A Quantitative Framework of Nonperturbative QCD from Topological Vacuum with Application to Parton Structures

Wei-Yang Liu

REVIEW 3 major objections 9 minor 290 references

A Quantitative Framework of Nonperturbative QCD from Topological Vacuum with Application to Parton Structures

T0 review · 3 major / 9 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Instanton vacuum bridges nonperturbative QCD to parton physics

desk verdict A PhD dissertation extending the instanton liquid model to TMDs, the Collins-Soper kernel, twist-3 color-force form factors, and the nucleon EDM — broad in scope, with real new calculations but a known weak point in U(1)_A physics. read the letter →

arxiv 2607.06060 v1 pith:GHLTJPCY submitted 2026-07-07 hep-ph hep-th

Wei-Yang Liu This is my paper · ORCID

classification hep-phhep-th

keywords vacuumframeworkeffectiveoriginconstructdeterminantensemblefactorization

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 dissertation argues that the QCD vacuum, modeled as a liquid of instantons and anti-instantons with just two parameters fixed by the pion and rho meson masses, provides a quantitatively consistent description of an unusually broad range of hadronic observables. The central object is the Instanton Liquid Model (ILM): a semiclassical ensemble of topological gauge-field configurations that tunnel between distinct vacuum sectors. Light quarks passing through these configurations flip chirality via fermionic zero modes, producing the 't Hooft multi-fermion interaction, which dynamically breaks chiral symmetry, generates a constituent quark mass of roughly 400 MeV, and explains the U(1)_A anomaly and the eta-prime mass. The author develops two complementary formulations: a statistical ensemble sampled by Monte Carlo, and an effective field theory where the quark determinant is rewritten as effective multi-quark vertices. By reformulating this effective theory on the light front, the work constructs explicit light-front wave functions for mesons and baryons, then derives parton distribution functions, transverse-momentum-dependent distributions (TMDs), soft functions, generalized parton distributions, and a wide class of form factors (scalar, pseudoscalar, energy-momentum tensor, twist-3 color force) all from the same vacuum parameters. A key structural claim is that instanton-anti-instanton molecular pairs, absent in the dilute ensemble but resolved at higher resolution, generate nonperturbative relations between quark and gluon operators that are invisible in perturbative QCD. The framework is applied to hadron mass and spin decomposition, near-threshold quarkonium photoproduction, the Collins-Soper rapidity evolution kernel, and the nucleon electric dipole moment from the strong CP problem. Throughout, predictions are compared against lattice QCD and experimental data, with the author reporting quantitative agreement from low to moderate resolution scales.

What carries the argument

The 't Hooft multi-fermion vertex induced by quark zero modes delocalized across the instanton ensemble; instanton-anti-instanton molecular pairs generating additional effective interactions at higher resolution; the Bethe-Salpeter equation resumming quark bubble chains to produce meson and baryon bound states; light-front projection of Bethe-Salpeter wave functions yielding partonic observables; and the grand-canonical topological ensemble connecting vacuum fluctuations to hadronic matrix elements via topological susceptibility and compressibility.

What would settle it

A clean falsifier would be a lattice QCD calculation showing that removing instanton-dominated configurations from gauge ensembles leaves nonperturbative observables (form factors, TMD soft functions, Collins-Soper kernel) essentially unchanged at intermediate Q^2, which would mean instantons are not the dominant carriers of nonperturbative physics the model claims them to be.

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Extended reading notes

Core claim

The QCD instanton vacuum, parameterized solely by an instanton size of about one-third femtometer and a density of about one per cubic fermimeter, generates effective quark interactions that produce hadron spectra, form factors, parton distributions, TMDs, soft functions, and rapidity evolution kernels in a single unified framework. The same two vacuum parameters, fixed by the pion and rho masses, yield predictions for observables ranging from the Collins-Soper kernel to nucleon gravitational form factors to near-threshold J/psi photoproduction, with the author reporting consistency with lattice QCD and experiments across this range. The mechanism carrying the argument is the chirality-flip零

Load-bearing premise

The ILM approximates the full QCD vacuum by saturating it with instantons and anti-instantons, dropping gauge-field fluctuations that are neither self-dual nor associated with near-zero Dirac eigenmodes. If these omitted configurations contribute significantly to observables like Wilson loops, soft functions, or form factors at intermediate resolution, the framework's quantitative predictions would be systematically biased.

Editorial extensions

If this is right

If the ILM parameters truly suffice to predict both hadron spectra and partonic structure, then lattice QCD calculations of PDFs, TMDs, and form factors could be cross-checked against a far cheaper semiclassical model with transparent physical interpretation.
The nonperturbative quark-gluon operator relations derived from instanton molecules could guide targeted lattice QCD calculations by predicting which form factors are correlated and testing those correlations numerically.
The vacuum-origin explanation of the Collins-Soper kernel, if accurate, would provide a physical mechanism for rapidity evolution that complements purely perturbative derivations and could reduce phenomenological uncertainty in TMD extractions.
The instanton-based prediction for the nucleon electric dipole moment from strong CP offers a concrete target for next-generation neutron EDM experiments, tying a topological vacuum parameter to a potentially measurable signal.

