation values of Wilson loops or Polyakov loops with chromoelectric field insertions. We discuss theoretical and technical aspects in detail, in particular, how to compensate large discretization errors with a multiplicative renormalizatio…
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We explore a novel approach to compute the force between a static quark and a static antiquark with lattice gauge theory directly. The approach is based on expectation values of Wilson loops or Polyakov loops with chromoelectric field insertions. We discuss theoretical and technical aspects in detail, in particular, how to compensate large discretization errors with a multiplicative renormalization factor and the evaluation using a multilevel algorithm. We also compare numerical results for the static force to corresponding results obtained in the traditional way, i.e., by computing first the static potential and then taking the derivative.
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Submitted 25 March, 2022; v1 submitted 3 June, 2021;
originally announced June 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…
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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 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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Lattice QCD constraints on the heavy quark diffusion coefficient
Authors:
Nora Brambilla,
Viljami Leino,
Peter Petreczky,
Antonio Vairo
Abstract:
We report progress towards computing the heavy quark momentum diffusion coefficient from the correlator of two chromo-electric fields attached to a Polyakov loop in pure SU(3) gauge theory. Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusion coefficient as a function of temperature in the wide range $1.1 < T / T_c < 10^4$ in order to compare it to pertur…
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We report progress towards computing the heavy quark momentum diffusion coefficient from the correlator of two chromo-electric fields attached to a Polyakov loop in pure SU(3) gauge theory. Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusion coefficient as a function of temperature in the wide range $1.1 < T / T_c < 10^4$ in order to compare it to perturbative expansions at high temperature. We find that within errors the lattice results are remarkably compatible with the next-to-leading order perturbative result.
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Submitted 10 November, 2020; v1 submitted 20 July, 2020;
originally announced July 2020.
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Inclusive Hadroproduction of $P$-Wave Heavy Quarkonia in Potential Nonrelativistic QCD
Authors:
Nora Brambilla,
Hee Sok Chung,
Antonio Vairo
Abstract:
We compute the color-singlet and color-octet nonrelativistic QCD (NRQCD) long-distance matrix elements for inclusive production of $P$-wave quarkonia in the framework of potential NRQCD. In this way, the color-octet NRQCD long-distance matrix element can be determined without relying on measured cross section data, which has not been possible so far. We obtain inclusive cross sections of $χ_{cJ}$…
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We compute the color-singlet and color-octet nonrelativistic QCD (NRQCD) long-distance matrix elements for inclusive production of $P$-wave quarkonia in the framework of potential NRQCD. In this way, the color-octet NRQCD long-distance matrix element can be determined without relying on measured cross section data, which has not been possible so far. We obtain inclusive cross sections of $χ_{cJ}$ and $χ_{bJ}$ at the LHC, which are in good agreement with data. In principle, the formalism developed in this Letter can be applied to all inclusive production processes of heavy quarkonia.
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Submitted 25 February, 2021; v1 submitted 15 July, 2020;
originally announced July 2020.
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FeynOnium: Using FeynCalc for automatic calculations in Nonrelativistic Effective Field Theories
Authors:
Nora Brambilla,
Hee Sok Chung,
Vladyslav Shtabovenko,
Antonio Vairo
Abstract:
We present new results on FeynOnium, an ongoing project to develop a general purpose software toolkit for semi-automatic symbolic calculations in nonrelativistic Effective Field Theories (EFTs). Building upon FeynCalc, an existing Mathematica package for symbolic evaluation of Feynman diagrams, we have created a powerful framework for automatizing calculations in nonrelativistic EFTs (NREFTs) at t…
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We present new results on FeynOnium, an ongoing project to develop a general purpose software toolkit for semi-automatic symbolic calculations in nonrelativistic Effective Field Theories (EFTs). Building upon FeynCalc, an existing Mathematica package for symbolic evaluation of Feynman diagrams, we have created a powerful framework for automatizing calculations in nonrelativistic EFTs (NREFTs) at tree- and 1-loop level. This is achieved by exploiting the novel features of FeynCalc that support manipulations of Cartesian tensors, Pauli matrices and nonstandard loop integrals. Additional operations that are common in nonrelativistic EFT calculations are implemented in a dedicated add-on called FeynOnium. While our current focus is on EFTs for strong interactions of heavy quarks, extensions to other systems that admit a nonrelativistic EFT description are planned for the future. All our codes are open-source and publicly available. Furthermore, we provide several example calculations that demonstrate how FeynOnium can be employed to reproduce known results from the literature.
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Submitted 30 November, 2020; v1 submitted 27 June, 2020;
originally announced June 2020.
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Decay and electromagnetic production of strongly coupled quarkonia in pNRQCD
Authors:
Nora Brambilla,
Hee Sok Chung,
Daniel Müller,
Antonio Vairo
Abstract:
We improve the pNRQCD approach to annihilation processes of heavy quarkonium and make first pNRQCD predictions for exclusive electromagnetic production of heavy quarkonium. We consider strongly coupled quarkonia, i.e., quarkonia that are not Coulombic bound states. Possible strongly coupled quarkonia include excited charmonium and bottomonium states. For these, pNRQCD provides expressions for the…
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We improve the pNRQCD approach to annihilation processes of heavy quarkonium and make first pNRQCD predictions for exclusive electromagnetic production of heavy quarkonium. We consider strongly coupled quarkonia, i.e., quarkonia that are not Coulombic bound states. Possible strongly coupled quarkonia include excited charmonium and bottomonium states. For these, pNRQCD provides expressions for the decay and exclusive electromagnetic production NRQCD matrix elements that depend on the wavefunctions at the origin and few universal gluon field correlators. We compute electromagnetic decay widths and exclusive production cross sections, and inclusive decay widths into light hadrons for $P$-wave quarkonia at relative order $v^2$ and leading order, respectively. We also compute the decay widths of $2S$ and $3S$ bottomonium states into lepton pairs and their ratios with the inclusive widths into light hadrons at relative order $v^2$.
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Submitted 17 April, 2020; v1 submitted 18 February, 2020;
originally announced February 2020.
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Heavy quark momentum diffusion coefficient from the lattice
Authors:
Nora Brambilla,
Viljami Leino,
Peter Petreczky,
Antonio vairo
Abstract:
We report progress towards computing the heavy quark momentum diffusion coefficient from the lattice correlator of two chromoelectric fields attached to a Polyakov loop in pure SU(3) gauge theory. Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusion coefficient as a function of temperature in the wide range $1.1 < T/Tc < 10^4$.
We report progress towards computing the heavy quark momentum diffusion coefficient from the lattice correlator of two chromoelectric fields attached to a Polyakov loop in pure SU(3) gauge theory. Using a multilevel algorithm and tree-level improvement, we study the behavior of the diffusion coefficient as a function of temperature in the wide range $1.1 < T/Tc < 10^4$.
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Submitted 5 December, 2019; v1 submitted 2 December, 2019;
originally announced December 2019.
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Static force from the lattice
Authors:
Nora Brambilla,
Viljami Leino,
Owe Philipsen,
Christian Reisinger,
Antonio Vairo,
Marc Wagner
Abstract:
We present a novel approach to compute the force between a static quark and a static antiquark from lattice gauge theory directly, rather than extracting it from the static energy. We explore this approach for SU(3) pure gauge theory using the multilevel algorithm and smeared operators.
We present a novel approach to compute the force between a static quark and a static antiquark from lattice gauge theory directly, rather than extracting it from the static energy. We explore this approach for SU(3) pure gauge theory using the multilevel algorithm and smeared operators.
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Submitted 8 November, 2019;
originally announced November 2019.
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Determination of the QCD coupling from the static energy and the free energy
Authors:
Alexei Bazavov,
Nora Brambilla,
Xavier Garcia i Tormo,
Péter Petreczky,
Joan Soto,
Antonio Vairo,
Johannes Heinrich Weber
Abstract:
We present two determinations of the strong coupling \(α_s\). The first one is from the static energy at three-loop accuracy, and may be considered an update of earlier determinations by some of us. The new analysis includes new lattice data at smaller lattice spacings, and reaches distances as short as \(0.0237\,{\rm fm}\). We present a comprehensive and detailed estimate of the error sources tha…
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We present two determinations of the strong coupling \(α_s\). The first one is from the static energy at three-loop accuracy, and may be considered an update of earlier determinations by some of us. The new analysis includes new lattice data at smaller lattice spacings, and reaches distances as short as \(0.0237\,{\rm fm}\). We present a comprehensive and detailed estimate of the error sources that contribute to the uncertainty of the final result, $α_s(M_Z)=0.11660^{+0.00110}_{-0.00056}$. The second determination is based on lattice data for the singlet free energy at finite temperature up to distances as small as \(0.0081\,{\rm fm}\), from which we obtain $α_s(M_Z)=0.11638^{+0.0009 5}_{-0.00087}$.
