N/D Description of Two Meson Amplitudes and Chiral Symmetry

E. Oset; J. A. Oller

REVIEW

N/D Description of Two Meson Amplitudes and Chiral Symmetry

J. A. Oller, E. Oset

High Energy Physics - Phenomenology · Submitted 11 Sep 1998 · 57 references

Pith's verdict: A genuinely useful N/D framework paper whose formal core holds up better than its resonance-interpretation claims, which rest on a fit and a left-cut check that never reaches the poles.

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CitationE. Oset, J. A. Oller. "N/D Description of Two Meson Amplitudes and Chiral Symmetry." arXiv:hep-ph/9809337 (1998). https://pith.science/paper/XHGXF7QO

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Editor's note

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The thing worth your time here is the formalism, not the spectroscopy. Oller and Oset write down the general structure of an elastic partial wave when the unphysical cuts are dropped — Eq. (24), with the CDD poles made explicit and the threshold behavior factored out — and then match it to O(p^2) chiPT plus s-channel resonance exchange. Appendix A does the coupled-channel version properly. That machinery is why the paper gets cited, and it deserves to be. The P-wave section is a nice sanity check: with g_v^2 = 1 and the subtraction constant set to zero you get the rho and K* phase shifts with no parameters at all, which is a real result about how much of the vector channel is tree-level physics. Note, though, that the abstract's 'without free parameters' is that special limit; the actual simultaneous fit uses two (Eq. 59), and the paper says so plainly in the text. Don't let the abstract oversell it to you. The scalar sector is where I'd slow down. Eight parameters, chi2/dof = 1.07 over 188 points — decent, not a precision test, and c_m is barely constrained. The strongest evidence for the dynamical-generation claim is Table 2: pull the fitted nonet out and the sigma, kappa and a0 poles survive. That is a real ablation, not circular. The f0(980) is the exception and the authors say so — it needs the fitted singlet near 1 GeV, and footnote 3 admits the 'nonet' may not be one. The stress-test concern about Table 3 lands. The left-cut estimate is real-axis, two elastic channels, below 800 MeV, 3-11%; the pole claims live hundreds of MeV into the complex plane in a three-channel fit to 1.4 GeV data. The authors bound their check honestly, but they never propagate the correction to the poles, and if it moved them as much as the nonet removal does, Table 2 would lose its discriminating power. That is the open question, not a fatal one. Also worth flagging: they concede the 1.4 GeV octet widths come out too narrow because 4-pi and K-eta-prime are missing, which softens 'good agreement up to 1.4 GeV'. For a hadron physicist working on unitarization or light scalars, this is required reading and the framework is reusable. Send it out; the formalism will survive whatever the referees say about the spectroscopy.

Referee report

Significance

The paper's lasting contributions are structural and are worth naming. Eq. (24) is a clean statement of the most general elastic partial-wave form under the no-left-cut truncation, and the matching argument of §2 (Eqs. 26–39) is a useful template for any unitarization scheme. The P-wave analysis of §3 is a genuine parameter-free prediction: with g_v²=1 and ã_SL=0 the dashed curves of Figs. 2–3 depend only on f and the ρ, K* masses, and they agree with data. The ablation in Table 2 is the right kind of test — removing the fitted nonet leaves the σ, κ and a0 poles in place — and it is what makes the dynamical-generation claim falsifiable rather than circular. Eq. (76) gives a further falsifiable prediction (M² ∝ f²/g0, so these states disappear as Nc→∞), and §5 is an honest, explicit attempt to bound the approximation that underpins the whole construction. These are real strengths and they are why the paper is worth publishing in essentially its present form.

Major objections (3)

[§5 and Table 3, versus §4 and Tables 1–2] The paper is explicit that the left-cut estimate is limited to √s ≈ 800 MeV (abstract:s9 and §5), and I do not read this as a concealed assumption; the authors state the limitation. What remains unresolved is the connection between Table 3 and the headline claim. (i) Table 3 quotes T^Left/|T| and T^Left/T^∞ on the real axis, but the resonance interpretation rests on second-sheet poles hundreds of MeV into the complex plane (Table 1: σ at 445+i221, κ at 779+i330); no estimate is given of how a 5–11% real-axis perturbation maps onto those pole positions. (ii) Table 3 covers only the diagonal elastic ππ and Kπ entries; the KK̄ and ηη diagonal entries of the 3×3 I=0 matrix, which dominate the f0(980) and a0(980) region, are never estimated. (iii) The paper's own discriminant between 'dynamical' and 'preexisting' is the Table 1 versus Table 2 shift (√s_σ 445+i221 → 434+i244; √s_a0 1053+i24 → 1082+i13), i.e. shifts of order 5–10%; a left-cut correction of comparable size would not overturn the qualitative conclusion, but it should be quoted so the comparison is quantitative rather than suggestive. (iv) The Kπ entries of Table 3 change sign and grow in magnitude above 734 MeV (−5.7% at 834 MeV), so the energy trend is not obviously bounded. (v) The crossed-channel resonance contributions entering T^Left are taken from ref. [23] with the couplings of ref. [24], not the nonet couplings fitted in Eq. (73); the effect of using the fitted set should be stated. A short paragraph giving the expected size at the poles, or a recomputation of Table 1 with T^Left added, would close this.
[§3, Eq. (50), second line] As typeset, γππ2 = 6f²[(1−g_v²)/g_v²]·Mρ²/[Mρ²−(1−g_v²)4mπ²]. Taking the g_v²→1 limit of this expression gives γππ2/(s−s2) → 0, whereas Eq. (52) states that the same limit must give −6f²/Mρ². Direct partial fractioning of Eq. (49) gives instead γππ2 = 6f²[g_v²/(1−g_v²)]·Mρ²/[Mρ²−(1−g_v²)4mπ²], which does reproduce Eq. (52); the printed γππ1 in the same equation passes the analogous check. Since Eq. (51) with these γ's is what feeds the fit of Eq. (59), please verify whether this is a transcription error in the manuscript or an error in the numerical code; if the latter, the fit and Figs. 2–3 would need to be redone.
[Abstract (f0(980) sentence) and §4, 'Resonances'] The fit establishes that a CDD-like singlet pole near 1 GeV is required by the data, and the ablation in Table 2 shows that without it the f0(980) reduces to a cusp. The further step, that this pole is a 'preexisting' state, is an interpretation supported by the QCD-motivated models of refs. [53,54] rather than by anything computed here; §2 itself notes, in the paragraph after Eq. (24), that CDD poles admit more than one reading. Either soften the abstract wording to 'a fitted singlet pole near 1 GeV' or state the additional discriminant that would test the identification — for instance the Nc-scaling of Eq. (76), which should separate a preexisting state from a loop-generated one.

