Pion-pion scattering at low energy

G. Colangelo; G. Ecker; J. Bijnens; J. Gasser; M.E. Sainio

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Pion-pion scattering at low energy

J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio

High Energy Physics - Phenomenology · Submitted 9 Jul 1997 · 49 references

Pith's verdict: The reference two-loop ππ amplitude: the machinery is the contribution, the numbers are explicitly placeholders, and the finite polynomial is unchecked by anything internal.

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CitationG. Colangelo, G. Ecker, J. Bijnens, J. Gasser, M.E. Sainio. "Pion-pion scattering at low energy." arXiv:hep-ph/9707291 (1997). https://pith.science/paper/LHJI5REY

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Editor's note

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This is the long paper behind the 1996 PLB letter, and it is what it says it is: the technical backup. What's actually new is the complete O(p^6) amplitude with explicit dependence on Mπ, Fπ, the four l_i and the six r_i, plus the renormalization procedure that gets you there. Nobody had the polynomial part before — Knecht et al. fixed only the absorptive piece through unitarity, and the paper is careful about that distinction in Sect. 4.3. The φ4 warm-up in Sect. 2 is useful, not filler, and the trick of using the renormalized off-shell one-loop amplitude as a single nonlocal vertex (Sect. 2.3.2, Fig. 5) is the kind of thing that saves a collaborator months. Appendix B, the off-shell one-loop four-point function in d dimensions, isn't in the literature elsewhere and is worth the paper by itself. Appendix A does the fish and sunset integrals in closed form with the subtraction procedure spelled out, which is more than most two-loop papers bother with. Where it's soft. The stress-test worry is fair and I'd repeat it: the internal checks — Eq. (3.15), the RGE (3.18), the comparison with Ref. [9] — constrain the divergent parts, the scale dependence and the absorptive part. Nothing constrains the finite rational coefficients in F^(2) and G^(2), or the k_i in (D.2). A factor error in a finite two-loop integral would sail through every check. That said, this is the normal state of affairs for a calculation this size, and the paper hands you App. A and B precisely so someone can redo it independently. Verification gap, not a suspicion. Second soft spot, and I'd weight it higher: the EOM-equivalence argument for L4 in the chiral case is deferred to Ref. [27], listed as in preparation. The φ4 version in Sect. 2.2.3 is done properly, so the claim is credible, but the chiral case is exactly where it bites, since b1–b4 shift under the ambiguity. The r_i estimates in (5.7) carry no error bars and rest on resonance saturation tested only at O(p^4). The authors say all of this themselves in Sects. 5.2 and 6 — the tables are illustrations, not results, and they ask you to read them that way. Who should read it: anyone doing two-loop CHPT, anyone fitting ππ data with Roy equations, anyone who needs the quark-mass dependence for lattice comparison. It's a reference, not a read-through. Solid, honest, worth referee time.

Referee report

Significance

If the finite parts are correct, this is the first field-theoretic two-loop ππ amplitude with complete quark-mass dependence, and it supplies precisely the polynomial part that unitarity-based dispersive constructions (Ref. [9]) leave undetermined. The paper is unusually explicit for its genre: the off-shell one-loop amplitude in d dimensions is given in Appendix B, the fish and sunset integrals are reduced to elementary functions in Appendix A, and the renormalization is organized so that the absence of mass singularities in the divergences (Eq. 3.15) and the scale dependence (Eq. 3.18) serve as internal consistency conditions. The threshold-parameter formulae of Appendix C and the b_i of Appendix D are parameter-free given the couplings, and the discussion in Sect. 5.3 of how effective-range and Rosselet-type approximations distort the extraction of a₀⁰ from phase-shift data is a genuinely useful, falsifiable statement that does not depend on the resonance estimates.

