Hot and Dense Medium Effects on the $B_s^*$ and $B^*$ Multiplets

J.Y. S\"{u}ng\"{u}; K. Azizi; N. Er

REVIEW 3 major objections 6 minor 53 references

Hot and Dense Medium Effects on the $B_s^$ and $B^$ Multiplets

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read In hot, dense matter, B* and B_s* mesons keep most of their mass but lose up to ~78% of their decay constant, while baryon density splits particles from antiparticles

desk verdict Competent six-state QCDSR study of B*/B_s* in hot dense matter with a real density-driven asymmetry claim, but the n=0 CPT inconsistency in Table VII makes the quantitative headline conditional. read the letter →

arxiv 2607.22419 v1 pith:AZNXKS3U submitted 2026-07-24 hep-ph hep-exhep-lat

K. Azizi , N. Er , J.Y. Süngü This is my paper · ORCID

classification hep-phhep-exhep-lat PACS 12.38.Lg14.40.Nd

keywords QCDsumrulesB*mesonsB_s*finitetemperaturebaryondensitydecayconstantsparticle-antiparticleasymmetryvectorself-energy

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 what happens to the beauty vector mesons — B_s*(5415), B*(5325), and their antiparticles — inside the hot, baryon-rich matter formed in heavy-ion collisions. Working with QCD sum rules at finite temperature and density, the authors claim a two-part pattern: the mass resists the medium, with no state losing more than ~13% of its vacuum value even at the extreme conditions T = T_c, n = 5n_0, while the decay constant tracks the condensate melting far more closely, falling by up to ~78% at that same point. The third claim carries the sharpest practical consequence: at zero baryon density all six states respond almost identically, but at finite density the vector self-energy acts with opposite signs on particles and antiparticles and drives a growing mass split — most visibly the antiparticle of B*0 losing 12.9% of its mass where B*0 loses 6.1%. A sympathetic reader cares because the paper hands heavy-ion physics a concrete diagnostic: the mass is a weak medium probe, while the decay constant and the particle-antiparticle gap carry the information.

What carries the argument

The load-bearing machinery is the in-medium two-point correlation function of vector currents, computed twice and matched: on the hadronic side, a pole ansatz with a medium-shifted momentum that introduces the vector self-energy; on the QCD side, an operator product expansion whose quark, gluon, and mixed condensates carry fitted temperature- and density-dependence (Eqs. 15 and 18). Borel transformation and continuum subtraction with an in-medium threshold s0(T,n) close the sum rules, with the threshold scaling linearly with the quark-condensate ratio (Eq. 23) and relaxing toward the heavy-quark mass squared as the condensate melts. The asymmetry mechanism is the vector self-energy itself: i

What would settle it

Recompute the same two-point correlator with a different in-medium threshold prescription — e.g., one tied to a thermal width or to the gluon condensate rather than the quark condensate — and check whether the antiparticle-of-B*0 versus B*0 mass-shift gap (12.9% vs 6.1%) survives; if it collapses or changes sign, the claim is an artifact of Eq. (23). A lattice calculation of the B* spectral function at finite baryon density, or a high-density heavy-ion measurement showing under ~30% decay-constant suppression, would falsify the quantitative predictions.

Watch

Extended reading notes

Core claim

The central discovery is a three-part answer to how beauty vector mesons behave in hot, dense matter. Heavy-quark decoupling protects the masses: shifts stay between -0.5% and -1.1% from temperature alone and reach at most ~13% at T = T_c, n = 5n_0. The decay constant gets no such protection and tracks the condensate suppression, falling to 14% of its vacuum value. Baryon density breaks particle-antiparticle symmetry through the vector self-energy, whose sign depends on the light-quark content: the antiparticle of B*0 loses 12.9% of its mass where B*0 loses 6.1%, while the strange pair splits by under a percent. This transplants the kaons' opposite-sign vector interaction into the beauty sec

Load-bearing premise

The load-bearing premise is that the in-medium threshold s0(T,n) scales with the quark-condensate ratio as in Eq. (23) and that the fitted condensates of Eqs. (15) and (18) hold up to n = 5n_0; every reported shift — the ≤13% mass bound, the ~78% decay-constant fall, the 12.9% vs 6.1% gap — flows through this channel, which the paper tests against no independent observable and itself cautions is unreliable near T ≈ T_c, n ≈ 5n_0.