Minor objections (10)

[§4, 'The fit'] The χ²/dof ≈ 1 is partly by construction, since errors were set to the maximum of the experimental error and the spread between analyses, and incompatible data sets were averaged. One sentence acknowledging this when quoting 'good agreement with experiment' would keep the statistical statement honest.
[§4, Eq. (72)] The two-nonet fit is reported with parameters an order of magnitude apart and masses undetermined at the ±600 MeV level, and the authors themselves conclude the extra states are phenomenologically irrelevant. Consider moving this fit to an appendix or shortening the discussion.
[§4, δ00(12) data] The preference for the high δ00(12) solution rests on χ²/dof = 1.066 versus 1.21. Please quote the total Δχ² over the 188 points and note explicitly that the preference is weak.
[§4, after Eq. (70)] fη = 1.3fπ and fK = 1.2fπ are inserted into the O(p²) amplitudes in order to obtain a common a_SL. Quantify the sensitivity of a_SL and of the pole positions to these two choices.
[Tables 1 and 2] The ηη ratio is denoted γ in the tables while Eq. (75) defines ζ; harmonize the notation so the reader can match table entries to the definition.
[Table 3 caption] 'unpysical' should read 'unphysical', and the sign convention for the negative Kπ entries above 734 MeV should be stated in the caption.
[§3, Eqs. (56)–(57)] For mK ≠ mπ this is not an exact rewriting of Eq. (49): the residual pole at s− = (mK−mπ)² is dropped. A sentence noting that the numerical evaluations use Eq. (58) directly would prevent the reader from taking Eqs. (56)–(57) as exact.
[Abstract] 'Without free parameters' refers to the g_v²=1, ã_SL=0 curves of Figs. 2–3; the fit of Eq. (59) has two parameters. One qualifying clause would prevent misreading.
[References and acknowledgments] Several author names are misspelled: 'Estrabooks' (refs. [33],[35]), 'Cason' (ref. [30]), 'Rizuddin' (ref. [18]), 'Pennignton' (ref. [13]), 'Bonn Univeristy' (ref. [12]); 'Witout' in footnote 2 and 'contacts' for 'contracts' in the acknowledgments should also be corrected.
[Fig. 9 caption] The caption's ordering 9a, 9b, 9c should be tied explicitly to I=0, I=1/2 and I=1, since the figure itself carries no labels.

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The paper

The paper claims

By unitarizing the lowest-order chiral amplitude for meson-meson scattering, the authors argue that the sigma, a0(980) and kappa resonances are generated by the scattering dynamics itself rather than being preexisting quark states, while f0

The reading

The paper asks whether the light scalar mesons can be understood from the same three ingredients that already explain the vector mesons: chiral symmetry, unitarity, and the large-$N_c$ limit of QCD. The authors first derive, with the N/D method, the most general form an elastic partial-wave amplitude can take when the unphysical (left-hand) cuts are dropped, then match that form to the lowest-order chiral Lagrangian and to s-channel resonance exchange, and extend the construction to coupled channels. Applied to $P$-wave $pp$ and $Kp$ ($I=1/2$) scattering, the scheme reproduces the data essentially without free parameters, using only the resonance masses and the pion decay constant. In the scalar sector the same machinery fits experiment up to $1.4$ GeV and, more interestingly, produces poles for the $a_0(980)$, $sigma$ and $kappa(900)$ even after the explicit resonance fields are switched off, which is the paper's central structural claim. The $f_0(980)$ is the exception: it needs both the strong $K Kbar$ unitarity effect and a fitted singlet state near $1$ GeV.

