{"id":"fe8539a6-10ec-404d-a420-7c285a68f1fa","arxiv_id":"2412.12266","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A flavor-universal GeV-scale vector boson's direct contribution to (g-2)_mu cancels against its on-shell contribution to the e+e- -> hadrons data when the data-driven HVP is used, redirecting the search to the lattice-vs-dispersion HVP test.","lead":"This paper studies how a new GeV-scale vector boson changes the comparison between two ways of computing the muon's anomalous magnetic moment: data-driven dispersion integrals and lattice QCD. It finds that in the data-driven version the new physics contribution largely cancels for hadronically decaying vectors, so the lattice-versus-data 'HVP test' becomes the sharper probe, and uses this to set new constraints on dark photons and baryon-number gauge bosons.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cancellation in Eq. (2.16) assumes the on-shell X resonance is fully captured in the e+e- -> hadrons data with the same kernel K(s); R-ratio undressing and finite-resolution effects are unquantified, so the claimed insensitivity of the data-driven a_mu test may be incomplete.","rationale":"The paper's main observation, Eq. (2.16), is a clean consequence of the optical theorem and the shared kernel K(s), and the renormalization-scheme analysis in Appendices A.2-A.3 supports its internal consistency. My stress-test read identifies the same load-bearing assumption as the reader: the cancellation is only as good as the degree to which the measured e+e- -> hadrons cross section used for the data-driven HVP captures the X resonance. I considered alternative concerns—the NWA validity for the parameter space, the off-shell Breit-Wigner tails, and the statistical framing of the 'arbitrarily small couplings' exclusions—but the first two are controlled (Gamma/m ~ 10^-3 for the couplings considered), while the third, while real, affects the presentation of constraints rather than the central cancellation. The experimental-capture concern is not an internal inconsistency; it is a boundary condition on the applicability of the central claim to real data. Since the paper does not quantify how the VP-undressing and acceptance corrections treat a hypothetical narrow resonance, the claimed insensitivity of the data-driven a_mu test should be regarded as provisional. A targeted analytical check of the NWA factorization and, ideally, an injection test through the ISR analysis chain would settle whether the cancellation is exact or only approximate. My read does not change the reader's CONDITIONAL verdict; it reinforces it. I agree with the reader that the 'arbitrarily small coupling' exclusions rest on the pre-existing 2.1 sigma HVP discrepancy and should be de-emphasized or clearly stated as a restatement of that tension.","tokens_in":38046,"tokens_out":36044,"duration_ms":296594,"concrete_test":"Analytically evaluate the integral in Eq. (2.1) using the full Breit-Wigner sigma_X(s) from Eq. (2.18) and the exact kernel K(s) from Eq. (2.7) for a test point (e.g., m_X = 0.9 GeV, g_q = 0.15, g_l/g_q = 0.02), including off-shell and finite-width terms, and compare the result to a_X^mu B(X -> had). If the difference exceeds a few percent, the NWA factorization underlying Eq. (2.16) is not exact. Additionally, simulate the same resonance through the BaBar ISR analysis chain (binning, efficiency, VP undressing) to check that the recovered (a_HVP^X)_data matches the NWA prediction to within the stated precision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the cancellation in Eq. (2.16), which follows from Eq. (2.12): (a_HVP^X)_e+e-->had = a_X^mu B(X -> had). This narrow-width factorization assumes the measured e+e- -> hadrons cross section used in the data-driven HVP includes the full on-shell X Breit-Wigner weighted by the same kernel K(s). Real R-ratio extractions apply a vacuum-polarization 'undressing' that divides by |1 - Pi_gamma_gamma(s)|^2, correct for detector efficiency and binning, and unfold radiative corrections. If the X resonance is partially smeared, vetoed by hadronic selection, or absorbed into the VP subtraction (e.g., if the undressing model is fit to the same data), the