N_f Dependence of the Quark Condensate from a Chiral Sum Rule

B. Moussallam (IPN; Orsay)

REVIEW

N_f Dependence of the Quark Condensate from a Chiral Sum Rule

B. Moussallam (IPN, Orsay)

High Energy Physics - Phenomenology · Submitted 8 Sep 1999 · 47 references

Pith's verdict: The sum rule and the Weinberg-type relation are real and cleanly derived, but the headline factor-of-two drop in the condensate leans heavily on a one-loop CHPT constant the author himself flags as unquantified.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 13, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-ph/9909292 (version 2)

Pith IDDSVPMCWW, Pith's permanent record of this paper and its review

CitationB. Moussallam (IPN, Orsay). "N_f Dependence of the Quark Condensate from a Chiral Sum Rule." arXiv:hep-ph/9909292 (1999). https://pith.science/paper/DSVPMCWW

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Worth your time, with one caveat that sits right on the headline number. The genuinely new pieces are a dispersive sum rule for L6 built on the correlator of (ūu+d̄d) with s̄s, and a Weinberg-type superconvergence relation for that correlator, derived from the OPE in eq. (20). Both are clean. The machinery in sec. 4 is careful work: the Muskhelishvili-Omnès system, the determinant check in sec. 4C that lets you verify the numerics against an analytic result, and the appendix algorithm using Legendre functions of the second kind with SVD for the homogeneous system. The low-energy spectral integrals are stable to about 10% between the Au and KLL two-channel fits (Table 1), and the form-factor observables reproduce ref. [12] and give sensible L4, L5. That part I trust. The positivity bound in sec. 3 is sketched rather than fully proved — the reality of Tr(iD̸+m)⁻¹ is asserted, and the argument is credited to a private communication — but the derivation is in the text and the logic is transparent, so I would call this minor. Where I part company with a purely phenomenological reading is the chain to R̃32. In eq. (11) the bracket is 16π²Π̄6(0) + 22/9. With the paper's own central value of about 3.4 for the first term, the finite one-loop constant 22/9 supplies roughly 42% of the deviation that becomes R̃32 ≈ 0.46. So the factor-of-two drop is not mostly a sum-rule output; a large chunk is a chiral counterterm. The Summary says plainly that two-loop corrections to this relation are unquantified, and the quoted ±0.27 is imported from the OZI estimate of ref. [9], so it does not cover a truncation error of unknown size. That is the load-bearing soft spot, more than the high-energy modelling, which eqs. (48)–(51) bound to roughly a factor of two. The Nf inference is by analogy from ms and is presented as such. Send it out; the sum rule and the WSR are citable and the low-energy analysis is solid, but the R̃32 number should be read as an indication, not a determination.

Referee report

Significance

The paper makes three genuinely useful contributions. First, the positivity bound of eqs. (14)–(18) is a clean, model-independent statement that the condensate must decrease with N_f, derived directly from the reality and positivity of the Euclidean fermion determinant; this is a real result independent of spectral modelling. Second, the derivation of the Weinberg-type superconvergence relation (eq. 19) from the OPE behaviour of eq. (20) is a nontrivial and checkable statement that constrains the scalar spectral function. Third, the numerical machinery in the Appendix — Legendre-function principal-value integration combined with singular-value decomposition, plus the analytic determinant check of §4C — is reproducible and is validated against the Truong–Willey analytic solution and against the earlier results of ref. [12]. The low-energy integrals I0 and I1 in Table 1 are stable at the ~10% level across the two-channel fits, which supports the robustness of the low-energy input. If the central estimate survives scrutiny of its error budget, the implied factor-of-two variation of the condensate is a physically significant result.

Major objections (3)

[§6C, eqs. (53)–(54)] The quoted uncertainty ±0.27 on R̃32 is not derived from this calculation. The text states explicitly that 'for the error, we have used the same value as that estimated in ref.[9]', i.e. the ±0.3×10^-3 of the OZI estimate in eq. (53), whereas the sum-rule determination itself gives the range 0.4–0.8 in eq. (52), corresponding to a spread of roughly ±0.2 around a central 0.6. The error bar therefore does not represent the uncertainty of the present analysis, and it is not obvious that it covers the model spread seen in Table 1 and Figs. 1–4. The author should recompute the uncertainty from the sum-rule range and the model-to-model variation, and should state separately the residual from the CHPT truncation discussed below.
[§2–3, eqs. (10)–(11), and Summary] The chain that produces the headline number passes through the one-loop CHPT expression of eq. (10) and then eq. (11), whose bracket is 16π²Π̄6(0) + 22/9. With the paper's own central value 16π²Π̄6(0) ≈ 3.4 from eq. (49), the finite one-loop piece 22/9 ≈ 2.44 is a substantial fraction of the total that drives R̃32 = 1 − 0.54. The Summary acknowledges that 'the relationship that we used between Π6(0) and the condensate ratio receives corrections from two-loop CHPT. It remains to be seen whether these are significant or not.' The authors are to be credited for stating this limitation, but no estimate of the two-loop size is attempted and the quoted error does not appear to cover it. A naive order-of-magnitude estimate, or at minimum an explicit statement of how much the bracket could shift, would materially strengthen the central claim. The same caveat applies to the 'model independent' lower bound of eq. (17), which is model-independent only within one-loop CHPT.
[§6C, eqs. (48)–(51)] The range 2 ≲ 16π²Π̄6(0) ≲ 6 in eq. (51) rests on the single-resonance ansatz of eq. (49) and on the assertion that 'there exist models which fit the data and can also accomodate the WSR constraint.' From §6B, among the three-channel parameter sets only set E approximately satisfies the WSR, while sets A and F violate it, so the range is supported by one three-channel set plus the two-channel KLL model. The bounds of eqs. (48) and (50) do confine the high-energy contribution to roughly a factor of two, which is a genuine strength, but the central value within that window is model-dependent. A sensitivity study of eq. (49) to the assumed resonance position and width, and an explicit statement of which models enter the quoted range, would make the estimate reproducible.