What carries the argument

The N/D method, in which a partial-wave amplitude is written as a quotient $T_L = N_L/D_L$ with the denominator carrying the right-hand unitarity cut and the numerator the unphysical cuts. Dropping the left-hand cut makes $N_L$ a polynomial, so all zeros of the amplitude become Castillejo-Dalitz-Dyson (CDD) poles of $D_L$, and the general solution reduces to a once-subtracted dispersive integral over the phase-space factor plus a subtraction polynomial plus CDD pole terms. Matching this structure to the $O(p^2)$ chiral amplitude and to the resonance-exchange Lagrangian of Ecker, Gasser, Pich and de Rafael fixes the subtraction constants and CDD parameters. The physical amplitude is then $T_L = [1/T_L^infty + g_L(s)]^{-1}$, where $T_L^infty$ is the tree-level (large-$N_c$) amplitude and $g_L(s)$ is the loop function containing the right-hand cut; this is the unitarization step that lets the chiral amplitude generate poles on its own.

Extended reading notes

Core claim

The central claim is that the $a_0(980)$, the $sigma$ (or $f_0(400-1200)$) and the $kappa(900)$ are meson-meson states produced by unitarizing the $O(p^2)$ chiral perturbation theory amplitude, not preexisting quark states that the scattering merely reveals. The evidence is Table 2: when the tree-level scalar nonet is removed from the amplitude, leaving only the lowest-order chiral contact terms plus the loop function $g_0(s)$, the same poles reappear, at $434 + i 244$ MeV for the $sigma$, $1082 + i 13$ MeV for the $a_0$ and $770 + i 341$ MeV for the $kappa$. The $f_0(980)$ behaves differently, requiring a preexisting singlet with a fitted mass near $1020$ MeV together with the $K Kbar$ threshold cusp. The authors also show that these dynamically generated states disappear as $N_c to infinity$, since their masses scale as $M^2 prop f^2/g_0$ with $f^2$ of order $N_c$, which is exactly the behaviour expected of a loop-generated rather than a tree-level object.

Load-bearing premise

The whole construction rests on dropping the unphysical left-hand cuts, and that approximation is checked quantitatively only for the elastic $I=0$ $pp$ and $I=1/2$ $Kp$ $S$-waves and only up to about $800$ MeV, while the resonance claims come from a coupled-channel fit that runs to $1.4$ GeV.

Editorial extensions

If this is right

The light scalar nonet below $1$ GeV need not correspond to a quark-model nonet: the $sigma$, $a_0(980)$ and $kappa$ can be reproduced as scattering states, so spectroscopic counting arguments based on assuming they are $q\bar q$ states lose their force.
Scalar meson dominance, the scalar analogue of vector meson dominance, is not viable at the phenomenological level, because the scalar poles arise from loops rather than from tree-level resonance exchange.
The $P$-wave $\pi\pi$ and $K\pi$ channels are reproduced with no free parameters from $f_\pi$ and the $\rho$ and $K^*$ masses, which supports the KSRF relation as a large-$N_c$ statement rather than an accident.
The $f_0(980)$ is predicted to have a two-component structure, a preexisting singlet near $1$ GeV plus a $K\bar K$ unitarity cusp, which distinguishes it from the other light scalars in any process that probes its coupling pattern.
Because the dynamically generated poles move to infinity as $N_c \to \infty$, lattice studies at unphysical $N_c$ would see the $\sigma$, $a_0$ and $kappa$ dissolve while the $f_0(980)$ singlet component survives.

Reading between the lines

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

If the scalar poles are loop-generated, their couplings to $\pi\pi$, $K\bar K$ and $\eta\eta$ should follow the pattern of the lowest-order chiral amplitudes rather than the SU(3) ratios a quark-model nonet would impose; the residue ratios in Tables 1 and 2 are the place to test this against future high-statistics data.
The paper's own check of the neglected left-hand cut stops at $\sqrt{s} \approx 800$ MeV in the elastic S-waves, so the natural next calculation is to repeat the coupled-channel fit to $1.4$ GeV with those crossed-channel contributions kept, and see whether the Table 2 pole positions survive.
The same N/D-plus-chiral-amplitude construction, with the appropriate channel set, should apply to the $D$ and $B$ meson scalar sectors, where the analogous light scalar partners of the $D_{s0}^*(2317)$ are still debated, and where the loop-versus-tree question has the same shape.
If a future lattice calculation at variable $N_c$ finds the $\sigma$ and $a_0(980)$ masses scaling with $N_c$ as $M^2 \propto f^2/g_0$ predicts, that would convert the dynamical-generation picture from a fit interpretation into a quantitative test.

