Decays of the light hybrid meson $1^{\mathrm{-+}}$

G. Daylan Esmer; H. Sundu; K. Azizi; S. T\"urkmen

REVIEW 2 major objections 4 minor 60 references

Decays of the light hybrid meson $1^{\mathrm{-+}}$

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Using QCD three-point sum rules, this paper predicts that the light isovector hybrid meson $H_V$ with $J^{PC}=1^{-+}$ has full width $(109.7\pm 16.0)$ MeV, with the $\rho\pi$ channel contributing about 61%.

desk verdict A competent six-channel sum-rule calculation gives a complete decay table for the 2.3 GeV 1^-+ hybrid, with genuinely new non-negligible eta-pi and eta'-pi widths, but the form-factor extrapolation to the pion pole is the soft spot and its model uncertainty is unquantified. read the letter →

arxiv 2501.11331 v3 pith:LO7OOHCZ submitted 2025-01-20 hep-ph hep-exhep-lat

G. Daylan Esmer , K. Azizi , H. Sundu , S. Türkmen This is my paper · ORCID

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

keywords lighthybridmesonexoticquantumnumbers1^-+QCDsumrulesthree-pointruledecaysrho-pidominancepi1(2015)candidate

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 settle what the lightest isovector hybrid meson—a state made of a quark, an antiquark, and a gluon field, with quantum numbers $J^{PC}=1^{-+}$ that ordinary mesons cannot have—actually decays into. Using QCD three-point sum rules, its authors compute the strong couplings at six hybrid-meson-meson vertices and find a full width $\Gamma[H_V]=(109.7\pm 16.0)$ MeV. The dominant channel is $H_V\to\rho\pi$, contributing about 67 MeV (61%), while $b_1\pi$, $f_1\pi$, $f_1'\pi$, $\eta\pi$, and $\eta'\pi$ give roughly 13, 10.7, 9.6, 5.3, and 4.7 MeV. If this is right, the 2.3 GeV hybrid is a moderate-width state whose main decay is $\rho\pi$, which would discriminate between competing sum-rule, flux-tube, and lattice predictions and support identifying it with the observed $\pi_1(2015)$ candidate.

What carries the argument

The working tool is the QCD three-point sum rule for each vertex $H_VM_1M_2$. One writes a correlation function of the hybrid interpolating current $J_\mu(x)=\frac{1}{\sqrt{2}}g_s\frac{\lambda^n_{ab}}{2}G^n_{\mu\theta}(x)[u_a\gamma^\theta u_b-d_a\gamma^\theta d_b]$ with the currents of the two final mesons, evaluates it once through hadronic matrix elements and once through quark and gluon propagators, and equates the two after double Borel transformations and continuum subtraction. The output is a sum rule for each strong coupling $g_i(q^2)$. Since sum rules are reliable at spacelike $Q^2=-q^2$, the authors fit the form factors by the ansatz $F_i(Q^2)=F_i^0\exp[c_1^i Q^2/m^2+c_2^i(Q^2/m^2)^2]$ over $Q^2=2$–$20$ GeV$^2$, then evaluate it at the pion mass shell $Q^2=-m_\pi^2$. Mixing in the $f_1$–$f_1'$ and $\eta$–$\eta'$ systems is treated in the quark-flavor basis, and partial widths follow from standard two-body phase-space formulas.

What would settle it

Look at a $J^{PC}=1^{-+}$ hybrid near 2.3 GeV in experiment or on the lattice and measure or compute its $\rho\pi$, $b_1\pi$, and $\eta\pi$ rates. The central claim would fail if $b_1\pi$ is found to dominate $\rho\pi$, if the total width is far from 110 MeV, or if $\Gamma(H_V\to\eta\pi)$ comes out well below about 2 MeV.

Watch

Extended reading notes

Core claim

On its own terms, the central result is Eq. (88): the full width of the isovector hybrid $H_V$ with $J^{PC}=1^{-+}$ and content $(\bar{u}gu-\bar{d}gd)/\sqrt{2}$ is $\Gamma[H_V]=(109.7\pm 16.0)$ MeV. It is composed of partial widths $\Gamma[H_V\to\rho^\pm\pi^\mp]\simeq 67$ MeV, $\Gamma[H_V\to b_1\pi]\simeq 13$ MeV, $\Gamma[H_V\to f_1(1285)\pi]=(10.7\pm 3.6)$ MeV, $\Gamma[H_V\to f_1(1420)\pi]=(9.6\pm 2.9)$ MeV, $\Gamma[H_V\to\eta\pi]=(5.3\pm 1.6)$ MeV, and $\Gamma[H_V\to\eta'\pi]=(4.7\pm 1.4)$ MeV. The authors express the decay fractions as approximately $61:12:10:9:5:3$ for $\rho\pi:b_1\pi:f_1\pi:f_1'\pi:\eta\pi:\eta'\pi$. They emphasize that $\eta\pi$ and $\eta'\pi$, previously found negligibly small or forbidden, together make up about 9% of the width, and they characterize $H_V$ as a moderately wide state whose dominant mode is $\rho\pi$—in line with earlier sum-rule analyses, but opposite to flux-tube and lattice expectations of $b_1\pi$ dominance.

Load-bearing premise

Everything rests on the assumption that the fitted exponential form factor, anchored only at spacelike $Q^2=2$–$20$ GeV$^2$, remains correct when continued to the pion mass shell $Q^2=-m_\pi^2$; if the true curve bends differently there, every partial width changes.

Editorial extensions

If this is right

The hybrid candidate $\pi_1(2015)$ acquires a concrete decay profile: a full width near 110 MeV, with $\rho\pi$ about three-fifths of it.
The predicted ratio $\Gamma(b_1\pi)/\Gamma(f_1\pi)\approx 1.2$ is a sharp discriminant against flux-tube and lattice pictures that put $b_1\pi$ several times above $f_1\pi$.
The sizable $\eta\pi$ and $\eta'\pi$ widths, roughly 5 MeV each, mean these two-ground-state-meson channels should be experimentally visible, contrary to flux-tube expectations that forbid them.
Adopting the input mass $m=2.30$ GeV places $H_V$ close to $\pi_1(2015)$; the paper is careful that including further channels could raise its width toward the experimental value.
The branching pattern $61:12:10:9:5:3$ gives a quantitative target for partial-wave analyses of $1^{-+}$ candidates.

Reading between the lines

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

The fit-and-continue step from spacelike $Q^2=2$–$20$ GeV$^2$ to the pion mass shell is the least controlled link; if the exponential ansatz misses timelike structure, all partial widths shift together rather than changing the relative pattern drastically.
Because every partial width uses the same hybrid mass and current coupling from Ref. [26], a future revision of those inputs would rescale the full width while largely preserving the reported branching ratios.
The same three-point machinery could be applied to other exotic channels, such as $H_V\to\omega\pi$ or strange hybrids, yielding predictions that future amplitude analyses could compare with the pattern found here.
A dedicated experimental search for a $\pi_1(2015)$-like signal in $\eta\pi$ and $\eta'\pi$ would directly test the non-negligible couplings: an upper limit well below 5 MeV would refute this part of the prediction.

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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

The genuinely new numbers are the eta-pi and eta'-pi widths, about 5 MeV each, and the combined full width of (109.7 ± 16.0) MeV. Prior sum-rule analyses had those eta-pi/eta'-pi channels below 1 MeV, so this is a real update, not a repackaging. The paper also does something useful that often gets skipped: it lists all six channels in one coherent framework and compares honestly with the conflicting lattice, flux-tube, and earlier sum-rule results. The rho-pi dominance and the moderate total width are defensible as a standard QCD sum-rule estimate.

The main soft spot is the one the stress-test note hits. All six couplings are extracted by fitting an exponential ansatz to sum-rule data in Q^2 = 2-20 GeV^2 and then evaluating at Q^2 = -m_pi^2. Because that point is extremely close to Q^2 = 0, each coupling is essentially the intercept F0_i of the fit. The intercept is fixed by backward extrapolation over a wide interval with no data near zero, and the paper assigns no systematic error to the choice of functional form. A 20% shift in an intercept becomes roughly a 40% shift in that partial width. The quoted ±16 MeV on the full width only reflects Borel-parameter and input uncertainties, not this model dependence. For a field where this kind of extrapolation is standard, the method is acceptable, but the central 109.7 MeV should be labeled as more uncertain than the error bar suggests until the ansatz dependence is quantified.

There is also a small but real typo: the printed width formulas for the b1, f1, and f1' channels use lambda^2 where the dimensions and the numerical values say it should be lambda. That is a correction, not a deep flaw. The reliance on the same group's mass and current coupling from Ref. [26] is a dependency, not circularity; the calculation does not fit the widths to data. The mixing and decay-constant inputs are externally sourced, and the citation pattern looks fair.