Minor objections (6)

[Abstract] The phrase 'allows a solid evaluation of L6' sits uneasily with the acknowledged uncertainty of the high-energy spectral region and the unquantified two-loop corrections. Consider softening to 'an estimate' to match the body of the paper.
[Abstract and §1] The abstract's 'strong N_f dependence' is an extrapolation from the variation of the condensate with m_s; the introduction correctly distinguishes R̃32 (eq. 3) from R32 (eq. 1) and states that the two are different. The abstract and Summary should be equally explicit that no direct N_f calculation is performed and that the m_s-variation is a proxy.
[§4D, eq. (41) and footnote 3] The hand-imposed asymptotic behaviour uses a cutoff E0 = 1.5 GeV and exponent m = 3, and the text notes only that 'changing these parameters will modify the details of the shape of the spectral function.' A numerical statement of the sensitivity of I1 and of Π̄6(0) to E0 and m would be useful.
[§6B and Table 2] Set E yields L5(M_ρ) ≈ 0.29×10^-3, which the text itself calls incompatible with F_K/F_π, yet set E is the one used to argue that the WSR can be satisfied. The tension between the low-energy and high-energy virtues of the same parameter set should be stated explicitly.
[Figs. 1–4] The vertical scales differ by more than an order of magnitude between figures (Fig. 4 reaches −90 while Fig. 1 spans ±15), which makes visual comparison of the spectral functions difficult. A common scale, or an inset, together with an indication of the model spread, would improve clarity.
[References] Reference [8] (Knecht and de Rafael) appears in the bibliography but I could not locate a citation in the body; please check. The title contains the typographical artifact 'CONDENSA TE', and there are numerous spacing artifacts throughout ('a t', 'con straints', 'sp ontaneously', 'l inearly', 'existe nce', 'communi cating', 'Flan nery'); 'Noordhof' should read 'Noordhoff'. These should be cleaned in proofs.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

A chiral sum rule links the quark condensate's size to how many flavors QCD has, and the answer comes out large.

The reading

The paper asks how the quark condensate — the vacuum expectation value $<0|$ $u$-quark pairs create in empty space, the standard order parameter of spontaneous chiral symmetry breaking — would change if the number of light quark flavors $N_f$ were varied. Nature only gives us $N_f = 2$ and $N_f = 3$, so the author uses a proxy: the dependence of the condensate on the strange quark mass, which chiral perturbation theory at one loop expresses through a single low-energy constant, $L_6$. The central move is to write $L_6$ as a dispersion integral over the correlation function of the scalar currents $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$, then evaluate that integral from measured $S$-wave $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$ scattering amplitudes. If the resulting value of $L_6$ is right, the condensate falls by roughly a factor of two as the strange quark mass is taken to zero, which would suggest that three-flavor QCD is not far from a chiral phase transition.

What carries the argument

The object is the connected two-point function $Pi_6(p^2) = i B_0^{-2} int d^4x e^{ipx} <T[(u-bar u + d-bar d)(x) s-bar s(0)]>_c$, whose value at zero is fixed by chiral perturbation theory at $O(p^4)$ in terms of $L_6$ and whose spectral function satisfies the superconvergence relation $int_0^infty Im Pi_6(s) ds = 0$. The spectral function below about 1 GeV is written exactly in terms of the pion and kaon scalar form factors $F_i(s)$, $G_i(s)$, which are in turn obtained by solving coupled Muskhelishvili-Omnes integral equations with experimental $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$ $S$-wave $T$-matrices; the $4pi$ continuum above 1.4 GeV is treated as an effective two-body channel. A numerical scheme expands the principal-value integrand in Legendre polynomials and uses Legendre functions of the second kind, with a singular-value decomposition to handle the near-homogeneous linear system.