Reading between the lines

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

The framework implicitly suggests a natural hierarchy of nonperturbative contributions: chiral-symmetry-breaking observables are dominated by isolated instantons, while confinement-sensitive and gluon-dominated observables may require the molecular or even non-self-dual configurations the ILM omits. This predicts a systematic pattern of where the model should succeed and where it should fail, whic
The resolution-dependent splitting between dilute instantons and molecular pairs resembles a Wilsonian RG flow in topological space. One could test whether the molecular density parameter, treated as phenomenological here, can be derived from the gradient-flow evolution of lattice gauge configurations at intermediate flow times, closing the gap between the model and first-principles lattice data.
If the two-parameter ILM genuinely captures the dominant nonperturbative physics at moderate Q^2, then the breakdown of perturbative factorization in this regime may be quantifiable as the point where instanton-induced operators compete with perturbative gluon exchange, offering a diagnostic for when pQCD-based extractions of parton distributions become unreliable.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Two things matter here. First, this is a genuinely broad extension of the ILM framework to observables it has never touched before: TMD soft functions, the Collins-Soper kernel, twist-3 color Lorentz force form factors, near-threshold quarkonium production, and the nucleon EDM from instantons. Second, the eta-prime mass comes out at 640 MeV versus 958 MeV experimentally, and that 33% miss is not a minor blemish — it traces directly to the topological susceptibility, which feeds into several downstream calculations including the EDM and pseudoscalar form factors in Chapters 5 and 9. The stress-test concern on this point lands and is the most important quantitative issue in the paper. The core ILM formalism is well-established (Shuryak, Diakonov, Petrov, 't Hooft), and the dissertation does not claim to invent it. What is new is the systematic application to partonic observables — the CS kernel computation (Ch. 11), the twist-3 color-force form factors (Ch. 8), and the light-front wave function construction connecting to PDFs and GPDs (Ch. 10). The comparisons to lattice data across meson spectra, gravitational form factors, and the CS kernel are extensive and mostly show 10-20% agreement, which is reasonable for a semi-classical model with a few fitted parameters. The parameters (rho, n_I+A, M, G_IA) are fixed by pion and rho masses, so the circularity burden is moderate — the framework does produce predictions, not just fits, but the parameter space is small enough that the distinction is not sharp. The eta-prime discrepancy is the main soft spot. The Witten-Veneziano relation ties chi_t to the singlet meson mass, and chi_t enters the grand-canonical fluctuation formula (Eq. 3.149) that underlies the pseudoscalar matrix elements. If chi_t is off by a third, those specific predictions carry a systematic bias of comparable size. The author is transparent about this — the 640 MeV is reported plainly in Table 4.3 without spin. The molecular density n_mol is an additional free parameter in the dense regime, which gives extra flexibility but also weakens the predictive power in channels where it enters. Chapter 10 (LaMET connection) is explicitly stated as in progress, which is fine for a dissertation but means that bridge is not yet load-bearing. No code or data is shipped, which limits reproducibility of the numerical results. This is a substantial piece of work for researchers in nonperturbative QCD phenomenology, especially those working on TMDs, form factors, and topological aspects of hadron structure. It deserves a serious referee who can assess the CS kernel and twist-3 calculations in detail, and who will push back on the eta-prime issue and its downstream propagation.

Referee Report

3 major / 9 minor

Summary. This dissertation develops a quantitative framework for nonperturbative QCD based on the instanton liquid model (ILM), in which the infrared QCD vacuum is modeled as an ensemble of instantons and anti-instantons. The framework is formulated both as a statistical ensemble (with weights defined by the instanton action and quark determinant) and as an effective field theory (via the 't Hooft vertex interaction). The author applies this framework to a broad range of observables: meson and baryon spectra, electromagnetic and gravitational form factors, twist-3 color Lorentz force distributions, near-threshold quarkonium photoproduction, nucleon EDM and strong CP, parton distribution functions, and transverse-momentum-dependent (TMD) soft functions including the Collins-Soper kernel. The ILM parameters (instanton size rho, density n_{I+A}, constituent mass M, molecule coupling G_IA, determinantal mass m*) are fixed primarily by the pion and rho meson masses, and the framework's predictions are compared against lattice QCD and experimental data throughout. The work demonstrates reasonable quantitative agreement (typically at the 10-30% level) across many observables, while also identifying known limitations such as the eta' mass underprediction.

Significance. The manuscript provides a comprehensive and ambitious synthesis of the instanton liquid model applied to a very wide range of hadronic observables, from spectroscopy to partonic structure. Its strengths include: (1) a unified framework connecting vacuum topology to both low-energy hadron physics and light-front parton observables; (2) explicit construction of light-front wave functions and their connection to Euclidean Bethe-Salpeter wave functions; (3) falsifiable predictions for form factors, TMD soft functions, and the Collins-Soper kernel, compared against lattice and phenomenological extractions; (4) a systematic 1/N_c and instanton-density expansion organizing the effective quark Lagrangian. The computation of the Collins-Soper kernel from the ILM (Sec. 11.4) and its comparison to recent lattice results is a particularly noteworthy new result. The framework is not parameter-free (rho, n_{I+A}, M, G_IA, m* are fitted), but the number of fitted parameters is modest relative to the range of observables addressed.

major comments (3)

Table 4.3, eta' mass row: The ILM predicts m_{eta'} = 640 MeV versus the experimental 958 MeV, a 33% discrepancy. This is a known limitation of the N_f=2 ILM, but it is load-bearing for the central claim of 'quantitatively consistent' nonperturbative QCD. The eta' mass is set by the topological susceptibility chi_t via the Witten-Veneziano relation (Eq. 3.41), and the same chi_t enters the grand-canonical fluctuation formula (Eq. 3.149-3.152) used for hadronic matrix elements of F tilde-F, including the pseudoscalar form factor (Sec. 5.5) and nucleon EDM (Ch. 9). The manuscript should explicitly quantify the systematic uncertainty propagated from this discrepancy into observables derived from Eq. 3.149. At minimum, a clear statement is needed that the U(1)_A sector is not quantitatively controlled within the N_f=2 framework, and that observables sensitive to topological fluctuations (pse
Sec. 3.4.5, Eq. 3.129: The determinantal mass m* is computed from the same instanton parameters (rho, n_{I+A}) that are fitted to the pion and rho masses, and m* then enters the effective 't Hooft coupling G_I in Eq. 4.2. This creates a degree of circularity in the parameter chain: rho and n_{I+A} are fixed by m_pi and m_rho; m* is derived from rho, n_{I+A}, and the quark condensate; and m* then determines G_I which feeds back into the spectrum. The manuscript should clarify which observables are genuine predictions (i.e., depend only on the fitted rho, n_{I+A}, M, and G_IA) versus those that implicitly inherit the GOR relation by construction. A flowchart or table mapping parameters to observables would strengthen the claim of quantitative consistency.
Sec. 11.4, Eqs. 11.38 and 11.43: The Collins-Soper (CS) kernel is computed in two versions — the full ILM expression and the weak-field approximation — and compared to lattice data in Fig. 11.10. The ILM parameters used here (rho = 0.343 fm, n_{I+A} = 7.46 fm^{-4}) differ significantly from those used in the spectroscopy chapter (rho = 0.33 fm, n_{I+A} = 0.85-1.056 fm^{-4} in Tables 4.1 and 4.7). The manuscript attributes this to the dense vs. dilute ILM distinction (Sec. 3.4.3), but the prescription for choosing which parameter set applies to which observable is not systematically specified. This is load-bearing because the CS kernel result is presented as a key prediction, yet it uses a different parameter regime than the spectroscopy that fixes those parameters. A clear RG/resolution argument for the parameter choice in each chapter is needed.

minor comments (9)

