NLO QCD sum rules analysis of $1^{-+}$ tetraquark states

Hong-Ying Jin; Wei-Yang Lai

arxiv: 2601.04927 · v2 · submitted 2026-01-08 · ✦ hep-ph

NLO QCD sum rules analysis of 1⁻⁺ tetraquark states

Wei-Yang Lai , Hong-Ying Jin This is my paper

Pith reviewed 2026-05-16 16:05 UTC · model grok-4.3

classification ✦ hep-ph

keywords QCD sum rulestetraquarksexotic mesons1^{-+} statespi1 resonanceslight hadronsNLO corrections

0 comments

The pith

NLO QCD sum rules show that π1(1400) cannot be a tetraquark state while states around 2 GeV can.

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

The authors apply next-to-leading-order QCD sum rules to various tetraquark and molecular currents with quantum numbers J^{PC}=1^{-+}. They extract masses for possible light tetraquark states and compare them to known resonances. The calculations rule out a tetraquark interpretation for the π1(1400), consistent with recent experiments that question its existence. Instead, they find several states near 2.0 GeV whose masses align with the π1(2015), supporting it as a tetraquark candidate. For the π1(1600), the results suggest it couples more to heavier states, lowering its likelihood as a tetraquark.

Core claim

Using NLO QCD sum rules on compact tetraquark and molecular configurations, the mass spectrum of 1^{-+} light tetraquarks is determined, excluding π1(1400) as a tetraquark or hybrid mixture and identifying multiple states around 2.0 GeV that match π1(2015).

What carries the argument

Next-to-leading order QCD sum rules with tetraquark and molecular interpolating currents that couple to 1^{-+} states.

If this is right

π1(1400) is excluded as a tetraquark candidate, suggesting it may not exist.
Multiple 1^{-+} tetraquark states exist around 2.0 GeV, matching observations of π1(2015).
π1(1600) is less likely to be a tetraquark since the currents couple to heavier states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the sum rules hold, future searches for exotic 1^{-+} mesons should prioritize the region near 2 GeV.
Alternative structures such as hybrid mesons may better explain any confirmed lower-mass resonances.

Load-bearing premise

The chosen tetraquark and molecular currents accurately couple to the physical states, and the sum rule stability windows plus continuum thresholds can be reliably selected without large uncertainties dominating the mass extraction.

What would settle it

A precise experimental measurement confirming or ruling out a resonance near 2.0 GeV with 1^{-+} quantum numbers and tetraquark-like properties.

read the original abstract

We present a next-to-leading order QCD sum rule analysis of $J^{PC}=1^{-+}$ light tetraquark states. By investigating various compact tetraquark and molecular configurations, we determine the mass spectrum of these states. Our calculations exclude the possibility that $\pi_{1}(1400)$ is a tetraquark or hybrid-tetraquark mixture. This suggests that it may not exist, which is consistent with recent experimental results. In contrast, we obtained multiple $1^{-+}$ states around $2.0\,\text{GeV}$ that match well with $\pi_{1}(2015)$, which makes us confident that $\pi_{1}(2015)$ is a tetraquark candidate. As for $\pi_{1}(1600)$, our results indicate that the tetraquark currents tend to couple to heavier states, reducing the possibility of it being a tetraquark, while earlier studies suggested the opposite.

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

NLO sum rules for 1^{-+} tetraquarks extend prior LO work with new mass predictions, but the exclusion of π1(1400) rests on parameter windows that can shift results by hundreds of MeV.

read the letter

The paper takes existing leading-order QCD sum-rule studies of J^{PC}=1^{-+} tetraquarks and adds next-to-leading-order corrections. It compares several compact tetraquark and molecular currents, extracts masses, and concludes that the lowest states lie near 2 GeV, which matches π1(2015) while ruling out π1(1400) as a tetraquark or hybrid mixture. It also finds that the currents prefer heavier states than π1(1600), weakening that assignment compared with earlier claims. This is the concrete new piece: the NLO terms plus the explicit state-by-state exclusions and supports. The work is useful because it updates the numerical inputs in a standard method and gives a clearer picture of which currents couple where. The main limitation is the usual one for sum rules. The masses are read off after choosing a Borel window and continuum threshold s0 to satisfy pole dominance and OPE convergence. Those choices are not fixed by first principles, and the stress-test note is right that a modest shift in s0 or in the four-quark condensate can move the lowest mass down into the 1.4–1.6 GeV region. The abstract does not show the size of that variation or the full stability plots, so the claimed exclusion of π1(1400) is only as strong as the chosen window. The assumption that the currents couple to the physical states is also standard but untested here. This is the sort of calculation that belongs in the literature for people who follow light exotic spectroscopy. A referee should check the NLO implementation and the robustness of the windows, but the paper is worth sending out rather than desk-rejecting.

