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Bootstrapping leading hadronic muon anomaly

T0 review · 3 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A bootstrap of unitarity and QCD sum rules puts the muon anomaly's hadronic floor at 688.4 x 10^-10.

desk verdict A solid bootstrap application to the hadronic muon anomaly, with a real caveat about unquantified OPE truncation at the chosen sum rule scale. read the letter →

arxiv 2412.00187 v2 pith:BNASPQNS submitted 2024-11-29 hep-th hep-exhep-lathep-ph

Ahmadullah Zahed This is my paper · ORCID
classification hep-thhep-exhep-lathep-ph
keywords muonanomalousmagneticmomenthadronicvacuumpolarizationbootstrapconstraintsfiniteenergysumrulesQCDcondensatesrhoresonanceunitaritycrossingsymmetry
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

The paper tries to establish that the leading-order hadronic contribution to the muon's anomalous magnetic moment can be bounded from below by a bootstrap: unitarity expressed as a matrix-positivity condition on the hadronic spectral density, pion partial wave, and vector form factor, together with analyticity, crossing symmetry, and QCD finite-energy sum rules. A sympathetic reader should care because this term carries the largest theoretical uncertainty in the muon g-2 discrepancy, and a rigorous lower bound would clarify whether the measured value points to new physics or to underestimated hadronic effects. The conservative result is a lower bound of $688.4^{+3}_{-3}\times10^{-10}$ on the leading hadronic contribution and $11659176.3^{+3}_{-3}\times10^{-10}$ on $a_\mu$ itself, consistent with the data-driven Standard Model value within errors. The extremal spectral density that saturates the bound develops a rho-meson-like peak near $\sqrt{s}=0.73\,\mathrm{GeV}$ without any resonance input, and varying the sum-rule tolerance moves the bound upward, with average-tolerance values matching lattice and experiment rather than the data-driven Standard Model.

What carries the argument

The central object is the $3\times3$ matrix $B(s)$ whose entries are the pion $P$-wave $S_1^1(s)$, the vector form factor $F_1^1(s)$, and the hadronic spectral density $\rho_1^1(s)$; the bootstrap demands $B(s)\succeq0$ for $s>4$, which encodes unitarity and implies the simple condition $\rho_1^1(s)\ge|F_1^1(s)|^2$ as one of its principal-minor constraints. Around this matrix the argument builds three further ingredients: the QCD finite-energy sum rules $F_n=\frac{1}{s_0^{1+n}}\int_4^{s_0}t^n\frac{\mathrm{Im}\Pi(t)}{(2\pi)^4}dt$ for $n=0,1,2$ at $s_0=1.19\,\mathrm{GeV}^2$, imposed as inequalities with tolerance $\epsilon$ times the quoted error; partial-wave unitarity $|S_\ell^I(s)|\le1$ for isospin $I=0,1,2$ and spins up to $\ell=9$; and tree-level chiral perturbation theory as a low-energy constraint. All functions are expanded in analytic basis sets suited to their crossing or dispersion properties, with truncation levels $P$ and $N$, and the minimization of $a_\mu^{\mathrm{LO\text{-}HVP}}$ over the expansion coefficients is carried out with a semidefinite-program solver, with convergence in $N,P,\ell$ checked numerically.

What would settle it

Recompute the three finite-energy sum rule moments $F_0,F_1,F_2$ at $s_0=1.19\,\mathrm{GeV}^2$ with a higher-order or resummed operator-product expansion and check whether they remain inside the quoted error intervals; if any moment moves outside, the bootstrap's feasible set changes and the lower bound must shift off $688.4\times10^{-10}$. A precise experimental $R(s)$ scan around $\sqrt{s}=0.73\,\mathrm{GeV}$ would also show whether the extremal single-peak spectral density is the true minimizer or whether additional near-peak strength is being missed.

