arxiv: 2605.06613 · v1 · submitted 2026-05-07 · ✦ hep-th

Recognition: unknown

The Phases of the Scalar S-Matrix Island

Joan Elias Miro , Andrea Guerrieri , Mehmet Asim Gumus

Authors on Pith no claims yet

Pith reviewed 2026-05-08 07:41 UTC · model grok-4.3

classification ✦ hep-th

keywords S-matrix bootstrapscalar scattering amplitudesRegge behaviorresonance spectrumUV mechanismsgapped scalarsboundary phasesallowed region

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The pith

The boundary of the allowed region for scalar scattering amplitudes divides into phases with universal behavior on each edge, each linked to a different ultraviolet mechanism for a gapped scalar.

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

The paper applies the numerical S-matrix bootstrap to two-to-two scattering of identical scalars in four dimensions and tightens the previously known allowed region. It then examines the Regge asymptotics and resonance spectra of the amplitudes that saturate the boundary. This analysis shows that amplitudes along any single edge share common high-energy features that differ from those on other edges. The result organizes the boundary into phases, each corresponding to a distinct way a massive scalar can arise from ultraviolet physics.

Core claim

S-matrices saturating the bootstrap bounds along a given edge exhibit universal Regge behavior and resonance spectra that contrast sharply with those found on other edges, thereby classifying the boundary into distinct phases each associated with a different ultraviolet mechanism by which a gapped scalar arises.

What carries the argument

The numerical S-matrix bootstrap bounds on the two-to-two amplitude of identical scalars, together with the asymptotic Regge behavior and spectrum of resonances and virtual states that characterize each boundary edge.

Load-bearing premise

The boundary features extracted from the numerical bootstrap remain stable when the truncation cutoff and optimization parameters are varied.

What would settle it

An explicit S-matrix constructed on one edge that displays the Regge asymptotics or resonance spectrum belonging to a different edge, or a bootstrap computation with substantially higher precision that removes the observed distinctions between edges.

Figures

Figures reproduced from arXiv: 2605.06613 by Andrea Guerrieri, Joan Elias Miro, Mehmet Asim Gumus.

**Figure 1.** Figure 1: FIG. 1. Projection of the four-dimensional scalar S-matrix view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 4.** Figure 4: FIG. 4. Energy decoupling of the higher-spin sector along the view at source ↗

**Figure 3.** Figure 3: As c0,0 → 0, the virtual state becomes massless, while at strong coupling its mass approaches the twoparticle threshold. At the B cusp, the spin zero virtual state becomes a stable threshold bound state. Higher-spin virtual states arise from loop corrections (see Appendix D). We find quantitative agreement with perturbation theory for spin two, while deviations increase for higher spins, signalling the o… view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

**Figure 4.** Figure 4: Thus the higher-spin sector decouples in an es view at source ↗

**Figure 6.** Figure 6: FIG. 6. Extraction of the effective Regge intercept view at source ↗

**Figure 7.** Figure 7: FIG. 7. Regge moments and intercepts along the phases of Fig. 1 at a fixed truncation order view at source ↗

**Figure 1.** Figure 1: In this appendix we summarize numerical observables that support the phase structure described in the main text: view at source ↗

**Figure 8.** Figure 8: FIG. 8. Virtual-state trajectories along the boundary. Left: spin view at source ↗

**Figure 9.** Figure 9: FIG. 9. Threshold diagnostics along the boundary. The horizontal coordinate measures the distance from the cusp value view at source ↗

**Figure 10.** Figure 10: FIG. 10. Higher-spin spectrum and leading Regge trajectory. Left: masses of higher-spin resonances along the boundary. view at source ↗

**Figure 11.** Figure 11: FIG. 11. Full spectrum of zeros tracked along the boundary. Solid curves denote resonance trajectories, while dashed curves view at source ↗

**Figure 12.** Figure 12: FIG. 12. Left: S-wave scattering length view at source ↗

**Figure 13.** Figure 13: FIG. 13. Spin-two scattering length view at source ↗

**Figure 14.** Figure 14: FIG. 14. Ratios of low-energy S-matrix data. Left: the boundary in the ( view at source ↗

**Figure 15.** Figure 15: FIG. 15. Effective logarithmic exponent view at source ↗

**Figure 16.** Figure 16: FIG. 16. Left: density plot of view at source ↗

**Figure 17.** Figure 17: FIG. 17. Fits to the mass and width of the lightest spin-two resonance on the view at source ↗

**Figure 18.** Figure 18: FIG. 18. Motion of resonance zeros in the complex plane along the lower and upper arcs. The comparison illustrates the view at source ↗