Extended reading notes

Core claim

The paper establishes a sum rule for the chiral coupling $L_6$, the one $O(p^4)$ constant that controls how the quark condensate responds to the strange quark mass, and uses it to argue that this response is large. Two ingredients carry the argument. First, positivity of the QCD measure in the equal-mass limit forces $16e^2$ times the correlator at zero momentum to be positive, giving a model-independent bound $10^3 L_6(M_eta) >= 0.35$ and hence already showing that the condensate must decrease with $N_f$. Second, the same correlator obeys a Weinberg-type superconvergence relation, $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$ being the lowest operator of the right chiral transformation property, so its spectral function integrates to zero and its value at zero is an unsubtracted dispersive integral. Feeding that integral with scalar pion and kaon form factors built from experimental $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$ $S$-wave $T$-matrix input, the author quotes $0.4 < 10^3 L_6(M_eta) < 0.8$, against the OZI-based estimate $0.0 +- 0.3$. Translated back through the one-loop relation, this gives a condensate ratio $R_32 = 1 - 0.54 +- 0.27$, i.e. roughly a halving of the condensate between $N_f = 3$ and $N_f = 2$.

Load-bearing premise

The estimate rests on how the spectral function is modeled above roughly 1 GeV, where the four-pion continuum is replaced by an effective two-body channel and the scattering matrix is forced by hand to have the asymptotic phase behavior needed for the form factors to fall off.

Editorial extensions

If this is right

The quark condensate is not approximately $N_f$-independent: the sum-rule value of $L_6$ implies $R_32 = 1 - 0.54 +- 0.27$, so the condensate falls by roughly a factor of two going from three flavors to two.
The OZI rule, which predicts $L_6 = 0$ and hence $R_32 = 0.96$, is badly violated in the scalar channel, consistent with the known strong coupling of the $f_0(980)$ to both $u$-bar $u$ + $d$-bar $d$ and $s$-bar $s$.
Because the condensate drops so steeply with $N_f$, three-flavor QCD may sit closer to a chiral phase transition than large-$N_c$ or quenched reasoning would suggest.
The superconvergence relation ties the low-energy $f_0(980)$ contribution to a compensating high-energy contribution around 1.5 GeV, so the scalar resonance spectrum above 1 GeV is constrained by the same sum rule.
The framework fixes one of the three constants ($L_6$, $L_7$, $L_8$) that the Kaplan-Manohar transformation mixes, opening a route to disentangling them individually rather than only in invariant combinations.

Reading between the lines

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

If lattice simulations could compute $d<u-bar u + d-bar d>/dm_s$ directly at physical quark masses, that single number would test the entire chain without needing the hadronic spectral modeling.
The same dispersion-plus-superconvergence template should apply to other $O(p^4)$ constants whose correlators are chirally protected, so $L_7$ and $L_8$ may be within reach of the same machinery once the corresponding spectral functions are built.
The steep dependence on $N_f$ suggests a concrete lattice program: scan $N_f = 2, 3, 4$ at fixed $N_c = 3$ and look for the condensate falling faster than linear in $N_f$, which would locate the critical flavor number.
Two-loop chiral corrections to the relation between $Pi_6(0)$ and $R_32$ are the natural next calculation; if they shift $L_6$ by the size of the quoted error band, the factor-of-two conclusion would soften considerably.

Original abstract

How fast does the quark condensate in QCD-like theories vary as a function of $N_f$ is inferred from real QCD using chiral perturbation theory at order one-loop. A sum rule is derived for the single relevant chiral coupling-constant, $L_6$. A model independent lower bound is obtained. The spectral function satisfies a Weinberg-type superconvergence relation. It is discussed how this, together with chiral constraints allows a solid evaluation of $L_6$, based on experimental $\pi\pi-K\bar K$ S-wave T-matrix input. The resulting value of $L_6$ is compatible with a strong $N_f$ dependence possibly suggestive of the proximity of a chiral phase transition

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,854 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.40 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

How fast does the quark condensate in QCD-like theories vary as a function of $N_f$ is inferred from real QCD using chiral perturbation theory at order one-loop. A sum rule is derived for the single relevant chiral coupling-constant, $L_6$. A model independent lower bound is obtained. The spectral function satisfies a Weinberg-type superconvergence relation. It is discussed how this, together with chiral constraints allows a solid evaluation of $L_6$, based on experimental $\pi\pi-K\bar K$ S-wave T-matrix input. The resulting value of $L_6$ is compatible with a strong $N_f$ dependence possibly suggestive of the proximity of a chiral phase transition