effective B(X->had) entering Eq. (2.16) is smaller than the true branching ratio and the cancellation is incomplete. The paper does not quantify these experimental effects. Relatedly, the 'arbitrarily small coupling' exclusions in Figs. 5 and 7 from the HVP test follow because zero NP fails the current 2.1 sigma HVP discrepancy; this is a restatement of the HVP puzzle, not a model-specific constraint, and it disappears if the discrepancy shifts. Both issues should be addressed before using the constraints as firm exclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers a generic flavor-universal GeV-scale vector boson X with independent lepton and quark couplings and studies its effect on two comparisons: the \"a_mu test\" (experimental vs theoretical muon g-2) and the \"HVP test\" (data-driven vs lattice hadronic vacuum polarization). The main formal result is Eq. (2.16): when the data-driven HVP is used in the a_mu test, the on-shell X contribution to e+e- -> hadrons cancels the direct 1-loop X contribution to a_mu up to a factor [1 - B(X -> had)], so a hadrophilic vector largely evades the data-driven a_mu constraint. The paper also emphasizes that the gamma-X interference contribution, Eq. (2.9), can be as important as the 1-loop term when g_q/g_l is large, and applies the formalism to the dark photon and to a baryon-number gauge boson. It presents exclusion plots using current TI 2020 and BMW 2020 central values, discusses future scenarios in which the HVP discrepancy disappears, and proposes an ISR bump hunt in the 3pi final state at B-factories.","tokens_in":38323,"tokens_out":17816,"duration_ms":166141,"significance":"If correct, Eq. (2.16) is an important conceptual point for the interpretation of (g-2)_mu anomalies and for dark/hadronic force searches: standard data-driven HVP evaluations already absorb the on-shell resonant NP contribution, so claims about the 1-loop X diagram as a resolution of the anomaly need revision for models with B(X -> had) close to 1. The complementary use of the HVP test and the lattice-based a_mu test, with the gamma-X interference included, is a useful framework. The detailed appendices (A.1-A.3) provide explicit proofs that longitudinal polarizations do not contribute to a_mu and that the combination a_X^mu + a_gamma-X^mu is renormalization-scheme independent; these are valuable and appear internally consistent. The proposed 3pi ISR search is a concrete, falsifiable suggestion. However, the strength of the phenomenological exclusions is limited by the issues in the major comments.","major_comments":[{"comment":"The cancellation in Eq. (2.16) relies on the narrow-width factorization (2.12): (a_HVP^X)_e+e-->had = a_X^mu B(X -> had), which in turn assumes that the e+e- -> hadrons cross sections used in the data-driven HVP contain the complete on-shell X resonance, integrated with the same kernel K(s) that defines a_X^mu. Section 2.2 briefly discusses the vacuum-polarization \"undressing\" around Eq. (2.19), but only asserts that the error is \"doubly suppressed\" because the NP cross section is a small correction; it does not quantify whether the on-shell X signal survives the undressing, energy-binning, efficiency-correction, and radiative-unfolding procedures of the BaBar, KLOE, SND, and CMD-3 measurements used in the TI 2020 average. If part of the X signal is smeared, vetoed, or absorbed into the VP-subtraction model, the effective B(X -> had) entering Eq. (2.16) is smaller than the true branching ratio, and the residual a_X^mu [1 - B_eff(X -> had)] is larger than claimed. This is load-bearing for the paper's central result that the data-driven a_mu test is insensitive to hadrophilic vectors; please either provide a quantitative argument (or an order-of-magnitude estimate) that standard R-ratio extractions preserve the integrated narrow-resonance contribution, or state the idealization as an explicit limitation and estimate the resulting correction.","section":"§2.1 (Eq. 2.12) and §2.2 (Eq. 2.19)"},{"comment":"The exclusions marked \"even for arbitrarily small coupling\" in §4.1 (Fig. 5) and §4.2 (Fig. 7) are a restatement of the current central-value discrepancies rather than a property of the NP