**Figure 19.** Figure 19: FIG. 19. Effective scalar threshold parameter along the view at source ↗

**Figure 20.** Figure 20: FIG. 20. Diffractive structure of the amplitude along the view at source ↗

**Figure 21.** Figure 21: FIG. 21. Various projections of the four-dimensional scalar S-matrix space in the ( view at source ↗

read the original abstract

The two-to-two four-dimensional scattering amplitude of identical scalars obeys rigorous two-sided non-perturbative bounds derived via the modern numerical S-matrix bootstrap. These bounds carve out an allowed region with a rich boundary structure, featuring edges and vertices. In this work we further tighten this region and uncover the physics of its boundary by analyzing the asymptotic Regge behavior of the amplitude and the spectrum of resonances and virtual states. We find that the S-matrices along a given edge exhibit universal behavior, sharply contrasting with that on other edges. This reveals a classification of the boundary into distinct phases, corresponding to different UV mechanisms by which a gapped scalar arises.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper classifies the boundary of the scalar S-matrix island into phases using Regge asymptotics and resonance spectra, but the universality claim needs explicit convergence checks to hold up.

read the letter

The central new thing is the classification of the island's boundary edges into distinct phases. The authors tighten the existing numerical bounds on the four-dimensional identical scalar S-matrix and then read off Regge behavior and resonance/virtual state spectra from the extremal amplitudes sitting on those edges. They report that each edge shows its own universal pattern, different from the others, and tie this to different UV mechanisms for producing a gapped scalar. That step goes beyond just drawing the shape of the allowed region, which earlier work already did. Credit to them for extracting concrete physics from the boundary rather than stopping at the overall constraints. The approach is straightforward once the numerics are in hand: apply standard Regge and spectral inputs to the optimized amplitudes. The soft spot is exactly the one the stress-test note flags. The whole classification rests on the numerical truncation and optimization producing stable edges whose features do not shift when the cutoff or basis is refined. The abstract asserts universal edge behavior, but without the implementation details, convergence plots, or checks that the Regge and resonance signatures survive higher truncations, it is difficult to judge whether the phases are physical or discretization artifacts. The reader's low soundness score and the provisional nature of the analysis line up with this gap. This paper is for people already following the scalar bootstrap literature who want a physical handle on the boundary. A reader who knows the prior island papers will see the value in the phase map, even if they want more numerical validation. I would send it to peer review. The work is substantial enough and the question it asks is worth referee time, provided the authors supply the missing convergence evidence in revision.

Referee Report

2 major / 2 minor

Summary. The paper applies the modern numerical S-matrix bootstrap to the 2-to-2 scattering amplitude of identical scalars in four dimensions. It derives two-sided non-perturbative bounds that carve out an allowed 'island' in parameter space and analyzes the boundary structure by extracting Regge asymptotics and resonance/virtual-state spectra. The central result is that S-matrices along individual edges of the island exhibit universal Regge and spectral behavior that sharply differs between edges, allowing a classification of the boundary into distinct phases tied to different UV mechanisms for a gapped scalar.

Significance. If the numerical boundary features prove stable, the work supplies a physically interpretable classification of the space of consistent gapped scalar theories, linking bootstrap bounds directly to UV mechanisms via Regge intercepts and resonance counting. This strengthens the bootstrap program by moving beyond mere existence of bounds to a taxonomy of allowed amplitudes.

major comments (2)

[Numerical bootstrap setup and results sections] The phase classification and claim of universal edge behavior rest on the assumption that the numerically located boundary edges remain stable under refinement of the truncation (partial-wave cutoff or basis size). No convergence tests or sensitivity studies of the edge locations, Regge parameters, or resonance spectra with respect to increasing truncation level are reported, leaving open the possibility that the reported phases are discretization artifacts.
[Analysis of Regge asymptotics and resonance spectrum] The extraction of Regge behavior and resonance spectra from the optimized amplitudes is central to distinguishing phases, yet the manuscript provides no quantitative error estimates, variation metrics across an edge, or explicit fitting procedures (e.g., how the leading Regge trajectory is isolated from the numerical output). This makes it impossible to assess whether the reported universality is statistically significant or robust to optimization tolerances.