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	How fast does the quark condensate in QCD-like theories vary as a function of $N_f$ is inferred from real QCD using chiral perturbation theory at order one-loop.	contribution 0.55	0.71	0.09	asserted 0.95
s1	A sum rule is derived for the single relevant chiral coupling-constant, $L_6$.	contribution 0.74	0.93	0.04	asserted 1.00
s2	A model independent lower bound is obtained.	result 0.94	0.91	0.11	asserted 1.00
s3	The spectral function satisfies a Weinberg-type superconvergence relation.	result 0.65	0.48	0.04	asserted 1.00
s4	It is discussed how this, together with chiral constraints allows a solid evaluation of $L_6$, based on experimental $\pi\pi-K\bar K$ S-wave T-matrix input.	method 0.97	0.65	0.09	asserted 0.59
s5	The resulting value of $L_6$ is compatible with a strong $N_f$ dependence possibly suggestive of the proximity of a chiral phase transition	interpretation 0.90	0.88	0.81	hedged possibility 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.82.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Spectral function computed from T-matrix models with two coupled channels. The solid line corresponds to the T-matrix of ref.[21], the dashed line corresponds to ref.[39] 21

**Figure 2.** Spectral function computed from the 3-channel T-matrix model of ref.[39]. The curve corresponds to the parameter set A of this reference 22

**Figure 3.** Spectral function computed from the 3-channel T-matrix model of ref.[40]. The curve corresponds to the parameter set E of this reference 23

**Figure 4.** Same as Fig.3. The curve corresponds to the parameter set F. 24

Figures reproduced from arXiv:hep-ph/9909292 by the authors.

References

Works cited (47)

[12]

J.F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B343 (1990) 341

View on Pith 1990
[9]

J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465

View on Pith 1985
[1]

N. Seiberg, Nucl. Phys. B435 (1995) 129

View on Pith 1995
[2]

T. Banks and A. Zaks, Nucl. Phys. B196 (1982) 189

View on Pith 1982
[3]

M. Knecht and J. Stern, in “ The second DAPHNE physics handbook ”, L. Maiani, G. Pancheri and N. Paver eds., p.169, hep-ph/9411253

View on Pith Pith review arXiv
[4]

J. Stern, Two alternatives of spontaneous chiral symmetry breaking in QCD, hep-ph/9801282

View on Pith Pith review arXiv
[5]

R.D. Mawhinney, Nucl. Phys. Proc. Supp. 60A (1998) 306, Y. Iwasaki et al., Prog. Theor. Phys. Suppl. 131 (1998) 415, hep-lat/9804005

View on Pith Pith review arXiv 1998
[6]

T. Sh¨ afer and E.V. Shuryak, Phys. Rev. D53 (1996) 6522

View on Pith 1996
[7]

T. Appelquist, J. Terning and L.C.R. Wijewardhana, Phys . Rev. Lett. 77 (1996) 1214

View on Pith 1996
[8]

M. Knecht and E. de Rafael, Phys. Lett. B424 (1998) 335

View on Pith 1998
[10]

J.F. Donoghue and E. Golowich, Phys. Rev. D49 (1994) 1513

View on Pith 1994
[11]

M. Davier, L. Girlanda, A. Hoecker and J. Stern, Phys. Re v. D58 (1998) 096014

View on Pith 1998
[13]

J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142

View on Pith 1984
[14]

G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafae l, Phys. Lett. B223 (1989) 425

View on Pith 1989
[15]

D.B. Kaplan and A.V. Manohar, Phys. Rev. Lett. 56 (1986) 2004

View on Pith 1986
[16]

J. Stern, private communication

View on Pith
[17]

S. Weinberg, “ The quantum theory of ﬁelds ”, Cambridge University Press (1996), vol.2, Chap.20

View on Pith 1996
[18]

C. Caso et al., Eur. Phys. Jour. C3 (1998) 1. 18

View on Pith 1998
[19]

H. Leutwyler, Masses of the light quarks , workshop “ Yukawa couplings and the origin of mass” p.166, Gainsville (1994) hep-ph/9405330

View on Pith Pith review arXiv 1994
[20]

J. Bijnens, J. Prades and E. de Rafael, Phys. Lett. B348 (1995) 226

View on Pith 1995
[21]

K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. D35 (1987) 1633

View on Pith 1987
[22]

A. Lanaro et al. (OBELIX coll.), Nucl. Phys. A558 (1993) 13C

View on Pith 1993
[23]

C. Amsler et al. (Crystal Barrel Coll.)Phys. Lett. B322 (1994) 431

View on Pith 1994
[24]

A. Abele at al. (Crystal Barrel Coll.), Phys. Lett. B380 (1996) 453

View on Pith 1996
[25]

D. Alde et al., Nucl. Phys. B269 (1986) 485

View on Pith 1986
[26]

S.J. Lindenbaum and R.S. Longacre, Phys. Lett. B274 (1992) 492

View on Pith 1992
[27]

S.J. Brodsky and G.P. Lepage, Phys. Rev. D22 (1980) 2150

View on Pith 1980
[28]

N.I. Muskhelishvili, ” Singular integral equations ”, P. Noordhof (1953), chap. 18 and 19

