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arxiv: 2606.17919 · v1 · pith:DNTSEVZ2new · submitted 2026-06-16 · ✦ hep-th

Holographic Schwinger-Keldysh effective action for heavy quarks in confinement and deconfinement phases

Shin Nakamura , Daichi Takeda This is my paper

Pith reviewed 2026-06-27 00:13 UTC · model grok-4.3

classification ✦ hep-th
keywords holographic Schwinger-Keldyshheavy quarksconfinement phasedeconfinement phaseeffective actionquark-antiquark pairnonequilibrium steady state
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The pith

The SvR holographic Schwinger-Keldysh prescription derives quadratic effective actions for heavy quark systems in both confinement and deconfinement phases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the holographic Schwinger-Keldysh framework of Skenderis and van Rees to heavy quarks. It derives the quadratic effective action for a quark-antiquark pair in the confinement phase. It also derives the quadratic effective action for a single heavy quark moving at constant velocity in a nonequilibrium steady state in the deconfinement phase. A sympathetic reader would care because the method works whether or not the gravity dual contains a black hole, extending real-time holographic descriptions to confined and nonequilibrium regimes.

Core claim

Taking advantage of the feature that the holographic Schwinger-Keldysh prescription is applicable whether or not the gravity dual contains a black hole, we derive the quadratic effective action for a quark-antiquark pair in the confinement phase within the holographic SK framework of SvR. We also apply the SvR prescription to derive the quadratic effective action for a single heavy quark moving at a constant velocity in a nonequilibrium steady state in the deconfinement phase.

What carries the argument

The SvR holographic Schwinger-Keldysh prescription, which enables derivation of real-time effective actions in holographic models without black holes.

Load-bearing premise

The SvR holographic Schwinger-Keldysh prescription remains valid and yields a reliable quadratic effective action when applied to these heavy-quark systems in both confinement and deconfinement phases.

What would settle it

If the derived quadratic effective action for the static quark-antiquark pair in the confinement phase fails to reproduce the expected linear confining potential, the application would be falsified.

Figures

Figures reproduced from arXiv: 2606.17919 by Daichi Takeda, Shin Nakamura.

Figure 1
Figure 1. Figure 1: The Schwinger-Keldysh contour CSK. The blue line represents the contour, whose initial and final points are periodically identified due to the cyclic property of the trace. 2.1 Thermal fields in the boundary theory Here, we explain the SK contour in QFT and see how various Green functions can be computed. For simplicity, we focus on a real scalar operator O(x) in the boundary theory, and later we discuss t… view at source ↗
Figure 2
Figure 2. Figure 2: The spacetime MSvR dual to the SK contour ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Static string in the AdS5 soliton phase. The red curve depicts the string profile. As inferred from (3.1), the neighborhood of r = rs is a Euclidean plane. represents the worldsheet. The worldsheet coordinates are gauge-fixed to the background coordinates (t, r), i.e., σ a = (t, r). The boundary condition ∂X = W means that the boundary of the worldsheet lies on the Wilson loop W. In this paper, we are inte… view at source ↗
Figure 4
Figure 4. Figure 4: Shape of the potential V (ρ) in (3.18), with L = 1, rs = 0.7, and r∗ = 0.5. 3.2.1 EOM and general solution in Lorentz signature We first analyze the EOM for Φf , which will then apply to the other segments of the SvR spacetime. With the Fourier transform Φ f (t, r) = Z dω 2π e −iωtΦ f ω (r), (3.17) the EOM becomes Schr¨odinger type:7  − d 2 dρ 2 + V (ρ)  Φ f ω = ω 2Φ f ω , V (ρ) := r(|ρ|) 2 L2 m2 (r(|ρ|)… view at source ↗
Figure 5
Figure 5. Figure 5: Configuration on a constant-t slice of the solution (4.6) with X1 = vt−ξ(r). 4 Deconfinement phase: a single trailing quark In this section, we consider the steady Nambu-Goto string solution [20,21] in the deconfinement phase (the AdS5 black brane). As the zeroth order of the perturbation, we first construct the trailing string solution [20, 21] on the black brane. We then deform the string endpoint and ev… view at source ↗
Figure 6
Figure 6. Figure 6: Mapping from the static solution to the perturbed solution. Since the [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