Editorial extensions

If this is right

If the paper is right, the in-medium mass is a weak diagnostic across the whole multiplet: even at T = T_c and n = 5n_0 no B*-family state loses more than ~13% of its vacuum mass, so mass shifts alone cannot separate temperature effects from density effects.
The decay constant is the sensitive channel: with suppressions between 69% and 86% at n = 5n_0, B*-state radiative and leptonic decay rates should be strongly suppressed in dense collisions, a measurable signature that tracks condensate melting rather than quark-mass kinematics.
The decay-constant ordering does not follow the mass ordering: the antiparticle of B*0 keeps the most decay constant (-69%) while suffering the largest mass shift (-12.9%), so the two observables carry independent information about the medium.
The particle-antiparticle mass gap is a baryon-density meter: the neutral pair splits by ~350 MeV at 5n_0 (shrinking toward ~300 MeV at T_c), the charged pair by ~240 MeV, the strange pair by only ~42 MeV — and the sign of the gap is tied to the light-quark content, with the antiparticle lighter in the non-strange sector and heavier in the strange sector.
At zero net baryon density the medium preserves particle-antiparticle degeneracy (uniform -0.5% to -1.1% mass shifts, -3.9% to -5.3% decay-constant shifts), so any observed asymmetry selects a baryon-rich environment.

Reading between the lines

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

Editorial extension: the same machinery predicts a flavor hierarchy — D* mesons, with a lighter charm quark, should lose a larger mass fraction than B* mesons under identical conditions, because the protection is kinematic (the heavy-quark mass); running the same sum rules on the charm sector would test this directly.
Editorial extension: since the asymmetry enters through the light-quark density operator of the medium, the model implies the mass-gap sign should flip in an antibaryon-rich environment — a qualitative prediction the paper does not state.
Editorial extension: the paper itself warns that results near T ≈ T_c and n ≈ 5n_0 are the least reliable, so the headline numbers (12.9% vs 6.1%) carry least confidence exactly at the advertised extremes; the safer experimental target is the density-driven trend at moderate n, not the endpoint values.
Editorial extension: the paper defers a term-by-term decomposition of the vector self-energy to future work, so the attribution of the entire asymmetry to that single term is established only at the level of the sum-rule extraction; a microscopic calculation of the vector self-energy in the same effective model would be the direct check.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Punchline: this is a standard thermal-dense QCD sum rule calculation, competently done, with a genuinely new piece — the full six-state map of the B*/B_s* multiplet and the claim of a density-driven particle-antiparticle splitting from the vector self-energy. The heavy-quark protection story is physically sensible, and the abstract's numbers match the tables. But there is a real internal inconsistency that makes the headline percentages conditional: at T=Tc, n=0, Table VII gives B*+ a mass shift of -0.6% and B*- -1.1%, with a similar split in the neutral pair. The text says the medium preserves particle-antiparticle degeneracy at zero net baryon density. CPT requires exact degeneracy. That ~30 MeV asymmetry at n=0 is a numerical artifact in the Sigma_v extraction, not physics. It does not by itself erase the large density-driven gap at n=5n0, but it means the 6.1% vs 12.9% split cannot be claimed as purely physical until the artifact is traced and removed. The paper also quotes all in-medium shifts without uncertainties, while the vacuum decay constants carry +/- 14-16 MeV; given the Borel-window sensitivity visible in Fig. 1, the precision of the ~80% suppressions is unquantified.