View on Pith 1953
[29]

R. Omn` es, Nuov. Cim. 8 (1958) 316

View on Pith 1958
[30]

B.R. Martin, D. Morgan and G. Shaw, “ Pion-pion interactions in particle physics ”, Academic Press (1976)

View on Pith 1976
[31]

B. Hyams et al., Nucl. Phys. B64 (1973) 134

View on Pith 1973
[32]

D.V. Bugg, A.V. Sarantsev and B.S. Zou, Nucl. Phys. B471 (1996) 59

View on Pith 1996
[33]

R. Kaminski, L. Lesniak and K. Rybicki, Zeit. Phys. C74 (1997) 79

View on Pith 1997
[34]

H. Becker at al., Nucl. Phys. B151 (1979) 46

View on Pith 1979
[35]

S.M. Roy, Phys. Lett. B36 (1971) 353

View on Pith 1971
[36]

V. Bernard, N. Kaiser and U.G. Meißner, Nucl. Phys. B357 (1991) 129

View on Pith 1991
[37]

A. Roessl, Pion Kaon scattering near the threshold in chiral SU(2) pert urbation the- ory, hep-ph/9904320

View on Pith Pith review arXiv
[38]

T.N. Truong and R.S. Willey, Phys. Rev. D40 (1989) 3635

View on Pith 1989
[39]

R. Kami´ nski, L. Le´ sniak and B. Loiseau, Phys. Lett. B413 (1997) 130

View on Pith 1997
[40]

R. Kami´ nski, L. Le´ sniak and B. Loiseau, Eur. Phys. Jou r. C9 (1999) 141

View on Pith 1999
[41]

D. Cohen et al., Phys. Rev. D22 (1980) 2595

View on Pith 1980
[42]

A. Etkin et al., Phys. Rev. D26 (1982) 1786

View on Pith 1982
[43]

J. Gasser and U.G. Meißner, Nucl. Phys. B357 (1991) 90. 19

View on Pith 1991
[44]

M. Knecht, B. Moussallam, J. Stern and N.H. Fuchs, Nucl. Phys. B457 (1995) 513

View on Pith 1995
[45]

J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sain io, Phys. Lett. B374 (1996) 210

View on Pith 1996
[46]

M. Abramowitz and I.A. Stegun, “ Handbook of mathematical functions ”, Dover (1972)

View on Pith 1972
[47]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flan nery, “ Numerical recipes”, Cambridge Univ. Press (1996). 20 -15 -10 -5 0 5 10 15 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Spectral function s1/2 (GeV) Two channels Au KLL Figure 1: Spectral function computed from T-matrix models with two co upled channels. The solid line corresponds to the T-matrix of ref.[21...

View on Pith 1996

Forward citations

Cited by 20 Pith papers

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

Dispersive analysis of the $J/\psi\to\pi^0 \gamma^\ast$ transition form factor with $\rho$-$\omega$ mixing effects
hep-ph 2025-11 unverdicted novelty 7.0 of 10

Dispersive analysis with ρ-ω mixing produces a two-parameter fit describing BESIII data on the J/ψ→π⁰γ* form factor from 0 to 2.8 GeV and extracts a (62 ± 21)° relative phase between strong and electromagnetic modes.
A new analysis of $\pi K$ scattering from Roy and Steiner type equations
hep-ph 2003-10 conditional novelty 7.0 of 10

Six coupled Roy-Steiner equations for pion-kaon scattering yield a narrow allowed region for the two S-wave scattering lengths and updated chiral low-energy constants.
Implications of two-channel rescattering for charm CP violation from precise dispersive data
hep-ph 2026-08 conditional novelty 6.0 of 10

A dispersive two-channel rescattering analysis, using only the ππ→KK phase shift and branching ratios, bounds the isospin-0 charm CP asymmetries below about 3×10^-4, far below LHCb's measured ΔA_CP.
Pion and kaon D terms from holographic QCD and coupled-channel dispersion relations
hep-ph 2026-08 conditional novelty 6.0 of 10

A holographic tensor form factor joined to a two-channel dispersive trace predicts kaon and pion D terms, with the kaon less negative than the pion at low Q^2.
Two-pion contribution to hadronic vacuum polarization
hep-ph 2018-09 conditional novelty 6.0 of 10

Constraining the pion form factor with Roy-equation phase shifts and a conformal inelasticity expansion yields a_mu^pipi|<=1GeV = 495.0(2.6)x10^-10 and a pion charge radius of 0.429(4) fm^2.

