Nonperturbative strong coupling at timelike momenta

Open Access

Nonperturbative strong coupling at timelike momenta

Jan Horak, Jan M. Pawlowski, Jonas Turnwald, Julian M. Urban, Nicolas Wink, and Savvas Zafeiropoulos

Phys. Rev. D 107, 076019 – Published 21 April 2023

Abstract

We compute the strong coupling constant of Landau gauge QCD in the full complex momentum plane, both directly and via spectral reconstruction. In particular, we consider the Taylor coupling given by the product of ghost and gluon dressing functions. Assuming spectral representations for the latter, we first show that also the coupling obeys such a representation. The subsequent spectral reconstruction of the coupling data, obtained from $2 + 1$ flavor lattice QCD results for the ghost and gluon, is based on a probabilistic inversion of this representation using Gaussian process regression with analytically enforced asymptotics. In contradistinction, our direct calculation relies on earlier reconstruction results for the ghost and gluon spectral functions themselves, as well as data obtained in functional QCD. Apart from its relevance for studies of resonances or scattering processes, the calculation also serves as a nontrivial benchmark of our reconstruction approach. The results show remarkable agreement, testifying to the reliability of the method.

Received 26 January 2023
Accepted 3 April 2023

DOI:https://doi.org/10.1103/PhysRevD.107.076019

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Lattice QCD QCD phenomenology Quantum chromodynamics

Functional renormalization group Machine learning

Particles & Fields

Authors & Affiliations

Jan Horak¹, Jan M. Pawlowski^1,2, Jonas Turnwald ³, Julian M. Urban ^4,5, Nicolas Wink³, and Savvas Zafeiropoulos⁶

¹Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany
²ExtreMe Matter Institute EMMI, GSI, Planckstr. 1, D-64291 Darmstadt, Germany
³Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
⁴Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
⁵The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
⁶Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 107, Iss. 7 — 1 April 2023

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
$q \bar{q}$ -scattering process with a one-gluon exchange. At sufficiently large timelike exchange momenta, this process plays an important role in its respective $S$ -matrix elements. Consequently, all internal quantities are dressed. Blue blobs represent full vertices, and the wiggly internal line is a full gluon propagator.
Reuse & Permissions
Figure 2
Spacelike Taylor coupling $α_{s}$ in QCD [Figure 2] and its spectral function $ρ_{α} (ω)$ [Fig. 2]. We compare the spectral function computed directly via (16) (red) to that obtained via reconstruction with GPR (blue). The direct calculation uses the reconstruction results for gluon and ghost spectral functions from [10]. For the reconstruction, we use the gluon and ghost propagator data in $2 + 1$ flavor lattice QCD from [32, 33]. Both the input spectral functions and the corresponding lattice data are displayed in Fig. 3. The coupling spectral functions obtained via these two complementary approaches share all qualitative features, such as peak positions and heights as well as asymptotic behavior. The peak structure can be connected to the respective peak structure of the gluon spectral function; see Fig. 3. The error band of the reconstruction result accounts for the change in the spectral function when varying the GP kernel parameters, whereas that of the direct calculation originates from propagating the uncertainty of the input. The Euclidean lattice data for the Taylor coupling $α_{s}$ are displayed as gray squares in Fig. 2. We compare it to the data from its spectral representation (14) (red) as well as the reconstruction result (blue), showing that the representation holds and that the reconstruction accurately describes the lattice data.
Reuse & Permissions
Figure 3
The continuous parts of the ghost [Fig. 3] and gluon [Fig. 3] spectral functions obtained in [3] (see also Sec. 2c), used here as input for the calculation of the coupling spectral function shown in Fig. 2 via its spectral representation (14). Shaded areas represent $1 σ$ -bands of the statistical error of the mean prediction based on the available observations and precision. Note that for the calculation of the gluon spectral function, the UV and IR asymptotic regimes are assumed to be maximally large. This leads to a small reconstruction error without accounting for systematics; see Appendix pp3-s3 for a detailed discussion.
Reuse & Permissions
Figure 4
Taylor coupling $α_{s} (ω)$ of $2 + 1$ flavor QCD defined in (10) in the complex frequency right half plane (positive real frequencies), real [Fig. 4] and imaginary part [Fig. 4]. The imaginary part explicitly shows the branch cut along the real frequency axis. The spectral function corresponds to the imaginary part of $α_{s}$ at the upper half plane boundary of the branch cut, divided by $ω^{2}$ . Both, the real and imaginary part, exhibit distinctive peaks which can be connected to the peak structure of the gluon spectral function; see Fig. 3. The coupling decays logarithmically for increasing $| ω |$ .
Reuse & Permissions
Figure 5
Behavior of the spectral function when varying the midpoints $μ_{IR / UV}$ of the transition kernels to the asymptotic IR [Fig. 5] and UV regimes [Fig. 5]. The respective values of the parameters are color-coded. The resulting scan of the spectral functions is compared to the final result with maximally enhanced asymptotics, displayed with a dashed blue line. The error band obtained by varying the parameters of the asymptotics—as indicated in Fig. 6—is given by the shaded blue area.
Reuse & Permissions
Figure 6
Scans of the bias parameters defined in (c6). We compare the quality of the dressing reconstruction—quantified by $χ^{2}$ —when varying the midpoint positions of the bias transition $μ_{IR / UV}$ [Fig. 6] as well as its steepness $ℓ_{IR / UV}$ [Fig. 6]. The values of the bias parameters chosen for the reconstruction are marked by crosses. This choice maximizes the size of the regions dominated by the coupling infrared and ultraviolet asymptotics, while producing small $χ^{2}$ reconstructions of the data. Additionally, the parameters are then scanned in the flat directions, indicated by the horizontal bars, in order to obtain the error estimation for the reconstruction results shown in Fig. 2.
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology