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Shear and bulk viscosity for a pure glue theory using an effective matrix model

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Pith's one-line read This paper claims that in a pure-glue matrix model with teen ghost fields, the bulk viscosity-to-entropy ratio peaks at the deconfinement transition at values comparable to shear viscosity, then falls rapidly to negligible by about twice…

desk verdict Careful new transport calculation in the semi-QGP matrix model with teen ghosts, but the headline ζ/s peak sits in a regime where the leading-log expansion is not controlled (α_s N_c ≈ 1.5 at T_d). read the letter →

arxiv 2504.20138 v2 pith:BRTHCUA4 submitted 2025-04-28 hep-ph hep-latnucl-th

Manas Debnath , Ritesh Ghosh , Najmul Haque , Yoshimasa Hidaka , Robert D. Pisarski This is my paper · ORCID
classification hep-phhep-latnucl-th
keywords shearviscositybulksemi-quark-gluonplasmaPolyakovloopmatrixmodelteenghostfieldsleadinglogarithmicorderpureglueQCD
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a pure-glue plasma made of gluons and effective two-dimensional adjoint ghost fields, called teens, has transport properties shaped by partial deconfinement. Using Boltzmann kinetic theory at leading logarithmic order in a background with nonzero Polyakov-loop holonomy, it claims that the shear viscosity to entropy ratio $\eta/s$ decreases toward the deconfinement temperature $T_d$ but stays above the conjectured $1/4\pi$ bound. In contrast, the bulk viscosity to entropy ratio $\zeta/s$ is largest at $T_d$, comparable to $\eta/s$, then falls off rapidly and is negligible by about $2T_d$. A sympathetic reader would care because a large $\zeta/s$ near the transition would affect hydrodynamic descriptions of heavy-ion collisions and would be a distinctive, testable signature of the semi-quark-gluon-plasma picture.

What carries the argument

The central object is the large-$N_c$ matrix model of deconfinement, whose eigenvalue density has the form $\rho(q)=1+b(T)\cos(d(T)q)$ and determines the Polyakov loops $\ell_k$ as well as the free energy. The load-bearing new ingredient is the teen field, defined as a two-dimensional adjoint ghost whose negative pressure cancels the leading gluon pressure near $T_d$; teen propagators and vertices enter the scattering amplitudes on the same footing as gluons. Transport is computed from the linearized Boltzmann equation at leading logarithmic order, keeping only $2\to 2$ forward scattering with a soft gluon exchanged at momentum $k\sim gT$, which produces the logarithm $\ln(\kappa/g^2 N_c)$ after angular integration. For the bulk viscosity, the source term is proportional to $2p^2(1/3-v_s^2)$, so the deviation of the speed of sound from the conformal value drives the result.

What would settle it

A next-to-leading-logarithmic computation of the same matrix-model Boltzmann equation would settle the leading-log prediction, but the sharpest test is a lattice extraction of the pure-glue $\zeta/s$ near $1.1T_d$: if the ratio remains below about $0.3$ instead of rising toward the model's $1.6$ to $1.7$, the central claim would be contradicted.

Watch

Extended reading notes

Core claim

The central claim is that, in an SU($N_c$) pure glue theory described by an effective matrix model for the thermal holonomy, the shear and bulk viscosities can be computed to leading logarithmic order in weak coupling using gluons and teen ghost fields as quasiparticles. The paper finds that $\eta/s$ is moderately suppressed toward $T_d$ but remains well above the conjectured lower bound of $1/4\pi$, whereas $\zeta/s$ is largest at $T_d$, around $1.6$ to $1.7$ for the central parameter choices, comparable to $\eta/s$, and then drops sharply so that bulk dissipation is negligible by roughly $2T_d$. The origin of the large bulk viscosity is the nonconformal equation of state, measured by $\Delta v_s^2 = 1/3 - v_s^2$, rather than thermal masses, so the result differs strongly from ordinary perturbative estimates where the bulk viscosity is strongly suppressed.

Load-bearing premise

The argument depends on the gauge coupling being moderate even at $T_d$, so that gluons and teen fields are well-defined quasiparticles whose leading-logarithmic $2\to 2$ scattering controls transport; if the effective coupling is large near $T_d$, the Boltzmann computation and the predicted large $\zeta/s$ do not follow.