Show 15 more citing papers

B_l4 decays and the extraction of |V_ub|
hep-ph 2013-12 conditional novelty 6.0 of 10

An Omnès-type, model-independent representation of the B → ππℓν form factors is built, with B* pole terms supplying the left-hand cut, but the |V_ub| extraction itself is deferred to future fits to data.
Lepton flavor violation in the Higgs sector and the role of hadronic tau-lepton decays
hep-ph 2013-09 conditional novelty 6.0 of 10

Dispersively determined two-pion form factors show that Higgs-mediated tau -> ell pi pi decays are dominated by the gluon and strange-quark couplings, yielding bounds on lepton-flavor-violating Higgs couplings competi...
A new global fit of the $L^r_i$ at next-to-next-to-leading order in Chiral Perturbation Theory
hep-ph 2011-03 conditional novelty 6.0 of 10

Updated next-to-next-to-leading-order chiral perturbation theory fits to the p^4 low-energy constants shift the values substantially and break the large-N_c relation 2L1 = L2 that the previous fit satisfied.
MO analysis of the high statistics Belle results on $\gamma\gamma\to \pi^+\pi^-,\pi^0\pi^0$ with chiral constraints
hep-ph 2010-06 conditional novelty 6.0 of 10

A modified Muskhelishvili-Omnes dispersive representation, fitted to Belle's charged and neutral two-photon pion-pair data with chiral constraints, yields updated pion dipole and quadrupole polarizabilities and p6 chi...
Pion and Kaon Electromagnetic Form Factors
hep-ph 2002-03 conditional novelty 6.0 of 10

The pion charge radius comes out at (0.452 +/- 0.013) fm^2 and L9^r(m_rho) = (5.93 +/- 0.43) x 10^-3 from a next-to-next-to-leading-order chiral perturbation theory fit to pion and kaon form factor data.
A dispersive method to study CP asymmetries in hadronic multi-body $B$ decays
hep-ph 2026-07 conditional novelty 5.0 of 10

The large localized CP asymmetry in B±→K±π+π− arises from a dip in the CP-conserving rate, not enhanced CP violation, and the non-resonant isospin-2 S-wave is essential — per a 13-parameter dispersive fit to LHCb's an...
Phenomenology of GeV-scale scalar portal
hep-ph 2019-04 accept novelty 5.0 of 10

The B-meson decay channels B to S K1(1270) and B to S K0*(700) contribute significantly to GeV-scale scalar production, and meson decays dominate at SHiP and LHC energies while bremsstrahlung and deep inelastic scatte...
Roy-Steiner-equation analysis of pion-nucleon scattering
hep-ph 2015-10 accept novelty 5.0 of 10

Solving Roy-Steiner equations for pion-nucleon scattering with pionic-atom scattering lengths imposed as constraints yields a pion-nucleon sigma-term of 59.1 +/- 3.5 MeV, about 14 MeV above the long-quoted canonical value.
Dispersive analysis of the scalar form factor of the nucleon
hep-ph 2012-04 conditional novelty 5.0 of 10

A two-channel dispersive analysis gives Delta_sigma = (13.9 +/- 0.3) MeV, with explicit linear dependence on the pion-nucleon coupling and subthreshold parameters.
QCD Isospin Breaking in Meson Masses, Decay Constants and Quark Mass Ratios
hep-ph 2001-01 conditional novelty 5.0 of 10

Including quark-mass isospin breaking at next-to-next-to-leading order in chiral perturbation theory leaves the low-energy constants essentially unchanged and yields m_u/m_d = 0.46 +/- 0.09.
Mesonic low-energy constants
hep-ph 2014-05 conditional novelty 4.0 of 10

A review of mesonic low-energy constants that adds new NLO and NNLO fits of the SU(3) constants L_i, with the NNLO values explicitly conditional on hand-chosen priors for the 34 higher-order constants.
Dispersion relations: foundations
hep-ph 2025-10 unverdicted novelty 2.0 of 10

Pedagogical review explaining how causality implies analyticity and its use in scattering amplitudes, form factors, and resonance extraction in hadronic physics.
Chiral perturbation theory
hep-ph 2006-11 accept novelty 2.0 of 10

Chiral perturbation theory supplies model-independent, systematically improvable formulas for the quark-mass dependence of low-energy hadron observables, and this review surveys its foundations, applications, and limi...
Precision tests of fundamental physics with $\eta$ and $\eta^\prime$ mesons
hep-ph 2020-07 accept novelty 1.0 of 10

The review argues that eta and eta-prime decays are a versatile, flavor-conserving laboratory for quark-mass, mixing, muon g-2, and beyond-Standard-Model physics, and maps the experiments that will exploit them.
Chiral Perturbation Theory Beyond One Loop
hep-ph 2006-04 conditional

A practitioner-written review of all order-p^6 ChPT calculations in the mesonic sector, summarizing the two-loop results and their experimental comparisons.

How the literature uses it

36 arXiv papers cite this in 76 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 citing it · 30 sentences without a confident label

Builds on it 1 paper

This is analogous to the approach in Refs. [ref] for the $S$-wave part of the matrix element but is also applied for the first time to the $D$-wave piece.