Who gets value from this: people working on exotic hadron spectroscopy, especially the 1^-+ puzzle around pi1(2015) and pi1(1600). It is a workmanlike application of an established method, not a breakthrough, but it is serious and citable. I would send it to a referee. The referee should ask for a quantified extrapolation uncertainty and the typo fix, then it can be published.

Referee Report

2 major / 4 minor

Summary. The paper computes the full width of the isovector light hybrid meson H_V (J^PC = 1^{-+}, mass 2.30 GeV taken from Ref. [26]) by evaluating six decay channels—H_V -> rho^± pi^∓, b_1^± pi^∓, f_1(1285) pi, f_1(1420) pi, eta pi, and eta' pi—using QCD three-point sum rules. Strong couplings are extracted by fitting a three-parameter exponential form factor F_i(Q^2) = F0_i exp[c1_i Q^2/m^2 + c2_i (Q^2/m^2)^2] to sum-rule data in the Euclidean region Q^2 = 2–20 GeV^2 and evaluating at the pion mass shell. The partial widths sum to Gamma[H_V] = (109.7 ± 16.0) MeV, with H_V -> rho pi contributing about 67 MeV (61%), b_1 pi about 13 MeV, f_1 pi and f_1' pi about 10.7 and 9.6 MeV, and eta pi / eta' pi about 5.3 and 4.7 MeV. The results are compared with earlier sum-rule, lattice, and flux-tube predictions, and the paper advocates a moderate-width picture for H_V with rho pi as the dominant mode.

Significance. If the central result is robust, it provides a concrete, testable sum-rule prediction for a 2.3 GeV hybrid candidate: a moderate total width dominated by rho pi, with non-negligible b_1 pi and f_1 pi modes and small but nonzero eta pi and eta' pi channels. This is interesting because previous lattice and flux-tube studies favored b_1 pi dominance, and earlier sum-rule studies differed widely on the rho pi width. The paper also presents a rare estimate of the formerly neglected eta pi and eta' pi modes. The manuscript is explicit about the sum rules, the interpolation ansatz, and the Borel/continuum windows, and the numerical check of the final width from the listed partial widths is internally consistent. The main weakness is that the quoted uncertainty does not include the dominant systematic error from the model-dependent extrapolation to the pion pole, nor the uncertainties of the input hybrid mass and current coupling.

major comments (2)

[Sec. II, Eq. (28); Figs. 2–5] All six strong couplings are obtained by fitting the exponential ansatz F_i(Q^2) = F0_i exp[c1_i Q^2/m^2 + c2_i (Q^2/m^2)^2] to sum-rule data over Q^2 = 2–20 GeV^2 and then evaluating at Q^2 = -m_pi^2. Because m_pi^2/m^2 ≈ 0.0036, the on-shell couplings are essentially the intercepts F0_i, which are fixed only by backward extrapolation across a region where no sum-rule data exist near Q^2 = 0. The paper propagates Borel-parameter and condensate uncertainties (e.g., ±19% for g1) but assigns no systematic uncertainty to the choice of functional form or to the fit interval. Since each partial width scales as g_i^2, a 20% shift in an intercept changes that partial width by about 40%, well outside the quoted ±16 MeV error in Eq. (88). I request an explicit estimate of this model dependence, e.g., by repeating the fits with different functional forms (polynomial, Padé, or exponential without the c2 term) and with different fit intervals, and by quoting the resulting spread in the on-shell couplings as an additional systematic uncertainty.
[Secs. II–IV and Eq. (88)] The errors quoted for the couplings and widths do not include the uncertainties of the input hybrid mass m = 2.30^{+0.18}_{-0.17} GeV and current coupling f = sqrt(2)·0.38^{+0.05}_{-0.04}·10^{-1} GeV^3 from Ref. [26]. These inputs appear directly in the normalization of every sum rule (1/f) and in the phase-space factors (lambda_i, m^2, |M|^2). For example, the rho pi phase-space factor lambda_1^3 changes by roughly -24% if m moves to its lower 2.13 GeV edge, and the 1/f^2 dependence of each width implies an additional ~26% uncertainty from the ±13% error in f. The quoted total uncertainty ±16 MeV therefore underestimates the error budget. I ask the authors to propagate m and f uncertainties into all couplings and partial widths, or to state clearly that these are treated as fixed central values and provide the resulting enlarged error estimate.