Editorial extensions

If this is right

  • At $T_d$, $\zeta/s \approx 1.6$ to $1.7$ for the central parameters, comparable to $\eta/s$; near $1.5T_d$, $\eta/s \approx 0.38$ to $0.4$ and $\zeta/s \approx 0.076$ to $0.087$, with the shear value at the high end of current lattice estimates.
  • $\zeta/s$ increases strongly as $T\to T_d$ and becomes negligible by about $2T_d$, so bulk dissipation is a near-transition phenomenon in this model.
  • The inclusion of gluon-teen and teen-teen scattering raises the shear viscosity near $T_d$ by about six percent relative to gluon-gluon scattering alone, because the ghost fields act as additional scatterers.
  • The temperature-dependent ratio $\zeta/\eta$ divided by $(1/3-v_s^2)^2$ lies around $56$ to $65$ at $T_d$, larger than the relaxation-time-approximation rule of thumb, and rises to a maximum near $2.5T_d$ before falling.
  • Even with large choices of the logarithmic parameter $\kappa$, the model keeps $\eta/s$ above the conjectured $1/4\pi$ bound throughout the deconfined phase.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If a large $\zeta/s$ near $T_d$ is real, hydrodynamic simulations of deconfined matter that currently employ small bulk viscosity near the transition may need revision; a testable signature would be enhanced entropy production or modified radial flow in the near-transition region.
  • The same machinery, applied to QCD with dynamical quarks where the Polyakov loop at the chiral transition is much smaller, should produce a substantially smaller $\eta/s$; computing $\zeta/s$ in that chiral semi-QGP is the natural next step.
  • The quoted values at $T_d$ span roughly $\zeta/s \approx 1.1$ to $7$ when the parameters $\kappa$ and $w$ are varied within the ranges considered, so a next-to-leading-logarithmic computation is the cleanest way to sharpen the prediction.
  • Because the model's bulk viscosity is driven by the equation of state rather than by scattering dynamics, lattice determinations of the pure-glue speed of sound near $T_d$ directly constrain the magnitude of the predicted $\zeta/s$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper extends a large-N_c effective matrix model of the SU(N_c) holonomy potential by adding adjoint two-dimensional ghost fields, called 'teens,' whose parameters are adjusted to reproduce the lattice Polyakov loop. The authors then compute the shear and bulk viscosities from a linearized Boltzmann equation with 2-to-2 scattering at leading logarithmic order in the presence of nonzero holonomy. The central reported result is that η/s remains above the AdS/CFT bound while ζ/s is largest at T_d, with values near ζ/s ≈ 1.6–1.7, and falls off rapidly by about 2T_d.

Significance. If the calculation were controlled, it would provide a concrete transport framework for the semi-quark-gluon plasma and a falsifiable prediction that the bulk viscosity peaks at deconfinement in pure-glue theory. The manuscript has genuine strengths: the matrix elements are written out explicitly, the sign structure for ghost fields is treated carefully in Appendix A, the cancellation of the δΓ terms through the equations of motion is a nontrivial consistency check around Eq. (59), and the zero-mode problem in the bulk-viscosity collision kernel is addressed numerically in Appendix C. However, the significance is conditional on several heuristic ingredients, and one of them—the validity of the weak-coupling leading-log expansion at T_d—is load-bearing for the main quantitative claim.

major comments (4)
  1. [Sec. V.C, Eq. (138) and Sec. VII, Eq. (196)] The leading-logarithmic expansion is not controlled at the temperatures where the headline prediction is made. With the values used for the figures, w = 0.5 and κ = 64, Eq. (165) gives g^2 N_c = 24π^2/[11 ln(π)] ≈ 18.8 at T = T_d, i.e., α_s N_c ≈ 1.5. The 'soft' momentum scale is then gT > T, so the HTL separation P ~ gT ≪ T used in Sec. IV is absent. Moreover, ln(κ/g^2 N_c) = ln(64/18.8) ≈ 1.2, so the logarithm that controls the truncation in Eq. (138) is not large and the leading-log approximation is not justified. The quoted value ζ/s ≈ 1.6–1.7 at T_d, and the associated claim that ζ/s peaks at T_d, are therefore not established by the calculation as presented.
  2. [Sec. VI, Eqs. (168), (27), and (196)] The bulk viscosity is largely inherited from lattice thermodynamics rather than derived from teen quasiparticle dynamics. The source term in Eq. (168) is proportional to Δv_s^2 = 1/3 − v_s^2, and v_s^2 is computed from the lattice pressure fit in Eq. (27). Consequently, the near-T_d enhancement of ζ/s is controlled by the input lattice equation of state. The manuscript should quantify how much of the peak survives if Δv_s^2 is instead computed self-consistently from the matrix model, and should avoid presenting the ζ/s peak as an independent prediction of the teen kinetic theory.
  3. [Appendix B and Sec. VI] The kinetic-theory and thermodynamic descriptions of the teen fields are not mutually consistent. Appendix B shows that kinetic theory gives E_kt = −T^2 T_d^2/24 and P_kt = −T^2 T_d^2/72, hence E_kt = 3P_kt, while direct thermodynamics gives E_th = P_th = −T^2 T_d^2/24. The bulk-viscosity calculation uses the thermodynamic Δv_s^2 while using teen quasiparticles with kinetic-theory dispersion in the collision kernel. This acknowledged mismatch means the computation is not a closed derivation from a single kinetic model; its quantitative output should be labeled accordingly.
  4. [Sec. IV.A and Sec. V.A] The teen-field action in Eq. (33) contains a transverse momentum cutoff at T_d that explicitly violates gauge invariance. The paper argues that the teen self-energy does not enter at leading logarithmic order, but the teen-gluon and teen-teen scattering amplitudes used in Secs. V and VI do rely on the teen propagator with this gauge-violating cutoff. A more explicit demonstration is needed that the gauge-violating terms cannot affect the leading-log results for η/s and ζ/s; otherwise the transport coefficients also rest on an unquantified model assumption.
minor comments (5)
  1. [Sec. VI.B, Fig. 15] The text near Fig. 15 states that the parameters are chosen with w = 1 and κ = 64, while the figure caption and Eq. (196) use w = 0.5 and κ = 64; these must be reconciled.
  2. [Eq. (165)] The running coupling is written in an ambiguous form; please display g^2 N_c explicitly and state the renormalization scheme and scale used.
  3. [Abstract and Sec. III] The abstract emphasizes the large-N_c limit, but several comparisons are made to lattice data for N_c = 3 and N_c = 5; the large-N_c status of the final results should be stated more precisely.
  4. [Sec. IV.A] The use of the Bengali character for the teen field is likely to be confusing to readers; a table of symbols or an alternative notation would improve readability.
  5. [Sec. VI.B and Fig. 3] The matrix model does not reproduce the sharp peak in the interaction measure near 1.1T_d, and this limitation should be explicitly connected to the reliability of ζ/s in that temperature region.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the viscosities are computed from a Boltzmann equation with matrix-model input, not fitted to transport data.

full rationale

The central claims (eta/s and zeta/s values and their temperature dependence) are outputs of the leading-log Boltzmann calculation in Secs. V and VI, and are never fit to viscosity data. The matrix-model parameters are calibrated to lattice Polyakov loops and pressure, but the transport coefficients are then computed from the resulting quasiparticle spectra and 2-to-2 scattering amplitudes. The bulk viscosity uses Delta v_s^2 taken from the lattice equation of state, Eq. (27), and from the fitted dA/dB ansatze; this is an external thermodynamic input used in the standard kinetic-theory source term, not a redefinition of the target quantity. Appendix B's acknowledged discrepancy between the kinetic-theory teen pressure and the thermodynamic pressure, and the zero-mode regularization of Appendix C, are internal consistency and approximation limitations rather than circular reductions. The parameters w and kappa are presented as uncertainties in the leading-log calculation, with ranges shown in Figs. 12, 13 and 16, rather than as fitted predictions. No equation in the paper reduces a claimed prediction to its input by construction, and no load-bearing premise is justified solely by a self-citation whose content is the target result.

Assumptions & free parameters 4 free parameters · 7 assumptions · 1 invented entities

The central claim rests on a matrix model calibrated to lattice data and on a set of modeling choices for the teen ghost fields and the transport calculation. The free parameters are the potential coefficient c, the Polyakov-loop fit functions d_A/d_B, the coupling scale w, and the logarithm constant kappa. The axioms include the non-perturbative B2 potential, the 2D teen ghost construction with a T_d cutoff, the quasiparticle assumption, the large-N_c solution, the kinetic-theory sign prescription, the neglect of teen self-energy, and the use of lattice-based Delta v_s^2 for the bulk viscosity. The only invented entity is the teen field itself, which has no independent experimental handle.

free parameters (4)
  • c (coefficient of V2 term) = not specified
    In Eq. (12), c T^2 T_d^2 sum P B2(qa-qb) is adjusted so the deconfinement transition occurs at T_d. This is a model input, not derived.
  • d(T) ansatz coefficients for d_A and d_B = d_A = 1.08 t + 5.2032; d_B = 0.26/t^3 + 1.105 t + 4.9182
    Eq. (28): the coefficients are fit so that the Polyakov loop l1 matches lattice values for N_c = 5. These are free parameters in the matrix model.
  • kappa (beyond-LO log scale) = chosen as 32, 64, 128
    Eq. (138) and Sec. V D: kappa parameterizes the log(kappa/g^2 N_c) in the t-channel scattering. Values are chosen so that eta/s is relatively small; not determined within leading-log order.
  • w (coupling scale) = 0.5, 1.0, 2.0
    Eq. (165): w sets the scale in the running coupling g^2 N_c = 4 pi * 6 pi / (11 log(w 2 pi T / T_d)). Varied to show sensitivity.
assumptions (7)
  • domain assumption Non-perturbative potential is ~ T^2 with second Bernoulli polynomial B2
    Eq. (12), motivated by pressure corrections ~ T^2 from Ref. [14] and to avoid first-order transition from perturbative phase. Not unique, as the paper states.
  • ad hoc to paper Teen fields are 2D adjoint ghosts with transverse momentum cutoff at T_d
    Eqs. (33)-(34): introduced to cancel leading N^2 pressure near T_d; cutoff at T_d is non-perturbative input.
  • domain assumption Gluons and teens are well-defined quasiparticles down to T_d; coupling is moderate
    Introduction: principal assumption for the leading-log transport calculation.
  • domain assumption Large N_c limit with spectral density rho(q) = 1 + b cos(d q)
    Sec. III: solution of the matrix model at large N_c from Refs. [33,37,43,51,52]; used for all color sums.
  • domain assumption Kinetic theory with sign functions for ghost absorption is valid for teens
    Sec. V B and Appendix A: the Boltzmann equation treats teens as periodic fermions; signs tracked by S_b. This is an assumption about the transport description.
  • domain assumption Teen self-energy and gauge-variant terms do not contribute at leading log
    Sec. IV C: power counting shows teen self-energy does not enter at leading log; the transverse cutoff violates gauge invariance but the violation is argued to be benign for the computed quantities.
  • ad hoc to paper Bulk viscosity uses Delta v_s^2 from thermodynamics (lattice fit) rather than from teen kinetic theory
    Sec. VI and Appendix B: kinetic theory gives wrong pressure for teens; the paper bypasses this by using lattice-based Delta v_s^2, an explicit modeling inconsistency.
invented entities (1)
  • Teen ghost fields (denoted by Bengali character 3)
    purpose: Two-dimensional adjoint ghost fields introduced to cancel the leading N_c^2 pressure as T approaches T_d, modeling partial deconfinement in the semi-QGP.
    No independent experimental or lattice handle; the teens are a phenomenological construct. Their parameters are fit to the lattice Polyakov loop, and they are used to compute viscosities.

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Pith. "Pith review of Shear and bulk viscosity for a pure glue theory using an effective matrix model." pith.science (2026). https://pith.science/paper/BRTHCUA4

@misc{pith2026250420138,
  author       = {Pith},
  title        = {Pith review of: Shear and bulk viscosity for a pure glue theory using an effective matrix model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BRTHCUA4}},
  note         = {Machine review of arXiv:2504.20138}
}
abstract

At nonzero temperatures, the deconfining phase transition can be analyzed using an effective matrix model to characterize the change in holonomy. The model includes gluons and two-dimensional ghost fields in the adjoint representation, or ``teens''. As ghosts, the teen fields are responsible for the decrease of the pressure as $T \rightarrow T_d$, with $T_d$ the transition temperature for deconfinement. Using the solution of this matrix model for a large number of colors, the parameters of the teen fields are adjusted so that the expectation value of the Polyakov loop is close to the values from the lattice. The shear, $\eta$, and bulk, $\zeta$, viscosities are computed in weak coupling but nonzero holonomy. In the pure glue theory, the value of the Polyakov loop is relatively large in the deconfined phase, $\approx 1/2$ at $T_d$. Consequently, if $s$ is the entropy density, while $\eta/s$ decreases as $T\rightarrow T_d$, it is still well above the conformal bound. In contrast, $\zeta/s$ is largest at $T_d$, comparable to $\eta/s$, then falls off rapidly with increasing temperature and is negligible by $\sim 2 T_d$.

