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Two-loop representations of low-energy pion form factors and pi-pi scattering phases in the presence of isospin breaking
High Energy Physics - Phenomenology · Submitted 27 Feb 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim It is shown that each of these phases consists of the sum of a "universal" part and a form-factor dependent contribution.
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The abstract, sentence by sentence
Dispersive representations of the pi-pi scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a "universal" part and a form-factor dependent contribution. The first one is entirely determined in terms of the pi-pi scattering amplitudes alone, and reduces to the phase satisfying Watson's theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the pi-pi amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and pi-pi scattering amplitudes in the presence of isospin breaking are provided.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Dispersive representations of the pi-pi scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. | contribution 0.98 | 0.91 | 0.24 | asserted 1.00 |
| s1 | Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. | result 0.64 | 0.93 | 0.11 | asserted 1.00 |
| s2 | It is shown that each of these phases consists of the sum of a "universal" part and a form-factor dependent contribution. | result 0.98 | 0.91 | 0.34 | asserted 1.00 |
| s3 | The first one is entirely determined in terms of the pi-pi scattering amplitudes alone, and reduces to the phase satisfying Watson's theorem in the isospin limit. | result 0.81 | 0.83 | 0.22 | asserted 1.00 |
| s4 | The second one can be sizeable, although it vanishes in the same limit. | result 0.86 | 0.86 | 0.12 | asserted 0.86 |
| s5 | The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the pi-pi amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. | contribution 0.98 | 0.91 | 0.08 | asserted 0.98 |
| s6 | In addition, partially analytical expressions for the two-loop form factors and pi-pi scattering amplitudes in the presence of isospin breaking are provided. | contribution 0.81 | 0.94 | 0.09 | asserted 1.00 |
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The paper
Original abstract
Dispersive representations of the pi-pi scattering amplitudes and pion form factors, valid at two-loop accuracy in the low-energy expansion, are constructed in the presence of isospin-breaking effects induced by the difference between the charged and neutral pion masses. Analytical expressions for the corresponding phases of the scalar and vector pion form factors are computed. It is shown that each of these phases consists of the sum of a "universal" part and a form-factor dependent contribution. The first one is entirely determined in terms of the pi-pi scattering amplitudes alone, and reduces to the phase satisfying Watson's theorem in the isospin limit. The second one can be sizeable, although it vanishes in the same limit. The dependence of these isospin corrections with respect to the parameters of the subthreshold expansion of the pi-pi amplitude is studied, and an equivalent representation in terms of the S-wave scattering lengths is also briefly presented and discussed. In addition, partially analytical expressions for the two-loop form factors and pi-pi scattering amplitudes in the presence of isospin breaking are provided.
Human review
How the literature uses it
10 arXiv papers cite this in 70 sentences
Papers citing it · 51 sentences without a confident label
Contests it 1 paper
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In the case of the scattering lengths, this can be achieved upon using the expressions (F.13) and the formulas given in Appendix E of [ref] [ We take this opportunity to correct three typos in Apps.
Isospin breaking in the phases of the Ke4 form factors · $ππ$ scattering amplitudes · contest 0.76
Uses its method or tool 3 papers
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These can be obtained from Eqs. (4.6), (4.10), (4.15) of Ref. [ref] , combined with the formulas (F.8)-(F.12) of that same reference.
Isospin breaking in the phases of the Ke4 form factors · $ππ$ scattering amplitudes · 0.90
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How this can be done has been described in Appendix F of Ref. [ref] .
Dispersive construction of two-loop $P \to \pi\pi\pi$ $(P=K,\eta)$ amplitudes · First iteration · 0.82
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[ref] and [ref] .
On the amplitudes for the CP-conserving $K^\pm(K_S)\to\pi^\pm(\pi^0)\ell^+\ell^-$ rare decay modes · The computation of $ψ_+,S (s)$ · 0.95
Uses its data or values 1 paper
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In the following, we consider the case [equation] We fix the subtraction constants of the form factors by requesting that the charged scalar form factors has the typical values $r_π^2=0.60$ fm$^2$ and $c_S^π=10$ GeV$^-4$ [ref] , leading to [equation] Using Ref. [ref] , one can compute the isospin-breaking shift between $a^π^0_S$ and $a^π_S$.