Exotic to standard bottomonium transitions · Introduction · 0.96

Uses its method or tool 19 papers

In [ref] the following iterative procedure for solving the system was suggested (further developed in [ref] ).

Hadronic decays of a light Higgs-like scalar · Two channels: numerical analysis · 0.84
The system of integral equations is solved by using the method presented in Refs. [ref] and we follow the implementation of Ref. [ref] .

Hadronic Decays of a Higgs-mixed Scalar · Numerical Solution · 1.00
There are two different approaches to the problem, the iterative one of Ref. [ref] and the one of Ref. [ref] , which is based on the singular value decomposition.

Uncertainties of hadronic scalar decay calculations · Two channel case · 0.88

Uses its data or values 2 papers

Their values are based on the dispersion analysis [ref] , [equation]

New method for fitting the low-energy constants in chiral perturbation theory · Observables, inputs and $χ^2$ · 0.93
This ensures the expected asymptotic $1/s$ fall-off behavior of the pion vector form factor.[We have checked that continuation prescriptions different than Eq. [ref], see Ref. [ref] , have small effects at low-energies, specially in the decay region of $ω→3π$ studied here.] Data from the A2 collaboration at MAMI [ref] and by the NA60 collaboration at SPS [ref] for $|f_ωπ^0(s) |^2$ (normalized at $s=0$) for low $ωπ^0$ invariant mass are shown in [ref].

Novel approaches in Hadron Spectroscopy · omega -> 3pi and psi to 3pi decays · 0.92

Compares with it 3 papers

Our results for the scalar pion radius are in good agreement with $⟨r^2⟩^S_π=(0.61±0.04) fm^2$ from [ref] and the range $⟨r^2⟩^S_π=(0.583…0.653) fm^2$ found in [ref] .

Dispersive analysis of the scalar form factor of the nucleon · Scalar pion and kaon form factors · 0.99
For the $S$-wave case we have checked our results agree with Refs. [ref] .

Exotic to standard bottomonium transitions · Dispersive representation for two meson production matrix elements · 1.00
In Ref. [ref] the Omnès functions were computed using the phase shifts parametrizations from Refs. [ref] and using a numerical solution also based on Refs. [ref] .

Hadronic Decays of a Higgs-mixed Scalar · Omnès function results · 0.83

Cites it as background 8 papers

The pion form factor $Γ_π$ has been considered, e.g., in Refs. [ref] in the context of the scalar radius of the pion $⟨r^2 ⟩_s^π$.

Uncertainties of hadronic scalar decay calculations · Scalar radius of the pion · 1.00
The Muskhelishvili-Omnès matrix is a class of solutions that embodies the effects of final-state rescattering in the inelastic case [ref] .

Implications of two-channel rescattering for charm CP violation from precise dispersive data · Introduction · 0.92
Would the SU(3) quark condensate be much smaller than the SU(2) one as discussed in Refs. [ref] would the results presented here be different.

Matching Chiral Perturbation Theory and the Dispersive Representation of the Scalar Kpi Form Factor · Introduction · 0.83

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [12]

J.F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B343 (1990) 341

View on Pith 1990

[2] [9]

J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465

View on Pith 1985

[3] [1]

N. Seiberg, Nucl. Phys. B435 (1995) 129

View on Pith 1995

[4] [2]

T. Banks and A. Zaks, Nucl. Phys. B196 (1982) 189

View on Pith 1982

[5] [3]

M. Knecht and J. Stern, in “ The second DAPHNE physics handbook ”, L. Maiani, G. Pancheri and N. Paver eds., p.169, hep-ph/9411253

View on Pith Pith review arXiv

[6] [4]

J. Stern, Two alternatives of spontaneous chiral symmetry breaking in QCD, hep-ph/9801282

View on Pith Pith review arXiv

[7] [5]

R.D. Mawhinney, Nucl. Phys. Proc. Supp. 60A (1998) 306, Y. Iwasaki et al., Prog. Theor. Phys. Suppl. 131 (1998) 415, hep-lat/9804005

View on Pith Pith review arXiv 1998

[8] [6]

T. Sh¨ afer and E.V. Shuryak, Phys. Rev. D53 (1996) 6522

View on Pith 1996

[9] [7]

T. Appelquist, J. Terning and L.C.R. Wijewardhana, Phys . Rev. Lett. 77 (1996) 1214

View on Pith 1996

[10] [8]

M. Knecht and E. de Rafael, Phys. Lett. B424 (1998) 335

View on Pith 1998

[11] [10]

J.F. Donoghue and E. Golowich, Phys. Rev. D49 (1994) 1513

View on Pith 1994

[12] [11]

M. Davier, L. Girlanda, A. Hoecker and J. Stern, Phys. Re v. D58 (1998) 096014

View on Pith 1998

[13] [13]

J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142

View on Pith 1984

[14] [14]