Figures

Figures reproduced from arXiv: 2504.20138 by the authors.

Figure 1
Figure 1. FIG. 1. Plots of the first four Polyakov loops as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Plot of the Polyakov loop using the solution [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plot of the interaction measure, divided by the number of gluons, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Results for the Polyakov loop with the ansatz of Eq. (28) versus the lattice for [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Results for the interaction measure, divided by the number of degrees of freedom, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Teen ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Teen self-energy diagrams [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Kinematics of 2-2 scattering, from Ref. [27] [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The diagrams involving the exchange of a virtual soft gluon: gluon-gluon scattering (a), gluon-teen scattering (b), and [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The diagrams involving the exchange of a soft, virtual teen particle: Compton scattering (a) and teen-teen annihilation [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Ratio of the shear viscosity to the entropy with the ansatzes of Eq. [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Ratio of the shear viscosity to the entropy for different parameter sets of [PITH_FULL_IMAGE:figures/full_fig_p027_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Ratio of the shear viscosity to the entropy for different parameter sets of [PITH_FULL_IMAGE:figures/full_fig_p028_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The green line denotes the ratio of the shear viscosity including teen-teen, teen-gluon, and gluon-gluon scattering, [PITH_FULL_IMAGE:figures/full_fig_p029_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Ratio of the bulk viscosity to the entropy density with the ansatz of Eq. [PITH_FULL_IMAGE:figures/full_fig_p034_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p035_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Variation of bulk viscosity with increasing basis number at [PITH_FULL_IMAGE:figures/full_fig_p040_18.png]

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Reference graph

Works this paper leans on

121 extracted references · 20 canonical work pages

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    9a, is Mgg→gg =ϵµ1 1 (Pa1b1 1 )ϵ∗µ3 3 (Pa3b3 3 ) Jµ,ab g,µ1µ3(Pa1b1 1 ,P a3b3 3 )Dba,dc µν (Kba) × Jν,cd g,µ2µ4(Pa2b2 2 ,P a4b4 4 )ϵµ2 2 (Pa2b2 2 )ϵ∗µ4 4 (Pa4b4 4 )

    Exchange of a soft, virtual gluon The amplitude for two gluons scattering into two gluons, Fig. 9a, is Mgg→gg =ϵµ1 1 (Pa1b1 1 )ϵ∗µ3 3 (Pa3b3 3 ) Jµ,ab g,µ1µ3(Pa1b1 1 ,P a3b3 3 )Dba,dc µν (Kba) × Jν,cd g,µ2µ4(Pa2b2 2 ,P a4b4 4 )ϵµ2 2 (Pa2b2 2 )ϵ∗µ4 4 (Pa4b4 4 ). (70) Here Pµ,ai,bi i are the momenta for particles i = 1,..., 4. The polarization tensor are ϵµ...

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    One contribution is where a teen plus a gluon scatters into a gluon plus a teen, Fig

    Exchange of a soft teen field There are two diagrams involving the exchange of a teen particle. One contribution is where a teen plus a gluon scatters into a gluon plus a teen, Fig. 10a. This is analogous to Compton scattering. The amplitude is M3 g→g3 =ϵµ(Pa1b1 1 )Jµ,ab 3 (Pa1b1 1 ,P a3b3 3 )Dba,dc 3 (Kba)Jν,cd 3 (Pa2b2 2 ,P a4b4 4 )ϵ∗ν(Pa4b4 4 ). (77) L...

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    Gluon-gluon scattering At nonzero holonomy, the matrix elements for gg → gg are Λgg nm = 2 g4 (2π)5 4Y i=1 X aibi ∞X n1,n2=1 ∞X n3,n4=0 G(Q,n 1,...,n 4)tab 13tba 24tcd 13tdc 24 Z ∞ 0 dpp 2e−βp(n1+n3) × Z ∞ 0 dp′p′2e−βp′(n2+n4) Z 1 −1 dx (F g nm(p,x ) +F g nm(p′,x ))cg ncg m Z π −π dϕ 2π Z ∞ 0 dkk 3 × Dab L (K) + (1−x2) cosϕDab T (K) Dcd L (K) + (1−x2) cos...