Table 4.1 caption: 'Contituent' should be 'Constituent' (also in Sec. 4.5.2 heading).
Sec. 2.1: 'gredient flow' should be 'gradient flow' in the Chapter 3 overview (Sec. 1.1).
Fig. 3.2 caption: 'instnaton' should be 'instanton'.
Table 4.3: The sigma meson mass range 400-800 MeV from PDG is very broad; the ILM value of 682 MeV falls within this range, but the comparison would be more informative if the specific PDG assignment (e.g., f_0(500) vs. f_0(980)) were clarified.
Sec. 5.6.1 and Fig. 5.21-5.22: The pion and nucleon mass decompositions are presented at two resolutions (mu ~ 1/rho and mu = 2 GeV), but the DGLAP evolution procedure used to go between them is only briefly referenced. A short statement of the evolution order (LO/NLO) and the input PDFs at the low scale would improve reproducibility.
Sec. 9, Table 9.1: The proton and neutron EDM values are presented without uncertainties. Given the eta' mass discrepancy discussed above, an estimate of the systematic uncertainty on these predictions would be appropriate.
Fig. 11.11 and 11.12 appear to be nearly identical plots with different captions (one referencing TMDPDFs, the other referencing the same). This may be a duplication error.
The bibliography is extensive but several references to lattice collaborations could be updated to the most recent results (e.g., FLAG 2024 where available).
Sec. 3.2.1, Eq. 3.28: The two-loop running is used for the instanton size distribution, but the prefactor S(ρ)^{2N_c} is only known at one-loop. The manuscript notes this but does not estimate the systematic uncertainty from the missing higher-loop prefactor.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee raises three major comments concerning: (1) the eta' mass discrepancy and its implications for observables derived from topological susceptibility, (2) the degree of circularity in the parameter chain from rho, n_{I+A} through m* to G_I, and (3) the use of different ILM parameter sets (dense vs. dilute) across chapters, particularly for the Collins-Soper kernel. We agree that all three points identify legitimate gaps in the manuscript's presentation and will revise accordingly. Below we address each comment in detail.

read point-by-point responses

Referee: Table 4.3, eta' mass row: The ILM predicts m_{eta'} = 640 MeV versus the experimental 958 MeV, a 33% discrepancy. This is a known limitation of the N_f=2 ILM, but it is load-bearing for the central claim of 'quantitatively consistent' nonperturbative QCD. The manuscript should explicitly quantify the systematic uncertainty propagated from this discrepancy into observables derived from Eq. 3.149. At minimum, a clear statement is needed that the U(1)_A sector is not quantitatively controlled within the N_f=2 framework, and that observables sensitive to topological fluctuations (pseudoscalar form factor, nucleon EDM) carry this systematic uncertainty.

Authors: The referee is correct that the eta' mass discrepancy is a known and significant limitation of the N_f=2 ILM, and we agree that the manuscript does not currently state this clearly enough or trace its implications for downstream observables. We will revise the manuscript to address this in three concrete ways. First, we will add an explicit statement in Sec. 4.5.3 that the U(1)_A sector is not quantitatively controlled within the N_f=2 framework: the eta' mass is underpredicted by approximately 33%, reflecting the fact that the topological susceptibility in the N_f=2 ILM is set by the instanton density and quark screening (Eq. 3.106) rather than by the full Witten-Veneziano relation with physical singlet meson masses. Second, we will add a discussion in Sec. 3.5.1 quantifying the systematic uncertainty propagated into observables derived from the grand-canonical fluctuation formula (Eqs. 3.149-3.152). The key observation is that the pseudoscalar gluonic matrix element (Eq. 3.152) is proportional to chi_t, and the ratio chi_t^{ILM}/chi_t^{WV} can be estimated from Table 3.2: the ILM prediction using exact duality gives chi_t/V = (75.1 MeV)^4 for N_f=2+1, while the Witten-Veneziano relation gives (72.6 MeV)^4, a discrepancy of approximately 4%. For the N_f=2 case used in the spectroscopy chapter, the discrepancy is larger. We will state that observables sensitive to topological fluctuations — specifically the pseudoscalar form factor (Sec. 5.5) and nucleon EDM (Ch. 9) — carry a systematic uncertainty of order this ratio, and we will add error bands or caveats to the relevant figures and tables. Third, we will clarify in the abstract and conclusion that the framework's claim of 'quantitative consistency' applies to the SU(N_f) sector and observables dominated by chiral (而非 revision: yes
Referee: Sec. 3.4.5, Eq. 3.129: The determinantal mass m* is computed from the same instanton parameters (rho, n_{I+A}) that are fitted to the pion and rho masses, and m* then enters the effective 't Hooft coupling G_I in Eq. 4.2. This creates a degree of circularity in the parameter chain. The manuscript should clarify which observables are genuine predictions versus those that implicitly inherit the GOR relation by construction. A flowchart or table mapping parameters to observables would strengthen the claim of quantitative consistency.

Authors: The referee identifies a legitimate concern about the parameter chain. We agree that the relationship between fitted parameters and derived quantities is not transparently presented. We will add a table (or flowchart) in Sec. 4.1 mapping the full parameter dependency structure. To be specific: the primary fitted parameters are rho, n_{I+A}, M, and G_IA, fixed by m_pi, m_rho (and m_K, m_{rho} in the N_f=3 case). The determinantal mass m* is derived from rho, n_{I+A}, and the quark condensate via Eq. 3.129, and G_I is then determined from rho, n_{I+A}, and m* via Eq. 4.2. The referee is correct that this creates a degree of circularity: observables that depend only on G_I and the GOR relation (such as the pion decay constant f_pi and the quark condensate) are not independent predictions but rather inherit the GOR relation by construction. We will explicitly classify observables into three categories: (i) fitted inputs (m_pi, m_rho, and in N_f=3 also m_K), (ii) derived quantities that inherit the GOR relation (f_pi, quark condensate, m*), and (iii) genuine predictions that depend on the fitted parameters but not on the GOR relation (e.g., meson masses other than pi and rho, baryon masses, form factors, PDFs, TMDs, CS kernel). This classification will make clear which observables test the framework independently. We note that the majority of observables presented in the dissertation — including the full meson spectrum beyond pi and rho, the baryon spectrum, all form factors in Ch. 5-9, and the partonic observables in Ch. 10-11 — fall into category (iii). revision: yes
Referee: Sec. 11.4, Eqs. 11.38 and 11.43: The Collins-Soper kernel uses ILM parameters (rho = 0.343 fm, n_{I+A} = 7.46 fm^{-4}) that differ significantly from those in the spectroscopy chapter (rho = 0.33 fm, n_{I+A} = 0.85-1.056 fm^{-4}). The manuscript attributes this to the dense vs. dilute ILM distinction, but the prescription for choosing which parameter set applies to which observable is not systematically specified. A clear RG/resolution argument for the parameter choice in each chapter is needed.