models. For the HVP test, zero NP gives Delta a_HVP^mu = (-144 ± 68) x 10^-11, a 2.1 sigma discrepancy, so no coupling, however small, can bring the prediction into the ±2 sigma band; this is a statement that the model cannot explain the HVP puzzle at the current central values. Similarly, the green \"a_mu test\" regions exclude small couplings because the model does not resolve the 5 sigma (g-2)_mu anomaly. These are \"disfavored as an explanation of the current anomaly\" statements, not conventional upper bounds: they would disappear if the central values shifted, as the paper itself shows in the future scenarios of Figs. 6 and 8 (DD = lattice). Please relabel these regions and state explicitly that they assume the X is the sole NP responsible for the observed discrepancy; the same wording should be softened in Section 6, where \"complete exclusion of these models\" overstates the status of an anomaly-fit incompatibility.","section":"§4.1 (Fig. 5) and §4.2 (Fig. 7)"},{"comment":"The derivation of Eq. (2.16) uses Eq. (2.10) to cancel the gamma-X interference term between a_mu and the data-driven HVP contribution, leaving only a_X^mu [1 - B(X -> had)]. This is a stronger statement than the cancellation of the pure X term alone, and it depends on the renormalization choice p0^2 = m_X^2 used in Eq. (2.10). The scheme-independence proof in Appendix A.3 covers a_X^mu + a_gamma-X^mu up to two VP insertions, but the main text does not state that Eq. (2.16) inherits this scheme choice; please add a sentence clarifying that the cancellation is exact only in that scheme and that the physical observable is the scheme-independent sum, as shown in Appendix A.3.","section":"§2.1, Eq. (2.15) and Eq. (2.16)"}],"minor_comments":[{"comment":"The caption of Fig. 20 refers to \"Fig. 9a and 9b\" when describing the panels; it should refer to Fig. 20a and 20b.","section":"Appendix B.3, Fig. 20"},{"comment":"In Eq. (5.2), the logarithm ln(m_X^2/Lambda^2) is negative for m_X = 0.6 GeV and Lambda ~ 1 GeV, whereas the numerical estimate 52 MeV x (g_d^X)^2 is positive; please state that the absolute value is used, or define Lambda below m_X.","section":"§5.1, Eq. (5.2)"},{"comment":"The coefficient 52 MeV in Eq. (5.2) is an order-of-magnitude estimate that depends on the unspecified logarithm and on the electromagnetic self-energy M_Omega^gamma; the conclusion that the NP scale-setting error is negligible at current precision should be presented as an estimate, not as a precise bound.","section":"§5.1, Eq. (5.2)"},{"comment":"The phrase \"the new physics contributions effectively cancels\" should be reworded for grammar and, more substantively, should specify that the cancellation applies to the on-shell X contribution in the data-driven a_mu test in the narrow-width limit, as stated in Eq. (2.16).","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The formal core is solid and the cancellation insight is publishable, but the phenomenological claims need to be made conditional on R-ratio extraction and on current central values. The paper overlaps with Refs. [22,23,30], but the cancellation point and the systematic treatment of gamma-X interference appear to be new. No scope concern; JHEP is an appropriate venue. With a focused revision that adds a quantitative discussion of experimental R-ratio effects and reframes the anomaly-driven exclusions, the paper should be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: the cancellation in Eq. (2.16) is real. If a GeV-scale vector boson decays mostly to hadrons, its on-shell contribution to e+e− → hadrons cancels the 1-loop muon g−2 diagram when the data-driven HVP is used. That is a clean, non-obvious observation and it matters for the dark photon and baryonic force search program.\n\nWhat is genuinely new here is the systematic treatment of γ–X mixing when g_q ≫ g_l, the explicit application to the baryon-number gauge boson with the 3π bump-hunt estimate, and the fine-grained window HVP test. The shift from the earlier framework of Refs. [22,30] is incremental but real; the paper cites those works and builds on them rather than ignoring them. The appendices on renormalization-scheme independence are careful and the claims are internally consistent.