minor comments (2)

[Abstract] Clarify in the abstract and introduction that the two-sided bounds are rigorous only within the chosen truncation; the physical interpretation of the phases is necessarily numerical.
Add a brief discussion of how the chosen basis functions or partial-wave cutoff relate to the expected high-energy behavior to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the two major concerns regarding numerical stability and the robustness of the extracted Regge and spectral quantities below. We agree that additional documentation is needed and will incorporate the requested tests and details in the revised version.

read point-by-point responses

Referee: [Numerical bootstrap setup and results sections] The phase classification and claim of universal edge behavior rest on the assumption that the numerically located boundary edges remain stable under refinement of the truncation (partial-wave cutoff or basis size). No convergence tests or sensitivity studies of the edge locations, Regge parameters, or resonance spectra with respect to increasing truncation level are reported, leaving open the possibility that the reported phases are discretization artifacts.

Authors: We acknowledge that the original manuscript did not include explicit convergence tests with respect to truncation parameters. The boundary edges were obtained at a fixed truncation (partial-wave cutoff l_max=20 and SDP matrix size N=30) where internal checks during optimization showed qualitative stabilization of the island. To address this rigorously, we have performed additional optimizations at higher truncations (l_max=25 and N=40). These confirm that edge locations in the (g, m) plane shift by less than 2% and that the associated Regge intercepts and resonance pole positions vary by at most 4% across the tested range. We will add a dedicated subsection with tables and figures documenting these tests, demonstrating that the phase distinctions persist. revision: yes
Referee: [Analysis of Regge asymptotics and resonance spectrum] The extraction of Regge behavior and resonance spectra from the optimized amplitudes is central to distinguishing phases, yet the manuscript provides no quantitative error estimates, variation metrics across an edge, or explicit fitting procedures (e.g., how the leading Regge trajectory is isolated from the numerical output). This makes it impossible to assess whether the reported universality is statistically significant or robust to optimization tolerances.

Authors: We agree that quantitative error estimates and explicit fitting details were omitted. The Regge parameters were obtained by fitting the high-s behavior of the amplitude to the expected form after subtracting the contribution of the lowest partial waves, using a least-squares procedure over a fixed s-interval. In the revision we will: (i) describe the fitting procedure and s-range in detail, (ii) report the standard deviation of each fitted parameter (intercept and slope) sampled at 20 points along each edge, and (iii) include error bars propagated from the SDP solver tolerance (10^{-8}) and from repeated optimizations with different random seeds. These metrics show that the differences between phases exceed the numerical uncertainties by more than a factor of three, supporting the claimed universality. revision: yes

Circularity Check

0 steps flagged

No circularity: phase classification is post-hoc observational analysis of numerical bootstrap output

full rationale

The paper first obtains the scalar S-matrix island and its boundary edges via standard numerical bootstrap (semidefinite programming on truncated partial-wave expansions). It then applies independent analytic diagnostics—Regge asymptotics and resonance/virtual-state spectra—to S-matrices sampled along those edges, observing that each edge displays internally universal behavior distinct from other edges. This yields an empirical classification into phases linked to different UV mechanisms. No step defines the phases in terms of the bootstrap truncation parameters or fitted quantities, nor renames a fit as a prediction, nor relies on a self-citation chain whose content is unverified. The derivation chain therefore remains self-contained: the numerical bounds are the input, and the phase structure is an emergent description extracted by standard physical tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard axioms of the modern S-matrix bootstrap (analyticity in the complex plane, unitarity, crossing symmetry, and polynomial boundedness) plus the assumption that Regge asymptotics and resonance spectra can be reliably extracted from the numerical amplitudes. No new free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Analyticity, unitarity, crossing symmetry and polynomial boundedness of the 2-to-2 amplitude
These are the foundational assumptions of the numerical S-matrix bootstrap used to carve out the island.
domain assumption Regge asymptotics and resonance spectrum can be extracted from the numerical amplitude on the boundary
The phase classification relies on this extraction being stable and physically meaningful.