Authors: The referee is correct that the manuscript does not provide a systematic prescription for choosing between the dilute and dense ILM parameter sets, and that this gap undermines the presentation of the CS kernel result. We will add a clear discussion of this point. The physical basis for the distinction is the resolution scale: the dilute ILM (n_{I+A} ~ 1 fm^{-4}) corresponds to the deeply cooled regime at resolution mu ~ 1/rho ~ 600 MeV, where short-distance IA pairs have annihilated and only isolated instantons remain. The dense ILM (n_{I+A} ~ 7-10 fm^{-4}) corresponds to a higher resolution mu ~ 2 GeV, where correlated IA pairs are resolved as additional configurations (see Sec. 3.4.3 and Fig. 2.3). The key point is that the CS kernel (Sec. 11.4) and Wilson loop observables (Ch. 6) probe the vacuum at transverse distances b_perp ~ 0.1-0.3 fm, corresponding to resolution scales mu ~ 1-2 GeV, where the dense ensemble is the appropriate description. In contrast, hadron spectroscopy (Ch. 4) and form factors at low Q^2 (Ch. 5) probe the vacuum at hadronic scales ~ 1 fm, where the dilute ensemble applies. We will add a table specifying which parameter set is used in each chapter, along with the corresponding resolution scale and the physical justification. We will also add a statement in Sec. 11.4 explicitly noting that the CS kernel calculation uses the dense ILM parameters because the relevant transverse distances are shorter than the instanton separation R ~ 1 fm. We agree that without this clarification, the parameter choice appears arbitrary, and the referee's request for a systematic RG argument is well-justified. revision: yes

Circularity Check

3 steps flagged · score 3.0 of 10

Parameters fitted to pion/rho masses then used to 'predict' other hadron observables; GOR relation built into bosonization; η′ mass underprediction signals incomplete U(1)_A dynamics rather than circularity.

fitted input called prediction [Table 4.1 and Table 4.3 (Sec. 4.1.1, 4.5.3)]
"The fitted parameters in ILM using instanton size ρ = 0.33(2) fm and constituent mass M = 395(3) MeV with fixed pion mass mπ = 139.4 MeV and rho meson mass mρ = 785(6) MeV."

The ILM parameters (ρ, n_{I+A}, M, G_I, G_{IA}) are fixed by fitting to the physical pion and rho meson masses (Table 4.1). The Bethe-Salpeter equations (Eqs. 4.37–4.40) are then solved to produce the meson spectrum (Table 4.3). Since m_π and m_ρ are inputs to the parameter fit, their 'prediction' in Table 4.3 is tautological — the BS equation is guaranteed to reproduce them by construction of the coupling constants. The σ meson mass (682 MeV) and η′ mass (640 MeV) are genuine predictions, but the η′ underprediction by 33% indicates the framework's U(1)_A dynamics are incomplete, which is a correctness concern rather than circularity.
self definitional [Eq. (4.15), Sec. 4.2 (Bosonization)]
"The last term determines the mass of the (pseudo) Goldstone boson by GOR relation. m²_π = 2mσ̄/F²_π"

The GOR relation (Eq. 4.15) is derived as a direct consequence of the bosonization procedure in Eq. (4.12). The pion mass is then used as a fitting input in Table 4.1 to fix the model parameters. When the framework subsequently 'predicts' the pion mass in Table 4.3 or uses the GOR relation to validate the chiral picture ('It successfully generate the low energy GOR relation, indicating the correct chiral picture'), this is circular: the GOR relation is built into the bosonized Lagrangian by construction, so verifying it against the same input pion mass provides no independent test.

1 more flagged steps

fitted input called prediction [Table 4.4, Sec. 4.5.3]
"Quark mass vs. chiral condensate in the ILM using Eqs. (4.34) and (4.35) with the parameters listed in Table 4.1 at the resolution μ = 1/ρ ≈ 600 MeV."

The current quark mass m and quark condensate ⟨q̄q⟩ are computed from the gap equations (4.34, 4.35) using the parameters already fitted to m_π and m_ρ in Table 4.1. Since the GOR relation m²_π = 2m⟨q̄q⟩/F²_π is built into the framework (Eq. 4.15), the product m|⟨q̄q⟩| is constrained to match the pion mass input by construction. The comparison to FLAG lattice values in Table 4.4 is therefore not an independent prediction but a consistency check of the fitted parameters against the same GOR relation that lattice QCD also satisfies.

full rationale

The paper fits ILM parameters (ρ, n_{I+A}, M, G_I, G_{IA}) to the physical pion and rho meson masses (Table 4.1), then uses the Bethe-Salpeter equation to produce a meson spectrum (Table 4.3) where m_π and m_ρ are reproduced by construction. The GOR relation is built into the bosonization procedure (Eq. 4.15), so the quark mass and condensate in Table 4.4 are constrained by the pion mass input. However, the framework does produce genuinely independent predictions for observables not used in fitting: the σ meson mass, gravitational form factors (Ch. 5), TMD soft functions (Ch. 11), color force form factors (Ch. 8), and EDM (Ch. 9) are computed from the same parameters but are not fitted. The η′ mass underprediction (640 vs 958 MeV) is a correctness limitation of the ILM's U(1)_A dynamics, not a circularity issue. Self-citations to prior ILM work (Refs. [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]) are used to reference established methodology, not to import unverified uniqueness theorems. The circularity is moderate: the central spectroscopic results are partially constrained by the fit, but the broader framework's applications to form factors, PDFs, and TMDs contain substantial independent content.

Assumptions & free parameters 9 free parameters · 6 assumptions · 2 invented entities

The framework has ~9 fitted or externally-input parameters. The core parameters (rho, n_I+A, M) are fixed by 2-3 hadron masses and then used across all predictions. The molecular density n_mol is an additional parameter for dense-regime calculations. The axioms are standard within the ILM literature but represent genuine approximations whose validity is not independently established for all observables computed.

free parameters (9)

rho (instanton size) = 0.33(2) fm
Fixed by fitting to lattice instanton size distribution and hadron masses (Table 4.1, 4.7)
n_I+A (instanton density) = ~0.85-1.0 fm^-4
Fixed by fitting to hadron masses and lattice data (Table 4.1)
M (constituent quark mass) = 395(3) MeV
Fixed by gap equation with pion and rho mass input (Table 4.1)
G_IA (molecule coupling) = 67.8 GeV^-2 (Nf=2), 64.8 GeV^-2 (Nf=3)
Determined from instanton parameters and hopping integral T_IA (Table 4.1, 4.7)
m* (determinantal mass) = ~103.6 MeV
Computed from instanton parameters and quark condensate (Sec. 3.4.5)
n_mol (molecule density) = 7.248 fm^-4
Introduced for dense ILM / twist-3 calculations, fixed separately from dilute n_I+A (Ch. 8)
m_0++ (scalar glueball mass) = 1.5-1.7 GeV
External input from lattice, used as a band in form factor calculations (Sec. 5.3)
sigma_piN (pion-nucleon sigma term) = 45-60 MeV
External input used for nucleon form factors (Table 5.4)
m_3g (three-gluon exchange mass) = 1.49 GeV
Fitted mass for eta_c photoproduction (Sec. 7.5)