Watch

Extended reading notes

Core claim

The paper claims that the bootstrap constraints are strong enough to fix a genuine lower bound on $a_\mu^{\mathrm{LO\text{-}HVP}}$. Imposing the full unitarity condition $B(s)\succeq0$, partial-wave unitarity up to spin 9, analytic and crossing-symmetric pion amplitudes, and the three FESRs at $s_0=1.19\,\mathrm{GeV}^2$ with a conservative tolerance equal to their quoted errors gives Min$[a_\mu^{\mathrm{LO\text{-}HVP}}]=688.4^{+3}_{-3}\times10^{-10}$; adding the charmonium and bottomonium contributions and the precisely computed QED and electroweak terms yields $a_\mu^{\mathrm{bootstrap\text{-}min\text{-}conservative}}=11659176.3^{+3}_{-3}\times10^{-10}$, which the data-driven Standard Model value saturates within errors. The extremal spectral density realizing this minimum is not fitted to data: it has essentially one peak near $\sqrt{s}=0.73\,\mathrm{GeV}$, matching the rho resonance, which the paper presents as a bootstrap prediction of the rho mass. When the tolerance parameter $\epsilon$ is scanned from 1 down to 0, the lower bound rises through values that disagree with the data-driven Standard Model prediction but remain consistent with the measured anomaly, until $\epsilon\lesssim0.2$ is excluded by measurement. The average over tolerance, $11659204.3^{+1.6}_{-1.6}\times10^{-10}$, is saturated by the lattice evaluation and lies within the measured value's error bars, though the paper notes this average is a heuristic benchmark rather than a statistically rigorous result.

Load-bearing premise

The load-bearing premise is that the QCD finite-energy sum rules evaluated at $s_0=1.19\,\mathrm{GeV}^2$, with their published central values and error bars, genuinely constrain the hadronic spectral density; the paper itself notes in Appendix C that the operator-product-expansion truncation error at this scale is not quantified, so if that error shifts the sum rules beyond the quoted uncertainties, the derived lower bound changes.

Editorial extensions

If this is right

  • If the conservative bound is correct, no admissible spectral density satisfying unitarity, analyticity, crossing, and the FESRs can lower the leading hadronic contribution below $688.4\times10^{-10}$; the data-driven Standard Model value already sits at that floor.
  • Full unitarity strengthens the light-quark bound from $630.7\times10^{-10}$ (positivity plus FESRs) to $680.0\times10^{-10}$, so the matrix positivity constraint is doing real work beyond simple spectral positivity.
  • The rho-like peak emerges dynamically from minimization, so the same bootstrap machinery can serve as a first-principles estimator of the rho mass ($\sqrt{s}\approx0.73\,\mathrm{GeV}$) without assuming any resonance shape.
  • Tightening the FESR tolerance moves the lower bound upward; tolerances below about $\epsilon=0.2$ are ruled out by the measured muon anomaly, which translates into constraints on the gluon condensate and vacuum-saturation parameter.
  • The average-tolerance bound is saturated by the lattice value and compatible with the measured anomaly, so within the FESR uncertainties the bootstrap can accommodate both the measured value and the lattice, but not simultaneously the data-driven Standard Model value.

Reading between the lines

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

  • Editorial extension: applying the same positivity matrix to the electron or tau anomalies would produce analogous lower bounds that could be checked against independent lattice and dispersion evaluations, giving a low-cost cross-check of the method.
  • Editorial extension: the extremal one-peak spectral density implies a sharp qualitative prediction that the low-energy hadronic spectral function is dominated by the rho; a bootstrap extended with coupled channels or more data would either preserve this peak or force additional structure, which is a testable difference.
  • Editorial extension: the excluded region in the ($\langle\alpha G^2\rangle,\kappa$) plane suggests that an independent determination of either condensate would convert the experimental constraint into a two-sided bound on the other, something the paper leaves open.
  • Editorial extension: comparing the bootstrap's Euclidean window observables with lattice window data, where the present one-peak solution overshoots the intermediate window, could provide a sharper lattice-versus-bootstrap test than the total lower bound alone.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 7 minor

