[Paper Review] pi pi Phase shifts from K to 2 pi

Vincenzo Cirigliano, C. Gatti+2 more|arXiv (Cornell University)|Jul 31, 2008

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper presents a refined extraction of the s-wave ππ scattering phase shift difference δ₀⁰ − δ₀² at the kaon mass scale using updated K→2π decay branching ratios from KLOE, incorporating experimental correlations and radiative corrections. The result, (57.5 ± 3.4)°, shows a 2.6σ discrepancy with Roy equation predictions, highlighting tensions in ππ scattering determinations from different experimental inputs.

ABSTRACT

We update the numerical results for the s-wave pi pi scattering phase-shift difference delta_0^0 - delta_0^2 at s = m_K^2 from a previous study of isospin breaking in K to 2 pi amplitudes in chiral perturbation theory. We include recent data for the K_S to pi pi and K^+ to pi^+ pi^0 decay widths and include experimental correlations.

Motivation & Objective

• To improve the precision of the s-wave ππ scattering phase shift difference δ₀⁰ − δ₀² at s = m_K² using updated experimental K→2π branching ratios.
• To account for electromagnetic radiative corrections and experimental correlations in the extraction, particularly from KLOE's inclusive measurement of K_S → π⁺π⁻(γ) and K_S → π⁰π⁰.
• To resolve discrepancies between phenomenological ππ scattering analyses and K→2π decay measurements by refining the theoretical framework in chiral perturbation theory with isospin-breaking effects.
• To quantify uncertainties from chiral scale dependence and π⁰–η mixing, and assess systematic effects on the phase shift difference.

Proposed method

• Uses chiral perturbation theory to relate K→2π amplitudes to ππ scattering phase shifts, incorporating isospin-breaking effects via electromagnetic corrections.
• Applies the formalism from Ref. 2 to express amplitudes A_1/2, A_3/2+A_5/2 in terms of physical decay rates and infrared-finite amplitudes.
• Employs a numerical minimization procedure to solve for χ₀ − χ₂, g₈, and g₂₇ using experimental inputs from KLOE and K+ lifetime measurements.
• Incorporates experimental correlations: a −0.9996 correlation between K_S → π⁺π⁻(γ) and K_S → π⁰π⁰ branching ratios, and a weak −0.032 correlation for K⁺ → π⁺π⁰.
• Uses Eqs. (5) and (7) to express A₀, A₂, and A₂⁺ in terms of measured branching ratios and phase differences, with coefficients dependent on chiral scale ν_χ and π⁰–η mixing angle ε⁽²⁾.
• Evaluates systematic uncertainties by varying ν_χ from 0.5 to 1.0 GeV and ε⁽²⁾ from 0.6×10⁻² to 1.5×10⁻², taking half the variation as uncertainty.

Experimental results

Research questions

• RQ1 What is the updated value of the s-wave ππ scattering phase shift difference δ₀⁰ − δ₀² at s = m_K² using recent KLOE data?
• RQ2 How do radiative corrections and experimental correlations affect the extraction of δ₀⁰ − δ₀² from K→2π decays?
• RQ3 Why does the extracted δ₀⁰ − δ₀² from K→2π decays differ significantly from predictions based on Roy equations and phenomenological ππ scattering analyses?
• RQ4 How sensitive is the phase shift difference to uncertainties in chiral scale and π⁰–η mixing angle?

Key findings

• The extracted phase shift difference is δ₀⁰ − δ₀² = (57.5 ± 3.4)°, with experimental uncertainty ±0.8° and systematic uncertainty ±3.0° from radiative corrections and ±1.4° from chiral scale and π⁰–η mixing variations.
• The result shows a 2.6σ discrepancy with the Roy equation prediction of (47.7 ± 1.5)°, indicating a significant tension between K→2π decays and ππ scattering data.
• The phase shift difference is insensitive to higher-order chiral couplings, but the extraction is highly sensitive to the isospin-breaking correction term (6.2 ± 3.0)° in Eq. (4).
• The value of g₈ is determined as 3.64 ± 0.20, and g₂₇ as 0.2987 ± 0.0030, with the latter dominated by experimental uncertainty.
• The systematic uncertainty from chiral scale and π⁰–η mixing is substantial, contributing ±1.4° to the total error on δ₀⁰ − δ₀².
• The result suggests that the discrepancy with phenomenological analyses may stem from unaccounted effects in K→2π data or in the modeling of isospin-breaking corrections.