Major objections (3)

[§3.2.3 and §5.2(ii)] The claim that Lagrangians of O(p⁴) differing by the equations of motion lead to generating functionals differing by local O(p⁶) actions, and that the δᵢ(μ) of Eq. (3.26) can be absorbed into the O(p⁶) couplings in a process-independent way, is load-bearing: it is what licenses the statement in §5.2(ii) that the L₄-form ambiguity shifts b₁–b₄ by less than the scale ambiguity, and it underlies the comparison of Table 1 with Table 3. As written, the argument is deferred to Ref. [27], which is listed as 'in preparation'. Section 2.2.3 does demonstrate the analogous mechanism explicitly in φ⁴ theory; a short explicit demonstration in CHPT at the level of the b_i, or at least a sketch of how the φ⁴ argument carries over to the derivative and χ₊ structures of L₄, would make the numerical statements in §5.2 self-contained. This is a support issue rather than a demonstrated error, but it should be closed before the b_i tables are used as reference values.
[Eqs. (4.15), (D.1), (D.2)] The finite, scale-independent rational coefficients in F⁽²⁾(s) and G⁽²⁾(s,t) — for instance the 503/108 s³ term — and the kᵢ entering Eq. (D.2) are the paper's central new output, yet the internal checks constrain only their divergent and scale-dependent parts: Eq. (3.15) fixes the pole structure, Eq. (3.18) fixes the running, and the comparison with Ref. [9] covers only the absorptive part. A finite error in a two-loop integral would pass all three. The paper supplies the machinery for an independent check (Appendices A and B), and one concrete test would materially strengthen it: recomputing a subset of the two-loop diagrams in a different parametrization of U (exponential rather than sigma-model) and verifying that the b_i agree, since parametrization independence is a nontrivial constraint on the finite parts only. A second, cheaper test is to verify that the threshold expansion of Eq. (4.14) reproduces Eqs. (C.1) analytically rather than numerically. I raise this as a correctness-risk concern with a concrete test, not as an assertion of error.
[§4.3, final paragraph] The comparison with Ref. [9] is stated as agreement 'as far as the absorptive part of the amplitude and the general structure of the real part is concerned'. The absorptive part is a real check; 'general structure of the real part' is not. Since the paper's stated advantage over Ref. [9] is precisely that it determines the six coefficients a₁–a₆ of Eq. (4.16), the comparison would be much more convincing if made quantitative: project the polynomial part of Eq. (4.14) onto the basis of Eq. (4.16) and exhibit the resulting aᵢ as functions of Mπ and the lⁱᵣ, rⁱᵣ, showing explicitly which contributions are absent in the S-matrix approach. This would also make the claim in the same paragraph about mass singularities in the aᵢ directly checkable by the reader.

Minor objections (8)

[§4.1] The sentence 'It is not needed in the following, and we no not display its explicit form [24] here' contains a typo ('no not') and should read 'do not'.
[Table 4] The column headed 'O(p⁶) kᵢ' is not defined in the caption; the reader has to infer from the text that it corresponds to keeping only the kᵢ contributions of Eq. (D.2). Please state this explicitly in the caption.
[Appendix C] The text notes that the terms in the last square brackets of Eqs. (C.1) generate O(p⁸) contributions and are removed in the numerical results. It would help to say precisely which terms these are, or to give the truncated formulae in a footnote, so that a reader reproducing Table 4 obtains the same numbers.
[Eq. (A.13)] The recursion relation used for H(1) and H′(1) is attributed to Ref. [48], which is listed as 'in preparation'. Since these quantities enter the explicit two-loop expressions of Sect. 4, a published cross-check or a one-line indication of the partial-integration identity would be preferable.
[§5.1.4 and §5.2] The authors acknowledge that the rⁱᴿ of Eq. (5.7) carry no error bars and that the b_i are dominated by the O(p⁴) couplings. Given that Tables 1, 3 and 4 are presented as reference numbers, it would be useful to attach at least a rough range to the rⁱᴿ (for instance from the spread between the vector-dominance and scalar-exchange pieces quoted in Eqs. 5.3 and 5.5), so that the reader can see how much of the spread between set I and set II is genuinely from the lⁱᵣ.
[Eqs. (2.13)] The subequation labels (2.13 a)–(2.13 e) are referenced throughout Sect. 2 but the corresponding diagrams in Fig. 1 are only loosely tied to them; a sentence in the caption mapping each panel to its subequation would help.
[Figs. 8 and 9] The figures would be easier to read if the O(p²) and O(p⁴) curves were distinguished by line style in addition to the legend, and if the energy range of the data sets from Refs. [42] and [43] were indicated.
[Eq. (5.17)] The expression for the effective range in terms of a₀⁰ and b₀⁰ should be checked for sign conventions against Eq. (5.16); as written the last two terms have the same sign structure as the first, which is not obviously consistent with the definition of b₀⁰ in Eq. (5.15).