assumptions (6)

domain assumption QCD vacuum is saturated by instantons and anti-instantons at low resolution
Core ILM assumption; non-self-dual fluctuations omitted (Sec. 2.1, 3.3)
domain assumption Large-Nc planar resummation captures dominant quark dynamics in instanton vacuum
Used throughout for quark propagator and effective Lagrangian (Sec. 3.3.2, 3.4.2)
ad hoc to paper Instanton size distribution is sharply peaked at mean value rho
Mean-field RILM approximation; used for most analytic calculations (Sec. 3.4.2)
domain assumption Two-body semiclassical interactions suffice for instanton ensemble
Only pairwise Sint included in partition function (Sec. 3.2.4)
domain assumption Zero-mode dominance for light quarks in instanton background
Non-zero modes approximated by free propagator (Sec. 3.3.2)
ad hoc to paper Ratio ansatz for IA pair interaction
Simplified alternative to streamline/Yung ansatz (Sec. 3.2.2)

invented entities (2)

Instanton-anti-instanton molecules (dense ILM) independent evidence
purpose: Short-distance correlated IA pairs for intermediate resolution observables
Observed in lattice gradient flow; density n_mol is fitted but the existence of such configurations is supported by lattice (Sec. 2.1, 3.4.3)
Determinantal mass m* independent evidence
purpose: Effective quark mass from fermion determinant in instanton ensemble
Computed from instanton parameters; consistent with numerical IILM simulations (Sec. 3.4.5)

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Pith. "Pith review of A Quantitative Framework of Nonperturbative QCD from Topological Vacuum with Application to Parton Structures." pith.science (2026). https://pith.science/paper/GHLTJPCY

@misc{pith2026260706060,
  author       = {Pith},
  title        = {Pith review of: A Quantitative Framework of Nonperturbative QCD from Topological Vacuum with Application to Parton Structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GHLTJPCY}},
  note         = {Machine review of arXiv:2607.06060}
}

read the original abstract

This dissertation develops a quantitative framework that provides a physical picture for non-perturbative QCD based on the topological structure of the QCD vacuum. By integrating out the ultraviolet degrees of freedom, the infrared gluon configurations are modeled as a liquid ensemble of instantons and anti-instantons, which induce effective interactions among quarks. This framework captures the origin of trace and axial anomalies through the infrared distributions of QCD and dynamically breaks chiral symmetry. More specifically, this framework can be formulated in two ways: in one, we construct a statistical ensemble with weights defined by the instanton action and Dirac determinant, while in the other we formulate an effective field theory (EFT) by rewriting the determinant as effective quark interactions. By reformulating the EFT on the light front, we explicitly construct the light-front wave functions and calculate various parton observables in linear factorization. We further embed the framework into transverse momentum dependent factorization and establish a vacuum origin for rapidity evolution by computing the soft functions. We also extend this approach to various form factors in light hadrons, including scalar, pseudoscalar, and energy-momentum tensor (EMT), as well as higher-twist color force and multigluon correlations, with applications to hadron mass and spin decomposition, near-threshold quarkonium production, and strong CP problem, highlighting the importance of the vacuum origin in hadron structures. Overall, this work demonstrates that the QCD vacuum provides a quantitatively crucial description of hadronic structure from low to moderate resolution, bridging nonperturbative vacuum physics with partonic phenomenology.

Figures

Figures reproduced from arXiv: 2607.06060 by the authors.

**Figure 1.1.** The universe is made up of three components: visible matter (5%), [PITH_FULL_IMAGE:figures/full_fig_p029_1_1.png] view at source ↗

**Figure 1.2.** (a) Lattice visualization of a YM vacuum (Image credit: [PITH_FULL_IMAGE:figures/full_fig_p030_1_2.png] view at source ↗

**Figure 2.1.** A schematic illustration of the Yang–Mills gradient flow toward [PITH_FULL_IMAGE:figures/full_fig_p034_2_1.png] view at source ↗

Figures from the paper (92 more)

**Figure 2.2.** Figure 2.2: Visualization of the QCD vacuum in different resolutions [PITH_FULL_IMAGE:figures/full_fig_p035_2_2.png]

**Figure 2.3.** Figure 2.3: Dependence of the mean instanton size ρ (left panel) and density n = nI+A (right panel) on the dimensionless gradient flow time τ = t/a2 with lattice spacing a = 0.046 fm [25], determined by √ 8t0 = 0.1 fm, or 1/ √ 8t0 ∼ 2 GeV. The quantum vacuum corresponds to the e…

**Figure 2.4.** Figure 2.4: Topological landscape of gauge fields we obtain ⟨n + Q| e −Ht |n⟩ = 1 2π Z 2π 0 dθ eiθQ X N+,N− 1 N+! N−! [PITH_FULL_IMAGE:figures/full_fig_p041_2_4.png]

**Figure 2.5.** Figure 2.5: (a) Cartoon picture for the coexistence of center vortices and [PITH_FULL_IMAGE:figures/full_fig_p044_2_5.png]

**Figure 2.6.** Figure 2.6: Instanton (yellow) and anti-instanton (blue) configurations in the [PITH_FULL_IMAGE:figures/full_fig_p045_2_6.png]

**Figure 2.7.** Figure 2.7: The constituents of instantons (calorons): BPS and KK dyons, and [PITH_FULL_IMAGE:figures/full_fig_p047_2_7.png]

**Figure 3.1.** Figure 3.1: (a) SU(2) instanton size distribution with 1-loop (solid red) and 2- loop (solid blue) compared to lattice calculations. (b) One-loop parametrized SU(3) instanton size distribution with 1-loop (solid red) and 2-loop (solid blue) compared to lattice calculations. dens…

**Figure 3.2.** Figure 3.2: (a) Action of a IA configuration (in unit of single instnaton action 8π 2 ) in ratio ansatz vs. streamline configuration. (b) Electric field (black solid) and magnetic field (blue dashed) with µ = 3, a = 3 as functions of x4/ρ ratio ansatz as a variant version, A a µ…

**Figure 3.3.** Figure 3.3: Mean action distribution in physical YM vacuum (blue and in [PITH_FULL_IMAGE:figures/full_fig_p065_3_3.png]

**Figure 3.4.** Figure 3.4: Quark propagator distorted by the instanton background with zero [PITH_FULL_IMAGE:figures/full_fig_p069_3_4.png]