\n\nThe soft spots are in the presentation of the exclusions. Figures 5 and 7 label regions as excluded “even for arbitrarily small coupling,” but those regions come from the current 2.1σ HVP discrepancy. If the lattice and data-driven central values move, as they already have with CMD-3, those bounds vanish — the authors actually show this in Figures 6 and 8. I would call these “current-discrepancy projection” and not claim them as robust model bounds. The 3π analysis is explicitly preliminary, which is fine, and the scale-setting estimate is a reasonable order-of-magnitude check.\n\nThe stress-test concern about experimental reconstruction is legitimate but not fatal. The cancellation assumes the full on-shell X peak is present in the measured hadronic cross section and is weighted by the same kernel K(s). Real experiments have binning, efficiencies, and VP undressing, and the paper does not quantify those. But this is a caveat on the exact numerical form of Eq. (2.16), not an invalidation of the qualitative conclusion. It should be flagged in the paper, not used as a reason to reject.\n\nWho is this for? Phenomenologists working on g−2, dark photons, and light vectors, plus lattice and HVP people who want to understand what the lattice-vs-data comparison can and cannot say about new physics. It deserves a serious referee. The central result is worth publishing even if the exclusion plots need reframing.\n\nI would send it to peer review with a request that the authors soften the “arbitrarily small coupling” language and add a short paragraph acknowledging the experimental assumptions behind the cancellation.","headline":"A real and useful cancellation result, but the 'arbitrarily small coupling' exclusions are just the current HVP discrepancy restated, and the data-driven cancellation assumes the resonance really is in the measured cross section.","tokens_in":38954,"tokens_out":2269,"would_cite":true,"duration_ms":25174,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A GeV-scale vector boson that decays mostly to hadrons cancels its own direct contribution in the data-driven muon $(g-2)$ test, so the lattice-vs-dispersion HVP comparison becomes the real probe.","keywords":["muon anomalous magnetic moment","hadronic vacuum polarization","data-driven dispersion relation","lattice QCD","dark photon","baryon number gauge boson","g-2 cancellation","e+e- annihilation to hadrons"],"falsifier":"Compute the data-driven $\\Delta a_{DD}^\\mu$ after injecting a narrow resonance with fixed $m_X$, $g_\\ell$, $g_q$ into the R-ratio data used for the data-driven HVP average, and compare the residual with $a_X^\\mu[1-\\mathrm{Br}(X\\to\\text{had})]$; agreement to the quoted precision confirms Eq. (2.16), while a mismatch from smeared or partially subtracted resonance contributions would show the cancellation is only approximate. A complementary experimental check is a dedicated ISR scan of $\\sigma(e^+e^-\\to 3\\pi)$ at the B-factories, where a narrow $B$-boson bump would either appear or be excluded down to $g_\\ell\\sqrt{\\mathrm{Br}(B\\to 3\\pi)} \\sim 2\\times 10^{-4}$.","tokens_in":37822,"feed_emoji":"🧲","tokens_out":16353,"duration_ms":127966,"temperature":0.7,"pith_summary":"The paper argues that for a flavour-universal vector boson $X$ with mass around $100$ MeV--$1$ GeV, the standard data-driven muon $(g-2)$ test is nearly blind to the new force. When $X$ decays mostly to hadrons, the on-shell $X$ contribution to the measured $e^+e^- \\to \\text{hadrons}$ cross section, which enters the data-driven hadronic vacuum polarization (HVP), cancels the direct one-loop $X$ contribution to $a_\\mu$; the residual is $a_X^\\mu\\,[1-\\mathrm{Br}(X\\to \\text{had})]$ as in Eq. (2.16). If correct, a hadrophilic light vector cannot explain or be probed by the $(g-2)_\\mu$ anomaly through the usual dispersive route, and the discriminating quantities become the HVP test (data-driven vs lattice) and the photon--$X$ interference term. The paper applies this to the dark photon and the baryon-number gauge boson $B$, producing complementary exclusions of previously uncharted parameter space, and proposes an ISR $B\\to 3\\pi$ bump-hunt