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The paper

The paper claims

The pion-pion scattering amplitude is carried to two loops, and the six coefficients that unitarity alone cannot fix are written out in full.

The reading

Elastic pion-pion scattering is the cleanest place to watch the strong interaction's approximate chiral symmetry at work, and for thirty years the standard tool for describing it has been chiral perturbation theory, an expansion in pion momenta. This paper supplies the technical machinery behind the first complete two-loop, order-$p^6$ evaluation of that amplitude: how the divergences are removed at two loops, how the counterterms may be reshuffled using the equations of motion without changing physics, and how every diagram reduces to logarithms and elementary functions. The payoff is that the polynomial part of the amplitude — six coefficients that a unitarity-based dispersive construction can only leave as free parameters — is now given explicitly as a function of the pion mass and of the low-energy constants of order $p^4$ and $p^6$. With estimates of the order-$p^6$ constants from meson exchange, the authors then produce threshold parameters and phase shifts for two candidate sets of order-$p^4$ couplings and compare them with data. A closing observation matters for how experimentalists read their own fits: the effective-range and single-parameter approximations used to extract scattering lengths from phase-shift data depart from the two-loop amplitude already at low energies.

What carries the argument

The load-bearing device is the treatment of the renormalized one-loop amplitude as a single nonlocal vertex inside a one-loop integral. Because the one-loop subdivergence has already been subtracted before the two-loop integral is written down, the nonfactorizable diagrams never produce one-loop subdivergences, and mass and wave-function renormalization inside the one-loop amplitude come for free. The remaining integrals are handled by a dispersive representation of the one-loop function $J$, which turns a two-loop diagram into a one-loop integral with three propagators and a variable mass to be integrated over. Around this sit two supporting pieces: a modified minimal subtraction scheme, in which the couplings are defined so that individual diagrams are scale independent and the constants $c$ of dimensional regularization are absorbed once and for all, and a careful bookkeeping of the equations of motion, showing that the ambiguity in the form of the order-$p^4$ Lagrangian only reshuffles the order-$p^6$ constants — a point the authors note affects $b_1$ through $b_4$ but not $b_5$ and $b_6$.

Extended reading notes

Core claim

The paper's central claim is that the elastic $ππ$ amplitude can be computed to order $p^6$ in chiral perturbation theory as a genuine Feynman-diagram calculation, and that the result is fully explicit. Concretely, the amplitude in Eq. (4.14) splits into a polynomial in the Mandelstam variables and a part built from the loop functions $J$ and $K_i$, all expressible in elementary functions, with the six polynomial coefficients $b_1$ through $b_6$ given in Appendix D as functions of the pion mass, the renormalized order-$p^4$ couplings $l_i^r$, and the order-$p^6$ couplings $r_i^r$. The authors emphasise what this adds beyond the dispersive treatment of the same order: unitarity fixes the absorptive part but not the polynomial, and diagrams with no absorptive part at all — tadpoles, butterflies, sunset — are invisible to that method, so their mass dependence and their dependence on the low-energy constants had to be obtained by direct evaluation. The paper also records the renormalization technology that makes this possible, including a demonstration in the $N$-component $φ^4$ model that different choices of one-loop counterterms, related by the equations of motion, shift the two-loop counterterms by a local action that can always be absorbed, leaving the generating functional unchanged.

Load-bearing premise

The numerical content rests on estimating the six order-$p^6$ constants from vector, scalar, kaon and eta exchange, with no error bars and no reference to which form of the order-$p^4$ Lagrangian is used.

Editorial extensions

If this is right

Any future dispersive or Roy-equation analysis of $ππ$ scattering can now be matched to an explicit polynomial at order $p^6$ rather than leaving six coefficients free.
The full quark-mass dependence of the amplitude allows the same expressions to be evaluated at unphysical pion masses and compared with lattice calculations, which a purely dispersive construction cannot do.
Threshold parameters and phase shifts follow as predictions once the order-$p^4$ couplings are chosen, so the two candidate sets can be discriminated by data on $δ^0_0-δ^1_1$ and on the $I=2$ $S$-wave.
The comparison in Fig. 10 shows that effective-range and single-parameter fits used to extract $a_0^0$ from phase-shift data can bias the extracted scattering length, which argues for re-extracting it against the two-loop amplitude.