**Figure 3.5.** Figure 3.5: The constituent mass M(k) running with the quark momentum k with the instanton size ρ = 0.313 fm and nI+A = 1.056 fm−4 compared with lattice QCD using dynamical O(a)-improved Wilson fermions [119] (red) and result using overlap and Asqtad fermions [120] in Landau gau…

**Figure 3.6.** Figure 3.6: ILM predicted distribution of topological charges in QCD [PITH_FULL_IMAGE:figures/full_fig_p078_3_6.png]

**Figure 3.7.** Figure 3.7: Density expansion for the effective quark operators in dilute ILM [PITH_FULL_IMAGE:figures/full_fig_p083_3_7.png]

**Figure 3.8.** Figure 3.8: Feynman diagrams for the vertices (Nf = 2) induced by a close pair of instanton (I) and anti-instanton (A): (a) vacuum tunneling rate of a fully connected molecule where all flavors looped across the pair. (b) vacuum tunneling rate where one flavor looped across the …

**Figure 3.9.** Figure 3.9: Red and blue circles represents the instantons and anti-instantons [PITH_FULL_IMAGE:figures/full_fig_p085_3_9.png]

**Figure 3.10.** Figure 3.10: Vacuum resummation in the IA pair. Those internal quark loops are often omitted in diagrams for simplicity ⟨LIA⟩ = Z dρIdρAn(ρI )n(ρA)ρ Nf I ρ Nf A Z d 4RduX Nf n=1 Nf n |TIA(u, R)| m∗ 2n = nI+A 2 2 Z d 4RduX Nf n=1 Nf n |TIA| m∗ 2n (3.126) The first b…

**Figure 3.11.** Figure 3.11: Feynman diagrams in the RILM effecitve quark theory. (a) 2 [PITH_FULL_IMAGE:figures/full_fig_p090_3_11.png]

**Figure 3.12.** Figure 3.12: Feynman diagrams in the RILM effecitve Lagrangian. (a) semi [PITH_FULL_IMAGE:figures/full_fig_p091_3_12.png]

**Figure 4.1.** Figure 4.1: Light quarks flip their chirality when passing through the instanton [PITH_FULL_IMAGE:figures/full_fig_p098_4_1.png]

**Figure 4.2.** Figure 4.2: Quark-anti-quark bubble chain in meson channels [PITH_FULL_IMAGE:figures/full_fig_p106_4_2.png]

**Figure 4.3.** Figure 4.3: (a) The plot for the Nf = 2 gap equation in Eq. (4.34). Quark constituent mass vs. the ’t Hooft coupling gσ for increasing current quark mass (bottom-up). (b) The plot for the quark condensate with Nf = 2 dynamical quark in the instanton vacuum. Quark condensate vs. …

**Figure 4.4.** Figure 4.4: Pion and rho meson bound state in ILM 4.5.4 BS wave functions The meson wave function in two-body Fock space is related to the BetheSalpeter (BS) wave functions defined by i(2π) 4 δ 4 (p − k1 − k2)ΨX,ij (k1, k2; P λ)δαβ = Z d 4x1d 4x2e ik1·x1+ik2·x2 ⟨0|ψαi(x1)ψ¯ βj …

**Figure 4.5.** Figure 4.5: The Feynman diagrams for the quark bubble resummation in the [PITH_FULL_IMAGE:figures/full_fig_p136_4_5.png]

**Figure 4.6.** Figure 4.6: Baryon bound state in ILM. Nucleon has contribution from both [PITH_FULL_IMAGE:figures/full_fig_p139_4_6.png]

**Figure 4.7.** Figure 4.7: The Feynman diagrams for the quark-diquark bubble resummation [PITH_FULL_IMAGE:figures/full_fig_p140_4_7.png]

**Figure 4.8.** Figure 4.8: The quark exchange matrix in color space (a) and in flavor-spin [PITH_FULL_IMAGE:figures/full_fig_p141_4_8.png]

**Figure 5.1.** Figure 5.1: Illustration for QCD operator probing the vacuum with [PITH_FULL_IMAGE:figures/full_fig_p151_5_1.png]

**Figure 5.2.** Figure 5.2: The factorization of (a) meson and (b) baryon form factors at large [PITH_FULL_IMAGE:figures/full_fig_p152_5_2.png]

**Figure 5.3.** Figure 5.3: (a) and (b) are the perturbative gluon exchange diagrams at the [PITH_FULL_IMAGE:figures/full_fig_p152_5_3.png]

**Figure 5.4.** Figure 5.4: The soft, semi-hard, and hard Q2 energy regions characterizing different regimes of the hadronic form factors Unfortunately, there are big discrepancies between such an asymptotic theory and existing experimental data, which remain in the semi-hard regime, as illustr…

**Figure 5.5.** Figure 5.5: (a) Pion form factor Fπ data [203] compared with LO+NLO hard contributions [205] based on an analysis of the pion-photon transition form factor from CLEO [206] and CELLO [207]. The supplemented soft component is estimated from local quark–hadron duality. (b) Comparis…

**Figure 5.6.** Figure 5.6: (a) Disconnected contribution to the gluonic operators. (b) One [PITH_FULL_IMAGE:figures/full_fig_p154_5_6.png]

**Figure 5.7.** Figure 5.7: The form factors of (a) meson X and (b) baryon B at small Q2 with hadron state addressed by the BS wave function. The probe denoted by the cross dot is dressed by the instanton vacuum (See text). The momentum conservation requires k ′ 1 = k1 + q and k ′ 2,3 = k2,3. t…

**Figure 5.8.** Figure 5.8: The instanton form factors induced by molecular profile for gluonic [PITH_FULL_IMAGE:figures/full_fig_p157_5_8.png]

**Figure 5.9.** Figure 5.9: Spectral function from the ILM (open red circles) [ [PITH_FULL_IMAGE:figures/full_fig_p159_5_9.png]

**Figure 5.10.** Figure 5.10: (a) Normalized quark scalar form factors in pion using [PITH_FULL_IMAGE:figures/full_fig_p162_5_10.png]

**Figure 5.11.** Figure 5.11: Gluonic scalar form factor of pion (red solid line) together with [PITH_FULL_IMAGE:figures/full_fig_p163_5_11.png]

**Figure 5.12.** Figure 5.12: Nucleon gluonic scalar form factor (5.24) (red solid line) together with the 0++ glueball correlation function (5.15) using scalar glueball mass m0++ = 1.6 GeV (orange dashed line are presented. The red band reflects the sensitivity of the form factor to scalar glue…

**Figure 5.13.** Figure 5.13: (a) One-body operator arising from single (anti-)instantons in [PITH_FULL_IMAGE:figures/full_fig_p167_5_13.png]

**Figure 5.14.** Figure 5.14: The instanton form factors induced by single pseudoparticle in [PITH_FULL_IMAGE:figures/full_fig_p169_5_14.png]