Summary. The paper presents a bootstrap calculation of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment. The author imposes unitarity (via a positive semi-definite matrix involving the spectral density, pion form factor, and pion partial waves), analyticity and crossing symmetry of pion amplitudes, and QCD finite-energy sum rules (FESRs) imposed as inequalities with a tolerance epsilon. Minimizing a_mu^LO-HVP over truncated basis expansions and extrapolating in the truncation order N gives Min[a_mu^LO-HVP] = 688.4(+3/-3) x 10^-10 for the conservative epsilon=1 case, and a combined lower bound a_mu^bootstrap-min-conservative = 11659176.3(+3/-3) x 10^-10, which is consistent with the SM data-driven value. The extremal spectral density exhibits a rho-like peak near sqrt(s)=0.73 GeV, which the paper labels a bootstrap prediction for the rho mass. Scanning epsilon from 1 to 0, the author reports an 'average' bound that is incompatible with the SM prediction but compatible with lattice and measurement, and uses this scan to exclude some QCD condensate parameter regions and to propose benchmark values for <alpha G^2> and kappa.

Significance. The paper is a serious attempt to bring S-matrix/bootstrap methods to bear on a precision observable, and it improves on the positivity-only bound of Li et al. [19] by imposing the full unitarity matrix. The numerical setup is described in unusual detail: convergence in N, P, and spin L is documented in Figs. 3-6, extrapolation procedures are stated, and the Step-1 comparison with Ref. [19] gives a useful calibration. If the reported lower bound could be certified as a rigorous bound, this would be a valuable non-perturbative constraint on the hadronic contribution to g-2. At present, however, the central bound is an extrapolated primal result rather than a dual certificate, and the FESR inputs at s0=1.19 GeV^2 carry an unquantified OPE truncation error that the paper itself acknowledges in Appendix C. These two issues are load-bearing for the main claims.

major comments (3)
  1. [Sec. III A, Eq. (A13), Figs. 3-5] The quantity reported as Min[a_mu^LO-HVP] is obtained by minimizing the objective over a truncated polynomial basis (N up to 95) and then extrapolating to large N with fitted models such as a + b/N^2 and a + b exp(-0.07N). This is a primal computation: each finite-N optimum is the minimum over a restricted function space, and is therefore an upper bound on the true infimum of a_mu^LO-HVP over the full admissible set, not a certificate that all physical spectral densities satisfy a_mu >= Min. The extrapolation and the +/-3 error bars reflect model spread, not a proof of exclusion. The abstract and conclusion state that the paper 'establish[es] a lower bound'; to support that claim the author should either provide a dual SDP bound (or an explicit rigorous relaxation) or rephrase the central result as an approximate lower bound whose systematic uncertainty is quantified.
  2. [Appendix C and Eq. (8)] The central result uses FESR inputs at s0 = 1.19 GeV^2 with central values and errors taken from Ref. [19]. The errors in Eq. (8) propagate uncertainties in QCD parameters such as <alpha G^2> and kappa, but do not include the error from truncating the OPE at dimension six. The paper explicitly states in Appendix C that at sufficiently low s0 the OPE may break down or its truncation error may become significant, and that a quantitative estimate is left to future work. Because s0 is only slightly above the rho mass, this is not a negligible effect: a shift in F1 or F2 by a few times 10^-7 from higher-dimensional operators or non-OPE contributions would move the minimized bound by more than the quoted +/-3. The epsilon-scan only rescales the quoted parameter errors and cannot probe shifts of the central values. The paper should either provide a quantitative estimate of the OPE truncation error, or show that the bound is stable under variations of s0 and under inclusion of dimension-eight terms, before presenting the result as a conservative lower bound.
  3. [Sec. II, Fig. 2, and Conclusion] The claim that the rho-like peak near sqrt(s)=0.73 GeV is a 'bootstrap prediction for the rho-meson mass' is overstated. The FESR moments F0, F1, F2 in Eq. (8) are themselves derived from QCD sum rules whose central values depend on quark masses and condensates, and their relative magnitudes effectively set the position of the resonance peak in the reconstructed spectral density. The minimization is therefore not independent of the rho scale; it is a consistency check at best. To make the prediction claim meaningful, the author should demonstrate that the peak position is stable when the F_i are varied within their assigned errors (or within the tolerance scan), and discuss how much of the peak location is inherited from the input sum rules rather than from the bootstrap constraints. The text should also address the fact that the peak appears at 0.73 GeV, about 6% below the physical rho mass.
minor comments (7)
  1. [Sec. III C] In Sec. III C, 'Once can be more precise' should be 'One can be more precise'.
  2. [Appendix B] In Appendix B, 'CDM-3 reported' should be 'CMD-3 reported'.
  3. [Sec. II] The notation ahad_mu appears without a definition; please define it consistently as aLO-HVP_mu or introduce it explicitly.
  4. [Fig. 1 caption] The phrase 'saturates our conservative lower bound' is ambiguous; the plot shows the SM value lying within error bars of the bound, not saturating it in the usual sense. Please rephrase.
  5. [Eq. (2) and Eq. (A13)] The relation between rho^1_1(s) and ImPi(s), namely rho^1_1(s) x (2pi)^4 / s = ImPi(s), is stated only once; spell out the normalization when the FESR integrals are defined in Eq. (A13), since the powers of s and (2pi)^4 appear confusing at first reading.
  6. [Table II] In Table II, the bootstrap row '63 335 284 682' sums to 682, but the conservative bound in the main text is 688.4 after adding charmonium and bottomonium contributions; clarify which total is being compared and why the table's total differs.
  7. [References] Reference [6] contains a second citation (F. Bhat et al.) inside the same bracket; please split or format it consistently.