Reading between the lines

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

The paper's insistence that the polynomial part carries mass singularities — logarithms of the pion mass — suggests that fits which treat the six coefficients as constants will drift systematically as lattice pion masses move toward the physical point.
Because the equations-of-motion ambiguity is shown to be absorbable only into order-$p^6$ constants, any model estimate of those constants that ignores the form of the order-$p^4$ Lagrangian is implicitly choosing a scheme; a model-independent estimate would need to state that choice explicitly.
The same nonlocal-vertex technique should transfer to any process with an elastic two-body subamplitude, so the bookkeeping demonstrated here is plausibly reusable well outside $ππ$ scattering.
The claim that resonance exchange dominates all six order-$p^6$ constants is testable in principle by a lattice determination of those constants at a few pion masses, which would either confirm the vector-meson pattern or show the estimates to be accidental.

Original abstract

We present technical details of the evaluation of the elastic pi-pi scattering amplitude to two loops in chiral perturbation theory. In particular, we elaborate on the renormalization procedure at the two-loop order and on the evaluation of the relevant Feynman diagrams that can all be expressed in terms of elementary functions. For the sake of clarity, we discuss these matters both in the N-component $\phi^4$ theory (in its symmetric phase) and in chiral perturbation theory. Estimates for the relevant low-energy constants of $O(p^6)$ are presented. Threshold parameters and phase shifts are then calculated for two sets of $O(p^4)$ coupling constants and compared with experiment. We comment on the extraction of threshold parameters from phase shift data.

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We present technical details of the evaluation of the elastic pi-pi scattering amplitude to two loops in chiral perturbation theory. In particular, we elaborate on the renormalization procedure at the two-loop order and on the evaluation of the relevant Feynman diagrams that can all be expressed in terms of elementary functions. For the sake of clarity, we discuss these matters both in the N-component $\phi^4$ theory (in its symmetric phase) and in chiral perturbation theory. Estimates for the relevant low-energy constants of $O(p^6)$ are presented. Threshold parameters and phase shifts are then calculated for two sets of $O(p^4)$ coupling constants and compared with experiment. We comment on the extraction of threshold parameters from phase shift data.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present technical details of the evaluation of the elastic pi-pi scattering amplitude to two loops in chiral perturbation theory.	contribution 1.00	0.94	0.15	asserted 0.99
s1	In particular, we elaborate on the renormalization procedure at the two-loop order and on the evaluation of the relevant Feynman diagrams that can all be expressed in terms of elementary functions.	method 0.82	0.93	0.07	asserted 0.99
s2	For the sake of clarity, we discuss these matters both in the N-component $\phi^4$ theory (in its symmetric phase) and in chiral perturbation theory.	method 0.95	0.82	0.03	asserted 0.95
s3	Estimates for the relevant low-energy constants of $O(p^6)$ are presented.	result 0.77	0.88	0.09	asserted 0.99
s4	Threshold parameters and phase shifts are then calculated for two sets of $O(p^4)$ coupling constants and compared with experiment.	result 0.95	0.94	0.27	asserted 1.00
s5	We comment on the extraction of threshold parameters from phase shift data.	interpretation 0.42	0.49	0.05	unclear 0.80

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.93.

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Figures

**Figure 1.** The two–loop graphs in φ 4 theory. Graph a corresponds to the contribution (2.13 a), and analogously for the diagrams b–e. The solid–dashed lines stand for the propagator (2.14). The black circles denote a vertex generated by the lagrangian L|Ci=0 in (2.1), and the filled squares stand for the contributions from the counterterm C1. Finally, the contributions from the counterterm C2 are indicated with an open square in graph e. 2.2.1 One–loop diagrams The singularities at one–loop order may easily be identified by writing hln(1 + D −1 0 σ)i = hD −1 0 σi − 1 2 hD −1 0 σD−1 0