What is actually new: the systematic treatment of all six charge/flavor states, the strange vs non-strange comparison, and the explicit separation of isospin splitting from C-violating splitting. The appendix gives the full OPE expression for one structure, which helps reproducibility. The citations are fair; the group is building on its own prior kaon and D* work, and Kumar's earlier B* calculation is acknowledged.

The soft spots beyond the n=0 problem: the results are largely inherited from the input channel. The continuum threshold scales with the fitted condensate ratio via Eq. (23), and the same 11-parameter fits supply the OPE terms. The authors themselves flag T≈Tc and n=5n0 as the reliability limit, yet those are exactly where the headline numbers live. That is acceptable as a stated limitation, but it should temper how the numbers are used.

The n=0 inconsistency is the load-bearing issue. The second-pass stress-test is right: the text's degeneracy claim and the table disagree. Fixing that, and adding error bars, would turn this into a useful baseline. It deserves a serious referee, but my own verdict would be major revision, not accept. If working on beauty-vector probes at RHIC/FAIR, I'd cite it with a caveat.

Referee Report

3 major / 6 minor

Summary. This paper presents a QCD sum-rule study of the in-medium masses and decay constants of the bottom-vector mesons B_s^*(5415) and B^*(5325), computed for all six charge/flavor states (B_s^{*0}, \bar B_s^{*0}, B^{*+}, B^{*-}, B^{*0}, \bar B^{*0}) at finite temperature and baryon density. The OPE includes thermal and density-dependent quark, gluon, and mixed condensates, with the medium dependence of the quark and gluon condensates taken from analytic fits to chiral SU(3) model results, and the effective continuum threshold s0(T,n) prescribed via the Dominguez-Loewe-Rojas scaling formula (Eq. 23). The main claims are that masses are remarkably resistant to the medium (maximum suppression ~13% at T=T_c, n=5n_0), decay constants are far more sensitive (suppression up to ~78% in the abstract; up to ~86% in the tables), and that a baryon-density-driven particle-antiparticle asymmetry emerges, driven entirely by the vector self-energy. The paper reports temperature alone does not distinguish particles from antiparticles at zero density.

Significance. If the results are correct, they provide a concrete set of predictions for beauty-vector mesons in heavy-ion conditions at RHIC, LHC, FAIR, and NICA, with the physically sensible message that masses are poorly sensitive to the medium while decay constants and the particle-antiparticle mass gap are the informative observables. The paper is comprehensive: it treats the full multiplet, presents explicit OPE expressions and numerical tables, and clearly identifies the role of the heavy quark in protecting the mass. The main caveats are internal consistency issues in the numerical extraction—notably violation of particle-antiparticle degeneracy at zero baryon density—and the strong dependence of all quantitative results on an untested continuum-threshold ansatz.

major comments (3)

[Table VII; Sec. III.d; Eq. (17)] At n=0 and T=T_c, Table VII reports (m̃−m0)/m0 = −0.6% for B*+ and −1.1% for B*−, and −0.5% for B*0 and −1.1% for B̄*0. In a charge-symmetric medium at zero baryon density, CPT requires equal masses for particle and antiparticle; the text in Sec. III.d explicitly states that 'the medium preserves the particle-antiparticle symmetry' and 'no asymmetry is induced,' so the table directly contradicts the narrative. The same inconsistency appears in isospin: Eq. (17) gives equal u and d densities, so B*0 and B*+ should have identical medium inputs, yet Table VII gives −6.1% vs −8.3% at T=0, n=5n0—a 126 MeV 'isospin splitting' (Eq. 34) with no physical origin in a symmetric bath. The paper acknowledges only a −3 MeV vacuum baseline offset for decay constants (Sec. III.e), not for masses. Because the headline density-driven asymmetry is extracted with the same procedure, these artifacts must be
[Eq. (23); Eq. (27)] All quantitative claims—the ~13% mass bound, the 78–86% decay-constant suppression, and the 12.9% vs 6.1% asymmetry—depend on the prescribed scaling s0(T,n)/s0 = (⟨q̄q⟩/⟨q̄q⟩0)(1−m_b^2/s0)+m_b^2/s0. The sum rule in Eq. (27) integrates up to s0(T,n), so every reported shift is channeled through this ansatz. The paper provides no sensitivity analysis (e.g., a fixed s0, a different interpolation, or propagation of the 11 fit parameters in Eqs. (15) and (18)). Given that the fitted condensates enter both the OPE and, through Eq. (23), the integration limit, the quantitative predictions are conditional on a prescription that is not independently tested. The authors should either benchmark the ansatz or report an uncertainty band from varying it.
[Abstract; Table VII] The abstract states that at T=0 and n=5n0 the decay constant is 'losing up to ~78%', but Table VII lists B*− = −86.2% and B*+ = −79.6%; the summary text itself says the decay constant drops to approximately 14% of its vacuum value (i.e., a 86% loss). The abstract understates the maximum suppression in the paper's own tables. This is a factual inconsistency in a headline quantitative claim and should be corrected.