Circularity Check

1 steps flagged · score 2.0 of 10

Central lower bound is not circular (it is a constrained extremum over external FESR inputs); the rho-peak 'prediction' is essentially forced by the input FESR moment ratio, but the paper is transparent about the FESR origin of the rho mass.

  1. other [Section II (Bootstrap strategy), Eq. (8); Section III.B / Fig. 2]
    "F0 = 0.0000416772+0.00000000260880 −0.00000000105807, F1 = 0.0000186454 ± 6.4034 × 10−8 ... For these data, the peak position is about √s = 0.73, corresponding to ρ mass."

    The peak quoted as a bootstrap prediction is fixed by the input FESR moment ratio: for a spectral function dominated by a single narrow peak at M, the moments in Eq. (8) satisfy F1/F0 = M^2/s0. Inserting the quoted central values gives M = sqrt(1.19 GeV^2 × 1.86454e-5 / 4.16772e-5) ≈ 0.73 GeV, exactly the quoted 'bootstrap prediction for rho mass'. Thus the rho peak is an arithmetic restatement of the FESR inputs taken from [19], not an independent bootstrap discovery. The paper itself notes that FESRs are already known to locate the rho when a resonance is assumed, so the dependence is acknowledged. This overstatement does not affect the main lower bound, which is a genuine constrained extremum over external inputs.

full rationale

The central derivation chain is: minimize a_mu^LO-HVP in Eq. (1) subject to the unitarity matrix condition (2), partial-wave unitarity, and the FESR inequalities (mean ± epsilon×error) with F0,F1,F2 from Eq. (8), whose central values and errors are taken from the external QCD sum-rule literature [19]. No parameter is fitted to the muon anomaly; the quoted ±3 uncertainties are numerical/methodological convergence uncertainties, while FESR uncertainties are handled by the inequality formulation. The comparison with the SM data-driven prediction, lattice results, and the measured value is post-hoc, not used to select the conservative bound. The bootstrap technology citations [8,9] are not self-citations and the paper demonstrates convergence in N, L, P; the author's own citations [6,15] are not load-bearing. The only notable overstatement is the rho-mass 'prediction', which is effectively fixed by the input FESR moment ratio as detailed above, although the paper is open about the known FESR extraction of the rho. The Appendix C caveat about OPE truncation error at low s0 is a real robustness concern, but it is an uncertainty in the external inputs, not a circularity. Overall, no significant circularity in the central claim; score 2 reflects only the minor rho-peak overstatement.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The result rests on external QCD sum rule inputs and the bootstrap framework. No parameter is fitted to the muon anomaly. The major choices are the tolerance epsilon and the scale s0, both clearly exposed. No new particles, forces, or entities are introduced.