On some aspects of isospin breaking in the decay $K^pm$ $\to$ $\pi^0 \pi^0 e^\pm$ $\stackrel{_{(-)}}{\nu_e}$ · Form factors and input parameters used for the generation of pseudo-data · 0.93
Compares with it 1 paper
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This is in contrast with the scalar form factors considered in Ref. [ref] , where the corresponding ratio was equal to unity at lowest order.
Isospin breaking in the phases of the Ke4 form factors · Isospin breaking in the phases of the two-loop form factors · 0.90
Cites it as background 5 papers
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These changes are not only meant to make the notation more compact, but also to make contact with Ref. [ref] , to which we will often refer in this section.
Isospin breaking in the phases of the Ke4 form factors · Isospin breaking in the phases of the two-loop form factors · 0.89
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The structure of the other amplitudes is similar, and we refer the interested reader to [ref] for details.
Isospin breaking in pion and $K_{e4}$ form factors · Two-loop representation of pion form factors with IB · 0.89
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Since the construction of the $ππ$ scattering amplitudes with $M_π≠M_π^0$ has already been discussed at some length in Refs. [ref] , our account will remain brief.
Dispersive construction of two-loop $P \to \pi\pi\pi$ $(P=K,\eta)$ amplitudes · $ππ$ scattering · 0.91
What it cites · 45 sentences without a confident label
Contests 1 paper
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Furthermore, we found a slight disagreement with the result of [ref] for $F_V(s)$: the radius $⟨r ⟩_V^π$ exhibits an infrared divergence proportional to $lnM_π^0^2$ as $M_π^0$ goes to zero, whereas we find that $⟨r ⟩_V^π$ remains finite in this limit, but diverges as $∼lnM_π^2$ if we send the charged pion mass to zero, keeping $M_π^0$ fixed.
Virtual photons in the pion form factors and the energy-momentum tensor · Expressions of the subtraction constants · contest 0.75
Builds on 2 papers
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[ref] to the situation where the difference between the masses of charged and neutral pions is taken into account.
What $\pi-\pi$ Scattering Tells Us About Chiral Perturbation Theory · INTRODUCTION · 1.00
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[ref] to the whole set of scattering amplitudes involving the mesons of the lightest pseudoscalar octet.
Dispersive Approach to Chiral Perturbation Theory · INTRODUCTION · 0.90
Uses the method or tool of 1 paper
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First, we make use of the relation between these two-flavour constants and their three-flavour counterparts $K_i^r$ [ref] , as worked out at one-loop level in ref.
Virtual Photons in Chiral Perturbation Theory · Numerical input values · 0.89
Compares with 5 papers
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[ref] from a Feynman graph calculation based on the low-energy effective lagrangian for QCD+QED, provided that one drops the contributions coming from virtual photons, while keeping the difference between charged and neutral pion masses [this procedure is discussed in greater detail in subsection [ref] and in appendix [ref]].
Virtual Photons in Low Energy pi-pi Scattering · One-loop representation of $ππ$ scattering amplitudes · 0.99
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[ref] from a Feynman graph calculation based on the low-energy effective lagrangian for QCD+QED, provided that one drops the contributions coming from virtual photons, while keeping the difference between charged and neutral pion masses [this procedure is discussed in greater detail in subsection [ref] and in appendix [ref]].
Virtual photons in SU(2) chiral perturbation theory and electromagnetic corrections to $\pi\pi$ scattering · One-loop representation of $ππ$ scattering amplitudes · 0.99
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In order to make a comparison with these one-loop calculations, we must therefore remove the contributions of virtual photons from the expressions given in these references, and only keep the effects due to the difference of the pion masses [ref] .
Isospin breaking in Kl4 decays · Determination of the subtraction parameters · 0.81
Cites as background 9 papers
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[ref] , and the explicit construction of the two-loop amplitude has subsequently been performed along these lines in detail in ref.
What $\pi-\pi$ Scattering Tells Us About Chiral Perturbation Theory · INTRODUCTION · 0.84
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Let us recall here that there exist explicit one-loop calculations of the various $ππ$ amplitudes [ref] and form factors [ref] considered here, obtained within the full QCD+QED effective theory [ref] .
Explicit generating functional for pions and virtual photons · Determination of the subtraction parameters · 0.87
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Let us recall here that there exist explicit one-loop calculations of the various $ππ$ amplitudes [ref] and form factors [ref] considered here, obtained within the full QCD+QED effective theory [ref] .
Virtual photons in the pion form factors and the energy-momentum tensor · Determination of the subtraction parameters · 0.87
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