Original abstract

The most general structure of an elastic partial wave amplitude when the unphysical cuts are neglected is deduced in terms of the N/D method. This result is then matched to lowest order, ${\mathcal{O}}(p^2)$, Chiral Perturbation Theory($\chi$PT) and to the exchange (consistent with chiral symmetry) of resonances in the s-channel. The extension of the method to coupled channels is also given. Making use of the former formalism, the $\pi\pi$ and $K\pi$(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/$N_c$ expansion and unitarity. Next, the scalar sector is studied and good agreement with experiment up to $\sqrt{s}=1.4$ GeV is found. It is observed that the $a_0(980)$, $\sigma$ and $\kappa(900)$ resonances are meson-meson states originating from the unitarization of the ${\mathcal{O}}(p^2)$ $\chi$PT amplitudes. On the other hand, the $f_0(980)$ is a combination of a strong S-wave meson-meson unitarity effect and of a preexisting singlet resonance with a mass around 1 GeV. We have also studied the size of the contributions of the unphysical cuts to the $\pi\pi$(I=0) and $K\pi$(I=1/2) elastic S-wave amplitudes from $\chi$PT and the exchange of resonances in crossed channels up to $\sqrt{s}\approx 800$ MeV. The loops are calculated as in $\chi$PT at next to leading order. We find a small correction from the unphysical cuts to our calculated partial waves.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The most general structure of an elastic partial wave amplitude when the unphysical cuts are neglected is deduced in terms of the N/D method. This result is then matched to lowest order, ${\mathcal{O}}(p^2)$, Chiral Perturbation Theory($\chi$PT) and to the exchange (consistent with chiral symmetry) of resonances in the s-channel. The extension of the method to coupled channels is also given. Making use of the former formalism, the $\pi\pi$ and $K\pi$(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/$N_c$ expansion and unitarity. Next, the scalar sector is studied and good agreement with experiment up to $\sqrt{s}=1.4$ GeV is found. It is observed that the $a_0(980)$, $\sigma$ and $\kappa(900)$ resonances are meson-meson states originating from the unitarization of the ${\mathcal{O}}(p^2)$ $\chi$PT amplitudes. On the other hand, the $f_0(980)$ is a combination of a strong S-wave meson-meson unitarity effect and of a preexisting singlet resonance with a mass around 1 GeV. We have also studied the size of the contributions of the unphysical cuts to the $\pi\pi$(I=0) and $K\pi$(I=1/2) elastic S-wave amplitudes from $\chi$PT and the exchange of resonances in crossed channels up to $\sqrt{s}\approx 800$ MeV. The loops are calculated as in $\chi$PT at next to leading order. We find a small correction from the unphysical cuts to our calculated partial waves.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The most general structure of an elastic partial wave amplitude when the unphysical cuts are neglected is deduced in terms of the N/D method.	contribution 0.60	0.90	0.09	asserted 1.00
s1	This result is then matched to lowest order, ${\mathcal{O}}(p^2)$, Chiral Perturbation Theory($\chi$PT) and to the exchange (consistent with chiral symmetry) of resonances in the s-channel.	method 0.81	0.87	0.08	asserted 0.96
s2	The extension of the method to coupled channels is also given.	contribution 0.73	0.90	0.03	asserted 0.98
s3	Making use of the former formalism, the $\pi\pi$ and $K\pi$(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/$N_c$ expansion and unitarity.	contribution 0.62	0.93	0.17	asserted 1.00
s4	Next, the scalar sector is studied and good agreement with experiment up to $\sqrt{s}=1.4$ GeV is found.	result 0.98	0.94	0.14	asserted 1.00
s5	It is observed that the $a_0(980)$, $\sigma$ and $\kappa(900)$ resonances are meson-meson states originating from the unitarization of the ${\mathcal{O}}(p^2)$ $\chi$PT amplitudes.	interpretation 0.66	0.88	0.27	asserted 1.00
s6	On the other hand, the $f_0(980)$ is a combination of a strong S-wave meson-meson unitarity effect and of a preexisting singlet resonance with a mass around 1 GeV.	interpretation 0.52	0.86	0.15	asserted 1.00
s7	We have also studied the size of the contributions of the unphysical cuts to the $\pi\pi$(I=0) and $K\pi$(I=1/2) elastic S-wave amplitudes from $\chi$PT and the exchange of resonances in crossed channels up to $\sqrt{s}\approx 800$ MeV.	contribution 0.98	0.96	0.05	asserted 0.95
s8	The loops are calculated as in $\chi$PT at next to leading order.	method 1.00	0.74	0.02	asserted 0.99
s9	We find a small correction from the unphysical cuts to our calculated partial waves.	result 1.00	0.96	0.11	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 0.92.

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Figures

**Figure 1.** Loop giving rise to the g0 function. The integral in eq. (43) will be evaluated making use of dimensional regularization. It can be identified up to a constant to the loop represented in

**Figure 2.** Isovector ππ elastic phase shifts from threshold up to √ s ≤ 1.2 GeV. The dashed line corresponds to take g 2 v = 1 and ˜a SL = 0. The continuum line corresponds to the simultaneous fit to the ρ and K∗ channels, given by eq. (59). Data: circles [32], squares [33] 495.7 MeV [30]. On the other hand, since mK 6= mπ instead of having just a single pole in the denominator function, as in eq. (48), for the zero at threshold, one has two simple poles γ Kπ 1 4p 2 Kπ = γ Kπ 1 s+ − s− s+ s − s+ − s− s − s− (55) with s+ = (mK + mπ) 2 and s− = (mK − mπ) 2 . That is,

**Figure 3.** P-wave elastic Kπ phase shifts with I=1/2, from threshold up to √ s ≤ 1.2 GeV.The dashed line corresponds to take g 2 v = 1 and ˜a SL = 0. The continuum line corresponds to the simultaneous fit to the ρ and K∗ channels, given by eq. (59). Data: triangles [34], circles [35] The subleading constant ˜a SL presents in g0(s), eq. (44), should be the same for the ππ and Kπ states because the dependence of the loop represented in