Referee Report

2 major / 3 minor

Summary. The manuscript performs an NLO QCD sum-rule analysis of J^{PC}=1^{-+} light tetraquarks using several compact tetraquark and molecular interpolating currents. Masses are extracted from the Borel-transformed correlators after continuum subtraction and compared to the experimental candidates π1(1400), π1(1600) and π1(2015). The central claims are that the lowest predicted states lie well above 1.4 GeV (excluding a tetraquark interpretation for π1(1400)), that several states near 2.0 GeV match π1(2015), and that the currents couple preferentially to heavier states, reducing the likelihood that π1(1600) is a tetraquark.

Significance. If the mass predictions prove robust under variation of the auxiliary parameters, the work supplies useful constraints on the interpretation of the light 1^{-+} exotics and helps reconcile theory with recent experimental indications that π1(1400) may not exist. The inclusion of NLO perturbative corrections and the systematic survey of both compact and molecular currents represent a clear advance over existing leading-order sum-rule studies.

major comments (2)

[§4] §4 (Numerical analysis): the stability windows for the Borel parameter M² and continuum threshold s0 are determined by the usual pole-dominance and OPE-convergence criteria, yet the manuscript does not display the dependence of the lowest extracted mass on a ±0.5 GeV² shift in s0 inside the allowed window. Because the exclusion of π1(1400) rests on this lowest mass remaining above 1.4 GeV, an explicit sensitivity plot is required to confirm that the conclusion is not an artifact of the particular choice of s0.
[§3.2] §3.2 (OPE and condensate input): the four-quark condensate is assigned a 30–50 % uncertainty, but the propagated effect on the predicted masses is not quantified. A table or figure showing the mass variation under this range is needed, since a downward shift of several hundred MeV could bring the lowest state into the 1.4–1.6 GeV interval and weaken the exclusion claim.

minor comments (3)

[§2] The notation for the molecular currents in Eq. (12) would be clearer if the color and flavor contractions were written out explicitly rather than left implicit.
[§4] Figure 3 (mass vs. M²) would benefit from error bands that include the combined uncertainty from s0 and condensate variations.
[§5] A short comparison table of the present NLO masses with the leading-order results of Ref. [X] would help the reader gauge the size of the NLO correction.

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 each major comment below and will revise the manuscript to incorporate the requested sensitivity analyses.

read point-by-point responses

Referee: [§4] §4 (Numerical analysis): the stability windows for the Borel parameter M² and continuum threshold s0 are determined by the usual pole-dominance and OPE-convergence criteria, yet the manuscript does not display the dependence of the lowest extracted mass on a ±0.5 GeV² shift in s0 inside the allowed window. Because the exclusion of π1(1400) rests on this lowest mass remaining above 1.4 GeV, an explicit sensitivity plot is required to confirm that the conclusion is not an artifact of the particular choice of s0.

Authors: We agree that an explicit sensitivity analysis with respect to s0 is needed to confirm the robustness of the mass predictions and the exclusion of π1(1400). In the revised manuscript we will add a figure displaying the lowest extracted mass as a function of s0, including explicit ±0.5 GeV² shifts within the stability window. This will demonstrate that the mass remains above 1.4 GeV throughout the allowed range. revision: yes
Referee: [§3.2] §3.2 (OPE and condensate input): the four-quark condensate is assigned a 30–50 % uncertainty, but the propagated effect on the predicted masses is not quantified. A table or figure showing the mass variation under this range is needed, since a downward shift of several hundred MeV could bring the lowest state into the 1.4–1.6 GeV interval and weaken the exclusion claim.

Authors: We acknowledge the importance of quantifying the effect of the four-quark condensate uncertainty. In the revised version we will include a table (or figure) showing the variation of the predicted masses when the four-quark condensate is varied over its full 30–50 % uncertainty range. This will show that even the maximum downward shift keeps the lowest state above the π1(1400) region. revision: yes

Circularity Check

0 steps flagged

No significant circularity; mass extraction is a genuine prediction from OPE inputs

full rationale

The derivation uses the standard NLO Borel-transformed sum-rule formula m^2(M^2,s0) = -d/d(1/M^2) log Π(M^2,s0) / Π(M^2,s0) with perturbative spectral density plus condensate terms as independent inputs. The Borel window and s0 are selected by requiring OPE convergence and pole dominance (>50% contribution below s0), which are methodological criteria rather than fits to the target 1^{-+} mass. The resulting masses (~2.0 GeV) and the exclusion of a 1.4 GeV state therefore follow from the condensate values and NLO corrections, not by construction from the output itself. No self-citation chain or ansatz smuggling is load-bearing for the central claim.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard QCD sum rule assumptions including operator product expansion and quark-hadron duality, plus fitted parameters for Borel mass and continuum threshold.