The holographic Schwinger-Keldysh (SK) prescription proposed by Skenderis and van Rees (SvR) has the advantage of being applicable whether or not the gravity dual contains a black hole. Taking advantage of this feature, we derive the quadratic effective action for a quark-antiquark pair in the confinement phase within the holographic SK framework of SvR. We also apply the SvR prescription to derive the quadratic effective action for a single heavy quark moving at a constant velocity in a nonequilibrium steady state in the deconfinement phase.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper applies the Skenderis-van Rees (SvR) holographic Schwinger-Keldysh prescription to derive the quadratic effective action for a quark-antiquark pair in the confinement phase and for a single heavy quark moving at constant velocity in a nonequilibrium steady state in the deconfinement phase. The work positions itself as a direct application of the existing SvR framework without modification.

Significance. If the derivations are free of gaps, the results would illustrate the reach of the SvR prescription into confining geometries and probe-brane nonequilibrium settings, supplying explicit quadratic actions that could be compared with lattice or phenomenological models of heavy-quark dynamics. The absence of new free parameters is a positive feature.

minor comments (3)
  1. The abstract and introduction should explicitly state the metric ansatz and the precise boundary conditions used for the probe branes in each phase so that the steps from the SvR contour to the quadratic action are traceable without consulting external references.
  2. Notation for the SK contour and the retarded/advanced propagators should be unified across the confinement and deconfinement sections to avoid reader confusion.
  3. A brief comparison table or paragraph relating the derived coefficients to known results in the literature (e.g., drag force or string tension) would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review of our manuscript and for the positive assessment of its significance. The report recommends minor revision but does not list any specific major comments. Accordingly, we have no individual points to address below. We remain available to incorporate any minor suggestions once they are provided.

Circularity Check

0 steps flagged

No significant circularity: direct application of external SvR prescription

full rationale

The paper states it derives the quadratic effective actions by applying the existing SvR holographic SK prescription to quark-antiquark pairs (confinement) and moving heavy quarks (deconfinement). No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The derivation chain is an application of an independent prior method (SvR by Skenderis-van Rees, distinct authors), with no equations or claims reducing by construction to the paper's own inputs. The central results are therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of free parameters, axioms, or invented entities; all such items remain unknown.

pith-pipeline@v0.9.1-grok · 5611 in / 1044 out tokens · 30066 ms · 2026-06-27T00:13:48.412326+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 31 linked inside Pith

  1. [1]

    Maldacena,The Large N limit of superconformal field theories and supergravity, Adv

    J.M. Maldacena,The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    Pith/arXiv arXiv 1998

  2. [2]

    Gubser, I.R

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov,Gauge theory correlators from noncritical string theory,Phys. Lett. B428(1998) 105 [hep-th/9802109]

    Pith/arXiv arXiv 1998

  3. [3]

    Witten,Anti-de Sitter space and holography,Adv

    E. Witten,Anti-de Sitter space and holography,Adv. Theor. Math. Phys.2(1998) 253 [hep-th/9802150]

    Pith/arXiv arXiv 1998

  4. [4]

    Son and A.O

    D.T. Son and A.O. Starinets,Minkowski space correlators in AdS / CFT correspondence: Recipe and applications,JHEP09(2002) 042 [hep-th/0205051]. 26

    Pith/arXiv arXiv 2002

  5. [5]

    Herzog and D.T

    C.P. Herzog and D.T. Son,Schwinger-Keldysh propagators from AdS/CFT correspondence,JHEP03(2003) 046 [hep-th/0212072]

    Pith/arXiv arXiv 2003

  6. [6]

    Son and D

    D.T. Son and D. Teaney,Thermal Noise and Stochastic Strings in AdS/CFT,JHEP07 (2009) 021 [0901.2338]

    Pith/arXiv arXiv 2009

  7. [7]