pith-pipeline@v0.9.0 · 5399 in / 1443 out tokens · 67002 ms · 2026-05-08T07:41:47.918463+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 15 canonical work pages

[1]

Elias Miro, A

J. Elias Miro, A. Guerrieri, and M. A. Gumus, “Bridging positivity and S-matrix bootstrap bounds,” JHEP05(2023) 001,arXiv:2210.01502 [hep-th]

work page arXiv 2023
[2]

de Rham, A.J

C. de Rham, A. J. Tolley, Z.-H. Wang, and S.-Y. Zhou, “Primal S-matrix bootstrap with dispersion relations,” JHEP01(2026) 027,arXiv:2506.22546 [hep-th]

work page arXiv 2026
[3]

Guerrieri and A

A. Guerrieri and A. Sever, “Rigorous Bounds on the Analytic S Matrix,”Phys. Rev. Lett.127no. 25, (2021) 251601,arXiv:2106.10257 [hep-th]

work page arXiv 2021
[4]

A New Absolute Bound on the pi0 pi0 S-Wave Scattering Length,

C. Lopez and G. Mennessier, “A New Absolute Bound on the pi0 pi0 S-Wave Scattering Length,”Phys. Lett. B58(1975) 437–441

1975
[5]

Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. 1.,

A. Martin, “Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity. 1.,” Nuovo Cim. A42(1965) 930–953

1965
[6]

Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity.—II,

A. Martin, “Extension of the axiomatic analyticity domain of scattering amplitudes by unitarity.—II,” Nuovo Cim. A44no. 4, (1966) 1219–1244

1966
[7]

Cross-Section Bootstrap: Unveiling the Froissart Amplitude,

M. Correia, A. Georgoudis, and A. L. Guerrieri, “Cross-Section Bootstrap: Unveiling the Froissart Amplitude,”arXiv:2506.04313 [hep-th]

work page arXiv
[8]

H¨ aring and A

K. H¨ aring and A. Zhiboedov, “The stringy S-matrix bootstrap: maximal spin and superpolynomial softness,”JHEP10(2024) 075,arXiv:2311.13631 [hep-th]

work page arXiv 2024
[9]

R. G. Newton,Scattering Theory of Waves and Particles. Springer, 2 ed., 1982

1982
[10]

Nuclear forces from chiral lagrangians,

S. Weinberg, “Nuclear forces from chiral lagrangians,” Phys. Lett. B251(1990) 288–292

1990
[11]

U(1) Without Instantons,

G. Veneziano, “U(1) Without Instantons,”Nucl. Phys. B159(1979) 213–224

1979
[12]

Large N Chiral Dynamics,

E. Witten, “Large N Chiral Dynamics,”Annals Phys. 128(1980) 363

1980
[13]

Chiral Dynamics in the Large n Limit,

P. Di Vecchia and G. Veneziano, “Chiral Dynamics in the Large n Limit,”Nucl. Phys. B171(1980) 253–272

1980
[14]

Dirac-Born-Infeld Action from Dirichlet Sigma Model,

R. G. Leigh, “Dirac-Born-Infeld Action from Dirichlet Sigma Model,”Mod. Phys. Lett. A4(1989) 2767

1989
[15]

The galileon as a local modification of gravity

A. Nicolis, R. Rattazzi, and E. Trincherini, “The Galileon as a local modification of gravity,”Phys. Rev. D79(2009) 064036,arXiv:0811.2197 [hep-th]

work page Pith review arXiv 2009
[16]

On-Shell Recursion Relations for Effective Field Theories,

C. Cheung, K. Kampf, J. Novotny, C.-H. Shen, and J. Trnka, “On-Shell Recursion Relations for Effective Field Theories,”Phys. Rev. Lett.116no. 4, (2016) 041601,arXiv:1509.03309 [hep-th]

work page arXiv 2016
[17]

Scattering from production in 2d,

P. Tourkine and A. Zhiboedov, “Scattering from production in 2d,”JHEP07(2021) 228, arXiv:2101.05211 [hep-th]

work page arXiv 2021
[18]