**Figure 2.** Diagrams that contribute to the two–point functio

**Figure 3.** Summing up the graphs d1 · · · d9 in

**Figure 4.** The elastic φφ → φφ scattering amplitude to two loops. The numbers on the external lines denote the momenta. Crossed diagrams and insertions on external lines are not displayed. A filled box (open box) denotes contributions from the counterterm C1(C2). The tree–level and one–loop results are A (1)(s, t, u) = −2 , A (2)(s, t, u) = − 2 16π 2 [a r 3 + (N + 8)a(M/µ)] + 2(N + 4)J¯(s) + 4 h J¯(t) + J¯(u) i , (2.47) with J¯(z)= 1 16π 2 σ ln σ − 1 σ + 1 + 2 ; σ = 1 − 4M2 /z 1 2 . (2.48) The calculation at two–loop order may be made more economical by using the

**Figure 5.** The notation is as follows: The double lines denote off–

**Figure 6.** Butterfly diagram contribution to the two–point fu

**Figure 7.** Contributions to elastic ππ scattering in CHPT. The graphs displayed have a different topology than the ones occurring in φ 4 theory. The filled box denotes a contribution proportional to the low–energy constant l3. Graph a1 is included by use of the formula indicated in

**Figure 8.** Phase shift difference δ 0 0 − δ 1 1 at O(p 2 ), O(p 4 ) and O(p 6 ) (set I and II). The data points are from Ref. [42]. the effective range is given by r I 0 = 1 Mπa I 0 − 2Mπb I 0 (a I 0 ) 2 − 2a I 0 Mπ . (5.17) A similar formula can be used for δ I 1 when we replace (a I 0 , bI 0 ) by (q 2a I 1 , q2 b I 1 ). In Ref. [42], the following approximation was used to extract the scattering length a 0 0 from the measured phase shifts: sin 2(δ 0 0 − δ 1 1 ) = 2s 1 − 4 s (a 0 0 + q 2 b) . (5.18) In addition, a relation between a 0 0 and b attributed to Basdevant et al. [45] was employed…

**Figure 9.** Phase shift δ 2 0 at O(p 2 ), O(p 4 ) and O(p 6 ) (set I and II). The data points are from Ref. [43]. The obvious conclusion is that both the effective range approximation and the approximation used in Ref. [42] deviate from CHPT to O(p 6 ) already at comparatively low energies

**Figure 10.** Comparison of the full two–loop phase shift differ

Figures reproduced from arXiv:hep-ph/9707291 by the authors.

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P. Post and J.B. Tausk, Mod. Phys. Lett. A11 (1996) 2115. 52

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Forward citations

Cited by 23 Pith papers

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

Cornering Large-$N_c$ QCD with Positivity Bounds
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Positivity and crossing symmetry in large-Nc pion scattering pin the boundary of the allowed chiral parameter space to a single light spin-1 pole, which explains Vector Meson Dominance.
$\eta \to 3 \pi$ at Two Loops In Chiral Perturbation Theory
hep-ph 2007-09 conditional novelty 7.0 of 10

A complete next-to-next-to-leading-order chiral perturbation theory calculation of eta to three pions finds corrections larger than dispersive estimates, giving R = 42.2 and Q = 23.2 from the measured charged rate.
Renormalization of chiral perturbation theory to order p^6
hep-ph 1999-07 accept novelty 7.0 of 10

The authors complete the renormalization of chiral perturbation theory at order p^6, giving the double- and single-pole divergences and the renormalization group equations for all O(p^6) low-energy constants in the me...
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hep-ph 2024-12 conditional novelty 6.0 of 10

A dispersive analysis of the eta and eta' transition form factors yields data-driven pole contributions to the muon g-2 of 14.7(9) x 10^-11 and 13.5(7) x 10^-11.
S-matrix Bootstrap for Effective Field Theories: Massless Pions
hep-th 2020-11 conditional novelty 6.0 of 10

Unitarity, crossing and analyticity alone confine the two leading massless-pion Wilson coefficients to a bounded region whose edge runs close to the QCD values.