free parameters (4)
  • tolerance factor epsilon = scanned from 1 to 0; conservative epsilon = 1, average epsilon = 0.5
    Controls how much of the FESR error bar is allowed; the lower bound varies from 681.6 to 742 x 10^-10 across this scan.
  • integration scale s0 = 1.19 GeV^2
    Chosen as the lowest scale at which the strange quark FESRs satisfy a Holder-based positivity inequality; lower s0 would strengthen the bound but weakens OPE control.
  • chiPT tolerance = 3 x 10^-2
    Ad hoc tolerance for matching tree-level chiral perturbation theory to the partial waves, taken from [9]; its variation does not change the bound at the fourth significant digit.
  • basis truncations = N = 95, P = 10, L = 9
    Convergence parameters for the spectral density, pion amplitude, and spin truncations; results are stable at the third significant digit beyond these values.
assumptions (4)
  • domain assumption The unitary matrix condition B(s) >= 0 for s > 4 is a valid non-perturbative constraint for QCD.
    Taken from [8,9]; this is the core bootstrap constraint relating the spectral density, pion partial wave, and form factor.
  • domain assumption The QCD FESR values for F0, F1, F2 at s0 = 1.19 GeV^2 with the quoted errors are reliable inputs.
    These are inputs from [19], based on the OPE with condensates; the paper acknowledges that OPE truncation uncertainty at low s0 is unquantified.
  • standard math The crossing-symmetric ansatz for pion amplitudes from [10] spans the relevant function space.
    Used to parameterize the partial waves; truncation P = 10 gives visible convergence.
  • domain assumption Tree-level chiral perturbation theory describes low-energy pion physics to within 30 percent.
    Used to impose low-energy constraints; this barely changes the bound and is imposed for completeness.

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Cite this review

Pith. "Pith review of Bootstrapping leading hadronic muon anomaly." pith.science (2026). https://pith.science/paper/BNASPQNS

@misc{pith2026241200187,
  author       = {Pith},
  title        = {Pith review of: Bootstrapping leading hadronic muon anomaly},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BNASPQNS}},
  note         = {Machine review of arXiv:2412.00187}
}
abstract

We bootstrap the leading order hadronic contribution to $a_\mu$ using unitarity, analytic properties, crossing symmetry and finite energy sum rules (FESR) from quantum chromodynamics (QCD), establishing a lower bound. Combining this lower bound with the remaining precisely calculated contributions from quantum electrodynamics and electroweak interactions, we achieve a lower bound on muon anomaly $a_\mu$. Since the FESRs have uncertainties, our bound depends on the choices of FESRs within these uncertainties. A conservative choice of the FESR gives a conservative lower bound, consistent with Standard Model (SM) data-driven prediction. We show that there are other valid choices of FESRs within the uncertainties that lead to lower bounds, which are inconsistent with SM data-driven prediction but consistent with the measured values of the muon anomaly. The bootstrapped spectral density shows a $\rho$-resonance peak similar to experimental hadronic cross-ratio data, providing a bootstrap prediction for $\rho$-meson mass.

Figures

Figures reproduced from arXiv: 2412.00187 by the authors.

Figure 1
Figure 1. FIG. 1: Comparison: The SM prediction (dark blue) [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Comparison of bootstrap spectral density with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Convergence of numerics with the number of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Convergence of numerics with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Convergence of the numerics with [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The green region corresponds to the known [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of bootstrap spectral density with [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 28 canonical work pages

  1. [19]

    QCD bounds on leading-order hadronic vacuum polarization contributions to the muon anomalous magnetic moment

    S. Li, T. G. Steele, J. Ho, R. Raza, K. Williams and R. T. Kleiv, “QCD bounds on leading-order hadronic vacuum polarization contributions to the muon anomalous magnetic moment,” Phys. Rev. D 110 (2024) no.1, 014046 [arXiv:2404.08591 [hep-ph]]

    work page Pith review arXiv 2024

  2. [1]