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Submitted 26 July, 2019;
originally announced July 2019.
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The $XYZ$ states: experimental and theoretical status and perspectives
Authors:
Nora Brambilla,
Simon Eidelman,
Christoph Hanhart,
Alexey Nefediev,
Cheng-Ping Shen,
Christopher E. Thomas,
Antonio Vairo,
Chang-Zheng Yuan
Abstract:
The quark model was formulated in 1964 to classify mesons as bound states made of a quark-antiquark pair, and baryons as bound states made of three quarks. For a long time all known mesons and baryons could be classified within this scheme. Quantum Chromodynamics (QCD), however, in principle also allows the existence of more complex structures, generically called exotic hadrons or simply exotics.…
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The quark model was formulated in 1964 to classify mesons as bound states made of a quark-antiquark pair, and baryons as bound states made of three quarks. For a long time all known mesons and baryons could be classified within this scheme. Quantum Chromodynamics (QCD), however, in principle also allows the existence of more complex structures, generically called exotic hadrons or simply exotics. These include four-quark hadrons (tetraquarks and hadronic molecules), five-quark hadrons (pentaquarks) and states with active gluonic degrees of freedom (hybrids), and even states of pure glue (glueballs). Exotic hadrons have been systematically searched for in numerous experiments for many years. Remarkably, in the past fifteen years, many new hadrons that do not exhibit the expected properties of ordinary (not exotic) hadrons have been discovered in the quarkonium spectrum. These hadrons are collectively known as $XYZ$ states. Some of them, like the charged states, are undoubtedly exotic. Parallel to the experimental progress, the last decades have also witnessed an enormous theoretical effort to reach a theoretical understanding of the $XYZ$ states. Theoretical approaches include not only phenomenological extensions of the quark model to exotics, but also modern non-relativistic effective field theories and lattice QCD calculations. The present work aims at reviewing the rapid progress in the field of exotic $XYZ$ hadrons over the past few years both in experiments and theory. It concludes with a summary on future prospects and challenges.
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Submitted 13 May, 2020; v1 submitted 17 July, 2019;
originally announced July 2019.
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Order $v^4$ corrections to Higgs boson decay into $J/ψ+ γ$
Authors:
Nora Brambilla,
Hee Sok Chung,
Wai Kin Lai,
Vladyslav Shtabovenko,
Antonio Vairo
Abstract:
The process $H \to J/ψ+ γ$, where $H$ is the Higgs particle, provides a way to probe the size and the sign of the Higgs-charm coupling. In order to improve the theoretical control of the decay rate, we compute order $v^4$ corrections to the decay rate based on the nonrelativistic QCD factorization formalism. The perturbative calculation is carried out by using automated computer codes. We also res…
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The process $H \to J/ψ+ γ$, where $H$ is the Higgs particle, provides a way to probe the size and the sign of the Higgs-charm coupling. In order to improve the theoretical control of the decay rate, we compute order $v^4$ corrections to the decay rate based on the nonrelativistic QCD factorization formalism. The perturbative calculation is carried out by using automated computer codes. We also resum logarithms of the ratio of the masses of the Higgs boson and the $J/ψ$ to all orders in the strong coupling constant $α_s$ to next-to-leading logarithmic accuracy. In our numerical result for the decay rate, we improve the theoretical uncertainty, while our central value is in agreement with previous studies within errors. We also present numerical results for $H \to Υ(nS) + γ$ for $n=1,2$, and 3, which turn out to be extremely sensitive to the Higgs bottom coupling.
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Submitted 26 September, 2019; v1 submitted 15 July, 2019;
originally announced July 2019.