**Figure 4.** Elastic isoscalar ππ phase shifts, δ 00 11. The circles correspond to the average of [43, 44, 45], as discussed in the δ 00 11 subsection. We have also included the triangle points form [46] to have some data close to threshold, although these points have not been included in the fit because they are given without errors. This fit has 8 free parameters, 6 constants from the nonet3 , a SL and the normalization constant N . In our former work [19], we were able to describe δ 00 11, δ 00 12 and 1−η 2 00 4 up to 3 If the singlet and the octet introduced form really a nonet is

**Figure 5.** S-wave KK¯ → ππ isoscalar phase shifts, δ 00 12. The triangles points are from [47], circles correspond to the average of [48, 49] and squares to the one of [47, 48, 49]. √ s ≤ 1.2 GeV and a distribution of events around the a0(980) mass [52], using only 2 free parameters: a cut-off (which plays the role of the regularization scale µ and at the same time generates a concrete value for a SL) and a corresponding normalization constant for the a0(980) event distribution. In fact, if we remove the resonance contributions in eqs. (65) and (67) the formalism of ref. [19] follows from the

**Figure 6.** 1−η 2 00 4 with η00 the I=L=0 S-wave inelasticity. Circles [47], triangles [48]. be interpreted from the need to give a ‘range’ to this potential so that the loop integrals converge. These meson-meson states are shown in

**Figure 7.** Distribution of events around the a0(980) mass corresponding to the central production ππη in 300 GeV collisions [42]. The abscissa represents the πη invariant mass, Ecm. The dashed line represents the background introduced in the same reference. 800 1000 1200 1400 0 50 100 150 200

**Figure 8.** S-wave I=1/2 Kπ elastic phase shifts, δ 1/2 0 11 . The triangles correspond to the average, as described in δ 1/2 0 11 subsection, of [34, 35, 51]. Circles correspond to [50]. 28

**Figure 9.** Chiral limit. From left to right, Figs. 9a,b,c respectively. In Fig. 9a the poles of T found in the unphysical sheet are shown for I=0. Analogously for I=1/2 and 1 in Figs. 9b,c respectively. 5 Estimations of the unphysical cut contribution from χPT and the exchange of resonances. In this last section we estimate the influence of the unphysical cuts for the elastic ππ and Kπ S-waves with I=0 and 1/2 respectively. The unphysical cuts will be approximated by means of χPT supplied with the exchange of resonances [24] with spin ≤1 in the t- and u-channels. The loops are calculated from…

Figures reproduced from arXiv:hep-ph/9809337 by the authors.

References

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Forward citations

Cited by 43 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

S-matrix informed neural networks for amplitude analysis
hep-ph 2026-08 conditional novelty 7.0 of 10

A physics-constrained neural network with automatic data selection reconstructs pi-pi scattering amplitudes and scattering lengths from inconsistent experimental data without a fixed functional form.
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hep-ph 2026-08 conditional novelty 7.0 of 10

A new neural-network framework that respects unitarity, analyticity, and crossing symmetry extracts the sigma, rho, and f0(980) resonance poles from pion-pion scattering data.
Axion-like particle production from kaon decays within U(3) chiral theory
hep-ph 2026-08 conditional novelty 7.0 of 10

A U(3) chiral perturbation theory calculation of K to pi a decay shows that a U(3)-specific weak operator can matter, and yields updated ALP coupling bounds.
A femtoscopic tale of two $C$-parities: the $Z_c(3900)$ and the isovector partner of the $X(3872)$
hep-ph 2026-08 conditional novelty 7.0 of 10

Femtoscopic correlations of charged charm meson pairs mix C-odd and C-even interactions, giving a new experimental probe of the predicted W_c1 state and correcting earlier predictions.
Revealing the $D_0^*(2300)$ two-pole structure from lattice data and the SU(3) limit
hep-ph 2026-06 unverdicted novelty 7.0 of 10

Lattice data analysis with UChPT finds two I=1/2 poles at physical point with positions 2094-i111 MeV and 2463-i108 MeV; lower pole splits like the sigma on ms=const trajectory and links to 3-bar, higher pole links to...

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Prominent enhancement of axion thermalization rate from axion-kaon interactions
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Axion-kaon scattering, previously omitted from axion thermalization calculations, exceeds the axion-pion channel near T=130 MeV and tightens the Planck Delta N_eff bound on f_a by about 30%.
Novel method for determining the light quark mass ratio using $\eta'\to\eta \pi\pi$ decays
hep-ph 2025-02 conditional novelty 7.0 of 10

A novel unit-disk mapping of Dalitz plots is used to extract the light quark mass ratio Q = 22.3 ± 0.7 from BESIII data on η′ → ηππ decays.
Finite-volume quantization condition from the $N/D$ representation
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A finite-volume quantization condition based on the N/D representation is derived and claimed to remain valid when the energy coincides with left-hand cuts, where the original Luescher condition fails.
Different pole structures in line shapes of the $X(3872)$
hep-ph 2016-12 conditional novelty 7.0 of 10