minor comments (6)

[Abstract; Sec. I] 'F AIR' and 'V FAIR' should be 'FAIR' (the Facility for Antiproton and Ion Research).
[Sec. II.F; Eq. (29)] The extraction of the physical mass m̃ from μ^2 = m̃^2 − Σ_v^2 + 2p0Σ_v is not explained. Please clarify how m̃^2 and Σ_v are obtained from the five sum-rule equations, especially the treatment of p0.
[Sec. III; Fig. 1] The paper states the vacuum Borel window M^2 ∈ [12,16] GeV^2 'remains reasonably stable' in medium, but no in-medium Borel-stability plot is shown. A representative stability check in medium would strengthen the extraction.
[Eq. (12)] The nonlocal quark background-field term ⟨χ_q^a(x) χ̄_q^b(0)⟩ is not defined. State explicitly how it reduces to the local condensates used in the OPE.
[Table V] Only decay constants are listed for the vacuum; the vacuum masses m0 that define the percentage shifts are not tabulated. Please provide them.
[General notation] The notation uses m̃ for the mass appearing in the current matrix element and μ for the shifted pole in Eq. (10). This distinction should be defined clearly and used consistently in Eqs. (29).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: in-medium parameters are solved from stated inputs, not re-labeled fits; the n=0 particle-antiparticle split is a correctness inconsistency, not a circular step.

full rationale

Walking the derivation chain: the hadronic side (Eqs. 3-10) is matched to the OPE (Eqs. 11-13 and Appendix) with explicitly stated in-medium condensate inputs (Eqs. 15-22) and the continuum-threshold ansatz (Eq. 23), and the observables are then solutions of the Borel-transformed sum rules in Eq. (29). The paper does not fit any parameter to the in-medium masses or decay constants it reports: the vacuum window (Eq. 30) is calibrated to known vacuum masses, and the in-medium threshold is taken from the published Dominguez-Loewe-Rojas prescription, not from the target observables. Although Eq. (23) ties s0(T,n) to the same quark-condensate fit that also enters the OPE, this is a transparent model input rather than a definitional or fitted-input-called-prediction circularity: the reported shifts are nontrivial outputs of the sum-rule inversion, and the paper explicitly warns that the extreme T,n results should be interpreted with caution. The cited density-dependent condensate corrections include prior work by the same authors (Ref. [46]), but this is normal self-citation of a separate calculation and is not load-bearing in the sense of a self-citation chain or imported uniqueness theorem. I separately flag a serious non-circular internal inconsistency: Table VII gives B*+ shift -0.6% vs B*- -1.1% and B*0 -0.5% vs \barB*0 -1.1% at T=Tc, n=0, while Sec. III.d states that 'the medium preserves the particle-antiparticle symmetry'; this ~30 MeV C-violating split at zero density is a correctness problem that should be resolved before interpreting the density-driven asymmetry, but it is not a circular step. The paper also admits a minor -3 MeV vacuum baseline offset for decay constants and cautions that the extreme-limit numbers are near the practical limit of the sum-rule approach; these limitations lower confidence but do not make the derivation circular.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The central results rest entirely on two imported input channels: (i) the T,n-dependence of the quark and gluon condensates, provided as 11-parameter analytic fits to the chiral SU(3) model of Kumar et al. (Eqs. 15, 18), and (ii) the prescription Eq. (23) that scales the continuum threshold with the fitted quark condensate. The density operators of Eqs. (17) and (22) come from Refs. [34, 46] and set the scale of the vector self-energy asymmetry. The paper adds no new entities; its contribution is the derivation of the OPE for this multiplet and the numerical mapping of these inputs onto masses and decay constants. Because the inputs are model fits rather than independent data, part of the 'prediction' is inherited from assumptions the reader did not see derived.