minor comments (4)

[Eq. (47)] Please check the printed exponent of lambda_2 in the width formula for H_V -> b_1 pi. The numerical result (6.4 ± 1.8) MeV and dimensional analysis require the factor lambda_2 (first power), not lambda_2^2, if the latter appears in the typeset version; if the manuscript already has only lambda_2, this comment is moot.
[Sec. II, Eq. (7)] The equality m_pi^2/(2 m_q) = -<q q>/f_pi^2 is not satisfied by the numerical inputs used later: with m_q = 3.49 MeV and f_pi = 130.2 MeV the left-hand side is about 2.79 GeV while the right-hand side is about 0.82 GeV. The second equality is not used in the calculation, but as written it is numerically inconsistent and should be corrected or clarified by specifying the renormalization scheme and scale.
[Sec. V, first paragraph] The sentence "we have performed these calculations without making any additional assumptions about the Borel parameters or imposing limits on the particles' momenta" is overstated; the calculation necessarily fixes Borel windows and continuum thresholds in Eqs. (25), (26), (44), (65), (69), and (82), and the extrapolation ansatz Eq. (28) is itself an additional assumption.
[Throughout] There are many typographical and formatting issues: the date line contains "ΩDated", the affiliation line has a space error in "3 4134", Fig. 3 axis labels are garbled, and various equations use inconsistent notation (e.g., "Pi'(p,p')" in Eq. (75) versus "Pi_mu" in the text). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the partial widths are computed from three-point sum-rule couplings, not fitted to the final width. The only overlapping-author dependency is the hybrid mass/current coupling taken from Ref. [26], which is a two-point sum-rule input rather than a re-use of the target result.

full rationale

The paper's derivation chain is self-contained for the widths. Each strong coupling g_i is obtained from a three-point QCD sum rule (e.g. Eq. (19) for g1), whose OPE side is computed from quark and gluon propagators and condensates; the widths in Eqs. (30), (47), (84) and (87) are then explicit functions of these couplings and known masses/decay constants. No width is used as an input to determine a coupling, so there is no fitted-input-called-prediction loop. The exponential ansatz Eq. (28) is fitted to sum-rule data in the Euclidean interval Q^2 = 2-20 GeV^2 and evaluated at Q^2 = -m_pi^2; because this is a long extrapolation and the on-shell value essentially equals the fitted intercept F0_i, the lack of a systematic error for the functional form is a genuine robustness concern, but it is a modeling uncertainty rather than a circular reduction. The mass and current coupling of H_V come from Ref. [26], whose authors overlap with the present paper; however, that prior result is a two-point sum-rule determination that does not contain the decay widths computed here, so it is independent support for the kinematics and Borel windows rather than a self-citation chain that forces Eq. (88). Therefore the central claim has independent content, and the paper should be flagged for dependency and extrapolation risk, not for circularity beyond a minor caveat.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The paper imports the mass and current coupling of H_V from the same group's previous sum rule, plus several decay constants and mixing angles from the literature. The only new numerical fitting in this paper is the three-parameter extrapolation of each form factor from Euclidean to timelike q^2. No new particles or degrees of freedom are introduced.

free parameters (7)

Mass of H_V = 2.30+0.18-0.17 GeV
Imported from Ref. [26] by overlapping authors (Azizi and Sundu). The widths scale roughly as the cubed center-of-mass momentum, so this input strongly controls the result.
Current coupling of H_V = sqrt(2)*0.38+0.05-0.04e-1 GeV^3
Also from Ref. [26]; appears in denominators of the sum rules for all g_i, so an error propagates into every partial width.
Form-factor fit parameters F0_i, c1_i, c2_i = F0, c1, c2 per channel, see Eqs. (28)-(29), (45), (66), (83), (86)
Fitted to the OPE sum rule data in Q^2=2-20 GeV^2 and then evaluated at Q^2=-m_pi^2. This is the main numerical step connecting the sum rules to the physical couplings.
Borel mass and continuum threshold windows = M1^2 in [2.5,3.5] GeV^2, s0 in [8,10] GeV^2; M2^2 and s0' per channel
Chosen to satisfy pole dominance and stability criteria; results have residual dependence on these windows, reflected in the quoted error bars.
f1-f1' mixing angle and decay constants = phi=24.0(+3.2,-2.7) deg, f1q=193(+43,-38) MeV, f1s=(230±9) MeV
Taken from LHCb and a covariant light-front quark model; these control the f1 and f1' couplings.
eta-eta' mixing parameters = phi=39.3±1.0 deg, hq=0.0025 GeV^3, hs=0.086 GeV^3
Taken from literature fits; control the eta and eta' channels.
Light quark mass and pion decay constant = mq=(3.49±0.07) MeV, fpi=(130.2±0.8) MeV
Inputs for PCAC and normalization of pion matrix elements.