    Unruh,Notes on black-hole evaporation,Phys

    W.G. Unruh,Notes on black-hole evaporation,Phys. Rev. D14(1976) 870

    1976

  8. [8]

    Glorioso, M

    P. Glorioso, M. Crossley and H. Liu,A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems,1812.08785

    Pith/arXiv arXiv

  9. [9]

    Skenderis and B.C

    K. Skenderis and B.C. van Rees,Real-time gauge/gravity duality,Phys. Rev. Lett.101 (2008) 081601 [0805.0150]

    Pith/arXiv arXiv 2008

  10. [10]

    Skenderis and B.C

    K. Skenderis and B.C. van Rees,Real-time gauge/gravity duality: Prescription, Renormalization and Examples,JHEP05(2009) 085 [0812.2909]

    Pith/arXiv arXiv 2009

  11. [11]

    van Rees,Real-time gauge/gravity duality and ingoing boundary conditions,Nucl

    B.C. van Rees,Real-time gauge/gravity duality and ingoing boundary conditions,Nucl. Phys. B Proc. Suppl.192-193(2009) 193 [0902.4010]

    Pith/arXiv arXiv 2009

  12. [12]

    Ammon, J

    M. Ammon, J. Germerodt, C. Sieling and J. Virrueta,A Holographic prescription for generalized Schwinger-Keldysh contours,2510.03404

    arXiv

  13. [13]

    Hawking and D.N

    S.W. Hawking and D.N. Page,Thermodynamics of black holes in anti-de Sitter space, Communications in Mathematical Physics87(1982) 577

    1982

  14. [14]

    Witten,Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,Adv

    E. Witten,Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,Adv. Theor. Math. Phys.2(1998) 505 [hep-th/9803131]

    Pith/arXiv arXiv 1998

  15. [15]

    Horowitz and R.C

    G.T. Horowitz and R.C. Myers,The AdS / CFT correspondence and a new positive energy conjecture for general relativity,Phys. Rev. D59(1998) 026005 [hep-th/9808079]

    Pith/arXiv arXiv 1998

  16. [16]

    Surya, K

    S. Surya, K. Schleich and D.M. Witt,Phase transitions for flat AdS black holes,Phys. Rev. Lett.86(2001) 5231 [hep-th/0101134]

    Pith/arXiv arXiv 2001

  17. [17]

    Botta-Cantcheff, P.J

    M. Botta-Cantcheff, P.J. Mart´ ınez and G.A. Silva,The Gravity Dual of Real-Time CFT at Finite Temperature,JHEP11(2018) 129 [1808.10306]

    Pith/arXiv arXiv 2018

  18. [18]

    Maldacena,Wilson loops in large N field theories,Phys

    J.M. Maldacena,Wilson loops in large N field theories,Phys. Rev. Lett.80(1998) 4859 [hep-th/9803002]

    Pith/arXiv arXiv 1998

  19. [19]

    Brandhuber, N

    A. Brandhuber, N. Itzhaki, J. Sonnenschein and S. Yankielowicz,Wilson loops, confinement, and phase transitions in large N gauge theories from supergravity,JHEP 06(1998) 001 [hep-th/9803263]

    Pith/arXiv arXiv 1998

  20. [20]

    Herzog, A

    C.P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L.G. Yaffe,Energy loss of a heavy quark moving through N=4 supersymmetric Yang-Mills plasma,JHEP07(2006) 013 [hep-th/0605158]. 27

    Pith/arXiv arXiv 2006

  21. [21]

    Gubser,Drag force in AdS/CFT,Phys

    S.S. Gubser,Drag force in AdS/CFT,Phys. Rev. D74(2006) 126005 [hep-th/0605182]

    Pith/arXiv arXiv 2006

  22. [22]

    Casalderrey-Solana and D

    J. Casalderrey-Solana and D. Teaney,Heavy quark diffusion in strongly coupled N=4 Yang-Mills,Phys. Rev. D74(2006) 085012 [hep-ph/0605199]

    Pith/arXiv arXiv 2006

  23. [23]