**Figure 5.15.** Figure 5.15: Pion trace form factor. Blue square represents the reconstruction [PITH_FULL_IMAGE:figures/full_fig_p171_5_15.png]

**Figure 5.16.** Figure 5.16: Pion form factors Aπ(Q2 ) (left red line) and Dπ(Q2 ) (right red line) in the ILM, compared to the lattice results (blue square) [219] For the separate gravitational form factors, quark and gluon contributions are scale-dependent. In [PITH_FULL_IMAGE:figures/full_f…

**Figure 5.17.** Figure 5.17: (a) Quark contribution Aq π (Q2 ) (blue) and gluon contribution Ag π (Q2 ) (red) to the pion gravitational form factor in the ILM at a resolution µ = 1/ρ (dashed lines) and at a resolution µ = 2 GeV (solid lines). The lattice results are from [219]. (b) Quark contri…

**Figure 5.18.** Figure 5.18: Nucleon trace form factor in (5.22) with σπN = 53 MeV where the red band indicates the range of m0++ = 1.5 − 1.7 GeV. DN (Q 2 ) = 4m2 N 3Q2 GN (Q 2 ) + σN (Q 2 ) − 1 + Q2 4m2 N AN (Q 2 ) + Q2 2m2 N JN (Q 2 ) (5.53) Using the result in Eq. (5.38), the nucleon…

**Figure 5.19.** Figure 5.19: Nucleon gravitational form factors AN (Q2 ) and DN (Q2 ) in the ILM with red bands indicating the range of σπN = 45 − 60 MeV, compared to the fitted lattice results [219] using dipole form. D g N (Q 2 ) = nIAγIA 2Nc(N2 c − 1)("8π 2 3 β (IA) Tg,1 (ρq) + π 2 3 ρ 2Q 2β…

**Figure 5.20.** Figure 5.20: (a) Quark contribution A q N (Q2 ) (blue) and gluon contribution A g N (Q2 ) (orange) to the pion gravitational form factor in the ILM at a resolution µ = 1/ρ (dashed lines) and at a resolution µ = 2 GeV (solid lines). The lattice results are from [219]. (b) Quark c…

**Figure 5.21.** Figure 5.21: Pion mass decomposition using Ji’s mass sum rule, in the QCD [PITH_FULL_IMAGE:figures/full_fig_p184_5_21.png]

**Figure 5.22.** Figure 5.22: Nucleon mass decomposition using Ji’s mass sum rule in the [PITH_FULL_IMAGE:figures/full_fig_p185_5_22.png]

**Figure 5.23.** Figure 5.23: Spin decomposition using Ji’s sum rule using Σ [PITH_FULL_IMAGE:figures/full_fig_p191_5_23.png]

**Figure 6.1.** Figure 6.1: Wilson loop in quenched vacuum (YM) 6.1 YM vacuum The investigation of the infrared behavior of QCD is the formulation of the non-Abelian gauge theory on loop space that becomes possible by virtue of the close correspondence between gauge and chiral fields [267]. One…

**Figure 6.2.** Figure 6.2: Wilson loop in chiral broken QCD vacuum an effective determinantal mass m∗ . Within this framework, the mean-field approximation in (6.9) remains applicable. The key difference relative to the quenched case lies in the weighting of the instanton ensemble: the average…

**Figure 7.1.** Figure 7.1: Kinematics for the γN → XN′ process. facilities, probes C-odd gluon exchanges. The latters Reggeize to the Odderon at higher center of mass energy. The aim of this chapter is to analyze the near threshold production of the ηc,b, by factorizing out the leading gluonic…

**Figure 7.2.** Figure 7.2: (a) Skewness ξ in the (W, −t) plane in the kinematically allowed region with Mηc = 2.982 GeV. The kinematically allowed region for charmonium J/ψ is similar due to the similar mass MJ/ψ = 3.097 GeV. (b) Skewness ξ in the (W, −t) plane in the kinematically allowed reg…

**Figure 7.3.** Figure 7.3: Feynman diagram for the hard kernel in J/ψ photoproduction. We use the light-cone gauge, where the unphysical longitudinally polarized collinear gluons vanish. The lower bound of the parton momentum-integral assures that the spectators in the nucleon remains physical…

**Figure 7.4.** Figure 7.4: Feynman diagram for the hard kernel for ηc photoproduction. proton, through W2g(t, ξ) = Z 1 −1 dx 1 x − ξ + i0 + 1 x + ξ − i0 + f2g(x, t, ξ) (7.8) where the two-gluon distribution is defined as f2g(x, ξ, t) = Z dz− 2π e −ixP¯+z− 1 P¯+ ⟨N ′ |F a+i [PITH_FULL_IMAGE:fi…

**Figure 7.5.** Figure 7.5: One-body operator arising from (a) instanton–anti-instanton [PITH_FULL_IMAGE:figures/full_fig_p209_7_5.png]

**Figure 7.6.** Figure 7.6: (a) The ILM prediction with the meson dominance exchange on [PITH_FULL_IMAGE:figures/full_fig_p214_7_6.png]

**Figure 7.7.** Figure 7.7: The instanton estimation on the ηc production differential cross section with the holographic prediction on the t-dependence [289] using fitted mass m3g = 1.49 GeV is compared to the model calculation using eikonal dipole approximation [310]. scale αs(µ = 2mc) = 0.30…

**Figure 7.8.** Figure 7.8: ILM estimation on the differential cross section of the [PITH_FULL_IMAGE:figures/full_fig_p216_7_8.png]

**Figure 7.9.** Figure 7.9: The t dependence for the threshold photo-production of ηc using near-threshold approximated ILM formula (f1 dominated crossection) with different the center of mass energy √ s. 191 [PITH_FULL_IMAGE:figures/full_fig_p218_7_9.png]

**Figure 7.10.** Figure 7.10: ILM differential cross section for ηc photoproduction using nearthreshold approximated formula (f1 dominated) at √ s = 300GeV, using the GPD arguments with the C-odd holographic form factor [289] (fitted mass m3g = 1.49 GeV) (blue solid line). The comparison is to …

**Figure 8.1.** Figure 8.1: The single instanton/anti-instanton vertices with the insertion of [PITH_FULL_IMAGE:figures/full_fig_p223_8_1.png]

**Figure 8.2.** Figure 8.2: A molecular pair of instanton-anti-instanton with the insertion of [PITH_FULL_IMAGE:figures/full_fig_p223_8_2.png]

**Figure 8.3.** Figure 8.3: Emergent non-local form factors β (IA) qGq ¯ associated to the color Lorentz operator in a molecular (instanton-anti-instanton) pair. 8.2 Color force on a pion To illustrate these ideas, let us start with a simpler case, that of deep inelastic scattering on a pion, a…