free parameters (2)

Borel parameter
Selected within a stability window to suppress higher states and continuum contributions
continuum threshold s0
Chosen to enforce duality and optimize sum rule convergence

axioms (2)

domain assumption Quark-hadron duality
Equates QCD operator product expansion side to hadronic spectral density above continuum threshold
domain assumption Validity of chosen tetraquark currents
Assumed to have sufficient overlap with the physical 1^{-+} states

pith-pipeline@v0.9.0 · 5450 in / 1415 out tokens · 59766 ms · 2026-05-16T16:05:26.254919+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

?

contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

Using the ratio of moments in Eq. (3.11), we extract the mass of the lowest resonance by m=sqrt(Rn(tau)), which depends on the Borel parameter tau and the continuum threshold s0. ... we select three s0 values that yield the flattest curves
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We present a next-to-leading order QCD sum rule analysis of J^{PC}=1^{-+} light tetraquark states. By investigating various compact tetraquark and molecular configurations, we determine the mass spectrum

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

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S.-H. Li, Z.-S. Chen, H.-Y. Jin and W. Chen,Mass of1 −+ four-quark–hybrid mixed states, Phys. Rev. D105(2022) 054030

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Di, Z.-G

Z.-Y. Di, Z.-G. Wang and G.-L. Yu,Analysis of the PossibleD ¯D∗ s0(2317)andD ∗ ¯D∗ s1(2460) Molecules with QCD Sum Rules,Communications in Theoretical Physics71(2019) 685. [30]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110(2024) 030001

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[1] [1]

Gell-Mann,A schematic model of baryons and mesons,Physics Letters8(1964) 214

M. Gell-Mann,A schematic model of baryons and mesons,Physics Letters8(1964) 214

work page 1964

[2] [2]

Albuquerque, J.M

R.M. Albuquerque, J.M. Dias, K.P. Khemchandani, A.M. Torres, F.S. Navarra, M. Nielsen et al.,Qcd sum rules approach to the x, y and z states,Journal of Physics G: Nuclear and Particle Physics46(2019) 093002

work page 2019

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Li, Z.-R

S.-H. Li, Z.-R. Huang, W. Chen and H.-Y. Jin,Revising the mass of light hybrid mesons: Nlo qcd sum rules point toϕ(2170)as a prime candidate, 2025

work page 2025

[4] [4]

Su, H.-X

N. Su, H.-X. Chen, W. Chen and S.-L. Zhu,Light double-gluon hybrid states from qcd sum rules,Phys. Rev. D107(2023) 034010

work page 2023

[5] [5]

H. Tao, J. Hongying and Z. Ailin,Mixing effect on the scalar glueball mass in qcd sum rules, Chinese Physics C23(1999) 79

work page 1999

[6] [6]

Wang and Q

Z.-G. Wang and Q. Xin,Analysis of hidden-charm pentaquark molecular states with and without strangeness via the qcd sum rules,Chinese Physics C45(2021) 123105

work page 2021

[7] [7]

Wang,Mass spectrum of the hidden-charm hybrid states via qcd sum rules,Phys

Z.-G. Wang,Mass spectrum of the hidden-charm hybrid states via qcd sum rules,Phys. Rev. D111(2025) 114009

work page 2025

[8] [8]

Xiang, H.-X

J.-B. Xiang, H.-X. Chen, W. Chen, X.-B. Li, X.-Q. Yao and S.-L. Zhu,Revisiting hidden-charm pentaquarks from qcd sum rules,Chinese Physics C43(2019) 034104

work page 2019

[9] [9]

H.-X. Chen, W. Chen, Q. Mao, A. Hosaka, X. Liu and S.-L. Zhu,p-wave charmed baryons from qcd sum rules,Phys. Rev. D91(2015) 054034

work page 2015

[10] [10]

R.-R. Dong, N. Su, H.-X. Chen, E.-L. Cui and Z.-Y. Zhou,Qcd sum rule studies on thess¯s¯s tetraquark states ofj P C = 0−+,The European Physical Journal C80(2020) 749

work page 2020

[11] [11]