    Gubser,Momentum fluctuations of heavy quarks in the gauge-string duality,Nucl

    S.S. Gubser,Momentum fluctuations of heavy quarks in the gauge-string duality,Nucl. Phys. B790(2008) 175 [hep-th/0612143]

    Pith/arXiv arXiv 2008

  24. [24]

    Casalderrey-Solana and D

    J. Casalderrey-Solana and D. Teaney,Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma,JHEP04(2007) 039 [hep-th/0701123]

    Pith/arXiv arXiv 2007

  25. [25]

    Giecold, E

    G.C. Giecold, E. Iancu and A.H. Mueller,Stochastic trailing string and Langevin dynamics from AdS/CFT,JHEP07(2009) 033 [0903.1840]

    Pith/arXiv arXiv 2009

  26. [26]

    Casalderrey-Solana, K.-Y

    J. Casalderrey-Solana, K.-Y. Kim and D. Teaney,Stochastic String Motion Above and Below the World Sheet Horizon,JHEP12(2009) 066 [0908.1470]

    Pith/arXiv arXiv 2009

  27. [27]

    Bu and B

    Y. Bu and B. Zhang,Schwinger-Keldysh effective action for a relativistic Brownian particle in the AdS/CFT correspondence,Phys. Rev. D104(2021) 086002 [2108.10060]

    arXiv 2021

  28. [28]

    de Boer, V.E

    J. de Boer, V.E. Hubeny, M. Rangamani and M. Shigemori,Brownian motion in AdS/CFT,JHEP07(2009) 094 [0812.5112]

    Pith/arXiv arXiv 2009

  29. [29]

    Lawrence and E.J

    A.E. Lawrence and E.J. Martinec,Black hole evaporation along macroscopic strings, Phys. Rev. D50(1994) 2680 [hep-th/9312127]

    Pith/arXiv arXiv 1994

  30. [30]

    Liu and P

    H. Liu and P. Glorioso,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,PoSTASI2017(2018) 008 [1805.09331]

    Pith/arXiv arXiv 2018

  31. [31]

    Ishigaki, S

    S. Ishigaki, S. Nakamura and K. Takasan,Patchwork Conditions for Holographic Nonlinear Responses: A Computational Method for Electric Conductivity and Friction Coefficient,PTEP2024(2024) 083B07 [2303.02633]

    arXiv 2024

  32. [32]

    Ishii and K

    T. Ishii and K. Murata,Turbulent strings in AdS/CFT,JHEP06(2015) 086 [1504.02190]

    Pith/arXiv arXiv 2015

  33. [33]

    Takeda,A Lindbladian for holographic Brownian motion,(2026)

    D. Takeda,A Lindbladian for holographic Brownian motion,(2026)

    2026

  34. [34]

    Ishii and D

    T. Ishii and D. Takeda,Lindblad dynamics in holography,Phys. Rev. D112(2025) 046020 [2504.17320]

    arXiv 2025

  35. [35]

    Akamatsu,Heavy quark master equations in the Lindblad form at high temperatures, Phys

    Y. Akamatsu,Heavy quark master equations in the Lindblad form at high temperatures, Phys. Rev. D91(2015) 056002 [1403.5783]

    Pith/arXiv arXiv 2015

  36. [36]

    Capovilla and J

    R. Capovilla and J. Guven,Geometry of deformations of relativistic membranes,Phys. Rev. D51(1995) 6736 [gr-qc/9411060]. 28

    Pith/arXiv arXiv 1995

  37. [37]

    Kiosses and A

    V. Kiosses and A. Nicolaidis,Second order perturbations of relativistic membranes in curved spacetime,Phys. Rev. D89(2014) 124016 [1404.4166]

    Pith/arXiv arXiv 2014

  38. [38]

    de Haro, S.N

    S. de Haro, S.N. Solodukhin and K. Skenderis,Holographic reconstruction of space-time and renormalization in the AdS / CFT correspondence,Commun. Math. Phys.217 (2001) 595 [hep-th/0002230]. 29

    Pith/arXiv arXiv 2001