in existing B-factory data.","feed_headline":"Hadron-decaying vector cancels its own muon g-2 signal","feed_subtitle":"In the data-driven (g-2) test, a vector that decays to hadrons leaves almost no trace; the lattice-vs-dispersion HVP test takes over.","key_machinery":"The machinery is the identity in Eq. (2.16), built on the narrow-width factorisation in Eq. (2.12): $(a_{\\mathrm{HVP}}^X)_{e^+e^-\\to \\text{had}} \\simeq a_X^\\mu\\,\\mathrm{Br}(X\\to \\text{had})$, together with the dispersive representation of the $\\gamma$--$X$ mixed vacuum polarisation that yields $(a_{\\mathrm{HVP}}^{\\gamma-X})_{e^+e^-\\to \\text{had}} \\simeq a^{\\gamma-X}_\\mu$ when the polarisation is renormalised at $k^2 = m_X^2$. The proof that the longitudinal pieces of the $X$ propagator and of the vacuum-polarisation tensors do not contribute to $a_\\mu$, and the demonstration that the sum of one- and two-loop contributions is renormalisation-scheme independent, complete the formalism needed to trust the cancellation.","core_discovery":"The central claim is Eq. (2.16): for a flavour-universal GeV-scale vector boson $X$, the data-driven $a_\\mu$ test receives a net new-physics shift $\\Delta a_{DD}^\\mu \\simeq a_X^\\mu\\,[1 - \\mathrm{Br}(X\\to \\text{had})]$, whereas the naive expectation would be $a_X^\\mu$. The cancellation follows from the narrow-width factorisation in Eq. (2.12), $(a_{\\mathrm{HVP}}^X)_{e^+e^-\\to \\text{had}} \\simeq a_X^\\mu\\, \\mathrm{Br}(X\\to \\text{had})$, which holds because the same kernel $K(s)$ weights the one-loop muon $g-2$ integral and the dispersion integral over $\\sigma(e^+e^-\\to \\text{hadrons})$. The remaining sensitivity to $X$ is carried by the photon--$X$ mixing term $a^{\\gamma-X}_\\mu$, which becomes comparable to $a_X^\\mu$ when $g_\\ell/g_q \\lesssim 10^{-3}$, and by the HVP test $\\Delta a_{\\mathrm{HVP}}^\\mu = (a_{\\mathrm{HVP}}^{\\gamma-X})_{e^+e^-\\to \\text{had}} + (a_{\\mathrm{HVP}}^X)_{e^+e^-\\to \\text{had}}$, which is positive and dominated by the on-shell $X$ term. Applied to the dark photon and the baryon-number gauge boson, this reshuffling converts the two precision tests into complementary new constraints, including regions near hadronic resonances where collider searches are blind.","pith_inferences":["The same $1-\\mathrm{Br}(X\\to\\text{had})$ suppression should appear in other observables that subtract the data-driven HVP, such as the running of $\\alpha$, $\\sin^2\\theta_W$, and muonium spectroscopy; the paper states this expectation but does not compute those observables.","Eq. (2.16) assumes the narrow-width on-shell $X$ resonance is fully captured by the measured R-ratio; real experiments' finite bins, efficiencies, and radiative-return corrections could partially remove it, which would make the residual larger than $1-\\mathrm{Br}(X\\to\\text{had})$, a complication the paper does not quantify.","If future data-driven averages migrate toward the recent high-statistics $e^+e^-\\to \\pi^+\\pi^-$ result, the same formalism predicts substantially weaker combined HVP and $a_\\mu$ exclusions, so the framework maps directly onto the ongoing consolidation of the $e^+e^-$ cross-section measurements.","A dedicated experimental analysis of the $3\\pi$ ISR spectrum with correlated systematic errors and a full background model would sharpen the preliminary bump-hunt reach and could close the remaining low-coupling window for the baryon-number gauge boson."],"forward_implications":["A hadrophilic GeV-scale vector with $\\mathrm{Br}(X\\to\\text{had})\\approx 1$ yields almost no net shift in the data-driven $a_\\mu$ test, so the residual sensitivity is set by $1-\\mathrm{Br}(X\\to\\text{had})$.","The HVP test and the $\\gamma$--$X$ interference term become the leading probes of such vectors; the paper shows the HVP test alone excludes arbitrarily small couplings when the current data-driven and lattice central values are used.","For the dark photon the lattice $a_\\mu$ test improves bounds by an order of magnitude near the $\\phi$ resonance, where visible-dark-photon collider searches have blind spots.","For the baryon-number gauge boson, the lattice $a_\\mu$ test deviates sharply from the naive one-loop