G. Ecker, J. Gasser, H. Leutwyler, A. Pich and E. de Rafae l, Phys. Lett. B223 (1989) 425

View on Pith 1989

[15] [15]

D.B. Kaplan and A.V. Manohar, Phys. Rev. Lett. 56 (1986) 2004

View on Pith 1986

[16] [16]

J. Stern, private communication

View on Pith

[17] [17]

S. Weinberg, “ The quantum theory of ﬁelds ”, Cambridge University Press (1996), vol.2, Chap.20

View on Pith 1996

[18] [18]

C. Caso et al., Eur. Phys. Jour. C3 (1998) 1. 18

View on Pith 1998

[19] [19]

H. Leutwyler, Masses of the light quarks , workshop “ Yukawa couplings and the origin of mass” p.166, Gainsville (1994) hep-ph/9405330

View on Pith Pith review arXiv 1994

[20] [20]

J. Bijnens, J. Prades and E. de Rafael, Phys. Lett. B348 (1995) 226

View on Pith 1995

[21] [21]

K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. D35 (1987) 1633

View on Pith 1987

[22] [22]

A. Lanaro et al. (OBELIX coll.), Nucl. Phys. A558 (1993) 13C

View on Pith 1993

[23] [23]

C. Amsler et al. (Crystal Barrel Coll.)Phys. Lett. B322 (1994) 431

View on Pith 1994

[24] [24]

A. Abele at al. (Crystal Barrel Coll.), Phys. Lett. B380 (1996) 453

View on Pith 1996

[25] [25]

D. Alde et al., Nucl. Phys. B269 (1986) 485

View on Pith 1986

[26] [26]

S.J. Lindenbaum and R.S. Longacre, Phys. Lett. B274 (1992) 492

View on Pith 1992

[27] [27]

S.J. Brodsky and G.P. Lepage, Phys. Rev. D22 (1980) 2150

View on Pith 1980

[28] [28]

N.I. Muskhelishvili, ” Singular integral equations ”, P. Noordhof (1953), chap. 18 and 19

View on Pith 1953

[29] [29]

R. Omn` es, Nuov. Cim. 8 (1958) 316

View on Pith 1958

[30] [30]

B.R. Martin, D. Morgan and G. Shaw, “ Pion-pion interactions in particle physics ”, Academic Press (1976)

View on Pith 1976

[31] [31]

B. Hyams et al., Nucl. Phys. B64 (1973) 134

View on Pith 1973

[32] [32]

D.V. Bugg, A.V. Sarantsev and B.S. Zou, Nucl. Phys. B471 (1996) 59

View on Pith 1996

[33] [33]

R. Kaminski, L. Lesniak and K. Rybicki, Zeit. Phys. C74 (1997) 79

View on Pith 1997

[34] [34]

H. Becker at al., Nucl. Phys. B151 (1979) 46

View on Pith 1979

[35] [35]

S.M. Roy, Phys. Lett. B36 (1971) 353

View on Pith 1971

[36] [36]

V. Bernard, N. Kaiser and U.G. Meißner, Nucl. Phys. B357 (1991) 129

View on Pith 1991

[37] [37]

A. Roessl, Pion Kaon scattering near the threshold in chiral SU(2) pert urbation the- ory, hep-ph/9904320

View on Pith Pith review arXiv

[38] [38]

T.N. Truong and R.S. Willey, Phys. Rev. D40 (1989) 3635

View on Pith 1989

[39] [39]

R. Kami´ nski, L. Le´ sniak and B. Loiseau, Phys. Lett. B413 (1997) 130

View on Pith 1997

[40] [40]

R. Kami´ nski, L. Le´ sniak and B. Loiseau, Eur. Phys. Jou r. C9 (1999) 141

View on Pith 1999

[41] [41]

D. Cohen et al., Phys. Rev. D22 (1980) 2595

View on Pith 1980

[42] [42]

A. Etkin et al., Phys. Rev. D26 (1982) 1786

View on Pith 1982

[43] [43]

J. Gasser and U.G. Meißner, Nucl. Phys. B357 (1991) 90. 19

View on Pith 1991

[44] [44]

M. Knecht, B. Moussallam, J. Stern and N.H. Fuchs, Nucl. Phys. B457 (1995) 513

View on Pith 1995

[45] [45]

J. Bijnens, G. Colangelo, G. Ecker, J. Gasser and M. Sain io, Phys. Lett. B374 (1996) 210

View on Pith 1996

[46] [46]

M. Abramowitz and I.A. Stegun, “ Handbook of mathematical functions ”, Dover (1972)

View on Pith 1972

[47] [47]

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flan nery, “ Numerical recipes”, Cambridge Univ. Press (1996). 20 -15 -10 -5 0 5 10 15 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Spectral function s1/2 (GeV) Two channels Au KLL Figure 1: Spectral function computed from T-matrix models with two co upled channels. The solid line corresponds to the T-matrix of ref.[21...

View on Pith 1996