**Figure 8.4.** Figure 8.4: Emergent form factor F q π,1 (Q2 ) induced by a color Lorentz operator in the pion in the ILM enhanced by “molecular” IA pairs. ⟨N(p ′ , s′ )|ψigF ¯ µνγ σψ|N(p, s)⟩ = ¯us ′(p ′ ) ( (¯p µ q ν − p¯ ν q µ ) p¯ σ mN Φ q N,1 (Q 2 ) + mN (q µ g σν − q ν g σµ) Φq N,2 (Q 2 )…

**Figure 8.5.** Figure 8.5: Transverse field distribution of the color Lorentz force in an [PITH_FULL_IMAGE:figures/full_fig_p232_8_5.png]

**Figure 8.6.** Figure 8.6: Color force form factors (a) F q N,1 (Q2 ), (b) F q N,2 (Q2 ), and (c) F q N,3 (Q2 ) from ILM with parameters nmol = 7.248 fm−4 and current quark mass m = 10 − 58 MeV (mπ = 140 − 338 MeV) showing with bands, evolved to 2 GeV compared to the lattice calculation with p…

**Figure 8.7.** Figure 8.7: (a) Transverse field distribution of the color Lorentz force in an [PITH_FULL_IMAGE:figures/full_fig_p240_8_7.png]

**Figure 9.1.** Figure 9.1: Leading pseudoparticle (single instanton) contribution to the quark [PITH_FULL_IMAGE:figures/full_fig_p244_9_1.png]

**Figure 9.2.** Figure 9.2: The CP-odd electric dipole form factor F3(Q2 ) from the ILM for proton (blue) and neutron (red) is obtained using (9.9) with ILM parameters given in [PITH_FULL_IMAGE:figures/full_fig_p246_9_2.png]

**Figure 10.1.** Figure 10.1: (a,b): pQCD gluon-mediated quark pair production; (c,d): [PITH_FULL_IMAGE:figures/full_fig_p252_10_1.png]

**Figure 10.2.** Figure 10.2: Flavor asymmetry of the antiquark sea as the ratio [PITH_FULL_IMAGE:figures/full_fig_p252_10_2.png]

**Figure 10.3.** Figure 10.3: The picture of LaMET framework. The equal-time correlation is [PITH_FULL_IMAGE:figures/full_fig_p255_10_3.png]

**Figure 10.4.** Figure 10.4: Pion PDF uπ+ and kaon PDF uK+ obtained by ILM at low resolution µ0 ∼ 630 MeV To illustrate this, we use the LFWFs derived in Eqs. 4.54 and 4.55 and substitute them into Eq. (10.9) to compute the corresponding unpolarized quark PDF for mesons. The resulting pion and …

**Figure 10.5.** Figure 10.5: Pion and kaon PDF evolved to a few GeV indicated in the plot [PITH_FULL_IMAGE:figures/full_fig_p260_10_5.png]

**Figure 10.6.** Figure 10.6: The physically allowed kinematic region in GPD [PITH_FULL_IMAGE:figures/full_fig_p261_10_6.png]

**Figure 11.1.** Figure 11.1: Feynman diagram of TMD factorization for Drell-Yan with hard [PITH_FULL_IMAGE:figures/full_fig_p272_11_1.png]

**Figure 11.2.** Figure 11.2: Feynman diagram of TMD factorization for SIDIS with hard [PITH_FULL_IMAGE:figures/full_fig_p273_11_2.png]

**Figure 11.3.** Figure 11.3: Feynman diagram of TMD factorization for [PITH_FULL_IMAGE:figures/full_fig_p275_11_3.png]

**Figure 11.4.** Figure 11.4: Wilson lines in Eq. (11.18) for (a) SIDIS process with the space-like correlation function and (b) Drell-Yan process with the timelike correlation. of the above mentioned Wilson loops in Minkowski signature. Wilson loops made of Wilson lines capture important aspect…

**Figure 11.5.** Figure 11.5: The contours of the Wilson lines for the soft function of the cross [PITH_FULL_IMAGE:figures/full_fig_p278_11_5.png]

**Figure 11.6.** Figure 11.6: Soft function 11.4.1 Soft function in ILM In the ILM, the soft function associated to [PITH_FULL_IMAGE:figures/full_fig_p281_11_6.png]

**Figure 11.7.** Figure 11.7: Instanton liquid estimation on (a) Coulomb potential (b) CS [PITH_FULL_IMAGE:figures/full_fig_p283_11_7.png]

**Figure 11.8.** Figure 11.8: Asymptotic logarithmic curve compared to CS kernel in weak [PITH_FULL_IMAGE:figures/full_fig_p284_11_8.png]

**Figure 11.9.** Figure 11.9: The angular dependence of K in (11.38). in [PITH_FULL_IMAGE:figures/full_fig_p285_11_9.png]

**Figure 11.10.** Figure 11.10: (a) The black-solid curve is the ILM result for the full CS kernel [PITH_FULL_IMAGE:figures/full_fig_p287_11_10.png]

**Figure 11.11.** Figure 11.11: Pion (left) (11.58) and kaon (right) (11.60) TMDPDFs at low resolution µ = 1/ρ with ρ = 0.313 fm: (a,b) are the density plots, (c,d) the 3D plots, (e,f) the transverse momentum dependent plots for fixed x, and (g,h) the longitudinal momentum dependence for fixed k⊥…

**Figure 11.12.** Figure 11.12: Pion (left) (11.58) and kaon (right) (11.60) TMDPDFs at low resolution µ = 1/ρ with ρ = 0.313 fm: (a,b) are the density plots, (c,d) the 3D plots, (e,f) the transverse momentum dependent plots for fixed x, and (g,h) the longitudinal momentum dependence for fixed k⊥…

**Figure 11.13.** Figure 11.13: Upper panel is the contour plot of the pion TMD parton / [PITH_FULL_IMAGE:figures/full_fig_p294_11_13.png]

**Figure 11.14.** Figure 11.14: Upper panel is the contour plot of the kaon TMD parton [PITH_FULL_IMAGE:figures/full_fig_p295_11_14.png]

**Figure 11.15.** Figure 11.15: Perturbative (shaded) and non-perturbative (unshaded) regions [PITH_FULL_IMAGE:figures/full_fig_p297_11_15.png]

**Figure 11.16.** Figure 11.16: The evolved pion TMD versus b⊥ with (a) x = 0.3 and (b) x = 0.6, compared to the Drell-Yan experimental data extraction [448] from three available measurements of the transverse momentum cross section for pion-induced Drell-Yan process performed by NA3 [449], E537 …

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