free parameters (7)

light-quark condensate fit parameters (A, B, kappa, D, phi, alpha, F, beta) = A=0.5316, B=0.1370, kappa=1.2262, D=0.1345, phi=0.3374, alpha=1.2516, F=0.0554, beta=20.0
Analytic fit to the T,n-dependence of ⟨¯qq⟩ taken from the graphical results of Kumar et al. [21] (chiral SU(3) model). This fitted ratio enters both the OPE and the continuum threshold Eq. (23), so it controls essentially all medium effects reported.
gluon condensate fit parameters (A3, A4, A5) = A3=-4.131e-3, A4=7.696e-3, A5=-8.530e-4 GeV^4
Fit to Kumar et al.'s numerical ⟨α_s G^2/π⟩(T,n) data (Eq. 18); sets the gluon-condensate medium response in the OPE.
vacuum continuum threshold s0 = 33-35 GeV^2 (window); s0(T,n) then follows Eq. (23)
Standard QCDSR auxiliary parameter, fixed by vacuum stability; the in-medium threshold is not extracted but prescribed via the quark-condensate ratio.
Borel window M^2 = 12-16 GeV^2
Chosen working interval where the vacuum mass is matched; kept fixed at all T,n, with the paper noting results near T_c, 5n0 are at the reliability limit.
strangeness suppression factor y = 0.05
Scales all strange-quark density operators in Eqs. (21)-(22), imported from Ref. [46].
m_0^2 (mixed-condensate scale) = 0.80 ± 0.10 GeV^2
Sets the density correction 3n GeV^2 in the mixed condensate Eq. (21); from Ref. [52].
nuclear density operator coefficients (⟨q†gsσGq⟩, ⟨q†iD0iD0q⟩, ⟨q†iD0q⟩) = -0.33n GeV^2, 0.031n GeV^2, 0.18n (linear in n)
Linear-in-density operator values from Ref. [46] that set the scale of the vector self-energy Sigma_v and hence the particle-antiparticle asymmetry; extrapolated linearly to n = 5n0.

assumptions (7)

domain assumption Quark-hadron duality and Borel-transformed dispersion relation connect the OPE to the hadronic spectrum (QCDSR paradigm).
Standard framework of the paper (Sec. IIF); the extracted m̃, f̃ are only as reliable as this duality and the continuum-subtraction scheme.
domain assumption In-medium condensates are given by the chiral SU(3) model outputs of Kumar et al., encoded in Eqs. (15) and (18).
All medium dependence of the OPE inputs is imported from this external model; no lattice or data cross-check at n > 0.
domain assumption The in-medium continuum threshold s0(T,n) scales with the quark condensate per Eq. (23) (Dominguez-Loewe-Rojas prescription).
This assumption directly converts condensate suppression into hadron-parameter shifts; a different prescription would change the quantitative results.
ad hoc to paper The vacuum Borel window M^2 ∈ [12,16] GeV^2 remains valid in medium up to T_c and 5n0.
Stated in Sec. III; the paper itself cautions that results near T_c, 5n0 mark the limit of reliability of the sum-rule calculation.
domain assumption Mean-field approximation: vector self-energy Sigma_v real, momentum-independent, Sigma'_v = 0.
Introduced in Eq. (6) and used to define the hadronic representation; the entire particle-antiparticle asymmetry comes from this single real number.
domain assumption Equal partitioning of the energy-momentum tensor between quark and gluon sectors, ⟨Θ_g⟩ = ⟨Θ_f⟩ = 1/2 ⟨Θ⟩.
Assumed after Eq. (19), following Refs. [28, 49]; affects the thermal contributions to the light-quark propagator.
domain assumption Baryon-density operators at linear order: ⟨q†q⟩ = 3/2 n, ⟨q†D0q⟩ = 0.18n, etc. (Eqs. 17, 22 from Refs. [34, 46]).
These linear-in-n operator values set the scale of the density response and hence of the asymmetry; extrapolated to n = 5n0.