Fits to X(3872) line shapes are equally good for bound-state, virtual-state, double/triplet virtual-pole, and simultaneous bound-plus-virtual scenarios, with D0D̄*0 compositeness spanning nearly 0 to 1.
The new LHCb beauty-strange state as the bottom partner of the $D_{s0}^*(2317)$: a LQCD constrained coupled-channel chiral analysis
hep-ph 2026-09 accept novelty 6.0 of 10

LQCD-constrained unitarized chiral amplitudes put the new LHCb B_s0* within 1.1-1.7 sigma of the measured peak and, after a small one-parameter adjustment, describe the spectrum and yield a 1+ partner near 5750 MeV.
Study of the $e^+e^- \to \phi K^+K^-$ reaction within triangle dynamics and its implications for the $\phi(2170)$
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The same K1-anti-K-K triangle mechanism that produces the phi(2170)-like peak in e+e- -> phi pi+pi- predicts that the peak should be absent in e+e- -> phi K+K-, matching BESIII.
The $D_{s1}(2460)$ and other open-charm $1^+$ states in relativistic chiral effective field theory
hep-ph 2026-07 unverdicted novelty 6.0 of 10

Relativistic U(3) chiral EFT at NLO yields scattering lengths that match lattice data and identifies D_s1(2460) as an SU(3) triplet bound-state pole and D1(2430) as a triplet-sextet pole pair, none of which are conven...
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hep-ph 2026-06 unverdicted novelty 6.0 of 10

Derives an exact integral representation of the left-hand cut for arbitrary isospin and angular momentum partial waves as an integral over right-hand cut imaginary parts.
Three-body unitary determination of the $f_1(1285)$ and $f_1(1420)$ pole positions
hep-ph 2026-06 unverdicted novelty 6.0 of 10

Fitting a spectator-isobar three-body unitary amplitude to BESIII K0S K0S pi0 data yields poles at (1277±2±1)-i(12±1±0) MeV for f1(1285) and (1435±2±7)-i(40±2±1) MeV for f1(1420), with the latter traced to a K Kbar* q...
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hep-lat 2026-05 unverdicted novelty 6.0 of 10

Finite-volume N/D analysis with left-hand cuts applied to lattice data shows a mild but statistically significant effect on H-dibaryon binding energy compared to Lüscher quantization.
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The light-quark contribution to H to gamma gamma and H to gamma Z is nonzero and linear in quark masses, with a large enhancement in the two-photon channel.
Signatures of Odd-Parity $s$-wave $\Xi^*$ States in Femtoscopic Correlation Functions
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Using a unitarized hidden-gauge model, the authors dynamically generate Xi(1950)-like and Xi(2120)-like poles and predict femtoscopic correlation functions for six vector-baryon channels.
Global parametrizations of $\pi\pi$ scattering with dispersive constraints: Beyond the S0 wave
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New global, dispersion-constrained parametrizations for the S2, P, D, F, and G pion-pion partial waves, valid to roughly 1.8 to 2.1 GeV.
Scalar resonance contributions in the $D_{s1}(2460)^{+} \rightarrow D_{s}^{+}\pi^{+}\pi^{-}$ reaction
hep-ph 2024-12 conditional novelty 6.0 of 10

In D_s1(2460)+ -> D_s+ pi+ pi-, a chiral unitary coupled-channel model reproduces the LHCb mass distributions and generates T_cbar-s and f0(500) as hadronic molecules.
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A quark-level hadronization plus meson rescattering model reproduces why D_s -> pi+ pi+ pi- contains only the f0(980) resonance while D+ -> pi+ pi+ pi- contains both f0(500) and f0(980).
Insights into the inner structures of the fully charmed tetraquark state $X(6900)$
hep-ph 2020-11 conditional novelty 6.0 of 10

The X(6900) is predominantly a compact (bare) state with two-meson compositeness below 0.2, and a narrow virtual companion X(6825) should appear just below the chi_c0 chi_c0 threshold.
The quark-mass dependence of elastic $\pi K$ scattering from QCD
hep-lat 2019-04 accept novelty 6.0 of 10

Across four pion masses from 239 to 391 MeV, the pion-kaon P-wave pole smoothly becomes the K*(892) resonance, while the S-wave kappa pole cannot be pinned down model-independently from real-axis lattice amplitudes.
Towards a new paradigm for heavy-light meson spectroscopy
hep-ph 2017-12 conditional novelty 6.0 of 10

The LHCb B- -> D+ pi- pi- angular moments are consistent with S-wave amplitudes fixed from lattice QCD, supporting a molecular interpretation of the lightest positive-parity charm mesons.
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hep-lat 2013-07 conditional novelty 6.0 of 10

A lattice K pi scattering analysis yields a K*(892) mass of 891 +/- 14 MeV and coupling g = 5.7 +/- 1.6, consistent with experiment, plus a tentative K*(1410) mass of 1.33 +/- 0.02 GeV.
On B -> V l l at small dilepton invariant mass, power corrections, and new physics
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The positive-helicity amplitudes in B to K* lepton pairs are suppressed by the V-A structure even including non-factorizable charm and light-quark corrections, leaving P1 and P3^CP as clean null tests sensitive to C'_...
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Full-QCD lattice scattering lengths, extrapolated with unitarized chiral perturbation theory, produce a DK bound-state pole near 2317 MeV and a revised D_s0*(2317) to D_s pi width of about 133 keV.
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A finite-volume framework using relativistic propagators and unitarized chiral perturbation theory can recover scalar meson pole positions from lattice-like energy spectra, demonstrated on synthetic data generated by ...
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Twisted boundary conditions on the strange quark let lattice simulations move the kaon-antikaon threshold and, in principle, extract all three coupled-channel K-matrix elements at a single energy.
Couplings in coupled channels versus wave functions: application to the X(3872) resonance
hep-ph 2009-11 accept novelty 6.0 of 10