    This positive semi- definite matrix implies all the minors should be positive including determinant

    Muon anomaly and the positive semi-definite matrix The leading hadronic muon anomaly is given by aLO-HVP µ = 4α2 π Z ∞ 4m2π K(t)ImΠ(t) t dt , (A1) 8 We want to determine the minimum of this integral imposing the unitary condition among ImΠ( t), pion partial wave S1 1 and form factor given by[8, 9] B(s) ≡   1 S1 1 (s) F 1 1 (s) S1∗ 1 (s) 1 F 1∗ 1 (s) F 1...

    work page

  3. [2]

    From the right-bottom minor: ρ1 1(s) ≥ |F1 1 (s)|2. (A3)

    work page

  4. [3]

    From the top-left minor: |S1 1 (s)| ≤1. (A4)

    work page

  5. [4]

    (A5) Note that the first condition is even stronger than sole positivity ρ1 1 > 0 or ImΠ > 0

    From the determinant of B(s): ρ1 1 1 − S1 1 2 − 2 F 1 1 2 + S1 1 F 1 1 ∗ 2 + S1 1 ∗(F 1 1 )2 ≥ 0. (A5) Note that the first condition is even stronger than sole positivity ρ1 1 > 0 or ImΠ > 0. Second condition is the usual partial wave unitarity of the pion partial waves. Third constraint very non-trivially relates ImΠ , F 1 1 and S1 1 with each other. The...

    work page

  6. [5]

    and iso-spins I = 0 , 1, 2

    Pion partial waves: crossing symmetry , analyticity and unitarity Pions have partial waves SI ℓ for spins ℓ = 0, 1, 2, 3 . . . and iso-spins I = 0 , 1, 2. Note that appearance of S1 1 is very non-trivial, it puts non-trivial constrains on ρ1

    work page

  7. [6]

    The S1 1 satisfies unitarity, and is related to other partial wave coefficients due to crossing and analyticity of pion amplitudes. The pion partial waves SI ℓ (s) = 1 + iπ q s−4 s f I ℓ (s) are given by f I ℓ (s) = 1 4 Z 1 −1 dxPℓ(x)M (I) s, t= (s − 4)(x − 1) 2 , (A6) where the isospin I channel amplitudes are M (0) = 3A(s|t, u) + A(t|s, u) + A(u|t, s) ,...

    work page

  8. [7]

    Note that the set of dn in these process that will provide ImΠ(s)

    Outcome The minimization process will optimizes the parameters dn, cn, anm, bnm until it satisfy all the unitary conditions and the FESRs then return the values of dn, cn, anm, bnm that minimised aLO-HVP µ [dn, N]. Note that the set of dn in these process that will provide ImΠ(s). As a final outcome SDPB solver will return MinaLO-HVP µ and the all paramet...

    work page

Show all 56 references
  1. [8]

    Following [8] we use the ansatz ρ1 1(s) = − NX n=1 dn sin n arccos 8 s − 1 , F (s) = NX n=0 bn √ 4 − √4 − sp 4 + √4 − s !n

    Implementing the positive semi-definite condition Usual practice to search for the functions ImΠ, F 1 1 and S1 1 that minimises ahad µ and satisfy (A2) is by writing ansatz for ImΠ , F 1 1 and S1 1 . Following [8] we use the ansatz ρ1 1(s) = − NX n=1 dn sin n arccos 8 s − 1 , ...

  2. [9]

    The minimization problem We want to solve the optimization problem: min dn,bn,anm,bnm aLO-HVP µ [dn, N] , (A13) subject to the constraints:

  3. [10]

    Hence |SI ℓ [s, anm, bnm, P]| ≤1, for spins ℓ = 0, 1, 2, 3

    for details. Hence |SI ℓ [s, anm, bnm, P]| ≤1, for spins ℓ = 0, 1, 2, 3 . . .and iso-spins I = 0, 1, 2. (A9) puts some further constrains on S1 1 [s, anm, bnm, P]

  4. [11]

    Spectral density unitarity B(s[j], dn, bn, anm, bnm) ⪰ 0 for all j = 1, . . . ,200