assumptions (6)

domain assumption Quark-hadron duality: excited and continuum spectral densities equal the OPE spectral density above thresholds s0 and s0'.
Invoked in Sec. II around Eqs. (10)-(12) and (18)-(20) for each channel; this is the standard QCD sum rule approximation and is not independently verified in this paper.
domain assumption The hybrid H_V is well described by the local interpolating current Eq. (2) and by the mass and current coupling from the two-point sum rule of Ref. [26].
The whole calculation assumes this current couples dominantly to a single 1^-+ ground state with m=2.30 GeV and f as used here.
ad hoc to paper The form factors g_i(Q^2) are represented by the three-parameter ansatz F_i(Q^2)=F0_i exp[c1_i Q^2/m^2 + c2_i (Q^2/m^2)^2], trusted for analytic continuation from Euclidean Q^2=2-20 GeV^2 down to the pion mass shell.
Eq. (28); the choice of this functional form is not derived and no systematic error for the extrapolation model is given.
domain assumption The two-gluon vacuum matrix element is approximated either as the free gluon propagator or as the local gluon condensate (Eqs. (16) and (17)), and higher-dimensional condensates are truncated.
This truncation defines the OPE accuracy; the paper keeps quark, gluon, mixed and four-quark condensates at leading order.
standard math PCAC relation mu_pi = m_pi^2/(2 m_q) = -<qq>/f_pi^2 (Eq. (7)) is used for pion normalization.
Standard chiral relation used without derivation.
domain assumption The quark-flavor mixing scheme for f1-f1' and eta-eta' with two mixing angles from external sources is valid.
Used in Secs. III B and IV; relies on Refs. [44,47-51].

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Cite this review

Pith. "Pith review of Decays of the light hybrid meson $1^{\mathrm{-+}}$." pith.science (2026). https://pith.science/paper/LO7OOHCZ

@misc{pith2026250111331,
  author       = {Pith},
  title        = {Pith review of: Decays of the light hybrid meson $1^\mathrm-+$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LO7OOHCZ}},
  note         = {Machine review of arXiv:2501.11331}
}

abstract

The full width of the light isovector hybrid meson $H_{\mathrm{V}}$ with spin-parities $1^{\mathrm{-+}}$ and content $(\overline{u}gu-\overline{d}gd)/ \sqrt{2}$ is evaluated by considering the decays $H_{\mathrm{V}} \to \rho^{\pm}\pi^{\mp}$, $b_1^{\pm}\pi^{\mp}$, $f_1(1285)\pi$, $f_1(1420)\pi$, $ \eta \pi$, and $\eta^{\prime} \pi$. To calculate the partial widths of these channels, we use QCD three-point sum rule method which is necessary to determine strong couplings at the corresponding hybrid-meson-meson vertices. It turns out that the main contribution to the full width $\Gamma[H_{\mathrm{ V}}]=(109.7 \pm 16.0)~\mathrm{MeV}$ of the hybrid meson comes from the processes $H_{\mathrm{V}} \to \rho^{\pm}\pi^{\mp}$ partial width of which amounts to $\approx 67~\mathrm{MeV}$. The effects of the decays $H_{\mathrm{V }} \to b_1\pi $ and $H_{\mathrm{V}} \to f_1\pi, f_1^{\prime} \pi$ are also sizeable: Their partial widths are equal to $13~\mathrm{MeV}$ and $20~ \mathrm{MeV}$, respectively. The decays to $\eta \pi$ and $\eta^{\prime} \pi$ mesons are subdominant reactions, nevertheless they form $\approx 9\%$ of the full width $\Gamma[H_{\mathrm{V}}]$. Results obtained in this work may be interesting to unravel the tangle of predictions about $H_{\mathrm{V}}$ existing in the literature, as well as useful in analyses of different resonances.

Figures

Figures reproduced from arXiv: 2501.11331 by the authors.