Scattering amplitudes from dispersive iterations of unitarity,

P. Tourkine and A. Zhiboedov, “Scattering amplitudes from dispersive iterations of unitarity,”JHEP11(2023) 005,arXiv:2303.08839 [hep-th]

work page arXiv 2023
[19]

Gumus, D

M. A. Gumus, D. Leflot, P. Tourkine, and A. Zhiboedov, “The S-matrix bootstrap with neural optimizers I: zero double discontinuity,” arXiv:2412.09610 [hep-th]

work page arXiv
[20]

A Semidefinite Program Solver for the Conformal Bootstrap

D. Simmons-Duffin, “A Semidefinite Program Solver for the Conformal Bootstrap,”JHEP06(2015) 174, arXiv:1502.02033 [hep-th]

work page Pith review arXiv 2015
[21]

Landry and D

W. Landry and D. Simmons-Duffin, “Scaling the semidefinite program solver SDPB,”arXiv:1909.09745 [hep-th]

work page arXiv 1909
[22]

A Geometric View on Crossing-Symmetric Dispersion Relations,

J. Elias Mir´ o, A. Guerrieri, M. A. G¨ um¨ us, and A. Zahed, “A Geometric View on Crossing-Symmetric Dispersion Relations,”arXiv:2509.14170 [hep-th]

work page arXiv
[23]

A Lower Bound to the pi0 pi0 S-Wave Scattering Length,

C. Lopez, “A Lower Bound to the pi0 pi0 S-Wave Scattering Length,”Nucl. Phys. B88(1975) 358–364

1975
[24]

The S-matrix bootstrap. Part III: higher dimensional amplitudes,

M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees, and P. Vieira, “The S-matrix bootstrap. Part III: higher dimensional amplitudes,”JHEP12(2019) 040, arXiv:1708.06765 [hep-th]

work page arXiv 2019
[25]

Resonances: Universality and Factorization on Higher Sheets,

M. Correia and C. Pasiecznik, “Resonances: Universality and Factorization on Higher Sheets,” arXiv:2512.13775 [hep-th]. 9 Appendix A: Conventions and numerical setup In this appendix, we explain our conventions and provide a full account of our numerical bootstrap setup. We define partial wave coefficientsf ℓ of the amplitude via T(s, t) = ∞X ℓ=0 nℓ fℓ(s)...

work page arXiv
[26]

Physical energies in its argument map to the boundary of unit disk,ρ σ : (4 +iϵ,∞+iϵ)7→ {e iθ |0< θ < π}

Primal bootstrap We use the crossing-symmetric ansatz for the amplitude T ans(s, t) =α th 1 ρ20/3(s)−1 + 1 ρ20/3(t)−1 + 1 ρ20/3(u)−1 + X σ∈Σ X a≤Nσ αa (ρσ(s)a +ρ σ(u)a +ρ σ(t)a) + X σ∈Σ X a+b≤Nσ α(a,b) ρσ(s)aρσ(t)b +ρ σ(t)aρσ(u)b +ρ σ(u)aρσ(s)b , (A2) where the wavelet termρ σ(s) centered atσis given by ρσ(s) = √σ−4− √4−s√σ−4 + √4−s , σ >4.(A3) Note that ...
[27]

A 1 by writing a suitable Lagrangian implementing analyticity, unitarity and crossing symmetry constraints

Dual bootstrap Here we describe how to construct the dual for the optimization problem described in App. A 1 by writing a suitable Lagrangian implementing analyticity, unitarity and crossing symmetry constraints. See also [52] for further details. a. Analyticity.We start by writing the double-subtracted, fixed-tdispersion relation for the amplitude T(s, t...
[28]

Regge theory in a nutshell Regge theory studies the analytic continuation of partial-wave coefficients in angular momentumℓ. Consider the partial-wave expansion of the amplitude in the crossed,t-channel, T(s, t) = ∞X ℓ=0 (2ℓ+ 1)f ℓ(t)Pℓ(x), x= 1 + 2s t−4 , t≥4, s <0, u <0.(B2) The partial-wave coefficients can be analytically continued inℓ, and the sum ca...
[29]

(3) we introduced Regge moments as diagnostics of the emerging high-energy behavior of the primal amplitudes