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Scattering of two and three physical pions at maximal isospin from lattice QCD
hep-lat 2020-08 conditional novelty 6.0 of 10

Two-flavor lattice QCD at the physical pion mass yields M_pi a_0 = -0.0481(86) for isospin-2 pion scattering and finds the energy-dependent three-pion contact term disagrees with leading-order chiral perturbation theory.
Direct CP violation and the $\Delta I=1/2$ rule in $K\to\pi\pi$ decay from the Standard Model
hep-lat 2020-04 conditional novelty 6.0 of 10

Lattice QCD with physical quark masses reproduces the Delta I = 1/2 rule and yields Re(epsilon-prime/epsilon) = 21.7(2.6)(6.2)(5.0) x 10^-4, consistent with the measured 16.6(2.3) x 10^-4.
Charged multi-hadron systems in lattice QCD+QED
hep-lat 2020-03 conditional novelty 6.0 of 10

Multi-meson energies from lattice QCD+QED are described by a non-relativistic effective theory with perturbative Coulomb exchange, giving charged-pion and neutral-kaon scattering lengths that differ from each other.
The I=2 pipi S-wave Scattering Phase Shift from Lattice QCD
hep-lat 2011-07 conditional novelty 6.0 of 10

Lattice QCD calculations of the I=2 pion-pion scattering phase shift yield threshold parameters that agree with Roy-equation determinations and with experiment.
Chiral expansions of the pi0 lifetime
hep-ph 2009-01 conditional novelty 6.0 of 10

The pi0 to two photon amplitude is computed to two loops in two-flavour chiral perturbation theory, shown to be renormalizable, and found to contain chiral logarithms that are absent at next-to-leading order.
pi/K -> e nu branching ratios to O(e^2 p^4) in Chiral Perturbation Theory
hep-ph 2007-07 accept novelty 6.0 of 10

A two-loop chiral perturbation theory calculation with a large-N_C matching of the counterterm yields R_pi = 1.2352e-4 and R_K = 2.477e-5 for the electron-to-muon leptonic decay ratios of pions and kaons.
Generalized positivity bounds on chiral perturbation theory
hep-ph 2020-04 conditional novelty 5.5 of 10

Generalized positivity bounds with t derivatives tighten the chiral perturbation theory constraints on l̄1 and l̄2 to l̄1+2l̄2 > 5.6±0.8 and l̄2 > 1.7±0.8, carve a convex region in the six b_i constants, and expose a ...
Improved Standard-Model predictions for $\eta^{(\prime)}\to \ell^+ \ell^-$
hep-ph 2025-12 accept novelty 5.0 of 10

Updated SM predictions yield Br(η→e⁺e⁻)=5.37(4)(2)[4]×10⁻⁹, Br(η→μ⁺μ⁻)=4.54(4)(2)[4]×10⁻⁶, Br(η'→e⁺e⁻)=1.80(2)(3)[3]×10⁻¹⁰, and Br(η'→μ⁺μ⁻)=1.22(2)(2)[3]×10⁻⁷, with a mild 1.6σ tension in the η→μ⁺μ⁻ channel.
Isospin-0 $\pi\pi$ s-wave scattering length from twisted mass lattice QCD
hep-lat 2016-12 conditional novelty 5.0 of 10

Using Osterwalder-Seiler valence quarks on three twisted-mass ensembles, the authors obtain the isospin-0 pion-pion scattering length M_pi a_0^{I=0} = 0.198(9)(6) at the physical pion mass.
Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: isospin-2 $\pi\pi$ scattering length
hep-lat 2015-06 conditional novelty 5.0 of 10

The first Nf=2+1+1 lattice QCD determination gives Mπa0 = -0.0442(2)stat(+4-0)sys for the isospin-2 ππ scattering length, consistent with chiral perturbation theory and prior lattice results.
Lattice QCD study of the s-wave $\pi\pi $ scattering lengths in the I=0 and 2 channels
hep-lat 2013-03 conditional novelty 5.0 of 10

Lattice QCD with MILC staggered configurations yields mπ a^{I=0} = 0.214(4)(7) and mπ a^{I=2} = -0.04430(25)(40) at the physical pion mass, with the I=0 value including the disconnected diagram.
Cusps in K --> 3pi decays: a theoretical framework
hep-ph 2011-03 conditional novelty 5.0 of 10