**Figure 2.** FIG. 2: The SR data for the form factor [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3: The sum rule’s data and the extrapolating function [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: QCD data and extrapolating functions [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: The SR data and the extrapolating functions [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

60 extracted references · 53 canonical work pages

[26]

Z. R. Huang, H. X. Chen, and S. L. Zhu, Phys. Rev. D 83, 014021 (2011)

work page 2011
[1]

R. L. Jaﬀe and K. Johnson, Phys. Lett. B 60, 201-204 (1976

work page 1976
[2]

It is worth to emphasize that the hybrid mesons can carry quantum numbers which are forbidden for the con- ventional mesons

(and references therein). It is worth to emphasize that the hybrid mesons can carry quantum numbers which are forbidden for the con- ventional mesons. For instance, they can have spin- parities J PC = 0 −−, 0+−, 1−+, 2−+ which are not al- lowed for the mesons with quark-antiquark structure. Therefore, discovery of the particles with such quantum numbers p...

work page 2015
[3]

where the pairs (M 2 1,s 0) and (M 2 2,s ′

work page
[4]

correspond to HV and ρ channels, respectively. The spectral density ρOPE(s,s ′,q 2) for the decay HV→ρπ is given by the formula ρOPE(s,s ′,q 2) = g2 smq 32 √ 2π4 ∫ 1 0 dα ∫ 1−α 0 dβ βα(1−α−β) × [ β2 +α2(α− 1) +β(α2 + 2α− 1) ] θ(N ) (21) where α and β are the Feynman parameters, and the argumentN of the function θ(N ) amounts to N = (sβ +s′α)(1−α−β) +q2βα....

work page
[5]

(45) Then, it is easy to evaluate g2 g2≡F 2(−m2 π ) = (4.0± 0.8)× 10−1 GeV−1

The ﬁt function F2(Q2) employed to calculate the coupling g2 is ﬁxed due to the parameters F 0 2 = 0.40 GeV −1,c 1 2 = 0.41,c 2 2 =−0.02. (45) Then, it is easy to evaluate g2 g2≡F 2(−m2 π ) = (4.0± 0.8)× 10−1 GeV−1. (46) 7 The width of the process HV → b− 1π+ is given by the expression Γ [ HV→b− 1π+] =g2 2 λ2 24πm2|M|2. (47) Here, |M|2 = 1 4m2 b [ m6− 2m2...

work page
[6]

The partial width of these chan- nels have been calculated using QCD three-point sum rule method

We have con- sidered the decays of this particle to ρπ, b1π, f1π, f ′ 1π, ηπ, and η′π mesons. The partial width of these chan- nels have been calculated using QCD three-point sum rule method. This approach is necessary to evaluate the strong couplings at the corresponding hybrid-meson- meson vertices. The three-point sum rules for the form factors gi(q2) ...

work page
[7]

We have used SRs for numerical computations of the QCD data which, by means of the ﬁtting functions, have been later extrapolated to regions of negative Q2 to extract strong couplings of interest. It is worth to emphasize that we have performed these calculations without mak- ing any additional assumptions about the Borel param- eters or imposing limits o...

work page 2015

Show all 60 references

[8]

C. A. Meyer and E. S. Swanson, Prog. Part. Nucl. Phys. 82, 21 (2015)

2015
[9]

Alde et al

D. Alde et al. [IHEP-Brussels-Los Alamos- Annecy(LAPP)], Phys. Lett. B 205, 397 (1988)

1988
[10]

Abele et al

A. Abele et al. [Crystal Barrel], Phys. Lett. B 423, 175 (1998)

1998
[11]

Kuhn et al

J. Kuhn et al. [E852], Phys. Lett. B 595, 109 (2004)

2004
[12]

Lu et al

M. Lu et al. [E852], Phys. Rev. Lett. 94, 032002 (2005)

2005
[13]

Aghasyan et al

M. Aghasyan et al. [COMPASS], Phys. Rev. D 98, 092003 (2018)

2018
[14]

M. G. Alexeev et al. [COMPASS], Phys. Rev. D 105, 012005 (2022)

2022
[15]

Rodas et al

A. Rodas et al. [JPAC], Phys. Rev. Lett. 122, 042002 (2019)

2019
[16]

B. Kopf, M. Albrecht, H. Koch, M. K¨ ussner, J. Pychy, X. Qin and U. Wiedner, Eur. Phys. J. C 81, 1056 (2021)

2021
[17]

Navas et al

S. Navas et al. [Particle Data Group], Phys. Rev. D 110, 030001 (2024)

2024
[18]