  5. [12]

    , I= 0, 1, 2 ,

    Partial wave unitarity |SI ℓ (s[j], anm, bnm, P)| ≤1, for all ℓ = 0, 1, 2, 3 . . . , I= 0, 1, 2 ,

  6. [13]

    Note that Fk = 1 s1+k 0 Z s0 4 tk−1ρ1 1(t) = 1 s1+k 0 Z s0 4 tk ImΠ(t) (2π)4 dt

    FESR sum rules (mean − ϵ × error) < Fn < (mean +ϵ × error), n = 0, 1, 2. Note that Fk = 1 s1+k 0 Z s0 4 tk−1ρ1 1(t) = 1 s1+k 0 Z s0 4 tk ImΠ(t) (2π)4 dt. There are three different sum rules, we choose one tolerance for all of them. Taking three different choices doesn’t change...

  7. [14]

    We verify whether the minimum remains unchanged beyond a certain N and P

    Checking for convergence with N, P Since our basis expansions are truncated at N and P , we check if our results stabilize as we increase the N and P . We verify whether the minimum remains unchanged beyond a certain N and P . If the value fluctuates significantly, we increase...

  8. [15]

    This translates 10 into the following semi-positive definite matrix condition: 1 F 1∗ 1 (s, bn, N) F 1 1 (s, bn, N) ρ1 1(s, dn, N) ⪰ 0

    Three steps We show details of three steps: Step 1\ Simplest condition for form factor and spectral density: At this stage, we impose the simplest constraint on the spectral density: ρ1 1(s) ≥ |F1 1 (s)|2, (A14) which is already stronger than positivity. This translates 10 int...

  9. [16]

    Spectral density unitarity B(s[j], dn, bn, anm, bnm) ⪰ 0 for all j = 1, . . . ,200. (A17)

  10. [17]

    , I= 0, 1, 2 , (A18)

    Partial wave unitarity |SI ℓ (s[j], anm, bnm, P)| ≤1, for all ℓ = 0, 1, 2, 3 . . . , I= 0, 1, 2 , (A18)

  11. [18]

    (A19) Convergence with N, L, Pare shown in figure (4) for the weakest choice of tolerance ϵ = 1

    FESR sum rules (mean − ϵ × error) < Fn < (mean + ϵ × error), n = 0, 1, 2 . (A19) Convergence with N, L, Pare shown in figure (4) for the weakest choice of tolerance ϵ = 1. Truncating the spin at L = 9 and P = 10 does not alter the third significant digits. Hence, throughout ou...

  12. [20]

    Window observables The formula for the HVP contribution as given in main text aLO-HVP µ = 4α2 π Z ∞ 4m2π K(s)ImΠ(s) s ds , (B1) The lattice QCD computation mostly uses the time- momentum representation [26] aHVP µ = α π 2 Z ∞ 0 dt ˜K(t)G(t), (B2) 11 aHVP µ SD aHVP µ int aHVP µ...

  13. [21]

    We can compare our bootstrap spectral density with CMD pion- pion form factor data[31, 32] in the energy range √s = 0.327 to 1.2 GeV

    Two pion contribution Since, the discrepancy between the theory and experiment lies mainly in the two-pion channel. We can compare our bootstrap spectral density with CMD pion- pion form factor data[31, 32] in the energy range √s = 0.327 to 1.2 GeV. Note that for comparison wi...

  14. [22]

    Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm,

    D. P. Aguillard et al. [Muon g-2], “Measurement of the Positive Muon Anomalous Magnetic Moment to 0.20 ppm,” Phys. Rev. Lett. 131 (2023) no.16, 161802 [arXiv:2308.06230 [hep-ex]]

    2023

  15. [23]

    Detailed report on the measurement of the positive muon anomalous magnetic moment to 0.20 ppm,

    D. P. Aguillard et al.[Muon g-2], “Detailed report on the measurement of the positive muon anomalous magnetic moment to 0.20 ppm,” Phys. Rev. D 110 (2024) no.3, 032009 [arXiv:2402.15410 [hep-ex]]

    2024

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