The paper gives the complete two-loop non-relativistic effective field theory amplitudes for K+ to pi0 pi0 pi+ and K+ to pi+ pi+ pi-, the framework used to extract the pion-pion S-wave scattering lengths from the cusp...
Precise Determination of the I=2 pipi Scattering Length from Mixed-Action Lattice QCD
hep-lat 2007-06 conditional novelty 5.0 of 10

Using domain-wall valence quarks on four MILC staggered ensembles and next-to-leading-order mixed-action chiral perturbation theory, the I=2 pion-pion scattering length is determined to be m_pi a_pipi = -0.04330 +/- 0.00042.
Direct CP violation in $K^0\to\pi\pi$: Standard Model Status
hep-ph 2017-12 accept novelty 4.0 of 10

The updated Standard Model prediction for direct CP violation in K0 to pi pi is Re(ε'/ε) = (15 ± 7) × 10^-4, in agreement with experiment.
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.
Dynamical Twisted Mass Fermions with Light Quarks: Simulation and Analysis Details
hep-lat 2008-03 accept novelty 4.0 of 10

The paper documents the tuning, correlator, error-analysis, and chiral-fit procedures behind ETMC's two-flavour maximally twisted mass lattice QCD results, and adds a variance-reduction identity for disconnected quark loops.
Novel approaches in Hadron Spectroscopy
hep-ph 2021-12 accept novelty 2.0 of 10

The review consolidates JPAC's methods for turning multibody hadron data into resonance pole positions with quantified model dependence, and applies them across the light, heavy, and three-body sectors.
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 later papers used it

56 later papers cite it

Evidence: substantial. 44 of the citing papers share no author with it, and 22 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 114 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

3 later papers by other authors are listed below: 2 used its method or tool and 1 used its data or values. 3 more papers by the same authors are listed: 2 built on it and 1 used its data or values.

Builds on it 2 of 3 papers with this label · 2 by the same authors

By the same authors

Extended the methods used for pion-pion scattering in chiral perturbation theory in this paper to all the present cases.

Meson-meson Scattering in QCD-like Theories (2011) (same authors) · 0.99 · paraphrased: the licence does not allow quoting
Employed an off-shell four-pion amplitude generalizing beyond N=3 the amplitude given in this paper.

Six-pion amplitude (2021) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Uses its method or tool 2 of 10 papers with this label

Used the two-loop pion-pion scattering calculation in this paper.

Chiral extrapolation of pion-pion scattering phase shifts within standard and unitarized Chiral Perturbation Theory (2011) · 0.93 · paraphrased: the licence does not allow quoting
Extrapolated its r_eff to the physical point using a formula based on chiral perturbation theory from this paper.

Quark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range (2019) · 0.93 · paraphrased: the licence does not allow quoting

Uses its data or values 2 of 14 papers with this label · 1 by the same authors

Used the expression for the chiral expansion of F_pi at order p^6 from this paper.

Chiral expansions of the pi0 lifetime (2009) · 0.87 · paraphrased: the licence does not allow quoting

By the same authors

Took the numerical values for the r_i^r from this paper.

The Pion Mass and Decay Constant at Three Loops in Two-Flavour Chiral Perturbation Theory (2017) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

52 arXiv papers cite this in 105 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 · 47 sentences without a confident label

Builds on it 2 papers

The reducible part is given in Eq. [ref], where we chose to employ the off-shell four-pion amplitude generalizing (beyond $N=3$) the amplitude given in Refs. [ref] .

Six-pion amplitude · Conclusions · 1.00
We have extended the methods used for $ππ$ scattering in ChPT [ref] to all the present cases.

Meson-meson Scattering in QCD-like Theories · Conclusions · 0.99

Uses its method or tool 10 papers

We use the two-loop $ππ$ scattering calculation in [ref] .

Chiral extrapolation of pion-pion scattering phase shifts within standard and unitarized Chiral Perturbation Theory · Two-loop ChPT · 0.93
The NLO $χ$PT expression for $M_πa_0$ written as a function of $(M_π/f_π)$ at a $χ$PT renormalisation scale $μ_R=f_π,phys$ reads [ref] [equation] with $ℓ_ππ$ related to the Gasser-Leutwyler coefficients $ℓ̅_i$ as follows [ref] [equation] Moreover, one can show in Twisted mass $χ$PT that the leading lattice artefacts to $M_πa_0$ are of order $𝒪(a^2M_π^2)$ [ref] .

Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: isospin-2 $\pi\pi$ scattering length · Chiral Extrapolation · 0.83
We use the standard renormalization procedure in ChPT [ref] with the extension to NNLO described in great detail in [ref] .

Meson-meson Scattering in QCD-like Theories · Renormalization · 0.95

Uses its data or values 12 papers

Further, using the following values of the LECs $l_1^r = -6.0 ·10^-3$ and $l_2^r = 2.7 ·10^-3$ at the renormalization scale equal to the $ρ$-meson mass that can be obtained from [ref] , we estimate for the coupling $g_2$ as $g_2≃1.4 ^-2=2.8·10^-2M_π^-2$ (error bars are not considered here, owing to a purely heuristic character of the above estimate).

Form factors of the $\rho$ meson from effective field theory and the lattice · Matching of the couplings $g_1,g_2,g_3$ to ChPT amplitude · 1.00
The numerical values for the $r_i^r$ are taken from [ref] [equation] The remaining $c_i^r$ and $r_M8^r$, $r_F8^r$ have been set to zero.

The Pion Mass and Decay Constant at Three Loops in Two-Flavour Chiral Perturbation Theory · Numerical study: mass dependence · 1.00
We will also need the expression for the chiral expansion of $F_π$ at order $p^6$ (from [ref] ) fpiexp F_πF= 1 +1F^2 [ M^2 l_4 -T_M ] +M^4F^4 [r_F + Ṫ_M^2 Q^F -Ṫ_M ∑_i=1^4 l_i Q_i^F] +8B^2(m_d-m_u)^2F^2(c_7+c_9) .

Chiral expansions of the pi0 lifetime · $π^0$ decay to NNLO in the two-flavour expansion · 0.87

Compares with it 10 papers

As the main result, the low-energy scattering lengths from the $LPA^'$-flow of Fig. [ref](a) [computed with Eqs. ([ref]) and ([ref])] simultaneously agree (within errorbars) with the experimental/theoretical ChPT values of Refs. [ref] , see columns six to eight.[The reference values in columns seven and eight are obtained by an interplay of experimental data and theoretical ChPT-input, leading to high-accuracy scattering lengths; for details see Refs. [ref] .

S-wave pion-pion scattering lengths from nucleon-meson fluctuations · Numeric results · 0.86
Our results for LLs for $N=3$ agree with two-loop calculations of $ππ$ amplitude [ref] and with five-loop results for the correlator of the scalar currents [ref] .

Chiral Logarithms Tamed · Introduction · 1.00
The one- and two-loop results agree with the earlier published ones [ref] [The two-loop coefficients agree with those of [ref] except for $b^2_0$.

The Massive O(N) Non-linear Sigma Model at High Orders · $ϕϕ$-scattering amplitude · 0.97

Cites it as background 17 papers

Then the chiral Lagrangian needed to calculate the pion scatterings up to $O(p^6)$, the next-to-next-to leading order, is [ref] ℒ_ChPT =ℒ_2+ ℒ_4 + ℒ_6 =F^2/4⟨u_μ u^μ+χ_+⟩+ l_1/4⟨u^μ u_μ⟩^2 + l_2/4⟨u_μ u_ν⟩⟨u^μ u^ν⟩+ l_3/16⟨χ_+⟩^2 +∑_i c_i Y_i , where $$ denotes taking the trace of the matrix in the flavor space, $l_i$ and $c_i$ are the LECs in the $O(p^4)$ and $O(p^6)$ Lagrangians, respectively, and the $Y_i$ operators can be found in Table 2 of Ref. [ref] .

Generalized positivity bounds on chiral perturbation theory · Chiral perturbation theory · 0.82
[ref] for previous studies along similar lines).

Light quark loops in $K^\pm \to \pi^\pm \nu\bar\nu$ from vector meson dominance and update on the Kaon Unitarity Triangle · Current status of long distance contributions to K+2pi+nunubar · 0.97
The leading logarithm for the masses for $N=3$ at two-loop was first calculated in [ref] and later to full two-loop order in [ref] .

The Massive O(N) Non-linear Sigma Model at High Orders · Mass · 0.88

Discussion

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