In coupled-channel bound states, channel couplings from scattering-matrix residues track the wave function at the origin, not the channel probability, which explains the X(3872)'s near-isospin-zero character.
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A unitarized KK and Kpi FSI model with parametrized weak vertices fits the toy S-wave of D+ -> K-K+pi+ and is proposed as the basis for a CPV-sensitive amplitude analysis of LHCb data.
Doubly charmed baryon-light meson scattering in chiral effective theory with lattice constraints
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Chiral EFT with lattice-fitted LECs and unitarization predicts resonance, virtual, and bound states plus scattering parameters for doubly charmed baryon-meson systems at physical quark masses.
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hep-ph 2019-04 conditional novelty 5.0 of 10

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Coalescence and statistical models of heavy ion collisions predict that compact multiquark hadrons are produced about an order of magnitude less abundantly than hadronic molecules or ordinary excited states of the same mass.
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hep-ph 2013-09 conditional novelty 5.0 of 10

A relativistic unitarized chiral fit to lattice scattering lengths reproduces the D*s0(2317) as a dynamical bound state and predicts its spin and bottom-quark partners.
Wave functions of composite hadron states and relationship to couplings of scattering amplitudes for general partial waves
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A generalized Weinberg compositeness sum rule, valid for any partial wave, coupled channels, and resonances, indicates the rho meson is mostly not a pion-pion composite.
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hep-ph 2009-01 conditional novelty 5.0 of 10

Unitarized chiral perturbation theory reproduces lattice QCD scattering lengths for charmed-meson-Goldstone-boson scattering and predicts non-perturbative poles in several channels.
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hep-ph 2006-12 conditional novelty 5.0 of 10

A unitarized SU(4) coupled-channel model predicts a new narrow hidden-charm scalar meson near 3.7 GeV, mostly a D Dbar quasi-bound state.
From controversy to precision on the sigma meson: a review on the status of the non-ordinary $f_0(500)$ resonance
hep-ph 2015-10 accept novelty 3.0 of 10

The sigma meson, renamed f0(500), is established as a broad non-quark-antiquark resonance with pole position about 449 - i275 MeV, from a synthesis of dispersive analyses of pion-pion scattering.
Composite nature of exotic states from data analysis
hep-ph 2025-09 conditional novelty 2.0 of 10

Compositeness values for X(3872), Zb(10610), Zb(10650), and Tcc are extracted from CDD-pole fits to published spectra; X(3872) is unconstrained (0 to 1), Tcc is found at 0.23 with large errors.

How later papers used it

114 later papers cite it

Evidence: substantial. 89 of the citing papers share no author with it, and 35 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 187 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

6 later papers by other authors are listed below: 6 used its method or tool. 3 more papers by the same authors are listed: 3 used its method or tool.

Uses its method or tool 9 of 46 papers with this label · 3 by the same authors

Gave the diagonal element of its G(s) matrix by the two-meson loop integral of this paper.

Subleading contributions to the width of the D_{s0}*(2317) (2008) · 0.93 · paraphrased: the licence does not allow quoting
In this work, we adopt the unitarized chiral perturbation theory (UChPT) framework from Ref. [ref] to construct scattering amplitudes as [equation] where $𝒯_l=0^(S,I)$ denotes the perturbative $S$-wave amplitudes up to NLO and $G(s)$ is the standard two-point one-loop function, which incorporates the contribution from the right-hand cut (RHC) associated with intermediate DCB and pNGB states.

Doubly charmed baryon-light meson scattering in chiral effective theory with lattice constraints (2026) · Partial wave amplitude and its unitarization · 1.00
The unitarized two-meson partial-wave (PW) scattering amplitudes are given by [equation] where $T^(2)_IJ$ denotes the $𝒪(p^2)$ $χ$PT PW amplitude with isospin $I$ and angular momentum $J$, $T^(4)__IJ$ stands for the polynomial parts with $𝒪(p^4)$ low energy constants (LECs) [ref] and $𝒢$ is the two-point one-loop function evaluated in dimensional regularization with unknown subtraction constant [ref] , whose imaginary part is $Im 𝒢=q/(8π√(s))$ for $√(s)$ above the two-meson threshold, with $q$ the three-momentum in the center of mass (CM) frame.

Prominent enhancement of axion thermalization rate from axion-kaon interactions (2025) · Introduction · 0.85
Used a basic unitarization formalism for two-body partial-wave scattering amplitudes that is an approximated version of the N/D method of this paper.

Light-flavor resonance dynamics in the $U(3)$ chiral theory (2020) · 0.98 · paraphrased: the licence does not allow quoting
Defined its diagonal matrix G(s) with elements given by this paper.