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Cite this review

Pith. "Pith review of Hot and Dense Medium Effects on the $B_s^*$ and $B^*$ Multiplets." pith.science (2026). https://pith.science/paper/AZNXKS3U

@misc{pith2026260722419,
  author       = {Pith},
  title        = {Pith review of: Hot and Dense Medium Effects on the $B_s^*$ and $B^*$ Multiplets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZNXKS3U}},
  note         = {Machine review of arXiv:2607.22419}
}

abstract

We present an extensive analysis of the in-medium masses and decay constants of the $B_s^*(5415)$ and $B^*(5325)$ multiplets, including both particles and antiparticles, using QCD sum rules at finite temperature and density. The OPE incorporates the full temperature- and density-dependent contributions from the quark, gluon, and mixed condensates. Computing the strange ($B_s^{*0}$, $\bar{B}_s^{*0}$), charged ($B^{*\pm}$), and neutral ($B^{*0}$, $\bar{B}^{*0}$) doublet properties allows us to study the effects of flavor symmetry breaking, strangeness, and heavy-quark decoupling on the beauty vector mesons in the medium. Our results indicate that the mass is remarkably resistant to the medium across the entire multiplet: no state loses more than $\sim 13\%$ of its vacuum value, even at $T = T_c$ and $n = 5n_0$, the extreme conditions explored here. The decay constant is far more sensitive, losing up to $\sim 78\%$ at the same point. Baryon density clearly dominates the medium response, while temperature plays a secondary role until the system approaches the deconfinement crossover. At zero density, every state loses almost the same fraction of its mass and decay constant: mass shifts lie between $-(0.5$-$1.1)\%$ and decay-constant shifts between $-(3.9$-$5.3)\%$, regardless of charge or flavor, so temperature alone does not distinguish a particle from its antiparticle. At finite baryon density, a clear particle-antiparticle asymmetry emerges: at $T = 0$ and $n = 5n_0$, the $\bar{B}^{*0}$ mass decreases by $12.9\%$, whereas the $B^{*0}$ mass shifts by only $6.1\%$, a gap of nearly seven percentage points driven entirely by the vector self-energy. This provides a theoretical basis for the future heavy-ion collision program at RHIC, LHC, FAIR, and NICA.

Figures

Figures reproduced from arXiv: 2607.22419 by the authors.

**Figure 2.** FIG. 2: In-medium mass shift of the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3: The same mass shift as in Fig [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: FIG. 4: In-medium shift of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5: The same decay constant shift as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: The in-medium mass ˜m [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7: The in-medium decay constant [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8: In-medium mass (left) and decay-constant (right) shifts of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 10.** Figure 10: FIG. 10: In-medium [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]

**Figure 11.** Figure 11: FIG. 11: In-medium [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]

**Figure 12.** Figure 12: FIG. 12: In-medium [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]

**Figure 13.** Figure 13: illustrates the density and thermal evolution of these differences. At the maximum evaluated density (n = 5n0, T = 0), the isospin splittings reach m˜ B∗0 − m˜ B∗+ ≈ +126 MeV, ˜fB∗0 − ˜fB∗+ ≈ −0.15 MeV. (34) As shown in [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]

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Pith tools

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