Ablikim et al

M. Ablikim et al. [BESIII], Phys. Rev. Lett. 129, 192002 (2022). [Erratum: Phys. Rev. Lett. 130, 159901 (2023)]

2022
[19]

I. I. Balitsky, D. Diakonov and A. V. Yung, Phys. Lett. B 112, 71 (1982)

1982
[20]

I. I. Balitsky, D. Diakonov and A. V. Yung, Z. Phys. C 33, 265 (1986)

1986
[21]

J. I. Latorre, S. Narison, P. Pascual and R. Tarrach, Phys. Lett. B 147, 169 (1984)

1984
[22]

Govaerts, L

J. Govaerts, L. J. Reinders and J. Weyers, Nucl. Phys. B 262, 575 (1985)

1985
[23]

K. G. Chetyrkin and S. Narison, Phys. Lett. B 485, 145 (2000)

2000
[24]

Narison, Phys

S. Narison, Phys. Lett. B 675, 319 (2009)

2009
[25]

H. X. Chen, Z. X. Cai, P. Z. Huang, and S. L. Zhu, Phys. Rev. D 83, 014006 (2011)

2011
[27]

Z. F. Zhang, H. Y. Jin and T. G. Steele, Chin. Phys. Lett. 31, 051201 (2014)

2014
[28]

Z. R. Huang, H. Y. Jin and Z. F. Zhang, JHEP 04, 004 (2015)

2015
[29]

Z. R. Huang, H. Y. Jin, T. G. Steele and Z. F. Zhang, Phys. Rev. D 94, 054037 (2016)

2016
[30]

Barsbay, K

B. Barsbay, K. Azizi and H. Sundu, Eur. Phys. J. C 82, 1086 (2022)

2022
[31]

Alaakol, S

A. Alaakol, S. S. Agaev, K. Azizi and H. Sundu, Phys. Lett. B 854, 138711 (2024)

2024
[32]

Barsbay, K

B. Barsbay, K. Azizi, and H. Sundu, Phys. Rev. D 109, 094034 (2024)

2024
[33]

Lacock et al

P. Lacock et al. [UKQCD], Phys. Lett. B 401, 308 (1997)

1997
[34]

McNeile et al

C. McNeile et al. [UKQCD], Phys. Rev. D 73, 074506 (2006)

2006
[35]

Burns, and F

T. Burns, and F. E. Close, Phys. Rev. D 74, 034003 (2006)

2006
[36]

C. J. Burden and M. A. Pichowsky, Few Body Syst. 32, 119 (2002)

2002
[37]

F. E. Close, and P. R. Page, Nucl. Phys. B 443, 233 (1995)

1995
[38]

Barnes, F

T. Barnes, F. E. Close and E. S. Swanson, Phys. Rev. D 52, 5242 (1995)

1995
[39]

Barnes, and F

T. Barnes, and F. E. Close, Phys. Lett. B 116, 365 (1982)

1982
[40]

M. S. Chanowitz, and S. R. Sharpe, Nucl. Phys. B 222, 211 (1983) [Erratum: Nucl. Phys. B 228, 588-588 (1983)]

1983
[41]

Bellantuono, P

L. Bellantuono, P. Colangelo and F. Giannuzzi, Eur. Phys. J. C 74, 2830 (2014)

2014
[42]

E. S. Swanson, Phys. Rev. D 107, 074028 (2024)

2024
[43]

Farina, and E

C. Farina, and E. S. Swanson, Phys. Rev. D 109, 094015 (2024)

2024
[44]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 385 (1979)

1979
[45]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Nucl. Phys. B 147, 448 (1979)

1979
[46]

R. M. Albuquerque, J. M. Dias, K. P. Khemchandani, A. Martinez Torres, F. S. Navarra, M. Nielsen and C. M. Zanetti, J. Phys. G 46, 093002 (2019)

2019
[47]

S. S. Agaev, K. Azizi and H. Sundu, Turk. J. Phys. 44, 95 (2020)

2020
[48]

B. L. Ioﬀe, and A. V. Smilga, Nucl. Phys. B 216, 373 12 (1983)

1983
[49]

Colangelo, and A

P. Colangelo, and A. Khodjamirian, hep-ph/00101765

arXiv
[50]

Feldmann, P

T. Feldmann, P. Kroll, and B. Stech Phys. Rev. D 58, 114006 (1998)

1998
[51]