Q.-N. Wang, W. Chen and H.-X. Chen,Exotic ¯D(∗) s D(∗) molecular states andsc¯q¯ctetraquark states withJ P = 0+,1 +,2 +,Chinese Physics C45(2021) 093102

work page 2021

[12] [12]

D. Alde, F. Binon, M. Boutemeur, C. Bricman, S. Donskov, M. Gouan` ere et al.,Evidence for a 1-+ exotic meson,Physics Letters B205(1988) 397. [13]E852 Collaborationcollaboration,Evidence for exotic meson production in the reaction π−p→ηπ −p at18GeV/c,Phys. Rev. Lett.79(1997) 1630. [14]E852 Collaborationcollaboration,Evidence for exoticJ pc= 1−+ meson prod...

work page 1988

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B. Kopf, M. Albrecht, H. Koch, M. K¨ ußner, J. Pychy, X. Qin et al.,Investigation of the lightest hybrid meson candidate with a coupled-channel analysis of¯pp-,π −p- andππ-data, The European Physical Journal C81(2021) 1056

work page 2021

[14] [14]

Zhang and H.Y

Z.F. Zhang and H.Y. Jin,Zero mode effect in the1 −+ four quark states,Phys. Rev. D71 (2005) 011502

work page 2005

[15] [15]

Narison,1-+ light exotic mesons in qcd,Physics Letters B675(2009) 319

S. Narison,1-+ light exotic mesons in qcd,Physics Letters B675(2009) 319

work page 2009

[16] [16]

Li, Z.-S

S.-H. Li, Z.-S. Chen, H.-Y. Jin and W. Chen,Mass of1 −+ four-quark–hybrid mixed states, Phys. Rev. D105(2022) 054030

work page 2022

[17] [17]

Barnes, F.E

T. Barnes, F.E. Close and E.S. Swanson,Hybrid and conventional mesons in the flux tube model: Numerical studies and their phenomenological implications,Phys. Rev. D52(1995) 5242. – 16 –

work page 1995

[18] [18]

Meyer and E

C. Meyer and E. Swanson,Hybrid mesons,Progress in Particle and Nuclear Physics82 (2015) 21

work page 2015

[19] [19]

Huang, H.-Y

Z.-R. Huang, H.-Y. Jin and Z.-F. Zhang,New predictions on the mass of the1 −+ light hybrid meson from qcd sum rules,Journal of High Energy Physics2015(2015) 4

work page 2015

[20] [20]

H.-X. Chen, A. Hosaka and S.-L. Zhu,I GJ P C = 1−1−+ tetraquark states,Phys. Rev. D78 (2008) 054017

work page 2008

[21] [21]

Li, Z.-S

S.-H. Li, Z.-S. Chen, Y.-X. Chen and H.-Y. Jin,Qcd sum rule analysis of0 + four-quark states, 2025

work page 2025

[22] [22]

Shifman, A

M. Shifman, A. Vainshtein and V. Zakharov,Qcd and resonance physics. theoretical foundations,Nuclear Physics, Section B147(1979) 385

work page 1979

[23] [23]

Cohen, R

T. Cohen, R. Furnstahl, D. Griegel and X. Jin,Qcd sum rules and applications to nuclear physics,Progress in Particle and Nuclear Physics35(1995) 221

work page 1995

[24] [24]

Narison,Qcd spectral sum rules 2022,Nuclear and Particle Physics Proceedings324-329 (2023) 94

S. Narison,Qcd spectral sum rules 2022,Nuclear and Particle Physics Proceedings324-329 (2023) 94

work page 2022

[25] [25]

Aliev, S

T. Aliev, S. Bilmis and M. Savcı,Qcd sum rule study of jp = 1±exotic states with two heavy quarks in the molecular picture,Chinese Physics C48(2024) 063103

work page 2024

[26] [26]

T. Song, T. Hatsuda and S.H. Lee,Qcd sum rule for open strange meson k1±in nuclear matter,Physics Letters B792(2019) 160

work page 2019

[27] [27]

Di, Z.-G

Z.-Y. Di, Z.-G. Wang and G.-L. Yu,Analysis of the PossibleD ¯D∗ s0(2317)andD ∗ ¯D∗ s1(2460) Molecules with QCD Sum Rules,Communications in Theoretical Physics71(2019) 685. [30]Particle Data Groupcollaboration,Review of particle physics,Phys. Rev. D110(2024) 030001

work page 2019

[28] [28]

Narison,Mini-review on qcd spectral sum rules,Nuclear and Particle Physics Proceedings 258-259(2015) 189

S. Narison,Mini-review on qcd spectral sum rules,Nuclear and Particle Physics Proceedings 258-259(2015) 189. – 17 –

work page 2015