expectation in the region $m_B \\gtrsim 0.6$ GeV, improving the reach by roughly a factor of 4 there.","An ISR bump-hunt in $e^+e^- \\to 3\\pi$ using existing B-factory data can probe $g_\\ell\\sqrt{\\mathrm{Br}(B\\to 3\\pi)} \\lesssim 2\\text{--}3\\times 10^{-4}$ for $m_B \\simeq 0.75\\text{--}1.1$ GeV."],"supporting_citations":[{"why":"Defines the kernel $K(s)$ and the dispersive HVP formula; the identity between this kernel and the one-loop muon $g-2$ kernel drives the cancellation in Eq. (2.16).","marker":"[5]"},{"why":"Supplies the data-driven HVP and $a_\\mu$ central values and errors used in both tests, plus the numerical evaluation of $K(m_X^2)$ in Eq. (2.7).","marker":"[10]"},{"why":"Provides the lattice HVP value used in the lattice $a_\\mu$ test and as the comparison point for the HVP test.","marker":"[21]"},{"why":"Introduced the comparison of data-driven and lattice HVP and the decomposition of $\\sigma(e^+e^-\\to \\text{had})$ into $\\sigma_\\gamma$, $\\sigma_{\\gamma-X}$ and $\\sigma_X$ that the present paper extends into the cancellation formula.","marker":"[22]"},{"why":"Gives the standard one-loop vector-boson contribution to $a_\\mu$ and the dark-photon coupling pattern used in the benchmark model.","marker":"[34]"},{"why":"Provides the $\\rho/\\omega/\\phi$ mixing amplitudes used to compute the $X$ contributions to $\\sigma(e^+e^-\\to \\text{had})$ and the collider-search exclusions applied to both models.","marker":"[42]"},{"why":"Supplies the $e^+e^- \\to \\pi^+\\pi^-\\pi^0$ ISR data used for the preliminary $B\\to 3\\pi$ bump-hunt reach estimate.","marker":"[25]"}],"fun_headline_variants":["Universal vector cancels its own g-2 trace","Dark photon hides from g-2, exposed by HVP clash","Lattice-vs-dispersion test probes hadronic-scale forces","Muon g-2: new vector leaves no trace in data-driven test","HVP test catches vector bosons that g-2 misses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The cancellation in Eq. (2.16) presupposes that the measured $e^+e^- \\to \\text{hadrons}$ cross sections used for the data-driven HVP contain the full on-shell $X$ resonance and weight it with the same kernel $K(s)$ that defines $a_X^\\mu$; if experimental binning, efficiency, or radiative corrections smear, veto, or subtract part of that resonance, the claimed blindness of the data-driven $(g-2)$ test is incomplete.","fun_headline_variants_meta":{"raw":{"variants":["Universal vector cancels its own g-2 trace","Dark photon hides from g-2, exposed by HVP clash","Lattice-vs-dispersion test probes hadronic-scale forces","Muon g-2: new vector leaves no trace in data-driven test","HVP test catches vector bosons that g-2 misses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000435,"raw_usage":{"total_tokens":2401,"prompt_tokens":1320,"completion_tokens":1081,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":936,"completion_tokens_details":{"reasoning_tokens":992}},"tokens_in":936,"tokens_out":1081,"duration_ms":10863,"temperature":1.0,"reasoning_tokens":992,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:16:59.995078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the data-driven $\\Delta a_{DD}^\\mu$ after injecting a narrow resonance with fixed $m_X$, $g_\\ell$, $g_q$ into the R-ratio data used for the data-driven HVP average, and compare the residual with $a_X^\\mu[1-\\mathrm{Br}(X\\to\\text{had})]$; agreement to the quoted precision confirms Eq. (2.16), while a mismatch from smeared or partially subtracted resonance contributions would show the cancellation is only approximate. A complementary experimental check is a dedicated ISR scan of $\\sigma(e^+e^-\\to 3\\pi)$ at the B-factories, where a narrow $B$-boson bump would either appear or be excluded down to $g_\\ell\\sqrt{\\mathrm{Br}(B\\to 3\\pi)} \\sim 2\\times 10^{-4}$.","supporting_citations":[{"cited_title":"New physics behind the new muon $g$-2 puzzle?","cited_arxiv_id":"2112.08312","evidence_quote":"Introduced the comparison of data-driven and lattice HVP and the decomposition of $\\sigma(e^+e^-\\to \\text{had})$ into $\\sigma_\\gamma$, $\\sigma_{\\gamma-X}$ and $\\sigma_X$ that the present paper extends into the cancellation formula."}],"review_version":1}