  4. [4]

    Gluon-teen scattering For gluon-teen scattering, Fig. 9b, Λg3 nm =−1 22 g4 (2π)5 4Y i=1 X aibi ∞X n1,n2=1 ∞X n3,n4=0 G(Q,n 1,...,n 4)tab 13tba 24tcd 13tdc 24 Z Td 0 p⊥dp⊥ Z ∞ 0 dp∥ × e−βp(n1+n3) Z ∞ 0 dp′p′2e−βp′(n2+n4) Z 1 −1 dx (F 3 nm(p,x )c3 nc3 m +F g nm(p′,x )cg ncg m) Z π −π dϕ 2π × Z ∞ 0 dkk 3 Dab L (K) + (1−x2) cosϕDab T (K) Dcd L (K) + (1−x2) co...

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    T 8X gg nm +T 6T 2 dX ′gg nm 0 0 T 4T 4 dX 33 nm +T 6T 2 dX ′33 nm #

    Teen-teen scattering We next turn to teen-teen scattering. The matrix elements in this case is Λ33 nm = 2 1 4 g4 (2π)5 4Y i=1 X aibi ∞X n1,n2=1 ∞X n3,n4=0 G(Q,n 1,...,n 4)tab 13tba 24tcd 13tdc 24 Z Td 0 p⊥dp⊥ Z ∞ 0 dp∥e−βp(n1+n3) × Z Td 0 p′ ⊥dp′ ⊥ Z ∞ 0 dp′ ∥e−βp′(n2+n4) Z 1 −1 dx (F 3 nm(p,x ) +F 3 nm(p′,x ))c3 nc3 m Z π −π dϕ 2π Z ∞ 0 dkk 3 × Dab L (K)...

  6. [6]

    Gluon-gluon scattering For the gluon-gluon scattering, at nonzero holonomy, the matrix element from Eq. (183) becomes ¯Λgg mn = g4N4 c 2(2π)5 log κ g2Nc ∞X m1,m2=1 m1X m3=1 m2X m4=1 ℓm1ℓm2 ℓm1,m2,m3,m4 Z ∞ 0 dpp 2 × Z ∞ 0 dp′p′2e−β(m1p+m2p′)¯cg m¯cg n[(ψg m(p))′− (ψg m(p′))′][(ψg n(p))′− (ψg n(p′))′] = g4N4 c 2(2π)5 log κ g2Nc ¯cg m¯cg nT 6Y gg mn , (185)...

  7. [7]

    Gluon-teen scattering Analogous to the gluon-gluon scattering, the matrix element for gluon-teen scattering at nonzero holonomy is ¯Λg3 mn = −1 2 g4N4 c 2(2π)5 log κ g2Nc ∞X m1,m2=1 m1X m3=1 m2X m4=1 ℓm1ℓm2 ℓm1,m2,m3,m4 T 2 d 2 Z ∞ 0 dp × Z ∞ 0 dp′p′2e−β(m1p+m2p′)[¯c3 m (ψ3 m(p))′− ¯cg m(ψg m(p′))′][¯c3 n(ψ3 n(p))′− ¯cg n(ψg n(p′))′] = g4N4 c 2(2π)5 log κ...

  8. [8]

    Teen-Teen scattering The matrix element for teen-teen scattering is ¯Λ33 mn = 1 4 g4N4 c 2(2π)5 log κ g2Nc ∞X m1,m2=1 m1X m3=1 m2X m4=1 ℓm1ℓm2 ℓm1,m2,m3,m4 T 2 d 2 Z ∞ 0 dp ×T 2 d 2 Z ∞ 0 dp′ e−β(m1p+m2p′)¯c3 m¯c3 n[(ψ3 m(p))′− (ψ3 m(p′))′][(ψ3 n(p))′− (ψ3 n(p′))′] = g4N4 c 2(2π)5 log κ g2Nc ¯c3 m¯c3 nT 4 dT 2Y 33 mn , (189) where Y 33 mn = 1 4 1 T 2 ∞X m...

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Reviewed August 16, 2026 · model on record in the stance chip above.