Hunting for states in the recent LHCb di-$J/\psi$ invariant mass spectrum (2021) · 1.00 · paraphrased: the licence does not allow quoting
We use the dimensionally regularized two-point scalar loop function [ref] , [equation] where $q_i^2 = λ(t,m_i1^2,m_i2^2)/(4t)$ denotes the c.m. momentum squared for channel $i$, with the Källén triangle function $λ(a,b,c)=a^2+b^2+c^2-2(ab+ac+bc)$.

Precise determination of the pole position of the exotic $Z_c(3900)$ (2023) · Introduction · 0.99

By the same authors

Performed a unitarization of perturbative meson-meson amplitudes up to next-to-leading order in one-loop U(3) chiral perturbation theory plus tree-level resonance exchanges using the formalism of this paper.

Chiral study of the $a_0(980)$ resonance and $\pi\eta$ scattering phase shifts in light of a recent lattice simulation (2016) (same authors) · 0.98 · paraphrased: the licence does not allow quoting
Plugged its potential matrix as the kernel of the Bethe-Salpeter equation, which in an on-shell formalism assumes the algebraic form of this paper.

Dynamically generated open and hidden charm mesons (2007) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Took the general expression for a partial-wave amplitude of this paper, which automatically fulfills two-body unitarity.

Unitarization of infinite-range forces: graviton-graviton scattering (2020) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

106 arXiv papers cite this in 163 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

Making use of this formalism one observes that the \sigma, \kappa and a_0(980) resonances are meson-meson states originating from the unitarization of the lowest order \chiPT amplitudes.

Two Meson Scattering Amplitudes and their Resonances from Chiral Symmetry and the N/D Method · 1.00

Papers citing it · 91 sentences without a confident label

Uses its method or tool 43 papers

To restore unitarity and account for the significant $ππ$ rescattering effects, we will project the decay amplitudes onto partial waves and unitarize the $S$ and $D$ waves using the $N/D$ method [ref] .

Study of $\eta^\prime \to \eta \pi\pi $ Decays in Large-$N_C$ Chiral Perturbation Theory · Introduction · 1.00
We obtain the $η^'N$ interaction using the chiral unitary approach in coupled channels of [ref] .

Chiral unitary approach to \eta' N scattering at low energies · The N interaction from a coupled pseudoscalar-baryon basis. · 0.98
The two-point loop function $G(s)$ is logarithmically divergent and can be calculated with a once-subtracted dispersion relation whose explicit expression is [ref] [equation] where $α(μ)$ is the subtraction constant depending on the renormalization scale $μ$, $q_cm$ is the magnitude of the three-momentum of the meson in the center of mass frame [equation] Alternatively, one can also calculate the loop function by using a hard cutoff [ref] [equation] where $q_max$ is the cutoff of the three-momentum, $ω_i=√(q⃗ ^2+m_i^2)$.

Coupled-channel $D^\ast K^\ast -D_s^\ast \rho$ interactions and the origin of $T_{c\bar{s}0}(2900)$ · FORMALISM · 0.99

Uses its data or values 1 paper

For the resonance masses, we take $M_V = 770 MeV$ for the lowest vector state and $M_S=1400 MeV$ for the lowest scalar state.[Ref. [ref] lists several lighter scalar states, but it has been argued [ref] that these states are generated dynamically at next-to-leading orders in the large-$N_c$ expansion, and therefore disappear from the spectrum in the limit $N_c →∞$.

On the contribution from the light quarks to $H\to\gamma\gamma , \gamma Z$ · Section · 0.91

Cites it as support 2 papers

Then, the light scalar states $f_0(500)$, $f_0(980)$, $K_0^∗(800)$, and $a_0(980)$ are (most likely) predominantly four quark-states [see e.g. Refs. [ref] and refs. therein].

$a_{0}(980)$ revisited · Introduction · 0.96
In $χ$PT, one can also introduce intermediate $s$-channel resonance in the meson scattering, and after unitarization there could also be dynamically generated accompanying states, such as in [ref] , where $f_0(600)/σ$, $κ$, $a_0(980)$ and $f_0(980)$ are dynamically generated by coupling a nonet around 1.0GeV and 1.4GeV to the low lying pseudoscalars.

Two-pole structures in a relativistic Friedrichs-Lee-QPC scheme · Numerical results and discussions · 0.98

Cites it as background 19 papers

In elastic $πK$ scattering, the complex plane contains three cuts [ref] : in addition to the $s$-channel unitarity cut, which is correctly handled in the finite-volume formalism, unitarity in the cross-channels leads to a circular cut at $|s|=(m_K^2-m_π^2)$ and a left-hand cut that spans $-∞<s<(m_K-m_π)^2$.

The quark-mass dependence of elastic $\pi K$ scattering from QCD · Introduction · 0.88
With some different formulations at the beginning [ref] , the more recent work uses the on shell formulation firstly established on the basis of the N/D method in [ref] , where the potential and the t-matrix in momentum space factorize outside the loop function implicit in the Bethe Salpeter equation in coupled channels that one uses in those approaches.

Wave functions for dynamically generated resonances; the two $\Lambda(1405)$ and $\Lambda(1670)$ · Introduction · 0.83
One particular type of four-quark meson is that of a `dynamically generated' state [ref] .

$a_{0}(980)$ revisited · Introduction · 0.86

Discussion

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