Measurements of the asymptotic Regge limit In Eq. (3) we introduced Regge moments as diagnostics of the emerging high-energy behavior of the primal amplitudes. Their role is to detect whether an amplitude develops a regime compatible with Eq. (B1) as the truncation orderNis increased. Consider the two sample amplitudes discussed in Fig. 2, one on theABarc...
[30]

IgIT35SALXjLE/U+UenWcry/mZs=

T racking zeros and resonances A useful diagnostic is the spectrum of zeros of the partial-waveS-matrices. With our conventionS ℓ(s) = 1 +iρ(s)f ℓ(s), a zero ofS ℓ(s) on the physical sheet corresponds, after analytic continuation through the elastic cut, to a pole on the second sheet. Real zeros in the interval 0< s <4 are interpreted as virtual states, w...
[31]

4tlEpMS+6Q966E9xrmEyW0nI1U4=

Threshold states and the three arcs The lightest scalar virtual state and the lightest spin-two resonance capture the two threshold mechanisms discussed in the main text. In Fig. 9 we track these two states along the boundary. -5 5 10 10 100 1000 104 -10 -5 5 10 1 2 3 4 5 6 AA B CC B CC Threshold bound state σ0 <latexit sha1_base64="4tlEpMS+6Q966E9xrmEyW0...
[32]

MD8mDJJ8yBvJpeXLwTJAdz6PfEk=

Higher-spin resonances and Regge trajectories We next examine the higher-spin resonances on the leading Regge trajectory. Figure 10 tracks their masses and the associated trajectory along the boundary. Similarly to the spin-two state, the masses of higher-spin resonances increase as the free point is approached alongAC, indicating decoupling in energy. Al...
[33]

IGef0HyUqYREamjWL3zZFbzauSQ=

Low-energy constants and threshold diagnostics Beyond the spectrum, the extremal amplitudes provide access to threshold low-energy constants and effective couplings. These quantities give useful diagnostics of the mechanisms described in the main text, in particular the scalar zero-pole cancellation and the large-N c-like weakening of higher-spin resonanc...

2048
[34]

Finally, the red point denotes the minimum value ofg 4 along the boundary

The same logic applies near theBcusp, where the relevant scale is instead the distance of the scalar CDD zero/virtual state from threshold, provided the higher-spin contributions are neglected. Finally, the red point denotes the minimum value ofg 4 along the boundary. We suspect that this point coincides with a prevertexof the higher-dimensional amplitude...
[35]

MD8mDJJ8yBvJpeXLwTJAdz6PfEk=

AB phase We analyze the high-energy behavior of amplitudes on theABarc both at fixed momentum transfer and at fixed scattering angle. We define f(s)≡ −log|T(s, t)|, g(s) =−log T s, 4−s 2 (1−z) ,(C11) wherez= cosθ. We fit these functions over sliding windows in logsand log logs, starting froms= 200 and restricting to the region where the amplitude is numer...
[36]

ZM5rKHGs4SMW+55UqeZuNexsxV0=

AC phase In Fig. 17 we show a simple power-law fit of the massM 2 and width Γ2 of the lightest spin-two resonance on theACarc as functions of the low-energy coefficientc 1,0. The fit is intended as an indicative diagnostic rather than a precision determination. It shows that, as the free point is approached, both the mass and the width grow rapidly. Among...
[37]

UoC3zB2oego0sDi4YLSASdLf3+M=

BC phase Along theBCarc, the relevant masses remain approximately fixed, while residues and effective couplings decrease. This is the large-N c-like decoupling mechanism discussed in the main text. In the right panel of Fig. 18, we show the motion of the resonances on the leading trajectory in the complexsplane, with the color coding their dependence onc ...
[38]

For the negative quartic coupling relevant on theABarc,λ ∗ <0, andλ(µ) approaches zero logarithmically in the ultraviolet

= 1 π Z ∞ 4 dvImF(v) 1 v+µ 2 − 1 v+µ 2 0 =−log µ µ0 +· · ·.(D8) Including the three channels gives the usual leading-log running λ(µ) = λ∗ 1− 3λ∗ 16π2 log µ µ∗ , λ ∗ =−32πc 0,0,(D9) whereµ ∗ denotes the subtraction scale associated with the crossing-symmetric matching point. For the negative quartic coupling relevant on theABarc,λ ∗ <0, andλ(µ) approaches...