S. S. Agaev, V. M. Braun, N. Oﬀen, F. A. Porkert and A. Sch¨ afer, Phys. Rev. D 90, 074019 (2014)

2014
[52]

S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D 92, 116010 (2015)

2015
[53]

Jiang, D

Z. Jiang, D. H. Yao, Z. T. Zou, X. Liu, Y. Li, and Z. J. Xiao, Phys. Rev. D 102, 116015 (2020)

2020
[54]

Aaij et al

R. Aaij et al. [LHCb], Phys. Rev. Lett. 112, 091802 (2014)

2014
[55]

R. C. Verma, J. Phys. G 39, 025005 (2012)

2012
[56]

Kaiser, and H

R. Kaiser, and H. Leutwyler, Eur. Phys. J. C 17, 623 (2000)

2000
[57]

Beneke, and M

M. Beneke, and M. Neubert, Nucl. Phys. B 651, 225 (2003)

2003
[58]

H. X. Chen, W. Chen, X. liu, Y. R. Liu, and S. L. Zhu, Rept. Prog. Phys. 86, 026201 (2023)

2023
[59]

J. J. Dudek, R. G. Edwards, M. J. Peardon, D. G. Richards, and C. E. Thomas, Phys. Rev. D 82, 034508 (2010)

2010
[60]

W. I. Eshraim, C. S. Fischer, F. Giacosa and D. Pargan- lija, Eur. Phys. J. Plus 135, no.12, 945 (2020)

2020

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [26]

Z. R. Huang, H. X. Chen, and S. L. Zhu, Phys. Rev. D 83, 014021 (2011)

work page 2011

[2] [1]

R. L. Jaﬀe and K. Johnson, Phys. Lett. B 60, 201-204 (1976

work page 1976

[3] [2]

It is worth to emphasize that the hybrid mesons can carry quantum numbers which are forbidden for the con- ventional mesons

(and references therein). It is worth to emphasize that the hybrid mesons can carry quantum numbers which are forbidden for the con- ventional mesons. For instance, they can have spin- parities J PC = 0 −−, 0+−, 1−+, 2−+ which are not al- lowed for the mesons with quark-antiquark structure. Therefore, discovery of the particles with such quantum numbers p...

work page 2015

[4] [3]

where the pairs (M 2 1,s 0) and (M 2 2,s ′

work page

[5] [4]

correspond to HV and ρ channels, respectively. The spectral density ρOPE(s,s ′,q 2) for the decay HV→ρπ is given by the formula ρOPE(s,s ′,q 2) = g2 smq 32 √ 2π4 ∫ 1 0 dα ∫ 1−α 0 dβ βα(1−α−β) × [ β2 +α2(α− 1) +β(α2 + 2α− 1) ] θ(N ) (21) where α and β are the Feynman parameters, and the argumentN of the function θ(N ) amounts to N = (sβ +s′α)(1−α−β) +q2βα....

work page

[6] [5]

(45) Then, it is easy to evaluate g2 g2≡F 2(−m2 π ) = (4.0± 0.8)× 10−1 GeV−1

The ﬁt function F2(Q2) employed to calculate the coupling g2 is ﬁxed due to the parameters F 0 2 = 0.40 GeV −1,c 1 2 = 0.41,c 2 2 =−0.02. (45) Then, it is easy to evaluate g2 g2≡F 2(−m2 π ) = (4.0± 0.8)× 10−1 GeV−1. (46) 7 The width of the process HV → b− 1π+ is given by the expression Γ [ HV→b− 1π+] =g2 2 λ2 24πm2|M|2. (47) Here, |M|2 = 1 4m2 b [ m6− 2m2...

work page

[7] [6]

The partial width of these chan- nels have been calculated using QCD three-point sum rule method

We have con- sidered the decays of this particle to ρπ, b1π, f1π, f ′ 1π, ηπ, and η′π mesons. The partial width of these chan- nels have been calculated using QCD three-point sum rule method. This approach is necessary to evaluate the strong couplings at the corresponding hybrid-meson- meson vertices. The three-point sum rules for the form factors gi(q2) ...

work page

[8] [7]

We have used SRs for numerical computations of the QCD data which, by means of the ﬁtting functions, have been later extrapolated to regions of negative Q2 to extract strong couplings of interest. It is worth to emphasize that we have performed these calculations without mak- ing any additional assumptions about the Borel param- eters or imposing limits o...

work page 2015